On-line part:Step 1. Using the wind-speed profile and the fuzzy Kalman filter, obtain the estimated states and the appropriate matrices ( ) at any entry wind-speed at each sample time.Step 2. At each sample time, calculate the matrices from Equations (18) and (23).Step 3. Solve the quadratic optimization problem as in Eq. (29) to find the optimal control action.Step 4. After solving the optimization problem, the final pitch angle is gotten from Eq. (30). 4. RESULTS AND DISCUSSIONS To test the superiority of the constrained FRHC on the wind-turbine system, two case studies are composed.
Firstly, stepwise varying wind-speed is first used on the mathematical model. Secondly, the benchmark wind-turbine simulator  is used to test controller superiority under turbulent wind-speed profile. The online quadratic optimization problem in (29) is solved using commercial solver Gurobi version 7.5.2 and YALMIP version R21081012 . For the pitch angle constraint ( ) is ( and the rate of change of pitch angle constraint ( ) is ( ). The state constraint ( ) is ( ). The state constraint ( ) is ( . According to the simulation, the wind-speed profiles changes each 0.
05 second, thus the model sampling time is chosen to equal 0.0125 seconds. To find the best performance of constrained FRHC, the optimization parameters are chosen as follows; the predicted control action penalty ( ), for the predicted state variables penalty ( ), prediction horizons ( ), and control horizon ( ).4.1 Case 1: Stepwise Wind-speed ProfileThe wind-turbine mathematical model of Eq. (8) is the plant which has six states, three output, and three inputs. In region 3, the first input is the torque which is assumed constant. The second input is the pitch angle is regulated using the proposed constrained FRHC. The three outputs are generator speed, rotor speed, and generator power. As previewed in Fig. 1, the generator speed represents the represents the main feedback signal and is controlled using the regulated pitch angle. The stepwise varying wind-speed, as presented in Fig. 3, is used to check the validation of the proposed constrained FRHC under a sudden wind change. Fig. 3. Stepwise wind-speed profile Fig. 4. Comparison between the constrained FRHC 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.The proposed constrained FRHC and the gain scheduled-PI controller is compared. The generator power, generator speed, rotor speed, and pitch angle control action for the two controllers can be shown in Fig. 4. The constrained FRHC has better performance and regulating to rated values much better than the gain scheduled-PI controller does according to the generator speed, the generator power, and the rotor speed. Also, the proposed controller better to respond to the step changes in the wind-speed rather than the baseline controller according to the generator speed, the generator power, and the rotor speed. Moreover, the proposed controller required less control effort than the gain-scheduled controller need.4.2 Validation using the FAST simulatorThe FAST (fatigue, dynamics, structure, and turbulence) simulator is used in simulations to test system’s validity and practicability under turbulent wind-speed with different case studies . It is developed by NREL organization (National Renewable Energy Laboratory). The variations in wind-speed are reflected while using IEC turbulence wind-speed profile applied. It can be produced using a software package founded by NREL called TurbSim . The FAST simulator can provide linearized models at the different operating points. So, same like in mathematical model design, seven linear models are generated form 12m/s to 24m/s with 2m/s step-up. Also, only the two dominant dynamics (generator and drive train DOFs) is enabled for control design. Then the rest control design is the same as mentioned before. Although the controllers are designed based on a reduced-order model (2 DOFs), the full order of the system (all 24 DOFs) is enabled in the FAST simulator to investigate the existence of the unstructured dynamics (unmeasured system states). In the control design, the unstructured dynamics is solved using fuzzy Kalman filter. The simulation results are performed using full system order (by enabling 24 DOF supported by FAST). This proposed controller is called CPC (Collective pitch controller).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 . 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 to CPC is as shown in Fig. 5 and presented in details in . Fig. 5. The pitch control synthesisIn Fig. 5, represents the flap-wise moments in each blade. The IPC design uses a PI controller as mentioned in details in . The total pitch angle ( ) is calculated by ). ( ) is the pitch input operating point. is calculated from the look-up table (specified pitch angle for each wind-speed in region 3), as reported in . The generator speed is controlled using CPC control action ( ). The reduction of the flap-wise moment of the blades is performed by ( ).The final CPC control action actuator constraints for the FAST model is the same for the mathematical model except for pitch angle constraint in Eq. (28), it is represented by the following form after taking the lookup table control action ( ) the IPC control action ( ) into account form the total control action: (31)For testing the fuzzy modeling, Fig. 6 shows the measured generator speed from the FAST simulator model and the estimated generator speed from the fuzzy Kalman filter. This test is based on IEC turbulence wind-speed profile in Case 2. As presented in Fig. 6, the measured and calculated output are close to being the same. This figure proves that the fuzzy Kalman filter can solve the problem of unmeasured system states. Fig. 6. Measured and calculated generator speed for wind-turbine modelCase 2: IEC Turbulence Wind-speed ProfileThe variations in wind-speed are verified here using an IEC turbulence wind-speed profile which generated using the TurbSim . The unstructured model dynamics is denoted by allowing all 24 DOFs in the FAST simulator. The constrained FRHC and the gain scheduled-PI controller is verified against the variation in the wind-speed, as shown in Fig. 7. The generator power, the generator-speed, the flapwise-moment, and the pitch-angle control action are shown in Fig. 8. As presented in Fig. 8, the constrained FRHC has better regulating the speed and power to the rated than the gain scheduled-PI controller does. As shown 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 29.2%, 80%, and for generator-speed, and generator power respectively. The proposed FRHC controller improves the mean value by 0.3%, 16.78% for generator-speed and generator power respectively. The proposed FRHC controller improves the error standard deviation (decreases the fluctuation) by 3%, 34.52% for generator-speed and generator power. Furthermore, the proposed FRHC controller improves the maximum value and standard deviation for the flapwise-moment by 946% and 44.62% respectively. Fig. 7. IEC turbulence wind-speed profile Fig. 8. Comparison between the constrained FRHC and the gain scheduled-PI controller using IEC turbulence profile: (a) the generator-speed (b) the generator power (c) the flapwise-moment of one blade (d) the pitch-angle control action of one blade.Table 1. Data analysis of simulation results in Fig. 8. Gain scheduled-PI Constrained FRHC Generator-speed (rpm) Max (abs(err.)) 383.5631 40.8512 Mean 1175.239 1171.7211 std(error) 37.7882 2.4532Electric Power (KW) Max (abs(err.)) 2262.263 1324.1663 Mean 4692.819 4889.7904 std(error) 501.1205 95.771Flap wise moment (KN.m) Max 18186 7076.7834 Mean 4505.344 4719.0633 std 1929.973 1406.20415. CONCLUSIONSThe constrained fuzzy-receding horizon control (FRHC) is proposed for power control problem of the wind-turbines. The proposed controller guarantees the nominal stability and converted to a quadratic optimization problem which solved with less computational time. The nonlinearities of the wind-turbine are represented using fuzzy modeling. An effective fuzzy Kalman filter is used for state estimation to overcome the unmeasured system states. The time-varying constraints of the wind-turbine are settled by solving a simple online quadratic optimization problem. The proposed controller is coupled with individual pitch controller for mechanical load reduction. Several case studies are made to prove the constrained fuzzy-receding horizon control effectiveness. The results (shown in Table 1) have confirmed significant improvements in speed regulation, power harvest, and mechanical load reduction. The results also have shown the proposed controller superiority over the baseline controller.