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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN

0976 – 6553(Online) Volume 3, Issue 3, October – December (2012), © IAEME

156

CALCULATION OF GROUNDING RESISTANCE AND EARTH

SURFACE POTENTIAL FOR TWO LAYER MODEL SOIL

Hatim Ghazi Zaini

Taif University, Faculty of Engineering, Electrical department

[email protected]

ABSTRACT

For two layer model soil, the calculation of apparent resistivity is considered very important

issue since the absent of a specified method to find it. Some empirical resistivity formula is

used in this paper to present the apparent resistivity of the two layer model soil. A current

simulation method technique which is a practical technique for calculating the grounding

resistance (Rg) as well as the Earth Surface Potential (ESP) of the grounding grids in two-

layer model soil which based upon the apparent resistivity of the two layer soil and

simulating current sources is used. it is analogous to the Charge Simulation Method. The

validation of the method is described by a comparison with the results in literatures.

Index terms--Grounding grids, two-layer soil, current simulation method, Computer methods

for grounding analysis, System protection.

I. NOMENCLATURE

Paij= Potential coefficient matrix related to apparent resistivity

P1ij, P2ij Potential coefficient matrix related to resistivity of layer 1 and 2 respectively

Ij =current source at point j

Vi= voltage at evaluation point i

ρa =apparent soil reistivity

ρ1 =soil reistivity of layer 1

ρ2 =soil reistivity of layer 2

d =distance between current source point and evaluation point in original grid

d' =distance between current source point and evaluation point in image grid

J =current density (A/m2)

F =field coefficient

zzi & zzj = the dimension of the contour point and current source in z direction respectively

Rg=grounding resistance

d0 = the depth to the boundary of the zones,

INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING

& TECHNOLOGY (IJEET)

ISSN 0976 – 6545(Print) ISSN 0976 – 6553(Online) Volume 3, Issue 3, October - December (2012), pp. 156-163 © IAEME: www.iaeme.com/ijeet.asp Journal Impact Factor (2012): 3.2031 (Calculated by GISI) www.jifactor.com

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157

K = the reflection factor (K=( ρ2- ρ1)/ ( ρ1+ ρ2))

z = the top layer depth

GPR=Ground potential rise (V)

Vtouch= touch voltage

II. INTRODUCTION

The knowledge of the grounding systems impulse characteristics has a great significance for

a proper evaluation of substation equipment stresses from lightning over-voltages and

lightning protection evaluation. As it is stated in the ANSI/IEEE a safe grounding design has

two objectives: the first one is the ability to carry the electric currents into earth under normal

and fault conditions without exceeding operating and equipment limits or adversely affecting

continuity of service. The second is how this grounding system ensures 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 difference between close points into the earth’s surface must

be kept under certain tolerances (step, touch, and mesh voltages). Analysis of substation

grounding systems, including buried grids and driven rods has been the subject of many

recent papers [1- 4].

Several publications [5-19] have discussed the analytical methods used when uniform and

two-layer soils are involved.

This paper uses a practical method to calculate the grounding resistance as well as the

earth surface potential for grounding grids which buried in uniform and two-layer soil. This

method is Current Simulation Method (CSM). The Current Simulation Method is analogous

to Charge Simulation Method.

The validation of a proposed method is explained by comparison between the results from

the proposed method and the other that formulated in [1].

III. CURRENT SIMULATION METHOD IN TWO-LAYER SOIL

The representation of a ground electrode based on equivalent two-layer soil is generally

sufficient for designing a safe grounding system. However, a more accurate representation of

the actual soil conditions can be obtained by using two-layer soil model [13].

As in the Current Simulation Method, the actual electric filed is simulated with a field

formed by a number of discrete current sources which are placed outside the region where the

field solution is desired. Values of the discrete current sources are determined by satisfying

the boundary conditions at a selected number of contour points. Once the values and

positions of simulation current sources are known, the potential and field distribution

anywhere in the region can be computed easily [20].

The field computation for the two-layer soil system is somewhat complicated due to the

fact that the dipoles are realigned in different soils under the influence of the applied voltage.

Such realignment of dipoles produces a net surface current on the dielectric interface. Thus in

addition to the electrodes, each dielectric interface needs to be simulated by fictitious current

sources. Here, it is important to note that the interface boundary does not correspond to an

equipotential surface. Moreover, it must be possible to calculate the electric field on both

sides of the interface boundary.

In the simple example shown in Fig. 1, there are N1 numbers of current sources and

contour points to simulate the electrode, of which NA are on the side of soil A and (N1- NA)

are on the side of soil B. These N1 current sources are valid for field calculation in both soils.

At the different soil interface there are N2 contour points (N1 +1,….., N 1+N2), with N2



158

current sources (N1+1,…..,N 1+N2) in soil A valid for soil B and N2 current sources (N1+N2

+1,…..,N1 +2N2) in soil B valid for soil A. Altogether there are (N1+N2) number of contour

points and (N1 + 2N2) number of current sources.

As in Fig. 1, h is the grid depth and z is the depth of top layer soil. In order to determine

the fictitious current sources, a system of equations is formulated by imposing the following

boundary conditions.

• At each contour point on the electrode surface the potential must be equal to the

known electrode potential. This condition is also known as Dirichlet’s condition on the

electrode surface.

• At each contour point on the dielectric interface, the potential and the normal

component of flux density must be same when computed from either side of the

boundary.

Thus the application of the first boundary condition to contour points 1 to N1 yields the

following equations.

11

,21

,

2

1,1

1,

,1.....

,1.....

21

!

1

21

2!

1

NNiVIPIP

NiVIPIP

A

NN

Njjji

N

jjjia

A

NN

NNjjji

N

jjjia

+==+

==+

∑∑

∑∑

+

+==

+

++== (1)

where,

+=

+=

+=

'2

,2

'1

,1',

11

4

11

4 ,

11

4

ddP

ddP

ddP

ji

jia

jia

π

ρ

π

ρ

π

ρ

Again the application of the second boundary condition for potential and normal current

density to contour points = N1+1 to N1+N2 on the dielectric interface results into the

following equations. From potential continuity condition:

211

2

1,1

1,2 ,1....0

21

2!

21

1

NNNiIPIPNN

NNjjji

NN

Njjji

++==− ∑∑+

++=

+

+=

(2)

From continuity condition of normal current density Jn:

( ) ( ) 21121 ,1 0 NNNiforiJiJ nn ++==− (3)

Eqn. (3) can be expanded as follows:

211

2

1,1

1

1,2

21,

21

,1..........01

111

21

2!

21

!

1

NNNiIF

IFIF

NN

NNjjji

NN

Njjji

N

jjjia

++==

+−

−

∑

∑∑

+

++=⊥

+

+=⊥

=⊥

ρ

ρρρ (4)

where,



159

( ) ( )

( ) ( )

( ) ( )

−+

−=

∂

∂−=

−+

−=

∂

∂−=

−+

−=

∂

∂−=

⊥

⊥

⊥

3'

'

322

,2

3'

'

311

,1

3'

'

3,

4

4

4

d

zzzz

d

zzzz

z

PF

d

zzzz

d

zzzz

z

PF

d

zzzz

d

zzzz

z

PF

jijiij

ji

jijiij

ji

jijiaija

jia

π

ρ

π

ρ

π

ρ

Fig.1. Fictitious current source with contour points for field calculation by current simulation

method in two-layer soil.

where, F┴,ij is the field coefficient in the normal direction to the soil boundary at the

respective contour point, ρa, ρ1 & ρ2 are the apparent resistivity nd resistivities of soil 1 and 2

respectively and zzi & zzj are the dimension of the contour point and current source in z

direction respectively. Equations 1 to 4 are solved to determine the unknown fictitious current

sources.

After solving 1 to 4 to determine the unknown fictitious current source points, the potential

on the earth surface can be calculated by using Eq. 1. Also, the ground resistance (Rg) can be

calculated using the following equation:

∑=

=1

1

N

j

g

I

VR (5)

where, V is the voltage applied on the grid which is assumed 1V.

The problem for the proposed method is how the apparent resistivity can be calculated. As

in [18], the apparent resistivity for two soil model calculates by the following formula;

( )

12

2

1

2

1

1 for

111 0

ρρ

ρ

ρ

ρρ <

−

−

+

=

+ zdK

a

e

(6)

( )12

2

1

1

22 for 111 0 ρρ

ρ

ρρρ >

−

−

+×= +

−

zdKa e (7)

where, d0 is the depth to the boundary of the zones, K is the reflection factor (K=( ρ2- ρ1)/ (

ρ1+ ρ2)) and z is the top layer depth.



160

Equations 6 and 7 are valid for the boundary depth greater than or equal the grid depth. But

in [19], Eq. 7 is modified because at very large depth of upper soil layer, resistivity ρa given

by Eq. 7 tends to ρ2. This is physically incorrect if the electrode lies in the upper soil layer, as

assumed in [18]. Therefore, Eq. 7 is modified [19] as follows:

( )12

2

1

1

21 for111 0 ρρ

ρ

ρρρ >

−

−

+×= +

−

zdKa e (8)

For finite h and very large d0, resistivity ρa given by Eq. 8 tends to ρ1, which is in

compliance with physical reasoning.

When the boundary depth is lower than the grid depth, the apparent resistivity tends to ρ2.

Therefore, by using Eq. 6 and 8 for calculating the grounding resistance by Current

Simulation Method, the large different between the proposed method results and the results in

[1] is observed for K<-0.5 and this shown in Fig.2.

If Eq. 8 is modified as in 9 the results by the proposed method are good agreement with the

results in [1].

( )12

15

1

1

21 for 111 0 ρρ

ρ

ρρρ >

−

−

+×= +

−

zdKa e (9)

Figures 3 and 4 present the comparison of the results calculated by the proposed method

with the results reported in [1] for a square 30m*30m, 4 and 16 meshes grids buried at 0.5 m

depth in various two layer structures. It is noticed from Figs. 3 and 4 that the proposed

method gives a good agreement with the results in [1].

IV. GROUNDING RESISTANCE AND EARTH SURFACE POTENTIAL

It is clear that the Ground Potential Rise (GPR) as well as distribution of the Earth Surface

Potential (ESP) during flow the impulse current into the grounding system is 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.

The maximum percentage value of Vtouch is given by:

Max 100_

% min ×=GPR

VGPRVtouch (10)

where, GPR is the ground potential rise, which equal the product of the equivalent resistance

of grid and the fault current and Vmin is the minimum surface potential in the grid boundary.

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.

Figure 5 explains the Earth Surface Potential per Ground Potential Rise (ESP/GPR) when

the case is square grid, 36 meshes, grid dimension 60m*60m, vertical rod length that connect

to grid 6m, grid conductor radius 0.005m, grid depth 0.7m, the top layer to lower layer

resistivity 1000/100 ohm.m and the top layer depth 3m. Fig. 6 illustrates that the maximum

touch voltage occurs at the boundary of the grid near the corner mesh but the maximum step

voltage occurs outside the boundary of the grid near the edge of it.



161

Fig. 2. Relation between 4 meshes grid resistance and the top layer depth



0.01

0.1

1

10

0.1 1 10 100

Res

ista

nce

(oh

m)

Top layer depth (m)

K=0.9 K=0.9-[1]

0.5 K=0.5-[1]

0 K=0-[1]

-0.5 K=-0.5-[1]

-0.9 K=-0.9-[1]

0.01

0.1

1

10

0.1 1 10 100

Res

ista

nce

(oh

m)

Top layer depth (m)

K=0.9 K=0.9-[1]

K=0.5 K=0.5-[1]

K=0 K=0-[1]

K=-0.5 K=-0.5-[1]

K=-0.9 K=-0.9-[1]

0.01

0.1

1

10

0.1 1 10 100

Res

ista

nce

(oh

m)

Top layer depth (m)

K=0.9 K=0.9-[1]

K=0.5 K=0.5-[1]

K=0 K=0-[1]

K=-0.5 K=-0.5-[1]

K=-0.9 K=-0.9-[1]



162

0

0.2

0.4

0.6

0.8

1

1.2

-80-60-40-20020406080

ES

P/G

PR

Distance from the grid center (m)

Fig. 5. ESP/ GPR for 36 meshes square grid

0

0.2

0.4

0.6

0.8

1

1.2

-80-60-40-20020406080

ES

P, S

tep

an

d T

ou

ch

Vo

lta

ge

s/G

PR

Distance from grid center (m)

ESP/GPR Step Voltage/GPR Touch Voltage/GPR

Fig. 6. ESP, Step and Touch voltages / GPR for 36 meshes square grid

V. CONCLUSIONS

This paper aims to calculate the Earth Surface Potential due to discharging current into

grounding grid in two-layer soil by using a traditional but practical method which is the

Current Simulation Method. The validation of the method is satisfying by a comparison

between the results from the method and the results in [1]. It is seen that a good agreement

between the proposed method results and the results in [1].

VI. REFERENCES

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[2] J. Nahman, and S. S Kuletich, “Irregularity correction factors for mesh and step voltages

of grounding grids”, IEEE Transactions on Power Apparatus and Systems, Vol. Pas-99,

No. 1, pp. 174-179, Jan/Feb: 1980.

[3] Substation Committee Working Group 78.1, Safe substation grounding, Part II, IEEE

Transactions on Power Apparatus and Systems, Pas-101, pp. 4006/4023, 1982.

[4] IEEE Guide for safety in AC substation grounding, IEEE Std.80-2000.

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