development of a pda model for water distribution systems using harmony search algorithm

KSCE Journal of Civil Engineering (2010) 14(4):613-625DOI 10.1007/s12205-010-0613-7

− 613 −

www.springer.com/12205

Water Engineering

Development of a PDA Model for Water Distribution Systems UsingHarmony Search Algorithm

Chun Woo Baek*, Hwan Don Jun**, and Joong Hoon Kim***

Received April 2, 2009/Revised September 9, 2009/Accepted November 16, 2009

···································································································································································································································

Abstract

Hydraulic analysis of water distribution systems can be divided into DDA (Demand-Driven Analysis) and PDA (Pressure-DrivenAnalysis). Many studies have reported the superiority of the PDA over the DDA in the realistic simulation of hydraulic conditions underabnormal operating conditions. Many of the developed PDA models rely on iterative processes to solve the equations, which is a time-consuming task and even worse it is not possible to solve them in some cases. To improve the efficiency of the PDA, the present studyproposes a new PDA model which interfaces a hydraulic simulator and an optimization algorithm with a customized searching scheme. Thesuggested model is applied to differently sized water distribution systems under abnormal operating conditions and its results are comparedwith ones by the DDA model and two other PDA models. As results, the DDA may generate unrealistic hydraulic results under the abnormaloperating conditions while the three PDA models produce more realistic results. Moreover, the suggested PDA model with the newoptimization process simulates the hydraulic conditions under the abnormal operating conditions in large water distribution systemsefficiently compared to the other PDA models.Keywords: pressure driven analysis, demand driven analysis, abnormal operating conditions, Harmony Search, EPANET, waterdistribution system

···································································································································································································································

1. Introduction

The DDA (Demand-Driven Analysis) and the PDA (Pressure-Driven Analysis) are two approaches widely used to solvesteady-state hydraulic conditions in a water distribution system.In the DDA, nodal demands are always assumed to be satisfiedregardless of the available nodal pressure heads. Most ofhydraulic simulators such as EPANET (Rossman, 1994) andKYPIPE (Wood, 1980) are based on the DDA. In most cases ofhydraulic simulations such as pump operation and nodal headassessment, this assumption of the DDA generally producesaccurate results under normal operating conditions. However,Gupta et al. (1996), Tanyimboh and Tabesh (1997a) and Mays(2003) reported that the DDA may generate unrealistic resultssuch as negative nodal pressure heads under abnormal hydraulicconditions such as pipe failure, temporal demand increase andfire. Under such operation conditions, the available demands atsome of the demand nodes may be less than the predefineddemands since available demand at a demand node is dependanton the available nodal pressure head at that node. Therefore, thesimulated nodal pressure heads may be negative or unacceptablylow when compared to actual nodal pressure heads.

On the contrary, the nodal demands in the PDA can be fullysatisfied only if the nodal pressure head at that node is greaterthan the minimum nodal pressure head. Otherwise nodal demandcan be partially satisfied and is dependant on available nodalpressure head. For this reason, the unknown nodal demand andnodal pressure head should be solved simultaneously for thePDA simulation of the hydraulic condition. It is why simulationresults for abnormal operating condition by the DDA and thePDA are different. However, it should be noted that they couldgive us exactly same hydraulic simulation results under thenormal operating condition since nodal demands at all nodes canbe satisfied when a water distribution network is being operatedwithout a problem. When an event which can paralyze a certainportion of a water distribution system has occurred, it is obviousand reasonable that some of the demand nodes cannot beprovided with their full demand and the failure impact area of awater distribution system is dependant on the magnitude andlocation of the event. The PDA can simulate the demand defici-encies and the affected range of a water distribution system.Therefore, the PDA is better than the DDA to simulate thehydraulic condition under abnormal operating conditions. Overallcomparison between the DDA and the PDA is shown in Table 1.

*Member, Research Associate, Centre for Ecohydrology and School of Environmental System Engineering, University of Western Australia, Crawley,WA 6009, Australia (E-mail: [email protected])

**Member, Assistant Professor, School of Civil Engineering, Seoul National University of Technology, Seoul 139-743, Korea (Corresponding Author, E-mail: [email protected])

***Member, Professor, School of Civil, Environmental and Architectural Engineering, Korea University, Seoul 136-713, Korea (E-mail: [email protected])

Chun Woo Baek, Hwan Don Jun, and Joong Hoon Kim

− 614 − KSCE Journal of Civil Engineering

Although the PDA produces more realistic solutions underabnormal operating conditions, it does require extensive fielddata to determine HOR (Head-Outflow Relationship) (Mays,2003) for application to real networks, which is very difficult toobtain. This is why many researchers such as Bhave (1981),Goulter and Coal (1986), Germanopoulos (1985), Reddy andElango (1989, 1991), Wagner et al. (1988) and Chandapillai(1991) have suggested theoretical relationships of nodal headand nodal flow. As a summary, Gupta and Bhave (1996) com-pared each of the suggested relationships.

The other obstacle to the easy application of the PDA to realnetworks is the development of a methodology to solve theunknown nodal demand and nodal head simultaneously andefficiently. Currently, the most of the proposed models requireiteration processes to reach the final solution, which is a time-consuming task and sometime is impossible to reach the finalsolution, especially for large water distribution systems.

To facilitate efforts to perform the PDA, this study proposes aconceptual structure of the PDA model consisting of anoptimization algorithm and a DDA hydraulic simulator. The newmodel’s optimization algorithm assumes a possible nodal head ateach demand node, after which the flow at each demand node isdetermined using HOR. The determined flow becomes thedemand flow rate for each demand node and using it thehydraulic simulator simulates hydraulic conditions to determinethe nodal head at each node. Then, the calculated nodal head iscompared with the assumed nodal head. Depending on thedifference between them, a new series of nodal heads is assumedby the optimization algorithm until the difference is minimized.This procedure obtains a partially satisfied flow at each demandnode under the abnormal operating conditions. The new PDAmodel is validated by applying it to water distribution systemsunder two abnormal operating conditions; pipe failure and firedemand. Moreover, results from three existing hydraulic simula-tion models and the new PDA are compared.

2. Concept of the PDA

2. 1 The Current PDA ModelsBhave (1981) firstly proposed the concept of the PDA consi-

dering both nodal flow and nodal head for the analysis of waterdistribution systems. He named the conventional hydraulicsolving methods as NHA (Node Head Analysis, identical toDDA). In the NHA, it is assumed that the demand at every nodeis always satisfied, after which flows in pipes and nodal heads atdemand nodes are estimated. However, he reported that thewater distribution system may fail to simultaneously provide thedemands at all the nodes if pressure heads are not adequate withthe NHA. To resolve this problem, a new method called NFA(Node Flow Analysis, identical to PDA) was suggested. with thefollowing two assumptions; (1) neither any additional heads atthe source nodes nor any additional boosting pressures anywherein the distribution system are provided; and (2) the flow at a nodeis available only if the available nodal head is equal to or morethan the minimum head. Otherwise, no flow is available at thenode. An iterative procedure is used to obtain solutions for awater distribution system in the NFA. After assuming initialconditions for all nodes, base demands are determined by theassumed condition. Then, the NHA is performed to examineactual conditions at all nodes and new conditions for all nodesare assumed. This iterative procedure continues until theassumed conditions equate to the calculated conditions for allnodes. He applied the NFA model to a virtual water distributionsystem and compared the results with those of the NHA.

Goulter and Coal (1986) also reported that available nodaldemand can be satisfied only if the possible nodal head is greaterthan the minimum nodal head. Otherwise no nodal demand isavailable. This concept is used as a constraint of the objectivefunction for reliability assessment of water distribution systems.

Germanopoulos (1985) suggested an empirical function for thepressure-consumption relationship and used the Newton-Raphsonprocedure and the Jacobian matrix to simulate water distributionsystems under pressure dependent demand conditions. Heshowed that the assumption of the DDA may not be valid while awater distribution system is being operated under failure condi-tion causing lower nodal heads at the demand nodes.

Wagner et al. (1988) reported that flow actually supplied to thenode is dependent on the nodal head at the node and that flow isproportional to the square root of nodal head. In this study, theminimum head and the service head for each node were adopted

Table 1. Comparison of the DDA and the PDA

DDA (Demand Driven Analysis) PDA (Pressure Driven Analysis)

Assumptions Demands of nodes are always fully satisfied Demands of nodes are dependant on available nodal head

Applications Normal operation condition Abnormal operation condition (leakage, failure, pump problem, fire fighting demand, etc)

Reliability for abnormal operating conditions Low High

Defects Negative nodal pressure heads may occur under an abnormal operating condition

Need of a relation equation between nodal heads and nodal flowsSolving nodal demand and head simultaneously is very difficult

Solving Method Iterative procedures to satisfy continuity and loop equations Iterative procedure using the DDA simulation

Development of a PDA Model for Water Distribution Systems Using Harmony Search Algorithm

Vol. 14, No. 4 / July 2010 − 615 −

and the flow conditions at each node were divided into threecategories: (1) no flow, (2) partial flow, and (3) full. Dependingon each flow category, the state of the network was also dividedinto the failure mode, the reduced mode, and the normal mode,respectively. Using SDP8 which is a hydraulic analysis programmade by Charles Howard and Associates, Ltd (1984), he simu-lated water distribution systems to determine the network state(mode).

Reddy and Elango (1989, 1991) also reported that the outflowand the available residual head are related to each other, so thatany simulated hydraulic conditions may be unrealistic if thisrelationship is not considered.

Gupta and Bhave (1996) compared the models proposedby Bhave (1981), Goulter and Coal (1986), Su et al. (1987),Germanopoulos (1985), Reddy and Elango (1989, 1991), Wagneret al. (1988) and Chandapillai (1991). Tanyimboh and Tabesh(1997a) found that pressure deficiency may cause seriousdiscrepancy between actual and simulated hydraulic conditionswhen a DDA model is applied to real water distribution systems.Also, Tanyimboh and Tabesh (1997b) suggested the SHM (SourceHead Method) that determines available nodal supply along withthe source head of a network to calculate the reliability of waterdistribution systems. Additionally the SHM was modified toISHM (Improved SHM) and ASHM (Advanced SHM) byTanyimboh et al. (1997, 2000) and to HDSEPRA (Head DrivenSimulation based Extended Period Reliability Analysis) byTabesh et al. (2001) for reliability analysis. However, the SHMcan be used only for a single source network.

Several studies used NLP (Non-Linear Programming) techni-ques to analysis pressure dependent demand conditions. Ackleyet al. (2001) and Tabesh et al. (2002) used the SQP (SequentialQuadratic Programming) and the modified Newton-Raphsonmethod, respectively, to develop a head-driven simulation model.Tabesh et al. (2004) modified his previous model to gain theability to perform extended period reliability analysis for a waterdistribution network considering the stochastic concept. Tanyimbohet al. (2003) developed a pressure-driven model using the equa-tion of Wagner et al. (1988) and Chandapillai (1991) and NRLSA(Newton-Raphson Line Search Algorithm).

Ozger (2003) suggested a semi-PDA (SPDA) model usingartificial reservoirs and iterative procedures. He defined theMRP (Minimum Required Pressure) as the pressure criterion. Inhis model, required demand cannot be satisfied if the nodal headis less than the MRP. Mays (2003) used the SPDA model to assessan water distribution system reliability. A similar PDA algorithm,called a PDNA (Pressure-Deficient Network Algorithm), wasdeveloped by Ang and Jowitt (2006) based on artificial reservoirsand iterative procedures.

Wu and Walski (2006) and Wu et al. (2006) used a powerfunction as the HOR and a modified GGA (Global GradientAlgorithm) to solve the PDD (Pressure Dependent Demand)condition, and developed a PDD model as one of the modelingfunctions in WaterCAD and WaterGEMS (Bentley, 2006). Wuand Walski (2006) and Todini (2006) mentioned that the emitter

function in EPANET which is the simplest way to handle PDAmay give unrealistic results when nodal pressure head is high ornegative.

As described above, the most of all PDA models developed tosimulate hydraulic conditions under abnormal operating condi-tions relied on an iteration process to reach the solution. Thisiteration process is a time-consuming task and sometime it isimpossible to reach the solution, especially, for large waterdistribution systems. Only the model of Wu and Walski (2006)and Wu et al. (2006) which used direct analytical methodshowed sufficient solving efficiency for PDA. It is clear that themodel suggested by Wu and Walski shows very efficient andstable results when it is applied to simulate abnormal operatingconditions. However, our model which is adopted the iterationprocedure to reach the solution demonstrates how it can beimproved by the heuristic optimization algorithm in terms ofsolving efficiency and how it reaches the solution stably,especially, when it is applied to a large water distribution system.

2.2 The Nodal Head-Outflow Relationship (HOR)For real networks, it is very difficult to define a HOR, the

relationship between the nodal head at a demand node and theavailable flow at that node (Mays, 2003), since a wide range offield data must be collected and calibrated for development ofthe relationship. However, many studies have defined the re-lationship theoretically such as Bhave (1981), Goulter and Coal(1986), Germanopoulos (1985), Reddy and Elango (1989, 1991),Wagner et al. (1988), Chandapillai (1991) and Fujiwara andGanesharajah (1993). Gupta and Bhave (1996) reviewed thoseHORs for predicting the performance of a water distributionsystem under the deficient condition and reported the perfor-mance superiority of the HORs by Wagner et al. (1988) andChandapillai (1991). Therefore, those HORs are used to cal-culate the available nodal flows depending on the nodal heads inthis study. Those equations are shown in Eq. (1) to Eq. (4).

(adequate-flow), if (1)

(partial-flow),

if (2) (no-flow), if (3)

(4)

where n is the coefficient of node j, Rj is the resistance constantand m is an exponent.

Fig. 1 shows the HOR suggested by Wagner et al. (1988) andChandapillai (1991).

3. The Suggested PDA Model

3.1 The Algorithm of the Suggested PDA ModelThe algorithms used for a DDA model such as the Linear

Method (KYPIPE, Wood, 1980), the Newton-Rapson Method

qjavi qj

req= Hjavi Hj

des≥

0 qjavi< qj

req Hjavi Hj

min–Hj

des Hjmin–

------------------------⎝ ⎠⎛ ⎞

1 n/

qjreq<=

Hjdes Hj

avi Hjmin> >

qjavi 0= Hj

min Hjavi≥

Hjdes Hj

min Rj qjreq( )m+=



(WADISO, Gessler and Walski, 1985) and the Gradient Method(EPANET, Rossman, 1994) can solve the equations only whenthe flow at each node is given. However, since nodal flows aredependant on unknown nodal heads, there is no direct solvingmethod for the PDA. For this reason, an iteration procedure isrequired for the PDA. As noted before, however, such a procedureacts as a bottle neck in the PDA because of its inefficiency andlow applicability. To improve the efficiency and the applicabilityof the PDA, especially, for large water distribution systems, thesimple iteration procedure is replaced with an optimizationprocedure in the suggested PDA. After assuming nodal heads(Hass) at demand nodes, the available nodal flows (Qavi) at thosenodes are calculated by the HOR. The calculated available nodalflows are assumed to be the demands for the DDA in order todetermine the calculated nodal heads (Hcal) and the differencebetween the assumed and the calculated nodal heads isestimated. Then, an optimization algorithm generates another setof the assumed nodal heads to minimize the difference. Thesesteps are explained in Fig. 2.

3.2 Harmony Search AlgorithmTo overcome the defects of traditional optimization algorithms,

heuristic optimization algorithms have been developed (Maysand Tung, 1992) many of which find very decent, near optimumsolution while consuming relatively less computation-time andcomputer memory. Moreover, the subtle nonlinear characteristicsand complex derivatives of the model are maintained. Also, theyare not sensitive to the initial value choices (Geem, 2000). Themotivation of the most heuristic algorithms is found in theparadigm of natural processes such as the metallic annealingprocess in SA (Simulated Annealing, Kirkpatrick, 1983) and theevolutionary process of the Darwin's natural selection theory inGA (Genetic Algorithm, Holland, 1975). The algorithm used inthis study is HS (Harmony Search) which is conceptualized fromthe artificial process represented by the search for a betterharmony involved in a musical performance process (Geem etal., 2001, 2002; Kim et al., 2001).

In HS, three basic parameters (HM, HMCR and PAR) are usedto find near-global optima. Conceptually, HM (Harmony Memory)is similar to the population size of GA. The best sets of experi-enced harmony are memorized in HM. HMCR (HM Considering

Rate) is introduced to escape from the local optima just like themutation probability used in GA. PAR (Pitch Adjusting Rate) isadopted to improve the solution by searching the adjacent region.HS works with those three basic parameters (HM, HMCR andPAR) and Fig. 3 shows flowchart of HS suggested by Geem etal. (2001)

Paik (2001), Kim et al. (2004) and Paik et al. (2005) reportedthat changes in the HS parameters may shorten the searchingtime for the global optimum. Paik (2001) and Paik et al. (2005)divided HS into three types, namely, type 1 (HS1), type 2 (HS2),

Fig. 1. Nodal Head-Outflow Relationship

Fig. 2. Flowchart for the PDA Model

Fig. 3. Flowchart of HS


Vol. 14, No. 4 / July 2010 − 617 −

type 3 (HS3) and suggested the MHS (Modified HS) having adifferent method for applying HMCR and PAR. Kim et al.(2004) suggest the ReHS (Revised HS) using non-fixed valuesof HMCR and PAR to improve the searching capability foroptimization of pipe rehabilitation and this approach is used inthis study.

3.3 A New Parameter for HS: SAM (SAMpling)Although the ReHS is the most advanced HS, it is not efficient

to determine proper values of heads at demand nodes, especially,when applied to large water distribution systems where the num-ber of decision variables are more than hundreds or thousands.Initial nodal heads assumed by HS process are quite importantfor efficiency of suggested PDA model. However, possible rangefor these initial values could be too wide depending on the size ofwater distribution system. If the nodal head can be assumedthrough the range close to actual nodal head, the efficiency of themodel will be improved. To improve the searching efficiencyof the ReHS, a new parameter of HS, which is called SAM(SAMpling) is suggested.

In brief, SAM uses results from the hydraulic simulation per-formed for the previous iteration step when it is constructing anHM for the next iteration to search for the optimum solution.SAM is based on the assumption that the values of the nodalheads calculated from the hydraulic simulation performed for theprevious iteration step are close to the actual values of the nodalheads. Thus, if the value of a head calculated at each node is usedas the assumed head at that node for the next iteration, the actualhead (the optimum solution for the optimization) can be obtainedeasily by the ReHS. This assumption is reasonable since flowrates in pipes and pressure heads of nodes are closely related toeach others, rather than independent, in order to satisfy the nodeand loop equations. In addition, the heads at nodes are deter-mined by the network characteristics such as the diameters andlengths of pipes, flow demands at the demand nodes and the totalenergy heads at the water sources. Those are the theoretical back-ground for the introduction of SAM to the optimization procedure.

Under this assumption, once an iteration has been completedand the values of the objective function of each HM have beenevaluated, the head calculated at each node from the best fit inHM is used as the assumed head at each node for the nextiteration. For example, as shown in Fig. 4, we have three nodeswhich are “node 1”, “node 2”, and “node 3” and ten HMS(Harmony Memory Size). From the previous iteration, the valueof the objective function of each HM is evaluated and based onthis value, each HM is ordered. In addition, we have the head ofeach node calculated by a hydraulic simulator using the assumedhead. SAM is applied to generate HM for the next iteration. Asmentioned before, the HM of Rank 9, where ‘rank’ is the orderaccording to the evaluation value of the objective function, issubstituted by the calculated heads of the HM of Rank 1 whichare 70.01 m, 68.24 m, and 66.21. Thus, the final HM for the nextiteration is shown in the latter part of Fig. 4. It should be notedthat the HM of Rank 10, which is the worst HM, is handled by

HMCR and that SAM revises the second worst HM. The SAMapplication achieves more feasible HM to search for the optimumsolution and the searching efficiency is therefore remarkablyimproved.

3.4 The Optimization ProcedureFor calculating the available nodal flows (Qavi) depending on

the nodal heads (Hass), the HOR equation suggested by Wagneret al. (1988) and Chandapillai (1991) is used in this study (Eqs.(1) to (4)) with 2.0, 0.1 and 2.0 being set as the coefficient nvalue in Eq. (2) and the Rj and m values in Eq. (4), respectively.Those values were originally used for Gupta and Bhave (1996)’sstudy and are applied here for a comparison.

To determine the possible nodal head range for assuming thenodal head (Hass), the minimum nodal elevation and themaximum surface elevation of the reservoirs or tanks in thesystem are used as the minimum range ( ) and the maximumrange ( ). If pumps are installed in a network, the maximumpumping head is added to the maximum surface elevation of thereservoir or tank for the maximum nodal head.

The sum of square error is used as an objective function inorder to enhance the initial approaching speed to the optimalsolution. Eq. (5) shows the objective function and the constraintof the optimization.

Minimization of

(N is the number of nodes in the system)(5)Subject to for all j

In HS, one iteration consists of applying the three parameterswhich are HMCR, PAR, and SAM. Fig. 5 shows the order ofapplying these parameters and the HM changes to determine theoptimum solution for a demand node as each parameter isapplied. Once initial HM is determined, HMCR is applied toavoid the situation of the searching process falling into the local

Hsysmin

Hsysmax

Hjass Hj

cal–( )2

j 1=

N

∑

Hsysmin Hj

ass Hsysmax≤ ≤

Fig. 4. Example of SAM to Generate HM for the Next Iteration



optima. In Fig. 5, the worst and second worst HM in the 20 HMSare replaced with new values since the value of HMCR is set to0.9 so that 10% of the total HM are replaced. Then, the ReHSestimates the fitness of the HM and ranks them by the fitness.Using the ordered HM, PAR is applied to adjust one of the HMto improve the fitness of the selected HM. After the PARapplication is completed, the ReHS re-estimates the fitness of theHM and ranks them by the fitness again. As the final process foran iteration, SAM is applied to improve the searching efficiencyof the ReHS by using the calculated nodal heads from the twobest-fitted HM (Rank 1 and Rank 2) and the ReHS re-estimatesthe fitness of the HM and ranks them by the fitness again. Then,the next iteration begins with applying HMCR. An example ofthese processes is shown in Fig. 5 with real values of assumedheads at a single node.

3.5 The Structure of the Suggested PDA ModelThe EPANET.DLL is used as the hydraulic simulator for the

suggested PDA and the GUI (Graphic User Interface) is writtenin Visual Basic. The suggested PDA consists of a main moduleand 4 sub-modules (shown in Fig. 6).

The Main Module controls the sub-modules and the Input-Output module handles input and output data using MS-Excel.An “.inp” file of EPANET is used to obtain water distribution

system data and the EPANET.DLL simulates various operatingconditions of the network in the Hydraulic Simulation Module.The Optimization Module adopted by the ReHS determines theoptimal solution with aid of the D/B Control Module and theHydraulic Simulation Module.

4. Application

4.1 Simple Serial NetworkAs the first application for validation of the suggested model,

we use the simple network of Gupta and Bhave (1996) shown inFig. 7 and its data are shown in Table 2. This network was used

Fig. 5. Example of ReHS Searching Process

Fig. 6. Structure of the PDA Model


Vol. 14, No. 4 / July 2010 − 619 −

for reviewing different HOR equations by Bhave (1981), Goulterand Coal (1986), Su et al. (1987), Germanopoulos (1985), Reddyand Elango (1989, 1991), Wagner et al. (1988) and Chandapillai(1991). The suggested model is validated with additional demandloading condition for fire fighting and the results from the DDA(by EPANET), the Gupta’s PDA (GPDA), and the suggestedPDA are compared as shown in Table 3.

As shown in Table 3, under the normal operating condition,nodal demands are 2.0, 2.0, 3.0 and 1.0 m3/min and the pressureheads calculated by the DDA are 7.30, 6.27, 1.24 and 6.01 m,respectively. The simulation results confirm the ability of thesystem to supply the required demand of each node. When anadditional 3.0 m3/min is required at Node 4 for fire fighting, thenodal pressure heads at Nodes 1 to 4 calculated by the DDA are5.14 m, 0.74 m, -9.84 m and -7.87 m, respectively. The negativenodal pressure heads at Nodes 3 and 4 are caused by the

additional 3.0 m3/min at Node 4. The DDA assumes that alldemands, including the additional 3.0 m3/min at Node 4, can besupplied; it incurs an additional hydraulic head loss which isgreater than the initial pressure head at Node 1. In reality,however, supplying all demands to every node is not possible if4.0 m3/min must be supplied at Node 4. Thus, the demands atNodes 1, 2, and 3 should be less than the initial demands for thiscondition to be simulated properly. The results of the suggestedmodel show that the actual demand at Node 3 should be reducedfrom 3.0 to 0.32 in order for Node 4 to receive 4.0 m3/min andthe demands at Nodes 1 and 2 to be maintained. The nodalpressure heads calculated at each node are 7.10, 5.77, 0.01 and2.24 m, respectively. Results from the developed PDA modeland the GPDA show good agreement in terms of availabledemands and nodal pressure heads at all the nodes, especially, atNode 3.

Fig. 7. Gupta and Bhave (1996)'s Network

Table 2. Data of the Gupta and Bhave (1996)’s network

Node ID Elevation(m)

Demand(m3/min) Pipe ID Length

(m)Diameter

(mm)Hazen-Williams

C Value

Node 1 90 2.00 Pipe 1 1000 400 130

Node 2 88 2.00 Pipe 2 1000 350 130

Node 3 90 3.00 Pipe 3 1000 300 130

Node 4 85 1.00 Pipe 4 1000 300 130

Table 3. Simulation Results (Gupta and Bhave (1996)'s network)

NodeID

Normal Operating Condition Additional demand loading condition for fire fighting

Elevation(m)

Demand(m3/min)

PressureHead byEPANET

(m)

RequiredDemand(m3/min)

DDA by EPANET GPDA DevelopedPDA Model

AvailableDemand(m3/min)

PressureHead(m)


PressureHead(m)


PressureHead(m)

Node 1 90 2.00 7.30 2.00 2.0 5.14 2.00 7.0390 2.00 7.10

Node 2 88 2.00 6.27 2.00 2.00 0.71 2.00 5.6170 2.00 5.77

Node 3 90 3.00 1.24 3.00 3.00 -9.84 0.37 0.0080 0.32 0.01

Node 4 85 1.00 6.01 4.00 4.00 -7.87 4.00 1.9455 4.00 2.24



4.2 Looped Network (Under pipe failure condition)Fig. 8 shows the water distribution system used in Mays

(2003)’s study consisting of 2 reservoirs, 13 nodes and 21 pipes,while Tables 4 and 5 show the network properties. The develop-ed model is applied to Mays (2003)’s network to compare theresults with the SPDA model (Ozger, 2003). The SPDA model isperformed by the following procedures. A water distributionsystem is simulated initially by the DDA to identify nodes withless nodal head than the MRP. Once pressure-deficient nonzerodemand nodes are identified, the properties of the pressure-deficient node are modified. The nodal elevation is temporallyreplaced with a value obtained by the sum of the real nodeelevation and the MRP, while the nodal demand is set to zero. Anartificial reservoir is then connected to the node by a very shortCV pipe that allows flow only from the node to reservoir. Theelevation of the artificial reservoir is set to the same level as the

node. Upon completion the modifications, the DDA is performeduntil none of the artificial reservoirs receive more water thanneeded. If some of artificial reservoirs receive more flow thanneeded, they are removed from the system and the correspondingnodal properties are restored.

The MRP in the SPDA is set at 15m, based on Mays (2003)’sstudy. In this application, values of nodal elevation with anadditional 15 m are used as for every demand node. Table 6shows the simulation results of the DDA, the SPDA and thesuggested PDA model when Pipe 3 fails.

As the result of the DDA by EPANET, the nodal pressureheads at nine nodes (nodes 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12) arebelow 15 m. Even though the nodal pressure heads at thesenodes are too low to provide the required demand ( ), it issimulated that the required demand at each node is provided,which is not possible in real. As results of the SPDA, four nodes(nodes 4, 6, 8 and 12) cannot be supplied with the requireddemand. The nodal pressure head is 15 m for those nodes andpartially satisfied demands of 165.77 m3/hr, 497.97 m3/hr, 274.74m3/hr and 66.25 m3/hr are calculated for each node. In case of thesuggested PDA, seven nodes (nodes 3, 4, 6, 7, 8, 11 and 12)

Hjdes

qjreq

Fig. 8 Mays (2003)’s Network

Table 5. Pipe Properties of the Mays (2003)'s Network

Pipe ID Length(m)

Diameter(mm) Roughness Pipe ID Length

(m)Diameter

(mm) Roughness

Pipe 1 609.60 762 130 Pipe 12 1371.60 381 108

Pipe 2 243.80 762 128 Pipe 13 762.00 254 106

Pipe 3 1524.00 609 126 Pipe 14 822.96 254 104

Pipe 4 1127.76 609 124 Pipe 15 944.88 305 102

Pipe 5 1188.72 406 122 Pipe 16 579.00 305 100

Pipe 6 640.08 406 120 Pipe 17 487.68 203 98

Pipe 7 762.00 254 118 Pipe 18 457.20 152 96

Pipe 8 944.88 254 116 Pipe 19 502.92 203 94

Pipe 9 1676.40 381 114 Pipe 20 883.92 203 92

Pipe 10 883.92 305 112 Pipe 21 944.88 305 90

Pipe 11 883.92 305 110

Table 4. Node Properties of the Mays (2003)'s Network

NodeID

Elevation(m)

RequiredDemand(m3/hr)

NodeID

Elevation(m)

RequiredDemand(m3/hr)

Node 1 27.43 0.00 Node 9 32.61 0.00

Node 2 33.53 212.40 Node 10 34.14 0.00

Node 3 28.96 212.40 Node 11 35.05 108.00

Node 4 32.00 640.80 Node 12 36.58 108.00

Node 5 30.48 212.40 Node 13 33.53 0.00

Node 6 31.39 684.00 Resvr. 1 60.96 N/A

Node 7 29.56 640.80 Resvr. 2 60.96 N/A

Node 8 31.39 327.60


Vol. 14, No. 4 / July 2010 − 621 −

cannot be supplied with the required demand. The pressureheads of those nodes are less than 15 m and the available supplydemands are less than the required demand. The discrepancybetween the SPDA and the suggested PDA is found in severalnodes. The required demand at Node 7 is satisfied by the SPDAbut not by the suggested PDA. However, the difference in theavailable demand calculated by the two models is not significant.Nevertheless, at Nodes 3, 7 and 11, the PDA simulates that therequired demands are not satisfied whereas the SPDA simulatesthat they are fully supplied. In the case of Nodes 4, 6, 8 and 12,both models simulate that they cannot be supplied fully, but theysimulate different levels of demand deficiency.

The total demands of the system are 2390.7 m3/hr and 2875.8m3/hr as calculated by the SPDA and the PDA, respectively.These differ from the DDA value of 3146.4 m3/hr, indicating thata lower total water supply is possible if Pipe 3 fails. In fact, Pipe3 was determined to be the 4th most important pipe out of the 21pipes in the system by the reliability analysis of Mays (2003).Thus, any change in the hydraulic condition due to failure of thecritical pipes such as Pipe 3 should be assessed properly tominimize the failure impact on a water distribution system. Forthe hydraulic simulation with Pipe 3 failure, the DDA may notproduce reliable results since it assumes that full demands can besupplied to those nine nodes whose nodal heads are lower thanthe MRP. If those unreliable results by the DDA are used forreliability assessment for each node, the reliabilities of nodes 3,4, 5, 6, 7, 8, 11 and 12 are determined at unrealistically low levelssince the DDA assumes that full demand supply at those nodes ispossible. However, the actual reliabilities of nodes 3, 5, 7, and 11

are high with the SPDA. Tanyimboh and Tabesh (1997a) statedthat when a network with locally insufficient heads is simulatedusing the DDA, the deficiency appears to be far more serious andwidespread than it is in reality, which confirms the necessity ofthe PDA.

4.3 Enhancement of Optimal Solution Search Speed bySAM

As mentioned above, since the real water distribution systemsrequiring simulation by the PDA model are usually composed ofseveral hundred or thousands pipes and nodes, a new ReHSparameter, SAM, is added to the developed PDA model toimprove its searching efficiency. Simulation results with andwithout SAM are compared to demonstrate SAM’s ability toimprove the searching efficiency. To demonstrate the effect ofSAM on the searching efficiency, “Net3”, which is one of thesample networks used in EPANET is used as shown in Fig. 9.The same network is used by Ozger (2003) to verify his SPDAmodel. “Net3” consists of 92 nodes, 117 pipes, 2 reservoirs, 3tanks, and 2 pumps as shown in Fig. 9. The parameters of theReHS for this problem are as follows:

- HMS (HM Size): 50 HMS- HMCR: 0.8- PAR: 0.2- SAM: 0.02 (which means that only one out of 50 HMS will

be replaced)

is set as the ground elevation and as the summationof and 103 ft (45 psi), which was the value used by Ozger

Hjmin Hj

des

Hjmin

Table 6. Simulation Results under Pipe 3 Failure Condition (Mays (2003)'s Network)

NodeID

Normal ConditionPipe 3 Failure

EPANET(DDA) SPDA PDA

NodeEl.(m)

OriginalDemand(m3/hr)

TotalHead(m)

Avail.Demand(m3/hr)

TotalHead(m)

PressureHead(m)

Avail.Demand(m3/hr)

TotalHead(m)

PressureHead(m)

Avail.Demand(m3/hr)

TotalHead(m)

PressureHead(m)

1 27.43 0.00 59.71 0.00 60.39 32.96 0.00 60.59 33.16 0.00 60.50 33.07

2 33.53 212.40 59.20 212.40 60.15 26.62 212.40 60.44 26.91 212.40 60.31 26.78

3 28.96 212.40 56.08 212.40 34.73 5.77 212.40 46.87 17.91 193.04 41.35 12.39

4 32.00 640.80 54.99 640.80 34.76 2.76 165.77 47.00 15.00 507.00 41.39 9.39

5 30.48 212.40 55.08 212.40 42.31 11.83 212.40 50.45 19.97 212.40 46.82 16.34

6 31.39 684.00 49.85 684.00 34.79 3.40 497.97 46.39 15.00 558.48 41.39 10.00

7 29.56 640.80 49.95 640.80 36.23 6.67 640.80 46.57 17.01 588.00 42.19 12.63

8 31.39 327.60 48.95 327.60 36.16 4.77 274.74 46.39 15.00 277.46 42.16 10.77

9 32.61 0.00 52.23 0.00 49.00 16.39 0.00 53.55 20.94 0.00 51.36 18.75

10 34.14 0.00 53.54 0.00 51.48 17.34 0.00 55.01 20.87 0.00 53.29 19.15

11 35.05 108.00 48.98 108.00 46.45 11.40 108.00 51.53 16.48 103.89 48.92 13.87

12 36.58 108.00 48.75 108.00 45.93 9.35 66.25 51.58 15.00 96.92 48.65 12.07

13 33.53 0.00 52.14 0.00 38.85 5.32 0.00 48.36 14.83 0.00 44.25 10.72

sum 3146.4 2390.7 2875.8



(2003). In other words, 45 psi is the available pressure head atthe demand nodes. If the available pressure head is less than 45psi, only partial, rather than full, demand is available at thedemand node.

As the termination condition, if the difference between theassumed and calculated nodal heads at every node is less than0.05 ft, ReHS is assumed to have reached the final solution.Under the normal operating condition (without pipe failure), theaverage of the nodal pressure heads is 60.7 psi (140.01 ft),implying the most of the demand nodes, except Node 34, have

sufficient nodal pressure heads to receive the full demands. Thus,Node 34 cannot be provided with full demand regardless of theoperating condition. 16 pipes out of 117 pipes with flow rates aregreater than 4,000 GPM are selected and a simulation is per-formed under pipe failure condition for each of these 16 pipes.Simulation results with and without SAM are shown in Table 7.

Except Pipes 49 and 70, pipe failures of the 16 pipes causedemand deficiency at several demand nodes and the level ofdemand deficiency over the entire network ranges from 1.93GPM to 520.56 GPM, which is less than 4% of the total demandof 12,219 GPM. For Pipes 49 and 70, the hydraulic condition ofthe network is significantly changed as 22 demand nodes cannotbe provided with their full demand and the demand deficiency atthe entire network rises to 1,080 GPM.

The improvement in the searching efficiency by SAM isshown in Table 7. With SAM, the change in the hydraulic condi-tion due to a pipe failure is simulated as being less than 3 secondsin the cases of 14 pipes, except Pipes 49 and 70 compared to morethan 8 minutes and 5000 iterations without SAM. In the case ofPipe 116, only one iteration consisting of applying the three HMparameters is required to reach the solution either with or withoutSAM, since its failure does not make significant change in thehydraulic condition of the network. Figs. 10 to 12 show thechange, with increasing number of iterations, in the values of theobjective function to simulate the failure of Pipes 49, 70, and 71.With SAM, the objective function value approaches the optimalvalue within 50 iterations, whereas more than 1000 iterationswithout SAM still fail to produce an objective function value close

Fig. 9. The Net3 Network

Table 7. Pipe Failure Analyses with and without SAM on the “Net3” Network

FailedPipe

Flows under Normal

condition

With SAM Without SAM

Iteration time(Min/ Sec)

No. of demandDeficient

Nodes

DeficientDemand(GPM)

Iteration time(Min/ Sec)

No. of demandDeficient

Nodes

DeficientDemand(GPM)

4 7759.98 1 /0 2 5.95 5,000 7/08

115 7759.98 1 /0 2 5.95 5,000 9/09

116 7759.98 1 /0 1 2.38 1 /1 1 2.38

117 7759.98 1 /1 2 5.95 5,000 9/12

45 6366.38 1 /0 3 2.51 5,000 8/09

39 6282.01 1 /1 4 5.61 5,000 7/21

42 6263.83 1 /1 4 3.53 5,000 8/01

112 6249.53 1 /1 4 3.48 5,000 9/04

21 6221.22 1 /1 5 33.45 5,000 7/09

40 6220.38 1 /0 4 5.35 5,000 7/22

48 6198.30 1 /0 2 1.93 5,000 7/13

41 6171.21 1 /1 4 5.15 5,000 7/11

22 5941.98 1 /1 1 4.65 5,000 8/01

49 4973.19 1,149 3/22 22 1080.54 5,000 7/15

70 4973.19 686 1/45 22 1081.78 5,000 7/10

71 4721.30 24 /3 5 520.56 5,000 8/19


Vol. 14, No. 4 / July 2010 − 623 −

to the optimal value. These study results have demonstrated theeffect of SAM on the searching.

In addition, only single pipe failure conditions were simulatedin this study. However, in reality, single pipe failure may affectnot only the pipe failed itself but more parts in the water distri-

bution system due to valve location or system topology. Theseaffected area can be estimated by using segment (Walski, 1993)or unintended isolation (Jun, 2005) concepts. As a future work,developed new model can be linked with these segment andunintended isolation concept to provide more realistic results.

5. Conclusions

This study proposes a conceptual structure of the PDA modelconsisting of HS as the optimization algorithm and EPANET asthe hydraulic simulator. Briefly, the PDA model is built on thebasis of the minimization of the difference between the assumedand the calculated heads at the demand nodes. Such minimi-zation of the difference makes it possible to simulate the hydrauliccondition properly under abnormal operating conditions. TheReHS is used to determine the proper head at each demand andthe HOR suggested by Wagner et al. (1988) and Chandapillai(1991) is used to determine the nodal demands which areavailable at the corresponding assumed head. For verification,the model is applied to the simple serial network described byGupta and Bhave (1996). To demonstrate its applicability to realnetworks and various operating conditions, the model is appliedto water distribution systems under abnormal operating condi-tions. The results obtained by the suggested model are comparedwith those by EPANET and the SPDA model of Ozger (2003).This comparison indicated that the suggested model may pro-duce more realistic results than the others. Moreover, a newparameter for the ReHS, SAM, is suggested in this study toimprove the searching efficiency of the ReHS and the studyresults support this improvement.

Notations

The following symbols are used in this paper:= Assumed nodal head = Calculated nodal head= Available head of node j= Desirable head of node j= Minimum head of node j= Minimum system head= Maximum system head

m = Exponent coefficientn = Exponent coefficient

Qavi = Available nodal demand assumed= Available demand of node j= Required demand of node j

Rj = Resistance constant of node

Acknowledgements

This study was supported by grant No. R01-2004-000-10362-0 from the Basic Research Program of the Korea Science andEngineering Foundation.

Hass

Hcal

Hjavi

Hjdes

Hjmin

Hsysmin

Hsysmax

qjavi

qjreq

Fig. 10. Changes in the Values of the Objective Function to Simu-late the Change of Hydraulic Condition due to the Failureof Pipe 49

Fig. 11. Changes in the Values of the Objective Function to Simu-late the Changes of Hydraulic Condition due to the Failureof Pipe 70

Fig. 12. Changes in the Values of the Objective Function to Simu-late the Changes of Hydraulic Condition due to the Failureof Pipe 71



References

Ackley, J. R. L., Tanyimboh, T. T., Tahar, B., and Templeman, A. B.(2001). “Head-driven analysis of water distribution system.” In :Ulanicki, B., Coulbeck, B. and Rance, J., editors, Water SoftwareSystems : Theory and Applications, Vol. 1, Research Studies PressLtd., Baldock, Hertfordshire, England, pp. 183-192.

Ang, W. K. and Jowitt, P. W. (2006). “Solution for water distributionsystems under pressure-deficient conditions.” J. of Water Resour.Plan. and Manage., Vol. 132, No. 3, pp. 175-182.

Bentley Systems, Incorporated (2006). WaterGEMS v8 User Manual,27 Siemon Co Dr, Suite200W, Watertown, CT06795, USA.

Bhave, P. R. (1981). “Node flow analysis of water distribution systems.”Transportation Engineering Journal of ASCE (Proc. of the ASCE),Vol. 107, No. TE4, pp. 457-467.

Chandapillai, J. (1991). “Realistic simulation of water distributionsystem.” J. of Transp. Eng., Vol. 117, No. 2, pp. 258-263.

Charles Howard and Associates, Ltd. (1984). Water distributionnetwork analysis: SPD8 user’s manual, Charles Howard andAssociates, Ltd., Victoria, B. C., Canada.

Fujiwara, O. and Ganesharajah, T. (1993). “Reliability assessment ofwater supply systems with storage and distribution networks.” WaterResour. Research, Vol. 29, No. 8, pp. 2917-2924.

Geem, Z. W. (2000). Optimal design of water distribution networksusing harmony search, PhD Dissertation, Dep. of Civil and Envir.Eng., Korea Univ., Seoul, Korea.

Geem, Z. W., Kim, J. H., and Loganathan, G. V. (2001). “A newheuristic optimization technique: Harmony search.” Simulation, Vol.14, No. 1, pp. 34-39.

Geem, Z. W., Kim, J. H., and Loganathan, G. V. (2002). “Harmonysearch optimization : Application to pipe network design.” Inter-national Journal of Modeling and Simulation, Vol. 22, No. 2, pp.125-133.

Germanopoulos, G. (1985). “A technical note on the inclusion ofpressure dependent demand and leakage terms in water supplynetwork models.” Civil Eng. System, Vol. 2, No. 3, pp. 171-179.

Gessler, J. and Walski, T. M. (1985). Water distribution systemoptimization, Technical Report EL-85-11, Envir. Eng. Lab., U.S.Army Engineer Waterways Experiment Station, Vicksburg, MS.

Goulter, I. C. and Coals, A. V. (1986). “Quantitative approaches toreliability assessment in pipe networks.” J. of Transp. Eng., Vol. 112,No. 3, pp. 287-301.

Gupta, R. and Bhave, P. R. (1996). “Comparison of methods forpredicting deficient-network performance.” J. of Water Resour.Plan. and Manage., Vol. 122, No. 3, pp. 214-217.

Holland, J. H. (1975). Adaptation in natural and artificial systems,Univ. of Michigan Press, Ann Arbor, Mich., USA.

Jun, H. D. and Loganathan, G. V. (2007) “Valve-controlled segments inwater distribution systems.” J. of Water Resour. Plan. and Manage.,Vol. 133, No. 2, pp. 145-155.

Kim, J. H., Zeem, J. W., and Kim, E. S. (2001). “Parameter estimationof the nonlinear muskingum model using Harmony search.” J. ofAmerican Water Res. Assoc., Vol. 37, No. 5, pp. 1131-1138.

Kim, J. H., Baek, C. W., Jo, D. J., Kim, E. S., and Park, M. J. (2004).“Optimal planning model for rehabilitation of water networks.”Water Science & Tech.: Water Supply, Vol. 4, No. 3, pp. 133-147.

Kirkpatrick, S., Gelatt Jr., C. D., and Vecchi, M. P. (1983). “Optimiza-tion by simulated annealing.” Science, Vol. 220, pp. 671-680.

Mays, L. W. (2003). Water supply systems security, McGRAW-HILL.Mays, L. W. and Tung, Y. K. (1992). Hydrosystems engineering and

management, McGraw-Hill.Ozger, S. S. (2003). A semi-pressure-driven approach to reliability

assessment of water distribution network, PhD Dissertation, Dep. ofCivil and Envir. Eng., Arizona State Univ., Tempe, Arizona.

Paik, K. R. (2001). Development of seasonal tank model and com-parison of optimization algorithms for parameter calibration, MaterDegree Dissertation, Dept. of Civil and Envir. Eng., Korea Univ.,Seoul, Korea.

Paik, K. R., Kim, J. H., Kim, H. S., and Lee, D. R. (2005). “Aconceptual rainfall-runoff model considering seasonal variation.”Hydrological Processes, Vol. 19, No. 19, pp. 3837-3850.

Reddy, L. S. and Elango, K. (1989). “Analysis of water distributionnetworks with head dependent outlets.” Civil Eng. System, Vol. 6,No. 3, pp. 102-110.

Reddy, L. S. and Elango, K. (1991). “A new approach to the analysis ofwater starved networks.” J. of Indian Water Works Assoc., Vol. 23,No. 1, pp. 31-38.

Rossman, L. A. (1994). EPANET users manual, Drinking WaterResearch Division, Risk Reduction Eng. Lab., Office of Researchand Development, U.S. Envir. Protection Agency, Cincinnati, OH.

Su, Yu-Chen, Mays, L. W., Duan, N., and Lansey, K. E. (1987). “Reli-ability-based optimization model for water distribution systems.” J.of Hydra. Eng., Vol. 114, No. 12, pp. 1539-1556.

Tabesh, M., Tanyimboh, T. T., and Burrows, R. (2001). “Extendedperiod reliability analysis of water distribution systems based onhead driven simulation method.” Proc., World Water Congress2001, ASCE, Orlando, Florida, USA.

Tabesh, M., Tanyimboh, T. T., and Burrows, R. (2002). “Head drivensimulation of water supply networks.” Int. J. of Eng., TransactionsA: Basics, Vol. 15, No. 1, pp. 11-22.

Tabesh, M., Tanyimboh, T. T., and Burrows, R. (2004). “Pressuredependent stochastic reliability analysis of water distributionnetworks.” Water Science and Technology: Water Supply, Vol. 4, No.3, pp. 81-90.

Tanyimboh, T. T. and Tabesh, M. (1997a). “Discussion comparison ofmethods for predicting deficient-network performance.” J. of WaterResour. Plan. and Manage., Vol. 123, No. 6, pp. 369-370.

Tanyimboh, T. T. and Tabesh, M. (1997b). “The basis of the source headmethod of calculating distribution network reliability.” Proc., 3rd

international conference on Water Pipeline System, BHR publica-tion, Hague, Netherlands.

Tanyimboh, T. T., Tabesh, M., and Burrows, R. (1997). “An improvedsource head method for calculation the reliability of water distri-bution networks.” Proc., The International Conference on Compu-ting and Control for the Water Industry, Brunel Univ., UK.

Tanyimboh, T. T., Tabesh, M., and Burrows, R. (2000). “Appraisal ofsource head methods for calculation reliability of water distributionnetworks.” J. of Water Resour. Plan. and Manage., Vol. 127, No. 4,pp. 206-213.

Tanyimboh, T. T., Tahar, B., and Tabesh, M. (2003). “Pressure-drivenmodeling of water distribution systems.” Water Science andTechnology: Water Supply, Vol. 3, No. 1-2, pp. 255-261.

Todini, E. (2006). “Toward realistic Extended Period Simulations (EPS)in looped pipe network.” Proc., the 8th Annual Water DistributionSystems Analysis Symposium, Univ. of Cincinnati, Cincinnati, USA.

Wagner, J. M., Shamir, U., and Marks, D. H. (1988). “Water distributionreliability : Simulation methods.” J. of Water Resour. Plan. andManage., Vol. 114, No. 3, pp. 276-294.

Walski, T. M. (1993). “Water distribution valve topology for reliabilityanalysis.” Reliability Engineering & System Safety, Vol. 42, No. 1,


Vol. 14, No. 4 / July 2010 − 625 −

pp. 21-27.Wood, D. J. (1980). User's manual-computer analysis of flow in pipe

network including extended period simulation, Dept. of Civil Eng.,Univ. of Kentucky, Lexington, KY.

Wu, Z. Y. and Walski, M. (2006). “Pressure-dependent hydraulicmodelling for water distribution systems under abnormal condi-tions.” Proc., IWA World Water Congress and Exhibition, IWA,

Beijing, China.Wu, Z. Y., Wang, R. H., Walski, T. M., Yang, S. Y., and Bowdler, D.

(2006). “Efficient pressure dependent demand model for large waterdistribution system analysis.” Proc., 8th Annual International Sym-posium on Water Distribution System Analysis, ASCE, Cincinnati,Ohio, USA.

development of a pda model for water distribution systems using harmony search algorithm

Documents