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Monte Carlo estimation of the rates of lightning strikes on power lines
Electric Power Systems Research 49 (1999) 201 – 210
Monte Carlo estimation of the rates of lightning strikes on power lines Roger Holt, Tam T. Nguyen * Energy Systems Centre, Department of Electrical and Electronic Engineering, The Uni6ersity of Western Australia, Nedlands, Perth 6907, Australia Received 2 May 1998; received in revised form 27 July 1998; accepted 9 September 1998
Abstract This paper reports the development of a general method for estimating the rates of lightning strikes on transmission lines using Monte Carlo simulation. Effects of towers, cross-arms, non-level ground, conductor sags and nearby structures are directly represented in the 3-dimension electrogeometric model (EGM). The method developed is a general one that is applicable to any transmission line configuration and is independent of the EGM used. Tedious analytical derivation of more than 100 equations for each configuration as required by analytical methods is avoided altogether. The formulation and a flow chart are detailed in the paper. The shortest distances from the lightning leader tip to individual structures that may be struck are evaluated and then compared with striking distances. The outcome of the comparison identifies the structure which will be struck in each simulation. The exposure area adopted in the simulation is determined on the basis of the transmission line route length and the maximum possible striking distance to ensure that the simulation results in a maximum possible number of strikes on the overhead line. Strokes outside the exposure area will always miss the transmission line and, therefore, have no effects on the results. The method proposed and its software implementation are verified on the basis of results from analytical methods of earlier work, and from field data. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Monte Carlo; Lightning; Shielding failures; Overhead lines; Electrogeometric model
1. Introduction Estimations of the rates of lightning strikes on transmission lines using the electrogeometric model [1] have been an area of active research for some time [2–12]. Early publications [1,3,6] concentrated on estimations using simplified transmission line configurations where shielded horizontal conductors above level ground were assumed, and the effects of towers and nearby structures were discounted. These simplifications led to an analytical method of estimation. More recently, the analytical method has been extended to include three-dimension electrogeometric model analysis to take account of effects of transmission line towers and conductor sags [12]. Each tower and line configuration requires detailed geometrical analysis that leads to a set of 168 equations. The analysis has been carried * Corresponding author. Tel.: +61-8-3802559; fax: 3803747; e-mail: [email protected].
+61-8-
out for three typical tower configurations [12]. The equations are for a system comprising one tower and two half spans on level ground. The drawback of the approach is that, for every new configuration, a detailed geometrical analysis needs to be carried out to give another set of 168 equations to be implemented in software for evaluation. Another approach which avoids a tedious geometrical analysis in each case is that based on Monte Carlo simulation. Research on the application of the Monte Carlo method for estimating transmission line lightning performance was first reported in reference [2]. Starting from this research, the Monte Carlo method has been refined and extended in further work [4] to represent variations in ground profile, trees and conductor sags. However, the analysis reported has been confined to the 2-dimension case only where the effects of towers and cross-arms were discounted in the estimations. Only a passing reference to the use of 3-dimension analysis to
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account for the shielding effects of towers and crossarms has been made in the open literature [5]. The detail of applying the Monte Carlo method in three dimensions for estimating transmission line lightning performance has not been given in the open literature. The objective of the present paper is to report the development of a general method of analysis for the 3-dimension case using Monte Carlo simulation. Availability of low-cost computer systems with highspeed processing capability makes Monte Carlo simulation a practical method for routine transmission line lightning performance estimation. The method developed in the paper together with its software implementation is validated using results from field data [3,13,14] and also those from the analytical method reported in previous published work [8,11,12].
2. Electrogeometric model The electrogeometric model is empirical. It is summarized in the following in relation to the assumptions on the path of the lightning leader tip and the striking distance [10].
2.1. Path of the lightning leader The path of the lightning leader is independent of any structure until the shortest distance between the lightning leader tip and a grounded structure, or the ground, is equal to the striking distance associated with that structure, or ground. When this occurs, the lightning leader follows a path that leads to that structure. In the cases where there are more than one grounded structure, the lightning strikes the first structure that comes within the striking distance of the lightning leader tip. The lightning leader can only strike one structure.
2.2. Striking distance
(1)
In Eq. (1), Km is a scaling coefficient which is dependent on the nature of the structure. Reference [12] gives representative values of Km for conductors, towers and the ground. The function f(I) is a function of the stroke current magnitude. Function f(I) has been determined empirically. Different forms of f(I) have been proposed. Recently an IEEE working group [10] recommended the following function:
(2)
Typical ranges of value for A and b were also given by the IEEE working group. The lightning stroke current magnitude is a non-deterministic variable that is generally expressed in the form of a probability distribution function. Different forms of the probability distribution function have been proposed in the open literature [5,6,8,10].
3. Monte Carlo method
3.1. Principle The factors that determine whether a lightning stroke will strike a transmission line are the line configuration, the ground profile, nearby objects or structures, the position of the lightning leader tip and the magnitude of the stroke current. The parameters relating to the line configuration, the ground profile and nearby structures are deterministic ones. However, the parameters of a lightning stroke expressed in terms of its leader tip position and the stroke current magnitude are random or non-deterministic ones. In Monte Carlo simulation, analysis is carried out for a large number of lightning strokes the parameters of which are selected on the basis of their probability distribution functions. For each lightning stroke, analysis is carried out using the electrogeometric model to determine which object that the lightning stroke will strike. The ground, conductors, towers or nearby structures are all objects which may be struck and are considered in the analysis. If Nt is the total number of simulations in a period t and A is the area over which the simulation is carried out, then the ground flash density, gs, is: gs =
The striking distance associated with a grounded structure depends on the form of the structure and the magnitude of the stroke current. The general equation for the striking distance, Sm, is given by [10,12]: Sm =Km f(I)
f (I)= A I b
Nt A·t
(3)
If Nc is the number of shield failures which are strikes to phase conductors, and l the length of the overhead line, then the shield failure rate, Frc, is: Frc =
100 Nc l·t
(4)
The unit for Frc in Eq. (4) is expressed in terms of shield failures per 100 km per year. The result given in Eq. (4) is for the ground flash density in Eq. (3). If the ground flash density gs is 1 strike/km2/year, then the normalised failure rate, Fc, is, from Eq. (3) and Eq. (4): N A Fc = 100 c Nt l
(5)
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Equations for the strike rates for towers and nearby objects can also be derived using procedures similar to that given in Eqs. (3) – (5).
3.2. Formulation A co-ordinate system (x, y, z) is first nominated. For convenience, the x– y plane is chosen to coincide with the level earth plane and the z-axis is perpendicular to it as shown in Fig. 1. The co-ordinate system is used to define the positions of the overhead line, natural shields, lightning leader tip and the ground profile. The Monte Carlo method is independent of the EGM and lightning current probability distribution function. The user may select any of the EGMs and lightning current probability distribution functions that are available in published work [7,9,10,12]. Based on a uniform probability distribution, the x –y co-ordinates of the lightning leader tip are randomly selected from the exposure area A. The exposure area A is determined on the basis of the transmission line route and the maximum value of the striking distance, Smax. Striking distances are calculated using the selected EGM. The maximum striking distance is found using the maximum value of the stroke current in the lightning distribution function and the individual elements of the overhead line. Selection of an exposure area in this manner ensures that the simulation process results in a maximum pos-
Fig. 1. Striking distances and distances from the leader tip to individual objects.
Fig. 2. Exposure area in Monte Carlo simulation.
sible number of strikes on the overhead line. Strokes outside this area will always miss the transmission line and therefore have no effects on the results. As an example, an exposure area used in the simulation is given in Fig. 2. In addition to a set of random (x, y) co-ordinates for the leader tip with uniform probability distribution, a random value for the corresponding lightning stroke current magnitude, In, is generated on the basis of its probability distribution function for the n th simulation. This sequence of generating random co-ordinates and stroke current magnitude is valid since their probability distribution functions are independent of each other. It then remains to define the z-co-ordinate of the lightning leader tip for use in the analysis. A number of evaluations will be executed. After each evaluation, the value of z is revised until a single object is struck. The initial value of z is arbitrary but good results have been obtained by setting it to the sum of the maximum striking distance, Smax, and the height of the highest structure. The striking distance for each object is calculated using the EGM with the appropriate coefficient Km in Eq. (1) for the object under consideration and the randomly-generated current magnitude. The striking distance for the m th object is denoted by Sm. With the co-ordinates of the lightning leader tip defined, the shortest distances from it to individual objects are calculated. In Appendix A are summarized the expressions for shortest distances from a point in space to elements with commonly-encountered geometries: linear segments; planes; and the tops of towers or poles. Conductor sags are represented by sub-dividing the conductor into a finite number of linear conductor segments. The shortest distance from the lightning leader tip to the m th object is denoted by dm. In Fig. 1 are shown the shortest distances from the lightning leader
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Fig. 3. Monte Carlo lightning performance estimation flow-chart.
tip to various elements of a transmission line. In Fig. 1 are also shown the striking distances for various elements of a transmission line.
A comparison between dm and Sm is made for each and every object. From the comparison, two outcomes are possible.
R. Holt, T.T. Nguyen / Electric Power Systems Research 49 (1999) 201–210 Table 1 Shield failure rates of 275 kV and 400 kV lines, correlation between simulation results and field data Line (kV)
275 400
4. Validation studies
4.1. Correlation with field data
Shield failure rates Field data
Simulation
Failure/100 km/year
Failure/100 km/year
0.44 0.51
0.45 0.55
3.2.1. Case 1 dm \Sm
205
m =1, 2,......, Ns
(6)
In inequality 6, Ns is the total number of objects. In this case, a strike will not occur and further evaluations have to be carried out. This situation occurs when the leader tip is too far from the objects and for the next evaluation z is halved.
3.2.2. Case 2 dm 5Sm for one or more objects
(7)
In this case, further checks are carried out as one or more strikes have occurred. If only one object satisfies inequality 7 for the n th simulation, the strike counter for the object which satisfies inequality 7 is incremented by 1 and a new set of random parameters are generated for the next simulation. If two or more objects satisfy inequality 7, it means that two or more objects could be struck as z is too low. The z-coordinate of the leader tip is then increased by 50%. A new set of shortest distances from the leader tip to individual objects is calculated and the comparison between the shortest distances and striking distances is repeated until only one object is struck. At the completion of the simulation process the contents of the strike counters for individual objects are transferred into an output module for presentation in a graphical format. The simulation sequence is summarized in the flowchart of Fig. 3.
The field data used in the validation study is from reference [13]. A validation study was carried out to ensure that the Monte Carlo method would give realistic results. An outage rate of 0.36 outage/100 km/year has been observed for 275 kV lines in the United Kingdom. For 400 kV lines, a higher outage rate of 0.41 outage/100 km/year has been observed. The outage rates due to lightning are for all of the 275 kV and 400 kV transmission lines which have several tower configurations and designs. Footing resistance, insulation strength, the magnitude of the strike current and secondary strikes are some of the factors that determine if a shield failure will lead to an outage. Brown and Whitehead published data collected in the Pathfinder project [3] on shield failures and line outages. Using the data it can be shown that about 80% of shielding failures lead to line outages. On this basis, the shield failure rates are derived from the field data in reference [13] and given in Table 1. They are about 1.25 times greater than the observed line outage rates. The results from simulation are compared with the field data in Table 1. Very close correlation is achieved. In the simulation, typical line configurations [15], tower heights and span lengths were used. The ground flash density is 2 strikes/km2/year [14]. The EGM used in the simulation is that given in Eq. (2) with A= 6.4 and b= 0.65 [10]. The probability distribution function for stroke current of magnitude I in kA is given in [11]: P (I)=
1 I 1+ IMed
2.6
(8)
In Eq. (8), the median current, IMed, is 31 kA. The EGM in Eq. (2) and the probability distribution function in Eq. (8) are used throughout in the analysis in this paper.
4.2. Correlation with results from analytical method
Fig. 4. Configuration for the validation study.
The Monte Carlo method was tested against an accepted analytical method developed by Mousa and Srivastava [12]. Numerous tests were carried out. A representative case is reported here. In Fig. 4 is shown a transmission line which was used for the test. The results of the analytical and Monte Carlo estimates are compared in Fig. 5. Pole and
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conductor heights were varied while all other parameters of the transmission line were held constant. A ground flash density of 1.0 strike/km2/year is used in the estimations. The accuracy of the Monte Carlo method increases with the number of simulations used. Increasing the number of simulations also increases the computing time. The effect of increasing the number of simulations is shown in Fig. 5 where one estimate was made using 10 ×103 simulations and another using 50× 103 simulations. Estimations based on 50×103 simulations compare closely with the analytical results. Results produced by using 10× 103 simulations give trends but are not sufficiently accurate for practical purposes. The computing time required of 50 ×103 simulations for each line configuration is about 2 min on a 200 MHz Pentium PC.
5. Representative studies The transmission line configuration of Fig. 6 and the ground flash density of 1.0 strike/km2/year are used in the following studies. The method allows estimates of the shield failure rate, line strike rate, average strike current and pole strike rates. The user can select any combination of these results. The studies presented in this paper are typical of those that can be carried out using the Monte Carlo method.
5.1. Effect of cross-arm length In Fig. 7 is shown the effect of increasing the cross-arm length on the shield failure rate. In the estimation, the shielding angle is kept constant at 45° by raising the shield conductor whilst the height of the cross-arms is kept constant. Increasing cross-arm
Fig. 5. Comparison of methods.
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5.2. Effect of pole height Fig. 8 shows the effect of increasing the pole height whilst keeping the spacings amongst phase conductors and earth wire constant. The shield failure rate and the average magnitude of the strike current both increase with the pole height. Two spans and three towers are represented in the analysis. 5.3. Effect of natural shields In this study, four natural shields were placed within a single transmission line span as shown in Fig. 9. The natural shields are modeled as towers or poles. The effect of natural shield height on the shield failure rate is shown in Fig. 10. Increasing the natural shield height decreases the shield failure rate significantly. However, the natural shield strike rate in Fig. 10 increases substantially with height. Fig. 6. Transmission line configuration used in representative studies.
length increases the shield failure rate. The average lightning stroke current associated with the shield failures is also shown. There is no appreciable increase in the average strike current. In the estimations, two spans and three towers are represented.
6. Conclusions The method developed in the paper offers an important advantage over the analytical method which was published previously. The Monte Carlo method does not require the user to derive numerous equations for
Fig. 7. Variation of shield failure rate and average stroke current with cross arm length.
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Fig. 8. Effects of pole height.
different transmission line configurations and then to implement these equations in software. The Monte Carlo method in three dimensions is a practical method applied in an iterative manner using powerful computers which are available at low cost. The EGM and lightning current probability distribution functions are implemented in the separate modules within the Monte Carlo simulation software. As improved EGMs and lightning current probability distribution functions become available, they can be incorporated into the Monte Carlo lightning performance estimation software developed in the present work.
Results from the Monte Carlo method compare favourably with accepted analytical methods for the same EGM. Estimates obtained by the Monte Carlo method also compare favourably with field data. The Monte Carlo method presented in this paper can be used to estimate the lightning performance of alternative transmission and distribution line designs with ease.
Acknowledgements The authors gratefully acknowledge the support of the Energy Systems Centre at The University of Western Australia for the research work reported in the paper. They express their appreciation to The University for permission to publish the paper.
Appendix A. Expressions for shortest distances [16] The shortest distance between two points (x1, y1, z1) and (x2, y2, z2) is given by Eq. (A1.1): Fig. 9. Natural shields.
D= (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2
(A1.1)
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Fig. 10. Effects of natural shield height.
Given a point (x1, y1, z1) and a plane ax+by + cz= d then the shortest distance between the line and the plane is given in Eq. (A1.2):
)
ax1 + by1 +cz1 −d
)
(A1.2)
a +b +c Given a point (x1, y1, z1) and a line as defined in Eq. (A1.3) then the shortest distance between the point and line is given in Eq. (A1.4): D=
2
2
2
x− x0 y − y0 z− z0 = = a b c D
=
Frc Fc
(A1.3) Sm dm
D
Ns P(I)
[b (z1 −z0)− c (y1 −y0)]2 +[c (x1 −x0)
−a (z1 −z0)]2 +[a (y1 −y0) − b (x1 −x0)]2
Smax l A t Nc Nt
(A1.4)
IMed
maximum striking distance (m) total length of overhead line (km) exposure area (km2) exposure period (years) number of conductor strikes (strikes) number of simulations in Monte Carlo analysis shield failure rate (shield failures/100 km/ year) normalized shield failure rate (shield failures/100 km/year) striking distance of the m th object (m) shortest distance of the m th object from lightning leader tip (m) total number of objects which may be struck probability that the lightning stroke current will be greater than I median current of lightning distribution (kA)
Appendix B. Nomenclature I Km gs
lightning stroke current (kA) scaling factor for general EGM lightning ground strike density (strikes/km2/year)
References [1] R.H. Golde, The Frequency of occurrence and the distribution of lightning flashes to transmission lines, AIEE Trans. 64 (1945) 902 – 910.
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[2] J.G. Anderson, Monte Carlo computer calculation of transmission line lightning performance, AIEE Trans. 80 (1961) 414 – 420. [3] G.W. Brown, E.R. Whitehead, Field and analytical studies of transmission line shielding: part II, IEEE Trans. Power Appar. Syst. PAS-88 (1969) 617–626. [4] J.R. Currie, Liew Ah Choy, M. Darveniza, Monte Carlo determination of the frequency of lightning strokes and shielding failures on transmission lines, IEEE Trans. Power Appar. Syst. PAS-90 (1971) 2305 –2313. [5] M. Darveniza, M. Sargent, G.J. Limbourne, Liew Ah Choy, R.O. Caldwell, J.R. Currie, B.C. Holcombe, R.H. Stillman, R. Frowd, Modelling for lightning performance calculations, IEEE Trans. Power Appar. Syst. PAS-98 (6) (1979) 1900–1908. [6] IEEE Working Group on Lightning Performance of Transmission Lines, A simplified method for estimating lightning performance of transmission lines, IEEE Trans. Power Appar. Syst. PAS-104 (4) (1985) 919–932. [7] A.J. Eriksson, An improved electrogeometric model for transmission line shielding analysis, IEEE Trans. Power Deliv. 2 (3) (1987) 871 – 886. [8] A.M. Mousa, K.D. Srivastava, The implications of the electrogeometric model regarding the effect of height of structure on the median amplitude of collected lightning strokes, IEEE Trans. Power Deliv. 4 (2) (1989) 1450–1460.
.
[9] F.A.M. Rizk, Modelling of transmission line exposure to direct lightning strikes, IEEE Trans. Power Deliv. 5 (4) (1990) 1983– 1997. [10] IEEE Working Group Report, Estimating lightning performance of transmission lines II — updates of analytical models, IEEE Trans. Power Deliv. 8 (3) (1993) 1254 – 1267. [11] P. Chowdhuri, S. Mehairijan, Alternative to Monte Carlo method for the estimation of lightning incidence to overhead lines, IEE Proc. Gener. Transm. Distrib. 144 (2) (1997) 129–131. [12] A.M. Mousa, K.D. Srivastava, Modelling of power lines in lightning incidence calculations, IEEE Trans. Power Deliv. 5 (1) (1990) 303 – 310. [13] M.J. Rawlins, Lightning performance of the CEGB 275 kV and 400 kV system 1977/8 to 1981/2, in: IEE Conference on Lightning and Power Systems, Conference Publication No. 236, 1984, pp. 146 – 152. [14] M.F. Stringfellow, Lightning incidence in the United Kingdom, in: IEE Conference on Lightning and the Distribution-System, Conference Publication No. 108, 1974, pp. 30 – 40. [15] GEC Measurements, Protective Relays Application Guide, 3rd Ed., June 1987. [16] A. Kean, S.A. Senior, Complementary Mathematics, 2nd ed., Science Press, Sydney, 1961.
Monte Carlo estimation of the rates of lightning strikes on power lines Roger Holt, Tam T. Nguyen * Energy Systems Centre, Department of Electrical and Electronic Engineering, The Uni6ersity of Western Australia, Nedlands, Perth 6907, Australia Received 2 May 1998; received in revised form 27 July 1998; accepted 9 September 1998
Abstract This paper reports the development of a general method for estimating the rates of lightning strikes on transmission lines using Monte Carlo simulation. Effects of towers, cross-arms, non-level ground, conductor sags and nearby structures are directly represented in the 3-dimension electrogeometric model (EGM). The method developed is a general one that is applicable to any transmission line configuration and is independent of the EGM used. Tedious analytical derivation of more than 100 equations for each configuration as required by analytical methods is avoided altogether. The formulation and a flow chart are detailed in the paper. The shortest distances from the lightning leader tip to individual structures that may be struck are evaluated and then compared with striking distances. The outcome of the comparison identifies the structure which will be struck in each simulation. The exposure area adopted in the simulation is determined on the basis of the transmission line route length and the maximum possible striking distance to ensure that the simulation results in a maximum possible number of strikes on the overhead line. Strokes outside the exposure area will always miss the transmission line and, therefore, have no effects on the results. The method proposed and its software implementation are verified on the basis of results from analytical methods of earlier work, and from field data. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Monte Carlo; Lightning; Shielding failures; Overhead lines; Electrogeometric model
1. Introduction Estimations of the rates of lightning strikes on transmission lines using the electrogeometric model [1] have been an area of active research for some time [2–12]. Early publications [1,3,6] concentrated on estimations using simplified transmission line configurations where shielded horizontal conductors above level ground were assumed, and the effects of towers and nearby structures were discounted. These simplifications led to an analytical method of estimation. More recently, the analytical method has been extended to include three-dimension electrogeometric model analysis to take account of effects of transmission line towers and conductor sags [12]. Each tower and line configuration requires detailed geometrical analysis that leads to a set of 168 equations. The analysis has been carried * Corresponding author. Tel.: +61-8-3802559; fax: 3803747; e-mail: [email protected].
+61-8-
out for three typical tower configurations [12]. The equations are for a system comprising one tower and two half spans on level ground. The drawback of the approach is that, for every new configuration, a detailed geometrical analysis needs to be carried out to give another set of 168 equations to be implemented in software for evaluation. Another approach which avoids a tedious geometrical analysis in each case is that based on Monte Carlo simulation. Research on the application of the Monte Carlo method for estimating transmission line lightning performance was first reported in reference [2]. Starting from this research, the Monte Carlo method has been refined and extended in further work [4] to represent variations in ground profile, trees and conductor sags. However, the analysis reported has been confined to the 2-dimension case only where the effects of towers and cross-arms were discounted in the estimations. Only a passing reference to the use of 3-dimension analysis to
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account for the shielding effects of towers and crossarms has been made in the open literature [5]. The detail of applying the Monte Carlo method in three dimensions for estimating transmission line lightning performance has not been given in the open literature. The objective of the present paper is to report the development of a general method of analysis for the 3-dimension case using Monte Carlo simulation. Availability of low-cost computer systems with highspeed processing capability makes Monte Carlo simulation a practical method for routine transmission line lightning performance estimation. The method developed in the paper together with its software implementation is validated using results from field data [3,13,14] and also those from the analytical method reported in previous published work [8,11,12].
2. Electrogeometric model The electrogeometric model is empirical. It is summarized in the following in relation to the assumptions on the path of the lightning leader tip and the striking distance [10].
2.1. Path of the lightning leader The path of the lightning leader is independent of any structure until the shortest distance between the lightning leader tip and a grounded structure, or the ground, is equal to the striking distance associated with that structure, or ground. When this occurs, the lightning leader follows a path that leads to that structure. In the cases where there are more than one grounded structure, the lightning strikes the first structure that comes within the striking distance of the lightning leader tip. The lightning leader can only strike one structure.
2.2. Striking distance
(1)
In Eq. (1), Km is a scaling coefficient which is dependent on the nature of the structure. Reference [12] gives representative values of Km for conductors, towers and the ground. The function f(I) is a function of the stroke current magnitude. Function f(I) has been determined empirically. Different forms of f(I) have been proposed. Recently an IEEE working group [10] recommended the following function:
(2)
Typical ranges of value for A and b were also given by the IEEE working group. The lightning stroke current magnitude is a non-deterministic variable that is generally expressed in the form of a probability distribution function. Different forms of the probability distribution function have been proposed in the open literature [5,6,8,10].
3. Monte Carlo method
3.1. Principle The factors that determine whether a lightning stroke will strike a transmission line are the line configuration, the ground profile, nearby objects or structures, the position of the lightning leader tip and the magnitude of the stroke current. The parameters relating to the line configuration, the ground profile and nearby structures are deterministic ones. However, the parameters of a lightning stroke expressed in terms of its leader tip position and the stroke current magnitude are random or non-deterministic ones. In Monte Carlo simulation, analysis is carried out for a large number of lightning strokes the parameters of which are selected on the basis of their probability distribution functions. For each lightning stroke, analysis is carried out using the electrogeometric model to determine which object that the lightning stroke will strike. The ground, conductors, towers or nearby structures are all objects which may be struck and are considered in the analysis. If Nt is the total number of simulations in a period t and A is the area over which the simulation is carried out, then the ground flash density, gs, is: gs =
The striking distance associated with a grounded structure depends on the form of the structure and the magnitude of the stroke current. The general equation for the striking distance, Sm, is given by [10,12]: Sm =Km f(I)
f (I)= A I b
Nt A·t
(3)
If Nc is the number of shield failures which are strikes to phase conductors, and l the length of the overhead line, then the shield failure rate, Frc, is: Frc =
100 Nc l·t
(4)
The unit for Frc in Eq. (4) is expressed in terms of shield failures per 100 km per year. The result given in Eq. (4) is for the ground flash density in Eq. (3). If the ground flash density gs is 1 strike/km2/year, then the normalised failure rate, Fc, is, from Eq. (3) and Eq. (4): N A Fc = 100 c Nt l
(5)
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Equations for the strike rates for towers and nearby objects can also be derived using procedures similar to that given in Eqs. (3) – (5).
3.2. Formulation A co-ordinate system (x, y, z) is first nominated. For convenience, the x– y plane is chosen to coincide with the level earth plane and the z-axis is perpendicular to it as shown in Fig. 1. The co-ordinate system is used to define the positions of the overhead line, natural shields, lightning leader tip and the ground profile. The Monte Carlo method is independent of the EGM and lightning current probability distribution function. The user may select any of the EGMs and lightning current probability distribution functions that are available in published work [7,9,10,12]. Based on a uniform probability distribution, the x –y co-ordinates of the lightning leader tip are randomly selected from the exposure area A. The exposure area A is determined on the basis of the transmission line route and the maximum value of the striking distance, Smax. Striking distances are calculated using the selected EGM. The maximum striking distance is found using the maximum value of the stroke current in the lightning distribution function and the individual elements of the overhead line. Selection of an exposure area in this manner ensures that the simulation process results in a maximum pos-
Fig. 1. Striking distances and distances from the leader tip to individual objects.
Fig. 2. Exposure area in Monte Carlo simulation.
sible number of strikes on the overhead line. Strokes outside this area will always miss the transmission line and therefore have no effects on the results. As an example, an exposure area used in the simulation is given in Fig. 2. In addition to a set of random (x, y) co-ordinates for the leader tip with uniform probability distribution, a random value for the corresponding lightning stroke current magnitude, In, is generated on the basis of its probability distribution function for the n th simulation. This sequence of generating random co-ordinates and stroke current magnitude is valid since their probability distribution functions are independent of each other. It then remains to define the z-co-ordinate of the lightning leader tip for use in the analysis. A number of evaluations will be executed. After each evaluation, the value of z is revised until a single object is struck. The initial value of z is arbitrary but good results have been obtained by setting it to the sum of the maximum striking distance, Smax, and the height of the highest structure. The striking distance for each object is calculated using the EGM with the appropriate coefficient Km in Eq. (1) for the object under consideration and the randomly-generated current magnitude. The striking distance for the m th object is denoted by Sm. With the co-ordinates of the lightning leader tip defined, the shortest distances from it to individual objects are calculated. In Appendix A are summarized the expressions for shortest distances from a point in space to elements with commonly-encountered geometries: linear segments; planes; and the tops of towers or poles. Conductor sags are represented by sub-dividing the conductor into a finite number of linear conductor segments. The shortest distance from the lightning leader tip to the m th object is denoted by dm. In Fig. 1 are shown the shortest distances from the lightning leader
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Fig. 3. Monte Carlo lightning performance estimation flow-chart.
tip to various elements of a transmission line. In Fig. 1 are also shown the striking distances for various elements of a transmission line.
A comparison between dm and Sm is made for each and every object. From the comparison, two outcomes are possible.
R. Holt, T.T. Nguyen / Electric Power Systems Research 49 (1999) 201–210 Table 1 Shield failure rates of 275 kV and 400 kV lines, correlation between simulation results and field data Line (kV)
275 400
4. Validation studies
4.1. Correlation with field data
Shield failure rates Field data
Simulation
Failure/100 km/year
Failure/100 km/year
0.44 0.51
0.45 0.55
3.2.1. Case 1 dm \Sm
205
m =1, 2,......, Ns
(6)
In inequality 6, Ns is the total number of objects. In this case, a strike will not occur and further evaluations have to be carried out. This situation occurs when the leader tip is too far from the objects and for the next evaluation z is halved.
3.2.2. Case 2 dm 5Sm for one or more objects
(7)
In this case, further checks are carried out as one or more strikes have occurred. If only one object satisfies inequality 7 for the n th simulation, the strike counter for the object which satisfies inequality 7 is incremented by 1 and a new set of random parameters are generated for the next simulation. If two or more objects satisfy inequality 7, it means that two or more objects could be struck as z is too low. The z-coordinate of the leader tip is then increased by 50%. A new set of shortest distances from the leader tip to individual objects is calculated and the comparison between the shortest distances and striking distances is repeated until only one object is struck. At the completion of the simulation process the contents of the strike counters for individual objects are transferred into an output module for presentation in a graphical format. The simulation sequence is summarized in the flowchart of Fig. 3.
The field data used in the validation study is from reference [13]. A validation study was carried out to ensure that the Monte Carlo method would give realistic results. An outage rate of 0.36 outage/100 km/year has been observed for 275 kV lines in the United Kingdom. For 400 kV lines, a higher outage rate of 0.41 outage/100 km/year has been observed. The outage rates due to lightning are for all of the 275 kV and 400 kV transmission lines which have several tower configurations and designs. Footing resistance, insulation strength, the magnitude of the strike current and secondary strikes are some of the factors that determine if a shield failure will lead to an outage. Brown and Whitehead published data collected in the Pathfinder project [3] on shield failures and line outages. Using the data it can be shown that about 80% of shielding failures lead to line outages. On this basis, the shield failure rates are derived from the field data in reference [13] and given in Table 1. They are about 1.25 times greater than the observed line outage rates. The results from simulation are compared with the field data in Table 1. Very close correlation is achieved. In the simulation, typical line configurations [15], tower heights and span lengths were used. The ground flash density is 2 strikes/km2/year [14]. The EGM used in the simulation is that given in Eq. (2) with A= 6.4 and b= 0.65 [10]. The probability distribution function for stroke current of magnitude I in kA is given in [11]: P (I)=
1 I 1+ IMed
2.6
(8)
In Eq. (8), the median current, IMed, is 31 kA. The EGM in Eq. (2) and the probability distribution function in Eq. (8) are used throughout in the analysis in this paper.
4.2. Correlation with results from analytical method
Fig. 4. Configuration for the validation study.
The Monte Carlo method was tested against an accepted analytical method developed by Mousa and Srivastava [12]. Numerous tests were carried out. A representative case is reported here. In Fig. 4 is shown a transmission line which was used for the test. The results of the analytical and Monte Carlo estimates are compared in Fig. 5. Pole and
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conductor heights were varied while all other parameters of the transmission line were held constant. A ground flash density of 1.0 strike/km2/year is used in the estimations. The accuracy of the Monte Carlo method increases with the number of simulations used. Increasing the number of simulations also increases the computing time. The effect of increasing the number of simulations is shown in Fig. 5 where one estimate was made using 10 ×103 simulations and another using 50× 103 simulations. Estimations based on 50×103 simulations compare closely with the analytical results. Results produced by using 10× 103 simulations give trends but are not sufficiently accurate for practical purposes. The computing time required of 50 ×103 simulations for each line configuration is about 2 min on a 200 MHz Pentium PC.
5. Representative studies The transmission line configuration of Fig. 6 and the ground flash density of 1.0 strike/km2/year are used in the following studies. The method allows estimates of the shield failure rate, line strike rate, average strike current and pole strike rates. The user can select any combination of these results. The studies presented in this paper are typical of those that can be carried out using the Monte Carlo method.
5.1. Effect of cross-arm length In Fig. 7 is shown the effect of increasing the cross-arm length on the shield failure rate. In the estimation, the shielding angle is kept constant at 45° by raising the shield conductor whilst the height of the cross-arms is kept constant. Increasing cross-arm
Fig. 5. Comparison of methods.
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207
5.2. Effect of pole height Fig. 8 shows the effect of increasing the pole height whilst keeping the spacings amongst phase conductors and earth wire constant. The shield failure rate and the average magnitude of the strike current both increase with the pole height. Two spans and three towers are represented in the analysis. 5.3. Effect of natural shields In this study, four natural shields were placed within a single transmission line span as shown in Fig. 9. The natural shields are modeled as towers or poles. The effect of natural shield height on the shield failure rate is shown in Fig. 10. Increasing the natural shield height decreases the shield failure rate significantly. However, the natural shield strike rate in Fig. 10 increases substantially with height. Fig. 6. Transmission line configuration used in representative studies.
length increases the shield failure rate. The average lightning stroke current associated with the shield failures is also shown. There is no appreciable increase in the average strike current. In the estimations, two spans and three towers are represented.
6. Conclusions The method developed in the paper offers an important advantage over the analytical method which was published previously. The Monte Carlo method does not require the user to derive numerous equations for
Fig. 7. Variation of shield failure rate and average stroke current with cross arm length.
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Fig. 8. Effects of pole height.
different transmission line configurations and then to implement these equations in software. The Monte Carlo method in three dimensions is a practical method applied in an iterative manner using powerful computers which are available at low cost. The EGM and lightning current probability distribution functions are implemented in the separate modules within the Monte Carlo simulation software. As improved EGMs and lightning current probability distribution functions become available, they can be incorporated into the Monte Carlo lightning performance estimation software developed in the present work.
Results from the Monte Carlo method compare favourably with accepted analytical methods for the same EGM. Estimates obtained by the Monte Carlo method also compare favourably with field data. The Monte Carlo method presented in this paper can be used to estimate the lightning performance of alternative transmission and distribution line designs with ease.
Acknowledgements The authors gratefully acknowledge the support of the Energy Systems Centre at The University of Western Australia for the research work reported in the paper. They express their appreciation to The University for permission to publish the paper.
Appendix A. Expressions for shortest distances [16] The shortest distance between two points (x1, y1, z1) and (x2, y2, z2) is given by Eq. (A1.1): Fig. 9. Natural shields.
D= (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2
(A1.1)
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Fig. 10. Effects of natural shield height.
Given a point (x1, y1, z1) and a plane ax+by + cz= d then the shortest distance between the line and the plane is given in Eq. (A1.2):
)
ax1 + by1 +cz1 −d
)
(A1.2)
a +b +c Given a point (x1, y1, z1) and a line as defined in Eq. (A1.3) then the shortest distance between the point and line is given in Eq. (A1.4): D=
2
2
2
x− x0 y − y0 z− z0 = = a b c D
=
Frc Fc
(A1.3) Sm dm
D
Ns P(I)
[b (z1 −z0)− c (y1 −y0)]2 +[c (x1 −x0)
−a (z1 −z0)]2 +[a (y1 −y0) − b (x1 −x0)]2
Smax l A t Nc Nt
(A1.4)
IMed
maximum striking distance (m) total length of overhead line (km) exposure area (km2) exposure period (years) number of conductor strikes (strikes) number of simulations in Monte Carlo analysis shield failure rate (shield failures/100 km/ year) normalized shield failure rate (shield failures/100 km/year) striking distance of the m th object (m) shortest distance of the m th object from lightning leader tip (m) total number of objects which may be struck probability that the lightning stroke current will be greater than I median current of lightning distribution (kA)
Appendix B. Nomenclature I Km gs
lightning stroke current (kA) scaling factor for general EGM lightning ground strike density (strikes/km2/year)
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