A Decentralized Load Frequency Control Method by Means of Frequency Measurement of Adjacent Power Systems

CopITight © IFAC POI"er S,'stems \foc\ellillg and Control Applica tions, Brusse ls, Belgium 1988

A DECENTRALIZED LOAD FREQUENCY

CONTROL METHOD BY MEANS OF FREQUENCY MEASUREMENT OF ADJACENT POWER SYSTEMS H. Sasaki*, H. Yorino*, T. Suizu*, S. Yurino*, R. Yokoyama** and Y. Tamura*** *Depa rtml'llt of [/('((rim/ Enginl'ning, Hiroshima ('ni"l'rsity. Shitllllli, SlIIju-d/() , H iga"hi-Hirushima i2-1, j apan **Dl'p" rtm elll of E/('(trim/ Engineering, Tokw .\letropulil(/n ( 'lIh'eni/y. 2 -/ - / Fllk,,:azl'a, Sl'lagaw-ku, Tokro J 58, j apan *** DI'/Jllrtllll' nt IIf E/('ctrim/ Enginerring, 1\ '({51'da L'ni,'l'nit\', 3--1 - 1 ()hkll/W , Silljllkll -kll, Tukw / 60, jll/)(lIl

Abstract. The presently used TBC and FFC methods have satisfactorily maintained power system frequency within a prescribed tolerance. With steady increase in power system capacity, however, undue fluctuations in tie-line power flow has become prominent; This fact suggests the sheer necessity of establishing a more sophisticated LFC system, Intensive research efforts have been concentrated on applications of modern control theory, above all the linear optimal control (LOC) using the state regulator theory. This paper presents a decentralized suboptimal LFC method which makes use of the frequencies of adjacent areas in addition to local information to calculate the control variables of each area. This artifice has made it possible to realize a practical yet almost optimal control scheme. Simulation studies on a 4 area system have demonstrated the effectiveness of the proposed method. Keywords. Load frequency control; discrete systems; computer control; optimal control .

decentralized control system;

In the second place, it is ne cessa ry to feedback all the state variables of a system under control to compute the optimal feedback Signal. This requirement must be a great obstacle when applying , to a power system. Although many power systems are interconnected to each other to realize stable and economical operations, each constituent system is still operated independentlY, thus being unable to gather the state variables of other companies to one specific point. Even if it is agreed to do this, a large scale communication link must be installed since a power system geographically spreads over a wide area.

INTROOOCfION The prime function of the load frequency control (LFC) is to maintain the frequency deviation of a power system arising from random load variations within a tolerated band. Another important duty is to reduce power fluctuations on tie-lines from their nominal values. The LFC dispatches control commands to generators dedicated to the LFC functions 50 as to adequately control their active power outputs. The conventional TBC and FFC methods based On the classic PI controller have been quite successful as to keeping the system frequency around its nominal value.

Inevitably, a decentralized LOC system should be constructpd for the application to power systems, in which each power system makes use of only locally available state variables, that is, the state variables of its own and tie-line powers (local state variables). However, when a centralized LOC system is decomposed into a set of subcontrol systems each of which utilizes the local state variables, there occurs a severe difficulty. That is, the coefficient matrix of the equation of a power system which interconnects to more than one system has zero eigenvalues, thus making it impossible to stabilize this system.

In recent years, however, unusually large tie-line power fluctuations have been observed as a result of increased system capacity and very close interconnection among power systems. This observation suggests a strong need of establishing a more advanced LFC scheme. Many novel approaches have been proposed 50 far to overcome this interesting issue. Above all, the most promising approach is the application of linear optimal control (LOC) theory based on state regulator to the LFC (Fosha & Elgerd, 1971). The LOC has excellent characteristics in that it is able to control a system with small transients and relatively short settling time. Nevertheless, the straightforward application of the LOC brings about some difficulties. The first point is that it cannot offset a sustained disturbance when its magnitude is not known. As disturbance in a power system is discrepancy between the generation and consumption, it is impossible to predict exactly how system load changes due to its random nature . This difficulty is first solved by Calovic (1972) by incorporating the integrals of the area control errors (ACE) as other state variables in order to cancel offset errors for a sustained disturbance. This idea stems from the fact that no offset errors exist in the classical integral control.

Mizutani (1978) has proposed to superimpose a small compensation matrix on the system matrix to overcome this difficulty, However, this devise is just for convenience to prevent the appearance of fixed modes and hence has no effect on improving the quality of control since this does not make use of no more than the local variables. In the present paper, linear regulator theory is first extended to include the integral of the ACE, and then a discrete-type optimal control system is constructed. In order to develop a practical decentralized LOC system, the authors propose that each area makes

387

388

H. Sasaki et al.

use of the frequency measurements of its adjacent areas in addition to the local state variables to independently compute feedback control signal. The proposed decomposition method does not incur difficulty of producing fixed modes, thus making all decomposed subsystems controllable. By usage of frequency measurements, each area can obtain more information on adjacent areas than merely using tie-line powers. Therefore, i t is expected that the proposed decentralized control system has nearly the optimal control effect, which can only obtained by a centralized LOC system. In the first place, we shall formulate the LOC on the LFC problem for' an interconnected power system as a whole. Next, the decentralized sUboptimal control system will be developed based on the LOC theory and the decomposition by use of frequency information. Finally, simulation studies have been carried out on a four area model system to assess the effectiveness of the proposed decentralized suboptimal control system, especially in comparison with the TBC and the centralized LOC. Simulation results for disturbances of step change and ramp change have demonstrated the excellence and practicality of the proposed approach against the current TBC method .

x(k)=xr(k)-x., u(k) = ur(k) - u" p(k)=pr(k)-p. Substituting the new variables into eq. (2) and taking into account eq. (3), we have the following expression:

x(k+ 1) = 4>x(k)+ 1fu(k)+~p(k) + 4>x.+ 1Fu.+~P.-x. Since p(k) = Pr(k) - Pe,= 0 for known step load changes, the above equatlon reduces to that devoid of the disturbance term: (4)

x(k+ 1)= 4>x(k)+
As already mentioned in the Introduction, the integrals of the area control errors (ACE) are incorporated as other state variables in this study to suppress the offset error. The ACE is defined as

ACE = BI1F+I1Ptie where B denotes the frequency bias, ,dF the frequency deviation, and ,dPtie the deviation of tieline power. Let us denote by get) the integral of the ACE and then we have

g(t)= ~(BI1F+ I1Ptie)dt DESIGN OF TIlE OPTIMAL CONTROL SYSTI'}1

Approximating derivative ference, we may have

Discretization of system equation Let us assume a power system under study consists of m areas. Then, its dynamic performance can be described by the following differential equation in state space form:

(S)

dg/dt by a finite

g(k+1)=g(k)+ TIBI1F(k)+I1Ptie(k)}

dif(6)

Here, the quantity inside the parentheses on the right-hand side signifies the ACE in the discrete version. In terms of an appropriate observation matrix D, the ACE can be expressed as

(1)

ACE(k)=D:e(k) where

:er Ur

Pr

A 11

r

Let denote by .i:(k) the new state variables as augmented by g(k):

n x 1 state ,vec tor m x 1 control vector m x 1 disturbance vector n x n system matrix n x m control matrix n x m disturbance matrix

.t(k) = Cx(k)

g(kW

(7)

By using eq. (7) and the ACE(k) above, eq.(6) can be expressed in the following matrix form:

The present LFC constitutes an integral part of the energy management system (EMS), the functions of which is exclusively processed by digital computer systems. Inevitably, the LFC becomes a discrete type of control in which control commands are given at some sampling rate. Therefore, the LFC must be formulated in discrete form to discuss its true control effects. Following the standard discretization procedure via sampler with sampling time T and zero-order hold circuit, we may obtain from eq. (1) the following difference equation:

g(k + 1) = CD

(8)

J).t(k)

This expression means that the ACE at the k+1 step can be computed from .i:(k). Thus, the augmented system equation in discrete form is obtained as

.t(k+ 1)= lD.t(k) + tuCk) where the coefficient matrices in the defined as

(9)

above

are

(2)

where '=exp (A T)

f'= (exp(A T)-I)A-'11 1/= (exp(A T)-I)A -Ir The Application of Linear Optimal Regulator In applying the LOC theory, equations describing a dynamical system should include no disturbance terms. That is, it is tacitly assumed that any disturbance is a step change of known magnitude. For known step load changes in some areas, eq. (2) becomes (3) where xe ' ue·and Pe denote the equilibrium points of the state, control and disturbance vectors, respectively. Now, let us define new variables as deviations from thp.ir p.qui1ibria:

In order to apply the optimal regulator theory to system equation (9), let us define the following quadratic performance equation: 00

J= 1: (.t(k)TQ.t(k)+u(k)TRu(k»)

(10)

1=0

where, Q denotes an n x n positive semi-definite matrix and R an m x m positive definite matrix. The optimal control input in the aense that it minimizes performance index (10) is given by

A Decentralized Load Frequency Control Method

389

u(k)= -Fx(k) =_(R+"TS7fr)-I"TS~x(k)

(11)

Matrix S involved in the above equation is a positive definite solution matrix of the following discrete type Ricatti equation:

Several remarks on the above opti mal control are now in order at this point . Although the control effect of the above scheme varies wid e ly depending on how matrices Q and R are selected, the optimal effect in th e sense that e q. ()O) is minimized will be gained only through a cent ralized control system. That is, all the system states must first be transf e rr ed t o a certai n control center insome area to compute the optimal contr ol output, which in turn e re sent back to each system to execute required controls. This implies that a highly reliable communication networks must be installed among the common control center and eac h systems. This is not totally acceptable from the point of view of economy and also of r eliability.

(a)

ui

DECENTRALIZED SUIJOPTIMAL CONTROL Difficulties in a Decentralized Control System The list of symbols: · In an interconnected power system, each area independently takes charge of the LFC functions, that is, it is operating its own LFC stations without any commitment from other interconnected systems . except the case of determining the amount of power exchange. Judging from this current practice, it is entirely impractical t o adopt a centralized LFC system mentioned in the previ ous section. Therefore, it is mandatory to construct a decentralized control system in which each system makes use of locally available inform ~ tion to compute control variables. However, a simple decomposition of a centralized LOP controller to a decentralized one brings about difficulty of some area becoming uncontrollable.

t..Pti e t..Fi llP ti t..XEi t..Pdi Tti Tgi Kpi Tpi Ri Tij aij

incremental change in tie-line power flow incremental frequency deviation incremental change in generation incremental governor valve position change incremental load demand change turbine time constant 0.3 sec. governor time constant 0.08 sec. system regulating energy 120 Hz/puMW power system time constant 20 sec self regulati on of generator 2.4 Hz/puMW synchronizing coe fficient 0.0707 sec. negati ve ratio between rated MW of areas i 10.2) and j (a12 = -1.5, a13 = -0.6, a14 E

The cause of this difficulty will be explained by using a 4 area system s hown in Fig. 1. In establishing a decentralized LOC contr oller for the LFC, let us ass ume that each subsystem utili zes its state variabl es and tie-line powers. In this example system, systems 2, 3 and 4 have only one tie-lines, and t herefo re the con tr ollability is preserved after the decomposition. On the other hand, system 1 has interconnectio ns to these three systems. Let the following eq ua ti on be the system equation to rlescribe the perf ormance of system 1 :

XI(t) = AIXI(t) + BI UI(t) where

(13)

XI = (LlFI, LlPTI, LlXEI, LIP tiel2, LIP tiell, LIP tieu)T

The system transiti on matrix A) is expressed as

1 -T' I 0

AI=

K. I

1 0 -Tg1RI

27r TI2 27r Tu 27r Tu

0

T,I 1 1 -TII TII 1

0 0 0

-Tgl

0 0 0

K, I

K, I

K,I

Fig . 1 (a) (b)

Four area model system The block diagram of area i

It is obvi ous that co lumns 4, 5 and 6 of matrix Al are dependent to each other. In other words, there appear zero e igenvalues (fixed modes) in this system, thu s becoming impossible to make it controllabl e. Mizutani(1978) proposed to superimpose a small compensation matrix on the system matrix to overcome this difficulty . That is, AAl + C is used as a new system matrix, where C has the same size as that of Al ' However, this devise is just to prevent from the appearance of fixed modes and hence has no effect on improving the quality of control since available information about interconnected systems are those associated with tie-line power. Utilization of Frequency of Adjacent Systems

-T,I -T,I -T,I 0

0

0

0

0

0

0

0

0

0 0

0 0

0

0

It is common that the central dispatching office of each system receives the frequency measurements of its interconnected or adjacent systems, as is the case for electric utilities in Japan. Even if this is not the case, it is a relatively easy task to have the frequencies of adjacent systems. The authors propose to use the frequency information in decomposing the optimal regulator control system. The system equation of system 1 becomes

H. Sasaki et al.

390

SIMULATION RFSULTS AND DISCUSSIONS

(14)

where

Model System

%1"'= (-dF1, -dPT1, -dXE1, -dPtie12, -dPtie13,

A four area radial system shown in Fig. l(a) is chosen a model system in the application of the proposed decentralized suboptimal LFC system. It is assumed further that each system is represented

-dPtieu, -dF" -dF3, -dF.)T The system transition matrix Al. is expressed by

1 -Tr, 0

AI'" =

1 T g2RI 2" TI2 2 1': TIJ 2r:Tu

K pl Tpl 1

0

1 Tu 1

Tn

K pl T pl

K pl -T p,

K p, -Tp,

0

0

0

0

0

0

0

0

0

0

0

0

-Tg]

0

0

0

0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

-2;rTI2

0

0 0

-2;r TI3

0 0

0

-2;rTu

0

0

0

0

0

0

1 T"

0

0

0

0

0

0

_ alZK p2 T p2

0

0

0

0

_a13K p3 Tpl

0

0

0

0

0

Obviously, there are no dependent vector set in this system. Furthermore, it is expected thut the incorporation of the frequency measurement greatly improves the · control performance of the proposed decentralized LFC control system.

au K p4

(l)i=exp(Ai T), Fi= (exp(Ai T)-J)Ai-IBi Note that xi in equation (15) contains the devi.ation of the frequency of systems conn ected to system i through tie-lines. The integral of area control error (ACE) of system i can be expressed in discrete form as

(li(k+l)=(li(k)+ TIBi-dFi(k) +-dPtiei(k)}

(16)

Adding gi(k) to xi(k) in eq.(15), we shall denote by %i(k) a new state variable. Then we have

.tl(k+ 1)= c6i.ti(k) + t i ui(k)

(17)

The following performance equation is associated ~'ith the above system equation :

x

r:

... 0

LlP tieu, LlP" LlPT., LlXE.) T

u = (LlUI, LlU2, LlU3, Llu,) T p= (LlPdl, LlPd2, LlPd" LlPd.)T Significance of Incorporating the Integral of ACE As stated previously, the optimal control law without the integral of the ACE has a defect that offset errors remain for a sustained disturbance of unknown magnitude. The effect of including the integral of the ACE into state variables will be demonstrated here. Providing that a step load change of 0.01 puMW occurs at t = 0 in area I, the following three cases are tested: (a) Without the integral of the ACE (disturbance of known magnitude) (b) Without the integral of the ACE (disturbance of unknown magnitude) (c) With the integral of the ACE (disturbance of unknown magnitude)

defi~ed

(.ti(WQi.ti(k)+Ui(W Riui(k))

The suboptimal control law which (18) may be given by

(LlP I , LlPT J , LlXE I , LlPtiel2, LlP2, LlPT2,

=

LlXE2, LlP tiell, LlP3, LlPT" LlXE3,

(a)

(b)

co

J I=

1

T p4

by the block diagram in Fig . l(b) . This modelling might seem impractical , but still is useful to evaluate the capability of the proposed method. Referring to this figure, state vector x, control vector u, and disturbance vector p of the total system may be defined as

We shall formulate a decentralized suboptimal LFC scheme in discrete form, which includes the integrals of the ACEs as state variables. First, the overall system equation should be decomposed into a set of subsystem equations. Discretizing each subsystem equation at sampling time T with the zero-order hold, we obtain the following:

where

T p2

-----y;:-

Formulation of Decentralized Suboptimal LFC System

(15)

1

(18)

g

(c)

c

minimizes

eq .

UI(k)= - FI.tI(k)

=-

(RI+tI T Sitl)-I'? sic6i.ti(k)

-·""'\r:mr----e:ov---rv:w--TIME (SEC)

(19) N

where Si is the s"lution matrix of the discrete type Ricatti equation:

SI=QI+'ITs,',-c6I TSltl(RI +'ITSltl)-ltIT'1

following

(20)

C

9

Fig.. 2. The effect of incorporating the integral of ACE.

391

A Decentralized Load Frequency Control Method It should be noted that in case (a) the magnitude of disturbance is ~ pri-ori given as an input to the control system. Fig. 2 displays the responses of 6F1 for the above 3 cases. First, case (a) shows excellent response judging from small transient deviations and fast settlement time. It is, however, impossible to know in advance the magnitudes of disturbances. In case (b), an offset error remains since the control system has no capability to suppress a sustained disturbance. In other words, the discrepancy is not handled by area 1 but by other related areas. Finally, it is shown that the control system in case (c) takes care of unknown magnitude disturbances, though the settling is rather delayed.

O.OlPUMWI O'L-------------~tl~·n-Ie~(-S~)

(a)

time(s) (b)

Fig. 3

Load disturbance model (a) a step change ; (b) a ramp change

Decentralized Suboptimal Control System In this section, the characteristics of the proposed decentralized system will be investigated in comparison with the current TBC system and the centralized LOC system. Throughout simulations in this section, these 3 LFC systems are labelled as (a) the present TBC system (b) The optimal LOC system (c) The proposed decentralized system In KI

=

the TBC system, its integral gain is fixed at 0.35 since this gives the best responses.

In constructing the decentralized LFC system, let us assume as mentioned in the previous section that each area could obtain the frequency of its adjacent areas. Then, the local state variables of each system become as follows: Xl=

(1) Response with respect to step disturbance Let assume that a step load change of 0.01 puMW as shown in Fig. 3(a) occurs in area 1. Fig. 4 shows the responses of 6F l ,and 6Ptie' which are chosen as representing typ1cal responses. The sampling time is T = 1.0 sec. This figure has brought about the following observations: While the TBC needs more than la sec. until transients have completely faded away, the proposed method has absorbed the step load change by 10 sec., providing significant improvement in the transient response of the LFC. Furthermore, its overall control effect is practically comparable with that of the centralized LOC. This must be attributable to the utilization of the frequency of adjacent systems.

(JFI , JPT I, JXEI, JPtielz, JPtjel3,

TIlC LOC

JPtiel., JFz, JF3, JF.JT

The proposed method.

X2= (JFz, JPT z, JXE z, JP tielz, JFI) T X3= (JF3, JPT3, JXE3, JPtiel3, JFI)T

\.A.oo

0.00

11.00

10.00

T I ME (SEC)

x.= (JF., JPT., JXE., JPtie14, JFI )T Table compares the solution time of the Ricatti equation for the centralized and decentralized systems. Though solution time for area 1 is the longest among the 4 areas, it is about 1/7 the solution time of the centralized LOC system, which must handle a large number of variables. This fact suggests that the centralized system is much more preferable for on-line use since the RicaTti equation must be solved every time system operating conditions change appreciably .

11.00

10.00

TIME (SEC)

Fig. 4

Responses under a step load disturbance.

(2) Response with respect to ramp disturbance TABLE 1

Comparison of the solution time of the Ricatti equation (double precision) LOC

solution time

14.31 s

decentralized area 1

2.10

s

area 2

0.54

s

area 3

0.51

s

area 4

0.51

s

Now, we shall assume a ramp disturbance in area 1 as in Fig. 3(b). Fig. 5 again shows the responses of 6F l and 6Ptie' In a practical sense, the proposed system has obtained as good control effects as the optimal control system (b) though it being somewhat oscillatory. It should be noted that the maximum excursions of the frequency and tie-line power deviations for the decentralized control are roughly half compared with the TBC. The TBC necessitates about 14 sec . until the frequency deviation recovers to a small value and yet is still in oscillation. In the proposed method, however, the frequency deviation becomes zero in about 10 sec and then gradually damps away.

392

H. Sasaki et al. TBC LOC c

................... ".... ----' - ........ '- ..... -

......

'-- - ......

llP t ie12 Fig. 5

Re sponses under a ramp l oad di s tu r bunce .

(3) Responses for different sampling times

(4) Se nsitivity with re s pec t t o mode l parameter

It is of prime importance to investigate how the response characteristics of the proposed method are affected by different sampling times . The sampling time is changed as 0.5, 1.0 1.5 and 2.0 sec. Fig. 6 shows the responses of llFl and llPtie for a step disturbance of 0.01 puMW in area 1 . In general, the shorter the sampling time, the better the responses since the control system approaches to a continuous system. For the sampling time of up to 1.0 sec., the system gives excellent control effects. When the sampling time is 1.5 sec . or longer, the responses become oscillatory.and n~ed longer settling times. As far as the s1mulat10n results are concerned, it seems that the samp11ng time of 1.0 sec. is favourable in this particular sample system. However, it is necessary to select as long a sampling time as possible in order to make the proposed system be practical. In view.of the present practice under the TBC, the samp11ng time of 2.0 sec. would be acceptable since transients diminishes in about 10 sec. o ---llFl _._._.- llP t iel3

Wh en a control system is desi gn ed on a linearized system, it is very rare that model parameters can be regarded to be constant du e t o modelling error and th e effect of nonlin ear e l ements . Therefore, it is ma ndatory to e xamine the parameter sensitivities o f the proposed dec e ntra lized system. Here, the se nsitivity with respec t to th e time constant of a turbine has been obtaine d since it has prominent influ ence on co nt ro l sy s t em performance. Simul a ti ons have been carried out on th e condition that th e time constant o f the turbine in area 1 is pe rturbe d by 30% fr om its tru e va lu e. TBC Right sy s t em para me t e r s Turbine tim e co nst ant : +30 % increase

8, 00'"

N

TIME

10,00

( SEC)

°1 ~g~--~~~--~~~~~==r=~~=;~=;==~~~==-=

>< -0 -g

-

0

; 0

-

°0

~

~

..... ~

;: 9

4 . 00

6 . 00

1;1.00

10 . 00

TIME

g ~

T

(SEC)

0.5 sec

7

0

o x

~ g~---r~~--,r~,-,f~~~~~~t'~'~t7~~~~~ \ ,y~OO , .. ' B 00 10.00

Cl. N

o

TIME

(SEC)

llP ti e 12

Th e pr oposed meth orl

T

= 1.0

sec

N

o

N

o

Right system parame ters .-----. Turbine time c onstant: +30 % increase

o

o 6 . 00

6.00

8 . 00

TIME N

w-

~o I- ,

8 . 00

TIME

10 , 00

10.00

(SEC)

(SEC)

N

o

';'

T

1.5 sec

Cl.

-~ o

o

o ><

~o

o

;0

4.00 6 . 00

Cl.

B.OO

10.00

TIME N

w-

~o I- ,

(SEC)

N

o

';'

T

2.0

sec

6 . 00

8 . 00

TIME

10.00

(SEC)

wo

~o I- ,

Cl.

Cl.

Fig. 6

Responses for different sampling time T.

Fig. 7

Responses under parame ter variation.

393

:\ Decentralized Load Frequency Control Method

Fig. 7 depicts the simulation results both for the TBC and the proposed method. It has made clear that the proposed method has less sensitivity to error of the turbine time constant than the TBC. That is, the TBC method oscillates significantly when the time constant is not exact, and therefore a longer settling time is needed. The authors can claim that the proposed decentralized system is quite robust for parameter variations.

REFERENCES Calovic, M.S. (1972). Power system load and frequency control using an optimum linear regulator with integral feedback. Proceedings of Fifth IFAC Congress, 1-9 Forsha, C.E. and Elgerd, 0.1. (1971). The megawatt frequency control problem, a new approach via optimal control theory. IEEE Trans. on PAS.,

90, 563-577 CONCLUDING IIDtARKS

In this paper, the authors have proposed a decentralized suboptimal LFC system based on the LOC theory to realize a more advanced LFC system. In order to prevent fixed modes appearing in a subcontrol system and also to pursue better control effects, we have advocated that each subsystem should utilize the frequency measurements of its adjacent systems in addition to locally available state variables. This requirement is technically very easy to implement in view of the current status of central dispatching offices . Through simulation results on a 4 area model system, it is demonstrated that the proposed system has clear advantages over the current TBC system, which will be summarized below. (1) When the frequency measurements of adjacent systems are used, the decentralized control system shows practically the same standard of performance as a centralized LOC system. Needless to say, it has much better responses than the TBC . (2) The solution time of the discrete type Ricatti equations for the decentralized system is much shorter than that of the optimal control law. (3) The proposed system is turned out to be quite robust for parameter variations such as generatorturbine time constant, compared with the TBC. Finally, the authors would like to express their sincere thanks to engineers of the Power System Operation Department of the Chugoku Electric Power Company for their suggestive comments and assistance extended on this study.

Mizutani, Y. (1978). A suboptimal control for load frequency control system using an area decomposition method (in Japanese). Trans . lEE of Japan, 98-B, 971-978