OR Spectrum

, Volume 38, Issue 3, pp 597–631 | Cite as

Valid inequalities for the topology optimization problem in gas network design

  • Jesco Humpola
  • Armin Fügenschuh
  • Thorsten Koch
Regular Article


One quarter of Europe’s energy demand is provided by natural gas distributed through a vast pipeline network covering the whole of Europe. At a cost of 1 million Euro per km extending the European pipeline network is already a multi-billion Euro business. Therefore, automatic planning tools that support the decision process are desired. Unfortunately, current mathematical methods are not capable of solving the arising network design problems due to their size and complexity. In this article, we will show how to apply optimization methods that can converge to a proven global optimal solution. By introducing a new class of valid inequalities that improve the relaxation of our mixed-integer nonlinear programming model, we are able to speed up the necessary computations substantially.


Network design Mixed-integer nonlinear programming  Cutting planes 



We are grateful to Open Grid Europe GmbH (OGE, Essen/Germany) and all members of the Forschungskooperation Netzoptimierung (ForNe) for supporting our work. Coauthor Armin Fügenschuh conducted parts of this research under a Konrad Zuse Junior Fellowship. Parts of this research have been supported by the German Ministry for Economic Affairs and Energy. We thank two anonymous referees for carefully reading our manuscript and their various comments helping us to improve its quality.


  1. Achterberg T (2009) SCIP: solving constraint integer programs. Math Program Comput 1(1):1–41CrossRefGoogle Scholar
  2. André J, Bonnans JF (2008) Optimal features of gas transmission trunklines. In Eng Opt 2008 - International Conference on Engineering Optimization, 2008. Rio de Janeiro, BrazilGoogle Scholar
  3. Berthold T, Heinz S, Vigerske S (2012) Extending a CIP framework to solve MIQCPs. In Lee J, Leyffer S (eds), Mixed integer nonlinear programming, vol 154. Part 6 of the IMA volumes in mathematics and its applications, pp 427–444. Springer. Also available as ZIB-Report 09–23Google Scholar
  4. Bonnans JF, André J (2009) Optimal structure of gas transmission trunklines. Technical Report 6791, Institut National de Recherche en Informatique et en AutomatiqueGoogle Scholar
  5. Boyd ID, Surry PD, Radcliffe NJ (1994) Constrained gas network pipe sizing with genetic algorithms. Technical Report EPCC-TR94-11, Edinburgh Parallel Computing CentreGoogle Scholar
  6. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University PressGoogle Scholar
  7. Bundesanstalt für Geowissenschaften und Rohstoffe (2013) Energiestudie 2013 - Reserven, Ressourcen und Verfügbarkeit von Energierohstoffen, 2013. Downloaded on August 26, 2014Google Scholar
  8. Castillo L, Gonzáleza A (1998) Distribution network optimization: finding the most economic solution by using genetic algorithms. Eur J Oper Res 108(3):527–537CrossRefGoogle Scholar
  9. Cerbe G (2008) Grundlagen der Gastechnik: Gasbeschaffung - Gasverteilung - Gasverwendung. Hanser, LeipzigGoogle Scholar
  10. CPLEX (2011) User’s Manual for CPLEX. IBM Corporation, Armonk, USA, 12.1 ednGoogle Scholar
  11. De Wolf D, Bakhouya B (2008) Optimal dimensioning of pipe networks: the new situation when the distribution and the transportation functions are disconnected. Technical Report 07/02, Ieseg, Université catholique de Lille, HEC Ecole de Gestion de l’ULGGoogle Scholar
  12. De Wolf D, Smeers Y (1996) Optimal dimensioning of pipe networks with application to gas transmission networks. Oper Res 44(4):596–608CrossRefGoogle Scholar
  13. De Wolf D, Smeers Y (2000) The gas transmission problem solved by an extension of the simplex algorithm. Manag Sci 46(11):1454–1465CrossRefGoogle Scholar
  14. Fügenschuh A, Geißler B, Gollmer R, Morsi A, Pfetsch ME, Rövekamp J, Schmidt M, Spreckelsen K, Steinbach MC (2014) Physical and technical fundamentals of gas networks. In Koch T, Hiller B, Pfetsch ME, Schewe L (eds) Evaluating gas network capacities, MOS-SIAM Series on Optimization, chapter 2. SIAM (To appear)Google Scholar
  15. GAMS Model Library (2013). General algebraic modeling system (GAMS) model library. Information available at URL.
  16. Geißler B, Martin A, Morsi A (2013) LaMaTTO++. Information available at URL.
  17. Humpola J (2014) Gas network optimization by MINLP. PhD thesis, Technical University BerlinGoogle Scholar
  18. Humpola J, Fügenschuh A (2013) A unified view on relaxations for a nonlinear network flow problem. ZIB-Report 13–31, Zuse Institute Berlin, Takustr. 7, 14195 Berlin, GermanyGoogle Scholar
  19. Humpola J, Fügenschuh A, Lehmann T (2014) A primal heuristic for optimizing the topology of gas networks based on dual information. EURO J Comput Optim pp 1–26Google Scholar
  20. Korte B, Vygen J (2007) Combinatorial optimization: theory and algorithms. Springer, BerlinGoogle Scholar
  21. Mariani O, Ancillai F, Donati E (1997) Design of a gas pipeline: optimal configuration. Technical report PSIG 9706, Pipeline Simulation Interest GroupGoogle Scholar
  22. Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization. Wiley, New YorkCrossRefGoogle Scholar
  23. O’Neill RP, Williard M, Wilkins B, Pike R (1979) A mathematical programming model for allocation of natural gas. Oper Res 27(5):857–873CrossRefGoogle Scholar
  24. Osiadacz AJ, Górecki M (1995) Optimization of pipe sizes for distribution gas network design. Technical Report PSIG 9511, Pipeline Simulation Interest GroupGoogle Scholar
  25. Pfetsch ME, Fügenschuh A, Geißler B, Geißler N, Gollmer R, Hiller B, Humpola J, Koch T, Lehmann T, Martin A, Morsi A, Rövekamp J, Schewe L, Schmidt M, Schultz R, Schwarz R, Schweiger J, Stangl C, Steinbach MC, Vigerske S, Willert BM (2014) Validation of nominations in gas network optimization: Models, methods, and solutions. Optimization Methods and SoftwareGoogle Scholar
  26. Schroeder DW (2001) A tutorial on pipe flow equations. Technical Report PSIG 0112, Pipeline Simulation Interest GroupGoogle Scholar
  27. SCIP: solving constraint integer programs (2013) URL.
  28. Smith EMB, Pantelides CC (1999) A symbolic reformulation/spatial branch-and-bound algorithm for the global optimization of nonconvex MINLPs. Comp Chem Eng 23:457–478CrossRefGoogle Scholar
  29. Tawarmalani M, Sahinidis NV (2004) Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math Program 99(3):563–591CrossRefGoogle Scholar
  30. Tawarmalani M, Sahinidis NV (2005) A polyhedral branch-and-cut approach to global optimization. Math Program 103(2):225–249CrossRefGoogle Scholar
  31. Vereinigung der Fernleitungsnetzbetreiber Gas e. V. Netzentwicklungsplan gas (2013) Konsultationsdokument der deutschen Fernleitungsnetzbetreiber. Downloaded on August 26, 2014Google Scholar
  32. Vigerske S (2012) Decomposition in multistage stochastic programming and a constraint integer programming approach to mixed-integer nonlinear programming. PhD thesis, Humboldt-Universität zu BerlinGoogle Scholar
  33. Wächter A, Biegler LT (2006) On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57CrossRefGoogle Scholar
  34. Weymouth TR (1912) Problems in natural gas engineering. Trans Am Soc Mech Eng 34(1349):185–231Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Jesco Humpola
    • 1
  • Armin Fügenschuh
    • 2
  • Thorsten Koch
    • 1
    • 3
  1. 1.Zuse Institute BerlinBerlinGermany
  2. 2.Helmut Schmidt UniversityUniversity of the Federal Armed Forces HamburgHamburgGermany
  3. 3.Technische Universität BerlinBerlinGermany

Personalised recommendations