## 1 Introduction  
With the rapid development of power electronic technology, the integration of the RES and energy storage system (ESS) has significantly transformed distribution systems [1, 2]. The proportion of distributed resources and DC-compatible technologies has increased dramatically, making AC/DC distribution systems particularly advantageous for accommodating diverse generation types and enhancing system flexibility [3, 4]. However, AC/DC distribution systems face multiple challenges: large-scale penetration of intermittent RESs, coordinated operation requirements, and the temporal-spatial imbalance of source-load relationships across subnetworks [1]. Conventional individual ESS configurations, while attempting to resolve source-load mismatches locally, often result in suboptimal RES utilization [4]. CSES emerges as a promising alternative that can leverage complementary characteristics to improve system-wide efficiency while addressing coordination challenges [5]. Developing effective optimal scheduling mechanisms that simultaneously enhance RES utilization and meet system coordination needs has become a crucial research direction for next-generation distribution systems. When solving complex optimization problems, distributed algorithms, particularly the ADMM, have shown significant advantages. Table 1 summarizes the state-of-the-art applications of ADMM in power electronics and power systems with their scale-time differences. The studies in [6; 7; 8; 9; 10; 11] employ the ADMM method to address optimization problems in AC/DC distribution networks or integrated energy systems, with scales ranging from buildings and 33-node distribution networks to larger, more complex systems, and time scales from minutes to hours. The studies in [12; 13; 14; 15; 16], on the other hand, focus on the control of power electronic devices at scales ranging from small subsystems, such as at the controller level, to wind farms with about 30 wind turbines (WT), and on time scales ranging from milliseconds to seconds. Based on the comparison in Table 1, ADMM offers distinct advantages for the optimization framework, especially for 15-min intraday coordination. As systems become more complex, centralized methods face computational burdens and privacy concerns. ADMM enables the framework to decompose the problem naturally along network boundaries, where subnetworks share only basic power information. This approach preserves operational privacy, reduces communication, and efficiently handles the coupling constraints. ADMM's mathematical structure works well for the optimization framework, providing good convergence properties while accommodating the time-scale requirements of distribution system operations. Energy storage is an essential component of modern distribution systems, playing a key role in improving system operational efficiency and economic benefits. Energy storage technologies for power systems can be categorized based on storage medium, ownership structure, and operational architecture. This paper focuses primarily on electrochemical battery energy storage systems (BESS) due to their rapid response characteristics, modular scalability, and declining cost trends [17]. While other storage technologies, such as chemical, solid, gravity, and heat storage, exist, they face geographical constraints or have lower energy density profiles that limit their application in distribution systems [18; 19]. In terms of operational architecture, conventional individual ESSs are independently operated and typically serve single network segments. However, traditional individual ESSs face limitations such as capacity surplus or shortage, low operational efficiency, and limited regulation capabilities [20]. SES enables multiple stakeholders to access storage capacity through various market mechanisms. CSES refers to the unified construction of ESSs designed for regulation and use by multiple users. SES involves individual users offering their idle ESS to meet the regulatory needs of other users [21].