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Under the background of carbon emission abatement worldwide, carbon trading is becoming an important carbon financing policy to promote emission mitigation. Aiming at the emerging coupling among various energy sectors, this paper proposes a bi-level scheduling model to investigate the low-carbon operation of the electricity and natural gas integrated energy systems (IES). Firstly, an optimal energy flow model considering carbon trading is formulated at the upper level, in which carbon emission flow model is employed to track the carbon flows accompanying energy flows and identify the emission responsibility from the consumption-based perspective, and the locational marginal price is determined at the same time. Then at the lower level, a developed demand-side management strategy is introduced, which can manage demands in response to both the dynamic energy prices and the nodal carbon intensities, enabling the user side to participate in the joint energy and carbon trading. The bi-level model is solved iteratively and reaches an equilibrium. Finally, case studies based on the IEEE 39-bus system and the Belgium 20-node system illustrate the effectiveness of the proposed method in reducing carbon emissions and improving consumer surplus.