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Abstract: With booming development of fifth-generation network technology and Internet of Things, the number of end-user devices (EDs) and diverse applications is surging, resulting in massive data generated at the edge of networks. To process these data efficiently, the innovative mobile edge computing (MEC) framework has emerged to enable low-latency and efficient computing capabilities close to the user traffic. Recently, federated learning (FL) has demonstrated its empirical success in edge computing due to its privacy-preserving advantages. Thus, it becomes a promising solution for analyzing and processing distributed data on EDs in various machine learning tasks, which are the major workloads in MEC. Unfortunately, EDs are typically powered by batteries with limited capacity, which brings challenges when performing energy-intensive FL tasks. To address these challenges, many strategies have been proposed to save energy in FL. Considering the absence of a survey that thoroughly summarizes and classifies these strategies, in this paper, we provide a comprehensive survey of recent advances in energy-efficient strategies for FL in MEC. Specifically, we first introduce the system model and energy consumption models in FL, in terms of computation and communication. Then we analyze the challenges regarding improving energy efficiency and summarize the energy-efficient strategies from three perspectives: learning-based, resource allocation, and client selection. We conduct a detailed analysis of these strategies, comparing their advantages and disadvantages. Additionally, we visually illustrate the impact of these strategies on the performance of FL by showcasing experimental results. Finally, several potential future research directions for energy-efficient FL are discussed.