This paper introduces sufficient schedulability tests for fixed-priority pre-emptive scheduling of a real-time system under energy constraints. In this problem, energy is harvested from the ambient environment and used to replenish a storage unit or battery. The set of real-time tasks is decomposed into two different types of task depending on whether their rate of energy consumption is (i) more than or (ii) no more than the storage unit replenishment rate. We show that for this task model, where execution may only take place when there is sufficient energy available, the worst-case scenario does not necessarily correspond to the synchronous release of all tasks. We derive sufficient schedulability tests based on the computation of worst-case response time upper and lower bounds. We show that these tests are sustainable with respect to decreases in the energy consumption of tasks, and increases in the storage unit replenishment rate. Further, we show that Deadline Monotonic priority assignment is optimal with respect to the derived tests. We examine both the effectiveness and the tightness of the bounds, via an empirical investigation.