A Self-Organizing Power System Stabilizer using ... - IEEE Xplore

442 IEEE Transactions on Energy Conversion, Vol. 11, No. 2, June 1996
A Self-organizing Power System Stabilizer using Fuzzy Auto-Regressive
Moving Average (FARMA) Model
Young-Moon Park, Senior Member, IEEE
Un-Chul Moon, Student Member, IEEE
Electrical Engineering Department
Seoul National University
Seoul 15 1-742, Korea
Kwang Y. Lee , Senior MemberJEEE
Electrical Engineering Department
The Pennsylvania State University
University Park, PA 16802, U.S.A.
Abstract - This paper presents a self-organizing power system
stabilizer (SOPSS) which use the fuzzy Auto-Regressive Moving
Average (FARMA) model. The control rules and the
membership functions of proposed the fuzzy logic controller are
generated automatically without using any plant model. The
generated rules are stored in the fuzzy rule space and updated
on-line by a self-organizing procedure. To show the effectiveness
of the proposed controller, comparison with a conventional
controller for one-machine infinite-bus system is presented.
1. INTRODUCTION
The high complexity and nonlinearity of power systems
have created a great deal of challenge to power system
control engineers for decades. One of the most important
problems in power systems is the damping of low-frequency
oscillation. If no adequate damping is available, the
oscillations may be sustained for minutes and grow to cause
system separation [ 1],[2].
Many kinds of stabilizers have been proposed to improve
the stability of synchronous generators. The gain settings of
power system stabilizers (PSS) are usually fixed at a certain
set of values which are determined based on a nominal
operating point [ 11-[5]. These fixed gain controllers are
96 WM 037-2 EC A paper recommended and approved by the IEEE Energy
Development and Power Generation Committee of the IEEE Power
Engineering Society for presentation at the 1996 IEEE/PES Winter Meeting,
January 21-25, 1996, Baltimore, MD. Manuscript submitted July 19, 1995;
made available for printing January 4, 1996.
always a compromise between the best settings for light and
heavy load conditions. It is impossible for these fixed gain
stabilizers to maintain the best damping performance when
there is a drastic change in system operating condition, such
as that resulting from a three-phase fault in power system. In
order to overcome this difficulty, a self-tuning stabilizer,
where the gain settings are adjusted in real time to
automatically track the variations in the operating condition,
was employed [6],[7].
As an alternative to these controls, the concept of fuzzy
logic was introduced by Zadeh [8], and since its introduction
by Mamdani [9], the Fuzzy Logic Control (FLC) method has
been successfully applied to various control problems
[14],[15]. Hassan, Malik and Hope applied it to PSS design.
[lo]. In this method, the output stabilizing signal was
calculated based on the representation of the alternator state
in the phase plain. Ramaswamy, Edwards, and Lee proposed
an automatic tuning method for a FLC and applied it to
control a nuclear reactor [l 11. In this method, the rules were
parameterized as functions of fuzzy input variables and the
parameters were tuned off-line through experiments.
Recently, Hiyama, Kugimiya and Satoh proposed a PID type
fuzzy logic PSS [12]. They took into account the PID
information of the generator speed. Additional parameters
were also tuned off-line to minimize the performance index.
In this paper, the self-organizing fuzzy logic controller
(SOFLC) proposed in [13], which was named as Fuzzy Auto-
Regressive Moving Average (FARMA) controller, is
modified to enhance the low frequency damping of a
synchronous machine. In [13]1, we proposed a complete
design method for an on-line SOFLC without using any plant
model. In the conventional FLC, the rule base and
membership functions are supplied by an expert or tuned offline
through experiment^ However, the FARMA FLC needs
no expert in making control rules. Instead, rules are
generated using the history of input-output pairs, and new
inference and defuzzification methods are developed. The
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443
generated rules are stored in the fuzzy rule space and updated
on-line by a self-organizing procedure.
Characteristics of the FARh4A FLC algorithm are discussed
in this paper from the point of view of its applicability to
PSS. A modified form of FARMA FLC which is more
suitable for PSS is developed and presented.
11. BRIEF REVIEW OF FARMA FLC [ 131
A. DeJinition of the FARMA Rule
In general, the output of a system can be described with a
function or a mapping of the plant input-output history. For a
single-input single-output (SISO) discrete-time system, the
mapping can be written in the form of a nonlinear autoregresisive
moving average (NARMA) as follows:
where y(k) and u(k) are respectively the output and input
variables at the k-th time step.
The objective of the control problem is to find a control
input sequence which will drive the system to an arbitrary
reference set point yIeF Rearranging (1) for control purpose,
the value of the input U at the k-th step that is required to yield
the reference output yref can be written as follows:
which is viewed as an inverse mapping of (1).
The proposed controller doesn't use rules pre-constructed by
experts, but forms rules with input and output history at every
sampling step. The rules generated at every sampling step are
stored in a rule base, and updated as experience is
accumulated using a self-organizing procedure.
The system (1) yields the last output value y(k+l) when the
output and input values, y(k), y(k- l), y(k-2), ..*, u(k), u(k- I),
u(k-2), ..., are given. This implies that u(k) is the input to be
applied when the desired output is yref as indicated explicitly
in (2). Therefore, a FARMA rule with the input and output
history is defined as follows:
IFYrefis Ali, y(k) is A2i, y(k-1) is A3i, ..., y(k-n+l) is A(,+iy
AND u(k-I) is Bli, ~(k-2) is B2i ,..., u(k-m) is B,i,
THEN u(k) is Ci, (for the i-th rule) (3)
where, n, m : number of output and input variables
Aij, Bij : antecedent linguistic values for the i-th rule
Ci : consequent linguistic value for the i-th rule.
The rule (3) is generated at (k+l) time step. Therefore, y(k+l)
is given value at &+I) step. The rule (3) explaines that " If
desired yref is y(k+l) with given input-output history, y(k),
y(k-l), y(k-2), -., u(k), u(k-I), u(k-2), -.., THEN u(k) is the
input to be applied".
In a conventional FLC, an expert usually determines the
linguistic values Ai+ Bij, and Ci by partitioning each universe
of discourse, and the formulation of fuzzy logic control rules
is achieved on the basis of the expert's experience and
knowledge. In this paper, however, these linguistic values are
determined from the crisp values of the input and output
history at every sampling step. Therefore, at the initial stage,
the assigned u(k) may not be a good control, but over time,
the rule base is updated using the self-organizing procedure,
and better controls are applied.
A fuzzification procedure for fuzzy values of (3) is
developed to determine A,,, -., A(,+,),, Bli, B2i , ..., Bmi,
and Ci from the crisp y(k+ l), y(k), y(k-1), -a,
y(k-n+l), u(k-
I), u(k-2), ..., u(k-m), and u(k), respectively. The fuzzification
is done with its base on a reasonably assumed input or output
ranges. When the assumed input or output range is [a, b], the
membership function for crisp x1 is determined in a triangular
shape as follows:
l o else
l+(x-xl)/(b-a) if alx

444
result of this inference, the net control range (NCR) is
determined as a subset [p, q] of [a, b] with the constant
membership value a.
Defuzzification is a procedure to determine a crisp value
from a consequent fuzzy set (C,). In FARMA controller,
defuzzification is to determine a crisp value from the net
control range (NCR) resulting from the inference. The NCR is
modified to compute a crisp value by using the prediction or
"trend" of the output response. The series of the last outputs is
extrapolated in time domain to estimate y(k+l) by the Newton
backward-difference formula. If the extrapolation order is I,
the estimate 9 (k+l) is calculated as follows:
1
i=O
Defuzzification is performed by comparing the two values,
the estimate 9 (k+l) and the reference output yref or the
temporary target yr(k+l), generated by
(7)
yr(k+l) = ~ (k) + a (Yref- ~(k)), (8)
where a is the target ratio constant (0 < a 5 1). The value of a
describes the rate with which the present output y(k)
, approaches the reference output value, and thus has a positive
value between 0 and 1. 'The value of a is chosen by the user to
obtain a dep:rable response.
When the :stimate exceeds the reference output, the control
has to slow down. On the other hand, when the estimate has
not reached the reference, the control should speed up. To
modi@ the control range, the sign of Vu(k) (= u(k)-u(k-1)) is
assumed to be the same as the sign of (yr(k+l)-jj(k+l))
without the loss of generality.
Thus, for the case of yr(k+l) > 9 (k+l), hence the sign of
Vu(k), is positive, u(k) has to be increased from the previous
input u(k-1). On the other hand, when the sign of Vu(k) is
negative and u(k) has to be decreased from the previous input
u(k-1). The final crisp control value u(k) is then selected as
one of the mid-points of the modified net control ranges as
follows:
(u(k- I)+q) / 2 for y,(k + I)> 9(k +I),
u(k) =
(9)
{ (p + u(k- I))/ 2 for y,(k+ I) < y(k + 1).
where p and q are the respective lower and upper limits of the
net control range (NCR) resulting from the inference.
(n+m+l)-dimensional rule space, i.e., (xli, x ~ ... ~ , x , (~+~+~)~).
To update the rule base, the following performance index is
defined:
J = /Y,(k+l) - Y(k+l)/, (10)
where y(k+l) is the real plant output, and y,(k+l) is the
reference output. Therefore, at the (k+l)-th step, the
performance index J is calculated with the real plant output
y(k+l) resulting from the k-th step control.
The fuzzy rule space is partitioned into a finite number of
domains and only one rule, i.e., a point, is stored in each
domain. If there are two rules in a given domain, the selection
of a rule is based on J. That is, if there is a new rule which has
the output closer to the reference output in a given domain,
the old rule is replaced by the new one. The self-oiganization
of the rule base, in other words "learning" of the object
system, is performed at each sampling time, Fig. 1. In Fig. 1,
the reference model block represents (8).
Yrei
a
_. --
Inference
Fuzzify
Fig. 1. The FARMA control system architecture
111. SELF-ORGANIZING POWER SYSTEM STABILIZER
A. Power System Model
The purpose of this paper is to demonstrate the selforganizing
feature of the SOPSS when there is no
mathematical model, either on the generator nor on the
power system to which the machine is connected. The"
Synchronous
Generator
I-
I
Vt Z R1+jX VO
C. Se'elf-Organization ofthe Rule Base
The FARMA rule defined in Section A is generated at every
sampling time. Each rule can be represented as a point in the
Fig. 2. A 0n.e-machine, infinite-bus power system

B. Self-organizing Power System Stabilizer
-.a
Fig. 3. A linearized one-machine, infinite bus model
Table 1. Constants of one-machine, infinite-bus model
Generator
Constants
Exciter Constants
Line Constants
Initial Constants
M= 9.26, D= 0, T,,'= 7.76
X ~ Z 0.973, ~d'= 0.19, xq= 0.55
K,= 50, TA= 0.05
R1=-0.051, XI= 1.49, R2=-0.102
X2 = 2.99, G = 0.249, B= 0.262
P,,= 1.0, Qeo= 0.015, V,,= 1.05
It is noted the system has a non-minimum phase zero and a
zero at the origin. The FARMA FLC can be interpreted as a
kind of inverse model trainer by (2). However, it is well
known that the non-minimum phase zero plant can't be
controlled by pole-zero cancellation because of the internal
instability. In other words, the control value U, can diverge
in order to maintain the Am to zero. Moreover, the system has
a zero at the origin, which means U, is not unique in the'
steady state. It is recommended the steady state control value,
U,, of PSS be zero since it is a supplementary control.
Therefore, a modification of the FARMA FLC is necessary to
prevent the divergence and non-zero steady state value of U,.
To overcome these problems, we directly limit the control
value according to the output error as follows:
SOPSS will be learning the system from the input-output data
as it stabilizes the unknown system. Consequently, the
SOPSS will be adapive from system to system, from one
operating condition to another operating condition. For
illustration purpose, the system considered in this paper is a
synchronous generator connected to an infinite bus through
two transmission lines, Fig. 2. During low frequency
oscillations, the linearized model can be drawn as Fig. 3 [2].
For the calculation of constants - K ~ the , initial currents,
voltages and torque angle of the system in a steady state must
be known. These initial values are found from a load flow
study. Since the real system is nonlinear, the parameters
K~ - p(, are changed with the load and the system conditions.
However, for demonstration purpose, we select few operating
conditions for the linearized model.
Table 1 shows the values of system parameters. The
negative RI and R2 stem from deriving the one-machine,
infinite bus model for a multimachine system by
equivalencing smaller generators by equivalent impedances
with negative resistances. Without supplementary excitation,
the system is unstable and has a non-minimum phase zero
and a zero at the origin.
The supplementary control U, is applied through the
T,, and K~ blocks in Fig. 3 to obtain the extra damping
AT, in Fig. 3. Since it is a linearized model, a conventional
PSS as a phase lead compensation is included by the
superposition principle. If Am is the control input, the control
including the reset block becomes as follows [2]:
where, uyk) : modified control value,
K : feedback constant,
u(k) : control value of FARMA FLC.
In (12), yref is the reference output, i. e., zero for speed
deviation in this paper. Then, the modified control value is
decreased with the output error (yref - y(k)). Moreover, in
steady state it always becomes zero.
Fig. 4 shows the modification of the FARMA FLC for
PSS. The plant input U' is U, and the output y is Am as in the
conventional method. The plant output and input values,
y(k + 1). y(k), . .') y(k - 3), u'(k), . . .,u'(k - 3) are used to form the
FARMA rule. The target ratio constant a and feedback
constant K are chosen by off-line. The sampling time is 0.02
sec. The proposed FARMA PSS doesn't assume a plant
model, instead it learns the behavior of the plant by inputoutput
history. Therefore, it can cope with unexpected load
conditions and faults.
I
I 1
Fig. 4. The Self-organizing Power System Stabilizer
i
I

446
IV. SIMULATION RESULTS
After setting the parameters for the conventional PSS and
the SOPSS, we consider four disturbances in simulations.
According to each disturbance, the plant parameters are
changed. To compare the closed loop chracteristics of two
systems, it is assumed that small torque angle deviations ( ~6)
are suddenly applied at 0 sec. for each disturbance.
A. Case I: Normal Loud Condition
Figs. 5 and 6 show the speed deviations of the conventional
PSS and the SOPSS, respectively. It is assumped that initial
torque angle deviations(A6) at 0 sec. are 0.06, 0.08 and 0.1
[radian] for a, b and c, respectively. The rising times are
similar, but the speed deviations of the SOPSS show smaller
overshoots and settling times than those of the conventional
method.
B. Case 2: Heavy Loud Condition
In case B, we consider different operating condition, i.e.,
the real power P is increased to 1.3 from 1. Figs. 7 and 8
show the speed deviations of the conventional PSS and the
SOPSS, respectively. The initial torque angle deviations ( ~ ) 6
are 0.06, 0.08 and 0.1 for a, b and c, respectively. Because
the conventional controller is designed for the normal
operating condition, the overshoots and the settling times are
increased somewhat than those in case A. On the contrary,
the undershoots and overshoots, in the case of SOPSS, are
improved from case A. This is because the SOPSS doesn't
assume any operating condition; instead, it constructs the rule
base of the system by on-line adaptation.
C. Case 3: isolation of a Transmission Line
In this case, transmission line 2 in Fig. 2 is isolated with
normal load condition. The isolation of line 2 may result
from three-phase fault or three-phase to ground fault, etc. To
simulate this case, the 2 2 is removed, and Y is changed in
Fig. 2 accordingly. Figs. 9 and 10 show the speed deviations
of the conventional PSS and the SOPSS, respectively. The
initial torque angle deviations are 0.06, 0.08 and 0.1 for a, b
and c, respectively. The settling times of the conventional
PSS are increased to almost 7 sec. On the other hand, the
SOPSS shows no significant difference from cases A and B.
D. Case 4: Different Inertia Constanl
Here, we consider different inertia constant for the
synchronous generator with normal load condition. The
purpose of this case is to consider the modeling error of the
synchronous generator. That is, the real values can be either
7, 9.26 or 12, respectively, while the assumed value is 9.26 in
the model. Figs. 1 I and 12 show the speed deviations of the
conventional PSS and the SOPSS, respectively. The inertia
-1. 2
--I timersec]
0 1 2 3 4 5 6 7 8 ' 3
Fig. 5. Conventional PSS. (case 1: normal load)
[pu*O.OOl]
0. a
1
-1 2c 0 ; 2 3 4 5 6 7 rT! timeIsec1
Fig. 8. Self-organizing PSS. (case 2: heavy load)

441
[pu*0.001]
constants are 7, 9.26 and 12 for a, b and c, respectively. The
initial torque angle deviation(A6) is 0.1. When the real
inertia constant is 12, the setling time of the conventional
PSS is increased to 6 sec (c of Fig. 11). On the other hand,
the SOPSS shows no significant difference.
V. CONCLUSIONS
Development of a self-organizing a power system stabilizer
(SOPSS) was described in this paper. The SOPSS doesn't use
any plant model or pre-constructed rule base of an expert.
Instead, the control rules are generated automatically with the
input-output history, and the rule base is updated on-line.
Simulations considered normal and heavy loads, isolation of
a transmission line, and different inertia constants for the
synchronous generator. Compared with the conventional
PSS, the SOPSS showed better performance. Especially, the
SOPSS maintained good performance for different operating
conditions, indicating adaptation and robustness properties.
However, simulation shows a similar stability margin for
both conventional PSS and SOPSS. Therefore, this behavior
has to be studied more.
-l. k i i i i i i i i d time[secl
Fig. 10. Self-organizing PSS. ( case 3: transmission loss)
VI. ACKNOWLEDGMENT
[pu*o.001]
0. 8
0.
0.
0.
Am
-0.
-0.
I
The work is supported in parts by Korea Science and
Engineering Foundation (KOSEF) and the National Science'
Foundation (NSF) under grants "U.S.-Korea Cooperative
Research on Intelligent Distributed Control of Power Plants
and Power Systems" (INT-9223030), and "Research and
Curriculum Development for Power Plant Intelligent
Distributed Control" (EID-92 1232).
-1. 2
[pu*0.001]
0.8
-1. 2
Fig. 1 1. Conventional PSS. ( case 4: modeling error)
Fig. 12. Self-organizing PSS. (case 4: modeling error)
I
VII. REFERENCE
[l] P. M. Anderson and A. A. Fouad, Power System Control
and Stability, IEEE Press, 1994.
[2] Y. N. Yu, Electric Power System Dynamics, Academic
Press, 1983.
[3] F. P. de Mello and C. A. Concordia, "Concepts of
Synchronous Machine Stability as Affected by Excitation
Control", IEEE Trans. on PAS, vol. 88, no. 4, pp. 316-
329, April 1969.
[4] R. J. Fleming, M. A. Mohan and K. Parvatisam,
"Selection of Parameters of Stabilizers in Multimachine
Power Systems", IEEE Trans. on PAS, vol. 100, no. 5,
pp. 2329-2333, May 1981.
[5] T. L. Huang, T. Y. Hwang and T. Yang, "Two-Level.
Optimal Output Feedback Stabilizer Design", IEEE
Trans. on PWRS, vol. 6, no. 3, August 1991.

448
[6] A. Ghosh, G. Ledwich, 0. P. Malik and G. S. Hope,
"Power System Stabilizer based on Adaptive Control
Technique", IEEE Trans. on PAS, vol. 103, no. 8, pp.
1983- 1989, August 1984.
[7] C. J. We and Y. Y. Hsu, "Design of Self-Tuning PID
Power System Stabilizer for Multimachine Power
System", IEEE Trans. on PWRS, vol. 3, no. 3, August
1988.
[8] L. A. Zadeh, "Fuzzy Sets", Inform. Contr., vol. 8, pp.
338-353, 1965.
[9] E. H. Mamdani and S. Assilian, "An experiment in
linguistic synthesis with a fuzzy logic controller", Int. J.
Man Mach. Studies, vol. 7, no. 1, pp. 1- 13. 1975.
[ 101 M. A. M. Hassan, 0. P. Malik and G. S. Hope, "A Fuzzy
Logic Based Stabilizer for a Synchronous Machine",
IEEE Trans. on Energy Conversion, vol. 6, no. 3, pp.
407-413, 1991.
[Ill P. Ramaswamy, R. M. Edwards, and K. Y. Lee, "An
automatic tuning method of a fuzzy logic controller for
nuclear reactors", IEEE Trans. Nuclear Science, vol. 40,
no. 4, pp. 1253-1262, August 1993.
[ 121 T. Hiyama, M. Kugimiya and H. Satoh, "Advanced PID
type Fuzzy Logic Power System Stabilizer", IEEE Trans.
on Energy Conversion, vol. 9, no. 3, pp. 514-520, 1994.
[13] Y. M. Park, U. C. Moon and K. Y. Lee, It A Self-
Organizing Fuzzy Logic Controller for dynamic systems
using a Fuzzy Auto-Regressive Moving
Average(FARMA) Model", IEEE Trans. on Fuzzy
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[14] W. Pedrycz, Fuzzy Control and Fuzzy Systems, John
Wiley & Sons, 1989.
[15] B. Kosco, Neural Network and Fuzzy Systems,
Englewood Cliffs, Prentice-Hall, 1992.
interests include fuzzy logic systems, artificial neural
networks, and their applications to control, operation, and
planning of power systems.
Kwang Y. Lee received the B.S. degree in electrical
engineering from Seoul National University, Seoul, Korea, in
1964, the M.S. degree in electrical engineering from North
Dakota State University, Fargo, ND, in 1967 and the Ph.D.
degree in system science from Michigan State University,
East Lansing, MI, in 1971. He has been on tne faculties of
Michigan State, Oregon State, University of Houston, and the
Pennsylvania State University, University Park, PA, where
he is a Professor of Electrical Engineering. His current
research interests include control theory, artificial neural
networks, fuzzy logic systems, and computational inteligence'
and their applications to power systems. Dr. Lee is a senior
member of IEEE.
VIII. BIOGRAPHY
Young-Moon Park was born in Masan, Korea on Aug. 20,
1933. He received his B.S., M.S., and Ph.D. in electrical
engineering from Seoul National University in 1956, 1959
and 197 1, respectively in electrical engineering. His major
research field includes power system operation and control,
and artificial intelligence applications to power systems.
Since 1959, he has been a faculty of Seoul National
University where he is currently a Professor of Eletrical
Engineering. He is also serving as the president of the
Electrical Engineering and Science Research Institute. Dr.
Park is a senior member of IEEE.
Un-Chul Moon received the B.S. and M.S. degrees in
electrical engineering from Seoul National University, Seoul,
Korea, in 1991 and 1994, respectively. He is currently a
Ph.D. candidate in Power System Laboratory, Electrical
Engineering, Seoul National University. His current research