Determining Currents of Cable Sheaths by means of
Current Load Factor and Current Reduction Factor
I. Sarajcev, M. Majstrovic, Member, IEEE, and R. Goic
Abstract- A new method for determining sheath currents as
the consequence of electromagnetic coupling during line-toground short circuit is described in this article. This method is
based on using both the current load factor and the current
reduction factor. The model cables are selected to represent
major constructions encountered in practice. The cable line
consists of three single-core cables laid in trefoil formation and
touching each other. Cable sheaths are grounded at both ends.
Current load factor and current reduction factor are
characteristic data of the analyzed cable line. The method
presented in this paper is easy to use.
p
I d , p I i - positive and negative sequence
components of the sheath current of the
single-core cable L1, respectively,
The factor k depends on the current that flows through the
earth. It is calculated as follows, [2]:
k=
Ie
3 Io
(2)
where
Key words- Cable, current, current factor, load factor, sheath,
sequence component
I. INTRODUCTION
Line-to-ground short circuit is an unsymmetrical fault.
Currents and voltages can be shown by sequence component
phasors. Sequence components of currents flow through phase
conductors and through the other active parts of the
transmission network. The cable line is an element of the
direct-grounded transmission network and consists of three
single-core power cables. Their conductive sheaths are
grounded at both ends. Currents through cable sheaths flow
during line-to-ground short circuit. These currents consist of
two components. The first component occurs as the
consequence of increased potential of grounding grids. The
second component occurs as the consequence of
electromagnetic couplings and is analyzed in this paper. So far
many methods have been developed to calculate these
currents. We propose a new method based on both the current
load factor of the cable sheath ( ξ ) and the current reduction
factor of the cable line ( k ). The proposed method is presented
in this paper.
Calculation of the factor ξ is presented in [1]. It is
calculated for symmetrical three phase currents (positive
and/or negative sequence components) of the analyzed cable
line. Three single-core cables in trefoil formation have equal
factor ξ . It is calculated as follows:
p
ξ =
Id
Id
p
=
Ii
I e - current that flows through the earth,
I o - zero sequence component of the current that flows
through the phase conductor during line-toground short circuit in the transmission network.
It is calculated as follows:
Io =
1
( I L1 + I L 2 + I L3 )
3
where, I L1 , I L 2 and I L3 are phase currents of single-core
cables L1, L2 and L3, respectively.
II. THEORETICAL BASIS
Three-phase cable line consisting of three single-core
cables laid in trefoil formation is shown in Fig.1. Their
conductive sheaths are grounded at both ends.
(1)
Ii
where
Id , Ii
- positive and negative sequence
components of the phase conductor
current of the single-core cable L1,
respectively,
(3)
Fig. 1. Single-core cables laid in trefoil formation
According to [1] and [2] factors ξ and k are calculated as
follows:
j
ξ=
ωµo
D
ln c
2π
rp
(4)
ωµo
D
R p1 + j
ln c
2π
rp
R p1
k=
3
ωµo
ωµo
658
+
+j
ln (
3
8
2π
3 r D2
p c
R p1
ρ
)
f
(5)
where
Rp1 - single-core cable sheath resistance per length unit,
Ω/m,
rp - mean radius of single-core cable sheath, m,
Dc - outer diameter of single-core cable, m,
ρ
- earth electrical resistivity, Ωm,
µo - air permeability, µo= 4π 10-7 Vs/Am,
ω
- angular frequency of the current in phase conductors
and cable sheaths. Angular frequency is calculated as
follows:
ω=2πf
(6)
where
Fig. 2. Currents of the cable line during line-to-ground short circuit
The current I L3 usually equals I L 2 .
I L3 = I L 2
Substituting from (8) into (7) yields:
Io =
1
( I L1 + 2 I L 2 )
3
Id = Ii =
f - current frequency, Hz.
The cable line connected with active networks A and B is
shown in Fig. 2. Networks A and B are direct-grounded. Lineto-ground short circuit is in the network B at the phase
conductor of L1. Currents flow through phase conductors
( I L1 , I L 2 and I L3 ), through cable sheaths ( p I L1 , p I L 2 and
(8)
(9)
1
( I L1 − I L 2 )
3
(10)
According to (1) and (10) positive and negative
components of the sheath current are as follows:
p
Id =
p
Ii =
ξ
( I L1 − I L 2 )
3
(11)
p
I L3 ) and through the earth ( I e ) during short circuit. They
are shown Fig. 2.
Currents I L1 , I L 2 and I L3 are known from the short
circuit analysis. They can be shown by sequence components
as follows [3]:
I o 
1 1
  1 
=
I
 d
1 a
 I  3 1 a 2

 i
1
a 2 
a 
 I L1 


I L 2 
I 
 L3 
(7)
Besides currents
ξ
( IL1 − IL 2 )
3
p
Id
and
p
(12)
Ii
the zero sequence
component of the sheath current p I o flows through the cable
sheath. . According to Fig. 2 it follows:
3 Io = Ie + 3 p Io
(13)
Substituting from (2) into (13) it becomes:
where a = e
j120o
.
p
I o = (1 − k ) I o
According to (9) and (14) it follows:
(14)
p
Io =
1− k
( I L1 + 2 I L 2 )
3
(15)
single-core cables and the current reduction factor of the cable
line are:
ξ = 0.238 / 76.2o
Currents p I L1 , p I L 2 and p I L3 can be calculated by the next
matrix equation, [3]:
 p I L1  1 1
 p
 
2
 I L 2  = 1 a
p
 I  1 a
 L3  
1
a 
a 2 
 p Io 
 p 
 Id 
pI 
 i 
(16)
k = 0.104 / -79.7
o
Phase currents of single-core cables, during line-to-ground
short circuit in the transmission network of 110 kV are known
from the short circuit analysis. They are as follows:
I L1 = 10500 / -80o
Substituting from (11), (12) and (15) into (16) yields:
A
I L 2 = I L3 = 2000 / 105o
p
p
I L1 =
1− k + 2ξ
2 (1 − k − ξ )
I L1 +
I L2
3
3
2 (1 − k ) + ξ
1− k −ξ
I L 2 = I L3 =
I L1 +
I L2
3
3
p
(17)
Substituting the above values into (17) and (18) yields:
p
p
(19)
Equations (17) and (18) become:
p
p
I L1 =
1− k + 2ξ
I L1
3
1− k − ξ
I L 2 = I L3 =
I L1
3
p
I L1 = 4722 / -37.7o
A
(18)
In case the network B is a passive network, phase currents
I L 2 and I L3 equal zero.
I L3 = I L 2 = 0
A
(20)
I L 2 = p I L3 = 3687 / 178.9o
A
IV. CONCLUSION
A new method for determining sheath currents as the
consequence of electromagnetic coupling during line-toground short circuit is described in this article. This method is
based on using both the current load factor and the current
reduction factor. These factors are the characteristic data of
the analyzed cable line. The method presented here can be
easily applied in practice.
V. REFERENCES
(21)
Currents p I L1 , p I L 2 and p I L3 , calculated by equations (17),
(18), (20) and (21), are an outcome of electromagnetic
coupling. They are calculated by both the current load factor
( ξ ) and the current reduction factor ( k ). The current load
factor includes the electromagnetic coupling of positive and
negative sequence components of the currents. The current
reduction factor includes the electromagnetic coupling of zero
sequence components of the currents.
III. A NUMERIC EXAMPLE
The method described in this paper can be applied in
transmission and distribution cable networks. The cable line of
110 kV is chosen for the numerical example. It consists of
three single-core cables of AXLJ 1x1000/95 mm2 , [4]. Cable
data are: DC = 85 mm, rP = 38 mm and RP1 = 0.206 mΩ/m.
Single-core cables are laid in trefoil formation and touch each
other. Their conductive sheaths are grounded at both ends.
The earth resistivity ρ = 500 Ωm. According to (4) and (5),
for frequency of f=50 Hz, the sheath current load factor of
[1]
I. Sarajcev, M. Majstrovic, E. Sutlovic, “Single-core Cable
Sheath Current Load Factor”, Proceedings of 12th
International DAAAM Symposium, Katalinic, B. (Ed.),
[2] I. Sarajcev, “The Cabke Transmission Power Losses”,
Ph.D. dissertation, Faculty of Electrical engineering,
University of Zagreb, Zagreb, 1985
[3] G. W. Stagg, A. H. El-Abiad, Computer Methods in
Power System Analysis,McGraww-Hill, New York, 1968
[4] ABB, “High Voltage Cables AB”, Catalogue Data
[5] G. J. Anders Rating of Electric Power Cables - Ampacity
Computations for Transmission, Distribution, and
Industrial Applications, IEEE PRESS, ISBN 0-78031177-9, New York , 1997
[6] G. J. Jonson et al. , The electric power engineeringhandbook, L.L. Grigsby, (Ed.), CRC Press, ISBN
0-8493-8578-4, Boca Raton, Florida, 2001
[7] L. Heinhold, Power cables and their application,
Siemens Aktiengsellschaft, Berlin and Munchen, 1979
[8] S. Y. King, Underground Power Cables, Longman,
London, 1983
[9] J. Nahman, Grounding neutral point in a distribution
network, Scientific book, ISBN 06-783/1, Beograd, 1980
VI. BIOGRAPHIES
Ivan Sarajcev was born in Split,
Croatia, on October 28, 1947. He
graduated BSEE from the University
of Split, Faculty of Electrical
Engineering. He obtained his MSEE
and Ph.D. in Electrical Engineering
from the University of Zagreb in
Croatia 1981 and 1985, respectively.
He is currently an associated
professor at University of Split,
Faculty of Electrical Engineering. His research interests
include Power System Analysis, Electromagnetic Phenomena,
and Protection in Electrical Power System. He is a member of
CIGRE, and Energy Association of Croatia.
Matislav Majstrovic was born in
Dragljane, Croatia, on December 24,
1949. He graduated B.S. degree in
Electrical Engineering from the
University of Split, Faculty of Electrical
Engineering in Croatia He received his
M.S. and Ph.D. degrees in Electrical
Engineering from the University of
Zagreb,
Faculty
of
Electrical
Engineering
1979
and
1986,
respectively. He is currently a senior
researcher at Energy Institute “ Hrvoje Pozar” Zagreb and full
professor at University of Split, Faculty of Electrical
Engineering. His research interests include Power System
Analysis, Implementation of fuzzy system theory and genetic
algorithm into Electrical Power System Analysis,
Restructuring of Electrical Energy Sector. He is a member of
IEEE, IASTED, CIGRE, WEC and Energy Association of
Croatia.
Ranko Goic was born on the island of
Brac, Croatia, on April 11, 1969. He
graduated B.S. degree from the Faculty
of Electrical Engineering, Mechanical
Engineering and Naval Architecture,
University of Split, where he also
received his M.S. and Ph.D. degree in
1997 and 2002, respectively. After
graduating, he has been working at the
same faculty, in the Power System department. His main
research interests are the power system network analysis and
power system planning and optimization. The great part of his
research and engineering interests include design and
modeling of software tools for network analysis and power
system planning. He is member of IEEE, CIGRE, and Energy
Association of Croatia.