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Optimized water distribution network design Smirfitt, Gary Robert 1977

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OPTIMIZED WATER DISTRIBUTION NETWORK DESIGN by GARY ROBERT SMIRFITT B . A . S c . , The University of Br i t i sh 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 C i v i l Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1977 © Gary Robert S m i r f i t t , 1977 In p resent ing 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 fo r an advanced degree at the U n i v e r s i t y of B r i t i s h ' C o l u m b i a , I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e fo r reference and study. I f u r t h e r agree t h a t permiss ion for e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted 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 or 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 ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Department of C i v i l Engineering The U n i v e r s i t y of 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 Thi s t h e s i s desc r ibes a study o f two approachs to the des ign o f water 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 a t minimum c o s t . One method i s based on an incrementa l i nc rease technique which f i r s t examines a l l p o s s i b l e "one - s i ze" pipe inc reases in the network, 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 pipe to inc rease one diameter s i z e . The second approach u t i l i z e s a computerized l i n e a r programming technique to r a p i d l y converge on an opt imal network des ign . Both techniques r e l y on the use of an e f f e c t i v e computerized network a n a l y s i s program. I t was found a f t e r s tudy ing severa l networks tha t the incremental inc rease technique 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 cos t s q u i c k l y become a l i m i t i n g f a c t o r i n the usefu lness o f t h i s approach. The l i n e a r programming based technique 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 to be f u l l y capable o f o p t i m i z i n g l a r g e networks i n i t s present developmental s t a t e . TABLE OF CONTENTS PAGE ABSTRACT i i i LIST OF TABLES v LIST OF FIGURES v i 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 i v 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 . . 29 6. TWELVE NODE OPTIMIZATION 30 v 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 . . 43 9. ORIGINAL TWELVE NODE NETWORK 44 10. OPTIMIZED TWELVE NODE NETWORK 45 ACKNOWLEDGEMENT The author wishes to express h i s a p p r e c i a t i o n to h i s 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 encouragement dur ing the w r i t i n g and research o f t h i s t h e s i s . He a l s o wishes to thank Dr. W.J . C a s t l e t o n f o r h i s help and guidance. v i i 1 Chapter 1 INTRODUCTION R e l i a b l e water supply systems are e s s e n t i a l to a l l modern r e s i d e n t i a l , commercial and i n d u s t r i a l communit ies. The d i s t r i b u t i o n network forms a major, and in.many cases the most expens ive , component i n any water supply system. S ince many communities are a c t i v e l y grow-i n g , water 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 to meet t h e i r needs. Expansion o f water 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 very l a r g e amounts o f c a p i t a l every y e a r . I t i s important t ha t the network design be as e f f i c i e n t as p o s s i b l e i n order to minimize the cos t and hence the need f o r c a p i t a l in p r o v i d i n g t h i s e s s e n t i a l s e r v i c e . This t h e s i s deals w i th the ques t ion o f des ign ing 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 cos t . The past 15 years have seen a r a p i d development i n the c a p a b i l i t i e s of computers and t h e i r use i n many eng inee r ing problems i n c l u d i n g water d i s t r i b u t i o n . A n a l y t i c a l programs can analyze and p r i n t o u t , w i t h i n seconds, network f low in fo rmat ion tha t would have taken months f o r a design engineer to c a l c u l a t e a shor 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 water d i s t r i b u t i o n networks , not j u s t a n a l y z i n g them. However, the problem i s very complex and s i m p l i f i c a t i o n s are necessary to make o p t i m i z a t i o n f e a s i b l e . The ma jo r i t y of the techniques developed to date apply themselves to the cont inuous pipe diameter case where i t i s assumed tha t the pipe d iameter -p ipe cos t f unc t ion i s con t inuous . Often there are s p e c i a l r e s t r i c t i o n s as to pressure pa t te rns i n the system and the l o c a t i o n and/or number o f s u p p l y i n g 2 and/or consuming nodes; some even r equ i r e tha t a l l demands be equal and one technique r equ i r e s tha t a l l p ipe lengths be e q u a l . In 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 techniques are too d i f f i c u l t or too s p e c i a l i z e d to be a p p l i e d to the ma jo 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 engineers i n t h i s f i e l d . Most o r g a n i z a t i o n s s t i l l r e l y on a s e n i o r engineer to sketch out a new system or an ex tens ion to an e x i s t i n g system, analyze i t , t r y a few a l t e r n a t e proposa ls based on h i s exper ience and recommend the best o f t h i s very l i m i t e d number o f a l t e r n a t i v e s . The procedure developed i n the present study u t i l i z e s a 1 inea r programming technique 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 lows in p ipe networks combined i n an i t e r a t i v e step by s tep procedure . 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 func t ion and l i n e a r c o n s t r a i n t s which can be r e a d i l y o p t i m i z e d . However, the r e s t r i c t i o n s p laced on the system by the bas ic 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 equ i r e tha t on ly small changes be made i n pipe diameters a t one t ime. Therefore the o p t i m i z a t i o n i s repeated i n a step by step process u n t i l a s t a b l e r e s u l t i s ach ieved . With the procedure developed the user can examine the c o s t s , f l o w s , pressure heads and o p t i m i z a t i o n t rends p r i o r to d e c i d i n g which pipes should be changed i n s i z e . Thus he can inpu t h i s judgement and exper ience dur ing the o p t i m i z a t i o n process . Chapter 2 reviews the work of o thers 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 to the present s tudy. The approach developed i n t h i s t h e s i s i s desc r ibed i n Chapter 3. Chapter 4 g ives the exper ience gained dur ing the development and presents the r e s u l t s . A shor t d i s c u s s i o n o f the r e s u l t s f o l l o w s in Chapter 5. Conclus ions and recommendations are made i n Chapter 6. 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 looped network of water d i s t r i b u t i o n pipes 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 imple two loop network w i t h one supply 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 minimize cos ts sub jec t to the demands f o r water and the p h y s i c a l c o n s t r a i n t s o f the network. The opt imal pipe diameters were sub-sequent ly rounded o f f to the neares t commerc ia l ly 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 to apply to compl ica ted systems (Rasmussen, 1976). Arun K. Deb has authored and co-authored seve ra l papers r e l a t e d to 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 f o r the cases o f a looped (Deb and S a r k e r , 1971; Deb, 1976) and a branched network (Deb, 1974). 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 pipe diameters and e q u i v a l e n t l eng ths and on the development o f an assumed p a r a b o l i c water pressure surface over the network. Th i s assumed p a r a b o l i c pressure surface i s based on known nodal h y d r a u l i c heads w i t h a l l pressures assumed to vary accord ing to a p a r a b o l i c r e l a t i o n s h i p which def ines pressure as a func t ion o f d i s t ance from a water supply p o i n t . His l a t e s t paper ( Deb, 1976) d e t a i l s a technique which r e q u i r e s tha t a l l p ipes be o f the same l eng th and a l l consumptive demands be equal p r i o r to any o p t i m i z i n g . These c o n d i t i o n s h inder the network des igner a t tempt ing to meet s p e c i f i e d demands at a minimum cos t by c r e a t i n g a f a l s e impress ion o f the ne twork ' s behaviour dur ing water d i s t r i b u t i o n . 4 In a technique more s u i t a b l e f o r a network d e s i g n e r , Watanatada (Watanatada, 1973) uses Lagrangian m u l t i p l i e r s to de r ive an opt imal water d i s t r i b u t i o n network. Incorporated i n h i s r e s u l t s 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 o f i n t e r e s t r a t e s and minimum pipe diameter . When us ing Watanatada's 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 rge compl ica ted networks the computer requirements have proven to be a l i m i t i n g f a c t o r . In h i s h e u r i s t i c approach to 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 inc reased pumping cos t s to inc reased c a p i t a l cos t s o f l a r g e r diameter p ipe s . Changes are made step by step to the network pipe diameters c o n s i d e r i n g on ly those p ipe diameters commerc ia l ly a v a i l a b l e . Inflow to the system i s a t a f i x e d ra te and a f i x e d head. I f more than one node i s an i n f l o w node Rasmussen found tha t the procedure became i n c r e a s i n g l y more compl i ca t ed . The s i m p l i c i t y o f t h i s approach however lends i t s e l f to the p r e l i m i n a r y stages o f a s tudy. In summary, past work has been too d i f f i c u l t from a computat ional p o i n t o f view or 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 apply to the vas t m a j o r i t y of 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 economica l ly f e a s i b l e . The m a j o r i t y o f the past work r e l i e d on cont inuous pipe func t ions and on ly i n the f i n a l stage do they face the r e a l i t y tha t water supply pipe has on ly a l i m i t e d number o f commerc ia l ly a v a i l a b l e d iameters . Th i s t h e s i s u t i l i z e s a d i s c r e t e pipe s i z e technique which lends i t s e l f to adap ta t ion o f any pipe d iameters . Th i s technique can a l so handle such problems as pressure reducing v a l v e s , check v a l v e s , 5 m u l t i p l e sources , v a r y i n g demands and va ry ing nodal e l e v a t i o n s which others have found to be unacceptable . 6 Chapter 3 PRESENT APPROACH A. I n t r o d u c t i o n The bas i c approach in the present study i n v o l v e s a number o f s t eps : (a) s p e c i f y i n g the demands and the supply (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 us ing a s tandard program and computing the cos t (d) " f i x i n g " the f lows and o p t i m i z i n g the network f o r the f i x e d f lows (e) i t e r a t i n g s teps (c) and (d) u n t i l the program converges to a s t a b l e s o l u t i o n , The r e s u l t a n t network may represent a l o c a l r a t h e r than a g l o b a l optimum but i t should s t i l l represent an improvement over the t r i a l and e r r o r approach g e n e r a l l y used a t p resent . The user o f t h i s technique i s f ree to choose whichever p ipe diameters he wishes to 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 . But the e x t r a support p rov ided by the l i n e a r program's r e s u l t s of ten are h e l p f u l i n a l l o w i n g the des igner to develop a " f e e l " f o r the system, and knowledge o f where the optimum l i e s p rovides him w i t h a bas i s f o r judg ing whether the mer i t s o f , say s t a n d a r d i z a t i o n of pipe s i z e s , j u s t i f i e s the c o s t . 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 study i n o p t i m i z i n g a water p ipe network i s t ha t f o r small changes i n a p i p e ' s d iameter , the f low through the pipe w i l l remain cons tan t . This a l l o w s the s tandard form o f the f r i c t i o n equa t ion : 2 h^ = constant x Q x k to be l i n e a r i z e d i n t o : h^ = new cons tan t x k For c a l c u l a t i o n purposes i t was decided tha t a small change i n diameter would be one s tandard p ipe s i z e l a r g e r o r sma l l e r than the present case. Continuous pipe diameters are not cons idered i n the present approach based on the l i n e a r i z e d f r i c t i o n fo rmula . 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 def ined i n order to op t imize a process o r eva lua te a l t e r n a t i v e s . The cos t of the water p ipe network was the o b j e c t i v e to be minimized f o r t h i s problem, sub jec t t o c o n s t r a i n t s l i m i t i n g minimum pipe diameters and pressures as w e l l as bas i c h y d r a u l i c f low laws. 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 occurrence i n B r i t i s h Columbia , on ly the g r a v i t y f low s i t u a t i o n was cons ide red i . e . p r o v i s i o n was not made f o r booster pumps or i n t ake pumps as cos t i t ems. However, m u l t i p l e sources o f water and m u l t i p l e demands f o r water are accep t ab l e . F igure 1 i n d i c a t e s the va r ious steps which occur i n the procedure employed h e r e i n . SYSDATA and LPDATA are s torage f i l e s and LIP i s a packaged l i n e a r programming program. Data f o r the p ipe 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 organized i n t o the r equ i r ed format f o r the f low a n a l y s i s made by a s tandard program developed by 8 Fowler and Epp (Epp and Fowler , 1970). Upon r e c e i v i n g the r e s u l t s o f t h i s a n a l y s i s and cos t d a t a , the user c a r r i e s out some bas i c c a l c u l a t i o n s i n p repa ra t ion f o r the o p t i m i z i n g program. A new data t ab leau (LPDATA) i s arranged f o r the l i n e a r programming computer package (LIP) and then fed i n . The r e s u l t s are eva lua ted by the user who may then change some pipe s i z e s and rerun the system once again or may decide tha t the network i s s a t i s f a c t o r y and h a l t the process . U l t ima te c o n t r o l r e s t s w i t h the user . 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 chap te r . C. SYSDATA F i l e For ease of opera t ion i t was found tha t the data f o r the water supply network to be ana lyzed should be handled i n a computer s torage f i l e r a the r than as data ca rds . At the present time t h i s f i l e i s used o n l y to s to re data r e q u i r e d f o r the network f low a n a l y s i s program. The f o l l o w i n g in fo rmat ion i s conta ined in the SYSDATA f i l e : (a) i npu t /ou tpu t u n i t s (b) cho ice of pipe f low a n a l y s i s formula (c) node data (d) pipe data (e) r e s e r v o i r data ( f ) booster pump data (g) check va lve data (h) pressure reducing va lve data ( D consumption and supply data ( j ) degree o f accuracy r e q u i r e d 9 At the present t i n e data f o r booster pumps, check va lves and pressure reducing va lves are not used s ince the present o p t i m i z a t i o n program cannot handle them. They a re , however, f u l l y acceptable to the a n a l y t i c a l program (Epp and Fowler , 1970). D. 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 lows i n each pipe and pressure heads a t the nodes of a given 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 Fowler , 1970). As s t a t ed by them, i t i s "an e f f i c i e n t computer program f o r the s o l u t i o n o f s t eady-s t a t e f lows i n water networks. I t s fea tures i n c l u d e : (a) The use o f Newton's method of s o l v i n g a system of s imultaneous n o n - l i n e a r equa t ions ; (b) a l o o p - o r i e n t a t e d network to reduce the number o f equat ions to be s o l v e d : (c) automatic loop numbering tha t produces a banded symmetic ma t r ix w i t h consequent r educ t ion i n computer memory requi rements ; and (d) the requirement of a minimum of input d a t a . " The program has been used by the author to analyze 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 output data from the program i n c l u d e s the f o l l o w i n g : (a) Pipe in fo rmat ion such as p ipe numbering, upstream and downstream nodes, pipe l e n g t h , p ipe d iamete r , pipe roughness c o e f f i c i e n t and the c a l c u l a t e d pipe r e s i s t a n c e . 10 (b) Pump or r e s e r v o i r data a t an inpu 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 a t t ha t p o i n t ; (c) C a l c u l a t e d pipe data i n c l u d i n g the d i r e c t e d f low i n a p ipe and the head l o s s through the p ipe . (d) Nodal data such as node e l e v a t i o n , demand, h y d r a u l i c grade l i n e and pressure . This in fo rmat ion i s then combined w i th cos t data in p repar ing an input tab leau (the LPDATA f i l e ) f o r the l i n e a r programming package. E. Formulat ion 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 def ined as f o l l o w s f o r a network as shown i n F igure 2: h-j =? head at node #1 = head a t node #2 = f low i n p ipe 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^ = cos t of pipe A k^ = r e s i s t a n c e o f pipe A Based on the assumption tha t on ly small changes i n p ipe diameters are to be c o n s i d e r e d , ( A C \ * / A C \ * * dC f l \Uk7 + Uk7 Where * represents a smal l inc rease i n p ipe diameter and ** represents a small decrease in pipe diameter . Th i s c rea tes a l i n e a r func t ion f o r d C / d k over the range o f changes a l lowed to pipe A at any p a r t i c u l a r o p t i m i z a t i o n s tage . The func t ion op t imized i n LIP i s the o b j e c t i v e func t ion and represents the network c o s t s . A t rue cos t o f the water supply network i s not used a t p resen t , o n l y a r e l a t i v e cos t f o r comparison purposes 11 i s r e q u i r e d . This o b j e c t i v e f u n c t i o n i s represented by: c t ; £ } , N where n = number o f p ipes i n the system k = v a r i a b l e to be op t imized ( r e s i s t a n c e to f low) When k inc reases i n value the amount o f f r i c t i o n i n the pipe decreases . The d e s i r e of the o p t i m i z a t i o n i s to minimize the value o f C, the t o t a l r e l a t i v e cos t of the pipe network. To main ta in proper 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 lower bounds to the a l l owab le pressure (head) e x i s t . S ince t h i s a n a l y s i s i s f o r g r a v i t y opera t ion 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 ha t o f the inpu t node w i t h the g rea te s t h y d r a u l i c e l e v a t i o n . The minimum pressure a l lowed can vary and i s based on , among other 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 . Into the LPDATA f i l e i s p laced a s e r i e s 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. Th i s then r e s t r i c t s the range of pressures m 3 r al lowed w i t h i n the pipe network. To def ine the pressure a t any node, a t l e a s t one s t a r t i n g nodal pressure must be inpu 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 the r op t imized pessures are then r e l a t e d back to t h i s e q u a l i t y which a t present 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 next 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 requirement t h a t : h /„ - h = h . ( i n tha t p ipe) u/s d/s f r r i 12 However, due to 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 necessary to r e v i s e t h i s t o : h , - h , , — h r u/s d/s f which i s e q u a l l y v a l i d f o r the f low o f water . This can be modi f i ed as f o l l o w s : h u /s - h d/s - h f ~ ° ~ h u/s + h d/s + h f ~ ° -h , + h ... + k Q 2 ^ 0 u/s d/s 2 where Q i s cons tant and de r i ved from the e a r l i e r a n a l y s i s . The d i r e c t i o n o f f lows in the system are c r i t i c a l i n ensur ing a proper p re sen ta t i on o f the network c h a r a c t e r i s t i c s to LIP and cau t ion must be e x e r c i s e d i n fo rmula t ing the i n e q u a l i t i e s . The f i n a l se t o f c o n s t r a i n t s l i m i t s the minimum pipe s i z e by l i m i t i n g the maximum value to which each o f the v a r i a b l e k ' s can r i s e . Based on the Manning equat ion where: k = r e s i s t a n c e C-|= cons tant n = Manning's roughness c o e f f i c i e n t L = l eng th o f pipe D = diameter i t can be de r i ved t h a t : k = C 1 n 2 L D " 5 Only the va lues o f "k" and "D" can vary and there i s a r e l a t i o n s h i p f o r one pipe t h a t : k new = k o l d / D o l d D new, S e t t i n g a minimum diameter of a watermain i n the network a l s o se ts a maximum r e s i s t a n c e value f o r each pipe a t a p a r t i c u l a r f low l e v e l . 13 For one a n a l y s i s l e v e l and on ly a l l o w i n g small changes i n diameter to be al 1 owed :• k max . = k f l ( °A \ 5 A A U m i n i The values o f and k^ are d e r i v e d from the Fowler a n a l y s i s output and thus a value f o r k n max f o r each k n i s e s t a b l i s h e d . (The l i n e a r program LIP does not a l l o w f o r nega t ive v a r i a b l e values thus l i m i t i n g k min to z e r o . ) When inpu t to the l i n e a r program v i a LPDATA, the k max c r i t e r i a l i m i t s the downward movement o f pipe diameters 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 water 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 probably r e s u l t i n the es tab l i shment o f a branched network, lower i n c o s t , but not acceptable f o r f i r e f i g h t i n g purposes. The o b j e c t i v e func t ion and i t s c o n s t r a i n t s are s u i t a b l y arranged i n the LPDATA f i l e to meet the data inpu t requirements o f L I P . F. LIP LIP i s a packaged computer program used to op t imize an o b j e c t i v e func t ion sub jec t to a se t o f c o n s t r a i n t s . I t was o r i g i n a l l y w r i t t e n and documented by D. O ' R e i l l y i n 1970 at the U n i v e r s i t y of B r i t i s h Columbia and p r e s e n t l y i t i s being 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 o r minimizes a l i n e a r o b j e c t i v e func t ion w i th n a c t i v i t i e s ( v a r i a b l e s ) o f the form: a, , X .. + a, 9 X „ + + a - i „ x = Z 1,1 1 1 ,d 2 I,n n sub jec t to the l i n e a r c o n s t r a i n t s : X , - ^ _ 0 j = 1,2 n •J and a 2 , l X l + a 2 , 2 X 2 + • • • + a 2 , n X n ~ a 2 ( n + l ) 14 a 3 , l X 1 + a 3 , 2 X 2 + + a 3 , n X n ^ a 3 ( n + l ) a ( m + l ) l X 1 + •••••• + a (m+1) ] X n ^ a (m+l ) (n+l ) where >-vx represents ^= or = " ( B i r d , 1975). ~ " Output from LIP covers a very broad scope o f a n a l y s i s but concern a t present i s d i r e c t e d to the pr imal s o l u t i o n v e c t o r . This vec to r represents the value o f each v a r i a b l e X, X 9 X ( k 1 5 , k^) tha t should occur f o r the o b j e c t i v e func t ion to be an optimum. Based on these opt imal v a r i a b l e v a l u e s , recommended values f o r the pipe diameters o f the network can be c a l c u l a t e d . G. User Dec i s ion Recommended diameters are not c a l c u l a t e d from the o p t i m i z i n g process r e s u l t s but the p ipe diameters may be changed up or down only one s tandard p ipe s i z e d i f f e r e n c e . They should on ly be changed i f the recommended diameter i s more t ha t 50% of the way to the next s tandard diameter . However, the user s t i l l has the op t ion to not 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 ipe diameters change then the process i s stopped a t t h i s s t a t e . A check i s made on the output from the Fowler a n a l y s i s to ensure tha t a l l pressures are g rea te r than the minimum a l lowed . I t may be h e l p f u l 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 repea t ing the a n a l y s i s - o p t i m i z a t i o n process aga in . I f changes are made to the ne twork ' s pipe d iameters , then these are i n s t i t u t e d i n the SYSDATA f i l e and the process i s repeated . 15 Chapter 4 EXPERIENCE AND RESULTS A. I n t r o d u c t i o n The present approach d e t a i l e d i n the p rev ious chapter i s based on the background de r ived 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 techniques f o r o p t i m i z i n g a network. Two o f these techniques 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 . Both are examined i n t h i s chapter f o r comparative purposes. I t was found t ha t a marked inc rease i n the investment r e tu rn f o r water supply networks can be ob ta ined i f a v a l i d o p t i m i z i n g procedure , such as put forward i n t h i s paper , i s u t i l i z e d . However, the exper ienced judgement o f a q u a l i f i e d network des igner i s s t i l l a most d e s i r a b l e inpu t to the des ign process f o r a c h i e v 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 a minimum of c o s t . B. One-step Technique Due to the d i s c r e t e nature of the diameters o f commerc ia l ly a v a i l a b l e water p i p e , the d i s t r i b u t i o n networks are favourab le to o p t i m i z a t i o n procedures t ha t a v a i l themselves o f t h i s f ea tu re . A "one-step t e chn ique" , was developed as a crude but e f f e c t i v e means of o b t a i n i n g an opt imal network a t a minimal p ipe cos t . The e f f e c t of i n c r e a s i n g each p i p e ' s diameter one s i z e and one a t a time i s examined based on the cos t and on movement towards a t ta inment of minimum l e v e l s o f f low and pressure throughout the network. The one-step pipe s i z e increase which has the h ighes t bene f i t to cos t r a t i o i s chosen as the best one-step toward an opt imal s o l u t i o n . Th i s opt imal change i s performed to the bas ic network and a l l op t ions are again re-examined 16 f o r i n c r e a s i n g pipe diameters one s tep , This process cont inues u n t i l the minimum d e s i r e d l e v e l s o f f low and pressure are a t t a i n e d . The one-step approach can be used most e f f e c t i v e l y on the improvement o f or a d d i t i o n s to an e x i s t i n g network but i t a l s o i s adaptable to the de s ign ing o f new proposed water d i s t r i b u t i o n networks. The a d d i t i o n to or 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 merely a s i m p l i f i e d case of the design o f a complete new network s ince the same procedure i s f o l l o w e d , but fewer pipes have to be cons ide red . Therefore on ly the development and r e s u l t s of the one-step technique 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 forward i n t h i s paper. The f i r s t s tep 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 one-step method i s the p r e l i m i n a r y l ayou t complete w i t h a l l the p ipe and nodal da ta . O r i g i n a l p ipe diameters are assumed to be the minimum s i z e a l l o w e d . The minimum a l l owab le nodal pressure i s determined and the d e s i r e d f low l e v e l s from each node are e s t a b l i s h e d . Cos t ing data i s assembled f o r each l eng th 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 value f o r the summa-t i o n o f the i n d i v i d u a l nodal pressure 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 out us ing the procedure i n d i c a t e d i n Figure 3. A f t e r one complete se t of a l t e r n a t e s are examined a new base system i s c r ea t ed . The one-step o p t i m i z a t i o n process i s repeated u n t i l the pressure d e f i c i t i s zero o r smal l enough t ha t the user i s s a t i s f i e d w i t h the r e s u l t s . This technique was found to be very simple to formulate 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 every p o s s i b l e a l t e r n a t e i s exp lo red on a one-step b a s i s . I t i s , however,an expensive technique due to l a r g e 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 step f o r each pipe in 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 con t i nua l a n a l y s i s of the poorer o t p i m i z i n g c h o i c e s . F igure 4 i n d i c a t e s the plan view of a proposed s i x node water supply network. For s i m p l i c i t y there i s on ly one nodal source o f water and one nodal demand f o r water . A l l e l e v a t i o n s are assumed to be e q u a l . The minimum pipe diameter i s se t a t s i x inches and t h i s i s the re fo re the o r i g i n a l s i z e o f each p ipe . I t i s d e s i r e d to have a minimum nodal pressure of 20 psi: throughout the network. Since there i s on ly one consumptive node, on ly i t need be examined f o r water pressure l e v e l . The pressure d e f i c i t can be def ined as the d i f f e r e n c e between our d e s i r e d minimum pressure (20 p s i ) and tha t found a t node #5. Based on the in fo rmat ion o f F igure 4 and the data i n Table 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 . The f i n a l p repa ra to ry s tep 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 lows i n the system i n i t s b a s i c form. This i n d i c a t e d a pressure d e f i c i t a t the zero o p t i m i z a t i o n l e v e l o f 414 p s i (PD n = 414) . Table 1 COST DATA FOR PIPES Diameter Cost to supply and i n s t a l l per l i n e a r foot 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 Pipe Cost ( D o l l a r s ) S i z e Change Pipe 1 . 2 3 4 5 6 7 6" - 8" 750 1500 3750 900 1500 1500 1500 8" -10" 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. A P . D . / A C O S T 1 0.04 2 0.07 3 0.04 4. 0.00 '5 0.08 6 0.01 7 0.01 21 Table 3 i s based on a one-step change f o r each of the ne twork ' s p i p e s . From t h i s t a b l e i t i s apparent t ha t p ipe #5 presents the best choice f o r a one-step increase i n diameter . F o l l o w i n g t h i s change the technique was repeated severa l t imes r e s u l t i n g i n : (a) a t o p t i m i z a t i o n #2; pipe #2 i s made 8" diameter (b) a t o p t i m i z a t i o n #3; p ipe #1 i s made 8" diameter (c) a t o p t i m i z a t i o n #4; pipe #5 i s made 10" diameter (d) a t o p t i m i z a t i o n #5; p ipe #2 i s made 10" diameter (e) a t o p t i m i z a t i o n #6; p ipe #1 i s made 10" diameter The r e s u l t a n t network i s shown i n F igure 5 and the pressure a t node #5 i s 18.8 p s i . The d e c i s i o n was made tha t t h i s r e s u l t was c l o s e enough to the o r i g i n a l goal o f 20 p s i to be cons ide red f i n a l . $86,000 was the es t imated cos t o f t h i s one-step op t imized system and to a t t a i n i t 43 separate analyses o f the va r ious s i x node one-step a l t e r n a t e networks were made. C. Development of Present Approach Chapter 2 presented a d e t a i l e d exp l ana t i on o f the present 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 . The f i r s t stage o f tha t development was based on a pr imary unders tanding o f the techniques and p r i n c i p l e s i n v o l v e d . Quite 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 technique was r e a l i z e d and as the c o m p l e x i t i e s of t h i s more s o p h i s t i c a t e d approach became apparent a computerized 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 subsequently made to data p r e s e n t a t i o n , the va r ious 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 developing the o p t i m i z a t i o n . 22 The r e s u l t s achieved bear wi tness to the p rogress ion o f the technique to a f u l l y o p e r a t i o n a l useful l e v e l f o r small networks. The f i r s t stage of development was based on h y d r a u l i c laws ( = kQ & "ST Q-j 0 0p = 0) and on the concept of minimum al 1 bwable pressure heads and pipe d iameters . As d i scussed i n Chapter 3 , the assumption o f f lows remaining constant f o r small changes i n pipe diameters i s made to 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 process fo l lowed the f low cha r t dep ic ted i n F igure 6. A t e s t case c o n s i s t i n g of a s imple three pipe system was chosen as shown i n F igure 7. A hand c a l c u l a t e d Hardy-Cross 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 ipes s i x inches i n d iameter ; the minimum acceptable diameter . S ince a l l p ipes were the same diameter the dC/dk values were e q u a l . The minimum pressure a t node B was set a t 200' and f o r node C i t was se t a t 100 ' . Based on the h y d r a u l i c law, H f = k Q 2 : (a) 2000' - (k 1 + k ^ Q ^ — 200' (b) 2000' - (k 3 + k 3) Q 3 2 — 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 of the values f o r k-| and Q-j: (c) 200' - (54.7 + A ^ ) 1180' — 200' which r e s u l t s i n Ak-j = -53.2 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 tha t the summation o f the d i r e c t e d f r i c t i o n los ses around a loop must equal z e r o ; t h e r e f o r e : (k 1 + A-k-j) Q}2 - (k 3 +Ak 3 ) Q 3 2 = 0 Based on t h a t , A m u s t equal - 5 6 . 1 . In summary: A k1 = -53 ,2 A k 2 = -56.1 A k 3 = -52 .9 23 For a pipe diameter change from 6" to 12" , A k = -53 .3 and based on t h i s value a l l three pipe diameters were inc reased to 12". A Hardy-Cross a n a l y s i s i n d i c a t e d tha t the r e s u l t s were s a t i s f a c t o r y when compared to the minimum pressure c r i t e r i a . A one-step o p t i m i z a t i o n was then c a r r i e d out on the o r i g i n a l system and i t a l s o r e s u l t e d i n the a l l 12" diameter system as the lowest cos t acceptable system. The next stage i n the development was the adapt ing of a computerized l i n e a r programming technique ( B i r d , 1975) to c a r r y out t h a t ~ p o r t i o n o f the o p t i m i z a t i o n process . The o b j e c t i v e f u n c t i o n , c o n s t r a i n t s and v a r i a b l e s r equ i r ed ca re fu l man ipu la t ion to r e t a i n t h e i r essence i n the data format to t h i s packaged program. P r e l i m i n a r y attempts a t o p t i m i z i n g a f a i r l y l a rge and complex network were made but wi th d i s a p p o i n t i n g r e s u l t s . More research and c o n s i d e r a t i o n was r e q u i r e d to achieve an o p e r a t i o n a l approach us ing the computerized l i n e a r program. For s i m p l i c i t y a s i x node system, as shown i n F igure 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 cos t s ( d k ' C 0 ' S ' t ^ sub jec t to the c o n s t r a i n t s o f : ' ( i ) ^ h = 0 loops ( i i ) a l l h n o d e s ^ 50' and to a l l o w the l i n e a r i z i n g of the f r i c t i o n formula : ( i i i ) a l l A k ' s i n the^/_ + A k fo r one pipe l i n e a r program — diamter change To s p e c i f y a minimum a l l owab le pressure head a t each node, i t i s necessary to i nc lude equat ions d e f i n i n g the pressure head a t each node. The a d d i t i o n o f the f o l l o w i n g form of equat ions was r e q u i r e d : ( i v ) ( h u / s •- h f - 5 0 ' ) ^ Q 2 : A k . However, t h i s 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 c o n s t a n t l y being made to h ^ s dur ing the o p t i m i z a t i o n . This was co r r ec t ed by adding the f o l l o w i n g c o n d i t i o n a l equa t ions : ^ h d / s = h u / s - k 0 - 2 - ^ A k -A C To c a l c u l a t e the f a c t o r f o r use i n the o b j e c t i v e func t ion i t was assumed tha t an increase o f one s tandard p ipe cou ld be used as a dC r ep re sen t a t i ve value f o r s ince the A k f a c t o r was qu i t e r e s t r i c t e d i n the range o f values accep tab le . This procedure cou ld be amended at a l a t e r date i f r e q u i r e d . 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 to be too harsh and was de le t ed from the l i n e a r program. Other 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 to c rea te 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 case : h - = 200' r e s e r v o i r 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 to take nega t ive values but A k must be a l lowed to take negat ive va lues . This r equ i r ed A k to be represented by x-j and Xg such t h a t : A k = Xi — x 2 When k was to be n e g a t i v e , x-j would take on a value o f zero and Xg a p o s i t i v e v a l u e . The reverse would be t rue f o r A k being p o s i t i v e . A f u r t h e r r e s t r i c t i o n was r equ i r ed to ensure tha t dur ing the o p t i m i z a t i o n no pressure heads were a l lowed to exceed the r e s e r v o i r l e v e l s ince ' there were no boos ter pumps. As a r e s u l t o f t h i s the input 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 . A l l network data and l i n e a r programming data were s to red on data ca rds . The convers ion of the network a n a l y s i s data to l i n e a r 25 programming data was done by hand. The f o l l o w i n g arrangement o f equat ions was used to program the computerized l i n e a r program: ( i ) the o b j e c t i v e func t ion (Z) 1 ' <*1 " *2> (f)l + <x3 " *4> § 2 + ; + ( x 1 3 - x ] 4 ) (^-) 7 ( i i ) a l l nodal pressure heads to be l e s s than the r e s e r v o i r l e v e l (200 ' ) x 1 5 ^ 200 x 1 6 ^ = 200 x 2 0 200 ( i i i ) a l l nodal heads to be g rea te r than minimum pressure o f 50' - x 1 5 ^ - 5 0 - x 1 6 £ - - 5 0 " X 20 - - 5 0 ( i v ) the summation o f the changes i n f r i c t i o n l o s se s around a loop are to equal zero - . 25 x 7 + .25 x 8 + 3.2 x g - 3.2 x 1 Q + = 0 5.24 x 1 - 5.24 x 2 + - . 25 x g = 0 (v) a l l v a r i a b l e s r ep re sen t ing nodal heads are to be def ined x 1 6 = x 1 5 " 1 4 3 , " 5 , 2 4 x l + 5 , 2 4 X 2 x 2 0 = X 19 " 3 7 ' 2 " 1 0 * 3 X 13 + 1 0 , 3 X 14 ( v i ) and a s t a r t i n g po in t f o r the nodal d e f i n i t i o n of pressure heads to 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 technique a l lows the v a r i a b l e s x-j to X^Q to vary w i t h i n the c o n s t r a i n t s imposed. The output o f the program i n d i c a t e s the values of these v a r i a b l e s which r e s u l t i n the h ighes t value o f the o b j e c t i v e f u n c t i o n . Due to the o r i g i n a l assumption o f on ly a smal l change i n pipe s i z i n g , the user must on ly a l l ow the pipes a one step change in pipe diameter a f t e r the l i n e a r programming r e s u l t s i n d i c a t e d a A k o f value g rea te r than 50% o f tha t to make a one s tep s i z e change. 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 changes. The p r o v i s i o n o f a minimum diameter 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 in the p o s i t i v e d i r e c t i o n ( e . g . x-j ^ 2 0 . 1 ) . Care i s r e q u i r e d to ensure tha t each nodal pressure head was def ined on ly once s ince f a i l u r e to do so cou ld again over c o n s t r a i n the system. Table 4 i s a summary of the progress o f t h i s development phase o f the re sea rch . r dC N -jj7J n value f o r the x 2 n and X2n-1 v a ^ u e s ™ t n e " z " f u n c t i o n . I t was proposed d C dC to use as obta ined f o r one s i z e l a r g e r pipe f o r X 2 n _ - | and as ob ta ined f o r one s i z e s m a l l e r pipe f o r X £ P . This proved u n s a t i s -f a c t o r y s ince sometimes both x 0 n and x 0 would take on values thus 2 n - l upse t t i ng the "mutual e x c l u s i v e n e s s " which should have e x i s t e d . No c l e a r reason f o r t h i s occurrence was obta ined and a re tu rn 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 stage the system data was converted to s torage f i l e s from data ca rds . The technique was automated such tha t a l i n e a r progrm would immediately be c a r r i e d out based on the data from the mod i f i ed f low a n a l y s i s program. A grea t r educ t ion i n user time was achieved and y e t the user s t i l l mainta ined 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 COST O r i g i n a l 1021 p s i ' $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 #10 #11 #12 - 12.8 96,150 #13 - 12.8 96,150 28 network and pipe cos t data was s to red i n SYSDATA w i t h the a n a l y s i s r e s u l t s being used as a bas i s f o r the c a c u l a t i o n of input data f o r the l i n e a r program to be s to red i n LPDATA. By us ing a c o n v e r s a t i o n a l t e rmina 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 . Changes to SYSDATA could a l so be r a p i d l y made. The s i x loop system was op t imized i n l e s s than four hours of design time to w i t h i n a few percentages o f the one-step lowest cos t a l t e r n a t i v e . The r e s u l t was a minimum pressure o f almost double that o f the one-step s o l u t i o n . Table 5 and Figure 8 summarize the r e s u l t s . To f u r t h e r study t h i s automated o p t i m i z a t i o n process a 12 node system (see Figure 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 to quicken the o p t i m i z a t i o n ra te being i n c l u d e d . These were: ( i ) The o r i g i n a l assumption f o r f low d i r e c t i o n be such tha t each node i s a downstream node at l e a s t once, ( i i ) A l l LPDATA statements be changed to to a v o i d over c o n s t r a i n t s . i . e . h u ^ s = h ^ s - h f - Q A k to become: A / s ^ h d / s + h f + ^ A k ( i i i ) L i m i t s be set on the x values i n the A k ' s such tha t x 2 n 1 n e v e r be g rea te r than "k o r i g i n a l " . Table 6 and F igure 10 i n d i c a t e the r e s u l t s of 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 ha t e s t a b l i s h i n g the l i m i t i n g optimum network would g r e a t l y a i d in the measuring of the performance o f these techniques . This l i m i t i n g optimum would be based on an u n l i m i t e d supply o f a l t e r n a t e pipe d iameters . The system cou ld 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 pipe diameters on any p a r t i c u l a r node to node pipe run. Based on t h i s continuous pipe diameter - cos t Table 5 AUTOMATED SIX NODE OPTIMIZATION PROGRESS STEP NUMBER SYSTEM PRESSURE DEFICIT COST O r i g i n a l 1021 p s i $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 COST O r i g i n a l 1100 p s i $35,250 O p t i m i z a t i o n #1 48 37,050 #2 1.4 39,050 #3 - 8.9 37,412 #4 - 4 .2 37,212 #5 0 37,250 #6 - 1.9 37,575 #7 3.5 36,950 #8 - 8.3 37,687 31 func t ion i t was hoped tha t a minimum p o s s i b l e expendi ture cou ld be de r ived to achieve the g o a l s . The pipe 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 changes i n d iameters . The r e s u l t s were not c o n c l u s i v e and showed no tendancy to 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 to develop a design 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 design 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 cos t . As d e t a i l e d i n the preceding chap te r , t h i s goal has been achieved us ing an expensive and crude technique . However, an automated technique based on a l i n e a r i z a t i o n o f the pipe f r i c t i o n 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 cos t i t 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 th small networks. Both o f these methods are c o n s i d e r a b l y more economic, i n cos t to achieve r e s u l t s and in the cos t o f the proposed network, than the t r i a l and e r r o r approach g e n e r a l l y used a t present . Using the one-step procedure to op t imize the s i x node example r e s u l t e d i n a network tha t cos t $86,000, had a minimum pressure o f 18.8 p s i and r e q u i r e d 15.0 seconds o f computer t ime to ach ieve . The automated technique used l e s s than one h a l f the computer time and produced a network tha t cos t 4% more ($89,250) but had a minimum pressure of 34.2 p s i . Th i s i n d i c a t e s tha t the l i n e a r i z a t i o n based automated technique 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 i s e f f e c t i v e . As s t a t ed e a r l i e r i n t h i s t h e s i s , the network des igner can s t i l l make a valued inpu t to the design process when us ing the automated type o f o p t i m i z a t i o n . Upon examining F igure 8 and no t ing the 14.2 p s i excess pressure i n tha t network, c o n s i d e r a t i o n may be given to reducing the diameter of p ipe #5 to 10" and i n c r e a s i n g pipe #1 to 33 12". This w i l l reduce cos t s by $1575 and s t i l l keep the pressure g rea te r than the 20 p s i minimum at demand f lows . As the network becomes more compl ica ted t h i s type o f obse rva t ion w i l l be i n c r e a s i n g l y more d i f f i c u l t to make. The twelve node s o l u t i o n which the automated technique a t t a i n e d i n 6.4 seconds o f computer time would take over 90 seconds to a t t a i n us ing the one-step procedure. This i n d i c a t e s the s u b s t a n t i a l r educ t ion in computer t ime cos t s which can be obta ined us ing t h i s procedure. The r e s u l t s of the twelve 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 in 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 tu rn to the chosen s o l u t i o n , optimum #4, i n d i c a t e d tha t t ha t optimum may be a l o c a l optimum o n l y . Conf i rmat ion o f t h i s would r equ i r e a one-step type o f o p t i m i z a t i o n of the network or a f u r t h e r c o n t i n u a t i o n of 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 to pipe s i z e s which cou ld prove accep tab le . Several of these were t r i e d dur ing the study 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 tha t the procedure can out perform the au tho r ' s modest c a p a b i l i t i e s in network d e s i g n . The r e s u l t s i n d i c a t e d tha t t h i s procedure can a l l o w the ne t -work des igner 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 exper ience to r e f i n e the network i f r e q u i r e d . The procedure has been shown to work but may need some f u r t h e r refinement to become f u l l y o p e r a t i o n a l . 34 Chapter 6 CONCLUSIONS AND RECOMMENDATIONS Money inves t ed 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 d i s t r i -but ion networks i s very hard to recover i f i t i s found tha t your cho ice o f pipe s i z e s was not the bes t . Therefore a water d i s t r i -but ion network designed to meet s p e c i f i e d demands at minimum cos t should be the bas i s f o r the o r i g i n a l c a p i t a l expendi ture on p ipe . The procedure developed i n t h i s t h e s i s should help the network des igner to p rov ide an op t imized network; one tha t combines adequate h y d r a u l i c performance w i t h a minimum of c o s t . Based on a " f l o w - f i x i n g " 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 technique and a r a p i d network a n a l y s i s program, the present approach r e s u l t s i n r a p i d convergence to an e f f i c i e n t network design when d e a l i n g w i t h small networks. Adjustments to the network by the des igner s t i l l remain an important c o n s i d e r a t i o n i n the p rocess . Fur the r research i n t h i s area i s recommended due to the l a rge sums o f money spent each yea 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 . A l o g i c a l ex tens ion o f t h i s research would be fu r t he r r e f i n i n g and t e s t i n g to p rov ide an o p e r a t i o n a l procedure tha t would work on very l a r g e networks , a step not y e t completed. Other areas of p o s s i b l e research expanding on t h i s paper a r e : -(a) m o d i f i c a t i o n s to the l i n e a r program to a l l o w g rea te r f l e x i b i l i t y i n input da ta ; (b) expansion o f the cos t data to i nc lude pumping cos t s and v a r i e d pipe i n s t a l l a t i o n c o s t s ; (c) o f g rea t a s s i s t a n c e would be the development 35 of a technique to determine how c l o s e the network i s to the g loba l optimum; (d) u t i l i z a t i o n o f a continuous "k vs . cos t " curve to supply dC/dk values f o r the o b j e c t i v e func t ion in the l i n e a r program; (e) comparison o f the cos t s and performance o f the present approach to the commonly used t r i a l and e r r o r technique based on expe r i ence ; ( f ) f u r t h e r o p t i m i z i n g o f l a rge s c a l e problems u t i l i z i n g , i f p o s s i b l e , ac tua l cases taken from c o n s u l t i n g engineers s t u d i e s . I t i s hoped tha t others w i l l be s t i m u l a t e d by t h i s research to f u r t h e r i n v e s t i g a t e t h i s area i n the hopes of b r i n g i n g to Canadian engineers a h igher a p p r e c i a t i o n o f o p t i m i z a t i o n techniques . FIG. I OPTIMIZATION F L O W C H A R T S U P P L Y * -D E M A N D D E M A N D F L O W F I G . 2 T Y P I C A L P I P E N E T W O R K S T A R T I A S S E M B L E B A S I C D A T A ± ' A S S E M B L E C O S T a N = 0 A N A L Y S E E X I S T I N G N E T W O R K I IF D E F E C I T E Q U A L S Z E R O , S T O P / o P T = N O P T \ ^ \ v a l = n H Y E S I N C R E A S E D I A . O F P I P E # V A L O N E S T E P YES/ DOES L A S T IN N E T W O N = \ NO i P I P E — RK/ MO / DOES N=l OR NOPT>OPT O L D D E F E C I T - N E W DEFEC IT = 'D ' H A N A L Y S E N E W N E T W O R K N = N + I INCREASE THE DIA.OF PIPE#N ONE STEP I N O P T = D / C C O S T O F C H A N G E = C F I G . 3 O N E S T E P O P T I M I Z A T I O N T E C H N I Q U E P I P E # 2 1000 ' 6 > P I P E * I 5 0 0 ' 6"4> P I P E # 3 2 5 0 0 6"<f) <D P I P E # 5 1 0 0 0 ' 6 > P I P E # 4 6 0 0 ' 6"<t> P I P E # 6 1 0 0 0 6"<f> PIPE #7 1 0 0 0 ' 6"<t> F I G . 4 O R I G I N A L SIX NODE N E T W O R K P I P E # 2 1000' I 0 > P IPE# I 5 0 0 ' I 0 > P I P E # 3 2 5 0 0 ' 6 > P I P E # 5 1000* I 0 > P I P E # 4 6 0 0 ' 6 > P I P E # 6 1 0 0 0 ' 6"<£ P I P E # 7 1 0 0 0 ' 6"<jb <•> F I G . 5 SIX NODE NETWORK A F T E R SIX O N E - S T E P O P T I M I Z A T I O N S . G I V E N •• C O S T D A T A E X I S T I N G N E T W O R K 8 D E M A N D S M I N I M U M A L L O W A B L E P R E S S U R E S A N A L Y S E E X I S T I N G N E T W O R K S T O P I I S N E T W O R K S A T I S F A C T O R Y ? C A L C U L A T E A C O S T A K I d C dC D O L . P . B A S E D O N ' M I N I M I Z I N G A K , ^ 7 ± A K 2 ^ - ± uK| dK2 S U B J E C T T O ' A L L P R E S S U R E S > M I N I M U M L E V E L A L L D I A M E T E R S > M I N I M U M S I Z E 2 ! A L L F L O W S I N L O O P = 0 I M A K E D I A M E T E R C H A N G E S B A S E D O N O P T I M I Z E D A K , , A K 2 r . . . . . A K j F I G . 6 P R E S E N T O P T I M I Z A T I O N P R O C E S S 7 0 c f s at 2 0 0 0 ' of head P IPE # I 1000' 4 5 c f s PIPE # 2 1 0 0 0 ' P IPE # 3 1 0 0 0 ' 2 5 c f s NOTE : A l l nodal e l e v a t i o n s are equaI . FIG.7 T H R E E NODE NETWORK E X A M P L E PIPE # 2 1 0 0 0 ' I 0 > P I P E # 5 1000' I2"d> P I P E * I 5 0 0 ' I0"d> PI P E # 3 2 5 0 0 ' 6"d> PIPE # 4 6 0 0 ' 6 > P I P E # 7 1 0 0 0 ' 6 > P I P E # 6 1 0 0 0 ' 6"<£ <s) F I G . 8 SIX NODE NETWORK A F T E R AUTOMATED OPTIMIZATION. © P I P E # 4 1 0 0 ' - 6 > PIPE#I 100 '-6"<£ © P I P E # 5 200' -6"<£ •PIPE* 2 IOO'-6"<£ P I P E # 6 200'-6"<£ P I P E * 3 • 3 P 0 - - 6 " * p | p E # 7 2 0 0 - 6 > <D PIPE*II 500'-6"<£ P I P E * 8 3 0 0 - 6"<£ p | p E # | 2 4cfs 400' -6"<£ P I P E * 9 2 0 0 , - 6 n * P , P E * I 3 3 0 0 - 6 > P I P E * I 0 100'- 6"<f> 8 P I P E * 14 2 0 0 ' - 6 > © P I P E * 15 1 0 0 ' - 6 > 10 P I P E * I 6 5 0 - 6 > © PIPE*I7 7 5 ' - 6 > 10 cfs FIG.9 ORIGINAL T W E L V E NODE NETWORK -p-I4cfs © P I P E * 4 I00'-6"d> © PIPE#I 1 0 0 ' - 6 > p | p E # 5 2 0 0 * - 6"<f> P I P E * 2 I 0 0 " 6 > P . P E * 6 2 0 0 ' - I0"d> P I P E # 3 3 0 0 - 6"4> P I P E * 7 200'-6"</> © PIPE*I I 500' -6"d> P I P E # 8 3 0 0 - 6 > 4c f s P I P E # 12 4 0 0 ' - 6 " d 5 © P I P E # 9 2 0 . 0 ' - 8 > p i p E # | 3 3 0 0 ' - 8"<f> P I P E * 10 IOO'-6Md> P I P E * 14 2 0 0 - 8 " d > FIG.10 OPTIMIZED TWELVE NODE N E T W O R K © PIPE #15 1 0 0 ' - 6 > ® P I P E * I 6 5 0 W < £ © P I P E * I 7 75' -8"dS lOcfs VJI 46 BIBLIOGRAPHY BIRD, C . j "A L inea r Programming Package", Computing Cen t re , U n i v e r s i t y o f B r i t i s h Columbia , January, 1975. DEB, A.K,, and SARKER, A . K . , " O p t i m i z a t i o n in Design o f H y d r a u l i c Network 1 1 , Journa l of the S a n i t a r y Engineer ing  D i v i s i o n , ASCE, V o l . 97, No. SA2, Proc . Paper 8032, A p r i l 1971, pp. 141-159 DEB, A . K . , "Leas t Cost Design o f Branched Pipe Network  System", Journa l o f the Environmental Engineer ing  D i v i s i o n , ASCE, V o l . 100, No. EE4, P roc . Paper 10711, August 1974, pp. 821-835. DEB, A . K . , " O p t i m i z a t i o n of Water D i s t r i b u t i o n Network Systems' Journal o f the Environmental Engineer ing D i v i s i o n , ASCE, V o l . 102, No. EE4, P roc . Paper 12343, August 1976, pp. 837-851. EPP, R. and FOWLER, A . G . , " E f f i c i e n t Code f o r Steady S ta te  Flows i n Networks" , Journa l of the H y d r a u l i c D i v i s i o n ASCE, V o l . 96, No, HY1, P roc . 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", Journa 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 roc . 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 of Water Supply Systems", Journa l o f the Environmental Engineer ing D i v i s i o n , ASCE, V o l . 102, No. EE2, Proc . Paper 12026, A p r i l 1976, pp. 313-327 47 WATANATADA, T , , "Leas t -Cos t Design o f Water D i s t r i b u t i o n  Systems", Journa l of the H y d r a u l i c s D i v i s i o n , ASCE, V o l . 99, No. HY9, Proc . Paper 9974, Sept . 1973, pp. 1497-1513. 

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