## 4 Conclusions  
The commonly used model of the grounding electrode as a PEC without any signs of corrosion or alteration of the electrode surface has been improved in this work by including realistic attributes such as the internal resistance of the conductors and the possible damage caused by partial oxidation or by coating with a conductive material. In order to evaluate the impact of changes in the properties of the electrode, some important magnitudes have been calculated that define its characteristics, such as the GPR or the potential of the different nodes introduced in the modeling of the electrode. The grounding resistance is now associated with the potential of the fault current input node. In view of the results, it can be concluded that there is a significant difference between an ideal PEC electrode and an equivalent one with real attributes. For the example of the transformation center shown in Figure 2, the difference in grounding resistance is greater than 10%, and in terms of GPR, differences of around 5% are found. The differences increase with the extension and degree of oxidation of the electrode. In summary, the calculations obtained from the presented model confirm that the attributes included in the modeling of a real electrode, in particular the internal resistance, must be taken into account at the design stage to adjust the size and shape of the electrode in order to meet the safety requirements of current regulations.  
Finally, although there may be many factors to consider that can affect ground resistance measurements, such as humidity, temperature, and other seasonal effects, oxidation is a systematic effect that can be detected in long-range measurements. The model presented in this paper would allow for an estimation of the degree of oxidation of an operational grounding electrode by comparing measurements of the real grounding resistance and its calculation using the model for a determined oxidation percentage, all without a high computational cost.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\rho_{e}=0\)\(\Omega\)m & PEC & \(\rho_{e}=72\times 10^{-8}\)\(\Omega\)m & \(\rho_{e}=72\times 10^{-8}\)\(\Omega\)m \\  & & & **(Non-Corroded)** & **(\(\rho_{ox}\) = 720 \(\Omega\)m)** \\ \hline
**R(\(\Omega\))** & 0.5567 & 0.5606 & 0.7063 & 0.7542 \\
**\%** & 0 & \textless{1}\% & 21.2\% & 26.2\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Grounding resistance of the deep-well electrode in different conditions.  
**Author Contributions:** Conceptualization, G.D. and E.F.; methodology, G.D., E.F. and G.A.; software, G.A. and J.M.; validation, G.D. and J.M.; formal analysis, G.D. and E.F.; investigation, G.D., E.F. and G.A.; data curation, J.M.; writing--original draft preparation, G.D. and E.F.; writing--review and editing, G.D., E.F., G.A. and J.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The authors would like to thank the Department of Electrical Engineering, Applied Mathematics and Applied Physics of the Escuela Tecnica Superior de Ingenieria y Diseno Industrial (ETSIDI) at Universidad Politecnica de Madrid (UPM) for their support to the undertaking of the research summarized here. Furthermore, the authors appreciate the collaboration with the firm INGESCO Lightning Solutions at Terrassa, Barcelona (Spain), for the technical support and data transfer from real cases studied in this work.  
The authors declare no conflict of interest.