## 1 Introduction
This nonlinear effect, in reality, reduces the effective length value of the grounding electrode, so in our analysis, we practically conservatively overestimate the effective length values. Due to this fact, the peak value of the lightning current will not influence the effective length in our approximation since the soil ionization is neglected. In addition to disregarding the soil ionization, we assumed a constant soil permittivity value in all considered cases since, in our preliminary analysis, we found that the effect of this parameter on the solution was less significant than the effect of the chosen input parameters. By varying the chosen parameters (soil resistivity, rise time of the lightning current, and the burial depth of the grounding system), we aim to generate a comprehensive dataset containing effective length values of the type-A grounding system for various input parameter combinations. This extensive dataset will serve as a foundation for applying various regression algorithms with the aim of developing predictive models that will enable users to accurately estimate the effective length of the type-A grounding system in most practical cases. To address this research goal, we explored multiple approaches. Initially, we tested various mathematical regression functions on the input dataset, utilizing nonlinear least-squares estimation to optimize the function parameters for a better fit. Through this process, we identified two regression functions that provided satisfactory results and are relatively easy to calculate. To further improve our approximations, we then applied a range of ML regression models, training them on the dataset. Our testing revealed that only a subset of these ML models delivered superior approximation results. In this paper, we will present only the best-performing ML regression model to avoid redundancy. We validated both the mathematical regression functions and the ML regression model using standard regression quality metrics. This will be elaborated on in the corresponding section. The paper is organized as follows. In the second section, following the Introduction, we provide a summary of the methodological approach used in this study. This includes a description of the observed type-A grounding system [16], the definition of the effective length used in the paper, and an explanation of the iterative procedure for determining the effective lengths of the type-A grounding system for various input parameters. The third section features an overview of the input parameters that were varied to generate the effective length dataset. The section includes a detailed description of each input parameter along with its observation interval and an initial feature analysis to identify which parameters most significantly impact the effective length. Additionally, this section describes the resulting dataset. In the fourth section, on the basis of the described dataset, we develop two simple mathematical regression functions followed by a detailed regression quality analysis. These formulas are intended for quick engineering use. It is important to emphasize here that the developed formulas take into account the burial depth of the grounding electrodes unlike the formulas available in the literature. In the final section, we explore the possibility of applying a specific ML regression model to our dataset in order to further increase the accuracy of the effective length approximation. The Gaussian process regression model was used in our paper since it performs extremely well on smooth functions, especially when a suitable kernel is selected. The model was trained and tested in the section and the reader was provided with the optimized values of the model hyperparameters so the results may be replicated without the training procedure.