# One index either for finite horizon-model predictive

One of the MPC approaches is receding-horizon control (RHC) that uses state space models to find an optimal state-feedback control by mini-maximization or minimization a specified performance index either for finite horizon-model predictive control (FH-MPC) or for infinite horizon-model predictive control (IH-MPC). According to 24, 25; FH-MPC may have nominal stability issues. Moreover, nominal stability requires a special tuning for the prediction horizons, control horizons, and the objective function weights. On the other hand, IH-MPC can guarantee closed-loop stability 25. Min-max FMPC is based on the IH-MPC.

The min-max MPC technique is first introduced for LPV systems in 26. The quasi-min-max MPC is first introduced for LPV systems in 27. This work is extended for the fuzzy systems by using the piecewise Lyapunov-function (PLF) 28. In 29, the FMPC controller with input constraint is considered based on quasi-min-max technique 29. In this paper, the MPC controller is used to overcome the wind turbine system physical constraints on control variables (limits on the pitch angle and the rate of change of 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 captured from the wind in case of gusts and rated power reached. So, the pitch angle constraints are added to the MPC optimization problem to keep the captured power at its rated value in high wind speed and also to prevent the saturation of the pitch actuator. The fuzzy modeling is used to overcome the wind speed variations. Since the fuzzy modeling constitutes a nonlinear mapping, it can efficiently denote the wind turbine models nonlinearities. Due to the nonlinearities and uncertainties of the wind turbine system, the quasi-min-max FMPC excellent characteristics inherited by combining MPC and fuzzy robust control causes it relatively suitable for wind turbine pitch control. The proposed quasi-min-max FMPC is converted to a convex optimization problem in the form of LMI constraints for online solution simplicity.

The online computational burden of the quasi-min-max FMPC required to solve the optimization problem may complicate the controller’s implementation. The partial offline quasi-min-max FMPC is proposed for the CPC of the wind turbine to overcome the computational time problem by partial offline (offline design and online synthesis). The partial offline quasi-min-max FMPC computational-burden complexity is analyzed in comparison with the online approach. Since the quasi-min-max FMPC is based on IH-MPC, it guarantees the closed-loop asymptotic stability and the solution recursive feasibility to the online and partial offline optimization problems. Augmenting the input variable as an extra state is proposed in this paper to show that the input and input rate constraints can be transformed to output and input constraints in order to handle both pitch angle constraints. Since the proposed controller is a state feedback controller and the only measurable states are the rotor speed, a Kalman filter is used to estimate the unmeasured system states to clarify the potential practical controller implementation rather than using the state measurements in simulations.

The proposed CPC is used for power regulation and it is decoupled with the IPC (designed by PI controller) for mechanical load reduction 7. The design of IPC takes into consideration preventing any effects in the generator speed tracking (power regulation). Different case studies are made to prove the effectiveness of the proposed controller.The contributions of this paper can be summed up as follows. First, the quasi-min-max FMPC is improved by proposing a way to handle the input and input rate constraints instead of only the input constraint in the previous studies. The partial offline quasi-min-max FMPC is introduced to satisfaction constraints, closed-loop asymptotic stability, the solution recursive feasibility to the optimization problem, and to overcome the computational burden when using online solution. Second, the partial offline quasi-min-max FMPC excellent characteristics inherited by combining MPC and fuzzy robust control causes it relatively suitable to overcome he nonlinearities and uncertainties of the wind turbine system.

Third, the proposed controller has an effective control performance for wind turbine speed and power regulation.The paper is prepared as follows: Section 2 presents an introduction of the nonlinear mathematical model differential equations and the wind turbine state space models are presented. In section 3, the proposed partial offline FMPC design on the mathematical model of wind turbine system is derived and the standard gain scheduled-PI controller is presented. In Section 4, the partial offline FMPC simulation results tested using a mathematical model of the wind turbine and a typical 5MW benchmark wind turbine simulator in comparison with gain scheduled-PI controller.

The conclusions are summarized in Section 5.