RELIABILITY-BASED OPTIMAL DESIGN FOR WATER DISTRIBUTION NETWORKS OF EL-MOSTAKBAL CITY, EGYPT (CASE STUDY)

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 1 RELIABILITY-BASED OPTIMAL DESIGN FOR WATER DISTRIBUTION NETWORKS OF EL-MOSTAKBAL CITY, EGYPT (CASE STUDY) Rham Ezzeldn *, Hossam A. A. Abdel-Gawad *, and Magdy Abou Rayan ** * Irrgaton and Hydraulcs Department, Faculty of Engneerng, Mansoura Unversty, El-Mansoura, Egypt E-mal: Rham_ezzeldn@hotmal.com ** Mechancal Power Engneerng Department, Faculty of Engneerng, Mansoura Unversty, El-Mansoura, Egypt E-mal: mrayan@mans.edu.eg ABSTRACT An approach to the Relablty-based optmzaton of water dstrbuton systems s presented and appled to a case study. The approach lnks a genetc algorthm (GA) as the optmzaton tool, the Newton method as the hydraulc smulaton solver wth the chance constrant combned wth the Monte Carlo smulaton to estmate network capacty relablty. The source of uncertanty analyzed s the future nodal external demands whch are assumed to be random normally dstrbuted varables wth gven mean and standard devatons. The performance of the proposed approach s tested on an exstng network. The case study s for El-Mostakbal Cty network, an extenson to an exstng dstrbuton network of Ismala Cty, Egypt. The applcaton of the method on the network shows ts capablty to solve such actual Relablty based-optmzaton problems. INTRODUCTION The complexty of (WDS) makes t dffcult to obtan least-cost desgn systems consderng other constrants such as relablty. A completely satsfactory water dstrbuton system (WDS) should supply water n the requred quanttes at desred resdual heads throughout ts desgn perod. How well a WDS can satsfy ths goal can be determned from water supply relablty. However, evoluton of WDS relablty s extremely complex because relablty depends on a large number of parameters, some of whch are qualty and quantty of water avalable at source; falure rates of supply pumps; power outages; flow capacty of transmsson mans; roughness characterstcs nfluencng the flow capacty of the varous lnks of the dstrbuton network; ppe breaks and valve falures; varaton n daly, weekly, and seasonal demands; as well as demand growth over the years.

2 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt There s currently no unversally accepted defnton of relablty of WDS. However, relablty s usually defned as the probablty that a system performs ts msson wthn specfed lmts for a gven perod of tme n a specfed envronment. For a large system, t s dffcult to analytcally compute relablty n a mathematcal form. Accurate calculaton of a mathematcal relablty requres knowledge of the exact relablty of the basc components of WDS and the mpact on system performance caused by possble falures n the components. Relablty models to compute system relablty have been developed snce 1980s. These models allow a modeler to determne the relablty of a system and account for such factors as the probablty and duraton of ppe and pump falure, the uncertanty n demands, and the varablty n the deteroraton of ppes. Some of these relablty models whch have been commonly used n lterature are cut-set method, Monte Carlo smulaton, chance constrants, sgnfcance ndex method, and frequency duraton analyss. Su et al. (1987) developed a relablty based optmzaton model that determned the least-cost desgn of water dstrbuton system subect to contnuty, conservaton of energy, nodal head bounds, and relablty constrants. The steady-state smulaton model (KYPIPE) by Wood (1980), was used to mplctly solve the contnuty and energy constrants and was used n the relablty model to defne mnmum cut sets. The relablty model, whch was based on a mnmum cut-set method, determned the values of system and nodal relablty. The optmzaton model was based on a generalzed reduced-gradent method (GRG2) by Lasdon and Waren (1979, 1984) whch solved an optmzaton problem wth a nonlnear obectve functon and nonlnear constrants. Lansey et al. (1989) were among the frst to present a chance constrant model for the least-cost desgn of water dstrbuton systems. The uncertanty n the requred demand, pressure heads, and ppe roughness coeffcent were explctly accounted for n the model. The generalzed reduced gradent (GRG2) technque was used to solve the nonlnear programmng sngle-obectve chance constraned mnmzaton model. The methodology assumed nodal heads to be random, normally dstrbuted varables wth gven mean and standard devaton. Snce head values are functons of many parameters, some of whch could be uncertan, they should be treated as a response functon rather than ndependent stochastc varables. Also, the generalzed reduced gradent method (GRG2) s a local search method whch could be easly trapped n the local mnmum (Savc and Walters, (1997)). Bao et al. (1990) presented a Monte Carlo smulaton model that estmated the nodal and system hydraulc relabltes of water dstrbuton systems that accounted for uncertantes. The model conssted of three maor components; random number generaton, hydraulc network smulaton, and computaton of relablty. The model could be appled n the analyss of exstng water dstrbuton systems or n the desgn of new or expandng systems.

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 3 Goulter et al. (1990) ncorporated relablty concept nto optmal desgn models for ppe network systems. The measure of the system relablty was used as a crteron to mprove the system dstrbuton. The chance constrants were the probablty of ppe falure for each lnk and the probablty of demand exceedng desgn values at each node n the network. Xu and Goulter (1998) developed an approach n whch a probablstc hydraulc model was used for the frst tme n the WDS desgn optmzaton. In the hydraulc model uncertantes were quantfed usng the analytcal technque known as the frst order second moment (FOSM) relablty method. Ths method assumes that a relatonshp between uncertan and response varables s very close to lnear, whch s often not the case for water dstrbuton systems. Xu and Goulter (1999) used the frst order relablty-method-based (FORM) algorthm that computed the capacty relablty of water dstrbuton networks. The senstvtyanalyss-based technque was used to derve the frst order dervatves. The (FORM) algorthm requred repettve calculaton of the frst order dervatves and matrx nverson whch was very computatonally demandng even n small networks and may lead to a number of numercal problems. Rayan et al. (2003) used the sequental unconstraned mnmzaton technque (SUMT) to solve the optmal desgn of El-Mostkbal cty whch s an extenson of Ismala cty (Egypt) combned wth the Newton-Raphson method for the hydraulc analyss of the network. Xu et al. (2003) ntroduced two algorthms for determnng the capacty relablty of ageng water dstrbuton systems consderng uncertantes n nodal demands and ppe capacty. The mean value frst order second moment (MVFOSM) method and the frst order relablty model (FORM) were used as a probablstc hydraulc models for relablty assessment. Both models provded reasonably accurate estmates of capacty relablty n cases that the uncertanty n the random varables was small. In cases nvolvng large varablty n the nodal demands and ppe roughness, FORM performed much better. Savc (2005) through the applcaton of varous approaches for optmal desgn and rehabltaton of urban water systems under the condton of nherent uncertanty; namely, the use of standard safety margns (redundant desgn methodology) and the stochastc robustness/rsk evaluaton models wth both sngle-obectve and multobectve optmzaton methods on the New York water supply tunnels problem and the Anytown network clearly demonstrated that neglectng uncertanty n the desgn process mght lead to serous under-desgn of water dstrbuton networks. Tolson et al. (2004) used GAs to solve the optmal water dstrbuton system desgn problems along wth the frst order relablty method (FORM) method to quantfy uncertantes. They demonstrated that the Monte Carlo Smulaton crtcal node capacty relablty approxmaton can sgnfcantly underestmate the true Monte

4 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt Carlo Smulaton network capacty relablty. Therefore, they developed a more accurate FORM approxmaton to network capacty relablty that consders falure events at the two most crtcal nodes n the network. Abdel-Gawad (2005) presented an approach for water network optmzaton under a specfc level of uncertanty n demand, pressure heads, and ppe roughness coeffcent. The approach depends on usng the chance constraned model to convert uncertantes n the desgn parameters to form a determnstc formulaton of the problem. The GA method was adopted to solve the nonlnear optmzaton problem settled n a determnstc form. A hypothetcal example was solved and compared wth prevous soluton from the gradent approach [3]. From the results t can be found that the constructon cost of the ppe system ncreases, wth an ncreasng rate, as the relablty requrement ncreases. Uncertantes n demand nodes or roughness coeffcents have a more pronounced effect on fnal constructon cost, than the effect of the requred mnmum pressure heads. Babayan et al. (2005) presented a methodology for the least cost desgn of water dstrbuton networks consderng uncertanty n node demand. The uncertan demand was assumed to follow both truncated Gaussan (normal) probablty densty functon (PDF) and unform probablty densty functon. The genetc algorthm was used to solve the equvalent determnstc model for the orgnal stochastc one to fnd relable and economc desgn for the network and the system relablty was then determned usng full Mont Carlo smulaton wth 100,000 samplng ponts. The model was tested on the New York tunnels and Anytown problems and then compared to avalable determnstc solutons. The results demonstrated the mportance of applyng the uncertanty concept n desgnng water dstrbuton systems. Babayan et al. (2006) developed two new methods to solve an optmzaton problem under uncertanty. Uncertanty sources used were both future water consumpton and ppe roughness. The stochastc formulaton after beng replaced by a determnstc one usng numercal ntegraton method, whle the optmzaton model was solved usng a standard genetc algorthm. The samplng method solved the stochastc problem drectly by usng the newly developed robust chance constrant genetc algorthm both methods had there own benefts and drawbacks. Nodal and System Relablty Bao and Mays (1990) defned nodal relablty R n as the probablty that a gven node receves suffcent flow rate at the requred pressure head. Theoretcally, therefore, the nodal relablty s a ont probablty of flow rate and pressure head beng satsfed at the gven nodes. They also stated that system relablty s such an ndex dffcult to defne because of the dependence of the computed nodal relabltes. Three heurstc defntons of the system relablty are therefore proposed:

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 5 (1) The system relablty R sm could be defned as the mnmum nodal relablty n the system R mn ( R ) = 1,2,, I (4.4) sm n where R n s the nodal relablty at node ; and I s the number of demand nodes of nterest. (2) The system relablty could be the arthmetc mean R sa, whch s the mean of all nodal relabltes. I Rn Rsa 1 (4.5) I (3) The system relablty s defned as a weghted average R sw, whch s a weghted mean of all nodal relabltes weghted by the water supply at the node. where R sw I R n 1 I 1 Q Q s s Q s the mean value of water supply at node. s Approaches for Assessment of Network Relablty (4.6) Two man approaches are avalable for assessment of relablty, (Goulter et al., 2000): Analytcal approach. A closed form of soluton for the relablty s derved drectly from the parameters whch defne the network demands and the ablty of network to meet these demands. Smulaton approach. The network s evaluated usng dfferent user defned scenaros or durng extended perod smulatons (Goulter et al., 2000). 4.6.1 Advantages and Dsadvantages of the Analytcal Approach (a) Advantages: 1. Consders the complete network rather than samples. 2. Less computatonal tme. (b) Dsadvantages: 1. Requres a smplfed descrpton of the water system. 2. Smplstc nterpretaton of relablty, e.g., connectvty versus hydraulc performance. 4.6.2 Advantages and Dsadvantages of the Smulaton Approach

6 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt (a) Advantages: 1. A number of relablty measures can be calculated. 2. Allows the analyss of a system wth complcated nteractons. 3. Allows the detaled modelng of the behavor of the system. (b) Dsadvantages: 1. Tme consumng n both terms of computer per tme per analyss and n terms of tme to set up and use such a program. 2. Its runs are hard to optmze and can be hard to generalze beyond a very specfc system. Thus perhaps the best approach to performng a relablty assessment s to use both smulaton and analytcal methods. The prevous lterature revew demonstrates that both analytcal and smulaton methods should be used together. Ths can be acheved by applyng the chance constrant method to take the uncertanty of dfferent ppe network parameters nto account, and a Monte Carlo smulaton to determne ts nodal and system relabltes more accurately. The present study of uncertanty-based optmzaton of water dstrbuton systems and for a specfed level of uncertanty ams to search the optmal dameters whch mnmze the cost and fulfll the pressure constrants at nodes. The uncertanty-based optmzaton was acheved by the chance constrant formulaton whch s dscussed later. The Monte Carlo smulaton s used to fnd the node and network relabltes for the optmal dameters of the network. In the present nvestgaton, (GACCnet)s used to solve the uncertanty based-optmal desgn of the network The optmzaton tool s the Genetc Algorthm (GA) whch s lnked n the present work wth the uncertanty formulaton. Expressed by the chance constrant method, and Monte Carlo Smulaton to estmate the nodal and network relabltes. The case study s a real network. It s an extenson to an exstng dstrbuton network of Ismala Cty named El-Mostkbal Cty. 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 the conservatons of mass and energy, and the constrants (e.g. hydraulc and desgn constrants). 1- Determnstc Model The formulaton of the optmzaton model for water dstrbuton system desgn can be generally wrtten n the followng form:[3]

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 7 Obectve functon: Mn. Cost = mn. C T = f D, M, (1) Model Constrants: q, Q 1,..., J h fn, n H (nodes) (2) 0 n 1,..., N (loops) (3) H 1,..., J (nodes) (4), mn D mn D, D max (5) The man obectve of the model, Eq. (1), s to mnmze the constructon cost of the water dstrbuton network as a functon of the ppe dameter, for the set of possble lnks, M, connectng nodes, n the system. connectng nodes,. Wllams formula: f D, q, s the flow rate n the ppe h s the head loss n the ppe and expressed by the Hazen- h f 1.852 K L, q, 1.852 4. 8704 C, D, H H (6) where K s a converson factor whch accounts for the system of unts used, (K = 10.6744 for q, n m 3 /s and D, and L, n m), C, s the Hazen-Wllams roughness coeffcent for the ppe connectng nodes,, L, s the length of the ppe connectng nodes,, and H, rate n the ppe s calculated as: H are the pressure heads at nodes,. Then, the flow q H H 0.54 0.54 2.63, K C, D, L, (7) Eq. (2) represents the law of conservaton of mass (contnuty equaton) whch states that the summaton of the flow rates n the ppes at node must be equal to the external demand, Q, at that node. It has to be notced that the contnuty constrant must be satsfed for each node,, n the network. Eq. (3) n the model constrants smply states that the algebrac summaton of the head loss, h f n, around each loop n = 1,, N must be equal to zero. The lower lmt, H, mn, of the pressure head, H, at each node,, s accounted for n the model by Eq. (4).

8 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt Fnally, Eq. (5) defnes the constrant on the ppes dameters n the network where D mn and D max are the mnmum and maxmum dameters, respectvely. Substtuton of the Hazen-Wllams formula, Eq. (7) back nto Eq. (2) automatcally satsfes Eq. (3), and whch n turn reduces the determnstc model constrants to equatons (4), (5), and (7) n combnaton wth (2). 2- Stochastc (Chance Constrant) Model The determnstc optmzaton model descrbed above s transformed nto a stochastc (chance constrant) formulaton by consderng that the future demand, Q, s uncertan because of the unknown future condtons of the system and can be consdered as an ndependent random varable. The chance constrant formulaton can now be expressed as Lansey et al. (1989): Obectve functon: Mnmum Cost = mn. f D Subect to the constrants:, M, (8) P. Q (9) 0.54 H H K 0.54 C D L 2,. 63,, H H, mn (10) D mn D, D max (11) Eq. (9) s the probablty, P ( ), that the node demands are equaled or exceeded wth probablty level,, The probablty level, s defned as the constrant performance relablty whch accounts for the effect of uncertanty of the future demand. 3- Determnstc Chance Constrant Model The chance constrant model s now transformed from a stochastc form nto a determnstc one through applyng the cumulatve probablty dstrbuton concept by consderng the future demand, to be represented by normal random varables wth mean, µ, and standard devaton, σ, as: Q~ N, ) ( Q Q

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 9 Smlarly, Eq. (9) s transformed nto a determnstc form as follows: W P Q D L H H C K P 1 0 0 2.63, 0.54,, 0.54 (12) Where W s a normal random varable wth mean: Q W D L H H C K 63 2,. 0.54,, 0.54 (13) and standard devaton: 2 1/ 2 2 2.63, 0.54,, 0.54. Q W D L H H C K (14) Eq. (12) can be rewrtten as: W W W W W P 1 0 (15) or n a smplfed form: W W 1 (16) where s the cumulatve dstrbuton functon and s the standard normal dstrbuton functon. The fnal determnstc form of the constrant Eq. (12) s now wrtten as: W W 1 1 (17) where W and W are determned usng Eqs. (13) and (14).

10 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt The fnal determnstc chance constrant model for water dstrbuton networks s gven by the obectve functon Eq. (8) subect to the constrants Eqs. (17) and (11). The model s nonlnear because of the nonlnear obectve functon Eq. (8) and the non lnear constrant Eq. (17) for every node. The other constrant gven by Eq. (11) for every ppe s consdered to be smple bound. The genetc algorthm (GA) wll be used as a technque to solve the determnstc chance constraned model for water dstrbuton networks. GACCnet PROGRAM: GACCnet program, t s conssted of: Ezzeldn (2007) 1. Genetc algorthm technque to produce the optmal dameters. The GA source code used s smlar to that used n Abdel-Gawad (2001). 2. Newton method to analyze the network usng The H-equatons soluton method. 3. Chance Constrant for the uncertantes. 4. Monte Carlo technque to compute the relablty of the optmal set of ppe dameters. CASE STUDY An actual water network has been selected to apply the developed program for the uncertanty-based optmzaton to evaluate the desgn of the network, also, to test the capabltes of the developed model n a real and large network. The network selected here as a case study s bult to serve a new resdental cty called El-Mostakbal. It s a new extenson to Cty of Ismala. The network was desgned as an extenson to the orgnal network of Ismala Cty. The data of ths network are taken from Herrck (2001) and Rayan et al. (2003). The layout of the network and the ndex numbers of the nodes and ppes are shown n Fgure 1. As the orgnal ndex numbers are great, the correspondng modfed ndex numbers are shown n Fgure 1(b). Smlarly, n Table 1, these modfed ndces are gven. The data for the studed network s shown n Table 2. It ncludes the new ndex (ID) for each node and ppe. The extenson network has 31 nodes (excludng node 32 whch s taken as the supplyng node, Fg. 1(b)) and 43 ppes. For the nodes, the elevaton and specfed demands are gven, whle for the ppes ther lengths and dameters are represented.

(a) (b) Fgure 1. El-Mostakbal Cty water dstrbuton network (a) Orgnal ID for nodes and ppes, Herrck (2001) (b) Modfed ID for nodes and ppes used n ths study Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 11

12 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt Table 1. El-Mostakbal Cty network Orgnal Node ID New Node ID Orgnal Ppe ID Start Node End Node New Ppe ID Start Node End Node 7001 32 7001 7001 7010 1 32 1 7010 1 7010 7010 7020 2 1 2 7020 2 7020 7020 7030 3 2 3 7030 3 7030 7030 7040 4 3 6 7032 4 7032 7030 7032 5 3 4 7034 5 7034 7032 7034 6 4 5 7040 6 7036 7034 7060 7 5 8 7050 7 7040 7040 7050 8 6 7 7060 8 7050 7050 7060 9 7 8 7070 9 7060 7060 7010 10 8 1 7075 10 7070 7040 7070 11 6 9 7080 11 7075 7070 7075 12 9 10 7085 12 7080 7070 7080 13 9 11 7090 13 7085 7075 7085 14 10 12 7100 14 7090 7080 7090 15 11 13 7110 15 7095 7085 7110 16 12 15 7120 16 7100 7090 7100 17 13 14 7130 17 7110 7100 7110 18 14 15 7140 18 7120 7110 7050 19 15 7 7150 19 7130 7080 7120 20 11 16 7160 20 7140 7120 7130 21 16 17 7165 21 7150 7130 7140 22 17 18 7170 22 7160 7140 7150 23 18 19 7175 23 7170 7150 7100 24 19 14 7180 24 7175 7090 7130 25 13 17 7190 25 7180 7120 7160 26 16 20 7195 26 7182 7165 7175 27 21 23 7200 27 7185 7160 7165 28 20 21 7205 28 7190 7140 7230 29 18 31 7210 29 7192 7175 7220 30 23 30 7220 30 7195 7165 7230 31 21 31 7230 31 7200 7160 7170 32 20 22 7205 7170 7205 33 22 28 7210 7170 7180 34 22 24 7215 7205 7195 35 28 26 7220 7180 7190 36 24 25 7225 7195 7190 37 26 25 7230 7190 7200 38 25 27 7240 7200 7210 39 27 29 7245 7195 7210 40 26 29 7250 7210 7220 41 29 30 7255 7175 7205 42 23 28 7260 7220 7230 43 30 31

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 13 Table 2. El-Mostakbal Cty network data (Orgnal desgn) (a) Nodes )b)ppes Node ID Elevaton (m) Demand (LPS) Ppe ID Length (m) Dameter (mm) 1 14.0 24.00 1 100.00 600 2 14.0 0 2 328.00 300 3 14.0 19.20 3 80.00 300 4 14.0 0 4 152.50 300 5 14.0 0 5 149.30 150 6 14.0 19.20 6 67.00 150 7 14.0 20.80 7 184.30 150 8 14.0 17.60 8 341.65 150 9 14.0 0 9 100.00 400 10 14.0 0 10 288.00 400 11 14.0 24.00 11 70.70 300 12 14.0 0 12 172.00 150 13 14.0 0 13 127.60 250 14 14.0 19.20 14 109.00 150 15 14.0 0 15 164.60 150 16 15.0 24.00 16 104.70 150 17 15.0 19.20 17 98.40 150 18 15.0 34.09 18 123.50 400 19 15.0 0 19 155.00 400 20 15.0 16.00 20 309.15 250 21 15.5 0 21 163.40 150 22 15.5 16.00 22 134.20 150 23 15.5 0 23 198.00 300 24 15.5 16.00 24 225.50 400 25 15.5 19.20 25 357.90 150 26 15.5 0 26 92.70 200 27 15.5 19.20 27 156.50 150 28 15.5 0 28 84.90 200 29 15.5 24.00 29 100.00 300 30 15.5 0 30 101.00 150 31 15.5 20.80 31 226.30 200 32 15.0 0 32 230.50 200 33 145.80 150 Total Demand = 352.49 LPS 34 370.60 150 35 109.90 150 36 184.00 150 37 257.40 150 38 120.00 150 39 181.90 150 40 114.90 150 41 262.60 200 42 185.00 150 43 217.00 300

14 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt The cost values used n the optmzaton problem are the real costs that are used n the Suez Canal Authorty water sector, Herrck (2001). There are 10 commercally avalable dameters for ductle ppes, Table 3. All ppes are selected from ductle although Rayan et al. (2003) gave other optons for ppes less than 6 nches whch s unpractcal n water dstrbuton networks. Table 3. Commercally avalable ppe szes and cost per meter Dameter (nches) Dameter (mm) Unt Cost (L.E./m) Ppe Type 6 150 188 Ductle 8 200 255 Ductle 10 250 333 Ductle 12 300 419 Ductle 16 400 570 Ductle 20 500 735 Ductle 24 600 1110 Ductle 30 800 1485 Ductle 40 1000 2505 Ductle 48 1200 3220 Ductle As mentoned n Rayan et al. (2003), the desgner of ths network chose node number 481 from the orgnal network of Ismala Cty to connect t wth the new extenson network. The average pressure head at ths node before connecton equals 25.5 meters (calculated from the hydraulc model analyss). The connecton ppe (Ppe 7000, Fg. 1(a)) between the two networks s 600 mm dameter wth 8692.7 meter long. To solve ths drawback, the node chosen to connect the old network wth the extenson s a dfferent node than that chosen n the orgnal desgn. The node chosen to connect the two networks by the optmzaton program s node number 456. Its average pressure head s 43.89 meters (calculated from the hydraulc model). The connecton ppe s 800 mm dameter wth length about 2463 meters long. Accordng to ths, ther study showed a decreasng n the total cost of ths ppe of LE 5,990,565. It s worth to menton that the orgnal exstng desgn of the extended network costs LE 11,868,999. On the other hand, the total cost of ppes for the exstng network wthout ncludng ppe 7000 s LE 2,220,879. The resulted new network under nvestgaton has node 32 as a source wth a total head of 50.856 m and the total demand for the network s 352.49 LPS. The mnmum acceptable pressure head requrements for all nodes of the network are set as 22 meters. Dfferences wth Prevous Studes The man dfferences between the present study and the study of Herrck (2001) and

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 15 Rayan et al. (2003) are: 1. The present study s uncertanty-based optmzaton whle the study of Herrck (2001) and Rayan et al. (2003) s optmzaton only. 2. In Herrck (2001) and Rayan et al. (2003), the Sequental Unconstraned Mnmzaton Technque (SUMT) was appled to solve the optmal desgn of network for the ppe network optmzaton. The SUMT was frst suggested by Carroll (1961) and thoroughly nvestgated by Facco and McCormck (1964). The explanaton of the optmzaton model formulaton s gven by Debedan et al. (2000). In the present study, the genetc algorthms are used for the ppe network optmzaton. 3. In Herrck (2001) and Rayan et al. (2003), the head loss h f n the ppe was expressed by the Darcy-Wesbach formula and the frcton factor f was calculated by the expresson proposed by Swamee and Jan (1975). In the present study, the Hazen-Wllams formula s used. Numercal tests for the frctonal losses calculated by Darcy-Wesbach and Hazen-Wllams formulae for El-Mostakbal network were done usng EPANET 2 and the correspondng approxmate Hazen-Wllams coeffcent was found to be 130 (.e. smooth ppe). As ths value decreases wth ppes ageng, the Hazen-Wllams roughness coeffcent s taken as 100 for all ppes throughout ths case study. Computatonal Results of Optmzaton The frst part of the present study s dedcated to fnd the optmal dameters and the correspondng total cost. For the studed network, t should be mentoned that for 43 ppes and a set of 10 commercal ppes, the total number of desgns s 10 43. Therefore, t s very dffcult for any mathematcal model to test all these possble combnatons of desgn and a very small percentage of combnatons can be reached. The optmal dameters found by GACCnet program are lsted n Table 4. The optmal cost s LE 2,234,046 compared to LE 2,220,879 for the orgnal desgn.

16 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt Table 4. Optmal ppe dameters for El-Mostakbal Cty network ( = 0.5) Ppe ID Dameter (mm) Ppe ID Dameter (mm) 1 500 26 250 2 150 27 150 3 150 28 150 4 150 29 300 5 150 30 150 6 150 31 150 7 150 32 250 8 150 33 150 9 500 34 150 10 500 35 200 11 150 36 150 12 150 37 200 13 200 38 150 14 150 39 200 15 200 40 150 16 150 41 250 17 250 42 150 18 400 43 300 19 500 20 250 21 150 22 150 23 400 24 400 25 150 Although the optmal cost s not less than the orgnal network cost, but the nodal pressure heads requrements are fulflled. The genetc algorthm parameters used for solvng ths case study are mentoned n Appendx D. The hydraulc analyss results of the case study network before and after optmzaton are shown n Table 5 and Fg. 2. For the network before optmzaton and as seen from Table 5 and Fg. 2, there are some nodes (22 and 24 to 29) wth pressure head values less than 22 m, whch s the mnmum pressure crteron. As expected, the nodal pressure heads n the extended network after optmzaton s hgher than that of the orgnal desgn. The pressure heads at all nodes of the optmzed network are greater than 22 meters whch s the mnmum acceptable pressure head requrements. Also, the average nodal pressure head n the optmzed network s greater than that of the orgnal desgn due to the well-known fact that decreasng the dameter of a ppe ncreases the frcton losses and consequently decreases the pressure head at the downstream node. It can be concluded that the optmzaton of the water dstrbuton system of El-Mostakbal Cty overcomes the drawback of low nodal pressure heads of the orgnal network. For the optmzed network, the utlzaton of optmzaton technque perhaps results n not mnmzng the cost but ncreasng the pressure heads at all nodes of the network to be greater than

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 17 the mnmum acceptable pressure head. Table 5. Results of hydraulc analyss of El-Mostakbal Cty network before and after optmzaton optmzed network Node ID Nodal Pressure Head (m) Before Optmzaton** After Optmzaton** 1 36.487 35.960 2 33.086 31.559 3 32.256 30.485 4 32.551 31.768 5 32.684 32.343 6 31.025 29.292 7 32.077 33.350 8 33.048 33.927 9 30.580 28.872 10 30.742 30.565 11 28.411 27.988 12 30.845 31.639 13 28.589 30.401 14 30.083 31.251 15 30.944 32.670 16 24.245 25.207 17 24.320 26.329 18 24.632 27.966 19 28.108 29.034 20 22.952 24.892 21 22.550 24.907 22 20.012 23.771 23 22.160 24.899 24 15.618 22.106 25 15.600 22.391 26 18.516 23.397 27 15.608 22.521 28 19.997 23.679 29 18.498 23.492 30 22.571 25.482 31 23.254 26.468 Average Pressure (m) 26.1951 28.020 Mnmum Pressure (m) 15.600 22.106 Maxmum Pressure (m) 36.487 35.960 * Orgnal desgn (Table 9.2), ** Optmal desgn (Table 9.4)

Pressure Head (m) 18 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 40 After Optmzaton Before Optmzaton 30 20 10 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Node ID Fgure 2. Comparson of nodal pressure heads between El-Mostakbal Cty orgnal desgn and optmzed network Computatonal Results of Uncertanty-Based Optmzaton The second part of the present study s dedcated to fnd the optmal dameters and the correspondng total cost for specfed levels of uncertanty. Table 6 lsts the optmum desgn of El-Mostakbal Cty network under sx levels of uncertanty for a coeffcent of varaton n nodal demands COV Q = 10%. The nodal pressure heads for these optmal networks are gven n Table 7. The results of nodal and system relabltes from the Monte Carlo smulaton assocated wth these sx dfferent levels of uncertanty are gven n Table 8.

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 19 Table 6. Optmal ppe dameters for El-Mostakbal Cty network for dfferent network uncertantes at COV Q = 10% Ppe Dameter (mm) ID = 0.5 = 0.6 = 0.7 = 0.8 = 0.9 = 0.99 1 500 500 500 500 500 500 2 150 150 150 500 400 500 3 150 150 200 500 300 500 4 150 150 150 500 300 400 5 150 150 150 150 150 200 6 150 150 150 150 200 150 7 150 150 150 150 150 150 8 150 150 150 150 150 150 9 500 500 500 300 500 300 10 500 500 500 300 500 400 11 150 150 200 500 400 400 12 150 150 150 150 200 150 13 200 200 200 400 300 500 14 150 150 150 150 150 200 15 200 200 300 150 150 150 16 150 150 150 150 150 150 17 250 250 400 150 250 150 18 400 500 500 200 400 300 19 500 500 500 200 400 400 20 250 200 400 400 400 400 21 150 200 150 150 150 150 22 150 150 150 400 150 150 23 400 400 300 150 300 250 24 400 400 300 150 300 250 25 150 150 150 150 150 150 26 250 250 300 400 300 400 27 150 150 150 200 200 250 28 150 150 150 250 150 250 29 300 300 250 150 300 200 30 150 150 150 150 150 200 31 150 150 150 150 150 150 32 250 200 300 300 250 300 33 150 150 200 200 150 250 34 150 200 150 200 200 250 35 200 150 250 400 200 300 36 150 150 150 150 150 150 37 200 200 200 150 150 200 38 150 150 150 150 200 150 39 200 200 200 250 400 150 40 150 150 200 200 150 150 41 250 250 200 200 250 250 42 150 150 150 150 200 250 43 300 300 250 200 200 200 Cost (LE) 2,234,046 2,240,746 2,330,245 2,380,332 2,517,901 2,584,412

20 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt Table 7. Nodal pressure heads of best solutons of El-Mostakbal Cty network for dfferent network uncertantes at COV Q = 10% Node Pressure Head (m) ID = 0.5 = 0.6 = 0.7 = 0.8 = 0.9 = 0.99 1 35.960 35.918 35.871 35.816 35.736 35.536 2 31.559 31.740 30.938 33.993 34.418 33.989 3 30.485 30.721 30.642 33.549 33.113 33.611 4 31.768 31.860 31.758 33.586 33.678 33.642 5 32.343 32.371 32.259 33.603 33.740 33.699 6 29.292 29.815 29.586 32.805 31.054 31.777 7 33.350 33.163 32.997 33.213 34.081 32.162 8 33.927 33.776 33.638 33.650 34.437 33.855 9 28.872 29.622 29.531 32.499 30.844 31.051 10 30.565 30.873 30.733 32.076 31.191 31.457 11 27.988 29.083 28.974 30.956 28.913 30.580 12 31.639 31.666 31.495 31.808 32.085 31.521 13 30.401 31.072 31.249 29.572 31.746 30.290 14 31.251 31.897 31.670 29.659 32.168 30.662 15 32.670 32.427 32.227 31.550 32.944 31.768 16 25.207 24.975 26.984 27.337 26.862 27.153 17 26.329 25.631 26.881 23.699 26.593 25.828 18 27.966 28.221 26.896 23.678 26.579 25.611 19 29.034 29.472 28.660 26.007 28.724 27.505 20 24.892 24.727 26.288 26.893 26.174 26.715 21 24.907 24.756 25.541 25.852 24.846 25.637 22 23.771 22.852 24.889 24.973 24.079 24.969 23 24.899 24.727 24.914 24.584 23.985 24.946 24 22.106 22.009 22.050 22.745 22.265 23.966 25 22.391 22.027 22.145 22.198 22.062 23.159 26 23.397 22.744 23.322 23.560 23.055 24.586 27 22.521 22.128 22.159 22.213 22.135 22.557 28 23.679 23.103 23.834 23.620 23.601 24.772 29 23.492 23.090 22.960 22.499 22.183 23.792 30 25.482 25.420 24.940 23.170 23.555 24.486 31 26.468 26.578 25.571 23.148 25.610 24.620 Average Pressure (m) 28.020 28.015 28.116 28.210 28.466 28.577 Mnmum Pressure (m) 22.106 22.009 22.050 22.198 22.062 22.557 Maxmum Pressure (m) 35.960 35.918 35.871 35.816 35.736 35.536

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 21 Table 8. Node and network relabltes of best solutons of El-Mostakbal Cty network for dfferent network uncertantes at COV Q = 10% Node Node Relablty, R n (%) ID = 0.5 = 0.6 = 0.7 = 0.8 = 0.9 = 0.99 1 100 100 100 100 100 100 2 100 100 100 100 100 100 3 100 100 100 100 100 100 4 100 100 100 100 100 100 5 100 100 100 100 100 100 6 100 100 100 100 100 100 7 100 100 100 100 100 100 8 100 100 100 100 100 100 9 100 100 100 100 100 100 10 100 100 100 100 100 100 11 100 100 100 100 100 100 12 100 100 100 100 100 100 13 100 100 100 100 100 100 14 100 100 100 100 100 100 15 100 100 100 100 100 100 16 100 100 100 100 100 100 17 100 100 100 100 100 100 18 100 100 100 100 100 100 19 100 100 100 100 100 100 20 100 100 100 100 100 100 21 100 100 100 100 100 100 22 99.92 98.54 100 100 100 100 23 100 100 100 100 100 100 24 51.04 80.21 96.61 99.98 100 100 25 62.93 81.36 97.57 99.87 99.99 100 26 95.94 97.56 99.98 100 100 100 27 69.63 85.21 97.75 99.88 99.99 100 28 99.92 99.48 100 100 100 100 29 97.78 99.35 99.93 99.95 99.99 100 30 100 100 100 100 100 100 31 100 100 100 100 100 100 Network Relablty, R sm (%) 51.04 80.21 96.61 99.87 99.99 100 Network Relablty, R sa (%) Network Relablty, R sw (%) 93.02 96.74 99.52 99.98 99.99 100 93.95 97.17 99.59 99.98 99.99 100 For maor values of uncertanty, Table 7 show that nodes 24, 25, 27 and 29 are the crtcal nodes n the network, whch have nodal pressure heads not far from the requred mnmum pressure head. Therefore, ther node relabltes and manly that of node 24 are affectng the network relablty, Table 8. It s worth to menton that the Monte Carlo smulaton wth 10,000 samples was performed to calculate the nodal and network relabltes of El-Mostakbal Cty network. Also, from Table 8 the weghted system relablty came out to be hgher than arthmetc system relablty results and the latter s greater than the mnmum nodal system relablty.

22 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt Smlar to the prevous study, the optmum desgns of El-Mostakbal Cty network under the same prevous levels of uncertanty for a coeffcent of varaton n nodal demands COV Q = 20% are gven n Table 9. The nodal pressure heads for these optmal networks are gven n Table 10. The calculated nodal capacty and system relabltes from the Monte Carlo smulaton assocated wth these sx dfferent levels of uncertanty are summarzed n Table 11. For ths COV Q, t s observed that the mnmum nodal pressure heads are at nodes 24, 25 and 27 dependng on the specfed level of uncertanty and that node 24 has very low nodal relablty compared to that for other nodes for = 0.5 and 0.6. Smlar to COV Q = 10%, the obtaned weghted system relablty s hgher than the arthmetc system relablty and the mnmum nodal system relablty.

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 23 Table 9. Optmal ppe dameters for El-Mostakbal Cty network for dfferent network uncertantes at COV Q = 20% Ppe Dameter (mm) ID = 0.5 = 0.6 = 0.7 = 0.8 = 0.9 = 0.99 1 500 500 500 500 500 600 2 150 150 400 500 250 400 3 150 150 500 400 200 400 4 150 150 500 500 250 400 5 150 150 150 150 150 150 6 150 200 150 150 150 150 7 150 150 150 150 150 200 8 150 150 150 150 250 150 9 500 500 400 400 500 500 10 500 500 400 400 500 500 11 150 200 500 500 400 400 12 150 150 150 200 150 150 13 200 150 500 500 400 400 14 150 150 150 150 150 150 15 200 200 150 150 150 150 16 150 150 150 150 150 250 17 250 250 150 150 200 150 18 400 500 250 300 500 500 19 500 500 250 250 400 500 20 250 200 400 400 400 400 21 150 150 150 250 150 150 22 150 150 150 150 200 150 23 400 400 200 200 400 400 24 400 400 200 200 400 400 25 150 150 150 150 150 150 26 250 200 500 500 400 250 27 150 150 200 150 200 250 28 150 150 250 200 150 200 29 300 400 200 150 300 400 30 150 150 200 150 150 200 31 150 200 150 150 150 150 32 250 200 300 300 250 250 33 150 150 150 400 150 200 34 150 150 250 150 200 200 35 200 150 150 250 250 150 36 150 150 200 150 150 200 37 200 200 150 250 250 150 38 150 150 150 150 200 200 39 200 200 200 200 200 250 40 150 150 150 200 150 150 41 250 300 200 150 250 250 42 150 150 150 150 250 200 43 300 300 150 200 250 300 Cost (LE) 2,234,046 2,251,260 2,386,508 2,490,224 2,586,316 2,755,927

24 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt Table 10. Nodal pressure heads of best solutons of El-Mostakbal Cty network for dfferent network uncertantes at COV Q = 20% Node Pressure Head (m) ID = 0.5 = 0.6 = 0.7 = 0.8 = 0.9 = 0.99 1 35.960 35.875 35.778 35.662 35.490 36.108 2 31.559 31.876 32.095 34.106 31.871 33.640 3 30.485 30.900 31.792 32.981 29.254 33.038 4 31.768 32.061 32.752 33.507 30.635 33.678 5 32.343 32.190 33.182 33.743 31.255 33.965 6 29.292 30.098 31.256 32.328 27.985 32.097 7 33.350 32.978 34.061 34.116 32.246 33.623 8 33.927 33.623 34.367 34.393 32.959 34.159 9 28.872 30.074 31.023 32.051 27.807 31.747 10 30.565 31.019 31.409 32.068 29.036 32.498 11 27.988 28.645 30.574 31.540 27.403 31.030 12 31.639 31.618 31.654 32.111 29.815 32.973 13 30.401 30.797 30.147 31.279 29.391 31.324 14 31.251 31.617 30.458 31.557 30.161 32.563 15 32.670 32.193 31.890 32.153 30.564 33.011 16 25.207 25.080 27.245 27.818 25.619 28.854 17 26.329 26.134 25.443 27.476 26.241 28.287 18 27.966 27.618 24.698 25.874 26.828 28.634 19 29.034 29.020 26.923 28.063 27.919 30.003 20 24.892 24.799 27.104 27.667 25.467 27.318 21 24.907 25.466 26.086 26.520 24.478 26.741 22 23.771 23.456 25.310 25.079 23.543 25.388 23 24.899 25.198 24.693 25.114 23.756 26.544 24 22.106 22.066 23.460 22.429 22.124 22.545 25 22.391 22.534 22.485 22.693 22.124 22.476 26 23.397 23.480 22.980 23.665 22.838 23.820 27 22.521 22.884 22.051 22.370 22.074 22.637 28 23.679 23.743 24.398 24.930 23.262 25.446 29 23.492 24.206 22.326 22.762 22.652 23.464 30 25.482 25.470 24.079 24.651 24.011 26.580 31 26.468 26.755 24.100 24.673 25.531 27.812 Average Pressure (m) 28.020 28.177 28.252 28.883 27.237 29.419 Mnmum Pressure (m) 22.106 22.066 22.051 22.370 22.074 22.476 Maxmum Pressure (m) 35.960 35.875 35.778 35.662 35.490 36.108

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 25 Table 11. Node and network relabltes of best solutons of El-Mostakbal Cty network for dfferent network uncertantes at COV Q = 20% Node Node Relablty, R n (%) ID = 0.5 = 0.6 = 0.7 = 0.8 = 0.9 = 0.99 1 100 100 100 100 100 100 2 100 100 100 100 100 100 3 100 100 100 100 100 100 4 100 100 100 100 100 100 5 100 100 100 100 100 100 6 100 100 100 100 100 100 7 100 100 100 100 100 100 8 100 100 100 100 100 100 9 100 100 100 100 100 100 10 100 100 100 100 100 100 11 100 100 100 100 100 100 12 100 100 100 100 100 100 13 100 100 100 100 100 100 14 100 100 100 100 100 100 15 100 100 100 100 100 100 16 99.68 99.99 100 100 100 100 17 100 100 100 100 100 100 18 100 100 99.98 100 100 100 19 100 100 100 100 100 100 20 99.53 99.94 100 100 100 100 21 99.64 99.99 100 100 100 100 22 94.92 98.06 100 100 100 100 23 99.61 99.99 99.99 100 100 100 24 49.05 80.58 99.81 99.84 100 100 25 55.98 89.59 98.16 99.91 99.99 100 26 80.29 98.05 99.33 99.99 100 100 27 60.21 93.70 95.73 99.81 99.99 100 28 94.40 99.01 99.99 100 100 100 29 84.57 99.59 97.13 99.93 100 100 30 99.86 99.99 99.96 100 100 100 31 100 100 99.94 100 100 100 Network Relablty, R sm (%) 49.05 80.58 95.73 99.81 99.99 100 Network Relablty, R sa (%) Network Relablty, R sw (%) 90.82 97.73 99.46 99.97 100 100 91.80 98.09 99.46 99.97 100 100 The nformaton on the trade-off between cost and uncertanty s shown n Fg. 3, whch gves the relatonshp or trade-off between cost and uncertanty requrements for a range of uncertanty requrements on the degraded network confguratons. It s evdent from Fgure 3 that, for a gven level of uncertanty, the cost of the desgn ncreases wth the ncrease n the coeffcent of demand varaton. As mentoned prevously, usng a Monte Carlo smulaton, the nodal relablty at every node s calculated and the network relablty s derved from them. The results

Cost (LE*10 6 ) Cost (LE*10 6 ) 26 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt of the network relablty gven n Tables 8 and 11 are plotted n Fg. 4. It s clear that the network relablty s 100% for = 0.99 whch means very relable network under uncertanty n nodal demands up to COV Q = 20%. 3 2.75 COV Q = 20%, COV C = COV H = 0% COV Q = 10%, COV C = COV H = 0% 2.5 2.25 2 0.5 0.6 0.7 0.8 0.9 1 Fgure 3. Total cost of network versus uncertanty for El-Mostakbal Cty network for COV Q = 10% and 20% 3 2.75 COV Q = 20%, COV C = COV H = 0% COV Q = 10%, COV C = COV H = 0% 2.5 2.25 2 40 50 60 70 80 90 100 Network Relablty, R sm (%) Fgure 4. Total cost of network versus network relablty for El-Mostakbal Cty network for COV Q = 10% and 20%

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 27 CONCLUSIONS The Relablty-based optmzaton of water dstrbuton networks s presented and appled to a case study. The approach lnks a genetc algorthm (GA) as the optmzaton tool, the Newton method as the hydraulc smulaton solver wth the chance constrant combned wth the Monte Carlo smulaton to estmate network capacty relablty. The source of uncertanty analyzed s the future nodal external demands. The results at two values of coeffcent of varaton reveal the well known relaton between the total cost and network relablty, that the hgher the relablty requrement, the greater the desgn cost. The hgh relablty of network ncreases the performance of the network at normal condtons. NOMENCLATURE C, Hazen-Wllams roughness coeffcent for ppe connectng nodes, C T total cost D, dameter of ppe connectng nodes, n the system (m) D max maxmum dameter, (m) D mn mnmum dameter, (m) H pressure head at node, (m) H mnmum requred pressure head at node, (m) h f,mn head loss due to frcton n a ppe, (m) K converson factor whch accounts for the system of unts used. L, length of ppe connectng nodes,, (m) M total number of nodes n the network N total number of ppes N s total number of Monte Carlo smulatons P ( ) probablty Q dscharges nto or out of the node, (m 3 /s) Q s mean value of water supply at node, (m 3 /s) q, flow n ppe connectng nodes,, (m 3 /s) R s R x Z N system relablty nodal capacty relablty Independent varable obectve functon Greek Symbols probablty level for the node demands

28 Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt cumulatve dstrbuton functon mean of random varable Q, (m 3 /s) Q standard devaton of random varable Q, (m 3 /s) Q REFERENCES 1. Abdel-Gawad, H.A.A., "Optmal Desgn of Ppe Networks by an Improved Genetc Algorthm," Proceedngs of the Sxth Internatonal Water Technology Conference IWTC 2001, Alexandra, Egypt, March 23-25, 2001, pp. 155-163. 2. Abdel-Gawad, H.A.A., "Optmal Desgn of Water Dstrbuton Networks under a Specfc Level of Relablty," Proceedngs of the Nnth Internatonal Water Technology Conference, IWTC9 2005, Sharm El-Shekh, Egypt, March 17-20, 2005, pp. 641-654. 3. Babayan, A.V, Kapelan, Z., Savć, D.A., and Walters, G.A., "Least Cost Desgn of Robust Water Dstrbuton Networks under Demand Uncertanty". Journal of Water Resources Plannng and Management, ASCE, 2005, Vol. 131, No. 5, pp. 375-382. 4. Babayan, A.V, Kapelan, Z., Savć, D.A., and Walters, G.A., "Comparson of two methods for the stochastc least cost desgn of water dstrbuton systems," Engneerng Optmzaton, Vol. 38, No. 03, Aprl 2006, pp. 281-297. 5. Bao, Y., and Mays, L.W., "Model for Water Dstrbuton System Relablty," Journal of Hydraulc Engneerng, ASCE, Vol. 116, No. 9, 1990, pp. 1119-1137. 6. Carroll, C.W., "The Created Response Surface Technque for Optmzng Nonlnear Restraned Systems," Operatons Research, Vol. 9, 1961, pp. 169-184. 7. Debedan, B., Herrck, A.M., and Rayan, M.M., 2000, "Modellng and Optmzaton of Potable Water Network," Internatonal Ppelne Conference (IPC 2000) October 1-5, Calgary, Canada. 8. Ezzeldn, R.M, ''Relablty-Based Optmal desgn model for Water Dstrbuton Networks '' M. Sc. Thess, Mansoura Unversty, El-Mansoura, Egypt, (2007). 9. Facco, A.V., and McCormck, G.P., 1964, "Computatonal Algorthm for the Sequental Unconstraned Mnmzaton Technque for Nonlnear Programmng," Vol. 10, pp. 601-617. 10. Goulter, C., and Bouchart, F., "Relablty-Constraned Ppe Network Model," Journal of Hydraulc Engneerng, ASCE, Vol. 116, No. 2, 1990, pp. 211-229. 11. Goulter, I., Thomas, M., Mays, L.W., Sakarya, B., Bouchart, F., and Tung, Y.K., "Relablty Analyss for Desgn," n Water Dstrbuton Systems Handbook, (Larry W. Mays, Edtor n Chef), McGraw-Hll, 2000, pp. 18.1-18.52. 12. Herrck, A.M., 2001, "Optmum Computer Aded Hydraulc Desgn and Control of Water Purfcaton and Dstrbuton Systems," M. Sc. Thess, Mansoura Unversty, Egypt. 13. Lansey, K., and Mays, L., 1989, "Optmzaton Model for Water Dstrbuton System Desgn," Journal of Hydraulc Engneerng, ASCE, Vol. 115, No. 10, pp. 1401-1418.

Twelfth Internatonal Water Technology Conference, IWTC12 2008 Alexandra, Egypt 29 14. Lasdon, L.S., and Waren, A.D., "Generalzed Reduced Software for Lnearly and Nonlnearly Constraned Problems," Desgn and Implementaton of Optmzaton Software, H. Greenberg, ed., Sgthoff and Noordoff, Netherlands, 1979. 15. Lasdon, L.S., and Waren, A.D., "GRG2 User s Gude," Unversty of Texas at Austn, Tex., 1984. 16. Rayan, M.A., Debedan, B., El-Hak, N.G., and Herrck, A., "Optmzaton of Potable Water Network (Case Study)," Seventh Internatonal Water Technology Conference, IWTC 2003, Aprl 1-3, 2003, Caro, Egypt, pp. 507-522. 17. Savc, D.A., "Copng wth Rsk and Uncertanty n Urban Water Infrastructure Rehabltaton Plannng," Acqua e Cttà - I Convegno Nazonale d Idraulca Urbana, Sant Agnello (NA), 28-30 September 2005. http://www.csdu.t/l_sto/pubblcazon/altre_pubbl/s'a_relazongeneral/ A-Relazone_Mem_Savc.pdf 18. 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. 123, No. 2, 1997, pp. 67-77. 19. Su, Y.C., Mays, L.W., Duan, N., and Lansey, K.E., "Relablty-Based Optmzaton Model for Water Dstrbuton Systems," Journal of Hydraulc Engneerng, ASCE, Vol. 114, No. 12, 1987, pp. 1539-1556. 20. Swamee, P. K., and Jan, A. K., 1975, "Explct Equatons for Ppe-Flow Problems," Journal of the Hydraulcs Dvson, ASCE, Vol. 102, No. HY5, May, pp. 657-664. 21. Tolson, B.A., Maer, H.R., Smpson, A.R., and Lence, B.J., "Genetc Algorthms for Relablty-Based Optmzaton of Water Dstrbuton Systems," Journal of Water Resources Plannng and Management, ASCE, Vol. 130, No. 1, 2004, pp. 63-72. 22. Wood, D.J., "Computer Analyss of Flow n Ppe Networks Includng Extended Perod Smulatons: User s Manual" College of Engneerng, Unversty of Kentucky, Lexngton, KY, 1980. 23. Xu, C., and Goulter, I.C., "Probablstc Model for Water Dstrbuton Relablty," Journal of Water Resources Plannng and Management, ASCE, Vol. 124, No. 4, 1998, pp. 218-228. 24. Xu, C., and Goulter, I.C., "Relablty-Based Optmal Desgn of Water Dstrbuton Networks," Journal of Water Resources Plannng and Management, ASCE, Vol. 125, No. 6, 1999, pp. 352-362. 25. Xu, C., Goulter, I.C., and Tckle, K.S., "Assessng the Capacty Relablty of Ageng Water Dstrbuton Systems," Cvl Engneerng and Envronmental Systems, Vol. 20, No. 2, 2003, pp. 119-133.