## 3 Chance-Constraint Two-Stage Stochastic Unit Commitment
### Scenario-Based Day-Ahead Unit Commitment
#### 3.2.1 Objective  
The objective of UC model is to reduce the system operating costs (\(Cost^{Sys}\)) including generation scheduling costs and economic dispatch costs. Generation scheduling costs (\(Cost^{GC}\)) are the first stage day-ahead costs, involving the fuel cost, the start-up cost and the reserve scheduling cost for the thermal units. Economic dispatch costs (\(Cost^{DC}\)) are the second stage costs, including the scenario-based spinning reserve deploying cost for thermal unit, the generating cost for the CSP and  
\[0\leq R_{g,t}^{G,up}\leq o_{g,t}^{G}R_{g,t}^{G,up,+},\ 0\leq R_{g,t}^{G,dn}\leq o_{g,t}^{G}R_{g }^{G,dn,+},\forall t,\forall g \tag{37}\]  
where, \(o_{g,t}^{G}\), \(o_{g,t-1}^{G}\) are the status variable of thermal unit at time \(t\) and \(t-1\) respectively; \(u_{g,t}^{G}\) represents the start-up variables of thermal unit; \(T_{gmin}^{G,on}\), \(T_{gmin}^{G,off}\) respectively denote the minimum on and off time; \(P_{g,t}^{G}\), \(P_{g,t-1}^{G}\) represent the power output of thermal unit at time \(t\) and \(t-1\) respectively; \(P_{g}^{G,max}\), \(P_{g}^{G,min}\) stand for the maximum and minimum power output of thermal unit respectively; \(\triangle P_{g}^{G,up}\), \(\triangle P_{g}^{G,dn}\) describe the up and down ramping limit for thermal unit respectively; \(R_{g,t}^{G,up}\), \(R_{g,t}^{G,dn}\) describe the scheduled up/down reserve for thermal unit respectively. \(R_{+,g}^{G,up}\), \(R_{+g}^{G,dn}\), denote the maximum up and down reserve for thermal unit to ensure the generation efficiency respectively. The start-up cost includes the hot start cost and the cold start cost. For each thermal unit, the start-up cost (\(ST_{g,t}^{G}\)) is classified by the idle time of unit before it is starts, namely start-up time. The set maximum hot start-up time \(m\), and the test time \(M\) determine the classification of the start process. Where, \(K_{g,n}\) is the coefficient of the start-up cost; \(hc_{g}\) represents the coefficient of the hot start cost; \(cc_{g}\) decides the coefficient of the cold start cost. The piecewise linearized Equation (38) is formulated in which the start-up time between \(0\) and \(m-1\) would be classified as hot start, and the time between \(m\) and \(M\) would be considered as a cold start. \[\left\{ \begin{array}{ll}ST_{g,t}^{G}\geq K_{g,n}(o_{g,t}^{G}-\sum\limits_{n =1}^{M}o_{g,t-n}^{G})&\\ ST_{g,t}^{G}\geq 0&,\forall t,\forall g\\ K_{g,1:m-1}=hc_{g},K_{g,m:M}=cc_{g}&\end{array} \right. \tag{38}\]  
The piecewise linearized Equation (39) calculates the fuel cost of thermal unit. \[C_{g,t}^{G}\geq a_{g,t}P_{g,t}^{G}+b_{g,t}o_{g,t}^{G},\forall t,\forall g \tag{39}\]  
where, \(a_{g,t}\) and \(b_{g,t}\) represent the coefficient of fuel cost. (2) CSP and EH constraints  
The operation constraints of the CSP and EH are described in the feasible operation space shown in Equations (1)-(23). When incorporated into the power system, CSP and EH provide energy services as well as reserve services. Equations (40)-(44) demonstrate the up/down spinning reserve scheduling constraints of TES system. When the \(PB\) system is on, the up spinning reserve (\(R_{c,t}^{PB,up}\)) is limited by the upper limit of \(PB\) operating range and the room of available power releasing of TES system. Similarly, the down spinning reserve (\(R_{c,t}^{PB,dn}\)) is constrained by the lower limit of operation space and the remaining space of the TES system.