Introduction and expansion of renewable power units (RPUs) in energy systems have significantly changed their operational and planning perspectives. In this regard, power systems could confront with flexibility ramp shortage (FRS) due to the variability of the output power of renewable power resources in the system. Utilities should consider the FRS while conducting operational and planning procedures in power systems. However, due to the FRS, system operators would rely on flexible local units (FLUs) to meet flexibility requirements in the system. Respectively, this paper investigates the planning of hydrogen storage units (HSUs) in a distribution system to meet the ramp constraints in the system. In consequence, the installation of HSUs in the distribution system is optimized while considering the ramp constraints as well as power loss and the operational constraints in the network. Moreover, stochastic programming is employed to address the uncertainty of RPUs in the system. As a result, the developed approach enables addressing the FRS in power systems utilizing hydrogen storage units, which finally facilitates the high integration of RPUs. Finally, this approach is employed to optimize the planning of HSUs in a 37-bus test system to facilitate the high integration of RPUs in the system.