Hydrogen production from renewable energy sources (RESs) is one of the effective ways to achieve carbon peak and carbon neutralization. In order to ensure the efficient, reliable and stable operation of the DC microgrid (MG) with an electric-hydrogen hybrid energy storage system (ESS), the operational constraints and static dynamic characteristics of a hydrogen energy storage system (HESS) needs to be fully considered. First, different hydrogen production systems, using water electrolysis are compared, and the modeling method of the electrolyzer is summarized. The operational control architecture of the DC MG with electric-hydrogen is analyzed. Combined with the working characteristics of the alkaline electrolyzer, the influence of hydrogen energy storage access on the operational mode of the DC MG is analyzed. The operational control strategies of the DC MG with electric-hydrogen hybrid ESS are classified and analyzed from four different aspects: static and dynamic characteristics of the hydrogen energy storage system, power distribution of the electric-hydrogen hybrid ESS and the efficiency optimization of hydrogen energy storage. Finally, the advantages of a modular hydrogen production system (HPS) are described, and the technical problems and research directions in the future are discussed.