Sparsity-Exploiting Moment-Based Relaxations of the Optimal Power Flow Problem
The paper "Sparsity-Exploiting Moment-Based Relaxations of the Optimal Power Flow Problem" authored by Daniel K. Molzahn and Ian A. Hiskens presents a sophisticated advancement in solving the Optimal Power Flow (OPF) problem using moment relaxations. The OPF problem aims to find the optimal operating conditions of an electric power system while satisfying various network constraints and is fundamentally non-convex due to the inherent non-linearities in power flow equations, posing significant computational challenges.
Traditional approaches to solving the OPF problem, such as interior point methods and other local optimization techniques, can fail either by not converging or by converging to local rather than global optima. This has prompted the exploration of convex relaxations, most notably through semidefinite programming (SDP). The SDP relaxations provide global solutions under specific conditions, but they commonly fall short in certain practical scenarios.
This paper introduces a stronger class of relaxations known as "moment relaxations," derived from the Lasserre hierarchy that approximates the global solutions of polynomial optimization problems, of which the OPF is a subset. The authors detail how to employ these relaxations by increasing the order in Lasserre's hierarchy to tighten the solution space, albeit at the cost of increased computational burden due to the expansion in dimension of the matrices involved in the relaxation.
A critical innovation in this work is the exploitation of sparsity present in power networks, which is pivotal in scaling the moment relaxation approach to real-world power systems consisting of hundreds of buses. By decomposing the moment relaxation using chordal extensions of the power network graph and applying sparse matrix techniques, the authors have extended computational tractability.
Moreover, the paper presents an iterative algorithm that selectively applies higher-order moment relaxations only at specific buses within the network where tighter relaxation is necessary according to a heuristic based on power injection mismatches. This targeted approach significantly reduces computation while maintaining global optimality guarantees, as evidenced by extensive testing on standard IEEE bus systems.
Results show that this method globally solves several OPF instances where existing SDP relaxations fail, engaging orders up to the third in the hierarchy for cases that were previously unsolvable. However, the method's efficacy and computational load are influenced by the specific problem structure and network topology, which motivate further research into heuristic and algorithm optimization.
In conclusion, this paper contributes a comprehensive and technically profound advancement for solving OPF problems, bridging a significant gap where traditional and even some recent convex approaches fall short. The method holds promising implications for enhancing the operational efficiency and stability of large-scale power networks. Future research directions may delve into improving heuristic strategies for selecting higher-order buses, broadening the relaxation framework to incorporate additional power system operational constraints, and extending the methodology to address other complex optimization problems in power systems and beyond.