Hydrogen is gradually becoming one of the important carriers of global energy transformation and development. To analyze the influence of the hydrogen storage module (HSM) on the operation of the gas-electricity integrated energy system, a comprehensive energy system model consisting of wind turbines, gas turbines, power-to-hydrogen (P2H) unit, and HSM is proposed in this paper. The model couples the natural gas network and power grid bidirectionally, and establishes a mixed integer nonlinear programming problem considering HSM. The linearization model of the natural gas pipeline flow equation and the generator set equation is constructed by piecewise linearization method to improve the efficiency of solving the model. And the energy flow distribution in the gas-electricity integrated energy system is finally solved. In Model 1, compared with not considering the installation of P2H units, when the hydrogen doping ratio is 10%, the operating cost can be reduced by 6.63%, and the wind curtailment cost can be reduced by 17.54%, and the carbon emission can be reduced by 298.7 tons. The optimization results of Model 2 reveal that compared with no HSM, the system operating cost is reduced by 5.96%, the hydrogen content level in the natural gas pipeline network is increased by 42.12%, and the carbon emission of the system is reduced by 117.6 tons, and the fluctuation of wind power is suppressed. This study demonstrates the feasibility of large-scale absorption of renewable energy through HSM.