A Comparison between Analytical Solutions for Lightning-Induced Voltage Calculation

An exact closed form solution for the calculation of lightning-induced voltages on overhead lines has been recently proposed by A. Andreotti et al. (2012). Predictions of this exact formulation are compared here to those based on approximate analytical solutions proposed in the literature. DOI: http://dx.doi.org/10.5755/j01.eee.20.5.2967


I. INTRODUCTION
Distribution lines are very sensitive to nearby lightning strike events, and this can cause power quality problems.For this reason, a study on lightning-induced voltages shall be carried out.
In recent years, remarkable progress has been made in the studying lightning effects.Numerical approaches (e.g., [1]- [7]) allow very good modeling of the problem.Return-stroke current waveshape, influence of ground conductivity, surge arresters and other non-linearities can be taken into account.On the other hand, analytical solutions (e.g., [8]- [16]) are also very important in dealing with the design of the line.Furthermore, many computer codes use analytical solutions for the evaluation of lightning induced effects [17].Finally, analytical solutions are not affected by instability problems [18].
A typical configuration that can be found in the calculation of lightning induced voltages is represented by a lossless, single conductor line, located over a perfectlyconducting ground and illuminated by a lightning field produced by a linearly-rising current which propagates according to the Transmission Line model [19], [20].The configuration is shown in Fig. 1.
Approximate analytical solutions to this problem have been proposed by Chowdhuri and Gross [21], [22], Liew and Mar [23], Sekioka [24], and Hoidalen [10].Andreotti et al. were able to find the exact solution [25].In this paper, this exact solution will be compared to those proposed by other authors.The paper is organized as follows: in Section II, a survey of the solutions presented by other authors is carried Manuscript received October 5, 2013; accepted December 14, 2013.out; in Section III, the Andreotti et al.'s solution is briefly reviewed; in Section IV, this solution is compared to the other solutions, and, finally, conclusions are drawn in Section V.

A. Chowdhuri-Gross Formula
Chowdhuri and Gross proposed a solution for a linearly- , where 0 I and f t are, respectively, the peak value and the front time of the lightning current.This solution, obtained starting with a coupling model developed by the authors themselves [26], was first published in [21].
The solution was modified in [22] on the basis of the suggestions given by Cornfield [27].We consider here the final expression specified for x = 0 Fig. 1.A typical configuration for lightning-induced voltage calculations.

A Comparison between Analytical Solutions for Lightning-Induced Voltage Calculation
Symbols' meaning can be found in [22].Formula (1) refers to the rising part of the lightning current, it is therefore necessary to add an additional contribution to complete the overall current waveshape, which corresponds to its postpeak part (constant-level or drooping tail).As shown in Fig. 2, where t t denotes the tail time of the lightning current, in the case of constant-level tail, this contribution consists of a time-delayed ramp of negative slope and the same magnitude as the positive ramp.For a drooping tail the second contribution has to be evaluated as explained in [25].

B. Liew-Mar Formula
The formula proposed by Liew and Mar was originally published in [23] and then revised in the discussion of [22].In fact, the original formula had several typographical errors.The formula, specified for x = 0, reads Symbols' meaning can be found in [22].In this case too, (4) refers to the first part of the current.Another term must be considered, as done in the previous subsection.

D. Hoidalen Formula
Hoidalen [10] proposed an approximate formula which allows one to evaluate the induced voltage along the line.This solution has been obtained by a numerical convolution of Rusck expression for the step current case.
As the solutions presented above, the Hoidalen solution is the sum of the two contributions, which account for both the rising part and the tail of the current.At x = 0, it reads with In this case, symbols' meaning can be found in [10].Note that the expression given in [10] refers to a finite length line.It can be easily extended to the case of an infinite length line.

III. ANDREOTTI ET AL. FORMULA
We will briefly review the formula proposed by Andreotti et al. [25] for the evaluation of the induced voltage at x = 0 in the case of a linearly-rising current (see Fig. 1).
This expression has been derived starting from the rigorous analytical solution presented by Andreotti at et al. for a step current [12] by using the Duhamel's integral.For a linearly-rising current of constant slope 0 / f I t   , the resulting expression is (see [25] for details) where 0  is the permeability of free space,  is the ratio of the return stroke speed to c (the speed of light in free space), t is the time ( 0 t  , return stroke inception), , h is the line conductor height above ground, d is the horizontal distance between the lightning channel and the line conductor, , and ( ) u  is the Heaviside function.The expression of 2 (0, ) v t can be derived from 1 (0, ) v t by changing the sign of h.Equations ( 8), (9) give the exact closed form expression for the linearly-rising portion of the lightning current.An additional contribution has to be added to obtain the voltage induced by the overall current waveshape, as done for previous models.

IV. COMPARISON OF THE EXACT CLOSED-FORM SOLUTION
WITH OTHER SOLUTIONS PRESENTED IN LITERATURE In this section, solution (8) will be compared to the other analytical solutions described in Section II.
For comparison purposes, we will assume a current at the base of the channel with a maximum value 0 I = 12 kA, a front time f t = 0.5 μs and a drooping tail with t t = 20 μs.
This specific channel-base current was selected since it represents the best fit to a typical measured channel base current [10].

A. Chowdhuri-Gross Formula
In this paragraph, we will compare the induced voltage waveform obtained by using the Chowdhuri-Gross formula (1) with the one obtained by using our exact solution (8).In Fig. 3, the comparison is shown for 10-meter height line placed at a distance d = 50 m away from the lightning channel; results have been obtained for the following values of the parameters, 0 I = 12 kA,  = 0.4, f t = 0.5 μs and t t = 20 μs.The same comparison, but with d = 100 meters, is shown in Fig. 4. We point out that, since the expression proposed by Chowdhuri-Gross is given for a lightning channel of finite length ( c h ), for comparison purposes we have assumed, for both graphs, c h → ∞ in (1).Furthermore, for the purpose of completeness, we have also considered a finite length channel having c h = 3 km (see Fig. 5 and Fig. 6).The comparison clearly shows that the Chowdhuri-Gross's formula, similar to the case of step current analysed in [12], cannot be considered correct.We notice a polarity inversion, more pronounced when c h is finite, which cannot be justified for the voltage induced by a linearly-rising current waveform with drooping tail in case of lossless ground.

B. Liew-Mar Formula
Now, we will compare results obtained by using our solution (8) and their counterparts obtained by using the Liew-Mar formula (4).In Fig. 7  As for the Chowdhuri-Gross's formula, the Liew-Mar solution also considers a finite channel length: for comparison we have assumed c h → ∞ in (4) for both graphs.To check the effects of a finite length channel, we have also considered the case c h = 3 km (see Fig.  10).As one can see from the comparison, the Liew-Mar formula predicts a lower peak value for d = 50 m, and a higher peak value for d = 100 m.In both cases, results obtained using (4) show a steeper front, a too rapid decay of the current tail, and a polarity inversion.The latter effect is seen both for finite and infinite lengths of the lightning channel.We conclude that the Liew-Mar solution cannot be considered correct.

C. Sekioka Formula
By comparing the results obtained using solution (8) and their counterparts obtained by means of the Sekioka formula (5), calculated for the same values of parameters proposed in the previous paragraphs ( h = 10 m, d = 50, 100 m, 0 I = 12 kA,  = 0.4, f t = 0.5 μs and t t = 20 μs), one can observe that the results are practically the same (overlapped).Some minor differences can be spotted by zooming in the graphs, as shown in Fig. 11 for the case of d = 50 m.We can therefore conclude that the Sekioka's formula is consistent with the exact solution and can be considered a useful tool for the analysis of power distribution lines, for which the height is relatively small.For higher lines, such as transmission lines, small differences between predictions of ( 8) and ( 5) can be detected.However, the Sekioka formula can still be considered a suitable tool for this kind of lines.In Fig. 12, a 30-meter height line is considered.

D. Hoidalen Formula
A comparison between (8) and Hoidalen formula ( 6) is shown in Fig. 13 and Fig. 14, where a 10 m and a 30 m lines have been considered, respectively.As in the case of Sekioka's expression, practically no differences are observed for a 10 m line (the graph was zoomed in to show some minor differences), whereas for the 30-m line, even if differences can be seen, Hoidalen's formula can still be considered a suitable approximation of the exact solution.

V. CONCLUSIONS
In this work, the rigorous analytical solution presented by Andreotti et al. [25] for the calculation of voltages induced on an overhead conductor located above a perfectlyconducting ground by a linearly-rising lightning current waveform has been reviewed and predictions of this formulation have been compared to those given by other analytical (approximate) solutions proposed in the literature.
The comparison shows that both the Chowdhuri-Gross and the Liew-Mar formulas predict results which are in disagreement with the Andreotti et al. solution.In particular, they predict incorrect peak values, steeper fronts, too rapid decays of the current tail, and physically unexplainable polarity inversions.Hence, they cannot be considered correct.Conversely, both the Sekioka solution and the Hoidalen's formula are consistent with the exact analytical Andreotti et al.'s formula Sekioka's formula solution, and can be considered suitable for lightning induced overvoltage analysis of distribution lines, i.e., lines with relatively small height.For such lines, differences between Andreotti et al.'s formula and both the Sekioka and the Hoidalen's expressions are within the 0.6 %.
, the results obtained for h = 10 m, d = 50 m, 0 I = 12 kA,  = 0.4, f t = 0.5 μs and t t = 20 μs are shown.In Fig. 8, the same comparison is shown, but for d = 100 m.
Fig.10).As one can see from the comparison, the Liew-Mar formula predicts a lower peak value for d = 50 m, and a higher peak value for d = 100 m.In both cases, results obtained using (4) show a steeper front, a too rapid decay of the current tail, and a polarity inversion.The latter effect is seen both for finite and infinite lengths of the lightning channel.