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.
We consider discrete-time dynamics, for cascading failure in DC networks, whose map is composition of failure rule with control actions. Supply-demand at the nodes is monotonically non-increasing under admissible control. Under the failure rule, a link is removed permanently if its flow exceeds capacity constraints . 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 the nodes, to a feasible state, i.e., a state which 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 . 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. When restricted to the class of one-shot control actions, the optimal solutions to the local problems possess a piecewise affine 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 polytopes. Illustrative simulations, including showing the effectiveness of a projection-based approximation algorithm, are also presented.
A
the
R
nonincreasing
non-increasing
R
its capacities
capacity constraints
A
the
R
that
which
D
, and provide time complexity analysis for these approaches
R
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
When restricted to the class of one-shot control actions, the optimal solutions to the
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linear
affine
D
convex