## 5 Sensitivity Analysis
### Conductor Diameter  
The effect of the conductor diameter is analysed, taking into account different values typical of overhead distribution lines (from 5 mm to 60 mm). The other line parameters have been described in the previous sections and in this framework we consider \(m=0.8\). Figure 5 shows that the influence of the corona discharge is negligible when the diameter is greater than 30 mm. Moreover, for very thin conductors (\(d=5\) mm), the corona discharge increases the number of flashovers even if the line CFO is very low (<100 kV): this can be ascribed to the low value of the critical electric field associated to such diameter. Figure 6 analyses the percentage variation (with respect to the case with \(d=30\) mm) of the number of flashovers as a function of the conductor diameter for different line CFO. Figure 5: Comparison of the lightning performance of an overhead distribution line with different diameters—the red dot line represents the case when the corona effect is not considered. \begin{table}
\begin{tabular}{c c c c c} \hline
**CFO** & \(\mathbf{p_{0}}\) & \(\mathbf{p_{1}}\) & \(\mathbf{p_{2}}\) & \(\mathbf{R^{2}}\) \\ \hline
50 & 0.00 & 0.00 & 0.00 &  
Figure 6 confirms that, if the line CFO is 50 kV, the enhancement is negligible for the same reason presented in the previous subsection. Considering a line CFO of 100 kV and 150 kV corresponds to a consistent enhancement of the percentage for low values of the diameter (<10 mm) and a flatness of the increase if the diameter is greater than 10 mm. As a consequence the two curves can be well described by an exponential. This behaviour is related to the high values of \(u_{th}\) when we consider a diameter larger than 10 mm: in these cases \(u_{th}>>\) 100 kV, thus every event that overcomes that threshold is already defined as dangerous. On the other hand, an increase of the line CFO (200, 250 kV) leads to a behaviour characterized by a low variation of its derivative. Equation (9) provides the general expression used for finding the fitting curve that expresses the percentage increase as a function of \(d\)  
\[P_{increase}=ae^{bd}. \tag{9}\]  
The values of the parameters \(a\) and \(b\) are reported in Table 4, where also the \(R^{2}\) index is shown, to quantify the reliability of the fitting. The values of the \(R^{2}\) indexes for CFO = 200, 250 kV are low as there are some deviations between the points and the curves of Figure 6. However, these deviations can be ascribed to the statistical procedure of the lightning performance and it is evident that the overall behaviour has a low variation in its derivative (low values of \(b\)). The sensitivity analysis on the diameter can help an user to evaluate, once that the CFO is set, how the number of flashovers due to corona increases with respect to diameter: for example, considering a line CFO of 100 kV, choosing a very thin diameter (5 mm) leads to a percentage increase of 25% but with a low cost for the conductor material. On the other hand increasing the diameter leads to higher costs but lower enhancement of dangerous events. It is important to notice that the cost is affected not only by the conductor diameter, but also by the ampac \begin{table}
\begin{tabular}{c c c c} \hline
**CFO** & \(a\) & \(b\) & \(R^{2}\) \\ \hline
100 & 258.90 & \(-\)0.46 & 0.95 \\
150 & 114.00 & \(-\)0.17 & 0.98 \\
200 & 72.25 & \(-\)0.1 & 0.92 \\
250 & 49.95 & \(-\)0.06 & 0.90 \\ \hline \end{tabular}
\end{table} 
Table 4: Fitting coefficients and \(R^{2}\) index. Figure 6: Percentage variation of the number of flashovers as a function of the conductor diameter. strength (RTS); as a consequence, also these two factors shall be taken into account while choosing the conductor. As a final remark, attention shall be dedicated to the bundled conductors as their installation is frequent in overhead distribution lines. Although, their geometry can be seen in some ways as an equivalent conductor with a larger diameter, thus its capability of mitigating the corona discharge can be discussed as previously presented.