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An indirect adaptive fuzzy-logic power system stabiliser
Electrical Power and Energy Systems 24 (2002) 837±842
www.elsevier.com/locate/ijepes
An indirect adaptive fuzzy-logic power system stabiliser N. Hossein-Zadeh a,*, A. Kalam b a
School of Engineering and Science, Monash University, Malaysia, P.O. Box 8975, 46780 Kelana Jaya, Selangor Daral Ehsan, Malaysia b Department of Electrical and Electronic Engineering, Victoria University, P.O. Box 14428, MCMC, Melbourne, Vic. 8001, Australia Received 2 April 2001; revised 1 October 2001; accepted 1 November 2001
Abstract An indirect adaptive fuzzy power system stabiliser (AFPSS) is developed using the concept of fuzzy basis functions. The power system is modelled using differential equations with nonlinear parameters which are functions of the state of the system. These nonlinear functions may not be known, however, some linguistic information is available about them. Utilising this information, fuzzy logic systems are designed to model the system behaviour. The control law is obtained using the uncertainty principle. Based on the Lyapunov's synthesis method, adaptation rules are developed to make the controller adaptive to changes in operating conditions of the power system. The simulation studies are carried out for an industrial cogenerator and utilise a one-machine in®nite-bus model. Nonlinear simulations reveal that the performance of AFPSS is better than the performance of a conventional (linear) power system stabiliser for a wide range of operating conditions. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Power system stabiliser; Adaptive fuzzy-logic power system stabiliser; Fuzzy basis function
1. Introduction One of the successful applications of fuzzy sets and systems theory to practical problems is fuzzy control. The present interest in fuzzy theory is largely due to the successful applications of fuzzy logic controllers (FLCs) to a variety of consumer products and industrial systems [1]. FLCs are very useful when an exact mathematical model of the plant is not available, however, experienced human operators are available for providing qualitative rules to control the system. Fuzzy control has been applied to the design of power system stabilisers (PSS) in a number of publications [2±7]. In all these papers, after the design is completed, the parameters of the fuzzy power system stabiliser (FPSS) are kept ®xed. This is in spite of the fact that the performance of the FPSS depends on the operating conditions of the power system, although it is less sensitive than conventional linear power system stabilisers (CPSS) [8]. * Corresponding author. E-mail addresses: [email protected] (N. HosseinZadeh), [email protected] (A. Kalam). Abbreviations: PSS, power system stabiliser; CPSS, conventional power system stabiliser; FPSS, fuzzy power system stabiliser; AFPSS, adaptive fuzzy power system stabiliser; IFPSS, initial fuzzy power system stabiliser; FBF, fuzzy basis function; FLC, fuzzy logic controller; FLS, fuzzy logic system
Wang and Mendel [9] introduced fuzzy basis functions (FBFs) which have the capability of combining both numerical data and linguistic information. These FBFs are quite general. Their exact mathematical structure depends on four choices that one must consider in order to design any fuzzy logic system, namely, type of fuzzi®cation, membership function, inference mechanism, and defuzzi®cation strategy. Further, Wang used the concept of FBFs to introduce stable adaptive fuzzy control of nonlinear systems [10]. His initial adaptive fuzzy controller is constructed from the fuzzy IF±THEN rules provided by human experts and some arbitrary rules. An adaptive law is then used to update the parameters of the adaptive fuzzy controller during the adaptation procedure. If the fuzzy IF±THEN rules from human experts provide good control strategies, then the adaptation procedure will converge very quickly. On the other hand, if there are no linguistic rules from human experts, then his adaptive fuzzy controller becomes a regular nonlinear adaptive controller, similar to the radial basis function adaptive controller [12]. He applied the adaptive fuzzy controller to an unstable ®rst-order nonlinear system and a second-order chaotic system [10]. He also applied a stable indirect adaptive fuzzy controller to an inverted pendulum tracking problem [11]. Following a similar approach, an indirect adaptive fuzzy power system stabiliser (AFPSS) is designed in this paper.
0142-0615/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0142-061 5(01)00093-X
838
N. Hossein-Zadeh, A. Kalam / Electrical Power and Energy Systems 24 (2002) 837±842
The power system under study is an industrial cogenerator plant. Some of its parameters are not known and others are not exact. Therefore, it is not possible to build a relatively exact mathematical model of the system. In order to design the AFPSS, ®rstly two fuzzy logic systems (FLSs) are designed which are the approximations for the unknown nonlinear functions present in the model of the power system. Then these FLSs are used to obtain the control law whose role is to stabilise the power system. The FLSs are adapted based on the Lyapunov's synthesis method. The adaptation law is such that the difference between the output of the plant and a desired track will be minimised. The AFPSS is applied to the power system. A onemachine in®nite-bus system is used for simulation studies, because the cogenerator plant is considered a small generating plant relative to the connected power network. The nonlinear simulation reveals that the adaptation procedure prevents large degradation of system performance for a wide range of operating conditions. This makes the AFPSS superior to a CPSS or a ®xed-parameter FPSS.
3. Fuzzy basis functions The basic con®guration of a fuzzy logic system is shown in Fig. 1 [14]. A FLS maps crisp inputs into crisp outputs. It contains four components: rules base, fuzzi®er, inference engine, and defuzzi®er. Once the rules have been established, the FLS can be viewed as a nonlinear mapping from crisp inputs to crisp outputs, and this mapping can be expressed mathematically as y f x: From Fig. 1, it is observed that such a mathematical relationship can be obtained [9] by following the signal x through the fuzzi®er, where it is converted to a fuzzy set Ax, into the inference block, where it is converted to a fuzzy set Av, and ®nally into the defuzzi®er, where it is converted to f x: In order to write such a relationship, speci®c choices should be made for fuzzi®er, membership functions, composition, inference, and defuzzi®er. For singleton fuzzi®cation, max-product composition, product inference, and height defuzzi®cation, leaving the choice of membership function open, it is easy to show that [15] N X
2. Direct and indirect adaptive fuzzy control In the adaptive control literature, adaptive controllers fall into two categories [13], direct and indirect. In direct adaptive control, the parameters of the controller are directly adjusted to reduce some norm of the output error between the plant and the reference model. In indirect adaptive control, the parameters of the plant are estimated and the controller is chosen assuming that the estimated parameters represent the true values of the plant parameters. In fuzzy control, linguistic information from human experts can be placed into two categories [10]: ² Fuzzy control rules which set forth the situations in which certain control actions should be taken (e.g. if the speed-deviation is negative medium and the acceleration is negative medium then the PSS output should be negative big). ² Fuzzy IF±THEN rules, which describe the behaviour of the unknown plant (e.g. if the control signal applied to the synchronous machine is negative, then the acceleration of the shaft will change in a positive direction). Adaptive fuzzy controllers which make use of these two classes of linguistic information correspond to the direct and indirect adaptive control schemes, respectively. More speci®cally, in the direct scheme, linguistic fuzzy control rules can be directly used to implement the adaptive fuzzy controller, whereas in the indirect scheme, linguistic rules are used to model the plant. The controller is then constructed assuming that the fuzzy logic system approximately represents the plant.
y f x
uk
k1 N X
L Y
i1 L Y
k1 i1
mFik xi
1
mFik xi
where N is the number of rules in the FLS and L is the number of inputs to the FLS. uk is the centre of gravity of the membership function of the output corresponding to the kth rule. Eq. (1) can be expressed as y f x; u
N X k1
uk j k x uT j x
2
where u u 1 ; ¼; uN T and j x j 1 x; ¼; j N xT : j k x are called fuzzy basis functions (FBFs) [9] and are given by L Y
j k x
mFik xi i1 N Y L X l1 i1
3
mFil xi
The FLS can be referred to as a fuzzy basis function expansion.
Fig. 1. Block diagram of a fuzzy logic system (FLS).
N. Hossein-Zadeh, A. Kalam / Electrical Power and Energy Systems 24 (2002) 837±842
4. Indirect adaptive fuzzy PSS In this section, the procedure for designing an indirect AFPSS for a synchronous machine is explained. Let x1 Dv speed deviation and x2 DP Pm 2 Pe accelerating power. It is possible to represent the machine with the following nonlinear equations x_1 ax2 ;
ax_2 g x1 ; x2 1 h x1 ; x2 u;
y x1 4
839
^ uh and g x; ^ ug are the FLSs with parameter where h x; vectors uh and ug ; respectively. ^ uh uc to both sides of Eq. (4) and using By adding h x; Eq. (9), the following equation is derived: ^ uh uc ^ ug 1 h x 2 h x; e 2kT e 1 g x 2 g x;
10
T
_ e ; Eq. (10) can be written as Since e_ e; ^ uh uc } 11 ^ ug 1 h x 2 h x; e_ Ac e 1 bc {g x 2 g x; where "
where a 1=2H and H is a constant parameter of the machine called per unit inertia constant. x x1 ; x2 T [ R2 is the state vector of the system and can be measured. g(´) and h(´) are nonlinear functions which are assumed to be unknown. u is the controlling signal which is the output of the PSS to be designed. Eq. (4) represents the machine during a transient period after a major disturbance has occurred in the system. It has been assumed that the two nonlinear functions g(´) and h(´) can be found such that
Ac is a stable matrix, because usI 2 Ac u s2 1 k1 s 1 k2 has its roots in the open left plane. Therefore, there exists a unique symmetric positive de®nite 2 £ 2 matrix P which satis®es the Lyapunov equation [16]
P_ e 22Hg x1 ; x2 1 h x1 ; x2 u
ATc P 1 PAc 2Q
5
This equation is based on the fact that the governor time constant is large compared to the time constants of the synchronous machine and its exciter, so that during the ®rst few seconds after the occurrence of a severe disturbance the governor function can be ignored. Therefore the mechanical input power is constant during the transient interval, say less than 5 seconds after the disturbance has occurred. Simulation studies show that a positive u will cause a positive change in Pe, i.e. P_ e . 0 whenever u . 0: This means that h may be selected to be a negative function, i.e. h x1 ; x2 , 0;
for all x1 ; x2
6
The control objective is to force y to follow a given desired signal ym. In the rest of this section the procedure to construct an indirect adaptive fuzzy controller [11] to achieve the above control objectives is discussed. _ T and k k2 ; k1 T [ R2 be such that the Let e e; e two roots of the polynomial h s s2 1 k1 s 1 k2 are in the open left half-plane. If the functions g and h are known, then the control law i 1 h up 2g x 1 ym 2 kT e 7 h x applied to Eq. (4), using u p instead of u, results in e 1 k1 e_ 1 k2 e 0
8
which implies that limt!1 e t 0: This is the main objective of control. However, g and h are not known. The objective is to design two FLSs close enough to g x and h x; then use them in Eq. (7). Therefore, the control law will be i 1 h ^ ug 1 ym 2 kT e uc 2g x; 9 ^ uh h x;
Ac
0
1
2k2
2k1
# ;
bc
" # 0
12
1
13
where Q is an arbitrary 2 £ 2 positive de®nite matrix. The next task is to replace g^ and h^ by FLSs represented by Eq. (2) and to develop an adaptation law to adjust the parameters in the FLSs in order to force the tracking error to converge to zero. Using the procedure suggested in ^ uh and Ref. [11], the parameter vectors of the FLSs h x; ^ ug will be adapted according to the following rules. g x; To make the parameter vectors bounded, the constraint sets V g and V h are de®ned as
V g {ug : iug i # Mg }
14
V h {uh : iuh i # Mh ; uhl # 21}
15
where Mg, Mh and 1 are positive constants speci®ed by the designer. If the parameter vectors are within the constraint sets or on the boundaries of the constraint sets but moving toward the inside of the constraint set, then the simple adaptation laws
u_ g g1 eT Pbc j x
16
u_ h g2 eT Pbc j xu c
17
will be used, where g 1 and g 2 are positive constants which will be used as the learning rate in the adaptation procedure. Otherwise, if the parameter vectors are on the boundary of the constraint sets and moving toward the outside of the constraint sets, the projection algorithm will be used to modify the adaptation law as
u_ g g1 eT Pbc j x 2 g 1 eT Pbc
ug uTg j x
18
uu g u2
u_ h g2 eT Pbc j xu c 2 g2 eT Pbc uc
uh uTh j x uu h u
2
19
840
N. Hossein-Zadeh, A. Kalam / Electrical Power and Energy Systems 24 (2002) 837±842
5. Design procedure 5.1. Step 1: off-line preprocessing 2 (1.1) Specify k1 1; k2 4 such p that the roots of s 1 kp 1 s1 k2 0 are s1 20:5 2 j 3=2 and s2 20:5 1 j 3=2: Other values for k1 and k2 are possible as long as roots s1 and s2 are in a position in the left half-plane with good damping characteristics. (1.2) Specify a diagonal positive de®nite 2 £ 2 matrix Q as " # 1 0 Q 0 1
There is a trade off between elements of Q and the learning rates g 1 and g 2. For bigger elements of Q, a smaller value should be chosen for learning rates g 1 and g 2. (1.3) Solve the Lyapunov Eq. (13) to obtain a symmetric positive de®nite matrix P. (1.4) Specify the design parameters g 1, g 2, Mg, Mh, and 1 based on the practical constraints. These values have been chosen to be g1 2; g2 20; Mg 3, Mh 100; and 1 0:5: There is a compromise in choosing these parameters. Small learning rates will result in a less adaptable system, which is more stable, compared with a system with higher learning rates. Mg and Mh are the upper limits on iuf i and iug i: The lower limit on the absolute value of the elements of the parameter vector ug is determined by 1 . 5.2. Step 2: initial fuzzy PSS construction (2.1) De®ne m1 fuzzy sets F1k1 for input Dv where k1 1; 2; ¼; m1 and m2 fuzzy sets F2k2 for input DP where k2 1; 2; ¼; m2 : Here m1 and m2 are chosen to be m1 m2 5: More fuzzy sets will result in a more smooth response. However, this will make the controller more complicated in terms of computational requirements. Five fuzzy sets for each input are suf®cient for the PSS to be designed. The fuzzy sets for input Dv are de®ned according to the membership functions shown in Fig. 2 and the fuzzy sets for input DP are de®ned according to the membership functions shown in Fig. 3. For the shape of membership functions, Gaussian functions have been chosen to make the calculations easier. The membership functions labelled ZR for input x2 has been chosen to be narrower than other labels. This will cause the oscillations in speed deviation response to damp faster when the speed deviation has reduced to small values around zero with a small acceleration. (2.2) Construct the fuzzy basis functions from the input membership functions as
j k 1 ;k2 x
mFk1 x1 mF k2 x2 1
m1 X m2 X
k1 1 k2 1
2
mF k1 x1 mFk2 x2 1
2
Fig. 2. Fuzzy membership functions for input x1.
and collect them into an m1 £ m2 dimensional vector j x in a natural ordering for k 1 1; 2; ¼; m1 and k2 1; 2; ¼; m2 : ^ ug (2.3) Construct the fuzzy rule base for the FLSs g x; ^ uh which consist of m1 £ m2 rules whose IF parts and h x; comprise of all the possible combinations of the fuzzy input sets. Speci®cally, the fuzzy rule base consists of rules ^ uh is x k1 ;k2 Rh k1 ;k2 : IF x1 is F1k1 and x2 is F2k2 ; THEN h x; 21 where x k1 ;k2 is the centre of gravity of the fuzzy output set. Construct vector uh as a collection of the values of x k1 ;k2 in the same ordering as j x: For example, for the speci®c synchronous machine under study, the fuzzy rule base and the parameter vector u h can be constructed from Table 1 (the elements of uh are in per unit and chosen to be negative as suggested in Eq. (6)). As an example, the eighth rule is Rh 2;3 : IF Dv is Negative Small and DP is Zero, ^ uh should be close to 24. From this rule, it is THEN h x; seen that the eighth element of the parameter vector is uh 8 u 2;3 24: In this way, the 25-element initial h
20 Fig. 3. Fuzzy membership functions for input x2.
N. Hossein-Zadeh, A. Kalam / Electrical Power and Energy Systems 24 (2002) 837±842
841
Table 1 ^ uh Decision table to construct the parameter vector uh for estimating h x; Speed deviation (Dv )
Accelerating power (DP) NB
NS
ZR
PS
PB
NB NS ZR PS PB
21 22 23 22 21
23 23 24 23 22
23 24 25 24 23
22 23 24 23 22
21 22 23 22 21
parameter vector uh will be obtained. Since ^ ug ; the initial enough information about g x; is chosen to be zero. ^ ug and (2.4) From (2.2) and (2.3) g x; obtained as
there is no value of ug ^ uh are h x;
^ ug uTg j x g x;
22
^ u h uTh j x h x;
23
Then the controlling signal will be obtained from Eq. (9). 5.3. Step 3: on-line adaptation (3.1) Apply the controlling signal to the plant. (3.2) Use the adaptive laws explained in Section 4 to adjust the parameter vectors u g and uh : 6. Simulation results The performance of the indirect AFPSS was evaluated by applying a large disturbance caused by a three-phase fault to ground on the transmission line. A power system model consisting of a synchronous machine connected to a constant voltage bus through a pair of three phase transmission lines is used in simulation studies. A simpli®ed schematic diagram of the model is shown in Fig. 4. A three-phase transformer is used between the synchronous machine and the transmission lines to boost the machine voltage. For comparison purposes, a CPSS was designed and its parameters were adjusted in the frequency domain [8]. The transfer function of the CPSS is K0 s t1 s 1 1t3 s 1 1 Gp s 24 t0 s 1 1 t2 s 1 1t4 s 1 1
Fig. 5. System speed deviation responses for Case 1.
fuzzy power system stabiliser before adaptation (FPSS) and indirect AFPSS, the system responses for three different conditions were obtained using nonlinear simulations. (The generator is represented by a set of ®rst-order differential equations as suggested in Ref. [17], chapter 13. The relationship between variables in the differential and algebraic equations is nonlinear and the system is not linearised in the simulation studies). Case 1: normal operating conditions. Operating point P0 0.9 pu, Q0 0.3 pu and Xe 0.2 pu. System responses to a fault, occurring at t 0:2 s with a duration of 0.06 s, are shown in Fig. 5. System responses with the initial FPSS is the worst. The reason is that there was not enough information about g x: If suitable information can be obtained either from an expert or through simulation, the FPSS can be improved. As is observed from the ®gure, the AFPSS has introduced more damping and the
The parameters of the CPSS were designed for the operating conditions P0 0:9 pu; Q0 0:3 pu and Xe 0:2 pu; where Xe is the equivalent reactance of the transmission line. For the plant with the three types of PSSs, i.e. CPSS,
Fig. 4. Power system con®guration for simulation studies.
Fig. 6. System speed deviation responses for Case 2.
842
N. Hossein-Zadeh, A. Kalam / Electrical Power and Energy Systems 24 (2002) 837±842
a simple ®xed-parameter FPSS. In a simple case, like the one-machine in®nite-bus model used, it may seem that it is not worthwhile to apply this algorithm. However, if a more complicated arrangement of a multimachine system is considered or the duration of the fault is longer than 0.06 s which was used in the simulation studies, then the improvement caused by the suggested scheme will be more apparent. The scheme is not suggested for all power systems, but it is recommended for the systems in which stability is of prime importance and the machines may be subjected to more severe faults.
References
Fig. 7. System responses for Case 3.
system response has less overshoot and settling time comparing to a CPSS. Case 2: heavy reactive load and weak connection. Operating point P0 0.9 pu, Q0 0.8 pu and Xe 0.45 pu. The system responses are shown in Fig. 6. The system response is more oscillatory in this case. Nevertheless, as in the previous case, the AFPSS has the best response. Case 3: importing reactive power and strong connection. Operating point P0 0:9 pu; Q0 20:3 pu and Xe 0:1 pu: In this case, the synchronous machine is absorbing reactive power and it can become unstable more easily. The system responses are shown in Fig. 7. As is observed from the ®gure, the system response for the AFPSS is optimum for this case also. 7. Conclusions An indirect adaptive fuzzy power system stabiliser was designed based on the approximation to the unknown nonlinear functions present in a model for a synchronous machine. The fuzzy logic systems were constructed using the concept of fuzzy basis functions. An adaptation algorithm was derived based on the Lyapunov's direct method to cause the system follow a desired response. The effectiveness of this scheme was investigated through nonlinear simulations. It was shown that the indirect adaptive fuzzy power system stabiliser designed in this way has the best response in a wide range of operating conditions, compared to the initial fuzzy power system stabiliser and a linear conventional power system stabiliser. Of course the proposed scheme is more complicated than
[1] Sugeno M. Industrial applications of fuzzy control. Amsterdam: North Holland, 1985. [2] Hsu YY, Cheng CH. Design of fuzzy power system stabilisers for multimachine power systems. IEE Proc, Part C 1990;137(3):233±8. [3] Hassan MAM, Malik OP, Hope GS. A fuzzy logic based stabiliser for a synchronous machine. IEEE Trans Energy Conversion 1991;6(3):407±13. [4] Hiyama T. Real time control of micro-machine system using microcomputer based fuzzy logic power system stabiliser. IEEE Trans Energy Conversion 1994;9(4):724±31. [5] El-Metwally KA, Malik OP. Fuzzy logic power system stabiliser. IEE Proc Generation, Transmission Distribution 1995;142(3):277±81. [6] Parniani M, Lesani H. Application of power system stabiliser at Bandar-Abbas power station. IEEE Trans Power Syst 1994;9(3):1366±70. [7] El-Metwally KA, Hancock GC, Malik OP. Implementation of a fuzzy logic PSS using a microcontroller and experimental test results. IEEE Trans Energy Conversion 1996;11(1):91±6. [8] Hosseinzadeh N, Kalam A, Lee WS. A fuzzy logic based power system stabiliser for a synchronous generator, EECON'95Ð Electrical Energy Conference, Adelaide, Australia; September 1995. p. 176±81. [9] Wang LX, Mendel JM. Fuzzy basis functions, universal approximation, and orthogonal least squares learning. IEEE Trans Neural Networks 1992;3(5):807±14. [10] Wang LX. Stable adaptive fuzzy control of nonlinear systems. IEEE Trans Fuzzy Syst 1993;1(2):146±55. [11] Wang LX. Stable adaptive fuzzy controllers with application to inverted pendulum tracking. IEEE Trans Syst, Man Cybern, Part B 1996;26(5):677±91. [12] Sanner RM, Slotine JE. Gaussian networks for direct adaptive control. Proceedings of the American Control Conference; 1991. p. 2153±9. [13] Narendra KS, Parthasarathy K. Identi®cation and control of dynamical systems using neural networks. IEEE Trans Neural Networks 1990;1(1):4±27. [14] Lee CC. Fuzzy logic in control systems: fuzzy logic controller, part I. IEEE Trans Syst, Man, Cybern 1990;20(2):404±18. [15] Mendel JM. Fuzzy logic systems for engineering: a tutorial. Proc IEEE 1995;83(3):345±77. [16] Slotine JE, Li W. Applied nonlinear control. Englewood Cliffs, NJ: Prentice-Hall, 1991. [17] Kundur P. Power system stability and control. New York: McGrawHill, 1994.
www.elsevier.com/locate/ijepes
An indirect adaptive fuzzy-logic power system stabiliser N. Hossein-Zadeh a,*, A. Kalam b a
School of Engineering and Science, Monash University, Malaysia, P.O. Box 8975, 46780 Kelana Jaya, Selangor Daral Ehsan, Malaysia b Department of Electrical and Electronic Engineering, Victoria University, P.O. Box 14428, MCMC, Melbourne, Vic. 8001, Australia Received 2 April 2001; revised 1 October 2001; accepted 1 November 2001
Abstract An indirect adaptive fuzzy power system stabiliser (AFPSS) is developed using the concept of fuzzy basis functions. The power system is modelled using differential equations with nonlinear parameters which are functions of the state of the system. These nonlinear functions may not be known, however, some linguistic information is available about them. Utilising this information, fuzzy logic systems are designed to model the system behaviour. The control law is obtained using the uncertainty principle. Based on the Lyapunov's synthesis method, adaptation rules are developed to make the controller adaptive to changes in operating conditions of the power system. The simulation studies are carried out for an industrial cogenerator and utilise a one-machine in®nite-bus model. Nonlinear simulations reveal that the performance of AFPSS is better than the performance of a conventional (linear) power system stabiliser for a wide range of operating conditions. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Power system stabiliser; Adaptive fuzzy-logic power system stabiliser; Fuzzy basis function
1. Introduction One of the successful applications of fuzzy sets and systems theory to practical problems is fuzzy control. The present interest in fuzzy theory is largely due to the successful applications of fuzzy logic controllers (FLCs) to a variety of consumer products and industrial systems [1]. FLCs are very useful when an exact mathematical model of the plant is not available, however, experienced human operators are available for providing qualitative rules to control the system. Fuzzy control has been applied to the design of power system stabilisers (PSS) in a number of publications [2±7]. In all these papers, after the design is completed, the parameters of the fuzzy power system stabiliser (FPSS) are kept ®xed. This is in spite of the fact that the performance of the FPSS depends on the operating conditions of the power system, although it is less sensitive than conventional linear power system stabilisers (CPSS) [8]. * Corresponding author. E-mail addresses: [email protected] (N. HosseinZadeh), [email protected] (A. Kalam). Abbreviations: PSS, power system stabiliser; CPSS, conventional power system stabiliser; FPSS, fuzzy power system stabiliser; AFPSS, adaptive fuzzy power system stabiliser; IFPSS, initial fuzzy power system stabiliser; FBF, fuzzy basis function; FLC, fuzzy logic controller; FLS, fuzzy logic system
Wang and Mendel [9] introduced fuzzy basis functions (FBFs) which have the capability of combining both numerical data and linguistic information. These FBFs are quite general. Their exact mathematical structure depends on four choices that one must consider in order to design any fuzzy logic system, namely, type of fuzzi®cation, membership function, inference mechanism, and defuzzi®cation strategy. Further, Wang used the concept of FBFs to introduce stable adaptive fuzzy control of nonlinear systems [10]. His initial adaptive fuzzy controller is constructed from the fuzzy IF±THEN rules provided by human experts and some arbitrary rules. An adaptive law is then used to update the parameters of the adaptive fuzzy controller during the adaptation procedure. If the fuzzy IF±THEN rules from human experts provide good control strategies, then the adaptation procedure will converge very quickly. On the other hand, if there are no linguistic rules from human experts, then his adaptive fuzzy controller becomes a regular nonlinear adaptive controller, similar to the radial basis function adaptive controller [12]. He applied the adaptive fuzzy controller to an unstable ®rst-order nonlinear system and a second-order chaotic system [10]. He also applied a stable indirect adaptive fuzzy controller to an inverted pendulum tracking problem [11]. Following a similar approach, an indirect adaptive fuzzy power system stabiliser (AFPSS) is designed in this paper.
0142-0615/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0142-061 5(01)00093-X
838
N. Hossein-Zadeh, A. Kalam / Electrical Power and Energy Systems 24 (2002) 837±842
The power system under study is an industrial cogenerator plant. Some of its parameters are not known and others are not exact. Therefore, it is not possible to build a relatively exact mathematical model of the system. In order to design the AFPSS, ®rstly two fuzzy logic systems (FLSs) are designed which are the approximations for the unknown nonlinear functions present in the model of the power system. Then these FLSs are used to obtain the control law whose role is to stabilise the power system. The FLSs are adapted based on the Lyapunov's synthesis method. The adaptation law is such that the difference between the output of the plant and a desired track will be minimised. The AFPSS is applied to the power system. A onemachine in®nite-bus system is used for simulation studies, because the cogenerator plant is considered a small generating plant relative to the connected power network. The nonlinear simulation reveals that the adaptation procedure prevents large degradation of system performance for a wide range of operating conditions. This makes the AFPSS superior to a CPSS or a ®xed-parameter FPSS.
3. Fuzzy basis functions The basic con®guration of a fuzzy logic system is shown in Fig. 1 [14]. A FLS maps crisp inputs into crisp outputs. It contains four components: rules base, fuzzi®er, inference engine, and defuzzi®er. Once the rules have been established, the FLS can be viewed as a nonlinear mapping from crisp inputs to crisp outputs, and this mapping can be expressed mathematically as y f x: From Fig. 1, it is observed that such a mathematical relationship can be obtained [9] by following the signal x through the fuzzi®er, where it is converted to a fuzzy set Ax, into the inference block, where it is converted to a fuzzy set Av, and ®nally into the defuzzi®er, where it is converted to f x: In order to write such a relationship, speci®c choices should be made for fuzzi®er, membership functions, composition, inference, and defuzzi®er. For singleton fuzzi®cation, max-product composition, product inference, and height defuzzi®cation, leaving the choice of membership function open, it is easy to show that [15] N X
2. Direct and indirect adaptive fuzzy control In the adaptive control literature, adaptive controllers fall into two categories [13], direct and indirect. In direct adaptive control, the parameters of the controller are directly adjusted to reduce some norm of the output error between the plant and the reference model. In indirect adaptive control, the parameters of the plant are estimated and the controller is chosen assuming that the estimated parameters represent the true values of the plant parameters. In fuzzy control, linguistic information from human experts can be placed into two categories [10]: ² Fuzzy control rules which set forth the situations in which certain control actions should be taken (e.g. if the speed-deviation is negative medium and the acceleration is negative medium then the PSS output should be negative big). ² Fuzzy IF±THEN rules, which describe the behaviour of the unknown plant (e.g. if the control signal applied to the synchronous machine is negative, then the acceleration of the shaft will change in a positive direction). Adaptive fuzzy controllers which make use of these two classes of linguistic information correspond to the direct and indirect adaptive control schemes, respectively. More speci®cally, in the direct scheme, linguistic fuzzy control rules can be directly used to implement the adaptive fuzzy controller, whereas in the indirect scheme, linguistic rules are used to model the plant. The controller is then constructed assuming that the fuzzy logic system approximately represents the plant.
y f x
uk
k1 N X
L Y
i1 L Y
k1 i1
mFik xi
1
mFik xi
where N is the number of rules in the FLS and L is the number of inputs to the FLS. uk is the centre of gravity of the membership function of the output corresponding to the kth rule. Eq. (1) can be expressed as y f x; u
N X k1
uk j k x uT j x
2
where u u 1 ; ¼; uN T and j x j 1 x; ¼; j N xT : j k x are called fuzzy basis functions (FBFs) [9] and are given by L Y
j k x
mFik xi i1 N Y L X l1 i1
3
mFil xi
The FLS can be referred to as a fuzzy basis function expansion.
Fig. 1. Block diagram of a fuzzy logic system (FLS).
N. Hossein-Zadeh, A. Kalam / Electrical Power and Energy Systems 24 (2002) 837±842
4. Indirect adaptive fuzzy PSS In this section, the procedure for designing an indirect AFPSS for a synchronous machine is explained. Let x1 Dv speed deviation and x2 DP Pm 2 Pe accelerating power. It is possible to represent the machine with the following nonlinear equations x_1 ax2 ;
ax_2 g x1 ; x2 1 h x1 ; x2 u;
y x1 4
839
^ uh and g x; ^ ug are the FLSs with parameter where h x; vectors uh and ug ; respectively. ^ uh uc to both sides of Eq. (4) and using By adding h x; Eq. (9), the following equation is derived: ^ uh uc ^ ug 1 h x 2 h x; e 2kT e 1 g x 2 g x;
10
T
_ e ; Eq. (10) can be written as Since e_ e; ^ uh uc } 11 ^ ug 1 h x 2 h x; e_ Ac e 1 bc {g x 2 g x; where "
where a 1=2H and H is a constant parameter of the machine called per unit inertia constant. x x1 ; x2 T [ R2 is the state vector of the system and can be measured. g(´) and h(´) are nonlinear functions which are assumed to be unknown. u is the controlling signal which is the output of the PSS to be designed. Eq. (4) represents the machine during a transient period after a major disturbance has occurred in the system. It has been assumed that the two nonlinear functions g(´) and h(´) can be found such that
Ac is a stable matrix, because usI 2 Ac u s2 1 k1 s 1 k2 has its roots in the open left plane. Therefore, there exists a unique symmetric positive de®nite 2 £ 2 matrix P which satis®es the Lyapunov equation [16]
P_ e 22Hg x1 ; x2 1 h x1 ; x2 u
ATc P 1 PAc 2Q
5
This equation is based on the fact that the governor time constant is large compared to the time constants of the synchronous machine and its exciter, so that during the ®rst few seconds after the occurrence of a severe disturbance the governor function can be ignored. Therefore the mechanical input power is constant during the transient interval, say less than 5 seconds after the disturbance has occurred. Simulation studies show that a positive u will cause a positive change in Pe, i.e. P_ e . 0 whenever u . 0: This means that h may be selected to be a negative function, i.e. h x1 ; x2 , 0;
for all x1 ; x2
6
The control objective is to force y to follow a given desired signal ym. In the rest of this section the procedure to construct an indirect adaptive fuzzy controller [11] to achieve the above control objectives is discussed. _ T and k k2 ; k1 T [ R2 be such that the Let e e; e two roots of the polynomial h s s2 1 k1 s 1 k2 are in the open left half-plane. If the functions g and h are known, then the control law i 1 h up 2g x 1 ym 2 kT e 7 h x applied to Eq. (4), using u p instead of u, results in e 1 k1 e_ 1 k2 e 0
8
which implies that limt!1 e t 0: This is the main objective of control. However, g and h are not known. The objective is to design two FLSs close enough to g x and h x; then use them in Eq. (7). Therefore, the control law will be i 1 h ^ ug 1 ym 2 kT e uc 2g x; 9 ^ uh h x;
Ac
0
1
2k2
2k1
# ;
bc
" # 0
12
1
13
where Q is an arbitrary 2 £ 2 positive de®nite matrix. The next task is to replace g^ and h^ by FLSs represented by Eq. (2) and to develop an adaptation law to adjust the parameters in the FLSs in order to force the tracking error to converge to zero. Using the procedure suggested in ^ uh and Ref. [11], the parameter vectors of the FLSs h x; ^ ug will be adapted according to the following rules. g x; To make the parameter vectors bounded, the constraint sets V g and V h are de®ned as
V g {ug : iug i # Mg }
14
V h {uh : iuh i # Mh ; uhl # 21}
15
where Mg, Mh and 1 are positive constants speci®ed by the designer. If the parameter vectors are within the constraint sets or on the boundaries of the constraint sets but moving toward the inside of the constraint set, then the simple adaptation laws
u_ g g1 eT Pbc j x
16
u_ h g2 eT Pbc j xu c
17
will be used, where g 1 and g 2 are positive constants which will be used as the learning rate in the adaptation procedure. Otherwise, if the parameter vectors are on the boundary of the constraint sets and moving toward the outside of the constraint sets, the projection algorithm will be used to modify the adaptation law as
u_ g g1 eT Pbc j x 2 g 1 eT Pbc
ug uTg j x
18
uu g u2
u_ h g2 eT Pbc j xu c 2 g2 eT Pbc uc
uh uTh j x uu h u
2
19
840
N. Hossein-Zadeh, A. Kalam / Electrical Power and Energy Systems 24 (2002) 837±842
5. Design procedure 5.1. Step 1: off-line preprocessing 2 (1.1) Specify k1 1; k2 4 such p that the roots of s 1 kp 1 s1 k2 0 are s1 20:5 2 j 3=2 and s2 20:5 1 j 3=2: Other values for k1 and k2 are possible as long as roots s1 and s2 are in a position in the left half-plane with good damping characteristics. (1.2) Specify a diagonal positive de®nite 2 £ 2 matrix Q as " # 1 0 Q 0 1
There is a trade off between elements of Q and the learning rates g 1 and g 2. For bigger elements of Q, a smaller value should be chosen for learning rates g 1 and g 2. (1.3) Solve the Lyapunov Eq. (13) to obtain a symmetric positive de®nite matrix P. (1.4) Specify the design parameters g 1, g 2, Mg, Mh, and 1 based on the practical constraints. These values have been chosen to be g1 2; g2 20; Mg 3, Mh 100; and 1 0:5: There is a compromise in choosing these parameters. Small learning rates will result in a less adaptable system, which is more stable, compared with a system with higher learning rates. Mg and Mh are the upper limits on iuf i and iug i: The lower limit on the absolute value of the elements of the parameter vector ug is determined by 1 . 5.2. Step 2: initial fuzzy PSS construction (2.1) De®ne m1 fuzzy sets F1k1 for input Dv where k1 1; 2; ¼; m1 and m2 fuzzy sets F2k2 for input DP where k2 1; 2; ¼; m2 : Here m1 and m2 are chosen to be m1 m2 5: More fuzzy sets will result in a more smooth response. However, this will make the controller more complicated in terms of computational requirements. Five fuzzy sets for each input are suf®cient for the PSS to be designed. The fuzzy sets for input Dv are de®ned according to the membership functions shown in Fig. 2 and the fuzzy sets for input DP are de®ned according to the membership functions shown in Fig. 3. For the shape of membership functions, Gaussian functions have been chosen to make the calculations easier. The membership functions labelled ZR for input x2 has been chosen to be narrower than other labels. This will cause the oscillations in speed deviation response to damp faster when the speed deviation has reduced to small values around zero with a small acceleration. (2.2) Construct the fuzzy basis functions from the input membership functions as
j k 1 ;k2 x
mFk1 x1 mF k2 x2 1
m1 X m2 X
k1 1 k2 1
2
mF k1 x1 mFk2 x2 1
2
Fig. 2. Fuzzy membership functions for input x1.
and collect them into an m1 £ m2 dimensional vector j x in a natural ordering for k 1 1; 2; ¼; m1 and k2 1; 2; ¼; m2 : ^ ug (2.3) Construct the fuzzy rule base for the FLSs g x; ^ uh which consist of m1 £ m2 rules whose IF parts and h x; comprise of all the possible combinations of the fuzzy input sets. Speci®cally, the fuzzy rule base consists of rules ^ uh is x k1 ;k2 Rh k1 ;k2 : IF x1 is F1k1 and x2 is F2k2 ; THEN h x; 21 where x k1 ;k2 is the centre of gravity of the fuzzy output set. Construct vector uh as a collection of the values of x k1 ;k2 in the same ordering as j x: For example, for the speci®c synchronous machine under study, the fuzzy rule base and the parameter vector u h can be constructed from Table 1 (the elements of uh are in per unit and chosen to be negative as suggested in Eq. (6)). As an example, the eighth rule is Rh 2;3 : IF Dv is Negative Small and DP is Zero, ^ uh should be close to 24. From this rule, it is THEN h x; seen that the eighth element of the parameter vector is uh 8 u 2;3 24: In this way, the 25-element initial h
20 Fig. 3. Fuzzy membership functions for input x2.
N. Hossein-Zadeh, A. Kalam / Electrical Power and Energy Systems 24 (2002) 837±842
841
Table 1 ^ uh Decision table to construct the parameter vector uh for estimating h x; Speed deviation (Dv )
Accelerating power (DP) NB
NS
ZR
PS
PB
NB NS ZR PS PB
21 22 23 22 21
23 23 24 23 22
23 24 25 24 23
22 23 24 23 22
21 22 23 22 21
parameter vector uh will be obtained. Since ^ ug ; the initial enough information about g x; is chosen to be zero. ^ ug and (2.4) From (2.2) and (2.3) g x; obtained as
there is no value of ug ^ uh are h x;
^ ug uTg j x g x;
22
^ u h uTh j x h x;
23
Then the controlling signal will be obtained from Eq. (9). 5.3. Step 3: on-line adaptation (3.1) Apply the controlling signal to the plant. (3.2) Use the adaptive laws explained in Section 4 to adjust the parameter vectors u g and uh : 6. Simulation results The performance of the indirect AFPSS was evaluated by applying a large disturbance caused by a three-phase fault to ground on the transmission line. A power system model consisting of a synchronous machine connected to a constant voltage bus through a pair of three phase transmission lines is used in simulation studies. A simpli®ed schematic diagram of the model is shown in Fig. 4. A three-phase transformer is used between the synchronous machine and the transmission lines to boost the machine voltage. For comparison purposes, a CPSS was designed and its parameters were adjusted in the frequency domain [8]. The transfer function of the CPSS is K0 s t1 s 1 1t3 s 1 1 Gp s 24 t0 s 1 1 t2 s 1 1t4 s 1 1
Fig. 5. System speed deviation responses for Case 1.
fuzzy power system stabiliser before adaptation (FPSS) and indirect AFPSS, the system responses for three different conditions were obtained using nonlinear simulations. (The generator is represented by a set of ®rst-order differential equations as suggested in Ref. [17], chapter 13. The relationship between variables in the differential and algebraic equations is nonlinear and the system is not linearised in the simulation studies). Case 1: normal operating conditions. Operating point P0 0.9 pu, Q0 0.3 pu and Xe 0.2 pu. System responses to a fault, occurring at t 0:2 s with a duration of 0.06 s, are shown in Fig. 5. System responses with the initial FPSS is the worst. The reason is that there was not enough information about g x: If suitable information can be obtained either from an expert or through simulation, the FPSS can be improved. As is observed from the ®gure, the AFPSS has introduced more damping and the
The parameters of the CPSS were designed for the operating conditions P0 0:9 pu; Q0 0:3 pu and Xe 0:2 pu; where Xe is the equivalent reactance of the transmission line. For the plant with the three types of PSSs, i.e. CPSS,
Fig. 4. Power system con®guration for simulation studies.
Fig. 6. System speed deviation responses for Case 2.
842
N. Hossein-Zadeh, A. Kalam / Electrical Power and Energy Systems 24 (2002) 837±842
a simple ®xed-parameter FPSS. In a simple case, like the one-machine in®nite-bus model used, it may seem that it is not worthwhile to apply this algorithm. However, if a more complicated arrangement of a multimachine system is considered or the duration of the fault is longer than 0.06 s which was used in the simulation studies, then the improvement caused by the suggested scheme will be more apparent. The scheme is not suggested for all power systems, but it is recommended for the systems in which stability is of prime importance and the machines may be subjected to more severe faults.
References
Fig. 7. System responses for Case 3.
system response has less overshoot and settling time comparing to a CPSS. Case 2: heavy reactive load and weak connection. Operating point P0 0.9 pu, Q0 0.8 pu and Xe 0.45 pu. The system responses are shown in Fig. 6. The system response is more oscillatory in this case. Nevertheless, as in the previous case, the AFPSS has the best response. Case 3: importing reactive power and strong connection. Operating point P0 0:9 pu; Q0 20:3 pu and Xe 0:1 pu: In this case, the synchronous machine is absorbing reactive power and it can become unstable more easily. The system responses are shown in Fig. 7. As is observed from the ®gure, the system response for the AFPSS is optimum for this case also. 7. Conclusions An indirect adaptive fuzzy power system stabiliser was designed based on the approximation to the unknown nonlinear functions present in a model for a synchronous machine. The fuzzy logic systems were constructed using the concept of fuzzy basis functions. An adaptation algorithm was derived based on the Lyapunov's direct method to cause the system follow a desired response. The effectiveness of this scheme was investigated through nonlinear simulations. It was shown that the indirect adaptive fuzzy power system stabiliser designed in this way has the best response in a wide range of operating conditions, compared to the initial fuzzy power system stabiliser and a linear conventional power system stabiliser. Of course the proposed scheme is more complicated than
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