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Pump scheduling in drinking water distribution networks with an LP/NLP-based branch and bound

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Abstract

This paper offers a novel approach for computing globally optimal solutions to the pump scheduling problem in drinking water distribution networks. A tailored integer linear relaxation of the original non-convex formulation is devised and solved by branch and bound where integer nodes are investigated through non-linear programming to check the satisfaction of the non-convex constraints and compute the actual cost. This generic method can tackle a large variety of networks, e.g. with variable-speed pumps. We also propose to specialize it for a common subclass of networks with several improving techniques, including a new primal heuristic to repair near-feasible integer relaxed solutions. Our approach is numerically assessed on various case studies of the literature and compared with recently reported results.

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Notes

  1. Available at http://www.or.deis.unibo.it/research_pages/ORinstances/ORinstances.htm.

  2. In Morsi et al. (2012), transitional regimes are taken into account through the hammer equation but, as pointed out in D’Ambrosio et al. (2015), it is yet unclear whether the dynamic hydraulic behavior needs to be described this accurately in the context of pump scheduling.

  3. With respect to Costa et al. (2016), we connect the water tanks 165 and 265 with a pipe of zero length to prevent non-physical behaviors induced by the discretization, especially with long time steps. This change is justified by the fact that the head levels in the two tanks are always very close, as shown in Figure 8 of Costa et al. (2016). The alternative used in Costa et al. (2016) is to run the extended period analysis on a smaller time step.

  4. To use our MINLP formulation, we made three modifications: 1) the minimal pressure level \(\overline{P}\) is only required for internal nodes with positive demands, 2) we modeled the complex operation of the water tanks (see Eq.(5)–(9) in Verleye and Aghezzaf (2013)) by preceding each water tank with a PRV, 3) we dropped the operating constraints related to raw water pump.

  5. The authors of Ghaddar et al. (2015) and Naoum-Sawaya et al. (2015) do not mention check valves in their mathematical formulation, but likely include them in the hydraulic model of EPANET. That could explain the inconsistency between the lower bounds reported in Ghaddar et al. (2015) and our results.

  6. Note that Menke et al. (2016b) and Verleye and Aghezzaf (2013) report approximated solutions that cannot be directly compared with ours.

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Acknowledgements

We would like to thank Bradley Eck, Joe Naoum-Sawaya, Bruno de Athayde Prata, Hanyu Shi and Derek Verleye for sharing with us some complementary elements concerning their respective data and for providing us technical details concerning their model. The work of Gratien Bonvin was partially supported by a Doc.Mobility Grant (P1SKP2_174858) from the Swiss National Science Foundation. We are indebted to three anonymous referees for a careful reading and some very useful suggestions.

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Authors and Affiliations

  1. Center for Applied Mathematics, Mines ParisTech, PSL Research University, Sophia-Antipolis, France

    Gratien Bonvin & Sophie Demassey

  2. CERC and École Polytechnique de Montreal, Montreal, Canada

    Andrea Lodi

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  1. Gratien Bonvin
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  2. Sophie Demassey
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  3. Andrea Lodi
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Correspondence to Andrea Lodi.

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Data Availability

The material related to the reproduction of the experiments is available under request to the second author. Parts of the code are also available through the GOPS (Global Optimisation for Pump Scheduling) project under the terms of the GPL license and can be downloaded at https://github.com/sofdem/gopslpnlpbb.

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Bonvin, G., Demassey, S. & Lodi, A. Pump scheduling in drinking water distribution networks with an LP/NLP-based branch and bound. Optim Eng 22, 1275–1313 (2021). https://doi.org/10.1007/s11081-020-09575-y

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