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With the increasing penetration of wind power into the power grid, maintaining system reliability has been a challenging issue for ISOs/RTOs, due to the intermittent nature of wind power. In addition to the traditional reserves provided by thermal, hydro, and gas generators, demand response (DR) programs have gained much attention recently as another reserve resource to mitigate wind power output uncertainty. However, the price-elastic demand curve is not exactly known in advance, which provides another dimension of uncertainty. To accommodate the combined uncertainties from wind power and DR, we allow the wind power output to vary within a given interval with the price-elastic demand curve also varying in this paper. We develop a robust optimization approach to derive an optimal unit commitment decision for the reliability unit commitment runs by ISOs/RTOs, with the objective of maximizing total social welfare under the joint worst-case wind power output and demand response scenario. The problem is formulated as a multi-stage robust mixed-integer programming problem. An exact solution approach leveraging Benders' decomposition is developed to obtain the optimal robust unit commitment schedule for the problem. Additional variables are introduced to parameterize the conservatism of our model and avoid over-protection. Finally, we test the performance of the proposed approach using a case study based on the IEEE 118-bus system. The results verify that our proposed approach can accommodate both wind power and demand response uncertainties, and demand response can help accommodate wind power output uncertainty by lowering the unit load cost.