A NEW ADAPTIVE PENALTY METHOD FOR CONSTRAINED GENETIC ALGORITHM AND ITS APPLICATION TO WATER DISTRIBUTION SYSTEMS

Transcription

1 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt 077 A NEW ADAPTIVE PENALTY METHOD FOR CONSTRAINED GENETIC ALGORITHM AND ITS APPLICATION TO WATER DISTRIBUTION SYSTEMS Berge Djebedjan *, Ashraf Yaseen **, and Magdy Abou Rayan * * Mechancal Power Engneerng Department, Faculty of Engneerng, Mansoura Unversty, El-Mansoura, Egypt ** Dametta Drnkng Water Company, Dametta, Egypt E-mal: bergedje@mans.edu.eg, ashrafsayedyas@yahoo.com, mrayan@mans.edu.eg ABSTRACT Ths paper presents a new adaptve penalty method for genetc algorthms (GA). External penalty functons have been used to convert a constraned optmzaton problem nto an unconstraned problem for GA-based optmzaton. The success of the genetc algorthm applcaton to the desgn of water dstrbuton systems depends on the choce of the penalty functon. The optmal desgn of water dstrbuton systems s a constraned non-lnear optmzaton problem. Constrants (for example, the mnmum pressure requrements at the nodes) are generally handled wthn genetc algorthm optmzaton by ntroducng a penalty cost functon. The optmal soluton s found when the pressures at some nodes are close to the mnmum requred pressure. The goal of an adaptve penalty functon s to change the value of the penalty drawdown coeffcent durng the search allowng exploraton of nfeasble regons to fnd optmal buldng blocks, whle preservng the feasblty of the fnal soluton. In ths study, a new penalty coeffcent strategy s assumed to ncrease wth the total cost at each generaton and nversely wth the total number of nodes. The applcaton of the computer program to case studes shows that t fnds the least cost n a favorable number of functon evaluatons f not less than that n prevous studes and t s computatonally much faster when compared wth other studes. Key Words: Genetc Algorthm, Penalty Functon, Water Dstrbuton Systems INTRODUCTION A water dstrbuton system conssts of elements such as ppes, tanks, reservors, pumps, and valves etc. They are an essental part of all water supply systems. The cost of ths porton of any szable water supply scheme amount to more than 60% of the entre cost of the project. Also, the energy consumed n a dstrbuton network suppled by pumpng may exceed 60% of the total energy consumpton of the system []. Water dstrbuton system desgn optmzaton s one of the most heavly researched areas n the hydraulcs professon. The optmzaton of ppe networks has been studed and

2 078 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt varous researchers have proposed the use of mathematcal programmng technques n order to dentfy the optmal soluton for water dstrbuton systems. Hundreds of papers and reports on approaches have been developed over the past few decades. The optmzaton technques can be categorzed as follows: - Determnstc optmzaton technques (lnear, non-lnear, dynamc and mxed nteger programmng ) - Stochastc optmzaton technques (Genetc Algorthms, Smulated Annealng, GLOBE, Shuffled Complex Evoluton and Shuffled Frog Leapng Algorthms). For the determnstc optmzaton technques, Alperovts and Shamr [2] proposed a lnear programmng gradent (LPG) n optmzng water dstrbuton network and Kessler and Shamr [] presented two stages LPG method. Eger et al. [4] used the same formulaton used n [] and solved the problem usng a nonsmooth branch and bound algorthms and dualty theory. Other developments n LPG are used n Sonak and Bhave [5] and Sârbu and Borza [6]. Nonlnear programmng (NLP) technque was developed and appled by Saman and Naeen [7], Djebedjan et al. [8] and Sârbu and Kalmár [9]. The stochastc optmzaton methods deal wth a set of ponts smultaneously n ts search for the global optmum. The search strategy s based on the objectve functon. Smpson et al. [0] used smple Genetc Algorthms (GA). The smple GA was then mproved by Dandy et al. [] usng the concept of varable power scalng of the ftness functon, an adjacency mutaton operator, and gray codes. Savc and Walters [2] also used smple GA n conjuncton wth EPANET network solver. Abdel-Gawad [] studed the effect of dfferent selecton, crossover and mutaton schemes of the GA on the network optmzaton. Instead of usng a sngle optmzaton algorthm, Abebe and Solomatne [4] appled GLOBE that comprses several search algorthms and dentfed that very few algorthms reach to optmal or near optmal solutons. Many other researches on the water dstrbuton network optmzaton usng GA s can be found n Lppa et al. [5], Gupta et al. [6], Varavamoorthy and Al [7] and Wu and Smpson [8]. Cunha and Sousa [9] ntroduced the Smulated Annealng (SA) that s based on the analogy wth the physcal annealng process wth Newton search method to solve the network equatons. Eusuff and Lansey [20] proposed the Shuffled Frog Leapng Algorthms (SFLA). Long and Atquzzaman [2] used the Shuffled Complex Evoluton (SCE) lnked wth EPANET network solver to dentfy the least cost of some water dstrbuton ppe networks. The orgnal SCE algorthm s modfed to accommodate hgher number decson varables; and the decson varables (ppe szes) are converted to commercally avalable dameters n determnng the cost of the network. In the present nvestgaton, a mcro-genetc algorthm s appled for ppe network optmzaton. An adaptve penalty functon s used to change the value of the penalty coeffcent durng the search allowng exploraton of nfeasble regons to fnd optmal

3 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt 079 buldng blocks, whle preservng the feasblty of the fnal soluton. The Newton- Raphson method s utlzed for the hydraulc analyss of the network. The approach s appled to two water dstrbuton networks to demonstrate ts effcency and effectveness. OPTIMIZATION MODEL FORMULATION The water dstrbuton network optmzaton ams to fnd the optmal ppe dameters n the network for a gven layout and demand requrements. The optmal ppe szes are selected n the fnal network satsfyng all mplct constrants (e.g. conservatons of mass and energy), and explct constrants (e.g. hydraulc and desgn constrants). The objectve functon s the total cost of the gven network. The total cost calculated as: C T s T N = = ( D ) L C = c. () where N s the total number of ppes, c ( D ) the cost of ppe wth dameter D per unt length and L s the length of ppe. The objectve functon s to be mnmzed under the mplct constrants and explct constrants. The mplct constrants are fulflled as follows. The conservaton of mass states that the dscharge nto each node must be equal to that leavng the node, except for storage nodes (tanks and reservors). Ths secures the overall mass balance n the network. For a total number of nodes M n the network, ths constrant can be wrtten as: where M j= Q = 0 (2) j Q j represents the dscharges nto or out of the node j (sgn ncluded). The second mplct constrant s the conservaton of energy accordng to whch the total head loss around any loop must equal to zero or s equal to the energy delvered by a pump f there s any: where h f = E p () h f s the head loss due to frcton n a ppe and E p s the energy suppled by a pump. Ths embeds the fact that the head loss n any ppe, whch s a functon of ts dameter, length and hydraulc propertes, must be equal to the dfference n the nodal heads.

4 080 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt Dfferent forms for the head loss formula have been developed for practcal ppe flow calculatons. In ths study, the head loss h n the ppe s expressed by the Hazen- Wllams formula: Q f L h f = (4) C D where Q s the ppe flow (m /s), C s the Hazen-Wllams coeffcent, dameter (m), and L s ppe length (m). D s ppe The explct constrants are the desgn and hydraulc constrants. The desgn constrants (the ppe dameter bounds (maxmum and mnmum)) and the hydraulc constrants (the flud velocty bounds and the pressure head bounds at each node) are gven respectvely as: D mn D D =,..., N (5) max V mn V V max =,..., N (6) H j,mn H j H j =,..., M (7) j, max where V s the flud velocty n ppe, H j s the pressure head at node j, and and H are the mnmum and maxmum allowable pressure heads at node j. j, max H j, mn The Newton-Raphson method, [22], s used to smulate hydraulcally the gven network. The technque used to solve the nonlnear set of equatons. The flow rates n each ppe are assumed whch satsfy contnuty, then they are corrected so that the sum of the head losses around each loop approaches zero. The equatons contanng the correcton factor are wrtten for each loop and ths nonlnear set of equatons s solved successvely for the fnal value of correcton factor n each loop. Then, the ntal flow rates n each ppe are adjusted to ther fnal values. NEW ADAPTIVE PENALTY METHOD The dea underlyng penalty functon methods s to transform the problem of mnmzng: z = f (x) (8) subject to certan constrants on x nto the problem of fndng the unconstraned mnmum of the objectve functon:

5 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt 08 Z = f ( x) + P( x) (9) where P( x ) s the penalty functon. It s desgned to penalze nfeasble soluton and to force the search towards the feasble soluton regon. For the network optmzaton, the functon f (x) s the total cost C T and the penalty cost C s used nstead of P( x ), so Eq. (8) yelds: P Z = C T + C P (0) The desgn constrants n ppe network optmzaton that wll be used n the penalty functon s the mnmum allowable hydraulc pressures at gven nodes as the dameter of each ppe s chosen from a specfed set of commercal ppes. When the pressure head condton at a node s not satsfed, then a penalty cost s added at that node, C P = M j= C Pj M ( H j,mn H j ) = c. () j= where c s the penalty draw-down coeffcent. Choosng the penalty coeffcent values for a penalty functon s often arbtrary, [2]. A small coeffcent wll mpose a smaller penalty than a large coeffcent for the same magntude of constrant volaton. In the GA, a large penalty can quckly elmnate nfeasble solutons from the search, whch may contan schemas that are key elements of the optmal soluton. Conversely, usng a small coeffcent may allow the survval of nfeasble desgns to the extent that the populaton converges at an nfeasble pont as the optmal ftness soluton. Clearly, a compromse must be struck between these two extremes. The goal of an adaptve penalty functon s to change the value of the penalty coeffcent durng the search allowng exploraton of nfeasble regons to fnd optmal buldng blocks, whle preservng the feasblty of the fnal soluton, [2]. Crossley and Wllams [2] defned three basc forms of draw-down coeffcent strateges: constant penalty coeffcent, generaton number-based strateges (ncreasng the value of c wth successve generatons) and populaton ftness-based strateges (usng the standard devaton and the varance of the populaton s ftness values). The penalty costs used n the lterature take many forms, [2], [4] and [8]. For C p. C Max H H P = max. j, mn j to M example, Abebe and Solomatne [4] defned t as: ( ) where p s the penalty cost coeffcent, C max s the maxmum possble cost that the network can have, t s calculated based on the largest commercal ppe avalable. The penalty cost coeffcent p must be selected carefully to provde a smooth transton from nfeasble to feasble desgns. In ths study, a new penalty coeffcent strategy s assumed. The dea s to use a coeffcent whch depends on the total cost of network calculated at each generaton j

6 082 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt and the total number of nodes. It s based on the mportant parameters havng effect on the penalty cost. These parameters are the total cost C T and the total number of nodes M. As the total cost s very large compared to the mnmum cost; the penalty cost should be ncreased by the same amount. Also, as the number of nodes ncreases n large networks ts mportance on the penalty cost should be decreased. Consequently, the coeffcent can be assumed to ncrease wth the total cost at each generaton and nversely wth the total number of nodes. Then, the fnal form of the coeffcent s c = C M. T The new form of the penalty coeffcent can be ntroduced n a dfferent manner. Neglectng the effect of ppe length and treatng t as a devce, then the average cost of each ppe s ( C T N ). Smlarly, assumng that each node plays the same role n the network, the average node cost s defned as ( C T M ). Ths term s multpled to the nodal pressure head condton to obtan the penalty cost. Applyng the prevous approach, the penalty cost s wrtten as: C P CT = M M ( H j,mn H j ). (2) j= and the objectve functon s calculated from: CT f H j,mn H j 0 Z = M () CT + ( H j,mn H j ) else M j= The penalty cost s appled at the nodes where the pressure head at node s less than the mnmum allowable pressure head at the same node. The present adaptve penalty method has several advantages such as: It does not contan any constant values. The penalty cost s functon of the total cost, the number of nodes, and the node pressure head. It s fast to reach the global optmzaton. It decreases the number of evaluatons. GENETIC ALGORITHMS Genetc algorthms are search technques based on the concepts of natural evoluton and thus ther prncples are drectly analogous to natural behavor, Gen and Cheng [24]. 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

7 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt 08 teratve manner the followng three processes (reproducton/selecton, crossover and mutaton) untl a desred crtera for stoppng s acheved. The mcro-genetc Algorthm (µga), Krshnakumar [25], s a "small populaton" GA. In contrast to the Smple Genetc Algorthm, whch requres a large number of ndvduals n each populaton (.e., 0-200), the µga uses a small populaton sze. The optmzaton program s wrtten n FORTRAN language and called (GANRnet) as t depends on GA and Newton-Raphson smulaton technques. A bref descrpton of the steps n usng GA for ppe network optmzaton s as follows, Smpson et al. [26]:. Generaton of ntal populaton. The GA randomly generates an ntal populaton of coded strngs representng ppe network solutons of populaton sze N. Each of the N strngs represents a possble combnaton of ppe szes. 2. Computaton of network cost. For each N 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.. Hydraulc analyss of each network. A steady state hydraulc network solver computes the heads and dscharges under the specfed demand patterns 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. The GA assgns a penalty cost for each demand pattern 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, whch s a measure of the cost of a defct of one unt of pressure head. 5. 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) plus the penalty cost (Step 4). 6. 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. 8. 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.

8 084 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt 0. Producton of successve generatons. The use of the three operators descrbed above produces a new generaton of ppe network desgns usng Steps 2 to 9. The GA repeats the process to generate successve generatons. The last cost strngs (e.g., the best 20) are stored and updated as cheaper cost alternatves are generated. These steps are llustrated n the flow chart of the GANRnet program, Fg.. Produce Optmzed Dameters No Convert Optmzed Dameters to Commercal Dameters Newton-Raphson Smulaton Analyze Gven Network Get Pressure Heads & Veloctes Maxmum Generaton Ftness Materal Cost Penalty Cost No If : H H mn V mn V V max Yes Yes Comprse between produced groups of dameters to select the group that has the lower dameters cost Fg.. Flow chart of the GANRnet program Best Soluton APPLICATION TO WATER DISTRIBUTION SYSTEMS The GANRnet program was appled to two case studes and the results were compared wth other optmzaton methods and GA's. The hydraulc analyss results of the GANRnet program were compared wth the EPANET (Rossman [27]) computer program. The EPANET program employs the "gradent method" (Todn and Plat [28]) whch was found to be the most sophstcated drect equaton solvng algorthm presented n the lterature. EPANET s avalable n the publc doman, so t s used to check the hydraulc soluton accuracy of the GANRnet. The genetc algorthm n the GANRnet 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. The GA parameters are gven n the two case studes.

9 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt 085 Case Study : Two-Loop Network The frst case study s gravty fed two-loop network wth 8 ppes, 7 nodes and one constant head reservor. The layout of the network, the lengths of ppes and the node data are shown n Fg. 2. The two-loop network problem s orgnally presented by Alperovts and Shamr [2] and taken as a model network by many researchers. All the ppes are 000 m long and the Hazen-Wllams coeffcent s assumed to be 0 for all the ppes. The demands are gven n cubc meters per hour and the mnmum acceptable pressure requrement for each node s 0 m above the ground level. There are 4 commercally avalable ppe dameters and Table 2 presents the total cost (n arbtrary unts) per meter of ppe length for dfferent ppe szes. 20 m 00 m /h (60 m) 2 2 L = 000 m 00 m /h (50 m) L = 000 m 270 m /h (50 m) 200 m /h (60 m) L = 000 m 20 m /h L = 000 m L = 000 m 6 L = 000 m 0 m /h L = 000 m (55 m) L = 000 m (65 m) Fg. 2. The two-loop network (Case )

10 086 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt Table. Cost data for the two-loop network (Case ) Dameter (n) Cost (unts) The GANRnet program was appled to the two-loop network usng the followng values for µga parameters: Npopsz = 2, Idum = 220, Maxgen = 62 and Nposbl = 2. The mutaton and crossover rates were set to 0.2 and 0.5, respectvely. Fgure depcts the evoluton of the soluton as the GANRnet develops n a sngle run. A rapd decrease n the cost value for the frst group of evaluaton then qute slow changes n the later evaluatons s observed. Cost (Unts) Evaluaton Number.. Fg.. Cost evoluton (Case )

11 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt 087 Table 2 lsts the optmal network solutons, total network cost, number of functon evaluatons (F.E.N.), and the run tme. It s mportant to note that some of the prevous studes dealt wth splt-ppe solutons. These solutons are not ncluded n ths table. The mnmum nodal head requrement s not volated n all cases mentoned n Table 2. The two-loop network contanng 8 ppes and wth 4 avalable commercal ppe szes has a total soluton space of 4 8 = dfferent network desgns, thus t s dffcult to optmze t. Usng the optmzaton technques, t can be observed from Table 2 that only a small fracton of the total search space s searched (.e. F.E.N.) by each algorthm to reach the optmal soluton. The GA solutons of the present study are smlar to the least cost solutons (49,000 unts) obtaned by Savc and Walters [2], Abebe and Solomatne [4], Cunha and Sousa [9], Eusuff and Lansey [20], and Long and Atquzzaman [2]. The comparson between the dfferent methods of optmzaton and the present study yelds that the present study reaches to the least cost solutons faster than the other methods. It converges only after 74 evaluatons wth a computatonal tme of 2 seconds. The pressure at each node calculated by GANRnet and the EPANET are shown n Table. The results from the GANRnet (Newton-Raphson technque) are lower than that of EPANET wthn the acceptable accuracy. Ppe Number Table 2. Results of the two-loop network (Case ) Savc and Walters [2] Abebe and Solomatne [4] Ppe Dameter (n) Cunha and Sousa [9] Eusuff and Lansey [20] Long and Atquzzaman [2] Present Study GA GA2 SA SFLA SCE GA Cost 49, ,000 49,000 49,000 49,000 49,000 49,000 F.E.N. * 65,000 65,000,7 25,000,2,09 74 Run Tme 0 mn 0 mn 7 mn 40 sec / 8 sec 2 sec ** * F.E.N.: Functon Evaluaton Number ** Run tme n present study s produced on a computer wth Pentum 4 (.7 GHz) processor

12 088 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt Table. Node pressure head (m) for the two-loop network (Case ) Nodes EPANET Smulaton Newton Raphson Smulaton Case Study 2: Hano Network The second case study s the water dstrbuton trunk network n Hano, Vetnam, Fg. 4. The data are gven by Fujwara and Khang [29] and summarzed n Table 4. Ths network conssts of one reservor (node ), demand nodes and 4 ppes. The mnmum pressure head requred at each node s 0 m. The set of commercally avalable ppe dameters (n nches) s (2, 6, 20, 24, 0, and 40) and ther unt cost s gven n [29] as: C =. D.5 n whch C s the cost per meter length n dollars and D s the ppe dameter n nches. The Hazen-Wllams coeffcent for all lnks s 0. Fg. 4. Hano network (Case 2)

13 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt 089 The GANRnet program was appled to Hano network. Two runs (trals) were performed usng dfferent values for µga parameters: Tral : Npopsz = 7, Idum = 400, Maxgen = 45 and Nposbl = 2. Tral 2: Npopsz = 2, Idum = 00, Maxgen = 7 and Nposbl = 2. The mutaton and crossover rates were set to 0.2 and 0.5, respectvely. Fgure 5 llustrates the cost evoluton for the two trals. In the second tral, the decrease n the cost value for the frst group of evaluaton s slower than that of the frst tral. Table 4. Data for Hano network (Case 2) Node Source Demand (m /h) 9, ,005, ,45 60, , Ppe Start Node End Node Length (m) 00.00, ,50.00, ,200.00, ,70.00, ,200.00, ,650.00,20.00, , ,000.00,

14 090 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt Cost ($) Cost ($) Evaluaton Number Evaluaton Number Tral Tral 2 Fg. 5. Cost evoluton (Case 2) It should be mentoned that some of the prevous studes dealt wth splt-ppe solutons. Ths knd of splt-ppe desgn s less realstc, [2] and are not allowed n the present study. Table 5 lsts solutons for the Hano network found n the lterature. The best network desgns obtaned by dfferent authors are gven n terms of cost (mllons of dollars) and selected dameters (nches). Alongsde these solutons, two solutons obtaned by GANRnet are presented for comparson. The total search space for ths network s 6 4 = 2.87 x 0 26 dfferent possble network desgns. Accordng to the number of functon evaluatons (F.E.N.) n Table 5, each algorthm reaches the optmal soluton n a very small percentage of all possble desgns. The comparson between the network hydraulc smulaton results of the present study wth that of other researchers can be acheved by dealng wth all the results of the dfferent algorthms by EPANET network solver. Ths method wll overcome the problem of the Hazen-Wllams formula constant, whch s a source of addtonal uncertanty assocated wth results obtaned, [2]. Takng the EPANET soluton as a reference soluton and usng the optmal dameters obtaned by each study n the EPANET solver, the resulted pressure heads are gven n Table 6. It can be observed that some of these solutons gve nfeasble solutons (H < 0m), such as Savc and Walters [2] - GA soluton and Cunha and Sousa [9]. These two solutons are dscarded, as the pressure head requrements are not fulflled although these pressure heads are near from feasblty.

15 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt 09 Ppe Number Table 5. Optmal dameters (n.) for Hano network (Case 2) Savc and Watters [2] Abebe and Solomatne [4] Ppe Dameter (n) Cunha and Sousa [9] Long and Atquzzaman Present Study [2] Tral Tral 2 GA GA2 GA ACCOL SA SCE GA GA Cost * F.E.N. / / 6,90,055 5,000 25,402 9,455 26,2 Run Tme hr hr 75 mn 5 mn 2 hr mn 4 sec** 45 sec** * Cost n mllons of Dollars. ** Run tme s produced on a computer wth Pentum 4 (.7 GHz) processor.

16 092 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt Node Number Table 6. Pressure head for Hano network (Case 2) Savc and Watters [2] Nodal Pressure (m) Abebe Cunha and and Solomatne Sousa [4] [9] Long and Atquzz aman [2] Present Study Tral Tral 2 GA GA2 GA ACCOL SA SCE GA GA * * * * * * * * * Infeasble soluton (pressure head s less than 0 m) when EPANET network solver s used Table 7 dsplays the correspondng nodal heads for the two trals of the present study obtaned as a result of smulaton by both EPANET and Newton-Raphson technque

17 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt 09 used n the GANRnet program. As prevously observed n Case, the pressure heads resulted from the Newton-Raphson technque are smaller than that of EPANET by 0.0 to 0.05 m whch s n the acceptable accuracy. Table 7. Pressure head for Hano network (Case 2) Node Pressure Head (m) Tral Tral 2 EPANET Newton- EPANET Newton- Raphson Raphson The results from Tables 5 to 7 yelds that usng the cost as the assessment of the qualty of the desgns, then the soluton by Cunha and Sousa [9] ($6.056 mllon) s superor among the results shown n Table 5. However, the pressure head requrement at nodes

18 094 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt, 6, 7, 27, 29 and 0 s not met. Savc and Walters [2] obtaned a slghtly greater network cost ($6.07 mllon) but the resultng pressure heads at nodes and 0 do not meet the head constrants, Table 6. Then accordng to the fulfllment of the pressure constrant, the best prevous soluton s the GA2 of Savc and Walters [2] ($6.95 mllon). It should be noted that the results of Varavamoorthy and Al [7] are smlar to that of Cunha and Sousa [9] and therefore are not shown n Table 6. The present study usng the GANRnet program and the new adaptve penalty method shows two new optmal costs. The fnal network cost, Tral ($6.27 mllon) requres 9,455 functon evaluatons and computatonal tme of only 4 sec and the other s Tral 2, ($6.20 mllon) and requres 26,2 functon evaluatons and computatonal tme of only 45 sec. These two optmal solutons whch are the lowest costs and wth the faster run tme compared to prevous studes make t possble to apply the GANRnet program to more complcated and actual networks. The mportance of the adaptve penalty coeffcent can be llustrated when t s compared wth the constant penalty coeffcent c, Eq. (). Fgure 6 compares the convergence rates of the GA optmzaton for two dfferent constant penalty coeffcent values (,000,000 and 0,000,000) and for the adaptve penalty coeffcent c = CT M. For these GA runs, all the parameters of the GA have the same values as that of Tral 2. The fgure shows that the optmalty of the fnal soluton depends on the penalty coeffcent value beng used. Also, the GA optmzaton wth the adaptve penalty coeffcent mproves the search process very quckly n the early stages of the optmzaton compared wth the constant penalty coeffcent. Ths ndcates that the adaptve penalty coeffcent guarantees a relable soluton and mproves the effcacy of GA optmzaton. CONCLUSIONS The determnaton of the optmal desgn for water dstrbuton networks s computatonally complex. In ths paper, a bnary-coded GA method coupled wth the Newton-Raphson technque (the hydraulc solver) s proposed for the optmal desgn of water dstrbuton systems. A new adaptve penalty functon s presented. It has many advantages manly t does not contan any constant values; the penalty cost s a functon of the total cost, the number of nodes, and the node pressure head. Also, t s fast to reach the global optmzaton and t decreases the computatonal tme. The method was tested on two networks, the two-loop and the Hano networks, and has been shown to be effcent and robust. For the two-loop network, the results compared wth that of other authors are favorable. For the Hano network, two solutons, whch are the lowest costs, are acheved.

19 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt Penalty Coeff. =,000,000 Penalty Coeff. = 0,000,000 Penalty Coeff. = CT/M Cost ($) Evaluaton Number Fg. 6. Comparson between the cost evoluton usng constant and adaptve penalty coeffcents (Case 2) Generally, the proposed method produces good results quckly although; t s not approprate to compare CPU tmes because of the dfferent computer platforms used n the prevous studes. The GANRnet program fnds the least cost n a favorable number of functon evaluatons f not less than that mentoned n the other researches. The adaptve penalty functon used n the GANRnet program helps to fnd least cost solutons n a small run tme. Ths makes t possble to apply t to more complcated and actual networks. NOMENCLATURE C Hazen-Wllams coeffcent of ppe C P penalty cost C T total cost c penalty draw-down coeffcent c ( D ) cost of ppe wth dameter D per unt length D dameter of ppe, (m) D max maxmum dameter, (m) D mn mnmum dameter, (m) E energy suppled by a pump, (m) p f (x) functon F.E.N. Functon Evaluaton Number H pressure head at node j, (m) j

20 096 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt H j,max maxmum allowable pressure head at node j, (m) H mnmum requred pressure head at node j, (m) h f j,mn Idum L M Maxgen N Npopsz Nposbl P( x ) Q Q V Z j head loss due to frcton n a ppe, (m) ntal random number seed for the GA run length of ppe, (m) total number of nodes n the network maxmum number of generatons to run by the GA total number of ppes populaton sze of a GA run array of nteger number of possbltes per parameter penalty functon flow n ppe, (m /s) dscharges nto or out of the node j, (m /s) flud velocty n ppe, (m/s) objectve functon ABBREVIATIONS ACCOL GA SA SCE SFLA Adaptve Cluster Coverng wth Local Search Genetc Algorthm Smulated Annealng Shuffled Complex Evoluton Shuffled Frog Leapng Algorthms REFERENCES [] Stephenson, D., Ppeflow Analyss, Elsever Scence Publshers B.V., 984. [2] Alperovts, E., and Shamr, U., "Desgn of Optmal Water Dstrbuton Systems," Water Resources Research, Vol., No. 6, 977, 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, 989, pp [4] Eger, G., Shamr, U., and Ben-Tal, A., "Optmal Desgn of Water Dstrbuton Networks," Water Resources Research, Vol. 0, No. 9, 994, pp [5] Sonak, V.V., and Bhave, P.R., "Global Optmum Tree Soluton for Sngle- Source Looped Water Dstrbuton Networks Subjected to a Sngle Loadng Pattern," Water Resources Research, Vol. 29, No. 7, 99, pp [6] Sârbu, I., and Borza, I., "Optmal Desgn of Water Dstrbuton Networks," Journal of Hydraulc Research, Vol. 5, No., 997, pp

21 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt 097 [7] Saman, H.M.V., and Naeen, S.T., "Optmzaton of Water Dstrbuton Networks," Journal of Hydraulc Research, Vol. 4, No. 5, 996, pp [8] Djebedjan, B., Herrck, A., and Rayan, M.A., "Modelng and Optmzaton of Potable Water Network," Internatonal Ppelne Conference and Technology Exposton 2000, October -5, 2000, Calgary, Alberta, Canada. [9] Sârbu, I., and Kalmár, F., "Optmzaton of Looped Water Supply Networks," Perodca Polytechnca Ser. Mech. Eng., Vol. 46, No., 2002, pp [0] Smpson, A.R., Dandy, G.C., and Murphy, L.J., "Genetc algorthms Compared to Other Technques for Ppe Optmsaton," Journal of Water Resources Plannng and Management, ASCE, Vol. 20, No. 4, July/August, 994, pp [] Dandy, G.C., Smpson, A.R., and Murphy, L.J., "An Improved Genetc Algorthm for Ppe Network Optmzaton," Water Resources Research, Vol. 2, No. 2, 996, pp [2] Savc, D.A., and Walters, G.A., "Genetc Algorthms for Least-Cost Desgn of Water Dstrbuton Networks," Journal of Water Resources Plannng and Management, ASCE, Vol. 2, No. 2, 997, pp [] Abdel-Gawad, H.A.A., "Optmal Desgn of Ppe Networks by an Improved Genetc Algorthm," Proceedngs of the Sxth Internatonal Water Technology Conference IWTC 200, Alexandra, Egypt, March 2-25, 200, pp [4] Abebe, A.J., and Solomatne, D.P., "Applcaton of Global Optmzaton to the Desgn of Ppe Networks," Proceedngs of the rd Internatonal Conference on Hydronformatcs, Copenhagen, August 998. [5] Lppa, I., Heaney, J.P., and Laguna, M., 999, "Robust Water System Desgn wth Commercal Intellgent Search Optmzers," Journal of Computng n Cvl Engneerng, Vol., No., pp [6] Gupta, I., Gupta, A., and Khanna, P., "Genetc Algorthm for Optmzaton of Water Dstrbuton Systems," Envronmental Modellng & Software, Vol. 4, 999, pp [7] Varavamoorthy, K., and Al, M., 2000, "Optmal Desgn of Water Dstrbuton Systems Usng Genetc Algorthms," Computer-Aded Cvl and Infrastructure Engneerng, Vol. 5, pp [8] Wu, Z.Y., and Smpson, A.R., "A Self-Adaptve Boundary Search Genetc Algorthm and ts Applcaton to Water Dstrbuton Systems," Journal of Hydraulc Research, Vol. 40, No. 2, 2002, pp [9] Cunha, M.D.C., and Sousa, J., "Water Dstrbuton Network Desgn Optmzaton: Smulated Annealng Approach," Journal of Water Resources Plannng and Management, ASCE, Vol. 25, No. 4, 999, pp [20] Eusuff, M.M., and Lansey, K.E., "Optmzaton of Water Dstrbuton Network Desgn Usng the Shuffled Frog Leapng Algorthm," Journal of Water Resources Plannng and Management, ASCE, Vol. 29, No., 200, pp

22 098 Nnth Internatonal Water Technology Conference, IWTC 2005, Sharm El-Shekh, Egypt [2] Long, S-Y., and Atquzzaman, Md., "Optmal Desgn of Water Dstrbuton Network usng Shuffled Complex Evoluton," Journal of the Insttuton of Engneers, Sngapore, Vol. 44, Issue, 2004, pp [22] Watters, G.Z., Analyss and Control of Unsteady Flow n Ppelnes, 2 nd Edton, Butterworth Publshers, 984. [2] Crossley, W.A., and Wllams, E.A., "A Study of Adaptve Penalty Functons for Constraned Genetc Algorthm Based Optmzaton," In AIAA 5 th Aerospace Scences Meetng and Exhbt, Reno, Nevada, January 997. AIAA Paper [ ] [24] Gen, M., and Cheng, R., Genetc Algorthms & Engneerng Optmzaton, John Wley & Sons, Inc., New York, [25] Krshnakumar, K., "Mcro-Genetc Algorthms for Statonary and Non- Statonary Functon Optmzaton," Proc. Soc. Photo-Opt. Instrum. Eng. (SPIE) on Intellgent Control and Adaptve Systems, Vol. 96, Phladelpha, PA, 989, pp [26] Smpson, A.R., Murphy, L.J., and Dandy G.C., "Ppe Network Optmzaton usng Genetc Algorthms," Paper presented at ASCE, Water Resources Plannng and Management Specalty Conference, ASCE, Seattle, USA, 99. [27] Rossman, L.A., EPANET, Users Manual. U.S. Envronmental Protecton Agency, Cncnnat, Oho, 99. [28] Todn, E., and Plat, S., "A Gradent Method for the Analyss of Ppe Networks," Proceedngs of the Internatonal Conference on Computer Applcatons for Water Supply and Dstrbuton, Lecester Polytechnc, Lecester, U.K., September 987. [29] Fujwara, O., and Khang, D.B., "A Two-Phase Decomposton Method for Optmal Desgn of Looped Water Dstrbuton Networks," Water Resources Research, Vol. 26, No. 4, 990, pp