Hydrogen produced from renewable electricity will play an important role in deep decarbonisation of industry. However, adding large electrolyser capacities to a low-carbon electricity system also increases the need for additional electricity generation from variable renewable energies. This will require hydrogen production to be variable unless other sources provide sufficient flexibility. Existing sources of flexibility in hydro-thermal systems are hydropower and thermal generation, which are both associated with sustainability concerns. In this work, we use a dispatch model for the case of Sweden to assess the power system operation with large-scale electrolysers, assuming that additional wind power generation matches the electricity demand of hydrogen production on average. We evaluate different scenarios for restricting the flexibility of hydropower and thermal generation and include 29 different weather years to test the impact of variable weather regimes. We show that (a) in all scenarios electrolyser utilisation is above 60% on average, (b) the inter-annual variability of hydrogen production is substantial if thermal power is not dispatched for electrolysis, and (c) this problem is aggravated if hydropower flexibility is also restricted. Therefore, either long-term storage of hydrogen or backup hydrogen sources may be necessary to guarantee continuous hydrogen flows. Large-scale dispatch of electrolysis capacity supported by wind power makes the system more stable, if electrolysers ramp down in rare hours of extreme events with low renewable generation. The need for additional backup capacities in a fully renewable electricity system will thus be reduced if wind power and electrolyser operation are combined in the system.