## 4 Outer Approximation Cutting Method for UFR  
To obtain the UFR, specifically the maximum acceptable fluctuation range of renewable energy, this paper employs an OAC method. Firstly, \(\mathbf{\Omega}_{\text{RG}}^{\text{FR}}\) is projected onto a low-dimensional space \(\mathbf{\Phi}_{\text{RG}}^{\text{FR}}(\mathbf{\Delta}\mathbf{w})\) that is only related to \(\mathbf{\Delta}\mathbf{w}\), as shown in Equation (18). \[\mathbf{\Phi}_{\text{RG}}^{\text{FR}}(\mathbf{\Delta}\mathbf{w})=\text{Pr}_{\mathbf{\Delta}\bm {w}}\Big{(}\mathbf{\Omega}_{\text{RG}}^{\text{FR}}\Big{)}=\{\,\mathbf{\Delta}\mathbf{w} \,|\,\exists\mathbf{y}:\mathbf{\Delta}\mathbf{x}+\mathbf{B}\mathbf{y}+\mathbf{C}(\mathbf{w}+\mathbf{\Delta} \mathbf{w})\leq\mathbf{b}\} \tag{18}\]  
For any point \(\mathbf{\Delta}\mathbf{w}\) in the low-dimensional space \(\mathbf{\Phi}_{\text{RG}}^{\text{FR}}(\mathbf{\Delta}\mathbf{w})\), there exists a corresponding intra-day adjustment \(\mathbf{y}\) that satisfies Equation (19). \[\mathbf{Y}=\{\,\mathbf{y}\,|\,\mathbf{B}\mathbf{y}\leq\mathbf{b}-\mathbf{\Delta}\mathbf{x}-\mathbf{C}(\mathbf{w }+\mathbf{\Delta}\mathbf{w})\} \tag{19}\]  
Here \(\mathbf{Y}\) represents the set of all intraday flexible resource adjustment capabilities. **Definition 1**. : _The uncertainty feasible region is defined as set \(\mathbf{\Phi}_{\text{RG}}^{\text{FR}}=\{\,\mathbf{\Delta}\mathbf{w}\,|\,\mathbf{\Delta}\mathbf{x}+ \mathbf{B}\mathbf{y}+\,\,\mathbf{C}(\mathbf{w}+\mathbf{\Delta}\mathbf{w})\,\,\,\leq\mathbf{b}\}\), satisfying \(\forall\mathbf{\Delta}\mathbf{w}\in\mathbf{\Phi}_{\text{RG}}^{\text{FR}}\), \(\mathbf{Y}\
eq\emptyset\) and \(\forall\mathbf{\Delta}\mathbf{w}\
otin\mathbf{\Phi}_{\text{RG}}^{\text{FR}}\), \(\mathbf{Y}=\emptyset\)._  
When the day-ahead scheduling is determined, Equation (17) defines a high-dimensional convex polytope composed of intra-day adjustable resource variables and renewable energy fluctuation variables, given that all constraints are linear. Equation (18) represents the projection of this high-dimensional convex polytope onto the direction of \(\mathbf{\Delta}\mathbf{w}\), resulting in another convex polytope, expressed as:  
\[\mathbf{\Phi}_{\text{RG}}^{\text{FR}}=\{\,\mathbf{\Delta}\mathbf{w}\,|\,\mathbf{D}\mathbf{\Delta}\bm {w}\leq\mathbf{d}\} \tag{20}\]  
The problem of feasible region analysis essentially involves solving for \(\mathbf{D}\) and \(\mathbf{d}\) in Equation (20), as these contain the boundary information of the UFR. To acquire \(\mathbf{D}\) and \(\mathbf{d}\), we introduce non-negative slack variables \(\mathbf{s}^{\text{I}}\) and \(\mathbf{s}^{\text{O}}\), leading to the following optimization problem (21). \[ \begin{split}& f=\min_{\mathbf{y},\mathbf{s}^{\text{I}}\geq\mathbf{0},\mathbf{s}^{ \text{O}}\geq\mathbf{0}}\,\mathbf{s}^{\text{I}}+\mathbf{s}^{\text{O}}\\ & s.t.\mathbf{B}\mathbf{y}+\mathbf{s}^{\text{I}}-\mathbf{s}^{\text{O}}\leq\mathbf{b}- \mathbf{\Delta}\mathbf{x}_{0}-\mathbf{C}(\mathbf{w}+\mathbf{\Delta}\mathbf{w}_{0})\end{split} \] (21  
feasible region analysis method proposed in this paper. The initial space can be determined through the following optimization problem:  
\[ \begin{split}&\max_{y,\Delta w}\mathbf{D}^{c}\mathbf{\Delta}w\\ & s.t.\mathbf{B}y+\mathbf{C}\mathbf{\Delta}w\leq\mathbf{b}^{{}^{ \prime}}\end{split}  \tag{26}\]  
Here, \(\mathbf{D}^{c}\) represents the matrix formed by the given search directions, and \(\mathbf{d}^{c}\) corresponds to the optimization result of Equation (26). Umbrella constraints refer to the minimal set of constraints that preserve the optimal solution of the original optimization problem. During the optimization process, it is observed that many constraints are inactive. The optimization problem (27) is designed to find whether the \(j\)-th constraint in the original problem qualifies as an umbrella constraint. In this study, the dimensionality of dual variables equals the number of MG constraints. By eliminating redundant constraints, the dimensionality of the dual variables is reduced, thereby decreasing computational complexity. \[