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We investigate the corrective unit commitment problem to deal with disruption in power system operations caused by an unforeseen unit breakdown with stochastic duration. Since the original unit schedule is no longer feasible when a unit breaks down during operation, a corrective scheduling that provides an immediate response to such a disruption is needed to update the original schedule in time. The objective of the corrective scheduling is to minimize the generation cost and the deviation from the original schedule. The corrective scheduling problem is formulated as a mixed integer nonlinear programming model where the stochastic duration is expressed by tree-structured duration scenarios. The proposed variable splitting-based Lagrangian relaxation algorithm decomposes the problem into multiple single-unit subproblems and a linear programming-type artificial variable subproblem. Each single-unit subproblem is solved by a two-stage procedure. In the first stage, the generating level of the unit in each committed period is determined optimally. In the second stage, before the dynamic programming is called, an effective pre-processing technique based on optimality conditions is applied to speed up the procedure. The dual problem is solved by a bundle method. The numerical results show that the algorithm can find solutions very close to optimums within a reasonable time.