The Price of Local Voltage Control in Distribution Systems

In this thesis, motivated by an optimization model for understanding the local voltage control in the distribution system, we introduce a minimum cost voltage control problem. We propose a minimum cost voltage control algorithm, and show that it cannot be implemented based on only local voltage. We further show that no local voltage control algorithm can achieve the minimum cost. However, in practice we may have the constraint or preference to adopt local controls. It is important to characterize the performance of the local control with respect to system-wide properties such as the aggregate cost. We thus introduce the notion of the price of local control (PoLC) to characterize the performance of local voltage control in terms of the aggregate cost minimization. Specifically, we use the gap between the minimum cost and the cost achieved by the network equilibrium of local voltage control as the metric for PoLC. We characterize how the PoLC scales with the size, topology, and heterogeneity of the power network for a few special cases. In particular, we identify a universal upper bound for PoLC that is independent of the power line reactance and the topology of the network; and for the tree network, we find that the PoLC saturates with the size of the network. Such results will be insightful to understanding the limitation of local controls and help deliberate the trade-off between informational constraint/requirement and system-wide efficiency in design choice