This paper tackles a complex variant of the unit commitment (UC) problem at Hydro-Quebec, referred to as the transient stability constrained unit commitment (TSCUC) problem. First, the complete problem is described as a mixed-integer linear program (MILP). Next, an investigation strategy is conducted to identify complexity sources. Then a matheuristic is proposed, taking advantage of the temporal dimension of the problem. Finally, the matheuristic is enhanced by tuning the solver's configuration and by relying on linearization techniques. The benefits of the matheuristic are highlighted by using real-life instances from Hydro-Quebec.