## 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
In a two-dimensional space example, the similarity between the approximate and original feasible regions is illustrated in Figure 1. In the figure, \(\mathbf{\Omega}\) represents the original feasible region of an individual DES, and \(\mathbf{Z}\) represents the approximate feasible region. Several unit normal vectors are selected to represent a specific direction of the feasible region, and the ratio of the widths of \(\mathbf{Z}\) and \(\mathbf{\Omega}\) in each direction is calculated to assess the approximation degree. \[\mathbf{G}=[g^{(1)},g^{(1)},\cdots,g^{(M)}]\in\mathbb{R}^{N\times M} \tag{9}\]0  
where \(N_{f}\) is the number of unit normal vectors; \(\Delta_{\mathbf{\Omega},k}\) and \(\Delta_{\mathbf{Z},k}\) are the widths of the original feasible domain and the approximate feasible domain in the direction of the \(k\)-th unit normal vector, respectively. The DES feasible domain consists of power boundary and energy boundary constraints, and each normal vector can describe the upper and lower bounds of a power or energy boundary constraint, so in the \(N\)-dimensional space, \(2N\)-\(1\) normal vectors are selected. \[\mathbf{G}=[g^{(1)},g^{(1)},\cdots,g^{(M)}]\in\mathbb{R}^{N\times M} \tag{9}\]1 Approximate similarity quantification of Chino polyhedra in two-dimensional space. For the calculation of the width of the zonotope in all unit normal vector directions \(\Delta_{\mathbf{Z}}\) is known from mathematical derivation:  
\[\mathbf{G}=[g^{(1)},g^{(1)},\cdots,g^{(M)}]\in\mathbb{R}^{N\times M} \tag{9}\]2  
where \(\mathbf{F}\) is the matrix consisting of the \(N_{f}\) normal vectors. Substituting Equation (18) into (17) to obtain the objective function is done as follows:  
\[\mathbf{G}=[g^{(1)},g^{(1)},\cdots,g^{(M)}]\in\mathbb{R}^{N\times M} \tag{9}\]3
2. Calculation of parameters related to the original feasible region approximation index Let \(\Delta_{\mathbf{\Omega}}\) be the maximum distance between two tangent planes of the original feasible domain in each unit normal vector direction, which can be solved by optimization methods. The primal feasible domain \(\mathbf{\Omega}\) is represented by a half-space form polyhedron as follows: \[\mathbf{G}=[g^{(1)},g^{(1)},\cdots,g^{(M)}]\in\mathbb{R}^{N\times M} \tag{9}\]4 (20) for a certain unit normal vector \(f_{k}=[a_{1},a_{2},\cdots,a_{N}]\), and its corresponding \(N\)-dimensional hyperplane is given by the following: \[\mathbf{G}=[g^{(1)},g^{(1)},\cdots,g^{(M)}]\in\mathbb{R}^{N\times M} \tag{9}\]5 (21) where \(x_{1},x_{2},\cdots,x_{N}\) is the position in each dimension and \(b_{l}\) is a constant. This unit normal vector \(f_{k}\) on \(\Delta_{\mathbf{\Omega},k}\) can be solved according to the following optimization model:  
\[\left\{ \begin{array}{rl}\max\Delta_{O,k}=\big{|}a_{1}(x_{1p}-x_{1q})+\cdots+ a_{N}(x_{Np}-x_{Nq})\big{|}\\ \text{s.t.}&\boldsymbol{A}\boldsymbol{X}_{P}\leq\boldsymbol{b}\\ \boldsymbol{A}\boldsymbol{X}_{Q}\leq\boldsymbol{b}\end{array} \right. \tag{22}\]  
where \(\boldsymbol{X}_{P}=[x_{1p},x_{2p},\cdots,x_{Np}]\) and \(\boldsymbol{X}_{Q}=[x_{1q},x_{2q},\cdots,x_{Nq}]\) denote a point outside the hyperplane and a point inside the hyperplane, respectively. 3. Constraints The constraint is that the approximated zonotope is inside the feasible domain of the original device. According to the mathematical derivation in Ref. [33], the constraints of this optimization problem are (23), i.e.,  
\[\boldsymbol{Z}(\mathbf{G},\boldsymbol{c},\overline{\boldsymbol{\beta}})\in \mathbf{\Omega}(\boldsymbol{A},\boldsymbol{b})\Leftrightarrow\boldsymbol{A} \boldsymbol{c}+|\boldsymbol{AG}|\overline{\boldsymbol{\beta}}\leq\boldsymbol{b} \tag{23}\]  
In summary, the zonotope-based device approximate feasible domain optimization model is as follows:  
\[\left\{