## 3 Chance-Constraint Two-Stage Stochastic Unit Commitment
### Chance-Constraintted Reserve Planning Model  
In the traditional reserve planning model, the scheduled reserve is usually set as a constant value at each time and thermal unit provides the only reserve service in the power system. Specially, in the robust reserve planning, a significant number of reserves are scheduled to satisfy any kind of renewable energy deviations. Nevertheless, some extreme renewable energy outputs have low possibilities and there is no need to meet these demands [40]. Besides, in the practical reserve planning model, the capacity of the scheduled reserve is set as a certain percentage of system load and the additional reserve is considered in accordance with the renewables uncertainty. However, this method is not applicable for the power system with high renewables penetration, which has increased power output uncertainty [41]. Therefore, to handle the problems above, a chance-constrained reserve planning model is proposed in this paper which is introduced as follows. Figure 3 illustrates the chance-constrained reserve scheduling strategy. From Equation (27), we know that the equivalent load obeys the Gaussian distribution in each hour, which is clearly shown in Equation (77). \[ \begin{array}{ll}L_{t,s}^{EQ,fore}&=\sum_{n=1}^{N_{N}}L_{n.t}^{0}-\sum_{w=1}^ {N_{W}}w_{w,t,s}^{WIND,fore}-\sum_{v=1}^{N_{V}}v_{v,ts}^{PV,fore}\\ &\sim N\Big{(}\sum_{n=1}^{N_{N}}L_{n,t}^{0}-\sum_{w=1}^{N_{W}}w_{w,t}^{WIND,0} -\sum_{v=1}^{N_{V}}v_{v,t}^{PV,0}\,\ (N_{W}\omega_{w,t}^{WIND,\rho})^{2}+(N_{V} \omega_{v,t}^{PV,\rho})^{2}\Big{)}\\ &\sim N\Big{(}L_{t}^{EQ,0}\,\sigma_{t}^{2}\Big{)}\end{array}  \tag{77}\]  
(1) Load shedding  
When the power output of renewable energy in the stochastic scenario is lower than the expected point forecast, there is a chance that the power from thermal units, renewable energy could not satisfy the power balance considering the overall system up reserves (thermal unit reserves and CSP reserves). The expected value of loss-of-load probability (LOLP) is defined as \"expected load not served\" (_ELNS_) which is shown in Equation (78). E represents the expected value. \[ELNS_{t}=E[F_{t}(s)\big{|}F_{t}(s)\geq 0] \tag{78}\]  
where, \(F_{t}(s)\) is shown in Equation (79). \[F_{t}(s)=L_{t,s}^{EQ,fore}-\sum_{g=1}^{N_{G}}\left(P_{g,t}^{G}+R_{g,t}^{G,up} \right)-\sum_{c=1}^{N_{C}}\left(P_{c,t}^{CSP}+R_{c,t}^{PB,up}-P_{c,t}^{EH}\right) \tag{79}\]  
Figure 3: Load shedding and renewable energy spillage. The probabilistic constraints concerning \(ELNS\) is depicts in Equation (80). Pr represents the probability value. \[\Pr\{ELNS_{t}\leq b_{t}^{l}\}\geq a^{l} \tag{80}\]  
(2) Renewables spillage  
When the stochastic renewable energy power appears higher than the expected point forecast, a chance occurs when the power from generators exceed the load demand after the utilization of the total system down reserves. The expected value of wind power and PV spillage probability is defined as \"expected wind and PV spillage\" (\(EWVS\)) which is shown in Equation (81). \[EWVS_{t}=-E[H_{t}(s)\big{|}H_{t}(s)\leq 0] \tag{81}\]  
where, \(H_{t}(s)\) is shown in Equation (82). \[H_{t}(s)=L_{t,s}^{EQ,forne}-\sum_{g=1}^{N_{G}}\left(P_{g,t}^{G}-R_{g,t}^{G,dn} \right)-\sum_{c=1}^{N_{C}}\left(P_{c,t}^{CSP}-R_{c,t}^{PB,dn}-P_{c,t}^{EH}\right) \tag{82}\]  
The probabilistic constraints concerning \(EWVS\) is depicts in Equation (83). \[\Pr\{EWVS_{t}\leq b_{t}^{wv}\}\geq a^{wv} \tag{83}\]  
Equations (80) and (83) should be converted into the equivalent deterministic formulations to be employed in the UC model [35]. Given the parameter \(\alpha^{l}\), \(\alpha^{wv}\), \(b_{t}^{l}\), \(b_{t}^{wv}\), the equivalent deterministic formulations could be described in Equations (84)-(86).