Federated learning (FL) on mobile edge devices suffers from heterogeneous computation and communication capabilities, time-varying wireless channels, and strict battery constraints. Existing client selection methods typically optimize per-round performance while ignoring heterogeneous energy consumption and per-device energy budgets, which may cause premature client dropout and degraded model accuracy. This paper proposes FEPCS, a Fine-grained Energy Planning Client Selection scheme for energy-efficient FL in mobile edge environments. At the client level, FEPCS dynamically adjusts local training workloads based on residual energy and estimates transmission energy using historical channel information. At the system level, it introduces a progressive participation rate and an energy budget deviation metric, and combines these with data contribution into a unified selection criterion. Extensive simulations under heterogeneous settings show that FEPCS achieves higher test accuracy, significantly extends client survival time, and effectively reduces client dropouts compared with existing schemes, while operating under the same energy constraints.