OPTIMAL POWER USE SURFACE FOR DESIGN OF WATER DISTRIBUTION
SYSTEMS
J. Saldarriaga, D. Páez, P. Cuero and N. León
1
1
Universidad de los Andes – Water Distribution and Sewer Systems Research Centre (CIACUA),
Bogotá, Colombia
ABSTRACT
This paper intends to bring forward a hydraulic based Water Distribution System (WDS) design
methodology named Optimal Power Use Surface (OPUS). Its objective is to reach least-cost
designs, executing a reduced number of iterations, and focusing on the setting-up of efficient ways
in which energy is dissipated and flow is distributed. For its validation, the proposed algorithm was
tested on three well known benchmark networks, frequently reported in the literature: Two-Loop,
Hanoi and Balerma. When compared to results obtained through other methodologies, OPUS stands
out for allowing designs with constructive costs very similar to those obtained in previous works
but requiring a number of iterations several orders of magnitude bellow, especially for real-size
networks. The methodology proved that following hydraulic principles is an excellent choice to
design WDS and also provides an alternative path to the tiresome search process undertaken by
metaheuristics.
INTRODUCTION
The optimized design of WDSs is a relevant problem at a global scale, this being due to the scarcity
of resources and the importance of drinking water for human life. This issue aggravates in the
context of developing countries, where millions of people still suffer the lack of an adequate
service. In this context minimum-cost design methodologies are essential.
Even though the design of WDSs is supposed to consider different criteria besides the construction
costs (e.g. reliability, environmental impact and water quality), the minimum cost as the only
objective is still used to validate and compare new design algorithms. This type of design consists
in determining the diameter size of each pipe of the system in such a way that flow demands are
satisfied with an adequate pressure and with a minimum capital cost. In spite of the fact that pipes
are usually manufactured in discrete-sized diameters, the amount of possible pipe configurations is
immense, which means that the problem is highly indeterminate. In fact, Yates et al. (1984) showed
that it is a NP-HARD problem and thus only approximate methods could be successful in finding
adequate solutions.
Initial approximations involved traditional optimization techniques such as enumeration, linear and
non-linear programming. But more recently these have been replaced by different metaheuristic
algorithms due to their ease of implementation and other advantages like their broader search of the
solution space, a relatively small reliance on the system’s initial configuration, and their capability
of incorporating the discrete-sized diameters restriction. Successful attempts include Genetic
Algorithms (Savic and Walters, 1997), Harmony Search (Geem, 2006), Scatter Search (Lin et al.,
2007), Cross Entropy (Perelman and Ostfeld, 2007), Simulated Annealing (Reca et al., 2007), and
Particle Swarm (Geem, 2009) among others.
These metaheuristics consist in bio-inspired algorithms that randomly generate a large number of
possible solutions and test their fitness in terms of quality and capital costs. Generic learning
functions are used to progressively improve the previous results. In the WDS design context, each
solution corresponds to an alternative design, which means a different set of pipe diameter sizes.
The evaluation of each of the alternative designs requires running static hydraulic simulations, thus
a large number of iterations is needed before convergence is reached. This makes metaheuristics
very demanding in terms of computational effort regardless their flexibility and their capability of
accomplishing near-optimal results. For this reason, apart from the cost of the final solution, the
number of hydraulic simulations (or iterations) is the main indicator used to measure and compare
the efficiency of the different methodologies. Even though the learning functions used in
metaheuristic algorithms involve testing the hydraulic performance of each of the candidate
solutions, neither of them make use of additional hydraulic criteria.
As a response to these tedious algorithms, some researchers have come through with new
approaches that seek to develop a hydraulic treatment of the problem. While metaheuristics intend
to optimize an objective function behaving towards the optimization variables simply as a series of
numbers that must follow certain logic, without any understanding of the machinery behind that
logic; these new approaches try to characterize the behaviour of the different hydraulic variables
and understand the underlying dynamics.
In 1975 I-Pai Wu carried out an analysis for the drip irrigation main line design problem,
considering the hydraulic principles that it follows. After setting up a minimum pressure (
) at
the end of the line, still a big number of configurations could be constructed. Wu discovered that
each of these configurations lead to a different way in which energy is spent. After analysing
numerous alternatives he concluded that the least-cost alternative was that with a parabolic
hydraulic gradient line (HGL) with a sag of 15% of the total head-loss (
). Thus, optimal designs
could be obtained by computing objective head-loss values for each pipe derived from the HGL
fabricated using Wu´s criterion.
Later in 1983, Professor Ronald Featherstone from Newcastle University in the United Kingdom
first proposed to extend Wu´s criterion to the optimization of looped networks. This idea seemed
like a sound possibility and was further developed by Saldarriaga (1998), who analysed hydraulic
gradient surfaces on several WDS designs obtained using metaheuristic algorithms. Based on Wu’s
criterion and Featherstone’s idea, the works of Villalba (2004) and Ochoa (2009) proved that
hydraulic criteria could be used as the basis of WDS design in order to replace the iteration-
intensive stochastic approach required by metaheuristics; obtaining promising results, not only in
performance, but also in the insight of the inner mechanics that govern WDS design.
Based on the works made by Ochoa (2009) and Villalba (2004), a first design methodology was
developed by the CIACUA (Water Distribution and Sewer Systems Research Centre), named
SOGH. It was tested on three well known benchmark networks (Two-Loop, Hanoi and Balerma).
This paper intends to bring forward an improvement to this WDS design methodology named
Optimal Power Use Surfaces (OPUS), which proposes a net hydraulic approach following the ideas
of the aforementioned authors (Takahashi et al., 2010). The objective of this methodology is to
reach least-cost designs with a reduced number of iterations especially for real-size networks. This
can be accomplished through the use of deterministic hydraulic principles drawn from the analysis
of flow distribution and the way energy is used in the systems. These principles provide an
alternative path to the tiresome search process undertaken by metaheuristics. Each of the steps that
make up the OPUS algorithm are explained in the following section. Three different benchmark
problems (Two-Loop, Hanoi and Balerma) are employed to test the methodology and the results are
presented and discussed in this paper. Finally, conclusions are drawn from the outcome obtained
and future work guidelines are stated.
METHODOLOGY
The developed OPUS design methodology consists in 6 basic sub-processes which are shown in
Figure 1 and explained below in this section.
Sump Search or Tree Structure. This step is based on two fundamental principles: The first one
states that a WDS of minimum cost should convey the water to each of the demand nodes from the
water sources, through a single route. This is drawn from the fact that redundancy is hydraulically
inefficient, even though it favors reliability. Therefore, open WDSs could be a lot cheaper than
looped networks, reason why this sub-process intends to decompose the looped system into an open
tree-like structure (a spanning tree), in order to identify the nodes in the original model that
correspond to the sumps of the open network (i.e., nodes with a lower head than that of all of its
neighbors).
Start
Sump Search
Optimal Power Use Surface
Optimal Flow Distribution
Diameter Calculation
Diameter Round-Off
Optimization
End
Figure 1: OPUS methodology BPMN diagram.
The second principle follows from the flow expression derived from the Darcy-Weisbach and
Colebrook-White equations. Leaving all the other parameters constant, the flow (
) presents a
relation approximately proportional with the diameter to a power of 2.6. Assuming a standard pipe
cost equation and replacing the diameter according to this proportion, the cost per length of a pipe
as a function of its design flow behaves as shown in Figure 2; which means that as the design flow
for a pipe increases, the marginal cost decreases.
Figure 2: Schematic relation between pipe cost and flow.
From the abovementioned principles, an algorithm was designed in order to obtain the tree
structure, aggregating flow values in the least number of main routes possible. The open network is
set up starting from the water sources and then adding adjacent pipe-node pairs, one at a time. The
group of available pairs in each iteration conform the ‘search front’ and each of these pairs are
assigned a cost-benefit value (
), making up a recursive process.
Figure 3: Layout of the Hanoi WDS. The labels show pipe and nodal identification numbers.
For example, take the Hanoi benchmark WDS shown in Figure 3: Starting from the source, the first
pair to be added is the one consisting in pipe 1 and node 2 (<1, 2>). Then, the pair <2, 3> is added.
At this point the pairs <3, 4>, <19, 19> and <20, 20> can be selected. These constitute the search
front. Figure 3 shows the result for the entire execution of the sub-process, where the pipes
highlighted (solid black) constitute the corresponding tree structure.
The pair in the front with the higher cost-benefit value is selected to be part of the tree structure.
The cost-benefit function of a pair is calculated by computing the quotient between the demand of
the new node and the marginal cost of connecting it to the source: This entails the addition of the
total cost of the pair’s pipe to the cost difference of transporting the additional flow through all of
the upstream pipes. It is worth noting that these are not actual costs but proportional values drawn
from the relation shown in Figure 2. The construction of the tree using this cost-benefit function has
an O(NN
2
) time complexity, where NN is the number of nodes.
The cost-benefit function is used because it favours the creation of few main routes that transport
the largest portion of the total water volume. The process concludes when all of the system nodes
have been added to the tree structure and at the end the leaf nodes in the tree structure are assigned
the status of ‘sumps’.
Optimal Power Use Surface. This sub-process gives the name to the entire methodology, being
essential to it due to the close relation that it has with the work developed by I-Pai Wu. In this step a
set of objective hydraulic heads (also understood as objective head losses) is established for each
pipe within the system. By analysing the behaviour of optimal designs, Ochoa (2009) corroborated
Wu’s suggestion, finding that the optimal hydraulic gradient line for pipe series was in effect a
parabola. Besides, Ochoa discovered that the sag of this parabola depended on the demand
distribution, the ratio between flow demands and pipe length, and the cost function; putting forward
a methodology for its calculation.
Contrarily to the approaches made by Villalba (2004) and Ochoa (2009), in the proposed
methodology, the optimal power use surface is computed in the tree structure instead of using a
graph algorithm on the original network. In this sense, Ochoa’s parabolic hydraulic gradient line is
applied to each of the branches in the open network. First, the minimum allowable pressure is
assigned to each of the sump nodes. Then, the topological distance for every node in the system to
its source is calculated. Knowing the head in the reservoir, the heads of the intermediate nodes in
each branch are calculated with a Parabolic Head Gradient Line (HGL) as a function of their
distance to the source, as shown in Figure 4.
Figure 4: I-Pai Wu's criterion for predefining the head on each node.
As the branches converge while going over the tree upstream, it is necessary to recalculate the sag
at each intersection by weighting the flow on each downstream route. It must be taken into account
that the objective parabola has to be modified in the upstream direction, in case of encountering
particularly elevated nodes, to make sure that the assigned head values meet the pressure
requirements in every instance. Once this sub-process is concluded, all nodes must have an
objective head value assigned, thus a flow is required in order to calculate the diameter of each pipe
in the network.
Figure 5 shows an example of the optimal power use surface for the Hanoi network. Note that for
the water source the HGL corresponds to the total head available in the reservoir and for the sump
nodes it was assigned the minimum pressure.
Figure 5: Assigned surface for the Hanoi network.
H
GL
(m
)
Optimal Flow Distribution. This step assigns a design flow to every pipe in the system.
Considering that in a looped network a specific hydraulic gradient surface could be obtained by an
infinite number of continuous diameter configurations (Saldarriaga et al., 2011), it is necessary to
predefine an objective flow for each pipe in order to obtain a configuration that minimizes costs.
Therefore, this sub-process pretends to find a unique flow distribution scheme that respects mass
conservation and conforms to the optimal power use surface previously obtained. In this case, the
process is executed using the original graph instead of the spanning tree.
Starting with the sumps, the flow demand is divided into the upstream pipes according to the
following criterion: Only one pipe will be assigned the largest flow value, while the others will be
assigned the flow that corresponds to the minimum diameter available (
. In order to determine
the principal pipe (i.e. the pipe that will have the largest portion of the total flow demanded), several
criterions can be used to evaluate their fitness. Same as in the tree structure step, the function
is used, which means that the pipe with the biggest value of this function will be defined as the
principal pipe. For non-sump nodes, the total demand is calculated adding its own flow demand and
the flow demanded downstream. An iterative-recursive algorithm (IRA) can be used to perform all
of the calculations with an O(NN) time complexity. At the end of the process, all of the pipes in the
system must have been assigned an objective flow value. Note that this step has a higher impact in
the case of looped systems.
As an example, Figure 6 shows the optimal flow distribution that corresponds to the Hanoi network.
Figure 6: Optimal flow distribution for the Hanoi network (Flow rates in m
3
/h).
Diameter Calculation. This sub-process assigns continuous diameter sizes to all pipes. Having
predefined the objective head losses and the design flow rate for each pipe in the system, the
continuous diameter needed is given by a straightforward calculation. This calculation is explicit
when the Hazen-Williams equation is used and iteratively for the Darcy-Weisbach and the
Colebrook-White equations. The resulting continuous design is in theory a full-operational WDS,
with a cost very close to the minimum. Due to the limited availability of diameter sizes, a next step
is required to transform this “optimal” design to a feasible one.
Diameter Round-Off. This step consists in approximating each continuous diameter to a discrete
value from the list of commercially available diameter sizes, which is represented by the set
{
}. It was found that rounding to the nearest equivalent flow value offers the best
results, even though it can be done following several criterions. This is done by elevating the
diameter values to a power of 2.6, as explained in the Tree Structure step. Unfortunately, this step
affects drastically the system’s hydraulic behaviour, especially if all the diameter sizes are rounded
up or down.
Optimization
This final sub-process has two main goals: The first one is to ensure every node has a pressure
higher than or equal to
; secondly, it seeks for possible cost reductions. Several criteria could be
used to establish the order in which pipes diameter values must be increased. It was found that the
pipes with larger unit head-loss difference between real and objective values should be changed
first. The process must continue until the whole system has acceptable pressures. The second part
executes a two-way sweep starting from the reservoirs going towards the sumps in the direction of
the flow, and then backwards: The reduction of each pipe’s diameter is considered twice. If any of
these changes entails a pressure deficit it must be reversed immediately, otherwise it holds. To
make sure minimum pressure is not being violated numerous hydraulic simulations are required.
In first place, the diameter size of one pipe is increased iteratively while there are nodes with
pressure deficit. Thus, this sub-process requires the most number of iteration of the whole
methodology, being necessary to run a hydraulic simulation per pipe, for each single diameter
modification. This sole heuristic can be used alone to obtain sound designs, in spite of this, it is
strongly dependant on the initial pipe configuration.
RESULTS
The OPUS methodology was used on three benchmark systems: Two-Loop, Hanoi and Balerma.
Different configurations of the parameters defined on each step of the methodology were tested,
looking for optimal designs with discrete diameters (
{
} contains only diameter
available at the local market) and, in some cases, continuous ones (
) as potentially near
optimal starting designs for the Round-off and Optimization sub-processes.
Two-Loop
Detailed information about this WDS can be found in Alperovits and Shamir (1977). The Hazen-
Williams head-loss equation was used with a roughness coefficient
, as specified in the
mentioned publication, and also the unit prices table for each diameter size was adopted. With these
values and with a minimum allowable pressure for every node of 30 m, the WDS was designed
using the SOGH algorithm which is the OPUS predecessor methodology.
The optimal discrete design was reached after 51 hydraulic simulations which leaded to a $419,000
network. The number of hydraulic simulations reported by other authors who also reached this cost
is shown in Table 1.
Table 1: Reported number of iterations before reaching a cost of $419.000 for the Two-Loop WDS.
Algorithm
Number of iterations
Genetic algorithms (Savic & Waters, 1997)
65,000
Simulated annealing (Cunha & Sousa, 1999)
25,000
Genetic algorithms (Wu & Simpson, 2001)
7,467
Shuffled frog leaping (Eusuff & Lansey, 2003)
11,155
Shuffled complex evolution (Liong & Atiquzzaman, 2004)
1,019
Genetic algorithms (Reca & Martínez, 2006)
10,000
Particle swarm optimization (Suribabu, 2006)
5,138
Harmony search (Geem, 2006)
1,121
Cross entropy (Perelman & Ostfeld, 2007)
35,000
Scatter search (Lin et al., 2007)
3,215
Particle swarm harmony search (Geem, 2009)
204
Differential evolution (Suribabu C. , 2010)
4,750
Honey-bee mating optimization (Mohan, 2010)
1,293
SOGH (Ochoa, 2009)
51
Hanoi
The Hanoi network was first presented by Fujiwara and Khang (1990) and similarly to Two-Loop
network, it has become a well-known benchmark WDS. The head-loss equation commonly used is
Hazen-Williams with a
, the minimum pressure for the design scenario is 30 m and the
pipes’ costs can be calculated using a potential function of the diameter with a unit coefficient of
$1.1/m and an exponent of 1.5.
The least cost continuous design reached for Hanoi presents a cost of $5,456,806, and is shown in
Figure 7. It is worth noting that the continuous design involves diameter values higher than 40
inches, which corresponds to the maximum allowable diameter according to Fujiwara and Khang
(1990). Therefore, two different discrete designs were performed: The first one being based on the
original diameter list, and the second one considering the availability of a 50 inches diameter.
Figure 7: Hanoi network least cost continuous design (diameters in inches).
The OPUS methodology for the first case reached a cost of $6’147,882.45 after 83 iterations.
Although this is not the least cost reported, the number of hydraulic simulations needed to reach this
result is three orders of magnitude smaller than that of other approaches, as can be seen in Table 2.
The pipe diameter sizes in inches for this configuration are: 40, 40, 40, 40, 40, 40, 40, 40, 40, 30,
24, 24, 16, 16, 12, 12, 20, 20, 24, 40, 20, 12, 40, 30, 30, 20, 16, 12, 16, 12, 12, 12, 16 and 30 (these
diameters are shown in order of pipe identification number).
Table 2: Reported costs and number of iterations for the Hanoi WDS.
Algorithm
Cost (millions)
Number of iterations
Genetic Algorithm (Savic and Walters, 1997)
$6.073
1,000,000
Simulated annealing (Cunha and Sousa, 1999)
$6.056
53,000
Harmony search (Geem, 2002)
$6.056
200,000
Shuffled frog leaping (Eusuff and Lansey, 2003)
$6.073
26,987
Shuffled complex evolution (Liong & Atiquzzaman, 2004)
$6.220
25,402
Genetic Algorithm (Vairavamoorthy, 2005)
$6.056
18,300
Ant colony optimization (Zecchin et al., 2006)
$6.134
35,433
Genetic Algorithms (Reca & Martínez, 2006)
$6.081
50,000
Genetic Algorithms (Reca et al., 2007)
$6.173
26,457
Simulated annealing (Reca et al., 2007)
$6.333
26,457
Simulated annealing with tabu search (Reca et al., 2007)
$6.353
26,457
Local search with simulated annealing (Reca et al., 2007)
$6.308
26,457
Harmony search (Geem, 2006)
$6.081
27,721
Cross entropy (Perelman & Ostfeld, 2007)
$6.081
97,000
Scatter search (Lin et al., 2007)
$6.081
43,149
Modified GA 1 (Kadu, 2008)
$6.056
18,000
Modified GA 2 (Kadu, 2008)
$6.190
18,000
Particle swarm harmony search (Geem, 2009)
$6.081
17,980
Heuristic based approach (Mohan S. a., 2009)
$6.701
70
Differential evolution (Suribabu C. , 2010)
$6.081
48,724
Honey-bee mating optimization (Mohan, 2010)
$6.117
15,955
Heuristic based approach (Suribabu C. , 2012)
$6.232
259
SOGH (Ochoa, 2009)
$6.337
94
Optimal power use surface (this study)
$6.173
83
Extrapolating the cost function for a 50” diameter it would have a unit cost of $388.91/m. Taking
this into account, the total cost of the design obtained following the OPUS algorithm was of only
$5’342,840.13, as the real hydraulic gradient surface resembled the optimal in a higher degree. The
diameter sizes in inches are: 50, 50, 40, 40, 40, 40, 40, 24, 24, 40, 20, 20, 12, 12, 12, 16, 20, 20, 20,
40, 16, 12, 40, 30, 24, 20, 12, 12, 12, 12, 12, 12, 12 and 20.
Balerma
Balerma corresponds to a WDS of an irrigation district in Almería, Spain. The pipe diameter sizes
commercially available for its design are manufactured exclusively in PVC, with an absolute
roughness coefficient of 0.0025 mm. The minimum pressure allowable is of 20 m and the pipes’
costs are calculated using a potential function, with a power of 2.06. Its topology is presented in
Figure 8.
Figure 8: Topology of the Balerma network.
As a result of implementing the OPUS methodology on this network, a €1.755 millions continuous
design was founded. Also, after executing the round-off and optimization processes, the optimal
discrete design was reached requiring 957 hydraulic simulations which leaded to a €2.106 millions
network. Table 3 presents other reported costs and their respective number of iterations.
Table 3: Reported costs and number of iterations for the Balerma WDS.
Algorithm
Cost (
€
millions) Number of iterations
Genetic algorithm (Reca & Martínez, 2006)
2.302
10.000.000
Harmony search (Geem, 2006)
2.601
45.400
Harmony search (Geem, 2006)
2.018
10.000.000
Genetic algorithm (Reca et al., 2007)
3.738
45.400
Simulated annealing (Reca et al., 2007)
3.476
45.400
Simulated annealing with taboo search (Reca et al., 2007)
3.298
45.400
Local search with simulated annealing (Reca et al., 2007)
4.310
45.400
Hybrid discrete dynamically dimensioned search
(Tolson, 2009)
1,940
30,000,000
Harmony search with particle swarm (Geem, 2009)
2.633
45.400
SOGH (Ochoa, 2009)
2.100
1.779
Memetic algorithm (Baños, 2010)
3,120
45,400
Genetic heritage evolution by stochastic transmission
(Bolognesi, 2010)
2,002
250,000
Differential evolution (Zheng, 2012)
1,998
2,400,000
Self-adaptive differential evolution (Zheng, 2012)
1,983
1,300,000
Optimal power use surface (this study)
2.106
957
CONCLUSIONS
The WDS least-cost design methodology known as Optimal Power Use Surface (OPUS) herein
introduced, considers optimal distribution pattern of flow in the network as a way of spending
energy properly. This approach differentiates it from metaheuristic algorithms that explore the
solution space without considering hydraulic principles.
The methodology significantly reduces the number of iterations and keeps the constructive costs of
the network significantly close to the minimum. In the case of Hanoi the difference results only of
1.9% with respect to the lowest cost reported in the literature and with a number of iterations three
orders of magnitude below.
OPUS is a methodology that worked well on large networks, making it possible to minimize
constructive costs in a very reduced number of iterations. On the other hand, when applied to small
WDSs, the hydraulics result very affected due to the relative difference between the tree structure
and the real looped network. Taking into account that the design is executed based on the open
network (spanning tree), the final result ends up being governed by the optimization process and not
by the steps based on hydraulic principles.
This methodology clearly proves that considering hydraulic bases allows the optimization of WDS
design to reduce significantly the number of iterations required. Theoretical networks with some
restrictions that limit the design possibilities were tested in this paper. For this reason, it is
recommended to test this methodology on real networks and including different additional
objectives such as maximizing reliability and minimizing leakage. Finally energy approaches can
also be applied to calibrate models and design system operation.
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