Optimization and Engineering

, Volume 20, Issue 2, pp 543–573 | Cite as

Maximizing the storage capacity of gas networks: a global MINLP approach

  • Robert BurlacuEmail author
  • Herbert Egger
  • Martin Groß
  • Alexander Martin
  • Marc E. Pfetsch
  • Lars Schewe
  • Mathias Sirvent
  • Martin Skutella
Research Article


In this paper, we study the transient optimization of gas networks, focusing in particular on maximizing the storage capacity of the network. We include nonlinear gas physics and active elements such as valves and compressors, which due to their switching lead to discrete decisions. The former is described by a model derived from the Euler equations that is given by a coupled system of nonlinear parabolic partial differential equations (\({\text{PDEs}}\)). We tackle the resulting mathematical optimization problem by a first-discretize-then-optimize approach. To this end, we introduce a new discretization of the underlying system of parabolic \({\text{PDEs}}\) and prove well-posedness for the resulting nonlinear discretized system. Endowed with this discretization, we model the problem of maximizing the storage capacity as a non-convex mixed-integer nonlinear problem (\({\text{MINLP}}\)). For the numerical solution of the \({\text{MINLP}}\), we algorithmically extend a well-known relaxation approach that has already been used very successfully in the field of stationary gas network optimization. This method allows us to solve the problem to global optimality by iteratively solving a series of mixed-integer problems. Finally, we present two case studies that illustrate the applicability of our approach.


Mixed-integer nonlinear programming Transient gas transport optimization Storage capacity maximization Power-to-gas First-discretize-then-optimize 



The authors gratefully acknowledge the compute resources and support provided by the Erlangen Regional Computing Center (RRZE). We are very grateful for the thorough reading of the paper by Martin Schmidt. We would also like to show our gratitude to Fabian Rüffler for fruitful discussions on the topic of discretization of \({\text{PDEs}}\). Finally, we thank Falk Hante for his helpful comments on the literature review of the general approaches for optimal control with discrete decisions.


  1. Allgor R, Barton P (1999) Mixed-integer dynamic optimization I: problem formulation. Comput Chem Eng 23(4):567–584. CrossRefGoogle Scholar
  2. Baumrucker B, Biegler L (2009) MPEC strategies for optimization of a class of hybrid dynamic systems. J Process Control 19(8):1248–1256. CrossRefGoogle Scholar
  3. Baumrucker B, Biegler LT (2010) Mpec strategies for cost optimization of pipeline operations. Comput Chem Eng 34(6):900–913CrossRefGoogle Scholar
  4. Belotti P, Kirches C, Leyffer S, Linderoth J, Luedtke J, Mahajan A (2013) Mixed-integer nonlinear optimization. Acta Numer 22:1–131. MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bock HG, Kirches C, Meyer A, Potschka A (2018) Numerical solution of optimal control problems with explicit and implicit switches. Optim Methods Softw 33(3):450–474. MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bonami P, Biegler LT, Conn AR, Cornuéjols G, Grossmann IE, Laird CD, Lee J, Lodi A, Margot F, Sawaya N, Wächter A (2008) An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optim 5(2):186–204. MathSciNetCrossRefzbMATHGoogle Scholar
  7. Brouwer J, Gasser I, Herty M (2011) Gas pipeline models revisited: model hierarchies, nonisothermal models, and simulations of networks. Multiscale Model Simul 9:601–623. MathSciNetCrossRefzbMATHGoogle Scholar
  8. Buchheim C, Kuhlmann R, Meyer C (2015) Combinatorial optimal control of semilinear elliptic PDEs. Technical report, Optimization Online.
  9. Burlacu R, Geißler B, Schewe L (2017) Solving mixed-integer nonlinear programs using adaptively refined mixed-integer linear programs. Preprint.
  10. Byrd RH, Nocedal J, Waltz RA (2006) Knitro: an integrated package for nonlinear optimization. In: Di Pillo G, Roma M (eds) Large-scale nonlinear optimization. Springer, New York, pp 35–59. CrossRefGoogle Scholar
  11. Devine MT, Gleeson JP, Kinsella J, Ramsey DM (2014) A rolling optimisation model of the UK natural gas market. Netw Spat Econ 14(2):209–244MathSciNetCrossRefzbMATHGoogle Scholar
  12. Domschke P, Geißler B, Kolb O, Lang J, Martin A, Morsi A (2011) Combination of nonlinear and linear optimization of transient gas networks. INFORMS J Comput 23(4):605–617. MathSciNetCrossRefzbMATHGoogle Scholar
  13. Domschke P, Hiller B, Lang J, Tischendorf C (2017) Modellierung von Gasnetzwerken: Eine Übersicht. Techreport.
  14. Ehrhardt K, Steinbach MC (2005) Nonlinear optimization in gas networks. In: Bock HG, Kostina E, Phu HX, Ranacher R (eds) Modeling, simulation and optimization of complex processes. Springer, Berlin, pp 139–148CrossRefGoogle Scholar
  15. Fügenschuh A, Geißler B, Gollmer R, Morsi A, Pfetsch ME, Rövekamp J, Schmidt M, Spreckelsen K, Steinbach MC (2015) Physical and technical fundamentals of gas networks. In: Koch et al. (2015), Chap 2, pp 17–43.
  16. Furey BP (1993) A sequential quadratic programming-based algorithm for optimization of gas networks. Automatica 29(6):1439–1450. MathSciNetCrossRefzbMATHGoogle Scholar
  17. Gamrath G, Fischer T, Gally T, Gleixner AM, Hendel G, Koch T, Maher SJ, Miltenberger M, Müller B, Pfetsch ME, Puchert C, Rehfeldt D, Schenker S, Schwarz R, Serrano F, Shinano Y, Vigerske S, Weninger D, Winkler M, Witt JT, Witzig J (2016) The SCIP optimization suite 3.2. Technical Report 15-60, ZIB, Takustr.7, 14195 BerlinGoogle Scholar
  18. GAMS (2017) General Algebraic Modeling System (GAMS) Release 24.8.3. Washington, DC, USA.
  19. Geißler B (2011) Towards globally optimal solutions of MINLPs by discretization techniques with applications in gas network optimization. PhD thesis, FAU Erlangen-NürnbergGoogle Scholar
  20. Geißler B, Martin A, Morsi A, Schewe L (2012) Using piecewise linear functions for solving MINLPs. In: Lee J, Leyffer S (eds) Mixed integer nonlinear programming. Springer, New York, pp 287–314CrossRefGoogle Scholar
  21. Geißler B, Martin A, Morsi A, Schewe L (2015a) The MILP-relaxation approach. In: Koch et al. (2015), Chap 6, pp 103–122.
  22. Geißler B, Morsi A, Schewe L, Schmidt M (2015b) Solving power-constrained gas transportation problems using an MIP-based alternating direction method. Comput Chem Eng 82:303–317. CrossRefGoogle Scholar
  23. Geißler B, Morsi A, Schewe L, Schmidt M (2018) Solving highly detailed gas transport MINLPs: block separability and penalty alternating direction methods. INFORMS J Comput 30(2):309–323. MathSciNetCrossRefGoogle Scholar
  24. Gerdts M (2006) A variable time transformation method for mixed-integer optimal control problems. Optim Control Appl Methods 27(3):169–182. MathSciNetCrossRefGoogle Scholar
  25. Gopalakrishnan A, Biegler LT (2013) Economic nonlinear model predictive control for periodic optimal operation of gas pipeline networks. Comput Chem Eng 52:90–99CrossRefGoogle Scholar
  26. Gu Z, Rothberg E, Bixby R (2015) Gurobi optimizer reference manual, version 6.0.4Google Scholar
  27. Gugat M, Leugering G, Martin A, Schmidt M, Sirvent M, Wintergerst D (2017) MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems. Comput Optim Appl
  28. Hahn M, Leyffer S, Zavala VM (2017) Mixed-integer PDE-constrained optimal control of gas networks. Techreport.
  29. Hante FM, Sager S (2013) Relaxation methods for mixed-integer optimal control of partial differential equations. Comput Optim Appl 55(1):197–225. MathSciNetCrossRefzbMATHGoogle Scholar
  30. Hante F, Leugering G, Martin A, Schewe L, Schmidt M (2017) Challenges in optimal control problems for gas and fluid flow in networks of pipes and canals: From modeling to industrial applications. In: Manchanda P, Lozi R, Siddiqi AH (eds) Industrial mathematics and complex systems, industrial and applied mathematics. Springer, New York, pp 77–122. CrossRefGoogle Scholar
  31. Jung MN, Reinelt G, Sager S (2015) The Lagrangian relaxation for the combinatorial integral approximation problem. Optim Methods Softw 30(1):54–80. MathSciNetCrossRefzbMATHGoogle Scholar
  32. Koch T, Hiller B, Pfetsch ME, Schewe L (eds) (2015) Evaluating gas network capacities. SIAM-MOS series on optimization. SIAM, Philadelphia. zbMATHGoogle Scholar
  33. LaMaTTO++ (2015) A framework for modeling and solving mixed-integer nonlinear programming problems on networks.
  34. Lee J, Leyffer S (eds) (2012) Mixed integer nonlinear programming. The IMA volumes in mathematics and its applications, vol 154. Springer, New York, selected papers based on the IMA Hot Topics Workshop “Mixed-integer nonlinear optimization: algorithmic advances and applications” held in Minneapolis, MN, November 17–21, 2008Google Scholar
  35. Lee H, Teo K, Rehbock V, Jennings L (1999) Control parametrization enhancing technique for optimal discrete-valued control problems. Automatica 35(8):1401–1407. MathSciNetCrossRefzbMATHGoogle Scholar
  36. Mahlke D, Martin A, Moritz S (2007) A simulated annealing algorithm for transient optimization in gas networks. Math Methods Oper Res 66(1):99–115. MathSciNetCrossRefzbMATHGoogle Scholar
  37. Mahlke D, Martin A, Moritz S (2010) A mixed integer approach for time-dependent gas network optimization. Optim Methods Softw 25(4):625–644. MathSciNetCrossRefzbMATHGoogle Scholar
  38. Osiadacz AJ (1994) Dynamic optimization of high pressure gas networks using hierarchical systems theory. In: Proceedings of PSIG annual meeting 1994, Pipeline Simulation Interest Group, San Diego, California, p 29.
  39. Osiadacz AJ (1996) Different transient models—limitations, advantages and disadvantages. Technical report, PSIG report 9606, Pipeline Simulation Interest GroupGoogle Scholar
  40. Rachford Jr HH, Carter RG et al (2000) Optimizing pipeline control in transient gas flow. In: PSIG annual meeting. Pipeline Simulation Interest GroupGoogle Scholar
  41. Rüffler F, Hante FM (2016) Optimal switching for hybrid semilinear evolutions. Nonlinear Anal Hybrid Syst 22:215–227. MathSciNetCrossRefzbMATHGoogle Scholar
  42. Ríos-Mercado RZ (2018) Metaheuristics for natural gas pipeline networks. In: Pardalos PM, Resende MGC, Marti R (eds) Handbook of heuristics. Springer, New York, pp 1103–1121Google Scholar
  43. Ríos-Mercado RZ, Borraz-Sánchez C (2015) Optimization problems in natural gas transportation systems: a state-of-the-art review. Appl Energy 147:536–555. CrossRefGoogle Scholar
  44. Sager S, Bock HG, Reinelt G (2009) Direct methods with maximal lower bound for mixed-integer optimal control problems. Math Program 118(1):109–149. MathSciNetCrossRefzbMATHGoogle Scholar
  45. Sager S, Jung M, Kirches C (2011) Combinatorial integral approximation. Math Methods Oper Res 73(3):363–380. MathSciNetCrossRefzbMATHGoogle Scholar
  46. Schewe L, Schmidt M (2018) Computing feasible points for minlps with mpecs. Math Program Comput.
  47. Schmidt M, Steinbach MC, Willert BM (2013) A primal heuristic for nonsmooth mixed integer nonlinear optimization. In: Jünger M, Reinelt G (eds) Facets of combinatorial optimization: Festschrift for Martin Grötschel. Springer, Berlin, pp 295–320. CrossRefGoogle Scholar
  48. Schmidt M, Aßmann D, Burlacu R, Humpola J, Joormann I, Kanelakis N, Koch T, Oucherif D, Pfetsch ME, Schewe L, Schwarz R, Sirvent M (2017) GasLib—a library of gas network instances. Data.
  49. Steinbach MC (2007) On PDE solution in transient optimization of gas networks. J Comput Appl Math 203(2):345–361MathSciNetCrossRefzbMATHGoogle Scholar
  50. Tawarmalani M, Sahinidis NV (2005) A polyhedral branch-and-cut approach to global optimization. Math Program 103:225–249. MathSciNetCrossRefzbMATHGoogle Scholar
  51. Westerlund T, Pörn R (2002) Solving pseudo-convex mixed integer optimization problems by cutting plane techniques. Optim Eng 3(3):253–280MathSciNetCrossRefzbMATHGoogle Scholar
  52. Zlotnik A, Chertkov M, Backhaus S (2015) Optimal control of transient flow in natural gas networks. In: 2015 54th IEEE conference on decision and control (CDC), pp 4563–4570.

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Discrete OptimizationFriedrich-Alexander-Universität Erlangen-Nürnberg (FAU)ErlangenGermany
  2. 2.Research Group Numerics and Scientific ComputingTechnische Universität DarmstadtDarmstadtGermany
  3. 3.Chair of Management ScienceRWTH AachenAachenGermany
  4. 4.Research Group OptimizationTechnische Universität DarmstadtDarmstadtGermany
  5. 5.Faculty II – Mathematics and Natural SciencesTechnische Universität BerlinBerlinGermany

Personalised recommendations