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The core of solving security-constrained unit commitment (SCUC) problems within the Lagrangian relaxation framework is how to obtain feasible solutions. However, due to the existence of the transmission constraints, it is very difficult to determine if feasible solutions to SCUC problems can be obtained by adjusting generation levels with the commitment states obtained in the dual solution of Lagrangian relaxation. The analytical and computational necessary and sufficient conditions are presented in this paper to determine the feasible unit commitment states with grid security constraints. The analytical conditions are proved rigorously based on the feasibility theorem of the Benders decomposition. These conditions are very crucial for developing an efficient method for obtaining feasible solutions to SCUC problems. Numerical testing results show that these conditions are effective.