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Polynomial time algorithms and extended formulations for unit commitment problems

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  • Yongpei Guan
  • Kai Pan
  • Kezhuo Zhou

Abstract

Recently, increasing penetration of renewable energy generation has created challenges for power system operators to perform efficient power generation daily scheduling, due to the intermittent nature of the renewable generation and discrete decisions of each generation unit. Among all aspects to be considered, a unit commitment polytope is fundamental and embedded in the models at different stages of power system planning and operations. In this article, we focus on deriving polynomial-time algorithms for the unit commitment problems with a general convex cost function and piecewise linear cost function, respectively. We refine an O(T3)$\mathcal {O}(T^3)$ time, where T represents the number of time periods, algorithm for the deterministic single-generator unit commitment problem with a general convex cost function and accordingly develop an extended formulation in a higher-dimensional space that can provide an integral solution, in which the physical meanings of the decision variables are described. This means the original problem can be solved as a convex program instead of a mixed-integer convex program. Furthermore, for the case in which the cost function is piecewise linear, by exploring the optimality conditions, we derive more efficient algorithms for both deterministic (i.e., O(T)$\mathcal {O}(T)$ time) and stochastic (i.e., O(N)$\mathcal {O}(N)$ time, where N represents the number of nodes in the stochastic scenario tree) single-generator unit commitment problems. We also develop the corresponding extended formulations for both deterministic and stochastic single-generator unit commitment problems that solve the original mixed-integer linear programs as linear programs. Similarly, physical meanings of the decision variables are explored to show the insights of the new modeling approach.

Suggested Citation

  • Yongpei Guan & Kai Pan & Kezhuo Zhou, 2018. "Polynomial time algorithms and extended formulations for unit commitment problems," IISE Transactions, Taylor & Francis Journals, vol. 50(8), pages 735-751, August.
  • Handle: RePEc:taf:uiiexx:v:50:y:2018:i:8:p:735-751
    DOI: 10.1080/24725854.2017.1397303
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    Cited by:

    1. Jianqiu Huang & Kai Pan & Yongpei Guan, 2021. "Multistage Stochastic Power Generation Scheduling Co-Optimizing Energy and Ancillary Services," INFORMS Journal on Computing, INFORMS, vol. 33(1), pages 352-369, January.
    2. Skolfield, J. Kyle & Escobedo, Adolfo R., 2022. "Operations research in optimal power flow: A guide to recent and emerging methodologies and applications," European Journal of Operational Research, Elsevier, vol. 300(2), pages 387-404.
    3. Kai Pan & Ming Zhao & Chung-Lun Li & Feng Qiu, 2022. "A Polyhedral Study on Fuel-Constrained Unit Commitment," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 3309-3324, November.
    4. Bernard Knueven & James Ostrowski & Jean-Paul Watson, 2020. "On Mixed-Integer Programming Formulations for the Unit Commitment Problem," INFORMS Journal on Computing, INFORMS, vol. 32(4), pages 857-876, October.
    5. Heng Yang & Ziliang Jin & Jianhua Wang & Yong Zhao & Hejia Wang & Weihua Xiao, 2019. "Data-Driven Stochastic Scheduling for Energy Integrated Systems," Energies, MDPI, vol. 12(12), pages 1-21, June.
    6. Farhan Hyder & Bing Yan & Mikhail Bragin & Peter Luh, 2024. "Convex Hull Pricing for Unit Commitment: Survey, Insights, and Discussions," Energies, MDPI, vol. 17(19), pages 1-20, September.

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