Research ArticleHydraulic Analysis of Water Distribution Network UsingShuffled Complex Evolution

Naser Moosavian1 and Mohammad Reza Jaefarzadeh2

1 Civil Engineering Department University of Torbat-e-Heydarieh Torbat-e-Heydarieh Iran2 Civil Engineering Department Ferdowsi University of Mashhad Mashhad Iran

Correspondence should be addressed to Naser Moosavian nasermoosavianyahoocom

Received 23 September 2013 Revised 16 November 2013 Accepted 1 December 2013 Published 16 January 2014

Academic Editor Prabir Daripa

Copyright copy 2014 N Moosavian and M R Jaefarzadeh This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Hydraulic analysis of water distribution networks is an important problem in civil engineering A widely used approach in steady-state analysis of water distribution networks is the global gradient algorithm (GGA) However when the GGA is applied to solvethese networks zero flows cause a computation failure On the other hand there are different mathematical formulations forhydraulic analysis under pressure-driven demand and leakage simulation This paper introduces an optimization model for thehydraulic analysis of water distribution networks using ametaheuristic method called shuffled complex evolution (SCE) algorithmIn this method applying if-then rules in the optimization model is a simple way in handling pressure-driven demand and leakagesimulation and there is no need for an initial solution vector whichmust be chosen carefully inmany other procedures if numericalconvergence is to be achieved The overall results indicate that the proposed method has the capability of handling various pipenetworks problems without changing inmodel or mathematical formulation Application of SCE in optimizationmodel can lead toaccurate solutions in pipes with zero flows Finally it can be concluded that the proposed method is a suitable alternative optimizerchallenging other methods especially in terms of accuracy

1 Introduction

A water distribution network is composed of an edge setconsisting of pumps pipes valves and a node set consistingof reservoirs and pipe intersections [1] The equations gov-erning the flows and heads in a water distribution system arenonlinear and often a Newton iterative solution algorithm isused in which a linearized set of equations is solved at eachiteration [2] The Newton-based global gradient algorithm(GGA) is a popular method used in solving the waterdistribution System (WDS) equations [3] Given the non-linearity of the system of equations the Newton-based com-putation of the solution involves an iterative two-step processThe first step includes computing the state variable updatewhich requires the solution of linear system derived from theJacobian of the WDS equations The second step deals withupdating estimates of the state variables The first step is typ-ically the most computationally expensive process within

the GGA [4] Furthermore some of the pipes in a networkin which the head losses are modeled by the Hazen-Williamsformula have zero flows In that case a key matrix in themethod becomes singular and the matrix to be invertedbecomes ill conditioned [2] As a result a failure occurs inthe computation On the other hand there are no optionsfor pressure-driven demand and leakage simulation in theEPANET program Meanwhile there is still a chance to de-velop a new method for water distribution network analysisin these conditions In this paper an optimization model isintroduced for hydraulic analysis of water distribution net-works using a metaheuristic algorithm called shuffled com-plex evolution (SCE) algorithm

The analysis of hydraulic networks should be treated asan optimization problem as shown by Arora [5] Hall [6]and Collins et al [1] Arora considered a simple two-pipedloop whereas Collins et al build the basis of their approachon rigorous theoretical background and developed nonlinear

Hindawi Publishing CorporationJournal of FluidsVolume 2014 Article ID 979706 12 pageshttpdxdoiorg1011552014979706

2 Journal of Fluids

Globe valve

Globe valve

KL1= 10

003m3sG

008m3s

008m3s

P2

P1170mH01

H02

200m

KL2= 10 KL3

= 2005m3sMeter

11 2

2

3

3

4

45

5

6

7

802ndash500

02ndash500

02ndash500

02

ndash300

02

ndash300

02ndash600

025ndash300

025ndash300

Figure 1 Schematic representation of the looped pipe network with5 unknown nodal heads

optimization models whose solutions yielded the hydraulicnetwork analysis [7] In this paper the Collins model isminimized through the application of shuffled complexevolution algorithm There is no need to solve linear systemsof equations in this methodology and handling of pressure-driven demand and leakage simulation can be done in asimple way so an initial solution vector which is sometimescritical to the convergence is not required Furthermore theproposed model does not entail any complicated mathemati-cal expression and operation The Collins model is describedin the following section

2 Cocontent Model Approach

Arora [5] is the first researcher who suggested an approachbased on the principle of conservation of energy Accordingto the principle ldquoFlow in the pipes of a hydraulic networkadjusts so that to minimize the expenditure of the systemenergyrdquo Then Collins et al [1] proposed a model termedthe cocontent model which is based on equations having theunknown nodal heads as the basic unknowns that is basedon 119867 equations The unknown pipe flows are expressed interms of the nodal heads and the known pipe resistances sothat the energy loss in pipe 119909 (119864

119909) is given by [7]

119864119909= 119876119909ℎ119909=

[119867119894minus 119867119895](1119899)+1

119877(1119899)

119909

(1)

in which ℎ119909is head loss in pipe 119909 119867

119894= 119867119900119894and 119867

119895= 119867119900119895

for source nodes119877119896is the characteristic parameter of the pipe

resistance which depends on roughness length diameter andunit of measurement For example if the Hazen-Williamsequation is used values 119899 and 119877 in SI units are defined as119877 = 1067119871119862

1852

119863487 119899 = 1852 119862 119863 and 119871 are the

Hazen-Williams coefficient (depending on the pipematerial)the diameter and its length respectively

Now consider the network of Figure 1 with the knownand unknown parameters as shown therein Let the unknownnodal heads at nodes 1 2 3 4 and 5 be119867

1119867211986731198674 and

1198675 respectively Herein also consider a ground node 119866 with

fixed known level1198670119866 as shown in Figure 1The nodes 1 2 3

and 5 are connected to the ground node 119866 with pseudopipes

carrying the known nodal outflows 1199021 1199022 1199023 and 119902

5as shown

in Figure 1The cocontent optimization model is expressed as

Min 119862 (119867) =1003816100381610038161003816100381611986701+ ℎ1199011minus ℎ1198711minus 1198671

10038161003816100381610038161003816

(1119899)+1

1198771119899

7

+

10038161003816100381610038161198674 minus 11986711003816100381610038161003816

(1119899)+1

1198771119899

4

+

100381610038161003816100381610038161198671+ ℎ1199012minus 1198672

10038161003816100381610038161003816

(1119899)+1

1198771119899

1

+

10038161003816100381610038161198672 minus 11986751003816100381610038161003816

(1119899)+1

1198771119899

5

+

10038161003816100381610038161198673 minus 11986721003816100381610038161003816

(1119899)+1

1198771119899

2

+

10038161003816100381610038161198673 minus 11986751003816100381610038161003816

(1119899)+1

1198771119899

6

+

10038161003816100381610038161198674 minus ℎ1198713 minus 11986731003816100381610038161003816

(1119899)+1

1198771119899

3

+

100381610038161003816100381611986702 minus ℎ1198712 minus 11986741003816100381610038161003816

(1119899)+1

1198771119899

8

+ (1

119899+ 1) 119902

1(1198671minus 1198670119866)

+ (1

119899+ 1) 119902

2(1198672minus 1198670119866)

+ (1

119899+ 1) 119902

3(1198673minus 1198670119866)

+ (1

119899+ 1) 119902

5(1198675minus 1198670119866)

(2)

where ℎ119871is local loss for valve and ℎ

119901is pump head

The first eight terms of the objective function repre-sent the energy loss in real pipes 1 8 of the networkrespectively and the last four terms show (1119899 + 1) timesthe energy loss in the pseudopipes [7] It should be notedthat there are no constraints and therefore an unconstrainedmodel in four decision variables is made For minimizationof optimization model which is partially differentiating inunknown heads the node-flow continuity equations arecreated Therefore the solution of the cocontent model givesthe values of the unknown heads such that the node-flowcontinuity relationships are satisfied [7]

By partially differentiating (2) with respect to119867111986721198673

1198674 and119867

5 we get

120597119862

1205971198671

= minus

1003816100381610038161003816100381611986701+ ℎ1199011minus ℎ1198711minus 1198671

10038161003816100381610038161003816

(1119899)

1198771119899

7

minus

10038161003816100381610038161198674 minus 11986711003816100381610038161003816

(1119899)

1198771119899

4

+

100381610038161003816100381610038161198671+ ℎ1199012minus 1198672

10038161003816100381610038161003816

(1119899)

1198771119899

1

+ 1199021= 0

120597119862

1205971198672

= minus

100381610038161003816100381610038161198671+ ℎ1199012minus 1198672

10038161003816100381610038161003816

(1119899)

1198771119899

1

+

10038161003816100381610038161198672 minus 11986751003816100381610038161003816

(1119899)

1198771119899

5

minus

10038161003816100381610038161198673 minus 11986721003816100381610038161003816

(1119899)

1198771119899

2

+ 1199022= 0

Journal of Fluids 3

120597119862

1205971198673

= +

10038161003816100381610038161198673 minus 11986721003816100381610038161003816

(1119899)

1198771119899

2

+

10038161003816100381610038161198673 minus 11986751003816100381610038161003816

(1119899)

1198771119899

6

minus

10038161003816100381610038161198674 minus ℎ1198713 minus 11986731003816100381610038161003816

(1119899)

1198771119899

3

+ 1199023= 0

120597119862

1205971198674

=

10038161003816100381610038161198674 minus 11986711003816100381610038161003816

(1119899)

1198771119899

4

+

10038161003816100381610038161198674 minus ℎ1198713 minus 11986731003816100381610038161003816

(1119899)

1198771119899

3

minus

100381610038161003816100381611986702 minus ℎ1198712 minus 11986741003816100381610038161003816

(1119899)

1198771119899

8

= 0

120597119862

1205971198675

= minus

10038161003816100381610038161198672 minus 11986751003816100381610038161003816

(1119899)

1198771119899

5

minus

10038161003816100381610038161198673 minus 11986751003816100381610038161003816

(1119899)

1198771119899

6

+ 1199025= 0

(3)

That is the purely head-based formulations of the networkequations So cocontentmodel not onlyminimizes the energyof flow but also preserves water balance in network Forsimplicity 119867

0119866can be taken as zero The general cocontent

model can be expressed as

Min 119862 (119867) = sum

119909

10038161003816100381610038161003816119867119894+ ℎ119901119909minus ℎ119871119909minus 119867119895

10038161003816100381610038161003816

(1119899)+1

1198771119899

119909

+ (1

119899+ 1)sum

119895

119902119895(119867119895)

(4)

Collins et al [1] suggested the solution of the NLPoptimization of the model The methods they used were (1)the Frank-Wolfe method (2) a piecewise linear approxima-tion and (3) the convex simplex method which are highlydependent on initial guesses and in some cases converge toan incorrect solution [1]

3 Head Dependent Analysis

In the common approaches it is presumed that the nodaldemands are always satisfied at all demand nodes irrespec-tive of the available HGL values at demand nodes [7] But inpractice when the head at a node is insufficient a reductionin the water flowing from the tap is expected and at worstthe discharge that can be drafted will be zero regardlessof the actual demand [8] There are several solutions forthese conditions in the literature Wagner et al [9] andChandapillai [10] suggested a parabolic relationship betweenrequired nodal head and minimum head Their relationshipsare

119902119895=

0 119867119895lt 119867min

119902119895(

119867119895minus 119867min

119867lowast minus 119867min)

1119901

119867min le 119867119895 lt 119867lowast

119902119895 119867

lowast

le 119867119895

(5)

119867lowast is the required nodal head In this situation applying

if-then rules in the optimization model is a simple way inhandling the above formulation

4 Leakage Simulation

Water losses via leakages constitute a major challenge to theeffective operation of municipal WDS since they representnot only diminished revenue for utilities but also underminedservice quality [11] and wasted energy resources [12] In orderto conduct a more accurate analysis of a WDS such as abetter estimate of flow through the network (with respectto both satisfied demand and losses through leakage) ahydraulic analysis capable of accounting for pressure-driven(also known as head-driven) demand and leakage flow atthe pipe level should prove invaluable To reach this goal aleakage model is expressed as follows [13]

119902119896-leak =

120573119896119897119896(119875119896)120572119896 if 119875

119896gt 0

0 if 119875119896le 0

(6)

where 119875119896is average pressure in the pipe computed as the

mean of the pressure values at the end nodes 119894 and119895 of the119896th pipe and 119897

119896is length of that pipe Variables 120572

119896and 120573

119896are

two leakage model parameters [14] The allocation of leakageto the two end nodes can be performed in a number of ways[15] Here the nodal leakage flow 119902

119895-leak is computed as thesum of 119902

119896-leak flows of all pipes connected to node 119895 as follows

119902119895-leak = sum

119896

1

2119902119896-leak = sum

119896

1

2120573119896119897119896(119875119896)120572119896 if 119875

119896gt 0

0 if 119875119896le 0

(7)

where 119875119896= (119875119894+ 119875119895)2 This formulation also easily applies

to cocontent model without any mathematical complexityNumerical example 4 demonstrates hydraulic analysis of areal pipe network in this situations

5 Application of Shuffled Complex EvolutionAlgorithm for Minimizing Cocontent Model

This study introduces the shuffled complex evolution (SCE)algorithm for the hydraulic analysis Since the algorithmwas originally developed to solve optimization problems thehydraulic network analysis was introduced into an optimiza-tion problem (cocontent model) One advantage of the SCEalgorithm is that it does not need an initial solution vectorwhich must be chosen carefully in many other proceduresif numerical convergence is to be achieved Furthermoreapplication of SCE algorithm in cocontent model does notrequire any complicated mathematical expression and opera-tion In this model pressure-driven demand and leakage canbe simulated easily and there is no failure in computation inzero flow conditions

51 Shuffled Complex Evolution (SCE) Shuffled complexevolution (SCE) is a simple powerful and population-basedstochastic optimization algorithm that outperforms manymetaheuristic algorithms in numerical single-objective opti-mization problems This method is based on a synthesis offour concepts (1) combination of deterministic and proba-bilistic approaches (2) systematic evolution of a ldquocomplexrdquoof points spanning the parameter space in the direction of

4 Journal of Fluids

f(x) objective function

xi decision variable (pipe diameter)

N number of decision variables

S number of solution vectors

P number of complexes

m = number of points in each complex

i = 1 S

Randomly generate the solutions

Evaluate objective function

Calculate cost function C(H)

Sort the solutions

Allocate solutions to complexes

For i = 1 B (number of inner iterations)

Select the subcomplexes from the complexesCompute the centroid of the worst point

Include new solution andexclude worst solution

Reflecting the worst pointthrough the centroid

Randomly generatea solution

ContractionCombine the solutions in the

evolved complexes into

a single sample population

Sort the solution vectors

Terminatecompetitions

Repeat Steps3 4 and 5

NI = number of iterations

NI

Step 1 initialize parameters Step 2 generate samples

Step 4 CCE algorithm

Step 3 shuffle complexes

Step 5 shuffle complexes

Step 6 check stopping criterion

Figure 2 SCE procedure for minimization of cocontent model

global improvement (3) competitive evolution (4) complexshuffling [16] The ldquocomplexrdquo is similar to the genetic poolin the GA The synthesis of these operators makes the SCEmethod effective and robust and also flexible and efficient[17]

In SCE method each individual represents a feasiblesolution for the problemThe searchwithin the feasible regionis conducted by first dividing the set of current feasible

trial solutions into several complexes each containing equalnumber of trial solutions Each complex represents a localarea of the whole domain Concurrent and independentsearches within each complex are conducted until eachconverges to its local optimal value Each of the complexeswhich are now defined by new trial solutions is collectedinto a common pool shuffled by ranking according to theirobjective function value and then further divided into new

Journal of Fluids 5

complexes The procedure is terminated when none of thelocal optima found among the complexes can improve onthe best current local optimum The SCE method usedthe downhill simplex method to accomplish local searchesSo shuffled complex evolution tries to balance between awide scan of a large solution space and deep search ofpromising locations It depends mainly on partitioning thesolution space into local communities and performing localsearch within these communitiesThen it shuffles these localcommunities to perform global search

The steps of the procedure of SCE as shown in Figure 2include the following

Step 1 initialize problem and algorithm parametersStep 2 samples generationStep 3 rank solutionsStep 4 partition into complexesStep 5 start Competitive Complex Evolution (CCE)Step 6 shuffle complexesStep 7 check the stopping criterion

511 Step 1 Initialize the Problem and Algorithm ParametersIn Step 1 the optimization problem is specified as follows

Min 119862 (119867) = sum

119909

10038161003816100381610038161003816119867119894+ ℎ119901119909minus ℎ119871119909minus 119867119895

10038161003816100381610038161003816

(1119899)+1

1198771119899

119909

+ (1

119899+ 1)sum

119895

119902119900119895(119867119895)

(8)

where 119862(119867) is an objective function 119867 is the set of eachdecision variable In this paper the objective function isthe cocontent model the unknown heads are the decisionvariables

512 Step 2 Samples Generation The initial population forthe DE is created arbitrarily by the following formula

119867(119894 119895) = 119867min (119894 119895) + 120591 (119867min (119894 119895) minus 119867max (119894 119895)) (9)

where 120591 denotes a uniformly distributed randomvaluewithinthe range [0 1] 119867min(119894 119895) and 119867max(119894 119895) are maximum andminimum limits of variable 119895 and node 119894Then the fitness val-ues 119862(119867) of all the individuals of population are calculatedThe position matrix of the population of generation 119866 can berepresented as

119875(119866)

=

[[[[[[[[[[[

[

1198621

1198622

119862119899Pop

]]]]]]]]]]]

]

=

[[[[[[[[[[[[[

[

1198671

11198671

2 119867

1

119873

1198672

11198672

2 119867

2

119873

119867119899Pop1

119867119899 Pop119873

]]]]]]]]]]]]]

]

(10)

in which119873 is the number of unknown nodes

513 Step 3 Rank Solutions In this step the s solutions aresorted in order of increasing criterion value so that the firstvector represents the smallest value of the objective functionand the last vector indicates the largest value

514 Step 4 Partition into Complexes The 119904 solutions arepartitioned into 119901 complexes each containing119898 points Thecomplexes are partitioned such that the first complex containsevery 119901(119896minus 1) + 1 ranked point the second complex containsevery 119901(119896 minus 1) + 2 ranked point and so on where 119896 = 1

2 119898 [16]

515 Step 5 Start Competitive Complex Evolution (CCE)CCE algorithm is based on the simplex downhill searchscheme and is one key component of SCE algorithm Thisalgorithm is presented as follows

(1) A subcomplex by randomly selecting 119902 solutions fromthe complex according to a trapezoidal probabilitydistribution is constructed The probability distri-bution is specified such that the best solution hasthe highest chance of being selected to form thesubcomplex and the worst point has the least chance

(2) Theworst solution of the subcomplex is identified andthe centroid of the subcomplex without including theworst solution is computed as follows

CR (119894) = (1 (119898 minus 1))

119898minus1

sum

119895=1

119904 (119894 119895) 119894 = 1 119875 (11)

(3) In this step reflection operator is used by reflectingthe worst point through the centroid according to thefollowing formula

119867new

(119894 119895) = 2 lowast CR (119894) minus 119867 (119894worst) (12)

If the newly generated solution is within the feasiblespace go to (4) otherwise randomly generate a pointwithin the feasible space by Equation (9) and go to (6)

(4) If the newly generated solution is better than theworst solution then it is replaced by the new solutionOtherwise go to (5)

(5) In this step contraction operator is applied by com-puting a solution halfway between the centroid andthe worst point

119867new

(119894 119895) =(CR (119894) minus 119867 (119894worst))

2 (13)

If the contraction solution is better than the worstsolution then it is replaced by the contraction solu-tion Otherwise go to (6) This step is imported fromcompetitive complex evolution (CCE)

(6) A solution within the feasible space is generatedrandomly and the worst solution is replaced by therandomly generated solution

(7) Steps (2)ndash(6) are repeated 120572 times and steps (1)ndash(7)are repeated 120573 times

6 Journal of Fluids

516 Step 6 Shuffle Complexes The solutions in the evolvedcomplexes into a single sample population are combinedand the sample population is sorted in order of increasingcriterion value and is shuffled into 119901 complexes

517 Step 7 Check the Stopping Criterion In this sectionSteps 3 4 and 5 are repeated until the termination criterionis satisfied

It should be noted that the competitive complex evolu-tion (CCE) algorithm is required for the evolution of eachcomplex Each point of a complex is a potential ldquoparentrdquowith the ability to participate in the process of reproducingoffspring A subcomplex functions like a pair of parents Useof a stochastic scheme to construct subcomplexes allows theparameter space to be searched more thoroughly The ideaof competitiveness is introduced in forming subcomplexeswhere the stronger survives better and breeds healthier off-spring than the weaker Inclusion of the competitive measureexpedites the search towards promising regions

A more detailed presentation of the SCE algorithm hasbeen given by Duan et al [17]

6 Numerical Examples

In this section the hydraulic analyses for several conditionsin some water distribution networks are performed All com-putations were executed inMATLAB programming languageenvironment with an Intel(R) Core(TM) 2Duo CPU P8700 253GHz and 400GB RAM In order to demonstrate theeffectiveness of SCE comparedwith othermethods this studyproposes the use of mass balance and energy balance in thenetworkThe average of mass and energy balance is shown by120575 and it is calculated by the following formula

120575 = mean(abs(119873

sum

119895=1

119910

sum

119894=119909

10038161003816100381610038161003816119867119894+ ℎ119901119896minus ℎ119871119896minus 119867119895

10038161003816100381610038161003816

(1119899)

1198771119899

119896

minus 119902119895))

(14)

For Figure 1 and (2) 120575 is calculated as follows

120575 = mean(10038161003816100381610038161003816100381610038161003816

120597119862

1205971198671

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198672

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198673

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198674

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198675

10038161003816100381610038161003816100381610038161003816

)

(15)

To check the performance of the SCE for the minimiza-tion of cocontent model in all examples ten optimizationrunswere performed using different random initial solutions

61 Numerical Example 1 In this part the verification ofthe above mentioned model was conducted via numericalsimulation based on an extremely simplified network scheme(5 nodes and 7 pipes) schematically shown in Figure 3 [18]The pipes resistances were 119877

1= 15625 119877

2= 50 119877

3= 100

1198774= 125 119877

5= 75 119877

6= 200 and 119877

7= 100 as reported in

Todini [18] for this networkThe SCE technique is applied to solve this problem

according to three cases The bound variables were setbetween 90 and 100m The problem is also solved using the

0Reservoir

11

1

2

2

2

3

3

3

4

4

5 6

7

800 ls

200 ls

200 ls

200 ls

300 ls

400 ls

400 ls

100 ls

100 ls

100 ls

100 ls

Figure 3 Schematic representation of the looped pipe network usedin numerical example 1

0 2 4 6 8 10 12 14 16 18 20

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

Figure 4 Convergence history of numerical example 1 (form 1)

global gradient algorithm (GGA) and the results are com-pared with those obtained by the SCE The best worst andaverage solutions of SCE algorithm in three cases are shownin Table 1 As it can be seen in Table 1 in all cases SCEthat found the optimal solution more accurately than GGAmethod The average number of function evaluation is about2000 in case 1 and about 100000 in case 3 This shows SCEcan converge to global optimum rapidly but reaching highaccuracy needs more operations The convergence process ofSCE algorithm has been shown in two forms in Figures 4and 5 The absolute value of 120575 is calculated for each iterationin Figure 4 and the amount of objective function 119862(119867) iscalculated for each iteration in Figure 5

62 Numerical Example 2 Example 2 considers the sym-metric network shown in Figure 6 It has 11 pipes sevenjunctions at which the head is unknown and one fixed headnode reservoir at 40m elevation and all other nodes areat zero elevation All pipes have diameters 119863 of 250mmand lengths 119871 of 1000m Node 8 has a demand of 80 lsand all other nodes have zero demands In the steady statethis network has zero flows in pipes 2 6 and 9 because of

Journal of Fluids 7

Table 1 Average of mass and energy balance for numerical example 1

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4460119864 minus 08 706119864 minus 07 304119864 minus 07 230119864 minus 07 203119864 + 03

Number of iterations in inner loop 4Number of complexes 9

119119864 minus 08 301119864 minus 08 204119864 minus 08 640119864 minus 09 100119864 + 05Number of iterations in inner loop 9Number of complexes 10

104119864 minus 08 419119864 minus 08 252119864 minus 08 828119864 minus 09 100119864 + 05Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy 640119864 minus 06

Table 2 Head and parameter 120575 in numerical example 2

Node number 119867 (m) [2] 119867 (m) 120575 [2] 120575 (SCE)1 40 40 0 02 366813 366853 0 18119864 minus 07

3 366813 366853 0 18119864 minus 07

4 333626 333706 65119864 minus 07 84E minus 085 333626 333706 65119864 minus 07 87E minus 086 30044 300559 65119864 minus 07 67E minus 087 30044 300559 65119864 minus 07 66E minus 088 267253 267411 52119864 minus 05 12E minus 10

0 2 4 6 8 10 12 14 16 18 20

1021718

10217179

10217178

10217177

10217176

10217175

10217174

10217173

Obj

ectiv

e fun

ctio

n

Number of iterations

Figure 5 Convergence history of numerical example 1 (form 2)

symmetry The head loss is modeled by the Hazen-Williamsequation and each pipe has a Hazen-Williams coefficient119862 = 120 [2] When the GGA is used in this network theiterates trend toward the solution and the flows in pipes 26 and 9 approach zero As this happens the Jacobian matrixbecomes more and more badly conditioned and the solutioncomputed becomes ill conditioned [2] Elhay and Simpson[2] proposed a regularization procedure for the GGA whichprevents failure of the solution process provided that a flowin the network is ultimately zero or near zero

The SCEparameters are set as follows number of decisionvariables = 7 number of points in each complex = 15 numberof complexes for case 1 = 4 case 2 = 9 and case 3 = 10 numberof iterations in inner loop for case 1 = 4 case 2 = 9 and

1

1

22

3

3

4

4

5

56

6

7

7

8

8

9

1011

Reservoir

Figure 6 Schematic representation of the looped pipe network usedin numerical example 2

case 3 = 15 The bound variables were set between 25 and40m The previous best solution for this network when it issimulated using the Elhay algorithm and the average solutionof SCE algorithm are shown in the second and third columnsof Table 2 respectively As can be observed in Table 2 massand energy balance (120575) in SCE are more accurate than theElhay algorithm Table 3 compares the results of applying theSCE algorithm in three casesThe convergence process of SCEalgorithm has been shown in two forms in Figures 7 and 8

8 Journal of Fluids

Table 3 Average of mass and energy balance for numerical example 2

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4221119864 minus 06 724119864 minus 06 465119864 minus 06 161119864 minus 06 419119864 + 03

Number of iterations in inner loop 4Number of complexes 9

105119864 minus 06 535119864 minus 06 346119864 minus 06 151119864 minus 06 929119864 + 03Number of iterations in inner loop 9Number of complexes 10

636119864 minus 08 178119864 minus 07 108119864 minus 07 389119864 minus 08 172119864 + 04Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy FailElhay algorithm Maximum accuracy 742119864 minus 06

Table 4 Head and parameter 120575 in numerical example 3

Node number 119885 (m) SCE 3 steps SCE 3 steps SCE 3 steps119867 (m) 119867 (m) 119867 minus 119885 119867 minus 119885 120575 120575

1 140 140 140 0 0 0 02 80 129304 13007 49304 5007 149E minus 07 310119864 minus 04

3 90 132288 13276 42288 4276 126E minus 07 000414 70 109587 11096 39587 4096 203E minus 07 000225 80 80000 8854 0000 854 0058 151E minus 046 90 90000 9145 0000 145 00069 602E minus 047 90 90000 9000 0000 000 0080 014218 100 88922 9043 minus11078 minus957 584E minus 08 00439

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations0 5 10 15 20 25 30 35 40 45 50

Figure 7 Convergence history of numerical example 2 (form 1)

The absolute value of 120575 is calculated for each iteration inFigure 7 and the value of head pressure in node 2 119867(2) iscalculated for each iteration in Figure 8

63 Numerical Example 3 The simplified water distributionnetwork shown in Figure 9 was used in order to demonstratethe advantages of the proposed model in pressure-drivendemand condition For the sake of simplicity the same

H(2)

Number of iterations0 5 10 15 20 25 30 35 40 45 50

365

366

367

368

369

37

371

372

373

Figure 8 Convergence history of head pressure (119898) in node 2 fornumerical example 2 (form 2)

Hazen-Williams roughness coefficient 119862 = 130 was assumedfor all the 14 pipes of identical length of 1000m while no locallosses have been added The following diameters have beenused in the example 500mm (P-2) 400mm (P-1) 300mm(P-4 P-7) 250mm (P-10) 200mm (P-3 P-5 P-6 and P-13)150mm (P-8 P-9 P-11 P-12 and P-14) The nodal demandsare listed in the following tables together with the groundelevation 119885

119894 Without loss of generality in this example the

Journal of Fluids 9

Reservoir

1 2

23

3

4

4

5

5

6

6

7

7

8

8

9

10 11 12 13

14

Figure 9 Schematic representation of the looped pipe network usedin numerical example 3

minimum head requirement 119867119894

lowast has been assumed to beequal to the ground elevation 119885

119894[8] So the relationship

between the required nodal head and minimum head is

119902119895=

0 119867119895lt 119885119895

119902119895 119885119895le 119867119895

(16)

Todini [8] proposed a three-step approach for solving thisnetwork and its solution is reported in the 4th column ofTable 4 In the proposed method pressure-driven model canbe applied in hydraulic analysis without any mathematicalformulation In this situation an if-then rule is added tococontent model and optimization process is conductedThenumber of decision variables in SCE algorithm is 7 the boundvariables were set between 50 and 140m ten optimizationruns were performed using different random initial solutionsfor all the cases and the results are illustrated in Table 5Results confirm that SCE is more accurate compared withTodini algorithm in case 2 and case 3 In Table 4 the bestresult is shown in bold and it is considered that the methodof SCE has calculated the best value of 120575 at 5 nodes whilethe Todini method has done it at 2 nodes The convergenceprocess of SCE algorithm has been shown in two forms inFigures 10 and 11The absolute value of 120575 is calculated for eachiteration in Figure 10 and the value of head pressure in node2119867(2) is calculated for each iteration in Figure 11

64 Numerical Example 4 The fourth considered networkis a real planned network designed for an industrial areain Apulian Town (Southern Italy) The network layout is

0 10 20 30 40 50 60 70 80 90

100

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

Figure 10 Convergence history of numerical example 3 (form 1)

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100

120

140

Number of iterations

H(2)

Figure 11 Convergence history of head pressure (119898) in node 2 fornumerical example 3 (form 2)

illustrated in Figure 12 and the corresponding data areprovided in Table 6 With respect to the leakages they havebeen assumed to be pressure-driven (see (6)) given that theyare implemented in the pressure driven network simulationmodel as described above [14] The parameter 120573 = 10632 times10minus7 and 120572 = 12 as reported in Giustolisi et al [14] for this

network Giustolisi et al [14] proposed a hydraulic simulationmodel which fully integrates a classic hydraulic simulationalgorithm such as that of Todini and Pilati [3] found inEPANET 2 with a pressure-driven model that entails amore realistic representation of the leakage They appliedtheir model in this network The results are demonstratedin Table 7 In this table the best result is shown in boldand it is considered that the method of SCE has calculatedthe best value of 120575 at 14 nodes while the Gistulishi methodhas done it at 9 nodes In the proposed method there is noneed to modify the mathematical formulation for hydraulic

10 Journal of Fluids

Table 5 Average of mass and energy balance for numerical example 3

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5332119864 minus 02 797119864 minus 02 595119864 minus 02 162119864 minus 02 430119864 + 04

Number of iterations in inner loop 5Number of complexes 9

207119864 minus 02 544119864 minus 02 252119864 minus 02 109119864 minus 02 627119864 + 04Number of iterations in inner loop 9Number of complexes 10

207119864 minus 02 207119864 minus 02 207119864 minus 02 217119864 minus 08 604119864 + 04Number of iterations in inner loop 15Three-step approach [8] Maximum accuracy 276119864 minus 02

Table 6 Hydraulic data relevant to numerical example 4

Pipe number 119871 (m) 119863 (mm)1 3485 3272 9557 2903 483 1004 4007 2905 7919 1006 4044 3687 3906 3278 4823 1009 9344 10010 4313 18411 5131 10012 4284 18413 419 10014 10231 10015 4551 16416 1826 29017 2213 29018 5839 16419 452 22920 7947 10021 7177 10022 6556 25823 1655 10024 2521 10025 3315 10026 500 20427 5799 16428 8428 10029 7926 10030 8463 18431 164 25832 4279 10033 3792 10034 1582 368

analysis An if-then rule is added to cocontent model andthe optimization process is performed easily As you can seein Table 8 SCE found the optimal solution more accurately

1

1

2 2

33

4

4

5

5

6

6

7

78

8

99

101011

11

12

12

1313

14

14

15

1516

1617

17

18

18

19

19

20

20

21

21

2222

23 2324

25

26

27

28

2930

31

32

33

34

Reservoir

Figure 12 Schematic representation of the looped pipe networkused in the numerical example 4

0 100 200 300 400 500 600 700 800 900 1000

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

10minus3

Figure 13 Convergence history of numerical example 4 (form 1)

than the Giustolisi algorithm The convergence process ofSCE algorithm has been shown in two forms in Figures 4 and5 The absolute value of 120575 is calculated for each iteration inFigure 13 and the amount of head pressure in node 20119867(20)is calculated for each iteration in Figure 14

Journal of Fluids 11

Table 7 Head and parameter 120575 in numerical example 4

Node number 119902 (ls) 119867 (m) (Giustolisi algorithm) 119867 (m) 120575 (Giustolisi algorithm) 120575 (SCE)1 10863 269 3329 0154743 00047382 17034 2481 3183 002131 00065323 14947 213 2739 0059002 00053644 1428 1722 2534 0001257 00050335 10133 2354 3089 0026184 00040186 1535 201 2902 0030898 00053997 9114 1891 2794 0017147 00030878 1051 179 2734 000227 00035459 12182 1785 2635 0002936 000386710 14579 1266 2324 0008277 000436211 9007 1623 2595 0031554 000272312 7575 1012 2205 0002732 000213813 152 1003 2245 00126 000432814 1355 1541 2595 0001318 000435215 9226 14 2417 0002199 000293416 112 1436 2405 0007089 000358617 11469 153 2542 0000103 00036118 10818 1883 2838 0011889 000390819 14675 1935 2839 543E minus 05 000509720 13318 1001 2379 0013059 000397421 14631 1148 2235 0003276 000414122 12012 14 2546 0003901 000367723 10326 1045 2011 0002551 000297924 mdash 3645 3645 mdash mdash

Table 8 Average of mass and energy balance for numerical example 4

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5400119864 minus 03 450119864 minus 02 410119864 minus 03 275119864 minus 04 659119864 + 04

Number of iterations in inner loop 5Number of complexes 9

390119864 minus 03 415119864 minus 03 400119864 minus 03 396119864 minus 05 115119864 + 05Number of iterations in inner loop 9Number of complexes 10

370119864 minus 03 410119864 minus 03 390119864 minus 03 253119864 minus 05 119119864 + 05Number of iterations in inner loop 15Giustolisi algorithm Maximum accuracy 181119864 minus 02

In general 4 different pipe networks were consideredin this paper and different mathematical formulations wereused for the hydraulic analysis of these networks Howeverthe overall results indicate that the proposed method hasthe capability of handling various pipe networks problemswith no change in the model or mathematical formulationApplication of SCE in cocontent model can result in findingaccurate solutions in pipes with zero flows and the pressure-driven demand and leakage simulation can be solved throughapplying if-then rules in cocontent model As a result itcan be concluded that the proposed method is a suitablealternative optimizer challenging othermethods especially interms of accuracy

7 Conclusions

The objective of the present paper was to provide an inno-vative approach in the analysis of the water distributionnetworks based on the optimization model The cocontentmodel is minimized using shuffled complex evolution (SCE)algorithm The methodology is illustrated here using fournetworks with different layouts The results reveal thatthe proposed method has the capability to handle variouspipe networks problems without changing in model ormathematical formulation The advantage of the proposedmethod lies in the fact that there is no need to solvelinear systems of equations pressure-driven demand and

12 Journal of Fluids

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

Number of iterations

H(20)

Figure 14 Convergence history of head-pressure (119898) in node 20 forexample 4 (form 2)

leakage simulation are handled in a simple way accu-rate solutions can be found in pipes with zero flowsand it does not need an initial solution vector which must bechosen carefully in many other procedures if numerical con-vergence is to be achieved Furthermore the proposedmodeldoes not require any complicated mathematical expressionand operation Finally it can be concluded that the proposedmethod is a viable alternative optimizer that challenges othermethods particularly in view of accuracy

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

References

[1] M Collins L Cooper R Helgason J Kenningston and LLeBlanc ldquoSolving the pipe network analysis problem usingoptimization techniquesrdquo Management Science vol 24 no 7pp 747ndash760 1978

[2] S Elhay and A R Simpson ldquoDealing with zero flows in solvingthe nonlinear equations for water distribution systemsrdquo Journalof Hydraulic Engineering vol 137 no 10 pp 1216ndash1224 2011

[3] E Todini and S Pilati ldquoA gradient algorithm for the analysis ofpipe networksrdquo in Computer Applications inWater Supply vol 1of System Analysis and Simulation pp 1ndash20 JohnWiley amp SonsLondon UK 1988

[4] A C Zecchin P Thum A R Simpson and C TischendorfldquoSteady-state behaviour of large water distribution systems thealgebraic multigrid method for the fast solution of the linearsteprdquo Journal ofWater Resources Planning andManagement vol138 no 6 pp 639ndash650 2012

[5] M L Arora ldquoFlow split in closed loops expending least energyrdquoJournal of the Hydraulics Division vol 102 pp 455ndash458 1976

[6] M A Hall ldquoHydraulic network analysis using (generalized)geometric programmingrdquo Networks vol 6 no 2 pp 105ndash1301976

[7] P R Bhave and R Gupta ldquoAnalysis of water distributionnetworksrdquo in Alpha Science International Technology amp Engi-neering 2006

[8] E Todini A More Realistic Approach to the ldquoExtended PeriodSimulationrdquo of Water Distribution Networks Advances in WaterSupplyManagement chapter 19 Taylor amp Francis London UK2003

[9] J M Wagner U Shamir and D H Marks ldquoWater distributionreliability analytical methodsrdquo Journal of Water ResourcesPlanning and Management vol 114 no 3 pp 253ndash275 1988

[10] J Chandapillai ldquoRealistic simulation of water distributionsystemrdquo Journal of Transportation Engineering vol 117 no 2 pp258ndash263 1991

[11] J Almandoz E Cabrera F Arregui E Cabrera Jr andR Cobacho ldquoLeakage assessment through water distributionnetwork simulationrdquo Journal of Water Resources Planning andManagement vol 131 no 6 pp 458ndash466 2005

[12] A F Colombo and B W Karney ldquoEnergy and costs ofleaky pipes toward comprehensive picturerdquo Journal of WaterResources Planning and Management vol 128 no 6 pp 441ndash450 2002

[13] G Germanopoulos ldquoA technical note on the inclusion ofpressure dependent demand and leakage terms in water supplynetworkmodelsrdquoCivil Engineering Systems vol 2 no 3 pp 171ndash179 1985

[14] O Giustolisi D Savic and Z Kapelan ldquoPressure-drivendemand and leakage simulation for water distribution net-worksrdquo Journal of Hydraulic Engineering vol 134 no 5 pp 626ndash635 2008

[15] L Ainola T Koppel K Tiiter and A Vassiljev ldquoWater networkmodel calibration based on grouping pipes with similar leakageand roughness estimatesrdquo in Proceedings of the Joint Conferenceon Water Resource Engineering and Water Resources PlanningandManagement (EWRI rsquo00) pp 1ndash9MinneapolisMinnUSAAugust 2000

[16] Q Duan S Sorooshian and V K Gupta ldquoOptimal use of theSCE-UA global optimization method for calibrating watershedmodelsrdquo Journal of Hydrology vol 158 no 3-4 pp 265ndash2841994

[17] Q Y Duan S Sorooshian and V K Gupta ldquoShuffled complexevolution approach for effective and efficient global minimiza-tionrdquo Journal of Optimization Theory and Applications vol 76no 3 pp 501ndash521 1993

[18] E Todini ldquoOn the convergence properties of the differentpipe network algorithmsrdquo in Proceedings of the Annual WaterDistribution Systems Analysis Symposium (ASCE rsquo06) pp 1ndash16August 2006

Submit your manuscripts athttpwwwhindawicom

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ThermodynamicsJournal of

2 Journal of Fluids

Globe valve

Globe valve

KL1= 10

003m3sG

008m3s

008m3s

P2

P1170mH01

H02

200m

KL2= 10 KL3

= 2005m3sMeter

11 2

2

3

3

4

45

5

6

7

802ndash500

02ndash500

02ndash500

02

ndash300

02

ndash300

02ndash600

025ndash300

025ndash300

Figure 1 Schematic representation of the looped pipe network with5 unknown nodal heads

optimization models whose solutions yielded the hydraulicnetwork analysis [7] In this paper the Collins model isminimized through the application of shuffled complexevolution algorithm There is no need to solve linear systemsof equations in this methodology and handling of pressure-driven demand and leakage simulation can be done in asimple way so an initial solution vector which is sometimescritical to the convergence is not required Furthermore theproposed model does not entail any complicated mathemati-cal expression and operation The Collins model is describedin the following section

2 Cocontent Model Approach

Arora [5] is the first researcher who suggested an approachbased on the principle of conservation of energy Accordingto the principle ldquoFlow in the pipes of a hydraulic networkadjusts so that to minimize the expenditure of the systemenergyrdquo Then Collins et al [1] proposed a model termedthe cocontent model which is based on equations having theunknown nodal heads as the basic unknowns that is basedon 119867 equations The unknown pipe flows are expressed interms of the nodal heads and the known pipe resistances sothat the energy loss in pipe 119909 (119864

119909) is given by [7]

119864119909= 119876119909ℎ119909=

[119867119894minus 119867119895](1119899)+1

119877(1119899)

119909

(1)

in which ℎ119909is head loss in pipe 119909 119867

119894= 119867119900119894and 119867

119895= 119867119900119895

for source nodes119877119896is the characteristic parameter of the pipe

resistance which depends on roughness length diameter andunit of measurement For example if the Hazen-Williamsequation is used values 119899 and 119877 in SI units are defined as119877 = 1067119871119862

1852

119863487 119899 = 1852 119862 119863 and 119871 are the

Hazen-Williams coefficient (depending on the pipematerial)the diameter and its length respectively

Now consider the network of Figure 1 with the knownand unknown parameters as shown therein Let the unknownnodal heads at nodes 1 2 3 4 and 5 be119867

1119867211986731198674 and

1198675 respectively Herein also consider a ground node 119866 with

fixed known level1198670119866 as shown in Figure 1The nodes 1 2 3

and 5 are connected to the ground node 119866 with pseudopipes

carrying the known nodal outflows 1199021 1199022 1199023 and 119902

5as shown

in Figure 1The cocontent optimization model is expressed as

Min 119862 (119867) =1003816100381610038161003816100381611986701+ ℎ1199011minus ℎ1198711minus 1198671

10038161003816100381610038161003816

(1119899)+1

1198771119899

7

+

10038161003816100381610038161198674 minus 11986711003816100381610038161003816

(1119899)+1

1198771119899

4

+

100381610038161003816100381610038161198671+ ℎ1199012minus 1198672

10038161003816100381610038161003816

(1119899)+1

1198771119899

1

+

10038161003816100381610038161198672 minus 11986751003816100381610038161003816

(1119899)+1

1198771119899

5

+

10038161003816100381610038161198673 minus 11986721003816100381610038161003816

(1119899)+1

1198771119899

2

+

10038161003816100381610038161198673 minus 11986751003816100381610038161003816

(1119899)+1

1198771119899

6

+

10038161003816100381610038161198674 minus ℎ1198713 minus 11986731003816100381610038161003816

(1119899)+1

1198771119899

3

+

100381610038161003816100381611986702 minus ℎ1198712 minus 11986741003816100381610038161003816

(1119899)+1

1198771119899

8

+ (1

119899+ 1) 119902

1(1198671minus 1198670119866)

+ (1

119899+ 1) 119902

2(1198672minus 1198670119866)

+ (1

119899+ 1) 119902

3(1198673minus 1198670119866)

+ (1

119899+ 1) 119902

5(1198675minus 1198670119866)

(2)

where ℎ119871is local loss for valve and ℎ

119901is pump head

The first eight terms of the objective function repre-sent the energy loss in real pipes 1 8 of the networkrespectively and the last four terms show (1119899 + 1) timesthe energy loss in the pseudopipes [7] It should be notedthat there are no constraints and therefore an unconstrainedmodel in four decision variables is made For minimizationof optimization model which is partially differentiating inunknown heads the node-flow continuity equations arecreated Therefore the solution of the cocontent model givesthe values of the unknown heads such that the node-flowcontinuity relationships are satisfied [7]

By partially differentiating (2) with respect to119867111986721198673

1198674 and119867

5 we get

120597119862

1205971198671

= minus

1003816100381610038161003816100381611986701+ ℎ1199011minus ℎ1198711minus 1198671

10038161003816100381610038161003816

(1119899)

1198771119899

7

minus

10038161003816100381610038161198674 minus 11986711003816100381610038161003816

(1119899)

1198771119899

4

+

100381610038161003816100381610038161198671+ ℎ1199012minus 1198672

10038161003816100381610038161003816

(1119899)

1198771119899

1

+ 1199021= 0

120597119862

1205971198672

= minus

100381610038161003816100381610038161198671+ ℎ1199012minus 1198672

10038161003816100381610038161003816

(1119899)

1198771119899

1

+

10038161003816100381610038161198672 minus 11986751003816100381610038161003816

(1119899)

1198771119899

5

minus

10038161003816100381610038161198673 minus 11986721003816100381610038161003816

(1119899)

1198771119899

2

+ 1199022= 0

Journal of Fluids 3

120597119862

1205971198673

= +

10038161003816100381610038161198673 minus 11986721003816100381610038161003816

(1119899)

1198771119899

2

+

10038161003816100381610038161198673 minus 11986751003816100381610038161003816

(1119899)

1198771119899

6

minus

10038161003816100381610038161198674 minus ℎ1198713 minus 11986731003816100381610038161003816

(1119899)

1198771119899

3

+ 1199023= 0

120597119862

1205971198674

=

10038161003816100381610038161198674 minus 11986711003816100381610038161003816

(1119899)

1198771119899

4

+

10038161003816100381610038161198674 minus ℎ1198713 minus 11986731003816100381610038161003816

(1119899)

1198771119899

3

minus

100381610038161003816100381611986702 minus ℎ1198712 minus 11986741003816100381610038161003816

(1119899)

1198771119899

8

= 0

120597119862

1205971198675

= minus

10038161003816100381610038161198672 minus 11986751003816100381610038161003816

(1119899)

1198771119899

5

minus

10038161003816100381610038161198673 minus 11986751003816100381610038161003816

(1119899)

1198771119899

6

+ 1199025= 0

(3)

That is the purely head-based formulations of the networkequations So cocontentmodel not onlyminimizes the energyof flow but also preserves water balance in network Forsimplicity 119867

0119866can be taken as zero The general cocontent

model can be expressed as

Min 119862 (119867) = sum

119909

10038161003816100381610038161003816119867119894+ ℎ119901119909minus ℎ119871119909minus 119867119895

10038161003816100381610038161003816

(1119899)+1

1198771119899

119909

+ (1

119899+ 1)sum

119895

119902119895(119867119895)

(4)

Collins et al [1] suggested the solution of the NLPoptimization of the model The methods they used were (1)the Frank-Wolfe method (2) a piecewise linear approxima-tion and (3) the convex simplex method which are highlydependent on initial guesses and in some cases converge toan incorrect solution [1]

3 Head Dependent Analysis

In the common approaches it is presumed that the nodaldemands are always satisfied at all demand nodes irrespec-tive of the available HGL values at demand nodes [7] But inpractice when the head at a node is insufficient a reductionin the water flowing from the tap is expected and at worstthe discharge that can be drafted will be zero regardlessof the actual demand [8] There are several solutions forthese conditions in the literature Wagner et al [9] andChandapillai [10] suggested a parabolic relationship betweenrequired nodal head and minimum head Their relationshipsare

119902119895=

0 119867119895lt 119867min

119902119895(

119867119895minus 119867min

119867lowast minus 119867min)

1119901

119867min le 119867119895 lt 119867lowast

119902119895 119867

lowast

le 119867119895

(5)

119867lowast is the required nodal head In this situation applying

if-then rules in the optimization model is a simple way inhandling the above formulation

4 Leakage Simulation

Water losses via leakages constitute a major challenge to theeffective operation of municipal WDS since they representnot only diminished revenue for utilities but also underminedservice quality [11] and wasted energy resources [12] In orderto conduct a more accurate analysis of a WDS such as abetter estimate of flow through the network (with respectto both satisfied demand and losses through leakage) ahydraulic analysis capable of accounting for pressure-driven(also known as head-driven) demand and leakage flow atthe pipe level should prove invaluable To reach this goal aleakage model is expressed as follows [13]

119902119896-leak =

120573119896119897119896(119875119896)120572119896 if 119875

119896gt 0

0 if 119875119896le 0

(6)

where 119875119896is average pressure in the pipe computed as the

mean of the pressure values at the end nodes 119894 and119895 of the119896th pipe and 119897

119896is length of that pipe Variables 120572

119896and 120573

119896are

two leakage model parameters [14] The allocation of leakageto the two end nodes can be performed in a number of ways[15] Here the nodal leakage flow 119902

119895-leak is computed as thesum of 119902

119896-leak flows of all pipes connected to node 119895 as follows

119902119895-leak = sum

119896

1

2119902119896-leak = sum

119896

1

2120573119896119897119896(119875119896)120572119896 if 119875

119896gt 0

0 if 119875119896le 0

(7)

where 119875119896= (119875119894+ 119875119895)2 This formulation also easily applies

to cocontent model without any mathematical complexityNumerical example 4 demonstrates hydraulic analysis of areal pipe network in this situations

5 Application of Shuffled Complex EvolutionAlgorithm for Minimizing Cocontent Model

This study introduces the shuffled complex evolution (SCE)algorithm for the hydraulic analysis Since the algorithmwas originally developed to solve optimization problems thehydraulic network analysis was introduced into an optimiza-tion problem (cocontent model) One advantage of the SCEalgorithm is that it does not need an initial solution vectorwhich must be chosen carefully in many other proceduresif numerical convergence is to be achieved Furthermoreapplication of SCE algorithm in cocontent model does notrequire any complicated mathematical expression and opera-tion In this model pressure-driven demand and leakage canbe simulated easily and there is no failure in computation inzero flow conditions

51 Shuffled Complex Evolution (SCE) Shuffled complexevolution (SCE) is a simple powerful and population-basedstochastic optimization algorithm that outperforms manymetaheuristic algorithms in numerical single-objective opti-mization problems This method is based on a synthesis offour concepts (1) combination of deterministic and proba-bilistic approaches (2) systematic evolution of a ldquocomplexrdquoof points spanning the parameter space in the direction of

4 Journal of Fluids

f(x) objective function

xi decision variable (pipe diameter)

N number of decision variables

S number of solution vectors

P number of complexes

m = number of points in each complex

i = 1 S

Randomly generate the solutions

Evaluate objective function

Calculate cost function C(H)

Sort the solutions

Allocate solutions to complexes

For i = 1 B (number of inner iterations)

Select the subcomplexes from the complexesCompute the centroid of the worst point

Include new solution andexclude worst solution

Reflecting the worst pointthrough the centroid

Randomly generatea solution

ContractionCombine the solutions in the

evolved complexes into

a single sample population

Sort the solution vectors

Terminatecompetitions

Repeat Steps3 4 and 5

NI = number of iterations

NI

Step 1 initialize parameters Step 2 generate samples

Step 4 CCE algorithm

Step 3 shuffle complexes

Step 5 shuffle complexes

Step 6 check stopping criterion

Figure 2 SCE procedure for minimization of cocontent model

global improvement (3) competitive evolution (4) complexshuffling [16] The ldquocomplexrdquo is similar to the genetic poolin the GA The synthesis of these operators makes the SCEmethod effective and robust and also flexible and efficient[17]

In SCE method each individual represents a feasiblesolution for the problemThe searchwithin the feasible regionis conducted by first dividing the set of current feasible

trial solutions into several complexes each containing equalnumber of trial solutions Each complex represents a localarea of the whole domain Concurrent and independentsearches within each complex are conducted until eachconverges to its local optimal value Each of the complexeswhich are now defined by new trial solutions is collectedinto a common pool shuffled by ranking according to theirobjective function value and then further divided into new

Journal of Fluids 5

complexes The procedure is terminated when none of thelocal optima found among the complexes can improve onthe best current local optimum The SCE method usedthe downhill simplex method to accomplish local searchesSo shuffled complex evolution tries to balance between awide scan of a large solution space and deep search ofpromising locations It depends mainly on partitioning thesolution space into local communities and performing localsearch within these communitiesThen it shuffles these localcommunities to perform global search

The steps of the procedure of SCE as shown in Figure 2include the following

Step 1 initialize problem and algorithm parametersStep 2 samples generationStep 3 rank solutionsStep 4 partition into complexesStep 5 start Competitive Complex Evolution (CCE)Step 6 shuffle complexesStep 7 check the stopping criterion

511 Step 1 Initialize the Problem and Algorithm ParametersIn Step 1 the optimization problem is specified as follows

Min 119862 (119867) = sum

119909

10038161003816100381610038161003816119867119894+ ℎ119901119909minus ℎ119871119909minus 119867119895

10038161003816100381610038161003816

(1119899)+1

1198771119899

119909

+ (1

119899+ 1)sum

119895

119902119900119895(119867119895)

(8)

where 119862(119867) is an objective function 119867 is the set of eachdecision variable In this paper the objective function isthe cocontent model the unknown heads are the decisionvariables

512 Step 2 Samples Generation The initial population forthe DE is created arbitrarily by the following formula

119867(119894 119895) = 119867min (119894 119895) + 120591 (119867min (119894 119895) minus 119867max (119894 119895)) (9)

where 120591 denotes a uniformly distributed randomvaluewithinthe range [0 1] 119867min(119894 119895) and 119867max(119894 119895) are maximum andminimum limits of variable 119895 and node 119894Then the fitness val-ues 119862(119867) of all the individuals of population are calculatedThe position matrix of the population of generation 119866 can berepresented as

119875(119866)

=

[[[[[[[[[[[

[

1198621

1198622

119862119899Pop

]]]]]]]]]]]

]

=

[[[[[[[[[[[[[

[

1198671

11198671

2 119867

1

119873

1198672

11198672

2 119867

2

119873

119867119899Pop1

119867119899 Pop119873

]]]]]]]]]]]]]

]

(10)

in which119873 is the number of unknown nodes

513 Step 3 Rank Solutions In this step the s solutions aresorted in order of increasing criterion value so that the firstvector represents the smallest value of the objective functionand the last vector indicates the largest value

514 Step 4 Partition into Complexes The 119904 solutions arepartitioned into 119901 complexes each containing119898 points Thecomplexes are partitioned such that the first complex containsevery 119901(119896minus 1) + 1 ranked point the second complex containsevery 119901(119896 minus 1) + 2 ranked point and so on where 119896 = 1

2 119898 [16]

515 Step 5 Start Competitive Complex Evolution (CCE)CCE algorithm is based on the simplex downhill searchscheme and is one key component of SCE algorithm Thisalgorithm is presented as follows

(1) A subcomplex by randomly selecting 119902 solutions fromthe complex according to a trapezoidal probabilitydistribution is constructed The probability distri-bution is specified such that the best solution hasthe highest chance of being selected to form thesubcomplex and the worst point has the least chance

(2) Theworst solution of the subcomplex is identified andthe centroid of the subcomplex without including theworst solution is computed as follows

CR (119894) = (1 (119898 minus 1))

119898minus1

sum

119895=1

119904 (119894 119895) 119894 = 1 119875 (11)

(3) In this step reflection operator is used by reflectingthe worst point through the centroid according to thefollowing formula

119867new

(119894 119895) = 2 lowast CR (119894) minus 119867 (119894worst) (12)

If the newly generated solution is within the feasiblespace go to (4) otherwise randomly generate a pointwithin the feasible space by Equation (9) and go to (6)

(4) If the newly generated solution is better than theworst solution then it is replaced by the new solutionOtherwise go to (5)

(5) In this step contraction operator is applied by com-puting a solution halfway between the centroid andthe worst point

119867new

(119894 119895) =(CR (119894) minus 119867 (119894worst))

2 (13)

If the contraction solution is better than the worstsolution then it is replaced by the contraction solu-tion Otherwise go to (6) This step is imported fromcompetitive complex evolution (CCE)

(6) A solution within the feasible space is generatedrandomly and the worst solution is replaced by therandomly generated solution

(7) Steps (2)ndash(6) are repeated 120572 times and steps (1)ndash(7)are repeated 120573 times

6 Journal of Fluids

516 Step 6 Shuffle Complexes The solutions in the evolvedcomplexes into a single sample population are combinedand the sample population is sorted in order of increasingcriterion value and is shuffled into 119901 complexes

517 Step 7 Check the Stopping Criterion In this sectionSteps 3 4 and 5 are repeated until the termination criterionis satisfied

It should be noted that the competitive complex evolu-tion (CCE) algorithm is required for the evolution of eachcomplex Each point of a complex is a potential ldquoparentrdquowith the ability to participate in the process of reproducingoffspring A subcomplex functions like a pair of parents Useof a stochastic scheme to construct subcomplexes allows theparameter space to be searched more thoroughly The ideaof competitiveness is introduced in forming subcomplexeswhere the stronger survives better and breeds healthier off-spring than the weaker Inclusion of the competitive measureexpedites the search towards promising regions

A more detailed presentation of the SCE algorithm hasbeen given by Duan et al [17]

6 Numerical Examples

In this section the hydraulic analyses for several conditionsin some water distribution networks are performed All com-putations were executed inMATLAB programming languageenvironment with an Intel(R) Core(TM) 2Duo CPU P8700 253GHz and 400GB RAM In order to demonstrate theeffectiveness of SCE comparedwith othermethods this studyproposes the use of mass balance and energy balance in thenetworkThe average of mass and energy balance is shown by120575 and it is calculated by the following formula

120575 = mean(abs(119873

sum

119895=1

119910

sum

119894=119909

10038161003816100381610038161003816119867119894+ ℎ119901119896minus ℎ119871119896minus 119867119895

10038161003816100381610038161003816

(1119899)

1198771119899

119896

minus 119902119895))

(14)

For Figure 1 and (2) 120575 is calculated as follows

120575 = mean(10038161003816100381610038161003816100381610038161003816

120597119862

1205971198671

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198672

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198673

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198674

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198675

10038161003816100381610038161003816100381610038161003816

)

(15)

To check the performance of the SCE for the minimiza-tion of cocontent model in all examples ten optimizationrunswere performed using different random initial solutions

61 Numerical Example 1 In this part the verification ofthe above mentioned model was conducted via numericalsimulation based on an extremely simplified network scheme(5 nodes and 7 pipes) schematically shown in Figure 3 [18]The pipes resistances were 119877

1= 15625 119877

2= 50 119877

3= 100

1198774= 125 119877

5= 75 119877

6= 200 and 119877

7= 100 as reported in

Todini [18] for this networkThe SCE technique is applied to solve this problem

according to three cases The bound variables were setbetween 90 and 100m The problem is also solved using the

0Reservoir

11

1

2

2

2

3

3

3

4

4

5 6

7

800 ls

200 ls

200 ls

200 ls

300 ls

400 ls

400 ls

100 ls

100 ls

100 ls

100 ls

Figure 3 Schematic representation of the looped pipe network usedin numerical example 1

0 2 4 6 8 10 12 14 16 18 20

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

Figure 4 Convergence history of numerical example 1 (form 1)

global gradient algorithm (GGA) and the results are com-pared with those obtained by the SCE The best worst andaverage solutions of SCE algorithm in three cases are shownin Table 1 As it can be seen in Table 1 in all cases SCEthat found the optimal solution more accurately than GGAmethod The average number of function evaluation is about2000 in case 1 and about 100000 in case 3 This shows SCEcan converge to global optimum rapidly but reaching highaccuracy needs more operations The convergence process ofSCE algorithm has been shown in two forms in Figures 4and 5 The absolute value of 120575 is calculated for each iterationin Figure 4 and the amount of objective function 119862(119867) iscalculated for each iteration in Figure 5

62 Numerical Example 2 Example 2 considers the sym-metric network shown in Figure 6 It has 11 pipes sevenjunctions at which the head is unknown and one fixed headnode reservoir at 40m elevation and all other nodes areat zero elevation All pipes have diameters 119863 of 250mmand lengths 119871 of 1000m Node 8 has a demand of 80 lsand all other nodes have zero demands In the steady statethis network has zero flows in pipes 2 6 and 9 because of

Journal of Fluids 7

Table 1 Average of mass and energy balance for numerical example 1

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4460119864 minus 08 706119864 minus 07 304119864 minus 07 230119864 minus 07 203119864 + 03

Number of iterations in inner loop 4Number of complexes 9

119119864 minus 08 301119864 minus 08 204119864 minus 08 640119864 minus 09 100119864 + 05Number of iterations in inner loop 9Number of complexes 10

104119864 minus 08 419119864 minus 08 252119864 minus 08 828119864 minus 09 100119864 + 05Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy 640119864 minus 06

Table 2 Head and parameter 120575 in numerical example 2

Node number 119867 (m) [2] 119867 (m) 120575 [2] 120575 (SCE)1 40 40 0 02 366813 366853 0 18119864 minus 07

3 366813 366853 0 18119864 minus 07

4 333626 333706 65119864 minus 07 84E minus 085 333626 333706 65119864 minus 07 87E minus 086 30044 300559 65119864 minus 07 67E minus 087 30044 300559 65119864 minus 07 66E minus 088 267253 267411 52119864 minus 05 12E minus 10

0 2 4 6 8 10 12 14 16 18 20

1021718

10217179

10217178

10217177

10217176

10217175

10217174

10217173

Obj

ectiv

e fun

ctio

n

Number of iterations

Figure 5 Convergence history of numerical example 1 (form 2)

symmetry The head loss is modeled by the Hazen-Williamsequation and each pipe has a Hazen-Williams coefficient119862 = 120 [2] When the GGA is used in this network theiterates trend toward the solution and the flows in pipes 26 and 9 approach zero As this happens the Jacobian matrixbecomes more and more badly conditioned and the solutioncomputed becomes ill conditioned [2] Elhay and Simpson[2] proposed a regularization procedure for the GGA whichprevents failure of the solution process provided that a flowin the network is ultimately zero or near zero

The SCEparameters are set as follows number of decisionvariables = 7 number of points in each complex = 15 numberof complexes for case 1 = 4 case 2 = 9 and case 3 = 10 numberof iterations in inner loop for case 1 = 4 case 2 = 9 and

1

1

22

3

3

4

4

5

56

6

7

7

8

8

9

1011

Reservoir

Figure 6 Schematic representation of the looped pipe network usedin numerical example 2

case 3 = 15 The bound variables were set between 25 and40m The previous best solution for this network when it issimulated using the Elhay algorithm and the average solutionof SCE algorithm are shown in the second and third columnsof Table 2 respectively As can be observed in Table 2 massand energy balance (120575) in SCE are more accurate than theElhay algorithm Table 3 compares the results of applying theSCE algorithm in three casesThe convergence process of SCEalgorithm has been shown in two forms in Figures 7 and 8

8 Journal of Fluids

Table 3 Average of mass and energy balance for numerical example 2

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4221119864 minus 06 724119864 minus 06 465119864 minus 06 161119864 minus 06 419119864 + 03

Number of iterations in inner loop 4Number of complexes 9

105119864 minus 06 535119864 minus 06 346119864 minus 06 151119864 minus 06 929119864 + 03Number of iterations in inner loop 9Number of complexes 10

636119864 minus 08 178119864 minus 07 108119864 minus 07 389119864 minus 08 172119864 + 04Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy FailElhay algorithm Maximum accuracy 742119864 minus 06

Table 4 Head and parameter 120575 in numerical example 3

Node number 119885 (m) SCE 3 steps SCE 3 steps SCE 3 steps119867 (m) 119867 (m) 119867 minus 119885 119867 minus 119885 120575 120575

1 140 140 140 0 0 0 02 80 129304 13007 49304 5007 149E minus 07 310119864 minus 04

3 90 132288 13276 42288 4276 126E minus 07 000414 70 109587 11096 39587 4096 203E minus 07 000225 80 80000 8854 0000 854 0058 151E minus 046 90 90000 9145 0000 145 00069 602E minus 047 90 90000 9000 0000 000 0080 014218 100 88922 9043 minus11078 minus957 584E minus 08 00439

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations0 5 10 15 20 25 30 35 40 45 50

Figure 7 Convergence history of numerical example 2 (form 1)

The absolute value of 120575 is calculated for each iteration inFigure 7 and the value of head pressure in node 2 119867(2) iscalculated for each iteration in Figure 8

63 Numerical Example 3 The simplified water distributionnetwork shown in Figure 9 was used in order to demonstratethe advantages of the proposed model in pressure-drivendemand condition For the sake of simplicity the same

H(2)

Number of iterations0 5 10 15 20 25 30 35 40 45 50

365

366

367

368

369

37

371

372

373

Figure 8 Convergence history of head pressure (119898) in node 2 fornumerical example 2 (form 2)

Hazen-Williams roughness coefficient 119862 = 130 was assumedfor all the 14 pipes of identical length of 1000m while no locallosses have been added The following diameters have beenused in the example 500mm (P-2) 400mm (P-1) 300mm(P-4 P-7) 250mm (P-10) 200mm (P-3 P-5 P-6 and P-13)150mm (P-8 P-9 P-11 P-12 and P-14) The nodal demandsare listed in the following tables together with the groundelevation 119885

119894 Without loss of generality in this example the

Journal of Fluids 9

Reservoir

1 2

23

3

4

4

5

5

6

6

7

7

8

8

9

10 11 12 13

14

Figure 9 Schematic representation of the looped pipe network usedin numerical example 3

minimum head requirement 119867119894

lowast has been assumed to beequal to the ground elevation 119885

119894[8] So the relationship

between the required nodal head and minimum head is

119902119895=

0 119867119895lt 119885119895

119902119895 119885119895le 119867119895

(16)

Todini [8] proposed a three-step approach for solving thisnetwork and its solution is reported in the 4th column ofTable 4 In the proposed method pressure-driven model canbe applied in hydraulic analysis without any mathematicalformulation In this situation an if-then rule is added tococontent model and optimization process is conductedThenumber of decision variables in SCE algorithm is 7 the boundvariables were set between 50 and 140m ten optimizationruns were performed using different random initial solutionsfor all the cases and the results are illustrated in Table 5Results confirm that SCE is more accurate compared withTodini algorithm in case 2 and case 3 In Table 4 the bestresult is shown in bold and it is considered that the methodof SCE has calculated the best value of 120575 at 5 nodes whilethe Todini method has done it at 2 nodes The convergenceprocess of SCE algorithm has been shown in two forms inFigures 10 and 11The absolute value of 120575 is calculated for eachiteration in Figure 10 and the value of head pressure in node2119867(2) is calculated for each iteration in Figure 11

64 Numerical Example 4 The fourth considered networkis a real planned network designed for an industrial areain Apulian Town (Southern Italy) The network layout is

0 10 20 30 40 50 60 70 80 90

100

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

Figure 10 Convergence history of numerical example 3 (form 1)

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100

120

140

Number of iterations

H(2)

Figure 11 Convergence history of head pressure (119898) in node 2 fornumerical example 3 (form 2)

illustrated in Figure 12 and the corresponding data areprovided in Table 6 With respect to the leakages they havebeen assumed to be pressure-driven (see (6)) given that theyare implemented in the pressure driven network simulationmodel as described above [14] The parameter 120573 = 10632 times10minus7 and 120572 = 12 as reported in Giustolisi et al [14] for this

network Giustolisi et al [14] proposed a hydraulic simulationmodel which fully integrates a classic hydraulic simulationalgorithm such as that of Todini and Pilati [3] found inEPANET 2 with a pressure-driven model that entails amore realistic representation of the leakage They appliedtheir model in this network The results are demonstratedin Table 7 In this table the best result is shown in boldand it is considered that the method of SCE has calculatedthe best value of 120575 at 14 nodes while the Gistulishi methodhas done it at 9 nodes In the proposed method there is noneed to modify the mathematical formulation for hydraulic

10 Journal of Fluids

Table 5 Average of mass and energy balance for numerical example 3

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5332119864 minus 02 797119864 minus 02 595119864 minus 02 162119864 minus 02 430119864 + 04

Number of iterations in inner loop 5Number of complexes 9

207119864 minus 02 544119864 minus 02 252119864 minus 02 109119864 minus 02 627119864 + 04Number of iterations in inner loop 9Number of complexes 10

207119864 minus 02 207119864 minus 02 207119864 minus 02 217119864 minus 08 604119864 + 04Number of iterations in inner loop 15Three-step approach [8] Maximum accuracy 276119864 minus 02

Table 6 Hydraulic data relevant to numerical example 4

Pipe number 119871 (m) 119863 (mm)1 3485 3272 9557 2903 483 1004 4007 2905 7919 1006 4044 3687 3906 3278 4823 1009 9344 10010 4313 18411 5131 10012 4284 18413 419 10014 10231 10015 4551 16416 1826 29017 2213 29018 5839 16419 452 22920 7947 10021 7177 10022 6556 25823 1655 10024 2521 10025 3315 10026 500 20427 5799 16428 8428 10029 7926 10030 8463 18431 164 25832 4279 10033 3792 10034 1582 368

analysis An if-then rule is added to cocontent model andthe optimization process is performed easily As you can seein Table 8 SCE found the optimal solution more accurately

1

1

2 2

33

4

4

5

5

6

6

7

78

8

99

101011

11

12

12

1313

14

14

15

1516

1617

17

18

18

19

19

20

20

21

21

2222

23 2324

25

26

27

28

2930

31

32

33

34

Reservoir

Figure 12 Schematic representation of the looped pipe networkused in the numerical example 4

0 100 200 300 400 500 600 700 800 900 1000

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

10minus3

Figure 13 Convergence history of numerical example 4 (form 1)

than the Giustolisi algorithm The convergence process ofSCE algorithm has been shown in two forms in Figures 4 and5 The absolute value of 120575 is calculated for each iteration inFigure 13 and the amount of head pressure in node 20119867(20)is calculated for each iteration in Figure 14

Journal of Fluids 11

Table 7 Head and parameter 120575 in numerical example 4

Node number 119902 (ls) 119867 (m) (Giustolisi algorithm) 119867 (m) 120575 (Giustolisi algorithm) 120575 (SCE)1 10863 269 3329 0154743 00047382 17034 2481 3183 002131 00065323 14947 213 2739 0059002 00053644 1428 1722 2534 0001257 00050335 10133 2354 3089 0026184 00040186 1535 201 2902 0030898 00053997 9114 1891 2794 0017147 00030878 1051 179 2734 000227 00035459 12182 1785 2635 0002936 000386710 14579 1266 2324 0008277 000436211 9007 1623 2595 0031554 000272312 7575 1012 2205 0002732 000213813 152 1003 2245 00126 000432814 1355 1541 2595 0001318 000435215 9226 14 2417 0002199 000293416 112 1436 2405 0007089 000358617 11469 153 2542 0000103 00036118 10818 1883 2838 0011889 000390819 14675 1935 2839 543E minus 05 000509720 13318 1001 2379 0013059 000397421 14631 1148 2235 0003276 000414122 12012 14 2546 0003901 000367723 10326 1045 2011 0002551 000297924 mdash 3645 3645 mdash mdash

Table 8 Average of mass and energy balance for numerical example 4

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5400119864 minus 03 450119864 minus 02 410119864 minus 03 275119864 minus 04 659119864 + 04

Number of iterations in inner loop 5Number of complexes 9

390119864 minus 03 415119864 minus 03 400119864 minus 03 396119864 minus 05 115119864 + 05Number of iterations in inner loop 9Number of complexes 10

370119864 minus 03 410119864 minus 03 390119864 minus 03 253119864 minus 05 119119864 + 05Number of iterations in inner loop 15Giustolisi algorithm Maximum accuracy 181119864 minus 02

In general 4 different pipe networks were consideredin this paper and different mathematical formulations wereused for the hydraulic analysis of these networks Howeverthe overall results indicate that the proposed method hasthe capability of handling various pipe networks problemswith no change in the model or mathematical formulationApplication of SCE in cocontent model can result in findingaccurate solutions in pipes with zero flows and the pressure-driven demand and leakage simulation can be solved throughapplying if-then rules in cocontent model As a result itcan be concluded that the proposed method is a suitablealternative optimizer challenging othermethods especially interms of accuracy

7 Conclusions

The objective of the present paper was to provide an inno-vative approach in the analysis of the water distributionnetworks based on the optimization model The cocontentmodel is minimized using shuffled complex evolution (SCE)algorithm The methodology is illustrated here using fournetworks with different layouts The results reveal thatthe proposed method has the capability to handle variouspipe networks problems without changing in model ormathematical formulation The advantage of the proposedmethod lies in the fact that there is no need to solvelinear systems of equations pressure-driven demand and

12 Journal of Fluids

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

Number of iterations

H(20)

Figure 14 Convergence history of head-pressure (119898) in node 20 forexample 4 (form 2)

leakage simulation are handled in a simple way accu-rate solutions can be found in pipes with zero flowsand it does not need an initial solution vector which must bechosen carefully in many other procedures if numerical con-vergence is to be achieved Furthermore the proposedmodeldoes not require any complicated mathematical expressionand operation Finally it can be concluded that the proposedmethod is a viable alternative optimizer that challenges othermethods particularly in view of accuracy

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

References

[1] M Collins L Cooper R Helgason J Kenningston and LLeBlanc ldquoSolving the pipe network analysis problem usingoptimization techniquesrdquo Management Science vol 24 no 7pp 747ndash760 1978

[2] S Elhay and A R Simpson ldquoDealing with zero flows in solvingthe nonlinear equations for water distribution systemsrdquo Journalof Hydraulic Engineering vol 137 no 10 pp 1216ndash1224 2011

[3] E Todini and S Pilati ldquoA gradient algorithm for the analysis ofpipe networksrdquo in Computer Applications inWater Supply vol 1of System Analysis and Simulation pp 1ndash20 JohnWiley amp SonsLondon UK 1988

[4] A C Zecchin P Thum A R Simpson and C TischendorfldquoSteady-state behaviour of large water distribution systems thealgebraic multigrid method for the fast solution of the linearsteprdquo Journal ofWater Resources Planning andManagement vol138 no 6 pp 639ndash650 2012

[5] M L Arora ldquoFlow split in closed loops expending least energyrdquoJournal of the Hydraulics Division vol 102 pp 455ndash458 1976

[6] M A Hall ldquoHydraulic network analysis using (generalized)geometric programmingrdquo Networks vol 6 no 2 pp 105ndash1301976

[7] P R Bhave and R Gupta ldquoAnalysis of water distributionnetworksrdquo in Alpha Science International Technology amp Engi-neering 2006

[8] E Todini A More Realistic Approach to the ldquoExtended PeriodSimulationrdquo of Water Distribution Networks Advances in WaterSupplyManagement chapter 19 Taylor amp Francis London UK2003

[9] J M Wagner U Shamir and D H Marks ldquoWater distributionreliability analytical methodsrdquo Journal of Water ResourcesPlanning and Management vol 114 no 3 pp 253ndash275 1988

[10] J Chandapillai ldquoRealistic simulation of water distributionsystemrdquo Journal of Transportation Engineering vol 117 no 2 pp258ndash263 1991

[11] J Almandoz E Cabrera F Arregui E Cabrera Jr andR Cobacho ldquoLeakage assessment through water distributionnetwork simulationrdquo Journal of Water Resources Planning andManagement vol 131 no 6 pp 458ndash466 2005

[12] A F Colombo and B W Karney ldquoEnergy and costs ofleaky pipes toward comprehensive picturerdquo Journal of WaterResources Planning and Management vol 128 no 6 pp 441ndash450 2002

[13] G Germanopoulos ldquoA technical note on the inclusion ofpressure dependent demand and leakage terms in water supplynetworkmodelsrdquoCivil Engineering Systems vol 2 no 3 pp 171ndash179 1985

[14] O Giustolisi D Savic and Z Kapelan ldquoPressure-drivendemand and leakage simulation for water distribution net-worksrdquo Journal of Hydraulic Engineering vol 134 no 5 pp 626ndash635 2008

[15] L Ainola T Koppel K Tiiter and A Vassiljev ldquoWater networkmodel calibration based on grouping pipes with similar leakageand roughness estimatesrdquo in Proceedings of the Joint Conferenceon Water Resource Engineering and Water Resources PlanningandManagement (EWRI rsquo00) pp 1ndash9MinneapolisMinnUSAAugust 2000

[16] Q Duan S Sorooshian and V K Gupta ldquoOptimal use of theSCE-UA global optimization method for calibrating watershedmodelsrdquo Journal of Hydrology vol 158 no 3-4 pp 265ndash2841994

[17] Q Y Duan S Sorooshian and V K Gupta ldquoShuffled complexevolution approach for effective and efficient global minimiza-tionrdquo Journal of Optimization Theory and Applications vol 76no 3 pp 501ndash521 1993

[18] E Todini ldquoOn the convergence properties of the differentpipe network algorithmsrdquo in Proceedings of the Annual WaterDistribution Systems Analysis Symposium (ASCE rsquo06) pp 1ndash16August 2006

Submit your manuscripts athttpwwwhindawicom

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Journal of Fluids 3

120597119862

1205971198673

= +

10038161003816100381610038161198673 minus 11986721003816100381610038161003816

(1119899)

1198771119899

2

+

10038161003816100381610038161198673 minus 11986751003816100381610038161003816

(1119899)

1198771119899

6

minus

10038161003816100381610038161198674 minus ℎ1198713 minus 11986731003816100381610038161003816

(1119899)

1198771119899

3

+ 1199023= 0

120597119862

1205971198674

=

10038161003816100381610038161198674 minus 11986711003816100381610038161003816

(1119899)

1198771119899

4

+

10038161003816100381610038161198674 minus ℎ1198713 minus 11986731003816100381610038161003816

(1119899)

1198771119899

3

minus

100381610038161003816100381611986702 minus ℎ1198712 minus 11986741003816100381610038161003816

(1119899)

1198771119899

8

= 0

120597119862

1205971198675

= minus

10038161003816100381610038161198672 minus 11986751003816100381610038161003816

(1119899)

1198771119899

5

minus

10038161003816100381610038161198673 minus 11986751003816100381610038161003816

(1119899)

1198771119899

6

+ 1199025= 0

(3)

That is the purely head-based formulations of the networkequations So cocontentmodel not onlyminimizes the energyof flow but also preserves water balance in network Forsimplicity 119867

0119866can be taken as zero The general cocontent

model can be expressed as

Min 119862 (119867) = sum

119909

10038161003816100381610038161003816119867119894+ ℎ119901119909minus ℎ119871119909minus 119867119895

10038161003816100381610038161003816

(1119899)+1

1198771119899

119909

+ (1

119899+ 1)sum

119895

119902119895(119867119895)

(4)

Collins et al [1] suggested the solution of the NLPoptimization of the model The methods they used were (1)the Frank-Wolfe method (2) a piecewise linear approxima-tion and (3) the convex simplex method which are highlydependent on initial guesses and in some cases converge toan incorrect solution [1]

3 Head Dependent Analysis

In the common approaches it is presumed that the nodaldemands are always satisfied at all demand nodes irrespec-tive of the available HGL values at demand nodes [7] But inpractice when the head at a node is insufficient a reductionin the water flowing from the tap is expected and at worstthe discharge that can be drafted will be zero regardlessof the actual demand [8] There are several solutions forthese conditions in the literature Wagner et al [9] andChandapillai [10] suggested a parabolic relationship betweenrequired nodal head and minimum head Their relationshipsare

119902119895=

0 119867119895lt 119867min

119902119895(

119867119895minus 119867min

119867lowast minus 119867min)

1119901

119867min le 119867119895 lt 119867lowast

119902119895 119867

lowast

le 119867119895

(5)

119867lowast is the required nodal head In this situation applying

if-then rules in the optimization model is a simple way inhandling the above formulation

4 Leakage Simulation

Water losses via leakages constitute a major challenge to theeffective operation of municipal WDS since they representnot only diminished revenue for utilities but also underminedservice quality [11] and wasted energy resources [12] In orderto conduct a more accurate analysis of a WDS such as abetter estimate of flow through the network (with respectto both satisfied demand and losses through leakage) ahydraulic analysis capable of accounting for pressure-driven(also known as head-driven) demand and leakage flow atthe pipe level should prove invaluable To reach this goal aleakage model is expressed as follows [13]

119902119896-leak =

120573119896119897119896(119875119896)120572119896 if 119875

119896gt 0

0 if 119875119896le 0

(6)

where 119875119896is average pressure in the pipe computed as the

mean of the pressure values at the end nodes 119894 and119895 of the119896th pipe and 119897

119896is length of that pipe Variables 120572

119896and 120573

119896are

two leakage model parameters [14] The allocation of leakageto the two end nodes can be performed in a number of ways[15] Here the nodal leakage flow 119902

119895-leak is computed as thesum of 119902

119896-leak flows of all pipes connected to node 119895 as follows

119902119895-leak = sum

119896

1

2119902119896-leak = sum

119896

1

2120573119896119897119896(119875119896)120572119896 if 119875

119896gt 0

0 if 119875119896le 0

(7)

where 119875119896= (119875119894+ 119875119895)2 This formulation also easily applies

to cocontent model without any mathematical complexityNumerical example 4 demonstrates hydraulic analysis of areal pipe network in this situations

5 Application of Shuffled Complex EvolutionAlgorithm for Minimizing Cocontent Model

This study introduces the shuffled complex evolution (SCE)algorithm for the hydraulic analysis Since the algorithmwas originally developed to solve optimization problems thehydraulic network analysis was introduced into an optimiza-tion problem (cocontent model) One advantage of the SCEalgorithm is that it does not need an initial solution vectorwhich must be chosen carefully in many other proceduresif numerical convergence is to be achieved Furthermoreapplication of SCE algorithm in cocontent model does notrequire any complicated mathematical expression and opera-tion In this model pressure-driven demand and leakage canbe simulated easily and there is no failure in computation inzero flow conditions

51 Shuffled Complex Evolution (SCE) Shuffled complexevolution (SCE) is a simple powerful and population-basedstochastic optimization algorithm that outperforms manymetaheuristic algorithms in numerical single-objective opti-mization problems This method is based on a synthesis offour concepts (1) combination of deterministic and proba-bilistic approaches (2) systematic evolution of a ldquocomplexrdquoof points spanning the parameter space in the direction of

4 Journal of Fluids

f(x) objective function

xi decision variable (pipe diameter)

N number of decision variables

S number of solution vectors

P number of complexes

m = number of points in each complex

i = 1 S

Randomly generate the solutions

Evaluate objective function

Calculate cost function C(H)

Sort the solutions

Allocate solutions to complexes

For i = 1 B (number of inner iterations)

Select the subcomplexes from the complexesCompute the centroid of the worst point

Include new solution andexclude worst solution

Reflecting the worst pointthrough the centroid

Randomly generatea solution

ContractionCombine the solutions in the

evolved complexes into

a single sample population

Sort the solution vectors

Terminatecompetitions

Repeat Steps3 4 and 5

NI = number of iterations

NI

Step 1 initialize parameters Step 2 generate samples

Step 4 CCE algorithm

Step 3 shuffle complexes

Step 5 shuffle complexes

Step 6 check stopping criterion

Figure 2 SCE procedure for minimization of cocontent model

global improvement (3) competitive evolution (4) complexshuffling [16] The ldquocomplexrdquo is similar to the genetic poolin the GA The synthesis of these operators makes the SCEmethod effective and robust and also flexible and efficient[17]

In SCE method each individual represents a feasiblesolution for the problemThe searchwithin the feasible regionis conducted by first dividing the set of current feasible

trial solutions into several complexes each containing equalnumber of trial solutions Each complex represents a localarea of the whole domain Concurrent and independentsearches within each complex are conducted until eachconverges to its local optimal value Each of the complexeswhich are now defined by new trial solutions is collectedinto a common pool shuffled by ranking according to theirobjective function value and then further divided into new

Journal of Fluids 5

complexes The procedure is terminated when none of thelocal optima found among the complexes can improve onthe best current local optimum The SCE method usedthe downhill simplex method to accomplish local searchesSo shuffled complex evolution tries to balance between awide scan of a large solution space and deep search ofpromising locations It depends mainly on partitioning thesolution space into local communities and performing localsearch within these communitiesThen it shuffles these localcommunities to perform global search

The steps of the procedure of SCE as shown in Figure 2include the following

Step 1 initialize problem and algorithm parametersStep 2 samples generationStep 3 rank solutionsStep 4 partition into complexesStep 5 start Competitive Complex Evolution (CCE)Step 6 shuffle complexesStep 7 check the stopping criterion

511 Step 1 Initialize the Problem and Algorithm ParametersIn Step 1 the optimization problem is specified as follows

Min 119862 (119867) = sum

119909

10038161003816100381610038161003816119867119894+ ℎ119901119909minus ℎ119871119909minus 119867119895

10038161003816100381610038161003816

(1119899)+1

1198771119899

119909

+ (1

119899+ 1)sum

119895

119902119900119895(119867119895)

(8)

where 119862(119867) is an objective function 119867 is the set of eachdecision variable In this paper the objective function isthe cocontent model the unknown heads are the decisionvariables

512 Step 2 Samples Generation The initial population forthe DE is created arbitrarily by the following formula

119867(119894 119895) = 119867min (119894 119895) + 120591 (119867min (119894 119895) minus 119867max (119894 119895)) (9)

where 120591 denotes a uniformly distributed randomvaluewithinthe range [0 1] 119867min(119894 119895) and 119867max(119894 119895) are maximum andminimum limits of variable 119895 and node 119894Then the fitness val-ues 119862(119867) of all the individuals of population are calculatedThe position matrix of the population of generation 119866 can berepresented as

119875(119866)

=

[[[[[[[[[[[

[

1198621

1198622

119862119899Pop

]]]]]]]]]]]

]

=

[[[[[[[[[[[[[

[

1198671

11198671

2 119867

1

119873

1198672

11198672

2 119867

2

119873

119867119899Pop1

119867119899 Pop119873

]]]]]]]]]]]]]

]

(10)

in which119873 is the number of unknown nodes

513 Step 3 Rank Solutions In this step the s solutions aresorted in order of increasing criterion value so that the firstvector represents the smallest value of the objective functionand the last vector indicates the largest value

514 Step 4 Partition into Complexes The 119904 solutions arepartitioned into 119901 complexes each containing119898 points Thecomplexes are partitioned such that the first complex containsevery 119901(119896minus 1) + 1 ranked point the second complex containsevery 119901(119896 minus 1) + 2 ranked point and so on where 119896 = 1

2 119898 [16]

515 Step 5 Start Competitive Complex Evolution (CCE)CCE algorithm is based on the simplex downhill searchscheme and is one key component of SCE algorithm Thisalgorithm is presented as follows

(1) A subcomplex by randomly selecting 119902 solutions fromthe complex according to a trapezoidal probabilitydistribution is constructed The probability distri-bution is specified such that the best solution hasthe highest chance of being selected to form thesubcomplex and the worst point has the least chance

(2) Theworst solution of the subcomplex is identified andthe centroid of the subcomplex without including theworst solution is computed as follows

CR (119894) = (1 (119898 minus 1))

119898minus1

sum

119895=1

119904 (119894 119895) 119894 = 1 119875 (11)

(3) In this step reflection operator is used by reflectingthe worst point through the centroid according to thefollowing formula

119867new

(119894 119895) = 2 lowast CR (119894) minus 119867 (119894worst) (12)

If the newly generated solution is within the feasiblespace go to (4) otherwise randomly generate a pointwithin the feasible space by Equation (9) and go to (6)

(4) If the newly generated solution is better than theworst solution then it is replaced by the new solutionOtherwise go to (5)

(5) In this step contraction operator is applied by com-puting a solution halfway between the centroid andthe worst point

119867new

(119894 119895) =(CR (119894) minus 119867 (119894worst))

2 (13)

If the contraction solution is better than the worstsolution then it is replaced by the contraction solu-tion Otherwise go to (6) This step is imported fromcompetitive complex evolution (CCE)

(6) A solution within the feasible space is generatedrandomly and the worst solution is replaced by therandomly generated solution

(7) Steps (2)ndash(6) are repeated 120572 times and steps (1)ndash(7)are repeated 120573 times

6 Journal of Fluids

516 Step 6 Shuffle Complexes The solutions in the evolvedcomplexes into a single sample population are combinedand the sample population is sorted in order of increasingcriterion value and is shuffled into 119901 complexes

517 Step 7 Check the Stopping Criterion In this sectionSteps 3 4 and 5 are repeated until the termination criterionis satisfied

It should be noted that the competitive complex evolu-tion (CCE) algorithm is required for the evolution of eachcomplex Each point of a complex is a potential ldquoparentrdquowith the ability to participate in the process of reproducingoffspring A subcomplex functions like a pair of parents Useof a stochastic scheme to construct subcomplexes allows theparameter space to be searched more thoroughly The ideaof competitiveness is introduced in forming subcomplexeswhere the stronger survives better and breeds healthier off-spring than the weaker Inclusion of the competitive measureexpedites the search towards promising regions

A more detailed presentation of the SCE algorithm hasbeen given by Duan et al [17]

6 Numerical Examples

In this section the hydraulic analyses for several conditionsin some water distribution networks are performed All com-putations were executed inMATLAB programming languageenvironment with an Intel(R) Core(TM) 2Duo CPU P8700 253GHz and 400GB RAM In order to demonstrate theeffectiveness of SCE comparedwith othermethods this studyproposes the use of mass balance and energy balance in thenetworkThe average of mass and energy balance is shown by120575 and it is calculated by the following formula

120575 = mean(abs(119873

sum

119895=1

119910

sum

119894=119909

10038161003816100381610038161003816119867119894+ ℎ119901119896minus ℎ119871119896minus 119867119895

10038161003816100381610038161003816

(1119899)

1198771119899

119896

minus 119902119895))

(14)

For Figure 1 and (2) 120575 is calculated as follows

120575 = mean(10038161003816100381610038161003816100381610038161003816

120597119862

1205971198671

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198672

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198673

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198674

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198675

10038161003816100381610038161003816100381610038161003816

)

(15)

To check the performance of the SCE for the minimiza-tion of cocontent model in all examples ten optimizationrunswere performed using different random initial solutions

61 Numerical Example 1 In this part the verification ofthe above mentioned model was conducted via numericalsimulation based on an extremely simplified network scheme(5 nodes and 7 pipes) schematically shown in Figure 3 [18]The pipes resistances were 119877

1= 15625 119877

2= 50 119877

3= 100

1198774= 125 119877

5= 75 119877

6= 200 and 119877

7= 100 as reported in

Todini [18] for this networkThe SCE technique is applied to solve this problem

according to three cases The bound variables were setbetween 90 and 100m The problem is also solved using the

0Reservoir

11

1

2

2

2

3

3

3

4

4

5 6

7

800 ls

200 ls

200 ls

200 ls

300 ls

400 ls

400 ls

100 ls

100 ls

100 ls

100 ls

Figure 3 Schematic representation of the looped pipe network usedin numerical example 1

0 2 4 6 8 10 12 14 16 18 20

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

Figure 4 Convergence history of numerical example 1 (form 1)

global gradient algorithm (GGA) and the results are com-pared with those obtained by the SCE The best worst andaverage solutions of SCE algorithm in three cases are shownin Table 1 As it can be seen in Table 1 in all cases SCEthat found the optimal solution more accurately than GGAmethod The average number of function evaluation is about2000 in case 1 and about 100000 in case 3 This shows SCEcan converge to global optimum rapidly but reaching highaccuracy needs more operations The convergence process ofSCE algorithm has been shown in two forms in Figures 4and 5 The absolute value of 120575 is calculated for each iterationin Figure 4 and the amount of objective function 119862(119867) iscalculated for each iteration in Figure 5

62 Numerical Example 2 Example 2 considers the sym-metric network shown in Figure 6 It has 11 pipes sevenjunctions at which the head is unknown and one fixed headnode reservoir at 40m elevation and all other nodes areat zero elevation All pipes have diameters 119863 of 250mmand lengths 119871 of 1000m Node 8 has a demand of 80 lsand all other nodes have zero demands In the steady statethis network has zero flows in pipes 2 6 and 9 because of

Journal of Fluids 7

Table 1 Average of mass and energy balance for numerical example 1

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4460119864 minus 08 706119864 minus 07 304119864 minus 07 230119864 minus 07 203119864 + 03

Number of iterations in inner loop 4Number of complexes 9

119119864 minus 08 301119864 minus 08 204119864 minus 08 640119864 minus 09 100119864 + 05Number of iterations in inner loop 9Number of complexes 10

104119864 minus 08 419119864 minus 08 252119864 minus 08 828119864 minus 09 100119864 + 05Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy 640119864 minus 06

Table 2 Head and parameter 120575 in numerical example 2

Node number 119867 (m) [2] 119867 (m) 120575 [2] 120575 (SCE)1 40 40 0 02 366813 366853 0 18119864 minus 07

3 366813 366853 0 18119864 minus 07

4 333626 333706 65119864 minus 07 84E minus 085 333626 333706 65119864 minus 07 87E minus 086 30044 300559 65119864 minus 07 67E minus 087 30044 300559 65119864 minus 07 66E minus 088 267253 267411 52119864 minus 05 12E minus 10

0 2 4 6 8 10 12 14 16 18 20

1021718

10217179

10217178

10217177

10217176

10217175

10217174

10217173

Obj

ectiv

e fun

ctio

n

Number of iterations

Figure 5 Convergence history of numerical example 1 (form 2)

symmetry The head loss is modeled by the Hazen-Williamsequation and each pipe has a Hazen-Williams coefficient119862 = 120 [2] When the GGA is used in this network theiterates trend toward the solution and the flows in pipes 26 and 9 approach zero As this happens the Jacobian matrixbecomes more and more badly conditioned and the solutioncomputed becomes ill conditioned [2] Elhay and Simpson[2] proposed a regularization procedure for the GGA whichprevents failure of the solution process provided that a flowin the network is ultimately zero or near zero

The SCEparameters are set as follows number of decisionvariables = 7 number of points in each complex = 15 numberof complexes for case 1 = 4 case 2 = 9 and case 3 = 10 numberof iterations in inner loop for case 1 = 4 case 2 = 9 and

1

1

22

3

3

4

4

5

56

6

7

7

8

8

9

1011

Reservoir

Figure 6 Schematic representation of the looped pipe network usedin numerical example 2

case 3 = 15 The bound variables were set between 25 and40m The previous best solution for this network when it issimulated using the Elhay algorithm and the average solutionof SCE algorithm are shown in the second and third columnsof Table 2 respectively As can be observed in Table 2 massand energy balance (120575) in SCE are more accurate than theElhay algorithm Table 3 compares the results of applying theSCE algorithm in three casesThe convergence process of SCEalgorithm has been shown in two forms in Figures 7 and 8

8 Journal of Fluids

Table 3 Average of mass and energy balance for numerical example 2

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4221119864 minus 06 724119864 minus 06 465119864 minus 06 161119864 minus 06 419119864 + 03

Number of iterations in inner loop 4Number of complexes 9

105119864 minus 06 535119864 minus 06 346119864 minus 06 151119864 minus 06 929119864 + 03Number of iterations in inner loop 9Number of complexes 10

636119864 minus 08 178119864 minus 07 108119864 minus 07 389119864 minus 08 172119864 + 04Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy FailElhay algorithm Maximum accuracy 742119864 minus 06

Table 4 Head and parameter 120575 in numerical example 3

Node number 119885 (m) SCE 3 steps SCE 3 steps SCE 3 steps119867 (m) 119867 (m) 119867 minus 119885 119867 minus 119885 120575 120575

1 140 140 140 0 0 0 02 80 129304 13007 49304 5007 149E minus 07 310119864 minus 04

3 90 132288 13276 42288 4276 126E minus 07 000414 70 109587 11096 39587 4096 203E minus 07 000225 80 80000 8854 0000 854 0058 151E minus 046 90 90000 9145 0000 145 00069 602E minus 047 90 90000 9000 0000 000 0080 014218 100 88922 9043 minus11078 minus957 584E minus 08 00439

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations0 5 10 15 20 25 30 35 40 45 50

Figure 7 Convergence history of numerical example 2 (form 1)

The absolute value of 120575 is calculated for each iteration inFigure 7 and the value of head pressure in node 2 119867(2) iscalculated for each iteration in Figure 8

63 Numerical Example 3 The simplified water distributionnetwork shown in Figure 9 was used in order to demonstratethe advantages of the proposed model in pressure-drivendemand condition For the sake of simplicity the same

H(2)

Number of iterations0 5 10 15 20 25 30 35 40 45 50

365

366

367

368

369

37

371

372

373

Figure 8 Convergence history of head pressure (119898) in node 2 fornumerical example 2 (form 2)

Hazen-Williams roughness coefficient 119862 = 130 was assumedfor all the 14 pipes of identical length of 1000m while no locallosses have been added The following diameters have beenused in the example 500mm (P-2) 400mm (P-1) 300mm(P-4 P-7) 250mm (P-10) 200mm (P-3 P-5 P-6 and P-13)150mm (P-8 P-9 P-11 P-12 and P-14) The nodal demandsare listed in the following tables together with the groundelevation 119885

119894 Without loss of generality in this example the

Journal of Fluids 9

Reservoir

1 2

23

3

4

4

5

5

6

6

7

7

8

8

9

10 11 12 13

14

Figure 9 Schematic representation of the looped pipe network usedin numerical example 3

minimum head requirement 119867119894

lowast has been assumed to beequal to the ground elevation 119885

119894[8] So the relationship

between the required nodal head and minimum head is

119902119895=

0 119867119895lt 119885119895

119902119895 119885119895le 119867119895

(16)

Todini [8] proposed a three-step approach for solving thisnetwork and its solution is reported in the 4th column ofTable 4 In the proposed method pressure-driven model canbe applied in hydraulic analysis without any mathematicalformulation In this situation an if-then rule is added tococontent model and optimization process is conductedThenumber of decision variables in SCE algorithm is 7 the boundvariables were set between 50 and 140m ten optimizationruns were performed using different random initial solutionsfor all the cases and the results are illustrated in Table 5Results confirm that SCE is more accurate compared withTodini algorithm in case 2 and case 3 In Table 4 the bestresult is shown in bold and it is considered that the methodof SCE has calculated the best value of 120575 at 5 nodes whilethe Todini method has done it at 2 nodes The convergenceprocess of SCE algorithm has been shown in two forms inFigures 10 and 11The absolute value of 120575 is calculated for eachiteration in Figure 10 and the value of head pressure in node2119867(2) is calculated for each iteration in Figure 11

64 Numerical Example 4 The fourth considered networkis a real planned network designed for an industrial areain Apulian Town (Southern Italy) The network layout is

0 10 20 30 40 50 60 70 80 90

100

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

Figure 10 Convergence history of numerical example 3 (form 1)

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100

120

140

Number of iterations

H(2)

Figure 11 Convergence history of head pressure (119898) in node 2 fornumerical example 3 (form 2)

illustrated in Figure 12 and the corresponding data areprovided in Table 6 With respect to the leakages they havebeen assumed to be pressure-driven (see (6)) given that theyare implemented in the pressure driven network simulationmodel as described above [14] The parameter 120573 = 10632 times10minus7 and 120572 = 12 as reported in Giustolisi et al [14] for this

network Giustolisi et al [14] proposed a hydraulic simulationmodel which fully integrates a classic hydraulic simulationalgorithm such as that of Todini and Pilati [3] found inEPANET 2 with a pressure-driven model that entails amore realistic representation of the leakage They appliedtheir model in this network The results are demonstratedin Table 7 In this table the best result is shown in boldand it is considered that the method of SCE has calculatedthe best value of 120575 at 14 nodes while the Gistulishi methodhas done it at 9 nodes In the proposed method there is noneed to modify the mathematical formulation for hydraulic

10 Journal of Fluids

Table 5 Average of mass and energy balance for numerical example 3

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5332119864 minus 02 797119864 minus 02 595119864 minus 02 162119864 minus 02 430119864 + 04

Number of iterations in inner loop 5Number of complexes 9

207119864 minus 02 544119864 minus 02 252119864 minus 02 109119864 minus 02 627119864 + 04Number of iterations in inner loop 9Number of complexes 10

207119864 minus 02 207119864 minus 02 207119864 minus 02 217119864 minus 08 604119864 + 04Number of iterations in inner loop 15Three-step approach [8] Maximum accuracy 276119864 minus 02

Table 6 Hydraulic data relevant to numerical example 4

Pipe number 119871 (m) 119863 (mm)1 3485 3272 9557 2903 483 1004 4007 2905 7919 1006 4044 3687 3906 3278 4823 1009 9344 10010 4313 18411 5131 10012 4284 18413 419 10014 10231 10015 4551 16416 1826 29017 2213 29018 5839 16419 452 22920 7947 10021 7177 10022 6556 25823 1655 10024 2521 10025 3315 10026 500 20427 5799 16428 8428 10029 7926 10030 8463 18431 164 25832 4279 10033 3792 10034 1582 368

analysis An if-then rule is added to cocontent model andthe optimization process is performed easily As you can seein Table 8 SCE found the optimal solution more accurately

1

1

2 2

33

4

4

5

5

6

6

7

78

8

99

101011

11

12

12

1313

14

14

15

1516

1617

17

18

18

19

19

20

20

21

21

2222

23 2324

25

26

27

28

2930

31

32

33

34

Reservoir

Figure 12 Schematic representation of the looped pipe networkused in the numerical example 4

0 100 200 300 400 500 600 700 800 900 1000

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

10minus3

Figure 13 Convergence history of numerical example 4 (form 1)

than the Giustolisi algorithm The convergence process ofSCE algorithm has been shown in two forms in Figures 4 and5 The absolute value of 120575 is calculated for each iteration inFigure 13 and the amount of head pressure in node 20119867(20)is calculated for each iteration in Figure 14

Journal of Fluids 11

Table 7 Head and parameter 120575 in numerical example 4

Node number 119902 (ls) 119867 (m) (Giustolisi algorithm) 119867 (m) 120575 (Giustolisi algorithm) 120575 (SCE)1 10863 269 3329 0154743 00047382 17034 2481 3183 002131 00065323 14947 213 2739 0059002 00053644 1428 1722 2534 0001257 00050335 10133 2354 3089 0026184 00040186 1535 201 2902 0030898 00053997 9114 1891 2794 0017147 00030878 1051 179 2734 000227 00035459 12182 1785 2635 0002936 000386710 14579 1266 2324 0008277 000436211 9007 1623 2595 0031554 000272312 7575 1012 2205 0002732 000213813 152 1003 2245 00126 000432814 1355 1541 2595 0001318 000435215 9226 14 2417 0002199 000293416 112 1436 2405 0007089 000358617 11469 153 2542 0000103 00036118 10818 1883 2838 0011889 000390819 14675 1935 2839 543E minus 05 000509720 13318 1001 2379 0013059 000397421 14631 1148 2235 0003276 000414122 12012 14 2546 0003901 000367723 10326 1045 2011 0002551 000297924 mdash 3645 3645 mdash mdash

Table 8 Average of mass and energy balance for numerical example 4

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5400119864 minus 03 450119864 minus 02 410119864 minus 03 275119864 minus 04 659119864 + 04

Number of iterations in inner loop 5Number of complexes 9

390119864 minus 03 415119864 minus 03 400119864 minus 03 396119864 minus 05 115119864 + 05Number of iterations in inner loop 9Number of complexes 10

370119864 minus 03 410119864 minus 03 390119864 minus 03 253119864 minus 05 119119864 + 05Number of iterations in inner loop 15Giustolisi algorithm Maximum accuracy 181119864 minus 02

In general 4 different pipe networks were consideredin this paper and different mathematical formulations wereused for the hydraulic analysis of these networks Howeverthe overall results indicate that the proposed method hasthe capability of handling various pipe networks problemswith no change in the model or mathematical formulationApplication of SCE in cocontent model can result in findingaccurate solutions in pipes with zero flows and the pressure-driven demand and leakage simulation can be solved throughapplying if-then rules in cocontent model As a result itcan be concluded that the proposed method is a suitablealternative optimizer challenging othermethods especially interms of accuracy

7 Conclusions

The objective of the present paper was to provide an inno-vative approach in the analysis of the water distributionnetworks based on the optimization model The cocontentmodel is minimized using shuffled complex evolution (SCE)algorithm The methodology is illustrated here using fournetworks with different layouts The results reveal thatthe proposed method has the capability to handle variouspipe networks problems without changing in model ormathematical formulation The advantage of the proposedmethod lies in the fact that there is no need to solvelinear systems of equations pressure-driven demand and

12 Journal of Fluids

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

Number of iterations

H(20)

Figure 14 Convergence history of head-pressure (119898) in node 20 forexample 4 (form 2)

leakage simulation are handled in a simple way accu-rate solutions can be found in pipes with zero flowsand it does not need an initial solution vector which must bechosen carefully in many other procedures if numerical con-vergence is to be achieved Furthermore the proposedmodeldoes not require any complicated mathematical expressionand operation Finally it can be concluded that the proposedmethod is a viable alternative optimizer that challenges othermethods particularly in view of accuracy

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

References

[1] M Collins L Cooper R Helgason J Kenningston and LLeBlanc ldquoSolving the pipe network analysis problem usingoptimization techniquesrdquo Management Science vol 24 no 7pp 747ndash760 1978

[2] S Elhay and A R Simpson ldquoDealing with zero flows in solvingthe nonlinear equations for water distribution systemsrdquo Journalof Hydraulic Engineering vol 137 no 10 pp 1216ndash1224 2011

[3] E Todini and S Pilati ldquoA gradient algorithm for the analysis ofpipe networksrdquo in Computer Applications inWater Supply vol 1of System Analysis and Simulation pp 1ndash20 JohnWiley amp SonsLondon UK 1988

[4] A C Zecchin P Thum A R Simpson and C TischendorfldquoSteady-state behaviour of large water distribution systems thealgebraic multigrid method for the fast solution of the linearsteprdquo Journal ofWater Resources Planning andManagement vol138 no 6 pp 639ndash650 2012

[5] M L Arora ldquoFlow split in closed loops expending least energyrdquoJournal of the Hydraulics Division vol 102 pp 455ndash458 1976

[6] M A Hall ldquoHydraulic network analysis using (generalized)geometric programmingrdquo Networks vol 6 no 2 pp 105ndash1301976

[7] P R Bhave and R Gupta ldquoAnalysis of water distributionnetworksrdquo in Alpha Science International Technology amp Engi-neering 2006

[8] E Todini A More Realistic Approach to the ldquoExtended PeriodSimulationrdquo of Water Distribution Networks Advances in WaterSupplyManagement chapter 19 Taylor amp Francis London UK2003

[9] J M Wagner U Shamir and D H Marks ldquoWater distributionreliability analytical methodsrdquo Journal of Water ResourcesPlanning and Management vol 114 no 3 pp 253ndash275 1988

[10] J Chandapillai ldquoRealistic simulation of water distributionsystemrdquo Journal of Transportation Engineering vol 117 no 2 pp258ndash263 1991

[11] J Almandoz E Cabrera F Arregui E Cabrera Jr andR Cobacho ldquoLeakage assessment through water distributionnetwork simulationrdquo Journal of Water Resources Planning andManagement vol 131 no 6 pp 458ndash466 2005

[12] A F Colombo and B W Karney ldquoEnergy and costs ofleaky pipes toward comprehensive picturerdquo Journal of WaterResources Planning and Management vol 128 no 6 pp 441ndash450 2002

[13] G Germanopoulos ldquoA technical note on the inclusion ofpressure dependent demand and leakage terms in water supplynetworkmodelsrdquoCivil Engineering Systems vol 2 no 3 pp 171ndash179 1985

[14] O Giustolisi D Savic and Z Kapelan ldquoPressure-drivendemand and leakage simulation for water distribution net-worksrdquo Journal of Hydraulic Engineering vol 134 no 5 pp 626ndash635 2008

[15] L Ainola T Koppel K Tiiter and A Vassiljev ldquoWater networkmodel calibration based on grouping pipes with similar leakageand roughness estimatesrdquo in Proceedings of the Joint Conferenceon Water Resource Engineering and Water Resources PlanningandManagement (EWRI rsquo00) pp 1ndash9MinneapolisMinnUSAAugust 2000

[16] Q Duan S Sorooshian and V K Gupta ldquoOptimal use of theSCE-UA global optimization method for calibrating watershedmodelsrdquo Journal of Hydrology vol 158 no 3-4 pp 265ndash2841994

[17] Q Y Duan S Sorooshian and V K Gupta ldquoShuffled complexevolution approach for effective and efficient global minimiza-tionrdquo Journal of Optimization Theory and Applications vol 76no 3 pp 501ndash521 1993

[18] E Todini ldquoOn the convergence properties of the differentpipe network algorithmsrdquo in Proceedings of the Annual WaterDistribution Systems Analysis Symposium (ASCE rsquo06) pp 1ndash16August 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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4 Journal of Fluids

f(x) objective function

xi decision variable (pipe diameter)

N number of decision variables

S number of solution vectors

P number of complexes

m = number of points in each complex

i = 1 S

Randomly generate the solutions

Evaluate objective function

Calculate cost function C(H)

Sort the solutions

Allocate solutions to complexes

For i = 1 B (number of inner iterations)

Select the subcomplexes from the complexesCompute the centroid of the worst point

Include new solution andexclude worst solution

Reflecting the worst pointthrough the centroid

Randomly generatea solution

ContractionCombine the solutions in the

evolved complexes into

a single sample population

Sort the solution vectors

Terminatecompetitions

Repeat Steps3 4 and 5

NI = number of iterations

NI

Step 1 initialize parameters Step 2 generate samples

Step 4 CCE algorithm

Step 3 shuffle complexes

Step 5 shuffle complexes

Step 6 check stopping criterion

Figure 2 SCE procedure for minimization of cocontent model

global improvement (3) competitive evolution (4) complexshuffling [16] The ldquocomplexrdquo is similar to the genetic poolin the GA The synthesis of these operators makes the SCEmethod effective and robust and also flexible and efficient[17]

In SCE method each individual represents a feasiblesolution for the problemThe searchwithin the feasible regionis conducted by first dividing the set of current feasible

trial solutions into several complexes each containing equalnumber of trial solutions Each complex represents a localarea of the whole domain Concurrent and independentsearches within each complex are conducted until eachconverges to its local optimal value Each of the complexeswhich are now defined by new trial solutions is collectedinto a common pool shuffled by ranking according to theirobjective function value and then further divided into new

Journal of Fluids 5

complexes The procedure is terminated when none of thelocal optima found among the complexes can improve onthe best current local optimum The SCE method usedthe downhill simplex method to accomplish local searchesSo shuffled complex evolution tries to balance between awide scan of a large solution space and deep search ofpromising locations It depends mainly on partitioning thesolution space into local communities and performing localsearch within these communitiesThen it shuffles these localcommunities to perform global search

The steps of the procedure of SCE as shown in Figure 2include the following

Step 1 initialize problem and algorithm parametersStep 2 samples generationStep 3 rank solutionsStep 4 partition into complexesStep 5 start Competitive Complex Evolution (CCE)Step 6 shuffle complexesStep 7 check the stopping criterion

511 Step 1 Initialize the Problem and Algorithm ParametersIn Step 1 the optimization problem is specified as follows

Min 119862 (119867) = sum

119909

10038161003816100381610038161003816119867119894+ ℎ119901119909minus ℎ119871119909minus 119867119895

10038161003816100381610038161003816

(1119899)+1

1198771119899

119909

+ (1

119899+ 1)sum

119895

119902119900119895(119867119895)

(8)

where 119862(119867) is an objective function 119867 is the set of eachdecision variable In this paper the objective function isthe cocontent model the unknown heads are the decisionvariables

512 Step 2 Samples Generation The initial population forthe DE is created arbitrarily by the following formula

119867(119894 119895) = 119867min (119894 119895) + 120591 (119867min (119894 119895) minus 119867max (119894 119895)) (9)

where 120591 denotes a uniformly distributed randomvaluewithinthe range [0 1] 119867min(119894 119895) and 119867max(119894 119895) are maximum andminimum limits of variable 119895 and node 119894Then the fitness val-ues 119862(119867) of all the individuals of population are calculatedThe position matrix of the population of generation 119866 can berepresented as

119875(119866)

=

[[[[[[[[[[[

[

1198621

1198622

119862119899Pop

]]]]]]]]]]]

]

=

[[[[[[[[[[[[[

[

1198671

11198671

2 119867

1

119873

1198672

11198672

2 119867

2

119873

119867119899Pop1

119867119899 Pop119873

]]]]]]]]]]]]]

]

(10)

in which119873 is the number of unknown nodes

513 Step 3 Rank Solutions In this step the s solutions aresorted in order of increasing criterion value so that the firstvector represents the smallest value of the objective functionand the last vector indicates the largest value

514 Step 4 Partition into Complexes The 119904 solutions arepartitioned into 119901 complexes each containing119898 points Thecomplexes are partitioned such that the first complex containsevery 119901(119896minus 1) + 1 ranked point the second complex containsevery 119901(119896 minus 1) + 2 ranked point and so on where 119896 = 1

2 119898 [16]

515 Step 5 Start Competitive Complex Evolution (CCE)CCE algorithm is based on the simplex downhill searchscheme and is one key component of SCE algorithm Thisalgorithm is presented as follows

(1) A subcomplex by randomly selecting 119902 solutions fromthe complex according to a trapezoidal probabilitydistribution is constructed The probability distri-bution is specified such that the best solution hasthe highest chance of being selected to form thesubcomplex and the worst point has the least chance

(2) Theworst solution of the subcomplex is identified andthe centroid of the subcomplex without including theworst solution is computed as follows

CR (119894) = (1 (119898 minus 1))

119898minus1

sum

119895=1

119904 (119894 119895) 119894 = 1 119875 (11)

(3) In this step reflection operator is used by reflectingthe worst point through the centroid according to thefollowing formula

119867new

(119894 119895) = 2 lowast CR (119894) minus 119867 (119894worst) (12)

If the newly generated solution is within the feasiblespace go to (4) otherwise randomly generate a pointwithin the feasible space by Equation (9) and go to (6)

(4) If the newly generated solution is better than theworst solution then it is replaced by the new solutionOtherwise go to (5)

(5) In this step contraction operator is applied by com-puting a solution halfway between the centroid andthe worst point

119867new

(119894 119895) =(CR (119894) minus 119867 (119894worst))

2 (13)

If the contraction solution is better than the worstsolution then it is replaced by the contraction solu-tion Otherwise go to (6) This step is imported fromcompetitive complex evolution (CCE)

(6) A solution within the feasible space is generatedrandomly and the worst solution is replaced by therandomly generated solution

(7) Steps (2)ndash(6) are repeated 120572 times and steps (1)ndash(7)are repeated 120573 times

6 Journal of Fluids

516 Step 6 Shuffle Complexes The solutions in the evolvedcomplexes into a single sample population are combinedand the sample population is sorted in order of increasingcriterion value and is shuffled into 119901 complexes

517 Step 7 Check the Stopping Criterion In this sectionSteps 3 4 and 5 are repeated until the termination criterionis satisfied

It should be noted that the competitive complex evolu-tion (CCE) algorithm is required for the evolution of eachcomplex Each point of a complex is a potential ldquoparentrdquowith the ability to participate in the process of reproducingoffspring A subcomplex functions like a pair of parents Useof a stochastic scheme to construct subcomplexes allows theparameter space to be searched more thoroughly The ideaof competitiveness is introduced in forming subcomplexeswhere the stronger survives better and breeds healthier off-spring than the weaker Inclusion of the competitive measureexpedites the search towards promising regions

A more detailed presentation of the SCE algorithm hasbeen given by Duan et al [17]

6 Numerical Examples

In this section the hydraulic analyses for several conditionsin some water distribution networks are performed All com-putations were executed inMATLAB programming languageenvironment with an Intel(R) Core(TM) 2Duo CPU P8700 253GHz and 400GB RAM In order to demonstrate theeffectiveness of SCE comparedwith othermethods this studyproposes the use of mass balance and energy balance in thenetworkThe average of mass and energy balance is shown by120575 and it is calculated by the following formula

120575 = mean(abs(119873

sum

119895=1

119910

sum

119894=119909

10038161003816100381610038161003816119867119894+ ℎ119901119896minus ℎ119871119896minus 119867119895

10038161003816100381610038161003816

(1119899)

1198771119899

119896

minus 119902119895))

(14)

For Figure 1 and (2) 120575 is calculated as follows

120575 = mean(10038161003816100381610038161003816100381610038161003816

120597119862

1205971198671

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198672

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198673

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198674

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198675

10038161003816100381610038161003816100381610038161003816

)

(15)

To check the performance of the SCE for the minimiza-tion of cocontent model in all examples ten optimizationrunswere performed using different random initial solutions

61 Numerical Example 1 In this part the verification ofthe above mentioned model was conducted via numericalsimulation based on an extremely simplified network scheme(5 nodes and 7 pipes) schematically shown in Figure 3 [18]The pipes resistances were 119877

1= 15625 119877

2= 50 119877

3= 100

1198774= 125 119877

5= 75 119877

6= 200 and 119877

7= 100 as reported in

Todini [18] for this networkThe SCE technique is applied to solve this problem

according to three cases The bound variables were setbetween 90 and 100m The problem is also solved using the

0Reservoir

11

1

2

2

2

3

3

3

4

4

5 6

7

800 ls

200 ls

200 ls

200 ls

300 ls

400 ls

400 ls

100 ls

100 ls

100 ls

100 ls

Figure 3 Schematic representation of the looped pipe network usedin numerical example 1

0 2 4 6 8 10 12 14 16 18 20

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

Figure 4 Convergence history of numerical example 1 (form 1)

global gradient algorithm (GGA) and the results are com-pared with those obtained by the SCE The best worst andaverage solutions of SCE algorithm in three cases are shownin Table 1 As it can be seen in Table 1 in all cases SCEthat found the optimal solution more accurately than GGAmethod The average number of function evaluation is about2000 in case 1 and about 100000 in case 3 This shows SCEcan converge to global optimum rapidly but reaching highaccuracy needs more operations The convergence process ofSCE algorithm has been shown in two forms in Figures 4and 5 The absolute value of 120575 is calculated for each iterationin Figure 4 and the amount of objective function 119862(119867) iscalculated for each iteration in Figure 5

62 Numerical Example 2 Example 2 considers the sym-metric network shown in Figure 6 It has 11 pipes sevenjunctions at which the head is unknown and one fixed headnode reservoir at 40m elevation and all other nodes areat zero elevation All pipes have diameters 119863 of 250mmand lengths 119871 of 1000m Node 8 has a demand of 80 lsand all other nodes have zero demands In the steady statethis network has zero flows in pipes 2 6 and 9 because of

Journal of Fluids 7

Table 1 Average of mass and energy balance for numerical example 1

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4460119864 minus 08 706119864 minus 07 304119864 minus 07 230119864 minus 07 203119864 + 03

Number of iterations in inner loop 4Number of complexes 9

119119864 minus 08 301119864 minus 08 204119864 minus 08 640119864 minus 09 100119864 + 05Number of iterations in inner loop 9Number of complexes 10

104119864 minus 08 419119864 minus 08 252119864 minus 08 828119864 minus 09 100119864 + 05Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy 640119864 minus 06

Table 2 Head and parameter 120575 in numerical example 2

Node number 119867 (m) [2] 119867 (m) 120575 [2] 120575 (SCE)1 40 40 0 02 366813 366853 0 18119864 minus 07

3 366813 366853 0 18119864 minus 07

4 333626 333706 65119864 minus 07 84E minus 085 333626 333706 65119864 minus 07 87E minus 086 30044 300559 65119864 minus 07 67E minus 087 30044 300559 65119864 minus 07 66E minus 088 267253 267411 52119864 minus 05 12E minus 10

0 2 4 6 8 10 12 14 16 18 20

1021718

10217179

10217178

10217177

10217176

10217175

10217174

10217173

Obj

ectiv

e fun

ctio

n

Number of iterations

Figure 5 Convergence history of numerical example 1 (form 2)

symmetry The head loss is modeled by the Hazen-Williamsequation and each pipe has a Hazen-Williams coefficient119862 = 120 [2] When the GGA is used in this network theiterates trend toward the solution and the flows in pipes 26 and 9 approach zero As this happens the Jacobian matrixbecomes more and more badly conditioned and the solutioncomputed becomes ill conditioned [2] Elhay and Simpson[2] proposed a regularization procedure for the GGA whichprevents failure of the solution process provided that a flowin the network is ultimately zero or near zero

The SCEparameters are set as follows number of decisionvariables = 7 number of points in each complex = 15 numberof complexes for case 1 = 4 case 2 = 9 and case 3 = 10 numberof iterations in inner loop for case 1 = 4 case 2 = 9 and

1

1

22

3

3

4

4

5

56

6

7

7

8

8

9

1011

Reservoir

Figure 6 Schematic representation of the looped pipe network usedin numerical example 2

case 3 = 15 The bound variables were set between 25 and40m The previous best solution for this network when it issimulated using the Elhay algorithm and the average solutionof SCE algorithm are shown in the second and third columnsof Table 2 respectively As can be observed in Table 2 massand energy balance (120575) in SCE are more accurate than theElhay algorithm Table 3 compares the results of applying theSCE algorithm in three casesThe convergence process of SCEalgorithm has been shown in two forms in Figures 7 and 8

8 Journal of Fluids

Table 3 Average of mass and energy balance for numerical example 2

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4221119864 minus 06 724119864 minus 06 465119864 minus 06 161119864 minus 06 419119864 + 03

Number of iterations in inner loop 4Number of complexes 9

105119864 minus 06 535119864 minus 06 346119864 minus 06 151119864 minus 06 929119864 + 03Number of iterations in inner loop 9Number of complexes 10

636119864 minus 08 178119864 minus 07 108119864 minus 07 389119864 minus 08 172119864 + 04Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy FailElhay algorithm Maximum accuracy 742119864 minus 06

Table 4 Head and parameter 120575 in numerical example 3

Node number 119885 (m) SCE 3 steps SCE 3 steps SCE 3 steps119867 (m) 119867 (m) 119867 minus 119885 119867 minus 119885 120575 120575

1 140 140 140 0 0 0 02 80 129304 13007 49304 5007 149E minus 07 310119864 minus 04

3 90 132288 13276 42288 4276 126E minus 07 000414 70 109587 11096 39587 4096 203E minus 07 000225 80 80000 8854 0000 854 0058 151E minus 046 90 90000 9145 0000 145 00069 602E minus 047 90 90000 9000 0000 000 0080 014218 100 88922 9043 minus11078 minus957 584E minus 08 00439

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations0 5 10 15 20 25 30 35 40 45 50

Figure 7 Convergence history of numerical example 2 (form 1)

The absolute value of 120575 is calculated for each iteration inFigure 7 and the value of head pressure in node 2 119867(2) iscalculated for each iteration in Figure 8

63 Numerical Example 3 The simplified water distributionnetwork shown in Figure 9 was used in order to demonstratethe advantages of the proposed model in pressure-drivendemand condition For the sake of simplicity the same

H(2)

Number of iterations0 5 10 15 20 25 30 35 40 45 50

365

366

367

368

369

37

371

372

373

Figure 8 Convergence history of head pressure (119898) in node 2 fornumerical example 2 (form 2)

Hazen-Williams roughness coefficient 119862 = 130 was assumedfor all the 14 pipes of identical length of 1000m while no locallosses have been added The following diameters have beenused in the example 500mm (P-2) 400mm (P-1) 300mm(P-4 P-7) 250mm (P-10) 200mm (P-3 P-5 P-6 and P-13)150mm (P-8 P-9 P-11 P-12 and P-14) The nodal demandsare listed in the following tables together with the groundelevation 119885

119894 Without loss of generality in this example the

Journal of Fluids 9

Reservoir

1 2

23

3

4

4

5

5

6

6

7

7

8

8

9

10 11 12 13

14

Figure 9 Schematic representation of the looped pipe network usedin numerical example 3

minimum head requirement 119867119894

lowast has been assumed to beequal to the ground elevation 119885

119894[8] So the relationship

between the required nodal head and minimum head is

119902119895=

0 119867119895lt 119885119895

119902119895 119885119895le 119867119895

(16)

Todini [8] proposed a three-step approach for solving thisnetwork and its solution is reported in the 4th column ofTable 4 In the proposed method pressure-driven model canbe applied in hydraulic analysis without any mathematicalformulation In this situation an if-then rule is added tococontent model and optimization process is conductedThenumber of decision variables in SCE algorithm is 7 the boundvariables were set between 50 and 140m ten optimizationruns were performed using different random initial solutionsfor all the cases and the results are illustrated in Table 5Results confirm that SCE is more accurate compared withTodini algorithm in case 2 and case 3 In Table 4 the bestresult is shown in bold and it is considered that the methodof SCE has calculated the best value of 120575 at 5 nodes whilethe Todini method has done it at 2 nodes The convergenceprocess of SCE algorithm has been shown in two forms inFigures 10 and 11The absolute value of 120575 is calculated for eachiteration in Figure 10 and the value of head pressure in node2119867(2) is calculated for each iteration in Figure 11

64 Numerical Example 4 The fourth considered networkis a real planned network designed for an industrial areain Apulian Town (Southern Italy) The network layout is

0 10 20 30 40 50 60 70 80 90

100

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

Figure 10 Convergence history of numerical example 3 (form 1)

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100

120

140

Number of iterations

H(2)

Figure 11 Convergence history of head pressure (119898) in node 2 fornumerical example 3 (form 2)

illustrated in Figure 12 and the corresponding data areprovided in Table 6 With respect to the leakages they havebeen assumed to be pressure-driven (see (6)) given that theyare implemented in the pressure driven network simulationmodel as described above [14] The parameter 120573 = 10632 times10minus7 and 120572 = 12 as reported in Giustolisi et al [14] for this

network Giustolisi et al [14] proposed a hydraulic simulationmodel which fully integrates a classic hydraulic simulationalgorithm such as that of Todini and Pilati [3] found inEPANET 2 with a pressure-driven model that entails amore realistic representation of the leakage They appliedtheir model in this network The results are demonstratedin Table 7 In this table the best result is shown in boldand it is considered that the method of SCE has calculatedthe best value of 120575 at 14 nodes while the Gistulishi methodhas done it at 9 nodes In the proposed method there is noneed to modify the mathematical formulation for hydraulic

10 Journal of Fluids

Table 5 Average of mass and energy balance for numerical example 3

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5332119864 minus 02 797119864 minus 02 595119864 minus 02 162119864 minus 02 430119864 + 04

Number of iterations in inner loop 5Number of complexes 9

207119864 minus 02 544119864 minus 02 252119864 minus 02 109119864 minus 02 627119864 + 04Number of iterations in inner loop 9Number of complexes 10

207119864 minus 02 207119864 minus 02 207119864 minus 02 217119864 minus 08 604119864 + 04Number of iterations in inner loop 15Three-step approach [8] Maximum accuracy 276119864 minus 02

Table 6 Hydraulic data relevant to numerical example 4

Pipe number 119871 (m) 119863 (mm)1 3485 3272 9557 2903 483 1004 4007 2905 7919 1006 4044 3687 3906 3278 4823 1009 9344 10010 4313 18411 5131 10012 4284 18413 419 10014 10231 10015 4551 16416 1826 29017 2213 29018 5839 16419 452 22920 7947 10021 7177 10022 6556 25823 1655 10024 2521 10025 3315 10026 500 20427 5799 16428 8428 10029 7926 10030 8463 18431 164 25832 4279 10033 3792 10034 1582 368

analysis An if-then rule is added to cocontent model andthe optimization process is performed easily As you can seein Table 8 SCE found the optimal solution more accurately

1

1

2 2

33

4

4

5

5

6

6

7

78

8

99

101011

11

12

12

1313

14

14

15

1516

1617

17

18

18

19

19

20

20

21

21

2222

23 2324

25

26

27

28

2930

31

32

33

34

Reservoir

Figure 12 Schematic representation of the looped pipe networkused in the numerical example 4

0 100 200 300 400 500 600 700 800 900 1000

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

10minus3

Figure 13 Convergence history of numerical example 4 (form 1)

than the Giustolisi algorithm The convergence process ofSCE algorithm has been shown in two forms in Figures 4 and5 The absolute value of 120575 is calculated for each iteration inFigure 13 and the amount of head pressure in node 20119867(20)is calculated for each iteration in Figure 14

Journal of Fluids 11

Table 7 Head and parameter 120575 in numerical example 4

Node number 119902 (ls) 119867 (m) (Giustolisi algorithm) 119867 (m) 120575 (Giustolisi algorithm) 120575 (SCE)1 10863 269 3329 0154743 00047382 17034 2481 3183 002131 00065323 14947 213 2739 0059002 00053644 1428 1722 2534 0001257 00050335 10133 2354 3089 0026184 00040186 1535 201 2902 0030898 00053997 9114 1891 2794 0017147 00030878 1051 179 2734 000227 00035459 12182 1785 2635 0002936 000386710 14579 1266 2324 0008277 000436211 9007 1623 2595 0031554 000272312 7575 1012 2205 0002732 000213813 152 1003 2245 00126 000432814 1355 1541 2595 0001318 000435215 9226 14 2417 0002199 000293416 112 1436 2405 0007089 000358617 11469 153 2542 0000103 00036118 10818 1883 2838 0011889 000390819 14675 1935 2839 543E minus 05 000509720 13318 1001 2379 0013059 000397421 14631 1148 2235 0003276 000414122 12012 14 2546 0003901 000367723 10326 1045 2011 0002551 000297924 mdash 3645 3645 mdash mdash

Table 8 Average of mass and energy balance for numerical example 4

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5400119864 minus 03 450119864 minus 02 410119864 minus 03 275119864 minus 04 659119864 + 04

Number of iterations in inner loop 5Number of complexes 9

390119864 minus 03 415119864 minus 03 400119864 minus 03 396119864 minus 05 115119864 + 05Number of iterations in inner loop 9Number of complexes 10

370119864 minus 03 410119864 minus 03 390119864 minus 03 253119864 minus 05 119119864 + 05Number of iterations in inner loop 15Giustolisi algorithm Maximum accuracy 181119864 minus 02

In general 4 different pipe networks were consideredin this paper and different mathematical formulations wereused for the hydraulic analysis of these networks Howeverthe overall results indicate that the proposed method hasthe capability of handling various pipe networks problemswith no change in the model or mathematical formulationApplication of SCE in cocontent model can result in findingaccurate solutions in pipes with zero flows and the pressure-driven demand and leakage simulation can be solved throughapplying if-then rules in cocontent model As a result itcan be concluded that the proposed method is a suitablealternative optimizer challenging othermethods especially interms of accuracy

7 Conclusions

The objective of the present paper was to provide an inno-vative approach in the analysis of the water distributionnetworks based on the optimization model The cocontentmodel is minimized using shuffled complex evolution (SCE)algorithm The methodology is illustrated here using fournetworks with different layouts The results reveal thatthe proposed method has the capability to handle variouspipe networks problems without changing in model ormathematical formulation The advantage of the proposedmethod lies in the fact that there is no need to solvelinear systems of equations pressure-driven demand and

12 Journal of Fluids

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

Number of iterations

H(20)

Figure 14 Convergence history of head-pressure (119898) in node 20 forexample 4 (form 2)

leakage simulation are handled in a simple way accu-rate solutions can be found in pipes with zero flowsand it does not need an initial solution vector which must bechosen carefully in many other procedures if numerical con-vergence is to be achieved Furthermore the proposedmodeldoes not require any complicated mathematical expressionand operation Finally it can be concluded that the proposedmethod is a viable alternative optimizer that challenges othermethods particularly in view of accuracy

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

References

[1] M Collins L Cooper R Helgason J Kenningston and LLeBlanc ldquoSolving the pipe network analysis problem usingoptimization techniquesrdquo Management Science vol 24 no 7pp 747ndash760 1978

[2] S Elhay and A R Simpson ldquoDealing with zero flows in solvingthe nonlinear equations for water distribution systemsrdquo Journalof Hydraulic Engineering vol 137 no 10 pp 1216ndash1224 2011

[3] E Todini and S Pilati ldquoA gradient algorithm for the analysis ofpipe networksrdquo in Computer Applications inWater Supply vol 1of System Analysis and Simulation pp 1ndash20 JohnWiley amp SonsLondon UK 1988

[4] A C Zecchin P Thum A R Simpson and C TischendorfldquoSteady-state behaviour of large water distribution systems thealgebraic multigrid method for the fast solution of the linearsteprdquo Journal ofWater Resources Planning andManagement vol138 no 6 pp 639ndash650 2012

[5] M L Arora ldquoFlow split in closed loops expending least energyrdquoJournal of the Hydraulics Division vol 102 pp 455ndash458 1976

[6] M A Hall ldquoHydraulic network analysis using (generalized)geometric programmingrdquo Networks vol 6 no 2 pp 105ndash1301976

[7] P R Bhave and R Gupta ldquoAnalysis of water distributionnetworksrdquo in Alpha Science International Technology amp Engi-neering 2006

[8] E Todini A More Realistic Approach to the ldquoExtended PeriodSimulationrdquo of Water Distribution Networks Advances in WaterSupplyManagement chapter 19 Taylor amp Francis London UK2003

[9] J M Wagner U Shamir and D H Marks ldquoWater distributionreliability analytical methodsrdquo Journal of Water ResourcesPlanning and Management vol 114 no 3 pp 253ndash275 1988

[10] J Chandapillai ldquoRealistic simulation of water distributionsystemrdquo Journal of Transportation Engineering vol 117 no 2 pp258ndash263 1991

[11] J Almandoz E Cabrera F Arregui E Cabrera Jr andR Cobacho ldquoLeakage assessment through water distributionnetwork simulationrdquo Journal of Water Resources Planning andManagement vol 131 no 6 pp 458ndash466 2005

[12] A F Colombo and B W Karney ldquoEnergy and costs ofleaky pipes toward comprehensive picturerdquo Journal of WaterResources Planning and Management vol 128 no 6 pp 441ndash450 2002

[13] G Germanopoulos ldquoA technical note on the inclusion ofpressure dependent demand and leakage terms in water supplynetworkmodelsrdquoCivil Engineering Systems vol 2 no 3 pp 171ndash179 1985

[14] O Giustolisi D Savic and Z Kapelan ldquoPressure-drivendemand and leakage simulation for water distribution net-worksrdquo Journal of Hydraulic Engineering vol 134 no 5 pp 626ndash635 2008

[15] L Ainola T Koppel K Tiiter and A Vassiljev ldquoWater networkmodel calibration based on grouping pipes with similar leakageand roughness estimatesrdquo in Proceedings of the Joint Conferenceon Water Resource Engineering and Water Resources PlanningandManagement (EWRI rsquo00) pp 1ndash9MinneapolisMinnUSAAugust 2000

[16] Q Duan S Sorooshian and V K Gupta ldquoOptimal use of theSCE-UA global optimization method for calibrating watershedmodelsrdquo Journal of Hydrology vol 158 no 3-4 pp 265ndash2841994

[17] Q Y Duan S Sorooshian and V K Gupta ldquoShuffled complexevolution approach for effective and efficient global minimiza-tionrdquo Journal of Optimization Theory and Applications vol 76no 3 pp 501ndash521 1993

[18] E Todini ldquoOn the convergence properties of the differentpipe network algorithmsrdquo in Proceedings of the Annual WaterDistribution Systems Analysis Symposium (ASCE rsquo06) pp 1ndash16August 2006

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Journal of Fluids 5

complexes The procedure is terminated when none of thelocal optima found among the complexes can improve onthe best current local optimum The SCE method usedthe downhill simplex method to accomplish local searchesSo shuffled complex evolution tries to balance between awide scan of a large solution space and deep search ofpromising locations It depends mainly on partitioning thesolution space into local communities and performing localsearch within these communitiesThen it shuffles these localcommunities to perform global search

The steps of the procedure of SCE as shown in Figure 2include the following

Step 1 initialize problem and algorithm parametersStep 2 samples generationStep 3 rank solutionsStep 4 partition into complexesStep 5 start Competitive Complex Evolution (CCE)Step 6 shuffle complexesStep 7 check the stopping criterion

511 Step 1 Initialize the Problem and Algorithm ParametersIn Step 1 the optimization problem is specified as follows

Min 119862 (119867) = sum

119909

10038161003816100381610038161003816119867119894+ ℎ119901119909minus ℎ119871119909minus 119867119895

10038161003816100381610038161003816

(1119899)+1

1198771119899

119909

+ (1

119899+ 1)sum

119895

119902119900119895(119867119895)

(8)

where 119862(119867) is an objective function 119867 is the set of eachdecision variable In this paper the objective function isthe cocontent model the unknown heads are the decisionvariables

512 Step 2 Samples Generation The initial population forthe DE is created arbitrarily by the following formula

119867(119894 119895) = 119867min (119894 119895) + 120591 (119867min (119894 119895) minus 119867max (119894 119895)) (9)

where 120591 denotes a uniformly distributed randomvaluewithinthe range [0 1] 119867min(119894 119895) and 119867max(119894 119895) are maximum andminimum limits of variable 119895 and node 119894Then the fitness val-ues 119862(119867) of all the individuals of population are calculatedThe position matrix of the population of generation 119866 can berepresented as

119875(119866)

=

[[[[[[[[[[[

[

1198621

1198622

119862119899Pop

]]]]]]]]]]]

]

=

[[[[[[[[[[[[[

[

1198671

11198671

2 119867

1

119873

1198672

11198672

2 119867

2

119873

119867119899Pop1

119867119899 Pop119873

]]]]]]]]]]]]]

]

(10)

in which119873 is the number of unknown nodes

513 Step 3 Rank Solutions In this step the s solutions aresorted in order of increasing criterion value so that the firstvector represents the smallest value of the objective functionand the last vector indicates the largest value

514 Step 4 Partition into Complexes The 119904 solutions arepartitioned into 119901 complexes each containing119898 points Thecomplexes are partitioned such that the first complex containsevery 119901(119896minus 1) + 1 ranked point the second complex containsevery 119901(119896 minus 1) + 2 ranked point and so on where 119896 = 1

2 119898 [16]

515 Step 5 Start Competitive Complex Evolution (CCE)CCE algorithm is based on the simplex downhill searchscheme and is one key component of SCE algorithm Thisalgorithm is presented as follows

(1) A subcomplex by randomly selecting 119902 solutions fromthe complex according to a trapezoidal probabilitydistribution is constructed The probability distri-bution is specified such that the best solution hasthe highest chance of being selected to form thesubcomplex and the worst point has the least chance

(2) Theworst solution of the subcomplex is identified andthe centroid of the subcomplex without including theworst solution is computed as follows

CR (119894) = (1 (119898 minus 1))

119898minus1

sum

119895=1

119904 (119894 119895) 119894 = 1 119875 (11)

(3) In this step reflection operator is used by reflectingthe worst point through the centroid according to thefollowing formula

119867new

(119894 119895) = 2 lowast CR (119894) minus 119867 (119894worst) (12)

If the newly generated solution is within the feasiblespace go to (4) otherwise randomly generate a pointwithin the feasible space by Equation (9) and go to (6)

(4) If the newly generated solution is better than theworst solution then it is replaced by the new solutionOtherwise go to (5)

(5) In this step contraction operator is applied by com-puting a solution halfway between the centroid andthe worst point

119867new

(119894 119895) =(CR (119894) minus 119867 (119894worst))

2 (13)

If the contraction solution is better than the worstsolution then it is replaced by the contraction solu-tion Otherwise go to (6) This step is imported fromcompetitive complex evolution (CCE)

(6) A solution within the feasible space is generatedrandomly and the worst solution is replaced by therandomly generated solution

(7) Steps (2)ndash(6) are repeated 120572 times and steps (1)ndash(7)are repeated 120573 times

6 Journal of Fluids

516 Step 6 Shuffle Complexes The solutions in the evolvedcomplexes into a single sample population are combinedand the sample population is sorted in order of increasingcriterion value and is shuffled into 119901 complexes

517 Step 7 Check the Stopping Criterion In this sectionSteps 3 4 and 5 are repeated until the termination criterionis satisfied

It should be noted that the competitive complex evolu-tion (CCE) algorithm is required for the evolution of eachcomplex Each point of a complex is a potential ldquoparentrdquowith the ability to participate in the process of reproducingoffspring A subcomplex functions like a pair of parents Useof a stochastic scheme to construct subcomplexes allows theparameter space to be searched more thoroughly The ideaof competitiveness is introduced in forming subcomplexeswhere the stronger survives better and breeds healthier off-spring than the weaker Inclusion of the competitive measureexpedites the search towards promising regions

A more detailed presentation of the SCE algorithm hasbeen given by Duan et al [17]

6 Numerical Examples

In this section the hydraulic analyses for several conditionsin some water distribution networks are performed All com-putations were executed inMATLAB programming languageenvironment with an Intel(R) Core(TM) 2Duo CPU P8700 253GHz and 400GB RAM In order to demonstrate theeffectiveness of SCE comparedwith othermethods this studyproposes the use of mass balance and energy balance in thenetworkThe average of mass and energy balance is shown by120575 and it is calculated by the following formula

120575 = mean(abs(119873

sum

119895=1

119910

sum

119894=119909

10038161003816100381610038161003816119867119894+ ℎ119901119896minus ℎ119871119896minus 119867119895

10038161003816100381610038161003816

(1119899)

1198771119899

119896

minus 119902119895))

(14)

For Figure 1 and (2) 120575 is calculated as follows

120575 = mean(10038161003816100381610038161003816100381610038161003816

120597119862

1205971198671

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198672

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198673

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198674

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198675

10038161003816100381610038161003816100381610038161003816

)

(15)

To check the performance of the SCE for the minimiza-tion of cocontent model in all examples ten optimizationrunswere performed using different random initial solutions

61 Numerical Example 1 In this part the verification ofthe above mentioned model was conducted via numericalsimulation based on an extremely simplified network scheme(5 nodes and 7 pipes) schematically shown in Figure 3 [18]The pipes resistances were 119877

1= 15625 119877

2= 50 119877

3= 100

1198774= 125 119877

5= 75 119877

6= 200 and 119877

7= 100 as reported in

Todini [18] for this networkThe SCE technique is applied to solve this problem

according to three cases The bound variables were setbetween 90 and 100m The problem is also solved using the

0Reservoir

11

1

2

2

2

3

3

3

4

4

5 6

7

800 ls

200 ls

200 ls

200 ls

300 ls

400 ls

400 ls

100 ls

100 ls

100 ls

100 ls

Figure 3 Schematic representation of the looped pipe network usedin numerical example 1

0 2 4 6 8 10 12 14 16 18 20

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

Figure 4 Convergence history of numerical example 1 (form 1)

global gradient algorithm (GGA) and the results are com-pared with those obtained by the SCE The best worst andaverage solutions of SCE algorithm in three cases are shownin Table 1 As it can be seen in Table 1 in all cases SCEthat found the optimal solution more accurately than GGAmethod The average number of function evaluation is about2000 in case 1 and about 100000 in case 3 This shows SCEcan converge to global optimum rapidly but reaching highaccuracy needs more operations The convergence process ofSCE algorithm has been shown in two forms in Figures 4and 5 The absolute value of 120575 is calculated for each iterationin Figure 4 and the amount of objective function 119862(119867) iscalculated for each iteration in Figure 5

62 Numerical Example 2 Example 2 considers the sym-metric network shown in Figure 6 It has 11 pipes sevenjunctions at which the head is unknown and one fixed headnode reservoir at 40m elevation and all other nodes areat zero elevation All pipes have diameters 119863 of 250mmand lengths 119871 of 1000m Node 8 has a demand of 80 lsand all other nodes have zero demands In the steady statethis network has zero flows in pipes 2 6 and 9 because of

Journal of Fluids 7

Table 1 Average of mass and energy balance for numerical example 1

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4460119864 minus 08 706119864 minus 07 304119864 minus 07 230119864 minus 07 203119864 + 03

Number of iterations in inner loop 4Number of complexes 9

119119864 minus 08 301119864 minus 08 204119864 minus 08 640119864 minus 09 100119864 + 05Number of iterations in inner loop 9Number of complexes 10

104119864 minus 08 419119864 minus 08 252119864 minus 08 828119864 minus 09 100119864 + 05Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy 640119864 minus 06

Table 2 Head and parameter 120575 in numerical example 2

Node number 119867 (m) [2] 119867 (m) 120575 [2] 120575 (SCE)1 40 40 0 02 366813 366853 0 18119864 minus 07

3 366813 366853 0 18119864 minus 07

4 333626 333706 65119864 minus 07 84E minus 085 333626 333706 65119864 minus 07 87E minus 086 30044 300559 65119864 minus 07 67E minus 087 30044 300559 65119864 minus 07 66E minus 088 267253 267411 52119864 minus 05 12E minus 10

0 2 4 6 8 10 12 14 16 18 20

1021718

10217179

10217178

10217177

10217176

10217175

10217174

10217173

Obj

ectiv

e fun

ctio

n

Number of iterations

Figure 5 Convergence history of numerical example 1 (form 2)

symmetry The head loss is modeled by the Hazen-Williamsequation and each pipe has a Hazen-Williams coefficient119862 = 120 [2] When the GGA is used in this network theiterates trend toward the solution and the flows in pipes 26 and 9 approach zero As this happens the Jacobian matrixbecomes more and more badly conditioned and the solutioncomputed becomes ill conditioned [2] Elhay and Simpson[2] proposed a regularization procedure for the GGA whichprevents failure of the solution process provided that a flowin the network is ultimately zero or near zero

The SCEparameters are set as follows number of decisionvariables = 7 number of points in each complex = 15 numberof complexes for case 1 = 4 case 2 = 9 and case 3 = 10 numberof iterations in inner loop for case 1 = 4 case 2 = 9 and

1

1

22

3

3

4

4

5

56

6

7

7

8

8

9

1011

Reservoir

Figure 6 Schematic representation of the looped pipe network usedin numerical example 2

case 3 = 15 The bound variables were set between 25 and40m The previous best solution for this network when it issimulated using the Elhay algorithm and the average solutionof SCE algorithm are shown in the second and third columnsof Table 2 respectively As can be observed in Table 2 massand energy balance (120575) in SCE are more accurate than theElhay algorithm Table 3 compares the results of applying theSCE algorithm in three casesThe convergence process of SCEalgorithm has been shown in two forms in Figures 7 and 8

8 Journal of Fluids

Table 3 Average of mass and energy balance for numerical example 2

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4221119864 minus 06 724119864 minus 06 465119864 minus 06 161119864 minus 06 419119864 + 03

Number of iterations in inner loop 4Number of complexes 9

105119864 minus 06 535119864 minus 06 346119864 minus 06 151119864 minus 06 929119864 + 03Number of iterations in inner loop 9Number of complexes 10

636119864 minus 08 178119864 minus 07 108119864 minus 07 389119864 minus 08 172119864 + 04Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy FailElhay algorithm Maximum accuracy 742119864 minus 06

Table 4 Head and parameter 120575 in numerical example 3

Node number 119885 (m) SCE 3 steps SCE 3 steps SCE 3 steps119867 (m) 119867 (m) 119867 minus 119885 119867 minus 119885 120575 120575

1 140 140 140 0 0 0 02 80 129304 13007 49304 5007 149E minus 07 310119864 minus 04

3 90 132288 13276 42288 4276 126E minus 07 000414 70 109587 11096 39587 4096 203E minus 07 000225 80 80000 8854 0000 854 0058 151E minus 046 90 90000 9145 0000 145 00069 602E minus 047 90 90000 9000 0000 000 0080 014218 100 88922 9043 minus11078 minus957 584E minus 08 00439

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations0 5 10 15 20 25 30 35 40 45 50

Figure 7 Convergence history of numerical example 2 (form 1)

The absolute value of 120575 is calculated for each iteration inFigure 7 and the value of head pressure in node 2 119867(2) iscalculated for each iteration in Figure 8

63 Numerical Example 3 The simplified water distributionnetwork shown in Figure 9 was used in order to demonstratethe advantages of the proposed model in pressure-drivendemand condition For the sake of simplicity the same

H(2)

Number of iterations0 5 10 15 20 25 30 35 40 45 50

365

366

367

368

369

37

371

372

373

Figure 8 Convergence history of head pressure (119898) in node 2 fornumerical example 2 (form 2)

Hazen-Williams roughness coefficient 119862 = 130 was assumedfor all the 14 pipes of identical length of 1000m while no locallosses have been added The following diameters have beenused in the example 500mm (P-2) 400mm (P-1) 300mm(P-4 P-7) 250mm (P-10) 200mm (P-3 P-5 P-6 and P-13)150mm (P-8 P-9 P-11 P-12 and P-14) The nodal demandsare listed in the following tables together with the groundelevation 119885

119894 Without loss of generality in this example the

Journal of Fluids 9

Reservoir

1 2

23

3

4

4

5

5

6

6

7

7

8

8

9

10 11 12 13

14

Figure 9 Schematic representation of the looped pipe network usedin numerical example 3

minimum head requirement 119867119894

lowast has been assumed to beequal to the ground elevation 119885

119894[8] So the relationship

between the required nodal head and minimum head is

119902119895=

0 119867119895lt 119885119895

119902119895 119885119895le 119867119895

(16)

Todini [8] proposed a three-step approach for solving thisnetwork and its solution is reported in the 4th column ofTable 4 In the proposed method pressure-driven model canbe applied in hydraulic analysis without any mathematicalformulation In this situation an if-then rule is added tococontent model and optimization process is conductedThenumber of decision variables in SCE algorithm is 7 the boundvariables were set between 50 and 140m ten optimizationruns were performed using different random initial solutionsfor all the cases and the results are illustrated in Table 5Results confirm that SCE is more accurate compared withTodini algorithm in case 2 and case 3 In Table 4 the bestresult is shown in bold and it is considered that the methodof SCE has calculated the best value of 120575 at 5 nodes whilethe Todini method has done it at 2 nodes The convergenceprocess of SCE algorithm has been shown in two forms inFigures 10 and 11The absolute value of 120575 is calculated for eachiteration in Figure 10 and the value of head pressure in node2119867(2) is calculated for each iteration in Figure 11

64 Numerical Example 4 The fourth considered networkis a real planned network designed for an industrial areain Apulian Town (Southern Italy) The network layout is

0 10 20 30 40 50 60 70 80 90

100

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

Figure 10 Convergence history of numerical example 3 (form 1)

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100

120

140

Number of iterations

H(2)

Figure 11 Convergence history of head pressure (119898) in node 2 fornumerical example 3 (form 2)

illustrated in Figure 12 and the corresponding data areprovided in Table 6 With respect to the leakages they havebeen assumed to be pressure-driven (see (6)) given that theyare implemented in the pressure driven network simulationmodel as described above [14] The parameter 120573 = 10632 times10minus7 and 120572 = 12 as reported in Giustolisi et al [14] for this

network Giustolisi et al [14] proposed a hydraulic simulationmodel which fully integrates a classic hydraulic simulationalgorithm such as that of Todini and Pilati [3] found inEPANET 2 with a pressure-driven model that entails amore realistic representation of the leakage They appliedtheir model in this network The results are demonstratedin Table 7 In this table the best result is shown in boldand it is considered that the method of SCE has calculatedthe best value of 120575 at 14 nodes while the Gistulishi methodhas done it at 9 nodes In the proposed method there is noneed to modify the mathematical formulation for hydraulic

10 Journal of Fluids

Table 5 Average of mass and energy balance for numerical example 3

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5332119864 minus 02 797119864 minus 02 595119864 minus 02 162119864 minus 02 430119864 + 04

Number of iterations in inner loop 5Number of complexes 9

207119864 minus 02 544119864 minus 02 252119864 minus 02 109119864 minus 02 627119864 + 04Number of iterations in inner loop 9Number of complexes 10

207119864 minus 02 207119864 minus 02 207119864 minus 02 217119864 minus 08 604119864 + 04Number of iterations in inner loop 15Three-step approach [8] Maximum accuracy 276119864 minus 02

Table 6 Hydraulic data relevant to numerical example 4

Pipe number 119871 (m) 119863 (mm)1 3485 3272 9557 2903 483 1004 4007 2905 7919 1006 4044 3687 3906 3278 4823 1009 9344 10010 4313 18411 5131 10012 4284 18413 419 10014 10231 10015 4551 16416 1826 29017 2213 29018 5839 16419 452 22920 7947 10021 7177 10022 6556 25823 1655 10024 2521 10025 3315 10026 500 20427 5799 16428 8428 10029 7926 10030 8463 18431 164 25832 4279 10033 3792 10034 1582 368

analysis An if-then rule is added to cocontent model andthe optimization process is performed easily As you can seein Table 8 SCE found the optimal solution more accurately

1

1

2 2

33

4

4

5

5

6

6

7

78

8

99

101011

11

12

12

1313

14

14

15

1516

1617

17

18

18

19

19

20

20

21

21

2222

23 2324

25

26

27

28

2930

31

32

33

34

Reservoir

Figure 12 Schematic representation of the looped pipe networkused in the numerical example 4

0 100 200 300 400 500 600 700 800 900 1000

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

10minus3

Figure 13 Convergence history of numerical example 4 (form 1)

than the Giustolisi algorithm The convergence process ofSCE algorithm has been shown in two forms in Figures 4 and5 The absolute value of 120575 is calculated for each iteration inFigure 13 and the amount of head pressure in node 20119867(20)is calculated for each iteration in Figure 14

Journal of Fluids 11

Table 7 Head and parameter 120575 in numerical example 4

Node number 119902 (ls) 119867 (m) (Giustolisi algorithm) 119867 (m) 120575 (Giustolisi algorithm) 120575 (SCE)1 10863 269 3329 0154743 00047382 17034 2481 3183 002131 00065323 14947 213 2739 0059002 00053644 1428 1722 2534 0001257 00050335 10133 2354 3089 0026184 00040186 1535 201 2902 0030898 00053997 9114 1891 2794 0017147 00030878 1051 179 2734 000227 00035459 12182 1785 2635 0002936 000386710 14579 1266 2324 0008277 000436211 9007 1623 2595 0031554 000272312 7575 1012 2205 0002732 000213813 152 1003 2245 00126 000432814 1355 1541 2595 0001318 000435215 9226 14 2417 0002199 000293416 112 1436 2405 0007089 000358617 11469 153 2542 0000103 00036118 10818 1883 2838 0011889 000390819 14675 1935 2839 543E minus 05 000509720 13318 1001 2379 0013059 000397421 14631 1148 2235 0003276 000414122 12012 14 2546 0003901 000367723 10326 1045 2011 0002551 000297924 mdash 3645 3645 mdash mdash

Table 8 Average of mass and energy balance for numerical example 4

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5400119864 minus 03 450119864 minus 02 410119864 minus 03 275119864 minus 04 659119864 + 04

Number of iterations in inner loop 5Number of complexes 9

390119864 minus 03 415119864 minus 03 400119864 minus 03 396119864 minus 05 115119864 + 05Number of iterations in inner loop 9Number of complexes 10

370119864 minus 03 410119864 minus 03 390119864 minus 03 253119864 minus 05 119119864 + 05Number of iterations in inner loop 15Giustolisi algorithm Maximum accuracy 181119864 minus 02

In general 4 different pipe networks were consideredin this paper and different mathematical formulations wereused for the hydraulic analysis of these networks Howeverthe overall results indicate that the proposed method hasthe capability of handling various pipe networks problemswith no change in the model or mathematical formulationApplication of SCE in cocontent model can result in findingaccurate solutions in pipes with zero flows and the pressure-driven demand and leakage simulation can be solved throughapplying if-then rules in cocontent model As a result itcan be concluded that the proposed method is a suitablealternative optimizer challenging othermethods especially interms of accuracy

7 Conclusions

The objective of the present paper was to provide an inno-vative approach in the analysis of the water distributionnetworks based on the optimization model The cocontentmodel is minimized using shuffled complex evolution (SCE)algorithm The methodology is illustrated here using fournetworks with different layouts The results reveal thatthe proposed method has the capability to handle variouspipe networks problems without changing in model ormathematical formulation The advantage of the proposedmethod lies in the fact that there is no need to solvelinear systems of equations pressure-driven demand and

12 Journal of Fluids

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

Number of iterations

H(20)

Figure 14 Convergence history of head-pressure (119898) in node 20 forexample 4 (form 2)

leakage simulation are handled in a simple way accu-rate solutions can be found in pipes with zero flowsand it does not need an initial solution vector which must bechosen carefully in many other procedures if numerical con-vergence is to be achieved Furthermore the proposedmodeldoes not require any complicated mathematical expressionand operation Finally it can be concluded that the proposedmethod is a viable alternative optimizer that challenges othermethods particularly in view of accuracy

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

References

[1] M Collins L Cooper R Helgason J Kenningston and LLeBlanc ldquoSolving the pipe network analysis problem usingoptimization techniquesrdquo Management Science vol 24 no 7pp 747ndash760 1978

[2] S Elhay and A R Simpson ldquoDealing with zero flows in solvingthe nonlinear equations for water distribution systemsrdquo Journalof Hydraulic Engineering vol 137 no 10 pp 1216ndash1224 2011

[3] E Todini and S Pilati ldquoA gradient algorithm for the analysis ofpipe networksrdquo in Computer Applications inWater Supply vol 1of System Analysis and Simulation pp 1ndash20 JohnWiley amp SonsLondon UK 1988

[4] A C Zecchin P Thum A R Simpson and C TischendorfldquoSteady-state behaviour of large water distribution systems thealgebraic multigrid method for the fast solution of the linearsteprdquo Journal ofWater Resources Planning andManagement vol138 no 6 pp 639ndash650 2012

[5] M L Arora ldquoFlow split in closed loops expending least energyrdquoJournal of the Hydraulics Division vol 102 pp 455ndash458 1976

[6] M A Hall ldquoHydraulic network analysis using (generalized)geometric programmingrdquo Networks vol 6 no 2 pp 105ndash1301976

[7] P R Bhave and R Gupta ldquoAnalysis of water distributionnetworksrdquo in Alpha Science International Technology amp Engi-neering 2006

[8] E Todini A More Realistic Approach to the ldquoExtended PeriodSimulationrdquo of Water Distribution Networks Advances in WaterSupplyManagement chapter 19 Taylor amp Francis London UK2003

[9] J M Wagner U Shamir and D H Marks ldquoWater distributionreliability analytical methodsrdquo Journal of Water ResourcesPlanning and Management vol 114 no 3 pp 253ndash275 1988

[10] J Chandapillai ldquoRealistic simulation of water distributionsystemrdquo Journal of Transportation Engineering vol 117 no 2 pp258ndash263 1991

[11] J Almandoz E Cabrera F Arregui E Cabrera Jr andR Cobacho ldquoLeakage assessment through water distributionnetwork simulationrdquo Journal of Water Resources Planning andManagement vol 131 no 6 pp 458ndash466 2005

[12] A F Colombo and B W Karney ldquoEnergy and costs ofleaky pipes toward comprehensive picturerdquo Journal of WaterResources Planning and Management vol 128 no 6 pp 441ndash450 2002

[13] G Germanopoulos ldquoA technical note on the inclusion ofpressure dependent demand and leakage terms in water supplynetworkmodelsrdquoCivil Engineering Systems vol 2 no 3 pp 171ndash179 1985

[14] O Giustolisi D Savic and Z Kapelan ldquoPressure-drivendemand and leakage simulation for water distribution net-worksrdquo Journal of Hydraulic Engineering vol 134 no 5 pp 626ndash635 2008

[15] L Ainola T Koppel K Tiiter and A Vassiljev ldquoWater networkmodel calibration based on grouping pipes with similar leakageand roughness estimatesrdquo in Proceedings of the Joint Conferenceon Water Resource Engineering and Water Resources PlanningandManagement (EWRI rsquo00) pp 1ndash9MinneapolisMinnUSAAugust 2000

[16] Q Duan S Sorooshian and V K Gupta ldquoOptimal use of theSCE-UA global optimization method for calibrating watershedmodelsrdquo Journal of Hydrology vol 158 no 3-4 pp 265ndash2841994

[17] Q Y Duan S Sorooshian and V K Gupta ldquoShuffled complexevolution approach for effective and efficient global minimiza-tionrdquo Journal of Optimization Theory and Applications vol 76no 3 pp 501ndash521 1993

[18] E Todini ldquoOn the convergence properties of the differentpipe network algorithmsrdquo in Proceedings of the Annual WaterDistribution Systems Analysis Symposium (ASCE rsquo06) pp 1ndash16August 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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6 Journal of Fluids

516 Step 6 Shuffle Complexes The solutions in the evolvedcomplexes into a single sample population are combinedand the sample population is sorted in order of increasingcriterion value and is shuffled into 119901 complexes

517 Step 7 Check the Stopping Criterion In this sectionSteps 3 4 and 5 are repeated until the termination criterionis satisfied

It should be noted that the competitive complex evolu-tion (CCE) algorithm is required for the evolution of eachcomplex Each point of a complex is a potential ldquoparentrdquowith the ability to participate in the process of reproducingoffspring A subcomplex functions like a pair of parents Useof a stochastic scheme to construct subcomplexes allows theparameter space to be searched more thoroughly The ideaof competitiveness is introduced in forming subcomplexeswhere the stronger survives better and breeds healthier off-spring than the weaker Inclusion of the competitive measureexpedites the search towards promising regions

A more detailed presentation of the SCE algorithm hasbeen given by Duan et al [17]

6 Numerical Examples

In this section the hydraulic analyses for several conditionsin some water distribution networks are performed All com-putations were executed inMATLAB programming languageenvironment with an Intel(R) Core(TM) 2Duo CPU P8700 253GHz and 400GB RAM In order to demonstrate theeffectiveness of SCE comparedwith othermethods this studyproposes the use of mass balance and energy balance in thenetworkThe average of mass and energy balance is shown by120575 and it is calculated by the following formula

120575 = mean(abs(119873

sum

119895=1

119910

sum

119894=119909

10038161003816100381610038161003816119867119894+ ℎ119901119896minus ℎ119871119896minus 119867119895

10038161003816100381610038161003816

(1119899)

1198771119899

119896

minus 119902119895))

(14)

For Figure 1 and (2) 120575 is calculated as follows

120575 = mean(10038161003816100381610038161003816100381610038161003816

120597119862

1205971198671

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198672

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198673

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198674

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

120597119862

1205971198675

10038161003816100381610038161003816100381610038161003816

)

(15)

To check the performance of the SCE for the minimiza-tion of cocontent model in all examples ten optimizationrunswere performed using different random initial solutions

61 Numerical Example 1 In this part the verification ofthe above mentioned model was conducted via numericalsimulation based on an extremely simplified network scheme(5 nodes and 7 pipes) schematically shown in Figure 3 [18]The pipes resistances were 119877

1= 15625 119877

2= 50 119877

3= 100

1198774= 125 119877

5= 75 119877

6= 200 and 119877

7= 100 as reported in

Todini [18] for this networkThe SCE technique is applied to solve this problem

according to three cases The bound variables were setbetween 90 and 100m The problem is also solved using the

0Reservoir

11

1

2

2

2

3

3

3

4

4

5 6

7

800 ls

200 ls

200 ls

200 ls

300 ls

400 ls

400 ls

100 ls

100 ls

100 ls

100 ls

Figure 3 Schematic representation of the looped pipe network usedin numerical example 1

0 2 4 6 8 10 12 14 16 18 20

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

Figure 4 Convergence history of numerical example 1 (form 1)

global gradient algorithm (GGA) and the results are com-pared with those obtained by the SCE The best worst andaverage solutions of SCE algorithm in three cases are shownin Table 1 As it can be seen in Table 1 in all cases SCEthat found the optimal solution more accurately than GGAmethod The average number of function evaluation is about2000 in case 1 and about 100000 in case 3 This shows SCEcan converge to global optimum rapidly but reaching highaccuracy needs more operations The convergence process ofSCE algorithm has been shown in two forms in Figures 4and 5 The absolute value of 120575 is calculated for each iterationin Figure 4 and the amount of objective function 119862(119867) iscalculated for each iteration in Figure 5

62 Numerical Example 2 Example 2 considers the sym-metric network shown in Figure 6 It has 11 pipes sevenjunctions at which the head is unknown and one fixed headnode reservoir at 40m elevation and all other nodes areat zero elevation All pipes have diameters 119863 of 250mmand lengths 119871 of 1000m Node 8 has a demand of 80 lsand all other nodes have zero demands In the steady statethis network has zero flows in pipes 2 6 and 9 because of

Journal of Fluids 7

Table 1 Average of mass and energy balance for numerical example 1

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4460119864 minus 08 706119864 minus 07 304119864 minus 07 230119864 minus 07 203119864 + 03

Number of iterations in inner loop 4Number of complexes 9

119119864 minus 08 301119864 minus 08 204119864 minus 08 640119864 minus 09 100119864 + 05Number of iterations in inner loop 9Number of complexes 10

104119864 minus 08 419119864 minus 08 252119864 minus 08 828119864 minus 09 100119864 + 05Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy 640119864 minus 06

Table 2 Head and parameter 120575 in numerical example 2

Node number 119867 (m) [2] 119867 (m) 120575 [2] 120575 (SCE)1 40 40 0 02 366813 366853 0 18119864 minus 07

3 366813 366853 0 18119864 minus 07

4 333626 333706 65119864 minus 07 84E minus 085 333626 333706 65119864 minus 07 87E minus 086 30044 300559 65119864 minus 07 67E minus 087 30044 300559 65119864 minus 07 66E minus 088 267253 267411 52119864 minus 05 12E minus 10

0 2 4 6 8 10 12 14 16 18 20

1021718

10217179

10217178

10217177

10217176

10217175

10217174

10217173

Obj

ectiv

e fun

ctio

n

Number of iterations

Figure 5 Convergence history of numerical example 1 (form 2)

symmetry The head loss is modeled by the Hazen-Williamsequation and each pipe has a Hazen-Williams coefficient119862 = 120 [2] When the GGA is used in this network theiterates trend toward the solution and the flows in pipes 26 and 9 approach zero As this happens the Jacobian matrixbecomes more and more badly conditioned and the solutioncomputed becomes ill conditioned [2] Elhay and Simpson[2] proposed a regularization procedure for the GGA whichprevents failure of the solution process provided that a flowin the network is ultimately zero or near zero

The SCEparameters are set as follows number of decisionvariables = 7 number of points in each complex = 15 numberof complexes for case 1 = 4 case 2 = 9 and case 3 = 10 numberof iterations in inner loop for case 1 = 4 case 2 = 9 and

1

1

22

3

3

4

4

5

56

6

7

7

8

8

9

1011

Reservoir

Figure 6 Schematic representation of the looped pipe network usedin numerical example 2

case 3 = 15 The bound variables were set between 25 and40m The previous best solution for this network when it issimulated using the Elhay algorithm and the average solutionof SCE algorithm are shown in the second and third columnsof Table 2 respectively As can be observed in Table 2 massand energy balance (120575) in SCE are more accurate than theElhay algorithm Table 3 compares the results of applying theSCE algorithm in three casesThe convergence process of SCEalgorithm has been shown in two forms in Figures 7 and 8

8 Journal of Fluids

Table 3 Average of mass and energy balance for numerical example 2

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4221119864 minus 06 724119864 minus 06 465119864 minus 06 161119864 minus 06 419119864 + 03

Number of iterations in inner loop 4Number of complexes 9

105119864 minus 06 535119864 minus 06 346119864 minus 06 151119864 minus 06 929119864 + 03Number of iterations in inner loop 9Number of complexes 10

636119864 minus 08 178119864 minus 07 108119864 minus 07 389119864 minus 08 172119864 + 04Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy FailElhay algorithm Maximum accuracy 742119864 minus 06

Table 4 Head and parameter 120575 in numerical example 3

Node number 119885 (m) SCE 3 steps SCE 3 steps SCE 3 steps119867 (m) 119867 (m) 119867 minus 119885 119867 minus 119885 120575 120575

1 140 140 140 0 0 0 02 80 129304 13007 49304 5007 149E minus 07 310119864 minus 04

3 90 132288 13276 42288 4276 126E minus 07 000414 70 109587 11096 39587 4096 203E minus 07 000225 80 80000 8854 0000 854 0058 151E minus 046 90 90000 9145 0000 145 00069 602E minus 047 90 90000 9000 0000 000 0080 014218 100 88922 9043 minus11078 minus957 584E minus 08 00439

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations0 5 10 15 20 25 30 35 40 45 50

Figure 7 Convergence history of numerical example 2 (form 1)

The absolute value of 120575 is calculated for each iteration inFigure 7 and the value of head pressure in node 2 119867(2) iscalculated for each iteration in Figure 8

63 Numerical Example 3 The simplified water distributionnetwork shown in Figure 9 was used in order to demonstratethe advantages of the proposed model in pressure-drivendemand condition For the sake of simplicity the same

H(2)

Number of iterations0 5 10 15 20 25 30 35 40 45 50

365

366

367

368

369

37

371

372

373

Figure 8 Convergence history of head pressure (119898) in node 2 fornumerical example 2 (form 2)

Hazen-Williams roughness coefficient 119862 = 130 was assumedfor all the 14 pipes of identical length of 1000m while no locallosses have been added The following diameters have beenused in the example 500mm (P-2) 400mm (P-1) 300mm(P-4 P-7) 250mm (P-10) 200mm (P-3 P-5 P-6 and P-13)150mm (P-8 P-9 P-11 P-12 and P-14) The nodal demandsare listed in the following tables together with the groundelevation 119885

119894 Without loss of generality in this example the

Journal of Fluids 9

Reservoir

1 2

23

3

4

4

5

5

6

6

7

7

8

8

9

10 11 12 13

14

Figure 9 Schematic representation of the looped pipe network usedin numerical example 3

minimum head requirement 119867119894

lowast has been assumed to beequal to the ground elevation 119885

119894[8] So the relationship

between the required nodal head and minimum head is

119902119895=

0 119867119895lt 119885119895

119902119895 119885119895le 119867119895

(16)

Todini [8] proposed a three-step approach for solving thisnetwork and its solution is reported in the 4th column ofTable 4 In the proposed method pressure-driven model canbe applied in hydraulic analysis without any mathematicalformulation In this situation an if-then rule is added tococontent model and optimization process is conductedThenumber of decision variables in SCE algorithm is 7 the boundvariables were set between 50 and 140m ten optimizationruns were performed using different random initial solutionsfor all the cases and the results are illustrated in Table 5Results confirm that SCE is more accurate compared withTodini algorithm in case 2 and case 3 In Table 4 the bestresult is shown in bold and it is considered that the methodof SCE has calculated the best value of 120575 at 5 nodes whilethe Todini method has done it at 2 nodes The convergenceprocess of SCE algorithm has been shown in two forms inFigures 10 and 11The absolute value of 120575 is calculated for eachiteration in Figure 10 and the value of head pressure in node2119867(2) is calculated for each iteration in Figure 11

64 Numerical Example 4 The fourth considered networkis a real planned network designed for an industrial areain Apulian Town (Southern Italy) The network layout is

0 10 20 30 40 50 60 70 80 90

100

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

Figure 10 Convergence history of numerical example 3 (form 1)

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100

120

140

Number of iterations

H(2)

Figure 11 Convergence history of head pressure (119898) in node 2 fornumerical example 3 (form 2)

illustrated in Figure 12 and the corresponding data areprovided in Table 6 With respect to the leakages they havebeen assumed to be pressure-driven (see (6)) given that theyare implemented in the pressure driven network simulationmodel as described above [14] The parameter 120573 = 10632 times10minus7 and 120572 = 12 as reported in Giustolisi et al [14] for this

network Giustolisi et al [14] proposed a hydraulic simulationmodel which fully integrates a classic hydraulic simulationalgorithm such as that of Todini and Pilati [3] found inEPANET 2 with a pressure-driven model that entails amore realistic representation of the leakage They appliedtheir model in this network The results are demonstratedin Table 7 In this table the best result is shown in boldand it is considered that the method of SCE has calculatedthe best value of 120575 at 14 nodes while the Gistulishi methodhas done it at 9 nodes In the proposed method there is noneed to modify the mathematical formulation for hydraulic

10 Journal of Fluids

Table 5 Average of mass and energy balance for numerical example 3

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5332119864 minus 02 797119864 minus 02 595119864 minus 02 162119864 minus 02 430119864 + 04

Number of iterations in inner loop 5Number of complexes 9

207119864 minus 02 544119864 minus 02 252119864 minus 02 109119864 minus 02 627119864 + 04Number of iterations in inner loop 9Number of complexes 10

207119864 minus 02 207119864 minus 02 207119864 minus 02 217119864 minus 08 604119864 + 04Number of iterations in inner loop 15Three-step approach [8] Maximum accuracy 276119864 minus 02

Table 6 Hydraulic data relevant to numerical example 4

Pipe number 119871 (m) 119863 (mm)1 3485 3272 9557 2903 483 1004 4007 2905 7919 1006 4044 3687 3906 3278 4823 1009 9344 10010 4313 18411 5131 10012 4284 18413 419 10014 10231 10015 4551 16416 1826 29017 2213 29018 5839 16419 452 22920 7947 10021 7177 10022 6556 25823 1655 10024 2521 10025 3315 10026 500 20427 5799 16428 8428 10029 7926 10030 8463 18431 164 25832 4279 10033 3792 10034 1582 368

analysis An if-then rule is added to cocontent model andthe optimization process is performed easily As you can seein Table 8 SCE found the optimal solution more accurately

1

1

2 2

33

4

4

5

5

6

6

7

78

8

99

101011

11

12

12

1313

14

14

15

1516

1617

17

18

18

19

19

20

20

21

21

2222

23 2324

25

26

27

28

2930

31

32

33

34

Reservoir

Figure 12 Schematic representation of the looped pipe networkused in the numerical example 4

0 100 200 300 400 500 600 700 800 900 1000

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

10minus3

Figure 13 Convergence history of numerical example 4 (form 1)

than the Giustolisi algorithm The convergence process ofSCE algorithm has been shown in two forms in Figures 4 and5 The absolute value of 120575 is calculated for each iteration inFigure 13 and the amount of head pressure in node 20119867(20)is calculated for each iteration in Figure 14

Journal of Fluids 11

Table 7 Head and parameter 120575 in numerical example 4

Node number 119902 (ls) 119867 (m) (Giustolisi algorithm) 119867 (m) 120575 (Giustolisi algorithm) 120575 (SCE)1 10863 269 3329 0154743 00047382 17034 2481 3183 002131 00065323 14947 213 2739 0059002 00053644 1428 1722 2534 0001257 00050335 10133 2354 3089 0026184 00040186 1535 201 2902 0030898 00053997 9114 1891 2794 0017147 00030878 1051 179 2734 000227 00035459 12182 1785 2635 0002936 000386710 14579 1266 2324 0008277 000436211 9007 1623 2595 0031554 000272312 7575 1012 2205 0002732 000213813 152 1003 2245 00126 000432814 1355 1541 2595 0001318 000435215 9226 14 2417 0002199 000293416 112 1436 2405 0007089 000358617 11469 153 2542 0000103 00036118 10818 1883 2838 0011889 000390819 14675 1935 2839 543E minus 05 000509720 13318 1001 2379 0013059 000397421 14631 1148 2235 0003276 000414122 12012 14 2546 0003901 000367723 10326 1045 2011 0002551 000297924 mdash 3645 3645 mdash mdash

Table 8 Average of mass and energy balance for numerical example 4

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5400119864 minus 03 450119864 minus 02 410119864 minus 03 275119864 minus 04 659119864 + 04

Number of iterations in inner loop 5Number of complexes 9

390119864 minus 03 415119864 minus 03 400119864 minus 03 396119864 minus 05 115119864 + 05Number of iterations in inner loop 9Number of complexes 10

370119864 minus 03 410119864 minus 03 390119864 minus 03 253119864 minus 05 119119864 + 05Number of iterations in inner loop 15Giustolisi algorithm Maximum accuracy 181119864 minus 02

In general 4 different pipe networks were consideredin this paper and different mathematical formulations wereused for the hydraulic analysis of these networks Howeverthe overall results indicate that the proposed method hasthe capability of handling various pipe networks problemswith no change in the model or mathematical formulationApplication of SCE in cocontent model can result in findingaccurate solutions in pipes with zero flows and the pressure-driven demand and leakage simulation can be solved throughapplying if-then rules in cocontent model As a result itcan be concluded that the proposed method is a suitablealternative optimizer challenging othermethods especially interms of accuracy

7 Conclusions

The objective of the present paper was to provide an inno-vative approach in the analysis of the water distributionnetworks based on the optimization model The cocontentmodel is minimized using shuffled complex evolution (SCE)algorithm The methodology is illustrated here using fournetworks with different layouts The results reveal thatthe proposed method has the capability to handle variouspipe networks problems without changing in model ormathematical formulation The advantage of the proposedmethod lies in the fact that there is no need to solvelinear systems of equations pressure-driven demand and

12 Journal of Fluids

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

Number of iterations

H(20)

Figure 14 Convergence history of head-pressure (119898) in node 20 forexample 4 (form 2)

leakage simulation are handled in a simple way accu-rate solutions can be found in pipes with zero flowsand it does not need an initial solution vector which must bechosen carefully in many other procedures if numerical con-vergence is to be achieved Furthermore the proposedmodeldoes not require any complicated mathematical expressionand operation Finally it can be concluded that the proposedmethod is a viable alternative optimizer that challenges othermethods particularly in view of accuracy

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

References

[1] M Collins L Cooper R Helgason J Kenningston and LLeBlanc ldquoSolving the pipe network analysis problem usingoptimization techniquesrdquo Management Science vol 24 no 7pp 747ndash760 1978

[2] S Elhay and A R Simpson ldquoDealing with zero flows in solvingthe nonlinear equations for water distribution systemsrdquo Journalof Hydraulic Engineering vol 137 no 10 pp 1216ndash1224 2011

[3] E Todini and S Pilati ldquoA gradient algorithm for the analysis ofpipe networksrdquo in Computer Applications inWater Supply vol 1of System Analysis and Simulation pp 1ndash20 JohnWiley amp SonsLondon UK 1988

[4] A C Zecchin P Thum A R Simpson and C TischendorfldquoSteady-state behaviour of large water distribution systems thealgebraic multigrid method for the fast solution of the linearsteprdquo Journal ofWater Resources Planning andManagement vol138 no 6 pp 639ndash650 2012

[5] M L Arora ldquoFlow split in closed loops expending least energyrdquoJournal of the Hydraulics Division vol 102 pp 455ndash458 1976

[6] M A Hall ldquoHydraulic network analysis using (generalized)geometric programmingrdquo Networks vol 6 no 2 pp 105ndash1301976

[7] P R Bhave and R Gupta ldquoAnalysis of water distributionnetworksrdquo in Alpha Science International Technology amp Engi-neering 2006

[8] E Todini A More Realistic Approach to the ldquoExtended PeriodSimulationrdquo of Water Distribution Networks Advances in WaterSupplyManagement chapter 19 Taylor amp Francis London UK2003

[9] J M Wagner U Shamir and D H Marks ldquoWater distributionreliability analytical methodsrdquo Journal of Water ResourcesPlanning and Management vol 114 no 3 pp 253ndash275 1988

[10] J Chandapillai ldquoRealistic simulation of water distributionsystemrdquo Journal of Transportation Engineering vol 117 no 2 pp258ndash263 1991

[11] J Almandoz E Cabrera F Arregui E Cabrera Jr andR Cobacho ldquoLeakage assessment through water distributionnetwork simulationrdquo Journal of Water Resources Planning andManagement vol 131 no 6 pp 458ndash466 2005

[12] A F Colombo and B W Karney ldquoEnergy and costs ofleaky pipes toward comprehensive picturerdquo Journal of WaterResources Planning and Management vol 128 no 6 pp 441ndash450 2002

[13] G Germanopoulos ldquoA technical note on the inclusion ofpressure dependent demand and leakage terms in water supplynetworkmodelsrdquoCivil Engineering Systems vol 2 no 3 pp 171ndash179 1985

[14] O Giustolisi D Savic and Z Kapelan ldquoPressure-drivendemand and leakage simulation for water distribution net-worksrdquo Journal of Hydraulic Engineering vol 134 no 5 pp 626ndash635 2008

[15] L Ainola T Koppel K Tiiter and A Vassiljev ldquoWater networkmodel calibration based on grouping pipes with similar leakageand roughness estimatesrdquo in Proceedings of the Joint Conferenceon Water Resource Engineering and Water Resources PlanningandManagement (EWRI rsquo00) pp 1ndash9MinneapolisMinnUSAAugust 2000

[16] Q Duan S Sorooshian and V K Gupta ldquoOptimal use of theSCE-UA global optimization method for calibrating watershedmodelsrdquo Journal of Hydrology vol 158 no 3-4 pp 265ndash2841994

[17] Q Y Duan S Sorooshian and V K Gupta ldquoShuffled complexevolution approach for effective and efficient global minimiza-tionrdquo Journal of Optimization Theory and Applications vol 76no 3 pp 501ndash521 1993

[18] E Todini ldquoOn the convergence properties of the differentpipe network algorithmsrdquo in Proceedings of the Annual WaterDistribution Systems Analysis Symposium (ASCE rsquo06) pp 1ndash16August 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Fluids 7

Table 1 Average of mass and energy balance for numerical example 1

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4460119864 minus 08 706119864 minus 07 304119864 minus 07 230119864 minus 07 203119864 + 03

Number of iterations in inner loop 4Number of complexes 9

119119864 minus 08 301119864 minus 08 204119864 minus 08 640119864 minus 09 100119864 + 05Number of iterations in inner loop 9Number of complexes 10

104119864 minus 08 419119864 minus 08 252119864 minus 08 828119864 minus 09 100119864 + 05Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy 640119864 minus 06

Table 2 Head and parameter 120575 in numerical example 2

Node number 119867 (m) [2] 119867 (m) 120575 [2] 120575 (SCE)1 40 40 0 02 366813 366853 0 18119864 minus 07

3 366813 366853 0 18119864 minus 07

4 333626 333706 65119864 minus 07 84E minus 085 333626 333706 65119864 minus 07 87E minus 086 30044 300559 65119864 minus 07 67E minus 087 30044 300559 65119864 minus 07 66E minus 088 267253 267411 52119864 minus 05 12E minus 10

0 2 4 6 8 10 12 14 16 18 20

1021718

10217179

10217178

10217177

10217176

10217175

10217174

10217173

Obj

ectiv

e fun

ctio

n

Number of iterations

Figure 5 Convergence history of numerical example 1 (form 2)

symmetry The head loss is modeled by the Hazen-Williamsequation and each pipe has a Hazen-Williams coefficient119862 = 120 [2] When the GGA is used in this network theiterates trend toward the solution and the flows in pipes 26 and 9 approach zero As this happens the Jacobian matrixbecomes more and more badly conditioned and the solutioncomputed becomes ill conditioned [2] Elhay and Simpson[2] proposed a regularization procedure for the GGA whichprevents failure of the solution process provided that a flowin the network is ultimately zero or near zero

The SCEparameters are set as follows number of decisionvariables = 7 number of points in each complex = 15 numberof complexes for case 1 = 4 case 2 = 9 and case 3 = 10 numberof iterations in inner loop for case 1 = 4 case 2 = 9 and

1

1

22

3

3

4

4

5

56

6

7

7

8

8

9

1011

Reservoir

Figure 6 Schematic representation of the looped pipe network usedin numerical example 2

case 3 = 15 The bound variables were set between 25 and40m The previous best solution for this network when it issimulated using the Elhay algorithm and the average solutionof SCE algorithm are shown in the second and third columnsof Table 2 respectively As can be observed in Table 2 massand energy balance (120575) in SCE are more accurate than theElhay algorithm Table 3 compares the results of applying theSCE algorithm in three casesThe convergence process of SCEalgorithm has been shown in two forms in Figures 7 and 8

8 Journal of Fluids

Table 3 Average of mass and energy balance for numerical example 2

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4221119864 minus 06 724119864 minus 06 465119864 minus 06 161119864 minus 06 419119864 + 03

Number of iterations in inner loop 4Number of complexes 9

105119864 minus 06 535119864 minus 06 346119864 minus 06 151119864 minus 06 929119864 + 03Number of iterations in inner loop 9Number of complexes 10

636119864 minus 08 178119864 minus 07 108119864 minus 07 389119864 minus 08 172119864 + 04Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy FailElhay algorithm Maximum accuracy 742119864 minus 06

Table 4 Head and parameter 120575 in numerical example 3

Node number 119885 (m) SCE 3 steps SCE 3 steps SCE 3 steps119867 (m) 119867 (m) 119867 minus 119885 119867 minus 119885 120575 120575

1 140 140 140 0 0 0 02 80 129304 13007 49304 5007 149E minus 07 310119864 minus 04

3 90 132288 13276 42288 4276 126E minus 07 000414 70 109587 11096 39587 4096 203E minus 07 000225 80 80000 8854 0000 854 0058 151E minus 046 90 90000 9145 0000 145 00069 602E minus 047 90 90000 9000 0000 000 0080 014218 100 88922 9043 minus11078 minus957 584E minus 08 00439

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations0 5 10 15 20 25 30 35 40 45 50

Figure 7 Convergence history of numerical example 2 (form 1)

The absolute value of 120575 is calculated for each iteration inFigure 7 and the value of head pressure in node 2 119867(2) iscalculated for each iteration in Figure 8

63 Numerical Example 3 The simplified water distributionnetwork shown in Figure 9 was used in order to demonstratethe advantages of the proposed model in pressure-drivendemand condition For the sake of simplicity the same

H(2)

Number of iterations0 5 10 15 20 25 30 35 40 45 50

365

366

367

368

369

37

371

372

373

Figure 8 Convergence history of head pressure (119898) in node 2 fornumerical example 2 (form 2)

Hazen-Williams roughness coefficient 119862 = 130 was assumedfor all the 14 pipes of identical length of 1000m while no locallosses have been added The following diameters have beenused in the example 500mm (P-2) 400mm (P-1) 300mm(P-4 P-7) 250mm (P-10) 200mm (P-3 P-5 P-6 and P-13)150mm (P-8 P-9 P-11 P-12 and P-14) The nodal demandsare listed in the following tables together with the groundelevation 119885

119894 Without loss of generality in this example the

Journal of Fluids 9

Reservoir

1 2

23

3

4

4

5

5

6

6

7

7

8

8

9

10 11 12 13

14

Figure 9 Schematic representation of the looped pipe network usedin numerical example 3

minimum head requirement 119867119894

lowast has been assumed to beequal to the ground elevation 119885

119894[8] So the relationship

between the required nodal head and minimum head is

119902119895=

0 119867119895lt 119885119895

119902119895 119885119895le 119867119895

(16)

Todini [8] proposed a three-step approach for solving thisnetwork and its solution is reported in the 4th column ofTable 4 In the proposed method pressure-driven model canbe applied in hydraulic analysis without any mathematicalformulation In this situation an if-then rule is added tococontent model and optimization process is conductedThenumber of decision variables in SCE algorithm is 7 the boundvariables were set between 50 and 140m ten optimizationruns were performed using different random initial solutionsfor all the cases and the results are illustrated in Table 5Results confirm that SCE is more accurate compared withTodini algorithm in case 2 and case 3 In Table 4 the bestresult is shown in bold and it is considered that the methodof SCE has calculated the best value of 120575 at 5 nodes whilethe Todini method has done it at 2 nodes The convergenceprocess of SCE algorithm has been shown in two forms inFigures 10 and 11The absolute value of 120575 is calculated for eachiteration in Figure 10 and the value of head pressure in node2119867(2) is calculated for each iteration in Figure 11

64 Numerical Example 4 The fourth considered networkis a real planned network designed for an industrial areain Apulian Town (Southern Italy) The network layout is

0 10 20 30 40 50 60 70 80 90

100

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

Figure 10 Convergence history of numerical example 3 (form 1)

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100

120

140

Number of iterations

H(2)

Figure 11 Convergence history of head pressure (119898) in node 2 fornumerical example 3 (form 2)

illustrated in Figure 12 and the corresponding data areprovided in Table 6 With respect to the leakages they havebeen assumed to be pressure-driven (see (6)) given that theyare implemented in the pressure driven network simulationmodel as described above [14] The parameter 120573 = 10632 times10minus7 and 120572 = 12 as reported in Giustolisi et al [14] for this

network Giustolisi et al [14] proposed a hydraulic simulationmodel which fully integrates a classic hydraulic simulationalgorithm such as that of Todini and Pilati [3] found inEPANET 2 with a pressure-driven model that entails amore realistic representation of the leakage They appliedtheir model in this network The results are demonstratedin Table 7 In this table the best result is shown in boldand it is considered that the method of SCE has calculatedthe best value of 120575 at 14 nodes while the Gistulishi methodhas done it at 9 nodes In the proposed method there is noneed to modify the mathematical formulation for hydraulic

10 Journal of Fluids

Table 5 Average of mass and energy balance for numerical example 3

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5332119864 minus 02 797119864 minus 02 595119864 minus 02 162119864 minus 02 430119864 + 04

Number of iterations in inner loop 5Number of complexes 9

207119864 minus 02 544119864 minus 02 252119864 minus 02 109119864 minus 02 627119864 + 04Number of iterations in inner loop 9Number of complexes 10

207119864 minus 02 207119864 minus 02 207119864 minus 02 217119864 minus 08 604119864 + 04Number of iterations in inner loop 15Three-step approach [8] Maximum accuracy 276119864 minus 02

Table 6 Hydraulic data relevant to numerical example 4

Pipe number 119871 (m) 119863 (mm)1 3485 3272 9557 2903 483 1004 4007 2905 7919 1006 4044 3687 3906 3278 4823 1009 9344 10010 4313 18411 5131 10012 4284 18413 419 10014 10231 10015 4551 16416 1826 29017 2213 29018 5839 16419 452 22920 7947 10021 7177 10022 6556 25823 1655 10024 2521 10025 3315 10026 500 20427 5799 16428 8428 10029 7926 10030 8463 18431 164 25832 4279 10033 3792 10034 1582 368

analysis An if-then rule is added to cocontent model andthe optimization process is performed easily As you can seein Table 8 SCE found the optimal solution more accurately

1

1

2 2

33

4

4

5

5

6

6

7

78

8

99

101011

11

12

12

1313

14

14

15

1516

1617

17

18

18

19

19

20

20

21

21

2222

23 2324

25

26

27

28

2930

31

32

33

34

Reservoir

Figure 12 Schematic representation of the looped pipe networkused in the numerical example 4

0 100 200 300 400 500 600 700 800 900 1000

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

10minus3

Figure 13 Convergence history of numerical example 4 (form 1)

than the Giustolisi algorithm The convergence process ofSCE algorithm has been shown in two forms in Figures 4 and5 The absolute value of 120575 is calculated for each iteration inFigure 13 and the amount of head pressure in node 20119867(20)is calculated for each iteration in Figure 14

Journal of Fluids 11

Table 7 Head and parameter 120575 in numerical example 4

Node number 119902 (ls) 119867 (m) (Giustolisi algorithm) 119867 (m) 120575 (Giustolisi algorithm) 120575 (SCE)1 10863 269 3329 0154743 00047382 17034 2481 3183 002131 00065323 14947 213 2739 0059002 00053644 1428 1722 2534 0001257 00050335 10133 2354 3089 0026184 00040186 1535 201 2902 0030898 00053997 9114 1891 2794 0017147 00030878 1051 179 2734 000227 00035459 12182 1785 2635 0002936 000386710 14579 1266 2324 0008277 000436211 9007 1623 2595 0031554 000272312 7575 1012 2205 0002732 000213813 152 1003 2245 00126 000432814 1355 1541 2595 0001318 000435215 9226 14 2417 0002199 000293416 112 1436 2405 0007089 000358617 11469 153 2542 0000103 00036118 10818 1883 2838 0011889 000390819 14675 1935 2839 543E minus 05 000509720 13318 1001 2379 0013059 000397421 14631 1148 2235 0003276 000414122 12012 14 2546 0003901 000367723 10326 1045 2011 0002551 000297924 mdash 3645 3645 mdash mdash

Table 8 Average of mass and energy balance for numerical example 4

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5400119864 minus 03 450119864 minus 02 410119864 minus 03 275119864 minus 04 659119864 + 04

Number of iterations in inner loop 5Number of complexes 9

390119864 minus 03 415119864 minus 03 400119864 minus 03 396119864 minus 05 115119864 + 05Number of iterations in inner loop 9Number of complexes 10

370119864 minus 03 410119864 minus 03 390119864 minus 03 253119864 minus 05 119119864 + 05Number of iterations in inner loop 15Giustolisi algorithm Maximum accuracy 181119864 minus 02

In general 4 different pipe networks were consideredin this paper and different mathematical formulations wereused for the hydraulic analysis of these networks Howeverthe overall results indicate that the proposed method hasthe capability of handling various pipe networks problemswith no change in the model or mathematical formulationApplication of SCE in cocontent model can result in findingaccurate solutions in pipes with zero flows and the pressure-driven demand and leakage simulation can be solved throughapplying if-then rules in cocontent model As a result itcan be concluded that the proposed method is a suitablealternative optimizer challenging othermethods especially interms of accuracy

7 Conclusions

The objective of the present paper was to provide an inno-vative approach in the analysis of the water distributionnetworks based on the optimization model The cocontentmodel is minimized using shuffled complex evolution (SCE)algorithm The methodology is illustrated here using fournetworks with different layouts The results reveal thatthe proposed method has the capability to handle variouspipe networks problems without changing in model ormathematical formulation The advantage of the proposedmethod lies in the fact that there is no need to solvelinear systems of equations pressure-driven demand and

12 Journal of Fluids

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

Number of iterations

H(20)

Figure 14 Convergence history of head-pressure (119898) in node 20 forexample 4 (form 2)

leakage simulation are handled in a simple way accu-rate solutions can be found in pipes with zero flowsand it does not need an initial solution vector which must bechosen carefully in many other procedures if numerical con-vergence is to be achieved Furthermore the proposedmodeldoes not require any complicated mathematical expressionand operation Finally it can be concluded that the proposedmethod is a viable alternative optimizer that challenges othermethods particularly in view of accuracy

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

References

[1] M Collins L Cooper R Helgason J Kenningston and LLeBlanc ldquoSolving the pipe network analysis problem usingoptimization techniquesrdquo Management Science vol 24 no 7pp 747ndash760 1978

[2] S Elhay and A R Simpson ldquoDealing with zero flows in solvingthe nonlinear equations for water distribution systemsrdquo Journalof Hydraulic Engineering vol 137 no 10 pp 1216ndash1224 2011

[3] E Todini and S Pilati ldquoA gradient algorithm for the analysis ofpipe networksrdquo in Computer Applications inWater Supply vol 1of System Analysis and Simulation pp 1ndash20 JohnWiley amp SonsLondon UK 1988

[4] A C Zecchin P Thum A R Simpson and C TischendorfldquoSteady-state behaviour of large water distribution systems thealgebraic multigrid method for the fast solution of the linearsteprdquo Journal ofWater Resources Planning andManagement vol138 no 6 pp 639ndash650 2012

[5] M L Arora ldquoFlow split in closed loops expending least energyrdquoJournal of the Hydraulics Division vol 102 pp 455ndash458 1976

[6] M A Hall ldquoHydraulic network analysis using (generalized)geometric programmingrdquo Networks vol 6 no 2 pp 105ndash1301976

[7] P R Bhave and R Gupta ldquoAnalysis of water distributionnetworksrdquo in Alpha Science International Technology amp Engi-neering 2006

[8] E Todini A More Realistic Approach to the ldquoExtended PeriodSimulationrdquo of Water Distribution Networks Advances in WaterSupplyManagement chapter 19 Taylor amp Francis London UK2003

[9] J M Wagner U Shamir and D H Marks ldquoWater distributionreliability analytical methodsrdquo Journal of Water ResourcesPlanning and Management vol 114 no 3 pp 253ndash275 1988

[10] J Chandapillai ldquoRealistic simulation of water distributionsystemrdquo Journal of Transportation Engineering vol 117 no 2 pp258ndash263 1991

[11] J Almandoz E Cabrera F Arregui E Cabrera Jr andR Cobacho ldquoLeakage assessment through water distributionnetwork simulationrdquo Journal of Water Resources Planning andManagement vol 131 no 6 pp 458ndash466 2005

[12] A F Colombo and B W Karney ldquoEnergy and costs ofleaky pipes toward comprehensive picturerdquo Journal of WaterResources Planning and Management vol 128 no 6 pp 441ndash450 2002

[13] G Germanopoulos ldquoA technical note on the inclusion ofpressure dependent demand and leakage terms in water supplynetworkmodelsrdquoCivil Engineering Systems vol 2 no 3 pp 171ndash179 1985

[14] O Giustolisi D Savic and Z Kapelan ldquoPressure-drivendemand and leakage simulation for water distribution net-worksrdquo Journal of Hydraulic Engineering vol 134 no 5 pp 626ndash635 2008

[15] L Ainola T Koppel K Tiiter and A Vassiljev ldquoWater networkmodel calibration based on grouping pipes with similar leakageand roughness estimatesrdquo in Proceedings of the Joint Conferenceon Water Resource Engineering and Water Resources PlanningandManagement (EWRI rsquo00) pp 1ndash9MinneapolisMinnUSAAugust 2000

[16] Q Duan S Sorooshian and V K Gupta ldquoOptimal use of theSCE-UA global optimization method for calibrating watershedmodelsrdquo Journal of Hydrology vol 158 no 3-4 pp 265ndash2841994

[17] Q Y Duan S Sorooshian and V K Gupta ldquoShuffled complexevolution approach for effective and efficient global minimiza-tionrdquo Journal of Optimization Theory and Applications vol 76no 3 pp 501ndash521 1993

[18] E Todini ldquoOn the convergence properties of the differentpipe network algorithmsrdquo in Proceedings of the Annual WaterDistribution Systems Analysis Symposium (ASCE rsquo06) pp 1ndash16August 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Journal of Fluids

Table 3 Average of mass and energy balance for numerical example 2

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 4221119864 minus 06 724119864 minus 06 465119864 minus 06 161119864 minus 06 419119864 + 03

Number of iterations in inner loop 4Number of complexes 9

105119864 minus 06 535119864 minus 06 346119864 minus 06 151119864 minus 06 929119864 + 03Number of iterations in inner loop 9Number of complexes 10

636119864 minus 08 178119864 minus 07 108119864 minus 07 389119864 minus 08 172119864 + 04Number of iterations in inner loop 15Global gradient algorithm Maximum accuracy FailElhay algorithm Maximum accuracy 742119864 minus 06

Table 4 Head and parameter 120575 in numerical example 3

Node number 119885 (m) SCE 3 steps SCE 3 steps SCE 3 steps119867 (m) 119867 (m) 119867 minus 119885 119867 minus 119885 120575 120575

1 140 140 140 0 0 0 02 80 129304 13007 49304 5007 149E minus 07 310119864 minus 04

3 90 132288 13276 42288 4276 126E minus 07 000414 70 109587 11096 39587 4096 203E minus 07 000225 80 80000 8854 0000 854 0058 151E minus 046 90 90000 9145 0000 145 00069 602E minus 047 90 90000 9000 0000 000 0080 014218 100 88922 9043 minus11078 minus957 584E minus 08 00439

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations0 5 10 15 20 25 30 35 40 45 50

Figure 7 Convergence history of numerical example 2 (form 1)

The absolute value of 120575 is calculated for each iteration inFigure 7 and the value of head pressure in node 2 119867(2) iscalculated for each iteration in Figure 8

63 Numerical Example 3 The simplified water distributionnetwork shown in Figure 9 was used in order to demonstratethe advantages of the proposed model in pressure-drivendemand condition For the sake of simplicity the same

H(2)

Number of iterations0 5 10 15 20 25 30 35 40 45 50

365

366

367

368

369

37

371

372

373

Figure 8 Convergence history of head pressure (119898) in node 2 fornumerical example 2 (form 2)

Hazen-Williams roughness coefficient 119862 = 130 was assumedfor all the 14 pipes of identical length of 1000m while no locallosses have been added The following diameters have beenused in the example 500mm (P-2) 400mm (P-1) 300mm(P-4 P-7) 250mm (P-10) 200mm (P-3 P-5 P-6 and P-13)150mm (P-8 P-9 P-11 P-12 and P-14) The nodal demandsare listed in the following tables together with the groundelevation 119885

119894 Without loss of generality in this example the

Journal of Fluids 9

Reservoir

1 2

23

3

4

4

5

5

6

6

7

7

8

8

9

10 11 12 13

14

Figure 9 Schematic representation of the looped pipe network usedin numerical example 3

minimum head requirement 119867119894

lowast has been assumed to beequal to the ground elevation 119885

119894[8] So the relationship

between the required nodal head and minimum head is

119902119895=

0 119867119895lt 119885119895

119902119895 119885119895le 119867119895

(16)

Todini [8] proposed a three-step approach for solving thisnetwork and its solution is reported in the 4th column ofTable 4 In the proposed method pressure-driven model canbe applied in hydraulic analysis without any mathematicalformulation In this situation an if-then rule is added tococontent model and optimization process is conductedThenumber of decision variables in SCE algorithm is 7 the boundvariables were set between 50 and 140m ten optimizationruns were performed using different random initial solutionsfor all the cases and the results are illustrated in Table 5Results confirm that SCE is more accurate compared withTodini algorithm in case 2 and case 3 In Table 4 the bestresult is shown in bold and it is considered that the methodof SCE has calculated the best value of 120575 at 5 nodes whilethe Todini method has done it at 2 nodes The convergenceprocess of SCE algorithm has been shown in two forms inFigures 10 and 11The absolute value of 120575 is calculated for eachiteration in Figure 10 and the value of head pressure in node2119867(2) is calculated for each iteration in Figure 11

64 Numerical Example 4 The fourth considered networkis a real planned network designed for an industrial areain Apulian Town (Southern Italy) The network layout is

0 10 20 30 40 50 60 70 80 90

100

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

Figure 10 Convergence history of numerical example 3 (form 1)

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100

120

140

Number of iterations

H(2)

Figure 11 Convergence history of head pressure (119898) in node 2 fornumerical example 3 (form 2)

illustrated in Figure 12 and the corresponding data areprovided in Table 6 With respect to the leakages they havebeen assumed to be pressure-driven (see (6)) given that theyare implemented in the pressure driven network simulationmodel as described above [14] The parameter 120573 = 10632 times10minus7 and 120572 = 12 as reported in Giustolisi et al [14] for this

network Giustolisi et al [14] proposed a hydraulic simulationmodel which fully integrates a classic hydraulic simulationalgorithm such as that of Todini and Pilati [3] found inEPANET 2 with a pressure-driven model that entails amore realistic representation of the leakage They appliedtheir model in this network The results are demonstratedin Table 7 In this table the best result is shown in boldand it is considered that the method of SCE has calculatedthe best value of 120575 at 14 nodes while the Gistulishi methodhas done it at 9 nodes In the proposed method there is noneed to modify the mathematical formulation for hydraulic

10 Journal of Fluids

Table 5 Average of mass and energy balance for numerical example 3

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5332119864 minus 02 797119864 minus 02 595119864 minus 02 162119864 minus 02 430119864 + 04

Number of iterations in inner loop 5Number of complexes 9

207119864 minus 02 544119864 minus 02 252119864 minus 02 109119864 minus 02 627119864 + 04Number of iterations in inner loop 9Number of complexes 10

207119864 minus 02 207119864 minus 02 207119864 minus 02 217119864 minus 08 604119864 + 04Number of iterations in inner loop 15Three-step approach [8] Maximum accuracy 276119864 minus 02

Table 6 Hydraulic data relevant to numerical example 4

Pipe number 119871 (m) 119863 (mm)1 3485 3272 9557 2903 483 1004 4007 2905 7919 1006 4044 3687 3906 3278 4823 1009 9344 10010 4313 18411 5131 10012 4284 18413 419 10014 10231 10015 4551 16416 1826 29017 2213 29018 5839 16419 452 22920 7947 10021 7177 10022 6556 25823 1655 10024 2521 10025 3315 10026 500 20427 5799 16428 8428 10029 7926 10030 8463 18431 164 25832 4279 10033 3792 10034 1582 368

analysis An if-then rule is added to cocontent model andthe optimization process is performed easily As you can seein Table 8 SCE found the optimal solution more accurately

1

1

2 2

33

4

4

5

5

6

6

7

78

8

99

101011

11

12

12

1313

14

14

15

1516

1617

17

18

18

19

19

20

20

21

21

2222

23 2324

25

26

27

28

2930

31

32

33

34

Reservoir

Figure 12 Schematic representation of the looped pipe networkused in the numerical example 4

0 100 200 300 400 500 600 700 800 900 1000

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

10minus3

Figure 13 Convergence history of numerical example 4 (form 1)

than the Giustolisi algorithm The convergence process ofSCE algorithm has been shown in two forms in Figures 4 and5 The absolute value of 120575 is calculated for each iteration inFigure 13 and the amount of head pressure in node 20119867(20)is calculated for each iteration in Figure 14

Journal of Fluids 11

Table 7 Head and parameter 120575 in numerical example 4

Node number 119902 (ls) 119867 (m) (Giustolisi algorithm) 119867 (m) 120575 (Giustolisi algorithm) 120575 (SCE)1 10863 269 3329 0154743 00047382 17034 2481 3183 002131 00065323 14947 213 2739 0059002 00053644 1428 1722 2534 0001257 00050335 10133 2354 3089 0026184 00040186 1535 201 2902 0030898 00053997 9114 1891 2794 0017147 00030878 1051 179 2734 000227 00035459 12182 1785 2635 0002936 000386710 14579 1266 2324 0008277 000436211 9007 1623 2595 0031554 000272312 7575 1012 2205 0002732 000213813 152 1003 2245 00126 000432814 1355 1541 2595 0001318 000435215 9226 14 2417 0002199 000293416 112 1436 2405 0007089 000358617 11469 153 2542 0000103 00036118 10818 1883 2838 0011889 000390819 14675 1935 2839 543E minus 05 000509720 13318 1001 2379 0013059 000397421 14631 1148 2235 0003276 000414122 12012 14 2546 0003901 000367723 10326 1045 2011 0002551 000297924 mdash 3645 3645 mdash mdash

Table 8 Average of mass and energy balance for numerical example 4

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5400119864 minus 03 450119864 minus 02 410119864 minus 03 275119864 minus 04 659119864 + 04

Number of iterations in inner loop 5Number of complexes 9

390119864 minus 03 415119864 minus 03 400119864 minus 03 396119864 minus 05 115119864 + 05Number of iterations in inner loop 9Number of complexes 10

370119864 minus 03 410119864 minus 03 390119864 minus 03 253119864 minus 05 119119864 + 05Number of iterations in inner loop 15Giustolisi algorithm Maximum accuracy 181119864 minus 02

In general 4 different pipe networks were consideredin this paper and different mathematical formulations wereused for the hydraulic analysis of these networks Howeverthe overall results indicate that the proposed method hasthe capability of handling various pipe networks problemswith no change in the model or mathematical formulationApplication of SCE in cocontent model can result in findingaccurate solutions in pipes with zero flows and the pressure-driven demand and leakage simulation can be solved throughapplying if-then rules in cocontent model As a result itcan be concluded that the proposed method is a suitablealternative optimizer challenging othermethods especially interms of accuracy

7 Conclusions

The objective of the present paper was to provide an inno-vative approach in the analysis of the water distributionnetworks based on the optimization model The cocontentmodel is minimized using shuffled complex evolution (SCE)algorithm The methodology is illustrated here using fournetworks with different layouts The results reveal thatthe proposed method has the capability to handle variouspipe networks problems without changing in model ormathematical formulation The advantage of the proposedmethod lies in the fact that there is no need to solvelinear systems of equations pressure-driven demand and

12 Journal of Fluids

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

Number of iterations

H(20)

Figure 14 Convergence history of head-pressure (119898) in node 20 forexample 4 (form 2)

leakage simulation are handled in a simple way accu-rate solutions can be found in pipes with zero flowsand it does not need an initial solution vector which must bechosen carefully in many other procedures if numerical con-vergence is to be achieved Furthermore the proposedmodeldoes not require any complicated mathematical expressionand operation Finally it can be concluded that the proposedmethod is a viable alternative optimizer that challenges othermethods particularly in view of accuracy

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

References

[1] M Collins L Cooper R Helgason J Kenningston and LLeBlanc ldquoSolving the pipe network analysis problem usingoptimization techniquesrdquo Management Science vol 24 no 7pp 747ndash760 1978

[2] S Elhay and A R Simpson ldquoDealing with zero flows in solvingthe nonlinear equations for water distribution systemsrdquo Journalof Hydraulic Engineering vol 137 no 10 pp 1216ndash1224 2011

[3] E Todini and S Pilati ldquoA gradient algorithm for the analysis ofpipe networksrdquo in Computer Applications inWater Supply vol 1of System Analysis and Simulation pp 1ndash20 JohnWiley amp SonsLondon UK 1988

[4] A C Zecchin P Thum A R Simpson and C TischendorfldquoSteady-state behaviour of large water distribution systems thealgebraic multigrid method for the fast solution of the linearsteprdquo Journal ofWater Resources Planning andManagement vol138 no 6 pp 639ndash650 2012

[5] M L Arora ldquoFlow split in closed loops expending least energyrdquoJournal of the Hydraulics Division vol 102 pp 455ndash458 1976

[6] M A Hall ldquoHydraulic network analysis using (generalized)geometric programmingrdquo Networks vol 6 no 2 pp 105ndash1301976

[7] P R Bhave and R Gupta ldquoAnalysis of water distributionnetworksrdquo in Alpha Science International Technology amp Engi-neering 2006

[8] E Todini A More Realistic Approach to the ldquoExtended PeriodSimulationrdquo of Water Distribution Networks Advances in WaterSupplyManagement chapter 19 Taylor amp Francis London UK2003

[9] J M Wagner U Shamir and D H Marks ldquoWater distributionreliability analytical methodsrdquo Journal of Water ResourcesPlanning and Management vol 114 no 3 pp 253ndash275 1988

[10] J Chandapillai ldquoRealistic simulation of water distributionsystemrdquo Journal of Transportation Engineering vol 117 no 2 pp258ndash263 1991

[11] J Almandoz E Cabrera F Arregui E Cabrera Jr andR Cobacho ldquoLeakage assessment through water distributionnetwork simulationrdquo Journal of Water Resources Planning andManagement vol 131 no 6 pp 458ndash466 2005

[12] A F Colombo and B W Karney ldquoEnergy and costs ofleaky pipes toward comprehensive picturerdquo Journal of WaterResources Planning and Management vol 128 no 6 pp 441ndash450 2002

[13] G Germanopoulos ldquoA technical note on the inclusion ofpressure dependent demand and leakage terms in water supplynetworkmodelsrdquoCivil Engineering Systems vol 2 no 3 pp 171ndash179 1985

[14] O Giustolisi D Savic and Z Kapelan ldquoPressure-drivendemand and leakage simulation for water distribution net-worksrdquo Journal of Hydraulic Engineering vol 134 no 5 pp 626ndash635 2008

[15] L Ainola T Koppel K Tiiter and A Vassiljev ldquoWater networkmodel calibration based on grouping pipes with similar leakageand roughness estimatesrdquo in Proceedings of the Joint Conferenceon Water Resource Engineering and Water Resources PlanningandManagement (EWRI rsquo00) pp 1ndash9MinneapolisMinnUSAAugust 2000

[16] Q Duan S Sorooshian and V K Gupta ldquoOptimal use of theSCE-UA global optimization method for calibrating watershedmodelsrdquo Journal of Hydrology vol 158 no 3-4 pp 265ndash2841994

[17] Q Y Duan S Sorooshian and V K Gupta ldquoShuffled complexevolution approach for effective and efficient global minimiza-tionrdquo Journal of Optimization Theory and Applications vol 76no 3 pp 501ndash521 1993

[18] E Todini ldquoOn the convergence properties of the differentpipe network algorithmsrdquo in Proceedings of the Annual WaterDistribution Systems Analysis Symposium (ASCE rsquo06) pp 1ndash16August 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Fluids 9

Reservoir

1 2

23

3

4

4

5

5

6

6

7

7

8

8

9

10 11 12 13

14

Figure 9 Schematic representation of the looped pipe network usedin numerical example 3

minimum head requirement 119867119894

lowast has been assumed to beequal to the ground elevation 119885

119894[8] So the relationship

between the required nodal head and minimum head is

119902119895=

0 119867119895lt 119885119895

119902119895 119885119895le 119867119895

(16)

Todini [8] proposed a three-step approach for solving thisnetwork and its solution is reported in the 4th column ofTable 4 In the proposed method pressure-driven model canbe applied in hydraulic analysis without any mathematicalformulation In this situation an if-then rule is added tococontent model and optimization process is conductedThenumber of decision variables in SCE algorithm is 7 the boundvariables were set between 50 and 140m ten optimizationruns were performed using different random initial solutionsfor all the cases and the results are illustrated in Table 5Results confirm that SCE is more accurate compared withTodini algorithm in case 2 and case 3 In Table 4 the bestresult is shown in bold and it is considered that the methodof SCE has calculated the best value of 120575 at 5 nodes whilethe Todini method has done it at 2 nodes The convergenceprocess of SCE algorithm has been shown in two forms inFigures 10 and 11The absolute value of 120575 is calculated for eachiteration in Figure 10 and the value of head pressure in node2119867(2) is calculated for each iteration in Figure 11

64 Numerical Example 4 The fourth considered networkis a real planned network designed for an industrial areain Apulian Town (Southern Italy) The network layout is

0 10 20 30 40 50 60 70 80 90

100

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

Figure 10 Convergence history of numerical example 3 (form 1)

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100

120

140

Number of iterations

H(2)

Figure 11 Convergence history of head pressure (119898) in node 2 fornumerical example 3 (form 2)

illustrated in Figure 12 and the corresponding data areprovided in Table 6 With respect to the leakages they havebeen assumed to be pressure-driven (see (6)) given that theyare implemented in the pressure driven network simulationmodel as described above [14] The parameter 120573 = 10632 times10minus7 and 120572 = 12 as reported in Giustolisi et al [14] for this

network Giustolisi et al [14] proposed a hydraulic simulationmodel which fully integrates a classic hydraulic simulationalgorithm such as that of Todini and Pilati [3] found inEPANET 2 with a pressure-driven model that entails amore realistic representation of the leakage They appliedtheir model in this network The results are demonstratedin Table 7 In this table the best result is shown in boldand it is considered that the method of SCE has calculatedthe best value of 120575 at 14 nodes while the Gistulishi methodhas done it at 9 nodes In the proposed method there is noneed to modify the mathematical formulation for hydraulic

10 Journal of Fluids

Table 5 Average of mass and energy balance for numerical example 3

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5332119864 minus 02 797119864 minus 02 595119864 minus 02 162119864 minus 02 430119864 + 04

Number of iterations in inner loop 5Number of complexes 9

207119864 minus 02 544119864 minus 02 252119864 minus 02 109119864 minus 02 627119864 + 04Number of iterations in inner loop 9Number of complexes 10

207119864 minus 02 207119864 minus 02 207119864 minus 02 217119864 minus 08 604119864 + 04Number of iterations in inner loop 15Three-step approach [8] Maximum accuracy 276119864 minus 02

Table 6 Hydraulic data relevant to numerical example 4

Pipe number 119871 (m) 119863 (mm)1 3485 3272 9557 2903 483 1004 4007 2905 7919 1006 4044 3687 3906 3278 4823 1009 9344 10010 4313 18411 5131 10012 4284 18413 419 10014 10231 10015 4551 16416 1826 29017 2213 29018 5839 16419 452 22920 7947 10021 7177 10022 6556 25823 1655 10024 2521 10025 3315 10026 500 20427 5799 16428 8428 10029 7926 10030 8463 18431 164 25832 4279 10033 3792 10034 1582 368

analysis An if-then rule is added to cocontent model andthe optimization process is performed easily As you can seein Table 8 SCE found the optimal solution more accurately

1

1

2 2

33

4

4

5

5

6

6

7

78

8

99

101011

11

12

12

1313

14

14

15

1516

1617

17

18

18

19

19

20

20

21

21

2222

23 2324

25

26

27

28

2930

31

32

33

34

Reservoir

Figure 12 Schematic representation of the looped pipe networkused in the numerical example 4

0 100 200 300 400 500 600 700 800 900 1000

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

10minus3

Figure 13 Convergence history of numerical example 4 (form 1)

than the Giustolisi algorithm The convergence process ofSCE algorithm has been shown in two forms in Figures 4 and5 The absolute value of 120575 is calculated for each iteration inFigure 13 and the amount of head pressure in node 20119867(20)is calculated for each iteration in Figure 14

Journal of Fluids 11

Table 7 Head and parameter 120575 in numerical example 4

Node number 119902 (ls) 119867 (m) (Giustolisi algorithm) 119867 (m) 120575 (Giustolisi algorithm) 120575 (SCE)1 10863 269 3329 0154743 00047382 17034 2481 3183 002131 00065323 14947 213 2739 0059002 00053644 1428 1722 2534 0001257 00050335 10133 2354 3089 0026184 00040186 1535 201 2902 0030898 00053997 9114 1891 2794 0017147 00030878 1051 179 2734 000227 00035459 12182 1785 2635 0002936 000386710 14579 1266 2324 0008277 000436211 9007 1623 2595 0031554 000272312 7575 1012 2205 0002732 000213813 152 1003 2245 00126 000432814 1355 1541 2595 0001318 000435215 9226 14 2417 0002199 000293416 112 1436 2405 0007089 000358617 11469 153 2542 0000103 00036118 10818 1883 2838 0011889 000390819 14675 1935 2839 543E minus 05 000509720 13318 1001 2379 0013059 000397421 14631 1148 2235 0003276 000414122 12012 14 2546 0003901 000367723 10326 1045 2011 0002551 000297924 mdash 3645 3645 mdash mdash

Table 8 Average of mass and energy balance for numerical example 4

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5400119864 minus 03 450119864 minus 02 410119864 minus 03 275119864 minus 04 659119864 + 04

Number of iterations in inner loop 5Number of complexes 9

390119864 minus 03 415119864 minus 03 400119864 minus 03 396119864 minus 05 115119864 + 05Number of iterations in inner loop 9Number of complexes 10

370119864 minus 03 410119864 minus 03 390119864 minus 03 253119864 minus 05 119119864 + 05Number of iterations in inner loop 15Giustolisi algorithm Maximum accuracy 181119864 minus 02

In general 4 different pipe networks were consideredin this paper and different mathematical formulations wereused for the hydraulic analysis of these networks Howeverthe overall results indicate that the proposed method hasthe capability of handling various pipe networks problemswith no change in the model or mathematical formulationApplication of SCE in cocontent model can result in findingaccurate solutions in pipes with zero flows and the pressure-driven demand and leakage simulation can be solved throughapplying if-then rules in cocontent model As a result itcan be concluded that the proposed method is a suitablealternative optimizer challenging othermethods especially interms of accuracy

7 Conclusions

The objective of the present paper was to provide an inno-vative approach in the analysis of the water distributionnetworks based on the optimization model The cocontentmodel is minimized using shuffled complex evolution (SCE)algorithm The methodology is illustrated here using fournetworks with different layouts The results reveal thatthe proposed method has the capability to handle variouspipe networks problems without changing in model ormathematical formulation The advantage of the proposedmethod lies in the fact that there is no need to solvelinear systems of equations pressure-driven demand and

12 Journal of Fluids

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

Number of iterations

H(20)

Figure 14 Convergence history of head-pressure (119898) in node 20 forexample 4 (form 2)

leakage simulation are handled in a simple way accu-rate solutions can be found in pipes with zero flowsand it does not need an initial solution vector which must bechosen carefully in many other procedures if numerical con-vergence is to be achieved Furthermore the proposedmodeldoes not require any complicated mathematical expressionand operation Finally it can be concluded that the proposedmethod is a viable alternative optimizer that challenges othermethods particularly in view of accuracy

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

References

[1] M Collins L Cooper R Helgason J Kenningston and LLeBlanc ldquoSolving the pipe network analysis problem usingoptimization techniquesrdquo Management Science vol 24 no 7pp 747ndash760 1978

[2] S Elhay and A R Simpson ldquoDealing with zero flows in solvingthe nonlinear equations for water distribution systemsrdquo Journalof Hydraulic Engineering vol 137 no 10 pp 1216ndash1224 2011

[3] E Todini and S Pilati ldquoA gradient algorithm for the analysis ofpipe networksrdquo in Computer Applications inWater Supply vol 1of System Analysis and Simulation pp 1ndash20 JohnWiley amp SonsLondon UK 1988

[4] A C Zecchin P Thum A R Simpson and C TischendorfldquoSteady-state behaviour of large water distribution systems thealgebraic multigrid method for the fast solution of the linearsteprdquo Journal ofWater Resources Planning andManagement vol138 no 6 pp 639ndash650 2012

[5] M L Arora ldquoFlow split in closed loops expending least energyrdquoJournal of the Hydraulics Division vol 102 pp 455ndash458 1976

[6] M A Hall ldquoHydraulic network analysis using (generalized)geometric programmingrdquo Networks vol 6 no 2 pp 105ndash1301976

[7] P R Bhave and R Gupta ldquoAnalysis of water distributionnetworksrdquo in Alpha Science International Technology amp Engi-neering 2006

[8] E Todini A More Realistic Approach to the ldquoExtended PeriodSimulationrdquo of Water Distribution Networks Advances in WaterSupplyManagement chapter 19 Taylor amp Francis London UK2003

[9] J M Wagner U Shamir and D H Marks ldquoWater distributionreliability analytical methodsrdquo Journal of Water ResourcesPlanning and Management vol 114 no 3 pp 253ndash275 1988

[10] J Chandapillai ldquoRealistic simulation of water distributionsystemrdquo Journal of Transportation Engineering vol 117 no 2 pp258ndash263 1991

[11] J Almandoz E Cabrera F Arregui E Cabrera Jr andR Cobacho ldquoLeakage assessment through water distributionnetwork simulationrdquo Journal of Water Resources Planning andManagement vol 131 no 6 pp 458ndash466 2005

[12] A F Colombo and B W Karney ldquoEnergy and costs ofleaky pipes toward comprehensive picturerdquo Journal of WaterResources Planning and Management vol 128 no 6 pp 441ndash450 2002

[13] G Germanopoulos ldquoA technical note on the inclusion ofpressure dependent demand and leakage terms in water supplynetworkmodelsrdquoCivil Engineering Systems vol 2 no 3 pp 171ndash179 1985

[14] O Giustolisi D Savic and Z Kapelan ldquoPressure-drivendemand and leakage simulation for water distribution net-worksrdquo Journal of Hydraulic Engineering vol 134 no 5 pp 626ndash635 2008

[15] L Ainola T Koppel K Tiiter and A Vassiljev ldquoWater networkmodel calibration based on grouping pipes with similar leakageand roughness estimatesrdquo in Proceedings of the Joint Conferenceon Water Resource Engineering and Water Resources PlanningandManagement (EWRI rsquo00) pp 1ndash9MinneapolisMinnUSAAugust 2000

[16] Q Duan S Sorooshian and V K Gupta ldquoOptimal use of theSCE-UA global optimization method for calibrating watershedmodelsrdquo Journal of Hydrology vol 158 no 3-4 pp 265ndash2841994

[17] Q Y Duan S Sorooshian and V K Gupta ldquoShuffled complexevolution approach for effective and efficient global minimiza-tionrdquo Journal of Optimization Theory and Applications vol 76no 3 pp 501ndash521 1993

[18] E Todini ldquoOn the convergence properties of the differentpipe network algorithmsrdquo in Proceedings of the Annual WaterDistribution Systems Analysis Symposium (ASCE rsquo06) pp 1ndash16August 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Journal of Fluids

Table 5 Average of mass and energy balance for numerical example 3

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5332119864 minus 02 797119864 minus 02 595119864 minus 02 162119864 minus 02 430119864 + 04

Number of iterations in inner loop 5Number of complexes 9

207119864 minus 02 544119864 minus 02 252119864 minus 02 109119864 minus 02 627119864 + 04Number of iterations in inner loop 9Number of complexes 10

207119864 minus 02 207119864 minus 02 207119864 minus 02 217119864 minus 08 604119864 + 04Number of iterations in inner loop 15Three-step approach [8] Maximum accuracy 276119864 minus 02

Table 6 Hydraulic data relevant to numerical example 4

Pipe number 119871 (m) 119863 (mm)1 3485 3272 9557 2903 483 1004 4007 2905 7919 1006 4044 3687 3906 3278 4823 1009 9344 10010 4313 18411 5131 10012 4284 18413 419 10014 10231 10015 4551 16416 1826 29017 2213 29018 5839 16419 452 22920 7947 10021 7177 10022 6556 25823 1655 10024 2521 10025 3315 10026 500 20427 5799 16428 8428 10029 7926 10030 8463 18431 164 25832 4279 10033 3792 10034 1582 368

analysis An if-then rule is added to cocontent model andthe optimization process is performed easily As you can seein Table 8 SCE found the optimal solution more accurately

1

1

2 2

33

4

4

5

5

6

6

7

78

8

99

101011

11

12

12

1313

14

14

15

1516

1617

17

18

18

19

19

20

20

21

21

2222

23 2324

25

26

27

28

2930

31

32

33

34

Reservoir

Figure 12 Schematic representation of the looped pipe networkused in the numerical example 4

0 100 200 300 400 500 600 700 800 900 1000

Aver

age o

f mas

s and

ener

gy b

alan

ce

Number of iterations

10minus1

10minus2

10minus3

Figure 13 Convergence history of numerical example 4 (form 1)

than the Giustolisi algorithm The convergence process ofSCE algorithm has been shown in two forms in Figures 4 and5 The absolute value of 120575 is calculated for each iteration inFigure 13 and the amount of head pressure in node 20119867(20)is calculated for each iteration in Figure 14

Journal of Fluids 11

Table 7 Head and parameter 120575 in numerical example 4

Node number 119902 (ls) 119867 (m) (Giustolisi algorithm) 119867 (m) 120575 (Giustolisi algorithm) 120575 (SCE)1 10863 269 3329 0154743 00047382 17034 2481 3183 002131 00065323 14947 213 2739 0059002 00053644 1428 1722 2534 0001257 00050335 10133 2354 3089 0026184 00040186 1535 201 2902 0030898 00053997 9114 1891 2794 0017147 00030878 1051 179 2734 000227 00035459 12182 1785 2635 0002936 000386710 14579 1266 2324 0008277 000436211 9007 1623 2595 0031554 000272312 7575 1012 2205 0002732 000213813 152 1003 2245 00126 000432814 1355 1541 2595 0001318 000435215 9226 14 2417 0002199 000293416 112 1436 2405 0007089 000358617 11469 153 2542 0000103 00036118 10818 1883 2838 0011889 000390819 14675 1935 2839 543E minus 05 000509720 13318 1001 2379 0013059 000397421 14631 1148 2235 0003276 000414122 12012 14 2546 0003901 000367723 10326 1045 2011 0002551 000297924 mdash 3645 3645 mdash mdash

Table 8 Average of mass and energy balance for numerical example 4

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5400119864 minus 03 450119864 minus 02 410119864 minus 03 275119864 minus 04 659119864 + 04

Number of iterations in inner loop 5Number of complexes 9

390119864 minus 03 415119864 minus 03 400119864 minus 03 396119864 minus 05 115119864 + 05Number of iterations in inner loop 9Number of complexes 10

370119864 minus 03 410119864 minus 03 390119864 minus 03 253119864 minus 05 119119864 + 05Number of iterations in inner loop 15Giustolisi algorithm Maximum accuracy 181119864 minus 02

In general 4 different pipe networks were consideredin this paper and different mathematical formulations wereused for the hydraulic analysis of these networks Howeverthe overall results indicate that the proposed method hasthe capability of handling various pipe networks problemswith no change in the model or mathematical formulationApplication of SCE in cocontent model can result in findingaccurate solutions in pipes with zero flows and the pressure-driven demand and leakage simulation can be solved throughapplying if-then rules in cocontent model As a result itcan be concluded that the proposed method is a suitablealternative optimizer challenging othermethods especially interms of accuracy

7 Conclusions

The objective of the present paper was to provide an inno-vative approach in the analysis of the water distributionnetworks based on the optimization model The cocontentmodel is minimized using shuffled complex evolution (SCE)algorithm The methodology is illustrated here using fournetworks with different layouts The results reveal thatthe proposed method has the capability to handle variouspipe networks problems without changing in model ormathematical formulation The advantage of the proposedmethod lies in the fact that there is no need to solvelinear systems of equations pressure-driven demand and

12 Journal of Fluids

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

Number of iterations

H(20)

Figure 14 Convergence history of head-pressure (119898) in node 20 forexample 4 (form 2)

leakage simulation are handled in a simple way accu-rate solutions can be found in pipes with zero flowsand it does not need an initial solution vector which must bechosen carefully in many other procedures if numerical con-vergence is to be achieved Furthermore the proposedmodeldoes not require any complicated mathematical expressionand operation Finally it can be concluded that the proposedmethod is a viable alternative optimizer that challenges othermethods particularly in view of accuracy

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

References

[1] M Collins L Cooper R Helgason J Kenningston and LLeBlanc ldquoSolving the pipe network analysis problem usingoptimization techniquesrdquo Management Science vol 24 no 7pp 747ndash760 1978

[2] S Elhay and A R Simpson ldquoDealing with zero flows in solvingthe nonlinear equations for water distribution systemsrdquo Journalof Hydraulic Engineering vol 137 no 10 pp 1216ndash1224 2011

[3] E Todini and S Pilati ldquoA gradient algorithm for the analysis ofpipe networksrdquo in Computer Applications inWater Supply vol 1of System Analysis and Simulation pp 1ndash20 JohnWiley amp SonsLondon UK 1988

[4] A C Zecchin P Thum A R Simpson and C TischendorfldquoSteady-state behaviour of large water distribution systems thealgebraic multigrid method for the fast solution of the linearsteprdquo Journal ofWater Resources Planning andManagement vol138 no 6 pp 639ndash650 2012

[5] M L Arora ldquoFlow split in closed loops expending least energyrdquoJournal of the Hydraulics Division vol 102 pp 455ndash458 1976

[6] M A Hall ldquoHydraulic network analysis using (generalized)geometric programmingrdquo Networks vol 6 no 2 pp 105ndash1301976

[7] P R Bhave and R Gupta ldquoAnalysis of water distributionnetworksrdquo in Alpha Science International Technology amp Engi-neering 2006

[8] E Todini A More Realistic Approach to the ldquoExtended PeriodSimulationrdquo of Water Distribution Networks Advances in WaterSupplyManagement chapter 19 Taylor amp Francis London UK2003

[9] J M Wagner U Shamir and D H Marks ldquoWater distributionreliability analytical methodsrdquo Journal of Water ResourcesPlanning and Management vol 114 no 3 pp 253ndash275 1988

[10] J Chandapillai ldquoRealistic simulation of water distributionsystemrdquo Journal of Transportation Engineering vol 117 no 2 pp258ndash263 1991

[11] J Almandoz E Cabrera F Arregui E Cabrera Jr andR Cobacho ldquoLeakage assessment through water distributionnetwork simulationrdquo Journal of Water Resources Planning andManagement vol 131 no 6 pp 458ndash466 2005

[12] A F Colombo and B W Karney ldquoEnergy and costs ofleaky pipes toward comprehensive picturerdquo Journal of WaterResources Planning and Management vol 128 no 6 pp 441ndash450 2002

[13] G Germanopoulos ldquoA technical note on the inclusion ofpressure dependent demand and leakage terms in water supplynetworkmodelsrdquoCivil Engineering Systems vol 2 no 3 pp 171ndash179 1985

[14] O Giustolisi D Savic and Z Kapelan ldquoPressure-drivendemand and leakage simulation for water distribution net-worksrdquo Journal of Hydraulic Engineering vol 134 no 5 pp 626ndash635 2008

[15] L Ainola T Koppel K Tiiter and A Vassiljev ldquoWater networkmodel calibration based on grouping pipes with similar leakageand roughness estimatesrdquo in Proceedings of the Joint Conferenceon Water Resource Engineering and Water Resources PlanningandManagement (EWRI rsquo00) pp 1ndash9MinneapolisMinnUSAAugust 2000

[16] Q Duan S Sorooshian and V K Gupta ldquoOptimal use of theSCE-UA global optimization method for calibrating watershedmodelsrdquo Journal of Hydrology vol 158 no 3-4 pp 265ndash2841994

[17] Q Y Duan S Sorooshian and V K Gupta ldquoShuffled complexevolution approach for effective and efficient global minimiza-tionrdquo Journal of Optimization Theory and Applications vol 76no 3 pp 501ndash521 1993

[18] E Todini ldquoOn the convergence properties of the differentpipe network algorithmsrdquo in Proceedings of the Annual WaterDistribution Systems Analysis Symposium (ASCE rsquo06) pp 1ndash16August 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Fluids 11

Table 7 Head and parameter 120575 in numerical example 4

Node number 119902 (ls) 119867 (m) (Giustolisi algorithm) 119867 (m) 120575 (Giustolisi algorithm) 120575 (SCE)1 10863 269 3329 0154743 00047382 17034 2481 3183 002131 00065323 14947 213 2739 0059002 00053644 1428 1722 2534 0001257 00050335 10133 2354 3089 0026184 00040186 1535 201 2902 0030898 00053997 9114 1891 2794 0017147 00030878 1051 179 2734 000227 00035459 12182 1785 2635 0002936 000386710 14579 1266 2324 0008277 000436211 9007 1623 2595 0031554 000272312 7575 1012 2205 0002732 000213813 152 1003 2245 00126 000432814 1355 1541 2595 0001318 000435215 9226 14 2417 0002199 000293416 112 1436 2405 0007089 000358617 11469 153 2542 0000103 00036118 10818 1883 2838 0011889 000390819 14675 1935 2839 543E minus 05 000509720 13318 1001 2379 0013059 000397421 14631 1148 2235 0003276 000414122 12012 14 2546 0003901 000367723 10326 1045 2011 0002551 000297924 mdash 3645 3645 mdash mdash

Table 8 Average of mass and energy balance for numerical example 4

SCE 120575 Average number of function evaluationsBest Worst Mean Std

Number of complexes 5400119864 minus 03 450119864 minus 02 410119864 minus 03 275119864 minus 04 659119864 + 04

Number of iterations in inner loop 5Number of complexes 9

390119864 minus 03 415119864 minus 03 400119864 minus 03 396119864 minus 05 115119864 + 05Number of iterations in inner loop 9Number of complexes 10

370119864 minus 03 410119864 minus 03 390119864 minus 03 253119864 minus 05 119119864 + 05Number of iterations in inner loop 15Giustolisi algorithm Maximum accuracy 181119864 minus 02

In general 4 different pipe networks were consideredin this paper and different mathematical formulations wereused for the hydraulic analysis of these networks Howeverthe overall results indicate that the proposed method hasthe capability of handling various pipe networks problemswith no change in the model or mathematical formulationApplication of SCE in cocontent model can result in findingaccurate solutions in pipes with zero flows and the pressure-driven demand and leakage simulation can be solved throughapplying if-then rules in cocontent model As a result itcan be concluded that the proposed method is a suitablealternative optimizer challenging othermethods especially interms of accuracy

7 Conclusions

The objective of the present paper was to provide an inno-vative approach in the analysis of the water distributionnetworks based on the optimization model The cocontentmodel is minimized using shuffled complex evolution (SCE)algorithm The methodology is illustrated here using fournetworks with different layouts The results reveal thatthe proposed method has the capability to handle variouspipe networks problems without changing in model ormathematical formulation The advantage of the proposedmethod lies in the fact that there is no need to solvelinear systems of equations pressure-driven demand and

12 Journal of Fluids

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

Number of iterations

H(20)

Figure 14 Convergence history of head-pressure (119898) in node 20 forexample 4 (form 2)

leakage simulation are handled in a simple way accu-rate solutions can be found in pipes with zero flowsand it does not need an initial solution vector which must bechosen carefully in many other procedures if numerical con-vergence is to be achieved Furthermore the proposedmodeldoes not require any complicated mathematical expressionand operation Finally it can be concluded that the proposedmethod is a viable alternative optimizer that challenges othermethods particularly in view of accuracy

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

References

[1] M Collins L Cooper R Helgason J Kenningston and LLeBlanc ldquoSolving the pipe network analysis problem usingoptimization techniquesrdquo Management Science vol 24 no 7pp 747ndash760 1978

[2] S Elhay and A R Simpson ldquoDealing with zero flows in solvingthe nonlinear equations for water distribution systemsrdquo Journalof Hydraulic Engineering vol 137 no 10 pp 1216ndash1224 2011

[3] E Todini and S Pilati ldquoA gradient algorithm for the analysis ofpipe networksrdquo in Computer Applications inWater Supply vol 1of System Analysis and Simulation pp 1ndash20 JohnWiley amp SonsLondon UK 1988

[4] A C Zecchin P Thum A R Simpson and C TischendorfldquoSteady-state behaviour of large water distribution systems thealgebraic multigrid method for the fast solution of the linearsteprdquo Journal ofWater Resources Planning andManagement vol138 no 6 pp 639ndash650 2012

[5] M L Arora ldquoFlow split in closed loops expending least energyrdquoJournal of the Hydraulics Division vol 102 pp 455ndash458 1976

[6] M A Hall ldquoHydraulic network analysis using (generalized)geometric programmingrdquo Networks vol 6 no 2 pp 105ndash1301976

[7] P R Bhave and R Gupta ldquoAnalysis of water distributionnetworksrdquo in Alpha Science International Technology amp Engi-neering 2006

[8] E Todini A More Realistic Approach to the ldquoExtended PeriodSimulationrdquo of Water Distribution Networks Advances in WaterSupplyManagement chapter 19 Taylor amp Francis London UK2003

[9] J M Wagner U Shamir and D H Marks ldquoWater distributionreliability analytical methodsrdquo Journal of Water ResourcesPlanning and Management vol 114 no 3 pp 253ndash275 1988

[10] J Chandapillai ldquoRealistic simulation of water distributionsystemrdquo Journal of Transportation Engineering vol 117 no 2 pp258ndash263 1991

[11] J Almandoz E Cabrera F Arregui E Cabrera Jr andR Cobacho ldquoLeakage assessment through water distributionnetwork simulationrdquo Journal of Water Resources Planning andManagement vol 131 no 6 pp 458ndash466 2005

[12] A F Colombo and B W Karney ldquoEnergy and costs ofleaky pipes toward comprehensive picturerdquo Journal of WaterResources Planning and Management vol 128 no 6 pp 441ndash450 2002

[13] G Germanopoulos ldquoA technical note on the inclusion ofpressure dependent demand and leakage terms in water supplynetworkmodelsrdquoCivil Engineering Systems vol 2 no 3 pp 171ndash179 1985

[14] O Giustolisi D Savic and Z Kapelan ldquoPressure-drivendemand and leakage simulation for water distribution net-worksrdquo Journal of Hydraulic Engineering vol 134 no 5 pp 626ndash635 2008

[15] L Ainola T Koppel K Tiiter and A Vassiljev ldquoWater networkmodel calibration based on grouping pipes with similar leakageand roughness estimatesrdquo in Proceedings of the Joint Conferenceon Water Resource Engineering and Water Resources PlanningandManagement (EWRI rsquo00) pp 1ndash9MinneapolisMinnUSAAugust 2000

[16] Q Duan S Sorooshian and V K Gupta ldquoOptimal use of theSCE-UA global optimization method for calibrating watershedmodelsrdquo Journal of Hydrology vol 158 no 3-4 pp 265ndash2841994

[17] Q Y Duan S Sorooshian and V K Gupta ldquoShuffled complexevolution approach for effective and efficient global minimiza-tionrdquo Journal of Optimization Theory and Applications vol 76no 3 pp 501ndash521 1993

[18] E Todini ldquoOn the convergence properties of the differentpipe network algorithmsrdquo in Proceedings of the Annual WaterDistribution Systems Analysis Symposium (ASCE rsquo06) pp 1ndash16August 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

12 Journal of Fluids

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

Number of iterations

H(20)

Figure 14 Convergence history of head-pressure (119898) in node 20 forexample 4 (form 2)

leakage simulation are handled in a simple way accu-rate solutions can be found in pipes with zero flowsand it does not need an initial solution vector which must bechosen carefully in many other procedures if numerical con-vergence is to be achieved Furthermore the proposedmodeldoes not require any complicated mathematical expressionand operation Finally it can be concluded that the proposedmethod is a viable alternative optimizer that challenges othermethods particularly in view of accuracy

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

References

[1] M Collins L Cooper R Helgason J Kenningston and LLeBlanc ldquoSolving the pipe network analysis problem usingoptimization techniquesrdquo Management Science vol 24 no 7pp 747ndash760 1978

[2] S Elhay and A R Simpson ldquoDealing with zero flows in solvingthe nonlinear equations for water distribution systemsrdquo Journalof Hydraulic Engineering vol 137 no 10 pp 1216ndash1224 2011

[3] E Todini and S Pilati ldquoA gradient algorithm for the analysis ofpipe networksrdquo in Computer Applications inWater Supply vol 1of System Analysis and Simulation pp 1ndash20 JohnWiley amp SonsLondon UK 1988

[4] A C Zecchin P Thum A R Simpson and C TischendorfldquoSteady-state behaviour of large water distribution systems thealgebraic multigrid method for the fast solution of the linearsteprdquo Journal ofWater Resources Planning andManagement vol138 no 6 pp 639ndash650 2012

[5] M L Arora ldquoFlow split in closed loops expending least energyrdquoJournal of the Hydraulics Division vol 102 pp 455ndash458 1976

[6] M A Hall ldquoHydraulic network analysis using (generalized)geometric programmingrdquo Networks vol 6 no 2 pp 105ndash1301976

[7] P R Bhave and R Gupta ldquoAnalysis of water distributionnetworksrdquo in Alpha Science International Technology amp Engi-neering 2006

[8] E Todini A More Realistic Approach to the ldquoExtended PeriodSimulationrdquo of Water Distribution Networks Advances in WaterSupplyManagement chapter 19 Taylor amp Francis London UK2003

[9] J M Wagner U Shamir and D H Marks ldquoWater distributionreliability analytical methodsrdquo Journal of Water ResourcesPlanning and Management vol 114 no 3 pp 253ndash275 1988

[10] J Chandapillai ldquoRealistic simulation of water distributionsystemrdquo Journal of Transportation Engineering vol 117 no 2 pp258ndash263 1991

[11] J Almandoz E Cabrera F Arregui E Cabrera Jr andR Cobacho ldquoLeakage assessment through water distributionnetwork simulationrdquo Journal of Water Resources Planning andManagement vol 131 no 6 pp 458ndash466 2005

[12] A F Colombo and B W Karney ldquoEnergy and costs ofleaky pipes toward comprehensive picturerdquo Journal of WaterResources Planning and Management vol 128 no 6 pp 441ndash450 2002

[13] G Germanopoulos ldquoA technical note on the inclusion ofpressure dependent demand and leakage terms in water supplynetworkmodelsrdquoCivil Engineering Systems vol 2 no 3 pp 171ndash179 1985

[14] O Giustolisi D Savic and Z Kapelan ldquoPressure-drivendemand and leakage simulation for water distribution net-worksrdquo Journal of Hydraulic Engineering vol 134 no 5 pp 626ndash635 2008

[15] L Ainola T Koppel K Tiiter and A Vassiljev ldquoWater networkmodel calibration based on grouping pipes with similar leakageand roughness estimatesrdquo in Proceedings of the Joint Conferenceon Water Resource Engineering and Water Resources PlanningandManagement (EWRI rsquo00) pp 1ndash9MinneapolisMinnUSAAugust 2000

[16] Q Duan S Sorooshian and V K Gupta ldquoOptimal use of theSCE-UA global optimization method for calibrating watershedmodelsrdquo Journal of Hydrology vol 158 no 3-4 pp 265ndash2841994

[17] Q Y Duan S Sorooshian and V K Gupta ldquoShuffled complexevolution approach for effective and efficient global minimiza-tionrdquo Journal of Optimization Theory and Applications vol 76no 3 pp 501ndash521 1993

[18] E Todini ldquoOn the convergence properties of the differentpipe network algorithmsrdquo in Proceedings of the Annual WaterDistribution Systems Analysis Symposium (ASCE rsquo06) pp 1ndash16August 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of