## 3 Two-Stage Stochastic Scheduling Model for the Hybrid AC/DC Distribution Network
### Constraints for the Two-Stage Stochastic Programming
#### 3.3.3 Relevant Constraints of Two Stages  
For the two-stage scheduling model, the relevant constraints of the first and second stages are described as follows:  
\[P_{i,t,s}^{\text{AC}}=P_{i,t,s}^{\text{Sub}}+P_{i,t,s}^{W}+P_{i,t}^{G}+P_{i,t,s}^ {\text{dis}}-P_{i,t,s}^{\text{ch}}-P_{i,t}^{L}\ \forall i\in\mathcal{N}_{\text{AC}}, \tag{44}\]  
\[Q_{i,t,s}^{\text{AC}}=Q_{i,t,s}^{\text{Sub}}+Q_{i,t,s}^{W}+Q_{i,t}^{G}+Q_{i,t}^ {\text{CB}}-Q_{i,t}^{L}\ \forall i\in\mathcal{N}_{\text{AC}}, \tag{45}\]  
\[P_{i,t}^{\text{DC}}=P_{i,t,s}^{W}+P_{i,t,s}^{\text{VSC(DC)}}-P_{i,t}^{L}\ \forall i\in\mathcal{N}_{\text{DC}}, \tag{46}\]  
where \(P_{i,t,s}^{\text{Sub}}\) and \(Q_{i,t,s}^{\text{Sub}}\) represent the active and reactive power from the substation connected with node \(i\) at time \(t\) under scenario \(s\); \(P_{i,t,s}^{W}\) and \(Q_{i,t,s}^{W}\) represent the active and reactive power from the wind generator under scenario \(s\); \(P_{i,t}^{G}\) and \(Q_{i,t}^{G}\) represent the active and reactive power output by the gas-fired units; and \(P_{i,t}^{L}\) and \(Q_{i,t}^{L}\) represent the active and reactive power load at node \(i\).  
In this study, the gas consumption plan by gas-fired units is determined before wind power uncertainty is observed and will not adjust in real time. Consequently, the power output of gas-fired units cannot be changed under different wind power scenarios at the second stage.  
Furthermore, the hybrid AC/DC distribution network model in Section 2 is established for the certain wind power scenario. In other words, all the variables in Equations (1)-(21) should be changed to the corresponding variables for different wind power scenarios. Taking the DC distribution network as an example, \(U_{i,t}^{\text{DC}}\), \(I_{k,t}^{\text{DC}}\), and \(P_{k,t}^{\text{DC}}\) should be replaced by \(U_{i,t,s}^{\text{DC}}\), \(I_{k,s}^{\text{DC}}\), and \(P_{k,t,s}^{\text{DC}}\), respectively. Additionally, the power flow model for the DC distribution network under scenario \(s\) can be expressed as follows:  
\[\left\{\begin{array}{l}P_{i,t,s}^{\text{DC}}=\sum\limits_{k\{i,c\}\in \mathcal{L}_{\text{DC}}}P_{k,t,s}^{\text{DC}}-\sum\limits_{l(i,j)\in\mathcal{ L}_{\text{DC}}}\left(P_{l,t,s}^{\text{DC}}-\overline{I}_{l,t,s}^{\text{DC}} \tau_{l}\right)\forall i\in\mathcal{N}_{\text{DC}}\\ \widetilde{U}_{i,t,s}^{\text{DC}}=\widetilde{U}_{i,t,s}^{\text{DC}}-2\tau_{R} \widetilde{U}_{k,t,s}^{\text{DC}}+\widetilde{U}_{k,t,s}^{\text{DC}}\tau_{k}^ {\text{W}}(j,i)\in\mathcal{L}_{\text{DC}}\\ \left\|\left[\ 2\widetilde{D}_{k,t,s}^{\text{DC}}\ \widetilde{I}_{k,t,s}^{\text{DC}}- \widetilde{U}_{j,t,s}^{\text{DC}}\ \right]^{\text{T}}\right\|_{2}\leq\widetilde{I}_{k,t,s}^{\text{DC}}+ \widetilde{U}_{j,t,s}^{\text{DC}}\\ \left(U_{i,\min}^{\text{DC}}\right)^{2}\leq\widetilde{U}_{i,t,s}^{\text{DC}} \leq\left(U_{i,\max}^{\text{DC}}\right)^{2}\ \forall i\in\mathcal{N}_{\text{DC}}\\ -S_{k,\max}^{\text{DC}}\leq P_{k,t,s}^{\text{DC}}\leq S_{k,\max}^{\text{DC}} \forall k(j,i)\in\mathcal{L}_{\text{DC}}\end{array}\right.. \tag{47}\]  
The power flow model for the AC distribution network and VSC under the wind power scenario, s, can be expressed in a similar way, but the corresponding formulas are not listed here to save page.