-
Abstract--Power system stability enhancement with
STATCOM-based damping stabilizers is investigated in this study.
Previously, the power system linearized model with STATCOM was
introduced by other authors. The linearizing method is considered
in this paper. With attention to presence of STATCOM changes
operating point, so the initial coefficients should be calculated
with new operating point. In this study, The Coordination among the
STATCOM internal AC and DC voltage controllers and AC-damping
stabilizers has been taken into consideration. This aims to enhance
both rotor angle stability and voltage regulation of the power
system. The design of STATCOM parameters is considered as an
optimization problem and GA is used for searching optimized
parameters. These studies are performed for five operating points.
The influence of STATCOM on stability is compared with eigenvalues
analysis and simulation, in two cases of without STATCOM and with
STATCOM. The nonlinear simulation results show the capability of
this method for power system dynamic stability improvement.
Index Terms-- Dynamic Stability, Optimization, Genetic
Algorithm, STATCOM
I. INTRODUCTION owadays, low frequency oscillations have become
the main problem for power system small signal stability.
They restrict the steady-state power transfer limits, which
therefore affects operational system economics and security. Using
PSS create change in oscillation stability. To increase power
system oscillation stability, the installation of supplementary
excitation control, power system stabilizer (PSS), is a simple,
effective and economical method [1]. The recent advances in power
electronics have led to the development of the flexible alternating
current transmission systems (FACTS).These devices in addition to
main control duties similar to voltage regulation and reactive
power injection should be able to damp power oscillations. From the
power system dynamic stability viewpoint, the STATCOM provides
better damping characteristics than the SVC as it is able to
transiently exchange active power with the system, so it can
improve oscillation stability better than SVC [2]. The method of
phase compensation [3] and damping torque analysis method [4] are
conventional methods for design and control of power system
stabilizers. Also multivariable design
The authors are with Electrical Engineering K.N.Toosi University
of
Technology,Tehran,Iran ( e-mail: [email protected] ;
[email protected] ).
method has been performed on coordination between internal AC
and DC voltage controllers [5]. In this study, to improve power
system dynamic stability and voltage regulation, coordination among
AC-damping stabilizer and internal AC and DC voltage controller of
STATCOM are performed. The studies are performed on a single
machine infinite-bus power system. From power system linearized
model is used for the studies [6]. The parameters design is
considered as an optimization problem and the genetic algorithm is
used for searching optimized parameters. The parameters should be
designed so that the eigenvalues of system move to the left part of
the j axis as much as possible. The participation factors method is
used for identifying of modes type. The eigenvalues analysis and
the nonlinear simulation results show the importance of the
parameters design for dynamic stability improvement of the
system.
II. POWER SYSTEM MODEL The power system is represented by a
single machine
infinite-bus system. The system has an installed STATCOM in
transmission line as shown in Fig.1.The STATCOM is used at the
middle point in transmission line for voltage regulation and power
oscillations damping. The system is modeled for low frequency
oscillations studies and the linearized power System model is used
for this purpose.
Fig. 1. Single machine infinite bus power system with
STATCOM
A. Generator The generator is represented by the third-order
model
comprising of the electromechanical swing equation and the
Analysis of Power System Linearized Model with STATCOM Based
Damping Stabilizer
Masoud Zarringhalami , and M.A.Golkar
N
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generator internal voltage equation. Swing equations can be
expressed as:
( ) ( )( ) dofdqq
em
b
TEEE
MDPP+=
==
1
B. Exciter The IEEE Type-ST1 excitation system is considered in
this
work. It can be described as:
( ) ( )2 1 ttoA
Afd
Afd VVT
KET
E +=
In equations (1) and (2): eP , qE and tV are related by the
following equations:
( )( ) ( )
( ) ( )223
tLqqtLddqt
tLdddqq
tLqtLddqtLqqe
IxIxEV
IxxEEIIxxIEP
+=
+=
+=
Where, AK and AT are the gain and time constant of the exciter
respectively; toV is the reference voltage. The terminal voltage tV
can be expressed as:
( )tLddqtq
tLqqtd
tqtdt
IxEVIxV
VVV
=
=
+=
4
222
C. STATCOM As shown in Fig.1, The STATCOM consists of a
three
phase gate turn-off (GTO) based voltage source converter (VSC)
and a DC capacitor. The STATCOM model used in this study is founded
well enough for the low frequency oscillation stability problem.
The STATCOM is installed through a step-down transformer with a
leakage reactance of SDTX . The voltage difference across the
reactance produces active and reactive power exchange between the
STATCOM and the transmission network. The STATCOM is one of the
important FACTS devices and can be used for dynamic compensation of
power systems to provide voltage support and stability improvement.
The VSC generates a controllable AC voltage given by:
( )5 )sin(cos jcVcVV DCDCO +== For PWM inverter mkc = , where m
is the modulation ratio defined by pulse width modulation (PWM), k
is the ratio between the AC and DC voltage depending on the
converter structure. DCV is the DC voltage, and is the phase
defined by PWM. The magnitude and the phase of oV can be controlled
through m and respectively. By adjusting the STATCOM
AC voltage oV , the active and reactive power exchange between
the STATCOM and the power system can be controlled. Capacitor
voltage dynamic has big influences on power system, so capacitor
voltage dynamic should be considered. If converter is assumed to be
lossless, the exchanged active power between converter and system
is equal to the active power that exchange among capacitor and
converter ( ACDC PP = ). So with these assumptions the relationship
between voltage and current of capacitor, it can be expressed
as:
{ } ( )6 ))(sin(cos)( LoqLodDCLooDCDC jIIjcVrealIVrealIV +==
Solving the above equation for DCI gives:
( )7 )sincos( LoqLodDC IIcI += With attention to (7) and the
equation between voltage and current of capacitor, we have:
( )8 )sincos( LoqLodDC
DC IICcV +=
Where, DCC is the capacitor voltage, DCI is the capacitor
current and LodI , LoqI are d and q axis of STATCOM current. From
(8) is used in part IV for modeling of capacitor voltage dynamic on
the Heffron-Phillips modified model. C.1. STATCOM control
system
As shown in Fig.1, the converter and step down transformer in
STATCOM can be modeled with a voltage source and a reactance for an
operating point. Changing modulation ratio can change the amplitude
of the output voltage of converter and so the active power absorbed
by the system. By changing the converter voltage angle, reactive
power exchanging with system can be controlled. In this study, the
P-I controllers are used for voltage regulation.
i. Terminal voltage controller AC voltage controller regulates
the voltage of terminal
according to requested reference that it accomplishes through
changing of converter output voltage magnitude. cu is input signal
for auxiliary damping stabilizer. The terminal voltage controller
is introduced in Fig.2.The proposed damping stabilizer of STATCOM
is shown in Fig. 3. This stabilizer has a structure similar to PSS.
In this stabilizer and cU are the stabilizer input and output
signals respectively. This stabilizer is used to create an
additional damping signal for STATCOM.
sKIKP ACAC +
refLV
LV
cu
maxVACu
minVACu
c
0c
cF
F
sTK+1
controllerPI dynamicConverter+
++
Fig. 2. AC voltage controller
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c
c
sTsT
2
1
11
++
cACKc
c
sTsT
4
3
11
++
maxcu
mincu
cuW
W
sTsT+1
GainrCompensatolaglead /
Fig. 3. AC-damping stabilizer
ii. Capacitor voltage controller
The inner DC voltage controller regulates the voltage of
capacitor. The converter phase angle is calculated according to
capacitor voltage of reference with capacitor voltage. The
capacitor voltage controller is introduced in Fig.4.
+
controllerPI dynamicConverter
sKIKP DCDC +
F
F
sTK+1
refDCV
DCV maxVDCu
minVDCu
0
++
Fig. 4. Capacitor voltage controller
III. LINEARIZED MODEL
A. Currents Calculation For creating power system linearized
model should be calculated the output currents of Generator and
STATCOM. In here, the output currents of Generator and STATCOM are
calculated respectively:
i. Generator Current : With attention to Fig.1 , It can be
written :
( )9 SDT
OtLtLttL
SDT
OLtLLOtLLB jx
VIjxVIjx
VVIIII ===
( )10 BLBLBtLtLt VIjxIjxV ++= The generator current ( tLI ), in
d-q is expressed as:
( )11 1
sincos1
dSDT
LB
SDT
LBtLLBtL
DCSDT
LBBq
SDT
LB
tLd
xxx
xxxxx
cVxxVE
xx
I
++++
+
=
( )12 1
cossin
qSDT
LB
SDT
LBtLLBtL
DCSDT
LBB
tLq
xxx
xxxxx
cVxx
VI
++++
+=
ii. STATCOM Current : With attention in Fig.1 , the LoI can be
calculated to this form :
( )13 SDT
OtLtLtLO jx
VIjxVI =
The STATCOM current ( LoI ), in d-q reference is expressed
as:
( )14 sinSDT
DCtLd
SDT
tLd
SDT
qLOd x
cVIx
xxxE
I +
=
( )15 cos tLqSDT
tLq
SDT
DCLOq Ix
xxx
cVI+
=
B. Heffron-Phillips Modified Model The power system linearized
model with STATCOM is
introduced in Fig.5. The operating point of system changes with
STATCOM connection to power system. Accordingly in this state, the
model should be linearized in new operating
point. In the design of stabilizer parameters, the linearized
model around operating point is usually employed. With linearizing
(3) and using (11), (12),(14) and (15), the differential equations
(16) can be calculated:
( )
++++=
++++=
++++=
++++=
ddcDCqDC
vvcDCvDCqt
qqcDCqDCqq
ppcDCpDCqe
KcKVKEKKV
KcKVKEKKVKcKVKEKKEKcKVKEKKP
987
65
34
21 16
With using equations (16) and to considerate, inputs
( MT , refV ) and converter controller signals ( c , ), the
state equation of power system can be extended to form of
below:
( )
00
0
00
10
0000
17
00
10
10
0
0000
987
65
34
21
+
=
cT
V
KKTKK
TKK
TK
TK
TK
MK
MK
M
VEE
KKKTKK
TTKK
TKK
MK
TTK
TK
MK
MK
MD
MK
VEE
m
ref
ddc
A
vA
A
vcA
A
A
dpo
q
dpo
qc
ppc
DC
fd
q
A
vDCA
AA
A
A
A
qDC
dpodpodpo
pDC
b
DC
fd
q
Using vector representation, the above equation can be written
as: uBxAx += The system has 5 state variables. 4 new state
variables are added to system with consideration of STATCOM
controllers. Three new state variables are added to the system with
consideration of damping stabilizer of STATCOM, so the system has
12 state variables totally. The AC damping stabilizer and STATCOM
internal controller parameters should be designed so that the
eigenvalues of system move to the left part of the j axis as much
as possible and the small signal stability of system has been
improved. In Table I the related state variables of power system is
introduced.
Fig. 5.The Power System Linearized Model with STATCOM
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TABLE I CLASSIFFICATION OF SYSTEM STATE VARIABLES
uBxAx xx += The State variables : [ ]Tfdq EEx = 1 Generator and
Exciter [ ]TDCVx = 2 Capacitor Dynamic [ ]Tc xcxx = 3 STATCOM
Internal Voltage Controllers [ ]Tccc uxxx = 214 AC Damping
Stabilizer [ ] 1124321 = xxxxx Total of State Variables
C. Initial Conditions Calculation Considering the operating
point of system, the initial
conditions can be calculated. Initial conditions are used for
calculating the coefficients of the Heffron-Phillips modified
model. In this study is used power flow for calculation of initial
conditions [7]. MATLAB program is used for the initial conditions
calculation [8]. In here the generator bus is considered as a PV
bus. These studies are performed for 5 operating points. The
studies are performed without STATCOM and with STATCOM in 5
operating points. The amount of Q in each operating point is
calculated with Power Flow program. The considered operating point
and the results of power flow have been shown in Table II. Also the
calculated initial conditions with STATCOM are shown in Fig.6.
Calculated Initial Conditions
0.11872
0.23023
0.329180.373 0.37352
0.99293 0.97314 0.944270.98152
1.0877
0.029708
0.11553
0.24892
0.50894
0.86449
0.19787
0.38371
0.548640.62166 0.62253
1.0018 1.0078 1.0189
1.1342
1.347
0.23908
0.47289
0.69741
0.836140.89942
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1 2 3 4 5Operating Point
Valu
e (P
U)
Vtd Vtq ItLd ItLq E'q delta
Fig.6. Calculated Initial Conditions in Operating Points (with
STATCOM)
TABLE II POWER FLOW RESULTS WITHOUT STATCOM AND WITH STATCOM
Operating Point Parameters 1 2 3 4 5
tV 1 1 1 1.05 1.15 tP 0.2 0.4 0.6 0.8 1
( )STATCOMWithouttQ 0.012 0.048 0.112 0.281 0.569
( )STATCOMWithtQ 0.006 0.024 0.054 0.267 0.708 ( )STATCOMWithoV
1.002 1.007 1.016 1.003 0.967
( ) radSTATCOMWith 0.061 0.120 0.181 0.242 0.304
IV. PROBLEM FORMULATION
A. Objective Function The objective function consists of two
individual objective
functions with weighed coefficients. The advantage of this
function is consideration of all modes in stability evaluation. In
other word, it can move the eigenvalues to the left part of j axis
with parameters optimization. Also, the damping of oscillation mode
can be increased. Flowchart of the proposed objective function is
introduced in Fig.7. The objective function can be expressed in 3
parts:
i.In the first part, two coefficients 1 and 2 are defined
( 21 > ). If thi eigenvalue of system became as an
electromechanical mode, the coefficient 1=K and else 2=K . The K is
a weight coefficient that is used in part of ii and iii for
Objective Function.
ii.In the second part, the real parts of eigenvalues are
evaluated. The area in the left j axis is divided to 1n vertical
parallel bands. The amount of 1n is calculated with (18). In here
01 =a , 51 =b and 1.01 =S are supposed. So if the real part of
eigenvalues become more negative with parameters changing, the
amount of 1J can be increased more.
iii. In the third part, the damping ratios of eigenvalues
are
evaluated. Like to second part, the left area j axis is divided
to 2n radial part. The amount of 2n is calculated with (18).In here
02 =a , 22 =b and 01.02 =S are supposed. To limit maximum
overshoot, the parameters of the stabilizer may be selected with
evaluation of 2J .
( )18 1,2i ==i
iii s
abn
iv. The single objective problem described maybe converted
to
a multiple objective problem by assigning distinct weights to
each objective [9]. With attention to flowchart in Fig.7 the
objective function consists of two individual objective functions
with weighted coefficient w, so a unified objective Function J is
created. . The parameters of stabilizer may be selected to minimize
the following objective function :
( ) ( )19 21 wJJJ +=
In Fig.8 the considered area for eigenvalues evaluation by
objective function has been shown. As shown in Fig.8 the left area
of j axis is divided to small areas. With this procedure we can
allocate a mark for each eigenvalue with attention to its place in
j plane. So the objective function can evaluate stability
easily.
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Fig.7 .Flow chart of proposed Objective Function
Fig.8.The S-plane divided regions via objective Function
B. Optimization Problem In this study, the proposed objective
function J is minimized. The problem constraints are the stabilizer
optimized parameter bounds. Therefore, the design problem can be
formulated as the following optimization problem:
{ } ( )
maxmin
maxmin
20 :
xxx
xxx
TTTKKK
JMinimize
The proposed approach employs genetic algorithm to solve this
optimization problem and search for optimal set of STATCOM
parameters.
C. Electromechanical Mode Identification The state equations of
the linearized model can be used for
eigenvalues calculation of matrix A. Out of these eigenvalues,
there is a mode of oscillation related to machine inertia. To
ensure an effective operation of the stabilizers, it is extremely
important to identify the eigenvalues associated with the
electromechanical mode. In this study, the participation factors
method [10] is used.
D. Genetic Algorithm The genetic algorithm is used for
optimization of system
parameters. This algorithm has been programmed in MATLAB
software based on continuous search algorithm [11]. The algorithm
considers many search points simultaneously, so the probability of
converging to local minimums is decreased. For all of operating
points the Generation and Population are considered, 50 and 1000
respectively. Also Crossover and Mutation are considered, 0.6 and
0.02 respectively.
V. RESULT AND DISCUSSION
A. Setting of Parameters In papers the analysis and assessment
of STATCOM for power system stability has been studied [12]. In
here the coordinated design of AC-damping stabilizer and internal
PI voltage controllers of STATCOM is performed and the results
illustrate that the power system stability have been improved. The
proposed approach has been implemented on a weakly connected power
system as shown in Fig.1. The detailed data of the power system
used in this study is given in Appendix. With attention to block
diagram in Fig.4, the damping stabilizer has 6 parameters. The
parameters of cT2 and cT4 are considered equal to 10 and only cT1 ,
cT3 and cACK are optimized. The STATCOM internal AC and DC voltage
controllers have 6 parameters. The parameters of FK and FT have
been supposed before and only 4 parameters are optimized in the
STATCOM internal AC and DC voltage controllers. In this study,
design of AC damping stabilizer and the STATCOM voltage controller
parameters is accomplished in coordinated form. The final setting
of the optimized parameters for five operating points has been
given in Table III. The eigenvalues of electromechanical
oscillation mode for each operating point have been shown in Table
IV.
TABLE III OPTIMAL PARAMETER SETTING
Operating Point Parameters 1 2 3 4 5
DCKP 4.26 4.03 3.73 3.51 3.69 DCKI 86.68 69.48 43.17 22.39 42.52
ACKP 7.51 13.59 4.89 3.82 5.20 ACKI 45.77 76.42 34.52 32.51
40.56
cACK 4.73 8.34 11.21 7.39 8.79
cT1 2.91 1.32 1.25 3.73 2.35
cT3 4.93 6.45 7.89 1.33 9.23
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TABLE IV EIGENVALUES OF SYSTEM
Without STATCOM With STATCOM Operating
Point Oscillation
Mode (EM) Oscillation
Mode (EM)
1 -0.166 7.249i 0.023 -0.770 7.535i 0.102 2 -0.161 7.271i 0.022
-0.818 6.531i 0.124 3 -0.142 7.242i 0.019 -1.004 4.732i 0.207 4
-0.115 7.419i 0.015 -1.261 3.012i 0.386 5 -0.098 7.812i 0.012
-1.267 2.072i 0.522
B. Nonlinear Simulation Results Nonlinear simulation has been
used for checking the design
accuracy. This work has been accomplished with a disturbance
creation in system. In here, a 10% pulse disturbance in input
mechanical torque is considered. The simulation time equal to 5
seconds is considered. The results in two cases, Without STATCOM
and with STATCOM for five operating points have been shown in Fig.9
to Fig.16. Case i. Without STATCOM: The time responses of variables
( , tV and eP ) for a disturbance in mechanical torque have been
introduced in Fig.9 to Fig.11. In here, with attention to Table IV,
the amounts of (damping ratio) and electromechanical oscillation
frequency can be calculated. For example in last operating point
the oscillation frequency is equal to 1.243 Hz and is equal to
0.012. Also we observe with increase in output active power
generator (changing operating point), the amount of is decreased.
Because amount of power angle is increased with active power
increasing .So Power System stability is decreased. Case ii. With
STATCOM: The system response ( , ,
tV , DCV and eP ) for a Pulse disturbance in mechanical torque
have been introduced in Fig.12 to Fig.16. In here, like to previous
case, with attention to Table IV, the amounts of (damping ratio)
and electromechanical oscillations frequency can be calculated. For
example in last operating point the oscillation frequency is equal
to 0.329 Hz and is equal to 0.522. In comparison to case of without
STATCOM, we can observe the amount of is increased and oscillation
frequency is decreased. So the STATCOM increases power system
stability, but the oscillation frequency is changed with STATCOM.
Also we can observe with increase in output active power generator,
the amount of is increased because the amount of power angle has
been increased with increase of active power .But the STATCOM can
control the amount of output reactive power of generator. So in
this case with attention to increase of power angle the amount of
generator output reactive power is controlled with STATCOM. In
result, the dynamic stability is improved. These results with
comparison two cases can be observed.
Fig.9. Time response of
Fig.10. Time response of
tV
Fig.11. Time response of
eP
Fig.12. Time response of
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Fig.13. Time response of
Fig.14. Time response of
tV
Fig.15. Time response of
eP
Fig.16. Time response of
DCV
VI. CONCLUSION In this study, the coordination among STATCOM
internal
AC and DC voltage controller and AC-damping stabilizer was
presented and discussed for power system dynamic stability
improvement. The design problem was formulated as an optimization
problem and the GA was used for searching optimized parameters. The
results of coordinated design show dynamic stability improvement.
Also In second case with nonlinear simulation has been shown that
the oscillations are damped properly.
VII. APPENDIX
TABLE V THE POWER SYSTEM DATA RELATING TO FIG.1
POWER SYSTEM GENERATOR EXCITER LINE
M
D do
T dx
dx
qx
AK
AT
tLx
LBx
6 2 5.044 1 0.3 0.6 10 0.1 0.3 0.3
STATCOM CAPACITOR,TRANSFORMER CONVERTER STABILIZER
DCC
DCV
SDTX
c
FK
FT WT
cT2 cT4
1 1 0.15 Power flow 1 0.05 5 10 10
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[5] H.F.Wang , "Interactions and Multivariable Design of STATCOM
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[8] Y.N.Yu , "Electric Power System Dynamics", Chapter 3 ,
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pp.177185.
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