Optimality Conditions for Distributed Primary Control in Energy State Space

This paper addresses the problem of designing optimal distributed control in the changing electric energy systems. We aim to improve the aggregated reactive power inefficiency of inverter-controlled and power-coupled electrical systems. To achieve this, we formulate an optimal control problem aimed at optimizing reactive power and interface mismatch using the energy state space model. Despite the general complexity of solving an optimal control problem, we demonstrate that our proposed model in energy state space admits a linear optimality condition in the variational form. We demonstrate that this condition can be decomposed into local state variables and measurements from adjacent connected components, facilitating a distributed approach to optimal control design. The variables exchanged with the neighboring components are technology agnostic and can be interpreted as costate variables with intuitive physical meaning. We illustrate the conditions and results of the distributed optimal control using a small system with controllable source serving a time-varying power load. Additionally, we explore voltage regulation in the proposed control design, which can be implemented using fast switching power electronics.