# Optimal Design of the Vertical Earthing with Electrodes Arranged in Line
## 4 Discussion  
VEGL sizing by applying optimal criteria offers the benefits of a simpler expression for the coefficient of use (4) and the fact that the number of the electrodes can be any natural number, compared to VEGL addressed in [15; 16; 17]. The calculation methodology, developed in relation to VEGR was adapted and applied in the same way to VEGL, for the electrode numbers corresponding to the range n\({}_{\rm e}\)\(\in\) {2, 3, 4,..., 12}, and the following values were adopted for the main parameters: d = 0.04 m, \(\rho_{\rm p}\) = 100 \(\Omega\)m, q = 0.8 m, and R\({}_{\rm Pn}\) = 4 \(\Omega\). The specificity of the calculation methodology consists of the preliminary calculation of the resistance to earth of one electrode r\({}_{\rm pv}\) (2), for each of the combinations: electrode length-electrode number-distance between adjacent electrodes. The range for the electrode length comprised 10 values (15), and the number of electrodes corresponded to 11 consecutive values (as mentioned above) so that the matrix with the total number  
Figure 3: The minimum areas corresponding to the optimal criteria in the (\(\ell\), a) plane, for VEGL. of technically appropriate solutions is 110. Therefore, the optimal solution of each optimal criterion was identified among 110 technically appropriate solutions. It would be expected that the optimal solutions are located at substantially equal values of the resistances to earth of all electrodes, and for the horizontal strip, that is for \(\rm R_{Pe}=R_{Pc}\approx 8\ \Omega\). However, the absolute minimum solution, for three criteria (\(\rm A_{P}\), \(\rm V_{P}\), and \(\rm I_{P}\)), having the coordinates (1.25; 9; 0.47), occurs in the conditions in which the two components of the total resistance to earth are very different, \(\rm R_{Pe}=5.56\ \Omega\) and \(\rm R_{Pc}=13.19\ \Omega\). The hypothesis stated above is fulfilled for the solution (3.5; 4; 2.11), when the values \(\rm R_{Pe}=8.09\ \Omega\) and \(\rm R_{Pc}=7.91\ \Omega\) are recorded, but this variant does not fit into the five criteria optimal areas. In contrast, the solution (0.8; 12; 0.72), for which \(\rm R_{Pe}=8.22\ \Omega\) and \(\rm R_{Pc}=7.77\ \Omega\), is part of the minimum bicriteria area, \(\rm A_{P}\) and \(\rm I_{P}\). Graphs, such as (\(\ell\), \(\rm n_{e}\)) (Figure 2), make it possible to directly determine the distance between adjacent electrodes, for any electrode lengths. Regarding the minimum areas, the essential aspect highlighted by the present research is their polarization. At one side is a three criteria minimum area, according to the \(\rm A_{P}\), \(\rm V_{P}\), and \(\rm I_{P}\) criteria, where the absolute minimum solution is located at the coordinates \(\ell=1.25\ \rm m\), \(\rm n_{e}=9\), and a = 2.11 m. On the other side, at the lower values of the length of the electrodes, \(\ell\in[0.5\)-0.8] m, the minimum areas corresponding to the other two criteria, \(\rm A_{D}\) and \(\rm M_{P}\), are in the vicinity. Comparing a solution proposed as \"a good practice\" [6], having the data \(\ell=3\ \rm m\), \(\rm n_{e}=5\), a = 6 m, and the other sizes identical to those considered when calculating the data in Table 1, we can find the followings: it occupies an area 90.5 times greater than the minimum, has a volume of 167.6 times larger, uses 4.9 times more metal, and costs 12.3 times more than the minimum variants, corresponding to the respective criteria. Conceptualization, S.G.P. and V.M. ; methodology, V.M. and H.G.B. ; validation, S.G.P., V.M., and C.C. ; formal analysis, H.G.B. and I.B. ; writing--original draft preparation, S.G.P. and V.M. ; writing--review and editing, C.C. and I.B. ; visualization, S.G.P. and H.G.B. ; supervision, V.M. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The results presented in this paper were obtained in the framework of the GNaC 2018 ARUT grant \"Consequences of Power Quality Issues for Electromechanical Receivers\", research Contract no.