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.

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

“…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, in some other cases, SDP relaxation leads to inexact solutions for the original problem . Thus, research efforts have been devoted to achieve SDP exactness, eg, Madani et al and Molzahn et al…”

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

“…In many cases, exact solutions can be found after dealing with the exactness condition. However, large gaps are still observed for special cases …”

ACOPF for three‐phase four‐conductor distribution systems: semidefinite programming based relaxation with variable reduction and feasible solution recovery

“…Inspired by this, Molzahn and Hiskens [23] proposed a sparse-moment relaxation approach to improve computational performance. Another research direction is to recover feasible solutions from an inexact solution of the relaxed model [24,25]. However, the iterative recovering process could be computationally inefficient [24] or require predefined coefficients that are hard to determine [25].…”

“…Another research direction is to recover feasible solutions from an inexact solution of the relaxed model [24,25]. However, the iterative recovering process could be computationally inefficient [24] or require predefined coefficients that are hard to determine [25]. (i) An SDP model is proposed to solve the ACOPF problem, which explicitly models neutral conductors and ground resistances to derive accurate power flow and bus voltage results.…”