## 2 Distributed Energy Storage Aggregation Model Based on Inner Approximation of Minkowski Sum
### Real-Time Optimization Scheduling Model
#### 3.3.2 Constraints  
In the real-time scheduling model, all constraints remain consistent with those in the day-ahead and intra-day models, except that current and voltage limit constraints are replaced with chance constraints [35]. These constraints are not elaborated further here. \[\Pr\Bigl{\{}\widetilde{l}_{\text{line,min}}\leq\widetilde{l}_{\text{line,f} }^{t}\leq\widetilde{l}_{\text{line,max}}\Bigr{\}}\geq 1-\xi_{1}\;,\;\;\;\; \forall L(i,j)\in\Omega_{\text{line}} \tag{48}\]  
\[\Pr\Bigl{\{}\widetilde{l}_{\text{min}}\leq\widetilde{l}_{t}^{u}\leq \widetilde{l}_{\text{max}}\Bigr{\}}\geq 1-\xi_{1}\;,\;\;\;\;\forall n\in \Omega_{\text{node}} \tag{49}\]  
where \(\Pr\{\cdot\}\) denotes the probability that the condition in parentheses holds; \(\xi_{1}\) is the node voltage crossing probability and branch current crossing probability, which characterizes the confidence level of the chance constraint equation. Since the voltage at each node and the current in each branch fluctuate within a certain range while satisfying a confidence level, the chance constraints in Equations (50) and (51), unlike wind and solar power forecasts, do not need to consider factors such as probability distributions or random variables. For the convenience of calculation and modeling, they can be transformed into the deterministic constraints shown in Equations (52) and (53). \[\left\{ \begin{array}{l}\widetilde{l}_{\text{line,min}}-a_{j,t}[\widetilde{l }_{\text{line,min}}-\widetilde{l}_{\text{line,min}}^{\text{lim}}]\leq \widetilde{l}_{\text{line,f}}^{t}\leq\widetilde{l}_{\text{line,max}}+a_{j,t}[ \widetilde{l}_{\text{line,max}}^{\text{lim}}-\widetilde{l}_{\text{line,max}} ]\\ \sum\limits_{j=1}^{N_{\text{line}}}\sum\limits_{t=1}^{T}a_{j,t}\leq TN_{line} \xi_{1}\end{array} \right. \tag{50}\]\[\left\{ \begin{array}{ll}\widetilde{U}_{\min}-b_{i,t}[\widetilde{U}_{\min}- \widetilde{U}_{\min}^{\lim}]\leq\widetilde{U}_{t}^{n}\leq\widetilde{U}_{\max}+b _{i,t}[\widetilde{U}_{\max}^{\lim}-\widetilde{U}_{\max}]\\ \sum\limits_{i=1}^{N_{bus}}\sum\limits_{t=1}^{T}b_{i,t}\leq TN_{bus}\xi_{1} \end{array} \right. \tag{51}\]  
where \(N_{bus}\) and \(N_{line}\) are the number of nodes and branches of the DN; \(\widetilde{U}_{\max}^{\lim}\) and \(\widetilde{U}_{\min}^{\lim}\) are the upper and lower limits of the overrun amplitude when the voltage of node \(i\) is overrun, taking 1.1 pu and 0.9 pu; \(\widetilde{I}_{\min,\max}^{\lim}\) and \(\widetilde{I}_{\min,\min}^{\lim}\) are the upper and lower limits of the overrun amplitude when the current of branch \(j\) is overrun; \(a_{i,t}\) and \(b_{i,t}\) are the Boolean decision variables for the current amplitude of branch \(j\) and the voltage amplitude of node \(i\) in the time period of \(t\). The value is set to 1 if overrun is found, and to 0 if overrun occurs. If there is an overrun condition, it is 1, and vice versa, it is 0. Power Allocation Method Within Distributed Energy Storage Aggregates Based on the Water-Filling Algorithm  
By solving the above optimization scheduling model, the charge and discharge commands of the DES cluster at each time period can be obtained. In this section, the WFA is adopted to allocate these commands. In the power control decomposition of the DES cluster, if the aggregator receives a charging command \(D_{t}^{c}\), the following problem is solved to determine the charging power \(p_{\text{DES},t}^{r}\) of each DES:  
\[\min-\sum\limits_{r\in N}\frac{E_{r}}{\eta_{c}^{r}}\log(\frac{E_{\text{DES},t- 1}^{r}}{E_{r}}+\frac{\eta_{c}^{r}p_{\text{DES},t}^{r}\Delta t}{E_{r}}) \tag{52}\]  
\[\text{s.t.}\quad 0\leq p_{\text{DES},t}^{r}\leq\hat{P}_{\text{DES},t}^{r},\forall r\]  
\[p_{\text{ABS},t}^{c}=\sum\limits_{r\in N}p_{\text{DES},t}^{r}\leq D_{t}^{c} \tag{53}\]  
where \(E_{r}\) is the capacity of the \(r\)-th DES; \(p_{\text{ABS},t}^{r}\) is the aggregated charging power at time \(t\); \(\hat{P}_{\text{DES},t}^{r}=\min\Bigl{\{}P_{\text{DES},t^{\prime}}^{r}\left(E_ {r}-E_{\text{DES},t-1}^{r}\right)/\left(\eta_{c}^{r}\Delta t\right)\Bigr{\}}\).