This paper studies a wireless powered communication network (WPCN) with non-orthogonal multiple access (NOMA) under successive interference cancellation (SIC) constraints, where the users first harvest energy from the power station and then transmit data to the information receiver simultaneously. Under this setup, we investigate the system throughput maximization problem. We first formulate an optimization problem for a general case, which is non-convex. To derive the optimal solution, new variables are introduced to transform the initial problem into a convex optimization problem. For a special case, i.e., two-user case, the optimal solution is derived as a closed-form expression. Simulations on the effect of SIC constraints show the importance of the distinctness among users' channels for the proposed model.