# On the Statistical Characterization of Lightning-Induced Voltages
## 4 Numerical Applications
### Step-Function Current  
The statistical analysis was carried out with reference to several values of \(\beta\) ranging from 0.2 to 0.4. In Figure 8, the distributions of \(V_{peak}\), corresponding to five different values of \(\beta\) are shown. Note that a few occurrences of the extracted data of \(V_{peak}\) take on values higher than 300 kV that are not shown in the figure.  
It is apparent in Figure 8 that the variation of \(\beta\) has a weak influence on the statistical behaviour of the induced voltages. For this reason, in what follows, the results of the statistical characterization of the induced voltages are reported with reference to \(\beta\) = 0.3. The following fitting analysis led to two distributions that, among those analyzed, seem the best fit: the LN distribution and the generalized extreme value (GEV).  
The LN distribution's pdfs and their parameters have been already presented in Table 1. As to the GEV, it is a three-parameter distribution (location parameter \(\mu\), scale parameter \(\sigma\), and shape parameter \(k~{}\
eq~{}0\)) whose pdf is as follows:  
\[y=f(x|k,~{}\mu,\sigma)=\tfrac{1}{\sigma}e^{(-(1+k\frac{x-y}{\sigma})^{-\frac{1 }{\delta}})}\left(1+k\frac{x-y}{\sigma}\right)^{-1-\frac{1}{\delta}}. \tag{5}\]  
The results of the fitting are summarized in Table 2, where the parameters' estimates, along with the result of the two methods used for evaluating the goodness of fit, are reported. Figure 9 shows the pdfs of the selected distributions as fitting functions, along with the distribution of the extracted data. Figure 10 shows the corresponding cumulative density functions (cdfs).  
Figure 8: Distributions of \(V_{peak}\) corresponding to different values of \(\beta\) (step function model).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Fitting Distribution**} & \\  & \multicolumn{2}{c}{**Generalized Extreme Value**} & \\ \hline Log Likelihood & Det. Coeff. & k & \(\sigma\) & \(\mu\) \\ \hline \(-\)119,149 & 0.9989 & 0.4865 & 23,409.2 & 35,382.7 \\ \hline \multicolumn{5}{c}{**Fitting Distribution**} & \\  & \multicolumn{2}{c}{**Lognormal**} & \\ \hline Log Likelihood & Det. [PERSON]. & \(\sigma\)\({}^{1}\) & \(\mu\)\({}^{1}\) \\ \hline \(-\)119,223 & 0.9940 & - & 0.7579 & 10.7806 \\ \hline \end{tabular} \({}^{1}\)\(\mu\) and \(\sigma\) are the mean and variance of the associated normal distribution, respectively.  
\end{table}
Table 2: Results of Fitting.  
Figure 10: Cumulative density functions (cdfs) related to the distribution of the extracted values of \(V_{peak}\) and of the selected fitting parametric distributions (step function model, \(\beta\) = 0.3).  
Figure 9: Distribution of the extracted values of \(V_{peak}\) and selected fitting parametric pdfs (step function model, \(\beta\) = 0.3).  
The analysis of the results reveals that both distributions seem to be suitable as reference parametric distributions for characterizing the lightning-induced voltage peak values; although, the LN underestimates the distribution's peak value. The log likelihood analysis gave similar values for the two distributions, and the determination coefficient is very close to 1 in both cases.