## 5 Simulation Results and Discussion
### Simulation Test for Three Sub-Objectives Based on Single-Level and Two-Level MPC  
Three sub-objectives were tested under the condition that \(w_{1}=0.9\), \(w_{2}=1.0\), and \(w_{3}=0.1\), where w\({}_{1}\), w\({}_{2}\), and w\({}_{3}\) are the weighting coefficients of three sub-objectives. Note that in Section 5.1, the weighting coefficients are constants; however, in Section 5.2, the values of these coefficients vary with time. Figure 9 shows the time histories of the power signals over the 24 h simulation time. Three power signals are shown: the plan power of the grid plan (solid, red), the single-level generation power of the wind-storage system \(p_{\_}{t}\)\" (dash-plus, black), and the two-level generation power of the wind-storage system \(p_{\_}{t}\)\" (dash-dot, gray). From Figure 9, it can be noted that the single-level ps follows the plan closely, except for some points. From the magnified views of different areas, ps clearly does not follow the plan clearly at 655, 680 or 710 min. Two possible reasons for the differences between the single-level ps and plan are as follows: (1) the capacity of fly wheel storage is close to saturation in these moments, strongly limiting \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Type**} & **Charging** & **Discharging** & **Initial** & **Minute** & **Maximum** & **Efficiency** & **Efficiency** \\  & **Power** & **Power** & **Energy** & **Energy** & **Energy** & **(\(\eta\)**, \(\eta\)**)** & **(\(\eta\)**, \(\eta\)**)** \\  & **(MW)** & **(MW)** & **(MWh)** & **(MWh)** & **(MWh)** & **(MWh)** & \\ \hline Pumped storage & 100 & 100 & 100 & 20 & 200 & 0.87 & 0.85 \\ Fly wheel & 10 & 10 & 2.5 & 0.0 & 5 & 0.95 & 0.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter representations of the pumped storage station and the fly wheel power station. the charging ability, and (2) the weight of wind curtailment (\(w_{2}=1.0\)) is greater than that of the difference between ps and plan (\(w_{1}=0.9\)). It seems that a difference between the plan and \(p_{{}_{t}}\)\({}^{{}^{\mathrm{w}}}\) should be permitted, for reducing wind curtailment. Moreover, the two-level ps also follows the plan closely, except for the time points of 640, 655, 670, 680, 685, and 710. However, the two-level ps fails to follow the plan completely at the 880-minute point. It is likely that the single-level MPC method performed better than the two-level MPC method in terms of following the plan. A possible explanation for this behavior is that the prediction horizon of the single-level MPC is longer than that of the two-level MPC, as shown in Figures 5 and 7. The two-level MPC method has the benefit of reducing the solution time. For the two-level method, the average computing time of a scheduling interval is approximately 12.1 ms, while for the single-level MPC method, the time is 33.6 ms. In other words, the two-level method significantly reduces the computing time by approximately 64%. The result of the second sub-objective is as follows. In Figure 10, two power signals are depicted: the single-level wind curtailment (wu) (solid-plus, black) and the two-level wind curtailment (dash-dot, gray). Figure 10 shows the results of the wind curtailment of the single-level MPC method _vs._ that of the two-level MPC method. For both methods, the time of occurrence of wind curtailment is nearly the same, and the amount of single-level wu is approximately the same as that of the two-level wu. The amount of single-level wu is 11.68 MW, corresponding to 0.3% wind energy during a day, while that of two-level wu is 10.35 MW, corresponding to 0.26% wind energy. The results suggest that for the second sub-objective, the two-level MPC method may be slightly better than the single-level MPC method. This waste of wind-sourced energy can be attributed to two main reasons: (a) the fly wheel is almost full, and (b) although the pumped hydro storage has a large amount of unused storage, it is constrained by Assumption 1, which does not allow it to use the excess wind energy.