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.
Received:
Revised:
Accepted:
Published online:
DOI:
10.5802/ojmo.17
Revised:
Accepted:
Published online:
Keywords:
ACOPF, Global Optimization, Semidefinite Programming, Mixed-Integer Linear Programming
Author's affiliations:
Antoine Oustry 1, 2
License: 
CC-BY 4.0
CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{OJMO_2022__3__A6_0,
author = {Antoine Oustry},
title = {AC {Optimal} {Power} {Flow:} a {Conic} {Programming} relaxation and an iterative {MILP} scheme for {Global} {Optimization}},
journal = {Open Journal of Mathematical Optimization},
eid = {6},
pages = {1--19},
publisher = {Universit\'e de Montpellier},
volume = {3},
year = {2022},
doi = {10.5802/ojmo.17},
language = {en},
url = {https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.17/}
}
TY - JOUR AU - Antoine Oustry TI - AC Optimal Power Flow: a Conic Programming relaxation and an iterative MILP scheme for Global Optimization JO - Open Journal of Mathematical Optimization PY - 2022 SP - 1 EP - 19 VL - 3 PB - Université de Montpellier UR - https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.17/ DO - 10.5802/ojmo.17 LA - en ID - OJMO_2022__3__A6_0 ER -
%0 Journal Article %A Antoine Oustry %T AC Optimal Power Flow: a Conic Programming relaxation and an iterative MILP scheme for Global Optimization %J Open Journal of Mathematical Optimization %D 2022 %P 1-19 %V 3 %I Université de Montpellier %U https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.17/ %R 10.5802/ojmo.17 %G en %F OJMO_2022__3__A6_0
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] The MOSEK optimization toolbox for MATLAB manual. Version 9.3, 2022 (http://docs.mosek.com/9.3/toolbox/index.html)
[2] The Power Grid Library for Benchmarking AC Optimal Power Flow Algorithms (2021) (https://arxiv.org/abs/1908.02788v2)
[3] , Combinatorial Optimization, Constraints and Applications (COCOA10) (Lecture Notes in Computer Science), Volume 6508 (2010), pp. 65-76 | DOI | MR | Zbl
[4] 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] Mathematical programming formulations for the alternating current optimal power flow problem, 4OR, Volume 18 (2020) no. 3, pp. 249-292 | DOI | MR | Zbl
[6] Contribution à l’étude du Dispatching Economique, Bulletin de la Société Française des Électriciens, Volume 3 (1962), pp. 431-447
[7] 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] , 2018 Power Systems Computation Conference (PSCC) (2018), pp. 1-8 | DOI
[9] , International conference on principles and practice of constraint programming (2015), pp. 39-57 | DOI
[10] 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] Introduction to algorithms, MIT Press, 2022
[12] , 2019 International Conference on Control, Decision and Information Technologies (2019), pp. 1227-1232 | DOI
[13] Benchmarking Large-Scale ACOPF Solutions and Optimality Bounds (2022) (https://arxiv.org/abs/2203.11328)
[14] Proving global optimality of ACOPF solutions, Electric Power Systems Research, Volume 189 (2020), 106688
[15] Positive definite completions of partial Hermitian matrices, Linear Algebra Appl., Volume 58 (1984), pp. 109-124 | DOI | MR | Zbl
[16] Pyomo: modeling and solving mathematical programs in Python, Math. Program. Comput., Volume 3 (2011) no. 3, pp. 219-260 | DOI | MR
[17] 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] V12.9: User’s Manual for CPLEX, 2018 (International Business Machines Corporation)
[19] 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] Strong SOCP relaxations for the optimal power flow problem, Oper. Res., Volume 64 (2016) no. 6, pp. 1177-1196 | DOI | MR | Zbl
[21] 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] Elementary proof for Sion’s minimax theorem, Kodai Math. J., Volume 11 (1988) no. 1, pp. 5-7 | DOI | MR | Zbl
[23] Global optimization with polynomials and the problem of moments, SIAM J. Optim., Volume 11 (2001) no. 3, pp. 796-817 | DOI | MR | Zbl
[24] Convergent SDP-relaxations in polynomial optimization with sparsity, SIAM J. Optim., Volume 17 (2006) no. 3, pp. 822-843 | DOI | MR | Zbl
[25] , 2018 Power Systems Computation Conference (2018), pp. 1-7 (Accessed 2021-06-22) | DOI
[26] 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] 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] 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] , 2016 Global Conference on Signal and Information Processing (2016), pp. 856-860 | DOI
[30] , 2022 Power Systems Computation Conference (2022) (https://hal.archives-ouvertes.fr/hal-03613385)
[31] A new reformulation-linearization technique for bilinear programming problems, J. Glob. Optim., Volume 2 (1992) no. 4, pp. 379-410 | DOI | MR | Zbl
[32] Optimization-Based Bound Tightening using a Strengthened QC-Relaxation of the Optimal Power Flow Problem (2019) (http://arxiv.org/abs/1809.04565v3)
[33] 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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