Power system reliability evaluation is highly complex and time-consuming, making it challenging to apply within short-term operational timescales. To enable power system operation that accounts for reliability, a reliability-constrained unit commitment (RCUC) model and its two-stage solution method are proposed based on load feasible region model. First, a load feasible region model is employed to formulate reliability constraints, enabling the integration and quantitative evaluation of system reliability during the unit commitment process. Then, a two-stage solution method is developed for the RCUC model to achieve efficient computation. Finally, case studies are performed in the IEEE30 and IEEE RTS-96 systems. The results demonstrate that the proposed RCUC model can effectively reduce the total operational costs, and the two-stage solution method can accurately and efficiently obtain the optimal unit commitment scheme.