*2.1. Indirect Lightning*

In this paper, the double exponential Equation (1) is used to generate standard 1.2/50 μs indirect lightning, where 1.2 μs stands for the front-time and 50 μs represents the time-to-half value. It is worth mentioning that this standard waveform is subject to a simplification since this representation assumes that the lightning current rises linearly until reaching the peak and neglects the influence of channel-base current [25,26]. In EMTP-RV, there exist a voltage surge device that takes advantage of a double exponential function to generate lightning impulses, see Figure 1.

$$V(t) = V\_m(\mathcal{e}^{at} - \mathcal{e}^{\mathcal{Y}t}) \tag{1}$$

where *Vm* is the maximum voltage of the source, and α and β are the coefficients to adjust the front time (e.g., 1.2 μs ± 30% ) and time-to-half value (e.g., 50 μs ± 20% ), respectively [27]. It is worth mentioning that, in EMTP-RV, this device has two options for start time and stop time to have more control over generating the surge impulses. However, generating indirect lightning impulse in software by using a double exponential function is not always an easy task. Determining the sensitive parameters of α and β for the double exponential function (1) is a challenging issue. In [28], an innovative approach was used to define the parameters of α and β, while for the sake of facilitating the parameter adjustments, an A factor was also used. Then, all the parameters A, α, and β were obtained via trial and error method, and the results show some violations compared to the standard 1.2/50 μs lightnng impulse. Therefore, adjusting these sensitive parameters requires utilizing some optimization model. In this work, the optimization model developed by Pourakbari-Kasmaei et al. in [29] is modified as (2)–(6).

$$\begin{aligned} \text{maxim} & \alpha + \beta\\ \text{s.t.} & \end{aligned} \tag{2}$$

$$V\_m(e^{\alpha \cdot t} - e^{\beta \cdot t}) \le V\_p; \quad \forall t < t\_p \tag{3}$$

$$V\_m(e^{\alpha \cdot t\_p} - e^{\theta \cdot t\_p}) = V\_p \tag{4}$$

$$V\_m(e^{\alpha \cdot t} - e^{\theta \cdot t}) \le V\_p; \quad \forall t > t\_p \tag{5}$$

$$\forall V\_{\text{lt}} \le V\_m (\boldsymbol{\varepsilon}^{a \cdot t} - \boldsymbol{\varepsilon}^{\otimes \cdot t}) \le \overleftarrow{V\_{\text{lt}}}; \ \forall t = t\_{\text{lt}} \tag{6}$$

where *Vp* is the peak value of the lightning impulse, *tp* is the frontier time, *th* is the time to half value, and *Vh* is the half value. In this paper, the A factor is melted into the *Vm* such that *Vm* = *<sup>A</sup>*·*Vp*, and more often than not, to determine the desired voltage, *Vm* is set a bit higher than *Vp*, i.e., A is greater than or equal to 1.

**Figure 1.** Voltage surge device to generate indirect lightning impulses.

To the best of our knowledge and according to practical experiences, to generate an indirect lightning impulse, parameters α and β take negative values. Therefore, maximizing the summation of these negative values guarantee to find the smallest values, which is extremely beneficial in enhancing the computational efficiency in transient studies via the EMTP-RV software [30]. It is noteworthy to mention that in order to support open access, the developed GAMS code used to obtain the parameters of the double exponential function for simulating an indirect standard 100 kV lightning impulse is available in [31].
