## 5 Results  
Taking a typical Transformer area MG as an example, it includes five controllable units, specifically micro gas turbine units, with their parameters listed in Table 1. The intra-day load curve, renewable generation forecast curve, and time-of-use electricity price are illustrated in Figure 3. The blue curve represents the time-of-use electricity price corresponding to the right \(y\)-axis, while the remaining curves correspond to the left \(y\)-axis. The day-ahead scheduling period is set as \(T=24\), with a time resolution of \(\Delta T=1\). For better visualization, the intra-day UFR analysis is conducted over a period of \(T^{\prime}=3\), with \(t_{start}\) selected at 14:00, a time characterized by high renewable energy generation.  
Figure 2: Flowchart of outside approximation method to solve the feasible region.  
The program is developed in MATLAB R2023b using YALMIP, running on a system with a 2.60 GHz processor and 16 GB of RAM. The proposed method is solved using IBM ILOG CPLEX 12.10 [27].  
Based on the above data, the day-ahead optimal economic scheduling problem is solved to obtain the day-ahead scheduling. The dark blue curve in Figure 4 represents the interaction power between the MG and the ULG, corresponding to the right \(y\)-axis, while the remaining curves represent the output of the controllable units, corresponding to the left \(y\)-axis.  
By inputting the day-ahead scheduling and intra-day parameters into the model, and setting the allowable IPDF \(\lambda_{ID}=0.1\), the model is solved step by step to obtain the following results. Figure 5a shows the initial space determined by the given search directions, which contains many fluctuation points of renewable energy that do not satisfy the MG constraints. Figure 5b represents the obtained UFR, where the yellow area and its boundary indicate that, within the next three consecutive time periods, no emergency measures \(\mathbf{s}^{1},\mathbf{s}^{\mathrm{O}}\) need to be applied by the MG. Instead, MG's own flexible resources can adjust to meet all security constraints without affecting the day-ahead scheduling in subsequent periods. Figure 5c combines both, showing the result of their integration.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No.** & **P\_min** & **P\_max** & **Ramp\_Minus** & **Ramp\_plus** & **Cost\_fuel/** & **Cost\_OM/** \\  & \(\mathbf{/kW}\) & \(\mathbf{/kW}\) & \(\mathbf{/kW}\) & \(\mathbf{/kW}\) & \(\mathbf{/kWh}\) & \(\mathbf{/kWh}\) \\ \hline
1 & 50 & 200 & 19 & 20 & 0.2594 & 0.089 \\
2 & 30 & 100 & 15 & 15 & 0.2943 & 0.054 \\
3 & 75 & 300 & 31 & 32 & 0.243 & 0.106 \\
4 & 50 & 150 & 19 & 20 & 0.289 & 0.06187 \\
5 & 60 & 250 & 21 & 22 & 0.2578 & 0.091 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Controllable generators’ parameters.  
Figure 3: Day-ahead parameters’ information.  
Figure 4: Day-ahead dispatch scheduling.  
Figure 5: Schematic Diagram of Initial Space and Feasible Region (**a**) Initial space; (**b**) Feasible region (lamda = 0.1): (**c**) Initial space and final feasible region.  
As can be observed, due to the various constraints of the intra-day MG, including coupling constraints of controllable units across time periods and flexible load constraints, the initial space is progressively cut, with feasible cuts acting as cutting planes (instead of the traditional approach of searching for points or lines), thereby improving computational efficiency.  
Next, we vary the allowable IPDF, setting it to 0.01, 0.1, and 0.2, respectively, and obtain the following figure. As shown in Figure 6, it can be observed that the range of UFR expands as the allowable IPDF increases. Although the geometries of feasible regions retain similarity, the cutting planes may vary significantly.