Optimal Tuning of PSS Using EA

OPTIMAL TUNING OF POWER SYSTEM STABILIZERS BASED ON EVOLUTION ALGORITHM

Prepared for Professor Professor K.A. Folly Foll y Associate professor Department Department of Electrical Engineering University of Cape Town

Prepared by Tshina Mulumba Electrical Electrical Engineering Student Department Department of Electrical Engineering University of Cape Town 13th May 2008 Prepared as a prerequisite and in partial fulfilment for the awarding of a Bachelor of Science (BSc) degree in Electrical Engineering at the University of Cape Town

DECALRATION

I, Tshina Mulumba, hereby declare that this thesis project, submitted for the fulfilment of the Bachelor of Science degree in Engineering, is my own work. I have not plagiarised from any sources. References and acknowledgments of sources are given and cited.

Date…..…………………………………………. Signature………………………………………..

DECALRATION

I, Tshina Mulumba, hereby declare that this thesis project, submitted for the fulfilment of the Bachelor of Science degree in Engineering, is my own work. I have not plagiarised from any sources. References and acknowledgments of sources are given and cited.

Date…..…………………………………………. Signature………………………………………..

ACKNOWLEDGMENTS

Firstly, I am grateful to The One who has given me life, supported me through hardship and good time, the source of my strength, The Almighty God. I dedicate this work to Him. I would like to thank my supervisor, Professor K.A. Folly, for his inspirational guidance and encouragement demonstrated throughout my research. He has influenced me to develop particular interests in the Power Systems stability and Control. A special word of thank to Professor J. Greene, for the time and help he gave me by answering all my questions. The biggest thank and gratitude goes to my parents, Mr. and Mrs. Mulumba, for their everlasting love and advices given, their supports and encouragements throughout my studies. I owe a special thank to my Aunts (Ivette and Nadine) and my big sister (Sophie), for their continual supports. My three brothers (Dilan, Glory and Christian), for being a source of inspiration, as well as my little sisters. Last but not least, a special word of gratitude to all my friends who have contributed in completing this report, I could not have done this without you.

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TERMS OF REFERENCE

Associate Professor K.A Folly of the Department of Electrical Engineering, University of Cape Town, initiated this research on the Optimal Tuning of Power System Stabilizer (PSS) using Evolution Algorithm. His specific directives were: •

To review the modelling of a single machine infinite bus (SMIB) equipped with a PSS.

•

To investigate the tuning method of a Power System Stabilizer (PSS) in SMIB system by using specifically the Differential Evolution (a variant of Genetic Algorithm).

•

To develop a program in MATLAB and implement the Differential Evolution (DE)

•

To chose an appropriate objective function for the DE.

•

To analyse and compare the simulations with the CPSS, the simple GA and the DE .

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SYNOPSIS The problem of damping low frequency oscillations in the range of 0.2 – 3 Hz observed in power systems, have been the subject of many researches over the past few years. These low frequencies are mainly caused by a heavy power transmission on weak transmission lines and the exciter’s high gain. The exciter, also known as the automatic voltage regulator (AVR), helps to improve the system voltage during faults conditions. However, the AVR adversely reduces the damping of the system causing oscillations. These oscillations limit the power transmission capability of a network and, sometimes, even cause a loss of synchronism and an eventual breakdown of the entire system [13]. To remediate this problem, the Power System Stabilizer (PSS) is used to improve the system stability by providing supplementary damping. The PSS is mainly constituted of a gain stage K, washout stage T w or high pass filter and a lead – lag compensator T1 – T4. These parameters are tuned at a particular operating condition with conventional techniques such as phase compensation and root locus to compensate for the system’s phase lag. Although many modern control techniques with different structures such as adaptive control have been developed, the conventional lead – lag PSS (CPSS) remains widely used by power industry because of its simple structure and reliability. However, the main problem that faces the CPSS is the nonlinearity of power systems of which the operating conditions change constantly. Under these conditions the CPSS performance becomes inadequate considering that it is only tuned for a particular operating condition. The above problem has led to many researchers to investigate methods to improve the PSS performance over the entire range of operating conditions. In the past 15 years, interests have been focused on the optimization of the PSS parameters to provide adequate performance for all conditions. Hence, many optimizations techniques such as Genetic Algorithms (GAs) have been used to find the optimum set of parameters to effectively tune the PSS. Recently, an optimization technique similar to GA, known as Differential Evolution (DE), has been proposed.

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DE is parallel direct search method that uses a differential mutation scheme and greedy selection process (the better one of new solution and its parents wins the competition) to direct its search toward the prospective regions of search space. DE is a new heuristic approach that present some advantages over GAs: •

Fast convergence

•

Finds true global minimum regardless of the initial parameters

•

Ease of use

•

Efficient memory utilization

•

Lower computational complexity

In this thesis, DE is used to tune the PSS’s parameters by optimizing a frequency domain objective function. The objective function consists of finding, shifting the poorly damped or unstable polesinto the left side of the s – plane (stability plane). The efficacy of the method was tested on a single machine infinite bus (SMIB) and compared with GA based PSS (GAPSS) and the CPSS. The results were as follows: •

DEPSS provides better damping than GAPSS and CPSS.

•

DEPSS is more robust than GAPSS and CPSS. As the stressed level increases, DEPSS provides better results.

•

DEPSS is able to restrain all eigenvalues in the desired region, indicate by the relative stability, for most of the operating conditions.

In order to contribute to the research in the area of PSS tuning strategies, the following recommendations were made: •

Investigate the application of DE to tune the PSS using an objective function based on the phase compensation technique and compare with DE based eigenvalue shifting method.

iv

•

Investigate the application of DE to tune the PSS in the area of multimachine system.

•

Investigate the application of other Evolution Algorithm to tune the PSS with faster convergence time.

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TABLE OF CONTENTS Page

DECALRATION ................................................................................................ i ACKNOWLEDGMENTS ................................................................................... i TERMS OF REFERENCE ............................................................................... ii SYNOPSIS ..................................................................................................... iii TABLE OF CONTENTS ................................................................................. vi LIST OF TABLES ............................................................................................ x LIST OF FIGURES ......................................................................................... xi 1

INTRODUCTION ................................................................................ 1

1.1

Subject of the thesis ......................................................................... 1

1.2

Problem definition ............................................................................ 1

1.3

Objectives of the thesis.................................................................... 3

1.4

Scope and Limitation........................................................................ 3

1.5

Thesis outline .................................................................................... 4

2

REVIEW OF POWER SYSTEM STABILITY ...................................... 5

2.1

Small signal stability ........................................................................ 7

2.2

Low Frequency Oscillations in Power System ............................... 8

2.2.1

Local modes .............................................................................................. 8 

2.2.2 

Inter – area modes .................................................................................... 9 

2.3

Power System Stabilizer .................................................................. 9

2.3.1

Conventional Power System Stabilizer structure and design.............. 10 

2.3.2 

Genetic Algorithm based PSS (GAPSS) ................................................ 12 

3

POWER SYSTEM MODELLING ...................................................... 15 vi

3.1

Small Signal Dynamic Modelling ................................................... 15

3.1.1

State – Space representation ................................................................. 15 

3.1.2 

Linearization ............................................................................................ 16 

3.2

Machine modelling ......................................................................... 17

3.2.1

Machine Modelling with Power System Stabilizer ................................ 18 

3.3

Stability analysis ............................................................................. 19

4

GENETIC ALGORITHM ................................................................... 22

4.1

Encoding of individuals ................................................................. 23

4.2

Objective and Fitness functions .................................................... 23

4.3

SGA operators ................................................................................ 24

4.3.1

Selection .................................................................................................. 24 

4.3.2 

Crossover ................................................................................................ 25 

4.3.3 

Mutation ................................................................................................... 25 

4.3.4 

Reinsertion .............................................................................................. 25 

4.3.5 

Convergence and termination ................................................................ 25 

5

DIFFERENTIAL EVOLUTION (DE) .................................................. 27

5.1

Advantage of DE over GA .............................................................. 27

5.2

Reason for using DE ...................................................................... 27

5.3

Population structure ....................................................................... 28

5.4

Initialization ..................................................................................... 29

5.5

Mutation ........................................................................................... 29

5.6

Recombination or crossover ......................................................... 30

5.7

Selection .......................................................................................... 31

6

PARAMETERS SETTING OF GA & DE .......................................... 32

6.1

Objective function .......................................................................... 32

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6.2

PSS tuning approach...................................................................... 35

6.2.1

Simple Genetic Algorithm ...................................................................... 35 

6.2.2 

Differential Evolution (DE) ...................................................................... 36 

6.2.3 

Tuning process ....................................................................................... 37 

7

SIMULATION RESULTS AND DISCUSSIONS ............................... 40

7.1

Power System to be investigated .................................................. 40

7.2

Operating conditions ...................................................................... 40

7.3

PSS parameters .............................................................................. 41

7.3.1

CPSS parameter selection ...................................................................... 41

7.3.2 

DEPSS and GAPSS parameters selection ............................................. 42 

7.4

Simulations results ......................................................................... 43

7.4.1

Eigenvalues analysis .............................................................................. 44 

7.4.2 

Time domain response ........................................................................... 50 

7.4.3 

Robustness tests of PSSs ...................................................................... 52 

8

CONCLUSION .................................................................................. 58

9

RECOMMENDATIONS .................................................................... 59

REFERENCES .............................................................................................. 60 APPENDIX A: SYSTEM EQUATIONS .......................................................... 66 A.1

EQUATIONS OF STATE SQUARE MATRIX VARIABLES ............. 66

A.2

HEFFRON – PHILLIPS DIAGRAM AND CONSTANTS .................. 67

APPENDIX B: TUNING GUIDELINES FOR A CPSS ................................... 69 APPENDIX C: SYSTEM OPERATION .......................................................... 70 C.1

SINGLE MACHINE INFINITE BUS DATA ....................................... 70

C.2

LOAD FLOW REPORTS .................................................................. 71

APPENDIX D: MATLAB CODES & SIMULATION TOOLS .......................... 73

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D.1

SIMULATION TOOLS ...................................................................... 73

Power System Toolbox ............................................................................... 73 Operations .................................................................................................... 73 smibDriver.m ......................................................................................................... 74  Model_maker & Step_testerMod.m ...................................................................... 74  pssOptimizer.m ..................................................................................................... 75  dePSS.m ................................................................................................................ 76  gaPSS.m ................................................................................................................ 76  eigenShiftDE.m and eigenShiftGA.m ................................................................... 77  Program flow chart................................................................................................ 77 

D.2

MATLAB CODES ............................................................................. 78

PSSOPTIMIZER.M ........................................................................................ 78 DEPSS.M....................................................................................................... 80 EIGENSHIFTDE.M ........................................................................................ 82 GAPSS.M ...................................................................................................... 84 EIGENSHIFTGA.M ........................................................................................ 86 APPENDIX E: SIMULATION RESULTS DATA & GRAPHS ........................ 88 E.1

DATA................................................................................................ 88

E.2

GRAPHS .......................................................................................... 94

APPENDIX F: SOFTWARE CD ................................................................... 101

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LIST OF TABLES Table 6.1 SGA parameters .................................................................................... 35 Table 6.2 DE parameters ....................................................................................... 37 Table 7.1 Operating conditions ............................................................................... 41 Table 7.2 CPSS parameters ................................................................................... 42 Table 7.3 Parameters boundaries .......................................................................... 43 Table 7.4 DEPSS and GAPSS parameters ............................................................. 43 Table 7.5 No PSS at minimum condition ................................................................ 44 Table 7.6 No PSS at nominal condition ................................................................... 45 Table 7.7 CPSS at nominal condition...................................................................... 45 Table 7.8 GAPSS at nominal condition .................................................................. 46 Table 7.9 DEPSS at nominal condition .................................................................. 47 Table 7.10 No PSS maximum condition .................................................................. 47 Table 7.11 CPSS at maximum condition ................................................................. 48 Table 7.12 GAPSS at maximum condition .............................................................. 49 Table 7.13 DEPSS at maximum condition ............................................................. 49 Table 7.14 CPSS eigenvalues and Damping ration under robust test .................... 52 Table 7.15 GAPSS eigenvalues and damping ratio under robust test ..................... 53 Table 7.16 DEPSS eigenvalues and damping ratio under robust test .................... 53 Table 7.17 Average PSSs damping ratio ............................................................... 54

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LIST OF FIGURES

Figure 2.1 Power system classification [2, 6] ........................................................... 6 Figure 2.2 CPSS structure ..................................................................................... 10 Figure 3.1 S – plane representation [50] ............................................................... 21 Figure 4.1 Chromosome structure constituted of 2 variables .................................. 23 Figure 4.2 The roulette wheel ................................................................................. 24 Figure 4.3 Single point crossover ........................................................................... 26 Figure 4.4 Mutated individual ................................................................................. 26 Figure 5.1 Differential Evolution cycles ................................................................... 28 Figure 5.2 Differential Evolution: the weighted differential, F ⋅ ( xr1, g to the

−

xr 2,g ) , is added

base vector,  xr 0, g , to produce a mutant, vi , g [48, 49]. ......................... 30

Figure 5.3 A flow chart of DE’s operation and test loop [48].................................... 31 Figure 6.1 Relative stability region on the left hand side of the line at −α  ............ 34 Figure 6.2 Flow chart representation of the tuning process..................................... 39 Figure 7.1: SMIB system ....................................................................................... 40 Figure 7.2: Step response at Nominal condition ..................................................... 50 Figure 7.3: Step response at Maximum condition ................................................... 51 Figure 7.4: Open - loop eigenvalues....................................................................... 55 Figure 7.5: CPSS closed - loop eigenvalues .......................................................... 55 Figure 7.6: GAPSS closed-loop eigenvalues .......................................................... 56 xi

Figure 7.7: DEPSS closed-loop eigenvalues .......................................................... 57 Figure A.9.1 Heffron – Philips 3 rd model of SMIB system with PSS included .......... 67 Figure D.9.2 Program flow chart ............................................................................ 77

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1 INTRODUCTION 1.1 Subject of the thesis This thesis explores the possibility of tuning the Power System Stabilizer (PSS) using the Differential Evolution (DE). The main objective focuses on finding the optimal parameters that ensures a robust system over a wide range of operating conditions.

1.2 Problem definition Small signal disturbances observed on the power system are caused by many factors such as heavy power transmitted over weak tie – line and the effect of fast acting, high gain automatic voltage regulator (AVRs) [6, 7, 52]. The main function of the AVR is to improve the transient stability during faults conditions. However, its high gain and fast acting effect, have an adverse effect on the system damping which is reduced to a negative value [2, 15, 52]. The underdamped

system

exhibits

low

frequency

oscillations

also

known

as

electromechanical oscillations. These oscillations limit the power transfer over the network and if not properly damped, they can grow in magnitude to cause system separation. To counteract the adverse effects of the AVRS, Power system stabilizer (PSS) is used in the auxiliary feedback to provide supplementary damping to the system to damp these low frequency oscillations on the rotor [5]. The PSS, also referred to as conventional PSS (CPSS), is made of gain stage K, a high pass filter and the lead – lag compensators, with T 1 – T4 as time constants. These parameters require fine tuning at a particular set of operating conditions, usually nominal, in order to improve the system damping. As the power system is extremely nonlinear, operating conditions are constantly changing. Therefore, the CPSS’s parameters may not provide adequate performance and may need to be retuned. Thus, finding a set of parameters that guarantee adequate damping and good system performance over the entire range of operating conditions is essential.

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To overcome this problem, several approaches based on modern control theory, such as Optimal control, Variable control and Intelligent control were simulated and tested with satisfactory results. But these stabilizers have been proved to be difficult to implement in real systems [2, 13]. Thus, CPSS remains widely used by power utilities for its simple structure and reliability [13]. Over the past 15 years, interests have been focused on the optimization of the PSS parameters to provide adequate performance for all operating conditions. Hence, many optimizations techniques such as Genetic Algorithms (GAs) have been used to find the optimum set of parameters to effectively tune the PSS. These optimization techniques have demonstrated to be slow when converging toward optimum values. They require complex computation and memory. Recently, an optimization technique similar to GA, known as Differential Evolution (DE), has been proposed. In fact, DE is new heuristic approach that uses differential mutation scheme, to direct its search toward the prospective regions of search space [47, 48], and present some advantages over GAs: •

Fast convergence

•

Finds true global minimum regardless of the initial parameters

•

Ease of use

•

Efficient memory utilization

•

Lower computational complexity

For these reasons, this thesis aims to investigate the application of DE to optimally tune PSS’s parameters in order to ensure robust system performance. The results will be compared to GA based PSS and the CPSS.

2

1.3 Objectives of the thesis The objectives of this research are formulated as follow: •

Review of the system modelling of single machine infinite bus (SMIB) equipped with conventional power system stabilizer (CPSS)

•

Select an appropriate objective function for the DE

•

Design and tune the PSS parameters using DE in MATLAB

•

Check for the robustness of the system over a wide range of operating conditions

•

Simulate and compare the results to the simple GA as well as the CPSS

1.4 Scope and Limitation This thesis examines the application of Differential Evolution (DE) to tune a power system stabilizer with a set of optimum parameters that ensures adequate performance for a robust operation. The PSS is installed on a single machine infinite bus (SMIB), which is implemented in MATLAB using the Power System Toolbox package (PST). The tuning of the PSS with optimal parameters is accomplished with the aid of the DEMAT package [48]. These optimum parameters are obtained by evaluating an eigenvalue based objective function. The DE based PSS performance is compared to the genetic algorithm based PSS as well as the conventional PSS over a wide range of operating conditions. This thesis is limited to the use of PST, DEMAT and GAOT toolbox, to optimize the parameters and simulate the results.

3

1.5 Thesis outline The thesis chapters are outlined as follows: •

Chapter 1 introduces the thesis topic and identifies the problem to be examined.

•

Chapter 2  reviews the power system stability with emphasis on the small signal stability problem and also present the conventional power system stabilizer structure. A review of the relevant work in the area of tuning of PSS is discussed. The present work tuning method is briefly introduced.

•

Chapter 3  reviews the mathematical modelling of a synchronous machine, the linearization of the small signal in a single machine infinite bus system. The system analysis based on the eigenvalues is also discussed.

•

Chapter 4  reviews the basic concepts of the genetic algorithms and the genetic operators.

•

Chapter 5  reviews the Differential Evolution, outlines the main differences with GAs and discusses in depth the DE operators.

•

Chapter 6 describes the implementation of DEPSS and GAPSS. The eigenvalue shift objective function is presented as well as the tuning procedure.

•

Chapter 7  describes the software tools with its main program files used to simulate the operation of the DEPSS.

•

Chapter 8  discusses the results obtained from the simulation and provides a comparative study between the DEPSS, GAPSS and the CPSS.

•

Chapter 9 summarizes the present work and draws conclusions on the proposed tuning method.

•

Chapter 10 covers the scope for the future work in the tuning area of PSSs.

4

2 REVIEW OF POWER SYSTEM STABILITY The stability of power system is one of the most important aspects in electric system operations. It determines whether or not the system can settle down to a new operating point after the occurrence of a disturbance [5 – 7]. Power system stability is defined in [9] as follows: “The ability of an electric power system, for a given initial operating condition, to  regain a state of operating condition after being subject to a physical disturbance, with most system variables bounded so that practically the entire system remains  intact.”  Over the past few decades, power system stability problems have received a great deal of attention. Many studies and techniques have been conducted and developed to help power systems maintain the frequency and the voltage level under any disturbances. These disturbances can occur due to a sudden increase in the load, loss of a generator, switching of a transmission line or a fault, etc [1, 2]. The rapid growth of power demand has been recorded since the advance of industrialization. Subsequently, more power generation infrastructures are needed. On the other hand, severe economical and environmental restrictions are also reinforced to preserve an ecological balance. These restrictions have limited the generation and the expansion of the power systems transmission networks. Consequently, modern power systems are more heavily loaded than before [3 – 5]. These constraints entail the power systems to be operated under intensive stress conditions and near their stability limits. This implies that the tight stability margins imposed on the power systems can be a limiting factor in the transmission of power [1, 3 – 6]. The problem of stability in power systems is classified into three categories [8]: i.

Angle stability – refers to the ability of a power system to maintain synchronism when subject to small and severe disturbances. These perturbations can cause a loss of generating capacity, a system separation or blackout if no proper actions are initiated such as load shading etc.

ii.

Voltage stability  – also referred to as load stability. The Voltage stability is concerned with the ability of a power system to maintain acceptable voltage at all 5

the buses under normal conditions and after perturbations [6, 8 – 10]. Voltage instability can lead to a low voltage profile observed in major parts of the power system. iii.

Frequency stability  – is the ability of a power system to maintain a steady frequency within an acceptable variation range following a disturbance. The Frequency instabilities can lead to a large generation – load imbalance [9].

The above classification provides a convenient system analysis and enhances the understanding of the nature of instability [2, 11]. This classification allows suitable grounds to develop solutions related to the disturbances in power system. The power system stability is further depicted into subclasses represented by the block diagram in Figure 2.1. The highlighted blocks show the areas of direct interest and significance to this research.

Power System Stability

Angle Stability

Voltage Stability

Frequency Stability

Transient Stability

Small disturbance Stability

Small signal Stability

Non-oscillatory instability

Large disturbance Stability

Oscillatory instability

Inter-area modes

Control modes

Local modes

Torsional modes

Figure 2.1 Power system classification [2, 6] 

This thesis emphasizes the Rotor angle stability by paying particular attention to the effects of small signal instability. 6

From the power system stability point of view, power systems are subjected to different types of disturbances. There are small scale disturbances, which occur for the vast majority of time, and large scale for the more severe ones. Therefore, rotor angle stability is subdivided into two categories known as transient stability and small signal stability. A system is said to be transiently stable if it can withstand large disturbances and remains stable after the perturbations. The transient stability problem is mainly concerned with the way the system responds to a severe disturbance such as short circuit on power line. The transient stability is related to t he short term or transient period which is usually limited to the first few seconds following the disturbance [2, 11]. On the other hand, the power system is said to be small signal stable if the generators are able to maintain synchronism with each other after being subject to small disturbances. These disturbances arise with the switching of capacitors, small and gradual generation changes [11] etc. Small signal stability is further discussed in the next point.

2.1 Small signal stability Small signal stability is defined in [14] as follows: “A power system is said to be small signal stable for a particular steady – state  operating condition if, following any small disturbance, it reaches a steady – state  operating condition which is identical or close to the pre – disturbance operating  condition.”  A disturbance is considered to be of small signal if the equations describing the system response can be linearized for the purpose of analysis [6]. The system instability resulting from small disturbances can be of two forms: (i) the steady increase in the rotor angle due to the lack of synchronizing torque , or (ii) the increase in rotor oscillation due to the lack of damping torque [6]. These disturbances are mainly caused by, the exciter’s high gain and the weak links in interconnected power systems [6, 52]. Therefore, the restructuring of the electric power industry as well as the complexity of the networks both contribute to the deterioration of the stability margin in power systems [6, 12 – 13]. Consequently, the systems are more vulnerable to small signal 7

disturbances nowadays than they were before. In fact, many studies on stability of small signals have been conducted to ensure sufficient stability margins in addition to system security and reliability. If the small signal oscillations are not damped properly, they can build up through out the network to cause transient instability [7]. Modern generators are equipped with high gain and fast response exciters. These exciters, also known as Automatic Voltage Regulators (AVR), enhance the transient stability and prevent voltage fluctuation [2, 15, 52]. It achieves the above mentioned by simply increase the synchronising torque which reduces the generator angle and avoid non – oscillatory instability [52] . The AVR, conversely, contributes to low frequency oscillations by decreasing the damping torque to a negative value [16, 52]. These oscillations lead to an unstable condition even without the existence of severe fault [2, 14 – 15].

2.2 Low Frequency Frequency Oscillations in Power System Low frequency oscillations are often observed when large power systems are connected with weak tie – line [18]. In fact, when bulk power is transmitted over long distances and weak transmission lines, oscillations of low frequency in the range of 0.2 to 3 Hz can be detected [14]. The AVRs also contribute to these low frequency oscillations. Large electric power systems usually have poorly damped electromechanical oscillations associated with the rotor angle of the synchronous machines [15]. The insufficient damping of electromechanical dynamics causes oscillations of low frequencies and negative damping to grow in magnitude [16 – 17]. Depending on the system, Low frequency oscillations are often classified into two modes.

2.2.1 Local modes For this mode, oscillations are in the range of 0.8 – 2.0 Hz, caused by one generator swinging against the rest of the system [17].

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2.2.2 Inter – area modes For the inter – area modes, groups of generators in different areas swing against each other with oscillations in the range of 0.2 – 0.7 Hz [17]. These oscillations are usually observed in a large interconnection between power systems with weak tielines. To remediate the small signal instabilities caused by the AVR and other factors, the Power System Stabilizer (PSS) was introduced to stabilize the system and increase the system’s security. PSS is further discussed in the next section.

2.3 Power System Stabilizer One problem that faces power systems nowadays is the low frequency oscillations arising from interconnected systems. Sometimes, these oscillations sustain for minutes and grow to cause system separation. The separation occurs if no adequate damping is available to compensate for the t he insufficiency of the damping torque in the synchronous generator unit [19]. This insufficiency of damping is mainly due to the AVR exciter’s high speed and gain and the system’s s ystem’s loading. In order to overcome the problem, PSSs have been successfully tested and implemented to damp low frequency oscillations [16, 28]. The PSS provides supplementary feedback stabilizing signal in the excitation system [5]. The feedback is implemented in such a way that electrical torque on the rotor is in phase with speed variations [20]. PSS parameters are normally fixed for certain values that are determined under particular operating conditions. Once the system’ operating conditions are changed, PSS may not produce adequate damping [18] into an unstable system. Since PSSs are tuned at the nominal operating point, the damping is only adequate in the vicinity of those operating points. But power systems are highly nonlinear systems, therefore, the machine parameters change with loading and time. The dynamic characteristics also vary at different points [21]. Hence, several approaches based on modern control theory have been applied to design different power system stabilizer structures [5, 22]. This includes optimal control, adaptive control, variable structure control and intelligent control which are further developed in [24 – 26].

9

In [23], S. Panda and N.P. Padhy presented a systematic procedure for modelling and designing of a power system equipped with PSS and Flexible AC transmission system (FACTS)-based controller. Further, Further, they evaluated the impact im pact of the PSS and FACTS – based controller on the power system. Alberto Del Rosso et al  examined in [27] the use of Thyristor Controlled series Capacitors (TCSC) for stability improvement of power systems. An appropriate appropriate model of TCSC was used to design a simple controller based primarily on the dynamics response of the power system. Despite the numerous approaches of modern control techniques with different structures, power system utilities still prefer the conventional lead – lag PSS (CPSS). The CPSS has a simple structure and is considered to be reliable for actual power system applications [5, 22].

2.3.1 Conventional Power System System Stabilizer Stabilizer structure structure and design The basic function of CPSS is to damp electromechanical oscillations. To achieve the damping, the CPSS proceeds by controlling the AVR excitation using auxiliary stabilizing signal. The CPSS’s CPSS’s structure is illustrated in Figure Figure 2.2.

V PSS_MAX  ∆ ω 

K PSS

Gain

s T w s T w

+ 1

Washout 

1 + s T 1

1 + s T 3

1 + s T 2

1 + s T 4

Lead Lead − Lag Lag Compensator 

V PSS

Filter  V PSS_ MIN 

 Limiter  Figure 2.2  CPSS structure 

10

2.3.1.1 CPSS input The CPSS classically uses the following inputs: •

The shaft speed deviation  ∆ω 

•

Active power output, ∆Pa (Change in accelerating power) and

•

∆Pe

•

Bus frequency  ∆ f  .

(change in electric power),

Since the main action of PSS is to damp electromechanical (or rotor) oscillations, thus ∆ω  is used as the input signal to the PSS. 2.3.1.2 Gain The gain determines the amount of damping introduced by the stabilizer. Therefore, increasing the gain can move unstable oscillatory modes into the left – hand complex plane [7]. Ideally, the gain should be set to a value corresponding to a maximum damping. However, in practice the gain K PSS is set to a value satisfactory to damp the critical mode without compromising the stability of other modes [2, 7]. 2.3.1.3 Washout The washout stage is a High Pass Filter (HPF) with purpose to respond only to oscillations in speed and block the dc offsets. The Washout filter prevents the terminal voltage of the generator to drift away due to any steady change in speed. 2.3.1.4 Phase compensation This stage consists of two lead – lag compensators as shown in Figure 1 (lead – lag  compensation stage). The lead stage is used to compensate for the phase lag introduced by the AVR and the field circuit of the generator [30, 29]. The lead – lag parameters T1 − T 4 are tuned in such as way that speed oscillations give a damping torque on the rotor [7, 29]. When the terminal voltage is varied, the PSS affects the power flow from the generator, which efficiently damps the local modes [29]. Larsen and Swann express in [30] the difficulty of tuning the Lead – Lag parameters to compensate the dynamic, which varies according to the operating points and the network reactance.

11

2.3.1.5 Torsional Filter This stage is added to reduce the impact on the torsional dynamics of the generator while preventing the voltage errors due to the frequency offset [7]. 2.3.1.6 Limiter The PSS output requires limits in order to prevent conflicts with AVR actions during load rejection. The AVR acts to reduce the terminal voltage while it increases the rotor speed and the bus frequency. Thus, the PSS is compelled to counteract and produce more positive output [31]. As described in by P. Kundur in [7], the positive and negative limit should be around the AVR set point to avoid any counteraction. The positive limit of the PSS output voltage contributes to improve the transient stability in the first swing during a fault. The negative limit appears to be very important during the back swing of the rotor. Indeed, after the initial acceleration is over, the system requires a large amount of synchronizing torque to return to equilibrium in the post – fault state [2, 7and 39]. The tuning of the PSS parameters remains a complex task. Kundur et al. presented in [9] a full analysis of CPSS and different effects of its parameters on the dynamic performance of the system. They demonstrated that appropriate selection of washout time, compensator parameters and PSS limits, provide satisfactory performance. Bikash Pal and & Balarko Chaudhuri in [31] outline criteria and guidelines, based on Larsen & Swann studies in [30], to choose the compensator parameters T 1 – T 4. 

2.3.2 Genetic Algorithm based PSS (GAPSS) GAs are global search techniques equipped with powerful tools used to solve optimization problems.

GAs are based on mechanics of natural selection and

genetics. They apply the principle of survival of the fittest on a population of potential solution to generate increasingly better approximations of solution until optimization is reached [2]. 2.3.2.1 GAPSS review Several techniques of tuning PSS have been developed and tested over the recent years. Komsa Hongesombut et al. in [20] incorporated the use of an analytical method known as phase control loop and intelligent method. This procedure was performed using micro – GA to select PSS parameters combined with Hierarchical GA (HGA) in the process of reinitialization. 12

M.A. Abida and Y.L. Abdel Magid in [5] employed Evolution Programming (EP) techniques to search for optimal setting of PSS parameters. These settings had shift the system eigenvalues associated with electromechanical modes to the left in the splane. Tested under different disturbances, loading conditions and system configurations, the methods were found to be effective. J. Lu et al. in [33] applied a selection of fuzzy rules used from the operating point settings,

to

tune

the

stabilizer

parameters

online

according

to

real-time

measurements. The membership functions of the fuzzy parameter tuner were optimized using a genetic algorithm (GA). In [34], Genetic Local Search (GLS) was presented. The proposed approach hybridized GA with heuristic local search and used to tune PSS parameters on different operating conditions. The simulation showed the effectiveness and robustness of the GLSPSS. M.A. Abido and Magid in [35] investigated the effect of tuning PSS using classical GA and compared the results with those of CPSS over a wide range of parameters. GAPSS demonstrated to be more robust over the CPSS. In [37], the authors presented a novel approach to combine GA with a new recurrent neural network (RNN). The method included the design of a genetic algorithm based on recurrent neural networks power system stabilizer (GARNNPSS) for multi machine power system. The GARNNPSS consists of a recurrent neural network identifier (RNNI) that tracks and identifies the power generator and a recurrent neural network controller (RNNC). It supplies an adaptive signal to the governor and exciter to damp the power system oscillation. Both RNNI and RNNC are firstly trained offline by GA to find the optimal learning rates, and then online to damp the oscillations. The simulation results demonstrated the effectiveness of the proposed GARNNPSS and its optimal performance. K.A. Folly presented in [40] a simplified version of GA called Population Based Incremental Learning (PBIL) to design a PSS for multimachine power system. The control problem was converted into a optimization problem solved with PBIL. The resulting controllers ensured robust stability and good performance for both the nominal and off-nominal operating conditions. Asante Phiri in [38] investigated the application of Breeder Genetic Algorithm (BGA) tuning PSS parameters. A comparative analysis between BGA and GA was 13

developed and simulated, which revealed that BGAPSS performed better than GAPSS. 2.3.2.2 Proposed optimization technique Although many researches have proposed GAs to tune the PSS, it still presents some disadvantages such as, the speed of convergence which remains time consuming and the parameters’ encoding that takes up a lot of memory [48]. In fact, classical GA use bit string, better suited for combinatorial optimization [48], to encode their parameters and to modify them with logical operators. This property requires heavy computational effort and memory [47, 48]. To overcome the above setbacks, the Differential Evolution (DE), a new heuristic approach that uses differential mutation and greedy selection process [47,48], is investigated in an attempt to tune PSS’s paramters. In fact, DE, classified as Evolutionary Algorithm, provides significant converging performance over GAs by using the principle of greedy selection: “the better one of new solution and its parents wins the competition” [47]. The resulting DE based PSS will then be compared to GAPSS, which uses Classical GA to tune the PSS parameters. The comparison will focus on the robustness of the system, speed of convergence toward the optimum solution and the overall performance of the system.

14

3 POWER SYSTEM MODELLING As mentioned in chapter two, power systems are highly nonlinear and consequently, difficult to analyse. They are also constantly subject to small signal instability for most of the time. These small disturbances can be linearized around the operating point and therefore analysed. The system analysis is accomplished by using control theories such as modal analysis, root locus etc. They provide valuable information about the inherent dynamic characteristics of the power system important for the system design.

3.1 Small Signal Dynamic Modelling 3.1.1 State – Space representation The state – space representation of a system is a fundamental concept in control theory. It gives some information about the system at any instant in time [6]. To achieve the state representation, the power system is represented by a set of first order nonlinear ordinary differential equations. They describe the behaviour of the dynamic system. These equations are of the form:

d  dt 

•

X (t ) = X

=

Where ‘  X  ‘ represents the state vector , ‘

f ( X , u, t) 

(3.1)

u ‘ is the vector of input to the system

and ‘ t  ‘ denotes the time. However, if the derivatives are not function explicit of time, the system is said to be autonomous. The equation (3.1) becomes •

X

=

f ( X , u)

(3.2)

The system output variables may be expressed in terms of the state vector and input vector as follows:

Y

=

g ( X ,u )

(3.3)

The variable ‘ Y  ’ is referred to as the output vector and ‘ g ’ is the vector of nonlinear functions relating the state and the input variable to output.

15

3.1.2 Linearization The equations describing the dynamic of the system can be linearized around the equilibrium point where the system is at rest. All the variables are constant and unvarying with time. Let X0 be the initial state vector and U0 the input vector corresponding to the equilibrium point that is under investigation. Equation (3.2) becomes •

X0

=

f ( X0 , u0 ) = 0

(3.4)

When the perturbation is introduced, the above state becomes: X = X0 + ∆ X  & u = u0 + ∆u Where ‘ ∆ ’ denotes a small deviation. Therefore, the new state is defined as: •

•

•

X = X0 + ∆ X   =

 

(3.5)

 

•

f  ( X0 + ∆ X ), ( u0 + ∆ u) 

Since the deviations are very small around the equilibrium point, the function f (X, u) •

can be developed into Taylor expression and solved for ∆  X  . Thus, the linearized form of the system equations 3.2 and 3.3 obtained are:

•

B∆ u

(3.6)

∆Y = C ∆X + D∆u

(3.7)

∆

X

=

A∆ X

+

Where ∆ X  is

the state vector of the system

∆Y 

is the output vector of the system

∆u

is the vector of input to the system

A

is the state square matrix

B

is the control matrix also called the input matrix 16

C

is the output matrix

D

is the feed-forward matrix, which defines the proportion of the input that appears directly in the output.

3.2 Machine modelling The system dynamics of the synchronous machine can be expressed as a set of four first order linear differential equations given in equations 3.8 – 3.11. These equations represent a fourth order generator model suggested by the IEEE 1986 task force [53]. Higher machine’s models are also proposed based on the varying degrees of complexity [13] which provide better results. But it is also adequate to use the fourth order machine with data correctly determined [13, 52]. The above cited machine is a two axis model, includes the AVR, PSS, turbine governors and excitation system necessary for this particular research. •

δ •

ω

=

=

•

Ed'

1

Tm

−

Te ) 

 − Eq' + ( xd T  

−

xd' ) id

2 H 

•

Eq'

(3.8)

= ω − ω 0

=

( − Dω +

1

' d 0

1

 − Ed' T   ' q0

+

(3.9)

+

Efd    

( x − x ) i  ' q

q

q

Where, id , iq = d-q components of armature current  E  fd  = voltage proportional to field voltage  E d ' = voltage proportional to damper winding flux  E q' = voltage proportional to field flux T d ' 0 = d-axis transient time constant

T q'0 = q-axis transient time constant .

17

(3.10)

(3.11)

Hence, the state space model of the synchronous machine can be expressed from •

the above equations to

∆

X

=

A∆ X

+

B∆.u.. In a more complete form, the state space

is as follows:

 •   ∆ ω    a a  •   11 12  ∆ δ    a21 0  • = 0 a 32  ∆ E q'    •   0 a42  ∆ E '   d  

a13

0 a33 a43

0   ∆ω  

b1     0 0 ∆δ    +   ∆T  m a34   ∆Eq'    0      a44   ∆Ed '    0 

(3.12)

The equations describing the variables a11 − a44 of the state square matrix A are given in Appendix A1. The power system model used for this thesis is the single machine infinite bus (SMIB), as in shown in f igure 8.1

3.2.1 Machine Modelling with Power System Stabilizer As mentioned in the precedent chapter, section 2.3.1, the PSS basic function is to provide supplementary damping torque to the system by controlling the excitation. Figure A.1 in appendix A, illustrates the role and shows the place of the PSS in the SMIB system from the Heffron – Phillips model. The function includes different stages of the PSS, also portrayed in Figure A.1, has been discussed in section 2.3.1. Hence, the state space representation obtained from the block diagram is:  •   ∆ ω    •   a11  ∆ δ     •   a21  ∆  E q'    •  0  ∆  E '  =  0 d    a •  ∆ν    51  1   a61  •    ∆ν 2   a71  •   ∆ u 

a12

a13

0

0

0

0   ∆ ω  

0 a12

0 a33

0 a34

0

0

0

0

a12

a12

a 44

0

a52

a53

0

0 a55

0   ∆ δ   ' 0   ∆E q    a47   ∆ E d ' 

a62

a63

0

a65

0 a66

a72

a73

0

a75

a76

18





0   ∆ ν 1 





0   ∆ ν 2  a 77   ∆ u 

(3.13)

•

It can be observed that three more variables ∆ν 1 ,

•

∆ν 2

•

and ∆u are added to the

state matrix. These variables represent respectively the output of the washout time and the phase compensation. a11 − a77 equations are given in Appendix A1. The state – space does not give any information on the stability of the system.

3.3 Stability analysis The stability of a system can be determined by analysing the state – space matrix properties such as eigenvalues, eigenvectors, modes shape, participation factor as well as controllability and observability. These concepts are detailed in [7]. Participation factor The participation factor is used as a measure of the association between the state variable and the mode [6]. It evaluates the relative participation of the i th eigenvalue or mode in the j th state [6]. Eigenvalues The eigenvalues of the state – space matrix are obtained from equation 3.14

det [ A− sI]  = 0

(3.14)

Where, A is the state matrix  I  is the identity matrix and ‘S ’ the eigenvalues

The resulting eigenvalues can either be real or complex. For a real eigenvalues, equation 3.15, the system is corresponding to a nonoscillatory mode. If the real part of the eigenvalue is positive, then the system is unstable. If the eigenvalues are negative, the system is stable in decaying mode [7]. The further the eigenvalues are in the negative s – plane, the faster will be the system response, see figure 3.3 below. s = ±σ  Where ‘ σ  ’is the real eigenvalue.

19

(3.15)

As for the complex form, like in equation 3.16, the eigenvalues occur in conjugate pairs [7]. The real component ,‘ σ  ’, depending on whether it is positive or negative, increases the oscillation amplitude to complete instability or damps out the oscillation. The imaginary part ‘ ω  ’ ,represents the frequency of oscillation.

s =σ

±

jω 

(3.16)

Thus, the frequency of oscillation can be express as follows

 f 

=

ω  2π 

(3.17)

The damping ratio of the oscillation is given by equation 3.17 determines how fast the oscillation is damped.

ς  =

Where the amplitude decay T  =

1

σ 

σ  σ 2 + ω 2

(3.18)

in seconds

Hence, applying appropriate control theory, the space – state matrix can be mapped onto the s – plane . This plane will determine whether the system is stable or not by simply mapping the eigenvalues to the satisfaction of figure 3.3 below. The eigenvalues represent the system poles.

20

Figure 3.1

S – plane representation [50] 

Hence, a stable system will have its eigenvalues restrained on the left hand side of the s-plane, as in figure 3.3 above. Furthermore, if the system has damped oscillatory modes (complex eigenvalues on left hand side of the origine), a damping ratio between 0.05 and 0.7071 is adequate for the power system [50] to operate normally under small perturbations.

21

4 GENETIC ALGORITHM Genetic Algorithms (GAs) are heuristic search procedures inspired by the mechanism of evolution and natural genetic. They combine the survival of the fittest principle with information exchange among individuals. GAs are simple yet powerful tools for system optimization and other applications [41]. This technique has been pioneered few decades ago by Holland, basing the approach on the Darwin’s survival of the fittest hypothesis. In GAs, candidates’ solutions to a problem are similar to individuals in a population. A population of individuals is maintained within the search space of GAs, each representing a possible solution to a given problem [42]. The individuals are randomly collected to form the initial population from which improvement is sought [26, 42]. The individuals are then selected according to their level of fitness within the problem domain and breed together. The breeding is done by using the operators borrowed from the natural genetic, to form future generations (offsprings). The population is successively improved with respect to the search objective. The least fit individuals are replaced with new and fitter offspring [2, 43] from previous generation. Over the recent years, GAs have been at the centre of researches. Especially in the optimization problems where GAs provide better solutions with simpler techniques than other optimization methods. According to Goldberg in [43], GAs differs from other optimization methods in four ways: •

GAs search from a population of points in parallel, no single point

•

GAs use probabilistic transition rules, not deterministic ones

•

GAs work on encodings encodings of parameters parameters set rather than the parameter set set itself (except where real – valued individuals are used)

•

GAs do not require derivative information or other auxiliary knowledge; only the objective function and corresponding fitness levels influence the directions of the search.

•

These differences give an edge to the GAs search techniques with respect to other methods. In addition, GAs can provide a number of potential solutions to a

22

particular problem that does not have a singular solution such as the Pareto –  optimal solutions. In this case, the final choice is left to the user [44]. There are many variations of GAs but the basic form is the Classical genetic algorithm (CGA). The SGA is considered in this research as well as the Differential Evolution, which is reviewed in the subsequent chapter. Thus, the working principle [13, 46] of CGA can be described as follows:

4.1 Encoding of individuals Individuals or current approximations are encoded as strings also called chromosomes . They are constituted of genes joined together. The chromosomes are represented in different manners that allow the genotypes (chromosome genotypes (chromosome value) to be uniquely mapped onto the decision variable (phenotypes  ( phenotypes ) domain [44]. In fact, CGA can be represented by different alphabets, such as binary alphabet {0, 1}, integers, real – valued etc [43, 44]. For example, if 10 bits are used to code each variable in a two-variable function optimization problem, the chromosome would contain two genes, and it would consist of 20 binary digits [14]. Figure 4.1 from [44] illustrates how the variables are mapped onto a chromosome structure.

Figure 4.1 Chromosome structure constituted of 2 variables 

where x 1 is encoded with 10 bits and x 2  2  with 15 bits, possibly reflecting the level of accuracy or range of the individual decision variables [44]. The above chromosome representation does not yield any information. Thus, to obtain valuable information, the chromosome has to be decoded into his phenotype value. value. The performance of an individual can then be assessed.

4.2 Objective and Fitness functions The objective function measures the ability of an individual to perform in the problem domain. The individuals are assigned values also called fitness values, from higher to lower relative to their performances. In the natural world, this can be interpreted as 23

an individual’s ability to survive in its present environment. Therefore, the objective function establishes the basis for selection of individuals that will be mated together during reproduction. The fitness function is used to transform the objective function value into measure of relative fitness [45]. Equation 4.1 reflects the operation of the fitness f itness function

(4.1)

F ( x ) = g ( f ( x ))

Where  f  is the objective function and g is the fitness function transforming the value from the objective function and F  is the relative fitness.

4.3 SGA operators The SGA works with a set of ‘N’ initial individuals constituting a randomly generated population. ‘N’ denotes the size of the population. This is usually achieved by generating the required number of individuals using a random number generator that uniformly distributes numbers in the desired range [44]. The individual’s fitness is then calculated for every member. These individuals will undergo a transformation in stages to form a new current population for the next iteration. To achieve the transformation, the following genetic operators are applied in sequence:

4.3.1 Selection In this stage, individuals are selected from the current population according to their fitness value, obtained from the objective function previously described. The purpose of the selection is to choose individuals to be mated. The selection can be performed in several ways. But many selection techniques employ a “roulette wheel”. It is a mechanism to probabilistically select individuals based on some measure of their performances [44]. Figure 4.2 clearly illustrates the wheel. The segment sizes sizes on the wheel correspond correspond to the individual individual fitness value. The larger the segment, the higher the fitness value for the individual. This wheel will be spun several times until enough offsprings are produced to populate the next generation.

24

Figure 4.2 Roulette wheel 

4.3.2 Crossover In this stage, the individuals retained (in pairs), from the above stage, exchange genetic information to form new individuals (offsprings). This process helps the optimization search to escape from possible local optima and search different zones of the search space [2, 43]. The combination or crossover is done by randomly choosing a cutting point where both parents are divided in two. Then the parents exchange information to form two offsprings that may replace them if the children are fitter. Figure 4.3 demonstrates how the single point crossover is done. There are other methods of combination, such as multi – point crossover, uniform crossover, discrete crossover and various other that are discussed in [43].

4.3.3 Mutation Mutation is a process where one random allele of the gene is randomly replaced to produce another new genetic structure [44]. This process increases the probability of a complete search that will allow an investigation in the vicinity of the local optima. The effect of the mutation, as shown in Figure 4.4, is applied with a low probability in the range of 0.001–0.1 [44].

4.3.4 Reinsertion It is in this process that children populate the next generation by replacing parents, if fitter. Reinsertion can be made partially or completely, uniformly (offspring replace parents uniformly at random) or fitness-based [2, 44].

4.3.5 Convergence and termination As the population evolves over successive generations, the best and average individual increase toward global optimum [14]. Therefore, the termination is set for a 25

fixed number of iterations after which, the best individual of the current population is taken as the optimum solution.

Figure 4.3 Single point crossover 

Figure 4.4  Mutated individual 

26

5 DIFFERENTIAL EVOLUTION (DE) The Differential Evolution (DE) was first proposed by Kenneth Price. He was attempting to solve the Chebyshev Polynomial fitting Problem that had been posed to him by Rainer Storm in 1994 [48]. The breakthrough occurred when Kenneth came up with the idea of using vector difference to perturb the vector population [48]. This method grew rapidly and made DE versatile and a robust optimization tool in today’s world. DE is a parallel direct search method that uses a population of points to search for a global minimum of a function over wide search space [55]. Like all GAs, DE is a population based genetic that uses similar operators; crossover, mutation and selection. However DE search methods differ from GAs in some aspects. The main difference between the two search methods is that, GAs rely on the crossover to escape from local optima and search in different zones of the search space. Whereas, DE relies on the mutation parameters as a search mechanism and selection operation to direct the search toward the prospective regions in the search space [47].

5.1 Advantage of DE over GA In DE, all solutions have the same chance of being selected as parents regardless of their fitness value. DE is known to use the greedy selection process whereby the better one of the new solution and its parents wins the competition. This principle provides a better convergence performance over GAs [47].

5.2 Reason for using DE DE encodes parameters in floating – point regardless of their type [48]. This encoding offers a great malleability with arithmetic operators and provides significant advantages over the other optimizations methods, including [48]: •

Ease of use

•

Efficient memory utilization

•

Lower computational complexity – scales better on large problems

27

•

Faster convergence

•

Greater freedom in designing a mutation distribution iteration

Nevertheless, DE is a very simple Evolutionary Algorithm (EA), which follows a sequence, presented in figure 5.1, until optimization is reached or termination occurs.

Figure 5.1 Differential Evolution cycles 

5.3 Population structure DE starts with a population of N P  vectors of D – dimensional real – valued parameters as represented in equation 5.1

Xg = ( xi,g ) where i= 1,2,..., Np g=1,..., gmax

(5.1)

xi, g = ( zj ,i ,g ) j=1,2,..., D

 X g

represents the population and the index

g denotes the generation. The

population is constituted of  Np vectors denoted by  xi , g

where the index i refers to

the vector within the population. The vector is also constituted of parameters  z   j, i, g where  j is the position of the parameter within the vector. In the mutation stage, DE creates an intermediate population V g of the same size as the initial population composed of vi , g vectors. The intermediate population proceeds to the next stage. DE also creates a second intermediate population U i , g which is also of the size Np  with u  j, i, g vectors. The population is created after the recombination stage.

28

5.4 Initialization The population in DE is initialized by specifying the Upper and Lower bound for each parameter of a vector. Equation 5.2 is used to generate the vector parameter so that

z Lj ≤ zj, i, g ≤ zU j . zj, i, g = randj (0,1) ⋅ ( Uzj − Lzj ) + Lzj

(5.2)

Where rand  j generates number in the range of [0, 1] for the  j th parameter.

5.5 Mutation After initialization, DE mutates the population to produce a population of trial vectors. As previously mentioned, DE relies on the mutation stage, also called differential  mutation , to expand the search space. It is worth highlighting that this operation is performed differently than in the conventional GAs where an allele was replaced. In DE, four vectors from the initial population are randomly sampled where one is chosen as the target vector mentioned in the next stage, and another as the base  vector . The difference of the remaining two vectors scaled by a factor is added to the base vector to form the trial vector. Equation 5.3 and Figure 5.2 show how the process of creating the intermediate vector is achieved. vi , g = xr 0, g + F ⋅ ( xr 1, g − xr 2,g )

(5.3)

The scale factor, F, is a positive real number that controls the rate at which the population evolves [48]. The base vector, denoted by r 0 , is randomly chosen, in such a way that r 0 ≠ r1 ≠ r 2 where r1& r 2 are also randomly chosen. V i,g is the trial vector

29

F.(Xr1 – Xr2)

Xr0 Vi (Xr1 – Xr2)

Xr1 Xr2

Figure 5.2 Differential Evolution: the weighted differential, F ⋅ ( xr1, g − xr 2, g ) , is added to the  base vector,  xr 0, g , to produce a mutant, vi , g [48, 49].

5.6 Recombination or crossover DE uses the crossover, also referred as discrete recombination , to complement the differential mutation strategy mentioned in previous section [48]. In this stage, DE crosses each vector with a mutant vector to form a second intermediate population as shown by Equation 5.4.

U i,g

= u j ,i , g =

v , , j i g     j, i, g z

≤ Cr or j if [rand (0,1) j

=

j

]

rand

 

(5.4)

. otherwise

Cr ∈ [0,1] is the crossover probability defined by the user within the specified range,

which control the parameter values that are copied from the mutants. Therefore, if rand  j , the random number generated, is lower than the crossover probability Cr  ,

the corresponding  j parameter is copied from the mutant vector. On the hand, if rand  j is higher than Cr  , the parameter will become  z j, i, g , the target vector. DE can

also apply different type of crossovers: uniform crossover, one – point crossover, N –  point crossover or exponential crossover detailed in [48]. 30

5.7 Selection The selection of vectors to populate the next generation is accomplished by comparing each vector ui , g of the second intermediate population U g to its target vector  xi , g from which it inherits parameters. The values of the vectors are obtained using the function as illustrated in equation 5.5.

 xi , g +1

=

ui , g   i , g x

if f (ui , g ) ≤ f ( xi , g )

. otherwise

(5.5)

As soon as the new population is installed, the cycle is repeated until the optimum is located or termination criterion is satisfied [48]. Figure 5.3 clearly illustrates all the steps and various populations used to achieve the cycle necessary to find the optimum values.

Figure 5.3 A flow chart of DE’s operation and test loop [ 48] 

31

6 PARAMETERS SETTING OF GA & DE The main objective of the PSS is to provide additional damping in order to stabilize an oscillatory unstable system. The stabilization is achieved by fine tuning the PSS’s parameters to optimum values, using various techniques stated in chapter two. The widely used conventional PSS (CPSS) is tuned by using two basic techniques: phase compensation and root locus. The phase compensation consists of adjusting the stabilizer to compensate for the phase lag through the generator, excitation system and the power system [30, 52].The root locus on the other hand consists of shifting the eigenvalues associated with the power system modes to the stable region [16, 30]. As mentioned before, the CPSS is tuned for a particular operating and system condition. As the condition changes, the CPSS can no longer preserve the quality of performance, therefore, it needs to be retuned. The alternative tuning methods investigated in this thesis are the Differential Evolution (DE) and the Classical Genetic Algorithms (CGA). They are implemented to optimize the PSS. Unlike the CPSS, the DE based PSS (DEPSS) and GA based PSS (GAPSS) parameters guarantee a minimum performance for all operating conditions. The processes of finding these parameters are described in later sections. Some factors, such as the choice of the objective function and the optimization problem discussed in the next sections, must be taken into account when tuning the PSS’s parameters in order to achieve the desired performance over the entire range of operating conditions.

6.1 Objective function The objective function, given by equations 6.1 & 6.2, was applied to determine the optimum values of the PSS’ parameters. The function is based on the eigenvalues that are associated with the unstable modes of oscillation. It consists of shifting the eigenvalues corresponding to the undamped mechanical modes to the stable region of the s – plane, therefore stabilizing the system. This approach gives considerable insight to the closed – loop poles performance of the system such as stability, damping and frequency. This function aims to restrain these closed – loop eigenvalues to lie to the stable region of the s – plane, at a specific damping factor. 32

6.2.2 Differential Evolution (DE) The DEPSS was implemented using the DeMat package, the MATLAB version developed by J.V. Zandt and A. Neumaier. DeMat [48] provides a framework for solving function optimization problems, which was adapted to suite the objective of this work. The structure is further discussed in the subsequent chapter. DEPSS employs genetic operators described in chapter five by following a sequence illustrated in figure 5.1. DE’s optimization is strongly influenced by the mutation scale factor ‘F’ and the mutation rate ‘Cr’. Thus, appropriate values of F and Cr will guarantee a good performance of the PSS. 6.2.2.1 Mutation scale factor DEPSS employs differential mutation technique as detailed in section 5.3. This technique consists of scaling the difference between two vectors by a factor F that is then added to the base vector to form the trial vector (equation 5.3). The scaling factor F is any number ranging from (0, 1+). This factor controls the rate at which the population evolves. As the selection operator has a tendency to reduce the diversity of population, the mutation on the other hand, increases it [48]. Therefore, to avoid premature convergence, it is imperative that F is properly selected to counteract the selection’s effect [48]. 6.2.2.2 Mutation rate The mutation rate also known as the crossover probability Cr, is the likelihood that a parameter will be inherited from a mutant as pointed out in section 5.4. The mutation rate ranges from [0, 1].

Accurate Cr value is crucial for the

performance of the DE. Strom and Price proposed in [48] after extensive test beds that, optimization is best achieved with 0 ≤ Cr  ≤ 0.2 or 0.9 ≤ Cr  ≤ 1 for all functions. DE parameters are summarized in Table 6.2. These parameters were set to values that ensure optimal performance of DE.

36

Table 6.2  DE parameters 

Population size DE step - size F Crossover probability Cr Number of parameters Minimum boundary Maximum boundary Maximum generation

100 0.8 1 5 K & T1-T4 K & T1-T4 100

6.2.3 Tuning process The tuning process of the PSS involves a number of steps, given below, to find the optimal parameters. Step 1. Set the minimum and maximum boundaries where the optimal values of the PSS’ parameters will be found. Step 2. Obtain the system operating conditions and select the desired relative stability. Step 3. Generate an initial population within the constraints given by the set of equations in 6.3, using DE and SGA. Step 4. Run the load flow for each individual to check if the system converges. If not, discards and then change the operating conditions. Step 5. Get the state space matrices; calculate the eigenvalues vector with their participation factors. Step 6. Check for system controllability. If the system is controllable, then the eigenvalue shift can be performed, therefore the system can be improved. If the system is not controllable, then the oscillation cannot be damped, therefore the system parameters need to be changed. Step 7. Identify the electromechanical modes and retain the eigenvalues associated to the above modes by using the participation factor for each individual. Step 8. Evaluate the objective function J defined by equation 6.2. Step 9. Check if any eigenvalues are on the right hand side of the s-plane. 37

Step 10. Check if the damping is within the specify range Step 11. Check if the electromechanical mode’s frequency is within the acceptable range of the undamped mode’ frequency. Step 12. Repeat step 4 to step 11 until maximum generation is reached. After intense performance checks (Step 1 – Step 12) on each individual, the best performing one is selected as the optimal set of PSS parameters. The flow chart in figure 6.2 summarized the above steps.

38

Figure 6.2 Flow chart representation of the tuning process 

39

7

SIMULATION RESULTS AND DISCUSSIONS

Simulations were performed using Power system Toolbox (PST), a MATLAB package further discussed in appendix D.

7.1 Power System to be investigated The system considered for the purpose of this thesis is the single machine connected to a large system or infinite bus (SMIB), via transmission lines. The general configuration is represented in Figure 8.1.

G

 X e

uuuuuuuv P + jQ

Infinite bus

Figure 7.1: SMIB system 

Where,

Pand Q  X e

are the power and reactive power at the generator terminal.

is the line reactance of the system.

7.2 Operating conditions The DEPSS’s performance was tested over a range of operating conditions defined by the power P and reactive power Q at the generator terminal along with the equivalent line reactance Xe. Three operating conditions, defined in Table 8.1, were set to encompass practically all system’s conditions, ranging from light to heavy loaded system.

40

Table 7.1 Operating conditions 

P

Q

Xe

Minimum

0.1

0.06

0.5

Nominal

1.0

0.44

0.7

Maximum

1.0

0.62

0.9

7.3 PSS parameters The performance analysis of the SMIB equipped with DEPSS is conducted for the all range of operating conditions in Table 8.1, to evaluate the efficacy of the proposed stabilizer. The results are compared with the GAPSS and the CPSS, all tuned with single set of parameters. The comparison includes the damping and frequency performance of each PSS including the system response to a disturbance of one percent (1%) in the input voltage. The desired damping in this research is as follows:

α  = −1.0 damping ratio = 0.2 ≤ ς  ≤ 0.5 Where ‘ α  ’ is the relative stability mentioned in chapter 6. The CPSS’s design is described in the subsequent section.

7.3.1 CPSS parameter selection The CPSS was designed at nominal operating condition using the phase compensation technique by following guidelines provided in Appendix B. The parameters are specified in Table 8.2.

41

Table 7.2 CPSS parameters 

K

29.5

T1

0.8

T2

0.1

T3

0.1

T4

0.3

The CPSS consists of a lead – lag compensator, washout stage and the gain, as described in chapter two. The selected CPSS’ parameters ensure a stable system at nominal operating condition and provide grounds to tune the DEPSS and GAPSS as discussed in the next section.

7.3.2 DEPSS and GAPSS parameters selection By minimizing the objective function,  J  = min[Re(λi ) − α ] described in chapter six, DEPSS and GAPSS, maximize the parameters. In fact, the CPSS’ parameters allow to define a search region where the optimum PSS’ parameters are expected to be found. Hence, the DEPSS and GAPSS parameters were obtained by solving the constraint given in equation 6.3 by setting the search boundaries as given in table 8.3. To provide a fair comparison between the different tuning methods, the DEPSS and GAPSS were also tuned at nominal operating condition. After some trial and error, and intensives tests, the two PSS (DEPSS & GAPSS) provided good results within the boundaries given in table 8.3 below. The washout time was chosen to be by following criteria in [52], Tw = 2.0

42

Table 7.3  Parameters boundaries 

Bounds

K

T1

T2

T3

T4

Min

15

0.6

0.02

0.02

0.02

Max

20

5

2

2

2

The optimum parameters found by DE and GA are given in Table 8.4 below. Table 7.4 DEPSS and GAPSS parameters 

DEPSS

GAPSS

K

19.811

18.6

T1

3.526

3.88

T2

0.409

0.801

T3

0.288

0.505

T4

0.901

0.787

7.4 Simulations results Having found the optimum parameters, the system was tested at various stress level by varying the generator power (P) and the reactance of the line (X e) independently from each other over the selected range defined in equation 7.1and 7.2 below. The following range was considered for the particular reason that the system is only unstable in the cited ranges. The left out ranges have been discarded because the system was able to stabilize after disturbances, as analysed in table 7.5 at minimum condition, with adequate damping. The selected testing range is as follows:

43

The power is varied from:

P = 0.6 −1.0 p.u

(7.1)

The line reactance as:

 X e =0.7−0.9 p.u.

(7.2)

The DEPSS is expected to provide adequate performance over the selected testing range. Load flow reports are provided in Appendix C for the nominal and maximum operating conditions.

7.4.1 Eigenvalues analysis 7.4.1.1 Minimal system condition The system’s performance of the open loop is given in Table 7.5 below. To track the electromechanical modes, as mentioned earlier, the participation factor, provided in Appendix E.1, was used to identify the highest participating eigenvalue(s) in the particular state. Table 7.5  No PSS at minimum condition 

Eigenvalue

Damping

Frequency (rad/s)

-0.422 + j5.037

0.084

5.055 

-0.422 - j5.037

0.084

5.055 

-1.205

1.00

1.205

-5.542 + j18.179

0.292

19.00

-5.542 - j18.179

0.292

19.00

-59.16

1.00

59.16

The system without PSS is stable and is able to damp oscillations of frequency 5.055 rad/s associated with the rotor angle and the speed, with associated eigenvalues

-0.422

±

j5.037 . Although the damping performance is just above to the lower

bound (0.05) but is acceptable for a power system. Therefore, No PSSs were tested for this operating condition.

Two eigenvalues, -5.542

±

j18.179 are associated

with the AVR, exhibit damped oscillations of 19 rad/s. Two non-oscillatory

44

eigenvalue, -1.205 is associated with the AVRS system and -59.16 is associated with the control mode. 7.4.1.2 Nominal condition The open - loop system in table 7.6 displays under damped electromechanical modes with eigenvalues 0.0212

±

therefore oscillatory at a frequency of 3.98

j3.98

rad/s. Table 7.6 No PSS at nominal condition

Damping

Frequency (Rad/s)

0.0212 + j3.98

-0.0053

3.98 

0.0212 - j3.98

-0.0053

3.98 

-1.728

1

1.728

-5.906 +j17.24

0.323

17.23

-5.9046 -j17.24

0.323

17.23

-58.5794

1

58.579

Eigenvalues

The CPSS tuned at this operating condition, stabilizes the system by improving the under damped electromechanical modes and moving the associated eigenvalues to

-0.64

±

j3.69

at a damped frequency of 3.7 rad/s, as shown in Table 7.7. Table 7.7 CPSS at nominal condition 

Eigenvalues

Damping

Frequency (Rad/s)

-0.245

1

0.245

-0.64 + j3.69

0.1705

3.70 

-0.64 - j3.69

0.1705

3.70 

-1.74

1

1.738

-3.97

1

3.978

-4.90 +j17.13

0.275

17.13

-4.90 -j17.13

0.275

17.13

-10

1

10

-58.61

1

58.61

45

The GAPSS in Table 7.8 considerably improves the damping factor of the mode associated to the speed and rotor angle and eigenvalues -1.49

±

j3.79 with a slight

increase in the frequency to 4.07 rad/s.

Table 7.8  GAPSS at nominal condition 

Eigenvalues

Damping

Frequency (Rad/s)

-0.45

1

0.45

-1.21 +j 0.57

0.905

1.34

-1.212 - j0.57

0.905

1.34

-1.49 + j3.79

0.364

4.07 

-1.49 - j3.79

0.364

4.07 

-1.77

1

1.77

-4.43 +j16.98

0.252

17.54

-4.43 -j16.98

0.252

17.54

-58.62

1

58.63

DEPSS, in Table 7.9, also has three oscillatory modes: one associated with rotor angle and speed or the electromechanical mode, with eigenvalues at -1.58

±

j3.93 with a frequency at 4.13 rad/s; the two other with frequencies of

1.588 and 17.53 respectively associated with the excitation system, -1.45

±

j0.65

and the PSS -4.51

±

j16.94 . DEPSS demonstrates better

damping performance.

46

Table 7.9 DEPSS at nominal condition 

Eigenvalues

Damping

Frequency (Rad/s)

-0.44

1

0.449

-1.45 +j0.65

0.912

1.588

-1.45 -j0.65

0.912

1.588

-1.58 +j3.93

0.374

4.130 

-1.581-j3.93

0.374

4.130 

-1.978

1

1.978

-4.51 +j16.94

0.257

17.53

-4.51-j16.94

0.257

17.53

-58.62

1

58.62

7.4.1.3 Maximum condition The open – loop system is unstable. The electromechanical modes are under damped as summarized in table 7.10. The associated eigenvalues 0.0885

±

j2.85

have further moved into the unstable region of the s – plane, thus, resulting in an unstable oscillatory mode at a frequency of 2.84 rad/s. Table 7.10 No PSS maximum condition 

Eigenvalues

Damping

Frequency(rad/s)

0.09 + j2.85

-0.031

2.84 

0.09 - j2.85

-0.031

2.84 

-1.77

1

1.76

-5.72 +j17.72

0.307

18.62

-5.72 -j17.72

0.307

18.62

-58.89

1

58.89

47

The CPSS, as expected, poorly performs for this condition due to its low damping. The electromechanical modes, with associated eigenvalues -0.233

±

j2.64 ,

are

damped at a frequency of 2.66 rad/s.

Table 7.11 CPSS at maximum condition 

Eigenvalues

Frequency (rad/s)

Damping

-0.2429

1

0.242

-0.33 + j2.64

0.121

2.66

-0.33 - j2.64

0.121

2.66

-1.765

1

1.76

-4.09

1

4.09

-4.93 +j17.65

0.268

18.32

-4.93 -j17.65

0.268

18.32

-10

1

10

-58.92

1

58.92

The GAPSS has a relatively good damping at maximum condition. The resulting system is stable. The oscillatory mode associate with the electromechanical has a frequency of 2.72 rad/s, with associated eigenvalues at -1.0681

48

±

j 2.50 .

Table 7.12 GAPSS at maximum condition 

Eigenvalues

Damping

Frequency (rad/s)

-0.429

1

0.429

-1.07 +j 2.50

0.393

2.72 

-1.07 - j2.50

0.393

2.72 

-1.30 + j0.685

0.885

1.47

-1.30 - j0.685

0.885

1.47

-1.75

1

1.75

-4.55 +j17.52

0.251

18.10

-4.55 –j17.52

0.251

18.10

-58.9294

1

58.93

With the addition of the DEPSS, the system becomes stable. The PSS exhibits a good damping of the electromechanical modes at a frequency of 2.7 rad/s and eigenvalues -1.2598

±

j2.498 . The system is expected to stabilize faster than

GAPSS and CPSS. Table 7.13  DEPSS at maximum condition 

Eigenvalues

Damping

Frequency (Rad/s)

-0.43

1

0.43

-1.25 + j2.498

0.450

2.70 

-1.25 - j2.498

0.450

2.70 

-1.58 + j1.035

0.836

1.88

-1.58 - j1.035

0.836

1.88

-1.70

1

1.70

-4.62+j17.49

0.255

18.09

-4.62 -j17.49

0.255

18.09

-58.93

1

58.93

49

7.4.2 Time domain response The comparison between the different PSSs is better illustrated in time domain response with 1% step change in the voltage as illustrated in figure 7.1 and 7.2, corresponding respectively to nominal and maximum conditions -3

Step response for op cond. P=1 and Q=0.43617 at Xe=0.7

x 10 13

DE PSS

12

GA PSS CPSS

11    ) 10   u  .   p    ( 9    t    V   n    i   e   g   n   a    h   c

8 7 6 5 4

3 0

1

2

3

4

5 time(s)

6

7

8

9

10

Figure 7.2 : Step response at Nominal condition 

The responses of all three stabilizers, CPSS, GAPSS and DEPSS are plotted simultaneously for the weak transmission with heavy power transfer. System is more stable in this case, following any disturbance. All three controllers are able to damp the oscillations thus improving the system dynamic stability significantly. DEPSS with its remarkable performance shows it superiority over the GAPSS and the CPSS by settling within 3 – 3.5 seconds. The GAPSS settles in 3.5- 4 seconds while the CPSS between 6 -7 seconds. DEPSS also presents some overshoot slightly higher than CPSS and more or less equal to the GAPSS.

50

-3

Step response for op cond. P=1 and Q=0.62222 at Xe=0.9

x 10 13

DEPSS

12

GAPSS CPSS

11    ) 10   u  .   p    ( 9    t    V   n    i   e 8   g   n   a    h 7   c

6 5 4 3 0

1

2

3

4

5 time(s)

6

7

8

9

10

Figure 7.3: Step response at Maximum condition 

DEPSS and GAPSS are able to damp the oscillations associated with the electromechanical mode, and stabilize the system. On the other hand, the CPSS is

less effective in damping these oscillations. DEPSS displays its efficacy by settling within 3-4 seconds. GAPSS settles within 5 to 6 seconds. The overshoots are also observed in the response where both DE-PSS and GAPSS present the biggest.

51

7.4.3 Robustness tests of PSSs The PSSs are tested over the selected range of the power system operating conditions in equations 7.1 and 7.2. The time domain responses for the robust test are provided in appendix E.2 7.4.3.1 Eigenvalues and Damping ratio The damping ratio associated with their eigenvalues were compared over a range of fifteen operating conditions by varying the reactance, Xe = 0.7, 0.8, 0.9 and the generator power, P = 0.6 – 1.0 p.u. The tables below summarize each PSS’ performances. The values in brackets are the damping ratio.

Table 7.14  CPSS eigenvalues and Damping ration under r obust test 

Xe

P

0.7

0.8

0.9

0.6

-0.71 ±4.20 (0.168)

-0.62 ±3.93 (0.156)

-0.54 ± 3.68 (0.146)

0.7

-0.72 ±4.11 (0.173)

-0.61 ±3.81 (0.159)

-0.52 ±3.53 (0.147)

0.8

-0.71 ±4.00 (0.176)

-0.59 ±3.66 (0.159)

-0.48 ±3.33 (0.144)

0.9

-0.68 ±3.86 (0.175)

-0.54 ± 3.47 (0.155)

-0.42 ±3.05 (0.136)

1

-0.64 ± 3.69 (0.17)

-0.48± 3.22 (0.146)

-0.32 ± 2.63 (0.121)

52

Table 7.15 GAPSS eigenvalues and damping ratio under robust test 

Xe

P

0.7

0.8

0.9

0.6

-1.2 ± j4.36 (0.27)

-1.09 ±j4.06 (0.25)

-1.00± j3.79 (0.25)

0.7

-1.29 ± j4.27 (0.29)

-1.16 ± j3.94 (0.28)

-1.05 ±j3.63 (0.27)

0.8

-1.35 ± j4.16 (0.31)

-1.20± j3.78 (0.3)

-1.08 ± j3.40 (0.31)

0.9

-1.39 ±4.01 (0.33)

-1.22 ± j3.56 (0.33)

-1.07± j 3.07 (0.34)

1

-1.42 ±j3.81 (0.34)

-1.21 ± j3.25 (0.35)

-1.06 ± j2.53 (0.39)

Table 7.16  DEPSS eigenvalues and damping ratio under robust test 

Xe

P

0.7

0.8

0.9

0.6

-1.27 ± j4.44 (0.275)

-1.16 ± j4.13 (0.271)

-1.08 ±j3.85 (0.27)

0.7

-1.37 ± j4.36 (0.300)

-1.25 ± j4.02 (0.297)

-1.15± j3.69 (0.298)

0.8

-1.46± j4.26 (0.323)

-1.31 ± j3.86i (0.323)

-1.20 ±j 3.47 (0.328)

0.9

-1.52 ± j4.12 (0.350)

-1.36± j3.65 (0.350)

-1.23 ± j3.12 (0.367)

1

-1.56± j3.93 (0.374)

-1.39± j3.33 (0.385)

-1.24 ± j2.51 (0.444)

53

DEPSS and GAPSS display better damping performance than CPSS for every condition. DEPSS outperforms GAPSS in all selected conditions. The average damping performance over the entire range is as follows:

Table 7.17  Average PSSs damping ratio 

Average damping

CPSS

GAPSS

DEPSS

0.155

0.318

0.330

DEPSS is 6.1% more robust than GAPSS and 53% than CPSS.

7.4.3.2 Eigenvalues plot under robust test The system’ eigenvalues were plotted on the s-plane. Figure 8.6 – 8.9 illustrate the effects on the system when the operating conditions change. Notice that, the fast non-oscillatory poles associated with the control mode, are not included in the s –  plane’ plots, since they are barely affected by the changes. Figure 7.3 shows the open – loop poles of the system for all conditions. The system is poorly damped and unstable for some of the conditions. The circled eigenvalues causing the instability are associated with the electromechanical modes. The relative stability line is also included in the plot.

54

20 15 10   s    i   x   a   y   r   a   n    i   g   a   m    I

5 0 -5 -10 -15 -20

-8

-7

-6

-5

-4 -3 Real axis

-2

-1

0

1

Figure 7.4: Open - loop eigenvalues 

Figure 7.4, shows the CPSS’ closed – loop poles. The CPSS achieve stability for all conditions. But it also presents poor damping for some operating conditions. Hence, CPSS does not guarantee robust performance for the selected set of conditions. 20 15 10 5   s    i   x   a   y   r 0   a   n    i   g   a   m    I -5 -10 -15 -20

-8

-7

-6

-5

-4 -3 Real axis

-2

-1

Figure 7.5: CPSS closed - loop eigenvalues 

55

0

1

Figure 7.5 shows the GAPSS closed – loop poles for the selected operating conditions. GAPSS has managed to restrict some eigenvalues to the left of the relative stability margin, at certain operating conditions. Hence, GAPSS is robust and guarantees stability for all conditions.

20

15

10

5

   s    i    x    a    y    r    a 0    n    i    g    a    m    I

-5

-10

-15

-20 -8

-7

-6

-5

-4

-3

-2

-1

Real axis

Figure 7.6: GAPSS closed-loop eigenvalues 

56

0

1

Figure 7.6 shows the DEPSS closed –loop poles. DEPSS is able to retrain most of the system’s eigenvalues to the left of the chosen stability margin. Therefore, DEPSS is the most robust stabilizer of all three. 20 15 10 5   s    i   x   a   y   r 0   a   n    i   g   a   m    I -5 -10 -15 -20

-8

-7

-6

-5

-4 -3 Real axis

-2

-1

Figure 7.7 : DEPSS closed-loop eigenvalues 

57

0

1

8 CONCLUSION The application of the Differential Evolution (DE) to optimally tune a Power System Stabilizer (PSS) capable of stabilizing a system over a wide range of operating conditions has been successfully investigated. An eigenvalue based objective function was implemented for the DE to find the optimum PSS’ parameters at a particular operating condition. The objective function consists, shifting and assigning the eigenvalues associated with the electromechanical modes, to the left side of the s – plane, at a specific location with damping ratio and frequency of oscillation. The DEPSS’ s performance was tested and simulated on a single machine infinite bus (SMIB) system. Results have been presented, in chapter 8 and appendix E, for a wide range of operating conditions, to establish the efficacy of the DEPSS. Therefore, the following conclusions are drawn: •

DEPSS is able to provide robust stabilization over the specified range of operating conditions.

•

The performance evaluation of the DEPSS compared to the GAPSS and the CPSS, revealed that both DE and GA based PSS outperform the CPSS in every system’s conditions. DEPSS surpasses GAPSS all selected conditions. Therefore DE is better and 6 % more robust than GA in terms of damping ratio and 53 % better than CPSS.

•

DEPSS is able to restrain the eigenvalues to the specific region, denoted by the relative stability, for most of the operating conditions.

•

The attractive feature of the eigenvalue based objective function is that it allows the relocation of the electromechanical modes to a desired region with a desired damping, unlike the phase compensation, eigenvalue shifting method gives considerable insights on the system performances.

DEPSS was implemented in MATLAB using DeMat package. As a result, DE’s simple, yet effective method offers much potential for a practical implementation.

58

9 RECOMMENDATIONS

Although considerable number of issues have been successfully investigated in the area of tuning PSSs, several problems remain unresolved. Based on the research carried out for this thesis, further issues need to be investigated. Therefore, the following recommendations are made: •

•

Investigate the application of DE to tune the PSS in a multimachine system. Investigate the application of DE in phase compensation method. The method consists of tuning the PSS parameters to compensate the phase lag through the generator, excitation system and the power system.

Then, conduct a

comparative study between DE based eigenvalue shifting method and DE based phase compensation. •

Investigate the application of other Evolutionary Algorithms (EAs) to tune a PSS with faster convergence than DE.

59

REFERENCES

[1] Jesus Fraile - Ardanuy, P.J. Zufiria: “Design and comparison of adaptive power  system stabilizer

based on neural fuzzy networks and genetic algorithms ”,

Polytechnic University of Madrid. [2] Adrian Andreou: “On power system stabilizers: Genetic algorithms based tuning  and Economic worth as ancillary services .” Department of Electrical Engineering, Chalmers University of Technology, Sweden 2004. [3] D. Menniti, A. Burgio, A. Pinnarelly, V. Principe, N. Scordino, N. Sorrentino “Damping oscillation improvement by fuzzy power system stabilizer tuned by  genetic algorithms ”. 14th PSCC, servilla, 24 – 28 June 2002. [4] Ali T. Al – Awami, Y.L. Abdel – Magid, M.A. Abido “Simultaneous stabilization of  power system using particle swarm – based robust coordinated design of PSS  and SVC ”, Electrical Engineering Departement, King Fahd University of petroleum and Minerals [5] M.A. Abido and Y.L. Abdel – Magid, Sentor member “Optimal design of power  system stabilizers using Evolutionary Programming ”. IEEE transaction on energy conversion, vol. 17, NO.4, December 2002. [6] P. Kundur, Power System Stability and Control , Mc Graw – Hill, New York, 1994. [7] K.R. Padiyar, Power System Dynamics Stability and Control , Wiley, Banga lore, 1996. [8] P. Kundur, “A course on power stability and control, “ ABB T & D University, Ludvika, sweden, April 2000. [9] P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose, C. Canizares, N. Hatziargyriou, D. Hill, A. Stankovic, C. Taylor, T. Van Cutsen and V. Vittal “Definition and classification of power system stability ”. IEEE/CIGRE joint task force on stability terms and definitions. Transaction on Power systems: Accepted for future publication.

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[10] C.W. Taylor. Power system voltage stability , McGraw – Hill, Inc., 1994 [11] Valeri JS Knazkins, “Stability of Power system with Large amounts of distributed  generation, doctoral thesis ,” Stockholm, Sweden 2004. [12] CIGRE Task force 38.02.16, “ impact of the interaction among power system  controls, technical report,” 2000 [13] C.L. Demarco, “The threat of predatory generation control: can ISO Police Fast  Time Scale Misbehavior ” in proceeding of Bank Power system Dynamics and control IV Restructuring, santorini Greece, August 1998, pp. 281 – 289. [14] Ravindra Singh, “A novel approach for tuning of Power system stabilizer using  genetic algorithm ”, Department of Electrical Engineering INDIAN INSTITUTE OF SCIENCE, Bangalore, thesis submitted for the degree of Master, July 2004. [15] Andre M.D. Ferreira, Jose A.L. Barreiros, Walter Barra Jr., Jorger Roberto Brito   – de – Souza, “A Robust adaptive LQG/LTR TCSC controller applied to damp  power system oscillations ”.Elsevier, Science Direct, Electric Power Research 77(2007) pp. 956 – 964. [16] F.P. deMello, C. Concordia, “Concept of synchronous machine stability as  affected by excitation control ”, IEEE trans. Power Appr. Syst. 88, 1969, 316 –  329. [17] M. Klein, G.J. Rogers, P. Kundur, “A fundamental study of inter – area oscillation  in power systems,” IEEE Trans. Power system, 1991, pp.914 – 921. [18] Sidhartha Panda, and N.P. Padhy, “Coordinated Design of TCSC Controller and  PSS employing Particle Swarm Optimization Technique ”, international journal of computer and information science and engineering volume 1 number 1 2007 pp.1307 – 4164. [19] Juan Shi, Herron L.H and Kalam A, “Application of Fuzzy Logic controller Power  System Stabilization ”, Victoria University of Technology, Australia, IEEE TENCON, 2002, Beijin. [20] Komsan Hongesombut, Yasunori Mitani, and Kiichiro Tsuji, “Power system  stabilizer tuning in multimachine power system based on minimum phase control 

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loop method and genetic algorithm ”, Osaka University, Graduate School of Engineering. [21] Y.L. Abdel – Magid, M.M. Dawoud, “Tuning of power system stabilizers using  genetic algorithms ”, Electrical Department, King Fhad University and Minerals, Saudi Arabia, 1996. [22] Komsan Hongesombut, Sanchai Dechaunupaprittha, Yasunori Mitani, and Issarachai Ngamroo, “Robust Power system stabilizer tuning based on multi-  objective design using hierarchical and parallel micro genetic algorithm ”, Department of Electrical Engineering , Kyushu Institute of Technology, Japan. [23] Sidhartha Panda and Narayana Prasad Padhy, “ Power system with PSS and  FACTS controller: Modelling, simulation and simultaneous tuning employing  Genetic Algorithm ”, international Journal of Electrical, computer and systems Engineering volume 1, num. 1, 2007, pp. 1307 – 5179. [24] D. Xia and G.T. Heydt, “Self – tuning controller for generator excitation control, ” IEEE Trans. Power Sys., vol PAS – 102, 1983, pp. 1877 – 1885, [25] V. Samarasinghe and N. Pahalawaththa, “Damping of multimodal oscillations in  power systems using variable structure control techniques,”  Proc. Inst. Elect. Eng. Gen. Transm. Distrib., vol. 144, Jan. 1997, pp. 323–331. [26] Y. L. Abdel-Magid, M. A. Abido, S. Al-Baiyat, and A. H. Mantawy, “Simultaneous  stabilization of multimachine power systems via genetic algorithms,” IEEE Trans. Power Syst., vol. 14, Nov. 1999, pp. 1428–1439. [27] Alberto Del Rosso, Claudia A. Canizares, Victor Quintana, Victor Dona, “Stability  Improvement using TCSC in Radial Power Systems,”  NAPS – 2000, Waterloo, ON, October 2000. [28] Y.N Yu, “Electric Power System Dynamics,” New York Academy, 1983. [29] Olof Samuelsson, “Power System Damping, structural aspect of Controlling  Active Power,”  Department of Industrial Electrical Engineering and Automation (IEA), Lundi Institute of Technology (LTH), Sweden 1997.

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[30] E.V. Larsen and D.A. Swann, ”Applying power system stabilizer,”  IEEE Transactions Power Apparatus and Systems PAS – 100 (1981), part I, II & III, pp. 3017 – 3046. [31] Bikash Pal, Balarko Chaudhuri, “Robust Control in Power System,”  Springer, 2005. [32] Komsan Hongesombut, Yasunori Mitani, and Kiichiro Tsuji, “An incorporated  use of Genetic Algorithm and Modelica library for simultaneous tuning of Power  System Stabilizers, ” Osaka University, Graduate school of engineering, Japan. [33] J. Lu, M.H. Nehrir, D.A. Pierre, “A fuzzy Logic – based Power System Stabilizer  optimized with Genetic Algorithm,”  Electrical and Computer Engineering Department, Montana State University, 27 august 2001. [34] M.A. Abido, “Parameters optimization of multimachine power system stabilizers  using genetic local search,”  Electrical Department, King Fhad University and Minerals, Saudi Arabia, November 2000. [35] M.A. Abida, Y.L. Abdel – Magid, “Genetic based power system stabilizer,”  Electrical Department, King Fhad University and Minerals, Saudi Arabia, November 2000. [36] Sidhartha Panda, and N.P. Padhy, “Power System with PSS and FACTS  controller: Modelling, simulation and simultaneous tuning employing Genetic  Algorithm” , international journal of computer and information science and engineering volume 1, number 1, 2007. [37] C. J. Chen and T. C. Chen, “Power System Stabilizer for multimachines using  Genetic Algorithm based on Recurrent Neural Network (RNN),”  Department of Engineering Science, national Cheng Kung, University Tainan, Taiwan. [38] A. Phiri, “Optimal tuning of Power System Stabilizer based on Evolution  Algorithm,”  Undergraduate thesis, Department of Electrical Engineering, University of Cape Town, October 2007. [39] Omer M. Awed – Badeeb, “Damping of electromechanical modes using Power  System Stabilizers (PSS) Case: Electrical Yemen Network.” Journal of Electrical Engineering, Vol.57, No 5, 291 – 295.

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[40] K.A. Folly, “Multimachine power system stabilizer design based on a simplified  version of Genetic Algorithm combined with Learning,”  Department of Electrical Engineering, University of Cape Town. [41] D. Menniti, A. Burgio, A. Pinnarelli, V. Principe, N. Scordino, N. Sorrento, “Damping oscillations improvement by fuzzy power system stabilizers tuned by  Genetic Algorithms” Department of Electronics, Computer and Systems Science, University of Calabria – Italy. [42] Hongesombut K., Mitani Y., Tsuji K. “An incorporated use of Genetic Algorithm  and Modelica library for simultaneous tuning of Power System Stabilizers,”  Second International Modelica conference, Proceedings, pp. 89 – 98. [43] Goldberg DE, “Genetic Algorithms in search optimization and machine learning”  Addison – Wesley, Reading, MA, 1989. [44] A. Chipperfield, A. Fleming, H. Pholheim, C. Fonseca, “Genetic Algorithm  ToolBox for MatLab use”  Department of Automatic Control and Systems Engineering, University of Sheffield. [45] K.A. De Jong, “Analysis of the behaviour of a class of Genetic Adaptive System,”  PhD thesis, Department of Computer and Communication Science, University of Michigan, AM Arbor, 1975. [46] D. Beasley, D.R. Bull and R.R. Martin, “An overview of Genetic Algorithms: Part  1 Fundamentals,” Universal Computing, Vol. 15, No. 2, 1993, pp 58 – 69. [47] D. Karaboga, S. Okdem, “A simple and Global Optimization Algorithm for  Engineering: Differential Evolution Algorithm,”  Turk Electrical Engineering, Vol. 12, No 1, 2004. [48] K.V. Price, R.M. Storn, J.A. Lampinen, “Differential Evolution: A practical  approach to global optimization” Springer – Verlag Berlin Heidelberg, 2005. [49] S. Das, A. Abraham and A. Konar, “Particle Swarm Optimization and Differential  Evolution Algorithms: Technical Analysis, Applications and Hybridization  perspectives,”  Department of Electronics and Telecommunication Engineering, Jadavpur University, India. Centre of Excellence for Quantifiable and Technology, Norway.

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[50] M. Braae, “Control Theory for Electrical Engineers,”  Department of Electrical Engineering, University of Cape Town, 1994. [51] “Power system Toolbox version 2: Dynamic tutorial and function”  Joe Chow/  Cherry Tree Scientific Software 1991 – 2003. [52] M.A Pai, D.P. Sen Gupta, K.R. Padiyar, “Small signal Analysis of Power  Systems,” Norosa publishing House, India, 2004. [53] IEEE Task Force, “Current Usage and Suggested Practices in Power System  Stability Simulations for Synchronous Machines" , IEEE Transactions on Energy Conversion, Vol.1, No. 1, pp 77 – 93, 1986. [54] G. Rogers, “Power System Oscillations” , Kluwer Academic Publishers,2000. [55] R. Storn and K. Price, “Differential evolution - a simple and efficient heuristic for  global optimization over continuous spaces,”  Journal of global optimization, vol. 11, 1997, pp. 341–359.

65

APPENDIX A: SYSTEM EQUATIONS A.1

Equations of state square matrix variables

The variables of the State matrix A without PSS from equation 3.12 is given as follows:

  D − M   2π  f   A =   0    0 

−

K1

−

M 0 −

M

K4

T



1

0  K 1 

Tdo ' K 3

Tdo '  

−

K A K 5

0 

 

0

Tdo ' −



K 2

−



K A K6   T

A

−

A

1



T  A

The variables in equation 3.13 which include the PSS are given as follow:

  D  − M   2π  f    0    0  A =   K  C  T2  KT − C 3  T2T4  − KCT3  T2T4

−

K1

−

K 2

M

M

0 −

0

K4

K A K5 T

−

KC K1T1 T2 M

−

KC K1TT 1 3 T2T4 M

−

KC K1TT 1 3 T2T4 M

K A K6   A

KC K2T1  

KC K2TT 1 3 T2T4 M

−

0

0

0

0 

0

0

0 

KC K2TT 1 3  T2T4 M

66

−

1 T

0 0 0

 

Tdo '  

T2 M −

0 

Tdo ' K3 T −

0

K 1

−

A

0

1

−

Tdo ' −

 



0

A

 K A   0 0 T A  1 0 0  − T2     T3   1 1 0  − − T4 T2T4 T4     T 3 1 1 1 − − − T4   T2T4 T4 TW   

 

A.2

Heffron – Phillips diagram and constants

rd 

Figure A.9.1 Heffron – Philips 3  model of SMIB system with PSS included 

The block diagram in Figure A.1 describes the dynamics of the SMIB system. The constants K1 − K 6 in the block diagram describe the dynamic characteristic of the system [7] known as the Heffron – Phillips K constants. K1 and K2 are derived from the electric torque, while (K 3 and K4) from the field winding circuit equations and (K 5, K6) from the terminal voltage. The relating equations are indexed in Appendix A2. The constants K2, K3, K4 and K6 are usually positive and they affect the system differently. K2 – K 4 influence the electric torque in different manner depending on the oscillation frequency. When K4 is positive, a positive damping torque component is introduced [7]. However, for negative value of K 4, the damping will be negative. K5 on the other hand is commonly negative in practice [7]. In the case where K 5 is positive, the AVR decreases the synchronizing torque and increases the damping

67

torque. For negative values of K 5, the AVR introduces a positive value of synchronizing torque and negative damping torque component [7]. The expression of the constants below are for a lossless network [16] where  Z  R = 0 and  Z I = xe . On the stator  Ra = 0 . ,  ( e x− d x) 2  sin δ 0  K1 =  E0 cos δ 0 + E 0 ( xe + xd, )  ( xe + xd, )     E 0 sin δ 0

E

0

K 2

=

,

( xe + xd  ) ,

K 3

( xe + xd  )

=

( xe + xd  ) =

( xe − xd , )

K 5

=

 E 0  xq vd 0 cos δ 0

K 6

=

K4

( xe + xd , )

E 0 sin δ 0



+

vt 0  ( xe + xq ) vq0

xd, vq 0 sin δ 0 



( xe + xd, )  

xe ,

vt 0 ( xe + xd  )

Where,

P0v t 0

iq 0 =

( Pe x q ) 2

+

( vt 0 2 + Q 0 x q )

2

v d 0 = iq 0 xq vq 0 = iq 0 =

vt 0 2 − vd  0 2 Q0 + i q2 x q vq 0

Eq 0 = vq 0 + id 0 xq E0 =

δ 0

=

2

( vd 0 + iq 0 xe ) tan −1

+

( vq 0 − id 0 xe )

2

v d 0 + iq 0 xe v q 0 − id 0 xe

68

APPENDIX B: TUNING GUIDELINES FOR A CPSS The objective of the PSS is to compensate for the phase lag of a signal introduced by the generator, exciter, and power system GEP(s). The phase lag strongly depends on the frequency of the signal ranging from 0.2 to 1.5 Hz, thus covering the inter-area and local mode as described in chapter two. In designing the CPSS, two main criteria critical for proper operation have to be satisfied; the time constants T1-T4 for the phase compensation and the gain to provide adequate damping [52]. Over compensation reduces synchronizing torque. Thus, the PSS(s) phase must balance the GEP(s) phase. The phase lead to be provided to compensate for the phase lag is different for different frequency [52]. The following tuning guidelines are recommended by [30, 52]:

1. Since the signal from the GEP(s) is passed through the washout stage before the phase lead, to eliminate the steady – state bias, it important to choose appropriate value for T w. According to [30, 52], it would be adequate to choose the time constant Tw between 1 – 2 sec if the damping of the local mode is the only concern. But Tw = 10 s when inter – area is considered.

2. As for the phase lag compensation, P ( s ) = GEP (s ) × PSS (s ) should pass through -90o at around 3.5 Hz.

3. The compensated phase lag at local mode should be below 45 o, most preferably at 20o.

4. The compensator gain at high frequencies, which is proportional to be minimized to reduce noise amplification through the PSS. 69

T1T 3 T2T 4

should

APPENDIX C: SYSTEM OPERATION C.1

Single Machine Infinite Bus data

C.1 Generator parameters 

MVA

H

 X d 

 X q

 X d '

 X q'

T d ' 0

T q' 0

225

3.53

1.81

1.76

0.3

0.6

7.8

0.9

C.2 AVR parameters 

Ka

T a

200

0.05

70

 E  fd max

 E  fd min

5.0

-5.0

C.2

Load flow reports

At minimum operating condition where P = 0.1 & Xe = 0.7, LOAD-FLOW STUDY REPORT OF POWER FLOW CALCULATIONS

10-May-2008 : BUS 2

SWING BUS

: 4

 NUMBER OF ITERATIONS

SOLUTION TIME

: 0.047 sec.

TOTAL TIME

: 0.141 sec.

TOTAL REAL POWER LOSSES

: 0.

TOTAL REACTIVE POWER LOSSES :

0.785345.

GENERATION

LOAD

BUS

VOLTS

ANGLE

REAL

REACTIVE

REAL

1.0000

1.0300

42.8133

1.00

0.4362

0

0

2.0000

1.0000

0

-1.00

0.3492

0

0

LINE FLOWS

LINE

FROM BUS

TO BUS

REAL

REACTIVE

1.0000

1.0000

2.0000

1.0000

0.4362

1.0000

2.0000

1.0000

-1.000

71

0.3492

REACTIVE

At maximum operating condition where P = 0.1 & X e = 0.9, LOAD-FLOW STUDY REPORT OF POWER FLOW CALCULATIONS

10-May-2008 : BUS 2

SWING BUS

: 5

 NUMBER OF ITERATIONS

SOLUTION TIME

: 0 sec.

TOTAL TIME

: 0.032 sec.

TOTAL REAL POWER LOSSES

: -1.11022e-016.

TOTAL REACTIVE POWER LOSSES :

1.17678.

GENERATION

LOAD

BUS

VOLTS

ANGLE

REAL

REACTIVE

REAL

1.000

1.0300

60.9017

1.00

0.6222

0

0

2.000

1.0000

0

-1.00

0.5546

0

0

LINE FLOWS

LINE

FROM BUS

TO BUS

REAL

REACTIVE

1.0000

1.0000

2.0000

1.0000

0.6222

1.0000

2.0000

1.0000

-1.0000

0.5546

72

REACTIVE

APPENDIX D: MATLAB CODES & SIMULATION TOOLS D.1

simulation tools

The tuning process described in chapter six was applied to design the PSS of a single machine infinite bus (SMIB) system represented in figure 8.1. The SMIB was designed and modelled using the power system toolbox (PST), depicted in the following section. The system’s parameters are given in Appendix C1

Power System Toolbox PST is a MATLAB based software that provides models of machines and control systems to perform transient stability simulations of a power system, and to build state variable models in small signal analysis and damping controller design [51]. PST is constituted of m-files representing models that can be assembled to tailor an application by following a set of rules described in [51]. Furthermore, PST offers the possibility to perform the load flow calculation in addition to system state space matrices needed for this thesis. The driver m-file is provided for small signal stability analysis (svm_mgen).This function provides an environment that requires only the system data to be specified and act much like stand-alone small signal stability programs [51]. The svm_mgen driver calculates the system’s state matrices, damping ratios, eigenvalues, eigenvectors and participation factors. In addition, the driver can perform the load flow. The svm_mgen was modified to accommodate the DEPSS and GAPSS as detailed in the following section.

Operations As in [38], the approach taken in programming the DEPSS and GAPSS was to have a driver program that would call the PST functions that provide linearized models of the power system, implement the DE and SGA to obtain the PSSs’ parameters and then perform step test responses. 73

Therefore, the structure used in [38] to perform the curve fitting exercise, was modified and adapted to achieve the eigenvalues’ shifting method, described in chapter six, by following the tuning procedure in figure 6.2. The mains m-files used to perform the simulation are described in subsequent sections and the codes are indexed in Appendix D.

smibDriver.m Purpose  Main file and driver for the power system analysis. Description smibDriver (Pmin, Pstep, Pmax, Xemin, Xestep, Xemax) calls other files to linearize the system and to tune the PSS. The function’ inputs are the system operating conditions where ‘P’ is the output power from the generator and ‘Xe’ the reactance of the line.

Pmin, Pstep, Pmax,

Xemin, Xestep and, Xemax  are respectively the minimum, the increment and the maximum power and line’s reactance of the system. smibDriver  outputs numerous graphs such as the open-loop system’ eigenvalues positions on the s-plane and the step response. This function is a modified version of model_developer used in [38].

Model_maker & Step_testerMod.m purpose These m-files are used to form the state matrices of a power system model, linearized about an operating point that is set by a load flow and then, perform modal analysis. Description These MATLAB script files calls the models of the PST to select data files, perform a load flow, form a linearized model by perturbing each state in turn and to do a modal analysis of the given system. These two m-files were developed in [38] by modifying the svm_mgen in the PST toolbox by removing the user prompt and setting the following parameters to default: 74

System frequency: 50Hz Datafiles: Model_maker: smib.m Step_testerMod

: smib2.m

Perform load flow: yes Further modifications were appended to these functions such as the participation matrix and the corresponding eigenvalues. Model_maker.m The function is called as followed, [A_mat, B_mat, C_mat, P_mat, L_vec] = Model_maker (P,Q), The outputs are the state space matrices (A_mat, B_mat, C_mat ), which are used to analyse the open – loop eigenvalues positions in the s-plane together with the damping ratio and the oscillation frequencies. P_mat, L_vec are  the participation matrix and vectors of eigenvalues to identify the most participating poles in the electromechanical modes. Step_testerMod.m The function is called as follows: [A_mat, B_mat, C_mat, P_mat, L_vec] = Step_testerMod (P,Q,T w,  k,T 1, T 2 , T 3,  T 4)  The inputs to this function are the operating conditions and the PSS parameters. The outputs are the state space matrices used to analyse the system closed – loop’ eigenvalues. P_mat and L_vec are also used for the same purpose as in Model_maker.

pssOptimizer.m purpose The purpose of this file is to call dePSS and gaPSS functions. Description

75

This function calls the dePSS and gaPSS to find the optimal PSS’ parameters to be returned to smibDriver for step testing. pssOtimizer further, plots the closed – loop eigenvalues on the s-plane. Invocation: [tw,k,t1,t2,t3,t4] = pssOptimizer (P,Xe).

dePSS.m Purpose Find the optimum parameters to tune the PSS that provide adequate performance over a wide range of operating conditions. Description This function implements DE. The population size and maximum number of generations, the mutation rate and the mutation scale factor are set in this file. This MATLAB script calls the DE toolbox to perform the optimization by evaluating the objective function’s file ‘eigenShiftDE’. Invocation: [tw, k1, t1, t2, t3, t4] = dePSS (Po, X) Where Po and X are the respectively the Power and the line reactance.

gaPSS.m Purpose Find the optimum parameters to tune the PSS to provide a minimum performance over a wide range of operating conditions. Description This function implements SGA. The population size and maximum number of generations, the crossover probability and the mutation scale factor are set in this file. This function uses the GAOT to optimize the objective function ‘eigenShiftGA’ by minimizing the fitness value. Invocation: [tw, k2, t12, t22, t32, t42] = gaPSS (Po, X).

76

eigenShiftDE.m and eigenShiftGA.m These MATLAB scripts are the objective function of DE and GA respectively. They are called to evaluate individuals by their ability to restrict the closed – loop eigenvalues to the left of the relative stability chosen by the user. These m-files call the Step_tester, described in section 7.2, to check for the load flow convergence and calculate the state matrices. The system matrices are then checked for controllability which enables the shifting of the unstable modes and achieve the desired damping.

Program flow chart The interaction between the cited m-files is summarized in figure 7.1 below.

Figure D.9.2  Program flow chart 

77

D.2

MATLAB codes

pssOptimizer.m function[tw,k,t1,t2,t3,t4,k2,t12,t22,t32,t42] = pssOptimizer(P,Xe) %syntax:[tw,k,t1,t2,t3,t4,k2,t12,t22,t32,t42] = pssDesign(Power,Reactance) %pssDesign %calls the functions dePSS and gaPSS to tune the parameters of two PSSs %using the phase compensation technique.Returns PSS parameters %k,t1-t4(from dePSS) and k2,t12-t24(from gaPSS). tw is set as 1 %(a typical value for local mode oscillations). % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% %T. MULUMBA (MLMTSH005) %University of Cape Town %Thesis Project:Optimal Tuning of Power System Stabilizers Based on % Genetic Algorithms %Supervisor: Assoc. Prof. K.A. Folly %13 May 2008 %Using structure from A.A Phiri," Optimal tuning of Power system stabilizer % using Evolution Algorithms" Departement of % Electrical Engineering UCT, Thesis 2007. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% global Xemin Xestep Xemax global Pmin Pstep Pmax P = P Xe = Xe % pause %Obtain parameters from DE [tw,k,t1,t2,t3,t4]=dePSS(P,Xe) %Obtain parameters from GA [tw,k2,t12,t22,t32,t42]= gaPSS(P,Xe) % pause %Calculate matrices [a,b,c,P_mat,L,L_idx]= Step_testerMod(1,0.5,k,tw,t1,t2,t3,t4); L_e = L; P_e = real(P_mat); damp(a); New_eigvalues = []; New_eigvalues1 = []; % pause figure hold for Xe=Xemin:Xestep:Xemax

78

for P=Pmin:Pstep:Pmax P Xe %******************** DIFFEENTIAL EVOLUTION*********************** % %run Step_tester to obtain closed - loop eigenvalues for DE [a_mat,b_mat,c_mat,bus2]=Step_tester2(P,Xe,k,tw,t1,t2,t3,t4);

%

% % % % %

%Store the open loop eigenvalues damp(a_mat); pause evalue1=eig(a_mat); New_eigvalues=[New_eigvalues;evalue1']; %Plot DE C/L eigenvalues subplot(2,1,1) y = -20:20; x = -1*ones(y); plot(x,y,'--r') drawnow plot(real(New_eigvalues),imag(New_eigvalues),'bx') title('System Closed-loop poles with DE') xlabel('Real axis'); ylabel('Imaginary axis');

%**************************************************************** %*********************GENETIC ALGORITHM************************** %run Step_tester to obtain closed - loop eigenvalues for SGA P Xe [a_mat1,b_mat1,c_mat1,bus21]=Step_tester(P,Xe,k2,tw,t12,t22,t32,t42); %Store the Closed-loop eigenvalues damp(a_mat1); evalue2=eig(a_mat1); New_eigvalues1=[New_eigvalues1;evalue2'];

%

%Plot DE C/L eigenvalues subplot(2,1,2) plot(x,y,'--r') drawnow plot(real(New_eigvalues1),imag(New_eigvalues1),'kx') title('System Closed-loop poles with GA') xlabel('Real axis'); ylabel('Imaginary axis');

end end hold off return

79

dePSS.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % T. MULUMBA MLMTSH005 %University of Cape Town %Thesis Project:Optimal Tuning of Power System Stabilizers Based on % Evolution Algorithms %Supervisor: Assoc. Prof. K.A. Folly %13 May 2008 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [tw,k,t1,t2,t3,t4] = dePSS(Power,Reactance) %******************************************************************** **** %syntax:[tw,k,t1,t2,t3,t4]=dePSS(Power,reactance) %dePSS: Differential Evolution for tuning power system stabilizer %Uses eigenvalues shifting methods % %Implemented using DeMat package by J.V. Zandt and A. Neumaier %K.V. Price, R.M. Storm and J.A. Lampinen,"Differential Evolution: % A practical approach to global optimization,"Springer 2005. %******************************************************************** **** % Operating condition at which the controller is optimally tuned F_P = Power; F_Xe= Reactance; %******************************************************************** % Script file for the initialization and run of the differential % evolution optimizer. %******************************************************************** % F_VTR

"Value To Reach" (stop when ofunc < F_VTR) F_VTR = -1;

% I_D

number of parameters of the objective function I_D = 5;

% FVr_minbound,FVr_maxbound vector of lower and bounds of initial population % the algorithm seems to work especially well if [FVr_minbound,FVr_maxbound] % covers the region where the global minimum is expected % *** note: these are no bound constraints!! *** FVr_minbound = [16 0.6 0.02 0.02 0.02]; FVr_maxbound = [20 4 2 2 2]; I_bnd_constr = 1; %1: use bounds as bound constraints, 0: no bound constraints % I_NP

number of population members I_NP = 100;

% I_itermax maximum number of iterations (generations) I_itermax = 100;

80

% F_weight DE-stepsize F_weight ex [0, 2] F_weight = 0.8; % F_CR

crossover probabililty constant ex [0, 1] F_CR = 1.0;

% I_strategy % % % % robustness % % % % % % robust. % % robust. % % %

1 --> DE/rand/1: the classical version of DE. 2 --> DE/local-to-best/1: a version which has been used by quite a number of scientists. Attempts a balance between and fast convergence. 3 --> DE/best/1 with jitter: taylored for small population sizes and fast convergence. Dimensionality should not be too high. 4 --> DE/rand/1 with per-vector-dither: Classical DE with dither to become even more 5 --> DE/rand/1 with per-generation-dither: Classical DE with dither to become even more Choosing F_weight = 0.3 is a good start here. 6 --> DE/rand/1 either-or-algorithm: Alternates between differential mutation and three-point-recombination.

I_strategy = 1; % I_refresh "I_refresh" % % I_refresh

intermediate output will be produced after iterations. No intermediate output will be produced if I_refresh is < 1 = 5;

% I_plotting Will use plotting if set to 1. Will skip plotting otherwise. I_plotting = 0; %******************************************************************** ******* % Problem dependent but constant values. For speed reasons these values are % defined here. Otherwise we have to redefine them again and again in the % cost function or pass a large amount of parameters values. %******************************************************************** ******* %------------------------------------------------S_struct.F_P = F_P; S_struct.F_Xe = F_Xe; S_struct.I_NP = I_NP; S_struct.F_weight = F_weight; S_struct.F_CR = F_CR; S_struct.I_D = I_D; S_struct.FVr_minbound = FVr_minbound;

81

S_struct.FVr_maxbound S_struct.I_bnd_constr S_struct.I_itermax S_struct.F_VTR S_struct.I_strategy S_struct.I_refresh S_struct.I_plotting

= = = = = = =

FVr_maxbound; I_bnd_constr; I_itermax; F_VTR; I_strategy; I_refresh; I_plotting;

%******************************************************************** % Start of optimization %******************************************************************** [FVr_x,S_y,I_nf] = deopt('eigenShiftDE',S_struct);

tw = 2; k = FVr_x(1); t1 = FVr_x(2); t2 = FVr_x(3); t3 = FVr_x(4); t4 = FVr_x(5);

eigenshiftDE.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Function: S_MSE= eigenShift(FVr_temp, S_struct) % Author: Rainer Storn % Description: Implements the cost function to be minimized. % Parameters: FVr_temp (I) Paramter vector % S_Struct (I) Contains a variety of parameters. % For details see Rundeopt.m % Return value: S_MSE.I_nc (O) Number of constraints % S_MSE.FVr_ca (O) Constraint values. 0 means % the constraints % are met. Values > 0 measure the % distance % to a particular constraint. % S_MSE.I_no (O) Number of objectives. % S_MSE.FVr_oa (O) Objective function values. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % T. MULUMBA MLMTSH005 %University of Cape Town %Thesis Project:Optimal Tuning of Power System Stabilizers Based on % Evolution Algorithms %Supervisor: Assoc. Prof. K.A. Folly %13 May 2008 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function S_MSE= eigenShiftDE(FVr_temp, S_struct)

82

%---Assign temporary values to parameters----------------------------------tw=2.0; k=FVr_temp(1); t1=FVr_temp(2); t2=FVr_temp(3); t3=FVr_temp(4); t4=FVr_temp(5); F_P = S_struct.F_P; F_Xe= S_struct.F_Xe; % %---State space matrices--------------------------------------[a,b,c,P_mat,L,L_idx]= Step_testerMod2(F_P,0.7,k,tw,t1,t2,t3,t4); a_mat = a; b_mat = b; d_mat = 0; L_vec = L; P_real = real(P_mat); L_idxVec = L_idx; %----Check for controllability--------------------------------mat_contr = ctrb(a_mat,b_mat); det_check = det(mat_contr); %Condition for controllability if det_check ~= 0 %find the electromechanical modes P_row = P_real(1,:); spd_mode = P_row - max(P_row); dummy = find(spd_mode == 0); lamnda = L_vec(dummy(1)); %Set conditions d = L_vec > 0; %objective function J = real(lamnda) + 2.0; sigma = -real(lamnda)/sqrt(real(lamnda)^2+... imag(lamnda)^2); %Check if individuals meet the conditions. Penalize if otherwise if sum(d)>0 J = J+100; end if (sigma < 0.2 || sigma > 0.4) J = J+100; end if (imag(lamnda)> 4.10 || imag(lamnda)< 3.85) J = J+100; end F_val =J+10; else disp('THE SYSTEM IS NOT CONTROLLABLE THUS CANNOT SETTLE') disp('THE EIGENVALUE SHIFT CANNOT BE PERFORMED')

83

disp('Change the system operating conditions') end %----strategy to put everything into a cost function-----------S_MSE.I_nc = 1;%no constraints S_MSE.FVr_ca = 0;%no constraint array S_MSE.I_no = 1;%number of objectives (costs) S_MSE.FVr_oa(1) = F_val;

gaPSS.m function [tw,k2,t12,t22,t32,t42]=gaPSS(Po,X) %syntax:[tw,k2,t12,t22,t32,t42]=gaPSS(Power,reactance) %gaPSS: genetic algorithm for tuning power system stabilizer %Uses phase compensation technique % %Implemented using Genetic Algorithm for Optimisation Toolbox(GAOT) by %C. Houck,J. Joines and M. Kay %http://www.ise.ncsu.edu/mirage/GAToolBox/gaot/ % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% %T. MULUMBA (MLMTSH005) %University of Cape Town %Thesis Project:Optimal Tuning of Power System Stabilizers Based on % Genetic Algorithms %Supervisor: Assoc. Prof. K.A. Folly %13 May 2008 %Modified structure from A.A Phiri," Optimal tuning of Power system stabilizer using Evolution Algorithms" Departement of Electrical Engineering UCT, Thesis 2007. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% global Pw global Xr Pw = Po; Xr = X; % Setting the seed back to the beginning for comparison sake rand('seed',0) % Crossover Operators xFns = 'simpleXover'; xOpts = [0.65]; % Mutation Operators mFns = 'binaryMutation'; mOpts = [0.1]; % Termination Operators termFns = 'maxGenTerm';

84

termOps = [100]; % 200 Generations % Selection Function selectFn = 'roulette'; selectOps = []; % Evaluation Function evalFn = 'eigenShiftGA'; evalOps = []; % Bounds on the variables bounds = [16 20;0.6 4;0.02 2;0.02 2;0.02 2]; % GA Options [epsilon gaOpts=[1e-6 0 1];

float1/binary0

display]

% Generate an intialize population of size 20 startPop = initializega(100,bounds,'eigenShiftGA',[],[1e-6 0]); % Run the GA [x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,... termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);

% Plot the best over time clf plot(trace(:,1),trace(:,2)); %Return variables tw=2; k2=x(1); t12=x(2); t22=x(3); t32=x(4); t42=x(5); return

85

eigenshiftGA.m function [sol,val] = eigenShiftGA(sol,options) %syntax:[sol,val] = eigenShiftEval(sol,options) %Fitness function for Gentic Algorithm %Calculates the fitness of the population,T. %The performance of each individual is calculated according to its ability %to restrain all eigenvalues to the left of the relative stability, thus %minimizing the objective function J. % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% %T. MULUMBA (MLMTSH005) %University of Cape Town %Thesis Project:Optimal Tuning of Power System Stabilizers Based on % Genetic Algorithms %Supervisor: Assoc. Prof. K.A. Folly %13 May 2008 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% global Pw global Xr tw=2; k=sol(1); t1=sol(2); t2=sol(3); t3=sol(4); t4=sol(5); %Check for load flow and get the system state space matrices [a,b,c,P_mat,L,L_idx]= Step_testerMod(1,0.7,k,tw,t1,t2,t3,t4); a_mat = a; b_mat = b; L_vec = L; P_real = real(P_mat); %real values of participation factor matrix L_idxVec = L_idx; [wn,z,e_v]=damp(a_mat); %Check for the controllability of the system p = ctrb(a_mat,b_mat); det_check = det(p); %Condition for controllability if det_check ~= 0 %get natural frequency, damping ratio and system poles %[Wn,z,P] = damp(a_mat); P_row = P_real(1,:); spd_mode = P_row - max(P_row); dummy = find(spd_mode == 0); lamnda = L_vec(dummy(1)); d = L_vec > 0;

86

%objective function J = real(lamnda) + 2.0; sigma = -real(lamnda)/sqrt(real(lamnda)^2+... imag(lamnda)^2);

%A = 0;B = 0; C = 0; if sum(d)>0 J = J+100; end if (sigma < 0.2 || sigma > 0.4) J = J+100; end if (imag(lamnda)> 4.10 || imag(lamnda)< 3.85) J = J+100; end val =J+10; else disp('System is not controllable') end

87

APPENDIX E: SIMULATION RESULTS DATA & GRAPHS E.1

Data

The participation matrices used to track the eigenvalues associated with the electromechanical modes are provided below relative to the operating condition. Each columns of the matrix represents an eigenvalue. Each row is associated with a state. Minimal condition The participation matrix of the open loop system is given below: No PSS: P_mat minimum condition -0.0393

0.5197

0.5197

0

0

0

∆ω 

-0.0393

0.5197

0.5197

0

0

0

∆δ 

0.0003

-0.0002

-0.0002

0.4441

0.4441

0.1119

∆ E q

1.0782

-0.0391

-0.0391

0

0

0

∆ E d 

0

0

0

0.1401

0.1401

0.7198

∆ Avr 

0

-0.0001

-0.0001

0.4159

0.4159

0.1683

∆ Avr 

eigen 1

eigen 2

eigen 3

eigen 4

eigen 5

eigen 6

Eigenvalues -1.205 -0.4222 - 5.0371i -0.4222 + 5.0371i -5.5423 - 18.1786i -5.5423 + 18.1786i -59.1579 + 0i

eigen 1 eigen 2 eigen 3 eigen 4 eigen 5 eigen 6

88

' '

Nominal system condition At this operating condition, the open loop exhibit undamped oscillatory behaviour. Therefore, the PSSs are included to improve stability. No PSS:  P_mat -0.0119 -0.0119

0.50702 0.50702

0.50702 0.50702

-0.0011 -0.0011

-0.0011 -0.0011

9.8E-05 9.8E-05

∆ω 

0.00379

0.00134

0.00134

0.4431

0.4431

0.10733

∆ E q

1.02051

-0.0101

-0.0101

-0.0002

-0.0002

-7E-07

∆ E d 

-0.0001

-0.0016

-0.0016

0.1366

0.1366

0.73011

∆ Avr 

-0.0003

-0.0037

-0.0037

0.42271

0.42271

0.16236

∆ Avr 

eigen 1

eigen 2

eigen 3

eigen 4

eigen 5

eigen 6

Eigenvalues 0.0212 + 3.9827i 0.0212 - 3.9827i -1.728 -5.9046 +17.2474i -5.9046 -17.2474i -58.5794

eigen 1 eigen 2 eigen 3 eigen 4 eigen 5 eigen 6

CPSS:  P_mat Eigenvalues -0.447 -1.844 -1.306 -1.644i -1.306+1.644i -2.775 -4.282i -2.775 +4.282i -4.284 -16.78i -4.284+16.78i -58.62

eigen 1 eigen 2 eigen 3 eigen 4 eigen 5 eigen 6 eigen 7 eigen 8 eigen 9

89

∆δ  '

'

GAPSS: P_mat  0.090173

-0.0707

-0.0707

0.084897

0.489728

0.489728

-0.00661

-0.00661

9.48E-05

∆ω 

-0.00127

-0.05016

-0.05016

-0.0089

0.591216

0.591216

-0.03641

-0.03641

0.000866

∆δ 

0.001821

0.01401

0.01401

0.004821

-0.0936

-0.0936

0.522185

0.522185

0.108168

∆ E q

0.003369

0.109517

0.109517

0.772733

0.002621

0.002621

-0.00019

-0.00019

-7.5E-07

∆ E d 

-0.00084

-0.0061

-0.0061

-0.00405

0.040243

0.040243

0.104655

0.104655

0.727305

∆ Avr 

-3.6E-05

-0.00199

-0.00199

-0.0005

0.002612

0.002612

0.417858

0.417858

0.163571

∆ Avr 

0.826683

0.105967

0.105967

-0.02777

-0.00542

-0.00542

-5.8E-06

-5.8E-06

-2.6E-08

∆v1

0.067885

0.435924

0.435924

0.533344

-0.24651

-0.24651

0.009959

0.009959

3E-05

∆v2

0.012214

0.463533

0.463533

-0.35458

0.219111

0.219111

-0.01144

-0.01144

-3.4E-05

∆v3

eigen 1

eigen 2

eigen 3

eigen 4

eigen 5

eigen 6

eigen 7

eigen 8

eigen 9

-0.0066 -0.0364

-0.00661 -0.03641

9.5E-05 0.00087

∆ω 

' '

Eigenvalues -0.4449 -1.2115 + 0.5666i -1.2115 - 0.5666i -1.4862 + 3.7973i -1.4862 - 3.7973i -1.772 -4.4269 +16.9785i -4.4269 -16.9785i -58.6272

eigen 1 eigen 2 eigen 3 eigen 4 eigen 5 eigen 6 eigen 7 eigen 8 eigen 9

DEPSS: P_matrix  0.09017 -0.0707 -0.0013 -0.0502

-0.0707 -0.0502

0.0849 -0.0089

0.48973 0.48973 0.59122 0.59122

0.00182

0.01401

0.01401

0.00482 -0.0936

-0.0936

0.52218

0.522185

0.10817

∆ E q

0.00337

0.10952

0.10952

0.77273 0.00262

0.00262 -0.0002

-0.00019

-7E-07

∆ E d 

-0.0008 -4E-05

-0.0061 -0.002

-0.0061 -0.002

-0.004 -0.0005

0.04024 0.04024 0.10465 0.104655 0.7273 0.00261 0.00261 0.41786 0.417858 0.16357

∆ Avr 

0.82668

0.10597

0.10597

-0.0278

-0.0054

-0.0054

-6E-06

-5.8E-06

-3E-08

∆v1

0.06788

0.43592

0.43592

0.53334 -0.2465

-0.2465

0.00996

0.009959

3E-05

∆v2

0.01221

0.46353

0.46353

-0.3546

0.21911

0.21911

-0.0114

-0.01144

-3E-05

∆v3

eigen 1

eigen 2

eigen 3

eigen 4

eigen 5

eigen 6

eigen 7

eigen 8

eigen 9

90

∆δ  '

'

∆ Avr 

Eigenvalues -4.49E-01 -1.448-0.652i -1.448-0.652i -1.98E+00 -1.581 -3.93i -1.581 -3.93i -4.507-16.94i -4.507-16.94i -58.62

eigen 1 eigen 2 eigen 3 eigen 4 eigen 5 eigen 6 eigen 7 eigen 8 eigen 9

Maximum system condition No PSS: P_matrix 

0.0061895

0.4978948

0.497895

-0.00103

-0.00103

8.26E-05

∆ω 

0.0061895

0.4978948

0.497895

-0.00103

-0.00103

8.26E-05

∆δ 

-0.001758

0.0031671

0.003167

0.442851

0.442851

0.109723

∆ E q

0.9891524

0.0053913

0.005391

3.32E-05

3.32E-05

-1.38E-06

∆ E d 

6.214E-05

-0.001294

-0.00129

0.138998

0.138998

0.724529

∆ Avr 

0.0001644

-0.003054

-0.00305

0.420179

0.420179

0.165585

∆ Avr 

eigenval1

eigenval2

eigenval3

eigenval4

eigenval5

eigenval6

Eigenvalues  -1.77

eigenval1

0.0885 - 2.8462i

eigenval2

0.0885 + 2.8462i

eigenval3

-5.7253 -17.7221i

eigenval4

-5.7253 +17.7221i

eigenval5

-58.8874

eigenval6

91

'

'

CPSS: P_matrix  0.0286773

0.0071147

0.522018

0.522018

-0.0748

-2.74E-16

-0.00256

-0.00256

8.18E-05

∆ω 

-0.000202

0.0048303

0.4379

0.4379

0.150617

3.54E-15

-0.01581

-0.01581

0.000579

∆δ 

0.0002217

-0.001409

-0.01612

-0.01612

-0.03432

2.24E-13

0.478746

0.478746

0.110254

∆ E q

-0.000161

0.9831655

0.006745

0.006745

0.003437

6.75E-14

3.48E-05

3.48E-05

-1.39E-06

∆ E d 

-0.000142

-3.4E-05

0.006648

0.006648

0.023019

-1E-13

0.12057

0.12057

0.722722

∆ Avr 

-3.45E-06

0.0001312

-0.0014

-0.0014

0.008521

-2.15E-13

0.413902

0.413902

0.166345

∆ Avr 

0.9694355

-0.001065

0.010284

0.010284

0.010964

6.86E-17

4.84E-05

4.84E-05

7.69E-08

∆v1

0.002893

0.005811

0.110177

0.110177

-0.48848

1.216204

0.021562

0.021562

9.13E-05

∆v2

-0.00071

0.0014554

-0.07626

-0.076 26

1.401041

-0.21624

-0.0 1649

-0.01649

-7.22E- 05

∆v3

eigenval1

eigenval2

eigenval3

eigenval4

eigenval5

eigenval6

eigenval7

eigenval8

eigenval9

' '

Eigenvalues -0.2429 -1.7654 -0.233 - 2.6355i -0.233 + 2.6355i -4.0869 -10 -4.9280 -17.6448i -4.928 +17.6448i -58.9166

eigenval1 eigenval2 eigenval3 eigenval4 eigenval5 eigenval6 eigenval7 eigenval8 eigenval9

GAPSS: P_mat ∆ω 

0.121986 -0.18468 -0.00263 -0.0267

-0.1847 -0.0267

0.00826 0.00216

0.6229 0.55176

0.6229 0.55176

-0.0034 -0.0252

-0.0034 -0.0252

8.1E-05 0.0008

0.00168

0.001176

0.00118

-0.0007

-0.0575

-0.05754

0.50064

0.50064

0.11049

-0.00158 -0.00109 -4.2E-05

-0.0382 -0.0013 -0.00076

-0.0382 -0.0013 -0.0008

1.0654 -0.0002 6.5E-05

0.00625 0.0283 0.00304

0.00625 0.0283 0.00304

3.6E-05 0.11266 0.41437

3.6E-05 0.11266 0.41437

-1.4E-06 0.72195 0.16668

∆ E d 

0.781547

0.122326

0.12233

-0.0036

-0.0113

-0.01135

2.4E-05

2.4E-05

3E-08

∆v1

0.67354

-0.0115

-0.2177

-0.21771

0.00341

0.00341

1.2E-05

∆v2

0.007139 0.454596 0.4546

-0.0599

0.07432

0.07432

-0.0025

-0.0025

-9E-06

∆v3

eigenval1

eigenval4

eigenval5

eigenval6

eigenval7

eigenval8

eigenval9

0.093033 0.67354

eigenval2

eigenval3

92

∆δ  '

∆ E q '

∆ Avr  ∆ Avr 

Eigenvalues -0.4288 -1.301 - 0.685i -1.301 + 0.685i -1.7526 -1.068 - 2.5i -1.068 + 2.5i -4.55 -17.52i -4.55 +17.52i -58.9294

eigenval1 eigenval2 eigenval3 eigenval4 eigenval5 eigenval6 eigenval7 eigenval8 eigenval9

DEPSS: P_mat 0.114404 -0.02554 -0.1949 -0.00257 0.003092 -0.1882

-0.1949 -0.1882

0.65373 0.7143

0.65373 0.7143

-0.0033 -0.0268

-0.0033 -0.0268

8.1E-05 0.00075

∆ω 

0.001595

-0.00045

0.02849

0.02849

-0.0834

-0.08345

0.49917

0.49917

0.11044

∆ E q

-0.00151

1.000717 -0.0023

-0.0023

0.00262

0.00262

3.6E-05

3.6E-05

-1.4E-06

∆ E d 

-0.00104 -4.0E-05

0.001025 -0.0189 4.38E-05 -0.0047

-0.0189 -0.0047

0.04272 0.00454

0.04272 0.00454

0.11511 0.41684

0.11511 0.41684

0.72213 0.16661

∆ Avr 

0.794598

0.010367

0.13178

0.13178

-0.0343

-0.03426

-5E-06

-5E-06

-2.2E-08

∆v1

0.081483 -0.10575

1.04665

1.04665

-0.5417

-0.54171

0.00717

0.00717

2.6E-05

∆v2

0.013073 0.116504 0.20198

0.20198

0.2415

0.2415

-0.0082

-0.0082

-2.9E-05

∆v3

eigenval1

eigenval4

eigenval5

eigenval6

eigenval7

eigenval8

eigenval9

eigenval2

eigenval3

Eigenvalues -0.2429 -1.7654 -0.3233 - 2.6355i -0.3233 + 2.6355i -4.0869 -10 -4.9280 -17.6448i -4.9280 +17.6448i -58.9166

eigenval1 eigenval2 eigenval3 eigenval4 eigenval5 eigenval6 eigenval7 eigenval8 eigenval9

93

∆δ  ' '

∆ Avr 

E.2

Graphs

This section includes the graphs tested for robustness. A step change of 0.001 in input voltage was applied for the all range of operating condition.

x 10

-3

Step response for op cond. P= 0.6 and Q=0.17203 at Xe=0.7

13

DEPSS GAPSS

12

CPSS

11    ) 10   u  .   p    ( 9    t    V   n    i 8   e   g   n   a    h 7   c

6 5 4 3 0

1

2

3

4

5 time(s)

6

7

8

9

10

Figure E.1: Xe = 0.7 and P =0.6 

-3

x 10

Step response for op cond. P=0.7 and Q=0.22131 at Xe=0.7

13

DEPSS GAPSS

12

CPSS

11    ) 10   u  .   p    ( 9    t    V   n    i   e 8   g   n   a    h 7   c

6 5 4 3 0

1

2

3

4

5 time(s)

6

7

Figure E.2 : Xe = 0.7 and P = 0.7 

94

8

9

10

x 10

-3

Step response for op cond. P= 0.8 and Q=0.28062 at Xe=0.7 DEPSS

13

GAPSS 12

CPSS

11    ) 10   u  .   p    ( 9    t    V   n    i   e 8   g   n   a    h 7   c

6 5 4 3 0

1

2

3

4

5 time(s)

6

7

8

9

10

Figure E.3 : X = 0.7 and P = 0.8 

x 10

-3

Step response for op cond. P= 0.9 and Q=0.35148 at Xe=0.7 DEPSS GAPSS CPSS

13 12 11    )   u  .   p    (    t    V   n    i   e   g   n   a    h   c

10 9 8 7 6 5 4 3 0

1

2

3

4

5 time(s)

6

7

Figure E.4 : X = 0.7 and P = 0.9 

95

8

9

10

x 10

-3

Step response for op cond. P=0.6 and Q=0.18698 at Xe=0.8 DEPSS

13

GAPSS 12

CPSS

11    ) 10   u  .   p    ( 9    t    V   n    i   e 8   g   n   a    h 7   c

6 5 4 3 0

1

2

3

4

5 time(s)

6

7

8

9

10

Figure E.5 : Xe = 0.8 and P = 0.6 

x 10

-3

Step response for op cond. P=0.7 and Q=0.24554 at Xe=0.8

13

DEPSS

12

GAPSS CPSS

11    ) 10   u  .   p    ( 9    t    V   n    i   e 8   g   n   a    h 7   c

6 5 4 3 0

1

2

3

4

5 time(s)

6

Figure E.6 : Xe = 0.8 and P = 0.7 

96

7

8

9

10

x 10

-3

Step response for op c ond. P=0.8 and Q=0.31734 at Xe=0.8

13

DEPSS

12

GAPSS CPSS

11    )   u  .   p    (    t    V   n    i   e   g   n   a    h   c

10 9 8 7 6 5 4 3 0

1

2

3

4

5 time(s)

6

7

8

9

10

Figure E.7:  X =0.8 and P = 0.8  -3

x 10

Step response for op cond. P=0.9 and Q=0.40544 at Xe=0.8 DEPSS GAPSS CPSS

13 12 11    )   u  .   p    (    t    V   n    i   e   g   n   a    h   c

10 9 8 7 6 5 4 3

0

1

2

3

4

5 time(s)

6

7

Figure E.8 : X =0.8 and P = 0.9 

97

8

9

10

x 10

-3

Step response for op cond. P=1 and Q=0.51516 at Xe=0.8 DEPSS GAPSS CPSS

13 12 11    ) 10   u  .   p    ( 9    t    V   n    i   e 8   g   n   a    h 7   c

6 5 4 3 0

1

2

3

4

5 time(s)

6

7

8

9

10

Figure E.9 Xe = 0.8 and P = 1.0 

x 10

-3

Step response for op cond. P=0.6 and Q=0.20423 at Xe=0.9 DEPSS

13

GAPSS 12

CPSS

11    ) 10   u  .   p    (    t 9    V   n    i   e 8   g   n   a    h 7   c

6 5 4 3 0

1

2

3

4

5 time(s)

6

7

Figure E.10 Xe = 0.9 and P = 0.6 

98

8

9

10

x 10

-3

Step response for op cond. P=0.7 and Q=0.27338 at Xe=0.9 DEPSS GAPSS CPSS

13 12 11    ) 10   u  .   p    ( 9    t    V   n    i   e 8   g   n   a    h 7   c

6 5 4 3 0

1

2

3

4

5 time(s)

6

7

8

9

10

Figure E.11: X = 0.9 and P = 0.7 

x 10

-3

Step response for op cond. P= 0.8 and Q=0.36039 at Xe=0.9 DEPSS GAPSS CPSS

13 12 11    )   u  .   p    (    t    V   n    i   e   g   n   a    h   c

10 9 8 7 6 5 4 3 0

1

2

3

4

5 time(s)

6

7

Figure E.12 : X = 0.9 and P = 0.8 

99

8

9

10

-3

x 10

Step response for op cond. P=0.9 and Q=0.47185 at Xe=0.9 DEPSS

13

GAPSS 12

CPSS

11    ) 10   u  .   p    ( 9    t    V   n    i   e 8   g   n   a    h 7   c

6 5 4 3 0

1

2

3

4

5 time(s)

6

7

Figure E.13: X = 0.9 and P = 0.9 

100

8

9

10