## 3 Two-Stage MG Operation Model Considering Interactive Power Deviation
### Intraday Model Considering Interactive Power Deviation  
Unlike transmission-distribution network models, the controllable units in distribution-microgrid systems do not possess reserve capacity. Instead, they are limited to a small degree of flexibility for adjustment, based solely on the day-ahead scheduling. (1) Power balance constraints  
\[p^{\text{RG}}(t)+\Delta P^{\text{RG}}(t)-\left(P^{\text{load}}_{\text{DA}}(t) +P^{\text{load,trans}}_{\text{ID}}(t)\right)+P^{\text{CG}}_{\text{ID}}(t)=P^{ \text{MG}}_{\text{DA}}(t)+\Delta P^{\text{MG}}_{\text{ID}}(t) \tag{9}\]where \(\Delta P^{\text{RG}}(t)\) is the variable of renewable generation fluctuation at the time \(t\), i.e., the uncertain variable; \(P_{\text{ID}}^{\text{load,trans}}(t)\) represents the transferable load at time \(t\) in the intra-day period; \(P_{\text{TD}}^{\text{CG}}(t)\) represents the total output of controllable units at time \(t\); and \(\Delta P_{\text{ID}}^{\text{MG}}(t)\) represents the allowable dispatch deviation of the interaction power between the MG and the ULG at time \(t\). (2) Controllable unit constraints  
\[P_{\text{ID}}^{\text{CG}}(t)=\sum_{i}P_{\text{ID},i}^{\text{CG}}(t) \tag{10}\]  
\[\underline{E}_{i}^{\text{CG}}\leq P_{\text{ID},i}^{\text{CG}}(t)\leq\overline {P}_{i}^{\text{CG}} \tag{11}\]  
\[-\underline{\delta}_{i}^{\text{CG}}\leq P_{\text{ID},i}^{\text{CG}}(t)-P_{ \text{ID},i}^{\text{CG}}(t-1)\leq\overline{\delta}_{i}^{\text{CG}} \tag{12}\]  
\[-\underline{\delta}_{i}^{\text{CG}}\leq P_{\text{ID},i}^{\text{CG}}(t_{start })-P_{\text{DA},i}^{\text{CG}}(t_{\text{start}}-1)\leq\overline{\delta}_{i}^ {\text{CG}} \tag{13}\]  
\[-\underline{\delta}_{i}^{\text{CG}}\leq P_{\text{DA},i}^{\text{CG}}(t_{\text {end}}+1)-P_{\text{ID},i}^{\text{CG}}(t_{\text{end}})\leq\overline{\delta}_{ i}^{\text{CG}}\]  
Here, \(t_{\text{start}}\) and \(t_{\text{end}}\) represent the start time and end time for intra-day UFR analysis, respectively. Equation (13) ensures that the output of controllable units satisfies the ramping constraints between adjacent time periods within the intra-day UFR analysis. (3) Power constraints between MG and ULG  
\[-\lambda_{\text{ID}}\Big{|}P_{\text{DA}}^{\text{MG}}(t)\Big{|}\leq\Delta P_{ \text{ID}}^{\text{MG}}(t)\leq\lambda_{\text{ID}}\Big{|}P_{\text{DA}}^{\text{ MG}}(t)\Big{|} \tag{14}\]  
Here, \(\lambda_{\text{ID}}\) represents the allowable IPDF. Equation (14) defines the range of allowable deviation. Practically, most MGs are constructed as retrofits to existing distribution infrastructure, with their interconnection power typically designed to fully cover internal loads. Nevertheless, the increasing renewable energy penetration exacerbates source-load mismatch within MGs. Sole reliance on ULG to absorb such fluctuations poses significant risks to system security and stability. To address this limitation, this work intentionally permits controllable power exchange deviations between MGs and ULG. This paradigm shift aims to alleviate regulation pressure on transmission networks while systematically enhancing the renewable energy hosting capacity of integrated power systems. (4) Flexible load constraints  
\[\sum_{t}^{T^{\prime}}P_{\text{ID}}^{\text{load,trans}}(t)=0 \tag{15}\]  
\[\underline{p}^{\text{load}}<P_{\text{ID}}^{\text{load,trans}}(t)<\overline{P}^{ \text{load}} \tag{16}\]  
Here, \(\overline{P}^{\text{load}}\) and \(\underline{p}^{\text{load}}\) represent the upper and lower bounds of the flexible load output, respectively. \(T^{\prime}\) denotes the time period for the intra-day UFR analysis. Equation (15) indicates that the flexible load has no impact on the load during the UFR analysis period. Equation (16) represents the output range of the flexible load.