## 1 Introduction
To verify the correctness of the presented method and the selected function \(U_{\mathrm{s}}=\textit{F}(\textit{fw})\), data were used from another transmission pole (TP2) with a different grounding system geometry and completely different soil data. Again, using an analytical soil model and the ABC method, we determined the soil parameters and confirmed the correctness of the selected function \(U_{\mathrm{s}}=\textit{F}(\textit{fw})\) using the FEM model. Our contributions in this work are as follows:  
* An analysis of the step voltage depending on the width of the used protective foil using an FEM model based on real data from the transmission line pole and the earthing system;
* A new procedure for determining the optimal width of the foil, with the minimum number of time-consuming FEM calculations;
* Determination of the most appropriate function \(U_{\mathrm{s}}=\textit{f}(\textit{fw})\) among the five selected functions;Selection of the most appropriate evolutionary method among DE using three different strategies, TLBO and ABC, to determine the coefficients of the function \(U_{\text{s}}=\text{F}(\textit{fw})\);
* Determination of potential differences between the potential above and below the protective foil on the basis of the FEM model, which are important for the selection of the appropriate material. Within a broader scope, our work can be related to [6], in which numerical analyses are also used for the safety design of grounding systems. The safety performance of typical configurations is also studied in [5]. Analytical methods for earth surface potential calculation for grounding grids are presented in [59]. Moreover, the optimal design of grounding systems is searched for in [60]. Numerical investigation of the use of electrically conductive concrete-encased electrodes is used in [61]. Protective measures for the earthing system of buildings near the transmission tower at ground contact currents are discussed in [62]. Potential elevations over transmission towers of transmission lines with multi-grounded shield wires are presented in [63]. The touch voltage over the earthing system caused by the earth fault current using FEM is shown in [64]. The calculation of the ground resistance of a power transmission line using the COMSOL (September 2019). program tool is presented in [65]. Due to the importance of the problem, inadequate protection can have catastrophic consequences, and many researchers are working on protection systems to reduce \(U_{\text{s}}\) and \(U_{\text{t}}\)[4; 5; 61; 62; 63; 64; 65; 66; 67]. Due to its applicability and universality, FEM is used increasingly for the analysis and dimensioning of grounding systems [6; 64; 65; 12; 6]. Also, modern evolutionary methods are used increasingly in electrical engineering due to their usability and robustness [68; 69; 20; 19; 52; 58]. Our work covers both the use of FEM and evolutionary methods in the dimensioning of the grounding system. In our work, an analysis of the step voltage as a function of the foil width is performed, the function \(U_{\text{s}}=\text{F}(\textit{fw})\) is determined, and the procedure for determining the optimal protective foil width with a minimum number of FEM calculations is presented, which is not the case in related works. To confirm the difference between article [58] by the same author and the presented article, a brief explanation is given to highlight the new contribution. Article [58] presents an innovative approach to determining soil parameters in the case of inhomogeneity in the soil and different results are obtained on different measurement lines using the Wener method. In the presented article, only the results from [58] are used regarding the determination of soil parameters required for the FEM model.