# On the Statistical Characterization of Lightning-Induced Voltages
## 1 Introduction  
Smart grids require a high level of surge protection, especially because of the presence of electronic-based equipment, as well as control and monitoring devices. Protection of renewable energy generation systems against lightning is also an emerging issue to be addressed [1; 2; 3; 4]. To this aim, a critical issue is the analysis of the severity of the induced voltages generated by the lightning. As the lightning phenomenon is stochastic, a statistical analysis of the induced voltage parameters is fundamental. Statistical analyses of the lightning current parameters have been proposed in the relevant literature with reference to peak values and to the parameters of waveshapes, such as the time intervals between 10% and 90% (or 30% and 90%) of peak value, the corresponding front duration, the average and maximum front steepness, and the maximum rate of rise in the front wave [5; 6; 7; 8; 9]. More specifically, the technical literature has proposed parametric probabilistic distributions for the statistical characterization of the aforementioned lightning parameters, which were also adopted in the IEEE Std. 1410 [10]. In the IEEE Std. 1410, a statistical procedure is proposed to infer the lightning performance in terms of the flashover rate, which is the expected number of annual insulation flashovers per 100 km per year occurring on the distribution line. In the IEEE Std. 1410 [10], the flashover rate is evaluated based on the Rusck formula [11]. This formula allows the evaluation of the peak values of the lightning-induced voltage only for the case of the lightning current modeled as a step-function current. In order to face more realistic current waveshapes, numerical models that make use of Monte Carlo techniques have been proposed [4]. They require many iterations to provide accurate results; hence, these tools are not easy to manage due to the required computational burden. Numerical methods, able to reduce the required computational effort related to the use of the Monte Carlo technique, have been recently proposed [12]. On the other hand, efforts have been made to propose accurate analytical solutions able to reproduce more accurate current waveshapes [13; 14; 15], thus overcoming the problems related to the computation effort.  
Aim of this paper is the statistical characterization of the lightning-induced voltages on distribution lines. To this aim, peak values of the induced voltages are evaluated by means of analytical solutions and, specifically, by means of the [PERSON] et al. exact solutions (AESs) [13; 14; 15]. The solutions refer to a horizontal conductor, placed over an infinite-conductivity ground, excited by an external field produced by either a step-function or a linearly-rising current, moving unattenuated and undistorted along a vertical lightning channel (Figure 1). The main contribution of this paper is the use of analytical formulations (more specifically, the exact ones [13; 14; 15]) for the statistical characterization of the peak-induced voltages, whose inputs were typical parameters of the lightning current. A further contribution is the identification of parametric distributions able to describe the statistical behavior of the induced voltage. Values of the parameters of such distributions are provided, which can be used as reference for future statistical evaluations. Thanks to the analytical formulations, a further contribution is given to statistically characterize the induced voltages in the presence of a mitigation procedure (use of an overhead ground wire) [16; 17; 18].  
The paper is organized as follows. The AESs are briefly reviewed in Section 2. Section 3 describes the method used for the statistical analysis. Section 4 reports the results of the numerical applications. Conclusions are drawn in Section 5.