# Computing Optimal Control of Cascading Failure in DC Networks

We consider discrete-time dynamics, for cascading failure in DC networks, whose map is composition of failure rule with control actions. Supply-demand at nodes is monotonically nonincreasing under admissible control. Under the failure rule, a link is removed permanently if its flow exceeds its capacities. We consider finite horizon optimal control to steer the network from an arbitrary initial state, defined in terms of active link set and supply-demand at nodes, to a feasible state, i.e., a state that is invariant under the failure rule. There is no running cost and the reward associated with a feasible terminal state is the associated cumulative supply-demand. We propose two approaches for computing optimal control, and provide time complexity analysis for these approaches. The first approach, geared towards tree reducible networks, decomposes the global problem into a system of coupled local problems, which can be solved to optimality in two iterations. In the first iteration, optimal solutions to the local problems are computed, from leaf nodes to the root node, in terms of the coupling variables. In the second iteration, in the reverse order, the local optimal solutions are instantiated with specific values of the coupling variables. Restricted to constant controls, the optimal solutions to the local problems possess a piecewise linear property, which facilitates analytical solution. The second approach computes optimal control by searching over the reachable set, which is shown to admit an equivalent finite representation by aggregation of control actions leading to the same reachable active link set. An algorithmic procedure to construct this representation is provided by leveraging and extending tools for arrangement of hyperplanes and convex polytopes. Illustrative simulations, including showing the effectiveness of a projection-based approximation algorithm, are also presented.

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