Convex relaxation of Sparse Tableau Formulation for the AC optimal power flow

Optimal power flow (OPF) approaches employing such methods as semi-definite programming (SDP) 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). Here, we consider a novel general power system modeling approach based on multi-port representation of individual components with Sparse Tableau Formulation (STF) of network constraints, which is more versatile than MNA for OPF requiring many monitored links with constrained flows. In STF, one is better able to exploit the fact that the vast majority of power grid network elements have voltage-current behavior that is well-modeled as linear. This opens the door to simple, engineering-based convex relaxations. We discuss two relaxations, admittance-based and current-based. The tightness of the relaxation is shown to improve when angle constraints and narrow bounds for active power generation are provided. Sequential bound tightening and reduced spatial branch-and-bound are discussed to obtain stronger relaxation solution. We conduct case studies to show the effectiveness of our relaxations with standard test cases.