# Optimal Design of the Vertical Earthing with Electrodes Arranged in Line
## 3 Results
### Minimum Zones  
The minimum zones, corresponding to the optimum criteria for VEGL, are represented in the (\(\ell\), a) plane, based on the data from Table 1. Thus, for each criterion, at least three of the minimum values where held, to which from one to three points with very close values were associated. The graph with the minimum zones is shown in Figure 3, where the minimum areas, corresponding to each optimal criterion, are identified by the notations introduced in their analytical definition: A\({}_{\rm P}\)--for the footprint area, V\({}_{\rm P}\)--for the VEGL total volume, A\({}_{\rm D}\)--for the VEGL dispersion surface area, M\({}_{\rm P}\)--for the metallic materials mass, and I\({}_{\rm P}\)--for the VEGL total investment cost. In order to simplify the identification of different solutions from Table 1, to which some points from Figure 3 are corresponding, it is proposed to identify the solution through the set of sizes (\(\ell\); n\({}_{\rm e}\); a), in this order. Thus, the solution representing the absolute minimum at the criterion of the footprint area, A\({}_{\rm P}\) = 1.99 m\({}^{2}\) (Table 1), has the coordinates (1.25; 9; 0.47). In the figure, the position of the absolute minimum, according to a specific criterion, is indicated by an arrow (Figure 3). As a general aspect, it can be observed that there is a bipolarization of the minimum areas, corresponding to the five optimal criteria: one at the small electrode lengths, \(\ell\) \(\in\) {0.5; 0.8} m, where the minimum areas for A\({}_{\rm D}\) and M\({}_{\rm P}\) criteria are positioned and the other at the smaller electrode distances, a \(\in\) {0.47-1.0} m, where the minimum areas for the A\({}_{\rm P}\), V\({}_{\rm P}\), and I\({}_{\rm P}\) criteria are positioned. In this latter situation, the common area of these three minimum zones, which is precisely represented by the V\({}_{\rm Pmin}\) area, defines a three-criteria minimum area. In addition, it should be noted that the solution (1.25; 9; 0.47) represents the absolute minimum for all these three minimum criteria, i.e., A\({}_{\rm P}\), V\({}_{\rm P}\), and I\({}_{\rm P}\). Figure 2: The graph of the function a(\(\ell\),ne), having as parameter the number of the electrodes n\({}_{\rm e}\), for \(\rho_{\rm P}\) = 100 \(\Omega\)m and R\({}_{\rm Pn}\) = 4 \(\Omega\). Regarding the minimum areas for the A\({}_{\rm D}\) and M\({}_{\rm P}\) criteria, even if they are located at the same abscissa from the lower limit of the range of the electrode length values, they are disjointed. The absolute minimum, according to A\({}_{\rm D}\) criterion, corresponds to the solution (0.5; 8; 1.70), while the one according to the M\({}_{\rm P}\) criterion corresponds to the solution (0.5; 2; 9.16). The latter appears as an extreme case, with only two electrodes of small length, \(\ell\) = 0.5 m, located at considerable distances, a = 9.16 m, so over 18 times the length of the electrodes. This case reveals a question regarding the validity ranges of mathematical models proposed for the dimensioning of different types and configurations of EG. By increasing the number of solutions included in the minimum areas, which means increasing the admitted deviation from the absolute minimum recorded in the A\({}_{\rm D}\) and M\({}_{\rm P}\) criteria, the two corresponding minimum areas generate an intersection, resulting in a bicriteria minimum area. The use of logarithmic scales for both variables, \(\ell\) and a, allowed a better highlighting of the minimum areas according to the A\({}_{\rm P}\), V\({}_{\rm P}\), and I\({}_{\rm P}\) criteria, located in the range of smaller values of the distance between adjacent electrodes, a \(\in\) [0.4-1] m.  
Finally, trying a comparison of the location of the minimum areas between the case of VEGL [12], it can be noticed a similar relative location of the minimum areas, even if the average coordinate values are different. Secondly, there is a significant reduction of the value ranges corresponding to the minimum areas according to the (0\(\ell\)) axis.