Computational Approaches for Optimal Load Shedding Control of DC Networks under Cascading Failure

We consider discrete-time dynamics, for cascading failure in DC power networks, whose map is composition of failure rule with control policy. Under feasible control actions, supply-demand at the nodes is monotonically nonincreasing in magnitude. Under the failure rule, a link is removed permanently if the flow on it exceeds given thermal capacity constraints. We consider a finite horizon optimal control problem to steer the network from an arbitrary initial state, defined in terms of active link set and supply-demand vector at the nodes, to a feasible state, i.e., a state which is invariant under the failure rule. There is no running cost and the value associated with a feasible terminal state is the associated cumulative supply-demand. We propose two novel computational frameworks for control synthesis. The first is a network decomposition approach which can be implemented in two iterations for tree reducible networks, and leads to a semi-analytical solution for the unit time horizon case. The second approach interprets optimal control synthesis as an optimal search problem. An algorithmic procedure is provided to compute the one-stage reachable set using arrangement of hyperplanes to facilitate this search. We outline a approximation strategy to reduce computations in the search.