An Efficient Method to Assess the Indirect Lightning

Performance of Overhead Lines

José Osvaldo S. Paulino, Ivan José da Silva Lopes,

Wallace do Couto Boaventura

Universidade Federal de Minas Gerais

Belo Horizonte, Brazil

josvaldo@cpdee.ufmg.br, wventura@cpdee.ufmg.br,

ivan@cpdee.ufmg.br

Celio Fonseca Barbosa

Centro de P&D em Telecomunicações (CPqD)

Campinas, Brazil

grcelio@cpqd.com.br

Abstract— This paper presents an approximate formula for

the evaluation of the peak value of lightning-induced voltages in

overhead lines over lossy ground, which considers the front-time

of the channel-base current. The formula is used in the

assessment of indirect lightning performance of overhead power

distribution lines, and its results are compared with results

available in the literature. The formula is then used to analyze

the influence of some parameters on the lightning performance of

overhead lines. The results show that the front-time influence

decreases as the soil resistivity increases, and that a higher front-

time leads to a lower flashover rate. It is shown that the use of a

fixed front-time T = 5.63 µs leads to results that matches well the

results obtained from Cigré's front-time probabilistic

distribution. Regarding the return stroke velocity, the results

show that, for soils with resistivity higher than 100 Ωm, the

flashover rate increases as the return stroke velocity increases.

The results also show that a relative velocity vr = 0.4 leads to

flashover rates that match well the results obtained considering

some correlations between return stroke velocity and peak

current proposed in the literature.

Keywords— lightning performance, induced voltage, overhead

distribution line, soil resistivity.

I. INTRODUCTION

The lightning performance of overhead lines can be estimated through probabilistic tools, such as Monte Carlo Method [1], and some guide-lines for such calculation are provided by the IEEE Guide [2]. The indirect lightning performance is usually evaluated with the aid of simulation software that computes the overvoltages induced in the line by nearby lightning flashes [3]. Although effective, this approach has the drawback of being very time consuming, as it requires the computation of induced voltages several hundred thousands of times.

In order to overcome this difficulty, this paper uses an alternative method for the assessment of the indirect lightning performance of aerial lines. The method is based on the evaluation of the induced-voltage peak value instead of its waveform, which is done by using a simple formula that can be easily computed. When used in Monte Carlo Method, the formula drastically reduces the simulation time.

The first part of the paper is dedicated to the description of the peak-value formula used in the simulations, which is based on previous work carried out by the authors [4, 5]. In order to validate the proposed formula and to establish its limits of validity, an extensive set of simulations were performed using the computer code described in [6]. The resulting formula is used with Monte Carlo Method in indirect lightning performance assessment and its results are compared with results from Borghetti et al. [3]. A sensitivity analysis is then carried out in order to analyze the effect of the stroke current front-time and the return stroke velocity on the indirect lightning performance of overhead power distribution lines.

II. LIGHTNING-INDUCED VOLTAGES

A. The Approximate Formula

Rusck [7] developed a formula for the peak value of

lightning induced voltages in an infinite aerial line considering

ideal soil, step current waveform and the transmission line

(TL) return-stroke model [8]. Darveniza [9] proposed an

empirical modification of Rusck's formula in order to take into

account the effect of soil resistivity. In a previous work [4],

the authors also presented a formula to compute the peak

value of induced voltages in an aerial line over lossy ground,

which provides better results than Darveniza's one. This

formula was later extended for a trapezoidal stroke current

with a fixed front time of 3.8 s [5]. Recently, Andreotti et al.

[10] developed an analytical approach for lightning induced

voltage calculation whose results are reasonably close to the

ones obtained by Paulino et al. formulas [4], [5]. In this paper,

the author's formula is further improved in order to take into

account an arbitrary stroke current front-time T in the range

1 µs ≤ T ≤ 12 µs and a return-stroke velocity v in the range

30 m/µs ≤ v ≤ 150 m/µs. The proposed formula for the

induced voltage peak value is:

SRP VVkV , (1)

where VR is the parcel that considers a perfect soil, VS includes the contribution of finitely conducting soil, and the factor k = 0.915 is needed to take into account the delay between the voltages VR and VS, as discussed in [4].

This work was financed in part by CNPq (Brazilian National Council for Scientific and Technological Development) and FAPEMIG (Minas Gerais

State Research Foundation).

2013 International Symposium on Lightning Protection (XII SIPDA), Belo Horizonte, Brazil, October 7-11, 2013.

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An expression for VS was developed by the authors [4] and

it is reproduced here for convenience:

yIvV rS

0

31

3 , (2)

where I0 is the peak value of the stroke current, is the soil resistivity, y is the closest distance between the lightning striking point and the line, vr is the relative velocity of the return stroke (vr = v/c), v is the return stroke velocity and c is the light velocity.

This paper uses an expression for VR that considers the current front-time, which was proposed by Rusck [7] for the calculation of the peak value of lightning induced voltage in an infinite aerial line over an ideal soil, and it is reproduced in (3). The peak value is calculated at the closest point along the line with respect to the flash, as the maximum induced voltage occurs at this point.

22

22

0

11

11ln15

y

hIVR

, (3)

where:

22 1121

rr

r

vv

v ,

y

Tvr150 ,

T is the return stroke current front time in s, and h is the line height in meters.

Expression (3) is an approximation but, when compared with computer code TIDA results [6], the errors are less than 5%. The simulations also show that the stroke current front-time has little influence on the contribution to the induced voltage from the finitely conducting soil (VS), so that (2) does not need to be changed. These results point in favor of using (1), (2), and (3) to assess the peak value of lightning induced voltages on aerial lines. The errors related to this approximate expression are systematically analyzed in the following.

B. Validation of the Proposed Formula

In order to validate the proposed formula and to establish its limits of validity, an extensive set of simulations were performed with the computer code TIDA [6], using a trapezoidal channel-base current defined by a peak value I0 and a front-time T. The return-stroke was represented by the transmission line (TL) model and the induced voltage was calculated at the center of the line shown in Fig. 1, which is matched at both ends, in order to avoid reflections. The difference between the results is calculated by:

100(%)

codeComputerPeak

FormulaPeakcodeComputerPeak

V

VVDifference (4)

Referring to Fig. 1, the induced voltages were calculated for a combination of the following set of parameters: I0 = 10 kA; h = 5 and 10 m; y = 50, 100, 200, 400, 600, 800 and

1000 m; r =10; ρ = 100, 500 and 1000 m; T = 1, 6, and 12 µs; vr = 0.1 and 0.5. For y = 50 m, the line was split into a

series of 7.5 m long segments and a time step of 0.025 s was used. For y > 50 m, the line was split into a series of 15 m

long segments and a time step of 0.05 s was used.

Fig. 1. Line parameters and lightning stroke relative position.

Figures 2 and 3 show the differences obtained using (4) for the outer range limits of resistivity, velocity and front- time.

For the range of parameters: 100 m < < 1000 m, 5 m < h < 10m, 50 m < y < 1000 m, 1 µs < T < 12 µs and 0.1 < vr < 0.5, the differences between the peak values obtained with the computer code and the proposed expression

are within 10%.

Fig. 2 . Assessment of the error in the total voltage VP as given by (1), (2) and (3), for h = 5 and 10 m, er = 10, ρ = 100 Ωm and vr = 0.1 and 0.5. Thin line: T = 1 µs and bold line: T = 12 μs.

Fig. 3 . Assessment of the error in the total voltage VP as given by (1), (2) and (3), for h = 5 and 10 m, er = 10, ρ = 1000 Ωm and vr = 0.1 and 0.5. Thin line: T = 1 µs and bold line: T = 12 µs.

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III. APPLICATION TO LIGHTNING PERFORMANCE OF LINES

A. Effect of Line Length

In order to illustrate the application of the proposed formula on the assessment of lightning performance of overhead lines, some calculations were carried out using a probabilistic approach similar to the one used by Borghetti et al. [3]. Monte Carlo Method was used as described in [5] and 10

6 simulations were performed for each case studied. In all

cases, it was considered the phase-to-ground induced voltage.

At this point, it is important to highlight that Borghetti et al. considered a relatively short line (2 km), while the development of (1) considered a 10 km long line. Although the influence of line length on the induced voltage reduces with the increase of line length, from 2 km to 10 km there is still a significant difference on the induced voltage, especially due to the contribution of finitely conducting soil.

An example of this difference is shown in Fig. 4, which shows the induced voltage at the center of the line shown in Fig. 1 for a 10kA stroke current at y = 500 m. The line is 10 m high and it is 2 km or 10 km long. It is clear that the increase in the line length leads to an increase in the induced voltage. Therefore, in order to allow a comparison between results from (1) and results from Borghetti et al. [3], a set of simulations was carried out with the computer code TIDA [6] and a correction factor of 0.80 applied to VS was introduced in (1) to convert it to a 2 km line. The adjusted equation reads:

SRP VVkV 8.0 . (5)

It shall be pointed out that (5) is only used here to compare its results with [3], and that (1) is used for all other results presented in this paper. It is clear that (1) provides more realistic results than (5), as long as long lines are concerned.

B. Comparison with Published Results

The adjusted equation (5) was used to assess the annual indirect lightning-induced voltage flashover rate of an overhead line, 2 km long, 10 m high, in a region with 1 ground flash per km

2 per year. The results are presented in Fig. 5, as a

function of the line critical flashover (CFO). The soil resistivity values used in the simulations were ρ = 0, 100, and 1000 Ωm, and the relative return stroke velocity was 0.4.

Fig. 4. Effect of line length on the induced voltage. I0 = 10 kA, h =10 m, vr = 0.4, ρ = 1000 Ω.m, and line length 2 km and 10 km.

Fig. 5. Annual indirect lightning induced voltage flashover rate. Solid line: results from Borghetti et al.[3]; Dashed line: results from this study using (5). NG = 1 flash/(km2 year), h =10 m and vr = 0.4.

On the same figure, the results from Borghetti et al. [3] are presented for the sake of comparison. As seen, there is a good correlation between the two sets of curves, for both perfectly conducting and finitely conducting soils. The slight change in the slope of the curves from [3] for low soil resistivity and high CFO may be due the dependency of these regions on very rare peak currents. This slope change is less pronounced in the dashed lines probably because each dashed line resulted from 10

6 simulations, while each solid line resulted from 1.2 × 10

5

simulations.

This is a key feature of the proposed method, i.e., as it is based on a simple formula to compute the peak value of the induced voltage, it requires very little computational effort and time. For instance, the 10

6 simulations shown in Fig. 5 took

only 19 seconds in a standard PC (Intel Core i3). This feature makes it possible to use this method to analyze the sensitivity of the flashover rate of an overhead line to some key parameters, which is carried out in the following section.

IV. SENSITIVITY ANALYSIS OF THE FLASHOVER RATE

A. Stroke Current Front Time

The influence of the stroke current front-time on the flashover rate is shown in Figs. 6 to 8, for different soil resistivity values. In these figures, the front-time value is obtained in three different ways: (i) fixed front-time value of 1 µs, (ii) fixed front-time value of 12 µs, and (iii) variable front-time value that follows the probabilistic distributions adopted by Cigré [11]. In this last case, a correlation coefficient of 0.47 between the current peak value and the front-time value was considered. In all simulations, the line height was h =10 m, the line length was L = 10 km, the ground flash density was NG = 1 flash/(km

2 year), and the relative

return stroke velocity was vr = 0.4.

Figure 6 shows the results for perfectly conducting soil, where it can be seen a strong dependency of the flashover rate with the current front-time. As expected, the line from Cigré model lies between the lines for the two fixed front-time values. It is also clear that a shorter front-time leads to a higher flashover rate.

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Fig. 6. Effect of the current front-time on the indirect lightning flashover rate for perfectly conducting soil.

Figure 7 shows the effect of the current front-time for a soil resistivity ρ = 100 Ωm. It can be seen that the current front-time still has an important influence on the flashover rate, although this influence is less pronounced than the one observed for perfectly conducting soil. Figure 8 shows the third case for a soil resistivity ρ = 1000 Ωm. It can be seen that, in this case, the current front-time has a very small influence on the flashover rate, when compared with the results observed for perfectly conducting soil.

From Figs. 6, 7, and 8, it is clear that the influence of the current front-time on the flashover rate reduces as the soil resistivity increases. This behavior can be explained if one considers that the component VS from finitely conducting soil becomes progressively dominant on the total induced voltage, as the soil resistivity increases. As this component is not significantly influenced by the current front-time, the total induced voltage also becomes less sensitive to the current front-time as the soil resistivity increases.

Fig. 7. Effect of the current front-time on the indirect lightning flashover rate for finitely conducting soil with ρ = 100 Ωm.

Fig. 8. Effect of the current front-time on the indirect lightning flashover rate for finitely conducting soil with ρ = 1000 Ω.m.

An observation of Figs. 6 to 8 suggests that a fixed front-time conveniently selected could reproduce the results obtained with the Cigré model. Considering that the IEEE Guide [2] provides a median value of the first stroke front

equal to 5.63 s, more simulations were carried out in order to compare the results obtained with this fixed front-time and the results from Cigré model. The results are shown in Fig. 9, where it can be seen that the curves obtained with the fixed

front time of T = 5.63 s match very well the curves obtained with the Cigré probabilistic distribution. Therefore, this fixed front-time value leads to results equivalent to the ones produced by the more complete Cigré model.

It should be pointed out that Borghetti et al. [3] also analyzed the influence of the stroke current front time in the lightning performance of distribution lines. The comparisons were made with the IEEE Guide curves [2] that uses a step current waveform, with curves using a trapezoidal current

waveform with front-time T = 1 s, and curves using the Cigré current front-time probabilistic distribution. The comparison results are in line with to the ones presented in this paper

Fig. 9. Effect of the current front-time on the indirect lightning flashover rate. Solid line: fixed current front-time T = 5.63 µs; Dashed line: current front-time according to Cigré [11] probabilistic model.

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B. Return Stroke Velocity

The influence of the return stroke velocity in the lightning performance of an overhead line is shown in Fig. 10, considering a line 10 km long, 10 m high in a region with 1 flash/(km

2 year). The current front-time followed the Cigré

[11] distribution. Two fixed relative return stroke velocity were used: 0.1 and 0.5, which are the validity limits of the proposed expression (1).

Figure 10 shows that, for soils with resistivities values

equal to 100 and 1000 m, the flashover rate increases as the relative velocity increases. However, for an ideal soil this behavior is in the opposite direction.

This behavior can be explained by the two components of the induced voltage, as shown in (1). For finitely conducting soil, the VS component seems to be the prevalent one, and it increases with the increase of the return stroke velocity, what can be clearly seen in (2). On the other hand, for perfectly conducting soil, the prevalent (and single) component is VR.

However, a look into (3) does not show the behavior of VR with the return stroke velocity, which requires some numerical computation. Fig.11 shows the induced voltage waveform calculated by the computer code TIDA for a 10 m high overhead line over an ideal soil, for a 10 kA peak stroke current with a relative velocity of 0.1 and 0.5, front time of

5.63 s, and striking point 50 m away from the line. It can be seen that, for the conditions considered, an increase in the return stroke velocity from 0.1 to 0.5 leads to a decrease in the induced voltage peak value. This result is consistent with the curves shown in Fig.10 for ideal soil.

As real soils are finitely conducting, one may expect that the flashover rate increases as the return stroke velocity increases. This is particularly true for soil resistivity equal to or higher than 100 Ωm.

Fig. 10. Effect of the return stroke velocity on the indirect lightning flashover rate. Stroke current front time following the Cigré [11] distribution.

Fig. 11. Induced voltage calculated with TIDA code. Perfectly conducting soil, I0 = 10 kA, T = 5.63 μs, y = 50 m.

An empirical expression relating the return stroke velocity to the current peak value of the first stroke was proposed by Lundholm [12] and modified by Rusck [7]:

2

1

05001

Ivr. (6)

Another expression obtained from field test data was proposed by the IEEE T&D Committee [13]:

)016.0(5.0 0Ierfvr , (7)

where I0 is in (kA) in (6) and (7).

Expressions (6) and (7) can be used in the assessment of lightning performance of overhead lines. The results are shown in Fig. 12 for stroke velocities obtained from (6), (7), and also for a fixed value of vr = 0.4, which is the value used in the IEEE Guide [2]. It can be seen that all three models lead to equivalent results.

Souza et al. [14] also analyzed the influence of the return stroke velocity in the distribution line lightning performance. Although they used a step stroke current waveform, the results obtained are similar to the ones presented in this paper.

Fig. 12. Effect of different models for the return stroke velocity on the indirect lightning flashover rate. Stroke current front time following the Cigré [11] distribution.

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V. DISCUSSION

According to IEEE Guide [2], for lightning performance assessment of power distribution lines, the relevant component of the lightning flash is negative first stroke. According to IEEE T&D Committee [13], 97% of the negative first strokes

have front time in the range between 1 s and 12 s, which means that the proposed expression covers nearly the entire range of interest.

According to IEEE T&D Committee [13], there is a strong correlation between the return stroke velocity and the current peak value. According to (7), vr = 0.10 leads to I0 = 11.2 kA, while vr = 0.50 leads to I0 = 180 kA. This current range comprises 98% of the negative first strokes, which would comprise an equivalent percentage of return stroke velocities. Therefore, the return stroke velocity range considered in this paper covers almost the entire range of interest.

The minimum distance from the line considered in this paper (y = 50 m) is linked to the maximum ground resistivity (ρ = 1000 Ωm), as for shorter distances the conductive coupling is relevant to the lightning induced voltages (see [15] and [16] for details). In principle, it would be possible to consider a shorter distance, as long as the ground resistivity is also reduced. However, it shall be considered that a flash closer than 50 m to an aerial line is likely to strike the line directly.

Section IV shows that the use of a fixed current front-time

of 5.63 s and a fixed relative stroke velocity of 0.4 lead to results that match very well the results obtained using the proposed probabilistic distributions for these parameters. Using these values, (2) and (3) can be simplified into:

yIVS

028.1 , (8)

22

22

0

11

11ln0568.0

hIVR

, (9)

where η = 337.8 / y.

VI. CONCLUSIONS

The proposed approximate formulas are suitable for sensitivity analysis in lightning induced voltage studies. The use of the formulas requires very little computational effort and time. For instance, it allows the calculation of 10

6

induced-voltage peak values in only 19 seconds, using a standard PC (Intel Core i3

TM). The results from the

methodology presented in this paper are consistent with results obtained with a more accurate methodology, such as the one presented by Borghetti et al. [3].

The analysis of the influence of the stroke current front-time on the flashover rate of an overhead power distribution line shows that the front-time influence decreases as the soil resistivity increases, and that a higher front-time leads to a lower flashover rate. It also shows that the use of a fixed front-time T = 5.63 µs leads to results that match well the results obtained from Cigré's front-time probabilistic distribution.

The analysis of the influence of the return stroke velocity on the flashover rate shows that, for finitely conducting soils with ρ ≥ 100 Ωm, the flashover rate increases as the return stroke velocity increases. It also shows that a relative velocity vr = 0.4 leads to flashover rates that are in close agreement with the results obtained considering Rusck-Lundholm's [7] and IEEE T&D Committee [13] correlations between return stroke velocity and peak current.

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[8] V. Rakov, "Engineering models of the lightning return stroke," Proc. VII Int. Symp. on Ligh. Protection, pp. 511-530, Curitiba, Brasil, 2003.

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[12] R. Lundholm, R. B. Finn, and W. S. Price, "Calculation of transmission line lightning voltages by field concepts," AIEE Trans. on Power App. and Systems, Part III, vol. 76, no. 3, pp. 1271-1281, Feb. 1958.

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[14] R. S. Souza, I. J. S. Lopes, and J. O. S. Paulino, "Influence of the return stroke velocity and the statistical method used in distribution lines indirect lightning performance estimation considering the ground resistivity," Proc. of the XI Int. Symp. on Lightning Protection, Fortaleza, Brazil, Oct. 2011.

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