multiobjective optimization for least cost design and resiliency of water distribution systems

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Multi-Objective Optimization for Least Cost Design and Resiliency of Water Distribution Systems

By Avi Ostfeld, Fellow ASCE1, Nurit Oliker2, and Elad Salomons3

Abstract

The multi-objective optimization model described in this study is aimed at exploring

the tradeoff between cost and resiliency for water distribution systems optimal design.

Many have dealt previously with minimizing cost where reliability was quantified as a

constraint. Fewer considered both cost and reliability as objectives. This work

suggests a methodology for least cost versus reliability (quantified as resiliency)

optimal design, introducing the following contributions: (1) a genetic algorithm multi-

objective formulation integrating a previous theoretical result of a possible maximum

of two adjacent discrete pipe diameters for a single pipe, (2) comparable results to

previous best least cost design solutions for the two looped and Hanoi networks, (3) a

real life sized example application analysis for pipes reinforcement, and (4) an

interpretation of resiliency through its comparison to two explicit reliability measures

involving demands increase and pipes failure, reconfirming that resiliency

improvement does not necessarily imply a reliability increase. Three example

applications are explored through base runs and sensitivity analyses for demonstrating

the study findings.

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1 Associate Professor (corresponding author), Faculty of Civil and Environmental Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel; PH: +972-4- 8292782; FAX: 972-4-8228898; E-mail: [email protected] 2 Graduate Student, Faculty of Civil and Environmental Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel; PH: +972-4-8292630; FAX: 972-4-8228898; E- mail: [email protected] 3 Director, OptiWater, 6 Amikam Israel St., Haifa, 34385, Israel; PH +972-54-2002050; FAX +972-15154-2002050; email: [email protected] Keywords: water distribution systems, reliability, resiliency, design, optimization, genetic algorithm

Journal of Water Resources Planning and Management. Submitted February 9, 2013; accepted August 28, 2013; posted ahead of print August 30, 2013. doi:10.1061/(ASCE)WR.1943-5452.0000407

Copyright 2013 by the American Society of Civil Engineers

J. Water Resour. Plann. Manage.

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Introduction

This study deals with least cost design and reliability (quantified through resiliency)

of water distribution systems which are the most explored topics of water distribution

systems management for almost five decades.

A water distribution system is an essential part of the urban infrastructure. As the

world population grows, together with a broad rise in living standards, there is a

constant demand for the establishment and development of such systems.

Finding the conjunctive least cost and reliable design of a water distribution system is

a multi-objective problem with a very broad solutions space. There is no straight-

forward algorithm to find the Pareto front of optimal solutions of this problem, neither

an agreed method for searching the solution space.

For any method, there is a fair likelihood that the solution of a single objective

problem would convergence to a local optimum. Several heuristic methods were

developed and applied for searching the global solution. Reca et al. (2008) showed

that on the whole, population based methods performed a better exploration of the

search space. One of those is a genetic algorithm (GA) (Holland 1975, Goldberg

1989), a potential method (Simpson et. al., 1994; Savic and Walters, 1997) that

already showed promising results for water distribution systems design and operation

in numerous studies.

This study suggests a multi-objective methodology for solving the least cost –

maximal reliability design problem through employing resiliency (Todini, 2000) as a




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reliability surrogate. The proposed method is demonstrated on the two looped network

(Alperovits and Shamir, 1977), the Hanoi (Fujiwara and Khang, 1990) network, and

on the EXNET reinforcement network problem (Farmani et al., 2005b).

Multi-objective optimal design of water networks

Gessler and Walski (1985) and Walski and Gessler (1988) were the first to suggest a

multi-objective optimization model for water distribution systems design entitled

WADISO. The model was developed to size pipes in water distribution systems under

multiple loading conditions and select optimal pipes for cleaning and lining. Halhal et

al. (1999) suggested a multi-objective procedure to solve a water distribution systems

management problem. Minimizing network cost versus maximizing the hydraulic

benefit served as the two conflicting objectives, with the total hydraulic benefit

evaluated as a weighted sum of pressures, maintenance cost, flexibility, and a measure

of water quality benefits. Kapelan et al. (2003) used a multi-objective genetic

algorithm to find sampling locations for optimal calibration. The problem was

formulated as a two-multi-objective optimization problem with the objectives been

the maximization of the calibrated model accuracy versus the minimization of the

total sampling design cost. Keedwell and Khu (2003) applied a hybrid multi-objective

evolutionary algorithm to the optimal design problem of a water distribution system.

Prasad and Park (2004) applied a non-dominated sorting genetic algorithm for

minimizing the network cost versus maximizing a reliability index. Vamvakeridou-

Lyroudia et al. (2005) employed a genetic algorithm multi-objective scheme to

tradeoff the least cost to maximum benefits of a water distribution system design

problem, with the benefits evaluated using fuzzy logic reasoning. The reader is

referred further to Nicklow et al. (2010) who provided a state of the art review on




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evolutionary computation for water resources systems, including multi-objective

optimization for water supply.

System reliability

Reliability is a stochastic measure of performance. A system is said to be reliable if it

functions properly for a given time interval and boundary conditions.

No system is perfectly reliable. In every system undesirable events - failures - may

cause a decline or interruption in system performance. Failures have a stochastic

nature, being the result of unpredictable events that occur in the system itself and/or in

its environment.

Most water supply networks are looped. The advantage of a looped layout resides in

the possibility of obtaining a modified flow regime in case of a pipe failure, without

disrupting the consumers supply. However, there is a significant difference between

the ability of various designs to overcome a failure. Systems with the same layout and

demand requirements, but with different designs might create systems with diverse

reliability levels.

There are two types of failure categories - mechanical (such as pipe breakage or pump

failure) and hydraulic (changes in supply demands, ageing of pipes, etc). The

probability of a network failure to occur is unknown, and stochastic simulation

models (e.g., Wagner et al., 1988; Ostfeld et al., 2002) were developed to explore

system reliability.




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Todini (2000) suggested the resilience index as a surrogate measure for reliability.

Instead of examining the likelihood of a failure to occur, the resilience index explores

the possibility of the system to endure a failure. It is thus a deterministic measure of

the system ability to cope with failures. Several authors (Prasad and Park, 2004;

Farmani et al., 2005a; Reca et al., 2008; Jayaram and Srinivasan, 2008; Raad et al.,

2010; Baños et al., 2011; Tanyimboh et al., 2011; Greco et al., 2012; Pandit and

Crittenden, 2012) suggested modifications to the resilience index of Todini and

compared its performance against other heuristic reliability surrogates for water

distribution systems reliability such as Entropy (Awumah et al., 1990). This study

employs the basic Todini’s resiliency index as a reliability surrogate.

The resiliency index (Todini, 2000) for the case of a gravitational system under a one

loading condition is defined as:

i

i

nmin

ii = 1

n

ii = 1

q hResiliency = 1 -

q h (1)

where: n = number of consumer nodes, and qi, hi, and i

minh = demand, pressure head,

and minimal pressure head required at the i-th node, respectively.

Expression (1) includes the ratio between the sum of the supply demand at each

consumer node multiplied by the minimum pressure required, and the sum of the

supply demand multiplied by the pressure at the nodes resulting from the network

design. The rational is such - if there is a lapse in the system, the flow regime in the

network will change. In the new flow regime the head loss will increase and the

network will necessarily consume more energy. Therefore, only if the system will




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have pressure surplus at the nodes, it could bear the failure and supply the minimum

head required. The resilience index makes it possible to evaluate the reliability of a

designed network in a comparable scale ranging between 0 and 1.

On the other hand, supplying the minimal pressure required will allow the designer to

use the smallest and cheapest possible diameters for the pipes. The cheapest network

design will necessarily be the one with the minimum surplus at the consumer nodes.

Therefore there is a tradeoff between the system cost and its reliability. The cheaper

the network is, the less it is reliable.

Methodology

The application of the GA for the multi-objective optimization utilized the "split pipe"

method, which enables more than one diameter for each pipe. It has already been

shown (Fujiwara et al., 1987) that even if multiple different diameters are allowed to

be selected for a single pipe, only one or two adjacent diameters will be chosen, given

that the cost-diameter function is convex, which is the case for the example

applications explored below. That is to say, a pipe segment will be composed either

from a single diameter or from a couple of subsequent diameters.

Therefore, the variables were formulated as the couple of subsequent diameters and

their length division for each of the pipes. That enabled receiving the "split pipes"

better solutions, yet with a limited increase of the solution space.

The vector space length is thus twice the pipe number in the network: two variables

for a pipe segment. The first section contains an index of the diameters couples, an




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integer variables, whose possible range is determined by the diameters suggested for

use. The last section determines the length of the first diameter in the couple. Those

are double type variables ranging from zero to the specific pipe length.

The only constraint for the given problems is associated with the pressure at the

consumer nodes which is required to maintain a minimum pressure head.

The two objective functions are:

M

i i i ii = 1

Min Cost = C d L d (2)

i

i

nmin

ii = 1

n

ii = 1

q hMin Resilience Complement =

q h (3)

where (2) is the length (Li) of each diameter (di) selected at the solution, multiplied by

its associated cost (Ci), M is the number of pipe diameters, and (3) is the complement

of Todini’s resilience index (Todini, 2000) defined in (1) [likewise expression (1)

could be maximized instead of minimizing (3)]. The expression (3) ranges from 0 to

1, where a lower value resembles a higher resiliency. The optimization is aimed at

simultaneously minimizing the two objectives.

Detailed description

A schematic flowchart of the proposed methodology is presented in Fig. 1. The

algorithm has four parameters (as will be further explained below): population size,

coupling fraction, and stopping conditions: number of subsequent generations for

which no additional non-dominated solutions are added or maximum number of

generations. The methodology consists of the following stages (follow Fig. 1):




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0. Initial feasible population, network solver

Generate an initial randomized feasible population. For a network having N links,

each string in the population is comprised of XN coupled or single diameters and LN

pipe lengths division. Each Xi (i = 1, …, N) is randomly selected from a set of

diameter indices, includes all possible diameters- single and coupled adjacent

diameters. The first indices are associated with the coupled adjacent diameters, and

the rest is the diameters list. Each Li (i = 1, …, N) is related to the length division of

the corresponding selected pipe diameters. If a coupled diameter is chosen, then Li is

randomly selected between zero and the link length, and is associated with the first

diameter in the couple, otherwise Li is equal to the corresponding link length.

For example: given that four diameters 4", 6", 10", and 14" are proposed for eight

links of a pipe network. The string will then have the form of: (X1, …, X8, L1, …, L8).

Xi is randomly selected from the vector: Y1 (coupled 4" and 6"), Y2 (coupled 6" and

10"), Y3 (coupled 10" and 14"), Y4 (4"), Y5 (6"), Y6 (10"), and Y7 (14"). If for X1, for

example, Y2 was randomly picked, then the pipes are a couple of 6" and 10", and the

length of the first coupled diameter (i.e., 6") will be randomly selected between zero

and the link length. The length of the 10" diameter will complement the length of the

chosen 6" pipe, up to the total link length.

Each of the randomized strings is checked for feasibility (i.e., whether all pressure

constraints are met) utilizing the Todini and Pilati (1987) network solver algorithm,

employing the Hazen-Williams head-loss equation.

The above process repeats until a population of feasible solutions is constructed.




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1. Evaluation

Each of the population strings is evaluated for its two objectives: cost [expression (2)]

and resilience complement [expression (3)].

2. Selection

2.1 Four strings from the population are randomly selected following a uniform

distribution.

2.2 Each string is given a weighted score equal to the square root of the sum of the

square of the normalized relative cost and the square of the resilience complement

values. For example: say that four strings were selected with the following figures of

cost, and resilient complement: 1000, 0.21; 2200, 0.12; 500, 0.32; and 750, 0.28. The

weights given to the strings are: [(1000/2200)2 + 0.212]0.5 = 0.501; 1.007; 0.392; and

0.441.

2.3 The string having the least weighted score is selected for the parent pool.

2.4 All four strings are returned to the population, and stage 2.1 repeats, if stopping

conditions are not met.

2.5. Check stopping conditions. The process ends once a predefined number of strings

is generated which corresponds to a selection fraction. The selection fraction

determines the number of parents required to generate a new population through the

coupling process, where for each two parents one child is formed. For example: if the

population holds 1000 strings and the selection fraction is set to 0.7, then 1400 strings

out of the 1000 population strings pool will be selected (some of which will obviously

repeat) for generating 700 new strings (i.e., 0.7 x 1000).




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3. Coupling and Mutation, New population, Elitism, Network solver

3.1 Coupling

3.1.1 Two parents are randomly selected from the parent’s pool.

3.2.1 Each gene of the diameters part of the strings is switched, with a probability of

0.5 (i.e., through "equal coin toggling"), with the corresponding other parent gene. If

moved, then its corresponding length attribute is also transferred.

3.3.1 Check feasibility of the new formed string through running a network solver. If

the solution is unfeasible, go to step 3.1.1, randomly generate strings until a feasible

string is constructed.

3.4.1 Check if the selection fraction of strings is reached. If not, go to step 3.1.1

3.2 Mutation

3.2.1 Randomly select a string from the current population.

3.2.2 Randomly select a gene from the diameters part of the string.

3.2.3 Randomly select a new index for the selected gene.

3.2.4 Assign a new diameter for the gene. If a diameter couple is selected, randomly

select its corresponding length. That is to say, for each mutated solution, the two

genes regarding one pipe segment are changed.

3.2.5 Check feasibility of the new formed string through running a network solver. If

the solution is unfeasible, go to step 3.2.1, randomly generate strings until a feasible

string is constructed.

3.2.6 Check if the mutation fraction (i.e., 1 – the coupling fraction) of strings is

reached. For example, for a population of 1000 strings with a coupling fraction of 0.7,




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the mutation fraction equals 0.3, thus 300 new strings need to be established through

mutation. If not, go to step 3.2.1

3.3 Elitism

Transform to the current population the two non-dominated extreme solutions of the

previous Pareto front (e.g., points A and B at the example applications section at Figs.

3 and 8).

4. Stopping

Check if stopping conditions are met: if either maximum number of generations is

reached, or the weighted average change in the Pareto front, over 10 generations is

negligible (less then 10-4). The average change is calculated by the distances between

all individuals on the front, favoring individuals that are relatively far away on the

front (Deb, 2001). If stopping conditions are met STOP and save the final Pareto front

(6), otherwise construct a new generation (5) through repeating from stage 1.

Example applications

Three example applications are explored: the two looped (Alperovits and Shamir,

1977), the Hanoi (Fujiware and Khang, 1990), and the EXNET (Farmani et al.,

2005b) networks. The later resembles a large reinforcement realistic water

distribution system case study.

The following assumptions are made for the example applications explored below: (1)

The design is for one loading gravitational systems with deterministic future demands,

assuming no additional demands growth thereafter, (2) All pipes are installed at time




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zero (i.e., no sequencing of construction), (3) No storage tanks are included, (4) Each

pipe can be isolated by valves, and (5) Demands are located at system nodes only. It

should also be noted that an additional assumption made herein is that a split pipe

solution will result in a cheaper solution than a single pipe diameter. This can be

questionable at different instances as the cost of the reducer connecting pipes of

different sizes is not considered, and demands are assumed to be perfectly known and

constant.

Example 1 – the two-looped network

The layout, demands, and elevations of the two looped network (Alperovits and

Shamir, 1977) are shown in Fig. 2. The system consists of eight links and six demand

nodes supplied by a single reservoir at a constant head of +210 (m). All link lengths

are 1000 (m); the minimum pressure head constraint at all demand nodes is 30 (m);

and the candidate pipe diameters and their associated unit costs are: 25 (mm), 2 (unit

cost/m); 51, 5; 76, 8; 102, 11; 152, 16; 203, 23; 254, 32; 305, 50; 356, 60; 406, 90;

457, 130; 508, 170; 559, 300; and 610, 550. Each pipe diameter is assumed to have a

Hazen-Williams friction coefficient of 130.

Figs. 3-5 and Tables 1, 2 and supplementary Table S1 summarize the results of

applying the proposed methodology on the two looped network. Fig. 3 describes the

initial and final Pareto fronts of the run attaining the least cost (point A) and most

resilient solutions (point B). The least cost outcome of 403485 is a slight

improvement over all previous published results (Table 1). It should be noted that the

comparison in Table 1 is to single objective optimization techniques were the current

study employed a multi-objective framework. Creaco and Franchini (2012a) utilized a




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genetic algorithm scheme coupled with a non linear optimization refinement problem

procedure with linear constraints, attaining a minimum cost of 403545 and a

resiliency of 0.145. Fig. 4 (part a) shows a comparison between the least cost and

most resilient solutions for the surplus pressures at the consumer nodes. It can be seen

from Fig. 4 (part a) that a very close proximity to the minimum pressure constraint of

30 (m) is attained at nodes 3, 5, 6, and 7 for the least cost solution, where for the most

resilient result surplus pressures are received at all nodes. Table 2 details the cheapest

and most resilient pipe diameter solutions. Table 2 shows an increase in all pipe

diameters for the most resilient solution compared to the cheapest. It should be noted

that for the least cost solution the model converged to a tree-like structure with a

minimum diameter of 1 (inch) for links 4 and 8. The convergence to a tree-like layout

for gravitational one loading networks was first observed by Fujiwara et al. (1987). In

reality links 4 and 8 wouldn’t have been constructed if the cheapest solution would

have been selected. The oversized pipe diameters level outcome for the highest

resiliency result, would probably not be used in reality. In Fig. 5 (part a) the cheapest

and most resilient solution performances are evaluated for an increase in the demands

multiplier. It can be seen from Fig. 5 (part a) that an infinitesimally increase of the

demands multiplier will instantaneously result a sharp drop in pressures feasibility for

the cheapest solution [i.e., four nodes promptly drop below 30 (m)], where for the

most resilient solution the system remains feasible up to an increase of about 4.8 in all

demands. Fig. 5 (part b) presents an analysis of the maximum number of unfeasible

nodes [i.e., nodes whose pressure drops below 30 (m)] in case of a one pipe outage, as

a function of resiliency. For example, the maximum number of unfeasible nodes for

the cheapest solution is five, where for the most resilient outcome, all nodes remain

feasible (supplementary Table S1). It can be seen from Fig. 5 (part b) that a reduction




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in the maximum number of unfeasible nodes (i.e., an increase in system reliability) is

not automatically guaranteed as resiliency increases. This was observed previously by

several authors (e.g., Creaco and Franchini, 2012b). This is attributed to the resiliency

nature of being a surrogate measure for reliability, not a direct indicator.

Example 2 – the Hanoi network

The second example application is the Hanoi network (Fujiware and Khang, 1990). It

consists of one reservoir, 34 pipes, and 31 consumer nodes (Fig. 6). The system is

subject to a one demand loading condition, and consists of 34 links and 32 demand

nodes (supplementary Table S2) supplied by a single reservoir at a constant head of

+100 (m). The minimum pressure head requirement at all nodes is 30 (m), and all

nodes are at zero elevation. Six candidate pipe diameters 12, 16, 20, 24, 30, 40 (inch)

with a Hazen-Williams coefficient of 130 are considered for each of the links. The

Cost ($) of installing a pipe of diameter d (inch) and length L (m) is:

1.5Cost = 1.1 d L (4)

The genetic algorithm vector length includes 68 variables (i.e., 34 links x 2).

Fig. 4 (part b), Figs. 7 and 8, Tables 3, 4, and supplementary Table S3 summarize the

results of applying the proposed methodology on the Hanoi network. Fig. 7 shows the

initial and final Pareto fronts for the run gaining the least cost (point A) and most

resilient solutions (point B). Fig. 4 (part b) details a comparison between the least cost

and most resilient solutions for the surplus pressures at the consumer nodes. It can be

seen from Fig. 4 (part b) that a very close proximity to the minimum pressure

constraint of 30 (m) is attained at nodes 13, 16, 22, 27, 29, 30, and 31 for the least

cost solution, where for the most resilient result surplus pressures are attained at all




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nodes. The least cost outcome is the second best solution attained over all previous

published results (Table 3). Table 4 describes the cheapest and most resilient pipe

diameter solutions, showing an increase in almost all pipe diameters for the most

resilient solution compared to the cheapest; and a minimum diameter of 12 (inch) for

links 15, 16, 22, 27, 28, 30, and 31 for the cheapest solution. In Fig. 8 (part a) the

cheapest and most resilient solution performances are evaluated for an increase in the

demands multiplier. It can be seen from Fig. 8 (part a) that an infinitesimally increase

of the demands multiplier will instantaneously result a drop in pressures feasibility for

the cheapest solution [i.e., from 31 to 23], where for the most resilient solution the

system remains feasible. Fig. 8 (part b) presents an analysis of the maximum number

of unfeasible nodes in case of a one pipe outage, as a function of resiliency. The

maximum number of unfeasible nodes for the cheapest solution is 27, where only 13

for the most resilient result (supplementary Table S3). Similar to the two looped

network (Fig. 5, part b), Fig. 8 (part b) shows a non-monotonic decrease in the

maximum number of unfeasible nodes as resiliency increase. Thus, an increase in

resiliency does not necessarily guarantee reliability improvement.

Results comparison between the two looped and the Hanoi networks

The following is a comparison between the results obtained for the two looped and

Hanoi networks example applications:

Pareto fronts [Fig. 3 (two looped network) versus Fig. 7 (Hanoi network)]

The distance between the initial and final Pareto fronts is higher at the Hanoi network

compared to the two looped system; the corresponding resiliency for the least cost

solution for the two looped network (point A) is lower than that of the Hanoi system




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(0.107 versus 0.299), and likewise for the most resilient outcome (0.403 versus

0.449); both Pareto fronts converge to a steady resilience value where an increase in

cost yields a negligible resiliency improvement; the relative increase in cost between

the least cost and the most resilient solutions is much higher for the two looped

network, than for the Hanoi system [9.229 = (4127090 - 403485)/403485 versus

0.503], and likewise for the corresponding resiliency values [2.766 = (0.403 –

0.107)/0.107 versus 0.502].

The higher resiliency values of the Hanoi network signify a higher reliable system

than the two looped network. On the other hand the relative lower increase in

resiliency at the Hanoi system compared to the two looped network indicate a less

flexible ability to improve reliability through pipe diameters enlargement.

Both of the above insights can be attributed to the higher redundancy and complexity

of the Hanoi network compared to the two looped system, and to the lower number of

candidate pipe diameter offered for the Hanoi, compared to the two looped network.

The Pareto fronts for the two looped and the Hanoi networks can be assessed using

the average distance (Raquel and Naval, 2005) and spread (Deb, 2001) measures.

The average distance measures the mean distance between adjacent solution on the

Pareto front by the "crowding" distance technique. The measure value is 0.218 and

0.066 for the Hanoi and the two looped networks, respectively. The measure of the

two looped network solution is lower, indicating a more uniform Pareto front. The

measure value is relatively low for both cases. The spread measure signifies the width

of the Pareto front over the objective space. The spread values of the Pareto fronts are




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0.583 and 0.496 for the Hanoi and the two looped networks, respectively. Those are

relatively high, indicating a wide fronts spread spanning of the objective space for the

two cases.

Surplus pressures [Fig. 4 (part a) (two looped network) versus Fig. 4 (part b) (Hanoi

network)]

For the least cost solution of the two looped network, four out of six nodes (i.e., 67%)

are almost at their minimal allowable pressures compared to only 23% (seven out of

31) of the Hanoi network. For the most resilient solutions, the Hanoi network depicts

much higher surplus pressures compared to the two looped network, which is aligned

with the higher corresponding resiliency values. The above are attributed to the layout

and elevation differences between the two systems.

Demand increase resiliency [Fig. 5 ( part a) (two looped network) versus Fig. 8 (part

a) (Hanoi network)]

A slight increase in the demand multiplier induces a sharp decrease in the number of

feasible nodes for the two looped network (two remain feasible at the two looped

network where 23 at the Hanoi system). A demands raise at the most resilient solution

to about 4.8 is possible at the two looped network until all node pressures become

infeasible [i.e., drop below 30 (m)], compared to only 1.02 for the Hanoi system.

This again [as evident in Figs. 5 (part b) and Fig. 8 (part b))] highlights the non-

uniqueness relationship between resiliency and reliability (i.e., one would expect that

the Hanoi higher system resiliency would induce a higher possible demand multiplier

increase to the stage where all node pressures become infeasible).




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Pipe failure resiliency [Fig. 5 (part b) (two looped network) versus Fig. 8 (part b)

(Hanoi network)]

For both the two looped and the Hanoi networks, the overall trend shows that the

maximum number of unfeasible nodes decreases as resiliency increases, in case of a

single (any) pipe outage. At the two looped network, however, the fluctuations are

much higher compared to the Hanoi system [i.e., Fig. 8 (part b) is much smoother

than Fig. 5 (part b)]. For both systems the non-uniqueness relationship between

resiliency increase and reliability is evident [e.g., Fig. 5 (part b) resiliency range

between 0.309 to 0.403; Fig. 8 (part b), 0.429 to 0.449]. A design with higher

resilience value may response worse to a failure occurrence.

Genetic algorithm parameters and sensitivity analysis

The default values for the statistic runs for both example applications were set as

follows: the population numbered 100 solutions and mutations comprised 20% of

them. The maximum number of generation was 500. However, convergence (with a

weighted average change in the Pareto front over 10 generations less then 10-4) was

attained for the two looped network on average (following 20 trials) after 138

generations with a standard deviation of 27.9 generations, and for the Hanoi system

after 113 generations with a standard deviation of 7.5 generations. The average

number of network solver evaluations for the two looped network was 32617

(standard deviation of 6003) and for the Hanoi system 34794 (7353). The average

number of network solver evaluations for a single non-dominated solution at the final

Pareto front was 808.3 for the two looped network, and 825.7 for the Hanoi system.




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It was found that for the presented algorithm and the above case studies, the influence

of the parameter values on the results was minor. Tests-run with a range of parameters

values, have shown narrow distribution. The optimal mutation fraction was around

0.35, yet the results with other values, were just slightly different. The population size

did not show clear impact, as long as it was set to at least 100. The results achieved

with a population of 500 solutions, was good as those achieved with 1000 solutions.

For the generation number there was a critical value in which the final population

stabilized. A figure of 100 generations was found as a minimum value required for the

GA to produce good results. After 100 generation the changes in the Pareto front was

rather small, and the solution improvement was almost negligible.

On the whole, it was found that the basic part of the GA performance is more in the

algorithm formulation (i.e., the method of creating initial population coupling and

mutating) and less in the GA parameter values set.

Example 3 – the EXNET network

The EXNET network (Farmani et al., 2005b) resembles a large reinforcement real life

water distribution system problem for a single loading gravitational system. The

system (Fig. 9) serves a population of approximately 400,000 people and consists of

two constant head sources, 1891 nodes, 2465 pipes, and five valves. The systems

consists of relatively small pipes and few transmission mains, making it highly

sensitive to demand increases .




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To continue providing water for future increased demands in the system at required

minimum pressures of 20 (m), a reinforcement plan for 567 of the pipes is suggested

(Fig. 9.).

Reinforcement using ten available discrete pipe diameters, as well as a decision of "no

action", are to be taken for each of the 567 candidate pipes. As with the two-looped

and Hanoi system examples, solutions of two diameters can be selected for each of

the reinforced pipes. Unit costs for pipe laying differ as a function of pipe diameter

and type of road. At major roads excavations are more difficult to undertake and thus

consequently are more expensive. Table 5 provides data for the available internal pipe

diameters, their corresponding Colebrook White friction factors, and their unit costs

for minor/major roads pipe laying. The rest of the data for EXNET (2013) is given in

the supplementary EPANET file of Exnet_network.inp

Fig. 10-13 summarize the model analysis for EXNET. The methodology (Fig. 1)

applied for the two looped and Hanoi networks failed to converge for EXNET. This is

most likely attributed to the strings decision variables length of the EXNET problem.

A different approach for generating the Pareto front was thus utilized, as described in

Fig. 10.

The methodology is based on running in parallel two single objective optimization

models for minimizing system cost and maximizing system resiliency. Results are

then accumulated and a Pareto front is constructed until no new members are

available or the maximum number of iterations is attained.




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For minimizing cost (or maximizing resiliency) a roulette based genetic algorithm

([email protected]/optiga.html) with a single location crossover, 1000

generation, a population of 100 strings, and a mutation of 0.005 was used. The

maximum total number of iterations (i.e., single objective optimization repetitions)

was set to ten. A single objective optimization running time was approximately 50

minutes on a 2.67GHz Lenovo with 8.00 GB of RAM, thus a single entire run with

ten iterations took ~ 17 (hrs).

Fig. 11 presents the resulted Pareto front for EXNET. This Pareto front is

representative and was attained at multiple model trials. The cost of the cheapest

solution is 16834280 ($) (~$17 million) with a resiliency of 0.356, where the cost of

the highest resiliency solution of 0.551, is 61932940 ($). It can be seen from Fig. 11

that the Pareto front is almost linear. This might be attributed to the EXNET example

application layout complexity and diversity (Fig. 9). Detailed solutions for the

cheapest and most resilient solutions are attached as supplementary EPANET files

entitled Cheapest_EXNET_solution.inp and Most_resilient_EXNET_solution.inp,

respectively.

Fig. 12 describes the pressure head map for the least cost and most resilient solutions,

showing a substantial excessive pressure head distribution of the most resilient

solution compared to the least cost result.

In Fig. 13 the diameter pipe length distribution for the least cost and most resilient

solutions is presented. For the least cost solution, 294 pipes out of the total 567

candidate pipes for reinforcement (i.e., ~ 52%) remained unchanged (i.e., a "no




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action" decision was selected), where for the most resilient solution only 16 pipes

(i.e., ~ 3%) remained unchanged. The total length of reinforced pipes for the least cost

and the most resilient solutions was 72.74 and 166. 84 (km), respectively. Most of the

16 pipes which were selected to stay as is at the most resilient solution were also

chosen at the cheapest solution.

Conclusions

This paper presented an application of a multi-objective GA for water distribution

system design and reinforcement, optimizing the solution for its cost and resiliency,

following the resilience index of Todini (2000).

The methodology generates Pareto fronts which can provide a beneficial planning tool

for the designer for selecting various options of optimal solutions, with each holding a

different compromise between cost and resiliency. Results showed that the systems

resiliency increased with cost. A selected design solution should thus be close to the

generated Pareto curve.

Three case studies were explored. For the two looped and Hanoi networks, the steep

slope of the Pareto front in the low-cost range allows a sharp increase in resiliency for

a small increase in cost. This deduction, if exists, can be useful when choosing a

system design. For the EXNET example the Pareto front was almost linear thus a

small increase in cost resulted in only a small increase in resiliency.

A non necessary uniqueness between resiliency increase and reliability improvement

is reconfirmed. Generally, the resilience gives a good notion of system reliability and




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an increase in resiliency will most likely also induce an increase in reliability.

However, the relationship is not monotonic. This was evident for the two looped and

Hanoi networks case studies when exploring the system’s performance capabilities in

case of demand increase and pipe failures. A possible outcome of this observation can

be a recommendation to explore the system reliability using also direct reliability

measures and methods, thus not solely relying on resiliency as a surrogate for

reliability.

The EXNET example application provided a real life large scale least cost design

reinforcement problem. For this case study the methodology implemented for the two

looped and Hanoi networks failed to converge. This is most likely attributed to the

decision variables large space of the EXNET problem. A heuristic procedure of

generating the Pareto front for EXNET was developed and implemented showing

stable outcomes of the resulted Pareto front. Construction of Pareto fronts for large

systems such as EXNET should however be further explored using more established

multi-objective methodologies such as the Non-Dominated Sorted Genetic

Algorithm–II (NSGA-II) (Deb et al., 2000).

Other extensions of this study may include model expansions to include systems with

pumping, storage, and multiple loading conditions, and the inclusion of demands

uncertainty.

Acknowledgements

This research was supported by the Fund for the Promotion of Research at the

Technion, and by the Technion Grand Water Research Institute (GWRI). We would




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also like to acknowledge reviewer 2 for his valuable comments and suggestions.




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List of Tables

Table 1: Least cost comparison results for the two looped network

Table 2: Two looped network cheapest (point A, see Fig. 3) versus most resilient

(point B, see Fig. 3) detailed solutions

Table 3: Least cost comparison results for the Hanoi network

Table 4: Hanoi network cheapest (point A, see Fig. 7) versus most resilient (point B,

see Fig. 7) detailed solutions

Table 5: Pipe rehabilitation costs for EXNET




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32

List of Figures

Fig. 1: Methodology flowchart

Fig. 2: The two looped network (adapted from Alperovits and Shamir, 1977)

Fig. 3: Resiliency versus cost tradeoff curve for the two looped network [A: cheapest

solution (see Table 1), B: most resilient solution (see Table 2)]

Fig. 4: Surplus pressures for the two looped (a) and Hanoi (b) networks

Fig. 5: (a) Demand increase resiliency analysis for the two looped network ; (b) Pipe

failure resiliency analysis for the two looped network (A and B refer to the

cheapest and most resilient solutions, respectively, see supplementary Table

S1, and Fig. 3)

Fig. 6: The Hanoi network (adapted from Fujiwara and Khang, 1990)

Fig. 7: Resiliency versus cost tradeoff curve for the Hanoi network [A: cheapest

solution (see Table 3), B: most resilient solution (see Table 4)]

Fig. 8: (a) Demand increase resiliency analysis for the Hanoi network ; (b) Pipe

failure resiliency analysis for the Hanoi network (A and B refer to the cheapest

and most resilient solutions, respectively, see supplementary Table S3, and

Fig. 7)

Fig. 9: EXNET network layout

Fig. 10: Methodology flowchart for EXNET example

Fig. 11: Resiliency versus cost tradeoff curve for the EXNET network (A: cheapest

solution, B :most resilient solution)

Fig. 12: Pressure head map for the least cost and most resilient solutions for EXNET

(see also Fig. 11)

Fig. 13: Diameter pipe length distribution for the least cost and most resilient

solutions for EXNET (see also Fig. 11)




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0. Initial feasible population

0. Network solver

1. Evaluation

2. Selection

3. Coupling; Mutation

3. New population; Elitism

3. Network solver

4. Stopping No

5. New generation

6. Final Pareto front

Yes

= repeat until a feasible population is generated Legend:

Fig. 1: Methodology flowchart



Accepted Manuscript Not Copyedited


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DQ6 = 330

Z6 = +165

DQ4 = 120

Z4 = +155

DQ2 = 100

Z2 = +150

Reservoir

8

6

DQ7 = 200

Z7 = +160

7

5

3

1 1

6

45 4

DQ5 = 270

Z5 = +150 7

23 2

DQ3 = 100

Z3 = +160 + 210

Legend

DQ2 = 100 – demand of 100 (m3/hr) at node 2 Z2 = + 150 – elevation of + 150 (m) at node 2 + 210 = reservoir total head of + 210 (m)

Fig. 2: The two looped network (adapted from Alperovits and Shamir, 1977)





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B: 4127090 0.403

A: 403485 0.107

0.06

0.10

0.14

0.18

0.22

0.26

0.30

0.34

0.38

0.42

2.2E+05 7.7E+05 1.3E+06 1.9E+06 2.4E+06 3.0E+06 3.5E+06 4.1E+06

Cost (unit cost)

Res

ilienc

e

Generation 1 Final generationA

B R

esilie

ncy

Fig. 3: Resiliency versus cost tradeoff curve for the two looped network [A: cheapest solution (see Table 1), B: most resilient solution (see Table 2)]



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(b)

(a)

Legend: = cheapest solution (point A, see Figs. 3 and 7), = most resilient solution (point B, see Figs. 3 and 7)

Fig. 4: Surplus pressures for the two looped (a) and Hanoi (b) networks



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(b)

(a)

Legend: = A: 403485 (Cost), 0.107 (Resiliency); = B: 4127090, 0.403

Fig. 5: (a) Demand increase resiliency analysis for the two looped network ; (b) Pipe failure resiliency analysis for the two looped network (A and B refer to the cheapest and most resilient solutions, respectively, see supplementary Table S1, and Fig. 3)



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22

21

2023

24

25 26 273231

30

29 28

16

1718

19

3 4 5 6

7

8

2

1

910

11

12

1415

13 12

11

109

13 1415 8

7

6

543

1617

18

19

2827 26

25

24

2023

21

22

2

1

2930

31

32

34 33

ReservoirLegend

Fig. 6: The Hanoi network (adapted from Fujiwara and Khang, 1990)

= node 5

5

20 = link 20

= node 5

= pipe 20



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A: 6053444 0.299

B: 9101734 0.449

0.250.270.290.310.330.350.370.390.410.430.45

5.8E+06 6.3E+06 6.8E+06 7.3E+06 7.8E+06 8.3E+06 8.8E+06 9.3E+06

Cost ($)

Res

ilienc

e

Generation 1 Final generation

A

B R

esilie

ncy

6057022

Fig. 7: Resiliency versus cost tradeoff curve for the Hanoi network [A: cheapest solution (see

Table 3), B: most resilient solution (see Table 4)]



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Legend: = A: 6057022 (Cost), 0.299 (Resiliency); = B: 9101734, 0.449

(b)

(a)

Fig. 8: (a) Demand increase resiliency analysis for the Hanoi network ; (b) Pipe failure resiliency analysis for the Hanoi network (A and B refer to the cheapest and most resilient solutions, respectively, see supplementary Table S3, and Fig. 7)



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Legend: = candidate pipe for reinforcement

Constant head source+ 62.4 (m)

+ 58.4 (m) Constant head source

Fig. 9: EXNET network layout



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Minimize COST

Store min COST for each generation and its resulted resiliency

Maximize RESILIENCY

Store max RESILIENCY for each generation and its resulted cost

GENERATE Pareto front (i.e., find all non-dominated solutions)

CHECK for new Pareto front members (i.e., new non-dominated solutions) OR maximum iterations

START

Sin

gle

obje

ctiv

e op

timiz

atio

n

Sin

gle

obje

ctiv

e op

timiz

atio

n

New members OR no maximum iterations encountered

No new members OR maximum iterations encountered

STOP and generate final Pareto front

Fig. 10: Methodology flowchart for EXNET example



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0.2000.2500.3000.3500.4000.4500.5000.5500.600

1.5E+07 2.0E+07 2.5E+07 3.0E+07 3.5E+07 4.0E+07 4.5E+07 5.0E+07 5.5E+07 6.0E+07 6.5E+07

Cost ($)

Res

ilien

cy (-

)

B: 61932940 0.551

A: 16834280 0.356

Fig. 11: Resiliency versus cost tradeoff curve for the EXNET network (A: cheapest solution, B: most resilient solution)



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MOST RESILIENT LEAST COST

Pressure (m)

28

32

35

42

A: 16834280 ($) 0.356 (-)

B: 61932940 ($) 0.551 (-)

Fig. 12: Pressure head map for the least cost and most resilient solutions for EXNET (see also Fig. 11)



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05

10152025303540

110 159 200 250 300 400 500 600 750 900Diameter (mm)

Tota

l pip

e le

ngth

(km

)

= A: 16834280 ($) 0.356 (-)

Legend: = B: 61932940 ($) 0.551 (-)

Fig. 13: Diameter pipe length distribution for the least cost and most resilient solutions for EXNET (see also Fig. 11)



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Table 1: Least cost comparison results for the two looped network

PIPE / NODE (SEE FIG. 2)

KESSLER AND SHAMIR (1991)

EIGER ET AL. (1994) KRAPIVKA AND OSTFELD

(2009) THIS STUDY

D (inch) [L (m)] P (m) D (inch) [L (m)] P (m) D (inch) [L (m)] P (m) D (inch) [L (m)] P (m) 1 18 (1000) 1 210 18 (1000) 1 210 18 (1000) 1 210 18 (1000) 1 210 2 10 (934) ; 12 (66) 53.25 10 (762) ; 12 (238) 53.25 10 (793) ; 12 (207) 53.25 10 (795) ; 12 (205) 53.25 3 16 (1000) 30.00 16 (1000) 30.28 16 (1000) 30.00 16 (1000) 30.00 4 2 (287) ; 3 (713) 43.63 1 (1000) 43.85 1 (1000) 43.85 1 (1000) 43.85

5 14 (164) ; 16 (836) 31.25 14 ( 371) ; 16 (629) 30.61 14 (307) ; 16 (693) 30.03 14 (309) ; 16 (691) 30.09

6 10 (891) ; 12 (109) 30.07 8 (11) ; 10 (989) 29.82

NF 8 (9) ; 10 (991) 30.00 8 (11) ; 10 (989) 30.00

7 8 (181) ; 10 (819) 30.11 8 (78) ; 10 (922) 29.82 NF

8 (96) ; 10 (904) 30.01 8 (91) ; 10 (909) 30.02

8 2 (80) ; 3 (920) NA 1 (1000) NA 1 (1000) NA 1 (1000) NA Cost (unit cost) 417500 402352 403572 403485 (see Fig. 3)

Legend: 1 210 = source total head of 210 (m); NF = not feasible [pressure less than 30 (m)]; NA = not available; D = pipe diameter [1 (inch) = 25.4 (mm)]; L = pipe length; P = pressure





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Table 2: Two looped network cheapest (point A, see Fig. 3) versus most resilient (point B, see Fig. 3) detailed solutions

Legend: D = pipe diameter [1 (inch) = 25.4 (mm)]; L = pipe length

Pipe (see Fig. 2)

Cheapest solution Most resilient solution

D (inch) [L (m)] D (inch) [L (m)] 1 18 (1000) 24 (1000) 2 10 (795) ; 12 (205) 24 (1000) 3 16 (1000) 24 (1000) 4 1 (1000) 22 (4) ; 24 (996) 5 14 (309) ; 16 (691) 24 (1000) 6 8 (11) ; 10 (989) 22 (861) ; 24 (139) 7 8 (91) ; 10 (909) 24 (1000) 8 1 (1000) 24 (1000)

Cost (unit cost) 403485 4127090





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Table 3: Least cost comparison results for the Hanoi network

PIPE / NODE (SEE FIG. 6)

PIPE LENGTH

(M)

SONAK AND BHAVE (1993) EIGER ET AL. (1994) PERELMAN ET AL. (2009) THIS STUDY

D (inch) [L (m)] P (m) D (inch) [L (m)] P (m) D (inch) [L (m)] P (m) D (inch) [L (m)] P (m) 1 100 40 (100) 1 100 40 (100) 1 100 40 (100) 1 100 40 (100) 1 100 2 1350 40 (1350) 97.14 40 (1350) 97.14 40 (1350) 97.14 40 (1350) 97.14 3 900 40 (900) 61.67 40 (900) 61.67 40 (900) 61.67 40 (900) 61.67

4 1150 40 (1150) 56.99 40 (1150) 57.21 40 (1150) 56.97 40 (1150) 56.97

5 1450 40 (1450) 51.19 40 (1450) 51.69 40 (1450) 51.14 40 (1450) 51.13

6 450 40 (450) 45.09 40 (450) 45.90 40 (450) 44.99 40 (450) 44.99 7 850 40 (850) 43.66 40 (850) 44.56 40 (850) 43.56 40 (850) 43.55 8 850 40 (850) 41.96 40 (850) 43.00 40 (850) 41.85 40 (850) 41.84 9 800 30 (140) ; 40 (660) 40.62 30 (641) ; 40 (159) 41.77 30 (75) ; 40 (725) 40.49 30 (62) ; 40 (738) 40.48

10 950 30 (950) 39.10 30 (950) 38.70 30 (950) 39.20 30 (950) 39.24 11 1200 24 (1200) 37.54 24 (1199) ; 30 (1) 37.14 24 (1200) 37.64 24 (1200) 37.69 12 3500 24 (3500) 34.34 24 (3500) 33.94 24 (3500) 34.21 24 (3500) 34.25 13 800 16 (160) ; 20 (640) 30.13 16 (800) 29.73 NF 16 (251) ; 20 (549) 30.00 16 (251) ; 20 (549) 30.05 14 500 16 (500) 34.49 12 (500) 31.90 16 (500) 33.66 16 (500) 33.70 15 550 12 (550) 33.05 12 (550) 29.57 NF 12 (550) 32.11 12 (550) 32.14 16 2730 12 (2730) 31.53 12 (2730) 29.53 NF 12 (2730) 30.31 12 (2730) 30.30 17 1750 12 (67) ; 16 (1683) 34.42 16 (634) ; 20 (1116) 40.04 16 (1750) 32.83 16 (1750) 32.85 18 800 20 (800) 53.66 24 (800) 52.74 20 (427) ; 24 (373) 49.74 20 (427) ; 24 (373) 49.74 19 400 20 (400) 58.92 24 (400) 58.61 24 (400) 58.94 24 (400) 58.94 20 2200 40 (2200) 50.49 40 (2200) 50.33 40 (2200) 50.52 40 (2200) 50.53 21 1500 16 (511) ; 20 (989) 34.88 16 (514) ; 20 (986) 34.69 16 (491) ; 20 (1009) 35.16 16 (491) ; 20 (1009) 35.16 22 500 12 (500) 29.72 NF 12 (500) 29.52 NF 12 (500) 30.00 12 (500) 30.00 23 2650 40 (2650) 44.31 40 (2650) 44.01 40 (2650) 44.37 40 (2650) 44.37 24 1230 30 (1230) 38.96 30 (1230) 38.11 30 (1230) 38.68 30 (1230) 38.69 25 1300 30 (1300) 35.58 30 (1300) 34.27 30 (1300) 35.02 30 (1300) 35.04 26 850 20 (850) 31.67 20 (850) 30.00 20 (850) 31.19 20 (850) 31.21 27 300 12 (15) ; 16 (285) 31.31 12 (7) ; 16 (293) 29.53 NF 12 (300) 30.02 12 (300) 30.04 28 750 12 (750) 36.25 12 (750) 38.30 12 (750) 38.69 12 (750) 38.65 29 1500 16 (1500) 32.08 16 (1500) 29.65 NF 16 (1500) 30.14 16 (1500) 30.00 30 2000 12 (1031) ; 16 (969) 31.54 12 (2000) 29.87 NF 12 (2000) 30.38 12 (1845) ; 16 (155) 30.21 31 1600 12 (1600) 31.68 12 (1600) 30.14 12 (1600) 30.65 12 (1600) 30.49 32 150 16 (150) 33.14 16 (150) 32.14 16 (150) 32.89 16 (150) 32.91 33 860 16 (860) NA 16 (633) ; 20 (227) NA 16 (751) ; 20 (109) NA 16 (860) NA 34 950 20 (247) ; 24 (703) NA 24 (950) NA 24 (950) NA 24 (950) NA

Cost ($) 6045500 6026660 6055246 6057022 (see Fig. 7)

Legend: 1 100 = source total head of 100 (m); NF = not feasible [pressure less than 30 (m)]; NA = not available; D = pipe diameter [1 (inch) = 25.4 (mm)]; L = pipe length; P = pressure



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Table 4: Hanoi network cheapest (point A, see Fig. 7) versus most resilient (point B, see Fig. 7) detailed solutions

Pipe (see Fig. 6)

Pipe length

(m)

Cheapest solution Most resilient solution

D (inch) [L (m)] D (inch) [L (m)]

1 100 40 (100) 40 (100) 2 1350 40 (1350) 40 (1350) 3 900 40 (900) 40 (900)

4 1150 40 (1150) 40 (1150)

5 1450 40 (1450) 40 (1450)

6 450 40 (450) 40 (450) 7 850 40 (850) 40 (850) 8 850 40 (850) 40 (850) 9 800 30 (62) ; 40 (738) 40 (800)

10 950 30 (950) 40 (950) 11 1200 24 (1200) 40 (1200) 12 3500 24 (3500) 40 (3500) 13 800 16 (251) ; 20 (549) 24 (800) 14 500 16 (500) 40 (500) 15 550 12 (550) 30 (550) 16 2730 12 (2730) 40 (2730) 17 1750 16 (1750) 40 (1750) 18 800 20 (427) ; 24 (373) 40 (800) 19 400 24 (400) 40 (400) 20 2200 40 (2200) 40 (2200) 21 1500 16 (491) ; 20 (1009) 30 (1500) 22 500 12 (500) 24 (500) 23 2650 40 (2650) 40 (2650) 24 1230 30 (1230) 40 (1230) 25 1300 30 (1300) 30 (1300) 26 850 20 (850) 24 (850) 27 300 12 (300) 30 (300) 28 750 12 (750) 40 (750) 29 1500 16 (1500) 24 (1500) 30 2000 12 (1845) 16 (155) 20 (2000) 31 1600 12 (1600) 12 (1600) 32 150 16 (150) 30 (150) 33 860 16 (860) 20 (685) ; 24 (175) 34 950 24 (950) 30 (950)

Cost ($) 6057022 9101734

Legend: D = pipe diameter [1 (inch) = 25.4 (mm)]; L = pipe length



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Table 5: Pipe rehabilitation costs for EXNET

Internal pipe diameter

(mm)

Colebrook White friction factors (mm)

Unit cost (/m) for minor roads (major

roads) 110 0.03 85 (100) 159 0.065 95 (120) 200 0.1 115 (140) 250 0.13 150 (190) 300 0.17 200 (240) 400 0.23 250 (290) 500 0.3 310 (340) 600 0.35 370 (410) 750 0.43 450 (500) 900 0.5 580 (625)





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opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

multiobjective optimization for least cost design and resiliency of water distribution systems

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