Global optimization of Optimal Power Flow using a branch & bound algorithm

Abstract

We propose two algorithms for the solution of the Optimal Power Flow (OPF) problem to global optimality. The algorithms are based on the spatial branch and bound framework with lower bounds on the optimal objective function value calculated by solving either the Lagrangian dual or the semidefinite programming (SDP) relaxation. We show that this approach can solve to global optimality the general form of the OPF problem including: generation power bounds, apparent and real power line limits, voltage limits and thermal loss limits. The approach makes no assumption on the topology or resistive connectivity of the network. This work also removes some of the restrictive assumptions of the SDP approaches [5]. We present the performance of the algorithms on a number of standard IEEE systems, which are known to have a zero duality gap. We also make parameter perturbations to the test cases that result in solutions that fail to satisfy the SDP rank condition and have a non-zero duality gap. The proposed branch and bound algorithms are able to solve these cases to global optimality. Allerton Confernce on Communication, Control and Computing (Allerton)This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. [5]. We present the performance of the algorithms on a number of standard IEEE systems, which are known to have a zero duality gap. We also make parameter perturbations to the test cases that result in solutions that fail to satisfy the SDP rank condition and have a non-zero duality gap. The proposed branch and bound algorithms are able to solve these cases to global optimality.

Global optimization of multi-period optimal power flow

“…Further, the above approaches provide no recourse when sufficient conditions are not satisfied. The work of Lesieutre et al [24] and Gopalakrishnan et al [1] provide examples that fail to satisfy the sufficient conditions.…”

“…In this work, we are interested in extending the algorithm in [1] for the solution of MOPF. The MOPF is time coupled due to the presence of storage at various buses in the network.…”

“…Recently the authors [1] proposed to solve the OPF problem using a branch and bound (B & B) algorithm with SDP based lower bounds. The proposed approach does not make assumptions on the network topology or the type of bounds.…”

Advanced optimization methods for power systems

“…These approaches include heuristics that attempt to find "nearby" local solutions (that may often be, in fact, the global solution) [30], [31] and the use of a branchand-bound method to eliminate the gap between the lower bound from a relaxation and the upper bound from a nonlinear programming solver [32]. We next discusses an alternative to these approaches that generalizes the semidefinite relaxation using the Lasserre hierarchy of moment relaxations.…”

A non-convex alternating direction method of multipliers heuristic for optimal power flow

“…To obtain a feasible optimal solution of the OPF, the polynomial optimization approach based on the moment relaxation has been proposed in [11] and [12], and the branch and bound method for OPF has been investigated in [13]. Although these global optimization approaches guarantee the global optimal solution which is also a feasible solution of OPF, but these algorithms may not be computationally tractable.…”