Empirical Investigation of Non-Convexities in Optimal Power Flow Problems

Abstract

Optimal power flow (OPF) is a central problem in the operation of electric power systems. An OPF problem optimizes a specified objective function subject to constraints imposed by both the non-linear power flow equations and engineering limits. These constraints can yield non-convex feasible spaces that result in significant computational challenges. Despite these non-convexities, local solution algorithms actually find the global optima of some practical OPF problems. This suggests that OPF problems have a range of difficulty: some problems appear to have convex or "nearly convex" feasible spaces in terms of the voltage magnitudes and power injections, while other problems can exhibit significant non-convexities. Understanding this range of problem difficulty is helpful for creating new test cases for algorithmic benchmarking purposes. Leveraging recently developed computational tools for exploring OPF feasible spaces, this paper first describes an empirical study that aims to characterize non-convexities for small OPF problems. This paper then proposes and analyzes several medium-size test cases that challenge a variety of solution algorithms. * :

Improving QC Relaxations of OPF Problems via Voltage Magnitude Difference Constraints and Envelopes for Trilinear Monomials

“…This section demonstrates the proposed improvements using test cases from the NESTA 0.7.0 archive [22] and four cases "nmwc14", "nmwc24," "nmwc57," and "nmwc118" from [24]. With large optimality gaps between the objective values from the best known local optima and the lower bounds from various relaxations, these test cases challenge a variety of solution algorithms and are therefore suitable for our purposes.…”

Tightening QC Relaxations of AC Optimal Power Flow Problems via Complex Per Unit Normalization

“…2a and 2b demonstrates the superiority of the proposed approach in providing tighter envelopes compared to those in [14], [22]. Note that (11) precludes the need for the linking constraint (5n) that relates the common term V l V m in the products V l V m sinpθ lm ´δlm ´ψl q and V l V m cospθ lm ´δlm ´ψl q.…”

“…Thus, local algorithms and relaxations can be used together in spatial branch-and-bound methods [6]. Power flow relaxations are also valuable for a range of other applications, including solving robust OPF problems [7], [8], calculating voltage stability margins [9], exploring feasible operating ranges [10], [11], computing optimal switching decisions [12], etc. Convex relaxation methods are under active research with ongoing efforts targeting to improve their tightness.…”

Verifying Global Optimality of Candidate Solutions to Polynomial Optimization Problems using a Determinant Relaxation Hierarchy

“…Any point that satisfies (10) provides values for dual variables that, in combination with the primal variables ŷ obtained from the candidate solution x, satisfy the KKT conditions corresponding to a convex relaxation of (1). Hence, any feasible point for (10) certifies global optimality of the candidate feasible point x for the polynomial optimization problem (1). 3 Since all values ŷ are fixed, we emphasize that ( 10) is linear in the KKT multipliers λ, λ 0,I , and λ 0,Ji .…”

“…OPF problems are known to be NP-hard [5], [6]. Accordingly, there exist various test cases that challenge many solution algorithms [7]- [10]. However, despite this challenging worstcase complexity, local solvers with physically justified initializations find the global optima for many practical OPF problems as verified by the small or zero optimality gaps obtained via a variety of convex relaxation techniques [11]- [16].…”