International Journal of Reviews in Computing
30th September 2011. Vol. 7
© 2009 - 2011 IJRIC & LLS. All rights reserved.
ISSN: 2076-3328
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IJRIC
E-ISSN: 2076-3336
UTILITIES OF DIFFERENTIAL ALGEBRAIC EQUATIONS
(DAE) MODEL OF SVC AND TCSC FOR OPERATION,
CONTROL, PLANNING & PROTECTION OF POWER
SYSTEM ENVIRONMENTS
BINDESHWAR SINGH
Assistant Professor, Kamla Nehru Institute of Technology (KNIT), Sultanpur-228118, U.P., India,
bindeshwar.singh2025@gmail.com
ABSTRACT
This paper presents the development of the differential Algebraic equation (DAE) model of various FACTS
controllers such as TCSC and SVC for operation, control, planning &protection of power systems. Also
this paper presents the current status on development of the DAE model of various FACTS controllers such
as TCSC and SVC for operation, control, planning &protection of power systems. Authors strongly believe
that this article will be very much useful to the researchers for finding out the relevant references in the
field of the DAE model of FACTS controllers for operation, control, planning & protection of power
systems.
Keywords:- Flexible AC Transmission Systems (FACTS), FACTS Controllers, Differential Algebraic
Equations (DAE) model, Thyristor Controlled Series Capacitor (TCSC), Static Var
Compensator (SVC), Power Systems.
1.
corridors, enhancement of transient stability,
mitigation of system oscillations and voltage
(reactive power) regulation. Performance analysis
and control synthesis of TCSC and SVC requires
its steady-state and dynamic models.
INTRODUCTION
Traditionally, the objective of the reactive power
(VAR) planning problem is to provide a minimum
number of new reactive power supplies to satisfy
only the voltage feasibility constraints in normal
and post-contingency states. Various researches
have been carried out for this subject [1] and [2].
Recently, due to a necessity to include the voltage
stability constraints, a few researches have been
reported concerning new formulations considering
the voltage stability problem [3] and [4], which
provides more realistic solutions for the VAR
planning problem. However, the obtained solutions
are sometimes too expensive since they satisfy all
of the specified feasibility and stability constraints.
In [5], a new formulation and solution method are
presented for the VAR planning problem including
FACTS devices, taking into account the issues just
mentioned. TCSC and SVC are used to keep bus
voltages and to ensure the voltage stability margin.
The TCSC model presented in [2]-[5] and SVC
model presented in [6]-[21]. TCSC [2]-[5] and
SVC [6]-[21] can be used for power flow control,
loop flow control, load sharing among parallel
This paper is organized as follows: Section II
discusses the DAE model of TCSC and SVC.
Section III presents the conclusions of the paper.
2.
DIFFERENTIAL
EQUATION (DAE)
and SVC
ALGEBRAIC
MODEL OF TCSC
A. DAE model of TCSC
1.
Fundamentals of TCSC:
Thyristor Controlled Series Capacitor (TCSC)
provides powerful means of controlling and
increasing power transfer level of a system by
varying the apparent impedance of a specific
transmission line. A TCSC can be utilized in a
planned way for contingencies to enhance power
system stability. Using TCSC, it is possible to
55
International Journal of Reviews in Computing
IJRIC
30th September 2011. Vol. 7
© 2009 - 2011 IJRIC & LLS. All rights reserved.
ISSN: 2076-3328
www.ijric.org
the TCSC can be modeled using the following
equations [4].
k σ + sin σ
⎡
⎤
+
⎢1 − k 2 − 1 .
⎥
π
⎥
X TCSC = X C ⎢ 2
2
⎢ 4.k . cos (σ / 2) .(k tan kσ − tan σ )⎥
⎢ π (k 2 − 1) 2
2
2 ⎥⎦
⎣
operate stably at power levels well beyond those
for which the system was originally intended
without endangering system stability [3]. Apart
from this, TCSC is also being used to mitigate SSR
(Sub Synchronous Resonance).The TCSC module
shown in Fig.5.
Where
α = Firing angle delay (after forward vale voltage)
σ = Conduction angle= 2(π − α ) and
α
k = TCSC ratio =
α
XC / XT
The TCSC can be continuously controlled in the
capacitive or inductive zone by varying firing angle
in a predetermined fashion thus avoiding steady
state resonance region.
Fig. 5.TCSC module
The steady-state impedance of the TCSC is that of
a parallel LC circuit, consisting of fixed capacitive
impedance, X C , and a variable inductive
impedance, X L (α ) , that is,
X TCSC (α ) =
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3.
Incorporation of TCSC in Multi-machine
Power Systems:
The block diagram representation of TCSC shown
in Fig. 7.
X C X L (α )
X L (α ) − X C
Where
X L (α ) = X L
π
, X L ≤ X L (α ) ≤ ∞
π − 2α − sin 2α
X L = ωL , and α is the delay angle measured from
the crest of the capacitor voltage (or, equivalently,
the zero crossing of the line current). The
impedance of the TCSC by delay is shown in Fig.
6.
Fig.7.Block diagram representation of TCSC
module
Let a TCSC be connected between bus k and bus m
as shown in Fig. It has been assumed that the
controller is lossless. The power-balance equation
and BTCSC are given as [4]
Pk = VkVm BTCSC sin(θ k − θ m )
(1)
2
Qk = Vk BTCSC − VkVm BTCSC cos(θ k − θ m )
(2)
Pm = VkVm BTCSC sin(θ m − θ k )
(3)
2
Qm = Vm BTCSC − VkVm BTCSC cos(θ m − θ k )
(4)
Fig.6.TCSC equivalent Reactance as a function of
firing angle
2.
TCSC Controller Model:
The structure of the TCSC is the same as that of a
FC-TCR type SVC. The equivalent impedance of
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BTCSC = −π (k − 2k + 1) cos k (π − α ) /
4
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2
⎤
⎡ X C (πk 4 cos k (π − α )
⎥
⎢
⎥
⎢− π cos k (π − α ) −
⎥
⎢2k 4α cos k (π − α )
⎥
⎢
2
⎥
⎢+ 2αk cos k (π − α )
⎥
⎢ 4
k
α
k
π
α
sin
2
cos
(
)
−
−
⎥
⎢
⎥
⎢+ k 2 sin 2α cos k (π − α )
⎥
⎢
3
2
⎥
⎢− 4k cos α sin k (π − α )
⎢− 4k 2 cos α sin α cos k (π − α ))⎥
⎦
⎣
Fig.8. Block diagram representation of TCSC with
PI controller
The controller equations are given as
K
X 2TCSC = I ( Pset − P)
s
•
X 2TCSC = K I Pset − K I P
(6)
(5)
(7)
Equation (21) is obtained from (16).
There are number of control strategies for TCSC
[4]
•
Reactance control: Bset − BTCSC = 0
•
Power control:
•
Current control:
Linearization of (6) and (7) yields
•
∆X 1TCSC =
•
Pset − P = 0
(8) & (9)
I set − I = 0
∆α
∑
KI
s
∑
αref
α0
−∆X 1TCSC ∆X 2TCSC K P ∆Pk
+
−
Tc1
Tc1
Tc1
∆X 2TCSC = − K I ∆Pk
As the TCSC controller is assumed lossless, the
• Transmission angle control: δ set − δ = 0
Where the subscript “set” indicates set point.
Any of the above mentioned control strategies can
be used to achieve the objectives of TCSC. In this
paper, the power control strategy has been used,
the block diagram of which is shown in Fig.
The line power is monitored and compared to
desired power Pset . The error is fed to proportionalintegral (PI) controller. The output of PI controller
is fed through a first order block to get the desired
α . The block diagram representation of TCSC
with PI controller shown in Fig.8.
∑
− X 1TCSC X 2TCSC K P Pset K P P α o
+
+
−
+
Tc1
Tc1
Tc1
Tc1
Tc1
•
X 1TCSC =
1
1 + sTC1
real power at the two ends of the bus is same. The
real power feedback is taken from bus k.
∆Pk = ∆VV
k mo BTCSCo sin(θko − θmo ) +Vko∆VmBTCSCo sin(θko − θmo ) +VkoVmo∆BTCSC sin(θko − θmo )
VkoVmoBTCSCo cos(θko −θmo )∆θk −VkoVmoBTCSCo cos(θko −θmo )∆θm
Substituting value of ∆Pk into (8) and (9)
⎡∆VV
⎤
k moBTCSCo sin(θko −θmo ) +Vko∆VmBTCSCo sin(θko −θmo )
−∆X1TCSC ∆X2TCSC KP ⎢
+
− ⎢+VkoVmo∆BTCSC sin(θko −θmo ) +VkoVmoBTCSCo cos(θko −θmo )∆θk ⎥⎥
Tc1
Tc1
Tc1
⎢⎣−VkoVmoBTCSCo cos(θko −θmo )∆θm
⎥⎦
⎡∆VV
⎤
k moBTCSCo sin(θko −θmo ) +Vko∆VmBTCSCo sin(θko −θmo )
•
∆X 2TCSC =−KI ⎢⎢+VkoVmo∆BTCSC sin(θko −θmo ) +VkoVmoBTCSCo cos(θko −θmo )∆θk ⎥⎥
⎣⎢−VkoVmoBTCSCo cos(θko −θmo )∆θm
⎦⎥
•
∆X1TCSC =
Writing above equations in matrix notation.
⎡ ∇θ k ⎤
⎢ ∇V ⎥
•
∇ X TCSC = ATCSC ∇X TCSC + BTCSC ⎢ k ⎥
⎢∇θ m ⎥
⎥
⎢
⎣∇Vm ⎦
α
⎡ ∇X
⎤
Where ∇X TCSC = ⎢ 1TCSC ⎥
X
∇
⎣ 2TCSC ⎦
X1TCSC
And ATCSC and BTCSC are given as
X 2TCSC
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30th September 2011. Vol. 7
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ISSN: 2076-3328
⎡ −1 K PVkoVmo ( FFCONS )sin(θ ko − θ mo )
−
Tc1
ATCSC = ⎢ Tc1
⎢
⎢⎣ − K IVkoVmo ( FFCONS )sin(θ ko − θ mo )
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1 ⎤
Tc1 ⎥
⎥
0 ⎥⎦
FF 2 = k 5 B TCSCo X cπ sin( kπ − kX 1TCSCo )
− kB TCSCo X cπ sin( kπ − kX 1TCSCo )
⎡ A B C D⎤
BTCSC = ⎢
⎥
⎣E F G H ⎦
−KP
A=
[VkoVmo BTCSCo cos(θ ko − θ mo )]
Tc1
B=
−KP
[Vmo BTCSCo sin(θ ko − θ mo )]
Tc1
C=
−KP
[ −VkoVmo BTCSCo cos(θ ko − θ mo )]
Tc1
−2k 4 B TCSCo X c cos( kπ − kX 1TCSCo )
−2k 5 X 1TCSCo B TCSCo X c sin( kπ − kX 1TCSCo )
+2k 2 B TCSCo X c cos( kπ − kX 1TCSCo )
+2k 3 X 1TCSCo B TCSCo X c sin( kπ − kX 1TCSCo )
−2k 4 cos2 ( X 1TCSCo ) B TCSCo X c cos( kπ − kX 1TCSCo )
+2k 4 sin 2 ( X 1TCSCo ) B TCSCo X c cos( kπ − kX 1TCSCo )
−2k 5 sin( X 1TCSCo )cos( X 1TCSCo ) B TCSCo X c sin( kπ − kX 1TCSCo )
−KP
D=
[Vko BTCSCo sin(θ ko − θ mo )]
Tc1
+8k 3 sin( X 1TCSCo )cos( X 1TCSCo ) B TCSCo X c sin( kπ − kX 1TCSCo )
+4k 4 cos2 ( X 1TCSCo ) B TCSCo X c cos( kπ − kX 1TCSCo )
E = − K I [VkoVmo BTCSCo cos(θ ko − θ mo )]
+4k 4 sin 2 ( X 1TCSCo ) B TCSCo X c cos( kπ − kX 1TCSCo )
F = − K I [Vmo BTCSCo sin(θ ko − θ mo )]
−4k 2 cos2 ( X 1TCSCo ) B TCSCo X c cos( kπ − kX 1TCSCo )
G = − K I [ −VkoVmo BTCSCo cos(θ ko − θ mo ) ]
−4k 3 sin( X 1TCSCo )cos( X 1TCSCo ) B TCSCo X c sin( kπ − kX 1TCSCo )
H = − K I [Vko BTCSCo sin(θ ko − θ mo ) ]
a.
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+2k 2 cos2 ( X 1TCSCo ) B TCSCo X c cos( kπ − kX 1TCSCo )
−2k 2 sin 2 ( X 1TCSCo ) B TCSCo X c cos( kπ − kX 1TCSCo )
Details of Constant FFCONS:
+2k 3 sin( X 1TCSCo )cos( X 1TCSCo ) B TCSCo X c sin( kπ − kX 1TCSCo )
Linearizing the equations (3.16) gives
Linearizing the equations (3.12-3.15) gives
∆BTCSC = ( FFCONS ) ∆X 1TCSC
Details of FFCONS are obtained as shown below.
Linearization of (3.11) gives
∆Pk = ∆VV
k mo BTCSCo sin(θko −θmo ) +Vko∆VmBTCSCo sin(θko −θmo ) +VkoVmo∆BTCSC sin(θko − θmo )
+VkoVmoBTCSCo cos(θko −θmo )∆θk −VkoVmoBTCSCo cos(θko −θmo )∆θm
∆BTCSC ( FF 1) + ∆X 1TCSC ( FF 2) = − kπ ( k 2 − 1) 2 sin( kπ − kX 1TCSCo ) ∆X 1TCSC
∆Qk = 2VkoBTCSCo∆Vk +V 2ko∆BTCSC −∆VV
k mo BTCSCo cos(θko − θmo ) −Vko∆VmBTCSCo cos(θko −θmo )
or
⎡ − FF 2 − kπ (k 2 − 1)2 sin(kπ − kX 1TCSCo ) ∆X 1TCSC ⎤
∆BTCSC = ⎢
⎥ ∆X 1TCSC
FF1
⎣
⎦
−VkoVmo∆BTCSC cos(θko −θmo ) +VkoVmoBTCSCo sin(θko −θmo )∆θk −VkoVmoBTCSCo cos(θko −θmo )∆θm
∆Pm = ∆VV
k mo BTCSCo sin(θmo −θko ) +Vko∆VmBTCSCo sin(θmo −θko ) +VkoVmo∆BTCSC sin(θmo − θko )
+VkoVmoBTCSCo cos(θmo −θko )∆θm −VkoVmoBTCSCo cos(θmo −θko )∆θk
or
∆BTCSC = ( FFCONS ) ∆ X 1TCSC
∆Qm = 2VmoBTCSCo∆Vm +V 2mo∆BTCSC −∆VV
k mo BTCSCo cos(θmo − θko ) −Vko∆VmBTCSCo cos(θmo − θko )
Where
−VkoVmo∆BTCSC cos(θmo −θko ) +VkoVmoBTCSCo sin(θmo −θko )∆θm −VkoVmoBTCSCo sin(θmo −θko )∆θk
⎡ − FF 2 − kπ ( k 2 − 1)2 sin(kπ − kX 1TCSCo ) ∆X 1TCSC ⎤
FFCONS = ⎢
⎥
FF1
⎣
⎦
The above equation can be written in matrix
notation
FF1 and FF2 are computed
⎡ ∇θ k ⎤
⎡ ∇Pk ⎤
⎢ ∇V ⎥
⎢ ∇Q ⎥
⎢ k⎥
⎢ k ⎥ = C ∇X
D
+
1TCSC
TCSC
TCSC
⎢∇θ m ⎥
⎢ ∇Pm ⎥
⎥
⎥
⎢
⎢
⎣∇Vm ⎦
⎣∇Qm ⎦
Incorporation of (25), (26), and (5) gives DAE
model of multi-machine power system with TCSC
incorporated in the system. After reordering, final
form of DAE model with TCSC is given as
FF 1 = X cπ k 4 cos( kπ − kX 1TCSCo )
− X cπ cos( kπ − kX 1TCSCo )
−2k 4 X 1TCSCo cos( kπ − kX 1TCSCo ) X c
+2k 2 X 1TCSCo )cos( kπ − kX 1TCSCo ) X c
−2k 4 sin( X 1TCSCo )cos( X 1TCSCo )cos( kπ − kX 1TCSCo ) X c
−4k 3 cos 2 ( X 1TCSCo )sin( kπ − kX 1TCSCo ) X c
−4k 2 sin( X 1TCSCo )cos( X 1TCSCo )cos( kπ − kX 1TCSCo ) X c
+2k 2 sin( X 1TCSCo )cos( X 1TCSCo )cos( kπ − kX 1TCSCo ) X c
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IJRIC
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⎡ ∇ X• ⎤ ⎡ A1mod P1t csc
A2new
Anew ⎤ ⎡ ∇X ⎤
⎥ ⎢
⎢ •
⎡E ⎤
⎥
Bt csc new ⎥ ⎢∇X TCSC ⎥ ⎢ ⎥
⎢∇ X TCSC ⎥ ⎢ P2t csc ATCSC Bt csc1new
⎢
⎥
+ 0 ∇U
⎥=⎢ K
⎢
P4t csc
K1new
C4new ⎥ ⎢ ∇z ⎥ ⎢ ⎥
⎢ 0 ⎥ ⎢ 2
⎥⎢
⎥ ⎢0⎥
⎢ 0 ⎥ ⎣⎢ G1 CTCSC D1new_ t csc D2new_ t csc ⎦⎥ ⎣ ∇v ⎦ ⎣ ⎦
⎦
⎣
Equation (27) can be written as
α
•
∇ X SYS _ TCSC = ASYS _ TCSC∇X SYS _ TCSC + ETCSC∇U
The System matrix with TCSC given as
α
ASYS _ TCSC = ATC 1 − ( ATC 2 * (inv ( ATC 4 )) * ATC 3 )
Where
⎡A
ATC 1 = ⎢ 1mod
⎣ P2 t csc
P1t csc ⎤
ATCSC ⎥⎦
Anew ⎤
⎡ A
ATC 2 = ⎢ 2 new
⎥
⎣ Bt csc1new Bt csc new ⎦
⎡ K P4t csc ⎤
ATC 3 = ⎢ 2
⎥
⎣ G1 CTCSC ⎦
ATC 4
⎡ K 1new
=⎢
⎣ D1new _ t csc
B.
DAE model of SVC
1.
Fig. 2(a) SVC firing angle model
α
C 4 new ⎤
D2 new _ t csc ⎥⎦
Fundamentals of SVC :
Fig. 2(b) SVC total susceptance model
Static VAR Compensator (SVC) is a shunt
connected FACTS controller whose main
functionality is to regulate the voltage at a given
bus by controlling its equivalent reactance.
Basically it consists of a fixed capacitor (FC) and a
thyristor controlled reactor (TCR). Generally they
are two configurations of the SVC.
a) SVC total susceptance model. A changing
susceptance Bsvc represents the fundamental
frequency equivalent susceptance of all shunt
modules making up the SVC as shown in Fig.
2(a).
b) SVC firing angle model. The equivalent
reactance XSVC, which is function of a
changing firing angle α, is made up of the
parallel combination of a thyristor controlled
reactor (TCR) equivalent admittance and a
fixed capacitive reactance as shown in Fig. 2
(b). This model provides information on the
SVC firing angle required to achieve a given
level of compensation.
Figure 3 shows the steady-state and dynamic
voltage-current characteristics of the SVC. In the
active control range, current/susceptance and
reactive power is varied to regulate voltage
according to a slope (droop) characteristic. The
slope value depends on the desired voltage
regulation, the desired sharing of reactive power
production between various sources, and other
needs of the system. The slope is typically1-5%. At
the capacitive limit, the SVC becomes a shunt
capacitor. At the inductive limit, the SVC becomes
a shunt reactor (the current or reactive power may
also be limited)
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linearization of the SVC power flow equations,
based on B SVC with respect to firing angle, will
exhibit a better numerical behavior than the
linearized model based on X SVC .
Bmin
Bmax
I min
I set
I max
Fig.3. steady-state and dynamic voltage/current
Characteristics of the SVC
SVC firing angle model is implemented in this
paper. Thus, the model can be developed with
respect to a sinusoidal voltage, differential and
algebraic equations can be written as
I SVC = − jBSVCVk
The fundamental frequency TCR
equivalent reactance X TCR
πX L
X TCR =
σ − sin σ
Where σ = 2(π − α ), X L = ωL
And in terms of firing angle
πX L
X TCR =
2(π − α ) + sin 2α
σ and α are conduction and firing angles
respectively.
At α = 90 0 , TCR conducts fully and the equivalent
reactance
XTCR becomes XL, while at α = 180 0 , TCR is
blocked and its
equivalent reactance becomes infinite.
The SVC effective reactance X SVC is determined
by the parallel combination of X C and X TCR
Fig.4. SVC equivalent susceptance profile
The initialization of the SVC variables based on
the initial values of ac variables and the
characteristic of the equivalent susceptance (Fig.),
thus the impedance is initialized at the resonance
point X TCR = X C , i.e. Q SVC =0, corresponding to
firing angle α = 115 0 , for chosen parameters of L
and C i.e. X L = 0.1134Ω and X C = 0.2267Ω .
2.
Proposed SVC power flow model:
The proposed model takes firing angle as the state
variable in power flow formulation. From above
equation the SVC linearized power flow equation
can be written as
⎡ ∇Pk ⎤
⎢
⎥
⎣∇Qk ⎦
(i )
⎡0
= ⎢0
⎢
⎣
(i )
⎤ ∇θ ( i )
⎥ ⎡ k⎤
2Vk
[cos 2α − 1]⎥ ⎢ ∇α ⎥
⎣
⎦
πX L
⎦
2
0
At the end of iteration i, the variable firing angle α
is updated according to
α ( i ) = α ( i −1) + ∇α ( i )
πX C X L
X SVC (α ) =
X C [2(π − α ) + sin 2α ] − πX L
Where X C = 1ωC
2 ⎧ X [ 2(π − α ) + sin 2α ⎫
Qk = −Vk ⎨ C
⎬
πX C X L
⎩
⎭
The SVC equivalent reactance is given above
equation. It is shown in Fig. that the SVC
equivalent susceptance ( BSVC = −1 / X SVC ) profile, as
function of firing angle, does not present
discontinuities, i.e., B SVC varies in a continuous,
smooth fashion in both operative regions. Hence,
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3.
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Above equation can be written as
SVC CONTROLLER MODEL:
•
∇ X SVC = AAVC ∇X SVC
Where
⎡ −1
⎢ T
⎢ m
ASVC = ⎢ − K I
⎢− KP
⎢ T
⎣ c
And
BSVC
Fig.5. Block diagram of SVC
The state equations of the SVC can be written from
above figure
•
X 1SVC =
4.
1
[VSVC (1 + KX 3SVC ) − X 1SVC ]
Tm
•
•
1
⎡ X 2 SVC + K P (Vref , SVC − X 1SVC ) − X 3SVC ⎤⎦
Tc ⎣
The reactive power QSVC supplied by the SVC can
be written as
QSVC = V 2 SVC X 3SVC
Linearization of above equations ( )-( ) yields
•
∆X 1SVC =
1
[ ∆VSVC (1 + KX 3SVCo ) + VSVCo K ∆X 3SVC ) − ∆X 1SVC ]
Tm
∆X 2 SVC = K I ( ∆Vref , SVC − ∆X 1SVC )
•
0
1
Tc
KV SVCo ⎤
Tm ⎥
⎥
0 ⎥
−1 ⎥
Tc ⎥⎦
⎡1
⎤
⎢ T (1 + KX 3SVCo ) ⎥
⎢ m
⎥
=⎢
0
⎥
⎢
⎥
0
⎢
⎥
⎣
⎦
INCORPORATION OF SVC IN MULTIMACHINE POWER SYSTEMS:
The SVC can be connected at either the existing
load bus or at a new bus that is created between
two buses. As DAE model is based on powerbalance, rewriting of the power-balance equations
at the buses with SVC connected in the system
requires modification of D2 new .When SVC is
connected at specified load buses, and gets
modified as given below
•
∆X 3SVC =
0
In its simplest form SVC is composed of FC-TCR
configuration as shown in Fig.2. The SVC is
connected to a coupling transformer that is
connected directly to the ac bus whose voltage is to
be regulated. The effective reactance of the FCTCR is varied by firing angle control of the
thyristors. The firing angle can be controlled
through a PI controller in such a way that the
voltage of the bus where the SVC is connected is
maintained at the desired reference value.
X 2 SVC = K I (Vref , SVC − X 1SVC )
X 3 SVC =
+ BSVC ∇VSVC
1
⎡ ∆X 2 SVC + K P ( ∆Vref , SVC − ∆X 1SVC ) − ∆X 3SVC ⎦⎤
Tc ⎣
∆QSVC = 2VSVCo ∆VSVC X 3SVCo + V 2 SVCo ∆X 3SVC
Where “ ∆ ” denotes perturbed value and subscript
“o” denotes the nominal value. The above
equations are linearized, reordered and then
expressed as
KV SVCo ⎤
⎡ −1
0
⎡ ∇ X• 1SVC ⎤ ⎢
Tm ⎥ ⎡ ∇X 1SVC ⎤
⎢ •
⎥ ⎢ Tm
⎥
⎢∇ X 2 SVC ⎥ = ⎢ − K I
0
0 ⎥ ⎢⎢∇X 2 SVC ⎥⎥ +
•
⎢
⎥ ⎢− K
− 1 ⎥ ⎢∇ X
1
⎥
P
⎢⎣∇ X 3 SVC ⎥⎦ ⎢
⎥ ⎣ 3 SVC ⎦
T
T
T
c
c
⎣ c
⎦
⎡1
⎤
⎢ T (1 + KX 3 SVCo ) ⎥
⎢ m
⎥
0
⎢
⎥[∇VSVC ]
⎢
⎥
0
⎢
⎥
⎣
⎦
n
PSVCi + PLi (Vi ) − ∑ ViVk Yik cos(θ i − θ k − α ik ) = 0
k =1
i = m + 1,................, n
n
QSVCi + QLi (Vi ) − ∑ ViVk Yik sin(θ i − θ k − α ik ) = 0
k =1
i = m + 1,......................n
Obtained state equations after linearization of
above equations
61
International Journal of Reviews in Computing
30th September 2011. Vol. 7
© 2009 - 2011 IJRIC & LLS. All rights reserved.
ISSN: 2076-3328
www.ijric.org
IJRIC
E-ISSN: 2076-3336
C SVC ∇Vl + DSVC ∇X SVC + D1∇V g + D 2 ∇Vl = 0
ACKNOWLEDGMENT
or
DSVC ∇X SVC + D1∇V g + D2 SVC ∇Vl = 0
The authors would like to thanks Dr. S. C.
Srivastava, and Dr. S. N. Singh, Indian Institute of
Technology, Kanpur, U.P., India, and Dr. K.S.
Verma, and Dr. Deependra Singh, Kamla Nehru
Institute of Technology, Sultanpur, U.P., India, for
their valuables suggestions regarding placement
and coordination techniques for FACTS controllers
form voltage stability, and voltage security point of
view
in
multi-machine
power
systems
environments.
Where
D2 SVC = C SVC + D2
The incorporation of the SVC into DAE model of
multi-machine power system is done on the same
lines as explained in [2] given as follows:
⎡ ∇ X• ⎤ ⎡A1mod P1SVC A2new
A3new ⎤⎡ ∇X ⎤ ⎡E⎤
⎢ • ⎥ ⎢
Bsvcnew ⎥⎥⎢∇XSVC⎥ ⎢ 0⎥
⎢∇ X SVC⎥ ⎢P2SVC ASVC P3svc
⎢
⎥ + ⎢ ⎥∇U
=
⎢
⎥ ⎢K
P4svc K1new
C4new ⎥⎢ ∇z ⎥ ⎢ 0⎥
2
0
⎢
⎥ ⎢
REFERENCES
⎥⎢
⎥ ⎢ ⎥
⎢⎣ 0 ⎥⎦ ⎣⎢ G1 DSVC D1new_ svc D2new_ svc ⎦⎥⎣ ∇v ⎦ ⎣ 0⎦
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International Journal of Reviews in Computing
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© 2009 - 2011 IJRIC & LLS. All rights reserved.
ISSN: 2076-3328
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AUTHOR PROFILE:
Bindeshwar Singh was born in Deoria, U.P.,
India, in 1975. He received the B.E.
degree in electrical engineering from
the Deen Dayal Uphadhayay
University
of
Gorakhpur,
Gorakhpur, U.P., India, in 1999, and
M. Tech. in electrical engineering
(Power Systems) from the Indian
Institute of Technology (IITR), Roorkee,
Uttaranchal, India, in 2001. He is now a Ph. D.
student at Gautam Buddha Technical University,
Lucknow, U.P., India. In 2001, he joined the
Department of Electrical Engineering, Madan
Mohan Malviya Engineering College, Gorakhpur,
as an Adoc. Lecturer. In 2002, he joined the
Department of Electrical Engineering, Dr. Kedar
Nath Modi Institute of Engineering & Technology,
Modinagar, Ghaziabad, U.P., India, as a Sr.
Lecturer and `subsequently became an Asst. Prof.
& Head in 2003. In 2007, he joined the Department
of Electrical & Electronics Engineering, Krishna
Engineering College, Ghaziabad, U.P., India, as an
Asst. Prof. and subsequently became an Associate
Professor in 2008. Presently, he is an Assistant
Professor with Department of Electrical
Engineering, Kamla Nehru Institute of Technology,
Sultanpur, U.P., India, where he has been since
August’2009. His research interests are in
Placement and Coordination of FACTS controllers
in multi-machine power systems and Power system
Engg.
Mobile: 09473795769
Email: bindeshwar.singh2025@gmail.com
63