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Conic relaxations of the unit commitment problem

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  • Fattahi, Salar
  • Ashraphijuo, Morteza
  • Lavaei, Javad
  • Atamtürk, Alper

Abstract

The unit commitment (UC) problem aims to find an optimal schedule of generating units subject to demand and operating constraints for an electricity grid. The majority of existing algorithms for the UC problem rely on solving a series of convex relaxations by means of branch-and-bound and cutting-planning methods. The objective of this paper is to obtain a convex model of polynomial size for practical instances of the UC problem. To this end, we develop a convex conic relaxation of the UC problem, referred to as a strengthened semidefinite program (SDP) relaxation. This approach is based on first deriving certain valid quadratic constraints and then relaxing them to linear matrix inequalities. These valid inequalities are obtained by the multiplication of the linear constraints of the UC problem, such as the flow constraints of two different lines. The performance of the proposed convex relaxation is evaluated on several hard instances of the UC problem. For most of the instances, globally optimal integer solutions are obtained by solving a single convex problem. For the cases where the strengthened SDP does not give rise to a global integer solution, we incorporate other valid inequalities. The major benefit of the proposed method compared to the existing techniques is threefold: (i) the proposed formulation is a single convex model with polynomial size and, hence, its global minimum can be found efficiently using well-established first-and second-order methods by starting from any arbitrary initial state, (ii) unlike heuristic methods and local-search algorithms that return local minima whose closeness to a global solution cannot be measured efficiently, the proposed formulation aims at obtaining global minima, (iii) the proposed convex model can be used in convex-hull pricing to minimize uplift payments made to generating units in energy markets. The proposed technique is extensively tested on IEEE 9-bus, IEEE 14-bus, IEEE 30-bus, IEEE 57-bus, IEEE 118-bus, and IEEE 300-bus systems with different settings and over various time horizons.

Suggested Citation

  • Fattahi, Salar & Ashraphijuo, Morteza & Lavaei, Javad & Atamtürk, Alper, 2017. "Conic relaxations of the unit commitment problem," Energy, Elsevier, vol. 134(C), pages 1079-1095.
    • Handle: RePEc:eee:energy:v:134:y:2017:i:c:p:1079-1095
    DOI: 10.1016/j.energy.2017.06.072
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    2. Zhou, Min & Wang, Bo & Li, Tiantian & Watada, Junzo, 2018. "A data-driven approach for multi-objective unit commitment under hybrid uncertainties," Energy, Elsevier, vol. 164(C), pages 722-733.
    3. Guanglei Wang & Hassan Hijazi, 2018. "Mathematical programming methods for microgrid design and operations: a survey on deterministic and stochastic approaches," Computational Optimization and Applications, Springer, vol. 71(2), pages 553-608, November.
    4. Liu, Yikui & Wu, Lei & Li, Jie, 2019. "Towards accurate modeling of dynamic startup/shutdown and ramping processes of thermal units in unit commitment problems," Energy, Elsevier, vol. 187(C).
    5. Chen, Honglin & Liu, Mingbo & Liu, Yingqi & Lin, Shunjiang & Yang, Zhibin, 2020. "Partial surrogate cuts method for network-constrained optimal scheduling of multi-carrier energy systems with demand response," Energy, Elsevier, vol. 196(C).
    6. Zohrizadeh, Fariba & Josz, Cedric & Jin, Ming & Madani, Ramtin & Lavaei, Javad & Sojoudi, Somayeh, 2020. "A survey on conic relaxations of optimal power flow problem," European Journal of Operational Research, Elsevier, vol. 287(2), pages 391-409.
    7. Kyu-Hyung Jo & Mun-Kyeom Kim, 2018. "Improved Genetic Algorithm-Based Unit Commitment Considering Uncertainty Integration Method," Energies, MDPI, vol. 11(6), pages 1-18, May.

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