1
New Method to Determine Optimum Impedance
of Fault Current Limiters for symmetrical and/or
asymmetrical faults in Power Systems
Seyed Mahmoud Modaresi1* and Hamid Lesani2
1
Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran.
2 The School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran.

Abstract—The selection of type and value of Fault Current
Limiters (FCLs) impedance is a problem for power network
designers. This paper introduces a new method to calculate the
optimum value for FCL impedance considering its position in the
network. Due to the complexity of this impedance, the costs of
both real and imaginary parts of FCL impedance are considered.
The optimization of FCL impedance is based on a goal function
that while minimizes the cost of FCL impedance; it maximizes
the fault current reduction. The FCL position in the network has
an effect on calculating the optimum impedance value however;
the selection of the FCL location is not the purpose of this paper.
The proposed method can be used for every network with
symmetrical and asymmetrical faults. However, for the purpose
of demonstration a 14-bus IEEE network is used to explain the
process. The optimum FCL impedance used in the above
mentioned network is calculated by taking into account the vast
range of costs for real and imaginary parts of FCL impedance.
Index Terms—Fault Current Limiter (FCL), FCL impedance,
Short circuit current, Fault Current, Power System.
I.
INTRODUCTION
I
NCREASING the fault currents has been a big challenge in
power system [1]–[8]. To decrease the fault current level, in
one method, Thevenin’s equivalent impedance is increased by
splitting Bus Ties of substations or adding series reactors to
network. These methods increase the system loss, and also can
decrease the voltage and the stability of the network [4]–[10].
In another method, a Fault Current Limiter (FCL) is used to
reduce the fault current as a result of adding impedance to the
system. The FCL impedance in a normal condition is nearly
zero. However, when a fault occurs in the network, the
impedance changes to ZFCL [4]–[13]. Fig. 1 shows the FCL
configuration before and after the fault occurrence.
Fig. 1. Circuital representation of the FCL at pre and post fault periods.
Depending on the mechanism and the time of adding or
removing FCL impedance, there are many FCL type such as
superconductor FCL (SFCL) [2]–[5], [9], [14]–[16], series or
parallel resonance FCL [8], [10], magnetic FCL [11], [12] and
solid state FCL [6], [13], [17]. In these types of FCL, the
*Corresponding author: (e-mail: [email protected])
impedance can be resistance (RFCL), inductive reactance
(XFCLL), capacitive reactance (XFCLC) or complex value that is a
combination of any or all of them [17].
During the fault occurrence, the FCL impedance can be
either a fixed value or a variable value. For the FCLs with
variable impedance, such as SFCLs, the impedance increases
rapidly until reaches its maximum value [3], [4], [9], [16],
[18]. In such cases, the maximum value of FCL impedance is
important for calculating optimum impedance [3], [16].
The FCL impedance increases when, the cost of FCL rises
[1]–[3] and the level of fault current decreases until it levels
off [9], [14], [15], [19]. In the power network, the fault current
of buses should be less than the maximum allowable current
(Im) of circuit breakers and other components of the network
[1]–[3], [20]. The selection of the value and the type of FCL
impedance is a big challenge for network designers.
In references [1]–[3], the impedance of FCL is selected
based on the lower value of FCL impedance, provided that,
the fault current of all buses of network be equal or less than
Im however, this approach cannot necessarily be an optimized
option because, if the fault current reduces further, it can cause
some positive effect on the network. For example it can reduce
the failure probability of the network components and
increases the system reliability [20]–[27]. It can also reduce
the level of electro mechanical and thermal tensions on
network components [2], [6], [7], [12]. Therefore more
reduction of fault current can be an important criterion in the
selection of FCL impedance.
In references [1]–[4], [28] the FCL impedance has been
simply chosen to be purely RFCL or purely XFCLL and then the
optimum value of FCL impedance is determined. The FCL
impedance can be generally a complex value, consisting of
real and imaginary parts [29], [30]. Therefore different prices
of the real and imaginary parts of FCL impedance can affect
the selection of the value and the type of FCL impedance.
In this paper, an approach is proposed to calculate optimum
FCL impedance for power network. To be neutral, the FCL
impedance can be assumed to be a complex value consisting
RFCL, XFCLL and XFCLC. Depending on the network data and the
type of faults three phase symmetrical (3ph) and single line to
ground (1ph), the fault current is calculated. By considering
both the higher fault current reduction and the lower cost of
FCL, the optimum value of FCL impedance is determined at
certain location of the network. The results are then verified
2
using a computer program developed for 14-bus IEEE system.
It should be mentioned that the FCL positioning is not the
main goal of this paper and it can be taken into account in the
future studies. In addition, the mechanism for FCL operation
is not the objective of this paper and only FCL impedance
value after the fault occurrence is considered.
This paper is organized as follows. Section II discusses the
fault current before and after installation FCL. In Section III,
locus of FCL impedance for decreasing the fault current equal
or less than Im is calculated and cost function of FCL is
defined. Method of FCL impedance optimization is probed in
Section IV. Numerical studies are examined in Section V.
Conclusions are given in Section VI.
II.
Fault current before and after FCL installation
Generally, different kinds of faults in a system include 3ph,
line to line, double line to ground and 1ph faults. But 3ph and
1ph fault are of great importance in network study [14], [20].
A.
Fault current before FCL installation
Assuming that the fault impedance is Zf, 3ph fault current
(If3ph) and 1ph fault current (If1ph) are calculated by the
following equations [31].
I 3f ph 
I 1f ph 
Vf
Z Bus
 Z11
 

Zi 1
 A
ZiB 1
 

 Z n1
new
 Z11

 
 Z new
new
A1
Z Bus
  inew
ZiB 1
 

 Z nnew
1
Z new
jk  Z jk 
Z ff  Z f
3V f
(2)
Z 0ff  Z ff  Z ff  3Z f
zero sequence, positive sequence and negative sequence to be
achieved from the impedance matrix of the network. To
calculate the worst fault current, Zf is assumed to be zero.
Fault current after FCL installation
If there is a need to install a FCL in bus i of a power
network, the bus i is divided into two parts as buses iA and iB
and then the FCL is placed between them. This concept is
shown in Fig. 2 where Comps. A and Comps. B are equivalent
of components connected to buses iA and iB.
Bus (i )
Comps. A
Comps. B
(a)
Bus (iA) Bus (iB)
Comps. A
Comps. B
(b)
Bus (iA)
Comps. A
FCL
Bus (iB)
Comps. B
(c)
Fig. 2. Bus (i), (a): before separating the bus, (b): after separating the bus and
before installation FCL, (c): after FCL installation.
In this case, the impedance matrix for n bus network after
separating buses and before FCL installation is represented by
Eq. (3), and the impedance matrix after the FCL installation is
represented by Eq. (4). Each matrix element in Eq. (4) is


Z1i A

Z1i B



 Z i Ai A
Z i Ai B

 ZiB i A


ZiB iB




Z ni B

Z ni A
 Z1new
iA


 Z inew
Ai A
 Z inew
BiA


 Z ninew
A
Z
ji A
Z1n 
 
Zi An 

ZiB n 
 

Z nn 

Z1new
 Z1new
iB
n



 
new
new 
Z i Ai B  Z i A n

Z inew
 Z inew

B iB
Bn


 

new

Z ninew
 Z nn
B

 Z ji B  Z i A k  Z i B k

Zi Ai A  Z i B i B  2 Z i Ai B  Z FCL
.
(3)
(4)
(5)
If bus f is the goal of fault current calculation and a FCL is
installed between buses iA and iB, therefore a general form of
If3ph and If1ph for bus f-th, is represented by Eqs. (6) and (7).
I 3f ph  A 
Z FCL  B
Z FCL  C
I 1f ph  D 
(1)
In the above equations Vf is voltage of bus f before the fault
occurrence, Z 0ff , Z ff and Z ff are Thevenin’s impedance of
B.
calculated by Eq. (5) [31].
(6)
2
Z FCL
 EZ FCL  F
(7)
2
Z FCL
 GZ FCL  H
Where, the variables A, B, C, D, E, F, G and H are
calculated by Eqs. (8) to (15) using the elements driven from
the impedance matrix of zero, positive sequence and negative
sequence and with j being equal to  1 .
A
Vf
 A1  jA2  A  A
Z ff
(8)
B  Z iAi A  Z iBiB  2Z iAiB  B1  jB 2
C  B
2Z
D
3V f
Z 0ff

fi A
Z

fi B
 2 Z ff
Z

fi A
Z
Z

ff

iA f
Z

fi B
Z

iB f
 D1  jD2  D  D
E  L  N  E1  jE2
(9)
 C1  jC 2
(10)
(11)
(12)
(13)
F  L  N  F1  jF2
GE
(K  M )
 G1  jG2
( Z 0ff  2Z ff )
(14)
H F
( K  N  M  L)
 H1  jH 2
( Z 0ff  2 Z ff )
(15)
In equations (12) to (15), the elements of K, L, M and N are
expressed by (16) to (19).
K  (Z 0fi A  Z 0fi B )  (Zi0A f  Zi0B f )
(16)
L  Zi0Ai A  Zi0B iB  2Zi0AiB
(17)
M  2  ( Z fi A  Z fi B )  ( ZiA f  ZiB f )
(18)
N
 ZiAi A
 ZiB iB
 2ZiAiB
.
(19)
The ZFCL is called FCL Impedance and according to Eq.
(20), it includes real part that is RFCL and imaginary part
(XFCL). The RFCL is FCL resistance and always positive but the
XFCL is FCL reactance and can be either positive or negative
3
depending on XFCLL or XFCLC.
(20)
Z FCL  RFCL  jX FCL
3ph
The current magnitude of 3ph fault (|If |) and the current
magnitude of 1ph fault (|If1ph|) for bus f-th can be calculated by
Eqs. (21) and (22) where O, P, T and U are expressed by Eqs.
(23) to (26).
I 3f ph  A 
RFCL  B1 2   X FCL  B2 2
RFCL  C1 2   X FCL  C2 2
(21)
I 1f ph  D 
O 2  P 2
T 2  U 2
(22)
O  RFCL 2  X FCL 2  RFCL E1  X FCL E2  F1
(23)
(24)
P  2RFCL X FCL  RFCL E2  X FCL E1  F2
T  RFCL 2  X FCL 2  RFCL G1  X FCL G2  H1
(25)
(26)
U  2RFCL X FCL  RFCL G2  X FCL G1  H 2
FCL Impedance locus regarding Im of network buses
2
 M 2 C1  B1  
   X

 M 2  1   FCL

 
where, M 
 M 2 C 2  B2

 M 2 1

2
  M
  
 
 


M 1
2
Eq. (27) represents a circle that is shown in Fig. 3 for a test
system. If M>1, the locus of RFCL and XFCL are outside of the
circle area and if M<1 they are inside of the circle area. In this
figure, RC is the radius of circle and DC is the distance from
the center to the origin. Note that RFCL is resistance and
actually cannot be a negative value.
(R-0.25)2+(X+0.5)2-1 = 0
1.5
0.5
0
X
RC
-1
-1.5
-1.5
-1
-0.5
0
0.5
R
1
C
(28)
Cost RFCL
Dividing both sides of Eq. (28) by
produces the cost
function of FCL (Z′) as shown in Eq. (29) and cost factors wL
and wC are calculated by Eqs (30) and (31).
L
C
(29)
Z   RFCL  wL  X FCL  wC  X FCL
Z 
FCL
Costeq
(30)
CostRFCL
wL 
wC 
Cost XFCL
L
(31)
Cost RFCL
Cost XFCL
C
Cost RFCL
(32)
Network designers can select the proper wL or wC depending
on their design priorities and not the cost of each parts of FCL
impedance. Simultaneous using of XFCLL and XFCLC is not
suitable because they cancel out each other’s effect. Hence,
the expression Z′ is modified and defined as Eq. (33).
L
L

 wL  X FCL
If ZFCL includes the RFCL and X FCL
R
(33)
Z    FCL
C
C

RFCL  wC  X FCL If ZFCL includes the RFCL and X FCL
IV. Optimization of FCL impedance
F 3 ph 
-2
-2.5
-2
C
Selection of FCL impedance based on the fault current of
buses is equal to Im does not necessarily produce optimized
results. It is possible to have impedance that greatly reduces
the fault current and be economically viable. Therefore
depending on |If3ph| and |If1ph|, functions F3ph and F1ph are
defined by Eqs. (34) and (35) that achieves FCL impedance
based on maximum reduction of fault current and minimum
cost of FCL.
1
DC
L
Per Unit (pu) for the RFCL, XFCLL and XFCLC and |RFCL|, |XFCLL|
and |XFCLC| are absolute value of the RFCL, XFCLL and XFCLC
respectively in Ω or pu, then the equivalent cost of a FCL (
FCL
) is obtained by Eq. (28).
Costeq
2
C1  B1 2  C 2  B2 2  (27)
Im
.
A
-0.5
The selection of FCL without considering the type and cost
of FCL impedance cannot produce the optimal results. For
example regarding to the Fig. 3, if the costs of RFCL, XFCLL and
XFCLC are the same, the purely XFCLL is the best selection due to
the lower impedance value compared to other types of FCL
impedance. But if the cost of XFCLL is three times the RFCL or
more, the RFCL is a better choice due to the lower cost of FCL.
The cost of a FCL includes both the cost of real and
imaginary parts of FCL impedance. Besides, the cost of each
part rises as the value of the impedance increases [1]–[3]. If
Cost RFCL , Cost XFCL and Cost XFCL are the costs of each ohm (Ω) or
L
The fault current passing through components of the
network should be less than or equal of the maximum
allowable current (Im) of the network components. The FCL
impedance locus is the acceptable ranges of FCL impedance
which causes the fault current magnitude of a specific bus to
be equal or less than Im for the same bus on that network.
The FCL impedance locus, for the 3ph fault is calculated by
|If3ph|=Im, which is shown by Eq. (27)

 R FCL

The cost function of FCL impedance
FCL
Costeq
 RFCL  Cost RFCL  X FCL  Cost XFCL
 X FCL  Cost XFCL
L
C
III. FCL impedance locus and its cost function
A.
B.
1.5
2
2.5
3
I 3f ph 0  I 3f ph
Z
I 1f ph 0  I 1f ph
(34)
Fig. 3. The FCL impedance locus for fault current equal to Im in a test system.
F 1 ph 
The FCL impedance locus for 1ph fault can be
mathematically calculated by |If1ph|=Im, however, to avoid
complicating this paper those calculations are not shown here.
Where the |If3ph0| and |If3ph| are the fault current absolute
value of bus f-th before and after FCL installation that are
expressed by Eq. (21) and the |If1ph0| and |If1ph| are the fault
Z
(35)
4
current absolute value of bus f-th before and after FCL
installation that are expressed by Eq. (22).
The functions F3ph and F1ph are including variables RFCL and
XFCL that can be different value, therefore assuming |Z| is the
absolute value of FCL impedance and Ɵ is its angle, the RFCL
and XFCL can be expressed by Eqs. (36) and (37).
RFCL  Z cos
(36)
X FCL  Z sin  .
X FCL 
Z  cos 
(39)

Z  sin 
(40)




 cos   wL sin   , 0    90
cos   wC sin   , 90    0


 
(41)
Let be noted that the function F1ph can be mathematically
calculated using Eq. (35), however, to avoid complicating this
paper, the process of further expanding related equations are
only shown for 3ph Fault that shown in Eq. (42).
F 3 ph 
A  B12  B2 2


Z   C12  C 2 2

Start




Z  2  2Z B1 cos   B2 sin     2 B12  B2 2 
2
2
Z  2  2Z C1 cos   C 2 sin     2 C1  C 2 

(42)
The extreme points of function F3ph can be calculated when
Ɵ and α determined a fix value and the derivative of F3ph with
respect to Z′ is set to zero. The number of extreme points (Z′)
for the Eq. (42) is 8. Among the calculated values of Z′, when
F3ph is higher it means that the FCL impedance is best. But the
maximum F3ph is not necessarily the optimum option, because
that impedance may not resolve the Im limitation of buses. If
none of the selected Z′ can decrease the fault current equal or
less than Im, therefore, in accordance with the FCL impedance
locus, this problem can be solved by increasing the Z′. In order
to find the new Z′ (Z′new) one must use Eq. (43) below, where
according to Fig. 3, the RC and DC are defined.
 2  2Z new
 DC cos   DC2  RC2  0
Z new
(43)
So far, the achieved Z′ is calculated by using the specific Ɵ
however, in order to use various Ɵ values, one can change Ɵ
at the range of  90    90 and repeating the FCL impedance
optimization using the new Ɵ. By doing so, the various values
of Z′ are calculated. Therefore, the optimal Z′ can be selected
by comparing the value of F3ph for all Z′ (See Eq. (42). It
should be noted that the selection of proper Ɵ can be done by
genetic algorithm or other methods, however in this paper to
simplify, the Ɵ is changed by steps ∆Ɵ=0.1 degree. The
smaller the ∆Ɵ increases, the more accurate will be the
calculations of optimum FCL impedance but this can impact
the computational time.
Finally, with the obtained values of Z′ and Ɵ, the real and
imaginary parts of optimal FCL impedance are calculated
using Eqs. (39) and (40). The computer algorithm (flowchart)
Z'=maximum valid point of F.
Input a location for FCL
instalation on network.
No
Input the Im and the cost
factors wL and wC.
No
Yes
FinalF=max (FinalF , F(Z',))
Yes
|If1ph|>|If3ph|
Fault Current(Z') > Im
Z'=Z'New (from Eq. (43))
Calculate the variables of A, B, C,
D, E, F and H by Eqs. (8) to (15).
(37)
In above equations, Ɵ can be varied from -90 to 90 degrees.
If Ɵ is greater than zero XFCL is equal to XFCLL and if Ɵ is less
than zero XFCL is equal to XFCLC. By uses the Eqs. (33), (36)
and (37), the Z′, RFCL and XFCL are expressed by Eqs. (38), (39)
and (40) respectively where, α is shown by Eq. (41).
Z  Z
(38)
RFCL 
for proposed FCL impedance optimization is shown in Fig. 4.
=+0.1
F=F1ph
F=F3ph
No
>90
Yes
=-90
End
Fig. 4. Flowchart of the proposed FCL impedance optimization.
If with respect to Eq. (38) and by replacing α|Z| instead Z′
in Eq. (42), the Eq. (44) below are created.
F 3 ph 

A  B1 2  B2 2

 Z  C1 2  C 2 2

Z
Z

sin    C
2
 2 Z B1 cos   B2 sin    B1  B2
2
2
 2 Z C1 cos   C 2
2
2
2
1
 C2
  (44)
 
The extreme points of function F3ph in Eq. (44), are
calculated when Ɵ is a fix value and the derivative of the
function F3ph with respect to |Z| is set to zero. With comparing
the extreme points of the function F3ph for Eq. (42) and Eq.
(44), it appears that the values of |Z| and Z′/α in extreme point
of the function F3ph are both equal.
The |Z| is independent of the wL and wC therefore, changing
wL and wC doesn’t affect the |Z| for optimum FCL impedance
when Ɵ is a fix value. It means that the value of Z′/α is
unchanged within a specific range of wL and wC in a fix Ɵ.
This is an interesting result that shows, although it is
possible that different range of wL and wC affect the type of
optimal FCL impedance but doesn’t affect the optimal value
of impedance in each case.
V.
A.
NUMERICAL STUDIES
Test system description
To achieve optimal FCL impedance, the proposed algorithm
can uses for each power network. For our purposes the IEEE
14 bus system as shown in Fig. 5 is used for calculating the
FCL impedance optimization [32]. The impedance data (Ra ,
X’d) of the Generators and Compensators are shown in Table I.
TABLE I
IMPEDANCE DATA OF GENERATORS AND COMPENSATORS IN PU
Component Type Gen.
Bus
No.
1
TType
Ra
0
X’d
0.23
Gen.
2
0.0031
0.23
Com.
3
0.0031
0.13
Com.
6
0.0014
0.12
Com.
8
0.0041
0.12
The |If3ph| before FCL installation are calculated for all
buses. Bus 2 has the highest value of |If3ph| that is equal to
15.37pu. Thus, according to Fig. 5, a FCL is placed on the bus
2. The variables |A|, B1, B2, C1 and C2 considering 3ph fault
occurrence in the bus 2 are calculated by Eqs. (8) to (10) and
recorded at Table II.
TABLE II
THE VARIABLES |A|, B1, B2, C1 AND C2.
|A|
10.84
B1
0.0145
B2
0.3257
C1
0.0031
C2
0.23
5
scenarios. In first scenario, the Im=11 pu and the acceptable
FCL impedance locus is shown in Fig. 8. In second scenario,
the Im=14 pu and its acceptable FCL impedance locus is
shown in Fig. 9. Different colors in those Figures shown
different fault current magnitude in pu.
Fig. 5. IEEE 14 bus test system
Fig. 6 shows the |If3ph| for bus 2 versus the different range of
FCL impedance including RFCL and XFCLL. It shows that for
FCL impedance with RFCL and XFCLL, the |If3ph| is decreased as
a homographic function. Such that, the minimum value for
fault current that is equal to |A| will occur at infinite value for
FCL impedance.
Fig. 8. The FCL impedance locus for Im=11 pu
16
15
f
|I3ph|
14
13
12
11
0
1
0.2
0.8
0.4
0.6
0.6
0.4
0.8
0.2
1
XFCLL
0
RFCL
XFCL
Fig. 9. The FCL impedance locus for Im=14 pu
3ph
Fig. 6. Differentiation of |If | versus to FCL impedance (RFCL and XFCLL).
Fig. 7 shows the |If3ph| for bus 2 versus the different range of
FCL impedance including RFCL and XFCLC. It is shows that for
FCL impedance with RFCL and XFCLC, the |If3ph| first reaches a
pick and then decreases to a minimum. It then rises to level off
and reaches |A| at infinite value for FCL impedance.
120
100
f
|I3ph|
80
60
40
20
1
0
-1
-0.8
0.5
-0.6
-0.4
-0.2
XFCLC
0
0
RFCL
XFCL
3ph
Fig. 7. Differentiation of |If | versus to FCL impedance (RFCL and XFCLC).
B.
Test result and discussion
Depending on Im, the acceptable FCL impedance locus is
different. The FCL impedance optimization calculates for two
The costs of real and imaginary parts of impedance for
different types of FCL can be different. The optimal FCL
impedance for different ranges of wL and wC is calculated. The
values of optimal RFCL, XFCL and |If3ph| for Im=11 pu is shown
in Figs. 10 to12 and for Im=14 pu is shown in Figs. 13 to 15.
Table III summarizes all related values used in Figs. 10 to 15.
TABLE III
THE OPTIMAL VALUE OF FCL IMPEDANCE AND |IF3PH| FOR IM=11PU AND
IM=14PU ACCORDING TO DIFFERENT RANGE OF WL AND WC.
Im=11 pu
wL
wL>0.2795
wL<0.2795
wL>(0.0165wC)
wC
wC>16.965
wC>(60.7wL)
wC<16.965
ZFCL
1.752 pu
6.258 i pu
-0.3227 i pu
|If3ph| 11 pu
11 pu
1.725 pu
Im=14 pu
wL
wL>1.6604
wL<1.6604
wL>(0.3255wC)
wC
wC>5.1012
wC>(3.0723wL)
wC<5.1012
ZFCL
0.266 pu
0.0992 i pu
-0.3227 i pu
|If3ph| 13.16 pu
14 pu
1.725 pu
Figs. 10 to 15 and the Table III show the following results:
1- At the optimal level of FCL Impedance for both RFCL and
XFCL, the |If3ph| is equal to Im where Im=11pu. But when
Im=14pu the |If3ph| for optimal resistive FCL (RFCL) is less than
Im. It shows that, the selection of FCL impedance solely based
on Im limitation is not necessarily the optimum solution.
6
2- The most decreasing of fault current for both Im=11pu
and Im=14pu are occurred when a XFCLC is selected for FCL.
However it is not necessarily the optimum solution due to
different ranges of wL and wC.
It should be noted that using the XFCLC might cause some
problems such as sub synchronous resonance and instability in
the power systems. Therefore in choosing the capacitive FCL,
one should study the power system stability that is out of the
scope of this paper and can be a base for future studies.
3- The optimal FCL impedance for different ranges of wL
and wC are only one of the cases of RFCL, XFCLL or XFCLC. Thus,
the combination of RFCL and XFCLL or RFCL and XFCLC for this
network is not optimized.
4- The optimum impedance value for pure RFCL, pure XFCLL
and pure XFCLC is a fix value for each of the above cases but a
different value among all of those cases. It means that
different range of wL and wC affect the type of optimal FCL
impedance but doesn’t affect the optimal value of impedance
in each case.
0.266 pu
0 pu
0 pu
Fig. 13. The value of RFCL for different range of wL and wC, when Im=14pu
0.0992 pu
0 pu
-0.3227 pu
1.752 pu
0 pu
0 pu
Fig. 14. The value of XFCL for different range of wL and wC, when Im=14pu
14 pu
13.16 pu
Fig. 10. The value of RFCL for different range of wL and wC, when Im=11pu
1.725 pu
6.258 pu
0 pu
-0.3227 pu
Fig. 11. The value of XFCL for different range of wL and wC, when Im=11pu
11 pu
Fig. 15. The value of |If3ph| for different range of wL and wC, when Im=14pu
Figs. 16 to 19 is shown the value of Z′ and Z′/α for various
range of wL and wC for both cases including Im=11 pu and
Im=14 pu. They show the value of Z′ varies but the value of
Z′/α is definite for different ranges of wL and wC. Thus, the
optimal FCL impedance is a fix value when Ɵ is a fix value.
11 pu
1.725 pu
Fig. 12. The value of |If3ph| for different range of wL and wC, when Im=11pu
Fig. 16.Value of Z′ for different range of wL and wC when Im=11kA
7
6.258 pu
1.752 pu
0.3227Ω
Fig. 17. Value of Z′/α for different range of wL and wC when Im=11kA
the fault current reaches the Im.
The cost of the FCL impedance includes the costs of RFCL,
XFCLL and XFCLC that can vary for different types of FCL. Thus,
the FCL impedance optimization in the 14 bus IEEE system is
calculated based on those cost variations.
The results show that the optimum FCL impedance for
different range of costs (including wL and wC) are different and
the FCL impedance can be any of the pure RFCL, pure XFCLL or
pure XFCLC depending of the range of costs. Be noted that in
the test system, the combination of RFCL, XFCLL and XFCLC does
not produce optimize results.
Because the Z′/α is a fix value for various costs of FCL
when the Ɵ is a fix value, the changes in the costs can only
affect the FCL impedance type and has no influence on the
optimum value of that FCL impedance.
As a result, one can select the optimum FCL impedance value
without concerning the costs of FCL impedance.
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In this paper, the importance of FCL impedance optimization in a power network has been pointed out. Generally
speaking the FCL impedance is a complex value and the FCL
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