# Earthing Calculation

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```					Calculation of earthing and screening effects of
compensation conductor laid alongside
underground multi-cable power lines

I. Medic1, M. Majstrovic2 & P. Sarajcev1
1
Faculty of Electrical Engineering, Mechanical Engineering
and Naval Architecture, University of Split, Croatia.
2
Energy Institute "Hrvoje Požar", Zagreb, Croatia.

Abstract

Power substations in an urban area are interconnected by high voltage
underground power cables, resulting also in interconnections of their earthing
grids through cable metallic screens/sheaths. Most modern power cables with
XLPE insulated core have metallic screens which are practically insulated from
ground. These screens can only be (and usually are) grounded at cable ends and
therefore do not act like extra earth electrodes extending from a station. In this
case, almost the same positive earthing effects of an extended electrode can be
obtained only by extending an auxiliary wire in the same trench and routing it
closely parallel to the power line. This bare conductor also provides better
screening for nearby telecommunication circuits against inductive interference.
This paper presents an analytical method to determine the line-to-ground fault
current distribution and related voltages in the earthing system which consists of
two substation earthings interconnected by insulated metal sheaths of modern
XLPE cables and one bare earthing wire laid in parallel in the same cable trench.
The analytical expressions, derived by solving a set of differential equations, are
based on the assumption of uniformly distributed conductance between the
earthing wire and earth. The proposed mathematical model takes into account
mutual inductive couplings between all conductors (phases, cable sheaths and the
bare conductor). In addition the paper presents a procedure for computing the
resulting nominal screening factor of multi-conductor connection between two
substations. The earthing and screening effects of one bare conductor laid
parallel to double cable power line are numerically illustrated and outlined.
206   Electrical Engineering and Electromagnetics VI

1 Introduction

Earthing grids of power substations in an urban area are interconnected through
underground cable metallic screens/sheaths. The older power cables usually have
paper-insulated core, lead sheath and external semiconducting jacket of
grounding electrode, which have finite and distributed conductance to ground.
During line-to-ground fault in power system, the zero-sequence fault current
returns to the source stations through the neutral conductors (metallic cable
sheaths, overhead ground wires, auxiliary earthing wires, etc.), or through the
earthing structures and earth. Parts of the fault current, flowing into the earth
through earthing structures raise the potential of the surrounding earth with
respect to remote earth. Therefore, with respect to both personnel safety and
equipment operational compatibility, the lead sheath cables have a positive
influence on a substation in reducing substation earthing impedance and earth
voltage rise. Most modern power cables have XLPE insulated core, copper wire
screen and insulated outer protecting covering (PVC or PE) and thereby do not
have distributed conductance to ground. In this case, almost the same positive
earthing effects of lead sheaths can be obtained only by extending an auxiliary
bare conductor in the same trench. At the same time this conductor will also
provide better screening for nearby telecommunication circuits and auxiliary
cables against inductive interference. The disturbances that result from the line-
to-ground fault current are of concern to both the power utility and to utilities
with nearby telecomunications facilities. Therefore, it is important to make an
accurate analysis of zero-sequence currents distribution among the earthing
system components. In the same time it is also difficult to achieve that goal,
mainly due to complexity and number of influencing factors. A number of
different analytical methods have been developed in the past three decades (e.g.
[1]-[4]) to accurately calculate fault current distribution. These methods yield
adequate results for overhead lines but cannot be applied directly to underground
cables. The method described by Guven and Sebo [5] deals with current
distribution along the single neutral conductor representing underground cable
sheath (or overhead shield-wire tower-footing chain). Therefore, it does not take
into account mutual coupling between neutral conductors. This paper introduces
an improved method of calculation of both fault current and voltage distributions
along a single earthing/compensation bare conductor and arbitrary number of
insulated neutral conductors buried in the same trench. The method includes
mutual inductive coupling among all conductors in the trench (phases, sheaths
and earthing wire).

2 Mathematical model

Consider two high-voltage power substations (”A” and ”B”) which are
interconnected by power cable lines with a set of M phase conductors. These
power cables have insulated metallic screens or sheaths that are connected at
Electrical Engineering and Electromagnetics VI                   207

both terminals on the appertained earthing grid. Let the number of insulated
screens/sheaths be N. Furthermore, we presume the existence of an earthing wire
which is laid in the same cable trench and parallel to cable sheaths. The total
number of connected neutral conductors is then N+1. To investigate this case,
the scheme of Figure 1 is constructed representing the zero-sequence equivalent
circuit for earthing system under consideration. Referring to Figure 1, the
following set of differential equations applies to neutral conductor circuits:

d
−   {i } = [g]⋅ {v}                                                     (1)
dx
d
− {v } = [ z ]⋅ {i }− [z f ]⋅ {I f }                                    (2)
dx

L1 L2 L3                                                   L1 L2 L3
IfM
I f M-1
I f M-2

If3
If2
If1

IN+1
IN
Ik

I2

“ A”           I1A                                          I1B          “ B”
i1(x)         i1(x+ ∆x)
J rA                                                                                   JrB
v1(x)                          v1(x+ ∆x)
IA                                                         IB

VA               ∆i1(x)                      VB

ZA                                                                        ZB

∆x
x                              L-x
x=0                                                        x=L

Figure 1: Equivalent scheme of multi-cable power lines earthing system.
208     Electrical Engineering and Electromagnetics VI

The axis of the power line is denoted as the x-axis, and the total line length is
denoted by L with the left end at x=0 and the right end at x=L. Denotations in
eqns (1) and (2) have the following meaning:

[z] - square (N+1, N+1) matrix of self and mutual unit-length serial
impedances of neutral conductors, with earth return.
[zf] - rectangular (N+1, M) matrix of unit-length mutual impedances
between neutral and phase conductors, with earth return.
[g] - (N+1, N+1) matrix of self and mutual unit-length shunt
{i} - the vector of unknown currents ik(x), k = 1, 2, ...,(N+1) of
neutral conductors at point x.
{v} - the vector of unknown voltages vk(x), k = 1, 2, ...,(N+1) of
neutral conductors at point x.
{If} - the vector of known phase conductors fault currents
If k, k = 1, 2, ..., M.

Matrices [z], [zf], {v} and {i} can be partitioned into blocks so that eqn (2) can
be rewritten as follows:

d  v1   z 11   z 1s   i 1   z f 1 
−       =                 ⋅       −          ⋅ {I }          (3)
dx  vs   z s1   z ss   I   z f s  f
              

where: [zss] is (N, N) submatrix related to impedances of insulated sheaths only.

The calculation method presented in this paper is based on the following
assumptions:

Only the fundamental frequency (e.g. 50 Hz) is considered.
Mutual admittances between all conductors are neglected, so the matrix [g]
is a diagonal one.
Uniformly distributed parameters (serial impedance z11 and shunt
admittance g11 are applied to the earthing wire circuit presentation.
Self admittances of insulated neutral conductors are neglected
(gkk = 0, k= 2, 3, ..,.N+1).
Serial unit-length impedances with earth return [z] and [zf], are calculated by
means of the Carson-Pollaczek [6], [7] equations.
The medium surrounding the earthing wire and cable lines is homogeneous
and characterized by soil resistivity ρ (Ωm).
Conductive coupling between the earthing wire and substation earthing grids
is neglected.
The magnitudes of phase conductor fault currents are known from system
studies.
Electrical Engineering and Electromagnetics VI                  209

Since the cable metal screens are insulated from ground, we may assume that
gkk=0 (for k = 2, 3,…,N+1) and eqn (1) yield solutions:

ik (x)= const. = Ik                    k = 2, 3, …, N+1.               (4)

Using eqns (1), (2) and (4), it follows:

d 2 v1 ( x )
− z11 ⋅g11 ⋅ v1 ( x ) = 0                              (5)
dx 2

The equation (5) may be solved by standard methods to obtain the general
solution which can be written in terms of two arbitrary constants. Using the
general solution for v1(x) and eqns (3) and (4), one can obtain the general
solutions for i1(x) in terms of the same two arbitrary constants plus the additional
one. Furthermore, integrating eqn (3) and using eqn (4) and the general solution
for i1(x), yields the set of general solutions for insulated neutral conductor
voltages vk(x), k=2, 3, …,N+1, each of them in terms of a different arbitrary
constant. So the total number of arbitrary constants are N+3. Next, if we assume
that the boundary conditions at x=0 (i.e. at the fault point "A") are known:

i1 (x=0) = I1A
vk(x=0) = VA                         (k = 1, 2…,N+1)

the arbitrary constants can be determined and the required solutions can be
written, as follows:

Z csh ( γx )  N +1           M              
v1 ( x ) = VA ch ( γx ) − I1A Z csh ( γx ) −                    ⋅  ∑ z1 k I k − ∑ z f 1 k I f k    (6)
z11                                      
 k=2          k =1            

VA                            ch ( γx ) − 1  N +1           M              
i1 ( x ) = −      sh( γx ) + I1A ch ( γx ) +              ⋅  ∑ z1 k I k − ∑ z f 1 k I f k          (7)
Zc                                z11                                      
 k =2         k =1            

z s1      N +1      z s1 ⋅ z1k      
vs ( x ) = v1 ( x )        − x ⋅ ∑  zsk −

I k ,

z11        k =2        z11          
(8)
M        z s1 ⋅ z f 1k                       
+ x ⋅ ∑  zf sk −                      I f + 1 − z s1  VA .
                               k           
k =1           z11                     z11 
s = 2, 3, …, N+1
where
210    Electrical Engineering and Electromagnetics VI

γ = z11g11
z11
Zc =
g11
Hence, at x=L (i.e. at point “B”):

Z csh( γL)  N +1           M              
VB = VA ch ( γL) − I1A Z csh( γL) −               ⋅  ∑ z1 k I k − ∑ z f 1 k I f k      (9)
z11                                    
 k=2          k =1            
VA                          ch ( γL) − 1  N +1           M              
I1B = −      sh( γL) + I1A ch ( γL) +             ⋅  ∑ z1 k I k − ∑ z f 1 k I f k    (10)
Zc                               z11                                    
 k =2         k =1            

Also, for the boundary condition at point "B" (x=L):

vk(x=L)=VB.                       (k = 1, 2…,N+1)
from eqn (8) we obtain:

N +1       z s1 ⋅ z1k                           z 
L⋅   ∑  zsk −
            z11 
 I k −( VA − VB ) ⋅ 1 − s1  =
                     z 
k =2                                              11 
(11)
M        z s1 ⋅ z f 1k 
L ⋅ ∑  zf sk −               If .
           z11  k
k =1                    
s = 2, 3, …, N+1

Usually, the known values are phase currents coming from all substation
transformers and power lines connected to buses L1, L2 and L3 of substations "A"
and "B" during the phase-to-ground fault. In order to achieve more realistic
results (as recommended in Seljeseth et al. [8]), the "reduced fault currents" JrA
and JrB should be calculated. These currents which actually enter the ground
through earthing structures connected at points "A" and "B" respectively, are
calculated separately using power line cross-section data and power line phase or
zero-sequence currents. Therefore, the boundary conditions should be modified
and the terminal impedances ZA and ZB incorporated as well. According to
Figure 1 these conditions can be written as:

VB − Z B I B = 0                                               (12)
VA − Z A I A = 0                                               (13)
N +1
I B − I1B −   ∑ I k = J rB                                    (14)
k =2
N +1
I A + I1A +    ∑ I k = J rA                                   (15)
k =2
Electrical Engineering and Electromagnetics VI    211

where ZA and ZB are the earthing impedances of substations "A" and "B",
respectively, with neutral interconection conductors excluded. Finally, to obtain
the solution for voltages and currents {X} at both terminals, one has to solve the
set of simultaneous linear eqns (9)-(15) which can be written in the following
matrix form:
[A]⋅ {X} = {B}                               (16)

{X} = {VA                                         {I }T }
T
where                 I1A   VB     I1B      IA     IB

{I}      – the column vector of sheath currents,
VA, VB, I1A, I1B – earthing wire terminal voltages and currents,
IA , IB – substation earthing structure currents,

3 Calculation of the resulting screening factor

The current in each neutral conductor has two components. One is caused by
inductive coupling between phase and neutral conductors and the other is caused
by voltage difference between the two earthing grids. The currents exclusively
caused by inductive coupling are of particular interest because they also reduce
negative electromagnetic influence of phase conductor fault currents on
neighbouring communication cables and other metallic installations that are
isolated in relation to the earth. They can be derived from eqns (16) as special
case when ZA = ZB = 0 or directly from eqns (2) substituting dv = 0. Thus for this
case the next matrix equation is valid:

[ z ]⋅ { I } = [ z f ]⋅ {I f }                   (17)

For a multi-conductor connection, the resulting nominal screening factor (also
known as the current reduction factor), can be defined as:

N +1
∑ Ii
r =1− i = 1                                      (18)
M
∑ If k
k =1

and using the solution for currents from (17) it can be finally expressed as:

N + 1 N + 1      M          
∑  ∑ Yi j ⋅ ∑ z f jk ⋅ If 
i =1  j=1
k
           k =1        
r =1−                                                      (19)
N +1
∑ If
k
k =1
212    Electrical Engineering and Electromagnetics VI

where Yij are elements of the inverse of the matrix [z]:

[Y] = [z]-1                                  (20)

The same procedure applied to the single line power cable conductors (phase and
neutral conductors) yields the value of the so called line screening factor that is
frequently used for substation earthing current calculations (e.q. Seljeseth et. al.
[8]).

4 Illustrative example

As an example of the mathematical model application we have developed,
consider the cable trench of Figure 2 in which the following conductors exist:

Two groups of three single-core 110 kV cables forming two power lines.
One earthing bare conductor (Cu 50 mm2) extended at depth 0,7 m.
Cable type: single-core 110 kV, (ABB AXKJ 1000Al mm2),
sheath 95 mm2 Cu, cable length L = 4000 m.
D = 0,086 m; S1 = 0,35 m; S = 0,7 m; H=0,5 m.

Other relevant data are:

Phase currents: If 1 = If 4 = 7050∠-79o A; If 2 = If 3 = If 5 = If 6 = 413∠119 o A.
Soil resistivity: 100 Ωm.
Earthing conductor shunt admittance is: g11=0,003 S/m.
Terminal reduced currents: JrA= 13503 ∠-78o A, JrB= 6308 ∠111o.
Substation earthing impedances: ZA=0,15 Ω, ZB=0,1Ω.

S1                   S

Cu 50 mm
3                           6
H

1                    2           4               5
D   D                       D       D

Figure 2: Example of cable trench with six 110 kV single-core
cables and an earthing conductor.
Electrical Engineering and Electromagnetics VI   213

1400
[V], [A]
1200

1000
| V1 |, | V2 |, | V20 |, | I1 |, | I1a |

800
V1
V2
600             V20
I1
I1a
400

200

0
0   500     1000    1500   2000   2500   3000   3500    4000
x [m]

Figure 3: Voltage and current distributions along the
earthing wire and power cable sheaths.

The main numerical results obtained using the derived mathematical model are
presented graphically in Figure 3, where.

V1=|V1(x)| and I1=|I1(x)| are distributions of voltage and current magnitudes
along the earthing conductor, respectively.
V2=|V2(x)| and V20=|V20(x)| are cable sheath voltage magnitude
distributions obtained with and without earthing conductor presence,
respectively.
I1a=| I1 |= 863 [A] is the earthing conductor constant current obtained in
case ZA = ZB = 0 Ω.

Numerical results obtained show a quantitative reduction of the voltage V1(x=0)
at fault point "A" as well as the reduction of cable sheath currents, caused by
earthing wire connection. In this example the magnitude of the remote terminal
voltage VB is lowered also. Both terminal voltages depend to a great extent on
magnitudes of corresponding reduced current Jr and substation earthing
impedance.

The resultant nominal screening factor and the earthing wire current (both
obtained in case ZA = ZB = 0 Ω) are respectively: r = 0,058∠-83,8o and
I1a = 863,7∠-120o A. In case that earthing wire is absent the nominal screening
factor is r = 0,076∠-78,3o. At the same time the screening factor calculated for
single power line is r1 = 0,152∠-73,4o.
214    Electrical Engineering and Electromagnetics VI

As differences in numerical results show, the calculations with individual line
reduction factors should not be used if more than one power line exist in the
same trench.

5 Conclusion

This paper has derived a set of equations providing a more accurate calculation
of the fault current distribution and related voltages in underground cable neutral
conductors and the accompanied parallel earthing wire interconnecting two
neighboring substations. These equations are based on the assumption of a
uniformly distributed conductance between earthing wire and earth. Furthermore,
the procedure for determining resulting nominal screening factor of multi-
conductor connection between two power substations has been presented. The
paper outlines the effects of inductive coupling between cable neutral conductors
and earthing wire on both current and voltage distributions in the earthing system
during phase-to-ground fault. The earthing wire connection is always in favor of
both the substation safety and cable thermal withstanding. It also provides better
screening for nearby auxiliary cables against inductive interference. The results
obtained show that method using individual line screening factors cannot be
considered as generally recommendable for practical applications when more
power lines are in the same trench.

6 References

[1] Endrenyi, J. Analysis of Transmission Tower Potentials During Ground
Faults, IEEE Trans. on Power Apparatus and Systems, 86(10), pp. 1274-
1283, 1967.
[2] Funk, G. Berechnung der Nullstromverteilung bei Erdkurzschluss einer
Freileitung, ETZ-A, 92(2), pp. 74-80, 1971.
[3] Dawalibi, F. Ground Fault Current Distribution between Soil and Neutral
Conductors, IEEE Trans. on Power Apparatus and Systems, 99(2), pp. 452-
461, 1980.
[4] Gooi, H. B. & Sebo, S. A. Distribution of ground fault currents along
transmission lines – an improved algorithm, IEEE Trans. on Power
Apparatus and Systems, 104(3), pp. 663-670, 1985.
[5] Guven, A. N. & Sebo, S. A. Analysis of Ground Fault Current Distribution
Along Underground Cables, IEEE Trans. on Power Delivery, 1(4), pp. 9-
18, 1986.
[6] Carson, J. R. Ground Return Impedance: Underground Wire with Earth
Return, Bell System Technical Journal, 8, pp. 94-98, 1929.
[7] Pollaczek, F. Über das feld einer unendlich langen wechsel-
stromdurchflossenen Einfachleitung, Elektr. Nachr. Technik, 3(9), pp. 339-
359, 1926.
[8] Seljeseth, H., Campling, A., Feist, K. H. & Kuussaari M. Station Earthing.
Safety and interference aspects, Electra, (71), pp. 47-69, 1980.

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