1
Tower Modeling Mod eling for f or Lightning Analysis of Overhead Transmission Transmission Lines Line s Juan A. Martinez, Member, IEEE , and Ferley Castro-Aranda, Student Member, IEEE
Abstract—A
sensitivity study of the lightning flashover rate of transmission lines is presented in this paper. The study is aimed at analyzing the influence of the tower model on the flashover rate derived from the application of a time-domain Monte Carlo procedure. The main conclusion from the simulation results presented in this paper is that the selected tower model can significantly affect the flashover rate and its influence will increase with the tower height. Index Terms—Transmission Lines, Lightning Overvoltages, Insulation Coordination, Modeling, Simulation.
I. INTRODUCTION
M
odeling for simulation of lightning overvoltages is a very difficult task. Although modern simulation tools, e.g. EMTP-like programs, allow users to implement very advanced models and reproduce transients very accurately, there is no full agreement on the most adequate representation for those parts of an overhead transmission line involved in lightning overvoltages. Due to the nature of the physical phenomena caused by lightning, an extremely complicated model would be needed to obtain accurate enough results. The experience has proved that some simplified models can be used to represent some parts of the line in many cases; e.g. a constant distributed-parameter representation of line conductors will usually suffice in these studies [1]. On the other hand, the random nature of lightning adds some uncertainties (e.g. no two lightning strokes are the same [2]), which makes difficult the validation of models using actual lightning records. One of the transmission line parts whose influence can be significant is the tower. As pointed out in a recent publication [3], the response of a tower is an electromagnetic problem, although its study is usually relied on a circuit approach, i.e. the tower is r epresented by means of one or several line/surge line/surge impedance sections that are assembled taking into account the tower structure [4] – [11]. The various models proposed to date for representing the tower response to a lightning stroke can be categorized in several ways, see [3] and [12]. Basically, two approaches have been applied: models have been derived from experimental results or from theoretical studies. Two important aspects mentioned in [3] are the different Juan A. Martinez and Ferley Castro-Aranda are with the Departament d’Enginyeria Elèctrica, Universitat Politècnica de Catalunya, 08028 Barcelona, Spain.
definitions definitions on which the calculation of the surge impedance is based and the different waveshapes used to represent the injected current. This paper presents the main results of a study aimed at analyzing the influence of the tower representation on the lightning flashover rate of overhead transmission lines. Several approaches were selected to represent a tower. Section II provides a summary of modeling guidelines for lightning simulations using a time-domain simulation tools, i.e. an EMTP-like program. A review of tower models is presented in Section III. The main principles of the Monte Carlo procedure used to obtain flashover rates are summarized in Section IV. The configuration of the test lines and the main simulation results derived with the various tower models used in this paper are presented in Section V. II. MODELING GUIDELINES Modeling guidelines used in this work are summarized below [1], [13]. • The transmission line is represented by several multi-phase multi-phase untransposed distributed parameter line spans at both sides of the point of impact. This representation is made by using either a constant-parameter model, whose whose parameters are calculated at 500 kHz. • A line termination is needed at each side of the above model to prevent reflections. In this work, this is achieved by adding a long enough section at each side. • Footing impedance modeling is one of the most critical aspects. The option chosen in this work is a nonlinear resistance RT given by RT =
Ro
(1)
1 + I / I g
being Ro the footing resistance at low current and low frequency, I g the limiting current to initiate sufficient soil ionization, I the stroke current through the resistance. The limiting current is given by I g =
E o ρ
2π Ro2
(2)
where ρ is the soil resistivity (ohm-m) and E 0 is the soil ionization gradient (400 kV/m). • The representation of insulator strings is based on the leader progression model. When the applied voltage exceeds the corona inception voltage, streamers propagate along the insulator string; if the voltage remains high enough, these streamers will become a leader channel. A
2
flashover occurs when the leader crosses the gap between the cross-arm and the conductor. The total time to flashover can be expressed as follows (3) t t = t c + t s + t l where t c is the corona inception time, t s is the streamer propagation time and t l is the leader propagation time. Usually t c is neglected, while t s is calculated as follows t s =
E 50
(4)
1.25 E − 0.95 E 50
where E 50 is the average gradient at the critical flashover voltage and E is the maximum gradient in the gap before breakdown. The leader propagation time, t l, can be obtainned from the following equation dl dt
= k lV (t )
V (t ) g −l
(5)
− E l 0
where V(t) is the voltage across the gap, g is the gap length, l is the leader length, E l0 is the critical leader inception gradient, and k l is a leader coefficient. The leader propagation stops if the gradient in the unbridged part of the gap falls below E l0. • Phase voltages at the instant at which the lightning stroke impacts the line must be included. For statistical calculations, they are deduced by randomly determining the phase voltage reference angle and considering a uniform distribution between 0º and 360º. • The lightning stroke is represented as a current source. Fig. 1 shows the concave waveform chosen in this work, it is the so-called Heidler model. Lightning stroke parameters are assumed independently distributed, being their statistical behavior approximated by a log-normal distribution, with the following probability density function [2] p( x) =
1 2π xσ ln x
exp − 0.5
ln x − ln xm
2
(6)
σ ln x
where σ lnx is the standard deviation of lnx, and xm is the median value of x. kA I100
IP
I90
I50
I30
t30 t90
th
time
Fig. 1. Parameters of a return stroke – Concave waveform ( I 100 = peak current magnitude, t f (= 1.67(t 90 – t 30))= rise time, t h = tail time).
III. TOWER MODELING. A REVIEW Although the lightning response of a transmission line tower is an electromagnetic phenomenon, the representation of a tower is usually made in circuit terms. There are some reasons to support this approach: such representation can be implemented in general purpose simulation tools, e.g. EMTPlike programs, and it is easy to understand by the practical engineer. Several tower models have been developed over the years, they were developed using a theoretical approach [4], [5], [9] - [11] or based on an experimental work [8]. The simplest representation is a lossless distributed-parameter transmission line, characterized by a surge impedance and a travel time. The first models were deduced by assuming a vertical leader channel that hits at the tower top. In fact, the response of a tower is different to horizontal stroke currents (the return stroke hits somewhere in midspan) from the response to vertical stroke currents (the return stroke hits at the tower top). In addition, the surge impedance of the tower varies as the wave travels from top to ground. To cope with this behavior, some corrections have been introduced into the first models and more complicated models have been developed: they are based on non-uniform transmission lines, [14], [15], or on a combination of lumped and distributed-parameter circuit elements [8] – [11]. The latter approach is also motivated by the fact that in many cases it is important to obtain the lightning overvoltages across insulators located at different heights above ground; this is particularly important when two or more transmission lines with different voltage levels are sharing the same tower. In this paper only constant-parameter line models are analyzed. Basically, the models based on a constant-parameter circuit representation can be classified into three groups: they were deduced by representing the tower as a single vertical lossless line, a multiconductor vertical line or a multistory model. A short summary of each approach is provided below. a) Single vertical lossless line models The first models were developed by using electromagnetic field theory, representing the tower by means of simple geometric forms, and assuming a vertical stroke to the tower top. Wagner and Hileman used a cylindrical model and concluded that the tower impedance varies as the wave travels down to ground [4]. Sargent and Darveniza used a conical model and suggested a modified form for the cylindrical model [5]. Chisholm et al. proposed a modified equation for the above models in front of a horizontal stroke current and recommended a new model for waisted towers [7]. This latter model was recommended by CIGRE [13], although the version presently implemented in the Flash program is a modified one [12]. The other models have been also implemented in the Flash program [16]. Although the surge propagation velocity along tower elements can be assumed that of the light, the multiple
3
paths of the lattice structure and the cross-arms introduce some time delays; consequently the time for a complete reflection from ground is longer than that obtained from a travel time whose value is the tower height divided by the speed of light. Therefore, the propagation velocity in the above models was reduced to include these effects in the tower response. b) Multiconductor vertical line models Each segment of the tower between cross-arms is represented as a multiconductor vertical line, which can be reduced to a single conductor. The tower model is then a single-phase line whose section increases from top to ground, as shown in Fig. 2. This representation has been analyzed in several references [9] - [11]. A modified model was presented in [10] which included the effect of bracings (represented by lossless lines in parallel to the main legs) and cross-arms (represented as lossless line branched at junction points), being the final representation that shown in Fig. 3. c) Multistory model It is composed of four sections that represent the tower sections between cross-arms. Each section consists of a lossless line in series with a parallel R-L circuit, included for attenuation of the traveling waves, see Fig. 4. The parameters of this model were deduced from experimental results. The values of the parameters, and the model itself, have been revised in more recent years [17]. The approach was originally developed for representing towers of UHV transmission lines. A study presented in [18] concluded that it is not adequate for representing towers of lower voltage transmission lines; according to this study, the tower model for shorter towers can be simpler than that assumed by the multistory model, i.e. four lossless lines with a smaller surge impedance would suffice. In any case, the propagation velocity is that of the light. For more details on models and parameter calculations the reader is referred to the original papers. Note that the overvoltages that can be obtained by means of digital simulation when the simplest models are used should be the same between terminals of all insulator strings, since these models do not distinguish between line phases. In fact, some differences will result due to the different c oupling between the shield wires and the phase conductors located at different heights above ground.
DT1 rT1 r1 r2
ZT2
DT3 rT3 r3
ZT3
D T4 rT4 h 1 h 2 h3 h 4 r4
ZT4
D’B DB
rB
Fig. 2. Multiconductor vertical line model. DT1 rT1
ZA1
DT2 rT2
ZA2
DT3 rT3
ZA3
DT4 rT4
ZA4
ZT1
ZL1
ZT2
ZL2
ZT3
ZL3
ZT4
ZL4
h1 h2 h3 h4
D’B DB
rB
Fig. 3. Multiconductor vertical line model, including bracings and cross-arms. r1 ZT1,h1,vt h1
R1
L1 ZT2,h2,vt
h2
R2
L2 ZT3,h3,vt
h3
R3
L3 ZT4,h4,vt
r2
R4
IV. MONTE CARLO PROCEDURE The following paragraphs detail the most important aspects of the procedure developed for the calculation of the lightning flashover rates of transmission lines [19]. a) The calculation of random values includes the parameters of the lightning stroke (peak current magnitude, rise time, tail time, and location of the leader channel), phase conductor voltages, the footing resistance and the insulator strength.
ZT1
DT2 rT2
h4
r3
Fig. 4. Multistory model.
L4
4
b) The last step of a return stroke is determined by means of the electrogeometric model, using the approach suggested in IEEE Std. 1243 [16]. c) Overvoltage calculations are performed once the point of impact has been determined. Overvoltages caused by nearby strokes to ground are not simulated, since their effect can be neglected for transmission insulation levels. d) If a flashover occurs in an insulator string, the counter is increased and the flashover rate updated. e) The convergence of the Monte Carlo method is checked by comparing the probability density function of all random variables to their theoretical functions; the procedure is stopped when they match within the specified error.
TABLE I CHARACTERISTICS OF WIRES AND CONDUCTORS
Phase conductors
Type
Diameter (mm)
Resistance (Ω /km)
CURLEW
31.63
0.05501
94S
12.60
0.642
Shield wires
10 m
14.05 m
40 cm 31.25m (21.25m) 10 m
V. SIMULATION RESULTS
A
10 m C
B
26.1m (14.1m)
5.1m
A. Test Lines
Fig. 5 shows the tower design of the lines tested in this paper. Main characteristics of phase conductors and shield wires are presented in Table I.
22.5m (10.5m) 17.2 m
B. Transmission Line and Lightning Parameters
A model of the test line was created using ATP capabilities and following the guidelines summarized in Section II. • The lines were represented by means of eight 400-m spans plus a 30-km section as line termination at each side of the point of impact. • The parameters used in the insulator equations were k l, = 1.3E-6 m2 /(V2s) and E l0, = 570 (kV/m) . The value of the average gradient at the critical flashover voltage, E 50, was assumed to be the same that E l0. The striking distance of insulator strings was 3.212 m. • Only negative polarity and single stroke flashes were considered. The following probability distributions were assumed for each random value: • The phase conductor reference angle had a uniform distribution, between 0 and 360 degrees. • A Weibull distribution was assumed for parameter E l0, which must be specified in the insulator equation. The mean values are those mentioned above, while the standard deviation was 5%. • A normal distribution was assumed for the footing resistance, being the mean value of the resistance at low current and low frequency 50 Ω and the standard deviation 5 Ω. The soil resistivity was 500 ohm-m. • The stroke location, before the application of the electrogeometric model, was generated by assuming a vertical path and a uniform ground distribution of the leader. No flashovers other than those across insulator strings, e.g. flashovers between conductors, have been considered. C. Simulation Results
The following models were used with each test line: • Towers of the test line 1 were represented using the twisted model presently implemented in the FLASH program [16],
7.164 m
a) Test line 1 17 m 2.2 m
A1 15.6 m
C2
13 m 40 cm
B1
B2 16 m 44.1 m (34.1 m)
C1
2.2 m A2
13.6 m
37.6 m (25.6 m) 29.8 m (17.8 m) 22 m (10 m)
25.5m
6.212 m
b) Test line 2 Fig. 5. 400 kV line configurations (Values within parenthesis are midspan heights).
[12], the model proposed by CIGRE [13], the multiconductor model presented in [10], and the multistory model [8], being the parameters in this latter model calculated according to the equations presented in [10] and [18].
5
• Towers of the test line 2 were represented using the CIGRE model, the conical model implemented in the FLASH program, the multiconductor model presented in [10], and the multistory model, whose parameters were calculated again according to [10] and [18]. A first study was performed to analyze the results that can be obtained with each tower model. The simulations were made without including insulator strings in the line model, i.e. they were represented as open switches. Fig. 6 shows some results derived with the two test lines; these plots depict the overvoltages caused by two different return strokes across the insulator strings of the outer phase of first test line tower and an upper phase of the second test line, respectively. One can observe that the calculated overvoltages can be very different when the return stroke waveshape has a very short rise time, being the differences derived with slow fronted waveshapes much smaller. On the other hand, the phase at which the higher overvoltages will be caused depends on the tower model for a configuration such as that of the test line 1, while it will be always the upper phase for double-circuits. The second study was aimed at deducing the flashover rate that could be obtained with each tower model and each test line by varying the median values of the peak current magnitude and the rise time of the return stroke current. In all cases the median value of the tail time was 77.5 µs, while the values of the standard deviation for each parameter of the return stroke waveshape, see Fig. 1, were as follows [2]: peak current magnitude ( I 100), 0.740 kA; rise time ( t f ), 0.494 µs; tail time (t h), 0.577 µs. Fig. 7 and 8 show the flashover rates that were derived with some tower models. The rates were obtained per 100 km/year and assuming N g = 1 fl/km2. The trend of the flashover rate is the same with all of them: it increases with the peak current magnitude, but it decreases as the median value of the rise time increases. As for the differences between models, they could be predicted fr om the depicted results: the highest and the lowest rates are derived from the multistory and the multiconductor models, respectively. Only when the median values of the peak current magnitude and the rise time are below 20 kA and above 3 µs, respectively, the rates obtained with different models are very similar.
0 -2 0 ) A k ( t n e r r u C
-4 0 -6 0 -8 0 -100
2500 Waist Multistory
) 2 0 0 0 V k (
Multiconductor
e 1 5 0 0 g a t l o V 1 0 0 0
500 Test Line 1 4000 Conical Multistory ) V k ( e g a t l o V
3000
Multiconductor
2000
1000 Test Line 2 0
0
1
2
3
Tim e (
a) I max = 120 kA, t f = 0.8 µs, t h = 77.5 µs 0 -2 0 ) A k ( t n e r r u C
-4 0 -6 0 -8 0 -100 1200 Waist M u l ti s t o r y
1000
Multiconductor ) V k ( e g a t l o V
800 600 400
VI. CONCLUSIONS
200
A statistical study of the lightning flashover rate of transmission lines was made. The main goal was to analyze the influence that some tower models can have on the flashover rate. Two test lines were used in calculations. The main conclusions derived from simulation results were that the tower representation can have a significant influence on the flashover rate, but this influence depends on the line configuration and the tower heights: the taller the tower the greater the differences obtained with different tower models. Some care is advisable when selecting the model and calculating its parameters, although these aspects are less critical when tower structures are about or less than 30 meters.
1500
Test Line 1
Conical M u l ti s t o r y
1200
Multiconductor ) V k ( e g a t l o V
900 600 300 Test Line 2 0
0
3
6
9
12
T im e (
b) I max = 120 kA, t f = 5 µs, t h = 77.5 µs Fig. 6. Tower model performance ( R0 = 50 Ω, ρ = 500 Ω.m).
15
6 2. 5
VII. ACKNOWLEDGEMENT CIGRE
2. 0 e t a r r e v o h s a l F
The second author wants to express his gratitude to the Universidad del Valle (Cali, Colombia) for the support received during the preparation of his P h.D.
Waist Multiconductor Multistory
1. 5 1. 0
VIII. REFERENCES [1]
0. 5 0. 0 10
20
30
40
50
Peak current magnitude (kA)
[2] [3]
a) Test line 1
[4]
5. 0 CIGRE
4. 0 e t a r r e v o h s a l F
[5]
Conical Multiconductor
3. 0
[6]
Multistory
2. 0
[7]
1. 0
[8] 0. 0 10
20
30
40
50
[9]
Peak current magnitude (kA)
b) Test line 2 Fig. 7. Sensitivity analysis: Flashover rate vs. peak current magnitude 2 (t f = 2 µs, t h = 77.5 µs, N g = 1 fl/km ).
[10]
[11] 1. 4 CIGRE
1. 2
[12]
Waist e t a r r e v o h s a l F
1. 0
Multiconductor Multistory
0. 8
[13] 0. 6
[14]
0. 4 0. 2
[15]
0. 0 1
2
3
4
5
Rise time (µs)
[16]
a) Test line 1
[17]
3. 5 CIGRE
3. 0
[18]
Conical e t a r r e v o h s a l F
2. 5
Multiconductor Multistory
2. 0
[19]
1. 5 1. 0 0. 5 0. 0 1
2
3
4
5
Rise time (µs)
b) Test line 2 Fig. 8. Sensitivity analysis: Flashover rate vs. r ise time ( I 100 = 34 kA, 2 t h = 77.5 µs, N g = 1 fl/km ).
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IX. BIOGRAPHIES Juan A. Martinez was born in Barcelona (Spain). He is Profesor Titular at the Departament d'Enginyeria Elèctrica of the Universitat Politècnica de Catalunya. His teaching and research interests include Transmission and Distribution, Power System Analysis and EMTP applications. Ferley Castro-Aranda was born in Tuluà ( Colombia). He is Profesor Asociado at the Universidad del Valle. He is currently pursuing his Ph.D. degree at the Universitat Politècnica de Catalunya. His research interests are focused on the areas of Insulation Coordination and System Modeling for Transient Analysis using EMTP.
