Federated Learning (FL) is a powerful paradigm for distributed learning, but its increasing complexity leads to significant energy consumption from client-side computations for training models. In particular, the challenge is critical in energy-harvesting FL (EHFL) systems where participation availability of each device oscillates due to limited energy. To address this, we propose FedBacys, a battery-aware EHFL framework using cyclic client participation based on users' battery levels. By clustering clients and scheduling them sequentially, FedBacys minimizes redundant computations, reduces system-wide energy usage, and improves learning stability. We also introduce FedBacys-Odd, a more energy-efficient variant that allows clients to participate selectively, further reducing energy costs without compromising performance. We provide a convergence analysis for our framework and demonstrate its superior energy efficiency and robustness compared to existing algorithms through numerical experiments.