Download Current Reduction Factor of Compensation Conductors Laid

Current Reduction Factor of Compensation Conductors Laid
alongside Three Single-Core Cables in Flat Formation
Ivan Sarajcev
University of Split
Faculty of Electrical
Engineering
Split, Croatia
Matislav Majstrovic
Energy Institute “H. Pozar”
Zagreb, Croatia
e-mail: [email protected]
Abstract: The three-phase cable line consists of three
single-core cables, which can have conductive sheaths,
grounded at either one end only, or at both ends. If they are
grounded at both ends, currents flow through sheaths of
single-core cables during both balanced and unbalanced
loads and the line-to-ground short circuit in a grounded
network. During the line-to-ground short circuit, currents
flow through cable sheaths and through the ground. When
cable sheaths are grounded at one end only, currents do not
flow through the sheaths and the influence of the short
circuit current on neighboring electrical circuits is greatest.
Compensation conductors are used to reduce this effect.
Reduction effect of compensation conductors laid alongside
three power single-core cables in flat formation is analyzed
in this paper. Calculation methods of current reduction
factor are also described.
Key words: Reduction factor, current, single-core cable,
sheath, compensation conductor
Introduction
The three-phase cable line consists of three single-core
cables, which can have conductive sheaths, grounded at
either one end only, or at both ends. If they are grounded at
both ends they are parts of the grounding system. During
line-to-ground short circuit in direct grounded networks,
currents flow through: line conductors of both overhead and
cable lines, windings of transformers, windings of
generators, elements of grounding systems, and earth.
Currents through conductive sheaths of single-core cables
occur as the consequence of both increased potential of
grounding grids and electromagnetic couplings. The current
through the earth is decreased by currents of single-core
cable sheathes. Its component, as the result of
electromagnetic coupling, can be written as follows:
Ie = k 3 Io
(1)
where
I e - current through the ground. It influences conductive
elements of neighboring devices,
I o - zero sequence-component of the short-circuit current
flowing through the cable line. It is as follows:
Io =
1
( IL1 + I L 2 + IL3 )
3
(2)
Ivan Medic
University of Split
Faculty of Electrical
Engineering
Split, Croatia
e-mail: [email protected]
where IL1, I L 2 and I L3 are currents of conductors of
single-core cables L1, L2 and L3, respectively,
k - current reduction factor of the cable line. It depends
on electromagnetic properties and geometrical
characteristics of single-core cables and geophysical
characteristics of the earth. This factor is less
than 1 and can be calculated according to [1, 2].
Currents do not flow through cable sheaths if they are
grounded at one end only. In this case current reduction
factor is:
k =1
(3)
In this case there is no reduction effect of sheath currents,
and 3 I o influences the neighboring electrical circuits.
Compensation conductors are used to reduce this
electromagnetic influence. A compensation conductor is a
separate conductor in parallel with power cables. It is laid
alongside these cables. Compensation conductors are
grounded at both ends. Currents through them flow during
the line-to-ground short circuit. They decrease the
electromagnetic influence of 3 I o on neighboring electrical
circuits. Reduction effect of compensation conductors laid
alongside single-core power cables in flat formation is
analyzed in this paper. The research results regarding their
best location are presented. The calculation of the current
reduction factor is shown in this paper as well.
Theoretical basis
A three-phase cable line consists of three single-core cables.
These cables are laid in flat formation. Their conductive
sheaths are grounded at one end only. Compensation
conductors C, C1, and C2 can be laid as shown in Figures 1
and 2. In the first case there is one compensation conductor
(C). In the second case there are two compensation
conductors (C1 and C2). Compensation conductors are
grounded at both ends.
According to Figures 1 and 2 s is as follows:
s = so + D
where
so - distance between neighboring cables
D - outside diameter of the single-core cable
(4)
Zc = ( R c1 +
ωµ o
ωµ o l
658
)l+ j
ln (
8
2π
rc,
ρ
)
f
(6)
'
=
Zm
ω µo
ω µo l
658
l+ j
ln ( 2
8
2π
s
ρ
)
f
(7)
''
=
Zm
ω µo
ω µo l
658
l+ j
ln ( 2
8
2π
3s
ρ
)
f
(8)
where
R c1 - resistance of the compensation conductor per unit
length, Ω/m,
Figure 1. Cross section of the cable line with a
compensation conductor
rc,
l
ω
- self geometric mean radius of the compensation
conductor, m,
- length of the compensation conductor, m. This
length is equal to the length of the cable line.
- angular velocity;
ω = 2 π f , where f – frequency, Hz,
µo - area permeability,
µ o = 4 π 10 −7 Vs / Am .
ρ
- mean earth resistivity, Ω m,
The current through the earth is as follows:
Ie = 3 I o − Ic
(9)
Combining (5) and (9) yields:
Figure 2. Cross section of the cable line with
two compensation conductors
Ie = ( 1 −
(10)
According to (1) the current reduction
compensation conductor C is as follows:
One compensation conductor
Electromagnetic coupling between single-core cables and
the compensation conductor creates currents through the
compensation conductor during the line-to-ground shortcircuit. The current through the compensation conductor (C)
is as follows:
'
''
+ Zm
I c Zc = I o ( 2 Z m
)
'
''
2 Zm
+ Zm
) 3 Io
3 Zc
k1 =1 −
R c1 + j
k1 =
Z c - self impedance of the compensation conductor
with earth return,
'
Zm
- mutual impedance between the compensation
conductor and phase conductors L1 and L2 with
earth return,
''
Zm
- mutual impedance between the compensation
conductor and the phase conductor L3 with
earth return.
'
''
According to [3, 4] Z c , Zm
and Zm
are calculated as
follows:
of
(11)
Substituting from (6), (7) and (8) into (11) yields:
(5)
where
I c - current through compensation conductor C,
'
''
2 Zm
+ Zm
3 Zc
factor
3
3s
ω µo
ln ( , )
2π
2 rc
658
ωµo
ω µo
R c1 +
+j
ln (
8
2π
rc,
ρ
)
f
(12)
Two compensation conductors
Compensation conductors C1 and C2 have equal
geometrical and electrical characteristics. Currents through
compensation conductors C1 and C2 during the line-toground short circuit, as a result of electromagnetic coupling,
are as follows:
 Zc

 Zm
Zm   Ic1 
'
'' 1
   = Io ( 2 Z m + Z m )  
Zc   Ic2 
1
(13)
where
Table 1. Current reduction factors
ρ
[Ωm]
Ic1, Ic 2 - currents through compensation conductors
C1 and C2, respectively
Zm
- mutual impedance between compensation
conductors C1 and C2 with earth return, Ω.
This impedance is:
ω µo
ω µo l
658
l+ j
ln (
Zm =
8
2π
s
ρ
)
f
so=7 cm
k2
so=25 cm
so=7 cm
so=25 cm
100
0.479
/-31.0o
0.509
/-25.7o
0.294
/-41.9o
0.323
/-35.8o
300
0.463
/-32.0o
0.492
/-26.7o
0.281
/-42.8o
0.308
/-36.7o
500
0.456
/-32.4o
0.484
/-27.2o
0.276
/-43.2o
0.302
/-37.1o
1000
0.446
/-33.0o
0.475
/-27.8o
0.268
/-43.7o
0.293
/-37.6o
2000
0.437
/-33.6o
0.465
/-28.3o
0.261
/-44.2o
0.285
/-38.1o
(14)
Solving (13) yields:
'
''
+ Zm
2 Zm
Ic1 = Ic 2 = Io
Zc + Z m
k1
(15)
The current through the ground is as follows:
I e = 3 I o − I c1 − I c 2
(16)
Conclusion
Substituting from (15) into (16) becomes:
'
''
2 2 Zm
+ Zm
Ie = ( 1 −
) 3 Io
3 Zc + Zm
(17)
According to (1) and (17), the current reduction factor of
compensation conductors C1 and C2 is as follows:
k2 = 1 −
'
''
2 2 Zm
+ Zm
3 Zc + Zm
(18)
Substituting from(6), (7), (8) and (14), it becomes:
k2 =
R c1
ω µo
+j
ln ( 3 3
2
2π
s
rc,
ω µo
R c1 ω µ o
658
+
+j
ln (
2
8
2π
rc,
)
ρ
)
f
(19)
References
A numerical example
The cable line of 110 kV is chosen for the numerical
example. It consists of three single-core cables of AXLJ
1x1000/95 mm2, [5]. The outside diameter D is 85 mm.
These cables are laid in flat formation. Copper stranding
conductors are used for compensation conductors C, C1, and
C2. Its data are: cross-sectional area = 50 mm2, RC1= 0.364
mΩ/m and rC=4.5 mm. Its self geometric mean radius is as
follows [6]:
rc, = 0.726 rc = 3.27 mm
Current reduction factors of compensation conductors are
described in this paper. Compensation conductors are laid
alongside single-core cables in flat formation. Cable sheaths
are grounded at one end only. There are no reduction effects
of cable sheaths during line-to-ground short circuit, thus
compensation conductors take over their roles. They are
grounded at both ends. The current reduction factor does not
depend on the length of the compensation conductor. It
depends on electromagnetic properties of compensation
conductors, geometrical characteristics of compensation
conductors and single-core cables and geophysical
characteristics of the earth. The current reduction factor
depends on the distance between single-core cables. If the
distance and/or the earth conductivity are smaller, the
current reduction factor is smaller too.
(20)
According to (12) and (19) current reduction factors are
calculated for different geophysical characteristics of the
earth and distances between cables. Frequency is 50 Hz..
Current reduction factors are shown in Table 1.
[1] Sarajcev,I.: The Cable Transmission Power Losses
(Dissertation, ETF, Zagreb, 1985)
[2] Heinhold, L.: Power Cable and their Application,
Siemens, Berlin, 1979
[3] J. R. Carson, Ground Return Impedance:
Underground Wire with Earth Return, Bell System
Tech. J., Vol. 8, pp. 94-98, 1929.
[4] Sarajcev, I., Majstrovic, M., Medic, I., Determining
currents of cable sheaths during unbalanced loads by
means of current load factor, DAAAM International
Scientific Book 2002, pp 517-524, DAAAM
International Vienna, ISBN: 3-901509-30-5,
Vienna 2002
[5] ABB, High Voltage Cables AB (Catalogue data)
[6] …., Electrical Transmission and Distribution
(Fourth Edition: Third Printing, Westinghouse
Electric Corporation, East Pittsburgh, Pennsylvania,
1950)
Biographical notes
Ivan Sarajcev 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 graduated B.S.
degree in Electrical Engineering from
the University of Split, Faculty of
Electrical Engineering in Croatia He
received his MSEE 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.
Ivan Medic graduated BSEE from the
University of Split in Croatia, Faculty
of Electrical Engineering. He obtained
his MSEE from the University of
Zagreb 1983 and Ph.D. from
University of Split 1999. He is
currently an assistant professor at
University of Split, Faculty of
Electrical Engineering. His research
interests
include
Grounding
System
Analysis,
Electromagnetic Phenomena, and Substation Design. He is a
member of IEEE, CIGRE, and Energy Association of
Croatia.