**6. Results**

In this section, the results for the test cases of Table 5 are presented, showing the voltage across the phase B insulator string ( *Vinsulator* in Figure 3) and the voltage difference occurring on the grounding system ( *Vgrounding* in Figure 3).

Figures 10–15 show the results for tests T1–T6, corresponding to a typical first stroke. The main differences in terms of voltage across the insulator can be observed considering a low soil conductivity (Figures 13–15) and near stroke locations (60 m). This is extremely important because the closer the stroke location, the higher (and the more dangerous) the induced voltage. For example, let us consider Test T4 (Figure 13). If we use the lowfrequency grounding resistance ( *RLF*) as grounding model, the maximum induced voltage across the insulator string is 115.12 kV, while if we consider the harmonic grounding impedance ( *<sup>Z</sup>*(*ω*)), which represents in a better way the reality, the voltage is 119.40 kV. This shows how the difference in the modeling could lead to either a fault or not across the insulator strings.

On the other hand, when the harmonic grounding impedance model presents a voltage across the insulator higher with respect to the *RLF* case, the voltage on the grounding system is lower. This can be explained as follows: let us consider Figure 3; the voltage difference occurring on the insulator string is

$$V\_{insulator} = V\_{conductor} - V\_{sv} \tag{8}$$

It is reasonable to assume that the voltage on the conductor does not change in a meaningful way. Considering the two different approaches (based on grounding system modeling), the only difference is the current flowing in the shield wire conductor causing a different coupling with the phase conductor. Even if not negligible, the coupling between conductors does not represent the dominant aspect in the lightning-induced voltages (which is the electric field illuminating the conductor). Consequently, *Vinsulator* + *Vsw* is almost constant. The shield wire voltage is:

$$V\_{sw} = V\_{tower} + V\_{\text{grounding}} \tag{9}$$

with the same current, *Vtower* is constant in the two cases but *Vgrounding* varies because the impedance varies according to Figures 4 and 5 for *σ* = 10 mS/m and 1 mS/m, respectively. Let us consider the most critical case, i.e., *σ* = 1 mS/m: from Figure 5 it is clear that for each considered frequency *<sup>Z</sup>*(*ω*) < *RLF*, thus with the same current the voltage on the grounding system is lower if we consider the harmonic impedance *<sup>Z</sup>*(*ω*) and consequently also *Vsw* is lower. Since *Vinsulator* + *Vsw* = *constant*, if *Vsw* decreases , *Vinsulator* increases. This aspect is confirmed in Tests T4-T5-T6, T10-T11-T12.

The results for subsequent strokes can be observed in Figures 16–21. The results are in agreemen<sup>t</sup> with the previous ones, confirming a significant increase of the maximum voltage if the equivalent circuit ( *<sup>Z</sup>*(*ω*)) is taken into account, especially if the soil conductivity is low. Moreover, the percentage increase considering the harmonic grounding impedance with respect to the low-frequency resistance is much more significant with respect to the first stroke case considering *σ* = 1 mS/m. It is expected since the subsequent strokes are faster. Thus, it has a higher frequency spectrum (the region where there are the highest differences between *RLF* and *<sup>Z</sup>*(*ω*)). Additionally, it is important to highlight that after a while, both overvoltages, considering *RLF* and *<sup>Z</sup>*(*ω*), tend to the same value (first and subsequent strokes). For instance, if we consider the T10 and use the *RLF* as grounding model, the maximum induced voltage across the insulator string is 66 kV, while if we consider the *<sup>Z</sup>*(*ω*), the voltage is 81 kV. This shows how the modeling difference could lead to either a fault or not across the insulator strings, especially for subsequent strokes.

**Figure 10.** Test T1—Voltage on the grounding system and on the insulator of phase B. Comparison between the two models.

**Figure 11.** Test T2—Voltage on the grounding system and on the insulator of phase B. Comparison between the two models.

**Figure 12.** Test T3—Voltage on the grounding system and on the insulator of phase B. Comparison between the two models.

**Figure 13.** Test T4—Voltage on the grounding system and on the insulator of phase B. Comparison between the two models.

**Figure 14.** Test T5—Voltage on the grounding system and on the insulator of phase B. Comparison between the two models.

**Figure 15.** Test T6—Voltage on the grounding system and on the insulator of phase B. Comparison between the two models.

**Figure 16.** Test T7—Voltage on the grounding system and on the insulator of phase B. Comparison between the two models.

**Figure 17.** Test T8—Voltage on the grounding system and on the insulator of phase B. Comparison between the two models.

**Figure 18.** Test T9—Voltage on the grounding system and on the insulator of phase B. Comparison between the two models.

**Figure 19.** Test T10—Voltage on the grounding system and on the insulator of phase B. Comparison between the two models.

**Figure 20.** Test T11—Voltage on the grounding system and on the insulator of phase B. Comparison between the two models.

**Figure 21.** Test T12—Voltage on the grounding system and on the insulator of phase B. Comparison between the two models.

Finally, Table 7 shows the percentage increase in the maximum voltage across the phase B insulator considering the harmonic grounding impedance (*Z*(*ω*)) with respect to the low-frequency grounding resistance (*RLF*). According to the previous considerations, the differences are almost negligible if the soil conductivity is high (tests T1–T3 and T7–T9), but they become consistent when the soil conductivity decreases (tests T4–T6 and T10–T12). This behavior is more evident for close stroke location (test T4 and T10).


**Table 7.** Maximum voltage across the insulator. Percentage increase considering the harmonic grounding impedance (*Z*(*ω*)) with respect to the low-frequency grounding resistance (*RLF*).
