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Strong SOCP Relaxations for the Optimal Power Flow Problem

Author

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  • Burak Kocuk

    (H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

  • Santanu S. Dey

    (H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

  • X. Andy Sun

    (H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

Abstract

This paper proposes three strong second order cone programming (SOCP) relaxations for the AC optimal power flow (OPF) problem. These three relaxations are incomparable to each other and two of them are incomparable to the standard SDP relaxation of OPF. Extensive computational experiments show that these relaxations have numerous advantages over existing convex relaxations in the literature: (i) their solution quality is extremely close to that of the standard SDP relaxation (the best one is within 99.96% of the SDP relaxation on average for all the IEEE test cases) and consistently outperforms previously proposed convex quadratic relaxations of the OPF problem, (ii) the solutions from the strong SOCP relaxations can be directly used as a warm start in a local solver such as IPOPT to obtain a high quality feasible OPF solution, and (iii) in terms of computation times, the strong SOCP relaxations can be solved an order of magnitude faster than the standard SDP relaxation. For example, one of the proposed SOCP relaxations together with IPOPT produces a feasible solution for the largest instance in the IEEE test cases (the 3375-bus system) and also certifies that this solution is within 0.13% of global optimality, all this computed within 157.20 seconds on a modest personal computer. Overall, the proposed strong SOCP relaxations provide a practical approach to obtain feasible OPF solutions with extremely good quality within a time framework that is compatible with the real-time operation in the current industry practice.

Suggested Citation

  • Burak Kocuk & Santanu S. Dey & X. Andy Sun, 2016. "Strong SOCP Relaxations for the Optimal Power Flow Problem," Operations Research, INFORMS, vol. 64(6), pages 1177-1196, December.
    • Handle: RePEc:inm:oropre:v:64:y:2016:i:6:p:1177-1196
    DOI: 10.1287/opre.2016.1489
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    References listed on IDEAS

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    1. Carleton Coffrin & Pascal Van Hentenryck, 2014. "A Linear-Programming Approximation of AC Power Flows," INFORMS Journal on Computing, INFORMS, vol. 26(4), pages 718-734, November.
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    8. Iacopo Savelli & Thomas Morstyn, 2020. "Electricity prices and tariffs to keep everyone happy: a framework for fixed and nodal prices coexistence in distribution grids with optimal tariffs for investment cost recovery," Papers 2001.04283, arXiv.org, revised Jun 2021.
    9. Subramanian, Vignesh & Feijoo, Felipe & Sankaranarayanan, Sriram & Melendez, Kevin & Das, Tapas K., 2022. "A bilevel conic optimization model for routing and charging of EV fleets serving long distance delivery networks," Energy, Elsevier, vol. 251(C).
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    13. 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.
    14. Reza Sabzehgar & Diba Zia Amirhosseini & Saeed D. Manshadi & Poria Fajri, 2021. "Stochastic Expansion Planning of Various Energy Storage Technologies in Active Power Distribution Networks," Sustainability, MDPI, vol. 13(10), pages 1-17, May.
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    16. 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.
    17. Jay, Devika & Swarup, K.S., 2021. "A comprehensive survey on reactive power ancillary service markets," Renewable and Sustainable Energy Reviews, Elsevier, vol. 144(C).
    18. Harsha Nagarajan & Mowen Lu & Site Wang & Russell Bent & Kaarthik Sundar, 2019. "An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs," Journal of Global Optimization, Springer, vol. 74(4), pages 639-675, August.
    19. Daniel Bienstock & Mauro Escobar & Claudio Gentile & Leo Liberti, 2022. "Mathematical programming formulations for the alternating current optimal power flow problem," Annals of Operations Research, Springer, vol. 314(1), pages 277-315, July.
    20. Puming Wang & Liqin Zheng & Tianyi Diao & Shengquan Huang & Xiaoqing Bai, 2023. "Robust Bilevel Optimal Dispatch of Park Integrated Energy System Considering Renewable Energy Uncertainty," Energies, MDPI, vol. 16(21), pages 1-23, October.
    21. Dan Bienstock & Mauro Escobar & Claudio Gentile & Leo Liberti, 2020. "Mathematical programming formulations for the alternating current optimal power flow problem," 4OR, Springer, vol. 18(3), pages 249-292, September.
    22. Kevin-Martin Aigner & Robert Burlacu & Frauke Liers & Alexander Martin, 2023. "Solving AC Optimal Power Flow with Discrete Decisions to Global Optimality," INFORMS Journal on Computing, INFORMS, vol. 35(2), pages 458-474, March.
    23. Wogrin, S. & Tejada-Arango, D. & Delikaraoglou, S. & Botterud, A., 2020. "Assessing the impact of inertia and reactive power constraints in generation expansion planning," Applied Energy, Elsevier, vol. 280(C).
    24. Bichler, Martin & Knörr, Johannes, 2023. "Getting prices right on electricity spot markets: On the economic impact of advanced power flow models," Energy Economics, Elsevier, vol. 126(C).
    25. Amir Ahmadi-Javid & Pooya Hoseinpour, 2022. "Convexification of Queueing Formulas by Mixed-Integer Second-Order Cone Programming: An Application to a Discrete Location Problem with Congestion," INFORMS Journal on Computing, INFORMS, vol. 34(5), pages 2621-2633, September.

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