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This study examines the use of bilevel programming to analyse the vulnerability of power systems under multiple contingencies. One of the main purposes of this study is to explain the state of the art of the subject matter. A minimum vulnerability model and a maximum vulnerability model are presented and discussed. In both models, the upper-level optimisation determines a set of simultaneous outages in the transmission network whereas the lower-level optimisation models the reaction of the system operator against the outages identified in the upper level. The system operator reacts by minimising the system load shed through an optimal operation of the power system. Two solution approaches for the resulting mixed-integer non-linear bilevel programs are analysed and compared. Both methodologies are based on the equivalent transformation of the lower-level problem into a set of constraints, so that the original bilevel programs, respectively, become a single-level optimisation problem. The first approach is based on the application of Karush-Kuhn-Tucker optimality conditions whereas the second procedure relies on duality theory. This study shows that both approaches are essentially equivalent from a rigorous mathematical viewpoint; however, the second method is more suitable for off-the-shell branch-and-cut software as corroborated by numerical simulations.