# Optimal Design of the Vertical Earthing with Electrodes Arranged in Line
## 1 Introduction  
From the outset, it is emphasized that all computational relations included in the mathematical model of the system of earthing (EG) used in the present research are based on current international standards [1; 2; 3; 4; 5; 6; 7; 8]. They have been used by specialists for over 40 years and are verified in practice. The calculation methods from the normative documents have been resumed and exemplified in many specialized manuals [9; 10; 11; 12; 13]. It is necessary to specify that the proposed developments, simplifications, and refinements for the updated design of the EG [14; 15; 16; 17] should not be interpreted as a reconsideration of their calculation methodology. Until the optimum criteria appeared, the EG designers first selected an electrode length suitable for the respective soil [1; 3; 5; 6; 7; 8], selected several values for the distance between adjacent electrodes, and then determined the number of electrodes to achieve the designed earth resistance. Usually, the distance between adjacent electrodes was chosen 1-3 times greater than the length of the electrodes, as indicated in the standard [5; 6], in order to obtain a higher value of the coefficient of use. Fromseveral calculated variants, the designer selected the most convenient one (e.g., a lower number of electrodes, lower earth resistance). For example, the standard [1] explicitly recommends the following: \"Pipes may be of cast iron of not less than 100 mm diameter, 2.5 m to 3 m long and 13 mm thick\". Examples, when given, are conducted at most until the design value resistance is obtained [1; 5; 6; 11; 12], and optimum selection criteria are not indicated. Previous publications concerning earthing optimum design refer to structural modifications of the earth grids or systems \"in order to equalize the leakage current distribution and the potential of ground surface\" [18] or to minimize the shielding effect [19]. These contributions do not apply optimal criteria to the calculation of the earthing system itself. In the absence of optimal criteria, each earthing system designer has proposed solutions to achieve the designed value resistance to earth. The evaluation of the proposed solution was also made according to the value of the coefficient of use, sufficiently large when the distance between the electrodes is 1-3 times greater than the length of the earth electrodes. The optimal criteria proposed and applied clearly show that there are solutions that minimize the surface or volume occupied by the earthing system, the amount of metal used, or the cost of the earthing system. Summarizing the achievements of the previous research on the application of optimum criteria in the design of EG [14; 15; 16; 17], the following important steps can be mentioned:  
* Defining five optimal criteria as follows: the footprint area (noted A\({}_{\text{P}}\)), the EG total volume (V\({}_{\text{P}}\)), the total dispersion surface (A\({}_{\text{D}}\)), the conducting materials mass (M\({}_{\text{P}}\)), and the investment costs (I\({}_{\text{P}}\)). * Identifying an analytical expression for the function f(n\({}_{\text{e}}\)), which introduces the dependence of the coefficient of use u\({}_{\text{P}}\) versus the number of the electrodes n\({}_{\text{e}}\), given in a tabular form in [5], and for which the following analytical expression was determined in [15]: \[\text{f(n}_{\text{e}})\;=\;1.45\cdot\text{lgn}_{\text{e}}\,+\,0.1.\] (1)
* From the eight characteristic variables of the earthing system with vertical electrodes (VEG), the length and number of electrodes are considered independent variables, and the distance between adjacent electrodes is considered as an intrinsic variable, determined by the condition of the design value resistance. * Creating an Excel program, which allows determining the distance between adjacent electrodes for all combinations of lengths and numbers of electrodes, achieving the design value resistance.