Stochastic, nonlinear and time-varying factors bring great challenges to accurate modeling and design of real-time (RT) energy management systems (EMSs) for modern energy systems with intermittent renewable energy sources. In this paper, we propose a novel model-free RT-EMS based on the predictive adaptive dynamic programming (ADPredictive) algorithm for energy scheduling problems in a home EMS environment. The proposed RT-ADPreditive EMS can minimize the total cost of electricity, reduce battery life loss, and maximize the utilization of renewable energy generation. In the proposed RT-ADPredictive EMS, the gated recurrent unit (GRU) neural network (NN) is employed to perform online prediction of renewable energy generation and load consumption of the household based on RT data. The RT-ADPreditive algorithm can approximate the performance index function and the optimal control law based on the RT data and predicted data for the next time step. Convergence of the proposed method is mathematically proven, and a hardware-in-the-loop (HIL) experimental platform comprising dSPACE and RT-lab is built to verify the effectiveness of the proposed RT-ADPreditive home EMS.