Resistive Network Optimal Power Flow: Uniqueness and Algorithms

Abstract

Abstract-The optimal power flow (OPF) problem minimizes the power loss in an electrical network by optimizing the voltage and power delivered at the network buses, and is a nonconvex problem that is generally hard to solve. By leveraging a recent development on the zero duality gap of OPF, we propose a second-order cone programming convex relaxation of the resistive network OPF, and study the uniqueness of the optimal solution using differential topology especially the Poincare-Hopf Index Theorem. We characterize the global uniqueness for different network topologies, e.g., line, radial and mesh networks. This serves as a starting point to design distributed local algorithms with global behaviors that have low complexity, computationally fast and can run under synchronous and asynchronous settings in practical power grids.

Optimization Decomposition of Resistive Power Networks With Energy Storage

“…However, by observing that ½ AE are matrices with nonpositive off-diagonal elements, the authors in [14], [15] show that (9) can be solved exactly by (10). Furthermore, it has been shown that this tight relaxation result also implies that the Lagrange duality gap of (9) is zero, and (9) can also be solved exactly by a second order cone programming relaxation [12], [13], [22]. A more general result that applies to the AC (alternating current) OPF problem can be found in [12], [13], [15].…”

“…In our recent work [22], we leverage this zero duality gap property to design low-complexity distributed algorithms to solve (9) directly instead of solving the SDP convex relaxation in (10) with a centralized SDP solver (e.g., interior-point method). In fact, the optimal solution of (9) can be unique (for different network topologies, e.g., line, radial and mesh networks).…”

“…Chapter 5 in [18]). Due to the zero duality gap, each decomposed subproblem (11) for a fixed Ø can be solved with the following message passing algorithm (for a fixed ´Øµ for all and Ø, and we use to index the iteration in the algorithm) in [22]. solution of (11) for each Ø ½ Ì [22].…”

“…Due to the zero duality gap, each decomposed subproblem (11) for a fixed Ø can be solved with the following message passing algorithm (for a fixed ´Øµ for all and Ø, and we use to index the iteration in the algorithm) in [22]. solution of (11) for each Ø ½ Ì [22]. The spectral property of the nonnegative matrix ´Øµ in Algorithm 1 can be used to characterize its convergence performance [22].…”

“…The problem formulation in (8) is a nonconvex quadratic constrained quadratic programming (QCQP) problem, which is generally hard to solve. In the following, we first review some results in [14], [15], [22] for a static OPF, and then leverage optimization decomposition techniques in Sections IV and V to solve (8).…”