Power system transients theory and applications-Ametani Akihiro, Baba Yoshihiro, Nagaoka Naoto, Ohno Teruo.pdf

POWER SYSTEM

TRANSIENTS Theory and Applications

AKIHIRO AMETANI • NAOTO NAGAOKA YOSHIHIRO BABA • TERUO OHNO

Power SyStem

tranSientS Theor y and Applications

Power SyStem

tranSientS theor y and Applications

Akihiro AmetAni nAoto nAgAokA Yoshihiro BABA teruo ohno

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2013 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20140428 International Standard Book Number-13: 978-1-4665-7786-2 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Contents Introduction............................................................................................................xv List of Symbols..................................................................................................... xix List of Acronyms.................................................................................................. xxi International Standards.................................................................................... xxiii 1. Theory of Distributed-Parameter Circuits and the Impedance/Admittance Formulas................................................................1 1.1 Introduction............................................................................................ 1 1.2 Impedance and Admittance Formula................................................. 2 1.2.1 Conductor Internal Impedance Zi..........................................3 1.2.1.1 Derivation of an Approximate Formula................3 1.2.1.2 Accurate Formula by Schelkunoff..........................6 1.2.2 Outer-Media Impedance Z0..................................................... 8 1.2.2.1 Outer-Media Impedance.......................................... 8 1.2.2.2 Overhead Conductor................................................9 1.2.2.3 Pollaczek’s General Formula for Overhead, Underground, and Overhead/Underground Conductor Systems.................................................. 14 1.2.3 Problems................................................................................... 16 1.3 Basic Theory of Distributed-Parameter Circuit............................... 17 1.3.1 Partial Differential Equations of Voltages and Currents........................................................................... 17 1.3.2 General Solutions of Voltages and Currents....................... 18 1.3.2.1 Sinusoidal Excitation.............................................. 18 1.3.2.2 Lossless Line............................................................ 21 1.3.3 Voltages and Currents on a Semi-Infinite Line.................. 23 1.3.3.1 Solutions of Voltages and Currents...................... 23 1.3.3.2 Waveforms of Voltages and Currents................... 24 1.3.3.3 Phase Velocity.......................................................... 25 1.3.3.4 Traveling Wave........................................................ 27 1.3.3.5 Wave Length............................................................ 28 1.3.4 Propagation Constants and Characteristic Impedance...... 28 1.3.4.1 Propagation Constants........................................... 28 1.3.4.2 Characteristic Impedance...................................... 31 1.3.5 Voltages and Currents on a Finite Line............................... 32 1.3.5.1 Short-Circuited Line............................................... 32 1.3.5.2 Open-Circuited Line............................................... 35 1.3.6 Problems................................................................................... 38

v

vi

Contents

1.4

1.5

1.6

Multiconductor System....................................................................... 38 1.4.1 Steady-State Solutions............................................................ 38 1.4.2 Modal Theory.......................................................................... 41 1.4.2.1 Eigenvalue Theory.................................................. 41 1.4.2.2 Modal Theory..........................................................44 1.4.2.3 Current Mode........................................................... 45 1.4.2.4 Parameters in Modal Domain............................... 46 1.4.3 Two-Port Circuit Theory and Boundary Conditions......... 48 1.4.3.1 Four-Terminal Parameter....................................... 48 1.4.3.2 Impedance/Admittance Parameters.................... 50 1.4.4 Modal Distribution of Multiphase Voltages and Currents............................................................................ 52 1.4.4.1 Transformation Matrix........................................... 52 1.4.4.2 Modal Distribution................................................. 53 1.4.5 Problems................................................................................... 55 Frequency-Dependent Effect.............................................................. 56 1.5.1 Frequency Dependence of Impedance................................ 56 1.5.2 Frequency-Dependent Parameters....................................... 58 1.5.2.1 Frequency-Dependent Effect................................. 58 1.5.2.2 Propagation Constant............................................. 59 1.5.2.3 Characteristic Impedance...................................... 61 1.5.2.4 Transformation Matrix...........................................63 1.5.2.5 Line Parameters in the Extreme Case.................. 68 1.5.3 Time Response........................................................................ 70 1.5.3.1 Time-Dependent Responses.................................. 70 1.5.3.2 Propagation Constant: Step Response................. 71 1.5.3.3 Characteristic Impedance...................................... 72 1.5.3.4 Transformation Matrix........................................... 74 1.5.4 Problems...................................................................................77 Traveling Wave.....................................................................................77 1.6.1 Reflection and Refraction Coefficients.................................77 1.6.2 Thevenin’s Theorem............................................................... 79 1.6.2.1 Equivalent Circuit of a Semi-Infinite Line........... 79 1.6.2.2 Voltage and Current Sources at the Sending End............................................................. 79 1.6.2.3 Boundary Condition at the Receiving End.......... 79 1.6.2.4 Thevenin’s Theorem................................................ 82 1.6.3 Multiple Reflection..................................................................84 1.6.4 Multiconductors...................................................................... 88 1.6.4.1 Reflection and Refraction Coefficients................. 88 1.6.4.2 Lossless Two Conductors....................................... 88 1.6.4.3 Consideration of Modal Propagation Velocities..........................................91 1.6.4.4 Consideration of Losses in a Two-Conductor System.......................................... 96 1.6.4.5 Three-Conductor System....................................... 99

Contents

vii

1.6.4.6 Cascaded System Composed of the Different Numbers of Conductors...................... 102 1.6.5 Problems................................................................................. 103 1.7 Nonuniform Conductors.................................................................. 104 1.7.1 Characteristic of Nonuniform Conductors....................... 105 1.7.1.1 Nonuniform Conductor....................................... 105 1.7.1.2 Difference from Uniform Conductors................ 108 1.7.2 Impedance and Admittance Formulas.............................. 109 1.7.2.1 Finite-Length Horizontal Conductor................. 109 1.7.2.2 Vertical Conductor................................................ 112 1.7.3 Line Parameters.................................................................... 114 1.7.3.1 Finite Horizontal Conductor............................... 114 1.7.3.2 Vertical Conductor................................................ 117 1.7.3.3 Nonparallel Conductor......................................... 119 1.7.4 Problems................................................................................. 119 1.8 Introduction of EMTP....................................................................... 122 1.8.1 Introduction........................................................................... 122 1.8.1.1 History of a Transient Analysis........................... 122 1.8.1.2 Background of EMTP........................................... 123 1.8.1.3 EMTP Development.............................................. 124 1.8.2 Basic Theory of EMTP.......................................................... 124 1.8.2.1 Representation of a Circuit Element by a Current Source and a Resistance........................ 126 1.8.2.2 Composition of Nodal Conductance.................. 128 1.8.3 Other Circuit Elements........................................................ 129 1.8.4 Solutions of the Problems.................................................... 131 References...................................................................................................... 136 2. Transients on Overhead Lines.................................................................. 141 2.1 Introduction........................................................................................ 141 2.2 Switching Surge on Overhead Line................................................. 142 2.2.1 Basic Mechanism of Switching Surge................................ 142 2.2.2 Basic Parameters Influencing Switching Surge................ 143 2.2.2.1 Source Circuit........................................................ 143 2.2.2.2 Switch...................................................................... 146 2.2.2.3 Transformer............................................................ 147 2.2.2.4 Transmission Line................................................. 147 2.2.3 Switching Surges in Practice............................................... 148 2.2.3.1 Classification of Switching Surges...................... 148 2.2.3.2 Basic Characteristic of Closing Surge: Field Test Results................................................... 149 2.2.3.3 Closing Surge on a Single-Phase Line................ 151 2.2.3.4 Closing Surges on a Multiphase Line................. 153 2.2.3.5 Effect of Various Parameters on Closing Surge..................................................... 162

viii

Contents

2.3

2.4

2.5

2.6

Fault Surge.......................................................................................... 166 2.3.1 Fault Initiation Surge............................................................ 166 2.3.2 Characteristic of a Fault Initiation Surge........................... 169 2.3.2.1 Effect of Line Transposition................................. 169 2.3.2.2 Overvoltage Distribution..................................... 169 2.3.3 Fault-Clearing Surge............................................................ 172 Lightning Surge.................................................................................. 175 2.4.1 Mechanism of Lightning Surge Generation..................... 177 2.4.2 Modeling of Circuit Elements............................................. 179 2.4.2.1 Lightning Current................................................. 179 2.4.2.2 Tower and Gantry................................................. 180 2.4.2.3 Tower Footing Impedance................................... 182 2.4.2.4 Arc Horn................................................................. 184 2.4.2.5 Transmission Line................................................. 185 2.4.2.6 Substation............................................................... 185 2.4.3 Lightning Surge Overvoltage.............................................. 185 2.4.3.1 Model Circuit......................................................... 185 2.4.3.2 Lightning Surge Overvoltage.............................. 187 2.4.3.3 Effect of Various Parameters............................... 188 Theoretical Analysis of Transients: Hand Calculations............... 194 2.5.1 Switching Surge on an Overhead Line.............................. 195 2.5.1.1 Traveling Wave Theory........................................ 195 2.5.1.2 Lumped-Parameter Equivalent with Laplace Transform................................................. 202 2.5.2 Fault Surge............................................................................. 206 2.5.3 Lightning Surge.................................................................... 208 2.5.3.1 Tower Top Voltage................................................. 208 2.5.3.2 Two-Phase Model.................................................. 208 2.5.3.3 No Back Flashover................................................. 210 2.5.3.4 Case of a Back Flashover...................................... 212 2.5.3.5 Consideration of Substation................................. 212 Frequency-Domain Method of Transient Simulations......................215 2.6.1 Introduction........................................................................... 215 2.6.2 Numerical Fourier/Laplace Transform............................. 215 2.6.2.1 Finite Fourier Transform...................................... 215 2.6.2.2 Shift of Integral Path: Laplace Transform.......... 217 2.6.2.3 Numerical Laplace Transform: Discrete Laplace Transform................................................. 218 2.6.2.4 Odd-Number Sampling: Accuracy Improvement.......................................................... 218 2.6.2.5 Application of FFT: Fast Laplace Transform (FLT)..................................................... 221 2.6.3 Transient Simulation............................................................ 228 2.6.3.1 Definition of Variables.......................................... 228

Contents

ix

2.6.3.2 Subroutine to Prepare F(ω)................................... 229 2.6.3.3 Subroutine FLT...................................................... 230 2.6.3.4 Remarks of the Frequency-Domain Method.. .......................................................... 230 References...................................................................................................... 230 3. Transients on Cable Systems..................................................................... 233 3.1 Introduction........................................................................................ 233 3.2 Impedance and Admittance of Cable Systems..............................234 3.2.1 Single-Phase Cable................................................................234 3.2.1.1 Cable Structure......................................................234 3.2.1.2 Impedance and Admittance................................234 3.2.2 Sheath Bonding..................................................................... 235 3.2.3 Homogeneous Model of a Cross-Bonded Cable............... 238 3.2.3.1 Homogeneous Impedance and Admittance....... 238 3.2.3.2 Reduction of Sheath.............................................. 243 3.2.4 Theoretical Formula of Sequence Currents...................... 246 3.2.4.1 Cross-Bonded Cable.............................................. 246 3.2.4.2 Solidly Bonded Cable............................................ 251 3.3 Wave Propagation and Overvoltages.............................................. 256 3.3.1 Single-Phase Cable................................................................ 256 3.3.1.1 Propagation Constant........................................... 256 3.3.1.2 Example of Transient Analysis............................ 258 3.3.2 Wave Propagation Characteristic....................................... 260 3.3.2.1 Impedance: R, L..................................................... 263 3.3.2.2 Capacitance: C........................................................ 264 3.3.2.3 Transformation Matrix......................................... 264 3.3.2.4 Attenuation Constant and Propagation Velocity............................................. 264 3.3.3 Transient Voltage................................................................... 265 3.3.4 Limitation of the Sheath Voltage........................................ 269 3.3.5 Installation of SVLs.............................................................. 271 3.4 Studies on Recent and Planned EHV AC Cable Projects............. 272 3.4.1 Recent Cable Projects........................................................... 273 3.4.2 Planned Cable Projects......................................................... 275 3.5 Cable System Design and Equipment Selection............................ 277 3.5.1 Study Flow............................................................................. 277 3.5.2 Zero-Missing Phenomenon................................................. 278 3.5.2.1 Sequential Switching............................................ 280 3.5.3 Leading Current Interruption............................................. 281 3.5.4 Cable Discharge....................................................................284 3.6 EMTP Simulation Test Cases............................................................ 285 References...................................................................................................... 287

x

Contents

4. Transient and Dynamic Characteristics of New Energy Systems..... 291 4.1 Wind Farm.......................................................................................... 291 4.1.1 Model Circuit of Wind Farm............................................... 291 4.1.2 Steady-State Analysis........................................................... 294 4.1.2.1 Cable Model........................................................... 294 4.1.2.2 Charging Current.................................................. 298 4.1.2.3 Load-Flow Calculation......................................... 301 4.1.3 Transient Calculation........................................................... 303 4.2 Power-Electronics Simulation by EMTP......................................... 306 4.2.1 Simple-Switching Circuit..................................................... 306 4.2.2 Switching-Transistor Model................................................ 307 4.2.2.1 Simple-Switch Model............................................308 4.2.2.2 Switch with Delay Model..................................... 312 4.2.3 MOSFET................................................................................. 314 4.2.3.1 Simple Model......................................................... 315 4.2.3.2 Modified Switching Device Model..................... 316 4.2.3.3 Simulation Circuit and Results........................... 321 4.2.4 Thermal Calculation............................................................. 329 4.3 Voltage Regulation Equipment Using Battery in a DC Railway System.................................................................................. 331 4.3.1 Introduction........................................................................... 331 4.3.2 Feeding Circuit...................................................................... 333 4.3.3 Measured and Calculated Results...................................... 336 4.3.3.1 Measured Results.................................................. 336 4.3.3.2 Calculated Results of Conventional System...... 336 4.3.3.3 Calculated Results with Power Compensator......................................... 340 4.4 Concluding Remarks.........................................................................343 References......................................................................................................344 5. Numerical Electromagnetic Analysis Methods and Their Applications to Transient Analyses.........................................................345 5.1 Fundamentals.....................................................................................345 5.1.1 Maxwell’s Equations.............................................................345 5.1.2 Finite-Difference Time-Domain Method..........................346 5.1.3 Method of Moments............................................................. 355 5.2 Applications........................................................................................ 363 5.2.1 Grounding Electrodes.......................................................... 363 5.2.2 Transmission Towers............................................................ 367 5.2.3 Distribution Lines: Lightning-Induced Surges................. 371 5.2.4 Transmission Lines: Propagation of Lightning Surges in the Presence of Corona....................................... 375 5.2.5 Power Cables: Propagation of Power Line Communication Signals....................................................... 379

Contents

xi

5.2.6 Air-Insulated Substations.................................................... 385 5.2.7 Wind Turbine Generator Towers........................................ 387 References...................................................................................................... 389 6. Electromagnetic Disturbances in Power Systems and Customers............................................................................................. 393 6.1 Introduction........................................................................................ 393 6.2 Disturbances in Power Stations and Substations.......................... 394 6.2.1 Statistical Data of Disturbances.......................................... 394 6.2.1.1 Overall Data........................................................... 394 6.2.1.2 Disturbed Equipments......................................... 395 6.2.1.3 Surge Incoming Route.......................................... 397 6.2.2 Characteristics of Disturbances.......................................... 397 6.2.2.1 Characteristics of Lightning Surge Disturbances............................................... 397 6.2.2.2 Characteristics of Switching Surge Disturbances............................................... 398 6.2.2.3 Switching Surge in DC Circuits.......................... 402 6.2.3 Influence, Countermeasures, and Costs of Disturbances........................................................... 403 6.2.3.1 Influence of Disturbances on Power System Operation.................................................. 403 6.2.3.2 Countermeasures Carried Out............................ 405 6.2.3.3 Cost of Countermeasures..................................... 406 6.2.4 Case Studies........................................................................... 407 6.2.4.1 Case No. 1...............................................................408 6.2.4.2 Case No. 2............................................................... 410 6.2.4.3 Case No. 3............................................................... 411 6.2.5 Concluding Remarks............................................................ 412 6.3 Disturbances in Customers and Home Appliances...................... 413 6.3.1 Statistical Data of Disturbances.......................................... 413 6.3.2 Breakdown Voltage of Home Appliances......................... 415 6.3.2.1 Testing Voltage....................................................... 415 6.3.2.2 Breakdown Test..................................................... 416 6.3.3 Surge Voltages and Currents into Customers due to Nearby Lightning................................................................. 416 6.3.3.1 Introduction........................................................... 416 6.3.3.2 Model Circuits for Experiments and EMTP Simulations................................................ 417 6.3.3.3 Experimental and Simulation Results................425 6.3.3.4 Concluding Remarks............................................ 429 6.3.4 Lightning Surge Incoming from a Communication Line............................................................ 429 6.3.4.1 Introduction........................................................... 429

xii

Contents

6.3.4.2 Protective Device...................................................430 6.3.4.3 Lightning Surge.....................................................430 6.3.4.4 Concluding Remarks............................................ 433 6.4 Analytical Method of Solving Induced Voltages and Currents...................................................................... 435 6.4.1 Introduction........................................................................... 435 6.4.2 F-Parameter Formulation for Induced Voltages and Currents.......................................................... 439 6.4.2.1 Formulation of F-Parameter................................. 439 6.4.2.2 Approximation of F-Parameters..........................440 6.4.2.3 Cascaded Connection of Pipelines.....................440 6.4.3 Application Examples.......................................................... 441 6.4.3.1 Single Section Terminated by R1 and R 2............ 441 6.4.3.2 Two-Cascaded Sections of a Pipeline (Problem 6.1)...........................................................446 6.4.3.3 Three-Cascaded Sections of a Pipeline.............. 453 6.4.4 Comparison with a Field-Test Result.................................454 6.4.4.1 Comparison with EMTP Simulations................454 6.4.4.2 Field-Test Result.....................................................454 6.4.5 Concluding Remarks............................................................ 459 Solution of Problem 6.1................................................................................ 460 Appendix 6.A.1 Test Voltage for Low-Voltage Control Circuits in Power Stations and Substations (JEC-0103-2004)............................................................461 6.A.2 Traveling Wave Solution................................................464 6.A.3 Boundary Conditions and Solutions of a Voltage and a Current...................................................................464 6.A.4 Approximate Formulas for Impedance and Admittance.............................................................. 465 6.A.5 Accurate Solutions for Two-Cascaded Sections......... 466 References...................................................................................................... 467 7. Problems and Application Limits of Numerical Simulations............ 471 7.1 Problems of Existing Impedance Formulas Used in Circuit Theory–Based Approaches.................................................. 471 7.1.1 Earth-Return Impedance..................................................... 471 7.1.1.1 Carson’s Impedance.............................................. 471 7.1.1.2 Basic Assumption of the Impedance.................. 472 7.1.1.3 Nonparallel Conductor......................................... 472 7.1.1.4 Stratified Earth....................................................... 473 7.1.1.5 Earth Resistivity and Permittivity...................... 473 7.1.2 Internal Impedance.............................................................. 473 7.1.2.1 Schelkunoff’s Impedance..................................... 473 7.1.2.2 Arbitrary Cross-Section Conductor.................... 473

Contents

xiii

7.1.2.3 Semiconducting Layer of Cable.......................... 474 7.1.2.4 Proximity Effect..................................................... 474 7.1.3 Earth-Return Admittance.................................................... 474 7.2 Existing Problems in Circuit Theory–Based Numerical Analysis........................................................................... 475 7.2.1 Reliability of a Simulation Tool........................................... 475 7.2.2 Assumption and Limit of a Simulation Tool.................... 475 7.2.3 Input Data.............................................................................. 476 7.3 Numerical Electromagnetic Analysis for Power System Transients............................................................................... 476 References...................................................................................................... 477

Introduction When lightning strikes a building or a transmission tower, an electric current flows into its structures, which are made of electrically conductive materials such as steel and copper. The electric current produces a high voltage called “overvoltage” (or abnormal voltage), which can damage or break electrical equipment installed in the building or in the power transmission system. The breakdown may shut down the electrical room of the building, resulting in a blackout of the whole building. If the breakdown occurs in a substation in a high-voltage power transmission system, a city where electricity is supplied by the substation can experience a blackout. An overvoltage can also be generated by switching operations of a circuit breaker or a load switch, which is electrically the same as a breaker in a house. A phenomenon during the time period in which an overvoltage occurs due to lightning or switching operation is called transient, while electricity being supplied under normal circumstances is called steady state. In general, a transient dies out and reaches a steady state within approximately 10 μs (10−6 s) in the lightning transient case and within approximately 10 ms (10−3 s) in the switching transient case. Occasionally, a transient sustains for a few seconds if it involves resonant oscillation of circuit parameters (mostly inductance and capacitance) or mechanical oscillation of the steel shaft of a generator (called subsynchronous resonance). In order to design the electrical strength of electrical equipment and to ensure human safety during a transient, it is crucial to perform a transient analysis, especially in the field of electric power engineering. Chapter 1 of this book describes a transient on a single-phase line from the physical viewpoint and how this is solved analytically by an electric circuit theory. The impedance and the admittance formulas of an overhead line are described. Simple formulas that can be calculated using a pocket calculator are also explained so that a transient can be analytically evaluated. Since a real power line is three-phase, theory that deals with multiphase lines is presented. Finally, the book describes how to tackle a real transient in a power system. Chapter 1 also presents the well-known simulation tool electromagnetic transients program (EMTP), originally developed by the US Department of Energy, Bonneville Power Administration, which is useful in dealing with a real transient in a power system. Chapter 2 describes wave propagation characteristics and transients in an overhead transmission line. The distributed-parameter circuit theory is applied to solve the transients analytically. The EMTP is then applied to calculate transients in a power system composed of an overhead line and a substation. Various simulation examples are demonstrated, together with comparison with field test results. xv

xvi

Introduction

Chapter 3 discusses transients in a cable system. A cable system is, in general, more complicated than an overhead line system, because one phase of the cable is composed of two conductors called the metallic core and metallic sheath. The former carries a current and the latter behaves as an electromagnetic shield against the core current. Another reason why a cable system is complicated is that most long cables are cross-bounded, that is, the metallic sheaths on phases a, b, and c in one cable section are connected to those of phases b, c, and a in the next section. Each section is called a minor section whose length ranges normally from approximately 100 m to 1 km. Three minor sections compose one major section. The sheath impedances of three phases thus become nearly equal to each other. Because of this, a transient on a cable system is quite different from that on an overhead line system. Similarly to Chapter 2, Chapter 4 analyzes the basic characteristic of wave propagation on a cable based on the distributed-parameter circuit theory, together with EMTP simulation examples. One of the most attractive subjects in recent years has been so-called clean energy (or sustainable energy) and smart grids. Wind farms and mega solar plants have become well known. The chapter describes transients in wind farms based on EMTP simulations. Since the output voltage of most wind generators is about 600 V, wind generators are connected to a low-voltage transmission (distribution) line. Also, as their generating capacity is small, a number of wind generators are connected together in a substation, which allows the voltage to be stepped up for power transmission, thus forming a wind farm. In the case of an off-shore wind farm, the generated power is sent to an on-shore connection point through submarine cables. A transient analysis in wind farms, mega solars, and smart grids requires a different approach in comparison to those in overhead lines and cables. A transient in an overhead line and cable is directly associated with traveling waves whose traveling time is in the order of 10 μs up to 1 ms; in most cases, the maximum overvoltage appears within a few milliseconds. In contrast, a transient in a wind farm involving power electronic circuits is affected by the dynamic behavior of power transistors/thyristors, which is a basic element of the power electronic circuit. In the case of photovoltaic (PV) generation, the output voltage and power generation vary depending on the amount of sunshine the photo cells are exposed to, which is based on the time of the day and the weather. A power conditioner and a storage system such as a battery are thus essential to operate a PV system. In the last section, voltage regulation on equipment in a dc railway is described when a lithium-ion battery is adopted, since this type of battery is used as a storage element for PV and wind farm generation systems. The first four chapters describe a transient analysis/simulation, which is based on a circuit theory derived by a transverse electro-magnetic (TEM) mode of wave propagation. When a transient involves a non-TEM mode of wave propagation, a circuit theory–based approach cannot provide an accurate solution. Typical examples include arcing horn flashover considering

Introduction

xvii

mutual coupling between power lines and tower arms, a transient in a grounding electrode, and an induced voltage from a lightning channel. Solving this type of transient requires the use of numerical electromagnetic analysis (NEA). Chapter 5 first discusses the basic theory of NEA and then describes various methods of NEA, for example, either in a frequency domain or in a time domain. It provides a brief summary of the methods and demonstrates application examples. Some of the examples compare field test results with EMTP simulation results. Chapter 6 further deals with electromagnetic compatibility (EMC)-related problems in a low-voltage control circuit in a power station and a substation. Electromagnetic disturbances experienced in Japanese utilities over a period of ten years are summarized and categorized based on the cause, that is, a lightning surge or a switching surge, and the incoming route. The influence of the disturbances on system operations and the countermeasures are explained together with case studies. Also, disturbances due to lightning in home appliances are explained based on collected statistical data, measured results, and EMTP/FDTD simulation results. Finally, an analytical method for evaluating electromagnetic-induced voltages on a telecommunication line or a gas pipeline from a power line is described. Nowadays, there are a number of numerical simulation tools that are used worldwide to analyze transients in power systems. The most well known among them is the EMTP. The accuracy and reliability of the original EMTP have been confirmed by a number of test cases since 1968. However, no simulation tool can be perfect. Any simulation tool will have its own application limits and restrictions. As mentioned previously, because the EMTP is based on a circuit theory under the assumption of TEM mode propagation, it cannot provide an accurate solution for a transient associated with a non-TEM mode propagation. Such application limits and restrictions are discussed in Chapter 7 for both circuit theory–based approaches and NEA methods. MATLAB® is a registered trademark of The MathWorks, Inc. For product information, please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 USA Tel: 508-647-7000 Fax: 508-647-7001 E-mail: [email protected] Web: www.mathworks.com

List of Symbols The symbols used in this book are listed together with the proper units of measurement, according to the International System of Units (SI). Angular frequency Conductance (admittance, susceptance) Conductivity Current density Decibel (dB)

Electric capacitance Electric current Electric field strength Electric resistance (impedance, reactance) Frequency Inductance: self, mutual Length

Magnetic field strength Magnetic flux Magnetic flux density Permeability Permittivity Potential difference, voltage, electric potential Power Resistivity Time, pulse rise time, pulse width Velocity

Radian per second (rad/s) Siemens (S)

ω

Siemens per meter (S/m) Ampere per square meter (A/m2) Decibel is a dimensionless number expressing the ratio of two power levels, W1 to W2: dB = 10 log (W1/W2) Further expressions of dB if both the voltages (U1, U2) or currents (I1, I2) are measured on the same impedance: dB = 20 log (U1/U2) dB = 20 log (I1/I2) Farad (F) Ampere (A) Volt per meter (V/m) Ohm (Ω)

σ

Hertz (Hz) Henry (H) Meter (m)

f L, M d, D, R, x (distance) r (radius) ℓ (length) h (height) δ (skin depth) λ (wavelength) H Φ

Ampere per meter (A/m) Weber (Wb) Tesla (T) Henry per meter (H/m) Farad per meter (F/m) Volt

G (Y, B)

J

C I E R (Z, X)

B μ ε V, U

Watt (W) Ohm meter (Ω·m) Second (s)

P ρ

Meter per second (m/s)

v

t, τ

xix

List of Acronyms The following list includes the acronyms frequently used in this book: AIS Air insulated substation ATP Alternative transients program CB Circuit breaker CM Common mode CT Current transformer DM Differential mode DS Disconnector EHV Extra-high voltage (330 kV ∼ 750 kV) EMC Electromagnetic compatibility EMF Electromotive force EMI Electromagnetic interference EMTP Electromagnetic transients program ESD Electrostatic discharge GIS Gas-insulated substation GPR Ground potential rise HV High voltage (1 kV ∼ 330 kV) IC Integrated circuit IEC International Electrotechnical Commission IKL Isokeraunic level LPS Lightning protection system LS Lightning surge SPD Surge protective device SS Switching surge TE Transverse electric TEM Transverse electromagnetic TL Transmission line TLM Transmission line model TM Transverse magnetic UHV Ultrahigh voltage (≥ 800 kV for ac and dc transmission) UNIPEDE International Union of Producers and Distributors of Electrical Energy VT Voltage transformer

xxi

International Standards

1. IEC 61000-4-5, Electromagnetic Compatibility (EMC)—Part 4-5: Testing and Measurement Technique—Surge Immunity Test, 2nd edn., 2005. 2. IEC 60364-5-54, Low-Voltage Electrical Installations—Part 5-54: Selection and Erection of Electrical Equipment—Earthing Arrangements and Protective Conductors, Edition 3.0, 2011. 3. IEC 61000-4-3, Electromagnetic Compatibility (EMC)—Part 4-3: Testing and Measurement Technique—Radiated, Radio Frequency Electromagnetic Field Immunity Test, Edition 3.1, 2008. 4. IEC 60050-161, International Electrotechnical Vocabulary—Chapter 161: Electromagnetic compatibility (EMC), 1st edn. (1990), Amendment 1 (1997), Amendment 2 (1988). 5. IEC 60050-604, International Electrotechnical Vocabulary—Chapter 604: Generation, transmission and distribution of electricity—Operation. Edition 1.0, 1987.

xxiii

1 Theory of Distributed-Parameter Circuits and the Impedance/Admittance Formulas

1.1 Introduction When investigating transient and high-frequency steady-state phenomena, all the conductors such as a transmission line, a machine winding, and a measuring wire show a distributed-parameter nature. Well-known lumpedparameter circuits are an approximation of a distributed-parameter circuit to discuss a low-frequency steady-state phenomenon of the conductor. That is, a current in a conductor, even with very short length, needs a time to travel from its sending end to the remote end because of a finite propagation velocity of the current (300 m/μs in a free space). From this fact, it should be clear that a differential equation expressing the behavior of a current and a voltage along the conductor involves variables of distance x and time t or frequency f. Thus, it becomes a partial differential equation. On the contrary, a lumped-parameter circuit is expressed by an ordinary differential equation since there exists no concept of the length or the traveling time. The aforementioned is the most significant differences between the distributedparameter circuit and the lumped-parameter circuit. In this chapter, a theory of a distributed-parameter circuit is explained starting from approximate impedance and admittance formulas of an overhead conductor. The derivation of the approximate formulas is described from the viewpoint of physical behavior of a current and a voltage on a conductor. Then, a partial differential equation is derived to express the behavior of a current and a voltage in a single conductor by applying Kirchhoff’s law based on a lumped-parameter equivalence of the distributed-parameter line. The current and voltage solutions of the differential equation are derived by assuming (1) sinusoidal excitation and (2) a lossless conductor. From the solutions, the behaviors of the current and the voltage are discussed.

1

2

Computation for Humanity: Information Technology to Advance Society

For this, the definition and concept of a propagation constant (attenuation and propagation velocity) and a characteristic impedance are introduced. As is well known, all the ac power systems are basically three-phase circuit. This fact makes a voltage, a current, and an impedance to be a 3D matrix form. A symmetrical component transformation (Fortescue’s and Clarke’s transformation) is well known to deal with the three-phase voltages and currents. However, the transformation cannot diagonalize an n by n impedance/admittance matrix. In general, a modal theory is necessary to deal with an untransposed transmission line. In this chapter, the modal theory is explained. By adopting the modal theory, an n-phase line is analyzed as n-independent single conductors so that the basic theory of a single conductor can be applied. To analyze a transient in a distributed-parameter line, a traveling-wave theory is explained for both single- and multiconductor systems. A method to introduce a velocity difference and attenuation in the multiconductor system is described together with a field test results. Impedance and admittance formulas of not ordinary conductors, such as a finite-length conductor and a vertical one, are also explained. Application examples of the theory described in this chapter are given so as to understand the necessity of the theory. Finally, the Electromagnetic Transients Program (EMTP), which has been widely used all over the world, is briefly explained. It should be noted that all the theories and formulas in this chapter are based on transverse electromagnetic (TEM) wave propagation.

1.2 Impedance and Admittance Formula In general, the impedance and admittance of a conductor are composed of the conductor internal impedance Zi and the outer-media impedance Z0. The same is applied to the admittance [1]:

Z = Zi + Z0 [ Ω/m ] (1.1)

Y = Yi + Y0 = jωP , P = Pi + P0 [ S/m ] (1.2)

where Zi is the conductor internal impedance Z0 is the conductor outer-media (space/earth-return) impedance Yi is the conductor internal admittance Y0 is the conductor outer-media (space/earth-return) admittance P is the potential coefficient matrix

Theory of Distributed-Parameter Circuits and the Impedance

3

It should be noted that the aforementioned impedance and admittance become a matrix when a conductor system is composed of multiconductors. Remind that a single-phase cable is, in general, a multiconductor system because the cable is consisting of a core and a metallic sheath or a screen. In an overhead conductor, there exists no conductor internal admittance Yi, except a covered conductor. 1.2.1 Conductor Internal Impedance Z i 1.2.1.1 Derivation of an Approximate Formula [1,2] Let’s obtain the impedance of a cylindrical conductor illustrated in Figure 1.1. We know that the dc resistance of the conductor is given in the following equation:

Rdc =

ρc [Ω/m ] (1.3) S

where S = π(r02 − r12 ) is the cross-sectional area[m 2 ] (1.4) r1 is the inner radius of the conductor [m] r0 is the outer radius of the conductor [m] ρc = 1/σe is the resistivity of the conductor [Ωm] σe is the conductor conductivity [S/m] μc is the permeability [H/m]

ρc , µc

ri

r0

FIGURE 1.1 A cylindrical conductor.

4


Also, it is well known that currents concentrate nearby the outer surface area of the conductor when the frequency of an applied (source) voltage (or current) to the conductor is high. This phenomenon is called “skin effect.” The depth “dc” of the cross-sectional area where most of the currents flow is given approximately as the (complex) penetration depth or the so-called skin depth in the following form: dc =

ρc = jωµc

1 [m ] (1.5) jωµc σc

The penetration depth is physically defined as the depth for an electromagnetic wave penetrating into a conductor when the wave hits the conductor surface. The physical concept of the penetration depth is very useful to explain the behavior of a current and a voltage on a conductor and also to derive impedance and admittance formulas of various conductor shapes and geometrical configuration. However, it should be reminded that the concept is based on TEM wave propagation and thus is not applicable to non-TEM propagation. Also, remind that it is just an approximation. By adopting the penetration depth, the internal impedance Zi in a highfrequency region can be derived in the following manner. Following the definition of the conductor resistance in Equation 1.3, the internal impedance is given by the ratio of the resistivity ρc and the crosssectional area S, which is evaluated as

{

S = π r02 − ( r0 − dc )

2

} = π(2r d − d ) 0 c

2 c

In a high-frequency region, dc is far smaller than the conductor outer radius r0. Thus, the following approximation is satisfied: S ≒ 2πr0 dc

for r0 dc

Substituting the earlier equation with Equation 1.5 into Equation 1.3,

Zhf ≒

ρc = 2πr0 dc

jωµcρc [Ω/m ] for a high frequency (1.6) 2πr0

The earlier formula can be found in many textbooks for a transmission line, power engineering, and a transient. Having known the low-frequency and high-frequency formulas, the internal impedance formula at any frequency is given as the following form by applying Rolle’s averaging theorem [2]:

 1 + jωµcS  2 Zi = Rdc + Zhf2 = Rdc  (1.7) 2   (Rdc ⋅ ) 


5

where S is the cross-sectional area of the conductor [m2] ℓ is the circumferential length of the conductor outer surface [m] It is easily realized that the earlier equation becomes identical to Equation 1.3 in a low-frequency region by assuming a small ω and to Equation 1.6 by assuming a large ω. It is noteworthy that the earlier equation is applicable to an arbitrary cross-sectional conductor, not necessarily a circular or a cylindrical conductor, because the equation is defined by the cross-sectional area “S” and the circumferential length of the conductor “ℓ,” but not by the inner and outer radii. For a low frequency, Rdc is much greater than Zhf in Equation 1.7. By adopting the approximation, 1+ x ≒ 1+

x 2

for x 1

Equation 1.7 is approximated in the following form: Zi ≒ Rdc +jωµ c /8π (1.8)

The earlier formula is well known as the conductor impedance at a power frequency. Example 1.1 Calculate the internal impedance (Rc and Lc) of a conductor with r1 = 1.974 mm, r2 = 8.74 mm, ρc = 3.78 × 10−8 Ω m, and μc = μ0 at frequency f = 50 Hz and 100 kHz. Solution

(

)

S = π r22 − r12 = π ( 8.74 + 1.974 ) ( 8.74 − 1.974 ) × 10 −6

= 10.714 × 6.766 × 10 −6 π = 72.49π × 10 −6 = 2πr2 = 17.48π × 10 −3 , Rdc =

ρc = 0.166 × 10 −3 Ω/m S

To calculate the square root of a complex number a + jb, it is better to rewrite the number in the following form so that we need only a real number calculation:

b a + jb = A ⋅ e jϕ , A = a 2 + b 2 , ϕ = tan −1   a

6


Thus, a + jb = A ⋅ e jϕ/2

In the internal impedance case, a = 1, b = ωμcS/Rdc · ℓ 2. At 50 Hz: b = 100π × 4π × 10−7 × 72.49 × 10−6 π/0.166 × 10−3 × 17.48 × 10−6 π2 =

4π × 72.49 × 10 −2 = 0.180 0.166 × 17.48 2

A = 1 + b 2 = 1.016,

Zc = 0.1673∠5.1° = 0.1667 + j14.87 × 10 −3 Ω/m

∴ Rc = 0.1667 Ω/km †, †Lc = 0.0473 †mH/km

A = 1.00797 , ϕ = 10.2°

If we adopt Equation 1.8, that is, Zc = Rdc + jωLc ∴ Rc = 0.166 Ω/km

Lc =

µc = 0.5 × 10 −7 H/m = 0.05 mH/km 8π

At 100 kHz: In the same manner as the aforementioned b = 359.19, Zc = 3.1452∠44.92°= 2.227 + j 2.221 Ω/km

Lc = 3.53 × 10 −3 mH/km m

In this case, b 1 ∴ Zc ≒ Rdc jb = ( 1 + j ) Rdc fµ c 2 [Ω/m ] = ( 1 + j ) 2.225 Ω/km It is clear that a further approximation for a low frequency and for a high frequency gives a satisfactory accuracy. The earlier results correspond to those given in Table 1.1 for a 500 kV transmission line in Figure 1.25. Because a phase wire is composed of four bundles, the earlier analytical results are four times of the internal impedance in the table. It is observed that the analytical results agree well with those in the table, which are calculated by the accurate formula in the following section (Equation 1.9).

1.2.1.2 Accurate Formula by Schelkunoff [3] The accurate formula of the internal impedance for a cylindrical conductor in Figure 1.1 was derived by Schelkunoff in 1934.

0.0416 0.0653 0.186 0.566 1.77

50 1k 10 k 100 k 1M

1.14E−02 8.56E−03 2.79E−03 8.85E−04 2.80E−04

Lc [mH/km]

3.57E−03 5.38E−02 0.175 0.556 1.76

Xc = ωLc [Ω/km] 0.048 0.883 7.28 46.9 218

Re

ln(2hp/rPe) = 5.2292, Ls = 1.058 mH/km, C = 10.62 nF/km

Rc [Ω/km]

f [Hz]

(a) Conductor Internal

0.739 0.455 0.261 0.117 0.0419

Le [mH/km] 0.232 2.86 16.4 73.7 263

Xe = ωLe [Ω/km]

(b) Earth Return

Self-Impedance of Phase a (Figure 1.25: hp = 16.67 m, he = 23.33 m)

TABLE 1.1

0.333 6.65 66.5 665 6650

Xs = ωLs [Ω/km]

(c) Space

0.0896 0.9483 7.466 47.47 219.77

R [Ω/km]

1.8110 1.5216 1.3218 1.1759 1.1002

L [mH/km]

(d) Total

0.5689 9.5604 83.051 738.84 6912.8

X = ωL [Ω/km]

Theory of Distributed-Parameter Circuits and the Impedance 7

8


1. Inner surface impedance zi  µ  {I 0 (x1 ) ⋅ K1(x2 ) + K 0 (x1 ) ⋅ I1(x2 )} z i = jω  c  ( x1D )  2π  2. Mutual impedance between the inner and the outer surfaces Zm zm =

ρc (2πr1r2D)

3. Outer surface impedance Z0 z0 = jω(µ c /2π{†I 0 (x2 ) ⋅ K1(x1 ) + K 0 (x2 ) ⋅ I1(x1 )} (x2D) (1.9) where x1 = ri /dc x2 = r0 /dc D = I1(x2 ) ⋅ K1 ( x1 ) − I1(x1 ) ⋅ K1(x2 )

In(x), Kn(x) is the modified Bessel function of kinds 1 and 2, respectively, with order n As is clear from the earlier equation, there exist three component impedances for a cylindrical conductor. In the case of a circular solid conductor, zi = zm = 0 for ri = 0, and z0 becomes

 µ  I (x ) z0 = jω  c  1 2 (1.10)  2π  I 2 ( x 2 )

When we say a conductor internal impedance (zi), it means z0, the outer surface impedance given in Equation 1.9 or in Equation 1.10 as far as an overhead line is concerned. However, in the case of a cable, zi is composed of a number of component impedances as in Equation 1.9 and also of an insulator impedance between metallic conductors, because the cable is, in general, composed of a core conductor carrying a current and a metallic sheath (shield or screen) for a current return path [4,5]. 1.2.2 Outer-Media Impedance Z0 1.2.2.1 Outer-Media Impedance The outer media of an overhead conductor are the air and the earth for the conductor is isolated by the air from the earth, which is a conducting medium. Therefore, the outer-media impedance Z0 of the overhead conductor is composed of the following two components:


Z0 = Zs + Ze

9

for an overhead line (1.11)

where Zs is the space impedance Ze is the earth-return impedance The outer-media impedance of an underground cable (insulated conductor) is the same as the earth-return impedance because the underground cable is surrounded only by the earth:

Z0 = Ze

for an underground cable (1.12)

When discussing the mutual impedance between an overhead conductor and an underground cable or a buried gas and/or water pipeline, the selfimpedance of the overhead conductor is given by Equation 1.11, while that of the underground conductor is given by Equation 1.12. The mutual impedance will be explained in Section 2.2.3. 1.2.2.2 Overhead Conductor 1.2.2.2.1 Derivation of an Approximate Formula By adopting the penetration depth “he” for the earth, the outer-media impedance of an overhead conductor is readily obtained based on the theory of image. Figure 1.2 illustrates a single overhead conductor and its image:

r

h

he

ρe , µe v=0 h + he

FIGURE 1.2 A single overhead conductor and its image.

10


he =

ρe (1.13) jωµ e

where ρe is the earth resistivity μe is the earth permeability In most cases, μe = μ0. Because the earth is not perfectly conducting, the earth surface is not the zero potential plane. Instead, the zero potential plane is located at the depth he from the earth surface. Then, the theory of image gives the following inductance Le [6]:  µ   2 ( h + he )  Le =  0  ln   (1.14) r  2π   

Thus, the outer-media impedance of the single overhead conductor is given by  2(h + he )  µ  Z0 = Ze = jωLe = jω  0  ⋅ ln   (1.15) r  2c   

For a multiconductor, illustrated in Figure 1.3, the outer-media impedance is obtained in the same manner as the aforementioned [6]:  Sij µ  Z0 ij = Zeij = jω  0  ⋅ Pij , Pij = ln  2 π    dij

  (1.16) 

where Sij2 = (hi + h j + 2he )2 + yij2 , dij2 = (hi − h j )2 + yij2 (1.17) hi, hj is the height of the ith and jth conductors, respectively yij is the horizontal separation between the ith and jth conductors Conductor “i” ρc “j”

r hi yij

ρe , εe

FIGURE 1.3 A multiconductor overhead line.

hj

11


Remind that the penetration depth is not a real value but a complex value and thus the zero potential plane at the depth he is just a concept and does not exist in physical reality. When the earth is perfectly conducting, that is, ρe = 0, then he = 0 in Equation 1.13. Therefore, Equation 1.16 becomes  Dij µ  Z0 ij = jω  0  ⋅ P0 ij , P0 ij = ln   2π   dij

  (1.18) 

where Dij2 = (hi + h j )2 + yij2 (1.19)

The earlier impedance is well known as the space impedance of an overhead conductor, that is,  µ   Dij Z0 ij = Z sij = jω  0  ln   2π   dij

  (1.20) 

For a single conductor, Dij = 2hi , dij = r

This is the reason why the space impedance is often mixed up with the earthreturn impedance. In fact, when the earth-return impedance is derived from Maxwell’s equation, the space impedance appears as a part of earth-return impedance [7,8]. Example 1.2 Calculate the earth-return impedance of a conductor with r = 0.1667 m, h = 16.67 m, and ρe = 200 Ωm at f = 50 Hz. Solution Similarly to Example 1.1, the impedance formula is rewritten in the following form so that only a real number calculation is necessary:  A Ze = 2πf × 10 −7 ϕ + j ln  2 d 

  

where

b A = a 2 + b 2 , ϕ = tan −1   , a = D2 + 2H1H e , b = 2H e ( H1 + H e ) , a

H1 = h1 + h2 , D2 = H12 + y 2 , d 2 = (h1 − h2 )2 + y 2 , H e =

2ρe ωµ 0

12


For a self-impedance, d = r, D = H1 = 2h, a = H1(H1 + 2He)

D = H1 = 33.34, H e = 1006.6, a = 68.232 × 10 3 , b = 2.094 × 10 6 ,

A = 2.095 × 10 6 ,

A ϕ = 88.13° = 1.538 rad, ln  2 d

 A   = ln  r 2  = 18.14   

∴ Ze = 0.0483 + j0.570 = 0.0483 + j(0.237 + 0.333)Ω/km

The result agrees well with that in Table1.1, which is calculated by the accurate Carson’s formula using EMTP Cable Constants (see Section 1.8, Table 1.19 (b) [5,24]). The earth-return impedance at a low frequency can be easily evaluated by an approximate formula derived from Equation 1.16 under the assumption that he ≫ h1, h2:  ρ    Ze = f + jf 8.253 + 0.628 ⋅ ln  e 2   [mΩ/km ] = 0.05 + j0.568 [mΩ/km ]  f ⋅ d    The earlier approximate result agrees well with that previously calculated by Deri’s formula.

1.2.2.2.2 Accurate Formula by Pollaczek [7] Pollaczek derived the following earth-return impedance in 1926: ∞

µ Zeij = jω 0  Poij + ( Q − jP )  , Q − jP = 2 F(x) ⋅ dx (1.21) 2π

∫

0

{

}

F ( x ) = exp − ( hi + h j ) cos ( yij ⋅ x )

( †x +

)

x 2 + m12 − m02 s (1.22)

where

m02 = jωµ0 ⋅jωε0 , m12 = jωµ e ( σe + jωε e ) , ε e :earth permitivity (1.23)

In the earlier equation, Q − jP is often called the correction term of the earthreturn impedance or the earth-return impedance correction. It should be clear that Poij gives the space impedance. m1 is called the intrinsic propagation constant of the earth. The infinite integral of the Pollaczek’s impedance is numerically very unstable and often results in numerical instability. However, the integral


13

can be numerically calculated by commercial software such as MAPLE and MATLAB® if special care is taken, for example, logarithmic integration [9]. 1.2.2.2.3 Carson’s Earth-Return Impedance [8] In the 1920s, there was no computer, and thus it was impossible to use Pollaczek’s impedance. Carson derived the same formula as the Pollaczek’s one neglecting the earth permittivity, that is, εe = ε0 in Equation 1.23, and further he derived a series expansion of the infinite integral in Equation 1.21. The detail of Carson’s expansion formula is explained in many publications, for example, Ref. [10]. 1.2.2.2.4 Admittance Almost always, the following well-known admittance is used in steady-state and transient analyses on an overhead line:

[Y ] = jω[C ] = jω[P0 ]−1 ,

P0 ij =

Dij 1 ln (1.24) dij 2πε0

For a single conductor, Y0 = jωC , C =

2πε0 ln(2h/r )

Wise derived the admittance formula considering an imperfectly conducting earth in 1948 [11]:

[Y ] = [Ye ] = jω[P]−1 (1.25)

∞

∫

Pij = P0 ij + M + jN , M + jN = 2 ( A + jB) dx (1.26)

0

{

} (a +1bx) . cos ( y ⋅ x ) (1.27)

A + jB = exp − ( hi + h j ) x

where

a 2 = x 2 + m12 − m02 , b =

m1 m0

ij

(1.28)

m02 = jωµ0 ·jωµ0 , m12 = jωµ e ( σe + jωε e ) same as Equation 1.23 Because of complicated infinite integral in Equation 1.26, similarly to Pollaczek’s impedance, the Wise’s admittance is, in most cases, neglected.

14


However, depending on the earth resistivity and the conductor height, the admittance for the imperfectly conducting earth should be considered especially in a high-frequency region, say, above some MHz. When a transient involves a transition between TEM wave and TM/TE waves, Wise’s admittance should be considered. Then, the attenuation constant differs significantly from that calculated by Equation 1.24. The numerical integration of Equation 1.26 can be carried out in a similar manner to that of the Pollaczek’s impedance by MAPLE or MATLAB. 1.2.2.2.5 Impedance and Admittance Formulation of an Overhead Conductor System Summarizing the earlier sections, the impedance and the admittance of an overhead conductor system are given in the following form: [ Z ] = [ Zi ] + [ Ze ][Ω/m ] [Y ] = [Y0 ][S/m ] (1.29) Zii: Equation 1.7 or the last equation of Equation 1.9 Zij = 0 Zeij: Equation 1.15, Equation 1.21, or Carson’s one Yij: Equation 1.24 or Equation 1.25 Remind that Equations 1.7 and 1.15 are an approximate formula for Zi and Ze, respectively. Also, Equation 1.24 is used almost always as an outer-media admittance. 1.2.2.3 Pollaczek’s General Formula for Overhead, Underground, and Overhead/Underground Conductor Systems Pollaczek derived a general formula that can deal with the earth-return impedances of overhead conductors, underground cables, and a multiconductor system composed of overhead and underground conductors in the following form [7,12]:

∞   µ  Ze = Z ( i ,†j ) = jω  0   K 0 ( mi d ) − K 0 ( mi D ) + F1 ( x ) ⋅ exp ( jyx ) ⋅ dx  (1.30)   2π   −∞  

∫

{−|h | x + m −|h | x + m } (1.31) F (s) = exp ( x +m + x +m ) 2

a

1

2

where m1 = jωµ 0 ( σ1 + jωε1 ) = jωµ 0σ1 − ω2µ 0ε1

m2 = jωµ0 ( σ1 + jωε 2 ) = jωµ0σ2 − ω2µ0ε 2 σ is the conductivity

2 i

2 1

b

2

2

2 2

2 j

15


ε is the permitivity μ0 is the permeability in free space y is the horizontal separation between conductors a and b h is the conductor height/depth, d2 = (ha − hb)2 + y2, D2 = (ha + hb)2 + y2 i, j are subscripts corresponding to media 1 and 2 in Figure 1.4 Assuming medium 1 is air, σ1 = 0 and ω2 μ0 ε1 = ω2 μ0 ε0 ≪ 1 yield m1 = 0. Thus, F2 (s) = exp

{−|h | a

x 2 + mi2 −|hb| x 2 + m 2j

(

x 2 + m2 + x

)

} (1.32)

where m2 = jωμ0 (σe + jωεe) ≅ jωμ0 σe = jα for soil. Equation 1.30 is rewritten depending on the position of conductors a and b. For example,

1. Overhead lines ha, hb ≥ 0; i = j = 1 (air)  2   2  D K 0 ( mi d ) − K 0 ( mi D ) = ln  − ln  = ln   for m1 = 0   d  rm1d   rm1D 

{

∞   exp jyx − ( ha + hb ) x  ∴ Ze = Z(1, 1) = jω(µ 0 /2π) ln ( D/d ) +    x 2 + m2 + x −∞  

∫

Conductor a y

Medium 1

ha ρ1, ε1

hb

Medium 2

ρ2, ε2

Conductor b FIGURE 1.4 A conductor system.

}  ⋅ dx  

 

(1.33)

16


2. Underground cable ha, hb ≤ 0; i = j = 2 (soil) µ  Ze = Z ( 2,†2 ) = jω  0   2π 

{

 ∞  exp jyx −|ha + †hb| x 2 + m 2   ×  K 0 ( md ) − K 0 ( mD ) +  x 2 + m2 + x  −∞   

∫

(

)

}  ⋅ dx   

  

(1.34)

3. Overhead/underground ha ≥ 0, hb ≤ 0; i = j = 2

K 0 ( mi d ) − K 0 ( mi D ) = 0 for i ≠ j ∞

∫

Ze = Z ( 1,†2 ) = jω ( µ 0 / 2π ) [exp{ jyx − ha |x|

−∞

+ hb x 2 + m 2 )}/ x 2 + m 2 +|x|] ⋅ dx (1.35) 1.2.3 Problems Calculate resistance Rc [Ω/km] and inductance Lc [mH/km] of a conductor with the radius r0 = 1 cm, r1 = 0, the resistivity ρc = 2 × 10−8 Ω m, and the permeability μc = μ0 = 4π × 10−7 [H/m] in Figure 1.1 at frequency f = 50 Hz and 100 KHz. 1.2 Calculate resistance Rc [Ω/km] and inductance Lc [mH/km] of a conductor with cross-sectional area S = 3.14 × 10−4[m 2], circumferential length ℓ = 6.28 cm, ρ c = 2 × 10−8 Ω m, and μ c = μ 0 at f = 50 Hz and 100 KHz. 1.3 Obtain an equivalent cylindrical conductor to a square conductor of 2 × 2 cm. 1.4 Calculate Re [Ω/km] and Le [mH/km] of the earth-return impedance for an overhead line with the radius r = 1 cm, h = 10 m, ρe = 100 Ω m, and μe = μ0 at f = 50 Hz and 100 KHz. 1.5 Discuss the difference between the conductor internal impedance and the earth-return impedance based on the results of Problems 1.1 and 1.4. 1.6 Derive a low-frequency approximate formula of the earth-return impedance from Equation 1.15 under the condition that |he| ≫ hi, hj. 1.7 Derive a high-frequency approximate formula of Equation 1.15 under the condition that |he| ≪ hi, hj by using the relation of ln(1 + x) ≒ x for x ≪ 1. 1.1

17


1.3 Basic Theory of Distributed-Parameter Circuit 1.3.1 Partial Differential Equations of Voltages and Currents Considering the impedance and the admittance explained in the previous sections, a single distributed-parameter line in Figure 1.5a is represented by a lumped-parameter equivalence as in Figure 1.5b. Applying Kirchhoff’s voltage law to the branch between nodes P and Q, the following relation is obtained: v − (v + ∆v) = R ⋅ ∆x ⋅ i + L ⋅ ∆x ⋅

di dt

Rearranging the earlier equation, the following result is given: −∆v di = R⋅i + L⋅ dt ∆x

By taking the limit of ∆x to zero, the following partial differential equation is obtained: −∂v ∂i = R ⋅ i + L ⋅ (1.36) ∂x ∂t

Similarly, applying Kirchhoff’s current law to node P, the following equation is obtained: −∂i ∂v = G⋅v + C ⋅ (1.37) ∂x ∂t

A general solution of Equations 1.36 and 1.37 can be derived in the following manner.

x

R ∆x

V

x + ∆x i + ∆i

P

i

C ∆x x

(a)

V + ∆V L ∆x

Q

x + ∆x

(b)

FIGURE 1.5 A single distributed-parameter line. (a) A distributed-parameter line. (b) A lumped-parameter equivalence.

18


1.3.2 General Solutions of Voltages and Currents 1.3.2.1 Sinusoidal Excitation Assuming v and i as sinusoidal steady-state solutions, Equations 1.36 and 1.37 can be differentiated with respect to time t. The derived partial differential equations are converted to ordinary differential equations, which makes it possible to obtain the solution of the earlier equations. By expressing v and i in polar coordinate, that is, in an exponential form, the derivation of the solution becomes straightforward. By representing v and i in a phasor form, V = V m exp( jωt), I = Im exp( jωt) (1.38)

Either real parts or imaginary parts of the earlier equations represent v and i. If imaginary parts are selected,

v = Im V  = Vm sin ( ωt + θ1 ) , (1.39)

I = Im  I  = I m sin ( ωt + θ2 ) ,

where V m = Vm exp( jθ1 ) Im = I m exp( jθ2 ) Substituting Equation 1.38 into Equation 1.36 and differentiate partially with respect to time t, the following ordinary differential equations are obtained: dV = RI + jωLI = (R + jωL)I = ZI dx (1.40) dI − = GV + jωCV = (G + jωC )V = YV dx −

where R + jωL = Z : series impedance of a conductor G + jωC = Y : shunt admittancce of a conductor

(1.41)

Differentiating Equation 1.40 with respect to x,

−

d 2 I d 2V dI dV = Z , − 2 = Y 2 dx dx dx dx

19


Substituting Equation 1.45 into the earlier equation,

d 2V , = ZYV dx 2

d 2 I = YZI (1.42) dx 2

where )1/2 : propagation constant with respect to volatge m −1  Γ v = (ZY  

(1.43)

)1/2 : propagation constant with respect to current m −1  Γ i = (YZ   When Z and Y are matrices, the following relation is given in general:

[Γv ] ≠ [Γi ]

[ Z ][Y ] ≠ [Y ][Z] (1.44)

since

Only when Z and Y are perfectly symmetrical matrices (symmetrical matrices whose diagonal entries are equal and nondiagonal entries are equal), [Γv] = [Γi] is satisfied. In case of a single-phase line, as Z and Y are scalars,

and Γ = ZY = YZ (1.45) Γ v2 = Γ i2 = Γ 2 = ZY

Substituting the earlier equation into Equation 1.42,

d 2V 2 d 2 I 2 = Γ V ,† = Γ I (1.46) 2 dx dx 2

A general solution is obtained solving one of the earlier equations. Once it is solved for V or I, Equation 1.40 can be used to derive the other solution. The general solution of Equation 1.46 with respect to voltage is given by

(

)

( )

V = A exp −Γ x + B exp Γ x (1.47)

where A, B are the integral constants determined by a boundary condition. The first equation of Equation 1.42 gives the general solution of current in the following differential form:

dV = Z −1Γ A exp −Γ x − B exp Γ x (1.48) I = −Z −1 dx

{

(

)

( )}

20


The coefficient of the earlier equation is rewritten as Γ YZ Y = = = Z Z Z

Y Y = = Y0 Γ ZY

where

Y 1 Y0 = = : characteristic admittance [ S ] (1.49) Z Z0 Z Z 0 = : characteristic impedance [ Ω ] Y

In general cases, when Z and Y are matrices,

[ Z0 ] = [Γv ]−1 [ Z ] = [Γv ][Y ]

−1

[Y0 ] = [Z0 ]

−1

= [Z]

−1

[Γv ] = [Y][Γv ]

(1.50)

−1

Substituting Equation 1.49 into Equation 1.48, the general solution of Equation 1.46 with respect to current is expressed as

{

(

( )} (1.51)

)

I = Y0 A exp −Γ x − B exp Γ x

Exponential functions in Equations 1.47 and 1.51 are convenient in order to deal with a line with an infinite length (infinite line), but hyperbolic functions are better preferred for treating a line with a finite length (finite line). To obtain an expression by the hyperbolic functions, new constants C and D are defined as C + D C − D A = , B = 2 2

Substituting the aforementioned into Equations 1.47 and 1.51,

( )

( )

( )

(

 exp Γ x − exp −Γ x  exp Γ x + exp(−Γ x)  V = C   + D  2 2   

(

  exp Γ x − exp −Γ x  I = −Y0 C    2  

) 

 

)  + D  exp ( Γ x ) + exp ( −Γ x )   

 

2

 

21


From the definitions of the hyperbolic functions,

(

)

V = C cosh Γ x + D sinh Γ x , I = −Y0 C cosh Γ x + D sinh Γ x (1.52)

Constants A, B, C, and D defined here are arbitrary constants and are determined by boundary conditions. 1.3.2.2 Lossless Line When a distributed line satisfies R = G = 0, the line is called “lossless line.” In this case, Equations 1.36 and 1.37 are written as −

∂v ∂i ∂i ∂v =L , − =C (1.53) ∂x ∂t ∂x ∂t

Differentiating the earlier equation with respect to x, −

∂ 2v ∂ 2i = L − =C ∂x 2 ∂x 2

Similarly to the sinusoidal excitation case, the following equations for the voltage and current are obtained: −

∂ 2v ∂(∂i/∂x) ∂(−C∂v/∂t) ∂ 2v =L =L = −LC 2 2 ∂x ∂t ∂t ∂t ∴

∂ 2v ∂ 2v = LC ∂x 2 ∂t 2

and

∂ 2i ∂ 2i = LC (1.54) ∂x 2 ∂t 2

where c0 = 1/ (LC ) [ m/s ] . From Equation 1.14 with he = 0 and (1.24), LC =

2h 1 µ 0 2h ln ⋅ 2πε0 / ln = µ 0ε0 = 2 r r c0 2π

Thus, c0 =

1 1 = = 3 × 108 [ m/s ] : light velocity in free space (1.55) LC µ 0ε0

22


Equations 1.54 are linear second-order hyperbolic partial differential equations and called wave equations. The general solutions of the wave equations are given by D’Alembert in 1747 [13] as

v = ve f ( x − c0t ) + eb ( x + c0t ) with variable of distance (1.56)

i = Y0 e f ( x − c0t ) − eb ( x + c0t )

  x    x  v = E f  t −    + Eb  t +    with variable of tim me (1.57) c   0    c0  

   x     x    i = Y0 E f  t −    − Eb  t +       c0       c0  

{

}

where c 0C = Z0 =

1 C = Y0 : surge admittance[S] C= L LC (1.58)

1 L = : surge impedance [Ω] Y0 C

Surge impedance Z0 and surge admittance Y0 in the earlier equation are extreme values of the characteristic impedance and admittance in Equation 1.49 for frequency f → ∞. The earlier solution is known as a wave equation and shows a behavior of a wave traveling along the x-axis by the velocity c0. It should be clear that the value of functions ef, eb, Ef, and Eb does not vary if x − c0t = constant and x + c0t = constant. Since ef and Ef show a positive traveling velocity, they are called “forward traveling wave”: c0 = x/t along x-axis to positive direction In contrast, eb and Eb are “backward traveling wave,” which means the wave travels to the direction of −x, that is, the traveling velocity is negative:

c0 =

−x t

Having defined the direction of the traveling waves, Equation 1.56 is rewritten simply by

v = e f + eb , i = Y0 ( e f − eb ) = i f − ib (1.59)

where ef, eb are the voltage traveling waves if, ib are the current traveling waves


23

The aforementioned is a basic equation to analyze traveling-wave phenomena, and the traveling waves are determined by a boundary condition. The detail will be explained in Section 1.6. 1.3.3 Voltages and Currents on a Semi-Infinite Line Here, we consider a semi-infinite line as shown in Figure 1.6. The ac constant voltage source is connected to the sending end (x = 0), and the line extends infinitely to the right-hand side (x = +∞). 1.3.3.1 Solutions of Voltages and Currents From the general solutions in Equations 1.47 and 1.51, solutions of voltages and currents on a semi-infinite line in Figure 1.6 are obtained by adopting the following boundary conditions: V = E at x = 0, and V = 0 at x = ∞ (1.60)

The boundary condition in the second equation in the aforementioned is obtained from the physical constraint in which all the physical quantities have to be zero at x → ∞. Substituting the condition into Equation 1.47,

(

)

( )

0 = A exp −Γ ∞ + B exp Γ ∞

) = ∞ , constant B has to be zero in order to satisfy the earlier Since exp(Γ∞ equation: B = 0

x=0

x=x I(x)

E

FIGURE 1.6 Semi-infinite line.

V(x)

24


Thus,

(

0 = A exp −Γ ∞

)

Substituting the first equation of Equation 1.60 into the earlier equation, constant A is found as A = E (1.61)

Substituting constants A and B into the general solutions, that is, Equations 1.47 and 1.51, voltages and currents on a semi-infinite line are given in the following form:

(

)

(

)

(

)

V = E exp −Γ x , I = Y0E exp −Γ x = I0 exp −Γ x (1.62)

where İ0 = Y·0Ė. 1.3.3.2 Waveforms of Voltages and Currents Since Γ is a complex value, it can be expressed as

Γ = α + jβ (1.63)

Substituting the aforementioned into the voltage of Equation 1.62,

V = E exp {−(α + jβ)x} = E exp(−α x)exp(− jβ x) (1.64)

If the voltage source at x = 0 in Figure 1.6 is a sinusoidal source,

E = Em sin(ωt) = Im {Em exp( jωt)} (1.65)

The voltage on a semi-infinite line is expressed by the following equation: v = Im(V ) = Im {Em exp( jωt)exp(−α x)exp(− jβ x)}

(1.66)

∴ v = Em exp(−α x)sin(ωtt − β x)

Figure 1.7 shows the voltage waveforms whose horizontal axis is set to time when the observation point is shifted from x = 0 to x1, x2, …. The figure illustrates as the observation point shifts in the positive direction, the amplitude of the voltage decreases due to exp(−αx), and the angle of the voltage lags due to exp(−jβ x).

25


x v

Em

)

αx

(–

p ex

–E

e

m

x)

Zero-potential line

–α

( xp

O

x=0

x = x3 x = x2 x = x1 t

FIGURE 1.7 Three-dimensional waveforms of the voltage.

The horizontal axis is changed to the observation point and look at the voltage waveforms at different times in Figure 1.8. Rewriting Equation 1.66,

 ωt  v = −Em exp(−α x)sin β  x −  (1.67) β  

Figure 1.8 illustrates that the voltage waveform travels in the positive direction of x as time passes according to Equation 1.67. 1.3.3.3 Phase Velocity The phase velocity is found from two points on a line whose phase angles are equal. For example, in Figure 1.8, x1 (point P1) and x2 (point Q1) determine the phase velocity. From Equation 1.67, the following relationship is satisfied as phase angles are equal:

x1 −

ωt1 ωt = x2 − 2 β β

26


v Em exp(–αx)

t2 O

x1

x2

x3

Q1

P1

P2

x

t1 t=0

λ Wave length

FIGURE 1.8 Voltage waveforms along x-axis at different times.

The phase velocity c is found from the earlier equation as c=

x2 − x1 ω = (1.68) t2 − t1 β

The earlier equation shows that the phase velocity is found from ω and β and is independent of the location and time. For a lossless line, Z = jωL, Y = jωC (1.69)

2πε0  µ   2h  † L =  0  ⋅ ln   , C = ln(2h/r )  2π   r 

From Equation 1.80,

= jω LC = jβ and β = ω LC (1.70) Γ = ZY

As a result, for a lossless line, the phase velocity is found from Equations 1.68 and 1.70 as Equation 1.55. The phase velocity in a lossless line is independent of ω.

27


1.3.3.4 Traveling Wave When a wave travels at constant velocity, it is called traveling wave. The general solutions of voltages and currents in Equations 1.56 and 1.57 are trav ) and exp(Γx ) in the general solueling waves. In a more general case, exp(−Γx tions, that is, Equations 1.47 and 1.51, also express traveling waves. The existence of traveling waves is confirmed by various physical phenomena around us. For example, when we drop a pebble in a pond, waves travel to all directions from the point where the pebble dropped. These waves are traveling waves. If a leaf is floating in a pond, it does not travel along with the waves. It only moves up and down according to the height of the waves. Figure 1.9a shows the movement of the leaf and water surface in xand y-axes. Here, x is the distance from the origin of the wave, and y is the height. Figure 1.9b illustrates the movement (past history) of the leaf along with time. Figure 1.9 demonstrates that the history of the leaf coincides with the shape of the wave. This observation implies that water in the pond does not travel along with the wave. What is traveling in the water is the energy given by the drop of the pebble, and water (medium) in the pond only carries the transmission of the energy. In other words, the traveling wave is the travel of energy, and the medium itself does not travel. Maxwell’s wave equations can thus be considered as the expression of the travel of energy, which means that the characteristics of energy transmission can be analyzed as those of traveling waves. For example, propagation velocity of the traveling wave corresponds to the propagation velocity of energy. Water surface

t=0

Leaf t1 t2 t3 t4 t5 y

t6 t7 y O (a)

t8 x1

x

0

t1

t3

t5

t7

t

(b)

FIGURE 1.9 Movement of a leaf on a water surface. (a) Position of a leaf as a function of x, y, and t. (b) y − t carve of leaf movement.

28


1.3.3.5 Wave Length The wave length is found from two points on a line whose phase angles are 360° apart at a particular time. For example, x1 (point P1) and x3 (point P2) in Figure 1.8 determine the wave length λ at t = 0: λ = x3 − x1 (1.71)

Since phase angles of the two points are 360° apart, the following equation is satisfied from Equation 1.66: (ωt1 − β x1 ) − (ωt1 − β x3 ) = 2π

∴ β(x1 − x3 ) = 2π

The wave length is found from the earlier equation and Equation 1.71 as λ=

2π (1.72) β

The earlier equation shows that the wave length is a function of β and independent of the location and time. For a lossless line, using Equation 1.70, λ=

2π 1 c = = 0 (1.73) f ω LC f LC

1.3.4 Propagation Constants and Characteristic Impedance 1.3.4.1 Propagation Constants The propagation constant Γ is expressed as follows:

= α + jβ (1.74) Γ = ZY

where α is attenuation constant [Np/m] β is the phase constant [rad/s] Let us consider the meaning of the attenuation constant using the semiinfinite line case as an example. From Equation 1.62 and the boundary conditions,

V 0 = V ( x = 0 ) = E at x = 0


29

V x = V ( x = x ) = E exp(−Γx) at x = x

The attenuation after the propagation of x is V x = exp −Γ x = exp(−αx)exp(− jβx), V 0

(

)

V x = exp(−αx) (1.75) V 0

From the earlier equation, αx = aT = −ln

V x  Np (1.76) V 0

The attenuation per unit length is α=

1 Vx αT = − ln [Np/m] x x V 0

The earlier equation shows that the attenuation constant gives the attenuation of voltage after it travels for a unit length. Now, we find propagation constants for a line with losses, that is, a line whose R and G are positive. From Equation 1.63,

= ( R + jωL ) ( G + jωC ) = α 2 − β2 + 2 jαβ Γ 2 = ZY

∴α 2 − β2 = RG − ω2LC , 2αβ = ω(LG + CR)

Also, α 2 + β2 = (R 2 + ω2L2 )(G 2 + ω2C 2 )

From earlier equations, the following results are obtained:

2α 2 =

(R

2β2 =

(R

)

2

+ ω2L2 (G 2 + ω2C 2 ) + (RG − ω2LC )†

2

+ ω2L2 G 2 + ω2C 2 − (RG − ω2LC )†

)(

)

30


Since αβ is positive, α and β have to have the same sign, both positive: α=

β=

{ (R + ω L )(G + ω C ) + (RG − ω LC)} 2  (1.77) 2

{

2 2

2

2

2

2

  (R + ω L )(G + ω C ) − (RG − ω LC) 2   2

2 2

2

2

2

2

}

Here, we find the characteristics of α and β defined in the preceding text. First, when ω = 0, α = RG , β = 0; ω = 0 (1.78)

For ω → ∞, using the approximation 1 + x ≈ 1 + x/2 for x << 1, R 2 + ω2L2 = ωL 1 +

 R2 R2  ω ≈ + L 1   ω2L2 2ω2L2  

 G2  G 2 + ω2C 2 ≈ ωC  1 + 2 2  2ω C  

Substituting the aforementioned into Equation 1.77,

α=

C/L R + L/C G , β = ω LC ; ω → ∞ (1.79) 2

Considering Equations 1.78 and 1.79, the frequency responses of α and β are found as in Figure 1.10. ω√ LC

α, β

1 2

C R+ L

α √ RG

β O

FIGURE 1.10 Frequency characteristics of α and β.

ω

L G C

31


Equation 1.79 shows that the propagation velocity at ω → ∞ is lim c =

ω→∞

1 = c0 (1.80) LC

The propagation velocity c0 in the earlier equation coincides with the propagation velocity for a lossless line in Equation 1.55. 1.3.4.2 Characteristic Impedance For a single-phase lossless overhead line in the air, the characteristic impedance is found from Equations 1.49 and 1.69: Z 0 =

L µ 0 /2π 2h 2h Z = ln = ≈ 60ln [ Ω ] (1.81) C 2πε0 r r Y

The earlier equation shows that the characteristic impedance becomes independent of frequency for a lossless line and it is called surge impedance as defined in Equation 1.58. For a line with losses, the characteristic impedance is found as (R + jωL)(G − jωC ) R + jωL = (1.82) G + jωC G 2 + ω2C 2

Z 0 =

The characteristic impedance is defined as Z 0 = r + jx (1.83)

The real part r and the imaginary part x of the characteristic impedance are found in the same way as we found α and β: r=

x=

{ ( R + ω L ) (G + ω C ) + ( RG + ω LC )} {2 (G + ω C )}  (1.84)  { ( R + ω L ) (G + ω C ) − ( RG + ω LC )} {2 (G + ω C )}  2

2 2

2

2

2

2

2

2

2

2

2 2

2

2

2

2

2

2

2

From the earlier equation,  R R , x = 0 that is Z0 = ; ω=0  G G  (1.85)   L L , x = 0 that is Z0 = ; ω → ∞ r= C C  r=

32


The earlier equation shows that the characteristic impedance for ω→∞ coincides with the surge impedance of a lossless line in Equation 1.81. 1.3.5 Voltages and Currents on a Finite Line 1.3.5.1 Short-Circuited Line In this section, we consider a line with a finite length (finite line) whose remote end is short-circuited to ground as illustrated in Figure 1.11. To deal with a finite line, the general solution in the form of hyperbolic functions as in Equation 1.52 is convenient. Boundary conditions in Figure 1.11 are V = E V = 0

x = 0  (1.86)  x = l 

at at

Substituting the earlier condition into Equation 1.52, the unknown constants C and D are determined as

E = C

0 = C cosh Γ l + D sinh Γ l

Substituting the aforementioned C and D into Equation 1.52, the following solutions are obtained:

(

E sinh Γ l cosh Γ x − cosh Γ l sinh Γ x cosh Γ l x = V = E cosh Γ x − E sinh Γ sinh Γ l sinh Γ l

)

E sinh Γ(l − x) (1.87) = sinh Γ l

x=0

x=l

l x

I0 E

FIGURE 1.11 A short-circuited line.

I1

I V

33


Similarly, Y0E cosh Γ ( l − x ) (1.88) I = sinh Γ l

The current at the sending end (x = 0) is Y E cosh Γ l = Y0E coth Γ l (1.89) I0 = I ( x = 0 ) = 0 sinh Γ l

The solution of the current in Equation 1.88 is rewritten by using I0:

I cosh Γ( l − x) I = 0 (1.90) cosh Γ l

The current at the remote end (x = 𝑙) is Il = I ( x = l ) =

Y0E I0 = (1.91) sinh Γ l cosh Γ l

The impedance of the finite line seen from the sending end is given as a function of the line length l: 1 E = Z 0 tanh Γ l (1.92) Z ( l ) = = I 0 Y0 coth Γ l

Figure 1.12 shows an example of |Ż(l)|. For l →∞, since tanh (∞) → 1, Z (l = ∞) = Z 0

For a lossless line, Z 0 =

L , Γ = jω LC C

Using the relationships sinh jx = j sin x and cosh jx = cos x, the solutions of the voltage and the current are expressed as

V =

{

} E ,

sin ω LC (l − x) sin(ω LCl)

{

}

C cos ω LC (l − x) I = − j E (1.93) L sin(ω LCl)

34


1600

|Z(l)| (Ω)

1200

800 |Z0|

r

400

h

0

λ/4

λ/2

3λ/4

λ

l FIGURE 1.12 Input impedance |Z(ℓ)| of a short-circuited line r = 1.3 mm, h = 30 cm, and f = 800 Hz.

In the earlier equations, the voltage and the current become infinite when the denominator is zero. This condition is referred to as the resonant condition. The denominators become zero when

(

)

sin ω LCl = 0 ∴ ω LCl = nπ; n : positive integers (1.94)

Therefore, natural resonant frequencies are found as

fSn =

ω nπ n (1.95) = = 2π 2π LCl 2 LCl

Infinite numbers of fSn exist for different n. The natural resonant frequency for n = 1 is called as the fundamental resonant frequency. Let’s define τ as the propagation time for the voltage and the current on a line with the length l. The propagation time τ is given by

τ=

l = LCl (1.96) c0

Using the propagation time τ, the natural resonant frequencies and the fundamental resonant frequency are expressed as

fSn =

n , 2τ

fS1 =

1 (1.97) 2τ

35

|Z(l)|=Z(l)/j


π/2

π

3π/2

2π

θ (l)

5π/2

FIGURE 1.13 |Z(ℓ)| − θ characteristic of a lossless short-circuited line.

The input impedance Z(l) of the finite line seen from the sending end is also rewritten for a lossless line as follows: Z (l) = j

L tan (ω LC l) (1.98) C

Figure 1.13 shows the relationship between |Ż(l)| and θ = LCl (or l) for a lossless line. The relationship coincides with Foster’s reactance theorem. The line is in a resonant condition for = nπ; n: positive integers, and the line is in an antiresonant condition for θ = (2n − 1)π/2. 1.3.5.2 Open-Circuited Line In this section, we consider a finite line whose remote end is opened as shown in Figure 1.14. x=0

I0

x=x

I1 = 0

I

V

FIGURE 1.14 An open-circuited line.

x=l

36


For this line, boundary conditions are defined as V = E at x = 0

I = 0 at x = l (1.99)

In a similar manner to a short-circuited line, the solutions of the voltage and the current are obtained in the following form: E cosh Γ (l − x) V = cosh Γ l

(1.100)

Y E sinh Γ (l − x) I = 0 cosh Γ l

The input impedance of the finite line seen from the sending end is expressed as E Z (l) = = Z 0 coth Γ l (1.101) I0

Figure 1.15 shows an example of the relationship between |Z(l)| and l. For a lossless line, the solutions of voltage and current are expressed as V =

{

} E ,

cos ω LC (l − x)

(

cos ω LCl

)

{

}

C sin ω LC (l − x) I = j E (1.102) L cos ω LCl

(

)

1600

|Z(l)| (Ω)

1200

800

|Z0|

400

0

λ/4

λ/2

3λ/4 l Between conductors

λ

FIGURE 1.15 Input impedance of an open-circuited line. r = 1.3 mm, y = 60 cm between conductors, and f = 800 Hz.

37


The line is in a resonant condition when the denominator of the earlier equations is zero: ω LCl =

(2n − 1)π ; n : positive integers (1.103) 2

Therefore, natural resonant frequencies are found as fOn =

ω (2n − 1)(π/2) (2n − 1)c0 2n − 1 (1.104) = = = 2π 4l 4τ 2π LCl

The fundamental resonant frequency is fO1 =

1 (1.105) 4τ

As fs1 = 1/2τ for a short-circuited line, fs1 = 2f01. The input impedance for a lossless line seen from the sending end is

(

)

L cot ω LC l (1.106) Z (l) = − j C

|Z(l)| = Z(l)/j

Figure 1.16 shows the relationship between |Ż(l)| and θ = LCl for a lossless line. As in a short-circuited line, the relationship coincides with Foster’s reactance theorem. The line is in a resonant condition for = (2n − 1)π/2; n: positive integers as in Equation 1.103.

0

λ 4

λ 2

3λ 4

λ

5λ 4

π 2

π

3 π 2

2π

5 π 2

Resonance

Anti-resonance

FIGURE 1.16 |Z(ℓ)| − θ characteristic of a lossless open-circuited line.

l θ = ω √ LCl

38


1.3.6 Problems 1.8 Prove α(ω = 0) < α(ω = ∞) in Figure 1.10. 1.9 Obtain the characteristic impedance Z0(ω) for ω → 0 in an R, L, C, and G line. 1.10 Obtain the sending-end current Is in an R, L, C, and G line for ω = 0 in a short-circuited (finite-length) line. In a real overhead line, G ≒ 0 in general. Then, what is the current Is? 1.11 Calculate Vr for E = 1000 · cos(ωt) [V] with f = 50 [Hz], and Z0 = L/C = 300[Ω] on a lossless line with the length ℓ = 300 km by using (a) F-parameter and (b) π-equivalent circuit in the following cases: (1) Zr = 1 [Ω], (2) Zr = 300 [Ω], (3) Zr = ∞.

1.4 Multiconductor System 1.4.1 Steady-State Solutions Equations 1.40 through 1.42 hold true for a multiconductor system shown in Figure 1.17, provided that all the coefficients Z, Y, R, L, G, and C are now matrices and variables V and I are vectors of the order n in an n-conductor system. The matrix P is defined as P = ZY (1.107)

where P = [P]: n × n matrix and in general P ≠ YZ. (V )

(V+∆V) (I)

(I-∆I) (Z)∆x

(Y)∆x

FIGURE 1.17 A multiconductor system.


39

Since Z and Y are both symmetrical matrices, the transposed matrix of P is found as Pt = (ZY )t = Yt Zt = YZ (1.108)

Here, the subscript t means the matrix is transposed, and Pt = [P]t: n × n matrix. From Equations 1.42 and 1.10,

d 2V = PV , dx 2

d2I = Pt I (1.109) dx 2

As in Equation 1.47, the general solution of the earlier equation is expressed as

V = exp(− P1/2x)Vf + exp( P1/2x)Vb (1.110)

where Vf and Vb are arbitrary n-dimension vectors. The first term of the right-hand side of Equation 1.110 expresses the wave propagation in the positive direction of x (forward traveling wave). The second term of the right-hand side corresponds to the wave propagation in the negative direction of x (backward traveling wave). Equation 1.110 shows that the voltage at any point of a line can be found by the sum of the forward and backward traveling waves. Since I = −Z−1dV/dx as in Equation 1.48, the current can be given as

{ (

)

}

I = Z −1P1/2 exp − P1/2x Vf − exp( P1/2x)Vb (1.111)

For a semi-infinite line, Equations 1.110 and 1.116 are simplified since Vb = 0 in the following form:

V = exp(− P1/2x)Vf

I = Z −1P1/2 exp(− P1/2x)Vf = Z −1P1/2V (1.112)

Equation 1.112 shows that the proportion of current to voltage at any point in a semi-infinite line, that is, the characteristic admittance matrix, is defined as follows:

Y0 = Z −1P1/2 (1.113)

Since Z0 = Y0−1, the characteristic impedance matrix is

Z0 = P −1/2Z (1.114)

40


The general solution of current can also be found from the second equation of Equation 1.109:

I = exp(− Pt1/2x) I f + exp( Pt1/2x) I b (1.115)

Using the second equation of Equation 1.40, the voltage in a semi-infinite line can also be found as follows since Ib = 0: V = Y −1Pt1/2 I (1.116)

From the earlier equation, the characteristic impedance and admittance matrices are

Z0 = Y −1Pt1/2 , Y0 = Pt −1/2Y (1.117)

In general, the characteristic impedance and admittance matrices are expressed by Equations 1.113 and 1.109 using P instead of Pt. Another way to express the characteristic impedance and admittance matrices can be found by integrating the second equation of Equation 1.109:

∫

I = −Y Vdx (1.118)

For a semi-infinite line, substituting the first one of Equation 1.112 into the earlier equation,

I = −Y(− P −1/2 )exp(− P1/2x)Vf = YP −1/2V (1.119)

Therefore, the characteristic impedance and admittance matrices are found as

Z0 = P1/2Y −1 , Y0 = YP −1/2 (1.120)

The earlier equation produces the same matrices as Equation 1.113. For example, for the characteristic admittance matrix, Y0 = ((YP −1/2 )−1 )−1 = ( P1/2Y −1 )−1 = ( P −1/2 PY −1 )−1 = ( P −1/2 (ZY )Y −1 )−1

= ( P −1/2Z)−1 = Z −1P1/2

(1.121)

The characteristic impedance and admittance matrices are symmetrical matrices. For example, for the characteristic impedance matrix,

Z0t = ( P1/2Y −1 )t = Yt −1Pt1/2 = Y −1Pt1/2 = Z0 (1.122)


41

Here, Y = Yt since Y is a symmetrical matrix. Therefore, Z0t = Z0 , Y0t = Y0 (1.123)

P is not a symmetrical matrix in general, but Z0 and Y0 are always symmetrical matrices. 1.4.2 Modal Theory The modal theory, which is established by L. M. Wedepohl in 1963 [14], provides the essential technique to solve for voltages and currents in a multiconductor system. Without the modal theory, propagation constants and characteristic impedances of a multiconductor system cannot be found precisely, except for an ideally transposed line. One may assume an ideally transposed line or perfectly conducting earth and find solutions of voltages and currents in a multiconductor system using symmetrical coordinate transformation [15,16]. However, it does not produce precise solutions of voltages and currents since an ideally transposed line and perfectly conducting earth do not exist in an actual system. Before the modal theory was established, propagation constants and characteristic impedances were found by expanding matrix functions to a series of polynomials. This section discusses propagation constants and characteristic impedances and admittance matrices in the modal domain after reviewing the modal theory. 1.4.2.1 Eigenvalue Theory Let us define matrix P as a product of series impedance matrix Z and shunt admittance matrix Y for a multiconductor system:

[ P ] = [ Z ][Y ] (1.124)

where [Z] and [Y] are n × n off-diagonal matrices. Applying the eigenvalue theory, off-diagonal matrix P can be diagonalized by the following matrix operation:

[ A] [ P ][ A] = [Q] = [U ] ( Q ) , [ A][Q][ A] −1

where [Q] is the n × n eigenvalue matrix of [P] [A] is the n × n eigenvector matrix of [P] (Q) is the eigenvalue vector [U] is the identity (unit) matrix

−1

= [ P ] , (1.125)

42


The notation of matrix [ ] and vector ( ) is, hereafter, omitted for simplification. Rewriting the earlier equation, PA = AQ , ∴ PA − AQ = 0 (1.126)

Since Q is the diagonal matrix, only the kth column of A is multiplied by the kth diagonal entry of Q when calculating AQ. Therefore, the following equation is satisfied for each k: Ak Qk = Qk Ak ; k = 1, 2, … , n (1.127)

The following equation is obtained for the kth column by substituting the earlier equation into Equation 1.126: ( P − QkU )Ak = 0 (1.128)

The earlier equation is a set of n equations with n unknowns. The determinant of (P − QkU) has to be zero in order to have the solutions Ak ≠ 0. det( P − QkU ) = 0 (1.129)

Equation 1.126 is the nth order polynomial with unknown Qk and is called characteristic equation. Eigenvalues of P (i.e., Qk) are found as the solutions of the characteristic equation. Eigenvector Ak is found from Equation 1.126 for each eigenvalue of P. Since the determinant of (P − QkU) is zero for the obtained Qk, eigenvector Ak is not uniquely determined. Thus, one element of Ak can take an arbitrary value, and the other elements are determined according to it, satisfying the proportional relationship. Eigenvectors Ak have to be linearly independent to each other. This is especially important when some eigenvalues of P are equal, that is, when the characteristic equation has repeated roots. As discussed in the previous sections, the analysis of a multiconductor system requires a number of computations of functions. The application of the eigenvalue theory makes it easy to calculate matrix functions. This is a major advantage of the eigenvalue theory. One way to calculate matrix functions without the eigenvalue theory is to use the series expansion. For example, the following series expansions are often used to calculate matrix functions: 1+ x ≈ 1+

x x3 x2 , sinh ( x ) ≈ x + , cosh ( x ) ≈ 1 + , 2 6 2

tanh ( x ) ≈ 1 −

x3 ; †x 1 2

43


exp ( x ) = 1 + x +

x2 xn ++ + , †x < ∞ (1.130) 2! n!

Using the aforementioned, the exponential function of matrix P, for example, is found as

[P]

2

exp ([ P ]) ≈ [U ] + [ P ] +

;

2!

P 1 (1.131)

Using the eigenvalue theory, a matrix function is given by f ([ P ]) = [ A] f ([Q ]) [ A] (1.132) −1

where Q and A are the eigenvalue matrix and eigenvector matrix of P, respectively. For example, [P]1/2 can be calculated simply by

[P]

1/2

Q11/2  0 1/2 where [Q ] =     0

0

Q21/2 0

= [ A][Q ]

1/2

[ A]

−1

    =     1/2 Qn    0 0

(1.133)

Q1

0

0

Q2

0

0

0   0    Qn 

The exponential function exp ([P]) can be calculated as exp ([ P ]) = [ A] exp ([Q ]) [ A] (1.134) −1

where exp([Q]) = [U]exp(Q); exp (Q) = (exp Q1, exp Q2, … , exp Qn)t Assuming eigenvalue matrix Q, eigenvector matrix A, and its inverse A−1 are found, the propagation constant matrix can be calculated as in Equation 1.133:

Γ = P1/2 = AQ1/2 A−1 = Aγ A−1 (1.135)

where Γ is the actual propagation constant matrix (off-diagonal) γ = α + jβ is the modal propagation constant matrix (diagonal) Here, α is the modal attenuation constant, and β is the modal phase constant. In Equation 1.135,

[ γ ] = [U ] ( γ ) = [U ] ( Q )

1/2

= [Q ]

1/2

44


or in another expression,

γ k = Qk 1/2 = Qk ; k = 1, 2, … , n (1.136)

The exponential function of the propagation constant matrix is found from Equation 1.136:

exp ( −Γx ) = Aexp(− γx)A−1 (1.137)

As a result, the voltage in a semi-infinite line given by Equation 1.112 can be calculated by

V = A exp(− γ x)A−1Vf (1.138)

Note that the computation of Equation 1.112 is not possible, but it is made possible as in Equation 1.138 using the eigenvalue theory. This section has discussed the method that directly applies the eigenvalue theory. However, it is not efficient in terms of numerical computations as it requires the product of off-diagonal matrices. The method is to be completed by the modal theory. 1.4.2.2 Modal Theory Equation 1.138 is rewritten as

A−1V = exp(− γ x)A−1Vf (1.139)

Define mode voltage (voltage in a modal domain) and modal forward traveling wave (forward traveling wave in a modal domain) as follows:

v = A−1V , v f = A−1Vf (1.140)

where lowercase letters are modal components (components in a modal domain) and uppercase letters are actual or phasor components (components in an actual or phasor domain). Using modal components, Equation 1.139 can be expressed as

v = exp(− γ x)v f (1.141)

In the earlier equation, all components are expressed in a modal domain including voltage vectors. It is to be noted that the earlier equation in a modal domain takes the same form as Equation 1.112 in an actual domain. Similarly, relationships in an actual domain, for example, Ohm’s law, are satisfied in a modal domain.


45

Using the relationships in a modal domain, the solutions in a modal domain are first derived. Once the solutions in a modal domain are found, they can be transformed to the solutions in an actual domain. For example, once the solution of Equation 1.141, that is, v, is found, the solution in an actual domain is found by V = Av (1.142)

Applying the modal theory, the solutions are derived by the earlier procedure. With the modal theory, since the coefficient matrix in Equation 1.141 is a diagonal matrix, the equation is also written as vk = exp(− γ k x)v fk ; k = 1, 2, … , n (1.143)

The earlier equation shows that each mode is independent of the other modes; therefore, a multiconductor system can be dealt as a single-conductor system in a modal domain. The solutions in a modal domain can be found by n operations, whereas solving Equation 1.138 in an actual domain requires time complexity of o(n2) since coefficient matrix is an n × n matrix. Matrix A is called voltage transformation matrix as it transforms the voltage in a modal domain to that in an actual domain. 1.4.2.3 Current Mode As the last section discussed the voltage in a modal domain, this section discusses the current in a modal domain. We first need to find the eigenvalues of Pt = YZ as the second equation of Equation 1.115 tells us. Since Pt ≠ P in general, we define Q′ as the eigenvalue matrix of Pt and B as the eigenvector matrix of Pt:

Pt = BQ′B−1 , Q′ = B−1PB t (1.144)

Since a matrix returns to the original matrix when it is transposed twice,

det( P − QkU ) = det( Pt − Q′kU )t = det ( Pt )t − (Q′kU )t  (1.145)  

Considering (Pt)t = P and (Qk′U)t = Qk′U, det( P − QkU ) = det( P − Q′kU )

(1.146)

∴ Qk = Q′k

The earlier equation shows that the eigenvalues for the voltage are equal to those for the currents. Since γ = Q , propagation constants for the voltage

46


are also equal to those for the currents. These are important characteristics when analyzing a multiconductor system and correspond to TEM mode propagation. However, the current transformation matrix B is not equal to the voltage transformation matrix A. Taking transpose of the first equation of Equation 1.144, P = Bt−1Q′Bt = Bt−1QBt (1.147)

At the same time, from Equation 1.125,

P = AQA−1 = AD−1DQA−1 = AD−1QDA−1 (1.148)

where D is an arbitrary diagonal matrix. Comparing Equations 1.147 and 1.148, Bt −1 = AD−1 , Bt = DA−1 (1.149)

The aforementioned shows that the current transformation matrix can be found from the voltage transformation matrix. In general, D is assumed as an identity matrix. Under this assumption, B = ( A−1 )t , B−1 = At (1.150)

1.4.2.4 Parameters in Modal Domain By applying modal transformation, differential equations in a multiconductor are given as

dV d( Av) dv  = =A = −ZI = −ZBi  dx dx dx  (1.151)  dI d(Bi ) di  = =B = −YV = −YAv  dx dx dx 

Modifying the earlier set of equations, dv = − zi , dx

di = − yv (1.152) dx

where z = A−1ZB : modal impedance   (1.153) y = B−1YA : modal admittance  or Z = AzB –1, Y = B · yA–1


47

Equation 1.152 in a modal domain takes the same form as that in a phase domain. In a modal domain, the impedance and admittance are defined by Equation 1.153. From Equation 1.152,

d 2v = zyv , dx 2

d 2i = yzi (1.154) dx 2

From previous discussions, we already know

zy = yz = Q = γ 2 , γ = ( zy)1/2 , γ k = zk yk (1.155)

In order for a product of two matrices to be a diagonal matrix, the two matrices have to be diagonal matrices. Since Q is a diagonal matrix, z and y are diagonal matrices (Wedepohl [14]). For a semi-infinite line, the following equation is satisfied: V = Z0 I (1.156)

Applying modal transformation to the earlier equation,

Av = Z0Bi ∴ v = A−1Z0Bi = z0i (1.157)

The characteristic impedance and admittance in a modal domain are defined as follows:

z0 = A−1Z0B :modal characteristic impedance  (1.158)  y0 = B−1Y0 A :modal characcteristic admittance 

Rewriting the earlier equation, the actual characteristic impedance and admittance (in phase domain) are given by

Z0 = A ⋅ z0 ⋅ B−1 , Y0 = B ⋅ y0 ⋅ A−1 (1.159)

From Equations 1.114 and 1.158,

z0 = A−1P −1/2ZB

Using the relationships in Equations 1.135 and 1.153,

z0 = A−1( AQ −1/2 A−1 )( AzB−1 )B = Q −1/2 z = γ −1z = y −1γ (1.160)

48


The earlier equation shows that z 0 is a diagonal matrix since γ and z are diagonal matrices. In the same way, it can be shown that y 0 is a diagonal matrix. Equation 1.160 also shows that z0 can be found from γ and z. Substituting Equation 1.155 into the earlier equation,

z0 = γ −1z = ( zy)−1/2 z = y −1/2 z1/2 = ( y −1z)1/2 = ( zy −1 )1/2 (1.161)

Therefore, modal characteristic impedance and admittance are found also by z0 k =

1 zk , y0 k = = yk z0 k

yk (1.162) zk

1.4.3 Two-Port Circuit Theory and Boundary Conditions The unknown coefficients Vf and Vb in the general solution expressed as Equation 1.110 are determined from boundary conditions. There are many approaches to obtain voltage and current solutions in a multiconductor system. The most well-known method is a four-terminal parameter (F-parameter) method of a two-port circuit theory. Also, an impedance parameter (Z-parameter) and an admittance parameter (Y-parameter) are well known. It should be noted that the F-parameter is not suitable for the application in a high-frequency region, while Z- and Y-parameter methods are not suitable to deal with low-frequency phenomena because of the nature of hyperbolic functions. 1.4.3.1 Four-Terminal Parameter The four-terminal parameter (F-parameter) of a two-port circuit illustrated in Figure 1.18 is expressed in the following form:

 Vs   F1  =  I s   F3

F2   Vf  (1.163)   F4   I f 

where Vs, Vr are the voltage vectors at the sending and receiving ends in a multiconductor system Is, Ir are the current vectors at the sending and receiving ends The coefficients F1 to F4 in a multiconductor system are obtained in the same manner as those in Equation 1.87 taking care of a matrix form from Equations 1.110 and 1.111:

49


Vs

Vr l

S Zs Is

Z0, Y0, Г

R Vfr

Ir

Vbr

E

Zr

Yin x=l

x=0

FIGURE 1.18 An impedance-terminated multiconductor system.

F1 = cosh(Γl), F2 = sinh(Γl) ⋅ Z0

(1.164)

F3 = Y0 sinh(Γl), F4 = Y0 cosh(Γl) ⋅ Z0

where Γ, Z0, Y0: n × n matrix for an n-conductor system. It should be noted that the order of the products in the earlier equation cannot be changed as has been done for a single conductor. That is,

F2 = Z0 ⋅sinh ( Γl ) , F4 = cosh(Γl) = F1 only for a single conductor (1.165)

Equation 1.163 cannot be solved directly from a given boundary condition unless the coefficients in Equation 1.164 are calculated. By applying the modal transformation explained in the previous sections, Equation 1.163 is rewritten as

A−1Vs = vs = A−1F1 A ⋅ A−1Vf + A−1F2B ⋅ B−1I f = f1v f + f2ir B−1I s = is = f3v f + f 4ir

In a matrix form,

 vs   f1  =  is   f 3

f2   vr  (1.166)   f 4   ir 

50


In the earlier equation, the modal F-parameters are given by

f2 = sinh( γl) ⋅ z0 = z0 sinh( γl) f3 = y0 sinh( γl) = sinh( γl) ⋅ y0 (1.167) f 4 = y0 cosh( γl) ⋅ z0 = y0 z0 cosh( γl) = cosh( γl) = f1

where z0, y0, and γ are defined in Section 4.2.4. The earlier modal parameters are easily obtained because every matrix, γ, z0, and y0 = 1/z0, is a diagonal matrix. Then, the parameters in an actual phase domain are evaluated by

F1 = Af1 A−1 , F2 = Af2B−1 , F3 = Bf3 A−1 , F4 = Bf 4B−1 (1.168)

It should be clear in the earlier equation that F1 is in the dimension of a voltage propagation constant, F4 in the dimension of a current propagation constant, F2 in the impedance dimension, and F3 in the admittance dimension. From Equations 1.167 and 1.168, the following relation is obtained: F2 = A ⋅ z0 sinh( γl)B−1 = Az0B−1B sinh( γl)B−1

{

}

{

= Z0 At −1 ⋅ sinh( γl)t At = Z0t A sinh( γl)A−1

}

t

= Z0t sinh(Γl)t = {sinh(Γl) ⋅ Z0 }t

In comparison with Equation 1.164,

F2 = F2t (1.169)

The earlier relation means that F2 is a symmetrical matrix. Similarly, F3 is symmetrical. F1 and F4 have the following relation:

F4 = F1t , F1 ≠ F4 ,

f1 = f 4 (1.170)

Remind that F1 is not the same as F4 in a multiconductor system, while those are the same in the case of a single conductor as is well known. 1.4.3.2 Impedance/Admittance Parameters The four-terminal parameter formulation in Equation 1.163 is rewritten taking care of matrix algebra in the following form: Vs = coth ( Γl ) Z0 I s − cosech ( Γl ) Z0 I r

Vr = cosech ( Γl ) Z0 I s − coth ( Γl ) Z0 I r

(1.171)

51


Zs

Vs

Vr

S

R

Is

Ir Zr

E

FIGURE 1.19 A multiconductor system for Z- and Y-parameters.

Up to now, the current has been positive when it flows in the positive direction of x. Here, it is more comprehensible to set the positive direction of the current to the direction of inflow (injection) to the finite line as shown in Figure 1.19. Since the positive direction of current has changed at the receiving end, Ir has to be changed to −Ir in Equation 1.171: Vs = coth ( Γl ) Z0 I s − cosech ( Γl ) Z0 I r   (1.172)  Vr = cosech ( Γl ) Z0 I s − coth ( Γl ) Z0 I r 

In a matrix form,

Vs   Z11 V  =  Z  r   12

Z12   I s   coth ( Γl ) Z0 = Z11   I r  cosech ( Γl ) Z0

cosech ( Γl ) Z0   I s  (1.173)  coth ( Γl ) Z0   I r 

Here, Zij (i, j = 1, 2) are called impedance parameters (Z-parameters). Taking inverse of the matrix,

 I s   Y11  I  =  −Y  r   12

−Y12  Vs   Y0 coth ( Γl ) = Y11  Vr   −Y0cosech ( Γl )

−Y0coseech ( Γl )  Vs  (1.174)  Y0 coth ( Γl )  Vr 

Here, Yij (i, j = 1, 2) are called admittance parameters (Y-parameters). Admittance parameters are more often used than impedance parameters since a voltage source is typically given as a boundary condition. Given the voltage source E in Figure 1.19, the voltage and current at the sending and receiving ends are found from Equation 1.174 and boundary conditions:

52


Vs = {Ys + Y11 − Y12 (Y11 + Yr )−1 Y12 }−1 YsE    Vr = (Y11 + Yr )−1 Y12Vs  (1.175)   I s = Ys (E − Vs )   I r = −YrVr

where Ys = Zs−1 Yr = Zr−1

The impedance and admittance parameter methods are stable for θ → ∞ since it is based on convergence functions, coth(θ) and cosech(θ). Thus, the method is suitable for the transient analysis. However, it should not be used for the analysis of low-frequency phenomena since cosech(θ) becomes infinite for θ → 0, that is, ω → 0. 1.4.4 Modal Distribution of Multiphase Voltages and Currents 1.4.4.1 Transformation Matrix When a three-phase transmission line is completely transposed or the impedance and admittance matrices are completely symmetrical, the following transformation matrices have been widely used for both voltages and currents:

a. Fortescue’s transformation [15] 1 [ AF ] = 1 1

1 a2 a

1 a , a2

1 1 = ⋅1 3 1

[ AF ]

−1

1 a a2

1 a2 , a

 j 2π  a = exp    3 

(1.176)

b. Clarke’s transformation [16] 1  [ AC ] = 1  1

1 −1/2 −1/2

0   3/2  ,  3/2 

[ AC ]

−1

1 1  = ⋅ 2 3  0

1 −1 3

1   −1  (1.177)  3


53

c. Karrenbauer’s transformation [10] 1  [ AK ] = 1 1

1 −2 1

1  1 , −2 

[ AK ]

−1

1 1  = ⋅ 1 3 1

1 −1 0

1  0  (1.178) −1

2 1  = ⋅ 3 6  1

2 0 −2

2  −3  (1.179) 1 

d. Traveling-wave transformation [1]

1  [ AT ] = 1 1

1 0 −1

1  −2  , 1 

[ AT ]

−1

Fortescue’s transformation is well known in conjunction with a symmetrical component theory [15]. Although it involves complex numbers, it has an advantage to generate only one nonzero modal voltage (positive-sequence voltage) if a source voltage is a three-phase symmetrical ac source. Clarke’s transformation is also related to the symmetrical component theory and is known as α − β − 0 transformation [16]. It involves only real numbers but generates positive and negative-sequence voltages. Karrenbauer’s transformation is adopted in the famous EMTP [10] and is easily extended to an “n” phase completely transposed line. This is also true for Fortescue’s transformation. There, however, exists no transposed line of which the phase number “n” is greater than three. The traveling-wave transformation is often used when analyzing traveling waves on a three-phase line [1]. Its advantage is that the transformation can deal with not only a completely transposed line but also an untransposed horizontal line. 1.4.4.2 Modal Distribution Let’s discuss the modal current (voltage) distribution on a completely transposed line. It should be noted that the modal voltage distribution is the same as that of the current in the completely transposed line case, because the impedance and admittance matrices are completely symmetrical. Assume phase currents are Ia, I b, and Ic as illustrated in Figure 1.20. Adopting the traveling-wave transformation, we obtain the following relation:

 i0  2   1 = i  1  6 3  i2   1  

2 0 −2

2   ia   2I a + 2I b + 2I c     1  −3  ⋅  ib  =  3 I a + 3 I b  (1.180) 6  I a − 2I b + I c  1   ic 

54


Phase

Ia

a

Ib

b

Ic

c

FIGURE 1.20 Actual phase current.

Assume Ia = Ib = Ic = I for simplicity, and then the following characteristic of the modal current is observed:

a. Mode 0 (earth-return mode) The current of I/3 following to the positive direction on each phase and the return current I have to flow back through the earth. Because the return current flows through the earth, the mode “0” component is called “earth-return” component. A circuit corresponding to the mode 0 can be drawn as Figure 1.21a. The mode 0 component involves the earth-return path of which the impedance is far greater than the conductor internal impedance as explained in Section 1.5.1, (i0/3) I/3

Phase a

I/3

b

I/3

c

I0=I

Earth

(a) I/6 (i2)

a I/2 (i1)

I/2 (i1) (b)

I/3 (2i2)

b

I/6 (i2)

c

a b c

(c)

FIGURE 1.21 Model current distribution for Ia = Ib = Ic = I. (a) Mode “0,” (b) mode “1,” and (c) mode “2.”


55

the mode 0 propagation constant is much greater, that is, the mode 0 attenuation is much greater, and the mode 0 propagation velocity is smaller than those of the other modes. b. Mode 1 (first aerial mode) The current of I/2 flows through phase “a” returning through phase “c,” and no current on phase “b.” Thus, the mode 1 circuit is composed of the phases a and c as shown in Figure 1.21b. Because the mode involves no earth-return path in ideal cases, the mode is called “aerial mode.” c. Mode 2 (second aerial mode) The current of I/6 flows through phases a and c, and the return current of I/3 flows back through phase b as illustrated in Figure 1.21c. The propagation characteristic of the mode 2 is identical to that of the mode 1 in the completely transposed line case. If the line is untransposed, those are different. The modal distribution can be also explained by applying the transformation matrix A rather than its inverse A−1 in the preceding text:

 ia  1     ib  = 1  ic  1   

1 0 −1

1   i0   1   1  1           −2  ⋅  i1  =  1  ⋅ i0 +  0  ⋅ i1  −2  ⋅ i2 (1.181)  1  1  1   i2   1     

For example, the mode 0 current is explained to have a distribution of the same amount of the actual current on each phase from the earlier equation. The same explanation as the aforementioned can be made by applying the transformation matrices given in Equations 1.176 through 1.178 in the case of a completely transposed line. When a line is untransposed, those transformation matrices are no more useful except that of Equation 1.179, which can be used as an approximation of a transformation matrix of an untransposed horizontal line. In the untransposed line case, the transformation matrix is frequency dependent as explained in Section 1.5.1, and thus the modal voltage and current distributions vary as frequency changes. Also, the current distribution differs from the voltage distribution. 1.4.5 Problems 1.12 Obtain a condition of reciprocity in Equation 1.163. 1.13 Obtain the eigenvalue and vector of the following matrices:

56


 P11 ( a ) † P21  P13

P12 P22 P12

P13   P21  P11 

 35 ( b )  8  5

8 32 8

5  8 35 

 35  ( c ) 5 3   5

5 3 35 5 3

5   5 3  35 

1.14 Explain why the modal propagation constants and the modal characteristic impedances for modes 1 and 2 (aerial modes) on a transposed three-phase line become identical. 1.15 Discuss how to obtain the inverse matrix of A when the transformation matrix A is singular. Remind that a numerical calculation by a computer can give its inverse matrix.

1.5 Frequency-Dependent Effect It is well known that a current distributes nearby a conductor surface when the frequency of the current is high. Under such a condition, the resistance (impedance) of the conductor becomes higher than that at a low frequency, because the resistance is proportional to the cross section of the conductor. This is called frequency dependence of the conductor impedance. As a result, the propagation constant and the characteristic impedance are also frequency dependent. 1.5.1 Frequency Dependence of Impedance Figure 1.22 illustrates a 500 kV horizontal transmission line, and Table 1.1 shows the frequency dependence of its impedance. It is observed that the resistance increases nearly proportional to f where f is frequency. On the contrary, the inductance decreases as f increases. The previously mentioned phenomena can be explained analytically based on the approximate impedance formula in Equations 1.7 and 1.15:

GW ho

PW hp

(a)

1

a

25 m b

14 m

0.4 rp1

2 c

1.974 mm

0.4

(b)

rp2

8.74 mm (c)

FIGURE 1.22 A 500 kV untransposed horizontal line. (a) Line configuration. (b) Phase wire (4 bundles). (c) Bundle conductor. rg = 6.18 mm, ρg = 5.36 × 10−8 Ω-m, ρ p = 3.78 × 10−8 Ω-m, ρe = 200 Ω-m.

57


1. For a low frequency: f ≪ fc (fc: critical frequency, which will be defined later) Z = R + jωL, R = Rdc =

ρ µ , L = 0 (1.182) S 8π

2. For a high frequency: f ≫ fc Z = (1 + j ) R, R =

ωµ0ρ 1 R (1.183) ∝ ω, L = ∝ ω 2 2πr † ω

where ωcμcS/Rdc · l2 = 1. Thus, ωc = 2πfc =

4ρ (1.184) µ 0r 2

For ρ ≒ 2 × 10 −8 [ Ωm ] , µ 0 = 4π × 10 −7 [H/m] : ωc = 0.2/πr 2 [rad/s] fc =

ωc 1 ≒ [Hz]† 2π 100r 2

For example, with r = 0.5 cm, fc =

106 = 400 Hz (1.185) 25 × 100

Considering the aforementioned, the frequency characteristics of R and L are drawn as in Figure 1.23. L (µH/m)

R (Ω/m)

L

R = ωL

µc 8π

Rdc 0

FIGURE 1.23 Frequency dependence of Z = R + jωL.

f

58


Similarly to the conductor internal impedance explained earlier, the earthreturn impedance in Equation 1.15 is frequency dependent as the penetration depth he is frequency dependent. Equation 1.15 is approximated considering ln(1 + x) for a small x by

Ze ≈ Re + jω(Le + L0 ) = jωL0 + (1 + j )Re (1.186)

Re = (ωµ0ρe )/2 2π h ∝ ω , Le = Re /ω ∝ 1/ ω

 µ   2h  L0 =  0  ln   : space inductance  2π   r 

1.5.2 Frequency-Dependent Parameters 1.5.2.1 Frequency-Dependent Effect The propagation constant Γ and characteristic impedance Z0 of a conductor are frequency dependent for those are a function of the impedance of the conductor as explained in the previous section. It should be noted that α and β in Section 1.3.4.1 (see Figure 1.10) are not frequency dependent in a sense discussed in this section. The frequency dependence of attenuation constant α(ω) and phase constant β(ω) in Section 1.3.4.1 comes from the definition of impedance Z and admittance Y of a conductor: Z = R + jωL, Y = jωC In this section, we discuss the frequency dependence, which comes from R = R(ω) and L = L(ω) as in Equation 1.183: Figure 1.24 shows an example of the frequency dependence of attenuation constant α and propagation velocity c for the earth-return mode and the self-characteristic impedance Z0 for a phase of a 500 kV overhead transmission line. It is observed that α increases remarkably as the frequency increases. Since a dominant factor of determining the attenuation constant is the conductor resistance, α is somehow proportional to f as explained in Section 1.5.1. The propagation velocity is converging to the light velocity c0 as the frequency increases. On the contrary, the characteristic impedance (absolute value |Z0|) decreases as the frequency increases. This is readily explained from Equation 1.183. Z0 =

ω 1 Z ∝ = (1.187) ω Y f

In a multiconductor system, the transformation matrix A is also frequency dependent. The frequency dependence is significant in the case of an

59


ln(α) 1

300

250

–1

–3 0.1 (a)

c

1

10

100

1000

200 0.1

1

(b)

f (kHz)

10

100

1000

f (kHz)

|Z0| 400 350 300 250 0.1 (c)

1

10

100

1000

f (kHz)

FIGURE 1.24 Frequency dependence of α, c, and Z0 of a 500 kV line. (a) α [dB/km], (b) c [m/μs], and (c) |Z0| [Ω].

untransposed vertical overhead line and of an underground cable. In the former, more than 50% difference is observed between Aij (i jth element of matrix A) at 50 Hz and 1 MHz. In an untransposed horizontal overhead line, the frequency-dependence is less noticeable. The frequency dependence is very significant when an accurate transient simulation on a distributed-parameter line, such as an overhead line and an underground cable, is to be carried out from the viewpoint of insulation design and coordination in a power system. However, a simulation can be carried out neglecting the frequency-dependence if a safer-side result is required, because the frequency dependence, in general, results in a lower overvoltage than that neglecting the frequency dependence. 1.5.2.2 Propagation Constant The frequency dependent effect is most noticeable in the propagation constant. Table 1.2 shows the frequency dependence of modal attenuation and velocities for untransposed and transposed lines. Figure 1.25 shows the frequency dependence of modal propagation constant for a vertical twin-circuit line illustrated in Figure 1.26. A large frequency dependence of the attenuation is clear from the tables and figure. The propagation velocity of mode 0 shows a significant frequency dependence, while the mode 1 velocity is not much frequency dependent.

60


TABLE 1.2 Frequency Responses of Modal Propagation Constant for a Horizontal Line in Figure 1.25 a. Untransposed Attenuation [dB/km]

Velocity [m/μs]

f [kHz]

α0

α1

α2

c0

c1

c2

0.1 1 10 100 1000

1.389E−3 8.166E−3 7.941E−3 0.5723 2.626

9.724E−5 3.041E−4 3.500E−3 5.446E−2 0.3932

6.314E−5 1.970E−4 6.798E−4 3.755E−3 3.000E−2

244.0 255.3 270.4 285.7 294.8

295.1 295.9 296.4 297.6 298.9

298.6 299.3 299.5 299.6 299.7

b. Transposed Attenuation [dB/km]

Velocity [m/μs]

f [kHz]

α0

α1 = α2

c0

c1 = c2

0.1 1 10 100 1000

1.386E−3 8.151E−3 7.930E−2 0.5723 2.628

8.191E−5 2.558E−4 2.204E−3 3.124E−2 0.2277

243.9 255.1 270.2 285.4 294.5

295.6 296.4 296.8 297.4 298.2

22.4 m

92.1 m

15.9 m

17.5 m

m 0.5

78.6 m

16.7 m

66.8 m 55.0 m (a)

0.5 m

Circuit No.1 No.2 PW: TACSR 810°×6 GW: AS 160° (b)

FIGURE 1.25 An untransposed vertical twin-circuit line. (a) Line configuration. (b) Phase wire.

When a line is transposed, all the aerial modes become identical. Thus, a number of different characteristic modes are reduced to two in the three-phase line case and to three in the twin-circuit line case. The different velocities of the aerial modes in an untransposed line cause a spike voltage on a transient voltage waveform, which is characteristic to the untransposed line.

61


1.5.2.3 Characteristic Impedance The definition of the characteristic impedance of a single-phase line is Z ∠ϕ ( R + j ωL ) Z = = =|z0|∠θ (1.188) Y †jωC ωC∠90°

z0 = where Z = R 2 + ω2L2 ϕ = tan −1 ( ωL/R ) z0 = |Z|/ωC θ = (ϕ − 90°)/2

Modes 1–5 attenuation (dB/km)

1 × 10–1

Modes 1–5 velocity (m/µs)

(a)

(b)

3

e0

od

M

3

1

1 × 10–2

100

1 4

10

2,5 1

1 × 10–3

1

5

1 × 10–4 0.01 2

300

0

1

2 2 5 4

3

1

5 0.1

1

10 Freq. (kHz)

100

2,4,5

290

280 0.01 2

0.1 1000

300

2 4

3

3

1

Mode 0

5 0.1

1

1

250

10

100

200 1000

Freq. (kHz)

FIGURE 1.26 Frequency responses of modal propagation constants for an untransposed vertical twin-circuit line in Figure 1.25. (a) Attenuation. (b) Velocity.

62


For R, ωL ≥ 0, 0 ≤ φ ≤ 90°. Thus,

−45° ≤ θ ≤ 0 or Re z0 ≥ 0, I m ( z0 ) ≤ 0 (1.189)

The previously mentioned fact should be noted. Table 1.3 shows the actual characteristic impedance of a horizontal line and Figure 1.27 for a vertical twin-circuit line illustrated in Figure 1.25. It is clear that the characteristic impedance is significantly frequency dependent. The variation reaches about 10% for the self-impedance and about 50% for the mutual impedance in the frequency range of 100 Hz to 1 MHz. The characteristic impedance decreases as frequency increases and tends to approach the value of the perfectly conducting earth and conductor case. In the vertical line case, it should be noted that the relation of magnitudes changes as frequency changes. For example, the self-impedance is in the following relation: Zcc > Zbb > Zaa

Zbb > Zcc > Zaa

for

f ≤ 5 kHz

for 20 kHz ≤ f ≤ 1MHz

TABLE 1.3 Frequency Responses of Characteristic Impedances for a Horizontal Line in Figure 1.22 a. Untransposed Freq. [kHz]

Zaa [Ω]

Zbb [Ω]

Zab [Ω]

Zac [Ω]

0.1 1 10 100 1000 ρe = 0 Ω m

348−j11 340−j6.4 331−j6.3 322−j4.7 317−j2.2 314.6

346−j11 338−j6.1 330−j5.9 322−j4.4 317−j2.1 314.4

87.3−j10 80.2−j5.9 71.5−j5.9 63.5−j4.3 59.1−j2.0 56.7

54.3−j9.7 47.2−j6.0 38.4−j5.7 30.9−j3.8 27.3−j1.6 25.5

b. Transposed Freq. [kHz] 0.1 1 10 100 1000 ρe = 0 Ωm

Zs [Ω]

Zm [Ω]

347−j11 339−j6.3 330−j6.1 321−j4.6 316−j2.2 314.5

76.5−j9.9 69.4−j5.9 60.6−j5.8 52.8−j4.1 48.6−j1.8 46.3

(a)  Zaa Z = [ 0 ] Zab  Zac 

Zab Zbb Zab

(b) Zac   Zs   Zab  [ Z0 ] =  Zm  Zm Zaa  

Zm Zs Zm

Zm   Zm  Zs 

63


Within a circuit

350

Zcc Zs

300 150

Zab

Zbb

Zaa

Zbc Zm

100

Zac 50 0.01 2

5 0.1

1

10 Freq. (kHz)

100

1000

Inter-circuit

150

Zbb΄

100 Zn

50 0.01 2

Zab΄

Zcc΄

Zaa΄

Zbc΄

Zac΄ 5 0.1

1

10 Freq. (kHz)

100

1000

FIGURE 1.27 Various current distribution modes on an untransposed vertical twin-circuit line.

Zbb > Zaa > Zcc

for 1MHz < f < some MHz

Zaa > Zbb > Zcc

for

f > some MHz

The previously mentioned phenomenon is due to the variation of return current distribution in the ground wires (GWs) and earth. At a very high frequency, all the currents returning through the earth surface and the earth surface potential becoming zero, the image theory is applicable. Thus, the magnitude of the self-impedance becomes proportional to the height of the conductor, that is, Zaa > Zbb > Zcc. The characteristic impedance of a transposed line shows about a mean value of those of an untransposed line. 1.5.2.4 Transformation Matrix The importance of the frequency-dependent transformation matrix was not well recognized until recently, although it has been pointed out for long [17,18].

64


TABLE 1.4 Frequency Responses of a Transformation Matrix for a Horizontal Line Shown in Figure 1.22 Freq.[kHz] 0.1 1 10 100 1000

A1

A2

0.996−j0.002 1.001−j0.006 1.015−j0.012 1.037−j0.014 1.055−j0.008

−2.294 + j0.010 −2.267 − j0.026 −2.216 − j0.044 −2.136 − j0.050 −2.074 − j0.026

1.5.2.4.1 Untransposed Horizontal Three-Phase Line Table 1.4 shows the frequency-dependent transformation matrix of the untransposed horizontal line of Figure 1.25 neglecting GWs. The transformation matrix has the following form: 1  [ A] =  A1  1

1 0 −1

1  A2  (1.190) 1 

It is observed that the frequency dependence of A2 in Table 1.4 is less than 10% for the range of frequencies from 100 Hz to 1 MHz. The change is small compared with the other parameters explained previously, and thus, the frequency-dependent effect of the transformation matrix in the untransposed horizontal line case can be neglected. Then, the following approximation is convenient because it agrees with the traveling-wave transformation of Equation 1.179 explained in Sections 1.4.4.1 and 1.4.4.2: A1 ≒ 1, A2 ≒ −2 (1.191)

In this case, the modal distribution is the same as that explained in Section 1.4.4.1. The current transformation matrix is given from Equation 1.179 by 2 1  [B] = [ A] = ⋅  2 6  2 −1 t

3 0 −3

1 1   −1 −2  , [B] = [ A]t = 1 1 1 

1 0 −2

1  −1 (1.192) 1 

It is observed from the earlier equation that the modal current distribution is basically the same as the voltage distribution. Therefore, the modal circuit given in Figure 1.21 is also applicable approximately to the untransposed horizontal line. Table 1.5 shows the transformation matrix of the untransposed horizontal line with GWs. It is observed no significant difference from Table 1.4 with the grounding wires.

65


TABLE 1.5 Frequency Responses of a Transformation Matrix for a Horizontal Line with No GW A1

Frequency [kHz] 0.1 1 10 100 1000

A2

Real

Imag

Real

Imag

0.9934 1.0007 1.0312 1.0643 1.0823

0.0074 0.0126 0.0198 0.0167 0.0079

−2.2504 −2.2280 −2.1885 −2.1339 −2.1043

0.0120 0.0206 0.0326 0.0275 0.0131

1.5.2.4.2 Untransposed Vertical Twin-Circuit Line The current transformation matrix B is given in the following form for the case of an untransposed vertical twin-circuit line shown in Table 1.6:  iI   [ BI I ] (i ) =   =   iII  [ BII I ]

[ BI ]  ⋅  ( I I ) = [B]−1 ⋅ (I ) (1.193)    − [ BII ]  ( I II )  I

II

where (i) is the modal current (I) is the actual current and  i0  ( iII ) =  i1  ,  i2   

 i3  ( iII ) =  i4  ,  i5   

 Ia    ( I I ) =  Ib  ,  Ic   

 I ′a    ( I IIII ) =  Ib′   I c′   

TABLE 1.6 Frequency Responses of Characteristic Impedances for a Vertical Line Kinds of Current Distribution Mode

Internal (aerial) modes

Same polarity

Intercircuit modes

Mode no. Opposite polarity

Mode no.

First

Second

Third

Fourth

a⚬+ +⚬a′ b⚬+ +⚬b′ c⚬+ +⚬c′ 0 ⚬+ −⚬ ⚬+ −⚬ ⚬+ −⚬ 3

⚬+ +⚬ ⚬+ +⚬ ⚬− −⚬ 1 ⚬+ −⚬ ⚬+ −⚬ ⚬− +⚬ 4

⚬+ +⚬ ⚬− −⚬ ⚬+ +⚬ 2 [1] ⚬+ −⚬ ⚬− +⚬ ⚬+ −⚬ 5

⚬+ +⚬ ⚬− −⚬ ⚬− −⚬ * [2] ⚬+ −⚬ ⚬− +⚬ ⚬− +⚬ *

66


It is clear from the aforementioned that modal currents (II), group I, are currents when there is only one circuit and is called “internal mode.” The currents (III) are generated by the existence of the second circuit that is called “intercircuit mode.” The internal mode has a plane of symmetry at the earth surface, while the intercircuit mode has a plane of symmetry at the vertical center of the two circuits. Therefore, the polarities of currents on a phase of the circuit I and the corresponding phase of the circuit II are the same in the internal mode, and the polarities are opposite though the amounts of the currents are the same in the intercircuit mode as illustrated in Table 1.6. Thus, the transformation matrix given in Equation 1.193 is obtained. In Table 1.6, the first kind distribution corresponds to the so-called zerosequence mode and is the mode “0” in the same polarity case and the mode “3” in the opposite polarity case. If the line is single circuit, there is no opposite polarity mode, and thus the first kind distribution is the same as the mode 0 distribution, which has been explained in Section 1.4.4.1. The mode 0 is often called “first zero-sequence mode” (earth-return mode), and the mode 3 is called “second zero-sequence mode” (intercircuit zero-sequence mode). The second to the fourth distributions correspond to an aerial mode. The second one is the positive-sequence mode, and the third is the negativesequence mode. However, the pattern of the current distribution varies as frequency changes, and thus, the fourth distribution can be the negativesequence mode at a certain frequency. In the single-circuit case, there exists no opposite polarity mode, and also the second and fourth distributions are the same. Thus, the number of the current distribution patterns, that is, natural modes, is reduced to three. Table 1.7 shows the frequency response of the submatrices BI and BII of the transformation matrix given in Equation 1.193 only for the real part because the imaginary part is much smaller. Rewriting Table 1.7 in the form of Equation 1.193 for f = 50 Hz, the following result is obtained:

  i0     0.751        i1     1   i2    0.272     =    i3    0.466        i4     1   i5    0.850     

0.878 0.158 −1

1   −0.570  0.361 

0.751 0.589 −1

1   −0.782  0.289 

 0.751   1 0.272  −0.466   −1  −0.850

  −0.570   0.361    −0.751 −1    −0.589 0.782   1 0.289  

0.878 0.158 −1

1

  Ia       Ib     Ic      I   a   I   b    I c    

From the earlier equation, it is clear that each mode has the following closed circuit:

67


TABLE 1.7 Frequency Responses of Current Transformation Matrix B−1 (Real Part) a. [BI] = BI( j, k) Mode 0

Mode 1

Mode 2

Freq. [kHz]

(1,1)

(1,2)

(2,2)

(2,3)

(3,1)

(3,3)

0.05 0.1 1 5 10 50 100 1000

0.751 0.725 0.691 0.665 0.653 0.627 0.618 0.600

0.878 0.866 0.848 0.832 0.825 0.808 0.802 0.790

0.158 0.215 0.364 0.410 0.427 0.466 0.488 0.511

−0.570 −0.558 −0.572 −0.546 −0.534 −0.506 −0.500 −0.478

0.272 0.325 0.501 0.542 0.561 0.601 0.621 0.645

0.361 0.339 0.276 0.264 0.258 0.246 0.240 0.236

b. [BII] = BII( j, k) Mode 3

Mode 4

Mode 5

Freq. [kHz]

(1,1)

(1,2)

(2,2)

(2,3)

(3,1)

(3,3)

0.05 0.1 1 5 10 50 100 1000

0.466 0.454 0.471 0.462 0.453 0.425 0.414 0.379

0.751 0.659 0.661 0.666 0.664 0.646 0.636 0.607

0.589 0.580 0.491 0.513 0.581 0.712 0.731 0.733

−0.782 −0.715 −0.671 −0.674 −0.699 −0.725 −0.717 −0.658

0.850 0.772 0.674 0.708 0.782 0.903 0.924 0.900

0.289 0.225 0.256 0.260 0.236 0.196 0.189 0.197

B(1,3) = B(2,1) = −B(3,2) = 1.

Mode 0: all the phases to earth Mode 3: first circuit to second circuit Mode 1: phases a and b to phase c Mode 2: phases a and c to phase b Mode 4: phases a, b, and c′ to phases a′, b′, and c Mode 5: phases a, c, and b′ to phases a′, c′, and b It is observed from Table 1.7 that the frequency dependence of the transformation matrix in the untransposed vertical twin-circuit case is significantly large. The largest frequency dependence is observed in BII(2,2) = B(5,2) = −B(5,5), and the variation reaches about 50% with reference to the smallest value, that is, the value at f = 1 kHz. Also, it should be noted that the value of the intercircuit

68


mode (modes 3–5), that is, the value of the submatrix BII, shows an oscillating nature. These may result in a difficulty of a transient calculation. 1.5.2.5 Line Parameters in the Extreme Case It has been already proved that line parameters at an infinite frequency are the same as those in the perfectly conducting earth and conductor case. It is quite useful to know the line parameters in such the extreme case, because the parameters are a good approximation to the line parameters at a finite frequency with imperfectly conducting earth and conductor. From the study in the previous sections, the following parameters have been known at the infinite frequency or in the perfectly conducting media case. 1.5.2.5.1 Line Impedance and Admittance µ

[ Z ] = [ Zs ] = jω  2π0 ⋅ [P]  

 (1.194)

[Y ] = [Ys ] = jω2πε0 ⋅ [P]−1

where  Dij Pij = ln   dij

  

 2h  †= ln  i   ri 

 for i ≠ j    modified potential coefficient (1.195)  for i = j  

and

Dij = (hi + h j )2 + yij2 , dij = (hi − h j )2 + yij2

From the previously mentioned impedance and admittance, we can derive the following line parameters in the actual phase domain. 1.5.2.5.2 Actual Propagation Constant

[Γ ] = ([ Z ][Y ])

1/2

{

}

−1 1/2

= jω(µ 0 /2π ⋅ [P] ⋅ jω2πε0 ⋅ [ P ] ω = j   [U ] = [ α ] + j [β]  c0 

= jωε0µ 0 [U ]

69


Therefore, ω  ⋅ [U ] (1.196)  0

[ α ] = [ 0 ] , [β ] =  c

Thus, the propagation velocity is

[c ] = c0 [U ]

or ci = c0

for phase “" i " (1.197)

From the earlier results, it is clear that the product Z·Y or the actual propagation constant matrix is diagonal and purely imaginary at the infinite frequency or in the perfect conductor case. This results in the fact that the attenuation is zero and the propagation velocity is a light velocity in free space on any phases. Also, it is noteworthy that the modal theory is not necessary as far as only the propagation constant is concerned, because it is already diagonal. 1.5.2.5.3 Actual Characteristic Impedance From Equation 1.159,

[ Z0 ] = [ A] ⋅ [ z0 ] ⋅ [B] − 1 = [ A] ([ A] − 1 ⋅ [Γ ] ⋅ [ Z ] ⋅ [B]) ⋅ [B] −1

−1

= [Γ]−1 ⋅ [Z]

c  µ  µ c  =  0  ⋅ [U ] ⋅ jω  0  [ P ] =  0 0  [ P ] = 60[P] (1.198) ω 2 π j    2π    or

 Dij Z0 ij = 60ln   dij

  for i = j 

 2h j  = 60ln   for i = j  ri 

It is clear that the actual characteristic impedance is constant independently of frequency. 1.5.2.5.4 Modal Parameters The line impedance, admittance, and characteristic impedance matrices involve nonzero off-diagonal elements or mutual coupling, although the propagation constant matrix is diagonal. If one needs to diagonalize the aforementioned matrices, the modal transformation is required.

70


In the case of completely transposed three-phase line, any of the transformation matrices explained in Section 1.4.4.1 can be used. The current transformation matrix is the same as the voltage transformation matrix. Let’s adopt the traveling-wave transformation:

[ Z ] = [ A]

−1

 Zs + 2Zm  ⋅ [ Z ] ⋅ [ A] =  0  0

0 Zs + Zm 0

0   0  Zs + Zm 

or Z0 = Zs + 2Zm, Z1 = Z2 = Zs − Zm In the same manner, y0 = Ys + 2Ym , y1 = y2 = Ys − Ym z00 = Z0 s + 2Z0 m , z01 = z02 = Z0 s − Z0 m (1.199)

γ 0 = γ1 = γ 2 =

jω c0

or c0 = c1 = c2

The same result as the aforementioned is obtained by applying the other transformation in Section 1.4.4.1. 1.5.2.5.5 Time-Domain Parameters The parameters explained in the aforementioned are in frequency domain. The parameters in time domain are the same as those in the frequency domain in the perfect conductor case, because those are frequency independent and thus time independent. In the case of imperfectly conducting earth and conductor, only the parameters at the infinite frequency are known analytically. Those are to correspond to the parameters at t = 0 in the time domain from the initial value theorem of Laplace transform, that is,

lim f ( t ) = lim {sF ( s )} (1.200)

t →+0

s →∞

Thus, we can obtain the time-domain parameters at t = 0 or in the perfect conductor case same as in Equations 1.198 and 1.199. 1.5.3 Time Response 1.5.3.1 Time-Dependent Responses The time response of the frequency dependence explained in the previous sections is calculated by a numerical Fourier or Laplace inverse transform in the following form [1,17]:


71

1. Propagation constant  {−Γ ( s ) x}  : step response of propagation constant e ( t ) = L−1 exp s  

(1.201) where s = α + jω is the Laplace operator L −1 is the Laplace inverse transform

2. Characteristic impedance  Z ( s)  Z0 (t) = L−1  0  : surge impedance response (1.202)  s  3. Transformation matrix  Aij (s)  Aij (t) = L−1   : time-dependent transformation matrix (1.203)  s 

1.5.3.2 Propagation Constant: Step Response The frequency dependence of the propagation constant appears as wave deformation in time domain. This is measured as a voltage waveform at distance x when a step (or impulse) function voltage is applied to the sending end of a semi-infinite line. The voltage waveform, which is distorted from the original waveform, is called “step (impulse) response of wave deformation,” and is defined in Equation 1.201. Figures 1.28 and 1.29 show modal step responses on the line of Figures 1.22 and 1.25, respectively. It is clear from the figures that the wave front is distorted especially in the mode 0, which has the largest attenuation and lowest velocity in the frequency domain. As time passes, the distorted waveform tends to reach 1 pu, the applied voltage. It is observed from Figure 1.28 that the wave deformation is greater when the line length is greater. This is reasonable for the increase of the line length results in a greater distortion. Also, a greater earth resistivity causes a greater wave deformation, because the line impedance becomes greater. A GW reduces the wave deformation of the mode 0 significantly. This is due to the fact that the return current through the earth is reduced by the GW, and thus the line impedance is reduced. The reason for the much smaller wave deformation in the aerial modes than that in the earth-return mode is that the conductor internal impedance that contributes mainly to the aerial modes is far smaller than the earthreturn impedance mainly contributing to the mode 0.

72


Voltage (pu)

1 Mode 0 0.5

0

100

200

300

1 Voltage (pu)

Mode 2 Mode 1

0.5

0

5

10

15

Time (µs) v(t) = L–1 exp(–γ(s) . x/s) x FIGURE 1.28 Modal step responses of wave deformation for a horizontal line.

The line transportation does not affect significantly to the mode 0 wave deformation. It, however, causes a noticeable effect on the aerial modes as observed from Figure 1.29. The difference between transposed and untransposed line is already clear in the frequency responses given in Table 1.2 and Figure 1.26. The significant difference in the mode 1 propagation velocity in Figure 1.26b results in the difference in the mode 1 wave deformation in Figure 1.29. 1.5.3.3 Characteristic Impedance It should be noted that the definition of Equation 1.202 proposed by the author in 1973 is effective only for a semi-infinite line or for a time period of 2τ, where τ is a traveling time of a line [17,18]. Also, the definition requires a further study in conjunction with the wave equation in time domain, because it has not been given the proof that the earlier definition expresses the physical behavior of the time-dependent characteristic impedance.

73


2,4,5

1.0

Voltage [pu]

1

1

3

0.5

Mode 0 3

Untransposed Transposed

0

0

5 50

Mode 1–5 Mode 0 Time (µs)

10 100

15 150

FIGURE 1.29 Modal step responses of wave deformation for a vertical line.

Table 1.8 shows the time response of the frequency-dependent characteristic impedance given in Table 1.3. The time-dependent characteristic impedance increases as the time increases. This is quite reasonable because of inverse relation of time and frequency. Figure 1.30 shows the time response of the characteristic impedance of a vertical twin-circuit line illustrated in Figure 1.25. The relation of magnitudes corresponds to that in frequency domain explained for Table 1.4 considering the inverse relation of time and frequency. It is observed from the figure compared with the frequency response of Table 1.4 that the time dependence is greater than the frequency dependence of the characteristic impedance. For example, the variation of Zcc is 8.5%, Zac 26.8%, and Zac′ 31.3% for 2 μs ≤ t ≤ 500 μs, while the variations of those in the frequency domain are 7.9%, 23.4%, and 27.8% for 500 kHz ≥ f ≥ 2 kHz. Also, it should be noted that the time-dependent impedance is greater by about 5–15 Ω than the frequency-dependent one in general. The calculated result agrees well with the measured result. TABLE 1.8 Time Response of the Characteristic Impedance of Table 1.3 Time [μs] 10 50 100 200 500

Zaa [Ω]

Zbb [Ω]

Zab [Ω]

Zac [Ω]

330.6 337.3 340.1 343.0 347.2

329.5 335.7 338.4 341.1 345.1

71.73 77.96 80.61 83.25 87.06

38.74 44.94 47.61 50.27 54.05


360

Zab

Self-impedance (Ω)

350

Zcc

330 320

Zbb

Zs

Zaa

310

290

110 100

Zm

90 80

Zaa

300

Inter-circuit impedance (Ω)

120

Zbc

340

130

1

70

Zac

2

5

Mutual impedance (Ω)

74

10

Time [µs]

60 1000

100

110 100 90

Zn

Zaa΄

80

Zcc΄

Zbc΄

Zbb΄

Zab΄

70 60 50

Zac΄ 1

2

5

10

Time (µs)

100

1000

FIGURE 1.30 Time response of the characteristic impedance of Figure 1.25.

1.5.3.4 Transformation Matrix The frequency dependence of the transformation matrix appears as time dependence in time domain. The time-dependent transformation matrix is defined in Equation 1.203. Table 1.9 shows the time response of the transformation matrix given in Equation 1.190 and Table 1.4 for the untransposed horizontal line of Figure 1.22 without GWs. The comparison of Table 1.9 with Table 1.4 shows that the time dependence is smaller than the frequency dependence as far as the results appeared in the table are concerned. Also, it is clear that the values of A1 and A2 in Table 1.9 are not much different from the real values of A1 and A2 in Table 1.4. The time dependence is inversely related to the frequency dependence, that is, A1 and A2 decrease as time increases, while those increase as frequency increases.

75


TABLE 1.9 Time Response of Transformation Matrix Given in Table 1.4 A1

A2

1.0290 1.0108 1.0047 1.0018 0.9998

−2.1850 −2.2151 −2.2248 −2.2296 −2.2326

Time [μs] 10 50 100 150 200

Figure 1.31 shows the frequency and time dependence of the voltage transformation matrix of an untransposed vertical single-circuit line. It is clear from the figure that the frequency dependence is greater than the time dependence of the transformation matrix. The maximum deviation of the time dependence from the average value is about 10%, while it is about 30% for the frequency dependence. Also, the frequency/time dependence is much greater in the vertical line case than that in the horizontal line case. Table 1.10 shows the time response of the transformation matrix given in Equation 1.203 and Table 1.7 for the untransposed vertical twin-circuit line of Figure 1.25. A

3 –A31

4

A32

3

1 A = A11 A12 –A22

2

1

–1

1 A21 A22

1 A31 A32

1 17 m

–A21 –A12

2

8m

A11

0

4m

8m

Time (µs), frequency (kHz)

500

Conductor 240 mm2 ASCR ρe = 3.78 × 10–8 Ω-m earth ρe = 100 Ω-m

FIGURE 1.31 Time/frequency dependence of transformation matrix of an untransposed vertical singlecircuit line.

76


TABLE 1.10 Time Response of Transformation Matrix Given in Table 1.7 a. [BI] = BI( j, k) Mode 0

Mode 1

Mode 2

Time [μs]

(1,1)

(1,2)

(2,2)

(2,3)

(3,1)

(3,3)

5 10 50 100 200 500

0.644 0.653 0.682 0.694 0.706 0.725

0.821 0.825 0.842 0.849 0.856 0.864

0.444 0.425 0.377 0.353 0.323 0.285

−0.524 −0.535 −0.562 −0.571 −0.578 −0.592

0.568 0.558 0.514 0.496 0.469 0.415

0.253 0.260 0.272 0.279 0.290 0.337

b. [BII] = BII( j, k) Mode 3

Mode 4

Mode 5

Time [μs]

(1,1)

(1,2)

(2,2)

(2,3)

(3,1)

(3,3)

5 10 50 100 200 500

0.443 0.452 0.467 0.470 0.472 0.480

0.657 0.660 0.658 0.647 0.631 0.588

0.468 0.478 0.500 0.499 0.503 0.481

−0.621 −0.642 −0.669 −0.666 −0.663 −0.662

0.659 0.670 0.685 0.672 0.678 0.653

0.288 0.278 0.251 0.241 0.214 0.180

B(1,3) = B(2,1) = −B(3,2) = 1 normalized.

The matrix deviation of each vector for 10 μs < t < 500 μs from the value at t = 10 μs is Mode 0: 11%, mode 1: 21%, mode 2: 30% Mode 3: 11%, mode 4: 5.2%, mode 5: 35% Assuming frequency is given as the inverse of time, the aforementioned time range corresponds to the frequency range of 100 kHz > f > 2 kHz. In this frequency range, the maximum deviation from the value at f = 100 kHz is Mode 0: 10%, mode 1: 21%, mode 2: 18% Mode 3: 13%, mode 4: 33%, mode 5: 40% From the earlier results, it can be said that the time dependence of the internal mode is greater and that of the intercircuit mode is smaller than the frequency dependence for an untransposed twin-circuit line. In general, the frequency dependence is greater than the time dependence of the transformation matrix.

77


1.5.4 Problems 1.16 Explain the reason why it is not easy to obtain a transformation matrix in the cases of an infinite frequency and of a perfectly conducting system. 1.17 Discuss a difference of modal components between three-phase transposed and untransposed horizontal lines.

1.6 Traveling Wave 1.6.1 Reflection and Refraction Coefficients [1,21] When an original traveling wave e1f (equivalent to a voltage source) comes from the left to node P along line 1 in Figure 1.32, the wave partially refracts to line 2, and the remaining reflects to the line 1 similarly to those of light at the surface of a water. Define the refracted wave as e2f and the reflected wave as e1b and also the characteristic (surge) impedance of the lines 1 and 2 as Z1 and Z2, respectively. Then, current I on the line 1 is given from Equation 1.56 as I = Y1(e1 f − e1b ) =

(e1 f − e1b ) (1.204) Z1

On the line 2, there being no backward wave, I=

e2 f (1.205) Z2 V

Line 1

Line 2

I

e2f

e1f

–∞ Z1

+∞ Z2

e1b P FIGURE 1.32 A conductor system composed of lines 1 and 2.

78


Voltage V at the node P on the line 1 is given from Equation 1.91 by V = e1 f + e1b (1.206)

On the line 2,

V = e2 f (1.207)

Substituting Equations 1.207 and 1.206 into Equation 1.205, I=

V (e1 f + e1b ) = Z2 Z2

Substituting the earlier equation into Equation 1.204, e1b is obtained as e1b = θ ⋅ e1 f (1.208)

where = (Z †2 −Z1 )†( / Z †2 + †Z1 ) : reflection coefficient. (1.209) Similarly, e2f is given as

e2 f = λ ⋅ e1 f (1.210)

where = 2Z2/Z2 + Z1 = 1 + θ : refraction coefficient. (1.211) It should be clear from Equations 1.208 and 1.210 that the reflected and refracted waves are determined from the original wave by the reflection and refraction coefficients, which represent the boundary condition at the node P between the lines 1 and 2 with the surge impedances Z1 and Z2. The coefficients θ and λ give a ratio of the original wave (voltage) and the reflected and refracted voltages. For example,

a. Line 1 open-circuited (Z2 = ∞): θ = 1, λ = 2, I = 0, V = 2e1f b. Line 1 short-circuited (Z2 = 0): θ = −1, λ = 0, I = 2e1f, V = 0 c. Line 1 matched (Z2 = Z1): θ = 0, λ = 1, I = e1f/Z1, V = e1f

The earlier results show that the reflected voltage e1b at the node P is the same as the incoming (original) voltage e1f and the current I becomes zero when the line 1 is open-circuited. On the contrary, under the short-circuited condition, e1b = −e1f, and the current becomes maximum. Under the matching termination of the line 1, there is no reflected voltage at the node P.

79


1.6.2 Thevenin’s Theorem 1.6.2.1 Equivalent Circuit of a Semi-Infinite Line In Figure 1.33a, the following relation is obtained from Equations 1.205 and 1.207: I=

V (1.212) Z2

The earlier equation is the same as Ohm’s law in a lumped-parameter circuit composed of resistance R. Thus, the semi-infinite line is equivalent to Figure 1.33b. 1.6.2.2 Voltage and Current Sources at the Sending End A voltage source at the sending end of a line illustrated in Figure 1.34a is equivalent to Figure 1.34b, because the traveling wave on the right in (b) is the same as that in (a). Then, (b) is rewritten as Figure 1.34c, that is, the voltage source at the sending end is represented by a voltage source at the center of an infinite line. Similarly, a current source in Figure 1.35a is represented by Figure 1.35b. Furthermore, by applying the result in Section 1.6.2.1, the voltage and current sources in Figures 1.34 and 1.35 are represented by Figure 1.36. 1.6.2.3 Boundary Condition at the Receiving End

1. Open-circuited line An open-circuited line Z0 with an incoming wave e(x − ct) from the left in Figure 1.37a is equivalent to an infinite line with the incoming wave from the left and another incoming wave e(x + ct) from the right with the same amplitude and the same polarity as in Figure 1.37b. 2. Short-circuited line A short-circuited line with an incoming wave e(x − ct) in Figure 1.38a is equivalent to an infinite line with e(x − ct) and −e(x + ct). V Z1

V Z2

I

I (a)

R = Z2

Z1

+∞ (b)

FIGURE 1.33 A semi-infinite line and its equivalent circuit. (a) A semi-infinite line. (b) Equivalent circuit.

80


e(t – x/c) +∞

Z0

–∞

e(t + x/c)

e(t) x=0 (a)

e(t)

V=0

e(t – x/c)

e(t)

V = –e(t)

V = +e(t)

x = –0

x = +0

+∞

(b) –∞ Z0

e(t + x/c)

e(t – x/c)

2e(t)

V = –e(t)

+e(t)

x = –0

x = +0

+∞ Z0

(c)

FIGURE 1.34 Equivalent circuit of a voltage source at the sending end. (a) A voltage source. (b) Equivalent circuit of (a). (c) Equivalent circuit of (b).

i (t – x/c) Z0

i(t )

+∞

–∞

i (t + x/c) Z0

i (t – x/c) Z0

+∞

2 i(t) x=0 x=0

(a)

(b)

FIGURE 1.35 Equivalent circuit of a current source at the sending end. (a) Current source. (b) Equivalent circuit.

e (t)

(a)

R

i (t)

R

(b)

FIGURE 1.36 Lumped-parameter equivalent of a source at the sending end. (a) Voltage source. (b) Current source.

81


e(x – ct)

e(x – ct) –∞

–∞

e(x + ct)

Z0

x=0

Z0

+∞

x=0

(a)

(b)

FIGURE 1.37 An open-circuited line. (a) Open-circuited line. (b) Equivalent circuit.

e(x – ct)

e(x – ct) Z0

–e(x + ct)

Z0

Z0 x=0

x=0 (a)

(b)

FIGURE 1.38 A short-circuited line. (a) Short-circuited line. (b) Equivalent circuit.

3. Resistance-terminated line A resistance is equivalent to a semi-infinite line of which the surge impedance is the same as the resistance as explained in Section 1.6.2.1 and in Figure 1.33. If the surge impedance of the semi-infinite line is taken to be the same as that of the line to which the resistance is connected, then a backward traveling wave eb(x + ct) = eb is to be placed on the semi-infinite line: er (t) = θ ⋅ e(t), θ =

(R − Z0 ) (1.213) (R + Z0 )

4. Capacitance-terminated line When a semi-infinite line Z0 is terminated by a capacitance C as in Figure 1.39, node voltage V and current I are calculated in the following manner: V = e0 + er ∴ er = V − e0

I=

( e0 − er ) dV = C⋅ Z0 dt

82


e0

er Z0

V

I

C

FIGURE 1.39 A capacitance-terminated line.

Substituting er into I and multiplying with Z0, 2e0 = Z0C ⋅

dV +V dt

Solving the earlier differential equation, the following solution is obtained:  −t  V = K ⋅ exp   + 2e0 , τ = Z0C  τ 

Considering the initial condition, V = 0 for t = 0,

   −t    −t   V = 2e0 1 − exp    , er = e0 1 − 2 exp    (1.214)  τ   τ   

In a similar manner, an inductance-terminated line either at the receiving end or at the sending end can be solved. 1.6.2.4 Thevenin’s Theorem When only a voltage and a current at a transition (boundary) point between distributed-parameter lines are to be obtained, Thevenin’s theorem is very useful. In Figure 1.40, the impedance seen from nodes 1 and 1′ to the right is Z0, and the voltage across the nodes is V0. S

Z

I

1

V0

1΄ FIGURE 1.40 Thevenin’s theorem.

Z0

83


e(t – x/c) Z0

S

S

P R

(a)

P

2 e(t)

Z0

R

(b)

FIGURE 1.41 A resistance-terminated line with a voltage traveling wave. (a) Original circuit. (b) Equivalent circuit.

When an impedance Z is connected to the nodes, a current I flowing into the impedance is given by Thevenin’s theorem as

I=

V0 (1.215) (Z0 + Z)

When an original traveling wave e comes from the left along a line Z0 in Figure 1.41a, voltage V and current I at node P are calculated in an equivalent circuit Figure 1.41b where a voltage source V0(t) is given as 2e(t) by Thevenin’s theorem. It is not straightforward to obtain a reflected traveling wave er when Thevenin’s theorem is applied to calculate a node voltage and a current. In such a case, the following relation is very useful to obtain the reflected wave er from the node voltage V and the original incoming wave e: er = V − e (1.216)

By applying the earlier relation, reflected waves in Figure 1.42 are easily evaluated:

e1b = V − e1 f , e2b = V − e2 f , e3 b = V − e3 f V e2b e2f e1f e1b

e3b e3f

FIGURE 1.42 Reflected waves at a node with three lines.

84


1.6.3 Multiple Reflection In a distributed-parameter circuit composed of three distributed lines as in Figure 1.43, node voltages V1 and V2 and currents I1 and I2 are evaluated analytically in the following manner. The refraction coefficients λ at the nodes 1 and 2 are given by

λ12 =

2Z2 2Z1 , λ 21 = (Z1 + Z2 ) (Z1 + Z2 )

λ 23 =

2Z3 2Z2 , λ 32 = (Z2 + Z3 ) (Z2 + Z3 )

1. 0 ≤ t < τ For simplicity, assume that a forward traveling wave e1f on line 1 arrives at node 1 at t = 0. Then, node voltage V1 is calculated by V1(t) = λ12e1 f (t)

The reflected wave er on the line 1 is evaluated by Equation 1.216 as er (t) = V1(t) − e1 f (t)

The same is applied to traveling waves on the line 2: e12 (t) = V1(t) − e2b (t)

For the moment, only an incoming wave from the line 1 is assumed and thus e2b (t) = 0, e12 (t) = V1(t)

V1 e1f Z1

V2 e12

er I e2b 1 Node 1˝

e2f Z2

τ FIGURE 1.43 A three-line system.

e21 I e3b 2 Node 2˝

Z3


Current I1 is evaluated by

I1 ( t ) =

{e ( t ) − e ( t )} = e ( t ) , 1f

r

Z1

12

Z2

I2 (t ) = 0

The refracted wave e12 travels to node 2 on the line 2. 2. τ ≤ t < 2τ At t = τ, e12 arrives at the node 2 and becomes e2 f (incoming wave to node 2): e2 f (t) = e12 (t − τ)

The e2f produces a voltage V2 at the node 2, a reflected wave e21 on the line 2, and a refracted wave e23, which never comes back to the node 2 because the line 3 is semi-infinite. Therefore, we can forget about e23: V2 (t) = λ 23 e2 f (t), e21(t) = V2 (t) − e2 f (t)

I 2 (t) =

{e2 f (t) − e21(t)} Z2

The reflected wave e21 travels to node 1. 3. 2τ ≤ t < 3τ e2b (t) = e21(t − τ)

V1(t) = λ12e1 f (t) + λ 21e2b (t), e21(t) = V (t) − e2b (t)

Repeating the earlier procedure, the node voltages V1 and V2 and the currents I1 and I2 are calculated. The procedure is formulated in general as follows [1,17]: a. Node equation for node voltages V1(t) = λ12e1 f (t) + λ 21e2b (t)

V2 (t) = λ 23 e2 f (t) + λ 32 e3 b (t) b. Node equation for traveling waves e12 (t) = V1(t) − e2b (t), e21(t) = V2 (t) − e2 f (t) c. Continuity equation for traveling waves e2 f (t) = e12 (t − τ), e2b = e21(t − τ)

85

86


d. Current equation I1(t) =

{e12 (t) − e2b (t)} ,

I 2 (t) =

Z2

{ e2 f (t) − e21(t)} Z2

The earlier procedure to calculate a traveling-wave phenomenon is called “refraction coefficient method,” which can easily deal with a multiphase line and require only a precalculation of the refraction coefficient [17]. “Lattice diagram method” [21] is well known, but the method requires both the refraction and reflection coefficients, and furthermore it is not easy to deal with the multiphase line. There is a more sophisticated approach called Schnyder–Bergeron (or simply “Bergeron”) method [22], which has been adopted in the well-known computer software EMTP [23,24] originally developed by the Bonneville Power Administration (BPA), U.S. Department of Energy. The method is very convenient for a numerical calculation by a computer but not convenient for a hand calculation with physical insight of the traveling-wave phenomenon. Example Let’s obtain voltages V1 and V2 and current I1 for 0 ≤ t < 6τ in Figure 1.44a. V1 Node 1

1

I2

I1 2 τ, Z2 = 200 Ω

(V)

V2

200

2

E = 100 V

0

Z1= 0

τ

2τ

(b)

(a)

3τ t

4τ

5τ

(A) 0.5 0

τ

2τ 3τ

4τ

5τ

t

–0.5 (c) FIGURE 1.44 Voltage and current responses on an open-circuited line. (a) An open-circuited line. (b) V2(t). (c) I1(t).


Solution λ12 = 2, λ 21 = 0, λ 23 = 2, e1 f ( t ) =

E = 50 [V] 2

1. 0 ≤ t < τ

V1 ( t ) = e12 ( t ) = λ12 ⋅ e2 f = E = 100 [ V ], I1 ( t ) =

e12 ( t ) = 0.5 [A] Z2

2. τ ≤ t < 2τ e2 f (t) = e12 (t − τ) = 100 [V], V2 (t) = λ 23 e12 (t − τ) = 200 [V]

e21(t) = V2 (t) − e2 f (t) = 100 [V], I 2 (t) =

{e2 f (t) − e21(t)} = 0 Z2

3. 2τ ≤ t < 3τ e2b (t) = e21(t − τ) = 100 [V], V1(t) = E + λ 21e2b (t) = E = 100 [V] e12 (t) = V1(t) − e2b (t) = 0 [V]

I1(t) =

{e12 (t) − e2b (t)} = −0.5 [A] Z2

4. 3τ ≤ t < 4τ

e2 f (t) = e12 (t − τ) = 0 [V], V2 (t) = 0 [V], I 2 (t) = 0 [A], e21(t) = 0 [V]

5. 4τ ≤ t < 5τ

e2b (t) = 0 [V], V1(t) = 100 [V], e12 (t) = 100 [V], I1(t) = 0.5 [A]

6. 5τ ≤ t < 6τ

e2 f (t) = 100 [V], V2 (t) = 200 [V], I 2 (t) = 0 [A], e21(t) = 100 [V]

Based on the earlier results, V1, V2, and I1 are drawn as Figure 1.44b and c.

87

88


1.6.4 Multiconductors 1.6.4.1 Reflection and Refraction Coefficients The refraction and reflection coefficient matrices are given in the following form in the circuit of Figure 1.45: −1 λ ij  = 2  Z j  ([ Zi ] + [Z j ])   (1.217)  Reflection coefficient θij  = λ ij  − [U ] 

Refraction coefficient

where i, j are for ith and jth lines [U] is the unit matrix 1.6.4.2 Lossless Two Conductors Let’s consider the lossless two-conductor system illustrated in Figure 1.46. The surge impedance matrices of the lines are given by R

[c ] =  0 

0 , ∞ 

 Zs

Zm  , Zs 

[ Z2 ] = [ Z0 ] =  Z 

m

(V1)

∞

[ Z3 ] =  0 

(V2)

Line 1

Line 2

Line 3

(Z1)

(Z2), (τ)

(Z3)

FIGURE 1.45 A multiconductor system. (V1)

E R

(V2) τ = l/c

Phase a

(I1)

Phase b

(Z0) = FIGURE 1.46 A lossless two-conductor system.

Zs Zm Zm Zs

0 (1.218) ∞ 


89

The incoming traveling wave is given by E (E) =   (1.219) 0

The refraction coefficient matrix at each node is 0 , 0 

 Zs

[λ12 ] = {2/R + Zs}  Z 

m



0 (1.220) 2 



0

 R

[λ 21 ] = {2/R + Zs}  −Z

2

[λ 23 ] = 0

m

 (R + Zs )

Applying the earlier refraction coefficients and the incoming wave, the node voltages in Figure 1.46 are calculated at every time step by using the refraction coefficient method in a similar manner to those in Sections 1.6.3.1 and 1.6.3.2: 1. 0 ≤ t < τ

(V1 ) = [λ12 ] ( E ) =

0E 2E Zs = = E12 0 0 R + Zs Zm

Zs 2 R + Zs Zm

( I1 ) = [ Z0 ] ( E12 ) = −1

Zs 1 2 Z − Zm −Zm 2 s

−Zm 2E Zs Zs R + Zs Zm

2E Zs2 − Zm2 2E = = R + Zs (R + Zs )(Zs2 − Zm2 ) −Zs Zm + Zs Zm 0

2. τ ≤ t < 2τ

{E ( )} = {E

12

2f t

(t − τ)}

(V2 ) = [λ 23 ] ( E2 f ) =

2 0

E2 fa 0 E2 fa 4E(t − †) τ Zs =2 = 2 ( E2 f ) = 2 {E12 (t − τ)} = E2 fb 2 E2 fb R + Zs Zm

{E ( )} = {V (t)} − {E ( )} = E 21 t

2

2f t

2 f (t )

= {E12 (t − τ)} =

2E ( t − †τ ) Zs R + Zs Zm

90


3. 2τ ≤ t < 3τ

{E ( t )} = {E ( t − τ )} = {E 2b

21

12

(V1 ) = [λ12 ]{E(t)} + [λ 21 ]{E2b ( t )} =

{E12 (t)} = (V1 ) − {E2b ( t )} =

2RE ( t − †2τ )  Zs 2  E ( t ) +  R + Zs  R + Zs  Zm

2  R − Zs  Zs E(t − 2τ) E ( t ) + R + Zs  R + Zs  Zm

( I1 ) = [Z0 ]−1 {E12 ( t )} − {E2b (t)} =

(t − 2τ)}

2  2Zs 1 E(t − 2τ) E ( t ) − R + Zs  R + Zs 0

Repeating the earlier procedure, voltages (V1) and (V2) and current (I1) are calculated. Example 1.1 Assuming that R = 100 Ω, Zs = 400 Ω, Zm = 100 Ω, and E = 1 pu, calculate (V1), (I1), and (V2) for 0 ≤ t < 5τ: 1. 0 ≤ t<τ

1.6

0 1 1.6 = = (E12 ) 0 0 0.4

( E2 f ) = 0.4

( I1 ) = [ Z0 ] ( E12 ) = −1

1 4 1500 −1

−1 1.6 4 × 10 −3 = 4 0.4 0

2. τ ≤ t < 2τ 1.6

0 1.6 3.2 1.6 ⋅ = , ( E21 ) = (V2 ) − ( E2 f ) = 0.4 2 0.4 0.8

2

( E2 f ) = 0.4 , (V2 ) = 0 3. 2τ ≤ t < 3τ 1.6

1.6

( E2b ) = 0.4 , (V1 ) = 0.4 0.64

( E12 ) = (V1 ) − ( E2b ) = 0.16 ,

0 1 0.4 + 0 0 −0.4

( I1 ) =

1 4 1500 −1

0 1.6 2.24 = 2 0.4 0.56 −1 0.96 2.4 × 10 −3 = 4 0.24 0


91

4. 3 ≤ t < 4τ

0.64

0.28

0.64

( E2 f ) = 0.16 , ( V2 ) = 0.32 , ( E21 ) = 0.16

5. 4 ≤ t < 5τ

0.64

1.856

( E2b ) = 0.16 , (V1 ) = 0.464 1.126

( E12 ) = 0.304 , ( I1 ) =

1.44 × 10 −3 0

Drawing the earlier results, Figure 1.47 is obtained.

1.6.4.3 Consideration of Modal Propagation Velocities In a real transmission line, the line impedance becomes frequency dependent due to the skin effects of the conductor and the earth as explained in Section 1.5. The propagation velocity is also frequency dependent, and furthermore the modal velocity differs from each other. The velocity difference causes a very significant effect on the voltage and current waveshapes along the line. The effect is included analytically in a calculation of the voltage and the current in the following manner:

1. Traveling wave at the sending end

(V1 ) = (E12 ) = [λ12 ] (E) =

E1a 2E Zs = (1.221) Z R + Zs m E1b

Transforming the earlier traveling waves in the phase domain into a modal domain,

e10 E1a = [ A]−1 (1.222) e11 E1b 2. Propagation of modal traveling waves to the receiving end The earlier modal traveling waves e10 and e11 propagate to the receiving end by the propagation velocities c0 and c1, respectively. Each traveling time is given by τ0 = where τ0 > τ1 because c0 < c1.

, τ1 = (1.223) c0 c1

92


I1(×10–3 pu) 6 I1a 4 2

I1b = 0

0 –2 –4

τ

2τ

V1(pu) 3

3τ

4τ

V1a

2 1

V1b

0 –1

τ

2τ

3τ

4τ

V2(pu) 4

V2a

3 2

V2b

1 0 –1

τ

2τ

3τ

4τ

FIGURE 1.47 Analytical voltage waveforms on a two-conductor system.

Thus, the modal traveling wave at the receiving end is

e20 = e10 ( t − τ0 ) , e21 = e1(t − τ1 ) (1.224) Transforming the earlier traveling wave back to the phase domain,

E2 a e20 = [ A] (1.225) E2b e21


3. Receiving-end voltage The receiving-end voltage (V2) is obtained using the aforementioned traveling waves in the same manner as that in Section 1.6.4.2: (V2 ) = [ λ 23 ]

E2 a (1.226) E2b

By repeating the earlier procedures, the difference of the modal velocities is included in transient voltage and current calculations. Example 1.2 Assuming that c0 = 250 m/μs, c1 = 300 m/μs, and the line length ℓ = 750 m in Example 1.1, calculate (V1), (V2) for 0 ≤ t < 2τ1. Solution

τ0 =

750 750 = 3 µs, τ1 = = 2.5 µs 250 300

Traveling wave at t = 0 at the sending end is 1.6

(V1 ) = ( E1 ) = 0.4

The voltage transformation matrix A for a symmetrical two-conductor system is given by 1

[ A] = 1

1 , −1

[ A]

−1

=

11 21

1 −1

The modal traveling waves e10 and e11 at the sending end are

 e10  −1 11   = [ A] ( E1 ) = 21  e11 

1 1.6 1.0 = −1 0.4 0.6

Modal traveling waves at the receiving end are given in the following form:

e20 ( t ) = e10 ( t − τ0 ) = 1.0u ( t − τ0 )

e21 ( t ) = 0.6u ( t − τ1 )

93

94


Transforming the previously mentioned modal components into an actual phase domain (phasor) component,

E2 a 1 = [ A] ( e 2 ) = E2b 1

1 u ( t − τ0 ) u ( t − τ0 ) + 0.6u ( t − τ1 ) = =c −1 0.6u ( t − τ1 ) u ( t − τ0 ) − 0.6u ( t − τ1 )

Thus, the receiving-end voltage (Vr) is given by V2 a 2u ( t − τ0 ) + 1.2u(t − τ1 ) = [λ 23 ] ( E2 f ) = 2 ( E2 f ) = V2b 2u ( t − τ0 ) − 1.2u(tt − τ1 )

Reflected waves at the receiving end are E21a = (V2 ) − ( E2 f ) = ( E2 f ) = ( E21 ) E21b

In the modal domain, e210 u ( t  τ0 ) = [A]−1 ( E21 ) = ( e21 ) e211 0.6u(t  τ1 )

Backward traveling waves at the sending end are given by

e1b 0 (t = e210 ( t − τ0 ) = u ( t − 2τ0 )

e1b1 (t = e211 ( t − τ1 ) = 0.6u ( t − 2τ )

Transforming into the actual phase domain, c u ( t  2τ0 ) + 0.6u ( t  2τ1 ) = [ A] ( e1b 0 ) = E1bb u ( t  2τ0 ) − 0.6u ( t  2τ1 )

Thus, the sending-end voltage is V1a 1.6 0.4 = ( E1 ) + [λ 21 ( E1b ) = + V1b 0.4 −0.4 =

0 u ( t  2τ0 ) + 0.6u ( t  2τ1 ) 2 u ( t  2τ0 ) − 0.6u ( t  2τ1 )

1.6 0.4u ( t  2τ0 ) + 0.24u ( t  2τ1 ) + 0.4 1.6u ( t  2τ0 ) − 1.44u ( t  2τ1 )

95


V 4 3 2 1

V1a

1.6

V1b

0.4

2.24

1.84

0.56

0 –1 –2

–1.04 0

1

2

3

t (µs)˝

4

V 4 3 2

1.2

1

5

6

V2a

3.2

V2b

0.8

7

0 –1 –2

–1.2 0

1

2

3

t (µs)˝

4

5

6

7

FIGURE 1.48 Analytical surge waveforms when considering modal velocities.

The earlier results are shown in Figure 1.48 by a real line in comparison with those in Figure 1.47 with the constant velocity c = 300 m/μs (τ = 2.5 μs) by a dotted line. It is observed in the figure that a negative voltage appears first at the receiving end on the induced phase (V b) because the mode 1 traveling wave arrives at the receiving end at t = τ1 = 2.5 μs. Then, 0.5 μs later, the mode 0 wave arrives, and the voltage V b becomes positive and equal to the voltage neglecting the modal velocity difference (dotted line). The negative voltage appears on the induced phase at the sending end when a refracted wave from the receiving end comes back to the sending end. The phenomena are clearly observed in the earlier analytical calculations, that is, the phase b voltage becomes negative when modal voltages, which are positive, are transformed into the actual phase domain (see the transformation matrices carefully). The phenomena were observed in field measurement on a 500 kV untransposed horizontal line [25], of which the line configuration is given in Figure 1.22. The measured result is shown in Figure 1.49.

96


R = 415 Ω

CB Phase a

(Vr)

(Is)

E = 1pu

b

(Vs)

c

(a)

l = 83.212 km

Phase a

Vr/E (pu)

0.5

A 0

B

Phase b Phase c Time (µs)

100

(b) FIGURE 1.49 A field test circuit and test result from Ref. [25]. (a) Test circuit. (b) Test result Vr.

The negative voltages on phases b and c in Figure 1.49 are explained by the earlier analytical evaluation. The distorted waveform observed in the measured result is caused by the frequency-dependent attenuation and propagation velocities as already explained in Figure 1.28, which is, in fact, a step response on the same line as that of Figure 1.49, that is, the 500 kV untransposed horizontal line in Figure 1.22. Similar but more complicated behaviors are observed in an untransposed vertical twin-circuit line as discussed in Refs. [26,27].

1.6.4.4 Consideration of Losses in a Two-Conductor System Traveling-wave deformation at the distance x from the sending end is defined in the frequency domain by

Ex ( ω) = exp {−Γ ( ω) x} E0 ( ω)

Or in Laplace domain with Laplace operator s = jω + α,

Ex ( s ) = exp {−Γ ( s ) x} E0 ( s ) (1.227)

97


where Ex(s) is the traveling wave at distance x E0(s) is the original wave at x = 0 Inverse Laplace transform of the aforementioned gives the following time response: t

  d x  L Ex ( s ) = L  s exp −Γ ( s ) ⋅  ⋅ E0 ( s )  =   s ( τ ) ⋅ e0 (t − τ) ⋅ dτ s     dt  0 −1

∫

−1

Or ex ( t ) = s ( t ) ∗ e0 (t) (1.228)

where ex ( t ) = L−1Ex ( s ) , e0 ( t ) = †L−1E0 ( s )

(1.229)

s(t) = L −1 exp{−Γ(s)x/s} is the step response of wave deformation * is the real-time convolution

Figure 1.50 illustrates the step responses of wave deformation. Figure 1.49 is a measured result of the step response on a three-phase line, when a step-function voltage e0(t) = 1 is applied to phase a at the sending end (x = 0) of the line. Let’s assume that the propagation constant Γ(ω) is given as a constant value at a frequency: Γ = α + jβ = const., β = ω/c : phase constant , c : velocity (1.230)

Then, s(t) is given by s ( t ) = L−1 exp(−αx) ⋅ exp(−sτ)/s) = exp(−αx) ⋅ u(t − τ) (1.231)

where τ=

x , u ( t ) : unit step function c Ex

E0 Γ, τ x FIGURE 1.50 Step response of wave deformation.

(1.232)

∞

98


Assuming e0(t) to be e0 · u(t), Equation 1.228 is rewritten by ex ( t ) = k ⋅ e0 ⋅ u ( t − τ ) (1.233)

where k = exp(−α ⋅ x) : attenuation ratio, α  Np/m  = α [dB/m ]/8.686 (1.234) For a multiconductor system, the earlier equation is applied to each modal wave. Example 1.3 In the same problem as Example 1.2, calculate voltages considering the attenuation k0 = 0.8 and k1 = 0.98. Solution From the solution of Example 2 at t = 0, V1a 1.6 e10 1.0 = †,††† = V1b 0.4 e11 0.6

Modal traveling waves at the receiving end are given considering the attenuation by

e20 = k0 e10U ( t − τ0 ) = 0.8U(t − τ0 )

e21 = k1e11U ( t − τ1 ) = 0.588U(t − τ1 )

In an actual phase domain,

E2 a 1 = E2b 1

1 0.8u(t − τ0 ) 0.8u ( t − τ0 ) + 0.588u(t − τ1 ) = −1 0.588u(t − τ1 ) 0.8u(t − τ0 ) − 0.588u(t − τ1 )

Thus, the receiving-end voltage is given as follows

V2 a E2 a 1.6u ( t − τ0 ) + 1.176u(t − τ1 ) =2 V2b E2b 1.6u(t − τ0 ) − 1.176u(t − τ1 )

The reflected wave at the receiving end is

E21a E2 a = E21b E2b


99

Transforming into a modal domain, e210 e20 0.8u(t − τ0 ) = = e211 e21 0.588u(t − τ1 )

At the sending end, the aforementioned traveling wave is attenuated as

e1b 0 (t) = k0 e210U ( t − τ0 ) = 0.64u(t − 2τ0 )

e1b1 ( t ) = 0.98 × 0.588u ( t − 2τ1 ) = 0.576u(t − 2τ1 )

Transforming into a phase or domain,

E1ba 1 = E1bb 1

1 e1b 0 (t) †. 0 64u ( t − 2τ0 ) + 0.576(t − 2τ1 ) = −1 e1b1 (t) 0.64u ( t − 2τ0 ) − 0.576(t − 2τ1 )

Thus, the sending-end voltage for 2τ1 ≤ t ≤ 4τ1 is obtained: V1a 1.6 0.4 = + V1b 0.4 −0.4 =

0 †. 0 64u ( t − 2τ0 ) + 0.576 ( t − 2τ1 ) 2 0.64u ( t − 2τ0 ) − 0.576 ( t − 2ττ )

1.6 0 25u ( t − 2τ0 ) + 0.23 ( t − 2τ1 ) †. + 0.4 1.024u ( t − 2τ0 ) − 1.382 ( t − 2τ )

The earlier results are illustrated in Figure 1.51.

1.6.4.5 Three-Conductor System Let’s consider the field test circuit in Figure 1.49a. The parameters of the line are given as Characteristic impedance: Zaa = Zcc ≅ Zbb = 331, Zab = Zbc = 72, Zac = 34[Ω] Modal velocity: c0 = 270.4, c1 = 296.4, c2 = 299.5[m/μs] Modal attenuation: α 0 = 7.94×10−2, α1 = 3.5×10−3, α2 = 6.8×10−4 [dB/km] Attenuation ratio: k0 = 0.468, k1 = 0.967, k2 = 0.994, (see Equation 1.234) Line length: ℓ = 83.212 km Modal traveling time: τ = ℓ/c : τ0 = 307.7, τ1 = 280.7, τ2 = 277.8[μs] Assume E = 1000 [V]. When a circuit breaker (CB: a switch) is closed at t = 0, the phase “a” voltage at the sending end is calculated as the ratio of source resistance R and the phase “a” characteristic impedance Zaa as

  Zaa   331 Vsa =  ⋅E =   × 1000 = 440 [V] ( R Z ) ( 415 331 ) + + aa    

100


V1(pu) 3 V1a

2 1

1.6

V1b

2.24

1.84

2.086

1.83

0.56

0.4

0 –1 –2

–0.038

–0.982 –1.04 0

1

2

3

t (µs)

V2(pu) 4

4

5

6

3.2

V2a

3

7

2.776

2 1.176

1

1.2

V2b

0.8

0.424

0 –1.176

–1 –2

–1.2 0

1

2

t (µs)

3

4

5

FIGURE 1.51 Analytical results of surge voltages on a two-phase line.

Then, the phase “a” current at the sending end is

I sa =

Vsa 440 = [A] Zaa 331

The phase “b” and “c” currents are zero because of the open-circuit condition:

I sb = I sc = 0

Thus, the sending-end voltage (Vs) is calculated by using the characteristic impedance matrix from the three-phase currents:

(Vs ) = [ Z ] ( I s ) (1.235)

101


Thus, the following results are obtained: Vsb = 97 , Vsc = 53 [ V ]

Assume that the voltage transformation matrix is given by

1  A = [ ] 1 1

1 0 −1

1  −2  , 1 

2 −1  1  A = [ ]  6  3    1

2 0 −2

2  −3  1 

Then, the modal traveling wave at the sending end is calculated as

es0 = 197 , es1 = 193.5, es 2 = 49.8 [ V ]

Each modal wave arrives at the receiving at different times (τi) and with different attenuations (ki):

er 0 = k0 ⋅ es0 ⋅ u ( t − τ0 ) = 92u ( t − τ0 ) , er 1 = k1 ⋅ es1 ⋅ u ( t − τ1 ) = 187 u ( t − τ1 ) er 2 = k2 ⋅ es 2 ⋅ u ( t − τ2 ) = 49.5u ( t − τ2 )

The refraction coefficient at the receiving end is given by a unit matrix multiplied by factor “2,” because the three phases at the receiving end are open-circuited. Thus,

vr 0 = 2er 0 = 184u ( t − τ0 ) , vr 1 = 374u ( t − τ1 ) , vr 2 = 99u ( t − τ2 )

Transforming the aforementioned modal voltages to phase voltage by V = Av,

Vra = vr 0 + vr 1 + vr 2 = 184u ( t − τ0 ) + 374u ( t − τ1 ) + 99u ( t − τ2 )

Vrb = vr 0 − 2vr 2 = 184u ( t − τ0 ) − 198u(t − τ2 )

Vrc = vr 0 − vr 1 + vr 2 = 184u ( t − τ0 ) − 374u ( t − τ1 ) + 99u ( t − τ2 )

Drawing the earlier results by considering the time difference τi, Figure 1.52 is obtained. It is observed in the figure that a positive spike voltage appears on the phase “c” for the time period of 0 ≤ t ≤ τ1 − τ2 = 2.9 μs. This explains the

102


1 Phase a

Vr (pu)

0.5 Voltage spike

Phase b

0 Phase c –0.5

0

10

20

Time (µs)

30

40

50

FIGURE 1.52 Analytical results of surge voltages on a three-phase line corresponding to Figure 1.49.

spike voltage A in the measured results of Figure 1.49b. Similarly, the negative voltage B on the phase “b” in Figure 1.49b is explained by the aforementioned analytical calculation as in Figure 1.52. 1.6.4.6 Cascaded System Composed of the Different Numbers of Conductors It is often observed in practice that the number of conductors changes at a boundary as in Figure 1.53 where the phases “a” and “b” are short-circuited at node 1. In the case of a cross-bonded cable, three-phase metallic sheaths are rotated at every cross-bonding point. In such a case, it is required to reduce the order of an impedance matrix and/or to rotate the matrix elements.

Node 1 Z1

V1

I1

Node 2 Va Vb

I2

V2

Vc

[Z] Ia Ib Ic

FIGURE 1.53 A system composed of a single conductor and three conductors.

103


In Figure 1.53, the following relations of voltages and currents are obtained:

 Va   1    (V ) =  Vb  =  1  Vc  0     I1   1 ( I′) =   =   I 2  0

0   V1  0    = [Tt ] (V ′ )  (1.236) V2 1    1 0

 Ia  0   I b = [T ] ( I ) (1.237) 1     Ic 

By adopting the relation V = Z·I, I′ in the earlier equation is rewritten as (I′) = [T](I) = [T][Z]−1 (V) = [T][Z]−1 [Tt](V)′ = [Z′]−1(V′) ∴ [ Z′] = [T ][ Z ] −1

−1

[Tt ] (1.238)

Z in the earlier equation is an original 3 × 3 matrix at the right of node 1, while Z′ is a matrix reduced to 2 × 2 considering the short circuit of the phases a and b. By using Z′, the refraction coefficient at node 1, for example, can be calculated. Remind the fact that there exists no inverse matrix for matrices T and Tt, and thus care should be taken to the sequence for calculating Equation 1.238. 1.6.5 Problems 1.18 Obtain voltage V, current I, and reflected voltage traveling wave er at node P when step-function voltage traveling wave e0 arrives at the node P at t = 0 in Figure 1.54 for I(0) = 0. 1.19 Obtain voltage V at node P when step-function voltage wave e0 arrives at the node P at t = 0 in Figure 1.55 for V(0) = 0. V

e0

I Z0

er

P L

FIGURE 1.54 Inductance L terminated line.

104


P

Z0

Z0

R1

C

R2

FIGURE 1.55 A tower model. Line 1

V1

S

e0

Z0 = 0

c1 = 300 m/µs e12

e1b

Z1 = 300 Ω l1 = 1.2 km τ1

Node 1

Line 2

V2 I2

V3

c2, Z2 e23

e2f e21

e2b 2

e3f l2 τ2

e32 3

FIGURE 1.56 Two cascaded lines.

1.20 When switch S is closed at t = 0 in Figure 1.56, obtain voltages V2 and V3 and current I2 in the following conditions (a) and (b), and draw the curves of V2, V3, and I2 for 0 0 ≤ t ≤ 15 μs: a. Z2 = 60 Ω, c2 = 150 m/μs, l2 = 150 m. b. c2 = c1, Z2 = Z1, l2 = 300 m. c. Discuss the effect of a cable connected to an overhead line knowing that the surge impedance 300 Ω is for the overhead line and 60 Ω is for the cable. 1.21 Obtain receiving-end voltage (Vr) when a source voltage is applied to phase b in Figure 1.49a.

1.7 Nonuniform Conductors There are a number of papers discussing a nonuniform line [28–47]. EMCrelated transients or surges in a gas-insulated substation and on a tower involve the nonuniform line such as a short-line, nonparallel, and vertical


105

conductors. Pollaczek’s [7], Caron’s [8], and Sunde’s [48] impedance formulas of an overhead line are well known and have been widely used in the analyses of the previously mentioned transients. It, however, has not been well understood that the formulas were derived assuming an infinitely long and thin conductor, that is, a uniform and homogeneous line. Thus, impedance formulas are restricted to the uniform line where the concept of “per-unitlength impedance” is applicable. This section explains impedance and admittance formulas of nonuniform lines, such as finite-length horizontal and vertical conductors based on a plane wave assumption. The formulas are applied to analyze a transient on a nonuniform line by an existing circuit theory–based simulation tool such as the EMTP [10,24]. The impedance formula is derived based on Neuman’s inductance formula by applying an idea of complex penetration depth explained already. Then, the admittance is obtained assuming the wave-propagation velocity being the same as the light velocity in free space in the same manner as an existing admittance formula, which is almost always used in steady-state and transient analyses on an overhead line. 1.7.1 Characteristic of Nonuniform Conductors 1.7.1.1 Nonuniform Conductor First of all, it is necessary to clarify a problem to be discussed in this section, that is, a nonuniform line or a nonhomogeneous line. Figure 1.57 shows a typical example of transient voltage responses measured on a vertical conductor with the radius r = 25 mm and the height h = 25 m [39,40]. (a) is the current, (b) the voltage at the top of the conductor, and (c) the voltage at the height of 12 m. It should be clear in the figure that the voltage waveforms of (b) and (c) are distorted before a reflection from the bottom (earth surface) comes back. Also, the waveform in (c) is different from that in (b). The reason for the earlier phenomena can be either the frequencydependent effect of the conductor or by reflection of the traveling wave due to discontinuities of the characteristic (surge) impedance along the vertical conductor other than the earth surface. There is another cause of the distortion, that is, radiation, in such a high-frequency region. Also, the electromagnetic field is not perpendicular to the conductor surface. This book, however, is restricted to a TEM mode of propagation, and the earlier phenomena can be translated or interpreted as the reflection/refraction of a traveling wave. It is hard to receive such a noticeable frequency-dependent effect only by less than 10 m between (b) and (c). Thus, it should be said that the voltage waveform at a vertical conductor is distorted due to the nonuniformity of the vertical conductor at every height (position). That is, the characteristic impedance (impedance and admittance in general) of the vertical conductor is position dependent.

106


2

200

100

V1 (V)

I1 (A)

150 1

0 (a)

0

40

80 120 t (ns)

160

200

0 (b) 200

150

150

40

80 120 t (ns)

160

0

40

80 120 t (ns)

160

200

V2 (V)

100 50

50 0

0

V3 (V)

200

100

(c)

50

0

40

80 120 t (ns)

160

200

0 (d)

200

FIGURE 1.57 Measured transient responses at various positions of an artificial tower (h = 15 m, r = 25 mm). (a) Current at the top. (b) Voltage at the top. (c) Voltage at the height of 12 m.

Figure 1.58 shows another example of a transient voltage at the top of an 1100 kV transmission tower (height 140 m, average radius 6.3 m) [41]. (a) is the measured result of an injected current and the voltage at the tower top, and (b) is the step response of the tower top voltage numerically evaluated from the measured current and voltage in (a). The step response is observed to be heavily distorted before reflection from the tower bottom, that is, time t less than 0.933 μs. Although there exist tower arms, its effect on the average distortion is estimated not as heavy as that in Figure 1.58 [40]. Most distortion is estimated due to the position-dependent surge impedance of the tower. Figure 1.59 shows a transient-induced voltage at the sending end of a horizontal nonparallel conductor when a steplike voltage is applied to the other conductor through a resistance nearly equal to the conductor surge impedance. All the other ends of the conductors are terminated by the surge impedance. The radius, length, and height of the conductor are 5 mm, 4 m, and 40 cm. The separation y2 at the receiving end between the conductors is 10 cm and that (y1) at the sending end is varied from 10 to 300 cm. A typical characteristic of a nonparallel conductor is observed in the figure due to a position (distance from the sending end)-dependent impedance [38]. The induced voltage is gradually increasing as time passes, because the mutual impedance increases as time passes and thus a positive reflection comes back to the sending end at every instance until a large negative reflection at about t = 27 ns from the receiving end.

107


200 V

2A Injected current

0

Current

Voltage

Tower-top voltage

0 2.5 µs

(a) 200 V

Voltage

126 V

0

2.5 µs

(b) FIGURE 1.58 Measured transient response at the top of a 1100 kV transmission tower. (a) Injected current and tower top voltage. (b) Step response of the tower top voltage.

20

Voltage [V]

y1 = 10 cm 20 50

10

100 200 300

0 0

10

20 Time (ns)

30

40

FIGURE 1.59 Measured transient-induced voltages on a horizontal nonparallel conductor (y2 = 10 cm, y1 = 10−300 cm).

108


1.7.1.2 Difference from Uniform Conductors Carson’s, Pollaczek’s, and Sunde’s formulas are well known and widely used as an impedance of an overhead line and an underground cable isolated from the earth (soil). It, however, seems not well understood that the formulas are applicable only to a uniform or a homogeneous line of which the impedance can be defined by per unit length. Those formulas are based on an infinitely long line or an assumption that the line length “x” is far greater than the line height “h” and the separation “y” between two lines and “h” and “y” are far greater than the radius “r.” The basic form of the earth-return impedance is given by x1 x2

Zij =

∫ ∫  f (x , x ) ⋅ d 1

2

x1

⋅ dx 2 (1.239)

0 0

Assuming x2 is infinite, the second integral is carried out, and the following expression is obtained: x1

Zij =

∫  g(x ) ⋅ d 1

x1

(1.240)

0

Again, assuming x1 being infinite, the following well-known Pollaczek’s, Carson’s, or Sunde’s impedance is obtained:

jωµ0 Zij = 2π

∞

∫ 0

exp {−(hi + hi )x} x 2 + γ 02 + x

⋅ cos( yij ⋅ x) ⋅ dx (1.241)

It should be noted that Equations 1.240 and 1.241 have already contained the effect of mutual coupling due to the infinitely long line 2. Or, in the singleline case, that is, in the self-impedance cases, the section of length dx1 has contained the mutual coupling effect due to the remaining part (infinitely long) of the line. Because Equation 1.241 has contained the mutual coupling effect of all the remaining part of the line on the line section, we can define the impedance per unit length. It should be clear from Equation 1.239 that the line impedance is a function of the line length x, and Z finite in Equation 1.239/x < Zinfinite in Equation 1.241 per unit length (1.242) Therefore, it is not possible to discuss the wave propagation on a finite line by Pollaczek’s, Carson’s, or Sunde’s one in Equation 1.241. The characteristic impedance seen from an arbitrary position (distance x from origin) is always the same on an infinitely long line as already


109

explained, while it cannot be defined on a finite line. If we define the characteristic impedance Z0(ω) as a ratio of voltage V(ω) and current I(ω) in a frequency domain at the sending end of the finite line within a time region of 2 travel times 2τ (τ = x/c, c = c(ω): velocity), we find a difference between the characteristic impedances Z1(ω) and Z2(ω) of two lines with the same configuration but different lengths x1 and x2, that is, length dependence, because the series impedance of the two lines is different as explained in the preceding text. The situation is more noticeable in a vertical conductor, because the finite length of the vertical line shows the height (distance)-dependent impedance/admittance and the nonhomogeneity is easily understood from the physical viewpoint as explained in Section 1.7.1.1. It should be noted that the “distance” (dependence) means the distance from the origin in a strict sense but not the distance from the earth surface. 1.7.2 Impedance and Admittance Formulas 1.7.2.1 Finite-Length Horizontal Conductor 1.7.2.1.1 Impedance A mutual impedance of a nonparallel conductor above an imperfectly conducting earth illustrated in Figure 1.60 is obtained in the following equation by applying the concept of the complex penetration depth to Neuman’s inductance formula [37,42]: θ µ  Zij = jω  0  Pij [ Ω/m ] Pij = ( Md − M j ) cos (1.243) 2  2π 

xi 2 x j 2

Md =

∫∫

xi 1 x j 1

xi 2 xi 2

 1   S  ds1 ⋅ ds 2 , M1 =  d x

1

∫ ∫  S  d i1

xi 1

i

s1

⋅ d′s 2 (1.244)

Where Sd = s12 + s22 − 2s1s′2cos θ + H12 Si = s12 + s′22 − 2s1s′2cos θ + H 22 H1 = hi − h j , H 2 = hi + h j + 2he , he =

ρe jωµ0

Integrating Equation 1.244, the following solution is derived: A  A  B  Md = xi 2 ⋅ ln  221  + x j 2 ⋅ ln  221  + xi1 ⋅ ln  111   A211   A121   B121 

B  + x j1 ⋅ ln  111  + H1(C121 + C211 − C221C111 )/sin θ (1.245)  B211 

110


Z

hi

ds1

xi 1

0 hi

xi 2 Conductor “i”

Sd

0 xi

–h0

Conductor “j” xi2

ds2

Si

X

Conductive place V = 0

Y –(hj + 2h0)

ds2΄ Image conductor “j ”

FIGURE 1.60 Nonparallel multiconductor system.

A  A  B  Mi = xi 2 ⋅ ln  222  + x j 2 ⋅ ln  222  + xi1 ⋅ ln  112   A212   A122   B122 

B  + x j1 ⋅ ln  112  + H 2 ( C122 + C212 − C222 − C112 ) sin θ (1.246)  B212 

Where Akmn = x jm − xik cos θ + Dkmn , Bkmn = xik − x jmcos θ + Dkmn 2 2  (xik †x jmsin θ + H n cos θ)  Ckmn = tan −1   ( H nsin θ ⋅ Dkmn )   Dkmn = xik2 + x 2jm − 2xik x jmcos θ + Hn2 , H 3 = hi + h j k, m = 1, 2; n = 1, 2, 3 In the case of perfectly conducting earth (ρe = 0), the substitution of he = 0 into Mi in the earlier equation gives the following expression:

P0 ij = ( Md − Mi 0 )cos θ/2 (1.247)


111

A  A  B  Mi 0 = xi 2 ⋅ ln  223  + x j 2 ⋅ ln  223  + xi1 ⋅ ln  113   A123   A213   B123  B + x j1 ⋅ ln  113  B213

  + H 2 (C123 + C213 − C223 − C113 )/sin θ 

Substituting xi1 = 0 and xi2 = xi and taking the limit of angle θ to zero with tan−1z = π/2 − 1/z (|z| > 1) in Equation 1.245, the impedance formula of a parallel horizontal conductor is obtained in the following form: Md = xi ln

xi − x j1 + (xi − x j1 )2 + dij2 x1 − x j 2 + (xi − x j1 )2 + dij2

− xi 2ln

− xi1ln

xi − x j 2 + (xi − x j 2 )2 + dij2 2 j2

2 ij

x + d − xj2

xi − x j1 + (xi − x j1 )2 + dij2 x 2j1 + dij2 − x j1

− (xi − x j1 )2 + dij2

+ (xi − x j 2 )2 + dij2 + x 2j1 + dij2 − x 2j 2 + dij2 Mi = xi ln

xi − x j1 + (xi − x j1 )2 + Sij2 x1 − x j 2 + (xi − x j1 )2 + Sij2

+ x j 2ln

− x j1ln

xi − x j 2 + (xi − x j 2 )2 + Sij2 x 2j 2 + Sij2 − x j 2

xi − x j1 + (xi − x j1 )2 + Sij2 x 2j1 + Sij2 − x j1

− (xi − x j1 )2 + Sij2

(1.248) + (xi − x j 2 )2 + Sij2 + x 2j1 + Sij2 − x 2j 2 + Sij2 When xj1 is taken to be zero, the earlier equation becomes identical to that given in Ref. [37]. Also, the substitution of xj1 = 0 and xj2 = xj into the earlier equation gives the same formula as that derived in Ref. [42]. In the case that xi1 = xj1 = 0 and xi2 = xj2 = x in Equations 1.245 and 1.246, that is, the case of a parallel horizontal conductor with the same length, the earlier formula is simplified as follows:

{ {

 1 + 1 + ( d / x )2 ij ( M d − Mi ) Pij = = x ln  2 2  1 + 1 + (Sij /x)

 Sij + x ln   dij

}  } 

 2 2 2 2  − x + dij + x + Sij + dij − Sij (1.249) 

112


Taking a limit of distance x to infinite, the earlier equation per unit length is reduced to only the second term, that is,

 µ   Sij Zij = jω  0  ln   2π   dij

  [ Ω/m ] (1.250) 

The earlier equation is identical to the impedance derived for an infinite horizontal conductor in Ref. [6], which is an approximation of Carson’s earth-return impedance as is well known, as already explained in Section 1.2.2.2. From the earlier observation, it should be clear that the impedance formula of Equations 1.245 and 1.246 is the most generalized form for a horizontal conductor, although it is an approximate formula based on the concept of the penetration depth. 1.7.2.1.2 Admittance The admittance of finite-length horizontal conductor is evaluated by the potential coefficient in the perfectly conducting earth [1]:

[Y ] = jω[C ] , [C ] = 2πε0 [ P0 ]

−1

(1.251)

The element P0ij of the earlier matrix [P0] is given in Equation 1.247. 1.7.2.2 Vertical Conductor Let us consider a vertical multiconductor system illustrated in Figure 1.61. In the same manner as the finite-length horizontal conductor, the following impedance formula is obtained: µ  Zij = jω  0  Pij (1.252)  2π 

Pij = ( 1 − 2Xij ) [ −h1lnA1 + h2lnA2 ] + h3lnA3 − h4lnA4 + a1 − a2 − a3 + a4

nA5 − h6lnA6 − h7 lnA7 + h8lnA8 − a5 + a6 + a7 − a8 ) − ( h5ln

(1.253)

where Xii = hn − hm , Xij = hs − hr , Xij = ( Xii + X jj ) 2 h1 = hn − hs , h2 = hm − hs , h3 = hn − hr h4 = hm − hr , h5 = hn + hs + 2he , h6 = hm + hs + 2he h7 = hm + hr + 2he , h8 = hm + hr + 2he ak = hk2 + d 2 , Ak = †ak + hk

(k = 1, 2, … , 8)

When the conductors i and j are at the same vertical position, that is, hs = hn, hr = hm, Equation 1.253 is simplified in the following form:

113


X

Conductor j

Conductor i

hs

ri

hn

rj

i

xi

dxi

xj hm hr 0

S dxj

S΄

–he

Y

yj

yi

ρe

h΄r

x΄j

dxj΄

h΄s

FIGURE 1.61 Vertical multiconductor system.

 Pij = ln  

(

d2 + X 2 + X

) { d + ( H − 2X ) 2

2

}

+ ( H − 2 X ) /d

{ d + ( H − X ) + ( H − X )} + ( H/2X )  × ln { d + ( H − X ) + ( H − X )} / ( d + H + H )   × { d + ( H − 2X ) + ( H − 2X )} + ( 1 / 2X )  2d + d + H  

×

2

2

2

2

2

2

2

2

2

2 2 + d 2 + ( H − 2X ) − 2 d 2 + X 2 − 2 d 2 + ( H − X )  

where H = 2(h + he) h = hn = hs h − X = hm = hr

2

2

(1.254)

114


If the earth is assumed to be perfectly conducting, the earlier equation is further simplified:   P0 ij = ln   

(

d2 + X 2 + X

){ d + 4 ( h − X ) 2

}

 + 2(h − X )    + ( 2h − X )  2

{ d + ( 2h − X ) }   d + ( 2h − X ) + ( 2h − X )}   { h  +   ln    X   d + 4 h + 2h d 4 + h − X + 2 h − X ( ) ( ) ) { } (    2

2

d

2

2

2

2

2

2

2

2  1  2 2 2 +  2d + d + 4 h + d + 4 ( h − X )  2X  

2 −2 d 2 + X 2 − 2 d 2 + ( 2h − X )  

(1.255)

When the bottom of a single conductor (d = r) is on the earth surface, that is, h − X = 0,

  P0 = ln   

{

{

}  + { + 2h} r  

r 2 + h2 + h r 2 + 4h 2

2

r 2 + 4h 2 + 3r/2 − 2 r 2 + h 2 h

} (1.256)

The admittance of the vertical conductor system is evaluated from Equation 1.251. 1.7.3 Line Parameters 1.7.3.1 Finite Horizontal Conductor Figure 1.62 shows measured results (exp.) of the self-impedance and capacitance of a conductor with the radius 1 cm and the length 4 m above a copper plate, when a step voltage with the rise time 2 ns is applied, as a function of the height h together with calculated results by the proposed formula (cal. fin.) of a finite-length conductor and by a Carson’s formula (cal. fin.) of a finite length conductor and by a Carson’s formula (cal. inf.) of an infinitely long conductor. In the figure, the calculated results by the proposed formula (average error 5.1%) show a better accuracy than those by Carson’s formula (average error 10.9%). It should be noticed that the accuracy of the proposed formula increases as the conductor height increases. The degree of finitelength is defined by the ratio of the conductor length x and the height h, “x/h.”

115


5

L (µH)

4

3

exp. Cal.-fin. Cal.-inf.

2

20

40

60

80

100

h (cm)

(a) 70

exp. Cal.-fin. Cal.-inf.

C (pF)

60

50

40 (b)

20

40

60

80

100

h (cm)

FIGURE 1.62 Self-inductance and capacitance of a finite horizontal conductor. exp.: experimental result. cal.fin.: calculated result of finite line impedance. cal.-inf.: calculated result of Carson’s infinite line impedance. (a) Inductance. (b) Capacitance.

As x/h decreases, the accuracy of the finite conductor formula increases. It should be noted that the inductance per unit length decreases as the length decreases. The reason for this is that the inductance of an infinitely long conductor includes the mutual inductance between the reference part of a conductor and the remaining part with infinite length as explained in Section 1.7.2.2. Figure 1.63 shows a mutual inductance between two conductors with different lengths x1 and x2.

116


Inductance (µH)

1.5

exp.

1

Cal.-fin. Cal.-inf.x = x2 Cal.x = (x1 + x2)/2

0.5

(a)

Cal.x = (x1 x2) 2

3 x2 (m)

4

Capacitance (pF)

30

exp.

20

Cal.-fin. Cal.-inf.x = x2 Cal.x = (x1 + x2)/2 Cal.x = (x1 x2)

(b)

2

3 x2 (m)

4

FIGURE 1.63 Mutual inductance between different-length horizontal conductors. (a) Inductance. (b) Capacitance.

Because Carson’s formula cannot deal with the different-length conductors, three approaches to determine an effective length x are investigated: (a) x = shorter length x2, (b) arithmetic mean distance x = (x1 + x2)/2, and (c) geometrical mean distance x = x1 ⋅ x2 . It is observed in the figure that the proposed formula has a satisfactory accuracy in general. In Carson’s formula case, approach (a) shorter length seems to be best among the three approaches. It should be noted that Carson’s inductance evaluated even by approach (a) is greater than the measured result. The reason for this is as explained in Section 1.7.2.2. The earlier observation is for a perfectly


117

TABLE 1.11 Surge Impedance of an Overhead Conductor with Length x = 4 m Surge Impedance [Ω] h [cm]

x/h

21.0 19.1 39.5 10.2 61.0 6.57 83.5 4.80 102 3.93 Error [%]

Measured

Finite

Infinite

201 233 256 278 295 ——

218 251 272 285 293 5.1

224 262 287 307 319 10.9

conducting earth, that is, for a high frequency. In power frequency region, Carson’s formula shows a rather poor accuracy. It has been pointed out in Ref. [42] that the average error of Carson’s formula was about 21%, while that of the proposed formula was 4%. Table 1.11 shows measured and calculated results of a surge impedance of a horizontal conductor. It should be clear that the proposed formula shows a better accuracy than Carson’s one. The accuracy of proposed formula increases as x/h decreases corresponding to the characteristic of the inductance. A similar observation has been made in different measurements in Ref. [42]. 1.7.3.2 Vertical Conductor Table 1.12 shows a comparison of measured and calculated surge impedances of a vertical single conductor with height h and radius r. Included in the table are results calculated by various formulas given in Refs. [29,30,32,39]. It is clear in the table that the accuracy of the proposed formula is the highest among the various formulas in comparison with the measured results. Jordan’s formula [29] also shows a high accuracy. It is worth noticing that the proposed formula, Equation 1.256 for a single conductor, is identical to Jordan’s formula. With r ≪ h, Equation 1.256 is further simplified:

 4h 2   2h  h h P0 ≈ ln   + h −   = ln   − 1 = ln   (1.257)  h  r  er   4hr 

where e = 2.71828 For the surge impedance case,

 h  Zs = 60 ln   − 1 = Jordan′ s formula (1.258)  r 

Height h [m]

Radius r [mm]

d

c

b

a

)

Zs = Zw − 60 = Zj + 62.4. Zh = Zw − 120 = Zj + 2.4.

(

Zj = 60 ln(h/r) − 60 = 60ln(h/er). Zw = 60 ln 2 2h /r = Z j + 122.4.

15.0 25.4 15.0 2.5 9.0 2.5 6.0 2.5 3.0 50.0 [13] 3.0 25.0 3.0 15.0 3.0 2.5 3.0 0.25 2.0 2.5 2.0 0.25 0.608 43.375 [5] 0.608 9.45 0.608 3.1125 0.608 1.1750 Average of absolute error [%]

Ref.

320.0 459.0 432.0 424.0 181.0 235.0 250.0 373.0 514.0 345.0 481.0 112.0 180.0 250.0 310.0

Measured Zmes [Ω] 59.5 59.6 60.1 62.5 58.5 62.3 58.2 61.2 61.2 60.7 60.2 68.3 56.9 58.5 59.1 (60.4)

Zmes ln(h / er ) 323.0 462.0 431.3 407.0 187.2 228.0 258.3 365.5 503.6 341.2 479.2 104.7 191.2 256.9 315.1 2.5

Proposed 322.9 462.0 431.3 407.0 185.7 227.2 257.9 365.4 503.6 341.1 479.2 98.4 189.8 256.5 314.9 2.7

Approx.a = Jordan Ref. [2]

Measured and Calculated Surge Impedances of Vertical Conductors

TABLE 1.12

445.2 584.4 553.7 529.4 308.0 349.6 380.3 487.8 625.9 463.5 601.6 220.8 312.2 378.9 437.3 44.8

Wagnerb Ref. [3] 385.2 524.4 493.7 469.4 248.0 289.6 320.3 427.8 565.9 403.5 541.6 160.8 252.2 318.9 377.3 22.6

Sargentc Ref. [5] 325.2 464.4 433.7 409.4 188.0 229.6 260.3 367.8 505.9 343.5 481.6 100.8 192.2 258.9 317.3 2.8

Harad Ref. [13]

118 Computation for Humanity: Information Technology to Advance Society


119

Results calculated by the formula of Ref. [40] also agree well with the measured results. This is quite reasonable because the formula has been derived empirically from the measured results. It is interesting that the empirical formula agrees well with Equation 1.258 by rewriting as follows: Zh = 60{ln(2 2h /r − 2) = 60 {ln(h/r ) − 0.96}

 h  = 60 ln   − 1 + 2.4  r 

(1.259)

The aforementioned fact, that is, the fact that the empirical formula differs from the proposed formula (1.258) only with 2.4 Ω, might prove the high accuracy of the proposed formula compared with the measured results. As is observed in Table 1.12, the measured surge impedance Zmes is roughly proportional to the parameter ln(h/er). This fact is another proof of the high accuracy of the proposed formula, because the formula is directly proportional to ln(h/er). 1.7.3.3 Nonparallel Conductor Figure 1.64 shows measured mutual inductance and capacitance of an overhead nonparallel conductor with the radius r1 = r2 = 1 cm, the length x1 = x2 = 4 m, and the height h1 = h2 = 0.4 m as a function of the separation y2 at the receiving end with the parameter of the separation y1 at the sending end. Included are calculated results by the proposed formula in Section 1.7.2.1. The calculated results agree satisfactorily with the measured results. Figure 1.65 shows the mutual impedance of a nonparallel conductor as a function of “y2 − y1.” In the figure, “fin” is the impedance evaluated by the formula proposed in this book, and “inf” is that of the nonparallel conductor impedance derived from an infinite-length impedance [35,49]. The mutual impedance decreases as the “y2 − y1” and “y1” increase. The finitelength impedance is far smaller than that of the infinite length. This agrees with the tendency observed between the impedances of finite and infinite conductors in the previous section. 1.7.4 Problems 1.22 Calculate the surge impedances of a horizontal conductor with length x = 4 m given in Table 1.11 by using the following approximation, and confirm the results in the table:

Zs = 60 Pij , Sij ≅ 2h, dij = r = 1 cm

120


y1: 10-fin. y1: 10-exp. 1.5

y1: 20-fin. y1: 20-exp. y1: 50-fin. y1: 50-exp.

1

y1: 100-fin.

L(µF)

y1: 100-exp.

0.5

0 (a)

0

20

40

60 y2 (cm)

80

100

y1: 10-fin.

30

y1: 10-exp. y1: 20-fin. y1: 20-exp. y1: 50-fin.

20

y1: 50-exp.

C(pF)

y1: 100-fin. y1: 100-exp. 10

0 (b)

0

20

40

60

80

100

y2 (cm)

FIGURE 1.64 Mutual inductance and capacitance of a nonparallel conduct. (a) Inductance. (b) Capacitance.

121


Rm (mΩ/km)

50 inf; y1 = 0 m

40

inf; 500

30 inf; 1000

20

fin: 0

10

fin: 500 fin: 1000

0

500

(a)

y2 – y1 (m)

1500

1000

0.6 fin: finite line inf : infinite line

Lm (mH/km)

0.5

0.3 0.2 0.1 0

(b)

inf; y1 = 0 m

0.4 fin: 0 fin: 500

inf; 1000

500

inf; 500

y2 – y1 (m)

fin: 1000 1000

1500

FIGURE 1.65 Mutual impedance of a nonparallel conductor. (a) Resistance. (b) Inductance.

Finite: Equation 1.249, infinite: Equation 1.250 1.23 Calculate the surge impedance of a vertical conductor given in Table 1.12 by using the following approximation, and discuss the results in comparison with measured results, Wagner’s and Sargent’s formulas, in the table:

 h  Zs = 60 ln   − 1  r 

1.24 Calculate the tower top voltage Vt in Figure 1.66 for 0 ≤ t < 40 ns, taking λ01 = λ10 = 1, and compare with a measured result shown in Figure 1.57b. 1.25 All the formulas in Section 1.7.2 are an approximation based on the penetration depth. Discuss theoretical problems and engineering advantages of the formulas. If possible, develop a new formula that is theoretically better than those in Section 1.7.2.

122


it

i0

λ01 Vt

λ10

Z1 Z1 = 320 Ω c0 = 300 m/µs

λ12

x1 = 3 m

it 0.5 A

λ1 = x1/c0 = 10 ns Z2 = 270 Ω

30 ns

t

c0 = 300 m/µs x2 = 12 m λ2 = 40 ns FIGURE 1.66 A tower model.

1.8 Introduction of EMTP 1.8.1 Introduction 1.8.1.1 History of a Transient Analysis As is well known, many physical phenomena are expressed mathematically as a second-order partial differential equation. Electrical transients associated with a wave-propagation characteristic are mathematically represented by a hyperbolic partial differential equation. Therefore, to solve electrical transients necessitates the solution of the differential equation with given initial and boundary conditions. The earliest solution of the partial differential equation was given by D’Alembert for the case of a vibrating string in 1750 [13]. At the same time, Bernoulli found a solution that was of quite different form from the D’Alembert solution. Bernoulli’s solution is based on the eigenfunction and is comparable with the Fourier series. Traveling-wave concepts and the theory have been well developed since D’Alembert’s solution. Allievi first applied the theory to the field of hydraulic engineering and established the general theory and the idea of a graphical method, which was a direct application of the traveling-wave concept to engineering fields [50]. At a later stage, Bewley developed the traveling-wave theory and its application to various electrical transients [21]. The propagation


123

of the traveling-wave has been well analyzed using the modified Heaviside transform and the Sylvester theorem, which is the same as the eigenvalue theory of matrix algebra by Hayashi [51]. The graphical method developed by Allievi has been applied to the analysis of a water hammer by Schnyder [52], Bergeron [53], and Angus [54]. This is originally called the Schnyder–Bergeron method in the electrical engineering field. Unfortunately, the name of the method was abbreviated, and it is nowadays called the Bergeron method, although Schnyder originated the method. The detail and application of the graphical method are well described by Parmakian in his book [55]. The graphical method corresponds to the method of characteristic to solve Maxwell’s equation mathematically [56]. Similarly, the lattice diagram method based on the traveling-wave concept was developed to solve electrical transients by Bewley [21]. At a later stage, both the graphical method and the lattice diagram method were implemented on a digital computer for calculating electrical transients [22,23,57–64]. These techniques are generally called the traveling-wave technique or a time-domain method. The numerical Fourier transform appeared in the electrical engineering field in late 1950s [65] although the basis of the method was given by Bernoulli in 1750. Gibbs’ phenomena and instability in a transform process, which are the inherent nature of the discrete Fourier transform, are greatly reduced by developing the modified Fourier (= Laplace) transform [66,67]. At a later stage, the modified Fourier/Laplace transform was applied to transient calculations by various authors [68–71]. Since the modified Fourier/ Laplace transform provides high accuracy for obtaining a time solution, the analysis of a partial differential equation is rather easy in the frequency domain, and the implementation of fast Fourier transform (FFT) procedure into the modified Fourier/Laplace transform greatly improves the computational efficiency [70], the method has become one of the most accurate and efficient computer techniques for transient calculations. 1.8.1.2 Background of EMTP The EMTP has been widely used all over the world as a standard simulation tool for a transient analysis not only in a power system but also in an electronic circuit. The BPA of the U.S. Department of Interior (later U.S. Department of Energy) started to develop a computer software for analyzing a power system transient, especially a switching overvoltage from the viewpoint of insulation design and coordination of a transmission line and a substation in 1966 by inviting Dr. H. W. Dommel from Germany as a BPA’s permanent staff. The EMTP development was a part of system analysis computerization including a power/load flow analysis program and a stability analysis program in the BPA System Analysis Department led by Dr. W. Tinny. Before the EMTP, a transient network analyzer (TNA) was used in the BPA.

124


The EMTP was based on Schnyder–Bergeron method [52,53] of travelingwave analysis in a hydraulic system, well known as a water hammer [50–55]. The Schnyder–Bergeron method was introduced to the field of an electrical transient by Frey and Althammer [22]. The method was incorporated with a nodal analysis method by representing all the circuit elements by a lumped resistance and a current source by Dommel [23]. This is the origin of the EMTP [10,24]. 1.8.1.3 EMTP Development Dommel developed a transient program at the Technical University of Munich based on reference [22] for a distributed line and on numerical integration of an ordinary differential equation for a lumped-parameter circuit. In 1966, he moved to the BPA to develop a generalized transient analysis program, and the original version of the EMTP called “transient program” was completed in 1968. In 1972, Scott-Meyer joined the BPA and Dommel left the BPA in 1973. Since then, the work on the EMTP was accelerated very much, and Semlyen [72], Ametani [5,73], Brandwajn [74], and Dube [75] joined the Scott-Meyer’s team, and the EMTP Mode 31, which is basically the same as the present EMTPs, was completed in 1981 [24]. Presently, there exist four versions of the EMTP: (1) Alternative Transients Program (ATP) (royalty free) originated by Scott-Meyer and developed/maintained by the BPA, European EMTP User Group (EEUG) [76], (2) EMTP RV (commercial) developed by Mahseredjian in Hydro-Quebec sponsored by EMTP Development Coordination Group (DCG)/EPRI [77], (3) EMTDC/PSCAD (commercial) developed by Woodford and Goni at Manitoba HVDC Research Center, which was founded by Wedepohl during his presidency of the Manitoba Hydro, and (4) original EMTP (in public domain) developed by the BPA, only source code available from Japanese EMTP Committee. As a similar software to the EMTP, SPICE is well known especially in the field of power electronics. The strength of the SPICE is that the physical parameters of the semiconductors are easily available as a part of the software. Table 1.13 summarizes the history of the EMTP from 1966 to 1991. Since the early 1990s, there are so many simulation tools related to or similar to the EMTP and too many publications related to the EMTP development, which cannot be covered in this book. 1.8.2 Basic Theory of EMTP The basic procedure of an EMTP transient simulation is shown in Figure 1.67 and is summarized as follows:

1. Represent all the circuit elements including a distributed-parameter line by a current source and a resistance.


125

TABLE 1.13 History of the EMTP 1961 1964 1966 1968 1973 1976

1976 1982 1978 1981 1982 1983

1984 1985 1986 1987 1991

Schnyder–Bergeron method by Frey and Althammer Dommel’s PhD thesis in Tech. Univ. Munich Dommel started EMTP development in BPA Transients Program (TP = EMTP Mode 0), 4000 statements Dommel moved to Univ. British Columbia Scott-Meyer succeeded EMTP development Universal Transients Program File (UTPF) and Editor/Translator Program (E/T) completed by Scott-Meyer. UTPF and E/T made by EMTP could be used in any computers because those solve machine-dependent problems and prepared a platform for any researchers able to join EMTP development, not necessary to visit BPA Japanese EMTP Committee founded Semlyen, Ametani, Brandwajn, Dube, and so on Marti joined EMTP development First EMTP workshop during IEEE PES meeting First EMTP tutorial course during IEEE PES meeting DCG proposed by BPA DCG 5 year project started, Chairman Vithayathil of BPA Members: BPA, Ontario Hydro, Quebec, U.S. Bureau of Reclamation, WAPA EPRI joined DCG, and DCG/EPRI started. Copyright of EMTP to be given to EPRI/DCG EMTP Mode 39 completed and distributed EMTP development in BPA terminated. Final version EMTP Mode 42 Scott-Meyer started to develop ATP–EMTP independently on BPA and EPRI/DCG ATP development transferred to Leuven EMTP Center (LEC) in Belgium ATP ver. 2 completed. BPA joined LEC DCG/EPRI EMTP ver. 1 completed DCG/EPRI 5 year project to be terminated, but DCG/EPRI project continued BPA resigned from DCG/EPRI ATP copyright transferred to Can/Am ATP User Group. BPA joined Can/Am User Group. DCG/EPRI ver. 2 completed

2. Produce a nodal conductance matrix representing the given circuit. 3. Solve the nodal conductance matrix from given voltage sources (or current sources). The aforementioned is the same for both a steady state and a transient. For the transient, the earlier procedure is repeated at every time step t = n · ∆t based on the known voltages and the currents at time t = t − ∆t in the circuit. When the circuit configuration is changed due to switching operation, for example, the node conductance matrix is reproduced from the new circuit configuration.

126


iL

ic

RL

JL(t – ∆t)

(a)

(b)

Rc

Jc(t – ∆t)

ik(t)

im(t)

Vk(t)

k

Vk(t)

Vm(t)

ik(t)

Zs

im(t)

Ik

Im

Zs

m

Vm(t)

(c) FIGURE 1.67 Representation of circuit elements by a resistance and a current source. (a) Inductance. (b) Capacitance. (c) Distributed line.

1.8.2.1 Representation of a Circuit Element by a Current Source and a Resistance Figure 1.67 illustrates representation of inductance L, capacitance C, and a distributed-parameter line Z0 by a current source and a resistance. The representation is derived in the following manner. The voltage and the current of the inductance are related by the following equation:

v = L⋅

di (1.260) dt


127

Integrating the earlier equation from time t = t − ∆t to t, t

∫

t − ∆t

v ( t ) ⋅ dt = L

t

 di 

∫  dt  ⋅ d = L [i(t)] t

t t − ∆t

= L {i ( t ) − i(t − ∆t)}

t − ∆t

By applying trapezoidal rule to the left-hand side of the equation,

∫ v (t ) ⋅ dt = {v (t ) + v (t − ∆t )} ∆t/2

From the earlier two equations,

v(t)  ∆t  i (t ) =  + J ( t − ∆t ) (1.261)  {v ( t ) + v ( t − ∆t )} − i ( t − ∆t ) = RL  2L 

where J(t − ∆t) = v (t − ∆t)/RL +i(t − ∆t) RL = 2L/∆t, ∆t: time step It is clear from the earlier equation that current i(t) at time t flowing through the inductance is evaluated by voltage v(t) and current source J(t − ∆t), which was determined by the voltage and the current at t = t − ∆t. Thus, the inductance is represented by the current source J(t) and the resistance RL as illustrated in Figure 1.67a. Similarly, Figure 1.67b for a capacitance is derived from a differential equation expressing the relation between the voltage and the current of the capacitance. Figure 1.67c is for a distributed-parameter line of which the voltage and the current are related in Equation 1.57:   x    x  v ( x , t ) + Z0 ⋅ i ( x , t ) = 2F1  t −    v ( x , t ) − Z0 ⋅ i ( x , t ) = 2F2  t +    (1.262) c      c  where Z0 is the characteristic impedance The earlier equation is rewritten at nodes 1 and 2 as v1 ( t − τ ) + Z0 i ( t − τ ) = v2 ( t ) − Z0 i2 (t),

v1 ( t ) − Z0 i ( t ) = v2 ( t − z ) + Z0 i2 (t − τ)

where τ = l/c is the traveling time from node 1 to node 2 l is the line length c is the propagation velocity

(1.263)

128


It is observed in Equation 1.263 that voltage v1(t) and current i1(t) at node 1, the sending end of the line, influence v2(t) and i2(t) at the receiving end for t ≥ τ, where τ is the traveling time from the node 1 to the node 2. Similarly, v2(t) and i2(t) influence v1(t) and i1(t) with time delay τ. In a lumped-parameter element, the time delay is ∆t as can be seen in Equation 1.261. In fact, ∆t is not a time delay due to traveling-wave propagation, but it is a time step for time discretization to solve numerically a differential equation describing the relation between the voltage and the current of the lumped element. From the earlier equation, the following relation is obtained:

i1 ( t ) =

v1(t) v (t) + J1(t − τ), i2 ( t ) = 2 + J 2 (t − τ) (1.264) Z0 Z0

where J1 ( t − τ ) = −v2 (t − τ)/†Z0 − i2 (t − τ) J 2 ( t − τ ) = −v1(t − τ)/†Z0 − i1(t − τ) The earlier results give the representation of a distributed-parameter line in Figure 1.67c. In Equation 1.263, Z0 is the characteristic impedance that is frequency dependent. When the frequency dependence of a distributed-parameter line explained in Section 1.5 is to be considered, a frequency-dependent line such as Semlyen’s and Marti’s line models is prepared as a subroutine in the EMTP. 1.8.2.2 Composition of Nodal Conductance In the EMTP, a nodal analysis method is adopted to calculate voltages and currents in a circuit. Figure 1.68 shows an example. By applying Kirchhoff’s current law to nodes 1–3 in the circuit,

At node 1: G1 (V1 − E1 − V3 ) + G2 (V1 + E2 − V3 ) + G3 (V1 − V2 ) = 0

At node 2: G3 (V1 − V2 ) + G5V2 + G6 (V2 − E6 ) = 0

At node 3: G4 (V3 − E4 ) + G2 (V3 − E2 − V1 ) + G1 (V3 + E1 − V1 ) = 0

where Gi = 1/Ri,i = 1 to 6 Rearranging the earlier equation and writing in a matrix form,

G1 + G2 + G3  −G3   −G1−G2

G2 G3 + G5 + G6 0

−G1−G2   V1   J 1− J 2      0 J6    V2  =  G1 + G2 + G4   V3   − J1 + J 2 + J 4 


I1

1

R3

I3

129

2

I2 R1

R6

R2 IA

E1

IB

R5

R4

I5

E2

3

4

I4

IC E6

E1 = 5 V, E2 = 2 V E4 = 14 V, E6 = 12 V R1 = 2 Ω, R2 = 3 Ω R3 = 5 Ω, R4 = 4 Ω R5 = 2 Ω, R6 = 3 Ω

I6

FIGURE 1.68 Nodal analysis.

or

[G ] (V ) = ( J ) (1.265)

where Ji = GiEi, i = 1, 2, 4, 6 [G] is the node conductance matrix It should be clear in Equation 1.265 that once the node conductance matrix is composed, the solution of the voltages is obtained by taking the inverse of the matrix, for current vector ( J) is known. In the nodal analysis method, the composition of the nodal conductance is rather straightforward as is well known in a circuit theory. In general, the nodal analysis gives a complex admittance matrix, because of jωL and jωC. However, in the EMTP, all the circuit elements being represented by a current source and a resistance, it becomes a real matrix. 1.8.3 Other Circuit Elements Table 1.14 shows circuit elements and subroutines prepared in the EMTP. As is observed from the table, most circuit elements are prepared for steadystate and transient simulations by the EMTP. TACS and MODELS are kinds of a computer language by which a user can produce a computer code as an input data of the EMTP. Those are, in a sense, a pioneering software before MATLAB, MAPLE, and so on. If a user needs to develop a model circuit, which is not available in the EMTP, it can be achieved by using TACS or MODELS. The usage of the circuit elements and supporting routines in the EMTP is explained in the EMTP Rule Books [24,76–78], and the theory is described in the theory book [10].

130


TABLE 1.14 Circuit Elements and Subroutines Prepared in the EMTP a. Circuit Elements Element Lumped R, L, C Line/cable

Transformer

Load, nonlinear

Arrester

Source

Rotating machine Switch

Semiconductor Control circuit

Model Series, parallel Multiphase π circuit Distributed line with constant parameters Frequency-dependent line Mutually coupled R–L element N winding, single phase Three-phase shell type Three-phase,•three-leg•core type Staircase R(t) (type-97) piecewise time varying R (type-91, type-94) Pseudo-nonlinear R (type-99) Pseudo-nonlinear L (type-98) Pseudo-nonlinear hysteretic L (type-96) Exponential function Zn0 Flashover-type multiphase R TACS-controlled arc model Step like (type-11) Piecewise linear (type-12, type-13) Sinusoidal (type-14) Impulse (type-15) TACS-controlled source Synchronous generator (type-59) Universal machine Time-controlled switch Flashover switch Statistic/systematic switch Measuring switch TACS-controlled switch (type-12, type-13) TACS-controlled switch (type-11) TACS MODELS

Remark Transposed, untransposed Overhead, underground Semlyen, Marti, Noda Single phase, three phase Saturation, hysteresis

With gap, gapless

Voltage source Current source

Synchronous, induction, dc CB

Diode, thyristor Transfer function Arithmetics, logics

b. Supporting Routines Name

Function

Line Constants

Overhead line parameters

Cable Constants

Overhead/underground cable parameters Overhead/underground cable parameters distributed Y, snaking Transformer parameters

CABLE PARAMETERS XFORMER

Input Data Frequency, configuration, physical parameters Frequency, configuration, physical parameters Frequency, configuration, physical parameters Configuration, rating, %Z

131


TABLE 1.14 (continued) Circuit Elements and Subroutines Prepared in the EMTP b. Supporting Routines Name

Function

BCTRAN SATURATION HYSTERESIS NETEQV

Input Data

Transformer parameters Saturation characteristics Hysteresis characteristics (type-96) Equivalent circuit

Configuration, rating, %Z Configuration, rating, %Z Configuration, rating, %Z Circuit configuration, Z, Y, frequency

1.8.4 Solutions of the Problems 1.1

S = πr2 = π × 10−4, ℓ = π × 2 × 10−2, Rdc =

ρ 2 × 10 −8  2  = =   × 10 −4 ≒ 0.637 × 10 −4 Ω/km S π × 10 −4  π 

At 50 Hz: Zc = Rdc + jωLc, Rdc = 0.0637 Ω/km

Lc =

µ 0 4π × 10 −7 = = 0.5 × 10 −7 H/m = 5 × 10 −2 mH/km 8π 8π

At 100 Hz: b =

ωµcS 2 2π × 10 5 × 4π × 10 −7 × π × 10 −4 ⋅ = = (10π)2 1 (2/π) × 10 −4 ⋅ 4π2 × 10 −4 Rdc

Zc ≒ Rdc b ⋅ j = 10π(2/π) × 10 −4 (1 + j )/ 2

= (20/ 2 ) × 10 −4 ( 1 + j ) Ω/m = 1.414 ( 1 + j ) Ω/km

1.2 1.3

Rc = 1.414 Ω/km , Lc =

1.414 × 10 3 = 2.25 × 10 −3 mH/km† 2π × 10 5

Same as 1.1 = 8 × 10 −2 = 2πr0 ∴ r2 = ( 4/π ) × 10 −2 = 1.273 cm , S = 4 × 10 −4 = r22 − r12

(

{

}

∴ r12 = ( 4/π)2 − 4/π × 10 −4

)

132


r1 = ( 4/π ) {1 − π/4} × 10 −2 = 0.5898 cm

1.4 For f = 50 Hz, a low-frequency approximation is good enough (see Example 1.2):

{

(

Ze = 50 + j 50 8.253 + 0.628ln 100/50 ⋅ 10 −4

)} = 50 + j723.6 [mΩ/km]

For f = 100 Hz, D = H1 = 20, He = 15.92,

a = 1.0368 × 10 3 , b = 1.6506 × 10 3 , D2 = 400, d 2 = r 2 = 1 × 10 −4 , A = 1.9492 × 10 3 , ϕ = 57.87° = 1.010 rad. Ze = 62.83 × 10 −3 ( 1.010 + j16.78 ) = 0.063 + j1.054 [Ω/m]

= 63 + j1054 Ω/km

1.6

The earth-return impedance is far greater than the conductor internal impedance, and thus the latter can be neglected. However, in a steadystate analysis such as fault and load flow calculations in a multiphase line, the positive-sequence (mode 1) component is important, and the conductor internal impedance is dominant for the positive-sequence component. he ≫ hi, Sij = 2he,

 2 Peij = ln   dij

1.5

  1   ρe   2  +   ln   = ln    2   jωµ0   dij

{ (

)

π   1    ρe  −j  2 ln e +  +   ln    2 ωµ   0      

(

= − jπ/4 + ln 2 + ln ρe /f ⋅ dij2 − ln 8π2 × 10 −7

{

(

= − jπ/4 + 0.6931 − ( −11.75 ) + ln ρe /f − dij2

{

)}

)}

2

2

(

∴ Ze = j ( 0.4πf ) − jπ/4 + 6.5677 + ( 1/2 ) ln ρe /f ⋅ dij2

{

}

≒ f + jf 8.253 + 0.628ln(ρe /f ⋅ dij2 )

)}


1.7

133

ln {2 ( h + he ) /r} = ln ( 2h/r ) + ln ( 1 + he /h ) ≒ ln ( 2h/r ) + he /h = P0 + he /h} ρe ρe = (1 − j ) jωµ 0 2ωµ 0

he =

Ze = jω ( µ 0 /2π ) ρe + ( j + 1) ρe /ωµ 0 /2 /2πh = Zs + ( 1 + j ) ρe ωµ 0 /2 /2πh

1.8 1.9

a 2 = R C/L , b 2 = G L/C ; a 2 + b 2 ≥ 2ab

In Section 1.3.4.1, ω = 2αβ/(LG + CR). For ω → 0 ( ω = 0 ) , α = RG .

Thus, c = ω/β = 2α ( LG + CR ) = 2 RG /(LG + CR)

1.10 Γ = RG ,†Y0 = G R for ω = 0. From Equation 1.124, Is = Y0 · E coth (Γℓ) = Y0E ⋅ coth ( θ ) .

For G → 0, θ → 0; exp ( θ ) → 1 + θ, coth ( θ ) → 1/θ Is ≒

E G/R

(

RG ⋅

)

≒

E : corresponding to Ohmsí law (R ⋅ )

1.11 Vr = (A − BC/A)E − (B/AZr)Vr = E − (B/AZr)Vr; A ≠ 0 ∴Vr = ZrE/(AZr + B)

(

)

1. A = coshΓ = cos ω LC = cos ( ω/c0 ) = cos ( ωτ ) = cos θ,

τ = /c0 , c0 = light velocity

B = Z0 sinh Γ = jZ0 ⋅ sin θ, θ ≠ ( 2n − 1) π/2,(n − 1)π/2

∴ Vr = Zr E/(Zr cos θ + jZ0sin θ)

2. A = 1 + zℓ · yℓ/2 = 1 − ω2LCℓ2/2 = 1 − θ2/2, B = Z · ℓ = jωLℓ = jZ0 θ, Z = j ωL , y = j ωC

Vr′ =

Zr E

{(1 − θ /2) Z + jZ θ} 2

r

0

134


a. Zr = 1 Ω : Vr = E (cos θ + jZ0sin θ), Vr′ = E

{(1 − θ /2) + jθZ } 2

0

b. Z1 = Z0:Vr = E/(cos θ + jsin θ)Z0 = E · e−jθ/Z0, Vr′ =

(E/Z0 )

{(1 − θ /2) + jθ} 2

Zr = ∞ : Vr = E/cos θ, Vr′ = E/(1 − θ2/2) c. f = 50 Hz, ω = 100π, = 300 km , τ = 1 ms = 10 −3 , θ = 0.1π = 0.3142, cos θ = 0.9511, sin θ = 0.3090, 1 − †θ2/2 = 0.9506 1.12 BC − A · Dt = U 1.13 Q2 = P11 − P13 ; Q1 , Q3 = P11 + P22 + P13 ± ( P11 − P22 + P13 )2 + 8 P12 P21 a.

{

}2

A1n = 1 ( n = 1 − 3 ) , A31 = A33 = 1, A32 = −1, A21 = †− P21 P22 − Q1 A22 = †, 0 A32 = −1, A23 = −2 b. Q1 = 48, Q2 = 30, Q3 = 24, An1 = 1(n = 1 − 3), A12 = A13 = A33 = 1, A22 = 0, A32 = −1, A23 = −2 c. Q1 = 50, Q2 = 30, Q3 = 25, A1n = 1, A31 = A33 = 1, A32 = −1, A21 = 2/ 3 , A22 = 0, A23 = − 3 1.14 The modal impedances and admittance for the modes 2 and 3 are identical, because all the off-diagonal elements of the series impedance and shunt admittance matrices are the same, and the diagonal elements are also the same in the transposed line. 1.15 Add a very small value to an element of the matrix A. In a numerical calculation by a computer, Aij ≠ Aji if one observes the values of Aij and Aji for more than a certain number of digits. 1.16 At an infinite frequency, an element of the impedance and admittance matrices becomes infinite. In a perfectly conducting system, Z · Y becomes diagonal. 1.17 Mode 0 (earth-return mode): no significant difference Modes 1 and 2 (aerial modes) are identical in the transposed line, while those are different in an untransposed line. 1.18 I = (2e 0/Z0){1 − exp(−t/τ)}, V = 2e 0 · exp(−t/τ), er = 2e 0{2 · exp(−t/τ)−1}, τ = L/Z0 1.19 V = (2e 0/a){R1 + R 2 − (Z0R1/b)exp(−t/τ)}, τ = CR1b/a, a = Z0 + 2(R1 + R 2) b = Z0 + 2R2

135


V2 (pu)

1.20 2 1

V3 (pu)

0

15

5

10

15

5

10 –2.43 A

15

1

6 I1 (A)

10

2

0

4

2.43 A

2 0

I2 (A)

5

4.05 A

4 2

2.43 A

0

5

(µs)

10

15 –2.43 A

(c) Because of lower surge impedance in a cable, the cable end voltage gradually increases with multiple reflection within the cable. 1.21 Vrb : similar waveform to Vra in Figure 1.52 Vra = Vrc : similar waveform to the average of Vrb and Vra in Figure 1.52 1.22 Results calculated by Equations 7.11 and 7.12 are given in Table. 1.11. 1.23 Calculated results are given in Table 1.12 as Approx. = Jordan. 1.24 V t

160

107

0

20

30

136


λ12 = 2 × 270/320 + 270 = 0.195, λ 01 = λ10 = 1

Vt ( t ) = Zi ⋅ it ( t ) , e12 ( t ) = Vt ( t ) for †t < τ1 = 10 ns for τ1 ≤ t < 3τ1: e2f (t) = e12 (t − τ1), V1 = λ12 · e2f = 0.915e12(t − τ1)

e21 ( t ) = V1 ( t ) − e2 f ( t ) = −0.085e12 ( t − τ1 )

2τ1 ≤ t < 4τ1 : Vt = Zi ⋅ it ( t ) + λ10 ⋅ e1b = 320it (t) − 0.085e12 ( t − 2τ1 )

= 320 {it ( t ) − 0.085it ( t − 2τ1 )}

at = 20 ns : Vt = 320 {0.5 × 20/30 − 0.085 × 0} = 107 V

= 30 ns : Vt = 320 × 0.5 ( 1 − 0.085/3 ) = 160 × 0.9717 = 155 V

= 40 ns : = 320 × 0.5 ( 1 − 0.085 × 2/3 ) = 160 × 0.9433 = 151 V

1.25 Formulas in Section 1.7.2 show a satisfactory accuracy in comparison with a number of measured results, and thus those are good enough from the viewpoint of engineering practice similarly to those in the so-called Electrician Handbook. However, the concept of the penetration depth is based on a theory of “electrostatics” within TEM mode propagation of electromagnetic waves. Thus, the formulas cannot be applied, in principle, to an electromagnetic phenomenon and nonTEM mode wave propagation.

References 1. Ametani, A. 1990. Distributed—Parameter Circuit Theory. Tokyo, Japan: Corona Pub. Co. 2. Ametani, A. and I. Fuse. 1992. Approximate method for calculating the impedances of multiconductors with cross sections of arbitrary shapes. Electr. Eng. Jpn. 111(2):117–123. 3. Schelkunoff, S. A. 1934. The electromagnetic theory of coaxial transmission line and cylindrical shields. Bell Syst. Tech. J. 13:523–579. 4. Wedepohl, L. M. and D. J. Wilcox. 1973. Transient analysis of underground power transmission systems. Proc. IEE 120(2):253–260.


137

5. Ametani, A. 1980. A general formulation of impedance and admittance of cables. IEEE Trans. Power App. Syst. 99(3):902–910. 6. Deri, A., G. Tevan, A. Semlyen, and A. Castanheira. 1981. The complex ground return plane: A simplified model for homogeneous and multi- layer earth return. IEEE Trans. Power App. Syst. 100(8):3686–3693. 7. Pollaczek, F. 1926. Über das Feld einer unendlich langen Wechselstromdurchflossenen Einfachleitung. ENT. 3(9):339–359. 8. Carson, J. R. 1926. Wave propagation in overhead wires with ground return. Bell Syst. Tech. J. 5:539–554. 9. Ametani, A. and K. Imanishi. 1979. Development of a exponential Fourier transform and its application to electrical transients. Proc. IEE. 126(1):51–59. 10. Dommel, H. W. 1986. EMTP Theory Book. Portland, OR: Bonneville Power Administration (B. P. A.). 11. Wise, W. H. 1948. Potential coefficients for ground return circuits. Bell Syst. Tech. J. 27:365–371. 12. Ametani, A. 2007. The history of transient analysis and the recent trend. IEE J. Trans. EEE (published by J. Wiley & Sons, Inc.). 2:497–503. 13. D’Alembert. 1747. Recherches sur la courbe que forme une corde tenduë mise en vibration. 14. Wedepohl, L. M. 1963. Application of matrix methods to the solution of travelling wave phenomena in poliphase systems. Proc. IEE 110(12):2200–2212. 15. Fortesque, C. L. 1918. Method of symmetrical coordinates applied to the solution of polyphase network. AIEE Trans. Pt. II 37:1027–1140. 16. Clarke, E. 1943. Circuit Analysis of A-C Power Systems, Vol. 1, Symmetrical and Related Components. New York: Wiley. 17. Ametani, A. 1973. Refraction coefficient theory and surge phenomena in power systems. PhD dissertation. Victoria University of Manchester, Greater Manchester, U.K. 18. Ametani, A. 1973. Refraction coefficient method for switching surge calculations on untransposed transmission line. IEEE PES 1973 Summer Meeting, C73-444-7, Vancouver, CA. 19. Ametani, A. 1980. Wave propagation characteristics on single core coaxial cables. Sci. Eng. Rev. Doshisha Univ. 20(4):255–273 (in Japanese). 20. Ametani, A. 1980. Wave propagation characteristics of cables. IEEE Trans. Power App. Syst. 99(2):499–505. 21. Bewley, L. V. 1951. Traveling Waves on Transmission Systems. New York: Wiley. 22. Frey, W. and P. Althammer. 1961. The calculation of electromagnetic transients on lines by means of a digital computer. Brown Boveri Rev. 48(5/6):344–355. 23. Dommel, H. W. 1969. Digital computer solution of electromagnetic transients in single- and multiphase networks. IEEE Trans. Power App. Syst. 88(4):388–398. 24. Scott-Meyer, W. 1982. EMTP Rule Book. Portland, OR: B. P. A. 25. Ametani, A., T. Ono, Y. Honaga, and Y. Ouchi. 1974. Surge propagation on Japanese 500 kV untransposed transmission line. Proc. IEE 121(2):136–138. 26. Ametani, A., A. Tanaka et al. 1981. Wave propagation characteristics on an untransposed vertical twin-circuit line. IEE Japan B-101(11):675–682. 27. Ametani, A., E. Ohsaki, and Y. Honaga. 1983. Surge characteristics on an untransposed vertical line. IEE Japan B-103(2):117–124. 28. Foster, R. M. 1931. Mutual impedance of grounded wires lying on or above the surface of the earth. Bell Syct. Tech. J. 10:408–419.

138


29. Jordan, C. A. 1934. Lightning computation for transmission line with overhead ground wires. G. E. Rev. 37(4):180–186. 30. Wagner, C. F. 1956. A new approach to calculation of lightning performance of transmission lines. AIEE Trans. 75:1233–1256. 31. Lundholm, R., R. B. Finn, and W. S. Price. 1957. Calculation of transmission line lightning voltages by field concepts. AIEE Trans. 76:1271–1283. 32. Sargent, M. A. and M. Darveniza. 1969. Tower surge impedance. IEEE Trans. Power App. Syst. 88:680–687. 33. Velzuez, R., P. H. Reynolds, and D. Mukhekar. 1983. Earth return mutual coupling effects in grounded grids. IEEE Trans. Power App. Syst. 102(6):1850–1857. 34. Okumura, K. and K. Kijima. 1985. A method for coupling surge impedance of transmission tower by electromagnetic field theory. Trans. IEE Jpn. 105-B(9):733–740 (in Japanese). 35. Ametani, A. and T. Inaba. 1987. Derivation of impedance and admittance of a nonparallel conductor system. Trans. IEE Jpn. B-107(12): 587–594 (in Japanese). 36. Sarimento, H. G., D. Mukhedkar, and V. Ramachandren. 1988. An extension of the study of earth return mutual coupling effects in ground impedance field measurements. IEEE Trans. Power Deliv. 3(1):96–101. 37. Rogers, E. J. and J. F. White. 1989. Mutual coupling between finite length of parallel or angled horizontal earth return conductors. IEEE Trans. Power Deliv. 4(1): 103–113. 38. Ametani, A. and M. Aoki. 1989. Line parameters and transients of a non-parallel conductor system. IEEE Trans. Power App. Syst. 4:1117–1126. 39. Hara, T. et al. 1988. Basic investigation of surge propagation characteristics on a vertical conductor. Trans. IEE Jpn. B-108:533–538 (in Japanese). 40. Hara, T. et al. 1990. Empirical formulas for surge impedance for a single and multiple vertical conductors. Trans. IEE Jpn. B-110:129–136 (in Japanese). 41. Yamada, T. et al. 1995. Experimental evaluation of a UHV tower model for lightning surge analysis. IEEE Trans. Power Deliv. 10:393–402. 42. Ametani, A. and A. Ishihara. 1993. Investigation of impedance and line parameters of a finite-length multiconductor system. Trans. IEE Jpn. B-113(8):905–913. 43. Ametani, A., Y. Kasai, J. Sawada, A. Mochizuki, and T. Yamada. 1994. Frequencydependent impedance of vertical conductor and multiconductor tower model. IEE Proc.-Gener. Transm. Distrib. 141(4):339–345. 44. Oufi, E., A. S. Aifuhaid, and M. M. Saied. 1994. Transient analysis of lossless singlephase nonuniform transmission lines. IEEE Trans. Power Deliv. 9:1694–1700. 45. Correia de Barros, M. T. and M. E. Aleida. 1996. Computation of electromagnetic transients on nonuniform transmission lines. IEEE Trans. Power Deliv. 11:1082–1091. 46. Nguyen, H. V., H. W. Dommel, and J. R. Marti. 1997. Modeling of single-phase nonuniform transmission lines in electromagnetic transient simulations. IEEE Trans. Power Deliv. 12:916–921. 47. Ametani, A. 2002. Wave propagation on a nonuniform line and its impedance and admittance. Sci. Eng. Rev. Doshisha Univ. 43(3):135–147. 48. Sunde, E. D. 1968. Earth Conduction Effects in Transmission Systems. New York: Dover. 49. IEE Japan and IEICE Japan. Study Committee of Electromagnetic Induction. 1979. Recent movements and problems for electromagnetic interference. IEE Japan and IEICE Japan (in Japanese).


139

50. Allievi, L. 1902. Teoria Generale del moto perturbato dell’acqua nei tubi in pressione. Annali della Societa degli Ingegneri ed Architette Italiani (English translation, Halmos, E. E. 1925. Theory of Water Hammer. American Society of Mechanical Engineering). 51. Hayashi, S. 1948. Operational Calculus and Transient Phenomena. Kokumin Kogaku-sha, Tokyo, Japan. 52. Schnyder, O. 1929. Druckstosse in Pumpensteigleitungen. Schweiz Bauztg 94(22):271–286. 53. Bergeron, L. 1935. Etude das variations de regime dans les conduits d’eau: Solution graphique generale. Rev. Generale de L’hydraulique 1:12–69. 54. Augus, R. W. 1938. Waterhammer in Pipes: Graphical Treatment. Bulletin 152. University of Toronto, Toronto, CA. 55. Parmakian, J. 1963. Waterhammer Analysis. New York: Dover. 56. Sommmerferd. A. 1964. Practical Differential Equations in Physics. New York: Academic Press. 57. Baba, J. and T. Shibataki. 1960. Calculation of transient voltage and current in power systems by means of a digital computer. J. IEE Jpn. 80:1475–1481. 58. Barthold, L. O. and G. K. Carter. 1961. Digital traveling wave solutions, 1-Single -phase equivalents. AIEE Trans. Pt. 3. 80: 812–820. 59. McElroy, A. J. and R. M. Porter. 1963. Digital computer calculations of transients in electrical networks. IEEE Trans. Power App. Syst. 82:88–96. 60. Arismunandar, A., W. S. Price, and A. J. McElroy. 1964. A digital computer iterative method for simulating switching surge responses of power transmission networks. IEEE Trans. Power App. Syst. 83:356–368. 61. Bickford, J. P. and P. S. Doepel. 1967. Calculation of switching transients with particular reference to line energisation. Proc. IEE 114:465–477. 62. Thoren, H. B. and K. L. Carlsson. 1970. A digital computer program for the calculation of switching and lighting surges on power systems. IEEE Trans. Power App. Syst. 89:212–218. 63. Snelson, J. K. 1972. Propagation of traveling waves on transmission linesFrequency dependent parameters. IEEE Trans. Power App. Syst. 91:85–90. 64. Ametani, A. 1973. Modified traveling-wave techniques to solve electrical transients on lumped and distributed constant circuits: Refraction-coefficient method. Proc. IEE 120(2):497–504. 65. Lego, P. E. and T. W. Sze. 1958. A general approach for obtaining transient response by the use of a digital computer. AIEE Trans. Pt. 1. 77:1031–1036. 66. Day, S. J., N. Mullineux, and J. R. Reed. 1965. Developments in obtaining transient response using Fourier integrals. Pt. I: Gibbs phenomena and Fourier integrals. Int. J. Elect. Eng. Educ. 3:501–506. 67. Day, S. J., N. Mullineux, and J. R. Reed. 1966. Developments in obtaining transient response using Fourier integrals. Pt. II: Use of the modified Fourier transform. Int. J. Elect. Eng. Educ. 4:31–40. 68. Battisson, M. J. et al. 1967. Calculation of switching phenomena in power systems. Proc. IEE 114:478–486. 69. Wedepohl, L. M. and S. E. T. Mohamed. 1969. Multi-conductor transmission lines: Theory of natural modes and Fourier integral applied to transient analysis. Proc. IEE 116:1553–1563. 70. Ametani, A. 1972. The application of fast Fourier transform to electrical transient phenomena. Int. J. Elect. Eng. Educ. 10:277–287.

140


71. Nagaoka, N. and A. Ametani. 1988. A development of a generalized frequencydomain transient program—FTP. IEEE Trans. Power Deliv. 3(4):1986–2004. 72. Semlyen, A. and A. Dabuleanu. 1975. Fast and accurate switching transient calculations on transmission lines with ground return using recursive convolution. IEEE Trans. Power App. Syst. 94:561–571. 73. Ametani, A. 1976. A highly efficient method for calculating transmission line transients. IEEE Trans. Power App. Syst. 95:1545–1551. 74. Brandwajn, V. and H. W. Dommel. 1977. A new method for interfacing generator models with an electromagnetic transients program. IEEE PES PICA Conf. Rec. 10:260–265. 75. Dube, L. and H. W. Dommel. 1977. Simulation of control systems in an electromagnetic transients program with TACS. IEEE PES PICA Conf. Rec. 10:266–271. 76. European EMTP Users Group (EEUG) 2007. ATP Rule Book. 77. DCG/EPRI. 2000. EMTP-RV Rule Book. 78. Ametani, A. 1994. Cable Parameters Rule Book. Portland, OR: B. P. A.

2 Transients on Overhead Lines

2.1 Introduction There are various kinds of transients in a power system. In general, the overvoltages caused by the transients are important in the power system because of its insulation against the overvoltages. Table 2.1 summarizes the overvoltages in the power system [1,2]. The temporary overvoltage is an overvoltage caused by an abnormal system condition such as a line fault and is evaluated by a steady-state solution of the abnormal system condition. Thus, the temporary overvoltage is not considered as a transient overvoltage. A typical example of a temporary overvoltage is shown in Figure 2.1 caused by a line-to-ground fault. Right after the initiation of the line-to-ground fault, a transient called “fault surge” occurs but is died out by few milliseconds. Then, a sustained dynamic overvoltage is observed, and this is called “temporary voltage.” A transient on a distributed-parameter line is called “surge,” because the transient is caused by traveling waves. An overvoltage due to the surge is, in general, much greater than the temporary overvoltage, and thus, the insulation of a transmission system is mainly determined by the surge overvoltage [3]. In Chapter 2, the following surges on overhead lines are explained: • Switching surges • Fault initiation and fault-clearing surges • Lightning surges Also, theoretical analyses of transients (hand calculation) are explained, and finally it describes a frequency-domain (FD) method of transient simulations of which a computer code is readily developed by a reader of this book.

141

142


TABLE 2.1 Classification of Overvoltages Dominant Frequency/ Time Period

Overvoltage

Temporary overvoltage

1. Power frequency 2. Harmonic

50, 60 Hz/10 ms s

3. Low frequency

Lower than 50 Hz/ some second 0.1–10 MHz/0.1–some 10 μs

−1

Some 100 Hz/1–100 ms

4. Lightning surge Surge overvoltage

5. Switching surge

Some kHz/0.1–20 ms

6. Fault surge

Some kHz/0.1–20 ms

Surge overvoltage

2.0 1.7 pu

v(t) [pu]

1.3 1.0

Remarks Sustained overvoltage Resonance Nonlinear, ferroresonance Sub-synchronous resonance Direct lightning BFO Induced lightning Closing Reclosing Line-to-ground fault Fault clearing

Commercial frequency overvoltage

0

t 10 ms

–1.0

–2.0 FIGURE 2.1 A temporary overvoltage due to a line-to-ground fault.

2.2 Switching Surge on Overhead Line 2.2.1 Basic Mechanism of Switching Surge In a single distributed-parameter line of which the remote end is open circuited as illustrated in Figure 2.2, the remote-end voltage V2 is given in the following equation as described in Section 6.1, when switch “S” is closed at t = 0:

143

Transients on Overhead Lines

S

Z0, l, τ V1

V2

E

FIGURE 2.2 An open-circuited single conductor.

V2 = λ 2 f ⋅ E = 2E for τ ≤ t < 3τ

The previously given voltage is called “switching (surge) overvoltage,” which reaches 2 pu (per unit voltage = V/E) in a single lossless line:

voltage Per unit voltage  pu  = (2.1) applied source ( steady-state ) voltage

In reality, the voltage is reduced to less than 2 pu because of traveling wave attenuation from the sending end to the remote end due to a resistance in the conductor (see Section 1.6.4.4). However, a real transmission line is of three phases, and a switching operation involves a switching of threephase switches (CB: circuit breaker). Then, a switching surge on one phase induces a switching surge to the other phases. In addition, traveling waves are reflected at a boundary, such as a transformer and a series capacitor, and overlap on original ones. Thus, the maximum switching overvoltage can reach higher than 3 pu, depending on CB operation sequence, phase angles of three-phase ac source voltages, line length, etc. A switching surge voltage on a lossless line is given by a solid line in Figure 2.3 for a long time period. When attenuation constant α due to a line resistance is considered, the surge waveform is distorted by exp(−αt) as time passes, and a dotted line in Figure 2.3 is obtained. After a certain time period, the oscillating surge voltage converges to the steady-state voltage 1 pu (= E). 2.2.2 Basic Parameters Influencing Switching Surge 2.2.2.1 Source Circuit 2.2.2.1.1 Source Impedance Zs The voltage source E in Figure 2.2 has no internal impedance as an ideal voltage source with infinite capacity. In practice, there exists no ideal source, and any source has its own internal impedance. Also, a transformer is connected with a generator to supply a higher voltage to a transmission line.

144


V2 [pu] 2

1

0

τ

3τ

5τ

7τ

Time

Lossless line

Line with loss ±exp(–αt)

FIGURE 2.3 Switching surge waveforms on a single conductor.

The transformer has an inductance and is often represented in a transient calculation by its leakage inductance L [H], which is evaluated by its capacity P [W], the rated voltage V [V], and the percent impedance (%Z) as X = ωL = (% Z) ⋅

V2 [Ω] (2.2) P

where ω = 2πf, f: source (power) frequency 50 or 60 [Hz]. For example, P = 1000 MW, V = 275 kV, %Z = 15, and f = 50 Hz:

X = 0.15 ×

2752 X = 11.34 †Ω †,† L = = 36.1 †mH 1000 ω

When there is a source inductance as illustrated in Figure 2.4, a switching surge waveform differs significantly from those shown in Figure 2.3. The sending- and receiving-end voltages in Figure 2.4 are obtained in a similar manner to Section 1.6.2.3(4) or Problem (1.18) in Chapter 1 when the source voltage e(t) = E, step function voltage, is applied to the sending end:

 E  dI  E = L   + Z0 I ∴ I ( t ) =   dt   Z0

  −t    1 − exp  τ     

  −t′   V1 ( t ) = Z0 ⋅ I ( t ) ,† V2 ( t ) = 2E 1 − exp   †† for 0 ≤ t < 2τ (2.3)  τ  

where τ = L/Z0, t′ = t − τ.

145


L

Bus CB

ZS

Z0, τ V1

V2

I

e(t)

FIGURE 2.4 A single-phase line with an inductive source e(t) = E cos(ωt).

V2 [pu]

V1 [pu] 2

2

1

1

0

2τ

4τ

Time

0

τ

3τ

5 τ Time

FIGURE 2.5 Effect of a source inductance on a switching surge.

For example, if τ = 0.12 ms for L = 36 mH and Z0 = 300 Ω, then

t = 0.12 ms t = 0.5†ms ,† V1 = 0.63E †V1 = 0.98E

Considering the previous result, V1 and V2 are drawn as in Figure 2.5 assuming 2τ = 0.5 ms. Included in the figure (– – – ) is the case of no source inductance, which is shown in Figure 2.3 by a solid line. Because of the inductance, the rise time of the wave front becomes longer, and the surge waveform is observed to be highly distorted. When a number of generators, transformers, and/or transmission lines are connected to the bus in Figure 2.4, the source impedance Zs ≒ jωL becomes very small. Such a bus is called “infinite bus,” and the source circuit becomes equivalent to an ideal source in Figure 2.2. A switching overvoltage generated in the infinite bus case is the severest as observed from Figure 2.3.

146


2.2.2.1.2 Sinusoidal AC Source Equation 2.3 shows that a switching overvoltage becomes higher as the source voltage E becomes greater. A sinusoidal ac source voltage is often approximated by its amplitude E as a step function voltage because the time period of a switching surge is much smaller than the oscillating period T of the ac voltage: e ( t ) = E ⋅ cos(ωt) (2.4)

where ω = 2πf, f = 50 or 60 [Hz]:

T=

1 [sec ] = 16.7 or 20 [ms] f

In such a time period, the ac voltage in Equation 2.4 can be regarded as a step voltage with the amplitude E. When a much longer time period of calculations is required, a source should be represented as a sinusoidal ac source. In the case of fault clearing and load rejection overvoltages, the observation time exceeds 1 s, and the mechanical and electrical characteristic of a generator is to be considered including generator control. In such a case, the sinusoidal ac source is not appropriate, and Park’s generator model is often used. 2.2.2.1.3 Impulse/Pulse Generator An impulse generator (IG) and a pulse generator (PG) are often used to measure a transient response of a transmission line/cable, a grounding electrode, or a machine. The IG is composed of capacitors, and the recent PG is composed of a coaxial cable, which is a kind of a capacitor. An impulse voltage and a pulse voltage are generated by charging the capacitors and thus become a capacitive source. The simplest model of the IG and PG is an ideal source voltage with a given waveshape, that is, the wave front time Tf and the tail Tt or the rise time Tf and the pulse width Tw. When the IG is to be modeled accurately, all the elements of the IG, that is, the capacitances, resistances, and inductances should be considered based on the circuit diagram. The PG is represented accurately as a charged cable in a transient calculation. 2.2.2.2 Switch There are various kinds and types of switches. In a transmission system, a CB to interrupt a current is most common. For an ordinary switching surge calculation, the CB is modeled as an ideal switch controlled by time.


147

When recovery and restriking transients are to be analyzed, the dynamic characteristic of a CB, especially for a vacuum CB or interrupter, has to be considered. The characteristic is very much dependent on the material of the electrode and the mechanism of the CB operation. A line switch (LS) or a disconnector (DS) is used in a substation to interrupt a voltage. Because the operating speed of LS/DS is slow in general, it cannot be modeled as a time-controlled switch like a CB. Instead, the LS/ DS is modeled as a voltage-controlled switch so as to be able to consider a restriking voltage during a transient. For an analysis of a very-fast-front surge (switching surge in a gas-insulated bus) or a lightning surge, stray capacitances of a CB between poles and to ground have to be considered. 2.2.2.3 Transformer As already explained in Section 2.2.2.1.1, a transformer is represented by its leakage inductance for most switching surge analyses. When dealing with a fault surge especially in a low-voltage system with high-resistance grounding or isolated neutral of the transformer, the transformer winding, either Y or ∆ connection, has to be taken into account. Occasionally, the magnetizing impedance of the iron core has to be considered. For a lightning surge analysis, stray capacitances between the primary and the secondary windings, a winding to ground and between phases, are to be considered especially in the case of a transferred surge. 2.2.2.4 Transmission Line As described in Section 2.1, attenuation of a traveling wave due to a conductor resistance along a transmission line affects a switching surge waveform (see Figure 2.3). It is explained in Section 1.5.2 that the series impedance of a transmission line is very much frequency-dependent, and the attenuation and the propagation velocity of the traveling wave are also dependent on frequency. As a result, the traveling wave is distorted as it travels along the line as shown in Figures 1.28 and 1.49. The frequency dependence causes a significant effect on the switching surge waveform. Therefore, the frequency-dependent effect should be included in an accurate calculation of a switching surge. Its inclusion is achieved as explained in Section 1.5.3.1. However, a source inductance also causes significant wave deformation as in Figure 2.5. Then, consideration of only a resistance of a transmission line can give a reasonable result of a switching surge waveform. This results in a safer side from the viewpoint of the insulation design/coordination of a transmission system and a substation, because the switching overvoltage is estimated more severe than that in an accurate calculation.

148


2.2.3 Switching Surges in Practice [1] 2.2.3.1 Classification of Switching Surges Switching surges are generated by a switching operation of a CB and are classified in the following manner:

a. Closing surge—CB closing—(1) energization of a line (3) re-energization (reclosing surge) b. Clearing surge—CB opening—(2) fault clearing

(1), (2), (3) is the sequence of a CB operation. The closing surge is caused by closing switch contact S1 of the CB as illustrated in Figure 2.6, and the resultant overvoltage is called “closing surge overvoltage.” The reclosing surge is generated, in practice, after fault clearing as explained earlier. That is, when a fault occurs in a power system, the faulty line has to be cleared from the source as soon as possible so as to avoid damage of the line and machineries in the system due to a large fault current. Thus, the CB connected to the faulty line is opened. In most cases, a line fault is so-called line-to-ground fault originally due to lightning and is sustained by an electrical arc of which the energy is supplied from a generator in the system. Therefore, when the energy from the generator is shut down by opening the CB, the arc is distinguished and fault itself is cleared. At this stage, the fault-clearing surge is generated. After the fault clearing, the system has, in general, to be returned to a normal operation as soon as possible. As a consequence, the CB may be reclosed. When the time period from the fault clearing to the reclosing of the CB is not long enough for the trapped charge on the faulted line to be discharged, there remains the “residual charge voltage” on the line. This means that the reclosing surge involves an initial condition of the line to be closed. In other words, a difference between the reclosing and closing surges is either there is an initial charge (voltage) or not.

CB Bus e(t)

Rs

Ls

S2

S1

Vs

RCB e(t) = Em sin(ωP t + θ) RS , LS = source impedance FIGURE 2.6 A model circuit of a switching surge analysis.

Vr l


149

As a common practice, the fault-clearing surge is not classified as a closing surge but is included in a fault surge since it occurs as a consequence of a fault. It is easily expected that the maximum closing overvoltage is 2 pu on an ideal lossless single-phase line as explained in Section 2.1. However, the reclosing overvoltage easily reaches 4 pu due to mutual coupling of the phases, CB closing time difference between the phases, and residual voltages in an actual multiphase line. 2.2.3.2 Basic Characteristic of Closing Surge: Field Test Results Figure 2.7b shows a field test result of a closing surge on an untransposed horizontal line in Figure 1.22 for the middle-phase closing as illustrated in Figure 2.7a [4,5]. The source voltage is of the waveform of 1/4000 μs. At the receiving end, phase b voltage rises very rapidly (A in the figure) right after the arrival of the traveling wave (t = 0 in the figure) and becomes nearly flat for about 20 μs (B). Then, it rises slowly (C) as observed in Figure 2.7b. The induced phase (a, c) voltage becomes negative at the beginning and slowly increases to a positive value. The previous condition is a typical characteristic of a closing surge at the wave front and is due to different propagation velocities of the earth return and aerial modes as explained in Section 1.6.4. Figure 2.7c shows analytical results obtained by the method in Section 1.6.4. The figure clearly explains the characteristic of the closing surge observed in Figure 2.7b. In Figure 2.7b, a dotted line is the calculated result by a frequency domain (FD) method based on the steady-state solution and numerical Laplace transform [6]. The calculated result agrees quite well with the field test result. A field test result for phase “a” application in the test circuit of Figure 2.7a has been explained in Section 1.6.4. Figure 2.8b to d shows another field test result on a test circuit illustrated in Figure 2.8a of an untransposed vertical double-circuit line [7,8]. The applied source voltage has the waveform of 1/5000 μs. In tests (b) and (c) in Figure 2.8, the source resistance R was 403 Ω and was 150 Ω in test (d) in which all the six phases were short-circuited at the sending end. It is observed in Figure 2.8b that twice of the traveling time of the fastest wave is about 681 μs, which results in the propagation velocity of 101.13 km/681 μs = 297 m/μs. Also it is observed that the rising part of the wave front during the time period of 680–700 μs has two steps of voltage increase. The first rise is due to the arrival of the fastest traveling wave at the sending end, which corresponds to modes 2, 4, and 5 explained in Section 1.5.2. Then, the second fastest wave corresponding to mode 1 arrives at the sending end and causes another rise of the sending-end voltage. The modal velocity for modes 2, 4, and 5 is estimated about 299.2 m/μs, and that of mode 1 is 298.4 m/μs at 1 kHz from Figure 1.26 in Section 1.5.2. The difference results in 1.8 μs delay of the mode 1 wave arrival time from those of modes 2, 4, and 5.

150


The delay is the cause of the stair case rise of the sending-end voltage observed in Figure 2.8b. Figure 2.8c shows the receiving-end voltages when the source voltage is applied to phase c. On phases a′ and b′, a spike voltage is observed just after the arrival of the fastest traveling wave. This phenomenon is the same as Vs

Vr a

R = 415 Ω

b c

V0 = 1 pu (a) V [pu] 1.0 Phase b

0.5

Phase a 0

0.6

0.8

Time (ms) a

1.0

1.2

V [pu] Phase b

0.5

B

C

A Phase a 0 Time (µs) b (b)

Measured

100

Calculated

FIGURE 2.7 Switching surge waveforms on a 500 kV untransposed horizontal line. (a) Test circuit: ℓ = 83.212 km (line configuration Figure 1.22). (b) Measured and calculated results.

151


1

vR0 vRb 0.42

Voltage, pu

2vR2

vR0

0.46 vRa 0 0.23

0.42

Time, µs

100

vR2 (c)

FIGURE 2.7 (continued) Switching surge waveforms on a 500 kV untransposed horizontal line. (c) Analytical waveforms corresponding to (b).

that explained in Section 1.6.4.5 for the induced phase (phase c) voltage on an untransposed horizontal line. An analytical result of the phase a′ voltage based on the parameters at f = 2 kHz of Figure 1.26 is given by

Vra′ = 0.285u ( t′ − τ′0 ) − 0.172u ( t′ − τ′1 ) − 0.115u ( t′ − τ′3 ) + 0.130u ( t′ ) (2.5)

where τ′0 = 64.6 µs, τ′1 = 0.8 †µs, τ′3 = 2.0 µs, t′ = t − τ2 = t − 337.8 †µs. The voltage waveform is illustrated in Figure 2.9, which clearly shows the spike voltage. No spike voltage appears on phase c′ because the voltages of modes 2, 4, and 5 are canceled out. Figure 2.8d indicates that there exist aerial mode voltages even in the case of the voltage application to all the phases. This cannot be explained by the conventional symmetrical component theory. The modal theory predicts the aerial mode components of 28% on the upper phase, 7.6% on the middle, and 13% on the lower, which agree well with the field test result of 20%, 4.3%, and 11%, respectively. 2.2.3.3 Closing Surge on a Single-Phase Line Figure 2.10 shows closing surges on a single-phase line. A broken line in the figure is the case of neglecting the frequency-dependent effect (wave deformation) explained in Sections 1.5.2 and 1.5.3. It is observed that an

152


R

Vs

Vr

V0

(a)

(b)

A

A΄

B

B΄

C΄

C (c)

Upper

Middle

Lower

(d)

FIGURE 2.8 Filed test results of closing surges on an untransposed vertical twin-circuit line in Figure 1.25 with the length 101.13 km. (a) Test circuit. (b) Phase a sending-end voltage for phase a application. (c) Receiving-end voltages for phase c application. (d) Receiving-end voltages for a source application to all the phases in short circuit.

153


V (pu)

0.1

0.13 (modes 2, 4, 5)

0

–0.1

2 –0.042 (mode 1)

0.128 (mode 0)

4

Time (µs)

62

64

66

68

–0.157 (mode 3) –0.2 FIGURE 2.9 Analytical voltage waveform of phase a′ corresponding to Figure 2.8c.

oscillating surge voltage is sustained for long and the maximum overvoltage reaches −2.67 pu at t = 10.2 ms. While, the case of including the wave deformation given by a dotted line shows that the oscillation is being damped as time increases, and the maximum voltage is reduced to about 2.0 pu. The earlier statement clearly indicates the significance of the frequency-dependent effect on a switching surge analysis. If it is neglected, the insulation level against the switching surge is overestimated and results in an economical inefficiency. In an extra-high-voltage (EHV) and ultrahigh-voltage (UHV) transmission systems, it is a common practice to control the closing surge overvoltage by means of an insertion resistor (closing resistor) or of synchronized switching of a CB. The effect of the closing resistor is shown by the dashed-dotted line (–.–.–) in Figure 2.10. It is quite clear how the closing resistor is effective to damp the overvoltage. The maximum overvoltage is reduced to 1.2 pu. The resistor is also used to damp a fault-clearing surge when necessary. The synchronized switching means that every phase is closed when the phase voltage is zero. Thus, no overvoltage appears in theory. Such a CB is widely used in Europe. 2.2.3.4 Closing Surges on a Multiphase Line 2.2.3.4.1 Wave Deformation Figure 2.11 shows a closing surge due to a sequential closing of a three-phase CB on an untransposed horizontal line. The effect of the wave deformation on the switching surge is also clear even in the multiphase line.

154


1Ω

50 mH

150 km

RCB

V

25 nF v0

100 Ω

(a) V (pu)

2

1

0

–1

5

10

Time (ms)

–2

(b) FIGURE 2.10 Single-phase closing surges. (a) Circuit diagram. (b) Calculated result.

2.2.3.4.2 Closing Angle Distribution It is assumed in Figure 2.11 that each phase of the CB is closed sequentially at electrical angles 90°, 150°, and 180°. In reality, it is not clear if a pole of any CB is closed at the time planned to be closed, because of a mechanical structure of the CB. Figure 2.12 shows a distribution of the CB closing angles. Figure 2.12a and b is an analytical distribution often used for a statistical

155


22 m 6.6 m 16.7 m

G.W.

14 m 1

2

3

P.W.

(a)

Voltage, pu

2 1 0

1 2 Time, ms

3

4

5

6

7

8

9

10

–1

(b)

–2

FIGURE 2.11 Switching surges at the receiving end for sequential closing. (a) Line configuration: phase wire (PW): radius = 0.1785 m, resistivity = 3.78 × 10−8 Ω-m; ground wire (GW): radius = 8.8 mm, resistivity = 5.36 × 10−8 Ω-m, line length = 150 km, source inductance = 50 mH. (b) Calculated results: phase 1 closed at 90° (t = 0, v = 1 pu), phase 2 at 150°, phase 3 at 180° no wave deformation; ——: phase 1, –.–.–: phase 3 wave deformation, earth resistivity = 100 Ω-m; ……: phase 1, × : phase 3.

analysis, and (c) shows measured distributions. (c-1) is a measured result of time dispersion of three phases of a CB. (c-2) is another measured result of a time delay of a following phase to the succeeding phase. Calculated results of the statistical distribution of closing overvoltages for the four previously mentioned distributions of the CB closing angles are shown in Figure 2.13. In the case of the uniform and Gaussian distributions, the maximum time delay of the first and third phases closed is assumed to be less than 7.5 ms. The total number of switching operations is taken as 100 for simplicity. For an insulation design, at least 1000 operations are required to obtain more detailed data of an overvoltage distribution. The calculated results indicate that the difference of the CB closing angle distribution causes noticeably different overvoltages, especially for the maximum and minimum overvoltages. For example, the maximum overvoltage, that is, an overvoltage that has the lowest probability of occurrence, in (c-1) is greater by about 0.3 pu than that in (a) and (c-2). In the case of the Gaussian distribution, the difference of the maximum overvoltage 0.12 pu is observed

156


Frequency

Frequency

σ: variance µ: mean value

(b)

30

15

20

10

Frequency

Frequency

µ

Time

(a)

10

Time

5

1 (c)

σ

2

3 4 Time (ms) (c)-1

5

0.04

1.05 Time (ms) (c)-2

FIGURE 2.12 Various distribution characteristics of closing angles. (a) Uniform distribution. (b) Gaussian distribution. (c) Measured distribution in a field test.

for 6σ = 7.5 ms and 5 ms, where σ is a standard deviation. The previous observation has made it clear that the distribution of the CB closing angles affects the maximum overvoltage significantly, and thus one has to be careful when choosing the distribution. Figure 2.14 shows a contour line expression of maximum overvoltages when the phase a closing angle θa is fixed to 90°. It is observed from the figure that higher overvoltages distribute along the axis of θc ≒ 150° and θb ≒ 210°. Also, it is found that higher overvoltages appear along the axis of θa = 90° when θa is varied. Thus, it is concluded that severe overvoltages appear along the axis ofθa ≒ 90°, θb ≒ 210°, and θc ≒ 150° as illustrated in Figure 2.15 in a 3D space. This is quite reasonable because the source voltage on each phase takes its peak at the angle, and the voltage difference across the CB terminals

157


Percent exceeding abscissa

100

50

0

1.5

(a)

2.0

Voltage, pu

2.5


100

3.5

6 σ = 7.5 ms 6 σ = 5.0 ms 3 σ = 7.5 ms 3 σ = 5.0 ms

50

0 (b)

3.0

1.5

2.0 2.5 Voltage, pu

3.0

3.5

FIGURE 2.13 Statistical distribution curves of maximum switching overvoltages (ℓ = 150 km, Ls = 50mH in Figure 1.22). (a) Uniform distribution (Figure 2.12a), × measured distribution ((c)-1), • measured distribution ((c)-2). (b) Gaussian distribution (Figure 2.12b).

of the source and line sides becomes largest before closing the CB of the phase if the mutual coupling and residual voltage are neglected. It is easily expected that the most severe overvoltage is generated when the CB is closed with the largest voltage difference across the CB. In reality, the closing angle that generates the most severe overvoltage is different to some degree from the earlier condition due to the effects of the mutual coupling between phases, the source inductance, and the residual voltage if any. 2.2.3.4.3 Resistor Closing It is a common practice to adopt a closing resistor into a CB on EHV and UHV systems in Japan. Figure 2.16 shows the effect of resistor closing on a closing surge on an untransposed vertical double-circuit line of Figure 1.25

158


Phase b, θb(deg)

120

2.4

2.6

2.4

90

2.2 pu

2.3

2.2

150 180 2.6

2.7 2.5

210 2.4

2.4

2.5

90

120

150

210

180

Phase c closing angle, θc(deg) : Maximum overvoltage 2.791 pu FIGURE 2.14 Distribution characteristics of maximum switching overvoltages (θa = 90°, ℓ = 150km, Ls = 50mH).

θb θa = 90° (90°, 210°, 150°) θc = 150°

θc θb = 210° θa FIGURE 2.15 3D expression of maximum switching overvoltages.

with line length of 101.13 km. It is quite clear that the overvoltage with no closing resistor is reduced to about a half with a closing resistor. Figure 2.17 shows the effect of closing resistors on maximum overvoltages for both closing and reclosing surges. It is observed that the surge overvoltage generated by resistor closing decreases as the value of resistor increases.

159


2.5 2.0 1.5

Voltage (pu)

1.0 0.5 0.0

0

5

10

15

20

25

30

35

40

45

30

35

40

45

–0.5 –1.0 –1.5 –2.0

(a)

–2.5

Time (ms)

1.5 1.0

Voltage (pu)

0.5 0.0

0

5

10

15

20

25

–0.5 –1.0

(b)

–1.5

Time (ms)

FIGURE 2.16 Three-phase closing surges on a 500 kV untransposed vertical line. (a) No resistor closing. (b) Resistor closing.

160


Voltage, pu

2.0 After short-circuiting the resistor 1.0

0 (a)

Before short-circuiting the resistor

1.0 Value of resistor, kΩ

2.0

4.0

Voltage, pu

3.0

2.0

1.0

0 (b)

After short-circuiting the resistor

Before short-circuiting the resistor

1.0

2.0

Value of resistor, kΩ

FIGURE 2.17 Effect of closing resistors on maximum overvoltages. (a) Closing surge. (b) Reclosing surge residual voltages (−1, −1, 1 pu).

161


However, the overvoltage generated due to short circuit of the resistor increases as the value of resistor increases. Thus, there exists an optimum value of the closing resistor. It is 300 Ω in the closing case in Figure 2.17a. The resistor closing is more effective in the reclosing case as observed from Figure 2.17b. The maximum overvoltage of 3.5 pu is reduced to 1.3 pu in the reclosing case, while it is 2.2–1.2 pu in the closing case. 2.2.3.4.4 Closing Surge Suppression by an Arrester An arrester is originally to protect power apparatuses from a lightning overvoltage. It is expected to apply arresters to suppress a switching overvoltage on a UHV system because the arrester characteristic is greatly improved by the development of a ZnO arrester. Figure 2.18a shows a typical characteristic of a ZnO arrester. Figure 2.19 shows an application example of the ZnO attester, which has the voltage–current characteristic of Figure 2.18b to control the closing surge overvoltage. The maximum overvoltage of 2.03 pu is suppressed to 1.56 pu by the arrester. V (kV) 1500

1000

1

10

100

(a)

1k

10 k

I (A) V (kV) 1370 1250 1140

(b)

0.1

100

I (A)

1k

I

FIGURE 2.18 Voltage–current characteristic of a ZnO arrester. (a) Actual characteristic. (b) Linear approximation.

162


CB Gas insul. cable

230 km ZnO arrester

(a) No arrester Arrester

1.5

Voltage, MV

1.0 0.5 0

4

12

8

Time, ms

–0.5 –1.0 –1.5

(b) 1.5

Voltage, MV

1.0 0.5 0

4

8

Time, ms

12

–0.5 –1.0 –1.5

(c) FIGURE 2.19 Suppression of closing surge overvoltage by a ZnO arrester. (a) An 1100 kV line. (b) Sending end. (c) Receiving end.

2.2.3.5 Effect of Various Parameters on Closing Surge As already explained in Section 2.2.3.4.2, the maximum overvoltage is dependent on the closing angle of each phase, which has a characteristic of probability in its appearance. Therefore, a deterministic analysis of the switching overvoltage is often found not good enough to investigate an insulation level. In these days, it is a common practice that the switching overvoltage is analyzed from the statistical viewpoint considering the probabilistic nature of the closing angles and also probabilistic nature of an insulation failure due to the overvoltage. Thus, a statistical analysis of

163



100 0 Ω-m 10 Ω-m 100 Ω-m 500 Ω-m 1000 Ω-m

50

0

1.5

2.0 2.5 Voltage, pu

3.0

3.5

FIGURE 2.20 Effect of earth resistivity on statistical overvoltage distribution curves.

the effect of various parameters on the closing overvoltage will be studied in this section. 2.2.3.5.1 Effect of Earth Resistivity Figure 2.20 shows statistical overvoltage distribution curves for various earth resistivities. It is clear that the highest overvoltages appear for the case of ρe = 0, that is, perfectly conducting earth corresponding to no wave deformation. Also, it is observed that the overvoltage becomes higher as the earth resistivity increases. The lowest overvoltage distribution is observed for ρe = 10 Ω-m. Thus, it should be clear that there exists a resistivity that gives the lowest overvoltage distribution. This is explained by the fact that an increase of an attenuation constant due to the increase of the earth resistivity decreases a current flowing into the earth and thus decreases the overall attenuation of the overvoltages. This results in a higher overvoltage. On the contrary, a decrease of the attenuation constant due to the decrease of the earth resistivity increases the current into the earth and increases the overall attenuation of the overvoltages. Thus, it results in a higher overvoltage as same as the case of increasing the earth resistivity. This means that there exists an earth resistivity that gives the lowest overvoltage distribution. 2.2.3.5.2 Effect of Line Length Figure 2.21 shows a statistical distribution of overvoltages for various line lengths. In general, the overvoltage increases as the line length increases. In practice, a long transmission line is compensated by a shunt reactor, and thus the overvoltage distribution differs from that shown in Figure 2.21. 2.2.3.5.3 Effect of Source Inductance Figure 2.22 shows the effect of the source inductance on a statistical distribution of overvoltages. No clear tendency is observed in the figure.

164



100 150 km 200 km 300 km 50

0

1.5

2.0 2.5 Voltage, pu

3.0

3.5

FIGURE 2.21 Effect of line length.


100

30 mH 50 mH 100 mH 300 mH

50

0

1.5

2.0 2.5 Voltage, pu

3.0

3.5

FIGURE 2.22 Effect of source inductance.

The effect of the line length is dependent on the source inductance, and the following observation is made:

a. Sending end i. The voltage tends to increase as the inductance increases. ii. For a small inductance, the overvoltage tends to decrease as the line length increases. For a large inductance, the opposite tendency is observed. b. Receiving end i. For a long line, the overvoltage tends to increase as the inductance increases. No clear tendency is found for a short line. ii. The overvoltage increase as the line length increases.

165



100 Untransposed Transposed

50

0

1.5

2.0 2.5 Voltage, pu

3.0

3.5

FIGURE 2.23 Effect of line transposition on statistical overvoltage distribution curves.

The reason why the overvoltage tends to increase as the source inductance and line length increase is that the natural oscillating frequency of the line approaches the power frequency by the increase of the inductance and line length. This results in a resonant oscillation of the system, and the overvoltage increases. This resonant overvoltage is a temporary overvoltage rather than the closing surge overvoltage. 2.2.3.5.4 Line Transposition An untransposed line shows a noticeable difference of a voltage waveform from the transposed line case for a very beginning of the waveform. A statistical distribution of the overvoltages, however, shows no significant difference as observed from Figure 2.23. The difference is more noticeable at the sending end than at the receiving end. If surge waveforms with a specific sequence of closing angles are compared, rather significant difference is often observed. 2.2.3.5.5 Reclosing Overvoltage Figure 2.24 shows a statistical distribution of reclosing overvoltages. It is clear that the reclosing generates a higher overvoltage distribution. In general, it is expected that the higher the residual charge voltage, the more severe the overvoltage distribution. The figure shows the polarity effect. This may vanish if the number of samples is large enough. 2.2.3.5.6 Effect of Closing Resistor Figure 2.25 shows the effect of a closing resistor on a statistical distribution. It is clear how the closing resistor is effective to reduce the overvoltage in both the closing and reclosing cases. The damping effect is especially noticeable in the region of higher overvoltages. The resistor of 500 Ω is effective to reduce the overvoltage by 1.7 pu.

166



100

50

0

1.5

2.0

No charge voltage (–1, –1, 1)

2.5 3.0 Voltage, pu

3.5

(0.5, 0.5, –0.5) (phase a: 1 pu, b: 1 pu, c: –1 pu)

4.0

4.5

(–0.5, –0.5, 0.5)

FIGURE 2.24 Effect of line charge voltage.

2.3 Fault Surge Fault surge is generated by a line fault. The fault surge is classified into a fault initiation surge (fault surge) and a fault-clearing surge. The most probable fault is a phase (single line)-to-ground (SLG) fault. In this section, the fault surge of the SLG will be explained. 2.3.1 Fault Initiation Surge [1] The fault initiation surge (fault surge) is generated on a sound phase due to a fault of one phase (faulty phase or line). The maximum overvoltage is, in general, much lower than that of a closing surge. Figure 2.26 shows a typical example of the fault surge. The maximum voltage observed is −1.32 pu, which is far smaller than the voltages discussed in the previous section. Figure 2.1 is another example of the fault surge measured in an actual EHV transmission line. The maximum overvoltage in the measurement was 1.7 pu as shown in the figure. Figure 2.27 shows calculated results of the fault surge on an 1100 kV transmission system. A single-phase-to-ground fault occurs at the midpoint of the first circuit of line 1 (node L1-13) as illustrated in Figure 2.27a. Figure 2.27b shows the faulty circuit voltages and (c) the sound circuit voltages. The maximum overvoltage is observed to be 1.67 pu (=1500 kV) on phase c of the faulty circuit. The result is for the case of arresters being installed. In general, the maximum fault overvoltage is expected to be less than 1.7 pu. This is for a rather simple system. In a complicated system such as

167



100

: R=0 Ω : R = 500 Ω : R = 1000 Ω

50

0

1.5

(a)

2.0

2.5 Voltage, pu


100

3.0

4.0

3.5

No resistor 500 Ω 1000 Ω

50

0

1.5

(b)

2.0 2.5 Voltage, pu

3.0

3.5

FIGURE 2.25 Effect of closing resistors on statistical overvoltage distribution curves. (a) Closing surges. (b) Reclosing surges (initial charge 0.5, 0.5, −0.5 pu).

150 km V3

1

2

CB1

CB2

V3 (pu) 2 a

1 0 –1 –2 FIGURE 2.26 A fault surge.

10 b

c –1.32 pu

Time (ms)

168


140 km L2-22 L2-12

230 km L1-23 L1-13

BUS-2

BUS-3

BUS-1

E0 = 2 × 1100/ 3 = 898 kV (a) 1.67 pu

V at L1-13 (pu)

1 0

00

5.00

10.00

15.00

20.00 25.00 Milliseconds

30.00

00

5.00

10.00

15.00

20.00 25.00 Seconds

30.00

35.00

40.00

45.00

–1

V at L2-13 (pu)

(b)

1 0

35.00

40.00

45.00

–1

(c)

FIGURE 2.27 Calculated results of fault surge on a 1100 kV circuit. (a) An 1100 kV model circuit. (b) Faulty circuit. (c) Sound circuit.

Figure 2.27a, the maximum overvoltage may reach 1.8 pu. The overvoltage being much lower than the closing overvoltages explained in the previous section, the fault surge has not been a dominant factor of determining the switching impulse withstanding level (SIWL) or switching impulse insulation level (SIL) of a conventional transmission line below 700 kV. It, however,


169

becomes quite significant in a UHV system since the SIWL or SIL is expected to be lower than 1.7 pu, which is easily achieved by adopting a closing resistor as far as the closing surge concerns. On the contrary the fault surge is quite difficult to control, because it is impossible to predict where and when a fault occurs, and also where the highest overvoltage appears. Only a possible way of controlling the fault overvoltage at present is the installation of so-called line arresters along the transmission line. 2.3.2 Characteristic of a Fault Initiation Surge 2.3.2.1 Effect of Line Transposition Figure 2.28 shows a comparison of the fault surge between untransposed and transposed double-circuit lines. The fault is initiated at the receiving end of one circuit as illustrated in the figure. The circuit configuration is equivalent to the model circuit illustrated in Figure 2.26 in which the fault occurs at the middle of the line. The figure clearly shows that there is no significant difference between the untransposed and transposed line. The observation is the same as that made for the closing surge case. As far as the surge overvoltage or line insulation concerns, the effect of line transposition is not significant. 2.3.2.2 Overvoltage Distribution Figure 2.29 illustrates a maximum overvoltage due to phase a-to-ground fault surge, when the fault position is changed from the sending end to the receiving end along the line, an untransposed 500 kV line with a length of 200 km. The case of 50 mH source inductances at both end, (a), shows that the highest maximum overvoltage 1.56 pu occurs at the middle of the line, and the overvoltage decreases symmetrically toward the ends of the lines. The reason for the earlier condition is estimated that the positive reflected waves from both ends arrive at the middle of the line at the same instance and are superposed. When an infinite source is connected to the right-hand side of the line, the highest overvoltage 1.56 pu same as case (a) appears at the right-hand side rather than the middle of the line. This is due to the asymmetry of the circuit, that is, the electrical center being shifted to the right of the line. Figure 2.30 shows a maximum overvoltage distribution along the line when the fault position is fixed. Figure 2.30a is the case of the fault at the middle and (b) at the 1/4 length from the sending end. In case (b), the highest overvoltage 1.53 pu appears at the fault point and its symmetrical position to the center of the line, and the overvoltage distribution is flat within the two points. While in case (a), the highest overvoltage 1.56 pu appears at the center, and the overvoltage decreases rapidly toward both ends of the line. The observation leads to the fact that the region of the line, where the

170


1 pu

101.3 km

50 mH

V Rs = 1 Ω (a) V (pu) 1.509 pu

1

0

5.00

10.00

15.00

20.00 25.00 30.00 Milliseconds

35.00

40.00

45

–1 (b) V (pu)

1.517 pu

1

0

5.00

10.00

15.00

20.00 25.00 30.00 Milliseconds

35.00

40.00

45

–1 (c) FIGURE 2.28 Effect of line transposition on a fault surge–fault circuit voltage. (a) A 500 kV model circuit. (b) Untransposed line. (c) Transposed line.

overvoltage is higher than a certain level, is wider in the case of the fault occurring at a point apart from the line center, while the highest overvoltage appears in the case of the middle point fault. This may be important from the protection viewpoint. Table 2.2 shows maximum overvoltage at various nodes of the system illustrated in Figure 2.27a. It is observed that the highest overvoltage appears at the fault point on the sound phase of the faulty circuit. The overvoltage decreases as the distance from the fault point increases.

171


2 pu

1.56 pu

1

0

100 km

50 mH

(a)

200

50 mH

2 pu

1.56 pu

1

100 km

0

200

50 mH

(b)

FIGURE 2.29 Overvoltages at the fault points. (a) 50 mH source. (b) 50 mH and infinite bus. 2 Pu

1.56 pu

1

0

200

100 km

50 mH

50 mH

(a) 2

1.53 pu

Pu

1.53 pu

1

0

50 mH

100 km

200

50 mH

(b) FIGURE 2.30 Overvoltage distribution along the line. (a) Fault at the middle. (b) Fault at the 1/4 point.

172


TABLE 2.2 Maximum Overvoltages at Various Points in the 1100 kV System of Figure 2.27a Phase Node

a

b

c

BUS-1 L1-13 L1-23 BUS-2 L2-12 L2-22 BUS-3

−1.06 [pu] — −1.07 −1.07 −1.07 −1.07 −1.14

1.29 1.55 1.49 1.31 1.32 1.32 1.29

1.37 1.67 1.64 1.41 1.44 1.44 1.39

1 pu = 2 × 1100

3 = 898 kV

2.3.3 Fault-Clearing Surge A fault-clearing surge is generated by clearing a fault by a CB. Generally, its overvoltage is smaller than a closing surge overvoltage and greater than a fault initiation surge overvoltage. Figure 2.31 shows an example of the fault-clearing surge. The maximum overvoltage is observed to be 1.5 pu, which is greater than that of the fault surge 1.32 pu in Figure 2.26. The overvoltage increases to about 1.6 pu in the two-phase-to-ground fault case and to about 2.0 pu in the three-phase-toground fault case. Figure 2.32 shows the effect of line transposition on the fault-clearing overvoltage. It is observed in both untransposed and transposed lines that the maximum overvoltage reaches higher than 2 pu, which is greater by about 0.6 pu than the fault initiation overvoltages observed in Figure 2.28. The line V2 (pu)

1.5

2 a

1

10

0 –1 –2

20 b

c Time (ms)

FIGURE 2.31 A fault-clearing overvoltage in the same circuit as Figure 2.26.

173


1 pu

50 mH

101.3 km

V RF = 1 Ω

V (pu) 2

1

0

5.00

10.00 15.00 20.00 25.00 30.00 35.00 40.00 45 Milliseconds

–1

–2 (a)

–2.147

V (pu) 2

2.151

1

0

5.00

10.00 15.00 20.00 25.00 30.00 35.00 40.00 45 Milliseconds

–1

–2 (b)

FIGURE 2.32 Effect of line transposition—sound circuit voltage. (a) Untransposed. (b) Transposed.

transposition does not significantly affect the overvoltage but affects the waveshape noticeably as is clear in the figure. The reason for this is that the attenuation of the aerial mode is greater, and the velocity is lower in a transposed line than in an untransposed line as explained in Section 1.5.2. As a result, the waveform is more distorted as time increases in the transposed

174


line case. The difference, however, is not noticeable if an opening resistor of a CB is adopted. Figure 2.33 shows the effect of the opening resistor of the CB on the faultclearing overvoltage on a Japanese 1100 kV system. At t = 0, the system is in a steady state of three-phase-to-ground fault at the middle of line 2 (140 km). At t = 1 ms, a resistor is inserted between the contacts of the CB, and the main contact is opened. Then, at t = 30 ms, the resistor is opened, and the threephase-to-ground fault is completely cleared. The fault-clearing surge voltages at the middle of line 1 (230 km) on the faulty circuit are shown in Figure 2.33. It is clear from the figure that the maximum overvoltage reaches 1.71 pu in the case of no opening resistor (a). The overvoltage is decreased to 1.31 pu by the opening resistor in Figure 2.33b. Also, the voltage waveform tends to reach rapidly a sinusoidal steady state. Table 2.3 shows maximum overvoltages at various positions of the circuit illustrated in Figure 2.33. It is observed from the table that the highest voltage appears at the middle of the line 1 other than the faulty line. The fact should be noted when a fault-clearing surge analysis is carried out. Otherwise, the highest overvoltage may be missed. Figure 2.34 shows a relation between the fault-clearing overvoltage and an opening (insertion) resistor on the 1100 kV system discussed earlier. It is observed that there exists an optimum resistor to control the overvoltage as same as the closing overvoltage case. The degree of the overvoltage reduction is less in the fault-clearing surge case than in the closing surge case shown in Figure 2.17. This is due to the fact that the highest overvoltage appears within the switched line in the closing surge case, while it may appear in the other line connected to the faulty line in the fault-clearing surge case. If this is the case, the effect of the insertion resistor is small. The overvoltage in Figure 2.34 is reduced to 1.42 pu, which is greater than 1.36 pu found in Table 2.3. The reason for this is that the overvoltage in Figure 2.34 involves all the voltages having appeared in the system studied. A significant difference of the opening resistor from the closing resistor is its thermal requirement. Since a large fault current flows through the opening resistor, the resistance cannot be too small. Therefore, an optimum value of the opening resistor has to be determined not only by the degree of the overvoltage reduction but also by the thermal requirement. Figure 2.35 shows an example of a possible combination of the opening and closing resistor. It is observed from the figure that about 400 Ω resistor is optimum for the closing surge (I-a, b) and the fault-clearing surge (II). In the reclosing case (I-a′ and I-b), the reclosing overvoltage being always higher than the fault-clearing overvoltage, the optimum resistor value may be determined by the resistor corresponding to the lowest reclosing overvoltage, that is, about 800 Ω for 1.6 pu.

175


140 km L2-22 L2-12

230 km L1-23 L1-13

Bus-3

Bus-2

Bus-1

E0 = 898 kV 2000

Voltage (kV)

1000 0

0.00

5.00

–1000

(a)

10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00

Time (µs)

–2000

1000

Voltage (kV)

500 0

0.00

5.00

10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00

–500 –1000

Time (µs)

(b) FIGURE 2.33 Effect of an opening resistor on a fault-clearing overvoltage. (a) No resistor. (b) Opening resistor.

2.4 Lightning Surge Among various surges, the cause of the highest overvoltage is lightning to a tower or line. An enormous amount of studies have been carried out [9–22], but there are still a number of unknown factors to analyze a

176


TABLE 2.3 Maximum Overvoltages at Various Points in the System Phase Node

a

b

c

(a) No resistor BUS-1 −1.52 [pu] L1-13 −1.70 L1-23 −1.70 BUS-2 −1.47 L2-12 — L2-22 −1.54 BUS-3 −1.44

1.42 1.48 1.48 1.40 — 1.55 1.43

1.53 1.71 1.71 1.52 — −1.58 −1.41

(b) RCB = 500 Ω BUS-1 1.13 [pu] L1-13 1.19 L1-23 1.19 BUS-2 1.13 L2-12 — L2-22 1.19 BUS-3 −1.12

1.11 1.12 1.12 1.11 — 1.11 1.10

−1.24 −1.31 −1.31 −1.23 — −1.36 −1.18

2.0

Overvoltage, pu

a: Overvoltages due to resistor insertion

1.5 b: Overvoltages due to resistor short-circuit

1.2

500

750 Insertion resistor, Ω

1000

FIGURE 2.34 Relation between fault-clearing overvoltages and an insertion resistor.

177


CB

50 mH

l = 150 km

1 pu

V

R

(a) V (pu) 4

3

2

1

0 (b)

I–a΄

I–b

I–a II

1 R (kΩ)

2

FIGURE 2.35 Reduction of surge overvoltages by an insertion resistor. I: closing surge. a: resistor insertion. b: resistor off. a′: resistor insertion, residual voltage. II: fault-clearing surge. (a) Model circuit. (b) Calculated results.

lightning surge in power systems, for example, a tower footing impedance model, which is time and current dependent, a lightning source, and a lightning channel impedance [23]. The lightning surge is still a fruitful field of research. 2.4.1 Mechanism of Lightning Surge Generation When lightning strikes a tower (or a ground wire (GW)) as illustrated in Figure 2.36, a lightning current flows into the tower and the GWs and causes a sudden increase of the tower voltage. When the voltage difference between the tower and a phase wire (PW) reaches its electrical withstand voltage, a flashover from the tower to the PW occurs. This is called “back flashover (BFO)” because the tower voltage is higher than the PW voltage on the contrary to a normal condition of power system operation. Then, the lightning current flows into the PW, and a traveling wave due to the current propagates to a substation. The traveling wave is partially reflected at and refracted into the substation. The traveling wave or its successive reflection generates a severe overvoltage at the substation.

eC

eB

eB

eC

eA AC source

Rf

Tower No. 5

8-phase line model

xt

FIGURE 2.36 A representative model system for a lightning surge simulation.

eA

Matching impedance

Rf

Tower No. 2 xt

I (t)

8-phase line model

Lightning current

Rf

xs

8-phase line model

Tower No. 1

Zt = 400 Ω

Rfg

Gantry

Arc horn

Substation


179


2.4.2 Modeling of Circuit Elements [23] 2.4.2.1 Lightning Current Lightning is often modeled by a current source with a parallel resistance, representing the lightning channel impedance as illustrated in Figure 2.36. Although the waveform of the current source has not been conclusively known, it is defined by the waveform shown in Figure 2.37. In the figure, Tf is the time from the origin to the peak called “wave front length (time),” and Tt is the time from the origin to a half of the peak voltage called “wavetail length.” In general, Tf is less than 10 μs, and Tt less than 100 μs. Often 1/40 μs (Tf = 1 μs, Tf = 40 μs) or 2/70 μs wave is used as a standard. It should be clear from the previous condition that the sustained time period of the lightning surge is in the order of microseconds as explained in Table 2.1. There are three different definitions of the current waveforms as shown in Figure 2.37. The first one is called “lump wave,” which is expressed by two linear lines. The second is “double exponential wave” expressed by i ( t ) = I m {exp ( − at ) − exp(−bt)} (2.6)

Im

Im

Im 2

Im 2

0 (a)

Tt

Tf

0 (b)

Tf

Im

Im 2

(c)

0

Tf

Tt

FIGURE 2.37 Waveforms of a lightning current. (a) Lump. (b) Double exponent. (c) CIGRE.

Tt

180


The exponential wave has been widely used especially when an insulation test of a power apparatus is carried out because an IG inherently produces the exponential waveform. However, the exponential wave gives an excessive overvoltage compared with the other if adopted in a lightning surge analysis. Thus, it is recommended not to use the double exponential wave in the lightning surge analysis or take Tf = 2 μs rather than 1 μs in the case of 1/40 μs waveform. The last one called “Conseil International Des Grands Réseaux Electriques (CIGRE) wave” is based on the study of the CIGRE and is characterized by its negative di/dt at the wave front. In contrast, di/dt = 0 for the lump wave, and di/dt>0 for the exponential wave. The following expression of the CIGRE wave has been proposed:   πt i(t) = I m 1 − cos   2Tf  = Im

(2Tt − Tf − t) 2(Tt − Tf )

=0

     

     for Tf ≤ t < 2Tt − T  (2.7)    for t ≥ 2Tt − Tf   for t < Tf

Measured examples of lightning current amplitudes vs. the occurrence are shown in Figure 2.38 [20–26]. It is observed from the figure that the frequency of occurrence of a current greater than 50 kA is about 20% and that of a current greater than 100 kA is about 5%. It is therefore estimated sufficient to assume 100 kA as a lightning current for an analysis on an EHV transmission line [22,26]. In Japan, the current of 200 kA has been adopted for an insulation design of an 1100 kV transmission system [26]. Table 2.4 shows recommended amplitudes of lightning currents for various voltage classes [26,27]. 2.4.2.2 Tower and Gantry A transmission tower is represented by four distributed-parameter lines [28] as illustrated in Figure 2.39 where t1 is the tower top to the upper phase arm = upper to middle = middle Z to lower. Zt4 is the lower to tower bottom. Table 2.4 gives a typical value of the surge impedance.

181


99.99 1 Guide book of lightning protection

99.9

Design (summer lightning) ( µ = 26.0 kA, σ|og| = 0.325)

1

99

2 CRIEPI (winter lightning, type A) 2.3 (P(I) = 1/{1 + (I/24) })

2

3

95

4

90 Cumulative frequency (%)

3 R.B. Anderson et al. ( I = 30.0 kA, σ|og| = 0.32)

4 EPRI

(P(I) = 1/{1 + (I/31)2.6}) 5 AIEE

80

(P(I) = 0.95 exp(–I/20) + 0.05 exp(–I/50))

70 60 50 40 30

5

20 10 5 2 1 0.5

2 4

0.1 0.05

3

0.01 0.001

5 1

2

3 4 5

10

20

30 40 50 70 100

200 300

1 500

1000

The peak value of lightning striking current (kA) FIGURE 2.38 Measured results of lightning current amplitudes. (From IEEE Guide for Improving the Lightning Performance of Transmission Lines, 1997; Ametani, A. et al., Power system transients and EMTP analyses, IEE Japan WG Report. Tech. Rep. 872, 2002; CRIPEI WG, Guide to transmission line protection against lightning, Report T72, 2003; Ametani, A. and Kawamura, T., IEEE Trans. Power Deliv., 20(2), 867, 2005; Anderson, R.B. and Eriksson, A.J., Electra, 69, 65, 1980.)

182


TABLE 2.4 Recommended Values of Lighting Parameters System Voltage [kV]

Lighting Current [kA]

1100 500 275 154(110)

200 150 100 60

107 79.5 52.0 45.8

12.5 7.5 9.0 6.2

18.5 14.5 7.6 4.3

18.5 14.5 7.6 4.3

130 220 220 220

90 150 150 150

10 10 10 10

30,40

28.0

3.5

4.0

3.5

220

150

10–20

77(66)

Tower Height/Geometry [m] h x1 x2 x3

Surge Imp. [Ω] Zt1 Zt4

Footing Res. [Ω] Rf

The propagation velocity c of a traveling wave along a tower is taken to be

c0 = 300 m/µs : light velocity in free space (2.8)

To represent traveling wave attenuation and distortion, an RL parallel circuit is added to each part as illustrated in Figure 2.39. The values of the R and L are defined in the following equation: Ri = ∆Ri ⋅ xi , Li = 2τRi ∆R1 = ∆R2 = ∆R3 = 2Zt1 ⋅ ln

∆R4 = 2Zt 4 ⋅ ln

(1/α1 ) (2.9)

(h − x 4 )

(1/α 4 ) h

where τ = h/c0 is the traveling time along the tower α1 = α4 = 0.89 is the attenuation along the tower h is the tower height The RL parallel circuits in Figure 2.39 can be neglected in most lightning surge analyses as explained later. A substation gantry is represented by a single distributed line with no loss. 2.4.2.3 Tower Footing Impedance A tower footing impedance is suggested in Japan to be modeled as a simple linear resistance Rf, although a current-dependent nonlinear resistance is recommended by the IEEE and the CIGRE [16,18,20]. The inductive and capacitive characteristics of the footing impedance as shown in Figure 2.40 are well known [10]. A recommended value of the resistance for each voltage class is given in Table 2.4.

183


h = h1

GW x1

Zt1, c0 R1, L1

h2 Upper x2

Zt1, c0 R2, L2

h3 Middle x3

Zt1, c0 R3, L3

h4 Lower Zt4, c0 R4, L4

x4 Rf

FIGURE 2.39 A tower model.

L

R1

R2

R2

R2

R1 + R2

R2

R1 + R 2

R2

R2 0 T (a)

R1

C

t

0 (b)

t

0 T

t

(c)

FIGURE 2.40 Footing impedance models and the step responses. (a) Inductive. (b) Resistive. (c) Capacitive.

184


2.4.2.4 Arc Horn The maximum voltage on a line and a substation due to lightning is highly dependent on the flashover voltage of an arc horn, which is installed between a tower arm and a PW along an insulator as illustrated in Figure 2.36. The purpose of the arc horn is to control the lightning overvoltage on the line and the substation and also to protect the insulator against its mechanical damage due to its electrical breakdown (=flashover). In general, the arc horn gap length is taken a little shorter than the length of the insulator, that is, the clearance between the tower arm and the PW, so that the arc horn gap flashovers before the voltage between the tower arm and the PW, that is, voltage across the insulator, reaches its breakdown voltage (=insulator withstand voltage). Thus, the insulator, in theory, never results in its own breakdown and is protected from a mechanical damage. Also, the voltages on the line and in the substation are dependent on the flashover voltage of the arc horn. Therefore, it should be noted that the so-called lightning overvoltage observed on a PW or in a substation is not the voltage of the lightning itself but is a voltage dependent on or controlled by the arc horn. If the gap length of the arc horn is too small, too many lightning surges appear in the substation and results in an excessive insulation of the substation apparatuses. On the contrary, if it is too large (but smaller than the clearance), the insulation of PWs from the tower, that is, clearance, becomes excessive. Therefore, it is a really difficult problem to determine an optimum gap length of an arc horn. It is entirely dependent on the overall cost of the transmission system construction and the philosophy of the overall insulation design and coordination of the system. An arc horn flashover is represented either by a piecewise linear inductance model with time-controlled switches as illustrated in Figure 2.41a or by a nonlinear inductance shown in Figure 2.41b based on a leader progression model [29,30]. The parameters Li (i = 1–3) and ti − ti − 1, assuming the initial time t0 = 0 in Figure 2.41a, are determined from a measured result of the V–I characteristic of an arc horn flashover. Then, the first simulation with no arc horn flashover is carried out in Figure 2.36, and the first flashover phase and the initial time SW1

L1

L2

L3/2 (a)

SW3

SW2

SW1

Ln

L0

SW2

L3/2 (b)

FIGURE 2.41 An arc horn flashover model. (a) A linear inductance model. (b) A nonlinear model.


185

t0 are determined from the simulation results of the voltage waveforms across all the arc horns. By adopting the previous parameters, the second simulation only with the first flashover phase is carried out to determine the second flashover phase. By repeating the previous procedure until no flashover occurs, a lightning surge simulation by the piecewise linear model is completed. Thus, a number of pre-calculations are necessary in the case of multiphase flashovers, while the nonlinear inductance model needs no pre-calculation and is easily applied to multiphase flashovers. The detail of the leader progression model is explained in Ref. [29] and that of the nonlinear inductance model in Ref. [30]. 2.4.2.5 Transmission Line Most transmission lines in Japan are of double-circuit vertical configuration with two GWs and thus are composed of eight conductors. It is recommended to represent the line by a frequency-dependent line model in a numerical simulation. But a distributed line model with a fixed propagation velocity, attenuation, and surge impedance, that is, fixed-parameter distributed line model explained in [23], is often used. 2.4.2.6 Substation 1. Gas-insulated bus and cable: A cable and a gas-insulated bus are represented either as three single-phase distributed lines with its coaxial mode surge impedance and propagation velocity or as a three-phase distributed line system. For a gas-insulated substation involves quite a number of gas-insulated buses/lines, the pipes are, in most cases, eliminated by assuming the voltage being zero. 2. CB, DS, transformer, and bushing: A CB and a DS are represented by lumped capacitances between the poles and to the earth. A transformer is also represented by a capacitance to the earth unless a transferred surge to the secondary circuit is needed to be calculated. A bushing is represented by a capacitance. Occasionally, it is represented by a distributed line. 3. Grounding mesh: A grounding mesh is in general not considered in a lightning surge simulation and is regarded as a zero-potential surface. When dealing with an incoming surge to a low-voltage control circuit, the transient voltage of the grounding mesh should not be assumed zero, and its representation becomes an important but difficult subject. 2.4.3 Lightning Surge Overvoltage 2.4.3.1 Model Circuit Figure 2.36 shows a representative model circuit for a lightning surge analysis [23]. Lightning strikes the top of tower No. 1 in the substation vicinity.

186


The lightning stroke is represented by a current source with a peak value of 200 kA and waveform 2/70 μs, which is the Japanese standard for a 1100 kV line, in the form of i ( t ) = I 0 K 0 exp ( − at ) − exp(bt) (2.10)

where I0 = 200 kA, K0 = 1.0224, a = 1.024 × 104 s−1 and b = 2.8188 × 106 s−1. Five towers are included in the model. The span distance of the transmission line between adjacent towers is 450 m, and that from tower No. 1 to the substation is 100 m. The end of the transmission line is terminated with the surge impedance matrix or approximately with matching resistances: Rp = 350 Ω for a PW and Rg = 560 Ω for a GW. The transmission line is of double-circuit vertical configuration with two GWs as shown in Figure 2.42. The total number of conductors is eight. The tower model is explained in the previous section; see Figure 2.39. It is divided at the cross-arm positions into four, and each section is modeled by a lossless distributed-parameter line neglecting the RL parallel circuit. Data for the elements are given in Table 2.5. The tower cross-arms are neglected. The tower footing resistance is taken as 10 Ω. Figure 2.43 is a model of a UHV substation for one phase. Cb and Cs in the figure are capacitances that GW 1 17 m 20 m 20 m

GW 2

39 m PWa

32 m

PWb

33 m

PWc

34 m

PWc΄ PWb΄ ACSR 810 mm2 × 8

PWa΄

109 m

50 cm

Earth resistivity 50 ohm-m FIGURE 2.42 An 1100 kV twin-circuit line.

ACSR 810 mm2 × 1

187


TABLE 2.5 Parameters of an 1100 kV Tower and a Structure Tower

Structure

Zt1 = 210 Ω Zt2 = 170 Ω c = 300 m/μs

Zt3 = 125 Ω Zt4 = 125 Ω c = 300 m/μs

200 m

Cb = 300 pF

RG = 70 Ω

Cs = 2000 pF

FIGURE 2.43 Single-phase expression of a substation model.

represent bushings and shunt reactors, respectively. A gas-insulated bus is represented by a lossless distributed-parameter line with a surge impedance of 70 Ω and a velocity of 270 m/μs. Figure 2.41a shows a lumped circuit model of an arc horn flashover proposed by Shindo and Suzuki [29]. Inductance and resistance values and closing times of switches are determined from a given voltage waveform across the arc horn gap based on a theory of a discharge mechanism. 2.4.3.2 Lightning Surge Overvoltage Figure 2.44 shows a typical result of the lightning surge. It should be clear from the figure that the overvoltage generated by a lightning surge is far greater than an insulation level estimated from a switching surge overvoltage. For example, the insulation level against the switching surge on the Japanese 1100 kV line is considered to be less than 1.7 pu. The lightning overvoltage on the PW observed in Figure 2.44a is 7.04 MV. For the nominal operating voltage of the line is 2 × 1100/ 3 = 898 †kV, the previous overvoltage is about 7.4 pu. If the lightning current is assumed to be 100 kA rather than 200 kA, the overvoltage is still 3.7 pu, which is much greater than the switching surge insulation level of 1.7 pu. It is observed in Figure 2.44b that the overvoltage at the substation entrance is about 4.6 MV, which is excessive to the insulation of the substation apparatuses such as a transformer. Thus, the apparatuses are protected by lightning arresters from the excessive overvoltage. If no flashover is assumed, the PW overvoltage at the substation entrance is reduced to about 0.4 MV = 400 kV, which is low enough from the viewpoint of substation insulation. However,

188


10

0 (a)

8

GW (tower top) PW

Voltage (MV)

Voltage (MV)

20

5

10

0

15

Time (µs)

4

(b)

5

15 Time (µs)

FIGURE 2.44 Lightning surge on the circuit of Figure 2.36. (a) No. 1 tower. (b) Substation entrance.

the voltage difference between the tower arm (or GWs) and the PW reaches about 7–8 MV. To insulate the PW from the tower and GWs against the overvoltage, the tower size might be too large to be economically feasible. Therefore, an analysis of the lightning surge is highly important to find an optimum and economically feasible insulation design of a power system. 2.4.3.3 Effect of Various Parameters 2.4.3.3.1 Frequency Dependence of Line Parameters It is well known that the frequency dependence of a transmission line due to an imperfectly conducting earth causes a significant effect on surges traveling through a long transmission line. The frequency-dependent effect on a lightning surge can be, in general, neglected because a line length is short. The effect will be investigated in this section. Calculated results using a frequency-dependent (distributed) line model and a frequency-independent model are shown in Figure 2.45 for the case of no flashover of an arc horn. In the latter model, line parameters are calculated at the dominant transient frequency given by

ft =

1 = 750 kHz 4 τ0

where τ0 = l/c, l is the distance from tower No. 1 to the substation = 100 m, and c is the velocity of light in free space. Table 2.6 shows the maximum voltages calculated by the frequencydependent Semlyen model and the frequency-independent distributedparameter line model of the electromagnetic transients program (EMTP). It is clear from the figure and the table that the results neglecting the frequencydependent effect show a minor difference from the results including the effect. Thus, it can be said that the frequency-dependent effect does not cause a significant effect on a lightning surge.

189


8000

Voltage [kV]

Phase a Phase b

4000

Phase c

0 0.00

2.50

5.00

7.50

10.00

12

Time [µs] (a)

Voltage [kV]

8000

Phase a Phase b Phase c

4000

0

1.00

2.50

5.00

7.50

10.00

Time [µs]

(b) FIGURE 2.45 Effect of frequency dependence on lightning surge for no flashover case. (a) Frequencydependent model. (b) Frequency-independent model.

The effect of various earth resistivities was also investigated, and it appears that there is no significant difference between calculated results with the earth resistivity of 50–1000 Ω-m. From the earlier observation, it is concluded that the frequency-dependent effect of a transmission line due to the imperfectly conducting earth is negligible in a lightning surge calculation. Thus, the lightning surge can be calculated with a reasonable accuracy by the frequency-independent distributed-parameter line model. By this approach, computation of the lightning surge becomes much more efficient, and also it becomes quite easy to explain a simulation result from the physical viewpoint.

190


TABLE 2.6 Maximum Voltages for the No Backflash Case FrequencyDependent Line Model

Constant Frequency Line Model

Line Model

[kV]

[μs]

[kV]

[μs]

Node 1 Tower top Upper arm Middle arm Lower arm Upper horn Middle horn Lower horn Upper phase Middle phase Lower phase

13217 12260 10348 8110 7180 6796 5692 5081 3682 2652

0.68 0.67 0.61 0.55 0.66 0.60 0.55 0.67 0.67 0.67

13026 12128 10258 8058 7301 6958 5901 4830 3392 2313

0.67 0.67 0.61 0.55 0.67 0.61 0.55 0.66 0.67 0.67

2.42 0.70 15.5 0.70 15.5 0.70

433 241 335 94 253 45

2.42 0.70 15.4 0.70 15.4 0.70

Node 2 (substation) Upper phase (First peak) Middle phase (First peak) Lower phase (First peak)

433 241 332 95 261 48

2.4.3.3.2 Tower Impedance and Footing Impedance It is well known that a tower surge impedance and a footing impedance affects lightning surges significantly. The tower surge impedance is a function of the height and the radius as explained in Section 1.7.2.2, but it ranges from 80 to 250 Ω as shown in Table 2.4 for real tower of various voltage classes. The tower footing impedance is always represented as a resistance as in Table 2.4 recommended by guide books for the insulation design and coordination of a transmission line and a substation [26]. It, however, is not a pure resistance but shows an inductive or a capacitive nature (see Figure 2.40) as investigated in many publications [10,15,23,31]. Figure 2.46 shows the effect of tower footing impedance on the tower top voltage in comparison with a measured result on a 500 kV transmission tower [23,31]. The inductive footing impedance shows a reasonable agreement with the measured result, but the resistive/capacitive impedance shows a far more oscillatory waveshapes. In fact, many measured results of the grounding electrode impedance show the inductive characteristic [32]. It should be noted that the effect of the tower surge impedance, that is, the effect of the

191


Voltage [V]

300 200 100 0 (2) Tower top voltage

(b)

300

300

200

200

200

100 0 0

1

2 3 Time [µs] (1) Resistive footing impedance

Voltage [V]

300

–100

(c)

(1) Applied current

Voltage [V]

Voltage [V]

(a)

100 0

–100

0

1

2 3 Time [µs] (2) Inductive footing impedance

0

1

2 3 Time [µs]

4

100 0

–100

0

3 2 Time [µs] (3) Capacitive footing impedance 1

FIGURE 2.46 Influence of a tower model on a tower top voltage. (a) Measured results. (b) Frequencydependent tower model with a resistive footing impedance. (c) Distributed line tower model with various footing impedances.

attenuation and the frequency dependence of the tower, is not significant if the footing impedance is inductive as observed in Figure 2.46, although a number of papers discussed modeling of the tower surge impedance. Figures 2.47 and 2.48 show the effect of the tower footing resistance as a function of a tower impedance on the tower top voltage. When the footing impedance is modeled as a resistance, the effect of the tower surge impedance is clear. It should be noted that the effect of the surge impedance is less noticeable when the wave front duration Tf of the lighting current is large and also when the footing resistance is high. Furthermore, the effect of the tower surge impedance on a surge voltage at a substation becomes less than that on the tower voltage as observed in Figure 2.49. 2.4.3.3.3 AC Source Voltage An ac source voltage is often neglected in a lightning surge simulation. It, however, has been found that the ac source voltage affects a flashover phase of an arc horn especially in the case of a rather small lightning current. Figure 2.50 is a measured result of arc horn flashover phases as a function of the ac source voltage on a 77 kV transmission line in Japan for a summer [33]. The measurements were carried out in two 77 kV substations by surge recorders installed in the substations. From the recorded voltages and currents, Figure 2.50 was obtained. The figure clearly shows that the arc horn flashover phase is quite dependent on the ac source voltage, that is, a flashover occurs at a phase of which the ac voltage is in the opposite polarity of a lightning current. Table 2.7 shows a simulation result of arc horn peak voltages (arc horn not operating) on (a) the 77 kV line and (b) a 500 kV line [34].

192


Vmax [kV/kA] 50 Tf = 1 µs

40

Tf = 2 µs

Tf = 10 µs

30 20 10 0

0

50

100

150

200

50

100

150

200

(a)

Z0 [Ω]

250

Vmax [kV/kA] 50 40 30 20 10 0 (b)

0

Z0 [Ω]

250

FIGURE 2.47 Effect of tower footing resistance Rf on the tower top voltage for a 66 kV line. (a) Rf = 10 Ω. (b) Rf = 50 Ω.

The simulation was carried out in a similar circuit to Figure 2.36, but another five towers were added instead of the gantry and the substation. The parameters are the same as those in Table 2.4 for a 77 kV system except the lightning current of 40 kA based on the field measurement [33]. The lower phase arc horn voltage is relatively smaller than the other phase arc horn voltages on the 500 kV (EHV) line compared with those on the 77 kV line. Thus, an arc horn flashover phase on an EHV line is rather independent from the ac source voltage, and the lower phase flashover is less probable than the other phase flashover. On the contrary, flashover probability is rather the same on each phase, and a flashover is dependent on the ac source voltage on a low-voltage line. Figure 2.51 shows simulation results of arc horn flashover phases by a simple distributed line “tower model,” that is, neglecting the RL circuit in

193


Vmax [kV/kA] 50 Tf = 1 µs Tf = 2 µs

40

Tf = 10 µs

30 20 10 0 (a)

0

50

100

150

200

50

100

150

200

Z0 [Ω]

250

Vmax [kV/kA] 50 40 30 20 10 0 (b)

0

Z0 [Ω]

250

FIGURE 2.48 Effect of tower footing resistance Rf on the tower top voltage for a 275 kV line. (a) Rf = 10 Ω. (b) Rf = 50 Ω.

Figure 2.39 with the parameters in Table 2.4, and by the recommended model illustrated in Figure 2.39. This figure should be compared with the field test result shown in Figure 2.50. It is clear that the recommended model cannot duplicate the field test result, while the simple distributed line model shows a good agreement with the field test result. The reason for the poor accuracy of the recommended model [28] is that the model was developed originally for a 500 kV line on which the lower phase flashover was less probable as explained in the previous section [23]. Thus, the recommended tower model tends to result in lower flashover probability of the lower phase arc horn. An R–L parallel circuit between two distributed lines in Figure 2.39 represents traveling wave attenuation and distortion along a tower. The R and L values were determined originally based on a field measurement (α in Equation 2.9), and thus those are correct only for the tower on which the measurement

194


Vmax [kV/kA] 50 Tf = 1 µs Tf = 2 µs

40

Tf = 10 µs

30 20 10 0

0

50

100

150

200

50

100

150

200

(a)

Z0 [Ω]

250

Vmax [kV/kA] 50 40 30 20 10 0 (b)

0

Z0 [Ω]

250

FIGURE 2.49 Effect of tower surge impedance Z0 on the substation overvoltage for a 66 kV line. (a) Rf = 10 Ω. (b) Rf = 50 Ω.

was carried out. Sometimes, the R–L circuit generates unreal high-frequency oscillation. This indicated a necessity of a further investigation of the R–L circuit, if the model is to be adopted.

2.5 Theoretical Analysis of Transients: Hand Calculations [2] In this section, examples of hand calculations of transients with a pocket calculator are explained by adopting (1) a traveling wave theory described in Section 1.6 and (2) Laplace transform by using a lumped-parameter circuit equivalent to the distributed line. Those two approaches are most powerful

195


80

Lower

60

Upper

Middle

Voltage (kV)

40 20 0 –20

Angle (deg) 0

30

60

90

120

150

180

210

240

270

300

330

360

–40 –60 –80 FIGURE 2.50 Measured results of arc horn flashover phases on a 77 kV transmission line.* Single-phase FO, × two-phase FO, ○ three-phase FO.

TABLE 2.7 Maximum Arc Horn Voltages and the Time of Appearance Transmission Voltage Upper Middle Lower

Maximum Voltage [kV]/ Time of Appearance [μs] 77 kV

500 kV

873.0/1.012 820.2/1.024 720.0/1.035

4732/1.025 4334/1.073 3423/1.122

to analyze a transient theoretically by hand and also correspond to the following representative simulation methods: 1. Time domain method: EMTP [35] 2. Frequency-domain method: frequency-domain transient analysis program (FTP) [36] 2.5.1 Switching Surge on an Overhead Line 2.5.1.1 Traveling Wave Theory Example 2.1 Obtain switching surge voltages at the sending end and the opencircuited receiving end of an untransposed horizontal line illustrated in Figure 2.52.

196


80

Lower

60

Middle

Upper

Voltage (kV)

40 20 0

Phase (deg) 0

30

60

90

120

150

180

210

240

270

300

330

360

–20 –40 –60

(a)

–80 80

Lower

60

Middle

Upper

Voltage (kV)

40 20 0 –20

Phase (deg) 0

30

60

90

120

150

180

210

240

270

300

330

360

–40 –60 (b)

–80

FIGURE 2.51 Simulation results of arc horn flashover phases corresponding to Figure 2.50. • Single-phase FO, × two-phase FO. (a) A simple distributed line model. (b) Recommended tower model.

Solution The surge impedance matrix [Zs] of the source circuit and [Zr] at the right of node r are

R [ Zs ] =  0  0

0 ∞ 0

0 ∞   0  ,† [ Zr ] =  0  0 ∞ 

0 ∞ 0

0  0 ∞ 

197


s

R

E = 1 kV

Node (Es)

(Ef )

(Eb)

(Er )

r

(I) (Z0), x, τ FIGURE 2.52 An untransposed horizontal line: x = 100 km.

Thus, the refraction coefficient matrices are

[λ s1 ] = 2[ Z0 ]([ Zs ] + [ Z0 ])

−1

 6/5

[λ1s ] = 2[U ] − [λ s1 ] =  −1/5

 −3/25

0 2 0

 4/5  =  1/5  3/25

0 0 0

0  0 0 

0 2   0  ,†††[λ1r ] = 0 0 2 

0 2 0

0  0 2 

The propagation time of the line τ0 =

x 10 5 = = 370.4 †µs †, †τ1 = 339.0 †µs†, τ2 = 333.3 †µs c0 270

Time delay τ12 = τ1 − τ2 = 5.7 †µs,† †τ02 = τ0 − τ2 = 37.1 †µs

Because of the symmetry of the line surge impedance [Z0], the voltage transformation matrices are 1

[ A] = 1 1

1 0 −1

1 2 −1  1   −2  , ††[ A] =   † 3  6  1  1

2 0 −2

2  −3  1 

400 E 1. = 0 : (Vs ) = [λ s1 ]   = 100 = (Es ) : phasor traveling waves 2 60

Modal traveling waves (es ) = †[ A]

−1

1120 1  1020 6 260

( Es ) = 

198


2. The modal traveling waves arrive at the receiving end at t = τn, and the modal waves at the receiving end are given considering the traveling time τn and attenuation kn by efn = kn · esn · (t − τn), where u(t − τ) is the unit step function with time delay τ, and k0 = 0.48, k1 = 0.90 and k2 = 0.96 for modes 0–2:

0.48 × 1120u ( t − τ0 ) 8.96u ( t − τ0 ) 89.6/τ0 1 (e f ) =   0.90 × 1020u ( t − τ1 ) = 153u ( t − τ1 ) = 153/τ1 6 41.6u ( t − τ2 ) 0.96 × 260u ( t − τ2 ) 41.6τ2

The receiving-end voltage 179.2/τ0 + 306/τ1 + 83.2/τ2 (Vr ) = [ A] ⋅ (Vr ) = †179.2/τ0 ††††††††††††−166.4/τ2 179.2/τ0 − 306/τ1 + 83.2/τ2 Reflected waves at the receiving end are (Er) = 2(Ef) − (Ef) = (Ef). Thus, (er) = (ef). 3. t = 2τn: 0.48 × 89.6u(t − 2τ0 ) 43.0/2τ0 ebn = kn ⋅ ern ⋅ u ( t − τn ) , †. i e. ( eb ) = 0.9 × 153u(t − 2τ1 ) = 137.7/2τ1 0.96 × 41.6u(t − 2τ2 ) 39.9/2τ2 43.0/2τ0 + 137.7/2τ0 + 39.9/2τ0 Thus, ( Eb ) = [ A] ( eb ) = 43.0/2τ†0†††††††††−79.8/2τ0 43.0/2τ0 − 137.7/2τ0 + 39.9/2τ0

The sending-end voltage is given in the following form: 400 + 51.6/2τ0 + 165.2/2τ1 + 47.9/2τ2 E λ E = 100 + 77.4/2τ0 − 27.5/2τ1 − 167.6/2τ2 + ( ) [ ] s 1 b  2 60 + 80.8/2τ0 − 291.9/2τ1 + 75.0/2τ2

(Vs ) = [λ s1 ]

The previous results are drawn as Figure 2.53. Example 2.2 In Example 2.1, consider transposition of the line with c1 = c2 = 298 m/μs. Solution Considering transposition, the surge impedance is given by

Z0 s =

Z0 aa + Z0 bb + Z0 cc Z + Z0 ac + Z0 bc = 300 Ω , Z0 m = 0 ab = 65 Ω 3 3

199


: x = 100 km c0 = 270, c1 = 295, c2 = 300(m/µs)

(Z0) =

300

75

45

75

300

75

45

75

300

572

(Ω)

600 (V)

568.4 389.2

389

400 Vrb Vrc

200

Untransposed

83.2

83.2 0

10

0

50

0

t΄ = t–τ2 (µs)

12.8 10

–11.0

–43.6 –166.4

(1) Phase a

–200 –222.8

(a) Vr

(2) Phase b and c 669.8 617 613.1

664.7

600 (V)

Vsa

447.9

400

100

400

200

135 t˝ = t–2τ2 (µs)

Vsc

100 –10

0

Vsb –124.3

(b) Vs

–95.1 –156.1

50 –75.3

–17.7 –43.9

0

–200

FIGURE 2.53 Analytical surge waveforms on a horizontal line. ― untransposed, - - - - transposed (V b=Vc). (a) Vr. (b) Vs.

200


4/5

[λ s1 ] = 13/75 13/75

0 0 0

0 0 , 0

[λ1s ] = 2[U ] − [λ s1 ]

Propagation time is τ0 = 370.4 μs, τ1 = τ2 = 335.6 μs, t01 = τ0 − τ1 = 34.8 μs. Transformation matrix is same as that in Example 2.1; see Section 1.4.4: 400 191 1. t = 0 : (Vs ) = 86.7 = ( Es ), ( es ) = 157 86.7 52.2 91.7/τ0 183.4/ττ0 + 389/τ1 2. t = τn : ( e f ) = 146/τ1 , (Vr ) = 2 ( e f ) ,(Vr ) = 183.4/τ0 − 194.5/τ1 48.5/τ1 183.4/τ0 − 194.5/τ1 44.0/2τ0 44/2τ0 + 181/2τ1 3. t = 2τn : ( eb ) = 136/2τ1 †,††††( Eb ) = 44/2τ0 − 90.5/2τ1 †,†† 45.1/2τ1 44/2τ0 − 90.5/2τ1 400 + 52.8/2τ0 + 217/2τ1 †(Vs ) = 86.7 + 80.4/2τ0 − 212/2τ1 86.7 + 80.4/2τ0 − 212/2τ1 The results are shown in Figure 2.53 by a dotted line considering difference of time delays in the untransposed line and the transposed line. Example 2.3 Calculate the receiving-end voltage (Vr) in an untransposed vertical twin-circuit line illustrated in Figure 2.42 for τ1 ≤ t < 3τ1 under the condition that the source voltage E = 1 pu is applied to phase c of the first circuit at t = 0 with c0 = 251.2, c1 = 298.4, c3 = 297.4, c2 = c4 = c5 = 299.2 [m/μs], x = 101.13 km, R = 403 Ω, and no attenuation. The surge impedance [Z0] and the voltage transformation matrix [A] are [Z1 ] [ Z0 ] = [Z ]  2

[Z2 ] †, [Z1 ]

[ A1 ] [ A] = [A ]  2

 311 Z = [ 1 ] 117  78

117 325 122

78   92   122  ,† [ Z2 ] =  83 67 325 

 0.7 [ A1 ]   †,†† [ A1 ] = 0.85 −[ A2 ]  0.1

1.0 0.36 −0.57

83 100 89

0.5   1.0  †† 0.28 

67   89  98 

201


0.47

[ A2 ] = 0.66  1.0

[ A]

−1

[ A3 ] = [ A3 ]

 1.267

[ A4 ] =  0.522

 −1.227

1.0 0.49 −0.167

0.67   1.0  0.26 

 0.762 [ A4 ]  †,† †[ A3 ] =  0.354 [ A5 ]  −0.775

−1.126 −0.301 1.511

−0.642 −0.0897 1.078

1.067   −0.112   −0.187  ,†† [ A5 ] =  −0.910  0.384 −0.726 

0.931   −0.312  † −0.679 

−0.0992 0.646 −0.762

−0.094 ?  −0.139  † 0.0166 

Solution When the phase c pole is closed at t = 0, the voltage Vsc = Z0ccE/ (R + Z0cc) = 0.466 pu. Thus, the current Isc/Z0cc = 1/728. The sending-end voltage (Vs) at t = 0 is obtained by using the surge impedance [Z0] as (Vs ) = †[ Z0 ] ( I s )

where the currents on the phases, except that on phase c, are zero for open circuited:

Vsa = 0.107,

†Vsb = 0.168,† Vsa′ = 0.092, Vsb′ = 0.122, Vsc′ = 0.135 †pu  

( Es ) = (Vs )†

for †t = 0

The modal traveling wave (es) = [A]−1 (Es) = [0.512†,†−0.130 †,†−0.232†,†. 0 379†,†−0.140 †,†−0.261]t

where t is for transposed matrix. The modal propagation time τ0 = x/co = 402.6[μs], τ1 = x/c1 = 338.9, τ2 = τ4 = τ5 = 338.0, τ3 = 340.0 Neglecting the attenuation of the line, the modal traveling waves at the receiving end is the same as those at the sending end.

( e (t − τ )) = ( e (t)) r

i

s

i = 0 to 5 for modal components

For receiving end is open circuited, the refraction coefficient matrix becomes a diagonal matrix of which all the diagonal elements equal to 2. Thus, the modal voltage Vr are

202


(Vr ) = 2(er ) Or Vr 0 = 1.024 ⋅ u ( t − τ0 ) = 1.024/τ0 ,††Vr 1 = −0.260/τ1 ,††Vr 2 = −0.464/τ2

Vr 3 = −0.758/τ2 ,††Vr 4 = −0.280/τ2 ,††Vr 5 = −0.522/τ2

Transforming the previous modal voltages into actual phasor voltages by using the transformation matrix [A], Vra = 0.7Vr 0 + 1.0Vr 1 + 0.5Vr 2 + 0.7Vr 3 + 1.0Vr 4 + 0.5Vr 5 = 0.717/τ0 − 0.260/τ1 − 0.232/τ2 + 0.531/τ3 − 0.280/τ2 − 0.261/τ2

= 0.717/τ0 − 0.260/τ1 − 0.773/τ2 + 0.531/τ3

In the same manner,

Vrb = 0.870/τ0 − 0.0936/τ1 − 1.087/τ2 + 0.644/τ3

Vrc = 0.102/τ0 + 0.148/τ1 − 0.116/τ2 + 0.758/τ3

Vra′ = 0.481/τ0 − 0.060/τ1 + 0.319/τ2 − 0.356/τ3

Vrb′ = 0.676/τ0 − 0.127/τ1 + 0.195/τ2 − 0.500/τ3

Vrc′ = 1.024/τ0 + 0.174/τ1 − 0.173/τ2 − 0.758/τ3

Figure 2.54 shows the previous analytical surge waveforms.

2.5.1.2 Lumped-Parameter Equivalent with Laplace Transform It is well known that a distributed-parameter line is approximated by a lumped-parameter circuit such as a PI equivalent and an L equivalent. For example, an open-circuited line in Figure 2.2 is approximated by Figure 2.55 with the L equivalent. Let us analyze switching surges in this circuit. 2.5.1.2.1 Single-Phase Line In an L equivalent of a single-phase line illustrated in Figure 2.55, current I, when switch “S” is closed at t = 0, is defined with Laplace operator s as

 sL + R + 1  E ( s) =  0  ⋅ I (s) sC   ∴ I ( s) =

E ( s) (2.11) (sL0 + R + 1/sC )

203


Vr(pu) 1.0

2.0

0 0.9

10

30

20

40

50

0.892

c

0.333 0.215

b a

64.6

Time (µs)

τ1 – τ2 = 0.9(µs) τ3 – τ2 = 2.0(µs) τ0 – τ2 = 64.6(µs)

–1.0

Vr (pu)

1.0 0.244 0

10

20

30

40

50

0.267 c΄ b΄ 0.184 a΄ Time (µs)

–1.0

FIGURE 2.54 Analytical surge waveforms at the receiving end on a vertical twin-circuit line.

LS

E

L I

R

Vr

C

FIGURE 2.55 An L equivalent of an open-circuited line.

where E(s) = E/s, L0 = LS + L LS is the source inductance (in most cases, transformer inductance) L, R, and C are the inductance, resistance, and capacitance of the line with length x

204


Then, the sending- and receiving-end voltages, VS(s) and Vr(s), are given by

Vs (s) = E ( s ) − sLS ⋅ I (s)†, †Vr ( s ) =

I (s) (2.12) sC

Solving the previous equations and transforming into time domain by using Laplace inverse transform, the following solutions are obtained:

( ω2t )

  (ω2L0 )     LS   vS ( t ) = E 1 −   exp(−αt) ⋅ sin ( ω2t − ϕ )  (2.13)   L0sinϕ      sin(ω2t + ϕ)  vr ( t ) = E 1 − exp(−αt)   sinϕ    i ( t ) = E ⋅ exp(−αt) ⋅ sin

where ϕ = tan −1 ( ω2/α ) ,††ω2 = ω12 − α 2 ,††α = R/2L0 ,††ω1 = 1/ L0C . 2.5.1.2.2 Single-Phase Line with a Residual (Line Charge) Voltage: Reclosing Surge When there is a residual voltage V0 (or charge Q 0) on the open-circuited line, a closing surge overvoltage becomes much higher than that with no residual voltage and is called “reclosing surge” as explained in Section 2.3.5(5); see Figure 2.24. The derivation of the reclosing surge voltage is similar to the case of the closing surge except that there exists an initial value of the line voltage, and the following results of vr(t) are obtained:

 sin ( ω2t + ϕ )  vr ( t ) = ( E − V0 ) 1 − exp ( −αt ) ⋅  + V0 (2.14) sinϕ  

In most real transmission lines, the following condition is satisfied:

R/2 Z0 = L/C : line surge impedance (see Chapter 1, Section 3.4.2.)

In such a case, Equations 2.13 and 2.14 are simplified by considering ω1 being far greater than α and nearly equal to ω2. For example, Equation 2.14 is rewritten as

vr ( t ) = E + (V0 − E) exp ( −αt ) ⋅ cos(ω1t) (2.15)


205

It is easily observed in the previous equation that

vr ( t ) ≒ E − 2Ecos ( ω1t ) ;†vrmax = 3E for v0 = −E    (2.16) vr ( t ) ≒ E − Ecos ( ω1t )†;†vrmax = 2E for v0 = 0 

The previous result is a proof of the reason why the reclosing surge overvoltage is much higher than the closing surge overvoltage. 2.5.1.2.3 Sinusoidal AC Voltage Source In the previous theoretical analysis, the source voltage was assumed to be a step function (or dc) voltage. This assumption is accurate enough as far as the observation time of a switching surge is less than one millisecond. If the observation time becomes greater than 5 ms, a sinusoidal ac voltage source should be taken into account. This makes Laplace transform quite complicated. Assume the following ac voltage source:

e0 ( t ) = E ⋅ sin ( ω0t + θ ) (2.17)

where ω 0 = 2πf0, f0 is the power frequency θ is the closing angle of a CB Considering the overall solution is given as superposition of the source voltage and a transient voltage at the instance of CB closing, the following result is obtained:

vr ( t ) ≒ Esin ( ω0t + θ ) + (V0 − Esinθ) ⋅ exp(−αt) ⋅ cos{ω1(t − τ)}    (2.18) vrmax ≒ Esin ( ω0t + θ ) − (V0 − Esinθ) 

where τ ≒ /c0 is the travel time of a line c0 is the light velocity ℓ is the line length 2.5.1.2.4 Three-Phase Line In the case of a three-phase line, Equation 2.11 becomes a matrix form, and we need to apply modal theory described in Section 1.4. Assuming that the line is transposed and all the phases are simultaneously closed at t = 0, the positive-sequence (aerial mode) voltage at the receiving end is given by

206


vr 1 = Ea − Esinθ ⋅ exp(−α1t) ⋅ cos {ω1 ( t − τ1 )} (2.19)

where α1 = R1/2L01 ,††ω1 = 1/ L01C1 ,††τ1 = /c1 ,††L01 = Ls + L1 R1, L1, C1 is the positive-sequence component of line resistance, inductance, and capacitance matrix τ1 is the propagation time of positive-sequence traveling wave c1 is the positive-sequence propagation velocity and  j 2π  Ea = Esin ( ω0t + θ ) ,† Eb = a 2Ea ,† Ec = aEa , †a = exp   (2.20)  3 

Transforming Equation 2.19 into a phase domain by (V) = [A](v), the following voltages for the three phases are obtained:     2 θ − π    Vrb = Eb − Esin  ⋅ − ⋅ − t t exp ( α )† cos ω τ ( ) { } 1 1 1  (2.21)   3     θ + 2π  Vrc = Ec − Esin  α t ) † ⋅ ω t − τ cos ⋅ exp ( − { 1 ( 1 )}  1   3   Vra = Ea − Esinθ ⋅ exp(−α1t)†⋅ cos {ω1 ( t − τ1 )}

An example of switching surges on a three-phase line calculated by the previous equation is shown in Figure 2.56. 2.5.2 Fault Surge A theoretical derivation of a fault surge voltage is, in principle, the same as that of a closing surge voltage, provided that the steady-state voltage at t = 0 is to be superposed to the transient voltage similarly to a reclosing surge. V [pu] 2

Sending Receiving

Phase a

1 0 –1 –2

2

10

20

30

Time [ms]

40

50

0

2

Phase b

1

20 10

40 30

Phase c

1 50

0

–1

–1

–2

–2

40

20 10

FIGURE 2.56 Switching surges due to simultaneous CB closing on a three-phase line.

30

50

207


Eb

x1

LS

x2 P

Eb

(a)

–Ea (b) LS

Lbb

Rbb

Z0 Cbb

LS

Laa –Ea

(c) FIGURE 2.57 A circuit for a fault surge analysis. (a) Original circuit. (b) An equivalent circuit. (c) An L equivalent of (b).

Let us consider a multiphase circuit illustrated in Figure 2.57a. Assume that phase a is short-circuited to ground (SLG) at node P at t = 0. Then the original circuit in (a) is represented by Figure 2.57b for a transient component. By applying an L-equivalent lumped-parameter circuit to the distributed line circuit, Figure 2.57c is obtained. In the circuit, [L], [R], and [C] are the inductance, resistance, and capacitance matrices of the original distributed line with length x. [Z0] is its surge impedance matrix. The circuit is similar to that of Figure 2.55 except that the source voltage −Ea is applied to node P. Thus, it is quite possible to obtain a transient voltage (Vp) at node P similarly to the switching surge on a three-phase line in the previous section.

208


For example, neglecting the source inductance Ls and the surge impedance [Z0] at the right of node P with the source voltage Ea = Ecos(ω 0t), the following phase b (sound phase) voltage is derived:  {cos ( ω0t ) − cos ( ω1t )} vb ( t ) = E C1 ( cosϕ − 1) (2C0 + C1 )  + ω0sinϕ ⋅

{cos ( ω t )/ω 0

− cos ( ω1t ) /ω1}   (2.22) {1 − (ω0 /ω1 )2  0

where ω1 = 1/τ,††ϕ = tan −1 ( ω1/α ) ≒ 2π/3,††α is the attenuation constant τ is the propagation time of a traveling wave for the line length x1 C0, C1 are zero- and positive-sequence capacitance of the line for x1 Considering (ω 0/ω1)2 is far smaller than 1 and φ = 2π/3, the following approximate solutions of the sound phases b and c voltages are obtained:

vb ( t ) = Eb ( t ) − kEa ( t ) − {Eb ( t = 0 ) − kEa ( t − 0 )} cos(ω1t)   (2.23) vc ( t ) = Ec ( t ) − kEa ( t ) − {Ec ( t = 0 ) − kEa ( t − 0 )} cos(ω1t) 

where = Lab/Laa is the ratio of the mutual and the self inductances. In Equations 2.22 and 2.23, a damping factor exp(−αt) is neglected. If this is included, the oscillating term in Equation 2.23 is dying out for t→∞. Then the equation becomes the steady-state voltage during the phase a-to-ground fault. For example, if k = 0.4, the maximum phase b voltage is given as

vbmax ≒ −1.48 pu at t = 1.0 ms ω1t = 2π/3

2.5.3 Lightning Surge 2.5.3.1 Tower Top Voltage A transient voltage at a tower top, to which lightning strikes, is easily calculated by applying a traveling wave theory. An example has been already explained in Section 1.7. See Problem 1.19. 2.5.3.2 Two-Phase Model [37] Lightning strikes a tower or a GW in most cases. Occasionally it strikes a PW when the lightning current is small. In field measurements of the lightning

209


strikes on an 1100 kV transmission system, it is found that lightning strikes a PW when the current is less than 35 kA [25]. When a tower is struck by the lightning, a large lightning current flows into the tower, and the tower voltage becomes much higher than the PW voltage, and “BFO” occurs. Then a part of the lightning current flows into the PW and travels toward a substation along the PW. This traveling wave produces a severer overvoltage in the substation. Therefore, the analysis of the BFO and the resultant overvoltages are the most significant subject from the viewpoint of the insulation design and coordination of the substation and the transmission line. For this, it is inherent to consider both the PW and a GW including the tower. Thus, the analysis involves the two-phase circuit composed of a PW and a GW as illustrated in Figure 2.58, where I0, R0 are the lightning current and channel impedance It, Vt, Zt, xt are the tower current, voltage, surge impedance, and height (length) Vf, Rf are the tower foot voltage and footing impedance Ig, Zg are the GW current and surge impedance Ia, Va, Za are the PW current, voltage, and surge impedance Zm is the mutual impedance between GW and PW I0 Il R0 Vt

GW

Ig΄

Ig΄ FO

Zm Ia

PW VP xt , Zt It Vf Rt

FIGURE 2.58 A two-phase model circuit for a BFO analysis.

Za

S/S

210


For an original 8 conductor (2 GWs and 6 PWs) system, there exists the following relation:

(Vg ) (V1 ) (V2 )

 [Z g ]  = [Zg 1 ]t [Zg 2 ]t

[Z g 1 ] [Z1 ] [Z12 ]t

[ Z g 2 ] ( I g )  [Z12 ]  ( I1 ) (2.24) [Z2 ]  ( I 2 )

where g for GWs, 1 for circuit-1 PWs, and 2 for circuit-2 PWs. Assuming a BFO occurs on phase a of the circuit-1, the following relation is derived from the previous equation:  Vg 1   Z11     Vg 2  =  Z12  Va   Z13   

Z12 Z11 Z23

Z13   I g 1    Z23   I g 2  (2.25) Z33   I a 

The previous equation is reduced to a 2 × 2 matrix for Vg1 = Vg2 = Vg and Ig1 = Ig2 = Ig:  Vg   Zg  =  Va   Zm

Zm   2I g   Zg  = Za   I a   Zm

Zm   Ig′    (2.26) Za   I a 

where Zg = (Z11 + Z12 )/2,††Zm = (Z13 + Z23 )/2,††Za = Z33 ,††I ′g = 2I g . The previous equation is an equation for a two-phase circuit model to analyze a lightning surge. 2.5.3.3 No Back Flashover The GW voltage Vg and the GW current Ig are easily obtained from the following relation, reminding that Zg is an equivalent impedance of two GWs as in Equation 2.26: I 0 = It + I + 2I ′g, † Vg = Zg I ′g = R0 I = Zt It (2.27)

Solving the equations,

I g′ =

R0 ⋅ Zt ⋅ I 0 †,† †Vg = Zin I 0 = Zg I g′ (R0 ⋅ Zg + 2R0 Zt + Zt Zg )

for t ≤ 2τ (2.28)

where 1/Zin = 1/R0 + 1/Zt + 2/Zg is the impedance seen from the current source I0 τ = xt/c is the traveling time along the tower

211


If there exists no channel impedance, that is, R0 = ∞, then,

I g′ =

Zt I 0 †, ††Vg = Zg I ′g (Zg + 2Zt )

for t ≤ 2τ (2.29)

At t = τ, a traveling wave eft generated at t = 0 arrives at the tower bottom and is reflected back to the tower top:

e ft = Vg

at t = 0 (2.30)

The refraction coefficient λb is given by

λb =

2R f (2.31) (R f + Zt )

Thus, the following voltage appears at the tower bottom, that is, at the tower footing resistance:

Vf = λ b ⋅ e ft =

2Vg R f (R f + Zt )

at

t = τ (2.32)

Then, the following reflected wave ebt at the bottom travels back to the tower top and arrives at the top at t = 2τ:

ebt = Vf − e ft (2.33)

The refraction coefficient at the top seen from the tower is

λt =

′ 2Zin (2.34) (Zt + Z′in )

where 1/Z′in = 1/R0 + 2/Zg . Thus, the tower top voltage Vg(t) is changed to the following value:

Vg ( t ) = Vg ( 0 ) + λ bt ⋅ ebt

for 2τ ≤ t < 4τ (2.35)

From Equations 2.28, 2.35, and 2.34, an analytical wave is drawn as shown in Figure 2.59. The waveform explains a numerical simulation result of the tower top voltage, when lightning strikes the tower top, published in many papers.

212


Vg(t)

0

t

2τ

lbt

FIGURE 2.59 Analytical waveform of the tower top voltage.

2.5.3.4 Case of a Back Flashover When a BFO occurs on phase a, the GW in Figure 2.58 is short-circuited to phase a. Then, the total impedance ZBF seen from the current source is given by

1 1 2 (2.36) = + ZBF Zin Za

where Zin is the total impedance in the case of no flashover; see Equations 2.28 and 2.29. The tower top voltage is obtained by

Vg = ZBF ⋅ I 0 (2.37)

2.5.3.5 Consideration of Substation When lightning hits a tower or a GW, traveling waves generated by the lightning currents propagate to a substation along the ground and PWs. When the waves arrive at the substation, those produce lightning surge overvoltages at the substation equipment. Let us analyze lightning surges at a substation. Assume that the lightning strikes the first tower next to the substation. The refraction coefficient matrix [λ s] from the line to the substation is given by

213


−1  Rg [λ s ] = 2 [ Zs ] ([ Zs ] + [Z0 ]) = 2  0 

 2   Rg (Rs + Za ) =    ∆   −Rs Zm

0   Rg + Zg Rs   Zm

Zm  Rs + Za 

−1

−Rg Zm  2Rs (Rg + Zg )

(2.38)

∆ = ( Rg + Zg ) ( Rs + Za ) − Zm2

where [Zs] is the substation impedance [Z0] is the line impedance defined in Equation 2.26 Rg is the surge impedance of substation gantry Rs is the phase a surge impedance of substation Then, the substation voltage (Vs) is calculated by  Vsg   Esg    = [ λ s ]   (2.39)  Vsa   Esa 

Esg and Esa are the traveling waves propagating from the tower and are given by Esg ( t ) = Etg u ( t − τ ) ,† †Esa ( t ) = Etau ( t − τ ) (2.40)

where Etg is the traveling wave on the GW at the tower Eta is the traveling wave on the PW at the tower u(t − τ) is the unit step function with time delay τ Example 2.4 Calculate the substation entrance voltage under the condition I0 = 100 kA step function, Zg = 332 Ω, Zm = 128 Ω, Za = 349 Ω, Zt = 210 Ω, Rg = 125 Ω, Rs = 70 Ω: gas-insulated bus, R0 = 400 Ω Solution

1. At the tower a. No BFO: Equation 2.28, Equation 2.26

 1 1 2 Zin =  + + R Z Z t g  0 Vg I ′g = = 22.67 kA , Zg

−1

  = 75.26 †Ω †,† †Vg = Zin ⋅ I 0 = 7.526 †MV ,  Va = Zm I ′g = 2.902 MV

214


b. BFO −1

 1 2  ZBF =  +  = 52.58 †Ω †,† Vg = †Va = ZBF I 0 = 5.258 †MV ,††  Zin Za  Ig′ = 15.84 †kA

2. At the substation: Equations 2.38 through 2.40 −0.18   0.6 Etg = Vg , Eta = Va , [λ s ] =  − 0 10 0.37  .  a. No BFO Vsg = 4.0 MV, Vsa = 0.32MV b. BFO Vsg = 2.2MV, Vsa = 1.42MV

It is observed from the refraction coefficient [λ s] at the substation that 37% of the incoming traveling wave Eta on the PW enters into the substation and determines overvoltages in the substation equipment. The remaining 63% reflects back to the transmission line. “−10%” of the traveling wave Etg on the GW is induced to the PW at the substation entrance and decreases the PW voltage. This effect of the PW voltage reduction has not been well realized but is very significant. If there is no negative induced voltage, the insulation of the substation equipment becomes much severe. This is the kind of an arrester gifted by God. Thanks to God! The earlier analysis is based on a step function current. In reality, the lightning current has a much slower rise time at the wave front, and thus the lightning overvoltage becomes mush lower. Such an analysis can be carried out considering the wave front but it is quite tedious as hand calculations. The earlier analytical results clearly show that the PW voltage in the BFO case is much higher than that in the case of no flashover. This is the reason why the insulation design/coordination of a substation is based on the result in the flashover case. The GW (=tower) voltage is certainly much higher in the no flashover case. In fact, because of this higher voltage, a BFO occurs in reality. Remind that no flashover in the analytical study here is just imaginary to understand the reality. It is noteworthy that a direct strike to a PW occasionally occurs. The magnitude of the lightning current is far smaller than that of the lightning strike to a tower or a GW. In a field test in Japanese 1100 kV line, it was found to be less than 30 kA [25]. Assume a direct strike with I0 = 30 kA to phase a, then the following results are obtained: −1

 2 1  I ′ = ′ ⋅ I 0 = 3.64 MV , Vg = Zm  0 + Zin  = 121.5 Ω, Va = Zin Z R  2 a 0  

Vsg = 0.497 MV , †Vsa = 1.155†MV

  =1.92†MV 

215


The results have indicated that the direct strike to a PW with 30 kA produces an overvoltage at a substation comparative to that in the BFO case with I0 = 100 kA. This fact should be carefully investigated, because this has not been considered in a standard of the insulation design/ coordination of a substation.

2.6 Frequency-Domain Method of Transient Simulations [2,6,36] 2.6.1 Introduction There exist powerful simulation tools such as the EMTP and SPICE. Those tools, however, involve a number of assumptions and the limit of the applications, which are not easy for a user to understand. Thus, it happens quite often that a simulation result is not correct due to the user’s misunderstanding of the application limits related to the assumptions of the tools. To avoid this kind of an incorrect simulation, the best way is to develop a user’s own simulation tool. For this purpose, an FD method of transient simulations is the easiest one, because the method is entirely based on the theory explained in Chapter 1 and Section 1.5, and requires only numerical transformation of a frequency response into a time response by adopting Fourier/Laplace inverse transform. The theory of a distributed-parameter circuit, a transient analysis in a lumped-parameter circuit, and the Fourier/Laplace transform is given as an undergraduate course in electrical engineering department in most universities all over the world. This section explains how to develop a computer code of the FD transient simulations. 2.6.2 Numerical Fourier/Laplace Transform A numerical calculation code of Fourier/Laplace transform is prepared in commercial softwares MATLAB®, MAPLE, etc., even in Excel. Therefore, it is easy to carry out an inverse transform provided that all the frequency responses are given by a user. Similarly, if a user can prepare a time response of a transient voltage, for example, as digital data of a measured result, then the user can easily obtain its frequency response by the software. However, it is better to understand the basic theory of the Fourier/Laplace transform. 2.6.2.1 Finite Fourier Transform Let us consider the following Fourier transform: ∞

 1  f ( t ) =   F ( ω) ⋅ exp ( jωt ) ⋅ dω (2.41)  2π  −∞

∫

216


A finite transform for [−Ω, Ω] is defined as Ω

 1  f1 ( t ) =   F ( ω) ⋅ exp ( jωt ) ⋅ dω (2.42)  2π  − Ω

∫

where F(ω) is the frequency responses for −Ω ≤ ω ≤ Ω f1(t) is the time response at time “t” evaluated by the previous equation, not accurate solution f(t) obtained by the original infinite integral f(t). Assume the following frequency function G(ω): G ( ω) = 1: |ω|≤ Ω (2.43)

0 : |ω|>Ω

By using G(ω), Equation 2.42 is rewritten as ∞

 1  f1 ( t ) =   G(ω) ⋅ F ( ω) ⋅ exp ( jωt ) ⋅ dω (2.44)  2π  −∞

∫

The time response g(t) of G(ω) is given by Ω

( Ωt ) (2.45)  1  g ( t ) =   1 ⋅ exp( jωt) dω = sin (πt)  2π  − Ω

∫

Expressing f1(t) by using time convolution (Duhamel’s integral) of f(t) and g(t) under the condition that f(t) = 0 for <0, f1 ( t ) =

 (Ωt) ′ ⋅ f (t − τ) ⋅ dt (2.46) sin  (πt)   −∞ t

∫

or replacing t − τ by ∞

 {Ω(u − t)}  f1 ( t ) = sin  ⋅ f ′(u) ⋅ du (2.47) {π(u − t)}   0 

∫

It is possible to estimate the error of the approximate time solution f1(t) defined by Equation 2.42 in comparison with the accurate one f(t) in Equation 2.41, which cannot be evaluated by numerical integration. When f(u) changes suddenly, noticeable oscillation called “Gibbs oscillation” appears in f1(t). This is the error caused by the finite Fourier transform. A countermeasure to this is to take an average for the time region [t − a, t + a] in the following form:

217


t+a

 1  f2 ( t ) =   f1 ( τ ) ⋅ dτ (2.48)  2a  t − a

∫

where a = π/Ω Substituting Equation 2.42 into the previous equation, and rearranging it, the following formula is obtained [38,39]: Ω

 1  f2 ( t ) =   F ( ω) ⋅ exp( jωt) ⋅ σ(ω) ⋅ dω (2.49)  2π  − Ω

∫

σ(ω) in the previous equation is called “sigma factor (weighting function)” and is expressed by

σ ( ω) =

sin ( ωa ) sin ( ωπ/Ω ) = (2.50) ωa (ωπ/Ω)

By adopting Equation 2.49 rather than Equation 2.42 as a finite Fourier transform, the Gibbs oscillation due to the finite interval in a numerical calculation of Fourier transform is reduced. 2.6.2.2 Shift of Integral Path: Laplace Transform In Fourier transform, integration is carried out along the imaginary axis “jω” as is clear from Equations 2.41 and 2.42. Thus, it happens that the integration hits a singular point along the jω axis. To avoid this, the integral path can be shifted to j(ω− jα) = α + jω rather than jω: Ω

 1  f3 ( t ) =   F ( ω − jα ) ⋅ exp j ( ω − jα ) t ⋅ dω  2π  − Ω

{

∫

Ω

}

(2.51)

 exp(αt)  =  F ( ω − jα ) ⋅ exp( jωt) ⋅ dω  2π  − Ω

∫

The earlier formulation is similar to Laplace transform and, thus, can be called “finite Laplace transform.” The following value has been known empirically optimum as the constant α in the earlier equation [40]:

where T is the observation time.

α=

2π (2.52) T

218


2.6.2.3 Numerical Laplace Transform: Discrete Laplace Transform Based on the explanation in the previous sections, the following form of Laplace transform is obtained [6,40]: Ω

 exp(αt)  f4 ( t ) =   F ( ω − jα ) ⋅ exp ( jωt ) ⋅ α(ω) ⋅ dω (2.53)  π 0

∫

For numerical calculations, the following equation is used:   exp(αt)  f 4 ( t ) = f ( k ⋅ t0 ) = Real    ⋅ ω0  π  ×

N −1



∑ F(nω − jα) ⋅ exp( jnω t) ⋅ 0

n =0

0

sin(nπ/N )   (2.54) (nπ/N ) 

where ω 0 = Ω/N, N is the total number of frequency (=time) samples t = k · t0, t0 = T/N, T is the observation time, k = 1, 2,…, N. The discretization of F(ω) by ω 0 in the numerical Laplace transform causes an error. The detail of the numerical discretization error is discussed in References [38,39]. 2.6.2.4 Odd-Number Sampling: Accuracy Improvement In principle, the numerical Fourier transform is a kind of numerical integration. Therefore, the accuracy of the numerical Fourier transform is greatly dependent on its integration method. In this section, various methods of integration, including odd-number sampling developed by Wedepohl, which gives quite a high accuracy but is not well known [6,40,41], are investigated, and a method with the highest accuracy is introduced into the discrete Laplace transform (DLT). The accuracy of various integration methods is investigated for the case of the conventional Fourier transform [41]. The conventional discrete Fourier transform (DFT) is given in the following form: Ω

Ω

1 1 f ( t ) =   F ( ω) exp ( jωt ) dω =   G(ω,†) t dω (2.55) π 0 π 0

∫

∫

Numerical evaluation of the previous equation is carried out by the following methods:

219


1. Method (a) 1 f (t ) =   π

N

∑ G(n∆ω,t)∆ω (2.56) n =1

This is shown in Figure 2.60a. If the sampling of G(ω, t) shown in Figure 2.60b is adopted, Equation 2.55 is evaluated by 2. Method (b)

1 f (t ) =   π

N

∑ G{(n − 1)∆ω,†}t ∆ω (2.57) n =1

Using Simpson’s method of integration shown in Figure 2.60c, Equation 2.55 is evaluated by 3. Method (c) 1 f (t ) =   π

N

∑ G ( n∆ω, t ) + 4G{( 2n − 1) ∆ω/2, t} n =1

+ G {( n − 1) ∆ω, t} ∆ω/6

(2.58)

The following method of odd-number sampling has been developed by Wedepohl [6,40,41]: 4. Method (d)

1 f (t ) =   π

N

∑ G{( 2n − 1) ∆ω,†t}2∆ω (2.59) n =1

This is shown in Figure 2.60d. This method can cover a frequency range twice as wide as methods (a) to (c) with the same number of frequency samples. If the maximum frequency is fixed to the same as methods (a) to (c), then in method (d) the number of samples halves. Calculated results of a unit step function by the previous integration methods are shown in Figure 2.61 [41]. In the calculations, the weighting function is included. Table 2.8 shows a comparison of accuracy between the various methods. From the results, it is obvious that the odd-number sampling method (d) is the most accurate and efficient.

220

G(ω)

G(ω)


n–1 (a)

n

n+1

N–1 N

ω

n–1 n (b)

N–1 N

∆ω

G(ω)

G(ω)

∆ω

n+1

n–1 (c)

n

N–1

N

∆ω

ω

2n–2 (d)

2n–1 2n 2n+1 2∆ω

Time response, p.u.

FIGURE 2.60 Various methods of integration. 1

1

0.5

0.5

0

0.01

Time response, p.u.

1

Time, ms

(a)

(c)

0.1

0

1

1

0.5

0.5

0

0.01

0.1 Time, ms

0.01

(b)

1

0 (d)

0.1

1

Time, ms

0.01

0.1 Time, ms

FIGURE 2.61 Calculated results of unit step response by various methods of integration.

1

ω

ω

221


TABLE 2.8 Comparison of Accuracy (Tmax = 5 ms)

Method a b c d

fmax

T1

T1/Tmax

T2

T2/Tmax

T3

T3/Tmax

Computation Time

N

kHz

ms

%

ms

%

ms

%

s

500 500 500 250

50 50 50 49.9

0.02 0.03 0.96 4.97

0.4 0.6 19.2 99.4

0.2 0.24 2.15 5.0

4 4.8 43.0 100

0.5 0.52 3.0 5.0

10 14 60 100

824 858 2393 215

T1 = time for accuracy higher than 99.8%. T2 = time for accuracy higher than 99%. T3 = time for accuracy higher than 95%. (except initial time) Computer HITAC 8350 (≃360/35).

5. Application of the odd-number sampling: modified Laplace transform (MLT) Integration method (d), explained previously, is introduced into Equation 2.54 in the following form: N   exp ( ατ )  − f ( t ) = Real   2 ω F {( 2n − 1) ω0 − jα} exp { j ( 2n − 1) ω0t} . 0  π    n =1

∑

sin {( 2n − 1) π/2N}/ ( 2n − 1) π/2N 

(2.60)

2.6.2.5 Application of FFT: Fast Laplace Transform (FLT) 2.6.2.5.1 Principle and Algorithms of the Fast Fourier Transform (FFT) [42] The complex DFT is defined in the following form: Fn =

N −1

∑ f ⋅ exp  k

k =0

− j 2πkn  ,† n = 0,†,† 1 … ,†N − 1 (2.61) N 

where Fn is the nth coefficient of the DFT and f k denotes the kth sample of the time series that consists of N samples. The inverse transform of the previous equation is

 1 fk =   N

N −1

∑F ⋅W n

n =1

kn

, k = 0, 1, … , N − 1†† (2.62)

222


where  j 2π  W = exp   (2.63)  N 

f k can be a complex number, but usually it is real in the field of electrical engineering Fn is almost always complex The principle and the algorithms of the FFT will be explained for the inverse DFT of Equation 2.62. Let us consider the case of N = 8 = 23, then W becomes  j 2π  π π W = exp   = cos   = jsin   (2.64) 4 4  8 

Using W8 = 1 and Wr = Ws, where s = r mod 8, and also W2 = j, W4 = −1, W5 = −W, etc., Equation 2.62 is rewritten in the following form:

 f0  1  f  1  1   f2  1     f3  = 1  f 4  1     f5  1  f  1  6   f7  1

1 1  1  1 = 1  1 1  1

1 W W2 W3 W4 W5 W6 W7

1 W2 W4 W6 W8 W 10 W 12 W 14

1 W3 W6 W9 W 12 W 15 W 18 W 21

1 W4 W8 W 12 W 16 W 20 W 24 W 28

1 W

1

1 W3 − W −1 −W 3

1 −1 1 −1 1 −1 1 −1

3

W −1 −W − −W 3

−1 − 1 −1 −

−W

1 W5 W 10 W 15 W 20 W 25 W 30 W 35 1 −W −W 3 −1 W − W3

1 W6 W 12 W 18 W 24 W 30 W 36 W 42 1 − −1 1 − −1

1   F0  W 7   F1  W 14   F2     W 21   F3  ⋅ W 28   F4     W 35   F5  W 42   F6     W 49   F7  1   F0  − W3   F1  −   F2     − W   F3  (2.65) ⋅ −1   F4     W3   F5   F    6 W   F7 

Changing columns rearranging for (F0, F2, F4, F6) and (F1, F3, F5, F7), the previous matrices can be rewritten as

223


1 1  1  1  fk  =  1  1 1  1

1 j −1 −j 1 j −1 −j

1 −1 1 −1 1 −1 1 −1

1  1  W −j  −1  F0   j   j   F2  W 3 ⋅ + 1   F4   −1     − j   F6   −W − j −1   j   −W 3

1 jW −j

1 −W

− jW 3 −1

j −W 3 −1

− jW j jW 3

W −j W3

1  − jW  − j   F1   jW 3   F3  ⋅ −1   F5     jW   F7   j  3 − jW 

(2.66) The following form of matrix is included in the previous matrices: 1 1 1 1  1 j −1 − j  [T ] = 1 −1 1 −1   1 − j 1 j 

Defining matrix [En] by

 F0   E0    E   1  = [T ] ⋅ F2  ,  F4   E2       F6  E3 

E4   F1  E     5  = [T ] ⋅  F3   E6   F5       E7   F7 

then matrix [f k] is expressed in the following form:

E0 + E4  E + WE  5  1  E2 + jE6    E3 + W 3E7   [ fk ] = E0 − E4    E1 − WE5  E − jE  6  2  3 E3 − W E7 

224


In the matrix [T], interchanging the second and third columns, then the following form of matrix is found: 1

[S] = 1 

1 −1

Using this orthogonal matrix, let us put  D0   F0   D2   F1   D4   F2   D6   F3   D  = [S]  F  ,  D  = [S]  F  ,  D  = [S]  F  ,  D  = [S]  F   1  4  3  5  5  6  7  7 

(2.67) The [En] matrix is expressed in the following form:

E0   D0 + D4  E   D + jD  5  1 =  1  , E2   D0 − D4      E3   D1 − jD5 

E4   D2 + D6  E   D + jD  7  5 =  3  E6   D2 − D6      E7   D3 − jD7 

Therefore, [f k] can be obtained by the following procedure: {Fn } → {Dn } → {En } → { fk } (2.68)

Expressing the subscripts of each matrix by the binary code, Fn = F(rqp)

then Dn is given from Equation 2.67 in the following form:

D ( qpo ) = F ( oqp ) + F ( 1qp )    q, p = 0 , 1 D ( qp1) = F ( oqp ) − F ( 1qp ) 

Secondly, En is given by

E ( por ) = D ( opr ) + D ( 1pr )    p = 0, 1, r = 0 E ( p1r ) = D ( opr ) − D ( 1pr ) 

E ( por ) = D ( opr ) + W 2D ( 1pr )    p = 0, 1, r = 1 E ( p1r ) = D ( opr ) − W 2D ( 1pr ) 

225


Lastly, f k is obtained from the previous equations:

f ( oor ) = E ( ooo ) + ( −1) W 0E(ooo)

f ( o1r ) = E ( o1o ) + ( −1) W 1E ( o11)

f ( 1or ) = E ( 1oo ) + ( −1) W 2E(1o1)

f ( 11r ) = E ( 11o ) + ( −1) W 03E(111)

r

r

r

r

The inverse Fourier transform of Equation 2.62 can be calculated in the previous manner. The more general description is described in many publications. 2.6.2.5.2 Computation Time [6] The total calculation units by repeated application of the FFT becomes TF = N ( p1 + p2 + + pn )

In the case of

p1 = p2 = = pn = m

n being given by

n = log m N

Then the total number of calculation units becomes

TF = N ⋅ m ⋅ log m N (2.69)

The ratio between TF and Tc is

TF ( N †m †log m N ) m †log m N  m   log 2 N  (2.70) = = =   Tc N2 N  log 2m   N 

Table 2.9 shows value of m/log2m. This becomes a minimum at m = 3. Table 2.10 shows Tc/TF with m = 2. From the table, it is obvious that the FFT is highly efficient compared with the conventional Fourier transform.

226


Table 2.9 m/log2m m

m/log2m

2 3 4 5 6 7 8 9 10

2.00 1.88 2.00 2.15 2.31 2.49 2.67 2.82 3.01

TABLE 2.10 Theoretical Comparison of Computation Time (m = 2) N

Tc/TF

32 64 128 256 512 1024 2048 4096 8192

3.2 5.2 9.1 16 28 51 92 170 315

2.6.2.5.3 Application of FFT to MLT [6] The application of the FFT to the MLT in Equation 2.60 may be rather difficult compared with the application to the DFT because the integer term (2k − 1) in the exponent of the previous equation is an odd number. Therefore, we need to modify this equation so that the integer term takes on sequential values 1, 2, 3, 4,.… Thus, the following form is obtained and then the FFT can be applied:

  exp(αkt0 )   jkπ   fk = Real    ⋅ exp   π   N  

n=1





∑ F ⋅ exp  j  N  (n − 1)k  n

N

2π

227


where fk = f ( k ⋅ t0 ) = f ( t ),† t0 =

π T †,† †ω0 = (2.71) N T

Fn = F{( 2n − 1) ω0 − jα} ⋅ sin

{( 2n − 1) π/2N}

( 2n − 1) π/2N ⋅ 2ω0

Figure 2.62 gives a complete program for computing the DLT in Equation 2.54 by the FFT. Figure 2.63 shows a calculation example in comparison with the exact solution. DIMENSION FV(N), FS(N) N2 = N/2 AN = N JO = N2 KEI = 1 CJ = CMPLX(0., 1.) PAI = 3.14159 CJP2 = CJ*2.*PAI/AN DW = 2.*PAI/TO DO 20 I = 1, N AI = FLOAT(1*2 − 1)2. WA = PAI*AI/AN SIGMA = SIN(WA)/WA 20 FV(1) = FV(I)*SIGMA*DW DO 50 K = 1, M KE2 = KE1*2 IL = 0 DO 41 J = 1, N, KE2 DO 40 I = 1, KE1 IL = IL+1 IR = IL+N2 JW = (I − 1)*JO + 1 AJW = JW−1 FV1=FV(IL) FV2=FV(IR)*CEXP(CJP2*AJW) JL = J+I−1 JR = L+KE1 FS(JL) = FV1 + FV2 40 FS(JR) = FV1 + FV2 41 CONTINUE KE1 = KE2 JO = JO/2 DO 45 I = 1, N 45 FV(I) = FS(I) 50 CONTINUE DO 60 I = 1, N AI = I−1 T=TO*AI/AN/2. EA=EXP(ALFA*T) VT=REAL(FV/1)*EA/PAI 60 WRITE T, VT FIGURE 2.62 Complete Program for Computing the DLT of Equation 2.54 by the FFT Method—Subroutine FLT.

228


Time 0.000E 00 0.500E-04 0.100E-03 0.150E-03 0.200E-03 0.250E-03 0.300E-03 0.350E-03 0.400E-03 0.450E-03 0.500E-03 0.550E-03 0.600E-03 0.650E-03 0.700E-03 0.750E-03 0.800E-03 0.850E-03 0.900E-03 0.950E-03 0.100E-02

EACT SOL. 0.196078E-01 0.105274E 00 0.159656E 00 0.174684E 00 0.152352E 00 0.102884E 00 0.412708E-01 −0.167940E-01 −0.585847E-01 −0.768302E-01 −0.706180E-01 −0.447274E-01 −0.775823E-02 0.303518E-01 0.607726E-01 0.776394E-01 0.789786E-01 0.666222E-01 0.452667E-01 0.210105E-01 −0.227285E-03

DLT SOL. 0.119409E-01 0.105109E 00 0.159499E 00 0.174562E 00 0.152281E 00 0.102846E 00 0.412929E-01 −0.167490E-01 −0.585381E-01 −0.768006E-01 −0.706177E-01 −0.447603E-01 −0.782012E-02 0.3028710E-01 0.606862E-01 0.775597E-01 0.789150E-01 0.665788E-01 0.452425E-01 0.209998E-01 −0.232818E-03

FLT SOL. 0.119409E-01 0.105109E 00 0.159499E 00 0.174562E 00 0.152281E 00 0.102846E 00 0.412929E-01 −0.167490E-01 −0.585381E-01 −0.768006E-01 −0.706177E-01 −0.447603E-01 −0.782012E-02 0.302710E-01 0.606862E-01 0.775597E-01 0.789150E-01 0.665788E-01 0.452425E-01 0.209998E-01 −0.232818E-03

Number of frequency samples = 128. Maximum observation time = 1.28 ms. Computation time : 1.0 by the DLT and 0.11 by the FLT. FIGURE 2.63 Calculated results by DLT and FLT in comparison with exact solution.

2.6.3 Transient Simulation A computer program for a transient simulation by an FD method can be easily produced by a university student. Figure 2.64 illustrates its flowchart. The program is composed of the three procedures [2,6]. 2.6.3.1 Definition of Variables N is the number of frequency samples for F(ω) = number of time samples f(t). DF is the frequency step; FMAX = N*DF is the maximum frequency. DT = 1/FMAX is the time step. TMAX = N*DF = 1/DF is the observation time corresponding to the sampling theorem DW = 2π*DF, W = n*DW. ALFA = 2π/TMAX, CJ = exp(jπ/2) is the symbol of imaginary variable S = CJ*W + ALFA.

229


1. Definition of variables N, DF, FMAX, DW, W, DT, TMAX ALFA, j, s = jW + ALFA 2. Subroutine FRESP for s = jW + α , W = n · DW, n = 1 to N Z(s) = E(s) = E/S I(s) = E(s)/Z(s) V(s) = E(s) – R1 · I (s)

2. Subroutine FLT Shown in Figure 2.62 i(t) = FLT [I(s)] ν(t) = FLT [V(s)]

FIGURE 2.64 Flowchart at FD method.

2.6.3.2 Subroutine to Prepare F(ω) F(ω) containing N samples of frequency responses is to be prepared by a user. For example, let us obtain transient (time) responses of v(t) in an RLC parallel circuit illustrated in Figure 2.65. Voltage V(s) in s-domain is given by

Y1 =

1 1 + + sC R sL

Z ( s ) = R1 + E ( s) = I ( s) =

1 + R2 Y1 E s

E(s) Z(s)

V ( s ) = E ( s ) − R1 ⋅ I (s)

The previous frequency responses are produced for n = 1 to N, where

s = jω + α ,† ω = n ⋅ ω0

230


R R1

L C

E

v(t)

R2

FIGURE 2.65 Switching of R–L–C parallel circuit. R1 = 500 Ω R = 1000 Ω L = 10 mH C = 1 μF. R 2 = 10 Ω E = Step function 1.

2.6.3.3 Subroutine FLT The frequency responses I(s) and V(s) are sent to subroutine FLT given in Figure 2.62. Then the FLT carries out the inverse Laplace transform and the time solutions are obtained. Figure 2.63 shows an example of a calculated result v(t) in comparison with the accurate solution. It is observed the accuracy of the FLT is quite high. 2.6.3.4 Remarks of the Frequency-Domain Method The advantage of the FD method is that any frequency-dependent effect is easily handled as it is based on the frequency response of a transient to be solved. Thus, the frequency-dependent effect of a transmission line/cable explained in Chapter 1 is very easily included in a simulation. On the contrary, a sudden change in a time domain such as switching causes a difficulty because the change involves an initial condition problem that requires repeated time/frequency transforms. A nonlinear element, for example, an arrester, requires a number of the time/frequency transforms. Thus, the FD method is often used to check the accuracy of the time domain method such as the EMTP on the frequency-dependent effect.

References 1. Ametani, A. 1987. Power System Transient Analysis. Kyoto, Japan: Doshisha University. 2. Ametani, A. 1990. Distributed—Parameter Circuit Theory. Tokyo, Japan: Corona Pub. Co.


231

3. Japanese Standard. 1994. High voltage testing (JEC-0102-1994). IEE Japan. 4. Ametani, A., T. Ono, Y. Honaga, and Y. Ouchi. 1974. Surge propagation on Japanese 500 kV untransposed transmission line. Proc. IEE. 121 (2):136–138. 5. Ametani, A. and T. Ono. 1975. Surge propagation characteristics on an untransposed horizontal line. IEE Japan. B-95 (12):591–598. 6. Ametani, A. 1972. The application of fast Fourier transform to electrical transient phenomena. Int. J. Elect. Eng. Educ. 10:277–287. 7. Ametani, A., A. Tanaka et al. 1981. Wave propagation characteristics on an untransposed vertical twin-circuit line. IEE Jpn. B-101 (11):675–682. 8. Ametani, A., E. Ohsaki, and Y. Honaga. 1983. Surge characteristics on an untransposed vertical line. IEE Jpn. B-103 (2):117–124. 9. Jordan, C. A. 1934. Lightning computation for transmission line with overhead ground wires. G. E. Rev. 37 (4):180–186. 10. Bewley, L. V. 1951. Traveling Waves on Transmission Systems. New York: Wiley. 11. Wagner, C. F. 1956. A new approach to calculation of lightning performance of transmission lines. AIEE Trans. 75:1233–1256. 12. Lundholm, R., R. B. Finn, and W. S. Price. 1957. Calculation of transmission line lightning voltages by field concepts. AIEE Trans. 76:1271–1283. 13. Ozawa, J. et al. 1985. Lightning surge analysis in a multiconductor system for substation insulation design. IEEE Trans. Power App. Syst. 104:2244. 14. Kawamura, T., A. Ametani et al. 1987. A new approach of a lightning surge analysis in a power system. IEE Japan. Technical Report No. 244. 15. Kawamura, T., A. Ametani et al. 1989. Various parameters and the effects on lightning surges in a substation. IEE Japan. Technical Report No. 301. 16. CIGRE SC33-WG01. 1991. Guide to procedures for estimating lightning performance of transmission lines. CIGRE. Tech. Brochure. 17. Kawamura, T., A. Ametani et al. 1992. A new method for estimating lightning surge in substations, IEE Japan WG report. Technical Report No. 446. 18. IEEE WG. 1993. Estimating lightning performance of transmission lines, II–update to analytical models. IEEE Trans. Power Deliv. 8:1254. 19. Kawamura, T., A. Ametani et al. 1995. An estimating method of a lightning surge for statistical insulation design of substations. IEE Japan WG Report. Technical Report No. 566. 20. IEEE Guide for Improving the Lightning Performance of Transmission Lines. 1997. 21. Ametani, A., T. Kawamura et al. 2002. Power system transients and EMTP analyses. IEE Japan WG Report. Technical Report No. 872. 22. CRIPEI WG. 2003. Guide to transmission line protection against lightning. Report T72. 23. Ametani, A. and T. Kawamura. 2005. A method of a lightning surge analysis recommended in Japan using EMTP. IEEE Trans. Power Deliv. 20 (2):867–875. 24. Anderson, R. B. and A. J. Eriksson. 1980. Lightning parameters for engineering application. Electra 69:65. 25. Takami, J. and S. Okabe. 2007. Characteristic of direct lightning strokes to phase conductors of UHV transmission line. IEEE Trans. Power Deliv. 22 (1):537–546. 26. Study Committee of Lightning Protection Design. 1976. Lightning protection design guide-book for power stations and substations. Central Research of Electric Power Industry (CRIEPI) Report 175034. 27. Elect. Reser. Assoc. 1988. Rationalization of insulation design. ERA Report. 44 (3).

232


28. Ishii, M. et al. 1991. Multistory transmission tower model for lightning surge analysis. IEEE Trans. Power Deliv. 6 (3):1372. 29. Shindo, T. and T. Suzuki. 1985. A new calculation method of breakdown voltage-time characteristics of long air gaps. IEEE Trans. Power App. Syst. 104:1556. 30. Nagaoka, N. 1991. An archorn flashover model by means of a nonlinear inductance. Trans. IEE Japan B-111 (5):529. 31. Nagaoka, N. 1991. Development of frequency-dependent tower model. Trans. IEE Japan B-111:51. 32. Ametani, A., H. Morii, and T. Kubo. 2011. Derivation of theoretical formulas and an investigation on measured results of grounding electrode transient responses. IEE Jpn. B-131 (2):205–214. 33. Ueda, T., M. Yoda, and I. Miyachi. 1996. Characteristic of lightning surges observed at 77 kV substations. Trans. IEE Jpn. B-116 (11):1422. 34. Ametani, A. et al. 2002. Investigation of flashover phases in a lightning surge by new archorn and tower models. In Proc. IEEE PES T&D Conf. 2002, Yokohama, Japan, pp. 1241–1426. 35. Meyer, W. S. 1973. EMTP Rule Book, 1st edn. Portland, OR: B.P.A. 36. Nagaoka, N. and A. Ametani. 1988. A development of a generalized frequencydomain transient program—FTP. IEEE Trans. Power Deliv. 3 (4):1986–2004. 37. Nagaoka, N. and A. Ametani. 1986. Lightning surge analysis by means of a twophase circuit model. TIEE Jpn. B-106 (5):403–410. 38. Day, S. J., N. Mullineux, and J. R. Reed. 1965. Developments in obtaining transient response using Fourier integrals. Pt. I: Gibbs phenomena and Fourier integrals. Int. J. Elect. Eng. Educ. 3:501–506. 39. Day, S. J., N. Mullineux, and J. R. Reed. 1966. Developments in obtaining transient response using Fourier integrals. Pt. II: Use of the modified Fourier transform. Int. J. Elect. Eng. Educ. 4:31–40. 40. Wedepohl, L. M. and S. E. T. Mohamed. 1969. Multi-conductor transmission lines: Theory of natural modes and Fourier integral applied to transient analysis. Proc. IEE. 116:1553–1563. 41. Ametani, A. and K. Imanishi. 1979. Development of a exponential Fourier transform and its application to electrical transients. Proc. IEE. 126 (1):51–59. 42. Cooley, J. 1967. What is the fast Fourier transform? Proc. IEEE. 55:1664–1674.

3 Transients on Cable Systems

3.1 Introduction This chapter focuses on transient phenomena peculiar to the cable. Transients on cable systems are characterized by the large charging capacity of cables and the existence of the metallic sheath around the phase conductor. Temporary overvoltages (TOVs), such as the overvoltage caused by the system islanding and the resonance overvoltage, observed on the cable system contain low frequency components due to the large charging capacity. Because of the low frequency, namely low damping, these TOVs can be sustained for an extended duration, posing challenges to insulation performance of related equipment. Examples of such studies are introduced in Chapter 4. Other issues, such as the zero-missing phenomenon, the leading current interruption, and the cable discharge, also stem from the large charging capacity of the cable. Their effects on the cable system design are discussed in Section 3.5. The discussion includes countermeasures to the problems and considerations in the equipment selection. Sheath bonding and grounding is another important issue in the cable system design. The sheath overvoltage requires careful studies not only to avoid failures of sheath voltage limiters (SVLs) and sheath interrupts but also to ensure the safety of maintenance crews. Sections 3.2 and 3.3 cover all major aspects of the sheath bonding and grounding, providing wide variety of information from fundamentals to applications. In addition, impedance calculations, wave propagation characteristics, and transient voltage behaviors discussed in Sections 3.2 and 3.3 form a base for transient phenomena discussed in later sections.

233

234


3.2 Impedance and Admittance of Cable Systems 3.2.1 Single-Phase Cable 3.2.1.1 Cable Structure The most significant difference of a cable from an overhead line is that the cable is composed of two conductors, in general, for one phase. Thus, a threephase cable consists of six conductors, while a three-phase overhead line consists of three conductors. Figure 3.1 illustrates the cross section of a typical coaxial cable. The core conductor carries a current in a way a phase conductor of an overhead line does. The metallic sheath is grounded at both ends of the cable so as to shield the core current. Thus, the metallic sheath is often called as “shield.” 3.2.1.2 Impedance and Admittance The impedance and admittance of a single-phase cable are given in a matrix form because it contains two conductors:  Zcc

[ Zi ] =  Zcs 

Zcs   Yc ,† [Yi ] =   Zss   −Yc

−Yc  for phase "i " (3.1) Ys 

Each element of the impedance matrix is composed of the cable internal impedance and the cable outer media (earth-return) impedance as explained in Chapter 1. In the overhead line case, the conductor internal impedance is composed only of one impedance that is the outer surface impedance of a conductor. The cable internal impedance consists of the following six components [1]:

R3

R4

R2 Core Insulation

Metallic sheath Outer cover

FIGURE 3.1 Cross section of a typical coaxial cable.

R5

235

Transients on Cable Systems

1. Core outer surface impedance (same as the internal impedance of an overhead line) 2. Core to sheath insulator impedance 3. Sheath inner surface impedance 4. Mutual impedance between the sheath inner and outer surfaces 5. Sheath outer surface impedance 6. Sheath outer insulator (outer cover shown in Figure 3.1) impedance

Now, it should be clear that the cable impedance is far more complicated than that of an overhead line [1,2]. The admittance matrix is expressed in the following form by using the potential coefficient matrix:

[ Y ] = jω [ C ] = jω [ P ]

−1

where  Pc

[ P ] =  Ps 

Ps   P12 + P23 = Ps   P23

†P12 =

1 R ln 3 , 2πεi1 R2

P23 =

R 1 ln 5 2πεi 2 R4

for one phase (3.2)

P23  , P23 

3.2.2 Sheath Bonding Before we discuss the impedance and admittance of a three-phase cable, it is necessary to discuss the sheath bonding. Underground cables, which are longer than 2 km normally adopt cross-bonding to reduce sheath currents and to suppress sheath voltages at the same time [3]. Figure 3.2 shows an example cross-bonding diagram of a cable. In the figure, one of three sheath circuits is highlighted with a dotted line. Starting from the left termination, the sheath circuit goes with the phase a conductor in the first minor section, the phase b conductor in the second minor section, and the phase c conductor in the third minor section. Theoretically, the vector sum of the induced voltage of the sheath circuit in these three minor sections becomes zero when three-phase currents in the phase conductors are balanced and three minor sections have the same length. As a result, the cross-bonding can reduce sheath currents and suppress sheath voltages at the same time. When lengths of three minor sections are different, it leads to unbalance of the induced voltages, which causes sheath currents. However, when there are more than a couple of major sections, it is a common practice to design

236


Termination

SSJ

SSJ

EJ

EJ

SSJ

SSJ

Phase a

Termination

Phase b Phase c

Core

Sheath sectionalizing joint (SSJ)

Sheath

Termination

Earthing joint (EJ)

FIGURE 3.2 Example of cross-bonding diagram of a cable.

cross-bonding, considering the best balance of the induced voltage. This results in so-called homogeneous nature of the cable impedance [4,5]. For submarine cables, it is more common to adopt solid bonding due to the difficulty in constructing joints offshore as shown in Figure 3.3. Hence, submarine cables have higher sheath currents compared with underground cables that are normally cross-bonded. In order to reduce the loss caused by higher sheath currents, the sheath conductors of submarine cables often have a lower resistance, that is, a larger cross-section. The single-point bonding has an advantage in terms of reducing the sheath currents. The sheath current loss can be reduced virtually to zero by applying the single-point bonding as shown in Figure 3.4. However, it can only be applied to short cables or short cable sections due to a limitation in the acceptable sheath voltage. In order to suppress the sheath voltage exceeding the limitation, SVLs are installed at the unearthed end of the sheath circuit. The installation of SVLs is further discussed in Section 3.3. Additionally, it is highly recommended to install the ECC in order to suppress the sheath overvoltage. Figure 3.5 shows an example where single-point bonding is employed in a long cable. As discussed earlier, cross-bonding is adopted for a long cable. In the figure, the first three minor sections from the left termination compose one major section of cross-bonding. Since the number of minor sections is four, which is not a multiple of three, the fourth minor section Termination


FIGURE 3.3 Solid-bonding diagram of a cable.

Termination Core Sheath Termination

237


Termination

Termination


SVL

ECC

Core

Termination

Sheath

Sheath voltage limiter (SVL)

Earth continuity cable (ECC) FIGURE 3.4 Single-point-bonding diagram of a cable.

Termination

SSJ

SSJ

EJ/SSJ

Termination

SVL

Core Sheath Earth continuity cable (ECC) Earthing joint (EJ)

ECC

Sheath sectionalizing joint (SSJ) Termination Sheath voltage limiter (SVL)

FIGURE 3.5 Single-point bonding as a part of a cross-bonded cable.

from the left termination cannot become a part of cross-bonding. In this situation, single-point bonding is applied to the remaining minor section as shown in Figure 3.5, as long as the sheath voltage allows it. This situation is often observed in actual installations as the number of minor sections is not determined considering the cross-bonding. Rather, it is determined to reduce the number of joints as much as possible, considering the cost.

238


The joint labeled as EJ/SSJ functions both as an earthing joint and as a sheath sectionalizing joint. The left side of the joint is solidly grounded as in an earthing joint. The left side and the right side of the joint are insulated as in a sheath sectionalizing joint, and the right side of the joint is unearthed. Since the grounding resistance at the EJ/SSJ is normally much higher than that at the termination (substation), this addition of the single-point-bonding section may significantly increase the zero sequence impedance of the cable without the earth continuity cable. 3.2.3 Homogeneous Model of a Cross-Bonded Cable 3.2.3.1 Homogeneous Impedance and Admittance Section 3.2.1 discussed the impedance and admittance of a single-phase cable. This section discusses the impedance and admittance of a crossbonded three-phase cable and how 6 × 6 impedance and admittance matrices can be reduced to 4 × 4 impedance and admittance matrices. Figure 3.6 illustrates a major section of a cross-bonded cable. The bold solid line and broken line express core and sheath, respectively. The sheaths are grounded through a grounding impedance Zg at both sides of the major section. The core and sheath voltages Vk and Vk′ and currents Ik and Ik′ at the kth cross-bonded node are related in the following equation: k

( I ′ ) = [R] ( I k

k

 (V )  (Vk ) =  kc  ,  (Vks ) 

0 (0΄)

1

k)

 Vkca  (Vkc ) =  Vkcb  ,  Vkcc    1΄

(I1) Zg

2

Minor section Major section

FIGURE 3.6 A major section of a cross-bonded cable.

(3.3)

 Vksa   (Vks ) =  Vksb  (3.4)  Vksc    2΄

(V2΄)

(V ′ ) = [ R ] (V )

3

Zg

239


The second subscripts “c” and “s” denote the core and sheath, respectively. And the third subscripts “a,” “b,” and “c” express the phases. The other voltage and current vectors (Vk ′), (Ik), and (Ik ′) have the same form as (Vk ). The sheath sectionalizing joint is mathematically expressed by a rotation matrix [R]:

[U ] [R] =   [0]

[0] 

[ R33 ]

0 [ R33 ] =  1 0

0 0 1

1  (3.5) 0  0 

where [0] and [U] denote 3 × 3 null and unit matrix, respectively. The rotation matrix has the following characteristics:

[ R ]3 = [U ] , [ R ]2 = [ R ]t = [ R ]−1 (3.6)

where the subscript t is for the transposed matrix. Defining voltage difference ∆Vk−1 between nodes k − 1 and k′, the following equation is obtained:

(Vk ) = (Vk −1′ ) + ( ∆Vk ) (3.7)

Voltage difference between the major section ∆V (between nodes 0 and 3) is given by

( ∆V ) = (V3 ) − (V0 ) (3.8)

From (3.3) to (3.8), ∆V is expressed by ∆Vk (k = 1, 2, 3) in the following form:

( ∆V ) = (V3 ) − (V0 ) = (V2′ ) + ( ∆V3 ) − (V0 ) = [ R ] (V2 ) + ( ∆V3 ) − (V0 ) = [ R ]{(V1′ ) + ( ∆V2 )} + ( ∆V3 ) − (V0 ) = [ R ]{[ R ] (V1 ) + ( ∆V2 )} + ( ∆V3 ) − (V0 )

{

}

= [ R ] [ R ]{(V0′ ) + ( ∆V1 )} + ( ∆V2 ) + ( ∆V3 ) − (V0 )

= [ R ] (V0′ ) + [ R ] ( ∆V1 ) + [ R ] ( ∆V V2 ) + ( ∆V3 ) − (V0 ) 2

2

(3.9)

The voltage and current deviations are expressed by using the cable impedance [Z] and the admittance [Y]:

( ∆Vk ) = − [ Z ] lk ( I k ) , ( ∆I i ) = − [Y ] lk (Vk ) (3.10)

where lk is the length of the kth minor section.

240


The voltage difference between the terminals of the major section (∆V) gives an equivalent impedance of a cross-bonded cable. It is obtained as (3.13) by applying the following relations:

( I k −1′ ) = ( I k ) = [ R ] ( I k −1 ) (3.11)

(V0 ) = [ R ]2 (V0′ ) , ∵ V0 sa = V0 sb = V0 sc = V0′sa = V0′sb = V0′sc (3.12) ( ∆V ) = − [ R ]2 [ Z ] l1 ( I1 ) − [ R ][ Z ] l2 ( I 2 ) − [ Z ] l3 ( I 3 ) 2 2 3 = − [ R ] [ Z ] l1 [ R ] ( I 0′ ) − [ R ][ Z ] l2 [ R ] ( I 0′ ) − [ Z ] l3 [ R ] ( I 0′ ) (3.13)

{

2 2 = − [ R ] [ Z ] l1 [ R ] + [ R ][ Z ] l2 [ R ] + [ Z ] l3

}( I

0′

)

If the lengths of the minor sections are identical (lk = l), an equivalent series impedance [Z′] can be obtained:

( ∆V ) = − [ Z′] 3l ( I 0′ )

[ Z′] = 1 ([ R ]2 [ Z ][ R ] + [ R ][ Z ][ R ]2 + [ Z ]) 3

=

1 [ ] [ ][ ] [ ][ ][ ] [ ] (3.14) ( R t Z R + R Z R t+ Z ) 3

′]  [ Zcc = [ Zcs′ ]t

[ Zcs′ ] [ Z′ss ]

The physical meaning of the aforesaid equation can be explained using the following calculation:

[ 0 ]   [ Zcc ]

[U ] [ R ]t [ Z ][ R ] =   [0]

[ R33 ] t [ Zcs ]t

 [ Zcc ] = [ R33 ]t [ Zcs ]t

[ Zcs ]  [U ] R [ 33 ]t [ Zss ]  [0]

 [ Zcc ] = [ R33 ]t [ Zcs ]t

[ Zcs ][ R33 ]  [ R33 ]t [ Zss ][ R33 ]

[ Zcc ] [ R33 ][ Zcs ]t 

[ R ][ Z ][ R ]t = 

[ Zcs ] [U ] [ Zss ]  [0]

[0] 

[ R33 ] [0] 

[ R33 ]

(3.15)

[ Zcs ][ R33 ]t  (3.16) [ R33 ][ Zss ][ R33 ]t 

where [Zcc], [Zcs], and [Zss] are the submatrices of the cable impedance matrix.

241


The submatrix for the cores [Zcc] is not changed by the operation shown in (3.14).

[ Zcc′ ] = [ Zcc ] (3.17)

Equations 3.18 and 3.19 show that the operation to the submatrix for the mutual impedance between cores and sheathes [Zcs] is averaging within the rows:

[ Zcs′ ] = 1 ([ Zcs ][ R33 ] + [ Zcs ][ R33 ]t + [ Zcs ]) = 1 [ Zcs ] ([ R33 ] + [ R33 ]t + [U ]) (3.18) 3

3

1 1 1 [ ] ([ R33 ] + [ R33 ]t + U ) = 3 1 3 1

1 1 1

1 1 (3.19) 1

The diagonal and mutual element (Z′sss and Z′ssm) of the submatrix for sheathes [Zss] is the mean of the self- and mutual-impedance of the sheath. The shape of the matrix is identical to that of a transposed overhead line: 0 1 1 [R33 ]t [Zss ][R33 ] = 0 3 3 1

Zss12 1 = Zss13 3  Zss11

0   Zss11 1 Zss12 0  Zss13 Zss 22 Zss 23 Zss12

Zss12 Zss 22 Zss 23

Zss 23  0 Zss 33  1 Zss13  0

Zss13  0 Zss 23  1 Zss 33  0 0 0 1

0 0 1

1 0  0 

1 Zss 22 1  0  = Zss 23 3  Zss12 0 

Zss 23 Zss 33 Zss13

Zss12  Zss13  Zss11  (3.20)

0 1 1 [R33 ][Zss ][R33 ]t = 1 3 3 0

1 0 0

0 0 1

Zss13 1 =  Zss11 3 Zss12

1  Zss11 0  Zss12 0  Zss13 Zss 23 Zss12 Zss 22

Zss12 Zss 22 Zss 23

Zss 33  0 Zss13  0 Zss 23  1

Zss13  0 Zss 23  0 Zss 33  1 1 0 0

1 0 0

0 Zss 33 1  1 =  Zss13 3 Zss 23 0 

0 1 0  Zss13 Zss11 Zss12

Zss 23  Zss12  Zss 22 

(3.21)

242


 Z′sss 1 [ Z′ss ] = ([ R33 ]t [ Zss ][ R33 ] + [ R33 ][ Zss ][ R33 ]t + [ Zss ]) =  Z′ssm 3  Z′ssm Z′sss = Z′ssm =

1 3 1 3

Z′ssm Z′sss Z′ssm

Z′ssm  Z′ssm  Z′sss 

3

∑Z

(3.22)

ssii

i =1 2

3

∑∑Z

ssij

i =1 j = i + 1

In the same manner, the equivalent admittance of a cross-bonded cable can be obtained from the current difference (∆I):

( ∆I ) = − [Y′] 3l (V0 )

[Y′] = [ R ]2 [Y ][ R ] + [ R ][Y ][ R ]2 + [Y ] = [ R ]t [Y ][ R ] + [ R ][Y ][ R ]t + [Y ]

(3.23)

The admittance matrix of the cable can be expressed in (3.24)–(3.26):  [Ycc ]  − [Ycs ]

[Y ] = 

 Cc 1 [Ycc ] = jω  0  0

Css1 + Csm12 + Csm13 −Csm12  −Csm13

[Yss ] = jω 

− [Ycs ]  [Ycc ] = [Yss ]   − [Ycc ] 0 Cc 2 0

−Csm12 Css 2 + Csm12 + Csm 23 −Csm 23

− [Ycc ] (3.24) [Yss ] 

0  0  (3.25) Cc 3  −Csm13   (3.26) −Csm 23  Css 3 + Csm 23 + Csm13 

The core admittance submatrix [Ycc] is a diagonal matrix determined by each capacitance between a core and a sheath, because a core is enclosed by a sheath. The admittance submatrix of the cores for the cross-bonded cable is identical to the solidly bonded cable:

[Ycc′ ] = [Ycc ] (3.27)

Equations 3.18 and 3.19 show that the operation to the submatrix for the mutual admittance between cores and sheathes [Ycs] is averaging within the rows:

243


[Ycs′ ] = 1 ([Ycs ][ R33 ] + [Ycs ][ R33 ]t + [Ycs ]) = 1 [Ycs ] ([ R33 ] + [ R33 ]t + [U ]) 3

3

 Cc 1 1 = j ω  Cc 2 3 Cc 3

Cc 1 Cc 2 Cc 3

Cc 1  Cc 2  Cc 3 

(3.28)

The diagonal and mutual element (Z′sss and Z′ssm) of the submatrix for sheathes [Zss] is the mean of the self- and mutual-impedance of the sheath: ′  Ysss 1  ′ [Yss′ ] = ([ R33 ]t [Yss ][ R33 ] + [ R33 ][Yss ][ R33 ]t + [Yss ]) = Yssm 3 Yssm ′ ′ = Ysss ′ = Yssm

1 3 1 3

3

∑Y

Ys′sm ′ Ysss ′ Yssm

′  Yssm ′  Yssm  ′  Ysss (3.29)

ssii

i =1 2

3

∑∑Y

ssij

i =1 j = i + 1

3.2.3.2 Reduction of Sheath [4,5] The lengths of minor sections can have imbalances due to the constraint on the location of joints. The imbalances are designed to be as small as possible since they increase sheath currents and raise sheath voltages. When a cable system has multiple major sections, the overall balance is considered to minimize sheath currents. As a result, when a cable system has more than a couple of major sections, sheath currents are generally balanced among three conductors, which allow us to reduce three metallic sheaths to one conductor. Since the three-phase sheath conductors are short-circuited and grounded in every major section as illustrated in Figure 3.6, the sheath voltages of three phases are equal at each earthing joint. Assuming sheath currents are balanced among three conductors, the sheath currents do not flow into the earth at each earthing joint: V1ssa = V1sb = V1sc ≡ V1s V4 ssa = V4 sb = V4 sc ≡ V4 s I1ssa + I1sb + I1sc ≡ I1s

I 4 ssa + I 4 sb + I 4 sc ≡ I 4 s

(3.30)

244


By applying connection matrix [T], the aforesaid equation is rewritten as

(Vk ) = [T ]t (Vk′′) , ( I ′′k ) = [T ] ( I k ) , k = 0, 3 (3.31)

where

1 0 [T ] =  0  0

0 1 0 0

0 0 1 0

0 0 0 1

( 0 )t = ( 0

0

0 ) , ( 1)t = ( 1  (Vkc )  ,  Vks 

(Vk′′) = 

0 0 0 1

0 0   [U ] = 0  ( 0 )t  1 1)

1

[0]  ( 1)t  (3.32)

 ( I kc )   (3.33)  I ks 

( I ′′k ) = 

From (3.14), (3.23), and (3.31), the following relation is obtained:

( ∆V ′′ ) = − [ Z′′] 3l ( I 0′′ ) where

( ∆I ′′ ) = − [Y′′] 3l (V0′′)

[ Z′′] = ([T ][ Z′]−1 [T ]t ) [Y′′] = [T ][Y′][T ]t

(3.34)

−1

(3.35)

The aforesaid impedance matrix [Z″] and the admittance matrix [Y″] are 4 × 4 matrices composed of three cores and reduced single sheath. In a high-frequency region in which the skin depth is smaller than the sheath thickness, the impedance matrix is composed by the following two submatrices, because the propagation mode of the cable can be expressed by a coaxial- and sheath-propagation mode:

[ Zcc ] = [ Zcd ] + [ Zss ] (3.36)

[ Zcs ] = [ Zss ] (3.37)

where

 Zcd [ Zcd ] =  0  0

0 Zcd 0

0  0  (3.38) Zcd 

245


The reduced impedance in a high-frequency region becomes  [ Z′′ ]

−1

( Zcs′′ ) (3.39)

[ Z′′] = ([T ][ Z′]−1 [T ]t ) =  cc ( Zcs′′ )t

 Z′′ss 

The core impedances including their mutual impedances are identical to the original impedances:

[ Z′′cc ] = [ Z′cc ] = [ Zcc ] = [ Zcd ] + [ Zss ] (3.40)

The mutual impedance between the kth core and the reduced sheath Z″kj is the average of the impedances between the core and the three-phase sheaths: ′′   Z14  ′′  ′′ Z = Z ( cs )  24   Z′′34    Z′′k 4 =

1 3

(3.41)

3

∑Z

ss kj

j =1

Finally, the sheath impedance is the average of all the elements of the original sheath impedance matrix: Z′′ss =

1 9

3

3

∑∑Z

ssij

i =1

(3.42)

j =1

In the same manner, the reduced admittance matrix becomes

[Ycc′′ ] = [Ycc′ ] = [Ycc ] (3.43)

 Y14′′  ( Ycs′′ ) =  Y24′′   Y34   ′′  Yk′′4 =

Yss′′ =

∑Y

cs kj

(3.44) = − jωCck

j =1

3

1 9

3

3

i =1

j =1

∑∑

3

∑C

Yssij = jω

ssii

(3.45)

i =1

 [Ycc′′ ] ( Ycs′′ )t

[Y′′] = [T ][Y′][T ]t = 

( Ycs′′ ) (3.46)  Yss′′ 

246


3.2.4 Theoretical Formula of Sequence Currents The sequence impedance/current calculation of overhead lines is well known and is introduced in textbooks [2]. For underground cables, theoretical formulas are proposed for the cable itself [6–9]. However, in order to derive accurate theoretical formulas, it is necessary to consider the whole cable system, including sheath bonding, since the return current of an underground cable flows through both metallic sheath and ground. Until now, there has existed no formula of sequence impedances or currents that considers sheath bonding and sheath grounding resistance at substations and earthing joints. As a result, it has been a common practice that those sequence impedances or currents are measured after the installation, as it is considered difficult to predict those values beforehand. As mentioned earlier, it is a common practice for underground cable systems which are longer than about 2 km, to cross-bond the metallic sheaths of three-phase cables to reduce sheath currents and to suppress sheath voltages at the same time [3]. Submarine cables, which are generally solidly bonded, are now becoming a popular type of cable due to the increase of offshore wind farms and cross-border transactions. Therefore, this section derives theoretical formulas of the sequence currents for a majority of underground cable systems, that is, a cross-bonded cable which has more than a couple of major sections. It also derives theoretical formulas for a solidly bonded cable, considering the increased use of submarine cables. 3.2.4.1 Cross-Bonded Cable • Impedance matrix One cable system corresponds to six conductor systems composed of three cores and three metallic sheaths. As in the last section, the 6 × 6 impedance matrix of the cable system is given by the following equation [1]:

 [ Zcc ] [Z] =  t [ Zcs]

[ Zcs] [ Zcc ] [ Zcs] (3.47) =  [ Zss] [ Zcs] [ Zss]

 Zcc11 [ Zcc ] =  Zcc12  Zcc13

Zcc12 Zcc 22 Zcc 23

Zcc13   Zss11  Zcc 23  , †Zss [ ] =  Zss12  Zss13 Zcc 33 

 Zcs11 [ Zcs] =  Zcs12  Zcs13

Zcs12 Zcs 22 Zcs 23

Zcs13   Zcs 23  Zcs33 

Zss12 Zss 22 Zss 23

Zss13   Zss 23  ,† Zss 33 

247


where c is the core s is the sheath m is the mutual coupling between the core and the sheath t is the transpose In (3.47), cable phase a is assumed to be laid symmetrical to phase c against phase b. The flat configuration and the trefoil configuration, which are typically adopted, satisfy this assumption. Reducing the sheath conductors, the six conductor system is reduced to four conductor system composed of three cores and one equivalent metallic sheath as shown in Figure 3.7. The 4 × 4 reduced impedance matrix can be expressed as  Zcc11 Z cc 12 [ Z′′] =  Z cc 13  ′′ Z  14

Zcc12 Zcc 22 Zcc 23 Z′′24

Zcc13 Zcc 23 Zcc 33 ′′ Z14

′′  Z14 Z′′24  (3.48) ′′  Z14  Z′′ss 

First major section Second major section

Rg

[Z]

Rg1

m-th major section

Rgn = Rg

(a) First major section

Second major section

Rg

[Z΄]

m-th major section

Rg

(b) FIGURE 3.7 Cross-bonded cable and its equivalent model: (a) cross-bonded cable system with m-major sections and (b) equivalent four conductor system.

248


Here, Z″(4, j) = Z″(j, 4) can be calculated from the 6 × 6 impedance ′′ = Z′′34 stands in the flat configuration matrix Z as shown in (3.41). Z14 and the trefoil configuration. • Zero sequence current The following equations are derived from Figure 3.8. Here, sheath grounding at earthing joints is ignored, but sheath grounding at substations can be considered through Vs:

(V1 ) = [ Z′] ( I1 ) (3.49)

where

(V1 ) = ( E ( I1 ) = ( Ia

E

E

Vs )

Ib

Ic

Is )

t t

Figure 3.8a shows the setup for measuring the zero sequence current for a cross-bonded cable. Assuming the grounding resistance at substations Rg, the sheath voltage Vs can be found by Vs = −2Rg Is (3.50)

The following equations can be obtained by solving (3.49) and (3.50): Ia = Ic =

( Z22 − Z12 ) E

∆0 ( Z11 − Z21 ) E Ib = ∆0

where ∆ 0 = Z11Z22 − Z12Z21 Z11 = Zcc11 + Zcc13 −

′′ 2 2Z14 ′′ ZSR

Z′′24 2 ′′ ZSR Z′′ Z′′ Z12 = Zcc12 − 14 24 , ′′ ZSR ′′ = Z′′ss + 2Rg ZSR Z22 = Zcc 22 −

Z21 = 2Z12 †

(3.51)

249


Ia

First major section

Ib


E

3I0

m-th major section

Ic Is

[Z΄]

Rg

Rg

(a) Ea

Ia

Eb

Ib

Ec

Ic

First major section


Is

[Z΄]

Rg

m-th major section

Rg

(b) FIGURE 3.8 Setup for measuring sequence currents for a cross-bonded cable: (a) zero sequence current and (b) positive sequence current.

The zero sequence current can be found from (3.51) in the following equation:

I0 =

2Ia + Ib E = ( Z11 + 2Z22 − 2Z12 − Z21 ) (3.52) 3 3∆ 0

When three-phase cables are laid symmetrical to each other, the following equations are satisfied:

Zcc11 = Zcc 22 = Zc , Zcc12 = Zcc13 = Zm ,

Zss11 = Zss 22 = Zs (3.53) ′′ = Z′′24 = Zn Z14

250


Using symmetrical impedances Zc, Zm, and Zn in (3.53), Z11, Z22, and Z12 can be expressed as 2Zn2 ′′ ZSR

Z11 = Zc + Zm − Z22 = Zc −

Zn2 ′′ ZSR Zn2 ′′ ZSR

Z12 = Zm −

(3.54)

Substituting Z11, Z22, and Z12 in (3.51) and (3.52) by the symmetrical impedances, Ia = Ib = Ic ≈ I0 ≈

E ∆1

E , ∆1

Is ≈

−3ZnE ′′ ∆1 (3.55) ZSR

′′ where ∆1 = Zc + 2Zm − 3Zn2/ZSR • Positive sequence current In Figure 3.8b, the equation Isa + Isb + Isc = 0 is satisfied at the end of the cable line. The following equations are obtained since Vs = 0:

(V1 ) = E

α 2E

( I1 ) =  Ia

Ib

0 

αE Ic

Is 

t

t

(3.56)

where α = exp(j2π/3) Solving (3.56) for Ia, Ib, and Ic yields  E   Z11  2    α E  =  Z12  αE   Z13   

 Ia   Z11    ∴  Ib  =  Z12  Ic   Z13   

Z12 Z22 Z12

2  Z11Z22 − Z12 1 =  Z12 (Z13 − Z11 ) ∆ 2  Z12 − Z13 Z22 

Z13   Z12  Z11 

−1

Z12 Z22 Z12

Z13   Ia    Z12   Ib  Z11   Ic 

 E   2  α E  αE   

Z12 (Z13 − Z11 ) 2 2 Z11 − Z13 Z12 (Z13 − Z11 )

2 Z12 − Z13 Z22   E    Z12 (Z13 − Z11 )  α 2E  2   Z11Z22 − Z12   αE 

(3.57)

251


Here, ′′ 2 Z14 Z′′ss Z′′ 2 Z22 = Zcc 22 − 24 Z′′ss ′′ Z′′24 ′ Z14 Z12 = Zcc12 − Z′′ss Z11 = Zcc11 −

Z13 = Zcc12 −

′′ 2 Z14 Z′′ss

The positive sequence current is derived from (3.57) I1 =

1 E Ia + αIb + α 2 Ic = 3 3∆ 2

(

)

{( Z

11

− Z13 ) ( Z11 + Z13 + 2Z12 )

2 + Z22 ( 2Z11 + Z13 ) − 3Z12

{

}

(3.58)

}

2 where ∆ 2 = (Z11 − Z13 ) Z22 ( Z11 + Z13 ) − 2Z12 When three-phase cables are laid symmetrical to each other, (3.58) can be further simplified using (3.53)

I1 =

E (3.59) Zc − Zm

3.2.4.2 Solidly Bonded Cable • Impedance matrix Figure 3.9 shows a sequence current measurement circuit for a solidly bonded cable. The following equations are given from the 6 × 6 impedance matrix in (3.47) and Figure 3.9:

( E ) = [ Zcc ] ( I ) + [ Zcs] ( Is ) (3.60)

(Vs ) = [ Zcs] ( I ) + [ Zss] ( Is ) = − 2  Rg  ( Is ) (3.61) Here, (I) = (Ia Ib Ia)t: core current (Is) = (Isa Isb Isa)t: sheath current 1   Rg  = Rg 1 1

1 1 1

1  1 1

252


Ea

Ia

Eb

Ib

Isa

Ec

Ic

Isb

Ia

E

Ib

Isa

Ic

Isb

Isc

3I0

Isc

[Z΄]

(a)

[Z΄]

(b)

FIGURE 3.9 Setup for measuring sequence currents for a solidly bonded cable: (a) zero sequence current and (b) positive sequence current.

From (3.61), sheath current (Is) is found by

( Is ) = − ([ Zss] + 2  Rg  ) [ Zcs] ( I ) (3.62) −1

Eliminating sheath current (Is) in (3.60), core current (I) can be derived as

(

)

−1

( I ) = [ Zcc ] − [ Zcs] ([ Zss] + 2  Rg  ) [ Zcs] ( E ) (3.63) −1

• Zero sequence current From Figure 3.9a, (E) and (I) are expressed as

( E ) = E

E

t

E  ,

( I ) =  Ia

Ib

t Ia  (3.64)

Core current (I) is obtained from (3.63) and (3.64), and then the zero sequence current is calculated as I0 = (Ia + Ib + Ic)/3. Since the relationship [Zcs] ≈ [Zss] generally stands, (3.60) and (3.61) can be simplified to (3.65) using (3.53):

(( E ) − (Vs )) = ([ Zcc ] − [ Zss]) ( I ) = ( Zc − Zs ) [U ] ( I ) (3.65) where [U]: 3 × 3 unit (identity) matrix.

253


Hence, I 0 = Ia = Ib = Ic =

1 ( E − Vs ) (3.66) Zc − Zs

Using (3.66), core current (I) in (3.61) can be eliminated, which yields

(Vs ) =

1 [ Zcs] ( ( E ) − (Vs ) ) − [ Zcs] ( Is ) (3.67) Zc − Zs

Adding all three rows in (3.67), 3Vs = 3

Zs + 2Zm Zs + 2Zm (3.68) Vs ( E − Vs ) − Zc − Zs 2Rg

Solving (3.68) for Vs and eliminating Vs from (3.66), the zero sequence current is found as I0 =

6Rg + Zs + 2Zm E (3.69) 6Rg ( Zc + 2Zm ) + ( Zc − Zs ) ( Zs + 2Zm )

• Positive sequence current From Figure 3.9b, (E) and (I) are expressed as

( E ) = E

t

( I ) =  Ia

α 2E αE  ,

t Ib Ia  (3.70)

Core current (I) is obtained from (3.63) and (3.70). Once the core current is found, the positive sequence current can be calculated as I1 = (Ia + αIb + α2 Ic)/3. The theoretical formula of the positive sequence current can also be simplified using (3.65): I1 =

{

}

1 E (3.71) ( E − Vs ) + α α 2E − Vs + α 2 ( αE − Vs ) = 3 ( Zc − Zs ) Zc − Zs

(

)

Equation 3.71 shows that the positive sequence current can be approximated by the coaxial mode current. It also shows that, similarly to a cross-bonded cable, the positive sequence current is not affected by substation grounding resistance Rg.

254


Example Figure 3.10 shows physical and electrical data of a 400 kV cable used for comparison. An existence of semiconducting layers introduces an error in the charging capacity of the cable. Relative permittivity of the insulation (XLPE) is converted from 2.4 to 2.729 according to (3.72), in order to correct the error and have a reasonable cable model [10]. ε′r =

ln(R3 / R2 ) ln(61.40 / 32.60) εr = ⋅ 2.4 = 2.729 (3.72) ln ( Rso / Rsi ) ln(59.50 / 34.10)

where Rsi is the inner radius of the insulation Rso is the outer radius of the insulation The total length of the cable is assumed to be 12 km. Figure 3.11 shows the layout of the cables. It is assumed that the cables are directly buried at the depth of 1.3 m with the separation of 0.5 m between phases. Earth resistivity is set to 100 Ω-m. Calculation process in case of a cross-bonded cable using proposed formulas is shown as follows. The 6 × 6 impedance matrix Z is found by CABLE CONSTANTS [11–13]: [Z′] (upper: R, lower: X, unit: Ω) 0.71646353 8.44986724

0.59136589 6.28523815

0.59136589 5.76261762

0.5913705 6.63566685

R3

R4

R5

R2

Core Metallic sheath Insulation Outer cover Core inner radius: 0.0 cm, R2 = 3.26 cm, R3=6.14 cm, R4 = 6.26 cm, R5 = 6.73 cm Core resistivity: 1.724 × 10–8 Ω-m, Metallic sheath resistivity: 2.840 × 10–8 Ω-m, Relative permittivity (XLPE, PE): 2.4 FIGURE 3.10 Physical and electrical data of the cable.

255


1.3 m

0.5 m

0.5 m

FIGURE 3.11 Layout of the cable.

0.59136589 6.28523815 0.59136589 5.76261762 0.5913705 6.63566685

0.71646353 8.44986724 0.59136589 6.28523815 0.5913705 6.80987369

0.59136589 6.28523815 0.71646353 8.44986724 0.5913705 6.63566685

0.5913705 6.80987369 0.5913705 6.63566685 0.83438185 6.63485268

Zero sequence current ∆ 0 = − 3.9605900 + j12.489448 Z11 = 4.0641578 + j 2.2224336 Z22 = 2.1959039 + j 2.1450759 Z12 = 2.0193420 + j0.1372637 Z21 = 4.0386840 + j0.2745275 I 0 ( rms ) = 81.814700 − j 31.778479 Positive sequence current ∆ 2 = − 4.8574998 − j1.8394591 Z11 = 0.3670612 + j1.8221556 Z22 = 0.3801952 + j1.4707768 Z12 = 0.2482475 − j0.5159115 Z13 = 0.2419636 − j0.8650940 I1 ( rms ) = 14.118637 − j 251.86277 Table 3.1 shows zero and positive sequence currents derived by proposed formulas. Grounding resistances at substations are assumed to be 1 Ω. In the calculations, the applied voltage is set to E = 1 kV/ 3 (angle: 0 degree) and the source impedance is not considered. Here, sequence currents are derived in accordance with the setups for measuring sequence currents shown in Figures 3.8 and 3.9. The assumptions on the applied voltage and the source impedance match a condition in actual setups for measuring sequence currents, since testing sets are generally used in the measurements.

256


TABLE 3.1 Comparison of Proposed Formulas with EMTP Simulations Zero Sequence

Positive Sequence

Amplitude (A)

Angle (deg)

Amplitude (A)

Angle (deg)

124.1

−21.23

356.7

−86.79

124.8

−22.50

722.7

−49.08

a. Cross-bonded cable Proposed formulas, Equation (3.8)/(3.14) b. Solidly bonded cable Proposed formulas, Equation 3.19, 3.20/3.26

The proposed formulas are known to have satisfactory accuracy for planning and implementation studies. An acceptable level of error is introduced by the impedance matrix reduction discussed earlier. Due to the matrix reduction, unbalanced sheath currents that flow into earth at earthing joints are not considered in proposed formulas. Table 3.1 shows that the positive sequence impedance is smaller for a solidly bonded cable than for a cross-bonded cable as the positive sequence current is larger for a solidly bonded cable. This is because the return current flows only through the metallic sheath of the same cable and earth in the solidly bonded cable whereas the return current flows through the metallic sheath of all the three-phase cables in a crossbonded cable (Zc − Zm > Zc − Zs). The impedance calculation in IEC 60909-2 assumes solid bonding. As a result, if the positive sequence impedance of a cross-bonded cable is derived based on IEC 60909-2, it might be smaller than the actual positive sequence impedance. The phase angle of the zero sequence current mentioned in Table 3.1 demonstrates that the zero sequence current is significantly affected by a grounding resistance at substations in both cross-bonded and solidly bonded cables. As a result, there is little difference in the zero sequence impedance of the cross-bonded cable and the solidly bonded cable. The result has indicated the importance of obtaining an accurate grounding resistance at substations, to derive accurate zero sequence impedances of cable systems.

3.3 Wave Propagation and Overvoltages 3.3.1 Single-Phase Cable 3.3.1.1 Propagation Constant As explained in Chapter 1, the evaluation of wave propagation related parameters necessitates eigenvalue/eigenvector calculations. Because of

257


the coaxial structure of a cable core and a metallic sheath, the propagationrelated parameters show the following characteristics in a high-frequency region [2].

1. Impedance matrix In a high-frequency region, the following relation is satisfied in (3.1):

Zcs = Zss = Zs or  Zc

Zs  ,† Zc = Zcc † (3.73) Zs 

[ Zi ] =  Zs 

2. Voltage transformation matrix

( v ) = [ A] (V ) , [ A] −1

−1

0 = 1

1 (3.74) −1

where (v) is the modal voltage (V) is the actual phase voltage

3. Modal propagation constant The modal propagation constant γ is given in the following equation from the actual propagation constant matrix [Γ] as explained in Chapter 1:

[ γ ] = [ A] [Γ ][ A] (3.75) −1

where [Γ]2 = [Z][Y] Considering (3.74) with (3.71) and (3.73), γe

[γ] =  0 

0 (3.76) γ c 

where γe = Zs(Ys − Yc) is the earth-return mode (mode 1) γc = Yc(Zs − Zc) is the coaxial mode (mode 2)

4. Characteristic impedance The modal characteristic impedance [z0] is given by

258


 z0 e

[ z0 ] =  0 

0 (3.77) z0 c 

where z0e = Z0s is the earth-return mode (mode 1) z0c = Z0c − Z0s is the coaxial mode (mode 2) The actual characteristic impedance [Z0] is obtained from the aforesaid equations in the following form:

[ Z0 ] = [ A][ z0 ][B]

−1

 Z0 c =  Z0 s

Z0 s  (3.78) Z0 s 

where [B]−1 = [A]t is the current transformation matrix From these equations, it should be clear that the coaxial mode current flows through the core conductor and returns through the metallic sheath in a high-frequency region. In fact, a communication signal cable and a measuring cable intentionally use the coaxial mode propagation for signal transmission because the propagation characteristic is entirely dependent on the insulator between the core outer surface and the sheath inner surface. In such a case, the propagation velocity cc and the characteristic impedance Z0c of the coaxial mode are evaluated approximately by cc =

60 c0 R ,†††Z0 c = ln 3 (3.79) εi1 εi1 R2

3.3.1.2 Example of Transient Analysis Figure 3.12 illustrates a circuit diagram of a single-phase coaxial cable. In the figure, the characteristic impedance of each section is defined as follows:  R0

[ Z1 ] =  0 

0  Z0 c , †[ Z2 ] =  Rs   Z0 s

Z0 s   Rc ,† [ Z3 ] =  Z0 s  0

0 (3.80) Rs 

where R0 is the source impedance Rs is the sheath grounding resistance Rc is the core terminating resistance The sheath grounding resistance Rs generally ranges from 0.1 to 20 Ω, depending on the earth resistivity and the grounding method. The source

259


(V1) Node 1 R0

(V2) Node 2 Rc

Core Sheath

E

Rs

Rs

Section 2 [Z2]

Section 1 [Z1]

Section 3 [Z3]

FIGURE 3.12 Circuit diagram of a single-phase coaxial cable.

impedance is either bus impedance or transformer impedance. If the cable is connected to an overhead line, R0 and Rc is the surge impedance of the overhead line. If the cable is extended beyond the node 2, Rc is the core selfcharacteristic impedance Z0c, or it can be the coaxial mode characteristic impedance z0c. As explained in Chapter 1, the refraction coefficient matrices at nodes 1 and 2 are given in the following form:

2 2  Z0 c ( Z0 s + Rs ) − Z0 s  λ1 f  =  ∆1  RsZ0 s

2  Rc ( Z0 s + Rs )  λ 2 f  =  ∆ 2  −Rs Z0 s

 R0 Z0 s  Z0 s ( Z0 c + R0 )  −Rc Z0 s   (3.81) Rs ( Z0 c + Rc ) 

where ∆1 = (Z0c + R0)(Z0s + Rs) − Z0s2, ∆2 = (Z0c + Rc)(Z0s + Rs) − Z0s2 Now, we consider a transient response of a coaxial cable to connect a pulse generator to a circuit, that is, a current lead wire. Then, the following condition is given assuming the receiving end of the core is open-circuited and the sheath is perfectly grounded:

R0 = z0 c ,† Rs = 0,† Rc = ∞ (3.82)

Then, (3.89) becomes

1  λ1 f  =  0

1 2 ,†  λ 2 f  =   2 0

−2  (3.83) 0 

260


Since the traveling wave voltage at node 1 is E 2 in Figure 3.12 from Thevenin’s theorem as explained in Chapter 1, the node 1 voltage (v1) at t = 0 is calculated as

E  E  2  =  2  = ( E12 ) at t = 0 (3.84)    0    0 

( v1 ) =  λ1 f  

Traveling wave (E12) from node 1 to node 2 is transformed to modal wave (e12) as follows:  0   (3.85)   2

( e12 ) = [ A] ( E12 ) =  E −1

Equation 3.85 indicates that a coaxial mode wave carries E 2 to the receiving end, that is, the cable works as a coaxial mode signal transfer system. The coaxial mode wave arrives at node 2 at t = ta. The wave is transformed to an actual phase domain wave (E2f) as follows: 1

( E2 f ) = [ A] ( e12 ) = 1 

1  0    0   E   2

E  at t = ta (3.86) 0

( v2 ) = λ 2 f  ( E2 f ) = 

The voltage E from the pulse generator therefore appears at the open-end of the coaxial cable at t = ta. 3.3.2 Wave Propagation Characteristic We now discuss the wave propagation characteristic of a three-phase singlecore cable. Figure 3.13 and Table 3.2 show a cross-section and parameters of a tunnel-installed cable. Assuming the tunnel as a pipe conductor, the propagation parameters are evaluated by using a pipe-type (PT) cable option of the electromagnetic transients program (EMTP) CABLE CONSTANTS. Table 3.3a shows calculated results of the impedance, the admittance, modal attenuation constant, and propagation velocity on the solidly bonded case and Table 3.3b on the cross-bonded case with the aforesaid homogeneous model at frequency f = 100 kHz.

261


y [m]

2

–2

1

–1

0

1 x [m]

2

–1

–2 (a)

r1

r2

r3

r4

r5

ρc, rc ε1 ρs, rs ε2 (b) FIGURE 3.13 Tunnel-installed cable represented by a PT cable: (a) configuration of a three-phase single-core cable and (b) cross-section of a single-core cable.

TABLE 3.2 Cable Parameters r1 r2 r3 r4 r5 ρe

0 30.45 mm 71.15 mm 74.80 mm 81.61 mm 100 Ω-m

ε1 ε2 μc μs ρc ρs

3.1 4.0 1.0 1.0 1.82 × 10−8 Ω-m 2.83 × 10−8 Ω-m

262


TABLE 3.3 Parameters of a Tunnel-Installed Cable at 100 kHz Core-1

Core-2

Core-3

(a) Solidly bonded cable a-1. Impedance [R] in (Ω/km) and [L] in (mH/km) R 83.150 81.660 82.120 L 1.172 0.801 0.800 R 81.660 82.690 81.890 L 0.801 1.172 0.800 R 82.120 81.890 83.620 L 0.800 0.800 1.171 R 82.200 81.660 82.120 L 0.989 0.801 0.800 R 81.660 81.740 81.890 L 0.801 0.990 0.800 R 82.120 81.890 82.670 L 0.800 0.800 0.989 a-2. Capacitance [C], in (nF/km) Core-1 191.0 0 Core-2 0 191.0 Core-3 0 0 Sheath-1 −191.0 0 Sheath-2 0 −191.0 Sheath-3 0 0 Mode 1 2 a-3. Current transformation matrix [Ti] Core-1 0 1.000 Core-2 1.000 0 Core-3 0 0 Sheath-1 0 −1.000 Sheath-2 −1.000 0 Sheath-3 0 0 a-4. Modal propagation constant Attenuation 0.134 0.134 (dB/km) Velocity (m/μs) 169.6 169.6

0 0 191.0 0 0 −191.0 3

Sheath-1

Sheath-2

82.200 0.989 81.660 0.801 82.120 0.800 82.200 0.989 81.660 0.801 82.120 0.800

81.660 0.801 81.740 0.990 81.890 0.800 81.660 0.801 81.740 0.990 81.890 0.800

−191.0 0 0 236.2 −19.0 −18.3 4

0 −191.0 0 −19.0 235.4 −18.7 5

Sheath-3

82.120 0.800 81.890 0.800 82.670 0.989 82.120 0.800 81.890 0.800 82.670 0.989 0 0 −191.0 −18.3 −18.7 237.1 6

0 0 1.000 0 0 −1.000

0 0 0 0.336 0.227 0.456

0 0 0 0.653 −0.140 −0.503

0 0 0 −0.229 0.768 −0.509

0.134

1.854

0.026

0.030

169.6

220.3

287.7

288.1

263


TABLE 3.3 (continued) Parameters of a Tunnel-Installed Cable at 100 kHz Core-1

Core-2

(b) Homogeneous cable b-1. Impedance [R] in (Ω/km) and [L] in (mH/km) R 83.150 81.660 L 1.172 0.801 R 81.660 82.690 L 0.801 1.172 R 82.120 81.890 L 0.800 0.800 R 81.990 81.760 L 0.863 0.864 b-2. Capacitance [C], in (nF/km) Core-1 191.0 Core-2 0 Core-3 0 Sheath −191.0 Mode 1

0 191.0 0 −191.0 2

b-3. Current transformation matrix [Ti] Core-1 −0.507 −0.339 Core-2 0.509 −0.335 Core-3 −0.002 0.673 Sheath 0 0 b-4. Modal propagation constant Attenuation (dB/km) 0.124 Velocity (m/μs) 118.8

0.125 118.8

Core-3

82.120 0.800 81.890 0.800 83.620 1.171 82.230 0.863 0 0 191.0 −191.0 3

Sheath

81.990 0.863 81.760 0.864 82.230 0.863 81.990 0.863 −191.0 −191.0 −191.0 596.7 4

0.333 0.333 0.333 −1.000

0 −0.001 0.001 1.000

0.134 169.6

1.860 220.5

3.3.2.1 Impedance: R, L In Table 3.3a-1, the upper 3 × 3 matrix expresses three-phase core impedance [Zcc], the right upper and the left lower matrices are three-phase core to sheath [Zcs] and sheath to core impedance [Zsc] = [Zcs]t, and the lower 3 × 3 matrix is three-phase sheath impedances [Zss]. The upper line is the resistance [Ω/km] and the lower one is the inductance [mH/km]. The [Zcc] matrix of the solidly bonded cable in Table 3.3a-1 is the same as that of the cross-bonded cable in Table 3.3b-1. The impedance in the last column, Z″4i (= Z″i4), in Table 3.3b is given as the average of [Zcs] in Table 3.3a-1, as shown in (3.41). This corresponds to the fact that the cross-bonding acts as a transposition of an overhead transmission line, and the three sheath conductors are reduced to an equivalent conductor explained in the previous section.

264


3.3.2.2 Capacitance: C The capacitance matrix is of a similar form as the impedance matrix in Table 3.3a-2. The capacitance between the core and sheath, Cck, of the homogeneous model is identical to that of the solidly bonded cable. The equivalent capacitance, Ck4″ , of the cross-bonded cable in Table 3.3b-2 is given as a sum of elements as shown in (3.44) and (3.45). 3.3.2.3 Transformation Matrix The transformation matrix [Ti] in Table 3.3a-3 and b-3, is to transform modal current (i) to phasor current (I), that is,

( I ) = [Ti ] ( i ) (3.87)

In the solidly bonded cable, the first three modes (columns) shown in Table 3.3a-3 express coaxial propagation modes, that is, “core to sheath” mode [2]. The other modes (columns) correspond to one of transformation matrices of an untransposed three-phase overhead line [2]. In this case, the mode 4 expresses an earth-return mode and the modes 5 and 6 correspond to aerial modes. The reduced transformation matrix of a cross-bonded cable is shown in Table 3.3b-3. The composition of the upper-left 3 × 3 matrix (the first three modes) is similar to that of an overhead transmission line. The current of the third mode returns from the equivalent sheath instead of the earth. The fourth mode expresses the equivalent earth-return mode of the crossbonded cable system. 3.3.2.4 Attenuation Constant and Propagation Velocity Modes 1–3 in the solidly bonded cable shown in Table 3.3a-4 are coaxial modes and are the same as mode 3 in the homogeneous cross-bonded cable model. Although the attenuations of the inter-core modes (modes 1 and 2) shown in Table 3.3b-4 are almost identical to that of the coaxial mode of the solidly bonded cable, the velocities are lower than that of the coaxial mode. The velocity of the coaxial mode vc is determined by the permittivity of the main insulator ε1 = 2.3 shown in Table 3.2.

vc =

c0 300 = ≈ 198 m/µs (3.88) ε1 2.3

where c0 is the speed of light. The velocity is converged to the aforesaid value as increases the frequency. The attenuation and velocity of the earth-return mode (mode 4) in both cable models are identical to each other.


265

Because the cable is installed in a tunnel, that is, the cable is in the air, the attenuation constant and the propagation velocity of mode 4 (earth-return) and modes 5 and 6 (first and second intersheath) in the solidly bonded cable show a similar characteristic to those of an overhead line [1]. The attenuation constants of the modes 5 and 6 are far smaller, and the propagation velocity is far greater in the solidly bonded cable than those in the other modes. 3.3.3 Transient Voltage Figures 3.14 and 3.15 show transient voltage waveforms at the sending-end core voltages and the first cross-bonding point of a cross-bonded cable system with five major sections (l1 = l2 = l3 = 400 m, total length l = 5 × 3 × 0.4 = 6 km) when a step voltage of 1 pu is applied to a sending-end core (phase a) through a resistor of Rs = 200 Ω, which models a backward surge impedance. The physical parameters of the cable are shown in Figure 3.6 and Table 3.2. Each receiving-end core is grounded through a resistor of Re = 200 Ω. The sheaths are short-circuited and grounded by a resistor of Rs = 0.1 Ω at both ends of each major section. The cable is represented by a constant parameter, Dommel’s model. The calculated results, when the minor sections are exactly modeled by multiphase distributed parameter lines, are shown in Figures 3.14a and 3.15a. The induced voltages on the cores are observed in Figure 3.14a. The voltage is generated by the reflections at the cross-bonded points. After 70 μs, the voltage on the applied phase is gradually increased. The time is determined by the round-trip time of the coaxial traveling wave:

T=

2lt 2 × 6000 = = 70.8 µs (3.89) vc 169.6

where lt and vc denote the total cable length and the traveling velocity, respectively, of the coaxial mode shown in Table 3.3a-4 or b-4. The time constant τ1 of the voltage increase is determined by the sendingend resistance Rs, terminating resistance Re, and the total cable capacitance Cclt. The capacitance is obtained from Table 3.3a-2 or b-2.

τ1 = ( Rs//Re ) Cc l = 100 × 0.191× 6 ≈ 110 µs (3.90)

The maximum sheath voltage at the first cross-bonded joint shown in Figure 3.15a becomes 0.05 pu and about 40% of the core voltage at the time. This voltage is generated by a reflection at the cross-bonded joint. This is an inherent characteristic of the cross-bonded cable. The high voltage is a key factor in the insulation design of a cross-bonded cable system.

266


Sending end core

800.0 1 2

1

3

Voltage [V] (10**–3)

00.0

3 0.0

50.0

150.0

250.0 Milliseconds (10**–3)

350.0

450.0

–400.0 Naoto 25-Nov-12 19:59:28 1 Plot type 4 From file: c:\ncat\crcbook\crossb\tunnel1.ps Node names : AIN 1 AIN 2 AIN 3

–800.0 (a)

Sending end core

800.0 1 2 00.0 Voltage [V] (10**–3)

1

3

0.0 3

50.0

150.0


350.0

450.0

–400.0

–800.0

Naoto 25-Nov-12 20:04:24 1 Plot type 4 From file: c:\ncat\crcbook\crossb\tunnel1.ps Node names : AIN 1 AIN 2 AIN 3

(b) FIGURE 3.14 Calculated transient core voltages on a tunnel-installed cable: (a) cross-bonded, (b) mixed.

267


Sending end core

800.0 1 2 3


00.0

3 0.0

50.0

150.0


350.0

450.0

–400.0

(c)

–800.0


Sending end core

800.0 1 2


00.0

0.0

1

3

50.0

150.0

250.0

350.0

450.0

3

Milliseconds (10**–3)

–400.0

(d)

–800.0


FIGURE 3.14 (continued) Calculated transient core voltages on a tunnel-installed cable: (c) homogeneous, and (d) solidly bonded.

268


1st Xbond sheath

80.00

1 2

3


0.00

3

2

0.00

10.0

30.0

50.0 Milliseconds (10** –3)

70.0

90.0

–40.00

(a)

–80.00

Naoto 25-Nov-12 19:59:28 5 Plot type 4 From file: c:\ncat\crcbook\crossb\tunnel1.ps Node names : A 114 A 115 A 116

1st Xbond sheath

80.00

1 3


0.00

0.00

2 3

2

10.0

30.0

50.0 Milliseconds (10** –3)

70.0

90.0

–40.00

(b)

–80.00

Naoto 25-Nov-12 20:04:24 5 Plot type 4 From file: c:\ncat\crcbook\crossb\tunnel1.ps Node names : A 114 A 115 A 116

FIGURE 3.15 Calculated transient sheath voltages on a tunnel-installed cable: (a) cross-bonded and (b) mixed.


269

Although the exact model of the cross-bonded cable is useful for the simulation of a simple grid, a simulation of a large-scale cable system becomes too complicated and difficult. For simplification, the homogeneous model explained in Section 3.2.3 is useful. Figures 3.14 and 3.15b illustrate the transient voltages when the first major section is exactly expressed and the other major sections are expressed by pi-equivalent circuits whose parameters are determined by the homogeneous model. By a comparison with the results from the exact model (a), the simplification is practical and is sufficient for an insulation design of the cable system. Figure 3.14c illustrates the result of the case where all major sections are expressed by homogeneous pi-equivalent circuits. It is clear from the figure that the calculated result has enough accuracy for the simulation of the switching surge, although the sheath voltages at the cross-bonded joints cannot be obtained. Figure 3.14d shows the transient response of the core voltage in a solidly bonded cable. It shows a stair-like waveform with 70 μs run of stairs length. The length is determined by the round-trip time shown in (3.89). Sheath voltages of the solidly bonded cable are far smaller than those of the crossbonded cable. The results indicate that any cross-bonded cable cannot be simplified by a solidly bonded cable from a viewpoint not only of the sheath voltages but also of the core voltages. 3.3.4 Limitation of the Sheath Voltage As mentioned in the previous section, the limitation of the sheath voltage is an important factor that decides sheath bonding and other cable system designs related to the sheath. There are two types of limitations in the sheath voltage: (1) the continuous voltage limitation and (2) the short-term voltage limitation. The continuous voltage limitation is the limitation of the sheath voltage induced by the normal load flow in phase conductors without any faults. It is enforced by government or district regulations in many countries and differs from country to country or district to district. The limitation is enforced for safety reasons as a maintenance crew may touch the sheath circuit. Even when the limitation is not enforced by these rules, utilities follow their own standards for the continuous voltage limitation. Since SVLs are designed not to operate by continuous sheath voltages, the continuous voltage limitation is maintained mainly by cable layouts, crossbonding designs, and grounding resistances. The cable span length (minor length) is more often limited by restrictions in the transportation, but it is also sometimes limited by the continuous voltage limitation. The short-term voltage limitation is specified in IEC 62067 Annex G (informative) as impulse levels [14]. Considering the short-term voltage limitation, the following phenomena are studied:

270


• Single-line-to-ground (SLG) faults (external to the targeted major section) • Three-phase faults (external to the targeted major section) • Switching surges • Lightning surges When only power-frequency components are considered, SLG faults and three-phase faults are studied with theoretical formulas. Some utilities study SLG faults and three-phase faults using EMTP in order to consider transient components of the sheath voltage. Switching surges rarely become an issue for the sheath overvoltage. Lightning surges have to be studied for a mixed overhead line/ underground cable as shown in Figure 3.16. A lightning strike on the ground wire can propagate into the sheath circuit since the transmission tower and cable sheath often share the grounding mesh or electrode at the transition site. The level of the sheath overvoltage is highly dependent on the grounding resistance at the transition site. The space of the transition site is sometimes limited, but it is necessary to achieve a low grounding resistance in order to lower the sheath overvoltage. A back flashover can occur when a lightning strikes the ground wire. In addition, a lightning strike can directly hit the phase conductor due to the shielding failure. In these cases, the lightning surge in the phase conductor can directly propagate into the cable core, which leads to the sheath

Ground wire Phase a Phase b

Phase conductor/ core Ground wire/ sheath

Phase c

Termination

Common ground FIGURE 3.16 Lightning surge in a mixed overhead line/underground cable.

271


OHL

Maintenance outage

Termination

Transformer Underground cable

FIGURE 3.17 Example of a substation with a limited number of feeders.

overvoltage. Since the lightning surge is not highly attenuated in this case, the voltage across the sheath interrupts at the first SSJ needs to be studied, in addition to the sheath-to-earth voltage at the transition site. Lightning surges are also studied when a limited number of feeders are connected to a substation together with a cable. In Figure 3.17, the substation has only two lines and a transformer considering the maintenance outage. The lightning surge on the overhead line can propagate into the cable core without a significant attenuation depending on the substation layout. 3.3.5 Installation of SVLs SVLs are installed at SSJs in order to suppress short-term sheath overvoltages. Figure 3.18 shows the connection of SVLs when the sheath circuit is cross-bonded in a link box. SVLs are often arranged in a star formation with its neutral point earthed. When study results show that the sheath overvoltage exceeds the TOV rating of SVLs, the ECC can be installed as shown in Figure 3.18. Other countermeasures include, changing the neutral point from solidly earthed to unearthed and changing the SVL connection from star formation to delta formation. When the link box is not installed, SVLs are located just next to sheathsectionalizing joints as shown in Figure 3.19. In this connection, SVLs are arranged in a delta formation. This connection has an advantage in terms of suppressing the short-term sheath overvoltage as bonding leads to SVLs can be much shorter than using the link box.

272


SSJ

SSJ

SVL

SVL

ECC

Link box

Link box

Sheath sectionalizing joint (SSJ) Sheath voltage limiter (SVL) Earth continuity cable (ECC) FIGURE 3.18 Connection of SVLs in a link box. SSJ SVL

Sheath sectionalizing joint (SSJ) Sheath voltage limiter (SVL)

FIGURE 3.19 Connection of SVLs without a link box.

3.4 Studies on Recent and Planned EHV AC Cable Projects This section introduces recent and planned extra high voltage (EHV) alternate current (ac) cable projects and cable system transients studied for the projects. In order to compensate large charging capacity of EHV ac cables, shunt reactors are often installed together with these cables. The large charging capacity and large shunt reactors lower the natural frequency of the network, which at times makes it necessary to conduct resonance overvoltage studies. Load shedding overvoltage and zero-missing phenomenon are the other power system transients peculiar to cable systems. Similarly to overhead line projects, switching transients, such as the cable energization, the ground fault and the fault clearing are also studied for EHV


273

ac cable projects as customary work. However, severe overvoltages related to these switching transients on cable systems have not been reported in literature. This section focuses on well-known long cable projects, which normally require shunt compensation, since the TOVs discussed in this section can only be observed with these cables. It therefore includes only cross-bonded land cables and submarine cables. It does not include short cables typically installed inside power stations and substations since power system transients peculiar to cable systems are normally not studied for these short cables. 3.4.1 Recent Cable Projects Table 3.4 lists long 500/400 kV cables, currently in operation. There have been a larger number of long 400 kV cable projects, compared to 500 kV cable projects. It is mainly due to the geographical area where the system voltage 400 kV is adopted—Europe and Middle East. The world’s first long 500/400 kV cable was installed in Canada by BC Hydro in 1984 [15–17]. The 500 kV ac submarine cable is a double-circuit line which connects the Vancouver Island to Mainland Canada through Texada Island. The route length between Vancouver Island and Texada Island is approximately 30 km, and the route length between Texada Island and Mainland Canada is approximately 8 km. In between, the line has an overhead section in Texada Island. Shunt reactors totaling 1080 MVar were installed to compensate the large charging capacity. The longest 500 kV cable, Shin-Toyosu line, was installed in Japan by Tokyo Electric Power Company in 2000. This double-circuit line has four 300 MVar shunt reactors for the compensation of the large charging capacity. This is the first 500 kV cable on which extensive power system transient studies are available in literature [18,19]. In addition to ordinal switching transients, the overvoltage caused by the system islanding, the series resonance overvoltage, the leading current interruption, and the zero-missing phenomenon were studied. Here, we introduce the overvoltage caused by the system islanding, studied on the Shin-Toyosu line. When one end of a long cable is opened, a part of the network can be separated from the main grid together with the long cable, which can lead to a severe overvoltage. Figure 3.20 illustrates the equivalent circuit when one end of the long cable is opened due to a bus fault. A cable fault will not lead to overvoltage since it results in the removal of the long cable from the equivalent circuit. From the aforesaid equivalent circuit, the overvoltage caused by the system islanding can be expressed using the following equations:

v ( t ) = Vmsinωt −

ω Vmsinω0t (3.91) ω0

274


TABLE 3.4 Installed Long 500/400 kV Cablesa Location 500 kV cables Vancouver, Canada Vancouver–Texada Island Texada Island–Mainland Mainland Japan–Shikoku Island, Japan Tokyo, Japan Shin-Toyosu Line Shanghai, China [20] Moscow, Russia [21] 400 kV cables Copenhagen, Denmark [22] Southern route Northern route Spain–Morocco [23] Berlin, Germany [24] Mitte–Friedrichshain Friedrichshain–Marzahn Madrid, Spain [25] San Sebastian de los Reyes–Loeches–Morata Line Jutland, Denmark [26] Trige–Nordjyllandsværket London Ring [27] Qatar Istanbul, Turkey Ikitelly–Davutpasa– Yenibosna London–West Ham Rotterdam, Netherlands Enecogen Dubai and Abu Dhabi, UAE Mushrif–Al Mamzar Abu Dhabi, UAE [28] Sadiyat–ADST Doha, Qatar Umm Al Amad Super–Lusail Development Super 2 a b

Route Length (km)

Number of Circuits

30 8 22.2 22.2

2 2 1 1

SCFFb

1984

SCFF

1994 2000

39.8

2

XLPE

2000

17 10.5

2 2

XLPE XLPE

2010 2012

22 12 28 31

1 1 1 1

XLPE

1997 1999 1997 2006

6.3 5.5

2 2

XLPE

1998 2000

12.8

2

XLPE

2004

2

XLPE

2004

1 2

XLPE XLPE

2005 2007

12.8 7.2 12.6 12.5

1 1 1 1

XLPE XLPE XLPE XLPE

2007–2011

11.5

1

XLPE

2011

13

2

XLPE

2011, 2012

22

3

XLPE

2012–2013

14 (4.5 + 2.5 + 7.0) 20 15

Include 500/525/550 kV cables and 380/400/420 kV cables. Self-contained fluid-filled.

Insulation

SCFF

Commissioned in

2008 2010

275


Underground cable with shunt reactors Fault clearance

Equivalent source L0 C

L

Em

FIGURE 3.20 Equivalent circuit of the overvoltage caused by system islanding.

Vm =

1 1 (3.92) Em L + ,††††ω0 = 2 CL0 CL L0 1 − ω CL + L

(

)

where L0 is the source impedance of the islanded system Em is the source voltage behind L0 Charging capacity of the long cable and inductance of the shunt reactors directly connected to the cable are expressed as C and L, respectively. Equation 3.91 shows that the overvoltage contains two frequency components, the nominal frequency ω and the resonance frequency ω 0. Since the overvoltage is caused by the superposition of two frequency components, the resulting overvoltage is oscillatory, and its level is often difficult to estimate before the simulation. The result of a simulation performed on the Shin-Toyosu line is shown in Figure 3.21. Most of the 500/400 kV cables shown in Table 3.4 are installed in highly populated areas, and hence the route lengths are limited to 10–20 km. These cables are equipped with shunt reactors for the compensation of the charging capacity, but their unit size and total capacity are not large due to their shorter route length. For these cables, only studies such as the reactive power compensation, the design of the cable itself, and the laying method are discussed in literature, and transient studies are not available. 3.4.2 Planned Cable Projects Table 3.5 lists lengthy 400 kV cable projects that are planned and which will be in operation within a couple of years. There is no mention of planned 500 kV cable projects in publicly available sources. Various transient studies have been performed on the 400 kV cable that will connect Sicily to Mainland Italy [29,30]. In addition to switching transients, the

276

1.000


2.23 pu

(V) 1.000 –0.500

(10×× 6) –0.000 0.500

1

00

09/03/92 16.4 Plot type Node names S3 1 A S3 1 B S3 1 C

1.0s

1.000

Without surge arresters 1.69 pu

(V) 1.000 –0.500

(10×× 6) –0.000 0.500

1

00

09/03/92 16.58 69 1 Plot type 4 Node names S3 1 A S3 1 B S3 1 C

1.0s

With surge arresters

FIGURE 3.21 Example of the overvoltage caused by system islanding.

TABLE 3.5 Planned Long 400 kV Cablesa Location

Route Length (km)

Number of ckts

Insulation

Planned Operation

38

2

SCOF

2013

13 25 21 13 42

1 3 2 3 2

Sicily—Mainland Italy Abu Dhabi Island, UAE [28] Sadiyat–ADST Bahia–Sadiyat SAS Al Nakheel–Mahawi Mahawi–Mussafah Messina strait, Italy Sorgente-Rizziconi a

Include 380/400/420 kV cables.

XLPE

XLPE

2014 2013 2013 2013 2015


277

studies include the harmonic overvoltage caused by the line energization, the leading current interruption, and the zero-missing phenomenon. The studies identified resonant condition at the second harmonic when the cable is energized from Sicily’s side under a particular condition. The harmonic overvoltage caused by the resonant condition is avoided by the operational constraint.

3.5 Cable System Design and Equipment Selection 3.5.1 Study Flow This section discusses the cable system design, except for the overvoltage analysis and the insulation coordination. The cable system design includes the selection and specification of the cable itself and related equipment such as circuit breakers and VTs. The cable system design related to the sheath is discussed in Section 3.2. During the planning stage, the transmission capacity and the reactive power compensation are normally studied. These studies mainly determine the cable route, the voltage level, the conductor size, and the amount and locations of shunt reactors. When the transmission development plan is set up for the cable, the cable route is studied further. One characteristic of cables, compared to overhead lines, is that the laying and soil conditions affect the planning studies in addition to the land availability. These factors affect the burial depth, soil thermal resistivity, and cable separation between phases, which may necessitate changes to the conductor size and the amount and locations of shunt reactors initially determined in the planning studies. Figure 3.22 illustrates the study flow and relationship of studies on the cable system design. In the figure, the boxes show the study items, and the bullets following them show the items mainly evaluated in these studies. The figure explains how the transmission capacity and the reactive power compensation studied during the planning stage affect the other studies conducted for cables. The amount of shunt reactors, in other words, the compensation rate of a cable, is a key figure that has a major impact on the following studies. The compensation rate close to 100% is often preferred, since it can eliminate the reactive power surplus created by the introduction of the cable. It also offers a preferable condition for the TOV but causes a severe condition for the zero-missing phenomenon. The ill effect on the zero-missing phenomenon is not a primary concern as there are countermeasures established for tackling the effect. Usually, shunt reactors for 500/400 kV cables are directly connected to the cables in order to mitigate the TOV when one end of the cable is opened.

278


Transmission capacity

Transient overvoltage - Insulation strength - Breaker duty - Energy absorption capability (surge arrester)

Zero-miss phenomenon

Reactive power compensation

Steady-state overvoltage - Insulation strength - Grid code

Cable discharge - Capability of VT

- Capability of line breaker Sequential switching Leading current interruption - Capability of line breaker FIGURE 3.22 Study flow and relationship of studies.

Shunt reactors for other cables are often connected to buses, as the area compensation is applied at these voltage levels. When shunt reactors are connected to buses, the zero-missing phenomenon does not occur. In this case, however, the inductive VT connected to the cable needs to have enough discharge capability, and the line breaker needs to have sufficient leading current interruption capability. 3.5.2 Zero-Missing Phenomenon A dc offset current (Zero-missing current) appears when an EHV underground cable is energized with its shunt reactors [31–36]. In this condition, an ac component of a charging current has the opposite phase angle to the ac component of a current flowing into the shunt reactors. If the compensation rate of the cable is 100%, the summation of these ac components becomes zero, and only the dc component remains. Since the dc component decays slowly with time, it can take more than 1 s, depending on the compensation rate, before a current that flows through the line breaker crosses the zero point. Figure 3.23 shows an example of current waveforms when an EHV cable is energized with its shunt reactors. It can be seen that the ac component of the energization current is very small since the compensation rate is close to 100%. The simulation was run for 0.2 s, but the energization current did not cross the zero point during this duration. Since the Zero-missing phenomenon is caused by a dc component of an energization current, it is most severe

279


5000 3750

Phase a

2500 1250 (A)

0 –1250 –2500 –3750 –5000 000

004

008

012

016

020

(s) FIGURE 3.23 Zero-missing current in the underground cable energization.

when the cable is energized and the maximum dc component is contained in the current. In order to realize this condition in phase a, the cable was energized at the voltage zero point of phase a. When a line breaker is to be operated to interrupt this current without the zero crossing, the arc current between the contacts cannot be extinguished within a couple of cycles and may continue for an extended duration. The extended duration may lead to the failure of the line breaker depending on the arc energy generated during this duration. The duration is mainly determined by the magnitude of the dc component and the relationship between arc resistance and arc current inside the line breaker. Typical durations for EHV cables can be several hundreds of milliseconds in severe conditions. The zero-missing phenomenon can theoretically be avoided by limiting the compensation rate to lower than 50%, but it is not a common way to address the problem. Normally, a compensation rate near 100% is adopted, especially for 500/400 kV cables, and the countermeasures listed in Table 3.6 are applied to the cables. All of these countermeasures, except for Countermeasure (4), have already been applied to the cable line in operation. Countermeasure (1), which in particular has a number of application records to long EHV cable lines, is discussed in detail later in this section. Countermeasure (3) will be applied to the 400 kV Sicily–Mainland Italy cable [30]. This countermeasure can be implemented rather easily as long as a cable line is installed together with single-phase circuit breakers and current differential relays. For this reason, Countermeasure (3) is more suited for EHV cable lines than high-voltage (HV) cable lines.

280


TABLE 3.6 Countermeasures of Zero-Missing Phenomenon Countermeasures 1.

Sequential switching

2.

Point-on-wave switching (synchronized switching)

3.

Delayed opening of healthy phases

4.

Breaker with preinsertion resistor

5.

Additional series resistance in shunt reactor for energization Energize shunt reactor after the cable

6.

Notes • Requires higher leading current interruption capability • Requires single-phase circuit breaker and current differential relay • May cause higher switching overvoltage • Requires single-phase circuit breaker • Requires single-phase circuit breaker and current differential relay • May not be possible to apply near generators • May be necessary to develop a new circuit breaker (expensive) • Requires special control to bypass series resistance after energization • Causes higher steady-state overvoltage and voltage step

In this countermeasure, the faulted phase is opened instantly, but healthy phases are opened about 10 s later when the dc component is decayed enough. When this countermeasure is applied near a generator, especially when the cable line offers radial path to the generator, it is necessary to evaluate the negative-sequence current capability of the generator, as Countermeasure (3) causes an unbalanced operation for a prolonged duration. In Countermeasure (5), a resistance is connected in series to shunt reactors when a cable line is energized. The resistance needs to be sized so that the dc component decays fast enough. After the cable line is energized, the resistance is bypassed in order to reduce losses. Considering the additional cost for the resistance, this countermeasure is more suited for HV cable lines than EHV cable lines. Countermeasure (6) cannot always be applied; especially, it is difficult to apply to long EHV cables due to the steady-state overvoltage. 3.5.2.1 Sequential Switching Figure 3.24 shows zero-missing current with a SLG fault in phase b. This assumes that a cable failure exists in phase b before energization, but it is not known to system operators until the cable is energized. The zero-missing current is observed only in a healthy phase (phase a). The current in the faulted phase (phase b) crosses zero point as it contains a large ac component due to the fault current. Hence, the line breaker of the faulted phase can interrupt the fault current.

281


15

Phase b

10 5 (kA) 0 –5 –10 –15

000

004

008

012

016

020

(s) FIGURE 3.24 Zero-missing phenomenon with a SLG fault.

Figure 3.25 shows the time sequence of sequential switching, when the cable line is energized from Substation A. The line breaker of phase b is opened, in Step 1, 60 ms after the fault, and the fault is cleared by this circuit breaker tripping. Since the fault is already cleared by the opening of phase b line breaker, there is no problem in taking some time before opening line breakers of other healthy phases. In Step 2, shunt reactors are tripped before line breakers of healthy phases. It is necessary to trip shunt reactors of only healthy phases. At this time, it is not necessary and recommended to trip shunt reactors of the faulted phase, since the current through shunt-reactor breakers of the faulted phase does not cross the zero point. It is recommended to trip half of shunt reactors of healthy phases or less as shown in Figure 3.25 as it will normally lower the compensation rate below 50% after the tripping. The remaining shunt reactors will be useful to maintain the charging current within the leading current interruption capability of line breakers. It is now possible to open the line breakers of the healthy phases in Step 3. Figure 3.26 shows that the current in healthy phases contains the ac component and crosses the zero point after tripping the shunt reactors. 3.5.3 Leading Current Interruption When the leading current is interrupted at current zero, it occurs at a voltage peak assuming that the current waveform is leading the voltage waveform

282


Substation B

Substation A

Step 1: Fault clearance

: Trip : Opened : Closed

Substation B

Substation A

Step 2: Shunt reactor-CB open

: Trip : Opened : Closed

Substation B

Substation A

Step 3: Line-CB open (load side)

: Trip : Opened : Closed FIGURE 3.25 Time sequence of sequential switching.

by 90°. After the interruption, the voltage on the source side of the circuit breaker changes according to the system voltage, whereas the voltage on the other side is fixed at the peak voltage E as shown in Figure 3.27. The most severe overvoltage occurs during a restrike after half cycle when the voltage on the source side becomes −E. As the voltage difference between the

283


15

10 Trip shunt reactors (Step 2)

5 (kA) 0

–5 Trip phase B line breaker (Step 1)

–10

–15

000

004

008

012

016

020

(s) FIGURE 3.26 Zero-missing phenomenon with sequential switching. Leading current interruption

Restrike

E 2E

Voltage

0 Source side

t

–E

–2E

The other side

–3E FIGURE 3.27 Overvoltage caused by leading current interruption and restrike.

source side and the other side is 2E, the overvoltage can go as high as −3E. The restrike can be repeated to cause a very severe overvoltage. Considering the severe overvoltage that can be caused by the leading current interruption, the leading current interruption capability of circuit breakers is specified in IEC 62271-100 (see Table 3.7).

284


TABLE 3.7 Leading Current Interruption Capability according to IEC 62271-100 Rated Voltage (kV) 420 550 a

Rated Capacitive Switching Currentsa (Cable) (A) 400 500

Preferred values, voltage factor: 1.4 pu.

When the charging capacity of a long EHV cable line is not compensated by shunt reactors directly connected to the cable, the leading current interruption capability requires careful attention [31]. Considering a typical capacitance of 0.2 μF/km, the maximum line length for 400 kV cable line is limited approximately below 26 km without shunt reactors directly connected to the cable. Here, it is assumed that the leading current is interrupted by one end, and the other end is open before the interruption. Usually, long EHV cable lines are compensated by shunt reactors directly connected to the cable. When the compensation rate is high enough, the leading current interruption capability is not a concern. If the sequential switching is applied to a cable line as a countermeasure to the zero-missing phenomenon, however, the tripping of shunt reactors makes the compensation rate lower. This is the only occasion that requires a careful attention. 3.5.4 Cable Discharge If a shunt reactor is directly connected to a cable, the cable line is discharged through the shunt reactor when disconnected from the network. In this case, the time constant of the discharge process is determined by the quality factor (Q factor) of the shunt reactor. Since Q factor ranges around 500 for EHV shunt reactors, the time constant of the discharge process is around 8 min. If the cable is disconnected from the network and energized again within a couple of minutes, there remains a residual charge in the cable line, and the residual charge can be highly dependent on the time separation between the disconnection and the reenergization. Under this condition, the reenergization overvoltage can exceed switching impulse with stand (SIWV) voltage, when the residual voltage has an opposite sign to the source voltage at the time of reenergization. This is usually an issue for overhead lines since auto-reclose is applied to them. For cables, it is not common to apply auto-reclose. If it is applied to cables, they may experience higher overvoltages because of their higher residual voltage. System operators are required to know that they need to wait for about 10 min (maybe more to be conservative) before reenergizing a cable, though it would not be a typical operation to reenergize a cable after a failure.

285


If a shunt reactor is not directly connected to the cable line, the cable line is discharged through inductive voltage transformers (VTs). In this case, the discharge process will be completed within several milliseconds. The inductive VTs need to have enough discharge capability for a cable line to be operated without a shunt reactor or if all the shunt reactors are tripped by sequential switching. It takes several hours for the inductive VTs to dissipate heat after dissipating the cable charge. If the inductive VTs are required to dissipate the cable charge twice within several hours, the required discharge capability will be doubled.

3.6 EMTP Simulation Test Cases Question 1

1. Assume that the example cable in Section 3.2.4 is buried as a singlephase cable, and find the impedance and admittance matrices for the single-phase example cable using EMTP. Use the Bergeron model and calculate the impedance and admittance matrices at 1 kHz. 2. From the impedance and admittance matrices found in (1), find the phase constants for the earth-return mode and the coaxial mode 1 1  using the voltage transformation matrix A =  . 1 0 

3. Find the propagation velocity for coaxial mode and calculate the propagation time when the cable length is 12 km. 4. Using the cable data created in (1), find the propagation time for a 12 km cable with EMTP and compare it with the propagation time theoretically found in (3). Assume that the sheath circuit is solidly grounded with zero grounding resistance at both ends. Solution 1  0.0010 + j 0.0111 Z= 1. 0.0010 + j 0.0102  j1.5068 × 10 −6 Y = −6  − j1.5068 × 10

0.0010 + j 0.0102  [Ω/m] 0.0010 + j 0.0102  − j1.5068 × 10 −6   [mho/m] j1.3095 × 10 −5 

βe = 3.4446 × 10 −4  Np/m  : earth-return mode 2. βc = j 3.5733 × 10 −5  Np/m  : coaxial mode 3. cc = 1.7584×108 [m/s], t = 0.0682 [ms]

286


2

Voltage [kV]

1.6 1.2 0.8

0.0682 ms

0.4 0

0

0.02

0.04

0.06 Time [ms]

0.08

0.1

0.12

FIGURE 3.28 Propagation time of the 12 km cable found by EMTP simulation.

4. A step voltage of 1.0 kV is applied at one end of the 12 km cable at 0 s. The coaxial mode arrives at the other end at 0.0682 ms. The propagation time found in EMTP exactly matches with that found in (3) as shown in Figure 3.28.

Question 2 Calculate zero- and positive-sequence currents using EMTP for the example cable in Section 3.2.4. Assume that the lengths of a minor section and a major section are 400 m and 1200 m, respectively. As the total length of the cable is 12 km, the cable will have 10 major sections. Set the grounding resistance at earthing joints to 10 Ω. Solution 2 (a) Cross-bonded cable Zero Sequence

EMTP simulation Proposed formulas, Equation 3.8/3.14

Positive Sequence

Amplitude (A)

Angle (deg)

Amplitude (A)

Angle (deg)

133.8 124.1

−21.42 −21.23

356.4 356.7

−86.35 −86.79

(b) Solidly-bonded cable Zero Sequence

EMTP simulation Proposed formulas, Equation 3.19, 3.20/3.26

Positive Sequence

Amplitude (A)

Angle (deg)

Amplitude (A)

Angle (deg)

121.6 124.8

−21.80 −22.50

694.9 722.7

−50.40 −49.08


287

References

1. Ametani, A. 1980. A general formulation of impedance and admittance of cables. IEEE Trans. Power Apparatus and Systems. PAS-99(3):902–910. 2. Ametani, A. 1990. Distributed-Parameter Circuit Theory. Tokyo, Japan: Corona Pub. Co. 3. CIGRE WG B1.19. 2004. General Guidelines for the Integration of a New Underground Cable System in the Network. CIGRE Technical Brochure 250. 4. Nagaoka, N. and A. Ametani. 1983. Transient calculations on crossbonded cables. IEEE Trans. Power Apparatus and Systems PAS-102(4). 5. Ametani, A., Y. Miyamoto, and N. Nagaoka. 2003. An investigation of a wave propagation characteristic on a crossbonded cable. IEEJ Trans. PE 123(3):395–401 (in Japanese). 6. IEC/TR 60909-2 ed. 2.0. 2008. Short-circuit currents in three-phase a.c. systems— Part 2: Data of electrical equipment for short-circuit current calculations. 7. Central Station Engineers. 1964. Electrical Transmission and Distribution Reference Book, 4th edn. East Pittsburgh, PA: Westinghouse Electric Corporation. 8. Lewis Blackburn, J. 1993. Symmetrical Components for Power Systems Engineering. Boca Raton, FL: CRC Press. 9. Vargas, J., A. Guzman, and J. Robles. 1999. Underground/submarine cable protection using a negative-sequence directional comparison scheme. 26th Annual Western Protective Relay Conference, Spokane, WA, October 25–28. 10. Gustavsen, B. 2001. Panel session on data for modeling system transients. Insulated cables. Proc. IEEE. Power Engineering Society Winter Meeting, Columbus, OH. 11. Ametani, A. 1980. A general formulation of impedance and admittance of cables. IEEE Transactions on Power Apparatus and Systems PAS-99 (3). 12. Ametani, A. 2009. On the impedance and the admittance in the EMTP cable constants/parameters. European EMTP-ATP Users Group Meeting, Delft, the Netherlands. 13. Scott-Meyer, W. 1982. ATP Rule Book, Can/Am EMTP User Group. 14. IEC 62067 ed. 2.0. 2011. Power cables with extruded insulation and their accessories for rated voltages above 150 kV (Um = 170 kV) up to 500 kV (Um = 550 kV) – Test methods and requirements. 15. Crowley, E., J. E. Hardy, L. R. Horne, and G. B. Prior. 1982. Development programme for the design, testing and sea trials of the British Columbia Mainland to Vancouver Island 525 kV alternating current submarine cable link. CIGRE Session 21-10. 16. Foxall, R. G., K. Bjørløw-Larsen, and G. Bazzi. 1984. Design, manufacture and installation of a 525 kV alternating current submarine cable link from Mainland Canada to Vancouver Island. CIGRE Session 21-04. 17. Cherukupalli, S., G. A. Macphail, J. Jue, and J. H. Gurney. 2006. Application of distributed fibre optic temperature sensing on BC Hydro’s 525 kV submarine cable system. CIGRE Session B1-203. 18. Momose, N., H. Suzuki, S. Tsuchida, and T. Watanabe. 1998. Planning and development of 500 kV underground transmission system in Tokyo Metropolitan Area. CIGRE Session 37-202.

288


19. Kawamura, T., T. Kouno, S. Sasaki, E. Zaima, T. Ueda, and Y. Kato. 2000. Principles and recent practices of insulation coordination in Japan. CIGRE Session 33-109. 20. Dubois, D. 2007. Shibo 500 kV cable project – Shanghai – China. Fall 2007 PESICC Meeting. Scottsdale, AZ. Presentation available at: http://www.pesicc.org/ iccwebsite/subcommittees/G/Presentations/Fall07/6.trans.Presentation_500kV_ Shibo_3.pdf 21. Kaumanns, J. 2012. The Skolkovo challenge: A 550 kV XLPE Cable System with 138 joints put into operation within 17 month after order intake. Proceedings of CIGRE Session 2012, Paris, France. 22. Andersen, P., M. Dam-Andersen, L. Lorensen et al. 1996. Development of a 420 kV XLPE cable system for the metropolitan power project in Copenhagen. CIGRE Session 21–201. 23. JICABLE/WETS Workshop. 2005. Long insulated power cable links throughout the world. Reactive power compensation achievement. Results of the WETS’05 study. 24. Henningsen, C. G., K. B. Müller, K. Polster, and R. G. Schroth. 1998. New 400 kV XLPE long distance cable systems, their first application for the power supply of Berlin. CIGRE Session 21–109. 25. Granadino, R., M. Portillo, and J. Planas. 2003. Undergrounding the first 400 kV transmission line in Spain using 2.500 mm2 XLPE cables in a ventilated tunnel: the Madrid “Barajas. Airport Project. JICABLE ‘03, A.1.2. 26. Argaut, P. and S. D. Mikkelsen. New 400 kV underground cable system project in Jutland (Denmark). JICABLE ‘03, A.4.3. 27. Sadler, S., S. Sutton, H. Memmer, and J. Kaumanns. 2004. 1600 MVA Electrical power transmission with an EHV XLPE cable system in the underground of London. CIGRE Session B1-108. 28. Abu Dhabi Transmission & Despatch Company. 2011 Seven year electricity planning statement (2012–2018). Available at http://www.transco.ae/media/ docs.htm (accessed on June 6, 2013). 29. Colla, L., S. Lauria, and F. M. Gatta. 2007. Temporary overvoltages due to harmonic resonance in long EHV cables. International Conference on Power System Transients (IPST) 233. 30. Colla, L., M. Rebolini, and F. Iliceto. 2008. 400 kV AC New submarine cable links between Sicily and the Italian Mainland. Outline of project and special electrical studies. CIGRE Session C4-116. 31. Ohno, T. 2010. Operation and protection of HV cable systems in TEPCO. Global Facts, Trends and Visions in Power Industry, Swiss Chapter of IEEE PES. Presentation available on the web: http://www.ieee.ch/assets/Uploads/pes/ downloads/1004/10042ohnoexperiencetepco.pdf 32. Leitloff, V., X. Bourgeat, and G. Duboc. 2001. Setting constraints for distance protection on underground cables. In Proc. 2001 7th IEE International Conference on Developments in Power System Protection. 33. Kulicke, B. and H. H. Schramm. 1980. Clearance of short-circuits with delayed current zeros in the Itaipu 550 kV-substation. IEEE Trans. Power Apparatus Syst. PAS-99(4).


289

34. Michigami, T., S. Imai, and O. Takahashi. 1997. Theoretical background for zeromiss phenomenon in the cable network and field measurements. IEEJ General Meeting 1459 (in Japanese). 35. Hamada, H., Y. Nakata, and T. Maekawa. 2001. Measurement of delayed current zeros phenomena in 500 kV cable system. IEEJ General Meeting, Nagoya, Japan. 36. Tokyo Electric Power Company. 2008. Joint feasibility study on the 400 kV cable line Endrup-Idomlund. Final Report.

4 Transient and Dynamic Characteristics of New Energy Systems New energy, or so-called “green” and “sustainable energy,” is becoming very significant, because of problems related to CO2 in thermal power generation and nuclear waste in nuclear power generation. At the same time, “smart grid” is becoming a very attractive research subject. In this chapter, transient and dynamic characteristics of a wind farm composed of many wind turbine generators are described first. The model circuit of the wind farm, the steady-state analysis, and transient calculation are described. Then, modeling of power-electronics circuit elements is described and the thermal calculations by electromagnetic transients program (EMTP) are explained. Photovoltaic generation and even a wind power generation necessitate energy storage, that is, battery. As an application example of a lithium-ion (Li-ion) battery, voltage-regulation equipment for a direct current (dc) railway system is developed based on EMTP simulations. EMTP simulation is explained in detail and a comparison with measured results is carried out. EMTP data lists are given in this chapter.

4.1 Wind Farm 4.1.1 Model Circuit of Wind Farm Figure 4.1 illustrates a model circuit of a wind farm. The plant has 10 generators with a capacity of 3 MW. Each generator is connected to a cable head through a step-up transformer, whose voltage ratio is 22/1 kV. Total capacity of the plant is 30 MW. The generated voltage is stepped up to 66 kV at Substation-L (S/S-L), and the plant is connected to a grid at S/S-K through a cable of 12 km length (Cable #1). The cable lengths in the plant are shown in Table 4.1. The parameters required for a circuit simulation by the EMTP are evaluated as follows:

291

292


S/S-K RB

KB66

LB

LB66

Backward impedance Cable #

66/22k

2

1

AH01

Tr_A

3

4

AH02

AH03

S/S-L

Bold line : cable 66/22k

7 BH01

Tr_B

6 AH04

AH05

Bank-A

AB22 1

5

8

9

BH02

BH03

10

11 BH04

BH05

Bank-B

BB22

FIGURE 4.1 Circuit diagram.

TABLE 4.1 Cable Length Cable #

Size mm2

Length km

1 2 3 4 5 6 7 8 9 10 11

600 250 150 150 60 60 250 150 150 60 60

12.00 5.00 1.50 0.50 0.50 1.00 2.50 0.50 1.50 1.00 0.50

The amplitude of the phase-to-ground voltage for the backward system is

Vm =

2 × 66 = 53.89 [kV] (4.1) 3

The rated terminal voltage of the generator is

Vm =

2 ×1 = 0.81650 [kV] (4.2) 3

The grid impedance % Z is assumed to be j2.5% (10 MVA Base). The inductance LB is obtained as follows:

Transient and Dynamic Characteristics of New Energy Systems

ZB =

% ZBV 2 j 2.5 × 66 2 = = j10.89 [Ω] 100 P 100 × 10

j10.89 Z Z LB = B = B = = 34.664 [mH] j 2π50 jω j 2πf

293

(4.3)

The capacity and impedance of each transformer installed in the substation S/S-L are 18 MVA and j10%, respectively. Those of the step-up transformer are assumed to be 3.5 MVA and j10%. The transformer model installed in the EMTP requires leakage inductances and winding resistances. Although the leakage inductances can be entered as winding data in theory, that of the secondary winding has to be nonzero in the EMTP. In this section, the winding resistances are combined and entered as a primary resistance. The leakage inductances are entered into a data column for the secondary winding. The leakage inductance referred to the low-voltage side is obtained as follows: ZTrAB =

LTrAB

j10 × 222 % ZV 2 = = j 2.6889 [Ω] 100 PTrAB 100 × 18

Z Z = TrB = TrB = 8.5590 [mH] jω j 2πf

(4.4)

The winding resistance is assumed to be 1% of the leakage reactance, that is, the resistive component is almost neglected in this simulation. The resistance referred to the high-voltage side is

RTrAB =

2

2

2.6889  66  ZTrAB  V1  =   = 0.242 [Ω] (4.5) 100  V2  100  22 

The leakage inductance of the step-up transformer for each generator referred to the low-voltage side is obtained as follows: ZTrG =

LTrG

j10 × 12 % ZV 2 = = j 28.571 [mΩ] 100 PTrAB 100 × 3.5

Z Z = TrB = TrB = 75.788 [µH] jω j 2πf

(4.6)

The winding resistance is assumed to be 1% of the leakage reactance. The resistance referred to the high-voltage side is

RTrG =

2

2

0.028571  22  ZTrAB  V1  =   = 0.13829 [Ω] (4.7)   100  V2  100  1 

294


4.1.2 Steady-State Analysis Voltage fluctuation caused by a charging current of cables is an important factor for the design of a wind farm. The voltage fluctuation can be simulated by a steady-state analysis. In the analysis, the cable can be approximately expressed by a lumped-parameter equivalent circuit. Since the steady-state behavior of a three-phase circuit is determined by its positive-sequence component, the wind farm can be expressed by a single-phase circuit. 4.1.2.1 Cable Model A three-phase cable system consisting of three single-core (SC) cables becomes a six-conductor circuit. If its sheath voltages can be neglected, the cable is can be expressed by a three-phase circuit. The voltage drop due to the cable-series impedance is expressed by the following equation:  VCA     Z11  VCB  Z  VCC  21 ∆ ( V )6 = ∆   = −  V SA     VSB   Z61    VSC   

Z12 Z22 Z62

 I CA    Z16   I CB  Z26   I CC    (4.8)   I SA   Z66   I SB       I SC 

= − [ Z ]66 ( I )6

Where the first subscripts “C” and “S” denote core and sheath, and the second subscripts “A,” “B,” and “C” express phases, respectively. The voltage and current vector in Equation 4.8 is defined as  VCA   I CA      (VC ) =  VCB  , ( I C ) =  I CB   VCC   I CC       VSA   VSA      (VS ) =  VSB  , ( I S ) =  VSB  (4.9)  VSC   VSC     

 (IC )   (VC )  (V )6 =   , ( I )6 =    (VS )   (IS ) 

If the sheath voltages are negligible ((VS) = (0)), the 6 by 6 impedance matrix [Z]66 can be reduced to a 3 by 3 matrix [Z]33.

295


 (VC )   (VC )   (IC )  ∆(V )6 = ∆   ≈ ∆  = −[Z]66    (VS )   (0 )   (IS )  (VC )   (IC )  −1    = −[Z]66 ∆    (IS )   (0 )  [ZCC ] = −  [ZSC ]

−1

[ZCS ]  (VC )  ∆   [ZSS ]   (0 ) 

(4.10)

−1  (VC )  = − ([ZCC ] − [ZCS ][ZSS ]−1[ZSC ])  ∆    (0 ) 

∆(VC ) = − ([ZCC ] − [ZCS ][ZSS ]−1[ZSC ]) ( I C ) = − ([ZCC ] − [ZCS ][ZSS ]−1[ZCS ]t ) ( I C )

= −[Z] ( I C )

In the same manner, the admittance matrix of the line can be reduced to a 3 by 3 matrix:  I CA    Y11  I CB  Y  I CC  21 ∆( I )6 = ∆   = −  I  SA    I SB  Y61    I SC   

Y12 Y22

Y62

 VCA    Y16   VCB  Y26   VCC    (4.11)   VSA   Y66   VSB       VSC 

= −[Y ]66 (V )6  (IC )   (VC )   (VC )  ∆( I )6 = ∆    = −[Y ]66   ≈ −[Y ]66   (IS )   (VS )   (0 )  [YCC ] = −  [YSC ]

[YCS ]  (VC )    [YSS ]   (0) 

(4.12)

∆( I C ) = −[YCC ]( I C ) = −[Y ]33 ( I C )

For a steady-state analysis, a cable can be expressed by a single or a cascaded π-equivalent circuit instead of a distributed parameter line. In the EMTP, even if a cable is represented by a constant-parameter line model (Dommel’s line model) or a frequency-dependent line model (Semlyen’s or

296


TABLE 4.2 Technical Data for a Cable Example Nominal cross section of the conductor Outer diameter of the conductor Insulation thickness Thickness of metallic sheath (screen) Thickness of corrosion-proof layer Outer diameter of cable Conductor resistance

S 2r2 di ds dc 2r5 Rdc

600 29.5 10 4 67 30.8

mm2 mm mm mm mm mΩ/km

Marti’s line model), the distributed parameter line is internally converted to a π-equivalent circuit and is passed to a steady-state analysis routine. If the cable impedance and admittance per unit length are provided by the cable manufacturer, the data of the π-equivalent circuit are easily calculated with the cable length. The cable impedance and admittance can be calculated by CABLE CONSTANTS or CABLE PARAMETERS installed in the EMTP using the physical parameters of the cable. In general, the parameters shown in Table 4.2 are provided by the manufacturer. Although the cross section of the conductor S is given, the radius of the conductor r2 is obtained from the outer diameter of the conductor (S ≠ πr22). The inner radius of the metallic sheath r3 is obtained from the radius of the conductor and the thickness of the insulator as r3 = r2+di. The outer radius of the metallic sheath r4 is obtained from the inner radius and the thickness of the sheath as r4 = r3+ds. If the sheath (screen) consists of metallic wires, the thickness of the metallic sheath is assumed to be the diameter of the wire screen. The outer radius of the cable r5 can be directly obtained from the diameter of the cable. If the thickness of the metallic sheath is not given and the thickness of the corrosion-proof layer is given, the outer diameter of the metallic sheath is obtained from the cable radius and the thickness as r4 = r5 − ds. The resistivity of the conductor ρc is obtained from the conductor resistance Rdc and the cross section of the conductor S (ρc = Rdcπr22 ≠ RdcS). In general, the resistivity is greater than that of the intrinsic resistivity of the conductor (e.g., copper: 1.8 × 10−8 Ω-m) because of the gap within the stranded conductor. If the resistivity of the metallic sheath is not provided by the manufacturer, it is obtained by the resistance and its cross section in the same manner as the main conductor. The relative permittivities of the main insulator and the corrosion-proof layer are determined by their materials. For example, the relative permittivity of a cross-linked polyethylene (XLPE) is 2.3. The permittivity of the corrosion-proof layer is widely ranging. However, the value has no effect on the positive-sequence impedance and admittance of the cable. The data for the cable impedance and admittance calculation for EMTP is shown in List 4.1. Table 4.3 shows calculated cable parameters of the XLPE


297

cables. The parameter of the positive sequence is employed for the steadystate voltage simulation. If the cable parameters are provided by the cable manufacturer, the parameter calculation by the CABLE CONSTANTS/ PARAMETERS is not required. From the parameters and cable length, the model parameters are obtained as shown in Table 4.4. List 4.1: Data for Cable Parameter Calculation 1 BEGIN NEW DATA CASE 2 CABLE CONSTANTS 3 PUNCH 4 C TY][SYS][NPC][EAR][KMO][ZFL][YFL][NPP][NGD] 5 2 -1 3 0 1 2 2 3 6 C NP][NCR][IRS][ XMAJOR ][ RSG ]! 7 1 0 0 1000. 1.E-1B 8 C N1][ N2][ N3][ N4][ N4][ N5][ N6][ N7][ N8][ N9][N10][N11][N12][N13][N14][N15] 9 2 2 2 10 C r1 ][ r2 ][ r3 ][ r4 ][ r5 ][ r6 ][ r7 ] 11 C 0.0 4.650E-03 1.165E-02 1.270E-02 1.500E-02 {22kV XLPE 60sqmm 12 C 0.0 7.350E-03 1.435E-02 1.550E-02 1.800E-02 {22kV XLPE 150sqmm 13 C 0.0 9.500E-03 1.650E-02 1.730E-02 2.000E-02 {22kV XLPE 250sqmm 14 0.0 1.475E-02 2.475E-02 2.950E-02 3.350E-02 {66kV XLPE 600sqmm 15 C rohc ][ muc ][ mui1 ][ epsi1 ][ rohs ][ mus ][ mui2 ][ epsi2 ] 16 2.100E-08 1.0 1.0 2.3 2.100E-08 1.0 1.0 3.3 17 C r1 ][ r2 ][ r3 ][ r4 ][ r5 ][ r6 ][ r7 ] 18 C 0.0 4.650E-03 1.165E-02 1.270E-02 1.500E-02 {22kV XLPE 60sqmm 19 C 0.0 7.350E-03 1.435E-02 1.550E-02 1.800E-02 {22kV XLPE 150sqmm 20 C 0.0 9.500E-03 1.650E-02 1.730E-02 2.000E-02 {22kV XLPE 250sqmm 21 0.0 1.475E-02 2.475E-02 2.950E-02 3.350E-02 {66kV XLPE 600sqmm 22 C rohc ][ muc ][ mui1 ][ epsi1 ][ rohs ][ mus ][ mui2 ][ epsi2 ] 23 2.100E-08 1.0 1.0 2.3 2.100E-08 1.0 1.0 3.3 24 C r1 ][ r2 ][ r3 ][ r4 ][ r5 ][ r6 ][ r7 ] 25 C 0.0 4.650E-03 1.165E-02 1.270E-02 1.500E-02 {22kV XLPE 60sqmm 26 C 0.0 7.350E-03 1.435E-02 1.550E-02 1.800E-02 {22kV XLPE 150sqmm 27 C 0.0 9.500E-03 1.650E-02 1.730E-02 2.000E-02 {22kV XLPE 250sqmm 28 0.0 1.475E-02 2.475E-02 2.950E-02 3.350E-02 {66kV XLPE 600sqmm 29 C rohc ][ muc ][ mui1 ][ epsi1 ][ rohs ][ mus ][ mui2 ][ epsi2 ] 30 2.100E-08 1.0 1.0 2.3 2.100E-08 1.0 1.0 3.3 31 C Vert1 ][ Horiz1 ][ Vert2 ][ Horiz2 ][ Vert3 ][ Horiz3 ][ Vert4 ][ Horiz4 ] 32 0.50 -.15 0.50 0.0 0.50 .15 33 C 0.50 -0.3 0.50 0.0 0.50 0.3 34 C rohe ][ Freq. ][DEC][PNT][ DIST ][ IPUN ] 35 100. 50. 1000. 36 BLANK ending frequency data 37 $PUNCH 38 BLANK ending cable constant case 39 BEGIN NEW DATA CASE 40 BLANK

TABLE 4.3 Cable Parameters Voltage kV

Size mm2

R Ω/km

L mH/km

C μF/km

66 22 22 22

600 250 150 60

0.059 0.143 0.202 0.388

0.182 0.509 0.490 0.619

0.247 0.232 0.191 0.139

298


TABLE 4.4 Cable Parameters for π-Equivalent Circuit Cable #

Size mm2

Length km

R Ω

L mH

C μF

Voltage kV

1 1 1 2 3 4 5 6 7 8 9 10 11

600 600 600 250 150 150 60 60 250 150 150 60 60

4.00 4.00 4.00 5.00 1.50 0.50 0.50 1.00 2.50 0.50 1.50 1.00 0.50

0.235 0.235 0.235 0.713 0.303 0.101 0.194 0.388 0.357 0.101 0.303 0.388 0.194

0.726 0.726 0.726 2.544 0.735 0.245 0.310 0.619 1.272 0.245 0.735 0.619 0.310

0.989 0.989 0.989 1.159 0.287 0.096 0.070 0.139 0.579 0.096 0.287 0.139 0.070

66 66 66 22 22 22 22 22 22 22 22 22 22

4.1.2.2 Charging Current Table 4.5a shows the analytical charging currents of the cables

I=

ωCV 2πfCV = (4.13) 3 3

where V is the system voltage, 22 or 66 kV C is the capacitance of the each cable f is the power frequency The charging current of Bank-A in S/S-L is 6.984 A (0.266 MVA) and that of Bank-B is 4.672 A (0.178 MVA). Their ratio is almost identical to the ratio of the cable length. The total charging current of the system is 35.513 A (4.060 MVA), which is mainly determined by the cable connecting S/S-K and S/S-L. The calculated charging currents shown in Table 4.5b are slightly larger than those derived from the analytical calculation. The difference comes from the voltage increase due to the charging, that is, leading current. From the calculated voltage shown in Table 4.6, the 1% voltage rise increases the charging current by 1%. The calculation is carried out by the EMTP using the data shown in List 4.2.

299


TABLE 4.5 Charging Current Length km

Charging Current A

Charging Capacity MVA

Voltage kV

12.0 5.0 1.5 0.5 0.5 1.0 8.5 2.5 0.5 1.5 1.0 0.5 6.0

35.513 4.624 1.145 0.382 0.278 0.556 6.984 2.312 0.382 1.145 0.556 0.278 4.672 11.657 39.399

4.060 0.176 0.044 0.015 0.011 0.021 0.266 0.088 0.015 0.044 0.021 0.011 0.178 0.444 4.504

66 22 22 22 22 22 22 22 22 22 22 22 22 22 66

Current (A)

Power (MW)

Reactive (MVA)

0.00 0.00 0.00

−0.273 −0.183 −4.609

Cable# (a) Analytical result 1 2 3 4 5 6 Subtotal Bank-A 7 8 9 10 11 Subtotal Bank-B Total 22 kV Total 66 kV Node-M

(b) Calculated result Subtotal Bank-A Subtotal Bank-B Total 66 kV

7.08 4.73 39.86

TABLE 4.6 Node Voltages Node KB66_A LB66_A AB22_A AH01_A AH02_A AH03_A AH04_A AH05_A BB22_A BH01_A BH02_A BH03_A BH04_A BH05_A

W/O Gen.

With Gen.

1.0114 1.0118 1.0133 1.0136 1.0136 1.0136 1.0136 1.0136 1.0128 1.0129 1.0129 1.0129 1.0129 1.0129

0.9951 0.9994 0.9901 1.0099 1.0168 1.0186 1.0209 1.0231 0.9904 1.0005 1.0028 1.0081 1.0127 1.0138

300


List 4.2: Voltage Distribution by Cable Charging 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

BEGIN NEW DATA CASE C WER FREQUENCY [STATFR] POWER FREQUENCY 50.0 C FIX SOURCE C DT][ TMAX ][ XOPT ][ COPT ][EPSILN][TOLMAT] 0.0 0.0 C IOUT][ IPLOT][IDOUBL][KSSOUT][MAXOUT][ IPUN ][MEMSAV][ ICAT ][NENERG][IPRSUP] 100 1 1 3 1 C ----------------------------------------------------------- Backward impedance C [BUS1][BUS2][BUS3][BUS4][ R ][ L ][ C ] ! SOUR_AKB66PA 34.664 { ZB, 2.5%, 10MVA } 0 C ------------------------------------------------------- Transformer A 66/22 kV C TRANSFORMER [BUS3] [Iste][Pste][BTOP][Rmag] ! TRANSFORMER LT66AA 9999 C [BUS1][BUS2] [ Rk ][ Lk ][Volt] ! 1LB66AA 0.242 0.0 66. { ! caution Y-Y } 0 2AB22_A 0.08.5590 22. { 18MVA, 10% } 0 C TRANSFORMER [BUS3] [Iste][Pste][BTOP][Rmag] ! TRANSFORMER LT66AA LT66BA 1LB66BA 0 2BB22_A 0 C --------------------------------------------------- Stepup Transformer 22/1 kV C TRANSFORMER [BUS3] [Iste][Pste][BTOP][Rmag] ! TRANSFORMER AT01_A 9999 C [BUS1][BUS2] [ Rk ][ Lk ][Volt] ! 1AH01_A .13829 0.0 22. { ! caution Y-Y 6.0% 2AG01_A .07579 1.0 { 3.5MVA, 10% } TRANSFORMER AT01_A AT02_A 1AH02_A 2AG02_A TRANSFORMER AT01_A AT03_A 1AH03_A 2AG03_A TRANSFORMER AT01_A AT04_A 1AH04_A 2AG04_A TRANSFORMER AT01_A AT05_A 1AH05_A 2AG05_A TRANSFORMER AT01_A BT01_A 1BH01_A 2BG01_A TRANSFORMER AT01_A BT02_A 1BH02_A 2BG02_A TRANSFORMER AT01_A BT03_A 1BH03_A 2BG03_A TRANSFORMER AT01_A BT04_A 1BH04_A 2BG04_A TRANSFORMER AT01_A BT05_A 1BH05_A 2BG05_A C ------------------------------------------------------------------- 22kV CABLE $VINTAGE, 1 C [BUS1][BUS2][BUS3][BUS4][ R11 ][ L11 ][ C11 ] 1KB66_AMB66_A 2.348088570E-01 7.260036593E-01 9.888645200E-01 1MB66_ANB66_A 2.348088570E-01 7.260036593E-01 9.888645200E-01 1NB66_ALB66_A 2.348088570E-01 7.260036593E-01 9.888645200E-01 1AH00_AAH01_A 7.134974733E-01 2.544378987E+00 1.158866150E+00 1AH01_AAH02_A 3.028793115E-01 7.347611338E-01 2.868727050E-01 1AH02_AAH03_A 1.009597705E-01 2.449203779E-01 9.562423500E-02 1AH03_AAH04_A 1.942236877E-01 3.095703631E-01 6.965880500E-02 1AH04_AAH05_A 3.884473753E-01 6.191407261E-01 1.393176100E-01


301

68 1BH00_ABH01_A 3.567487367E-01 1.272189493E+00 5.794330750E-01 69 1BH01_ABH02_A 1.009597705E-01 2.449203779E-01 9.562423500E-02 70 1BH02_ABH03_A 3.028793115E-01 7.347611338E-01 2.868727050E-01 71 1BH03_ABH04_A 3.884473753E-01 6.191407261E-01 1.393176100E-01 72 1BH04_ABH05_A 1.942236877E-01 3.095703631E-01 6.965880500E-02 73 $VINTAGE, 0 74 BLANK ending BRANCH 75 C [BUS1][BUS2] MEASURING ! 76 AB22_AAH00_A MEASURING 1 77 BB22_ABH00_A MEASURING 1 78 LB66_ALB66AA MEASURING 1 79 LB66_ALB66BA MEASURING 1 80 KB66PAKB66_A MEASURING 1 81 BLANK ending SWITCH 82 C ----------------------------------------------------------------- 66 kV system 83 C [BUS1][][ AMP. ][ FREQ. ][ ANGLE ] [ TSTART ][ TSTOP ] 84 14SOUR_A 53.88878E3 50.0 -90.0 -1. 85 BLANK ending Source 86 C [BUS1][BUS2][BUS3][BUS4][BUS5][BUS6][BUS7][BUS8][BUS9][BUSA][BUSB][BUSC][BUSD] 87 KB66_ALB66_A 88 AB22_A 89 AH01_AAH02_AAH03_AAH04_AAH05_A 90 BB22_A 91 BH01_ABH02_ABH03_ABH04_ABH05_A 92 BLANK ending OUTPUT Specification 93 BLANK ending PLOT 94 BEGIN NEW DATA CASE 95 BLANK

4.1.2.3 Load-Flow Calculation It is easily estimated that the voltage distribution within a wind farm depends on the operation of the generators. The effective power of each generator is mainly determined by the phase difference, and the reactive power is determined by the amplitude of the voltage. However, the correct values cannot be obtained by a linear calculation. The EMTP has a load flow feature called “FIX SOURCE”. In this section, an example is shown assuming that every generator operates at its rated capacity and the reactive power is controlled to be zero. The key word “FIX SOURCE” should be inserted before the miscellaneous data (uncomment Line 4 in List 4.2). The source data expressing the generators (List 4.3) are inserted before Line 85 in List 4.2. The amplitudes and angles of the sources are automatically corrected by the FIX SOURCE routine. The entered values are used as initial values of the nonlinear calculation in the FIX SOURCE routine. In general, each amplitude of the source is specified as the amplitude of the phase-to-ground voltage given in equation (4.2). The phase angle is given by the angle of the source of the backward grid specified by Line 84 in List 4.2, because the system is expressed by a single-phase system, in this case. If the system is expressed by a three-phase system, the phase shift by a transformer winding, that is, Y-∆ or ∆-Y connection should be taken into account. List 4.3: Source Data for Generators 1 2 3

C [BUS1][][ AMP. ][ 14AG01_A 8.1650E+2 14AG02_A 8.1650E+2

FREQ. ][ 50.0 50.0

ANGLE ] -90.0 -90.0

[ TSTART ][ -1. -1.

TSTOP ]

302

4 5 6 7 8 9 10 11 12


14AG03_A 8.1650E+2 14AG04_A 8.1650E+2 14AG05_A 8.1650E+2 14BG01_A 8.1650E+2 14BG02_A 8.1650E+2 14BG03_A 8.1650E+2 14BG04_A 8.1650E+2 14BG05_A 8.1650E+2 BLANK ending Source

50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0

-90.0 -90.0 -90.0 -90.0 -90.0 -90.0 -90.0 -90.0

-1. -1. -1. -1. -1. -1. -1. -1.

The generator powers have to be specified by the data shown in List 4.4, followed by the BLANK line for terminating the source data (after Line 85 in List 4.2). The power is one third of the generated power, because the system is expressed by a single-phase model, in this case. The power of the 3 MW generator is specified by Line 3 of the data shown in List 4.4. List 4.4: Additional Data for FIX SOURCE 1 2 3 4 5 6 7 8 9 10 11 12 13 14

BLANK ending Source C [BUS1][BUS2][BUS3][ Pk ][ Qk ][ VMIN ][ VMAX ][THMI][THMA] AG01_A 1.000E+6 0.0 AG02_A 1.000E+6 0.0 AG03_A 1.000E+6 0.0 AG04_A 1.000E+6 0.0 AG05_A 1.000E+6 0.0 BG01_A 1.000E+6 0.0 BG02_A 1.000E+6 0.0 BG03_A 1.000E+6 0.0 BG04_A 1.000E+6 0.0 BG05_A 1.000E+6 0.0 C [NNNOUT][NITERA][NFLOUT][NPRINT][RALCHK][CFITEV][CFITEA][VSCALE][KTAPER] 1 6000 1 0.001 0.005 0.10 2

The miscellaneous data for the load-flow calculation (Line 14 in List 4.4) should be specified, followed by the generator powers. The calculated result is shown in List 4.5. The outputs of the generators are converged to the specified data within 1% error. List 4.5: Calculated Load Flow Exit the load flow iteration loop with counter preceding line, convergence was attained. Row Node Name Voltage magnit Degrees 2 36 AG01_A 8.23225827E+02 -75.33539 3 37 AG02_A 8.28950371E+02 -75.04581 4 38 AG03_A 8.30382307E+02 -74.97387 5 39 AG04_A 8.32251107E+02 -74.91679 6 40 AG05_A 8.34117694E+02 -74.85980 7 41 BG01_A 8.15497500E+02 -75.93822 8 42 BG02_A 8.17424393E+02 -75.83980 9 43 BG03_A 8.21754221E+02 -75.61970 10 44 BG04_A 8.25520356E+02 -75.50343 11 45 BG05_A 8.26461175E+02 -75.47438

NEKITE = 1366.

If no warning on the

Real power P 9.99043389E+05 9.99017237E+05 9.99010832E+05 9.99005868E+05 9.99000958E+05 9.99146319E+05 9.99137898E+05 9.99119250E+05 9.99109641E+05 9.99107270E+05

Reactive power 7.08178870E+01 6.16391826E+01 5.93467166E+01 5.62211791E+01 5.31057736E+01 7.91157970E+01 7.62948678E+01 6.99421381E+01 6.41407115E+01 6.26963934E+01

In general, the load-flow calculation requires a long computational time. If the initial voltages and angles are specified by the data shown in List 4.5, the calculation time of subsequent calculations will be fairly reduced. Figure 4.2 illustrates a calculated result of the cable energization, that is, when all generators are disconnected (w/o gen., Table 4.6) and when the generators

303


1.04 w/o gen.

Voltage (pu)

1.03

with gen.

1.02 1.01 1.00 0.99 0.98

_A _A 1_A 2_A 3_A 4_A 5_A 2_A 1_A 2_A 3_A 4_A 5_A 0 0 0 0 0 0 0 0 0 0 B2 66 B22 B L A BH BH BH BH BH AH AH AH AH AH B

A

6_

6 KB

FIGURE 4.2 Calculated steady-state voltage.

are operated at their rated capacity (with gen.). The voltage increase of the case without the generators is due to the leading current for the cable charging. The minor voltage differences within the plant indicate that the voltage is increased by the cable between the grid and the substation S/S-K. When generators are operated, each voltage increases as the distance between the substation and the generator increases. It is due to the voltage drop caused by the cable impedance. 4.1.3 Transient Calculation Figure 4.3 illustrates the calculated transient responses when the circuit breaker at Bank-B is closed for charging the cables in Bank-B while the generators on Bank-A are fully operated. The initial conditions are determined by the load-flow feature of the EMTP by specifying each output power of the Bank-A generator to be 3 MW, that is, 1 MW for each phase. Although the effect of the closing on the 66 kV bus voltage is minor, the maximum bus voltage of Bank-B is increased to 2 pu and oscillates with a frequency of 1.6 kHz. The oscillation is principally generated by a resonance between the leakage inductance of the transformer LTrB shown in Equation 4.4 and the total cable capacitance of Bank-B shown in Table 4.4 (cable No. 7–11). f = = =

1 2π LTrBC 1

2π ( 8.559 + 8.559 ) mH × ( 0.579 + 0.096 + 0.287 + 0.139 + 0.070 ) µF 1 1 = = 1.590 [kHz] 2π 17.118 mH × 1.171 µF 0.6290 ms

(4.14)

304


80.00

66 kV bus 1 2

2

Voltage (V) (10** 3)

40.00

0.00

12.00

4.00


36.00

–40.00

–80.00

Naoto 12-Nov-12 09:25:22 1 Plot type 4 from file: c:\ncat\crcbook\windg\transing1.ps Node names: KB66_A LB66_A

(a) 40.00

22 kV bus

2

1 2 1


20.00

0.00

4.00

12.00


–20.00

(b)

–40.00

Naoto 12-Nov-12 09:25:22 2 Plot type 4 from file: c:\ncat\crcbook\windg\transing1.ps Node names: AB22_A BB22_A

FIGURE 4.3 Calculated transient voltage: (a) 66 kV bus voltages, (b) 22 kV bus voltages.

36.00

305


A generators

40.00


20.00

0.00

1 2 3

3

4.00

12.00


36.00

–20.00

–40.00

Naoto 12-Nov-12 09:25:22 3 Plot type 4 from file: c:\ncat\crcbook\windg\transing1.ps Node names: AH01_A AH03_A AH05_A

(c) 40.00

B generators

3

1 2 3


20.00

0.00

4.00

12.00


–20.00

(d)

–40.00

Naoto 12-Nov-12 09:25:22 4 Plot type 4 from file: c:\ncat\crcbook\windg\transing1.ps Node names: BH01_A BH03_A BH05_A

FIGURE 4.3 (continued) Calculated transient voltage: (c) Bank-A voltages, and (d) Bank-B voltages.

36.00

306


4.2 Power-Electronics Simulation by EMTP Simulation of a switching circuit is important for the design of equipment using power-electronics technique. For a simulation of an electronics circuit, numerical simulators specialized in the circuit, such as simulation program with integrated circuit emphasis (SPICE), are widely used. Although they have accurate semiconductor models and high functionality to simulate the behavior of the electronics circuit, they have no model of power-system equipment, such as accurate multiphase transmission line models, synchronous machines, etc. The programs cannot be applicable to the power-system analysis including power-electronics equipment, such as an inverter. The solutions are the following: (1) expansion of the electronics simulator to power-system analysis by developing some modes of power apparatuses and (2) expansion of the power-system simulator, such as the EMTP, by developing some modes of semiconductor devices. In this section, the latter method is employed. Models of a bipolar transistor and of a metal oxide semiconductor fieldeffect transistor (MOSFET), along with the simulation techniques by the EMTP are explained with EMTP data in this section. The techniques of these devices are applicable to model an insulated-gate bipolar transistor (IGBT). Transient analysis of controlled systems (TACS) or MODELS installed in the EMTP is indispensable for modeling. These features are originally developed for modeling a control circuit of a power system. They can be applicable to express the characteristics of the semiconductor because their functionality and generality are quite high. In general, the required accuracy of the semiconductor model is lower than that of an electronics-circuit simulator for the power-system simulation. The model should be as simple as possible if the requirement to the accuracy of the power-system simulation is satisfied. 4.2.1 Simple-Switching Circuit Figure 4.4 illustrates a logical inverter (NOT) circuit using a bipolar transistor. The source voltage VCC and the collector resistance RC are assumed to be 5 V and 4.7 kΩ, respectively. The collector current IC becomes

IC =

5 [V] VCC = ≈ 1 [mA] (4.15) RC 4.7 [kΩ]

If the current gain of the transistor hFE is 100, the base current should be greater than 10 μA:

IB ≥

1 [mA] IC = = 10 [µA] (4.16) hFE 100

307


vcc

Sigin

Sigout Rsis 50

Signal square GND

Vb1 Rb1 47 k

Vc1

Rb2 47 k

GND

Rc 4.7 k

Q1 2sc2458

GND

FIGURE 4.4 Switching circuit.

An input signal is applied by a signal source with an internal impedance of 50 Ω (Rsig). The base current IB is obtained from the amplitude of the input voltage VSigout and the base resistance Rb1: IB ≈

VSigin − VBE VSigout − VBE V = , I B BE (4.17) RSig + Rb1 Rb1 Rb 2

Equation 4.17 gives the maximum base resistance Rb1:

Rb1 =

VSigout − VBE VSigout − VBE (5 − 0.7 ) [V] ≤ = = 430 [k kΩ] (4.18) IB IC hFE 1 [mA] 100

In this section, the base resistance Rb1 is assumed to be 47 kΩ. Rb2 is re quired for discharging the charge remaining in the transistor. The resistance is 47 kΩ. 4.2.2 Switching-Transistor Model In this section, a simulation method of basic switching circuit by the EMTP is explained. The technique can be applicable to a power-switching device. The base-emitter characteristic can be expressed by a nonlinear resistor model installed in the EMTP. Although both TYPE-92 and -99 models accept point-by-point data expressing its current–voltage characteristic, the true nonlinear resistor model (TYPE-92) is suitable for the simulation from the viewpoint of stability. The characteristic is easily obtained from the data sheet of the transistor.

308


4.2.2.1 Simple-Switch Model A simplest model of a switching transistor of the EMTP is the TACScontrolled switch model (TYPE-13) illustrated in Figure 4.5. If saturation voltage between the collector and the emitter cannot be negligible, a resistor Ron is inserted in series with the switch. The resistor Rib and the capacitor Cib represent a base spreading resistance and a base input capacitance, respectively. The “OPEN/CLOSE” signal of the TYPE-13 switch is synthesized within the TACS from the base-emitter voltage of the transistor. Figure 4.6 shows the flow chart of the control signal. List 4.6 and Figure 4.7 show the input data for the simple model. The EMTP data are given by a text file, that is, by character user interface (CUI). The ATP-Draw is developed for the data input by graphical user interface (GUI). Although the latter method is easy to use and to grasp the configuration of the circuit, the circuit B

C

Vbe

Rib

SW TYPE–13

Vbeq

Cib

Ron E FIGURE 4.5 Simplest-switching-transistor model.

VBE, VCE

VBE >VBEon

VCE > 0 On FIGURE 4.6 Flowchart for simplest-switching-transistor model.

No

No

Off


309

parameters cannot be obtained from the graphics. In this section, the data are explained using the original data format, CUI. List 4.6: EMTP Data for Simple Switching Circuit (Conventional Format, CUI) 1 BEGIN NEW DATA CASE { -------------------------------------------- C2458NOT1.dat 2 C Logical NOT (Inverter) circuit simulation using 2SC2458 3 C DT][ TMAX ][ XOPT ][ COPT ][EPSILN][TOLMAT] 4 40.E-09 60.E-06 5 C IOUT][ IPLOT][IDOUBL][KSSOUT][MAXOUT][ IPUN ][MEMSAV][ ICAT ][NENERG][IPRSUP] 6 100 2 1 0 1 0 0 1 7 TACS HYBRID 8 C ========================================================== Signal source model 9 C [VAR1] [ Amp. ][ T(s) ][width(s)] [ T-START][ T-END ] 10 23TY23SO 5.0 25.00E-06 12.50E-06 11 C [NAME] +[ IN1] +[ IN2] +[ IN3] +[ IN4] +[ IN5] [GAIN][F-LO][F-HI][N-LO][N-HI] 12 1SIGIN_ +TY23SO 13 1.0 14 1.0 0.050E-06 15 C D0 ][ D1 ][ D2 ][ D3 ][ D4 ][ D5 ][ D6 ][ D7 ] 16 C ============================================================================== 17 C [NODE] 18 90VB1___ { Base voltage 19 90VC1___ { Collector voltage 20 C VE1___ { Emitter voltage 21 C [NAME] =---------------------------- FREE FORMAT ---------------------------22 99VBE1__ = VB1___ { - VE1___ } { Vbe = Vb - Ve 23 99VCE1__ = VC1___ { - VE1___ } { Vce = Vc - Ve 24 C 25 99VBEON_ = 0.62 { threshold Vbe on-voltage 26 C $DISABLE { ====================================== comment out for simple model 27 C [NAME]60+[ IN1] +[ IN2] +[ IN3] [CNST] [SIG1][SIG2] 28 99VBEIF_60+ZERO +ZERO +PLUS1 VBE1__VBEON_ 29 98SW1CTL60+ZERO +ZERO +VBEIF_ VCE1__ZERO 30 C $ENABLE { ====================================== comment out for simple model 31 $DISABLE { ====================================== comment out for delay model 32 C [NAME]53+[ IN1] +[ IN2] +[ IN3] +[ IN4] +[ IN5] [hist][FDel][MaxD][NDel][Nhis] 33 99VBE1DF53+VBE1__ {transport_delay, tdoff} 1.0E-6 34 C [NAME]63+[ IN1] +[ IN2] +[ IN3] +[ IN4] +[ IN5] [CNTL] 35 99VBE1IN63+VBE1__ +VBE1DF {instantaneous max} 1.0 36 C [NAME]60+[ IN1] +[ IN2] +[ IN3] [CNST] [SIG1][SIG2] 37 99VBEIF_60+ZERO +ZERO +PLUS1 VBE1INVBEON_ 38 98SW1CTL60+ZERO +ZERO +VBEIF_ VCE1__ZERO 39 $ENABLE { ====================================== comment out for delay model 40 BLANK ENDING TACS 41 C [BUS1][BUS2][BUS3][BUS4][NFLS] 4444. ! 42 92VB1INT 4444. {Rbe} 43 C R-lin ][ V-flash ][ V-zero ] 44 0.0 0.0 0.0 45 C Current ][ Voltage ] 46 1.00000E-06 -1.00 47 0.00 0.00 48 2.37704E-07 0.539007092 49 8.80783E-07 0.567375887 50 2.35231E-06 0.588652482 51 4.52805E-06 0.602836879 52 8.71621E-06 0.617021277 53 2.04206E-05 0.635460993 54 5.10802E-05 0.655319149 55 0.000105998 0.673758865 56 0.000309901 0.723404255 57 0.000731080 0.763120567 58 9999 59 C SIGOUT VB1___ IB1___ VB1INT VC1___ VCC 60 C +--Rb1--+--Rib--+---+ +--RC--+

310


61 C | | | SW | 62 C Rb2 Cib Reqb | Vcc 63 C | | | Ron | 64 C +-------+-------+---+------+------+ GND (Emitter) 65 C [BUS1][BUS2][BUS3][BUS4][ R ][ L ][ C ] ! 66 SIGIN_SIGOUT 50. 67 SIGOUTVB1___ 47.E3 {Rb1} 1 68 VB1___ 47.E3 {Rb2} 69 VCC VC1___ 4.7E3 {RC} 1 70 RON___ 10.0 {Ron} 71 VB1___IB1___ 50.0 {Rib} 72 IB1___ 0.0 20.E-6 {Cib} 73 BLANK ENDING BRANCH CARDS 74 C [BUS1][BUS2] MEASURING ! 75 IB1___VB1INT {Base current sensor} MEASURING 1 76 C [BUS1][BUS2] CLOSED [CLMP] !! 77 13VC1___RON___ {Collecter-Emitter } SW1CTL 0 78 BLANK ENDING SWITCH CARDS 79 60SIGIN_ {Signal voltage} 80 C [BUS1][][ AMP. ] [ TSTART ][ TSTOP ] 81 11VCC 5.0 {Power supply} 82 BLANK ENDING SOURCE CARDS 83 SIGOUTVB1INTVC1___VB1___IB1___ 84 BLANK ENDING OUTPUT SPECIFICATION CARDS 85 C !![H][ST]END][MIN]MAX][BUS1][BUS2][BUS3][BUS4][ HEADING LABEL][ VERTICAL AXIS] 86 145 5. 10. 60. -10. 10.SIGOUTVB1___VC1___ VOLTAGE (V) 87 195 5. 10. 60. SIGOUTVB1___ BASE CUR CURRENT (A) 88 BLANK ENDING PLOT CARDS 89 BEGIN NEW DATA CASE 90 BLANK

The data of the EMTP is divided into two parts, a TACS or MODELS part for controlling circuit and an electrical part. At first, a square-wave signal source, whose amplitude of 5 V and frequency of 40 kHz, is defined in Line 10 in List 4.6, just after “TACS HYBRID” declaration. The signal is defined by the TYPE-23 built-in source and a first-order transfer function (s-block) (Lines 12–14) to represent its rise and fall time. The output “SIGIN_” is sent to the electrical part of the EMTP and is expressed as a voltage source by a TACS-controlled source (TYPE-60, Line 79). The base and collector voltages (VB1___ and VC1___) in the electrical part are sent to the TACS using TYPE-90 TACS sources (Lines 18 and 19). In this case, the voltages are identical to the base-emitter and the collector-emitter voltage (VBE1__ and VCE1__) because the emitter is directly grounded. If a circuit has an emitter resistor, that is, the emitter voltage is different from zero, the definitions should be modified by a subtraction of the emitter voltage, VE1___, from the base and collector voltages, respectively (Lines 20, 22, and 23). The threshold voltage VBeon in Figure 4.6 can be obtained from the IB-VBE characteristic provided by the manufacturer or an experimental result. The threshold value (Line 25) is determined as the voltage where the base current becomes the minimum base current given in Equation 4.16. The IF-Devices (Device 60) of the TACS (Lines 28 and 29) are used for the logical judgments illustrated in the flowchart (Figure 4.6). The output SW1CTL is used for the “OPEN/CLOSE” signal of TYPE-13 switch in the electrical part (Line 77).

311


V

VCC

VB1

IB1

V

I

V

I

SIGOUT

V

RON

M

VB1INT

VC1

R(i)

V

SIGIN

SW1CTL

T

TY23SO SIGIN_

K 1+T.s

SW1CTL

VB1 F

VBE1 ZERO ZERO PLUS1

T

(a)

T

VC1

F

if

VBEON VBEIF

ZERO ZERO

if

60

VBEIF_

60

VCC

V

VC1

IB1

VB1 I

V

V

I

SIGOUT

T

(b)

SIGIN_

RON

VBE1 F

SW1CTL

VBE1DF

M

K 1+T.s

T

VB1INT

VB1 TY23SO

V M

R(i)

V

SIGIN

SW1CTL

MIN MAX

53

VBE1IN

63

ZERO ZERO PLUS1

F

if

60

ZERO VC1

VBEON ZERO VBEIF ZERO VBEIF_

T if

60

FIGURE 4.7 EMTP data for simple-switching circuit (ATP-Draw, GUI): (a) without delay and (b) with delay.

The next comment-outed data by $DISABLE and $ENABLE (Lines 31–39) are for a delay model, which will be described in the next section. The nonlinear characteristic between the base and emitter of the transistor is expressed by a nonlinear resistor Rbeq. This characteristic can be expressed by a TYPE-92 ZnO arrester model installed in the EMTP (Lines 42–58).

312


The current–voltage characteristic is almost identical to that of a diode, and is given by point-by-point data. The resistors and the capacitor illustrated in Figures 4.4 and 4.5 are specified as RLC branches (Lines 66–72). In this simulation, the base spreading resistance Rib and the base input capacitance Cib are assumed to be 50 Ω and 20 pF, respectively. The MEASURING switch between nodes IB1___ and VB1INT is used for detecting the base current (Line 75). The TYPE-13 switch expresses the switching operation of the transistor (Line 77). The “OPEN/CLOSE” signal SW1CTL is defined in the TACS (Line 29). The TYPE-60 source (SIGIN_) expresses the signal source and the TYPE-11 source is for the power source, VCC (Lines 79 and 81). Figures 4.8 and 4.9 illustrate the measured and the calculated results of the switching circuit. The switching operation can be simply expressed by the simple model, although the time delay at the turn off cannot be reproduced by the model. The model is accurate enough if the switching frequency is much lower than the transition frequency f T of the transistor. 4.2.2.2 Switch with Delay Model The accuracy of the previous model decreases as the frequency of the signal source is increased due to the delay of the transistor. The delay at the turn off is generally greater than that at the turn on. The turn-off delay is easily included into the model using a pulse-delay device (Device 53) and

6 5

Voltage (V)

4 3

Vin

Vc Vbe

2 1 0 –1 –25

–15

–5

5

Time (micros) FIGURE 4.8 Measured result.

15

25

313


5 (V) 4 3 2 1 0

0

10

20

30

40

50

(µs)

60

50

(µs)

60

(file c2485 not 1. pl4; x-vart) v:SIGOUT v:VB1_ v:VC1_

(a) 5 (V) 4 3 2 1 0

0

10

20

30

40


(b)

FIGURE 4.9 Calculated result by simple-switch model: (a) Cib = 20 pF and (b) Cib = 0.

an instantaneous-maximum device (Device 63) of the TACS. Figure 4.10 and List 4.7 (Lines 33–38 in List 4.6) illustrate the control algorithm. The transistor model represented by a switch cannot express the fall time of the collector–emitter voltage VCE, although it reproduces the switching delay time (Figure 4.11). The fall time is quite important for a thermal design of a switching circuit. A more generalized transistor model can be expressed by nonlinear resistances as shown in Figure 4.12.

314


VBE

t

VBE1__ tdoff Delayed VBE

t

Max(Delayed VBE, VBE )

t

t

VCE tdoff

FIGURE 4.10 Block diagram for off-delay representation.

List 4.7: Turn -Off Delay by Device 53 1 2 3 4 5 6 7

C [NAME]53+[ IN1] 99VBE1DF53+VBE1__ C [NAME]63+[ IN1] 99VBE1IN63+VBE1__ C [NAME]60+[ IN1] 99VBEIF_60+ZERO 98SW1CTL60+ZERO

+[ IN2] +[ IN3] +[ IN4] +[ IN5] [hist][FDel][MaxD][NDel][Nhis] {transport_delay, tdoff} 1.0E-6 +[ IN2] +[ IN3] +[ IN4] +[ IN5] [CNTL] +VBE1DF {instantaneous max} 1.0 +[ IN2] +[ IN3] [CNST] [SIG1][SIG2] +ZERO +PLUS1 VBE1INVBEON_ +ZERO +VBEIF_ VCE1__ZERO

The nonlinear characteristic between the collector and emitter cannot be expressed by any conventional nonlinear resistor model, such as TYPE-92 resistor, because its characteristic depends not only on its terminal voltage VCE but also on the base current IB. The resistor Rceq is expressed by the TYPE-91 TACS-controlled resistor. Its resistance is calculated in the TACS according to the collector–emitter voltage VCE, base current IB, and transient characteristic of a transistor. This modeling technique is explained in the next section. 4.2.3 MOSFET MOSFET is widely used as a switching device for high-frequency operation. The drain current of the MOSFET is controlled by its gate-source voltage VGS. Common MOSFET has a reverse diode within its package. Two MOSFET models are introduced in the following sections.

315


5 (V) 4

3

2

1

0

0

10

20

30

40

50

(µs)

60


FIGURE 4.11 Calculated result with turn-off delay.

B

C

Vbeq

Vceq

E FIGURE 4.12 Nonlinear resistor model of transistor.

4.2.3.1 Simple Model The simplest model of a MOSFET with a reverse diode can be expressed by a switch and a diode as illustrated in Figure 4.13. The switch status is controlled by the gate-source voltage VGS. If the voltage is greater than its threshold voltage VP, the switch is closed. The control signal is easily produced by the TACS. If the gate-source voltage VGS is smaller than the threshold voltage VP, the MOSFET acts as a diode for bypassing its reverse current. The model, which consists of a switch and a diode, can be simply expressed by a Type-11 switch (diode) model of the EMTP. The diode model has “OPEN/CLOSE” signal for controlling the switch status. If the signal is positive, the switch is closed as long as the signal is active. If the signal

316


VGS VGS

D

SW D MOS–FET Ron

FIGURE 4.13 MOSFET simple model.

is zero, the switch acts as an ideal diode. The series connected resistor Ron expresses the on-resistance of the MOSFET. However, the model cannot be used from the ATP-Draw, because there is no input column for “OPEN/CLOSE” signal. The switch and the diode have to be separately entered. In this case, the switch and the diode are required to have their own on-resistance, because the EMTP cannot handle parallel connected switches. 4.2.3.2 Modified Switching Device Model The transistor model described in the previous section is simple and useful for low-frequency switching circuit. However, the accuracy of the model becomes insufficient as the switching frequency increases. A modified model of the MOSFET is illustrated in Figure 4.14. The input (gate-source) circuit of a MOSFET is simply illustrated by a capacitor Ciss. The output (drain-source) circuit is expressed by an equivalent resistor Rdseq and a capacitor Coss. The resistance of the equivalent resistor Rdseq is controlled by the gate-source voltage, VGS. The static coupling between the gate and drain is expressed by the reverse transfer capacitance Crss.

G

D Crss Ciss

S FIGURE 4.14 MOSFET model.

Rdseq

Coss

317


ID–VGS

60

COMMON SOURCE VDS = 10 V PULSE TEST

Drain current ID (A)

50 40 30 20

100

10 0

0

1

2

Tc = –55°C 25 3

4

5

6

Gate-source voltage VGS (V) FIGURE 4.15 ID –VGS characteristic of MOSFET 2SK2844.

Figure 4.15 illustrates an example of drain-current versus gate-source voltage (ID –VGS) characteristic. The characteristic should be expressed as accurate as possible for a precise simulation. A function approximation of the characteristic is useful for the accurate representation of the characteristic. In a high-voltage region (VGS ≥ Vlh), the ID –VGS characteristic can be expressed by the following linear equation:

I D = ahVGS + bh , VGS ≥ Vlh (4.19)

In a low-voltage region (VGS < Vlh), the characteristic is approximated by a quadratic function:

I D = al (VGS − Vp )2 , Vp < VGS < Vlh (4.20)

where Vp is the threshold voltage (ID = 0 at VGS = VP) The coefficient al is obtained by a condition in which the both curves are in contact at VGS = Vlh:

al = −

ah 2 (4.21) 4( ahVP + bh )

2b  Vlh = −  Vp + h ah 

  (4.22) 

318


Figure 4.16 illustrates a drain-current versus drain-source voltage (ID –VDS) characteristic. Although the saturating characteristic has no significant effect on switching operations, the on-resistance is an important factor. The resistance is expressed as the inverse of the slope of the ID –VDS characteristic in a low-voltage region. IC –VDS

20

10

3.5 3.25

8

16

Drain-current ID (A)

4

6

12

3

8

COMMON SOURCE Tc = 25°C PULSE TEST 2.75

4 0

VGS = 2.5 V 0

0.2

0.6

0.4

0.8

1.0

Drain–source voltage VDS (V) ID–VDS

100

10

8

80

Drain-current ID (A)

COMMON SOURCE Tc = 25°C PULSE TEST

6

5.5 5

60 4.5

40

4

1/Ro 20 0

3.5

VGS = 3 V 0

2

4

6

Drain–source voltage VDS (V)

FIGURE 4.16 ID –VDS characteristic of MOSFET 2SK2844.

8

10


319

TABLE 4.7 Drain–Source on Resistance Ron VGS = 4 V, ID = 18 A VGS = 10 V, ID = 18 A

Typ.

Max.

26 16

35 20

mΩ mΩ

The on-resistance taken from the data sheet of the MOSFET is shown in Table 4.7. It is clear from Table 4.7 and Figure 4.16 that the on-resistance depends on the gate-source voltage and it decreases as the voltage increases. The resistance can be approximated by the following equation: − Ronτ Ron = Ron 0VGS (4.23)

If the saturation on ID –VDS characteristic has to be expressed, the characteristic can be approximated by the following function involving an exponential function:

  V − Ron I D   I D = I D max 1 − exp  − DS   (4.24) Vτ   

where IDmax is the maximum drain current given by Equation 4.19 or 4.20. Table 4.8 shows the parameters of 2SK2844 MOSFET for its static characteristics. Transient (dynamic) characteristic of a MOSFET is determined by capacitors and by behavior of carriers in the device. In general, the capacitance is larger than that of a bipolar transistor. Table 4.9 shows the capacitances taken from the data sheet of 2SK2844 MOSFET. TABLE 4.8 Model Parameters al Vp Vlh Ron0

12.2 A/V2

ah

28.3 A/V

1.93 V 3.09 V 60 mΩ

bh Vτ Ronτ

−71.1 A 0.15 V 0.59

TABLE 4.9 Typical Capacitance Input capacitance Reverse-transfer capacitance Output capacitance VDS = 10 V, VGS = 0 V, f = 1 MHz

Ciss Crss Coss

980 pF 270 pF 580 pF

320


TABLE 4.10 Typical Switching Time Rise time Turn on time Fall time Turn off time

tr ton tf toff

14 ns 23 ns 64 ns 190 ns

The capacitance cannot be negligible in a high frequency switching operation. In most cases, transient overvoltage generated in a switching circuit is caused by a resonance between the capacitors and stray inductors in the switching circuit. A switching characteristic of a MOSFET is expressed by parameters shown in Table 4.10. Although the physical behavior of a MOSFET is too complicated for an EMTP simulation, the operational characteristic can be reproduced with satisfactory accuracy from the viewpoint of a numerical simulation of a power system including power-electronics apparatuses. A simple representation method of the dynamic characteristic is proposed in this section. Figure 4.17 illustrates a schematic diagram of signals for representing the transient-switching characteristic of a MOSFET. The gate-source voltage VGS is delayed by ton and by toff using transport delay

VGS

t

VGS1d

t ton

VGS2d

t toff

Max(VGS1d, VGS1d)

t

t

VDS ton

FIGURE 4.17 Control signal for Rdseq.

toff

321


devices (Device 53). The rise time tr is expressed by an s-block F(s) from the delayed signals: F(s) =

1 (4.25) 1 + sτ

The time constant τ is obtained as a solution of the following equations:  t  0.1 = 1 − exp  − 1   τ  t  t   t +t  0.9 = 1 − exp  − 1 r  = 1 − exp  − 1  exp  − r τ τ      τ

  (4.26) 

The time constant τ becomes τ=

tr = 0.455tr (4.27) ln(9)

In the same manner, the time constant for the fall time tf is obtained. An instantaneous maximum device (Device 63) gives an equivalent gatesource voltage from the deformed signals (VGS1d and VGS2 d). 4.2.3.3 Simulation Circuit and Results Figure 4.18 illustrates a switching circuit using MOSFET 2SK2844, and List 4.8 and Figure 4.19 show the input data for an analysis using the switch (diode)

15 V VCC

Rg 15

Vin

Signal square GND FIGURE 4.18 Switching circuit.

Rsis 50

Rg1 50

Q3 2SK2844 Rg2 50 GND

GND

D

322


model. A square-wave voltage, whose amplitude is 10 V and frequency is 200 kHz (Lines 10–14, and 63 in List 4.8), is applied to the gate through the resistors Rsig and Rg1 (Lines 48 and 49 in List 4.8). A voltage source (Vcc) of 15 V amplitude is applied to the drain through the resistor Rd of 15 Ω (Lines 54 and 65 in List 4.8). Thus, the drain current becomes 1 A. The sum of the stray inductance of the drain resistor Rd and the source Vcc is 1.6 μH (Line 54 in List 4.8). List 4.8: MOSFET Switching Circuit (Switch Model, Conventional Format, CUI) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

BEGIN NEW DATA CASE { ------------------------------------------- k2844notsw.dat C Switching circuit simulation (device MOSFET 2SK2844) C DT][ TMAX ][ XOPT ][ COPT ][EPSILN][TOLMAT] 2.E-09 50.E-6 C IOUT][ IPLOT][IDOUBL][KSSOUT][MAXOUT][ IPUN ][MEMSAV][ ICAT ][NENERG][IPRSUP] 100 2 1 0 1 0 0 1 TACS HYBRID C ========================================================== Signal source model C [VAR1] [ Amp. ][ T(s) ][width(s)] [ T-START][ T-END ] 23TY23SO 10.0 5.000E-06 2.500E-06 { 200kHz, square wave C [NAME] +[ IN1] +[ IN2] +[ IN3] +[ IN4] +[ IN5] [GAIN][F-LO][F-HI][N-LO][N-HI] 1SIGIN_ +TY23SO 1.0 1.0 0.010E-06 { freq. characteristic of signal source C D0 ][ D1 ][ D2 ][ D3 ][ D4 ][ D5 ][ D6 ][ D7 ] C ============================================================================== C [NODE] ===================================================== voltage sensors 90VG1___ { Gate voltage 90VD1___ { Drain voltage C ============================================================================== C [NAME] =---------------------------- FREE FORMAT ---------------------------99VGS1__ = VG1___ { VGS = VG - VS, source is grounded (VS=0) 99VDS1__ = VD1___ { VDS = VD - VS, source is grounded (VS=0) 99NZERO = 1.E-9 {small non-zero value 99R1ONM_ = 16.E-3 {minimum on resistance} 99VGSON_ = 1.930434783 {threshold VGS on-voltage C $DISABLE { comment out for simple sw model C ===================================== MOS-FET = sw (diode) model without delay C [NAME]60+[ IN1] +[ IN2] +[ IN3] [CNST] [SIG1][SIG2] 99SW1CTL60+ZERO +ZERO +PLUS1 VGS1__VGSON_ C $ENABLE { comment out for simple sw model $DISABLE { comment out for sw with delay model C ======================================== MOS-FET = sw (diode) model with delay C [NAME]53+[ IN1] +[ IN2] +[ IN3] +[ IN4] +[ IN5] [hist][FDel][MaxD][NDel][Nhis] 99VGS1DF53+VGS1__ {transport_delay, tdr} .25E-6 C [NAME]63+[ IN1] +[ IN2] +[ IN3] +[ IN4] +[ IN5] [CNTL] 99VGS1IN63+VGS1__ +VGS1DF {instantaneous max} 1.0 99SW1CTL60+ZERO +ZERO +PLUS1 VGS1INVGSON_ $ENABLE { comment out for sw with delay model BLANK ENDING TACS C SIGOUT VG1___ VD1___ VCC C +--Rg1---+-----------+ +--RD--+ C | | | | C Rg2 Cgs Rds Vcc C | | | | C +--------+-----------+------+------+ GND (Emitter) C [BUS1][BUS2][BUS3][BUS4][ R ][ L ][ C ] ! SIGIN_SIGOUT 50. {Rsig} SIGOUTVG1___ 50. {Rg1} 1 VG1___ 50. {Rg2} 1 VG1___ .98E-3 {Cgs} C VG1___VD1___ .27E-3 {Cgd} VD1___ .58E-3 {Cds} VCC___VD1___ 15.1.6E-3 {Rd} 1

323


55 C [BUS1][BUS2][BUS3][BUS4][ R ][ L ][ C ] ! 56 RON___ 30.E-3 57 BLANK ENDING BRANCH CARDS 58 C [BUS1][BUS2] MEASURING ! 59 P1DS__HSINK1 MEASURING 60 C [BUS1][BUS2] CLOSED [CLMP] !! 61 11RON___VD1___ {Source-Drain} SW1CTL 0 62 BLANK ENDING SWITCH CARDS 63 60SIGIN_ {Signal voltage} 64 C [BUS1][][ AMP. ] [ TSTART ][ TSTOP ] 65 11VCC___ 15.0 {Power supply} 66 BLANK ENDING SOURCE CARDS 67 SIGOUTVB1INTVD1___VG1___HSINK1 68 BLANK ENDING OUTPUT SPECIFICATION CARDS 69 C !![H][ST]END][MIN]MAX][BUS1][BUS2][BUS3][BUS4][ HEADING LABEL][ VERTICAL AXIS] 70 145 1. 5.0 15. SIGOUTVG1___VD1___ VOLTAGE (V) 71 BLANK ENDING PLOT CARDS 72 BEGIN NEW DATA CASE 73 BLANK

The comment-outed data by $DISABLE and $ENABLE (Lines 32–39 in List 4.8) expressing the turn off delay of the MOSFET. The technique is identical to that of the switching transistor explained in the previous section. List 4.9 and Figure 4.20 show an input data for the switching circuit using the nonlinear resistor model. In this data, the dynamic on-resistance (Equation 4.23) and the saturation characteristic (Equation 4.24) are neglected because there are minor effects on the result. The data also includes a thermal model described in the next section.

V

SIGIN_

I

VCC VG1_ V

I

V

VD1_

SIGOUT RONSW

RON

TY23SO K

1+T.s T

SIGIN_

F

T

VGS1_ VGS1DF M

VG1

T

T

MIN MAX

53

T

VGS1IN

63

ZERO ZERO PLUS1 FIGURE 4.19 MOSFET switching circuit (switch model, ATP-Draw, GUI).

F VGSON_

if

60

SW1CTL

324


List 4.9: MOSFET Switching Circuit (Nonlinear Model with Heat Sink, CUI) 1 BEGIN NEW DATA CASE { ------------------------------------------k2844NOTnonl.dat 2 C Switching circuit simulation (device MOSFET 2SK2844) 3 C DT][ TMAX ][ XOPT ][ COPT ][EPSILN][TOLMAT] 4 2.E-09 50.E-6 5 C IOUT][ IPLOT][IDOUBL][KSSOUT][MAXOUT][ IPUN ][MEMSAV][ ICAT ][NENERG][IPRSUP] 6 100 2 1 0 1 0 0 1 7 TACS HYBRID 8 C ========================================================== Signal source model 9 C [VAR1] [ Amp. ][ T(s) ][width(s)] [ T-START][ T-END ] 10 23TY23SO 10.0 5.000E-06 2.500E-06 { 200kHz, square wave 11 C [NAME] +[ IN1] +[ IN2] +[ IN3] +[ IN4] +[ IN5] [GAIN][F-LO][F-HI][N-LO][N-HI] 12 1SIGIN_ +TY23SO 13 1.0 14 1.0 0.010E-06 { freq. characteristic of signal source 15 C D0 ][ D1 ][ D2 ][ D3 ][ D4 ][ D5 ][ D6 ][ D7 ] 16 C ============================================================================== 17 C [NODE] ===================================================== voltage sensors 18 90VG1___ { Gate voltage 19 90VD1___ { Drain voltage 20 C ============================================================================== 21 C [NAME] =---------------------------- FREE FORMAT ---------------------------22 99VGS1__ = VG1___ { VGS = VG - VS, source is grounded (VS=0) 23 99VDS1__ = VD1___ { VDS = VD - VS, source is grounded (VS=0) 24 99NZERO = 1.E-9 {small non-zero value 25 99R1ONM_ = 16.E-3 {minimum on resistance} 26 99VGSON_ = 1.930434783 {threshold VGS on-voltage 27 C =============================== MOS-FET = nonlinear resistor model based on gm 28 99GM1INM = 12.15776308*(VGS1__-VGSON_)**2 / VDS1__ {Id-Vgs in small Vgs region 29 99GM1INH = (28.3*VGS1__-71.1) / VDS1__ {Id-Vgs in large Vgs region 30 C [NAME]60+[ IN1] +[ IN2] +[ IN3] [CNST] [SIG1][SIG2] 31 99GM1INL60+NZERO +GM1INM +GM1INM VGS1__VGSON_ 32 99GM1IN_60+GM1INL +GM1INH +GM1INH 3.0943 VGS1__ZERO 33 C [NAME]53+[ IN1] +[ IN2] +[ IN3] +[ IN4] +[ IN5] [hist][FDel][MaxD][NDel][Nhis] 34 99GM1DON53+GM1IN_ {transport_delay, ton} 23.E-9 35 1GM1TR_ +GM1DON 36 1.0 37 1.0 6.37E-09 { 0.455*tr } 38 99GM1DOF53+GM1IN_ {transport_delay, toff} .19E-6 39 1GM1TF_ +GM1DOF 40 1.0 41 1.0 29.12E-09 { 0.455*tf } 42 C [NAME]63+[ IN1] +[ IN2] +[ IN3] +[ IN4] +[ IN5] [CNTL] 43 99GM1INT63 GM1TR_ GM1TF_ {instantaneous max} 1.0 44 99R1EQIS = 1 / GM1INT {eq. R between D & S} 45 C 46 99MAXR__ = 10000. 47 99R1EQIT63 MAXR__ R1EQIS {instantaneous max} -1.0 48 99R1EQIU63 R1ONM_ R1EQIT {instantaneous max} 1.0 49 1R1EQDS +R1EQIU 50 1.0 51 1.0 0.020E-06 52 C [NAME]63+[ IN1] +[ IN2] +[ IN3] +[ IN4] +[ IN5] [CNTL] 53 99ID1EST = VDS1__ / R1EQDS {estimated collector current} 54 C $ENABLE { comment out for nonlinear model 55 C ====================================================== Temperature calculation 56 99P1DS__ = VDS1__**2/R1EQDS 57 90HSINK1 58 1TEMP1_ +HSINK1 59 1.0 60 1.0 5.0E-6 61 C [VAR1][VAR2][VAR3][VAR4][VAR5][VAR6][VAR7][VAR8][VAR9][VARA][VARB][VARC][VARD] 62 33R1EQDS 63 33ID1EST 64 33GM1TR_GM1TF_GM1IN_GM1INT 65 33P1DS__HSINK1TEMP1_


325

66 BLANK ENDING TACS 67 C SIGOUT VG1___ VD1___ VCC 68 C +--Rg1---+-----------+ +--RD--+ 69 C | | | | 70 C Rg2 Cgs Rds Vcc 71 C | | | | 72 C +--------+-----------+------+------+ GND (Emitter) 73 C [BUS1][BUS2][BUS3][BUS4][ R ][ L ][ C ] ! 74 SIGIN_SIGOUT 50. 75 SIGOUTVG1___ 50. {Rg1} 1 76 VG1___ 50. {Rg2} 1 77 VG1___ .98E-3 {Cgs} 78 C $DISABLE 79 VG1___VD1___ .27E-3 {Cgd} 80 C $ENABLE 81 VD1___ .58E-3 {Cds} 82 VCC___VD1___ 15.1.6E-3 {Rd} 1 83 C [BUS1][BUS2]TACS [BUS4] ! 84 91VD1___ TACS R1EQDS {Rce} 4 85 C ============================================== Thermal model for TO220 package 86 CH1___HSINK1 2.08 { K/W channel to case, Thermal resistance 87 CH1___ 83.30 { K/W channel to ambient, Thermal resistance 88 HSINK1 .16663 { J/K *10^6 1/40 Thermal capacity 89 $DISABLE { comment out to take into account the heat-sink 90 HSINK1 .01666 { J/K *10^7 1/40 Thermal capacity 91 C ============================================================== Heat-Sink model 92 C [BUS1][BUS2][BUS3][BUS4][ R ][ L ][ C ] ! 93 HSINK1 17.30 { K/W case to ambient, Thermal resistance 94 HSINK1 .66652 { J/K *10^7 Thermal capacity 95 C ============================================================================== 96 $ENABLE { comment out to take into account the heat-sink 97 BLANK ENDING BRANCH CARDS 98 C [BUS1][BUS2] MEASURING ! 99 P1DS__HSINK1 MEASURING 100 BLANK ENDING SWITCH CARDS 101 60SIGIN_ {Signal voltage} 102 C [BUS1][][ AMP. ] [ TSTART ][ TSTOP ] 103 11VCC___ 15.0 {Power supply} 104 C 105 60P1DS__-1 {Thermal 106 BLANK ENDING SOURCE CARDS 107 SIGOUTVB1INTVD1___VG1___HSINK1 108 BLANK ENDING OUTPUT SPECIFICATION CARDS 109 C !![H][ST]END][MIN]MAX][BUS1][BUS2][BUS3][BUS4][ HEADING LABEL][ VERTICAL AXIS] 110 145 1. 5.0 15. SIGOUTVG1___VD1___ VOLTAGE (V) 111 195 1. 5.0 15. TACS R1EQDS EQUIV. R RESISTANCE (OHM) 112 195 1. 5.0 15. -1.0 1.0TACS ID1ESTVCC___VD1___ ID CURRENT (A) 113 195 1. 5.0 15. BRANCH CONDUCTANCE CONDUCTANCE (S) 114 TACS GM1TR_TACS GM1TF_TACS GM1IN_TACS GM1INT 115 195 1. 5.0 15. TACS P1DS__ POWER POWER (W) 116 195 5. 0.0 50. TACS HSINK1TACS TEMP1_ TEMP. RISE TEMPERATURE (K) 117 BLANK ENDING PLOT CARDS 118 BEGIN NEW DATA CASE 119 BLANK

Figure 4.21a illustrates a measured result of the switching circuit. Highfrequency oscillations are observed on the drain voltage just after the MOSFET is turned off. Its frequency is determined by the stray inductance of the drain circuit and the capacitance of the MOSFET. The oscillation is induced on the gate voltage through the transfer capacitance Crss. Figure 4.21b shows the results obtained by the simplified switch model. The amplitude of the high-frequency oscillation is far greater than that of the measured result, because the model cannot correctly represent the resistance

F GM1INH

F

F

GM1INM

SIGIN_

I

VDS1

F

F

NZERO

VGS1

F

VGSON_

K 1+T.s

V

VG1

T

if

60

D1

GM1INL

V

if

60

FIGURE 4.20 MOSFET switching circuit (nonlinear model with heat sink, GUI).

VD1

VG1

T

I

SIGOUT

V

TY23SO

SIGIN_

T

T

I

VCC

F

CH1

K 1+T.s

GM1TR_

K 1+T.s

P1DS_ _

M

GM1DON

T

P1DS

GM1TF_ GM1DOF

53

53

GM1IN_

ZERO

M M

y

x y

GM1NT

63

MIN MAX

PLUS1 x

HSINK1 T

63

MAX

K 1+T.s

F

63

R1EQIU R1ONM_

MIN MAX

K 1+T.s

R1EQDS

TEMP1_

T

MAXR_ R1EQIT MIN R1EQIS

F

HSINK1

T


327


within the MOSFET, which attenuates the oscillation. The switching delay is not represented in this case, because the Type-11 switch is directly controlled by the gate-source voltage VGS. If the delayed signal explained in the previous section is used as the control signal, the delay could be approximately introduced. In the simulation case, the transfer capacitor Crss is neglected for a stable calculation. If the high-frequency oscillations are induced on the

25 VDS

Voltage (V)

20

Vin VGS

15 10 5 0 –5

(a)

–5

–4

–3

–2

–1

1 0 Time (µs)

2

3

80.00

4

5

1 3

2 3

Voltage (V)

40.00

0.00

1 2 6.00

8.00

10.00 12.00 Microseconds

14.00

–40.00

–80.00 (b)

ATPWNT 29-JUN-06 14.34.55 2 PLOT TYPE 4 FROM FILE: C:\NCAT\EMTP\EEUG2006\SWTR\K2844NOT3.PS NODE NAMES : SIGOUT VG1__VD1__

1 µs/div, 20 V/div, 1:Vin, 2: VGS, 3: VDS

FIGURE 4.21 Responses of switching circuit: (a) measured result, (b) calculated result by switch model. (continued)

328


80.00

1 2

3

3

Voltage (V)

40.00

0.00

1 2 6.00

8.00


14.00

–40.00

–80.00

ATPWNT 19-JUN-06 12.06.56 1 PLOT TYPE 4 FROM FILE: C:\NCAT\EMTP\EEUG2006\SWTR\K2844NOTSW.PS NODE NAMES : SIGOUT VG1__VD1__

(c) 40.00

1 2

3

3

Voltage (V)

20.00 1 0.00

6.00

2 8.00


14.00

–20.00

–40.00 (d)

ATPWNT 29-JUN-06 00.54.11 2 PLOT TYPE 4 FROM FILE: C:\NCAT\EMTP\EEUG2006\SWTR\K2844NOT3.PS NODE NAMES : SIGOUT VG1__VD1__

1 µs/div, 20 V/div, 1:Vin, 2: VGS, 3: VDS

FIGURE 4.21 (continued) Responses of switching circuit: (c) calculated result by switch model (with delay), and (d) calculated result by proposed model.


329

gate-source voltage at around the threshold voltage Vp through the transfer capacitor, the switching operation becomes unstable. Figure 4.21c illustrates the results by the switch model with the turnoff delay. Even if the switching delay is included into the simulation, the accuracy is not improved. Figure 4.21d illustrates the results by the accurate model. The highfrequency oscillation as well as the switching delay is accurately reproduced. The loss of the MOSFET reduces the oscillation. 4.2.4 Thermal Calculation The EMTP has been widely used for estimating transient overvoltages on a power system. In a power-electronics field, prediction of temperature rise and the overvoltages in the switching circuit is one of the important aims of numerical simulations. The result provides valuable information for designing power-electronics apparatuses. Thermal equation is analogous to electrical-circuit equation. Electrical power consumption P corresponds to a current source. Static heat-transfer properties are usually specified using a thermal resistance Rθ that defines a relation between heat flow per unit time Q and temperature difference θ = T – T0: Q=

1 θ (4.28) Rθ

The thermal capacitance Cθ is specified to model the dynamical properties of the heat transfer: Q = Cθ

dθ (4.29) dt

An equivalent equation can be derived using electrical quantities instead of thermal quantities. The relations between the electrical and thermal quantities are given in Table 4.11.

TABLE 4.11 Relations between Thermal and Electrical Quantities Thermal Quantity Heat flux Q Temperature difference θ Thermal resistance Rθ Thermal capacitance Cθ

Electrical Quantity Current I Voltage V Resistance R Capacitance C

330


The equivalent voltage V is proportional to the difference between an absolute temperature T and a reference temperature T0. Usually, T0 is selected to be an ambient temperature. The equivalent current source I transfers heat to the thermal circuit, and its value is proportional to power dissipation P in an electrical component. Time constant of a thermal-equivalent circuit is usually far greater than that of an electrical circuit. An accelerating coefficient at is introduced to compress the difference. Equation 4.30 is transformed into “accelerated domain”: t = att′ (4.30)

Q = Cθ

dθ Cθ dθ = (4.31) dt at dt′

In the accelerated domain, the thermal capacitance is inversely proportional to the accelerating coefficient at. Figure 4.22 illustrates a thermal-equivalent circuit for a switching device. The resistor Rthca expresses the heat resistance between the channel of the MOSFET and the ambient air. The resistor Rthch is for between the channel and the package. The resistor Rthrh expresses the heat radiation from the package to the ambient air. The resistance is generally determined by a heat sink. The capacitor Cthh corresponds to the thermal capacitance of the heat sink. If dynamic heat characteristic is not important, the capacitor is negligible. The current source Pth is power dissipation within the channel. The voltage across the capacitor V expresses the temperature rise of the MOSFET. In this section, two cases are investigated: (a) without heat sink and (b) with heat sink. Tables 4.12 and 4.13 show the circuit parameters used in this section. Figure 4.23 illustrates calculated results by the proposed model.

Rthch

Pth

Rthca

Rthrh

Cthh

FIGURE 4.22 Thermal-equivalent circuit.

TABLE 4.12 Thermal Characteristics of 2SK2844 MOSFET Characteristics Channel to case Channel to ambient

Symbol

Max

Unit

Rthch Rthca

2.08 83.3

°C/W °C/W

V


331

TABLE 4.13 Thermal Characteristics of Heat Sink Characteristics Thermal resistance Surface area Weight Specific heat of aluminum Thermal capacity

Symbol

Max

Unit

Rthha Sh Mh

17.3 42 7.6 0.877 6.67

°C/W cm2 g J/g°C J/°C

Cths

The voltage across the thermal capacitor with a saw-tooth oscillation is smoothed by a first-order s-block with a time constant of 5 μs. Figure 4.23a shows the result when the heat sink is removed and the acceleration coefficient at of 106 is applied. Although the maximum observation time Tmax is 50 μs, the smoothed waveform expresses the change of the temperature up to 50 s (=Tmax × at). The result with the heat sink is illustrated in Figure 4.23b and is obtained with at = 107. The power consumption of the MOSFET is determined by its switching loss, because its on-resistance is small enough for the application. The periodic power consumption causes the oscillations in the results. The oscillations are not observed in practical situations. The difference is caused by the acceleration introduced for saving computational time. The error can be easily suppressed by a smoothing filter. The calculated results show that the temperature rise is 20°C when the heat sink is removed. The measured surface temperature of the MOSFET is 46.2°C when the ambient temperature is 25.7°C. The difference is 20.5°C, which agrees with the calculated result. Figure 4.23b shows that the heat sink reduces the temperature rise to 3°C and the temperature converges at about 500 s (=Tmax × at).

4.3 Voltage Regulation Equipment Using Battery in a DC Railway System 4.3.1 Introduction A numerical simulation method of a voltage regulator using Li-ion battery in a dc railway system using EMTP with TACS is proposed in this section. A couple of TACS-controlled resistors are used for representing each line resistance within a feeding section for expressing a train operation. In addition, a power compensator, which regulates the line voltage, can be expressed by the functions installed in the TACS. The calculated result of the system including the voltage regulator agrees well with the measured result of a practical train system. The proposed method indicates the

332


25 20

15

10

5 0

0

10

20

30

40

(µs)

50

40

(µs)

50

(File k2844notnonl.pl4; x-vart) t: P1DS_t

(a) 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 (b)

0

10

20

30

(File k2844notnonl.pl4; x-vart) t: P1DS_t 5 s/div, 5 K/div

50 s/div, 0.5 K/div

FIGURE 4.23 Calculated results of temperature rise: (a) without heat-sink (at = 106) and (b) with heat-sink (at = 107).

optimal installing position and capacity of the compensator. The numerical simulation using the EMTP enables a computer-aided design of feeding circuits of a dc train system. A voltage distribution along the feeding line of a dc railway system is determined by output voltages of substations and by line voltage drops. The voltage drop is proportional to the train current, resistance of the feeding


333

line, and line length between the substation and the train, that is, the train position. The line voltage is lowered by the current of a powering train and is increased by the regenerative current of a braking train. In recent years, line-voltage fluctuation of the feeding system has become larger because the train current is increased for increasing transportation capacity. The line voltage should be kept at its rating voltage as much as possible. The voltage drop caused by a train operation is proportional to the distance between a substation and a train. Some apparatuses for stabilizing the line voltage have been proposed. A substation with a voltage regulator, which consists of thyristors and a controller, is effective for compensating the voltage drop caused by the powering train. Recently, power compensators with some kind of electrical storage device have been developed to stabilize the voltage and to increase the efficiency of dc train systems [9]. The regenerative energy is stored in Li-ion batteries or electrical double-layer capacitors. Power compensators can be installed on the ground or in a train. A verification test of a prototype of the storage system installed on the ground is currently being carried out [10–13]. The compensator releases the stored energy to powering trains and has the capability to store the regenerative energy of a braking train. To design an efficient feeding system including the voltage regulator and/ or the various compensators, an estimation of the voltage and current distributions is indispensable. Numerical simulation is one of the solutions for estimation. For an accurate simulation of the train feeding system, movement of trains has to be taken into account. Thus, the line resistance should be expressed by time-dependent resistors for expressing the train operation. A flexible modeling capability is also required for a circuit-simulation program for expressing various characteristics of compensators. The EMTP has been widely used as a standard transient-analysis program in the field of power-system simulation. TACS, which is installed in the EMTP as a modeling tool of the control system, is suitable for the simulation of train feeding system. The line resistance taking into account the moving train is expressed by a TACS-controlled resistor. In addition, the power compensators can be expressed by the functions installed in the TACS. 4.3.2 Feeding Circuit Figure 4.24 illustrates a feeding circuit investigated in this chapter. The circuit has five substations. The substations S/S2 and S/S3 have conventional voltage regulators, which consist of thyristors. A power compensator is installed between these substations. In the figure, the line length is expressed by lk and the train position is represented by distance lt from the substation S/S1 as a function of time. The line lengths are shown in Table 4.14. These values are taken from a practical feeding system. The total length of the system is 47.4 km. The resistance of the feeding line Rl, including the rail resistance, is assumed to be 40 mΩ/km.

334


Compensator S/S2

S/S1 l1

l2

i0 #0

#1

S/S3 l3

l4

S/S4 l5

l6

S/S5 l7

l8 l9

lt

#2

#3

#4 #5 Node number

#6

#7

#8

FIGURE 4.24 Feeding circuit.

TABLE 4.14 Distance between Substations Section Distance

l1, 2 6.3 km

l3, 4 7.4 km

l5, 6 5.05 km

l7, 8 4.95 km

The rated voltage of the system Vr is 1500 V, and the capacity of the substation is assumed to be 4000 kW. The rated current of the substation Ir is Ir =

Pr 4000 [kW] = ≈ 2.7 [kA] (4.32) Vr 1.5 [kV]

Figure 4.25 illustrates a numerical model of the single section of the feeding circuit. The substation is expressed by a series circuit of a diode Dm, an internal resistor Rm and a voltage source Em. If the backward impedance, that is, the voltage-fluctuation ratio of the substation is 9% (= Zpu), the internal voltage Em of the substation model becomes Em = 1.09Vr = 1.09 × 1500 = 1635 [V] (4.33)

lk Rka

lk+1 Rkb

Rk+1a

Rk+1b

Dm

Dm+1

Rm

Rm+1

Em

Ik S/Sm

FIGURE 4.25 Feeding-circuit model.

Ik+1

Train current

Em+1 S/Sm+1


335

The internal resistance Rm is obtained from the backward impedance: Rm =

ZpuVr 2 0.09 × 1.52 = = 0.051 [Ω] (4.34) Pr 4

If a voltage regulator is installed into a substation, the substation can be simply modeled with a small internal resistor Rm = 1 mΩ and an ideal source Em of 1690 V, which is the target voltage of the voltage regulator. A diode is inserted in series for preventing the reverse current of substations S/S2, S/S3, and S/S4. There are no diodes in the substations at both ends (S/S1 and S/S5) to approximately express the currents flowing out from the model sections (i0 and i9 in Figure 4.24) as reverse currents. The train operation is modeled by current sources and nonlinear resistors illustrated in Figure 4.25. The train position lt is obtained from the train velocity v(t) by an integrator (1/s) illustrated in Figure 4.26: t

∫

lt (t) = v(t) dt (4.35)

0

The comparator in the figure determines the section where the train operates at the time. The comparator is represented by a nonlinear function (Device 56) and a truncation function TRUNC() installed in the TACS. The resistances of the feeding line are determined by Equation 4.36. These resistances Rka and Rkb are modeled by TACS-controlled resistors (Type-91) as time-varying resistors: k −1   Rka (t) =  lt (t) − ln  Rl   n =1   (4.36)   k Rkb (t) =  ln − lt (t)  Rl    n =1 

∑

∑

lt(t)

Train velocity ν(t)

1/s lk

FIGURE 4.26 Train model.

comp. select k

k

cal. Rka,b Ik

Rka,b(t) Ik(t)

336


The movement of a train is modeled by current sources Ik(t) with the aforementioned nonlinear resistors. A current source in the section, where the train is running, is activated, and the other sources are inactivated: I k (t) = It (t) I n (t) = 0

where It(t) is the train current

(n ≠ k )

(4.37)

4.3.3 Measured and Calculated Results 4.3.3.1 Measured Results Figure 4.27 illustrates the measured feeding-line voltages at node #4 and the output current of the substation S/S3 for Cases-a and -b shown in Table 4.15. The voltage regulators installed in substations S/S2 and S/S3 are turned on in Case-a, and the regulator in S/S2 at node #2 only operates in Case-b. A pulse-like current waveform is observed in both results. The time region where a high current is observed expresses that the train is powering, and the low current region expresses that the train is coasting. The difference between the current waveforms in Figure 4.27a and b mainly comes from the variation in the operation of the train operator. A base current of about 100 A observed in Figure 4.27a expresses a power sent to a train operating in the other sections. If the voltage regulator at the observation node #4 is turned on, the feeding line voltage is stabilized by the regulator and the output voltage of the substation becomes 1690 V (Figure 4.27a). The maximum current of the substation is 2.1 kA. The current linearly increases as a train comes close to the substation and decreases after the train passes the substation, as shown by broken lines. The decreasing rate is greater than the increasing rate because the length between the substations on the left-hand side of the substation S/S3 (l3+l4) is greater than the length between the substations on the right-hand side (l5+l6). The slope is determined by the resistances between the substations. Figure 4.27b illustrates the result when the voltage regulator at the observation node (#4) is turned off. The output voltage is fluctuated due to the train current even if the other regulator operates. The maximum current is 1.56 kA, and it is smaller by 25% compared to the previous result. The maximum current depends on the voltage regulator as well as the train operation. No current is observed at the coasting operation periods, although a current of 100 A flows when the voltage regulator at the substation operates. The feeding line voltage at the coasting operation is about 1.64 kV. 4.3.3.2 Calculated Results of Conventional System Investigation on the electrical characteristics of the feeding system using numerical simulations is helpful rather than that using measurements, because the voltage and current are affected by many parameters and

337

1700

2500

1600

2000

1500

1500

1400

1000

1300

500

1200 10:30

10:35

10:40

10:45

10:50

10:55

Current [A]

Voltage (V)


0 11:00

Time

Voltage (V)

Current

1700

2500

1600

2000

1500

1500

1400

1000

1300

500

1200 12:45

12:50

12:55

13:00

13:05

13:10

Current [A]

Voltage

(a)

0 13:15

Time 1690V

(b)

Voltage

Current

FIGURE 4.27 Measured feeding line voltages at node #4 and output current of substation S/S3: (a) with voltage regulators (Case-a) and (b) with single voltage regulator (Case-b).

TABLE 4.15 Circuit Conditions Case-a Case-b Case-c Case-d

Voltage Regulator

Compensator

S/S2 & S/S3 (#2 & #4) S/S2 (#2) No No

No No No #3

338


2.1 kA t 75 s

45 s

75 s

45 s

FIGURE 4.28 Train-current waveform.

simultaneous measurements of these parameters are quite difficult. Some numerical simulations by the proposed method are carried out using the EMTP in this section. The train-current waveform is assumed as illustrated in Figure 4.28, in the simulations. Its amplitude, duty ratio, and period are assumed to be 2.1 kA, 5/8, and 120 s, respectively. The turn-off period expresses that the train runs in coasting operation. This waveform is easily generated using a squarewave source (Type-23) installed in the TACS. The rise- and fall-time of the current are assumed to be 2.2 s and they are represented by an s-block with a time constant of 1 s. The regenerative current is neglected in this simulation. If the current is required, it is simply realized by including negative current pulses into the waveform. In this chapter, a constant train speed of 54 km/h is assumed in the simulations. Figure 4.29 illustrates the calculated results at node #4 (S/S3). It is clear from Figure 4.29a that there is no voltage fluctuation, if the conventional voltage regulator installed in S/S3 operates. The maximum current is 2.1 kA and is identical to the maximum train current. The minimum current is 65 A, and it is observed while the train is in its coasting operation. This result expresses that the substation S/S3 feeds power to trains running in other sections. No pulse current is observed before 0:15, because the current while the train is running in the first section (between nodes #0 and #2) is fed by S/S1 and S/S2, which has a voltage regulator. On the contrary, pulse currents are observed after 0:43 when the train is running the last section (between nodes #6 and #8), because the feeding line voltage at node #4 (S/S3) is kept high by the voltage regulator and no voltage regulator is installed in the right-hand section (S/S4 and S/S5). It is clear from Figure 4.29b that the maximum current of the substation is reduced from 2.1 to 1.63 kA (−22%), if the voltage regulator at S/S3 is turned off. The remaining current (0.47 kA) is fed from S/S2, which has a voltage regulator. No current is observed while in the coasting periods, although some current is observed in the previous result. There is no pulse current observed before 0:15, because the substation S/S2 with the voltage regulator feeds to the train running in the first stage. On the contrary, small pulse

339

1700

2500

1600

2000

1500

1500

1400

1000

1300

500

1200 0:10

0:15

0:20

0:25

0:30

0:35

0:40

0:45

Current (A)

Voltage (V)


0 0:50

Time

Voltage (V)

(a)

Current

1700

2500

1600

2000

1500

1500

1400

1000

1300

500

1200 0:10

0:15

0:20

0:25

0:30

0:35

0:40

0:45

Current (A)

Voltage

0 0:50

Time

Voltage (V)

Current

1700

2500

1600

2000

1500

1500

1400

1000

1300

500

1200 0:10

0:15

0:20

0:25

0:30

0:35

0:40

0:45

Current (A)

Voltage

(b)

0 0:50

Time (c)

Voltage

Current

FIGURE 4.29 Calculated results of conventional system: (a) with voltage regulators (Case-a), (b) with single voltage regulator (Case-b), and (c) without voltage regulators (Case-c).

340


TABLE 4.16 Minimum Voltages Node #

#3

#4 (S/S3)

#5

Case-a Case-b Case-c Case-d

1386 V 1340 V 1284 V 1451 V

1688 V 1553 V 1548 V 1548 V

1433 V 1379 V 1376 V 1376 V

currents are observed after 0:43, that is, the train is running in the last section (between nodes #6 and #8). The substation S/S3 feeds a minor current to the train running in the next section, if the next substation has no voltage regulator. The voltage while the train runs in coasting period is 1.67 kV, which is greater than the open-circuited voltage 1.64 kV shown in Equation 4.33. This result also shows that the feeding line voltage is kept by the voltage regulator operating at the adjoining substation. The minimum voltage is 1.55 kV, and is determined by the voltage drop mainly caused by the internal impedance of the substation S/S3. Figure 4.29c indicates that the current flowing from the substation S/S3 is increased by a disconnection of the voltage regulator at S/S2. The maximum current is 1.72 kA. The difference in the train current (0.38 kA = 2.1–1.72) is fed by the adjoining substations (S/S2 and S/S4). For the same reason, in Case-b the pulse currents are observed both before 0:15 and after 0:43 in Case-c. The minimum voltage is almost identical to that of Case-b. Table 4.16 shows the minimum voltages obtained by simulations. The nodes #3 and #5 denote halfway points of the feeding sections. The minimum voltage in the table is observed at node #3 in Case-c (without voltage regulators). The voltage at node #3 (1.28 kV) is lower than that at node #5 (1.38 kV), because the line resistance between substations S/S2 and S/S3 is higher than that between S/S3 and S/S4. The minimum voltage at the substation is 1.55 kV, and is almost identical to the result of Case-b. If the regulator at node #4 (S/S3) is also turned on (Case-a), the middle-point voltages at node #3 and #5 are increased. Comparisons between the results of Cases-b and -c and between the results of Cases-a and -b indicate that the voltage regulator is effective in increasing the voltage of the feeding line connected to the regulator. 4.3.3.3 Calculated Results with Power Compensator A power compensator for the dc railway system has been developed to compensate the voltage drop by the line resistance [10–13]. The power compensator installed on the ground is composed of some parallelly connected units, and each unit consists of a bidirectional dc/dc converter and a Li-Ion battery


341

TABLE 4.17 Specification of Power Compensator Maximum battery bank voltage Nominal battery bank voltage Minimum battery bank voltage Number of cells in battery bank Battery capacity Maximum discharging current Maximum discharging capacity Internal resistance of battery cell

746 V 655 V 564 V 182 60 Ah 600 A (10C) 393 kW 0.8 mΩ

570 Battery Current

1470 1520

[A] –300 =–IBd

1630

1700

Voltage [V]

FIGURE 4.30 Control characteristic of compensator unit.

bank. Table 4.17 shows the specification of the unit. The battery bank consists of 182 cells connected in series. The capacity of a cell is 60 Ah and the maximum discharging current is 600 A. In this simulation, the number of units Nu is assumed to be 8. The maximum power of the compensator is 3145 kW (=393 × 8) and is 79% of that of the substation. The ground-based compensator cannot directly know the train operation. The operational characteristic is determined according to the line voltage at the installed point as shown in Figure 4.30. The vertical axis is scaled in the charging current into a battery bank installed in a unit of the compensator. Because the voltage conversion ratio of the dc/dc converter γ is 2.3 (≈1500/655), the maximum injecting current to the feeding line becomes 130 A (=300/2.3 = IBd/γ) per unit. All conventional voltage regulators are turned off in the following simulation. Figure 4.31 illustrates a result using the proposed model [10–11], when the compensator is installed at node #3 where the minimum voltage is observed in Case-c. Pulse currents are injected into the feeding line from the compensator at node #3 as illustrated in Figure 4.31a, and the current is determined by the control characteristic illustrated in Figure 4.30. The maximum injected current Iimax is about 1 kA, which is calculated

1700

2500

1600

2000

1500 1400

1500

Compensator current

1000

1300

500

1200 0:10

Current (A)


Voltage (V)

342

0:15

0:20

0:25

0:30

0:35

0:40

0:45

0 0:50

Time Voltage

(a)

Current

Com. Cur.

0:30

0:40

1700

Voltage (V)

1600 1500 1400 1300 1200 0:10

0:15

0:20

0:25

0:35

0:45

0:50

Time Case-c

(b)

Case-d

FIGURE 4.31 Calculated results with compensator: (a) with compensator (Case-d) and (b) pantograph voltage.

from the maximum discharge current IBd (300 A) illustrated in Figure 4.30, voltage conversion ratio γ, and the number of the units Nu installed in the compensator:

I imax = N u

300 I Bd =8 = 8 × 130 = 1.0 [kA] (4.38) 2.3 γ

A comparison between the results shown in Figures 4.29c and 4.31a indicates that the minimum voltage at node #4 and the maximum current flowing from the substation S/S3 are almost identical. The voltage and current waveforms in the period from 0:17 to 0:27 are however slightly different from the


343

results without the compensator. The voltage drop and the current flowing from the substation are reduced by the compensator. Figure 4.31b shows a comparison of the pantograph voltages. The figure clearly indicates that the voltage fed to the train is improved by the compensator. The improvement is also observed in Table 4.16. The numerical simulation shows that the parameters used in this chapter are optimal from the viewpoint of voltage equalization. If further reduction of the feeding voltage fluctuation is required, the maximum discharging current (IBd) should be increased and also the compensators should be installed in the other sections.

4.4 Concluding Remarks Numerical simulations of a wind farm by the EMTP are explained in this chapter. The voltage increase due to the charging current of the cables is easily obtained by EMTP’s steady-state analysis routine. The load-flow calculation option of the EMTP called “FIX SOURCE” enables an estimation of the steady-state behavior of the wind farm, which has plural generators. These techniques are applicable to a simulation of conventional grids. Simulation models of a switching transistor and a MOSFET are also explained in this chapter. The model parameters of the device are easily obtained from a data sheet supplied by its manufacturer or from a simple experiment without complicate semiconductor physical parameters. The proposed model also enables temperature estimation. The accuracy of the models is satisfactory for the design of a switching circuit, such as a dc/dc converter and an inverter. A numerical simulation model of a train feeding system for the EMTP is proposed in this chapter. The feature of TACS installed in the EMTP is suitable for the simulation. The TACS-controlled nonlinear resistor is used for representing the movement of a train. The calculated results of the system including the voltage regulator agree with the measured results of a practical train system. This proves the accuracy of the proposed method. Both the voltage regulator and the power compensator installed for stabilizing the line voltage are effective for regulating the feeding line voltage. If the compensator is installed at the middle of a feeding section, its effectiveness is greater than that of the conventional voltage regulator. For an optimal design of a feeding system, the proposed simulation method is useful to confirm the effectiveness and to determine the parameters of the control characteristic. Although the conventional regulator is one of the solutions for increasing the feeding line voltage, it cannot decrease the voltage rise caused by the regenerative brake because a substation generally has no reverse power flow capability. The line voltage has to be kept below the maximum rating voltage

344


of the feeding system. The voltage rise by the regenerative brake is proportional to the distance between the braking and powering train, which consumes the regenerated power. The proposed simulation model is also useful for an analysis of the voltage rise, because the model has a capability to take the regenerative current into account. These techniques are useful for numerical simulations in new energy system and for expanding the applicable fields of the EMTP.

References 1. Thompson, M. T. 2006. Intuitive Analog Circuit Design. Oxford, U.K.: Newnes. 2. Nagaoka, N. 1988. Large-signal transistor modeling using the ATP version of EMTP. EMTP News 1(3). 3. Nagaoka, N. and E. Yamamoto. 1988. Numerical analysis of transistor circuit by EMTP, Part-1 Switching circuit. Record of the 1988 Kansai-section joint convention of Inst. of Elec. Eng., Japan, G1-11. 4. Nagaoka, N. and E. Yamamoto. 1988. Numerical analysis of transistor circuit by EMTP, Part-2 Amplifier. Record of the 1988 Kansai-section joint convention of Inst. of Elec. Eng., Japan, G1-12. 5. Nagaoka, N. and Y. Kimura. 1990. Numerical analysis of FET circuit by EMTP, Record of the 1988 Kansai-section joint convention of Inst. of Elec. Eng., Japan, G3-18. 6. Nagaoka, N. and Y. Kimura. 1990. Development of numerical model of switching FET. Proc. 28th Convention of Science and Engineering Research Institute of Doshisha Univ. Science and Engineering Research Institute of Doshisha Univ. 7. Nose, N., A. Sugiyama, I. Okada, N. Nagaoka, and A. Ametani. 1998. Transistor model including temperature dependent characteristic for EMTP. National convention record, IEEJ 709. 8. Nagaoka, N. and A. Ametani. 2006. Semiconductor device modeling and circuit simulations by EMTP. Joint convention of Power Eng. and Power System Eng. IEEJ, PE-06-74/PSE-06-74. 9. Nagaoka, N. 2006. Chapter 3 DC/DC Converter, Textbook of EEUG 2006 Course, pp. III-1–III-20. 10. Nagaoka, N., H. Oue, M. Sadakiyo et al. 2006. Power compensator using lithium-ion battery for DC railway and its simulation by EMTP. 63rd IEEE Vehicular Technology Conference, Conf. CD, 6P-9. 11. Nagaoka, N., M. Sadakiyo, N. Mori, A. Ametani, S. Umeda, and J. Ishii. 2006. Effective control method of power compensator with lithium-ion battery for DC railway system. Proc. 41st International Universities Power Engineering Conference. pp. 1067–1071. 12. Umeda, S., J. Ishii, and H. Kai. 2004. Development of hybrid power supply system for DC electric railways. JIASC04 3-84. 13. Umeda, S., J. Ishii, N. Nagaoka, H. Oue, N. Mori, and A. Ametani. 2005. Energy storage of regenerated power on DC railway system using lithium-ion battery. Proc. The 2005 International Power Electronics Conference, pp. 455–460.

5 Numerical Electromagnetic Analysis Methods and Their Applications to Transient Analyses

5.1 Fundamentals 5.1.1 Maxwell’s Equations Before numerical electromagnetic analysis (NEA) methods are explained, Maxwell’s equations, which are fundamentals of electromagnetics, are shown. They are stated in the time domain as

∇× E(r , t) = − ∇× H (r , t) =

∂B(r , t) (5.1) ∂t

∂D(r , t) + J (r , t) (5.2) ∂t

∇⋅ D(r , t) = ρ(r , t) (5.3)

∇⋅ B(r , t) = 0 (5.4)

where E(r, t) is the electric field H(r, t) is the magnetic field D(r, t) is the electric flux density B(r, t) is the magnetic flux density J(r, t) is the conduction current density ρ(r, t) are the volume charge density, each at the space point r and at the time t Equation 5.1 represents Faraday’s law, and Equation 5.2 represents Ampere’s law. Equations 5.3 and 5.4 represent Gauss’ law for electric and magnetic fields, respectively. 345

346


In the frequency domain, Maxwell’s equations are expressed as

∇ × E(r ) = − jωB(r ) (5.5)

∇× H (r ) = J (r ) + jωD(r ) (5.6)

∇⋅ D(r ) = ρ(r ) (5.7)

∇⋅ B(r ) = 0 (5.8)

where j is the imaginary unit ω is the angular frequency The relations of the electric and magnetic-flux densities to the electric and magnetic fields are given as

D(r ) = εE(r ) = εr ε0 E(r ) (5.9)

B(r ) = µH (r ) = µrµ0 H (r ) (5.10)

where ε0 is the permittivity of vacuum (8.854 × 10−12 F/m) μ0 is the permeability of vacuum (4π × 10−7 H/m) εr is the relative permittivity of medium μr is the relative permeability of medium ε is the permittivity of medium μ is the permeability of medium In a conductive medium, the following relation of the electric field to the conduction current density, known as Ohm’s law, is fulfilled:

J (r ) = σE(r ) (5.11)

where σ is the conductivity of the medium. 5.1.2 Finite-Difference Time-Domain Method The finite-difference time-domain (FDTD) method [1] is one of the most frequently used techniques in electromagnetics. It involves the space and time discretization of the whole working space and the finite-difference approximation to Maxwell’s differential equations. For the analysis of the

347

Numerical Electromagnetic Analysis Methods

electromagnetic response of a structure in an unbounded space using the FDTD method, an absorbing boundary condition, which suppresses unwanted reflections from the surrounding boundaries that truncate the unbounded space, needs to be applied. For avoiding numerical instabilities or spurious resonances, the time increment or step ∆t needs to be determined following the Courant–Friedrichs–Lewy (CFL) criterion [2]: ∆t < ∆s/ 3 c in three-dimensional (3D) computations, where ∆s is the (cubic) cell-side length and c is the speed of light. Advantages of this method are summarized as follows: (1) it is based on a simple procedure in electric- and magnetic-field computations, and therefore its programming is also relatively easy; (2) it is capable of treating complex geometrical shape and inhomogeneity; (3) it is capable of incorporating nonlinear effects and components; and (4) it can yield wide-band data from one run with the help of a time-to-frequency transforming tool. Its disadvantages are given as follows: (1) it is inefficient compared with the method of moments (MoM); (2) it cannot deal with oblique boundaries that are not aligned with the Cartesian grid; and (3) it would require a complex procedure for implementing dispersive materials (materials with frequencydependent constitutive constants). The method requires the whole working space to be divided into cubic or rectangular cells. The cell size should not exceed one-tenth of the wavelength corresponding to the highest frequency in the excitation. The electromagnetic field components, Ex, Ey, Ez, Hx, Hy, and Hz are located in each cell in the 3D Cartesian coordinate system as shown in Figure 5.1. Time-updating equations for electric and magnetic fields are derived next.

( )

z

Ez

Hy

∆z

Hx Ey

y Ex

x

Hz

∆x

∆y

FIGURE 5.1 FDTD cell with x-, y-, and z-directed components of electric and magnetic fields.

348


Considering Equations 5.9 and 5.11, Ampere’s law, Equation 5.2 is rewritten as ∇× H n −(1/2) = ε

∂E n −(1/2) ∂E n −(1/2) + J n −(1/2) = ε + σE n −(1/2) (5.12) ∂t ∂t

where n − (1/2) is the present time step. If the time-dependent, partialdifferential term in Equation 5.12 is approximated by its central finite difference, Equation 5.12 is expressed as

ε

E n − E n −1 E n + E n −1 ∂E n −(1/2) + σE n −(1/2) ≈ ε +σ ≈ ∇× H n −(1/2) (5.13) ∂t ∆t 2

where ∆t is the time increment. Note that E n−(1/2) in Equation 5.13 is approximated by its average value, (E n +E n−1)/2. Rearranging Equation 5.13, the updating equation for electric field at time step n from its one-time-step previous value En−1 and half-time-step previous magnetic-field rotation Hn−(1/2) is obtained as follows:  1 − ( σ∆t 2ε )  n −1   ∆t ε ∇× H n −(1/2) (5.14) En =  E +   1 + ( σ∆t 2ε )   1 + ( σ∆t 2ε )     

∇ × H is given by i ∂ ∇× H = ∂x Hx

j ∂ ∂y Hy

k  ∂H z ∂H y ∂ = i − ∂z ∂z  ∂y Hz

  ∂H x ∂H z − + j ∂x   ∂z

 ∂H y ∂H x   −   + k ∂y    ∂x

(5.15) where i, j, and k are x-, y-, and z-directed unit vectors, respectively. From Equations 5.14 and 5.15, the updating equation for Exn at the space point (i + (1/2), j, k), for example, is expressed as  1  Exn  i + , j , k  2  

 1 − ( σ∆t 2ε )  n −1  1  = E i + , j, k   1 + ( σ∆t 2ε )  x  2   


349

 n −(1/2)  1 1   n −(1/2 )   i + , j, k    i + , j , k  ∂H y    ∂H z ∆t ε  2   2 −  +  1 + ( σ∆t 2ε )   ∂ z ∂ y       1 − ( σ∆t 2ε )  n −1  1  ≈ i + , j, k  E  1 + ( σ∆t 2ε )  x  2     n −(1/2)  1 1  1 1   n − ( 1/2 )   i + , j + , k  − Hz i + , j − ,k   Hz 2 2 2 2       ∆ y    ∆t ε +   1 + ( σ∆t 2ε )     1 1 1 1  n − ( 1/2 )  n − ( 1/2 )   i + , j, k +  − Hy  i + , j, k −    Hy 2 2   2  2 − ∆z  

(5.16)

where the spatial, partial-differential terms in the first equation are approximated by their central finite differences. In the same manner, updating equations for Eyn and Enz can be derived. Considering Equation 5.10, Faraday’s law, Equation 5.1 is rewritten as ∇ × E n = −µ

∂H n (5.17) ∂t

where n is the present time step. If the time-dependent, partial-differential term in Equation 5.17 is approximated by its central finite difference, Equation 5.17 is expressed as

µ

H n +(1/2) − H n −(1/2) ∂H n ≈µ ≈ −∇ × E n (5.18) ∂t ∆t

Rearranging Equation 5.18, the updating equation for magnetic field at time step n + (1/2) from its one-time-step previous value Hn−(1/2) and half-time-step previous electric-field rotation En is obtained as follows:

∆t n + 1/2 n − 1/2 H ( ) = H ( ) − ∇ × E n (5.19) µ

From Equations 5.19 and 5.15, the updating equation for Hn+(1/2) at the location x (i, j + 1/2, k + 1/2), for example, is expressed as follows:

350


 n 1 1   ∂Ez  i , j + 2 , k + 2       ∂ y   n+1/2  n − 1 / 2 1 1 1 1  ∆t  Hx  i, j + , k +  = Hx  i, j + , k +  −   2 2 2 2 µ    1 1  n  ∂E y  i , j + 2 , k + 2     − ∂z   n−1/2

≈ Hx

1 1   i, j + , k +  2 2 

 n 1 1  n  E z  i , j + 1, k + 2  − E z  i , j , k + 2         ∆ y  ∆t  (5.20) −   µ  1 1   n n  Ey  i , j + 2 , k + 1  − Ey  i , j + 2 , k      −  ∆ z   where the spatial, partial-differential terms in the first equation are approximated by their central finite differences. In the same manner, updating equations for Hyn+1/2 and Hn+1/2 can be derived. z Updating equations for Exn, Eyn, Enz Hxn+1/2, Hyn+1/2, and Hn+1/2 are summarized z in the following:  1  Enx  i + , j , k   2  σ ( i + 1/2, j , k ) ∆t 2ε ( i + 1/2, j , k ) n −1  1  = Ex  i + , j , k  σ ( i + 1/2, j , k ) ∆t  2  1+ 2ε ( i + 1/2, j , k ) 1−

∆t ε ( i + 1/2, j , k ) + σ ( i + 1/2, j , k ) ∆t 1+ 2ε ( i + 1/2, j , k )

1  1   1  1 H nz −1/2  i + , j + , k  − H nz −1/2  i + , j − , k  2  2   2  2 ∆

1 1  1  1 H n −1/2  i + , j , k +  − H n −1/2  i + , j , k −  2 2 2 2   − ∆ z (5.21)


351

1   Eny  i , j + , k  2   σ ( i , j + 1/2, k ) ∆t 2ε ( i , j + 1/2, k ) n −1  1  Ex  i , j + , k  = σ ( i , j + 1/2, k ) ∆t 2   1+ 2ε ( i , j + 1/2, k ) 1−

∆t ε ( i , j + 1/2, k ) + σ ( i , j + 1/2, k ) ∆t 1+ 2ε ( i , j + 1/2, k )

1 1 1 1   H nx −1/2  i , j + , k +  − H nx −1/2  i , j + , k −  2 2 2 2   ∆z

1  1   1  1 H nz −1/2  i + , j + , k  − H nz −1/2  i − , j + , k  2  2 2  2   − ∆ x (5.22)

1  Enz  i , j , k +  2  σ ( i , j , k + 1/2 ) ∆t 2ε ( i , j , k + 1/2 ) n −1  1 Ez  i , j , k +  = σ ( i , j , k + 1/2 ) ∆t 2  1+ 2ε ( i , j , k + 1/2 ) 1−

∆t ε ( i , j , k + 1/2 ) + σ ( i , j , k + 1/2 ) ∆t 1+ 2ε ( i , j , k + 1/2 )

1 1  1  1 H ny −1/2  i + , j , k +  − H ny −1/2  i − , j , k +  2 2  2  2 ∆ 1 1 1 1   H n −1/2  i , j + , k +  − H n −1/2  i , j − , k +  2 2 2 2   − ∆y

(5.23)

n+1/2

Hx

1 1   i, j + , k +  2 2  n−1/2

= Hx

1 1   i, j + , k +  2 2 

352


 n 1 1  n  Ez  i , j + 1, k + 2  − Ez  i , j , k + 2         ∆ y   ∆t −   (5.24) µ ( i , j + 1/2, k + 1/2 )   1 1     n n  Ey  i , j + 2 , k + 1  − Ey  i , j + 2 , k      −  ∆z  

n+1/2

Hy

1  1  i + , j, k +  2  2 n−1/2

= Hy

 n 1  n 1    Ex  i + 2 , j , k + 1  − Ex  i + 2 , j , k         z ∆   (5.25) ∆t −  µ ( i + 1/2, j , k + 1/2 )  n   Ez i + 1, j , k + 1  − Enz  i , j , k + 1        2 2    −  ∆x  

n+1/2

Hz

1   1 i + , j + ,k 2   2 n−1/2

= Hz

1  1  i + , j, k +  2  2

1   1 i + , j + ,k 2 2  

 n 1  n 1    E y  i + 1, j + 2 , k  − E y  i , j + 2 , k         x ∆   (5.26) ∆t −   µ ( i + 1/2, j + 1/2, k )  n  1 1   n E i + , j + 1, k  − Ex  i + , j , k   x  2   2  −  ∆y  

By updating Equations 5.21 through 5.26 at every point, transient electric and magnetic fields throughout the working space are obtained. For analyzing lightning surges on power systems, it is necessary to appropriately represent thin wires such as overhead transmission-line conductors, distribution-line conductors, and steel frames of towers and buildings.


353

Noda and Yokoyama [3] have found that a thin wire in air has an equivalent radius of r0 = 0.23∆s (∆s is the side length of cubic cells used in FDTD simulations), in the case that the electric field along the axis of the thin wire is set to zero in an orthogonal and uniform Cartesian grid for FDTD simulations. They further showed that a thin wire having an arbitrary radius r could be equivalently represented by placing a zero-radius wire in an artificial rectangular prism, coaxial with the thin wire, having a cross-sectional area of 2∆s × 2∆s and the modified permittivity mε0 and permeability μ0/m given by

mε 0 ,

ln ( 1/0.23 ) µ0 (5.27) , m= m ln ( ∆s/r )

For example, in representing a thinner wire having the radius r (

1  V (n∆t)  (5.28) Ezn  i , j , k +  = z ∆z 2 

A lumped-current source in the z direction at the space point (i, j, k + 1/2), which generates a time-varying current Iz [(n − 1/2)∆t], is realized by 1 1   Ezn  i , j , k +  = Ezn−1  i , j , k +  2 2    n−1/2  1 1 1 1  n−1/2   H y  i + 2 , j, k + 2  − H y  i − 2 , j, k + 2         x ∆   ∆t +   ε(i , j , k + 1/2)  1 1 1 1   H nx −1/2  i , j + , k +  − H xn−1/2  i , j − , k +    2 2 2 2    −  ∆ y   ∆t (5.29) I z [(n − 1/2)∆t ] − ε(i , j , k + 1/2)∆x∆y Voltage and current sources in x and y directions can be represented similarly.

354


Representation of lumped elements, a resistor R, a capacitor C, and an inductor L are described here. For the voltage Vz [(n − 1/2)∆t] across a resistor R in the z direction, the current flowing through it, Iz [(n − 1/2)∆t], has the relation Iz [(n − 1/2)∆t] = Vz [(n − 1/2)∆t]/R ≈ (Ezn + Ezn−1)∆z/2R. If this relation is substituted in Equation 5.29, the following updating equation for a z-direct resistor located at the space point (i, j, k + 1/2) is obtained: 1  Ezn  i , j , k +  2  ∆t∆z 2Rε(i , j , k + 1/2)∆x∆y n−1  1 = Ez  i , j , k +  ∆t∆z 2  1+ 2Rε(i , j , k + 1/ 2)∆x∆y 1−

 n−1/2  1 1 1 1 n−1/2   H y  i + 2 , j, k + 2  − H y  i − 2 , j, k + 2      ∆t  ∆x  ε(i , j , k + 1/2) +  ∆t∆z 1 1 1 1 n−1/2  n−1/2   1+ 2Rε(i , j , k + 1/2)∆x∆y  H x  i , j + 2 , k + 2  − H x  i , j − 2 , k + 2  − ∆y  (5.30) For the voltage Vz [(n − 1/2)∆t] across a capacitor C in the z direction, the current flowing through it, Iz [(n − 1/2)∆t], has the relation Iz [(n − 1/2)∆t] = C ∂Vz [(n − 1/2)∆t]/∂t ≈ C (Ezn − Ezn−1)∆z/∆t. If this relation is substituted in Equation 5.29, the following updating equation for a z-direct capacitor at the space point (i, j, k + 1/2) is obtained: 1  Ezn  i , j , k +  2  1  = Ezn −1  i , j , k +  2   n −1/2  1 1 1 1  n −1/2   i + , j, k +  − H y  i − , j, k +    Hy 2 2 2 2      ∆t  ∆x   ε(i , j , k + 1/2) +   C ∆z  H n −1/2  i , j + 1 , k + 1  − H n −1/2  i , j − 1 , k + 1   1+ x x     ε(i , j , k + 1/2)∆x∆y  2 2 2 2    −  ∆y  

(5.31)


355

For the voltage Vz [(n − 1/2)∆t] across an inductor L in the z direction, the current flowing through it, Iz [(n − 1/2)∆t], has the relation

∑

( n −1)

I z [(n − 1/2)∆t] = ∫ Vz (t)dt/L ≈ ∆z∆t Ezm L. If this relation is substituted in m =1 Equation 5.29, the following updating equation for a z-direct inductor at the space point (i, j, k + 1/2) is obtained: 1 1   Ezn  i , j , k +  = Ezn−1  i , j , k +  2 2     n−1/2  1 1 1 1  n−1/2   H y  i + 2 , j, k + 2  − H y  i − 2 , j, k + 2         x ∆   ∆t +   ε(i , j , k + 1/2)  1 1 1 1  n−1/2  n−1/2  H x  i, j + , k +  − Hx  i, j − , k +   2 2 2 2    −  ∆y   n−1 ∆z(∆t)2 1  Ez  i , j , k +  − (5.32) ε(i , j , k + 1/2) ∆x∆y =1 2 

∑

Lumped elements in x and y directions can be represented similarly. 5.1.3 Method of Moments The method of moments (MoM) [4] is also frequently employed in transient electromagnetic computations. This method is based on an electric-field integral equation in either frequency or time domain, which relates the induced current on a conductor to the incident electric field. Only the conducting structure to be analyzed needs to be represented as a combination of short cylindrical segments. Advantages of this method are summarized as follows: (1) it is computationally more efficient than the FDTD method; (2) it requires no absorbing boundary condition; (3) it can represent oblique conductors easily without any staircase approximation; (4) it is capable of considering dispersive materials in the frequency-domain MoM; and (5) it is capable of incorporating nonlinear effects and components in the time-domain MoM. Its disadvantages are listed as follows: (1) it cannot deal with complex boundaries, compared with the FDTD method; (2) in the time-domain MoM, it would require a complex procedure for considering dispersive materials; and (3) in the frequency-domain MoM, it would be essentially impossible to incorporate nonlinear effects and components.

356


In the following, an electric-field integral equation in the frequency domain is described first. Then, the corresponding electric-field integral equation in the time domain is explained. The electric field at the space point r is generally expressed in the frequency domain as

E(r ) = − jωA(r ) − ∇ϕ(r ) (5.33)

where A(r) is the magnetic vector potential φ(r) is the electric scalar potential If the Lorenz gauge given as

∇⋅ A(r ) = − jωµεϕ(r ) (5.34)

is applied to Equation 5.33, the following wave equations for each potential are obtained:

∇ 2 A(r ) + ω2µεA(r ) = −µJ (r ) (5.35) ∇ 2ϕ(r ) + ω2µεϕ(r ) = −

ρ(r ) (5.36) ε

Each of these wave equations, excited by a source at the location r′ in the form of Dirac delta function δ (r, r′), can be expressed as

∇ 2 g(r , r ′) + k 2 g(r , r ′) = −δ(r , r ′) (5.37)

If the source is assumed to locate at the origin r′ = 0, Equation 5.37 is rewritten in the spherical coordinate as

1 ∂  2 ∂  g(r )  + k 2 g(r ) = −δ(r ) (5.38) r r 2 ∂r  ∂r 

The solution of Equation 5.38 is expressed as g(r) = e−jkr/(4πr). Thus, the solution at the location r for the arbitrary source point r′ is expressed as

g(r, r′) =

− jk r − r ′

1 e (5.39) 4π r − r′

On the basis of the superposition principle, the potentials generated by arbitrary sources can be written as

357


ϕ(r ) =

∫ V

ρ(r′) g(r , r ′)dV (5.40) ε

∫

A(r ) = µJ (r′) g(r , r ′)dV (5.41)

V

The total electric field E at the space point r is the sum of the incident electric field Ei and the scattered electric field E s there. This relation can be written as E(r ) = E i (r ) + E s (r ) (5.42)

Also, the total electric field follows Ohm’s law, which is E( r ) = E i ( r ) + E s ( r ) =

J (r ) (5.43) σ

where σ is the conductivity of the material of interest. If Equations 5.40 and 5.41 are substituted in Equation 5.33, the scattered electric field E s at the space point r due to the current J at the space point r′ is expressed as

∫

E s (r ) = − jωA(r ) − ∇ϕ(r ) = − jωµ g(r , r' ) J (r' ) dV −

V

∇ ε

∫ g(r , r' )ρ(r ) dV (5.44) V

If Equation 5.44 is substituted in Equation 5.43, the relation of the incident electric field to the induced current density and volume charge density is obtained as E i (r ) =

∇ J (r ) + jωµ g(r , r' ) J (r′) dV + σ ε

∫ V

∫ g(r , r' )ρ(r' ) dV (5.45) V

The volume charge density ρ is related to the current density J via the charge conservation equation, which is as follows: ∇ ⋅ J (r′) = − jωρ(r ′) (5.46)

If Equation 5.46 is substituted in (5.45), the following form is obtained: E i (r ) =

∇ J (r ) + jωµ g(r , r′) J (r′) dV − σ jωε

∫ V

∫ g(r , r′)∇ ⋅ J(r′) dV (5.47) V

358


When a perfect conductor is analyzed, the tangential component of the total electric field on the conductor surface is zero. Therefore, the following relation is fulfilled on the conductor surface: n ×  E i (r ) + E s (r ) = n ×

J (r ) = 0 (5.48) σ

Furthermore, the current and charge are distributed only on the surface of a perfect conductor. Thus, Equation 5.48 is rewritten as n × E i (r ) = −n × E s (r )

 ∇ = n ×  jωµ g(r , r′) J s (r′) dS − jωε  S 

∫

∫ S

 g(r , r ′ )∇ ⋅ J s (r ′)) dS (5.49)  

where n is a unit normal vector on the conductor surface Js is the surface current density When the radius of a perfectly conducting wire is much smaller than the wavelength of interest, the current I and charge q could be assumed to be confined to the wire axis as shown in Figure 5.2. This assumption is called thin-wire approximation. With this assumption, the electric-field integral equation, Equation 5.47 or 5.49, for a perfectly conducting thin wire in air is simplified to

∫

E i (r ) = jωµ0 I (r ′)g(r , r ′)s′ds′ −

C

∇ jωε0

∫ g(r , r′)∇ ⋅ [ I(r′)s′] ds′ (5.50) C

where s is the unit tangential vector along the wire surface C(r) s′ is the unit tangential vector on the wire axis

s(r) = s΄

C(r)

s(r) r

r΄ Origin point FIGURE 5.2 Thin wire approximated conductor.

359


The incident electric field, which is tangential to the wire surface and is parallel with the wire axis, is given as a dot product, s · Ei, as follows:

∫

s ⋅ E i (r ) = jωµ0 s ⋅ s′I (r′)g(r , r ′) ds′ − s ⋅

C

∇ jωε0

∫ g(r , r′)∇ ⋅ [ I(r′)s′] ds′ (5.51) C

or

∫

s ⋅ E i (r ) = jωµ0 s ⋅ s′I (r ′)g(r , r′) ds′ − C

=

1 jωε0

∫ C

∂  I (r′) g(r , r′) ds′ ∂s∂s′

jη  2 ∂   k s ⋅ s′ −  I (r ′)g(r , r ′) ds′ k  ∂s∂s′  C

∫

(5.52)

where k = ω (μ0 ε0)1/2 and η (μ0/ε0)1/2 Now, an electric-field integral equation in the time domain is described. Electric scalar potential and magnetic vector potential are expressed in the time domain as follows: ϕ(r, t) = A(r , t) =

1 4πε

µ 4π

∫ V

ρ(r′, t′) dV (5.53) R

1 ∂J (r′, t′) dV (5.54) ∂t′

∫R⋅ V

where R = |r − r′| t′ = t − R/v v is the speed of electromagnetic wave in the medium of interest The electric field at the space point r is expressed as E( r , t ) = −

∂A(r , t) − ∇ϕ(r , t) (5.55) ∂t

If Equations 5.53 and 5.54 are substituted in Equation 5.55, the scattered electric field E s at the space point r due to the current J at the space point r′ is expressed as E s (r , t) = −

µ 4π

∫ V

1 ∂J (r′, t′) ∇ dV − ⋅ ∂t′ 4πε R

∫ V

ρ(r ′, t′) dV (5.56) R

360


The volume charge density in (5.56) is evaluated by the following continuity equation: t′

∫

ρ(r ′, t′) = − ∇ ⋅ J (r′, τ) dτ (5.57)

−∞

When a perfect conductor is analyzed, the current and charge are distributed only on the surface of a perfect conductor, and the tangential component of the total electric field on the conductor surface is zero. Therefore, the following relation is fulfilled on the conductor surface:  µ 1 ∂J (r′, t′) ∇ ρS (r′, t′)  dS n × E i ( r , t ) = − n × E s ( r ′, t ) = n ×  ⋅ S dS + R 4πε ∂t′  4π R  S S   (5.58)

∫

∫

where ρs is the surface charge density, which is evaluated by t′

∫

ρS (r′, t′) = − ∇ ⋅ J S (r′, τ) dτ (5.59)

−∞

The electric-field integral equation in the time domain for a perfectly conducting thin wire in air is obtained similarly to that in the frequency domain as follows: s ⋅ E i (r , t) =

µ0 1 ∂I (r′, t′) ∇ ds′ + s ⋅ s ⋅ s′ ⋅ 4π ∂t′ 4πε0 R

∫ C

∫ C

q(r′, t′) ds′ (5.60) R

t′

∂I (r′, τ) dτ ∂s′ −∞ The last term of Equation 5.60 is converted to

where q(r′, t′) = −

∫

1 ∂q ∂(1/R) 1 ∂q 1  q(r′, t′)  ∂  q(r′, t′)  R ∇ =   = q ∂R + R ∂R = − R 2 q + R ∂s′  R  ∂R  R  R

=−

1 1 ∂q q− R2 R c∂t′

(5.61)

With the continuity equation in one-dimensional form, which is given as

∂q ∂I (5.62) =− ∂s′ ∂t′


361

the last term is expressed as 1 1 ∂I  q(r ′, t′)  ∇  = − R 2 q + Rc ∂s′ (5.63)  R 

If Equation 5.63 is substituted in (5.60), the electric-field integral equation in the time domain for a thin-wire perfect conductor is obtained as s ⋅ E i (r , t) =

1 ∂I (r ′, t′) µ0  s ⋅ R ∂I (r ′, t′) 2 s ⋅ R q(r′, t′)  +c 2 −c ds′ s ⋅ s′ ⋅ 4π  ∂t′ ∂s′ R3 R  R R C

∫

(5.64) In solving Equation 5.52 or 5.64, a mathematical function for approximating the distribution of current along the wire axis is employed. The function is usually expressed as a linear combination of basis functions having unknown coefficients, and the unknown coefficients are evaluated numerically. Equation 5.52 can be written as follows: L( f ) = E (5.65)

where L is a linear operator E is the excitation function f is the unknown current function The unknown function f(x) can be expanded as f (x) =

N

∑ a f (x) (5.66) n n

n =1

where an is an unknown coefficient fn(x) is a known basis function that is illustrated in Figure 5.3 If Equation 5.66 is substituted in Equation 5.65, the following equation is obtained: N

∑ a L ( f (x)) ≈ E (5.67) n

n =1

n

362


a4 f4(x) a1 f1(x)

x1

a2 f2(x)

x2

a3 f3(x)

an fn(x)

x3

x4

xn–2

xn–1

xn

xn+1

(a) a4 f4(x) a1 f1(x)

x1

a2 f2(x)

x2

an fn(x)

a3 f3(x)

x3

x4

xn–2

xn–1

xn

xn+1

(b) FIGURE 5.3 MoM typical basis functions for approximating the distribution of current along the wire axis. (a) Piecewise triangular function and (b) piecewise sinusoidal function.

To solve Equation 5.67, the dot product of weight function fm is applied to Equation 5.67. Then, the following equation is obtained: N

∑a

n

fm , L ( fn ) ≈ fm , E (5.68)

n =1

Equation 5.68 can be written in a matrix form:

Za = b (5.69)

with

Zmn = fm , L ( fn ) (5.70)

and

bmn = fm , E (5.71)

where Zmn is an element of the matrix Z at row m and column n bm is an element of the vector b at row m The unknown coefficients of the current function are obtained by solving Equation 5.69. The widely known computer program, the numerical electromagnetic code (NEC) [5,6], is based on the MoM in the frequency domain. The thinwire time domain (TWTD) code [7] is based on the MoM in the time domain.

363


5.2 Applications 5.2.1 Grounding Electrodes

Unit: m z Reference line: 2 mm2

x y

1.5 1.5

Ground surface

50 Remote potential electrode (six pieces of copper rods)

Cu (0. rrent 135 lin m e in dia m

ete

r)

The role of grounding electrodes is to drain fault currents effectively into the soil, and thereby to mitigate damage of installations of telecommunication systems and electrical power systems. Thus, the performance of such a system is influenced by the transient characteristics of its grounding electrodes. It is, therefore, important to study the transient characteristics of grounding electrodes. Recently, NEA methods have been applied successfully in analyzing the transient responses of grounding electrodes. Tanabe [8] has analyzed the transient response of a vertical grounding electrode of 0.5 m × 0.5 m × 3 m, shown in Figure 5.4, using the FDTD method [1]. For FDTD computations, the conductor system shown in the figure is accommodated in a working volume of 27.5 m × 61 m × 55 m, which is divided uniformly into cubic cells of 0.25 m × 0.25 m × 0.25 m. The working volume is surrounded by six planes of Liao’s second-order absorbing boundary condition [9] to minimize unwanted reflections there. The time increment is set to 0.481 ns, which is determined following the CFL criterion [3]. The conductivity, relative permittivity, and relative permeability of the ground are set to σ = 1.9 to 2.7 mS/m (based on their low-frequency measurement), εr = 50, and μr = 1, respectively. Figure 5.5 shows the FDTD-computed voltage and current waveforms for the vertical grounding electrode and the corresponding measured waveforms [8]. The FDTD-computed waveforms are in good agreement with the corresponding measured ones. Tanabe et al. [10] have studied the transient response of a horizontallyplaced square-shape grounding electrode of 7.5 m × 7.5 m, buried 0.5 m in

1

1.5 Remote current electrode (three pieces of copper rods) Series resistance (800 ohms) Pulse generator (voltage source) 20

Electrode being tested

Ground: Kanto loam

FIGURE 5.4 Configuration of a 3 m long vertical grounding electrode and its auxiliary wires for measurement of its surge response. (Reprinted with permission from Tanabe, K., Novel method for analyzing dynamic behavior of grounding systems based on the finite-difference time-domain method, IEEE Power Eng. Rev., 21(9), 55–577. Copyright 2001 IEEE.)


Transient electric potential (V), resistance (f)

80

0 TGR

60

–0.4

40

Transient current

20

–0.6

0

–0.8

–20 –40

(a)

–0.2

–1

Transient electric potential

0

1

Transient current (A)

2

–1.2

Time (µs)

Transient electric potential (V), resistance (Ω)

80

0 TGR

60 40

–0.4

Transient current

20

–0.6

0

–0.8

–20 –40

(b)

–0.2

–1

Transient electric potential

0

1

Transient current (A)

364

2

–1.2

Time (µs)

FIGURE 5.5 (a) Measured waveforms of voltage and current for the 3 m long vertical grounding electrode and (b) the corresponding FDTD-computed waveforms. (Reprinted with permission from Tanabe, K., Novel method for analyzing dynamic behavior of grounding systems based on the finite-difference time-domain method, IEEE Power Eng. Rev., 21(9), 55–57. Copyright 2001 IEEE.)

depth using the FDTD method. For FDTD computations, the conductor system, which includes a 50 m long, horizontal voltage-reference wire, a 26.25 m long horizontal current-lead wire, and the aforementioned grounding electrode, is accommodated in a working volume of 83.75 m × 67.5 m × 30 m, which is divided uniformly into cubic cells of 0.25 m × 0.25 m × 0.25 m. The working volume is surrounded by six planes of Liao’s second-order absorbing boundary condition. The time increment is set to 0.481 ns, which is determined on the basis of the CFL criterion. The conductivity, relative permittivity, and relative permeability of the ground are set to σ = 3.8 mS/m (based on

365

1

Current Electric potential

0.2

0.4

0.6

0.8

1

0

0

0

0.2

0.4

Time (µs) 2

Current

1

Electric potential

0

0

0.2

0.4

0.6

0.8

1

200 Current (A)

NW

100

0

0.2

0.4

0.6

Electric potential

0

0

0.2

0.4

0.8

1

0

(a)

0.2

0.4

0.6

Time (µs)

0.8

1

0

Electric potential (V)

1

Current

0

1

0

2

100

1


0

0

200 Current (A)


2

Electric potential

0

0.8

0.2

0.4

0.6

0.8

1

0

Time (µs)

SW

100

0.6

SE

Time (µs) 200

1

Current

200 Current (A)

1

Current

0

0

2

100



SE

Electric potential

0

1

Time (µs) 2

100

0.8

NW

Time (µs) 200

0.6 Time (µs)



200

1


Current (A)

0

100

Current (A)

0

2

NE Grounding resistance

2

SW

100

(b)

1


0

0

0.2

0.4

0.6

0.8

1

Current (A)

100

200

Current (A)

2

NE Grounding resistance

Current (A)

200

Electric potential (V) Resistance (Ω)

Electric potential (V) Resistance (Ω)


0

Time (µs)

FIGURE 5.6 (a) Measured waveforms of voltage at each corner of a horizontally-placed square-shape grounding electrode of 7.5 m × 7.5 m and injected current, and (b) the corresponding FDTDcomputed waveforms. (Reprinted from Tanabe, K. et al., IEE J. Trans. Power Energy, 123(3), 358, 2003. With permission from IEEJ.)

their low-frequency measurement), εr = 50, and μr = 1, respectively. Figure 5.6 shows the FDTD-computed voltage and current waveforms for the squareshape electrode and the corresponding measured waveforms [10]. The overall waveforms of voltage and current computed using the FDTD method agree reasonably well with the measured ones. Note that Miyazaki and Ishii [11] have reasonably well reproduced the measured waveforms, shown in Figure 5.6a, using the MoM in the frequency domain. Ala et al. [12] have considered the soil ionization around a grounding electrode in their FDTD computations. The ionization model is based on the dynamic soil-resistivity model of Liew and Darveniza [13]. Figure 5.7 shows the resistivity profile in the dynamic model proposed by Liew and Darveniza [13] and employed by Ala et al. [12]. In the model, the resistivity of each soil-representing cell is controlled by the instantaneous value of the electric field and time. When the instantaneous value of the electric field E at

366


ρ

ρ0

Ionization Deionization ρi Ec

Emax

E

FIGURE 5.7 Resistivity profile in the dynamic model proposed by Liew and Darveniza [13] and employed by Ala et al. [12] for FDTD computations.

a soil-representing cell is lower than the critical electric field Ec, the resistivity ρ is equal to its steady-state value ρ 0:

ρ = ρ0 (5.72)

When E at a soil-representing cell exceeds the critical electric field Ec, ρ begins to decrease with time as follows:

 t  ρ = ρ0 exp  −  (5.73)  τ1 

where t is the time defined so that t = 0 at the instant of E = Ec τ1 is the ionization time constant This decreasing resistivity with time represents soil-ionization process. When E at a cell in the ionized-soil region falls below Ec, ρ begins to increase with time as follows: 2

 E  t   ρ = ρi + (ρ0 − ρi ) 1 − exp  −    1 −  (5.74) Ec   τ2    

where ρi is the minimal value reached by the ionization process t is the time defined so that t = 0 at the instant of E = Ec τ2 is the de-ionization time constant

367


PEC boundary

y

z

Current lead wires

x

Air

Rod under test

Soil

PML

0.61 m 2.12 m

Auxiliary rod PML PEC boundary

FIGURE 5.8 Configuration of a 0.61 m long vertical grounding electrode and its auxiliary wires for measurement of its surge response. (Reprinted from Ala, G. et al., IET Sci., Meas. Technol., 2(3), 134, 2008. With permission from IET.)

This increasing resistivity with time from ρi to ρ 0 represents de-ionization process of soil. Figure 5.8 shows a 0.61 m vertical grounding rod buried in homogeneous soil of resistivity ρ 0 = 50 Ω-m, relative permittivity εr = 8, and relative permeability μr = 1, to be analyzed using the FDTD method. The vertical grounding rod is energized by a lumped current source whose other terminal is connected to four auxiliary grounding electrodes via overhead wires. The current source generates a current having a magnitude of about 3.5 kA and a risetime of about 5 μs, as shown in Figure 5.9a. The working volume is divided uniformly into 61 mm × 61 mm × 61 mm cubic cells and is surrounded by six planes of an absorbing boundary condition to minimize unwanted reflections there. The equivalent radius of the vertical grounding rod is 14 mm (=0.23∆s = 0.23 × 61 mm) [4,14]. Figure 5.9b shows the waveform of voltage at the top of the vertical grounding rod, computed using the FDTD method with the soil-ionization model. In this computation, Ec = 110 kV/m, τ1 = 2.0 μs and τ2 = 4.5 μs were employed for the soil-ionization model. Also shown in Figure 5.9b is the voltage waveform computed without the soil-ionization model. The peak voltage computed with the soil-ionization model is about 40% smaller than that computed without the soil-ionization model. 5.2.2 Transmission Towers Lightning overvoltages in overhead power-transmission systems are mainly caused by back-flashovers of tower insulations. The electromagnetic field around a transmission tower hit by lightning changes dynamically while electromagnetic waves make several round-trips between a shield wire and the ground.

368


4000

Current (A)

3000

2000

1000

0

0

10

(a)

20

30

Time (µs)

Potential to remote ground (kV)

150

100

50 No ionization Ionization 0

(b)

0

5

10

15

20

Time (µs)

FIGURE 5.9 Waveform of the current injected into the top of the 0.61 m long vertical grounding rod, and waveforms of the voltage at the top of the grounding rod, computed using the FDTD method with and without the soil ionization model. (Reprinted from Ala, G. et al., IET Sci., Meas. Technol., 2(3), 134, 2008. With permission from IET.) (a) Current injected and (b) voltages computed with and without soil ionization.

During this interval, the waveforms of insulator voltages vary complexly. For a tall structure such as an extra-high-voltage, double-circuit transmission tower, the contribution of the tower-surge characteristic to the insulator voltages becomes dominant, because the travel time of a surge along the tower is comparable to the risetime of a lightning current. Therefore, it is important to investigate the surge characteristics of tall towers.

369


Mozumi et al. [15] have computed, using the MoM in the time domain [7], voltages across insulators of a 500 kV double-circuit transmission-line tower with two overhead ground wires located above a flat perfectly conducting ground, in the case that the tower top is struck by lightning and thereby the back-flashover occurs across the insulator of one phase. In order to analyze back-flashover using the MoM in the time domain, they incorporated a flashover model developed by Motoyama [16] in it. For the computations, the lightning return-stroke channel is represented by a vertical perfectly conducting wire in air. The lightning channel and the tower are excited by a lumped voltage source in series with 5 kΩ lumped resistance inserted between them. Figure 5.10 shows the structure of the tower to be analyzed. This conductor system is divided into cylindrical thin segments of about 4 m in length. The time increment is set to 20 ns. Figure 5.11 shows waveforms of insulator voltages computed using the MoM in the time domain and using the electromagnetic transients program (EMTP) [17], when an upper-phase back-flashover occurs for a current of magnitude 150 kA and risetime 1 μs is being injected. Note that, in the EMTP 29.0 m r = 20 mm 8.0 m

16.0 m

12.0 m 80.0 m

11.2 m

4.0 m

11.6 m

4.0 m

12.0 m

4.0 m r = 0.373 m

44.0 m

FIGURE 5.10 Structure of a 500 kV transmission-line tower to be analyzed using the MoM in the time domain. (Reprinted with permission from Mozumi, T., Baba, Y., Ishii, M., Nagaoka, N., and Ametani, A., Numerical electromagnetic field analysis of archorn voltages during a back-flashover on a 500 kV twin-circuit line, IEEE Trans. Power Deliv., 18(1), 207–213. Copyright 2003 IEEE.)

370


Voltage (MV)

4 2

6 Voltage (MV)

Upper phase Middle phase Lower phase

6

4 2 0

0 (a)

Upper phase Middle phase Lower phase

0

1 Time (µs)

2

(b)

0

1 Time (µs)

2

FIGURE 5.11 Waveforms of insulator voltages computed using the MoM in the time domain and using EMTP, in the case of an upper-phase back-flashover for a current having a magnitude 150 kA and a risetime 1 μs being injected. (Reprinted with permission from Mozumi, T., Baba, Y., Ishii, M., Nagaoka, N., and Ametani, A., Numerical electromagnetic field analysis of archorn voltages during a back-flashover on a 500 kV twin-circuit line, IEEE Trans. Power Deliv., 18(1), 207–213. Copyright 2003 IEEE.) (a) TWTD and (b) EMTP.

computation, a multistory transmission-line tower model [18] is employed, and its top part characteristic impedance is set to 245 Ω and its bottom part characteristic impedance is set to 180 Ω. MoM-computed waveforms are reasonably well reproduced by the corresponding EMTP-computed waveforms. Noda [19] has computed, using the FDTD method [1], voltages across insulators of a 500 kV double-circuit transmission-line tower, located above a flat ground having a conductivity of 10 mS/m, in the case that the tower top is struck by lightning. In his computation, the lightning returnstroke channel is represented by a vertical perfectly conducting wire having additional distributed series inductance of 10 μH/m, and the resultant speed of current wave propagating along the wire is 0.33c. The lightning channel and the tower are excited by a lumped current source inserted between them. Figure 5.12 shows the structure of the tower to be analyzed using the FDTD method. This conductor system is accommodated in a working volume of 250 m × 250 m × 150 m, which is divided uniformly into cubic cells of 1 m × 1 m × 1 m. The working volume is surrounded by six planes of Liao’s second-order absorbing boundary condition [9] to minimize unwanted reflections there. Figure 5.13 shows waveforms of insulator voltages computed using the FDTD method and using the EMTP when a ramp current having a magnitude 1 A and a risetime 1 μs is injected. Note that, in his EMTP computation, a new circuit model for a tower [19] is employed, and its characteristic impedance is set to 192 Ω. FDTD-computed waveforms are reasonably wellreproduced by the corresponding EMTP-computed waveforms. Also, Noda [19] has shown that his FDTD-computed waveforms of tower-top voltage and tower current for a similar tower agree reasonably well with the corresponding measured waveforms.


371

Absorbing boundary

Lightningchannel model

133 m

Current source representing a return stroke

Insulator-string voltages

VH1 VH2 VH3

77 m

FIGURE 5.12 Structure of a 500 kV transmission-line tower to be analyzed using the FDTD method. (Reprinted from Noda, T., IEE J. Trans. Power Energy, 127(2), 379, 2007. With permission from IEEJ.)

5.2.3 Distribution Lines: Lightning-Induced Surges In order to optimize lightning-protection means of telecommunication and power distribution lines, one needs to know voltages that can be induced on overhead wires by lightning strikes to ground or to nearby grounded objects. NEA methods have recently been employed in analyzing lightning-induced voltages on overhead telecommunication and power distribution lines. Using the MoM in the frequency domain [2], Pokharel et al. [20] have reproduced lightning-induced voltages on an overhead horizontal wire of radius 0.25 mm, length 25 m, and height 0.5 m, measured by Ishii et al. [21]. Figure 5.14 shows the configuration of Ishii et al.’s small-scale experiment. In the experiment, a vertical return-stroke channel is represented by a coiled wire along which a current wave propagates upward at a speed of about 125 m/μs. The close (to the simulated channel) end of the overhead horizontal wire is either terminated in a 430-Ω resistor or left open, and the remote end is terminated in a 430-Ω resistor. The lightning-induced voltages at both ends of the wire are measured using voltage probes having 20-pF input capacitance. Figure 5.15 shows MoM-computed and measured waveforms of induced voltages. Note that, in the MoM computations, the lightning channel is represented by a vertical wire having 1-Ω/m series-distributed resistance and 3 μH/m

372


VH1 VH2

Voltage (V)

40

VH3

20

0 (a)

0

1

Time [µs]

2

VH1 VH2 VH3

Voltage (V)

40

20

0

3

0

2

1

(b)

3

Time [µs]

FIGURE 5.13 Waveforms of insulator voltages computed using FDTD and EMTP, in the case of a ramp current having a magnitude 1 A and a risetime 1 μs being injected. (Reprinted from Noda, T., IEE J. Trans. Power Energy, 127(2), 379, 2007. With permission from IEEJ.) (a) FDTD and (b) EMTP.

Simulated lightning channel

28 m Simulated overhead line

1.5 m

P.G. 7.5 m

0.5 m 25 m

FIGURE 5.14 Configuration of a small-scale experiment for measuring lightning-induced voltages. (Reprinted with permission from Ishii, M., Michishita, K., and Hongo, Y., Experimental study of lightning-induced voltage on an overhead wire over lossy ground, IEEE Trans. Electromag. Compatib., 41(1), 39–45. Copyright 1999 IEEE.)

373


Induced voltage (V)

2.0

1.0

0.0 Measured Calculated (conductivity - 0.06 S/m) Calculated (perfect ground) –1.0

0

100

0.8

Induced voltage (V)

300

Time [ns]

(a)

Measured Calculated (conductivity - 0.06 S/m) Calculated (perfect ground)

0.0

–0.8

–1.6 (b)

200

0

100

200

300

Time [ns]

FIGURE 5.15 MoM-computed and measured waveforms of lightning-induced voltages at both ends of the overhead wire; each end is terminated in a 430 Ω resistor. (Reprinted with permission from Pokharel, R.K., Ishii, M., and Baba, Y., Numerical electromagnetic analysis of lightninginduced voltage over ground of finite conductivity, IEEE Trans. Electromag. Compatib., 45(4), 651–656. Copyright 2003 IEEE.) (a) Voltages at the close end and (b) voltages at the remote end.

series-distributed inductance, and the ground conductivity and its relative permittivity are set to σ = 0.06 S/m and εr = 10, respectively. The conductor system is modeled as the combination of cylindrical segments of either 1 m or 0.5 m in length. Computation is carried out over the frequency range from 195.3 kHz to 50 MHz with an increment step of 195.3 kHz. This corresponds to the time interval from 0 to 5.12 μs with a time increment of 10 ns. In Figure 5.15, MoMcomputed voltage waveforms for the case of perfectly conducting ground are also shown for reference. Owing to the finitely conducting ground, the polarity of the remote-end voltage is opposite to that of the close end.

374


Note that Ishii et al. [21] have well-reproduced lightning-induced voltages measured in their experiment with Agrawal et al.’s field-to-wire electromagnetic coupling model [22], and Baba and Rakov [23] have reproduced the same measured waveforms using the FDTD method [1]. Using the MoM in the time domain [7], Pokharel et al. [24] have computed lightning-induced voltages on a 500 m long overhead wire, which is located above a flat perfectly conducting ground. The overhead wire is terminated in a 540 Ω resistor at each end, and is connected to ground in the middle of the wire via a surge arrester. The surge arrester is represented by a nonlinear resistor whose characteristics are shown in Figure 5.16. Figure 5.17 shows computed waveforms of lightning-induced voltage in the middle of the wire with or without the surge arrester and the arrester current. Note that the simulated lightning channel was located 100 m away from the middle of the overhead wire, the magnitude of lightning current is set to 10 kA, and its risetime is set to 1 μs. Using the same time-domain method, Moini et al. [25] have performed a computation of lightning-induced voltages on vertically arranged and horizontally arranged multiphase conductors above a flat perfectly conducting ground.

V Vd

Rt Ω 1 MΩ

–I

I

Rt Ω

–Vd

–V

FIGURE 5.16 Approximate voltage versus current characteristics of nonlinear resistance representing a surge arrester, employed by Pokharel et al. [24] in their computations using the MoM in the time domain. Vd is set to 30 kV. (Reprinted with permission from Pokharel, R.K. and Ishii, M., Applications of time-domain numerical electromagnetic code to lightning surge analysis, IEEE Trans. Electromag. Compatib., 49(3), 623–631. Copyright 2007 IEEE.)

375


Induced voltage (kV)

Without arrester With arrester Arrester current

30

30

20

20

10

10

0

0

1

2

3

4

5

–10

6

Arrester current (A)

40

40

0 –10

Time (µs) FIGURE 5.17 Waveforms of lightning-induced voltage in the middle of a 500 m long overhead wire with or without a surge arrester and the arrester current, computed using the MoM in the time domain. (Reprinted with permission from Pokharel, R.K. and Ishii, M., Applications of timedomain numerical electromagnetic code to lightning surge analysis, IEEE Trans. Electromag. Compatib., 49(3), 623–631. Copyright 2007 IEEE.)

5.2.4 Transmission Lines: Propagation of Lightning Surges in the Presence of Corona When an overhead shield wire of transmission line is struck by lightning, corona discharge occurs on this wire. Corona discharge around a shield wire reduces its characteristic impedance and increases the coupling between the shield wire and phase conductors. The reduced characteristic impedance of the shield wire results in a smaller tower current, and the increased coupling to the phase conductors increases phase-conductor voltages. As a result, corona discharge gives reduced insulator voltages. Also, it distorts the wavefronts of propagating lightning-surge voltages. Thus, it is important to consider corona effects in computing lightning surges on transmission lines and in designing their lightning protection. Thang et al. [26] have proposed a simplified corona-discharge model for FDTD computations. They represent the radial progression of corona streamers from energized wire by the radial expansion of cylindrical conducting region. The critical electric field E0 on the surface of cylindrical wire of radius r0 for initiation of corona discharge is given by equation of Hartmann [27], which is reproduced as follows:

0.1269   E0 = m ⋅ 2.594 × 106  1 + 0.4346  [ V/m ] (5.75) r0  

where m is the coefficient depending on the wire-surface conditions. Note that this coefficient was not employed by Hartmann, but was apparently

376


later introduced by Guillier et al. [28]. Since radial electric-field computation points closest to the wire are located not at 0.23∆x and 0.23∆z (which are equal to the equivalent wire radius) from the wire axis, but at 0.5∆x and 0.5∆z, they assume that corona streamers start emanating from the wire when the radial electric field at 0.5∆x (and 0.5∆z) exceeds 0.46E0 (=E0 × 0.23∆x/0.5∆x). They set the critical background electric field necessary for streamer propagation (which determines the maximum extent of the radially expanding corona region) for positive, Ecp, and negative, Ecn, polarity as follows: Ecp = 0.5 [MV/m]  (5.76)  Ecn = 1.5 [MV/m] 

The corona radius rc was obtained using analytical expression (5.77), based on Ec (0.5 or 1.5 MV/m, depending on polarity; see Equation 5.76) and the FDTD-computed charge per unit length (q). Then, the conductivity of the cells located within rc was set to σcor = 20 or 40 μS/m:

Ec =

q q [V/m] (5.77) + 2πε0rc 2πε0 (2h − rc )

Simulation of corona discharge implemented in the FDTD procedure is summarized in the following:

1. If the FDTD-computed electric-field, Ezbn, at time step n and at a point located below and closest to the wire (at 0.5∆z from the wire axis shown in Figure 5.18a) exceeds 0.46E0, the conductivity of σcor = 20 or 40 μS/m is assigned to x- and z-directed sides of the four cells closest to the wire. 2. The radial current In per unit length of the wire at y = j∆y from the excitation point at time step n is evaluated by numerically integrating radial conduction and displacement current densities as follows:

(

)

(

)

n n n I n ( j∆y) = σ  Exln + Exr ∆z + Eza + Ezb ∆x  ∆y  

n n −1  Exln − Exln−1 Exr   − Exr +   ∆z  ∆t ∆t    +ε0   ∆y (5.78) n n −1 n n −1     Eza Ezb − Eza − Ezb +  ∆xx  +  ∆t ∆t    

where Exl, Exr, Eza, and Ezb are radial electric fields closest to the wire shown in Figure 5.18b.

377


n+1

Time step: n

Wire z E

n zb> 0.46E0

σcor = 20 or 40 µS/m

x

y (a)

n+1

Time step: n rcn+1

From Ec and q n

E nza E nxl

E nxr

z

E nzb σcor = 20 or 40 µS/m

y

x

(b)

FIGURE 5.18 FDTD representations of (a) inception of corona discharge at the wire surface and (b) radial expansion of corona discharge. (Reprinted with permission from Thang, T.H., Baba, Y., Nagaoka, N. et al., FDTD simulation of lightning surges on overhead wires in the presence of corona discharge, IEEE Trans. Electromag. Compatib., 54(6), 1234–1243. Copyright 2012 IEEE.)

The total charge (charge deposited on the wire and emanated corona charge) per unit length of the wire at y = j∆y from the excitation point at time step n is calculated as follows:

qn ( j∆y) = qn −1( j∆y) +

I n −1( j∆y) + I n ( j∆y) ∆t (5.79) 2

From qn yielded by Equation 5.79 and Ec given by Equation 5.76, the corona radius rcn + 1 at time step n + 1 is calculated using Equation 5.77. The conductivity of σcor = 20 or 40 μS/m is assigned to x- and z-directed sides of all cells located within rcn + 1. Figure 5.19a shows 3D view of a 12.65 mm radius, 1.4 km long overhead horizontal single perfectly conducting wire located 22.2 m above ground of conductivity 10 mS/m and a 1.4 km long bundled perfect conductor (four conductors in the bundle) located 14 m above the same ground and horizontally 2 m

378


1.4 km

z y x

22.2 m 2m

PG

490 Ω

14 m

(a) ∆z = 100 cm ∆z = 20 cm ∆z = 10 cm ∆z = 5.5 cm

8.2 m

40 cm

∆z = 5.0 cm

z

y (b)

x

40 cm 2m

FIGURE 5.19 (a) 3D and (b) cross-sectional views of a horizontal single wire of radius 12.65 mm and length 1.4 km, located 22.2 m above ground of conductivity 10 mS/m and a four-conductor bundle of length 1.4 km, located 14 m above the ground and horizontally 2 m away from the single wire [30]. One end of the single wire is energized by a lumped voltage source and the other end is connected to the ground via a 490 Ω resistor. (Reprinted with permission from Thang, T.H., Baba, Y., Nagaoka, N. et al., A simplified model of corona discharge on an overhead wire for FDTD computations, IEEE Trans. Electromag. Compatib., 54(3), 585–593. Copyright 2012 IEEE.)

away from the single wire. This configuration represents an experiment of Inoue [29]. The radius of each conductor of the bundle is 11.5 mm and the distance between conductors is 0.4 m. One end of the single wire is energized by a lumped voltage source and the other end is connected to the ground via a 490 Ω (matching) resistor. For FDTD computations, this conductor system is


379

accommodated in a working volume of 60 m × 1460 m × 80 m, which is divided nonuniformly into rectangular cells and is surrounded by six planes of Liao’s second-order absorbing boundary condition [9] to minimize unwanted reflections there. At each ground connection point, a perfectly conducting grounding electrode of 20 m × 20 m × 10 m is employed. The side length in y direction of all the cells is 1 m (constant). Cell sides along x- and z-axes are not constant: 5.5 cm in the vicinity (220 cm × 220 cm) of the horizontal single wire, increasing gradually (to 10, 20, and 100 cm) beyond that region, except for a region around the bundled conductor, and 5 cm in the vicinity (80 cm × 80 cm) of the bundled conductor, except for a region around the horizontal single wire, increasing gradually (to 10, 20, and 100 cm) beyond that region, as shown in Figure 5.19b. The equivalent radius of the horizontal single wire used in this paper is r0 ≈ 12.65 mm (=0.23∆x = 0.23∆z = 0.23 × 5.5 cm), which is equal to those used in the corresponding experiment of Inoue [29]. The time increment was set to ∆t = 1.75 ns. Figure 5.20 shows waveforms of positive surge voltage at d = 0, 350, 700, and 1050 m from the energized end of the horizontal single wire above ground whose conductivity is 10 mS/m, computed using the FDTD method for corona-region conductivity σcor = 40 μS/m [30]. The critical electric field for corona onset on the wire surface was set to E0 = 2.4 MV/m (for m = 0.5). The corresponding measured waveforms [29] are also shown in this figure. FDTD-computed waveforms agree reasonably well with the corresponding measured ones. Both FDTD-computed and measured waveforms of surge voltage suffer distortion, which becomes more significant with increasing applied voltage peak and the propagation distance. Maximum corona radii for positive voltage peaks of 1580, 1130, and 847 kV are 66, 44, and 27.5 cm, respectively. Figure 5.21 shows FDTD-computed waveforms of surge voltages without considering corona discharge for 847 kV positive voltage application. The measured waveforms [29] with corona discharge are also shown in this figure. In the absence of corona, the FDTD-computed surge voltages suffer little distortion with propagation, and significantly differ from the corresponding measured waveforms with corona discharge. Figure 5.22 shows FDTD-computed waveforms of induced voltage at d = 700 and 1050 m on the 1.4 km long horizontal four-conductor bundle, which is located horizontally 2 m away from the energized horizontal wire and 14 m above flat ground [30]. The corresponding measured waveforms [29] are also shown in Figure 5.22. The computed waveforms of voltages induced on the bundled conductor also agree fairly well with the corresponding measured waveforms. 5.2.5 Power Cables: Propagation of Power Line Communication Signals Power line communication (PLC) systems use power distribution lines and cables for data communications in a frequency range up to 30 MHz. Within

380


2000 Applied (measured at d = 0) Voltage (kV)

1500 Computed

1000 500 0

(a)

Measured by Inoue d = 350 m 700 m 1050 m 0

1

Time (µs)

2

3

1500 Applied (measured at d = 0)

Voltage (kV)

1200 900

Computed

600 Measured by Inoue

300 0 (b)

0

1000

1

Time (µs)

2

3

Applied (measured at d = 0)

Voltage (kV)

800 Computed

600 400

Measured by Inoue

200 0 (c)

0

1

Time (µs)

2

3

FIGURE 5.20 FDTD-computed (for σcor = 40 μS/m and E0 = 2.4 MV/m) and measured waveforms of surge voltage at d = 0, 350, 700, and 1050 m from the energized end of the 12.65 mm radius, 1.4 km long horizontal wire, located 22.2 m above ground of conductivity 10 mS/m [30]. The applied voltage is positive and Ecp = 0.5 MV/m. Applied voltage peaks are (a) 1580 kV, (b) 1130 kV, and (c) 847 kV. (Reprinted with permission from Thang, T.H., Baba, Y., Nagaoka, N. et al., A simplified model of corona discharge on an overhead wire for FDTD computations, IEEE Trans. Electromag. Compatib., 54(3), 585–593. Copyright 2012 IEEE.)

381


1000 Computed (without corona)

Voltage (kV)

800

Applied (measured at d = 0)

600 400 Measured by Inoue 200 0

0

1

2

3

Time (µs) FIGURE 5.21 Same as Figure 5.20c, but computed without corona discharge. (Reprinted with permission from Thang, T.H., Baba, Y., Nagaoka, N. et al., A simplified model of corona discharge on an overhead wire for FDTD computations, IEEE Trans. Electromag. Compatib., 54(3), 585–593. Copyright 2012 IEEE.)

a power cable, semiconducting layers are usually incorporated between the core conductor of the power cable and the insulating layer and between the insulating layer and the sheath conductor. Since power cables are not designed for effectively transmitting the PLC signals, they might attenuate significantly along the cables owing to the presence of semiconducting layers. Okazima et al. [31] have investigated the propagation characteristics of a PLC signal of frequency 30 MHz along a single-core power cable having two 3 mm thick semiconducting layers, using the FDTD method [1] in the twodimensional (2D) cylindrical-coordinate system. Figure 5.23 shows a 130 m long single-core power cable to be analyzed using the FDTD method. The radius of the core conductor is 5 mm, and the inner radius of the sheath conductor is 25 mm. Both the core and sheath conductors are perfectly conducting. Figure 5.23 also shows a 14 mm thick insulating layer with semiconducting layers of 3 mm thickness on both inner and outer surfaces. The relative permittivity εr of the insulating layer and of each semiconducting layer is set to εr = 3. The conductivity of each semiconducting layer is set to a value in a range from σ = 10−5 to 105 S/m. Note that this power cable is rotationally symmetric around its axis and has a circular cross section. It is represented without staircase-approximated contour in the FDTD method using a 2D cylindrical coordinate system with a working space of 130 m × 27 mm rectangles, contoured by the thick black line shown in Figure 5.23. At one of the ends of the cable, a 10 V positive half-sine pulse of frequency f = 30 MHz is applied between the core and sheath conductors. The other end of the cable model is terminated using Liao’s second-order absorbing boundary.

382


Voltage (kV)

600

400

Measured by Inoue

200

0

Computed

0

1

(a)

2

3

Time (µs)

Voltage (kV)

600

400

200

0 (b)

Computed

Measured by Inoue

0

1

2

3

Time (µs)

FIGURE 5.22 FDTD-computed (for σcor = 40 μS/m and E0 = 2.4 MV/m) and measured waveforms of voltage induced on the nearby four-conductor bundle at d = 700 and 1050 m, located 14 m above ground of conductivity 10 mS/m [30]. The applied voltage is positive and Ecp = 0.5 MV/m. The applied voltage peak is 1580 kV. (Reprinted with permission from Thang, T.H., Baba, Y., Nagaoka, N. et al., A simplified model of corona discharge on an overhead wire for FDTD computations, IEEE Trans. Electromag. Compatib., 54(3), 585–593. Copyright 2012 IEEE.) (a) d = 700 m (b) d = 1050 m.

The working space of 130 m × 27 mm for the FDTD computation is divided into 1 mm × 1 mm square cells. The time increment is set to 2.3 ps. Figure 5.24 shows waveforms of the voltage between the core and sheath conductors of the power cable at different distances of 20, 40, 60, 80, and 100 m from the excitation point. Figure 5.25 shows how σ affects the magnitude of the voltage between the core and sheath conductors at distance 100 m from the excitation point. It can be noticed from Figures 5.24 and 5.25 that the magnitude of a voltage pulse decreases with increasing propagation distance in all cases considered. However, the dependence of the signal attenuation on σ is not monotonic; the attenuation is significant around

383


27 mm Z Sheath conductor

Absorbing boundary

3-mm thick semiconducting layers

Working space of 130 m × 27 mm 130 m

14-mm thick insulation Core conductor

Voltage source r

FIGURE 5.23 130 m long single-core power cable with semiconducting layers, to be analyzed using the FDTD method. The 130 m × 27 mm rectangle space, contoured by a thick black line, is the actual working space for the present FDTD computations in the 2D cylindrical coordinate system. (Reprinted with permission from Okazima, N., Baba, Y., Nagaoka, N., Ametani, A., Temma, K., and Shimomura, T., Propagation characteristics of power line communication signals along a power cable having semiconducting layers, IEEE Trans. Electromag. Compatib., 52(3), 756–759. Copyright 2010 IEEE.)

σ = 10−3 and 103 S/m, while it is not when σ is lower than about 10−5 S/m or σ is around 1 S/m. When σ = 10−3 S/m and 103 S/m, dispersion is also marked. Therefore, it is quite difficult to conduct PLC signals in a power system cable with semiconducting layers of conductivity about σ = 10−3 or 103 S/m, but there are more possibilities if σ ≤10−5 S/m or σ = 1 S/m. The signal attenuation around σ = 10−3 S/m is caused by capacitive charging and discharging of the semiconducting layers in radial direction. For σ = 103 S/m, axial current propagation in the semiconducting layers is the dominant cause of attenuation. These are quantified as follows: The time constant τ of each semiconducting layer is given by

τ = CR =

2πε0εr ln(r2/r1 ) ε0εr (5.80) = ln(r2/r1 ) 2πσ σ

where C = 2πεrε0/ln(r2/r1) is the per-unit-length capacitance of each semiconducting layer R = ln(r2/r1)/(2πσ) is its radial direction per-unit-length resistance r2 is the outer radius of the semiconducting layer r1 is its inner radius ε0 is the permittivity of vacuum εr is the relative permittivity of the semiconducting layer σ is its conductivity

384


12

σ = 10–5 S/m

80 m 100 m

6 4

0

200

400 Time (ns)

600

800

Applied 10 voltage20 m 8

(b)

σ = 1 S/m 40 m

60 m

80 m

100 m

6 4

40 m 60 m 200

0

200

400 Time (ns)

σ = 10–3 S/m

6 4

20 m

(d)

40 m

200

0

12

Voltage (V)

Applied 10 voltage 8

60 m

80 m

400 Time (ns)

600

100 m 800

σ = 105 S/m

20 m

6

800

8

0

800

600

100 m

600

Applied 10 voltage

2 0

80 m

400 Time (ns)

12

2 0

20 m

4

0

12

(c)

6

2 0

σ = 10–3 S/m

8

2

(a)

Voltage (V)

Applied 10 voltage Voltage (V)

8

12

Voltage (V)

Voltage (V)

Applied 10 voltage 20 m 40 m 60 m

40 m 60 m

80 m 100 m

4 2 0

(e)

0

200

400 Time (ns)

600

800

FIGURE 5.24 Waveforms of the voltage between the core and sheath conductors at different distances from the excitation point when a 30 MHz and 10 V half-sine pulse is injected and the semiconductinglayer conductivity σ is (a) 10−5, (b) 10−3, (c) 1, (d) 103, and (e) 105 S/m. (Reprinted with permission from Okazima, N., Baba, Y., Nagaoka, N., Ametani, A., Temma, K., and Shimomura, T., Propagation characteristics of power line communication signals along a power cable having semiconducting layers, IEEE Trans. Electromag. Compatib., 52(3), 756–759. Copyright 2010 IEEE.)

For εr = 3 and σ = 10−3 S/m, the time constant is τ = 27 ns. Charging and discharging processes in the radial direction of the semiconducting layer with σ = 10−3 S/m will have a strong effect on a 30 MHz signal with 17-ns half cycle, leading to significant attenuation and distortion. Note that, in this condition, the magnitude of the radial conduction current across the semiconducting layer is close to that of the radial displacement current. In other words, the conductance of the semiconducting layer is close to its susceptance. At high conductivity of σ = 103 S/m, the depth d of penetration for an electromagnetic wave of frequency f into a medium of conductivity σ and permeability μ0 is relevant for loss calculations. This depth is given by

385


12

Voltage (V)

10 8 6 4 2 0 0.00001

0.001

0.1

10

1,000

100,000

Conductivity (S/m) FIGURE 5.25 Dependence of the magnitude of the voltage between the core and sheath conductors at distance 100 m from the excitation point on the semiconducting-layer conductivity σ when a 30 MHz and 10 V half-sine pulse is injected. (Reprinted with permission from Okazima, N., Baba, Y., Nagaoka, N., Ametani, A., Temma, K., and Shimomura, T., Propagation characteristics of power line communication signals along a power cable having semiconducting layers, IEEE Trans. Electromag. Compatib., 52(3), 756–759. Copyright 2010 IEEE.)

d=

1 (5.81) 2πfσµ 0

For μ0 = 4π × 10−7 H/m, f = 30 MHz and σ = 103 S/m, Equation 5.81 yields d = 2 mm, which is close to the thickness of the semiconducting layer (3 mm). Therefore, most of the axial current flows in the semiconducting layers rather than on the core and sheath conductor surfaces. This results in the significant signal attenuation and dispersion. 5.2.6 Air-Insulated Substations For appropriately estimating the lightning impulse withstand voltage level of substations, lightning overvoltages that would be generated in 3D complexstructure substations need to be known. Using the FDTD method [1], Watanabe et al. [32] have computed surge voltages on an air-insulated substation, to which an impulse voltage is applied, and compared the FDTD-computed voltage waveforms with the corresponding waveforms measured on a 1/10-scale model shown in Figure 5.26. Figure 5.27 shows the plan view of the FDTD model and the reduced-scale experimental model. An impulse voltage is applied to the terminal of line B-2, and surge voltages are measured at the voltage application point and at the number-2 transformer.

386


FIGURE 5.26 1/10-scale model of an air-insulated substation. (Reprinted from Watanabe, T. et al., The measurement and analysis of surge characteristics using miniature model of air insulated substation, Paper presented at IPST 2005, Montreal, Quebec, Canada, 2005. With permission from IPST.)

Line B-2

Line B-1

Line A-2

Line A-1

Busbar 1A

Busbar 2

Busbar 1B

No.2 TR

No.1 TR

Measuring point

FIGURE 5.27 Plan view of the small-scale model and the FDTD simulation model: An impulse voltage is applied to the terminal of line B-2, and surge voltages are measured at the voltage application point and at the number-2 transformer. (Reprinted from Watanabe, T. et al., The measurement and analysis of surge characteristics using miniature model of air insulated substation, Paper presented at IPST 2005, Montreal, Quebec, Canada, 2005. With permission from IPST.)

387


1000 500

(a)

0 0.0

Induction phase 1.0 Time (µs)

Voltage (V)

Voltage injection phase

1000

(b)

500



1000 500 0 0.0

2.0

1500

0 0.0

1500 Voltage (V)



2.0

1500 Voltage (V)

Voltage (V)

1500

2.0


1000 500 0 0.0


2.0

FIGURE 5.28 Measured and FDTD-computed waveforms: The left-side plots show voltages at the voltage application point, and the right-side plots show voltages at the number-2 transformer. (Reprinted from Watanabe, T. et al., The measurement and analysis of surge characteristics using miniature model of air insulated substation, Paper presented at IPST 2005, Montreal, Quebec, Canada, 2005. With permission from IPST.) (a) Measured voltages (b) FDTD-computed voltages.

Figure 5.28 shows measured and FDTD-computed voltage waveforms. FDTD-computed waveforms agree well with the corresponding measured waveforms. Note that Oliveira and Sobrinho [33] have performed a similar FDTD computation for an air-insulated substation struck by lightning. 5.2.7 Wind Turbine Generator Towers Wind turbine generator towers are frequently struck by lightning. In order to optimize lightning protection means of wind turbine generator systems, it is important to know the mechanism of lightning overvoltages generated in the systems. Yamamoto [34] and Yamamoto et al. [35] have investigated the lightning protection of wind turbine generator systems using the FDTD method [1], and experimentally with a small-scale model of a wind turbine generator tower struck by lightning. Figure 5.29 shows a 3/100-scale model of a 50 m high wind turbine tower with 25 m long blades, one of which is connected to a lightning-channel-representing vertical current lead wire or the core conductor of a vertical coaxial cable. The grounding resistance is set to 9.4 Ω. In the FDTD simulation, this conductor system is accommodated in a working volume of 6 m × 5 m × 7.5 m, which is divided uniformly into cubic cells of 25 mm × 25 mm × 25 mm. The working volume is surrounded by six planes of Liao’s second-order absorbing boundary condition [9]. V11 to V14 in Figure 5.30 represent the voltage differences between the incoming conductor connected to a distant point and an equipment in the tower foot. The voltage difference is generated by the voltage rise at the tower foot.

388


Current lead wire (coaxial cable for summer and winter lightning)

3.0 m

Current lead wire (coaxial cable for summer lightning, insulated copper wire for winter lightning)

6.5 m

Voltage measuring wire I

3.0 m V Aluminum plate

Pulse generator Rg

Wind turbine model Grounding resistance

FIGURE 5.29 3D view of a 3/100-scale model of a wind turbine generator tower, blades, nacelle, current lead wire, and a voltage reference wire. (Reprinted from Yamamoto, K., A study of overvoltages caused by lightning strokes to a wind turbine generation system, PhD thesis, Doshisha University, Kyoto, Japan, 2007. With permission.)

V11 Conductor

Tower body

V12 V13

Core Sheath V14 Core Sheath

Tower body

Tower body

Incoming conductor (4.5 m) Aluminum plate (which has a thickness of 2 mm)

FIGURE 5.30 Configuration of incoming conductors to the wind turbine tower foot for measuring and computing overvoltages between the incoming conductors and equipment in the tower foot. (Reprinted from Yamamoto, K., A study of overvoltages caused by lightning strokes to a wind turbine generation system, PhD thesis, Doshisha University, Kyoto, Japan, 2007. With permission.)

389


(a)

50 40 30 20 10 0 –10

Voltage (V)

Voltage (V)

50 40 30 20 10 0 –10

0

5

10

15 20 Time (ns)

25

30

35

(b)

0

5

10

15 20 Time (ns)

25

30

35

FIGURE 5.31 Voltage differences between the incoming conductor and equipment in the tower foot. The grounding resistance is 9.4 Ω and the wave front of the injected current is 4 ns. (Reprinted from Yamamoto, K., A study of overvoltages caused by lightning strokes to a wind turbine generation system, PhD thesis, Doshisha University, Kyoto, Japan, 2007. With permission.) (a) Voltage V11–V14 of measured results and (b) Voltage V11 of FDTD analytical results.

This voltage difference might become an overvoltage between the power line and the power converter or transformer installed inside the tower. Figure 5.31 shows measured and FDTD-computed voltages for an injected current having a risetime of 4 ns. The FDTD-computed waveforms agree well with the corresponding measured ones.

References

1. Yee, K. S. 1966. Numerical solution of initial value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propagation 14(3):302–307. 2. Courant, R., K. Friedrichs, and H. Lewy. 1965. On the partial difference equations of mathematical physics. IBM Journal 11(2):215–234. 3. Noda, T. and S. Yokoyama. 2002. Thin wire representation in finite difference time domain surge simulation. IEEE Trans. Power Deliv. 17(3):840–847. 4. Harrington, R. F. 1968. Field Computation by Moment Methods. New York: Macmillan Co. 5. Burke, G. and A. Poggio. 1980. Numerical electromagnetic code (NEC)method of moment. Technical document 116. Naval Ocean Systems Center, San Diego, CA. 6. Burke, G. 1992. Numerical electromagnetic code (NEC-4)-method of moment. UCRL-MA-109338. Lawrence Livermore National Laboratory, Livermore, CA. 7. Van Baricum, M. and E. K. Miller. 1972. TWTD—A computer program for timedomain analysis of thin-wire structures. UCRL-51-277. Lawrence Livermore Laboratory, Livermore, CA. 8. Tanabe, K. 2001. Novel method for analyzing dynamic behavior of grounding systems based on the finite-difference time-domain method. IEEE Power Eng. Rev. 21(9):55–57. 9. Liao, Z. P., H. L. Wong, B. P. Yang, and Y. F. Yuan. 1984. A transmission boundary for transient wave analysis. Sci. Sin. A27(10):1063–1076.

390


10. Tanabe, K., A. Asakawa, M. Sakae, M. Wada, and H. Sugimoto. 2003. Verifying the computational method of transient performance with respect to grounding systems based on the FD–TD method. IEE J. Trans. Power Energy 123(3):358–367. 11. Miyazaki, S. and M. Ishii. 2005. Analysis of transient response of grounding system based on moment method. Paper presented at ISH 2005, Beijing, China. 12. Ala, G., P. L. Buccheri, P. Romano, and F. Viola. 2008. Finite difference time domain simulation of earth electrodes soil ionisation under lightning surge condition. IET Sci., Meas. Technol. 2(3):134–145. 13. Liew, A. C. and M. Darveniza. 1974. Dynamic model of impulse characteristics of concentrated earth. Proc. IEE 121(2):123–135. 14. Baba, Y., N. Nagaoka, and A. Ametani. 2005. Modeling of thin wires in a lossy medium for FDTD simulations. IEEE Trans. Electromag. Compatib. 47(1):54–60. 15. Mozumi, T., Y. Baba, M. Ishii, N. Nagaoka, and A. Ametani. 2003. Numerical electromagnetic field analysis of archorn voltages during a back-flashover on a 500 kV twin-circuit line. IEEE Trans. Power Deliv. 18(1):207–213. 16. Motoyama, H. 1996. Development of a new flashover model for lightning surge analysis. IEEE Trans. Power Deliv. 11(2):972–979. 17. Scott-Meyer, W. 1977. EMTP Rule Book. Portland, OR: Bonneville Power Administration. 18. Ishii, M., T. Kawamura, T. Kouno et al. 1991. Multistory transmission tower model for lightning surge analysis. IEEE Trans. Power Deliv. 6(3):1327–1335. 19. Noda, T. 2007. A tower model for lightning overvoltage studies based on the result of an FDTD simulation. IEE J. Trans. Power Energy 127(2):379–388 (in Japanese). 20. Pokharel, R. K., M. Ishii, and Y. Baba. 2003. Numerical electromagnetic analysis of lightning-induced voltage over ground of finite conductivity. IEEE Trans. Electromag. Compatib. 45(4):651–656. 21. Ishii, M., K. Michishita, and Y. Hongo. 1999. Experimental study of lightninginduced voltage on an overhead wire over lossy ground. IEEE Trans. Electromag. Compatib. 41(1):39–45. 22. Agrawal, A. K., H. J. Price, and H. H. Gurbaxani. 1980. Transient response of muliticonductor transmission lines excited by a nonuniform electromagnetic field. IEEE Trans. Electromag. Compatib. 22(2):119–129. 23. Baba, Y. and V. A. Rakov. 2006. Voltages induced on an overhead wire by lightning strikes to a nearby tall grounded object. IEEE Trans. Electromag. Compatib. 48(1):212–224. 24. Pokharel, R. K. and M. Ishii. 2007. Applications of time-domain numerical electromagnetic code to lightning surge analysis. IEEE Trans. Electromag. Compatib. 49(3):623–631. 25. Moini, R., B. Kordi, and M. Abedi. 1998. Evaluation of LEMP effects on complex wire structures located above a perfectly conducting ground using electric field integral equation in time domain. IEEE Trans. Electromag. Compatib. 40(2):154–162. 26. Thang, T. H., Y. Baba, N. Nagaoka et al. 2012. FDTD simulation of lightning surges on overhead wires in the presence of corona discharge. IEEE Trans. Electromag. Compatib. 54(6):1234–1243. 27. Hartmann, G. 1984. Theoretical evaluation of Peek’s law. IEEE Trans. Indust. Appl. 20(6):1647–1651.


391

28. Guillier, J. F., M. Poloujadoff, and M. Rioual. 1995. Damping model of travelling waves by corona effect along extra high voltage three phase lines. IEEE Trans. Power Deliv. 10(4):1851–1861. 29. CRIEPI Report 114 by Central Research Institute of Electric Power Industry Report. 30. Thang, T. H., Y. Baba, N. Nagaoka et al. 2012. A simplified model of corona discharge on an overhead wire for FDTD computations. IEEE Trans. Electromag. Compatib. 54(3):585–593. 31. Okazima, N., Y. Baba, N. Nagaoka, A. Ametani, K. Temma, and T. Shimomura. 2010. Propagation characteristics of power line communication signals along a power cable having semiconducting layers. IEEE Trans. Electromag. Compatib. 52(3):756–759. 32. Watanabe, T., K. Fukui, H. Motoyama, and T. Noda. 2005. The measurement and analysis of surge characteristics using miniature model of air insulated substation. Paper presented at IPST 2005, Montreal, Quebec, Canada. 33. Silva de Oliveira, R. M. and C. L. S. Souza Sobrinho. 2009. Computational environment for simulating lightning strokes in a power substation by finite-difference time-domain method. IEEE Trans. Electromag. Compatib. 51(4):995–1000. 34. Yamamoto, K. 2007. A study of overvoltages caused by lightning strokes to a wind turbine generation system. PhD thesis, Doshisha University, Kyoto, Japan (in Japanese). 35. Yamamoto, K., T. Noda, S. Yokoyama, and A. Ametani. 2009. Experimental and analytical studies of lightning overvoltages in wind turbine generator systems. Electr. Power Syst. Res. 79(3):436–442.

6 Electromagnetic Disturbances in Power Systems and Customers

6.1 Introduction A number of analog control circuits have been replaced by digital control circuits in power stations and substations in Japan since the middle of the 1980s. The same is for any industrial products such as an automobile, and for home appliances, because of the advancement of digital circuit technologies. At the same time, electromagnetic compatibility (EMC) environments are becoming a significant problem for the digital circuits that are very sensitive to high-frequency electromagnetic waves such as switching and lightning surges. Considering the aforementioned, Electro-technical Research Association in Japan had carried out investigations of disturbances of digital control circuits in generator stations and substations experienced by all the Japanese utilities for 10 years from 1990 [1]. The total number of the disturbances is 330, and one-third is for the protective relays. For protective relays show the largest number, the detail of disturbances of the protective relays is presented in [2,3]. Similarly to the aforementioned, electromagnetic disturbances in customers have become a significant problem and a number of surveys were carried out by utilities and public organizations related to customers’ and home appliances [4–9]. For example, it was reported that there were more than 1000 cases of damages in the home appliances in one year in one Japanese utility. This chapter summarizes the disturbances experienced in Japan including disturbed equipments, surge incoming routes, and the characteristics of the disturbances [1,10]. The influence of the disturbances to power system operation including countermeasures and costs is explained. Some case studies are demonstrated. Also, lightning surges to customers due to nearby lightning and from a communication line are described with measured and simulation results. Finally, an analytical method is explained for calculating an induced voltage to a pipeline or a communication line from a power line. 393

394


6.2 Disturbances in Power Stations and Substations 6.2.1 Statistical Data of Disturbances 6.2.1.1 Overall Data Altogether 330 cases of disturbances were collected in substation and generator stations for about 10 years from 1990 [1]. Table 6.1 summarizes the number of disturbances classified by troubled equipments (a) and by the causes (b). It is observed in Table 6.1a that one-third of the disturbances out of 330 cases are for protection equipments, one-quarter for telecontrol equipments and the other quarter for control equipments. Table 6.1b shows that two-thirds of the disturbances are caused by lightning surges, one-sixth by switching surges in main circuits (high-voltage side), and one-twelfth by switching surges in dc circuits that are a part of control circuits (low-voltage side). TABLE 6.1 Total Number of Disturbances and Failures Collected for 10 Years (a) Disturbed equipments Equipment Protection Telecontrol Supervisory control Communication

Number

Equipment

Number

105 73 5

Controla Measuring Automatic processing Others Total

73 49 2

15

8 330

(b) Causes Basic Cause Lightning Main-circuit switching operation DC circuit switching operation Others

Total

Kind of Surges

Number

Lightning surge DS switching surge CB switching surge Capacitor bank SS DC circuit switching surge Fault surge CPU switching noise Welder noise Not clear

220 21 24 2 21

220 (0.72) 47 (0.15)

Subtotal

2 2 2 13

19 (0.06)

21 (0.07)

307b (ratio 1.0)

Source: Working Group (Chair: Agematsu, S.) Japanese Electrotechnical Research Association, Technologies of countermeasure against surges on protection relays and control systems, ETRA Report , 57(3), 2002 (in Japanese). a Control: control board, station power circuit board, generator sequencer. b Communication and others in Table 6.1a deleted. DS, disconnector; CB, circuit breaker; SS, switching surge; LS, lightning surge.

Electromagnetic Disturbances in Power Systems and Customers

395

The disturbances because of lightning surges (LS/220 cases) and main- circuit switching surges (SS/47) in Table 6.1 were found in the following manner: A. Malfunction and no reaction of equipments lightning surge (LS): 56 cases, switching surge (SS): 21 cases B. Indication of abnormal supervision (alarm etc.) LS: 124 cases, SS: 18 cases C. Routine maintenance LS: 20 cases, SS: 3 cases D. Others LS: 20 cases, SS: 6 cases The disturbed control equipments in Table 6.1 are installed in the following manner:

1. Control room (centralized) 275 cases (0.896) 2. Dispersed 21 cases (0.068) 3. Unknown 11 cases (0.036)

Total 307 cases (1.0) This shows that the electromagnetic disturbances are far smaller in the dispersed control in the centralized one. The reason for this is estimated, that is, the incoming surge route is limited in the dispersal case while a surge can cause many disturbances and also the number of the surge incoming route is larger in the centralized case. 6.2.1.2 Disturbed Equipments The details of the disturbed equipments in Table 6.1a are summarized in Table 6.2. It is observed in Table 6.2a that more than 80% of the disturbed equipments because of lightning are protection, control, and telecommunication circuits, and another 15% are measuring circuits and indicators. This fact suggests that those circuits should be well protected against surge and also that an incoming route of the surge, which will be explained in the following section, should be designed to reduce the surges. Similarly to the lightning, more than 70% of the disturbed equipments because of switching are protection and measuring/indicator circuits, but only few disturbances are observed in control and telecommunication circuits contrary to the lightning. The difference may be caused by a mechanism and a characteristic of a switching surge differed from these of a lightning surge is explained later. Also, a comparison of Table 6.2b which shows clearly that the number of disturbances in digital-type control equipments is far greater than that in analog ones, the ratio is 4:1 as in Table 6.2c in the switching surge case, whereas the ratio is 1:1 in the lightning surge case as in Table 6.2b. Also, the number of disturbances of mechanical control circuits reaches nearly 10%. This observation leads to a conclusion that a lightning surge can cause a disturbance to any type of control equipments because

66 (0.30) 24 (0.51)

64 (0.29) 3 (0.06)

Control 50 (0.23) 9 (0.06)

Telecoma

20 (0.09) 2 (0.03)

79 (0.36) 26 (0.45)

Analog 73 (0.33) 21 (0.37)

Digital 48 (0.22) 9 (0.15)

Unknown

9 (0.19) 3 (0.11)

Analog

154

Digital

36

Aux Relay

36 (0.76) 22 (0.89)

a

37

Distant supervision.

Number

Digital Process

35

Digital Input

(e) Details of IC board disturbances

Number

IC Board

(d) Disturbed elements

All the data Protection

20

Transmit.

29

Lamp, Fuse, Switch

2 (0.05) 0

Unknown

15

Analog Input

29

Source

47 (1.0) 25 (1.0)

Total (Ratio)

(c) Types of control equipments disturbed by switching surges

All the data Protection

Mechanical

5 (0.02) 0

14

Source

20

10

Measuring

8

Arrester

220 (1.0) 58 (1.0)

6

Analog

5

24

Others

32

Others

2 (0.01) 0

Automatic Process

Wiring Terminal

Centralized Superv./Control

Total (Ratio)

Relay

33 (0.15) 11 (0.23)

Measuring, Indicator

(b) Types of control equipments disturbed by lightning surges

LS SS

Protection

(a) Disturbed equipments

Details of Disturbed Equipments

TABLE 6.2

6

Unknown

18

Unknown

220 (1.0) 47 (1.0)

Total (Ratio)

154

Total

331

Total


397


of its high overvoltage and current. On the contrary, a switching surge causes more disturbances in digital ones because of its high-frequency component. The earlier observation suggests that overvoltage/current reduction, that is, an arrester and a surge absorber, is more important against a lightning surge, whereas reduction of high frequency, that is, a surge capacitor, is effective against a switching surge. Table 6.2d shows that nearly half of disturbed equipments are IC boards, of which digital processors and input circuits are most likely disturbed. 6.2.1.3 Surge Incoming Route Table 6.3 summarizes surge incoming routes that were made clear from the corresponding disturbances. It is not easy to find the incoming route of a surge resulting in the disturbances. However, the most possible route is a control cable both for lightning and switching surges. Also, a communication circuit and a voltage transformer (measuring VT) show a high ratio as the surge incoming route in case of a lightning surge. The table suggests a part where a surge predictive device is to be installed. From Table 6.3 and [11,12], the incoming route of surge can be drawn as in Figure 6.1. 6.2.2 Characteristics of Disturbances 6.2.2.1 Characteristics of Lightning Surge Disturbances Table 6.4 categorizes the isokeraunic level (IKL) of substations where disturbances were found. It is observed from the earlier table that the number of disturbances tends to be proportional to IKL in the case of substations of 154 kV and below. On the contrary, no co-relation to IKL is observed in substations of 187 kV and above. The reason for this is estimated, that is, countermeasures against lightning surges are well done for the substations of 187 kV and above. Table 6.5 shows types of disturbances due to lightning surges. It is observed in Table 6.5a that more than 70% of the disturbances are permanent, that is, breakdown of the equipments, and thus results in permanent halt/lock and malfunctions. TABLE 6.3 Surge Incoming Routes Resulting in Disturbances LS SS LS SS

Control Cable

DC Power

Communication

VT

CT

40 20

36 8

24 2

15 5

2 2

Direct to Panel

Others

Subtotal

Unknown

Total

4 0

8 1

130 38

90 9

220 47

VT, measuring voltage transformer; CT, current transformer.

398


Switching surges (main-circuit)

Bus

VT/CT transfer

Control cable core

Pipe (tank) Mutual coupling

Mutual coupling Control cable metallic sheath

Phase wire

Arrestor discharge

Lightning surges (to transmission line)

Mutual coupling

Grounding mesh

Ground wire Ground potential rise (GPR)

Control cable sheath grounding both ends grounding

Lightning surges to S/S

Sheath voltage rise

Circulating current

FIGURE 6.1 Incoming routes of switching and lightning surges to a control cable.

TABLE 6.4 Number of Disturbances and the Ratio Categorized by IKL IKL 187 kV/above 154 kV/below Total

Higher than 20

Less than 20

9 (0.056) 108 (0.671) 117

7 (0.043) 37 (0.230) 44

6.2.2.2 Characteristics of Switching Surge Disturbances One of the dominant causes of electromagnetic disturbances against control circuits in a gas-insulated substation (GIS) is a switching surge due to a disconnect or (DS) (or occasionally a circuit breaker [CB]) operation. Because of a complex combination of short gas-insulated buses and lines in the GIS, multiple reflection and refraction of traveling waves at the boundaries of the buses and the lines generate a high-frequency surge, which invades via a capacitor voltage transformer (CVT) and a current transformer (CT) into low-voltage control circuits and results in malfunction and occasionally insulation failure of digital elements of the control circuits [13,14].

63

NonRepetitive

2

Unknown

27

Malfunction

a

67 35

29 2

Aux. Relay

21

Not Operated

Include multiple disturbance.

Permanent Non-repetitive

IC Board

(c) Disturbed elements

Number

Type

220

25 1

Lamp

20

17 8

DC Converter

116

Halt Lock

Total

Erroneous Display

(b) Types of disturbances due to lightning surges

155 cases

Permanent (Breakdown)

(a) Aspects of disturbances

Types of Disturbances due to Lightning Surges

TABLE 6.5

16 4

Others

29

7

Unknown

13 —

Relay Element

Others

5 —

Wiring

220

Total

4 4

Arrester

0 2

Meter

2 7

Unknown

178a 63

Total

Electromagnetic Disturbances in Power Systems and Customers 399

400


TABLE 6.6 The Number of Disturbances due to Switching Surges Number of Disturbances Cause

Operating

A

DS

DS in GIS Non-GIS CB in GIS Non-GIS Unknown Non-GIS Unknown

4 5 2 9

CB

Capacitor bank Subtotal

20

B 4 5 5 1 1 1 15(2)

Not Clear

Subtotal

2 1 2 3 1 1 8(2)

10 11 9 13 2 1 1 43(4)

The total number of GISs: 11,102, non-GISs: 27,456. A: control/ protection circuits of the operating CB/DS. B: control/protection circuits independent of the operating CB/DS. () for unknown and capacitor bank switching.

Table 6.6 shows a relation between operating CB/DS and disturbances. For the total number of GISs is 11,102 and that of non-GISs is 27,456, the ratio of the disturbances in the GISs is 0.0017, which is higher than that in the nonGISs, 0.00087. Types of disturbances are categorized as follows: Malfunction 15 (7), malfunction (not operated) 3 (2), halt/lock 14 (11), erroneous indication 8 (3), others 6 (1), unknown 1 (1), total 47 (25) cases, () for protection relays Aspects of disturbances are summarized as follows: DS operation: permanent failure 7, temporary failure 14 CB operation: permanent failure 7, temporary failure 17 Figure 6.2 summarizes overall results of a voltage–frequency characteristic of switching surges at CT secondary circuits measured in 13 different GIS in Japan (58 cases). It is observed in the figure that the frequency of the switching surges ranges from 2 to 80 MHz and the peak-to-peak voltage from 10 to 600 V. It is distinctive that no frequency component from 20 up to 40 MHz is observed. Thus, there exist two average values of the frequency, about 10 and 60 MHz. The results have suggested a necessity to revise oscillating frequency of a test wave in existing standard IEC 61000-4-12 [15]. The average of the peak-to-peak voltage is 100–200 V. Figure 6.3 shows surge voltages at various parts of control circuits, that is, (1) VT secondary, (2) CT secondary, (3) source circuit dc 110 V P-E, and (4) CB control (pallet) circuit. In general, the following trend of the voltage amplitude is observed in the figure.

401


Surge voltage (Vp-p)

10,000

1,000

100

10 0.1

1

10 Frequency (MHz)

100

FIGURE 6.2 Voltage–frequency characteristic of GIS surges.

Surge voltages (Vp-p)

800 700

P-E voltage source

600

CT secondary circuit

VT secondary circuit Control circuit

500 400 300 200 100 0

3–1 3–2 3–3 3–5 3–6 5–3 8–1 8–2 8–3 8–4 8–5 12–1 12–2 12–3 12–4 12–5 12–6 13–1 13–2

Measuring case number 1. VT secondary 2. CT secondary 3. DC circuit 4. CB pallet FIGURE 6.3 Voltages at various parts of control circuits.

CT secondary > VT secondary > source circuit > CB control circuit

The earlier trend is reasonable, since the CT and VT circuits are connected directly to the main (high voltage) circuit of a GIS, and the number of turns of the CT is smaller than that of the VT.

402


Frequency (MHz)

100

50

0

CB1 66 kV

CB2

CB3 275 kV

CB4

DS2 DS1 66 kV 275 kV

FIGURE 6.4 Frequency of surge voltages due to CB and DS.

Figure 6.4 shows the frequency of surge voltages due to a DS operation (o) and a CB operation (x). The frequency due to the DS operation tends to be higher than that due to the CB operation. It is said in general that a disconnector (DS) produces a surge voltage of which the frequency is higher than that produced by a CB. The reason for this is that the operating speed of the DS is slow and the polarity of the source side voltage can become opposite to the disconnected side. Thus, a discharge across the poles occurs due to a quite high voltage across the poles. Such a phenomenon cannot occur in the CB case, since its operating speed is high and the voltage across the poles is reduced by a stray capacitance across the poles. Also, the circuit length is shorter in the DS operation than that in the CB operation. 6.2.2.3 Switching Surge in DC Circuits 1. Basic characteristics: total 21 cases: due to SS within dc circuits 13 2. Disturbed equipments a. Types of equipments: digital 9, analog 3, unknown 1: total 13 cases It is clear that more disturbances (3 times) in the digital one compared with that in the analog one. The ratio (3: 1) is similar to that due to main-circuit switching surges (digital 36, analog 9) in Section 2.1.2. b. Voltage class All the disturbances are in the voltage class of 154 kV and lower. The reason for this is estimated due to more countermeasures to reduce the surges in a higher voltage class. c. Disturbed equipments: telecommunication 7, protection 3, control 3 The reason for more disturbances in the telecommunications is that the equipments have more connection with CB control


403

circuits and auxiliary relays, which are the source of the dc switching surges. d. Relation to standards The reason for the seven cases in the telecommunication equipments is estimated due to switching overvoltages higher than the voltage defined in a standard or the parts that are not defined in a standard [15,16]. e. Disturbed elements: IC base plate 6, aux. relay 3, lamp 1, source circuit 1, others 2, total 13 The aforementioned ratio of the disturbed elements is similar to that in the disturbances due to main-circuit switching surges except the auxiliary relays, which is characteristic to the dc switching surges. 3. Types of disturbances a. Types of disturbances Malfunction 3, malfunction (not operated) 1, halt/lock 3, erroneous indication 1, others 4, unknown 1, total 13 b. Aspects of disturbances: permanent 3, non-repetitive 10 4. Surge voltage and frequency a. Surge voltage: 3–3.6 kV b. Surge frequency: lower than the frequency of main-circuit switching surge 6.2.3 Influence, Countermeasures, and Costs of Disturbances This section investigates disturbances that influenced power system operation. Some of the disturbances cause no trouble in the power system operation, although a control circuit element itself was broken down. Therefore, the total number of the disturbances is not the same as that explained in the previous sections. In fact, 43 cases out of the total 330 cases in Table 6.1a resulted in a trouble in power system operation. 6.2.3.1 Influence of Disturbances on Power System Operation Types of disturbances for each equipments are categorized in Table 6.7a. It is clear from the table that halts and locks (freeze) of equipments are about half of the total disturbances, and those including malfunctions (not operated too) reach 70% of the total disturbances. The aforementioned disturbances of control equipments in Table 6.1a result in troubles of power supply and system operation as in Figure 6.5, which shows that more than 10% of the control circuit disturbances cause power system disturbances. As far as a control equipment disturbance is closed within a control circuit, the disturbance costs not

48 (0.46) 31 (0.42) 42 (0.58) 20 (0.41) 141 (0.47)

Protection Control Telecom Measure./indicat. Total

25 (0.24) 12 (0.16) 2 (0.03) 13 (0.27) 52 (0.17)

Malfunc.

Lightning Switching Others Generator station Substation Control circuit Signal transmission Source circuit Others (CT/VT etc.) Protection Control Measure./indicat. TC/distant. control

Type of surge

(): ratio.

Cause

Incoming route

Place

55 (Overlapped 12)

Total 25 4 3 18 14 9 6 5 12 13 10 7 2

32 (Overlapped 2)

Generation/ Power Supply

(b) Causes and incoming routes of power system troubles

Holt/Lock

Equipment

(a) Types of disturbances of each control equipment

Syst. Operation

8 (0.08) 11 (0.15) 3 (0.04) 4 (0.08) 26 (0.09)

Not Operat.

14 6 3 — 23 7 3 3 10 15 3 4 1

23 (Overlapped12)

Influences of Control Circuit Disturbances on Power Systems

TABLE 6.7

Others 24 (0.22) 19 (0.27) 26 (0.35) 12 (0.24) 81 (0.27)

Total 105 (1.0) 73 (1.0) 73 (1.0) 49 (1.0) 300 (1.0)



405

A 8 1

B 11

1 4

7

C 11

FIGURE 6.5 Power system troubles resulting from Table 6.10a: (A) generation: 17 cases; (B) power supply: 17 cases; (C) system operation: 23 cases.

so much. However, if it affects the power system operation, it becomes an urgent and very significant problem. It is noted that more than one-third of control equipment disturbances cause troubles at the same time of generation, power supply, and system operation. The cause and the incoming route of the disturbances in Table 6.7a are categorized in Table 6.7b. Although a lightning surge is dominant to cause the disturbances of power generation and supply, a switching surge also results in the disturbances. It is reasonable that disturbances because of switching surge are only in substations. As an incoming route, CT/VT and the other are 40% and control circuits show 29%. To make a countermeasure, those incoming route is very significant. 6.2.3.2 Countermeasures Carried Out The total number of countermeasure carried out is 153 cases (51%) out of the 279 total number of disturbance cases as in Figure 6.6 except 28 cases that were not clear if a countermeasure or repair was carried out. In the case of main-circuit switching surges, the countermeasures carried out reach 39 cases that are 85% of all the disturbances because of the maincircuit switching surges. The dominant reason for this is that the disturbances tend to be repeated. Also, the maximum switching overvoltage in the main circuit can be predicted by a numerical simulation, and the incoming route can be estimated. It should be noted that the countermeasures against the switching surge involve modification of software used for a digital control equipment. For example, a software to refresh contents of memories is added against a malfunction of a CPU board, and recovery processing is added against a freeze of a keyboard controller. In the lightning surge case, the number of countermeasures carried out is 94, which is only 43% of the total disturbances (220 cases), because a disturbance because of the lightning is mostly permanent, that is, breakdown of the disturbed circuit (element). Thus, the circuit is replaced by a

406


180

Total N = 279

160

Coutermeasure Repair (replace)

Number of cases

140 120 100 80 60 40 20 0

Total

Lightning

Main-circuit switching

DC-circuit switching

FIGURE 6.6 Countermeasures carried out.

new one or repaired to keep the function as in Figure 6.6, that is, repair (replace) reaches 118 cases. 6.2.3.3 Cost of Countermeasures 6.2.3.3.1 Flow of Countermeasures The primary stage in a flow of countermeasures is for restoration of disturbed equipments carried out in a utility. Only when the restoration is not effective, a manufacturer is involved in the restoration. The secondary stage is carried out when (a) similar disturbances occur repeatedly, (b) the influence of the disturbances is large, or (c) the cause and the incoming route of the surge are not clear. The countermeasure is investigated in parallel by a utility and a manufacturer. The third stage is taken place when a number of similar disturbances are estimated and/or the disturbance is estimated to occur repeatedly. Occasionally, the third stage results in revision of an existing standard [16] of which the summary is added as Appendix 6.A.1. There were 25 cases that went to the third stage out of the 153 cases of countermeasures carried out. Figure 6.7 shows the ratio of the numbers of the third stage countermeasures to the number of all the disturbances. It is observed that the third stage countermeasures are carried out against an equipment that significantly affects power system operation. A large part of the countermeasures involve IC board of CPU, digital input/output circuits, and A/D conversion circuits. It should be noted that restoration was dominant in the case of disturbances because of lightning, whereas disturbances because of main-circuit switching often require a countermeasure because of the repetitive nature.


407

16 14

Percent

12 10 8 6 4 2 0

Protection

Distance supervision

Control

Measuring

FIGURE 6.7 Ratio of the third stage countermeasures to the number of disturbances.

A 38 43

B 0

174 0

26

C 0

FIGURE 6.8 Manpower and material costs in utilities and manufacturer: (A) utilities, (B) manufacturers, and (C) material costs.

6.2.3.3.2 Manpower and Cost of Countermeasures It appears that the average manpower spent for the countermeasures was 50 man-day per countermeasure. The average cost per countermeasure was US$ 10,000 (Figure 6.8). 6.2.4 Case Studies [1] This section presents case studies on the disturbances experienced. Each case study is categorized by the cause and involves the following items:

1. Simulation of disturbances a. The voltage class of a generator station or a substation where the disturbance was found

408


b. The cause of the disturbance, that is, lightning and switching surge c. The influence of the disturbance such as breakdown of a diode and freeze of a control equipment 2. Disturbed equipment 3. Surge incoming route to the disturbed equipment 4. Countermeasure carried out against the disturbance 5. Investigation and analysis of the disturbance if necessary

6.2.4.1 Case No. 1 When lightning strikes the nearest tower to a substation, a diode for a surge protection of a relay of a CB enclosure equipment, 52X1 in Figure 6.9a, is broken, and an auxiliary relay contact is melted. Also, it is found that an auxiliary relay coil of a telecontrol equipment for indication, 79SX1 in Figure 6.9a, is broken. A. (1) 77 kV, (2) lightning, (3) dc short circuit, erroneous indication B. 77 kV line CB enclosure equipment 1. Diode for surge protection broken 2. Auxiliary relay contact melted 3. A telecontrol equipment’s auxiliary relay coil for indication broken C. Lightning to the nearest tower to the substation A lightning current flowed into a ground mesh and induced a surge voltage to a control cable. A differential mode voltage was induced to the diode. D. Countermeasures Varistor installation for auxiliary relays in reclosure. Independent molded case circuit breakers (MCCB) installation for dc circuits in reclosure. E. Reclosure terminal Assuming that a lightning current of 30 kA strikes a tower of a 66 kV system, a current flowing into the mesh is estimated as in Table 6.8. A lightning current of 6 kA assuming a lightning strike to the ground wire (GW) from Table 6.8 flowing into a ground mesh results in a surge voltage at the reclosure terminal that is estimated from Table 6.9 as

CVV cable case 0.24V/A × 6 kA =1440 V

CVVS cable (metallic sheath) case 0.036 V/A × 6 kA = 220 V


409

PC 52a

Usual

52 X1 Diode broken

Error operation

79 S1X

79 SX1

8TX

NC

(a)

43TL

NCT

Contact melted

79S Test

CB

Cable

Reclosure PC

52a Tower

52X

NC

Surge current (b) PC 52a

Adding capacitor

52 X1 Adding varistor

(c) FIGURE 6.9 A disturbed circuit for Case No. 1: (a) disturbed circuit, (b) flow of the surge voltage, and (c) countermeasures.

410


TABLE 6.8 Recommended Value of Lightning Currents Flowing into a Ground Mesh Lightning to

S/S Tower

Distance from S/S Mesh current

10.9 kA

GW 150 m 6.6 kA

Tower 1 300 m 2.8 kA

2nd.

3rd.

600 m 1.7 kA

900 m 1.4 kA

Standard lightning current waveform 1.2/50 μs, estimated wavefront of the above 0.5 μs.

TABLE 6.9 Induced Voltage to a Control Cable due to Lightning Induced Voltage (V: Peak Value) Distance from Applied Node

The Number of Cables

0 m (A)

8 4 4

14 m (B) Applied point

Control cable (A,B)

Current Applied Node 16,007 12,717 3,179

End Side 11,082 5,813 2,634

End side 14 m 14 m

20 m Model of substation ground mesh.

200 m

Lightning current: 10.9 kA, 1.2/50 µs, CVV cable: 8 mm2

The voltage exceeds the surge strength of the broken diode that was manufactured in 1975. A lightning overvoltage induced at a relay terminal, that is, at the end of a control cable is given in Table 6.9. An attached figure shows a schematic diagram of an experiment of which the results are given in the table. With the cable length of 200 m, it is estimated to be 11,082 V = 2.42 V/A × 10.9 kA × 0.5 μs/1.2 μs under the worst condition in an experiment. By considering the real length of the cable of 150 m in Figure 6.9b, the estimated voltage at the relay terminal becomes

11, 082 V × 150 m/200 m = 8422 V

6.2.4.2 Case No. 2 Figure 6.10 shows circuit configuration in a GIS for case No. 2 studied. CB1 and CB2 were opened in the substation where Line 1 and Line 2 were charged from the other end in Figure 6.10. When DS1 (or DS2) was opened


Line1

411

Line2

Ry

Opened DS1

DS2

CB1

CB2

Closed

66 kV GIS FIGURE 6.10 Circuit configuration for Case No. 2: 66/6.6 kV distribution substation. CB: circuit breaker and DS: disconnector.

(or closed), the relay (for a 66 kV line protection equipment) malfunction occurred, and no CB was tripped. The surge was transferred through the control cable via a CT. As countermeasure, the CVV (no metallic shield) cable for the CT circuits was replaced by a CVVS (metallic shield) cable with both ends of the shield grounded. Then the frequency of the malfunction decreased. Also, ferrite cores were installed at the secondary circuit of an internal auxiliary transformer for CT. Since then, no mal-operation appears. Figure 6.11 shows a test result of surge propagation along a 201 m CVVS cable of which both the terminals of the metallic shield are grounded [1]. It is observed in the test result that the transferred surge through a CT is quite attenuated and the oscillating frequency is lowered during its propagation along the CVVS cable, and the surge at the relay terminal become low enough to cause a disturbance as expected in the earlier countermeasure. 6.2.4.3 Case No. 3 A. (1) 275 kV, (2) dc switching surge, (3) flicker of LEDs for system state indicator and measurement B. Numerical control equipment (6.6 kV distribution system) C. 1. 6.6 kV VCB closing 2. DC switching surge appeared in the dc 110 V circuit. (700 V/0-peak, 10–30 kHz) 3. LEDs connected to the dc 5 V circuits were affected by the dc switching surge

Voltage (V)

412

Voltage (V)

(a)


3000 2000 1000 0 –1000 –2000 –3000 –0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Time (µs) 1500 1000 500 0 –500 –1000 –1500 –0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Voltage (V)

(b)

(c)

Time (µs) 6 4 2 0 –2 –4 –6 –0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Time (µs)

FIGURE 6.11 A test result of surge propagation along a CVVS control cable: (a) CT secondary terminal, (b) GIS control box, and (c) relay terminal.

D. 1. Separation of dc 110 V circuit from the dc 5 V circuit 2. Replacement of wires used in the dc 5 V circuit by twist-pair type 3. Installation of noise filters in the dc 5 V power circuit 6.2.5 Concluding Remarks This section has summarized experiences of electromagnetic disturbances of control circuits in Japanese power stations and substations collected for 10 years from 1990, and the collected data have categorized by the cause, the incoming route, disturbed equipments, and elements. Also, the characteristics of disturbances, case studies, the influence of the power system operation, countermeasures, and the costs are presented.


413

It appears that the average manpower and material cost for the countermeasures are 50 man-days and US$ 10,000 per countermeasure. It should be noted that a lightning surge often results in a permanent failure such as breakdown and burnt of a control element such as a diode (155 permanent failures out of 220 cases) and thus it is rather easy to find the disturbance. On the contrary, switching surge tends to be a non-repetitive fault such as freeze (31 temporary failures out of 45 cases) because the surge overvoltage is low but the oscillating frequency of the surge is high. Although the number of disturbances because of the lightning surge (220 cases) is much more higher than that because of the switching surge (68 cases including dc circuit), the high ratio of the non-repetitive nature of the switching surge causes a difficulty of finding the disturbances and this fact suggests that such a disturbance may result in a severe trouble such as a system shutdown. Having considered the increase of digital control equipments and compact substations, a switching surge is estimated to become very significant for an EMC disturbance of control circuits in power stations and substations.

6.3 Disturbances in Customers and Home Appliances Insulation design and coordination of high-voltage system are quite well and the system is protected sufficiently against various surge voltages. Lowvoltage distribution and service systems to customers, however, are not well protected against the overvoltages, and disturbances in customers’ and home appliances have often been informed [4–9]. Increasing usage of digital appliances also makes the protection of the appliances against the overvoltages but also electromagnetic disturbances a quite important problem. 6.3.1 Statistical Data of Disturbances Figure 6.12 shows statistical data for the number of damaged home electric appliances (HEAs) in a Japanese utility, (a) from 1987 to 1991 and (b) for 1996 to 2006 [8,9]. The data were based on information from monitors more than 2000. The thunderstorm days in the area for Figure 6.12a were 21–34 days per year from 1987 to 1991. The thunderstorm days for Figure 6.12b were 1996–1997: 15–23 2004–2005: 24 It was informed that 228 HEAs were damaged in 129 houses from 1987 to 1991. Among them, 49% for TV/video antennas, 18% for communication


Number of damaged HEAs

16

s O th

m

St

er

ac hi ne

eo er

ne di tio

hi ng W as

(a)

Number of damaged HEAs (1996–1997)

14

Number of damaged HEAs (2004–2005)

12

Number of damaged HEAs (2006)

10 8 6 4 2 0

L

TE

/ AX

F

(b)

r

r ea te

Ai rc on

W at e

rh

VT

Te

R

l

100 90 80 70 60 50 40 30 20 10 0 TV

Number of damaged HEAs

414

PC

TV e

at

W

er

at

e rh

r Ai

n

co

er

on

ti di

R

VT

IH

er

at

he

p

m

Pu

na

en

t An a

W

ng

i sh

m

ne

hi

ac

s

er

th

O

FIGURE 6.12 Number of damaged home electric appliances (HEAs): (a) 1987–1991 and (b) 1996–1997, 2004–2005, and 2006.

equipment, and 16% for grounding. In the data from 1996 to 1997, 32% for antennas, 40% for communication equipment, and 30% for grounding. The results indicate that the protection of the TV/video against lightning has been well improved in these years. On the contrary, because the number of digital circuits used in the communication equipment increases and those are very weak and sensitive to a lightning surge, the ratio of the damaged communication equipment increases. Figure 6.13 shows the ratio of the connecting circuits of damaged HEAs. Figure 6.14 shows the expenses paid to customers from an insurance company [17]. It is observed in the figure that the number of the cases paying

415


60

[%]

1987–1991 [%] 1996–1997 [%] 2004–2005 [%]

50

2006 [%]

40 30 20 10 0

Both AC and com-line

Both AC and earth line

Both AC and ANT line

AC line

FIGURE 6.13 Ratio of connecting circuits of damaged home electric appliances.

Indemnity (million yen)

100~ 50~100 10~50 5~10 1~5 0~1 0

200

400

600

800

1000

Number FIGURE 6.14 Expenses (insurance) paid to damages. (From IEEJ WG, The fact of lightning disturbances in a highly advanced ICT society and the subject to be investigated, IEE J. Tech. Report 902, 2002.)

more than 10 million Japanese yen is 82 among the total 1417 cases for 13 years from 1987. 6.3.2 Breakdown Voltage of Home Appliances 6.3.2.1 Testing Voltage A voltage waveform with 1.2/50 μs (wavefront Tf = 1.2 μs, wavetail Tt = 50 μs) and a current waveform with 8/20 μs are adopted as a lightning impulse to test a withstand voltage of home appliances or electronic equipments

416


in general. The impulse voltage is mainly used for a high impedance equipment and its maximum value is taken as 6 kV. On the other hand, the impulse current is mainly for a low impedance equipment and its peak value is taken as 3 kA [18,19]. In addition to the aforementioned, ring waves of 6 kV/500 A with 0.5 μs/100 kHz and 0.6 kV/120 A with 1.5 μs/5 kHz are recommended for an insulation test of home appliances in the IEEE [4,18]. As for an induced lightning surge on a communication line, an impulse voltage with 10/200 μs is also used [18] and an impulse voltage of 1.5 kV with 10/700 μs is internationally recommended for a lightning impulse test of a home communication equipment [20]. Testing voltage of home appliances surveyed in the chapter is summarized in Table 6.10a. In general, the testing voltage of a home appliance ranges from 1 to 1.5 kV [21]. 6.3.2.2 Breakdown Test An experiment was carried out to find an actual breakdown voltage of a home appliance. An impulse voltage with about 1.0/23 μs was applied to the home appliance. The test result is summarized in Table 6.10b. It is observed in the table that the breakdown voltage of the home appliances is greater than 5 kV for line to earth, and 7 kV between lines, although the recommended testing voltage is about 1.5 kV. The test result agrees with those carried out in Ref. [4], where the breakdown voltage was found to be 4–6 kV. 6.3.3 Surge Voltages and Currents into Customers due to Nearby Lightning 6.3.3.1 Introduction This section investigates experimentally an incoming path of a lightning surge into a customer due to lightning nearby the customer [12,22]. There exist four incoming paths: (1) low-voltage distribution line (feeder) through a distribution pole, (2) telephone line, (3) customer’s TV antenna, and (4) grounding electrode of a customer’s electrical equipment. A lightning strike to a distribution line and an induced lightning voltage to the distribution line result in the path (1). Similarly a lightning strike to a telephone line and an induced lightning voltage result in the path (2), and the same is applied to the path (3). A lightning strike to a ground, a wood, or a distribution pole nearby a customer results in a ground potential rise (GPR) due to a lightning current flowing into the ground [23,24]. The current causes a potential rise to a grounding electrode of a customer’s equipment, which is the path (4). Based on the measured results, modeling of electrical elements related to the aforementioned paths for a lightning surge simulation is developed, and

417


TABLE 6.10 Withstand Voltages of Home Appliances (a) Testing voltages surveyed from standards and guides Rated Voltage Electric equipments

Home appliances

Information and communication appliance Audio visual

Incandescent lamp, Discharge lamp Air conditioner Microwave oven Telephone Fax Personal computer TV Video

∼30 ∼150 ∼300 ∼1000 100 V 200 V 100 V 200 V 100 V

Voltage

Applied Time

500

1 min

1000 1500 2E + 1000 AC 1000 V AC 1500 V AC 1000 V AC 1500 V

1 min

AC 1000 V

1 min

1 min

100 V 100 V

(b) Experimental results of breakdown voltages Breakdown Voltage [kV] Line to Earth Ventilating fan Electric rice cooker Refrigerator Electric fan 1 Electric fan 2 Electric fan 3 Cassette deck Hair dryer

6.0 5.0 11.0 7.0 8.0 8.0 8.0 10.0

Line to Line 8.0 20.0 13.0 7.0 7.0 7.0 12.0 7.0

electromagnetic transients program (EMTP) simulations are carried out by the model. The simulation results are compared with the measured results and the accuracy of the modeling method is discussed. 6.3.3.2 Model Circuits for Experiments and EMTP Simulations 6.3.3.2.1 Experimental Conditions Kansai Electric Power Co. (KEPCO) has carried out a number of experiments to investigate lightning currents flowing into a house [25,26]. An impulse current representing a lightning current is injected into an antenna, a distribution pole, and a ground from an impulse current generator (IG, maximum

418


(b)

LA

PW

GW

(d)

Messenger Telephone line Tr

NTT pole Concrete pole

(a)

NTT pole

END pole

(c) Tr grounding

Matching resistance

FIGURE 6.15 Lightning and its path to a house. Lightning to (a) antenna, (b) pole, (c) ground, and (d) telephone (messenger) line.

voltage 3 MV, maximum current 40 kA). An impulse current from 500 [A] to 5000 [A] is applied to a. An antenna: Figure 6.15a b. A distribution pole: Figure 6.15b c. A structure nearby a house: Figure 6.15c d. A messenger wire of a telephone line: Figure 6.15d In a test yard of the KEPCO, 3 distribution poles and 2 telephone company (NTT) poles are constructed and 6600/220/110 [V] distribution lines with a pole transformer are installed as in Figure 6.15. Also, a telephone line and the messenger wire are installed. A model house is built nearby the pole transformer, and a feeder line from the transformer is led in. In the model house, model circuits of an air conditioner and a fax machine are installed as shown in Figure 6.16. Those involve a surge protective device (SPD), a kind of a surge arrestor (air conditioner, operating voltage 2670 V; fax, 2800 V; NTT SPD, 500 V). The poles, the transformer, the telephone line (NTT), and the air conditioner (home appliance) have their own grounding as in Figure 6.16. Table 6.11 summarizes the measured results of maximum voltages and currents through the grounding resistances.

419


6.3.3.2.2 Modeling for EMTP Simulations The distribution line, the pole, and home appliances in the house in Figure 6.16 can be readily represented by horizontal and vertical distributed line models and lumped-parameter circuits [27,28]. The grounding electrodes of the pole, the telephone line SPD, and the home appliances if those are grounded Vo Tr Ip

KI

lo lant

lt lh

VTR

WHM

Air conditioner

Vp

Vt

Va

Fluorescent lamp

lnp

la

Matching resistance

Vnp Rnp Rn1

Ra

R p Rt

ln 1

NTT SPD

FAX

ELB

(a) Vo lo

KI

Tr

lant

Ip lh

VTR

WHM

Air conditioner

Vp

NTT SPD

FAX

ELB

lt Vt

la Va

Rp Rt

Ra

Fluorescent lamp

Vnp

ln 1 Matching resistance

lnp Rnp

Rn1

(b) FIGURE 6.16 Experimental circuit and lightning current path: (a) lightning to antenna, (b) lightning to a distributed line. (continued )

420


Tr lt lh

VTR

WHM ELB

FAX

lf Air conditioner

Vp

la

NTT SPD

ln 1

lnp

KI lo

Vt

Va

Vnp

Vo

R p Rt

Ra

Rnp

Ro

Matching resistance

Rn1

(c) FIGURE 6.16 (continued) Experimental circuit and lightning current path: (c) lightning to the ground.

are modeled by a combination of a distributed line and a lumped-parameter circuit to simulate the transient characteristic [29]. This section, however, adopts a simple resistance model of which the resistance value is available from the experiments explained in Refs. [29,30], because the vertical grounding electrode used for a home appliance is short and the transient period is far smaller in the phenomenon investigated in this chapter. A protective device (PD) installed in a home appliance and the NTT SPD is represented simply by a time-controlled switch prepared in the EMTP [31]. A model circuit for an EMTP simulation is shown in Figure 6.17. In the figure, Zp is the surge impedance of a distribution pole, which is represented by a lossless distributed line with the propagation velocity of 300 m/μs. The surge impedance is evaluated by the following formula of a vertical conductor [28]:

 Zp = 60 ln 

 hp   rp

   − 1 [Ω] (6.1)  

where hp is the height of the pole rp is the radius of the pole rp = 17 to 19 Ω is the grounding resistance of the pole. The value is given by a utility [25,26]. Tr is a pole mounted distribution transformer, which is

— — 34.4 247 —

— — 72 500 —

— 7.5

— 3.5

Vnp Vp

Current (A) Ia Inp It Ip If

25.9 7.7 —

11.6 3.8 —

Voltage (kV) Vo Vt Va

739

17 89 830 900 153

306

Grounding resistance (Ω) Rp 17 Rt 89 Ra 830 Rnp 900 Rn1 153

Appl. Current I0 [A]

Test Conditions and Results

TABLE 6.11

A

— — 120 888 —

— 12.2

41.5 12.9 —

17 89 830 900 153

1247

— — 159 1150 —

— 16

56.1 17.5 —

17 89 830 900 153

1660

0 3.6 3.4 — —

1.3 —

10 3.8 0.8

19 75 100 300 150

602

44 5.9 −27 — —

2.9 —

19.3 5.6 5.1

19 75 100 300 150

1183

B

177 52 174 — —

16.7 —

31.5 21.2 19.6

19 75 100 300 150

1977

Case

206 68 197 — —

19.1 —

37.6 24.6 22.8

19 75 100 300 150

2416

0 13.2 — — 0

1.3 —

55.4 — 1.8

19 85 140 380 100

676

27.5 31.2 — — 0

2.9 —

88 — 2.2

19 85 140 380 100

1532

C1

58.2 56.6 — — 0

5 —

122 2.5 3.1

19 85 140 380 100

2815

83.1 80.2 — — 0

6.3 —

146.3 — 4.4

19 85 140 380 100

4134

55.7 — — — 31.8

0.8 —

122 2.5 3.1

19 85 — 380 100

2771

C2


422


Rp

POLE1T

0.001[Ω]

IG

DC

400[k]

Zpw 530[Ω] PWA Zpw

PW

50[m]530[Ω]

Tr

Cm 23.9[pF]

TRAH

TRN

Zpw

Fax

520[Ω] 2[m]

TRCH

660[Ω]

Air1 Air2

520[Ω] 1[m]

3000[V] Antenna 333[Ω] 6[m]

520[Ω] 2[m]

2400[V] 150[Ω]8[m]

AIRN

2870[V]

TRN3 NP4

ANT

CW

40[Ω]

TRNB POLE1B Rp 16.8[Ω]

TRN4 10

2800[V] 40[Ω]

NP1

SO

520[Ω]4[m]

520[Ω]4[m]

2

1

12[m]227[Ω]

0.1[µF]

520[Ω]4[m]

TRCL

Distribution pole

520[Ω]5[m] 520[Ω] 1[m]

520[Ω] 2[m]

TRBL

Cm

Air

560[Ω] 20[m]

Cm

TRBH

PWB PWC

TRAL

Feeder

50[m] 560[Ω]

NP

Rt 89[Ω]

Ra 830[Ω]

NTT1

NPN Rnp 900[Ω]

(a)

487[Ω]

480[Ω]

Rn1 153[Ω]

G 800[Ω] 762[kV] 0.1[µF]DC Rp

POLE1T Zpw PW 530[Ω] 50[m]530[Ω] PWA

0.001[Ω]

Tr

Cm 23.9[pF]

TRAH

TRBH

TRAL

Feeder

AIR

560[Ω] 20[m]

520[Ω] 1[m]

520[Ω] 2[m]

TRN

520[Ω] 1[m]

AIR1 AIR2

1

Distribution pole 12[m]227[Ω]

150[Ω]8[m]

POLE1B Rp 19[Ω]

(b)

560[Ω] 5[m]

2400[V] TRNB Rt 75[Ω] GRND

FAX

520[Ω] 2[m]

2800[V] 520[Ω] 2[m]

2

NP1

40[Ω] 40[Ω]

10

NP4

AIR3 AIRN Ra 50[Ω]

CW 50[m] 560[Ω]

NPN Rnp 300[Ω]

NTT1

NTT2 Rn1

150[Ω]

Rm 10[Ω]

FIGURE 6.17 Model circuits for EMTP simulations: (PW) power line and (CW) communication (NTT) line: (a) Lightning to an antenna, (b) lightning to a pole.

423


POLE1T Zpw

530[Ω]

TRAL Feeder

Tr

PW 50[m]530[Ω]

Cm 23.9[pF]

TRAH

PWA TRBH

Air

560[Ω] 20[m]

520[Ω] 1[m]

520[Ω] 2[m]

TRN

520[Ω] 1[m]

Distribution pole

150[Ω]8[m]

520[Ω]5[m]

AIR1 AIR2

1

10[m]200[Ω]

2400[V]

POLE1B 19[Ω] Rp

TRNB 85[Ω] Rt

520[Ω] 2[m]

NTT2

FAX

520[Ω] 2[m]

CW 50[m] 560[Ω]

Rn1

150[Ω]

40[Ω]

2800[V] 40[Ω] 2 NP4 NP1

10

AIR3 NPN

AIRN

140[Ω] Ra

GRND Rm 4[Ω]

Rnp 380[Ω]

IG

300[Ω] 0.1[µF] 900[kV]

DC

(c) FIGURE 6.17 (continued) Model circuits for EMTP simulations: (PW) power line and (CW) communication (NTT) line: (c) lightning to a ground.

represented by an ideal transformer with the voltage ratio 6600:110 V and stray capacitances. TRN-TRNB is a grounding lead of the transformer represented by a lossless distributed line of which the surge impedance is explained in Ref. [32]. Rt is the grounding resistance of the transformer grounding lead and is given as in Table 6.1 by a utility [25,26]. PW is a phase wire of a distribution line and is modeled by a lossless distributed line with the surge impedance Zpw = 530 Ω. The other end of PW1 is terminated by the matching impedance Zpw. TRAL-FAX and TRAL-HOME1 are a feeder line from the pole transformer to a house. For the transformer, A is for phase A and N for neutral. The steadystate ac voltage between the phase A and the neutral is 114 V in Japan. In the case of lightning to an antenna, case A, a flashover between the transformer grounding lead and the steel of a distribution pole was observed. The flashover is modeled by short-circuiting those by resistance Rp in Figure 6.18a. The feeder line is represented by a lossless distributed line with the surge impedance of 560 Ω and the velocity of 300 m/μs. Ra is the grounding resistance of an air conditioner of which the value is given is Table 6.11. Za represents the air conditioner expressed by a lead wire inductance 1 μH and by a time-controlled switch representing a surge arrester operating when the voltage exceeds 2670 V in parallel with the resistance. Zb represents a fax machine. It is expressed in the same manner as Za except that the other terminal of Zb is connected to a telephone line through an NTT SPD, a surge arrester valve with the operating voltage of 500 V represented by a time-controlled switch. The telephone line is

424


25,000

800

Vo Vo (simulation)

20,000

600

Voltage (V)

Current (A)

500

15,000

400

10,000

300 200

5,000 0

100

0

5

10

15

20 25 Time (µs)

10,000

30

35

0

40

Vt Vt (simulation)

Voltage (V)

6,000 4,000

5

10

15

20 25 Time (µs)

30

35

40

35

40

35

40

It It (simulation)

80 60 40 20

2,000 0 0

5

10

15

20 25 Time (µs)

10,000

30

35

0

40

0

5

10

15

600

Vp Vp (simulation)

8,000

20 25 Time (µs)

30

Ip Ip (simulation)

500 400

Current (A)

Voltage (V)

0

100

Current (A)

8,000

6,000

300

4,000

200

2,000

(a)

Io Io (simulation)

700

0 0

100 5

10

15

20 25 Time (µs)

30

35

40

0

(b)

0

5

10

15

20 25 Time (µs)

30

FIGURE 6.18 Experimental and simulation results in the case of lightning to antenna: (a) voltage and (b) current.

represented by a lossless distributed line with the surge impedance of 560 Ω and the propagation velocity of 200 m/μs and the other end is terminated by the matching resistance with the grounding resistance of 153 Ω. IG is an impulse current source used in the experiment and is represented by a charged capacitance and a resistance [25,26]. Rm is a mutual grounding impedance between various grounding electrodes, and the value of 2–10 Ω is adopted [24,30].

425


6.3.3.3 Experimental and Simulation Results 6.3.3.3.1 Experimental Results Figures 6.18 through 6.20 show experimental and simulation results in the case of lightning to (a) antenna, (b) pole, and (c) ground. From the experimental results, the following observations have been made.

25,000

250

Vt (simulation) Vt

20,000

200 Current (A)

15,000

Voltage (V)

150 100

10,000 5,000 0

50

0

5

10

15

20 25 Time (µs)

25,000

30

35

0

40

10,000

5

10

15

30

35

40

Ia (simulation) Ia

150 100

5,000

50

0

5

10

15

20

25

30

35

0

40

0

5

10

15

Time (µs) 20,000

20

25

30

35

40

Time (µs) 80

Vnp (simulation) Vnp

Inp (simulation) Inp

70

15,000

60 Current (A)

Voltage (V)

20 25 Time (µs)

200 Current (A)

Voltage (V)

15,000

0

250

Va (simulation) Va

20,000

0

It (simulation) It

10,000

5,000

50 40 30 20 10

0

(a)

0

5

10

15

20 25 Time (µs)

30

35

40

0

(b)

0

5

10

15

20 25 Time (µs)

30

35

40

FIGURE 6.19 Experimental and simulation results in the case of lightning to a pole: (a) voltage and (b) current.

426


8,000

Va

0

Va (simulation)

–10 Current (A)

Voltage (V)

6,000 4,000 2,000

–30 –40 –50 –60

0 0

5

10

7,000

15 20 25 Time (µs) Vnp

6,000

30

–70

35 40

3,000 2,000

15 20 25 Time (µs)

0

5

10

15 20 25 Time (µs)

0

5

10

15 20 25 Time (µs)

Ia (simulation)

30

35

40

–20 –30 –40

–60

0 0

5

10

6,000

15 20 25 Time (µs) Vt

30

35

40

–70

Inp

10

Vt (simulation)

5,000

Inp (simulation)

30

35

40

5

4,000

Current (A)

Voltage (V)

10

–50

1,000

3,000 2,000

0 –5

1,000 0

5

–10 Voltage (V)

4,000

Ia

0

0

Vnp (simulation)

5,000 Voltage (V)

–20

0

5

10

15 20 25 Time (µs) 70

30

35 40

Ifax

Ih

60 Current (A)

–10

Ih (simulation)

50 40 30 20 10 0

0

5

10

15 20 25 Time (µs)

30

35

40

FIGURE 6.20 Experimental and simulation results in the case of lightning to a ground.

If (simulation)

30

35

40


427

a. Lightning to antenna When lightning strikes the antenna of a house in Figure 6.16a, the lightning current flows out (1) to a distribution line I1 (In, It) through feeders in the house, and (2) to the earth I2 (Ia, Inp) through grounding electrodes of home appliances. The ratio of I1 and I2 is dependent on the grounding resistances: (1) seen from the house feeder to the distribution line, (2) seen from the house to the telephone line, and (3) of the grounding electrodes of the home appliances. In the experimental results, one of which is given in Figure 6.18 with the applied current I0 = 739 A, about 85% of the current into the antenna flows out to the distribution line. The remaining 15% is estimated to flow into the other houses. A large current 543 A flowing into the distribution pole is caused by a flashover between the transformer grounding lead and the steel of the pole and the low grounding resistance of the pole. b. Lightning to distribution pole Figure 6.19 shows a measured result when I0 is 2416 A. In this case, 8.1% of I0 flows into the grounding resistances Rt of the transformer neutral, 8.5% flows into the grounding resistance Ra of the air conditioner, and 2.8% to the grounding resistance Rnp of the telephone line SPD. c. Lightning to grounding A part of the applied current I0 flows into the air conditioner (Ia max = 58.2 A for I0 = 2815 A) and the NTT SPD (Inp max = 56.6 A) in Figure 6.20, case C1. The current flowing into the air conditioner flows out to a distribution line. The current into the NTT-SPD protector flows out through the telephone line. No current flows into the fax, because the operating voltage 2800 [V] of a surge arrestor within the fax is higher than the voltage across the fax, that is, the voltage difference between the feeder line in the house and the telephone line. Compared with case C2, in the case of no air conditioner grounding, the currents flowing into the house increase heavily. 6.3.3.3.2 Simulation Results Table 6.12 summarizes simulation conditions and results, and Figures 6.18 through 6.20 show a comparison of simulated voltage and current waveforms with the measured results. From the figures, the following observations are made.

a. In Figure 6.18 when lightning strikes the antenna, simulation results of transient voltages and currents agree well with the measured results except the wavefront of the applied current and voltage at the top of the pole and the voltage at the primary winding on the transformer.

306

87.2 17.6 — — 22.4

— — 198 240 —

Appl. Current I0 [A]

Voltage (kV) Vo Vt Va Vnp Vp

Current (A) Ia Inp It Ip If

— — 395 480 —

174.3 35.2 — — 44.8

739

A

— — 593 721 —

261.4 52.8 — — 67.1

1247

— — 791 961 —

348.5 70.5 — — 89.5

1660 16.2 13.6 5.9 9.8 —

602

61 23.7 143 189.4 —

Simulation Results Corresponding to Table 6.11

TABLE 6.12

114.9 44.7 269.3 356.7 —

30.6 25.6 11.2 18.4 —

1183

B

186.5 72.5 437.2 579.2 —

49.6 41.5 18.2 29.9 —

1977

Case

225.6 87.8 528.8 700.5 —

60 50.2 22 36.2 —

2416

−19.8 −8 — — −2.4

50.8 1.7 4.3 2.9 —

676

−31.3 −12.7 — — −3.9

81.24 2.7 6.9 4.6 —

1532

C1

−47.6 −19 — — −6

121.9 4 103.3 6.9 —

2815

−62.6 −25.4 — — −7.9

162.4 5.3 137.8 9.3 —

4134

−53.9 — — — 7.3

121.9 3.4 103.8 6.5 —

2771

C2



429

b. In the case of lightning to the pole top in Figure 6.19, voltage differences between the top and the bottom of the pole and at the air conditioner grounding electrode are observed to be smaller than those of the measured results. Otherwise the simulation results show a reasonable agreement with the measured results. c. When lightning hits ground nearby a house, the simulation results in Figure 6.20 show a qualitatively satisfactory agreement with the measured results, and thus, the proposed approach to deal with a ground voltage rise by means of a mutual impedance is confirmed to be adequate. It, however, requires a further improvement of the overall simulation method to achieve a quantitative agreement with a measured result. For example, a surge protecting device has to be carefully represented based on its circuit and the nonlinear characteristic. Also, a grounding impedance should be modeled considering its transient characteristic rather than a simple resistance adopted in this section. 6.3.3.4 Concluding Remarks This section has shown experimental and EMTP simulation results of a lightning surge incoming into a house due to lightning nearby the house. The simulation results agree qualitatively with the experimental results, and thus, the simulation models in the chapter are said to be adequate. The measurements being carried out in different time periods for 3 years, the experimental condition such as the earth resistivity, a voltage probe used, and a reference voltage line for each measurement are different, and the difference is not considered in the simulations. Some oscillations observed in the measured result are estimated to be caused mutual coupling between the measuring wires and feeder lines in the experiments. Also, a grounding electrode may have coupling to the other electrodes. A further improvement for modeling the experimental circuits is required to achieve a higher accuracy in comparison with the measured results. Lightning surge voltages in a customer and lightning currents flowing out to a distribution line can be made clear based on the experimental and simulation results. The results are expected to be applied to protection coordination of SPDs for customers and a telephone line, and to investigate the necessity of home appliance grounding. 6.3.4 Lightning Surge Incoming from a Communication Line 6.3.4.1 Introduction Nowadays, a number of digital appliances like a PC have been used by customers. This makes a protection of appliances against a lightning surge very important. However, the lightning surge characteristic in a customer device coming from a communication line is not well investigated. Therefore, the

430


incoming lightning surge is investigated, and when the incoming surge is large, the installation of a PD is required [33]. In this section, characteristics of PD and incoming lightning surges are investigated based on experiments and finite-difference time-domain (FDTD) simulations [34]. Also, the optimum method to protect customer appliances and adequacy of the FDTD simulation are investigated. 6.3.4.2 Protective Device 6.3.4.2.1 Experiment Figure 6.20 shows the experimental circuit. A step-like voltage from a pulse generator (PG) is applied to a discharge tube through a 3D2V cable and a resistance. The circuit is grounded on an aluminum plate. Figure 6.21 shows measured results of the V–t and the V–I characteristics of PD, and their approximate curves. The V–t characteristic is approximated by three equations. The V–I characteristic is approximated by six equations. 6.3.4.2.2 Simulation Figure 6.22 shows a simulation circuit. The analytical space of an FDTD simulation is 31 cm × 21 cm × 23 cm, and the cell size is 1 cm. The time step is 19 ps, and an absorbing boundary condition of Berenger’s perfectly matched layer (PML) is used [34]. The model circuit of PD is developed based on measured results. Until the PD flashovers, the PD’s performance follows the V–t characteristic, and its resistance is 1 M Ω. Firstly, the threshold voltage is calculated based on the V–t characteristic in Figure 6.21a. When the PD’s voltage reaches the threshold voltage, the PD flashovers, and the calculation based on the V–t characteristic is shifted to that based on the V–I characteristic in Figure 6.21b. The value of the variable resistance is the gradient of the linear approximation indicated as Vvi1 to Vvi6 in Figure 6.21b. For example, the resistance is 333 Ω corresponding to Vvi1 when Vdis = 180 V. The resistance is 12.4 Ω corresponding to Vvi6 when Vdis = 250 V. Figure 6.23 shows an applied voltage and a comparison of measured and simulation results. It is observed in the figure that the initial rise and the steady-state value are in good agreement. However, the waveform before the steady state shows a difference. The difference is caused by the approximation of the discharge characteristic. 6.3.4.3 Lightning Surge 6.3.4.3.1 Model Circuit Figure 6.24 shows the experimental circuit. The voltage from PG is applied to the circuit through a 3D2V cable. In the case of current source, a voltage is applied through a resistance. The circuit is grounded on an aluminum plate. Experimental cases include an individual grounding, a common grounding,

431


Idis

3D2V cable P.G.

Vdis

100 Ω

50 m

0.1 m

(a) Experiment Vvt1 Vvt2 Vvt3

Voltage (V)

1000 900 800 700 600 500 400 300 200 100 0

Vvt2

Vvt1

Vvt3 0

50

100

150

(b) 300

Voltage (V)

250

300

350

400

Vvi5

Vvi4

250

Experiment

200 150

Vvi3

100

0

Vvi1 Vvi2

Vvi6 Vvi2

Vvi3 Vvi4

Vvi1

50

(c)

200

Time (ns)

Vvi5 Vvi6

0

0.5

1

1.5

2

2.5

3

3.5

4

Current (A)

FIGURE 6.21 Experimental circuit and results for PD characteristics: (a) experimental circuit for a PD characteristic, (b) measured results of V–t characteristic, and (c) measured results of V–I characteristic.

and a proposed method, which is a kind of individual grounding with a bypass installed on an appliance. 6.3.4.3.2 Measured Results Figure 6.25 shows experimental results. In the case of a current source, the results of the common and individual methods are equal, so only the result of individual is shown. It is observed in Figure 6.25a that the

432


0.23 m

100 Ω 0.12 m 0.1 m

0.11 m

0.31 m

(a)

0.1 m

Perfect conductor

0.21 m

3 pF

(b) FIGURE 6.22 Simulation circuit for a PD characteristic: (a) simulation circuit and (b) PD model.

maximum and steady-state values of the common and proposed methods are lower than the individual one. Also, the maximum voltage of the proposed method is lower than the common one. In the case of common, a bigger current flows to the resistance, since PD has characteristic of discharging lag. However, in the case of the proposed method, the current flowing to the resistance decreases quickly because the varistor works quickly. Therefore, the proposed method can reduce the maximum voltage better than the common one. It is observed in Figure 6.25b that the maximum and steady-state values of the proposed method are lower than the individual grounding. The varistor has hundreds of picofarad capacitance, so the initial rise slows down. Because of that, the proposed method can reduce a lightning surge better than the individual grounding. 6.3.4.3.3 Simulation Figure 6.26 shows the model circuit. The analytical space of the FDTD simulation is 5.85 m × 0.2 m × 0.21 m for the individual and common groundings, and 5.87 m × 0.2 m × 0.21 m for the proposed method. The cell size is 1 cm. The time step is 15 ps, and an absorbing boundary condition is the same as that in Section 3.4.2.

433


700

Voltage (V)

600

Applied voltage Experiment FDTD

500 400 300 200 100 0

0

50

100

150

(a)

200

250

300

350

400

Time (ns) 700

Voltage (V)

600 500

Applied voltage Experiment FDTD

400 300 200 100 0

0

50

100

(b)

150

200

250

300

350

400

Time (ns)

FIGURE 6.23 Experiment and simulation results of PD voltages: (a) applied voltage 400 V with the wavefront of 50 ns and (b) applied voltage 600 V with the wavefront of 50 ns.

Figures 6.27 and 6.28 show simulation results. It is observed in the figures that the initial rise and steady-state values are in good agreement. However, the waveform before the steady state shows a difference. The characteristic of this part was not simulated well enough, since it could not be developed from results, which caused the difference. 6.3.4.4 Concluding Remarks The characteristic of PD is developed based on the experimental results. The FDTD simulation result based on that characteristic agrees reasonably with the measured result. Therefore, the PD model can be used to simulate communication system to a customer appliance. The experiment of lightning surge characteristic is carried out. At the steady-state voltage, the common grounding and proposed method can reduce the lightning voltage equally well. However, at the maximum voltage,

434


1.5 m

PTC thermistor

V1mA: VR V1mA: 100 V 100 V 1.5 m 1.5 m

P.G.

Ω

TIV

0 or 2 kΩ TOV

1.0 m VVF 66 Ω V1mA: 100 V 241 Ω

VVF

Communication device VS:350 V

P.D.

66 Ω 0.1 m

100 Ω

100 Ω (a)

1.5 m

PTC thermistor

V1mA: VR V1mA: 100 V 100 V 1.5 m 1.5 m

P.G. 0 or 2 kΩ

TOV

Ω

TIV

VVF

Communication device

P.D.

VS:350 V

1.0 m VVF V1mA: 66 Ω 100 V 241 Ω

66 Ω 0.1 m

100 Ω (b) VR PTC V1mA: thermistor 1.5 m 1.5 m 100 V P.G. 0 Ω or 2 kΩ TOV

TIV P.D. 100 Ω

V1mA: 100 V

V1mA: 100 V 1.5 m

Ω Communication device

VVF VS:350 V

1.0 m VVF 66 Ω V1mA: 100 V 241 Ω

66 Ω 0.1 m

100 Ω

(c) FIGURE 6.24 Experimental circuit for a transient: (a) individual grounding, (b) common grounding, and (c) proposed grounding method.

the proposed method can reduce better than the common grounding. Therefore, the proposed method is well suited for protecting communication appliances. The FDTD simulation result agrees reasonably well with the measured result. The FDTD simulation can be used to study lightning surge characteristics from a communication line.

435


2500

Voltage (V)

2000 1500

Individual Common Proposed

1000 500 0 –500

0

50

100

150

200

250

300

Time (ns)

(a) 900 800

Voltage (V)

700 600 Individual Proposed

500 400 300 200 100 0

0

(b)

50

100

150

200

250

300

Time (ns)

FIGURE 6.25 Experimental results of VR in Figure 6.24: (a) voltage source case and (b) current source case.

6.4 Analytical Method of Solving Induced Voltages and Currents 6.4.1 Introduction Numerical simulation softwares are widely used by engineers, researchers, and university students and are very powerful to solve various problems. At the same time, it often happens that a user of a software is not well versed in its use. Also, physically nonexistent input data may give erroneous result, which cannot be realized if a user does not understand the practical problems that may arise. This partially comes from the fact that software technology is highly advanced and does not require a physical understanding of the phenomenon

436


Perfect 0 or conductor 2 kΩ

PTC thermistor 0.1 m

0.21 m

1 MΩ

P.D. 100 Ω

0.11 m

V1mA: 100 V Vs : 350 V

347 Ω 0.1 m

100 Ω

5.64 m

0.1 m

0.11 m 0.2 m

5.85 m

(a) Perfect 0 or conductor 2 kΩ


0.21 m 0.11 m

VR 1 MΩ

P.D. 100 Ω

V1mA: 100 V Vs : 350 V

347 Ω 0.1 m

0.11 m 0.1 m

5.64 m

0.2 m

5.85 m

(b) Perfect conductor

0 or 2 kΩ


0.21 m 0.11 m

(c)

VR

VR 1 MΩ

P.D. 100 Ω 5.65 m

V1mA: 100 V

Vs : V1mA: 350 V 100 Ω 100 V

347 Ω 0.1 m

0.11 m 0.1 m

0.2 m

5.87 m

FIGURE 6.26 Model circuits for transient simulations: (a) individual grounding, (b) common grounding, and (c) proposed grounding method.

to be solved, and furthermore, software technology is far beyond physical and engineering theory that explains the phenomenon analytically. Considering the earlier situation, an analytical method that gives an insight into the practical aspects of a phenomenon might be valuable and useful. This section concerns analytical formulation and investigation of induced voltages and currents to conductors such as a telephone line and a pipeline from power lines. There are a number of works that discuss induced voltages and currents. Carson and Sunde [35,36] are pioneers in this field, especially of induced voltages to telephone line. Taflov and Dabkowski [37] developed an analytical method to predict induced voltages to a buried pipeline based on a reflection coefficient method, the detailed theory of which is described in [38,39]. The method was adopted in the CIGRE Guide [39]. There are many text books explaining the theory of electromagnetic coupling [40–45]. Also, a number of publications describe numerical simulations of the voltages and

437


2500

Voltage (V)

2000 1500

Experiment Simulation

1000 500 0 –500

0

50

100

(a)

150

200

250

300

Time (ns) 2500

Voltage (V)

2000 1500


1000 500 0 –500

0

50

100

(b)

150

200

250

300

Time (ns) 2500

Voltage (V)

2000 1500


1000 500 0 –500

(c)

0

50

100

150

200

250

300

Time (ns)

FIGURE 6.27 Effect of grounding on VR for the voltage source case: (a) individual grounding, (b) common grounding, and (c) proposed method.

currents either by a circuit theory–based simulation tool such as the EMTP, or by a recent electromagnetic analysis method such as a finite-element method (FEM) and an FDTD method [31,46–60]. Among the publications, Ametani [45] and Chirstoforidis et al. [58] have proposed an approach combining numerical electromagnetic analysis to calculate the impedance and the


Voltage (V)

438

900 800 700 600 500 400 300 200 100 0

Voltage (V)

(a)

(b)

900 800 700 600 500 400 300 200 100 0


0

50

100

150 Time (ns)

200

250

300


0

50

100

150

200

250

300

Time (ns)

FIGURE 6.28 Effect of grounding on VR for the current source case: (a) individual grounding and (b) proposed method.

admittance of a given circuit and the circuit theory–based simulation tool to calculate induced voltages and currents in the circuit. This hybrid approach is able to solve problems in which the impedance and admittance are not known, nor easy to obtain, and allows researchers to keep in mind the physical dimensions of the phenomenon. This section describes an analytical method of calculating induced voltages and currents in a complex induced circuit, such as a cascaded pipeline with a few power lines, based on a conventional four-terminal parameter (F-parameter) formulation. The F-parameter formulation itself is well known, and it is straightforward to write a theoretical formula of the F-parameter for a multiphase circuit [38,45]. Thus, the calculation of the induced voltages and currents require a computer, that is, a software such as the EMTP. The method explained here replaces the multiphase F-parameter by a singlephase parameter by introducing an artificially induced current. The method is applied to a cascaded pipeline where the circuit parameters, the induced currents, and the boundary conditions are different in each part of the pipeline. The basic characteristic of the induced voltage and current distribution

439


along the pipeline is explained based on real use analytical results. Calculated results are compared with the EMTP simulations with an example given in Taflov’s and Dabkowski’s paper [37], with the CIGRE Guide [39] and also with a field result [60]. 6.4.2 F-Parameter Formulation for Induced Voltages and Currents 6.4.2.1 Formulation of F-Parameter There exist two basic approaches to deal with electromagnetic induction from a power line (inducing circuit) to a pipeline (induced circuit) [35,36,40–45]. The first is to represent the inducing and induced circuits as a multiphase line system. The other is to take into account the induction from the inducing circuit as a voltage source [35,37] or as a current source [61] so that the system becomes a single-phase circuit source approach. A multiphase line approach becomes too complicated, as pointed out in [46]. To overcome the difficulty, it is proposed to apply an F-parameter formulation with the current source [45,62]. The F-parameter can easily handle the cascaded line that appears very often in a real pipeline when discussing electromagnetic induction. In a distributed-parameter line composed of a power line and a pipeline parallel to each other as illustrated in Figure 6.29, the following solution of voltage Vx and current distance x from the sending end of the pipeline in Appendices 6.A.2 and 6.A.3: Vx = cosh(Γ ⋅ x) ⋅ V1 − Z0 ⋅ sinh(Γ ⋅ x) ⋅ ( I1 − I 0 ) = Ax ⋅ V1 − Bx ⋅ ( I1 − I 0 )

  (6.2) I − I = −Y0 sinh(Γ ⋅ x) ⋅ V1 + cosh(Γ ⋅ x) ⋅ ( I1 − I 0 ) = −Cx ⋅ V1 + Dx ⋅ ( I1 − I 0 ) x 0 where (6.3) I = E/Z = − zm I p /z : artificially induced current 0 Γ = z ⋅ y : propagation constant, Y0 = y/z = 1/Z0: characteristic admittance z: series impedance of pipeline [Ω/m] Ip Power line Z0

V1

I1

I2

V2

Z0

Pipe line Z0 R1

R2

x=0 FIGURE 6.29 A pipeline parallel to a power line for 0 ≤ x ≤ l.

x=l

440


zm: mutual impedance between: power line and pipeline [Ω/m] y: shunt admittance of pipeline [S/m] E = −zmIp: electromagnetically induced voltage [V/m] Ip: inducing current = power line current [A] Rewriting the earlier equation in a matrix form, the following equation is given:  Vx   Ax  =  I x − I 0   −Cx

−Bx   V1    (6.4)  Dx   I1 − I 0 

where Ax = Dx = cosh(Γ · x), Bx = Z0 sinh(Γ · x), Cx = Y0 sinh(Γ · x) Rewriting the earlier equation for x = l, the following form is obtained.

 A  V2  =  I 2 − I 0   −C  

  V1  A −B   V1 =   or    I1 − I 0   C D   I1 − I 0     

 B   V2   (6.5)  D  I2 − I0   

where A = D = cosh(Γ ⋅ l), B = Z0 sinh(Γ ⋅ l), C = Y0 sinh(Γ ⋅ l) (6.6) The earlier formulation is identical to the well-known F-parameter equation except that current I is replaced by I − I0 taking into account induction from a power line. 6.4.2.2 Approximation of F-Parameters Analytical evaluations of sinh and cosh functions in Equation 6.6 are not easy, and it is useful to adopt approximate F-parameters that are easily evaluated by hand calculations. In the same manner as conventional F-parameter approximation assuming |Γ · l| to be far smaller than 1, the following result is given:

A = D ≒ 1, B ≒ Z0 ⋅ Γ ⋅ l = z ⋅ l = z , C ≒ 0 for|Γ ⋅ l| 1 (6.7)

6.4.2.3 Cascaded Connection of Pipelines Assume a system of two-cascaded sections of a pipeline illustrated in Figure 6.30. In Section 6.1, a pipeline is parallel to power line 1 with current IP1 for the parallel length x1. In Section 2, the same pipeline is parallel to power line 2 with current IP2. The mutual impedance is zm1 and zm2 in Sections 6.1 and 6.2, respectively. Then, the following F-parameter equations are given:

441


Ip1, x1

Ip2, x2

V1

R1

V2

Section 6.1, Zm1 Z0, x1 (Z1)

I1

I΄2

R2

V3

Section 6.2, Zm2 I2 Ir

Z0, x2 (Z2)

I3 R3

FIGURE 6.30 Cascaded connection of a pipeline.

 V1   A1  =  I1 − I 01   C1  

  V  A B1   V2 2 2  ,  =  D1   I ′ − I   I 2 − I 02   C2 01   2

B2   V3    (6.8) D2   I 3 − I 02 

and

V1 = −R1I1 , V2 = R2 ( I 2′ − I 2 ), V3 = R3 I 3 (6.9)

where I01 = E1/z = zm1 IP1/z = Zm1 IP1/Z1, I02 = E2/z = zm2 IP2/z = Zm2 IP2/Z2, Z1 = z · x1, Zm1 = zm1 · x1, Z2 = z · x2, Zm2 = zm2 · x2 Ai = cosh(Γ · xi), Bi = Z0 · sinh(Γ · xi), Ci = Y0 · sinh(Γ · xi) Di = Ai; i = 1, 2 Voltage Vi and current Ii along the pipeline are obtained by solving Equations 6.8 and 6.9. 6.4.3 Application Examples 6.4.3.1 Single Section Terminated by R1 and R2 In a circuit illustrated in Figure 6.31, by applying the same boundary conditions as Equation 6.9, the following relation is given:

−R1I1 = AR2 I 2 + B( I 2 − I 0 ), I1 − I 0 = CR2 I 2 + A( I 2 − I 0 )

Solving the earlier simultaneous equations, the following result is easily obtained:

442


Node 1 V1

V2

Pipeline z

I1

R1

(a)

Node 2

Power line Ip I2

R2

x=0

x=l

R2ZI0 R1+R2+Z

V2 0

V1=

x=l

–R1ZI0 R1+R2+Z

(b) V2=

ZI0 Z0I0 {1–exp (–Гl)} ≈ 2 2

x=0

V1= – (c) R2ZI0 R2+Z

R1ZI0 R1+Z

ZI0 Z0I0 {1–exp (–Гl)} ≈ – 2 2

ZI0 R1 = 0 0

–

x=l

RZI0 x = 0 2R+Z R2 = 0

RZI0 2R+Z

R2 = ∞

R1 = R2 = R

x=l

R1 = ∞

–ZI0

(d)

FIGURE 6.31 Voltage profile along a gas pipeline. (a) Circuit configuration (b) R1 > R 2. (c) R1 > R 2 = Z0. (d) Effect of grounding resistance R1 and R 2.


443

I1 = {B + R2 ( A − 1)}I 0 /K , I 2 = {B + R1( A − 1)}I 0 /K

  V1 = −R1I1 , V2 = R2 I 2  (6.10)  Vx = −(Bx + R1 ⋅ Ax )I1 + Bx ⋅ I 0 , I x = ( Ax + R1 ⋅ Cx )I1 − ( Ax − 1)I 0 

where K = (R1 + R 2) A + B + R1R 2C Substituting Equations 6.3 and 6.6 into the earlier equations and rearranging, Vx = −Z0 sinh(Γ ⋅ x) ⋅ ( I1 − I 0 ) − R1 cosh(Γ ⋅ x) ⋅ I1

      R1  I x = cosh(Γ ⋅ x) +  sinh(Γ ⋅ x) I1 − {cosh(Γ ⋅ x) − 1}I 0    Z0     (6.11)  I1 = [Z0 sinh(Γ ⋅ l) + R2 {cosh(Γ ⋅ l) − 1}] ⋅ I 0 /K   I 2 = [Z0 sinh(Γ ⋅ l) + R1 {cosh(Γ ⋅ l) − 1}] ⋅ I 0 /K 

where K = (Z0 + R1R 2/Z0) sinh(Γ · l) + (R1 + R 2) cosh(Γ · l) 6.4.3.1.1 Approximation of Vx and Ix It is hard to observe the characteristic of Vx and Ix based on Equation 6.11. By applying Equations 6.7 through 6.10, the following approximate solution is obtained:

Vx = {(R1 + R2 ) ⋅ z ⋅ x − R1Z}I 0 /(R1 + R2 + Z)  I x = I1 = I 2 = ZI 0 /(R1 + R2 + Z)(18)  (6.12)  for R1 ≠ 0, R2 ≠ 0 

where z·l = Z [Ω], z [Ω/m], l [m] The voltage along the pipeline in Figure 6.31a is drawn as in Figure 6.31b where

V1 =

−R1ZI 0 (R1 + R2 + Z)

V2 =

R2ZI 0 (R1 + R2 + Z)

Vmax = V1

for R1 > R2 , Vmax = V2

(6.13) for R2 > R1

It should be noted that the voltage profile along the pipeline appears linear because of the approximation given in Equation 6.7. If Equation 6.11 is

444


hg r1

i

r2

r3

ρg , µg

FIGURE 6.32 −7 Configuration of a pipeline ρ g = 1.5×10 Ω -m , µ g = 280 , ε i = 2.3 , ρ e = 50 Ω -m r1 = 19.13 cm , r2 = 20.32 cm , r3 = 20.64 cm , hg = − 1.8 m.

adopted, the profile becomes not linear as in Figure 6.31c because of the characteristic of hyperbolic functions or exponential functions, but it is continuous in the region of 0 ≤ x ≤ l. Thus, there exists a position x where the voltage of the pipeline becomes zero. This fact has been shown in Figure 4 (a) of [37], which is the same as Figure 6.31b in this chapter. However, the fact seems to be remembered by engineers. It might be better to consider the phase angle, or the polarity at both ends, of the pipeline voltage from the viewpoint of realizing the physical nature of the induced voltage corresponding to the theoretical analysis, although it is conventionally neglected. The applicable range of Equation 6.7 needs to be discussed. Figure 6.32 illustrates a typical gas pipeline used in Japan [59,60]. Its impedance and admittance are easily evaluated by hand based on approximate formulas [45,59,63], as given in Appendix 6.A.4, and are given here as

z = 0.613∠78.6° [Ω/km], y = 2.572∠90° [mS/km]

Γ·l is evaluated for l = 10 km as

Γ ⋅ l = z ⋅ y ⋅ l = 0.03902 1 (6.14)

It is clear that the approximation of |Γ · l| ≪ 1 is valid up to a few kilometers in general. If a pipeline is very long, say 10 km, then it is divided into sections, each section of which satisfies the earlier condition. In practice, a pipeline is divided into many sections corresponding to different power line and groundings. 6.4.3.1.2 R1 = R2 = Z0 R1 = R2 = Z0 corresponds to the matching condition (Z0 = characteristic impedance) of a pipeline and is identical to semi-infinite pipeline connected at nodes 1 and 2. The power line is parallel to the pipeline only for the section


445

between the nodes. Although it is described that the pipeline extends for a few km beyond the parallel route in CIGRE Guide [39], “a few km” should be replaced by a semi-infinite pipeline, or by the length l of the parallel route being far smaller than a few km. With R1 = R2 = Z0, Equation 6.11 becomes quite simple:

Z I  Vx =  0 0  exp{−Γ(l − x) − exp(−Γ ⋅ x) for 0 ≤ x ≤ 1  2 

Vx = Vmax ⋅ exp(− γ ⋅ x) for x 〈0, x〉 l

I Ix =  0 2

   2 − exp{−Γ(l − x)} − exp(−Γ ⋅ x) 

ZI  I  Vmax = −V1 = V2 =  0 0  ⋅ {1 − exp(−Γ ⋅ l)} , I1 = I 2 = −  0  {1 − exp(−Γ ⋅ l)} 2   2 The voltage profile along the pipeline is shown in Figure 6.31c. Equation 6.11 gives the same results as those given in [37] and the CIGRE Guide [39] although the Guide does not deal with the effect of grounding resistances of a pipeline. By the way, the earlier results are simplified by adopting the approximation in Equation 6.7 or exp(−Γ ⋅ x) ≒ 1 − Γ ⋅ x as

ZI 0 ZI  l  Vx = −  , − V1 = V2 = 0  zI 0 , I x = 2Z0 2  2−x  Vmax =

ZI 0 2

It should be noted that the maximum induced voltage in the earlier approximation is nearly the same as that of the accurate solution for exp (−Γ · l) ≪ 1. Also, it should be pointed out that the same assumption was adopted in [37] for a short pipeline and exp (−Γ · l) ≅ 0.1 for a long line so as to make an analysis possible. 6.4.3.1.3 Effect of Grounding Resistances of a Pipeline Formulas in Table 6.13 for various grounding resistances are derived from the general solution of Equation 6.11. It is noted that the voltages in Tables 6.13 and 6.14 are normalized by I0. It is hard to observe the voltage and current characteristic along the pipeline from the earlier formulas without numerical calculations. By applying the approximation in Equation 6.7, simple results in Table 6.14 together with

446


TABLE 6.13 Voltages and Currents for Various Grounding Resistances: Exact Solution R1 0 0 R R ∞ R

R2 0 R 0 R R ∞

−Vx/I0 0 –(Z0R/K1)·sinh(θ) (Z0R/K1)·sinh(θ0 – θ) (2Z0RK4/K3)·sinh(θ0/2 – θ) [Z0 sinh(θ0–θ) – R{cosh(θ) – cosh(θ0 – θ)}]Z0/K2 [Z0 sinh(θ) – R{cosh(θ) – cosh(θ0 – θ)}]Z0/K2

Ix/I0 1 1 − (R/K1)cosh(θ) 1 − (R/K1)cosh(θ0 − θ) 1 − (2 RK4/K3)cosh(θ0/2 − θ) I2 = (Z0/K2){cosh(θ0) − 1}, I1 = 0 I1 = (Z0/K2){cosh(θ0) − 1}, I2 = 0

K1 = Z0 sinh(θ0) + R cosh(θ0), K2 = Z0 cosh(θ0) + R sinh(θ0), K3 = Z0 K1 + R K2, θ0 = Γ l, θ = Γ x K4 = Z0 cosh(θ0/2) + R sinh(θ0/2).

TABLE 6.14 Voltages and Currents for Various Grounding Resistances: Approximate Solution R1 0 R R ∞ R ∞

R2 R 0 R R ∞ ∞

−Vx/I0 −R z x/(R + Z) Rz(l – x)/(R + Z) 2Rz(l/2 – x)/(2R + Z) Z(l–x) −z x Z(l/2–x)

−V1/I0 0 RZ/(R + Z) RZ/(2R + Z) Z 0 Z/2

−V2/I0 −RZ/(R + Z) 0 −RZ/(2R + Z) 0 −Z −Z/2

Vmax/I0 RZ/(R + Z) RZ/(R + Z) RZ/(2R + Z) Z Z Z/2

maximum voltage Vmax are obtained. The analytical results given in Table 6.14 are shown in Figure 6.31d. It is clear in the table and Figure 6.31 that the severest voltage, ZI0, appears when one end of a pipeline is open-circuited and the other end is grounded by resistance R, which is neither zero nor infinite. Taking into account the fact that R = 0 is almost impossible in practice, both ends of a pipeline should be grounded so that the maximum voltage becomes less than half of the severest case as given by the ratio R2/(R1 + R2 + Z). If the pipeline length l is greater than 10 km, the ratio becomes less than 1/3 with R1 = R2 ≒ 10 Ω. 6.4.3.2 Two-Cascaded Sections of a Pipeline (Problem 6.1) An induced voltage and a current in the system of Figure 6.30 can be obtained by solving the simultaneous equations given by Equations 6.8 and 6.9. The solution is given in Appendix 6.A.4. By adopting the approximation in Equation 6.7, 6.8 becomes very simple, and the following equations are given in the system of Figure 6.31:


 V1  1  =  I1 − I 01  0  

  V1 Z1   V2  1  ,  =  1   I 2' − I 01   I 2 − I 02  0  

 V2   1  ′ =  I 2  1/R2

0   V2    1   I 2 

447

Z2   V3    1   I 3 − I 02    (6.15)    

where Z1 = z · x1, Z2 = z · x2 Solving Equations 6.15 and 6.9, the following results are easily obtained: I1 = I ′2 = {(R2 + R3 + Z2 )Z1I 01 + R2Z2 I 02 }/K1 , V1 = −R1I1

  I 2 = I 3 = {R2 Z1I 01 + (R1 + R2 + Z1 )Z2 I 02 }K1 , V3 = R3 I 3  (6.16)  I r = I ′2 − I 2 = {(R3 + Z2 )Z1I 01 − (R1 + Z1 )Z2 I 02 }/K1 , V2 = R2 I r 

where K1 = R1R 2 + R 2 R3 + R3R1 + R3Z1 + R1Z2 + R 2Z + Z1Z2, Z = Z1 + Z2 One of the reasons for developing the F-parameter approach to deal with cascaded connection of pipeline sections is to find effective grounding resistances and separation distance between them. Those become a significant subject in practice in Japanese gas pipelines. Based on the earlier approximate results, the characteristic of induced voltages and currents and the effect of grounding resistances can be discussed. 1. I01 = I02 = I0: Equation 6.16 is further simplified as follows:

I1 = I ′2 = {(R3 + Z2 )Z1 + R2Z}I 0 /K1   I 2 = I 3 = {(R1 + Z1 )Z2 + R2Z}I 0 /K1  (6.17)  I r = (R3 Z1 − R1Z2 )I 0 / K1 

a. R 2 = ∞ I1 = I 2 = I 3 =

V1 =

ZI 0 (R1 + R3 + Z)

−R1ZI 0 R3 ZI 0 , V3 = (R1 + R3 + Z) (R1 + R3 + Z) V2 =

(R3 Z1 − R1Z2 )I 0 (R1 + R3 + Z)

The earlier result is the same as Equation 6.13 for a single-section pipeline considering that node 2 in Equation 6.13 is node 3 here.

448


b. R3 = ∞

I1 =

Z1I 0 = Ir , I2 = I3 = 0 (R1 + R2 + Z1 ) V1 =

V2 =

−R1Z1I 0 (R1 + R2 + Z1 )

R2Z1I 0 , V3 = ZI 0 (R1 + R2 + Z1 )

The earlier results shows similar characteristic to that in Section 4.3.1(3) as far as V1 and V3 are concerned. Equation 6.17 corresponds to the case of grounding an intermediate position of a pipeline of which both ends are grounding through resistances R1 and R3, and thus, the effect of the grounding resistance R 2 on an induced voltage can be discussed in comparison with no grounding, that is,

R2 = ∞ : V3 = −V1 =

RZI 0 = Vmax ( 2R + Z )

Assumption of R1 = R 2 = R3 = R leads to the following result: R2 ≠ ∞ : V1 = −R {R(Z1 + Z) + Z1Z2 } I 0 /K 2 ,

V3 = −R {R(Z2 + Z) + Z1Z2 } I 0 /K 2 K 2 = 3R 2 + 2RZ + Z1Z2

Furthermore assume that x1 = x2 = l/2, then

V3 = −V1 =

RZI 0 , V2 = 0 for R2 ≠ ∞ , ( 2R + Z )

The earlier result is the same as that in the case of a single pipeline section in Section 4.3.1. This fact indicates that a grounding resistance (R 2 in Figure 6.30) at the middle of a pipeline is not effective to reduce the induced voltage as far as the circuit parameter and the inducing current are constant. This is readily understood because the voltage at the center of the pipeline V(l/2) is zero when R1 = R3 = R. Even in the case of R1 ≠ R3, the grounding resistance R 2 shows no reduction of the maximum voltage.


449

2. I02 = 0: (R2 + R3 + Z2 )Z1I 0 (R + Z2 )Z1I 0  , Ir = 3  K1 K1   R2Z1I 0  (6.18) I2 = I3 =  K1   V1 = −R1I1 , V2 = R2 I r , V3 = R3 I 3  I1 =

This case is a more general condition of Section 3.1.2 and of the calculation examples discussed in the CIGRE Guide [39], where a pipeline extends beyond the zone of influence but no grounding resistance is considered. Figure 6.33 shows the system diagram and analytical results of the induced voltages. Figure 6.33b is the result of Equation 6.18 where it should be noted that V2 > V3. −R1Z1I 0 a. R2 = ∞ : V1 = (R1 + R3 + Z)

b. R2 = ∞ : V1 =

V2 =

(R3 + Z2 )Z1I 0 (R1 + R3 + Z)

V3 =

R3 Z1I 0 (R1 + R3 + Z)

−R1Z1I 0 (R1 + R2 + Z1 ) V2 = V3 =

R2Z1I 0 (R1 + R2 + Z1 )

As has been explained in Section 4.3.1(1) and in Figure 6.31c, all the curves shown in Figure 6.33 are not linear but are to be exponential or hyperbolic. If this is considered, V3 should be expressed as

V3 = V2 exp(−Γx) for x > x1 in the case of R3 = ∞

It should be easily realized that the result for R 2 = R3 = ∞ becomes the same as Figure 6.31c if R1 = Z0, which corresponds to a semiinfinite pipeline to the left of node 1.

450


Power line Ip V3

V2

V1 Z1, x1

I1 R1

Z2, x2

I2

R2 x1

x=0

x=l

(a) Vi V1 R2(R3 + Z2) V3 R2R3 0

x1

l

V1 — R1(R2 + R3 + Z2)

(b) Vi [× Z1I0/(R2 + R3 + Z)] R3 + Z2

0

x1

R3

l

–R1

(c) R2

0

x1

I3 R3

Ir

R2 exp(– x2)

l

–R1

(d) FIGURE 6.33 Case of I02 = 0. (a) System diagram, (b) R1 ≠ R 2 ≠ R 3 ≠ 0, (c) R 2 = ∞, (d) R 3 = ∞.


451

3. I01 ≠ I02: When I01 is far greater than I02, similar results as those in the previous section for I02 = 0 are obtained. Figure 6.34 shows analytical results in the case of (a) I01 = 0.1I02 = 0.5I0 and (b) I01 = 0.5I02 under the condition of x1 = x2 = l/2 that is, Z1 = Z2 = Z/2. Similar characteristics of the induced voltage to those in the previous section are observed when I01 = 0.1I02, that is, I02 is far greater than I01. On the contrary, the induced voltage characteristics for I02 > I01 > 0.5I01 are significantly different from those in the case of I01 or I02 is zero as shown in Figure 6.33. It is clear that the proposed approach is very effective to observe a basic and qualitative characteristic of an induced voltage. If one needs a quantitative characteristic of the induced voltage, it is quite easy to produce a computer code based on the proposed approach. 4. Effect of I01 relative to I02 = I0: Assume that R1 = R 2 = R3 = R, x1 = x2 and I01 = mI02 where m < 1. Then, the following results of induced voltages V1 to V3 are derived from Equation 6.16:

V1 = −R{2R(1 + 2m) + mZ}ZI 0 /(6R + Z)(2R + Z)  V2 = −R(1 − m)ZI 0 /(6R + Z)  (6.19)  V3 = R{2R(2 + m) + Z}ZI 0 /(6R + Z)(2R + Z) 

The earlier analytical results are illustrated in Figure 6.34c. The result for m = 0 is the same as that in Equation 6.18, and that for m = 1 is identical to that in Figure 6.37d. It is quite reasonable that the induced voltages |V1| and V3 increase as the ratio m increase. On the contrary, the voltage V2 reaches to zero corresponding to the characteristic explained in Section 4.3.1. The simplified formula in Equation 6.19 is very useful to investigate the effect of inducing currents of different power lines observed quite often in practice. 5. Effect of x1 relative to x2: Assuming that R1 = R 2 = R3 = R, I01 = 0.5I02 = 0.5I0, and x1 = nx2, that is, Z1 = nZ2 = nZ′, the following results are obtained from Equation 6.16: V1 = −R{(1 + n)R + nZ′/2}Z′I 0 /K 2

  V2 = −R{(1 − n/2)R + nZ′/2}Z′I 0 /K 2  (6.20)  V = R{(2 + n/2)R + nZ′}Z′I 0 /K 2 

where K 2 = 3R 2 + 2R(n + 1)Z′ + n2Z′

452


Ip1

Ip2

I01

I02

R1

R2

R3

V, × Z’I0 1/2

R3 = ∞, R2 = R3 = ∞

1/3 1/4

1/2 1/3 1/4

R1 = R2 = R3 = R

–1/6 –1/4 –1/2

R2 = ∞, R1 = R2 = ∞

(a) V, × Z’I0 1/2 1/3

R3 = ∞ 0 R1 = R2 = R3 = R

R2 = ∞

–1/3

(b) 1/2 V3

1/3 1/6

0.5

1

0

k V2

–1/6 –1/3

(c)

V1

–1/2

FIGURE 6.34 Case of I01 ≠ I02 = I0. (a) I01 = 0.1I02, (b) I01 = 0.5I02, (c) Effect of I01, relative to I01I02 = mI02.

453


V, × Z΄I0 1

V3

0

V2

–1

2

3

V1

FIGURE 6.35 Effect of x1 relative to x2. Z1 = nZ2 = nZ′, Z = (n + 1)Z′.

The earlier analytical result is illustrated in Figure 6.35. It is clear from Equation 6.20 and Figure 6.35 that the induced voltages are linearly proportional to the length of x1. The result is readily understood for the induced voltages is proportional to the total mutual impedance Zm1 = zm1 · x1. 6.4.3.3 Three-Cascaded Sections of a Pipeline Obtain currents I1, I2, I3, Ir1, Ir2 and voltage V1 in Figure 6.36, and draw voltage profiles along the line for (a) I02 = 0, I01 = I03 = I0, R1 = R 2 = R3 = R4 = R and (b) R 2 = R3 = ∞, I01 > I02 > I03. Figure 6.36 illustrates three-cascaded sections of a pipeline where each section is parallel to a power line different from the other power lines and has an induced current I0i(i = 1 to 3). In this case, we adopt the approximation in Equation 6.7 so that the solution of the induced voltages and the currents on each section are easily obtained. Based on the same equations as those of Equation 6.15 with two more equations for nodes 3 and 4, the solutions of the currents and the voltage are easily obtained.

1

I1

I01

Ip1 I2΄

I2

2

Z1

I02

Ip2

Z2

R1

R2 Ir1

I3΄ I3

I3

3 I2 R3

Ir2

FIGURE 6.36 Three-cascaded sections of pipeline. Z1 = z · x1 = Z′, Z2 = z · x2, Z3 = z · x3.

I03

Ip3 4

I4

Z3 R3

454


6.4.4 Comparison with a Field-Test Result 6.4.4.1 Comparison with EMTP Simulations Table 6.15 shows a comparison of analytical results evaluated by Equation 6.11, an accurate formula, and by Equation 6.13, an approximate one, for the case of including (power line) current I P = 1000∠0 °, separation distance y = 50–500 m and pipeline length x = 1 and 10 km in Figure 6.31a. Grounding resistance R1 and R2 are varied from 0 to infinite, which corresponds to the highest induced voltage on the pipeline. The highest voltage is essential to investigate the effect of pipe grounding. The cross-section of the pipeline is z = 0.613∠ 78.6° (Ω/km), and the mutual impedance to the power line is Zm = 0.0592∠49.9° for y = 500 m. Thus, the artificially induced current in this case is evaluated as I 0 = 96.57 ∠28.7 ° (A/km) by Equation 6.4. The analytical results show the maximum error of less than 5% in comparison with the EMTP simulation results in Table 6.15, and thus, the accuracy of the analytical formula is said to be satisfactory. Table 6.16 shows a comparison for the case of three-cascaded section of a pipeline in Figure 6.36 with the separation distance y = 500 m. Case 10 in the table corresponds to Figure 6.38a, and case 32 to Figure 6.38b in the solution (6.1) of the problem (6.1). The maximum error of the analytical result is observed to be 3.3% in case 30. From the earlier observation, it is concluded that the accuracy of the analytical method proposed in this chapter is satisfactory enough, and thus, the method is useful in practice. It should be noted that the node voltages in Tables 6.15 and 6.16 are the maximum voltage on the pipeline, and thus, the maximum voltage is accurately calculated even with the approximation. 6.4.4.2 Field-Test Result A field-test result of induced voltages on an underground gas pipeline in Japan is shown in Ref. [60]. Figure 6.37a illustrates the system configuration. The real line is the pipeline and dotted lines are an overhead transmission line. Figure 6.37b shows the configuration of a 500 kV vertical twin-circuit line. The system is simplified by 5 cascaded sections as in Figure 6.37c. In a section where the separation distance y between the power line and the pipeline exceeds 300 m, the power line is neglected. Thus, the power line and its inducing current are taken into account only in Sections 6.2 and 6.4. Both ends (nodes 1 and 6) of the pipeline are grounded by R1 = R 2. The pipeline cross-section is given in Figure 6.32. The 500 kV power line is of twin-circuit vertical configuration with two GWs as in Figure 6.37b. The earth resistively along the line ranges from 50 to 200 Ω-m. In the same manner as explained in Section 6.3, the node voltages are given in the following equation:

y [m]

50 100 200 500 500 500 500 500 500

x [km]

1 1 1 1 10 1 1 1 1

10 ∞

10

0

10 10 10 10 10 ∞

R2 [Ω]

10 10 10 10 10 10 ∞ ∞

R1 [Ω]

95.8 75.8 55.8 30.6 281.9 0.79 61.6 30.7 0

V1 73.7 69.8 63.3 45.4 31.7 136.3 47.8 47.8

θ1 95.8 75.8 55.8 30.6 281.9 61.6 0.79 30.7 60.7

V2

Accurate Formula

−106.3 −110.2 −116.7 −134.6 −148.3 −132.2 −43.7 −132.2 −135.6

θ2 95.8 75.8 55.9 30.6 279.1 0 61.5 30.8 0

V1

47.8 47.8

73.7 69.8 63.3 46.1 32.0

θ1 95.8 75.8 55.9 30.6 279.1 61.5 0 30.8 60.7

V2

−132.2 −135.6

−106.3 −110.2 −116.7 −133.9 −148.0 −132.2

θ2

Approximate Formula

Comparison with EMTP Simulation Results: Single Section (Ip = 1000∠0 A)

TABLE 6.15

92.9 73.5 53.8 29.4 277.5 0.76 58.9 29.5 0.0

V1

79.0 75.2 68.9 53.8 38.8 138.4 54.8 55.6 52.8

θ1

92.9 73.5 53.8 29.4 277.5 58.9 0.76 29.5 58.5

V2

EMTP

−101.0 −104.8 −111.1 −126.2 −141.2 −125.2 −41.6 −124.4 −127.9

θ2


10 20 21 30 31 32

Case No.

IP1/IP2/IP3 [A]

1000/0/1000 1000/100/1000 1000/100/1000 1000/500/1000 1000/500/1000 1000/500/1000

1/1/1 1/1/1 1/10/1 1/1/1 1/10/1 1/1/1

V1

θ1 V2

θ2 V3

θ3

Approximate Formula V4

θ4 V1

θ1 V2

θ2

V3

EMTP Result θ3

V4

θ4

10 60.1 42.8 1.8 −60.3 1.8 119.7 60.1 −137.2 58.4 51.3 0.89 −44.6 0.89 135.4 58.4 −128.7 10 63.1 42.7 3.1 9.8 3.1 −170.2 63.1 −137.3 61.1 50.3 3.2 18.8 3.2 −161.2 61.1 −129.7 10 75.2 27.8 24.7 −14.4 24.7 165.6 75.2 −152.2 74.1 33.7 26.4 −7.7 26.4 172.3 74.1 −146.3 10 75.1 42.5 14.8 37.6 14.8 −142.4 75.1 −137.5 72.7 50.1 14.5 45.2 14.5 −134.8 72.7 −129.9 10 169.0 21.7 117.1 13.4 117.1 −166.6 169.0 −158.3 169.3 27.2 119.6 18.7 119.6 −161.3 169.3 −152.8 10/∞/∞10 75.2 42.8 14.9 39.3 14.9 −140.7 75.2 −137.2 72.8 50.4 14.5 46.9 14.5 −133.1 72.8 −129.6

x1/x2/x3 R1/R2/R3/ [km] R4 [Ω]

Comparison with EMTP Simulation Results: Three Sections (y = 500 m)

TABLE 6.16


457


GP13

Gas pipeline

12

A

Insulating joint Transmission line 07 05

03 GP01

04

66 kV line 11 09 10

06

B

08

02

GP00

A

500 kV line B

(a)

z

d

hgw h1 h2

d

a

c’

b

b’

c

a’

Phase wire

y h3 y hg Gas pipe d = 7[m], h1 = 31, h2 = 23, h3 = 15, hgw = 39 rp = 16.77[cm], ρp = 3.18 × 10–8[Ωm], ρe = 50[Ωm]

(b) Power line

V1

Z1 R1

x1 = 9.9[km] Section 6.1

V2

y2

V3

x2 = 1.1 y2 = 105 m

Z2 x3 = 1.5

V4

y4

V5

x4 = 1.1 y4 = 66 m

Z5 x5 = 5.8 Section 6.5

V6 R2

(c) FIGURE 6.37 System of configuration of a field measurement. (a) Field test circuit, (b) vertical twin-circuit 500 kV line, (c) model circuit for an analytical calculation.

458


V1 = −R1(Z2 I 01 + Z4 I 02 )/(R1 + R2 + Z)

   V2 = −(R1 + Z1 )(Z2 I 01 + Z4 I 02 )/(R1 + R2 + Z)  V3 = {(R2 + Z3 + Z4 + Z5 )Z2 I 01 − (R1 + Z1 + Z2 )Z4 I 02 }/(R1 + R2 + Z)  (6.21) V4 = {(R2 + Z4 + Z5 )Z2 I 01 − (R1 + Z1 + Z2 + Z3 )Z4 I 02 }/(R1 + R2 + Z)  V5 = (R2 + Z5 )(Z2 I 01 + Z4 I 02 )/(R1 + R2 + Z)    V6 = R2 (Z2 I 01 + Z4 I 02 )/(R1 + R2 + Z)

where Z = Z1 + Z2 + Z3 + Z4 + Z5. In the earlier equation, the original induced currents I01 and I02 are given by the induced voltage E as in Equation 6.3. The original induced voltage E is determined by the vector sum of the induced voltages due to the phase currents of the transmission line [60], or approximately by the zero-sequence current of a transmission line and the GW currents as explained in the CIGRE Guide [39]. The mutual impedance between a transmission line and a pipeline is calculated either numerically by the EMTP cable parameters [63] or analytically by approximate formulas in Appendix 6.A.4. The following is the EMTP result for the section 4 in Figure 6.37c, with the earth resistivity of 50 Ω-m, where the separation between the center of the 500 kV power line and the pipeline is 66 m. The mutual impedance estimated from the curve in the CIGRE Guide, Figure 3.4 [39], agrees well with the EMTP result.

Zm1 = 0.1428∠71.7° [Ω/km] : phase 1 to pipeline

Zm 2 = 0.1446∠71.7°, Zm 3 = 0.1459∠71.0°, Zm 4 = 0.1530∠72.2°, Zm 5 = 0.1557∠72.4° Zm 6 = 0.1578∠72.4°, Zm 7 = 0.1407∠71.0° : GW 1 to pipeline, Zm 8 = 0.1498∠72.1° The current on each phase of the 500 kV line is 1000 A. The GW currents are calculated in the same manner as those for a pipeline. The original induced voltage E in Equation 6.3 is analytically calculated using the earlier mutual impedances in the following manner: Ei = zmi ⋅ I P1i

For example, E1 is calculated as E1 = zm1 · IP1 = 0.1428∠71.7 × 1000 = 142.8∠71.7 [V/km]. The total induced voltage E is evaluated as the vector sum of the earlier voltages. E=

8

∑ E = 2.029∠54.5° [V/km] i

i =1


459

Then, the original induced current I02 in the section 4 of Figure 6.37b is given by

I a2 =

E = 3.427 ∠ − 23.7° [A] z

where z = 0.592∠78.2°[Ω/km]: self impedance of pipeline with ρe = 50 [Ω-m] In the same manner, the current I01 in the section 2 is calculated as

I 01 =

E = 0.778∠ − 106.6° [A] Z

Substitution of the earlier I01 and I02 into Equation 6.22 yields the following results: a. R1 = Z0 = 15.1 Ω, R 2 = ∞: V1 = V2 = 0,

V3 = V4 = 0.5047 ∠151.6°

V5 = V6 = 2.340∠ − 137.9°

b. R1 = R 2 = Z0: V6 = −V1 = 1.026∠23.0°,

V2 = −1.17 ∠42.4°, V3 = −1.159∠68.9°

V4 = 1.2101∠ − 109.6°, V5 = 1.092∠ − 144.9°

The measured result was 1.17–2.48 V in the system in Figure 6.37a [60]. The analytical results are observed to agree reasonably with the measured result, even with the approximation explained earlier. There exist a number of distribution lines (6.6 kV, 3.3 kV) near the pipeline, and the distribution line might have contributed to the discrepancy. 6.4.5 Concluding Remarks An analytical method to calculate induced voltages and currents on a complex pipeline system is described in this section by applying an F-parameter. The approach makes it easy to handle the system parameters, inducing currents and boundary conditions that are different in each section of the pipeline system. The profile of the induced voltages and currents along the pipeline is easily evaluated by hand calculations, especially by adopting a well-known approximation of hyperbolic functions of the F-parameter. Even with the approximation, the maximum induced voltage is accurately calculated. An analytical test result has been compared with EMTP simulation test results and a field test result, and it is observed that even manual calculation gives a reasonable agreement with the test result.

460


Analytical results of the induced voltages show clearly that the maximum voltage appears at one end of a pipeline when the other end is grounded, and both ends when both ends are grounded by the same resistance, as pointed out in various texts. Also, there exists a position where the voltage of the pipeline to ground becomes zero. The facts should be remembered by engineers in the field. The effect of the grounding resistances and inducing currents on the induced voltages has been explained based on the analytical results. The approach based only on such knowledge of electrical circuit theory, as would be taught in a university undergraduate course, is expected to be very useful to a qualitative and predictive analysis of induced voltages and currents.

Solution of Problem 6.1 {(B ⋅ C − R32 )Z1I 01 + CR2Z2 I 02 + R2R3 Z3 I 03 } = I 2′ K1 (CR2Z1I 01 + ACZ2 I 02 + AR3 Z3 I 03 ) + I 3′ I2 = K1

      2  {R2R3 Z1I 01 + AR3 Z2 I 02 + ( AB − R2 )Z3 I 03 } I3 = = I4  (6.22) K1  2 [{C(R3 + Z2 ) − R3 }Z1I 01 − C(R1 + Z1 )Z2 I 02 − R3 (R1 + Z1 )Z3 I 03 ]   Ir1 = K1  [R2 (R4 + Z3 )Z1I 01 + A(R4 + Z3 )Z2 I 02 − { A(R2 + Z2 ) − R22 }Z3 I 03 ]   Ir 2 = K1   V1 = −R1I1 , V2 = R2 I r 1 , V3 = R3 I r 2 , V4 = R4 I 4  I1 =

where K1 = A(BC − R32 ) − CR22 = C( AB − R22 ) − AR32 A = (R1 + R2 + Z1 ), B = (R2 + R3 + Z2 ), C = (R3 + R4 + Z3 ) Based on the earlier equation, various investigations on the induced voltages and currents can be made. It is easily observed that the assumption of I01 = I02 = I03 = I0, R 2 = R3 = ∞, and Z1 + Z2 + Z3 = Z gives the same formula as Equation 6.13, and the assumption of I01 = I02, R 2 = ∞, and Z1 + Z2 replaced by Z2 gives the identical result to Equation 6.16. Some examples are demonstrated in Figure 6.36b and c. 1. I02 = 0, I01 = I03 = I0: In Figure 6.38a, −V1 = V4 ≒ −Z′I 0 for R ≫ Z′. When I02 = 0, a similar result to Figure 6.36b is obtained even in the case of R2 = R3 = ∞,


461

V, × Z’I0 V4 x1 + x2 0

V1

x1

x1 + x2 + x3

Z1 = Z2 = Z3 = Z’

(a) V, × Z’I0

(b) FIGURE 6.38 Solutions for three-cascaded sections of a pipeline in Figure 6.36. (a) I02 = 0, I01 = I03 = I0, R1 = R 2 = R 3 = R4 = R. (b) R 2 = R 3 = ∞, I01 > I02 > I03.

because of the finite length of the section 2, that is, x2, and of the fact that the polarity of the voltage at node 2 induced by the current I01 is opposite to that at node 3 induced by the current I03. This clearly indicates the significance of the induced voltage polarity that has been neglected in most conventional studies [39]. When x2 is very long or semi-infinite, the inducing currents IP1 (correspondingly I01) and IP3 (I03) do not affect each other, and thus, a well-known characteristic explained in Section 4.3.1 for the single-section case is observed. 2. R1 = R4 = ∞: Figure 6.38b shows an analytical result in the case of R1 = R4 = ∞ with I01 = I0 > I02 > I03. The gradient of the voltage profile along the x-axis in the section i is proportional to the current I0i.

Appendix 6.A.1 Test Voltage for Low-Voltage Control Circuits in Power Stations and Substations (JEC-0103-2004) [16]

2 2 2 1.5 2

2 2 2 2 1.5 —

1 2–1 2–2 2–3 3

4 5 6 7-1 7-2 8

Circuit Categorya

2 — — — — —

2 —— — — —

4 4 4 — — —

7 7 5 5 3

4.5 3 — — — —

4.5 3 3 3 3 3

4.5

3 — — – –

3 3 3 3

DC/AC VT/CT Circuit

Between Contacts/Coil Terminals

Lightning Impulse Withstand Voltage

To Between To Between Earth Terminals Earth Terminals

AC Withstand Voltage

2.5 2.5 — — – –

I/O Signal Source Circuit Circuit

To Earth

Immunity

—— 2.5 – — – —

1 1 0.5 0.5 0.5 —

2 1 1 1

Rectangular Impulse Waveb

2 2 – – — –

1 1 – – — —

1 1 — – — —

—— 1 — – — –

only in Japanb

To Between To Between Earth Terminals Earth Terminals

Surge Immunity

same as IEC 0225-22-1, 4 and 5

To Between Earth Terminals

Oscillatory Wave

EMC Test Voltages (Unit: kV)


Category: main-circuit side 1 → control box side 8 (greater the number, lower the voltage) Category 1: Secondary circuit and tertiary circuit of an instrument transformer (VT or CT), which is equipped with the main circuit. Category 2: Control circuit of a circuit breaker and a DS. Category 2-1: Control circuit in case of the high dielectric strength required In this case, 7 kV is applied as the test voltage stated in the earlier table. Category 2-2: Control circuit that has surge reduction countermeasures or no possibility for excessive lightning surge, in case of the high dielectric strength required. In this case, 5 kV is applied as the test voltage stated as in the earlier table. For example, if control cables with metallic sheath (CVVS) are adopted, both ends/terminals are to be grounded. Category 4: Secondary circuit and tertiary circuit of VT/CT in a direct control board, a protective relay board, a remote supervisory control board, other control devices, etc. In this case, 4 kV is applied as the test voltage stated in Table 6.16 b See Refs. [2,3,16]. Note: Breakdown accidents in control boards have occurred, even the control circuits were tested and confirmed according to the category 4, 4 kV to earth. The surge voltage at the equipment side is higher than the one at the board side. As a result, the category 2–1, 7 kV to earth, is applied considering the dielectric strength capability of equipment.

a


464


6.A.2 Traveling Wave Solution In a distributed-parameter line system composed of a power transmission line and a pipeline parallel to each other, as illustrated in Figure 6.1, the following differential equation is given, assuming that current IP along the power line is constant [37,39,40]: −dvx −dI x = z ⋅ I x − E, = y ⋅ Vx (6.23) dx dx

where z: series impedance of pipeline (Ω/m) zm: mutual impedance between the power line and pipeline (Ω/m) y: shunt admittance of pipeline (S/m) E = −zmIP: electromagnetically induced voltage (V/m) IP: inducing current = power line current (A) Vx = V(x), Ix = I(x): pipeline voltage and current at position x In the same manner as an ordinary distribution line, the following traveling wave solutions for the voltage and the current are easily obtained [45]: Vx = k1 ⋅ exp ( Γ ⋅ x ) + k2 ⋅ exp ( −Γ ⋅ x )

  (6.24)  †I x − ††I 0 = −Y0 {k1 ⋅ exp ( Γ ⋅ x ) − k2 ⋅ exp ( −Γ ⋅ x ) 

where k1 and k2: constant to be determined by boundary conditions

I 0 =

E − zm I P = : artificial induced content (6.25) z z

Γ = √(z·y): propagation constant, Y0 = √(y/z) = 1/Z0: characteristic admittance.

6.A.3 Boundary Conditions and Solutions of a Voltage and a Current Boundary conditions in Figure 6.29 are given in general by

x = 0 : V ( x = 0 ) = V1 †,†I ( x = 0 ) = I1 (6.26)

x = l : V (x = 1) = V2 †,†I (x = 1) = I 2 (6.27)


465

Substituting Equation 6.26 into 6.24, the unknown constants k1 and k2 are given as a function of V1 and I1. k1 =

k2 =

{V1 − Z0 (I1 − I0 )} 2

{V1 − Z0 (I1 − I0 )} (6.28) 2

Similarly, k1 and k2 are defined as a function of V2 and I2 by applying Equation 6.27. When there exists no pipeline to the left of node 1 and the right of node 2, Equations 6.26 and 6.27 are rewritten by V (x = 0) = −R1 ⋅ I1 †,†V (x = 1) = R2 ⋅ I 2 (6.29)

The earlier condition leads to the following results of k1 and k2: k1 =

k2 =

{(1 + T2 )exp(Γ ⋅ l) − (1 + T1 )T2} Z0 ⋅ I0 2k 3

{(1 + T2 )T1 exp(Γ ⋅ l) − (1 + T1 )exp(2Γ ⋅ l)} × Z0 ⋅ I 0 (6.30) 2k 3

where k1 = exp(2Γl) − †T1 ⋅ T2 T1 = (R 1 − Z 0 )/(R 1 + Z 0 ) : reflection coefficient to the left at node 1 T2 = (R 2 − Z 0 )/(R 2 + Z 0 ) : reflection coefficient to the left at node 2 Equation 6.30 agrees with those given in [37] and the CIGRE Guide [39], although there exist errors in derivation of the formula in the appendix of the Guide. Substituting Equation 6.28 into 6.24 and rewriting as a function of V1 and I1, the following equation is obtained:

Vx = cosh(Γ ⋅ x) ⋅†V1 − Z0 ⋅ sinh(Γ ⋅ x) ⋅ ( I1 − I 0 ) = Ax ⋅ V1 − Bx ⋅ ( I1 − I 0 ) (6.31) I x − I 0 = −Y0 sinh(Γ ⋅ x) ⋅†V1 + cosh(Γ ⋅ x) ⋅ ( I1 − I 0 ) = −Cx ⋅ V1 + Dx ⋅ ( I1 − I 0 )

6.A.4 Approximate Formulas for Impedance and Admittance

1. Conductor internal impedance: See, Equation 1.7 2. Earth-return impedance of an underground cable [45,64]

466


  µ    S  2(hi + h j ) Z eij = jω  0  ln  ij  + − 0.077  [Ω /m] (6.32)  2π    dij  3he   where Sij = (hi + h j + 2he )2 + yij2 , dij = (hi − h j )2 + yij2 dij = r1 = r2 hi, hj: buried distance between cable i and j yij: separation distance between cable i and j he = ρe /jωµ 0 : complex penetration depth ρe: earth resistivity, μe: free space permeability for earth

3. Mutual impedance between overhead and underground cables [65]  µ   Sij*  Z m = jω  0  ⋅ ln   [Ω/m] (6.33)  2π   dij*  where Sij* = (hi − h j + 2he )2 + yij2 †,† dij* = (hi − h j )2 + yij2 4. Pipeline admittance Y = G + j ωC =

jω2πε* [S/m] (6.34) ln(r3 /r2 )

where ε* = εi + 1/jωρi: complex permittivity εi: coating permittivity, ρi: coating resistivity

6.A.5 Accurate Solutions for Two-Cascaded Sections

I1 =

(m1I 01 + n1I 02 ) (m′ I + n′2 I 02 ) , I ′2 = 2 01 , K K

I2 =

(m2 I 01 + n2 I 02 ) (m I + n3 I 02 ) , I 3 = 3 01 K K

m1 = {R2R3 ( A1C2 + C1 A2 ) + (R2 + R3 )A1 A2 + (R2C1 + A1 )B2 }

× (B1 + R1 − R1 A1 )/K K + 1 − A1


467

n1 = R2 {R3C2 + A2 )B4 + (R3 A2 + B2 )(1 − A2 )}/K m2′ = {(R2 + R3 )A2 + B2 + R2R3C2 } (B1 + R1 − R1 A1 )/K

n2′ = R2 (R1C1 + A1 ){(R3C2 + A2 )B2 + (R3 A2 + B2 )(1 − A2 )}/K

m2 = R2 (R3C2 + A2 )(B1 + R1 − R1 A1 )/K

n2 = {R1 A1 + B1 ) + R2 (R1C1 + A1 )}{(R3C2 + A2 )B2 + (R3 A2 + B2 )(1 − A2 )}/K

m3 = R2 (B1 + R1 − R1 A1 )/K

n3 = (R1 A1 + B1 ){B2 − R2 (1 − A2 )} + R2B2 (R1C1 + A1 ) /K

K = (R1 A1 + B1 ){(R3 A2 + B2 ) + R2 (R3C2 + A2 )} + R2 (R 1 C1 + A1 )(R3 A2 + B2 )

Considering the characteristic of hyperbolic functions, for example, B1C2 = C1B2 and A1B2 + B1 A2 = Z0 sinh θ, the earlier equations are rewritten in the form of the hyperbolic functions. For example, K = (R1R2 + R2R3 + R3 R1 )cosh θ1 ⋅ cosh θ2 + R1Z0 cosh θ1 ⋅ sinh θ2 + R3 Z0 sinh θ1 ⋅ cosh θ2 + (Z02 + R1R2 + R2R3 )sinh θ1 ⋅ sinh θ2

+ R2 (Z0 + R1R3 /Z0 )sinh θ

where θ1 = Γ · x1, θ2 = Γ · x2, θ = Γ (x1 + x2) = Γ · l The earlier solutions become identical to those in Equation 6.16 when the approximation in Equation 6.7 is adopted.

References 1. Working Group (Chair: Agematsu, S.) Japanese Electrotechnical Research Association. 2002. Technologies of countermeasure against surges on protection relays and control systems. ETRA Report 57 (3) (in Japanese). 2. Matsumoto, T., Y. Kurosawa, M. Usui, K. Yamashita, and T. Tanaka. 2006. Experience of numerical protective relays operating in an environment with high-frequency switching surges in Japan. IEEE Trans. Power Deliv. 21(1):88–93. 3. Agematu S. et al. 2006. High-frequency switching surge in substation and its effects on operation of digital relays in Japan. CIGRE 2006, General Meeting, Paris, France, Paper C4-304.

468


4. Smith B. and B. Standler. 1992. The effects of surge on electronic appliances. IEEE Trans. Power Deliv. 7(3):1275. 5. Imai, Y., N. Fujiwara, H. Yokoyama, T. Shimomura, K. Yamaoka, and S. Ishibe. 1993. Analysis of lightning overvoltages on low voltage power distribution lines due to direct lightning hits to overhead ground wire. IEE Japan Trans. PE 113-B: 881–888. 6. Kawahito, M. 2001. Investigation of lightning overvoltages within a house by means of an artificial lightning experiment. R&D News Kansai Electric Power: 32–33. 7. Nagai, Y. and H. Sato. 2005. Lightning surge propagation and lightning damage risk across electric power and communication system in residential house. IEICE Japan. Research Meeting. Tokyo, Japan, EMC-05-18. 8. Hosokawa, T., S. Yokoyama, and T. Yokota. 2005. Study of damages on home electric appliances due to lightning. IEE Japan Trans. PE. 125-B (2): 221–226. 9. Hosokawa, T., S. Yokoyama, and M. Fukuda. 2009. Trend of damages on home appliances due to lightning and future problems, IEEJ Trans. PE 129-B (8): 1033–1038. 10. Ametani, A., H. Motoyama, K. Ohkawara, H. Yamakawa, and N. Sugaoka. 2009. Electromagnetic disturbances of control circuits in power stations and substations experienced in Japan. IET GTD 3 (9): 801–815. 11. Sonoda, T., Y. Takeuchi, S. Sekioka, N. Nagaoka, and A. Ametani. 2003. Induced surge characteristics from a counterpoise to an overhead loop circuit. IEEJ Trans. PE 123 (11):1340–1349. 12. Ametani, A. 2006. EMTP study on electro-magnetic interference in low-voltage control circuits of power systems. EEUG 2006, Dresden, Germany. Paper D-3 (EEUG-Proc.), pp. 24–27. 13. Ametani, A., T. Goto, S. Yoshizaki, and H. Motoyama. 2006. Switching surge characteristics in gas-insulated substation. UPEC 2006, Newcastle, U.K. Paper 12–19. 14. Ametani, A., T. Goto, N. Nagaoka, and H. Omura. 2007. Induced surge characteristics on a control cable in a gas-insulated substation due because of switching operation. IEEJ Trans. PE 127(12):1306–1312. 15. International Electrotechnical Commission 2001. Standard for Electromagnetic Compatibility, IEC-61000-4. 16. Japanese Electrotechnical Commission. Test voltage for low-voltage control circuits in power stations and substations. JEC-0103-2004. IEE Japan (in Japanese). 17. IEEJ WG. 2002. The fact of lightning disturbances in a highly advanced ICT society and the subject to be investigated. IEEJ Tech. Report 902. 18. IEEE. 1980. Guide for Surge Voltages in Low-Voltage AC Power Circuits. ANSI/ IEEE C62.41. 19. Murota, N. 1993. Characteristic of the lightning surge suppressors at low voltage responses. IEICE. Japan Research Meeting, Tokyo, Japan, EMCJ93-72. 20. Ideguchi, T. and M. Hatori. 1993. Measures for lightning protection of telecommunication equipment in the premises. IEICE Japan Trans. 13(1):16. 21. Study report on insulation management of low voltage circuits. 1993. J. Elect. InstIn. Eng. Japan 13 (11):1165. 22. Ametani, A., K. Matsuoka, H. Ohmura and Y. Nagai. 2009. Surge voltages and currents into a customer due to nearby lightning. Electric Power Syst. Res. 79:428–435.


469

23. Yokoyama, S. and H. Taniguchi. 1997. The third cause of lightning faults on distribution lines. IEE Japan Trans. PE 117-B (10):1332–1335. 24. Ametani, A., K. Hashimoto, N. Nagaoka, H. Omura, and Y. Nagai. 2005. Modeling of incoming lightning surges into a house in a low-voltage distribution system. EEUG 2005. Warsaw, Poland. 25. Nagai, Y. and N. Fukusono. 2004. Lightning surge propagation on an electric power facility connected with feeder lines from a pole transformer. KEPCO Research Committee of Insulation Condition Technologies, Osaka, Japan. 26. Nagai, Y. 2005. Lightning surge propagation into a model house from various places, in KEPCO Research Committee of Insulation Condition Technologies. Osaka,Japan. 27. Ametani, A., K. Kasai, J. Sawada, A. Mochizuki, and T. Yamada. 1994. Frequencydependent impedance of vertical conductors and a multiconductor tower model. IEE Proc. GTD 141 (4):339–345. 28. Ametani, A., K. Shimizu, Y. Kasai, and N. Mori. 1994. A frequency characteristic of the impedance of a home appliance and its equivalent circuit. IEE Japan. Annual Conference 1405, Tokyo, Japan. 29. Soyama, D., Y. Ishibashi, N. Nagaoka, and A. Ametani. 2005. Modeling of a buried conductor for an electromagnetic transient simulation. ICEE 2005. Kunming, China. SM1-04. 30. Nayel, M. 2003. A study on transient characteristics of electric grounding systems. PhD thesis, Doshisha University, Kyoto, Japan. 31. Scott Meyer, W. 1982. EMTP Rule Book. B. P. A. Portland, Ore. 32. Mozumi, T., T. Ikeuchi, N. Fukuda, A. Ametani, and S. Sekioka. 2002. Experimental formulas of surge impedance for grounding lead conductors in distribution lines. IEEJ Trans. PE 122-B (2): 223–231. 33. Asakawa, S. et al. 2008. Experimental study of lightning surge aspect for the circuit mounted distribution and telecommunication and customer systems. CRIEPI Research Report H07011. 34. IEEJ WG (Covenor Ametani, A.). 2008. Numerical transient electromagnetic analysis method. IEEJ. ISBN 978-4-88686-263-1. 35. Carson, J. R. 1926. Wave propagation in overhead wires with ground return. Bell System Tech. J. 5: 539–554. 36. Sunde, E. D. 1951. Earth Conduction Effect in Transmission System. New York: Wiley. 37. Taflov, A. and J. Dabkowski. 1979. Prediction method for buried pipeline voltages due to 60 Hz AC inductive coupling, Part I: analysis’. IEEE Trans. Power App. Syst. 98(3):780–787. 38. Wedepohl, L.M. 1963. Application of matrix methods to the solution of traveling wave phenomenon in polyphase systems. Proc. IEE 110 (12):2200–2212. 39. CIGRE WG.36.02. 1995. Guide on the Influence of High Voltage AC Power System on Metallic Pipeline. Paris, France: CIGRE Publication 95. 40. Sakai, H. 1971. Induction Interference and Shielding. Tokyo, Japan: Nikkan Kogyo Pub. 41. Rickets, L. W., S. E. Bridges, and S. Mileta. 1976. EMP Radiation and Protective Techniques. New York: Wiley. 42. Degauque, P. and J. Homelin. 1993. Electromagnetic Compatibility. Oxford: Oxford Univ. Press. 43. Paul, C. R. 1994. Analysis of Multiconductor Transmission Lines. New York: Wiley.

470


44. Koike, T. 1995. Transmission and Distribution Engineering. Tokyo, Japan: Yokohama Publication Company (in Japanese). 45. Ametani, A. 1990. Distributed-Parameter Circuit Theory. Tokyo, Japan: Corona Pub. Co. 46. Dommel, H. W. 1986. EMTP Theory Book. B.P.A. Portland, Ore. 47. Tesche, F. M., M. V. Ianoz, and T. Karlsson, 1997. EMC Analysis Methods and Computational Models. New York: Wiley. 48. IEE Japan WG (Covenor Ametani, A.). 2002. Power system transients and EMTP analysis. IEE Japan Technical Report 872. Tokyo (in Japanese). 49. Harrington, R. F. 1968. Field Computation by Moment Methods. New York: Macmillan Company. 50. Uno. T. 1998. Finite Difference Time Domain Method for Electromagnetic Field and Antenna. Tokyo, Japan: Corona Pub. Co. (in Japanese). 51. IEE Japan WG (Covenor Ametani, A.). 2006. Recent trends of power system transient analysis—a numerical electromagnetic analysis. J. IEE Jpn. 126(10):654–673 (in Japanese). 52. Frazier. M.J. 1984. Power line induced AC potential on natural gas pipelines for complex rights-off-way configurations. EPRI. Report EL-3106. AGA. Cat. L51418. 53. Dawalibi, F. P. and R. D. Southey. Analysis of electrical interference from power lines to gas pipelines. Part 1 Computation methods, IEEE Trans. Power Deliv. 1989, 4(3):1840–1846, Part 2, 1990. 5 (1):415–421. 54. Southey, R. D., F. P. Dawalibi and W. Vukonichi. 1994. Recent advances in the mitigation of AC voltages occurring in pipelines located close to electric transmission lines. IEEE Trans. Power Deliv. 9 (2):1090–1097. 55. Dawalibi, F. P. and F. Dosono. 1993. Integrated analysis software for grounding EMF and EMI. IEEE Comput. Appl. Power 6 (2):19–24. 56. Haubrich, H. J., B. A. Flechner and W. A. Machsynski. 1994. A universal model for the computation of the electromagnetic interference on earth return circuits. IEEE Trans. Power Deliv. 9 (3):1593–1599. 57. Safe Eng. Service & Tec. HIFREQ User Manual. 2002. Montreal, CA. 58. Chirstoforidis, G. C., D. P. Labridis, and P. S. Dokopoulos. 2005. Hybrid method for calculating the inductive interference caused by faulted power lines to nearby pipelines. IEEE Trans. Power Deliv. 20 (2):1465–1473. 59. Ametani, A., J. Kamba, and Y. Hosokawa. 2003. A simulation method of voltages and currents on a gas pipeline and its fault location. IEE Jpn. Trans PE 123 (10):1194–1200 (in Japanese). 60. Isogai, H., A. Ametani, and Y. Hosakawa. 2006. An investigation of induced voltages to an underground gas pipeline from an overhead transmission line. IEE Jpn. Trans. PE 126 (1):43–50 (in Japanese). 61. Boker, H. and D. Oeding. 1996. Induced voltage in pipelines on right-of-way to high voltage lines. Elektrizitatswirtschaft 65:157–170. 62. Ametani, A. 2008. Four-terminal parameter formulation of solving induced voltages and currents on a pipeline system. IET Sci. Meas. 2. Technol. 2(2):76–87. 63. Ametani, A. 1994. EMTP Cable Parameters Rule Book. B. P. A. Portland, Ore. 64. Wedepohl, L. M. and D. J. Wilcox. 1973. Transient analysis of underground power transmission systems. Proc. IEE 120123:253–260. 65. Ametani, A., T. Yoneda, Y. Baba, and N. Nagaoka. 2009. An investigation of earth-return impedance between overhead and underground conductors and its approximation. IEEE Trans. EMC 51(3):860–867.

7 Problems and Application Limits of Numerical Simulations Because the electromagnetic transients program (EMTP) is based on a circuit theory assuming transverse electro-magnetic (TEM) mode propagation, it cannot give an accurate solution for a high-frequency transient which involves non-TEM mode propagation. Also, the EMTP cannot deal with a circuit of unknown parameters. On the contrary, a numerical electromagnetic analysis (NEA) method can deal with a transient associated with both TEM and non-TEM mode propagation. Furthermore, it requires no circuit parameter. However, it results in numerical instability if the analytical space, the boundary conditions, the cell size, etc., are not appropriate. Also, it requires a large amount of computer resources, and existing codes are not general enough to deal with various types of transients, especially in a large network.

7.1 Problems of Existing Impedance Formulas Used in Circuit Theory–Based Approaches 7.1.1 Earth-Return Impedance 7.1.1.1 Carson’s Impedance [1] The reason why Carson’s impedance is very popular and has been widely used is simply due to its asymptotic expression. During the early days of computing, the calculation of Pollaczek’s infinite integral [2] was very hard because of the limited computation capability. Therefore, Carson’s asymptotic formula was the only possible way to evaluate the earth-return impedance [3]. However, the asymptotic expression inherently necessitates formulas for a small variable, that is, a low frequency, and for a large variable, and this results in a discontinuity of the calculated impedance as a function of frequency. Also, the accuracy in the boundary region is not high enough. The same is true with Schelkunoff’s formula for internal impedance of a conductor [4]. Nowadays, the advancement in computing capabilities makes it possible to calculate an infinite integral, and various methods of evaluating Pollaczek’s 471

472


impedance have been proposed. A typical example is the work by Noda [5]. This author, however, doubts Pollaczek’s formula itself. 7.1.1.2 Basic Assumption of the Impedance Pollaczek’s and Carson’s formulas were derived under the assumption that

length x height h radius r (7.1)

It should be noted that most formulas of capacitance and inductance of conductors given in textbooks are based on the aforesaid condition. It is easily confirmed that any capacitance formula gives an erroneously large value when the radius reaches the height. Correspondingly, the inductance of an infinite conductor becomes larger than that of a real finite conductor [6,7]. Furthermore, the formulas neglect displacement currents, that is,

1 ωε e ρe

or

f

1 (7.2) 2πε eρe

where ρe is the earth resistivity εe is the permittivity ω = 2πf For example, the applicable range of a frequency in the case of ρe = 1000 Ω-m and εe = ε0 is given by

f 18 MHz or t 50 ns

Even in the case of ρe = 100 Ω-m, a transient of 10 ns time region cannot be simulated by Pollaczek’s and Carson’s impedance [8–11]. It should be noted that most of the frequency-dependent line models are not applicable to the aforesaid cases, because those models are based on Pollaczek’s and Carson’s impedances. Under conditions where Equations 7.1 and 7.2 are not satisfied, only Kikuchi’s and Wedepohl’s impedance formulas are applicable at present [8–10]. This requires a far advanced numerical integration than that applied to Pollaczek’s formula. 7.1.1.3 Nonparallel Conductor Pollaczek’s and Carson’s impedances are for a horizontal conductor. In reality, there are a number of nonhorizontal conductors such as vertical and inclined. Although many papers have been published on the impedance of the vertical conductor such as a transmission tower, it is still not clear if the

Problems and Application Limits of Numerical Simulations

473

proposed formulas are correct. An empirical formula in Ref. [12] is almost identical to an analytical formula [13], which agrees quite well with measured results. However, the analytical formula requires further investigation if the derivation is correct. Impedance formulas for inclined and nonparallel conductors have been proposed in Refs. [6,7,14]. Since the formulas have been derived by the idea of complex penetration depth [15] using Neumann’s inductance formula, those require a further theoretical analysis. 7.1.1.4 Stratified Earth Earth is stratified, as is well known, and its resistivity varies significantly at the top layer depending on the weather and climate. The earth-return impedance of an overhead conductor above the stratified earth was derived in Ref. [16], and the stratified-earth effect was investigated in Ref. [17]. The stratified-earth effect might be far more significant than the accurate evaluation of the homogenous earth-return impedance of Pollaczek and Carson, and this requires a further investigation. 7.1.1.5 Earth Resistivity and Permittivity Earth resistivity, as mentioned earlier, is quite weather/climate dependent. The resistivity after rain is lower than that measured during dry days. Also, it may be frequency dependent. The frequency dependence of the earth permittivity might be far more significant than that of the earth resistivity. Furthermore, water (H2O), which is a dominant factor of the earth permittivity, is extremely temperature dependent [18]. As a result, an error due to uncertainty of the earth resistivity and permittivity might be far greater than that due to the incompleteness of the earth-return impedance derived by Carson and Pollaczek. This fact should be reminded as a physical reality and engineering practice. 7.1.2 Internal Impedance 7.1.2.1 Schelkunoff’s Impedance Schelkunoff’s impedance was derived under the condition that a conductor was in a free space corresponding to Equation 7.1. Therefore, the impedance is not applicable to finite-length conductors with proximity. This fact suggests that an internal impedance of the finite length with the proximity effect is to be developed. 7.1.2.2 Arbitrary Cross-Section Conductor Schelkunoff’s impedance assumes that a conductor is circular or cylindrical. In reality, there exist many conductors of which the cross-section is not

474


circular or cylindrical. Reference [19] derived the internal impedance of a conductor with an arbitrary cross-section, which has been implemented into the EMTP Cable Parameters program [20]. Reference [21] shows an approximation of a conductor with T or hollow rectangular shape by a cylindrical shape conductor. Although the internal impedance of a conductor with an arbitrary cross-section can be accurately evaluated by a finite-element method of numerical calculation, it requires a lot of time and memory. Either an analytical formula or an efficient numerical method needs to be developed. 7.1.2.3 Semiconducting Layer of Cable It is well known that there exists a semiconducting layer on the surface of a cable conductor, which occasionally shows a significant effect on a cable transient. The impedance of the semiconducting layer was derived in Ref. [22] and may be implemented into a cable-impedance calculation. It should be noted that the admittance of the semiconducting layer is far more important than the impedance, from the viewpoint of a transient analysis. 7.1.2.4 Proximity Effect The significance of the proximity effect on conductor impedance is well known, and there are a number of papers that derive a theoretical formula of the impedance and admittance [23–29] and discuss the impedance variation due to the proximity based on numerical simulations [30–33]. The proximity effect might be very important in a steady-state power system performance from the viewpoint of power loss, and some measured results in a power frequency were published [34–37]. It has been pointed out that the proximity effect is also significant in a transient state for a surge waveform is noticeably distorted by an increase of a conductor resistance due to the proximity effect. Unfortunately, there exists almost no measured data investigating the proximity effect on a transient [33]. There exists a formula that considers the proximity or the eccentricity of a conductor enclosed within a conducting-pipe enclosure [24], that is, a pipetype cable [20,38]. However, there is no formula that considers the proximity between two conductors above the earth. 7.1.3 Earth-Return Admittance The earth-return impedance has been well discussed, and its effect on the wave-propagation characteristic and the transient waveform is well known, as is clear from a number of publications. The earth-return admittance [8,9,39,40], however, is neglected in most studies on wave propagation and surge characteristics, and its significant effect is not well realized [8,9,40–43].


475

It has been pointed out in Refs. [8,9,40,43] that the attenuation starts to decrease at a critical frequency, which is inversely proportional to the earth resistivity and the conductor height. This phenomenon is caused by a negative conductance, and corresponds to transition between TEM mode propagation called “earth-return wave” and transverse magnetic (TM) mode propagation called “surface wave,” as discussed by Kikuchi in 1957. When the earth-return admittance is neglected as usual, the attenuation increases monotonously as frequency increases. The wave-propagation velocity and the characteristic impedance become greater when the earth-return admittance is considered. A study on the earth-return admittance might be another challenging and fruitful field of a transient analysis including transition between TEM, TM, and TE modes of propagation [44].

7.2 Existing Problems in Circuit Theory–Based Numerical Analysis 7.2.1 Reliability of a Simulation Tool Quite often a problem appears unexpectedly from a user, but not from developers of a simulation tool, and it is hard for the developers to predict the problem at the development stage. The problem is dependent quite often on the user’s misuse of the tool. Therefore, reliability and severity tests of a simulation tool become very significant. For example, the EMTP Cable Constants had taken nearly 10 years to carry out the reliability and severity tests with more than tens of thousands of cases. It should be noted that the reliability of a tool, that is, the probability of trouble occurrence, is proportional to the number of elements, that is, the number of subroutines and options, even though each element keeps very high reliability. Also, the input data often cause numerical instability when the data physically do not exist. This trouble relates to the assumption of formulas adopted in the simulation tool as explained in the previous section. To avoid such trouble, a “KILL CODE” is prepared in the EMTP. The kill code is to judge whether the input data are beyond the assumption and the limit. It might be noteworthy that nearly half of the EMTP codes are kill codes. This might be considered in another simulation tool. 7.2.2 Assumption and Limit of a Simulation Tool It should be noted that most of the existing or well-known formulas of conductor impedances and admittances are derived based on the assumption of an infinity long conductor. The frequency of an interested electrical phenomenon is increasing year by year corresponding to the advancement in measuring equipment. For example, the sampling frequency of an oscilloscope, which is 1 GHz today, was approximately 10 MHz 10 years ago. The length is inversely

476


proportional to the frequency, and therefore it becomes necessary to deal with a transient on a 1 m conductor, of which the natural resonant frequency is in the order of 100 MHz. Then, Schelkunoff’s, Pollaczek’s, and Carson’s impedances adopted in any circuit theory–based simulation tool such as the EMTP may not be applied [3]. The earlier-explained assumption and the limit should be clearly explained in a rule book of a simulation tool, and the kill codes corresponding to the limit of the impedance and the admittance should be prepared in the tool. The aforementioned problem often appears when the user adopts a commercial software, unless a developer or a user group gives a guide for the usage. Even in the case of a publicized simulation tool such as the EMTP, it happened many times. To avoid the problem, we have to realize that we are electrical engineers. The best solution that overcomes the aforementioned problem is the physical understanding of the phenomenon to be simulated. That is engineering. We are not computer engineers, nor IT engineers. 7.2.3 Input Data Corresponding to what is mentioned in the earlier section, a user of a simulation tool should be careful of input data. Quite often, input data beyond the assumption and the limit of the tool are used, and the user complains that the tool gives erroneous output—this is the author’s experience as a developer of the original EMTP since 1976. At the same time, both the user and the developer should recognize that there are a number of uncertain physical, typically the earth resistivity parameters, which vary along a transmission line and also along the depth of the earth [16,17]. The stratified-earth effect on a transient may be far more influential than the accuracy of numerical calculations of Pollaczek’s and Carson’s earth-return impedance, assuming a homogenous earth. It is interesting to state the fact that the stratified-earth option of the EMTP Cable Constants has never been used since 1978. Also, stray capacitances and residual inductances of a power apparatus are, in general, not available from a manufacturer. The same is the case with the nonlinear characteristic of the apparatus, and the resistivity and permittivity of a cable insulator and a semiconducting layer [18].

7.3 Numerical Electromagnetic Analysis for Power System Transients The numerical electromagnetic analysis (NEA) method [45–50] is becoming one of the most promising approaches to solve transient phenomena that are very hard to be solved by existing circuit theory–based simulation tools such as the EMTP. Existing circuit theory–based approaches cannot solve a three-dimensional (3D) transient and a transient involving sphere-wave


477

propagation and a scattered field, such as a transient across an archon, a wavefront transient at a transmission tower due to lightning, and the voltage and current at the corner or across a spacer of a gas-insulated bus due to a switching surge. Also, the circuit theory–based approach has the difficulty of solving a transient in a complex medium, such as the transient on a grounding electrode and that on a semiconducting layer of a cable. Furthermore, the circuit theory approach cannot be applied if circuit parameters are not known. NEA can solve such problems, because it calculates Maxwell’s equation directly. A working group of the IEE Japan was founded in April 2004, and was carrying out an investigation on NEA and its application examples. The results derived by the working group were published as a book from the IEE Japan [49]. Also, CIGRE WG C4. 501 was established [50] in 2009, and a CIGRE technical brochure (TB) has been completed and will be published soon. The NEA method is powerful to deal with power system transients, for example, in the following subjects: • Surge characteristics of overhead transmission-line towers • Surge characteristics of vertical grounding electrodes and horizontally-placed square-shape grounding electrodes • Surge characteristics of air-insulated substations • Lightning-induced surges on overhead distribution lines • Surge characteristics of a wind-turbine tower struck by lightning and its inside transient magnetic field • Very fast transients in gas-insulated switchgears • Three-dimensional electromagnetic-field analysis The details of NEA have been explained in Chapter 5. In summary, NEA methods can provide better accuracy in comparison with simulation results obtained using circuit theory–based approaches. However, as large computation resources are, in general, required, NEA methods can be considered useful tools to set reference cases and study specific problems. Also, a perfect conductor assumption in a finite-difference time-domain (FDTD) method, for example, results in a difficulty to analyze TEM, TM, and TE transition of wave propagation along a lossy conductor above a lossy earth [8,9,43,44].

References 1. Carson, J. R. 1926. Wave propagation in overhead wires with ground return. Bell Syst. Tech. J. 5:539–554.

478


2. Pollaczek, F. 1926. Uber das Feld einer unendlich langen wechselstromdurchflossenen Einfachleitung. ENT 9(3):339–359. 3. Dommel, H. W. 1986. EMTP Theory Book. Portland, OR: B.P.A. 4. Schelkunoff, S. A. 1934. The electromagnetic theory of coaxial transmission line and cylindrical shields. Bell Syst. Tech. J. 13:532–579. 5. Noda, T. 2006. Development of accurate algorithms for calculating groundreturn and conductor-internal impedances CRIEPI Report H05003 by Central Research Institute of Electric Power Industries in Japan (CRIPEI Tokyo) Report H05003. 6. Ametani, A. and A. Ishihara. 1993. Investigation of impedance and line parameters of a finite-length multiconductor system. Trans. IEE Jpn. 113-B(8):905–913. 7. Ametani, A. and T. Kawamura. 2005. A method of a lightning surge analysis recommended in Japan using EMTP. IEEE Trans. Power Deliv. 20(2):867–875. 8. Kikuchi, H. 1955. Wave propagation on the ground return circuit in high frequency regions. J. IEE Jpn. 75(805):1176–1187. 9. Kikuchi, H. 1957. Electro-magnetic field on infinite wire at high frequencies above plane-earth. J. IEE Jpn. 77:721–733. 10. Wedepohl, L. M. and A. E. Efthymiais. 1978. Wave propagation in transmission line over lossy ground-A new complete filed solution. IEEE Proc. 125(6):505–510. 11. Ametani, A., T. Yoneda, Y. Baba, and N. Nagaoka. 2009. An investigation of earth-return impedance between overhead and underground conductors and its approximation. IEEE Trans. EMC 51(3): 860–867. 12. Hara, T., O. Yamamoto, M. Hayashi, and C. Uenosono. Empirical formulas of surge impedance for single and multiple vertical conductors. Trans. IEE Jpn. 110-B:129–136. 13. Ametani, A., Y. Kasai, J. Sawada, A. Mochizuki, and T. Yamada. 1994. Frequencydependent impedance of vertical conductors and a multiconductor tower model. IEE Proc. Generat. Transm. Distrib. 141(4):339–345. 14. Ametani, A. 2002. Wave propagation on a nonuniform line and its impedance and admittance. Sci. Eng. Rev. Doshisha Univ. 43(3):135–147. 15. Deri, A. et al. 1981. The complex ground return plane: A simplified model for homogeneous and multi-layer earth return. IEEE Trans. Power App. Syst. 100(8): 3686. 16. Nakagawa, M., A. Ametani, and K. Iwamoto. 1973. Further studies on wave propagation in overhead lines with earth return—Impedance of stratified earth. Proc. IEE 120(2):1521–1528. 17. Ametani, A. 1974. Stratified effects on wave propagation—Frequency-dependent parameters. IEEE Trans. Power App. Syst. 93(5):1233–1239. 18. Ametani, A. 2000. Problems and countermeasures of cable transient simulations. EMTP J. 5:3–11. 19. Ametani, A. and I. Fuse. 1992. Approximate method for calculating the impedances of multi conductors with cross-section of arbitrary shapes. Elect. Eng. Jpn. 111(2):117–123. 20. Ametani, A. 1994. Cable Parameters Rule Book. Portland, OR: B.P.A. 21. Ametani, A., N. Nagaoka, R. Koide, and T. Nakanishi. 1999. Wave propagation characteristics of iron conductors in an intelligent building. Trans. IEE Jpn. B-120(1): 271–277. 22. Ametani, A., Y. Miyamoto, and N. Nagaoka. 2004. Semiconducting layer impedance and its effect on cable wave-propagation and transient characteristics. IEEE Trans. Power Deliv. 19(4):523–531.


479

23. Tegopoulos, J. A. and E. E. Kriezis. 1971. Eddy current distribution in cylindrical shells of infinite length due to axial currents, Part II—Shells of infinite thickness. IEEE Trans. Power App. Syst. 90:1287–1294. 24. Brown, G. W. and R. G. Rocamora. 1976. Surge propagation in three-phase pipe-type cables, Part I—Unsaturated pipe. IEEE Trans. Power App. Syst. 95:88–95. 25. Dugan, R. C. et al. 1977. Surge propagation in three-phase pipe-type cables, Part II—Duplication of filed test including the effects of neutral wires and pipe saturation. IEEE Trans. Power App. Syst. 96:826–833. 26. Schinzinger, R. and A. Ametani. 1978. Surge propagation characteristics of pipe enclosed underground cables. IEEE Trans. Power App. Syst. 97:1680–1687. 27. Dokopoulos, P. and D. Tampakis. 1984. Analysis of field and losses in threephase gas cable with thick walls: Part I. Field analysis. IEEE Trans. Power App. Syst. 103(9):2728–2734. 28. Dokopoulos, P. and D. Tampakis. 1985. Part II Calculation of losses and results. IEEE Trans. Power App. Syst. 104(1):9–15. 29. Poltz, J., E. Kuffel, S. Grzybowski, and M. R. Raghuveer. 1982. Eddy-current losses in pipe-type cable systems. IEEE Trans. Power App. Syst. 101(4):825–832. 30. Fortin, S., Y. Yang, J. Ma, and F. P. Dawalibi. 2005. Effects of eddy current on the impedance of pipe-type cables with arbitrary pipe thickness. ICEE 2005, Gliwice, Poland. Paper TD2-09. 31. Chien, C. H. and R. W. G. Bucknall. 2009. Harmonic calculation of proximity effect on impedance characteristics in subsea power transmission cables. IEEE Trans. Power Deliv. 24(2):2150–2158. 32. Gustavsen, B., A. Bruaset, J. J. Bremnes, and A. Hassel. 2009. A finite element approach for calculating electrical parameters of umbilical cables. IEEE Trans. Power Deliv. 24(4):2375–2384. 33. Ametani, A., K. Kawamura et al. 2013. Wave propagation characteristics on a pipe-type cable in particular reference to the proximity effect. IEE J. High Voltage Eng. Conf. Kyoto, Japan, Paper HV-13-005. 34. Ishikawa, T., K. Kawasaki, and I. Okamoto. 1976. Eddy current losses in cable sheath (1). Dainichi Nihon Cable J. 61:34–42. 35. Ishikawa, T., K. Kawasaki, and O. Okamoto. 1977. Eddy current losses in cable sheath (2). Dainichi Nihon Cable J. 62:21–64. 36. Kawasaki, K., M. Inami, and T. Ishikawa. 1981. Theoretical consideration on eddy current losses on non-magnetic and magnetic pipes for power transmission systems. IEEE Trans. Power App. Syst. 100(2):474–484. 37. Mekjian, A. and M. Sosnowski. 1983. Calculation of altering current losses in steel pipe containing power cables. IEEE Trans. Power App. Syst. 102(2): 382–388. 38. Ametani, A. 1980. A general formulation of impedance and admittance of cables. IEEE Trans. Power App. Syst. 99(3):902–910. 39. Wise, W. H. 1948. Potential coefficients for ground return circuits. Bell Syst. Tech. J. 27:365–371. 40. Nakagawa, M. 1981. Further studies on wave propagation along overhead transmission lines: Effects of admittance correction. IEEE Trans. Power App. Syst. 100(7):3626–3633. 41. Rachidi, F., C. A. Nucci, and M. Ianoz. 1999. Transient analysis of multiconductor lines above a lossy ground. IEEE Trans. Power Deliv. 14(1):294–302.

480


42. Hashmi, G. M., M. Lehtonen, and A. Ametani. 2010. Modeling and experimental verification of covered conductors for PD detection in overhead distribution networks. IEE J. Trans. PE 130(7):670–678. 43. Ametani, A., M. Ohe, Y. Miyamoto, and K. Tanabe. 2012. The effect of the earthreturn admittance on wave propagation along an overhead conductor in a highfrequency region. EEUG Proceedings, Zwickau, Germany 1:6–22. 44. Sommerfeld, A. 1964. Partial Differential Equation in Physics. New York: Academic Press. 45. Yee, K. S. 1966. Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propagat. 14(3):302–307. 46. Uno, T. 1998. FDTD Method for Electromagnetic Fields and Antennas. Tokyo, Japan: Corona Pub. Co. 47. Taflove, A. and S.C. Hagness. 2000. Computational Electromagnetics. The FiniteDifference Time-Domain Method. Norwood, MA: Artech House. 48. CRIEPI by Central Research Institute of Electric Power Industry, Tokyo, Japan, 2007. Visual Test Lab. (VSTL). http://cripei.denken.or.jp/jp/electric/ substance/09.pdf 49. IEE Japan WG. Working Group of Numerical Transient Electromagnetic Analysis (Convenor: A. Ametani) 2008. Numerical Transient Electromagnetic Analysis Methods. IEE Japan, ISBN 978-4-88686-263-1. 50. Ametani, A., T. Hoshino, M. Ishii, T. Noda, S. Okabe, and K. Tanabe. 2008. Numerical electromagnetic analysis method and its application to surge phenomena. CIGRE 2008 General Meeting, Paris, France. Paper C4-108.

Power Engineering

POWER SYSTEM

TRANSIENTS Theory and Applications

AKIHIRO AMETANI • NAOTO NAGAOKA YOSHIHIRO BABA • TERUO OHNO As a transient phenomenon can shut down a building or an entire city, transient analysis is crucial to managing and designing electrical systems. Power System Transients: Theory and Applications discusses the basic theory of transient phenomena—including lumped- and distributed-parameter circuit theories— and provides a physical interpretation of the phenomena. It covers novel and topical questions of power system transients and associated overvoltages. Using formulas simple enough to be applied using a pocket calculator, the book presents analytical methods for transient analysis. It examines the theory of numerical simulation methods such as the EMTP (circuit theory–based approach) and numerical electromagnetic analysis. The book highlights transients in clean or sustainable energy systems such as smart grids and wind farms, since they require a different approach than overhead lines and cables. Simulation examples provided include arcing horn flashover, a transient in a grounding electrode, and an induced voltage from a lightning channel.

an informa business www.taylorandfrancisgroup.com

6000 Broken Sound Parkway, NW Suite 300, Boca Raton, FL 33487 711 Third Avenue New York, NY 10017 2 Park Square, Milton Park Abingdon, Oxon OX14 4RN, UK

K16810

Power system transients theory and applications-Ametani Akihiro, Baba Yoshihiro, Nagaoka Naoto, Ohno Teruo.pdf

Recommend Documents