## 1 Introduction  
With the energy crisis and global environmental issues becoming increasingly prominent, it is necessary to develop multi-energy complementarity and optimization of an integrated energy system to achieve low-carbon environmental operations [1]. The development of gas turbines [2] and power to gas (P2G) [3] has enabled a bidirectional energy flow between the power system and the natural gas system, which promotes the formation of integrated electricity-gas system (IEGS). As an important part of Energy Internet, IEGS is also the development direction of Smart Grid [4; 5; 6]. Recently, a considerable amount of research has been made on the optimization of IEGS [7; 8; 9; 10; 11; 12; 13; 14; 15]. A unified modeling framework for IEGS is proposed in [7]. Based on this model, the effect of a natural gas system on the operation of a power system in accordance with the security-constrained unit commitment is discussed. Based on mixed integer linear programming, paper [8] presents an integrated electricity-gas system model that can be used to research interdependencies and identify critical vulnerabilities. Paper [9] studies the electricity and steady-state natural gas flow. The above studies are directed at steady-state models. However, for the network characteristics of IEGS, different systems have different dynamic characteristics. Natural gas flow operates at slower rates than electric power. Some natural gas is stored in the pipeline, which is called the \"line pack\". In order to describe the natural gas network model more accurately, it is necessary to consider the line pack. Papers [10; 11; 12; 13] have proved the improvement effect of the management of IEGS on the flexibility and reliability by considering the line pack. Paper [10] proposes a mixed-integer linear programing (MILP) model that takes into account the dynamic characteristics of natural gas system to solve the optimal energy flow of IEGS. Considering the different response times of the gas and power systems, the dynamic optimal energy flow in the IEGS is discussed in [11; 12]. Paper [13] establishes the optimal operation model of IES considering heat-gas dynamic characteristics and demonstrates the efficiency of the proposed method for improving the flexibility and economy of IES operation. Paper [14] proposes a hybrid multi-energy load forecasting (MELF) method for IEGS considering temporal dynamic and coupling characteristics. An interdependent electricity-natural gas system considering electricity grid steady-state and gas network dynamics is established in [15]. Based on the model, the effect of P2 Gs and gas turbines on the operational economics and robustness of IEGS is discussed. On the other hand, in the above research works, the optimal scheduling model is mainly solved by centralized algorithms, which requires a central computer to collect global data to perform centralized calculations and control. However, the power and natural gas systems belong to different interest groups, and the information and operating conditions of each system have a certain privacy, which makes centralized optimization scheduling no longer applicable. Decentralized algorithms can solve the abovementioned limitations of centralized algorithms and can reduce the amount of information transmitted and reduce the risk of information blocking. Therefore, in the background of the energy internet, it is necessary to research the decentralized OMEF of IEGS [16; 17; 18]. However, the OMEF problem is a non-convex optimization problem that is difficult to solve. The inherent non-convexity makes traditional decentralized algorithms unable to converge or obtain ideal results. The non-convexity of the OMEF problem comes from the second-order power flow equation and natural gas flow equation. To handle this issue, extensive optimization methods have been presented. There are two main convex methods for the second-order power flow equation. One is the linear DC power flow model [19], the other is the second-order cone relaxation for the radial grid model [20].