## 3 Geometry of Test Transmission Poles and Determination of Soil Parameters
### Transmission Pole TP1  
The geometry of TP1 is presented in Figure 2.  
Figure 2 shows the initial and final depth of the rods. The depth of individual parts of the grounding system is defined, and their positions are determined based on the geometric grid. Wenner's method of measurement was used to determine the soil data [53; 54; 55; 56; 57]. The measured data are presented in Table 2.  
The four-electrode Wenner method [53; 54; 55; 56; 57] was used, as defined by IEEE 81-1983 and IEEE 81-2012 [66; 70]. Four electrodes were placed with equal spacing \(d\). The measured potential difference \(U\) between the internal electrodes is caused by the current \(I\) imposed between the external electrodes. The apparent resistivity \(\rho\) is defined with (1) [66]. \[\rho=\frac{4\cdot\pi\cdot d\cdot R}{1+\frac{2d}{\sqrt{d^{2}+4b^{2}}}-\frac{d}{ \sqrt{d^{2}+b^{2}}}} \tag{1}\]  
Figure 2: Geometry of TP1. In practice, the depth of the electrodes \(b<0.1d\), and (1) can be written as (2) [56,66]. \[\rho=2\cdot\pi\cdot d\cdot R=\frac{2\cdot\pi\cdot d\cdot U}{I} \tag{2}\]  
The soil parameters were determined using an analytical horizontal soil model [55, 56, 57, 67, 71]. Based on our experience with previous works, the focus was on a three- and four-layer model. The two-layer soil model is easy to model but is not precise enough. If we compare three-, four-, five-, and six-layer models, the precision increases slightly with the number of layers, but the complexity of modeling with the finite element method increases significantly, which is the reason why we decided to use three- and four-layer models [68]. To determine the parameters, the artificial bee colony (ABC) optimization method was used which in previous works [58] proved to be better for determining soil parameters than differential evolution (DE) or teaching-learning-based optimization (TLBO). The objective function (OF) used for the determination of the soil parameters is written in (3). \[\text{OF}=\frac{1}{n}\sum_{i=1}^{n}\left|\frac{\rho_{\text{c,i}}-\rho_{\text{m, i}}}{\rho_{\text{m,i}}}\right|\cdot 100\quad(\%) \tag{3}\]  
The parameters of the three- and four-layer models are shown in Figure 3.  
Based on Figure 3, it can be seen that all the layers except the deepest are defined by a finite width and specific resistivity. Theoretically, the deepest one extends to an unlimited depth. The analytic expression of apparent resistance is a function of distance between electrodes \(d\), and it is defined with (4) [72]. \[\rho=\rho_{1}\left(1+2d\int_{0}^{\infty}f(\lambda)[J_{0}(\lambda d)-J_{0}(2 \lambda d)]\ d\lambda\right) \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(d\) (m) & \(\rho\) (\(\Omega\)m) & \(d\) (m) & \(\rho\) (\(\Omega\)m) & \(d\) (m) & \(\rho\) (\(\Omega\)m) \\ \hline
0.5 & 45.5 & 5 & 105 & 10 & 137 \\ \hline
1 & 58.3 & 6 & 115 & 12 & 138 \\ \hline
2 & 72.8 & 7 & 124 & 15 & 136 \\ \hline
3 & 84.2 & 8 & 130 & 20 & 132 \\ \hline
4 & 95 & 9 & 134 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Measured soil data for TP1. Figure 3: (a) Three-layer soil model; (b) four-layer soil model. \(\rho_{1}\) is the specific resistivity of the first layer, \(J_{0}\) is the Bessel's function of the first kind, and \(d\) is the distance between the electrodes. Equation (4) contains function \(f(\lambda)\), which was calculated using (5). \[f(\lambda)=\ \alpha_{1}(\lambda)-1 \tag{5}\]  
For the three-layer model, \(\alpha_{1}\) was calculated with the equations presented in (6). \[ \begin{array}{ll}K_{1}(\lambda)=\frac{\rho_{2}\alpha_{2}(\lambda)-\rho_{1}}{ \rho_{2}\alpha_{2}(\lambda)+\rho_{1}};&\alpha_{1}(\lambda)=1+\frac{2K_{1}e^{-2 \lambda t_{1}}}{1-K_{1}e^{-2\lambda t_{1}}}\\ K_{2}(\lambda)=\frac{\rho_{3}-\rho_{2}}{\rho_{3}+\rho_{2}};&\alpha_{2}(\lambda) =1+\frac{2K_{2}e^{-2\lambda t_{2}}}{1-K_{2}e^{-2\lambda t_{2}}}\end{array}  \tag{6}\]  
For the four-layer model, \(\alpha_{1}\) was calculated with the equations presented in (7). \[