## 2 Prediction of Wind-Power probability Density Based on Quantile Regression LSTM
### Constraint Condition  
The constraints included in the model are as follows:  
\[P_{Gi}+P_{uris}-\sum_{l=1}^{m}c_{ils}P_{Lil}-P_{dresi}=\sum_{i,j}P_{ij},i=1,2, \ldots,N \tag{22}\]  
\[Q_{Gi}+Q_{uris}-\sum_{l=1}^{m}c_{ils}Q_{Lli}-Q_{dresi}=\sum_{i,j}Q_{ij},i=1,2, \ldots,N \tag{23}\]  
\[P_{ij}=U_{i}^{2}G_{ij}-U_{i}U_{j}(G_{ij}\cos\theta_{ij}+B_{ij}\sin\theta_{ij}) \tag{24}\]  
\[Q_{ij}=-U_{i}^{2}(B_{ij}+B_{ij}^{\text{C}})-U_{i}U_{j}(G_{ij}\sin\theta_{ij}- B_{ij}\cos\theta_{ij}) \tag{25}\]  
\[P_{gi0}\leq P_{gi}\leq P_{gi}^{\text{max}} \tag{26}\]  
\[Q_{gi0}\leq Q_{gi}\leq Q_{gi}^{\text{max}} \tag{27}\]  
\[\mathcal{U}_{i}^{\text{min}}\leq U_{i}\leq U_{i}^{\text{max}} \tag{28}\]  
\[\Big{|}P_{ij}\Big{|}\leq P_{ij}^{\text{max}} \tag{29}\]  
\[|\theta_{mn}|\leq\theta_{set} \tag{30}\]  
\[\Delta f(c_{il}P_{Lil})\leq\Delta f_{\text{max}} \tag{31}\]  
Formulas (22)-(31) are the constraint conditions to be met in the process of closing angle regulation. Equations (22) and (23) are constraints of active power balance and reactive power balance of nodes, where \(P_{dresi}\), \(Q_{dresi}\) are the restored loads of node \(i\); \(P_{vis}\), \(Q_{vis}\) are the active output and reactive output of the wind farm, respectively, connected to node \(i\) under different probabilities, \(P_{si}\); \(P_{Gi}\), \(Q_{Gi}\) are the active output and reactive output of the conventional unit connected to node \(i\), respectively. Equations (24) and (25) are the active and reactive power calculation formulas of the line \((i,j)\) flowing through the \(i\) end, where \(U_{i}\),\(U_{j}\) is the voltage of the node \(i\), \(j\); \(G_{ij}\), \(B_{ij}\) and \(\theta_{ij}\) are the conductance and susceptance of the line \((i,j)\) and the phase angle difference of the nodes at both ends of the line, respectively; \(B_{ij}^{C}\) is the susceptance parameter of the \(i\) end to the ground branch of the \(\Pi\) type equivalent circuit of the line \((i,j)\). Equations (26)-(29) are inequality constraints for safe operation of the power system, which are unit active output constraints, unit reactive output constraints, node voltage constraints, and tie line transmission power constraints; here, \(P_{gi0}\) and \(Q_{gi0}\) are the initial active output and reactive output of conventional units, respectively. During system recovery, the system has a large power loss load, and most units are in an upward-climbing state. To accelerate the recovery process, It is required that all units are not allowed to operate with reduced output; \(P_{gi}^{\max}\) and \(Q_{gi}^{\max}\) are the maximum active output and reactive output allowed for unit operation, respectively; \(U_{i}^{\min}\) and \(U_{i}^{\max}\) are the allowable lower and upper limits of the node voltage, respectively; \(P_{ij}^{\max}\) is the maximum power allowed to flow through the line \((i,j)\). Equation (30) is the phase angle difference constraint at both ends of the line to be paralleled and \(\theta_{set}\) is the maximum setting value allowed in system recovery. Equation (31) is the constraint of system frequency change caused by load input and \(\Delta f_{\max}\) is the maximum frequency drop value caused.