*5.1. 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).

**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.

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.** Percentage variation of the number of flashovers as a function of the surface state coefficient

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 *uth*, 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 *uth*. 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

for CFO ∈ {100, 150, 200} kV is substantially linear, as shown by the fitting provided in Figure 4. This is in agreemen<sup>t</sup> with the linear variation of the critical electric field with respect to *m*. A decrease of the surface coefficient determines a linear decrease of *uth*; the low value of *uth* 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*

$$P\_{\text{increase}} = p\_0 + p\_1 m + p\_2 m^2. \tag{8}$$

The values of the parameters *p*0, *p*1 amd *p*2 are reported in Table 3, where also the *R*<sup>2</sup> index is shown, for quantify the reliability of the fitting.


**Table 3.** Fitting coefficients and *R*<sup>2</sup> index.

### *5.2. 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.

**Figure 6.** Percentage variation of the number of flashovers as a function of the conductor diameter.

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 *uth* when we consider a diameter larger than 10 mm: in these cases *uth* >> 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*

$$P\_{\text{increase}} = ae^{hd}.\tag{9}$$

The values of the parameters *a* and *b* are reported in Table 4, where also the *R*<sup>2</sup> index is shown, to quantify the reliability of the fitting.

The values of the *R*<sup>2</sup> 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*).


**Table 4.** Fitting coefficients and *R*<sup>2</sup> index.

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 ampacity and the rated tensile

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.
