optimization algorithms of operative control

8/14/2019 Optimization Algorithms of Operative Control

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ELSEVIER Journal of Computational and Applied Mathematics 84 (1997) 81-99

JOURNAL OFCOMPUTATIONAL ANDAPPLIED MATHEMA TICS

O ptimization algorithm s o f operative controlin water distribution systems

R y s z a r d K l e r n p o u s * , J e r z y K o t o w s k i , J a n N i k o d e m , J ~ d rz e j U - ta s ie w i czInstitute o f Technical Cybernetics, The W roctaw Technical University, 27 W ybrze2e Wyspiah skiego St.,50-370 Wro ctaw, Poland

Received 5 October 1994; received in revised form 6 May 1997

Abs t r a c tThis paper discusses a mult ilevel algorithm for finding optimal control in a static distribut ion system based on the ideaof aggregation technique. We present mathematical model of this system with its elements as well as two basic algorithms.

The first is a simulation algorithm of the pipeline network and the other is an algorithm for finding an optimal control atthe pumping station. This paper discusses the static problem (Kotowski and Olesiak, 1980) of energy wastes minimizationin the water network and also describes an algorithm for solving it. Finally, an algorithm of operative control of the waterdistribution systems is presented.

Keywords: Distribution networks; Flows in networks; Simulation; Optimization; Optimal controlA M S classif ication: 65K10; 65Y20; 76M25

I . I n t r o d u c t i o n

T h e m a i n g o a l o f w a t e r d i s tr i b u t io n n e t w o r k s i s to f u lf il l t h e d e m a n d o f r e c e iv e r s . T o a c h i e v et h a t it is n e c e s s a r y d o d e l i v e r t h e a p p r o p r i a t e a m o u n t o f w a t e r ( w i t h t h e d e t e r m i n e d q u a l i t y a n d i nt h e s p e c i f i e d t i m e i n t e r v a l s) . T h e n e t w o r k c o n s i s ts o f p i p e l i n e s w h i c h c o n n e c t s o u r c e s ( e .g . , p u m ps t a t i o n s ) w i t h c o n s u m e r s .T h e m u l t i l e v e l c o n t r o l s tr u c t u re o f s u c h a s y s t e m h a s b e e n d i s c u s s e d i n [ 2 - 4 ] . M o r e o v e r , t h e m o s to f t e n u s e d o p t i m i z a t i o n c r i t e r io n i s th a t o f e l e c t ri c a l e n e r g y c o s t m i n i m i z a t i o n , b e c a u s e i ts o p e r a t i o na n d m a i n t e n a n c e m a y b e i n c l u d e d i n t h e c a p i t a l c o s t s . T h e m u l t i l e v e l c o n t r o l s t r u c t u r e f o r t h i s k i n do f s y s t e m h a s b e e n d i s c u s s e d i n [ 2 - 4 ].

T h e p r o p o s e d m o d e l o f a c o n t r o l s y s t e m a s s u m e s [ 1, 2, 7 ] t h a t th e i m p o r t a n t a i m o f c o n tr o l i st o s a t is f y th e r e q u i r e m e n t s o f w a t e r c o n s u m e r s . W a t e r d i s tr i b u ti o n s y s t e m s a r e d e s i g n e d to d e l i v e r

* Corresponding author.0377-0427/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reservedP H S 0 3 7 7 - 0 4 2 7 ( 9 7 ) 0 0 1 1 3 - 1


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82 R. Klempous et al./Journal of Computational and Applied Mathematics" 84 (1997) 81 99w a t e r f r o m p u m p s t a t i o n s t o w a t e r c o n s u m e r s t h r o u g h p i p e l i n e n e t w o r k s e q u i p p e d w i t h a v a r i e t y o fc o m p o n e n t s - p u m p s t a t i o n s , v a l v e s , r e s e r v o i r s .

A c c o r d i n g t o [ 9 ,1 0 ,1 3 ] w e a s s u m e t h a t in o u r m o d e l o f w a t e r d i s tr i b u ti o n s y s t e m , c o s t m i n i m i z a t i o nis t h e b a s ic o p t i m i z a t io n c r i te r io n . I n f r a m e w o r k t h e c o n t r o l s y s t e m p r o p o s e d f o r i m p l e m e n t a t i o n i sb a s e d o n a t h r e e l e v e l s t r u c t u r e [ 3 , 1 3 ] .

T h e f ir st le v e l is t h a t o f p u m p u n i t s d i re c t c o n t ro l , re g u l a t i n g v a l v e s , h e a d s a n d f l o w o f a n e t w o r k .O n t h i s l e v e l, b a s e d o n a c o n t r o l a l g o r i th m , t h e a c t u a l n u m b e r o f w o r k i n g p u m p u n i ts a s w e l l a st h e d e s i re d p o s it io n o f r e g u la t in g v a l u e s a r e d e t e r m i n e d , w h e r e a s t h e r e c o m e n d e d v a lu e s o f h e a da n d f l o w f r o m p u m p i n g s t a ti o n s a r e r e c e i v e d (a s p a r a m e t e r s o f c o n t r o l a l g o r i t h m ) f r o m t h e s e c o n d( u p p e r ) l e v e l .

T h i s o n e d e t e r m i n e s t h e a b o v e m e n t i o n e d p a r a m e t e r s w h i c h e n s u r e s t h e i m p l e m e n t a t i o n o f re -c e i v e r s ' d e m a n d . T h e a c t u a l v a l u e o f th i s p a r a m e t e r s w e o b t a i n f r o m e l e c t r i c i ty c o s t m i n i m i z a t i o n .A s a r e su l t, w e o b t a i n a g r a p h i c s c h e d u l e w h i c h i l lu s t ra t e t h e c o o p e r a t i o n b e t w e e n p u m p s t at io n sa n d r e s e r v o i r s .

T h e h i g h e s t , t h i r d l e v e l d e t e r m i n e s a n o p t i m a l g r a p h i c s c h e d u l e f o r f i l l i n g t h e r e s e r v o i r s . T h i s i sb a s e d o n t h e f o r e c a s te d w a t e r h is t o g r a m o f th e c o n s u m e r s ' d e m a n d s a s w e l l a s th e p a r a m e t e r s o f t h ea g g r e g a t e d w a t e r d i s tr ib u t i o n s y s t e m . T h e a l g o r i t h m e n s u r e s t h e fu l f i lm e n t o f th e c o n s u m e r d e m a n d sa n d t h e m i n i m i z a t i o n o f t h e e n e r g y c o s t s t a k i n g i n t o c o n s i d e r a t i o n t h e v a r y i n g p r i c e s f o r e l e c t r ic a le n e r g y .

T h e r e s u l t s o f o u r i n v e s t i g a t i o n s in t h i s s t u d y a r e r e l a t e d t o t h e s e c o n d l e v e l o f t h e c o n t r o l s y s t e m .T h e a i m s o f t h i s l e v e l is to m a i n t a i n t h e w a t e r d i st r ib u t i o n s y s t e m i n a n o p t i m a l r e g i m e o f a c t i o np r e s u m i n g t h a t th e d e s i r e d d e m a n d s o f t h e c o n s u m e r s w i ll b e f u l fi ll e d .

T h e m a t h e m a t i c a l m o d e l o f o p t im a l c o n t r o l in t h e w a t e r d i s tr i b u ti o n n e t w o r k s h o u l d t a k e i n t oa c c o u n t t h e f o l l o w i n g a s s u m p t i o n s : r a n d o m d e m a n d s o f c o n s u m e r s , d y n a m i c s t r u c tu r e c a u s e d b y t h e p a r a m e t e r s o f r e s e rv o i r , s t ru c t u r e a n d t o p o g r a p h y o f t h e n e t w o r k , n o n l i n e a r c h a r a c t e r i st i c s o f p u m p s a n d p i p e l in e s , t i m e - v a r y i n g p ri c e s o f e le c t r ic a l e n e r g y .

T h e o p t i m i z a t i o n p r o b l e m i n c l u d e s t h e v a l u e s o f f lo w s i n p i p e li n e , t h e h e a d s i n t h e n e t w o r k n o d e s ,t h e p o s it io n s o f v a lv e s a n d p u m p s t o b e o p e r a t e d in e v e r y m o m e n t o f c o n s i d e r e d p e ri o d o f t im e .T h i s y i e l d s to t h e d y n a m i c a n d n o n l i n e a r p r o b l e m o f h i g h d i m e n s i o n , w h i c h i s p r a c t ic a l l y u n s o l v a b l e .T h e r e f o r e , i n o u r p r e v i o u s p a p e r s w e h a d t o u s e s o m e s p e c i a l t e c h n i q u e s b a s e d o n t h e a g g r e g a t i o na n d d e c o m p o s i t io n m e t h o d s , c o n n e c t e d w i th s o m e n e c e s s a r y s i m p l if ic a ti o n s o f th e m a t h e m a t i c a lm o d e l [ 13 ], t o a c h i e v e f i n a l l y t h e t h r e e - l e v e l o p t i m i z a t io n a l g o r it h m .

2 . M o d e l d e s c r ip t io nA m a t h e m a t i c a l m o d e l o f w a t e r d i st r ib u t i o n s y s t e m [ 5, 7 , 8, 1 2] c o n s i st s o f t h e f o l l o w i n g e l e m e n t s :

m o d e l o f t h e p u m p s t a ti o n s, m o d e l o f th e p i p e li n e , m o d e l o f t h e p i p e l i n e s n e t w o r k , m o d e l o f th e r e se r v o i r.


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R. Klempous et al./ Journal of Com putational and Applied Mathem atics 84 (1997) 81- 99 83N o w w e d e s c r ib e th e a b o v e e l e m e n t s o f th e m a t h e m a t i c a l m o d e l i n d e ta il s. L e t IP b e a n u m b e r o f

t h e p u m p i n g s t at io n s . W e h a v e a s s u m e d t h a t a t e a c h i s ta t io n 2 ;, p u m p s w i t h i d e n ti c a l c h a r a c te r i s ti c sa r e s et u p . W h e n o n e o f t h e s e p u m p s w o r k s i t c o n s u m e s e l e c tr ic i ty a n d

P ( y ) = 7 + f l y , c t , f i > ~ O ( 1 )i s t h e g e n e r a l f o r m u l a t h a t c o n n e c t s th e c o n s u m e d e l e c t r ic i t y w i t h t h e o u t p u t f l o w y . F o r p u m ps t a t i o n s o n e o b t a i n s t h e f o l l o w i n g f o r m u l a s :

P i o ( Y i ~ , z i o ) = o ~ i p Z i p + f l i, Y i p ; ip = 1 , . . . , I P ( 2 )w h i c h i s t ru e , w h e n z;o p u m p s w o r k a t i t, th u s t h e s u m m a r i z e d o u t p u t f l o w i s y i p . L e t H ;r d e n o t e st h e w o r k c h a r a c t e r i s t i c s :

/ "x2G i o t Y i p - y i ) ; ip ---= 1 . . . . IP (3 )I'-lip(Yip'Z ip)---- l'~Oip -- \Z ip

w h i c h a r e th e r e la t io n s b e t w e e n t h e n u m b e r o f w o r k i n g p u m p s z , , , t h e s u m m a r i z e d o u t p u t f lo w Y ioa n d t h e h e a d o f w a t e r H i , . I n f o r m u l a ( 3 ) w e h a v e H i , G i , > 0 . I t i s c l e a r t h a t O < ~ z i o ~


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8 4 R . K l e m p o u s e t a l . / J o u r n a l o f C o m p u t a t io n a l a n d A p pl i ed M a t h e m a t i c s 8 4 ( 1 9 9 7 ) 8 1 - 9 9

3 . S i m u l a t i o n a l g o r i t h m o f w a t e r d i s t r i b u t i o n n e t w o r kO n e o f t h e m o s t i m p o r t a n t p r o b l e m s i n t h e o p e r a t i v e c o n t r o l o f th e w a t e r d i st r ib u t i o n s y s t e m s

[6 , 1 2 ] is t h e d e t e r m i n a t i o n o f f l o w s a n d h e a d s i n th e w a t e r s u p p l y n e t w o r k . O u r a l g o r i t h m i si n t e r a c ti v e a n d th e r e f o r e it is e s se n t i a l t o s o l v e t h e p r o b l e m ( 6 ) - ( 8 ) m a n y ti m e s f o r v a r y i n g v a l u e so f o ic ic = 1 , . . . , I C. T h e r e a r e s e v e r a l w e l l - k n o w n m e t h o d s f o r s o l v i n g t h e f lo w a n d t h e h e a d i nd i s tr ib u t i o n n e t w o r k s [ 1, 6 - 8 , 1 2]. W e h a v e d e v e l o p e d a m e t h o d b a s e d o n t h e t h e o r y o f n e t w o r k f lo w .F o r t h e g i v e n v a l u e s o i c, ic = 1 . . . , I C E q s . ( 6 ) - ( 8 ) m a y b e t r a n s f o r m e d i n to a p r o b l e m o f e n e r g yw a s t e s m i n i m i z a t i o n i n t h e w a t e r s u p p l y n e t w o r k . T h e a l g o r i t h m e l a b o r a t e d b y u s a n d p r e s e n t e d h e r ei s d e s i g n e d t o s o l v e t h e f o l l o w i n g t a s k o f a s t a ti c o p t i m i z a t i o n w i t h l i n e a r c o n s tr a in t s :

1Af ( Y ) = ~ f ,~ ( Y i. ) ( 9 )ia = 1

s u b j e c t t oA y = a . ( 1 0 )

T h e o b j e c t i v e f u n c t i o n in ( 9 ) h a s i ts i n t e r p re t a t io n a s t h e p o w e r w a s t e s i n t h e a n a l y z e d n e t w o r kf o r t h e w a t e r d i s tr ib u t io n . I ts d e t a i l e d c o m p o n e n t s a r e d e f i n e d a s f o l lo w s :

f . ( Y i . ) = k ia Y i3 o s g n ( y i o ) + di , Y i . ; i a = 1 . . . . , I A . (11)F o r m u l a ( 1 1 ) r e su l ts f r o m B e m o u l l i 's l aw . O n e m a y s e e t h a t f ~ ( Y i , ) = x i, Y i, o r u s i n g v e c t o r

n o t a t i o n , f ( y ) = x V y . T h e a l g o r i t h m d e s i g n a t e d t o s o l v e t h e t a s k ( 9 ) - ( 1 0 ) c o n s i s ts o f t h r e e b a s i cp a r ts . T h e i n t r o d u c t o r y p a r t i s d e s i g n e d t o t r a n s f o r m t h e i n i ti a l p r o b l e m i n to t h e p r o b l e m w i t h o u tc o n s t ra i n ts . T h e s e c o n d p a r t d e a l s w i t h t h e p r o b l e m o f d e t e r m i n i n g t h e s e a r c h i n g d i re c t i o n a n d i nt h e t h ir d w e c o p e w i t h t h e p r o b l e m o f m i n i m i z i n g a t t h e g i v e n d i r e c t io n .3 .1 . T r a n s f o r m a t i o n o f t h e w a t e r d i s tr i b u ti o n n e t w o r k m o d e l

T h e t ra n s f o rm a t i o n o f th e p r o b l e m ( 9 ) - ( 1 0 ) i n to t h e p r o b l e m w i t h o u t c o n s tr a in t s is b a s e d o nd i s e n t a n g l e m e n t s o f th e c o n s t ra i n t s ( I 0 ) . F o r m a l l y , t h e f in a l re s u l ts m a y b e p r e s e n t e d a s f o l lo w s :

y = B V y z + D a . ( 1 2 )T h e B m a t r i x c o n s i s t s o f t h e e l e m e n t s 0, l , - 1 . O n e m a y p r o v e t h a t it s d e t a i l e d f o r m i s a l w a y s

g i v e n a s f o l l o w s :B = [ I, BI] ( 1 3 )

w h e r e I i s a u n i t y m a t r i x . S o , ( 1 3 ) i s a d e f i n i t i o n o f BI a n d ( 1 2 ) i s a d e fi n i ti o n o f D . T h e a l g e b r a o fm a t r i c e s s h o w s t h a t s u c h a D a n d BI m u s t e x i st . In o u r p r o g r a m a c o r r e s p o n d i n g s e g m e n t e x e c u t e st h e r e n u m e r a t i o n o f t h e v a r i a b le s s o t h a t t h e c o m p o n e n t s o f t h e v e c t o r 3'1 b e c o m e t h e f i rs t c o m p o n e n t so f t h e v e c t o r y . I t d e t e r m i n e s t h e D a v e c t o r ' s c o m p o n e n t s a n d i n f o r m s t h e c o m p a c t f o r m a b o u t t h ep o s i t io n o f t h e n o n z e r o e l e m e n t s o f t h e m a t r i x B . T h u s , t h e d e t e r m i n a t i o n o f a ll th e c o m p o n e n t so f v e c t o r y , o n t h e b a s i s o f (1 2 ) , r e q u i r e s o n l y t h e a d d i n g a n d s u b t r a c ti n g o p e r a t i o n to b e c a r r i e d


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R. Klem pous et al./Journal of Com putational and Applied Mathem atics 84 (1997) 81-99 85o u t . T a k i n g i n t o a c c o u n t ( 1 2 ) i n t h e o b j e c t iv e f u n c t i o n ( 9) , t h e i ni ti a l p r o b l e m ( 9 ) - ( 1 0 ) s h o u l d b er e p l a c e d b y t h e t a s k o f s t a t i c o p t i m i z a t i o n w i t h o u t c o n s t r a i n t s i n t h e f o r m o f

f ( B T y i + D a ) ---+m i n . ( 1 4 )T h e d i m e n s i o n o f t h e p r o b l e m ( 1 4 ) , i .e ., t h e n u m b e r o f c o m p o n e n t s o f th e v e c t o r y l i s t h e

d i f fe r e n c e b e t w e e n t h e n u m b e r o f v a ri a b l e s a n d t h e n u m b e r s o f c o n s tr a i n t s in t h e i n i ti a l p r o b l e ma n d w i l l b e d e n o t e d b y IR . A c c o r d i n g t o ( 9 ) - ( 1 0 ) t h e o b j e c ti v e fu n c t i o n in t h e re d u c e d p r o b l e m i sa s t r i c t l y c o n v e x a n d a t w i c e - d i f f e r e n t i a b l e f u n c t i o n . I t m a y b e e a s i l y s h o w n t h a t t h e b g r a d i e n t a n dH m a t r i x o f H e s s e o f th e o b j e c t i v e f u n c t i o n in ( 1 4 ) a re e x p r e s s e d a s f o ll o w s :

b = B . ~ 7 f ( y ) ( 1 5 )H = B A B z (1 6 )

w h e r e A i s d i a g o n a l m a t r i x i n t h e f o r mA = d i a g { 6 k ~ , l y ~ l ; i a = I , . . . , I A } . ( 1 7 )

T h e f o l l o w i n g p a r ts o f t h e a l g o r i t h m a r e t h e p r o c e d u r e s f o r d e t e r m i n i n g t h e s e a r c h e d d i r e c t i o na n d f o r m i n i m i z a t i o n o f th e o b j e c t i v e f u n c t i o n a t th e g i v e n d i r e c t io n . A d e t a i le d d e s c r i p t i o n o f t h e s ep r o c e d u r e s i s p r e s e n t e d i n t h e s u b s e q u e n t s e c t i o n s .3 .2 . A l g o r i t h m f o r d e t e r m i n i n 9 a n e w d i re c ti o n o f s ea r c h in 9 b a s e d o n t h e m o d i f ie d N e w t o n ' sm e t h o d

T h e p r o b l e m o f d e t e r m i n a t i o n o f a n e w f e a s i b l e d i r e c ti o n f o r se a r c h i n t h e m o d i f i e d N e w t o n ' sm e t h o d i s s i m p l i f ie d t o s o l v e th e l i n e a r -e q u a t i o n s y s t e m

H . q - - - b ( 18 )w h e r e H i s t h e H e s s i a n o f re d u c e d o b j e c t i v e f u n c t i o n a n d b t h e g r a d i e n t d e t e r m i n e d a t th e p o i n t yi s t h e a c t u a l o p t i m a l s o l u t io n . T h e p e c u l i a r i ty o f Eq . ( 1 8 ) i n d u c e d t h e a u t h o r s t o t r y s u c h a m e t h o df o r i ts s o l u t io n , w h i c h c o u l d e n a b l e t h e e l i m i n a t i o n o f t h e o p e r a t i o n s o f m u l t i p l i c a t io n b y z e r o i nt h e c o m p u t a t i o n a l p r o g r a m . F i n a l l y , t h e p r o b l e m ( 1 8 ) h a s b e e n r e p l a c e d b y t h e e q u i v a l e n t t a s k o fo p t i m i z a t i o n i n t h e f o r m :

1 bVF ( q ) = ~ . q T H ' q + q - -~ m i n ( 1 9 )a n d t h i s p r o b l e m h a s b e e n s o l v e d u s i n g t h e F l e t c h e r - R e e v e s ' s m e t h o d . T h e e f f ec t iv i ty o f s u c h a na p p r o a c h d e p e n d s o n p a r t i c u la r p r o p e r ti e s o f ta s k ( 1 9 ) . T h e g e n e r a l o u t l in e o f th e F l e t c h e r - R e e v e s ' sa l g o r i t h m i s a s s h o w n b e l o w :A l g o r i t h m 1

S t e p 1 . L e t i = 1 , q 0 = 0 , q i = - b .S t e p 2 . D e t e r m i n e t h e m i n i m u m o f t h e o b j e c t iv e f u n c t i o n i n t h e d i r e c t io n q 0 + r q i:

F ( q o + ~ b~ qi) = m i n F ( q 0 + r q i ) ( 2 0 )"r>0N o w let q0 = q0 + ~b~qi.


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86 R. Klempous et al . / Journal of Com putational and Applied Mathem atics 84 (1997 ) 81-99S t e p 3 . I f i > I R t h e n S t o p . O t h e r w i s e d e t e r m i n e t h e g r a d i e n t o f th e f u n c t i o n F a t t h e p o i n t q 0:

gi z ~TF(qo) = Hqo + b . ( 2 1 )S t e p 4 . D e t e r m i n e a n e w d i r e c t io n q i +~ a c c o r d i n g t o t h e e q u a t i o n :

(gi, gi)q i = -g i + " q i -1 ( 2 2 )( g i - l , g i - I )p u t i = i l a n d g o to S te p 2 .

T h e s o l u ti o n o f t h e p r o b l e m ( 2 0 ) a c c o r d i n g t o ( 2 2 ) i s b r o u g h t to t h e s o l u t i o n o f t h e e q u a t i o nd-d-~F(qo + ~ " q i ) - - 0 . ( 2 3 )

H e n c e , w e o b t a i nq ~ . H "qi + z . q y . H . qi + b T . q i = O . ( 2 4 )

B e c a u s e q 0 a n d q ~ a r e c o n j u g a t e d i r e c t i o n s , i t i s o b t a i n e d f r o m t h e d e f i n i t i o n :z i = - b Tqi/qVi H qi ( 2 5 )

T h e d e t e r m i n a t i o n o f a n e w f e a s ib l e d i r e c t i o n f o r s e a r c h i n g q 0 f o r o u r i n i ti a l ta s k ( 1 9 ) r e q u i re s , i nc a s e o f a p p l ic a t i o n o f t h e F l e t c h e r - R e e v e s ' s a l g o r it h m , t o b e a r c e r ta i n c o m p u t a t i o n a l e x p e n d i t u r e s .T h e o p e r a ti o n s c o n n e c t e d w i t h t h e d e t e r m i n a t io n s o f p r o d u c t s ( 2 5 ) a s w e l l a s H q o i n ( 2 1 ) a r ec r i ti c a l . A p p l i c a t i o n o f f o r m u l a ( 1 6 ) r e s u l t s i n t h e f i rs t c a s e :

IAqTiHq~ = (BTqi ) T A . ( B T q i ) - ~ ~ _ a "~jV2ij i = 1 , . . . , I R ( 2 6 )

j= lw h e r e vi = BTqi; i = 1 . . . , I R . B e c a u s e B i s a m a t r i x w i t h e l e m e n t s 0, 1 , - 1 t h e d e t e r m i n a t i o no f s u c c e s si v e c o m p o n e n t o f t h e v e c t o r vi re q u i re s o n l y t h e o p e r a t i o n o f a d d i ti o n a n d s u b tr a ct io n .I n o r d e r t o c a l c u l a t e t h e p r o d u c t q T H q a c c o r d i n g t o ( 2 6 ) o n l y 2 I A m u l t ip l i c a t io n s a r e r e q u i r e d ,w h e r e I A is t h e n u m b e r o f a rc s o f t h e a n a l y z e d n e t w o r k . A s i m i la r a n a ly s i s m a y b e c a r ri e d o u t f o rE q . ( 2 2 ) :

H q 0 - - B " A B T q0 = B " w 0 ( 2 7 )w h e r e w 0 = ABTvo; Vo = B rqo. B e c a u s e A i s a d i a g o n a l m a t r i x f o r th e d e t e r m i n a t i o n o f p r o d u c t H q oi t r e q u i r e s , a c c o r d i n g t o ( 2 7 ) , o n l y I A m u l t i p l i c a t i o n s .3 .3 . A n a l y s i s o f t h e p ro ced u re p ro p er t i e s f o r s ea rch i n g t h e m i n i m u m i n t h e g i ven d i r ect io n

T h e s u c c e s si v e s e a r c h in g s t a g e f o r c u r r e n t m i n i m u m o f t h e o b j e c t iv e f u n c t i o n f ( y ) i s b a s e d o nt h e m i n i m i z a t i o n p r o c e s s o f th e o n e v a r i a b l e f u n c t i o n G ( t ) d e f i n e d b y t h e e q u a t i o n

G ( t) = f ( y ( t ) ) ( 2 8 )


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R . Klem pous et aL / Journal of Com putational and Applied Mathematics 84 (1997) 81-99 87w h e r e

y( t) = yo + t "qo ( 2 9 )i s t h e p a r a m e t r i c e q u a t i o n o f a s t ra i g h t l i n e i n R IA. A c c o r d i n g t o t h e d i f fe r e n t i a ti o n p r i n c i p l e s o f af u n c t i o n w i t h m a n y v a r i a b l e s o n e o b t a i n s

1 A I AG '( t) = ~_~ diaqoi. + 3 Z kio(Yoi. + t . qoi~)2sgn(Yoi. + t . qoi.) "qoi. . ( 3 0 )

i a = l i a = l

I n a n s i m i l a r w a y a s i n t h e c a s e o f f o r m u l a ( 3 0 ) o n e m a y d e t e r m i n e t h e s e c o n d d e r i v a t iv e o f t h ef u n c t i o n G ( t ) :

I AG" (t) = 6 ~ -~ k i, lYoi , + t , . q0/,I" q~i,. ( 3 1 )

i a = 1

A s a r e s u l t o f f o r m u l a ( 3 1 ) t h e c h a r t o f f u n c t i o n G " ( t ) i s a p o i n t w i s e l i n e a r f u n c t i o n . M o r ep r e c i s e l y , G ( t ) i s a s to c h a s t i c p r o c e s s : G ( t ) = G ( t , w ) , w h e r e w i s a n e l e m e n t f r o m t h e r a n d o ms p a c e 12 : w E f2 . S p a c e f2 r e s p o n d f o r t h e r a n d o m f o r m o f t h e c o n s u m e r s ' d e m a n d . O n e m a y s h o wt h a t c o n d i t i o n G " ( t ) > 0 i s s a t is f ie d f o r a n y t w i t h a p r o b a b i l i t y e q u a l t o 1 ( i. e ., P ( A c ) = 1 , w h e r eA c = { w E f 2 : G " ( t , w ) > O } a n d P i s a p r o b a b i l i t y m e a s u r e o n f 2) . I ts d e r i v a t i v e G ' ( t ) i s a s t r i c t l yi n c r e a si n g f u n c t i o n c o n s is t in g o f p a r a b o li c s e g m e n t s . F r o m ( 3 0 ) t h e f o l lo w i n g c o n c l u s i o n m a y b ed r a w n :

l i m G ' ( t ) = + e ~ . ( 3 2 )t ----*~H e n c e a n d f r o m ( 2 2 ) i t r e s u l t s t h a t G ' ( t ) h a s e x a c t l y o n e r o o t to w h i c h i s th e s e a r c h e d m i n i m u m

o f t h e f u n c t io n G ( t ) . A d d i t io n a l l y , to > 0 . T h e a b o v e d e s c r i b e d p r o p e r t ie s e n a b l e t h e e l a b o r a t i o n o fa n e f fi c ie n t a l g o r i th m f o r t h e d e t e r m i n a t i o n o f t h e p o s i ti o n o f m i n i m u m t o w a r d s t .T h e i d e a o f t h e a l g o r i t h m i s b a s e d o n t h e d e t e r m i n a t i o n o f t w o s e q u e n ti a l ru p t u r e s o f f u n c t io nG ' ( t ) b e t w e e n w h i c h i s t h e m i n i m u m a s w e l l a s t h e v a l u e s o f t h e f u n c ti o n G ' ( t ) a t t h e s e p o i n t s t 'a n d t " . T h e n o n e m u s t d e t e r m i n e th e v a l u e o f th e f u n c t i o n G ' ( t ) a t a n y p o i n t , e . g . , t = ( t ' + t") /2 .T h r o u g h t h e s e t h r e e p o i n t s o n e s h o u l d c a r r y o u t a p a r a b o l a a n d d e t e r m i n e it s r o o t s. T h e s k e t c h e da l g o r i t h m s e n a b l e s t o d e f i n e w i t h c o m p u t e r a c c u r a c y t h e p o s i ti o n o f t h e p o i n t t o. T h e p r o c e d u r ef o r d e t e r m i n a t i o n o f p o i n t s t ' a n d t " i s b a s e d o n t h e i d e a o f t h e s e t o f p o i n ts d i s c o n t i n u i t y o f t h ef u n c t i o n G ' " ( t ) . A c c o r d i n g t o E q . ( 3 0 ) , t ' i s th e p o i n t o f d i sc o n t i n u i t y o f t h e f u n c t i o n G ' " ( t ) , i f

( 3ia = 1 , . . . , I A ) (qoi~ # 0 , Yi, + t'qoi~. = 0 ) . ( 3 3 )S u c h a m e t h o d c a n b e i m p l e m e n t e d b y c o n s t r u c t i n g t h e fi rs t s e t N 1 d e f i n e d b y

N , = { 3i a = 1 , . . . , I A ; qoia 0, Yjqoi+ < 0 } ( 3 4 )b e c a u s e , t h e e x t r e m e G ( t ) s h o u l d be s e a r c h e d f o r t > 0 . H a v i n g d e t e r m i n e d to, t h e n e w c u r r e n ts o l u t i o n i s f o u n d :

Yio = yio + to qoi, ia = 1 . . . . I A . ( 3 5 )T h i s f i n is h e s th e p r o c e s s o f d e t e r m i n a t i o n o f t h e s u c c e s si v e m i n i m u m o f t h e o b j e c t i v e f u n c t io n

f ( Y ) .


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88 R . Klem pous et al. / Journal of Com putational and Applied Mathem atics 84 (1997) 81-993 .4 . N u m e r i c a l a n a l y s is

T h e c o m p l e x i t y o f th e F l e t c h e r - R e e v e s a l g o r it h m b a s e d o n t h e p r i n c ip l e s p re s e n t e d a b o v e , m e a -s u r e d b y t h e n u m b e r o f m u l t i p l ic a t i o n i s o f o r d e r o f IR I A . T h e c l a s s ic a l a p p r o a c h r e l y i n g o n t h es o lu t io n o f s y s t e m ( 1 8 ) is c h a r a c t e r i z e d b y t h e c o m p l e x i t y o f o r d e r o f ( I R ) 3. T h e v e r s io n w h i c he n a b l e s t h e o p t i m i z a t i o n o f t h e n e t w o r k c o n s i s t in g o f a b o u t 1 00 a r c s h a s n o m o r e t h a n 1 9 k b y t e s o fm e m o r y . F o r o u r m o d e l c o n s i s t in g o f IN = 7 6 n o d e s a n d I A = 9 6 a r c s ( a l s o o n e r e s e r v o i r a n d t h r e ep u m p s t a ti o n s ) w e r e q u i r e n o m o r e t h a n a f e w s e c o n d s f o r d e t e r m i n i n g t h e o p t i m a l f lo w w i t h t h eI B M c o m p a t i b l e P C .

4 . O p e r a t i v e c o n t r o l i n w a t e r d i s t r i b u t i o n s y s t e m sA g g r e g a t e d n e t w o r k m o d e l s h a v e b e e n d e s c r ib e d b y m a n y a u t h o rs [8, 1 3]. I n t h e m o d e l o f th e

w a t e r d i s t r i b u t i o n s y s t e m t h e f o l l o w i n g p r o p e r t i e s h a v e b e e n t a k e n i n t o a c c o u n t : T h e p r o p e r w o r k i n g s y s t e m d e p e n d s u p o n t h e r e a l iz a t i o n o f c e r t a i n r e l a ti o n s b e t w e e n h e a d s a tn e t w o r k n o d e s . T h e p o s s i b i li t y o f f il li n g o r e m p t y i n g r e s e r v o i r s i s p a r t ic u l a r l y i m p o r t a n t . T h e p u m p s ta t io n s a r e c o n t r o l l e d d i sc r e t e l y b y t u r n i n g o n a n d o f f t h e p u m p s o n e a f t e r a n o t h e r . T h e

p u m p s h a v e t o o p e r a te w i t h a m a x i m u m e f f ic i e nc y . T h e c o s t o f e l e c t r ic i t y c o n s u m e d b y p u m p s ta t io n s i n a g i v e n p e r i o d is a g o a l fu n c t i o n .

T h e o p t i m i z a t i o n p r o b l e m s t h u s o b t a i n e d a r e s o c o m p l i c a t e d th a t , in p r a c t ic e , t h e a t te m p t s t os o l v e l a r g e r s y s t e m s a r e p r a c t i c a l ly i m p o s s i b le [ 1 ,8 ,1 2 ] . T h e d i f fi c u lt ie s c a n b e o v e r c o m e b y u s i n gp r o b l e m d e c o m p o s i t i o n a n d t h e a g g r e g a t i o n o f t h e m o s t c o m p l e x e l e m e n t s o f t h e s y s t e m ( e .g . , th ep i p e l i n e n e t w o r k ) . T h e a p p r o a c h p r e s e n t e d h e r e , d i f f e r s f r o m t h e o t h e r f o r m u l a t i o n s [ 2 , 8 ] b e c a u s e i tt a k e s i n t o a c c o u n t t h e h e a d s r e l a t io n s i n t h e n e t w o r k , f u r t h e r m o r e , t h e d i s c r e t e c h a r a c t e r o f p u m ps t a ti o n s y i e l d s t o a m o r e p r e c i s e d e s c r i p t i o n o f t h e s y s t e m , b u t t h i s c r e a t e s a d d i t io n a l c o m p u t a t i o n a lp r o b l e m s .

T h e o p t i m i z a t i o n a l g o r i t h m h a s t w o l e v e ls . A t t h e u p p e r l e v e l a d y n a m i c p r o b l e m i s so l v e d .I ts r e s u l t i s a s c h e d u l e o f t h e r e s e r v o i r s ' e x p l o i t a t i o n . T h e d a t a a b o u t f l o w s to t h e r e s e r v o i r s a r es u b m i t te d t o t h e l o w e r le v e l , w h e r e a s ta t ic p r o b l e m i s so l v e d . It s s o lu t i o n d e t e r m i n e s h o w m a n yp u m p s s h o u l d b e t u r n e d o n i n t h e p u m p s t at io n s a n d t h e i r c u r r e n t y i e l d s . T h e s u b j e c t o f th i s p a r t i st h e d e s c r i p t i o n o f th e a l g o r i t h m f o r s o l v i n g t h e s t a ti c p r o b l e m . I t e x p l o i t s b r a n c h - a n d - b o u n d m e t h o d[11 , 13 ] as we l l .4 .1 . F o r m u l a t i o n o f t h e s t a t i c p r o b l e m

T h e s y s t e m u n d e r c o n s i d e r a t i o n i n c l u d e s I P p u m p i n g s t a t i o n s , I S r e s e r v o i r s a n d I C c o n s u m e r s .V e c t o r r h a s t h e f o l l o w i n g f o r m : r = ( - y , q , a ) . O p t i m i z a t i o n p r o b l e m o f t h e l o w e r l e v e l c o n s is t s o fm i n i m i z a t i o n o f t h e e n e r g y c o s ts u s e d u p b y p u m p i n g s ta ti on s :

F ( q , a ) = m i n ( a - z + f l- y ) ( 3 6 )z , ys u b j e c t t o

Hip(yip,zio)>~vip(r), i p - - 1 , . . . , I P , ( 3 7 )


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R. Klempous et al . /Journal of Computat ional and Applied Mathematics 84 (1997) 81-99 89h i , - k i , q i2 > ~ v i , ( r ) i f q i s < 0 ,hi, + ki, . qi2, > ~ vi~(r) i f qi~ ~ 0 ,Y - ip " z i p ~ Y i p ~ Y i p " Z i p ,

0 ~


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90 R . Klempous et al./Jou rnal of Com putational and Applied Mathem atics 84 (1997 ) 81-9 94.2. A m eth od fo r solving the opt im izat ion prob lem o f the water dis tr ibution sys tem

T h e p r o b l e m f o r m u l a t e d o n th e b a s is o f th e a g g r e g a t e d m o d e l h a s a n u m b e r o f v a ri a b le s a n dc o n s t ra i n t s th a t a r e p r o p o r t io n a l t o t h e n u m b e r o f p u m p s t a ti o n s a n d r e s e r v o i rs i n t h e s y s t e m . U s u a l l yt h is i s s m a l l e r th a n t h e n u m b e r o f v a ri a b l e s a n d c o n s t r a in t s t h a t c a n o c c u r w h e n t h e o r i g in a l s y s t e mi s i n v e s t i g a t e d .

T h e a l g o r i t h m i s b a s e d o n t h e c o n c e p t i o n o f th e b r a n c h - a n d - b o u n d m e t h o d [ 1 1] . T h e c o n s t r a in t sI-I1P ~"1c a u s e a s e t o f s o l u t i o n s t h a t a re i n c o h e r e n t a n d m a y c o n t a i n n o m o r e t h a n 1 1 ip = ~ + ~ i p ) s u b a r e a s .I t c a n n o t b e a s s u m e d t h a t t h e q u a d r a t i c f o r m s ( 4 3 ) a r e s e m i n e g a t i v e . T h e f u n c t i o n s c a n n o t b ec o n c a v e .

T h i s p r o p e r t y o f p r o b l e m ( 4 4 ) - ( 4 8 ) m a k e s i t s s o l u t i o n m o r e d i ff ic u lt . A s i t is w e l l k n o w n t h e r ea r e n o g e n e r a l a n d e f f e ct iv e m e t h o d s o f s o l v i n g t h e p r o b l e m s o f t h e t y p e d e s c r i b e d a b o v e . T h i si n v o l v e d t h e a u t h o r s t o d e v e l o p a n a l g o r i t h m s p e c i a l ly d e s i g n a t e d f o r s o l v i n g th e p r o b l e m s o f t h isc l a ss [ 5 , 6 , 1 3]. I n a n a l y z i n g p r o b l e m ( 4 4 ) - ( 4 8 ) o n e m a y o b s e r v e t h a t w h e n t h e v e c t o r o f i n te g e rv a r i a b l e s z i s f i x e d t o a n a c c e p t a b l e v a l u e , t h e o b t a i n e d p r o b l e m i s d e p e n d e n t o n l y o n c o n t i n u o u sv a r i a b l e s y . T h e p r o b l e m w i l l b e d e n o t e d a s P ( z ) :

P (z ) = ~ eip .zip + myin flip" Yip (4 9)i p = l ~ i p = l

s u b j e c t t oy T . C k ( z ) . y + D i k ( z ) . y + E g k ( Z ) ~ > O ; i = l , . . . , I K , ( 5 0 )Y-ip "zip< ~ Yip~ Y ip " z ip ; ip = 1 , . . . , I P , ( 5 1 )

IPZ Yip = Yo. ( 5 2 )i p = l

S u m m i n g u p b o t h s i d e s o f c o n s tr a i n ts ( 5 1 ) , o n e o b t a i n sIP 1P IPZ Y-ipZiP ~ Y ip~ Z YipZip" ( 5 3 )

i p --1 i p = l i p = lA n d t a k i n g i n t o c o n s i d e r a t i o n r e l a t i o n ( 5 2 ) r e s u l t s i n t h e f o l l o w i n g i n e q u a l i t i e s :

IP IPy_jzj


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R. Klempous et al./Journal of Computational and Applied Mathematies 84 (1997) 81 99 91L e t Y ( z ) b e a s e t o f f e a s ib l e s o l u t i o n s o f t h e p r o b l e m P(z) . Y(z) i s d e f i n e d b y c o n s t r a in t s

( 5 0 ) - ( 5 2 ) . L e t Yc(z), i n t u r n , b e a s e t o b t a i n e d f r o m Y ( z ) b y l e a v i n g o u t t h e n o n l i n e a r c o n s t r a i n t s( 5 0 ) .Y L(z) = { y E ~ r' l y_j_zj


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92 R Klem pous e t a l . / Journ al o f Computa tional and Appl ied Mathem at ics 84 (1997) 81 99

U s i n g t h e f e a s i b i l i t y a n d o p t i m a l i t y t e s t s w e m a y s t a t e t h a t c e r t a i n P(z) p r o b l e m s c a n n o t g i v eo p t im a l s o lu t io n w i th o u t t h e n e c e s s i t y o f s o lv i n g t h e m . T h e b r a n c h - a n d - b o u n d m e t h o d p r e s e n t e d h e r ec o n s i s t s o f t w o s t a g e s. I n t h e f i r s t s t a g e a l i s t o f a ll p r o b l e m s P(z), w h i c h m a y p o t e n t i a l l y g i v e a no p t i m a l s o l u ti o n , is c r e a t e d . T h e p r o b l e m s P(z) t h a t a r e a s s i g n e d t o t h e l is t, a r e s e l e c t e d o n t h e b a s i so f th e f e a s i b i l it y te s t s ( 5 5 ). I n t h e s e c o n d s t a g e w e f i n d t h e s o l u t i o n o f t h e p r o b l e m w i t h t h e b e s te s t i m a t i o n v a l u e . N e x t w e d e l e t e f r o m t h e l i s t t h e s e p r o b l e m s f o r w h i c h , t h e r e i s n o d o u b t t h a t , t h e i rs o l u t i o n c a n n o t b e b e t t e r .

5. Conclus ions

I t m u s t b e n o t e d t h a t t h e p r o b l e m p r e s e n t e d a b o v e i s q u it e d i ff e r e n t f r o m t h e o n e s k n o w n a s i n v e n -t o r y p r o b l e m s [ 1 0] . T h e c l a s s ic a l m o d e l o f th i s p r o b l e m c o n c e m s th e i s o l a te d r e s e r v o i r o r a s y s t e mo f t h e m . I n w a t e r d i s t ri b u t io n s y s t e m s s u c h a n i s o l a t io n w o u l d b e r e g a r d e d a s a n o v e r s i m p l i f i c a t io n

i i

Fig. 1. Exemplary water distribution system.


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R . K l e m p o u s e t a l . / J o u r n a l o f C o m p u t a t i o n a l a n d Ap p l i e d M a t h e m a t i c s 8 4 (1 9 9 7 ) 8 1 9 9 93

o f t h e p r o b l e m b e c a u s e t h e re a r e s t ro n g c o n n e c t i o n s b e tw e e n w a t e r l e v e ls i n a r e s e r v o i r a n d f lo w sa n d h e a d s i n t h e n e t w o r k . T h i s s h o u l d n o t b e n e g l e c t e d , o th e r w i s e i t l e a d s t o t h e a c h i e v e m e n t o fs o m e p h y s i c a l l y u n r e a l i z e d v a r ia n t s.

T h e r e s u lt s o f t h e t e s ts c o n f i r m th e u s e f u l n e s s o f th e p r e s e n t e d a g g r e g a t e d n e t w o r k m o d e l . T h i sa l g o r i th m c a n b e u s e d a s a l o w e r s t a g e o f t h e d y n a m i c p r o b l e m s o l v i n g p r o c e d u r e f o r t h e d e t e r-m i n a t i o n o f a n o p t i m a l s c h e d u l e q(t) f o r f i l l i n g t h e r e s e r v o i r . W e u s e d t h e d y n a m i c p r o g r a m m i n gm e t h o d f o r th e s y s t e m s w i t h o n e r e s e rv o i r. F o r s y s t e m s w i t h m o r e t h a n o n e r e s e r v o i r t h e a l g o r i th mb a s e d o n t h e W o l f e m e t h o d is m o r e a p p r o p ri a te .

AppendixI n o r d e r t o v e r i f y t h e n u m e r i c a l e f fe c t iv e n e s s o f t h e d e v e l o p e d a l g o r i t h m s o f o p e r a t i v e c o n t r o l,

s e r ie s o f 4 0 t e s t s h a v e b e e n s o l v e d . I n a d d i t i o n , s o m e s i m u l a t i o n t e s t s fo r w a t e r d i s t r ib u t i o n n e t w o r ke q u i p p e d w i t h th r e e p u m p s t at io n s a n d t w o r e s e r v o i rs h a v e b e e n a l so c a r ri e d o u t.

T h e e l a b o r a t e d t e s ts o f fe a s i b i li t y a n d o p t i m a l i t y e n a b l e d th e r e j e c t i o n o f n e a r l y 9 0 % o f t h ep r o b l e m s P(z) w i t h o u t t h e n e c e s s i ty o f s o l v i n g th e m . T h e o b t a i n e d n o n l in e a r p r o g r a m m i n g p r o b l e mP(z) i s s o l v e d b y m e a n s o f a m o d i f ie d c u t t in g p l a n e s K e l l e y ' s m e t h o d d e v e l o p e d b y t h e a u t h o r s [ 9].T h e a i m o f t h e m o d i f i c a ti o n w a s t o t a k e i n t o c o n s i d e r a t io n th e p o s s i b il it y o f n o n c o n v e x i t y o f t h ef e a s i b l e s o l u t i o n s s e t .

E x e m p l a r y s y s t e m ( F ig . 1 ) c o n s i s t o f t h r e e p u m p s t at io n s ( P ~ , P z, / 3 ) , t w o r e s e r v o i r s ( Z I , Z 2 ) , 2 4n o d e s a n d 2 6 a r c s .

T h e p a r a m e t e r s o f th e n e t w o r k a re g i v e n f r o m T a b l e 1 w h e r e a s t h e p u m p s t a t io n s a n d r e s e rv o i r sp a r a m e t e r s f r o m T a b l e 2 .

T a b l e 1N e t w o r k p a r a m e t e r sN o d e s I N = 2 4 P i p e l in e s I A = 2 6i hi qi i ki di

1 0 .0 - 5 7 9 1 0 . 1 0 0 E - 0 62 4 . 0 - 5 7 8 2 0 . 1 4 0 E - 0 53 1 3. 0 - 5 7 8 3 0 . 2 5 0 E - 0 54 1 8 .0 0 4 0 . 6 0 0 E - 0 55 2 1 . 0 0 5 0 . 1 0 0 E - 0 46 4 , 0 1 7 4 6 0 . 9 0 0 E - 0 47 1 5 .0 1 4 7 7 0 . 1 0 0 E - 0 38 1 5 .0 1 6 8 0 . 5 0 0 E - 0 49 1 5 .0 1 8 0 9 0 . 1 9 0 E - 0 3

1 0 1 4 .0 4 5 1 0 0 . 1 4 0 E - 0 51 1 1 4 .0 5 0 1 1 0 . 5 0 0 E - 0 51 2 1 7 .0 1 5 4 1 2 0 . 1 0 0 E - 0 31 3 1 5 .0 9 6 1 3 0 . 5 0 0 E - 0 41 4 1 3. 0 7 4 1 4 0 . 1 0 0 E - 0 31 5 1 3 .0 8 7 1 5 0 . 1 6 0 E - 0 4

4 . 05 .00 .0

- 2 . 0- 2 . 0

1 1 . 07 .00 . 00 .01 .00 . 03 .01 .01 .05 .0


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94 R . K l e m p o u s e t a l . / J o u r n a l o f C o m p u t a t io n a l a n d A p p l i ed M a t h e m a t i c s 8 4 ( 1 9 9 7 ) 8 1 - 9 9Table 1 ( C o n t d . )16 15.0 38 16 0.790E-03 2.017 7.0 128 17 0.100E-02 -6. 018 7.0 57 18 0.370E-04 0.019 9.0 37 19 0.220E-03 7.020 10.0 74 20 0.180E-03 1.021 10.0 96 21 0.100E-05 1.022 1.0 164 22 0.3 70E-04 -3 .023 16.0 38 23 0.6 50E-04 -9 .024 11.0 80 24 0.540E-03 10.0

25 0.410E-03 -2.026 0.110E-03 4.0

Note: hi - - the altitude of ith node,qi -- the desired flow in ith node,ki - - the resistance of ith pipeline,di - - the difference of ith pipeline ends altitude.

Table 2Pump station and reservoir parametersPump stations IP = 3i ~.i y i Yi Hi ki ~i ~i

Reservoirs IS = 2i si hi ki

1 2 192 292 67.4 3 x 10 -4 159 991 197 12 4 178 444 72.9 1.6 x 10 -4 137220 380 23 2 132 192 59.8 5.6 10 -4 79920 107

7500 30 10 -67500 30 10 -6

Note: The symbols are the same as in the Section 2.~i - - the capacity of ith reservoir,hi -- the height of ith reservoir in relation to connection node,ki - - the resistance of ith reservoir connection,For the other symbol interpretation see formulas (2), (3) and (6).

The resul ts of the s imu lat ion algori thm are given from Tabl e 3.The resul ts of the abov e-me ntio ned tests (Sect ion 4) are shown in Table 4 (where reservoirs carry

a funct ion of sources) and Table 5 (where reservoirs carry a funct ion of receivers) .The name s of the co lumn s (Tables 4 and 5) correspo nd to the not ions in Sect ion 2. So we not icea0 - - the consu mer ' s demand,qi -- the reservoirs flows,Yo -- the total output flow from the pum p stations,z* -- the vector of opt imal pumps configurat ion,


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R . K l e m p o u s e t a l . / J o u r n a l o f C o m p u t a t io n a l a n d A p p li e d M a t h e m a t i c s 8 4 ( 1 9 9 7 ) 8 1 - 9 9 95T a b l e 3R e s u l t s o f n e t w o r k s i m u l a t io nN o d e s I N = 2 4 P i p e l in e s I A = 2 6i vi i y, xi

1 55.9 1 577.0 4.02 51.6 2 573.0 5~53 48.6 3 579.0 0 ,84 3 3. 1 4 0 . 0 - 2 . 05 3 5 .4 5 0 .0 - 2 . 06 51.9 6 222.7 15.57 36.4 7 180.4 10.38 36.1 8 75.8 0.39 30.0 9 179.9 6.1

1 0 3 7 . 9 1 0 - 3 . 4 1. 011 37.4 11 299.8 0 .412 32.0 12 153.9 5 .413 35.9 13 95.9 1.514 47 .8 14 29 8 .6 9 .915 47.1 15 206.4 0 .716 43.9 16 38.0 3 .117 46.5 17 81.5 0.61 8 4 6 . 5 1 8 - 4 6 . 5 - 0 . 119 46.1 19 87.5 8.720 44 .1 20 74 .0 2 .021 44.9 21 467.1 1.222 51.8 22 191.0 - 1 .723 3 5 .1 23 18 0 .2 - 6 .924 41.6 24 16.3 10.12 5 - 4 1 . 4 - 2 . 7

26 116.7 5 .5N o t e : yg - - t h e f l o w i n i t h p i p e l i n e ,

x , - - t h e p r e s s u r e h e a d d i f f e r e n c e i n i t h p i p e l i n e ,v i - - t h e p o t e n t i a l o f t h e i t h n o d e .

y * , y * , y ~ ' - - t h e o p t i m a l o u t p u t f l o w f r o m e a c h p u m p s t a t i o n s ,Foo t - - t h e m i n i m a l e n e r g y , w h i c h e n a b l e s th e r e a l i z a t i o n o f c o n s u m e r ' s d e m a n d ,D L - - t h e n u m b e r o f p r o b l e m s L l i s t , a f t e r f e a s i b i l i t y t e s t ,L R Z - - t h e n u m b e r o f n o n e l i m i n a t e d p r o b l e m s f r o m L l i s t ,T o - - t h e c o m p u t i n g t i m e .

H e r e a c c u r a c y o f l 0 - 5 i s a s s u m e d . A n a l y z i n g t h e t i m e s T o o f s o l v i n g t h e t e s t , o n e c a n s a y t h a t o u ra l g o r i t h m s c a n b e o n - l i n e i m p l e m e n t e d t o t h e s t a t ic o p t i m i z a t i o n o f w a t e r d i s t r i b u t i o n s y s t e m .


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96 R. Klem pous et al . / Journal of Computational and Applied Mathem atics 84 (1997) 8 1-99T a b l e 4R e s u l t s o f 2 0 t e s t s f o r o p t i m i z a t i o n a l g o r i th m s o f o p e r a t i v e c o n t r o l ( r e s e r v o i r s c a r ry a f u n c t i o n o f s o u r c e s )

* Fopt DLr a o q l q 2 y o z * Y l( I / s ) ( l / s ) ( l / s ) ( I / s ) y ~ ( k W ),Y3

L R Z To( s)

1 5 0 0 1 00 - 1 0 0 3 0 0 ( 0 0 2 )

2 7 5 0 - 1 0 0 - 1 0 0 5 50 ( 0 1 1 )

3 1 00 0 - 1 0 0 - 1 0 0 8 00 ( 1 1 1 )

4 1 50 0 - 2 5 0 - 2 5 0 1 00 0 ( 1 1 2 )

5 1 50 0 - 3 0 0 - 3 0 0 9 00 ( 1 1 1 )

6 7 5 0 - 1 5 0 - 1 5 0 4 5 0 ( 1 0 1 )

7 1 00 0 - 1 5 0 - 1 5 0 7 0 0 ( 0 1 2 )

8 1 80 0 - 3 0 0 - 3 0 0 1 2 00 (1 2 1 )

1 0 9 0 0 - 1 0 0 - 2 0 0 6 0 0 ( 0 1 1 )

1 1 2 0 0 0 - 3 0 0 - 3 0 0 1 4 00 ( 2 2 1 )

1 2 2 0 0 0 - 4 0 0 - 4 0 0 1 2 00 - -1 3 1 0 00 - 3 0 0 - 3 0 0 4 0 0 ( 0 1 0 )

1 4 3 0 0 - 5 0 0 2 5 0 ( 1 0 0 )

1 5 3 0 0 - 1 0 0 0 2 0 0 ( 1 0 0 )

1 6 3 0 0 - 1 5 0 0 1 50 ( 0 0 1 )

0 191 2 1 90

3 000 3 7 4 6 1 1 0

3 62188292 575 9 2 123 1 6192292 679 11 1 33 2 738 12 9 2 6 1 3 1 0 1 1 24 1 6192259 311 3 1 14

0192

0 473 8 1 153 7 43 2 6290 865 13 3 427 2 21880 396 6 1 174 2 4176488 1068 16 7 417 34178

- - 13 13 450 289 3 1 5

4 0 00

2 5 0 2 0 9 1 1 300

200 199 1 1 3000 95 1 1 30

150


17/19


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98 R. K lempous et al . / Journal of Computat ional an d Applied Mathem atics 84 (1997) 8 1-99Table 5 (Contd.)11 300 400 0 700 (210) 413 647 8 8 12287

012 500 100 250 850 (112) 240 660 10 5 11

33427513 750 100 100 950 (211) 435 766 11 8 1336O15614 500 250 250 100 (212) 422 846 11 10 253O8270

15 1000 100 100 1200 (221) 437 1007 13 13 23606157

16 500 100 300 900 (121) 195 780 10 10 2955014917 500 200 200 900 (211) 419 761 10 10 3333015218 300 100 250 650 (111) 219 542 7 6 17278153

19 1000 200 200 1400 - - - - - - 13 13 3720 1000 300 300 1600 - - - - - - 16 16 38

Re f e r en c e s

[1] G. Cohen, Optimal control of water supply networks, in: S.G. Tzafestas (Ed.), Optimization of Dynamics OperationalResearch Models, North-Holland, Amsterdam, 1982, pp. 251-276.[2] B. Coulbeck, M. Sterling, Optimized control of water distribution systems, Proc. IEEE, vol. 122, No. 2, 1978.[3] R. Klempous, J. Kotowski, J. Nikodem, System approach to the water distribution problems, Proc. 12th European

Meeting on Cybernetics and Systems Research, Vienna, Austria, 1994, pp. 957-963.[4] R. Klempous, J. Kotowski, J. Nikodem, Supervisory control and data acquisition system for the water distributionnetworks, Proc. IASTED Intemat. Conf., Gold Coast, Australia, 1996, pp. 32-34.[5] R. Klempous, J. Kotowski, J. Nikodem, M. Olesiak, J. Utasiewicz, Some models for water distr ibution systems,

J. Comput. Appl. Math. 21 (1988) 257-269.[6] J. Kotowski, M. Olesiak, The optimization of energy wastes in the water complex-supply systems, Proc. sixthIFAC/IFIP Intemat. Conf. o f Digital Computer Applications to Process Control, Diisseldorf, 1980, pp. 386--395.[7] Z. Mahjoub, Contribut ion d'etude l'optimization des reseaux mailles, These d'Etat-Informatique-Mecanique desFluides, Institute Nationale de Tolouse'76, 1983.[8] S. Miyaoka, M. Funubashi, Optimal control of water distribution systems by network flow theory, IEEE Trans.Automat. Control AC-29 (4) (1984) 303-311.[9] J. Nikodem, J. Utasiewicz, Generalization of the Kelley's algorithm to problems with nonconvex set of feasiblesolutions, Archiwum Automatyki i Telemechaniki 33 (I ) (1988) 105-115 (in Polish).

[10] A.A. Piervozvanski, Mathematical Models in Production Processes, Izdatielstwo Nauka, Moscow, 1975 (in Russian).


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R. Klempous et al./Journal of Computational and Applied Mathematics 84 (1997) 81-99 99[11] H.M. Salkin, Integer Programming, Addison-Wesley, Reading, MA, 1975.[12 ] U. Sham ir , Opt imizat ion in wate r dist r ibut ion system s engineering, Mathem at ical Programm ing Study , vol . 11,Nor th -Holl and , Am ste rdam, 1979 , pp . 65 -84 .[13] J . Utasiewicz , M inimal energy control in nonl inear dist r ibut ion systems b y netwo rk mod el aggregat ion method,Repo rt of Insti tute o f Technical Cybernet ics, W roclaw, Po land, 1984 ( in Pol ish) .

optimization algorithms of operative control

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