# American Institute of Mathematical Sciences

2012, 2(4): 695-711. doi: 10.3934/naco.2012.2.695

## Towards globally optimal operation of water supply networks

 1 Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany 2 Siemens AG, Corporate Technology (CT RTC AUC SIM-DE), Otto-Hahn-Ring 6, 81739 Munich, Germany 3 Technische Universität München, International School of Applied Mathematics, Boltzmannstr. 3, 85748 Garching b. Munich, Germany 4 Humboldt-Universität, Department of Mathematics, Unter den Linden 6, 10099 Berlin, Germany

Received  March 2012 Revised  October 2012 Published  November 2012

This paper is concerned with optimal operation of pressurized water supply networks at a fixed point in time. We use a mixed-integer nonlinear programming (MINLP) model incorporating both the nonlinear physical laws and the discrete decisions such as switching pumps on and off. We demonstrate that for instances from our industry partner, these stationary models can be solved to $\epsilon$-global optimality within small running times using problem-specific presolving and state-of-the-art MINLP algorithms.
In our modeling, we emphasize the importance of distinguishing between what we call real and imaginary flow, i.e., taking into account that the law of Darcy-Weisbach correlates pressure difference and flow along a pipe if and only if water is available at the high pressure end of a pipe. Our modeling solution extends to the dynamic operative planning problem.
Citation: Ambros M. Gleixner, Harald Held, Wei Huang, Stefan Vigerske. Towards globally optimal operation of water supply networks. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 695-711. doi: 10.3934/naco.2012.2.695
##### References:
 [1] Tobias Achterberg, "Constraint Integer Programming," PhD thesis, Technische Universität Berlin, 2007. [2] Tobias Achterberg, SCIP: Solving constraint integer programs, Mathematical Programming Computation, 1 (2009), 1-41, doi: 10.1007/s12532-008-0001-1. [3] Pietro Belotti, Jon Lee, Leo Liberti, Francois Margot and Andreas Wächter, Branching and bounds tightening techniques for non-convex MINLP, Optimization Methods and Software, 24 (2009), 597-634. doi: 10.1080/10556780903087124. [4] Timo Berthold, "Primal Heuristics for Mixed Integer Programs," Master's thesis, Technische Universität Berlin, 2006. [5] Timo Berthold and Ambros M. Gleixner, Undercover - a primal heuristic for MINLP based on sub-MIPs generated by set covering, in "Proceedings of the EWMINLP" (eds. Pierre Bonami, Leo Liberti, Andrew J. Miller and Annick Sartenaer), (2010), 103-112. [6] Timo Berthold and Ambros M. Gleixner, Undercover - a primal MINLP heuristic exploring a largest sub-MIP, ZIB-Report 12-07 (2012), Zuse Institute Berlin, http://vs24.kobv.de/opus4-zib/frontdoor/index/index/docId/1463/. [7] Timo Berthold, Stefan Heinz, Marc E. Pfetsch and Stefan Vigerske, Large neighborhood search beyond MIP, in "Proceedings of the 9th Metaheuristics International Conference" (eds. Luca Di Gaspero, Andrea Schaerf, and Thomas Stützle), (2011), 51-60. [8] Timo Berthold, Stefan Heinz and Stefan Vigerske, Extending a CIP framework to solve MIQCPs, in "The IMA volumes in Mathematics and its Applications" (eds. Jon Lee and Sven Leyffer), 154 (2012), 427-444. doi: 10.1007/978-1-4614-1927-3_15. [9] Cristiana Bragalli, Claudia D'mbrosio, Jon Lee, Andrea Lodi and Paolo Toth, On the optimal design of water distribution networks: a practical MINLP approach, Optimization and Engineering, 13 (2012), 219-246. doi: 10.1007/s11081-011-9141-7. [10] Jens Burgschweiger, Bernd Gnädig and Marc C. Steinbach, Optimization models for operative planning in drinking water networks, ZIB-Report 04-48 (2004), Zuse Institute Berlin, http://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/823. [11] Björn Geißler, Oliver Kolb, Jens Lang, Günter Leugering, Alexander Martin and Antonio Morsi, Mixed integer linear models for the optimization of dynamical transport networks, Mathematical Methods of Operations Research, 73 (2011), 339-362, doi: 10.1007/s00186-011-0354-5. [12] Wei Huang, "Operative Planning of Water Supply Networks by Mixed Integer Nonlinear Programming," Master's thesis, Freie Universität Berlin, 2011. [13] Kathrin Klamroth, Jens Lang, Günter Leugering, Alexander Martin, Antonio Morsi, Martin Oberlack, Manfred Ostrowski and Roland Rosen, "Mathematical Optimization of Water Networks," International Series of Numerical Mathematics, 162 (2012), Birkhäuser-Science, Basel. [14] Oliver Kolb, "Simulation and Optimization of Gas and Water Supply Networks," PhD thesis, Technische Universität Darmstadt, 2011. [15] Ailsa H. Land and Alison G. Doig, An automatic method for solving discrete programming problems, Econometrica, 28 (1960), 497-520. [16] Youdong Lin and Linus Schrage, The global solver in the LINDO API, Optimization Methods & Software, 24 (2009), 657-668. doi: 10.1080/10556780902753221. [17] Hanif D. Sherali and Ernest P. Smith, A global optimization approach to a water distribution network design problem, Journal of Global Optimization, 11 (1997), 107-132. doi: 10.1023/A:1008207817095. [18] Mohit Tawarmalani and Nikolaos V. Sahinidis, Global optimization of mixed-integer nonlinear programs: A theoretical and computational study, Mathematical Programming, Ser. A, 99 (2004), 563-591. doi: 10.1007/s10107-003-0467-6. [19] Stefan Vigerske, "Decomposition of Multistage Stochastic Programs and a Constraint Integer Programming Approach to Mixed-Integer Nonlinear Programming," Ph.D thesis, Humboldt-Universität zu Berlin, 2012. [20] Andreas Wächter and Lorenz T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Mathematical Programming, Ser. A, 106 (2006), 125-157. doi: 10.1007/s10107-004-0559-y. [21] CppAD, "A Package for Differentiation of C++ algorithms," Available from: http://www.coin-or.org/CppAD/. [22] EPANET, "A software that models water distribution piping systems," Available from: http://www.epa.gov/nrmrl/wswrd/dw/epanet.html. [23] Ipopt, "Interior Point Optimizer," Available from: http://www.coin-or.org/Ipopt/. [24] SCIP, "Solving Constraint Integer Programs," Available from: http://scip.zib.de/. [25] SoPlex, "Sequential Object-oriented SimPlex," Available from: http://soplex.zib.de/. [26] Zimpl, "Zuse Institute Mathematical Programming Language," Available from: http://zimpl.zib.de/.

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##### References:
 [1] Tobias Achterberg, "Constraint Integer Programming," PhD thesis, Technische Universität Berlin, 2007. [2] Tobias Achterberg, SCIP: Solving constraint integer programs, Mathematical Programming Computation, 1 (2009), 1-41, doi: 10.1007/s12532-008-0001-1. [3] Pietro Belotti, Jon Lee, Leo Liberti, Francois Margot and Andreas Wächter, Branching and bounds tightening techniques for non-convex MINLP, Optimization Methods and Software, 24 (2009), 597-634. doi: 10.1080/10556780903087124. [4] Timo Berthold, "Primal Heuristics for Mixed Integer Programs," Master's thesis, Technische Universität Berlin, 2006. [5] Timo Berthold and Ambros M. Gleixner, Undercover - a primal heuristic for MINLP based on sub-MIPs generated by set covering, in "Proceedings of the EWMINLP" (eds. Pierre Bonami, Leo Liberti, Andrew J. Miller and Annick Sartenaer), (2010), 103-112. [6] Timo Berthold and Ambros M. Gleixner, Undercover - a primal MINLP heuristic exploring a largest sub-MIP, ZIB-Report 12-07 (2012), Zuse Institute Berlin, http://vs24.kobv.de/opus4-zib/frontdoor/index/index/docId/1463/. [7] Timo Berthold, Stefan Heinz, Marc E. Pfetsch and Stefan Vigerske, Large neighborhood search beyond MIP, in "Proceedings of the 9th Metaheuristics International Conference" (eds. Luca Di Gaspero, Andrea Schaerf, and Thomas Stützle), (2011), 51-60. [8] Timo Berthold, Stefan Heinz and Stefan Vigerske, Extending a CIP framework to solve MIQCPs, in "The IMA volumes in Mathematics and its Applications" (eds. Jon Lee and Sven Leyffer), 154 (2012), 427-444. doi: 10.1007/978-1-4614-1927-3_15. [9] Cristiana Bragalli, Claudia D'mbrosio, Jon Lee, Andrea Lodi and Paolo Toth, On the optimal design of water distribution networks: a practical MINLP approach, Optimization and Engineering, 13 (2012), 219-246. doi: 10.1007/s11081-011-9141-7. [10] Jens Burgschweiger, Bernd Gnädig and Marc C. Steinbach, Optimization models for operative planning in drinking water networks, ZIB-Report 04-48 (2004), Zuse Institute Berlin, http://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/823. [11] Björn Geißler, Oliver Kolb, Jens Lang, Günter Leugering, Alexander Martin and Antonio Morsi, Mixed integer linear models for the optimization of dynamical transport networks, Mathematical Methods of Operations Research, 73 (2011), 339-362, doi: 10.1007/s00186-011-0354-5. [12] Wei Huang, "Operative Planning of Water Supply Networks by Mixed Integer Nonlinear Programming," Master's thesis, Freie Universität Berlin, 2011. [13] Kathrin Klamroth, Jens Lang, Günter Leugering, Alexander Martin, Antonio Morsi, Martin Oberlack, Manfred Ostrowski and Roland Rosen, "Mathematical Optimization of Water Networks," International Series of Numerical Mathematics, 162 (2012), Birkhäuser-Science, Basel. [14] Oliver Kolb, "Simulation and Optimization of Gas and Water Supply Networks," PhD thesis, Technische Universität Darmstadt, 2011. [15] Ailsa H. Land and Alison G. Doig, An automatic method for solving discrete programming problems, Econometrica, 28 (1960), 497-520. [16] Youdong Lin and Linus Schrage, The global solver in the LINDO API, Optimization Methods & Software, 24 (2009), 657-668. doi: 10.1080/10556780902753221. [17] Hanif D. Sherali and Ernest P. Smith, A global optimization approach to a water distribution network design problem, Journal of Global Optimization, 11 (1997), 107-132. doi: 10.1023/A:1008207817095. [18] Mohit Tawarmalani and Nikolaos V. Sahinidis, Global optimization of mixed-integer nonlinear programs: A theoretical and computational study, Mathematical Programming, Ser. A, 99 (2004), 563-591. doi: 10.1007/s10107-003-0467-6. [19] Stefan Vigerske, "Decomposition of Multistage Stochastic Programs and a Constraint Integer Programming Approach to Mixed-Integer Nonlinear Programming," Ph.D thesis, Humboldt-Universität zu Berlin, 2012. [20] Andreas Wächter and Lorenz T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Mathematical Programming, Ser. A, 106 (2006), 125-157. doi: 10.1007/s10107-004-0559-y. [21] CppAD, "A Package for Differentiation of C++ algorithms," Available from: http://www.coin-or.org/CppAD/. [22] EPANET, "A software that models water distribution piping systems," Available from: http://www.epa.gov/nrmrl/wswrd/dw/epanet.html. [23] Ipopt, "Interior Point Optimizer," Available from: http://www.coin-or.org/Ipopt/. [24] SCIP, "Solving Constraint Integer Programs," Available from: http://scip.zib.de/. [25] SoPlex, "Sequential Object-oriented SimPlex," Available from: http://soplex.zib.de/. [26] Zimpl, "Zuse Institute Mathematical Programming Language," Available from: http://zimpl.zib.de/.
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