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17th International Conference on Electricity Distribution
Barcelona, 12-15 May 2003
REDUCTION FACTOR OF FEEDING LINES THAT HAVE A CABLE AND AN OVERHEAD
SECTION
Ljubivoje Popovic
J.P. Elektrodistribucija - Belgrade - Yugoslavia
[email protected]
SUMMARY
The paper presents an analytical procedure which enables a
quick and, at the design stage, correct evaluation of the
ground fault current distribution, for a fault in a substation
supplied by a line composed of two or more mutually different
(overhead and underground) sections. The advantages of the
method are based on the simplicity and accuracy of the
formulae for solving uniform lumped parameter ladder
circuits of any size and under any terminal conditions.
INTRODUCTION
During ground faults in an HV installation, raised potentials
appear in places where they do not exist under normal
operating conditions. The magnitudes of these potentials, as
well as all of the potential differences significant for the
safety conditions (touch and step voltages) are proportional to
the current emanating from the substation grounding system
into the surrounding earth.
It is known that only a part of the total fault current flows into
the earth, and the rest returns to its sources (power system)
through various metal conductors (ground wire(s) of the
overhead line, the cable sheath(s) and grounding
connections), so that it does not contribute to the creation of
the elevated potentials in the grounding system of the
substation [2]. Therefore, when designing grounding systems
of high voltage plants the primary task is to evaluate as
precisely as possible the part of the fault current flowing
through the grounding system into the earth. This is the only
way to be sure that the problem is solved according to actual
requirements, which practically means that the safety
conditions are reached without excessive expenditures.
The evaluation of the distribution of the ground fault current
for faults in substations supplied by transmission lines
homogenous through their entire length is performed by a
relatively simple procedure. This case has been studied,
systematized and presented in [7] for various types of stations
and for different types of transmission towers. However, the
problem gains in complexity if the feeding line has two or
more sections with one or more mutually different relevant
parameters. Examples for such feeding lines are encountered
in distribution networks of various voltage levels and are a
consequence of different degree of urbanization of the areas
where the feeding line passes. Regarded as a whole, these
lines are usually a combination of cable and overhead
sections with different lengths. As a rule, the overhead section
is closer to the source station, and the cable section is closer
to the supplied station. When ground faults appear, such lines
form a very complex electrical circuit with a large number of
conductively and inductively coupled elements.
References [5] and [8] point to the problem and give some
analytical expressions enabling its solution in some practical
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Session 3 Paper No 55
situations. However, these papers do not encompass all the
cases of importance for current engineering practice. Thus it
may be said that this work represents a logical continuation of
the efforts to solve this kind of problems.
The possibility to simplify the problem is found in the
reduction of the number of the elements of the equivalent
circuit. To this purpose, we have derived formulae for an
exact representation of lumped parameter ladder circuits of
any size (from one pi to an infinite number of pis) by only one
pi. This is achieved using the “General equations of uniform
ladder circuits” [3], [4], [5].
Starting from the usual and unavoidable approximations at the
design stage and using the mentioned solution for ladder
circuits, the paper derives analytical expressions for
evaluating the ground fault current distribution when the
feeding line contains a cable section of an arbitrarz length.
With a permanent elevation of the ground fault current level
and with an increasing deficiency of free urban areas for the
building of large substations, a correct solution of the problem
of ground fault current distribution in various practical
situations gains in importance.
PROBLEM DESCRIPTION
If the feeding line is solely a cable or an overhead line along
the whole of its length from the source station to the supplied
station, the calculation of the ground fault current distribution
in supplied substation is a routine engineer’s job [7].
Problems appear when a substation is supplied directly by an
underground cable representing the continuation of an
overhead line, as shown in Fig. 1.
Fig. 1. Typical situation
The situation represented in Fig. 1 is typical for the feeding
lines in distribution networks and appears as a consequence of
the following facts. The large source stations supplying
distribution networks of big cities require large areas and are
thus built outside of urban zones. Lines emerge from them
and form a radial network supplying high-voltage distribution
stations (e.g. 110/10 kV), situated deep within urban
surroundings (the center of electric energy consumption).
Each of the outgoing lines usually supplies two or three such
stations in a succession (or in the so-called radial distribution
system).
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The current which appears due to the inductive coupling
faulted phase conductor/ground wire (or cable/sheath) in such
situations (Fig. 1) cannot be determined by standard
calculation procedures [5]. One part of the component of the
ground fault current at the point of discontinuity D flows
towards the supply source through the ground. Because of this
the reduction of the fault current through the grounding
system of the supplied station is smaller than in the case when
the feeding line is a cable along the whole of its length, but is
larger than in the case when it is an overhead line along the
whole length. In practice, however, when determining the
fault current such a line is treated either as a cable only, or as
a overhead line only. This may result in serious errors when
estimating and ensuring the safety conditions in the supplied
station.
Because of the strong inductive coupling of the cable/sheath,
most of the fault current can be injected into the ground at the
place of discontinuity rather than at the fault location. When
the place of discontinuity is in a transition station, this
phenomenon is described as the “fault application transfer
effect” [8].
EQUIVALENT CIRCUIT AND NECESSARY
ANALYTICAL EXPRESSIONS
General case
Let us assume that a distribution substation, B, is supplied
from the station, A, by a cable which is an elongation of an
overhead line. We will adopt the most complex case when the
cable has a non-insulated sheath, and the overhead line has a
ground wire. The following theoretical foundation
• Method of symmetrical components,
• Driving point technique,
• Decoupling technique [9] and
• Reduction of ladder circuits [5]
is used to represent such a line under the conditions of a
ground fault at the substation B by the equivalent circuit in
Fig. 2.
If
A
Za
Qa
Pa Pa
ZB
Iba
Iia
~
F
~
D
RN
In the case when the station B has a grounded neutral
point(s), the equivalent circuit in Fig. 2 should have an
additional branch containing the zero sequence impedance of
the local transformer and connects the points F and B (Fig 2).
According to [5], the elements representing the cable section
of the line in the given equivalent circuit are determined by
the following expressions
I ib = (1 − rc )I f
Qb =
Pb =
k c− L
−
2
k cL + 1
k cL − 1
kc = 1 +
Ie
Zb
k cL
Fig. 2. Equivalent circuit
The symbols in the circuit have the following meaning
Ffault position (substation B)
Vf auxiliary driving source at fault location with the
voltage equal to the voltage at the source station
If total ground fault current
Ie current injected into the earth through the substation
grounding grid and through the connected long
Session 3 Paper No 55
(1)
Z ' c∞
(2)
Z ' c∞ .
(3)
In the given expressions rc is the reduction factor of the cable
(usually given as factory data,) and L the relative length of the
cable section (expressed relative to a one meter length), while
the other parameters are defined by the following relations
B
G
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external grounding conductor (metal cable sheath(s),
neutral conductors of the low voltage network, metal
pipelines, etc).
Iib current source which replaces the influence of the
inductive coupling between the faulted phase
conductor and the metal cable sheath(s)
Iia current source which replaces the influence of the
inductive coupling between the faulted phase
conductor and ground wire of the overhead line
ZB impedance of the power system at fault location
Qb, Pb - impedances of the equivalent pi replacing the
influence of galvanic coupling of metal cable
sheath(s) on the cable section
Qa, Pa - impedances of the equivalent pi replacing the
influence of galvanic coupling of ground wire(s) of
the overhead line
Za(Zb) - impedance of the grounding system of station A (B),
not incorporating the grounding effects of the
feeding line
RN footing resistance of the tower at the point of
discontinuity D of the line
Gremote ground.
~
Qb
Pb Pb
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Z c∞ =
Z c∞
Rc
(4)
Zc
Z2
+ Rc Z c + c
2
4
(
Z ' c∞ = Z c−∞1 + Rc−1
)
−1
(5)
.
(6)
The parameters Zc and Rc represent the elements of the
uniform ladder circuit obtained by the discretization of the
cable sheath as a grounding conductor. By this, the
impedance Zc represents the self-impedance of the sheath on a
length of one meter, while Rc represents the grounding
resistance of the sheath on a length of one meter. According
to [5] the resistance Rc is given by
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Rc =
L'
ρ
ln
,
π
dh
(7)
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rcIf
r (1–rc)If
D
where ρ is the specific soil resistivity along the cable in Ωm,
L' is the cable length in m, d is the outer cable diameter in m
and h is the burial depth of the cable, also expressed in
meters.
For the part of the equivalent circuit of the line overhead
section, according to [4], [5] the following relations are valid
I ia = (1 − r )I f
Qa =
k 2N −1
Z∞
k + k N +1
k N +1
Pa = N
Z∞
k −k

Z 
r = 1 − m  .
Zs 

N
(8)
(11)
Z
k = 1+ ∞
R
(12)
Zs
Z2
+ RZ s + s ,
2
4
(13)
where R is the towers' footing resistance (the average value).
The impedances Zc, Zs and Zm are calculated using the
formulae based on the Carson’s theory of ground fault current
return path (e. g. [1]).
Since to estimate the safety conditions it is necessary to
determine the maximum of the current Ie, the impedance Zm
should be determined under an assumption that the fault
occurred on the phase conductor which is the farthest one
from the ground wire(s).
The equivalent circuit does not comprise the “proximity effect
of the cable and the grounding grid” [6] of the supplied
station B. However, it is necessary to bear in mind that under
the considered circumstances thus effect is very weak. The
reason for this is the fact that due to the strong inductive
coupling cable-metal sheath the most part of the ground fault
current goes into the earth at the point of discontinuity which
is in the case of any larger cable section far enough from the
station B.
In practical situations the influence of the impedance Za can
be neglected (Za≈0). Thus to determine the current If
distribution we can use the equivalent circuit shown in Fig. 3.
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Session 3 Paper No 55
B
Zb
Pb
Ie
Fig. 3. Simplified equivalent circuit
According to the circuit shown in Fig. 3 and using the
Kirchhoff’s rules we can write a closed system of equations.
Its solution gives us
rt =
In the given expressions Zs is the self-impedance of the
ground wire(s) per span and Zm is the mutual impedance
between the ground wire(s) and the faulted phase conductor
per span, while N is the total number of spans of the overhead
section.
The parameters k and Z∞ are defined by the following
relations
Z∞ =
ZD
(9)
(10)
Qb
Ie
r (1 − rc ) Z D + rc (Qb + Z D )
=
,
If

Z 
1 + b (Qb + Z D ) + Z b
Pb 

(14)
where the impedance Zd and the grounding impedance of the
ground wire at the pint of discontinuity, ZN, are given by
 1
1
1 
Z D = 
+
+ 
 RN Z N Pb 
Z N ( Z a = 0) =
k 2N −1
k 2N + k
−1
(15)
Z∞ .
(16)
The coefficient rt defines the ground fault current distribution
including the grounding effects of the feeding line.
At the design stage the impedance Zb is often calculated to
include the grounding effects of the feeding line. In such
cases the reduction factor should exclude these effects. We
will eliminate them if we adopt Zb=0. Then according to (14)
we obtain
ri = rc +
r (1 − rc ) Z D
.
Qb + Z D
(17)
The coefficient ri expresses only the influence of the inductive coupling cable/metal sheath(s) and phase conductor/grounding wire(s) to the distribution of the ground fault
current If.
The second addend in the equation (17) obviously represents
the increment of the reduction factor of the feeding line in
comparison to its value when the feeding line is a cable along
the whole of its length. If we assume that the length of the
cable section tends to infinity (Qb→∞), the value of the
reduction factor ri tends to rc. Since under real conditions the
reduction factor of a cable is lower than the reduction factor
of an overhead line (rc<r), it can be said that the value of ri is
situated between two limiting values (rc<ri<r). In practice,
however, when designing a grounding system of a supplied
station, the line under consideration is often treated as if it
were solely a cable or an overhead line along the whole of its
length. In the first case it produces results on the unsafe side
(more favorable than in reality), while in the second case the
estimation of the safety conditions is too severe. What is the
value of the deviation from the reduction factor of the real
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feeding line will be seen later from the result of quantitative
analysis.
According to the expression (17) it can be also seen that the
reduction factor ri depends on the value of the grounding
impedance at the place of discontinuity. With the decrease of
this impedance, if other relevant parameters remain
unchanged (17), the value of the reduction factor ri also
decreases. If the value of this impedance becomes negligible,
the value of the reduction factor of the feeding line becomes
practically equal to the reduction factor of the cable (ri≈rc).
Special Cases
The presented equivalent circuit and the necessary analytical
expression can be simply modified for various special cases
encountered in practice.
When the length of the overhead section is such that we can
consider it as infinite, instead of (15) we can use
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corresponding parameters. However, since the active length
La depends on Rc and, according to (7), Rc is a function of L',
this expression cannot be directly applied. Its purpose is to
define x (by testing a large number of numerical examples) in
the following semi-empirical formula
La ( ε = const ) ≈
xρ
,
| Z c | / 1m
(23)
where x is an unknown, dimensionless number, different for
different cable cross-sections and burial depths, determined
from the condition that (19) should be approximately satisfied
for different values of ρ. At that, we apply in (19) the
appropriate parameters (instead on N, Na, Zs and R we have
L', La, Zc and Rc, respectively).
When the length of the cable section tends to its active length,
according to (2) and (3), Qb tends to infinite and Pb to Z∞, so
that according to (22) it follows
−1
 1
1
1 
+
+  .
Z D = 
R
Z
P
a 
∞
 N
(18)
Qb → ∞
Pb → Z C∞
A line or its section with N spans can be considered as infinite
from the point of view of grounding effects if, according to
[4], the following condition is satisfied
 1
N > N a = ln1 + 
 ε

| Z s | 
ln1 +

R 

(19)
active line length expressed in the number of spans
desired relative accuracy, |Z∞–ZN|/Z∞ (an a priori
adopted arbitrarily small number)
ZN grounding impedance of ground wire seen from the
point D.
When the place of discontinuity is a transition station, the
impedance ZD is equal to the grounding impedance of this
station, or
rc Qb + Pb

Z 
1 + b (Qb + Pb ) + Z b
Pb 

ri =
rc Qb + Pb
Qb + Pb
(21)
(24)
(25)
As it can be seen from (25), the reduction factor of a
sufficiently long cable section becomes equal to the reduction
factor of the feeding line that is cable along its whole length.
Cables with insulated outer sheath
This type of cables, because the outer sheath is insulated,
does not have grounding properties (the fault current is not
conducted via the sheath into the surrounding ground). This is
the reason why it is necessary to modify the circuit shown in
Fig. 3. The impedance Pb is omitted (Pb=∞), and for the impedances Qb and ZD the following relations are valid
Q b = L' z c
(
Z D = R N−1 + Z N−1
(26)
)
−1
(27)
where zc is the self-impedance of the cable sheath per meter.
Then, on the basis of (22), (26) and (27) we have
ri =
rc L' z c + Z D
.
L' z c + Z D
(28)
In the common case when the overhead section regarding the
grounding effects may be treated as infinitely long, we have
(22)
Active cable length can be estimated by using (19) and the
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Qb → ∞
Pb → Z C∞
rc Qb + Pb
= rc .
Qb + Pb
On the basis of (24) we can write
(20)
where Zgt denotes the grounding impedance of the transition
station.
In the case of a ground wire of steel the expressions (14) and
(17) can be simplified by introducing the approximation r≈1.
When the overhead section is without a ground wire (r=1 and
ZD≈Pb), instead of (14) and (17), we have
rt =
ri = lim
ri (L' ≥ La) = rc .
where
Na ε-
Z D = Z gt
lim
Session 3 Paper No 55
(
Z D = R N−1 + Z ∞−1
)
−1
(29)
and when a place of a discontinuity is a transition station,
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discontinuity [8].
then it is
ZD=Zgt .
(30)
The last case is especially important for the substations in the
high voltage (e.g. 110 kV) distribution networks. The real
values of the relevant magnitudes (L’ and Zgt) in these
networks are such that the condition Zgt/L’zc<0.05 is satisfied
in many cases and the approximation ri≈rc can be used.
If the discontinuity point is on the line itself, the reduction
effect of this type of cables is less pronounced, because the
grounding impedance ZD is larger in that case (Pb=∞).
The parameters k and Z∞ are defined by the following
relations
In the mid-voltage distribution networks it is common case
that two or more substations are successively supplied
through the same cable section. If the cable is with an
insulated outer sheath, the reduction factor for these
substations is also defined by (28). However, in that case L'
represents the length from D to the considered substation,
while the grounding impedance ZD includes the influence of
the galvanic connections (through the cable sheath(s)) with
the grounding systems of all of these substations.
Now we shall assume that one cable of a voltage 110 kV
supplies a distribution station with a transformation
110/X kV, while, at the other end, the cable is connected to
the overhead section of the feeding line. The cable has three
lead sheaths and their external and internal diameters are
equal: 54.6 mm and 51.4 mm, while the value of the reduction
factor of the cable, according to the manufacturer’s data, is
equal to 0.22. For the overhead part of the line we assume that
in one case it is equipped by a steel ground wire, so that it can
be taken that r≈1, while in the other case we have a ground
wire ACSR 95/55 mm2 with a reduction factor of r=0.675.
The calculation results of the feeding line reduction factor
versus cable length for the first and the second case are shown
in Figs. 5 and 6, respectively.
|ri|
1.0
0.8
|Zgt|=2 Ω
0.6
|Zgt|=1 Ω
QUANTITATIVE ANALYSIS
|Zgt|=0.5 Ω
0.4
At first we shall assume that the substation is directly fed by a
cable with an uninsulated outer sheath which is connected at
the other end to an overhead line without any ground wire (ZN
and RN are treated as infinitely large). This situation is quite
common in mid-voltage distribution network with oil-filled
cables. After the discretization of the cable sheath based on
the following cable data: outer diameter d=44 mm and
experimentally determined Zm/Zs≈0.8 and Zs=(0.0007 + j
0.002) Ω, we performed the calculations and for differential
soil resistivity obtained the results displayed in Fig. 4.
0.2
|Zgt|=0.2 Ω
|Zgt|=0.1 Ω
0.0
0
1
2
3
4
5
L’ (km)
Fig. 5 Cable with insulated outer sheath in the case r=1
|ri|
|Zgt|=2 Ω
0.8
|ri|
|Zgt|=1 Ω
100 Ωm
0.6
|Zgt|=0.5 Ω
60 Ωm
0.8
40 Ωm
0.6
0.4
0.2
|Zgt|=0.2 Ω
|Zgt|=0.1 Ω
0.4
0.0
0
0.2
0.0
20 Ωm
0.1
0.2
2
3
4
5
L’ (km)
Fig. 6 Cable with insulated outer sheath in the case r=0.675
0.3
0.4
L' (km)
Fig. 4. Cable with uninsulated outer sheath in the case r=1, ZN=∞, RN=∞.
As the cable length L' increases, it is evident that the
reduction factor ri tends to the value it would have if the cable
were laid along the whole length of the feeding line.
Certainly, for the equal cable length, the effect would be still
more pronounced if a transition station were at the point of
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Session 3 Paper No 55
If we assume that the length of the cable section is 3 km (the
average length of the cable lines in the 100 kV network of
Belgrade is approximately 4 km), the increase of the
reduction factor ri in comparison to rc is for the first case
(Fig. 5) 6.4%, 13%, 49%, 110% and 195%; while for the
second case (Fig. 6) it is 4%, 10%, 32%, 71% and 128%.
The absolute value of the grounding impedance of stations in
110 kV distribution networks larger than 0.1 Ω (|Zgt|>0.1 Ω) is
rare and may be expected only in the cases of unfavorable
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natural conditions (high soil resistivity). However, according
to the presented calculation results one can gain a global
insight into the reduction factor ri in comparison to the cable
reduction factor when the impedance Zgt has different values.
On the basis of the given results the following can be
concluded. The safety conditions based on the cable reduction
factor rc would give more favorable results (lower values of
touch and step voltages). This means that it could happen that
the required safety conditions in the supplied station would
not be met. It should be stressed that the standard
measurements of safety conditions, which are mandatory
before a newly built station is put in operation, do not
comprise check of the reduction factor value of the feeding
line (e.g. [2]).
CONCLUSIONS
The paper presents a directly applicable and practical method
for the determination of the ground fault current distribution
in substations supplied by a cable section of the feeding line
containing at least one overhead section as well. The
calculations are very simple and take into account all relevant
data available at the design stage.
REFERENCES
[1] “Current During Two Separate Simultaneous Single
Phase Line-To-earth Short Circuits and Partial ShortCircuit Currents Flowing Through Earth,” International
Standard, ref. No. CEI/IEC 909-3, 1995.
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Session 3 Paper No 55
Barcelona, 12-15 May 2003
[2] IEEE Guide for Safety in Substation Grounding St. d.
80-1986.
[3] Lj.M. Popović, 2000, “Efficient Reduction of Fault
Current Through the Grounding Grid of Substation
Supplied by Cable Line”, IEEE Trans. Power Delivery,
15, 556-561.
[4] Lj.M. Popović, 1998, “Practical Method for Evaluating
Ground Fault Current Distribution in Station, Towers
and Ground Wire,” IEEE Trans. Power Delivery, 13,
123-128.
[5] Lj.M. Popović, 1997, “Practical Method for Evaluating
Ground Fault Current Distribution in Station Supplied
by an Inhomogeneous Line,” IEEE Trans. Power
Delivery, 12, 722-727.
[6] Lj.M. Popović, 1993, “Practical Method for the
Analysis of Earthing Systems with Long External
Electrodes,” IEE Proc. C, 140, 213-220.
[7] B. Thapar, S. Madan, 1981, “Current for Design of
Grounding Systems,” IEEE Trans. Power Apparatus
and Systems, 100, 342-345.
[8] J. Villas, D. Mukhedkar, V. Fernandes, A. Magalhaes,
1990, “Grounding Grid Design of a Transition Station
System - a Typical Example of Fault Transfer,” IEEE
Trans. Power Delivery, 5, 124-129.
[9] S. Sobral, V. Costa, M. Campas, D. Mukhedkar, 1988,
“Dimensioning of Nearby Substations Interconnected
Ground System”, IEEE Trans. Power Delivery, 3, 16051614.
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