## 4 Detailed Multi-Time-Scale Coordinated Operation Model for the Combined WPTHB System
### Hour-Level Rolling Corrective Schedule  
The hour-level rolling corrective scheduling is performed every hour, with an optimization cycle of 3 h and an interval of 15 min. The corrective power of thermal generators and hydro-pumped storage units in the next 3 h will be optimized in this time scale. (1) Objective of hour-level rolling corrective schedule:  
\[\min F_{hours}=\sum_{t=t_{0}+1}^{T0}(\sum_{i=1}^{NG}f_{\mathrm{C}}\bigg{(}U_{g,i}^{day}(t),P_{g,i}^{hours}(t)\bigg{)}+\sum_{j=1}^{NH}\bigg{(}U_{h,j}^{gen,hours }(t),U_{h,j}^{pnum,hours}(t)\bigg{)}), \tag{18}\]  
where \(F_{hours}\) is the corrective power output cost of thermal power generators and hydro-pumped storage units; \(t_{0}\) is the current time point. Since the interval is 15 min, there are 12 time periods for the next 3 h, so \(T0=t_{0}+12\); \(U_{g,i}^{day}(t)\) is the on-off state of the thermal power unit \(i\) at the \(t\)-th time period; \(P_{g,i}^{hours}(t)\) is the active power output adjustment of thermal power unit \(i\) with the on-off state solved in the day-ahead scheduling; \(U_{h,j}^{gen,hours}(t)\) and \(U_{h,j}^{pnum,hours}(t)\) are, respectively, the power generation and water pumping states of the hydro-pumped storage unit \(j\) in the hour-level rolling corrective scheduling. (2) Constraints of the hour-level rolling corrective scheduling  
The ramping-up and down limits of thermal power unit in 15 min:  
\[\Delta P_{g,i,A}^{-}\leq P_{g,i}^{hours}(t)-P_{g,i}^{hours}(t-1)\leq\Delta P_{g,i,A}^{+}, \tag{19}\]  
where \(P_{g,i}^{hours}(t)\) and \(P_{g,i}^{hours}(t-1)\) are the amount of active power output adjustment of the thermal power unit \(i\) in the \(t\)-th time period and \(t-1\) time period (note: each time period is 15 min); \(\Delta P_{g,i,A}^{-}\), \(\Delta P_{g,i,A}^{+}\) are, respectively, the lower and the upper limits of power adjustment for thermal power unit \(i\) in 15 min. The power output upper and lower limits of the thermal generators in 15 min:  
\[U_{g,i}^{day}(t)P_{g,i}^{min}\leq P_{g,i}^{hours,0}(t)\leq U_{g,i}^{day}(t)P_{ g,i}^{max}. \tag{20}\]  
The power output upper and lower limits of the hydro-pumped storage unit in 15 min:  
\[P_{h,j}^{min}\leq P_{h,j}^{hours,0}(t)\leq P_{h,j}^{max}, \tag{21}\]where \(P_{g,i}^{hours,0}(t)\) is the sum of power output in day-ahead scheduling and the adjustment power output in hour-level rolling corrective scheduling for thermal power unit \(i\), that is \(P_{g,i}^{hours,0}(t)=P_{g,i}^{hours}(t)+P_{g,i}^{day}(t)\); similarly, \(P_{h,j}^{hours,0}(t)\) is the sum of day-ahead scheduling power output and hour-level rolling corrective scheduling adjustment power output \(P_{h,j}^{hours}(t)\) of hydro-pumped storage unit \(j\), that is \(P_{h,j}^{hours,0}(t)=P_{h,j}^{hours}(t)+P_{h,j}^{day}(t)\). Coupling constraints for water pumping and the power generation state of hydro-pumped storage unit in 15 min:  
\[U_{h,j}^{sen,hours}(t)\times U_{h,j}^{pnum,hours}(t)=0. \tag{22}\]  
The reservoir energy constraints of pumped storage station:  
\[E_{t}^{hours}=E_{t0}^{day}+\sum_{x=1}^{t-0}\sum_{j\in H_{s}}P_{h,j}^{hours,0}(t 0+x)\times\Delta t, \tag{23}\]  
\[E_{min}\leq E_{t}^{hours}\leq E_{max}, \tag{24}\]  
where \(E_{t}^{hours}\) is the reservoir energy in the \(t\)th 15-min time period. \(E_{t0}^{days}\) is the reservoir energy at the current time point \(t_{0}\) under the day-ahead scheduling. \(x\) is the number of time intervals between the predicted time point and the current time point. The constraint for continuous utilization of the hydro-pumped storage station in the next day:  
\[E_{24}^{hours}=E_{0}, \tag{25}\]  
where \(E_{0}\) is the energy reservoir at the beginning of the day; \(E_{24}^{hours}\) is the reservoir energy at the end of the day in the hour-level rolling corrective scheduling.