Energy Systems

, Volume 2, Issue 3–4, pp 209–242 | Cite as

Modeling and solving a large-scale generation expansion planning problem under uncertainty

  • Shan Jin
  • Sarah M. Ryan
  • Jean-Paul Watson
  • David L. Woodruff
Original Paper

Abstract

We formulate a generation expansion planning problem to determine the type and quantity of power plants to be constructed over each year of an extended planning horizon, considering uncertainty regarding future demand and fuel prices. Our model is expressed as a two-stage stochastic mixed-integer program, which we use to compute solutions independently minimizing the expected cost and the Conditional Value-at-Risk; i.e., the risk of significantly larger-than-expected operational costs. We introduce stochastic process models to capture demand and fuel price uncertainty, which are in turn used to generate trees that accurately represent the uncertainty space. Using a realistic problem instance based on the Midwest US, we explore two fundamental, unexplored issues that arise when solving any stochastic generation expansion model. First, we introduce and discuss the use of an algorithm for computing confidence intervals on obtained solution costs, to account for the fact that a finite sample of scenarios was used to obtain a particular solution. Second, we analyze the nature of solutions obtained under different parameterizations of this method, to assess whether the recommended solutions themselves are invariant to changes in costs. The issues are critical for decision makers who seek truly robust recommendations for generation expansion planning.

Keywords

Generation expansion planning Stochastic programming Scenario generation Multiple replication procedure Solution stability 

References

  1. 1.
    Ahmed, S.: Introduction to stochastic integer programming. http://www.stoprog.org/SPIntro/intro2sip.html (2009)
  2. 2.
    Ahmed, S., Sahinidis, N.: An approximation scheme for stochastic integer programs arising in capacity expansion. Oper. Res. 51(3), 461–471 (2003) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Ahmed, S., King, A., Parija, G.: A multi-stage stochastic integer programming approach for capacity expansion under uncertainty. J. Glob. Optim. 26, 3–24 (2003) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Albornoz, V., Benario, P., Rojas, M.: A two-stage stochastic integer programming model for a thermal power system expansion. Int. Trans. Oper. Res. 11, 243–257 (2004) MATHCrossRefGoogle Scholar
  5. 5.
    AMPL: A modeling language for mathematical programming. http://www.ampl.com (2010)
  6. 6.
    Anderson, T.: An Introduction to Multivariate Statistical Analysis, 3rd edn. Wiley, New York (2003) MATHGoogle Scholar
  7. 7.
    Annual Energy Outlook 2009: Tech. rep., Energy Information Administration, Department of Energy (2009) Google Scholar
  8. 8.
    Booth, R.: Optimal generation planning considering uncertainty. IEEE Trans. Power Appar. Syst. PAS-91, 70–77 (1972) CrossRefGoogle Scholar
  9. 9.
    Chuang, A., Wu, F., Varaiya, P.: A game-theoretic model for generation expansion planning: problem formulation and numerical comparisons. IEEE Trans. Power Syst. 16(4), 885–891 (2001) CrossRefGoogle Scholar
  10. 10.
    COIN-OR: COmputational INfrastructure for Operations Research. http://www.coin-or.org (2010)
  11. 11.
    CPLEX: http://www.cplex.com (2010)
  12. 12.
    DeMeo, E., Grant, W., Milligan, M., Schuerger, M.: Wind plant generation. IEEE Power Energy Mag. 3(6), 38–46 (2005) CrossRefGoogle Scholar
  13. 13.
    Denny, E., O’Malley, M.: Wind generation power system operation, and emissions reduction. IEEE Trans. Power Syst. 21(1), 341–347 (2006) CrossRefGoogle Scholar
  14. 14.
    Dentcheva, D., Romisch, W.: Optimal power generation under uncertainty via stochastic programming. Stoch. Program. Methods Tech. Appl., pp. 22–56 (1998) Google Scholar
  15. 15.
    Doherty, R., Outhred, H., O’Malley, M.: Establishing the role that wind generation may have in future generation portfolios. IEEE Trans. Power Syst. 21(3), 1415–1422 (2006) CrossRefGoogle Scholar
  16. 16.
    Dupacova, J., Consigli, G., Wallace, S.: Scenarios for multistage stochastic programs. Ann. Oper. Res. 100, 25–53 (2000) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Eichhorn, A., Heitsch, H., Romisch, W.: Stochastic optimization of electricity portfolios: scenario tree modeling and risk management. In: Rebennack, S., Pardalos, P., Pereira, M., Iliadis, N. (eds.) Handbook of Power Systems, vol. II, pp. 405–432. Springer, Berlin (2010) CrossRefGoogle Scholar
  18. 18.
    Electric Generation Expansion Analysis System (EGEAS): Tech. rep., Electric Power Research Institute (2009) Google Scholar
  19. 19.
    Firmo, H., Legey, L.: Generation expansion planning: an iterative genetic algorithm approach. IEEE Trans. Power Syst. 17(3), 901–906 (2002) CrossRefGoogle Scholar
  20. 20.
    Fukuyama, Y., Chiang, H.: A parallel genetic algorithm for generation expansion planning. IEEE Trans. Power Syst. 11(2), 955–961 (1996) CrossRefGoogle Scholar
  21. 21.
    Garcia-Gonzalez, J., de la Muela, R., Santos, L., Gonzalez, A.: Stochastic joint optimization of wind generation and pumped-storage units in an electricity market. IEEE Trans. Power Syst. 23(2), 460–468 (2008) CrossRefGoogle Scholar
  22. 22.
    Growe-Kruska, N., Heitsch, H., Romisch, W.: Scenario reduction and scenario tree construction for power management problems. In: IEEE Bologna Power Tech. Conference, Bologna, Italy (2003) Google Scholar
  23. 23.
    Holmes, D.: A collection of stochastic programming problems. Tech. rep. (1994) Google Scholar
  24. 24.
    Høyland, K., Wallace, S.: Generating scenario tree for multistage decision problems. Manag. Sci. 47(2), 295–307 (2001) CrossRefGoogle Scholar
  25. 25.
  26. 26.
  27. 27.
  28. 28.
  29. 29.
  30. 30.
  31. 31.
  32. 32.
  33. 33.
  34. 34.
  35. 35.
    International Energy Outlook 2009: Tech. rep., Energy Information Administration, Department of Energy (2009) Google Scholar
  36. 36.
    Johnson, N., Kotz, S.: Continuous Univariate Distributions, 2nd edn. Wiley Series in Probability and Mathematical Statistics, vol. 1. Wiley, New York (1994). Chap. 3: Lognormal distribution MATHGoogle Scholar
  37. 37.
    Joint Coordinated System Planning Report 2008: Tech. rep. (2009) Google Scholar
  38. 38.
    Kanna, S., Slochanal, S., Padhy, N.: Application and comparison of metaheuristic techniques to generation expansion planning problem. IEEE Trans. Power Syst. 20(1), 466–475 (2005) CrossRefGoogle Scholar
  39. 39.
    Karaki, S., Chaaban, F., Al-Nakhl, N., Tarhini, K.: Power generation expansion planning with environmental consideration for Lebanon. Int. J. Electr. Power Energy Syst. 24, 611–619 (2002) CrossRefGoogle Scholar
  40. 40.
    Karki, R., Billinton, R.: Cost-effective wind energy utilization for reliable power supply. IEEE Trans. Energy Convers. 19(2), 435–440 (2004) CrossRefGoogle Scholar
  41. 41.
    Laurent, A.: A scenario generation algorithm for multistage stochastic programming: application for asset allocation models with derivatives. Ph.D. thesis, University of Lugano, Lugano, Switzerland (2006) Google Scholar
  42. 42.
    Lund, H.: Large-scale integration of wind power into different energy systems. Energy 30(13), 2402–2412 (2005) CrossRefGoogle Scholar
  43. 43.
    Mahalanobis, P.: On the generalised distance in statistics. In: Proceedings of the National Institute of Sciences of India, vol. 2, pp. 49–55 (1936) Google Scholar
  44. 44.
    Mak, W., Morton, D., Wood, R.: Monte Carlo bounding techniques for determining solution quality in stochastic programs. Oper. Res. Lett. 24, 47–56 (1999) MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Malcolm, S., Zenios, S.: Robust optimization for power systems capacity expansion under uncertainty. J. Oper. Res. Soc. 45(9), 1040–1049 (1994) MATHGoogle Scholar
  46. 46.
    Marathe, R., Ryan, S.: On the validity of the geometric Brownian motion assumption. Eng. Econ. 50(2), 159–192 (2005) CrossRefGoogle Scholar
  47. 47.
    McCalley, J.: Introduction to electric systems expansion planning. http://home.eng.iastate.edu/~jdm/ee590/PlanningIntro.pdf (2008)
  48. 48.
    McLachlan, G.: Discriminant Analysis and Statistical Pattern Recognition. Wiley Interscience, New York (1992) CrossRefGoogle Scholar
  49. 49.
    Meza, J., Yildirim, M., Masud, A.: A model for the multiperiod multiobjective power generation expansion problem. IEEE Trans. Power Syst. 22(2) (2007) Google Scholar
  50. 50.
    Milligan, M.: Measuring wind plant capacity value. Tech. rep., National Renewable Energy Laboratory, Colorado (1996) Google Scholar
  51. 51.
    Milligan, M.: Variance estimates of wind plant capacity credit. Tech. rep., National Renewable Energy Laboratory, Colorado (1996) Google Scholar
  52. 52.
    Milligan, M.: Modeling utility-scale wind power plants. Part 2: Capacity credit. Wind Energy 3, 106–206 (2000) CrossRefGoogle Scholar
  53. 53.
    Mo, B., Hegge, J., Wangensteen, I.: Stochastic generation expansion planning by means of stochastic dynamic programming. IEEE Trans. Power Syst. 6(2), 662–668 (1991) CrossRefGoogle Scholar
  54. 54.
    Mulvey, J., Vanderbei, R., Zenios, S.: Robust optimization of large-scale systems. Oper. Res. 43(2), 264–281 (1995) MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Paulun, T., Haubrich, H.J.: Long-term and expansion planning for electrical networks considering uncertainties. In: Rebennack, S., Pardalos, P., Pereira, M., Iliadis, N. (eds.) Handbook of Power Systems, vol. I, pp. 391–408. Springer, Berlin (2010) CrossRefGoogle Scholar
  56. 56.
    PySP: PySP: Python-based stochastic programming. https://software.sandia.gov/trac/coopr/wiki/PySP (2011)
  57. 57.
    Rockafellar, R.: Coherent approaches to risk in optimization under uncertainty. In: Tutorials in Operation Research. INFORMS Annual Meeting (2007) Google Scholar
  58. 58.
    Rockafellar, R., Uryasev, S.: Optimization of Conditional Value-at-Risk. J. Risk 2, 21–41 (2000) Google Scholar
  59. 59.
    Rockafellar, R.T., Wets, R.J.B.: Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16(1), 119–147 (1991) MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Ross, S.: Brownian motion and stationary processes. In: Introduction to Probability Models, 9th edn. Elsevier, Amsterdam (2007). Chapter 10 Google Scholar
  61. 61.
    Ryan, S., McCalley, J., Woodruff, D.: Long term resource planning for electric power systems under uncertainty. Technical report, Iowa State University (2011) Google Scholar
  62. 62.
    Sahinidis, N.: Optimization under uncertainty: state-of-the-art and opportunities. Comput. Chem. Eng. 28, 971–983 (2004) CrossRefGoogle Scholar
  63. 63.
    Schultz, R., Tiedemann, S.: Conditional Value-at-Risk in stochastic programs with mixed-integer recourse. Math. Program. 105(2–3), 365–386 (2005) MathSciNetGoogle Scholar
  64. 64.
    Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming: Modeling and Theory. Society for Industrial and Applied Mathematics, Philadelphia (2009) MATHCrossRefGoogle Scholar
  65. 65.
    Slyke, R.M.V., Wets, R.J.: L-shaped linear programs with applications to optimal control and stochastic programming. SIAM J. Appl. Math. 17, 638–663 (1969) MathSciNetMATHCrossRefGoogle Scholar
  66. 66.
    Takriti, S., Ahmed, S.: On robust optimization of two-stage systems. Math. Program., Ser. A 99, 106–126 (2004) MathSciNetCrossRefGoogle Scholar
  67. 67.
    Voorspools, K., D’haeseleer, W.: An analytical formula for the capacity credit of wind power. Renew. Energy 31, 45–54 (2006) CrossRefGoogle Scholar
  68. 68.
    Voropai, N., Ivanova, E.: Multi-criteria decision analysis techniques in electric power system expansion planning. Int. J. Electr. Power Energy Syst. 24, 71–78 (2002) CrossRefGoogle Scholar
  69. 69.
    Watson, J., Murray, R., Hart, W.: Formulation and optimization of robust sensor placement problems for drinking water contamination warning systems. J. Infrastruct. Syst. 15(4), 330–339 (2009) CrossRefGoogle Scholar
  70. 70.
    Watson, J.P., Woodruff, D., Hart, W.: Modeling and solving stochastic programs in Python. Math. Program. Comput. (2011) (to appear) Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Shan Jin
    • 1
  • Sarah M. Ryan
    • 1
  • Jean-Paul Watson
    • 2
  • David L. Woodruff
    • 3
  1. 1.Department of Industrial and Manufacturing Systems EngineeringIowa State UniversityAmesUSA
  2. 2.Discrete Math and Complex Systems DepartmentSandia National LaboratoriesAlbuquerqueUSA
  3. 3.Graduate School of ManagementUniversity of California DavisDavisUSA

Personalised recommendations