The Intentional Controlled Islanding (ICI) and the Black Start Allocation (BSA) are two examples of problems in the power systems literature that have been formulated as Mixed Integer Programs (MIPs). A key consideration in both of these problems is that each island must have at least one energized generator. In this paper, we provide three alternative MIP formulations for this restriction, show their equivalence, and prove that two of them are stronger in terms of their linear programming relaxation than the one most commonly used in the power systems literature. Since the time to solve MIPs can vary significantly between equivalent formulations, we also present computational experiments on IEEE test systems for the ICI and BSA problems. We observe that a polynomially separable, exponential in size, strong formulation yields the best performance for the BSA problem and exhibits a comparable performance to a linear in size, weak formulation for the ICI problem.
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