Discrete-mode automatic generation control of a two-area reheat thermal system with new area control error

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730

IEEE Transactions on Power Systems, Vol. 4, No. 2, May 1989

DISCRETE-MODE AUTOMATIC GENERATION CONTROL OF A TWO-AREA REHEAT THERMAL SYSTEM WITH NEW AREA CONTROL ERROR

M . L . KOTHARI J . NANDA D . P . KOTHARI D . DAS INDIAN INSTITUTE OF TECHNOLOGY

NEW D E L H I - 1 1 0 0 1 6 , I N D I A

Keywords: Power system c o n t r o l , Automatic g e n e r a t i o n c o n t r o l . Discrete-mode c o n t r o l Abstract - This paper deals with discrete-mode au-

tonatic generation control of an interconnected reheat thermal system considering a new area control error

(ACEN) based on tie-power deviation, frequency deviation, time error and inadvertent interchange. Optimum

integral and proportional integral controllers using the concept of stability margin and ISE technique have been obtained with conventional ACE and new ACE, and their dynamic performances compared for a step load perturbation. Results reveal that regulator based on

the new ACE concept always guarantees zero steady state time e r r o r and i n a d v e r t e n t i n t e r c h a n g e u n l i k e in the case of a c o n t r o l l e r based on conven- t i o n a l ACE. The s e t t l i n g time for tie-power and frequency d e v i a t i o n s is however, somewhat more with t~he c o n t r o l l e r based on new ACE.

NOMENCLATURE

aPtiei AF.

nominal system frequency

subscript referring to area i (i=1,2) incremental change in t i e - l i n e power incremental frequency deviation AP • i n c r e m e n t a l g e n e r a t i o n change

AXg. i n c r e m e n t a l governor v a l v e p o s i t i o n change APjy i n c r e m e n t a l l o a d demand change

H. inertia constant

D^ load frequency constant (K .=1/Di,T .=2H./fD.) K . high pressure turbine power fraction

T . reheat time constant

T12 T .

T

ti

R.

synchronising coefficient speed governor time constant

steam chest time constant (control valves to HP exhaust)

rated area power (a-2=-P VP 2)

self regulation parameter for the governor of the ith area

83 SM 7 25-4 A paper recommended and approved by the IEEE Power System Engineering Committee of the IEKE Power Engineering Society Eor presentation at the IEEE/PES 198R Summer Meeting, Portland, Oregon, July 2 4 - 2 9 , 1938. Manuscript submitted September 1, 1987; aade available Eor printing May 11, 1988.

tiemax

frequency bias constant

proportional and integral gains respectively.

time error bias setting in MW/sec(m=1,2) inadvertent interchange bias setting in s e c o n d "1. (m=1,2)

nominal phase angle of voltage(s ] =5.-sJ conventional area control error

(ACE.=AP . .+B.AF.) i tiei i i' maximum t i e - l i n e power

area frequency response c h a r a c t e r i s t i c , sampling period

time error (i=1,2)

inadvertent interchange accumulation (i=i,2) ACEN. new area control error

' prime notation stands for transpose Introduction

Most of the work reported in the literature pertaining Automatic Generation Control (AQC) of interconnected power systems is centered around tie-line frequency bias control strategy. Supplementary controllers are desig- ned to regulate the area control errors to zero effectively. Several modern design techniques have been used to optimize the parameters of the supplementary contro- l l e r s . Supplementary controllers regulate the generation so as to match load variation. As the generation change chases the load variation, the frequency and tie-power deviate from the scheduled values. This would result in accumulations of time error and inadvertent interchange. Time error and inadvertent interchange accumulations would also occur due to errors in measurement of frequencies and tie-powers, scheduled frequency and tie-power settings or intentional offsets in scheduled settings. It is expected that individual areas will make a l l reasonable efforts to minimize time error and inadvertent interchange accumulations by minimizing or eliminating source causes. Accumulations will nev- ertheless occur and there is a need for correcting them.

Such corrections are achieved by making appropriate offsets in system frequency schedules to compensate for time error accumulations and offsets in area net interchange schedules to compensate for inadvertent interchange accumulations. Detailed literature survey shows that the above mentioned two-step correction scheme has been used by u t i l i t i e s , inspite of practical difficul- t i e s . In order to avoid such practical difficulties the u t i l i t i e s are looking forward for a control strategy that not only maintains constancy of system frequency and desired tie-power flow but also achieves

?ero steady s t a t e time e r r o r and i n a d v e r t e n t i n t e r c h a n g e . I t i s e s s e n t i a l l y i n t h i s d i r e - c t i o n , the i n v e s t i g a t i o n s are c a r r i e d out i n t h i s work.

Nathan Cohn [1] had proposed a new technique of coordinated, system-wide correction of time error and

inadvertent interchange. The modified expression for area control error for mth area % is

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(1) set T' measured value of net interchange of area m in MW.

1" scheduled value of net interchange of area m in MW.

cm

H = time of correction in .: -.:•..,

F' =measured value of frecuency of area m in Hz.

F1 =scheduled frequency of area m in Hz.

b =time error bias setting in Hz/sec.

e =time error in sees.

The general objective of using above expression for area control error is to provide for coordinated, system-wide correction of time error and inadvertent interchange accumulations. Additionally, it provides for achieving normal system frequency by creating and maintaining a counterbalancing sustained time error when there is unaccounted for inadvertent interchange. When all areas regulate their ACEs (as defined above) to zero, inadvertent interchange and time error will also be regulated to zero. Not much work is available in the literature using the modified expression for ACE. Fouad and Kwon [2] have investigated AGC problem of a 3-area system using the modified expression for ACE. They have, however, not mentioned how frequently the terms inadvertent interchange and time error are updated. In a dynamic sense the 'time error fem) accumulation of each area is different. It is worth investigating the AGC problem of a two area reheat thermal system using a new ACE defined as

or ACENm= where.

V TO

M F

m

d t : = / A Ptiemd t

(3)

(4)

^ m = A Ptiem + VFm (2)

This equation is different from Eq.(l) since £ and I are, respectively, the time error and inadvertent inter- charge of area m and are updated at every sampling ins- tant. If all the areas regulate A C E Nm to zero, then obviously AF

m' "tiem New area control error, ACEN

ACEN = AP. . +B AF m tiem m m

e_ and I

wifl

reduce to zero.

may be rewritten as

Thus the new area control error ACENT is sum of conventional ACE and integral of conventiohal ACE. By setting a

— = 60 B , it is ensured that the necessary assistance a m

m

is provided to a d e f i c i e n t area as per the programmed b i a s s e t t i n g . The main objectives of the p r e s e n t work a r e :

1) To obtain optimum value of a and i n t e g r a l gain s e - t t i n g KT of the supplementary c o n t r o l l e r considering new area c o n t r o l e r r o r .

2) To obtain optimum i n t e g r a l and optimum p r o p o r t i o n a l - i n t e g r a l c o n t r o l l e r s using new area c o n t r o l e r r o r and to compare t h e i r performances with those of the corresponding c o n t r o l l e r s using t h e conventional area c o n t r o l e r r o r .

System Investigated

The AGC system comprises of two equal area reheat thermal systems provided with supplementary c o n t r o l l e r s . A s t e p load p e r t u r b a t i o n of 1% of nominal loading has been considered in a r e a - 1 . The nominal parameters of the system are given in Appendix-A. A sampling period of T = 2 s i s considered. Generation r a t e c o n s t r a i n t has been neglected.

Transfer Function Model

Small p e r t u r b a t i o n t r a n s f e r function block diagram of a two-area reheat thermal system is shown in Fig. 1 [ 3 ] . The c o n t r o l vector in continuous mode can be given as

t

u (t) =-K, .ACENm(t)-KI / AC&Jm(x) dx (5)

AF|(S)

F i g . 1 : Transfer function model of a two-area reheat thermal system.

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732

The discrete-time equivalent'of Eq. (5) is k

u (k)=-K. .ACEN (k)-KT Z ACENm(p).T

m P m J- p 0 m

(6)

p

k=0,1,2, ,etc. T is the sampling period. Unless otherwise stated u (k) and ACEN (k) indicate u (kT) and

A C E N

m

( k T ) m m

Discrete-Time Dynamic Model

The continuous-time dynamic model in the state space form is

X = AX + BU +rp (7) X , uand p are the state, control and disturbance vectors

respectively. A,B and r are compatible matrices and depend on system and controller parameters. The discrete-time behaviour of the continuous-time system is modelled by the system of first-order linear difference equations:

= »X(k) + ¥U(k) + \>p(k)

(8)

X(k), u(k) and p(k) have the same dimensions as in the continuous-time description and are specified at t=kT, k=0,1,2,—,etc. T is the sampling period. Unless otherwise stated X(k) indicates X(kT). That is X(k) implies the vector X(t) at t=kT, the sampling instant.

4, * and v are the state-transition, control transition and disturbance transition matrices. * , ¥ and v can be evaluated using the following relations.

eA T; ^ A T ^ ^{A T —1}

3: v=(e - I ) A r

(9) In Eq.(9) I is an identity matrix.

Matrices * , * and v are the function of sampling period T. Once T is fixed these matrices become time

invariant-' The introduction of the discretization requires that the inputs be piecewise constant function of time and changes in the values of u and p occur only at the sampling instants. The determination of the discrete-time model from the continuous model requires the evaluation of A~1 , therefore, the continuous-time dynamic model must be of minimal order.

Stability and Stability Margin The characteristic equation of discrete-time dynamic system is written as

Determination of The Discrete-Time S t a t e Space Model

For d i s c r e t e - t i m e analysis it is convenient to obtain the d i s c r e t e - t i m e s t a t e equations from continuous-time s t a t e equations.

This requires the continuous-time s t a t e model to be of minimal order [ 3 ] . For a two area reheat thermal s y s - tem the minimal 9

^{t n}

order s t a t e vector can be defined as

'

= [AF

1

AX

E1

i?

R1

AP

g1

AF

2

A X

E2

iP

R2

AP

g2

A P

tie1 With a non-singular system A Matrix.

For designing t h e i n t e g r a l or p r o p o r t i o n a l - i n t e g r a l c o n t r o l l e r s for AGC, the s t a t e vector in Eq.(11) is augmented by four additional s t a t e variables defined as

x

10

= / ( A P

t i e 1

+ B

1

A F

1

) d t : X

11 = '

( a

1 2

A P

t i e 1

+ B

2

A F

2

) d t

(12)

The first order derivatives of the additional variables are

10 =

3

1

i F

1

^{A P}

tie1

(14)

(15)

^ , ) d t

2 5 1 ^ 9 2 1 1

Discrete time equivalent of Eqs (13) to (16) are xl o(k+l) = x1Q(k) +. BjT Xj(k) + T xg(k)

x

n

(k+1) = x

n

( k ) + B

2

T x

5

(k) + T a

12

x

g

(k)

= X1 2( k )

x1 3(k+1) = ;k) + B2T x5(k) + T a1 2Xg(k)+a2T Now defining t h e augmented s t a t e vector as

X' = [AFi AX£1 APR1 APg1 A F2 / iXE 2 APR 2 4 Pg 2

(16)

(17) ( I B )

(19)

(20)

| z - * I | = 0 • ( 1 0 )

z=z-transform parameter.

* = system state transition matrix I = identity matrix.

For stability a l l the roots of Eq. (10) must be inside the unit circle. The system becomes unstable if any of the roots lies outside the unit circle and/or any multi- ple root lies on the unit circle. It may be stated that the discrete-time system is asymptotically stable if and only if a l l i t s poles l i e strictly within the unit cir- cle in the Z-plane. i . e .

Where,

I X 1 < 1 , ( m = 1 , 2 , . . . , N )

x , m=1,2,...,N are the roots of Eq.(10) ifjx j <p<1

for a l l m, we may conclude t h a t each mode of the t r a n s - ient response diesaway a t l e a s t as fast as p and the system is said to have s t a b i l i t y margin P, where k i-

s

sampling count. The smaller t h e radius P, larger is the s t a b i l i t y margin and hence b e t t e r is the dynamic r e s - ponse.

/ACE2dt /ACEN.jdt /ACEK^dt] (21) Eqns. (17), (18), (19), and (20) can be w r i t t e n in t h e compact form

[x

1Q

(k+D

D1 = B ^

D2=

x

n

(k+1) x

1 2

(k+1)

0

0 0 0

1 0 dT 0

0 1 0 dT

B2T _{a 1 2 T}

X(k) (22)

and

(4)

Where, ct|=ou= a, for two equal area system.

The augmented set of difference equations for the AGC system are

where,

= *X(k) + * u(k) + vp (k)

and

(23)

p first decreases (stability margin increases) and then increases (stability margin decreases). It is seen that for a=0.04, stability margin is maximum.. Further, it is seen that the optimum value GfKj=0.2 and is inde- pendent of a. The optimum value of Kj as obtained with conventional ACE has also the same value of 0.2 [3] . Another very powerful tool for obtaining optimum gain settings is integral squared error (ISE) technique. A cost function

Time error and inadvertent interchange at each sampling instant are evaluated using

( 2 4 )

I

2

(k+1)= I

2

(k)

T.al2.APtie1(.k)

Investigations are carried out considering simple integral and proportional-integral controllers.

1) Optimization of Integral Controller Gain Settings and a.

Dynamic performance of the AGC system would obviously depend on the value''.of a and K . In order to optimize

a and K the concept of maximum stability margin is used, evaluated by analyzing the eigenvalues of the closed loop system •

For the two equal area reheat thermal system, conside- ring integral type supplementary controllers in both the areas.

0 K

]

0 0 K _ is the integral gain of the controller.

Substituting for u(k) from Eq. (25) in Eq. (23) , the closed loop system matrix becomes

w h e r e . G= I

lo

•ACE>

0 0

]

1

d t

0 0

: u

0 0

2 = ^

0 0

I /AC

0 0

It or

0 0

u(k) =

0 0

* = *- v G and Eq.(23) reduces to X (k+1) = * X(k) + vp(k) The matrix t is function of Kj and a.

T - 2 5

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J =

k=0

e

k 0

minimized to obtain optimum values of Kj and a.

12

10 xiO-3

0 0 4

0 0 02 0-4 0-6

Fig.3: Cost function (J) ft Kj for various values of a.

Fig. 3 shows the plot of J as a function of Kj. for several values of a. For any'a as K

T

increases j first decreases and then increases as""^ is further increased. It is clearly seen that a=0.04 and Kj=0.35 are the optimum values of these parameters. It is seen that the optimum value ofuC obtained using the two^

alternative techniques is the sa-ne. Hence a=o.O4 is chosen for further investigations.

0 0 2r

KpO-2 Conventional ACE 0-2 New ACE

Fig.2: Stability margin (p) V

s

Kj for various values of a.

Fig.2 shows the plot of stability margin p as a function of K for several values of a. For any a as i< T increases

Fig. 4: Dynamic responses for various control strategies.

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734

Fig. 4 shows the dynanic responses (forAF. and A p .) for 1% step load perturbation in area-1 with K =0.20 and K =0.35 using new area control error. Dynamic responses with optimum integral controller considering conventional ACE are also plotted for the purpose of comparison. It is seen that the maximum transient deviations in frequency and tie-power are same in all the three cases. However, the settling time is somewhat more with new area control error. Dynamic responses with K,=0.20 are found to be superior to those with K=0.35 and hence selection of optimum KT for the controller considering new ACE can be based on the concept of maximum stability margin.

T

.2s

t= 0 2 Conventional ACE 0-2 New ACE 0 3 5 » •'

Time (sees)—^

20 -100 - -

0 0

-002

-0-04

10 \

Time

/

(sees)

20 ^

/ /

- \ 30

• * .

•

^ ~

Fig.5: Dynamic responses for and I, for various control strategies. ' '

Fig. 5 shews the plot of time error and inadvertent interchange using new ACE and conventional ACE. It is seen that the time error and inadvertent interchange are effectively regulated to zero with the new ACE while these quantities settle to their steady-state values with conventional ACE. It may thus be inferred that by using new ACE the steady state frequency, tie-power, time error and inadvertent interchange errors are regu- lated to zero simultaneously following a step load per- turbation, while with the use of conventional ACE, a finite value of time error and inadvertent interchange are produced requiring further correction.

0 0 1

4

00

001

I

L oo

M

i -0 01

T : 2 S

10

20 30

- K ] = 0 2 0 Conventional ACE 0-20 New ACE

.. 0-35 New ACE

^ 20 ^ 10

Time (sees)

Fig.6: Generation responses for various control stra- tegies.

Fig. 6 shows the plot of generation responses (AP^, AP

?

) . It is seen that with the use of new ACE

9

settling time is more as compared to those with conven- tional ACE. This is obvious since the controller forces the net energy transfer following a step load pertur- bation to zero between the two areas.

2. Proportional-Integral Controllers

Literature survey shows that many utilities use P-I controller to achieve improved dynamic performance.

An attempt is therefore made to study AGC problem with P-I controllers considering new ACE.

When P-I type supplementary controllers are used in both the areas, the control signals can be written as

u

1

=-K

p

.ACEN

1

-K

I

ACE^dt =- (

U2=-K

j-Kj f or u(k) = -G

1

X(k) where,—

(27)

(28) (29)

B

1

K P

° °

0 0 0

0 0 0 0 B

2

K

p

0 K

p

aK

p

0 oK

p

0 K and K are proportional and integral gains respecti- vely. Substituting for u(k) from Eq. (29) in Eq. (23) the closed loop system matrix becomes

and Eq. (23) reduces to

vp(k)

⁽³⁰⁾

*. is function of K and K...

0-90 1 10

& 0-0^

00 0 2 0 4 0 6 0 8 10

K

Fig.7: Stability margin (p) Vs K for various values of K .

Fig.7 shows the plot of s t a b i l i t y margin as a function of K for several values of K . The optimum P-I gain settings are K =0.90 and fe

K =

Optimum P-I gain settings using cost function J = 2

k=0 to be

+AP

tie12

(k) + are found

K = 1.20 and ^0.78 (Fig.8).

Fig. 9 shows the dynamic responses (A F.. andAP . .) for

the two optimum P-I gain settings (i.e. K =1.2,

K =0.78 and K =0.90, K =0.50). Dynamic rispenses with

optimum integral controller are also plotted for the

purpose of comparison. The dynamic responses with

optimum P-I controllers are seen to be much superior to

those with simple optimum integral controller. However,

there is not much difference in the dynamic responses

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with the two different sets of P-I gain settings.

0 0 0-4 0-8 12

F i g . 8 : Cost function (J) V» . K for various values of K .

P

<

001 -

-002L 00021- 00

-0004|- V _ ^ ' K,= O2O

K

p=

1 -20,Kj =0-78 Kp=0.90,K

1 =

0-50

Fig.9: Dynamic responses for various control strategies K]=0-20

K

p

= 1-20

f

Kj= 0-78

—

\

\ X

_

xiO

-

i

r

I

\ u

/

\ .. K

V

^ • • * " •

/

P =

101

/

090 , K p

y , ' Time

0-50

20 30

(sees)—

o-o

-10 0

I 00

-0-02 -

- 0 0 4 -

Fig.10: Dynamic respaises for

^e^ and I. for various

control strategies.

_

\ \

10

1

y

\

20 1

Time ( s e e s ) - •

30 LJTT- /

Fig. 10 shows the plot of time error and inadvertent interchange with optimum P-I and optimum integral gain settings. It is seen that with optimum P-I controllers peak deviations and settling time of time error and inadvertent interchange are much less as compared to those obtained with optimum integral controller.

K

p

=0.80,Ki= OJSjConventional ACE _K

p

=0-90, Kj= 0-50,New ACE

T i m e ( s e e s )

t 0-0

- 0 0 2 -

Fig.1 1:

Dynamic responses with two different optimum P-I controllers.

Fig. 11 shows the dynamic responses for optimum P-I controller setting considering conventional ACE

(K =0.80 and K =0.35) and optimum P-I controller

P J-

considering new ACE (K =0.90 and KI=0.50), settings being based on maximum margin of stability. It is seen that the maximum transient deviations in frequency and tie-power and settling time are sane for both the control strategies. However, the time error and inadvertent interchange reduce to zero with new ACE while these quantities attain steady state values with conventional ACE. Thus it can be inferred that the dynamic performance of optimum P-I controller considering new ACE is comparable to those with P-I controller using conventional ACE and moreover, it regulates the time error and inadvertent interchange to zero following a step load perturbation.

Conclusions

1. AGC problem of a two' equal area reheat thermal system has been analyzed considering a new area control error in discrete mode.

2. A simple approach for obtaining optimum integral and optimum proportional-integral controllers using the concept of stability margin aid ISE technique has been demonstrated. :

3. Investigations reveal that the settling time of the dynamic responses with new ACE is somewhat more as compared to those with conventional ACE.

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736

However, the regulator based on new ACE always guarantees zero steady state error of inadvertent

interchange and time error accumulations which is so desired, unlike in the case of controller based on conventional ACE.

4. P-I controllers provide superior dynamic responses as compared to those with simple integral controllers and should be preferred.

References

1. Nathan Cohn, "Techniques for improving the control of bulk power transfers on interconnected systems", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-90, pp.2409-2419, November/December 1971.

2. A.A. Fouad and S.H. Kwcn, "Effect of coordinated correction of tie-line bias control in interconn- ected power system operation", IVFF. Transactions on Power Apparatus and Systems, vol. PAS-101, pp. 1134-1143, May 1982.

3. M.L. Kothari, P.S. Satsangi and J. Nanda,

"Sampled-data automatic generation control of interconnected reheat.thermal systems considering generation rate constraint", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, pp. 2334-2342, May 1981.

Appendix - A The nominal system parameters are

f = 60Hz, Pr1=Pr2=2000MW, H:=H2 = 5 Sees, T. , = T.n=0.3 sec, T =T n=0.08 sec, P_. =200 MW

tl t2 g1 g2 ' tiemax 12 = 3 0 ° ,

= 0-50, T

^{r 1}

=T

^{r 2}

=10.0 s e e s .

0^02=8.33x10 puMW/Hz, Bi = 6^1=1,2)

M.L. Kothari, received the B.E. degree in Electrical Engineering from University of Jodhpur in 1964, M.E.

degree in Power Systems from University of Rajasthan in 1970 and Ph.D. degree in 1981 from Indian Institute of Technology, New Delhi.

He has held teaching appointements at the University of Jodhpur, Malaviya Regional Engineering College, Jaipur and G.B. Pant University of Agriculture and Technology Pantnagar. Currently, he is Assistant Professor at I.I.T. Delhi. He has published many papers in several areas of power systems. His current research activities are in the areas of automatic generation control and stability analysis of large

interconnected systems.

He is a member of the Institution of Engineers (India).

During 1978-79 he was a Visiting Professor in the Department of Electrical Engineering at West Virginia University, U.S.A. He served as the Head of Electrical Engineering Department at I . I . T . Delhi during 1984 to

1987 and currently he is the Dean of Undergraduate Studies at I . I . T . Delhi. Professor Nanda's field of interest comprises of Power system planning, analysis, s t a b i l i t y , computer control, optimization and energy conservation. He has been a consultant to many power u t i l i t i e s and has been Chairman of National Expert Committees pertaining to power system planning in the country. He has many publications in several areas of Power systems in International and National Journals of repute. Professor Nanda is a Bellow of the National Science Academy.

P.P. Kothari, was born in Bikaner, Rajasthan (India) in 1944. He obtained his B.E., M.E., and Ph.D.degrees frcm the Birla Institute of Technology and Science, Pilani, and after teaching at Pilani and Kurukshet- ra, has been involred in tea- ching and research, since 1977 at IIT Delhi. He is a Professor of Electrical Engineering at the Centre of Energy Studies, IIT Dalhi. He was a Visiting Fellow in 19 82-83 at the Royal Melbourne Institute cf Technology, Australia. He has published and presented o\er 135 papers in various national and international journals and conferences of repute. Along with Prof. Nagrath, he has ccauthored

"Modern Pover System Analysis" (Tata-McGraw-Hill, 1980).

and "Electric Machines" (Tata-McGraw-Hill, 1985). His research interests include power system control, opti- mization and r e l i a b i l i t y .

D. Das, was born in Kharagpur, India, in 1961. He obtained the B.E. degree in Electrical Engineering from Calcutta Uni\aersity in 1982 and M. Tech.

frcm I.I.T. Kharagpur in 1984, India. During 1984-85 he was lecturer at B. I. T. Ranchi, India. Since July, 1985, he has been working towards the Ph.D. degree in the area of Automatic Generation Control at I . I . T . Delhi, India.

J. Nanda, received his oh.D.

degree in Electrical Power System engineering 'in 1964 from Moscow Power I n s t i t u t e . In 1965, he joined Indian Institute of Technology, Delhi as an Assistant Pro-

fessor. He served as a visiting faculty in the Depa- rtment of Electrical Engin- eering at Imperial College, London during 1969-70. Since

1973 he is a Professor in the

Deoartment of Electrical Engineering at IIT Delhi.

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737

Discussion

O. P. Malik (The University of Calgary, Calgary, Alberta, Canada): The authors have analyzed the discrete-mode automatic generation control of a power system. In such a system, the controller is in the discrete domain and the power system being controlled is in the analogue or continuous-time domain as shown in Fig. A.

Simulation of the model of a continuous system on a digital computer entails discretization of the system with a sampling period equal to the integration step. Discretization of the model, that is in the continuous domain, must be done in accordance with Shannon's sampling theorem.

Otherwise an error, proportional to the amount of aliasing, occurs. This requires that the integration step (sampling period) must be less than one half the smallest time constant in the system model.

The authors have chosen a sampling period of 2 s for the discrete-mode automatic generation controller and discretized the entire system, including the power system, shown in Fig. A with that sampling period. To avoid aliasing errors, the discretization interval for the power system must be chosen according to the time constants involved in the power system model and must conform to the sampling theorem. The power system parameters given in the paper show that the smallest time constant is 0.08 s. For this case, the discretization period must be 0.04 s or less and not 2 s. Using a single sampling interval of 2 s for both the controller and the power system not only means that the time constants T, = 0.3 s and Tg = 0.08 s are ineffective, but it also is a fundamental error in the modelling and analysis described in the paper.

Instead of correcting the results presented in these papers, authors have been able to present another paper with the same fundamental error. I would request the authors to read my comments given in Ref. [C], and repeat their analyses using one of the methods described in the same reference since all the results shown in the present paper are wrong and misleading to the readers involved in AGC area.

References

[A] J. Nanda, M. L. Kothari, and P. S. Satsangi, "Automatic generation control of an interconnected hydrothermal system in continuous and discrete modes considering generation rate constraints," IEE Proc.

D, Control theory and application, 1983, 130, (1), pp. 17-27.

[B] M. L. Kothari, P. S. Satsangi, and J. Nanda, "Sampled-data automatic generation control of interconnected reheat thermal systems considering generation rate constraints", IEEE Trans., 1981, PAS- 100, pp. 2334-2342

[C] A. Kumar and O. P. Malik, "Discrete analysis of load-frequency control problem", IEE Proceedings, Vol. 131, Pt. C, No. 4, July 1984, pp. 144-145.

Manuscript received August 12, 1988.

CLOSURE

»} CONTROLLER 1 ZOH POWER SYSTEM

Fig. A Power system and controller representation

—— Continuous domain

• • • Discrete domain

Studies [A] have shown that discretizing the power system with a wrong sampling period gives inaccurate and misleading results. Have the authors tried to model the system correctly and checked if they get the same results as reported in the paper?

Use of a two area system in the analysis of AGC and LFC has been carried on for over 20 years now. It is, however, a trivial example. To be meaningful these studies should be performed on a reasonably practical multi-area interconnected system [B] and also should include non-linearities such as generation rate constraint, and dead-band. Authors' comments on these aspects are invited.

References

[A] A. Kumar and O. P. Malik, "Discrete analysis of load-frequency control problem", IEE Proceedings, Vol. 113, Part C, No. 4, 1984, pp. 144-145.

[B] O. P. Malik, Ashok Kumar and G. S. Hope, "A load frequency control algorithm based on a generalized approach," Paper #87 WM 068-0, presented at IEEE Power Engineering Society 1987 Winter Meeting.

Manuscript received August 5, 1988.

Ashok Kumar, (North Carolina A&T State University, Greensboro, NC):

The authors of this paper have analysed a two-area reheat thermal system with New Area Control Error in DISCRETE MODE. The authors are to be complimented for using New Area Control Error for their AGC analyses.

Similar analyses have been presented by the authors in references [A] and [B] using classical area control error. Since a fundamental error was made in their analyses, a correspondence was published in IEE Proceeding [3].

ivi.U KOTHARI, J. NANDA, D.P. KOTHARI, AND D.

DAS : iVe thank the discussers for their keen i n t e r e s t in our work, and for their c o m m e n t s .

We have the following reply to offer to the discussers' c o m m e n t s :

The discussers have c o m m e n t e d t h a t our model is e r r o n e - ous. One of the discussers suggests us to refer to their work {_l ] for relevant c o r r e c t i o n . We have gone through Reference 1, v/here the discussers have c o m m e n t e d on our earlier works [2 \ 3 } . >'v'e a r e sorry to s t a t e t h a t the discussers have made vvrong s t a t e m e n t s / o b s e r v a r t i o n s about our works [ 2 4 3 ] without devoting enough t i m e to careful reading to our works £ V i 3 3 and have jumped to wrong inferences on our work.

vVe will like to reinforce t h a t our model is valid and all results a r e c o r r e c t and meanirigful. We have considered a sampling period T = 2 sees', for both the controller and the power s y s t e m . The justification for choosing T = 2 s e e s is as follows :

Several values of T were tried. Investigation reveals t h a t for T = 0.1 sec for buth the controller and the power system, the responses for the d i s c r e t i z e d mode e-wctly match the responses for the continuous mode.

In fact up to T = 0.5 sees the respunses for both continuous and d i s c r e t e mudes a r e found to be practically the s a m e . Thus t h e r e is no need to choose a very small T (O.;1Q3 to 0.02 sec) as suggested by the discussers [1 ] ivhich tells on the effort involved on the sampler

as alsu u:i the c o m p u t a t i o n a l t i m e .

beyond T = 0.5 sec, the t i m e c o n s t a n t s , "Vn = 0.08 sec and T = 0.3 sec a r e ineffective and thus fjie only r e l e - vant l i m e c o n s t a n t s to be considered in t'.ic problem under investigation a r e r e h e a t t i m e c o n s t a n t T = 10 sees and power system t i m e c o n s t a n t T = 20 s e e s . Thus the sai.iplino periou of I seconds "chosen in the paper is ^uite in order, l.i reality the primary controller is very fast, an-.! a c t s initially fur the first cou,jle of seconds and t'len the secondary control - YGC (the slow co.itrol) manifests. The ^hiliosophy as discusse-i in Refer- ences 2&3 is to go for as I a r6e a T as possible without jeopardising the stability ,,iuroiii and responses. It is lelieved t h a t a h i ^ i e r sampling joriod may be preferred as this requires less frequent Sj...pliiig and ^le.ice reduces the effurt involved in the sampling operation.

(9)

738

\ sampling period of 2 seconds chosen in the paper is

realistic and practical. Oei.'lello et al [4 ] in their work on Automatic Generation Control : Digital Control Techni- ques, have also chosen a sampling period of % seconds.

In a recent book on Computer Aided Power System Analy- sis by Kusifc 15 3 » it has -icon .mentioned (pa

^o

e 9) that the present practice is to scan the tie-flows every 2 Secou.ls and the turoine - ^oiicroturs arc often commanded to new power levels every 4 seconds sharing the load adjustment iaased o.i each units response capability in iviW/min. 3ose and Atiyyah £6 3 in their vork on Regulation uror in Load frequency Control have considers' sa...plins periods U,J to 30 seconds for the controller.

We could have for our problem chosen f = 2 sees for the controller and the model for the power system as continuous or a different smaller samplin

o

period for the power system model. However, as stated earlier for a T = 0.5 sec for both the controller and the power system, the responses in the discrete mode are practically the same as for the continuous mode. ve did not choose two different campling periods for the controller and the power sytsem but instead chose a T = 2 seconds fui 'xnh of the.;., so as to make the computations fast enough without incurring any appreciable error in the responses.

In tue present paper we have not considered the generat- ion rate constraint ( j ^ C ) and deaJba.id since .ve wanted to demonstrate the effectiveness of the use of .iew area control error (AC;-,| in a oimpic way. rve have mean- while considered GRC in our model and the results would b~ reported soon.

4.

A. Kumar and O.P. i/ialik, 'Discrete analysis of load-frequency control problem

¹

, Vol. 113, Part C, Ko. 4, 1334, pp. 144-145.

J. Wanda, ivi.L. Kothari and P.S. Satsangi, 'Auto- matic Generation control of an interconnected hydro- t^rmcil system in co.itiiiuoas mid Jiscrete uiodes coiisidering jje.ieratio,. rate constraints', I£E Proc.

D, :^ntroi theory ^nj duplication, 19'i3, 130, (1), .,p. 17-27.

M.L. Kothari, P.S. Satsangi and J. Nanda, 'Sampled- data automatic generation control of an i n t e r - connected reheat thermal systems considering generation r a t e c o n s t r a i n t s ' , IEEE Trans., 1981, PAS-100 pp. 2334-2342.

F.P. de.viello, n . j . .*ulls, and vv.F.'-i'Rells, 'Auto-

matic tienerution ooiitrol part II - digits! co.itrol techniques' I \

^r.ri Tr^us. on 'over A^par^tttS- and

iystei,

l£