## Appendix A The Derivation Procedure of Equations (A13)-(A17)  
The actual generation output of the AGC units is adjusted by the participation factor \(\beta\) in Equation (12) to maintain the power balance, such that  
\[\left\{ \begin{array}{c}P_{a}^{act}=P_{a}^{sch}+\beta_{a}\varepsilon\ \forall a\\ P_{a}^{act}=P_{r}^{sch}\ \forall r\end{array} \right.,\ \varepsilon=\sum_{m=1}^{N_{m}}E_{m} \tag{12}\]  
Here, we consider that the wind power output is within the forecasted interval \([E_{m}^{\min},E_{m}^{\max}]\), \(k=1,\cdots,N_{w}\). We will have the following constraints:  
\[\left\{ \begin{array}{c}\sum_{j=1}^{N_{d}}D_{j}-\sum_{m=1}^{N_{m}}\left(E_{m} +W_{m}^{f}\right)=\sum_{i=1}^{N_{g}}P_{i}^{act}\\ P_{a}^{sch}-A_{a}^{+}\leq P_{a}^{act}\leq P_{a}^{sch}+A_{a}^{+}\ \forall a\\ -P_{l_{k}\max}\leq\sum_{i=1}^{N_{g}}H_{kl}P_{i}^{act}+\sum_{m=1}^{N_{m}}H_{ km}\left(E_{m}+W_{m}^{f}\right)-\sum_{j=1}^{N_{g}}H_{kj}D_{j}\leq P_{l_{k}\max} \end{array} \right. \tag{13}\]  
Substituting Equation (12) into the first constraint of Equation (13) yields Equation (14). This constraint shows that the power flow balance can be satisfied under the given forecasted wind power and the uncertainties will be adjusted by the participation factor \(\beta\). \[\sum_{j=1}^{N_{d}}D_{j}-\sum_{m=1}^{N_{m}}W_{m}^{f}=\sum_{i=1}^{N_{g}}P_{i}^{sch} \tag{14}\]  
Substituting Equation (12) into the second constraint of Equation (13) yields  
\[-A_{a}^{-}\leq\beta_{a}\sum_{m=1,m\in\Omega_{m}}^{N_{m}}E_{m}\leq A_{a}^{+} \tag{15}\]  
Taking Equation (12) into the third constraint of Equation (13) leads to  
\[-P_{l_{k}\max}\leq\sum_{i=1}^{N_{g}}H_{kl}P_{i}^{sch}-\sum_{j=1}^{N_{d}}H_{kj} D_{j}+\sum_{m=1}^{N_{m}}H_{km}W_{m}^{f}+\sum_{a=1}^{N_{a}}\sum_{m=1}^{N_{m}} \left(H_{ka}\beta_{a}+H_{km}\right)E_{m}\leq P_{l_{k}\max} \tag{16}\]  
Note that \(E_{m}\) is a random parameter, which varies within the interval \([E_{m}^{\min},E_{m}^{\max}]\). In order to fully guarantee the constraints to be satisfied, i.e., constraints are satisfied for any given value with the interval, a robust optimization model is set up, where the constraints should be satisfied under the worst-case scenario. Specifically, the worst case of Equation (16) can be written as  
\[-P_{l_{k}\max}\leq\sum_{i=1}^{N_{g}}H_{kl}P_{i}^{sch}-\sum_{j=1}^{N_{d}}H_{kj} D_{j}+\sum_{m=1}^{N_{m}}H_{km}W_{m}^{f}+\sum_{a=1}^{N_{a}}\sum_{m=1}^{N_{m}} \min_{E_{m}^{\min}\leq E_{m}\leq E_{m}^{\max}}(H_{ka}\beta_{a}+H_{km})E_{m} \tag{17}\]  
\[\sum_{i=1}^{N_{g}}H_{kl}P_{i}^{sch}-\sum_{j=1}^{N_{d}}H_{kj}D_{j}+\sum_{m=1}^{ N_{m}}H_{km}W_{m}^{f}+\sum_{a=1}^{N_{a}}\sum_{m=1}^{N_{m}}\max_{E_{m}^{\min} \leq E_{m}\leq E_{m}^{\max}}(H_{ka}\beta_{a}+H_{km})E_{m}\leq P_{l_{k}\max} \tag{18}\]  
It can be observed that the above two constraints contain an optimization model, which leads to a bi-level model. To simplify the above two constraints, we introduce a general system (14) without the loss of generality. It can be split into two parts as shown in Equations (15) and (16) according to [27]. \[\max_{x^{\min}\leq x\leq x^{\max}}f(\beta)x\leq a,\min_{x^{\min}\leq x\leq x ^{\max}}f(\beta)x\geq b \tag{19}\]  
\[\max(f(\beta),0)x^{\max}+\min(f(\beta),0)x^{\min}\leq a \tag{20}\]\[\max(f(\beta),0)x^{\min}+\min(f(\beta),0)x^{\max}\geq b \tag{10}\]  
By use of Equations (A8)-(10), the constraints--Equations (A6) and (A7) can be equivalently transformed into Equations (A11)-(A13) as  
\[\sum_{i=1,i\in\Omega_{g}}^{N_{g}}H_{ki}P_{i}^{sch}-\sum_{j=1,j\in\Omega_{d}}^{ N_{d}}H_{kj}D_{j}+\sum_{m=1,m\in\Omega_{w}}^{N_{w}}H_{km}W_{m}^{f}+\sum_{m=1,m \in\Omega_{w}}^{N_{w}}\left(y_{k,i}^{+}E_{m}^{\max}+y_{k,i}^{-}E_{m}^{\min} \right)\leq P_{i_{k}\max} \tag{11}\]  
\[\sum_{i=1,i\in\Omega_{g}}^{N_{g}}H_{ki}P_{i}^{sch}-\sum_{j=1,j\in\Omega_{d}}^{ N_{w}}H_{kj}D_{j}+\sum_{m=1,m\in\Omega_{w}}^{N_{w}}H_{km}W_{m}^{f}+\sum_{m=1,m \in\Omega_{w}}^{N_{w}}\left(y_{k,i}^{+}E_{m}^{\max}+y_{k,i}^{-}E_{m}^{\min} \right)\geq-P_{i_{k}\max} \tag{12}\]  
With \(y_{k,i}^{+}\) and \(y_{k,i}^{-}\) being dummy matrices and they can be written as  
\[\left\{