A heuristic algorithm for optimal cost design of gravity-fed water distribution networks. a real case study

This paper presents cross entropy (CE) optimization for optimal design of water distribution networks (WDN) under demand uncertainty. In design of WDNs, it is desired to achieve a minimum cost WDN that provides higher reliability in meeting the demands. To achieve these goals, an optimization model is formulated for design of WDNs with an objective of minimizing the total cost of WDN subject to meeting the nodal demands at a specified system reliability, mass conservation and other physical constraints. The uncertainty in future water demands is modeled using the theory of fuzzy random variable (FRV). The water demand at each node is assumed to be following a normal distribution with a fuzzy mean, and 10 % (or 20 %) of the fuzzy mean as its standard deviation. The water demand is represented as a triangular fuzzy number with the random demand as its kernel, and the interval of ±5 % (or ±10 %) variation of the random demand as its support for two scenarios. The fuzzy random system reliability (R) of WDNs is defined on the basis of necessity measure to assess system performance under fuzzy random demands and crisp head requirements. The latin hypercube sampling method is adopted for sampling of uncertain demands. The methodology is applied to two WDNs, and optimization models are solved through cross entropy optimization for different levels of reliability, and generated tradeoffs between the cost and R. On comparing the solutions obtained with the proposed methodology with earlier reported solutions, it is noted that the proposed method is very effective in producing robust optimal solutions. On analyzing the tradeoffs between reliability and costs, the results show that negligence of uncertainty can lead to under design of the WDNs, and the cost increases steeply at higher levels of reliability. The results of the two case studies demonstrate that the presented CE based methodology is effective for fuzzy-probabilistic design of WDNs.

Multi-objective meta-heuristics are used to optimize water distribution networks (WDNs) as they can achieve a near-optimal balance between cost and resilience in a unified platform. Most of these algorithms include tuning of algorithm-specific control parameters for higher optimization efficiency, leading to an increased computational effort. The current study is inspired by the desire to address the above problem. The goal is to formulate a multi-objective Rao algorithm (MORao) considering an existing modified resilience index (MRI) in the optimal design of WDNs. The model is demonstrated to attain Pareto-optimal solutions to complex WDN problems without exclusive parameter tuning. The algorithm is written in Python and is linked to a hydraulic model of a WDN implemented in EPANET 2.2 using pressure-driven demand (PDD) analysis. The method is demonstrated on three widely used networks: Two-loop, Goyang, and Fossolo. The Pareto-optimal solutions examine a tradeoff between two objectives to recognize competitive solutions. The network's resilience is increased 2.5 times by only 0.8 times increase in least-cost of Two-loop network. This research indicates that this method can achieve a satisfactory level of performance with a limited number of function evaluations.

This study presents an innovative data-driven approach to optimally design water distribution networks (WDNs). The methodology comprises five key stages: Generation of 600 synthetic WDNs with diverse properties, optimized to determine optimal component diameters; Extraction of 80 topological and hydraulic features from the optimized WDNs using graph theory; preprocessing and preparing the extracted features using established data science methods; Application of six feature selection methods (Variance Threshold, k-best, chi-squared, Light Gradient-Boosting Machine, Permutation, and Extreme Gradient Boosting) to identify the most relevant features for describing optimal diameters; and Integration of the selected features with four machine learning models (Random Forest, Support Vector Machine, Bootstrap Aggregating, and Light Gradient-Boosting Machine), resulting in 24 ensemble models. The Extreme Gradient Boosting-Light Gradient-Boosting Machine (Xg-LGB) model emerged as the optimal choice, achieving R2, MAE, and RMSE values of 0.98, 0.017, and 0.02, respectively. When applied to a benchmark WDN, this model accurately predicted optimal diameters, with R2, MAE, and RMSE values of 0.94, 0.054, and 0.06, respectively. These results highlight the developed model’s potential for the accurate and efficient optimal design of WDNs.

Global performance of metaheuristic optimization tools for water distribution networks

The paper describes the development of a DENET computer model that involves the application of an evolutionary optimization technique, differential evolution, linked to the hydraulic simulation solver, EPANET, for optimal design of water distribution networks. A model is formulated with the objective of minimizing cost and this formulation is applied to two benchmark water distribution system optimization problems—New York water supply system and Hanoi water distribution network. The study yielded promising results as compared with earlier studies in the literature and encouraged to reformulate the model for a new objective of maximizing network resilience. The results of the analysis demonstrate that DENET can be considered as a potential alternative tool for economical and reliable water distribution network planning and management.

Water distribution networks (WDNs) evolve continuously over time. Changes in water demands and pipe deterioration require construction upgrades to be performed on the network during its entire lifecycle. However, strategically planning WDNs, especially for the long term, is a challenging task. This is because parameters that are essential for the description of WDNs in the future, such as climate, population and demand transitions, are characterized by deep uncertainty. To cope with future uncertainty, and avoid overdesign or costly unplanned and reactive interventions, research is moving away from the static design of WDNs. Dynamic design approaches, aim to make water networks adaptive to changing conditions over long planning horizons. A promising, dynamic design approach is the staged design of WDNs, in which the planning horizon is divided into construction phases. This approach allows short-term interventions to be made, while simultaneously considering the expected long-term network growth outcomes. The aim of this paper is to summarize the current state of the art in staged design of water distribution networks. To achieve that, we critically examined relevant publications and classified them according to their shared key characteristics, such as the nature of the design problem (new or existing network design, expansion, strengthening, and rehabilitation), problem formulation (objective functions, length of planning horizon), optimization method, and uncertainty considerations. In the process, we discuss the latest findings in the literature, highlight the major contributions of staged design on water distribution networks, and suggest future research directions.

Optimization of water distribution networks is a NP-hard problem that researchers have tried to deal with using different formulations and algorithmic approaches. Among these, multi-objective heuristic algorithms are interesting because of their capacity for dealing with separate objectives that allow us to choose a posteriori the best compromise, but one of their main drawbacks is the long time required to obtain good solutions. Parallel processing is the most promising way to reduce the computing time and can make the convergence to adequate solutions faster. This paper intends to investigate the possibility of improving the efficacy and efficiency of an NSGA-II algorithm by parallelization of the optimization process at the same time. Results of different parallel implementations of NSGA-II applied to optimal design of small- and medium-size water distribution networks are presented. Good speed-up can be reached with a global model, hence improving the algorithm efficiency. Unlike the global model, the island model (or the hierarchical parallelization) can also improve its efficacy because it introduces fundamental changes in the algorithm exploration method. Possibilities offered by parallel island models have been investigated showing that some parameter configurations can find better solutions compared with the serial version of the algorithm.

Decision Support for Optimal Design of Water Distribution Networks: A Real Options Approach

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.

For optimally designing water distribution networks (WDNs), the nondeterministic polynomial hard problem, a novel hybrid model, is introduced with the combined features of evolutionary and swarm intelligence techniques. An evolutionary algorithm with better exploration properties, differential evolution (DE), and the swarm intelligence technique with better exploitation properties, namely the krill herd algorithm (KHA), is considered for this purpose. Because exploration and exploitation are the essential features of the metaheuristic algorithms, the hybrid algorithm with a combination of the DE and KHA features, the DE-KHA, resulted in a balanced search methodology. The results on the application of the proposed model on well-studied benchmark problems have demonstrated its enhanced search behavior, converging faster to the promising results with considerable robustness. Moreover, compared with other competing algorithms reported for optimally designing the WDNs, the DE-KHA outperforms with better computational efficiency. Additionally, considering the few control parameters that have to be calibrated for their optimal values, the computational burden will be less for performing the sensitivity analysis. As a result, considering the solution precision, quick convergence ability, and robustness of DE-KHA, the study suggests the algorithm for efficiently handling real-life case studies.

The paper treats looped networks wich have concentrated outflows or uniform outflow along the length of each pipe. An improved model is presented for optimal desing of new or partially extended water distribution networks, which operate either by means of gravity or a pump system.The model is based on the method of linear programming and allows the determination of an optimal distribution of commercial diameters for each pipe in the network and the length of the pipes which correspond to these diameters. Also, it is possible to take into account the various functional situations characteristic found during exploitation.

The design of water distribution networks (WDNs) is addressed by using a variant of the particle swarm optimization (PSO) algorithm. This variant, which makes use of a discrete version of PSO already considered by the authors, overcomes one of the PSO's main drawbacks, namely its difficulty in maintaining acceptable levels of population diversity and in balancing local and global searches. The performance of the variant proposed here is investigated by applying the model to solve two standard benchmark problems: the Hanoi new water distribution network and the New York Tunnel water supply system. The results obtained show considerable improvements in both convergence characteristics and the quality of the final solutions, and near-optimal results are consistently achieved at reduced computational cost.

We developed a hybrid algorithm for multi-objective design of water distribution networks (WDNs) in the present study. The proposed algorithm combines the global search schemes of differential evolution (DE) with the local search capabilities of harmony search (HS) to enhance the search proficiency of evolutionary algorithms. This method was compared with other multi-objective evolutionary algorithms (MOEAs) including NSGA2, SPEA2, MOEA/D and extended versions of DE and HS combined with non-dominance criteria using several metrics. We tested the compared algorithms on four benchmark WDN design problems with two objective functions, (i) the minimization of cost and (ii) the maximization of resiliency as reliability measure. The results showed that the proposed hybrid method provided better optimal solutions and outperformed the other algorithms. It also exhibited significant improvement over previous MOEAs. The hybrid algorithm generated new optimal solutions for a case study that dominated the best-known Pareto-optimal solutions in the literature

A novel two-hybrid optimization model of particle swarm optimization (FAPSO) and firefly algorithm with genetic algorithm (FAGA) are introduced to improve the performance of the conventional firefly algorithm for the least cost design of water distribution networks. The performance of the models is tested through application to three of the well-known benchmark networks available in the literature and also to the real case study of the El-Mostakbal City network, Ismailia, Egypt. The performance of the different algorithms was determined by evaluating the minimum, maximum, mean and standard deviation of costs, the function evaluation number, the consumed computational time for 1000 evaluations and the success rate calculated using the fuzzy logic concept for different optimal solutions slightly greater than the known optimal solution (by about 1.0% and 2.0%) were utilized for testing the convergence and search capabilities of the models. It was found that the FAGA model is superior to the standard firefly and FAPSO models in exploring the search space, exploiting the promising areas and convergence to the optimal solution and can be considered as a reasonable optimization technique for the management of water distribution networks.

Optimal design of the water distribution network is a multi-dimension complex optimization problem. In this paper, an auxiliary individual oriented crossover genetic algorithm is adopted to optimize the problem. Auxiliary individual oriented crossover operator adopts oriented crossover framework and assistant individual technique which is beneficial to multidimensional network problem. The simulation experiment is designed to test the performance of the improved algorithm. In the experiment, the improved algorithm is used to optimize the two classical pipeline networks. The results show that the algorithm can obtain the optimal engineering cost with low computational cost.