Optimization of Distribution Power Networks: from Single-Phase to Multi-Phase

Citation

Zhou, Fengyu (2022) Optimization of Distribution Power Networks: from Single-Phase to Multi-Phase. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/tg26-9857. https://resolver.caltech.edu/CaltechTHESIS:06012022-005449566

Abstract

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 economic dispatch, battery displacement, unit commitment, and voltage control can be formulated as an OPF. A paradoxical observation is the problem's complexity in theory but simplicity in practice. On the 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 test cases 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 simplicity in practice 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 distributed energy resources (DERs) can be connected in either Wye or Delta configurations.

The main results of this thesis are two-fold. In the first half, we propose a class of sufficient conditions for a non-convex problem to simultaneously have exact relaxation and no spurious local optima. Then we apply the result to single-phase system and conclude that if all buses have no injection lowerbounds, then both properties (exactness and global optimality) can be achieved. While the same condition is already known to be sufficient for exactness, our work is the first to extend it to global optimality. In the second half, we focus on the exactness property for multi-phase systems. For systems without Delta connections, the exactness can be guaranteed if 1) the binding constraints are sparse in the network at optimality; or 2) all nodal prices fall within a narrow range. Using the DC model as an approximation, we further analyze the OPF sensitivity and explain why nodal prices tend to be close to each other. In the presence of Delta connections, we conclude that the inexactness can be resolved by either postprocessing an optimal solution, or adding a new regularization term in the cost function. Both methods achieve global optimality for IEEE standard test cases.

Thesis Files