## 2 Materials and Methods
Since these methods expand outward from within, they may fail to fully cover the entire feasible region in some complex polytopes or non-convex sets, especially when the region's boundary exhibits non-convexity or discontinuities, which can lead to missed boundary information [24]. In high-dimensional problems, inner approximation methods may require a significant number of iterations to approach the boundary, resulting in an unacceptable computation time, especially when high precision is needed. In multi-objective optimization scenarios, inner approximation methods may struggle to simultaneously satisfy the boundary description requirements for all objectives, potentially leading to suboptimal solutions. To address the limitations of inner approximation methods in terms of global coverage, computational efficiency, and the ability to handle complex boundaries, researchers have increasingly explored complementary outer approximation methods in recent years [25]. The core idea of outer approximation is to construct the boundary of the feasible region from the outside by progressively eliminating infeasible areas or constrained spaces, thereby approaching the actual feasible region. This method effectively overcomes the shortcomings of inner approximation, such as the inability to fully cover the feasible region, and shows significant advantages in handling non-convex feasible regions or complex high-dimensional polytopes. Additionally, outer approximation leverages geometric properties to construct more precise boundary descriptions from a global perspective, providing optimization algorithms with a more comprehensive solution space. Reference [25], drawing inspiration from robust optimization [26], proposed an outer approximation feasible region solving method applied to transmission-distribution system coordination. However, from the perspective of distribution-microgrid coordination, no specific studies have been conducted. The dispatch schedules for tie-line power between transmission and distribution networks must be executed with zero deviation once finalized. However, such strict constraints can be relaxed at the MG level. MGs allow for deviations between planned output and the actual power exchanged with the ULG, which differs significantly from the interactions in transmission and distribution systems. Therefore, this paper proposes an OAC method for intra-day UFRs considering dispatch deviations. The main contributions of this paper are as follows:  
1. An MG operation model is established. The intraday UFR of the MG is solved using the OAC method, and umbrella constraint identification is employed to eliminate redundant constraints, which accelerates the solution speed compared to traditional methods. 2. Power deviations between the MG and the ULG are allowed in the intraday stage. Under this condition, the size of the intraday MG's UFR is evaluated. The structure of this paper is as follows. Section 1 of this paper introduces the background of the study. Section 2 presents the current state of research. Section 3 presents the two-stage operation model of the MG. Section 4 discusses the OAC method used to solve the UFR of MG. Section 5 provides the case study analysis, and Section 6 concludes the paper.