**3. Results**

In order to demonstrate the superiority of the proposed method, some simulations were performed in MATLAB on a PC with i7 CPU. To focus on the difference between the two methods, the discretization step along the lightning channel, *dz*-, was adopted as the variable, while the other parameters are listed in Table 2. The main purpose of this paper was to deal with the calculation problem of the horizontal electric field and it is noted that the horizontal electric field has grea<sup>t</sup> attenuation when it is more than 1 km away from the lightning return channel, so the lightning striking distance *r* is restricted to 0~1 km. According to the geometric formula, the height difference caused by the curvature of the earth is below 8 cm, which is very small compared with the distance of 1 km [26], so the influence of the curvature of the earth is ignored in this paper.

**Table 2.** Basic parameters used in the simulation examples.


Since both of the methods include numerical integration with respect to *dz*-, the value of *dz*- would inevitably influence the accuracy. Theoretically, the smaller the value of *dz*-, the more accurate the result. However, there must be a trade-off between the adoption of *dz*- and the efficiency.

Firstly, the effect of *dz*- on the calculation accuracy of the two methods was studied, which is shown in Figure 3. By comparing Figure 3a with b, it can be seen that *dz*- has a grea<sup>t</sup> influence on the horizontal electric field's calculation of the conventional method, while it has basically no influence on that of the proposed method. When *dz*- ≥ 0.05 m, the horizontal electric field obtained by the conventional method will deviate from the real value. However, no matter what value of *dz*- is taken by the proposed method, the final calculation results are consistent. Therefore, the proposed method is more accurate when the *dz*- is the same.

**Figure 3.** The time-domain evolutions of the horizontal electric field at the observation point under different *dz*-; (**a**) calculated by the conventional method; (**b**) calculated by the proposed method.

The calculation formulae of the proposed method are composed of two parts: The arithmetic part as shown in Equation (13) and the integral part as shown in Equation (14). In order to better explain the reason why the proposed method is not sensitive to *dz*-, these two parts were analyzed respectively. For the arithmetic part, it is an error-free analytical result completely independent of *dz*- in the proposed method (analytical calculation) while an inexact numerical result sensitive to *dz*- in the conventional method (numerical calculation), as shown in Figure 4a,b. This indicates that the proposed method is

more accurate in this part than the conventional method. For the integral part, its accuracy would be dependent on *dz*-. However, as can be seen in Figure 4c, the results are coincident together, although *dz*- takes different values. That is to say, an adoption of *dz*- = 2 m is enough to achieve the accurate result. This is why the proposed method can guarantee the accuracy with a relatively larger *dz*-.

**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.

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*- = 2 m is close to that of the conventional method with *dz*- = 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 5.** The horizontal electric field curves at the observation point calculated by the proposed method and the conventional method under different *dz*-.


**Table 3.** Comparison of the computation time between two methods.

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 e fficient 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 6b that the horizontal electric field calculated by the conventional method with *dz*- = 2m is close to the one with *dz*- = 0.05 m when *r* reaches more than 500 m. The e fficiency 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*- is still required, which can be examined in Figure 6a. In a word, as for the conventional method, small *dz*- should be adopted for the observation point near the lighting channel, while large *dz*- can be used for the observation point far away. However, in the proposed method, small *dz*- 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*- without considering the di fferent 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.

**Figure 6.** The horizontal electric field curves at the di fferent observation points calculated by the proposed method and the conventional method under di fferent *dz*-. (**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.

**Table 4.** Comparison of the computation time between the two methods.


It should also be pointed out that the horizontal component of the lightning electric field attenuates greatly with the distance, as shown in Figures 4–6. Considering the lightning striking distances of more than 500 m, the overvoltage induced by the lightning electromagnetic field on the power line is generally small, which poses little threat to the insulation. Therefore, the example of the horizontal component of the lightning electric field at a longer distance is not discussed in this paper.

As for the calculation of the magnetic field near the lightning channel, its formula can be derived in the same manner as the horizontal electric field. It is not described in this paper for the sake of brevity.

### **4. Discussion and Conclusions**

For the calculation of lightning electromagnetic fields over a perfectly conducting ground, the common method is to evaluate the integral by means of numerical integration. In the proposed method, the formula for calculating the lightning electromagnetic field is divided into two parts: One that can be solved analytically, and the other that can be solved numerically only by integral operations.

For the conventional method, the results are sensitive to *dz*- at a close distance (tens of meters), which must be small enough to ge<sup>t</sup> an accurate numerical solution. However, when *dz*- is too small, the amount of data required in the program is huge and the calculation steps are tedious, resulting in a lengthy time and slow calculation speed. As for the proposed method, *dz*- can be adopted as a relatively large value, so the computational e fficiency can be improved inevitably. Two calculation methods were simulated in MATLAB, of which the simulation results showed that under the same accuracy, the calculation time of the proposed method is about 1/40 of that of the original method. As for large lightning striking distances (hundreds of meters), the conventional method's results are not sensitive to *dz*- anymore, and relative larger *dz*- can be adopted to ge<sup>t</sup> an accurate numerical solution. In other words, for the conventional method, small *dz*- should be adopted for the observation point near the lighting channel while large *dz*- can be used for the observation point far away. However, the proposed method has better generality because it can calculate the horizontal electric fields at di fferent lightning striking distances with a uniform *dz*-, which makes its implementation more convenient and easily programmed. Additionally, as can be seen, the only item required to be calculated numerically is very simple and other items can be implemented in an analytical way, so the proposed method is very easy to be programmed, compared with the original method.

**Author Contributions:** Conceptualization, X.L.; methodology, X.L. and T.G.; software, X.L. and T.G.; validation, X.L. and T.G.; formal analysis, T.G. and X.L.; writing—original draft preparation, T.G.; writing—review and editing, X.L.; visualization, T.G.; supervision, X.L.; project administration, X.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
