Formulations for designing robust networks. An application to wind power collection

We are interested in the design of survivable capacitated rooted Steiner networks. Given a graph G = (V,E), capacity and cost functions on E, a root r, a subset T of V of terminals and an integer k, we search for a minimum cost subset E′ ⊂ E, covering T and r, such that the network induced by E′ is k-survivable: after the removal of any k edges, there still exists a feasible flow from r to T . We also consider the possibility of protecting a given number of edges. We propose three different formulations: a cut-set, a flow and a bi-level formulation where the second-level is a min-max problem (with an attacker and a defender). We propose algorithms for each problem formulation and compare their efficiency.