Tightening McCormick Relaxations Toward Global Solution of the ACOPF Problem

We demonstrate that a strong upper bound on the objective of the alternating current optimal power flow (ACOPF) problem can significantly improve the effectiveness of optimization-based bounds tightening (OBBT) on a number of relaxations. We additionally compare the performance of relaxations of the ACOPF problem, including the rectangular form without reference bus constraints, the rectangular form with reference bus constraints, and the polar form. We find that relaxations of the rectangular form significantly strengthen existing relaxations if reference bus constraints are included. Overall, relaxations of the polar form perform the best. However, neither the rectangular nor the polar form dominates the other. Ultimately, with these strategies, we are able to reduce the optimality gap to less than 0.1% on all but 5 NESTA test cases with up to 300 buses by performing OBBT alone.

[1]  Nikolaos V. Sahinidis,et al.  Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming , 2002 .

[2]  Carl D. Laird,et al.  Strengthened SOCP Relaxations for ACOPF with McCormick Envelopes and Bounds Tightening , 2018 .

[3]  Santanu S. Dey,et al.  Strong SOCP Relaxations for the Optimal Power Flow Problem , 2015, Oper. Res..

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

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

[6]  R. Jabr,et al.  A Conic Quadratic Format for the Load Flow Equations of Meshed Networks , 2007, IEEE Transactions on Power Systems.

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

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

[9]  Alberto Caprara,et al.  Global optimization problems and domain reduction strategies , 2010, Math. Program..

[10]  J. Momoh.  Electric Power System Applications of Optimization , 2000 .

[11]  WächterAndreas,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006 .

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

[13]  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..

[14]  Daniel Bienstock,et al.  On linear relaxations of OPF problems , 2014, 1411.1120.

[15]  Jianfeng Liu,et al.  A multitree approach for global solution of ACOPF problems using piecewise outer approximations , 2017, Comput. Chem. Eng..