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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

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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)

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10.1109/TIA.2015.2429644, IEEE Transactions on Industry Applications

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.