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Parametric and statistical investigations of lightning-induced voltages on overhead lines by exact analytical solutions
Electric Power Systems Research 178 (2020) 106044
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Parametric and statistical investigations of lightning-induced voltages on overhead lines by exact analytical solutions
T
Amedeo Andreotti , Fabio Mottola, Antonio Pierno, Daniela Proto, Antonio Sforza, Claudio Sterle ⁎
Department of Electrical Engineering and Information Technology, University of Naples Federico II, Naples, Italy
ARTICLE INFO
ABSTRACT
Keywords: Lightning Lightning-induced voltages Statistical analysis
In this paper, which is an extended version of the paper presented by the authors at the 34th International Conference on Lightning Protection (ICLP 2018), a parametric and a statistical analysis of lightning-induced voltages on overhead lines has been carried out by means of exact analytical solutions, for both step-function and linearly rising channel-base currents. The role played by the various parameters affecting the induced voltages has been analyzed. Further, the statistical investigations has allowed us to identify the parametric statistical distributions which best fit the data obtained by simulations. Based on this statistical investigation, an analysis of the lightning performance of an overhead line has been carried out too.
1. Introduction Assessment of the severity of the induced voltages generated by indirect lightning on overhead lines is a critical task to identify the best protection measures. Various parameters can affect the induced voltages, and the role played by each of them should be clearly identified. Further, as the phenomenon is stochastic, a statistical analysis is very important too. Two different approaches have been used to carry out parametric and statistical analysis: one is based on the so-called numerical models, while the other on the analytical ones. Numerical models (e.g. Refs. [1–3]) are very popular, as they can model complex configurations of the power network; however, analytical solutions, less powerful in that respect as they are limited to more simple configurations (e.g. Refs. [4–11]), are important as well, since they can give an insight that numerical models cannot give, do not suffer from numerical instabilities and/or from lack of convergence [12]; they are also more practical to carry out parametric and statistical analysis (e.g. Refs. [13,14]). In this paper, which is an extended version of the paper presented by the authors at the 34th International Conference on Lightning Protection (ICLP 2018) [15], we will focus on the Andreotti et al. exact analytical solutions [9,11,16]. We will refer to an horizontal conductor, placed over an infinitely conducting ground, and excited by a lightning electromagnetic field produced by either a step-function or a linearly rising current, moving unattenuated and undistorted along a vertical channel. This configuration is depicted in Fig.1. By means of these exact solutions, a parametric and statistical
analysis of lightning-induced voltages will be carried out with the aim of analyzing the role played by the various parameters affecting the induced voltages; in addition, the statistical investigations will allow us to identify the parametric statistical distributions which best fit the simulated data. In order to widen the statistical investigation proposed in Ref. [15], the paper has been completed with the introduction of further statistical analyses. More specifically, parametric distributions were identified to describe distributions of the peak induced-voltage and the goodness of fit was evaluated; the lightning performance evaluation has been carried out too. Note that this analysis refers to the case of perfectly conducting ground. Parametric and statistical treatment in the case of finitely conducting ground has been developed by the authors in Refs. [17] and [18]. The paper is organized as follows. In Section 2, the exact solutions are briefly recalled; in Section 3, a parametric analysis is carried out, while in Section 4 a statistical analysis is developed. Section 5 is devoted to the lightning performance analysis, while Section 6 presents some concluding remarks. 2. Exact analytical solutions In this section, the Andreotti et al. exact analytical solutions are briefly recalled.1 In particular, subsection 2.1 is devoted to the case of the step-function channel-base current, while subsections 2.2 to the case of a linearly rising channel-base current.
Corresponding author. E-mail address: [email protected] (A. Andreotti). 1 By the term exact we mean analytical solutions that, for the assumed models, have been obtained without approximations. ⁎
https://doi.org/10.1016/j.epsr.2019.106044 Received 28 March 2019; Received in revised form 12 September 2019; Accepted 19 September 2019 Available online 07 October 2019 0378-7796/ © 2019 Elsevier B.V. All rights reserved.
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Nomenclature c d
v x
Speed of light in free space. Horizontal distance between the lightning channel and the line conductor (shortly called “strike point distance”). Line height. x 2 + d2 . x 2 + d2 + h2 . Time (t = 0 corresponds to return stroke onset). r0/ c . d 2 + h2 / c . Front time.
h r r0 t t0 t tf
H I Vp
0 0
Return stroke speed. Abscissa along the line (x = 0 corresponds to the point on the line that is closest to the lightning channel). Heaviside step function. Return stroke current magnitude. Induced voltage peak value. Current rate-of-rise for linearly-rising current waveform (I tf ). v /c . 2 (Lorentz factor). 1/ 1 Intrinsic impedance of free space 377 . 0 /(8 ) .
2.1. Induced voltage due to a step-function channel-base current The contribution to the exact solution due to the scalar potential is given by [9,16]:
vs (x , t ) =
0I [
v1 (h) + v1 ( h) + v2 ( ct , h) + v2 (
v2 (
ct ,
ct , h)
v2 ( ct ,
h) (1)
h)]
with
h) + x ]2 + d 2},
v1 (h) = ln{[ ( ct
v2 ( ct , h) = ln [ ( ct
h) + x
2
+
h) 2 + (d2 + x 2)/
( ct
2 ],
while the contribution due to the vector potential is given by [9,16]:
v v (x , t ) =
0I
[v3 ( ct ) + v3 (
ct )
(2)
v4]
with
v3 ( ct ) = 2 ln[
ct + h +
(
ct + h) 2 + (d 2 + x 2)/
Fig. 2. 3D plot of the induced voltage using Eq. (3) (x is the point along the line where the induced voltage is evaluated, h = 10 m, d = 100 m, I = 10 kA, β = 0.4).
2 ],
v4 = 2 ln[(d 2 + x 2 )/ 2].
v 1 (h ) =
Hence, the exact analytical solution (valid for t > t 0 ) for the induced voltage reads [16]:
v (x , t ) =
0I [
v2 (
v1 (h) + v1 ( h) + v2 ( ct , h) + v2 ( ct ,
h)+ v3 ( ct ) + v3 (
ct )
v4].
ct , h)
v2 ( ct ,
v1 ( h)]
h) +
( ct
h)2 + d 2 /
ln [( ct
h) +
( ct
h) 2 + d2 /
2] 2 ].
We show in Fig. 2 a 3D plot of the induced voltage against time and position x along the line.
h) (3)
2.2. Induced voltage due to a linearly rising channel-base current
For the special case of x = 0 (the point closest to the lightning channel), Eq. (3) can be simplified as [9,16]:
v (0, t ) = 2 0 I [v1 (h)
( ct
ln[
The exact analytical solution in the case of a linearly rising current is given by [11]:
(4)
v (0, t ) = 2
with
0
c
[v1 (t , h)
v 1 (t , h )
v1 (t ,
h ) + v 1 (t ,
h)],
(5)
with v1 (t , h) given by
v 1 (t , h ) =
( ct + +
1 d
[ ln( atan
( ct
h) ln ( ct ( ct ct
h+
h) + h
h)2 + d 2
( ct atan
d 2
( ct
h) 2 + d 2
h) 2 + d 2
2)
2)
1]
d ( ct
h) 2 + d 2
2
.
This solution is valid only for the linearly rising part of the channelbase current. A second component must be added to account for either constant (trapezoidal shape) or linearly-decreasing (triangular shape) part of the current waveform. Details can be found in Ref. [11].
Fig. 1. Lightning channel traversed by a return stroke current wave near an infinitely long overhead line: (a) step-function current; (b) linearly rising current. 2
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Fig. 3. Induced voltage peak values against line height h, for different values of: strike point distance d (a), return stroke current magnitude I (b), ratio β (c). 3-D plot of the overall waveforms for different values of line height h is shown in (d) (where not specified, d = 100 m, I = 10 kA, β = 0.4).
Fig. 4. Induced voltage peak values against strike point distance d, for different values of: line height h (a), ratio β (b), return stroke current magnitude I (c). 3D plot of the overall waveforms for different values of strike point distance d is shown in (d) (where not specified, d = 100 m, I = 10 kA, β = 0.4).
3
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Fig. 5. Induced voltage peak values against ratio β, for different values of: line height h (a), strike point distance d (b), return stroke current magnitude I (c). 3D plot of the overall waveforms for different values of β is shown in (d) (where not specified, d = 100 m, I = 10 kA, β = 0.4).
Fig. 6. Induced voltages peak values against line height h, for different values of: strike point distance d (a), return stroke current magnitude I (b), front time tf (c). 3D plot of the overall waveforms for different values of line height h is shown in (d) (where not specified, d = 100 m, I = 10 kA, β = 0.4, tf = 1 μs). 4
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Fig. 7. Induced voltages peak values against strike point distance d, for different values of: line height h (a), return stroke current magnitude I (b), front time tf (c). 3D plot of the overall waveforms for different values of strike point distance d (d). (Where not specified, d = 100 m, I = 10 kA, β = 0.4, tf = 1 μs).
Fig. 8. Induced voltages peak values against ratio β, for different values of: line height h (a), strike point distance d (b), front time tf (c). 3D plot of the overall waveforms for different values of ratio β (d. (where not specified, d = 100 m, I = 10 kA, β = 0.4, tf = 1 μs). 5
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Fig. 9. Induced voltages peak values against front time tf, for different values of: ratio β (a), line height h (b), strike point distance d (c). 3D plot of the overall waveform for different values of the front time tf (d) (where not specified, d = 100 m, I = 10 kA, β = 0.4, tf = 1 μs).
3. Parametric analysis
vapprox (0, t ) = 2 0 Ih
In this section, a parametric analysis of the lightning-induced voltages is carried out. Analyzed parameters are: line height (h), strike point distance (d), return stroke current magnitude (I), front time (tf), and the ratio β. The two cases of a step-function and linearly rising channel-base currents with constant tail (trapezoidal shape) are analyzed; hence, induced voltages are evaluated by means of Eqs. (3) and (5).
According to Eq. (3), the induced voltage is directly proportional to the return stroke current magnitude. In Fig. 3(a)–(c) the induced voltages against line height for different parameters (d, I, β) are analyzed and almost direct proportionality can be observed. This behavior is not immediately inferable from Eq. (3), where h appears in most of the logarithmic terms. However, when the first order approximation of Eq. (3) (i.e., first term of the Taylor's expansion about h = 0) is considered:
Parameters
I
Lognormal
d
Uniform
T-10
Lognormal
Median = 31.1 σ = 0.48 p1 = 50 p2 = 500 Median = 4.5 σ = 0.58
( ct ) 2 + d 2 /
2)
(6)
3.2. Linearly rising channel-base current Similar to the case of the step-function channel-base current, the induced voltage is directly proportional to the current magnitude, as inferable from Eq. (5). The influence of the various parameters on the induced voltage peak values can be observed in Figs. 6–9. The relationship between induced voltage and line height (Fig. 6), strike point distance (Fig. 7), and ratio β (Fig. 8), show a similar behaviour of that observed in the case of the step-function channel-base current; the dependence on the front time is also shown (Figs. 6(c), 7 (c), and 8 (c)). Fig. 8(c) shows an interesting result, namely that for different front time values, different behaviours of the induced voltages compared to ratio β can be seen: more specifically, for low front times, the induced voltage increases as return stroke speed increases; for higher front times the opposite can be seen. This result can be explained as follows [19]: at close distances and long rise times, the static/induction terms dominate for low return stroke speed (therefore the decreasing trend); for short
Table 1 Pdf parameters of the input variables. Distribution
) 2ct (ct
the proportionality is apparent. The overall waveform is shown in Fig. 3(d). Fig. 4 shows the induced voltages peak values against strike point distance d. In particular, used parameters are line height h (a), ratio β (b), and return stroke current magnitude I (c). Fig. 4(d) is a 3D plot of the overall waveforms for different values of the strike point distance. The effect of β on the induced voltage peak values is presented in Fig. 5 where a return stroke speed varying between 20% and 40% of the speed of light has been considered. Observing Fig. 5(a)–(c), this effect is relatively weak; however, an influence of β on the overall waveform can be seen (Fig. 5(d)).
3.1. Step-function channel-base current
Parameter
ct d2 + (
6
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Fig. 10. Step-function channel-base current. Distribution of Vp for β = 0.2 (a), β = 0.25 (b), β = 0.3 (c), β = 0.35 (d), β = 0.4 (e); an overlap is shown in (f) for comparison purposes.
rise times, the radiation term dominates and hence the increase. The dependence of induced voltage on the front time can be better analyzed in Fig. 9. Consistently with the trend shown in Fig. 8(c), Fig. 9(a) shows that the induced voltage peak values decrease with increasing front time, at a rate that is higher for higher values of return stroke speed. The 3D plot of Fig. 9(d) shows how the induced voltage waveform is related to the front time. 4. Statistical analysis In this section, a statistical characterization of peak values of the Table 2 Average values of Vp distributions.
Fig. 11. Distributions of Vp for all the considered values of β and corresponding LN pdfs in the case of step-function channel-base currents.
Average value [kV] 0.2 0.25 0.3 0.35 0.4
61.6 63.6 65.6 67.7 69.8
induced voltages is carried out. For the analysis, a Monte Carlo procedure is applied, in which all the lightning current parameters are assumed as statistically independent. Peak values (Vp) are evaluated by Eq. (3) in the case of a step-function channel-base current, and by Eq. 7
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(5), in the case of linearly rising one; in both cases, the considered line height is 10 m. The Monte Carlo procedure considers the statistical variations of the lightning input parameters in Eqs. (3) and (5): more specifically, the statistical characterization of Vp is carried out assuming, as random, the return stroke current magnitude and the strike point distance. Further, in the case of linearly rising current, the front time is assumed as random input variable too. The probabilistic characterization of Vp is derived by evaluating Eqs. (3) and (5) within an iterative procedure whose inputs are drawings of the aforementioned random variables (I and d, in Eq. (3) and I, d and tf, in Eq. (5)) from their distributions: the statistical behavior of the lightning parameters has been approximated by parametric probability distributions: I and tf are lognormally distributed [16], and d can be assumed uniformly distributed. In this paper tf is derived through the parametric probability distribution provided in terms of T-10 [20] (In this case tf = T − 10/0.8). Table 1 shows the values of the parameters of the probability density functions (pdfs) of the input variables. Note that the minimum distance which discriminates between direct and indirect hits is strongly related to the assumed model [21]. As we are carrying out an investigation based on the statistical variation of the various parameters, we have preferred not to link the investigation to a specific model, and the typical short distance of 50 m has been selected as a minimum distance (similarly, 500 m is distance beyond which induced effects are considered not relevant). The outputs of the Monte Carlo procedures, which involved 10,000 iterations, are distributions of Vp. These are then fitted by parametric distributions, namely Lognormal (LN) and Generalized Extreme Value (GEV) which in previous research [22–24] have been proven to well fit the distributions of the simulated data. The fitting is evaluated by means of the method of coefficient of determination, which is one of the most used to assess goodness-of-fit [25]: this test allows analyzing the fitting by evaluating the value assumed by the coefficient of determination R2 . R2 can range between 0 and 1, specifically, the better the fit, the more R2 approximates 1. In the following two subsections, we will analyze the step-function channel-base current case and the linearly rising one.
Table 3 Parameters of fitting distributions. Parameters Lognormal fitting distribution μ 0.2 3.817 0.25 3.849 0.3 3.880 0.35 3.911 0.4 3.942
Median 45.505 46.968 48.460 49.987 51.558
σ 0.7608 0.7608 0.7608 0.7608 0.7608
GEV fitting distribution k 0.2 0.4832 0.25 0.4832 0.3 0.4832 0.35 0.4832 0.4 0.4832
Σ 22.279 22.995 23.726 24.474 25.243
μ 33.524 34.602 35.7013 36.827 37.984
Fig. 13. cdfs of the simulated Vp data and those obtained by LN fitting for all β values in the case of step-function current.
4.1. Step-function channel-base current The iterative evaluation of Eq. (3) in the Monte Carlo procedure has been carried out for different values of ratio β (i.e. β = 0.2/0.25/0.3/ 0.35/0.4), leading to the outputs in Fig. 10. We can see that β has a weak influence on the statistical behavior of the peak voltages: this is particularly apparent in Fig. 10(f), where we show the overlap. Table 2 gives the average values of the distributions for the different values of β. The maximum variation of the average values is about 13%. It can be observed that the average values of Vp increase with increasing values of β. This behavior is consistent with the results of the parametric analysis reported in Section 3.
Fig. 14. CDFs of the empirical Vp data and those obtained by GEV fitting for all β values in the case of step-function current.
To identify parametric distributions able to describe distributions of the simulated data, the latter have been fitted with LN and GEV distributions. The fitting was performed for all values of β. Figs. 11 and 12 show the results. Both figures give a qualitative evidence of the better performance of GEV. Table 3 shows the parameters of the fitting distributions for all values of β. In the case of GEV, the coefficient of determination, R2 is about 0.999 for all β values, in the case of LN, R2 is about 0.994 for all β values. This better value of R2 confirms the qualitative appraisal of better performance of the GEV. This result can be further explained by comparing the cumulative distribution functions (cdfs) of the simulated and fitting distributions of the peak voltages (Figs. 13 and 14).
Fig. 12. Distributions of Vp for all the considered values of β and corresponding GEV pdfs in the case of step-function channel-base currents. 8
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Fig. 15. Distributions of Vp for a) β = 0.2 b) β = 0.25, c) β = 0.3, d) β = 0.35, e) β = 0.4 and f) for all of the considered values of β in the case of linearly rising channel-base current.
4.2. Linearly rising channel-base current The distributions of Vp are shown in Fig. 15 for different values of β. Again, the distributions are quite similar for all considered β; their average values are given in Table 4. In this case, the maximum variation of the average values is about 3%. Compared to the step-function channel-base current, an opposite trend of the average values can be observed. Indeed, larger values of β lead to smaller average values of Vp. This result is consistent with that obtained in the parametric analysis of Section 3. Identical investigations to those of the step-function channel-base current are carried out here. The results are shown in Figs. 16 and 17 in terms of distributions of Vp and corresponding LN (Fig. 16) and GEV Fig. 16. Distributions of Vp for all the considered values of β and corresponding LN pdfs in the case of linearly rising channel-base current.
Table 4 Average values of Vp distributions. Mean of the distribution [kV] 0.2 0.25 0.3 0.35 0.4
(Fig. 17) fitting pdfs. The analysis of Figs. 16 and 17 essentially confirms the found results for the case of step-function channel-base current. In Table 5 the values of the parameters of the fitting distributions for all β are reported. For all β values, in the case of GEV the coefficient of determination,
50.3 51.2 49.7 48.6 48.6
9
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Table 5 Parameters of fitting distributions. Parameters Lognormal fitting distribution μ 0.2 3.674 0.25 3.693 0.3 3.677 0.35 3.663 0.4 3.665 GEV fitting distribution 0.2 0.25 0.3 0.35 0.4
Fig. 17. Distributions of Vp for all the considered values of β and corresponding GEV pdfs in the case of linearly rising channel-base current.
K 0.373 0.359 0.340 0.331 0.323
Median 39.425 40.185 39.559 38.985 39.087
σ 0.682 0.681 0.660 0.652 0.644
Σ 18.534 19.088 18.348 17.953 17.869
μ 30.448 31.194 30.962 30.632 30.813
R2 is about 0.999; in the case of LN is about 0.998: we only have a slightly better performance of GEV compared to LN. In Figs. 18 and 19, the performance of the fitting is shown in terms of cdfs. We can see that, in the case of linearly rising channel-base current (which is more realistic than the step-function), the LN distribution can be considered a sufficiently accurate solution for fitting the simulated data. In a way, we could state that the lognormality of the lightning parameters is preserved in producing the peak voltages. An additional interesting result is that the outcomes are practically insensitive to , i.e. to the return stroke velocity. 5. The lightning performance of an overhead distribution line The lightning performance of overhead distribution lines quantifies the number of flashovers per year per 100 km a line can experience due to indirect hits. It is represented by plots representing this quantity as a function of its insulation level. A large number of lightning events must be randomly generated to obtain these plots: each lightning event is generally characterized by three parameters: channel-base current peak value, front time, and distance between lightning channel and overhead line. As explained before, the first two parameters are generally assumed to follow a lognormal distribution. We will again use the parameters reported in Table 1. The distance is assumed uniformly distributed between two values: a minimum distance ymin, below which the lightning is assumed as direct, according to the electrogeometric lightning incidence model [20], and a maximum distance ymax, beyond which it is assumed that none of the lightning events could cause a flash on the line. We will investigate the case of linearly rising current by using (5) along with the parameters of Table 1. Following a similar procedure to that proposed in Refs. [26,27] the lightning performance has been obtained as follows. A lightning current magnitude and a current front time are randomly generated according to their lognormal distributions, and a strike point distance, according to its uniform distribution. For each indirect lightning event the maximum induced voltage is calculated. A large number of lightning events have been generated, from which, only indirect events have been selected. Being ntot the number of indirect lightning events and n the number of events generating induced voltages larger than the insulation level2, the number of annual insulation flashovers per 100 km of distribution line has been obtained as [28]
F = 200
n Ng ymax , ntot
Fig. 18. cdfs of the simulated peak voltages and those obtained by LN fitting for all considered β values in the case of linearly rising channel-base current.
Fig. 19. cdfs of the simulated peak voltages and those obtained by GEV fitting for all considered β values in the case of linearly rising channel-base current.
suggested procedure and the procedure suggested in Ref. [28], based on Rusck's analytical solution used in a simplified statistical approach (see Fig. 5 in Ref. [28]). A significant difference between the two solutions can be seen, this is essentially due to the approximation used in the 1410 for the channel-base current, represented as step-function current.
(7)
6. Conclusions
where Ng is the annual lightning ground flash density. In Fig. 20, a comparison between the lightning performance calculated by means of
In this paper, parametric and statistical analyses of the lightning induced overvoltage have been carried out, based on the application of exact analytical formulas. The results obtained in the parametric analysis have showed the different role played by the various parameters
2 The insulation level is assumed being equal to the line critical flashover voltage (CFO), multiplied by a factor equal to 1.5, as suggested in [28].
10
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[4] [5] [6] [7] [8] [9] [10] [11]
Fig. 20. Comparison between the lightning performance presented in Ref. [28] (solid line) and that obtained by the proposed procedure (dashed line).
[12] [13]
affecting the induced voltages. Particularly, small variations of induced voltage peak values and of the average values of their statistical distributions with different values of the parameter β were observed. In the statistical analysis, two parametric fitting distributions have been checked in terms of their ability to fit induced voltages, namely the LN and GEV. The GEV distribution has showed a better performance, however, the LN is more than acceptable. The identification of parametric distributions describing the statistical behaviour of lightning induced voltage could represent an interesting mean to analyze lightning-induced voltages. Exact formulas have been used to evaluate the lightning performance The comparison between the lightning performance evaluated by means of the Rusck's analytical formula (IEEE Std. 1410-2004) and by the procedure developed in this paper showed a difference between the two solutions, which is essentially due to the approximation used in the Standard, in particular due to the channelbase current assumed as step-function current. Future work will examine more complex and realistic configurations, e.g. distribution lines with load, branches and transformers.
[14]
[15]
[16] [17] [18]
[19] [20] [21]
Declaration of Competing Interest [22]
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper
[23]
Acknowledgments The research activity of some of the authors was developed in the framework of the research project “Optimization Approaches for designing and protecting Electric Power Grid – OPT - APP for EPG”.
[24]
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Assante, F. Mottola, L. Verolino, An exact closed-form solution for lightning-induced overvoltages calculations, IEEE Trans. Power Deliv. 24 (July (3)) (2009) 1328–1343. S. Sekioka, An equivalent circuit for analysis of lightning-induced voltages on multiconductor system using an analytical expression, Proc. Int. Conf. Power Syst. Trans, Montreal, Canada, June, 2005. A. Andreotti, A. Pierno, V.A. Rakov, L. Verolino, Analytical formulations for lightning-induced voltage calculations, IEEE Trans. Electromagn. Compat. 55 (February (1)) (2013) 109–123. N.J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM Press, Philadelphia, PA, 2002. P. Chowdhuri, Parametric effects on the induced voltages on overhead lines by lightning strikes to nearby ground, IEEE Trans. Power Deliv. 4 (April (2)) (1989) 1185–1194. A. Andreotti, F. Mottola, M. Pagano, L. Verolino, Lightning induced voltages on power lines: a new statistical approach, Proc. 2008 Int. Symp.on Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM), Ischia, Italy, June 11–13, 2008, pp. 445–451. A. Andreotti, F. Mottola, A. Pierno, D. Proto, A. Sforza, C. Sterle, Parametric and statistical analysis of lightning-induced voltages by exact analytical solutions, Proc. 2018 34th International Conference on Lightning Protection (ICLP), Rzeszow, Poland, 2.7 September, 2018, pp. 1–7. A. Andreotti, V.A. Rakov, L. Verolino, Exact and approximate analytical solutions for lightning-induced voltage calculations, IEEE Trans. Electromagn. Compat. 60 (December (6)) (2018) 1850–1856. A. Andreotti, F. Mottola, A. Pierno, D. Proto, V.A. Rakov, Statistical analysis of lightning-induced voltages in case of lossy Ground, 19th EEEIC Int. Conference on Environment and Electrical Engineering, Genoa (Italy), 11–14 June, 2019. A. Andreotti, F. Mottola, A. Pierno, D. Proto, Statistical analysis of lightning-induced voltages in case of linearly-rising channel-Base currents and lossy Ground, IEEE 5th Int. Forum on Research and Technology for Society and Industry (RTSI) (Accepted), September, 2019. C.A. Nucci, F. Rachidi, M.V. Ianoz, C. Mazzetti, Lightning-induced voltages on overhead lines, IEEE Trans. Electromagn. Compat. 35 (February (1)) (1993) 75–86. R.B. Anderson, A.J. Erikkson, Lightning parameters for engineering applications, Electra 69 (1980) 65–102. A. Borghetti, C.A. Nucci, M. Paolone, M. Bernardi, Effect of the lateral distance expression and of the presence of shielding wires on the evaluation of the number of lightning induced voltages, Proc. 25th Inte. Conf. on Lightning Protection (ICLP 2000), Rhodes, GR, Sep. 18-22, 2000, pp. 340–345. A. Andreotti, A. Masone, F. Mottola, D. Proto, V.A. Rakov, Statistical analysis of lightning-induced voltages in the case of lossy ground, Accepted at the 2019 IEEE Conference on Environment and Electrical Engineering and I&CPS Europe, Genoa, Italy, June 11–14, 2019. A. Andreotti, F. Mottola, A. Pierno, D. Proto, Statistical features of lightning-induced voltages, Proc. 2018 Int. Symp. on Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM), Amalfi, Italy, June 20–22, 2018, pp. 1037–1042. A. Andreotti, F. Mottola, A. Pierno, D. Proto, On the statistical characterization of lightning-induced voltages, Appl. Sci. 8 (April (4)) (2018) 651–665. J.A. Carta, P. Ramirez, S. Velazquez, Influence of the level of fit of a density probability function to wind-speed data on the WECS mean ower output estimation, Energy Convers. Manage. 49 (Octuber (10)) (2008) 2647–2655. A. Borgetti, C.A. Nucci, M. Paolone, An improved procedure for the assessment of overhead line indirect lightning performance and its comparison with the IEEE Std. 1410 method, IEEE Trans. Power Deliv. 22 (January (1)) (2007) 684–692. J.O.S. Paulino, C.F. Barbosa, I.J.S. Lopes, W.C. Boaventura, Assessment and analysis of indirect lightning performance of overhead lines, Electr. Power Syst. Res. 118 (January) (2015) 55–61. "IEEE guide for improving the lightning performance of electric power overhead distribution lines, in: IEEE Std 1410-2004, pp. 1–50, 12 July 2004.
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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Parametric and statistical investigations of lightning-induced voltages on overhead lines by exact analytical solutions
T
Amedeo Andreotti , Fabio Mottola, Antonio Pierno, Daniela Proto, Antonio Sforza, Claudio Sterle ⁎
Department of Electrical Engineering and Information Technology, University of Naples Federico II, Naples, Italy
ARTICLE INFO
ABSTRACT
Keywords: Lightning Lightning-induced voltages Statistical analysis
In this paper, which is an extended version of the paper presented by the authors at the 34th International Conference on Lightning Protection (ICLP 2018), a parametric and a statistical analysis of lightning-induced voltages on overhead lines has been carried out by means of exact analytical solutions, for both step-function and linearly rising channel-base currents. The role played by the various parameters affecting the induced voltages has been analyzed. Further, the statistical investigations has allowed us to identify the parametric statistical distributions which best fit the data obtained by simulations. Based on this statistical investigation, an analysis of the lightning performance of an overhead line has been carried out too.
1. Introduction Assessment of the severity of the induced voltages generated by indirect lightning on overhead lines is a critical task to identify the best protection measures. Various parameters can affect the induced voltages, and the role played by each of them should be clearly identified. Further, as the phenomenon is stochastic, a statistical analysis is very important too. Two different approaches have been used to carry out parametric and statistical analysis: one is based on the so-called numerical models, while the other on the analytical ones. Numerical models (e.g. Refs. [1–3]) are very popular, as they can model complex configurations of the power network; however, analytical solutions, less powerful in that respect as they are limited to more simple configurations (e.g. Refs. [4–11]), are important as well, since they can give an insight that numerical models cannot give, do not suffer from numerical instabilities and/or from lack of convergence [12]; they are also more practical to carry out parametric and statistical analysis (e.g. Refs. [13,14]). In this paper, which is an extended version of the paper presented by the authors at the 34th International Conference on Lightning Protection (ICLP 2018) [15], we will focus on the Andreotti et al. exact analytical solutions [9,11,16]. We will refer to an horizontal conductor, placed over an infinitely conducting ground, and excited by a lightning electromagnetic field produced by either a step-function or a linearly rising current, moving unattenuated and undistorted along a vertical channel. This configuration is depicted in Fig.1. By means of these exact solutions, a parametric and statistical
analysis of lightning-induced voltages will be carried out with the aim of analyzing the role played by the various parameters affecting the induced voltages; in addition, the statistical investigations will allow us to identify the parametric statistical distributions which best fit the simulated data. In order to widen the statistical investigation proposed in Ref. [15], the paper has been completed with the introduction of further statistical analyses. More specifically, parametric distributions were identified to describe distributions of the peak induced-voltage and the goodness of fit was evaluated; the lightning performance evaluation has been carried out too. Note that this analysis refers to the case of perfectly conducting ground. Parametric and statistical treatment in the case of finitely conducting ground has been developed by the authors in Refs. [17] and [18]. The paper is organized as follows. In Section 2, the exact solutions are briefly recalled; in Section 3, a parametric analysis is carried out, while in Section 4 a statistical analysis is developed. Section 5 is devoted to the lightning performance analysis, while Section 6 presents some concluding remarks. 2. Exact analytical solutions In this section, the Andreotti et al. exact analytical solutions are briefly recalled.1 In particular, subsection 2.1 is devoted to the case of the step-function channel-base current, while subsections 2.2 to the case of a linearly rising channel-base current.
Corresponding author. E-mail address: [email protected] (A. Andreotti). 1 By the term exact we mean analytical solutions that, for the assumed models, have been obtained without approximations. ⁎
https://doi.org/10.1016/j.epsr.2019.106044 Received 28 March 2019; Received in revised form 12 September 2019; Accepted 19 September 2019 Available online 07 October 2019 0378-7796/ © 2019 Elsevier B.V. All rights reserved.
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Nomenclature c d
v x
Speed of light in free space. Horizontal distance between the lightning channel and the line conductor (shortly called “strike point distance”). Line height. x 2 + d2 . x 2 + d2 + h2 . Time (t = 0 corresponds to return stroke onset). r0/ c . d 2 + h2 / c . Front time.
h r r0 t t0 t tf
H I Vp
0 0
Return stroke speed. Abscissa along the line (x = 0 corresponds to the point on the line that is closest to the lightning channel). Heaviside step function. Return stroke current magnitude. Induced voltage peak value. Current rate-of-rise for linearly-rising current waveform (I tf ). v /c . 2 (Lorentz factor). 1/ 1 Intrinsic impedance of free space 377 . 0 /(8 ) .
2.1. Induced voltage due to a step-function channel-base current The contribution to the exact solution due to the scalar potential is given by [9,16]:
vs (x , t ) =
0I [
v1 (h) + v1 ( h) + v2 ( ct , h) + v2 (
v2 (
ct ,
ct , h)
v2 ( ct ,
h) (1)
h)]
with
h) + x ]2 + d 2},
v1 (h) = ln{[ ( ct
v2 ( ct , h) = ln [ ( ct
h) + x
2
+
h) 2 + (d2 + x 2)/
( ct
2 ],
while the contribution due to the vector potential is given by [9,16]:
v v (x , t ) =
0I
[v3 ( ct ) + v3 (
ct )
(2)
v4]
with
v3 ( ct ) = 2 ln[
ct + h +
(
ct + h) 2 + (d 2 + x 2)/
Fig. 2. 3D plot of the induced voltage using Eq. (3) (x is the point along the line where the induced voltage is evaluated, h = 10 m, d = 100 m, I = 10 kA, β = 0.4).
2 ],
v4 = 2 ln[(d 2 + x 2 )/ 2].
v 1 (h ) =
Hence, the exact analytical solution (valid for t > t 0 ) for the induced voltage reads [16]:
v (x , t ) =
0I [
v2 (
v1 (h) + v1 ( h) + v2 ( ct , h) + v2 ( ct ,
h)+ v3 ( ct ) + v3 (
ct )
v4].
ct , h)
v2 ( ct ,
v1 ( h)]
h) +
( ct
h)2 + d 2 /
ln [( ct
h) +
( ct
h) 2 + d2 /
2] 2 ].
We show in Fig. 2 a 3D plot of the induced voltage against time and position x along the line.
h) (3)
2.2. Induced voltage due to a linearly rising channel-base current
For the special case of x = 0 (the point closest to the lightning channel), Eq. (3) can be simplified as [9,16]:
v (0, t ) = 2 0 I [v1 (h)
( ct
ln[
The exact analytical solution in the case of a linearly rising current is given by [11]:
(4)
v (0, t ) = 2
with
0
c
[v1 (t , h)
v 1 (t , h )
v1 (t ,
h ) + v 1 (t ,
h)],
(5)
with v1 (t , h) given by
v 1 (t , h ) =
( ct + +
1 d
[ ln( atan
( ct
h) ln ( ct ( ct ct
h+
h) + h
h)2 + d 2
( ct atan
d 2
( ct
h) 2 + d 2
h) 2 + d 2
2)
2)
1]
d ( ct
h) 2 + d 2
2
.
This solution is valid only for the linearly rising part of the channelbase current. A second component must be added to account for either constant (trapezoidal shape) or linearly-decreasing (triangular shape) part of the current waveform. Details can be found in Ref. [11].
Fig. 1. Lightning channel traversed by a return stroke current wave near an infinitely long overhead line: (a) step-function current; (b) linearly rising current. 2
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Fig. 3. Induced voltage peak values against line height h, for different values of: strike point distance d (a), return stroke current magnitude I (b), ratio β (c). 3-D plot of the overall waveforms for different values of line height h is shown in (d) (where not specified, d = 100 m, I = 10 kA, β = 0.4).
Fig. 4. Induced voltage peak values against strike point distance d, for different values of: line height h (a), ratio β (b), return stroke current magnitude I (c). 3D plot of the overall waveforms for different values of strike point distance d is shown in (d) (where not specified, d = 100 m, I = 10 kA, β = 0.4).
3
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Fig. 5. Induced voltage peak values against ratio β, for different values of: line height h (a), strike point distance d (b), return stroke current magnitude I (c). 3D plot of the overall waveforms for different values of β is shown in (d) (where not specified, d = 100 m, I = 10 kA, β = 0.4).
Fig. 6. Induced voltages peak values against line height h, for different values of: strike point distance d (a), return stroke current magnitude I (b), front time tf (c). 3D plot of the overall waveforms for different values of line height h is shown in (d) (where not specified, d = 100 m, I = 10 kA, β = 0.4, tf = 1 μs). 4
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Fig. 7. Induced voltages peak values against strike point distance d, for different values of: line height h (a), return stroke current magnitude I (b), front time tf (c). 3D plot of the overall waveforms for different values of strike point distance d (d). (Where not specified, d = 100 m, I = 10 kA, β = 0.4, tf = 1 μs).
Fig. 8. Induced voltages peak values against ratio β, for different values of: line height h (a), strike point distance d (b), front time tf (c). 3D plot of the overall waveforms for different values of ratio β (d. (where not specified, d = 100 m, I = 10 kA, β = 0.4, tf = 1 μs). 5
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Fig. 9. Induced voltages peak values against front time tf, for different values of: ratio β (a), line height h (b), strike point distance d (c). 3D plot of the overall waveform for different values of the front time tf (d) (where not specified, d = 100 m, I = 10 kA, β = 0.4, tf = 1 μs).
3. Parametric analysis
vapprox (0, t ) = 2 0 Ih
In this section, a parametric analysis of the lightning-induced voltages is carried out. Analyzed parameters are: line height (h), strike point distance (d), return stroke current magnitude (I), front time (tf), and the ratio β. The two cases of a step-function and linearly rising channel-base currents with constant tail (trapezoidal shape) are analyzed; hence, induced voltages are evaluated by means of Eqs. (3) and (5).
According to Eq. (3), the induced voltage is directly proportional to the return stroke current magnitude. In Fig. 3(a)–(c) the induced voltages against line height for different parameters (d, I, β) are analyzed and almost direct proportionality can be observed. This behavior is not immediately inferable from Eq. (3), where h appears in most of the logarithmic terms. However, when the first order approximation of Eq. (3) (i.e., first term of the Taylor's expansion about h = 0) is considered:
Parameters
I
Lognormal
d
Uniform
T-10
Lognormal
Median = 31.1 σ = 0.48 p1 = 50 p2 = 500 Median = 4.5 σ = 0.58
( ct ) 2 + d 2 /
2)
(6)
3.2. Linearly rising channel-base current Similar to the case of the step-function channel-base current, the induced voltage is directly proportional to the current magnitude, as inferable from Eq. (5). The influence of the various parameters on the induced voltage peak values can be observed in Figs. 6–9. The relationship between induced voltage and line height (Fig. 6), strike point distance (Fig. 7), and ratio β (Fig. 8), show a similar behaviour of that observed in the case of the step-function channel-base current; the dependence on the front time is also shown (Figs. 6(c), 7 (c), and 8 (c)). Fig. 8(c) shows an interesting result, namely that for different front time values, different behaviours of the induced voltages compared to ratio β can be seen: more specifically, for low front times, the induced voltage increases as return stroke speed increases; for higher front times the opposite can be seen. This result can be explained as follows [19]: at close distances and long rise times, the static/induction terms dominate for low return stroke speed (therefore the decreasing trend); for short
Table 1 Pdf parameters of the input variables. Distribution
) 2ct (ct
the proportionality is apparent. The overall waveform is shown in Fig. 3(d). Fig. 4 shows the induced voltages peak values against strike point distance d. In particular, used parameters are line height h (a), ratio β (b), and return stroke current magnitude I (c). Fig. 4(d) is a 3D plot of the overall waveforms for different values of the strike point distance. The effect of β on the induced voltage peak values is presented in Fig. 5 where a return stroke speed varying between 20% and 40% of the speed of light has been considered. Observing Fig. 5(a)–(c), this effect is relatively weak; however, an influence of β on the overall waveform can be seen (Fig. 5(d)).
3.1. Step-function channel-base current
Parameter
ct d2 + (
6
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Fig. 10. Step-function channel-base current. Distribution of Vp for β = 0.2 (a), β = 0.25 (b), β = 0.3 (c), β = 0.35 (d), β = 0.4 (e); an overlap is shown in (f) for comparison purposes.
rise times, the radiation term dominates and hence the increase. The dependence of induced voltage on the front time can be better analyzed in Fig. 9. Consistently with the trend shown in Fig. 8(c), Fig. 9(a) shows that the induced voltage peak values decrease with increasing front time, at a rate that is higher for higher values of return stroke speed. The 3D plot of Fig. 9(d) shows how the induced voltage waveform is related to the front time. 4. Statistical analysis In this section, a statistical characterization of peak values of the Table 2 Average values of Vp distributions.
Fig. 11. Distributions of Vp for all the considered values of β and corresponding LN pdfs in the case of step-function channel-base currents.
Average value [kV] 0.2 0.25 0.3 0.35 0.4
61.6 63.6 65.6 67.7 69.8
induced voltages is carried out. For the analysis, a Monte Carlo procedure is applied, in which all the lightning current parameters are assumed as statistically independent. Peak values (Vp) are evaluated by Eq. (3) in the case of a step-function channel-base current, and by Eq. 7
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(5), in the case of linearly rising one; in both cases, the considered line height is 10 m. The Monte Carlo procedure considers the statistical variations of the lightning input parameters in Eqs. (3) and (5): more specifically, the statistical characterization of Vp is carried out assuming, as random, the return stroke current magnitude and the strike point distance. Further, in the case of linearly rising current, the front time is assumed as random input variable too. The probabilistic characterization of Vp is derived by evaluating Eqs. (3) and (5) within an iterative procedure whose inputs are drawings of the aforementioned random variables (I and d, in Eq. (3) and I, d and tf, in Eq. (5)) from their distributions: the statistical behavior of the lightning parameters has been approximated by parametric probability distributions: I and tf are lognormally distributed [16], and d can be assumed uniformly distributed. In this paper tf is derived through the parametric probability distribution provided in terms of T-10 [20] (In this case tf = T − 10/0.8). Table 1 shows the values of the parameters of the probability density functions (pdfs) of the input variables. Note that the minimum distance which discriminates between direct and indirect hits is strongly related to the assumed model [21]. As we are carrying out an investigation based on the statistical variation of the various parameters, we have preferred not to link the investigation to a specific model, and the typical short distance of 50 m has been selected as a minimum distance (similarly, 500 m is distance beyond which induced effects are considered not relevant). The outputs of the Monte Carlo procedures, which involved 10,000 iterations, are distributions of Vp. These are then fitted by parametric distributions, namely Lognormal (LN) and Generalized Extreme Value (GEV) which in previous research [22–24] have been proven to well fit the distributions of the simulated data. The fitting is evaluated by means of the method of coefficient of determination, which is one of the most used to assess goodness-of-fit [25]: this test allows analyzing the fitting by evaluating the value assumed by the coefficient of determination R2 . R2 can range between 0 and 1, specifically, the better the fit, the more R2 approximates 1. In the following two subsections, we will analyze the step-function channel-base current case and the linearly rising one.
Table 3 Parameters of fitting distributions. Parameters Lognormal fitting distribution μ 0.2 3.817 0.25 3.849 0.3 3.880 0.35 3.911 0.4 3.942
Median 45.505 46.968 48.460 49.987 51.558
σ 0.7608 0.7608 0.7608 0.7608 0.7608
GEV fitting distribution k 0.2 0.4832 0.25 0.4832 0.3 0.4832 0.35 0.4832 0.4 0.4832
Σ 22.279 22.995 23.726 24.474 25.243
μ 33.524 34.602 35.7013 36.827 37.984
Fig. 13. cdfs of the simulated Vp data and those obtained by LN fitting for all β values in the case of step-function current.
4.1. Step-function channel-base current The iterative evaluation of Eq. (3) in the Monte Carlo procedure has been carried out for different values of ratio β (i.e. β = 0.2/0.25/0.3/ 0.35/0.4), leading to the outputs in Fig. 10. We can see that β has a weak influence on the statistical behavior of the peak voltages: this is particularly apparent in Fig. 10(f), where we show the overlap. Table 2 gives the average values of the distributions for the different values of β. The maximum variation of the average values is about 13%. It can be observed that the average values of Vp increase with increasing values of β. This behavior is consistent with the results of the parametric analysis reported in Section 3.
Fig. 14. CDFs of the empirical Vp data and those obtained by GEV fitting for all β values in the case of step-function current.
To identify parametric distributions able to describe distributions of the simulated data, the latter have been fitted with LN and GEV distributions. The fitting was performed for all values of β. Figs. 11 and 12 show the results. Both figures give a qualitative evidence of the better performance of GEV. Table 3 shows the parameters of the fitting distributions for all values of β. In the case of GEV, the coefficient of determination, R2 is about 0.999 for all β values, in the case of LN, R2 is about 0.994 for all β values. This better value of R2 confirms the qualitative appraisal of better performance of the GEV. This result can be further explained by comparing the cumulative distribution functions (cdfs) of the simulated and fitting distributions of the peak voltages (Figs. 13 and 14).
Fig. 12. Distributions of Vp for all the considered values of β and corresponding GEV pdfs in the case of step-function channel-base currents. 8
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Fig. 15. Distributions of Vp for a) β = 0.2 b) β = 0.25, c) β = 0.3, d) β = 0.35, e) β = 0.4 and f) for all of the considered values of β in the case of linearly rising channel-base current.
4.2. Linearly rising channel-base current The distributions of Vp are shown in Fig. 15 for different values of β. Again, the distributions are quite similar for all considered β; their average values are given in Table 4. In this case, the maximum variation of the average values is about 3%. Compared to the step-function channel-base current, an opposite trend of the average values can be observed. Indeed, larger values of β lead to smaller average values of Vp. This result is consistent with that obtained in the parametric analysis of Section 3. Identical investigations to those of the step-function channel-base current are carried out here. The results are shown in Figs. 16 and 17 in terms of distributions of Vp and corresponding LN (Fig. 16) and GEV Fig. 16. Distributions of Vp for all the considered values of β and corresponding LN pdfs in the case of linearly rising channel-base current.
Table 4 Average values of Vp distributions. Mean of the distribution [kV] 0.2 0.25 0.3 0.35 0.4
(Fig. 17) fitting pdfs. The analysis of Figs. 16 and 17 essentially confirms the found results for the case of step-function channel-base current. In Table 5 the values of the parameters of the fitting distributions for all β are reported. For all β values, in the case of GEV the coefficient of determination,
50.3 51.2 49.7 48.6 48.6
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Table 5 Parameters of fitting distributions. Parameters Lognormal fitting distribution μ 0.2 3.674 0.25 3.693 0.3 3.677 0.35 3.663 0.4 3.665 GEV fitting distribution 0.2 0.25 0.3 0.35 0.4
Fig. 17. Distributions of Vp for all the considered values of β and corresponding GEV pdfs in the case of linearly rising channel-base current.
K 0.373 0.359 0.340 0.331 0.323
Median 39.425 40.185 39.559 38.985 39.087
σ 0.682 0.681 0.660 0.652 0.644
Σ 18.534 19.088 18.348 17.953 17.869
μ 30.448 31.194 30.962 30.632 30.813
R2 is about 0.999; in the case of LN is about 0.998: we only have a slightly better performance of GEV compared to LN. In Figs. 18 and 19, the performance of the fitting is shown in terms of cdfs. We can see that, in the case of linearly rising channel-base current (which is more realistic than the step-function), the LN distribution can be considered a sufficiently accurate solution for fitting the simulated data. In a way, we could state that the lognormality of the lightning parameters is preserved in producing the peak voltages. An additional interesting result is that the outcomes are practically insensitive to , i.e. to the return stroke velocity. 5. The lightning performance of an overhead distribution line The lightning performance of overhead distribution lines quantifies the number of flashovers per year per 100 km a line can experience due to indirect hits. It is represented by plots representing this quantity as a function of its insulation level. A large number of lightning events must be randomly generated to obtain these plots: each lightning event is generally characterized by three parameters: channel-base current peak value, front time, and distance between lightning channel and overhead line. As explained before, the first two parameters are generally assumed to follow a lognormal distribution. We will again use the parameters reported in Table 1. The distance is assumed uniformly distributed between two values: a minimum distance ymin, below which the lightning is assumed as direct, according to the electrogeometric lightning incidence model [20], and a maximum distance ymax, beyond which it is assumed that none of the lightning events could cause a flash on the line. We will investigate the case of linearly rising current by using (5) along with the parameters of Table 1. Following a similar procedure to that proposed in Refs. [26,27] the lightning performance has been obtained as follows. A lightning current magnitude and a current front time are randomly generated according to their lognormal distributions, and a strike point distance, according to its uniform distribution. For each indirect lightning event the maximum induced voltage is calculated. A large number of lightning events have been generated, from which, only indirect events have been selected. Being ntot the number of indirect lightning events and n the number of events generating induced voltages larger than the insulation level2, the number of annual insulation flashovers per 100 km of distribution line has been obtained as [28]
F = 200
n Ng ymax , ntot
Fig. 18. cdfs of the simulated peak voltages and those obtained by LN fitting for all considered β values in the case of linearly rising channel-base current.
Fig. 19. cdfs of the simulated peak voltages and those obtained by GEV fitting for all considered β values in the case of linearly rising channel-base current.
suggested procedure and the procedure suggested in Ref. [28], based on Rusck's analytical solution used in a simplified statistical approach (see Fig. 5 in Ref. [28]). A significant difference between the two solutions can be seen, this is essentially due to the approximation used in the 1410 for the channel-base current, represented as step-function current.
(7)
6. Conclusions
where Ng is the annual lightning ground flash density. In Fig. 20, a comparison between the lightning performance calculated by means of
In this paper, parametric and statistical analyses of the lightning induced overvoltage have been carried out, based on the application of exact analytical formulas. The results obtained in the parametric analysis have showed the different role played by the various parameters
2 The insulation level is assumed being equal to the line critical flashover voltage (CFO), multiplied by a factor equal to 1.5, as suggested in [28].
10
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[4] [5] [6] [7] [8] [9] [10] [11]
Fig. 20. Comparison between the lightning performance presented in Ref. [28] (solid line) and that obtained by the proposed procedure (dashed line).
[12] [13]
affecting the induced voltages. Particularly, small variations of induced voltage peak values and of the average values of their statistical distributions with different values of the parameter β were observed. In the statistical analysis, two parametric fitting distributions have been checked in terms of their ability to fit induced voltages, namely the LN and GEV. The GEV distribution has showed a better performance, however, the LN is more than acceptable. The identification of parametric distributions describing the statistical behaviour of lightning induced voltage could represent an interesting mean to analyze lightning-induced voltages. Exact formulas have been used to evaluate the lightning performance The comparison between the lightning performance evaluated by means of the Rusck's analytical formula (IEEE Std. 1410-2004) and by the procedure developed in this paper showed a difference between the two solutions, which is essentially due to the approximation used in the Standard, in particular due to the channelbase current assumed as step-function current. Future work will examine more complex and realistic configurations, e.g. distribution lines with load, branches and transformers.
[14]
[15]
[16] [17] [18]
[19] [20] [21]
Declaration of Competing Interest [22]
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper
[23]
Acknowledgments The research activity of some of the authors was developed in the framework of the research project “Optimization Approaches for designing and protecting Electric Power Grid – OPT - APP for EPG”.
[24]
References
[26]
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