This paper presents a stochastic mixed-integer nonlinear programming model (MINLP) for the security-constrained energy management system (EMS) of microgrids under demand and renewable generation uncertainties. Considering a balanced single-phase representation of the network, the proposed MINLP model is transformed into a mixed-integer second-order cone programming (MISOCP) model that can be solved via off-the-shelf convex programming solvers. The proposed formulation considers photovoltaic (PV) generation, energy storage systems (ESS), direct load control (DLC) and a diesel generator (genset), which can be turned on when the microgrid is operating in isolated mode. The proposed model minimizes the average operational costs for the day-ahead scheduling. Constraints consider the operation in either grid-connected or isolated mode due to a predefined set of plausible contingencies, hence the term security-constrained. The proposed model is tested using data of the real microgrid Laboratory of Intelligent Electrical Networks (LabREI), located at the UNICAMP facilities. Results show that the proposed model is suitable for real-world applications since it provides cost-efficient and contingency-robust solutions.