## 1 Introduction
Numerous studies have explored the feasibility and advantages of SES in distribution systems. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**References** & **Application Area** & **Time Resolution** & **Scale** & **Key Features** & **Advantages** \\ \hline \multirow{3}{*}{[6; 7; 8]} & AC/DC & \multirow{3}{*}{Minutes to hours} & 33-node or larger & \multirow{3}{*}{Decentralized dispatch} & Privacy, reduced \\  & distribution network & & distribution network & & communication \\ \cline{1-1}  & Integrated energy & \multirow{3}{*}{Minutes to hours} & Buildings to & \multirow{3}{*}{Decoupled} & Handles temporal \\  & systems (IES) & & large-scale systems & & energy sector & couplings, \\ \cline{1-1}  & & & & & scalability \\ \cline{1-1}  & & & Small subsystems to & Reduced & \\ \cline{1-1}  & Power electronics & \multirow{3}{*}{Miniseconds} & wind farms & communication and & \multirow{3}{*}{Real-time control, scalability} \\ \cline{1-1}  & & & (around 30 WTs) & & computation burden & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: State-of-the-art applications of ADMM in power electronics and power systems. Previous research on SES has extensively explored economic efficiency and market mechanisms. In [4], Nash bargaining game theory is employed to model interactions in multi-microgrid systems with SES. Other studies have explored multi-time-scale resource allocation, combining long-term contracts with real-time rentals to enhance profitability [22]. While these studies advance economic operation, the challenge of integrating diverse storage types across various timescales persists. The study in [23] detailed a comprehensive three-stage multi-timescale optimization for IES with various generalized energy storage (GES). While our study specifically addresses CSES in AC/DC distribution systems, our parameters focus on AC subnetworks, a DC ring, and CSES SOC within a dual-time-scale (day-ahead/intraday) framework, optimizing power flow and RES utilization. This differs from the multi-energy, multi-GES, three-stage robust optimization in [23] as we concentrate on CSES challenges and ADMM-based RES coordination in AC/DC contexts. Recognizing the need for coordination in SES operations, research has also focused on optimal scheduling strategies. Innovative mechanisms such as energy sharing clouds for smart microgrids [24] and peer-to-peer energy sharing frameworks for buildings [25] have also been developed. Furthermore, decentralized schemes leveraging blockchain for SES market evaluation have been proposed [26]. Cooperative game-based models have been formulated for local integrated energy systems [27]. To address distributed coordination challenges, modified ADMM algorithms have been proposed in [28]. Optimal scheduling of clustered micro-energy grids with SES has also been investigated using cooperative game models for balanced profit allocation [29]. However, these coordination-centric approaches, while effective in their specific contexts, often inadequately utilize the temporal and spatial complementarity of RES across interconnected AC/DC subnetworks. This limits the potential for maximizing system-wide RES utilization. In summary, the parameters, performance, and interrelationships of different energy storage operation architectures can be derived as shown in Table 2:  
As shown in Table 2, conventional individual ESS primarily serves the owner and lacks a sharing mechanism. SES, while enabling sharing, often aggregates decentralized resources. Users are limited by pre-allocated capacity; individuals participating in the SES can share storage capacity but not the energy stored, thus preventing optimal system coordination. This limitation in sharing capability restricts its application within the framework proposed in this paper. However, in CSES, a unified entity operates a centralized energy storage system, enabling dynamic sharing of both capacity and energy.