## 3 CSES Optimization Framework with Dual-Time-Scale Coordination
### Dual-Time-Scale Coordination
#### 3.3.2. ADMM-Based Distributed Algorithm  
The intraday adjustment problem is complex as it involves multiple AC subnetworks interconnected through a DC ring network, each with its own operational issues and privacy concerns. The ADMM algorithm is particularly suitable as it allows decomposition based on the natural structure of the AC/DC distribution system. The key coupling variables are the power exchanges through VSCs, which must satisfy global power balance constraints. Let \(P^{\text{VSC},e}_{j,t}\) denote the actual VSC power exchange from the perspective of AC subnetwork \(e\), \(P^{\text{VSC},DC}_{j,t}\) denote the actual VSC power exchange from the DC ring network perspective, and \(\hat{P}^{\text{VSC}}_{j,t}\) denote the global VSC power exchange variables. The coupling constraints can be expressed as Equation (45):  
\[P^{\text{VSC},e}_{j,t}=P^{\text{VSC},DC}_{j,t},\forall j\in B^{\text{VSC}}, \forall e\in E,\forall t\in T \tag{45}\]  
Figure 2: Dual-time-scale coordination process. The augmented Lagrangian function is formulated as Equations (46) and (47):  
\[\mathcal{L}_{e}=C_{\text{AC},e}+\sum_{e\in\mathbb{E}}\sum_{j\in\mathbb{B}^{ \text{VSC}}}\sum_{t\in\mathbb{T}}\lambda_{j,t,e}^{\text{AC}}(P_{j,t}^{\text{VSC },e}-\hat{P}_{j,t}^{\text{VSC}})+\frac{\rho_{k}}{2}\sum_{e\in\mathbb{E}}\sum_{j \in\mathbb{B}^{\text{VSC}}}\sum_{t\in\mathbb{T}}\Big{|}P_{j,t}^{\text{VSC},e}- \hat{P}_{j,t}^{\text{VSC}}\Big{|}_{2}^{2} \tag{46}\]  
\[\mathcal{L}_{\text{DC}}=C_{\text{DC}}+\sum_{e\in\mathbb{E}}\sum_{j\in\mathbb{ B}^{\text{VSC}}}\sum_{t\in\mathbb{T}}\lambda_{j,t}^{\text{DC}}(P_{j,t}^{\text{VSC },\text{DC}}-P_{j,t}^{\text{VSC},e})+\frac{\rho_{k}}{2}\sum_{e\in\mathbb{E}} \sum_{j\in\mathbb{B}^{\text{VSC}}}\sum_{t\in\mathbb{T}}\Big{|}P_{j,t}^{\text{ VSC},\text{DC}}-P_{j,t}^{\text{VSC},e}\Big{|}_{2}^{2} \tag{47}\]  
where \(\lambda_{j,t,e}^{\text{AC}}\) is the Lagrangian multiplier associated with the power consistency constraint for VSC \(j\) in AC subnetwork \(e\) at time \(t\); and \(\lambda_{j,t}^{\text{DC}}\) is the Lagrangian multiplier associated with the power consistency constraint for VSC \(j\) in DC subnetwork at time \(t\). The iterative process of the ADMM algorithm is shown in Figure 3:  
1. Initialize parameters \(k=0\), \(\rho_{0}\), \(\lambda_{j,t}^{\text{DC},0}\), \(\lambda_{j,t,e}^{\text{AC},0}\), \(\hat{P}_{j,t}^{\text{VSC},0}\);
2. Each subnetwork solves its local optimization problem, as shown in Equations (48) and (49): \[P_{j,t}^{\text{VSC},e,k+1}=\text{argmin}\mathcal{L}_{e}(\hat{P}_{j,t}^{\text{ VSC},k},\lambda_{j,t,e}^{\text{AC},k})\] (48)  
Figure 3: A flowchart of the ADMM-based distributed algorithm. \[P_{j,t}^{\text{VSC,DC,$k+1$}}=\text{argmin}\mathcal{L}_{\text{DC}}(P_{j,t}^{\text{ VSC,$k+1$}},\lambda_{j,t,e}^{\text{DC,$k$}}) \tag{49}\]
3. The DC ring network coordinator updates the global VSC power exchange variables: \[\hat{P}_{j,t}^{\text{VSC,$k+1$}}=\frac{P_{j,t}^{\text{VSC,$\ell$},k+1}+P_{j,t}^ {\text{VSC,DC,$k+1$}}}{2}\] (50)
4. The Lagrangian multipliers are updated: \[\lambda_{j,t,e}^{\text{AC,$k+1$}}=\lambda_{j,t,e}^{\text{AC,$k$}}+\rho(P_{j,t }^{\text{VSC,$\ell$},k+1}-\hat{P}_{j,t}^{\text{VSC,$\ell$},k+1})\] (51)
5. The primal residual \(r_{j,t}^{e,k}\) and dual residual \(s_{j,t}^{e,k}\) are calculated: Figure 2:0 (53) Figure 2:1 (54)
6. An adaptive penalty parameter update strategy is adopted to accelerate convergence, with the penalty factor updated via Equation (55): \[\rho_{k+1}= \begin{cases}0.5\rho_{k},&s_{j,t}^{e,k}\geq 10r_{j,t}^{e,k},\\ 2\rho_{k},&r_{j,t}^{e,k}\geq 10s_{j,t}^{e,k},\\ \rho_{k},&\text{otherwise}\end{cases} \] (55)  
Initial penalty parameter value (\(\rho_{0}=1\times 10^{3}\)) is selected based on empirical tuning, with considerations of problem scale and data magnitude. 7. The convergence criterion is checked: \[\max\{ \begin{subarray}{c}|r_{j,t}^{e,k}|\\ s_{j,t}^{e,k}|\\ \end{subarray}