optimal design of water distribution 1

7/31/2019 Optimal Design of Water Distribution 1

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Optimal Design of Water Distribution System by

Multiobjective Evolutionary Methods

Klebber T. M. Formiga1, Fazal H. Chaudhry1, Peter B. Cheung1 and Luisa F. R.Reis1

1So Carlos School of Engineering, University of So PauloAv. do Trabalhador So Carlense 400, C.P. 359, So Carlos - SP - Brazil 13560-250

{[email protected] [email protected] [email protected] [email protected]}

Abstract. Determination of pipe diameters is the most important problem indesign of water supply networks. Several authors have focused on the methodscapable of sizing the network considering uncertainty and other importantaspects. This study presents an application of multiobjective decision makingtechniques using evolutionary algorithms to generate a series of nondominatedsolutions. The three objective functions considered here include investmentcosts, entropy system and system demand supply ratio. The determination ofPareto frontier employed the public domain library MOMHLib++ and a hybridhydraulic simulator based on the method of Nielsen. This technique is found tobe quite promising, the nondominated region being identified in a reasonablysmall number of iterations.

1. Introduction

Water distribution networks are hydraulic systems consisting of components suchas: pipes, pumps, valves and reservoirs among others, in order to supply water indesired quantity and quality within preestablished pressure limits. These systems canbe represented by a graph in which the nodes may symbolize points of consumptionor sources and the links are the pipes, pumps or valves.

The planning process of water supply networks, in general, consists of three steps:definition of the layout, design of the components and the systems operation. Thispaper deals with the design phase which has been the object of the study for morethan 30 years by researchers from various disciplines. One considers a waterdistribution network composed of pipes, nodes and circuits where the topology and

topography are known, water supply is guaranteed by one or more sources and wheredemands at various node points are assumed known. The classical design problem ofsuch systems involves finding the least cost network that meets the required demandssatisfying some constraints. Within this framework, various researchers developeddesign procedures making use of different optimization methods such as LinearProgramming ([1], [2]), Nonlinear Programming ([3]-[4]), and Genetic Algorithms([5]-[6]). The problem posed in these studies had as unknowns pipe diameters invarious links and constraints were placed on pressures.


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2 Klebber T. M. Formiga et al.

The above design problem dealt with by classical optimization methods faces twodifficulties [7]: 1) the decision involves pipe diameters, pumps and valvesspecification which are discrete in nature and must me chosen among sizes which arecommercially available; 2) any realistic formulation of the problem involvesnonlinear and nonconvex equations. Further, it is observed that the solutions obtainedby these methods may not be feasible in practice as they tend towards branchingnetworks, instead of looped network, choosing null pipe diameters and discharges.

In an effort to circumvent these difficulties, some researches ([8]-[11]) introducednew constraints, such as reliability, into the design problems. The use of traditionalreliability measures as constraints in the optimization of networks, particularly thoseof medium and large sizes, did not prove to be fruitful despite the addedcomputational complexity of exponential nature. In the search for a simple andefficient reliability measure, heuristic concepts ([12]-[15]), such as the concept ofentropy of system, have been experimented with good results [15].

The traditional optimization problems look for a single optimal solution. However,this search considering a single aspect of design may not be convenient for watersupply systems. For example, when one optimizes such a system, one obtains a leastcost configuration. However, there may be a layout with slightly higher cost but ahigher reliability. Thus, a more convenient solution may be non-optimal with respectto cost but more advantageous with respect to other aspects.

An alternative is to consider this problem in the multiobjective framework whereinthe evolutionary techniques have differentiated themselves by their capacity to dealwith a variety of problems. This paper evaluates the use of multiobjectiveevolutionary methods in the design of water supply systems.

2. Hydraulic Model

The hydraulic analysis of a pipe network determines two kinds of unknowns:hydraulic heads at the nodes (hi) and discharges in various pipe reaches (qj). Let therebe a pipe network composed ofn nodes, nrsources nodes (reservoirs and tanks) and mpipes connecting the nodes. The hydraulic head loss, between two nodes i andj, canbe expressed as:

= ii1,ij2,ij qKhh (1)

in which Ki is the hydraulic resistance coefficient of link i, and is the exponent ofthe head loss formula.

Mass conservation in the system is given by:

n1j0Qqafm

1ijiijj K===

= (2)

where Qj = demand concentrated at the nodej, aij = 1 if the discharge is towards nodej, aij = -1 if discharge is away from node j, and aij = 0 if the nodes i and j are notconnected.

In the case of solution in terms of node heads, equation (2), can be written as:


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Optimal Design of Water Distribution System by Multiobjective Evolutionary

Methods 3

( )n1j0Q

K

hhaf

m

1ij/1

i

/1

1,ij2,ij

ijj K==

= =

(3)

The incidence matrix (m x n+nr) formed by the elements aij can be divided intwo submatrices A (m x n) and Ar (m x nr). The first matrix corresponds to theinterior nodes and the second to the source nodes.

The system of equations in (3) is nonlinear of order n. Iterative numerical methodssuch as Newton-Raphson (N-R) or its variations can be employed to solve this systemfor nodal heads.

2.1. Newton-Raphson Method

The N-R method for solution of nonlinear system of equations can be described as,

kk1k XXX +=+ (4)

in whichXis the vector of variablesxin iteration kand Xkcan be obtained from theexpression:

kkk FXJ = (5)

where the Jacobian matrix function is given by,

kxxi

i

k x

f

J=

= (6)

The system for finding the hydraulic characteristics of the network can beexpressed as:

kk1k HHH +=+ (7)

The Jacobian of the function f in (3) is given by,

AC'AJ Tk = (8)

where,

=

/1m

1a1

1,mj2,mj

/12

1a1

1,2j2,2j

/11

1a1

1,1j2,1j

K

hh1,,

K

hh1,

K

hh1diagC' K (9)

Putting

C=C (10)


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one obtains,

CAA1

JT

k =

(11)

The vector Fk is expressed mathematically as [16]:

])HAC(AHAQ[F rrkT

k ++= (12)

Hrbeing the vector of reservoir elevations.Thus the equation (7) becomes:

])hAC(AhA[QCA]A[hh rrkT-1T

k1k ++=+ (13)

3. Objective Functions

This study considers three objectives, namely, the network cost, system entropy,and the water supply capacity of the network. Water supply capacity was incorporatedinto the problem to avoid any kind of constraint. In the order words, the constraintsare transformed into objectives.

3.1. Network Cost

The cost function includes expenses for the main hydraulic components of thewater distribution system. These are divided into: investment costs and operationcosts. The investment costs cover the acquisition costs and the costs of installation ofthe pipes, pumps and reservoirs. The operation costs are expressed in terms of energyconsumption by motor-pumps. These costs are distributed along the useful life of thenetwork, thus requiring the use of present value factor.

This paper considers the minimization of investment costs expressed in terms ofdiameters. Thus, the first objective function is:

==

=

m

1iii1 L)D(CTMin]Cost[MinF

(14)

where Costrefers to the investment cost, CT(D)i express the unit cost associated withthe pipe reach i with diameter Di andLi is the length of reach i.

3.2. Entropy

The basic requirement of a looped water distribution network is that water demandat a given node is met not only by a preferential path, but also through a number ofalternative paths directly connected to this node. Such paths would transport water indifferent amounts to meet the nodal demands. However, in order to increase thereliability of the network, it is desirable that this distribution of flow be as even as


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Methods 5

possible. If the distribution were very uneven, the failure of a reach that supplies thelargest discharge to the node would cause a major impact on the water supply in thenetwork.

Consider a network of N nodes. Let the entropy (or redundancy) of node j be Sj.The axioms proposed by [17] that make possible the calculation of reliability of waterdistribution system on the basis of entropy are:1. Sj will be a function of the discharge fractions X1j, X2j, X3j, ... Xn(j)j, given that:

1X)j(n

1iij =

= (15)

where n(j) is the number of pipes that are incident on node j;

2. Sj will be zero ifn(j) is 1;3. for given fixed number of nodes j, the redundancy will be maximum when thedischarges in the incident pipes are equal.4. the value ofSj is incremented by an increase in n(j) which implies that, for equaldischarges, redundancy can be increased by increasing the number of incident pipes.

There are many functions that help express entropy. However, we prefer thedefinition in [17], given that it produced consistent results for different types ofnetworks (communication networks [17], electrical networks [18], hydraulic networks[12] ). That expression for entropy is:

=

=I

1iii )Pln(P

K

S

(16)

where K is an arbitrary positive constant generally taken to be 1, Pi is a systemparameter andIis the number of subsystems.

This entropy function is the basis of all other functions used in the water supplysystems ([12]-[15]). It can be seen that this equation satisfies all the axioms.

To begin with, it is necessary to define the parameter P for the network. Let thisnetwork be composed of N nodes which are its subsystems. For a given demandpattern, let the ith pipe incident on nodej transport discharge qij then,

j

ijij

Q

qX = (17)

where Qj is the sum ofn(j) discharges incident on nodej.The parameterXij represents the relative contribution of pipe i to the total discharge

that passes through node j and serves as a measure of its relative capacityincorporated in the entropy function. This measure is an indicator of the potentialcontribution of a pipe to the systems possibility of meeting the extra demandimposed by the failure of another pipe in this system. Thus, the entropy for the node jcan be written as:

=

=

)j(n

1i j

ij

j

ijj

Q

qln

Q

qS (18)


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Maximizing the function Sj in equation (18) is equivalent to maximizingredundancy at the node j because the highest value of Sj will happen when all qijvalues at j are equal, which is desirable discharge distribution. The expression for Sjin equation (18) gives entropy of one node only. The development of entropyequation for whole network follows similar procedure.

One might think that the entropy of a system is the sum of entropies at all thenodes in the network. However, this description does not take into account thedischarge carried by given pipe in the general context because, in the event of afailure, it is not just the pipes adjacent to the node that will be exacted. Thus, theimportance of a pipe with respect to the total discharge, and not just relative to thedischarges incident on the node, is relevant to the determination of network entropy.

Let Qo be the sum all the discharges incident on all the nodes of the network for a

given demand pattern. It is important to observe that the value ofQ

o shall be equal toor greater than the total demand on the network (QT), in view of the fact that a certainportion of water circulates through more than one pipe. Thus, the parameter P used tocompute network entropy should be qij/Qo. This definition leads to the followingexpression for entropy of the system [12]:

= =

=

N

1j

)j(n

1i o

ij

o

ij

o

j

Q

qln

Q

q

Q

QS)

(19)

This expression can also be expressed as:

==

=

N

1j o

j

o

jN

1jj

o

j

Q

Qln

Q

QS

Q

QS)

(20)

Due to the presence of the logarithm, the entropy of the system presents very closevalues. We opted for working with the exponential of the value of the entropy of thenetwork. Then, the second objective function will be:

= S2 eMaxF

(21)

3.3. Demand Supply Ratio

The amount of water supplied by a given node depends upon the pressure at thatnode. In order to establish a correspondence between pressure and demand, it is

necessary to determine a functional relationship between these two variables. Thedescription used for this purpose is that proposed by [19], and modified by [11], is asfollows.

minjj

demj hhse0Q


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Methods 7

des

jj

req

j

dem

j hhQQ >= se

Here, Qjdem is the demand met at nodej, Qj

req is the demand required at the nodej,hj

des is the hydraulic head required at the nodej to satisfy the demand completely, hjmin

corresponds to the zero demand satisfaction and j is the exponent associated with thenodal head set equal to 2.

On the basis of discharge supplied at the node, one can establish a function thatdetermines the failure in the system. Thus F3, the third objective function, is themaximization of average demand supply ratio which was proposed by [10], as,

=

=

n

1i

dem

j

reqj

3

Q

Q

n

1MaxF (23)

4. Multiobjective Evolutionary Methods

The multiobjective evolutionary methods appeared in the middle of the 1980sdecade starting with the work of Schaffer [20]. Since then dozens of new suchmethods have been proposed in the order to obtain the Pareto frontier for diverseproblems.

This study employs the method NSGAII - Elitist Non-Dominated Sorting GeneticAlgorithm - [21] based on elitism implementing the code developed by Andrzej

Jaskiewicz [22] in the MOMHLib++ - Multiobjective Methods Metaheuristic Libraryfor C++.The crossover method BLX- [23], was used with crossover probability 1. The

Random Mutation [24] was adopted with rate of 5%.

5. The Problem

The NSGAII multiobjective method was applied to two test networks used inliterature. The first is a two-loop fictitious network [1], and the second is Hanoi-Vietnam water distribution system [25]. These networks were chosen with thepurpose of evaluating the adaptation characteristics of networks of different sizes.

5.1. Example Problem 1 - Two-loop Network

This network was proposed by [1] who applied Linear Programming to find itsleast cost solution. Later various authors ([2]-[7]) used this problem as a basis forcomparison of their formulations. This network is shown in Fig 1 and its hydraulicdata can be found in the original paper [1].


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1

23

45

67

12

3

4

5

7

8

6

1000 m

1000

m

1000m

1000m

1000m

1000 m

1000 m 1000 m

7

7

Legend:

Link

Node

Fig. 1. Two-loop network [1]

5.2. Example Problem 2 - Hanoi Network

The Hanoi Network presented in Fig.2 was studied by [25]. This is a medium size

network composed of 35 pipes and 32 nodes, including one reservoir. Its hydraulicdata are available in the above paper.The original data limited the diameters to 1000 mm (40 in) size. This diameter is

considered insufficient to transport the design discharge of about 5.5 m/s, as velocityin this pipe would be greater than 7 m/s, which is much higher than the usualmaximum tolerance of 3 m/s. Thus, the diameter upper bound of 2000 mm (80 in)was used in this study.

5.3. Performance Measures

According to [26], at least two types of performance measures should be used inthe multiobjective analysis. The first measure considers the proximity of the

calculated solution to the Pareto region. The second evaluates the manner ofdistribution of solutions along the Pareto frontier.

5.3.1 Measure of Proximity of Solutions to the Pareto Front. The method used forthis measure was proposed by [27]. This measure is well-suited to the comparisonbetween 2 sets of solutions. Expressed as C(A,B), it determines the proportion ofsolutions ofB that are dominated by the setA. It is given by,


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Methods 9

{ }B

ba:Aa|Bb)B,A(C

p= . (24)

The value ofC(A,B) shall be 1 if all the members ofB are dominated byA and 0, if nomember ofA dominates B. As the measure is not symmetrical, it follows that, inmajority of the cases, C(B,A)1- C(A,B), then, the calculation of both C(A,B) andC(B,A) becomes necessary.

1

2 3 4 5

6

7

8

9

10

1112

1314

16

15

17

18

19

20

222728

2

1

29

30

30

31

23

24 25 26

21

32

3 4 5

6

7

8

9

10

11

12

131415

16

17

18

19

20

21

22

23

24

25

26 27 28

2930

32

33

33

34

Legend:

Link

Node

Fig. 2. Hanoi Network [24]

5.3.2. Measure of Diversity among Non-Dominated Solutions. The measure of thediversity of solutions, proposed by [28], is calculated on the basis of relative distance

between the solutions expressed as,

( )=

=B

1i

2

iB ddB

1S (25)

where di is the lowest Euclidean distance from the nearest solution, d is the average ofdi values and |B| is the number of solutions in the setB.


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6. Results

To study the influence of the population size and the number of generations on thenondominated solutions, a 30-population set was generated by NSGAII method fordifferent populations sizes (50, 100, 250 and 500) and generations (50 and 100).

Table 1. Average run time (s) for various sizes of initial solutions and generations

Configuration Number of InitialSolutions

Number ofGenerations

Number of SolutionsGenerated

Network 1 Network 2

1 50 50 2500 23 632 50 100 5000 46 1123 100 50 5000 47 113

4 100 100 10000 89 2025 250 50 12500 121 2886 250 100 25000 233 5217 500 50 25000 251 6108 500 100 50000 488 1253

The average processing time for each type of generated set of solutions using anAMD Athlon Dual 1800+ is presented in Table 1. The average run time per evaluatedindividual was 0.009s for Two-loop network and 0.025s for Hanoi network showing anearly linear relation between number of solutions sought and processing time in bothnetworks.

In case of network in Example Problem 1, some solution results are similar to theoptimal values considering single objective F1 only. Savic and Walters [7] found theoptimal value of $419.000, considering one diameter in a pipe reach, for F3=1. This

result corresponds to a low value of entropy (F2) of 5.13. For slightly higher solutioncost (F1), as compared to the above solution, a much higher value for F2 is indicated,as for example the solution: F1 = $421.000 (only 0,5% higher) and F2 = 5.84 (14%higher).

Fig. 3 presents a set of nondominated solutions found for configuration #8. It isseen that the relation between the functions F1 and F3 has a much concentrateddistribution of the solutions along the Pareto frontier as compared to the relations thatcontain F2. This happens because F1 and F3 are directly proportional. In others works,more expensive the network, greater is its tendency towards meeting demands.However, entropy does present a similar relation to cost especially for its intermediatevalues. One can also observe that for large capacity of demand satisfaction (F 3), equalto 1, entropy does not present large values which are of the order of 5.9. There is aninverse relation between F2 and F3 for large values of F2.

In the Example Problem 2 (Fig. 4), one observes, in general, the same behavior asabove for Example Problem 1. However some differences can be noted as theobjective functions F1 and F3, now show a much neater relationship amongthemselves. Also, one can observe that as entropy, and consequently reliabilityincreases, the network becomes more expensive. Another important point is that themaximum values of entropy now occur corresponding to the maximum demandsupply ratio.


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Methods 11

Fig. 3. Scatter-matrix of a generated set of nondominated solutions after 100 generations inExample Problem 1 starting with an initial set of 500 individuals.

The difference in the behavior of entropy in the two problems is due to differentdegrees of redundancy in the two networks, given that the network in ExampleProblem 2 has higher number of loops and, at the same time, has greater capacity toface pipe failures.

The results of comparison of generated solutions in terms of the measure ofproximity to the nondominated frontier are presented in Fig. 5 as Boxplots. Themiddle line represents the medium value and the upper and lower edges of therectangles indicate the upper and lower quartiles, respectively, of the distribution ofsolutions. The upperworst and lowerworst lines indicate the maximum and theminimum values in the sample.

The results in Fig. 5 show that, in Example Problem 1, the initial population sizehas greater importance than the number of generation. This is obvious when onecompares C values of the solutions for the same number of evaluations (2x3 and 6x7).This happens due to the fact that, in generation 50, the population presents higherdegree of maturation because when one doubles the number of generations, there isno significant evolution of the solutions. This becomes evident when one comparesthe solutions obtained with the same population size.


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Fig. 4. Scatter-matrix of a generated set of nondominated solutions after 100 generations inExample Problem 2 starting with an initial set of 500 individuals.

In Example Problem 2, one observes (Fig. 6)that the number of generations is asimportant a factor as the population size, as shown clearly in the comparison ofconfigurations 6 and 7. Nearly half of the final solutions of the sets 7 are dominated atleast by one solution of the sets 6. Once again, it is observed that the degree ofevolution of the population is relevant. The 50th generations in Hanoi networkproblem, are not sufficiently mature as can be seen from the comparison of thesolutions corresponding to initial population sizes and different number ofgenerations.

The measure of performance for the problems under study points to a morepredictable behavior. It is seen that the initial population size has greater importancefor the diversity of the final population.

7. Conclusions

This paper employs an elitist multiobjective evolutionary method (NSGAII) for thedetermination of the nondominated region, considering the objectives ofminimizations of hydraulic network costs and maximization of entropy and capacityof network to meet water demand. The MOMHLib++ Library of Andrzej Jaskiewiczand a hydraulic simulator based on the method of Nielsen [16] were used to model the


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Methods 13

multiobjective problem for two classic literature networks: a two-loop network andthe Hanoi network.

Fig. 5.Values of the metric C(A,B) for different initial population and generation sizes. Thesolution A and B refer to the solutions along the rows and columns respectively.

Fig. 6. Values of the measure of diversity of the nondominated solutions for different initialpopulation and generation sizes.

It was found that the two-loop network problem shows greater sensitivity to theevolutionary parameter of initial population size, whereas the Hanoi network problemis more sensitive to the number of generations. Thus it is concluded that the numberof generations is important to a certain point where there is no significant change in


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the Pareto frontier. When the number of generations reaches this point, the initialpopulation size becomes more relevant.

In order to obtain satisfactory results using multiobjective evolutionary methodsfor water distribution networks, it is important to choose appropriate size of initialpopulations and the number of generations necessary for arriving at a stable Paretofrontier.

Acknowledgements

The present paper has resulted from current research grant Tools for the Rationaland Optimized Use of Urban Water (CT-HIDRO 01/2001 - 550070/200-8) funded

by Brazilian National Research Council (CNPq). The authors wish to express theirgratitude also to FAPESP and CAPES for concession of scholarships to the first andthird authors of this work, respectively. We are grateful to anonymous reviewers fromEMO 2003 whose comments greatly improved the manuscript.

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optimal design of water distribution 1

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