INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 721302, DECEMBER 27-29, 2002
421
Modeling of Multiple FACTS Controllers for
Multimachine Power System, A Modular
Approach
Nikhlesh Kumar Sharma, Arindam Ghosh, and Rajiv K. Varma
Abstract-- This paper presents a systematic modular
approach of modeling multiple FACTS devices in multimachine
power system environment using power balance form. In
particular, two key FACTS devices, viz., SVC, TCSC and their
combination are modeled and incorporated in the multimachine
DAE (Differential Algebraic Equations) model of the power
system. The models developed are modular and flexible in
nature and any type of associate control can be added with
these FACTS devices. The models developed have been
utilized for eigenvalue analysis of a 9-bus power system.
Index Terms—DAE, Eigenvalue analysis, SVC,TCSC.
1.
models developed have been utilized for eigenvalue analysis
of 9-bus power system.
II.
DAE MODEL OF MULTIMACHINE POWER SYSTEM
WITHOUT FACTS CONTROLLERS
The methodology given in [1] describes dynamic modeling
of a general m-machine, n-bus system. This model
represents each machine by a two -axis model and the
excitation system is chosen as the IEEE type-I rotating
exciter. The transmission system has been modeled by static
equations. DAE model utilizes power balance form. The
equations are written as
INTRODUCTION
A
study of the implication of adding various FACTS
Controllers in multimachine environment requires an
appropriate mathematical model of the system and the
FACTS Controllers. There are many commercial packages
available for transient simulation and analysis of power
systems. The transient simulation packages (e.g.
EMTDC/PSCAD) allow incorporation of FACTS Controller
models. This facility is however not available in the small
signal stability analysis packages. The objective of this paper
is to develop a methodology to incorporate FACTS
Controllers in a modular fashion to facilitate eigenvalue and
voltage stability analysis using MATLAB.
The DAE (Differential Algebraic Equation) methodology
for multimachine system has been presented in [1] is used in
this paper. Even tough the SVC model has been incorporated
in DAE model [2], the other FACTS Controllers and their
combinations have not been incorporated. So the purpose of
this paper is to derive models of various FACTS Controllers
such that these can be incorporated in DAE model. Further
in a large power system there may be more than one FACTS
Controllers, therefore it is important to develop a
combination of series and shunt FACTS Controllers that can
be incorporated in the DAE model in modular fashion. The
Nikhlesh Kumar Sharma is Lecturer in Electrical Engg. Deptt., Madan Mohan
Malaviya Engineering College, Gorakhpur, UP, India. (telephone: 0551-273500,
e-mail: nikhlesh@mailcity.com).
Arindam Ghosh is Professor of Electrical Engg at Indian Institute of
Technology, Kanpur, UP, India. (telephone: 0512-598799, e-mail:
aghosh@iitk.ac.in).
Rajiv K. Varma is Associate Professor of Electrical & Computer Engineering
at University of Western Ontario London, Ontario, Canada
x& = f ( x, y , u)
x (0) = xo
(1)
0 = g (x , y , u )
y (0) = y o
(2)
where x is a vector of state variables, y is vector of algebraic
variables and u is a vector of inputs and parameters. Equation
(1) consists of the differential equations of the mechanical
system, field winding, q-axis damper winding, and the
electrical equations of the exciter. Equation (2) consists of
the stator algebraic equations and the network power balance
equations. Various vectors are defined as
′ , E fdi , V Ri , R Fi ]
x = [δ i , ω i , E qi′ , E di
T
T
y = [V j ,θ j , I di , I qi ]
T
u = [TMi , V REFi , PLj , Q Lj ] i = 1, KK ; j = 1, K K , n
(3)
(4)
Based on the methodology described in [1], the linearized
model is shown in (5).
 ∆X& 
 
 0 =
 0 
 
(5)
 A1mod
 K
 2
 G1
A2 new
K 1new
D1new
A3new   ∆X   E 
C 4new   ∆z  +  0  ∆U
D2new   ∆v   0 
422
NATIONAL POWER SYSTEMS CONFERENCE, NPSC 2002
where D2 new is the load flow Jacobian J LF and
K
J AE =  1new
 D1new
C 4 new 
is the algebraic Jacobian.
D 2new 
The vectors [∆zT] and [∆v T] are
 T
 ∆z  = [∆θ 1 , ∆V1 , K K , ∆Vm ]
base MVA is 100, and system frequency is 60 Hz. Table 1
shows the eigenvalues of WSCC system. Column 1 of Table
1 shows the eigenvalues reported in [1] while column 2
depicts the eigenvalues obtained from developed MATLAB
program. It is evident that eigenvalues obtained from
developed MATLAB program correlate very well with those
reported in [1]. This validates the developed MATLAB
program.
Table 1 Eigenvalues of WSCC system
 T
 ∆v  = [∆θ 2 , ∆θ 3 ,KK, ∆θ n , ∆Vm +1,K, ∆Vn ]
The system matrix Asys can be obtained as
∆X& = Asys ∆X + E ∆U
where
Asys = A1mod − [A2 new
A3new ][ J AE ]
−1
K 2 
G 
 1
The details of DAE model are given in [1]. This DAE
model for multimachine system can be used for studying
steady state stability, voltage stability and low frequency
electromechanical oscillations. Based on this methodology,
a small signal stability program has been developed using
MATLAB.
III. INCORPORATION OF SVC IN MULTIMACHINE MODEL
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 FC-TCR 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.
Fig.1. Three machine 9-bus system
Fig. 2. Simplified one line diagram of SVC
In order to ensure that the developed small signal stability
program gives satisfactory results, eigenvalue analysis is
performed for the Western System Coordinating Council
(WSCC) 9-bus system (Fig. 1). This WSCC system
comprises three generators and nine buses. Loads are
connected at buses 5, 6 and 8 as shown in Fig. 1. At base
case loading condition of the system, the generator 2 and 3
are supplying 163MW and 85MW power respectively. The
The incorporation of the SVC into DAE model of
multimachine system is done on the same lines as explained
in [2]. Table 2 shows the eigenvalues reported in [2] and
from developed MATLAB program.
IV. THYRISTOR CONTROLLED SERIES CAPACITOR
INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 721302, DECEMBER 27-29, 2002
Thyristor Controlled Series Capacitor (TCSC) provides
powerful means of controlling and increasing po wer 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 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).
Table 2. Eigenvalues of WSCC system with SVC
423
Let a TCSC be connected between bus k and bus m as
shown in Fig. 3. It has been assumed that the controller is
lossless. The power-balance equation and BTCSC are given
as [4]
Pk = Vk Vm BTCSC sin( θ k − θ m )
(7)
Qk = V k2 BTCSC − Vk Vm BTCSC cos(θ k − θ m )
(8)
Pm = Vk Vm BTCSC sin( θ m − θ k )
(9)
Qm = Vm2 BTCSC − Vk Vm BTCSC cos(θ m − θ k )
(10)
BTCSC = −π (κ 4 − 2κ 2 + 1) cosκ (π − α ) / [ X c (πκ 4 cosκ (π − α ) −
π cosκ (π − α ) − 2κ 4α cosκ (π − α )
(11)
+2α κ 2 cosκ (π − α ) − κ 4 sin 2α cosκ (π − α ) +
κ 2 sin 2α cosκ (π − α ) − 4κ 3 cos2 α sin κ (π − α )
−4κ 2 cosα sin α cosκ (π − α ))]
Equation (11) is obtained from (6).
There are number of control strategies for TCSC [4]
•
•
A. Controller Model
The structure of the TCSC is the same as that of a FC-TCR
type SVC. The equivalent impedance of the TCSC can be
modeled using the following equation [4].
κ σ + sin σ
X TCSC = X c [ 1 − 2
+
π
κ −1
4 κ 2 cos 2 (σ 2)
(κ tan
κσ
σ
− tan )]
2
2
π (κ − 1)
(6)
where
α = Firing angle delay (after forward valve voltage)
σ = Conduction angle = 2 (π − α) and
2
2
κ = TCSC ratio = X c X L
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.
B. Incorporation of TCSC in Multimachine Power System
•
•
Reactance Control: Bset − BTCSC = 0
Power Control: Pset − P = 0
Current Control: I set − I = 0
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. Of these, the power
control strategy has been used here, the block diagram of
which is shown in Fig. 4.
Transmission angle control:
Fig. 4. Block diagram of TCSC power controller
The line power is monitored and compared to desired
power Pset. The error is fed to proportional-integral (PI)
controller. The output of PI controller is fed through a first
order block to get the desired α .
The controller equations are given as
K
X 2TCSC = I ( Pset − P )
s
(12)
X& 2TCSC = K I Pset − K I P
(13)
− X 1TCSC X 2TCSC K P Pset K P P α o
X& 1TCSC =
+
+
−
+
Tc1
Tc1
Tc1
Tc1
Tc1
Fig. 3. TCSC representation
(14)
424
NATIONAL POWER SYSTEMS CONFERENCE, NPSC 2002
In order to get the linearized model of TCSC, (7-11), (13)
and (14) are linearized. The linearized TCSC model in matrix
notation can be written as
 ∆θ k 
 ∆V 
k 
∆X& TCSC = ATCSC ∆X TCSC + BTCSC 
 ∆θ m 


 ∆Vm 
(15)
 ∆X&   A1mod
 &
 
 ∆X SVC   P2svc
∆X& TCSC  =  P2tcsc

 
 0   K2
 0   G

  1
E 
0 
 
+  0  ∆U
 
0 
 0 
 ∆X 1TCSC 

∆X 2TCSC 
where ∆X TCSC = 
 ∆θ k 
 ∆Pk 
 ∆V 
 ∆Q 
 k
 k=C
∆
+
X
D
TCSC
TCSC
1TCSC
 ∆θ m 
 ∆Pm 




 ∆Vm 
 ∆Qm 
(16)
Incorporation of (15), (16) and (5) gives DAE model of
multimachine system with TCSC incorporated in the system.
After reordering, final form of DAE model with TCSC is
given as
A2new
A3 new   ∆X 
 ∆X&   A1mod P1tcsc
 &
 P


 ∆X TCSC  =  2tcsc ATCSC Btcsc1new Btcscnew   ∆X TCSC 
 0   K 2 P4tcsc
K1new
C4 new   ∆ z 

 


 0   G1 C TCSC D1new _ tcsc D2new− tcsc   ∆v 
E 
 0
+   ∆U
 0
 
 0
(17)
Equation (17) can be written as
∆X& SYS −TCSC = ASYS −TCSC ∆ X SYS −TCSC + ETCSC ∆U
(18)
P1svc P1tcsc A2new
A3new   ∆X 
ASVC tcsv1 Bsvc1new Bsvcnew   ∆X SVC 
tcsv2 ATCSC Btcsc1new Btcscnew  ∆XTCSC 


P4svc P4tcsc K 1new C 4new   ∆z 
DSVC CTCSC D1newsvtc D2newsvtc   ∆v 
(19)
VI. CASE STUDY
After incorporating FACTS Controllers individually and in
combination into DAE model of multimachine system,
voltage stability of 9-bus system is carried out at various
loading conditions. However results are presented for
maximum loading condition.
At maximum loading condition, there is an SVC connected
at bus 5 and TCSC is connected between buses 7 and 5. Table
3 shows eigenvalues of the 9-bus system at maximum
loading condition for four different cases - without any
FACTS device, with an SVC connected at bus 5 and with a
TCSC connected between buses (7-5) and combination of
SVC-TCSC (The locations of SVC and TCSC are same as
considered individually). The TCSC controller parameters
are kept as KP =0.1 and KI =10 and the SVC controller
parameters are chosen as KP =0.3 and KI =100. When SVCTCSC combination is used, the controller parameters are
kept the same.
Table 3 shows that without any FACTS device the system
is unstable, where unstable eigenvalues are highlighted.
However the system becomes stable when either SVC or
TCSC or their combintion are connected.
VII.
CONCLUSIONS
V. INCORPORATION OF MULTIPLE FACTS DEVICES
Incorpo ration of multiple FACTS devices can be done on
the same lines as described for TCSC. Any of these models
or a combination of them can be incorporated in the DAE
multimachine model to form the model of the overall
system. For example when an SVC-TCSC combination is
required, the overall system in compact form will emerge as
This paper presents a systematic modular approach to
incorporate series and shunt FACTS Controllers. The
approach is general and can be applied to any large power
system. With the proposed approach it is possible to connect
any number and any type (Series and Shunt and their
combination) of FACTS Controllers. The results of the
proposed modular approach are illustrated for 9-bus WSCC
system.
VIII.
[1]
[2]
REFERENCES
Peter W Sauer and M. A. Pai, Power System Dynamics and Stability,
Prentice Hall, 1998.
M. J Laufenberg, M. A. Pai and K. R. Padiyar, “Hopf Bifurcation
Control in Power System with Static Var Compensators,” Electric
Power & Energy Systems, Vol. 19, No. 5, pp. 339-347, 1997.
INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 721302, DECEMBER 27-29, 2002
[3]
E. V. Larsen, C. Bowler, B. Damsky and S. Nilsson, “Benefits of
Thyristor Controlled Series Compensation,” CIGRE, 14/37/38-04,
Paris, 1992.
[4]
425
C. A. Cañizares and Z. T. Faur, “Analysis of SVC and TCSC
Controllers in Voltage Collapse,” IEEE Trans. on Power Systems, Vol.
14, No. 1, pp. 158-165, February 1999.
Table 3 Eigenvalues of the 9-bus system at maximum loading condition