IEEE Transactions on Power Systems, Vol. 14, No. 1, February 1999

Adaptive Fuzzy Gain Scheduling for Load Frequency Control Jawad Talaq and Fadel Al-Basri

University of Bahrain Department of Electrical Engineering

P.O. Box 32038. Bahrain

145

ABSTRACT: An adaptive fuzzy gain scheduling scheme for conventional PI and optimal load frequency controllers has been proposed. A Sugeno type fuzzy inference system is used in the proposed controller. The Sugeno type fuzzy inference system is extremely well suited to the task of smoothly interpolating linear gains across the input space when a very non-linear system moves around in its operating space. The proposed adaptive cont,roller requires much less training patterns than a neural net based adaptive scheme does and hence avoiding excessive training time. Results of simulation show that the proposed adaptive fuzzy controller offers better performance than fixed gain controllers at different operating conditions.

INTRODUCTION

Large scale power systems are normally composed of control areas or regions representing coherent groups of generators. The various areas are interconnected through tie- lines. The tie-lines arc utilised for contractual energy exchange between areas and provide inter-area support in case of abnormal conditions. Area load changes and abnormal conditions, sucbh as outages of generation, lead to mismatches in frequency and scheduled power interchanges between areas. These mismatches have to be corrected via supplementary control. ILoad Frequency Control (LFC) of interconnected systems isl defined as the regulation of power output of generators within a prescribed area, in response to change in system frequency, tie-line loading, or the relation of these to each other; so as to maintain scheduled system frequency and/or established interchange with other areas within predetermined limits [ 11. Many investigations in the area of LFC problem of interconnected power systems have been reported over the Fast six decades [l-81. A number of control strategies have been employed in the design of load frequency controllers in order to achieve better dynamic performance. Among the various types of load frequency controllers, the most widely employed is the conventional proportional integral (P [) controller [ 1-3,8,9,25]. The PI

PE-966-PWRS-0-07-1997 A paper recommended and approved by the iEEE Power System Analysis, Computing and Economics Committee of the IEEE Power Engineering Society for publication in the IEEE Transactions on Power Systems. Manuscript submitted January 3, 1997; made available for printing August 19, 1997.

controller is simple for implementation but generally gives large frequency deviations. A number of state feedback controllers based on linear optimal control theory have been proposed to achieve better performance [ 1042,261. Alternative and more practical forms of feedback controllers such as output feedback and decentralised controllers have been the subject of extensive investigations [ 13,141. Fixed gain controllers are designed at nominal operating conditions and fail to provide best control performance over a wide range of operating conditions. So, to keep system performance near its optimum, it is desirable to track the operating conditions and use updated parameters to compute the control. Adaptive controllers with self-adjusting gain settings have been proposed for LFC [15-181. Despite the promising results achieved by these adaptive controllers, the control algorithms are complicated and require on-line system model identification. Artificial neural networks and fuzzy systems have been successfully applied to the LFC problem with rather promising results [ 19-23]. The salient feature of these techniques is that they provide a model-free description of control systems and do not require model identification. In this paper, an adaptive fuzzy gain scheduling scheme for conventional PI and optimal controllers has been proposed and tested for off-nominal operating conditions. The proposed controller offers better performance than fixed gain controllers.

THE TWO-AREA MODEL

A block diagram of a two-area interconnected system for the uncontrolled case is shown in Figure 1. The state variables model for the system is

(1) where A is system matrix, B is input distribution matrix, L is disturbance distribution matrix, x(t) is state vector, u(t) is control vector and d(t) is disturbance vector of load changes. x(t) = [ A f l , A P g l , A P v l , A P t i e , l Z , A f Z , A P g 2 , A p ~ 2 ] ~

x(t) = Ax(t) + Bu(t) + Ld(t)

T T u(t) = [Ul,U2] = [APCl,AF%Z]

d(t) = [ M d l , hPd2!IT

Since, the effectiveness of LFC is judged in terms of area control errors, ACE, the system output is given by

0885-8950/99/$10.00 0 1997 IEEE

146

and

where y(t) is the output vector, ACEi is area i control error, bi is area i frequency bias constant, Afi is area i frequency change, APtie,i is the change in tie-line power and C is the output matrix.

ACEi = APtie, i + bidfi (3)

Fig. 1 A Two Area Interconnected System

The two area model parameters of Figure 1 are defined in the appendix.

CONVENTIONAL PI CONTROLLER

The task of load frequency controller is to generate a control signal u that maintains system frequency and tie-line interchange power at predetermined values. The block diagram of the PI controller is shown in Figure 2. The control inputs uI and u2 are constructed as follows

z z

ui = -&I (ACEi)dt = -E! (APtie, i + blafi)dt 0 0

Taking the derivative of equation (4) yields

6 = -&(ACE;) = -I&( &tie, i + biafi) or in matrix form

U = -KICX where

L

combining equations (1) and (6) yields

xc = Acxc + Lcd where

xc =[:I; A, = [ -KIC A "1; 0 Lc =[:I (9)

$. I- ; I bi I Integral Control : A

Scheme

..--- I

APtie, i Fig. 2 Conventional PI Controller Installed on i" Area

OPTIMAL PI CONTROLLER

As proposed by Porter [24] and Reddoch et. al. [ll], the integral of area control errors are considered as states. This will drive the area control errors to zero at steady state. Defining these new states as vector q(t) yields

k a Aaxa + Bau + L a d (10)

Ya(t) = Caxa(t) (1 1) where

The control vector u is generated by feeding back all states through a constant gain matrix K.

u = -Kxa(t) (14) The gain matrix K is determined by minimising the

following quadratic performance index .a 1

J = ,j( xaTQ x, + uTRu )dt (15) & O

where Q and R are positive semidefinite and positive definite constant matrices defined through the minimisation of ACEi, ACEi.dt and control vector u excursions [11,21-221. This is the state-space optimal regulator problem [lo]. K is obtained from the solution of the algebraic matrix Riccati equation. The control law of equation (14) may be partitioned into its proportional and integral components as follows:

(16) u = Kpx + K I q

where Kp and KI are tl gain matrices, respecti1 to as Optimal PI State E

FUZZY (

Fuzzy set theory ha! problem. Hsu et. al. I which the change of fre inputs. Indulkar et. al. which he used the ar inputs. The block diag shown in Figure 3. Th some linguistic variabl large positive (LP), m (SP), very small (VS), s (MN), and large negati expressed in linguistic input signals to the ou rules, which are summi membership functions, operations performed b the decision rules that €

I Calculating A 4

FUZZY CONTROLLER

Fig. 3 Fuzzy Load Fr

Table 1 Dt

- LN

Frequency M P SN Deviation SP MN

SN LN MN LN LN LN

( A f ) VS MN

* Rule 1

For example, rule 1 in ' IF Af is LP a

Once the membershil been computed, a suit; determine the contr membership values.

proportional and integral feedback y. That is why this LFC is referred :dback Controller.

INTROL SCHEMES

3een recently applied to the LFC '1 proposed a fuzzy controller in ency and its rate have been used as 01 proposed another fuzzy LFC in control error and its change as

m of Hsu fuzzy controller [19] is input signals are first expressed in using fuzzy set notations such as

ium positive ( M P ) , small positive all negative (SN), medium negative (LN). A set of decision rules, also

miables, are established to relate ut (control) signal. These decision sed by a fuzzy relation matrix using orm the basis of the fuzzy logic lie fuzzy controller. Table 1 shows 1 been proposed by Hsu [ 191.

CONTROL AREA ( i )

A P t i e , i 1 uency Controller Proposed by Hsu

sion Rules of Hsu [ 191

.equency Deviation Rate ( A i ) MN SN VS SP M P LP

SP M P LP LP LP LP vs SP M P M P LP LP SN VS SP SP M P LP MN SN VS SP MP M P MN SN SN VS SP M P LN MN MN SN VS SP LN LN LN MN SN VS

ble 1 is: I AfisLNTHENU isVS (17)

ralues for the controller output have le algorithm must be employed to ler output signal from these

147

NUERAL NETWORK BASED ADAPTIVE GAIN SCHEDULLING

The main drawback of fixed gain controllers is that their performance deteriorates as a result of changes in system operating conditions. In order to maintain desired quality of the system dynamics over a wide range of operating conditions, it is highly desirable to be able to adapt the controller gains according to the on-line information. Djukanovic et. al. [21-221 used Artificial Neural Networks (ANN) for the adaptation of elements of the gain matrix K of the state space optimal load frequency controller. In his approach, three parameters that represent system operating conditions have been monitored and used as inputs to the neural net based adaptive controller. These parameters are power system time constant, Tp, synchronising power coefficient, T12 and frequency bias setting b. The parameters depend on the load frequency characteristic, D (see Appendix), which is load dependent that must be determined at every load pattern to precisely monitor the parameters. These parameters activate many individual neural-nets trained with supervised learning. The training set of patterns are generated by solving the state-space optimal regulator problem for different values of parameters Tp, T12 and b. Network training is slow if it contains a large number of patterns. Furthermore, a criterion for selecting the optimum number of nodes in the hidden layers of the net should be developed, which significantly affects learning rate and classification performance.

ADAPTIVE FUZZY GAIN SCHEDULLING

A Sugeno type fuzzy inference system to replace the neural nets of [22] is proposed in this paper. The advantage is that the design is simpler and fewer training patterns are needed. Sugeno type fuzzy inference system is extremely well suited to the task of smoothly interpolating linear gains across the input space. So, we could build a fuzzy system that switches smoothly between several linear controllers as a non-linear system moves around in its operating space. Consequently, two fuzzy based adaptive load frequency controllers are proposed in this paper. The first is an optimal load frequency controller with fuzzy gain scheduling and the second is a conventional integral load frequency controller with fuzzy gain scheduling.

Optimal Controller with Fuzzy Gain Scheduling The linear optimal PI controller is modified in such a way

that its feedback gain matrix K elements are adapted by means of a fuzzy gain scheduler according to on-line system information (Tp, T12 and b) as shown in Figure 4. Unlike the neural-net approach wherein a large number of training patterns are required, we need fewer number of patterns in the fuzzy approach (only 27 combinations of system monitored parameters are used). Each monitored parameter

148

is divided into three fuzzy sets, Small, Medium and Large, represented by the membership functions shown in Figure 5 .

u x I A POWER SYSTEM 1 I I P I * I I MONITORING T12, Tp, b I I I. I FUZZYGAINSCHEDULER I

I1

Fig. 4 Optimal Controller with Fuzzy Gain Scheduling I* F P

1

TP TI 2 b Fig. 5 Membership Functions of The Monitored Parameters

Patterns are represented by Sugeno w e fuzzy rules such as: IF Tp is AI' and T12 is A i and b is A3'

THEN optimal gain = K' (18) where T ~ , T ~ ~ , b: monitored parameters representing a particular

operating condition. AI' , A i , A3': fuzzy sets of the ith rule. K': optimal gain matrix of the i rule. th

The fuzzy gain scheduler output is determined as follows: 27

w'K'

(19) i = l

scheduled = 27

W '

i = 1

w' = min (P ( AI' ), P ( A i ), P ( Aji 1) (20) where w' is the firing strength of the i'" rule calculated as the minimum of membership values associated with that particular rule. p ( AI' ), p ( A i ) and p ( A3' ) are the membership values of the fuzzy variables T,, T12 and b respectively Conventional Integral Control With Fuzzy Gain Scheduling

The same concept of fuzzy gain scheduling of the optimal controller is applied for adaptation of the conventional integral controller gain according to changes in system operating conditions. The design of the fuzzy gain scheduler block of this controller is exactly similar to that designed for the optimal controller except that the calculated optimal gain

matrices are replaced by integral gains. The calculated integral gains for the conventional integral controller corresponding to the 27 patterns of operating conditions are listed in Table 2. To demonstrate the method of calculating the integral gain using equation (19), consider the following example. Assume an off-nominal operating condition represented by the parameters Tp-16s, T12-0 .2~~ and b-O.4pdHz. If we consider rule 12, that is 'If T, is M, T12 is S and b is L THEN K=0.52' as can be seen from Table 2, then the membership values calculated from Figure 5 for this particular case are p ("T, is W)=0.6, p ("T12 is s")=0.725, and p ("b is L")=0.834. The firing strength in this case is:

w12 = min ( 0.6, 0.725,0.834 ) = 0.6 If we complete the process for the 27 rules, then the

scheduled integral gain using equation (19) is 0.5458.

14 I20 IO345 I O 275 I 0 7 8 I SIMULATION AND RESULTS

Two identical areas have been assumed with the following

H-5.0s, R=2.40Hz/pu, D=O.O0833pu/Hz, Kp-12OHz/pu, parameters.

b=0.425pu/Hz, T12=0.545p~, Tp=20s, Tg=O.O&, Tt=0.3~. The following abbreviations are used.

CLF: Conventional Integral Load Frequency Controller. OPT: Optimal PI Load Frequency Controller. CLFGS: Conventional Integral Load Frequency Controller with Fuzzy Gain Scheduling. OPTGS: Optimal PI Load Frequency Controller with Fuzzy Gain Scheduling. Figure 6 shows the response of the conventional integral

coiitrvller (CLF) and the conventional integral controller with fuzzy gain scheduling (CLFGS) following a step load change in area 1 while the system is operating at off-nominal conditions (T,=12, T12=.5, b=. 12). Optimal PI controller gain matrix K calculated at nominal operating conditions using Riccati equation and off-nominal operating conditions using the fuzzy gain scheduler are shown in Table 4. Also shown in Table 4 are svsteiii eieeiivalues for the off-nominal

operating conditions 1

Figure 7 shows the n (OPT) and the optin scheduling (OPTGS) fc while system is operatii and integral of squared line power (pu) chant conditions are shown ii

Table 3. Average and Inte

Feedback Gain Matrix KT Obtained From

Riccati Equation Fuzzy

5 -0.03 ‘

Fig. 6-a. AI

0.005 1-

System Eigenvalues Obtained From

Riccati Equation Fuvly

-0.025- 0 5

Fig. 6-b. A

-0 012-

-0 016-

Fig. 6-c. ’ Fig. 6 CLF

; 2

ing the calculated gain matrices. ponse of the optimal PI controller 11 PI controller with fuzzy gain lowing a step load change in area 1 ; at off-nominal conditions. Average rror of area 1 frequency (Hz) and tie 5 obtained at different off-nominal I’able 3. 11 of Squared Error of Af(Hz) and AFTie(pu)

i

, I I

f”

- 4 -

i--ij -CLFGS

, - 10 15 20

Time(s) . 1 Frequency Deviation (Hz)

149

I Nominal I OE-Nominal I OE-Nominal I Off-Nominal I

Fig. 7-a. Area 1 Frequency Deviation (Hz)

Fig. 7-b Area 2 Frequency Deviation (Hz)

x163 ’

0 5 10 15 20 Time(s)

Fig. 7-c. Tie-Line Power Deviation (pu) Fig. 7 OPT and OPTGS for APdl=.olpU.

1.50

CONCLUSIONS An adaptive fuzzy gain scheduling scheme for

conventional PI and optimal load frequency controllers has been proposed, The controllers have been simulated on a two area interconnected system. A comparison between conventional controllers and the proposed adaptive fuzzy controllers, using fuzzy gain scheduling, reveals the effectiveness of fuzzy gain scheduling used for off nominal operating conditions.

REFERENCES

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[lo] Gopal, M., ‘‘Modern Control System Theory,” Wiley Eastem Ltd., Znd

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[13] Bekhouche, N., and Feliachi, A, “Decentralised Estimation for the Automatic Generation Control Problem in Power Systems,” First IEEE Conference on Control Applications, Vol. 2, 1992, pp 626-63 1.

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[ 151 Kanniah, J., et. al., “Microprocessor-Based Adaptive Load-Frequency Control,” IEE Proceeding&, Vol. 131, No, 4, July 1984.

[16] Vajk, I., et. al., “Adaptive Load-Frequency Control of The Hungarian Power System,” Automatica, Vol. 21, No. 2, 1985, pp 129-137.

[17] Yamashita, K., and Miyagi, H., “Multivariable SeFTuning regulator for Load Frequency Control System with Interaction of Voltage on Load Demand, ” IEE Proceedings-D, Vol. 138, No. 2, March 1991.

[18] Pan, C. T., and Liaw, C. M., “An Adaptive Controller for Power System Load-Frequency Control,” IEEE Trans. On Power Sys., Vol. 4, No. 1, Feb. 1989, pp 122-128.

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Vol. 1, 1993, ~ ~ 3 7 4 - 3 7 9 .

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APPENDIX: Two Area Model Parameters

f System Frequency (Hz). H Inertia Constant (sec). R Regulation (Hdpu). D Load Frequency Characteristic (pu/Hz). Tg Tt Turbine Time Constant (sec). Tp

Speed Governor Time Constant (sec).

Power System Time Constant (sec).

2H

Df (A. 1) T =-

Kp Power System Gain (Hdpu).

b Frequency Bias Constant (pmz) . Kp= 1/D (A.2)

04.3) b=D+l/R

T12 Synchronising Power Coefficient (pu). 2n v1v2

T12 =- cos (61 - 6,) XI,

1L

where x12 is the tie-line reactance in pu, VI and V 2 are area 1 and 2 voltages in pu respectively.

Jawscl Tslnq, received a bachelor’s degree in Electrical Engineering from the University of Technology at Baghdad in 1981, a master’s degree in Power Engineering from the University of Strathclyde in 1987, and a Ph.D. in Electrical Engineering from the Technical University of Nova Scotia in 1993. Between 1981 and 1989 he was with the Electricity Directorate of Bahrain. Since 1989 he has been on Faculty in the Department of Electrical Engineering at the University of Baluain where he is Assistant Professor.

Fadel Al-Bnsri, received a hachelor’s degree in Electrical Engineering from the University of Bahrain in 1992, He joined the University of Bahrain as a research assistant in 1994. Currently, he is a candidate in the master’s degree program at the University of Bahrain.