A Self-Organizing Power System Stabilizer using ... - Semantic Scholar

IEEE Transactions on Energy Conversion, Vol. 11, No. 2, June 1996

442

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

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.

Kwang Y. Lee , Senior MemberJEEE Electrical Engineering Department The Pennsylvania State University University Park, PA 16802, U.S.A.

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 AutoRegressive 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

0885-8969/96/$05.00 0 1996 IEEE

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 yrefcan 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:

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

l+(x-xl)/(b-a)

p A ,= l - ( x - x l ) / ( b - a )

if a l x < x , , if x1 5 x 5 b,

(4)

else

A FARMA rule is generated at each sampling step, and stored in a rule base. This means that every experience is regarded initially as a fuzzy logic control rule. As the run continues, the experience will be accumulated and the FARMA rule is updated for each domain in the nile space. B. Inference and Defuzzijkation

To attain the consequent linguistic value, it is necessary to determine a "truth value" of the input fuzzy set with respect to each rule [14],[15]. The consequent linguistic value, that is the net linguistic control action, C,, is deduced with the cpoperation [ 141 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),

where, C, : net linguistic control action, wi : truth value of the i-th rule, pci : membership degree of the consequent linguistic value Ci in the i-th rule. With the C,, we take the a-cut of the C, where a = max p(C,), to find a control range for the highest possibility. As a

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

(7)

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 yr(k+l) = ~ ( k+) a (Yref- ~ ( k ) ) ,

{

(u(k- I ) + q ) / 2 ( p + u(k- I ) ) / 2

for y,(k + I ) > 9(k + I ) , for y,(k+ I ) < y(k + 1).

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). a Yrei

(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) =

(n+m+l)-dimensional rule space, i.e., (xli, x ~ ...~, x, ( ~ + ~ + ~ ) ~ ) . To update the rule base, the following performance index is defined:

Inference

--I

I -

_.

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 i s connected. The"

(9)

where p and q are the respective lower and upper limits of the net control range (NCR) resulting from the inference.

Fuzzify

Vt

Z

R1+jX

Synchronous Generator

C. Se'elf-Organizationofthe 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

VO

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, X I = 1.49, R2=-0.102 X2 = 2.99, G = 0.249, B= 0.262 P,,= 1.0, Qeo= 0.015, V,,= 1.05

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]. , initial currents, For the calculation of constants - K ~ the 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]:

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:

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

iI

Fig. 4. The Self-organizing Power System Stabilizer

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 -1.

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.

2

--I3 timersec] 1 2 3 4 5 6 7 8 ' Fig. 5. Conventional PSS. (case 1: normal load)

0

[pu*O.OOl] 0. a

1

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. 1I and 12 show the speed deviations of the conventional PSS and the SOPSS, respectively. The inertia

-1

2 c 0

;

2

3

4

5

6

7

r T !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.

k

-l.

i

i

i

i

i

i

i

i

d

time[secl

Fig. 10. Self-organizing PSS. ( case 3: transmission loss) [pu*o.001] 0. 8

I

0. 0. 0.

Am -0. -0.

-1.

Fig. 11. Conventional PSS. ( case 4: modeling error) [pu*0.001]

-1.

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).

VII. REFERENCE

2

0.8

VI. ACKNOWLEDGMENT

I

2

Fig. 12. Self-organizing PSS. (case 4: modeling error)

[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. 316329, 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.

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[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. [ I l l 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, A SelfOrganizing Fuzzy Logic Controller for dynamic systems using a Fuzzy Auto-Regressive Moving Average(FARMA) Model", IEEE Trans. on Fuzzy Systems, vol. 3, no. 1, February, pp. 75-82, 1995. [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. It

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

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.