The scenario approach is a general data-driven algorithm to chance-constrained optimization. It seeks the optimal solution that is feasible to a carefully chosen number of scenarios. A crucial step in the scenario approach is to compute the cardinality of essential sets, which is the smallest subset of scenarios that determine the optimal solution. This article addresses the challenge of efficiently identifying essential sets. For convex problems, we demonstrate that the sparsest dual solution of the scenario problem could pinpoint the essential set. For nonconvex problems, we show that two simple algorithms return the essential set when the scenario problem is nondegenerate. Finally, we illustrate the theoretical results and computational algorithms on security-constrained unit commitment (SCUC) in power systems. In particular, case studies of chance-constrained SCUC are performed in the IEEE 118-bus system. Numerical results suggest that the scenario approach could be an attractive solution to practical power system applications.