350
IEEE TRANSACTIONSON CIRCUITSAND SYSTEMS,
VOL. CAS-33, NO. 3, MARCH 1986
proposed extended version. In both the cases we use the ideal
autocorrelation samples. We see from this figure the superiority
of the extended Levinson recursion, which is able to recover
exactly the poles of both the clutters.
REFERENCES
PI G. Martinelli and G. Orlandi, “ARMA spectrum estimation by extended
lattice,” in Proc. ICASSP, Paper 13.3, San Diego, CA, Mar. 1984.
VI J. A. Cadzow, “Spectral estimation: an overdetermined rational model
equation approach,” Proc. IEEE, vol. 70, pp. 907-939, Sept. 1982.
[31 G. Martinelli, P. Burrascano and G. Orlandi, “Simultaneous cancellation
of two clutters by the extended lattice oredictor.” in Proc. Int. Svmo.
Noise Clutter Rejdfion in Radars and Im&ing Sensors, Tokyo, Japan, p’p.
274-278. Oct. 1984.
141 S. Haykin, B. W. Curie and S. B. Kesler, “Maximum entropy spectral
analysis of radar clutters,” Proc. IEEE, vol. 70, pp. 953-962, Sept. 1982.
A Unique Method for Coordination of Excitation and
Governor Controls for Stability Enhancement of a
Power System
CHOO-MIN LIM AND S. ELANGOVAN
It should be noted that the first element of X is chosen to be the
system only output or the rotor speed [4], [5], and uE and uc are
the scalar excitation and governor control signals, respectively.
It has been shown in [2], [3] that the feeding back of rotor
acceleration, SAO, into the excitation and governor loops is
suitable for improving the damping characteristics of power
systems. However, the measurement of rotor acceleration is not a
simple matter and alternative means have been proposed [2], [6].
Here, the power system rotor speed is chosen to be its output and
the structure of the excitation and governor stabilizers are preselected to be [6]
FE = K,s/(
1 + ST)
(2.1)
FG = K,s/(
1 + ST)
(24
where T is prespecified and (K,, KG) are constant parameters
which remain to be determined. It is interesting to note that in
[7], it has been shown that irrespective of the values of (K,, KG)
the minimizing value of, T with respect to a quadratic performance index is always zero. However, to ensure the physical
implementation of each stabilizer, T should be chosen as small as
possible but within physical limit [7]. Here, this approach is
adopted and stabilizers thus designed will yield optimal performance.
Abstract -A simple and unique method for coordinating the excitation
and governor stabilizers for stability enhancement of power systems is
presented. The required stabilizers are simple in structure and, most
importantly, are readily realizable.
I.
INTRODUCTION
The coordination of excitation and governor stabilizers for
enhancing the stability of power systems has received much
attention [l]-[3]. A primary design objective in the coordination
of excitation and governor controls is to provide additional
damping to a power system rotor oscillations [l]-[6], which are
governed predominantly by a pair of complex-conjugate eigenvalues. Here, a simple and unique method for coordinating the
excitation and governor stabilizers of a power system based on
eigenvalue assignment is presented. Specifically, the parameters
of both stabilizers are so chosen that the eigenvalues associated
with rotor oscillations have desired values or specific damping [l],
[4], [5]. As in [2], [4]-[6], the required stabilizers are simple in
structure and readily realizable.
II.
Consider a single-machine infinite-bus power system whose
dynamics may be expressed in the complex frequency domain as
sX=AX(s)-tBU(s)
(1.1)
y=Ao
(1.2)
where
UG]l
0
..
.
b,
...
0
...
...
0
0
bc
1 1
A 1 1 42
A=
and
A-:-T
211
2
(3.1)
where O,,O,,,O,
AA,,=[O
The composite
given by
are null submatrices and
0
...
0
closed-loop
bEFE 0
...
0
1
t
bGFGlr.
system characteristic
A(s)=IsZ-(A+AA)I=O.
COORDINATION OF EXCITER AND GOVERNOR
STABILIZERS
u=[u,
The design objective may now be formulated as the determination of (K,, Z&) such that the complex-conjugate eigenvalues
associated with rotor oscillations have desired values or specific
damping [l], [4], [5]. To this end, the composite closed-loop
system equation is obtained and expressed in the form sX=
(A + AA) X where the A and AA matrices are as follows:
(3.2)
equation is
(4)
Let the desired eigenvalues associated with rotor oscillations be
(sd, s:) or equivalently preselected to be (- [a,, k jw,J(lE2)).
Now for (sd, s:) to be a pair of complex-conjugate roots of the
closed-loop system characteristic equation, the following results
hold 141,151
A(Q)
= A(s$)
= 0.
(5)
In order to reduce (5) into an equivalent and manageable form,
partition A and AA matrices as shown in (3) such that the size of
submatrix A, is 1 X 1 [4], [5]. Then it can be readily shown that
A(s,)=~s,Z-A,-A12(~dZ-A2)-1(A21+AA21)~=0.
Upon expansion and simplification
Manuscript received May 6, 1985.
The authors are with Ngee Ann Polytechnic, Electrical Engineering Department, Singapore 2159.
IEEE Log Number 8406166.
009%4094/86/0300-0350$01.00
of (6) yields
sd - cl = c,b,F,(s,) + c~b&(sct)
=(c,b,K,+c,b,K,)s,/(l+s,T)
01986 IEEE
(6)
(7)
351
IEEE TRANSACTIONSONCIRCUITSAND SYSTEMS,VOL.CAS-33,N0. 3,MARCH 1986
+o-
TIME
IN
SEC.
Fig. 1. Rotor angle response of study system with and without coordinated
stabilizers: Cl = open loop; C2 = with stabilizers, (5 = 0.5); C3 = with
stabilizers, (5 = 0.3).
TABLE I
A SAMPLEOFRESULTSOBTAINED
Stabilizer
Parameters
Desired
Eigenvalues
( x0:1
Closed-loop
)
KE = 0.0064
(-1.51
f j4.83,
-0.759,
-1.52
k j2.41,
(t=.3,f~J,=5.06)
KG = 0.0009
-5.14
t j6.08,
-10.0
, -12.7,
-16.0
-2.03
-1.51
t j4.83
KK = 0.0080
(-2.03
t j4.64,
-0.757,
-1.42
2 j2.47,
lk.4,W,.5.06)
KG = 0.0010
-4.44
t j6.44,
-10.0
, -12.7,
-16.7
-2.53
KE q 0.0095
l-2.53
? j4.38,
-0.752,
-1.26
t j2.47,
KG i 0.0011
-3.85
t j6.93,
-10.0
, -12.8,
-17.3
* j4.64
+ j4.38
(k.5,Wnz5.06)
where ( c1, c2, cs) are known constants [4], [5]. By equating the
real and imaginary parts of both sides of (7), two equations are
obtained from which (K,, K,) are readily computed for a given
T, and no interative scheme is required.
III.
NUMERICAL EXAMPLES
The effectiveness of coordinated excitation and governor
stabilizers with rotor acceleration as input signal for enhancing
the damping of turbogenerator has been demonstrated [2], [3].
Thus it would be interesting to examine the effectiveness of such
a coordinated scheme in a hydroelectric power system. To this
end, consider a hydro power system which consists of a singlemachine connected to an infinite-bus [l], [4].
For controllers design purposes, the system is represented by a
fifth-order synchronous machine model equipped with a firstorder exciter [4] and a third-order governor without a dashpot [l].
The system equations are expressed in the form of (1) where
x=
I Aw
AS
A#,
A$kd
A#,,
AErd
Ah
Ag
A~]‘:
The open-loop system eigenvalues are
(0.119+ j4.64, -0.805,
-8.47-t
System
Eigenvalues
j5.17,
-2.39+
j1.80,
-13.1,
-27.3).
Clearly, the system is unstable due to the‘ first pair of complex
, -26.8)
, -26.7)
, -26.6)
eigenvalues associated with rotor oscillations [l], [2], [4].
In order to improve the system stability, excitation and governor stabilizers of the form given by (2) with T set to 0.1 are
introduced and then coordinated using the above proposed
method. Several sets of (sd, ~2) or (5, w,) are considered and in
each case w, is kept constant [4], [5]. A sample of results
obtained is summarized as shown in Table I, and in all cases
considered exact eigenvalue assignment is always achieved.
The rotor angle responses of the system with and without a
coordinated excitation and governor stabilizers in service to small
disturbance are shown in Fig. 1. Clearly, excitation and governor
stabilizers coordinated using the proposed method are effective in
improving the hydroelectric power system damping of its rotor
oscillations.
The effects of different values of T of both stabilizers are next
examined using a performance index Z given by
Z = jmAwZ dt.
0
A sample of results obtained is shown in Fig. 2. Clearly, the
results obtained here are in good agreement with that of [7].
The robustness of stabilizers designed using the proposed
method is also investigated by examining the eigenvalues of the
overall power system over a wide range of operating conditions.
In this study, the power system is represented by a third-order
IEEE TRANSACTIONSON CIRCUITSAND SYSTEMS,VOL.CAS-33,N0. 3, MARCH 1986
352
Fig. 2. Performance index against time constant of stabilizers C2.
TABLE II
SYSTEMEIGENVALUESASSOCIATEDWITHROTOROSCILLATIONS
(**EXACTASSIGNMENTOFTHESEEIGENVALUES)
Heavy
open
loop
Closed-loop
load
Nominal
Light load
P = 0.952
P
=
a
i 0.45
a
= 0.015
P
= -0.225
Y t = 1.05
"t
= 1.05
"t
=
0.82
f j3.02
0.18
+ j4i61
-0.14
f j4.95
-1.05
t jl.81
-0.90
f j6.25
-0.86
t j6.07
0.500
1.05
l *
synchronous machine model equipped with a first-order exciter
and a 3rd-order governor without a dashpot. It is observed that in
order to guarantee an adequate amount of damping to the system
over a wide range of operating conditions, both the stabilizer
parameters should be selected using the system linearized equations corresponding to the heavy load condition. A sample of
results obtained for such a case is shown in Table II which shows
only the system eigenvalues associated with rotor oscillation. It
should be noted that all of the remaining eigenvalues of the
system not shown in Table II have negative real part.
IV.
load
P = 1.25
CONCLUSION
A unique method for coordinating the excitation and governor
stabilizers for stability enhancement of power systems has been
presented and demonstrated using a numerical example.
The method is easy to apply since no interative scheme is
required and moreover, it yields stabilizers which are simple in
structure and physically realizable.
REFERENCES
WI H. A. M. Moussa and Y. N. Yu, “Optimal power system stabilization
through excitation and/or governor control,” IEEE Trans. Power App.
Sysr., vol. PAS-91, pp. 1166-1174,1972.
121 S. I. Ahson, B. W. Hogg, and R. T. Pullman, “Integrated control system
for turbogenerator designed by Nyquist Array method,” IEEE Trans.
Power App. Sysr., vol. PAS-98, pp. 543-553, 1979.
[31 H. Lu, P. A. Hazel& and A. R. Daniels, “Co-ordinated single-variable
excitation control and governing of turboalternators,” Proc. Inst. Elect.
Eng., vol. 129, Pt. C, pp. 278-284, 1982.
[41 C. M. Lim and S. Elangovan, “Pole assignment of SISO system using
dynamic compensator with prespecificed poles,” IEEE Trans. Circuits
Syst., vol. CAS-31, pp. 990-991, Nov. 1984.
“Pole assignment of a class of multivariable system using localized
[51 -1
dynamic output compensators with prespecified poles,” IEEE Trans.
Circuits Sysr., accepted for publication.
161 Y. N. Yu and C. Siggers, “Stabilization and optimal control signals for a
power system,” IEEE Trans. Power App. Syst., vol. PAS-90, pp.
1469-1481,198O.
[71 V. H. Quintana, M. A. Zodhy, and J. H. Anderson, “Design of output
feedback controller for power system stabilization,” IEEE Trans. Power
App. Syst., vol. PAS-95, pp. 954-961, 1976.
A Two-Stage Pole-Zero Predictor
W. B. MIKHAEL, A. S. SPANIAS, G. KANG,
ANDL.F%%NSEN
Ahstruct -A two-stage pole-zero predictor is proposed which is capable
of predicting minimum phase Auto Regressive Moving Average (ARMA)
processes accurately with a reduced number of parameters. This is achieved
by cascading the classical predictor (1st stage), and the pole-zero recursive-like structure (2nd stage). The constraint on the order of the first stage
is relaxed since the unknown process is actually represented by the
parameters of the second stage. Computer simulations for both low- and
high-order ARMA processes are given to demonstrate the excellent performance of the proposed technique.
I.
INTRODUCTION
The classical linear prediction problem can be stated as follows: Let y(n) be a (scalar) discrete-time stationary zero-mean
process. We are interested in predicting the current value y(n),
of this process from past measurements, where n is the discrete
time index. A linear predictor of order N, will have the form
NP
F(n)= c PkYtn-k)
(1)
k=l
where j(n) is the predicted value of y(n) based on data up to
time (n - 1) and { pk, k = 1,2,. . . , NJ,} are the predictor coefficients. The difference between the actual value of the process and
Manuscript received July 23, 1985. This work was supported by the Naval
Research Laboratory under Grant SFRC 82 K 2047.
W. B. Mikhael and A. S. Span& are with the Department of Electrical
Engineering, West Virginia University, Morgantown, WV 26506.6101.
G. Kang and L. Fransen are with the Naval Research Laboratory, Waahington, DC 20375.
0098-4094/86/0300-0352$01.00
01986 IEEE