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We present a perfect formulation for a single generator in the unit commitment problem, inspired by the dynamic programming approach taken by Frangioni and Gentile. This generator can have characteristics such as ramp up/down constraints, time-dependent start-up costs, and start-up/shut-down limits. To develop this perfect formulation we extend the result of Balas on unions of polyhedra to present a framework allowing for flexible combinations of polyhedra using indicator variables. We use this perfect formulation to create a cut-generating linear program, similar in spirit to lift-and-project cuts, and demonstrate computational efficacy of these cuts in a utility-scale unit commitment problem.

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