Under the background of "dual-carbon" goal, aiming at the contradiction between operation economy and low-carbon of rural integrated energy system (IES), IES models of power-to-gas (P2G), ground source heat pump (GCHP) and energy storage devices are established. According to the perceived fuzziness of time-of-use electricity price, time-of-use gas price and heat load, the demand response model of electricity-heat-gas load is put forward. Combining the power purchase, gas purchase, carbon emission intensity of gas turbine operation and carbon dioxide absorption coefficient of P2G operation, the IES carbon emission mechanism is established. A multi-objective optimization function with improved customer satisfaction, economy and carbon emission is constructed, and MAPSO (multi-group adaptive particle swarm optimization) algorithm is used for optimization analysis, and the Pareto optimal frontier solution sets in different scenarios are obtained. The simulation results show that P2G, GCHP and other units can improve the coupling degree of energy on the supply side, adjust the energy supply mode and promote the consumption of clean energy. Multi-objective optimization with improved user satisfaction plays a positive role in improving the practicability of IES.