Abstract—A rotor speed of remote generator is used as wide
area measurement signal in this paper. The signal is detected and
transmitted by the phasor measurement unit (PMU). The wide
area signal inputs to the local power system stabilizer (PSS) to
regulate the generator excitation and enhancing the power
system damping. The gains of the wide area controller are
determined by solving linear matrix inequalities (LMI). The LMI
solving method is introduced for the wide area control of power
systems. A proportional plus derivative network is used to
compensate the communication delay. Kundur's four-machine
two-area system is used to test the performance of the wide area
damping control. The simulation results show that the wide area
control can improve power system stability.
Index Terms–Linear matrix inequalities, power system
stability, time delay, wide area control
I. INTRODUCTION
HE stability of power systems is a very important and
common problem. Considerable research effort goes into
investigating the stability of power systems. The power
system stabilizer (PSS) was invented to improve the power
system damping. Power systems are highly nonlinear, large
scale, and multivariable. Conventional power system
stabilizers are not always able to guarantee stability in large
interconnected power system.
Djukanovic and Khammash presented a systematic
procedure for the design of decentralized controllers for
multimachine power systems [1]. The shaft speed deviation is
selected as an input to the controller. But the sequential
synthesis based on μ-approach is complicated for the design of
decentralized controllers.
Befekadu and Erlich introduced a linear matrix inequalities
(LMI) based robust decentralized dynamic output feedback
controller design for power systems [2]. An algorithm based
on iterative the LMI programming method is proposed to
solve for the controller design. However, solving LMIs
iteratively is a challenging problem.
The phasor measurement unit (PMU) can provide wide
area measurement signals. The signals can be used to enhance
H. Wu, Q. Wang and X. Li are with the School of Electrical and
Automation Engineering, Nanjing Normal University, 210042 Nanjing, China
(e-mail: huarenwu68@yahoo.com.cn).
978-1-4244-1762-9/08/$25.00 ©2008 IEEE
the wide area damping characteristics of a power system. Hui
Ni and Heydt proposed a supervisory level power system
stabilizer (SPSS) using wide area measurements [3]. An SPSS
agent is composed of agent communications, a fuzzy logic
controller switch, and robust controller loops. An LMI-based
method is applied to controller design. The time delay for the
simulation studies falls in range of 10~30 ms. The effect of the
time delay on the performance of the controller was not
analyzed.
Hongxia Wu and Heydt suggested a centralized control
method using system-wide data to enhance the dynamic
performance of a large interconnected power system [4]. This
parameter-dependent controller ensures closed-loop system
stability with a given bound from the disturbance to the
output. The parameters of power systems are measured in real
time and the controller is adjusted accordingly. The system
response is tested by introducing a unit step input disturbance
to the reference voltage. The effects of communication delays
on the controller design and performance was examined.
These delays can be up to 0.25 seconds long.
Hiyama presented a wide area stabilization control system
using power system stabilizers [5]. The input signal to the PSS
is the real power flow signal on one of the trunk lines in the
system. He proposed communication delay compensation in
[5].
This paper introduces wide area damping control of
excitation systems. A rotor speed signal from a remote
generator is used to damp speed oscillation between
generators with feedback control. The wide area controller
includes a compensation network for the communication
delay. A LMI-based design method of wide area damping
controller for interconnected power systems is discussed.
Kundur's four-machine two-area system is used to test the
performance of the wide area damping controller [6].
II. WIDE AREA CONTROLLER DESIGN
This section discusses the controller structure, the
determination of the controller parameters based on LMI, and
a communication delay compensation network.
A. Power System Model
The two-axis model is used to represent the synchronous
machine. Therefore, generator i in the power system is
represented by a fourth order model [7]
'
' ' '
0 di ( )
q i di qi qi qi
T dE E X X I
dt
= − − −
PMU-Based Wide Area Damping Control of
Power Systems
Huaren Wu, Member, IEEE, Qi Wang, and Xiaohui Li
T
fdi qi di di di
qi
d i E E X X I
dt
dE
T ' ( ' )
'
'
0 = − + −
mi di di qi qi i i
i
ji T I E I E D
dt
τ dω = − ( ' + ' ) − ω
i n
in
dt
dδ =ω −ω
where di I , qi I , '
di E , and '
qi E represent the currents, the
damping-winding flux linkage, and the field flux voltage in
the d-q reference frame, respectively. i
ω is the rotor speed.
in δ is the rotor angle difference between generator i and
generator n. fdi E represents the field voltage and Mi T
represents the mechanical torque. '
d 0i T and '
q0i T are the direct
and quadrature-axis open circuit time constants, respectively.
di X , qi X , '
di X , and '
qi X represent the d- and q-axis
synchronous and transient reactances, respectively. i D is the
damping power coefficient. The time constant ji τ depends on
the rotating inertia.
The block diagram of the automatic voltage regulator
(AVR) and the PSS of generator i is shown in Fig. 1. The
conventional PSS uses local signals for feedback control.
Fig. 1. Block diagram of the AVR and PSS.
If two-stage lead-compensated stabilizers are used, the nmachine
system can be described by 7n-1 first-order
differential equations. The linearized state space equations for
the n-machine system are of the form:
•
x = Ax +Bu (1)
T
1 2 [ , , , ]n = = Δω Δω Δω L L y Cx
T
1 2 [ , , , ] n n nn = = ω ω ω W W y C x
= + L L W W u K y K y
= + L L W W K C x K C x
where x is state vector, u is control vector, in i n ω =ω −ω ,
and KL and KW are the following diagonal matrices:
KL=diag(KL1, KL2, …, KLn)
KW=diag(KW1, KW2, …, KWn)
nn ω in output equation may be deleted.
B. Determination of the feedback gain KW
The conventional PSS included in (1) is designed according
to the guidelines in [6]. The rotor speed of generator n is used
as a wide area measurement signal. n-1 generators receive the
rotor speed signal from generator n. The time delay of this
signal is not considered. The determination of the gain factor
KW based on LMI is described in the following analysis.
The Lyapunov function is selected so that V = xTPx , and
P > 0 . From
T
V( ) T 0
• • •
x = x Px + x Px < , the following
LMI is derived:
( + )+( + )T + + T T <0 L L L L W W W W P A BKC A BKC P P C C P (2)
where W W P = PBK .
The matrix A, B, CL, CW and KL can be obtained from the
parameters of power systems. The matrix variables P and Pw
in (2) can be obtained by solving the LMI using the MATLAB
LMI toolbox [8].
KW is a n× n diagonal matrix. P is a (7n −1)×(7n −1)
block diagonal matrix. In order to obtain KW, P should have
the structure P=diag(P1, P2, …, Pn), where Pi is a 7×7
symmetric matrix for i=1, 2, …, n-1 and Pn is 6× 6
symmetric matrix.
PW is a (7n −1)× n block diagonal matrix of the form
PW=diag(PW1, PW2, …, PWn), where PWi is a 7 ×1 matrix for
i=1, 2, …, n-1 and PWn is 6×1 matrix.
The structures of P and PW were declared using the Matlab
lmivar(3,struct) function and the LMIs were solved using the
feasp( ) Matlab function to obtain P and PW . KW can be
calculated from W W K = (PB) \ P .
If a LMI is established for each interesting operating point
and these LMIs are solved simultaneously to obtain the matrix
variables P and Pw, the wide area controller will have robust
performance for a large range of system operating conditions.
C. Compensation for Communication Delay
The wide area measurement signal from the PMU is used
for feedback control. The wide area signal without time delay
could greatly improve the dynamic performance of the
interconnected power system; the communication delay is not
beneficial for power system stability. Thus, the feedback
control signal time delay should be considered in the design of
wide area controllers. The communication delay is expressed
by the Laplace transformation of e−sTd , where Td is the
communication delay. e−sTd may be compensated by esTd .
esTd can be approximated by d
esTd ≈1+ sT . Therefore, the
proportional plus derivative d 1+ sT (PD) network is used to
compensate the time delay for simplification. A filter is
introduced to avoid the influence of the noise in the wide area
signal. Finally, the communication delay is compensated by
the compensator )
1
1 (
f
d
T sT
K sT
+
+ , where KT is the
i Δω
KL Σ (1 )(1 )(1 )
(1 )(1 )
2 4
1 3
sT sT sT
sT sT sT
W
W
+ + +
+ +
Eti
Σ
R 1+ sT
1 +
+
Vrefi
νsi Efdi
ui
uwi +
+
–
KA
compensation factor and Tf is the time constant [5]. The
compensation network is shown in Fig. 2. ωin (t −Td ) in Fig.
2 is the following:
( ) ( ) ( ) in d i d n d ω t −T =ω t −T −ω t −T
Fig. 2. Time delay compensated communication.
The communication delay Td can be obtained from the wide
area signal. The compensation factor KT depends on the time
delay Td. KT is calculated with the following formula obtained
by simulation numerically.
2
2 2
1000 10 1 0.1
1 0.1
d d
T
d
d
T T s
K
T s
T
α
α
⎧⎛’ ‘⎞’ − + < ‘⎜’ ‘⎟’ ‘⎪⎪
= ‘⎝’ ‘⎠’ ‘⎨⎪
≥
⎪⎩
where α is chosen to get the best power system damping at
time delay Td =0.1 s.
III. SIMULATION AND RESULTS
The design method of the wide area controller presented
above was applied to the four-machine two-area system
shown in Fig. 3. The generators, transformers, and lines
parameters are given in [6]. The parameters of the AVR and
PSS, selected from [6], are as follows:
1 2 200, 0.01, 0.05, 0.02 A R K = T = T = T =
3 4 3, 5.4, 20 L T = T = K =
Fig. 3. Four-machine two-area system.
All four generators are equipped with the local PSS. The
wide area measurement signal is the rotor speed of generator 4.
Generator 1, 2 and 3 have wide area damping controllers and
receive the wide area signal from generator 4.
Two operating conditions are considered. One is the same
as the MATLAB example "Performance of Three PSS for
Interarea Oscillations" [8]. The other has half of the real
power load of the MATLAB example. The 27th-order
linearized state space equations for the test system without
time delay compensation is obtained for each operating
condition. That is, matrix A and B can be obtained for each
operating condition.
P is a 27× 27 matrix., P1, P2 and P3 are declared as 7×7
symmetric matrix, and P4 is defined as a 6× 6 symmetric
matrix. PW is a 27× 4 matrix where PW=diag(PW1, PW2, PW3,
0), and PW1, PW2, and PW3 are 7 ×1 matrices. A LMI of (2) is
created for each operating point. These LMIs were solved
using feasp(LMIs, [0,0,0,0,0],-4000) to obtain P and PW. KW is
calculated from W W K = (PB) \ P . The results of these
analyses are Kw1=140.7, Kw2=95.6’，’and Kw3=61.1.
A three-phase fault at the center of one of tie-lines was
created to verify the performance of controllers under transient
conditions. The fault was cleared by opening the circuit
breakers at both ends of the faulted tie-line. The fault-clearing
time was 0.12 s and re-closure was completed in 0.5 s after the
fault-cleared. Fig. 4 shows swing curves with the AVR and
PSS but without wide area control. The swing curves indicate
that the system is stability.
Fig. 4. Swing curves with the PSS.
Fig. 5 presents the swing curves with the PSS and the wide
area control for the time delay of 0.1 s. KT=13.2 is selected for
time delay compensation. Comparing Fig. 5 with Fig. 4 shows
that the wide area control can greatly improve power system
stability.
Fig. 5. Swing curves with the PSS, wide area control,
and time delay compensation.
Fig. 6 depicts swing curves with the PSS and wide area
control for a time delay of 0.1 s. The time delay compensation
was not used with the controller in this simulation. Fig. 6
shows that the communication delay is detrimental to the
power system stability. Fig. 5 and Fig. 6 demonstrate the
effectiveness of the compensator for power system stability
enhancement.
~ ~
~ ~
L7
G1 1 5 6 7 8 9 10 11 3 G3
G2 G4
2 4
L9
C7 C9
( ) in d ω t −T (1 )
1
d
T
f
K sT
sT
+
+ W K wi u
Fig. 6. Swing curves without time delay compensation.
Curves 1 and 2 in Fig. 7 show the swing curves with wide
area control for time delays of 0.1 and 0.25 s, respectively.
Curve 3 shows the swing curve without wide area control. Fig.
7 indicates that wide area control with time delay
compensation can improve power system stability when the
time delay is 0.25 s.
Fig. 7. Effect of time delay compensation on the damping.
The effectiveness of wide area control was also tested at
another operating point with the same fault. Curve 1 in Fig. 8
shows the swing curve with wide area control for a time delay
of 0.1 s. Curve 2 shows the swing curve without wide area
control. Fig. 8 indicates that wide area control has good
robustness at the different operating points.
Fig. 8. Swing curves for another operating condition.
IV. CONCLUSIONS
T