A laplacian-based approach for finding near globally optimal solutions to OPF problems

Abstract

Abstract-A semidefinite programming (SDP) relaxation globally solves many optimal power flow (OPF) problems. For other OPF problems where the SDP relaxation only provides a lower bound on the objective value rather than the globally optimal decision variables, recent literature has proposed a penalization approach to find feasible points that are often nearly globally optimal. A disadvantage of this penalization approach is the need to specify penalty parameters. This paper presents an alternative approach that algorithmically determines a penalization appropriate for many OPF problems. The proposed approach constrains the generation cost to be close to the lower bound from the SDP relaxation. The objective function is specified using iteratively determined weights for a Laplacian matrix. This approach yields feasible points to the OPF problem that are guaranteed to have objective values near the global optimum due to the constraint on generation cost. The proposed approach is demonstrated on both small OPF problems and a variety of large test cases representing portions of European power systems.

Scheduled-Asynchronous Distributed Algorithm for Optimal Power Flow

“…This criteria is justified by the observation that if γ t + i = 0, ∀i ∈ N , then W t + N is the optimal solution for both (12) and (10). Using the continuity of the cost function, f i , having γ t + i sufficiently small for all i ∈ N guarantees that the solution of Algorithm 1 is reasonably close to the optimum.…”

“…Conditions on the exact convex relaxation have been further established in [8], [9]. Several recent works [10], [11] further develop SDP-based algorithms for a near-global optimal solution for OPF problems where the SDP convex relaxation does not provide a feasible solution. Relatively few works consider solving the OPF problem in a distributed way.…”

Rank‐1 positive semidefinite matrix‐based nonlinear programming formulation for AC OPF

“…9 Although it has been studied that SDP relaxation can give global optimum for many IEEE test systems while the solutions are feasible to the original AC OPF problems (termed as "SDP exact") in Lavaei and Low,13 in some other cases, SDP relaxation leads to inexact solutions for the original problem. 8,14,15 Thus, research efforts have been devoted to achieve SDP exactness, eg, Madani et al 16 and Molzahn et al 17 The exactness conditions for SDP and SOCP relaxations are presented in Low. 9 Some researches have been conducted to achieve exactness for convex relaxation through exploiting the exactness conditions.…”

“…9 Some researches have been conducted to achieve exactness for convex relaxation through exploiting the exactness conditions. In Madani et al 16 and Molzahn et al, 17 objective functions are modified to include penalty related to the rank-1 constraint. You and Peng 18 treat an AC OPF problem as an SDP relaxation problem and a nonconvex rank-1 feasible region mapping problem.…”

“…However, large gaps are still observed for special cases. 17 In this paper, we rely on nonlinear programming formulation with a PSD matrix as the decision variable. We reformulate the rank-1 constraint as a set of quadratic minor constraints.…”