## 4 System Model
### Single-Level MPC Method  
Before the rolling optimization process is analyzed, some basic information must be given. We assume that the prediction horizon is equal to the control horizon and each of them are two hours. The sample time is 5 min. Let \(t_{s}=0\) when MPC controller begins to work, where \(t_{s}\) represents the serial number of the scheduling interval. Two hours are equal to twenty-four scheduling intervals. The principle of rolling optimization is as follows:  
First, at the current time, the MPC controller samples the current interval information, and predicts the information for the next two hours. Second, MPC performs on-line optimization. Finally, MPC makes the current time decisions. The above process is repeated on the arrival of the next scheduling interval. Although the principle is easy and clear, it will become relatively complex, once it is implemented to hybrid energy storage, which has significantly different response speeds. For understanding the principle, we will first analyze a simple example, and then we will explain the complex process. The simple example is that the pumped hydro storage is replaced by a battery. From Table 1, the response speed of the fly wheel is almost equal to that of the battery; therefore, at any scheduling interval, the battery and fly wheel are ready to change their work states according to the optimal results. In contrast, according to assumption 1, the pumped storage is not ready to change its work state at any scheduling interval. In other words, at any scheduling interval, we have the control right to change the fly wheel and battery, but we have almost lost the control right for pumped storage. Therefore, when the principle is applied to the battery and the fly wheel, it is a simple example, and the analysis process is as follows:  
MPC controller samples, forecasts, execute on line optimization and produces the results. The MPC controller repeats the above process on the next scheduling interval. Therefore, the MPC controller always performs the same set of actions at any scheduling interval. The process is shown in Figure 4.  
In Figure 4, at any scheduling interval, the context of MPC controller performing is similar. When the principle is applied to pumped storage and fly wheel, the process will become relatively complex, due to the control right of the pumped storage. According to Assumption 1, the pumped storage is under control at every starting half an hour, consequently, when \(t_{s}\) is 0, 6, 12, 18, 24..., both the pumped storage and the fly wheel are under control. The control right situation is similar to that of battery and fly wheel. Any other time, the two situations are different. The details are shown in Figure 5.  
Figure 5 schematically shows the process by which the single-level MPC performs the rolling optimization. \"Change\" means that, at the beginning of this half-hour period, the pumped storage can change operating state once; \"hold on\" means that, in the scheduling interval, pumped hydro storage cannot change its operating state; and \"Case 0-5\" denotes the difference conditions. The difference between Figure 4 and Figure 5 is significant. The main reasons for this difference are as follows:  
Figure 4: Single-level MPC performing rolling optimization (based on battery and fly wheel). When \(t_{s}=0\) (Case 0), the MPC controller starts working, samples and predicts information, performs on-line optimization, and then sends its optimization decisions to the pumped hydro storage and the fly wheel. The pumped hydro storage system receives and executes the decisions. Once the pumped storage changes its operating state, it must remain in the state until the half-hour period ends. In other words, when \(t_{s}=1\), \(2\ldots 5\), the pumped hydro storage cannot change its operating state for \(x\) min, where \(x\) is 25, 20, 15, 10, and 5, respectively.