COST OPTIMIZATION OF WATER DISTRIBUTION SYSTEMS SUBJECTED TO WATER HAMMER

Transcription

1 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt COST OPTIMIZATION OF WATER DISTRIBUTION SYSTEMS SUBJECTED TO WATER HAMMER Berge Djebedjan *, Mohamed S. Mohamed *, Abdel-Gawad Mondy **, and Magdy Abou Rayan * * Mechancal Power Engneerng Department, Faculty of Engneerng, Mansoura Unversty, El-Mansoura, Egypt ** Gulf of Suez Petroleum Co. Gupco, Egypt E-mals: bergedje@mans.edu.eg, msafwat@mans.edu.eg, MondyA@gupco.net, mrayan@mans.edu.eg ABSTRACT The paper presents the water dstrbuton systems optmzaton by selectng the optmal ppe dameters for water hammer transents. The optmzaton method used s the Genetc Algorthm (GA). The GA s have been used n solvng the water network optmzaton for steady state condtons. The GA s ntegrated wth the steady state hydraulc analyss program and a transent analyss program to mprove the search for the optmal dameters under certan constrants. These nclude the mnmum allowable pressure head constrants at the nodes for the steady state flow, and the mnmum and maxmum allowable pressure heads constrants for the water hammer. Three cases are studed ncludng the followng causes of water dstrbuton transents: changes n water demands at the nodes, sudden valve closure and pump power falure. The applcaton of the computer program to the studed cases shows the sutablty of the method to fnd the least cost n a favorable number of functon evaluatons. Ths technque can be used n the frst stages of the desgn of water dstrbuton networks to protect t from the water hammer damages. The technque s very economcal as the network desgn can be acheved wthout usng hydraulc devces for water hammer control. Keywords: Water Hammer, Flud Transents, Genetc Algorthm, Ppe Networks 1. INTRODUCTION The cost optmzaton of ppe networks under steady flow condtons was the subject of varous researches. The optmzaton technques were used to dentfy the optmal soluton for water dstrbuton systems. These technques are classfed nto determnstc (lnear, non-lnear and dynamc programmng) and stochastc technques (smulated annealng and genetc algorthm). The determnstc methods can not guarantee a global optmal soluton and requre that the objectve functons satsfy certan restrctve condtons (e.g., contnuty, dfferentablty to the second order, etc.) that cannot be generally guaranteed for a water dstrbuton system. Stochastc

2 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt technques and especally the genetc algorthm became a popular technque for optmzaton. The optmzaton of ppng network desgns have been addressed by a range of researchers wthout consderng the occurrence of water hammer event, they addressed the same n dfferent ways durng the past decades, Kessler and Shamr (1989), Djebedjan et al. (2) and Sârbu and Kalmár (22). The percepton of network optmzaton n steady state analyss lnked to the consequences of water hammer s recently examned. Few water network optmzaton approaches have been acheved. Lane and Karney (1997) studed the event of water hammer n a smple ppelne. Zhang (1999) studed the flud transents and ppelne optmzaton usng Genetc Algorthms. Kaya and Güney (2) studed the same for sprnkler rrgaton systems. Jung and Karney (23) studed the optmum selecton of hydraulc devces for water hammer control n the ppelne systems usng Genetc Algorthm. Jung and Karney (24a) studed the optmal selecton of ppe dameters n a network consderng steady state and transent analyss n water dstrbuton systems by usng Genetc Algorthm (GA) and Partcle Swarm Optmzaton (PSO). Jung and Karney (24b) studed the ppelne optmzaton by selectng, szng and placement for hydraulc devces n ppelne systems consderng the occurrence of water hammer event. Djebedjan et al. (25a) studed the water dstrbuton systems n both steady and transent (water hammer) states. They developed a numercal technque to analyze the network n the steady and transent states and select the optmum soluton to overcome the dfferent water hammer events usng genetc algorthms. They lnked between the hydraulc network solver (Newton-Raphson), transent analyzer and genetc algorthm as an optmzaton tool. The model was appled successfully on a network under water hammer event caused by pump staton power falure. The model was based on selectng the proper (optmum) ppes szes from a range of the avalable ppe szes, whch satsfed the network requrements such as pressure heads, demands and pressure lmts for water hammer. Ths approach provded the opportunty for potental savngs n costs. Djebedjan (26) studed the relablty-based optmzaton of potable water networks by selectng the optmal ppe dameters for water hammer under hydraulc relablty. Genetc Algorthm (GA) as an optmzaton tool has been lnked wth the Monte Carlo Smulaton for estmatng network capacty relablty and node capacty relablty. The prevous lterature revew demonstrates the sutablty of the Genetc Algorthms as an optmzaton technque to handle small and large-scale water dstrbuton systems

3 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt and to mnmze the cost n the steady state. From the prevously mentoned lterature revew, t appears that there were no studes carred out consderng the followng extremes altogether, water dstrbuton systems desgn, optmzaton usng genetc algorthms and water hammer as a major rsk, all networks should be desgned to elmnate t. 2. OPTIMIZATION OF PIPELINE SYSTEMS Water dstrbuton system (WDS) desgn problem s formulated and solved here as a sngle-objectve optmzaton problem wth the selecton of ppe dameters as the decson varables. The man parameter s subject to mnmzaton whch s the cost of the network desgn and constructon. The optmzaton problem s solved usng a sngle-objectve genetc algorthm (GA). The proposed robust desgn method s appled to a case study wth three causes of water hammer. The objectve of the optmum desgn model presented here s to mnmze total desgn costs under the constrant of mnmum head requrements n steady state condton and mnmum and maxmum heads requrements n transent condton (water hammer). The later s ncluded n order to protect the system from negatve or postve transent pressures. More specfcally, the optmzaton problem s to mnmze the objectve functon Z. It s the summaton of the network cost and penalty cost n both cases: steady state and water hammer: Z = C + C C (1) T P SS + P WH Network ppe cost s descrbed as follows: T N = =1 ( D ) L C = c. (2) Penalty cost n case of steady state s descrbed as follows: f H mn, ST H j C = M P SS C (3) T ( H mn, ST H j ) f H mn, ST H j > M j= 1 Equaton (3) was proposed by Djebedjan et al. (25b, 26). The total penalty cost n case of water hammer s descrbed as follows: C P -WH = C C (4) P WH MAX + P WH MIN

4 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt f H j,max H max, Tr C = M P -WH -MAX (5) C T ( H j,max H max, Tr ) f H j,max H max, Tr > j = 1 f H mn, Tr H j,mn C = M P WH -MIN (6) C T ( H mn, Tr H j,mn ) f H mn, Tr H j,mn > j= 1 where the parameters are as follows: c ( D ) : Cost of ppe per unt length C P-SS : Penalty cost n case of steady state C P-WH : Penalty cost n case of water hammer C P-WH-MAX : Penalty cost n case of water hammer when the pressure head exceeds the maxmum allowable pressure head lmt C P-WH-MIN : Penalty cost n case of water hammer when the pressure head decreases below the mnmum allowable pressure head lmt C T : Network total cost D : Dameter of ppe H : Pressure head at node j j H : Maxmum pressure head at node j under water hammer j,max H : Mnmum pressure head at node j under water hammer j,mn H max, TR ST : Maxmum allowable pressure head for water hammer H mn, : Mnmum allowable pressure head for steady state H mn, : Mnmum allowable pressure head for water hammer L M N Z TR : Length of ppe : Total number of nodes : Total number of ppes : Total cost of the network (desgn and penalty) Generally, the penalty cost s a functon of mnmum allowable pressure head at each node, pressure at each note and number of nodes volatng the crtera. The mnmzaton of the objectve functon Z n Eq. (1) s subject to: (a) Mass balance constrant: M j = 1 where Q = (7) j Q represents the dscharges nto or out of the node j (sgn ncluded). j

5 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt (b) Energy balance constrant: h f = E p (8) The conservaton of energy states that the total head loss around any loop must equal to zero or s equal to the energy delvered by a pump, E, f there s any. The head loss due to frcton n a ppe h f s expressed by the Hazen-Wllam formula: p L Q h f = (9) C D where Q s the ppe flow (ft 3 /s), D s ppe dameter (ft), L s ppe length (ft) and C s the Hazen-Wllams coeffcent. (c) Decson varables constrant: The desgn constrants (the ppe dameter bounds (maxmum and mnmum)) and the hydraulc constrants are gven respectvely as: D mn D D = 1,..., N (1) max where D s the dscrete ppe dameters selected from the set of commercally avalable ppe szes. (d) The hydraulc constrants for steady state and water hammer are gven as: H j H mn j = 1,..., M (11) ST where H j s the pressure head at node j, pressure head at node j for the steady state. H j mn ST, s the mnmum allowable H mn, TR H k H, TR k = 1,..., M (12) max p where H mn TR and H max TR are the mnmum and maxmum allowable pressure heads at node k for the transent condtons, and M p s the number of parts nto whch the ppe s dvded. From the prevous constrants, tem (a) through (d), t s noted that only ppes are consdered for the desgn. Mnor losses are neglected wth respect to frcton losses due to ppe length. Therefore; pumps, valves, and other specal hydraulc appurtenances are not ncluded for purposes of dscusson and smplcty of the model development. Usng GA to solve the optmzaton problem n Equaton (1), constrants (a), (b), (c) and (d) can be satsfed by lnkng GA to the determnstc WDS solver such as Newton-Raphson method and transent analyzer.

6 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt The Newton-Raphson method s used to smulate hydraulcally the gven network for the steady state and the water hammer analyss s mplemented by a method of characterstcs, Wyle et al. (1993). Constrant (c) can also be automatcally satsfed by usng the approprate GA codng. Transents Analyss n Ppng Networks Larock et al. (2) stated that the process of obtanng an unsteady soluton for a specfc problem n whch the demand or heads are specfed functons of tme conssts of the followng tasks: 1. The tme span, over whch the unsteady soluton s to be obtaned, s dvded nto NT tme ncrements. 2. The dscharges n all ppes and the heads at all nodes are assgned ntal values that are chosen from steady state soluton that has the same demands, and all other data. As the unsteady soluton has at tme zero. 3. All demands over each tme ncrement must be specfed. 4. Over each new tme ncrement, defne and evaluate the functon and the Jacobean matrx of dervatves of these functons. 5. Solve the resultng lnear equaton system. The soluton of ths equaton system s then subtracted from the set of unknown values, accordng to the Newton method. 6. Steps 4 and 5 are repeated teratvely, untl the specfed convergence crteron has been satsfed. 7. Wrte the soluton for the dscharges and the nodal for ths tme ncrement, and then repeat steps 3 through 7 untl the unsteady soluton spans the entre tme perod. The steps from 1 through 7 are the general method for analyzng an unsteady ppng system. Equatons Descrbng Unsteady Flow n Ppes The analyss of the unsteady flow n ppng networks s based on the characterstc method. A par of equatons can be developed to fnd the pressure head H and the velocty V n a ppe dvded nto N segments at the nteror pont P startng from pont 2 to pont N (pont 1 s related to the boundary condton), Larock et al. (2): V a 2 g ( V + V ) + ( H H ) P = Le R Le R H g a ( V V ) + ( H + H ) P = Le R Le R f t 2D a g ( V V + V V ) Le f t 2D Le R R ( V V V V ) Le Le R R (13) (14) Le and R are consdered as the left and rght ponts on the characterstc grd wth respect to a certan pont P and at the same dstance from t.

7 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt 3. IMPLEMENTATION OF GENETIC ALGORITHMS OVER PIPE NETWORK Genetc algorthms (GAs) are adaptve methods whch may be used to solve search and optmzaton problems. They are based on the genetc processes of bologcal organsms. Over many generatons, natural populatons evolve accordng to the prncples of natural selecton and "survval of the fttest. By mmckng ths process, genetc algorthms are able to "evolve" solutons to real world problems, f they have been sutably encoded. The basc prncples of GAs were frst lad down rgorously by Holland (1975) and Glodberg (1989). The flow chart n Fg. 1 shows the sequence of the basc operators used n genetc algorthms. We start out wth a randomly selected frst generaton. Every strng n ths generaton s evaluated accordng to ts qualty, and a ftness value s assgned. Next, a new generaton s produced by applyng the reproducton operator. Pars of strngs of the new generaton are selected and crossover s performed. Wth a certan probablty, genes are mutated before all solutons are evaluated agan. Ths procedure s repeated untl a maxmum number of generatons s reached. Whle dong ths, the all tme best soluton s stored and returned at the end of the algorthm. As one notces from the flow chart, the genetc algorthm serves as a framework whch provdes the outer cycle of the search or optmzaton process. An mportant part of the loop s the evaluaton functon whch determnes the ftness value of a specfc strng. Wthn ths method, the strng has to be mapped to a realstc soluton, and the objectve functon has to be evaluated. For ths, heurstc methods mght be necessary. The bref dea of GA s to select populaton of ntal soluton ponts scattered randomly n the optmzed space, then converge to better solutons by applyng n teratve manner the followng three processes (reproducton/selecton, crossover and mutaton) untl a desred crtera for stoppng s acheved. Mutaton Frst Generaton Crossover Evaluaton Reproducton No Max. Generaton? Yes Return best Soluton Fg. 1 Genetc algorthm flow chart

8 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt The optmzaton program GASTnet (Genetc Algorthm Steady Transent network) s wrtten n FORTRAN language and t lnks the GA, the Newton-Raphson smulaton technque for the steady state hydraulc smulaton and the transent analyss. The Newton-Raphson and transent analyzer are consdered as subroutnes n the man code genetc algorthms. A bref descrpton of the steps n usng GA for ppe network optmzaton, and ncludng water hammer s as follows: 1. Generaton of ntal populaton. The GA randomly generates an ntal populaton of coded strngs representng ppe network solutons of populaton sze Npopsz. Each of the Npopsz strngs represents a possble combnaton of ppe szes. 2. Computaton of network cost. For each Npopsz strng n the populaton, the GA decodes each substrng nto the correspondng ppe sze and computes the total materal cost. The GA determnes the costs of each tral ppe network desgn n the current populaton, as descrbed n Equaton (2). 3. Hydraulc analyss of each network. A steady state hydraulc network solver computes the heads and dscharges under the specfed demands for each of the network desgns n the populaton. The actual nodal pressures are compared wth the mnmum allowable pressure heads, and any pressure defcts are noted. In ths study, the Newton-Raphson technque s used. 4. Computaton of penalty cost for steady state. The GA assgns a penalty cost for each demand f a ppe network desgn does not satsfy the mnmum pressure constrants. The pressure volaton at the node at whch the pressure defct s maxmum, s used as the bass for computaton of the penalty cost. The maxmum pressure defct s multpled by a penalty factor ( C T / M ) as descrbed n Equaton (3). 5. Transent analyss of each network. A transent analyss solver computes the transent pressure heads resultng from the pump power falure, sudden valve closure or sudden demand change. The mnmum and maxmum pressure heads are estmated n each ppe of the network and compared wth the mnmum and maxmum allowable pressure heads, and any pressure defcts are noted. 6. Computaton of penalty cost for transent state. The GA assgns a penalty cost f a ppe desgn does not satsfy the mnmum and maxmum allowable pressure heads constrants. The penalty cost s estmated as the pressure volaton multpled by a penalty factor equals to the cost of the specfed ppe c ( D). L. 7. Computaton of total network cost. The total cost of each network n the current populaton s taken as the sum of the network cost (Step 2), the penalty cost (Step 4), plus the penalty cost (Step 6), ths step s an expresson to Eq. (1). 8. Computaton of the ftness. The ftness of the coded strng s taken as some functon of the total network cost. For each proposed ppe network n the current populaton, t can be computed as the nverse or the negatve value of the total network cost from Step Generaton of a new populaton usng the selecton operator. The GA generates new members of the next generaton by a selecton scheme.

9 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt 1. The crossover operator. Crossover occurs wth some specfed probablty of crossover for each par of parent strngs selected n Step The mutaton operator. Mutaton occurs wth some specfed probablty of mutaton for each bt n the strngs, whch have undergone crossover. 12. Producton of successve generatons. The use of the three operators descrbed above produces a new generaton of ppe network desgns usng Steps 2 to 11. The GA repeats the process to generate successve generatons. The last cost strngs (e.g., the best 2) are stored and updated as cheaper cost alternatves are generated. Generate Intal Populaton for Dameters (GA) Produce Optmzed Dameters Convert Optmzed Dameters to Commercal Dameters Newton Smulaton Analyze Gven Network Get Pressure Heads & Veloctes Materal Cost Penalty Cost Yes If : H H mn-st No Transent Analyss Get Mnmum and Maxmum Pressure Heads No Penalty Cost Yes If : H mn H mn-tr H max H max-tr Maxmum Generaton Ftness No Yes Comprse between produced groups of dameters to select the group that has the lower dameters cost Best Soluton Fg. 2 Flow chart of the GASTnet program

10 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt These steps for the optmzaton of water network consderng both steady state and transent condtons are llustrated n the flow chart of the GASTnet program, Fg. 2. Ths program s an extenson of the GANRnet computer program, Djebedjan et al. (25b). It has been developed to optmze ppe networks for steady state usng the genetc algorthm approach. The genetc algorthm n the GASTnet program has several parameters that enable movng to dfferent search regons to approach the global soluton; these parameters are: Npopsz: the populaton sze of a GA run, Idum: the ntal random number seed for the GA run, and t must equal a negatve nteger, Maxgen: the maxmum number of generatons to run by the GA, and Nposbl: the array of nteger number of possbltes per parameter. Case Study As llustrated n Fg. 3, a pre-defned water supply ppng network, Larock et al. (2), conssts of three reservors at nodes 1, 6 and 1, ten nodes, two pump statons and twelve ppes. The demands at nodes (3, 4, 5, 8 and 9) are (13, 9, 18, 45 and 13 gpm), respectvely. The lengths, dameters of ppes and Hazen-Wllams roughness coeffcents are gven n Table 1. The pumps data are gven n Table 2. The case study used n ths study s subjected to the followng causes of water hammer: pump power falure, valve sudden closure and sudden demand change. The set of commercally avalable ppe dameters are (6, 8, 1, 12, and 15) nches and the correspondng cost per foot length s (15, 25, 35, 45, and 65) unts, Table 3. Fg. 3 Typcal ppng network wth two pump statons

11 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt Table 1 Ppes data for the network wth two pump statons Ppe ID Start Node End Node L (ft) D (n.) C Wave Speed (ft/s) Table 2 Pumps data for the network wth two pump statons Pump Staton 1 Pump Staton 2 No. of Parallel Pumps 2 6 No. of Stages 2 1 N (r.p.m.) Rotatonal Moment of Inerta (lb.ft 2 ) Q (gpm) H (ft) Power (hp) Table 3 Network ppes unt cost Dameter (n.) Cost (Unts)

12 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt All calculatons were produced on a computer wth Pentum 4 (3. GHz) processor and 512 MB of RAM. Theoretcally, the requred mnmum pressure head at all nodes was assumed to be 8 ft for the steady state and for the transent condtons, the mnmum and maxmum pressure heads were consdered as 8 ft and 18 ft, respectvely. The GA parameters used n GASTnet optmzaton program n ths case were: Npopsz = 5, Idum = 5, Maxgen = 1 and Nposbl = 16. Mutaton and crossover rates were set to.2 and.5, respectvely. For the steady state calculatons, the accuracy was.1 ft 3 /s. The tme of the transent flow smulaton was taken as 4 s and the hydraulc tme step t was.4 s. The GASTnet program has the capablty to run under the followng modes: 1- Steady State-Smulaton Mode: It uses the hydraulc analyss of network to obtan the flows n ppes and the heads t nodes under steady state condtons. 2- Transent-Smulaton Mode: After the applcaton of the steady state condtons, the water hammer cause s appled and the transent smulaton s carred out gvng the pressure head aganst tme varaton at nodes. 3- Transent-Optmzaton Mode: In ths mode, the GA chooses a set of dameters and the prevous two smulatons are appled and checked by the pressure head requrements. Ths procedure s repeated to a maxmum number of generatons, Maxgen, and the best set of dameters gvng the least cost s selected. The subject network s predefned one, meanng that network has ts own characterstcs (ppe dameters, layout ). The GASTnet program was appled on the predefned network n a transent-smulaton mode, to delneate the effcency of the GASTnet program before optmzaton, then the GASTnet program was appled n the transent-optmzaton mode. All the program outputs (pressure aganst tme) were plotted on one dagram for each node before and after optmzaton. 1. Two Pumps Power Falure The network of Fgure 3 s used to demonstrate the effect of water hammer event by pump staton power falure n the two statons. The results of the water hammer event ntated separately by shuttng down each one can be found n AbdelBary (28). The GASTnet optmzaton program was appled and the network wth the new optmal dameters was found. Fgure 4 depcts the evoluton of the soluton as the program develops n a sngle run. A qute slow decrease n the cost value for the frst group of evaluaton then fast changes n the later evaluatons s observed.

13 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt 6 56 Cost (unt) Evaluaton Number Fg. 4 Cost unts versus evaluaton number for the two pump power falure The network contans 12 ppes and wth 5 avalable commercal ppe szes, the avalable number of solutons s 5 12 = Applyng the GASTnet program, t s found that the number of functon evaluatons was 4669 to reach the optmal soluton. It s very mnor value compared to the total soluton space (.19%). Table 4 shows the optmal dameters for the network aganst the orgnal ones. The least cost s 48,5. unts after optmzaton aganst 437,5. unts, whch ndcates.934 of the orgnal ppe network cost. Table 5 dsplays the subsequent nodal pressure heads for the steady state. These values are fulfllng the mnmum pressure constrant of 8 ft at all nodes except the reservors nodes. The three reservors at nodes 1, 6 and 1 have heads of 65, 2 and 35 ft, respectvely.

14 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt Table 4 Optmal aganst orgnal dameters (n.) and assocated cost for the two pumps power falure Table 5 Pressure heads at nodes for the steady usng the optmal dameters Ppe Number Orgnal Dameter Optmal Dameter (n.) (n.) Node Pressure Head (ft) Pump 1 n Ppe 9: Dscharge = gpm, Head = ft Cost (unts) 437,5. 48,5. Pump 2 n Ppe 1: Run Tme (mn) 7 3 Dscharge = gpm Head = ft The applcaton of the GASTnet program n transent-smulaton mode usng the orgnal network ppes dameters, Table 1, whch subjected to the same water hammer cause reveals the dashed curves n Fg. 5. It s apparent that the pressure heads at nodes are quanttatvely affected by the two pump statons power falure. The pressure fluctuatons exceed the maxmum pressure (18 ft) at nodes 8 and 9 and decrease below the lower lmt (8 ft) n node 6. The effect of the two pump statons power falure n the ppe network wth the optmal dameters s realzed after the applcaton of the GASTnet program n transentoptmzaton mode. The contnuous curves n Fg. 5 show the pressure head versus tme response at all nodes ncludng the reservor nodes. After optmzaton, the pressure fluctuatons at nodes 8 and 9 became wthn the acceptable range (8 18 ft).

15 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt Node 1 Before Optmzaton Node 2 Before Optmzaton Node 3 Before Optmzaton Node 6 (Pump Dscharge) Before Optmzaton Node 4 Before Optmzaton Node 5 Before Optmzaton Node 7 Before Optmzaton Node 8 Before Optmzaton Node 9 Before Optmzaton Fg. 5 Pressure head versus tme for varous nodes for the two pumps power falure 2. Sudden Valve Closure One of the most mportant water hammer causes s the sudden valve closure. In ths secton, the effect of a sudden valve closure located at the downstream end of ppe 2 n the network n Fg. 3 s studed. The same values for steady state, transent condtons, and GA parameters were used. As mentoned above, the mnor loss due to the valve s neglected.

16 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt Fgure 6 depcts the evoluton of the soluton as the GASTnet program develops n transent-optmzaton mode. The cost s decreased gradually over the fnal evaluaton number tll reachng to the least cost. The total soluton space s 5 12 = dfferent network desgns. Usng the GA optmzaton technques, the number of functon evaluatons was 1,8 to reach the optmal soluton and ths s only a very small fracton of the total search space (.44%) Cost (unt) Evaluaton Number Fg. 6 Cost unts versus evaluaton number for the sudden valve closure case Table 6 shows the optmal dameters for the network aganst the orgnal ones. The least cost s 495,5. unts after optmzaton aganst 437,5. unts, whch s equal tmes the orgnal cost. Here, the cost s not consdered as a domnant factor as t s meanngless to desgn a cheap non-relable network; the optmum cost exceeded the orgnal cost desgn, that n order to overcome the water hammer event. From Table 6, t can be noted that the dameter found by the genetc algorthm for ppe number 1 s 6 n. whch s very small. If the ppes connected to the reservors at nodes 1, 6 and 1 were not ncluded n the optmzaton process by gvng ther dameters a pre-specfed value and by treatng them as constrants; then these ppe dameters reman as specfed and the pressure head can be kept at hgher values. In ths network optmzaton, all dameters were searched and the optmal ppe dameters for some ppes whch are connected to the reservors were small. Table 7 dsplays the correspondng nodal pressure heads for the steady state. These heads fulfll the network requrements.

17 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt Table 6 Optmal aganst orgnal dameters (n.) and assocated cost for the sudden valve closure Table 7 Pressure heads at nodes for the steady state usng the optmal dameters Ppe Number Orgnal Dameter Optmal Dameter (n.) (n.) Node Pressure Head (ft) Pump 1 n Ppe 9: Dscharge = gpm, Head = ft Cost (unts) 437,5. 495,5. Pump 2 n Ppe 1: Dscharge = gpm Run Tme (mn) 2 25 Head = ft The GASTnet program was appled twce: n transent-smulaton mode and transentoptmzaton mode to demonstrate the dfferences n smulaton before and after optmzaton. For the case before optmzaton, the results of smulaton were not plotted as dashed curves n Fg. 7. The GASTnet program ceases the runnng operaton at tme t =.4 s due to the non stablty of the network under the water hammer event caused by the sudden valve closure downstream ppe 2. The operaton halt has occurred as an evdence of the nstablty and non-operablty of the network wth the orgnal set of dameters that n case of sudden valve closure. As depcted n Fgure 7, the pressure head versus tme response at all nodes are plotted. The convergence to steady state caused by sudden valve closure s rapd. The pressure heads at the nodes are quanttatvely affected by the sudden valve closure; the more quanttatvely affected node s node 2. The choce of the tme of the transent flow smulaton as 4 s was suffcent to obtan nearly steady state condton.

18 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt Node 1 (Pump Dscharge) Node Node Node Node 5 4 Node 6 (Pump Dscharge) Node Node Node 9 Fg. 7 Pressure head versus tme for varous nodes for the sudden valve closure 3. Sudden Demand Change Sudden demand change event s one of the water hammer causes. It s ntroduced on the network of Fg. 3. The demand was changed at node 5 from 18 GPM to 2 GPM to meet a sudden need for more water for fre suppresson. For ths example and as gven n the prevous examples for ths network, the requred mnmum pressure head at all nodes was 8 ft for the steady state and for the transent condtons, the mnmum and maxmum pressure heads were 8 ft and 18 ft, respectvely. The accuracy of the steady state calculatons was.1 ft 3 /s. The tme

19 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt of the transent flow smulaton was taken as 4 s and the hydraulc tme step t was.4 s. The evoluton of the soluton s depcted n Fgure 8. A rapd decrease n the cost value for the frst group of evaluaton then slow changes n the later evaluatons s observed Cost (unt) Evaluaton Number Fg. 8 Cost unts versus evaluaton number for the sudden demand change The total soluton space s 5 12 = dfferent network desgns. The number of functon evaluatons was 4327 to reach the optmal soluton whch s.177% of the total search space. Table 8 shows the optmal dameters for the network aganst the orgnal ones. The least cost s 334,5. unts after optmzaton aganst 437,5. unts for the orgnal network, whch s equal.765 tmes the orgnal cost. Table 9 dsplays the correspondng nodal pressure heads for the steady state. These heads fulfll the mnmum pressure constrant of 8 ft at all nodes except the reservors nodes.

20 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt Table 8 Optmal aganst orgnal dameters (n.) and assocated cost for the sudden demand change Table 9 Pressure heads at nodes for the steady state usng the optmal dameters Ppe Number Orgnal Dameter Optmal Dameter (n.) (n.) Node Pressure Head (ft) Pump 1 n Ppe 9: Dscharge = gpm Head = ft Cost (unts) 437,5. 334,5. Pump 2 n Ppe 1: Dscharge = gpm Run Tme (mn) 6 25 Head = ft The operablty and relablty of the orgnal network was checked usng the GASTnet optmzaton program n transent-smulaton mode usng the orgnal network ppes dameters, Table 1. The program ceased ts runnng operaton after tme t =.4 s therefore the results before optmzaton were not plotted n Fg. 9. The GASTnet optmzaton program results n transent-optmzaton mode are llustrated n Fgure 9. The pressure head versus tme response at all nodes are shown and t can be observed that the convergence to steady state caused by sudden demand change s rapd. It s obvous that the pressure heads at the nodes are affected by the sudden demand change. The smulaton tme was suffcent to obtan nearly steady state condton. It s concluded that for the orgnal ppe network, the ppng network s not operatonal under sudden demand change whch means that ppng could not sustan such changes although the demand changed only from 18 to 2 gpm,.e. by ncreasng about 11%. The applcaton of the GA technques converted the non-operatonal ppe network to operatonal one by the proper ppes dameters selecton.

21 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt Node 1 (Pump Dscharge) Node Node Node Node 5 4 Node 6 (Pump Dscharge) Node Node Node 9 Fg. 9 Pressure head versus tme for varous nodes for the sudden demand change 4. CONCLUSIONS The optmzaton of water dstrbuton systems under transent state s a complex problem. In ths paper the optmzaton s acheved by obtanng the optmal ppe dameter consderng both steady and water hammer. The orgnal ppng networks are not operatonal or sufferng from excessve pressure fluctuatons under water hammer crcumstances. The genetc algorthm (GA) s utlzed to fnd optmal ppe dameters n a case study wth encountered water hammer causes. The case study shows that

22 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt consderng steady state desgn only s nsuffcent when there s a possblty for water hammer events. For that reason and dependng on the encountered transent condtons, the approprate szng of ppe dameters can be selected for preventng water hammer. The selecton of optmal ppe dameters s mportant to system performance under steady and transent states and decreasng costs. The applcaton of the GASTnet optmzaton program reveals the followng benefts: GASTnet optmzaton program has been successfully appled on a case study wth 3 causes of water hammer provng hgh performance wth least cost avodng usng of water hammer arrestors. The applcaton of GASTnet program to the examples demonstrates the capablty of the GA to fnd the optmal ppe dameters n a small fracton of the total search space and a reasonable run tme n spte of the complcated behavor of flud transents. The applcaton of GA technque n GASTnet optmzaton program fnds the optmal dameters for the transent state and the nodal pressure fluctuatons fallen between the water hammer predetermned lmts and converges rapdly to the steady state case. Ths proves the valdaton of the ntegrated program water hammer analyzer and GA technque. The cost savng has been ncreased by 6.6, 13.3 and 23.5% n the theoretcal Examples 1 through 3, respectvely. Not necessarly the optmum cost to be the lowest, such as Example 2. The cost was slghtly ncreased, but ths s not a factor as the orgnal network was not functonal. REFERENCES AbdelBary, A.G.M., Optmzaton of Water Dstrbuton Systems Subjected to Water Hammer Usng Genetc Algorthms, M.Sc. Thess, Faculty of Engneerng, Mansoura Unversty, Egypt, 28. Djebedjan, B., "Relablty-Based Water Network Optmzaton for Steady State Flow and Water Hammer," Proceedngs of IPC26, 6th Internatonal Ppelne Conference, September 25-29, 26, Calgary, Alberta, Canada, IPC Djebedjan, B., Herrck, A., and Rayan, M.A., "Modelng and Optmzaton of Potable Water Network", Internatonal Ppelne Conference and Technology Exposton 2, October 1-5, 2, Calgary, Alberta, Canada, Paper IPC Djebedjan, B., Mondy, A., Mohamed, M.S., and Rayan, M.A., "Network Optmzaton for Steady Flow and Water Hammer usng Genetc Algorthms," Proceedngs of the Nnth Internatonal Water Technology Conference IWTC 25, Sharm El-Shekh, Egypt, 17-2 March, 25a, pp Djebedjan, B., Yaseen, A., and Rayan, M.A., "A New Adaptve Penalty Method for Constraned Genetc Algorthm and ts Applcaton to Water Dstrbuton Systems,"

23 Thrteenth Internatonal Water Technology Conference, IWTC13 29, Hurghada, Egypt Proceedngs of the Nnth Internatonal Water Technology Conference IWTC 25, Sharm El-Shekh, Egypt, 17-2 March, 25b, Djebedjan, B., Yaseen, A., and Rayan, M.A., "A New Adaptve Penalty Method for Constraned Genetc Algorthm and ts Applcaton to Water Dstrbuton Systems," Proc. of Internatonal Ppelne Conference and Exposton 26, Calgary, Alberta, Canada, September, 26, IPC Goldberg, D.E., Genetc Algorthms n Search, Optmzaton and Machne Learnng, Addson-Wesley Readng, Mass., Holland, J.H., Adaptaton n Natural and Artfcal Systems, MIT Press, Jung, B.S., and Karney, B.W., "Flud Transents and Ppelne Optmzaton usng GA and PSO: the Dameter Connecton," Urban Water Journal, Vol. 1, No. 2, June, 24a, pp Jung, B.S., and Karney, B.W., "Transent State Control n Ppelnes usng GAs and Partcle Swarm Optmzaton," Proc. of 6th Internatonal Conference on Hydronformatcs, Sngapore, June, 24b. Jung, B.S., and Karney, B.W., "Optmum Selecton of Hydraulc Devces for Water Hammer Control n the Ppelne Systems usng Genetc Algorthm," 4th ASME_JSME Jont Fluds Engneerng Conference, Honolulu, Hawa, USA, July 6-11, 23, Paper FEDSM Kaya, B., and Güney, M.S., "An Optmzaton Model and Water-Hammer for Sprnkler Irrgaton Systems," Turksh Journal of Engneerng and Envronmental Scences, Vol. 24, 2, pp Kessler, A., and Shamr, U., "Analyss of the Lnear Programmng Gradent Method for Optmal Desgn of Water Supply Networks," Water Resources Research, Vol. 25, No. 7, 1989, pp Lane, D.A., and Karney, B.W., "Transent Analyss and Optmzaton n Ppelne A Numercal Exploraton," 3rd Internatonal Conference on Water Ppelne Systems, Edted by R. Chlton, 1997, pp Larock, B.E., Jeppson, R.W., and Watters, G.Z., Hydraulcs of Ppelne Systems, CRC Press LLC, New York, 2. Sârbu, I., and Kalmár, F., "Optmzaton of Looped Water Supply Networks," Perodca Polytechnca Ser. Mech. Eng., Vol. 46, No. 1, 22, pp Wyle, E.B., Streeter, V.L., and Suo, L., Flud Transents n Systems, Englewood Clffs, New Jersey, U.S.A., Zhang, Z., "Flud Transents and Ppelne Optmzaton usng Genetc Algorthms," Master Thess, Unversty of Toronto, Canada,