## 5 Simulation Results and Discussion
### Simulation Test for Three Sub-Objectives Based on Single-Level and Two-Level MPC
We know that the \(p_{i}^{\text{w}}\) weighting coefficient \(w_{2}\) (\(w_{2}=1.0\)) is larger than the \(p_{i}^{\text{wd}}\) weighting coefficient \(w_{3}\) (\(w_{3}=0.1\)), and the difference between \(w_{2}\) and \(w_{3}\) forms a decision principle; that is, for reducing wind curtailments, sacrificing \(p_{i}^{\text{wd}}\) is permitted. Therefore, as shown in Figure 11, when \(p_{i}^{\text{wind}}>p_{i}^{\text{plan}}\), \(p_{i}^{\text{wd}}\) are equal to \(p_{i}^{\text{plan}}\), except for some scheduling intervals in which \(p_{i}^{\text{wd}}<p_{i}^{\text{plan}}\). From the above analysis, we remark that there was a significant possibility of the single-level \(p_{i}^{\text{wd}}\) being equal to the two-level \(p_{i}^{\text{wd}}\). Moreover, exchanging \(w_{2}\) and \(w_{3}\) yields the following results. After exchanging \(w_{2}\) and \(w_{3}\), the \(p_{i}^{\text{w}}\) weighting coefficient \(w_{2}\) (\(w_{2}=0.1\)) is less than the \(p_{i}^{\text{wd}}\) weighting coefficient \(w_{3}\) (\(w_{3}=1.0\)), and the new decision principle for maximizing \(p_{i}^{\text{wd}}\), _i.e._, sacrificing wind curtailments (\(p_{i}^{\text{w}}\)), is permitted. Therefore, \(p_{i}^{\text{wd}}\) is equal to \(p_{i}^{\text{plan}}\), and the wind curtailments (\(p_{i}^{\text{w}}\)) increase greatly when \(p_{i}^{\text{wind}}>p_{i}^{\text{plan}}\). In other words, the single-level \(p_{i}^{\text{wd}}\) is equal to the two-level \(p_{i}^{\text{wd}}\). Although exchanging \(w_{2}\) and \(w_{3}\) is infeasible, this result implies that \(p_{i}^{\text{wd}}\) is influenced by the sub-objective, Equation (17), \(w_{2}\) and \(w_{3}\). From the above-mentioned information, for the third sub-objective, the effect of the single-level MPC method is equal to that of the two-level MPC method. The operating states of pumped storage are as follows:  
Figure 13 shows the one-day and four-day test results for pumped storage; from Figure 13a, it can be seen that the \(E_{i}^{\text{w}}\) (Ep) of the single-level MPC method is equal to that of the two-level MPC method. To show the differences between the single-level and the two-level MPC, one-day simulations were performed, and the results are depicted in Figure 13b. As shown in the figure, the values of \(E_{i}^{\text{w}}\) for the two cases are equal, except at some points. For example, at minute 1920, the two \(E_{i}^{\text{w}}\) values begin to differ. Moreover, the two \(p_{i}^{\text{w}}\) (pa) values also start to display differences. A possible explanation for these differences between the pairs of \(E_{i}^{\text{w}}\) and \(p_{i}^{\text{w}}\) values is as follows:  
Figure 11: Simulation result for the third sub-objective. the capacity of the fly wheel. This observation explains why the pumped storage operating state of the single-level MPC method is always similar to that of the two-level MPC method. Figure 16 shows the one-day and four-day operating states of the fly wheel. The differences for the fly wheel are greater than those for the pumped storage, and the differences for former often occurs when the fly wheel is nearly full or empty. The differences may be attributed to the length of the prediction horizon. In the single-level MPC method, the length is two hours, while it is less than half an hour in the two-level MPC method. Figure 14: Minute 1920 of the wind power plan and operating states of the fly wheel. (**a**) Four days wind power and plan; (**b**) Operating states of the fly wheel. Figure 15: Sensitivity analysis for the capacity of the fly wheel and \(p_{t}^{\text{\tiny{pa}}}\) of the pumped storage.