International Journal of Engineering & Technology IJET-IJENS Vol:09 No:10
37
TCSC Control of Power System oscillation and
Analysis using Eigenvalue Techniques
M.W. Mustafa. MIEEE ,Nuraddeen Magaji, IEEE Student Memberand Z. bint Muda
Universiti Teknologi Malaysia, Department of Power Engineering, Johor Bahru, Malaysia nmagaji2000@yahoo.com
Abstract— TCSC devices are used to improve real power an d
eliminate line loses in ac systems. An additional task of TCSC
is to increase transmission capacity as result of power
oscillation damping. In this paper eigenvalue-based methods
for analysis and control of power system oscillations using
TCS C have been developed. The characterization of power
system oscillations using the eigenvalues and eigenvectors of
the state matrix is detailed. Design of power system damping
controllers using residue method is addressed for two area four
machine systems. The result shows the effectiveness of the
method used
Index Term-- TCS C, Power system oscillations, linear models,
eigenvalues, eigenvectors, participation factors and residue.
I.
INT RODUCT ION
The concept of flexible ac transmission systems is made
T possible by the application of high power electronic
devices for power flow and voltage control [l]. In addition
a number o f TCSC devices have already been installed to
aid power system dynamics wh ich help to mitigation a lo w
frequency oscillations often arise between areas in a large
interconnected power network [2].
Eigenvalue sensitivities are one important outcome of the
modal analysis and control of oscillatory behaviour and
dynamic stability in power systems. The pioneering work
[3] considers the local oscillation of a single machine by
means of a transfer function model. The usually co mplex
pattern of oscillations in a large power system can be
studied through linear, t ime invariant, state-space models
based on the perturbations of the system state variables fro m
their nominal values at a specific operating point
Power system oscillations occur due to the lack of damp ing
torque at the generators rotors. The oscillation of the
generators rotors cause the oscillat ion of other power system
variables (bus voltage, bus frequency, transmission lines
active and reactive powers, etc.). Po wer system oscillations
are usually in the range between 0.1 and 2 Hz depending on
the number of generators involved in [4]. Local oscillations
lie in the upper part of that range and consist of the
oscillation of a single generator or a group of generators
against the rest of the system. In contrast, inter-area
oscillations are in the lower part of the frequency range and
comprise the oscillations among groups of generators. In
addition, power system oscillations exh ibit low damping
compared to oscillations found in other dynamic systems: an
oscillation of 10% damp ing is commonly accepted as well
damped. To imp rove the damping of oscillations in power
systems, supplementary control laws can be applied to
existing devices. These supplementary actions are referred
to as power oscillation damping (POD) control
This paper reviews the basic concepts of eigenvalue analysis
of linear systems. The physical meaning of eigenvalues,
eigenvectors,
participation
factors,
residues
and
controllability and observability indices will be introduced
and illustrated in small scale power systems. This technique
has been successfully used in location and tuning of power
system stabilizers [5] and FACTS devices.
The application of sensitivity measures to the design of
power system damping (POD) controllers has been applied
to TCSC. The design method utilizes the residue approach;
this presented approach solves the optimal sitting o f the
TCSC device, selection of the proper feedback signals and
the controller design problem [6].
II. LINEAR SYST EM ANALYSIS T OOLS IN POWER SYST EMS
Low frequency electro mechanical oscillations range fro m
less than 1 Hz to 3 Hz other than those with subsynchronous resonance (SSR) [6,15]. Mu lti-machine power
system dynamic behavior in this frequency range is usually
expressed as a set of non-linear differential and algebraic
(DA E) equations. The algebraic equations result from the
network power balance and generator stator current
equations. The init ial operating state of the algebraic
variables such as bus voltages and angles are obtained
through a standard power flow solution. The in itial values of
the dynamic variables are obtained by solving the
differential equations
A.
Eigenvalues, Eigenvectors and Modes
Let us start from the mathematical model a dynamic system
expressed in terms of a system of non-linear d ifferential
equations:
(1)
x  F ( x, t )
If th is system of non-linear d ifferential equations is
Linearized around an operating point of interest x=x0 , it
results in:
x  Ax(t )
(2)
A mean ingful solution method of (2) is based on the
eigenvalues and eigenvectors of the state matrix A. An
eigenvalue i of the state matrix A and the associated right
vi and left wi eigenvectors are defined accord ing to:
Avi  i vi
In a mat rix with all distinct eigenvalues (not a necessity but it
is easier to understand when it is so), all the right
eigenvectors and eigenvalues can be expressed as a compact
matrix expression
AV  VA
(3)
Where,
V  ( v1 ,v2 ..... vn 1 vn , )
(4)
= diag ( 1 2 ...n-1 ,n )
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-1
Pre-multiplying both sides of (3) by V gives
(5)
V 1 AV  
A similar expression holds for the left eigenvectors W such
that
(6)
WA  W
Where
(7)
W  [ 1t ,2t ..... nt 1 nt ]t
Post-multiplying both sides of (6) by W -l , gives
(8)
WAW 1  
The transformed physical state variables (x) can be put into
modal variables (2) with the help of eigenvector matrices V
and W
x  Vz
(9)
z  Wx
In power system literature, the right eigenvector matrix v is
known as the mode shape matrix, that is, eigenvector vi is
known as the ith mode shape, corresponding to eigenvalue λi .
The mode shape provides important in formation on the
participation of an individual machine or a group of
machines in one particular mode.
solution of (2) can be expressed in terms of the eigenvalues
and eigenvectors of the state matrix as:
N
x(t)  Vet Wx(0)   vi ei t [w iT x(0)] (10)
i 1
The analysis of equation (10) allo ws drawing the fo llo wing
conclusions:
i.
The system response is the comb ination of the
system response to each of the N modes.
ii.
The eigenvalues determine the system stability. A
real positive (negative) eigenvalue determines
exponentially increasing (decreasing) behavior. A
complex eigenvalue of positive (negative) real part
results in a increasing (decreasing) oscillatory
behavior.
iii.
The components of the right eigenvector vi
measure the relative activity of each variable in the
ith mode.
iv.
The components of the left eigenvector wi weight
the initial conditions in the i-th mode
B.
Participation factors
It is natural to suggest that the significant state variables
influencing a particular mode are those having larg e entries
corresponding to the right eigenvector of λi . The
participation factor of the j-th variable in the k-th mode is
defined as the product of the j-th's co mponents of the right
vjk and left wki eigenvectors corresponding to the k-th mode
[7]
Pjk =W jkVkj
(11)
The product W jkVkj is a dimensionless measure which is
called participation factor. In other words, they are
independent on the units of the state variables. In addition,
both the sum of the participation factors of all variab les in a
mode and the sum of the participation of all modes in a
variable are equal to one. Other interesting measure is the
subsystem participation. The subsystem participation is the
38
magnitude of the sum of the part icipation factors of the
variables that describe a subsystem in a mode.
C.
Modal controllability and observability factors
The effectiveness of control in power system can be
indicated through controllability and observability indices.
This is important as control cost is influenced to a great deal
by the controllability and observability of the plant. These
issues are addressed through modal controllab ility and
observability
1)
Controllability index
Assume that an input Δu(t) and an output Δy(t) of the linear
dynamic system (2) have defined
x(t)  Ax(t)  Bu(t)
(12)
y(t)  Cx(t)
The applicat ion of a linear transformation defined by the
eigenvectors of the state matrix to the system as described
by (12) results in: equation (13):
Let v and w be the right and left eigen vector matrices of A,
respectively. If eigenvalues of A are distinct, then w Tv = I,
where w T is conjugate transpose of w and I is the identity
matrix. Substituting Δx =wΔz in (12), we obtain
z(t)  w T Awz(t)  w T Bu(t)
(13)
y(t)  cwz(t)
Equation (13) can be written for kth eigen mode as
m
z( t )   k z k ( t )   w Tk Bi vi ( t )
(14)
i 1
Where wk is the left eigenvector corresponding to kth mode
and Bi is the ith column vector of matrix B. Fro m (14), one
can find the controllability of kth eigen mode with respect to
the ith input. The controllability index (CI) of an ith input to
the kth mode [8] is defined as
(15)
CIi = w Tk Bi
The input i, for wh ich the value of wkT Bi is maximu m, is
considered the suitable parameter to be controlled for
affecting the kth eigen mode to maximu m extent.
2)
Observability index
The observability index (cv i) o f an ith input to the kth
mode is defined as
(16)
OIi = Ci wk
The study of equations (15) and (16) leads to the follo wing
conclusions:
CIi Measures the controllability of the mode
i.
x i ( t ) fro m the input
u( t ) .In other words, if the mode  i can be
controlled from the input u( t )
OIi Measures the observability of the mode
associated to the variable
ii.
x i ( t ) form the
output y( t ) . In other words, if the mode  i can
be observed from the variable y( t )
associated to the variable
Therefore, a mode can be controlled if only if it is controllab
D.
Residues
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International Journal of Engineering & Technology IJET-IJENS Vol:09 No:10
Considering (12) with single input and single output (SISO)
and assuming D = 0, the open loop transfer function of the
system can be obtained by
y( s )
G( s ) 
u( s )
(17)
 C( sI  A )1 B
The transfer function G(s) can be expanded in partial
fractions of the Laplace transform of y in terms of C and B
matrices and the right and left eigenvectors as
N
C B
G( s )   i i
i 1 ( s  i )
(18)
N
Ri

i 1 ( s  i )
Each term in the denominator, Ri , of the summation is a
scalar called residue. The residue Ri of a particu lar mode i
Yref ( s )
e
+
G(s)
Y ( s)
u
-
H (s)
Fig.3 closed system with POD controller
gives the measure of that mode‘s sensitivity to a feedback
between the output y and the input u; it is the product of the
mode‘s observability and controllability. Fig. 4 shows a
system G(s) equipped with a feedback control H(s). When
applying the feedback control, eigenvalues of the initial
system G(s) are changed. It can be proven, that when the
feedback control is applied, the shift of an eigenvalues can
be calculated by
The model to be adopted for any device in power systems
analysis must be in accordance with the type of study
involved and the tools used for simu lation. Since th is work
is concerned with the application of the TCSC for stability
improvement, the TCSC model used must rely in the
assumptions that are typically adopted for transient stability
analysis, i.e., voltages and currents are sinusoidal, balanced,
and operate near fundamental frequency.
In [9], a TCSC model suitable fo r voltage and angle stability
applications and power flo ws studies is discussed. In that
model, the equivalent impedance Xe of the device is
represented as a function of the firing angle α, based on the
assumption of a sinusoidal steady-state controller current.
The TCSC is modeled here as a variab le capacitive reactance
within the operating reg ion defined by the limits imposed by
the firing angle α. Thus, Xemin ≤ Xe ≤ Xemax, with Xemax =
Xe(αmin ) and Xemin = Xe(180o ) = XC , where XC is the
reactance of the TCSC capacitor. (In this paper, the
controller is assumed to operate only in the capacitive
region, i.e. αmin > αr, where αr corresponds to the resonant
point, as the inductive region associated with 90o < α < αr
induces high harmonics that cannot be properly modelled in
stability studies [10]. The dynamic model characteristics of
the TCSC are assumed to be modeled by a set of d ifferential
equations as follows [11] and model in fig. 2.
(20)
x1 = ( 0 + K r vPOD -x1 )/Tr
(21)
x2 = K I (Pkm -Pref )
Where 0 = K P (Pkm -Pref ) + x2
(22)
The state variable x 1 =α0 , fo r firing angle model of TCSC.
The PI controller is enabled only for the constant power
flow operation mode [11]. According to D Jovic [12 ] the
vc
(19)
i = Ri H( i )
It can be observed from (19) that the shift of the eigenvalue
caused by the controller is proportional to the magnitude of
the corresponding residue. For a certain mode, the same
type of feedback controls H(s), regardless of its structure
and parameters can be tested at different locations. For the
mode of the interest, residues at all locations have to be
calculated. The largest residue then indicates the most
effective location to apply the feedback control.
III.
TCSC M ODEL
Thyristor-controlled series capacitor (TCSC) is a series
FACTS device wh ich allows rapid and continuous changes
of the transmission line impedance It has great application
potential in accurately regulating the power flo w on a
transmission line, damping inter-area power oscillations,
mitigating sub synchronous resonance (SSR) and imp roving
transient stability.
A typical TCSC modu le consists of a fixed series capacitor
(FC) in parallel with a thyristor controlled reactor (TCR) as
shown in fig. 1. The TCR is formed by a reactor in series
with a bi-d irect ional thyristor valve that is fired with an
angle ranging between 900 and 1800 with respect to the
capacitor voltage [9]
39
itcr
c
ltcr
TCR 
Fig. 1. T CSC Model
v POD
Kr
 Pkm

KP 
Pref
 max

KI
0
s
1
Tr s+1


B( xc , )
B
 min
Fig. 2. small-signal dynamic model of T CSC
value of susceptance B is given as:



B(  ) =  k x4 -2k x2  1 cos k x     /  xC  k x4 cos k x    
- cos k x      2k x4 cos k x      2k x2 cos k x    
(23)
-k sin 2 cos k x      k sin 2 cos k x    
4
x
2
x

 4k x3 cos 2  sin k x     -4k x2 cos  sin  cos k x     
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Where k X=XC/XL., The limits of the controller are g iven by
the firing angle limits, which are fixed by design.
A
TCSC POD Controller Design
Supplementary control action applied to TCSC devices to
increase the system damp ing is called Power Oscillation
Damping (POD). Since TCSC controllers are located in
transmission systems, local input signals are always
preferred, usually the active or reactive power flow through
TCSC device or TCSC terminal voltages. Fig. 3 shows the
considered closed-loop system where G(s) represents the
power system including TCSC devices and H(s) TCSC POD
controller
In order to shift the real co mponent of λi to the left, SVC
POD controller is emp loyed. That movement can be
achieved with a transfer function consisting of an
amp lification block, a wash-out block and mc stages of leadlag blocks. We adapt the structure of POD controller given
in [13, 9] , i.e. the transfer function of the TCSC POD
controller is
sTw  1  sTlead
1
H ( s)  K *
*

1  sTm 1  sTw  1  sTlag



mc
40
IV. EIGENVALUE ANALYSIS OF POWER SYST EMS
The concepts detailed in the previous section will be
illustrated considering two small-scale power systems. The
size of these systems allo ws the computation of all
eigenvalues and eigenvectors of the state matrix without
emp loying advance techniques due to small sizes of the
system.
A Analysis of single machine connected to an infinite
bus with TCSC
The case of a single generator connected to an infin ite bus is
considered first with and without TCSC. The generator
model contains accurate representations of the synchronous
mach ine, the excitation and the speed-governing systems. It
has been assumed that the generator is equipped with a static
excitation system [14]. A thyristor controlled series
capacitor is connected between bus 2 and 3 as shown in fig.
5.
The linear model of this system is described by 11 state
1
1  sTm
Kp
KP
1  sTlead
1  sTlag
sTw
1  sTw
1  sTlead
1  sTlag
Fig.4 POD Controller structure
 KH1 ( s )
(24)
Where K is a positive constant gain and H1 is the transfer
function of the wash-out and lead-lag blocks. The washout
time constant, Tw, is usually equal to 5-10 s. The lead –lag
parameters can be determined using the following
equations:
comp  1800  arg( Ri )
(25)


1-sin  comp 
m
T
c 

 c  lead 
Tlag


1+sin  comp 
m
c


1
Tlag 
, Tlead = cTlag
i  c
Where arg(Ri) denotes phase angle of the residue Ri ,
(26)
(27)
i
is
the frequency of the mode of oscillat ion in rad/=sec, mC is
the number if co mpensation stages (usually mC = 2). The
controller gain K is computed as a function of the desired
eigenvalue location λ ides according to equation 26:
i  d
(28)
K
Ri H1 (i )
variables. The synchronous machine, the TCSC and the
exciter are described respectively by 6, 3 and 2 state
variables. The eigenstructure of the state matrix contains 3
pairs of co mp lex eigenvalues and 5 real eigenvalues which
are detailed in tables I and I1 respectively.
Eigenvalues accurately determine linear systems stability:
this system is close to instability due to the presence of a
poorly damped oscillatory mode. Ho wever, if the
connections between eigenvalues and state variables are
sought, participation factors have to be used. Table III
details the participation of the generator subsystems: rotor
dynamics, synchronous machine, exciter, and TCSC in all
modes. Table III clearly indicates that the poorly damped
oscillatory mode (eigenvalues 1 and 2) is associated to the
rotor dynamics and that the other oscillatory mode
(eigenvalues 3 and 4) describes the interaction between the
synchronous mach ine and the exciter. The mode associated
to the rotor dynamics is also known as electro mechanical
mode. Table III also shows that three exponential modes are
associated to the machine (damper windings), other two to
TCSC and the remaining mode to the exciter. The slower
1 j0.15
2
j0.5
4
3
2200MVA
Et
P
Q
TCSC
j0.93
EB
Fig.5 SMIB with TCSC device
modes correspond to the TCSC dynamics whereas the
fastest mode is associated to the exciter
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International Journal of Engineering & Technology IJET-IJENS Vol:09 No:10
Power system stabilizer
Egenvalue of two area four machine test system
8
Design of power system stabilizers or power oscillat ion
damper (POD) in case of FA CT can also be addressed using
eigenvalue methods. Eigenvalue sensitivities with respect to
the parameters of the stabilizer provide a first order
approximation o f the eigenvalue movement in the co mp le x
plane when those parameters are varied. Precisely, the
residue of the transfer function between the stabilizers
output (reference o f the excitation system, ΔVr) and the
stabilizer input (speed, Δ w, terminal voltage, ΔVt , electric
power, ΔPg) indicates the magnitude and direction of the
eigenvalue movement in the co mplex p lane when a static
controller is considered able. Table IV contains the residues
of transfer functions relevant in stabilizer design
corresponding to the electromechanical mode. The phase of
the residue informs about the phase compensation required
at the eigenvalue frequency so the phase of the eigenvalue
sensitivity becomes 1800 and the magnitude of the residue
determines the gain required to achieve the desired
damping. A speed stabilizer requires almost 900 of phase
compensation whereas accelerating power or electric power
stabilizers do not require phase compensation. The gain of
the speed stabilizer will be greater than the gains of either
accelerating or electric power stabilizers.
C
Analysis of two areas four machine with TCSC
In this study, a two area interconnected four machine power
system shown in Fig.6 is considered. The system consists of
four mach ines arranged in two areas inter-connected by a
weak tie line [14].
Fig. 7 contains a plot of the eigenvalues in the comp lex
plane. Three pairs of poorly damped eigenvalues are found.
They result to be associated to the rotor dynamics. The
slowest eigenvalues are associated to the speed-governing
systems whereas the fastest are associated to the excitation
systems. The synchronous machine modes are in between.
Fro m the table V, we see that the system is stable. There are
four rotor angle modes. There mode shapes are described by
the component of the right eigenvector corresponding to the
generator speed
V.
DESIGN OF TCSC POD CONTROLLER USING
RESIDUE M ETHOD
The uncontrolled system, Fig.6, has one inter-area
oscillatory mode characterized by λ = -0.1211 ± j3.7559
with damping ratio ζ= 3.22%. According to Table VII, the
bus 8 has the largest residue and therefore the most effective
location of the SVC and to apply the feedback control.
Using the method presented in
1
5
G1
6
7
9
10
11
8
L7
2
L9
G2
Fig. 6 Two area test system with TCSC
4
2
0
-2
-4
-6
-8
-80
-70
-60
-40
-30
-20
-10
0
Real
T ABLE I
COMP LEX EIGENVALUES OF SMIB WIT H T CSC
Mode
No.
Eigenvalue
Frequency
(Hz)
1,2
-13.494
±17.304i
-0.257±6.772i
2.7541
3,4
1.0777
Dampi
ng %
61.5
3.8
T ABLE II
REAL EIGENVALUES OF SMIB WITH T CSC
MODE
Eigenvalue
T IME CONSTANT (S)
14658.0
-1000.0
-78.9
-22.5
-1.9
-0.2
-1.0
0.0001
-0.0010
-0.0127
-0.0445
-0.5382
-5.0531
-1.0000
NO
5
6
7
8
9
10
11
T ABLE III
EIGENVECTOR AND NORMALIZED PARTICIPATION FACTOR CORRESPONDING
P OOR MODE -0.29835+J7.8548
S/
Right
Left
Particip Participation
N
eigenvect
eigenvector
ation
state
or
factor
1
-0.09 –
-1.8*10-7
0.4715
Machine
j0.55
+j1.8*10-17
9
angle 1
2
0.012 –
-2140.4715
Machine
j0.0015
j2.25*10-16
9
speed 1
3
0.016 j0.016
-151 +j1.66
*10-15
0.0095
9
q-axis
-0.028 j0.008
5.81 + j0.0
0.0188
6
d-axis
5
0.026+
j0.022
34.58
+j45.61
0.0029
2
q-axis
damper eq
6
-0.047j0.07
34.58 -j45.61
0.0246
6
d-axis
7
0.16 -j0.067
0.0006
Exiter vm
8
-0.004 j0.00
0
0.16 +j0.067
0
Exiter vr1
9
0.7553
0.12 + j0.0
Exiter vf
10
-0.029 j0.32
11
-0.0096 +
0.0003i
-2.28*10-6
+j2.09*1019
0
0.0004
5
0.0001
4
4
G4
-50
Fig. 7. Eigenvalue of t wo area test system
3
G3
1
6
Imag
B
41
94310-6767 IJET-IJENS © December 2009 IJENS
0.0000
4
damper eq
damper ed
damper ed
x1 of Tcsc
x2 of Tcsc
IJENS
International Journal of Engineering & Technology IJET-IJENS Vol:09 No:10
Section 3, POD controller parameters are calcu lated in o rder
to shift the real part of the oscillatory mode, to the left half
complex p lane. The obtained transfer function for the SVC
POD controller is
1
10 s 1  0.1329 s 1  0.1329 s
H ( s)  K *
*
*
1  0.1s 1  10 s 1  0.4325s 1  0.4325s
Eigenvalue of our interest moves form the original location
λ = -0.1211 ± j3.7559 to the desired location λ d = -0.745 ±
j3.638 to give about 20% damping as:
K
i  d
 25.8963
Ri H1 (i )
VI. SIMULAT ION RESULTS
The effectiveness of the proposed method of POD designed
was tested on two- area four -machine systems. The analysis
results for the two systems are presented in tables I to IX.
A three phase fault is applied for second test model at the
bus 8 and cleared after 74ms. The original system is restored
upon the fault clearance. The transient stability
performances of the system with TCSC without POD and
TCSC with POD controller are shown in fig.s 8-11. The
TCSC with damping controller stabilizes system as can be
seen from fig. 8-11.The oscillations of the system fro m fig.
8 to 11 also are well damped with POD controller.
T ABLE V
COMPLEX EIGENVALUE OF T WO AREA FOUR MACHINE
T EST
Frequen
cy
Damping
Mode
ratio %
No.
Complex Eigenvalue (Hz)
1,2
-12.3267±j 20.5784
0.08
99.99
3,4
-12.0224 ±j19.9823
0.08
99.99
5,6
-15.2167±j15.8377
0.53
97.97
7,8
-14.8232 ±j 5.6141
0.43
-98.63
9,10
11,12
-1.7779 ±j 6.4726
-1.9176±j 6.7494
1.20
1.16
10.05
10.23
13,14
-0.11727±j 3.6383
0.60
3.22
15,16
-5.1493±j 0.04188
0.10
99.72
17,18
-0.07742±j .22111
0.09
99.76
T ABLE VI
P ARTICIP ATION OF THE GENERATORS IN THE ELECTROMECHNICAL
MODES OF THE TWO AREA TEST SYSTEM
Mode
No.
Eigenvalue
G1
G2
G3
G4
9,10
0.7647
±7.5680i
0.011
45
0.042
9
0.41
007
0.7514
±7.3036i
0.1211
±3.7559i
0.411
04
0.246
13
0.536
42
0.141
99
0.02
767
0.34
591
0.55
216
0.01
232
11,12
A CKNOWLEDGMENT
The authors would like to express their appreciation to the
Universiti Teknologi Malaysia (UTM) and Min istry of
Science Technology and Innovation (MOSTI) for funding
this research
CONCLUSION
This paper has reviewed methods for analysis and control of
power system oscillations with TCSC device based on the
eigenstructure of the state matrix of the linear model of the
power system. Residue-based methods also provide valuable
informat ion on how to design power system damping
controllers. Although eigenvalue based methods are very
powerful, the co mplexity of the power system stability
problem requires the complementary use of other methods
such as non-linear time do main simulation. A ll the
simu lations were done with PST toolbo x in Matlab
environment.
13,14
TEST
Mode residues of the transfer function
ΔP/Δkc
TCSC
Normalised Residue
location
|Ri|
Fault at bus 8
300
250
200
Vr
Vr
50
0
without POD controller
with POD controller
-100
-150
0
5
10
15
20
25
Time(s)
Fig. 8. Active power flow with and without POD in line 7 -8
-0.1252 + 0.0585i
-25.044
Fault at bu 8
20
10
0
0.0429 - 0.0087i
Vr
100
-50
-11.46
Angle deviaton (G1-G3)
V1
150
Active power in MW
MIN
RESIDUE CORRESPONDING TO LOCAL MODE -0.257- J6.772
T ransfer function
Residue
Phase
angle
-56.15- 133.27
67.15 0
P

0.001
0.735
1.000
0.181
0.000
0.788
6-7
5-6
7-8
9-8
10-9
11-10
T ABLE IV
g
0.24
344
T ABLE VII
SIT ING INDICES OF TCSC FOR TWO AREA FOUR MACHINE
A PPENDIX
TCSC data
Tr = 10 ms, XL = 0.2, XC=0.1,.Kc=50%
Kr=10, TW = 10 s, α =3.1416, α =- 0.314
MAX
42
-10
-20
-30
-40
without POD
with POD
-50
-60
0
2
4
6
8
10
12
14
16
18
20
Tim e (s)
Fig. 9. Angle response of G1
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International Journal of Engineering & Technology IJET-IJENS Vol:09 No:10
43
Faut at bus8
50
Reactive power in MVar
0
-50
-100
Without POD
With POD
-150
-200
0
2
4
6
8
10
12
14
16
18
20
Time (s)
Fig. 10. Reactive power response for Bus 7
Fault is applied at bus 8
10
Speed deiation in rad/s
5
0
-5
-10
-15
-20
0
Without POD
With POD
5
10
15
20
Time(s)
Fig. 11. speed response of G1.
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