Wind generation (WG) units as renewable energy sources (RESs) are increasing in the world due to environmental functions and lack of conventional energy sources. Also, hydrogen storage system (HSS) as an energy storage system (ESS) is used to cope with variable nature of RESs in which the concepts of power to hydrogen (P2H) and hydrogen to power (H2P) are defined. In this work, a risk-averse stochastic operation of HSS and WG is modeled using a scenario-based stochastic approach by considering price-responsive demand response (DR) program. All uncertainties are modeled via a scenario-based stochastic approach while the risk related uncertainties are modeled via the downside risk constraints (DRC) to capture the risk-averse operation of the HSS and WG. In order to investigate the impact of DRC implementation, a risk-averse strategy is compared versus risk-neutral strategy. Compared results show that the risk-in-cost (RIC) is reduced while the expected operation cost (EOC) is raised to deal with the risk of the uncertainty.