Hydrogen-fired gas turbines can play an important role in carbon-neutral energy and industry sectors. However, the required demand-oriented supply of -neutral hydrogen is technically and economically challenging. These challenges arise due to interdependencies between the volatility of renewable power generation, available hydrogen production capacities, available hydrogen storage capacities and the operational demands to be met by gas turbines. The present study aims to quantify these interdependencies by conducting a model-based analysis of an exemplary CHP system featuring a hydrogen-fired industrial gas turbine with on-site hydrogen production via electrolysis and on-site hydrogen storage. To identify the sought-after interdependencies, simulations featuring various system parameterizations are analyzed. If only local power surpluses are utilized for the operation of electrolyzers, the results show a nonlinear impact of both the hydrogen production capacity and the hydrogen storage capacity on the hydrogen-based gas turbine operation. Furthermore, the results indicate that an exclusive utilization of local power surpluses leads to limited periods of hydrogen-based gas turbine operation and low utilization rates of the hydrogen production and storage capacities. If additional power for the operation of electrolyzers is supplied by the grid, increased utilization rates and prolonged periods of hydrogen-based gas turbine operation can be achieved. However, to realize an overall reduction of emissions, this mode of operation requires the supply of large quantities of renewable power by the grid. The results of an additional economic assessment reveal that both investigated operational modes are not economically viable within the considered economic framework.