Framework for optimal grounding system design concerning IEEE standard

Abstract

This paper investigates the optimal grounding grid design using artificial intelligence techniques based on the empirical formula for grounding resistance (R_g), touch and step voltages (E_t and E_s), which are addressed in IEEE Std. 80-2013/Cor. 1-2015. The objective function is formulated based on the grid conductors’ material cost and the installation cost. Particle swarm optimization (PSO) and genetic algorithm (GA) are individually utilized to search for and confirm the global best solution of minimizing the grounding system cost with considering all operation constraints including the safety criteria. A modified objective function is constructed based on self-adaptive penalization utilized to account for constraint violations during the optimization process. The selection strategy for modifying the local best positions of particles and the global best position of the swarm is proposed to enhance the ability of PSO for fast convergence. Results proved that either PSO or GA settles on the global best solution, successfully. To perform the space domain investigation, the optimal grounding grid design is implemented using a finite element method, where the COMSOL Multiphysics program is selected to verify the settled optimal global solution. Using COMSOL, the grounding resistance and safety criteria are evaluated over the grid diagonally. The COMSOL results ensure the operating constraints satisfaction of the optimal grounding grid design. The performance evaluation reveals that the grounding potential rise limit, the available grid area, and the number of vertical rods greatly affect the optimization of the grounding system design. The results also illustrate a great significant effect of the fault current and upper surface resistivity on the optimal grid design.

Appendix A

The grounding system parameter Eqs. (1) to (6) presented in Sect. 2 are as follows, where the complete description is reported in IEEE Std 80-2013/Cor 1-2015 [3]. The cross-sectional area A (mm²) of the conductors is calculated by [27];

$$ A = I_{\text{G}} /\sqrt {\frac{{{\text{TCAP}} \cdot 10^{ - 4} }}{{t_{\text{C}} \cdot \alpha_{\text{r}} \cdot \rho_{\text{r}} }}\ln \left( {\frac{{K_{0} + T_{\text{m}} }}{{K_{0} + T_{\text{a}} }}} \right)} $$

(A1)

where I_G is the maximum fault current in kA, T_a and T_m are the ambient and maximum temperatures in °C; respectively. The thermal coefficient of resistivity α_r and the resistivity of the ground conductors ρ_r are at the reference temperature T_r, K₀ is a constant equal to (1/α₀) or (1/α_r) at T_r and eventually, the TCAP refers to the thermal capacity per unit volume that is obtained from Table 1 in [3] in (J/cm³ °C). The I_G is defined by:

$$ I_{\text{G}} = D_{\text{f}} \times I_{\text{g}} $$

(A2)

where D_f is the decrement factor that determines the rms equivalent of the asymmetrical current wave at the entire duration of fault (t_f) that is given in s. The D_f can be computed by:

$$ D_{\text{f}} = \sqrt {1 + \frac{{T_{\text{a}} }}{{t_{\text{f}} }}\left( {1 + e^{{\frac{{ - 2t_{\text{f}} }}{{T_{\text{a}} }}}} } \right)} $$

(A3)

D_f can be calculated for specific X/R ratios and fault duration. T_a is the dc offset time constant in s (T_a = X/120πR for 60 Hz).

The effective rms value of the approximate asymmetrical current I_F for the entire duration of the fault in A is computed by:

$$ I_{\text{F}} = D_{\text{f}} \times I_{\text{f}} $$

(A4)

where I_f represents the rms symmetrical current which is 3I₀, where I₀ is the zero-sequence fault current in A.

The rms symmetrical current I_g that flows between the grounding grid and the surrounding earth is given by:

$$ I_{\text{g}} = S_{\text{f}} \times I_{\text{f}} $$

(A5)

where S_f is the fault current division factor which represents the relation between the symmetrical fault current to that portion of the fault current that flows between the grounding grid and the surrounding earth ($S_{\text{f}} = I_{\text{g}} /I_{\text{f}}$).

The difference between one-layer and two-layer soil is the resistivity. In case of two-layer soil, the apparent resistivity (ρ_a) of the two-layer soil is computed based on (ρ₁, ρ₂) as follows [15, 28]:

$$ \rho_{\text{a}} = \frac{{\rho_{1} }}{{\left[ {1 + \left( {\left( {{\raise0.7ex\hbox{${\rho_{1} }$} \!\mathord{\left/ {\vphantom {{\rho_{1} } {\rho_{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\rho_{2} }$}}} \right) - 1} \right)\left( {1 - e^{{\frac{1}{{K\left( {d_{0} + 2h} \right)}}}} } \right)} \right]}}(\rho_{2} < \rho_{1} ) $$

(A6)

$$ \rho_{\text{a}} = \rho_{1} \times \left[ {1 + \left( {\left( {{\raise0.7ex\hbox{${\rho_{1} }$} \!\mathord{\left/ {\vphantom {{\rho_{1} } {\rho_{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\rho_{2} }$}}} \right) - 1} \right)\left( {1 - e^{{\frac{1}{{K\left( {d_{0} + 2h} \right)}}}} } \right)} \right](\rho_{1} < \rho_{2} ) $$

(A7)

where K is the reflection factor, h is the grid depth, d₀ is the thickness of the top layer, and ρ₁ and ρ₂ refer to the resistivity of the upper and bottom layer, respectively.

The parameters of E_m defined by (1) are formulated as follows. Sverak [29] proposed the geometrical factor K_m;

$$ K_{\text{m}} = \frac{1}{2\pi }\left( {\ln \left( {\frac{{D^{2} }}{{16 \cdot h \cdot d_{\text{c}} }} + \frac{{\left( {D + 2h} \right)^{2} }}{{8 \cdot D \cdot d_{\text{c}} }} - \frac{h}{{4d_{\text{c}} }}} \right) + \frac{{K_{\text{ii}} }}{{K_{\text{h}} }}\left( {\frac{8}{{\pi \left( {2n - 1} \right)}}} \right)} \right) $$

(A8)

where D is the spacing between parallel conductors (m), K_ii is the corrective weighting factor that adjusts for the effects of inner conductors on the corner mesh, K_h is the corrective weighting factor that emphasizes the effects of the grid depth and n is a geometric factor composed of factors n_a, n_b, n_c, and n_d. If the vertical rods are at the perimeter or at the corner of the grid, then K_ii is equal 1. Otherwise, it is computed by:

$$ K_{\text{ii}} = \frac{1}{{\left( {2 \cdot n} \right)^{2/n} }} $$

(A9)

The factor K_h is calculated by:

$$ K_{\text{h}} = \sqrt {1 + \frac{h}{{h_{0} }}} $$

(A10)

where h₀ is the reference depth and equal to 1. In [3, 30], the effective number of parallel conductors n is:

$$ n = n_{\text{a}} \cdot n_{\text{b}} \cdot n_{\text{c}} \cdot n_{\text{d}} $$

(A11)

where $n_{\text{a}} = \frac{{2 \cdot L_{\text{C}} }}{{L_{\text{p}} }}$, n_b = 1 for square grids, n_c = 1 for square and rectangle grids, and n_d = 1 for square, rectangle and L-shaped grids. L_p is the peripheral length of the grid. If the conditions of the grounding grid differ, the next relationships $n_{\text{b}} = \frac{{L_{\text{p}} }}{4 \cdot \sqrt A }$, $n_{\text{c}} = \left[ {\frac{{L_{x} \cdot L_{y} }}{A}} \right]^{{\frac{0.7A}{{L_{x} \cdot L_{y} }}}}$, and $n_{\text{d}} = \frac{{D_{\text{m}} }}{{\sqrt {L_{x} .L_{y} } }}$ are utilized, where L_x and L_y are the maximum length of the side length of the grid in x and y directions, respectively, and D_m is the maximum distance between any two points on the grid in m, i.e., the length of the diagonal of the grid. The irregularity factor, K_i, is determined by:

$$ K_{\text{i}} = 0.644 + 0.148 \cdot n $$

(A12)

If there were a few vertical rods at locations neither the corner nor the perimeter, the effective conductor burial length L_m can be calculated as follows,

$$ L_{\text{m}} = L_{\text{C}} + L_{\text{r}} $$

(A13)

where, L_C is the total length of grid conductor. Otherwise, the L_m was computed as follows;

$$ L_{\text{m}} = L_{\text{C}} + \left[ {1.55 + 1.22\left( {\frac{{L_{\rm{r}}^{ '} }}{{\sqrt {L_{x}^{2} + L_{y}^{2} } }}} \right)} \right]L_{\rm{r}} $$

(A14)

where, $L_{\rm{r}}^{\prime }$ is the length of each ground rod in m.

The second parameter for the safety criterion is the maximum step voltage (E_s) that is defined in (2), where the effective buried length, L_s, is computed by;

$$ L_{\text{s}} = 0.75 \cdot L_{\text{C}} + 0.85 \cdot L_{\text{r}} $$

(A15)

where L_r is the total length of all vertical rods connected to the grid and it is

$$ L_{\text{r}} = N_{\text{r}} \times L_{\text{r}}^{\prime } $$

(A16)

where N_r is the number of vertical rods and $L_{\text{r}}^{\prime }$ is the vertical rod length.

The factor K_s for calculating the step voltage can be determined by:

$$ K_{\text{s}} = \frac{1}{\pi }\left[ {\frac{1}{2 \cdot h} + \frac{1}{D + h} + \frac{1}{D}\left( {1 - 0.5^{n - 2} } \right)} \right] $$

(A17)

The above equation is valid when the grid depth h subjects to the following constraint:

$$ 0.25\,{\text{m}} \le h \le 2.5\,{\text{m}} $$

(A18)

For the safe limits of E_t and E_s defined in (3) and (4), the factor C_s is obtained by the following formula:

$$ C_{\text{s}} = 1 - \frac{{0.09\left( {1 - \frac{\rho }{{\rho_{\text{s}} }}} \right)}}{{2h_{\text{s}} + 0.09}} $$

(A19)

where h_s is the thickness of the protective surface and if there is no protective surface, then, C_s = 1 and ρ = ρ_s.

As defined in (6), the grounding resistance is a function of R₁, R₂ and R_m that are formulated as follows:

$$ R_{1} = \frac{\rho }{{\pi L_{\text{C}} }}\left[ {\ln \left( {\frac{{2L_{\text{C}} }}{{a^{\prime}}}} \right) + \frac{{k_{1} \times L_{\text{C}} }}{\surd A} - k_{2} } \right] $$

(A20)

$$ a^{\prime} = \left\{ {\begin{array}{*{20}l} {\sqrt {d_{\text{c}} \times 2h} } \hfill &\quad {{\text{conductors }}\,{\text{buried}}\,{\text{at}}\,{\text{depth}}\,\,h} \hfill \\ {d_{\text{c}} } \hfill &\quad {{\text{conductors}}\,{\text{on}}\,{\text{the}}\,{\text{earth}}\,{\text{surface}}} \hfill \\ \end{array} } \right. $$

(A21)

where ρ is the soil resistivity, L_C is the total length of grid conductors, A is the grid area, d_c is the grid conductor diameter, h is the grid depth, and the coefficients k₁ and k₂ are determined in [3].

$$ R_{2} = \frac{\rho }{{\pi N_{\text{r}} \cdot L_{\text{r}} }}\left[ {\ln \left( {\frac{{2L_{\text{r}} }}{{0.5 \cdot d_{\text{r}} }}} \right) - 1 + \frac{{2k_{1} \times L_{\text{r}} }}{\surd A} \times \left( {\sqrt {N_{\text{r}} } - 1} \right)^{2} } \right] $$

(A22)

where L_r is the total length of rods, d_r is the rods diameter, and N_r is the rods number.

The value of the mutual resistance between the ground resistance of the grid conductors R₁ and the ground resistance due to the effect of vertical rods R₂ is obtained by:

$$ R_{\text{m}} = \frac{\rho }{{\pi L_{\text{C}} }}\left[ {\ln \left( {\frac{{2L_{\text{C}} }}{{L_{\text{r}} }}} \right) + \frac{{k_{1} \times L_{\text{C}} }}{\surd A} - k_{2} + 1} \right] $$

(A23)