Solution of optimal power flow problems using moment relaxations augmented with objective function penalization

Abstract

Abstract-The optimal power flow (OPF) problem minimizes the operating cost of an electric power system. Applications of convex relaxation techniques to the non-convex OPF problem have been of recent interest, including work using the Lasserre hierarchy of "moment" relaxations to globally solve many OPF problems. By preprocessing the network model to eliminate lowimpedance lines, this paper demonstrates the capability of the moment relaxations to globally solve large OPF problems that minimize active power losses for portions of several European power systems. Large problems with more general objective functions have thus far been computationally intractable for current formulations of the moment relaxations. To overcome this limitation, this paper proposes the combination of an objective function penalization with the moment relaxations. This combination yields feasible points with objective function values that are close to the global optimum of several large OPF problems. Compared to an existing penalization method, the combination of penalization and the moment relaxations eliminates the need to specify one of the penalty parameters and solves a broader class of problems.

A Laplacian-Based Approach for Finding Near Globally Optimal Solutions to OPF Problems

“…To further demonstrate the capabilities of the SDP relaxation, 1000 modified versions were created for each of the IEEE 14-, 30-, 39-, and 57-bus systems using normal random perturbations (zero-mean, 10% standard deviation) of the load demands and power generation limits. The SDP relaxation was exact (or proved infeasibility) 2 To obtain satisfactory convergence of the SDP solver, these systems are pre-processed to remove low-impedance lines (i.e., lines whose impedance values have magnitudes less than 1 × 10 −3 per unit) as in [15]. 3 These relaxation gaps are calculated using the objective values from the SDP relaxation (9) and solutions obtained either from the second-order moment relaxation [14] (where possible) or from MATPOWER [24].…”

“…The combination of the moment relaxations with the penalization methods enables the computation of near-globally-optimal solutions for a broader class of OPF problems than either method achieves individually. See [15] for further details on this approach.…”

“…By exploiting network sparsity and selectively applying the computationally intensive higher-order constraints, the moment relaxations are capable of globally solving larger OPF problems [14], including problems with several thousand buses representing portions of European power systems [15]. Ongoing efforts include further increasing computational speed.…”

“…Penalty parameters in the literature range over several orders of magnitude for various test cases, and existing literature largely lacks systematic algorithms for determining appropriate parameter values. Recent work [15] proposes a "moment+penalization" approach that eliminates the need to choose apparent power flow penalization parameters, but still requires selection of a penalty parameter associated with the total reactive power injection. This paper presents an iterative algorithm that builds an objective function intended to yield near-globally-optimal solutions to OPF problems.…”

“…Section IV describes the Laplacian objective function approach that is the main contribution of this paper. Section V demonstrates the effectiveness of the proposed approach through application 1 The moment+penalization method in [15] applies the Lasserre hierarchy for polynomial optimization to OPF problems that are augmented with a reactive power penalty. For sufficiently high relaxation orders, the approach in [15] is guaranteed to yield a feasible solution.…”