Energy Systems

, Volume 5, Issue 3, pp 449–473 | Cite as

Mathematical optimization for challenging network planning problems in unbundled liberalized gas markets

  • Armin Fügenschuh
  • Björn Geißler
  • Ralf Gollmer
  • Christine Hayn
  • René Henrion
  • Benjamin Hiller
  • Jesco Humpola
  • Thorsten Koch
  • Thomas Lehmann
  • Alexander Martin
  • Radoslava Mirkov
  • Antonio Morsi
  • Jessica Rövekamp
  • Lars Schewe
  • Martin Schmidt
  • Rüdiger Schultz
  • Robert Schwarz
  • Jonas Schweiger
  • Claudia Stangl
  • Marc C. Steinbach
  • Bernhard M. Willert
Original Paper

Abstract

The recently imposed new gas market liberalization rules in Germany lead to a change of business of gas network operators. While previously network operator and gas vendor were united, they were forced to split up into independent companies. The network has to be open to any other gas trader at the same conditions, and free network capacities have to be identified and publicly offered in a non-discriminatory way. We discuss how these changing paradigms lead to new and challenging mathematical optimization problems. This includes the validation of nominations, that asks for the decision if the network’s capacity is sufficient to transport a specific amount of flow, the verification of booked capacities and the detection of available freely allocable capacities, and the topological extension of the network with new pipelines or compressors in order to increase its capacity. In order to solve each of these problems and to provide meaningful results for the practice, a mixture of different mathematical aspects have to be addressed, such as combinatorics, stochasticity, uncertainty, and nonlinearity. Currently, no numerical solver is available that can deal with such blended problems out-of-the-box. The main goal of our research is to develop such a solver, that moreover is able to solve instances of realistic size. In this article, we describe the main ingredients of our prototypical software implementations.

Keywords

Gas market liberalization Entry–exit model Gas network access regulation Mixed-integer nonlinear nonconvex stochastic optimization 

Mathematics Subject Classification (2000)

90B10 90C11 90C30 90C90 

References

  1. 1.
    Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    André, J., Bonnans, F., Cornibert, L.: Optimization of capacity expansion planning for gas transportation networks. Eur. J. Oper. Res. 197(3), 1019–1027 (2009)Google Scholar
  3. 3.
    Babonneau, F., Nesterov, Y., Vial, J.-P.: Design and operations of gas transmission networks. Oper. Res. 60(1), 34–47 (2012)Google Scholar
  4. 4.
    Arne Stolbjerg Drud. CONOPT—a large-scale GRG code. INFORMS J. Comput. 6(2), 207–216 (1994)Google Scholar
  5. 5.
    Baumrucker, B.T., Biegler, L.T.: MPEC strategies for cost optimization of pipeline operations. Comput. Chem. Eng. 34(6), 900–913 (2010)CrossRefGoogle Scholar
  6. 6.
    Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex MINLP. Opt. Methods Softw. 24(4–5), 597–634 (2009)CrossRefMATHGoogle Scholar
  7. 7.
    Ben-Tal, Aharon, El Ghaoui, Laurent, Nemirovski, Arkadi: Robust Optimization, Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)Google Scholar
  8. 8.
    Benedetti, M., Lallouet, A., Vautard, J.: Quantified constraint optimization. In CP, Lecture Notes in Computer Science, pp. 463–477. Springer, Berlin (2008)Google Scholar
  9. 9.
    Berthold, T., Heinz, S., Vigerske, S.: Extending a CIP framework to solve MIQCPs. In: Lee, J., Leyffer, A. (eds.) Mixed Integer Nonlinear Programming, volume 154, part 6 of The IMA Volumes in Mathematics and its Applications, pp. 427–444. Springer, New York (2012) (also available as ZIB-Report 09–23)Google Scholar
  10. 10.
    Bixby, R.E.: Solving real-world linear programs: a decade and more of progress. Oper. Res. 50(1), 1–13 (2002)Google Scholar
  11. 11.
    Borráz-Sánchez, C., Ríos-Mercado, R.Z.: A non-sequential dynamic programming approach for natural gas network optimization. WSEAS Trans. Syst. 3, 1384–1389 (2004)Google Scholar
  12. 12.
    Byrd, R., Nocedal, J., Waltz, R.: Knitro: an integrated package for nonlinear optimization. In: Pillo, G., Roma, M., Pardalos, P. (eds.) Large-Scale Nonlinear Optimization, volume 83 of Nonconvex Optimization and Its Applications, pp. 35–59. Springer, New York (2006)Google Scholar
  13. 13.
    CPLEX. User’s Manual for CPLEX. IBM Corporation, Armonk, 12.4 edition (2011)Google Scholar
  14. 14.
    De Wolf, D., Bakhouya, B.: The gas transmission problem when the merchant and the transport functions are disconnected. Technical Report 01/01, Ieseg, Université catholique de Lille, HEC Ecole de Gestion de l’ULG (2007)Google Scholar
  15. 15.
    de Wolf, D., Smeers, Y.: Optimal dimensioning of pipe networks with application to gas transmission networks. Oper. Res. 44(4), 596–608 (July 1996)Google Scholar
  16. 16.
    DeMiguel, A.-V., Friedlander, M.P., Nogales, F.J., Scholtes, S.: A two-sided relaxation scheme for mathematical programs with equilibrium constraints. SIAM J. Optim. 16(1), 587–609 (2005)Google Scholar
  17. 17.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1(1), 269–271 (1959)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Egging, R.G., Gabriel, S.A.: Examining market power in the European natural gas market. Energy Policy 34(17), 2762–2778 (2006)Google Scholar
  19. 19.
    Ehrhardt, K., Steinbach, M.C.: Nonlinear optimization in gas networks. In: Bock, H.G., Kostina, E., Phu, H.X., Ranacher, R. (eds.) Modeling, Simulation and Optimization of Complex Processes, pp. 139–148. Springer, Berlin (2005)Google Scholar
  20. 20.
    Ellis, A., Bowitz, E., Roland, K.: Structural change in Europe’s gas markets: three scenarios for the development of the European gas market to 2020. Energy Policy 28, 297–309 (2000)Google Scholar
  21. 21.
    European Parliament and Council of the European Union. Directive 2003/55/EC of the European Parliament and of the Council of 26 June 2003 concerning common rules for the internal market in natural gas and repealing Directive 98/30/EC. Offic. J. of Europ. Union, pp. L 176, 57–78 (2003)Google Scholar
  22. 22.
    Feistauer, M.: Mathematical methods in fluid dynamics, volume 67 of Pitman Monographs and Surveys in Pure and Applied Mathematics Series. Longman Scientific & Technical, Harlow (1993)Google Scholar
  23. 23.
    Finnemore, E.J., Franzini, J.E.: Fluid Mechanics with Engineering Applications, 10 th edn. McGraw-Hill (2002)Google Scholar
  24. 24.
    Friedl, H., Mirkov, R., Steinkamp, A.: Modeling and forecasting gas flow on exits of gas transmission networks. Int. Stat. Rev. 80(1), 24–39 (2012)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Fügenschuh, A, Hiller, B., Humpola, J., Koch, T., Lehman, T., Schwarz, R., Schweiger, J., Szabó, J.: Gas network topology optimization for upcoming market requirements. IEEE Proc. 8th Intern. Conf. on EEM 2011, pp. 346–351 (2011)Google Scholar
  26. 26.
    Furey, B.P.: A sequential quadratic programming-based algorithm for optimization of gas networks. Automatica 29(6), 1439–1450 (1993)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Geißler, B.: Towards globally optimal solutions for MINLPs by discretization techniques with applications in gas network optimization. PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg (2011)Google Scholar
  28. 28.
    Geißler, B., Martin, A., Morsi, A., Schewe, L.: Using piecewise linear functions for solving MINLPs. In: Lee, J., Leyffer, A. (eds.) Mixed Integer Nonlinear Programming, volume 154 of The IMA Volumes in Mathematics and its Applications, pp. 287–314. Springer, New York (2012)Google Scholar
  29. 29.
    Gilmour, B.J., Luongo, C.A., Schroeder, D.W.: Optimization in Natural Gas Transmission Networks: A Tool to Improve Operational Efficiency, vol. 4. Technical report, Stoner Associates Inc. (1989)Google Scholar
  30. 30.
    Gu, Z., Rothberg, E., Bixby, R.: Gurobi Optimizer Reference Manual, Version 5.0. Gurobi Optimization Inc., Houston (2012)Google Scholar
  31. 31.
    Hamam, Y.M., Brameller, A.: Hybrid method for the solution of piping networks. Proc. Inst. Electr. Eng. 118(11), 1607–1612 (1971)CrossRefGoogle Scholar
  32. 32.
    Hansen, C.T., Madsen, K., Nielsen, H.B.: Optimization of pipe networks. Math. Program. 52(1–3), 45–58 (1991)Google Scholar
  33. 33.
    Heitsch, H., Römisch, W.: Scenario reduction algorithms in stochastic programming. Comput. Optim. Appl. 24, 187–206 (2003)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Heren, P.: Removing the government from European gas. Energy Policy 27, 3–8 (1999)Google Scholar
  35. 35.
    Holz, F., von Hirschhausen, C., Kemfert, C.: Perspectives of the European natural gas markets until 2025. Energy J. 30, 137–150 (2009)Google Scholar
  36. 36.
    Hu, X.M., Ralph, D.: Convergence of a penalty method for mathematical programming with complementarity constraints. J. Optim. Theory Appl. 123, 365–390 (2004)Google Scholar
  37. 37.
    Jeníček, T.: Steady-state optimization of gas transport. In: SIMONE: [56], pp. 26–38Google Scholar
  38. 38.
    Koch, T., Leövey, H., Mirkov, R., Römisch, W., Wegner-Specht, I.: Szenariogenerierung zur Modellierung der stochastischen Ausspeiselasten in einem Gastransportnetz. VDI-Berichte: Opt. i. d. Energiewirt. 2157, 115–125 (2011)Google Scholar
  39. 39.
    Kráik, J.: Compressor stations in SIMONE. In: SIMONE [56], pp. 93–117Google Scholar
  40. 40.
    Lall, H., Percell, P.: A dynamic programming based Gas Pipeline Optimizer. In: Bensoussan, A., Lions, J. (eds.) Analysis and Optimization of Systems, volume 144 of Lecture Notes in Control and Information Sciences, pp. 123–132. Springer, New York (1990)Google Scholar
  41. 41.
    Li, C., Jia, W., Yang, Y., Wu, X.: Adaptive genetic algorithm for steady-state operation optimization in natural gas networks. J. Softw. 6, 452–459 (2011)Google Scholar
  42. 42.
    Lorenz, U., Martin, A., Wolf, J.: Polyhedral and algorithmic properties of quantified Linear Programs. In: Algorithms—ESA, volume 6346 of Lecture Notes in Computer Science, pp. 512–523 (2010)Google Scholar
  43. 43.
    Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical programs with equilibrium constraints. Cambridge University Press, Cambridge (1996)Google Scholar
  44. 44.
    Lurie, M.V.: Modeling of Oil Product and Gas Pipeline Transportation. Wiley-VCH, Weinheim (2008)Google Scholar
  45. 45.
    Mahlke, D., Martin, A., Moritz, S.: A simulated annealing algorithm for transient optimization in gas networks. Math. Methods Oper. Res. 66(1), 99–116 (2007)CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Mahlke, D., Martin, A., Moritz, S.: A mixed integer approach for time-dependent gas network optimization. Opt. Methods Softw. 25(4), 625–644 (2010)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Mallinson, J., Fincham, A.E., Bull, S.P., Rollet, J.S., Wong, M.L.: Methods for optimizing gas transmission networks. Ann. Oper. Res. 43, 443–454 (1993)CrossRefMATHGoogle Scholar
  48. 48.
    Markowitz, H.M., Manne, A.S.: On the solution of discrete programming problems. Econometrica 25, 84–110 (1957)CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    Martin, A., Möller, M., Moritz, S.: Mixed integer models for the stationary case of gas network optimization. Math. Program. 105(2), 563–582 (2006)CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Meeus, Leonardo, Purchala, Konrad, Belmans, Ronnie: Development of the internal electricity market in Europe. Electr. J. 18(6), 25–35 (2005)CrossRefGoogle Scholar
  51. 51.
    Modisette, J.L.: Equation of state tutorial. Technical Report 0008, PSIG—Pipeline Simulation Interest, Group (2000)Google Scholar
  52. 52.
    Möller, M.: Mixed Integer Models for the Optimisation of Gas Networks in the Stationary Case. PhD thesis, Technische Universität Darmstadt, Fachbereich Mathematik (2004)Google Scholar
  53. 53.
    Moritz, S.: A Mixed Integer Approach for the Transient Case of Gas Network Optimization. PhD thesis, Technische Universität Darmstadt, Fachbereich Mathematik (2006)Google Scholar
  54. 54.
    Percebois, J.: The gas deregulation process in Europe: economic and political approach. Energy Policy 27, 9–15 (1999)Google Scholar
  55. 55.
    Pfetsch, M.E., Geißler, A.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, M.C., Vigerske, S., Willert, B.M.: Validation of nominations in gas network optimization: Models, methods, and solutions. ZIB-Report 12–41, Zuse Institute Berlin, Berlin (2012)Google Scholar
  56. 56.
    Proceedings of 2nd International Workshop SIMONE on Innovative Approaches to Modeling and Optimal Control of Large Scale Pipeline Networks, vol. 9, Prague (1993)Google Scholar
  57. 57.
    Prékopa, A., Stochastic Programming. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar
  58. 58.
    Radetzki, M.: European natural gas: market forces will bring about competition in any case. Energy Policy 27, 17–24 (1999)Google Scholar
  59. 59.
    Ríos-Mercado, R.Z., Wu, S., Scott, L.R., Boyd, E.A.: A reduction technique for natural gas transmission network optimization problems. Ann. Oper. Res. 117(1), 217–234 (2002)CrossRefMATHGoogle Scholar
  60. 60.
    Schmidt, M.: A Generic Interior-Point Framework for Nonsmooth and Complementarity Constrained Nonlinear Optimization. PhD thesis, Gottfried Wilhelm Leibniz Universität Hannover (2013)Google Scholar
  61. 61.
    Schmidt, M., Steinbach, M.C., Willert, B.M.: High detail stationary optimization models for gas networks—part 1: model components. IfAM Preprint 94, Inst. of Applied Mathematics, Leibniz Universität Hannover (2012, submitted)Google Scholar
  62. 62.
    Schmidt, M., Steinbach, M.C., Willert, B.M.: A primal heuristic for nonsmooth mixed integer nonlinear optimization. IfAM Preprint 95, Inst. of Applied Mathematics, Leibniz Universität Hannover (2012, submitted)Google Scholar
  63. 63.
    Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11(4), 918–936 (2001)Google Scholar
  64. 64.
    Smith, E.M.B., Pantelides, C.C.: A symbolic reformulation/spatial branch-and-bound algorithm for the global optimization of nonconvex MINLPs. Comput. Chem. Eng. 23, 457–478 (1999)CrossRefGoogle Scholar
  65. 65.
    Steinbach, M.C.: On PDE solution in transient optimization of gas networks. J. Comput. Appl. Math. 203(2), 345–361 (2007)CrossRefMATHMathSciNetGoogle Scholar
  66. 66.
    Subramani, K.: On a decision procedure for quantified linear programs. Ann. Math. Artif. Intell. 51(1), 55–77 (2007)CrossRefMATHMathSciNetGoogle Scholar
  67. 67.
    Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer Academic Publishers, Dordrecht (2002)Google Scholar
  68. 68.
    Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99, 563–591 (2004)CrossRefMATHMathSciNetGoogle Scholar
  69. 69.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)CrossRefMATHMathSciNetGoogle Scholar
  70. 70.
    Vigerske, S.: Decomposition in Multistage Stochastic Programming and a Constraint Integer Programming Approach to Mixed-Integer Nonlinear Programming. PhD thesis, Humboldt-Universität zu Berlin (2012)Google Scholar
  71. 71.
    Verordnung über den Zugang zu Gasversorgungsnetzen (Gasnetzzugangsverordnung-GasNZV) (2005)Google Scholar
  72. 72.
    Vostrý, Z.: Transient optimization of gas transport and distribution. In: SIMONE [56], pp. 53–62Google Scholar
  73. 73.
    Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)CrossRefMATHMathSciNetGoogle Scholar
  74. 74.
    Waverman, Leonard, Sirel, Esen: European telecommunications markets on the verge of full liberalization. J. Econ. Perspect. 11(4), 113–126 (1997)CrossRefGoogle Scholar
  75. 75.
    Weymouth, T.R.: Problems in natural gas engineering. Trans. Am. Soc. Mech. Eng. 34, 185–231 (1912)Google Scholar
  76. 76.
    Wong, P.J., Larson, R.E.: Optimization of natural-gas pipeline systems via dynamic programming. IEEE Trans. Autom. Control 15(5), 475–481 (1968)CrossRefGoogle Scholar
  77. 77.
    Wright, S., Somani, M., Ditzel, C.: Compressor station optimization. Technical Report PSIG 9805, Pipeline Simulation Interest Group (1998)Google Scholar
  78. 78.
    Zhang, J., Zhu, D.: A bilevel programming method for pipe network optimization. SIAM J. Optim. 6(3), 838–857 (1996)CrossRefMATHMathSciNetGoogle Scholar
  79. 79.
    Zheng, Q.P., Rebennack, S., Iliadis, N.A., Pardalos, P.M.: Optimization models in the natural gas industry. In: Handbook of Power Systems I, pp. 121–148. Springer, New York (2010)Google Scholar
  80. 80.
    Zimmer, H.I.: Calculating Optimum Pipeline Operations. Technical Report SAND2009-5066C, El Paso Natural Gas Company (1975)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Armin Fügenschuh
    • 1
  • Björn Geißler
    • 3
  • Ralf Gollmer
    • 4
  • Christine Hayn
    • 3
  • René Henrion
    • 5
  • Benjamin Hiller
    • 2
  • Jesco Humpola
    • 2
  • Thorsten Koch
    • 2
  • Thomas Lehmann
    • 2
  • Alexander Martin
    • 3
  • Radoslava Mirkov
    • 6
  • Antonio Morsi
    • 3
  • Jessica Rövekamp
    • 7
  • Lars Schewe
    • 3
  • Martin Schmidt
    • 8
  • Rüdiger Schultz
    • 4
  • Robert Schwarz
    • 2
  • Jonas Schweiger
    • 2
  • Claudia Stangl
    • 4
  • Marc C. Steinbach
    • 8
  • Bernhard M. Willert
    • 8
  1. 1.Helmut-Schmidt-UniversitätHamburgGermany
  2. 2.Konrad Zuse Zentrum für InformationstechnikBerlinGermany
  3. 3.Friedrich-Alexander Universität Erlangen-NürnbergErlangenGermany
  4. 4.Universität Duisburg-EssenDuisburgGermany
  5. 5.Weierstrass InstitutBerlinGermany
  6. 6.Humboldt Universität BerlinBerlinGermany
  7. 7.Open Grid Europe GmbHEssenGermany
  8. 8.Leibniz Universität HannoverHannoverGermany

Personalised recommendations