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With the increasing penetration of variable renewable generations, independent system operators (ISOs)/regional transmission organizations (RTOs) are faced with new challenges for the secure and economic operation of power systems. This paper proposes an effective approach for deriving robust solutions to the security-constrained unit commitment (SCUC) problem, which considers load and wind uncertainties via interval numbers. Different from most robust optimization-based SCUC approaches in literature which explore robust unit commitment (UC) solutions for immunizing against the worst economic scenario in terms of the highest minimum dispatch cost, the proposed robust SCUC model minimizes operation cost for the base case while guaranteeing that the robust UC and dispatch solutions could be adaptively and securely adjusted in response to uncertain intervals. Thus, the proposed model achieves smaller unit commitment costs while maintaining the solution robustness as compared with literature. In addition, the proposed model describes base case dispatches and corrective actions in uncertain intervals, which is more consistent with the current day-ahead and real-time markets. Furthermore, besides budget constraints used in literature, this paper also considers load and wind variability correlations in constructing uncertain intervals, which would eliminate unlike-to-happen scenarios and further limit the level of conservatism of the robust solution. The proposed robust SCUC model is solved by Benders decomposition, which decomposes the original problem into a master UC problem for the base case and subproblems for the base case network evaluation and the security checking for uncertain intervals. Feasibility cuts are generated and fed back to the master problem for further iterations when violations are identified in subproblems. Numerical case studies on the modified IEEE 118-bus system illustrate the effectiveness of the proposed robust SCUC model for the secure and ec- nomic operation of power systems under various uncertainties.