With the maturity of hydrogen storage technologies, hydrogen-electricity coupling energy storage in green electricity and green hydrogen modes is an ideal energy system. The construction of hydrogen-electricity coupling energy storage systems (HECESSs) is one of the important technological pathways for energy supply and deep decarbonization. In a HECESS, hydrogen storage can maintain the energy balance between supply and demand and increase the utilization efficiency of energy. However, its scenario models in power system establishment and the corresponding solution methods still need to be studied in depth. For accelerating the construction of HECESSs, firstly, this paper describes the current applications of hydrogen storage technologies from three aspects: hydrogen production, hydrogen power generation, and hydrogen storage. Secondly, based on the complementary synergistic mechanism of hydrogen energy and electric energy, the structure of the HECESS and its operation mode are described. To study the engineering applications of HECESSs more deeply, the recent progress of HECESS application at the source, grid, and load sides is reviewed. For the application of the models of hydrogen storage at the source/grid/load side, the selection of the solution method will affect the optimal solution of the model and solution efficiency. As solving complex multi-energy coupling models using traditional optimization methods is difficult, the paper therefore explored the advantages of deep reinforcement learning (DRL) algorithms and their applications in HECESSs. Finally, the technical application in the construction of new power systems supported by HECESSs is prospected. The study aims to provide a reference for the research on hydrogen storage in power systems.