## 4 Scenario Based Wind Curtailment Scheduling
### Decomposition of the Stochastic UC Problem  
The paper decomposes the original UC problem in Section 4 into a deterministic UC problem and a flexibility-checking sub-problem using Benders decomposition. The deterministic UC problem solves the optimization problem using the objective function in (13) subject to the constraints in (14)-(17). In the deterministic UC problem, forecasted wind power is substituted by (24). Under a specific scenario for uncertain wind and load, the deterministic UC solution may fail to adjust the generation level due to insufficient spinning reserve and ramping capability. In such cases, the system flexibility needs to be increased by deploying an additional spinning reserve or by scheduling wind power curtailment by following a flexibility-checking sub-problem for all scenarios shown below:  
\[R_{t}^{s}=Minimize\ e_{usr,t}^{s}+e_{dsr,t}^{s}+e_{urge,t}^{s}+e_{dreg,t}^{s}\,\] (28a) Subject to: \[\sum_{i=1}^{N_{G}}p_{G,i,t}^{s}+\sum_{i=1}^{N_{W}}p_{W,i,t}^{s}\ =\sum_{i=1}^{N_{B}}p_{D,\ i,t}^{s}-e_{usr,t}^{s}+e_{dsr,t}^{s}\, \tag{28b}\] \[p_{W,i,t}^{s}=\sum_{m=1}^{N_{m}}\big{\{}(s_{i,t}(m)-s_{i,t}(m-1))\cdot u_{ AW,i,t}^{s}(m)\ \cdot u_{L,i,t}^{s}(m)\big{\}}\, \tag{28c}\]\[ \begin{array}{c}e_{urg,t}^{s}+\sum\limits_{i=1}^{N_{G}}reg_{up,i,t}^{s}\geq uv_{D }+\sum\limits_{i=1}^{N_{W}}p_{W,i,t}^{s}\\ -\sum\limits_{i=1}^{N_{W}}\sum\limits_{m=1}^{N_{m}}\left\{(s_{i,t}(m)-s_{i,t}(m -1))\cdot u_{dv,i,t}^{s}(m)\cdot\hat{u}_{L,i,t}(m)\right\}\\ \\ e_{drg,t}^{s}+\sum\limits_{i=1}^{N_{G}}reg_{du,i,t}^{s}\geq dv_{D}-\sum\limits _{i=1}^{N_{W}}p_{W,i,t}^{s}\\ +\sum\limits_{i=1}^{N_{W}}\sum\limits_{m=1}^{N_{m}}\left\{(s_{i,t}(m)-s_{i,t}( m-1))\cdot u_{uv,i,t}^{s}(m)\cdot\hat{u}_{L,i,t}(m)\right\}\end{array}  \tag{28d}\]  
and also subject to the constraints (20) and (21), where the variables with caret on their top are the solutions of the deterministic UC. If an objective value is greater than zero, then the solution of the deterministic UC problem cannot ensure operational security against both the forecast error and the short-term variations for a specific scenario. In this case, the following reliability cuts will be generated and added to the deterministic problem for the next iteration:  
\[ \begin{array}{c}\bar{e}_{usr,t}^{s}+\bar{e}_{urg,t}^{s}+\sum\limits_{i=1}^{N _{W}}\sum\limits_{m=1}^{N_{m}}m_{t}^{s}\left(\hat{u}_{L,i,t}(m)\right)\cdot \left(u_{L,i,t}(m)-\hat{u}_{L,i,t}(m)\right)\\ +\sum\limits_{i=1}^{N_{G}}m_{t}^{s}(\hat{p}_{G,i,t})\cdot\left(p_{G,i,t}-\hat{ p}_{G,i,t}\right)\;+\sum\limits_{i=1}^{N_{G}}m_{t}^{s}(\hat{r}_{up,\;i,t})\cdot \left(r_{up,i,t}-\hat{r}_{up,\;i,t}\right)\leq 0\\ \\ \bar{e}_{ds,t}^{s}+\bar{e}_{drg,t}^{s}+\sum\limits_{i=1}^{N_{W}}\sum\limits_{m =1}^{N_{m}}m_{t}^{s}\left(\hat{u}_{L,i,t}(m)\right)\cdot\left(u_{L,i,t}(m)- \hat{u}_{L,i,t}(m)\right)\\ +\sum\limits_{i=1}^{N_{G}}m_{t}^{s}(\hat{p}_{G,i,t})\cdot\left(p_{G,i,t}-\hat{ p}_{G,i,t}\right)\;+\sum\limits_{i=1}^{N_{G}}m_{t}^{s}(\hat{r}_{dn,\;i,t})\cdot \left(r_{dn,i,t}-\hat{r}_{dn,\;i,t}\right)\leq 0\end{array}  \tag{29}\]  
where \(m_{t}^{s}(\cdot)\) is a marginal value of the objective function (28a) with respect to a corresponding variable at the solutions of the deterministic UC problem. After checking the flexibility of the power system for all scenarios, the original probability density function (pdf) of wind and load will be updated using the following formula of cross-entropy method [19]:  
\[\mu_{n}=\;\sum\limits_{s=1}^{N_{S}}I(R_{t}^{s}\geq\hat{\gamma}_{n})\cdot W^{ s}\cdot P_{t}^{s}\left/\sum\limits_{s=1}^{N_{S}}I(R_{t}^{s}\geq\hat{\gamma}_{n}) \cdot W^{s}\right., \tag{31}\]  
\[\sigma_{n}\;=\;\sum\limits_{s=1}^{N_{S}}I(R_{t}^{s}\geq\hat{\gamma}_{n})\cdot W ^{s}\cdot(P_{t}^{s}-\mu_{n})^{2}\left/\sum\limits_{s=1}^{N_{S}}I(R_{t}^{s} \geq\hat{\gamma}_{n})\cdot W^{s}\right., \tag{32}\]  
where the function \(I(m_{t}^{s}\geq 0)\) is equal to one if \(m_{t}^{s}\geq 0\), is equal to zero otherwise.