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Distributionally robust chance constrained optimization for economic dispatch in renewable energy integrated systems

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Abstract

Distributionally robust optimization (DRO) has become a popular research topic since it can solve stochastic programs with ambiguous distribution information. In this paper, as the background of economic dispatch (ED) in renewable integration systems, we present a new DRO-based ED optimization framework (DRED). The new DRED is addressed with a coupled format of distribution uncertainty for objective and chance constraints, which is different from most existing DRO frameworks. Some approximation strategies are adopted to handle the complicated DRED: the data-driven approach, the approximation of chance constraints by conditional value-at-risk, and the discrete scheme. The approximate reformulations are solvable nonconvex nonlinear programming problems. The approximation error analysis and convergence analysis are also established. Numerical results using an IEEE-30 buses system are presented to demonstrate the approach proposed in this paper.

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Notes

  1. Note that the Matlab solver “fminimax” is often used to solve smooth problem. In numerical tests, we applied the smoothing method in [36, Formulation (1.6)] and [37, Formulation (4.2)] to nonsmooth objective function in (3.15) and (3.26). Note also that the smoothing method is well known and it is not our main focus, we omit the details. For more details about the smoothing method, see [36,37,38].

References

  1. Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. NERC (North American Electric Reliability Corporation), Integrated bulk power system risk assessment concepts. NERC Whitepaper (2010)

  3. 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)

    Article  Google Scholar 

  4. Varaiya, P., Wu, F., Bialek, J.: Smart operation of smart grid: risk-limiting dispatch. Proceed. IEEE 99(1), 40–57 (2011)

    Article  Google Scholar 

  5. 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(1), 52–63 (2013)

    Article  Google Scholar 

  6. Street, A., Moreira, A., Arroyo, J.M.: Energy and reserve scheduling under a joint generation and transmission security criterion: an adjustable robust optimization approach. IEEE Trans. Power Syst. 29(1), 3–14 (2014)

    Article  Google Scholar 

  7. Wei, W., Liu, F., Mei, S.: Robust and economical scheduling methodology for power systems: part one theoretical fundation (Chinese). Autom. Electr. Power Syst. 37(17), 37–43 (2013)

    Google Scholar 

  8. Wang, Q., Guan, Y., Wang, J.: A chance-constrained two-stage stochastic program for unit commitmen with uncertain wind power output. IEEE Trans. Power Syst. 27(1), 206–215 (2012)

    Article  MathSciNet  Google Scholar 

  9. Gu, Y., Xie, L.: Stochastic look-ahead economic dispatch with variable generation resources. Accept IEEE Trans. Power Syst. (2016) doi:10.1109/tpers.2016.2520498

  10. Zhao, C., Guan, Y.: Unified stochastic and robust unit commitment. IEEE Trans. Power Syst. 28(3), 3353–3361 (2013)

    Article  Google Scholar 

  11. Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58, 595–612 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Scarf, H.: A min-max solution of an inventory problems. In: Arrow, K.S., Karlin, S., Scarf, H.E. (eds.) Studies in Mathematical Theory of Inventory and Production, pp. 201–209. Stanford University Press, Stanford (1958)

    Google Scholar 

  13. Zhu, S.S., Fukushima, M.: Worst-case conditional value-at-risk with application to robust portfolio management. Oper. Res. 57, 1155–1168 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Delage, E.: Distributionally robust optimization in context of data-driven problems, Stanford University, Ph.D. Thesis (2009)

  15. Sun, H., Xu, H.: Convergence analysis for distributionally robust optimization and equilibrium problems. Math. Oper. Res. 41(2), 377–401 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wiesemann, W., Kuhn, D., Sim, M.: Distributionally robust convex optimization. Oper. Res. 62(6), 1358–1376 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zymler, S., Kuhn, D., Rustem, B.: Distributionally robust joint chance constraints with second-order moment information. Math. Prog. 137, 167–198 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Xin, L., Goldberg, D.A., Shapiro, A.: Time (in) consistency of multistage distributionally robust inventory models with moment constraints, Optimization Online, (2014)

  19. Tong, X.J., Wu, F., Qi, L.: Worst-case CVaR based portfolio optimization models with applications to scenario planning. Opt. Methods Softw. 24(6), 933–958 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tong, X.J., Wu, F.: Robust reward-risk ratio optimization with application in allocation of generation asset. Optimization 63(11), 1761–1779 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhou, R., Min, X., Tong, X., et al.: Distributional robust optimization under moment uncertainty of environmental and economic dispatch for power system. Proceed. CSEE 35(3), 3248–3256 (2015)

    Google Scholar 

  22. Zhang, J., Xu, H., Zhang, L.: Quantitative stability analysis for distributionally robust optimization with moment constraints. SIAM J. Optim. 26(3), 1855–1882 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Anderson, E., Xu, H., Zhang, D.: Confidence levels for CVaR risk measures and minmax limits, Submitted to Oper. Res., (2014). (see http://www.personal.soton.ac.uk/hx/research/publication.htm.)

  24. Rockafellar, R.T., Royset, J.O.: On buffered failure probability in design and optimization of structures. Reliab. Eng. Syst. Saf. 95, 499–510 (2010)

    Article  Google Scholar 

  25. Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2(3), 21–41 (2000)

    Article  Google Scholar 

  26. Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17, 969–996 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hong, L., Jiang, G.: Gradient and hessian of joint probability function with applications on chance-constrained programs. J. Oper. Res. Soc. China, First online (2017)

  28. Sun, H., Xu, H., Wang, Y.: Asymptotic analysis of sample average approximation for stochastic optimization problems with joint chance constraint via conditional value at risk and difference of convex functions. J. Optim. Theory Appl. 161, 257–284 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shapiro, A.: On duality theory of conic linear problems. In: Goberna, M.A., López, M.A. (eds.) Semi-Infinite Programming: Recent Advances. Kluwer Academic Publishers, Dordrecht (2001)

    Google Scholar 

  30. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  31. Xu, H., Liu, Y., Sun, H.: Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods. Math. Program, First online (2017)

  32. Xu, H.: Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming. J. Math. Anal. Appl. 368, 692–710 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhou, J.L., Tits, A.L.: An SQP algorithm for finely discretized continuous minmax problems and other minmax problems with many objective functions. SIAM J. Optim. 6(2), 461–487 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pee, E.Y., Royset, J.O.: On solving large-scale finite minimax problems using exponential smoothing. J. Optim. Theory Appl. 148, 390–421 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  36. Peng, J.: A smoothing function and its applications. In: Fukushima, M., Qi, L. (eds.) Reformulation: Nonsmooth Piecewise Smooth, Semismooth and Smoothing Methods, pp. 293–361. Kluwer Academic Publisher, Dordrecht (1998)

    Chapter  Google Scholar 

  37. Tong, X.J., Xu, H., Wu, F.F., Zhao, Z.: Penalized sample average approximation methods for stochastic programs in economic and secure dispatch of a power system. Comput. Manag. Sci. 13(3), 393–422 (2016)

    Article  MathSciNet  Google Scholar 

  38. Chen, X.: Smoothing methods for nonsmooth, nonconvex minimization. Math. Program. 134, 71–99 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Panigrahi, B.K., Pandi, V.R., Das, S., et al.: Multi-objective fuzzy dominance based bacterial foraging algorithm to solve economic emission dispatch problem. Energy 35(12), 4761–4770 (2010)

    Article  Google Scholar 

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Acknowledgements

The authors thank Professor Huifu Xu and Dr Yun Shi for valuable discussion on technique details and the presentation of the paper. The authors are also grateful to the three anonymous referees for careful reading and constructive comments that improved the quality of this paper significantly.

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Authors and Affiliations

  1. Department of Mathematics, Hunan First Normal University, Changsha, 410205, China

    Xiaojiao Tong

  2. School of Economics and Management, Nanjing University of Science and Technology, Nanjing, 210049, China

    Hailin Sun

  3. Hunan Electric Power Research Institute, Changsha, 410007, China

    Xiao Luo

  4. Hunan Province Key Laboratory of Smart Grids Operation, Changsha, 410114, China

    Quanguo Zheng

Authors
  1. Xiaojiao Tong
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  2. Hailin Sun
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  3. Xiao Luo
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  4. Quanguo Zheng
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Corresponding author

Correspondence to Hailin Sun.

Additional information

This work is supported by National Natural Science Foundation of China (11671125, 71371065, 51707013, 11401308) and Natural Science Foundation of Jiangsu Province, China (BK20140768).

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Tong, X., Sun, H., Luo, X. et al. Distributionally robust chance constrained optimization for economic dispatch in renewable energy integrated systems. J Glob Optim 70, 131–158 (2018). https://doi.org/10.1007/s10898-017-0572-3

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  • DOI: https://doi.org/10.1007/s10898-017-0572-3

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