Robustness of DC Networks With Controllable Link Weights

We study robustness of dc power networks with controllable link susceptances, or
weights
, toward balanced disturbances to a nominal supply–demand vector. Consider the set of balanced disturbances, for which there exist feasible link weights so that the resulting link flows are within specified capacity bounds. The margin of robustness is defined as the radius of the largest
$\ell _1$
ball around the origin, which belongs to this set. Computation of this margin can be posed as a nonconvex optimization problem. We propose an equivalent multilevel programming approach for a class of networks, in which the nodes with nonzero demand or supply are relatively sparse. This approach is based on recursive application of equivalent bilevel formulation for a relevant class of, possibly nonconvex, optimization problems. The lower level problem in each recursion corresponds to replacing a subnetwork by a (virtual) link with equivalent weight and capacity functions. The equivalent capacity function for a
link-reducible network
possesses a strong quasi-concavity property. This property facilitates an easy solution to the multilevel programming formulation for
tree-reducible networks
. The computational gain from such a reduction procedure is illustrated via simulations on a benchmark IEEE network.