# Wind ?_r, the rotor blade pitch angle

Wind Turbine System ModelingThe wind turbine is a device which can convert the kinetic energy from the wind to electrical energy. The wind turbine energy conversion (WECS) block diagram contains three subsystems: the mechanical subsystem (drive-train and structure), the rotor aerodynamics subsystem, and the electrical generator subsystem. The WECS structure is presented in Fig. 2. There are many different wind turbine dynamics can be taken into concern when deriving the wind turbine model. The most important of them include the rotor aerodynamics, generator unit, drivetrain shaft flexibility 30. Fig.

2. Wind turbine energy conversion system (WECS) structure2.1 Rotor Aerodynamic Model The rotor aerodynamic power is the wind kinetic energy which depends on the rotor angular velocity ?_r, the rotor blade pitch angle ?, and wind speed on the rotor ? as presented in Fig. 2. The aerodynamic power captured from the wind turbine P_r and the aerodynamic torque T_r are described by:P_r=1/2 ??R^2 ?^3 C_p (?,?) , T_r=P_r/?_r (1)where ? is the density of air; ? is the wind speed; R is the length of the blade (rotor radius); ?=?_r R/? refers to the tip speed ratio (TSR).

The power coefficient C_p (?,?) is a nonlinear function, which represents the power captured efficiency of the wind turbine and modeled as in 21:C_p=0.5176(116/?_i -0.4?-5) e^((-21)/?_i )+0.

0068? (2)where ?_i is specified using: 1/?_i =1/(?+0.08?)-0.035/(?^3+1)For the pitch angle ?, the pitch actuator is used to rotate the turbine blades around its axis. Increasing the pitch angle is used to decrease the drift force applied by the wind onto the rotor blades. It limits the power extracted from the wind in case of rated power achieved and gusts. The pitch actuator can be modeled by a 2nd order differential equation as shown below: ? ?=-w_n^2 ?-2?w_n ? ?+w_n^2 ?_ref (3)where ?_ref is the demanded pitch angle, w_n is the natural frequency, ? is the damping coefficient.2.

2 Drivetrain Shaft ModelFor the drivetrain shaft, the drivetrain system is accountable for transferring the mechanical power, which produced at the rotor, to the electrical parts (generator). It consists of two speed shafts (low and high), which are connected via a gearbox. The low-speed shaft (on rotor side) is assumed flexible while the second one (on generator side) is assumed rigid. The gearbox presents an increase in speed from the turbine rotor to the high-speed shaft to suitable values to drive the generator. Based on the previous assumptions, the two mass drivetrain model is used with the following equations:J_r (?_r ) ?=T_r-K_s ?_?-D_s ?_r+D_s/N_g ?_gJ_g N_g (?_g ) ?=-N_g T_g+K_s ?_?+D_s ?_r-D_s/N_g ?_g? ?_?=?_?=?_r-?_g/N_g (4)where ? N?_(g )refers to the gear ratio; ?_r is the angular positions of the rotor shaft; ? ??_(g ) is the is the angular positions the generator shaft;? ??_(g ) is the generator speed;? T?_(g ) is the generator torque; J_r are the rotor moment of inertia; J_g generator-moment of inertia; D_s is the drivetrain spring constant; K_s is drivetrain dampening constant. The drivetrain system has been clarified in Fig. 3.

Fig. 3. Dynamics of the drivetrain system2.3 Generator ModelThe electric generator utilizes a torque T_g on the drivetrain system. The generator torque dynamics is expressed by a 1st order system as shown below:T ?_g=-1/?_g T_g+1/?_g T_(g_ref ) (5)where ?_g is generator time constant; T_g is the generator torque; T_(g_ref ) is the torque actuator reference. The electric generator converts the mechanical power P_m=T_g ?_g to electrical power P_e.

The delivered electrical power is expressed as below:P_m=??T?_g ?_g (6)where is the power transferring efficiency.2.4 State space modelsBased on the above equations (1-6) for the wind turbine subsystems, the overall nonlinear state space model has six states x=??(?_r&?_g&?(?_?&T_g&?(?&? ? )))?^T, two inputs , u=??(T_(g_ref )&?_ref )?^T, and three outputs y=??(?_g&?(?_r&P_e ))?^T. The state space model can be summarized and given in a more general form as shown below:x ?=f(x,u,?) (7)For J_g ,J_r ,N_g ,K_s ,?_g ,?_n ,? & D_s values, they are presented in Table 1 from a typical 5MW offshore wind turbine 31.

The partial derivatives of T_r are obtained using BEM (blade element momentum) theory, which is used to get the nonlinearity in aerodynamic torque numerically 7, 32. The procedure of getting these models is illustrated in details in 30. In region 3, the wind speed range is between 11.4 m/s and 25 m/s. Seven linear state space models are derived from 12 m/s to 24 m/s with step up 2m/s. This models are presented in the continuous form as below:x ?_d (t)=?A_c?_i x_d (t)+?B_c?_i u(t)y_d (t)=?C_c?_i x_d (t) i=1,2,…..

,7 (8)Table 1. The of the offshore 5MW wind turbine model parameters 31parameter symbol unit valueRated generator speed ?_g rpm 1173.7Rated power P_e kW 5000Rated generator torque T_g N.m 43093.55cut in, rated rotor speed ?_r rpm 6.9, 12.1cut in, rated, cut out wind speed ? m/s 3, 11.

4, 25drivetrain spring constant K_s N.m/rad 867636000drivetrain dampening constant D_s N.m/(rad/s) 6215000generator inertia J_g kg.

m^2 534.116rotor inertia J_r kg.m^2 38768000blades radius R m 63Hub-height – m 90Blade diameter – m 126number of blades – – 3Torque actuator time constant ?_g s 0.1Torque actuator natural frequency w_n rad/s 0.

88Pitch actuator damping coefficient ? – 0.1Sampling time T_s s 0.05Generator efficiency ? – 94.4%gear ratio N_g – 973. Collective Pitch Controller Design3.1 Problem FormulationThe wind turbine dynamic and the control strategy are relatively dependent on wind speed.

The key task control of wind turbine in region 3 is used for regulating the generator speed and power despite wind speed variations. The control system problems are the power regulating control problem of the wind turbines, mitigation of the mechanical loads of wind turbines, the system dynamics nonlinearities, operating points changes during operation caused by wind speed variations and unmeasured system states. The power control problem of the wind turbines is regulated using the proposed partial offline FMPC as presented in Fig.

1. Fig. 4. Block diagram of the wind turbine collective pitch control systemHowever, all previous studies of quasi-min-max MPC are dealing only with input constraint, this study proposed a way to deal with the pitch actuator and its rate of change constraints using LMI constraints form by converting these constraints to input and output constraints. The operating points changes are solved using fuzzy modeling. The time-varying constraints of the wind turbine are solved using the MPC controller. The nonlinearities in the wind turbine model are solved by combining the fuzzy modeling within the LMI constraints of the MPC controller.

In the next subsection, the proposed partial offline FMPC controller is derived. Finally, the gain scheduled-PI controller is presented.