Sparse tableau relaxation for the optimal power flow problem

Optimal power flow (OPF) approaches employing such methods as semi-definite programming have garnered considerable interest in the literature of the last decade. The OPF formulations for these approaches have almost universally relied on Y bus admittance matrix representations, which derive from nodal analysis, and restrict allowable network elements to be voltage-controlled only. Limitations of nodal analysis long been recognized, and to overcome these, commercial power system software often employs modified nodal analysis (MNA). However, with OPF requiring many monitored links with constrained flows, the work here argues for a formulation even more versatile than MNA: Sparse Tableau Formulation (STF) of network constraints, with multi-port representation of individual components. The vast majority of transmission network components (e.g., transmission lines, transformers) have multi-port voltage-current behavior that is well-modelled as linear. Therefore, nonlinearities appear only in equations associated with generation and load at each bus. Utilizing STF, these sources of nonconvexity are confined to constraints local to each bus. This opens the door to simple, engineering-based convex relaxations. We introduce a constant current source at each bus, with these sources' values bounded in a manner that contains any feasible current produced by the actual generator or load model. The tightness of the relaxation is shown to improve when angle constraints are provided, and a DCOPF-based heuristic method is proposed to obtain such angle bounds.