## 4 Detailed Multi-Time-Scale Coordinated Operation Model for the Combined WPTHB System
### Objective Function  
The objective of day-ahead scheduling model is to minimize the combined system total generation cost. According to the forecasted wind-photovoltaic power load in the next 24 h, the power generation cost of thermal power units, hydro-pumped storage units, and batteries is taken into account as follows:  
\[ \begin{array}{rl}\min F_{day}&=\sum\limits_{t=1}^{T}\left[\sum\limits_{i=1}^{ NG}f_{\text{G}}\bigg{(}U_{g,i}^{day}(t),P_{g,i}^{day}(t)\bigg{)}+\sum\limits_{j=1}^{ NH}f_{\text{fi}}\bigg{(}U_{h,j}^{pun,day}(t),U_{h,j}^{gen,day}(t)\bigg{)}\\ &\hskip 142.26378 pt+\sum\limits_{l=1}^{NB}f_{\text{B}}(P_{b,j}^{day}(t))\big{]} \,\end{array}  \tag{1}\]  
where \(F_{day}\) is the total generation cost of combined system; \(T\) is the number of optimization time periods; \(NG\) is the number of thermal power units; \(f_{\text{G}}\bigg{(}U_{g,i}^{day}(t),P_{g,i}^{day}(t)\bigg{)}\) is the power generation costs of thermal power unit \(i\); \(NH\) is the number of hydro-pumped storage units; \(f_{\text{fi}}\bigg{(}U_{h,j}^{pun,day}(t),U_{h,j}^{gen,day}(t)\bigg{)}\) is  
Figure 1: Multi-time-scale coordinated scheduling framework for the combined WPTHB system. the power generation costs of hydro-pumped storage unit \(j\); \(NB\) is the number of batteries; \(f_{B}(P_{b,l}^{day}(t))\) is the power generation costs of battery \(l\). \[f_{G}\Big{(}U_{g,i}^{day}(t),P_{g,i}^{day}(t)\Big{)}=a_{g,j}P_{g,i}^{day}(t)^{2}+ b_{g,i}P_{g,i}^{day}(t)+c_{g,i}+S_{g,i}U_{g,i}^{day}(t)\Big{(}1-U_{g,i}^{day}(t-1) \Big{)}, \tag{2}\]  
\[f_{H}\Big{(}U_{h,j}^{pump,day}(t),U_{h,j}^{gen,day}(t)\Big{)} =C_{h,j}^{gen}U_{h,j}^{gen,day}(t)\Big{[}U_{h,j}^{gen,day}(t)-U_{h,j}^{gen,day}(t-1)\Big{]} \tag{3}\] \[+C_{h,j}^{pum}U_{h,j}^{pum,day}(t)\Big{[}U_{h,j}^{pum,day}(t)-U_{h,j}^{pum,day}(t-1)\Big{]},\] \[f_{B}(P_{b,l}^{day}(t))=C_{B,l}P_{b,l}^{day}(t)^{2}, \tag{4}\]  
where \(U_{g,i}^{day}(t)\) is the on-off status of the thermal power unit \(i\) at the \(t\)-th hour, 1 means on and 0 means off; \(P_{g,i}^{day}(t)\) is active power output of the thermal power unit \(i\) at the \(t\)-th hour; \(a_{g,i}\), \(b_{g,i}\), \(c_{g,i}\) are generation cost coefficients of the thermal power unit \(i\); \(S_{g,i}\) is the starting cost of the thermal power unit \(i\); \(U_{h,j}^{pum,day}(t)\) is the water pumping state of hydro-pumped storage unit \(j\) at the \(t\)-th hour, 1 means in the pumping state and 0 means in the non-pumping state; \(U_{h,j}^{gen,day}(t)\) is the power generation state of the hydro-pumped storage unit \(j\) at the \(t\)-th hour, 1 means in the power generation state, 0 means in the non-power generation state; \(C_{h,j}^{gen}\) is the starting cost for the power generation state of hydro-pumped storage unit \(j\); \(C_{h,j}^{pum}\) is the starting cost for the pumping state of hydro-pumped storage unit \(j\); \(C_{B,l}\) is the power generation cost coefficient of battery \(l\); \(P_{b,l}^{day}(t)\) is the active power output of battery \(l\) at the \(t\)-th hour and the positive value represents the discharging power and the negative represents the charging power. b. Operation Constraints  
The hourly ramping-up and down limits of the thermal power unit:  
\[\Delta P_{g,i}^{-}(t)P_{g,i}^{day}\leq P_{g,i}^{day}(t)-P_{g,i}^{day}(t-1)\leq \Delta P_{g,i}^{+}, \tag{5}\]  
where \(\Delta P_{g,i,hour}^{-}\) and \(\Delta P_{g,i,hour}^{+}\) are the lower and upper ramping limits of thermal power units in 1 h, respectively. The power output upper and lower limits of thermal power unit:  
\[U_{g,i}^{day}(t)P_{g,i}^{min}\leq P_{g,i}^{day}(t)\leq U_{g,i}^{day}(t)P_{g,i}^ {max}. \tag{6}\]  
The minimum on-off time constraints:  
\[\left\{ \begin{array}{l}\Big{(}T_{g,i}^{on,day}(t-1)-MT_{g,i}^{on,day}\Big{)} \bigg{[}U_{g,i}^{day}(t-1)-U_{g,i}^{day}(t)\Big{]}\geq 0\\ \Big{(}T_{g,i}^{off,day}(t-1)-MT_{g,i}^{off,day}\Big{)}\bigg{[}U_{g,i}^{day}(t)- U_{g,i}^{day}(t-1)\Big{]}\geq 0\end{array}