# Open Journal of Mathematical Optimization

AC Optimal Power Flow: a Conic Programming relaxation and an iterative MILP scheme for Global Optimization
Open Journal of Mathematical Optimization, Volume 3 (2022), article no. 6, 19 p.

We address the issue of computing a global minimizer of the AC Optimal Power Flow problem. We introduce valid inequalities to strengthen the Semidefinite Programming relaxation, yielding a novel Conic Programming relaxation. Leveraging these Conic Programming constraints, we dynamically generate Mixed-Integer Linear Programming (MILP) relaxations, whose solutions asymptotically converge to global minimizers of the AC Optimal Power Flow problem. We apply this iterative MILP scheme on the IEEE PES PGLib [2] benchmark and compare the results with two recent Global Optimization approaches.

Revised:
Accepted:
Published online:
DOI: 10.5802/ojmo.17
Keywords: ACOPF, Global Optimization, Semidefinite Programming, Mixed-Integer Linear Programming
Antoine Oustry 1, 2

1 LIX CNRS, Ecole polytechnique, Institut Polytechnique de Paris, Palaiseau, France
2 Ecole des Ponts, Marne-La-Vallée, France
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Antoine Oustry. AC Optimal Power Flow: a Conic Programming relaxation and an iterative MILP scheme for Global Optimization. Open Journal of Mathematical Optimization, Volume 3 (2022), article  no. 6, 19 p. doi : 10.5802/ojmo.17. https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.17/

[1] Mosek Aps The MOSEK optimization toolbox for MATLAB manual. Version 9.3, 2022 (http://docs.mosek.com/9.3/toolbox/index.html)

[2] Sogol Babaeinejadsarookolaee; Adam Birchfield; Richard D. Christie; Carleton Coffrin; Christopher DeMarco; Ruisheng Diao; Michael Ferris; Stephane Fliscounakis; Scott Greene; Renke Huang; Cédric Josz; Roman Korab; Bernard Lesieutre; Jean Maeght; Terrence W. K. Mak; Daniel Molzahn; Thomas J. Overbye; Patrick Panciatici; Byungkwon Park; Jonathan Snodgrass; Ahmad Tbaileh; Pascal Van Hentenryck; Ray Zimmerman The Power Grid Library for Benchmarking AC Optimal Power Flow Algorithms (2021) (https://arxiv.org/abs/1908.02788v2)

[3] Pietro Belotti; Sonia Cafieri; Jon Lee; Leo Liberti Feasibility-based bounds tightening via fixed points, Combinatorial Optimization, Constraints and Applications (COCOA10) (Lecture Notes in Computer Science), Volume 6508 (2010), pp. 65-76 | DOI | MR | Zbl

[4] Pietro Belotti; Jon Lee; Leo Liberti; François Margot; Andreas Wächter Branching and bounds tightening techniques for non-convex MINLP, Optim. Methods Softw., Volume 24 (2009) no. 4-5, pp. 597-634 | DOI | MR | Zbl

[5] Dan Bienstock; Mauro Escobar; Claudio Gentile; Leo Liberti Mathematical programming formulations for the alternating current optimal power flow problem, 4OR, Volume 18 (2020) no. 3, pp. 249-292 | DOI | MR | Zbl

[6] Jean Carpentier Contribution à l’étude du Dispatching Economique, Bulletin de la Société Française des Électriciens, Volume 3 (1962), pp. 431-447

[7] Chen Chen; Alper Atamturk; Shmuel S. Oren A spatial branch-and-cut method for nonconvex QCQP with bounded complex variables, Math. Program., Volume 165 (2017) no. 2, pp. 549-577 | DOI | MR | Zbl

[8] Carleton Coffrin; Russell Bent; Kaarthik Sundar; Yeesian Ng; Miles Lubin PowerModels.jl: An Open-Source Framework for Exploring Power Flow Formulations, 2018 Power Systems Computation Conference (PSCC) (2018), pp. 1-8 | DOI

[9] Carleton Coffrin; Hassan Hijazi; Pascal Van Hentenryck Strengthening convex relaxations with bound tightening for power network optimization, International conference on principles and practice of constraint programming (2015), pp. 39-57 | DOI

[10] Carleton Coffrin; Hassan Hijazi; Pascal Van Hentenryck Strengthening the SDP Relaxation of AC Power Flows With Convex Envelopes, Bound Tightening, and Valid Inequalities, IEEE Trans. Power Syst., Volume 32 (2017) no. 5, pp. 3549-3558 | DOI

[11] Thomas Cormen; Charles Leiserson; Ronald Rivest; Clifford Stein Introduction to algorithms, MIT Press, 2022

[12] Hadrien Godard; Sourour Elloumi; Amélie Lambert; Jean Maeght; Manuel Ruiz Novel Approach Towards Global Optimality of Optimal Power Flow Using Quadratic Convex Optimization, 2019 International Conference on Control, Decision and Information Technologies (2019), pp. 1227-1232 | DOI

[13] Smitha Gopinath; Hassan Hijazi Benchmarking Large-Scale ACOPF Solutions and Optimality Bounds (2022) (https://arxiv.org/abs/2203.11328)

[14] Smitha Gopinath; Hassan Hijazi; Tillmann Weisser; Harsha Nagarajan; Mertcan Yetkin; Kaarthik Sundar; Russell Bent Proving global optimality of ACOPF solutions, Electric Power Systems Research, Volume 189 (2020), 106688

[15] Robert Grone; Eduardo Johnson; Henry Wolkowicz Positive definite completions of partial Hermitian matrices, Linear Algebra Appl., Volume 58 (1984), pp. 109-124 | DOI | MR | Zbl

[16] William Hart; Jean-Paul Watson; David Woodruff Pyomo: modeling and solving mathematical programs in Python, Math. Program. Comput., Volume 3 (2011) no. 3, pp. 219-260 | DOI | MR

[17] Hassan Hijazi; Guanglei Wang; Carleton Coffrin Gravity: A Mathematical Modeling Language for Optimization and Machine Learning, 2018 (Machine Learning Open Source Software Workshop at NeurIPS 2018, available at www.gravityopt.com)

[18] IBM ILOG CPLEX V12.9: User’s Manual for CPLEX, 2018 (International Business Machines Corporation)

[19] Burak Kocuk; Santanu Dey; Andy Sun Inexactness of SDP relaxation and valid inequalities for optimal power flow, IEEE Trans. Power Syst., Volume 31 (2015) no. 1, pp. 642-651 | DOI

[20] Burak Kocuk; Santanu Dey; Andy Sun Strong SOCP relaxations for the optimal power flow problem, Oper. Res., Volume 64 (2016) no. 6, pp. 1177-1196 | DOI | MR | Zbl

[21] Burak Kocuk; Santanu Dey; Andy Sun Matrix Minor Reformulation and SOCP-based Spatial Branch-and-Cut Method for the AC Optimal Power Flow Problem, Math. Program. Comput., Volume 10 (2018) no. 4, pp. 557-596 | DOI | MR | Zbl

[22] Hidetoshi Komiya Elementary proof for Sion’s minimax theorem, Kodai Math. J., Volume 11 (1988) no. 1, pp. 5-7 | DOI | MR | Zbl

[23] Jean-Bernard Lasserre Global optimization with polynomials and the problem of moments, SIAM J. Optim., Volume 11 (2001) no. 3, pp. 796-817 | DOI | MR | Zbl

[24] Jean-Bernard Lasserre Convergent SDP-relaxations in polynomial optimization with sparsity, SIAM J. Optim., Volume 17 (2006) no. 3, pp. 822-843 | DOI | MR | Zbl

[25] Mowen Lu; Harsha Nagarajan; Russell Bent; Sandra Eksioglu; Scott Mason Tight Piecewise Convex Relaxations for Global Optimization of Optimal Power Flow, 2018 Power Systems Computation Conference (2018), pp. 1-7 | DOI

[26] Garth P. McCormick Computability of global solutions to factorable nonconvex programs: Part I–Convex underestimating problems, Math. Program., Volume 10 (1976) no. 1, pp. 147-175 | DOI | Zbl

[27] Daniel Molzahn; Ian Hiskens Sparsity-exploiting moment-based relaxations of the optimal power flow problem, IEEE Trans. Power Syst., Volume 30 (2014) no. 6, pp. 3168-3180 | DOI

[28] Daniel Molzahn; Ian Hiskens A Survey of Relaxations and Approximations of the Power Flow Equations, Foundations and Trends in Electric Energy Systems, Volume 4 (2019) no. 1, pp. 1-221 | DOI

[29] Daniel Molzahn; Cédric Josz; Ian Hiskens Moment relaxations of optimal power flow problems: beyond the convex hull, 2016 Global Conference on Signal and Information Processing (2016), pp. 856-860 | DOI

[30] Antoine Oustry; Claudia D’Ambrosio; Leo Liberti; Manuel Ruiz Certified and accurate SDP bounds for the ACOPF problem, 2022 Power Systems Computation Conference (2022) (https://hal.archives-ouvertes.fr/hal-03613385)

[31] Hanif Sherali; Amine Alameddine A new reformulation-linearization technique for bilinear programming problems, J. Glob. Optim., Volume 2 (1992) no. 4, pp. 379-410 | DOI | MR | Zbl

[32] Kaarthik Sundar; Harsha Nagarajan; Sidhant Misra; Mowen Lu; Carleton Coffrin; Russell Bent Optimization-Based Bound Tightening using a Strengthened QC-Relaxation of the Optimal Power Flow Problem (2019) (http://arxiv.org/abs/1809.04565v3)

[33] Andreas Wächter; Lorenz Biegler On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., Volume 106 (2006) no. 1, pp. 25-57 | DOI | MR | Zbl

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