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

Transcription

1 Twelfth Internatonal Water Technology Conference, IWTC 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 2 Twelfth Internatonal Water Technology Conference, IWTC 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.

3 Twelfth Internatonal Water Technology Conference, IWTC 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 4 Twelfth Internatonal Water Technology Conference, IWTC 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:

5 Twelfth Internatonal Water Technology Conference, IWTC 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) 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 Advantages and Dsadvantages of the Smulaton Approach

6 6 Twelfth Internatonal Water Technology Conference, IWTC 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]

7 Twelfth Internatonal Water Technology Conference, IWTC 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 K L, q, C, D, H H (6) where K s a converson factor whch accounts for the system of unts used, (K = 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 , 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 8 Twelfth Internatonal Water Technology Conference, IWTC 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

9 Twelfth Internatonal Water Technology Conference, IWTC Alexandra, Egypt 9 Smlarly, Eq. (9) s transformed nto a determnstc form as follows: W P Q D L H H C K P , 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/ , 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 10 Twelfth Internatonal Water Technology Conference, IWTC 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.

11 (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, IWTC Alexandra, Egypt 11

12 12 Twelfth Internatonal Water Technology Conference, IWTC 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

13 Twelfth Internatonal Water Technology Conference, IWTC 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) Total Demand = LPS

14 14 Twelfth Internatonal Water Technology Conference, IWTC 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 Ductle Ductle Ductle Ductle Ductle Ductle Ductle Ductle Ductle 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 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 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 m and the total demand for the network s 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

15 Twelfth Internatonal Water Technology Conference, IWTC 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 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 16 Twelfth Internatonal Water Technology Conference, IWTC Alexandra, Egypt Table 4. Optmal ppe dameters for El-Mostakbal Cty network ( = 0.5) Ppe ID Dameter (mm) Ppe ID Dameter (mm) 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

17 Twelfth Internatonal Water Technology Conference, IWTC 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** Average Pressure (m) Mnmum Pressure (m) Maxmum Pressure (m) * Orgnal desgn (Table 9.2), ** Optmal desgn (Table 9.4)

18 Pressure Head (m) 18 Twelfth Internatonal Water Technology Conference, IWTC Alexandra, Egypt 40 After Optmzaton Before Optmzaton 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.

19 Twelfth Internatonal Water Technology Conference, IWTC 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 = Cost (LE) 2,234,046 2,240,746 2,330,245 2,380,332 2,517,901 2,584,412

20 20 Twelfth Internatonal Water Technology Conference, IWTC 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 = Average Pressure (m) Mnmum Pressure (m) Maxmum Pressure (m)

21 Twelfth Internatonal Water Technology Conference, IWTC 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 = Network Relablty, R sm (%) Network Relablty, R sa (%) Network Relablty, R sw (%) 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 22 Twelfth Internatonal Water Technology Conference, IWTC 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.

23 Twelfth Internatonal Water Technology Conference, IWTC 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 = Cost (LE) 2,234,046 2,251,260 2,386,508 2,490,224 2,586,316 2,755,927

24 24 Twelfth Internatonal Water Technology Conference, IWTC 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 = Average Pressure (m) Mnmum Pressure (m) Maxmum Pressure (m)

25 Twelfth Internatonal Water Technology Conference, IWTC 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 = Network Relablty, R sm (%) Network Relablty, R sa (%) Network Relablty, R sw (%) 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

26 Cost (LE*10 6 ) Cost (LE*10 6 ) 26 Twelfth Internatonal Water Technology Conference, IWTC 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% COV Q = 20%, COV C = COV H = 0% COV Q = 10%, COV C = COV H = 0% Fgure 3. Total cost of network versus uncertanty for El-Mostakbal Cty network for COV Q = 10% and 20% COV Q = 20%, COV C = COV H = 0% COV Q = 10%, COV C = COV H = 0% Network Relablty, R sm (%) Fgure 4. Total cost of network versus network relablty for El-Mostakbal Cty network for COV Q = 10% and 20%

27 Twelfth Internatonal Water Technology Conference, IWTC 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 28 Twelfth Internatonal Water Technology Conference, IWTC 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 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 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 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 Bao, Y., and Mays, L.W., "Model for Water Dstrbuton System Relablty," Journal of Hydraulc Engneerng, ASCE, Vol. 116, No. 9, 1990, pp Carroll, C.W., "The Created Response Surface Technque for Optmzng Nonlnear Restraned Systems," Operatons Research, Vol. 9, 1961, pp 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 Goulter, C., and Bouchart, F., "Relablty-Constraned Ppe Network Model," Journal of Hydraulc Engneerng, ASCE, Vol. 116, No. 2, 1990, pp 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 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

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