## 1 Introduction  
Since the Industrial Revolution, human society has entered a period of rapid development, and the reliance on high-quality energy has increased significantly. However, traditional fossil energy sources have been facing challenges such as over-exploitation and inefficient utilization. Meanwhile, the environmental crises triggered by unreasonable energy structures have become increasingly severe. In the context of Carbon Neutrality, sustainable development technologies have emerged as a central focus for countries worldwide [1]. Multi-energy systems (MESs), integrating electricity, heat, and gas, enable the hierarchical utilization and flexible regulation of heterogeneous energy sources, breaking the barriers of traditionally independent energy supply systems [2, 3]. This provides new pathways for promoting low-carbon emissions and improving energy efficiency. With continuous breakthroughs in clean energy technologies, an increasing number of coupling devices, such as power-to-gas (P2G), have been applied in MESs, making the coupling relationships among subsystems more complex [4]. Therefore, collaborative analysis of MESs from a holistic perspective has become a research hotspot in recent years. Energy flow calculation (EFC) is an essential tool for steady-state analysis of MESs. By utilizing network parameter information and specified boundary conditions, it calculates the operating state variables of subnetworks and coupling devices within the MES. As such, it serves as a critical foundation for various MES studies, including the exploration of multi-energy interaction characteristics, optimization of scheduling, collaborative planning, and static security assessment [5; 6]. Currently, research on multi-energy flow calculation in MESs primarily focuses on model development and solution strategies. In terms of multi-energy flow modeling, research on the modeling of the district electricity network (DEN) has become highly advanced, with the classical alternative current (AC) power flow model being widely adopted [7]. For the electricity-gas coupled system, an energy flow model for this type of MES was developed in [8], which highlighted the impact of temperature on the operating state of the MES. By introducing P2G devices and considering the per-unit normalization of heterogeneous state variables, a bidirectional energy flow model for electricity-gas MESs was proposed in [9]. Due to the physical property differences between various gas sources, such as synthetic natural gas (SNG), hydrogen, etc., an energy flow model for electricity-gas MESs that is suitable for multiple slack buses and alternative gas source injections was established in [10], reflecting the impact of gas quality differences on the operating state of the district gas network (DGN). In the research on the district heat network (DHN), the hydraulic and thermal models for DHN were constructed in [11]; based on this, an energy flow model for the electricity-heat MES was designed. Building upon [11], studies [12; 13; 14] further developed MES models encompassing the electricity, heat, and gas subnetworks. Subsequently, energy flow models for electricity-heat-gas MESs, which consider the diversity of compressor operation modes [15] and nodal pressure-based DHN models [16], were proposed, further enriching MES modeling theories. However, due to the varying focuses, the aforementioned models are evidently not interchangeable. Additionally, the formulation of these models followed the traditional recursive approach, which is not conducive to efficient implementation in engineering applications. In response, matrix-based energy flow modeling methods for DHNs were explored in [17; 18; 19; 20], providing insights for the practical modeling of MESs. Overall, achieving a matrix-based and concise expression of the MES energy flow model while ensuring accuracy and universality remains a critical challenge for further research.