Instruction


FACTA UNIVERSITATIS  
Series: Electronics and Energetics Vol. 27, N

o
 1, March 2014, pp. 25 - 39 

DOI: 10.2298/FUEE1401025P 

 

DETERMINATION OF ACTUAL REDUCTION FACTOR OF HV 

AND MV CABLE LINES PASSING THROUGH URBAN AND 

SUBURBAN AREAS

 

Ljubivoje M. Popović 

School of Electrical Engineering, Beograd, Serbia 

Abstract: The paper presents the method for determination of ground fault current 

distribution in the cases when feeding cable lines are passing through urban and/or 

suburban areas, or when many relevant data are uncertain, or completely unknown. 

The problem appears as a consequence of the fact that many of surrounding urban 

installations are situated under the surface of the ground and cannot be visually 

determined or verified. On the basis of on site measurements, the developed method 

enables compensation of all deficiencies of the relevant data about metal installations 

involved with the fluctuating magnet field appearing along and around of a power line 

during an unbalanced fault. The presented analytical procedure is based on the fact 

that certain measurable quantities cumulatively involve the inductive effects of all, 

known and unknown surrounding metal installations. The performed quantitative 

analysis points on at the significance of taking into account the existence of 

surrounding metal installations. 

Key words: substation, grounding system, ground fault current, inductive influence  

1. INTRODUCTION 

The fault current during an earth fault in a power network has at least two alternative 

paths for returning to the source which feeds the fault. Because of that, each ground fault 

current has at least two fractions. One of them is injected into surrounding earth through 

the grounding system of a supplied substation, while the other is returning to the source 

of origin through the neutral conductor(s) of the feeding line [1]. The first one produces 

all potentials and potential differences (touch and step voltages) relevant for the safety 

conditions on the grounding system of a supplied substation, while other causes the 

thermal stress on the neutral conductor(s) of the feeding line. Thus, the correct estimation 

of the ground fault distribution is of prime importance for correct designing of the 

grounding system of a supplied substation and correct selection of the feeding line neutral 

conductor(s). 

                                                           

 Received December 24, 2013 

Corresponding author: Ljubivoje M. Popović 

School of Electrical Engineering, Beograd, Serbia 
(e-mail: ljubivoje@beotel.net) 



26 LJ. POPOVIĆ 

 

With the aim of defining the influence of available return paths on the ground fault 

current distribution, a special parameter of feeding line is introduced in professional 

literature, including technical standards [2]. This parameter is called the reduction factor 

of the feeding line and is defined as the ratio of the part of the ground fault current 

returning through earth and the total ground fault current. By this, it is assumed that the 

grounding impedances at both line ends are negligible (e.g. [2]). Under such assumption 

the fault current(s) in the line neutral conductor(s) is a consequence solely of inductive 

coupling between this/these conductor(s) and the phase conductor through which the total 

ground fault current passes.  

With the aim of solving the problem of determination of ground fault current 

distribution, an extensive and continuous research work has been done in the last at least 

five decades.  

The firstly developed methods for solving this problem relate to the case when the 

feeding line is constructed as an overhead line, e.g. [3]-[5]. Somewhat later, the papers 

considering this problem in special cases, i.e. when a feeding line appears as a 

longitudinal combination of one cable and one overhead section, have been published, 

e.g. [6], [7]. Also, several methods have been developed for solving the problem in cases 

where, because if high local soil resistivity and/or a high short-circuit level, special 

measurements (e.g. bare copper conductor laid in the same trench as the cable returning 

line, counterpoises, etc.) in the aim of reducing the part of ground fault current returning 

exclusively through the earth are considered indispensable [8]-[10]. Then, the papers 

[11], [12] present the method developed for determination of the ground fault current 

distribution when a feeding cable line is constructed of three single-core cables. The 

method enables taking into account the participation of all three metal sheaths on ground 

fault current distribution for the fault at any place along the line.  

The common characteristic of the mentioned methods is that the value of the 

reductions factor depends only on design/constructive characteristics of a feeding line 

and on characteristics of the surrounding earth as a conductive medium. Thus, none of the 

mentioned methods enables obtaining the solution of this problem when it appears in 

urban and suburban conditions, when many other metal installations participate in the 

ground fault current returning to the power system. In such surroundings each power-

cable line represents a very complex electrical circuit with many conductively and 

inductively coupled elements having uncertain or completely unknown parameters. 

The problem of determination of the influence of metal installations surrounding a 

feeding cable line on ground fault current distribution through the grounding system of a 

supplied substation is considered and solved for the first time in [14]. The solution is 

achieved by substituting all surrounding metal installations by one, from the standpoint of 

the ground fault current distribution, equivalent conductor of cylindrical form placed 

around and along each considered cable line. Somewhat later, the achieved solution 

extended to include the overhead distribution lines [15].  

The investigation results presented in [14], [15] show that part of the ground fault 

current flowing through the earth, in typically urban environments, is three to five times 

smaller than it has been considered earlier. The developed method enables a new insight 

into the whole grounding problem of urban HV/MV substations and dramatically changes 

our perception about the magnitude of this problem. It can be seen in realistic 

frameworks and solved in each concrete case without any redundant expenditure.  



 Determination of Actual Reduction Factor of HV and MV Cable Lines Passing Through Urban and... 27 

 

The developed method simultaneously gives possibility of solving another problem of 

the current engineering practice that has also not been solved in the past. This is the 

problem of determination of the feeding line series impedance without ignoring the fact 

that the surrounding metal installations are attendant. Namely, on the basis of the 

imagined physical appearance of the introduced equivalent conductor, it is not difficult to 

see that this conductor acts as an additional neutral conductor of each distribution line 

passing through urban and suburban areas and, in accordance with this, improves its 

transfer characteristics [16].  

This paper presents the theoretical foundations of the mentioned method in the more 

transparent and complete manner and introduces certain improvement in the development 

procedure of the calculative part of the method. This improvement enables a more direct 

and easier determination of the actual ground fault current part dissipated through the 

grounding system of the supplied substation into the surrounding earth and returning to 

the power system only through the earth.  

2. PROBLEM DESCRIPTION 

The increasing sizes of modern distribution networks, as well as the higher operating 

and short-circuit currents of these networks, have been matched by over-spreading 

networks of earth return circuits (different pipelines, different cable and overhead line 

neutrals, etc.) close to HV and MV distribution lines. Space dispositions of all of these 

installations, determined mainly by dispositions of city streets, and small mutual 

distances result in an inductive and, in the vicinity of substations, conductive coupling of 

different network types. 

The usage of common routes (mainly street pavements) for positioning various supply 

networks (electricity, water, gas, oil, telecommunications, etc.) unavoidably leads to the 

appearance of their mutual interaction that should be determined in many concrete cases. 

The whole problem has at least three different aspects important for the current 

engineering practice. They are: 

 influence of the surrounding metal installations on the fault current dissipated into 
the surrounding earth through the grounding system of a supplied substation [14], 

[15],  

 influence of the surrounding metal installations on the transfer characteristics of 
power lines passing through urban and suburban areas [16], and 

 inductive influence of an HV power line on each of the surrounding metal installations 
considered separately. 

One of the main parameters for estimation of these mutual interactions, soil resistivity 

of the surrounding area, can not be determined exactly. Although there are several 

methods to measure the soil resistivity [1], no one of them can be practically applied in 

urban conditions. The reason stems from the fact that the surface of urban areas are 

already covered/occupied by buildings, streets, pavements and many other permanently 

constructed objects; while under the ground surface many known and unknown metal 

installations already exist.  

Many urban metal installations of different basic functions are usually situated in a 

relatively small space, like: sheaths of different types of cable lines, neutral conductors of 

the low voltage network, steel water pipes, building foundations, etc. Some of them are 



28 LJ. POPOVIĆ 

 

not in direct contact with the earth, while the others are in an effective and continuous 

contact with the earth. They are interconnected and their spatial dispositions are different 

in each concrete case and vary along any of the distribution lines. Also, most of them are 

laid under the street pavements and many relevant data about them cannot be notified and 

visually determined or verified.  

Grounding system of a distribution HV/MV substation consists of the substation 

grounding electrode and many outgoing MV cable lines acting as external grounding 

electrodes, and/or conductive connections with the grounding systems of the supplied 

MV/LV substations [17]. The spontaneously formed grounding system involves a large 

urban area around an HV/MV substation. Such grounding system includes, through the 

terra-neutral (TN) grounding system in the low-voltage (LV) network and consumer 

installations, many, known and unknown, metallic installations typical for an urban area. 

As a consequence, the outgoing cable lines simultaneously become the conductive 

connections with the metal installations laid along the same street(s) as the feeding line. 

Thus, it is not difficult to notice that in the case of unbalanced operating conditions two, 

essentially different, currents flow out of the power system. One of them is dissipated 

into the surrounding earth through the grounding system of the supplied substation, while 

the other is induced in the metal installations surrounding the feeding line. As an 

illustration, these two, mutually separated fault currents, when a ground fault occurs in 

the substation supplied by a three-phase line, are presented in Fig. 1.  

 

Fig. 1 Current fractions passing through the elements of the grounding systems  

The used notation has the following meaning: 

A – supply substation, 

F  – ground fault place (supplied substation), 

If  – ground fault current, 

Ii  – ground fault current component induced in metal installations surrounding the 

feeding line, and  

Ie  – ground fault current component injected into the earth. 

Both of the presented ground fault current fractions leave power system through the 

elements of the grounding system of the supplied substation, F. The current, Ie, is injected 

into the earth, while the other, Ii, circulates only through the surrounding metal 

installations that are foreseen for some others purposes. Because of that all potentially 

dangerous and harmful influences of power - cable lines and supplied substations on their 

environment emanate from these two ground fault current components. Accordingly, 



 Determination of Actual Reduction Factor of HV and MV Cable Lines Passing Through Urban and... 29 

 

determination of these two currents is of prime importance for estimating the safety 

conditions within, and in the vicinity, of a supplied substation and inductive influence of 

a feeding line on the neighboring parallel circuits (pipeline, telecommunication line, etc).  

Since the process of splitting to these two fractions occurs along many external 

grounding electrodes and under the surface of the ground, none of these components can 

be separated and determined by direct or indirect measurements [14], [15]. 

Each of the metal return paths, together with earth as the common return path, forms 

an electrical circuit, while all together these paths form a large number of conductively 

and inductively coupled circuits. Since these circuits can be represented by the 

corresponding system of equations and since the analytical expressions necessary for the 

self and mutual impedances of metal conductors are known [2], it can be said that the 

considered problem has been in principle solvable long ago. However, it is not possible 

because of many practical difficulties and limitations in collecting for calculations 

necessary data. Thus, the problem can be defined as follows: How to find the method 

enabling the compensation of the lack of huge number of relevant data? 

3. EXPERIMENTAL INVESTIGATION 

The experimental investigations of the influence of metal installations, typical of 

urban areas, on value of the feeding line reduction factor are performed on a cable line 

that supplies, in series, two substations of the 110 kV distribution network in Belgrade, 

Serbia [14]. Length of the line to the closer of the two supplied substations, measured 

from the supply substation, is 2320 m, while the feeding line length to the more distant 

substation is 6590 m. The line is realized by XHLP cables having mutually identical 

design parameters, laid in a triangular formation over the entire line length. The cross-

bounding technique necessary for reduction of circulating currents was not applied. 

In the areas through which the line is passing the specific soil resistivity is estimated 

on the basis of the main geological characteristics of the involved area, the only possible 

way for doing this in urban areas. The roughly estimated equivalent soil resistivity of the 

entire area is within the range from 30 to 50 Ωm. The line section between the supply and 

the transit (nearer) substation passes through the area with a lower degree of urbanization 

compared with the rest of the line. The phase conductors are made of aluminum of a 

cross-section of 1000 mm
2
, while the metallic sheaths are made of copper strings of a 

total cross-section of 95 mm
2
 and a medium diameter of 91 mm. The main elements of 

the grounding system in both supplied substations are the station building foundation and 

the 44 MV outgoing cable lines performed by cables with uncovered metal sheaths. The 

grounding impedances of the supply substation and of both of the supplied substations, 

determined by standard measurements, were found to be between 0.02 and 0.03 Ω. Since 

these impedances are very small compared to the other parameters influencing measured 

values of the reduction factor, they are completely disregarded. 

The described line is used to obtain the experimental results for two different feeding cable 

lines, one 2320 m and the other 6590 m long. This was achieved in the following manner. The 

longer line is obtained as a continuous cable line along its entire length by disconnecting its 

metal sheaths from the grounding electrode of the transit substation. The necessary 

measurements are performed by using the test circuit schematically represented in Fig. 2. 



30 LJ. POPOVIĆ 

 

 

Fig. 2 Measurement circuit  

The used notation has the following meaning: 

A and B – substations connected through the tested cable line, 

Ua  – auxiliary voltage source,   

It  – test current, 

Is  – current induced in the cable sheath, 

Ic  – total current induced in the surrounding metal installations, 

Ie  – current injected into the earth through the grounding system of substation B, 

ZA (ZB )  – impedance of the grounding system of substation A(B), and 

G  – remote ground. 

The influence of the surrounding metal installations has been observed by measuring 

value of the self - impedance of one of the line phase conductors and by using the known 

analytical expressions for this impedance that is, according to e.g. [2], given by 

 
Ω/km,;ln

428
'' 00
















ph

r
phph

r
jRZ











 (1) 

where 

R'ph – phase conductor resistance per unit length, /km, 

rph  – outer radius of the phase conductor, m,

  – angular frequency = 2f, 

0  – magnetic permeability of vacuum, 4∙10
–7

 Vs/Am, and 

r  – relative magnetic permeability. 
(Prime (') denotes values per unit length). 

The equivalent earth penetration depth  is determined by the following expression 

 ,m;658
f


   (2) 

where 

 – equivalent soil resistivity along the cable line in Ωm, and 
f – test circuit frequency. 



 Determination of Actual Reduction Factor of HV and MV Cable Lines Passing Through Urban and... 31 

 

Here, it should be mentioned that these expressions are based on Carson's theory of 

the current return path through the earth. They have been derived under the assumptions 

that the power line is laid in a homogeneous soil of a resistivity equal to the equivalent 

resistivity of the normally heterogeneous (multilayer, with each layer having different 

resistivity) soil and that there are no other metal installations in the vicinity of the line. 

However, these assumptions do not correspond to the described actual situation. 

Since the considered cable lines pass through areas covered by a spontaneously 

formed network of different underground metal installations, the measured quantities are 

affected by the conductive and inductive couplings of all, known and unknown, 

surrounding metal installations. Thus, measurements can give only an apparent value of 

the self impedance of the line phase conductor.  

By simulating single-phase ground fault in the supplied substation and by disconnecting 

all three cable sheaths from the grounding electrode at one of the line ends (Fig. 2), the 

following values for the apparent phase conductor self-impedance were obtained 

 Zpha = (0.1819 + j1.0243) Ω, for the line 2320 m, and 

 Zpha = (0.5167 + j2.6260) Ω, for the line 6590 m. 

The values of the line reduction factor obtained only on the bases of the measured values 

of the currents appearing in the cable sheaths were the following 

 r = 0.0473 – j 0.1565, for the shorter line, and 

 r = 0.0637 – j 0.1724, for the longer line.  

When the apparent values of impedance Z'pha are determined, the only unknown 

parameters in the given expressions, (1) and (2),  and δ can be obtained by using these 

equations. However, these parameters in the considered cases involve, besides local earth 

characteristics, the constructive characteristics and mutual space disposition of all others 

available return paths. According to this, they should be defined in somewhat different 

way. A new definition of the parameter is that it represents the apparent equivalent soil 

resistivity, because it involves the conductive and inductive influences of all return paths, 

while the parameter δ is the equivalent penetration depth all return paths because it 

involves the conductive and inductive influences of all return paths. Thus, the new 

notation which expresses this new meaning is  and δa.   

The corresponding values of the newly defined parameters are: 

 δa = 20.9 m, or ρa = 0.052 Ωm, for the shorter line and 

 δa = 10.88 m, or ρa = 0.0137 Ωm, for the longer line. 

The obvious difference between the two values can be explained by the fact that the 

shorter line passes through an area of a lower degree of urbanization, e. i. along lower 

number of surrounding metal installations. 

Since the design data concerning the tested lines, as well as the newly defined 

parameters  or δa are known, the analytical expression, for the reduction factor in the 

case of single-core cables laid in a triangular formation, can be tested. This expression 

has, according to [13], the following mathematical form 



32 LJ. POPOVIĆ 

 

 

3 2

00 ln
2

3
8

3'

'

dr
jR

R
r

S

a
S

S









 


 (3) 

where 

R'S  metal sheath resistance per unit length, /km, 

d   distance between two adjacent cables (or, diameter of the single-core cable) in the 

case of triangular formation, and 

rS   medium radius of the cable sheath. 

By applying the above expression to the considered cable lines one obtains: 

 r = 0.0471 – j0.1537, for the shorter and 

 r = 0.0589 – j0.1693, for the longer line. 

It can be seen that the results obtained by the applied expression and the results obtained 

by the measurements are in good agreement, practically identical. In this way the 

experimental proof has been obtained for the accuracy of the given analytical expression, 

but under the unreal assumption that the surrounding metal installations do not exist.  

Based on the experimental results, the following facts can be still noted. The apparent 

equivalent soil resistivity obtained in this way is drastically lower compared to the realistic 

soil resistivity (between 30 and 50 Ωm). The corresponding values of the reduction factor 

are by 52,1% and 69.6 % higher than the value obtained with the approximately estimated 

equivalent specific soil resistivity, r( =30 Ωm)= 0.0204 – j0.1037. This can be explained 
by the fact that the presence of the nearby underground metal installations reduces not only 

of the fault current flowing through the earth, but also the currents flowing through the 

cable line sheaths.  

For obtaining a more complete insight into the influence of metal installations 

surrounding a feeding line, currents through the cable sheaths were also experimentally 

investigated. At first, a ground fault current distribution was observed when only the 

sheath of the cable with the ground fault current was connected to both grounding 

electrodes at the line ends. Then another situation was considered, i.e. when one more 

cable sheath was connected to both of the grounding electrodes at the line ends. Finally, 

normal operating conditions have been observed, i.e. when all three metal sheaths were 

grounded at the line ends.  

The measurement results show that the successive increase of the number of grounded 

metal sheaths (Fig. 2) reduces not only the current flowing through the earth, but also the 

relative participation of each of the already connected sheaths in reducing the fault 

current through the earth. These relative reductions in the case of the sheath of the cable 

with simulated ground fault are from 60.66% to 46.31% and from 46.31% to 37.49%, 

respectively. These results are helpful for understanding the influence of the surrounding 

metal installations. 

Since the reduction factor is defined as the ratio Ie/If, for obtaining the actual value of 

the reduction factor the presence of the surrounding metal installations has to be taken 

into account.  



 Determination of Actual Reduction Factor of HV and MV Cable Lines Passing Through Urban and... 33 

 

4. METHOD DEVELOPMENT 

On the basis of the former analysis, it is clear that each power line passing through 

urban and/or suburban areas during a ground fault represents a very complex electrical 

circuit. This circuit consists of a large number of mutually conductively and inductively 

coupled electrical circuits with common return path through the earth. The number of 

these circuits, if the line phase conductors are excluded, is equal to the number of the line 

neutral conductors enlarged by the number of surrounding metal installations.  

If again the cable line presented in Fig. 2 is considered, but now, with all metal 

sheaths grounded at both line ends, and if it is assumed that the total number of 

surrounding metal installations, including the neutral conductor, is equal to an arbitrary 

number N, then this line can be represented by the equivalent circuit shown in Fig. 3. 

An arbitrary current in Fig. 3, In, induces in an also arbitrary (m
th

) current circuit, 

voltage, Umn, determined by 

 
nmnmn

IZU   (4) 

where 

Zmn – mutual impedance between two arbitrary (n
th 

and m
th
) surrounding metal installations 

(circuits), Fig. 3. 

It is well known that distribution substations are located in areas occupied by many 

underground metal installations, acting as perfect grounding electrodes [17]. Thus, 

grounding impedances ZA and ZB can be neglected (ZA ≈ 0 and ZB ≈ 0). Because of that, 

the fault currents appearing in the cable line metal sheaths are a consequence solely of the 

inductive influence of the ground fault current appearing in the faulted phase conductor 

[15]. This is in accordance with the formerly mentioned reduction factor definition. Also, 

for further considerations it is necessary to mention that the current directions shown in 

Fig. 3 are taken arbitrarily. 

On the basis of the equivalent circuit presented in Fig. 3 it is possible to write the 

system of (N + 1) equations and, for the known the values of Ua and all self and mutual 

impedances, determine in the given equivalent circuit each of the presented currents. 

Unfortunately, because of the previously mentioned practical difficulties and limitations 

the parameters of the surrounding metal installations necessary for determination of these 

impedances should be treated as unknown quantities. Thus, for solving the problem of 

determining the current circulating through the earth, Ie, and reduction factor of the 

considered line a completely new approach is necessary. 

The problem is solved in [14] by measuring currents It and I1 (Fig. 2) and by 

substituting all surrounding metal installations by one equivalent conductor that is 

imagined as a cylinder placed around and along the entire feeding line. Here, for the sake 

of simplifying the necessary calculation procedure, it will be assumed that this conductor 

also involves 



34 LJ. POPOVIĆ 

 

 

Fig. 3 Complete line equivalent circuit 

The used notation has the following meaning 

Ua – auxiliary voltage source, 

Un0, Un1, Un2, …, UnN – voltages induced in an arbitrary (n-th) circuit by the current in 

each of the surrounding circuits (metal installations), 

It – test current through the phase conductor of the considered line, 

I1,– current through the sheath of the cable with the current It, 

I2, I3 – currents through the metal sheaths of the other two single-core cables, 

I4, I5, I6 , … ,In, …, IN - currents induced in the individual surrounding metal installations 

Z1, Z2, Z3 – self- impedances of cable metal sheaths, and 

Z4, Z5, Z6, … , ZN – self-impedances of the individual surrounding metal installations. 

two sheaths of the remaining single-core cables through which the ground fault current 

do not circulate. Under such assumption the considered cable line is transformed into the 

physical model whose cross-section can be represented as shown in Fig. 4. 



 Determination of Actual Reduction Factor of HV and MV Cable Lines Passing Through Urban and... 35 

 

 

Fig. 4 Cross section of the introduced line model 

For the equivalent conductor imagined in such manner, the corresponding equivalent 

circuit of the entire cable line has the appearance as shown in Fig. 5. 

 

Fig. 5 Equivalent circuit of the introduced line model 

The used notation has the following meaning 

U0C , U1C – voltages that current IC induces in the phase conductor and its metal sheath, 

UC0 , UC1 – voltages that currents It and I1 induce in the equivalent conductor. 

The relevant parameters of the assumed equivalent conductor will be determined 

under condition that currents It and I1 in Fig. 3 remain unchanged after introducing the 

equivalent conductor, Figs. 4 and 5. By using the equivalent circuit presented in Fig. 5 

this condition can be expressed by the following system of equations 

 
0.2

0.1

110

11101





CCCtC

CCt

IZIZIZ

IZIZIZ
 (5) 



36 LJ. POPOVIĆ 

 

where 

ZC  – self - impedance of the equivalent conductor, and 

Z1C  – mutual impedance between the equivalent conductor and the cable sheath. 

Impedances Z1 and Z01 are determined, according to e.g. [14], by the following 

expressions: 

 

S

S
r

jRZ








 ln

28

00
1

 ;  Ω/km (6) 

 

S
r

jZ








 ln

28

00
01

  ;     Ω/km (7)  

For the adopted physical appearance of the equivalent conductor, the analytical 

expressions for impedances ZC, Z0C, and Z1C are  

 

C

CC
r

jRZ








 ln

28

00   ;  Ω/km (8) 

 

C

CC
r

jZZ








 ln

28

00
10

  ;  Ω/km (9) 

where 

rc  – medium radius of the cylinder representing the equivalent conductor, and 

R'C – equivalent conductor resistance per unit length, /km. 

Since currents It and I1 represent the known quantities, obtained by measurements (Fig. 

2), the condition given by (5) can be modified to the folowing simpler form 

 

tCC

CCC

I

I

ZZZ

ZZZZ
1

2

11

0110 


  (10) 

In the given expression the only unknown quantities, according to (6), (7), (8), and 

(9), are R'C and rc. Since (10) gives the relationship between complex quantities, it can be 

presented in the form of the following system of two equations  

 
 

tCCC
IZZZZ

0110
Re 

 =
 

1

2

11
Re IZZZ

CC


  

  (11) 

  
 

tCCC
IZZZZ

0110
Im 

 =
 

1

2

11
Im IZZZ

CC


 

After determining the relevant parameters of the equivalent conductor (R'C and rC) 

relations (8) and (9) can be used for obtaining: Z1C, Z0C and Z1C, as well as:  Ic and Ie. 

Then, in accordance with the equivalent circuit in Fig. 5 and the reduction factor 

definition, the feeding line reduction factor is determined by 

    
2

11

0110111
CC

CCCC

t

e

ZZZ

ZZZZZZ

I

I
r




 , (12) 

or, according to (8), and (9), in somewhat more compact form 



 Determination of Actual Reduction Factor of HV and MV Cable Lines Passing Through Urban and... 37 

 

  
2

11

01101

'

1
CC

CCC

ZZZ

ZZZZR
r




  (13) 

Condition defined by (5) means at the same time:  

 



N

n

nnCCC
IZIZU

2

000
, and (14) 

 



N

n

nnCCC
IZIZU

2

111
 (15)  

On the basis of (14) and (15) it is clear that the introduced equivalent conductor substitutes 

many known and unknown relevant parameters. 

Data about actual reduction factor of a distribution line, or about actual ground fault 

current distribution in a supplied substation, are usually necessary before the line has 

been constructed. In these cases, for performing of the previously defined measurements 

one can utilize a provisory cable line posed on the surface of the soil along the foreseen 

path of the future line. This can be done by using any single-core cable suitable for this 

purpose in different, practically possible conditions and by using calculation procedure 

described here. Since the metal installations in urban areas are mainly under the soil 

surface, the considered inductive influence will be slightly smaller [16].  

According to the developed analytical procedure, the presented method enables 

determination of the reduction factor of any type of cable lines and takes into account all 

relevant factors and parameters, including even those whose contribution is negligibly 

small. It means that this method enables a correct problem solution for each, including 

extremely complex, practical situation. Some inaccuracy can appear only as a consequence 

of the inductive influence of the nearby power distribution lines. This influence can be 

efficiently avoided by using the test current of somewhat higher frequency that can easily 

be discriminated from the omnipresent power frequency (e.g. [15]). The introduced error is 

small and gives the final results that are slightly on the safe side.  

Although there are several methods of measurement of soil resistivity (e.g. [1]), no 

one of them is applicable in urban conditions (e.g. [15]). The reason stems from the fact 

that the surfaces of urban areas are already covered/occupied by buildings, streets, 

pavements, and many other permanently constructed objects; while under the ground 

surface many known and unknown metal installations already exist. Thus, one is forced 

to adopt an approximate value of equivalent soil resistivity, based on the main geological 

characteristics of the relevant area, and use it in the calculation part of the developed 

method. Here, the favorable circumstance is that the self- and mutual impedances are, 

according to the given equations, only slightly dependent on the equivalent soil resistivity 

and a more accurate data about this parameter does not bear any practical significance. It 

is sufficient to know that, within the framework of actually possible values, the lowest 

one should be preferred because it gives final results that are slightly on the safe side.  



38 LJ. POPOVIĆ 

 

5. QUANTITATIVE ANALYSIS 

By using the previously described method we obtain that the reduction factor taking 

into account the influence of the surrounding metal installations in the cases of the 

considered lines is: 

– r = – 0.0267 – j0.0245, or | r | ≈ 0.036, for the shorter, and  

– r = – 0.0225 – j 0.0170, or | r |≈ 0.029, for the longer line. 

It can be seen that its effective values are by 65.9% and 72.6% lower than the value 

obtained without taking into account the surrounding metal installations ( | r |  ≈ 0.105). If 

it is assumed that the cables in the considered cases are laid in a flat formation and at a 

distance of d = 0.5 m, one obtains 

r = – 0.0275 – j0.0297, or | r | = 0.0405 for the shorter and 

r = – 0.0232 – j 0.0202, or | r | = 0.0308, for longer line. 

The differences are still greater in comparison with the previously given results of 

measurements; the obtained values are lower in this case: 78.0% and 84.2%, respectively. 

Obviously, disregarding the influence of metal installations surrounding a feeding line, as 

well as determining the reduction factor only by measuring currents through the cable 

sheaths give results that are excessively conservative.    

Bearing in mind that this also means the reduction by the same ratio of all potentials 

appearing on the grounding systems of the supplied substations, one can conclude that the 

results of this analysis throw a completely new light on the grounding problem of the 

supplied substations. Also, having in mind the similarity of the urban conditions all over 

the world, this conclusion can be treated as generally valid for the safety conditions of the 

distribution substations supplied by cable lines.  

Certainly, greater economical effects can be expected in cases where, because of a 

high soil resistivity and/or a high short-circuit current level, special measurements (e.g. 

bare copper conductor laid in the same trench as the cable feeding line, counterpoises, 

etc.) were considered necessary. Besides, one can expect elimination of the strict 

requirement for the application of expensive MV cables acting as grounding conductors 

(cables with an uncovered sheath), as was the case with the MV network of Beograd. The 

only difficulty arises from the fact that the actual ground fault current distribution 

depends on the metal installations laid in the area through which the feeding line passes. 

It practically means that for obtaining actual ground fault current distribution, each 

concrete distribution line should be considered separately. 

6. CONCLUSIONS 

The presented method enables taking into account the favorable influence of urban 

metal installations surrounding HV and MV distribution cable lines on the ground fault 

current distribution in the supplied substation(s).  

Since the cable lines are almost without any exception applied in urban areas and 

since the effect of the surrounding metal installations are considerable, the presented 

method can by used as a foundation for the revision of the current version of the 

international technical standard [2]. 



 Determination of Actual Reduction Factor of HV and MV Cable Lines Passing Through Urban and... 39 

 

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