## 3 Problem Formulation
### Constraints  
(1) Minimum up and down time.  
For ease of calculation, we need to linearize the minimum up and down time constrains, which are only related to the generators' commitment states. According to [33], the minimum up and down time constraints can be formulated by the equivalent mixed-integer linear expressions relying on binary variables, which are associated with the different statuses of generating units (e.g., start-up and shutdown). Therefore, the expressions for the minimum up-time constraints can be described as follows:  
\[\sum_{t=1}^{G_{i}}(1-S_{G,i,t})=0 \tag{18}\]  
\[\sum_{n=t}^{t+T_{Gi,m}-1}S_{G,i,d}\geq T_{Gi,on}(S_{G,i,t}-S_{G,i,t-1})\quad t =G_{i}+1,\ldots,T-T_{Gi,on}+1 \tag{19}\]  
\[\sum_{n=t}^{T}[S_{G,i,n}-(S_{G,i,t}-S_{G,i,t-1})]\geq 0\quad t=T-T_{Gi,on}+2, \ldots,T \tag{20}\]  
where \(G_{i}\) refers to the number of initial periods that unit \(i\) must be online, \(G_{i}=\min\{T\), \((T_{Gi,on}-T_{Gi,on,0})\)\(S_{G,i,0}\}\), \(T_{Gi,on}\) is the minimum up-time of the \(i\)th generation unit, \(T_{Gi,on,0}\) is the time that the \(i\)th generation unit has been online, and \(S_{G,i,0}\) is the initial state of the \(i\)th generation unit. The constraints in Equation (18) are influenced by the beginning status of the units, which are defined by \(G_{i}\). The constraints in Equation (19) are used for the following time intervals to meet the requirements of the minimum up-time constraint in all the possible sets of consecutive periods of size \(T_{Gi,on}\). The constraints of Equation (20) simulate the last \(T_{Gi,on}-1\) periods in which, if unit \(i\) is activated, it is kept online until the end of the time interval.  
The minimum down-time constraints are described as follows:  
\[\sum_{t=1}^{L_{i}}(1-S_{G,i,t})=0 \tag{21}\]  
\[\sum_{n=t}^{t+T_{Gi,off}-1}(1-S_{G,i,n})\geq T_{Gi,off}(S_{G,i,t-1}-S_{G,i,t}) \quad t=L_{i}+1,\ldots,T-T_{Gi,off}+1 \tag{22}\]  
\[\sum_{n=t}^{T}[1-S_{G,i,n}-(S_{G,i,t-1}-S_{G,i,t})]\geq 0\quad t=T-T_{Gi,off}+2, \ldots,T \tag{23}\]  
where \(L_{i}\) refers to the number of initial periods that unit \(i\) must be offline, \(L_{i}=\min\{T\), \((T_{Gi,off}-T_{Gi,off,0})\)\((1-S_{G,i,0})\)\(T_{Gi,off}\) is the minimum down-time of the \(i\)th generation unit, and \(T_{Gi,off,0}\) is the time that the \(i\)th generation unit has been offline.  
(2) The power balance is  
\[\sum_{i=1}^{N_{G}}P_{G,i,t,s}+\sum_{j=1}^{N_{W}}P_{W,j,t,s}=P_{L,t}-\sum_{n=1} ^{N_{AC}}P_{AC,n,t,s} \tag{24}\]  
where \(P_{W,j,t,s}\) refers to the real power of wind power at time \(t\) in scenario \(s\), and \(P_{L,t}\) to the total system loads at time \(t\).  
(3) The generation constraints for the power unit are  
\[P_{G,i,\min}\leq P_{G,i,t,s}\leq P_{G,i,\max} \tag{25}\]where \(P_{G,j,min}\) and \(P_{G,i,max}\) refer respectively to the upper and lower limits of power from the \(i\)th generation unit.  
(4) The ramping constraints of the power unit are  
\[P_{G,j,t,s}-P_{G,i,t-1,s} \leq\gamma_{ui}T \tag{26}\] \[P_{G,j,t-1,s}-P_{G,i,t,s} \leq\gamma_{di}T \tag{27}\]  
where \(\gamma_{ui}\) and \(\gamma_{di}\) refer to the ramping-up and -down power rates of the \(i\)th generation unit, respectively, and \(T\) is equal to \(t_{c}\) minutes, which is the control cycle for the ACL.  
(5) The reserve constraints for the power unit are  
\[\sum_{i=1}^{N_{G}}(P_{G,i,\max}-P_{G,i,t,s})\geq R_{t} \tag{28}\]  
where \(R_{t}\) refers to the traditional spinning reserve capacity at time \(t\).  
(6) The constraints for the scenario are given by  
\[\left|P_{G,i,t,s}-P_{G,i,t,bs}\right|\leq\epsilon_{i} \tag{29}\]  
where \(P_{G,i,t,bs}\) is the power of the \(i\)th generation unit in the baseline scenario, and \(\epsilon_{i}\) is the spinning reserve capacity of unit \(i\).  
(7) The air-conditioning unit constraint is  
\[P_{AC,n,t,s}\leq\overline{P}_{AC,n} \tag{30}\]  
Our paper selects mixed-integer programming (MIP) to solve the proposed SUC model.