## 1 Introduction  
Because of the continuous increase of greenhouse gas emissions, traditional production and lifestyles must be changed to meet the needs of low-carbon energy transformation [1]. In this context, the integrated energy system (IES) has become an important form of efficient energy use [2]. Analyzing the traits of the power-gas-heat network, carbon measurement methods, market mechanism, and scheduling strategy to improve the economy and low carbon of IES is greatly meaningful [3]. The community integrated energy system (CIES), which takes the coordination of various links of source, grid, and load as the main feature, helps promote the large-scale development of new energy [4, 5]. The CIES schedules and optimizes the generation, transmission, conversion, consumption, and trading of various energy sources [6]. The CIES has many new characteristics, such as multi-energy flow coupling and a multi-time scale, which complicates its operation scheduling [7]. Currently, the centralized optimization operation strategy of the CIES is the main content of the research. Studies [8, 9, 10] investigated the multi-objective optimal operation strategy of IES with different structures. Another study [11] compared and analyzed the economy of two modes of independent operation and the multi-network coordinated operation of the combined cooling, heating, and power (CCHP) micro-energy network. However, these studies only optimized the energy supply side and ignored the load side. IES carbon emissions are usually used as part of the optimization goal or as a constraint condition to reduce carbon emissions [12]. Under the constraint of the carbon emission reduction index, reference [13] proposed a carbon capture power plant scheduling strategy to achieve emission reduction. Some studies have verified that introducing a carbon trading mechanism can reduce IES carbon emissions [14; 15; 16]. From the perspective of promoting the consumption of renewable energy and reducing carbon emissions, an optimization scheduling strategy to stabilize the fluctuation in loads and increase the wind power consumption of the IES is proposed in references [17; 18]. However, the above studies all adopted the fixed carbon quota price trading method, without considering the non-performance penalty measures in the actual carbon emission process. The carbon reduction effect of a single carbon trading price is not obvious, and the enthusiasm of each subject to participate in the carbon trading market is low. With the development and reform of the CIES, the interactive coupling between source and load is changing from the traditional vertical integration structure to the interactive competition structure [19]. A park IES based on the master-slave game method was proposed in [20], aiming at maximizing the interests of each subject and achieving game equilibrium. This method needs to discuss the change in carbon emissions in the system and ensure the low-carbon economic operation of IES. Reference [21] proposed a maximum power point tracking scheme based on an artificial neural network algorithm to maximize photovoltaic power generation. The resulting strategy reduced electricity costs without affecting consumer satisfaction and promoted environmental friendliness by reducing carbon dioxide emissions. The traditional centralized scheduling strategy cannot demonstrate the interaction between energy prices and user demand. Moreover, CIES optimization scheduling has the characteristics of multiple participating entities, wide data dimensions, and complex calculations. Traditional centralized scheduling strategies have low solving efficiency and cannot achieve balanced decision-making among various entities. Therefore, the study of CIES distributed optimization is a more appropriate choice. In view of the problems existing in the above research, the research questions, main contributions, and expected results of this paper are as follows:  
1. A multi-agent stage feedback carbon trading model is proposed to calculate the carbon trading cost in stages.