Hydrogen is emerging as a crucial component for the advancement and integration of renewable energy sources (RESs) within modern power systems. It plays a vital role as an energy storage system (ESS), ensuring stability and reliability in the power grid. Due to its high energy density, large storage capacity, and fast operational capabilities, hydrogen is rapidly gaining popularity as a preferred choice among other ESSs. In a hydrogen energy system, hydrogen stored in the hydrogen storage system is converted into direct current (DC) power by a hydrogen fuel cell during energy shortages in the power system. Conversely, an electrolyzer converts excess power from the grid into hydrogen, which is stored in the hydrogen storage system. These operations necessitate the implementation of various control systems to facilitate the connection of fuel cells and electrolyzers to the power grid. These control systems include power electronic converters and DC link regulation control systems. This review article presents a comprehensive examination of the current challenges related to the control aspects of integrating hydrogen energy storage technologies into modern power grids. It provides an in-depth analysis of the advancements and deployments of control systems associated with these technologies, along with the barriers and prospects they entail. By focusing on the integration of renewable energy sources, this study aims to assist researchers, engineers, and policymakers in gaining a comprehensive understanding of the current state and future possibilities of hydrogen energy storage systems.