Distributed Optimal Resource Allocation with Time-Varying Quadratic Cost Functions and Resources over Switching Agents

This paper proposes a distributed solution for an optimal resource allocation problem with a time-varying cost function and time-varying demand. The objective is to minimize a global cost, which is the summation of local quadratic time-varying cost functions, by allocating time-varying resources. A reformulation of the original problem is developed and is solved in a distributed manner using only local interactions over an undirected connected graph. In the proposed algorithm, the local state trajectories converge to a bounded neighborhood of the optimal trajectory. This bound is characterized in terms the parameters of the cost and topology properties. We also show that despite the tracking error, the trajectories are feasible at all times, meaning that the resource allocation equality constraint is met at every execution time. Our algorithm also considers the possibility of some generators going out of production from time to time and adjusts the solution so that the remaining generators can meet the demands in an optimal manner. Numerical examples demonstrate our results.