## 1 Introduction  
Grounding systems come in various forms and sizes, with their primary function being the safe dissipation of electric current into the surrounding soil. This dissipated current may originate from human activities (including direct current, alternating current, or transient current) or natural phenomena (such as lightning currents). Lightning currents [1], or transient currents in general, including those generated by switching operations [2], uniquely impact grounding systems. Specifically, when a grounding system is subjected to a transient current, only a portion of the grounding system effectively dissipates the current into the surrounding soil, while the remaining part of the grounding system plays a passive role. Throughout the years, numerous researchers have extensively studied this phenomenon in grounding systems of varying complexity. For example, effective lengths of simple horizontal or vertical grounding electrodes are examined in [3; 4] and, more recently, in [5; 6; 7; 8], to name a few. More complex grounding electrode configurations and their effective areas are analyzed in [9] and, more recently, in [10; 11; 12]. In addition to analyzing the effect of this phenomenon, some researchers have developed approximation formulas to predict the area (or length) of the grounding system that participates in the dissipation of transient current depending on several input parameters such as soil resistivity, the rise time of the dissipated lightning function, and the peak value of the lightning current [9; 13; 14; 15]. These approximation formulas basically utilize the least-squares method to fit the selected formula to the analytically computed data. In this paper, we will improve upon this approach by providing our own approximation formulas with the addition of utilizing advanced machine learning (ML) regression models to achieve a much better fit. In the scope of this paper, only one type of grounding system will be observed--a radial-based grounding system as described in the international standard IEC 62305-3 [16]. This grounding system, the so-called type-A grounding system, is characterized by a set of horizontal electrical ground electrodes, buried parallel to the Earth's surface, all radiating from a central point. The simplest form of this grounding system, which consists of two grounding electrodes, will be the object of observation in this paper. This configuration is mainly used in lightning protection of relatively low dislocated objects such as meteorological stations that have only one conductor. The standard [16] states that in the type-A arrangement, the minimum number of ground electrodes should be one for each down-conductor and at least two for the entire lightning protection system. Note that this grounding electrode with a central point lightning injection is also observed and analyzed in [11; 15]. In this paper, it is our intention to model the mentioned type-A grounding system using a well-tested frequency-domain-based algorithm [17] and to subject the grounding system model to a number of various lightning strikes with the purpose of finding the effective length of the grounding system. To further refine our effective length analysis, we will modify two additional input parameters alongside the lightning current: the burial depth of the grounding system and the soil resistivity. Note that the influence of the burial depth of the grounding electrode on its effective length was not analyzed in any of the available references. This is to be expected, since the electrodes are almost always buried at a standard prescribed depth. However, as in all practical cases, a scenario may occur where the actual burial depth will deviate from the prescribed depth, so we included this as an input parameter to our model. It is also important to note here that in our analysis we disregarded the frequency dependence of the soil parameters, and we disregarded the beneficial effect of the soil ionization phenomenon [18].