We propose a generalized N - k security criterion, which is called N - alpha k, and applies it to a robust contingency-constrained unit commitment (CCUC) model. In the previous works, the N - k security criterion takes into account the simultaneous loss of up to k components. However, a robust solution of N - k CCUC model may be over-conservative in most cases. To mitigate this over-conservativeness, we propose a robust N - alpha k CCUC model which guarantees N - 2 security criterion throughout the service hours and adopts N - k security criterion only for a few hours. The operational information is used to define an uncertainty set of security criterion. In addition, we introduce a constant, which we denote as Delta to control the conservativeness of the N - alpha k security criterion. We propose a mixed-integer linear programming formulation for the robust N - alpha k CCUC problem. The proposed N - alpha k CCUC model can provide cost savings as well as the robustness of the solutions with a proper criterion setting. Benders decomposition (BD) type cutting plane algorithm is used to solve the proposed model, and maximum feasible subsystem cuts are additionally generated at each BD iteration to improve the performance of the BD algorithm. Numerical case studies on a modified IEEE 118-bus system illustrate the effectiveness of the proposed robust N - alpha k CCUC model and provide some guidelines for alpha criterion setting.