OR Spectrum

, Volume 40, Issue 1, pp 265–310 | Cite as

Two-stage stochastic, large-scale optimization of a decentralized energy system: a case study focusing on solar PV, heat pumps and storage in a residential quarter

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Abstract

The expansion of fluctuating renewable energy sources leads to an increasing impact of weather-related uncertainties on future decentralized energy systems. Stochastic modeling techniques enable an adequate consideration of the uncertainties and provide support for both investment and operating decisions in such systems. In this paper, we consider a residential quarter using photovoltaic systems in combination with multistage air-water heat pumps and heat storage units for space heating and domestic hot water. We model the investment and operating problem of the quarter’s energy system as two-stage stochastic mixed-integer linear program and optimize the thermal storage units. In order to keep the resulting stochastic, large-scale program computationally feasible, the problem is decomposed in combination with a derivative-free optimization. The subproblems are solved in parallel on high-performance computing systems. Our approach is integrated in that it comprises three subsystems: generation of consistent ensembles of the required input data by a Markov process, transformation into sets of energy demand and supply profiles and the actual stochastic optimization. An analysis of the scalability and comparison with a state-of-the-art dual-decomposition method using Lagrange relaxation and a conic bundle algorithm shows a good performance of our approach for the considered problem type. A comparison of the effective gain of modeling the quarter as stochastic program with the resulting computational expenses justifies the approach. Moreover, our results show that heat storage units in such systems are generally larger when uncertainties are considered, i.e., stochastic optimization can help to avoid insufficient setup decisions. Furthermore, we find that the storage is more profitable for domestic hot water than for space heating.

Keywords

Large-scale energy system optimization Stochastic programming Uncertainty modeling Markov process 
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Notes

Acknowledgements

The authors acknowledge support by the state of Baden-Württemberg through bwHPC and the Germany Research Foundation (DFG) through Grant No INST 35/1134-1 FUGG. This research has been supported by KIC InnoEnergy. KIC InnoEnergy is a company supported by the European Institute of Innovation and Technology (EIT), and has the mission of delivering commercial products and services, new businesses, innovators and entrepreneurs in the field of sustainable energy through the integration of higher education, research, entrepreneurs and business companies. Valentin Bertsch acknowledges funding from the Energy Policy Research Centre of the Economic and Social Research Institute.

References

  1. Ahmed S (2010) Two-stage stochastic integer programming: a brief introduction. In: Cochran JJ, Cox LA, Keskinocak P, Kharoufeh JP, Smith JC (eds) Wiley encyclopedia of operations research and management science. Wiley, HobokenGoogle Scholar
  2. Alonso-Ayuso A, Escudero LF, Teresa Ortuño M (2003) BFC, a branch-and-fix coordination algorithmic framework for solving some types of stochastic pure and mixed 0–1 programs. Eur J Oper Res 151(3):503–519.  https://doi.org/10.1016/S0377-2217(02)00628-8 CrossRefGoogle Scholar
  3. Alonso-Ayuso A, Escudero LF, Garın A, Ortuño MT, Pérez G (2005) On the product selection and plant dimensioning problem under uncertainty. Omega 33(4):307–318.  https://doi.org/10.1016/j.omega.2004.05.001 CrossRefGoogle Scholar
  4. Altmann M, Brenninkmeijer A, Lanoix J-C, Ellison D, Crisan A, Hugyecz A, Koreneff G, Hänninen S (2010) Decentralized energy systems. Technical report European Parliament’s Committee (ITRE). http://www.europarl.europa.eu/document/activities/cont/201106/20110629ATT22897/20110629ATT22897EN.pdf. Zugegriffen 29 Sept 2016
  5. Amato U, Andretta A, Bartoli B, Coluzzi B, Cuomo V, Fontana F, Serio C (1986) Markov processes and Fourier analysis as a tool to describe and simulate daily solar irradiance. Sol Energy 37(3):179–194.  https://doi.org/10.1016/0038-092X(86)90075-7 CrossRefGoogle Scholar
  6. Beale EML (1955) On minimizing a convex function subject to linear inequalities. J R Stat Soc B 17(2):173–184Google Scholar
  7. Beck T, Kondziella H, Huard G, Bruckner T (2017) Optimal operation, configuration and sizing of generation and storage technologies for residential heat pump systems in the spotlight of self-consumption of photovoltaic electricity. Appl Energy 188:604–619.  https://doi.org/10.1016/j.apenergy.2016.12.041 CrossRefGoogle Scholar
  8. Bedford T, Cooke RM (2001) Probabilistic risk analysis. Foundations and methods. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  9. Ben-Tal A, El Ghaoui L, Nemirovskiĭ AS (2009) Robust optimization. Princeton series in applied mathematics. Princeton University Press, PrincetonCrossRefGoogle Scholar
  10. Bertsch V, Schwarz H, Fichtner W (2014) Layout optimisation of decentralised energy systems under uncertainty. In: Huisman D, Louwerse I, Wagelmans AP (Hrsg) Operations research proceedings 2013: selected papers of the international conference on operations research, OR2013, organized by the German Operations Research Society (GOR), the Dutch Society of Operations Research (NGB) and Erasmus University Rotterdam, Springer, Cham, S 29–35, 3–6 Sept 2013Google Scholar
  11. Birge JR (1982) The value of the stochastic solution in stochastic linear programs with fixed recourse. Math Program 24(1):314–325.  https://doi.org/10.1007/BF01585113 CrossRefGoogle Scholar
  12. Birge JR, Louveaux F (1997) Introduction to stochastic programming. Springer series in operations research and financial engineering. Springer, New YorkGoogle Scholar
  13. Carøe CC, Schultz R (1999) Dual decomposition in stochastic integer programming. Oper Res Lett 24(1–2):37–45.  https://doi.org/10.1016/S0167-6377(98)00050-9 CrossRefGoogle Scholar
  14. Carøe CC, Tind J (1998) L-shaped decomposition of two-stage stochastic programs with integer recourse. Math Program 83(1–3):451–464.  https://doi.org/10.1007/BF02680570 CrossRefGoogle Scholar
  15. Conn AR, Scheinberg K, Vicente LN (2009) Introduction to derivative-free optimization. Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefGoogle Scholar
  16. Connolly D, Lund H, Mathiesen BV, Leahy M (2010) A review of computer tools for analysing the integration of renewable energy into various energy systems. Appl Energy 87(4):1059–1082.  https://doi.org/10.1016/j.apenergy.2009.09.026 CrossRefGoogle Scholar
  17. Dantzig GB (1955) Linear programming under uncertainty. Manag Sci 1:197–206CrossRefGoogle Scholar
  18. Dantzig GB, Infanger G (2011) A probabilistic lower bound for two-stage stochastic programs. In: Infanger G (Hrsg) Stochastic programming. The state of the art; in honor of George B. Dantzig, Bd 150. Springer, New York, NY, S 13–35Google Scholar
  19. Diagne M, David M, Lauret P, Boland J, Schmutz N (2013) Review of solar irradiance forecasting methods and a proposition for small-scale insular grids. Renew Sustain Energy Rev 27:65–76.  https://doi.org/10.1016/j.rser.2013.06.042 CrossRefGoogle Scholar
  20. Dupačová J, Gröwe-Kuska N, Römisch W (2003) Scenario reduction in stochastic programming. Math Program 95(3):493–511.  https://doi.org/10.1007/s10107-002-0331-0 CrossRefGoogle Scholar
  21. Ehnberg JS, Bollen MH (2005) Simulation of global solar radiation based on cloud observations. ISES Solar World Congr 2003 78(2):157–162.  https://doi.org/10.1016/j.solener.2004.08.016 Google Scholar
  22. Escudero LF, Garín A, Merino M, Pérez G (2007) A two-stage stochastic integer programming approach as a mixture of branch-and-fix coordination and benders decomposition schemes. Ann Oper Res 152(1):395–420.  https://doi.org/10.1007/s10479-006-0138-0 CrossRefGoogle Scholar
  23. Escudero LF, Garín MA, Merino M, Pérez G (2010) An exact algorithm for solving large-scale two-stage stochastic mixed-integer problems: some theoretical and experimental aspects. Eur J Oper Res 204(1):105–116.  https://doi.org/10.1016/j.ejor.2009.09.027 CrossRefGoogle Scholar
  24. Evins R, Orehounig K, Dorer V, Carmeliet J (2014) New formulations of the ‘energy hub’ model to address operational constraints. Energy 73:387–398.  https://doi.org/10.1016/j.energy.2014.06.029 CrossRefGoogle Scholar
  25. Forrest S, Mitchell M (1993) Relative building-block fitness and the building-block hypothesis, vol 2. Elsevier, Amsterdam, pp 109–126Google Scholar
  26. French S (1995) Uncertainty and imprecision. Modelling and analysis. J Oper Res Soc 46(1):70.  https://doi.org/10.2307/2583837 CrossRefGoogle Scholar
  27. Göbelt M (2001) Entwicklung eines Modells für die Investitions- und Produktionsprogrammplanung von Energieversorgungsunternehmen im liberalisierten Markt [Development of a model for investment and production program planning of energy supply companies in liberalized markets]. Dissertation, Karlsruher Institut für TechnologieGoogle Scholar
  28. Goldstein M (2012) Bayes linear analysis for complex physical systems modeled by computer simulators. In: Dienstfrey AM, Boisvert RF (Hrsg) Uncertainty quantification in scientific computing. 10th IFIP WG 2.5 Working Conference, WoCoUQ 2011, Boulder, CO, USA, 1–4 Aug 2011, Revised selected papers, Bd 377. Springer, Berlin, S 78–94Google Scholar
  29. Growe-Kuska N, Heitsch H, Romisch W (2003) Scenario reduction and scenario tree construction for power management problems. In: 2003 IEEE Bologna powertech conference proceedings, 23–26 June 2003. Faculty of Engineering, University of Bologna, Bologna, Italy. IEEE, Piscataway, NJ, S 152–158Google Scholar
  30. Haneveld WK, van der Vlerk MH (1999) Stochastic integer programming: general models and algorithms. Ann Oper Res 85:39–57.  https://doi.org/10.1023/A:1018930113099 CrossRefGoogle Scholar
  31. Hawkes A, Leach M (2005) Impacts of temporal precision in optimisation modelling of micro-combined heat and power. Energy 30(10):1759–1779.  https://doi.org/10.1016/j.energy.2004.11.012 CrossRefGoogle Scholar
  32. Hayn M, Zander A, Fichtner W, Nickel S, Bertsch V (2018) The impact of electricity tariffs on residential demand side flexibility: Results of bottom-up load profile modeling, Energy Systems (Accepted) Google Scholar
  33. Heitsch H (2007) Stabilität und Approximaton stochastischer Optimierungsprobleme [Stability and approximation of stochastic optimization problems]. DissertationGoogle Scholar
  34. Heitsch H, Römisch W (2011) Scenario tree generation for multi-stage stochastic programs. In: Bertocchi M, Consigli G, Dempster MAH (eds) Stochastic optimization methods in finance and energy. New financial products and energy market strategies, vol 163. Springer, New York, pp 313–341CrossRefGoogle Scholar
  35. Helgason T, Wallace SW (1991) Approximate scenario solutions in the progressive hedging algorithm. Ann Oper Res 31(1):425–444.  https://doi.org/10.1007/BF02204861 CrossRefGoogle Scholar
  36. Hurink J, Schultz R, Wozabal D (2016) Quantitative solutions for future energy systems and markets. OR Spectr 38(3):541–543.  https://doi.org/10.1007/s00291-016-0449-8 CrossRefGoogle Scholar
  37. Huyer W, Neumaier A (2008) SNOBFIT—stable noisy optimization by branch and fit. ACM Trans Math Softw 35(2):1–25.  https://doi.org/10.1145/1377612.1377613 CrossRefGoogle Scholar
  38. Jochem P, Schönfelder M, Fichtner W (2015) An efficient two-stage algorithm for decentralized scheduling of micro-CHP units. Eur J Oper Res 245(3):862–874.  https://doi.org/10.1016/j.ejor.2015.04.016 CrossRefGoogle Scholar
  39. Jones PA (1992) Cloud-cover distributions and correlations. J Appl Meteorol 31(7):732–741.  https://doi.org/10.1175/1520-0450(1992) 031<0732:CCDAC>2.0.CO;2
  40. Jones TW, Smith JD (1982) An historical perspective of net present value and equivalent annual cost. Account Hist J 9(1):103–110CrossRefGoogle Scholar
  41. Kalvelagen E (2003) Two-stage stochastic linear programming with GAMS. GAMS CorporationGoogle Scholar
  42. Kanngießer A (2014) Entwicklung eines generischen Modells zur Einsatzoptimierung von Energiespeichern für die techno-ökonomische Bewertung stationärer Speicheranwendungen [Development of a generic model for operation optimization of energy storages for a techno-economic evaluation of stationary storage applications]. UMSICHT-Schriftenreihe, Bd. Nr. 69. Laufen, K M, Oberhausen, RheinlGoogle Scholar
  43. Kaschub T, Jochem P, Fichtner W (2016) Solar energy storage in German households. Profitability, load changes and flexibility. Energy Policy 98:520–532.  https://doi.org/10.1016/j.enpol.2016.09.017 CrossRefGoogle Scholar
  44. Kelman R, Barroso LAN, Pereira MV (2001) Market power assessment and mitigation in hydrothermal systems. IEEE Power Eng Rev 21(8):57.  https://doi.org/10.1109/MPER.2001.4311542 CrossRefGoogle Scholar
  45. Kiureghian AD, Ditlevsen O (2009) Aleatory or epistemic? Does it matter? Risk Accept Risk Commun 31(2):105–112.  https://doi.org/10.1016/j.strusafe.2008.06.020 Google Scholar
  46. Kobayakawa T, Kandpal TC (2016) Optimal resource integration in a decentralized renewable energy system. Assessment of the existing system and simulation for its expansion. Energy Sustain Dev 34:20–29.  https://doi.org/10.1016/j.esd.2016.06.006 CrossRefGoogle Scholar
  47. Korpaas M, Holen AT, Hildrum R (2003) Operation and sizing of energy storage for wind power plants in a market system. Int J Electr Power Energy Syst 25(8):599–606.  https://doi.org/10.1016/S0142-0615(03)00016-4 CrossRefGoogle Scholar
  48. Kovacevic RM, Paraschiv F (2014) Medium-term planning for thermal electricity production. OR Spectr 36(3):723–759.  https://doi.org/10.1007/s00291-013-0340-9 CrossRefGoogle Scholar
  49. Kovacevic RM, Pichler A (2015) Tree approximation for discrete time stochastic processes. A process distance approach. Ann Oper Res 235(1):395–421.  https://doi.org/10.1007/s10479-015-1994-2 CrossRefGoogle Scholar
  50. Lorenzi G, Silva CAS (2016) Comparing demand response and battery storage to optimize self-consumption in PV systems. Appl Energy 180:524–535.  https://doi.org/10.1016/j.apenergy.2016.07.103 CrossRefGoogle Scholar
  51. Märkert A, Gollmer R (2016) User’s Guide to ddsip—AC package for the dual decomposition of two-stage stochastic programs with mixed-integer recourse. Department of Mathematics, University of Duisburg-Essen, Duisburg. https://github.com/RalfGollmer/ddsip/ddsip-man.pdf. Accessed 1 May 2016
  52. Metaxiotis K (2010) Intelligent information systems and knowledge management for energy. Applications for decision support, usage, and environmental protection. Information Science Reference, Hershey PAGoogle Scholar
  53. Morf H (1998) The stochastic two-state solar irradiance model (STSIM). Sol Energy 62(2):101–112.  https://doi.org/10.1016/S0038-092X(98)00004-8 CrossRefGoogle Scholar
  54. Morgan MG, Henrion M (1992) Uncertainty. A guide to dealing with uncertainty in quantitative risk and policy analysis. Cambridge University Press, CambridgeGoogle Scholar
  55. Möst D, Keles D (2010) A survey of stochastic modelling approaches for liberalised electricity markets. Eur J Oper Res 207(2):543–556.  https://doi.org/10.1016/j.ejor.2009.11.007 CrossRefGoogle Scholar
  56. Mustajoki J, Hämäläinen RP, Lindstedt MR (2006) Using intervals for global sensitivity and worst-case analyses in multiattribute value trees. Eur J Oper Res 174(1):278–292.  https://doi.org/10.1016/j.ejor.2005.02.070 CrossRefGoogle Scholar
  57. Nürnberg R, Römisch W (2002) A two-stage planning model for power scheduling in a hydro-thermal system under uncertainty. Optim Eng 3(4):355–378.  https://doi.org/10.1023/A:1021531823935 CrossRefGoogle Scholar
  58. Owens B (2014) The rise of distributed power. General electric (ecomagination). https://www.ge.com/sites/default/files/2014%2002%20Rise%20of%20Distributed%20Power.pdf. Zugegriffen 30 Sept 2016
  59. Pagès-Bernaus A, Pérez-Valdés G, Tomasgard A (2015) A parallelised distributed implementation of a branch and fix coordination algorithm. Eur J Oper Res 244(1):77–85.  https://doi.org/10.1016/j.ejor.2015.01.004 CrossRefGoogle Scholar
  60. Pflug GC, Römisch W (2007) Modeling, measuring and managing risk. World Scientific, HackensackCrossRefGoogle Scholar
  61. Rios LM, Sahinidis NV (2013) Derivative-free optimization—a review of algorithms and comparison of software implementations. J Glob Optim 56(3):1247–1293.  https://doi.org/10.1007/s10898-012-9951-y CrossRefGoogle Scholar
  62. Ritzenhoff P (2006) Erstellung eines Modells zur Simulation der Solarstrahlung auf beliebig orientierte Flächen und deren Trennung in Diffus-und Direktanteil. Forschungszentrum Jülich, ZentralbibliothekGoogle Scholar
  63. Rockafellar RT, Wets RJ-B (1991) Scenarios and policy aggregation in optimization under uncertainty. Math OR 16(1):119–147.  https://doi.org/10.1287/moor.16.1.119 CrossRefGoogle Scholar
  64. Ruszczynski A (1999) Some advances in decomposition methods for stochastic linear programming. Ann Oper Res 85:153–172.  https://doi.org/10.1023/A:1018965626303 CrossRefGoogle Scholar
  65. Ruszczyński A, Świȩtanowski A (1997) Accelerating the regularized decomposition method for two stage stochastic linear problems. Eur J Oper Res 101(2):328–342.  https://doi.org/10.1016/S0377-2217(96)00401-8 CrossRefGoogle Scholar
  66. Schermeyer H, Bertsch V, Fichtner W (2015) Review and extension of suitability assessment indicators of weather model output for analyzing decentralized energy systems. Atmosphere 6(12):1871–1888.  https://doi.org/10.3390/atmos6121835 CrossRefGoogle Scholar
  67. Schicktanz MD, Wapler J, Henning H-M (2011) Primary energy and economic analysis of combined heating, cooling and power systems. Energy 36(1):575–585.  https://doi.org/10.1016/j.energy.2010.002 CrossRefGoogle Scholar
  68. Schlesinger M, Hofer P, Kemmler A, Kirchner A, Strassburg S, Lindenberger D, Lutz C (2010) Energieszenarien für ein Energiekonzept der Bundesregierung. Projekt Nr. 12/10 des Bundesministeriums für Wirtschaft und Technologie; Studie. Prognos, Basel, Köln, OsnabrückGoogle Scholar
  69. Schultz R (1995) On structure and stability in stochastic programs with random technology matrix and complete integer recourse. Math Program 70(1–3):73–89.  https://doi.org/10.1007/BF01585929 Google Scholar
  70. Schultz R (2003) Stochastic programming with integer variables. Math Program 97(1):285–309.  https://doi.org/10.1007/s10107-003-0445-z CrossRefGoogle Scholar
  71. Sen S, Sherali HD (2006) Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming. Math Program 106(2):203–223.  https://doi.org/10.1007/s10107-005-0592-5 CrossRefGoogle Scholar
  72. Shang C, Srinivasan D, Reindl T (2017) Generation and storage scheduling of combined heat and power. Energy 124:693–705.  https://doi.org/10.1016/j.energy.2017.02.038 CrossRefGoogle Scholar
  73. Shapiro A, Dentcheva D, Ruszczynski AP (2009) Lectures on stochastic programming: modeling and theory, vol 9. MPS-SIAM Series on Optimization, SIAM, PhiladelphiaCrossRefGoogle Scholar
  74. Sherali HD, Fraticelli BM (2002) A modification of benders decomposition algorithm for discrete subproblems: an approach for stochastic programs with integer recourse. J Glob Optim 22(1/4):319–342.  https://doi.org/10.1023/A:1013827731218 CrossRefGoogle Scholar
  75. Sherali HD, Smith JC (2009) Two-stage stochastic hierarchical multiple risk problems. Models and algorithms. Math Program 120(2):403–427.  https://doi.org/10.1007/s10107-008-0220-2 CrossRefGoogle Scholar
  76. Shirazi E, Jadid S (2017) Cost reduction and peak shaving through domestic load shifting and DERs. Energy 124:146–159.  https://doi.org/10.1016/j.energy.2017.01.148 CrossRefGoogle Scholar
  77. Silveira JL, Tuna CE (2003) Thermoeconomic analysis method for optimization of combined heat and power systems. Part I. Prog Energy Combust Sci 29(6):479–485.  https://doi.org/10.1016/S0360-1285(03)00041-8 CrossRefGoogle Scholar
  78. Syed A (2010) Australian energy projections to 2029-30. ABARE, Canberra, A.C.TGoogle Scholar
  79. Taborda D, Zdravkovic L (2012) Application of a hill-climbing technique to the formulation of a new cyclic nonlinear elastic constitutive model. Comput Geotech 43:80–91.  https://doi.org/10.1016/j.compgeo.2012.02.001 CrossRefGoogle Scholar
  80. Till J, Sand G, Urselmann M, Engell S (2007) A hybrid evolutionary algorithm for solving two-stage stochastic integer programs in chemical batch scheduling. Comput Chem Eng 31(5–6):630–647.  https://doi.org/10.1016/j.compchemeng.2006.09.003 CrossRefGoogle Scholar
  81. van Slyke RM, Wets R (1969) L-shaped linear programs with applications to optimal control and stochastic programming. SIAM J Appl Math 17(4):638–663.  https://doi.org/10.1137/0117061 CrossRefGoogle Scholar
  82. VDI 4655 guideline (2008) Reference load profiles of single-family and multi-family houses for the use of CHP systems. VDI Guideline 4655. Verein Deutscher Ingenieure (VDI), DüsseldorfGoogle Scholar
  83. Ventosa M, Baıllo Á, Ramos A, Rivier M (2005) Electricity market modeling trends. Energy Policy 33(7):897–913.  https://doi.org/10.1016/j.enpol.2003.10.013 CrossRefGoogle Scholar
  84. Verderame PM, Elia JA, Li J, Floudas CA (2010) Planning and scheduling under uncertainty: a review across multiple sectors. Ind Eng Chem Res 49(9):3993–4017.  https://doi.org/10.1021/ie902009k CrossRefGoogle Scholar
  85. Vögele S, Kuckshinrichs W, Markewitz P (2009) A hybrid IO energy model to analyze CO\(_{2}\) reduction policies: a case of Germany. In: Tukker A, Suh S (eds) Handbook of input-output economics in industrial ecology, vol 23. Springer, Dordrecht, pp 337–356CrossRefGoogle Scholar
  86. Wald A (1945) Statistical decision functions which minimize the maximum risk. Ann Math 46(2):265.  https://doi.org/10.2307/1969022 CrossRefGoogle Scholar
  87. Wallace SW, Fleten S-E (2003) Stochastic programming models in energy stochastic programming, vol 10. Elsevier, Amsterdam, pp 637–677Google Scholar
  88. Yazdanie M, Densing M, Wokaun A (2016) The role of decentralized generation and storage technologies in future energy systems planning for a rural agglomeration in Switzerland. Energy Policy 96:432–445.  https://doi.org/10.1016/j.enpol.2016.06.010 CrossRefGoogle Scholar
  89. Yuan Y, Sen S (2009) Enhanced cut generation methods for decomposition-based branch and cut for two-stage stochastic mixed-integer programs. INFORMS J Comput 21(3):480–487.  https://doi.org/10.1287/ijoc.1080.0300 CrossRefGoogle Scholar
  90. Zhou M (1998) Fuzzy logic and optimization models for implementing QFD. Comput Ind Eng 35(1–2):237–240.  https://doi.org/10.1016/S0360-8352(98)00073-4 CrossRefGoogle Scholar
  91. Zhu X (2006) Discrete two-stage stochastic mixed-integer programs with applications to airline fleet assignment and workforce planning problems. Dissertation, Virginia Polytechnic Institute and State UniversityGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Karlsruhe Institute of Technology (KIT), Institute for Industrial Production (IIP), Chair of Energy EconomicsKarlsruheGermany
  2. 2.Economic and Social Research Institute (ESRI)Dublin 2Ireland
  3. 3.Department of EconomicsTrinity College DublinDublin 2Ireland

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