# Impact of Grounding Modeling on Lightning-Induced Voltages Evaluation in Distribution Lines
## 5 Test Cases
It is worth mentioning that these equivalent circuits are mathematical models that have a frequency response very close to \(Z(\omega)\), but their electrical parameters do not have physical consistency, it is also important mentioning that although some elements may have negative values, the circuit is passive in overall. Hence the existence of negative values for resistance, inductance and capacitance in Tables 1-4 do not mean that the circuit is not passive. The i-index appearing in Tables 1-4 refers to the electrical branch. Figure 4: Grounding harmonic impedance \(Z(\omega)\) with \(\sigma=10\) mS/m. Adjusted \(Z\) refers to the impedance obtained with the equivalent circuit. Figure 3: Equivalent circuit of the power system, tower and grounding. Figure 5: Grounding harmonic impedance \(Z(\omega)\) with \(\sigma=1\) mS/m. Adjusted \(Z\) refers to the impedance obtained with the equivalent circuit. Figure 6: General layout of the grounding circuit for the harmonic grounding impedance. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**i** & **Resistance [\(\Omega\)]** & **Inductance [mH]** & **Capacitance [μF]** \\ \hline
0 & \(3.33\times 10^{15}\) & - & \(3.00\times 10^{-10}\) \\
1 & \(-6.35\times 10^{3}\) & \(-5.67\times 10^{3}\) & - \\
2 & \(-8.44\times 10^{3}\) & \(-1.19\times 10^{3}\) & - \\
3 & \(-1.30\times 10^{3}\) & \(-3.66\times 10^{1}\) & - \\
4 & \(-2.75\times 10^{3}\) & \(-1.74\times 10^{1}\) & - \\
5 & \(-2.78\times 10^{2}\) & \(-4.23\times 10^{-1}\) & - \\
6 & \(-8.96\times 10^{2}\) & \(-3.39\times 10^{-1}\) & - \\
7 & \(-3.13\times 10^{1}\) & \(-1.98\times 10^{-3}\) & - \\
8 & \(1.77\times 10^{-5}\) & \(5.00\times 10^{-5}\) & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Elements of Complex Poles of the grounding circuit. \(\sigma=1\) mS/m. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**i** & **Resistance [\(\Omega\)]** & **Inductance [mH]** & **Capacitance [μF]** \\ \hline
0 & \(3.33\times 10^{15}\) & - & \(3.00\times 10^{-10}\) \\
1 & \(-6.35\times 10^{3}\) & \(-5.67\times 10^{3}\) & - \\
2 & \(-8.44\times 10^{3}\) & \(-1.19\times  
with parameters reported in Table 6. The representation of the channel-base current is proposed in Figures 8 and 9. \[I_{0}(t)=\frac{l_{01}}{\eta_{1}}\frac{\left(\frac{t}{\tau_{11}}\right)^{n_{1}}}{1+ \left(\frac{t}{\tau_{11}}\right)^{n_{1}}}e^{-\frac{t}{\tau_{12}}}+\frac{l_{02}} {\eta_{2}}\frac{\left(\frac{t}{\tau_{12}}\right)^{n_{2}}}{1+\left(\frac{t}{ \tau_{21}}\right)^{n_{2}}}e^{-\frac{t}{\tau_{22}}} \tag{6}\]  
being  
\[\eta_{i}=\exp\left(-\frac{\tau_{11}}{\tau_{22}}\left(n_{i}\frac{\tau_{22}}{ \tau_{11}}\right)^{\frac{1}{n_{i}}}\right) \tag{7}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test** & \(\sigma\) [S/m] & **Stroke Distance [m]** & **Stroke Type** \\ \hline T1 & 0.01 & 60 & First \\ T2 & 0.01 & 200 & First \\ T3 & 0.01 & 2000 & First \\ T4 & 0.001 & 60 & First \\ T5 & 0.001 & 200 & First \\ T6 & 0.001 & 2000 & First \\ T7 & 0.01 & 60 & Subsequent \\ T8 & 0.01 & 200 & Subsequent \\ T9 & 0.01 & 2000 & Subsequent \\ T10 & 0.001 & 60 & Subsequent \\ T11 & 0.001 & 200 & Subsequent \\ T12 & 0.001 & 2000 & Subsequent \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Test details. Figure 7: Sketch that illustrates the distance between the tower under study and the lightning stroke Location. Figure 8: Channel-base current: [PERSON]’s current-first stroke. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **First** & **Subsequent** \\ \hline \(I_{01}\) [kA] & 28.0 & 10.7 \\ \(\tau_{11}\) [us] & 1.8 & 0.22 \\ \(\tau_{12}\) [us] & 95.0 & 2.5 \\ \(n_{1}\) & 2 & 2 \\ \(I_{02}\) [kA] & - & 6.5 \\ \(\tau_{21}\) [us] & - & 2.1 \\ \(\tau_{22}\) [us] & - & 230.0 \\ \(n_{2}\) & - & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: [PERSON]’s current parameters. Figure 9: Channel-base current: Heidler’s current-subsequent stroke.