## 3 Chance-Constraint Two-Stage Stochastic Unit Commitment
### Scenario-Based Day-Ahead Unit Commitment
#### 3.2.3 The Second-Stage Constraint  
The first stage utilizes the point forecasts to estimate the UC decisions and the scheduled reserve decisions. In the second stage, the uncertainty of renewables is considered and \(N_{s}\) pre-determined scenarios are generated using the method proposed in Section 3.1. In each pre-decided scenario, the second stage constraints guarantee the first stage decisions to be appropriately deployed to satisfy the power deviations of renewable energy.  
1. Thermal unit constraints  
Equation (55) formulates the dispatched power output of each unit which equals to the day-ahead power output in the first stage plus the reserve deployment in the second stage.  
\[P_{g,t,s}^{G}=P_{g,t}^{G}+r_{g,t,s}^{G,up}-r_{g,t,s}^{G,dn},\ \forall t,\forall g,\forall s \tag{55}\]  
Equations (56)-(58) demonstrate that the deployed reserve is constrained within the capacity of the scheduled reserve and the up/down reserve deployment would not happen simultaneously. Equations (59)-(61) describe the other thermal unit constraints.  
\[0\leq r_{g,t,s}^{G,up}\leq o_{g,t,s}^{G,up}\,\mathit{dp\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!  
\[0\leq r^{PB,dn}_{c,t,s}\leq o^{PB,dn}_{c,t,s}R^{PB,dn}_{c,t},\ \forall t,\forall c, \forall s \tag{64}\] \[o^{PB,up}_{c,t,s}+o^{PB,dn,deploy}_{c,t,s}\leq 1,\ \forall t, \forall c,\forall s\] (65) \[P^{EH}_{c,t,s}=P^{EH}_{c,t}-r^{EH,up}_{c,t,s}+r^{EH,dn}_{c,t,s}, \ \forall t,\forall c,\forall s\] (66) \[0\leq r^{EH,up}_{c,t,s}\leq o^{EH,up}_{c,t,s}R^{EH,up}_{c,t}, \ \forall t,\forall c,\forall s\] (67) \[0\leq r^{EH,dn}_{c,t,s}\leq o^{EH,dn,deploy}_{c,t,s}R^{EH,dn}_{c,t },\ \forall t,\forall c,\forall s\] (68) \[o^{EH,up}_{c,t,s}+o^{EH,dn,deploy}_{c,t,s}\leq 1,\ \forall t, \forall c,\forall s\] (69) \[o^{PB}_{c,t,s}=o^{PB}_{c,t},\ \forall t,\forall c,\forall s\] (70) \[o^{PB}_{c,t,s}P^{PB,min}_{c,t,s}\leq p^{CSP}_{c,t,s}\leq o^{PB}_{ c,t,s}P^{PB,max}_{c},\ \forall t,\forall c,\forall s\] (71) \[o^{EH}_{c,t,s}P^{EH,min}_{c,t,s}\leq p^{EH}_{c,t,s}\leq o^{EH}_{ c,t,s}P^{EH,max}_{c},\ \forall t,\forall c,\forall s\] (72) \[\text{(Other constraints are the modified function of 1-20)} \tag{73}\]  
where, Equations (63), (64), (67) and (68) should be linearized by Big-M method; \(P^{CSP}_{c,t,s}\), \(P^{EH}_{c,t,s}\) stand for the power output of the CSP and EH respectively in the second stage; \(r^{PB,up}_{c,t,s}\), \(r^{PB,dn}_{c,t,s}\) represent the up/down deployed reserve for CSP in the stochastic scenario; \(r^{EH,up}_{c,t,s}\), \(r^{EH,dn}_{c,t,s}\) represent the up/down deployed reserve for EH in the stochastic scenario; \(o^{PB,up}_{c,t,s}\), \(o^{PB,dn,deploy}_{c,t,s}\), \(r^{EH,up}_{c,t,s}\), \(o^{EH,dn,deploy}_{c,t,s}\) stand for the on/off variable of the up reserve deployment and down reserve deployment for the CSP and EH system respectively. \(o^{PB}_{c,t,s}\), \(o^{EH}_{c,t,s}\) stand for the on/off state variable of the CSP in the second stage. (3) Wind power and PV constraints The second stage power output of wind power and PV station are guaranteed in Equation (74).  
\[\text{(Modified function of 48-49)} \tag{74}\]  
(4) System constraints The second stage power balance is demonstrated in Equations (75) and (76). Load shedding is permitted in the second stage.  
\[\sum_{w=1}^{N_{W}}P^{WIND}_{w,t,s}+\sum_{v=1}^{N_{V}}P^{PV}_{v,t,s}+\sum_{g=1 }^{N_{G}}P^{G}_{g,t,s}+\sum_{c=1}^{N_{C}}(P^{CSP}_{c,t,s}-P^{EH}_{c,t,s})=\sum _{n=1}^{N_{N}}(L^{0}_{n,t}-L^{cur}_{n,t,s}),\forall t,\forall s \tag{75}\]  
\[0\leq L^{cur}_{n,t,s}\leq L^{0}_{n,t},\ \forall t,\forall n,\forall s \tag{76}\]  
\(P^{WIND}_{w,t}\), \(P^{PV}_{v,t,s}\) stand for the power output of wind and PV respectively in the second stage. \(L^{cur}_{n,t,s}\) represents the loss of load in the stochastic scenario.