Infeasibility Maps: Application to the Optimization of the Design of Pumping Stations in Water Distribution Networks
2. Materials and Methods
- A new constraint was added to the mathematical optimization model. For each PS, this equation allows us to discard all pump models that, due to their specifications, did not manage to supply at least the maximum flow rate during the analysis period.
- The method used network preprocessing to determine in advance the maximum and minimum flow that each PS could provide. This procedure made it possible to limit the search space for solutions to the problem, thus eliminating areas of total infeasibility. An area of infeasibility is limited by ranges of the decision variable where it is impossible to meet the hydraulic constraints of the model. Our proposed method maps these ranges before starting the optimization process, accelerating the convergence of the algorithm using infeasibility maps (IMs).
- This study combined the use of the SC with the mapping of infeasibility zones to rule out unfeasible solutions during the evaluation process in any period, thus avoiding unnecessary hydraulic simulations when it was detected that part of the solution was not viable. Consequently, the IMs reduce the search space of the optimization algorithm. Reducing the search space to increase computational efficiency is a significant challenge faced when optimizing water networks
2.1. Mathematical Model
2.1.1. Mathematical Notation
- Nt: total number of time steps in the optimization process.
- Nps: total number of PSs in the network.
- Nb: total number of pump models available in the data set.
- Fa: amortization factor.
- r: interest rate.
- Np: total number of project life periods.
- H0,i, Ai: characteristic coefficients of the pump head installed in PSi.
- Ei, Fi: characteristic coefficients of the performance curve of the pump installed in PSi.
- Qi,j,k: represents the discharge of pump k during time step j in PSi.
- pi,j: energy cost in PSi during the time step j.
- ϒ: specific gravity of water.
- Δtj: discretization interval of the optimization period.
- mi,j: the number of FSPs running in PSi at time step j.
- ni,j: the number of VSPs running in PSi at time step j. These values depend on the selected pump model and the system selected to control the operation point.
- NB,i: total number of total pumps.
- HBmax: maximum head of the largest pump available in the data set.
- Hmax,i: maximum head supplied by PSi during time analysis.
- Cpump, i: purchase cost of a pump installed in PSi.
- ni: number of frequency inverters in PSi.
- Cfacility,i: cost of accessories including pipes in PSi.
- Ccontrol,i: sum of a pressure transducer, flowmeter, and programmable logic controller cost for PSi.
2.1.2. Decision Variables
- Xij: percentage of the flow supplied from PSi at each time step j.
- mi: number of fixed speed pumps in PSi.
- bi: ID of the pump model to be installed in PSi in the range [1,Nb].
2.1.3. Objective Function
2.2. Infeasibility Maps
- The distribution of flow generates sectors of the network where it is not possible to reach the minimum required pressures.
- Some of the PSs must provide a pressure greater than the maximum head of the largest pump that exists in the available catalog.
- The sum of the flows supplied is greater than the demand.
2.3. Case Study
2.4. Optimization Method
- The exhaustive construction of the IMs required a significant number of hydraulic simulations. However, this procedure only needed to be done once, representing only 2% of the total number of simulations.
- When IMs were not used, the search space was too large, and the algorithm took a long time to find feasible regions, which were usually local minima. The use of IMs allowed for accelerating the convergence of the optimization algorithms, rapidly evolving toward better solutions. Specifically, the number of simulations required by the IM-guided algorithm managed to reduce the number of hydraulic simulations necessary to achieve convergence in the case study by 60%.
- The use of IMs in the case study achieved savings of 71% compared with the solutions obtained by the optimization algorithm when considering the complete search space. Additionally, the 100 experiments ran using IMs had better solutions than the best solution obtained using the PGA when no IMs were used. An inadequate exploration of the solution space implies unnecessary cost overruns and non-optimal solutions for a given problem.
Data Availability Statement
Conflicts of Interest
- Mala-Jetmarova, H.; Sultanova, N.; Savic, D. Lost in optimisation of water distribution systems? a literature review of system design. Water 2018, 10, 307. [Google Scholar] [CrossRef][Green Version]
- Mala-Jetmarova, H.; Sultanova, N.; Savic, D. Lost in optimisation of water distribution systems? a literature review of system operation. Environ. Model. Softw. 2017, 93, 209–254. [Google Scholar] [CrossRef][Green Version]
- Gupta, A.; Kulat, K.D. A selective literature review on leak management techniques for water distribution system. Water Resour. Manag. 2018, 32, 3247–3269. [Google Scholar] [CrossRef]
- Bordea, D.; Pro, O.; Filip, I.; Drăgan, F.; Va, C. Modelling, Simulation and Controlling of a Multi-Pump System with Water Storage Powered by a Fluctuating and Intermittent Power Source. Mathematics 2022, 10, 4019. [Google Scholar] [CrossRef]
- León-Celi, C.F.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Savic, D. Operation of multiple pumped-water sources with no storage. J. Water Resour. Plan Manag. 2018, 144, 04018050. [Google Scholar] [CrossRef]
- Blinco, L.J.; Simpson, A.R.; Lambert, M.F.; Marchi, A. Comparison of pumping regimes for water distribution systems to minimize cost and greenhouse gases. J. Water Resour. Plan Manag. 2016, 142, 04016010. [Google Scholar] [CrossRef][Green Version]
- Gutiérrez-Bahamondes, J.H.; Mora-Meliá, D.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Salgueiro, Y. Pumping station design in water distribution networks considering the optimal flow distribution between sources and capital and operating costs. Water 2021, 13, 3098. [Google Scholar] [CrossRef]
- Martin-Candilejo, A.; Santillan, D.; Iglesias, A.; Garrote, L. Optimization of the design of water distribution systems for variable pumping flow rates. Water 2020, 12, 359. [Google Scholar] [CrossRef][Green Version]
- Oshurbekov, S.; Kazakbaev, V.; Prakht, V.; Dmitrievskii, V. Improving reliability and energy efficiency of three parallel pumps by selecting trade-off operating points. Mathematics 2021, 9, 1297. [Google Scholar] [CrossRef]
- Nagkoulis, N.; Katsifarakis, K.L. Minimization of Total Pumping Cost from an Aquifer to a Water Tank, Via a Pipe Network. Water Resour. Manag. 2020, 34, 4147–4162. [Google Scholar] [CrossRef]
- Makaremi, Y.; Haghighi, A.; Ghafouri, H.R. Optimization of Pump Scheduling Program in Water Supply Systems Using a Self-Adaptive NSGA-II; a Review of Theory to Real Application. Water Resour. Manag. 2017, 31, 1283–1304. [Google Scholar] [CrossRef]
- Gutiérrez-Bahamondes, J.H.; Salgueiro, Y.; Silva-Rubio, S.A.; Alsina, M.A.; Mora-Meliá, D.; Fuertes-Miquel, V.S. jHawanet: An open-source project for the implementation and assessment of multi-objective evolutionary algorithms on water distribution networks. Water 2019, 11, 2018. [Google Scholar] [CrossRef][Green Version]
- Fecarotta, O.; McNabola, A. Optimal location of pump as turbines (pats) in water distribution networks to recover energy and reduce leakage. Water Resour. Manag. 2017, 31, 5043–5059. [Google Scholar] [CrossRef]
- Carpitella, S.; Brentan, B.; Montalvo, I.; Izquierdo, J.; Certa, A. Multi-criteria analysis applied to multi-objective optimal pump scheduling in water systems. Water Sci. Technol. Water Supply 2019, 19, 2338–2346. [Google Scholar] [CrossRef][Green Version]
- Weber, J.B.; Lorenz, U. Optimizing Booster Stations. In Proceedings of the Genetic and Evolutionary Computation Conference Companion, Berlin, Germany, 15–19 July 2017; pp. 1303–1310. [Google Scholar] [CrossRef]
- Predescu, A.; Truică, C.O.; Apostol, E.S.; Mocanu, M.; Lupu, C. An advanced learning-based multiple model control supervisor for pumping stations in a smart water distribution system. Mathematics 2020, 8, 887. [Google Scholar] [CrossRef]
- Gil, F.A.A.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Cortes, J.V.L.; Mora-Meliá, D. Methodology for Projects Of Pumping Stations Directly Connected To The Network Considering The Operation Strategy. In Proceedings of the 22nd International Congress on Project management and Engineering, Madrid, Spain, 11–13 July 2018; 2018; pp. 551–563. Available online: http://dspace.aeipro.com/xmlui/handle/123456789/1728 (accessed on 20 March 2023).
- León-Celi, C.F.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Mora-Melia, D. The Setpoint Curve as a Tool for the Energy and Cost Optimization of Pumping Systems in Water Networks. Water 2022, 14, 2426. [Google Scholar] [CrossRef]
- Briceño-León, C.X.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Creaco, E. Integrating Demand Variability and Technical, Environmental, and Economic Criteria in Design of Pumping Stations Serving Closed Distribution Networks. J. Water Resour. Plan Manag. 2023, 149, 4023002. [Google Scholar] [CrossRef]
- Mora-Melia, D.; Iglesias-Rey, P.L.; Martinez-Solano, F.J.; Fuertes-Miquel, V.S. Design of water distribution networks using a pseudo-genetic algorithm and sensitivity of genetic operators. Water Resour. Manag. 2013, 27, 4149–4162. [Google Scholar] [CrossRef]
- Rossman, L.A. EPANET 2.0 User’s Manual (EPA/600/R-00/057); National Risk Management Research Laboratory: Cincinnatti, OH, USA, 2000. [Google Scholar]
- Lučin, I.; Lučin, B.; Čarija, Z.; Sikirica, A. Data-driven leak localization in urban water distribution networks using big data for random forest classifier. Mathematics 2021, 9, 672. [Google Scholar] [CrossRef]
- Negrete, M. Modelación Computacional en EPANET de un Sector de la Red de Abastecimiento de Agua Potable de Curicó; Universidad de Talca, Facultad de Ingeniería: Curicó, Chile, 2021. [Google Scholar]
- Mora-Melia, D.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Muñoz-Velasco, P. The efficiency of setting parameters in a modified shuffled frog leaping algorithm applied to optimizing water distribution networks. Water 2016, 8, 182. [Google Scholar] [CrossRef][Green Version]
- Benítez-Hidalgo, A.; Nebro, A.J.; García-Nieto, J.; Oregi, I.; Del Ser, J. jMetalPy: A Python framework for multi-objective optimization with metaheuristics. Swarm Evol. Comput. 2019, 51, 100598. [Google Scholar] [CrossRef][Green Version]
|OPEX||CAPEX||Fa ● CAPEX + OPEX|
|PGA without IMs||PGA with IMs||PGA without IMs||PGA with IMs||PGA without IMs||PGA with IMs|
|PS1||EUR 121||EUR 80.7||EUR 190,756||EUR 94,470||EUR 54,710||EUR 34,002|
|PS2||EUR 34||EUR 31||EUR 34,544||EUR 37,077||EUR 13,754||EUR 13,054|
|PS3||EUR 16||-||EUR 85,618||-||EUR 12,107||-|
|Total||EUR 80,571||EUR 47,056|
|Model ID||GNI 50-13/7.5||GNI 50-13/7.5|
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Gutiérrez-Bahamondes, J.H.; Mora-Melia, D.; Valdivia-Muñoz, B.; Silva-Aravena, F.; Iglesias-Rey, P.L. Infeasibility Maps: Application to the Optimization of the Design of Pumping Stations in Water Distribution Networks. Mathematics 2023, 11, 1582. https://doi.org/10.3390/math11071582
Gutiérrez-Bahamondes JH, Mora-Melia D, Valdivia-Muñoz B, Silva-Aravena F, Iglesias-Rey PL. Infeasibility Maps: Application to the Optimization of the Design of Pumping Stations in Water Distribution Networks. Mathematics. 2023; 11(7):1582. https://doi.org/10.3390/math11071582Chicago/Turabian Style
Gutiérrez-Bahamondes, Jimmy H., Daniel Mora-Melia, Bastián Valdivia-Muñoz, Fabián Silva-Aravena, and Pedro L. Iglesias-Rey. 2023. "Infeasibility Maps: Application to the Optimization of the Design of Pumping Stations in Water Distribution Networks" Mathematics 11, no. 7: 1582. https://doi.org/10.3390/math11071582