## 3 Chance-Constraint Two-Stage Stochastic Unit Commitment
### Scenario-Based Day-Ahead Unit Commitment
#### 3.2.1 Objective
\[R_{c,t}^{PB,up}\leq o_{c,t}^{PB}\cdot(P_{c}^{PB,max}-P_{c,t}^{CSP})\, \forall t,\forall c \tag{40}\]  
\[R_{c,t}^{PB,up}\leq o_{c,t}^{PB}\cdot(E_{c,t}^{TES}-E_{c}^{TES,-})\eta_{c}^ {PB},\forall t,\forall c \tag{41}\]  
\[R_{c,t}^{PB,dn}\leq o_{c,t}^{PB}\cdot(P_{c,t}^{CSP}-P_{c}^{PB,min})\, \forall t,\forall c \tag{42}\]  
\[R_{c,t}^{PB,dn}\leq o_{c,t}^{PB}\cdot(E_{c}^{TES,+}-E_{c,t}^{TES})\eta_{c}^ {PB},\forall t,\forall c \tag{43}\]  
\[0\leq R_{c,t}^{PB,up}\leq o_{c,t}^{PB}R_{c}^{PB,up,+},0\leq R_{c,t}^{PB,dn} \leq o_{c,t}^{PB}R_{c}^{PB,dn,+}\,\forall t,\forall c \tag{44}\]  
where, \(R_{c}^{PB,up,+}\), \(R_{c}^{PB,dn,+}\) denote the maximum up and down reserve for each unit to ensure the generation efficiency. Equations (40)-(43) are composed of bilinear terms which belong to the non-deterministic polynomial hard (NP-Hard) problem. This kind of problem could not be calculated directly only if it is linearized. In this paper, the McCormick convex envelope method, namely Big-M method, is utilized to linearize the bilinear terms [39]. The detailed method is shown in Appendix A.  
Equations (45)-(47) demonstrate the up/down spinning reserve scheduling constraints of EH equipment. By contrast with the CSP plant and the traditional thermal unit, the EH equipment consumes the excess electricity in the power system and responds quickly without set ramping rate. The EH provides up spinning reserve (\(R_{c,t}^{EH,up}\)) and down spinning reserve (\(R_{c,t}^{EH,dn}\)) through decreasing and increasing electricity consumptions. \[P_{c,t}^{EH}-R_{c,t}^{EH,up}\geq o_{c,t}^{EH}P_{c}^{EH,min}\,\forall t, \forall c \tag{45}\]  
\[P_{c,t}^{EH}+R_{c,t}^{EH,dn}\leq o_{c,t}^{EH}P_{c}^{EH,max}\,\forall t, \forall c \tag{46}\]  
\[0\leq R_{c,t}^{EH,up}\leq o_{c,t}^{EH}P_{c}^{EH,up,+},0\leq R_{c,t}^{EH,dn} \leq o_{c,t}^{EH}P_{c}^{EH,dn,+}\,\forall t,\forall c \tag{47}\]  
where, \(o_{c,t}^{EH}\) is the status variable of EH; \(R_{c}^{EH,up,+}\), \(R_{c}^{EH,dn,+}\) denote the maximum up and down reserve for each EH unit. (3) Wind power and photovoltaic (PV) constraints  
Equations (48) and (49) formulate the power output range of wind power and PV station. The wind power (\(P_{w,t}^{WIND}\)) and the PV power (\(P_{v,t}^{PV}\)) are definitely lower than the point forecast of wind power and PV power (\(w_{w,t}^{WIND,0}\)\(w_{v,t}^{PV,0}\)) and allow for the renewables spillage. \[0\leq P_{w,t}^{WIND}\leq w_{w,t}^{WIND,0}\,\forall t, \forall w \tag{48}\] \[R_{c,t}^{PB,up}\leq o_{c,t}^{PB}\cdot(P_{c}^{PB,max}-P_{c,t}^{CSP})\, \forall t,\forall c \tag{40}\]0  
(4) System constraints  
The system power balance is formulated in Equation (50) to guarantee the power output to meet the load requirement at each time. In the first stage, load shedding is not permitted. \[R_{c,t}^{PB,up}\leq o_{c,t}^{PB}\cdot(P_{c}^{PB,max}-P_{c,t}^{CSP})\, \forall t,\forall c \tag{40}\]1  
Equations (51) and (52) set the scheduled up reserve range is [\(R_{t}^{up,-}\), \(R_{t}^{up,+}\)], and the scheduled down reserve range is [\(R_{t}^{dn,-}\), \(R_{t}^{dn,+}\)]. \[R_{c,t}^{PB,up}\leq o_{c,t}^{PB}\cdot(P_{c}^{PB,max}-P_{c,t}^{CSP})\, \forall t,\forall c \tag{40}\]2  
\[R_{c,t}^{PB,up}\leq o_{c,t}^{PB}\cdot(P_{c}^{PB,max}-P_{c,t}^{CSP})\, \forall t,\forall c \tag{40}\]3  
(5) Chance-constraints  
The reliability requirements at time \(t\) is described in the chance-constrained manner and are shown in Equations (53) and (54). Under the circumstances that the CSP reserve and the thermal unit reserve are considered, the load shedding case Equation (53) at time \(t\) is bounded by \(b_{t}^{t}\) with the probability greater than \(\alpha_{t}^{l}\). A similar formulation in Equation (54) could be used for the renewables spillage limit. The chance-constraints must be converted into the equivalent deterministic formulations to be employed in the UC model. The detailed introduction of chance-constrained reserve planning as well as the linearized conversion method is described in Section 3.3.