*6.1. Method*

The simulation was performed for the transition region and the rated power region where pitch control was used. The wind speed calculated through the wind speed estimator was used as the input wind speed for the simulation. For this, the measured generator speed, generator torque, and pitch angle from the target 100 kW wind turbine were used.

Figure 13 shows the wind speed estimated by the measured data, and the nacelle wind speed measured by an anemometer on top of the nacelle. Figure 13a shows the wind speed of the transition region, and Figure 13b shows the wind speed of the rated power region. In the case of the nacelle wind speeds, although the speeds actually had a higher frequency, they appeared to be similar to the estimated wind speed because the data measuring device collected data with a sample rate of 1 Hz. Although the nacelle wind speed was affected by the rotor rotation, the estimated wind speed was found to be similar to the nacelle wind speed, and as expected, it was found to be slightly higher than the nacelle wind speed. The mean value and the standard deviation of the input wind speed were 10.58 m/s and 0.98 m/s, respectively, for the transition region, and they were 15.21 m/s and 1.73 m/s, respectively, for the rated power region.

**Figure 13.** Comparison between actual nacelle wind speed and estimated wind speed: (**a**) transition region; (**b**) rated power region.

The simulation was performed with three different control algorithms including the conventional PI, the LQR, and the proposed LQR-PI. The simulation was also performed with and without noise to evaluate the controller performance in the presence of noise in the measured signal. In the simulation with noise, randomly mixing Gaussian noise was added to the generator speed.

#### *6.2. Results without Noise*

Figure 14a,b show the simulation results without considering noise in two different wind speed regions, i.e., the transition region and the rated power region. They compare the results with the conventional PI, the LQR, and the proposed LQR-PI controller. The simulation was performed for 600 s, but for visibility purposes, only the results from 0 to 100 s are presented. In Figure 14, the black line for wind speed represents the input wind speed obtained from the previous section. The subplot of wind speed also includes the wind speeds obtained from the wind speed estimators in the simulations with three different controllers. The estimated wind speed obtained by three different controllers showed a difference of less than 1% in mean wind speed compared with input wind speed, but the standard deviations were 4.08% and 3.47% higher for transition and rated power regions, respectively.

**Figure 14.** Simulation results according to the control method applied. (**a**) Transition region; (**b**) rated power region.

In the transition region, the PI control showed the largest overshoot of generator speed. This can be explained in relation to the pitch angle. The PI control regulates the pitch angle from the moment it exceeds the rated generator speed, but the LQR and the LQR-PI controls started the pitch angle in advance to attenuate the generator speed increase. Although the LQR and the LQR-PI controls used the estimated wind speed delayed by about 1 second for their control command calculation, the standard deviation of the generator speed was reduced compared with that obtained with the PI control. Sudden dips in the generator torque and power were observed in the simulation with all three controllers, but the greatest one was obtained with the PI controller. The results with the LQR controller were the best, and that those the LQR-PI were intermediate. On the basis of these results, it was concluded that the overshoot of the power was mostly due to the generator speed, and the dip was mostly due to the generator torque.

For the rated power region, the results with three different controllers were similar, but the lowest standard deviation of the generator speed was achieved when the LQR-PI control was used. Quantitative comparisons of the simulation results are shown in Tables 2 and 3.


**Table 2.** Quantitative comparison of performance data in the transition region without noise.

**Table 3.** Quantitative comparison of performance data in the rated power region without noise.


Tables 2 and 3 show the simulation results for 600 seconds. The most notable performance indicators in the results presented are the standard deviations of the generator speed, which can represent the operating stability of the wind turbine. The estimated wind speed in Tables 2 and 3 represents the estimated wind speed from the wind speed estimator. These were about the same with three different controllers, although the operating points were slightly different.

The results given in Table 2 indicate that the LQR control reduced the standard deviation of the generator speed by 26.58% compared to the PI control. In the case of the LQR-PI control, the standard deviation of the generator speed was even 16.7% lower than that for the LQR control, and 38.85% less than that for the PI control. However, as a side effect, the mean power with the LQR-PI control was reduced by 0.57% with respect to that measured with the LQR control.

As can be seen in Table 3, the LQR control and LQR-PI control had less than a 1% difference in all average performance indices compared with the PI control. For the standard deviation, the LQR control had a lower generator speed of 1.15% and a higher pitch angle of 17.63% compared to the PI control. A higher standard deviation of the pitch angle means that the pitch control was busier. The generator torque and power generation increased by 7.41% and 4.9%, respectively. The LQR-PI control reduced the standard deviation of the generator speed by 26.86% compared with the PI control, and the standard deviations of the pitch angle, generator torque, and power increased by 23.13%, 11.63%, and 1.53%, respectively.

As a result, the LQR control was able to increase the stability of wind turbines by reducing the standard deviation of the generator speed. However, in regions where the pitch control was continually used, the effect was reduced. On the other hand, the LQR-PI control was able to reduce the standard deviation of the generator speed in the two wind speed regions compared with the PI control, and its effect was the greatest in the rated control region, where the pitch control was used continually.

#### *6.3. Results with Noise*

The LQR control can improve the stability of the generator speed, but a practical problem is that it relies on the accuracy of the wind speed estimators. Noise in the feedback signal causes the wind speed estimator to become inaccurate, which causes the controller to send abnormal commands to the actuator. Figure 15a,b show the simulation results in the transition and power controlled regions, respectively, when noise was taken into consideration. To simulate the noise, white noise was introduced into the generator speed. Compared with Figure 14a,b, the dip in the generator torque by mode switch occurred more frequently, and the pitch angle movement was more active.

**Figure 15.** Simulation results according to the control method applied in the presence of noise. (**a**) Transition region; (**b**) rated power region.

Figure 15a shows the input wind speed (black line) as well as the wind speeds estimated in the simulations with three different controllers. Unlike the results without noise, the estimated wind speeds were now oscillatory with high-frequency components. However, this oscillation in the estimated wind speed is not visible in Figure 15b.

In the transition region, the PI control had the largest overshoot in the generator speed, similar to the results obtained without noise. However, the LQR control showed unstable behavior, much differently from the results obtained without noise. This is because the input wind speed to the LQR control which was obtained from the wind speed estimator was distorted by the noise of the generator speed. In addition, the oscillations of the LQR and LQR-PI controls were reflected in the behavior of pitch angle and were more clearly detected than when using the PI control. The generator torque command was determined using the generator speed, so the noise component was still present, and showed unstable behavior, which also affected the electrical power.

In the rated power region, the LQR-PI control yielded the lowest standard deviation in the generator speed, similar to the results without noise. The difference in the pitch angles with the three different control techniques was not significant.

The dip in the generator torque affected the overall electrical power. The LQR control reduced the frequency of the dip in the generator torque and resulted in an increase of the electrical power.

Tables 4 and 5 show a quantitative comparison of the simulation results in the presence of noise. The wind speed estimated for the three different controllers was found to differ more compared to the estimated wind speed without noise as a consequence of the noise added to generator speed. Similar to the condition without noise, the mean wind speed did not show a significant difference, but the standard deviation decreased or increased by 23.70% and 2.98% for the transition and rated power regions, respectively.


**Table 4.** Quantitative comparison of the performance data in the transition region in the presence of noise.

**Table 5.** Quantitative comparison of the performance data in the rated power region in the presence of noise.


Based on Table 4, the LQR and LQR-PI controls used average pitch angles smaller than those of the PI control by 4.56% and 8.21%, respectively, and achieved power increases of 1.69% and 0.60%, respectively. For the standard deviation in the generator speed, it increased by 2.61% with the LQR control, while it decreased by 35.29% with the LQR-PI control.

Table 5 lists the simulation results in the rated power region. The average values show differences within 2%. However, the standard deviation of the generator speed increased by 6.04% with the LQR compared with the PI and decreased by 21.48% with the LQR-PI. When the noise was taken into consideration, the standard deviation in the generator speed increased with the LQR control compared with the PI control for both transition and rated power regions. In the case of the LQR-PI control, on the other hand, the standard deviation of the generator speed was reduced compared with that of the PI control, even though noise was introduced.

Overall, the LQR control was better in performance compared with other controllers without any noise; however, when noise was considered, the LQR-PI was the best. Also, the LQR-PI controller showed better performances than the PI controller in both situations, with and without noise. Especially, the target 100 kW wind turbine in this study has a much lower rotor inertia compared with MW wind turbines, and power shutdowns are often encountered because of the generator overspeeding. The proposed LQR-PI controller reduced the standard deviation of the generator speed substantially and is expected to reduce the occurrence of shutdowns in the target wind turbine.
