A STATISTICAL METHOD FOR LIGHTNING INCIDENCE CALCULATIONS IN TRANSMISSION LINES

P. N. Mikropoulos* and Th. E. Tsovilis High Voltage Laboratory, School of Electrical & Computer Engineering,

Faculty of Engineering, Aristotle University of Thessaloniki, Building D, Egnatia St., 54124 Thessaloniki, Greece

*Email: Abstract: Lightning is the main cause of transmission line outages affecting reliability of power supply thus, consequently, resulting in economic losses. A statistical method for lightning incidence calculations in transmission lines is introduced. Simple expressions for the estimation of an expected range of lightning strikes to a transmission line depending on interception probability distribution have been obtained, based on a recently proposed statistical lightning attachment model derived from scale model experiments. The expected number of lightning strikes depends, besides transmission line geometry, on lightning stroke current distribution and interception probability. The results of the statistical method have been compared with those obtained by employing other models from literature, including that suggested by the IEEE Std. 1243:1997, in lightning incidence calculations, and with field observation data; a satisfactory agreement has been shown to exist. Results on lightning incidence calculations are further discussed through an application to typical 150 kV and 400 kV lines of the Hellenic transmission system.

1. INTRODUCTION

Lightning is the main cause of transmission line outages affecting reliability of power supply thus, consequently, resulting in economic losses. Therefore, shielding against direct lightning strokes to phase conductors of transmission lines is provided by shield wires, which are metallic elements that are able to, by physical means, launch a connecting upward discharge that intercepts the descending lightning leader from a distance termed striking distance within a capture radius commonly termed attractive radius or lateral distance.

Lightning interception by an air terminal depends on the total probability for a certain lightning stroke current and a connecting upward discharge initiated at the air terminal. Thus, lightning incidence calculations or shielding analysis in overhead transmission lines should take into account, besides lightning stroke current distribution, the interception probability distribution by considering the striking distance and attractive radius of a conductor as statistical quantities. However, the IEEE Standard [1], suggesting for lightning incidence calculations the use of Erikssons method [2], does not consider interception probability.

In the present study, a statistical method for lightning incidence calculations in transmission lines is introduced, which yields an expected range of lightning strikes depending on interception probability distribution. This task is accomplished by simple expressions, which have been derived on the basis of scale model experiments [3-6], and take into account besides transmission line geometry, the lightning stroke current and interception probability distributions. The results according to the statistical method are compared with those obtained from the IEEE Standard [1], previously reported models [2, 7-10] and with field

observation data [2, 11]; a satisfactory agreement exists. Results on lightning incidence calculations are discussed also through an application to typical 150 kV and 400 kV lines of the Hellenic transmission system.

2. STATISTICAL LIGHTNING ATTACHMENT MODEL

Lightning interception is a stochastic phenomenon, thus to analyse it statistically knowledge of the lightning stroke current and interception probability distributions is required. Although, the former distribution is available in [12] the latter, being dependent upon several geometrical and physical parameters, is almost impossible to be estimated in practice. However, this becomes feasible when simplifying the real case into laboratory investigation on the discharge interception efficiency of a simple earthed rod inserted in a rod-plane gap (Figure 1). Although such scale model experiments can be considered as a rough approximation of the natural lightning flash, they were proved useful in investigating the relation between striking distance, interception radius and interception probability, and to extend this experimental work to the lightning environment [3-6].

In previous work [4], by using the three electrode arrangement shown in Figure 1, at applied voltages always causing breakdown interception probability distributions were obtained, for several electrode configurations differing in D and h, by varying R; typical such distributions are shown in Figure 2. The interception probability distributions were found to be well approximated with the normal distribution, thus both striking distance and interception radius can be treated as statistical quantities with their distribution described by a mean value and a corresponding standard deviation [4]. Actually, the interception radius R is

ISBN 978-0-620-44584-9Proceedings of the 16th International Symposium on High Voltage Engineering

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Figure 1: Schematic diagram of the electrode arrangement; D striking distance to earth surface; S striking distance to earthed rod; R interception radius; h height of the earthed rod.

Figure 2: Interception probability distribution of an earthed rod when inserted in a 75 cm rod-plane gap; fitting curves are drawn according to normal distribution; (a) positive polarity, (b) negative polarity.

well expressed at any interception probability by the following expression:

2,k

ciR hch D

= [6] (1)

where h, D are defined in Figure 1 and values for the coefficients c2, k and for , in formula form, are given in Table I [6]. Equation (1) is valid for 01 the asymptotic values of (1) at h/D unity i.e. the equal coefficient c2 may be adopted [6].

Table 1: Coefficients c2, k and to be used in (1).

For negative lightning, a widely used expression in literature for the striking distance to earth surface D is: 0.6510D I= [13] (2) with D in meters and I in kA; expression (2) is used hereafter in (1) for interception radius calculations. Figure 3 shows the variation of interception radius with air terminal height at 2.5% (failure), 50% (critical) and 97.5% (attractive) interception probabilities obtained by using the statistical lightning attachment model. Figure 3 also shows the interception radii obtained from those lightning attachment models suggested by [1] and [14] to be applied for lightning incidence calculations. It can

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60 70Air terminal height (m)

Inte

rcep

tion

radi

us (m

) .

Attractive Critical Failure

[10]

[2]

[9]

Figure 3: Interception radius as a function of air terminal height; negative lightning peak current 30 kA.

be deduced that all models yield an increase of interception radius with increasing air terminal height; however, the variation of interception radius with interception probability, yielded by the statistical model, indicates that the expected lightning strikes to an air terminal should vary with interception probability.

3. LIGHTNING INCIDENCE CALCULATIONS

3.1. Equivalent interception radius

In an analogous way to the overall attractive distance in [10], the equivalent interception radius Req of a conductor with height h is defined as:

( )0

, ( )eqR R I h f I dI= (3)

where f(I) is the probability density function of the lightning stroke current distribution and R is the interception radius.

According to [12] the probability density function of the stroke current is lognormally distributed as:

( ) ( )2

2

ln ln1exp

22 ii

I If I

I =

(4)

where I is the median current and i is the standard deviation of the natural logarithm of the current amplitudes.

According to the statistical model, by using the equations (1)-(4) the variation of the equivalent interception radius with interception probability for negative lightning can be estimated by:

222

0.32 2

ln lnexp 0.455

2 26.21

c

ieq

i i

I IhR = + +

(5a)

0.18 0.27 0.43(%) 13.3 ie I h = (5b)

ISBN 978-0-620-44584-9Proceedings of the 16th International Symposium on High Voltage Engineering

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where h is in meters and I is in kA. Expression (5b) has been obtained by solving equation (3) with the aid of a mathematical software package by considering the distribution of interception radius. For the lightning crest current distribution with I = 30.1 kA and i = 0.76, suggested in [12], (5a) and (5b) become:

0.331ceq

hR = , 0.43(%) 38.2h = . (6)

The general expression (7) can be used for the estimation of the equivalent interception radius according to the lightning attachment models suggested for lightning incidence calculations in [14]: eq rhR

= (7)

where factors r and E, derived from [14], are listed in Table 2: Table 2: Factors r and E to be used in (7).

The variation of the equivalent interception radius with conductor height, obtained by using (6) as well as (7) and Table 2 is shown in Figure 4 together with field data reported in [2] and derived from [11]. There is a large variability in equivalent interception radius among lightning attachment models, however all of them yield an increase of Req with increasing conductor height.

0

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180

0 10 20 30 40 50 60 70Conductor height (m)

Equ

ivql

ent i

nter

cept

ion

radi

us (m

)

Attractive Critical Failure[2] [11]

TL

[9]

[10]

[1], [2][7], [8]

PA

Figure 4: Equivalent interception radius as a function of conductor height; points depict field data.

In all models, the equivalent interception radius tends to saturate with increasing conductor height; an exception to this is model [7], which, as noted in [14], tends to concave upwards and seems to underestimate the equivalent interception radius at lower conductor heights. In general, there is satisfactory agreement between the present work and previously reported models with the field data in [2] considering also that the latter are relatively uncertain and have a statistical

dispersion, as was recognized by Eriksson [2]. Apart from [7], all models agree well with the data points TL from [2], which is regarded as the most reliable according to Eriksson [2], and PA which was derived from [11] where direct flashes were recorded together with the local ground flash density.

It is important to note that expressions (6) and (7) do not take into account the variation of equivalent interception radius with lightning stroke current distribution. This is considered in expressions (5a) and (5b) and the result of their application for three different lightning stroke current lognormal distributions (Table 3) is shown in Figure 5. From Figure 5 it is obvious that with increasing I the equivalent interception radius increases, however this is more evident for lower interception probabilities and higher conductors.

Table 3: Stroke current distribution parameters.

33

0

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120

140

0 10 20 30 40 50 60 70Conductor height (m)

Equ

ival

ent I

nter

cept

ion

radi

us (m

) .

Median current 24 kA, = 0.72Median current 30.1 kA, = 0.76

Attractive

Critical

Failure

0

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60

80

100

120

140

0 10 20 30 40 50 60 70Conductor height (m)

Equ

ival

ent I

nter

cept

ion

radi

us (m

) .

Median current 39kA, = 0.76

Median current 30.1 kA, = 0.76

Attractive

Critical

Failure

Figure 5: Equivalent interception radius as a function of conductor height calculated according to (5a) and (5b) for different lightning stroke current distributions.

3.2. Application to transmission lines

The annual number of lightning strikes to shield wires per 100 km of a transmission line, NS, is given as:

( )0.1 2ReqS gN N b= + (8) where Req is in meters, Ng is the ground flash density (strikes km-2yr-1) and b is the separation distance between the outer shield wires.

The expected annual number of lightning strikes to shield wires of typical 150 kV and 400 kV lines of the Hellenic transmission system, tower geometries are shown in Figure 6, according to different models employed in lightning incidence calculations is shown in Tables 4 and 5, respectively. In these calculations Req has been estimated by putting the average height of shield wires over the span in equations (6) and (7) and assuming Ng = 4 km-2yr-1. From Tables 4 and 5 it can be deduced that there is variability in NS among models, however all of them yield an increase of NS with increasing transmission line height. The range of NS yielded by the statistical method depending on

ISBN 978-0-620-44584-9Proceedings of the 16th International Symposium on High Voltage Engineering

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Figure 6: Typical towers of the 150 kV (a), (b) and 400 kV (c), (d) lines of the Hellenic transmission system.

Table 4: NS of 150 kV transmission lines.

Table 5: NS of 400 kV transmission lines.

interception probability distribution, agrees well with the values of NS obtained from the models suggested by [1], [14] for lightning incidence calculations.

Table 6 shows the variation of NS with lightning stroke current distribution obtained by using in the set of equations (5a), (5b) and (8) the lightning stroke current distribution parameters shown in Table 3. There is a variability in NS with lightning stroke current distribution; a maximum increase of about 30% is found at the lowest interception probability.

Table 6: NS of double circuit 150 kV and 400 kV lines.

4. DISCUSSION

The expected annual number of lightning strikes to shield wires of a transmission line depends upon the ground flash density and equivalent interception radius of the shield wires. From (3) it can be deduced that the equivalent interception radius of a conductor varies with the lightning attachment model used for interception radius calculation, as also shown in Figure 4, and depends on the conductor height (Figure 4) and lightning stroke current distribution. Both IEEE Std [1] and [14] suggest for lightning incidence calculations in transmission lines the use of height dependent expressions for NS, which do not consider the variation of the latter with lightning stroke current distribution; this is important when considering that the lightning stroke current distribution varies seasonally and geographically [12].

Lightning incidence calculations performed according to the present statistical method, that is by putting in (8) the equivalent interception radius given by (5a), take into account, besides transmission line height, the lightning stroke current distribution, as shown in Figure 5 and Table 6. In addition, employing also (5b) according to the statistical method the variation of equivalent interception radius with interception probability may be considered; this results, rather than in a deterministic value, in an expected range of NS (Tables 4-6), which seems more realistic when considering the stochastic nature of lightning interception phenomenon. Actually, the expected range of NS of a transmission line calculated according to the present statistical method, agrees well with the values of NS obtained from other lightning attachment models [2, 7-10], when considering also that there is variability in NS among models (Tables 4 and 5). It must be noted that the Erikssons method [2], which is suggested by the IEEE Std [1] for lightning incidence calculations, yields generally the highest values of NS among models (Tables 4 and 5).

It is well established that S is related to the expected backflashover rate of a transmission line [17]. The latter together with the shielding failure flashover rate determine the lightning performance of the transmission line, hence also its expected outage rate due to lightning strokes. Thus, careful selection of the lightning attachment model and of the lightning stroke current distribution necessitates when evaluating the lightning performance of overhead transmission lines.

ISBN 978-0-620-44584-9Proceedings of the 16th International Symposium on High Voltage Engineering

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5. CONCLUSIONS

A statistical method for lightning incidence calculations has been introduced which takes into account besides transmission line geometry, lightning stroke current distribution and interception probability. Simple expressions are proposed to estimate the distribution of the equivalent interception radius of a conductor with interception probability. Hence, rather than in a deterministic value, an expected range of annual number of lightning strikes to shield wires of a transmission line is calculated; this seems more realistic when considering the stochastic nature of lightning interception phenomenon.

The results of the proposed statistical method have been compared with those obtained by employing other models from literature, including that suggested by the IEEE Std. 1243:1997, in lightning incidence calculations, and with field observation data; a satisfactory agreement has been shown to exist. Results on lightning incidence calculations are further discussed through an application to typical 150 kV and 400 kV lines of the Hellenic transmission system.

6. ACKNOWLEDGMENTS

Th. E. Tsovilis wishes to thank the Research Committee of Aristotle University of Thessaloniki for the support provided by a merit scholarship.

7. REFERENCES

[1] IEEE Guide for improving the Lightning performance of Transmission Lines, IEEE Std. 1243-1997, Dec. 1997.

[2] A. J. Eriksson, The Incidence of lightning strikes to power lines, IEEE Trans. Power Delivery, vol. PWRD-2, no. 3, pp. 859-870, Jul. 1987.

[3] P. N. Mikropoulos and Th. E. Tsovilis, Experimental investigation of the Franklin rod protection zone, in Proc. 15th International Symposium on High Voltage Engineering, Ljubljana, Slovenia, paper 461, pp.1-5.

[4] P. N. Mikropoulos and Th. E. Tsovilis, Striking distance and interception probability, IEEE Trans. Power Delivery, vol. 23, no. 3, pp. 1571-1580, Jul. 2008.

[5] P. N. Mikropoulos and Th. E. Tsovilis, Interception radius and shielding against lightning, 29th Int. Conf. Lightning Protection, Uppsala, Sweden, 2008, paper 4-10, pp. 1-11.

[6] P. N. Mikropoulos and Th. E. Tsovilis, Interception probability and shielding against lightning, IEEE Trans. Power Delivery, vol. 24, no. 2, pp. 863-873, Apr. 2009.

[7] A. G. Anderson, Transmission line reference book 345 kV and above, Second Edition, 1982, chapter 12, Electric Power Research Institute, Palo Alto, California.

[8] IEEE Working Group, A simplified method for estimating lightning performance of transmission lines, IEEE Transactions on Power Apparatus and Systems, vol. PAS-104, no. 4, pp. 919-932, Apr. 1985.

[9] A. M. Mousa and K. D. Srivastava, Modelling of power lines in lightning incidence calculations, IEEE Trans. Power Delivery, vol. 5, no. 1, pp. 303-310, Jan. 1990.

[10] F. A. M. Rizk, Modelling of transmission line exposure to direct lightning strokes, IEEE Trans. Power Delivery, vol. 5, no. 4, pp. 1983-1997, Oct. 1990.

[11] E. Pyrgioti, D. Agoris, C. Menemenlis and P. Stavropoulos, Recording lightning activity in Patras Greece and correlation with the outages of distribution lines, 25th Int. Conf. Lightning Protection, Rhodes, Greece, 2000, pp. 547-552.

[12] Lightning and Insulator Subcommittee of the T&D Committee, Parameters of lightning strokes: A Review, IEEE Trans. Power Delivery, vol. 20, no. 1, pp. 346-358, Jan. 2005.

[13] E. R. Love, Improvements in lightning stroke modeling and applications to design of EHV and UHV transmission lines, M.Sc. thesis, Univ. Colorado, Denver, CO, 1973.

[14] IEEE Working Group, Estimating lightning performance of transmission Lines II updates to analytical models, IEEE Trans. Power Delivery, vol. 8, no. 3, pp. 1254-1267, Jul. 1993.

[15] A. M. Mousa and 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 Delivery, vol. 4, no. 2, pp. 1450-1460, Apr. 1989.

[16] T. Narita, T. Yamada, A. Mochizuki, E. Zaima and M. Ishii, Observation of current waveshapes of lightning strokes on transmission towers, IEEE Trans. Power Delivery, vol. 15, no. 1, pp. 429-435, Jan. 2000.

[17] CIGRE Working Group 33.01, Guide to procedures for estimating the lightning performance of transmission lines, Technical Bulletin 63, Oct. 1991.

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