Smart grid applications, such as predicting energy consumption, grid user behavior analysis and predicting energy theft, etc., are data-driven applications that require machine learning with a wealth of data generated from Internet of Things (IoT) based metering devices. However, traditional methods of uploading this huge data to the remote cloud for data analytics may be low efficient due to the non-negligible network transmission delay. By deploying a number of computing-enabled devices at the network edge, edge computing supports the implementation of machine learning close to the power grid environment. Considering the limited computing resources of edge devices and non-independent and identical (non-IID) data source, federated learning is a feasible edge computing based machine learning model. In federated learning, distributed mobile clients and a federated server collaborate to perform machine learning. Generally, the more clients to join the federated learning, the faster to obtain learning convergence and the higher resource utility. However, the communications between clients and the server in training rounds of federated learning may fail due to time-varying link reliability properties in a wireless network of smart grid, which not only slows down the model convergence rate but also wastes resources, such as energy consumption for invalid local training. This paper studies a dynamic federated learning problem in a power grid mobile edge computing (GMEC) environment, considering the high dynamic of link reliability. We design a delay deadline constrained federated learning framework to avoid extremely long training delay, and then formulate a dynamic client selection problem for computing utility maximization in such learning framework. Two online client selection algorithms, including climax greedy and uti-positive guarantee, are proposed to address the problem. The theoretical analysis and simulation results are conducted to illustrate the efficiency of the proposal. INDEX TERMS Mobile edge computing, machine learning, federated learning, smart grid, link reliability.