## 2 Distributed Energy Storage Aggregation Model Based on Inner Approximation of Minkowski Sum
### Characterization of Feasible Region for Distributed Energy Storage Clusters Based on Zonotope  
To address the intractability of the Minkowski Sum problem mentioned above, an efficient aggregation method based on Zonotope approximation is adopted in this paper. Specifically, the convex polytope shown in Equation (6) is approximated using the Zonotope form presented in Equation (8), and then the approximated Zonotope is aggregated. Unlike conventional polyhedra represented in half-space form (H-representation) or vertex form (V-representation), Zonotope is a polyhedron with a specific mathematical expression (Z-representation) [12], which is formulated as follows:  
\[\mathbf{Z}=\left\{\mathbf{x}\in\mathbb{R}^{N}:\mathbf{x}=\mathbf{c}+G\beta,-\overline{\beta} \leq\beta\leq\overline{\beta}\right\} \tag{8}\]  
\[\mathbf{G}=[g^{(1)},g^{(1)},\cdots,g^{(M)}]\in\mathbb{R}^{N\times M} \tag{9}\]  
\[\left\|g^{(m)}\right\|_{2}=1\ \ \ m=1,2,\cdots,M \tag{10}\]  
\[\beta=\left[\beta_{1},\beta_{1},\cdots,\beta_{M}\right]^{\mathsf{T}} \tag{11}\]  
where \(N\) is the spatial dimension; \(\mathbf{c}\) is the coordinates of the centroid of the polyhedron; \(\mathbf{G}\) is the generator matrix, consisting of \(M\) generator vectors \(g^{(m)}\), representing the extension of the polyhedron from the centroid in all directions; \(\mathbf{\beta}\) is the extension length of the generator in all directions, limited by the maximum extension length \(\overline{\beta}\); \(\left\|\cdot\right\|_{2}\) denotes the 2-paradigm operation. In the problem of zonotope approximation for the feasible region of DES, the generator matrix \(\mathbf{G}\) is considered as a known parameter. The corresponding generator matrices are designed for the power and energy constraints of DES devices, as shown below:  
\[g^{(m)}=\left[0,\cdots,0,\underbrace{1}_{m}0,\cdots,0\right]^{\mathsf{T}}\ \ m=1,2,\cdots,N \tag{12}\]  
\[g^{(N+m)}=\left[0,\cdots,0,\ \underbrace{\frac{-1}{\sqrt{2}}}_{m},\ \frac{1}{ \underbrace{\frac{\sqrt{2}}{m+1}}},0,\cdots,0\right]^{\mathsf{T}}\ \ m=1,2,\cdots,N-1 \tag{13}\]  
where Equation (12) comprises the first \(N\) generators, corresponding to the \(N\) power constraints of the DES; Equation (13) includes \(N\)-1 generators, corresponding to the \(N\)-1 energy constraints of the DES. Consequently, the resulting generator matrix is \(\mathbf{G}\in\mathbb{R}^{N\times(2N-1)}\). For zonotopes with the same generator matrix \(\mathbf{G}\) selected, the Minkowski Sum process for these zonotopes is simplified due to their identical extension directions:  
\[\mathbf{Z}^{qgg}=\mathbf{Z}_{1}\oplus\mathbf{Z}_{1}\cdots\oplus Z_{n} \tag{14}\]\[e^{agg}=\sum_{r=1}^{N_{c}}\mathbf{c}_{r} \tag{15}\]  
\[\beta_{m}^{agg}=\sum_{r=1}^{N_{c}}\beta_{m,r} \tag{16}\]  
where \(\mathbf{Z}^{agg}\) is the total zonotope after aggregation; \(N_{c}\) is the number of feasible domains for aggregation, i.e., the number of aggregated resources; \(\mathbf{c}^{agg}\) is the centroid coordinates of the aggregated zonotope; \(\mathbf{c}_{r}\) is the coordinates of the \(r\)-th centroid; \(\mathbf{\beta}_{m}^{agg}\) is the total extension length in the direction of the \(m\)-th generator of the aggregated zonotope; and \(\beta_{m,r}\) is the extension length of the \(r\)-th zonotope in the direction of the \(m\)-th generator before aggregation. It can be observed that the Minkowski Sum process for Zonotopes is a simple linear superposition process, which is attributed to its specific mathematical representation. Thus, the expression of the zonotope is employed to approximate the characterization of the device's feasible region, transforming from the form of Equation (6) to that of Equation (8). This issue can be modeled as an optimization problem [33], as outlined in the following modeling process:  
1. Objective functions  
A feasibility approximation index \(\Lambda\) is defined to measure the difference between the approximate and original feasible regions of individual DES resources.