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47
Analytical Methods for Earth Surface Potential
Calculation for Grounding Grids
Sherif S. M. Ghoneim1,3, 2,3Kamel A. Shoush
1
Suez University, Suez, Egypt, 2Al-Azhar University, Cairo, Egypt
3
Taif University, 21974, KSA,
ghoneim_sherif2003@yahoo.com, kamel.shoush@yahoo.com
1
Abstract— The grounding resistance (Rg) as well as step and
touch voltages determine the performance and quality of
grounding grids. The estimation of grounding resistance values
and the touch and step voltages is usually carried out by means
of formulas and algorithms that take into account the mutual
influence between the grid electrodes. Two methods analytical
methods are used For Rg and earth surface potential (ESP)
calculations. The first one is the charge simulation method
(CSM) and the other is the boundary element method (BEM)
from commercial software TOTBEM. The validation of two
methods is explained by comparing their results and the other
results that formulated in IEEE Guide for Safety in AC
Substation Grounding (ANSI/IEEE std 80-1986). The soil is
assumed to be homogeneous. Vertical rods that added to the
grounding grids play an important role in improving the
performance of it by reducing not only the grid resistance but
also the step and touch voltages to values that safe for human.
The paper investigates the effect of vertical rods location on Rg
and ESP. A comparison between the addition of horizontal
rods and vertical rods to the grounding grids is investigated.
Also the paper focuses on the effect of profile location on some
parameters on Rg and ESP.
Keywords— Grounding grids, Step voltage, Touch voltage,
Boundary element approach, and vertical rod location.
1. INTRODUCTION
A safe grounding design has two main objectives:
1.
To carry the electric currents into earth under
normal and fault conditions without exceeding
operating and equipment limits or adversely
affecting continuity of service.
2. To ensure that the person in the vicinity of
grounded facilities is not exposed to the danger of
electric shock.
To attain these targets, the equivalent electrical resistance
(Rg) of the system must be low enough to assure that fault
currents dissipate mainly through the grounding grid into the
earth, while maximum potential different between close
points into the earth’s surface must be kept under certain
tolerances (step, touch, and mesh voltages) [1]-[2]. The
basic design quantities of the grounding grid are the
resistance (Rg) (or ground potential rise (GPR), which is the
product of the resistance by the grid current), touch (V t) and
step (Vs) voltages. In a uniform soil, the resistance can be
calculated with an acceptable accuracy using several
simplifying assumptions [3-7]. Touch and step voltages are
1
difficult to calculate by simplified method but it determined
by analytical expressions [2], [8-10].
A “Boundary Element Approach” technique for
calculating the earth surface potential and then knowing the
step and touch voltages, is developed [11].
In the last decades, some intuitive techniques for
grounding grid analysis such as the Average Potential
Method (APM) have been developed.
A new Boundary Element Approach has been recently
presented [12-14] that includes the above mentioned
intuitive techniques as particular cases. In this kind of
formulation the unknown quantity is the leakage current
density, while the potential at an arbitrary point and the
equivalent resistance for grounding grids must be computed
subsequently.
Vertical ground rods are considered the most suitable
method for earth termination of electrical and lightning
protection systems. The addition of the vertical ground rods
to the grounding grid achieve a convenient design for
grounding system by decreasing the grid resistance, the step
and touch voltage to a safe values for human and public.
Studying the effect of addition of vertical grounding rods
at different locations on earth surface potential (ESP) is
investigated in this paper. This study describes how can
improve a design of grounding grid with vertical rods that
achieve the convenient values of (Rg, Vt and Vs) with the
minimal cost of design.
Studying the effect of profile locations on earth surface
potential (ESP) is also investigated. This study describes the
role of profile locations on the values of (R g, Vt and Vs).
Studying the volubility to add a horizontal wire or a
vertical rod to improve the grounding resistance Rg, the step
and touch voltages.
This paper will present the comparison between to
analytical methods that used for calculating the grounding
resistance and earth surface potential, the first method is the
Boundary Element Method that have been implemented in a
computer aided design (CAD) system for grounding grids of
electrical substations called TOTBEM [12], the second one
is the Charge Simulation Method which is considered a
practical method for calculating the fields and from its
simplicity in representing the equipotential surfaces of the
electrodes, its application to unbounded arrangements whose
boundaries extend to infinity and its direct determination to
the electric field [15].
Figure 1 explains the grounding system with grounded
electrical equipment and shows the very important
parameters in grounding system design.
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current and Vmin is the minimum surface potential in the grid
boundary.
Fig. 1: Illustration of the grounding system
2.
BOUNDARY ELEMENT METHOD
In this paper a computer aided design (CAD) system for
grounding grids of electrical substations called TOTBEM
[14] is presented to get the grounding resistance and earth
surface potential. The effect of the vertical rods location on
the earth surface potential and then on Vt and Vs using this
technique is presented in this section.
The characteristics of the grid are 50m*50m, and
75m*31.25m to achieve the equal area between two cases,
the number of meshes is 16, the radius of the grid rods (r) is
0.005 m, the length of vertical rods (Lvr) is 2m, the radius of
it (rvr) is (0.005m), the grid depth (h) is 0.5 m, the resistivity
of the soil is 2000 ohm.m, and the total ground potential rise
(GPR) is defined as 1V. The cases under study are shown in
Figure 2.
Figure 3 illustrates the 3 dimensions (3D) and contour
profile for the case R014P16.
It is clear that the ground potential rise (GPR) as well as
distribution of the earth surface potential (ESP) during the
current flow in the grounding system are important
parameters for the protection against electric shock. The
distribution of the earth surface potential helps us to
determine the step and touch voltages, which are very
important for human safe.
By definition, the touch voltage is the difference between
the ground potential rise (GPR) and the surface potential at
the point where a person is standing while at the same time
having his hands in contact with a grounded structure, and
the mesh voltage is defined as the maximum touch voltage
to be found within a mesh of a ground grid. The maximum
touch voltage is the difference between the GPR and the
lowest potential in the grid boundary [1]. The maximum
percentage value of Vtouch is given by
Max touch voltage% 
Vgrid  Vt m in
GPR
 100
(1)
Where, Vgrid is the ground potential rise (GPR), which equal
the product of the equivalent resistance of grid and the fault
Fig. 2: Grounding grids with different meshes and different locations of the
vertical rods. (a. without vertical rods, b. with vertical rods at all point, c.
the vertical rods at the points across the diagonal and center lines, d. the
vertical rods at the perimeter of the grid only)
Fig. 3: The distribution of EPS per GPR above the grounding grid (case
R04P16 in two cases for rectangle and square grids)
Furthermore, the maximum step voltage of a grid will be
the highest value of step voltages of the grounding grid. The
maximum step voltage can be calculated by using the slope
of the secant line. The max step voltage occurs outside the
grid boundary, where the slope of the recorded surface
potential against distance is a maximum.
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Effect of the locations of the vertical rods on the
grounding resistance and earth surface potential is presented
in Figure 4.
250000
R04_4
R04P_4
R04P16_4
ESP(V)
200000
150000
Table I explains that if the number of meshes increases
the grid resistance, step and touch voltages will decrease but
we should take into account the total length of the
conductors used in the grid (the cost of the grounding
system), for example the case R08P gives us the good
results but the length of copper used is 1062 m in this case,
then the cost is much higher but we can choose the case
R04P16 as the optimum case of design because it produces a
suitable results and give economical cost.
TABLE I
GROUND GRID RESISTANCE, STEP AND TOUCH VOLTAGES AT
DIFFERENT VERTICAL RODS LOCATIONS WITH GROUNDING
GRIDS [ IF= 1000 A]
100000
50000
0
-50
49
0
50
100
150
Distance along the diagonal of the grid from its
center (m)
Fig. 4: The effect of vertical rod location on the earth surface potential for
the case study of square grid
It is seen from Figure 4 that the minimum earth surface
potential doesn’t vary significantly since the vertical rods
are connected to grid in the homogenous soil. The vertical
rods that connect to the grid should be used in the multi-soil
to reach to the lower resistivity soil. In contrast the grid
potential rise decreases considerably when the fault current
is considered as 10kA. As a result the touch voltage
decreases with increase the number of vertical rods to the
grid at the same number of meshes. A small effect in the
earth surface potential when changing the vertical rods
locations at the same number of meshes then the economical
cost plays a great part for choosing the suitable design for
the square grids.
Figure 5 shows the variation of the touch voltage from the
center of grid along the diagonal and it is clear that the
maximum touch voltage is at the corner mesh of the grid at
different soil resistivity.
Case
lt (m)
Rg
(ohm)
GPR
(kV)
R02
R04
R08
R02P
R04P
R08P
R08P57
R02P08
R04P16
R08P32
300
500
900
318
550
1062
1014
316
532
964
23.1
20.3
18.6
22.7
19.9
18.1
18.1
22.7
19.9
18.2
23.1
20.3
18.6
22.7
19.9
18.1
18.1
22.7
19.9
18.2
Vt max
% of
GPR
41.4
29
20
40.9
28
17.81
17.92
40.92
28
18.08
Vs max
% of
GPR
17
16
12
20
16
15
15
18
16
15
Figure 6 shows that an increase in the number of meshes
makes the curve of earth surface potential much flatter and
then a reduction in the grid resistance, touch and step
voltages, also it is seen that the max touch voltage moves
towards the corner mesh in the grid.
ESP/
GPR.1
0.9
1 Me sh
4 Me sh
16 Mesh
36 Mesh
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-20
-10
0
10
Distance from the center of grid (m ) "Diagonal
Profile".
20
Fig. 6: Effect of the number of meshes on the earth surface potential
Fig. 5: Variation of touch voltage from the center of grid along the diagonal
case (R04P16).
Table II explains that an addition of ground rods with 2m
lengths to the 4 meshes grounding grid for the case of study
gives nearest results when add the horizontal conductors to
the same grid. The difference in the max touch voltage
between the two cases is 26.919 V but the difference in the
total length of conductors is 62 m as shown from the table
and hence, an increase in the cost of design occurs when add
horizontal rods. Therefore the addition of vertical rods plays
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an important part to get the same results and decreases the
cost of design.
The effect of profile location of the cases shown in Figure
7 will be studied.
Figures 8, 9 and 10 illustrate the effect of profile location,
vertical rods and number of meshes for both earth surface
potential and ground potential rise. It is noted that the
number of meshes and the addition of vertical rods to the
grid have a great effect for decreasing grid resistance and
then both of ESP and GPR.
TABLE II
COMPARISON BETWEEN THE ADDITION OF VERTICAL RODS OR
HORIZONTAL RODS TO A GROUNDING GRID
No of meshes
36
4
No. of vertical rods
0
9
Vertical rods length (m)
0
2
Total grid length (m)
140
78
Resistance (Ω)
4.26
4.29
GPR (V) at 100 A
426
429
Max touch voltage % GPR
25.0
31.1
Max touch voltage (V)
106.5
133.419
Max step voltage % GPR
16
16
Fig. 8: The earth surface potential (ESP) and ground potential rise (GPR) at
specified profile location and different number of meshes, with and without
vertical rods for square grids
It is seen from the results that the minimum earth surface
potential doesn’t vary significantly when the number of
meshes increases above 16 meshes, but in contrarily the grid
potential rise decreases considerably. It is also seen in the
pervious figures that the earth surface potential of 16 meshes
is above that of 36 meshes because the profile location of
the 16 meshes lies above one conductor of the grid which
causes the earth surface potential is high in this region.
Fig. 9: The earth surface potential (ESP) and ground potential rise (GPR) at
specified profile location and different number of meshes, without vertical
rods for square grids
Fig. 10: The earth surface potential (ESP) and ground potential rise (GPR)
at specified profile location and different number of meshes, with vertical
rods for square grids.
Fig. 7: Grounding grids with different profile locations. (a. square grids
(20m*20m) without vertical rods, b. square grids (20m*20m) with vertical
rods at all, , c. rectangle grids (10x40m) without vertical rods, d. rectangle
grids (10x40m) with vertical rods at all point)
Table III shows the values of ground grid resistance, step
and touch voltages for the different profile locations of
different configurations of grounding grids for fault current
(If) of 100 A.
It is clear from the table that if the number of meshes
increases the grid resistance as well as the step and touch
voltages will decrease. The rectangular grid offers good
values of the parameters (grid resistance, step and touch
voltages) and then represents an improved design for
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grounding systems. The profile location also plays a great
deal for reducing the values of these parameters, the touch
voltage decreases when the profile location comes near to
the center line of grid.
TABLE III
THE EFFECT OF PROFILE LOCATION ON THE STEP AND TOUCH
VOLTAGE FOR DIFFERENT RECTANGLE AND SQUARE GRIDS
Case
R02_1
R02_2
R02P_1
R02P_2
R04_1
R04_2
R04P_1
R04P_2
R06_1
R06_2
R06P_1
R06P_2
S02_2
S02P_2
S04_2
S04P_2
S06_2
S06P_2
R
(ohm)
2.273
2.273
2.179
2.179
2.079
2.079
1.965
1.965
2.004
2.004
1.868
1.868
2.626
2.491
2.363
2.20
2.260
2.072
GPR
(V)
227.34
227.34
217.97
217.97
207.94
207.94
196.56
196.56
200.47
200.47
186.85
186.85
262.68
249.16
236.30
220.07
226.07
207.28
Vsmax
/GPR
0.096
0.096
0.091
0.100
0.091
0.12
0.073
0.093
0.099
0.114
0.082
0.101
0.111
0.1
0.144
0.133
0.110
0.096
Vtmax/
GPR
0.2566
0.2918
0.2567
0.2751
0.1622
0.17
0.1567
0.168
0.1184
0.132
0.116
0.1162
0.3557
0.336
0.1346
0.1143
0.151
0.1214
If at
GPR=1
0.4398
0.4398
0.4398
0.4398
0.4808
0.4808
0.5087
0.5087
0.4988
0.4988
0.5351
0.5351
0.3806
0.4013
0.4231
0.4539
0.4423
0.4824
Pij 
1
4
51
1 1 
  
 d ij d ' ij 
(3)
where, dij is the distance between contour point i and charge
point j and d’ij is the distance between the contour point i
and image charge point j’ as shown in Figure 11.
As in Figure 11, the fictitious charges are taken into
account in the simulation as point charges. The position of
each point charges and each contour point are determined in
X, Y and Z coordinates where the distance between the
contour (evaluation) points are calculated as the following ;
d ij  X j  X i   Y j  Yi   Z j  Z i 
2
2
2
Where, Xj, Yj and Zj are the dimensions of the point charge
and Xi, Yi and Zi are the dimensions of the contour point.
After solving 2 to determine the magnitude of simulation
charges, a number of checked points located on the
electrodes where potentials are known, are taken to
determine the simulation accuracy. As soon as an adequate
charge system has been developed, the potential and field at
any points outside the electrodes can be calculated.
3.
FIELD COMPUTATION WITH EQUIVALENT
CHARGES
In the charge simulation method, the actual electric filed
is simulated with a field formed by a number of discrete
charges which are placed outside the region where the field
solution is desired. Values of the discrete charges are
determined by satisfying the boundary conditions at a
selected number of contour points. Once the values and
positions of simulation charges are known, the potential and
field distribution anywhere in the region can be computed
easily [15].
The basic principle of the charge simulation method is
very simple. If several discrete charges of any type (point,
line, or ring, for instance) are present in a region, the
electrostatic potential at any point C can be found by
summation of the potentials resulting from the individual
charges as long as the point C does not reside on any one of
the charges. Let Qj be a number of n individual charges and
Φi be the potential at any point C within the space.
According to superposition principle;
n
i   Pij Q j
(2)
j 1
where Pij are the potential coefficients which can be
evaluated analytically for many types of charges by solving
Laplace or Poisson’s equations, Φi is the potential at contour
(evaluation) points, Qj is the charge at the point charges.
Because of the ground surface is flat, the method of
images can be used with the charge simulation method and
the potential will be characterized for being constant on the
grounding grids and its symmetry [16]. The potential
coefficients will be as in the following equation;
Fig. 11: Illustration of the charge simulation technique
The grid is divided into equal segments by the point
charges distribution along the axis of grid conductors.
Figure 12 shows the distribution of the point charges (dots)
for the grounding grid (1 mesh), the number of point charges
is distributed on the axis of the grid conductors equally and
also the evaluation points distributed on each conductor as
shown in Figure 12. The meshes of the grid are always
symmetrical.
The charge simulation technique is used to get the ground
resistance (Rg), ground potential rise (GPR) and then the
surface potential on the earth due discharging impulse
current into ground grid is known. The touch and step
voltages are calculated from surface potential. The duality
expression is used to calculate the ground resistance Rg
from the next equation;
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n
C
Qj
j 1
V
Rg  C    
(4)
where, V is the GPR that is defined 1 V, Qj is the charge of
point charge j that used for the calculation, ρ is the soil
resistivity and ε is the soil permittivity.
In this section, some figures explain the earth surface
potential along diagonal profile for the square grid with
different number of meshes.
The characteristics of the grid are 50m*50m, the radius of
the grid rods (r) is 8 mm, the grid depth (h) is 0.5 m, the
resistivity of the soil () is 100 Ω.m, and the total ground
potential rise (GPR) is defined as 1 V.
Figure 13 shows the Earth surface potential in 3D and the
contour map of this case.
52
4. COMPARISON BETWEEN THE BEM AND CSM
The following case of study is taken to compare between
the results by BEM and CSM, the input data about the grid
configuration:
Number of meshes (N) = 64, side length of the grid in X
direction (X) = 75 m, side length of the grid in Y direction
(Y) = 75 m, grid conductor radius = 5 mm, vertical rod
length (Z) = 0 (no vertical rod), depth of the grid (h) = 0.5
m, resistivity of the soil (ρ) = 2000 Ω.m and the permittivity
of the soil is 9.
The following table I explains that the result from the
proposed method is close to the other formula in [1] and also
the values of resistance that calculated by [11-14].
TABLE IV
GROUNDING RESISTANCE BETWEEN THE BEM AND CSM AND
THE OTHER FORMULAS THAT USED IN IEEE STANDARDS [1]
R (Ω)
11.8
13.29
13.23
12.67
11.11
11.87
Formula
Dwight [1]
Laurent [1]
Sverak [1]
BEM [11-14]
Schwarz [1]
CSM 4320 Points
Figure 14 explains that the proposed method satisfies an
agreement with the other method that used to calculate
surface potential for example Boundary Element Method
[11-14].
14
12
Fig. 12: Distribution of point charges on the grid (1 mesh)
10
ESP(kV)
8
6
4
BEM
2
CSM
0
-150
-100
-50
0
50
100
150
Distance from grid center (m)
Fig. 14: Comparison between proposed method and Boundary Element
Method for 64 meshes grid
Fig. 13a: ESP/GPR for 64 meshes
Fig. 13b: Contour map for 64 mesh grid
5. CONCLUSIONS
The vertical rods play an important role for reducing the
grid resistance, the step and touch voltages. The number of
meshes is an effective parameter for reducing the pervious
values but it needs more copper then increases the cost. The
study explains a small effect in the earth surface potential
when changing the vertical rods locations at the same
number of meshes hence the economical cost plays a great
part for choosing the suitable design for the square grids,
then the optimal case of the grids is that consists of 16
meshes with the vertical rods in the perimeter (case
R04P16), it gives suitable results for the grid resistance, step
and touch voltages and in the same time it presents an
economical cost.
Additional vertical ground rods gives nearest results to
that results when add the horizontal rods to the same grid,
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International Journal of Engineering & Computer Science IJECS-IJENS Vol:13 No:03
then the addition of vertical rods to the grounding grid gives
a good performance and decrease the cost of design. Both
vertical rods and number of meshes are considered effective
parameters for reducing the grid resistance, the step and
touch voltages. The paper demonstrates that another
important parameter that effects in the pervious values is the
profile location. The man location in case of fault
determines the value of step and touch voltage that he will
experiences. The figures illustrate that the dangerous point is
at the side of the grid and comes in the corner mesh in the
grid.
The proposed methods (BEM and CSM) that used to
calculate the earth surface potential and grounding
resistance due to discharging current into grounding grid are
efficient. The validation of these methods is satisfying by a
comparison between the results from it and the results from
the formula in IEEE standard. The proposed methods give a
good agreement with the IEEE standard. The two methods
give the closest results to each other althought the different
techniques that applied in each method.
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BIOGRAPHIES
Sherif S. M. Ghoneim Received B.Sc.
and M.Sc. degrees from the Faculty of
Engineering at Shoubra, Zagazig
University, Egypt, in 1994 and 2000,
respectively. Starting from 1996 he was a
teaching staff at the Faculty of Industrial
Education, Suez Canal University,
Egypt. Since end of 2005 to end of 2007,
he is a guest researcher at the Institute of
Energy Transport and Storage (ETS) of
the University of Duisburg-EssenGermany. In 2008, he got Ph.D Degree
in Electrical power and machines,
Faculty of Engineering-Cairo University
(2008). He joins now the Taif University
as an assistant professor in the Electrical Engineering Department,
Faculty of Engineering. His research focuses in the area of high
voltage engineering.
Dr. Kamel A. Shoush (1961) is associate
professor in the Electrical Engineering
Department- Faculty Of EngineeringAL-Azhar University- Cairo – Egypt. He
received his B. Sc. And M. Sc. Degrees
from AL-Azhar University- Cairo –
Egypt in 1986 and 1993 respectively. And
his Ph. D. degree from AL-Azhar
University- Cairo – Egypt in 1998 after
having worked for two years in “GerhardMercator-Universität–esamthochschula
Duisburg” - Duisburg/Germany.
Now, he is an Associate Prof in the Taif
University, College of Engineering,
Electrical Engineering Department, Saudi
Arabia. His Areas of interest Intelligent Systems Applications For
Power Systems Optimization And Control.
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