## 3 Implementation of Corona Discharge  
This section shows the implementation of the corona effect in the procedure proposed in Reference [5] following the one in Reference [12]. The main idea of this approach is to consider a dynamic capacitance in the Agrawal model. As in Reference [12], the dynamic capacitance has been described through the following equation, which aims at reproducing the experimental _q-v_ characteristics. \[C_{dyn}(x,t)= \begin{cases}C_{0}&\text{for }u(x,t)<u_{th}(x,t)\\ C_{0}\frac{(k_{1}+k_{2}(u(x,t)-u_{th}(x,t)))}{u_{th}(x,t)}&\text{for }u(x,t)\geq u _{th}(x,t)\text{and }\frac{du(x,t)}{dt}>0,\end{cases}  \tag{4}\]  
where \(k_{1}>1\) is related to the sudden change of the capacitance when the voltage exceeds the corona threshold \(u_{th}\) and \(k_{2}>0\) is related to the gradual increase of the capacitance when the voltage is rising above the threshold. \(C_{0}\) is the geometrical capacitance of the line and \(u(x,t)\) is the voltage on a generic conductor at time instant \(t\) and at a distance \(x\) from the beginning of the line. The voltage threshold \(u_{th}\) is related to \(C_{0}\), to the conductor radius \(r\) and to the critical electric field in (1), according to  
\[u_{th}=\frac{2\pi\varepsilon_{0}r}{C_{0}}E_{c}. \tag{5}\]  
Here, as usual, \(\varepsilon_{0}\) is the electric permittivity in vacuum. **4. Corona Effect Influence on the Lightning Performance**  
This section shows the enhancement of the number of dangerous lightning events due to the corona effect. Let us consider a single-phase overhead distribution line whose details are available in Table 1. In order to avoid reflections, the line extremities are matched. Moreover, let us suppose a ground conductivity of 1 mS/m and a ground permittivity of 10. According to Reference [6] and supposing a lightning channel height of 8 km and a propagation velocity along the lightning channel of \(c_{0}/2\) (being \(c_{0}\) the light speed in vacuum ), the lightning performance of the overhead distribution line is computed as follows (for further details check Figure 1):  
1. A counter \(n\) is initialized to 0. 2. A large number of events, able to guarantee the convergence of the Monte-Carlo procedure (here \(n_{tot}=\) 10,000), is generated. Each one is characterized by a stroke location extracted from a uniform distribution, peak current and front duration extracted from the log-normal distributions proposed in Reference [1]. Please note that the channel-base current is assumed to be the typical Heidler's first stroke waveform with variable front duration [23]. 3. Each event is classified as direct or indirect one according to the EGM criterion [24]. In any case the maximum induced voltage on the line is computed through the procedure in Reference [5]. 4. For each event, the maximum voltage is compared with the line CFO. If it is greater than 1.5 CFO [1], the counter n is increased by one. 5. Once all the considered events have been evaluated, the total number of dangerous events \(n\) is obtained and the number of flashovers per year per 100 km of line is computed according to References [1,7] \[F=200\frac{n}{n_{tot}}GFDy_{max},\] (6) where \(GFD\) is the ground flash density expressed as number of flashes per square kilometer per year ad \(y_{max}\) is the maximum value of the y-coordinate where the events are extracted. According to Reference [1], \(y_{max}\) is a function of the CFO and it is computed through the extended [PERSON]'s formula [25], choosing as lightning current the maximum value (\(I_{max}\)) obtainable from the probabilistic density function. \[y_{max}=\frac{38.8I_{max}\left(h+\frac{0.15}{\sqrt{\sigma_{g}}}\right)}{CFO}, \tag{7}\]  
where \(\sigma_{g}\) is the ground conductivity. For what concerns the corona effect, in the following we assume \(m=0.8\), \(k_{1}=1.2\) and \(k_{2}=4.8\)[12]. Consequently, \(E_{c}=35.49\) kV/cm and \(u_{th}=150\) kV.