# An Estimator of the Resistance of Large Grounding Electrodes from Its Geometric Characterization
## 2 Basics of the Method
The result of the estimation was R = (0.112 \(\pm\) 0.014) \(\Omega\) while the theoretically calculated one was R = 0.105 \(\Omega\), which represents a percentage difference of less than 7% between them. However, the confidence interval of the estimate contained the theoretically calculated value. As a second test of the estimation formula Equation (4), a large square electrode as a regular grid of a 100 m side and composed by 840 rods of 5 m long and 4 mm radius, will be considered. The electrode is buried at 0.5 m in a homogeneous soil of \(\rho\) = 100 \(\Omega\)m resistivity, so that the theoretical calculation is not computationally very expensive, since it involves a considerable number of rods, the segmentation is set at 2 segments per rod, clearly poor but necessary if we want the theoretical calculation to be carried out in a reasonable time. This calculation results in R = 0.448 \(\Omega\). Estimation through Equation (4) requires the calculation of \(i_{\rm s}\) with N = 2/5 segments/m or any other segmentation and L = 4200 m. This is done in a simple and very quick way, resulting in \(i_{\rm s}=0.0064\). Finally, the estimation from the Equation (4) results in R = (0.391 \(\pm\) 0.126) \(\Omega\), which represents a percentage difference of around 13% between the representative values, although it is observed that the calculated value was within the confidence interval of the estimator. The difference was large, but taking into account that \(i_{\rm s}\) will not change in value if the segmentation is refined and that an increase in segmentation always produces a decrease in the theoretically calculated resistance, we could conclude that this percentage difference will decrease if a more precise theoretical calculation, with a greater segmentation, of the electrode resistance is carried out. Next, the estimate of the grounding resistance of a real operating electrode will be obtained from Equation (4). Specifically, the resistance of the grounding electrode of the Balaidos electrical substation located in Vigo, Galician Community (Spain), will be estimated. Figure 4 shows the geometric structure of the electrode, which is a non-regular grid formed by rods of variable length with a total conductor length of 289.77 m.  
The electrode is buried at a depth of 0.8 m and the parameter \(k_{\rm R}=\frac{\rm R}{\rho}\) is previously known. Its value is \(k_{\rm R}\) = 0.028 \(\Omega\)/\(\Omega\)m, which represents the grounding resistance of the electrode, buried in a homogeneous soil of resistivity \(\rho\) = 1\(\Omega\)m. The resistance estimated by Equation (4) for a soil of such a resistivity with a segmentation of N = 2.5 segments/m and a stretching index of \(i_{\rm s}\) = 0.030 resulted in a value of R = (0.021 \(\pm\) 0.005) \(\Omega\), an interval whose upper end almost reached the real value that the electrode would have. Figure 4: The Balaidos substation grounding grid. In summary, the estimation of R for a specific electrode requires knowledge of its total length L, and its precise geometric shape in order to calculate \(\mathrm{i_{s}}\), just the same information needed to theoretically calculate the resistance. With this information available, a suitable segmentation as a choice of N segments per meter of straight conductor can be carried out and thus determine the stretching index \(\mathrm{i_{s}}\) by simple calculation. Equation (4) was then applied and the estimate of R was thus obtained. As a final comment, the numerical values of the constants that appear in Equation (4) can be refined from the knowledge of the resistance of other electrodes than those used here for the fitting, which has led to Table 2. In this way, by having a larger number of data available to determine the fitting parameters of Equation (4), the confidence intervals could be reduced, gaining in the estimator precision.