## 2 Materials and Methods for Coupled Thermal-Electrical Modeling of the Grounding Electrode
### The Finite Differences Method for Heat Equation (4)
In Code (9) to calculate the final temperature, the time interval will be divided into a few temporal segments or intervals, in each of which the currents \(I_{c}(t)\) and \(I_{s}(t)\) are considered constant. At the end of each interval, a temperature will be obtained, which will be used to recalculate the values of these currents, which will influence the intensity of the sources. These new source values will be used in the next interval, and the cycle will repeat until the final time is reached. As a final note, it is necessary to consider that, if there is a temperature profile in an electrically conductive medium such as soil, the electrical resistivity will also have a profile dependent on the location. This is partially reflected in the second equation in (18) and in (19). It is also important to clarify that no significant differences are observed between the simple and improved approximations at the surface temperature of the conductor, i.e., the boundary condition at \(R_{1}\). Therefore, unless further refinements are made, we will continue working at first order \(o(\Delta r)\) in boundary conditions. Some final comments on the impact of the variation of soil resistivity with temperature are provided below. Since the temperature increase in the soil leads to a decrease in its electrical resistivity around the conductor, this may cause a reduction in its electric potential. The effect would be similar to surrounding the conductor with a layer of soil made of a material with a lower resistivity than the rest of the soil. It can also be thought of as equivalent to increasing the radius of the conductor, as happens in ionization phenomena due to high leakage currents. The magnitude of this increase can be estimated in several ways. In this paper, the decrease in the resistivity of the soil near the conductor is considered through the following reasoning. Since soil resistivity as a function of the radial distance \(r\) from the original radius of the conductor \(R_{1}\) to the boundary \(R_{2}\) is available, the theoretical resistance \(R_{G}\) of a cylindrical layer [22] with radii \(R_{1}\) and \(R_{2}\) can be approximately calculated by \(R_{G}=\int\limits_{\dot{R}_{1}}^{R_{2}}\frac{\rho(r)}{2\pi Lr}dr\). This resistance is equated to that calculated for a cylindrical layer composed of two juxtaposed layers with constant resistivities \(\rho_{1}\) (close to the conductor resistivity) surrounding the conductor with radii \(R_{1}\) and \(r_{b}\), and a second layer with resistivity \(\rho_{2}\), equal to that of the original soil, with radii \(r_{b}\) and \(R_{2}\), as it is shown in Figure 3.  
Through this operation, the optimal value of \(r_{b}\) is determined, which will be assigned to the new virtual radius of the conductor. As previously established, the analysis is conducted on a cylindrical segment of bare conductor with radius \(R_{1}\) and length \(L\), encapsulated by a surrounding soil layer extending to an outer radius \(R_{2}\). It can be easily demonstrated that  
\[r_{b}=\left[\frac{\exp(2\pi LR_{\Omega})\cdot R_{1}^{\rho_{1}}}{R_{2}^{\rho_{ 2}}}\right]^{\frac{1}{\rho_{1}-\rho_{2}}} \tag{20}\]  
Consequently, in the computation of electric potentials, current distributions, and other temperature-dependent electrical parameters, the actual electrode radius should be substituted with the effective (or virtual) \(r_{b}\). It is important to consider that the increase in the radius of the conductor segments should be subordinated to the size of the segments themselves. We know that potential calculations are acceptable when \(\delta>4r\), where \(\delta\) is the segment length and \(r\) is its radius. Therefore, the radius cannot be increased beyond this value, as the potential obtained  
Figure 3: Radial resistivity profile from cable surface to outer soil boundary. would become an upper bound to the real value, since increasing the radius results in a decrease in potential.