Abstract
The AC optimal power flow (AC OPF) problem is considered and five convex relaxations for solving this problem—the semidefinite, chordal, conic, and moment-based ones as well as the QC relaxation—are overviewed. The specifics of the AC formulation and also the nonconvexity of the problem are described in detail. Each of the relaxations for OPF is written in explicit form. The semidefinite, chordal and conic relaxations are of major interest. They are implemented on a test example of four nodes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Cain, M., O’Neill, R., and Castillo, A., History of Optimal Power Flow and Formulations, Federal Energy Regulatory Commission, 2012, vol. 1, pp. 1–36.
Stott, B. and Alsac, O., Optimal Power Flow—Basic Requirements for Real-Life Problems and Their Solutions, Proc. SEPOPE XII Sympos., Rio de Janeiro, Brazil, 2012, vol. 1, pp. 1866–76.
Venikov, V.A. and Sukhanov, R.P., Kiberneticheskie modeli elektricheskikh sistem (Cybernetic Models of Electrical Power Systems), Moscow: Energoizdat, 1982.
Stott, B., Jardim, J., and Alsac, O., DC Power Flow Revisited, IEEE Transact. Power Syst., 2009, vol. 24, no. 3, pp. 1290–1300.
Momoh, J., Electric Power System Applications of Optimization, Boca Raton: CRC Press, 2009.
Zhifeng, Q., Deconinck, G., and Belmans, R., A Literature Survey of Optimal Power Flow Problems in the Electricity Market Context, IEEE/PES Power Syst. Conf. Expos., 2009, vol. 1, pp. 1–6.
Gan, L., Li, N., Topcu, U., and Low, S., Exact Convex Relaxation of Optimal Power Flow in Radial Networks, IEEE Transact. Autom. Control, 2015, vol. 60, no. 1, pp. 72–87.
Zorin, I., Vasilyev, S., and Gryazina, E., Fragility of the Semidefinite Relaxation for the Optimal Power Flow Problem, IEEE Int. Conf. Sci. Electr. Eng. (ICSEE), 2016, pp. 1–5.
Boyd, L. and Vandenberghe, S., Semidefinite Programming, SIAM Rev., 1996, vol. 38, no. 1, pp. 49–95.
Bose, S., Low, S., Teeraratkul, T., and Hassibi, B., Equivalent Relaxations of Optimal Power Flow, IEEE Transact. Autom. Control, 2015, vol. 60, no. 3, pp. 729–742.
Lavaei, J. and Low, S., Zero Duality Gap in Optimal Power Flow Problem, IEEE Transact. Power Syst., 2011, vol. 27, pp. 92–107.
Capitanescu, F., Critical Review of Recent Advances and Further Developments Needed in AC Optimal Power Flow, Electr. Power Syst. Res., 2016, vol. 136, pp. 57–68.
Lesieutre, B., Molzahn, D., Borden, A., and DeMarco, C., Examining the Limits of the Application of Semidefinite Programming to Power Flow Problems, Proc. Allerton Conf., 2011, vol. 1, pp. 1492–1499.
Robert, G., Johnson, C., Sa, E., and Wolkowicz, H., Positive Definite Completions of Partial Hermitian Matrices, Linear Algebra Appl., 1984, vol. 58, pp. 109–24.
Woerdeman, H., Minimal Rank Completions for Block Matrices, Linear Algebra Appl., 1989, vol. 121, pp. 105–22.
Mituhiro, F., Kojima, M., Murota, K., and Nakata, K., Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework, SIAM J. Optim., 2001, vol. 11, pp. 647–74.
Low, S., Convex Relaxation of Optimal Power Flow—Part I: Formulations and Equivalence, IEEE Transact. Control Net. Syst., 2014, vol. 1, no. 1, pp. 15–27.
Low, S., Convex Relaxation of Optimal Power Flow—Part II: Exactness, IEEE Transact. Control Net. Syst., 2014, vol. 1, no. 2, pp. 177–189.
Molzahn, D. and Hiskens, I., Moment-Based Relaxation of the Optimal Power Flow Problem, Power Syst. Comput. Conf. (PSCC), 2014, vol. 1, pp. 1–7.
Josz, C., Maeght, J., Panciatici, P., and Gilbert, J., Application of the Moment-SOS Approach to Global Optimization of the OPF Problem, IEEE Transact. Power Syst., 2015, vol. 30, no. 1, pp. 463–470.
Lasserre, J., Global Optimization with Polynomials and the Problem of Moments, SIAM J. Optim., 2006, vol. 11, no. 3, pp. 796–817.
Lasserre, J., Moments, Positive Polynomials and Their Applications, London: Imperial College Press, 2010.
Coffrin, C., Hijazi, H., and Van Hentenryck, P., The QC Relaxation: A Theoretical and Computational Study on Optimal Power Flow, IEEE Transact. Power Syst., 2016, vol. 31, no. 4, pp. 3008–3018.
Hijazi, H., Coffrin, C., and Van Hentenryck, P., Convex Quadratic Relaxations for Mixed-Integer Nonlinear Programs in Power Systems, Math. Program. Comput., 2017, vol. 9, no. 3, pp. 321–367.
Zimmerman, R., Murillo-Sanchez, C., and Thomas, R., MATPOWER: Steady-State Operations, Planning and Analysis Tools for Power Systems Research and Education, IEEE Transact. Power Syst., 2011, vol. 26, no. 1, pp. 12–19.
Grant, M., Boyd, S., and Ye, Y., CVX: Matlab Software for Disciplined Convex Programming, 2008. http://cvxr.com/cvx
Mosek Optimization Solver. www.mosek.com
University of Washington, Power Systems Test Case Archive. www.ee.washington.edu/research/pstca
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2019, published in Avtomatika i Telemekhanika, 2019, No. 5, pp. 32–57.
Rights and permissions
About this article
Cite this article
Zorin, I.A., Gryazina, E.N. An Overview of Semidefinite Relaxations for Optimal Power Flow Problem. Autom Remote Control 80, 813–833 (2019). https://doi.org/10.1134/S0005117919050023
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117919050023