INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, VOL. 9, 151-162 (1995)

A DESIGN METHOD OF ADAPTIVE LOAD FREQUENCY CONTROL WITH DUAL-RATE SAMPLING

KATSUMI YAMASHITA, MITSURU HIRAYASU, KEI OKAFUJI AND HAYAO MlYAGl Department of Electronics and Information Engineering, Faculty of Engineering, Ryukyu University, I Senbaru,

Nishihara. Okinawa 903-01, Japan

SUMMARY

An adaptive control strategy for load frequency control, the purpose of which is to reduce the transient errors in the frequency and scheduled tie-line power deviations and to ensure zero steady state errors of these two quantities, is described. A dual-rate sampling self-tuning regulator whereby data sampling and control output are performed at different rates is designed for interconnected power systems with unknown deterministic load disturbances. The proposed control scheme is applied to a two-area power system provided with reheat thermal turbines and the control effects of the regulator are examined using digital simulation.

KEY WORDS load frequency control; adapth e control; digital control; dual-rate sampling

1. INTRODUCTION

Load frequency control (LFC) is a very important factor in power system operation. It aims at controlling the output power of each generator to minimize the transient errors in the frequency and scheduled tie-line power deviations and to ensure zero steady state errors of these quantities. ' For the past few decades a considerable research effort has been devoted to the development of control strategies for LFC based on advanced topics of modern control theory.2 In these control techniques the controller design is normally based on a fixed parameter model of a system derived by linearization around a normal operating point. However, because of the inherent characteristics of changing loads, the operating point of power systems may change greatly during a daily cycle. As a result, the system performance with controllers designed for a specific operating point will not stay optimum in another status. From this point of view, some author^^-^ have applied self-tuning controllers to the LFC problem. Since a self-tuning controller consists of an identifier and a controller, the system performance is always kept near its optimum. In these adaptive control techniques it is assumed that data sampling and control are done at the same frequency and the sampling frequency is then selected higher than the Nyquist frequency so as to satisfy the sampling theorem.' However, the practical control frequency used for LFC8 cannot be acceptable for adaptive regulator design, because this control frequency does not satisfy the sampling theorem. Therefore the control interval must be short enough to satisfy the sampling theorem, but a smaller control interval may be undesirable since it will require more frequent sampling and hence lead to excessive effort in the control operation.

This paper was recommended f o r publication by editor E. Mosca

CCC 0890-6327/95/09015 1 - 12 0 1995 by John Wiley & Sons, Ltd.

Received 1 October 1993 Revised 6 June 1994

K. YAMASHITA ET AL. 152

The purpose of this paper is to derive a dual-rate sampling self-tuning regulator by modifying the theory of the algorithm discussed in our earlier paper, based on the algorithm proposed in Reference 10, and to construct a load frequency controller via a speed governor control for an interconnected power system. The proposed control scheme should be very realistic, because this control strategy is implemented using the practical control interval for LFC.

In this paper we apply the newly designed dual-rate sampling regulator to a two-area power system provided with reheat thermal turbines and the control effects of this regulator are examined using digital simulation.

2. PROBLEM FORMULATION

The general structure of a self-tuning controller consists of an identifier and a controller. The system to be controlled is represented by a linear discrete model of a preassigned order. Here it is assumed that the system model of the ith controlling plant in a two-area power system can be described by the linear differential equation

(1)

where y ( t ) is the area control error (ACE) given as f l p f ( t ) + A P , i , ( t ) , u ( t ) is the control variable (APc( t ) ) , q-' is the backward time shift operator, the time interval of which is assumed to be T, (s), k is the time delay, the time interval of which is given by multiplying a control interval Tj = j x T, (s) by an integer d as shown in Figure 1, and

A ( q - ' ) y ( t ) = B ( q - ' ) u ( t - k) + C ( q - ' ) w ( t )

A ( q - ' ) = l + ~ l q - ' + ~ 2 ~ - ~ + . . . + ~ n q - ~

E ( q - ' ) = blq-' + b2qW2 + + b,q." C(q-1) = co + c1q-'+ c2q-2 + ... + c p q - p

The detailed information on the plant model is given in the Appendix. It is assumed that the disturbance w ( t ) given as A Pd(t) is the unknown deterministic load disturbance described by an ndth-order polynomial function of time. Then w ( t ) can be mathematically modelled by the expression

w ( t ) = W l t " d - ' + ~ 2 t ~ ~ - ~ + I.. + Wnd (2) To avoid unnecessary complication in notation, the subscript i implying the ith controlling plant is dropped in (1).

In order to decouple the effect of the load disturbance from the plant output, (1) may be

I ! I

( a ) O u t p u t y ( t ) ( b ) C o n t r o l u ( t ) Figure 1 . Dual-rate sampling

ADAPTIVE LOAD FREQUENCY CONTROL 153

rewritten at time t + j + k by replacing t by t + j + k to give

A ( q - ' ) y ( t + j + k ) = B ( q - ' ) u ( t + j ) + C ( q - ' ) w ( t + j + k ) ( 3 )

Then, by subtracting (1) from (3) and forming the difference of ndth order for the resultant equation, we obtain

A(q- ' )Andy( t + j + k ) = B(q-')Andu(t + j ) (4)

where A n d = ( 1 - q - j - k n d 1

in which the term of the load disturbance is eliminated. Notice that the decision of the control strategy for (4) is much simpler than that for (3).

Next a plant model with dual-rate sampling is derived based on the modified model of (4). We define a polynomial P ( q - ' ) aslo

( 5 ) - j - k + l P ( q - ' ) = 1 + p1q-l + p24-' + " * + P i + & - 14

such that the coefficients of q - l , q-', ...,q-J-'+' are set to zero in

P ( q - l ) A ( q - ' ) = A(q- ' ) (6)

where - j - & - I + ... + i n q - j - k - n + l A(4-1) = 1 + i1q-J-& + i24

On the other hand, multiplying the right-hand side of (4) by P ( q - ' ) and using the following equation in which the control with the assumed control interval Tj is changed only every j ,

A%(t - ( i - 2) j - 1) = Andu(f - ( i - 2)j - 2) = -.. = Andu(t - ( i - 1)j) ( i = 1,2, ..., s) (7)

where s is the smallest integer greater than or equal t o the value of (m + j + k - l ) / j , we obtain

P(q-')B(q-')Andu(t + j ) = B(q-j)A"'Uu(t) (8)

where - ( s - l ) j B(q-') = 61 + 82q-J+ 63q-2J+ *.. + 6,q

Therefore the plant model with dual-rate sampling is expressed as

A(q- ' )Andy( f + j + k) = B(q-J)Andu(t) (9)

Furthermore, since the time delay k is defined by d x j and the term And in (4) can be split into the two components

(10) A n d = And + and

where

2, = ( 1 - q - j ) n d , a n d = (1 - q - j ) n d [ ( 1 + q - j + q - 2 j + ... + q - d j 1 - 1 1 nd

(9) can be rewritten as

A (q-')A""y(t + ( d + 1)j) = biLndU(t) + 6iPdu(t) + [B(q-') - di]Andu(t) ( 1 1 )

The primary objective of LFC is to minimize the transient errors in the frequency and scheduled tie-line power deviations and to ensure zero steady state errors of these two quantities under a load disturbance. To achieve this objective and enforce a constraint on the

154 K . YAMASHITA ET A L .

control effort, we may try to minimize a cost function of the form

J = b(t + ( d + 1)j)I + A[A%d~(t)]

where X is a positive scalar constant. The cost function J is minimized by setting its gradient to zero:

J -=2A1( [ l - A(q-’ )And]y(t + ( d + l)j)+ 8 1 6 n d ~ ( f ) + [B(q - j ) - 6l1Andu(t)) Andu ( t )

+ 2(6: + X)A”d~(t) (13) Tht optimal control law is then given by

6, &’%(?)= -2- ( [ l - k ( q - ’ ) A n d ] y ( t + ( d + l ) j ) + 6 1 6 ~ ~ ~ ( f ) + [B(q-’)- 61]Andu(t)] (14) bl+X

This control is physically realizable, since the term [ l - A(q- l )And]y(t + ( d + 1)j) is represented by powers under t, as

Then, substituting the control strategy (14) in ( l ) , the block diagram of the closed loop system shown in Figure 2 is obtained and the transfer function G,(q-’) of w ( t ) to y ( t ) is given using (6), (8) and (10) as

Therefore, if the denominator of G,(q-’) is stable, since the relations ~ w ( l ) = Andu(t) = 0 are satisfied under the disturbances defined by (2), it can be guaranteed that the output y ( t ) converges to zero as t -+ 00.

In the self-tuning regulator the model parameters G I , Gz, .... f in and 61, 62, .... 6, in (9) must be estimated on-line by an adaptive algorithm. The estimated values 61, 62, .... 6, and

..........................................................................

c ( 9 - 1 ) I + I

I + I I

! ........................................................................... ; I

P L A N T

Figure 2. Block diagram of closed loop system

ADAPTIVE LOAD FREQUENCY CONTROL 155

61, 62, ..., 6, of these parameters are then used to calculate the control strategy of (14). To track the model parameters, (9) may be rewritten at instant t by setting k equal to d x j and replacing t by t - (d + 1 ) j to give

(17) Andy(t) = eTt(t - ( d + 1 ) j )

eT= [ n l , ii2, ..., ii,, h l , h2, ..., h,] where

tT( t - (d+ l ) j ) = [ -A"'y ( t - (d+ l)j), ..., - A n d y ( t - ( d + 1 ) j - n + l ) , A"'u(~- ( d + l)j), ..., AndU(t - ( d + s ) J ) ]

and T denotes the transpose of a vector.

algorithm" The best estimate 6 of the control parameters €3 is obtained using a standard recursive

[A"'y(t) - W ( t - l)[(t - ( d + l)j)] r( t - l ) € ( t - ( d + 1)j) I + t T ( t - ( d + i)j)r(t- i)[(t-(d+ i)j)

e ( r ) = e(t- 1 ) +

The correlation matrix r(t) can be calculated using

Then, if the denominator of G,(q-') is stable for the value of the estimate 6, it can be guaranteed that the output y ( t ) converges to zero as t + 00.

3. NUMERICAL RESULTS

A typical two-area interconnected power system provided with reheat thermal turbines is shown in Figure 3. This model is used to evaluate the effectiveness of the proposed load frequency self-tuning regulator. The values of the system parameters are given in Table I. We investigate the system performance of the adaptive regulator for the following load disturbance using digital simulation:

0.01 puMW (0 < t < 40) 0.02 puMW (t 2 40) A p d l ( t ) = [

Since the load disturbance in this assumption consists of a step load change, the value of n d

is considered to be one. Therefore the control law is defined from (14) as

156 K. YAMASHITA ET A L .

Steam turbine A P A I

Steam turbine tw Table I. System parameters ( i = 1,2)

Hi 5 . 0 s Di 8 .33 x 1 0 - ~ PUMW HZ-' Ri 2 . 4 Hzpu-'MW-' Tri 10.0 s K h i 0.5 T, i 0.3 s

Ti2 0,545 PU MW Hz-' f * 60 Hz Tgi 0.08 s p i = Di + 1/Ri 0.425 PUMW Hz-'

where

i u ( t ) = u ( f ) - u(t - j ) , A y ( t ) = y ( t ) - y ( t - (d+ 1)j)

and the initial values are taken to be zero. In the estimation of model parameters the initial parameters are taken to be zero except for

b1(0), the initial matrix r(0) which is 1031 and h ( t ) and X2(t) which are taken as one. A quantitative measure of the performance under the load disturbance shown in (20) is

obtained in terns of the integral squares of the frequency deviations in the two-area system and of the tie-line power deviation for a period of 90 s after the simulation starting transients have died out. The corresponding cost functions are defined as

A u ( t ) = u ( t ) - u( t - ( d + l)j) ,

where

APtie( t )=APtie l ( t )= -APtiez(t)

The decision on the quality of the regulator is related to the lowest value of the integral squares IPI and IP2. In this study the value of the control interval Tj is considered to be 2 .0 s, since

ADAPTIVE LOAD FREQUENCY CONTROL 157

the practical control intervals used for LFC are in the range 1-5-2.5 s.' Also, the value of the integer d is considered to be one, since a control with a delay of one control interval is sufficient to consider the time delay due to identification and control computations and the transmission time of the system data over the telemetric links to the controlling plant. The effects of the following parameters on the performance of the power system have been investigated under the load disturbance given in (20) and the above-mentioned conditions:

(a) the order of the model by which each area is modelled in the regulator, i.e.the choice

(b) the value of the control weighting X (c) the sampling interval T, under the control interval Tj = 2 . 0 s.

of n and m

3. I . Effects of model order and control weighting

The effect of the control weighting A on the system performance is studied for three different orders of the system model used in the regulator. The model order is assumed to the same order for both areas, i.e. m = n. Also, the value of the sampling interval T, is chosen as 0.2 s so as to satisfy the sampling theorem and considering the computation time for parameter estimation. Studies have been conducted for the models of order n = 2 , 4 and 6 under the load disturbance given in (20); the corresponding cost functions IP1 and IP2 are shown in Figures 4-6. It can be seen from these figures that the lowest values of IP1 and IP2 for the three different orders of the system model are approximately the same in the neighbourhood of X = 0.2. Therefore we conclude that the most adequate model is of second order and the optimal value of X is 0 . 2 for this example.

3.2. Sampling interval

The integral squares IP1 and IPz as a function of the sampling interval T, are shown in Figure 7. All results are given for n = 2, X = 0 . 2 and Tj = 2.0 s. These investigations show that the lowest values of IPI and IP2 are approximately constant with the sampling interval up to

0 . li(l 23

0. 19 0. 00 0. 20 0. 40 0. 60 0. EO 1 . 0 3

Control weighting k Figure 4. Integral squares of different variables versus control weighting X for n = 2

158 K. YAMASHlTA ET AL.

T, = 2.0 (corresponding to single-rate sampling for Tj = T, = 2 - 0 s) . We consider that the high values of IP1 and IPz for Tj = T, arise because the denominator of Goy(q-l) in (16) becomes an unstable polynomial. Thus we conclude that the value of the most adequate sampling interval is T, = 0 . 2 s considering the sampling theorem and the computation time for parameter estimation.

3.3. Time response

Figure 8 shows the control effects achieved by the proposed self-tuning regulator under the load disturbance given in (20). In these studies we assumed n = 2 , X = 0.2, T, = 0.2 s and Tj=2*Os . The curves in Figures 8(a)-8(c) show the performances of A f ~ ( t ) , A f z ( t ) and A Pti,(t) respectively for the self-tuning system (solid curves) and the uncontrolled system

0. 23 0 . 2 4 1

0. 00 0 . 20 0. 40 0. 60 0. 80 I . 00

Control weighting I Figure 5. Integral squares of different variables versus control weighting A for n = 4

0. 0 '241 23

0.19 1, 0. 00 0. 20 0. 4 0 0. 60 0. 80 I . 00

Control weighting A Figure 6. Integral squares of different variables versus control weighting A for R = 6

ADAPTIVE LOAD FREQUENCY CONTROL 159

-3.00 - -0 .15 -

-0.20 -

(broken curves). Also, the control performances of AP,l and A PCz are shown in Figure 8(d). It can be seen from these figures that the transient errors in the frequency and scheduled tie-line power deviations are much reduced and zero steady state errors of these quantities are ensured by the self-tuning regulator.

3.4. Hydrothermal system

Here we consider a different two-area power system with steam turbines and hydroturbines as shown in Figure 9. This model is used to show the effectiveness of the proposed regulator.

0. "'I 23

a - 0.

0.

0 .19 I I I 0.01 0. 10 I . 00 10.00

Sampling period Ts [sec] Figure 7. Integral squares of different variables versus sampling interval T, for n = 2, X = 0 .2 and T, = 2 . 0 s

160 K. YAMASHITA ET A L .

hydrogovernor I hydroturbine Area 2 _...__....____________

1 tO.STrs I . . .___._._____._____--- I

APc I

Figure 9. Block diagram of two-area hydrothermal system

0. 06 - 0. 06 buu G - N

a Y

0 .03

2 0.03 0. 00 Q

G 0. 00

-0. 03 __._____.-.-.... 0. 00

0. 0 2. -0. 06

:-, Uncontrolled sys tern -0.09 -0. 03

(d)

Figure 10. System responses for No. 1 and No. 2

ADAPTIVE LOAD FREQUENCY CONTROL 161

The detailed description and parameter values of the system are given in Reference 12. We investigate the system performances of the proposed regulator No. 1 and a conventional discrete-type integral regulator No. 2 with optimal integral gains K : = 0 - 18 and K: = 0.14 designed for a sampling period T, = 2.0 s. l 2 Figure 10 shows the results for No. 1 and No. 2 under the load disturbance of (20) and changing the inertia constants H ; from their nominal values to 170% of the values for 2 2 40. It can be seen from these figures that No. 1 can always act satisfactorily, whereas No. 2 cannot act satisfactorily for t 2 40. This means that the optimal integral gain of No. 2 designed for the nominal values is not optimum for t 2 40. It may thus be inferred that the proposed control scheme can be acceptable for the LFC of interconnected power systems.

4. CONCLUSIONS

In this paper a new method of designing a dual-rate sampling self-tuning regulator via a speed governor control has been proposed. This method has been applied to two-area power systems with reheat thermal turbines and with steam turbines and hydroturbines and the effectiveness of the proposed regulator has been investigated using digital simulations. The major contributions of this paper are as follows.

(a) The effect of load disturbances is decoupled from the plant output and a large value of control is discouraged indirectly by penalizing the rate of change of the control.

(b) The proposed control scheme is very realistic, because the data sampling is implemented using the interval which satisfies the sampling theorem and the control is implemented using the practical control interval used for LFC.

(c) The realization of this regulator is very easy because of its required data, i.e. frequency and scheduled tie-line power deviations, which are used by most utilities nowadays.

REFERENCES

1. Elgerd, 0. I., Electric Energy System Theory: An Introduction, McGraw-Hill, New York, 1983. 2. Bohn, E. V., and S. M. Miniesy ‘Optimum load-frequency sampled-data control with randomly varying system

3. Sheirah, M. A., and M. M. Fattah, ‘Improved load-frequency self-tuning regulator’, Int. J. Control, 39, 143-158

4. Kanniah, J . , S. C. Tripathy, 0. P. Malik and G. S. Hope, ‘Microprocessor-based adaptive load-frequency

5 . Yamashita, K., and H. Miyagi, ‘Load frequency self-tuning regulator for interconnected power systems with

6. Yamashita, K., and H. Miyagi, ‘Multivariable self-tuning regulator for load frequency control system with

7. Kuo, B. C., Digital control systems, Holt-Saunders, 1980. 8. deMello, F., R. J . Mills and W. F. B’rells, ‘Automatic generation control. Part 11: Digital control techniques’,

IEEE Trans. Power Appar. Sysl.. PAS-92, 716-724 (173). 9. Kothari, M. L., P. S. Satsangi and J . Nanda, ‘Sampled-data automatic generation control of interconnected

reheat thermal systems considering generation rate constraints’, IEEE Trans. Power. Appar. Syst., PAS-100,

10. Kanniah, J . , 0. P. Malik and G. S. Hope, ‘Self-tuning regulator based on dual-rate sampling’, IEEE Trans.

1 I . Landau, I . E., and M. Tomizuka, Theory and Practice of Adaptive Control Systems, Ohmusha, Tokyo, 1981. 12. Nanda, J.. M. L. Kothari and P. S. Satsangi, ‘Automtic generation control of an interconnected hydrothermal

system in continuous and discrete modes considering generation rate constraints’ Proc. IEE, 130, 17-27 (1983).

disturbances’, IEEE Trans. Power Appar. Syst., PAS-91, 1916-1922 (1972).

( 1984).

control’, Proc. IEE, 131, 121-128 (1984).

unknown deterministic load disturbances’, Int. J. Control, 49, 1555-1568 (1989).

interaction of voltage on load demand’, Proc. IEE, 138, 177-183 (1991).

2334-2342 (1981).

Automatic Control, AC-29, 755-759 (1984).

162 K. YAMASHITA ET AL.

APPENDIX: LIST OF PRINCIPAL SYMBOLS

f * i

Hi Di

Kh i Tr i

nominal system frequency subscript referring to area i inertia constant load frequency constant high-pressure turbine power fraction reheat time constant synchronizing coefficient steam chest time constant speed governor time constant self-regulation parameter for governor frequency bias parameter sampling interval control interval incremental frequency deviation incremental generation change incremental generation change during steam reheat incremental change in governor valve position incremental change in tie-line power incremental change in speed changer position incremental load demand change small deviation of state variable Laplace operator