Optimal cost design of water distribution networks using harmony search

This study presents a cost minimization model for the design of water distribution networks. The model uses a recently developed harmony search optimization algorithm while satisfying all the design constraints. The harmony search algorithm mimics a jazz improvisation process in order to find better design solutions, in this case pipe diameters in a water distribution network. The model also interfaces with a popular hydraulic simulator, EPANET, to check the hydraulic constraints. If the design solution vector violates the hydraulic constraints, the amount of violation is considered in the cost function as a penalty. The model was applied to five water distribution networks, and obtained designs that were either the same or cost 0.28–10.26% less than those of competitive meta-heuristic algorithms, such as the genetic algorithm, simulated annealing and tabu search under similar or less favorable conditions. The results show that the harmony search-based model is suitable for water network design.     

An adaptive heuristic cross-entropy algorithm for optimal design of water distribution systems

The optimal design problem of a water distribution system is to find the water distribution system component characteristics (e.g. pipe diameters, pump heads and maximum power, reservoir storage volumes, etc.) which minimize the system's capital and operational costs such that the system hydraulic laws are maintained (i.e. Kirchhoff's first and second laws), and constraints on quantities and pressures at the consumer nodes are fulfilled. In this study, an adaptive stochastic algorithm for water distribution systems optimal design based on the heuristic cross-entropy method for combinatorial optimization is presented. The algorithm is demonstrated using two well-known benchmark examples from the water distribution systems research literature for single loading gravitational systems, and an example of multiple loadings, pumping, and storage. The results show the cross-entropy dominance over previously published methods.     

Self-adaptive multi-objective harmony search for optimal design of water distribution networks

In multi-objective optimization computing, it is important to assign suitable parameters to each optimization problem to obtain better solutions. In this study, a self-adaptive multi-objective harmony search (SaMOHS) algorithm is developed to apply the parameter-setting-free technique, which is an example of a self-adaptive methodology. The SaMOHS algorithm attempts to remove some of the inconvenience from parameter setting and selects the most adaptive parameters during the iterative solution search process. To verify the proposed algorithm, an optimal least cost water distribution network design problem is applied to three different target networks. The results are compared with other well-known algorithms such as multi-objective harmony search and the non-dominated sorting genetic algorithm-II. The efficiency of the proposed algorithm is quantified by suitable performance indices. The results indicate that SaMOHS can be efficiently applied to the search for Pareto-optimal solutions in a multi-objective solution space.     

Particle-swarm harmony search for water network design

The optimal design of water distribution networks is a non-linear, multi-modal, and constrained problem classified as an NP-hard combinatorial problem. Because of the drawbacks of calculus-based algorithms, the problem has been tackled by assorted stochastic algorithms, such as the genetic algorithm, simulated annealing, tabu search, shuffled frog-leaping algorithm, ant colony optimization algorithm, harmony search, cross entropy, and scatter search. This study proposes a modified harmony search algorithm incorporating particle swarm concept. This algorithm was applied to the design of four bench-mark networks (two-loop, Hanoi, Balerma, and New York City networks), with good results.     

Optimal cost design of water distribution networks using a decomposition approach

Water distribution network decomposition, which is an engineering approach, is adopted to increase the efficiency of obtaining the optimal cost design of a water distribution network using an optimization algorithm. This study applied the source tracing tool in EPANET, which is a hydraulic and water quality analysis model, to the decomposition of a network to improve the efficiency of the optimal design process. The proposed approach was tested by carrying out the optimal cost design of two water distribution networks, and the results were compared with other optimal cost designs derived from previously proposed optimization algorithms. The proposed decomposition approach using the source tracing technique enables the efficient decomposition of an actual large-scale network, and the results can be combined with the optimal cost design process using an optimization algorithm. This proves that the final design in this study is better than those obtained with other previously proposed optimization algorithms.     

Minimizing residence times by rerouting flows to improve water quality in distribution networks

Management of water quality is a major issue for water companies, especially as many systems are old and have excess capacity. A methodology which uses an evolutionary algorithm to minimize water age, and hence improve water quality, is presented in this article. A steady-state model is used to find the water age at various nodes of a network. Three parameters are derived from these nodal age values to represent quality for the entire network. The evolutionary algorithm reconfigures the network by selecting a set of pipes for closure. The optimal network configuration is achieved when the chosen water age parameter is minimized subject to maintaining connectivity and hydraulic feasibility in the network. The methodology is applied to an example network to identify the age parameter that best represents quality over the entire network. The evolutionary model is then applied to re-route flows in a real water distribution network and the results are compared with those from the unmodified network. The validity of the use of steady-state hydraulics is tested by conducting an extended period simulation (EPS) on these results.     

Path entropy method for multiple-source water distribution networks

Over the past decade, one of the key research areas for water distribution networks has been the quantification of network reliability. An interesting approach in this research topic is the use of informational entropy as a surrogate measure for the reliability of water distribution network. Research on water distribution network entropy has progressed to the stage where practical applications are possible, but the actual meaning of the network entropy has never been fully elucidated. Recently, an alternative approach to calculate network entropy was proposed, which was termed the path entropy method (PEM). This alternative method was shown to be useful for the case of single-source water distribution networks. The purpose of this article is to explore the use of the PEM on multiple-source networks. A two-source and two-demand water distribution network is analysed to gain insight into the differences between single-source and multiple-source networks in terms of maximum-entropy flow distribution. This leads to the formal proof of the principle governing maximum-entropy flow ratios in multiple-source water distribution networks.     

Some new insights on informational entropy for water distribution networks

Several researchers have studied the use of informational entropy as a surrogate measure for the reliability of water distribution networks. The hypothesis is that the numerical value of network entropy in some way reflects the reliability of water distribution networks, and this appears to be supported by the analysis of some example water distribution networks. However, the precise relationship between the entropy value and some measures of reliability has not been formally established. The primary objective in this paper is to present an alternative methodology to calculate the informational entropy of water distribution networks. This methodology is termed as the Path Entropy Method (PEM), which provides some insights into the entropy of branching-tree networks and maximum-entropy flows of single-source networks. In addition, a quick method of computing the maximum-entropy value of single-source networks is presented and termed as the Simplified Path Entropy Method (SPEM).     

Optimal long-term design,rehabilitation and upgrading of water distribution networks

Given a limited budget, the choice of the best water distribution network upgrading strategy is a complex optimization problem. A model for the optimal long-term design and upgrading of new and existing water distribution networks is presented. A key strength of the methodology is the use of maximum entropy flows, which reduces the size of the problem and enables the application of linear programming for pipe size optimization. It also ensures the reliability level is high. The capital and maintenance costs and hydraulic performance are considered simultaneously for a predefined design horizon. The timing of upgrading over the entire planning horizon is obtained by dynamic programming. The deterioration over time of the structural integrity and hydraulic capacity of every pipe are explicitly considered. The upgrading options considered include pipe paralleling and replacement. The effectiveness of the model is demonstrated using the water supply network of Wobulenzi town in Uganda.     

Optimal design of water distribution networks by a discrete state transition algorithm

In this study it is demonstrated that, with respect to model formulation, the number of linear and nonlinear equations involved in water distribution networks can be reduced to the number of closed simple loops. Regarding the optimization technique, a discrete state transition algorithm (STA) is introduced to solve several cases of water distribution networks. Firstly, the focus is on a parametric study of the ‘restoration probability and risk probability’ in the dynamic STA. To deal effectively with head pressure constraints, the influence is then investigated of the penalty coefficient and search enforcement on the performance of the algorithm. Based on the experience gained from training the Two-Loop network problem, a discrete STA has successfully achieved the best known solutions for the Hanoi, triple Hanoi and New York network problems.     

A jazz-based approach for optimal setting of pressure reducing valves in water distribution networks

This study presents a model for valve setting in water distribution networks (WDNs), with the aim of reducing the level of leakage. The approach is based on the harmony search (HS) optimization algorithm. The HS mimics a jazz improvisation process able to find the best solutions, in this case corresponding to valve settings in a WDN. The model also interfaces with the improved version of a popular hydraulic simulator, EPANET 2.0, to check the hydraulic constraints and to evaluate the performances of the solutions. Penalties are introduced in the objective function in case of violation of the hydraulic constraints. The model is applied to two case studies, and the obtained results in terms of pressure reductions are comparable with those of competitive metaheuristic algorithms (e.g. genetic algorithms). The results demonstrate the suitability of the HS algorithm for water network management and optimization.     

Applying tabu search and simulated annealing to the optimal design of sewer networks

Optimizations of sewer network designs create complicated and highly nonlinear problems wherein conventional optimization techniques often get easily bogged down in local optima and cannot successfully address such problems. In the past decades, heuristic algorithms possessing robust and efficient global search capabilities have helped to solve continuous and discrete optimization problems and have demonstrated considerable promise. This study applied tabu search (TS) and simulated annealing (SA) to the optimization of sewer network designs. For a case study, this article used the sewer network design of a central Taiwan township, which contains significantly varied elevations, and the optimal designs from TS and SA were compared with the original official design. The results show that, in contrast with the original design's failure to satisfy the minimum flow-velocity requirements, both TS and SA achieved least-cost solutions that also fulfilled all the constraints of the design criteria. According to the average performance of 200 trials, SA outperformed TS in both robustness and efficiency for solving sewer network optimization problems.     

A heuristic design support methodology based on graph theory for district metering of water supply networks

The management of existing water supply networks can be substantially improved by permanent water district metering (WDM) which is one of the most efficient techniques for water loss detection and pressure management. However, WDM may compromise water system performance, since some pipes are usually closed to delimit districts in order not to have too many metering stations, to decrease costs and simplify water balance. This may reduce the reliability of the whole system and not guarantee the delivery of water at the different network nodes. In practical applications, the design of district meter areas (DMAs) is generally based on empirical approaches or on limited field experiences. In this work a design support methodology (DSM) is proposed, which helps to identify the position of flow meters and of boundary valves needed to define permanent DMAs. The DSM is based on graph theory and is applied to a test case.     

The harmony search heuristic algorithm for discrete structural optimization

Many methods have been developed and are in use for structural size optimization problems, in which the cross-sectional areas or sizing variables are usually assumed to be continuous. In most practical structural engineering design problems, however, the design variables are discrete. This paper proposes an efficient optimization method for structures with discrete-sized variables based on the harmony search (HS) heuristic algorithm. The recently developed HS algorithm was conceptualized using the musical process of searching for a perfect state of harmony. It uses a stochastic random search instead of a gradient search so that derivative information is unnecessary. In this article, a discrete search strategy using the HS algorithm is presented in detail and its effectiveness and robustness, as compared to current discrete optimization methods, are demonstrated through several standard truss examples. The numerical results reveal that the proposed method is a powerful search and design optimization tool for structures with discrete-sized members, and may yield better solutions than those obtained using current methods.     

Optimal operation of multiquality water distribution systems: unsteady conditions

This paper describes the methodology and application of a genetic algorithm scheme tailor-made to EPANET, for optimizing the operation of a water distribution system under unsteady water quality conditions. The water distribution system consists of sources of different qualities, treatment facilities, tanks, pipes, control valves, and pumping stations. The objective is to minimize the total cost of pumping and treating the water for a selected operational time horizon, while delivering the consumers the required quantities at acceptable qualities and pressures. The decision variables for each of the time steps that encompass the total operational time horizon include: the scheduling of the pumping units, settings of the control valves, and treatment removal ratios at the treatment facilities. The constraints are: head and concentrations at the consumer nodes, maximum removal ratios at the treatment facilities, maximum allowable amounts of water withdrawals at the sources, and returning at the end of the operational time horizon to a prescribed total volume in the tanks. The model is explored through two example applications.     

A scatter search algorithm for stacking sequence optimisation of laminate composites

This paper explores the metaheuristic approach called scatter search for lay-up sequence optimisation of laminate composite panels. Scatter search is an evolutionary method that has recently been found to be promising for solving combinatorial optimisation problems. The scatter search framework is flexible and allows the development of alternative implementations with varying degree of sophistication. The main objective of this paper is to demonstrate the effectiveness of the proposed scatter search algorithm for the combinatorial problem like stacking sequence optimisation of laminate composite panels. Preliminary investigations have been carried out to compare the optimal stacking sequences obtained using scatter search algorithm for buckling load maximisation with the best known published results. Studies indicate that the optimal buckling load factors obtained using the proposed scatter search algorithm found to be either superior or comparable to the best known published results.

Later, two case studies have been considered in this paper. Thermal buckling optimisation of laminated composite plates subjected to temperature rise is considered as the first case study. The results obtained are compared with an exact enumerative study conducted on the problem to demonstrate the effectiveness and performance of the proposed scatter search algorithm. The second case study is optimisation of hybrid laminate composite panels for weight and cost with frequency and buckling constraints. The two objectives are considered individually and also collectively to solve as multi-objective optimisation problem. Finally the computational efficiency of the proposed scatter search algorithm has been investigated by comparing the results with various implementations of genetic algorithm customised for laminate composites. It was shown in this paper through numerical experiments that the scatter search is capable of finding practical solutions for optimal lay-up sequence optimisation of composite laminates and results are comparable and sometimes even superior to genetic algorithms.     

A local search heuristic for bounded-degree minimum spanning trees

The bounded-degree minimum spanning tree (BDMST) problem has many practical applications. Unlike the unbounded case, the BDMST problem is NP-hard, and many attempts have been made to devise good approximation methods, including evolutionary algorithms. Inspired by recent applications to wireless communications, the present article focuses on the geometric version of the problem, i.e. the weights assigned to links (u, v) are equal to the Euclidean distance between u and v, but no grid geometry is used as an underlying structure. The proposed genetic local search procedure for BDMST-approximations utilizes a specific edge crossover operation, and the local search in-between applications of crossover performs alternating sequences of descending and ascending steps for each individual of the population. The length of a sequence with uniform direction is controlled by the estimated value of the maximum depth of local minima of the associated fitness landscape. The computational experiments were executed on ten synthetic networks, and a comparison to two recently published BDMST algorithms is presented.     

A stochastic framework for hydraulic performance assessment of complex water distribution networks: Application to connectivity detection problems

This paper is concerned with the hydraulic performance assessment of large scale water distribution networks in presence of uncertainty. In particular, the associate connectivity detection problem is examined in detail. For this purpose, a Bayesian system identification methodology is combined with an efficient hydraulic simulation model. A number of hydraulic model classes are defined as potential connectivity events. Based on information from flow rates in the pipes, the proposed updating technique provides estimates of the most probable connectivity scenarios. Such scenarios correspond to the model classes that maximize their evidences or posterior probabilities. The effectiveness of the proposed identification framework is illustrated by applying the connectivity detection approach to a real water distribution system.     

Linear program-based algorithm for the optimal design of wastewater treatment systems

This paper presents a new approach for the optimal design of distributed wastewater treatment networks with multiple contaminants. It consists of a two-stage solution strategy. In the first stage, a decomposition method is employed that replaces the general non-linear program (NLP) by a succession of linear programs, one for each treatment unit. In the second stage, the resulting network is used as a starting point for the solution of the general NLP by a local optimization solver. The decomposition process considers a specific substructure, where it is assumed that the wastewater streams go through the treatment units in sequence. To consider all combinations, the two-stage solution strategy is applied as many times as the number of possible sequences. This allows considering multiple and structurally different starting points, thus increasing the probability of finding global optimal solutions. The results have shown that the proposed approach can find better solutions than other approach reported in the literature, however with a drawback of being more demanding computationally. An erratum to this article can be found at