## 4 Detailed Multi-Time-Scale Coordinated Operation Model for the Combined WPTHB System
### Objective Function
\right., \tag{7}\]  
where \(T_{g,i}^{on,day}(t-1)\) is the hours of continuous \"on\" status of the thermal power unit \(i\) until the hour \(t-1\); \(T_{g,i}^{off,day}(t-1)\) is the hours that the thermal power unit \(i\) is continuously in the \"off\" status until the hour \(t-1\); \(MT_{g,i}^{on,day}\) is the minimum \"on\" time in hours of the thermal power unit \(i\); \(MT_{g,i}^{off,day}\) is the minimum \"off\" time in hours of the thermal power unit \(i\). The power output upper and lower limits of the hydro-pumped storage unit:  
\[P_{h,j}^{min}\leq P_{h,j}^{day}(t)\leq P_{h,j}^{max}, \tag{8}\]where \(P_{h,j}^{min}\) is the maximum pumping power of the hydro-pumped storage unit \(j\) (negative value); \(P_{h,j}^{max}\) is the maximum power generation of the hydro-pumped storage unit \(j\) (positive value); \(P_{h,j}^{day}(t)\) is the day-ahead power output of the pumped storage unit \(j\) at the hour \(t\), and the positive value represents in the power generation state, and a negative value represents in the pumping state. Coupling constraints for water pumping and power generation state of hydro-pumped storage units:  
\[U_{h,j}^{gen,day}(t)\times U_{h,j}^{pum,day}(t)=0. \tag{9}\]  
The reservoir energy balance constraint of hydro-pumped storage power station:  
\[E_{t+1}^{day}=E_{t}^{day}+\sum_{j\in Hs}P_{h,j}^{day}(t)\times\Delta t, \tag{10}\]  
where \(E_{t}\) is the reservoir energy of hydro-pumped storage station at the \(t\)-th hour, \(Hs\) is the collection of hydro-pumped storage units contained in the hydro-pumped storage station. The upper and lower limits of reservoir energy of the hydro-pumped storage station:  
\[E_{min}\leq E_{t}^{day}\leq E_{max}, \tag{11}\]  
where \(E_{min}\), \(E_{max}\) are respectively the minimum and maximum reservoir energy of hydro-pumped storage station. The reservoir energy of hydro-pumped storage station at the end of each day shall be ready for continuous utilizations in the next day:  
\[E_{24}^{day}=E_{0}, \tag{12}\]  
where \(E_{0}\) is the reservoir energy at the beginning of the day; \(E_{24}\) is the reservoir energy at the end of the day under the day-ahead scheduling. The power output upper and lower limits of the battery \(l\):  
\[P_{b,l}^{min}\leq P_{b,l}^{day}(t)\leq P_{b,l}^{max}, \tag{13}\]  
where \(P_{b,l}^{min}\), \(P_{b,l}^{max}\) are, respectively, the maximum charging power (negative value) and the maximum discharging power (positive value) of the battery \(l\). The upper and lower limits of the battery energy:  
\[\left\{ \begin{array}{l}W_{b,l}^{min}\leq W_{b,l}^{day}(t)\leq W_{b,l}^{max}\\ W_{b,l}^{day}(24)=W_{b,l}^{day}(0)\\ W_{b,l}^{day}(t+1)=W_{b,l}^{day}(t)+P_{b,l}^{day}(t)\end{array} \right., \tag{14}\]  
where \(W_{b,l}^{day}(t)\) is the remaining capacity of the battery \(l\) at hour \(t\); \(W_{b,l}^{min}\) and \(W_{b,l}^{max}\) are, respectively, the minimum and the maximum energy threshold of the battery unit \(l\). \(W_{b,l}^{day}(0)\) is the initial energy of the battery \(l\) at the beginning of the day. \(W_{b,l}^{day}(24)\) is the remaining energy of the battery \(l\) at the end of the day in the day-ahead scheduling. Power balance constraint:  
\[P_{day,w}(t)+P_{day,x}(t)+\sum_{i=1}^{NG}P_{g,i}^{day}(t)+\sum_{j=1}^{NH}P_{h,j}^{day}(t)+\sum_{l=1}^{NB}P_{b,l}^{day}(t)=P_{day,Load}(t), \tag{15}\]  
where \(P_{day,w}(t)\), \(P_{day,y}(t)\), and, \(P_{day,Load}(t)\) are the day-ahead forecast data of wind power, photovoltaic power, and load demand in the next 24 h, respectively. System spinning reserve capacity constraint at hour \(t\):  
\[\sum_{i=1}^{NG}(P_{g,i}^{max}\times U_{g,i}^{day}(t))\geq P_{day,Load}(t)+r(t), \tag{16}\]  
where \(r(t)\) is the system spinning reserve margin at the \(t\)-th hour.