Results and Discussions To test the superiority of the partial offline FMPC on the wind turbine system

Results and Discussions
To test the superiority of the partial offline FMPC on the wind turbine system, four case studies are composed. Firstly, a stepwise varying wind speed is first used to check the validation of the proposed partial offline FMPC control under a sudden wind change for the full nonlinear mathematical model mentioned before. Secondly, the benchmark wind turbine model 31 is used to test the system’s validity and practicability under turbulent wind speed with another three different case studies. The proposed partial offline FMPC was simulated on a 5MW wind turbine system. The system parameters based on the typical 5MW wind turbine are previewed in Table 3.
4.1 Case 1: Stepwise wind speed on the mathematical model
The torque input is constant for each wind speed. The wind turbine mathematical model of Eq. (9) is the plant which has six states, three output, and three inputs. The first input is the torque which is assumed constant. The second input is the pitch angle depend on control strategy (the gain scheduled-PI or the proposed FMPC). The third input is the wind speed. The three outputs are generator speed, rotor speed, and generator power. The pitch control for the mathematical model input consists of one component as presented in Fig. 1, where w_g represents the generator speed (which represents the main feedback signal). The generator speed is controlled by the CPC control action (u_cpc). For the pitch angle constraints (???u?_max=?u?_max^Tot,?u_max=u?_max^Tot) is (8 deg./s,90 deg.).
According to the simulation, the model sampling time is chosen to equal 0.05 second. In designing the proposed partial offline FMPC controller for a wind turbine system, the cost function parameters are fine-tuned to get the best performance with guaranteed stability. The cost function parameters are Q=0.1*I_(n*n), and n is the system order and R=0.1 for all the following case studies. The initial states? x?_h=?B_??_d*??_h is calculated based on is solved using ??_h=2 m/s. Based on the? x?_h, a bank of six invariant-sets is built.
The proposed partial offline FMPC and the gain scheduled-PI controller is compared using a stepwise wind speed as displayed in Fig. 2. The generator power, generator speed, rotor speed, and pitch angle control action for the two controllers can be previewed in Fig. 3. From this figure, the partial offline FMPC controller has better performance and regulating the generator power to rated value better than the gain scheduled-PI controller does according to stepwise wind speed profile. Also, the proposed controller faster to respond to the step changes in the wind speed rather than the gain scheduled-PI.

Fig. 7. Stepwise wind speed profile

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Fig. 8. Comparison between the partial offline FMPC and the gain scheduled-PI control on the mathematical model using stepwise wind speed profile: (a) the generator speed (b) the generator power (c) the pitch angle control action (d) the rotor speed.
4.2 Validation using the FAST simulator
The FAST simulator 31 is used in simulations to test system’s validity and practicability under turbulent wind speed with different case studies. The variations in wind speed are reflected while using an International Electrotechnical Commission (IEC) turbulence wind speed profile applied. IEC turbulence wind speed profile can be produced using a software package founded by NREL called TurbSim 39. TurbSim generates a 2-dimensional wind speed profile that shields the whole wind turbine body with its tower. TurbSim Wind speed profiles have effects on the turbine at each sample time. The unstructured dynamics is denoted by enabling all 24 DOFs in the FAST simulator. Although the controllers are designed based on a reduced-order model (2 DOFs), the unstructured dynamics (unmeasured system states) are solved using a Kalman observer with fuzzy modeling. The simulation results are performed using full system order (by enabling 24 DOF supported by FAST).
When the blades’ turbine sweeps, it faces wind speed changes due to tower shadow, wind shear, turbulence, and yaw misalignment. These variations lead up to once-per-revolution (1P) huge component in the blades’ turbine loads, it’s necessary to design an IPC (individual pitch control) to cancel this component. An IPC key task is to mitigate the flap-wise moment of the blades 4.
The total pitch control for the FAST simulator is contained of three components as presented in Fig. 4, where w_g represents the generator speed (the main feedback signal). The total pitch angle (u) is calculated by (u= u_cpc+u ?+ u_cpc) as proposed in 14. (u ?) is the pitch input operating point. u ? is calculated from the look-up table (wind speed range in region 3 has specified pitch angle) 7. The generator speed is regulated using CPC control action (u_cpc). The reduction of the flap-wise moment of the blades is performed by (u_ipc).

Fig. 9. The pitch control synthesis
The FAST simulator can provide the measurements for the blade loads which can be used for designing an individual pitch controller used for mitigating the reducing the mechanical loads (flap-wise moment) by canceling 1P frequency. The new proposed control strategy after adding IPC is as shown in Fig. 4. M_1,2,3 are the flap-wise moments in each blade. The IPC design uses a PI controller as mentioned in details in 4, 14.
As we design partial offline FMPC for only CPC. So, we need to get the pitch actuator related only to the CPC, not the total control action. For the pitch input constraint (?u_min^Tot,u_max^Tot) is (8 deg./s,90 deg.). The final CPC control action actuator constraints for the FAST model:
u_max=u_max^Tot-u ?-u_ipc
??u?_max=??u?_min^Tot+?u ?-??u?_ipc (39)
where u ?,u_ipc refer to the look-up table and individual pitch control actions respectively. u ? is calculated from the look-up table (wind speed range in region 3 has specified pitch angle) 7. Twelve operating points in the look-up table are used from 11 m/s to 25 m/s with 1 m/s step to represent the look-up table as presented in details in 7. According to the simulation, the turbulence wind speed profile changes each 0.05 second, thus the model sampling time is chosen to equal 0.0125 seconds. The cost function parameters are the same as in previous case study.
Fig. 10. Measured and calculated generator speed for wind turbine model
After getting the linearized models from the FAST simulator, the T-S fuzzy rules will take the form as in the mathematical model discussed before in section 3.2. For testing the fuzzy modeling, Fig. 5 shows the measured generator speed from the FAST simulator model and the estimated generator speed based for Case-3 (IEC turbulence wind speed profile). As presented in Fig. 5, the measured and calculated output are close to being the same. This figure can prove that the fuzzy modeling with the Kalman filter can solve the problem of unmodeled system dynamics. For the following subsection, the FAST simulator is used with a full order of wind turbine (24 DOFs) with different case studies. First, is the standard wind speed profile is used for robustness test. Finally, the IEC turbulence wind speed profile to examine the controller under wind speed variations below the rated wind speed and all the comparisons made against the standard gain scheduled-PI controller.
4.2.1 Case 2: Robustness test
The variations in wind speed are verified here using a standard wind speed profile. Online and partial offline FMPC and the gain scheduled-PI controller is verified against a step-change in the wind speed. This test is applied using the full order of the wind turbine FAST Model (24 DOF), which used for design in the controllers. Fig. 6 shows the step-change wind speed profile. The generator power, generator speed, flap-wise moment, and the pitch angle control action are shown in Fig. 7. As presented in Fig. 7, partial offline FMPC has better disturbances rejection and regulating the rated generator speed than the gain scheduled-PI controller does. As in Table 1, the comparison depends on the maximum absolute-error, the average value, and the standard deviation. Moreover, it advances the maximum absolute-error by 30.2%, 73.8%, and 22.6% for generator speed, generator power, and the flap-wise moment respectively. The proposed partial offline FMPC controller improves the mean value by 0.269%, 0.295%, and 0.937% for generator speed, generator power, and the flap-wise moment respectively. The proposed partial offline FMPC controller improves the standard-deviation of error (decreases the fluctuation) by 78.8%, 39%, and 4.7% for generator speed, generator power, and the flap-wise moment respectively. As a preview in Fig. 7 and Table 1, partial offline FMPC has almost the same performance of online FMPC.

Fig. 11. Comparison between the gain scheduled-PI controller, online FMPC and partial offline FMPC using standard wind speed profile: (a) the generator speed (b) the generator power (c) the flap-wise moment on the 1st blade (d) the pitch angle control action of one blade (e) the collective pitch angle control action of one blade.
Table 2. Comparison between gain scheduled-PI, online FMPC and partial offline FMPC using standard wind speed profiles
Gain scheduled-PI Online FMPC Partial offline FMPC
Generator speed (rpm) Max (abs(error)) 62.28233 43.71293 43.25628
Mean 1177.178 1173.843 1173.961
std(error) 13.04717 2.545758 2.813893
Electric Power (KW) Max (abs(error)) 1347.01 1351.953 1347.528
Mean 4974.748 4983.965 4986.928
std(error) 125.2988 68.77914 67.20053
Flap wise moment (KN.m) Max 11865.9 9312.164 9316.536
Mean 5217.901 5215.574 5216.112
std 1746.557 1685.137 1680.102
4.2.2 Case 3: IEC turbulence wind speed profile
The variations in wind speed are verified also while using an IEC turbulence wind speed. The unstructured model dynamics is denoted by allowing all 24 DOFs in the FAST simulator. partial offline FMPC and the gain scheduled-PI controller is verified against large variations in the wind speed. This test is applied to the full order of the wind turbine FAST model (24 DOF). Fig. 8 shows that wind speeds may vary below the rated speed. So, two controllers are needed in region-2 to extract maximum captured power and in region 3 to keep the rated output power. Our proposed partial offline FMPC here is designed for working within region 3. If the wind speed drops under the rated speed, the pitch controller should be automatically disconnected (input equal zero). As displayed in Fig. 8, IEC turbulence wind speed profile contains wind speed below the rated wind speed between 25 s and 35 s. The corresponding pitch angle control action and collective pitch control action is disconnected as presented in Fig. 9. The generator power, generator speed, flap-wise moment, and the pitch angle control action for one blade are shown in Fig. 9. As presented in Fig. 9, partial offline FMPC has better disturbances rejection and tracking the rated generator speed better than the gain scheduled-PI controller does. As a preview in Table 2, the comparison depends on the maximum absolute-error, the average value, and the standard deviation. Moreover, it advances the maximum absolute-error by 83.1%, 77.6%, and 55.96% for generator speed, generator power, and the flap-wise moment respectively. The proposed partial offline FMPC controller improves the mean value by 0.61% and 2.36% for generator speed and generator power respectively. The proposed partial offline FMPC controller improves the standard-deviation of error (decreases the fluctuation) by 89.8%, 70% and 35.1% for generator speed, generator power, and flap-wise moment respectively.
The flapwise moment power spectral density when using the partial offline FMPC without and with IPC is presented in Fig. 5. As shown in Fig. 5, the IPC succeeded to reduce the (1p) component (which is denoted at frequency 0.2 Hz). As a result, the fatigue moment influencing the wind turbine blades and the nacelle is mitigated significantly.

Fig. 12. Comparison between the constraint partial offline FMPC and the gain scheduled-PI controller using IEC turbulence wind speed profile: (a) the generator speed (b) the generator power (c) the flap-wise moment on the first blade (d) the pitch angle control action of one blade.
Table 3. Comparison between gain scheduled-PI, online FMPC using IEC turbulence wind speed profiles
Gain scheduled-PI Online FMPC Partial offline FMPC
Generator speed (rpm) Max (abs(error)) 280.97 41.17 40.77
Mean 1175.24 1172.50 1171.55
std(error) 29.04 2.38 2.40
Electric Power (KW) Max (abs(error)) 1538.14 1327.27 1323.34
Mean 4672.39 4920.21 4881.26
std(error) 493.91 87.09 98.78
Flap wise moment (KN.m) Max 17826.32 7364.56 7156.11
Mean 4452.15 4696.75 4659.81
std 1774.51 1476.70 1456.23

Fig. 13. The flapwise moment power spectral density when using the partial offline FMPC without and with IPC (zoomed vision from the full frequency range for 0.25Hz).
4.2.3 Discussion for the computational burden and performance
The online and partial offline FMPC schemes require to solve LMIs series at each sample time. The algorithms (as ex: interior point algorithm) have polynomial-time complexity O(MN^3) 40, where M is the total rows number in the LMIs series, and N is the total decision variables number. In the online FMPC as in Eq. (25), the computation complexity is O((4M^3+2M+4M+4) N_1^3) where M=2, N_1=38. For the proposed partial offline FMPC as in convex-optimization problem (27) and (30), the bisection searches number at each sample time is log_2?n. It is observed that matrix multiplication is required for each search, and then the computation complexity is O(n_s^2 log_2?n) for the bisection searches, where n_s the wind turbine model states number 41. With LMIs series to be solved, the total complexity for the partial offline FMPC is O(n_s^2 log_2?n+(4M+4)N_2^3), where n_s=3, N_2=2. Since the log_2?n value is normally not big, and it holds that N_1;N_2, the partial offline FMPC is much less computational burden consuming than the online FMPC. This result is confirmed in simulations for different case studies. As shown in Fig. 9b (step-up wind profile) and Fig. 9d (for turbulence wind profile), the summation of CPU computational time slope in online FMPC is much higher than in partial offline FMPC. Also as shown in Table 5, the total summation of CPU time for online FMPC is greater more than five times using partial offline FMPC. As shown in Fig. 9a, Fig. 9c, and Table 5, the summation of the cost performance of the online FMPC is greater than using partial offline FMPC.