Fractional-order sliding-mode control of a wind turbine for tracking maximum power point

. This study examines a nonlinear control technique based on fractional-order sliding mode theory to track the highest power point of a wind energy conversion system. The proposed method achieves chatter-free fractional sliding-mode control by using a continuous control strategy. The main advantages of the proposed fractional-order SMC law are accurate reference tracking, elimination of the chattering problem and minimization of overshoot. To demonstrate the boundedness and convergence characteristics of the closed-loop signals, Lyapunov's stability theory is used. Extensive simulations results are presented to evaluate the robustness and effectiveness of the proposed control under various uncertainty.


Introduction
The sliding-mode control (SMC) method, grounded in Lyapunov principles [1], [2], offers substantial advantages over conventional methodologies.It stands out as one of the most potent design approaches applicable to a wide array of practical systems.Its versatility extends to both linear and nonlinear systems [3], [4].This technique is adaptable to both continuous-time and discrete-time systems.Given its capacity to address system uncertainties and external disturbances, SMC has maintained its status as a robust and effective technique within the field of control systems.
However, one of the most common and significant undesirable phenomena that SMCs experience is chattering, which is a type of high-frequency oscillation that can damage the system and cause instability [5]- [7].To address and alleviate the chattering phenomena within a sliding mode controller, significant research has been conducted by researchers [8]- [11].Among the various methods explored, fractional-order sliding mode control has emerged as an effective approach to mitigate the chattering issue [12]- [15].Nonetheless, it's worth noting that the wind turbine model does not adequately encompass the pertinent system dynamics, resulting in suboptimal control performance.
In this particular study, a fractional-order sliding mode control structure is devised for wind turbine systems.This controller is meticulously crafted based on a two-mass drive-train model, which accurately captures the principal dynamics inherent to the nonlinear wind turbine system.
The rest of this paper is organized as follows: The two-mass nonlinear model is described in Section 2. Section 3 is the control of the system which presents the proposed FOSMC with detailed analysis.The results of the simulation are provided and discussed in Section 4. Finally, section 5 concludes the paper.

Wind turbine modelling
The dynamic model of a wind turbine with variable speed is presented in this section.The aerodynamic properties, the mechanics of the turbine, and the dynamics of the generator are all included in the wind turbine model.In the next subsections, comprehensive descriptions of the various dynamic sub-models are provided.

Aerodynamics
The aerodynamic power captured by the rotor is as given below [16]: where the tip speed ratio: Where   is the rotor speed,  is the wind speed,  is the rotor radius, and  represents the air density.
The power coefficient   depends on both the tip speed ratio  and the blade pitch angle , which is defined as follows: The aerodynamic power can also be given as follows: Where   is the aerodynamic torque given by: With the torque coefficient shown below:

Mechanics of turbines
The mechanical model of the two-mass wind turbine (Fig. 1) can be explained as follows [17]: Eq. ( 7) illustrates the dynamics of the rotor speed   with the rotor inertia   .being propelled by the aerodynamic torque   .
Low speed shaft torque   , which serves as a breaking torque on the rotor, can be derived from the stiffness and damping factor of the low speed shaft provided in Eq. ( 8) The dynamics of the generator speed   as represented by Eq. ( 9) include the generator's inertia   being driven by high-speed shaft torque  ℎ and brake electromagnetic torque   .
The gearbox ratio is defined as follows: Using ( 7)-(10), the system below is derived: The WT electrical system has a substantially faster temporal reaction than the rest of the WT.This enables the decoupling of the generator and aero turbine control concepts, defining a two control loops are encircled by a cascaded control framework [18]. 1) Power converters and an electrical generator make up the inner control loop.
2) The control of the aero turbine in the outer loop serves as a reference point for the inner loop.

Control objective
The main objective is torque control for optimum power extraction.To achieve that objective, the blade pitch angle (  ) and tip speed ratio (  ) are both set to their ideal values.The rotor speed must be adjusted to the reference/optimal rotor speed (  ) by adjusting the control input, i.e. the generator torque, to achieve the optimal tip speed ratio.The reference/optimal rotor speed is specified in Eq. (12).

Fractional-order sliding-mode control
A fractional-order sliding-mode controller is suggested in this section and used to control a wind turbine to get maximum power possible.The FOSMC controls the nonlinear system using the Lyapunov stability approach.To achieve optimal power extraction, the control law must act on the   to drive the speed of the WECS to it reference value.In the following, an improved fractional order sliding mode technique for this task.Let defined the tracking error defined as: where the reference speed is given by:  − =       , with   is the optimal speed ratio.The sliding surface is defined as: where  represents the fractional operator order,  ∈ (0,1), and µ > 0 is a setting parameter.This method is applied in this work under the following conditions: The control is chosen to satisfy Lyapunov stability criteria for the following candidate function [19].Lyapunov is defined as follows  = This shows that the defined control law can follow the generator speed reference, which is a time-varying reference signal.
The fractional-order sliding surface is appealing and invariant because the Lyapunov equation provides the convergence condition.The generator should asymptotically follow the ideal generator speed at steady state.
̇≤ 0 satisfies the following condition The control variable values should be set so that the system is stable.The control variables are given below: In order to derive the control input   , the following conversion was performed.substituting the ̇ eq (9) and eq (10) in the eq (18), we have the following result: The control structure is defined as Typically, the SMC consists of two parts: the equivalent control   and the switching control   .
() =   () +   () The control of switching is defined in two ways The control structure of the torque is given in eq (25):

Results of simulation
The proposed control approach is applied to a wind turbine using Matlab/Simulink.The results of the simulation demonstrate the effectiveness of the control approach used in this paper.The parameters of the wind turbine are given in Table 1.The profile of wind speed applied, with an average value of 8 m/s, is shown in (Fig. 2).The generator speeds, both obtained with the fractional-order SMC method and the reference speed, and the fractional-order sliding surface are shown in (Fig. 3) and (Fig. 10).
Respectively, Good tracking capability has been achieved and high-frequency oscillations are eliminated.
From the results of the simulation, we can conclude that the proposed fractional-order SMC avoids chattering and reference tracking.

Conclusion
In this paper, we have proposed an approach to maximum power point control based on a fractional-order sliding mode for a variable wind turbine.From the simulation results, it can be concluded that the proposed control algorithm has a good ability to track maximum power points.The main advantages of the proposed fractional-order SMC law are accurate reference tracking, elimination of the chattering problem and minimization of overshoot.Simulation results demonstrate the effectiveness, robustness to external disturbances and uncertainty parameters of the proposed control technique.
fig 1: Two-mass model of a wind turbine