1
Analysis of Faulted Power Systems in Three Phase
Coordinates–A Generic Approach
Rajeev Kumar Gajbhiye, Student Member, IEEE, Pushpa Kulkarni, Student Member, IEEE, S. A. Soman
Abstract— Analysis of faulted power systems in 3-phase coordinates is desirable when dealing with unbalanced networks
and complex faults. Thevenin’s model in 3-phase domain is wellknown for shunt faults. However, lack of generic circuit model for
series and simultaneous faults is an impediment, as it sacrifices
both simplicity and computational efficiency. In this paper, we
extend this approach to arbitrarily complex faults. Case studies
with complex faults on two example systems demonstrate the
claims made in the paper.
3−φ Thevenin’s Voltage 3−φ Coupled Thevenin’s Impedence
3−φ Fault Model
l
Zf
a
Vtha
Vthb
Vthc
ab
Z
th
aa
Z
th
bc
Z
th
bb
Z
th
l
Zf
b
ca
Z
th
cc
Z
th
Fig. 1.
l
Zf
g
l
Zf
c
Thevenin’s equivalent circuit for shunt fault
Index Terms— Fault Circuit Analysis, Phase Coordinates,
Thevenin’s approach
where,
−1
I. I NTRODUCTION
NALYSIS of standard shunt faults in sequence domain
is based upon construction of Thevenin’s equivalent
circuit in sequence domain. However, analysis of shunt, series
and simultaneous faults in sequence domain is cumbersome
[1]. Further, sequence domain analysis requires a balanced
network assumption. More precisely, network elements should
have circulant symmetry. Unfortunately, distribution systems
as well as transmission systems with untransposed lines are
unbalanced. This motivates analysis in 3-phase co-ordinates
which does not require balanced network assumption [2], [3].
Even in the 3-phase domain, Thevenin’s equivalent circuit
representation is available for standard shunt faults [4]–[6].
We now briefly describe 3-phase Thevenin’s model for the
analysis of shunt faults
Let Yfsh represent a 3 × 3 shunt fault admittance matrix
for a fault at the ith bus of a network and 3-phase pre-fault
admittance matrix model be given by the following equation
A
old
old old
= Ybus
V
Iinj
(1)
Then, under the assumption that inverse of the matrix (I3 +
old
(i, i)Yfsh ) exists, where I3 is an identity matrix of size
Zbus
3 × 3, Thevenin’s equivalent circuit in 3-phase coordinates is
given by the following equation:
sh sh
If + Vfsh
Viold = Zth
(2)
Ifsh = Yfsh Vfsh
(3)
This research work has been funded by CPRI project titled ”Coordination
of Overcurrent and Distance Relays Considering Power Swings in Object
Oriented Paradigm”.
R. K. Gajbhiye, P. Kulkarni and S. A. Soman are with the Indian Institute of Technology, Bombay 400076, India (e-mail: [email protected];
[email protected]; [email protected]).
sh
Zth
old
old
Zbus
= Ybus
 aa
ab
zii zii
old
ba
bb

= Zbus (i, i) = zii zii
ca
cb
zii zii
T
a b
sh
If = if if icf
Viold = [ viaold
vibold
Vfsh = Vinew = [ vianew

ac
zii
bc 
zii
cc
zii
vicold ]T
vibnew
vicnew ]T
Superscript old and new refer to pre and post fault network
quantities respectively. This result can be proved by application of compensation theorem [5]. An alternative proof using
Woodbury’s generalization of Sherman- Morrison formula has
been presented in [6]. Fig 1 provides the visualization of this
model. The representation is simple and straightforward. This
simplifies the analysis of standard shunt faults.
However, series faults do not have such simpler version
of analysis. Typically, analysis of complex faults involves
specifying additional constraint equations like Ia = 0 for
phase a open circuit, Va = 0 for bolted short circuit on phase a
etc. Elimination of such constraints leads to a reduced system
of equations [7]. An alternative approach is to approximate
bolted shunt and series faults by approximate small and large
impedances and modify the admittance matrix [8]. Either way,
we obtain a modified network admittance matrix formulation.
Solution of the resulting system of equations provides the post
fault system voltages. Such an approach is computationally
inefficient, as a new matrix refactorization is required for a
fault. Yet another strategy is to use hybrid approach [9], [10]
where in mixed phase and sequence coordinate representation
is used. This leads to computational efficiency. However, in
our opinion, such an approach is also complex. In [11], Tinney
has presented a compensation approach, wherein opening of
a branch is modeled by the addition of a parallel current
source of appropriate value. For the case of mutually coupled
branches, with one branch being opened, compensating current
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2
Bus i
Bus i
Network
∆Yf
Network
sh
∆Yf
sr
Bus j
(a)
Shunt
fault (Type 1)
(b)
Series
fault (Type 2)
Bus i
Network
∆Yf
sh
∆I
(c) Type 3 fault
Fig. 2.
Schematic for various fault models
source has to be attached in parallel to both the branches. The
advantage of this approach is that original Ybus need not to
be modified and hence no need of refactorization. In [12], the
idea of compensated Thevenin’s impedance is developed to
accommodate line outage in fault analysis. The key idea is to
express network changes as low-order matrix product of the
form BRC T . Fault analysis of simultaneous shunt faults using
Thevenin’s model is also discussed in [13].
To conclude, a considerable amount of literature exists on
fault analysis, both in sequence and 3-phase domain. However,
its study reveals lack of an integrated yet simplified treatment.
One of the goals of this paper is to provide such an unified
treatment that includes generalization of Thevenin’s model to
accommodate series and simultaneous fault. To the best of our
knowledge, no such consolidation of existing literature into an
integrated yet simple approach is available.
This paper is organized as follows:
1) Section II presents the fault models in three phase
coordinates.
2) Section III generalizes Thevenin’s model for series and
simultaneous faults.
3) Section IV presents case studies for varies types of
faults.
4) Section V concludes the paper.
The nomenclature followed in this paper is as follows:
1) The variables in capital letters are blockmatrices or
“vector of vectors”.
2) The variables in capital letters which are indexed, either
in bracket like Z(i, i) or by subscripts e.g. Vi , represent
either a matrix of size 3 × 3 or a vector of size 3.
3) Small letters are reserved for scalars.
Thevenin’s equivalent circuit in sections III and III-B. A fault
can alter the network admittance as well as current / voltage
injections. Therefore, we classify faults in to following types:
1) A fault which alters a 3 × 3 block diagonal entry in the
3-phase Ybus . Let ∆Yfsh be the alteration in ith block
diagonal element of Ybus . Such faults will be referred
as shunt faults. All standard faults, like bolted 3-phase
fault, line to line (LL), single line to ground (SLG),
etc. belong to this class. For standard shunt faults,
∆Yfsh = Yfsh , which is the fault admittance matrix
itself. Schematically, such a fault can be represented by
fig 2(a).
2) A fault which alters both 3 × 3 block diagonal entries
and off diagonal entries in the 3-phase Ybus . Such faults
will be referred as series faults, e.g. open circuit fault
due to opening of any phase conductor of a transmission
line belongs to this class. Schematically, such a fault can
be represented by fig 2(b). For phase open circuit on a
line, ∆Yfsr is the alteration in the line admittance.
3) A fault which also alters the injection of current / voltage
in the system. Open circuit of a generator phase belongs
to this class. It results in alteration in current injection
and shunt admittances. Schematically, such a fault can
be represented by fig 2(c).
A fault which involves multiple instances of either the same
or different fault types (1–3) will be referred as simultaneous
fault. In what follows, we show that faults of type 1 and 2 can
be effectively handled through Thevenin’s equivalent circuit
in 3-phase coordinates. Faults of type 3 in addition, require
application of superposition theorem.
III. T HEVENIN ’ S E QUIVALENT IN P HASE C OORDINATES
A. Analysis of Series Faults
Series fault refers to fault between busses, like opening of
phase a conductor. Such faults are common in distribution
systems. Let a series fault be created in a branch connected
between nodes i and j. Correspondingly, let the 3×3 primitive
admittance of branch change from Ylold to Ylnew . The resulting
change will modify the four block entries namely (i, i), (i, j),
old
. Let,
(j, i) and (j, j) in Ybus
∆Yfsr = Ylnew − Ylold
(4)
The modification of block diagonal entries will involve addition of ∆Yfsr where as the modification of block non-diagonal
entries will involve subtraction of ∆Yfsr . So the modified
block entries in new admittance matrix will be given by,
new
old
(i, i) = Ybus
(i, i) + ∆Yfsr
Ybus
new
old
Ybus
(i, j) = Ybus
(i, j) − ∆Yfsr
new
old
Ybus
(j, i) = Ybus
(j, i) − ∆Yfsr
II. FAULT M ODEL IN 3- PHASE C OORDINATES
new
old
Ybus
(j, j) = Ybus
(j, j) + ∆Yfsr
A fundamental principle that we have used in fault modeling
is that the fault model should depict the change perceived by
the network. This notion is fundamental to development of
This modification can be represented as the rank-6 update,
T
new
old
Ybus
= Ybus
+ Ei Ej Ŷfsr Ei Ej
(5)
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3
I f1
Thevenin’s Voltage Thevenin’s Impedence
Fault Side
Z ii
old
i
V
I f2
sr
Z
& ij
Z ji
If
Z jj
old
j
Vth1
sr
Vth2
∆Y f
Z th
Yf
sr
V
If
Vthm
I fm
I f =Yf Vf
Vth =Zth If
Fig. 3. Block diagram representation of Thevenin’s equivalent circuit for
series fault
where,
Ŷfsr =
∆Yfsr
−∆Yfsr
−∆Yfsr
∆Yfsr
(6)
Ei is a block vector such that,
Ei (j) = O3
i = j
= I3
i=j
with O3 being zero matrix of size 3 × 3 and I3 is identity
matrix of the same size.
The key result of this work is stated by the following
theorem.
Theorem 1 (Series Fault). For a series fault created in
a branch between bus i and j, post-fault voltages and fault
currents at the faulted buses are given by the following
relationships,
new −1 V old Vi
sr sr
i
Ŷ
+
Z
=
I
(7)
6
th f
Vjnew
Vjold
(8)
Ifl = Ylnew Vinew − Vjnew
where, Ŷfsr is given by (6) and
old
old
Zbus (i, i) Zbus
(i, j)
sr
Zth =
old
old
Zbus
(j, i) Zbus
(j, j)
Fig. 4.
Block representation of Thevenin’s model for multiple faults
S. Let the fault admittance Yfsh
k represent a shunt admittance
to be connected on bus k and Yfsr
k,l represents series fault
admittance required between bus k and l to capture the postnew
old
can be obtained from Ybus
fault changes in the network. Ybus
by the following sequence of steps:
1) Set ∆Y to n × n zero block matrix.
2) Identify subset Ssh of busses in set S on which shunt
fault occurs. Similarly, identify subset Ssr of busses
from set S involved in series fault. Note that S =
Ssh ∪ Ssr and Ssh ∩ Ssr need not be empty.
sh
3) For all the shunt faults Yfsh
is , is ∈ Ssh , add Yf is to
∆Y (is , is ).
4) For all the series faults Yfsr
is ,it , is , it ∈ Ssr , add the
said admittance to the diagonal entries ∆Y (is , is ) and
∆Y (it , it ). Also subtract the same admittance from
∆Y (is , it ) and ∆Y (it , is ).
Finally,
new
old
= Ybus
+ ∆Y
(10)
Ybus
In matrix ∆Y , all the entries except those belonging to
faulted nodes, are zero (or more precisely O3 ). Eqn 10 can
old
,
also be expressed as rank-3m update of Ybus
new
old
Ybus
= Ybus
+ [Ei1 :im ] Yfsim [Ei1 :im ]
(9)
Proof. The result is derived by the application of Woodbury’s
formula to eqn (5). It is detailed in appendix I.
Equation 7 describes the Thevenin’s equivalent circuit for the
sr
given by equation (9). Fig 3 visualizes
series fault with Zth
this equation. The Thevenin’s equivalent circuit consists of
old
old
(i, i) and Zbus
(j, j) mutually
two 3-phase impedances Zbus
old
old
(j, i). Current
coupled with each other by Zbus (i, j) and Zbus
in the faulted line (Ifl ) is given by eqn 8. The current Ifsr is
a fictitious current associated with the sub-network depicting
the change due to the fault.
B. Simultaneous Faults
We now consider the case of simultaneous fault where one
or more busses may be involved in more than one fault. The simultaneous fault is composed of shunt and series faults. Let the
set of busses involved in the fault be S = {i1 , i2 , . . . , im }.
Let us assume that msh number of shunt faults and msr
number of series faults occur on busses that belong to the set
T
(11)
where, [Ei1 :im ] = [Ei1 Ei2 · · · Eim ]. An entry (k, l) in Yfsim
is given by Yfsim (k, l) = ∆Y (ik , il ) k, l ∈ {1, 2, · · · , m}
i.e., Yfsim is a compressed representation of ∆Y , in which all
zero rows and columns of ∆Y have been eliminated.
Proceeding in a similar fashion to the case of series fault,
one can verify that post fault bus voltages at faulted busses
can be calculated using following equation
T
Vf = V new (i1 )T V new (i2 )T · · · V new (im )T
sim sim −1 sim
= (I3m + Zth
Yf ) Vth
(12)
where, resultant Thevenin’s voltage source vector and
impedance matrix are given by the following relationships:
T
sim
= V old (i1 )T V old (i2 )T · · · V old (im )T
(13)
Vth
sim
old
= Zbus
(i1 : im , i1 : im )
Zth
 old
old
Zbus (i1 , i1 ) Zbus
(i1 , i2 )
old
old
 Zbus
(i
,
i
)
Z
(i2 , i2 )
2
1
bus

=
..
..

.
.
old
(im , i1 )
Zbus
···
···
..
.
old
Zbus
(im , i2 ) · · ·
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
old
Zbus
(i1 , im )
old
Zbus
(i2 , im ) 


..

.
old
Zbus
(im , im )
(14)
4
e1
Z ( i1 , i1 )
i1
Iffict
i2
Iffict
Remark 1. Knowing the postfault voltages, postfault current
can be easily computed for each component in the system from
their primitive 3-phase admittance matrix.
1
Viold
1
sr
e2
V
Z ( i2 , i2 )
∆ Yf
1
2
old
IV. T EST C ASES
i2
sh
Yf
sr
∆ Yf
e3
V
1
2
i3
Iffict
old
In this section, we consider faults whose analysis in sequence component is cumbersome and complex. Further, we
show how their analysis in 3-phase coordinates is simplified.
3
i3
Z ( i3 , i3 )
e4
A. Analysis of Open Conductor Falling on Ground
3
i4
Iffict
Viold
4
The single line diagram of the system [1] along with
sequence data for base values of 30M V A, 34.5KV is shown
in figure 6. The positive and negative sequence impedances
of load are 1.0∠25.84◦ p.u, 0.60∠29◦ p.u. respectively. Load
voltage is kept at 1.0 p.u.
4
Z ( i4 , i4 )
4
sh
Yf
2
Thevenin’s
Voltage
Fault
Network
Thevenin’s
Impedance
Fig. 5. Figure shows the Thevenin’s equivalent circuit for simultaneous fault
analysis. Faults are at busses i1 , i2 , i3 and i4 . The faults consists of two
shunt faults on bus i2 and i4 , and two series faults, one between bus i1
and i3 , another between i1 and i4 . The expressions for mutual couplings in
Thevenin’s impedance are not shown for clarity.
G
50MVA
X1s = X2s= 15%
V
=V
+
old
Zbus
[Ei1
Ei2 · · ·
67
m
(17)
Eim ][−Iff ict ]
(18)
30MVA
34.5: 13.8KV
XTH
= 8%
Single line diagram of system
Consider the case of a simultaneous fault. Conductor in
phase a of transmission line between bus G and bus H
breaks and it gets grounded on the load side. Now we
apply the Thevenin’s equivalent method, for simultaneous fault
(section III-B). Block diagram of Thevenin’s equivalent circuit
sim
is represented in figure 7. Here, Thevenin’s impedance Zth
Thevenin’s Voltage
Thevenin’s Impedence
Fault Side
V
old
Z GG
Z
Z
Z
Now, post fault voltages can now be calculated by applying
superposition theorem.
old
0.64
&
can be calculated as follows:
 new

V
(i1 )
 V new (i2 ) 

Iff ict = Yfsim 


···
new
(im )
V
new
=
G
is as shown in fig 3.
3) Simultaneous Fault: Let the fictitious current vector associated with sub-network depicting the change be represented
as
T
T
T T
Iff ict = Iffiict
Iffiict
· · · Iffiict
2
Z 0GH
30MVA
LOAD
90 %P.F
Per Unit at 30 MVA,
34.5 KV Base
V
Ifsr
1
71
XTG= 7%
where, i and j are the faulty busses, and
sr new If
sr Vi
= Ŷf
Vjnew
−Ifsr
Iff ict
0.21
50MVA
115 : 34.5 KV
Fig. 6.
2) Series Fault: Fictitious current in fig 3 represents the
compensation current required to model the post fault system.
Hence, by the application of superposition theorem
sr If
old
Ei E j −
V new = V old + Zbus
(16)
−Ifsr
=
Fault
Fig 4 illustrates the scheme to analyze simultaneous fault using
Thevenin’s equivalent circuit. Figure 5 illustrates Thevenin’s
equivalent circuit for simultaneous fault at four busses in a
system.
C. Calculation of Post Fault Voltages
It can be shown by the application of superposition theorem
that, post fault voltages are given by the following equations.
1) Shunt Fault:
sh old
Ei −If
(15)
V new = V old + Zbus
H
Z 1GH
old
sr
∆Y f
GH
HG
HH
If
H
sh
Yf
Fig. 7.
Block diagram representation of Thevenin’s equivalent circuit for
simultaneous fault on system shown in figure 6.
and fault admittance Yfsim are block matrices of size 2 × 2.
Entries of Yfsim are given as follows:
∆Yfsr
−∆Yfsr
sim
Yf
=
−∆Yfsr ∆Yfsr + Yfsh
where, ∆Yfsr is the series admittance (see equation 4) representing the change due to opening of phase a of transmission
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5
line between busses G and H. Yfsh is shunt fault admittance
representing single line to ground (SLG) fault on phase a of
bus H. Though Yfsh can not be represented for bolted fault,
it is constructed by assuming very small value of ground
impedance (10−6 p.u.). Thus, fault bus voltages is calculated
using eqn 12. This leads to the following results:
old new VG
VG
sim sim −1
+
Z
Y
)
=
(I
(19)
6
th
sh
VHnew
VHold


1.26199, ∠−58.809◦
(20)
If =  1.01855, ∠5.175◦ 
1.68279, ∠−91.727◦
Thevenin’s Impedence
Thevenin’s Voltage
V
(1)
Ifict
f
Z 22
old
Z
2
&
V
Fault Side
Z
sh
Yf
24
42
Z 44
old
SLG fault
(2)
I fict
f
∆I
4
sh
∆Y f
Generator Fault
The result is consistent with sequence component analysis [1].
Fig. 9.
Thevenin’s equivalent circuit for system in fig 8
B. Simultaneous fault on system with mutually coupled lines
TABLE I
P OST- FAULT VOLTAGES ( IN KV)
1
Bus
1
2
3
4
2
4
E=345 kV
X1=17.5 Ω
X0=12.0 Ω
E=345 kV
X1=17.5 Ω
X0=12.0 Ω
3
•
Fig. 8.
System with Mutually Coupled Lines
In this example, taken from [8], we analyze 345 kV double
circuit transmission line for simultaneous fault on it. T-type
equivalent of the 100 mi double circuit 345 kV transmission
line with equivalent impedances of the receiving and the
sending end systems is reproduced here in fig 8. In [14], a
old
in sequence domain considering
procedure to calculate Ybus
the mutual couplings has been presented. Similar steps are
followed in 3-phase domain to obtain 3-phase Ybus .
The simultaneous fault considered for this system is a SLG
fault on phase a of bus 2 and open phase fault on phase b of
generator on bus 4. Open circuit fault on a generator bus is a
shunt fault. It has a peculiar nature that it not only modifies the
Ybus but it also alters the current injection. Hence, this fault is
combination of type 1 and type 3 faults. The effect of change
in current injection on post fault system is analyzed by using
superposition theorem. Fault analysis is done as follows:
sim
old
• Zth
is formed by picking proper entries from Zbus
corresponding to buses involved in fault. Figure 9 shows
Thevenin’s equivalent circuit for simultaneous fault.
Shunt admittance Yfsh is connected at bus two and open
circuit of generator is shown by a current injection ∆I
with shunt admittance ∆Yfsh .
• ∆I is calculated as
∆I
new
old
− Igen
= Igen
2
2
new
old
= Ygen
Egen2 − Ygen
Egen2
2
2
=
∆Yfsh Egen2
Define for the faulted busses,
∆Iinj
O3
=
∆I
•
Phase A
157.9
0.0
92.1
156.8
Phase B
201.4
248.9
239.9
209.1
Phase C
200.4
255.0
248.9
199.7
Using eqn (12), post fault voltages at faulted busses are
calculated as follows:
new V2
sim sim −1
Yf )
= (I6 + Zth
V4new
old V2
sim
×
+ Zth ∆Iinj (21)
V4old
sim
∆Iinj in above equation is
The additional term Zth
a consequence of superposition effect of ∆I current
injection.
Fictitious current vector (see fig 9) is calculated by using
the following equation
Yfsh V2new
Iff ict =
∆Yfsh V4new − ∆I
The post fault voltages are calculated by eqn (18). These
are tabulated in table I. They are consistent with results
in [8], which validates the proposed approach.
V. C ONCLUSIONS
Three phase analysis is suitable for fault analysis of unbalanced systems and complex faults. In this paper, we have
proposed generic Thevenin’s circuit representation for shunt,
series and simultaneous faults. It’s advantage is conceptual
simplicity and computational efficiency as repeated matrix
factorization is not required. Two test systems with complex
faults were analyzed to validate the approach.
ACKNOWLEDGMENT
The authors acknowledge the valuable suggestions and
feedback by Prof. M. A. Pai, University of Illinois at Urbana
Champaign, Prof. D. Thukaram, Indian Institute of Science,
Bangalore and Profs S. A. Khaparde and A. M. Kulkarni of
Indian Institute of Technology, Bombay.
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6
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A PPENDIX I
P ROOF OF T HEOREM 1
The Woodbury’s formula [15] for calculating inverse of a
rank-k updated matrix A is given by,
−1
−1 T −1
A + UV T
= A−1 − A−1 U Ik + V T A−1 U
V A
(22)
where, A is n × n matrix, U and V are n × k matrices and Ik
is k × k identity matrix. It is assumed that necessary inverse
in eqn (22) exists.
For the case of series fault, from equation (5), we set
T −1
new
old
= Ybus
+ Ei Ej Ŷfsr Ei Ej
Zbus
where, Ŷfsr is defined in the eqn (6)
old
formula
(eqn
22) with A = Ybus , U =
Applying Woodbury’s
sr
Ei Ej Ŷf and V = Ei Ej
T old
new
old
old
Ei Ej Ŷfsr I6 + Ei Ej Zbus
⇒ Zbus
= Zbus
− Zbus
−1 T old
Ei Ej Ŷfsr
Ei Ej Zbus
old
old
old
(:, i) Zbus
(:, j) Ŷfsr
= Zbus
− Zbus
old
−1 old
old
Z (i, i) Zbus
(i, j)
Zbus (i, :)
sr
Ŷ
I6 + bus
old
old
old
Zbus
Zbus
(j, i) Zbus
(j, j) f
(j, :)
(23)
old
and sampling ith and j th row and
Post-multiplying by Iinj
old
old
old
old
substituting Zbus (i, :)Iinj = Viold and Zbus
(j, :)Iinj
= Vjold ,
new old old
old
Vi
Vi
(i, j)
Zbus (i, i) Zbus
Ŷ sr
=
−
old
old
Vjnew
Vjold
Zbus
(j, i) Zbus
(j, j) f
old
−1 old old
Vi
Zbus (i, i) Zbus
(i, j)
sr
Ŷ
I6 +
old
old
Vjold
Zbus
(j, i) Zbus
(j, j) f
(24)
Let,
T
T T
= Vth
T
Vinew T Vjnew T
= Vf
old
old
Zbus (i, i) Zbus (i, j)
sr
= Zth
old
old
Zbus
(j, i) Zbus
(j, j)
Viold Vjold
(25)
(26)
(27)
Then we have
−1
sr sr
sr sr
Ŷf I6 + Zth
Ŷf
Vf =Vth − Zth
Vth
−1
sr sr
sr sr
sr sr
I6 + Zth
Ŷf − Zth
Ŷf
Ŷf
= I6 + Zth
Vth
−1
sr sr
Ŷf
⇒ Vf = I6 + Zth
Vth
(28)
Rajeev Kumar Gajbhiye is currently working towards Ph.D. degree in
Department of Electrical Engineering at Indian Institute of Technology,
Bombay, India. His research interests include power system analysis and
object oriented analysis.
Pushpa Kulkarni is currently pursuing M.Tech degree in Indian Institute of
Technology, Bombay, India.
S. A. Soman is Associate Professor in the Department of Electrical Engineering at Indian Institute of Technology Bombay, India. Along with Prof. S.
A. Khaparde, he has authored a book on Computational Methods for Large
Sparse Power System Analysis: An Object Oriented Approach, published by
Kluwer Academic Publishers in 2001.
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