## 4 Detailed Multi-Time-Scale Coordinated Operation Model for the Combined WPTHB System
### Real-Time Corrective Scheduling  
The real-time corrective scheduling utilizes the feedback and rolling optimization of model predictive control (MPC) to accurately track the wind-photovoltaic power load demand. Based on the updated wind-photovoltaic power load forecast data in the next 15 min and the real-time actual power output of thermal power units, hydro-pumped storage units, and batteries; power adjustments of thermal power units, hydro-pumped storage units, and batteries are optimized for tracking load fluctuations in the next 15 min. The proposed MPC-based real-time corrective scheduling model is composed of the following general steps: (1) Select the state variables \(X_{t}=\left[P_{g,i}(t),P_{h,j}(t),P_{b,j}(t),W_{b,j}(t)\right]\), \(i\in NG\), \(j\in NH,l\in NB\), which include the active power outputs of thermal power units, hydro-pumped storage units, and batteries, and the remaining energy of batteries; select the real-time power adjustment of thermal power units, hydro-pumped storage units, and batteries as the control variable \(\Delta U_{t}\), \(\Delta U_{t}=\left[\Delta P_{g,i}(t),\Delta P_{h,j}(t),\Delta P_{b,j}(t) \right]\), \(i\in NG\), \(j\in NH,l\in NB\). The real-time corrective scheduling is performed every 5 min with an optimization horizon of 15 min. The discrete-state space model is established as (29) for the above parameters. (2) Based on the discrete-state space model and the forecasted wind-photovoltaic power load in the next 15 min, whilst considering the operation constraints of the thermal power units, hydro-pumped storage units, and the battery in 5 min and the constraints of system power balance (as shown in formula (31)), the power correction of thermal units, hydro-pumped storage units, and the battery unit in the next 15 min is solved for the control variables sequences at time \(t+1\Delta t,\ t+2\Delta t,\ t+3\Delta t\). (3) Only the control variables in the first 5 min period are applied as the real-time corrective power. (4) Repeat the above procedures at time point \(t+1\Delta t\) to complete the real-time corrective scheduling. The discrete space state model is as follows:  
\[X_{t+\Delta t}=\left[ \begin{array}{c}P_{g,i}(t+\Delta t)\\ P_{h,j}(t+\Delta t)\\ P_{h,j}(t+\Delta t)\\ W_{b,j}(t+\Delta t)\end{array} \right]=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&1&1\end{array} \right]\left[ \begin{array}{c}P_{g,i}(t)\\ P_{h,j}(t)\\ P_{b,j}(t)\\ W_{b,j}(t)\end{array} \right]+\left[ \begin{array}{cccc}1&0&0\\ 0&1&0\\ 0&0&1\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}\Delta P_{g,i}(t)\\ \Delta P_{h,j}(t)\\ \Delta P_{b,j}(t)\\ \Delta P_{b,j}(t)\end{array} \right] \tag{29}\]  
where \(i\in NG\) is the index of thermal power units; \(j\in NH\) is the index of hydro-pumped storage units; \(l\in NB\) is the index of batteries. Taking the state variable \(X_{t0}\) at the current time of \(t0\) as the initial value, the discrete-state space model is iterated three times to obtain the state variable for the 15 min time period with a time resolution of 5 min. The MPC-based real-time corrective scheduling model is detailed in the following. (1) Objective function  
Because day-ahead and hour-level scheduling basically guarantee the power generation economy of thermal power generators and pumped storage units, real-time corrective scheduling needs to correct the power output of thermal power units, pumped storage units and batteries in the presence of prediction errors, and ensure the total power output of the whole combined WPTHB system is accurately tracking prediction errors and that it reduces power regulation cost by (30). \[\min\text{Y}=\sum_{\tau=1}^{P}\left[\left(X_{t0+\tau\Delta t}^{r}-X_{t0+\tau \Delta t}\right)W\left(X_{t0+\tau\Delta t}^{r}-X_{t0+\tau\Delta t}\right)^{T} +\Delta U_{t0+\tau\Delta t}Q_{u}\Delta U_{t0+\tau\Delta t}^{T}\right], \tag{30}\]  
where \(W\) and \(Q_{u}\) are the coefficient matrices; \(t_{0}\) is the current time; \(\Delta t\) is 5 min; \(P\) is the number of prediction steps.