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Energy Systems

, Volume 9, Issue 2, pp 223–256 | Cite as

Risk-averse capacity planning for renewable energy production

  • Bo Sun
  • Pavlo Krokhmal
  • Yong Chen
Original Paper
  • 154 Downloads

Abstract

This paper considers the problem of capacity planning and operation of energy grids where the power demands are served from renewable energy sources, such as wind farms, and the transmission network is represented by the high-voltage direct current (HVDC) lines. The principal question considered in this work is whether a risk-averse design of the grid, including the selection of wind farm locations and assignment of power delivery from wind farms to customers, would allow for effective hedging of the risks associated with uncertainties in power demand and production of energy from renewable sources. To this end, the problem is formulated in the general context of supply chain/facility location, with both the supply and the demand being stochastic variates. Several stochastic optimization models are presented and analyzed, including the traditional risk-neutral, or expectation-based model and risk-averse models based on linear and nonlinear coherent measures of risk. Exact solutions algorithms that employ Benders decomposition and polyhedral approximations of nonlinear constraints have been proposed for the obtained linear and nonlinear mixed-integer programming problems. The conducted numerical experiments illustrate the properties of the constructed models, as well as the efficiency of the developed algorithms.

Keywords

Capacity planning Facility location Stochastic supply Coherent measures of risk Benders decomposition Mixed integer p-order cone programming 

Notes

Acknowledgements

This work was supported in part by the DTRA Grant HDTRA1-14-1-0065 and NSF Grant DMI 0457473.

References

  1. 1.
    Abbey, C., Joós, G.: A stochastic optimization approach to rating of energy storage systems in wind-diesel isolated grids. Power Syst. IEEE Trans. 24(1), 418–426 (2009)CrossRefGoogle Scholar
  2. 2.
    Aksoy, H., Toprak, Z.F., Aytek, A., Ünal, N.E.: Stochastic generation of hourly mean wind speed data. Renew. Energy 29(14), 2111–2131 (2004)CrossRefGoogle Scholar
  3. 3.
    Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Mathematical finance 9(3), 203–228 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Azaron, A., Brown, K., Tarim, S., Modarres, M.: A multi-objective stochastic programming approach for supply chain design considering risk. Int. J. Prod. Econ. 116(1), 129–138 (2008)CrossRefGoogle Scholar
  5. 5.
    Baghalian, A., Rezapour, S., Farahani, R.Z.: Robust supply chain network design with service level against disruptions and demand uncertainties: A real-life case. Eur. J. Oper. Res. 227(1), 199–215 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baron, O., Milner, J., Naseraldin, H.: Facility location: a robust optimization approach. Prod. Oper. Manag. 20(5), 772–785 (2011)CrossRefGoogle Scholar
  7. 7.
    Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization: analysis, algorithms, and engineering applications, MPS/SIAM Series on Optimization, vol 2. SIAM (2001)Google Scholar
  8. 8.
    Ben-Tal, A., Nemirovski, A.: On polyhedral approximations of the second-order cone. Math. Oper. Res. 26(2), 193–205 (2001b)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Numerische mathematik 4(1), 238–252 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Berman, O., Drezner, Z.: The p-median problem under uncertainty. Eur. J. Oper. Res. 189(1), 19–30 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Berman, O., Krass, D., Menezes, M.B.: Facility reliability issues in network p-median problems: strategic centralization and co-location effects. Oper. Res. 55(2), 332–350 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming, 2nd edn. Springer, New York (2011)Google Scholar
  13. 13.
    Burke, D.J., O’Malley, M.: Optimal wind power location on transmission systems-a probabilistic load flow approach. In: Probabilistic Methods Applied to Power Systems, 2008. PMAPS’08. Proceedings of the 10th International Conference, pp. 1–8. IEEE (2008)Google Scholar
  14. 14.
    Burton, T., Jenkins, N., Sharpe, D., Bossanyi, E.: Wind energy handbook. Wiley, New York (2011)Google Scholar
  15. 15.
    Carpentier, J.: Contribution a letude du dispatching economique. Bulletin de la Societe Francaise des Electriciens 3(1), 431–447 (1962)Google Scholar
  16. 16.
    Chen, G., Daskin, M.S., Shen, Z.-J.M., Uryasev, S.: The \(\alpha \)-reliable mean-excess regret model for stochastic facility location modeling. Nav. Res. Logist. (NRL) 53(7), 617–626 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Clauset, A., Shalizi, C.R., Newman, M.E.: Power-law distributions in empirical data. SIAM Rev. 51(4), 661–703 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Conserve Energy Future.: Cost of Wind Energy (2013). http://www.conserve-energy-future.com/WindEnergyCost.php. Online. Accessed 12 Nov 2013
  19. 19.
    Cui, T., Ouyang, Y., Shen, Z.-J.M.: Reliable facility location design under the risk of disruptions. Oper. Res. 58(4-part-1), 998–1011 (2010)Google Scholar
  20. 20.
    Daskin, M.S., Hesse, S.M., Revelle, C.S.: \(\alpha \)-reliable p-minimax regret: a new model for strategic facility location modeling. Locat. Sci. 5(4), 227–246 (1997)CrossRefzbMATHGoogle Scholar
  21. 21.
    Dommel, H.W., Tinney, W.F.: Optimal power flow solutions. IEEE Trans. Power Apparatus Syst. 10, 1866–1876 (1968)CrossRefGoogle Scholar
  22. 22.
    Drezner, Z.: Heuristic solution methods for two location problems with unreliable facilities. J. Oper. Res. Soc. 38(6), 509–514 (1987)Google Scholar
  23. 23.
    Ekren, O., Ekren, B.Y.: Size optimization of a PV/wind hybrid energy conversion system with battery storage using simulated annealing. Appl. Energy 87(2), 592–598 (2010)CrossRefGoogle Scholar
  24. 24.
    Garcia, A., Torres, J., Prieto, E., De Francisco, A.: Fitting wind speed distributions: a case study. Sol. Energy 62(2), 139–144 (1998)CrossRefGoogle Scholar
  25. 25.
    Ghiani, G., Laporte, G., Musmanno, R.: Introduction to logistics systems planning and control. Wiley, New York (2004)zbMATHGoogle Scholar
  26. 26.
    Glass, R., Beyeler, W., Stamber, K., Glass, L., LaViolette, R., Conrad, S., Brodsky, N., Brown, T., Scholand, A., Ehlen, M.: Simulation and analysis of cascading failure in critical infrastructure. Technical report, Sandia National Laboratories (2005)Google Scholar
  27. 27.
    Katsigiannis, Y., Georgilakis, P.: Optimal sizing of small isolated hybrid power systems using tabu search. J. Optoelectron. Adv. Mater. 10(5), 1241 (2008)Google Scholar
  28. 28.
    Kouvelis, P., Yu, G.: Robust discrete optimization and its applications. Kluwer Academic Publishers, Dordrecht (1997)CrossRefzbMATHGoogle Scholar
  29. 29.
    Krokhmal, P., Zabarankin, M., Uryasev, S.: Modeling and optimization of risk. Surv. Oper. Res. Manag. Sci. 16(2), 49–66 (2011)Google Scholar
  30. 30.
    Krokhmal, P.A.: Higher moment coherent risk measures. Quant. Finance 7(4), 373–387 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Krokhmal, P.A., Soberanis, P.: Risk optimization with \(p\)-order conic constraints: a linear programming approach. Eur. J. Oper. Res. 201(3), 653–671 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kuznia, L., Zeng, B., Centeno, G., Miao, Z.: Stochastic optimization for power system configuration with renewable energy in remote areas. Ann. Oper. Res. 210(1), 411–432 (2013)Google Scholar
  33. 33.
    Lopez, A., Roberts, B., Heimiller, D., Blair, N., Porro, G.: U.S. renewable energy technical potentials: a GIS-based analysis. Technical report, National Renewable Energy Laboratory (2012)Google Scholar
  34. 34.
    Louveaux, F.: Discrete stochastic location models. Ann. Oper. Res. 6(2), 21–34 (1986)CrossRefGoogle Scholar
  35. 35.
    Lu, M., Ran, L., Shen, Z.-J.M.: Reliable facility location design under uncertain correlated disruptions. Manuf. Serv. Oper. Manag. 17(4), 445–455 (2015)CrossRefGoogle Scholar
  36. 36.
    Mei, S., Zhang, X., Cao, M.: Power grid complexity. Springer, Berlin (2011)Google Scholar
  37. 37.
    Melo, M.T., Nickel, S., Saldanha-Da-Gama, F.: Facility location and supply chain management—a review. Eur. J. Oper. Res. 196(2), 401–412 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Mirchandani, P.B., Oudjit, A.: Localizing 2-medians on probabilistic and deterministic tree networks. Networks 10(4), 329–350 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Mirchandani, P.B., Oudjit, A., Wong, R.T.: ’Multidimensional’ extensions and a nested dual approach for the m-median problem. Eur. J. Oper. Res. 21(1), 121–137 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Morenko, Y., Vinel, A., Yu, Z., Krokhmal, P.: On \(p\)-cone linear discrimination. Eur. J. Oper. Res. 231, 784–789 (2013)CrossRefzbMATHGoogle Scholar
  41. 41.
    Nesterov, Y., Nemirovskii, A., Ye, Y.: Interior-point polynomial algorithms in convex programming, Studies in Applied Mathematics, vol. 13. SIAM, Philadelphia (1994)Google Scholar
  42. 42.
    Owen, S.H., Daskin, M.S.: Strategic facility location: a review. Eur. J. Oper. Res. 111(3), 423–447 (1998)CrossRefzbMATHGoogle Scholar
  43. 43.
    Peng, P., Snyder, L.V., Lim, A., Liu, Z.: Reliable logistics networks design with facility disruptions. Transp. Res. Part B Methodol. 45(8), 1190–1211 (2011)CrossRefGoogle Scholar
  44. 44.
    Prékopa, A.: Stochastic Programming. Kluwer Academic Publishers, Chichester (1995)Google Scholar
  45. 45.
    Rahmaniani, R., Crainic, T.G., Gendreau, M., Rei, W.: The benders decomposition algorithm: a literature review. Eur. J. Oper. Res. 259(3), 801–817 (2017)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)CrossRefGoogle Scholar
  47. 47.
    Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26(7), 1443–1471 (2002)CrossRefGoogle Scholar
  48. 48.
    Roques, F., Hiroux, C., Saguan, M.: Optimal wind power deployment in Europea portfolio approach. Energy Policy 38(7), 3245–3256 (2010)CrossRefGoogle Scholar
  49. 49.
    Santoso, T., Ahmed, S., Goetschalckx, M., Shapiro, A.: A stochastic programming approach for supply chain network design under uncertainty. Eur. J. Oper. Res. 167(1), 96–115 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Schütz, P., Tomasgard, A., Ahmed, S.: Supply chain design under uncertainty using sample average approximation and dual decomposition. Eur. J. Oper. Res. 199(2), 409–419 (2009)CrossRefzbMATHGoogle Scholar
  51. 51.
    Schwartz, M., Heimiller, D., Haymes, S., Walt, M.: Assessment of offshore wind energy resources for the United States. Technical report, National Renewable Energy Laboratory (2010)Google Scholar
  52. 52.
    Scott, W.R., Powell, W.B.: Approximate dynamic programming for energy storage with new results on instrumental variables and projected Bellman errors. Oper. Res. (2012, under review)Google Scholar
  53. 53.
    Senjyu, T., Hayashi, D., Yona, A., Urasaki, N., Funabashi, T.: Optimal configuration of power generating systems in isolated island with renewable energy. Renew. Energy 32(11), 1917–1933 (2007)CrossRefGoogle Scholar
  54. 54.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM-MPS, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  55. 55.
    Shapiro, J.F.: Modeling the Supply Chain. Duxbury Press, Pacific Grove (2001)Google Scholar
  56. 56.
    Shen, Z.-J.M., Zhan, R.L., Zhang, J.: The reliable facility location problem: formulations, heuristics, and approximation algorithms. INFORMS J. Comput. 23(3), 470–482 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Sheppard, E.: A conceptual framework for dynamic location-allocation analysis. Environ. Plan. A 6(5), 547–564 (1974)CrossRefGoogle Scholar
  58. 58.
    Simchi-Levi, D., Kaminsky, P., Simchi-Levi, E.: Managing the Supply Chain: The Definitive Guide for the Supply Chain Professional. McGraw-Hill, New York (2004)zbMATHGoogle Scholar
  59. 59.
    Snyder, L.V.: Facility location under uncertainty: a review. IIE Trans. 38(7), 547–564 (2006)CrossRefGoogle Scholar
  60. 60.
    Snyder, L.V., Atan, Z., Peng, P., Rong, Y., Schmitt, A.J., Sinsoysal, B.: OR/MS models for supply chain disruptions: a review. IIE Trans. 48(2), 89–109 (2016)Google Scholar
  61. 61.
    Snyder, L.V., Daskin, M.S.: Reliability models for facility location: the expected failure cost case. Transp. Sci. 39(3), 400–416 (2005)CrossRefGoogle Scholar
  62. 62.
    Snyder, L.V., Daskin, M.S.: Stochastic p-robust location problems. IIE Trans. 38(11), 971–985 (2006)CrossRefGoogle Scholar
  63. 63.
    Vielma, J.P., Ahmed, S., Nemhauser, G.L.: A lifted linear programming branch-and-bound algorithm for mixed-integer conic quadratic programs. INFORMS J. Comput. 20(3), 438–450 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Vinel, A., Krokhmal, P.: Polyhedral approximations in \(p\)-order cone programming. Optim. Methods Softw. 29(6), 1210–1237 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Vinel, A., Krokhmal, P.: Mixed integer programming with a class of nonlinear convex constraints. Discrete Optim. 24, 66–86 (2017)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Weaver, J.R., Church, R.L.: Computational procedures for location problems on stochastic networks. Transp. Sci. 17(2), 168–180 (1983)CrossRefGoogle Scholar
  67. 67.
    World Wind Energy Association.: 2014 Half-year Report. Technical report, World Wind Energy Association (2014)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringUniversity of IowaIowa CityUSA
  2. 2.Department of Systems and Industrial EngineeringUniversity of ArizonaTucsonUSA

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