An Algorithm for Estimating the Grounding Resistance of Complex Grounding Systems Including Contact Resistance

Abstract

In cases where the grounding system is buried in soils characterized by poor contact with the electrodes (e.g. karst and sandy terrains), the contact resistance frequently represents a dominant component of the total grounding resistance. In such cases, estimation of the grounding resistance by conventional formulas given in the literature is useless, because they do not take into account the contact resistance. An algorithm for estimating the total grounding resistance of complex grounding systems, with the contact resistance included, was developed and presented in this paper. The algorithm is applied to a grounding system of a typical 110 kV transmission line tower used in the Serbian transmission power system. Simple formulas by which the total grounding resistance of the analyzed grounding system can easily be calculated are also derived. The obtained results are validated using 3D FEM modeling and a practical method from the literature. It was shown that the total grounding resistances determined by the proposed algorithm deviate less than 4% from those obtained by FEM calculations. Since the proposed algorithm is general and can be applied to any grounding system, it represents a powerful tool for estimating the grounding resistance in an early stage of the design process.

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An algorithm for estimating the grounding resistance of complex grounding systems
including contact resistance
Jovan Trifunovic
Faculty of Electrical Engineering
University of Belgrade
Bulevar kralja Aleksandra 73
11000 Belgrade, Serbia
jovan.trifunovic@etf.rs
Miomir Kostic
Faculty of Electrical Engineering
University of Belgrade
Bulevar kralja Aleksandra 73
11000 Belgrade, Serbia
kostic@etf.rs

Abstract – In cases where the grounding system is buried in
soils characterized by poor contact with the electrodes (e.g. karst
and sandy terrains), the contact resistance frequently represents
a dominant component of the total grounding resistance. In such
cases, estimation of the grounding resistance by conventional
formulas given in the literature is useless, because they do not
take into account the contact resistance. An algorithm for
estimating the total grounding resistance of complex grounding
systems, with the contact resistance included, was developed and
presented in this paper. The algorithm is applied to a grounding
system of a typical 110 kV transmission line tower used in the
Serbian transmission power system. Simple formulas by which
the total grounding resistance of the analyzed grounding system
can easily be calculated are also derived. The obtained results
are validated using 3D FEM modeling and a practical method
from the literature. It was shown that the total grounding
resistances determined by the proposed algorithm deviate less
than 4% from those obtained by FEM calculations. Since the
proposed algorithm is general and can be applied to any
grounding system, it represents a powerful tool for estimating
the grounding resistance in an early stage of the design process.
Index Terms – Complex grounding systems, Contact
resistance, Electrical engineering, Finite element method,
Grounding loop, Grounding resistance, Modeling, Power
transmission lines, Soil, Transmission line tower
I. INTRODUCTION
In order to ensure that an adequate grounding system is
designed, its grounding resistance should be estimated in an
early stage of the design process 1, because its value is
needed for calculating the ground fault loop impedance and
ground potential rise 2–4. This is not always easy to
obtain, especially in troubled environments 4–6. It was
noticed in 7, and confirmed by 3D FEM modeling in 8,
that in cases where the grounding system is buried in soils
which form poor contact with the electrodes (e.g. karst and
sandy terrains), the contact resistance becomes a dominant
component of the total grounding resistance. In such cases,
conventional formulas for estimating the grounding
resistance, given in standards [9] and [10], as well as in many
scientific engineering papers [11]–[21], are useless because
they do not include the contact resistance.
Analyzing the experimentally obtained data presented in
7, related to a grounding loop embedded in a former
This research was partially supported by the Ministry of Education,
Science and Technological Development of the Republic of Serbia (project
TR 36018).
stonebed, a general method for quantitative estimation of soil
properties related to the soil contact with the electrodes, was
developed 8. Although it was stated in 8 that the method
could not be used for precise calculations of the grounding
resistance of loops buried in soils characterized by poor
contact with the electrodes, further research showed that it
actually could. An algorithm for estimating the grounding
resistance of complex grounding systems, with the contact
resistance included, based on the methods given in 8 and
22, was developed and presented in this paper. To the best
of the authors’ knowledge, such an algorithm does not exist in
the available literature.
It was shown in 23–26 that the grounding resistances
obtained by the measurements at the steady-state conditions
are similar to those obtained by the FEM simulations.
Therefore, 3D FEM modeling can be used for the validation
of new algorithms and formulas for the calculation of relevant
grounding system parameters, as done in the research
presented in this paper (as well as in 27 and 28). The
algorithm was tested applying 3D FEM modeling to the
grounding system of a typical 110 kV transmission line tower
used in the Serbian transmission power system 22, assuming
several types of soil. It was also validated using a suitable
practical method from [29]. The FEM modeling of the
considered complex grounding system and the surrounding
soil which forms imperfect contact with the electrodes was
described in 8 and 26.
II. CALCULATION OF THE GROUNDING SYSTEM RESISTANCE
OF A TYPICAL 110 KV TRANSMISSION LINE TOWER USED IN
THE SERBIAN TRANSMISSION POWER SYSTEM
A frequently used grounding system of a typical 110 kV
transmission line tower is presented in Fig. 1. The grounding
system contains 5 electrically connected square loops. The
upper loop (of dimensions L1 x L1) is buried at depth h1, while
each of the 4 identical lower loops (of dimensions L2 x L2) is
placed around the tower’s footing concrete foundation at
depth h2. The dimensions L1 and L2 are conditioned by the
tower height (H) and soil bearing capacity (σ), representing
the input parameters needed for determining the dimensions
of the footing foundation, as well as the span of footings,
through the tower construction stability calculations. The

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Fig. 1. The grounding system of a typical 110 kV transmission line tower
used in the Serbian transmission power system.
ranges of relevant construction parameters for towers used in
practice, obtained by examining the design documentation of
72 different towers used in the Serbian transmission power
system, are presented in Table I 22 (r = L2/L1 and p is the
perimeter of the cross-section of the strips forming the loop).
While L1, r, h1, h2 and p have a direct influence on the
grounding system resistance, H and σ have an indirect
influence on its value through L1 and r, which is why
variations of H and σ were not analyzed in this research.
First, an algorithm for fast calculation of the grounding
system resistance will be presented, assuming a perfect
contact between its electrodes and the surrounding soil. Such
a grounding resistance represents the theoretical value of the
real grounding system resistance.
The following, very accurate formula for quick calculation
of the theoretical grounding resistance of square loops whose
construction parameters belong to the ranges given in Table I
was recommended in 22 (see Appendix I):
ph
L
L
RT
T
SL
2
0
85.8
ln
2


 (1)
(ρ is the soil resistivity, LT the loop perimeter and R
0SL its
theoretical grounding resistance).
In 22 an algorithm and approximate formulas for fast
calculation of the theoretical grounding resistance of the
grounding system shown in Fig. 1 were also given. It was
shown that for any combination of the input parameters
belonging to the ranges given in Table I and for any value of
the uniform soil resistivity, the theoretical grounding
resistance of the grounding system sketched in Fig. 1 can be
calculated using (1) and the following two equations:
)
11
(
1
040
4
0LLUL
LLUL
GS RRR  

, and (2)
LL
LL
LLLLLLLL
LL
LL RRRRRR 0
4
0000
4
40
4
)
1111
(
1

, (3)
where:
TABLE I
RANGES OF RELEVANT STRUCTURE PARAMETERS OF A TYPICAL 110 KV
TRANSMISSION LINE TOWER USED IN THE SERBIAN TRANSMISSION POWER
SYSTEM 22
Parameter Range
H 12 – 30 m
σ 100 – 300 kPa
L1 5 – 10 m
r 0.20 – 0.44
h1 0.7 m (fixed value)
h2 2 m (fixed value)
p 0.044 – 0.088 m
- R0GS represents the theoretical grounding resistance of the
whole grounding system,
- R0UL is the theoretical grounding resistance of the upper
grounding loop,
- R04LL is the theoretical grounding resistance of all four
lower loops,
- ηUL–4LL is the coefficient reflecting the mutual (vicinity)
effect existing between the upper and the four lower
loops,
- R0LL is the theoretical grounding resistance of a solitary
lower loop, and
- η4LL is the coefficient reflecting the vicinity effect among
the four lower loops.
The η (ηUL–4LL and η4LL) coefficients can be calculated by
the following equation:
4312111 ),( CrCLCrLCrL






, (4)
where C1, C2, C3 and C4 are the constants. For the ranges of
the input parameters given in Table I, the following values of
the constants C1, C2, C3 and C4 were determined 22 (see
Appendix I):
(C1, C2, C3, C4) = (0.005185, -0.005094, -0.1656, 0.7299) for
ηUL–4LL, and
(C1, C2, C3, C4) = (-0.005264, 0.004609, -0.7925, 0.8793) for
η4LL.
The values of R0UL and R0LL can be calculated using (1), the
ηUL–4LL and η4LL coefficients by incorporating the
corresponding constants (C1, C2, C3, C4) into (4), R04LL by (3),
and R0GS by (2). If the upper loop is not an element of the
grounding system, R0GS is reduced to R04LL.
III. A NEW ALGORITHM FOR ESTIMATING THE TOTAL
GROUNDING RESISTANCE
A. Formulas for Estimating the Total Grounding Resistance
of a Square Loop
Analyzing the experimentally obtained values of the square
loop grounding resistance presented in 7, it was noticed in
8 that the soil properties (concerning the soil contact with
the electrodes) can be described by size, number and position
of air gaps placed between the grounding loop electrodes and
the surrounding soil. The analyzed square loop (5 m × 5 m),
made of zinc-protected steel strips with a rectangular cross-

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Fig. 2. Schematic representation of the concept of air gaps sequentially
placed along the grounding strip.
section (30 mm × 4 mm), was installed at a depth of 0.5 m. A
schematic representation of the concept of air gaps
sequentially placed along the grounding strip is given in
Fig. 2. The sequential distribution of air gaps along a part of
the strip is shown in yz plane, while the cross-section of the
modeled air gap surrounding a strip is shown in xz plane. The
sequences of length T are uniformly distributed along the
grounding loop perimeter. Each of them contains one air gap
(b represents the length of a part of the strip sequence
completely surrounded by the air gap (the rest of the strip
sequence forms a perfect contact with the surrounding soil),
and d is the thickness of the air gap).
The quality of the contact between the grounding
electrodes and the surrounding soil is quantitatively described
by a parameter F, representing the fraction of the electrode
surface which is not in contact with the surrounding soil. This
parameter can be calculated as
100(%) 
T
b
F. (5)
The actual (total) grounding resistance (RTSL) of the
considered square loop, as a function of F, can be
decomposed by the following expression:
)()( 0FRRFR CSLSLTSL  , (6)
where:
- R0SL is the theoretical grounding loop resistance (obtained
assuming a perfect contact), and
- RCSL(F) is the contact resistance.
It was shown in 8 that RCSL(F) can be approximated by
the following expression:










1
100
100
)()( 2
210 F
FKFKK
Lp
d
FR
T
CSL

, (7)
where K0, K1 and K2 are correction coefficients. For the
analyzed loop and the surrounding soil described by
d0 = 16·10-3 m and T0 = 25·10-2 m, the following values of K0,
K1 and K2 were determined (see Appendix I):
(K0, K1, K2) = (0.857573, 0.017936, -0.000184). (8)
TABLE II
COMPARISON OF THE RESULTS OBTAINED BY FEM AND BY (1) AND (7)
LT
(m)
F0
(%)
RTSL
(Ω)
FEM
RTSL
(Ω)
(1) and (7)

RTSL
(%)
4 0 32.84 32.32 -1.57
4 50 40.31 42.51 5.45
4 80 59.63 60.08 0.76
4 92 98.56 97.20 -1.38
4 96 155.78 157.33 1.00
4 98 262.93 276.94 5.33
40 0 5.26 5.06 -3.74
40 50 6.04 6.08 0.70
40 80 8.04 7.84 -2.42
40 92 12.02 11.55 -3.88
40 96 18.21 17.57 -3.53
40 98 29.43 29.53 0.34
B. Application of the Previously Derived Equations for
Estimating the Total Grounding Resistance of Square Loops
Buried in Soils Characterized by Unknown Parameters
Further investigation showed that (1) and (7) are
characterized by very high accuracy for all square loop
dimensions given in Table I. As can be seen from Table II,
assuming the above values of d0 and T0, the differences
between the total grounding resistances obtained by 3D FEM
modeling and those obtained by (1) and (7), related to the
minimum (LT = 4 m) and maximum (LT = 40 m) loop
perimeters from Table I for six different values of parameter
F0, amount up to 5.45%.
Taking into account the stated in the above paragraph, it
was assumed that it is possible to calculate, with reasonable
accuracy, the total grounding resistance of a square loop
(dimensions of which belong to the ranges given in Table I)
buried in soil characterized by unknown parameters (dx, Tx
and Fx), using the information obtained by measurement of
the grounding resistance of a small square loop buried in that
soil. The idea was to make the “unknown” soil equivalent to
the previously analyzed one (characterized by d0 and T0, as
well as by the corresponding F
0), and to calculate the total
grounding resistance of a square loop using (1) and (7).
This idea was validated by 3D FEM modeling of both the
surrounding soil and square loops similar to those belonging
to the grounding systems sketched in Fig. 1. The upper and
lower loops representing elements of complex grounding
systems denoted by GS1 and GS2, were analyzed. Those
loops are characterized by the minimum and maximum
dimensions of both L1 and r given in Table I, respectively. All
of the relevant constructional and soil input parameters
characterizing both GS1 and GS2 are given in Table III. It
was assumed that these grounding systems were buried in
soils described by the parameters ρ, dx and Tx, given in
Table III, as well as by 6 different values of the parameter
Fx(%), given in Table IV.
A small square loop (1 m × 1 m) made of steel strips with a
rectangular cross-section (30 mm × 4 mm), installed at a
depth of 0.5 m, was used as a simple test grounding system.
Assuming ρ = 100 Ωm, the theoretical value of the loop

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TABLE III
INPUT PARAMETERS OF THE ANALYZED GROUNDING SYSTEMS AND THE MODELED SOILS, AS WELL AS THE CORRESPONDING VICINITY EFFECT COEFFICIENTS
CALCULATED BY (4)
GS L1
(m) r p
(10-3 m)
h1
(m)
h2
(m)
ηUL–4LL
(4)
η4LL
(4)
ρ
(Ωm)
dx
(10-3 m)
Tx
(10-2 m)
GS1 5 0.2 68 0.7 2 0.677 0.739 100 20 5
GS2 10 0.44 68 0.7 2 0.629 0.554 100 25 40
TABLE IV
COMPARISON OF THE RESULTS OBTAINED BY THE PROPOSED PROCEDURE, BASED ON (1)–(11), AND BY THE FEM CALCULATIONS
GS Fx
(%)
RTtest
(Ω)
FEM

RTtest
(%)
F0
(%)
RTLL
(Ω)
(6)
RTLL
(Ω)
FEM

RTLL
(%)
RTUL
(Ω)
(6)
RTUL
(Ω)
FEM

RTUL
(%)
RT4LL
(Ω)
(10)
RT4LL
(Ω)
FEM

RT4LL
(%)
RTGS
(Ω)
(11)
RTGS
(Ω)
FEM

RTGS
(%)
GS1 0 32.84 -0.97 3.28 27.32 28.32 -3.53 8.86 9.12 -2.88 9.27 9.57 -3.21 6.71 6.83 -1.80
GS1 50 38.46 15.99 34.00 32.95 33.99 -3.06 9.99 10.28 -2.82 10.68 10.95 -2.45 7.35 7.41 -0.92
GS1 80 54.08 63.07 74.19 48.56 49.51 -1.92 13.11 13.42 -2.31 14.59 14.95 -2.42 9.09 9.23 -1.50
GS1 92 90.28 172.24 90.97 84.76 85.71 -1.10 20.35 20.59 -1.18 23.64 24.06 -1.76 13.12 13.37 -1.81
GS1 96 149.23 350.00 95.71 143.71 144.72 -0.70 32.14 32.59 -1.37 38.37 39.12 -1.91 19.68 20.05 -1.82
GS1 98 270.87 716.83 97.95 265.36 266.59 -0.46 56.47 58.60 -3.64 68.79 69.91 -1.61 33.21 33.68 -1.41
GS2 0 32.84 -0.97 3.28 8.89 8.99 -1.07 4.98 5.12 -2.63 4.03 4.06 -0.71 3.55 3.59 -1.20
GS2 50 42.76 28.94 50.83 11.14 11.06 0.72 5.97 6.09 -1.92 4.59 4.63 -0.86 3.92 3.81 2.90
GS2 80 68.22 105.73 84.84 16.93 16.49 2.68 8.52 8.54 -0.22 6.04 6.00 0.70 4.86 4.71 3.23
GS2 92 120.08 262.09 94.20 28.72 28.13 2.08 13.71 13.72 -0.09 8.99 9.00 -0.19 6.76 6.64 1.83
GS2 96 197.21 494.69 97.00 46.25 45.04 2.68 21.42 20.77 3.11 13.37 13.20 1.28 9.57 9.26 3.39
GS2 98 339.53 923.87 98.42 78.59 75.91 3.53 35.65 35.60 0.15 21.46 21.48 -0.11 14.74 14.22 3.69
grounding resistance, calculated using (1), amounts to
R0test = 33.16 Ω. The total grounding resistances (RTtest) of the
test loop, obtained by FEM calculations, as well as their
relative errors (

RTtest) related to the theoretical grounding
resistance (R0test), are given in Table IV for 12 different types
of the surrounding soil. It is obvious that in cases where the
loop is buried in soils characterized by poor contact with the
electrodes the contact resistance represents a dominant
component of the total grounding resistance, making
estimation of the grounding resistance by using conventional
formulas impossible (relative errors are up to 924%).
Assuming that the unknown surrounding soil (dx, Tx and
Fx) can be described using the parameters of the known
(analyzed) soil (described by d0 = 16·10-3 m, T0 = 25·10-2 m
and the corresponding F
0), for the considered test loop (6)
becomes:










 1
100
100
)(
0
2
02010
0
0F
FKFKK
Lp
d
RR
T
testTtest

, (9)
emphasizing that the coefficients K0, K1 and K2 are equal to
those given in (8). Obtaining the value of RTtest by
measurement, the only unknown parameter in (9) is F0(%),
which can easily be calculated, e.g. using the iterative
calculation method (its initial value could be F0 = 50). The
obtained values of F0 for the 12 considered types of the
surrounding soil are also given in Table IV. Note that the
values of F0 and Fx differ (because the known and unknown
soils have different values of d, T, K0, K1 and K2). In further
calculations, based on the use of the derived approximate
formulas, unknown soils will be characterized by d0, T0, K0,
K1 and K2 (which describe the known soil analyzed in 8), as
well as by the calculated values of F0 given in Table IV.
The theoretical grounding resistances of the square loops
selected in the third paragraph of subsection B (R0UL and R0LL)
were calculated using (1). Their contact resistances (RCUL and
RCLL) were calculated by (7), using d0, the values of K0, K1
and K2 given in (8) and the values of F0 given in Table IV.
For the 12 considered types of the surrounding soil the total
grounding resistances of the considered square loops (RTUL
and RTLL) were calculated by (6) and given in Table IV. The
total grounding resistances were also obtained by FEM
calculations (the surrounding soil was modeled by the
previously adopted parameters dx, Tx and Fx). They are given
in Table IV, together with the relative errors (

RTUL and

RTLL), taking the values obtained by FEM calculations as
referent. The fact that the results obtained by the proposed
procedure, based on (1) and (7), deviate less than 4% from
those obtained by FEM calculations confirms the starting
assumption that it is possible to determine the total grounding
resistance of a square loop buried in the soil characterized by
unknown parameters (dx, Tx and Fx) using the information
obtained by the measurement of the grounding resistance of a
small square loop buried in that soil.
C. Calculation of the Total Grounding Resistance of the
Considered Complex Grounding System Shown in Fig. 1
The final step of the validation of the new algorithm was
related to the total grounding resistance of the grounding
systems GS1 and GS2 buried in different types of soil. Both
theoretical and contact resistances of the square loops
analyzed in Table IV were used in further analysis.
Conventional formulas for estimating the grounding
resistance of complex grounding systems contain coefficients
reflecting the mutual (vicinity) effect existing between their
elements 11, 12, 22. For the considered complex
grounding systems GS1 and GS2, assuming a perfect contact
between the electrodes and the surrounding soil, (2) and (3)
apply. Incorporating the sets of the corresponding constants
(C1, C2, C3, C4) given in Section II into (4), the corresponding

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η coefficients were obtained for both GS1 and GS2 (see
Table III).
The hypothesis used in further analysis was that the mutual
(vicinity) effect existing between the elements of the
considered grounding system only influences their theoretical
grounding resistances, and not their contact resistances. This
assumption was based on the finding reported in 8 that in
cases with imperfect contact the distribution of the potential
becomes uniform very close to the grounding electrode (at a
distance of around 0.1 m). This means that imperfect contact
does not influence the distribution of the current outside this
narrow region, and, therefore, does not significantly affect the
grounding resistances of other “distant” elements of the
grounding system. Hence, it was assumed that the total
grounding resistances of all four lower loops (RT4LL) and of
the whole grounding system (RTGS) can be obtained using the
following formulas:








LL
LL
CLLLLT RRR 0
4
4
1
4
1

, and (10)
LLULLL
LL
CLL
LLUL
UL
CUL
TGS R
R
R
R
R
44
0
4
0
411
 





. (11)
The total grounding resistances RT4LL and RTGS for the
considered grounding systems GS1 and GS2, buried in the
selected types of the surrounding soil, were calculated by (10)
and (11), respectively, and given in Table IV. The grounding
resistances were also obtained by FEM calculations (the
surrounding soil was modeled by the previously adopted
parameters dx, Tx and Fx). They are also given in Table IV,
together with the relative errors (

RT4LL and

RTGS), taking the
values obtained by FEM calculations as referent. The fact that
the proposed procedure, together with (10) and (11), produces
results deviating less than 4% from those obtained by FEM
calculations confirms that this procedure allows estimation of
the total grounding resistance of the considered complex
grounding system with good accuracy. It also confirms
validity of the hypothesis that mutual (vicinity) effect existing
between the elements of the considered grounding system
only influences their theoretical grounding resistances.
D. Steps of the Proposed Algorithm
1) While conducting the soil resistivity measurements at
the site where the complex grounding system is going
to be installed, additional measurement of the
grounding resistance of a small square loop should be
performed,
2) Using the measured soil resistivity and (1), the
theoretical grounding resistance of the small loop
should be calculated. Using its value, the measured
grounding resistance and (9), the soil properties
(regarding the soil contact with the electrodes) should
be quantitatively described by d0, T0 and the calculated
value of F0),
3) Using the value of F0 and (7), the contact resistances of
both (upper and lower) loops of the complex grounding
system should be calculated,
4) Using (1), the theoretical grounding resistances of those
loops should also be calculated,
5) Using (2) and (3), coefficients ηUL–4LL and η4LL,
reflecting the mutual (vicinity) effect existing between
the grounding system elements, should be calculated,
and
6) Using the calculated theoretical and contact resistances,
the coefficients ηUL–4LL and η4LL, as well as (10) and
(11), the total grounding system resistance should be
calculated.
IV. DISCUSSION
A method for the derivation of simple formulas for
quantitative estimation of soil properties related to the soil
contact with the electrodes, which is given in 8, is general.
Therefore, it can be applied to any type of elements of a
complex grounding system surrounded by any type of soil
characterized by imperfect contact with the electrodes. A
method for generating approximate formulas intended for fast
calculations of the grounding resistance of any type of
complex grounding systems, based on the coefficients
reflecting the mutual (vicinity) effect existing between their
elements, which is given in 22, is also general. Therefore,
being based on the methods given in 8 and 22, the
presented method for deriving a set of simple formulas for
estimating the total grounding resistance of complex
grounding systems, with the contact resistance included, is
general and can be applied to any type of complex grounding
system, as well as for any type of the surrounding soil.
Without using the proposed algorithm, the designer would
have to rely only on conventional formulas for estimating the
grounding resistance ((1)–(4) in the considered case). This
way the designer equalizes the previously defined total
grounding resistances of the considered complex grounding
system (RTLL, RTUL, RT4LL, RTGS) with their theoretical values
(R0LL, R0UL, R04LL, R0GS), which, calculated using (1)–(4), for
ρ = 100 Ωm amount to (27.65, 8.93, 9.36, 6.75) Ω for GS1,
and (8.96, 5.01, 4.05, 3.56) Ω for GS2. However, if GS1 or
GS2 is buried in a soil characterized by imperfect contact with
the electrodes, the real grounding resistances (RTLL, RTUL,
RT4LL, RTGS) are much higher, which will be confirmed by the
mandatory measurements performed after installing the
grounding system. In such cases, additional grounding
electrodes would have to be installed in order to achieve the
designed (required) value of the grounding system resistance.
However, without using the proposed procedure the designer
would not be able to determine the quantity of the additional
electrodes needed. Besides, the total labor and equipment
costs would be considerably higher than those specified in the
design.
Comparing the total grounding resistances (RTLL, RTUL,
RT4LL, RTGS) given in Table IV, calculated applying FEM on
the twelve considered types of the surrounding soil, with their
theoretical values (R0LL, R0UL, R04LL, R0GS), it was obtained that

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the relative errors amount up to (864.3, 556.5, 647.1,
398.8) % for GS1, and up to (746.9, 609.9, 430.7, 299.2) %
for GS2. Taking into account that the experimentally obtained
values of the square loop grounding resistance presented in
7 differ from the theoretical ones from 238% to 1354%, it
can be concluded that the cases analyzed in this paper are
quite realistic.
Let us emphasize that the presented algorithm, applied
using (1)–(11), produces results deviating less than 4% from
those obtained by the FEM calculations for all of the
considered cases and each of the following grounding
resistances: RTLL, RTUL, RT4LL and RTGS.
In order to validate the proposed algorithm, as well as the
derived formulas (1)–(11), the algorithm was applied to the
grounding system presented in Fig. 1 once again, but now
using the method from [29] (see Appendix II). Comparing the
total grounding resistances (RTLL, RTUL, RT4LL, RTGS) given in
Table V, calculated applying FEM to the twelve considered
types of the surrounding soil, with the corresponding
theoretical grounding resistances (R0LL, R0UL, R04LL, R0GS),
calculated by (12)–(14), it was obtained that the relative
errors amount up to (879.3, 499.5, 667.6, 390.1) % for GS1,
and up to (737.3, 545.0, 412.4, 265.7) % for GS2. However,
comparing the total grounding resistances (RTLL, RTUL, RT4LL,
RTGS), calculated using the presented algorithm and (15)–(17),
with their values calculated applying FEM, it was obtained
that the relative errors amount up to (20.78, 17.68, 29.45,
13.86) % for GS1, and up to (31.10, 31.97, 22.34, 18.18) %
for GS2. These results confirm that the use of the presented
algorithm can significantly reduce the possible errors when
estimating the grounding system resistance in an early phase
of the design process, even using the modified formulas from
[29] (the corresponding initial formulas are not intended for
calculating the total grounding resistance in cases where the
grounding system is buried in soils characterized by poor
contact with the electrodes). Note that appropriate formulas
from [29] (modified as done in this research) can be used with
the proposed algorithm for an approximate calculation of the
grounding resistance for practically all grounding systems
(not only for the one presented in Fig. 1). However, for cases
where high accuracy is required, the method presented in this
paper should be used for the derivation of the designer
oriented formulas.
The authors are also dealing with an experimental
validation of the proposed algorithm and the derived formulas
(1)–(11). Small scale wire models of the grounding system
presented in Fig. 1 have been constructed, with the controlled
size of the isolating material sequentially placed along the
wires (it simulates the air gaps). Their electrical behavior is
being examined in an electrolytic tank filled with water.
Although the initial results of the measurements are highly
encouraging, some additional adjustments of the models,
electrolytic tank and measuring equipment are necessary
before the details regarding the experimental setup and the
obtained results can be published.
V. CONCLUSIONS
An algorithm for estimating the total grounding resistance
of complex grounding systems, including both the theoretical
and contact resistances, characterized by high accuracy, is
presented in this paper. It is based on additional measurement
of the grounding resistance of a simple test grounding system,
and the methods given in 8 and 22 intended for fast
calculations of the total grounding resistance of any type of
complex grounding system.
The proposed method was applied on a grounding system
of a typical 110 kV transmission line tower used in the
Serbian transmission power system. A set of simple formulas
which represent the core of the algorithm were derived and
given in the paper. Applying them the total grounding
resistance of the grounding system sketched in Fig. 1 can
easily be calculated for any combination of the input
parameters belonging to the ranges given in Table I and for
any type of the surrounding soil.
The results obtained by the application of the proposed
algorithm were validated using 3D FEM modeling and a
suitable practical method from [29]. It was shown that they
deviated less than 4% from those obtained by FEM
calculations, confirming high accuracy of the proposed
algorithm.
Both the algorithm and the method for deriving simple
formulas are general and can be applied to any type of
complex grounding system. They represent a powerful tool
for estimating the grounding resistance in an early stage of the
design process.
APPENDIX I
THE PROCEDURES FROM [8] AND [22] INTENDED FOR
DETERMINING FORMULAS AND COEFFICIENTS
Equations (1)–(4) and coefficients C1–C4 adopted from
[22], as well as (6)–(8) adopted from [8], were derived by
numerical analyses of the results obtained by the 3D FEM
modeling of the considered grounding systems.
A. Equation (1)
A large number of square grounding loops, the parameters
of which belong to the ranges given in Table I, were modeled
applying 3D FEM. Analyzing the obtained results, it was
concluded that logarithmic dependence should be included in
function R(ρ,LT,h,p), adopting only one constant in the
fraction within the logarithm. This constant was determined
varying its value by Microsoft Excel Solver [30] (the adopted
constant corresponds to the minimum value of the sum of
squares of differences between the results obtained by 3D
FEM and the R(ρ,LT,h,p) function values). Equation (1) was
obtained as a final result of the described procedure.
B. Equations (2)–(4) and the Coefficients C1–C4
A simple method for determining the theoretical grounding
resistance of a complex grounding system is presented in [11]
and [12]. The method, based on the principles of
superposition and reciprocity, results in a simple formula for
the calculation of the grounding system resistance. Applying

0093-9994 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See
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this method to the grounding system shown in Fig. 1, (2) and
(3) are obtained.
The values of R, RUL, R4LL and RLL for a large number of
grounding systems such as the one shown in Fig. 1,
parameters of which belong to the ranges given in Table I,
were obtained by the 3D FEM calculations. By incorporating
those values into (2) and (3), the η (ηUL–4LL and η4LL)
coefficients were calculated for each considered grounding
system. It was noticed that the obtained values of the η
coefficients are mainly dependent of L1 and r. Therefore, it
was convenient to determine the function η(L1, r), which was
realized by fitting several functions through the obtained data
(again, using both the method of least squares and the
iterative calculation method through Microsoft Excel Solver).
The most precise approximation was achieved by (4), where
C1–C4 are the constants, the values of which were varied until
the minimum value of the sum of squares of differences
between the results obtained by 3D FEM and (4) was
achieved.
C. Equations (6)–(8)
A grounding loop (the parameters of which are given in
the subsection III.A.), with air gaps (characterized by
d0 = 16·10-3 m and T0 = 25·10-2 m) sequentially placed along
its steel strips, was modeled applying 3D FEM for several
different values of the parameter F. By examining the values
of the grounding loop resistance, it was noticed that the
resistance, as a function of F, can be approximated by (6) and
(7). Again, using Microsoft Excel Solver, the coefficients K0–
K2 given in (8) were obtained, providing the minimum value
of the sum of squares of differences between the results
obtained by 3D FEM and (6)–(8).
APPENDIX II
THE METHOD FROM [29] FOR CALCULATING THE GROUNDING
RESISTANCE OF THE GROUNDING SYSTEM PRESENTED IN FIG. 1
A. Calculation of the Theoretical Grounding Resistance of
Complex Grounding Systems
According to the method presented in [29], any complex
grounding system can be approximated by an electrode that
extends vertically to the soil surface and envelops all parts of
the grounding system (the first terms in (12)–(14)),
additionally taking into account the fact that the grounding
resistance of any wire structure is higher than that of the solid
electrode occupying the same volume (the second terms in
(12)–(14)). Applying equations 11.8–11.13 and Table 11.1
from [29], the grounding resistance of many differently
shaped electrodes can be calculated.
For a square loop with the side length L and the perimeter
of the strip cross-section p, buried in a uniform soil at a depth
h, (12) applies (suitable for calculating R0LL and R0UL). If the
grounding system consists of only 4 lower grounding loops
presented in Fig. 1, (13) can be used. For the whole
grounding system presented in Fig. 1, (14) applies.
For ρ = 100 Ωm, the theoretical grounding resistances of
the considered complex grounding system (R0LL, R0UL, R04LL,
R0GS), calculated using (12)–(14), amount to (27.22, 9.78,
9.11, 6.87) Ω for GS1, and (9.07, 5.52, 4.19, 3.89) Ω for GS2.
These values are very close to the values calculated using (1)–
(4), given in Section IV.
B. Modified Formulas for Calculating the Total Grounding
Resistance of the Grounding System Presented in Fig. 1
Although formulas (12)–(14) were not initially intended
for calculating the total grounding resistance in cases where
the grounding system is buried in soils which form poor
contact with the electrodes, the second term in each of them
can be modified in order to provide their additional purpose.
Simply by including the parameter F, (12)–(14) become (15)–
(17), respectively.
The proposed algorithm for estimating the total grounding
resistance can also be applied with these formulas,
emphasizing that in this case the steps of the proposed
algorithm (given in subsection III.D. for the derived formulas
(1)–(11)) become:
1) While conducting the soil resistivity measurements at the
site where the complex grounding system is going to be
installed, additional measurement of the grounding
resistance of a small square loop should be performed,
2) Using the measured soil resistivity and (15), the soil
properties (regarding the soil contact with the electrodes)
should be quantitatively described by the calculated
value of F, and
3) Using the value of F and (15)–(17), all total grounding
resistances of the considered complex grounding system
(RTLL, RTUL, RT4LL, RTGS) should be calculated.
The results of the application of the proposed algorithm
using formulas (15)–(17), compared with those obtained by
3D FEM, are presented in Table V. Note that the theoretical
value of the loop grounding resistance, calculated using (12)
and assuming ρ = 100 Ωm, amounts to R0test = 33.38 Ω.





























p
hL
LhLL
hL
hL
RSL 8
4
ln
4
1
42
217
ln
2
2
2
22
22
0


(12)





























pr
hL
rLhLL
hL
hL
RLL 32
4
ln
16
1
42
217
ln
2
2
2
21
1211
2
2
2
1
2
2
2
1
04


(13)

 






























rp
hL
rLhLL
hL
hL
RGS 418
4
ln
414
1
42
217
ln
2
2
2
21
1211
2
2
2
1
2
2
2
1
0


(14)

0093-9994 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
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
































Fp
hL
FLhLL
hL
hL
RTSL 100
100
8
4
ln
100
100
4
1
42
217
ln
2
2
2
22
22


(15)

































Fpr
hL
FrLhLL
hL
hL
RLLT 100
100
32
4
ln
100
100
16
1
42
217
ln
2
2
2
21
1211
2
2
2
1
2
2
2
1
4


(16)

 


































Frp
hL
FrLhLL
hL
hL
RTGS 100
100
418
4
ln
100
100
414
1
42
217
ln
2
2
2
21
1211
2
2
2
1
2
2
2
1


(17)
TABLE V
COMPARISON OF THE RESULTS OBTAINED BY THE PROPOSED ALGORITHM, BASED ON (15)–(17), AND BY THE FEM CALCULATIONS
GS Fx
(%)
RTtest
(Ω)
FEM

RTtest
(%)
F
(%)
RTLL
(Ω)
(15)
RTLL
(Ω)
FEM

RTLL
(%)
RTUL
(Ω)
(15)
RTUL
(Ω)
FEM

RTUL
(%)
RT4LL
(Ω)
(16)
RT4LL
(Ω)
FEM

RT4LL
(%)
RTGS
(Ω)
(17)
RTGS
(Ω)
FEM

RTGS
(%)
GS1 0 32.84 -1.63 -5.37 26.45 28.32 -6.60 9.63 9.12 5.52 8.89 9.57 -7.17 6.79 6.83 -0.62
GS1 50 38.46 15.21 30.55 34.22 33.99 0.70 11.13 10.28 8.27 11.12 10.95 1.51 7.61 7.41 2.60
GS1 80 54.08 61.99 61.48 54.89 49.51 10.86 15.13 13.42 12.73 16.96 14.95 13.48 9.79 9.23 6.10
GS1 92 90.28 170.42 79.45 101.01 85.71 17.86 24.09 20.59 17.00 29.83 24.06 23.97 14.70 13.37 9.94
GS1 96 149.23 346.99 87.67 174.14 144.72 20.33 38.35 32.59 17.68 50.02 39.12 27.85 22.50 20.05 12.26
GS1 98 270.87 711.37 92.91 321.98 266.59 20.78 67.24 58.60 14.74 90.50 69.91 29.45 38.35 33.68 13.86
GS2 0 32.84 -1.63 -5.37 8.88 8.99 -1.21 5.44 5.12 6.21 4.15 4.06 2.18 3.86 3.59 7.48
GS2 50 42.76 28.08 43.80 12.20 11.06 10.24 6.91 6.09 13.37 4.94 4.63 6.64 4.31 3.81 13.27
GS2 80 68.22 104.36 71.64 20.23 16.49 22.66 10.46 8.54 22.50 6.87 6.00 14.57 5.43 4.71 15.38
GS2 92 120.08 259.67 84.73 35.89 28.13 27.59 17.39 13.72 26.78 10.66 9.00 18.36 7.65 6.64 15.22
GS2 96 197.21 490.72 90.52 58.56 45.04 30.02 27.42 20.77 31.97 16.15 13.20 22.34 10.89 9.26 17.66
GS2 98 339.53 917.03 94.21 99.51 75.91 31.10 45.52 35.60 27.88 26.10 21.48 21.49 16.80 14.22 18.18
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Jovan Trifunovic was born in Belgrade, Serbia, in
1979. He received the Dipl.Ing.El. degree and the
M.Phil. degree from the Faculty of Electrical
Engineering, University of Belgrade (Serbia), in 2003
and 2009, respectively. He is now a Ph.D. candidate at
the same University, where he has been working as an
assistant since 2005. His areas of interest currently
include grounding systems, low-voltage electrical
installations and energy efficiency.
Miomir B. Kostic was born in Vranje, Serbia, in
1956. He received the Dipl.Ing.El., M.Sc. and Ph.D.
degrees from the University of Belgrade, Serbia, in
1980, 1982 and 1988, respectively, all in electrical
engineering. In 1980 he joined the University of
Belgrade, where he is presently employed as
Professor. His current research interests include
grounding systems, low-voltage electrical
installations, energy efficiency in public lighting and
architectural lighting.