# Impact of Grounding Modeling on Lightning-Induced Voltages Evaluation in Distribution Lines
## 4 Electrical Grounding Modeling  
In this paper, the grounding transient behavior is modeled by HEM [18; 19]. This model is an electromagnetic computational method developed for the numerical solution of lightning problems and, according to CIGRE [30], it is classified as a hybrid electromagnetic-circuit approach. The main motivations for using HEM are as follows: (i) it is accurate and flexible, i.e., it can be used in different types of grounding configuration; (ii) its results have been extensively validated experimentally, such as measurements in TS [18; 19], horizontal electrodes [18; 31], vertical rods [31; 32], and typical substation grounding grids [33] and (iii) it is faster than traditional full-wave methods (without losing accuracy). It is worth mentioning that the usage of HEM has increased significantly recently [30; 34; 35]. Basically, HEM consists of subdividing the actual system (in this case, electrical grounding) into N small conductive cylindrical segments and, for each segment, the electromagnetic theory is applied. After that, by using the circuit theory, it is possible to obtain a matrix system that computes the wideband response of the electrical grounding. For the sake of clarity, we present a brief overview of HEM below. More details about HEM are described in [18; 19]. It is worth commenting on the fact that HEM corresponds to an electromagnetic model developed in the frequency domain. Thus, it is necessary first to determine the frequency spectrum (depending on the phenomenon of interest). After that, the electrical grounding is divided into N segments, where the length of each segment is equal to 10 times its radius (thin wire approximation). A discussion about segmentation length is presented in [36]. Each segment is considered a source of two currents, one longitudinal that flows along the electrode (\(I_{L}\)) and another transversal that flows from the electrode to the surrounding soil (\(I_{T}\)). It is worth noting that \(I_{L}\) generates a non-conservative electric field and \(I_{T}\) a conservative one. With the aid of the magnetic vector and electric scalar potentials, both voltage drops (\(\Delta V\)) and electric potentials (\(V\)) in each pair of segments (transmitter and receiver) are determined. Additionally, double integral equations are established for \(\Delta V\) and \(V\). These integrals depend on the frequency, geometry, soil parameters and \(I_{T}\) and \(I_{L}\) distributions. However, the distributions of \(I_{T}\) and \(I_{L}\) are not known and are integrands of the integrals. From this point on, the Method of Moments (MoM) is applied to solve these integral equations [37]. The effect of the air-soil interface is included using the method of images, similar to [38; 39]. The \(I_{T}\) and \(I_{L}\) distributions considered in this paper are of the piecewise-constant function type [18; 19]. Mod makes it possible to transform integral equations into algebraic ones, the solution of which allows determining all the quantities of interest (in the frequency domain), such as \(I_{T}\), \(I_{L}\Delta V\) and \(V\) distributions; transverse (capacitive and conductive couplings) and longitudinal (resistive and inductive couplings) impedances (self and mutual); electromagnetic field; harmonic grounding impedance (\(Z(\omega)\)); low-frequency grounding resistance (\(R_{LF}\)), etc. Also, it has been documented in the literature, over almost one hundred years [40], that the soil is a dispersive medium, i.e., the response is not instantaneous. Several researchers have presented a numerical solution to consider this dispersivity in the frequency domain, such as [31; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49]. According to [50], for the values of conductivity considered in this paper, the frequency-dependent soil electric parameters can be neglected in the computation of the EM fields that illuminate the line. On the other hand, according to [17], the impact ofconsidering the frequency-dependence of the soil in the grounding modeling generates sensible differences.