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UBC Theses and Dissertations

Optimized water distribution network design Smirfitt, Gary Robert 1977

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OPTIMIZED WATER DISTRIBUTION NETWORK DESIGN  by  GARY ROBERT SMIRFITT B.A.Sc.,  The University of B r i t i s h Columbia, 1972  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE DEGREE REQUIREMENTS  FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  in  The Faculty Of Graduate Studies (Department  of  Civil  Engineering)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA April,  ©  1977  Gary Robert S m i r f i t t ,  1977  In p r e s e n t i n g t h i s t h e s i s  in p a r t i a l  f u l f i l m e n t o f the requirements f o r  an advanced degree at the U n i v e r s i t y o f B r i t i s h ' C o l u m b i a , I agree the L i b r a r y  s h a l l make i t f r e e l y  that  a v a i l a b l e f o r r e f e r e n c e and study.  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e  copying o f t h i s  thesis  f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s r e p r e s e n t a t i v e s .  It  i s understood that copying o r p u b l i c a t i o n  o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my written  permission.  Department o f  C i v i l Engineering  The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date  A p r i l 1,  1977  11  ABSTRACT  T h i s t h e s i s d e s c r i b e s a s t u d y o f two approachs t o  the  d e s i g n o f w a t e r d i s t r i b u t i o n networks t o meet s p e c i f i e d demands  at  minimum c o s t . One method i s based on an i n c r e m e n t a l i n c r e a s e  technique  which f i r s t examines a l l p o s s i b l e " o n e - s i z e " p i p e i n c r e a s e s  i n the  n e t w o r k , then based on a b e n e f i t / c o s t a n a l y s i s a d e c i s i o n i s made on which p i p e t o i n c r e a s e one d i a m e t e r s i z e .  The second  approach u t i l i z e s a c o m p u t e r i z e d l i n e a r programming t e c h n i q u e r a p i d l y converge on an o p t i m a l network d e s i g n .  Both  to  techniques  r e l y on the use o f an e f f e c t i v e c o m p u t e r i z e d network a n a l y s i s program. I t was found a f t e r  s t u d y i n g s e v e r a l networks t h a t  the  i n c r e m e n t a l i n c r e a s e t e c h n i q u e i s o p e r a t i o n a l f o r any s i z e o f network.  However, computer c o s t s q u i c k l y become a l i m i t i n g  i n the u s e f u l n e s s o f t h i s a p p r o a c h .  factor  The l i n e a r programming based  t e c h n i q u e was c o n s i d e r a b l y l e s s c o s t l y but d i d not prove i t s e l f be f u l l y c a p a b l e o f o p t i m i z i n g l a r g e networks i n i t s developmental  state.  present  to  TABLE  OF  CONTENTS  PAGE ABSTRACT  iii  LIST OF TABLES  v  LIST OF FIGURES  vi  CHAPTER  1.  INTRODUCTION  1  2.  LITERATURE REVIEW  3  3.  PRESENT APPROACH  6  4.  EXPERIENCE AND RESULTS  15  5.  DISCUSSION OF RESULTS  32  6.  CONCLUSIONS AND RECOMMENDATIONS  34  FIGURES  36  BIBLIOGRAPHY  46  iv  LIST OF TABLES  TABLE  PAGE  1.  COST DATA FOR PIPES  18  2.  ONE-STEP COSTS  19  3.  ONE-STEP OPTIMIZATION OF ORIGINAL SIX NODE EXAMPLE - OPTIMIZATION ONE  20  4.  SIX NODE OPTIMIZATION PROGRESS  27  5.  AUTOMATED SIX NODE OPTIMIZATION PROGRESS  6.  TWELVE NODE OPTIMIZATION  v  . .  29  30  LIST OF FIGURES  FIGURE  PAGE  1.  OPTIMIZATION FLOWCHART . .  36  2.  TYPICAL PIPE NETWORK  37  3.  ONE-STEP OPTIMIZATION TECHNIQUE.  38  4.  ORIGINAL SIX NODE NETWORK  39  5.  SIX NODE NETWORK AFTER SIX ONE-STEP OPTIMIZATIONS..  40  6.  PRESENT OPTIMIZATION PROCESS  41  7.  THREE NODE NETWORK EXAMPLE  42  8.  SIX NODE NETWORK AFTER AUTOMATED OPTIMIZATION  9.  ORIGINAL TWELVE NODE NETWORK  44  OPTIMIZED TWELVE NODE NETWORK  45  10.  . .  43  ACKNOWLEDGEMENT  The a u t h o r w i s h e s to e x p r e s s h i s a p p r e c i a t i o n his  s u p e r v i s o r , Dr. S.O. R u s s e l l , f o r h i s guidance and  d u r i n g the w r i t i n g and r e s e a r c h o f t h i s t h e s i s .  encouragement  He a l s o w i s h e s to  thank Dr. W . J . C a s t l e t o n f o r h i s help and g u i d a n c e .  vii  to  1  Chapter 1 INTRODUCTION  R e l i a b l e water s u p p l y systems are e s s e n t i a l r e s i d e n t i a l , commercial and i n d u s t r i a l c o m m u n i t i e s . network forms a m a j o r ,  to a l l modern The d i s t r i b u t i o n  and in.many c a s e s the most e x p e n s i v e ,  i n any water s u p p l y system.  component  S i n c e many communities are a c t i v e l y grow-  i n g , w a t e r d i s t r i b u t i o n networks must c o n t i n u a l l y expand t o meet t h e i r needs.  E x p a n s i o n o f w a t e r d i s t r i b u t i o n networks i n v o l v e s the expend-  i t u r e o f v e r y l a r g e amounts o f c a p i t a l e v e r y y e a r .  It is  important  t h a t the network d e s i g n be as e f f i c i e n t as p o s s i b l e i n o r d e r m i n i m i z e the c o s t and hence the need f o r c a p i t a l essential  service.  This thesis  to  in providing t h i s  d e a l s w i t h the q u e s t i o n o f d e s i g n i n g  a water d i s t r i b u t i o n network to meet s p e c i f i e d demands a t minimum c o s t . The p a s t 15 y e a r s  have seen a r a p i d development i n the  c a p a b i l i t i e s o f computers and t h e i r use i n many e n g i n e e r i n g i n c l u d i n g water d i s t r i b u t i o n .  Analytical  problems  programs can a n a l y z e and  p r i n t o u t , w i t h i n s e c o n d s , network f l o w i n f o r m a t i o n t h a t would have taken months f o r a d e s i g n e n g i n e e r t o c a l c u l a t e a s h o r t time ago.  With  the advent o f more r a p i d and v e r s a t i l e computers, attempts have been made a t o p t i m i z i n g w a t e r d i s t r i b u t i o n n e t w o r k s , n o t j u s t a n a l y z i n g them.  However, the problem i s v e r y complex and s i m p l i f i c a t i o n s are  necessary  t o make o p t i m i z a t i o n f e a s i b l e .  The m a j o r i t y o f the  developed t o date a p p l y themselves t o the c o n t i n u o u s p i p e  techniques  diameter  case where i t i s assumed t h a t the p i p e d i a m e t e r - p i p e c o s t f u n c t i o n continuous.  Often t h e r e are s p e c i a l r e s t r i c t i o n s as t o p r e s s u r e  p a t t e r n s i n the system and the l o c a t i o n a n d / o r number o f s u p p l y i n g  is  2  a n d / o r consuming nodes; some even r e q u i r e t h a t a l l demands be equal and one t e c h n i q u e r e q u i r e s t h a t a l l p i p e l e n g t h s be e q u a l . g e n e r a l , p r e s e n t l y a v a i l a b l e t e c h n i q u e s are too d i f f i c u l t  In o r too  s p e c i a l i z e d t o be a p p l i e d to the m a j o r i t y o f water d i s t r i b u t i o n systems and thus have not come i n t o common use among e n g i n e e r s this field.  Most o r g a n i z a t i o n s s t i l l  in  r e l y on a s e n i o r e n g i n e e r  to  s k e t c h out a new system o r an e x t e n s i o n t o an e x i s t i n g s y s t e m , a n a l y z e i t , t r y a few a l t e r n a t e  p r o p o s a l s based on h i s e x p e r i e n c e  and recommend the b e s t o f t h i s v e r y l i m i t e d number o f The procedure developed i n the p r e s e n t  alternatives.  study u t i l i z e s a  1 i n e a r programming t e c h n i q u e and an e x i s t i n g e f f i c i e n t computer program f o r a n a l y z i n g steady s t a t e f l o w s i n p i p e networks combined i n an i t e r a t i v e s t e p by s t e p p r o c e d u r e .  The problem i s reduced to  one i n v o l v i n g a l i n e a r o b j e c t i v e f u n c t i o n and l i n e a r c o n s t r a i n t s can be r e a d i l y o p t i m i z e d .  which  However, the r e s t r i c t i o n s p l a c e d on the  system by the b a s i c assumptions r e q u i r e d f o r t h i s l i n e a r i z a t i o n r e q u i r e t h a t o n l y s m a l l changes be made i n p i p e d i a m e t e r s a t time. until  T h e r e f o r e the o p t i m i z a t i o n i s r e p e a t e d a stable  r e s u l t i s achieved.  i n a s t e p by s t e p  process  With the procedure developed the  u s e r can examine the c o s t s , f l o w s , p r e s s u r e trends  one  heads and o p t i m i z a t i o n  p r i o r to d e c i d i n g which p i p e s s h o u l d be changed i n s i z e .  Thus he can i n p u t h i s judgement and e x p e r i e n c e d u r i n g the o p t i m i z a t i o n process. Chapter 2 r e v i e w s the work o f o t h e r s on the o p t i m i z a t i o n o f water d i s t r i b u t i o n networks as background t o the p r e s e n t approach developed i n t h i s t h e s i s  study.  i s d e s c r i b e d i n Chapter 3.  Chapter  4 g i v e s the e x p e r i e n c e g a i n e d d u r i n g the development and p r e s e n t s results.  A s h o r t d i s c u s s i o n o f the r e s u l t s  follows  i n Chapter 5.  C o n c l u s i o n s and recommendations are made i n Chapter 6.  The  the  3  Chapter 2 LITERATURE REVIEW  One o f the f i r s t documented attempts a t o p t i m i z i n g a l o o p e d network o f water d i s t r i b u t i o n p i p e s was c a r r i e d out by Jacoby (Jacoby, 1968).  In o p t i m i z i n g a s i m p l e two l o o p network w i t h one  s u p p l y node and f i v e comsumptive nodes he u t i l i z e d l i n e a r programming to m i n i m i z e c o s t s s u b j e c t constraints  to the demands f o r water and the p h y s i c a l  o f the network.  The o p t i m a l p i p e d i a m e t e r s were  sub-  s e q u e n t l y rounded o f f t o the n e a r e s t c o m m e r c i a l l y a v a i l a b l e s i z e . The approach Jacoby developed has been found to be too d i f f i c u l t a p p l y to c o m p l i c a t e d systems  (Rasmussen,  to  1976).  Arun K. Deb has a u t h o r e d and c o - a u t h o r e d s e v e r a l  papers  r e l a t e d t o the 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 networks f o r  the  cases o f a l o o p e d (Deb and S a r k e r , 1971; Deb, 1976) and a branched network (Deb, 1974). pipe diameters  A major p o r t i o n o f h i s work i s based on e q u i v a l e n t  and e q u i v a l e n t l e n g t h s and on the development o f an  assumed p a r a b o l i c water p r e s s u r e s u r f a c e o v e r the network. assumed p a r a b o l i c p r e s s u r e s u r f a c e  This  i s based on known nodal h y d r a u l i c  heads w i t h a l l p r e s s u r e s assumed t o vary a c c o r d i n g to a p a r a b o l i c r e l a t i o n s h i p which d e f i n e s p r e s s u r e as a f u n c t i o n o f d i s t a n c e a water s u p p l y p o i n t .  His l a t e s t  from  paper ( Deb, 1976) d e t a i l s a  t e c h n i q u e which r e q u i r e s t h a t a l l p i p e s be o f the same l e n g t h and a l l consumptive demands be equal p r i o r t o any o p t i m i z i n g .  These c o n d i t i o n s  h i n d e r the network d e s i g n e r a t t e m p t i n g t o meet s p e c i f i e d demands a minimum c o s t by c r e a t i n g a f a l s e  i m p r e s s i o n o f the  b e h a v i o u r d u r i n g water d i s t r i b u t i o n .  network's  at  4 In a t e c h n i q u e more s u i t a b l e f o r a network Watanatada  (Watanatada,  1973) uses L a g r a n g i a n m u l t i p l i e r s to d e r i v e  an o p t i m a l water d i s t r i b u t i o n network.  Incorporated  was a s e n s i t i v i t y a n a l y s i s o f the e f f e c t s minimum p i p e d i a m e t e r .  designer,  in his  of interest  When u s i n g Watanatada's  r a t e s and  approach, a l a r g e  and f a s t computer i s r e q u i r e d due to the s o p h i s t i c a t e d  mathematics.  With l a r g e c o m p l i c a t e d networks the computer r e q u i r e m e n t s to be a l i m i t i n g  results  have proven  factor.  In h i s h e u r i s t i c approach t o water d i s t r i b u t i o n network o p t i m i z a t i o n Rasmussen (Rasmussen, 1976) compared i n c r e a s e d pumping c o s t s to i n c r e a s e d c a p i t a l c o s t s o f l a r g e r diameter p i p e s . are made s t e p by s t e p t o the network p i p e diameters o n l y those p i p e d i a m e t e r s c o m m e r c i a l l y a v a i l a b l e . system i s a t a f i x e d r a t e and a f i x e d head.  Changes  considering Inflow to  the  I f more than one node  is  an i n f l o w node Rasmussen found t h a t the procedure became i n c r e a s i n g l y more c o m p l i c a t e d . itself  The s i m p l i c i t y o f t h i s approach however l e n d s  to the p r e l i m i n a r y s t a g e s o f a s t u d y . In summary, p a s t work has been too d i f f i c u l t  from a  c o m p u t a t i o n a l p o i n t o f view o r too s p e c i a l i z e d o r w i t h too many r e s t r i c t i v e assumptions  to a p p l y to the v a s t m a j o r i t y o f cases where  o p t i m i z i n g a water d i s t r i b u t i o n network would prove e c o n o m i c a l l y feasible.  The m a j o r i t y o f the p a s t work r e l i e d on c o n t i n u o u s  f u n c t i o n s and o n l y i n the f i n a l  pipe  stage do they face the r e a l i t y t h a t  water s u p p l y p i p e has o n l y a l i m i t e d number o f c o m m e r c i a l l y a v a i l a b l e diameters. lends i t s e l f  This thesis  u t i l i z e s a d i s c r e t e p i p e s i z e t e c h n i q u e which  to a d a p t a t i o n  o f any p i p e d i a m e t e r s .  This technique  can a l s o handle such problems as p r e s s u r e r e d u c i n g v a l v e s , check v a l v e s ,  5 m u l t i p l e s o u r c e s , v a r y i n g demands and v a r y i n g nodal e l e v a t i o n s which o t h e r s have found t o be u n a c c e p t a b l e .  6  Chapter 3 PRESENT APPROACH  A.  Introduction The b a s i c approach i n the p r e s e n t study i n v o l v e s a number  of  steps: (a)  s p e c i f y i n g the demands and the  (b)  making a p r e l i m i n a r y network l a y o u t  (c)  a n a l y z i n g the network u s i n g a s t a n d a r d and computing the  (d)  " f i x i n g " the f l o w s and o p t i m i z i n g the flows  i t e r a t i n g s t e p s ( c ) and (d) u n t i l program converges to a s t a b l e  The r e s u l t a n t  program  cost  network f o r the f i x e d (e)  supply  network may r e p r e s e n t a l o c a l  optimum but i t s h o u l d s t i l l  the  solution,  r a t h e r than a g l o b a l  r e p r e s e n t an improvement o v e r the  and e r r o r approach g e n e r a l l y used a t p r e s e n t .  trial  The u s e r o f t h i s  technique  i s free  to choose w h i c h e v e r p i p e d i a m e t e r s  use a f t e r  the l i n e a r programming i s completed depending on p r a c t i c a l  c o n s i d e r a t i o n s such as advantages o f s t a n d a r d i z a t i o n . s u p p o r t p r o v i d e d by the l i n e a r program's r e s u l t s  he wishes to  But the  o f t e n are  extra  helpful  i n a l l o w i n g the d e s i g n e r t o develop a " f e e l " f o r the s y s t e m , and knowledge o f where the optimum l i e s p r o v i d e s him w i t h a b a s i s j u d g i n g whether the m e r i t s o f , justifies  the  cost.  say s t a n d a r d i z a t i o n o f p i p e  for  sizes,  7 B.  D e s c r i p t i o n o f the Technique The b a s i c assumption o f the approach taken i n t h i s  i n o p t i m i z i n g a w a t e r p i p e network i s t h a t f o r s m a l l changes p i p e ' s diameter,  the f l o w through the p i p e w i l l  T h i s a l l o w s the s t a n d a r d  form o f the f r i c t i o n  remain  study in a  constant.  equation:  2  h^ = c o n s t a n t  x Q  x k  to be l i n e a r i z e d i n t o : h^ = new c o n s t a n t x k For c a l c u l a t i o n purposes  i t was d e c i d e d t h a t a s m a l l change i n  d i a m e t e r would be one s t a n d a r d p i p e s i z e l a r g e r o r s m a l l e r than present case.  Continuous p i p e d i a m e t e r s  the  are not c o n s i d e r e d i n the  p r e s e n t approach based on the l i n e a r i z e d f r i c t i o n  formula.  An o b j e c t i v e f u n c t i o n must be c l e a r l y d e f i n e d i n o r d e r optimize a process or evaluate a l t e r n a t i v e s .  The c o s t o f the  to  water  p i p e network was the o b j e c t i v e t o be m i n i m i z e d f o r t h i s p r o b l e m , subject to constraints  l i m i t i n g minimum p i p e d i a m e t e r s and p r e s s u r e s  as w e l l as b a s i c h y d r a u l i c f l o w l a w s . For the sake o f s i m p l i c i t y and c o n s i d e r i n g i t s common o c c u r r e n c e i n B r i t i s h C o l u m b i a , o n l y the g r a v i t y flow s i t u a t i o n was c o n s i d e r e d i . e . p r o v i s i o n was not made f o r b o o s t e r pumps o r i n t a k e pumps as c o s t i t e m s .  However, m u l t i p l e s o u r c e s o f w a t e r and m u l t i p l e  demands f o r water are  acceptable.  F i g u r e 1 i n d i c a t e s the v a r i o u s s t e p s which o c c u r i n the procedure employed h e r e i n .  SYSDATA and LPDATA are s t o r a g e f i l e s  LIP i s a packaged l i n e a r programming program.  and  Data f o r the p i p e network  under c o n s i d e r a t i o n i s c o l l e c t e d and then o r g a n i z e d i n t o the  required  format f o r the f l o w a n a l y s i s made by a s t a n d a r d program developed by  8  Fowler and Epp (Epp and F o w l e r , 1970). o f t h i s a n a l y s i s and c o s t d a t a ,  Upon r e c e i v i n g the  the u s e r c a r r i e s out some b a s i c  c a l c u l a t i o n s i n p r e p a r a t i o n f o r the o p t i m i z i n g program. tableau  (LPDATA)  results  A new data  i s arranged f o r the l i n e a r programming computer  package ( L I P ) and then fed i n .  The r e s u l t s are e v a l u a t e d by the  user  who may then change some p i p e s i z e s and r e r u n the system once a g a i n or may d e c i d e t h a t the network i s s a t i s f a c t o r y and h a l t the U l t i m a t e c o n t r o l r e s t s w i t h the u s e r .  The d e t a i l s o f each component  i n t h i s system are reviewed i n the remainder o f t h i s C.  process.  chapter.  SYSDATA F i l e For ease o f o p e r a t i o n i t was found t h a t the data f o r  the  water s u p p l y network t o be a n a l y z e d s h o u l d be handled i n a computer storage file  file  r a t h e r than as data c a r d s .  A t the p r e s e n t  time t h i s  i s used o n l y to s t o r e data r e q u i r e d f o r the network f l o w a n a l y s i s  program.  The f o l l o w i n g  i n f o r m a t i o n i s c o n t a i n e d i n the SYSDATA  (a)  input/output  units  (b)  choice of pipe flow a n a l y s i s formula  (c)  node data  (d)  p i p e data  (e)  reservoir  (f)  b o o s t e r pump data  (g)  check v a l v e data  (h)  pressure  (D  consumption and s u p p l y data  (j)  degree o f a c c u r a c y r e q u i r e d  data  r e d u c i n g v a l v e data  file:  9  A t the p r e s e n t t i n e d a t a f o r b o o s t e r pumps, check v a l v e s and p r e s s u r e r e d u c i n g v a l v e s are n o t used s i n c e the p r e s e n t o p t i m i z a t i o n program cannot handle them. analytical D.  They a r e ,  however, f u l l y a c c e p t a b l e  to  the  program (Epp and F o w l e r , 1970). Network A n a l y s i s A n a l y s i s o f the network i . e .  computing f l o w s i n each p i p e  and p r e s s u r e heads a t the nodes o f a g i v e n network i s done by a program developed a t the U n i v e r s i t y o f B r i t i s h Columbia by Fowler and Epp (Epp and F o w l e r , 1 9 7 0 ) .  As s t a t e d by them, i t i s "an  efficient  computer program f o r the s o l u t i o n o f s t e a d y - s t a t e f l o w s i n water networks.  Its (a)  features  include:  The use o f Newton's method o f s o l v i n g a system of simultaneous  (b)  equations;  a l o o p - o r i e n t a t e d network to reduce the of equations  (c)  non-linear  t o be s o l v e d :  a u t o m a t i c l o o p numbering t h a t produces symmetic m a t r i x w i t h consequent memory r e q u i r e m e n t s ;  (d)  number  the requirement  a banded  r e d u c t i o n i n computer  and  o f a minimum o f i n p u t  data."  The program has been used by the a u t h o r t o a n a l y z e water  networks  f o r the B r i t i s h Columbia communities o f C r e s t o n , Vernon and Kamloops. P r i n t e d o u t p u t data from the program i n c l u d e s the (a)  P i p e i n f o r m a t i o n such as p i p e numbering, and downstream nodes,  pipe l e n g t h , pipe  p i p e roughness c o e f f i c i e n t and the pipe  resistance.  following:  upstream diameter,  calculated  10  (b)  Pump o r r e s e r v o i r d a t a a t an i n p u t node, such as. the flow i n t o the node and the h y d r a u l i c grade l i n e at t h a t p o i n t ;  (c)  C a l c u l a t e d p i p e data i n c l u d i n g the d i r e c t e d f l o w i n a p i p e and the head l o s s through the  (d)  pipe.  Nodal data such as node e l e v a t i o n , demand,  hydraulic  grade l i n e and p r e s s u r e . T h i s i n f o r m a t i o n i s then combined w i t h c o s t data i n p r e p a r i n g an input tableau E.  (the  LPDATA f i l e )  f o r the l i n e a r programming  package.  F o r m u l a t i o n o f Data f o r LIP The f o l l o w i n g symbols w i l l  be used i n t h i s s e c t i o n and are  d e f i n e d as f o l l o w s f o r a network as shown i n F i g u r e 2: h-j =? head a t node #1 = head a t node #2 = flow i n pipe A w i t h d i r e c t i o n i n d i c a t e d being p o s i t i v e C^ = c o s t o f p i p e A k^ = r e s i s t a n c e  of pipe A  Based on the assumption t h a t o n l y s m a l l changes i n p i p e  diameters  are to be c o n s i d e r e d , (AC\ *  dC \Uk7 + fl  /AC\**  Uk7  Where * r e p r e s e n t s a s m a l l i n c r e a s e  i n p i p e diameter and ** r e p r e s e n t s  a small decrease i n pipe d i a m e t e r .  This creates a l i n e a r  for  d C  function  / d k o v e r the range o f changes a l l o w e d t o p i p e A a t any  optimization  particular  stage.  The f u n c t i o n o p t i m i z e d i n LIP i s the o b j e c t i v e f u n c t i o n and r e p r e s e n t s the network c o s t s . is  A t r u e c o s t o f the water s u p p l y network  not used a t p r e s e n t , o n l y a r e l a t i v e c o s t f o r comparison purposes  11  is required.  c where  This objective function i s represented  t ;£}, N  by:  n = number o f p i p e s i n the system k = v a r i a b l e t o be o p t i m i z e d ( r e s i s t a n c e  When k i n c r e a s e s i n v a l u e the amount o f f r i c t i o n decreases.  to  flow)  i n the p i p e  The d e s i r e o f the o p t i m i z a t i o n i s t o m i n i m i z e the v a l u e  o f C, the t o t a l  r e l a t i v e c o s t o f the p i p e network.  To m a i n t a i n p r o p e r s e r v i c e i n a water d i s t r i b u t i o n system upper and l o w e r bounds t o the a l l o w a b l e p r e s s u r e  (head)  exist.  S i n c e t h i s a n a l y s i s i s f o r g r a v i t y o p e r a t i o n systems o n l y , the maximum h y d r a u l i c e l e v a t i o n i n the system i s t h a t o f the i n p u t node w i t h greatest hydraulic elevation.  the  The minimum p r e s s u r e a l l o w e d can v a r y  and i s based o n , among o t h e r c r i t e r i a , the F i r e U n d e r w r i t e r ' s A s s o c i a t i o n Standards.  I n t o the LPDATA f i l e  i s placed a series of  c o n s t r a i n t s , one p a i r f o r each node such t h a t : h  ^=  h max  -h  4=.  - h min  where h = head a t node m. m  T h i s then r e s t r i c t s the range o f 3  pressures r  a l l o w e d w i t h i n the p i p e network. To d e f i n e the p r e s s u r e a t any node, a t l e a s t one s t a r t i n g nodal p r e s s u r e must be i n p u t as an e q u a l i t y i n t o the o p t i m i z a t i o n program.  A l l o t h e r o p t i m i z e d pessures a r e then r e l a t e d back to  e q u a l i t y which a t p r e s e n t  this  i s a r e s e r v o i r ' s water surface e l e v a t i o n .  The n e x t s e r i e s o f c o n s t r a i n t s on the o p t i m i z a t i o n i s based on the r e q u i r e m e n t t h a t : h u //„ s - h d / s = h f.  ( i n t h a t rp i p e) r i  12  However, due t o problems a s s o c i a t e d w i t h o v e r c o n s t r a i n i n g  the  o p t i m i z a t i o n program, i t has been n e c e s s a r y t o r e v i s e t h i s h  , u/s  -  h ,, d/s  —  h f  r  which i s e q u a l l y v a l i d f o r the f l o w o f w a t e r . as  to:  T h i s can be m o d i f i e d  follows: h  ~  h  u/s  -h  , u/s  2 where Q  u/s -  d/s  h  +  d/s  h  +  +  f  h  h  f  h ... + d/s  ~ ° ~ °  k Q  2  ^0  i s c o n s t a n t and d e r i v e d from the e a r l i e r a n a l y s i s .  d i r e c t i o n o f f l o w s i n the system are c r i t i c a l presentation  o f the network c h a r a c t e r i s t i c s  be e x e r c i s e d i n f o r m u l a t i n g the The f i n a l  The  i n ensuring a proper  t o LIP and c a u t i o n must  inequalities.  set of constraints  l i m i t s the minimum p i p e s i z e  by l i m i t i n g the maximum v a l u e t o which each o f the v a r i a b l e k ' s can rise.  Based on the Manning e q u a t i o n k =  where:  resistance  C-|= c o n s t a n t n = M a n n i n g ' s roughness c o e f f i c i e n t L = length of pipe D = diameter i t can be d e r i v e d t h a t : k = C n LD" 2  5  1  Only the v a l u e s o f "k" and "D" can v a r y and t h e r e i s a r e l a t i o n s h i p f o r one p i p e t h a t : k new = k o l d  / D old D new,  S e t t i n g a minimum d i a m e t e r o f a watermain i n the network a l s o s e t s a maximum r e s i s t a n c e  v a l u e f o r each p i p e a t a p a r t i c u l a r f l o w  level.  13  For one a n a l y s i s l e v e l  and o n l y a l l o w i n g s m a l l changes  in  diameter  to be al 1 owed :• k max . = k ( °A  \  fl  A  The v a l u e s o f  A  5  Umini  and k^ a r e d e r i v e d from the F o w l e r a n a l y s i s  and thus a v a l u e f o r k  max f o r each k  n  is established.  n  output  (The l i n e a r  program LIP does not a l l o w f o r n e g a t i v e v a r i a b l e v a l u e s thus k min to z e r o . )  limiting  When i n p u t to the l i n e a r program v i a LPDATA,  k max c r i t e r i a l i m i t s the downward movement o f pipe diameters  the thus  m a i n t a i n i n g the looped c h a r a c t e r i s t i c o f most w a t e r d i s t r i b u t i o n networks.  F a i l u r e to do t h i s would p r o b a b l y r e s u l t i n the  o f a branched network, lower i n c o s t , but not a c c e p t a b l e  establishment  for f i r e f i g h t i n g  purposes. The o b j e c t i v e f u n c t i o n and i t s c o n s t r a i n t s arranged F.  i n the LPDATA f i l e  are  suitably  to meet the data i n p u t requirements  of LIP.  LIP LIP i s a packaged computer program used t o o p t i m i z e an  objective function subject  to a s e t o f c o n s t r a i n t s .  I t was  originally  w r i t t e n and documented by D. O ' R e i l l y i n 1970 a t the U n i v e r s i t y o f B r i t i s h Columbia and p r e s e n t l y i t i s b e i n g handled by C. B i r d a t  the  U . B . C . Computing Centre ( B i r d , 1975). " B r i e f l y , the package maximizes or minimizes a l i n e a r o b j e c t i v e function with n a c t i v i t i e s ( v a r i a b l e s ) of  the form: a,  , X .. + a,  1,1 subject  and  1  X„ +  9  1 ,d  to the l i n e a r  +  2  2,l  X  x  I,n  n  j = 1,2  l  + a  = Z  constraints:  X,-^_0 •J a  - i „  a  2,2  X  2  +  •••  +  a  2,n  X  n  n  ~ 2 ( n a  +  l )  14  a  3,l  a  (m+l)l  X  1  +  X  where >-vx r e p r e s e n t s  a  3,2  1  X  2  ••••••  +  ^=  +  +  +  a  a  3  ,  (m+1)  n  X ]  X  n ^  n^ 3(n+l) a  a  (m+l)(n+l)  o r = " ( B i r d , 1975).  ~  "  Output from LIP covers a v e r y broad scope o f a n a l y s i s but concern a t p r e s e n t i s d i r e c t e d to the p r i m a l s o l u t i o n v e c t o r . This vector represents (k  the v a l u e o f each v a r i a b l e X,  X  X  9  , k^) t h a t s h o u l d o c c u r f o r the o b j e c t i v e f u n c t i o n to be  1 5  an optimum.  Based on these o p t i m a l v a r i a b l e v a l u e s , recommended  v a l u e s f o r the p i p e diameters o f the network can be c a l c u l a t e d . G.  User D e c i s i o n Recommended diameters are n o t c a l c u l a t e d from the o p t i m i z i n g  p r o c e s s r e s u l t s but the p i p e d i a m e t e r s may be changed up o r down o n l y one s t a n d a r d p i p e s i z e d i f f e r e n c e .  They s h o u l d o n l y be changed i f  the recommended d i a m e t e r i s more t h a t 50% o f the way to the n e x t standard diameter.  However, the u s e r s t i l l  has the o p t i o n t o n o t  f o l l o w the program's recommendations i f he so d e s i r e s .  I f no p i p e  d i a m e t e r s change then the p r o c e s s i s stopped a t t h i s s t a t e .  A check  i s made on the o u t p u t from the Fowler a n a l y s i s t o ensure t h a t a l l pressures  are g r e a t e r than the minimum a l l o w e d .  I t may be  helpful  to t e s t the s e n s i t i v i t y o f the r e s u l t by changing one p i p e ' s  diameter  and r e p e a t i n g the a n a l y s i s - o p t i m i z a t i o n p r o c e s s a g a i n . I f changes are made to the n e t w o r k ' s p i p e d i a m e t e r s , these are i n s t i t u t e d i n the SYSDATA f i l e  and the p r o c e s s i s  then  repeated.  15  Chapter 4 EXPERIENCE AND RESULTS  A.  Introduction The p r e s e n t  approach d e t a i l e d i n the p r e v i o u s c h a p t e r  based on the background d e r i v e d w h i l e i n v e s t i g a t i n g a l t e r n a t e for  o p t i m i z i n g a network.  is  techniques  Two o f these t e c h n i q u e s were found to be  r e l i a b l e and capable o f d e l i v e r i n g a s a t i s f a c t o r y r e s u l t . examined i n t h i s c h a p t e r f o r c o m p a r a t i v e  Both  are  purposes.  I t was found t h a t a marked i n c r e a s e i n the  investment  r e t u r n f o r w a t e r s u p p l y networks can be o b t a i n e d i f a v a l i d  optimizing  p r o c e d u r e , such as put f o r w a r d i n t h i s p a p e r ,  However,  is utilized.  the e x p e r i e n c e d judgement o f a q u a l i f i e d network d e s i g n e r i s  still  a most d e s i r a b l e i n p u t t o the d e s i g n p r o c e s s f o r a c h i e v i n g a w a t e r d i s t r i b u t i o n network t o meet s p e c i f i e d demands a t a minimum o f c o s t . B.  One-step Technique Due to the d i s c r e t e n a t u r e o f the diameters o f c o m m e r c i a l l y  a v a i l a b l e w a t e r p i p e , the d i s t r i b u t i o n networks are f a v o u r a b l e  to  o p t i m i z a t i o n procedures  A  "one-step  that avail  themselves o f t h i s f e a t u r e .  t e c h n i q u e " , was developed as a crude but e f f e c t i v e  o f o b t a i n i n g an o p t i m a l network a t a minimal p i p e c o s t .  means  The e f f e c t o f  i n c r e a s i n g each p i p e ' s d i a m e t e r one s i z e and one a t a time i s examined based on the c o s t and on movement towards a t t a i n m e n t o f f l o w and p r e s s u r e  t h r o u g h o u t the network.  o f minimum l e v e l s  The o n e - s t e p  pipe s i z e  i n c r e a s e which has the h i g h e s t b e n e f i t t o c o s t r a t i o i s chosen as b e s t one-step  toward an o p t i m a l s o l u t i o n .  T h i s o p t i m a l change  the  is  performed to the b a s i c network and a l l o p t i o n s are a g a i n re-examined  16  f o r i n c r e a s i n g p i p e d i a m e t e r s one s t e p ,  This process continues  the minimum d e s i r e d l e v e l s o f f l o w and p r e s s u r e are a t t a i n e d .  until The  o n e - s t e p approach can be used most e f f e c t i v e l y on the improvement o f o r a d d i t i o n s t o an e x i s t i n g network but i t a l s o i s a d a p t a b l e  to  the  d e s i g n i n g o f new proposed w a t e r d i s t r i b u t i o n n e t w o r k s . The a d d i t i o n to o r improvement o f an e x i s t i n g water d i s t r i b u t i o n network i s m e r e l y a s i m p l i f i e d case o f the d e s i g n o f a complete new network s i n c e the same procedure have t o be c o n s i d e r e d .  i s followed,  but fewer p i p e s  T h e r e f o r e o n l y the development and r e s u l t s o f  the o n e - s t e p t e c h n i q u e as u t i l i z e d i n the o p t i m i z a t i o n o f a proposed system w i l l  be put f o r w a r d i n t h i s  paper.  The f i r s t s t e p i n the development o f a new water d i s t r i b u t i o n network v i a the o n e - s t e p method i s the p r e l i m i n a r y l a y o u t complete w i t h a l l the p i p e and nodal d a t a . to be the minimum s i z e a l l o w e d .  O r i g i n a l p i p e d i a m e t e r s are  The minimum a l l o w a b l e nodal p r e s s u r e  i s determined and the d e s i r e d f l o w l e v e l s from each node established.  assumed  are  C o s t i n g data i s assembled f o r each l e n g t h o f p i p e .  A  p r e l i m i n a r y network a n a l y s i s i s c a r r i e d out and a v a l u e f o r the summat i o n o f the i n d i v i d u a l  nodal p r e s s u r e d e f i c i e n c i e s i s c a l c u l a t e d .  One-step o p t i m i z a t i o n i s c a r r i e d o u t u s i n g the procedure F i g u r e 3.  indicated in  A f t e r one complete s e t o f a l t e r n a t e s are examined a new base  system i s c r e a t e d .  The o n e - s t e p o p t i m i z a t i o n p r o c e s s i s repeated  the p r e s s u r e d e f i c i t i s z e r o o r s m a l l enough t h a t the u s e r s a t i s f i e d w i t h the  until  is  results.  T h i s t e c h n i q u e was found to be v e r y s i m p l e to f o r m u l a t e and the r e s u l t s  t o t a l l y r e l i a b l e as e v e r y p o s s i b l e a l t e r n a t e i s e x p l o r e d  on a o n e - s t e p b a s i s .  I t i s , however,an e x p e n s i v e t e c h n i q u e due to  large  17  computer time requirements  i f run e x a c t l y as proposed h e r e i n ; one  a n a l y s i s a t each s t e p f o r each p i p e i n the network.  Improvement  can be r e a l i z e d by u t i l i z i n g a p r e l i m i n a r y s o r t i n g o f a l t e r n a t e s  to  e l i m i n a t e c o n t i n u a l a n a l y s i s o f the p o o r e r o t p i m i z i n g c h o i c e s . F i g u r e 4 i n d i c a t e s the p l a n view o f a proposed s i x node water s u p p l y network.  For s i m p l i c i t y t h e r e i s o n l y one nodal  o f w a t e r and one nodal demand f o r w a t e r . to be e q u a l . i s therefore  A l l e l e v a t i o n s are  source assumed  The minimum p i p e d i a m e t e r i s s e t a t s i x i n c h e s and t h i s the o r i g i n a l  s i z e o f each p i p e .  minimum nodal p r e s s u r e o f 20 psi: throughout  I t i s d e s i r e d to have a the network.  Since there  i s o n l y one consumptive node, o n l y i t need be examined f o r water pressure l e v e l .  The p r e s s u r e d e f i c i t can be d e f i n e d as the  difference  between our d e s i r e d minimum p r e s s u r e (20 p s i ) and t h a t found a t #5.  Based on the i n f o r m a t i o n o f F i g u r e 4 and the d a t a  i n T a b l e 1;  Table 2 i s developed to a i d i n e v a l u a t i n g the a l t e r n a t i v e s . final  preparatory  node  The  s t e p i s the p r e l i m i n a r y a n a l y s i s o f the f l o w s i n the  system i n i t s b a s i c form. zero o p t i m i z a t i o n l e v e l  This i n d i c a t e d a pressure d e f i c i t at  o f 414 p s i ( P D = 4 1 4 ) . n  the  Table 1 COST DATA FOR PIPES  Diameter  Cost t o s u p p l y and i n s t a l l per l i n e a r f o o t  6"  $10.00  8"  $11.50  10"  $14.00  12"  $17.25  14"  $20.75  16"  $25.00  18"  $29.00  Table 2 ONE-STEP COSTS  Cost  Pipe  (Dollars)  Size Change  Pipe 1 . 2  3  4  5  6  7  6" - 8"  750  1500  3750  900  1500  1500  1500  8" - 1 0 "  1250  2500  6250  1500  2500  2500  2500  10 -12"  1625  3250  8125  1950  3250  3250  3250  12"-14"  1750  3500  8750  2100  3500  3500  3500  ,!  Table 3 ONE-STEP OPTIMIZATION OF ORIGINAL SIX NODE EXAMPLE - OPTIMIZATION ONE  PIPE NO.  AP.D./ACOST  1  0.04  2  0.07  3  0.04  4.  0.00  '5  0.08  6  0.01  7  0.01  21  T a b l e 3 i s based on a o n e - s t e p change f o r each o f the network's pipes. presents  From t h i s t a b l e i t i s apparent t h a t p i p e #5  the b e s t c h o i c e f o r a o n e - s t e p  increase i n diameter.  F o l l o w i n g t h i s change the t e c h n i q u e was repeated s e v e r a l  times  resulting i n : (a)  a t o p t i m i z a t i o n #2; p i p e #2 i s made 8" d i a m e t e r  (b)  a t o p t i m i z a t i o n #3; p i p e #1 i s made 8" d i a m e t e r  (c)  a t o p t i m i z a t i o n #4; p i p e #5 i s made 10" d i a m e t e r  (d)  a t o p t i m i z a t i o n #5; p i p e #2 i s made 10" d i a m e t e r  (e)  a t o p t i m i z a t i o n #6; p i p e #1 i s made 10" d i a m e t e r  The r e s u l t a n t network i s shown i n F i g u r e 5 and the p r e s s u r e a t node #5 i s 18.8 p s i .  The d e c i s i o n was made t h a t t h i s r e s u l t was c l o s e  enough t o the o r i g i n a l  goal o f 20 p s i t o be c o n s i d e r e d f i n a l .  $86,000  was the e s t i m a t e d c o s t o f t h i s o n e - s t e p o p t i m i z e d system and to i t 43 s e p a r a t e a n a l y s e s o f the v a r i o u s s i x node o n e - s t e p  attain  alternate  networks were made. C.  Development o f P r e s e n t Approach Chapter  2 p r e s e n t e d a d e t a i l e d e x p l a n a t i o n o f the  approach to network o p t i m i z a t i o n as developed i n t h i s t h e s i s . first  present The  s t a g e o f t h a t development was based on a p r i m a r y u n d e r s t a n d i n g  o f the t e c h n i q u e s and p r i n c i p l e s i n v o l v e d .  Q u i t e e a r l y the need  f o r u t i l i z i n g a l i n e a r programming t e c h n i q u e was r e a l i z e d and as  the  c o m p l e x i t i e s o f t h i s more s o p h i s t i c a t e d approach became apparent a c o m p u t e r i z e d l i n e a r programming procedure was i n t e g r a t e d i n t o the technique.  Changes were s u b s e q u e n t l y made to data p r e s e n t a t i o n ,  the  v a r i o u s l i n e a r programming and network a n a l y s i s computer programs, and to the b a s i c c h a r a c t e r i s t i c s u t i l i z e d i n d e v e l o p i n g the o p t i m i z a t i o n .  22  The r e s u l t s a c h i e v e d bear w i t n e s s to the p r o g r e s s i o n o f the to a f u l l y operational useful  level  technique  f o r small networks.  The f i r s t stage o f development was based on h y d r a u l i c laws (  = kQ  pressure  & "ST Q-j p  0) and on the concept o f minimum a l 1 bwable  =  00  heads and p i p e d i a m e t e r s .  As d i s c u s s e d i n Chapter 3 ,  assumption o f f l o w s r e m a i n i n g c o n s t a n t diameters  f o r s m a l l changes  the  in pipe  i s made t o a l l o w f o r the l i n e a r i z i n g o f the dC/dk f u n c t i o n .  The o p t i m i z a t i o n p r o c e s s f o l l o w e d the f l o w c h a r t d e p i c t e d i n F i g u r e 6. A t e s t case c o n s i s t i n g o f a s i m p l e t h r e e pipe system was chosen as shown i n F i g u r e 7.  A hand c a l c u l a t e d H a r d y - C r o s s a n a l y s i s was c a r r i e d  out on t h i s system w i t h a l l p i p e s s i x i n c h e s i n d i a m e t e r ; acceptable  diameter.  S i n c e a l l p i p e s were the same d i a m e t e r  dC/dk v a l u e s were e q u a l .  The minimum p r e s s u r e  a t 200' and f o r node C i t was s e t a t 1 0 0 ' . law,  the minimum  a t node B was  the set  Based on the h y d r a u l i c  H = kQ : 2  f  (a)  2000' - ( k  1  +  k^  Q^ —  (b)  2000' - (k  3  +  k) Q 3  —  2 3  200' 100'  Equation (a) reduces to the f o l l o w i n g upon i n s e r t i o n o f the v a l u e s  for  k-| and Q-j: (c)  200' - (54.7 + A ^ ) 1180' —  which r e s u l t s  200'  i n Ak-j = - 5 3 . 2 a t a maximum and s i m i l a r l y A k^ = - 5 2 . 9 .  C o n t i n u i t y s t a t e s t h a t the summation o f the d i r e c t e d f r i c t i o n around a loop must equal z e r o ; (k  + A-k-j) Q  1  Based on t h a t , A m u s t A  k  Ak A k  1  = -53,2  2  = -56.1  3  = -52.9  2 }  - (k  therefore: 3  +Ak ) Q 3  equal - 5 6 . 1 .  2 3  = 0  In summary:  losses  23  For a p i p e diameter change from 6" t o 1 2 " , A k = - 5 3 . 3 and based on t h i s v a l u e a l l t h r e e pipe diameters were i n c r e a s e d to 12". A H a r d y - C r o s s a n a l y s i s i n d i c a t e d t h a t the r e s u l t s were when compared to the minimum p r e s s u r e c r i t e r i a . was then c a r r i e d o u t on the o r i g i n a l  satisfactory  A one-step o p t i m i z a t i o n  system and i t a l s o r e s u l t e d i n  the a l l 12" d i a m e t e r system as the l o w e s t c o s t a c c e p t a b l e  system.  The n e x t stage i n the development was the a d a p t i n g o f a c o m p u t e r i z e d l i n e a r programming t e c h n i q u e that~portion constraints  o f the o p t i m i z a t i o n p r o c e s s .  (Bird,  1975) to c a r r y out  The o b j e c t i v e  function,  and v a r i a b l e s r e q u i r e d c a r e f u l m a n i p u l a t i o n to  retain  t h e i r essence i n the data format to t h i s packaged program. attempts at o p t i m i z i n g a f a i r l y  Preliminary  l a r g e and complex network were made  but w i t h d i s a p p o i n t i n g r e s u l t s .  More r e s e a r c h  r e q u i r e d t o a c h i e v e an o p e r a t i o n a l  and  c o n s i d e r a t i o n was  approach u s i n g the c o m p u t e r i z e d  l i n e a r program. For s i m p l i c i t y a s i x node s y s t e m , as shown i n F i g u r e 3 , was taken as the base network on which a l l o p t i m i z i n g was to be done. The o b j e c t i v e o f the o p t i m i z a t i o n remained the m i n i m i z i n g o f (dk' ' ' C 0  S  ^  t  s u b j e c t t o the c o n s t r a i n t s  ' (i)  ^ h loops  (ii)  all h  n  of:  = 0  o  d  e  s  ^  50'  and t o a l l o w the l i n e a r i z i n g o f the f r i c t i o n (iii)  formula:  a l l A k ' s i n the^/_ + A k f o r one p i p e l i n e a r program — d i a m t e r change  To s p e c i f y a minimum a l l o w a b l e p r e s s u r e head a t each node, i t necessary node.  costs  to include equations  d e f i n i n g the p r e s s u r e head a t  The a d d i t i o n o f the f o l l o w i n g form o f e q u a t i o n s  is each  was r e q u i r e d :  (iv) However, t h i s made to h ^  s  (h  •- h  u / s  -  f  50')^Q :Ak. 2  f a i l e d to a l l o w f o r the a l t e r a t i o n s d u r i n g the o p t i m i z a t i o n .  the f o l l o w i n g c o n d i t i o n a l ^  h  To c a l c u l a t e the  d/s  =  AC  h  constantly  T h i s was c o r r e c t e d  by adding  equations: u / s -  k  0  -  2  - ^  A  k  -  f a c t o r f o r use i n the o b j e c t i v e f u n c t i o n  assumed t h a t an i n c r e a s e  being  o f one s t a n d a r d  i t was  p i p e c o u l d be used as a  dC representative  value f o r  s i n c e the A k f a c t o r was q u i t e  i n the range o f v a l u e s a c c e p t a b l e . a t a l a t e r date i f  T h i s procedure  restricted  c o u l d be amended  required.  During the f i r s t o p t i m i z a t i o n the l i m i t imposed by ( i i i ) proved t o be too harsh and was d e l e t e d from the l i n e a r program. m o d i f i c a t i o n s i n c l u d e d the i n s e r t i o n o f an e q u a l i t y c o n s t r a i n t c r e a t e a base l e v e l f o r the head c a l c u l a t i o n s ; i n t h i s h - = 200' reservoir  Other to  case:  The l i n e a r program as developed does not a l l o w the v a r i a b l e s t o take n e g a t i v e v a l u e s but A k must be a l l o w e d to t a k e n e g a t i v e  values.  This  r e q u i r e d A k t o be r e p r e s e n t e d by x-j and Xg such t h a t :  A k = Xi — x When  2  k was t o be n e g a t i v e ,  Xg a p o s i t i v e v a l u e . A further  x-j would take on a v a l u e o f z e r o and  The r e v e r s e would be t r u e f o r A k b e i n g p o s i t i v e .  r e s t r i c t i o n was r e q u i r e d to ensure t h a t d u r i n g the o p t i m i z a t i o n  no p r e s s u r e heads were a l l o w e d t o exceed t h e r e s e r v o i r l e v e l t h e r e were no b o o s t e r pumps.  since '  As a r e s u l t o f t h i s the i n p u t data  to  the l i n e a r programming assumed a f a i r degree o f c o m p l e x i t y . All data cards.  network data and l i n e a r programming d a t a were s t o r e d on The c o n v e r s i o n o f the network a n a l y s i s d a t a t o  linear  25  programming d a t a was done by hand. The f o l l o w i n g arrangement o f e q u a t i o n s was used t o program the computerized l i n e a r program: (i)  the o b j e c t i v e f u n c t i o n (Z)  (ii)  <*1 " *2>  (f)l  (x  ) (^-)  '  1  x x  +  ;  +  7  1  ^  5  200 200 200  2 0  a l l nodal heads t o be g r e a t e r than minimum p r e s s u r e o f 50' - x  1  -x  1 6  "  X  5  ^ - 5 0  £--50  20  - -  5  0  the summation o f the changes i n f r i c t i o n a loop are t o equal -.25 x  7  5.24 x (v)  2  (200')  16^=  x  (iv)  ] 4  §  x  a l l nodal p r e s s u r e heads t o be l e s s than the r e s e r v o i r level  (iii)  -x  1 3  < 3 " *4>  +  1  + .25 x - 5.24 x  8  2  l o s s e s around  zero  + 3.2 x  - 3.2 x  g  +  1 Q  -.25 x  g  +  =0  = 0  a l l v a r i a b l e s r e p r e s e n t i n g nodal heads are t o be defined  (vi)  x  16  =  x  15 "  1  4  3  ,  "  5  x  20  =  X  19 "  3  7  '  2  "  1 0  ,  2  *  4  3  x  l  X  13  +  5  +  ,  2  4  1  0  ,  X  3  2  X  14  and a s t a r t i n g p o i n t f o r the nodal d e f i n i t i o n o f p r e s s u r e heads t o be based on x  1 5  = 200  26  The l i n e a r o p t i m i z a t i o n t e c h n i q u e a l l o w s the v a r i a b l e s x-j to X^Q t o vary w i t h i n the c o n s t r a i n t s  imposed.  The o u t p u t o f the program  indicates  the v a l u e s o f these v a r i a b l e s which r e s u l t i n the h i g h e s t v a l u e o f the objective function.  Due to the o r i g i n a l  assumption o f o n l y a s m a l l  change i n p i p e s i z i n g , the u s e r must o n l y a l l o w the p i p e s a one step change i n p i p e diameter a f t e r  the l i n e a r programming r e s u l t s  indicated  a A k o f v a l u e g r e a t e r than 50% o f t h a t t o make a one s t e p s i z e The f i r s t o p t i m i z a t i o n r e s u l t e d i n s a t i s f a c t o r y  change.  changes.  The  p r o v i s i o n o f a minimum d i a m e t e r was added as a c o n s t r a i n t by l i m i t i n g the movement o f A k i n the p o s i t i v e d i r e c t i o n ( e . g . x-j ^ i s r e q u i r e d to ensure t h a t each nodal p r e s s u r e  20.1).  Care  head was d e f i n e d o n l y  once s i n c e f a i l u r e t o do so c o u l d a g a i n o v e r c o n s t r a i n the  system.  T a b l e 4 i s a summary o f the p r o g r e s s o f t h i s development phase o f the research.  f o r the x to use  r 2 n  dC  and X2n-1  v  a  ^  u  e  s  ™  t  n  e  " " function. z  P  f a c t o r y s i n c e sometimes both x  0  n  and x  0  value  n  I t was proposed  as o b t a i n e d f o r one s i z e l a r g e r p i p e f o r  as o b t a i n e d f o r one s i z e s m a l l e r p i p e f o r X £ .  dC N -jj7J  X2 _-| n  dC  and  T h i s proved  unsatis-  would take on v a l u e s thus  2n-l u p s e t t i n g the "mutual e x c l u s i v e n e s s " which s h o u l d have e x i s t e d .  No  c l e a r reason f o r t h i s o c c u r r e n c e was o b t a i n e d and a r e t u r n was made to the o r i g i n a l  concept.  F o l l o w i n g t h i s s t a g e the system data was c o n v e r t e d storage f i l e s  from data c a r d s .  to  The t e c h n i q u e was automated such t h a t  a l i n e a r progrm would i m m e d i a t e l y be c a r r i e d out based on the  data  from the m o d i f i e d flow a n a l y s i s program.  A great reduction i n user  time was a c h i e v e d and y e t the u s e r s t i l l  m a i n t a i n e d complete c o n t r o l  o f the o p t i m i z a t i o n o f the d i s t r i b u t i o n network.  Data d e s c r i b i n g the  27  Table 4 SIX NODE OPTIMIZATION PROGRESS  STEP NUMBER  SYSTEM PRESSURE DEFICIT  Original  1021 p s i  '  COST  $102,000  O p t i m i z a t i o n #1  222  86,750  #2  260  86,000  #3  27  96,000  #4  3.0  102,750  #5  4.4  120,500  #6  -  0,7  101,450  #7  - 10.1  105,650  #8  - 17.0  99,400  #9  - 13.1  107,650  #12  - 12.8  96,150  #13  - 12.8  96,150  #10 #11  28  network and p i p e c o s t data was s t o r e d i n SYSDATA w i t h the a n a l y s i s r e s u l t s b e i n g used as a b a s i s f o r the c a c u l a t i o n o f i n p u t data the l i n e a r program t o be s t o r e d i n LPDATA.  By u s i n g a c o n v e r s a t i o n a l  t e r m i n a l the r e s u l t s were e a s i l y and r a p i d l y s t u d i e d . SYSDATA c o u l d a l s o be r a p i d l y made. in  for  Changes t o  The s i x loop system was o p t i m i z e d  l e s s than f o u r hours o f d e s i g n time t o w i t h i n a few percentages  o f the o n e - s t e p l o w e s t c o s t a l t e r n a t i v e .  The r e s u l t was a minimum  p r e s s u r e o f almost double t h a t o f the one-step s o l u t i o n . and F i g u r e 8 summarize the To f u r t h e r node system (see  Table 5  results.  s t u d y t h i s automated o p t i m i z a t i o n p r o c e s s a 12  F i g u r e 9) was s e l e c t e d f o r o p t i m i z a t i o n .  A review of  some o f the p r i n c i p l e s was made and r e s u l t e d i n c e r t a i n m o d i f i c a t i o n s t o q u i c k e n the o p t i m i z a t i o n r a t e b e i n g i n c l u d e d . (i)  The o r i g i n a l  These were:  assumption f o r flow d i r e c t i o n be such t h a t  each node i s a downstream node a t l e a s t (ii)  A l l LPDATA statements be changed t o over  to a v o i d  constraints.  i.e.  h ^ u  = h^  s  A/s^ (iii)  once,  h  s  d/s  - h +  h  f  f  - Q +  ^  Ak A  to become:  k  L i m i t s be s e t on the x v a l u e s i n the A k ' s such t h a t x  2n  1  n  e  v  e  r  be g r e a t e r than "k o r i g i n a l " .  Table 6 and F i g u r e 10 i n d i c a t e the r e s u l t s o f t h i s o p t i m i z a t i o n . F o l l o w i n g t h i s i t was f e l t t h a t e s t a b l i s h i n g the optimum network would g r e a t l y a i d i n the measuring o f the o f these t e c h n i q u e s .  limiting performance  T h i s l i m i t i n g optimum would be based on an  u n l i m i t e d supply of a l t e r n a t e pipe diameters.  The system c o u l d be  d u p l i c a t e d i n the f i e l d  by v a r y i n g p i p e d i a m e t e r s on any p a r t i c u l a r  node t o node p i p e r u n .  Based on t h i s c o n t i n u o u s p i p e d i a m e t e r - c o s t  Table 5 AUTOMATED SIX NODE OPTIMIZATION PROGRESS  STEP NUMBER  SYSTEM PRESSURE DEFICIT  Original  1021 p s i  COST  $102,000  O p t i m i z a t i o n #1  400  76,000  #2  63  85,000  #3  91  89,250  #4  -  8.7  97,000  #5  - 12.8  92,150  #6  -  4.0  92,750  #7  - 14.2  89,250  #8  -  3.1  91,250  #9  - 14.2  89,250  #10  -  3.1  91,250  #11  - 14.2  89,250  Table 6 TWELVE NODE OPTIMIZATION  STEP NUMBER  TOTAL PRESSURE DEFICIT  Original  1100 p s i  O p t i m i z a t i o n #1  48  #2  COST  $35,250 37,050  1.4  39,050  #3  -  8.9  37,412  #4  -  4.2  37,212  0  37,250  1.9  37,575  3.5  36,950  8.3  37,687  #5 #6  -  #7 #8  -  31 function  i t was hoped t h a t a minimum p o s s i b l e e x p e n d i t u r e  d e r i v e d to a c h i e v e the g o a l s .  The p i p e s i z e s recommended by the  l i n e a r program were s t r i c t l y adhered to r e s u l t i n g i n l a r g e in diameters. tendancy t o  c o u l d be  changes  The r e s u l t s were not c o n c l u s i v e and showed no converge.  32  Chapter 5 DISCUSSION OF RESULTS  The  i n t e n t o f t h i s t h e s i s was t o develop a d e s i g n  o p t i m i z a t i o n procedure which would r e s u l t i n the a b i l i t y to d e s i g n water d i s t r i b u t i o n networks to meet s p e c i f i e d demands a t minimum cost.  As d e t a i l e d i n the p r e c e d i n g c h a p t e r ,  t h i s goal has been  a c h i e v e d u s i n g an e x p e n s i v e and crude t e c h n i q u e .  However, an  automated t e c h n i q u e based on a l i n e a r i z a t i o n o f the p i p e  friction  formula has shown t h a t , a t a small p o r t i o n o f the f o r e g o i n g ' s it  cost  can produce high q u a l i t y r e s u l t s when d e a l i n g w i t h small n e t w o r k s .  Both o f t h e s e methods are c o n s i d e r a b l y more economic, i n c o s t to a c h i e v e r e s u l t s and i n the c o s t o f the proposed n e t w o r k , than trial  and e r r o r approach g e n e r a l l y used a t  the  present.  U s i n g the o n e - s t e p procedure to o p t i m i z e the s i x node example r e s u l t e d i n a network t h a t c o s t $ 8 6 , 0 0 0 , had a minimum p r e s s u r e o f 18.8 p s i and r e q u i r e d 15.0 seconds o f computer time to achieve.  The automated t e c h n i q u e used l e s s than one h a l f the computer  time and produced a network t h a t c o s t 4% more ($89,250) but had a minimum p r e s s u r e o f 34.2 p s i .  T h i s i n d i c a t e s t h a t the l i n e a r i z a t i o n  based automated t e c h n i q u e o f o p t i m i z i n g water d i s t r i b u t i o n networks is  effective. As s t a t e d e a r l i e r i n t h i s t h e s i s , the network d e s i g n e r can  still  make a v a l u e d i n p u t to the d e s i g n p r o c e s s when u s i n g the  automated type o f o p t i m i z a t i o n . the 14.2 p s i e x c e s s p r e s s u r e  Upon e x a m i n i n g F i g u r e 8 and n o t i n g  i n t h a t n e t w o r k , c o n s i d e r a t i o n may be g i v e n  to r e d u c i n g the d i a m e t e r o f p i p e #5 to 10" and i n c r e a s i n g p i p e #1 to  33  12".  This w i l l  reduce c o s t s by $1575 and s t i l l  g r e a t e r than the 20 p s i minimum a t demand f l o w s .  keep the  As the network  becomes more c o m p l i c a t e d t h i s type o f o b s e r v a t i o n w i l l more d i f f i c u l t The attained  pressure  be i n c r e a s i n g l y  to make. t w e l v e node s o l u t i o n which the automated  technique  i n 6.4 seconds o f computer time would take o v e r 90 seconds  to a t t a i n u s i n g the o n e - s t e p  procedure.  T h i s i n d i c a t e s the  substantial  r e d u c t i o n i n computer t i m e c o s t s which can be o b t a i n e d u s i n g t h i s procedure. The  results  o f the t w e l v e node o p t i m i z a t i o n , as shown i n  Table 6, i n d i c a t e an u n c e r t a i n t y  i n the chosen s o l u t i o n .  The  f a i l u r e o f the o p t i m i z a t i o n to r e t u r n to the chosen s o l u t i o n , optimum #4,  i n d i c a t e d t h a t t h a t optimum may be a l o c a l optimum o n l y .  C o n f i r m a t i o n o f t h i s would r e q u i r e a o n e - s t e p the network o r a f u r t h e r  type o f o p t i m i z a t i o n o f  c o n t i n u a t i o n o f the o p t i m i z a t i o n .  Examination  o f the network suggests m o d i f i c a t i o n t o p i p e s i z e s which c o u l d prove acceptable.  S e v e r a l o f these were t r i e d d u r i n g the s t u d y but  the  r e s u l t a n t network was not as e f f i c i e n t as the o r i g i n a l ; an i n d i c a t i o n t h a t the procedure can out perform the a u t h o r ' s modest c a p a b i l i t i e s in network d e s i g n . The  results  i n d i c a t e d t h a t t h i s procedure can a l l o w the  net-  work d e s i g n e r to r a p i d l y converge on a f e a s i b l e network and p o s s i b l y use h i s e x p e r i e n c e t o r e f i n e the network i f r e q u i r e d . has been shown t o work but may need some f u r t h e r fully  operational.  The procedure  r e f i n e m e n t t o become  34  Chapter 6 CONCLUSIONS AND RECOMMENDATIONS  Money i n v e s t e d i n r e s i d e n t i a l and i n d u s t r i a l water  distri-  b u t i o n networks i s v e r y hard to r e c o v e r i f i t i s found t h a t y o u r c h o i c e o f pipe s i z e s was not the b e s t .  T h e r e f o r e a water  distri-  b u t i o n network d e s i g n e d to meet s p e c i f i e d demands a t minimum c o s t s h o u l d be the b a s i s f o r the o r i g i n a l  c a p i t a l e x p e n d i t u r e on p i p e .  The procedure developed i n t h i s t h e s i s s h o u l d help the network d e s i g n e r t o p r o v i d e an o p t i m i z e d network; one t h a t combines adequate h y d r a u l i c performance w i t h a minimum o f c o s t .  Based on a  "flow-fixing"  approach which u t i l i z e d a l i n e a r programming o p t i m i z a t i o n t e c h n i q u e and a r a p i d network a n a l y s i s program, the p r e s e n t approach  results  i n r a p i d convergence t o an e f f i c i e n t network d e s i g n when d e a l i n g with small networks. still  Adjustments t o the network by the  remain an i m p o r t a n t c o n s i d e r a t i o n i n the Further research  designer  process.  i n t h i s area i s recommended due to  the  l a r g e sums o f money spent each y e a r i n Canada on water d i s t r i b u t i o n network c o n s t r u c t i o n . further  A l o g i c a l e x t e n s i o n o f t h i s r e s e a r c h would be  r e f i n i n g and t e s t i n g t o p r o v i d e an o p e r a t i o n a l procedure  would work on v e r y l a r g e n e t w o r k s , a s t e p not y e t c o m p l e t e d . areas o f p o s s i b l e r e s e a r c h expanding on t h i s paper (a)  m o d i f i c a t i o n s t o the l i n e a r program t o a l l o w greater f l e x i b i l i t y  (b)  are:-  in input  data;  e x p a n s i o n o f the c o s t data to i n c l u d e pumping c o s t s and v a r i e d p i p e i n s t a l l a t i o n c o s t s ;  (c)  of great assistance  would be the  development  that  Other  35  o f a t e c h n i q u e t o determine how c l o s e  the  network i s t o the g l o b a l optimum; (d)  u t i l i z a t i o n o f a c o n t i n u o u s "k v s . c o s t " curve t o s u p p l y dC/dk v a l u e s f o r the o b j e c t i v e f u n c t i o n i n the l i n e a r program;  (e)  comparison o f the c o s t s and performance o f the p r e s e n t approach to the commonly used t r i a l  and  e r r o r t e c h n i q u e based on e x p e r i e n c e ; (f)  f u r t h e r o p t i m i z i n g o f l a r g e s c a l e problems utilizing,  i f p o s s i b l e , a c t u a l cases taken  c o n s u l t i n g engineers I t i s hoped t h a t o t h e r s w i l l further  i n v e s t i g a t e t h i s area  from  studies.  be s t i m u l a t e d by t h i s r e s e a r c h  to  i n the hopes o f b r i n g i n g t o Canadian  engineers a higher appreciation of o p t i m i z a t i o n techniques.  FIG. I  OPTIMIZATION  FLOWCHART  SUPPLY * -  DEMAND  DEMAND  FLOW  FIG.2  TYPICAL  PIPE  NETWORK  S T A R T  /oPT=  I  \  v a l = n  YES/  INCREASE  A S S E M B L E  DIA.OF  BASIC DATA  PIPE  # VALONE  STEP  DOES \ LAST PIPE IN N E T W WORK/ O N  =  'ASSEMBLE N =  0  YES  I  NOPT=  a  OLD  ANALYSE EXISTING  H  DEFECIT  I  = 'D'  EQUALS  N = N + I  DEFECIT  ZERO,STOP  FIG.3  ONE  STEP  D/C  ANALYSE NEW  - N E W DEFECIT  NETWORK  IF  H  MO i/ D O E S N = l NO — OR NOPT>OPT  ±  COST  N O P T \ ^  NETWORK  INCREASE THE DIA.OF PIPE#N ONE STEP  OPTIMIZATION  TECHNIQUE  COST  OF  CHANGE = C  PIPE # 2 1000' 6>  PIPE # 5 1000' 6 >  <D PIPE* I 500'  6"4>  PIPE#4 600'  PIPE #7 1000'  6"<t>  6"<t>  PIPE#3 2500  PIPE # 6 1000  6"<f)  6"<f>  FIG.4  ORIGINAL  SIX N O D E  NETWORK  PIPE# 2 1000' I0>  PIPE# I 500' I0>  PIPE# 3 2500' 6 >  PIPE # 5 1000* I0>  PIPE#4 600' 6>  PIPE#7 1000' 6"<jb  PIPE#6 1000' 6"<£  <•>  FIG.5  SIX NODE NETWORK A F T E R SIX O N E - S T E P  OPTIMIZATIONS.  GIVEN  ••  C O S T  D A T A  EXISTING MINIMUM  A N A L Y S E  NETWORK  8  D E M A N D S  A L L O W A B L E  EXISTING  PRESSURES  N E T W O R K  I IS  N E T W O R K  S A T I S F A C T O R Y  ?  S T O P  ACOST  C A L C U L A T E  AK  I DO  L.P. B A S E D  O N '  dC  A K , ^ 7  MINIMIZING  ±  A K  uK| S U B J E C T  T O '  M A K E  D I A M E T E R  FIG.6  C H A N G E S  P R E S E N T  ^ - ±  dK2  A L L  P R E S S U R E S  >  MINIMUM  A L L  D I A M E T E R S  >  M I N I M U M  2!ALL  I  dC 2  B A S E D  ON  O P T I M I Z A T I O N  F L O W S  IN  OPTIMIZED  P R O C E S S  L E V E L SIZE  L O O P = 0  A K,, AK  2  r  .....AKj  PIPE # I 1000'  45cf s  PIPE # 2 1000'  70cfs at 2 0 0 0 ' of head PIPE # 3 1000'  NOTE  FIG.7  THREE  :  25cfs  A l l nodal e l e v a t i o n s are e q u a I .  NODE N E T W O R K  EXAMPLE  PIPE # 2 1000' I0>  PIPE* I 500' I0"d>  PIPE # 4 600' 6> PI P E # 3 2500' 6"d>  FIG.8  PIPE #5 1000' I2"d>  PIPE#6 1000' 6"<£  SIX NODE N E T W O R K A F T E R A U T O M A T E D  PIPE#7 1000' 6 >  <s)  OPTIMIZATION.  PIPE#4 100'-6>  © PIPE#I 100 '-6"<£  ©  PIPE*II 500'-6"<£  <D  ©  PIPE*8 3 0 0 - 6"<£  PIPE#5 200'-6"<£  4cfs  p|p  E  #  P I P E * 15 100'-6>  |2  400'-6"<£ 10  •PIPE* 2 IOO'-6"<£  PIPE*9 PIPE#6 200'-6"<£  2  PIPE* 3 •3P0--6"*  p  |  p  E  #  200-6  0  0  ,  -  6  n  *  PIPE*I0 100'- 6"<f>  7  >  P,PE*I3 300-6>  P I P E * 14 200'-6 >  8  F I G . 9 O R I G I N A L T W E L V E NODE N E T W O R K  PIPE*I6 50-6>  © PIPE*I7 75'- 6 > 10 cfs  -p-  PIPE*4 I00'-6"d>  ©  ©  PIPE#I  ©  100'-6>  p  |  p  E  #  PIPE#8 300-6>  5  2 0 0 * - 6"<f>  4cf s  PIPE* 2 I  0  0  "  6  >  P I P E # 12 400'-6"d5  © PIPE #15 100'-6 >  ®  PIPE#9 P.PE*6 2 0 0 ' - I0"d>  I4cfs PIPE#3 3 0 0 - 6"4>  PIPE*II 500'-6"d>  PIPE*7 200'-6"</>  ©  20.0'-8>  p  i  p  E  #  |  3  3 0 0 ' - 8"<f>  P I P E * 10 IOO'-6 d> M  P I P E * 14 200-8"d>  PIPE*I6 50W<£  © PIPE*I7 75'-8"dS lOcfs VJI  FIG.10 OPTIMIZED T W E L V E NODE  NETWORK  46  BIBLIOGRAPHY  BIRD, C . j "A L i n e a r Programming Package", Computing C e n t r e , U n i v e r s i t y o f B r i t i s h C o l u m b i a , J a n u a r y , 1975. DEB, A.K,, and SARKER, A . K . , " O p t i m i z a t i o n i n Design o f H y d r a u l i c Network , J o u r n a l 11  o f the S a n i t a r y  Engineering  D i v i s i o n , ASCE, V o l . 97, No. SA2, P r o c . Paper 8032, April  1971, pp.  141-159  DEB, A . K . , " L e a s t Cost Design o f Branched P i p e Network S y s t e m " , J o u r n a l o f the Environmental  Engineering  D i v i s i o n , ASCE, V o l . 100, No. EE4, P r o c . Paper 10711, August 1974, pp.  821-835.  DEB, A . K . , " O p t i m i z a t i o n o f Water D i s t r i b u t i o n Network Systems' Journal  o f the Environmental E n g i n e e r i n g D i v i s i o n ,  V o l . 102, No. EE4, P r o c . pp.  Paper  ASCE,  12343, August 1976,  837-851.  EPP, R. and FOWLER, A . G . , " E f f i c i e n t Code f o r Steady Flows i n N e t w o r k s " , J o u r n a l  State  o f the H y d r a u l i c D i v i s i o n  ASCE, V o l . 9 6 , No, HY1, P r o c . Paper 7002, January 1970, pp.  43-56,  JAC0BY, S . L . S , ,  "Design o f Optimal H y d r a u l i c Networks",  J o u r n a l o f the H y d r a u l i c s D i v i s i o n , ASCE, V o l . 94, No.  HY3, P r o c .  Paper  5930, May 1968, pp.  641-661.  RASMUSSEN, H . J . , " S i m p l i f i e d O p t i m i z a t i o n o f Water Supply Systems", J o u r n a l o f the Environmental  Engineering  D i v i s i o n , ASCE, V o l . 102, No. EE2, P r o c . Paper April  1976, pp.  313-327  12026,  47  WATANATADA, T , , " L e a s t - C o s t Design o f Water D i s t r i b u t i o n Systems", J o u r n a l o f the H y d r a u l i c s D i v i s i o n , ASCE, V o l . 9 9 , No. HY9, P r o c . Paper 9974, S e p t . pp.  1497-1513.  1973,  

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