## 5 Sensitivity Analysis
### Surface Conditions  
The effect of the surface conditions is taken into account by varying the surface state coefficient \(m\). A minimum value of \(m=0.5\) and a maximum value of \(m=1\) are here considered [26]. Figure 3 shows the number of dangerous events per 100 km per year, computed for six different values for \(m\) in the selected range. For sake of completeness, the case treated in Figure 2 is here reported as the black line. The corresponding values of critical electric field and voltage can be found easily from Equations (1) and (5).  
\begin{table}
\begin{tabular}{c c c} \hline
**CFO [kV]** & **Enhancement Due to Lossy Ground [\%]** & **Enhancement Due to Corona [\%]** \\ \hline
50 & 166.58 & 0 \\
100 & 185.39 & 0 \\
150 & 97.50 & 19.03 \\
200 & 50.61 & 27.27 \\
250 & 24.93 & 27.79 \\ \hline \end{tabular}
\end{table}
Table 2: Percentage enhancement—the second column describes how the lossy ground enhances the number of flashovers with respect to the PEC ground, while the third column describes how the corona effect enhances the number of flashovers with respect to the lossy ground case.  
Figure 2: Number of dangerous events on an overhead distribution line. Comparison between perfect ground, lossy ground and lossy ground with corona discharge.  
The number of dangerous events increase with the decrease of the surface coefficient since it corresponds to a linear decrease of the critical electric field.  
Figure 4 shows the percentage variation (with respect to \(m=1\)) of the number of flashovers as a function of the surface state coefficient for different line CFO.  
Figure 4 shows that the percentage increase is negligible for \(CFO=50\) kV: this is obvious as the corona effect contributes to the enhancement of the lightning-induced voltages only when the travelling wave overcomes \(u_{th}\), which is usually really higher than 50 kV. As a consequence, when \(CFO=50\) kV, those cases are already defined as dangerous whatever is the value of \(m\). On the other hand, when \(CFO=250\) kV the curve can be well-described by a second-order polynomial because there is a sort of saturation for low values of \(m\). This can be related to the fact that low values of \(m\) correspond to a decrease of \(u_{th}\). In other words, the corona discharge contributes to the enhancement of the lightning-induced voltages also when the voltage on the line is low. However, the increase due to corona is not sufficient to overcome the threshold set with \(CFO=250\) kV, thus its effect on the enhancement of the flashovers number is less evident. Finally, the behaviour of the percentage increase  
Figure 4: Percentage variation of the number of flashovers as a function of the surface state coefficient  
Figure 3: Comparison of the lightning performance of an overhead distribution line with different surface state conditions—the red dot line represents the case when the corona effect is not considered.  
for CFO \(\in\{100,150,200\}\) kV is substantially linear, as shown by the fitting provided in Figure 4. This is in agreement with the linear variation of the critical electric field with respect to \(m\). A decrease of the surface coefficient determines a linear decrease of \(u_{th}\); the low value of \(u_{th}\) causes an higher occurrence of the corona discharge, causing an enhancement of the lightning-induced voltages and of the line flashovers. The higher increases can be noticed for \(CFO=150\) and 200 kV.  
Equation (8) provides the general expression used for finding the fitting curve, that express the percentage increase as a function of \(m\)  
\[p_{increase}=p_{0}+p_{1}m+p_{2}m^{2}. \tag{8}\]  
The values of the parameters \(p_{0}\), \(p_{1}\) amd \(p_{2}\) are reported in Table 3, where also the \(R^{2}\) index is shown, for quantify the reliability of the fitting.