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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
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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
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
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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
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
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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.
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
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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
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
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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
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
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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):
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
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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.
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
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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).
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
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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,
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
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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
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
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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
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
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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
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
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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
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
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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
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
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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
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
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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).
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
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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.
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
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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 .
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
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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.
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
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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
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
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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
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
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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
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
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also like to acknowledge reviewer 2 for his valuable comments and suggestions.
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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
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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
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
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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)
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
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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
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
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)
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
Accepted Manuscript Not Copyedited
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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)]
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
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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
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
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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)
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
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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
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
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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)]
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
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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)
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
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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
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
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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
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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
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
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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)
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
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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)
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
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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)
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
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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
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
Accepted Manuscript Not Copyedited
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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
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
Accepted Manuscript Not Copyedited
J. Water Resour. Plann. Manage.
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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 ($) 6045500 6026660 6055246 6057022 (see Fig. 7)
Legend: 1 100 = source total head of 100 (m); NF = not feasible [pressure less than 30 (m)]; NA = not available; D = pipe diameter [1 (inch) = 25.4 (mm)]; L = pipe length; P = pressure
Journal of Water Resources Planning and Management. Submitted February 9, 2013; accepted August 28, 2013; posted ahead of print August 30, 2013. doi:10.1061/(ASCE)WR.1943-5452.0000407
Copyright 2013 by the American Society of Civil Engineers
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Table 4: Hanoi network cheapest (point A, see Fig. 7) versus most resilient (point B, see Fig. 7) detailed solutions
Pipe (see Fig. 6)
Pipe length
(m)
Cheapest solution Most resilient solution
D (inch) [L (m)] D (inch) [L (m)]
1 100 40 (100) 40 (100) 2 1350 40 (1350) 40 (1350) 3 900 40 (900) 40 (900)
4 1150 40 (1150) 40 (1150)
5 1450 40 (1450) 40 (1450)
6 450 40 (450) 40 (450) 7 850 40 (850) 40 (850) 8 850 40 (850) 40 (850) 9 800 30 (62) ; 40 (738) 40 (800)
10 950 30 (950) 40 (950) 11 1200 24 (1200) 40 (1200) 12 3500 24 (3500) 40 (3500) 13 800 16 (251) ; 20 (549) 24 (800) 14 500 16 (500) 40 (500) 15 550 12 (550) 30 (550) 16 2730 12 (2730) 40 (2730) 17 1750 16 (1750) 40 (1750) 18 800 20 (427) ; 24 (373) 40 (800) 19 400 24 (400) 40 (400) 20 2200 40 (2200) 40 (2200) 21 1500 16 (491) ; 20 (1009) 30 (1500) 22 500 12 (500) 24 (500) 23 2650 40 (2650) 40 (2650) 24 1230 30 (1230) 40 (1230) 25 1300 30 (1300) 30 (1300) 26 850 20 (850) 24 (850) 27 300 12 (300) 30 (300) 28 750 12 (750) 40 (750) 29 1500 16 (1500) 24 (1500) 30 2000 12 (1845) 16 (155) 20 (2000) 31 1600 12 (1600) 12 (1600) 32 150 16 (150) 30 (150) 33 860 16 (860) 20 (685) ; 24 (175) 34 950 24 (950) 30 (950)
Cost ($) 6057022 9101734
Legend: D = pipe diameter [1 (inch) = 25.4 (mm)]; L = pipe length
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
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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)
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
Accepted Manuscript Not Copyedited
J. Water Resour. Plann. Manage.
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