Mathematical Methods of Operations Research

, Volume 89, Issue 2, pp 223–255 | Cite as

A multilevel model of the European entry-exit gas market

  • Veronika Grimm
  • Lars Schewe
  • Martin SchmidtEmail author
  • Gregor Zöttl
Original Article


In entry-exit gas markets as they are currently implemented in Europe, network constraints do not affect market interaction beyond the technical capacities determined by the TSO that restrict the quantities individual firms can trade at the market. It is an up to now unanswered question to what extent existing network capacity remains unused in an entry-exit design and to what extent feasible adjustments of the market design could alleviate inefficiencies. In this paper, we offer a four-level modeling framework that is capable of analyzing these issues and provide some first results on the model structure. In order to decouple gas trading from network congestion management, the TSO is required to determine technical capacities and corresponding booking fees at every entry and exit node up front. Firms book those capacities, which gives them the right to charge or discharge an amount of gas at a certain node up to this capacity in every scenario. Beyond these technical capacities and the resulting bookings, gas trade is unaffected by network constraints. The technical capacities have to ensure that transportation of traded quantities is always feasible. We assume that the TSO is regulated and determines technical capacities, fees, and transportation costs under a welfare objective. As a first step we moreover assume perfect competition among gas traders and show that the booking and nomination decisions can be analyzed in a single level. We prove that this aggregated model has a unique solution. We also show that the TSO’s decisions can be subsumed in one level as well. If so, the model boils down to a mixed-integer nonlinear bilevel problem with robust aspects. In addition, we provide a first-best benchmark that allows to assess welfare losses that occur in an entry-exit system. Our approach provides a generic framework to analyze various aspects in the context of semi-liberalized gas markets. Therefore, we finally discuss and provide guidance on how to include several important aspects into the approach, such as network and production capacity investment, uncertain data, market power, and intra-day trading.


Entry-exit system Gas market Multilevel modeling 

Mathematics Subject Classification

90-XX 90C35 91B15 91B16 91B24 



This research has been performed as part of the Energie Campus Nürnberg and is supported by funding of the Bavarian State Government and by the Emerging Field Initiative (EFI) of the Friedrich-Alexander-Universität Erlangen-Nürnberg through the project “Sustainable Business Models in Energy Markets”. The authors acknowledge funding through the DFG Transregio 154, subprojects A05, B07, and B08. We also thank Alexander Martin, Julia Grübel, and Jonas Egerer for many fruitful discussions on the topic of this paper. Finally, we are very grateful to two anonymous reviewers, whose comments on the manuscript greatly helped to improve the quality of the paper.


  1. Abada I, Gabriel S, Briat V, Massol O (2013) A generalized Nash-Cournot model for the northwestern European natural gas markets with a fuel substitution demand function: the GaMMES model. Netw Spat Econ 13(1):1–42. MathSciNetzbMATHCrossRefGoogle Scholar
  2. Abada I, Massol O (2011) Security of supply and retail competition in the European gas market: some model-based insights. Energy Policy 39(7):4077–4088. Special Section: Renewable energy policy and developmentCrossRefGoogle Scholar
  3. Alonso A, Olmos L, Serrano M (2010) Application of an entry-exit tariff model to the gas transport system in Spain. Energy Policy 38(5):5133–5140. CrossRefGoogle Scholar
  4. Baltensperger T, Füchslin RM, Krütli P, Lygeros J (2016) Multiplicity of equilibria in conjectural variations models of natural gas markets. Eur J Oper Res 252(2):646–656. MathSciNetzbMATHCrossRefGoogle Scholar
  5. Boots MG, Rijkers FA, Hobbs BF (2004) Trading in the downstream European gas market: a successive oligopoly approach. Energy J 25(3):73–102.
  6. Borraz-Sánchez C, Bent R, Backhaus S, Hijazi H, van Hentenryck P (2016) Convex relaxations for gas expansion planning. INFORMS J Comput 28(4):645–656. MathSciNetzbMATHCrossRefGoogle Scholar
  7. Boucher J, Smeers Y (1985) Gas trade in the European community during the 1970s. Energy Econ 7(2):102–116. CrossRefGoogle Scholar
  8. Boucher J, Smeers Y (1987) Economic forces in the European gas market–a 1985 prospective. Energy Econ 9(1):2–16. CrossRefGoogle Scholar
  9. Brouwer J, Gasser I, Herty M (2011) Gas pipeline models revisited: model hierarchies, nonisothermal models, and simulations of networks. Multiscale Model Simul 9(2):601–623. MathSciNetzbMATHCrossRefGoogle Scholar
  10. Chyong CK, Hobbs BF (2014) Strategic Eurasian natural gas market model for energy security and policy analysis: formulation and application to south stream. Energy Econ 44:198–211. CrossRefGoogle Scholar
  11. Commission: First benchmarking report on the implementation of the internal electricity and gas market. Tech. rep., European Commission SEC (2001) 1957 (2001).
  12. European Commission: Competition. Energy and environment. Gas (2012).
  13. Cremer H, Laffont JJ (2002) Competition in gas markets. Eur Econ Rev 46(4–5):928–935. CrossRefGoogle Scholar
  14. Crew MA, Fernando CS, Kleindorfer PR (1995) The theory of peak-load pricing: a survey. J Regul Econ 8(3):215–248. CrossRefGoogle Scholar
  15. Dempe S (2002) Foundations of bilevel programming. Springer, BerlinzbMATHGoogle Scholar
  16. 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. MathSciNetzbMATHCrossRefGoogle Scholar
  17. Egging R (2013) Benders decomposition for multi-stage stochastic mixed complementarity problems–applied to a global natural gas market model. Eur J Oper Res 226(2):341–353. MathSciNetzbMATHCrossRefGoogle Scholar
  18. Egging R, Gabriel SA, Holz F, Zhuang J (2008) A complementarity model for the European natural gas market. Energy Policy 36(7):2385–2414. CrossRefGoogle Scholar
  19. Egging R, Holz F, Gabriel SA (2010) The world gas model: a multi-period mixed complementarity model for the global natural gas market. Energy 35(10):4016–4029. CrossRefGoogle Scholar
  20. European Parliament and Council of the European Union (1998) Directive 98/30/EC of the European Parliament and of the Council of 22 June 1998 concerning common rules for the internal market in natural gasGoogle Scholar
  21. European Parliament and Council of the European Union (2003) 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/ECGoogle Scholar
  22. European Parliament and Council of the European Union (2009) Directive 2009/73/EC of the European Parliament and of the Council concerning common rules for the internal market in natural gas and repealing Directive 2003/55/ECGoogle Scholar
  23. European Parliament and Council of the European Union (2009) Regulation No 715/2009 of the European Parliament and of the Council on conditions for access to the natural gas transmission networks and repealing Regulation No 1775/2005Google Scholar
  24. Facchinei F, Fischer A, Piccialli V (2007) On generalized nash games and variational inequalities. Oper Res Lett 35(2):159–164. MathSciNetzbMATHCrossRefGoogle Scholar
  25. Facchinei F, Kanzow C (2007) Generalized nash equilibrium problems. 4OR 5(3):173–210. MathSciNetzbMATHCrossRefGoogle Scholar
  26. Fiacco AV, Kyparisis J (1986) Convexity and concavity properties of the optimal value function in parametric nonlinear programming. J Optim Theory Appl 48(1):95–126. MathSciNetzbMATHCrossRefGoogle Scholar
  27. Fodstad M, Midthun KT, Tomasgard A (2015) Adding flexibility in a natural gas transportation network using interruptible transportation services. Eur J Oper Res 243(2):647–657. MathSciNetzbMATHCrossRefGoogle Scholar
  28. 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 T, Hiller B, Pfetsch ME, Schewe L (eds) Evaluating gas network capacities. SIAM-MOS series on optimization, chap. 2. SIAM, Philadelphia, pp 17–43.
  29. Gabriel SA, Conejo AJ, Fuller JD, Hobbs BF, Ruiz C (2012) Complementarity modeling in energy markets, vol 180. Springer, BerlinzbMATHGoogle Scholar
  30. Gabriel SA, Kiet S, Zhuang J (2005) A mixed complementarity-based equilibrium model of natural gas markets. Oper Res 53(5):799–818. MathSciNetzbMATHCrossRefGoogle Scholar
  31. Gabriel SA, Zhuang J, Egging R (2009) Solving stochastic complementarity problems in energy market modeling using scenario reduction. Eur J Oper Res 197(3):1028–1040. MathSciNetzbMATHCrossRefGoogle Scholar
  32. Geißler B, Morsi A, Schewe L (2013) A new algorithm for MINLP applied to gas transport energy cost minimization. In: Jünger M, Reinelt G (eds) Facets of combinatorial optimization. Springer, Heidelberg, pp 321–353.
  33. Geißler B, Morsi A, Schewe L, Schmidt M (2015) Solving power-constrained gas transportation problems using an MIP-based alternating direction method. Comput Chem Eng 82:303–317. CrossRefGoogle Scholar
  34. Geißler B, Morsi A, Schewe L, Schmidt M (2017) Solving highly detailed gas transport MINLPs: block separability and penalty alternating direction methods. INFORMS J Comput. CrossRefGoogle Scholar
  35. Glachant JM, Hallack M, Vazquez M (2013) Building competitive gas markets in the EU. Edward Elgar Publishing, CheltenhamCrossRefGoogle Scholar
  36. Grimm V, Grübel J, Schewe L, Schmidt M, Zöttl G (2017) Nonconvex equilibrium models for gas market analysis: failure of standard techniques and alternative modeling approaches. Tech. rep.
  37. Grimm V, Kleinert T, Liers F, Schmidt M, Zöttl G (2017) Optimal price zones of electricity markets: a mixed-integer multilevel model and global solution approaches. Optim Methods Softw. zbMATHCrossRefGoogle Scholar
  38. Grimm V, Martin A, Schmidt M, Weibelzahl M, Zöttl G (2016a) Transmission and generation investment in electricity markets: the effects of market splitting and network fee regimes. Eur J Oper Res 254(2):493–509. MathSciNetzbMATHCrossRefGoogle Scholar
  39. Grimm V, Martin A, Weibelzahl M, Zöttl G (2016b) On the long run effects of market splitting: why more price zones might decrease welfare. Energy Policy 94:453–467. CrossRefGoogle Scholar
  40. Grimm V, Schewe L, Schmidt M, Zöttl G (2017) Uniqueness of market equilibrium on a network: a peak-load pricing approach. Eur J Oper Res 261(3):971–983. MathSciNetzbMATHCrossRefGoogle Scholar
  41. Grimm V, Zöttl G (2013) Investment incentives and electricity spot market competition. J Econ Manag Strategy 22(4):832–851. CrossRefGoogle Scholar
  42. Gugat M, Leugering G, Martin A, Schmidt M, Sirvent M, Wintergerst D (2018) Towards simulation based mixed-integer optimization with differential equations. Networks 72(1):60–83. MathSciNetzbMATHCrossRefGoogle Scholar
  43. Hallack M, Vazquez M (2013) European union regulation of gas transmission services: challenges in the allocation of network resources through entry/exit schemes. Util Policy 25 25(5):23–32.
  44. Harker PT (1991) Generalized nash games and quasi-variational inequalities. Eur J Oper Res 54(1):81–94. MathSciNetzbMATHCrossRefGoogle Scholar
  45. Hirschhausen C (2006) Reform der Erdgaswirtschaft in der EU und in Deutschland: Wieviel Regulierung braucht der Wettbewerb? Perspektiven der Wirtschaftspolitik 7(1):89–103. CrossRefGoogle Scholar
  46. Hobbs BF, Helman U (2004) Complementarity-based equilibrium modeling for electric power markets. In: Bunn D (ed) Modeling prices in competitive electricity markets. Wiley, New YorkGoogle Scholar
  47. Holz F, von Hirschhausen C, Kemfert C (2008) A strategic model of European gas supply (GASMOD). Energy Econ 30(3):766–788. CrossRefGoogle Scholar
  48. Hubert F, Ikonnikova S (2011) Investment options and bargaining power: the Eurasian supply chain for natural gas. J Ind Econ 59(1):85–116. CrossRefGoogle Scholar
  49. Hunt P (2008) Entry–exit transmission pricing with national hubs. Can it deliver a Pan-European wholesale market in gas? Tech. rep., Oxford Institute of Energy StudiesGoogle Scholar
  50. Huppmann D (2013) Endogenous production capacity investment in natural gas market equilibrium models. Eur J Oper Res 231(2):503–506. MathSciNetzbMATHCrossRefGoogle Scholar
  51. Ikonnikova S, Zwart GT (2014) Trade quotas and buyer power, with an application to the E.U. natural gas market. J Eur Econ Assoc 12(1):177–199. CrossRefGoogle Scholar
  52. Jansen T, van Lier A, van Witteloostuijn A, von Ochssée TB (2012) A modified Cournot model of the natural gas market in the European union: mixed-motives delegation in a politicized environment. Energy Policy 41:280–285. Modeling Transport (Energy) Demand and Policies
  53. Joskow P, Tirole J (2007) Reliability and competitive electricity markets. RAND J Econ 38(1):60–84. CrossRefGoogle Scholar
  54. Kema for European Commission (2013) Country factsheets. entry–exit regimes in gas, a project for the european commission—dg ener under the framework service contract for technical assistance tren/r1/350-2008 lot 3. Tech. rep., European Commission.
  55. Keyaerts N, D’haeseleer W, (2014) Forum shopping for ex-post gas-balancing servies. Energy Policy 67:209–221.
  56. Keyaerts N, Hallack M, Glachant JM, D’haeseleer W, (2011) Gas market distorting effects of imbalanced gas balancing rules: inefficient regulation of pipeline flexibility. Energy Policy 39(2):865–876.
  57. Kleinert T, Schmidt M (2018) Global optimization of multilevel electricity market models including network design and graph partitioning. Tech. rep., FAU Erlangen-Nürnberg.
  58. Koch T, Hiller B, Pfetsch ME, Schewe L (eds.) Evaluating gas network capacities. SIAM-MOS series on optimization. SIAM (2015).
  59. Krebs V, Schmidt M (2018) Uniqueness of market equilibria on networks with transport costs. Oper Res Perspect 5:169–173. MathSciNetCrossRefGoogle Scholar
  60. Krebs V, Schewe L, Schmidt M (2018) Uniqueness and multiplicity of market equilibria on DC power flow networks. Eur J Oper Res 271(1):165–178. MathSciNetzbMATHCrossRefGoogle Scholar
  61. Le Veque RJ (1992) Numerical methods for conservation laws. Birkhäuser, BaselGoogle Scholar
  62. Le Veque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, CambridgeGoogle Scholar
  63. Mahlke D, Martin A, Moritz S (2010) A mixed integer approach for time-dependent gas network optimization. Optim Methods Softw 25(4):625–644MathSciNetzbMATHCrossRefGoogle Scholar
  64. Mangasarian OL (1988) A simple characterization of solution sets of convex programs. Oper Res Lett 7(1):21–26. MathSciNetzbMATHCrossRefGoogle Scholar
  65. Mehrmann V, Schmidt M, Stolwijk JJ (2018) Model and discretization error adaptivity within stationary gas transport optimization. Vietnam J Math.
  66. Meran G, von Hirschhausen C, Neumann A (2010) Access pricing and network expansion in natural gas markets. Zeitschrift für Energiewirtschaft 34(3):179–183. CrossRefGoogle Scholar
  67. Midthun KT, Bjorndal M, Tomasgard A (2009) Modeling optimal economic dispatch and system effects in natural gas networks. Energy J 30(4):155–180.
  68. Midthun KT, Fodstad M, Hellemo L (2015) Optimization model to analyse optimal development of natural gas fields and infrastructure. Energy Procedia 64:111–119. CrossRefGoogle Scholar
  69. Moritz S (2007) A mixed integer approach for the transient case of gas network optimization. Ph.D. thesis, Technische Universität DarmstadtGoogle Scholar
  70. Murphy FH, Smeers Y (2005) Generation capacity expansion in imperfectly competitive restructured electricity markets. Oper Res 53(4):646–661. zbMATHCrossRefGoogle Scholar
  71. Nagayama D, Horita M (2014) A network game analysis of strategic interactions in the international trade of Russian natural gas through Ukraine and Belarus. Energy Econ 43:89–101. CrossRefGoogle Scholar
  72. Oliver ME, Mason CF, Finnoff D (2014) Pipeline congestion and basis differentials. J Regul Econ 46(3):261–291. CrossRefGoogle Scholar
  73. 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 (2015) Validation of nominations in gas network optimization: models, methods, and solutions. Optim Methods Softw 30(1):15–53. MathSciNetzbMATHCrossRefGoogle Scholar
  74. Rømo F, Tomasgard A, Hellemo L, Fodstad M, Eidesen BH, Pedersen B (2009) Optimizing the norwegian natural gas production and transport. Interfaces 39(1):46–56. CrossRefGoogle Scholar
  75. Ríos-Mercado RZ, Borraz-Sánchez C (2015) Optimization problems in natural gas transportation systems: a state-of-the-art review. Appl Energy 147(1):536–555. CrossRefGoogle Scholar
  76. Rose D, Schmidt M, Steinbach MC, Willert BM (2016) Computational optimization of gas compressor stations: MINLP models versus continuous reformulations. Math Methods Oper Res. MathSciNetzbMATHCrossRefGoogle Scholar
  77. Rövekamp J (2015) Background on gas market regulation. In: Koch T, Hiller B, Pfetsch ME, Schewe L (eds) Evaluating gas network capacities. SIAM-MOS series on optimization. SIAM, Philadelphia, pp 325–330.
  78. Schmidt M (2013) A generic interior-point framework for nonsmooth and complementarity constrained nonlinear optimization. Ph.D. thesis, Gottfried Wilhelm Leibniz Universität HannoverGoogle Scholar
  79. Schmidt M, Sirvent M, Wollner W (2018) A decomposition method for MINLPs with Lipschitz continuous nonlinearities. Math Program. CrossRefGoogle Scholar
  80. Schmidt M, Steinbach MC, Willert BM (2015) High detail stationary optimization models for gas networks. Optim Eng 16(1):131–164. MathSciNetzbMATHCrossRefGoogle Scholar
  81. Schmidt M, Steinbach MC, Willert BM (2016) High detail stationary optimization models for gas networks: validation and results. Optim Eng 17(2):437–472. MathSciNetzbMATHCrossRefGoogle Scholar
  82. Shaw D (1994) Pipeline system optimization: A tutorial. Tech. Rep. PSIG 9405, Pipeline Simulation Interest GroupGoogle Scholar
  83. Siddiqui S, Gabriel SA (2017) Modeling market power in the U.S. shale gas market. Optim Eng 18(1):203–213.
  84. Smeers Y (2008) Gas models and three difficult objectives. Tech. rep., Core Discussion Paper.
  85. Vazquez M, Hallack M, Glachant JM (2012) Designing the European gas market: More liquid and less natural? Econ Energy Environ Policy. CrossRefGoogle Scholar
  86. Weymouth TR (1912) Problems in natural gas engineering. Trans Am Soc Mech Eng 34(1349):185–231Google Scholar
  87. Wogrin S, Hobbs BF, Ralph D, Centeno E, Barquín J (2013) Open versus closed loop capacity equilibria in electricity markets under perfect and oligopolistic competition. Math Program 140(2):295–322. MathSciNetzbMATHCrossRefGoogle Scholar
  88. Yang Z, Zhang R, Zhang Z (2016) An exploration of a strategic competition model for the European Union natural gas market. Energy Econ 57:236–242. CrossRefGoogle Scholar
  89. Zheng Q, Rebennack S, Iliadis N, Pardalos P (2010) Optimization models in the natural gas industry. In: Pardalos PM, Iliadis NA, Pereira MV, Rebennack S (eds) Handbook of power systems I. Springer, Berlin, pp 121–148CrossRefGoogle Scholar
  90. Zhuang J, Gabriel SA (2008) A complementarity model for solving stochastic natural gas market equilibria. Energy Econ 30(1):113–147. CrossRefGoogle Scholar
  91. Zöttl G (2010) A framework of peak load pricing with strategic firms. Oper Res 58(4):1637–1649. MathSciNetzbMATHCrossRefGoogle Scholar
  92. Zöttl G (2011) On optimal scarcity prices. Int J Ind Organ 29(5):589–605. CrossRefGoogle Scholar
  93. Zwart G, Mulder M (2006) NATGAS: a model of the European natural gas market. Tech. Rep. 144, CPB Netherlands Bureau for Economic Policy Analysis.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Economic TheoryFriedrich-Alexander-Universität Erlangen-Nürnberg (FAU)NürnbergGermany
  2. 2.Energie Campus NürnbergNürnbergGermany
  3. 3.Discrete OptimizationFriedrich-Alexander-Universität Erlangen-Nürnberg (FAU)ErlangenGermany
  4. 4.Industrial Organization and Energy MarketsFriedrich-Alexander-Universität Erlangen-Nürnberg (FAU)NürnbergGermany

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