PhD Thesis Defense
Distributed energy resources play an important role in today's distribution power system. The Optimal Power Flow (OPF) problem is fundamental in power systems as many important applications such as economical dispatch, battery displacement, unit commitment, and voltage control, can be formulated as an OPF. A paradoxical observation is the problem's theoretical complexity and its practical simplicity. On one hand, the problem is well known to be non-convex and NP-hard, so it is likely that no simple algorithms can solve all problem instances efficiently. On the other hand, there are many known algorithms which perform extremely well in practice for both standard testcases and real-world systems. This thesis attempts to reconcile this seeming contradiction.
Specifically, this thesis focuses on two types of properties that may underlie the practical simplicity of OPF problems. The first property is the exactness of relaxations, meaning that one can find a convex relaxation of the original non-convex problem such that the two problems share the same optimal solution. This property would allow us to convexify the non-convex problem without altering the optimal solution and cost. The second property is that all locally optimal solutions of the non-convex problem are also globally optimal. This property allows us to apply local algorithms such as gradient descent without being trapped at some spurious local optima. We focus on distribution systems with radial networks (i.e., the underlying graphs are trees). We consider both single-phase models and unbalanced multi-phase models, since most real-world distribution systems are multi-phase unbalanced, and DERs can be connected in either Wye or Delta configurations.