To achieve low-carbon economic operation of integrated energy system, a distributed optimal dispatch model is proposed. Firstly, green certificate and stepped carbon trading models are established, followed by two comprehensive demand response models. An optimization model is developed with the objective of minimizing the total system cost. To protect the privacy of electricity and gas networks, the electricity and gas subsystems are decoupled. A distributed co-planning method of energy system is constructed based on the alternating direction method of multipliers (ADMM). Simulation experiments on the modified IEEE33 power grid and Belgium 20 gas network model demonstrate that this optimization method enables decentralized autonomy, reducing operation costs and carbon emissions effectively.