Calculation of Grounding Grids Parameter at

Arbitrary Geometry
Carlos L. B. Silva, Thyago G. Pires, Wesley P. Calixto, Diogo N. Oliveira, Luis A. P. Souza and
Antonio M. Silva Filho
 Some results of grounding grids will be presented in
Abstract—This paper deals with the computation of ground standard formats, which are compared with traditional
resistance, surface voltage, touch voltage and step voltage, to methods. Results of a ground grid of unconventional geometry
mesh with horizontal wires arranged in different angles. The are also presented.
computer program implemented used in the mathematical
modeling is based on the method proposed by Heppe, which
allows obtaining the grounding parameters for homogeneous soil II. METHODOLOGY
and soil stratified in two layers. The results obtained with the The grid conductors are conceptually divided in rectilinear
proposed method will be compared with other methods in segments in order to discretize the system. The accuracy of the
literature. Also will be presented the results of a grounding grid
modeling is associated with the number of segments used. The
using wires at various angles.
greater the number of segments, the more precise is the
Index Terms— Grounding grids parameters, Heppe, soil modeling.
stratified in two layers. In each segment, it is considered that the distribution of
leakage current is constant throughout its length, but distinct
I. INTRODUCTION from segment to segment. It is assumed that all segments have

T HE study and analysis of grounding grids brings great the same voltage, which is equal to the ground potential rise
concern to engineers, as is the initial step in the process of (GPR).
building a substation. The main purpose of the grounding grid After the division, the leakage current of each segment and
design is to keep the step voltages, touch and electrical GPR are calculated. Then, the leakage current is used to
resistance to earth within tolerable limits [1]. calculate the ground resistance and the voltage at the ground
The classic method of grounding grid design [2] is a method surface at any desired point. To find the leakage current (i) in
that does not require computing resources and its intended to each segment the linear equation shown in (1) must be solved.
be easy to use. However, it has some limitations for Where m is the number of segments.
heterogeneous soil, to the analysis of potential on the ground’s
surface and the geometry of the ground grid. It can only be R11  i1  R12  i2  R13  i3    R1m  im  v1
used in cases where the wires are equidistant and in grounding R21  i1  R22  i2  R23  i3    R2 m  im  v2
grids with the following shapes: square, rectangular, L-shape
R31  i1  R32  i2  R33  i3    R3m  im  v3 (1)
and T-shape.
The geometry of the grounding grid depends on the area of 
the substation [3] and several studies prove a greater Rm1  i1  Rm 2  i2  Rm 3  i3    Rmm  im  vm 
effectiveness of the unequally spaced grounding grids as
regards the trend the touch voltages [4]. The above system can be written in matrix form as:
The methodology used in this paper to obtain the ground
resistance and the potential on the soil surface is based on  R11 R12 R13  R1m   i1   v1 
Heppe [5] using the method of images and the average R
potential method. The examples shown in [5] used only grids  21 R22 R23  R2 m   i2   v2 
containing conductors placed in parallel and perpendicular to  R31 R32 R33  R3m    i3    v3  
each other, deployed on homogeneous soil. However, our      
          
method enables the use of meshes in any relative positions  Rm1 Rm 2 Rm 3  Rmm  im  vm 
with conductors placed in soil stratified in two layers. 
The computer program was developed to implement the
mathematical model and allows the calculation of the The total current injected into the grid ( i g ) is equal to the
grounding potential rise, the potential on the soil surface and sum of leakage current of all segments, as shown in (3).
the ground resistance. The touch voltages and the step
voltages obtained from de surface potential. m
  ik  ig  
k 1

TRANSACTIONS ON ENVIRONMENT AND ELECTRICAL ENGINEERING ISSN 2450-5730 Vol 2, No 2 (2017)

© Carlos L. B. Silva, Thyago G. Pires, Wesley P. Calixto, Diogo N. Oliveira, Luis A. P. Souza and Antonio M. Silva Filho
 Appending (3) in (2), we have: Where K is the reflection factor.

 R11 R12 R13  R1m  1  i1   0   2  1 

R   K 
 21 R22 R23  R2 m  1  i2   0   2  1
 R31 R32 R33  R3 m  1  i3   0 
     The term M is given by (8), for 0     .
           
 R m1 Rm 2 Rm 3  Rmm  1  im   0 
       BF  B ' F   
 1 1 1  1 0  GPR  i g 
 M (CG )  CB  ln    CA  ln  AF  A' F 
 BE  B ' E   AE  A' E 
   
Thus the GPR becomes a system variable, because the total  BF  F ' B   
GF  ln    GE  ln  BE  E ' B   
current injected into the grid is usually a project information  AF  F ' A   AE ' E ' A 
   
and not the potential of electrodes. CG  
Next, computation of mutual and self-resistance of (4), the
sin  
ground resistance and voltages will be explained. All terms are
calculated for each individually segment, without any
symmetry of the grid as used in [5]. To calculate the mutual
The term  is the following equation:
resistance and the voltage at the ground surface the method of
images is used.  CG CB GF  sin  
  tan 1    
 BF  tan  CG BF 
A. Mutual Resistance
 CG CB GE  sin  
The mutual resistance (Rjk) is the ratio of the voltage  tan 1    
produced on the segment k by leakage current of segment j.  BE  tan  CG BE 

The symmetry of mutual resistance allows. The self-resistance  CG CA GF  sin  
 tan 1    
(Rjj) is the ratio between the voltages produced on the segment  AF  tan  CG AF 
by its own leakage current.  CG CA GE  sin  
Considering a soil composed of two layers with the upper  tan 1    
 AE  tan  CG AE 
layer having resistivity ρ1 and depth H, and lower layer having
resistivity ρ2 and extending to a great depth. The mutual
In the case of parallel segments, when θ decrease towards
resistance between a segment j and a segment k, and their
zero, the term CG.Ω /sin θ approaches BE+AF-BF-AE.
images, buried at the same depth (D) in the upper layer of soil
To compute the self-resistance a hypothetical segment
is given by (5) and in the bottom layer is given by (6).
parallel and identical to the original segment separated by a
Considering a soil composed of two layers with the upper
distance equal to the radius of the conductor is considered.
layer having resistivity ρ1 and depth H, and lower layer having
resistivity ρ2 and extending to a great depth. The mutual B. Ground Resistance
resistance between a segment j and a segment k, and their The ground resistance (Rg) is the ratio between the GPR,
images, buried at the same depth (D) in the upper layer of soil computation with (4), and the total current injected into the
is given by (5) and in the bottom layer is given by (6). grid.
Fig. 1 is the corresponding diagram to the terms of (8) and
(9). The images of segment are in different planes. The point Rg  GPR it 
C is in the same plane of segment AB and point G is in the 
same plane of segment EF.
C. Voltage on Soil Surface
1 Once the leakage currents in each segment is found, the
R jk  
4    L j  Lk voltage at a point on the soil surface due to the contribution of
 n a leakage current of a segment located in a upper layer is
 K M (2  n  H )  M 2  n  H  2  D   
 n 0

 F
 K  M 2  n  H   M (2  n  H  2  D)
n
B

n 1 
E
2 
R jk   M 0  K  M 2  H  2  D  
4    L j  Lk  A
y


1  K   K

2 n
M 2  n  H  2  D  C,G
 x
n 0   A E
F
B
Lj
Fig. 1. Angled segments.
calculated by (11) and of a segment located in a bottom layer with the method presented in this paper. The values in
is calculated by (12). parentheses are the percentage differences from the values
calculated by VCM [15].
i  1    x2  y2  D2  x  The grounding grid features used as program inputs are:
V  ln  
2    L   ( x  L) 2  y 2  D 2  x  L 
 
d = 0.01 m (diameter of the conductor)

D = 0.5 m (depth of burial)
   x 2  y 2  2  n  H  D   x
2 
K n
 ln  

ρ = 100 Ωm (soil resistivity)
  ( x  L) 2  y 2  2  n  H  D 2  x  L 
n 1
  
TABLE I
 x 2  y 2  2  n  H  D   x
2  
ln    GROUND RESISTANCE
 ( x  L) 2  y 2  2  n  H  D 2  x  L   Square Rectangular
  
  (20mx20m) (40mx10m)
Method
4 16 4 16
meshes meshes meshes meshes
 
ρ1  i  (1  K )   x p  L   y 2p  D 2  L  x p 
2
Dwight
2.2156 2.2156 2.2156 2.2156
V  ln    (15.9%) (6.4%) (6.8%) (3.2%)
2π  L   x 2p  y 2p  D 2  x p
   3.0489 2.7156 2.9848 2.6918
 Laurent
  
 x  L
2 
 y  2  n  H  D   L  x p  
2 2 (15.7%) (14.7%) (25.5%) (25.4%)
 K n  ln 
p p

    2.9570 2.6236 2.8929 2.5998

x 2p  y 2p  2  n  H  D   x p
2
  Sverak
n 1
   (12.2%) (10.8%) (21.6%) (21.1%)
2.8084 2.6035 2.4690 2.3211
Schwarz
Therefore, the voltage at a point on the soil surface is (6.6%) (10.0%) (3.8%) (8.15%)
calculated by superposition, by the sum of the contribution of 3.6367 3.1491
Nahman - -
all segments. (38.1%) (33.0%)
4.8017 3.2621
Chow - -
(82.3%) (37.8%)
D. Touch, Mesh and Step Voltages
2.6269 2.3631 2.2734 2.0795
With the surface voltages, the other voltages can be BEM
(0.3%) (0.2%) (4.4%) (3.1%)
determined. The touch voltages is the potential difference
VCM 2.6343 2.3669 2.3784 2.1461
between the GPR of a ground grid and the surface potential at
the point where a person could be standing while at the same
time having a hand in contact with a grounded structure. B. Case Study 2
Furthermore, the mesh voltage is the maximum touch voltage This case study compares VCM with traditional method [6]
within a mesh of ground grid. Moreover, the step voltage is for two grids in a soil stratified in two layers, rectangular grid
the difference in surface potential that could be experienced by and L-shape grid. To calculate the classic method was used the
a person a distance of 1m with the feet without contacting any methodology of [13] to find the apparent resistivity. The
grounded object. features of the soil and of two ground grids used as program
inputs are:
III. RESULTS ρ1 = 200 Ωm (upper layer resistivity)
Three case studies are presented. The case studies 1 and 2 ρ2 = 400 Ωm (bottom layer resistivity)
perform the validation of the proposed method by comparing H = 8 m (depth of the upper layer)
VCM with traditional methods. Case study 1 compare the D = 0.5 m (depth of burial of ground grid)
values of the ground resistance of the grids with square mesh d = 5 mm (wire diameter)
by other methods. Case study 2 compare the ground ∆L = 5 m (distance between parallel conductors)
resistance, mesh voltage and step voltage with the design ig = 1000 A (total current injected into the grid)
procedure in [6]. Finally, case study 3 show the results for an Fig. 2 show a rectangular grid with dimensions 35m x 20m
unconventional grid. containing 28 meshes. The apparent resistivity seen by grid is
253.33Ωm. For the classic method the ground resistance was
4.87Ω, the mesh voltage (Vm) was 1019.95V and the step
voltage (Vs) was 687.77V. With VCM the ground resistance
A. Case Study 1 was 4.66 Ω, the mesh voltage was 927.92V in the corners, the
Table I shows the ground resistance values for a square grid maximum step voltage within the grid was 250.32V and the
(20m x 20m) and a rectangular grid (40m x 10m) in step voltage in the corners was 509.82V.
homogeneous soil. The ground resistance values are calculated Assuming a T-shaped grid as show in Fig. 3 with 18 meshes
using the simplified calculations provided in the ANSI-IEEE and dimensions 30m x 25m, the apparent resistivity seen by
Std. 80/2013: Dwight [7], Laurent and Nieman [6], Sverak [8] grid is 246.67Ωm. According IEEE Std. 80-2013 [6], the
and Schwarz [9]. In addition to the calculations presented by ground resistance was 6.00 Ω, the mesh voltage was 1278.20V
Nahman [10] and Chow [11]. The BEM method (Boundary and the step voltage was 830.82V. Calculating by VCM the
Element Method) is obtained from [12] and VCM is computed
ground resistance was 5.40 Ω, the mesh voltage was 1168.56V d = 8.75 mm (wire diameter)
and the step voltage was 330.63V within the grid and ig = 10000 A (total current injected into the grid)
639.80V in the top corners.

Fig. 2. Rectangular grid – 35m x 20m.

Fig. 7 Grounding grid with different spacing.

Figure 8 shows the potential on the soil surface profile

obtained.

Fig. 3 T-shape grid – 30m x 25m.

Table II show the results found to the grids above with the
difference of VCM to ANSI-IEEE Std. 80/2013.
TABLE II
PARAMETERS WITH IEEE STD. 80 AND VCM
Method
Grid Data Difference
Std. 80 VCM
Fig. 8 Profiles on the soil surface, results obtained by the proposed method.
Rg (Ω) 4.87 4.66 4.31%
Rectangular
The potential on the soil surface with geographic location of
Vm (V) 1019.95 927.92 9.02% coordinates x = 1.25m and y = 2.0m, obtained in the work of
35mx20m
Vs (V) 687.77 509.82 25.87% Huang [1] is 10.37kV while by the proposed method is
10.40kV. The result obtained for the soil surface potential with
Rg (Ω) 6.00 5.40 10.00% geographic location of coordinates x = 52.5m and y = 32.5m
T-Shape in the work of Huang [1] is 10.23kV and by the proposed
Vm (V) 1278.20 1168.56 8.58%
30mx25m method is 10.34kV.
Vs (V) 830.82 639.80 22.99% Figure 9 shows the distribution of the equipotential through
isolines. Potential peaks observed at the intersections of the
C. Case Study 3 electrodes, except at the border of the grid where potential
Figure 7 show a grounding grid of 120m x 80m, with reduction occurs. The maximum potential at the soil surface
variable spacing between the conductor. The profiles of the occurs in coordinate x = 60m and y = 40m, with a value of
potential at the soil surface in the lines indicated by A,B,C and 11.33kV.
D obtained by the method proposed in this work are compared
with the results of Huang [1]. The following input data used:

ρa = 200 Ωm (apparent ground resistivity)

D = 0.6 m (depth of burial of ground grid)
Fig. 9 Equipotential distributed on the soil surface.

The maximum surface potential obtained at the central Fig. 11 Surface Potential.
point of the grid due to the symmetrical distribution of the
electrodes around the point. All voltages calculated for points on the surface located
within the perimeter of the mesh. The value obtained for the
Case Study 4
ground resistance was 8.0Ω, for mesh voltage was 2075.98V
It presented a grid composed of conductors at different at the coordinates x = 0m and y = 10m; and the maximum step
angles and different lengths as show in the Fig. 10. The grid voltage was 925.04V between the point of coordinates
has 16 meters in the x-axis and 17 meters in the y-axis [14]. x1 = 16m and y1 = 17m, and the point of coordinates
The following input data were used: x2 = 15.36m and y2 = 16.23m. The GPR was 9595.60V and
the maximum surface voltage (Vsurf) is 9245.45V at the
ρ1 = 200 Ωm (upper layer resistivity) coordinates x = 9.8m e y = 10.0m.
ρ2 = 400 Ωm (bottom layer resistivity)
H = 8 m (depth of the upper layer) D. Study Case 5
D = 0.5 m (depth of burial of ground grid) The study case presented to verify the influence of the depth
d = 5 mm (wire diameter) of the grounding grid, the ground grid used shown in Figure
ig = 1200 A (total current injected into the grid) 10, and the depth varied from 0.5m to 3.5m. The potential
profiles on the surface were obtained from the cut at y = 11m
8.92 7.08 in the grounding grid shown in Figure 10. Table III show the
values obtained for the resistance of the grounding grid, GPR,
the maximum potential at the ground surface, the touch
voltage and the maximum step voltage for different depths of
5.00

the ground grid. The following input data used:

2.00

ρ1 = 200 Ωm (upper layer resistivity)

ρ2 = 400 Ωm (bottom layer resistivity)
17.00

H = 2 m (depth of the upper layer)

D = 0.5m – 3.5m (depth of burial of ground grid)
d = 5 mm (wire diameter)
ig = 1200 A (total current injected into the grid)
y
The Table IV show the coordinate maximum of the surface
0 x potential and step voltage.
Fig.12 and Fig.13 shows the elevation of the ground
6.00
resistance values and the GPR of the ground grid, which are
directly proportional.
Fig. 10 Unconventional grid.

Fig. 11 shows the voltage profile in three dimensions and

contour of the soil surface potential inside the perimeter of the
ground grid.
 TABLE III
GROUNDING GRID PARAMETERS AT DIFFERENT DEPTHS
D(m) Rg (Ω) GPR(V) Vs (V) Vtouch (V) Vstep (V)
0.5 9.93 11916.08 11626.38 2377.19 1103.1
1.0 9.64 11565.18 11223.86 2771.30 984.57
1.5 9.48 11371.17 10984.02 3040.44 847.52
1.6 9.46 11350.29 10948.32 3094.98 826.73
1.7 9.45 11337.78 10917.37 3154.29 808.28
1.8 9.45 11337.74 10892.96 3223.90 792.26
1.9 9.47 11363.41 10881.04 3320.62 779.05
2.0 9.68 11620.32 10965.79 3687.02 773.77
2.1 16.89 20270.30 11231.81 12577.11 762.37
2.2 16.85 20219.40 11140.67 12611.32 739.89
2.3 16.78 20138.91 11047.19 12610.62 718.45
2.4 16.71 20050.57 10953.58 12598.41 698.02
2.5 16.63 19960.81 10860.20 12581.82 678.55
3.0 16.29 19549.73 10396.24 12501.50 593.77
3.5 16.02 19224.64 9937.84 12459.86 525.35

TABLE IV
COORDINATE MAXIMUM OF THE SURFACE POTENTIAL AND STEP VOLTAGE.
Parameter Coordinates
x = 16.0m and y = 17.0m Fig. 13 GPR versus depth (D).
Vstep
x = 12.4m and y = 16.3m
Vs x = 0m and y = 20m

Fig. 14 Superficial potential (Vs) versus depth (D).

Fig. 12 Resistance (Rg) versus depth (D).

The boundary between the first and second soil layers

occurs exactly in D = 2m. The potential on the soil surface
increases in the depths just below to this border (Figure 14).
Figure 15 shows the increase of the touch voltage near the
boundary between the soil layers, since the grounding grid
when positioned in the second soil layer, which has a higher
resistivity (400Ω.m) in relation to the first layer that has lower
resistivity (200Ω.m), produces higher touch potential.
Figure 16 shows that the pitch voltage decreases smoothly
with increasing depth, having a level in the depths near the
boundary between the layers.

Fig. 15 Touch potential (Vt) versus depth (D).

 V. ACKNOWLEDGEMENT

The authors thank the National Counsel of Technological

and Scientific Development (CNPq), Coordination for the
Improvement of Higher Level Personnel (CAPES) and the
Research Support Foundation for the State of Goias (FAPEG)
for financial assistance to this research.

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[14] Pires, Thyago G. ; Nerys, Jose W. L. ; Silva, Carlos L. B. ; Oliveira,
voltage of grids composed by horizontal wire electrodes in Diogo N. ; Filho, Antonio M. Silva ; Calixto, Wesley P. ; Alves, Aylton
shapes that are more complex. Wire segments can have any J., Computation of resistance and potential of grounding grids in any
position or displacement among them. geometry. In: 2016 IEEE 16th International Conference on
Environment and Electrical Engineering (EEEIC), 2016, Florence.
The difference between the results obtained with this 2016 IEEE 16th International Conference on Environment and
method and those of the ANSI-IEEE Std. 80/2013 for the Electrical Engineering (EEEIC), 2016.
grounding resistance was up to 25.5%. For grid voltage was [15] Pires, Thyago G. ; Silva, Carlos L. B. ; Oliveira, Diogo N. ;Nerys, Jose
W. L. ; Alves, Aylton J.; Calixto, Wesley P. Computation of grounding
up to 16.6% and 41.9% for step voltage. The individual grids parameter on unconventional geometry. In: 2015 CHILEAN
calculation of the leakage current for each segment leads to a Conference on Electrical, Electronics Engineering, Information and
greater precision of the method. Communication Technologies (CHILECON), 2015, Santiago. 2015.
This method also proves to be useful for allowing a precise
analysis of the voltage on the soil surface, it is possible to
calculate the voltage at any desired point. Also, the detailed
study of any grounding grid at any depth in the soil is possible.