American Institute of Mathematical Sciences

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

Towards globally optimal operation of water supply networks

Ambros M. Gleixner 1, , Harald Held 2, , Wei Huang 3, and  Stefan Vigerske 4,

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.
Keywords: water supply networks., operative planning, global optimization, Nonconvex MINLP.
Mathematics Subject Classification: Primary: 90C26; Secondary: 90C11, 90C9.
Citation: Ambros M. Gleixner, Harald Held, Wei Huang, Stefan Vigerske. Towards globally optimal operation of water supply networks. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 695-711. doi: 10.3934/naco.2012.2.695
References:
[1]

Tobias Achterberg, "Constraint Integer Programming,", PhD thesis, (2007).   Google Scholar

[2]

Tobias Achterberg, SCIP: Solving constraint integer programs,, Mathematical Programming Computation, 1 (2009), 1.  doi: 10.1007/s12532-008-0001-1.  Google Scholar

[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.  doi: 10.1080/10556780903087124.  Google Scholar

[4]

Timo Berthold, "Primal Heuristics for Mixed Integer Programs,", Master's thesis, (2006).   Google Scholar

[5]

Timo Berthold and Ambros M. Gleixner, Undercover - a primal heuristic for MINLP based on sub-MIPs generated by set covering,, in, (2010), 103.   Google Scholar

[6]

Timo Berthold and Ambros M. Gleixner, Undercover - a primal MINLP heuristic exploring a largest sub-MIP,, ZIB-Report 12-07 (2012), (2012), 12.   Google Scholar

[7]

Timo Berthold, Stefan Heinz, Marc E. Pfetsch and Stefan Vigerske, Large neighborhood search beyond MIP,, in, (2011), 51.   Google Scholar

[8]

Timo Berthold, Stefan Heinz and Stefan Vigerske, Extending a CIP framework to solve MIQCPs,, in, 154 (2012), 427.  doi: 10.1007/978-1-4614-1927-3_15.  Google Scholar

[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.  doi: 10.1007/s11081-011-9141-7.  Google Scholar

[10]

Jens Burgschweiger, Bernd Gnädig and Marc C. Steinbach, Optimization models for operative planning in drinking water networks,, ZIB-Report 04-48 (2004), (2004), 04.   Google Scholar

[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.  doi: 10.1007/s00186-011-0354-5.  Google Scholar

[12]

Wei Huang, "Operative Planning of Water Supply Networks by Mixed Integer Nonlinear Programming,", Master's thesis, (2011).   Google Scholar

[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).   Google Scholar

[14]

Oliver Kolb, "Simulation and Optimization of Gas and Water Supply Networks,", PhD thesis, (2011).   Google Scholar

[15]

Ailsa H. Land and Alison G. Doig, An automatic method for solving discrete programming problems,, Econometrica, 28 (1960), 497.   Google Scholar

[16]

Youdong Lin and Linus Schrage, The global solver in the LINDO API,, Optimization Methods & Software, 24 (2009), 657.  doi: 10.1080/10556780902753221.  Google Scholar

[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.  doi: 10.1023/A:1008207817095.  Google Scholar

[18]

Mohit Tawarmalani and Nikolaos V. Sahinidis, Global optimization of mixed-integer nonlinear programs: A theoretical and computational study,, Mathematical Programming, 99 (2004), 563.  doi: 10.1007/s10107-003-0467-6.  Google Scholar

[19]

Stefan Vigerske, "Decomposition of Multistage Stochastic Programs and a Constraint Integer Programming Approach to Mixed-Integer Nonlinear Programming,", Ph.D thesis, (2012).   Google Scholar

[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, 106 (2006), 125.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

[21]

CppAD, "A Package for Differentiation of C++ algorithms,", Available from: , ().   Google Scholar

[22]

EPANET, "A software that models water distribution piping systems,", Available from: , ().   Google Scholar

[23]

Ipopt, "Interior Point Optimizer,", Available from: , ().   Google Scholar

[24]

SCIP, "Solving Constraint Integer Programs,", Available from: , ().   Google Scholar

[25]

SoPlex, "Sequential Object-oriented SimPlex,", Available from: , ().   Google Scholar

[26]

Zimpl, "Zuse Institute Mathematical Programming Language,", Available from: , ().   Google Scholar

show all references

References:
[1]

Tobias Achterberg, "Constraint Integer Programming,", PhD thesis, (2007).   Google Scholar

[2]

Tobias Achterberg, SCIP: Solving constraint integer programs,, Mathematical Programming Computation, 1 (2009), 1.  doi: 10.1007/s12532-008-0001-1.  Google Scholar

[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.  doi: 10.1080/10556780903087124.  Google Scholar

[4]

Timo Berthold, "Primal Heuristics for Mixed Integer Programs,", Master's thesis, (2006).   Google Scholar

[5]

Timo Berthold and Ambros M. Gleixner, Undercover - a primal heuristic for MINLP based on sub-MIPs generated by set covering,, in, (2010), 103.   Google Scholar

[6]

Timo Berthold and Ambros M. Gleixner, Undercover - a primal MINLP heuristic exploring a largest sub-MIP,, ZIB-Report 12-07 (2012), (2012), 12.   Google Scholar

[7]

Timo Berthold, Stefan Heinz, Marc E. Pfetsch and Stefan Vigerske, Large neighborhood search beyond MIP,, in, (2011), 51.   Google Scholar

[8]

Timo Berthold, Stefan Heinz and Stefan Vigerske, Extending a CIP framework to solve MIQCPs,, in, 154 (2012), 427.  doi: 10.1007/978-1-4614-1927-3_15.  Google Scholar

[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.  doi: 10.1007/s11081-011-9141-7.  Google Scholar

[10]

Jens Burgschweiger, Bernd Gnädig and Marc C. Steinbach, Optimization models for operative planning in drinking water networks,, ZIB-Report 04-48 (2004), (2004), 04.   Google Scholar

[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.  doi: 10.1007/s00186-011-0354-5.  Google Scholar

[12]

Wei Huang, "Operative Planning of Water Supply Networks by Mixed Integer Nonlinear Programming,", Master's thesis, (2011).   Google Scholar

[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).   Google Scholar

[14]

Oliver Kolb, "Simulation and Optimization of Gas and Water Supply Networks,", PhD thesis, (2011).   Google Scholar

[15]

Ailsa H. Land and Alison G. Doig, An automatic method for solving discrete programming problems,, Econometrica, 28 (1960), 497.   Google Scholar

[16]

Youdong Lin and Linus Schrage, The global solver in the LINDO API,, Optimization Methods & Software, 24 (2009), 657.  doi: 10.1080/10556780902753221.  Google Scholar

[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.  doi: 10.1023/A:1008207817095.  Google Scholar

[18]

Mohit Tawarmalani and Nikolaos V. Sahinidis, Global optimization of mixed-integer nonlinear programs: A theoretical and computational study,, Mathematical Programming, 99 (2004), 563.  doi: 10.1007/s10107-003-0467-6.  Google Scholar

[19]

Stefan Vigerske, "Decomposition of Multistage Stochastic Programs and a Constraint Integer Programming Approach to Mixed-Integer Nonlinear Programming,", Ph.D thesis, (2012).   Google Scholar

[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, 106 (2006), 125.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

[21]

CppAD, "A Package for Differentiation of C++ algorithms,", Available from: , ().   Google Scholar

[22]

EPANET, "A software that models water distribution piping systems,", Available from: , ().   Google Scholar

[23]

Ipopt, "Interior Point Optimizer,", Available from: , ().   Google Scholar

[24]

SCIP, "Solving Constraint Integer Programs,", Available from: , ().   Google Scholar

[25]

SoPlex, "Sequential Object-oriented SimPlex,", Available from: , ().   Google Scholar

[26]

Zimpl, "Zuse Institute Mathematical Programming Language,", Available from: , ().   Google Scholar

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