Torsional vibration characteristics analysis and adaptive fixed‐time control of wind turbine drivetrain

The drivetrain has an important impact on the stability and reliability of wind turbines operating in complex conditions, which belongs to the electromechanical coupling system. This work studies the two domains of research on drivetrain vibration analysis and control method in detail. The electromechanically coupled torsional vibration model is first established by considering the air gap magnetic field energy of the PMSG. Preliminary analysis in terms of Hamiltonian energy indicates that there is sufficient energy to sustain the exciting oscillation behaviors. Moreover, to reduce the vibration amplitude caused by wind speed or torsional stiffness, a novel adaptive fixed‐time control method is proposed to solve the issue of wind turbine drivetrain chaotic oscillation with uncertainties and disturbances. Furthermore, the proposed method not only can prevent the jumping and bifurcation of torsional vibration from happening, but also there are the advantages of the control scheme with high accuracy, fast convergence rate and strong robustness. For the fair comparison, three different control methods have been used to demonstrate the superiority of the control scheme by the simulation results. The research results can provide a theoretical basis for the parameter design and control of wind turbine drivetrain.


| INTRODUCTION
With the rapid increase in the size of wind power generation installations over the past decade, the reliability and safety of wind turbines have already been attached more and more importance. 1,2Wind power is usually transformed from blades through a gear drivetrain to a generator.Wind turbine drivetrain could be classified into two main categories: direct-drive, semidirect drive gear transmission. 3,4Due to random fluctuations of wind speed, especially for large-scale turbines rotor with high-torque, the torsional vibration, inevitable by-generates during the operation of wind turbines, directly influence the stability and reliability of drivetrains, and could lead to more fatigue damage from the drivetrain components. 5In this case, the dynamic characteristic of torsional vibration becomes really complex in the wind turbine transmission system.Moreover, the unsuitable control strategy is also an incentive factor for torsional vibration. 6,7Therefore, it is very essential to study the dynamic characteristics and suppression strategy of torsional vibration.
In general, the wind turbine drivetrain could be described as a typical electromechanical coupling system between the mechanical transmission component and the generator.Because of the existence of nonlinear factors, complex dynamic behavior (i.e., bifurcation, chaos motion) may be exhibited in the electromechanical coupling system operating particular conditions.Some scholars have studied the dynamic characteristics of wind turbine drivetrain. 8,9Tan et al. 10 constructed a rigid flexible coupling dynamic model of the wind turbine drivetrain, and revealed the effect of electromagnetic stiffness on the dynamic characteristics.Moghadam et al. 11 further established the analytical model to estimate the dynamic properties of drivetrain, and tested the validity of this method by using 10 MW simulated and 1.75 MW operational drivetrains.Golnary et al. 12 established an electromechanical model with two lumped mass blocks, and then studied the dynamic characteristics and robustness of the system by using the comparison of three high order sliding mode methods.All of these studies indicates that in the condition of the wind speed variation, the strong nonlinearity and the coupling dynamics, the drivetrain will be continually subjected to multiples excitation sources, which will damage the reliability of the drivetrain and reduce the service life of drivetrain components.Thus, it is necessary to establish an accurate model to reflect the vibration dynamic characteristics of the electromechanical coupling drivetrain, and to reveal the influencing mechanism on internal parameters and external wind speed excitation change.
With the deepening of the research, an increasing number of scholars are focusing on this field of analyzing the vibration dynamics of electromechanical transmission systems.Although there are some studies on drivetrain vibration of wind turbine, but the related studies can only be divided into two categories from the perspective of the separate analysis: mechanical system research and electrical system research, such as condition monitoring of the gearbox, 13 angular velocity measurements of the drivetrain 14 and so on.Several scholars simplify the wind turbine drivetrain as two lumped masses: the generator as a mass block or a load torque, the mechanical system with a gearbox as a mass block.6][17] At this point, Zhao et al. 18 investigated the nonlinear dynamics of a wind turbine gearbox with torsional vibration considering time-varying mesh stiffness, damping and static transmission error.Li et al. 19 presented a flexible dynamic model of the drivetrain including the full coupling of gear meshing, the flexibilities of planet carrier and ring gear, and dynamic characteristics of the drivetrain were analyzed under the time-varying mesh stiffness.Moghadam et al. 20 continued the research on the wind turbine drivetrain system fault diagnosis by using torsional vibrations and modal estimation.Vamsi and colleagues 21,22 studied the effect of multiple defects and multicomponent failure on the dynamic behavior of a wind turbine gearbox, and further provided the comparison of condition monitoring techniques in assessing fault severity.The dynamic model of the multistage gear system with electric drive was established and electromechanical interaction mechanism of the system was investigated by Yi et al. 23 However, all of these studies ignored the effect of the generator and electrical system in wind turbine.
Nevertheless, in large-scale rotating machines, the effects of the electromagnetic stiffness and the damping coefficient should be investigated by considering the electromechanical coupling between the rotating mechanical drive system and the motor. 24,25Against this background, it is important and necessary to study the electromechanical coupling model of the drivetrain and to analyze the torsional vibration in various situations.To acquire the precise interaction between the mechanical and electrical components, Chen et al. 26 proposed an electromechanical coupling dynamic model of the torsional vibration in PMSM driven system.Besides, the influences of mechanical parameters and internal parameters of the motor on the dynamic characteristics were studied.Zhang et al. 27 established a coupled electromechanical dynamic model of wind turbine drivetrain, explored the effect of the drivetrain system parameters on the current harmonics and dynamic response.Chen and colleagues 28,29 continued to establish a mechanical-electrical coupling nonlinear dynamic model of the wind turbine main drive system, and further revealed the influence of magnetic saturation coefficient and external variable-load excitation change on the dynamic characteristics.Form the above mentioned research, torsional vibration characteristics and the electromechanical coupling effect have been comprehensively investigated and discussed in wind turbine.Whereas, the influence problem of sensitive parameters and the varying excitation in wind turbine drivetrain YANG ET AL.
| 4667 have less been studied, especially for multivalue characteristics and the dynamic behaviors such as bifurcation and chaos motion.
In addition, to enhance the stability of the electromechanical coupling system, it is important to study the torsional vibration suppression and chaos control of the drivetrain.1][32] Toha et al. 33 discussed on the problem of the torsional vibration reduction in wind turbine drivetrain, and a generator torque control used the particle swarm optimization technique to search strict parameters of the controller is investigated.Kavil Kambrath et al. 34 proposed a novel control technique to reduce torsional vibration of the wind turbine driveline based on the derived two-inertia system model.Liu et al. 35 studied the torsional vibration of the wind turbine drive chain caused by the disturbance wind and the DPC strategy, and an appropriate damping and stiffness compensation control method was proposed to suppress the torsional vibration.
From the above studies, a common feature of several vibration suppression techniques is to track the desired trajectory, such as generator torque or speed feedback, stress damping control, and virtual inertia damping control.Thus, the trajectory tracking control is very important to solve the stability of the nonlinear torsional system. 36,37For instance, in the robotic manipulators, Jing et al. 38 proposed a new adaptive sliding mode control to guarantee the transient and steady-state properties of the trajectory tracking process.Boukattaya et al. 39 continued to develop robust and adaptive nonsingular fast terminal sliding-mode (NFTSM) control method for the tracking problem, and the advantages of the proposed controllers are singularity avoidance, rapid convergence speed, and strong robustness in opposition to uncertainties and external disturbances.Zhang and Shi 40 studied a novel adaptive fast fixed-time sliding mode control scheme, which could obtain the expected tracking effect with a faster convergence speed and high precision through the proposed fixed-time stability lemma.All of above works have paid attention to the issue of high accuracy sliding mode tracking control.Whereas, few studies have been performed on investigating the trajectory tracking issue of the vibration dynamics in the wind turbine drivetrain.
Motivated by the aforementioned discussion, the major objective of this paper is to study dynamic characteristics and suppression strategy of torsional vibration for the wind turbine drivetrain with electromechanical coupling, which provides a theoretical basis for further revealing the dynamic parameter design and control.Therefore, it is significant to explore torsional vibration characteristics and adaptive fixed-time control of wind turbine drivetrain.The major contributions of this paper are summarized as follows: 1. Electromechanical coupled model of wind turbine drivetrain is established to analyze the torsional vibration characteristics comprehensively.2. A generalized Hamiltonian form can be used for Homoclinic or Heteroclinic bifurcation analysis, and the system is divided into a few categories according to the shapes of potential functions and phase portraits.3. A new fast fixed-time sliding mode surface and adaptive updated law are developed, which can acquire more rapid convergence speed and parameter estimation velocity, by applying the comparison of different control algorithms, the effectiveness and superiority of the theoretic results are demonstrated.4. A novel adaptive fixed-time control method is proposed, which is free of chattering, high precision and practical, and will solve the stability issue for a wide range of dynamical systems with uncertainties and disturbances.
The structure of this paper is organized as follows.In Section 2, the dynamic model of wind turbine drivetrain is established.In Section 3, torsional vibration analysis of the wind turbine drivetrain is investigated including Hamilton energy analysis and dynamic analysis, and research results are given.Then, torsional vibration control of electromechanical coupling system is presented in Section 4, including problem statement, controller design, stability analysis and the application and comparison of the developed control scheme.Finally, some useful conclusions of this paper are provided in Section 5.

| DYNAMIC MODEL OF WIND TURBINE DRIVETRAIN
Due to wind speed uncertainties, the drive chain system may cause the excited vibration under the nonperiodic impact interference, which leads to torsional vibration for wind turbine.While the torsional vibration angle becomes larger, which directly influences the stability and operating reliability of wind turbine.Taking into account the influence of the torsion angle on the magnetomotive force of PMSG, the model of electromagnetic excitation with the nonlinear torsional vibration can be derived.
As shown in Figure 2, from the view of the mechanical principle, the electromechanical coupling flexible drive chain of wind turbine could be simplified into a twoinertia torsional vibration system, which is consisted of a wheel hub through a connecting shaft and a PMSG.In Figure 1, T t and T e denote the wind turbine and the PMSG torque, respectively.J t and J e are the equivalent moment of inertia of the wind turbine and the PMSG, respectively.ϕ t and ϕ e denote the torsional vibration angles of the shaft and the PMSG under different excitation, respectively; C s is a damping coefficient, K s is the equivalent stiffness of the shaft.
Base on the theory of Bates, aerodynamic torque T t of wind turbine can be written as where, ρ R , , θ and V w represent the air density, turbine rotor radius, the pitch angle and wind speed, respectively.And C λ θ ( , ) P denotes the power coefficient of the wind turbine, λ is the tip speed ratio (TSR).Figure 2 shows the power coefficient versus tip speed ration and pitch angle.It is clear from the Figure that when the pitch angle is zero, the optimal tip-speed ratio is λ opt = 8.1 and the power coefficient is C P max = 0.4801.To maintain its generality and simplify the calculation, these aerodynamic parameters are directly adopted in this paper.
According to the Newton's law, the mathematical expression of the simplified two-inertia torsional vibration system could be described as where, T sh represents the mechanical torque of the flexible shaft.C t and C e are the self-damping coefficient of the wind turbine and the PMSG, respectively.Due to they are usually small, and the effects of them are generally neglected.Taking x ϕ ϕ = − t e , By using Equation (2), it can be easy to obtain From Equation (3), to obtain the value of the electromagnetic torque T e , the effect of the torsion angle on the magnetomotive force of PMSG should be taken into account.Therefore, a detailed analysis about electromechanical coupling problem of PMSG is presented below.

| Coupling nonlinear model of PMSG
The electromagnetic driving torque is generated by the air gap magnetic field, which is determined by the magnetomotive force between the stator and rotor and air gap permeance.Therefore, it is necessary to analyze the magnetomotive force and air gap permeance of PMSG.
The position relationship between the stator armature and the permanent magnet is shown in Figure 3.
P versus tip speed ratio and pitch angle.
YANG ET AL.
| 4669 Here, φ denotes the relative position angle between the rotor and stator.When φ = 0, it's located just at the center line of the pole.
According to the magnetic circuit principle for permanent magnet motors, the PMSG is regard as a constant source of magnetic motive force (MMF).Then, the magnetomotive force of the air gap synthesis, the fundamental analytic expression of MMF and MMF of a three-phase symmetrical stator winding can be expressed as follows: , where, ω e is the angular frequency, p is the number of pole pairs, ψ is the inner power factor angle, and ϕ e is the rotor torsional angle.F rm denotes the amplitude of the fundamental MMF, F sm denotes the amplitude of the combined magnetomotive force of a three-phase, B r is the magnetic remanence of the permanent magnet material, μ 0 is the air permeability, , h m is the thickness of the perma- nent magnet magnetization direction.α p is the polar arc coefficient of the permanent magnet, N is the number of series turns for each phase winding, k w is the fundamental winding distribution factor, I m is the rated current.
The PMSG often operates in a magnetic saturation region under magnetic or electric loads.Thus, the effect of magnetic saturation should be taken into account by involving the saturation coefficient of the magnetic circuit.Then, the air gap magnetic field energy of the PMSG can be shown as where, and k μ is the saturation coefficient of the magnetic circuit, δ φ t ( , ) is the air gap length, R is the mean radius of the air gap and l is the effective length of the rotor.
After simplified calculation, it is easy to derive that On the basis of the previous description, considering the rotor torsion angle ϕ e , the electromagnetic torque acting on the rotor can be expressed as follows: , e e e m e e 0 sm rm (7)   where ( ) . According to Equation ( 7), one can get where are electromagnetic parameters relating to the operational state of the motor, which are the time-varying parameters.The main parameters of the PMSG used in this paper are arranged in Table 1.
F I G U R E 3 Spatial position of stator armature and permanent magnet.
When the system is in equilibrium state, in other words, wind turbine aerodynamic torque and electromagnetic torque are equal.The mechanical angular velocity Ω of the rotor is constant.Assume , and it can be easy to obtain the following equation by Equation (2).
The electromagnetic and mechanical excitation are regarded as harmonic excitation.The torsional dynamic equation of the electromechanical coupling system can be rewritten as follows: x ϖ x μx αx βx γx νx where 2 .Significantly，ω n is the natural oscillation frequency without considering the electromechanical coupling effect.From Equation (3), it can be obtained as , and damping ratio is denoted as According to Table 1, then the natural oscillation frequency can be calculated as ω = 15.198rad/sec0 ( f Hz = 2.419 n ), and damping ratio is ς = 0.0085.Due to the damping ratio is much less than 1, the system is a typical underdamped system, and the torsional vibration of the electromechanical coupling system is easily excited, which is also prone to occur complex nonlinear dynamic behavior.12) can be described accordingly as: where μ ξμ = ˜, f ξf = ˜, ξ 0 < 0 ≪ .When ξ = 0, the damping and disturbance terms are irrespective.Equation ( 13) can be transformed into an unperturbed system as follow:

| Hamilton energy analysis of the system
To analyze the continuous dynamic system, a vector field X can be utilized to discuss energy problems and a generalized Hamiltonian form is further used for Homoclinic or Heteroclinic orbit analysis. 41This equation can be given as follow: where, the matrix J X ( ) and R X ( ) can be obtained as the following form: and a dissipative vectors fields f X ( ) . According to Equation (15), the conservative and dissipative vector fields are given as the following Hamiltonian condition: Thus, the energy consumption of the electromechanical coupling system is commonly associated with a Hamiltonian function.This function can be transformed into a generalized Hamiltonian form as follow: Notice that the corresponding potential energy is given by According to the above expression，the potential energy plays a critical role in studying different nonlinear dynamical behaviors.When Equation ( 14) equal to zero (y y = ̇= 0), for γ ν = = 0, the equilibrium points of the system can be obtained as P (0, 0) 0 , and , among those points, Without regard to the actual physical meaning of the parameters, obviously, the stability of equilibrium points depends on the parameters α, β, η and state variables in Hamiltonian systems.Subsequently, preliminary analysis based on asymmetric potential energy function and Homoclinic or Heteroclinic orbit could be organized as follow: Case A: The Hamilton system has only one equilibrium point P (0, 0) 0 .When η > 1, P 0 is a saddle point; When η < 1, P 0 is a center point.In this case, there is no such thing as Homoclinic or Heteroclinic orbit.
Case B: and β η 4 (1 − ) > 0, The Hamilton system have three equilibrium points P (0, 0) , and P x ( , 0) . When β > 0, P 1 is a saddle point, P 0 and P 2 are the center point.As results, the overview of equilibrium point and phase trajectory are shown in Figure 4A,B for α = 0.8, η = 0.3, and β = 0.2.From this figure, it shows that there is Homoclinic orbit.
Also, when β < 0, P 1 is the center point, P 0 and P 2 are the saddle point.Similarly, the overview of equilibrium point and phase trajectory are shown in Figure 4C,D for α = 0.8, η = 1.6, and β = −0.2.From this figure, it shows that there is Heteroclinic orbit.
When β > 0, P 0 is a saddle point, P 1 and P 2 are the center point.As results, the overview of equilibrium point and phase trajectory are shown in Figure 4E,F for α = −0.2,η = 1.5, and β = 0.1.From this figure, it shows that there is Homoclinic orbit.
Further, Also, when β < 0, P 1 is the center point, P 0 and P 2 are the saddle point.Accordingly, the overview of equilibrium point and phase trajectory are shown in Figure 4G,H for α = 0.2, η = −0.1, and β = −0.9.From this figure, it shows that there is Heteroclinic orbit.As shown in Figure 4, P 1 and P 2 are both on the side of P 0 in Case B. Whereas P 1 and P 2 are on opposite sides of P 0 in Case C. With respect to Homoclinic orbit, the Hamilton system has a closed orbit at the center point, and small perturbation amplitude of the system may lead to Homoclinic chaos.Further for Heteroclinic orbit, increasing the excitation amplitude of the system may lead to Heteroclinic chaos.Consequently, Torsional vibration of the electromechanical coupling system will exhibit abundant nonlinear dynamic behaviors.

| Dynamic analysis of the drivetrain
To study the effect of the external excitation on the drivetrain, a numerical investigation of the model is comprehensively analyzed, using the typical nonlinear analysis tools involving Lyapunov exponent, bifurcation diagrams, Time domain waveform and phase portraits.The main parameters of the wind power system in this study are shown in Table 1.Unless otherwise specified, the mechanical characteristic values used in the numerical analysis are as follows: , and the other parameters can be computed by the formulae.
According to Equation ( 12), the amplitude excitation f is determined through the electromagnetic parameters and the mechanical parameters.Thus, to demonstrate the dynamical behaviors, trying to change the value of the amplitude excitation within the design and operation range.As shown in Figure 5, the complex dynamics of the established model are illustrated by bifurcation diagram and Lyapunov exponent.Obviously, varying in the range   f 1 2.5, a chaotic attractor can be found from the bifurcation diagram.And it can also be verified by the Lyapunov exponent.Looking closely at the Figure 5, the chaotic area is characterized by the positive values of the maximum Lyapunov exponent.When the maximum Lyapunov exponent remains negative, the periodic oscillations can be clearly observed such as   f 0.65 0.9.To better understand the dynamic behaviors in the above-mentioned area, it can be seen from the above discussion that with increasing the amplitude excitation such as torsional stiffness of rotor and wind speed, the potential possibility of exciting phenomenon (i.e., periodic states, bifurcation or chaotic) can be predictable.And some existing attractors can be obtained by the bifurcation diagram.As shown in Figure 6, time domain waveform and phase portrait of the drivetrain are illustrated for f = 1.65.This graph reveals that the complex torsional vibration can be clearly observed in cases of sudden wind speed variation or torsional stiffness failure.These torsional vibration could cause serious consequences such as the working inefficiency of the motor and mechanical failure, which will affect the stable operation of the system.Therefore, it is essential to design an appropriate oscillation suppression controller and an observer of uncertainties and external disturbances, which is of great significance and practical value for the stable operation in wind turbine drivetrain.

| ADAPTIVE FIXED-TIME CONTROL OF THE WIND TURBINE DRIVETRAIN
In this section, to suppress the chaotic behaviors caused by torsional vibration of the wind turbine drivetrain, an adaptive control method based on fixed-time theory is proposed.First, the control objective of the system is determined.Subsequently, some useful lemmas is presented.The proof of the stability, the application and comparison of the control scheme are discussed in the next subsection below.
F I G U R E 5 Bifurcation diagram and the corresponding Lyapunov exponent when   f 0 5.

| Preliminaries and problem description
，the torsional dynamic model of the system with electromechanical coupling (12) can be described as the following form of the second-order uncertain nonlinear dynamical system where x 1 and x 2 are the state variables of the system, y is the output signal, b is positive constant, f x τ , f x τ Δ ( , ), and d τ ( ) denote uncertainties and external disturbances of the system, and u τ ( ) is the control input.To achieve the control objective, the tracking error e y y = − d 1 and its derivative e y y = ̇− ̇d 2 are defined, respectively.y d is the desired signal.Then, the tracking error of the system can be expressed as follows: where ) is regarded as the lumped uncertainty of the system.
Assumption 1: The uncertainties f x τ Δ ( , ) and external disturbances d τ ( ) are bounded, that is, there exist a lumped uncertainty δ, which is bounded by positive function as follows where, a 0 , a 1 , and a 2 are all positive constants.
Remark 1: The torsional vibration suppression problem of wind turbine is transformed into the stabilization problem of the tracking error system.The control objective of this article can be formulated as designing a continuous fixed-time control scheme to realize the trajectory tracking of the error system.Nevertheless, in many practical systems, it is difficult to get the accurate values for external disturbances and uncertainties.Owing to the upper bound of external disturbances and uncertainties can be commendably estimated according to Assumption 1.Therefore, a novel composite controller combining adaptive techniques will be proposed, which can not only allow the tracking error to stabilize within a fixed time, but also higher precision, faster convergence rate and weaker chattering of the system states.
Then, some useful lemmas need to be introduced for the design and analysis of the controller.Lemma 1 42 : z i N , = 1, 2 i ∈ ⋯ are arbitrary real numbers, the following inequalities satisfy: Lemma 2 43 : Consider the differential equation system x f x ̇= ( )，assuming that V t ( ) is a continuous, positive definite function, and it satisfies where a > 0, b > 0, and c 0 < < 1 are three constants.Then, it meets the following inequality: ,   t T, the convergence time T is given by ( ) , it is easy to verify the settling time Lemma 3 40 : Consider the differential equation system x f x ̇= ( ), if there exists a function V x ( ) that is continuous and positive-definite, and one has where a > 0, b > 0, and ρ > 0 are three constants.Then, the system can realize fixed-time stability.When  x V x = 0 ( ) = 0, and the fixed time T is estimated as T = πρ ab .

| Controller design and stability analysis
From the error Equation ( 21), to improve the error convergence accuracy and speed of the control system, a novel fast terminal sliding-mode (FTSM) surface is designed as r e e ( ) = + ( + )sign( ), where，r 1 and r 2 are positive constants，c = 1 − ρ 1 1 ，and sign() is signum function.
To simplify the process of the proof, the following Lyapunov function is designed as (26) , by transforming Equation ( 26), calculating the time derivative of V t ( ) , which yields V t e e e r e r e e r e r e r V r V where .
Then, it holds that According to Lemma 3, taking integral of both sides of Equation ( 28) from 0 to t r , one can obtain Remark 2: According to the results of the above proof, it shows that the proposed sliding-mode (SM) surface has the ability to realize fixed time stabilization.Further, it is very clear that the convergence time of Equation ( 29) is upper bounded by a constant independent of initial condition, and is only related to design parameters of SM surface.Besides, the superiority of the proposed control strategy is observed by comparing with the existing finite-time stabilization and asymptotic stabilization.Therefore, to obtain an ideal control effect, adjusting the parameters size need to be considered.
After designing the suitable sliding surface, by using Then, combining with the error dynamic Equation ( 21) and without regard to uncertainty and disturbance, one can get When the sliding surface derivative satisfying s t ( ) = 0, the equivalent control law u eq can be expressed by To make the system obtain stronger robustness against unmodelled uncertainties and external disturbances in practical applications, and satisfy the sliding condition.The switching control law u sw can be described as where，r 3 and r 4 are positive constants，c > 1 , a ˆ0, a ˆ1, and a ˆ2 are the estimates of a 0 , a 1 , and a 2 , respectively.Thus, the total control law can be obtained as follows The adaptive law of parameter estimation corresponding to Equation ( 35) is designed as Remark 3: To overcome the structural complexity of the uncertainties and disturbances, the above-proposed adaptive update law is utilized to estimate the upper bounds of the system lumped uncertainty δ in Equation (22).It is worth mentioning that the adaptive law can not only make the tracking error converge to zero rapidly, but also obtain a better rate of convergence for the estimated parameters a ˆ0, a ˆ1, and a ˆ2.Furthermore, as for the upper bound of the system uncertainty, the prior knowledge of the adaptive tuning law does not require.Nevertheless, increasing the convergence rate of the parameter estimation may weaken the control performance.Thus, adjusting the control parameter size can be comprehensively considered.
As shown in Figure 7, the general structure block diagram of the proposed control scheme is drawn as follow.
On the basis of the above results，to demonstrate the process of the stability analysis, the Lyapunov function is selected as where, a a a i − ˆ= ˜, = 0, 1, 2 . Taking the time derivative of (37), one can obtain r c e r c e e a a a By substituting the error dynamic ( 21) and the total control law ( 35) into (38), it is easy to verify that Considering the adaptive laws Equations (36a-c), it holds that YANG ET AL.
Then, after simple calculation, it is easy to deduce that According to Lemma 3，taking integral of both sides of Equation ( 42) from 0 to t r , one can obtain , after simple calculation, it is easy to derive that The structure of the proposed control scheme.
Remark 4: On the basis of the above analyses, it is clear that the total convergence time of the control system should be  T T T + 1 2 , which is independent of the initial conditions and relies only on the size of control parameter.What's more, the proposed scheme has an advantageous performance, such as high accuracy, less chattering, strong robustness, and fast convergence rate.Consequently, the contribution of this paper is demonstrated.The detailed application and comparison of the proposed algorithm is presented below.

| The application and comparison of control strategy
To demonstrate the effectiveness and advantage of the proposed control strategies in this study, first of all, three various control strategies, called C1,C2, 44 and C3, 45 are presented in this section.Second, two illustrative cases are provided to compare the different control schemes.The nominal parameters and initial values of the system are referred to the above discussions.A detailed analysis is given as follow.
As regards the C1 control strategy the SM surface and the control input are Equation (25) and Equation (35), respectively.
With regard to the C2 control strategy, according to Lemma 2, the SM surface is constructed as: r e r e e = + + sign( ).
The control inputs are given as With respect to the C3 control strategy, a linear SM surface is selected as s e r e = + .(48) In light of the above expressions, it is obvious that the SM surface and the reaching law of the three control strategies are adjusted, respectively.Nonetheless, the adaptive law of all control strategies is unchanged.To make a fair comparison, the same control parameter is chosen in the four control schemes after lots of repeated experimental adjustments.As mentioned in Assumption 1, the external disturbances are chosen as δ t πt = 0.5 sin(3 ) + 0.5 cos (2 ).The expected trajectories of the system is described in two cases as follow, respectively.
Case 1: The desired signal is chosen as y t t ( ) = 2 sin(2 ). .Under the effect of matching disturbance, applying the above control strategy, the results are drawn in Figures 8-10.From the Figures, when the control input is activated at t s = 3 , it is clear that the error of four control strategies can converge to zero in a finite time.As shown in Figure 8, Figure 8B,D is a partial enlargement of Figure 8A,C, respectively.It is obvious from the figures that the system states can track the desired trajectories quickly.Furthermore, it can be clear that the actual state of the system tracks the desired trajectory at the fastest convergence rate and achieves the best tracking performance under the function of the C1 control strategy.And the order of the other three control schemes is C2, C3, respectively.
Figure 9 reveals that the error trajectories are tend to zero. Figure 9C,D is a partial enlargement of Figure 9A,C, respectively.Obviously, one can observe that the chattering phenomenon emerges at the beginning of activating the control input in Figure 9B,D.By comparing with the other control strategies, it is obvious that the error fluctuation range of the proposed strategy C1 is the smallest, the control convergence time is shortest and the chattering phenomenon is also very well weakened, which further implies the effectiveness and advantage of the proposed strategy.Meanwhile, to make a quantitative comparison, the root mean square error (RMSE) and the absolute average error (AAE) are applied to analyze the numerical measure of the error tracking performance, and the corresponding calculation results are shown in Table 2.The calculated results from the table verify the superiority of the C1 control strategy.Thus, it is concluded that the proposed strategy is of high tracking accuracy, quick response, weak chattering.From the results in Figure 10, the external disturbances of all four control schemes are handled through the uniform adaptive law.It can be seen from the figure that all the parameters converge to constant values, which can be verified that the designed adaptive law can deal with the demand for prior knowledge of disturbances appropriately.It is remarkable that the control strategy C1 maintain a faster parameter estimation speed in all the parameter estimation, which is more beneficial to use in practice.Besides, there is almost no chattering phenomenon and strong robustness against uncertainties and external disturbances.The results verify the superiority of the proposed control strategy.
Case 2: The desired signal is set as y t t ( ) = 0.5 cos(2 ).The control input is applied at t s = 5 .The tracking trajectories are depicted in Figure 11.The tracking error is shown in Figure 12.It is clear from these Figures that the tracking error converges to zero immediately, which implies that the nonlinear torsional vibration of the uncertain electromechanical coupling system is efficiently suppressed in a finite time, and it is clear that the C1 control scheme has the fastest convergence speed.As shown in Figure 11B,D, from the comparison curves of the three control strategies, the order of control performance is C1, C2, C3.Similarly, from Figure 12B,D, the proposed control scheme C1 is further proved the validity and superiority.For the same quantitative comparison purpose, the RMSE and the AAE of the tracking error are given in Table 2. From the quantitative analysis in Table 2, it is confirmed that the presented strategy provides lower RMSE and AAE values compared with the other control methods.

Method
Figure 13 shows the convergence procedures of the all estimation parameters and the comparison curves of the three control strategies, it is clear that the identified parameters converge to a constant and are estimated successfully, which intuitively verifies the effectiveness and superiority of the proposed scheme in this paper.
To summarize, the proposed control strategy C1 provides better tracking accuracy, faster convergence rate, less chattering and stronger robustness than the other methods under different conditions.Thus, it can be said that the validity and superiority of the proposed control strategy C1 is fully verified.And it is successfully applicable to a wide range of systems.

| CONCLUSION
To reduce the vibration amplitude and enhance the stability and reliability of wind turbine drivetrain, this paper studies the dynamic characteristics of wind turbine transmission system with electromechanical coupling in detail.The nonlinear model with coupled quadratic and cubic terms is established by considering the effect of electromagnetic excitation.The mechanism and reasons of torsional vibration were further revealed.Subsequently, as for the suppression strategy of torsional vibration, a novel adaptive fast fixed-time control scheme is developed.In summary, the findings of this research work mainly includes the following: 1. Based on the Helmholtz theorem, combined with the Hamilton energy function, the Homoclinic or Heteroclinic orbit and phase portraits of the state variables are illustrated, which is divided into several categories.2. When the torsional stiffness failure occurs or wind speed suddenly changes, the drivetrain may operate in a chaotic state, from which the instability of the system will increase, then may cause a serious damage of the component and endanger the safety.So, trying The curve of parameter estimations.
to increase the value of the torsional stiffness is important for the design of power trains.3. To enhance the accuracy and rapidity of suppression strategy, considering uncertainties and disturbances, a novel fixed-time control method combining adaptive techniques is proposed.Afterwards, the three different control schemes are applied to make a fair comparison analysis.The result verify the effectiveness and superiority of the proposed control algorithm, such as high accuracy, fast convergence rate, and strong robustness and so on.In addition, it is worth mentioning that the proposed control scheme in this work is also applied to other nonlinear systems.

| LIMITATIONS AND FUTURE WORK
This work illustrates dynamic characteristics and suppression strategy of torsional vibration in wind turbine.This provides a reference for the structural parameters and the electromagnetic parameters optimization design of wind turbine.Based on that, the research of wind turbine torsional vibration can be further explored in depth.For instance, when electromechanical coupled model of wind turbine drivetrain is established, nonlinear stiffness, and fractional damping could be considered.Additionally, dynamic analysis of torsional vibration characteristics could also apply Melnikov theory to study the Homoclinic bifurcation and chaos prediction.
Moreover, the control effect on the wind turbine torsional vibration needs to be examined in the future.Especially for smarter control schemes, such as machine learning, neural network approach.

F I G U R E 4
Phase trajectories and potential energy of the Hamilton system in various cases.

F I G U R E 6
Motion state of the drivetrain when stiffness failure occurs (A) time domain waveform; (B) phase portrait.f = 1.65.
the time derivative of the sliding surface (25) can be obtained as

d 1 , 3 . 8 3,
To achieve better control effect, the FTSM surface parameters are set as r = 6 The reaching law parameters are selected as r =

8
The comparison curve of state tracking under different control schemes.F I G U R E 9 The comparison curve of tracking error under different control schemes.

d
To further illustrate the validity of the proposed control scheme C1 in this paper.To make the F I G U R E 10 The curve of parameter estimations.T A B L E 2 Comparison of tracking error RMSE value and AAE value of different control scheme.

1 , r = 7 2 ,
comparison results fair and convincing, all three control schemes are performed with the same initial conditions.It's similar to the Case 1, just change the expected signal.After repeated experimental adjustments, the FTSM surface parameters are set as r = 3

F
I G U R E 11 The comparison curve of state trajectory tracking under different control schemes.F I G U R E 12 The comparison curve of tracking error under different control schemes.