## 3 Results
The most fundamental reason why the proposed method can improve the efficiency can be well explained by Figure 5. It can be seen that the horizontal electric field calculated by the proposed method under a condition of \(dz^{\prime}\) = 2 m is close to that of the conventional method with \(dz^{\prime}\) = 0.05 m, and the computation time is 0.143 and 5.776 s, respectively, as shown in Table 3. That is to say, the acceleration of the proposed method is more efficient than that of the conventional method with the same accuracy. Figure 4: The two parts of the horizontal electric field at the observation point: (**a**) the arithmetic parts calculated by the conventional method; (**b**) the arithmetic parts calculated by the proposed method; (**c**) the integral part calculated by the proposed method. Figure 5: The horizontal electric field curves at the observation point calculated by the proposed method and the conventional method under different \(dz^{\prime}\). Considering the calculation of lightning-induced voltages of distributed overhead lines, it is often necessary to calculate a large number of observation points. Taking a one-kilometer three-phase line network as an example, it is generally necessary to calculate at least 300 observation points. According to Table 3, the computation time of the proposed method will be about 1 min. It can be imagined that the more complex the power networks studied, the more efficient the proposed method. Additionally, compared with the conventional method, the proposed method is easier to be programmed because the only formula required to be calculated numerically is very simple. We also considered the cases of \(r=250\) and 500 m. In these examples, the static E-field component of the horizontal electric field will gradually decrease, but the advantages of the proposed method still remain. The comparison of the two methods is shown in Figure 6 and Table 4. It can be seen from Figure 6 that the horizontal electric field calculated by the conventional method with \(dz^{\prime}=2\)m is close to the one with \(dz^{\prime}=0.05\) m when \(r\) reaches more than 500 m. The efficiency of the two methods is almost the same, as the computation time listed in Table 4. However, as for the relative closer observation points, such as \(r=250\) m, small \(dz^{\prime}\) is still required, which can be examined in Figure 6. In a word, as for the conventional method, small \(dz^{\prime}\) should be adopted for the observation point near the lighting channel, while large \(dz^{\prime}\) can be used for the observation point far away. However, in the proposed method, small \(dz^{\prime}\) can be adopted no matter the distance from the observation point and the lightning striking point, which indicates that the proposed method is more general. In other words, the proposed method can adopt uniform \(dz^{\prime}\) without considering the different lightning striking distances. Therefore, its implementation is more convenient. Additionally, as the statement above, the proposed method is easier to be programmed because the formulation of the item for numerical integration is simpler, which is superior to the proposed method. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(dz^{\prime}\) & \multicolumn{2}{c}{**Conventional Method**} & \multicolumn{2}{c}{**Proposed Method**} \\ \hline - & \(r=250\) m & \(r=500\) m & \(r=250\) m & \(r=500\) m \\
0.05 m & 4.688 s & 4.673 s & - & - \\
2 m & 0.203 s & 0.163 s & 0.150 s & 0.142 s \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the computation time between the two methods. Figure 6: The horizontal electric field curves at the different observation points calculated by the proposed method and the conventional method under different \(dz^{\prime}\). (**a**) the horizontal electric field at \(r=250\) m calculated by two methods; (**b**) the horizontal electric field at \(r=500\) m calculated by two methods.