Minimum variance strategy for load-frequency control

Minimum variance strategy for load-frequency control M A Sheirah

Qatar University, Doha, Qatar O P Malik and G S Hope

The University of Calgary, Calgary, Alberta, Canada

A minimum variance strategy for load-frequency control o f interconnected power systems is described, lnstead o f using fixed weights on the frequency error and the tie-line power deviation to compute the area control error, the relative weights on these variables are changed online according to their respective variances. This allows the variances o f the individual variables to approach their minimum. The control is computed by the self-tuning regulator algorithm using the area control error with the time-varying weights as the regulator input variable. Studies show that with the proposed algorithm, the system response is noticeably improved.

I. Introduction Normal procedure used to design a load-frequency controller by the techniques discussed in the literature 1-1s is to construct a linear system model with fixed parameters. The parameters are obtained by linearizing the system around an operating point. In general, system response characteristics tend to be nonlinear. Therefore, as the operating conditions change, system performance with controllers designed for a specific operating point will not stay optimal. To keep the system performance near its optimum, it is necessary to track the operating conditions and update the system parameters continuously. Control should then be computed based on updated parameters. Also, the load changes are random in magnitude and duration. It is thus more appropriate to consider the system as a stochastic system and for better performance design an adaptive stochastic controller. Such an adaptive controller combines a parameter-estimation algorithm to update online the parameters of a discrete noisy model of the system and a control algorithm. A well-known controller of this type, called the self-tuning regulator 16, combines the recursive least-squares estimation and minimum-variance control strategy. Application of the self-tuning algorithm to the load-frequency control of an interconnected power system is described in Received: 1 February 1984, Revised: 7 August 1985

120

this paper with suitable modifications to suit this problem. Area control error (ACE), formed as the weighted sum of the frequency error and the deviation of the total tie-line power, is considered as the regulated variable. To minimize the variances of the frequency error and the deviation of the tie-line power individually, it is proposed to use timevarying weights on these variables in obtaining ACE. Locally available measurements, i.e. the area frequency error and the algebraic sum of the power in all tie-lines connected to that area, are used as the data required for the algorithm described in this paper. Studies with the proposed controller on a two-area interconnected system show good performance. II. Problem formulation Each area in an interconnected system is a complicated nonlinear system in itself. In a self-tuning regulator 16, a system is modelled by a linear-discrete finite-order model with time-varying parameters. The system operating conditions are tracked by identifying the system model parameters every sampling period using the actual input and output of the system. Based on the updated parameters, control that minimizes the variance of the output is computed. Let area i be modelled as ni ni yi(t) = -- ~ atlYi(t --j) + ~ btlut(t - - k i - - j ) 1=i i=1 ni

+ ¢Ji ~ c i i e i ( t - - / ) /=o

(1)

where yt(t) is the area control error (ACE), ui(t ) is the control variable [APa(t)] , et(t ) represents the uncertainty in the model, ail , . . . , ainl; btl . . . . , blni; ~i; cfl . . . . . Clnt are the model parameters, Cio = 1 without loss of generality, n i is the order of the model, (t --j) represents the time (t --iT), and T is the sampling period. For a system modelled by equation (1), the minimum variance control strategy, as described in Appendix 1, is

0142-0615/86/020120-07 © 1986 Butterworth & Co (Publishers) Ltd

Electrical Power & Energy Systems

ing weight to the earlier values of error squared. Note that p = 1 is a special case that weights all values equally.

1

ul(O = - - [&tl(t) yi(t) + . . . + &ini(t) y i ( t - n t + 1)l {3to -

f3tl(t ) u t(t

-

l) --...

--/Jt~tu (t -- £i)

(2)

The values in equations (5) and (6) can be computed using the following recursive relations:

where £l = nt + k t - - 1, and &q, ~q coefficients are computed from the identified value of the parameters at! and bq. The area control error,yt(t), of area i can be defined as3 Yt(t) = (1 --Pt) APtie t(t) + Pt Aft(t)

(3)

where APtie t is the deviation in area i tie-line power, Af/is the area i frequency error, and Pt is the assigned relative weight for Afi. The objective is to determine the control, APa(t), which minimizes the individual variances of A.f/and APtie i. Consider initially that Pt is a previously selected constant. Then the variance, Vyt, ofyt(t) is given by 16

Vy, = =

e

Ezxpf~i(t ) = pEzxl,tiei(t -- 1) + [APtiet(t)] 2

(7)

Ea~(t) = pEzih(t - 1) + [Afi(t)] 2

(8)

The weight, pi(t), may now be calculated as

Pt(t ) =

EA[i(t) Ea[i(t) + ~.iE,x/,tiel(t)

(9)

where k i, a constant, determines the relative importance of A P ~ i and Aft. The parameter Xi will make one of the variances closer to the minimum than the other. As shown in the studies an optimum value of Xi can be determined. The problem may now be defined as the determination of control, ui(t ), which minimizes the variance ofyi(t ) where

)

+ 2 ( 1 - p l ) p i E ( A P e e i(t) + Aft(t))

(10)

yi(t) = [1 --pi(t)] APtiei(t ) + pi(t) Afi(t)

(1--pi) 2 VAp~t+p~Vzx.1, i (4)

Using yt(t) as the input, the self-tuning regulator calculates the control that minimizes the variance ofyi(t). Since A P ~ t and Aj~ are not independent, the third term in equation (4) will not vanish. Minimizing the variance of ofyt(t), therefore, does not lead to the minimum variance of A P ~ t and A [ t individually. To minimize the individual variances of Af t and APtlet as much as possible, it is proposed to use time-varying relative weight according to the following: 1 Assign Pi a nominal value at the start according to the system requirements under steady-state conditions. 2 Change Pt dynamically according to the variances of frequency error and tie-line power deviation.

and pt(t) is given by equation (9). The proposed regulator is shown in Figure 1. It can be seen that measurements available locally in area i are the only requirements for the area i regulator. A low-pass filter and limiter are added to smoothen the control signal - a desirable feature for economic operation. III.

System studies

A two-area interconnected power system has been represented as shown in Figure 2 to study the performance of the proposed regulator. The system representation includes governor dead-band, power generation limit and generation rate constraint as recommended in an IEEE Committee Report 17. Each area has its own regulator as shown in Figure 1. A quantitative measure of the performance is obtained in

Changing the weight as above will change dynamically the value of ACE. To be near the minimum variance of a variable, the corresponding weight has to be increased as the variance of that variable increases. Hence Pt can be considered as a relative value related to the integral of the weighted frequency error and the integral of the weighted tie-line power deviation.

r-

~

M in.imum I Y; (t) JComputaton variance ~ ACE regulator I J I •

°,:',

1to,"" I

of

/0,

ILl

|

Define the weighted integral of frequency error and tie-line power deviation, Ezxt,faet(t) and Ea[i(t ) respectively, as t

Ezxvtiet(t)= ~ [APmi(j)]2p t-i 1=o

(5)

I ~'~ I Load-frequency regulator

A/D "----I -u

t

EA~(t)= ~ [Ah(j)]2pt-i l=0

(6)

where p is an assigned discount factor, such that 0 < p <<,1. The weighting function, p t-/, will assign a weight equal to 1 to the latest value at time t and an exponentially decreas-

Vol 8 No 2 April 1986

J Control area i APci (t ) "1

Af i (n

I

-

I

t APtiei (t) Figure 1. Control area i with load-frequency regulator

121

AP~, I Deed- b a n d

Generation

I

rate

AP¢I

r°,

1

1 Trl +

I

1 .T,~, I

s

~t P

~ ue!

Turbine Governor Control oree I

Control area 2 l

Limiter

Limiter

l

AP,2

Afz

1 1 • Trz=

1 * Tp==

]

Turbine

t

I

T

AP~2

I

Figure 2. Model for a two-area system

terms of the integral squares of the frequency deviations in the two areas and of tie-line power deviation over a period of 500 s. The integral squares are defined as

For these studies area 1 was subjected to a load disturbance of APal = r/(t) + 0.05~'(t)

(11)

t o + 500

Af~(t)

dr;

i = 1, 2

where r/(t) consists of step load changes, and ~'(t) is a superimposed random load variation.

to

The disturbance considered is shown in Figure 3. to+ 500

Ip = f

AP~.e(t ) dt

to

where to = 10 s. Decision on the quality (goodness) of the regulator is related to the lowest value of the integral square. In these studies it was assumed without loss of generality that/~io = 1 in equation (2).

I V . E f f e c t o f controller parameters The effects of the following parameters on the performance of the power system have been studied: 1 the order of the model by which each area is modelled in the regulator, i.e. the choice o f n i in equation (1),

IV.1 Mode/order To study the effect of the order of the system model used in the regulator (equation (1)), the ratio pi/(1 -- Pi) was assumed constant at the optimum value found in References 2 and 15, i.e. Pi/(1 --Pi) = 0.21. Thus the weights for the two areas were taken as Pl = Pz = 0.173. The sampling period for the purposes of control was chosen as 1 s. Since there are no pure delays in the system, kz and k2 were considered to be zero. Regulators in both areas considered their corresponding systems by models of the

0.50

-L

~0.25.

2 the value of the coefficient Xi, equation (9), 3 the sampling period, T, 4 the effect of the discount factor,p.

122

0

120

240 360 Time, s

480

600

Figure 3. APdl used in the study

Electrical Power & Energy Syitems

the power dispatch requirements, the value o f )` can, however, be changed dynamically to suit each area. 0.075

2.5

If~

2.4

IV.3

0.070 a.

2.3

0.065

2.2

0.060 I 2

I 3

I I I 4 5 6 Model order, M

I 7

I 8

Figure 4. Integral squares of different variables versus the m o d e l order m

Samplingperiod

Integral square values o f different variables for various values o f the sampling period are shown in Figure 6. All results are given for n = 3, )` = 15 and p = 1.0. These studies show that for a sampling period below 0.6 s overall system response deteriorates. The system response is not too sensitive to the sampling period in the range 0.6 s to 1.5 s and thus the choice o f T is not too critical in this range. IV.4 Discount factor The effect o f the discount factor, p , on the system performance is shown in Figure 7 for n = 3, ;k = 15 and T = 1 s. Studies were conducted for p varying from 0.05 to 1.0. F r o m Figure 7 the optimal value o f p is found to be in the neighbourhood o f 0.7.

0.10

2.5-

V.

0.09

2.4 ,. 2.3

o.o84. 0.07

Time

response

System response to a load disturbance in area 1 having a step change o f 0.15 p.u. for Pt constant at 0.173 as determined from References 2 and 15, and for dynamically adjustable Pi, is shown in Figures 8 and 9 respectively. In these studies, n = 3, )` = 15, T = 1 s and p = 0.7. Comparing the results in these two figures it can be seen that the

0.06 2.1~1 1

I 5

I 10

X

I 15

I 20

I25 lO 0.16

Figure 5. Integral squares of d i f f e r e n t variables versus )` 8

0.12 6

same order, i.e. nx = n2 = n and £1 = £2 = £ = (n - - 1). Studies have been conducted for model orders n varying from 2 to 8 inclusive, and the values of various integral squares are shown in Figure 4.

O.OS

zp 4

0.04

#, 2

9.02

It can be seen that as the order o f the model is increased, the system performance first improves and then starts to deteriorate. A perusal o f the results in Figure 4 shows that the representation o f an area by a third or fourth order model in the regulator is adequate. Increasing the order o f the model also increases the computation time, which is an important consideration from the point o f view o f real-time implementation. Using a third or fourth-order model and a sampling period o f 1 s, the proposed algorithm can be easily implemented on many o f the microcomputers available today. IV.2 Effect of ~i The effect o f the parameter )`i on system performance was studied b y varying )` (=)`1 = ),2) over the range I to 25. In these studies, each area was modelled in the regulator as a third-order system. The sampling period was again taken as 1 s, and the discount factor was selected to be 1.

C

Vol 8 No 2 April 1986

0.25

0

I

I

I

I

0.50 0.75 1.00 Sampling period T, s

1.25

1.50

Figure 6. Integral squares of d i f f e r e n t variables versus sampling period T

2.8

O.lO

2.6

Z.,;

0.09

2.4

o.o8

2.2

O.O7

2.0

It can be seen from the integral square values plotted in Figure 5 that the best compromise value o f ) ` for this example, from the point o f view of individual area frequency error, lies in the range o f 10 to 15. Depending upon

I

I O

I I I 0.2 0.4 0.6 Discount factor, p

I 0.8

I 1.0

0.06

Figure 7. Integral squares of d i f f e r e n t variables versus the discount factor p

123

0.25

.50

0

-C .50

-1 .00

V

,o

~oV%^~^~---^--4o Time,s

-

-

.d -5O

o

I

I

I

I

10

20

30

40

•

50

Time, s

d

-0.25

a

C .5C

.0.25[

/•

0

0

20

30 Time,

40

50

~ 4:3

s . . . .

( .5C

-0.25

°Le

16

I

I

I

24

32

40

I

I

I

32

40

Time, s

1.0C ~. 0 . 2 5 [

5 0.25

01 I - I \ ^ /~/k./VV I vv~o L -0.25

~ IA ,~, A /'%1A ~ J ' ~ ' ~ - ~ ~ ~ I v ~ - 20 30 .o 50 Time, s

~ 5°

0

I

1

8

16

-0.25

C

24 Time, s

Figure 8. Response w i t h f i x e d p

N

0.50[

="0.50 20

30

46

50

Time,s

m~0.25

_o.5oLa

d

I

I

I

I

I

~0

2o

30

40

50

I 30

I 40

Time, s 0.50

~"

0

20

<3

30

-

-

740-

50

Time, s

-0.50

\ I X

I 10

I 20 Time, s

,.s

<~-0,25

b _o.5o Le

"25I .~ ~'

0.25

0 /

I

_o.2c

10

I __

_

20

t

I

I

30

40

5O

Time, s

D ¢L •

0


Time, s

50

_f

Figure 9. Response w i t h a u t o m a t i c a l l y adjustable p

124

Electrical Power & Energy Systems

1.00

3

0.75

4

5 ¢L 0.50

6 0.25

7 0 0

I 10

I 20

I 30

I 40

50

Time, s

8

Figure 10. Variation of p 9

frequency deviation with automatically adjustable p is considerably reduced in both areas as compared to the case of constant p. Undesirable power flow on the tie-line is also reduced. The settling time with variable p is reduced to about one-half of that with constant P. For the case of the variable weighting factor, p was assigned a nominal value of 0.5 at the start. With the onset of disturbance, p varied dynamically. Variation of p corresponding to the study given in Figure 9 is shown in Figure 10.

VI. Conclusions A regulator based on a minimum variance strategy for the load-frequency control of interconnected power systems has been proposed in this paper. The control criterion is to track and minimize the variances of the area frequency error and the total tie-line power deviation simultaneously using time-varying weights on these two variables employed in computing the area control error.

10

11

12

13

14

15

16

17

The proposed regulator is two-way adaptive. Firstly, there is online continuous updating of the area model used in the regulator. Secondly, the weights assigned to the frequency error and tie-line power deviations are changed periodically. Studies illustrate that the modified regulator shows a noticeable improvement in system response. Based on the implementation of the self-tuning algorithm for other applications, e.g. synchronous generator control is, it can be said that the proposed regulator can be implemented easily using microprocessors.

18

19

Fo=ha,C E Jr and Elgard, O I 'The megawatt frequency control problem: a new approach via optimal control theory' IEEE Trans. PowerAppar. & Syst. Vol PAS-89 No 4 (1970) pp 563577 Caloric, M 'Linear regulator design for a load and frequency control' IEEE Trans. Power Appar. & Syst. Vol PAS-91 (1972) pp 2271-2281 Glover, 3 D and Schweppe, F C 'Advanced load frequency control' IEEE Trans. Power Appar. & Syst. Vol PAS-91 (1972) pp 2095-2103 Miniesy, S M and Bohn, E V 'Optimum load-frequency control with unknown deterministic demand' IEEE Trans. Power Appar. & Syst. Vol PAS-91 (1972) pp 1910-1915 Bohn, E V and Miniuy, S M 'Optimum load-frequency sampleddata control with randomly varying system disturbances' IEEE Trans. Power Appar. & Syst. Vol PAS-91 (1972) pp 1916-1923 Glavitsch, H and Gallana, F D 'Load-frequency control with particular emphasis on thermal power stations' Real time control o f electric power systems (1972) Elsevier Publishing Company, New York, pp 115-145 Kwatny, H G, Kalnitsky, K C and Bhatt, A 'An optimal tracking approach to load-frequency control' IEEE. Trans. PowerAppar. & Syst. Vol PAS-94 No 5 (1975) pp 1635-1643 Premakumaran, N and Midtra, K L P 'Design of load-frequency controller via invariance principle' Proc. IFAC Symposium on Automatic Control and Protection of Electric Power Systems, Melbourne (February 1977) Doraiswami, R 'A nonlinear load-frequency control design' IEEE Trans. Power Appar. & Syst. Vol PAS-97 No 4 (1978) pp 1278-1284 Bose, A and Atiyyah, J 'Regulation error in load frequency control' IEEE Trans. Power Appar. & Syst. Vol PAS-99 No 2 (March/April 1980) Hiyama, T 'Optimisation of discrete-type load-frequency regulators considering generation rate constraints' Proc. lEE Vol 129 Pt C No 6 (November 1982) pp 285-289 Nanda, J 'Automatic generation control of an interconnected hydrothermal system in continuous and discrete modes considering generation rate constraints' Proc. lEE Vo1130 Pt D No 1 (January 1983) pp 17-27 Calovic, M 'Automatic generation control: decentralized areawise optimal solution' Electric Power System Research Vol 7 (1984) pp 115-139 Astr6m, K J, Borisson, U, Ljung, L and Wittenmark, B 'Theory and application of self-tuning regulators' Automatica Vo113 (1977) pp 457-476 Anon 'Dynamic models for steam and hydro turbines in power system studies' IEEE Committee Report IEEE Trans. PowerAppar. & Syst. Vol PAS-92 No 6 (1973) pp 1904-1915 Shairah, M A, Malik, O P and Hope, G S 'Self-tuning voltage regulator - implementation and test results' IEEE Winter Meeting, New York (February 1979) Paper No A79 060-5 Mendel, J M Discrete techniques of parameter estimation (1973) Marcel Dekker, New York

A p p e n d i x 1: M i n i m u m variance c o n t r o l It is proposed to minimize the variance ofyi, (ki + 1) sampling periods ahead of time. Then, at time (t + k i 4- 1), the model given by equation (1) can be written as y i ( t + k i + 1) + a n Y i ( t + ki) + . . . + ainiyi(t + k i + 1 - - h i )

VI I. Acknowledgements The authors wish to thank the Qatar University, Doha, Qatar and The Natural Sciences and Engineering Research Council of Canada for financial support of this work.

VIII. References 1

2

Concordia, C and Kirchmayer, L K 'Tie-line power and frequency control of electric power systems' AIEE Trans. Power Apparatus Systems Pt IliA Vol 72 (1953) pp 562-572 Elgard, O I and Fosha, C E Jr 'Optimum megawatt-frequency control of multiarea electric energy systems' IEEE Trans. PowerAppar. & Syst. Vol PAS-89 No 4 (1970) pp 556-563

V o l 8 No 2 A p r i l 1986

= bilUi(t) + . . . + biniUt(t + 1 - - n i )

+ ~i[ei(t + k i + 1) + cilei(t + kt) + . . . + Ciniei(t + k i + 1 --n)]

(12)

By writing equation (1) at times (t + kt), (t + k i -- 1). . . . . (t + 1), and substituting in equation (12) to eliminate y t ( t + k i ) , y i ( t + k i -- 1) . . . . . y~(t + 1), equation (12) may be modified to y i ( t + k i + 1) + ctilyi(t) + . . . + aimiYi(t -- m i + 1)

= #~o [ui(t) +

#ilui(t

-

1) + ...

125

Jr ~i~iUi(t -- £i)] Jr ei(t Jr k s Jr 1)

(13)

where m s = ns, £s = ns + ks -- 1, ~so is a previously selected constant parameter, as] and/~s! coefficients are computed from the parameters aq and bs! in equation (1) for all cil = 0, and the disturbance es(t ) is a moving average of order k s of the driving noise ei(t ). For a system modelled by equation (13), if the parameters of the model are constant and known, the minimum variance control strategy from well-known techniques 16 is simply

1

hi(t) = [--Yi(t -- k s -- 1) . . . . . --ys(t

-- k i --

mi ) ,

1)] T

~ s o U i ( t - - k i - - 2 ) . . . . . [JoUs(t--ki - - ~ i -

Oi

=

[O~sl . . . . .

Otimi, ~il . . . . . ~i~i ]T

and [ ]T denotes the transpose of a matrix. A number of recursive-parameter estimation algorithms can be used to obtain an estimate, 0s(t), for the parameter vector 019. Using the more commonly employed recursive least-squares technique, the estimate, Os(t), of 0 s can be obtained a s 1 6 ' 1 9

ui(t) = ~io [C~tlYs(t) + "'" + C~SmsYi(t-- ms + 1)] Os(t) = Os(t -- 1) + Ks(t ) [zs(t ) -- hT(t) es(t -- 1)]

--/3Sl Us (t -- 1) - - . . . -- ~i~sU i(t -- ~i)

(16)

(14) The correction vector, Ks(t), can be calculated as

For systems with unknown parameters, the model parameters need to be identified. The estimated values of the model parameters can then be used to calculate the control. Estimation of the model parameters may be required every sampling period. In the self-tuning regulator the parameters 0 i n , . . . , O t i m i and {3sl. . . . . {Js~s in the assumed model, equation (13), are estimated online at any sampling instant t. The estimated values, &il . . . . , ~Sm I and/~tl, • - . , fli~ i, of these parameters are then used to calculate the minimum variance control strategy, equation (4). To estimate the model parameters, equation (13) may be rewritten at the instant t, by replacing t by (t - - k i -- 1) in equation (13), in the following form: z i ( t ) = hT(t) @S

where zs(t ) = Yi(t) -- [JioUs(t -- k i -- 1)

126

(15)

K/(t) =

Ps(t -- 1) hi(t ) 1 + h T ( t ) P i ( t -- 1) h/(t)

(17)

where P/(t), the covariance matrix of estimation error, can be obtained using the recursive relation Pi(t) = [I -- K/(t) hT(t)] Ps(t -- 1)

(18)

Initial values of Oi(t ) and Pt(t) can be chosen arbitrarily. A recommended choice is 0t(0 ) = 0 and Ps(0) = 3'1 where 3' is a relatively large number and I is the identity matrix. Once the estimate 0s(t) is obtained, us(t ) can be calculated as

1

u t ( t ) =-2-- [&il(t)yi(t) + . . . ~so

+ &tmi(t)yi(t--mi

-- ~il(t) u i ( t -- 1) - - . . . -- ~i~i(t ) u i ( t - - ~ i )

+ 1)]

(19)

Electrical Power & Energy Systems