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Enhancement of Voltage Stability by
Coordinated Control
of Multiple FACTS Controllers in Multi
-
Machine Power
System Environments


Bindeshwar Singh, N. K. Sharma,

and
A. N. Tiwari,




Abstract
-
This
paper
presents

the implication of adding
various FACTS controllers in multi
-
machine power system
environment

in coordinated control manner

for
enhancement of voltage stability
requires
an appropriate
m
athematical model of the
power
system and the FACTs
co
ntrollers

such as a Static Var Compensator (SVC) and
Thyristor Controlled Series Capacitor (TCSC)
.

The DAE
(Differential

Algebraic Equation)
methodology for multi
-
machine system has been

is

used in this paper. Event tough
the SVC model has been incorporated in DAE
model,

TCSC model has not been incorporated. So the purpose of
this paper is to derive a TCSC model such that it ca
n be
incorporated in DAE model

of power system
.
Further in a
l
arge 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 models developed have been utilized f
or e
i
gen
-
value
analysis of IEEE 9
-
bus

3
-
machine

power systems.

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 eigen
-
value and voltage
stability a
nalysis using MATLAB toolbox.



Index Terms
-

F
lexible
AC

T
ransmission
S
ystems (FACTS)
,
FACTS Controllers,

SVC, TCSC
,
Power Systems
.


I.
INTRODUCTION


HE
DAE (Differential Algebraic Equation) methodology
for multi
-
machine system has been presented in
[1
]
is

used
in this paper.

Event tough the SVC model has been
incorporated in DAE model [2], TCSC model has not been
incorporated. So the purpose of this paper is to derive a TCSC
model such that it 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
models developed have been utilized for egen
-
value analysis
of IEEE 9
-
bu
s power systems.

This paper is

organized as follows: Section II

discusses the
DAE model of multi
-
ma
chine power system without FACTS

controllers
.

Section III

introduces

the
DAE model of multi
-
machine power system with FACTs controllers.

Section IV

introduce
s

the
results and discussions
.
Section V
presents the
conclusions of the paper
.


II.

DAE MODEL OF MULTI
-
MACHINE POWER SYSTEM
WITHOUT FACTS CONTROLLERS


T
he methodology given in [1] describes dynamic modeling of
a general m
-
machine, n
-
bus system. This mode
l 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. The DAE model
utilizes power balance form. The equations are written as:



(1)



(2)

Where

is a vector of state variables

is vec
tor of
algebraic variables and

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 e
quations of the exciter. Equation (2) consists of
the stator algebraic equations and the network power balance
equations. Various vectors are defined as

[1]



(3)






(4)

Based on the methodology described in [1], the linearized
model is given as



(5)

Where
is the load flow Jacobian

and

T


is the algebraic Jacobian.

The vectors

and

are



The system matrix
can be obtained as


Where


(6)


The details of DAE model
are given in [1]. This DAE model
for multi
-
machine system can be used for studying steady state
stability, voltage stability and low frequency
electro
-
mechanical oscillations. Based on this methodology, a small
signal stability program has been developed u
sing MATLAB.
The developed program is tested for 9
-
bus WSCC test system
and its results are corrected with the results published in [1] as
shown below.


III. DAE MODEL OF MULTI
-
MACHINE POWER
SYSTEM WITH FACTS CONTROLLERS


A.

Case Study(WSCC
9 bus System)
:


In

order to ensure that the developed small signal stability
program gives satisfactory results, eigen
-
value analysis is
performed for the Western System Coordinating Council
(WSCC) 9
-
bus system

shown in Fig. 1. This WSCC system
comprises th
r
ee 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 163 MW and 85MW power respectively. The base
MVA is 100, and system frequency is 60Hz. Table 1 sho
ws
the eigen
-
values of WSCC system. Column 1 of table 1 shows
the eigen
-
values reported in [1] while column 2 depicts the
eigen
-
values obtained from developed MATLAB program. It
is evident that eigen
-
values obtained from developed
MATLAB program correlate
very well with those reported in
[1]. This validates the developed MATLAB program.


Fig.1. WSCC (9
-
bus, 3
-
machine) power system


B.

Mathematical model of SVC :


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 changin
g susceptance
Bsvc represents the fundamental frequency equivalent
susceptance of all shunt modules maki
ng 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.




Fig.

2
(a)

SVC firing angle model



Fig. 2
(b) SVC total susceptance model


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).


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


The fundamental freq
uency TCR equivalent reactance


Where

And in terms of firing angle


(7)


and
are conduction and firing angles respectively.

At
,

TCR conducts fully and the equivalent reactance

XTCR becomes XL,

while at
,

TCR is blocked and its

equivalent
reactance becomes infinite.

The SVC effective reactance

is determined by the

parallel combination of

and


(8)

Where


(9)

The SVC equivalent reactance is given

above equation
. It is
shown in

Fig.
that the SVC equivalent susceptanc

profile
,
as function of firing angle, does not
present

discontinuities, i.e.,

varies in a continuous,
smooth

fashion in both operative regions. Hence, linearization
of the

SVC power flow equations, based on
with respect
to

firing angle, will exhibit a better

numerical behavior than
the

l
inearized model based on

.



Fig.
4.

SVC equivalent susceptance profile


The initialization of the SVC variables based on the initia
l

values of ac variables and the characteristic of the equivalent

susceptance (Fig.), thus the impedance is initialized at the

resonance point
,
i.e.

=0, corresponding to

firing angle

, for chosen parameters of L and C i.e.
and
.

Proposed SVC power flow model:

The proposed model takes firing angle as the state variable in

power flow formulat
ion. From above equation the SVC
linearized power flow equation can be written as


(10)

At the end of iteration i, the variable firing angle α is updated

according to



SVC

Controller Model
:



Above equation can be written as




(11)

Where


And

`


Incorporation of SVC in multi
-
machine power systems:


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 co
ntroller in
such a way that the voltage of the bus where the SVC is
connected is maintained at the desired reference value.


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 o
n power
-
balance, rewriting of the power
-
balance
equations at the buses with SVC connected in the system
requires modification of
.When SVC is connected at
specified load buses, and
gets

modified as given below


Obtained state

equations
after linearization
of above

equations



(12)

Where


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:

Incorporation of (11), (12), and (5) gives DAE model of multi
-
machine power system with SVC incorporated in the system.
After reordering, final form of DAE model with SVC is given
as


The state equation for th
e system with SVC is then given as
follows:


(13)

The System matrix with SVC given as


(14)

Where



c)

Mathematical model 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 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 S
SR (Sub
Synchronous Resonance).
The TCSC
module

shown in Fig.
5.



Fig.
5.
TCSC module


The steady
-
state impedance of the TCSC is that of a parallel
LC circuit, consisting of
fixed

capacitive impedance,
, and
a variable inductive impedance,
, that is,


(15)

Where


(16)

,

and

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.
6
.
TCSC equivalent Reactance as a function of firing

angle


TCSC
Controller Model
:


The structure of the TCSC

is the same as that of a F
C
-
TCR
type SVC. The equivalent impedance of the TCSC can be
modeled using the following equations [4].


(17)



Where

Firing angle delay (after forward vale voltage)

Conduction angle=
and

TCSC

ratio =

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.


Incorporation of TCSC in Multi
-
machine Power
Systems:


The block diagram representation of TCSC shown in Fig. 7.





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
are given as [4]







Equation (
21) is obtained from (16).

There are number of control strategies for TCSC [4]




Reactance c
ontrol:



Power control:



Current control:



Transmission angle control:

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 blo
ck diagram of which is
shown in Fig.

The line power is monitored and compared to desired power
. The error is fed to proportional
-
integral (PI) controller.
The output of PI controller is fed through a first order block to
get the desir
ed
.

The block diagram representation of TCSC
with PI controller shown in Fig.
8.



Fig.
8.

Block diagram representation of TCSC with PI
controller


The controller equations are given as
( from fig.)


(18)


(19)


(20)

In order to get the linearized model of
TCSC, (
18

), (
19

)
, and
(
20
) are linearized. The linearized TCSC model in matrix
notation can be written as


(21)

Where



(22
)

Inc
orporation of
(
21), (22
), 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

Equation (27) can be written as


(2
3
)

The System matrix with TCSC given as


(2
4
)

Where






C.

Mathematical model of SVC and
TCSC :


Incorporation of
Multiple FACTS controllers (SVC and
TCSC
)

in Multi
-
machine Power Systems
:


The

matrix
equations given as



(25
)

The System matrix with SVC+TCSC given as



(26
)

Where



IV
.
RESULTS&DISCUSSIONS


After incorporating FACTS controllers individually and in
combination into DAE model of multi
-
machine system,
voltage stability of 9
-
bus system is
carried out at various
loading conditions. However results are presented for
maximum loading condition.

Table 3

show

that without any
FACTS controllers the system is unstable, where unstable
eigen
-
values are highlighte
d


Table 1 Eigen
-
values of WSCC (9
-
bus
, 3
-
machine) power
system


Eigen
-
values from [1]

Eigen
-
values from
developed MATLAB
program




























Table 2 Eigen
-
values of WSCC (9
-
bus,
3
-
machine) power
system with SVC


Eigen
-
values from [2]

Eigen
-
values from
developed MATLAB
program


























However the system become stable when
SVC

or TCSC or
SVC

and
TCSC are connected. At maximum loading
condition, there is a need for
a shunt device at bus 5. Table 2

shows e
i
gen
-
values of the 9
-
bus system at maximum loading
conditions for three different cases
-
without any FACTS
device, with an SVC connected at bus 5 a
nd with a TCSC
connected between
lines

(7
-
5). Whereas TCSC controller
parameters are same as those used for base case loading
condition, the SVC controller

parameters are chosen as
and
.

Table 3 Eigen
-
values of WS
CC

(9
-
bus
, 3
-
machine) power
system with
only
SVC,

or only TCSC, or
SVC

and
TCSC

at
maximum loading condition
.


Without any
FACTS
device

With SVC

With TCSC

With
SVC+TCSC










































































In the similar fashion multiple FACTS controllers can also be
added to DAE model of multi
-
machine power systems for
enhancement of voltage stability of the systems

in coordinated
control manner
.


V.CONCLUSIONS


This
paper presents a systematic modular approach to
incorporate series and shunt FACTS controllers

in DAE model
of multi
-
machine power systems

in coordinated
control
manner

for enhancement of voltage stability of the systems.
This proposed

approach is general and can be ap
plied to any
large power system environments
.

With the proposed
approach it is possible to connect any number and any
type
(
series and shunt) of FACTS cont
rollers. The results of the
proposed modular approach are illustrated for 9
-
bus

3
-
machine

WSCC system.


ACKNOWLEDGMENT


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
in

regarding with
control
coordination
of

multiple
FACTS
controllers
in

multi
-
machine power systems for enhancement
of

voltage stability
.


REFERENCES


[1]

Peter W. Sauer and M. A. P
ai, Power System Dynamics and Stability,
Prentice Hall, 1998.

[
2
] 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.

[
3
]
E. V. Larsen, C. Bowl
er, B. Damsky and S. Nilsson, “Benefits

of Thristor
Controlled Series Compensation,
“CIGRE
, 14/37/
-
04
,
Paris,1992.

[
4
]
C. A. Canizares 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.

BIOGRAPHIES


Bindeshwar Singh

received the M.Tech. in electrical engineer
ing from the Indian
Institute of Technology, Roorkee, in 2001.He is now a Ph. D. student at UPTU,
Lucknow, India. His research interests are in Coordination of FACTS controllers in
multi
-
machine power systems and Power system Engg.. Currently, he is an
Ass
istant Professor with Department of Electrical Engineering, Kamla Nehru
Institute of Technology, Sultanpur,

U.P., India, where he has been since
August’2009.

Mobile: 09473795769, 09453503148

Email
:
bindeshwar_singh2006@rediffmail.com

,
bindeshwar.singh2025@gmail.com



Nikhlesh Kumar Sharma
received the Ph.D. in electrical engineering from the
Indian Institute of Technology, Kanpur, in 2001. Currently, he is a
Prof.&Head

with
, R
aj
K
umar
G
oel
Institute of Technology
, Ghaziabad,

U.P.,

India, where he
has been since June’2009. His interests are in the areas of FACTS control and
Power systems.

Mobile: 09654720667, 09219532281

Email:
drnikhlesh@gmail.com



A.N.Tiwari

received the Ph.D. in electrical engineering from the Indian Institute of
Technology, Roorkee, in 2004. Currently, he is a
n

Asst. Prof. with Department of
Electrical Engineering, M
adan Mohan
M
alviya
E
ngineering
C
ollege
,
Gorakhpur,
U.P.,

India, where he has

been since June’1998. His interests are in the
areas of Electrical Drives and Application of Power Electronics.

Mobile: 09451215400

Email:
amarndee@reffimail.com


APPENDIX


SYSTEM DATA FOR WSCC 3
-
MACHINES, 9
-
BUS SYSTEM


Base MVA 100MVA


Machine Data


Parameters

M/C1

M/C2

M/C3


23.6400

6.4000

3.0100


0.14600

0.8958

1.3125


0.06080

0.1198

0.1813


0.09690

0.8645

1.2578


0.09690

0.1969

0.2500


8.96000

6.0000

5.8900


0.31000

0.5350

0.6000



Exciter Data


Parameters

M/C1

M/C2

M/C3


20.0

20.0

20.0


0.20

0.20

0.20


1.0

1.0

1.0


0.314

0.314

0.314


0.063

0.063

0.063


0.35

0.35

0.35


0

0

0


0.0039

0.0039

0.0039


1.555

1.555

1.555


Line Data


Line
number

Bus

Impedance

From

To

R(pu)

X(pu)

Y/2(pu)

1

2

7

0

0.0625

0

2

1

4

0

0.0576

0

3

3

9

0

0.0586

0

4

4

6

0.0170

0.0920

0.0790

5

4

5

0.0100

0.0850

0.0880

6

5

7

0.0320

0.1610

0.1530

7

6

9

0.0390

0.1700

0.1790

8

9

8

0.0119

0.1008

0.1045

9

8

7

0.0085

0.0720

0.0745


Load Flow Results for Base Case of WSCC 9Bus System


Bus

Type

Angles

Voltages

PL

QL

PG

QG

1

SL

0

1.0400

0

0

0.7164

0.2705

2

PV

9.2800

1.0250

0

0

1.6300

0.0665

3

PV

4.6648

1.0250

0

0

0.8500

-
0.1086

4

PQ

-
2.2168

1.0258

0

0

0

0

5

PQ

-
3.9888

0.9956

1.2500

0.5000

0

0

6

PQ

-
3.6874

1.0127

0,9000

0.3000

0

0

7

PQ

3.7197

1.0258

0

0

0

0

8

PQ

0.7275

1.0159

1.0000

0.3500

0

0

9

PQ

1.9667

1.0324

0

0

0

0



SVC data


K

Tc

Tm

Kp

KI

0.1

0.02

0.02

0.
3

100



TCSC data