Proving global optimality of ACOPF solutions

We present our latest contributions in terms of mathematical modeling and algorithm development for the global optimization of the ACOPF problem. These contributions allow us to close the optimality gap on a number of open instances in the PGLIB and NESTA benchmark libraries. This is achieved by combining valid cut generation with semidefinite programming-based bound tightening. The mathematical formulations along with the solution algorithms are implemented in the modeling framework Gravity (www.gravityopt.com), an open-source platform for reproducible numerical experiments.

[1]  Carleton Coffrin,et al.  NESTA, The NICTA Energy System Test Case Archive , 2014, ArXiv.

[2]  Russell Bent,et al.  An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs , 2017, J. Glob. Optim..

[3]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[4]  Masakazu Muramatsu,et al.  Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity , 2004 .

[5]  Pascal Van Hentenryck,et al.  Convex quadratic relaxations for mixed-integer nonlinear programs in power systems , 2016, Mathematical Programming Computation.

[6]  Ian A. Hiskens,et al.  Sparsity-Exploiting Moment-Based Relaxations of the Optimal Power Flow Problem , 2014, IEEE Transactions on Power Systems.

[7]  Paul A. Trodden,et al.  Local Solutions of the Optimal Power Flow Problem , 2013, IEEE Transactions on Power Systems.

[8]  Timo Berthold,et al.  Three enhancements for optimization-based bound tightening , 2017, J. Glob. Optim..

[9]  Carleton Coffrin,et al.  The QC relaxation: A theoretical and computational study on optimal power flow , 2016, 2017 IEEE Power & Energy Society General Meeting.

[10]  Garth P. McCormick,et al.  Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems , 1976, Math. Program..

[11]  Russell Bent,et al.  PowerModels.J1: An Open-Source Framework for Exploring Power Flow Formulations , 2017, 2018 Power Systems Computation Conference (PSCC).

[12]  Pascal Van Hentenryck,et al.  Strengthening Convex Relaxations with Bound Tightening for Power Network Optimization , 2015, CP.

[13]  Pascal Van Hentenryck,et al.  Polynomial SDP cuts for Optimal Power Flow , 2015, 2016 Power Systems Computation Conference (PSCC).

[14]  Carleton Coffrin,et al.  Gravity: A Mathematical Modeling Language for Optimization and Machine Learning , 2018 .

[15]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

[16]  Russell Bent,et al.  Tight Piecewise Convex Relaxations for Global Optimization of Optimal Power Flow , 2018, 2018 Power Systems Computation Conference (PSCC).

[17]  Daniel K. Molzahn,et al.  Lasserre Hierarchy for Large Scale Polynomial Optimization in Real and Complex Variables , 2017, SIAM J. Optim..

[18]  Pascal Van Hentenryck,et al.  Strengthening the SDP Relaxation of AC Power Flows With Convex Envelopes, Bound Tightening, and Valid Inequalities , 2017, IEEE Transactions on Power Systems.

[19]  Jean Charles Gilbert,et al.  Application of the Moment-SOS Approach to Global Optimization of the OPF Problem , 2013, IEEE Transactions on Power Systems.

[20]  K. Mani Chandy,et al.  Inverter VAR control for distribution systems with renewables , 2011, 2011 IEEE International Conference on Smart Grid Communications (SmartGridComm).

[21]  Hanif D. Sherali,et al.  A new reformulation-linearization technique for bilinear programming problems , 1992, J. Glob. Optim..

[22]  Ignacio E. Grossmann,et al.  A Branch and Contract Algorithm for Problems with Concave Univariate, Bilinear and Linear Fractional Terms , 1999, J. Glob. Optim..

[23]  Javad Lavaei,et al.  Convexification of optimal power flow problem , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[24]  R. Jabr.  Radial distribution load flow using conic programming , 2006, IEEE Transactions on Power Systems.

[25]  J. Gallier.  Quadratic Optimization Problems , 2020, Linear Algebra and Optimization with Applications to Machine Learning.

[26]  Santanu S. Dey,et al.  Matrix minor reformulation and SOCP-based spatial branch-and-cut method for the AC optimal power flow problem , 2017, Math. Program. Comput..

[27]  Carl D. Laird,et al.  Tightening McCormick Relaxations Toward Global Solution of the ACOPF Problem , 2019, IEEE Transactions on Power Systems.