Optimization and Engineering

, Volume 16, Issue 1, pp 131–164 | Cite as

High detail stationary optimization models for gas networks

  • Martin Schmidt
  • Marc C. Steinbach
  • Bernhard M. Willert


Economic reasons and the regulation of gas markets create a growing need for mathematical optimization of natural gas networks. Real life planning tasks often lead to highly complex and extremely challenging optimization problems whose numerical treatment requires a breakdown into several simplified problems to be solved by carefully chosen hierarchies of models and algorithms. This paper presents stationary NLP type models of gas networks that are primarily designed to include detailed nonlinear physics in the final optimization steps for mid term planning problems after fixing discrete decisions with coarsely approximated physics.


Gas networks Stationary flow High-detail modeling Nonsmooth nonlinear discrete-continuous optimization Smoothing techniques 

Mathematics Subject Classification

90B10 90C06 90C30 90C90 



This work has been supported by the German Federal Ministry of Economics and Technology owing to a decision of the German Bundestag. The responsibility for the content of this publication lies with the authors. We would also like to thank our industry partner Open Grid Europe GmbH and the project partners in the ForNe consortium; Friedrich-Alexander-Universität Erlangen-Nürnberg, Konrad Zuse Zentrum für Informationstechnik Berlin (ZIB), Universität Duisburg-Essen, Weierstraß Institut für Angewandte Analysis und Stochastik (WIAS), Humboldt Universität zu Berlin, and Technische Universität Darmstadt. At last, we are also very grateful to four anonymous referees, whose comments greatly helped to improve the quality of the paper.


  1. Bales P (2005) Hierarchische Modellierung der Eulerschen Flussgleichungen in der Gasdynamik. Master’s thesis, Technische Universität Darmstadt Google Scholar
  2. Banda MK, Herty M (2008) Multiscale modeling for gas flow in pipe networks. Math Met Appl Sci 31:915–936. doi: 10.1002/mma.948 CrossRefMATHMathSciNetGoogle Scholar
  3. Banda MK, Herty M, Klar A (2006) Gas flow in pipeline networks. Netw Heterogen Media 1(1):41–56. doi: 10.3934/nhm.2006.1.41 CrossRefMATHMathSciNetGoogle Scholar
  4. Borraz-Sánchez C, Ríos-Mercado RZ (2004) A procedure for finding initial feasible solutions on cyclic natural gas networks. In: Proceedings of the 2004 NSF Design, Service and Manufacturing Grantees and Research Conference, DallasGoogle Scholar
  5. Borraz-Sánchez C, Ríos-Mercado RZ (2005) A hybrid meta-heuristic approach for natural gas pipeline network optimization. In: Blesa M, Blum C, Roli A, Sampels M (eds.) Hybrid Metaheuristics, Lecture Notes in Computer Science, vol. 3636, pp. 54–65. Springer. doi: 10.1007/11546245_6
  6. Boyd EA, Scott LR, Wu S (1997) Evaluating the quality of pipeline optimization algorithms. In: PSIG 29th Annual Meeting, Tucson, Arizona. Pipeline Simulation Interest Group. Paper 9709Google 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(2):601–623CrossRefMATHMathSciNetGoogle Scholar
  8. Burgschweiger J, Gnädig B, Steinbach MC (2008) Optimization models for operative planning in drinking water networks. Optim Eng 10(1):43–73 doi: 10.1007/s11081-008-9040-8. Published in print 2009Google Scholar
  9. Burgschweiger J, Gnädig B, Steinbach MC (2009) Nonlinear programming techniques for operative planning in large drinking water networks. Open Appl Math J 3:14–28. doi: 10.2174/1874114200903010014 CrossRefMathSciNetGoogle Scholar
  10. Carter RG (1996) Compressor station optimization: Computational accuracy and speed. In: 28th Annual Meeting. Pipeline Simulation Interest Group. Paper 9605Google Scholar
  11. Carter RG (1998) Pipeline optimization: Dynamic Programming after 30 years. In: 30th Annual Meeting. Pipeline Simulation Interest Group Paper 9803Google Scholar
  12. Carter RG, Schroeder DW, Harbick TD (1994) Some causes and effects of discontinuities in modeling and optimizing gas transmission networks. Tech. rep., Stoner Associates, CarlisleGoogle Scholar
  13. Cerbe G (2008) Grundlagen der Gastechnik. Hanser, MunichGoogle Scholar
  14. Chodanowitsch IJ, Odischarija GE (1964) Analiž žavisimosti dlja koeffizienta gidravličeskogo soprotivlenija (analysis of the relations of friction coefficients). gazovaya promyšlennost 11:38–42Google Scholar
  15. Cobos-Zaleta D, Ríos-Mercado RZ (2002) A MINLP model for a problem of minimizing fuel consumption on natural gas pipeline networks. In: Proceedings of the XI Latin-Ibero-American Conference on Operations Research, pp. 1–9. Paper A48-01Google Scholar
  16. Colebrook CF (1939) Turbulent flow in pipes with particular reference to the transition region between smooth and rough pipe laws. J Inst Civil Eng 11:133–156. doi: 10.1680/ijoti.1939.13150 CrossRefGoogle Scholar
  17. 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. doi: 10.1287/ijoc.1100.0429 CrossRefMATHMathSciNetGoogle Scholar
  18. Ehrhardt K, Steinbach MC (2004) KKT systems in operative planning for gas distribution networks. Proc Appl Math Mech 4(1):606–607CrossRefGoogle Scholar
  19. Ehrhardt K, Steinbach MC (2005) Nonlinear optimization in gas networks. In: Bock HG, Kostina E, Phu HX, Rannacher R (eds.) Modeling, Simulation and Optimization of Complex Processes. Springer, Berlin, pp 139–148CrossRefGoogle Scholar
  20. Feistauer M (1993) Mathematical Methods in Fluid Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics Series, vol. 67. Longman Scientific & Technical, HarlowGoogle Scholar
  21. Finnemore EJ, Franzini JE (2002) Fluid mechanics with engineering applications, 10th edn. McGraw-Hill, New YorkGoogle Scholar
  22. Fügenschuh A, Geißler B, Gollmer R, Hayn C, Henrion R, Hiller B, Humpola J, Koch T, Lehmann T, Martin A, Mirkov R, Morsi A, Rövekamp J, Schewe L, Schmidt M, Schultz R, Schwarz R, Schweiger J, Stangl C, Steinbach MC, Willert B (2013) Mathematical optimization for challenging network planning problems in unbundled liberalized gas markets. Energy Syst. pp. 1–25. doi: 10.1007/s12667-013-0099-8
  23. GasLib: a library of gas network instances.
  24. Geißler B (2011) Towards globally optimal solutions for MINLPs by discretization techniques with applications in gas network optimization. Ph. D. dissertation, University of Erlangen-Nuremberg, GermanyGoogle Scholar
  25. Gill PE, Murray W, Saunders MS (2002) SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM J. Optim. 12(4):979–1006CrossRefMATHMathSciNetGoogle Scholar
  26. Gugat M (2003) Boundary controllability between sub- and supercritical flow. SIAM J Control Optim 42:1056–1070CrossRefMATHMathSciNetGoogle Scholar
  27. Gugat M, Leugering G (2003) Global boundary controllability of the de St. Venant equations between steady states. Annales de l’Institut Henri Poincare 20(1):1–11. doi: 10.1016/S0294-1449(02)00004-5 CrossRefMATHMathSciNetGoogle Scholar
  28. Gugat M, Leugering, G, Schittkowski K, Schmidt EJPG (2011) Modelling, stabilization, and control of flow in networks of open channels. In: Grötschel M, Krumke SO, Rambau J (eds.) Online optimization of large scale systems. Springer, Berlin, pp 251–270Google Scholar
  29. Hackländer P (2002) Integrierte Betriebsplanung von Gasversorgungssystemen. Ph. D. dissertation, Universität Wuppertal (2002)Google Scholar
  30. Hofer P (1973) Beurteilung von Fehlern in Rohrnetzberechnungen (error evaluation in calculation of pipelines). GWF Gas 11:113–119Google Scholar
  31. Jeníček T (1993) Steady-state optimization of gas transport. In: Proceedings of 2nd international workshop SIMONE on innovative approaches to modeling and optimal control of large scale pipeline networks, pp. 26–38Google Scholar
  32. Jeníček T, Králik J, Štěrba J, Vostrý Z, Záworka J (1993) Study to analyze the possibilities and features of an optimization system (optimum control system) to support the dispatching activities of Ruhrgas. Vertrauliche Dokumentation, LIWACOM Informationstechnik GmbH EssenGoogle Scholar
  33. Katz DLV (1959) Handbook of natural gas engineering. McGraw-Hill series in chemical engineering, McGraw-Hill, New YorkGoogle Scholar
  34. Koch T, Bargmann D, Ebbers M, Fügenschuh A, Geißler B, Geißler N, Gollmer R, Gotzes U, Hayn C, Heitsch H, Henrion R, Hiller B, Humpola J, Joormann I, Kühl V, Lehmann T, Leövey H, Martin A, Mirkov R, Möller A, Morsi A, Oucherif D, Pelzer A, Pfetsch ME, Schewe L, Römisch W, Rövekamp J, Schmidt M, Schultz R, Schwarz R, Schweiger J, Spreckelsen K, Stangl C, Steinbach MC, Steinkamp A, Wegner-Specht I, Willert BM, Vigerske S (2014) Evaluating gas network capacities, In preparationGoogle Scholar
  35. Králik J (1993) Compressor stations in SIMONE. In: Proceedings of 2nd international workshop SIMONE on innovative approaches to modeling and optimal control of large scale pipeline networks, pp. 93–117Google Scholar
  36. Králik J, Stiegler P, Vostrý Z, Závorka J (1984a) A universal dynamic simulation model of gas pipeline networks. IEEE Trans Syst Man Cybern 14(4):597–606CrossRefGoogle Scholar
  37. Králik J, Stiegler P, Vostrý Z, Záworka J (1984b) Modeling the dynamics of flow in gas pipelines. IEEE Trans Syst Man Cybern 14(4):586–596CrossRefGoogle Scholar
  38. Králik J, Stiegler P, Vostrý Z, Záworka J (1988) Dynamic modeling of large-scale networks with application to gas distribution, Studies in Automation and Control, vol. 6. Elsevier Sci. Publ., New YorkGoogle Scholar
  39. Kunz O, Klimeck R, Wagner W, Jaeschke M (2007) The GERG-2004 wide-range equation of state for natural gases and other mixtures. No. 557 in Fortschritt-Berichte VDI, Reihe 6. VDI Verlag, Düsseldorf (2007). GERG Technical Monograph 15Google Scholar
  40. Leugering G, Schmidt EJPG (2002) On the modelling and stabilization of flows in networks of open canals. SIAM J Control Optim 41(1):164–180CrossRefMATHMathSciNetGoogle Scholar
  41. LIWACOM Informations GmbH and SIMONE Research Group s.r.o.: Gleichungen und Methoden Benutzerhandbuch (2004)Google Scholar
  42. Lurie MV (2008) Modeling of oil product and gas pipeline transportation. Wiley-VCH, HeidelbergCrossRefGoogle Scholar
  43. Martin, A., Geißler B, Hayn C, Hiller B, Humpola J, Koch T, Lehmann T, Morsi A, Pfetsch M, Schewe L, Schmidt M, Schultz R, Schwarz R, Schweiger J, Steinbach MC, Willert BM (2011) Optimierung Technischer Kapazitäten in Gasnetzen. In: Optimierung in der Energiewirtschaft, VDI-Berichte, 2157:105–114Google Scholar
  44. Martin A, Mahlke D, Moritz S (2007) A simulated annealing algorithm for transient optimization in gas networks. Math Methods Oper Res 66(1):99–115. doi: 10.1007/s00186-006-0142-9 CrossRefMATHMathSciNetGoogle Scholar
  45. Martin A, Möller M (2005) Cutting planes for the optimization of gas networks. In: Bock HG, Kostina E, Phu HX, Rannacher R (eds.) Modeling, simulation and optimization of complex processes. Springer, Berlin, pp 307–329Google Scholar
  46. Martin A, Möller M, Moritz S (2006) Mixed integer models for the stationary case of gas network optimization. Math Program 105(2–3, Ser. B):563–582. doi: 10.1007/s10107-005-0665-5 CrossRefMATHMathSciNetGoogle Scholar
  47. Mischner J (2012) Notices about hydraulic calculations of gas pipelines. J-GWF Gas 4:158–273Google Scholar
  48. Möller M (2004) Mixed integer models for the optimisation of gas networks in the stationary case. Ph. D. dissertation. Technische Universität DarmstadtGoogle Scholar
  49. Moritz S (2007) A mixed integer approach for the transient case of gas network optimization. Ph. D. dissertation. Technische Universität DarmstadtGoogle Scholar
  50. de Nevers N (1970) Fluid mechanics. Addison-Wesley series in chemical engineering. Addison-Wesley Pub. Co. URL
  51. Nikuradse J (1933) Strömungsgesetze in rauhen Rohren. Forschungsheft auf dem Gebiete des Ingenieurwesens. VDI-Verlag, DüsseldorfGoogle Scholar
  52. Nikuradse J (1950) Laws of flow in rough pipes, Technical Memorandum. vol. 1292. National Advisory Committee for Aeronautics, WashingtonGoogle Scholar
  53. Odom FM, Muster GL (2009) Tutorial on modeling of gas turbine driven centrifugal compressors. Tech. Rep. 09A4, Pipeline Simulation Interest GroupGoogle Scholar
  54. Osiadacz A (1980) Nonlinear programming applied to the optimum control of a gas compressor station. Int J Numer Methods Eng 15(9):1287–1301. doi: 10.1002/nme.1620150902 CrossRefMATHMathSciNetGoogle Scholar
  55. Oucherif D (2009) Approximation der stationären Impulsgleichung realer Gase in Pipelines. Diploma thesis, Leibniz Universität HannoverGoogle Scholar
  56. Papay I (1968) OGIL Musz. Tud. KozlGoogle Scholar
  57. 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 (2012) Validation of nominations in gas network optimization: Models, methods, and solutions. ZIB Report 12-41, Zuse Institute Berlin, Takustr.7, 14195 Berlin. SubmittedGoogle Scholar
  58. Pratt KF, Wilson JG (1984) Optimization of the operation of gas transmission systems. Trans Inst Meas Control 6(5):261–269. doi: 10.1177/014233128400600411 CrossRefGoogle Scholar
  59. Redlich O, Kwong JNS (1949) On the Thermodynamics of Solutions. V. An equation of state. Fugacities of Gaseous Solutions. Chem Rev 44(1):233–244 . doi: 10.1021/cr60137a013. PMID: 18125401Google Scholar
  60. Ríos-Mercado RZ, Kim S, Boyd AE (2006) Efficient operation of natural gas transmission systems: A network-based heuristic for cyclic structures. Comput Oper Res 33(8):2323–2351. doi: 10.1016/j.cor.2005.02.003 CrossRefMATHGoogle Scholar
  61. Ríos-Mercado RZ, Wu S, Scott LR, Boyd AE (2002) A reduction technique for natural gas transmission network optimization problems. Ann Oper Res 117:217–234CrossRefMATHGoogle Scholar
  62. Saleh, J (eds) (2002) Fluid Flow Handbook. McGraw-Hill Handbooks, New YorkGoogle Scholar
  63. 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, pp. 295–320. Springer, Berlin, Heidelberg (2013). doi: 10.1007/978-3-642-38189-8_13
  64. Schmidt M, Steinbach MC, Willert BM (2014) High detail stationary optimization models for gas networks—Part 2: Validation and results, In preparationGoogle Scholar
  65. Sekirnjak E (1998) Mixed integer optimization for gas transmission and distribution systems. Presentation manuscript, INFORMS-Meeting, SeattleGoogle Scholar
  66. Sekirnjak E (1999) Transiente Technische Optimierung (TTO-Prototyp). Vertrauliche Dokumentation, PSI AG, BerlinGoogle Scholar
  67. SIMONE software:
  68. Starling KE, Savidge JL (1992) Compressibility factors of natural gas and other related hydrocarbon gases. Transmission Measurement Committee report. American Gas Association, New YorkGoogle Scholar
  69. Steinbach MC (2007) On PDE solution in transient optimization of gas networks. J Comput Appl Math 203(2):345–361. doi: 10.1016/ CrossRefMATHMathSciNetGoogle Scholar
  70. van der Hoeven T (2004) Math in gas and the art of linearization. Ph.D. thesis, Rijksuniversiteit GroningenGoogle Scholar
  71. Villalobos-Morales Y, Cobos-Zaleta D, Flores-Villarreal HJ, Borraz-Sánchez C, Ríos-Mercado RZ (2003) On NLP and MINLP formulations and preprocessing for fuel cost minimization of natural gas transmission networks. In: Proceedings of the 2003 NSF Design, Service and Manufacturing Grantees and Research Conference. BirminghamGoogle Scholar
  72. Vostrý, Z (1993) Transient optimization of gas transport and distribution. In: Proceedings of 2nd international workshop SIMONE on innovative approaches to modeling and optimal control of large scale pipeline networks, pp 53–62Google Scholar
  73. Wächter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57. doi: 10.1007/s10107-004-0559-y CrossRefMATHMathSciNetGoogle Scholar
  74. Weimann A (1978) Modellierung und Simulation der Dynamik von Gasnetzen im Hinblick auf Gasnetzführung und Gasnetzüberwachung. Ph. D. dissertation, Technische Universität MünchenGoogle Scholar
  75. Weymouth TR (1912) Problems in natural gas engineering. Transactions of the American Society of Mechanical Engineers 34:185–231 URL
  76. Wikipedia: Redlich-Kwong equation of state (2012). URL Accessed 11 April 2012
  77. de Wolf D, Smeers Y (2000) The gas transmission problem solved by an extension of the simplex algorithm. Manag Sci 46(11):1454–1465CrossRefMATHGoogle Scholar
  78. Wong PJ, Larson RE (1968) Optimization of natural-gas pipeline systems via dynamic programming. IEEE Trans Autom Control 13:475–481. doi: 10.1109/TAC.1968.1098990 CrossRefGoogle Scholar
  79. Wong PJ, Larson RE (1968) Optimization of tree-structured natural-gas transmission networks. J Math Anal Appl 24:613–626CrossRefMathSciNetGoogle Scholar
  80. Wright S, Somani M, Ditzel C (1998) Compressor station optimization. In: 30th Annual Meeting. Pipeline Simulation Interest Group. Paper 9805Google Scholar
  81. Wu S (1998) Steady-state simulation and fuel cost minimization of gas pipeline networks. ProQuest LLC, Ann Arbor, MI (1998). URL Thesis (Ph.D.)-University of Houston
  82. Wu S, Ríos-Mercado RZ, Boyd AE, Scott LR (2000) Model relaxations for the fuel cost minimization of steady-state gas pipeline networks. Math Comput Model 31(2-3):197–220. doi: 10.1016/S0895-7177(99)00232-0 CrossRefGoogle Scholar
  83. Záworka J (1993) Project SIMONE—Achievements and running development. In: Proceedings of 2nd international workshop SIMONE on innovative approaches to modeling and optimal control of large scale pipeline networks, pp. 1–24Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Martin Schmidt
    • 1
  • Marc C. Steinbach
    • 1
  • Bernhard M. Willert
    • 1
  1. 1.Leibniz Universität Hannover, Institute for Applied MathematicsHannoverGermany

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