## 1 Introduction  
Breakdowns occur in power transmission systems. In the event of a malfunction or lightning strikes, a fault current flows through the grounding system, causing a potential funnel in the vicinity of the ground. As a result, a potential appears on the surface of the earth that can be dangerous to humans and animals. The maximum values of the touch voltage (\(U_{\mathrm{J}}\)) and step voltage (\(U_{\mathrm{s}}\)) are determined depending on the standard [1, 2, 3]. The earthing system must be dimensioned properly so that the touch and step voltages do not exceed the permissible values [4, 5, 6]. In this article, we focus on the problem of grounding systems for transmission poles that are buried in the ground in the vicinity of the pole. Different methods for grounding system implementation are used so that adequate protection is achieved, and, at the same time, the touch and step voltages do not exceed permissible values. One approach is to use grounding rings, but probes are also used. In the case of high touch and step voltage values, protective foil is used, which is grounded in the soil around the pole. It is important that the correct size of protective foil is used to ensure adequate protection but at the same time, that it is not larger than necessary. Our work focuses on protection against excessive voltages, specifically the determination of the appropriate size of protective foil. The goal is to determine the process for identifying the optimal size of protective film in a simple and time-saving manner. The process of defining the appropriate size of protective foil is presented using a model of the earthing system made by the finite element method (FEM) [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. Since the modeling process is demanding, an approach is shown whereby the number of models with foils of different sizes is minimized. The process is based on only three FEM calculations for three different foil sizes, with the foil widths denoted by _fw_. On the basis of these, the function \(U_{\mathrm{s}}=\textit{f}(\textit{fw})\) is written, from which the optimal size of protective foil is obtained. Five different functions are tested to determine the most suitable for the presented problem. To determine the coefficients of the function, \(U_{\mathrm{k}}=\mathrm{F}(\textit{fw})\), we tried to use the Newton-Raphson procedure, but this was not possible because of the singularity of the Jacobi matrix. Therefore, evolutionary methods were used to determine the coefficients. They were compared with each other, and the most appropriate one was determined. The used evolutionary methods were differential evolution (DE) [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31] with three different strategies (DE/rand/1/exp, DE/rand/2/exp, and DE/best/1/bin), teaching-learning-based optimization (TLBO) [32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42], and artificial bee colony (ABC) [43; 44; 45; 46; 47; 48; 49; 50; 51; 52]. The tests were conducted on the basis of real data obtained from ELES, d.o.o. (combined transmission and distribution system operator of the Republic of Slovenia), and ENS, d.o.o. (New Electric Systems, a member of the Elnos Group). The data obtained from ELES, d.o.o., and ENS, d.o.o., were used exclusively in the present analysis. The basis for the presentation of the proposed method for optimal foil width determination is the data from the first transmission pole (TP1), i.e., the data from soil measurements using the Wenner method [53; 54; 55; 56; 57] and the geometry of the grounding system in the vicinity of the pole. Based on the measurements using the Wenner method, we determined the soil data needed for the FEM models using an analytical soil model and the ABC evolutionary method. ABC was used based on our experience from previous research [58]. The FEM was conducted to determine the function of \(U_{\mathrm{s}}=\textit{F}(\textit{fw})\).