## 2 Distributed Energy Storage Aggregation Model Based on Inner Approximation of Minkowski Sum
### Day-Ahead Optimal Scheduling Model
#### 3.1.2 Constraints  
1. Power Balance Constraint: \[\sum_{i=1}^{N_{\rm G}}P_{{\rm G}i,t,s}+\sum_{i=1}^{N_{\rm BC}}P_{{\rm D}{\rm G }i,t,s}+\sum_{i=1}^{N_{\rm ESS}}P_{{\rm ESS},i,s}+P_{{\rm buy},t}=P_{{\rm load },t,s}\] (32)  
where \(P_{{\rm load},t,s}\) is the load at moment \(t\) for scenario s.
2. GT Operation Constraints: \[P_{Gi}^{\rm min}\leq P_{Gi,t,s}\leq P_{Gi}^{\rm max}\] (33) \[\left\{\begin{array}{ll}P_{{\rm G}i,t,s}-P_{{\rm G}i,t-1,s}&\leq&n_{Gi,t}R _{i}\\ P_{{\rm G}i,t-1,s}-P_{{\rm G}i,t,s}&\leq&n_{Gi,t-1}R_{i}\end{array}\right.\] (34)  
where \(P_{{\rm G}i}^{\rm min}\) and \(P_{{\rm G}i}^{\rm max}\) are the upper and lower output limits of the \(i\)-th GT; \(R_{i}\) is the creep rate of the \(i\)-th GT.
3. PV Output Constraints: \[0\leq P_{{\rm D}{\rm G}i,t,s}\leq P_{{\rm D}{\rm G}i,t,s}^{\rm Pre}\] (35)  
where \(P_{{\rm D}{\rm G}i,t,s}^{\rm Pre}\) is the predicted value of the output of the \(i\)-th PV at the moment \(t\) under the \(s\) scenario.
4. ESS Constraints: \[\left\{\begin{array}{ll}P_{{\rm ESS},i}^{\rm dis}u_{{\rm ESS},i,t,s}\leq P _{{\rm ESS},i,t,s}\leq P_{{\rm ESS},i}^{\rm cha}u_{{\rm ESS},i,t,s}\\ S_{{\rm SOC},i}^{\rm min}\leq S_{{\rm SOC},i,t,s}\leq S_{{\rm SOC},i}^{\rm max },\end{array}\right.\] (36)  
where \(P_{{\rm ESS},i}^{\rm dis}\) and \(P_{{\rm ESS},i}^{\rm cha}\) are the rated charging power and rated discharging power of the ESS \(i\), respectively; \(S_{{\rm SOC},i,t,s}\) is the charging state of the ESS \(i\) at the moment \(t\) under the \(s\)-scenario; \(S_{{\rm SOC},i}^{\rm min}\) and \(S_{{\rm SOC},i}^{\rm max}\) are the upper and lower limits of the charging state of the ESS \(i\).
5. DN Power Flow Constraints: \[\sum_{p\in\Omega_{+}^{+}}P_{{\rm line},t}^{p}-\sum_{q\in\Omega_{ -}^{-}}(P_{{\rm line},t}^{q}-\widetilde{I}_{{\rm line},t}^{q}R_{{\rm line}}^{ q})=P_{{\rm out},t}^{n}\] (37) \[\widetilde{U}_{t}^{j}=\widetilde{U}_{t}^{i}-2P_{{\rm line},t}^{L}R _{{\rm line}}^{L}+\widetilde{I}_{{\rm line},t}^{L}(R_{{\rm line}}^{L}  
\[\left\|\frac{2P_{\text{line},t}^{L}}{\tilde{l}_{\text{line},t}^{L}-\tilde{U}_{t}^ {i}}\right\|_{2}\leq\tilde{l}_{\text{line},t}^{L}+\tilde{U}_{t}^{i},\quad\forall L (i,j)\in\Omega_{\text{line}} \tag{41}\]  
\[\left|P_{\text{line},t}^{L}\right|\leq P_{\text{line},\text{max}},\quad\forall L (i,j)\in\Omega_{\text{line}} \tag{42}\]  
where \(\Omega_{n}^{+}\) and \(\Omega_{n}^{-}\) represent the sets of branches starting from and ending at node \(n\) in the DN; \(\Omega_{\text{node}}\) and \(\Omega_{\text{line}}\) denote the sets of nodes and branches in the DN; \(P_{\text{out},t}^{n}\) is the output power of node \(n\) at time \(t\); \(U_{t}^{n}\), \(I_{\text{line},t}^{L}\), \(P_{\text{line},t}^{L}\), and \(R_{\text{line}}^{L}\) represent the voltage at node \(n\), the current, power, and the resistance in branch \(L\) at time \(t\); \(\tilde{l}_{\text{line},\text{max}}\), \(\tilde{U}_{\text{max}}\) and \(P_{\text{line},\text{max}}\) are the upper limits of \(\tilde{l}_{\text{line},t}^{L}\), \(\tilde{U}_{t}^{n}\) and \(P_{\text{line},t}^{L}\).  
6. DES Cluster Constraints: \[P_{\text{des},n}=c_{n}^{agg}u_{\text{des},n}+G\beta_{n,t}-\overline{\beta}_{ n}^{agg}u_{\text{des},n}\leq\beta_{n}\leq\overline{\beta}_{n}^{agg}u_{ \text{des},n}\] (43)  
where \(\mathbf{P}_{\text{des},n}\) is the power matrix of the DES cluster at node \(n\); \(G\) is the generator matrix; \(\beta_{n}\) is the extension length of the generator in each direction; \(c_{n}^{agg}\) and \(\overline{\beta}_{n}^{agg}\) represent the center point coordinates of the aggregated zonotope and the maximum extension length of the generators, respectively.