Energy Systems

, Volume 10, Issue 3, pp 517–541 | Cite as

Robust optimization vs. stochastic programming incorporating risk measures for unit commitment with uncertain variable renewable generation

  • Narges KazemzadehEmail author
  • Sarah M. Ryan
  • Mahdi Hamzeei
Original Paper


Unit commitment seeks the most cost effective generator commitment schedule for an electric power system to meet net load, defined as the difference between the load and the output of renewable generation, while satisfying the operational constraints on transmission system and generation resources. Stochastic programming and robust optimization are the most widely studied approaches for unit commitment under net load uncertainty. We incorporate risk considerations in these approaches and investigate their comparative performance for a multi-bus power system in terms of economic efficiency as well as the risk associated with the commitment decisions. We explicitly account for risk, via Conditional Value at Risk (CVaR) in the stochastic programming objective function, and by employing a CVaR-based uncertainty set in the robust optimization formulation. The numerical results indicate that the stochastic program with CVaR evaluated in a low-probability tail is able to achieve better cost-risk trade-offs than the robust formulation with less conservative preferences. The CVaR-based uncertainty set with the most conservative parameter settings outperforms an uncertainty set based only on ranges.


Unit commitment Renewable energy Stochastic programming Robust optimization CVaR 



Funding was provided by Iowa Energy Center (Grant no. OG-14-014).


  1. 1.
  2. 2.
    Renewable Energy Analysis Laboratory, Department of Electrical Engineering, University of Washington.
  3. 3.
    van Ackooij, W.: A comparison of four approaches from stochastic programming for large-scale unit-commitment. EURO J. Comput. Optim., 5, 1–29 (2015).
  4. 4.
    An, Y., Zeng, B.: Exploring the modeling capacity of two-stage robust optimization: variants of robust unit commitment model. IEEE Trans. Power Syst. 30(1), 109–122 (2015)Google Scholar
  5. 5.
    Asensio, M., Contreras, J.: Stochastic unit commitment in isolated systems with renewable penetration under CVaR assessment. IEEE Trans. Smart Grid 7(3), 1356–1367 (2016)Google Scholar
  6. 6.
    Bertsimas, D., Brown, D.B.: Constructing uncertainty sets for robust linear optimization. Oper. Res. 57, 1483–1495 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bertsimas, D., Litvinov, E., Sun, X.A., Zhao, J., Zheng, T.: Adaptive robust optimization for the security constrained unit commitment problem. IEEE Trans. Power Syst. 28, 52–63 (2013)Google Scholar
  8. 8.
    Bouffard, F., Galiana, F., Conejo, A.: Market-clearing with stochastic security. Part I: formulation. IEEE Trans. Power Syst. 20(4), 1818–1826 (2005)Google Scholar
  9. 9.
    Bukhsh, W., Papakonstantinou, A., Pinson, P.: A robust optimisation approach using CVaR for unit commitment in a market with probabilistic offers.
  10. 10.
    Carrion, M., Arroyo, J.M.: A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem. IEEE Trans. Power Syst. 21, 1371–1378 (2006)Google Scholar
  11. 11.
    Cheung, K., Gade, D., Silva-Monroy, C., Ryan, S.M., Watson, J.P., Wets, R.J.B., Woodruff, D.L.: Toward scalable stochastic unit commitment. Part 2: solver configuration and performance assessment. Energy Syst. 6(3), 417–438 (2015)Google Scholar
  12. 12.
    Dai, C., Wu, L., Wu, H.: A multi-band uncertainty set based robust SCUC with spatial and temporal budget constraints. IEEE Trans. Power Syst. 31, 4988–5000 (2016)Google Scholar
  13. 13.
    Dvorkin, Y., Pandzic, H., Ortega-Vazquez, M.A., Kirschen, D.S.: A hybrid stochastic/interval approach to transmission-constrained unit commitment. IEEE Trans. Power Syst. 30(2), 621–631 (2015)Google Scholar
  14. 14.
    Feng, Y., Rios, I., Ryan, S.M., Spürkel, K., Watson, J.P., Wets, R.J.B., Woodruff, D.L.: Toward scalable stochastic unit commitment. Part 1: load scenario generation. Energy Syst. 6(3), 309–329 (2015)Google Scholar
  15. 15.
    Feng, Y., Ryan, S.M.: Solution sensitivity-based scenario reduction for stochastic unit commitment. Comput. Manag. Sci. 13(1), 29–62 (2016)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gourtani, A., Xu, H., Pozo, D., Nguyen, T.D.: Robust unit commitment with \(n-1\) security criteria. Math. Methods Oper. Res. 83(3), 373–408 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Guan, Y., Wang, J.: Uncertainty sets for robust unit commitment. IEEE Trans. Power Syst. 29(3), 1439–1440 (2014)MathSciNetGoogle Scholar
  18. 18.
    Heitsch, H., Römisch, W.: Scenario reduction algorithms in stochastic programming. Comput. Optim. Appl. 24(2–3), 187–206 (2003)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Huang, Y., Zheng, Q.P., Wang, J.: Two-stage stochastic unit commitment model including non-generation resources with Conditional Value-at-Risk constraints. Electr. Power Syst. Res. 116, 427–438 (2014). Google Scholar
  20. 20.
    Jiang, R., Guan, Y., Watson, J.P.: Risk-averse stochastic unit commitment with incomplete information. IIE Trans. 48(9), 838–854 (2016)Google Scholar
  21. 21.
    Jiang, R., Wang, J., Guan, Y.: Robust unit commitment with wind power and pumped storage hydro. IEEE Trans. Power Syst. 27(2), 800–810 (2012)Google Scholar
  22. 22.
    Jiang, R., Zhang, M., Li, G., Guan, Y.: Two-stage network constrained robust unit commitment problem. Eur. J. Oper. Res. 234(3), 751–762 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kazemzadeh, N.: Risk consideration in electricity generation unit commitment under supply and demand uncertainty. Ph.D. thesis, Iowa State University (2016)Google Scholar
  24. 24.
    Lagos, G., Espinoza, D., Moreno, E., Amaya, J.: Robust planning for an open-pit mining problem under ore-grade uncertainty. Electron. Notes Discret. Math. 37, 15–20 (2011)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Lorca, A., Sun, X.A.: Adaptive robust optimization with dynamic uncertainty sets for multi-period economic dispatch under significant wind. IEEE Trans. Power Syst. 30(4), 1702–1713 (2015)Google Scholar
  26. 26.
    Lorca, A., Sun, X.A., Litvinov, E., Zheng, T.: Multistage adaptive robust optimization for the unit commitment problem. Oper. Res. 64(1), 32–51 (2016)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Martinez, G., Anderson, L.: Toward a scalable chance-constrained formulation for unit commitment to manage high penetration of variable generation. In: 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 723–730. IEEE (2014)Google Scholar
  28. 28.
    Martinez, G., Anderson, L.: A risk-averse optimization model for unit commitment problems. In: 2015 48th Hawaii International Conference, pp. 2577–2585. IEEE Computer Society (2015)Google Scholar
  29. 29.
    Murillo-Sanchez, C.E., Zimmerman, R.D., Anderson, C.L., Thomas, R.J.: A stochastic, contingency-based security-constrained optimal power flow for the procurement of energy and distributed reserve. Decis. Support Syst. 56, 1–10 (2013)Google Scholar
  30. 30.
    Naoum-Sawaya, J., Elhedhli, S.: An interior-point benders based branch-and-cut algorithm for mixed integer programs. Ann. Oper. Res. 210(1), 33–55 (2013)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Ozturk, U., Mazumdar, M., Norman, B.: A solution to the stochastic unit commitment problem using chance constrained programming. IEEE Trans. Power Syst. 19(3), 1589–1598 (2004)Google Scholar
  32. 32.
    Pandzic, H., Dvorkin, Y., Qiu, T., Wang, Y., Kirschen, D.S.: Toward cost-efficient and reliable unit commitment under uncertainty. IEEE Trans. Power Syst. 31(2), 970–982 (2016)Google Scholar
  33. 33.
    Peralta, J., Perez-Ruiz, J., de la Torre, S.: Unit commitment with load uncertainty by joint chance-constrained programming. In: PowerTech (POWERTECH), 2013 IEEE Grenoble, pp. 1–6 (2013).
  34. 34.
    Rockafellar, R.T., Uryasev, S.: Conditional Value-at-Risk for general loss distributions. J. Bank. Financ. 26, 1443–1471 (2002)Google Scholar
  35. 35.
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2006)Google Scholar
  36. 36.
    Ruiz, P., Philbrick, C., Zak, E., Cheung, K., Sauer, P.: Uncertainty management in the unit commitment problem. IEEE Trans. Power Syst. 24(2), 642–651 (2009)Google Scholar
  37. 37.
    Sun, X., Fang, C.: Interval mixed-integer programming for daily unit commitment and dispatch incorporating wind power. In: 2010 International Conference on Power System Technology (POWERCON), pp. 1–6 (2010)Google Scholar
  38. 38.
    Tahanan, M., van Ackooij, W., Frangioni, A., Lacalandra, F.: Large-scale unit commitment under uncertainty. 4OR 13(2), 115–171 (2015)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Takriti, S., Birge, J., Long, E.: A stochastic model for the unit commitment problem. IEEE Trans. Power Syst. 11(3), 1497–1508 (1996)Google Scholar
  40. 40.
    Wang, J., Botterud, A., Bessa, R., Keko, H., Carvalho, L., Issicaba, D., Sumaili, J., Miranda, V.: Wind power forecasting uncertainty and unit commitment. Appl. Energy 88(11), 4014–4023 (2011)Google Scholar
  41. 41.
    Wang, J., Shahidehpour, M., Li, Z.: Security-constrained unit commitment with volatile wind power generation. IEEE Trans. Power Syst. 23(3), 1319–1327 (2008)Google Scholar
  42. 42.
    Wang, Q., Guan, Y., Wang, J.: A chance-constrained two-stage stochastic program for unit commitment with uncertain wind power output. IEEE Trans. Power Syst. 27(1), 206–215 (2012)MathSciNetGoogle Scholar
  43. 43.
    Wang, Y., Xia, Q., Kang, C.: Unit commitment with volatile node injections by using interval optimization. IEEE Trans. Power Syst. 26(3), 1705–1713 (2011)Google Scholar
  44. 44.
    Wong, P., Albrecht, P., Allan, R., Billinton, R., Chen, Q., Fong, C., Haddad, S., Li, W., Mukerji, R., Patton, D., Schneider, A., Shahidehpour, M., Singh, C.: The IEEE reliability test system-1996. IEEE Trans. Power Syst. 14(3), 1010–1020 (1999)Google Scholar
  45. 45.
    Wood, A., Wollenberg, B.: Power Generation, Operation, and Control. Wiley, New York (1996)Google Scholar
  46. 46.
    Wu, L., Shahidehpour, M., Li, Z.: Comparison of scenario-based and interval optimization approaches to stochastic SCUC. IEEE Trans. Power Syst. 27(2), 913–921 (2012)Google Scholar
  47. 47.
    Xiong, P., Jirutitijaroen, P., Singh, C.: A distributionally robust optimization model for unit commitment considering uncertain wind power generation. IEEE Trans. Power Syst. 32(1), 39–49 (2017)Google Scholar
  48. 48.
    Zhao, C., Guan, Y.: Unified stochastic and robust unit commitment. IEEE Trans. Power Syst. 28, 3353–3361 (2013)Google Scholar
  49. 49.
    Zhao, C., Guan, Y.: Data-driven stochastic unit commitment for integrating wind generation. IEEE Trans. Power Syst. 31(4), 2587–2596 (2016)Google Scholar
  50. 50.
    Zheng, Q.P., Wang, J., Liu, A.L.: Stochastic optimization for unit commitment—a review. IEEE Trans. Power Syst. 30(4), 1913–1924 (2015)Google Scholar
  51. 51.
    Zugno, M., Morales, J.M., Madsen, H.: Commitment and dispatch of heat and power units via affinely adjustable robust optimization. Comput. Oper. Res. 75, 191–201 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Narges Kazemzadeh
    • 1
    Email author
  • Sarah M. Ryan
    • 1
  • Mahdi Hamzeei
    • 2
  1. 1.Department of Industrial and Manufacturing Systems EngineeringIowa State UniversityAmesUSA
  2. 2.University of MarylandBaltimoreUSA

Personalised recommendations