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Water Resour ManageDOI 10.1007/s11269-010-9701-1

Effect of Breakage Level One in Design of WaterDistribution Networks

Mohammadjafar Soltanjalili · Omid Bozorg-Haddad ·Migual A. Mariño

Received: 7 January 2010 / Accepted: 11 July 2010© Springer Science+Business Media B.V. 2010

Abstract Design of water distribution networks (WDNs) that do not consider per-formance criteria would possibly lead to less cost but it could also decrease waterpressure reliability in abnormal conditions such as a breakage of pipes of the net-work. Thus, awareness of the situation of consumption nodes, by considering waterpressures and the amount of water that is being supplied, could be an effective sourceof information for designing high performance WDNs. In this paper, Two-loop andHanoi networks are selected for least-cost design, considering water pressures andthe amount of water supplied on each consumption node under breakage levelone, using the honey-bee mating optimization (HBMO) algorithm. In each stateof design, a specific pressure is defined as the minimum expected pressure under

M. Soltanjalili (B) · O. Bozorg-HaddadDepartment of Irrigation and Reclamation, Faculty of Agricultural Engineeringand Technology, College of Agriculture and Natural Resources, University of Tehran,Karaj, Tehran, Irane-mail: [email protected]

O. Bozorg-Haddade-mail: [email protected]

M. A. MariñoDepartment of Land, Air and Water Resources, University of California,139 Veihmeyer Hall, Davis, CA 95616-8628, USAe-mail: [email protected]

M. A. MariñoDepartment of Civil and Environmental Engineering, University of California,139 Veihmeyer Hall, Davis, CA 95616-8628, USA

M. A. MariñoDepartment of Biological and Agricultural Engineering, University of California,139 Veihmeyer Hall, Davis, CA 95616-8628, USA

M. Soltanjalili et al.

breakage level one which holds the pressure reliability in the considered range. Also,variations of some criteria such as reliabilities of pressure and demand, vulnerabilityof the network, and flexibility of the design are analyzed as a tool for choosing theappropriate state of design. Results show that a minor increase in the cost of designcould lead to a considerable improvement in reliabilities of pressure and demandunder breakage level one.

Keywords Water distribution networks · Design · Performance criteria ·Honey-bee mating optimization (HBMO) algorithm · Breakage level one

1 Introduction

Traditionally, the optimal design of WDNs considered minimization of economiccost. Nowadays, however, minimization of design cost is considered as the objectivefunction of a WDN design problem and diameters of the network’s pipes as itsdecision variables. In a discrete optimization problem with 10 locations for the pipesand 10 commercial diameters available for each pipe, the decision space for such aproblem would be 10

∧10 different states.

It is clear that searching in such an expanded discrete decision space, without usingan optimization tool is not possible. In addition, calculating a solution in a reasonabletime is an important factor. Linear programming (LP) and nonlinear programming(NLP) are optimization tools capable of determining optimal solutions. Alperovitsand Shamir (1977), Quindry et al. (1981), Goulter et al. (1986), Kessler and Shamir(1989), and Fujiwara and Khang (1990) used LP and NLP to design WDNs.

The time-consuming process of direct search methods and the inefficiency ofgradient methods to solve discrete problems decrease the efficiency of traditionalalgorithms such as LP and NLP. Thus, evolutionary algorithms, especially geneticalgorithm (GA), have been used to solve optimization problems. Simpson et al.(1994) used the GA to solve discrete and nonlinear problems. Pipe diameters arediscrete variables which cause more application of the GA. Dandy et al. (1996), Savicand Walters (1997), and Montesinos et al. (1999) used different methods based onthe GA to solve WDN problems. Cunha and Sousa (1999) used simulated annealing(SA) algorithm for optimal design of WDNs (Hanoi and New York networks).Lippai et al. (1999) applied Opt-Quest software in the optimal design of the NewYork WDN. Geem (2005) proposed harmonic search (HS) algorithm in the designof two benchmark problems (Two-loop and Hanoi networks) in WDN design.

Walski (2001) indicated that concentration on cost minimization without usingperformance criteria, such as reliability of supplying demand or nodal pressure andnetwork reliability are important reasons for the inefficiency of proposed models inthe design of WDNs. Thus, design of WDNs should include minimization of projectcost subject to supplying acceptable pressure. Farmani et al. (2005) considered theeffects of different types of failure, such as pump being out of service and pipebreakage, on the performance of WDN and also the influence of those considerationson the reliability and the cost of design. They stated that reliability in a WDN is theprobability of meeting a desirable operation in the system, as it had been expected inthe designing period. Fujiwara and De Silva (1990) proposed a method based oncost minimization considering a specific reliability. Todini (2000) considered cost

Effect of Breakage Level One in Design of Water Distribution Networks

minimization and system flexibility maximization as objective functions and theoptimal trade-off between those two objectives was found. Prasad and Park (2004)modified Todini’s (2000) criteria and reported that an increase in flexibility criteriais not enough reason for increasing reliability criteria.

In recent years, simulation of social behavior of insects such as ants and bees hasbeen developed as algorithms to solve optimization problems. Afshar et al. (2007)used the honey-bee mating optimization (HBMO) algorithm to solve nonlinearconstrained and unconstrained problems. They showed the capability of HBMO indetermining near-optimal solutions compared to GA and considerable proximityto optimal solutions compared to Lingo 8. Bozorg Haddad et al. (2008) used theHBMO algorithm to extract an optimal policy for rehabilitation of WDN for periodsof 30 and 100 years. The HBMO algorithm was also used by Ghajarnia et al. (2009) todetermine the least-cost design of WDN with respect to different levels of reliabilityduring an operational period. They also identified critical nodes of the network byconsidering different nodal pressure reliabilities.

In this paper, the HBMO algorithm is used as the optimization tool for optimaldesign of Two-loop and Hanoi networks considering pressure reliability conditionswhen breakage level one occurs. In most existing investigations, the reliabilitycriterion is calculated after least-cost design. In this paper, however, the least-costdesign of WDNs subject to pressure reliability is considered as the optimizationproblem. Results show that an acceptable increase in the amount of cost could leadto a considerable improvement in the reliability of WDNs.

2 Breakage, Its Levels and Their Probabilities in WDNs

Breakage in WDNs means failure of pipes in a period of time which is specifiedto their repairs. Different levels of breakage show how many pipes are breaking inone period simultaneously. Breakage level zero means that no breakage occurs in aperiod irrespective of the useful life operational period of the WDN. Breakage levelone means that in a period of time just one breakage happens, irrespective of totalnumber of breakages in the useful life operational period of WDN. Thus, breakagelevel two means that two breakages happen simultaneously in one period. UrbanWDNs are mostly designed in looped shape. Usually in looped WDNs, includingcase studies considered in this paper, there are some nodes which are terminatedwith only two pipes. Breakage level two may cause them to be out of any supplyat all, because no pipes are terminating and supplying them. As the result of thissituation, the hydraulic simulator would not be able to analyze the network anymore,unless the solitary nodes are eliminated from the process of hydraulic simulation. Asstated previously, the aim of this paper is to strengthen the network so as to be ableto satisfy the demands of all nodes in a breakage situation. To be able to analyzethe network even under breakage level two and have an economic approximationof the cost which it imposes to the design of the network, this paper considers theanalyzable states of breakage level two in a Two-loop network to design the networkto be reliable with respect to simultaneity of every analyzable breakage level two.The statement “analyzable breakage level two” means the states of simultaneity oftwo breakages which do not eliminate the connection of any consumption node withthe pipes of the network. These analyzable states are shown in Table 1.


Table 1 Analyzability and non-analyzability of simultaneous breakage of any couple of the net-work’s pipes

Number Number of the network’s pipes

of the 2 3 4 5 6 7 8network’spipes

2 NA** A* A* A* NA** A*3 A* A* A* NA** A*4 A* A* A* A*5 NA** A* NA**6 A* NA**7 A*8

A* Analyzable, NA** Non analyzable

It has been determined by a statistical calculation that the occurrence of breakagelevel two is unlikely to happen in comparison with breakage level one. Equation 1achieves the probability of occurrence of breakage with different levels:

P r =(

PL

) (B

Y × P

)L (1 − B

Y × P

)P−L

(1)

in which: Pr = probability of occurrence of breakage level L; L = number ofsimultaneous breakages (level of breakage); P = number of the network’s pipes;B = total probable number of breakages in the network during an operationalperiod; and Y = number of periods for occurrence of breakage in the network duringoperational period, which is achieved dividing the length of the operational periodby the time required to repair each breakage. Figure 1 shows changes in probabilityof occurrence of breakage with different levels considering different total numberof breakages in the network during an operational period with Y = 25 and P = 7.It is seen in the figure that in a few number of breakages, the probability of facingperiods which do not experience any breakages (level zero) is high while there isa low probability of facing periods in which breakage level one happens. Also, byincreasing the total number of breakages in an operational period of the network, asthe level of breakage increases, the difference between its occurrence probabilityand the probability of breakage level one also increases. For a total number ofbreakages less than 15, the occurrence probability of breakages level 3 and higherare achieved nearly zero and also there is a considerable difference between theoccurrence probability of breakage levels 1 and 2. Although moving ahead alongthe horizontal axis shows that the probabilities of breakage levels 1 and 2 are gettingcloser to each other, when one considers the useful life operational period of theWDNs, the parameter Y could have values much greater than 25. Thus, regardingEq. 1, an increase in the value of parameter Y would cause the difference betweenthe probabilities of breakage level one and higher levels to remain stable for a greaternumber of total breakages.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35 40 45 50Number of Breakages

Pro

babi

lity

Breakage Level 0 Breakage Level 1 Breakage Level 2 Breakage Level 3

Breakage Level 4 Breakage Level 5 Breakage Level 6 Breakage Level 7

Fig. 1 Changes in probability of occurrence of breakage with various levels, considering totalnumber of breakages in the network during operational period with Y = 25 and P = 7

3 HBMO Algorithm

The HBMO algorithm is an evolutionary algorithm based on bee behavior that worksas a hybrid tool including GA, SA, and local search (LS) algorithms which improvethe capability of the preceding individual algorithms by combining them into oneprocedure. The HBMO algorithm includes three repetitive stages: (1) selection; (2)reproduction; and (3) improvement. Naturally, in each hive there is a queen which isin fact the best bee of the colony. In the mating season, the queen flies to a properplace and attracts the attention of drones with a special dance. Afterwards with afast sudden move, it flies towards the sky, which is known as mating flight, and inthis manner only some of the best drones are able to mate with the queen. Afterthe mating flight, the queen returns to the hive and starts laying eggs. The workersthen breed the broods and the best broods are selected for becoming a new queen.Selected broods are fed by a special nutrient called royal jelly. After growing up,the best nurtured brood will replace the present queen provided that it is better thanthe queen. Hereby successive generations of honey-bees in the hive will move towardevolution. In this algorithm, firstly a random set of decision variables is generated andis assumed to be the queen. Other random answers are then generated and the bestones are combined with the queen (queen generation). Afterwards, the new answers(broods) are improved by some predefined functions, and the best answer is thenreplaced by the queen if it is better. One of the advantages of the HBMO algorithmis its ability to weighing the operators and functions based on their performanceand desirability observed in previous generations. That is to say, an evolutionaryfunction which has done well in the previous generation will be allowed to generatea bigger portion of the initial population of the next generation. This rule has the


Start

Define algorithm and model input parameters

Generating a set of initial random solutions, keeping the best, based on the objective function, as the queen and the other ones as the drones

Simulated annealing

(The limits of mating flight)

Utilizing previous trail solutionsSimulated annealing (generatingnew trail solutions and replacing

the previous ones with them)

Generating new solutions (breeding process) using cross over operators and heuristic functions (workers) between the best solution (queen) and trail solutions according to

their fitness value

Improving the newly generated set of solutions (feeding selected broods and queen with the royal jelly) utilizing heuristic functions and mutation operators

Sorting new solutions in accordance with their fitness value,selecting the best new solutions and the best new trail solutions

Is the new best solution better

than the previous one?

Substituting the best solution

Examination of

termination criterion

Finish

No

No

No

Yes

Yes

Yes

Fig. 2 Computational flowchart of the algorithm

same justification about the values of the algorithm’s parameters. That is, if the valueof the probability for the crossover function in a generation improves the solutionand objective function, then the algorithm must change their values for the next


generation using previous nearby values. Thus, initial values of parameters do nothave a great influence on the progress of the algorithm. The computational flowchartof the algorithm is presented in Fig. 2. More information about HBMO algorithm isincluded in Bozorg Haddad et al. (2008).

4 Optimal Design Model of WDNs in Breakage Level One Condition

In this model, pipe diameters are considered as decision variables to be selectedfrom commercially available diameters. The objective function is the total cost ofnetwork pipes. Constraints include the pressure provided in each consumption nodeafter hydraulic analysis of the set of solutions in which every node has to be at leastmore than the minimum pressure requirement under normal conditions and alsomore than the minimum expected pressure when breakage level one occurs. Thereare two penalty functions in the optimization model: (1) one related to providing lesspressure of PMin under normal conditions and (2) another related to less pressure ofPexp when breakage level one happens. Penalty functions are defined so as to preventthe best infeasible solution to be better than any feasible one. Thus, the optimizationmodel can be expressed as:

Min. OF =NI∑i=1

Costi + PF1 + PF2 (2)

Costi = Ci(Di, Li) (3)

PF1 =N J∑j=1

K × (Pmin − Pj + 1)2 , i f P j < Pmin (4)

PF2 =N J∑j=1

K × (Pexp − Pj + 1)2 , i f P j < Pexp (5)

in which: Costi = cost of ith pipe; PF(1 and 2) = penalty functions for provid-ing respectively minimum pressures in normal and breakage level one conditions;C(Di, Li) = cost function of ith pipe with diameter of D and length of L; Pj =pressure of jth node; Pexp = minimum expected pressure in breakage level oneconditions which varies under each state of design; NJ = number of consumptionnodes; and NI = number of pipes. Hydraulic calculations are done using EPANET2software (Rossman 2000), which was linked to the HBMO algorithm. The algorithmwas run with 110 number of drones in the hive (sample solutions), and allowed to runfor 5000 mating flights.

5 Performance Criteria in WDNs

Five criteria are defined that present performance of the system under different costsand conditions.


5.1 Reliability (Re)

This criterion contains the concept of providing minimum required pressure (Pmin)under different hydraulic simulation states and is calculated as:

R e = NS × N J − NP≺Pmin

NS × N J(6)

where NS = number of different states for design and consequently hydraulicsimulations of the network, which would lead to a set of pressures achieved for eachconsumption node of the network. The aforementioned states involve simulationsof looped designed network without considering pressure reliability under breakagecondition and reliable looped designed network under normal condition. Again, thepreceding reliable network under breakage of just one pipe could be any pipe ofthe network except pipe number 1. NP≺P min = total number of nodes in which theirpressures are less than the minimum required under all aforementioned designs andsimulations states.

The minimum pressure required is a fixed value for all the states of design. Onthe other hand, the penalty functions are added to the objective function in: (1)least-cost design when the pressure provided in each node is less than PMin and (2)reliable design of WDN when the pressure is less than Pexp under breakage level onecondition.

5.2 Vulnerability (Vu)

In breakage level one condition, some nodal pressure possibly could be provided lessthan PMin. Thus, vulnerability presents the maximum difference between providedpressures in all nodes and PMin among different reliable designs under breakagelevel one:

Vu = PMin − Min(P)

PMin(7)

where Min(P) = minimum pressure among all consumption nodes considering aspecific value for minimum expected pressure in each state of simulation (NS).

5.3 Flexibility (Fl)

This criterion involves both reliability and vulnerability of the system and is calcu-lated as:

Fl = R e × (1 − Vu) (8)

5.4 Hydraulic Benefit of Pressure (HBP)

This criterion was proposed by Carrijo and Reis (2004). To calculate HBP, twopositive values are considered as the minimum and maximum permissible pressuresfor each node. The pressure of each node and the pressure hydraulic benefit of it (ψ j)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90 100 110 120 130

Fig. 3 Changes in � with respect to changes in pressure

are then calculated with Eqs. 9 and 10. Finally, HBP would be the sum of ψ j in allconsumption nodes:

HBP =NJ∑j=1

ψ j (9)

ψ j ={(

P j−Pmin

Pmax−Pmin

)0.5i f Pmin < Pj < Pmax

0 i f P j < Pmin or Pj > Pmax

}(10)

in which Pmax is the maximum allowable nodal pressure in the network. The greaterthe amount of ψ j, the greater the reliability on node j. Figure 3 shows an increasingrelation between Pj and ψ j. If the existing pressure of a node falls outside of theallowable boundaries, its ψ j will be equal to zero. It seems not to be a right optionfor ψ j to increase at the same rate as the amount of Pj along the interval betweenminimum and maximum defined allowable pressures. This is because both very smalland excessive values for the pressure provided at nodes would affect reversely theperformance of WDNs. Thus, it is preferable to define HBP so that its maximumamount would be achieved between minimum and maximum allowable pressures.

5.5 Hydraulic Benefit of Demand (HBD)

Also proposed by Carrijo and Reis (2004), this criterion is similar to HBP and isweighed considering the amount of demand in each specific node. So, the more


demand in a node, the higher the priority of supplying the demand for the samenode.

HBD =N J∑j=1

(Pj − Pmin

Pmax − Pmin

)0.5

×

⎛⎜⎜⎜⎝

QD j

N J∑j=1

QD j

⎞⎟⎟⎟⎠ (11)

where QD j = demand of jth node.

6 Case Study

The Two-loop network is a benchmark network that is used as the first case studyin this paper. Alperovits and Shamir (1977) presented this simple network whichdoes not have any pump and storage tank. Figure 4 shows a schematic of the Two-loop network with eight pipes and seven nodes. Length of all pipes is 1,000 m andthe Hazen–Williams coefficient is assumed to be 130 for all pipes. The minimumrequired pressure in all nodes (PMin) is equal to 30 m-H2O and Pexp has discretevalues between 0 and 30 with 3 m-H2O as intervals.

123

45

7 6

7

12

3

4

58

6

150

160 165

155150

160

Reservoir

Node

Pipe

Elevation

Demand

100

120

330

100

270

200

Fig. 4 Schematic of two-loop network


Reservoir

Node

Pipe

Demand

Length

26

25

10051275

615900

865

525

550

500

940

10

11

1213

9

8

7

6 5 4 3

2

1

22

1415162726253231

30 24

29 28 23

17

18

19

20

21

1211

10

913141528273433

32

31 24

2930 23 20

16

17

18

19

8

7

6

543

221

22 1

560

525280 310370 170 805105

360 820 1345

60

850 1045290

360

930

485

890

130 725

1350

100

1350900 1150 1450

4

850

850

800

950

1200 3500

800 500550

2730

1750

800 400

2200

1500

500

2650

1230

1300

850 300 750

1500 2000

1600

150 860 950

Fig. 5 Schematic of Hanoi network

The Hanoi network in Vietnam is used as the second case study. This simplenetwork involves 34 pipes and 32 nodes without any pump and storage tank.Elevation of all nodes is zero and here the Hazen–Williams coefficient of all pipesis 130. Figure 5 shows a schematic of the Hanoi network.

7 Results and Discussion

In this paper, optimal design of WDNs (Two-loop and Hanoi) with the aim ofminimizing their total cost are designed so as to provide minimum required pressuresin normal conditions and minimum expected ones in breakage level one conditions.

7.1 Two-Loop Network

Under each reliable state of design, which guarantees minimum expected pressuresof 0, 3, ..., 30 m-H2O under breakage level one condition, the diameter of each pipevaries in a specific range (Fig. 6). For instance, it is expected to achieve relativelyless values of diameters for different pipes of the network under unreliable loopedstate of design in comparison with the reliable state of design which guarantees 30 m-


0

5

10

15

20

25

Tre

e sh

aped

WD

N

Unr

elia

ble

loop

ed

WD

N

0 3 6 9 12 15 18 21 24 27 30

Minimum Expected Pressure (m-H2o)

Dia

met

er (

Inch

es)

pipe 1 pipe 2 pipe 3 pipe 4 pipe 5 pipe 6 pipe 7 pipe 8

Fig. 6 Changes of each pipe’s diameter for various states of design of two-loop network

H2O in breakage level one condition. Table 2 presents statistical measures relevantto the mentioned variability for each pipe. For example, the value located at thejunction of row 2 (numbered “1”) and column 2 (minimum, in inches) illustratesthat under different states of design which are previously mentioned in detail, theleast achieved value for pipe number 1 is 18 in. This and other statistical parameterscalculated herein for each pipe, considering different design states, could give thedesigners a viewpoint that will be helpful to identify the more important pipes ofthe network considering the influence of the size of their diameter on the hydraulicperformance of the network. Because the aim of this paper is to achieve the least-costdesign subject to different levels of reliability (different design states), the resultinginformation could differentiate the effect of enlarging and reducing the diameterof different pipe sizes. Figure 6 shows changes of each pipe’s diameter for variousstates of design. In the first state (tree-shaped network on the horizontal axis),

Table 2 Statistical measures for diameters achieved for each pipe of two-loop network under variousreliable design states

Pipe Minimum Average Maximum ST. DEV Variance C.V. Median Modenumbers (inches) (inches) (inches) (inches) (inches)

1 18 20 20 0.93 0.87 0.05 18 182 16 20 20 1.62 2.62 0.08 16 163 14 18 18 1.35 1.82 0.07 16 144 10 16 16 1.96 3.85 0.12 12 125 12 14 14 0.93 0.87 0.07 14 146 10 14 14 1.75 3.05 0.12 12 107 14 18 18 1.40 1.96 0.08 16 168 12 16 16 0.89 0.80 0.06 14 14


the model is allowed to select a zero value for diameters. Thus, a least-cost designleads to a tree-shaped network. In the second state (unreliable looped network),the model is not allowed to select zero value for diameters and also the networkis not expected to provide discrete values of minimum expected pressures underbreakage level one condition. This has led to a looped network but unreliable toprovide specific amounts of pressure under breakage level one conditions. In bothmentioned states of design, providing minimum required pressure (30 m-H2O) hasbeen considered necessary. Other states guarantee providing minimum requiredpressure under normal condition and also minimum expected pressures 0 to 30m-H2O (each discrete value on the horizontal axis) under breakage level one con-dition. As it is seen in Fig. 6, diameters of most of the network’s pipes have been in-creased with respect to the rise in the amount of minimum expected pressure. This ismore obvious looking at the curves related to pipe numbers 4 and 8. Also, under twotree shaped and unreliable looped states, diameters of these pipes are zero and nearlyzero, respectively. Considering the next values for diameter of these pipes along thehorizontal axis, it could be inferred that increasing their diameter would lead to arise in the amount of reliability under breakage level one. On the other hand, whenthe minimum expected pressure in breakage condition is 21 m-H2O, diameters ofdifferent pipes are closer to each other in comparison with other states. This wouldpractically ease preparation and installation of the network’s elements. Figure 7shows the relation between performance criteria and total cost under differentdesign states. It shows the ascending trend of reliability and flexibility criteria anddescending trend of the vulnerability criterion with rise in the cost of network. Thereliability of the system in the state of 21 m-H2O as the minimum expected pressure

0

10

20

30

40

50

60

70

80

90

100

419

(unr

elia

ble

loop

ed W

DN

)

514

(Pex

p=0)

524

(Pex

p=3)

542

(Pex

p=6)

554

(Pex

p=9)

572

(Pex

p=12

)

610

(Pex

p=15

)

650

(Pex

p=18

)

680

(Pex

p=21

)

740

(Pex

p=24

)

790

(Pex

p=27

)

870

(Pex

p=30

)

Cost (×10^3 $)

Per

form

ance

Cri

teri

a (%

)

Reliablity Vulnerability Flexibility

Fig. 7 Relation between performance criteria and total cost under various design states of two-loopnetwork


500

600

700

800

900

0 3 6 9 12 15 18 21 24 27 30


Cos

t (×

10^

3 $)

Fig. 8 Relation between total costs with respect to rise in minimum pressure expected in two-loopnetwork

is equal to 0.85, which can be presented as an alternative design state. The rea-son is closeness of the diameters of pipes connecting to a consumption node. In thissituation, if one of the pipes were broken, the other could considerably cover itsportion of the node’s demand. Otherwise, if the larger one breaks, the smaller one isnot able to supply any critical node. Figure 8 presents the ascending relation amongthe total costs with respect to a rise in minimum pressure expected. It is seen hereagain that after point 21 on the horizontal axis, the change in cost curve is occurringwith steeper slope. Also, Figs. 7 and 8 show that an increase in the minimum expectedpressure from 0 to 21 under breakage condition has increased the cost of design from$514 to $680 × 10∧3 (33% rise in cost) and also has increased the reliability of thesystem by 15%. A cost increase of $ 190 × 10∧3 occurs between points 21 and 30on the horizontal axis of Fig. 8. This again confirms the desirability of this pointfor selecting diameters of the network’s pipes. Figures 9 and 10 respectively presentpressure and demand hydraulic benefit in all nodes under breakage of each pipesolely. It is shown in the figures that as minimum expected pressure increases, thevalue of these criteria increases as well. Also, it could be inferred from the figuresthat in each state of hydraulic simulation of the network, breakage of pipes 2 and3 have the most influence on the sum of these criteria in the network. This couldbe rationalized by considering the location of these pipes at the beginning of thenetwork. It is shown in Fig. 6, which introduces suggested diameters for differentpipes under each state of design, that in most of the reliable design states, suggesteddiameters for pipes 2 and 3 are close to each other. Because these two pipes play therole of distributors of the water delivered from the reservoir, and also consideringtheir nearly equal diameters, it would be expected that breakage of each of themwould make a serious hardship for the other one and will cause a shortage of pressureprovided on consumption nodes. Also, it is seen that in the most of reliable design


0

1

2

3

4

5

6

2 3 4 5 6 7 8

Network's Pipes Number

Hyd

raul

ic B

enef

it a

t A

ll N

odes

(P

ress

ure)

Pmin=0 Pmin=3 Pmin=6 Pmin=9Pmin=12 Pmin=15 Pmin=18 Pmin=21Pmin=24 Pmin=27 Pmin=30 Unreliable looped WDN

Fig. 9 Pressure hydraulic benefit in all nodes under breakage of each pipe solely in two-loop network

states, breakage of pipe 2 has a larger influence on reliability criteria of the network.This is because of the bigger diameter for this pipe suggested by the model. Whenpipe 2 does not exist, it will force pipe 3 to supply the network, while its diameteris smaller than that of pipe 2. Naturally, this would negatively affect demand and

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 3 4 5 6 7 8


Hyd

raul

ic B

enef

it a

t A

ll N

odes

(D

eman

d)


Fig. 10 Demand hydraulic benefit in all nodes under breakage of each pipe solely in two-loopnetwork


0

1

2

3

4

5

6

7

8

9

10

2 3 4 5 6 7

Number of Consumption Nodes

Hyd

raul

ic B

enef

it a

t ea

ch N

ode

(Pre

ssur

e)


Fig. 11 Sum of pressure hydraulic benefits in each consumption node with respect to breakage ofvarious pipes in two-loop network

pressure reliability criteria. Figures 11 and 12 show the sum of pressure and demandhydraulic benefits criteria in each consumption node with respect to breakage ofdifferent pipes. The values of pressure hydraulic benefit criterion in nodes 6 and7 and also in node 2 have been achieved more than in other nodes. The reasonwhich causes node 5 to be located at the top of the demand hydraulic benefit curve

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2 3 4 5 6 7


Hyd

raul

ic B

enef

it a

t ea

ch N

ode

(Dem

and)


Fig. 12 Sum of demand hydraulic benefits in each consumption node with respect to breakage ofvarious pipes in two-loop network


is the high value of pressure and supply in comparison with other nodes. Node 6 hasthe greatest portion of total demand of the network. But because of low pressuresprovided in this node in the most of design states (especially in the states that theirminimum expected pressures are less than 24 m-H2O), node 6 has been located at thelowest level in Fig. 12. Figure 13 shows the cumulative number of nodes in which thepressure provided in them is less than 30 m-H2O under breakage of different pipesare delineated for different design states. Thus, each curve is showing the resultsof its own design state with a specific value for minimum expected pressure or theminimum pressure which is guaranteed by applying that state of design. The curvesare cumulative for more clarity. Therefore, the vertical distance between each twoconsecutive curves shows the number of nodes with pressure provided less than 30 m-H2O, under the design state related to the upper curve. For example, the verticaldistance between two lowermost curves at point 2 on the horizontal axis equals 5 (10− 5). It means that at the state of design which guarantees 3 m-H2O in breakage levelone condition, in the condition of breakage of pipe 2, the number of nodes in whichtheir pressure provided would be less than 30 m-H2O is 5. The curve which is locatedbefore the uppermost one guarantees 30 m-H2O in breakage level one condition. Asit is clear from the figure, as we go down from this curve, the distance between curveshas an ascending trend. That means the less minimum pressure expected, the morenodes with the minimum pressure provided less than the minimum required one.

Reliable design regarding breakage level two in the Two-loop network has beendone for only one minimum expected pressure, which is the state of design thatguarantees the pressure equal to 30 m-H2O in breakage level two conditions. Asmentioned previously in Section 2, by neglecting the non analyzable occurrenceconditions of breakage level two in the two-loop network, the network is reliable tosupply the demand of all consumption nodes with 30 m-H2O pressure in analyzable

0

5

10

15

20

25

30

35

40

45

50

2 3 4 5 6 7 8


Cum

ulat

ive

Num

ber

of N

odes

wit

h L

ess

Pre

ssur

e P

rovi

ded

than

M

inim

um R

equi

red


Fig. 13 Cumulative number of nodes with pressure provided less than 30 m of water under breakageof various pipes, for various design states in two-loop network


Table 3 Statistical measures for diameters achieved for each location’s pipes of Hanoi networkunder various reliable design states


1 0 20 40 23 533 1 20 00 28 40 19 358 1 35 40

40 40 40 0 0 0 40 402 40 40 40 0 0 0 40 40

0 10 40 20 400 2 0 00 30 40 20 400 1 40 40

3 0 10 40 20 400 2 0 00 10 40 20 400 2 0 00 20 40 23 533 1 20 40

4 0 25 40 17 300 1 30 300 14 30 16 249 1 12 00 3 12 6 36 2 0 0

5 0 20 40 23 533 1 20 00 10 40 20 400 2 0 00 10 40 20 400 2 0 0

6 0 18 40 21 425 1 15 00 13 40 19 356 1 6 00 8 30 15 225 2 0 0

7 0 0 0 0 0 – 0 00 10 40 20 400 2 0 00 25 40 17 300 1 30 30

8 0 11 24 13 164 1 10 00 19 30 13 169 1 220 11 24 13 164 1 10 0

9 0 23 30 15 225 1 30 300 10 40 20 400 2 0 00 0 0 0 0 – 0 0

10 20 26 30 5 24 0 27 300 0 0 0 0 – 0 00 3 12 6 36 2 0 0

11 0 3 12 6 36 2 0 00 9 20 11 111 1 8 0

16 20 24 3 11 0 20 2012 0 16 24 11 117 1 20 20

0 5 20 10 100 2 0 00 3 12 6 36 2 0 0

13 0 10 40 20 400 2 0 00 8 30 15 225 2 0 00 15 30 17 300 1 15 30

14 0 23 30 15 225 1 30 300 0 0 0 0 – 0 00 10 40 20 400 2 0 0

15 0 20 40 23 533 1 20 400 10 40 20 400 2 0 00 10 40 20 400 2 0 0

breakages level two. Ignoring pipe number 1 which is the only connection betweenthe reservoir and the network, there are 21 states of breakage level two which ispossible to happen in the two-loop network. As it is stated in Section 2, some states


Table 3 (continued)


16 0 10 40 20 400 2 0 00 30 40 20 400 1 40 400 0 0 0 0 – 0 0

17 0 10 40 20 400 2 0 00 10 40 20 400 2 0 00 20 40 23 533 1 20 40

18 0 10 40 20 400 2 0 00 30 40 20 400 1 40 400 0 0 0 0 – 0 0

19 0 20 40 23 533 1 20 00 20 40 23 533 1 20 400 3 12 6 36 2 0 0

20 0 15 40 19 367 1 10 00 18 40 21 425 1 15 00 8 30 15 225 2 0 0

21 0 5 20 10 100 2 0 00 12 24 14 192 1 12 240 5 20 10 100 2 0 0

22 0 9 20 11 111 1 8 00 0 0 0 0 – 0 00 7 16 8 68 1 6 0

23 0 18 40 21 425 1 15 00 3 12 6 36 2 0 00 18 40 21 425 1 15 0

24 30 38 40 5 25 0 40 400 0 0 0 0 – 0 00 0 0 0 0 – 0 0

25 0 20 40 23 533 1 20 00 8 30 15 225 2 0 00 8 30 15 225 2 0 0

26 0 10 40 20 400 2 0 00 10 40 20 400 2 0 00 20 40 23 533 1 20 0

27 0 20 40 23 533 1 20 00 0 0 0 0 – 0 00 20 40 23 533 1 20 40

28 0 8 30 15 225 2 0 00 10 40 20 400 2 0 00 20 40 23 533 1 20 0

29 0 14 20 10 91 1 18 200 0 0 0 0 – 0 00 8 20 10 96 1 6 0

30 0 0 0 0 0 – 0 00 14 20 10 91 1 18 200 8 20 10 96 1 6 0

31 0 9 20 11 111 1 8 00 9 20 11 111 1 8 00 3 12 6 36 2 0 0


Table 3 (continued)


32 0 6 12 7 48 1 6 120 7 16 8 68 1 6 00 3 12 6 36 2 0 0

33 0 8 16 9 85 1 8 00 3 12 6 36 2 0 00 8 20 10 96 1 6 0

34 0 12 16 8 64 1 16 160 0 0 0 0 – 0 00 4 16 8 64 2 0 0

would lead to disconnection of some nodes of the network. Ignoring these statesof breakage level two, there would remain 15 states of breakage level two whichare analyzable. Thus, the two-loop network is designed in a reliable manner whichguarantees 30 m-H2O under analyzable breakage level two states and the cost ofdesign is $1,160.

7.2 Hanoi Network

Reliable design of the Hanoi network is impossible using available commercialdiameters. Thus, the model is allowed to select more than one pipe for each location,up to three pipes. If it is necessary to install more than one pipe between twonodes, it is assumed that they are installed in parallel. Table 3 presents statisticalmeasures relevant to diameters of pipes selected for different locations of thenetwork. Figure 14 shows changes of diameter of equivalent pipe for each location

0

20

40

60

80

100

120

Tree shaped WDN Unreliable looped WDN

0 10 20 30


Dia

met

er (

Inch

es)

Location 1 Location 2 Location 3 Location 4 Location 5 Location 6





Location 31 Location 32 Location 33 Location 34

Fig. 14 Changes of each location’s equivalent pipe diameter for various states of design of Hanoinetwork


of the network for various states of design. As it is seen in Fig. 14, diameters ofmost of the network’s locations have been increased with respect to the rise inthe amount of minimum expected pressure. This is more obvious looking at curvesrelated to the diameter of equivalent pipes for locations 1, 2, 14, and 31, thatshows a prominent increase in reliability of the network with respect to rise in theirdiameters, when breakage level one happens. On the other hand, when the minimumexpected pressure in breakage condition is 20 m-H2O, diameters of equivalent pipesare closer to each other in comparison with other states. This would practicallyease preparation and installation of the network’s elements. Figure 15 shows therelation between performance criteria and total cost for various design states.It is seen that the difference between reliability criteria under different states is atits uppermost level between unreliable state and reliable ones in comparison withthe difference between each two reliable states. Because in an unreliable state thenetwork has not been designed for breakage condition, facing this condition wouldshock the network and will cause lots of nodes to fail in supplying the demands andproviding required pressures. As it is expected with an increase in cost, the trendof flexibility and vulnerability are opposite. Considering the equations for flexibilityand vulnerability, low (nearly zero) pressures would cause the vulnerability criteriato reach an undesirable value of 1 and so the value of 0 (or nearly zero) for flexibilityof network under the two states of design which are unreliable looped network andminimum expected pressure of zero under breakage condition. Figure 16 presentsthe ascending relation between total costs with respect to rise in minimum expectedpressure. It is seen that the slope of the curve decreases after point 20 on thehorizontal axis. By considering Fig. 14, it is clear that the diameter of pipes has not

0

10

20

30

40

50

60

70

80

90

100

6.1 (unreliable looped WDN)

7.9 (Pexp=0) 8.2 (Pexp=10) 8.6 (Pexp=20) 8.7 (Pexp=30)

Cost (×10^6 $)

Per

form

ance

Cri

teri

a (%

)

Reliablity Vulnerability Flexibility

Fig. 15 Relation between performance criteria and total cost under various design states of Hanoinetwork


7.8

8.1

8.4

8.7

0 10 20 30


Cos

t (×

10^

6 $)

Fig. 16 Relation between the total costs with respect to rise in minimum pressure expected in Hanoinetwork

been increased considerably between points 20 and 30. Figures 17 and 18 respectivelypresent pressure and demand hydraulic benefit in all nodes under breakage of eachpipe solely. As it is seen in the figures, these criteria have the least value under

0

5

10

15

20

25

30

3 4 5 6 7 8 9 13 14 15 16 17 18 19 20 23 24 25 26 27 28 29 30 31 32 33 34

Network's Location Number

Hyd

raul

ic B

enef

it a

t A

ll N

odes

(P

ress

ure)

Pmin=0 Pmin=10 Pmin=20 Pmin=30 Unreliable looped WDN

Fig. 17 Pressure hydraulic benefit in all nodes under failure of each location’s pipes in Hanoinetwork


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1H

ydra

ulic

Ben

efit

at

All

Nod

es (

Dem

and)


3 4 5 6 7 8 9 13 14 15 16 17 18 19 20 23 24 25 26 27 28 29 30 31 32 33 34


Fig. 18 Demand hydraulic benefit in all nodes under breakage of each location’s pipes in Hanoinetwork

failure in locations 3, 4, 18, 19, and 20, which shows the importance and sensitivity ofthese pipes in supplying demands and providing required pressures of consumptionnodes. In Fig. 19, the sum of pressure hydraulic benefits for each consumption node

0

5

10

15

20

25

30

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32


Hyd

raul

ic B

enef

it a

t ea

ch N

ode

(Pre

ssur

e)


Fig. 19 Sum of pressure hydraulic benefits in each consumption node with respect to breakage ofvarious location’s pipes in Hanoi network


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2H

ydra

ulic

Ben

efit

at

each

Nod

e (D

eman

d)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32


Fig. 20 Sum of demand hydraulic benefits in each consumption node with respect to breakage ofvarious location’s pipes in Hanoi network

has been delineated with respect to breakage of different pipes. Clearly, nodes 8–14 and 28–32 are more sensitive than others. This is because the sum of the valuesof hydraulic benefit of pressure under breakage of different pipes in each state ofdesign in aforementioned nodes is relatively less than other consumption nodes ofthe network. In Fig. 20, the sum of demand hydraulic benefits for each consumptionnode has been delineated with respect to breakage in different pipes. This criteriongives specific weights to pressure hydraulic benefit in the nodes regarding the amountof their demand. It means that the importance of providing the required pressure ineach node depends on its value of required demand. For example, by consideringnodes 4 and 5, at the point of 20 m-H2O, as minimum expected pressure, the pressureprovided in node 4 is six units more than its value in node 5. But because of thedemand of node 5 in comparison with node 4, the value of demand hydraulic benefitof node 5 is almost five times more than its value in node 4. As going ahead alongFig. 20, it is seen that the peaks of the curves are related to the nodes in which theirdemand are more than others. By comparing nodes 3 and 7, which are almost thesame in their value of demand, it is seen that providing more pressure in node 3has caused more value for demand hydraulic benefit for this node. In Fig. 21, thecumulative number of nodes in which the pressure provided in them is less than30 m-H2O under breakage of the pipes of different locations have been delineatedfor different design states (as in the two-loop network). It is seen here again, as wecome down along the figure, the vertical distance between the curves is increasing,especially under the failure of locations 3, 4, 18, and 19. Also, it is seen that thecurve related to minimum expected pressure of 30, conforms to the curve relatedto minimum expected pressure of 20, meaning that under these states no node hasprovided the minimum pressure less than 30.


0

10

20

30

40

50

60

70

80C

umul

ativ

e N

umbe

r of

Nod

es w

ith

Les

s P

ress

ure

Pro

vide

d th

an

Min

imum

Req

uire

dPmin=0 Pmin=10 Pmin=20 Pmin=30 Unreliable looped WDN

3 4 5 6 7 8 9 13 14 15 16 17 18 19 20 23 24 25 26 27 28 29 30 31 32 33 34


Fig. 21 Cumulative number of nodes with pressure provided less than 30 m of water under breakageof various location’s pipes, for various design states in Hanoi network

8 Concluding Remarks

This paper presented the optimization of reliable design of two-loop and HanoiWDNs, using the HBMO algorithm, under breakage level one condition. Theobjective function consisted of minimization of the total cost of design. Constraintsconsidered providing defined minimum expected pressures under breakage of eachpipe in the two-loop network and failure of each pipe location in the Hanoi network,pressure and demand hydraulic benefits, and other performance criteria. Resultsof the two-loop network showed that the composition of pipes that guarantees21 m-H2O with breakage level one condition is acceptable, considering performancecriteria and the cost of design. The interesting point about the states of design whichguarantee 21 m-H2O in the two-loop and 20 m-H2O in the Hanoi networks was thecloseness of diameters achieved for the networks that considerably ease preparationand installation of the network’s elements. It could be inferred that the number ofnodes, with provided pressure less than 30 m-H2O, are maximum under breakage ofpipes, which will cause the reliability of the network to reach to its minimum value.The reason for pressure and demand hydraulic benefits under unreliable states ofdesign to be more than their values under some states of reliable design, is thatbreakage of some pipes would lead to incalculable pressures as the result of hydraulicsimulation of the network. Thus, it seems to be necessary to do further research todevelop methods which could help to make these situations known.

The two-loop network was designed to be reliable to supply all demands of thenetwork with a pressure of 30 m-H2O in analyzable breakage level two states as iden-tified in Table 1. The cost of design was considerably more than the reliable state ofdesign, guaranteeing 30 m-H2O in breakage level one condition. The aforementionedcost was achieved for designing the network under analyzable states of breakage level


two. However, there are some states of non-analyzable states of breakage level twowhich have the same chance of occurrence as analyzable ones. Thus, the achievedcost of design for breakage level two was not the one which strengthened the networkto be able to supply all the nodes under all possible breakage level two conditions.The scope of this research, however, was to strengthen the network in order to supplyall the consumption nodes in breakage conditions.

By a statistical computation it was determined that the higher breakage levelsare much less probable than breakage level one. It has been understood that byincreasing the total number of breakages in the operational period of the networkas the level of breakage increases, the difference between its occurrence probabilityand the probability of breakage level one increases. Although moving ahead alongthe horizontal axis showed that the probability of breakage levels one and two aregetting closer to each other, but by considering the useful life operational period ofthe WDNs, the parameter Y could have values much greater than 25. Thus, from theEq. 1, it could be inferred that an increase in the value of parameter Y would causea considerable difference between the probabilities of breakage level one and higherlevels remain stable for a greater number of total breakages.

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