Optimal design of water distribution systems based on entropy and topology

A new multi-objective evolutionary optimization approach for joint topology and pipe size design of water distribution systems is presented. The algorithm proposed considers simultaneously the adequacy of flow and pressure at the demand nodes; the initial construction cost; the network topology; and a measure of hydraulic capacity reliability. The optimization procedure is based on a general measure of hydraulic performance that combines statistical entropy, network connectivity and hydraulic feasibility. The topological properties of the solutions are accounted for and arbitrary assumptions regarding the quality of infeasible solutions are not applied. In other words, both feasible and infeasible solutions participate in the evolutionary processes; solutions survive and reproduce or perish strictly according to their Pareto-optimality. Removing artificial barriers in this way frees the algorithm to evolve optimal solutions quickly. Furthermore, any redundant binary codes that result from crossover or mutation are eliminated gradually in a seamless and generic way that avoids the arbitrary loss of potentially useful genetic material and preserves the quality of the information that is transmitted from one generation to the next. The approach proposed is entirely generic: we have not introduced any additional parameters that require calibration on a case-by-case basis. Detailed and extensive results for two test problems are included that suggest the approach is highly effective. In general, the frontier-optimal solutions achieved include topologies that are fully branched, partially- and fully-looped and, for networks with multiple sources, completely separate sub-networks

GALAXY: A new hybrid MOEA for the Optimal Design of Water Distribution Systems

This is the final version of the article. Available from American Geophysical Union via the DOI in this record.The first author would like to appreciate the financial support given by both the University of Exeter and the China Scholarship Council (CSC) toward the PhD research. We also appreciate the three anonymous reviewers, who help improve the quality of this paper substantially. The source code of the latest versions of NSGA-II and ε-MOEA can be downloaded from the official website of Kanpur Genetic Algorithms Laboratory via http://www.iitk.ac.in/kangal/codes.shtml. The description of each benchmark problem used in this paper, including the input file of EPANET and the associated best-known Pareto front, can be accessed from the following link to the Centre for Water Systems (http://tinyurl.com/cwsbenchmarks/). GALAXY can be accessed via http://tinyurl.com/cws-galaxy

On the Complexities of the Design of Water Distribution Networks

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.This work has been developed with the support of the project IDAWAS, DPI2009-11591, of the Direccion General de Investigacion of the Ministerio de Educacion y Ciencia, and ACOMP/2010/146 of the Conselleria d'Educacio of the Generalitat Valenciana. The first author is also indebted to the Universitat Politecnica de Valencia for the sabbatical leave granted during the first semester of 2011. The use of English in this paper was revised by John Rawlins.Izquierdo Sebastián, J.; Montalvo Arango, I.; Pérez García, R.; Matías, A. (2012). On the Complexities of the Design of Water Distribution Networks. Mathematical Problems in Engineering. 2012:1-25. https://doi.org/10.1155/2012/9479611252012Goulter, I. C., & Coals, A. V. (1986). Quantitative Approaches to Reliability Assessment in Pipe Networks. Journal of Transportation Engineering, 112(3), 287-301. doi:10.1061/(asce)0733-947x(1986)112:3(287)Goulter, I. C., & Bouchart, F. (1990). Reliability‐Constrained Pipe Network Model. Journal of Hydraulic Engineering, 116(2), 211-229. doi:10.1061/(asce)0733-9429(1990)116:2(211)Kleiner, Y., Adams, B. J., & Rogers, J. S. (2001). Water Distribution Network Renewal Planning. Journal of Computing in Civil Engineering, 15(1), 15-26. doi:10.1061/(asce)0887-3801(2001)15:1(15)Dandy, G. C., & Engelhardt, M. O. (2006). Multi-Objective Trade-Offs between Cost and Reliability in the Replacement of Water Mains. Journal of Water Resources Planning and Management, 132(2), 79-88. doi:10.1061/(asce)0733-9496(2006)132:2(79)Izquierdo, J., Pérez, R., & Iglesias, P. L. (2004). Mathematical models and methods in the water industry. Mathematical and Computer Modelling, 39(11-12), 1353-1374. doi:10.1016/j.mcm.2004.06.012Giustolisi, O., Savic, D., & Kapelan, Z. (2008). Pressure-Driven Demand and Leakage Simulation for Water Distribution Networks. Journal of Hydraulic Engineering, 134(5), 626-635. doi:10.1061/(asce)0733-9429(2008)134:5(626)Montalvo, I., Izquierdo, J., Pérez, R., & Tung, M. M. (2008). Particle Swarm Optimization applied to the design of water supply systems. Computers & Mathematics with Applications, 56(3), 769-776. doi:10.1016/j.camwa.2008.02.006Montalvo, I., Izquierdo, J., Pérez, R., & Iglesias, P. L. (2008). A diversity-enriched variant of discrete PSO applied to the design of water distribution networks. Engineering Optimization, 40(7), 655-668. doi:10.1080/03052150802010607Montalvo, I., Izquierdo, J., Pérez-García, R., & Herrera, M. (2010). Improved performance of PSO with self-adaptive parameters for computing the optimal design of Water Supply Systems. Engineering Applications of Artificial Intelligence, 23(5), 727-735. doi:10.1016/j.engappai.2010.01.015Martínez, J. B. (2010). Cost and reliability comparison between branched and looped water supply networks. Journal of Hydroinformatics, 12(2), 150-160. doi:10.2166/hydro.2009.080Goulter, I. C. (1992). Systems Analysis in Water‐Distribution Network Design: From Theory to Practice. Journal of Water Resources Planning and Management, 118(3), 238-248. doi:10.1061/(asce)0733-9496(1992)118:3(238)Park, H., & Liebman, J. C. (1993). Redundancy‐Constrained Minimum‐Cost Design of Water‐Distribution Nets. Journal of Water Resources Planning and Management, 119(1), 83-98. doi:10.1061/(asce)0733-9496(1993)119:1(83)Khomsi, D., Walters, G. A., Thorley, A. R. D., & Ouazar, D. (1996). Reliability Tester for Water-Distribution Networks. Journal of Computing in Civil Engineering, 10(1), 10-19. doi:10.1061/(asce)0887-3801(1996)10:1(10)Tanyimboh, T. T., Tabesh, M., & Burrows, R. (2001). Appraisal of Source Head Methods for Calculating Reliability of Water Distribution Networks. Journal of Water Resources Planning and Management, 127(4), 206-213. doi:10.1061/(asce)0733-9496(2001)127:4(206)Kalungi, P., & Tanyimboh, T. T. (2003). Redundancy model for water distribution systems. Reliability Engineering & System Safety, 82(3), 275-286. doi:10.1016/s0951-8320(03)00168-6Morgan, D. R., & Goulter, I. C. (1985). Optimal urban water distribution design. Water Resources Research, 21(5), 642-652. doi:10.1029/wr021i005p00642Walters, G. A., & Knezevic, J. (1989). Discussion of « Reliability‐Based Optimization Model for Water Distribution Systems » by Yu‐Chun Su, Larry W. Mays, Ning Duan, and Kevin E. Lansey (December, 1987, Vol. 113, No. 12). Journal of Hydraulic Engineering, 115(8), 1157-1158. doi:10.1061/(asce)0733-9429(1989)115:8(1157)LOGANATHAN, G. V., SHERALI, H. D., & SHAH, M. P. (1990). A TWO-PHASE NETWORK DESIGN HEURISTIC FOR MINIMUM COST WATER DISTRIBUTION SYSTEMS UNDER A RELIABILITY CONSTRAINT. Engineering Optimization, 15(4), 311-336. doi:10.1080/03052159008941160Bouchart, F., & Goulter, I. (1991). Reliability Improvements in Design of Water Distribution Networks Recognizing Valve Location. Water Resources Research, 27(12), 3029-3040. doi:10.1029/91wr00590Gupta, R., & Bhave, P. R. (1994). Reliability Analysis of Water‐Distribution Systems. Journal of Environmental Engineering, 120(2), 447-461. doi:10.1061/(asce)0733-9372(1994)120:2(447)Xu, C., & Goulter, I. C. (1999). Reliability-Based Optimal Design of Water Distribution Networks. Journal of Water Resources Planning and Management, 125(6), 352-362. doi:10.1061/(asce)0733-9496(1999)125:6(352)Su, Y., Mays, L. W., Duan, N., & Lansey, K. E. (1987). Reliability‐Based Optimization Model for Water Distribution Systems. Journal of Hydraulic Engineering, 113(12), 1539-1556. doi:10.1061/(asce)0733-9429(1987)113:12(1539)Cullinane, M. J., Lansey, K. E., & Mays, L. W. (1992). Optimization‐Availability‐Based Design of Water‐Distribution Networks. Journal of Hydraulic Engineering, 118(3), 420-441. doi:10.1061/(asce)0733-9429(1992)118:3(420)Vamvakeridou-Lyroudia, L. S., Walters, G. A., & Savic, D. A. (2005). Fuzzy Multiobjective Optimization of Water Distribution Networks. Journal of Water Resources Planning and Management, 131(6), 467-476. doi:10.1061/(asce)0733-9496(2005)131:6(467)Montalvo, I., Izquierdo, J., Schwarze, S., & Pérez-García, R. (2010). Multi-objective particle swarm optimization applied to water distribution systems design: An approach with human interaction. Mathematical and Computer Modelling, 52(7-8), 1219-1227. doi:10.1016/j.mcm.2010.02.017Izquierdo, J., Montalvo, I., Pérez, R., & Fuertes, V. S. (2008). Design optimization of wastewater collection networks by PSO. Computers & Mathematics with Applications, 56(3), 777-784. doi:10.1016/j.camwa.2008.02.007Dong, Y., Tang, J., Xu, B., & Wang, D. (2005). An application of swarm optimization to nonlinear programming. Computers & Mathematics with Applications, 49(11-12), 1655-1668. doi:10.1016/j.camwa.2005.02.006Jin, Y.-X., Cheng, H.-Z., Yan, J., & Zhang, L. (2007). New discrete method for particle swarm optimization and its application in transmission network expansion planning. Electric Power Systems Research, 77(3-4), 227-233. doi:10.1016/j.epsr.2006.02.016Arumugam, M. S., & Rao, M. V. C. (2008). On the improved performances of the particle swarm optimization algorithms with adaptive parameters, cross-over operators and root mean square (RMS) variants for computing optimal control of a class of hybrid systems. Applied Soft Computing, 8(1), 324-336. doi:10.1016/j.asoc.2007.01.010Izquierdo, J., Montalvo, I., Pérez, R., & Fuertes, V. S. (2009). Forecasting pedestrian evacuation times by using swarm intelligence. Physica A: Statistical Mechanics and its Applications, 388(7), 1213-1220. doi:10.1016/j.physa.2008.12.008Herrera, M., Izquierdo, J., Montalvo, I., García-Armengol, J., & Roig, J. V. (2009). Identification of surgical practice patterns using evolutionary cluster analysis. Mathematical and Computer Modelling, 50(5-6), 705-712. doi:10.1016/j.mcm.2008.12.026Molina, J., Santana, L. V., Hernández-Díaz, A. G., Coello Coello, C. A., & Caballero, R. (2009). g-dominance: Reference point based dominance for multiobjective metaheuristics. European Journal of Operational Research, 197(2), 685-692. doi:10.1016/j.ejor.2008.07.01510.1029/89WR02879. (2010). Water Resources Research. doi:10.1029/89wr02879Savic, D. A., & Walters, G. A. (1997). Genetic Algorithms for Least-Cost Design of Water Distribution Networks. Journal of Water Resources Planning and Management, 123(2), 67-77. doi:10.1061/(asce)0733-9496(1997)123:2(67)Zecchin, A. C., Simpson, A. R., Maier, H. R., & Nixon, J. B. (2005). Parametric Study for an Ant Algorithm Applied to Water Distribution System Optimization. IEEE Transactions on Evolutionary Computation, 9(2), 175-191. doi:10.1109/tevc.2005.844168Yurong Liu, Zidong Wang, Jinling Liang, & Xiaohui Liu. (2009). Stability and Synchronization of Discrete-Time Markovian Jumping Neural Networks With Mixed Mode-Dependent Time Delays. IEEE Transactions on Neural Networks, 20(7), 1102-1116. doi:10.1109/tnn.2009.2016210Jinling Liang, Zidong Wang, & Xiaohui Liu. (2009). State Estimation for Coupled Uncertain Stochastic Networks With Missing Measurements and Time-Varying Delays: The Discrete-Time Case. IEEE Transactions on Neural Networks, 20(5), 781-793. doi:10.1109/tnn.2009.2013240Zidong Wang, Yao Wang, & Yurong Liu. (2010). Global Synchronization for Discrete-Time Stochastic Complex Networks With Randomly Occurred Nonlinearities and Mixed Time Delays. IEEE Transactions on Neural Networks, 21(1), 11-25. doi:10.1109/tnn.2009.2033599Bo Shen, Zidong Wang, & Xiaohui Liu. (2011). Bounded $H_{\infty}$ Synchronization and State Estimation for Discrete Time-Varying Stochastic Complex Networks Over a Finite Horizon. IEEE Transactions on Neural Networks, 22(1), 145-157. doi:10.1109/tnn.2010.209066

Multiobjective evolutionary optimization of water distribution systems : exploiting diversity with infeasible solutions

This article investigates the computational efficiency of constraint handling in multi-objective evolutionary optimization algorithms for water distribution systems. The methodology investigated here encourages the co-existence and simultaneous development including crossbreeding of subpopulations of cost-effective feasible and infeasible solutions based on Pareto dominance. This yields a boundary search approach that also promotes diversity in the gene pool throughout the progress of the optimization by exploiting the full spectrum of non-dominated infeasible solutions. The relative effectiveness of small and moderate population sizes with respect to the number of decision variables is investigated also. The results reveal the optimization algorithm to be efficient, stable and robust. It found optimal and near-optimal solutions reliably and efficiently. The real-world system based optimisation problem involved multiple variable head supply nodes, 29 fire-fighting flows, extended period simulation and multiple demand categories including water loss. The least cost solutions found satisfied the flow and pressure requirements consistently. The cheapest feasible solutions achieved represent savings of 48.1% and 48.2%, for populations of 200 and 1000, respectively, and the population of 1000 achieved slightly better results overall

Informational entropy : a failure tolerance and reliability surrogate for water distribution networks

Evolutionary algorithms are used widely in optimization studies on water distribution networks. The optimization algorithms use simulation models that analyse the networks under various operating conditions. The solution process typically involves cost minimization along with reliability constraints that ensure reasonably satisfactory performance under abnormal operating conditions also. Flow entropy has been employed previously as a surrogate reliability measure. While a body of work exists for a single operating condition under steady state conditions, the effectiveness of flow entropy for systems with multiple operating conditions has received very little attention. This paper describes a multi-objective genetic algorithm that maximizes the flow entropy under multiple operating conditions for any given network. The new methodology proposed is consistent with the maximum entropy formalism that requires active consideration of all the relevant information. Furthermore, an alternative but equivalent flow entropy model that emphasizes the relative uniformity of the nodal demands is described. The flow entropy of water distribution networks under multiple operating conditions is discussed with reference to the joint entropy of multiple probability spaces, which provides the theoretical foundation for the optimization methodology proposed. Besides the rationale, results are included that show that the most robust or failure-tolerant solutions are achieved by maximizing the sum of the entropies

A multiobjective optimization framework for multicontaminant industrial water network design.

The optimal design of multicontaminant industrial water networks according to several objectives is carried out in this paper. The general formulation of the water allocation problem (WAP) is given as a set of nonlinear equations with binary variables representing the presence of interconnections in the network. For optimization purposes, three antagonist objectives are considered: F1, the freshwater flow-rate at the network entrance, F2, the water flow-rate at inlet of regeneration units, and F3, the number of interconnections in the network. The multiobjective problem is solved via a lexicographic strategy, where a mixed-integer nonlinear programming (MINLP) procedure is used at each step. The approach is illustrated by a numerical example taken from the literature involving five processes, one regeneration unit and three contaminants. The set of potential network solutions is provided in the form of a Pareto front. Finally, the strategy for choosing the best network solution among those given by Pareto fronts is presented. This Multiple Criteria Decision Making (MCDM) problem is tackled by means of two approaches: a classical TOPSIS analysis is first implemented and then an innovative strategy based on the global equivalent cost (GEC) in freshwater that turns out to be more efficient for choosing a good network according to a practical point of view

A robust approach based on time variable trigger levels for pump control

AbstractAn approach for the control of a pumping plant feeding a tank at the inlet of a water distribution system is presented. The approach is aimed at minimizing the energy costs by maximizing pumping during off-peak electricity tariff periods. It is based on trigger levels which are variable during the day according to a prefixed pattern in order to ensure that the water level in the elevated tank is at its minimum and maximum values at the end of the peak and off-peak tariff periods, respectively. The pattern of the trigger levels is defined by solving a multi-objective problem aimed at minimizing the energy costs and the number of pump switches. The approach was applied to a couple of real cases with a single tank. The approach was compared with other methodologies typically used for pump control, i.e. fixed trigger levels (FTLs) and pump scheduling (PS). The results show for the two particular cases that the proposed approach achieves energy costs that are lower than those obtainable by using FTLs, and comparable with those obtainable by using PS. This is based on achieving a similar number of pump switches

Scalable Pareto set generation for multiobjective co-design problems in water distribution networks: a continuous relaxation approach

In this paper, we study the multiobjective co-design problem of optimal valve placement and operation in water distribution networks, addressing the minimization of average pressure and pressure variability indices. The presented formulation considers nodal pressures, pipe flows and valve locations as decision variables, where binary variables are used to model the placement of control valves. The resulting optimization problem is a multiobjective mixed integer nonlinear optimization problem. As conflicting objectives, average zone pressure and pressure variability can not be simultaneously optimized. Therefore, we present the concept of Pareto optima sets to investigate the trade-offs between the two conflicting objectives and evaluate the best compromise. We focus on the approximation of the Pareto front, the image of the Pareto optima set through the objective functions, using the weighted sum, normal boundary intersection and normalized normal constraint scalarization techniques. Each of the three methods relies on the solution of a series of single-objective optimization problems, which are mixed integer nonlinear programs (MINLPs) in our case. For the solution of each single-objective optimization problem, we implement a relaxation method that solves a sequence of nonlinear programs (NLPs) whose stationary points converge to a stationary point of the original MINLP. The relaxed NLPs have a sparse structure that come from the sparse water network graph constraints. In solving the large number of relaxed NLPs, sparsity is exploited by tailored techniques to improve the performance of the algorithms further and render the approaches scalable for large scale networks. The features of the proposed scalarization approaches are evaluated using a published benchmarking network model

Maximum entropy based evolutionary optimization of water distribution networks under multiple operating conditions and self-adaptive search space reduction method

Previously held under moratorium from 1st December 2016 until 1st December 2021.One of the complexities in designing WDN is evaluation of network performance. The accurate network performance measures such as reliability or failure tolerance are very time consuming calculations, thus surrogate measures are used for water distribution network (WDN) design optimization. Entropy is particularly advantageous since it involves only the flow in the pipe and the demands at the nodes. This thesis developed efficient new computational methods based on the maximum entropy formalism for the optimization of water distribution systems. Thus the maximum entropy based design approach has been extended here to include multiple operation conditions. Also, the path-related properties of the flow entropy have been exploited to develop a new self-adaptive approach for solution space reduction in multiobjective evolutionary optimization of water distribution systems that resulted in a significant reduction in the number of function evaluations required to find optimal and near optimal solutions. The novelty and originality of the current research are presented next. A new penalty-free multi-objective evolutionary optimization approach for the design of WDNs has been developed. It combines genetic algorithm with least cost design and maximum entropy. The approach can handle single operating conditions (SOC) as well as multiple operating conditions (MOC) for any given network. Previously, most of the work has been done for single loading patterns and it was assumed that nodal demands are constant. In reality nodal demand vary over the time so network designed to satisfy one operating condition might not be able to satisfy other loading patterns (i.e. pressure constraints might not be meet). The model has been applied to three well known water distribution networks. The approach has also been implemented on a large real-world network in the literature. Three different methods of designing for multiple loading patterns were investigated. Extensive testing proved that MOC outperform SOC in terms of hydraulic feasibility, pipe size distribution and reliability. The approach is computationally efficient and robust. The above mentioned penalty-free approach has been extended to form a module that would improve the convergence criteria of the GA by reducing its search space. For large real-world network GA might require extremely large number of function evaluations which could lead to delayed convergence. By reducing the search space, the GA’s effectiveness and efficiency will increase as the algorithm will identify the solutions in smaller number of function evaluations. The search space reduction method presented herein is based on entropy and uses the importance of every path through network, which is an inherent property of the entropy function. The developed algorithm is dynamic, self-adaptive and does not require pre-defining the reduced sets of candidate diameters for each pipe. The method has been applied to a large network from the literature. Two cases were studied, one based on full search space and one for reduce search space (RSS) approach. Rapid stabilization was observed for the results obtained using RSS.One of the complexities in designing WDN is evaluation of network performance. The accurate network performance measures such as reliability or failure tolerance are very time consuming calculations, thus surrogate measures are used for water distribution network (WDN) design optimization. Entropy is particularly advantageous since it involves only the flow in the pipe and the demands at the nodes. This thesis developed efficient new computational methods based on the maximum entropy formalism for the optimization of water distribution systems. Thus the maximum entropy based design approach has been extended here to include multiple operation conditions. Also, the path-related properties of the flow entropy have been exploited to develop a new self-adaptive approach for solution space reduction in multiobjective evolutionary optimization of water distribution systems that resulted in a significant reduction in the number of function evaluations required to find optimal and near optimal solutions. The novelty and originality of the current research are presented next. A new penalty-free multi-objective evolutionary optimization approach for the design of WDNs has been developed. It combines genetic algorithm with least cost design and maximum entropy. The approach can handle single operating conditions (SOC) as well as multiple operating conditions (MOC) for any given network. Previously, most of the work has been done for single loading patterns and it was assumed that nodal demands are constant. In reality nodal demand vary over the time so network designed to satisfy one operating condition might not be able to satisfy other loading patterns (i.e. pressure constraints might not be meet). The model has been applied to three well known water distribution networks. The approach has also been implemented on a large real-world network in the literature. Three different methods of designing for multiple loading patterns were investigated. Extensive testing proved that MOC outperform SOC in terms of hydraulic feasibility, pipe size distribution and reliability. The approach is computationally efficient and robust. The above mentioned penalty-free approach has been extended to form a module that would improve the convergence criteria of the GA by reducing its search space. For large real-world network GA might require extremely large number of function evaluations which could lead to delayed convergence. By reducing the search space, the GA’s effectiveness and efficiency will increase as the algorithm will identify the solutions in smaller number of function evaluations. The search space reduction method presented herein is based on entropy and uses the importance of every path through network, which is an inherent property of the entropy function. The developed algorithm is dynamic, self-adaptive and does not require pre-defining the reduced sets of candidate diameters for each pipe. The method has been applied to a large network from the literature. Two cases were studied, one based on full search space and one for reduce search space (RSS) approach. Rapid stabilization was observed for the results obtained using RSS

Self-adaptive solution-space reduction algorithm for multi-objective evolutionary design optimization of water distribution networks

An effective way to improve the computational efficiency of evolutionary algorithms is to make the solution space of the optimization problem under consideration smaller. A new reliability-based algorithm that does this was developed for water distribution networks. The objectives considered in the formulation of the optimization problem were minimization of the initial construction cost and maximization of the flow entropy as a resilience surrogate. After achieving feasible solutions, the active solution space of the optimization problem was re-set for each pipe in each generation until the end of the optimization. The algorithm re-set the active solution space by reducing the number of pipe diameter options for each pipe, based on the most likely flow distribution. The main components of the methodology included an optimizer, a hydraulic simulator and an algorithm that calculates the flow entropy for any given network configuration. The methodology developed is generic and self-adaptive, and prior setting of the reduced solution space is not required. A benchmark network in the literature was investigated, and the results showed that the algorithm improved the computational efficiency and quality of the solutions achieved by a considerable margin