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IEEE/PES T&D 2010 LATIN AMERICA - SAO PAULO - 8 A 10 DE NOVEMBRO 1

Influence of Electrogeometric Model and Statistical

Current Distribution in Distribution Lines Indirect

Lightning Performance Estimation Considering the

Ground ResistivityRosembergue P. Souza, Ivan J. S. Lopes, and Jose O. S. Paulino

Abstract—The estimation of overhead distribution lines light-ning performance is very important for insulation design. Dueto the random nature of lightning, different aspects of thephenomenon, such as the discharge points of incidence andthe statistical distribution of the current intensity, are modeledin performance evaluation studies. This paper presents a com-parative study of different models proposed in the literatureto estimate the indirect lightning flashover rates of overheaddistribution lines. The electrogeometric model, that defines thelightning striking point, and the statistical current distributionare evaluated using Monte Carlo simulations. The results showthe importance of considering the most accurate information onthese parameters for a more realistic performance estimation.

Index Terms—Overhead distribution lines, eletrogeometricmodel , lightning-induced overvoltage, Monte Carlo Method.

I. INTRODUCTION

D ISTURBANCES to power distribution lines are mainly

caused by lightning [1],[2]. This happens because their

insulation level is relatively low, with typical values from

90kV to 250kV .Many factors affect the induced voltages caused by indirect

lightning strokes. Among them, one could emphasize the

current return stroke, intensity, velocity and front time, the

distance from the striking point to the line, line height and the

ground resistivity. In recent years, many authors recognized

that the soil resistivity plays an important role in the induced

voltages [3], [4], [5], [9]. Nowadays, there is a general

agreement that the overall effect of the soil resistivity is to

increase the induced voltage on a line when compared to a

perfectly conductive soil [3].

Considering that the annual number of flashovers caused by

indirect lightning strokes is very important for the insulation

design of a distribution line, this paper investigates how thechoice of the electrogeometric model and the statistical current

distribution affect the estimation of the indirect lightning

performance of a distribution line.

I I . METHODOLOGY

The procedure used to estimate the lightning performance

is based on Monte Carlo method. The distribution line is

Rosembergue P. Souza is a Graduate Student (Master’s Degree in ElectricalEngineering), Ivan J. S. Lopes and Jose O. S. Paulino are with the ElectricalEngineering Department at Universidade Federal de Minas Gerais, BeloHorizonte, Minas Gerais, Brazil.

considered located in an open area. The formula for peak

value of the induced voltages, presented by Paulino et al in

[5], is used. Three electrogeometric models and two statistical

lightning current distributions, proposed in the literature, are

considered.

For clarity’s sake, the models used in the proposed proce-

dure are described in this section.

A. Formula for the peak value of the induced voltages

Recently, Paulino et al [5] proposed a formula to estimate

the peak value of the lightning induced voltage in an infinite

line, taking into account the soil resistivity. The formula, pre-

sented in Eq. (1), shows that the induced voltage is a function

of the lightning return stroke and propagation velocity, the soil

resistivity, the line distance to the striking point and the line

height.

V peak = kC (ρ, h) ·√

3 · (vr)1/3 · I 0 ·

ρ

y +

30 · I 0 · h

y

·

1 +

1√ 2· vr · 1

1 − 1

2 · v2

r

(1)

where,

kC (ρ, h) = 1, if ρ = 0 or h = 0 (2a)

kC (ρ, h) = 0.85, if ρ = 0 or h = 0 (2b)

h height of the line (m);

I 0 current peak value (kA);

ρ soil resistivity (Ω.m);

vr current relative propagation velocity;y distance between the striking point and the line(m).

B. The electrogeometric model

The electrogeometric model (EGM) is used to establish

the minimum distance in which the lightning flash will not

strike the line but the ground nearby. Several researchers have

contributed to the EGM for determining the last step of the

lightning flash. The investigation performed in this paper has

considered three of them: the model proposed by Love in [7],

the model of Whitehead and Armstrong in [8] and the model

used in IEEE Std. 1410-2004 [10]. The striking distances

458 2010 IEEE/PES Transmission and Distribution Conference and Exposition: Latin Ameri

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to the line and to the ground are calculated by (3) and (4)

respectively. Equation (5) gives the shortest distance in which

lightning will not strike the line. Table I shows the parameters

of the three EGMs considered in this study.

rs = α × I β0

(3)

rg = k × rs (4)

ymin =

r2s − (rg − h)

2(5)

where:

rs striking distance to the conductor (m);

rg striking distance to the ground (m);

I 0 lightning-peak current (kA);

ymin minimum distance for which lightning will not be

attracted by the line (m).

TABLE IPARAMETERS OF THE E LECTROGEOMETRIC M ODELS

Electrogeometric Models α β k

IEEEstd1410-2004 10 0.65 0.9

Love 10 0.65 0.8

Whitehead-Armst rong 6.7 0.8 0.9

C. Statistical distributions of the peak values return strokes

The statistical distribution of the current return stroke peak

value can be approximated with a log-normal distribution

[14]. A simple equation representing a log-normal distribution

for the current return stroke with quite reasonable accuracy

between 5 kA and 200 kA can be given by two equations [6].

For Popolansky’s curve:

P c(> I 0) = 1

1 +I 025

2 (6)

and for Anderson-Erikson’s curve:

P c(> I 0) = 1

1 +I 031

2.6 (7)

where,


P c probability that the peak current in any flash will

exceed I 0 in kA.

D. Monte Carlo simulations

Monte Carlo method randomly selects parameters for each

stroke from the probabilistic distributions. Once the lightning

parameters are selected, the calculation is performed [12],[13].

An empirical relationship, well known from the literature

[17], between the velocity and the peak value of the current

return stroke is used (Eq.(8)):

v = c 1 + 500

I 0

(m/s) (8)

where:

c velocity of light in free space (m/s);


The current relative velocity can be calculated by (9).

vr = v

c (9)

Combining (1), (8) and (9) leads to Eq. (10). This is theformula to evaluate the peak value of the induced voltage used

in this paper:

V p = kC (ρ, h) ·√

3 ·

1 + 500

I 0

−1/6

· I 0 ·

ρ

y +

30 · I 0 · h

y

·

1 +

1√ 2·

1 + 500

I 0

−1/2

· 1 1− 1

2 ·

1 + 500

I 0

−1

(10)

1) Amplitude of the current return stroke: In order to createa set of random samples of lightning current parameters, the InverseTransformation Method [15] is used to generate a cumulative distri-bution function for the current return stroke P d(≤ I o). Therefore,

P d(≤ I 0) = 1− P c(> I 0) (11)

where:

P d probability that the peak current in any flash will be lowerthan or equal to I 0 in kA.

For Popolansky’s curve:

P d(≤ I 0) =

I 025

2

1 + I 0252 (12)

Equation (12) solved for the current peak to any desired probabilitylevel leads to:

I 0 = 25

P d

1− P d

1/2(13)

For Anderson-Erikson’s curve, one can write:

P d(≤ I 0) =

I 031

2.6

1 +I 031

2.6

(14)

and

I 0 = 31

P d

1− P d

1/2.6

(15)

Making P d random number uniformly distributed between 0 and1 in (13) and (15) generates a random probability according to thecurrent distribution curve.

The peak values of the current return stroke higher than 100 kArarely happen - for instance, equation (7) shows that values higherthan 100 kA have 4% of chance to happen - therefore, for sim-plification, the method has adopted a condition where every currentamplitude larger than 100kA receives another random number lowerthan or equal to 100kA. Figure 1 shows the adjusted sample of datapoints using Eq. (15), for a set of 200000 samples generated by thisprocedure.

PEREIRA DE SOUZA et al.: INFLUENCE OF ELECTROGEOMETRIC 4

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10 20 30 40 50 60 70 80 90 100 110 120 130 140 1500

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

4

Amplitude of current return stroke (kA)

F r e q u e

n c y

Histogram of sampled Io with adjustment

Fig. 1. Histogram of sampled lightning peak current with adjustment

2) Lightning striking distance: In order to select the maximumvalue of the lightning striking distance - ymax - that includes allthe lightning events that may produce a flashover for the lowerCFO value of interest, the procedure has considered a peak valueof 100kA . Afterwards, Equation (10) has been made equal to1.5 × (lower CF O of interest) and ymax has been calculatedusing Newton-Raphson’s method.

The distance between the striking point and the line has receiveda uniform random number between 0 to ymax.

With the random observation from the stroke distance and theprobability distribution of the current return stroke, the method hasevaluated if the stroke has been a direct stroke or an indirect stroke.This decision is based on the following: if the stroke distance is lowerthan ymin, a direct stroke happens, otherwise if the stroke distanceis higher than ymin, an indirect stroke happens. A direct stroke to anunprotected distribution line generally causes a flashover [11], [16]and it is not the point of this work. The peak value of the inducedvoltage has been calculated by (10). The number of discharges forwich the peak value of the induced voltage has exceeded 1.5×CF Ohas been considered in the procedure.

Finally, the flashover rate is calculated with Equation 16, also usedby Borghetti et al in Ref. [16]. The annual number of insulationflashovers per 100 km of distribution line is obtained by:

F p = 200 · n

ntot· N g · ymax (16)

where,

n number of events generating induced voltages larger thanthe insulation level here assumed being equal to the linecritical flashover voltage (CFO), multiplied by a factor

equal to 1.5.ntot number of the total events.N g annual lightning ground flash density in (km−2yr−1).

Figure 2 shows a simplified flow chart of the proposed method.

Fig. 2. Simplified flow chart of the proposed method

50 100 150 200 250 30010

−3

10−2

10−1

100

101

102

Critical Flashover Voltage − CFO (kV)

F l a s h o v e r / 1 0 0 k m / y r

Borghetti − LIOV

Proposed procedure

Fig. 3. Comparison between the line flashover rate for the proposedmethod and the procedure Borghetti - LIOV from [16], for the case of a single conductor 10-m-high infinitely long line for ground conductivityρ = 100 Ω.m


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50 100 150 200 250 30010

−1

100

101

102

103


F l a s h o v e r / 1 0 0 k m / y r

Borghetti − LIOV

Proposed procedure

Fig. 4. Comparison between the line flashover rate for the proposedmethod and the procedure Borghetti - LIOV from [16], for the case of a single conductor 10-m-high infinitely long line for ground conductivityρ = 1000 Ω.m

III. VALIDATION OF THE PROCEDURE TO ESTIMATE THE

INDIRECT PERFORMANCE OF THE DISTRIBUTION LINE

Using the method proposed in the previous section with EGMproposed in IEEE Std 1410-2004 [10], a computer code was writ-ten in MATLAB R to calculate the flashover rates. The follow-ing parameters were used ntot = 200000, line height 10 m,N g = 1 km−2yr−1. Two different resistivity values were considered100 Ω.m and 1000 Ω.m.

Figures 3 and 4 show the comparison between the estimated lineflashover rates presented by Borghetti in [16] using the LIOV-MCprocedure for the case of a infinite long line consisting of a single

conductor above a lossy ground and the proposed method using theEGM proposed in IEEE std 1410-2004 [10]. As seen, the annualflashover rates in Fig. 3 and 4 are in reasonably good agreement.Therefore, the proposed procedure is considered acceptable to per-form the sensitivity analysis that follows.

IV. COMPARATIVE STUDY

A sensitivity analysis about the influence of the EGMs on theindirect lightning performance estimation has been performed. Theproposed procedure used three types of EGMs for performing thecomparison. The coefficients for the three EGM models are describedin Table I. In the simulation the following parameters were usedntot = 200000, line height 10 m, N g = 1 km−2yr−1. Anderson-Erikson’s curve for statistical current distribution was used and5 diferent resistivity values, varying from 50 to 5000 Ω.m were

considered.Figures 5 to 9 show the results for the different soil resistivities.

It can be seen that the flashover rate calculated using the proposedprocedure with the EGM proposed in IEEE Std 1410-2004 [10] ishigher than the flashover rates obtained using the other ones whenthe soil resistivity is low. For higher resistivity, the results obtainedfor the proposed procedure using the three EGMs are very similar. Inaddition, it can be seen that the higher the soil resistivity, the higherare the obtained flashover rates.

The influence of the statistical current distribution curve on theestimation of the indirect lightning performance was investigatedusing the proposed procedure with the EGM of IEEE Std 1410-2004and two types of current distribution curve, Anderson-Erikson’s curveand Popolansky’s curve. Figures 10 to 14 show the results for five

different soil resistivities. As seen, for all five cases, the flashoverrate calculated using the two statistical current distributions is quitesimilar. In addition, the higher the soil resistivity, the higher are theobtained flashover rates.

50 100 150 200 250 30010

−2

10−1

100

101

102


F l a s h o v e r / 1 0 0 k m / y r

Proposed Procedure with EGM IEEE Sdt1410

Proposed Procedure with EGM Love

Proposed Procedure with EGM Whitehead

Fig. 5. Comparison between electrogeometric models presented in Love[7], Whitehead and Armstrong [8] and IEEE Std. 1410-2004 [10] using theproposed procedure with ρ = 50 Ω.m.

50 100 150 200 250 30010

−3

10−2

10−1

100

101

102


F l a s h o v e r / 1 0 0 k m / y r






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50 100 150 200 250 30010

−1

100

101

102

103


F l a s h o v e r / 1 0 0 k m / y r





50 100 150 200 250 30010

0

101

102

103


F l a s h o v e r / 1 0 0 k m / y r





50 100 150 200 250 30010

0

101

102

103


F l a s h o v e r / 1 0 0 k m / y r





50 100 150 200 250 30010

−2

10−1

100

101

102


F l a s h o v e r / 1 0 0 k m / y r

Proposed procedure with Anderson curve

Proposed procedure with Popolansky curve

Fig. 10. Comparison between current distributions Popolansky’s curve andAnderson-Erikson’s curve using the proposed procedure with ρ = 5 0 Ω.mand the EGM presented in IEEE Std. 1410-2004 [10].


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50 100 150 200 250 30010

−2

10−1

100

101

102


F l a s h o v e r / 1 0 0 k m / y r



Fig. 11. Comparison between current distributions Popolansky’s curve andAnderson-Erikson’s curve using the proposed procedure with ρ = 100 Ω.mand the EGM presented in IEEE Std. 1410-2004 [10].

50 100 150 200 250 30010

−1

100

101

102

103


F l a s h o v e r / 1 0 0 k m / y r




50 100 150 200 250 30010

0

101

102

103


F l a s h o v e r / 1 0 0 k m / y r




50 100 150 200 250 30010

0

101

102

103


F l a s h o v e r / 1 0 0 k m / y r





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V. CONCLUSION

This paper presented a comparative study of different modelsproposed in the literature to estimate the indirect lightning flashoverrates of overhead distribution lines considering the soil resistivity.

The electrogeometric model, that defines the lightning strikingpoint, and the statistical current distribution were evaluated usinga procedure based on Monte Carlo method.

The results from the comparative study show that the EGM directlyaffects the results of the estimation of indirect lightning performance

for low soil resistivity,while the flashover rates obtained by thismethod using the Popolansky’s curve and Anderson-Erikson’s curveare very similar.

ACKNOWLEDGMENT

The authors are grateful to CNPq (Brazilian National Council forScientific and Technological Development) for the financial support.

REFERENCES

[1] P. Chowdhuri, “Estimation of Flashover Rates of Overhead Power Distri-bution Lines by Lightning Strokes to Nearby Ground,” IEEE Transactionson Power Delivery, vol. 4, no. 3, pp. 1982-1989, Jul. 1989.

[2] F. H. Silveira and S. Visacro, “Lightning Induced Overvoltages: Effectson Consumer Service Entrance”, presented at IEEE Bologna PowerTech

Conference, 2003, Jun. 23-26, Bologna, Italy.[3] M. Darveniza,“A Practical Extension of Ruscks Formula for Maximum

Lightning-Induced Voltages That Accounts for Ground Resistivity,” IEEE Transactions on Power Delivery,vol. 22, no. 1, pp. 605-612, Jan. 2007.

[4] E. Perez and J. Herrera and H. Torres,“Sensitivity Analysis of InducedVoltages on Distribution Lines”, paper accepted for presentation at 2003IEEE Bologna PowerTech Conference, Jun. 23-26, Bologna, Italy.

[5] J. O. S. Paulino and C. F. Barbosa and I. J. S. Lopes and C. Boaventura,“An Approximate Formula for the Peak Value of Lightning-InducedVoltages in Overhead Lines,” IEEE Transactions on Power Delivery ,vol. 25, no. 2, pp. 843-851 Apr. 2010.

[6] J. G. Anderson, Transmission Line Reference Book 345 kV and Above,2nd Edition,Electric Power Research Institute, 1982, p.549.

[7] R. R. Love, “Improvements on Lightning Stroke Modelling and Applica-tions to the Design of EHV an UHV Transmission Lines,” M.SC. Thesis,University of Colorado,1973.

[8] H. R. Armstrong and E. R. Whitehead,“Field and Analytical Studies of

Transmission Line Shielding,” IEEE Transactions on Power Apparatusand Systems,vol. 87, pp. 270-281, Jan. 1968.

[9] C. A. Nucci and F. Rachidi,“Lightning-Induced Overvoltages ,”presentedat IEEE Transmission and Distribution Conference, New Orleans, Apr.14, 1999.

[10] “Guide for improving the lightning performance of electric poweroverhead distribution lines,” IEEE Std. 1410, IEEE Working Group onthe Lightning Performance of Distribution Lines, 2004.

[11] “Working group report: calculating the ligtning performance of distri-bution lines,” IEEE transactions on Power Delivery, vol. 5, no. 3, pp.1408-1417, Jul. 1990.

[12] J. G. Anderson, “Monte Carlo Computer Calculation of TransmissionLine Lightning Performance,” AIEE Trans., vol. 80, pp. 414-20, Aug.1961.

[13] M. A. Sargent, “Monte Carlo Simulation of the Lightning Performanceof Overhead Shielding Networks of High Voltage Stations,” IEEE Transactions on Power Apparatus and Systems, vol. 91, no. 4, pp. 1651-

1656, Jul./Aug. 1972.[14] Lightning and Insulator Subcommittee of the T&D Committee , “Pa-

rameters of Lightning Strokes: A Review”, Transmission and DistributionConference and Exhibition, 2005/2006 IEEE PES, p. 465 470.

[15] F. S. Hillier and G. J. Lieberman, Introduction to Operations Research,7rd ed. McGraw-Hill Higher Education, New York, 2001.

[16] A. Borghetti and C. A. Nucci and M. Paolone,“An Improved Procedurefor the Assessment of Overhead Line Indirect Lightning Performance andIts Comparison with the IEEE Std. 1410 Method,” IEEE Transactions onPower Delivery,vol. 22, no. 1, pp. 684-692, Jan. 2007.

Rosembergue P. Souza graduated in electrical en-gineering at Federal University of Vicosa (UFV),Vicosa, Brazil, in 2009. Currently, he is a graduatestudent at the Federal University of Minas Gerais(UFMG), Electrical Engineering Graduate Programpursuing his Master’s Degree. His interests includeinsulation coordination, power quality and electro-magnetic compatibility.

Ivan J. S. Lopes received the B.Sc. and M.Sc.

degrees in electrical engineering from the FederalUniversity of Minas Gerais (UFMG), Belo Hori-zonte, Brazil, in 1987 and 1990, respectively, andthe Ph.D. degree in electrical and computer engi-neering from the University of Waterloo, Waterloo,ON, Canada, in 2001. Currently, he is an AssociateProfessor at the Electrical Engineering Departmentof UFMG, where he has been since 1992. Hisinterests include high voltage engineering and itsapplications, dielectrics and electrical insulation, and

electromagnetic compatibility

Jose Osvaldo Saldanha Paulino received the B.Sc.and M.Sc. degrees in electrical engineering from theFederal University of Minas Gerais, Belo Horizonte,Brazil, in 1979 and 1985, respectively, and theD.Sc. degree in electrical engineering from the StateUniversity of Campinas, Campinas, Brazil, in 1993.In 1980, he joined the Department of Electrical

Engineering, Federal University of MinasGerais, asa Professor. His current research interests includehigh voltage and electromagnetic compatibility.


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