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Water supply is one of the most recognizable and important public services contributing to quality of life. Water distribution networks (WDNs) are extremely complex assets. A number of complex tasks, such as design, planning, operation, maintenance, and management, are inherently associated with such networks. In this paper, we focus on the design of a WDN, which is a wide and open problem in hydraulic engineering. This problem is a large-scale combinatorial, nonlinear, nonconvex, multiobjective optimization problem, involving various types of decision variables and many complex implicit constraints. To handle this problem, we provide a synergetic association between swarm intelligence and multiagent systems where human interaction is also enabled. This results in a powerful collaborative system for finding solutions to such a complex hydraulic engineering problem. All the ingredients have been integrated into a software tool that has also been shown to efficiently solve problems from other engineering fields.

Water distribution networks (WDNs) are important and dynamic systems. Most people are unaware of their importance, despite the fact that they routinely open the water tap every day. The existence of WDNs is generally ignored, except when disruption occurs. Politicians tend to neglect them because they are buried assets. Nevertheless, WDNs provide citizens with an essential public service. They supply drinking water, which is a crucial requirement for the normal development of most basic activities of life, such as feeding and hygiene [

For mosto of the people, perhaps with the exception of those directly involved in their management, WDNs are simply static infrastructures. However, WDNs are dynamic and living beings. They are born, when they are designed and built; they grow, as a consequence of increasing urban development and the appearance of new demands; they age and deteriorate, since they suffer a number of operational and environmental conditions that cause progressive and insidious deterioration; they need care, preventive care but also sometimes surgery; they are expected to work properly, despite the great amount of uncertainty involved and defective information about the state and operation of these systems due to their geographical dispersion and the fact that they are hidden assets; they have to meet basic requirements even under adverse circumstances, hence they must show resilience; and so on.

Our aim is to achieve a quality long-lasting life for WDNs. Nevertheless, it is a fact that with the passing of time these systems become gradually, but substantially, impaired. Some of the reasons include increasing loss of pressure triggered by increasing roughness of the inner pipes; breakage or cracking of pipes provoked by corrosion and mechanical or thermal charges; loss of water (leaks) and pathogen intrusion due to pipe breaks and cracks, with their corresponding economic loss, third party damage, and risk of contamination.

Several complex tasks, such as design, planning, operation, maintenance, and management, are inherently associated with WDNs. These tasks are not simple at all and require considerable investment. Efficiency and reduction of costs have been compelling reasons for practitioners to progressively move away from manual design based on experience and support the development of suitable automatic or semiautomatic tools. Support, of course, is increasingly multidisciplinary and receives contributions from various scientific areas. Mathematics is one of the areas contributing most effectively with suitable and efficient methods and tools.

We focus on the design of a WDN, a wide and open problem in hydraulic engineering that involves the addition of new elements in a system; the rehabilitation or replacement of existing elements; decision making on operation; reliability and protection of the system; among other actions. Designs are necessary to carry out new configurations or enlarge and improve existing configurations to meet new conditions. The design of a WDN involves finding acceptable trade-offs among various conflicting objectives: finding the lowest costs for layout and sizing using new components and rehabilitating, reusing, or substituting existing components; creating a working system configuration that fulfils all water demands, including water quality; adhering to the design constraints; guaranteeing a certain degree of reliability for the system [

The formulation of the optimal WDN design problem has traditionally been influenced by the available mathematical support for solving the problem; formulations have been adapted (restricted and/or simplified) to the available mathematical techniques [

In this paper, we provide a detailed description of the various objectives involved in the WDN optimal design problem. We argue that classical optimization techniques are unable to satisfactorily solve these problems and describe an evolutionary, multiagent approach to tackle this task. We then provide the solution for some real-world water distribution networks.

The layout of a network is usually conditioned by urban plans and/or various other reasons. Street positions, location of sources, main consumption points, and so forth are perfectly known before starting the design. Therefore, design does not generally consider the layout and is usually restricted to sizing the necessary network components. Figure

A WDN.

Various objectives may be considered in the WDN optimal design problem. In this section, we describe these objectives, namely, cost of components; adherence to hydraulic constraints; satisfaction of water demand quality; resilience of the system during stressed conditions.

Apart from the basic variables of the problem, which are the diameters of the new pipes, additional variables that depend on the design characteristics of the system may be required: storage volume, pump head, type of rehabilitation to be carried out for various parts of the network, and so forth. The estimation of individual costs will always depend on these variables. The correct approach to assess the costs for each element is important when defining the objective function, which has to be fully adapted to the problem under consideration in terms of design, enlargement, rehabilitation, operational design, and so forth. For example, in the network shown in Figure

In addition, it is important that the objective function reflects with utmost reliability the total cost of the system during its lifetime [

Many authors have used, in their optimization, an objective function that only considers the cost of pipelines, new and/or additional, duplicated pipelines, while others have taken into account other various costs [

The cost of the pipes is expressed as
^{t} is the vector of the pipe diameters. The costs per meter, depending on the diameter of pipe

Typically, the unitary cost of a new pipe, including purchase, transport, and installation costs, takes the form

There are various rehabilitation options: no rehabilitation, relining, duplication, and replacement are the most usual. Relining costs may be evaluated by the nonlinear function

The working conditions of a WDN depend on the values adopted by two types of variables, namely, demand models and roughness coefficient values. Typically, independent random variables are used to model both types of variables. Under the assumption that design is made to work for

In the case of inclusion of the operational costs of the network along a certain temporal horizon, the necessary amortization rates must be considered. In this case, the objective cost function, omitting for simplicity the independent variables, may be represented by

In this case, values

When modeling physical processes where the underlying functions are known, the deterministic equations can be solved to forecast the model output to a certain degree of accuracy. In hydraulic modeling, different governing laws, [

Analyzing pressurized water systems is a mathematically complex task that hydraulic engineers must face, especially for the large systems found in even medium-sized cities, since it involves solving many nonlinear simultaneous equations. Several formulations are available (see, for example, [

_{k}_{1} and _{k}_{2}, piezometric heads at nodes

Since most water systems involve a huge number of equations and unknowns, the system (

To be integrated in the algorithm later described, we have modified the EPANET Toolkit to support pressure-driven demands as described in [

WDN design is typically performed subject to several performance constraints in order to achieve an adequate service level. The most used constraint requires a certain minimum pressure level at each node of the system. Other constraints may include maximum pipe flow velocities and minimum concentrations of chlorine, for example. For many years, nodal pressure constraints have been considered as strong constraints in the sense that they should be strictly satisfied. Nevertheless, the possibility of violating by a small degree some of these constraints opens the door to various strategies for adopting suboptimal designs or soft solutions that may be more convenient from other (global or political) perspectives. This fact has been openly favored by multiobjective approaches such as the one we present in this paper.

In many studies, these constraints have been included as penalty terms in the cost function. However, in this paper, we consider the satisfaction of demand quality as a new objective that must be fulfilled.

There are various ways of expressing lack of compliance with pressure, velocity, disinfectant, and so forth conditions. For example, an objective function considering nodal and velocity constraints given by minimum values of node pressures and pipe velocities may be given by

Here,

WDNs have almost always been designed with loops so as to provide alternative paths from the source to every network node or junction. This reduces the number of affected consumers when a pipe is withdrawn from service for various reasons. Under normal operating conditions, there is no need for loops, meaning that a looped network is redundant. Redundancy is the capacity of the network to distribute water to users using alternative routes. Redundancy is only needed to maintain service, reduce deficit, and minimize the number of affected consumers when a pipe is withdrawn from service. Redundancy includes two important concepts: firstly, the connectivity necessary to provide alternative flow paths to each node; secondly, the provision of an adequate flow capacity (diameter) for those paths [

The concept of redundancy is closely related to reliability [

The reliability calculation in looped water supply networks is formulated in the literature as a function of the causes affecting consumer demand. The causes usually considered include real demand exceeding the design demand (e.g., a fire demand); growth of population served by the network; pipe aging; pipe failure.

An explicit formulation of all these causes in probabilistic terms and their further integration implies considerable mathematical and algorithmic complexity. Thus, although numerous WDN reliability quantification schemes exist [

This paper considers a simple reliability formulation as in [

Accordingly, it is accepted that the probability of simultaneous pipe failures is practically zero. In this case, the probability _{0}_{k}

The value of _{k}

Considering an average time for the duration of pipe failure, reliability ^{k}

After this definition, it can be seen that reliability

By calling

As WDNs should behave satisfactorily under normal conditions when there are no failures (

In this expression, the variables are related to the whole network, and

This tolerance to failure represents the expected

From (

Kalungi and Tanyimboh [

In short, the reliability concept, as defined above, can be regarded as simply a measure of the behavior of the network under normal conditions: meaning that reliability is not a good measure of behavior under failure situations. This is because the term

An additional factor considered by various authors (see, for example, [

Given the complexity of the problem described above, we now develop suitable tools to handle the problem. Our approach is a synergetic combination of swarm intelligence principles together with multiagent system properties aimed at solving the above multiobjective problems. Let us first briefly introduce the necessary details around multiobjective optimization.

In multiobjective optimization, the goal is to find the vector, ^{d}^{m}

Both the decision and the objective spaces are multidimensional, and a change in the decision variables producing a positive increment in one of the components of

Each solution in the Pareto optimal set is optimal because improvements in one of the components are not possible without impairing at least one of the other components. Two solutions are compared based on the concept of

The design of a WDN is a large-scale combinatorial, nonlinear, multiobjective optimization problem, involving various types of decision variables and many complex implicit constraints, such as the hydraulic constraints already mentioned. There is no single search algorithm for solving such real-world optimization problems without compromising solution accuracy, computational efficiency, and problem completeness.

Classical methods of optimization involve the use of gradients or higher-order derivatives of the fitness function. But these methods are not well suited for many real-world problems since they are unable to process inaccurate, noisy, discrete, and complex data. Robust methods of optimization are often required to generate suitable results. Several works (e.g., [

Many researchers have shifted direction and embarked on the implementation of various evolutionary algorithms: genetic algorithms, ant colony optimization, particle swarm optimization, simulated annealing, shuffled complex evolution, harmony search, and memetic algorithms, among many others. These derivative-free global search algorithms have been shown to obtain better solutions for large network design problems. Recent examples of the use of evolutionary algorithms for multiobjective design of WDN include [

The advantages of the growing use of evolutionary algorithms in the optimal design of WDN include [

evolutionary algorithms can deal with problems in a discrete manner, which unlike other optimization methods, enables the use of naturally discrete variables and the use of the binary variables in yes/no decisions so frequently in many real-world problems;

evolutionary algorithms work with only the information of the objective function, and this prevents complications associated with the determination of the derivatives and other auxiliary information;

evolutionary algorithms are generic optimization procedures and can directly adapt to any objective function, even if it is not described by closed expressions, but by whole, complex procedures;

because evolutionary algorithms work with a population of solutions, various optimal solutions can be obtained, or many solutions can be obtained with values close to the optimal objective function, and this can be of great value from an engineering point of view;

an analysis of systems with various loading conditions or forcing terms can be performed within the optimal design process.

A particle swarm optimization-based environment has been developed by the authors that mimics the judgment of an engineer. It was built by using various prior features and improvements regarding swarm intelligence, multiagent systems, and the necessary adaptation to multiobjective performance, including human interaction.

The first feature derives from the philosophy behind PSO (particle swarm optimization) [

A swarm of ^{d}

current position,

best position,

flight velocity

The particle which is in the best position,

In each generation, the velocity of each particle is updated as in (

For discrete variables, we use

Expressions (

In each dimension, particle velocities are clamped to minimum and maximum velocities, which are user-defined parameters,

There is, however, a singular aspect regarding velocity bounds that must be taken into consideration so that the algorithm can treat both continuous and discrete variables in a balanced way. In [

Finally, the position of each particle is updated every generation. This is performed by adding the velocity vector to the position vector,

Thus, each particle or potential solution moves to a new position according to expression (

The main drawback of PSO is the difficulty in maintaining acceptable levels of population diversity while balancing local and global searches [

Frequent collisions of birds in the search space, especially with the leader, can be detected—as shown in [

The role of inertia,

In the variant, we propose that, the acceleration coefficients and the clamping velocities are neither set to a constant value, as in standard PSO, nor set as a time-varying function, as in adaptive PSO variants [

By using expressions (

Although the authors have applied this algorithm mainly to WDN design, it has proven very efficient in solving optimization problems in other fields [

The emergent behavior of a PSO swarm is strongly reminiscent of the philosophy behind the multiagent (MA) paradigm [

For the optimal design of WDNs, an MA system offers considerable added value because of the introduction of several agents with different visions of the evaluation of solutions for the same problem, so enabling a multiobjective optimization that is qualitatively much closer to reality. From a practical standpoint, the development of a multiobjective optimization process enables the combination of economic, engineering, and policy viewpoints when searching for a solution.

Taking into account the desirability of solving the optimal design of WDNs with a multiobjective approach and the benefits offered by MA systems, a departure from the standard behavior of particles in PSO must be performed. In addition to using the concept of dominance, various other aspects must also be restated.

Firstly, the concept of leadership in a swarm must be redefined. The most natural option is to select as leader the closest particle to the so-called

Secondly, because each objective may be expressed in different units, it is necessary to make some regularization for evaluating distances in the objective space. Once a regularization mechanism has been enforced, to establish the distance between any two objective vectors, the Euclidean distance between them is calculated. Note that the worst and best objective values are not usually known a priori; they are updated while the solution space is explored.

Thirdly, arguably, the most interesting solutions are located near the singular point and not too far from the peripheral areas of the Pareto front. Therefore, instead of seeking a complete and detailed Pareto front, we may be more interested in precise details around the singular point. Nevertheless, situations can occur when unbalanced Pareto fronts develop with respect to the singular point. Consequently, poorly detailed sections on the Pareto front may appear to be worth exploring. It seems plausible that problem complexity is the cause of this asymmetry in many real-world, multiobjective optimization problems.

It is not easy to find a general heuristic rule for deciding which parts of the Pareto front should be more closely represented and how much detail the representation of the Pareto front should contain. Various ways of favoring the completeness of a Pareto front may be devised. We describe one possible approach based on dynamic population increases to raise the Pareto front density [

In the first approach, during the search process, swarms can increase their population when needed in order to better define the Pareto front; a particle whose solution already belongs to the Pareto front may, on its evolution, find another solution belonging to the front. In this situation, a new clone of the particle is placed where the new solution is found, thus increasing the density of particles on the Pareto front. Greater densities on the Pareto front must be restricted to the case where the new clone has at least one of its neighbors located further away than some minimal permissible distance in the objective space. It must be noted that two particles are considered to be neighbors when no other particle is located between them for at least one of the objectives considered in the problem.

In the second approach, users are allowed to add new swarms for searching in the desired region of the objective space. The concept of a singular point is now extended to any desired point in the objective space for particles to search around.

Decisions are strongly dependent on the people solving the problem and on the problem itself. The user can specify additional points where the algorithm should focus the search and specify how much detail a region should contain. This must be achieved in real time during the execution of the algorithm. Once a new singular point is added, a new swarm is created with the same characteristics as the first created swarm. Swarms will run in parallel, but they share (and can modify) the information related to the Pareto set. Particles from any swarm can be added to the Pareto set. If the user changes the fixed values for a singular point, then the corresponding swarm selects a new leader considering the location of the new singular point.

Human interaction with the algorithm in real time also enables the incorporation of human behavior, so that the human becomes another member of the swarm by proposing new candidate solutions. Eventually, such a solution can be incorporated into the Pareto front or lead the behavior of a group of particles. User solutions will always be evaluated in the first swarm created. If a particle is being evaluated, then the user request waits until the evaluation of the particle is finished. If a solution proposed by the user is being evaluated, then any particle belonging to the first swarm should wait for evaluation. Once any solution is evaluated, the algorithm checks whether it could be incorporated in the Pareto front. Synchronization is effected among all the swarms in order to open access for managing the Pareto front. Proposed solutions could even become leaders of the swarm(s) if they are good enough. At this point, human behavior begins to have a proactive role during the evolution of the algorithm.

The participation of several human agents with different perspectives on a problem is very close to what happens in the practice of engineering decision making, where politicians, economists, engineers, and environmental specialists are involved in final decisions. The idea of incorporating user experience into the search process is a step forward in the development of computer-aided design.

The combination of various swarms within the same algorithm is efficient because it conducts a neighborhood search in which each of the swarms specializes, and the best improvement step in terms of Pareto optimality is followed to yield a new solution. The practice of incorporating different search mechanisms also reduces the probability of the search becoming trapped in local optima.

In the approach presented, as explained above, new particles are used that are based on the behavior of particles in PSO. Swarms running in parallel may be distributed in different computers, and it must be ensured that the swarms can communicate amongst themselves—a peer-to-peer scenario would be a good choice for this task. The steps of the algorithm for every swarm may be summarized in Pseudocode

particle equal to its current location.

(a) Execute from

Start

(ii) Calculate the new fitness function vector for particle

(iii) If the new fitness function vector for particle

(iv) If particle

if the new fitness function vector may also be a point on the

Pareto front and this new position has at least one of its

neighbors located further than the minimal permissible distance

from any of the objectives, then add a new particle

with

else

try to add (if possible) the particle

Pareto front; if the particle is added, remove from the list any

dominated solution; dominated clones are eliminated from the swarm.

(v) If particle

(vi) If particle

End

(b) Increase the iteration number.

The algorithm and its connection with EPANET2 (modified with the pressure-driven demand feature explained above) were implemented in a software program called WaterIng (

This software has been used to perform the design of the case studies shown below. The multiobjective model implemented by this software has shown robustness and good explanatory outcomes. Decision makers are provided with a set of informed solutions to select the best design with regard, for example, to available resources and/or other criteria.

Some parameters of the algorithm were established a priori for running the case studies; the initial population size was set equal to 20. The inertia weight was calculated using (

This is a well-known problem, one of the most explored systems in the research literatura, and it has been included in this paper for comparison purposes. In this problem, the engineer seeks the minimum investment cost for a water network, constrained to have at least a minimum pressure value at demand nodes. This constraint was turned into an objective for considering a multiobjective approach: to find solutions minimizing the investment and minimizing the lack of pressure at demand nodes. More information about the original problem can be found in [

Hanoi network. Approximated Pareto front regarding investment and lack of pressure.

This problem was also run considering as objectives the minimization of the investment costs and the maximization of the minimum expected pressure in the network (see Figure

Approximated Pareto front regarding investment and minimum pressure in the network.

All approximated Pareto fronts were obtained in less than 25 seconds (in a Pentium core2duo, 2 GB RAM). Although images were taken when the algorithm was still running (without reaching the termination condition), some similarities with results obtained in other works may be identified (see [^{6} $. After a zoom in the Pareto front it can be seen that the algorithm found a solution without lack of pressure and with a cost of

Finally, in Figure

Approximated Pareto front regarding investment and reliability.

The second case study is a real-world design with three objectives: minimizing the investment cost, minimizing the lack of pressure at demand nodes, and minimizing additional costs caused by reliability problems. This case is a modification of a network designed in collaboration with Wasser SL, a company located at Madrid, Spain.

Results obtained in this design showed a good tolerance to failure conditions. Consideration of tolerance made it possible to find a solution (Figure

Lima sector, a modified real case example.

Figure

Approximated Pareto front regarding cost and tolerance to failure condition.

This case was taken from studies developed in San José de las Lajas, Havana, Cuba. A modified variant of the network of this town (see Figure

San José town, a modified real case example.

In this network, a significant investment is necessary for increasing the minimum pressure at demand nodes. The layout and configuration of the network do not contribute to good pressure distribution throughout the network. This system is currently working without distributing water 24 hours a day; various zones have been determined that receive water at specific moments of the day. Even if this system was designed anew, with the presented configuration and the proposed commercial pipes, the network would have pressure problems at various points. The addition of another tank and different layouts were proposed for analysis in future works. Figure

San José town. Approximated Pareto front regarding investment and minimum pressure in the network.

In this paper, one of the aspects that reflects the huge complexity of WDNs, namely, the design of these infrastructures, is addressed. We have described the various design elements that render the problem into a large-scale combinatorial, nonlinear, nonconvex, multiobjective optimization problem, involving various types of decision variables and many complex implicit constraints. We then argued that this type of problem is not accessible for classical optimization techniques and discussed the great interest and impact produced by the many evolutionary algorithms.

Based on PSO, we have developed the necessary tools to build an MA algorithm that has proven to be efficient for solving the stated problem. We described some features that substantially improve the searching ability of PSO, paying special attention to two aspects: increase in diversity and self-management of parameters. We then provided the necessary adaptations to deal with multiobjective problems. We proposed the possibility that various swarms could explore different areas of the Pareto front. And finally, we have integrated human interaction into the process. This offers a special platform for finding solutions in a human-computer cooperative framework. Integrating the search capacity of algorithms with the ability of specialists to redirect the search towards specific points of interest—based on their experience in solving problems—results in a powerful collaborative system for finding solutions to complex engineering problems. Agents can profit from the creativity and ideas of human experts to improve their own solutions; and in turn, human experts can profit from the speed and search capabilities of artificial agents to explore broader solution spaces.

WaterIng is just a first step for engineers working in WDN decision making. The development of the software has not stopped with the contents of this paper; it continues to meet new challenges. Extensions to this work are possible for solving more complex problems, especially in the field of hydraulic engineering. The ideas presented in this paper can also be applied to other engineering fields or, in general, to problems where the search for optimal solutions needs the use of a computer.

One of the aspects we have not explicitly addressed in this paper is the partial stochastic character of the considered problem due to demand variations and the probability, or even the possibility from a fuzzy theory point of view, of the evolution (due to aging) of the roughness within the pipes. Future research topics could explore this aspect, perhaps using recently developed ideas regarding stochastic complex networks, as in [

This work has been developed with the support of the project IDAWAS, DPI2009-11591, of the Dirección General de Investigación of the Ministerio de Educación y Ciencia, and ACOMP/2010/146 of the Conselleria d‘Educació of the Generalitat Valenciana. The first author is also indebted to the Universitat Politècnica de València for the sabbatical leave granted during the first semester of 2011. The use of English in this paper was revised by John Rawlins.