In this paper we discuss and compare a few methods combining mathematical optimization and heuristics to reduce the search space of the MIP problem when solving the Unit Commitment (UC) problem. Solving the UC problem is already today often slow and can be expected to become even more complex with the increasing number of small and scattered renewable generation units with associated energy storages. We show that by analyzing the problem before solving the full MIP we can remove (fix) a set of binary variables, leading to a speed-up between 2 and 5 times, empirically without compromising optimality.