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This paper elaborates compact MIP formulations for a discrete unit commitment problem with minimum stop and ramping constraints. The variables can be defined in two different ways. Both MIP formulations are tightened with clique cuts and local constraints. The projection of constraints from one variable structure to the other allows to compare and tighten the MIP formulations. This leads to several equivalent formulations in terms of polyhedral descriptions and thus in LP relaxations. We analyse how MIP resolutions differ in the efficiency of the cuts, branching and primal heuristics. The resulting MIP implementation allows to tackle real size instances for an industrial application.