DWESDrinking Water Engineering and ScienceDWESDrink. Water Eng. Sci.1996-9465Copernicus PublicationsGöttingen, Germany10.5194/dwes-11-67-2018Algorithms for optimization of branching gravity-driven water networksAlgorithms for optimization of branching gravity-driven water networksDardaniIanian.dardani@villanova.eduJonesGerard F.College of Engineering, Villanova University, Villanova, PA 19085, USAIan Dardani (ian.dardani@villanova.edu)15May2018111678531January201724February20173February20187March2018This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://dwes.copernicus.org/articles/11/67/2018/dwes-11-67-2018.htmlThe full text article is available as a PDF file from https://dwes.copernicus.org/articles/11/67/2018/dwes-11-67-2018.pdf
The design of a water network involves the selection of pipe diameters that
satisfy pressure and flow requirements while considering cost. A variety of
design approaches can be used to optimize for hydraulic performance or reduce
costs. To help designers select an appropriate approach in the context of
gravity-driven water networks (GDWNs), this work assesses three
cost-minimization algorithms on six moderate-scale GDWN test cases. Two
algorithms, a backtracking algorithm and a genetic algorithm, use a set of
discrete pipe diameters, while a new calculus-based algorithm produces a
continuous-diameter solution which is mapped onto a discrete-diameter set.
The backtracking algorithm finds the global optimum for all but the largest
of cases tested, for which its long runtime makes it an infeasible option.
The calculus-based algorithm's discrete-diameter solution produced slightly
higher-cost results but was more scalable to larger network cases.
Furthermore, the new calculus-based algorithm's continuous-diameter and
mapped solutions provided lower and upper bounds, respectively, on the
discrete-diameter global optimum cost, where the mapped solutions were
typically within one diameter size of the global optimum. The genetic
algorithm produced solutions even closer to the global optimum with
consistently short run times, although slightly higher solution costs were
seen for the larger network cases tested. The results of this study highlight
the advantages and weaknesses of each GDWN design method including closeness
to the global optimum, the ability to prune the solution space of infeasible
and suboptimal candidates without missing the global optimum, and algorithm
run time. We also extend an existing closed-form model of Jones (2011) to
include minor losses and a more comprehensive two-part cost model, which
realistically applies to pipe sizes that span a broad range typical of GDWNs
of interest in this work, and for smooth and commercial steel roughness
values.
Introduction
A gravity-driven water network (GDWN) is commonly used to deliver potable
water from a source at a high elevation, such as a natural spring or
reservoir, to households or public tap stands (Fig. 1). When feasible,
gravity-driven water networks are attractive in comparison to pumped
networks because of their simplicity and lower capital, operational, and
maintenance costs. In addition, in many locations where GDWN are considered,
there may be little or no access to reliable grid-based electrical power for
pumps. To improve reliability, networks may be designed with loops or
multiple water sources, although often material cost considerations restrict
attention to single-source branched networks.
Water networks are modeled as a collection of nodes, each representing a
point of water demand or supply, which are connected with links representing
pipes. The geometrical layout of the site is known and fixed, including water
source and demand locations and elevations of all nodes. For the present
work, design flow rates are determined from community survey data, which are
extrapolated for future population growth. Networks in this category are
referred to as “demand-driven” designs. Bhave (1978, 1983) refers to these as
“Q-specified” designs. Thus, to design a network of this type, pipe
diameters for each link must be chosen such that acceptable but arbitrary
minimum (positive) pressure heads are maintained at each node given the design
flow rate at the node. Furthermore, application of the energy equation to
each link in the network demonstrates that the design problem is nonunique;
i.e., choosing different pressure heads at the nodes will result in a
different pipe diameter solution for the network, and thus different network
costs. Minimizing network cost will produce a unique solution to the design
problem, i.e., unique link diameters and nodal pressure heads.
In practice, gravity-driven water networks are commonly designed by a
marching method, where diameters for each link of the network are chosen
sequentially. After selecting a reasonable diameter for each link, the
designer calculates the pressure head at the link outlet, and proceeds to
the next link if this result is acceptable. In this way, the designer
marches through the network until all pipe diameters have been selected.
This method produces a feasible solution, but not a cost optimized one. As
noted by Bhave (2003), cost savings of 20–30 % can result from the use of
optimization techniques. In developing regions, the cost of a water network
can be prohibitive, adding to the importance of optimizing network design.
Within the provided framework, the global optimum can be found through an
exhaustive search of the solution space, known as complete enumeration,
although this is infeasible when considering networks with many links and
diameter choices (Kadu et al., 2008; González-Cebollada et
al., 2011). To reduce the
computational time required by enumeration, authors have proposed various
partial enumeration methods which prune the search space (Kadu et al., 2008),
although some of these techniques may remove the global optimum (Simpson et
al., 1994). The most common types of algorithms that have been applied to
optimize water network design include traditional deterministic methods,
heuristic methods, metaheuristic methods, multi-objective methods, and
decomposition methods (Zhao et al., 2016).
Element schematic of a GDWN.
Deterministic methods include linear programming (LP), dynamic programming,
and nonlinear programming (NLP), and typically involve rigorous mathematical
approaches (Zhao et al., 2016). A brief overview and comparison of these
algorithms is given in Kansal et al. (1996), who use a single-part cost
correlation for metric pipe diameters between 100 and 350 mm. Linear
programming techniques have relatively low computational complexity and
allow each link to be composed of two diameters, called a split-pipe
solution, although these may not always be practical to implement (Bhave,
1983; Kessler and Shamir, 1989; Swamee and Sharma, 2000; Samani and Mottaghi,
2006). LP can also get stuck in a local optimum (Zhao et al., 2016), although
combining LP with metaheuristic techniques can help with the problem's
non-smoothness properties (Krapivka and Ostfeld, 2009). Dynamic programming
has been used by Yang et al. (1975) and Martin (1980) to optimize networks
in stages. This approach begins at the discharge nodes, proceeding to select
feasible diameters and joints for upstream stages and storing these partial
candidates in memory until the source node is reached. At this point, the
algorithm reviews the feasible segment design options and selects a
combination of stage solutions producing the lowest cost overall solution.
This method, however, requires the designer to allow a relatively narrow
range for the design pressure of each node, or otherwise store a large set
of feasible candidate solutions in memory and also allow adjoining branches
to arrive at different heads at the same node.
Nonlinear programming, a calculus-based method, deals with each link's
diameter as a continuous variable. Using Lagrange multipliers and a one-part,
pipe-cost model with minor-lossless flow, Swamee and Sharma (2000) developed
systems of equations for both continuous and discrete pipe diameters for
branch networks, assuming a constant friction factor. When solved, the
solution gives diameter values that minimize distribution main cost, not
network cost. In carrying out the solution, iteration is required to update
the value of the friction factor. For the discrete diameter case, large
computational times were noted by Swamee and Sharma because of the stiffness
of the mathematical system. Cases where one or more nodal pressure heads are
not acceptable need to be treated manually by the designer in various ways as
discussed by the authors. For branching networks, Jones (2011) showed that by
restricting the focus to smooth-turbulent (turbulent flow in a smooth pipe)
minor-lossless flow, and the use of a one-part, pipe-cost model, a simple
nonlinear algebraic equation for each internal node in the distribution main
could be developed. The development of this algorithm, as well as solution
methodology, differs from that of Bhave (1978), which assumes constancy in
several terms and thus requires iteration to solve. The Jones algorithm has
been extended in the present work to include minor losses and rough pipe.
When solved simultaneously with the energy equation for each link, a unique
solution for all link diameters and nodal pressure head values is obtained
which produces minimum network cost, as opposed to the distribution main cost
as in Swamee and Sharma (2000). The method of Jones also applies to serial
and loop networks because of its generality.
Heuristic methods follow specific rules to incrementally build better
solutions, although the rules are not strictly formulated to trend towards
local or global optima. An approach by Monbaliu et al. (1990) sets all network
pipes to their minimum size, where the pipe that has a maximum head loss
gradient is incremented to its next-highest size until all nodal head
requirements are satisfied. Similarly, an algorithm by Keedwell and
Khu (2006) selects an initial solution and iteratively responds to nodal head
deficits and surpluses by incrementing or decrementing pipe sizes accordingly
until a feasible solution is found. Suribabu (2012) proposed a heuristic that
identifies pipes to increment or decrement in size based on flow velocity and
alternative metrics such as proximity to the source node, achieving
acceptable cost results with computational efficiency. While these algorithms
are typically computationally efficient, they do not guarantee a global
optimum.
Metaheuristic optimization methods allow for a set of solutions to evolve
through random processes that are guided with an objective function which
rewards low network costs and penalizes hydraulic insufficiencies. Examples
include evolutionary algorithms, which are most commonly genetic algorithms
(Krapivka and Ostfeld, 2009; Simpson et al., 1994; Kadu et al., 2008; Prasad
and Park, 2004), simulated annealing (Vasan and Simonovic, 2010;
Tospornsampan et al., 2007), ant colony optimization (Maier et al., 2003),
and differential evolution (Vasan and Simonovic, 2010). As reviewed by
Nicklow et al. (2010), evolutionary algorithms are an emerging popular
alternative to the deterministic methods, and they offer the opportunity to
accommodate unique constraints and multiple design objectives. The main
challenges for evolutionary algorithms are the difficulty of incorporating
constraints into objective functions, the optimum selection of parameters,
and a relatively large amount of computational effort. In addition to
optimizing for cost, multi-objective methods, often based on evolutionary
algorithms, allow the designer to choose from a Pareto optimal front of
objectives, such as cost and reliability (Prasad and Park, 2004). In addition
to water network design, metaheuristic algorithms have been used for a range
of problems in water resources engineering, such as rainfall and runoff
modeling (Taormina and Chau, 2015).
Decomposition methods involve the partitioning of networks into smaller
sub-networks which are each optimized using one of many types of techniques
and then combined into an overall solution. In some cases, the loops in the
sub-networks are removed, producing branching trees which are then optimized
individually. Techniques used to optimize the sub-networks can involve
multiple methods, including linear programming (Saldarriaga et
al., 2013) and differential
evolution (Zheng et al., 2013), with a later stage optimizing the network as a whole
using the sub-network solutions as inputs. Note that another distinct use of
the term “decomposition” refers to the approach of iteratively solving
“inner” and “outer” mathematical problem formulations, and has been used
in the literature by Krapivka and Ostfeld (2009) who traces its use in this
context back to Alperovits and Shamir (1977).
In the present study, we present three algorithms, each from one of three
major categories of methods applied to the cost optimization of water
distribution networks, and compare their performance on five cases adapted
from real GDWNs. These algorithms include (1) the calculus-based (CB)
optimization model of Jones (2011), an NLP method; (2) backtracking (BT), a
partial enumeration method; and (3) a genetic algorithm (GA), a metaheuristic
method. Major distinguishing features of these algorithms include their
working use of continuous diameters (CB) versus discrete diameters (BT and
GA), their deterministic (CB and BT) versus stochastic (GA) search process,
and their relative scalability as better (CB, GA) and worse (BT) for larger
networks. In terms of their ability to find a global optimum solution for the
problem formulation, CB finds a global optimum for continuous diameters but
cannot guarantee a discrete diameter global optimum in its mapped solution,
BT can guarantee a discrete global optimum, and GA cannot guarantee an
optimum. For a direct comparison of techniques, the pipe costs used for all
algorithms are found by interpolating a two-part cost formula based on a
curve fit of real cost data for available diameter values. The three
algorithms are tested against networks adapted from field data on five actual
GDWNs installed in Panama, Nicaragua, and the Philippines.
Within the broader context of water network problem formulations, this paper
is concerned with finding cost-optimal single-diameter solutions to
branching water distribution networks with steady-state demand flows and
pre-specified pipe locations. By implication of being gravity-driven, the
problem does not involve the use of pumping stations. This problem
formulation is directly applicable to typical gravity-driven water networks,
and is also useful for multi-objective algorithms, the consideration of
sub-networks in a decomposition technique, pumped networks, and looped
system optimization, which can involve reformulating the problem into a
branching configuration.
The results of this study highlight the advantages and weaknesses of each
GDWN design method including closeness to the global optimum, the ability to
prune the solution space of infeasible and suboptimal candidates without
missing the global optimum, and also computational time. We present two
pre-processors
which discrete-diameter search methods can use to reduce
the search space without pruning the global optimum. To the authors'
knowledge, this is the first implementation of “pre-processor 1” in
enumeration methods and the first implementation of “pre-processor 2” in
any water network design method. We also extend the Jones closed-form model
to include minor losses, a more comprehensive two-part cost model, which
realistically applies to pipe sizes that span a broad range typical of GDWNs
of interest in this work, and for smooth and commercial steel roughness
values.
Problem formulation
Branching networks are considered (Fig. 1), where all branches connect a
distribution main node with a delivery node, shown as tap stands or houses.
For each link in a network of NL links, pipe length (L) and the net
elevation change (Δz) are considered known and fixed.
Steady-state flow rates (Q) are prescribed for each link based on the
demand flow data at delivery nodes. As noted above, demand flows are
determined by community surveys and extrapolated in time to quantitatively
account for population growth. Minor losses are accounted for through a
minor loss coefficient K or a dimensionless equivalent pipe length,
(Le/D, or in symbol form, LebyD), where Le is the pipe length
of diameter D whose frictional loss results in the corresponding minor loss.
An optimal solution is obtained by selecting pipe diameters (D) from a set
of commercially available diameters such that the network's material cost is
minimized. With ND choices of diameters for NL links, the problem
has NDNL candidate solutions.
For all nodes, pressure head, h, is greater than or equal to a chosen
minimum, hmin. The value for hmin is selected to eliminate
possible leakage of contaminated ground water into the network should the
operating conditions change in an unanticipated way. The change in pressure
head, Δh, across each link is calculated with the energy
equation for pipe flow as follows:
Δh=-Δz+α+K+fLD+LebyD8Q2π2gD4,
where for each link, α is the kinetic energy correction factor and
f is the Darcy friction factor, calculated with the Colebrook–White equation
(Colebrook and White, 1937) or Churchill correlation (Churchill, 1977), and
g is acceleration of gravity. The kinetic energy correction factor, α, is considered only in the first link, where acceleration from a
zero-velocity source is sometimes non-negligible for the smallest of GDWNs
that have been encountered. Thus,
α=2Re≤23001.05Re>2300,
where Re is the Reynolds number for pipe flow, 4Q/πνD, and ν
is the kinematic viscosity of water. The possibility of laminar flow
(Re≤2300) is permitted since branches from the smallest GDWN
observed in practice have been in this regime.
The pressure upper bound is not incorporated into the optimization process.
Worst-case pressure conditions occur under hydrostatic conditions, which are
directly related to the maximum elevation change in the network and where no
flow occurs. Therefore, before the optimization process is undertaken, the
selections of appropriate pressure ratings for the pipe and, if needed,
break-pressure tanks are left to the correct judgment of the designer under
no-flow conditions. In addition, precautions against water hammer are left
to the designer.
New calculus-based algorithm
In this section we develop a new calculus-based algorithm for pipe diameters
that minimize overall pipe cost for the network. First appearing in Jones (2011), this algorithm is solved simultaneously with the energy
equation for each link to produce unique solutions for D and nodal pressure
head values that minimize network pipe cost, as opposed to only the
distribution main cost as in Swamee and Sharma (2000). The method also
applies to serial and loop networks but the focus for the present work is on
branching networks.
We assume continuous pipe diameters in this section; values that result from
the solution of the energy equation. Mapping between continuous diameters and
the discrete nominal sizes, required to complete the design, will be
addressed below.
Consider the three-pipe network shown in Fig. 2. Pipes 1–2, 2–3, and 2–4
meet where head h2 is unknown. Each pipe has a prescribed volume flow
rate and length and unknown diameter D as shown. The change in elevation
between the top and bottom of each pipe is Δz, and
Δh is the change in pressure head. There is a prescribed
head at each outlet for pipes 2–3 and 2–4.
Three-pipe branch network.
To facilitate insight, we at first assume turbulent flow (which can be
verified post-calculation if necessary) in smooth pipe and that minor losses
are negligible. Two sources for the friction factor for smooth-turbulent
flow are considered, namely the classical Blasius equation (reported in
Streeter et al., 1998), f=0.316Re-1/4, and the Swamee–Jain
correlation (Swamee and Jain, 1976), f=0.175Re-0.1923 (though not
explicitly appearing in this reference, f from the Swamee–Jain correlation is
obtained by writing it for smooth pipe and comparing this with the energy
equation, where f is assumed to be in the form
aRen). The Blasius equation has higher accuracy (2 % for low Re and
3 % for high Re) in the range 104 < Re < 105,
over which most of the GDWNs in this work operate, compared with the
Swamee–Jain correlation of +8 %/-3 %, thus the Blasius
equation is chosen for this work. A combination of the Blasius equation
with the energy equation gives explicit formulas for D for the three links
in Fig. 2. For simplicity, and to reduce the number of free parameters, the
conditions for pipes 2–3 and 2–4 are assumed to be identical without loss of
generality. We therefore obtain
D12=0.741Δz12+Δh12L1-4/19Q12ν1/7g4/77/19D23=D24=0.741Δz23+Δh23L2-4/19Q23ν1/7g4/77/19.
With our assumptions and inspection of Fig. 2, Δh12=-h2 and Δh23=Δh24=h2-h3=h2-h4, we furthermore obtain
D12=0.741Δz12-h2L1-4/19Q12ν1/7g4/77/19D23=D24=0.741Δz23-h3+h2L2-4/19Q23ν1/7g4/77/19.
The single-part pipe-cost model can be assumed to follow a power-law
relationship (Swamee and Sharma, 2008)
C=aDDub,
where C is cost per unit length of pipe, a is a constant coefficient, b
is a constant exponent, and Du an assumed unit diameter. A more
robust, two-part model, valid for a greater range of pipe sizes than that of
Swamee and Sharma (2008), will be used below. The use of pipe material cost
as the objective function was assumed because of relevance. In most GDWNs of
interest in this work, installation labor comes from the local community and
has no well-defined associated cost, such that the material cost for the
network is of prime importance. For a more general case, the economics of a
GDWN may be more encompassing and include materials, labor, operation and
maintenance, depreciation, taxes, and salvage, among others. The time value
of money may also need to be considered, which includes interest rates and
estimation of the network lifetime.
With Eq. (4) the general expression for the total cost for the pipe
material, CT, is obtained by summing over all links
ij,
CT=a∑ijLijDijDub,
which, for the present problem, becomes
CT=aL12D12Dub+L23D23Dub+L24D24Dub=aL12D12Dub+2L23D23Dub.
The mathematical basis for a unique solution for h2 with cost
minimization is now presented. In addition to the fixed pipe lengths, the
total cost depends on the diameters for all pipes in the network. For the
case of Fig. 2, where we now allow pipe 2–3 and pipe 2–4 to be different, we
get
CT=CTD12h2,D23h2,D24h2.
Using the chain rule from the calculus, the total differential of Eq. (7) is
dCT=∂CT∂D12∂D12∂h2dh2+∂CT∂D23∂D23∂h2dh2+∂CT∂D24∂D24∂h2dh2.
The minimum value of CT is found once dCT=0
(and once it is verified that the second derivative of CT is
positive thus indicating that CT is indeed a minimum). Requiring
this, we obtain
0=∂CT∂D12∂D12∂h2+∂CT∂D23∂D23∂h2+∂CT∂D24∂D24∂h2.
The cost CT is from Eq. (5), so the derivatives like ∂CT/∂D12 in Eq. (9) are written in general as
∂CT∂Dij=abDijb-1DubLij
for any link ij.
The derivatives like ∂D12/∂h2 in Eq. (9) are
obtained by taking the partial derivative of the pipe diameter with respect
to the relevant pressure head in the appropriate energy equation. For the
full energy equation, where D appears in a nonlinear way in more than one
location, this would be done using numerical methods. However, if we assume
minor-lossless, smooth-turbulent flow as noted above, we can use the energy
equations like Eq. (3). We therefore obtain the following for
pipe 1–2:
∂D12∂h2=0.156Δz12-h2L12-2319ν1/7Q12g4/7L1219/7719;
for pipe 2–3, we get
∂D23∂h2=-0.156Δz23+h2-h3L23-2319ν1/7Q23g4/7L2319/7719;
and for pipe 2–4,
∂D24∂h2=-0.156Δz24+h2-h4L242319ν17Q24g47L24197719.
Equations (10)–(13) are combined with Eq. (9) to produce a single algebraic
equation that depends on h2, as well as D12, D23, and
D24. Introducing D12,D23, and D24 from Eq. (3) into this
algebraic equation, we get
0=Q127b/19Δz12-h2L12-(1+4b/19)-Q237b/19Δz23+h2-h3L23-(1+4b/19)-Q247b/19Δz24+h2-h4L24-(1+4b/19).
The general form of Eq. (14), written at any internal node is
0=∑ij,inQij7b/19Sij-(1+4b/19)-∑ij,outQij7b/19Sij-(1+4b/19),
where the hydraulic gradient, Sij, is
Sij=Δzij+ΔhijLij.
In Eq. (15) the indices ij,in and ij,out on the summations refer to inflows and outflows
at the node (e.g., in Fig. 2, ij,in = 12 and ij,out = 23 and 24). Equation (15), the
new CB algorithm proposed in this work, is written for each internal node in
the network and solved simultaneously with the energy equation for each link
to obtain unique and optimal values of Dij for all links and
hj for all internal nodes. It is understood that the nodal
pressure heads determined from the solution of this system must be greater
than or equal to the hmin prescribed for the network. For nodes that do
not satisfy this condition, the pressure head is set equal to hmin, as
part of the CB algorithm. Thus, hj≥hmin.
Minor losses using the equivalent-length method can be included in the above
developments by artificially extending the length of the link by Le in
which minor loss occurs, thus contributing a non-zero LebyD term in Eq. (1). We also extend the cost model of Eq. (5) from Swamee and Sharma (2008)
to encompass two different ranges of pipe diameters having two different
coefficients a and exponents b. The link between the two ranges starts at
discrete pipe size Dco, at and below which the cost model for the small
(subscript s) pipe sizes applies, and discrete pipe size Dco+1, at and
above which the cost model for the large (subscript l) pipe sizes applies.
The cutoff diameter, Dco is chosen by the designer based on inspection
of cost vs. diameter data. Thus,
Cij=LijasDijDubs,Dij≤Dcoc1+c2DijDu+c3DijDu2+c4DijDu3,Dco<Dij<Dco+1alDijDubl,Dij≥Dco+1.
In Eq. (17), as and al are the coefficients for the
small (designated by subscript s) and large (subscript l) pipe size regions, respectively, and bs and
bl are the exponents for the small and large pipe size regions,
respectively. A cubic spline is fit between pipe sizes Dco and
Dco+1 to complete the transition between small and large pipe
sizes. The coefficients of this polynomial are c1, c2, c3, and
c4 as seen in Eq. (17). These coefficients are evaluated by matching the
cubic polynomial and pipe data at Dco and Dco+1 and
the first derivative of the polynomial with respect to Dij/Du
to
asbsDcoDubs-1
at Dij=Dco and to
alblDco+1Dubl-1
at Dij=Dco+1. An example of data for polyvinyl chloride (PVC)
pipe and the curve fit is shown in Fig. 3. The results of the curve fit are
as follows: Dco=2.067 in., Dco+1=2.469 in.,
as= USD 1.349 m-1, bs=1.157,
al= USD 1.381 m-1, bl=1.344,
c1= USD 237.516 m-1, c2= USD 316.125 m-1,
c3= USD 140.450 m-1, c4=-USD 20.499 m-1.
It is clear from inspection of Fig. 3 that a one-part cost model would not
have produced an acceptable curve-fit to pipe-cost data.
With the inclusion of the two-part cost model and minor loss term, Eq. (15)
becomes
0=∑ij,inCij′Aij4191+ϵij419Sij-2319Qij7νg4Du191191-BAij419ϵij′1+ϵij-1519Sij-419Qij7νg4Du19119-∑ij,outCij′Aij4191+ϵij419Sij-2319Qij7νg4Du191/191-BAij419ϵij′1+ϵij-1519Sij-419Qij7νg4Du191/19,
where B=0.1989 and
ϵij=∑kLeDk,ijDijLijϵij′=∑kLeDk,ijDuLijAij=0.318,smoothpipe0.420,steelpipe
and A accounts for the effect of pipe roughness (smooth and commercial
steel). The term Cij′ is the derivative of the cost
function per unit length with respect to D/Du. For the two-part cost
model from above, we obtain
Cij′=asbsDijDubs-1,Dij≤Dcoc2+2c3DijDu+3c4DijDu2,Dco<Dij<Dco+1alblDijDubl-1,Dij≥Dco+1.
PVC pipe cost from 2011 data.
Equation (18), and its simpler form Eq. (15) for minor-lossless flow and a
single-part pipe-cost model (it is easy to show that Eq. (18) regresses to
Eq. (15) for these conditions), is the root of the calculus-based
optimization in this work and is applied at all internal nodes to uniquely
determine hj. Equation (18) is valid over the range of
∼ 4000 < Re < ∼ 300 000. Algorithms
to solve a general set of independent, nonlinear algebraic equations using,
for example, the Levenberg–Marquardt, quasi-Newton, Newton–Raphson, or
conjugate gradient methods are available in most commercial math packages
including Matlab (1 Apple Hill Drive, Natick, MA USA 01760) and Mathcad
(http://www.ptc.com 31 January 2018). We used the package Mathcad in
the present work. Thus, compared with an iterative solution procedure, a
solution flowchart is not relevant here.
Bhave (1978) first proposed an algorithm like Eq. (15) using slightly
different notation than here. For clarity, we re-present Eq. (15) using
Bhave's notation as
0=∑Qij7b/19Sij-(1+4b/19)-∑Qjk7b/19Sjk-(1+4b/19),
where the ij and jk notation are shown in Fig. 4. Index j spans all internal nodes along the distribution main. A
quantitative comparison between Eq. (18) and the method of Bhave is
presented below.
Backtracking algorithm and genetic algorithm
Bhave (1978) index notation at an internal node, j.
Backtracking (BT) and genetic algorithm (GA) assess candidate solutions
composed of discrete diameters from a commercially available set. These
candidates are represented by a vector of NL elements where each
element corresponds to a commercially available diameter of a network link.
To reduce the computational time associated with these evaluations, the
constraints imposed by the energy equation and cost minimization may be more
efficiently evaluated through lookup tables. With fixed L,
Δz, K, LebyD, and α, the change in
pressure head Δh is evaluated for all ND×NL
combinations of pipe diameter and link index:
Δh=Δh11⋯Δh1NL⋮⋱⋮ΔhND1⋯ΔhNDNL.
While an algorithm evaluates a candidate solution, the pressure head at each
node is sequentially calculated by “marching” through the network. Starting
with the fixed source pressure head, the algorithm finds the pressure head
hi for a given node by adding the head at the upstream node, hi-1
to the change in head for that link iL and the diameter
iD under consideration. Thus,
hi=hi-1+ΔhiD,iL.
Along with the hydraulic evaluation of a candidate solution, the cost of the
partial candidate is found through the use of a lookup table C:
C=C11⋯C1NL⋮⋱⋮CND1⋯CNDNL,
where C(iDiL) returns the additional cost of assigning a
diameter with index iD to link iL. In this way, the candidate
solution's hydraulic performance and cost are incorporated into the genetic
algorithm and backtracking approaches. In contrast to GA, the backtracking
algorithm evaluates pressure head and cost upon consideration of each
partial candidate, where GA calculates these values on full candidates as
part of the objective function.
BT and GA pre-processor 1: maximum available diameter
To increase the efficiency of BT and GA, it is advantageous to limit the
number of pipe diameters in the available set, especially those outside of
the range of the optimal solution. For the BT algorithm in particular,
larger diameters can require considerable computational effort, since they
tend not to violate static head requirements and require multiple-link
partial candidates for the algorithm to reject them once their cost exceeds
that of an already-found viable candidate. Therefore, a pre-processor is
used to provide a maximum diameter (Dmax) that should be considered
during the optimization process. This procedure, which produces a
conservative estimate, finds the smallest diameter at which a network with a
single pipe diameter choice produces no nodes with a pressure head below
hmin, similar to the technique used by Mohan and Jinesh Babu (2009).
After this diameter is found, the next larger diameter in the set is
selected as Dmax to allow the algorithm to select a
larger than necessary diameter if this is able to save cost elsewhere. It
worth noting that Kadu et al. (2008) presents another method to further
prune the search space with the critical path concept, where Dongre and
Gupta (2011) noted the computational advantages of having just four diameter
choices per link. This method, however, may prune the global optimum and may
not produce feasible head values at intermediate nodes, as in the case of
networks with a local high point.
BT and GA pre-processor 2: adjusted minimum pressure head
A second pre-processor adjusts the minimum pressure head requirement for
each internal node by considering the total head required at downstream
nodes. It can be recognized that, without the use of a pump, the total head
cannot increase at nodes downstream of a given node i. Furthermore, the
total head must decline at a minimum grade that is determined by the demand
volume flow rate and the largest pipe diameter available (Dmax) for
selection. This energy constraint is utilized to reduce the number of
candidates to be considered by increasing the minimum pressure head at nodes
where these rules produce a higher minimum head than the original hmin.
For example, nodes upstream of a local network high point can have their
minimum pressure head increased beyond the normal minimum, since the
pressure head must be great enough to ensure adequate flow to the
higher elevation downstream node. To begin this process, each node i is
initialized with a baseline minimum total head:
thmin,i=zi+hmin.thmin,i is thus initialized by considering only the node's hydraulic requirements in
isolation, i.e., without acknowledging the neighboring downstream nodes. The
pre-processor then considers updating thmin,i by checking the following
condition, which is false when the minimum pressure head at downstream nodes
produces further constraints on an upstream node i. Thus, for all nodes
i which are upstream of some node j, the following inequality can be
evaluated:
thmin,i-thmin,j≥αi-j+Ki-j+fi-jLi-jDi-j+LebyDi-j8Qi-j2π2gDmax4.
Also, consider that when flow rate Qi-j is small and Dmax is
large, the right-hand side of Eq. (26) approaches zero, representing the
simple statement that upstream total head must always be greater than
downstream total head. When the condition in Eq. (26) is false, the minimum
total head can be updated in node i such that the maximum diameter size in
link i-j is able to meet the downstream node's minimum total head, or
thmin,i=thmin,j+αi-j+Ki-j+fi-jLi-jDi-j+LebyDi-j8Qi-j2π2gDmax4.
In this way, thmin,i may be updated for each node until the condition
in Eq. (26) is true for all nodes i with a downstream node j connected
by a single link.
After the values for thmin,i are updated, they are converted back into
minimum pressure head values by subtracting the elevation zi from
thmin,i. This pre-processor serves to narrow the search for viable
candidate solutions by potentially increasing the minimum pressure head.
Since backtracking and GA consider network links in the downstream
direction, these algorithms are otherwise blind to future downstream
pressure head requirements. This limitation is alleviated by the
pre-processor, which allows these algorithms some implicit information about
what local diameter choices will be viable for the full network solution.
Note that both pre-processors discussed will not prune the global optimum
from the solution.
Backtracking algorithm (BT)
The backtracking algorithm is a partial enumeration method that employs a
systematic search of candidate solutions to find a global optimum. The
algorithm works recursively to incrementally build candidate solutions while
checking the candidates for hydraulic and cost acceptability. The strength
of the BT is that, upon discovery of an infeasible partial candidate, all
extensions of that candidate can be eliminated from consideration. In this
way, many solutions can be pruned from the solution tree to achieve greater
computational efficiency.
Two backtracking methods in the literature are those by Gessler (1985) and
González-Cebollada et al. (2011). The algorithm proposed by Gessler
proposes a pipe-grouping strategy which speeds up the algorithm but risks
pruning the global optimum. Additionally, pipe grouping represents its own
optimization problem (Raad, 2011). The González-Cebollada algorithm does
not include such pipe-grouping criteria, although it does halt its search
after finding the first feasible solution, thus it does not guarantee a
global optimum. The present study's BT algorithm, once run to completion,
does guarantee a global optimum. It operates similarly to the method
presented by González-Cebollada et al. (2011), with the major
differences being that the algorithm continues searching once it has found
its first feasible solution and uses pre-processors 1 and 2 to further
reduce the search space. This implementation of BT, however, scales poorly
with larger network sizes and would not be appropriate for use on large
urban networks. Its appropriateness is shown here for many of the GDWNs
encountered in practice, as evidenced by its use on real-world GDWN test
cases in this paper. Moreover, it serves as a benchmark against which other
algorithms can be compared.
BT uses two rejection criteria to discard candidate solutions from further
consideration. The first rejection criterion is that when a candidate
violates pressure head constraints, all candidates with equal or lesser
diameter sizes can be discarded. This condition is leveraged even more
effectively with pre-processor 2 above, which can increase pressure heads at
individual nodes by anticipating the head requirements at surrounding nodes.
The second rejection criterion is that once a feasible candidate has been
found, all other partial candidates with a higher cost can also be
discarded. The BT algorithm further extends this second criterion by
considering that the links yet to be considered in a partial candidate, an
“extension” to the partial candidate, will cost at a minimum that of the
entire extension being composed of the smallest available diameter.
The backtracking algorithm begins its search of the solution tree by
considering the partial candidate with the smallest diameter size assigned
to the first network link. The pressure head and the partial candidate cost
at the outlet node are calculated with the Δh and C lookup tables. If this partial candidate
meets pressure head and cost requirements, the algorithm extends this
partial candidate by assigning the smallest diameter to the downstream link.
If a partial candidate produces a node that is rejected on the basis of
pressure head, the next largest larger diameter is chosen for the link
upstream of the node. If no diameter satisfies the pressure head condition,
the algorithm backtracks to the upstream link and assigns a larger diameter
to the link. In this way, the algorithm continues to extend and reject
candidate solutions until a full candidate satisfies the pressure head
requirements. Once a working solution has been found, candidate solutions
may be rejected based on cost. For each new candidate, cost is calculated by
adding the cost of diameters that have already been assigned to the cost of
assigning all downstream links with the smallest diameter available. If this
cost exceeds the cost of the running optimum, the partial candidate is
rejected. While the minimum pressure head criterion tends to prune
candidates with diameters that are too small, the cost-based criterion tends
to prune candidates of diameters that are too large.
Modified backtracking algorithm (BT-NoUp)
A modification to the BT algorithm was made to further improve its
computational speed, although at the risk of pruning the global optimum from
the search. This modified algorithm (BT-NoUp) rejects all candidates that
feature a smaller diameter that is upstream of a larger diameter when an
equal or smaller flow rate is present in the downstream link. Typically, a
network designer would not consider such designs, and in cases where a
single source feeds into a network with constant-length links, it is
advantageous (or equivalent) to place larger diameters upstream of smaller
diameters. However, due to the discrete nature of diameter choices and link
lengths, an optimization problem may, in fact, have an optimal candidate
with a larger diameter downstream from smaller ones. For this reason, the
BT-NoUp algorithm, unlike the BT algorithm, may miss the global optimum at
the expense of its greater computational efficiency.
Genetic algorithm (GA)
Genetic algorithms are stochastic optimization techniques that mimic the
process of natural selection, and numerous variations of GAs have
demonstrated acceptable performance on WDN design (Nicklow et al., 2010).
Given their popularity, the GA included in this study is meant to provide a
point of comparison to the BT and CB algorithms when applied to GDWNs.
When implemented in water network design, each candidate solution represents
a selection of pipe diameters. The algorithm is initialized with a
population of candidates of size Nc that repeatedly undergoes the
processes of mutation, crossover, and selection
ci=D1,iD2,i…DNL,i,
where each candidate in the population ci contains NL diameters.
In the present work, candidates are represented as a string of natural
numbers, which is used over a binary representation to improve the ease of
encoding (Vairavamoorthy and Ali, 2000). The mutation operator replaces pipe
diameters with a diameter from a uniform random distribution, where each
link diameter has a probability of pmut of mutating on each generation.
The crossover operator randomly pairs individuals in the population with
probability pxover and performs a single-point crossover of the two
individuals, where the point of crossover is randomly chosen. The fitness,
fi, of each candidate is assessed with penalties associated with the
solution's pipe cost, Cpipe,i, and hydraulic cost, Chyd,i, which
is assigned when violations of the pressure head requirements occur:
fi=1Cpipe,i+Chyd,i.
The hydraulic cost is found for each individual by identifying nodes in
which the pressure head is less than hmin and multiplying the total
amount of head violation by a hydraulic penalty coefficient, ahyd:
Chyd,iC=ahyd∑1NLhmin-hiN|hiN<hmin.
To allow for a hydraulic penalty coefficient to produce similar results in
both small-scale (inexpensive) network and a large-scale (more expensive)
cases, the hydraulic penalty coefficient is made directly proportional to
the average solution cost. With each generation, ahyd is updated by
multiplying the normalized penalty coefficient, ahyd,norm, by the
average pipe cost of the population,
ahyd=ahyd,norm∑1NcCpipe,iCNc.
The algorithm then selects candidates to be carried into the next generation
with a tournament selection method, where Nc groups of s individuals
are randomly assigned and the fittest candidate among each group is
selected, thus replacing the previous population with an equally sized
population of Nc individuals.
Characteristics of test cases.
Test caseTypeNumber ofNumber ofQtotLtotdiameter choiceslinks(L s-1)(km)(1) Kiangan, PhilippinesBranching894.370.82(2) Los Modulos, NicaraguaSerial4130.391.24(3) Cañazas, PanamaBranching10236.2915.2(4) San Miguel, NicaraguaSerial9100.401.18(5) El Guabo, NicaraguaBranching121717.74.71(6) Los Mangos, NicaraguaBranching7591.922.64
In this study, the genetic algorithm parameters used were Nc=200,
pmut=0.05, pxover=1, Ngen=500,
ahyd,norm=0.05, and s=10. These parameters were chosen by
systematically varying parameter values until the optimum cost of a network,
case 2, could no longer be significantly improved. The first four of these
values are in line with typically used values from Simpson et al. (1994) of
Nc (30–200), pmut (0.01–0.05), pxover
(0.7–1.0), and Ngen (100–1000).
Cases studied
Six cases were studied based on actual GDWN in Panama, Nicaragua, and the
Philippines. Global characteristics of each network are presented in Table 1
and the details of each network are presented in Table 4a–f. Each
network is a branching type without loops. The total lengths of the networks
range from less than 1 to over 15 km. Two serial networks are tested to
demonstrate the effect of a local high point on the algorithm solutions.
Elevation plots for each case are shown in Fig. 5.
The choice of hmin is not standardized, and should appropriately
balance the risk of negative pressure in pipes and the increase in network
cost due to the requirement of using larger diameters. The choice of
hmin in GDWN design is typically in the range of 5–20 m (Arnalich,
2010; Bouman, 2014; Swamee and Sharma, 2008). In the present study hmin= 7 m, although this requirement was reduced at selected nodes at the
beginning of a network where changes in elevation are still small (case 2,
where the pressure head at node 2 is relaxed to 2 m). At the source node,
the pressure head is fixed at atmospheric pressure. All cases assumed
minor-lossless flow, although all algorithms (e.g., Eq. 18 for CB-Theor)
are capable of handling minor loss coefficients through the equivalent
length method as described above. All algorithms were run in a late-version
of MATLAB (or Mathcad for CB) on an Intel i5 processor at 2.50 GHz.
Network elevation (z) and hydraulic grade lines (HGLs) of
algorithm solution for main distribution links.
Mapping the theoretical D to discrete pipe sizes
The mapping between continuous diameters and the discrete nominal pipe sizes
was accomplished in our solution in one of the following ways:
For small and moderate size networks, the designer may manually adjust the
pipe sizes (downward, normally one pipe size) starting from the first link
downstream from the source and continuing along the rest of the distribution
main to the end in a step-by-step manner. A nearby plot of the pressure
heads compared with the theoretical Dij from the CB approach (e.g., on
the same Mathcad page for our solution) will highlight the acceptability or
unacceptability of any change. This exercise also gives the designer
valuable understanding of the sensitivity of the design to small changes in
pipe sizes.
Based on the theoretical Dij from the CB approach, a split-pipe can be
created for each link. That is, the lengths for the two discrete pipes sizes
that bound the theoretical Dij from above and below are calculated such
that the pressure drop between two consecutive nodes in the distribution
main matches between the composite pipeline and the CB approach. This also
provides discrete pipe sizes that nearly match the CB solution in terms of
cost.
Results
The current study evaluated three types of algorithms that optimize the
design of gravity-driven water networks (GDWN). The algorithms include the
calculus-based (CB) algorithm (Eq. 18), a backtracking algorithm (BT) and
its modified version (BT-NoUp), and a genetic algorithm (GA). The algorithms
were applied to six test cases that are based on real GDWNs. Our results
show that the CB, GA, and BT-NoUp algorithms could find solutions to the
GDWNs within 25 % of the BT global optimum. All cases assume
minor-lossless flow and a two-part pipe-cost model. Solution costs from each
algorithm are shown in Table 2 and runtime statistics are shown in Table 3.
BT could run to completion in < 1 min in all but the largest case
(case 6 with 59 links), which did not complete after 7 days. As such,
cost comparisons to BT are not made for case 6.
Solution costs for each algorithm.
CaseSolution cost (USD) Percentage cost increase over Percentage cost increase BT (global optimum) over CB-Theor BTBT-NoUpCB-TheorCB-DiscGABT-NoUpCB-TheorCB-DiscGABTBT-NoUpCB-DiscGA(1) Kiangan, Philippines233123312257259423370-3.211.30.33.33.314.93.5(2) Los Modulos, Nicaragua144114721404176714452.1-2.622.60.32.74.825.92.9(3) Cañazas, Panama72 19072 44368 24584 44173 9640.4-5.517.02.55.86.223.78.4(4) San Miguel, Nicaragua541854185172562754220-4.53.90.14.84.88.84.8(5) El Guabo, Nicaragua61 44561 44559 50673 88663 1130-3.220.22.73.33.324.26.1(6) Los Mangos, Nicaragua∗4082367044054339∗∗∗∗∗11.220.018.2
∗ Note: BT did not complete after 7 days of
runtime.
Runtime and size of solution space for each algorithm.
a BT, BT-NoUp, and GA algorithm run times do not
include approximately 2 s of pre-processing time. b BT did not
complete case 6 after 7 days of runtime.
Diameter sizes from calculus-based (CB-Disc) solutions compared with
global optimum solutions (from backtracking, BT). A global optimum for
case 6, Los Mangos, is not included since BT did not complete after 7 days of runtime.
The CB algorithm based on Eq. (18), unlike the other algorithms in this
work, finds a solution with theoretical diameters that are drawn from a
continuous domain (CB-Theor). For all test cases, the costs of the CB-Theor
solutions was less when compared with the BT discrete-diameter global
optimum (5.5 to 2.6 % lower cost than BT). In fact, because of the
discrete pipe sizes needed for an actual network, the continuous model will always produce the smallest theoretical network cost. The CB algorithm then
maps this solution to a commercially-available discrete set (CB-Disc). The
mapping process used in this study simply mapped each theoretical diameter
to the nearest available diameter of a larger size, thus producing a
solution which still satisfies static head requirements but with a higher
associated material cost. This tended to oversize the diameters, although
the CB-Disc solutions were always within two diameters of the known global
optimum solutions, as shown in Fig. 6. From all the combined test cases with
known global optima, all but one (71 out of 72) of the diameter selections
were within one diameter of the global optimum. More sophisticated mapping
schemes, like independently adjusting D for each link in the distribution main
in a step-by-step manner starting with the source while ensuring all
pressure head constraints are satisfied, would be more likely to produce
results identical to the global optimum (see Sect. 6). This was performed
in the current study but the results are not presented because of space
constraints. The CB-Disc solution costs were, in all cases, larger than the
global optimum, with costs ranging from 3.9 to 22.6 % above the global
optimum. Thus, for all cases, the calculus-based algorithm bounded the cost
of the global optima with a lower-cost CB-Theor solution and a higher-cost
CB-Disc solution. This trend is a result of the additional constraints
imposed by the finite set of diameter choices. If the algorithm is allowed a
greater number of discrete diameter choices, i.e., through adding a
less-common nominal diameter size to the available set, the cost of the
CB-Disc solution would approach the CB-Theor solution. For all but case 6,
the CB algorithm converged on a solution in 3 min or less.
Case network properties, diameter (D) results (inch nominal sizes,
with CB-Theor in inches), and nodal h (in meters) for (a) Case 1, Kiangan, (b) Case 2,
Los Modulos, (c) Case 3, Cañazas, (d) Case 4, San Miguel, (e) Case 5, El
Guabo, and (f) Los Mangos.
BT-NoUp, a modified version of BT which does not consider smaller diameters
upstream of large diameters, completed itself within 4 s for all
cases, and found solutions which matched or came very close to the BT global
optimum. BT-NoUp missed the global optimum in cases 2 and 3, although by a
small percentage increase in cost (2.1 and 0.4 % respectively).
BT-NoUp, however, finished its search in a shorter amount of time in
comparison to BT, a benefit that becomes relevant on problems with larger
solution spaces, such as cases 3 (1.0 × 1023 candidate solutions) and
case 6 (7.3 × 1049 candidate solutions).
GA was run on each case a total of 100 times, each run itself evolved
200 candidates for 500 generations. The lowest-cost candidate amongst the
final population that did not violate the pressure head condition was chosen
as the GA solution. Because GA is a stochastic search algorithm producing
different results from run-to-run, the costs of the optima from all 100 runs
were averaged, with this averaged value presented in Table 2. Overall, GA
costs came close to the global optima (within 3 %) for cases 1–5 where
the global optimum was known from BT. GA solution costs increased with larger
network sizes, with its solution cost 18 % higher than CB-Theor for
case 6, the largest case run. Each GA run finished consistently within
1–5 s, not including about 2 seconds of pre-processor time. We note that
variations of GAs have been reported in the literature and several of these
may improve upon the GA results obtained in this study. Potential
improvements to the GA a self-adapting penalty function (Wu and Walski,
2005), the use of elitism to preserve the best solutions (Kadu et al., 2008),
and a reduction in the search space (Kadu et al., 2008). One reported
improvement, the scaling of the fitness function to magnify the rewards
towards slightly fitter candidates at later generations (Dandy et al., 1996),
was attempted for case 2 but did not result in a noticeable effect on
performance.
Optimization results from Bhave (1978) algorithm. LHS sum and RHS
sum are the left and right sides of his Eq. (19), which should be equal.
To visually compare the algorithm solutions, the hydraulic grade lines from
BT, BT-NoUp, CB-Theor, and CB-Disc are presented in Fig. 5 along with the
network elevation for each test case. For clarity, the hydraulic grade lines
of branch links are omitted from the figure. In addition, the GA solutions
are omitted since 100 solutions were obtained for each test case.
Collectively, the hydraulic grade lines reveal a close alignment of the BT
solution (the global optimum) with the CB-Theor solution which utilizes a
continuous diameter set. Furthermore, the mapping scheme used to generate a
CB-Disc solution is shown to increase pipe sizes in some cases far beyond
the limit imposed by hmin, which was set to 7 m in the present work.
We compared the CB results for the Los Mangos network with those from the
Bhave (1978) optimization algorithm (see Table 5). Like Eq. (15) in the
present work, Bhave's optimality equation (his Eq. 19) equates the sum of
a weighted term for all links entering and leaving each internal node in the
distribution main. In the present work the term is proportional to the
hydraulic gradient and the weighting factor is proportional to flow rate. In
Bhave's case the term is the ratio of pipe cost to head loss, where the
weighting factor is pipe-cost exponent b. There are 60 nodes for this
network, including 30 nodes in the distribution main. The rest are delivery
nodes (note there are 2 branches from node 30 of the distribution main). The
terms required for the calculations include b, pipe cost, and head loss in
the main and branches. The designation LHS refers to nodes in the
distribution main entering, and RHS to those leaving, the node at the far-left
side of Table 5. The exponent b comes from curve fitting pipe-cost data to
the two-part pipe-cost model. Linear interpolation was used between
diameters Dco and Dco+1 to obtain b in this range. Except for a few
nodes, agreement between the two CB algorithms is very good. Although Bhave's
Eq. (19) and Eq. (18) in the present work, appear quite different due to
the different ways each was developed, both produce optimality for the
networks considered in this paper. The key distinction between the two
developments is the assumption of constancy in terms that comprise the
coefficient A in Bhave (his Eq. 13), mainly the exponent b (m in his paper).
In general, b depends on pipe diameter, thus making b=b(D) for multi-part cost
models. When taking derivatives to obtain the final algorithms in both
works, this dependence must be included, which produces additional terms in
the optimization equation (see our Eq. 18 above). However, if the system
of equations is solved by an iterative method, as Bhave proposed, the
dependency may be neglected (though issues with convergence of the numerical
solution may arise because of this). It is very important to note that if a
non-iterative method is used to solve the system of equations as done in the
present work (using a commercial program like Mathcad), all terms in the
governing equations must be treated as continuous, not discrete, and the b=b(D)
dependence must be explicitly included. It should also be noted that the
optimization algorithm of Eq. (18) in this paper includes minor loss, which
is not included in the Bhave (1978) work.
Conclusions
Algorithms to optimize the cost of branching gravity-driven water networks
are evaluated on six test cases from real networks in the Philippines,
Nicaragua, and Panama. A calculus-based algorithm produced a solution
composed of theoretical diameters from a continuous set (CB-Theor), which
are then mapped onto discrete commercially available diameters (CB-Disc).
Backtracking (BT), a recursive algorithm, systematically searches discrete
candidate solutions and, when run to completion, is guaranteed to find the
global optimum by following rules that prune only higher-cost or
hydraulically infeasible candidates. The BT algorithm was modified (BT-NoUp)
to improve computational speed by rejecting all candidates that included a
small diameter directly upstream of a larger diameter but allowed for the
possibility of missing the global optimum. The third type of algorithm
evaluated was a genetic algorithm (GA) that used single-point crossover and
tournament selection.
BT could find the global optimum in most test cases with relatively little
computational effort, although its poor scaling to larger networks is
evidenced by its inability to find a solution to case 6, a network with 60 nodes and 59 links. The BT-NoUp completed its search in less time than BT
and could find a solution to case 6. Based on case 1–5 results, the extra
pruning condition adopted in BT-NoUp sacrificed only small cost increases.
Both BT and BT-NoUp, however, could become prohibitively time-consuming when
dealing with networks with significantly more links, diameter choices, or an
unfavorable layout. While the test cases represent the range of GDWN sizes
encountered in the authors' experience, future work would be needed to
verify the suitability of the BT and BT-NoUp algorithms on larger GDWNs. The
calculus-based algorithm produced consistently good results for the networks
tested, although a more robust mapping scheme from theoretical diameters to
discrete diameters would further improve on these results as discussed
above. In potential future work, the CB-Theor solutions could be used to
prune the BT search space, like Kadu et al. (2008), by only including the
two diameters above and below the CB-Theor diameters, producing four
diameter choices per link. The calculus-based methodology provides an
additional benefit to the designer by explicitly revealing the sensitivities
to cost for a design. The calculus-based algorithm requires greater
computational effort than backtracking for smaller networks, however, this
effort scales more linearly with the number of network links, while
backtracking scales exponentially. Furthermore, backtracking's computational
time is sensitive to the number of available diameters. Still, when applied to the present study's GDWN test cases with a modest number of links (23), backtracking quickly found a global optimum. In addition, because it is guaranteed to
find the global optimum, it can be useful for benchmarking the performance
of other algorithms which scale better with more network links. While the
genetic algorithm produced solutions with good proximity to the global
optimum, its solution costs tended to be further from the global optimum in
cases with more links.
For all test cases, the calculus-based algorithm's theoretical diameter
solutions (CB-Theor) produced a lower cost than the discrete-domain global
optimum. This result is made possible because it is not constrained to a
discrete set of diameters. As such, the CB-Theor results represent a
lower-bound on the optimum solution within the problem formulation, which
could be approached with a finer selection of pipe diameters. We also
demonstrated good agreement between the CB-based optimization algorithm
developed here and that of Bhave (1978). Though Bhave's algorithm and
Eq. (18) in the present work appear quite different due to the different
ways each was developed, both produce optimality for the networks considered
in this paper. The key distinction between the two developments is that Bhave
assumed exponent b constant in the pipe-cost model, which was justified
based on his iterative method of solution. In the present work, which uses a
commercial program to solve the nonlinear governing equations for D and
h, b(D) dependence is explicitly included for multi-part cost models.
Contrasted with Bhave, minor losses are included in the CB optimization
algorithm in the present work.
All survey data from the network cases tested are available in Table 4.
The authors declare that they have no conflict of
interest.
Acknowledgements
This work was partially supported by the Villanova Undergraduate Research Fellowship Program and the Goldwater Foundation.
Edited by: Luuk Rietveld
Reviewed by: three anonymous referees
ReferencesAlperovits, E., and Shamir, U.: Design of optimal water distribution systems,
Water Resour. Res., 13, 885–900, 10.1029/WR013i006p00885, 1977.
Arnalich, S.: How to design a Gravity Flow Water System, Arnalich – Water
and Habitat, 2010.
Bhave, P. R.: Optimization of Gravity-Fed Water Distribution Networks:
Theory, J. Environ. Eng.-ASCE, 109, 189–205, 1983.
Bhave, P. R.: Non-computer optimization of single source networks, J.
Environ. Eng.-ASCE, 104, 799–814, 1978.
Bhave, P. R.: Optimal design of water distribution networks, Alpha Science
International Ltd., Pangbourne, UK, 2003.
Bouman, D.: Hydraulic design for gravity based water schemes, Aqua for All,
Den Haag, the Netherlands, 2014.
Churchill, S. W.: Friction factor equation spans all regimes, Chem. Eng. J.,
84, 91–92, 1977.Colebrook, C. F. and White, C. M.: Experiments with fluid friction in
roughened pipes, P. R. Soc. London, 161, 367–381,
10.1098/rspa.1937.0150, 1937.Dandy, G. C., Simpson, A. R., and Murphy, L. J.: An improved genetic
algorithm for pipe network optimization, Water Resour. Res., 32, 449–458,
10.1029/95WR02917, 1996.Dongre, S. R. and Gupta, R.: Discussion of `Recursive Design of Pressurized
Branched Irrigation Networks' by César González-Cebollada, Bibiana
Macarulla, and David Sallán, J. Irrig. Drain Eng., 138, 697–697,
10.1061/(ASCE)IR.1943-4774.0000441, 2011.
Gessler, J.: Pipe Network Optimization by Enumeration, Proceedings of the
Specialty Conference on Computer Applications in Water Resources, American
Society of Civil Engineers, New York, USA, 572–851, 1985.González-Cebollada, C., Macarulla, B., and Sallán, D.: Recursive
Design of Pressurized Branched Irrigation Networks, J. Irrig. Drain Eng.,
137, 375–382, 10.1061/(ASCE)IR.1943-4774.0000308, 2011.
Jones, G. F.: Gravity-driven Water Flow in Networks: Theory and Design,
Wiley, Hoboken, NJ, USA, 2011.Kadu, M. S., Gupta, R., and Bhave, P. R.: Optimal design of water networks
using a modified genetic algorithm with reduction in search space, J. Water
Res. Plan. Man., 134, 147–160, 10.1061/(ASCE)0733-9496(2008)134:2(147),
2008.
Kansal, A., Gupta, R., and Bhave, P. R.: Optimization algorithms for design
of branching water distribution networks, J. Indian Water Works Association,
28, 135–140, 1996.
Keedwell, E. and Khu, S.: Novel Cellular Automata Approach to Optimal Water
Distribution Network Design, J. Comput. Civil. Eng., 20, 49–56,
10.1061/(ASCE)0887-3801(2006)20:1(49), 2006.Kessler, A. and Shamir, U.: Analysis of the Linear Programming Gradient
Method for Optimal Design of Water Supply Networks, Water Resour. Res., 25,
1469–1480, 10.1029/WR025i007p01469, 1989.Krapivka, A. and Ostfeld, A.: Coupled genetic algorithm – Linear programming
scheme for least cost design of water distribution systems, J. Water Res.
Plan. Man., 135, 298–302, 10.1061/(ASCE)0733-9496(2009)135:4(298), 2009.Maier, H. R., Simpson, A. R., Zecchin, A. C., Foong, W. K., Phang, K. Y., Seah, H. Y., and Tan, C. L.: Ant colony
optimization for design of water distribution systems, J. Water Res. Plan.
Man., 129, 200–209, 10.1061/(ASCE)0733-9496(2003)129:3(200), 2003.
Martin, Q. W.: Optimal design of water conveyance systems, J. Hydr. Eng.
Div.-ASCE, 106, 1415–1433, 1980.Mohan, S. and Jinesh Babu, K. S.: Water distribution network design using
heuristics-based algorithm, J. Comput. Civil. Eng., 23, 249–257,
10.1061/(ASCE)0887-3801(2009)23:5(249), 2009.
Monbaliu, J., Jo, J., Fraisse, C. W., and Vadas, R. G.: Computer aided design of pipe networks, Proc. Int. Symp. On Water Resource
Systems Application, Friesen Printers, Winnipeg, Canada, 1990.Nicklow, J., Reed, P., Savic, D., Dessalegne, T., Harrell, L., Chan-Hilton,
A., Karamouz, M., Minsker, B., Ostfeld, A., Singh, A., Zechman, E., and ASCE
Task Committee on Evolutionary Computation in Environmental and Water
Resources Engineering: State of the Art for Genetic Algorithms and Beyond in
Water Resources Planning and Management, J. Water Res. Plan. Man., 136,
412–432, 10.1061/(ASCE)WR.1943-5452.0000053, 2010.Prasad, T. D. and Park, N. S.: Multiobjective genetic algorithms for design
of water distribution networks, J. Water Res. Plan. Man., 130, 73–82,
10.1061/(ASCE)0733-9496(2004)130:1(73), 2004.
Raad, D. N.: Multi-objective optimisation of water distribution systems
design using metaheuristics, PhD thesis, University of Stellenbosch, South
Africa, 2011.
Saldarriaga, J., Páez, D., Cuero, P., and León, N.: Optimal Design of
Water Distribution Networks Using Mock Open Tree Topology, World
Environmental and Water Resources Congress, 19–23 May 2013, Cincinnati,
Ohio, USA, 869–880, 2013.Samani, H. M. V. and Mottaghi, A.: Optimization of water distribution
networks using integer linear programming, J. Hydraul. Eng., 132, 501–509,
10.1061/(ASCE)0733-9429(2006)132:5(501), 2006.Simpson, A. R., Dandy, G. C., and Murphy, L. J.: Genetic Algorithms Compared
to Other Techniques for Pipe Optimization, J. Water Res. Plan. Man., 120,
423–443, 10.1061/(ASCE)0733-9496(1994)120:4(423), 1994.
Streeter, V. L., Wylie, E. B., and Bedford, K. W.: Fluid Mechanics,
McGraw-Hill, New York, NY, USA, 1998.Suribabu, C. R.: Heuristic-based pipe dimensioning model for water
distribution networks, J. Pipeline Syst. Eng., 3, 115–124,
10.1061/(ASCE)PS.1949-1204.0000104, 2012.
Swamee, P. K. and Jain, A. K.: Explicit equations for pipe flow problems, J.
Hydr. Eng. Div.-ASCE, 102, 657–664, 1976.
Swamee, P. K. and Sharma, A. K.: Gravity low water distribution network
design, J. Water Supply Res. T., 49, 169–179, 2000.
Swamee, P. K. and Sharma, A. K.: Design of Water Supply Pipe Networks, Wiley,
Hoboken, NJ, USA, 2008.
Taorimina, R. and Chau, K. W.: Data-driven input variable selection for
rainfall–runoff modeling using binary-coded particle swarm optimization and
Extreme Learning Machines, J. Hydrol., 529, 1617–1632, 2015.
Tospornsampan, J., Kita, I., Ishii, M., and Kitamura, Y.: Split-pipe design
of water distribution network using simulated annealing, International
Journal of Computer, Information, and Systems Science, and Engineering, 1.3,
153–163, 2007.Vairavamoorthy, K. and Ali, M.: Optimal design of water distribution systems
using genetic algorithms, Comput. Aided Civ. Infrastruct. Eng., 15, 374–382,
10.1111/0885-9507.00201, 2000.Vasan, A. and Simonovic, S. P.: Optimization of water distribution network
design using differential evolution, J. Water Res. Plan. Man., 136, 279–287,
10.1061/(ASCE)0733-9496(2010)136:2(279), 2010.Wu, Z. Y. and Walski, T.: Self-adaptive penalty approach compared with other
constraint-handling techniques for pipeline optimization, J. Water Res. Plan.
Man., 131, 181–192, 10.1061/(ASCE)0733-9496(2005)131:3(181), 2005.
Yang, K. P., Liang, T., and Wu, I. P.: Design of conduit system with
diverging branches. J. Hydr. Eng. Div.-ASCE, 101, 167–188, 1975.Zhao, W., Beach, T., and Rezgui, Y.: Optimization of Potable Water
Distribution and Wastewater Collection Networks: A Systematic Review and
Future Research Directions, IEEE T. Syst. Man Cyb., 46, 659–681,
10.1109/TSMC.2015.2461188, 2016.Zheng, F., Simpson, A. R., and Zecchin, A. C.: A decomposition and multistage optimization approach applied to the
optimization of water distribution systems with multiple supply sources, Water Resour. Res., 49, 380–399,
10.1029/2012WR013160, 2013.