BIBLIOGRAPHY N. -...

BIBLIOGRAPHY

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RUNG0 t::UTTa METHOD

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B ( 3 > =B<2)

C < 3 ) = A < 3 >

- A ( 4 > = A C 1 ) / 3 -

B < 4 ) = F ( l )

C ( 4 ) - C <I)

I N I T I A L I S A T I O N OF PROGR A M

DO 4 L = 1 ,NEQ

Y ( L > = Y O < L )

DO 1 I = 1,NEQ

Q C I 3 = 0 -

RG4 fiS MODIFIED B Y G I L L

DO 3 k< = 1 , NH

DO 2 3 = 1 , 4

C G L L D E R V I S ( Y , A t < )

DO 2 f = 1 ,NEP

SLUF'E ( I ) -- <I-\+< * H

Q = Q C Y )

CONT INUE

RETURN

E N D

SUFjFtOUl- I N E DERVIS < X , D X )

D I M E N S I O N X < 4 ) , W X < 4 )

COT?t?f)bJ TP ,R l iP ,R ,TG,TT .PD,U

D X ( I ) = I -0

DX ( 2 ) = X ( 3 )

D X ( 3 ) ' X ( 4 )

nX(4)=(1-/T3)*(-A!tP+PD-( ( A P . : P + R ) / R ) * X ( 2 )

l - T l * X ( 3 ) - T 2 * X (4).+fif:.P*!J)

RETURN

E N D

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NEW CONTROL STRATEGY FOR LOAD FREQUENCY CONTROL PROBLEM

k. B.SUBRAMANYAMt Membeh, IEEE I k V . C . V E E R A R E D D Y

P L= b I S.V.U.College of Englneer lng

1 Ti ru a t 1 - 517 502, I N D I A a 5: 5 USTRACT:- This paper d e a i s w i th a new

c o n t i ; o l t e g y of quenching t r a n s i e n t s of a I M , ~ frequency problem. The l oad f requency pmbl" is represented by a new s t a t e space

for a s i n g l e a r e a E l e c t r i c power ". pho s t a t e v a r i a b l e s s e l e c t e d i n t h i s 1 are f r e q u e ~ c y , f i r s t and second ratives. o t f r equenc i e s . The , s t e a d y s t a w ,

wprNting p i n t s b e f o r e and. a f t e r t h e l oad disturbance a r e named a s i n i t i a l and f i n a l dates of the system. Now t h e LFC problem is restructured a s a s t a t e t r a n s i t i o n problem [initial t o f i n a l s t a t e s ) us ing a s u i t a b l e antrul parameter- With t h e h e l p of pontryajins Maximum P r i n c i p l e (PMP) t h e LFC pmbleap~is proved t o be bang-bang c o n t r o l problem by minimizing t h e t ime of s t a t e transfer. The c o n t r o l pa rame te r is taken t o be the position of speed change r which is not an external parameter.

A single area power system is cons ide red and the optimal c o n t r o l l e r s a r e syn thas i zed for an increase i n l oad p o s i t i o n of t h e rjatem. The swi t ch ing i n s t a n t s f o r t h e control s t r a t egy a r e e v a l u a t e d . I t i s berved tha t t he f r equency t r a n s i e n t s a r e quenched a t much f a s t e r r a t - - w i thou t any oscillations.

INTRODUCTION

The development of de s ign t echn iques f o r load frequency c o n t r o l of l a r g e interconnected sys tems i s an impor tant control problem i n power sys tems. C l a s s i c a l ~ontrol techniques a r e a p p l i e d f o r t h e uprovewent of t h - dynamic r e sponse and stability of power sys tems. However t h e clusical techniques a r e having t h e i r own

backs. In r ecen t l i t e r a t u r e many people applied leodern c o n t r o l t h e o r y f o r s o l v i n g LFC problems [1-71.

This paper d e a l s w i th t h e des ign of trollers f o r a s i n g l e a r e a power sys tem "in9 BANG-BANG c o n t r o l of speed changer Paition. The system dynamics is de r ived i n Stat* variable form us ing change i n f requency a its phase v a r i a b l e s . The s o c a l l e d LFC Problem is r e s t r u c t u r e d a s a t ime op t ima l mntrol problem and t h e op t ima l c o n t r o l ( t h e

t , I ; .

speed change? p o s i t i o n ) is proved t o : b e

\ s y n t h e s i z e d 5 '' 4ANG- BkNG i n na u re The op t ima l c o n t r ~ l l e r s , , a r e

f o r p r a c t i c a l s i n g l e . a r e a power., syatem. The system dynamic response f o r t h e s e

- op t ima l c o n t r o l l e r s is s imu la t ed . $ .

i , . % b ; @

.'. . A NEW STATE-SPACE MODEL ' F ~ R PRdBLEM ' ''

7- The s i n g l e a r e a power system using block

diagram approach is shown in appendix. A new s t a t e v a r i a b l e model , cons ide r ing f requency d e v i a t i o n and phase v e r i a b l e s 1s de r ived i n appendix . The system equa t ions a r e g iven by

where ( x i , x 2 , x3) = ( A f , ~ f t ~ f O

TIME OPTIMAL STATE CONTROL ---- I n t h e s o c a l l e d LFC probien b ~ c ~ r e t h e

load d i s t u r b a n c e , the change in f requency i s zero ( I n i t i a l s t a t e ) . I f t he re i s a l oad d i s t u r b a n c e , the system frequency v a r i e s and t h i s v a r i a t i o n should be wi th in the l imits. A f t e r cons ide rab l e t ime the system once aga in should have ze ro frequency deviation ( f i n a l s t . a t e I l 31 .

Th i s probiem of LFC can be r e s t r u c t u r e d a s an opt imal c o n t r o l problem a s d i s c u s s e d below.

The system equa t ions governing t h e dynamics of a s i n g l e a r ea power sys tem t ak ing speed changer p c s i t i o n a s c o n t r o l parameter a r e g iven i n e q n s . ( l ) I

The i n t e g r a l performance t o be minimized i s eqn. (2) chosen a s t ime of s t a t e t r a n s f e r from an i n i t i a l s t a t e t o f i n a l s t a t e , w i t h i n t h e limits of frequelicy d e v i a t i o n s .

The c o n t r o l p roces s would be such t h a t t he system d e s c r i b e d by e q w ( 1 ) would be t r a n s f e r r e d from an i n l t i a l s t a t e t o t h e f i n a l s t a t e i n a minimum time. The c o n t r o l parameter is assumed t o l i e between two l i m i t i n g va lues ul and u2.

Using pon t ryag ins nbaximurn p r i n c i p l e , . t i l e Hamiltonian func t ion ii is g lven by

The c o m p o n e n t s o f a d j o i n t v e c t o r s a r e o b t a i n e d f r o m

R e w r i t i n g the H a m i l t o n i a n f u n c t i o n a s two e x p r e s s i o n s

H = QD u + t e r m s n o t i n v o l v i n g u ( 7 )

where Q,, 1s g i v e n by

The c o n t r o l u is t o be s e l e c t e d a s t o maximize H a t e v e r y p i n t a l o n g t h e t r a j e c t o r y u s i n g t h e s e t o f a d m i s s i b l e c o n t r o l s t h a t s a t i s f y e q n . ( 3 ) .

H a x i m i z i n g H o f e q n s . ( l ) w i t h r e s p e c t t o u, o n e o b t a i n s t h e o p t i m a l v a l u e o f u a s f o l l o w s

Uopt a u1 i f s i g n QD > 0

= U 2 i f s i g n QD < 0

= u n d e f i n e d i f QD = 0

The o p t i m a l c o n t r o l is found t o b e BANG- BANG i n n a t u r e f o r t h i s c o n t r o l p a r a m e t e r .

The o p t i m a l c o n t r o l is proved t o b e !BANG-

BANG, t h i s i m p l i e s t h a t t h e c o n t r o l would

have e i t h e r a v a l u e o f ' u l o r u2. The p l l y s i c s

of . t h e p r o b l e m a n d t i l e d e s c r i p t i o n o f t h e

c o n t r o l p r o c e s s would c l e a r l y i n d i c a t e

w h e t h e r u l o r u2 t o b e used i n t h e b e g i n i n y

and a l s o a t t h e e n d . w:ien o n c e t h e

i n f o r m a t i o n is known :t o n l y r e m a i h s t o

d e t e r m i n e the i n s t a n t when t h e s w i t c h i n g

t a k e s p l a c e f r o m u, t o u2 o r from u2 t o ul-

l h i s i n f o r m a t i o n may b e o b t a i n e d by

. ~ n t e g r a t i n g t h e s y s t e m - d i f f e r e n t i a l e q u a t i o n s . . w i ' q o p t i m a l v a l u e o f ul f o r w a r d z i n t i m e

. . *3i ,nb- - . . in i t i s l . c o n d l t i o n s . S i m i l a r l y t h e

s y s t e m e q u a t i o n s may b e i n t e g r a t e d b a c k w a r d s

i n . t i m e u s i n g t h e o p t i m a l c o n t r o l u2 s t a r t i n g

from t h e t a r g e t state. The i n t e r s e c t i o n o f

t h e s e t r a j e c t o r i e s :ill --g,g i n f o r m a t i o n a b o u t t h e s u l t c h l ~

.:.p

SYbiTliESIS OF OPTIMAL CONT!

1 4 Ttie o p t i m a l c o n t r o l l e r s at,

TT 3 0.3, APD = 0.01

and t h e t a r g e t s t a t e i s

The s y s t e m e q n s . (1)

f o r w a r d and backward i n :i

c o n t r o l ,param

-@re -012 !n-ON Change of f r c y

Fig.1 Phase plane l r o j ~ t o r i t ~ d. power system with. A %it . . . . - d

Time, T 5

Fig.2 Chonpe in Irwuency 0' power system with ' S

TAt! l i. D A T A F. .< THE BANG-UP.! ;C CONTROLLERS

0.011 0 . 8 4 0 0 . 2 4 0

0.012 0 . 8 4 5 0 .230

0.015 0 . 8 5 0 0 . 2 2 0

i v e s t h e d a t a f r o m F i g . ¶ . f o r

values of ul a n d u2 a n d s w i t c h i n g

ants.

nor the T a b l e i t c a n b e o b s e r v e d t h a t

optimal c o n t r o l l e r s a r e ul = 0.01 and

#0.015 and t h e c o r r e s p o n d i n g A f i s e q u a l

-0.0003 Hz. The s i m i l a r o b s e r v a t i o n c a n

be seen f o r 'ul = 0 . 0 1 5 . The r e s p o n s e of

@en f o r ~ n c o n t r o l l e d a n d f o r c o n t r o l l e d

r is depic ted i n F i g - 2

CONCLUSIONS

i new m a t h e m a t i c a l model i n s t a t e ble form u s i n g p h a s e v a r i a b l e s is d. The s o c a l l e d LFC p r o b l e m is bred as t i m e o p t i m a l c o n t r o l problem t h ~ c o n t r o l p a r a m e t e r is p r o v e d t o be LUG using PMP. The o p t i m a l c o n t r o l l e r s agothesized f o r a s i n g l e a r e a p r a c t i c a l ayaten.

APPENDIX

The s t a t e v a r i a b l e model i s d e r i v e d by means. of b l o c k d i a g r a m r e p r e s e n t a t i o n f o r a s i n g l e a r e a power s y s t e m shown In F i g . b e l o w .

( u - A f / R ) = ( l + s ( T +T ) + S L x ' r G ~ , ) ~ P , ( 1 . 1 ) , T G

( A P - L P G ) = A f ( l + s T p ) / K p G (1.21

S u b s t i t u t i n g t h e v a l u e of A P G from e q n . (1 .2) i n t o eqn. ( 1 . 1 ) .

( u - A f / R ) = ( l + s ( T T + T G ) + s 2 TT 'PG)

( A P , + A ~ ( ~ + s T,) j K~ j

Assuming & P c o n s t a n t , t h e t e r m s 5PD, S2 APD a r e n e g l e c t g d .

K ( u - A E / R ) = K p ~ P ~ + d f + Tl A f + P

T$f + T3 A £ ) 1

~ f = l / ~ ~ [ ( - ( k ~ + R ) / R ) A ~ - T ~ A ~ - -2 ~ f

- k p A P D + K U l ( 1 . 3 ) P

NOTATIONS where T1 = (Tp + TG + TT) ' Generated power d e v i a t i o n , pu MW

T2 = (TG TT + TG TF + Tp TT) Change i n power demand, pu M W

T3 = (TG Tp TT) @+viation i n f r e q u e n c y , Hz

Let A f = X1 Change i n a p e e d c h a n g e r p o s i t i o n ( u ) r pl nw AI? = x 2

Static g a i n o f p o w e r s y a t e m i n t e r t i a A £ = X3 dynamic b l o c k , HZ/PU nw

~ h ~ system of e q n . (1 .31 is r e p r e s e n t e d

Yime c o n s t a n t of power system inertia i n s t a t e v a r i a b l e f o r m u s i n g Phase dynamic b l o c k , s e c

i l . = X 2

Governor t i m e c o n s t a n t i n s e c X Z = x3

hrbine ( n o n r e h e a t t y p e ) t i m e mnatant , s e c 1 (K +R)

y. = ---- L-xl --e,-- - , 1 . 2 -T ?, 3 - b p D - u ) K p ]

r e g u l a t i o n p a r a m e t e r , Hz/Pu HW 3 .

T3 R

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