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In order to study the stability of the wind turbine drivetrain in further depth, we present a nonlinear relative rotation mathematical model considering the nonlinear time-varying stiffness and the nonlinear damping force. Meanwhile, the nonlinear dynamics of the model under combined harmonic excitation are studied in detail. And some interesting dynamic phenomena are observed visually. Furthermore, to suppress chaotic oscillation within bounded time independent of initial conditions, a novel adaptive fixed-time terminal sliding mode controller is proposed. The stability of the final closed loop system is guaranteed according to Lyapunov stability theory. Rigorous mathematical analyses are used to prove the validity of the presented approach. Finally, compared with the existing finite-time stability method, simulation results are given to highlight the effectiveness and superiority of the proposed method and verify the theoretical analyses.

Two basic goals for wind energy conversion system (WECS) can be summarized as increasing its annual energy yield and extending its service life. Generally, in order to achieve the more annual energy production in a wide range of wind speeds, most of the research works regarding maximum power point tracking (MPPT) algorithms have been devoted to the main control strategies [

The effective control of uncertain nonlinear dynamic systems has been a hot issue of academia. The various advanced control methods including feedback control [

However, an important limitation of finite-time control is that its stable time depends mainly on the initial condition of the system in the practical application. To compensate this drawback, Polyakov [

The main innovations of this paper are summarized as follows. First, the wind turbine drivetrain model with the nonlinear time-varying stiffness and the nonlinear damping force is demonstrated in detail. Second, stability of wind turbine drivetrain is analyzed with and without considering external excitation according to Lyapunov stability theory. Dynamic characteristics of the system are depicted concretely by the aid of nonlinear dynamical theory. Third, an adaptive fixed-time terminal sliding mode controller is proposed for the suppression of chaotic oscillation in wind turbine drivetrain with combined harmonic excitation. The proposed control scheme can guarantee the system stabilization within fixed-time independent of initial value and have advantages in convergence rate and chattering problem. Finally, simulation results are given to demonstrate the effectiveness of the proposed approach.

The rest of this paper is organized as follows. In Section

The purpose of this section is to introduce the two-mass model based wind turbine drivetrain. Then, the wind turbine system including the blade, wind rotor, drivetrain, and generator is illustrated in Figure

Two-mass drivetrain scheme.

A typical configuration of variable-speed wind turbine drivetrain is illustrated in Figure

Two-mass model of drivetrain.

where

Therefore, the inertia moment of turbine rotor

where superscript

From (

Considering (

The drivetrain of the 2 MW turbine generator has the following partial parameters [

In the existing literatures [

In order to further study the dynamic characteristics, nonlinear time-varying stiffness and damping are used by equivalent principle in Figure

where

Combining (

where

From (

When the wind turbine operates in rated conditions, an external disturbance excitation, such as the fluctuation of the random energy frequency or amplitude, will affect the stability of the system. Subsequently, the system will enter a new operating state. And there will be a complex transient process because of the nonlinearity of every part. The nonlinear dynamical behaviors along with the fluctuation of the amplitude of the shaft twist angle are particularly focused on.

From (

According to the properties of the autonomous system [

For system (

According to system (

Consider the following Lyapunov positive definite function as

If

For system (

According to system (

The Jacobi matrix of system (

The corresponding characteristic equation of Jacobi matrix J is

As shown in Figures

The running state of system (

Number | Parameter | Eigenvalue | Root | Notes |
---|---|---|---|---|

1 | | | Two positive real roots | Instability |

2 | | | Two positive real roots | Instability |

3 | | | Two negative real roots | Stability |

4 | | | Two negative real roots | Stability |

Time domain diagrams and phase portrait when

Time domain diagrams and phase portrait when

Considering the combination harmonic excitation with unequal frequency, assume

Adding a small parameter

The system (

where

where

where

In order to simplify the calculation process without loss of generality, substituting (

The secular term can be removed by setting the coefficient to zero; then the real part and the imaginary part of the derivative for the complex function are opened and set to zero separately; the response amplitude equation is finally expressed as

where

From (

The Bifurcation diagram and Lyapunov exponent are used to observe the dynamical characteristics of the nonlinear system as the system parameter varies. For a periodic steady state, all Lyapunov exponents of the nonlinear dynamical system are less than zero, whereas at least one more than zero is the signature of a chaotic behavior. Transparently, it is known that, as the amplitude

Bifurcation diagram and Lyapunov exponential spectrum.

In accordance with the setting values of the abovementioned system parameters, various numerical computations of the timing diagrams and phase portraits were procured as illustrated in Figure

Time domain waveform and phase portrait for different

For the global stability analysis, we introduce some necessary lemmas in advance.

Assume that there exists a continuous positive definite and radially unbounded function

Based on Lemma

From Lemma

If

The second-order nonlinear dynamical system is adopted as follows:

where, according to (

In order to stabilize state variable

where

Then the proposed nonsingular terminal sliding mode manifold can be described as

where

According to the sliding mode control theory, the sliding mode manifold and its derivative must satisfy

By substituting (

From (

In order to satisfy the sliding condition in the presence of combination harmonic excitation, a switching adaptive law

where

The estimations of the parameters

where

Control flowchart diagram of closed loop system.

In this section, some main results of the proposed adaptive fixed-time terminal sliding mode control are analytically proved in the following theorem.

The adaptive controller of system (

The Lyapunov function candidate is constructed as follows:

Then the time derivative of

Substituting the designed control law

Combining the adaptive updating laws (

To simplify the calculation, the above equation is modified as

According to Lemma

It follows from Lemma

When the error variables are on the sliding manifold, their dynamics will meet (

The sliding mode dynamics (

The Lyapunov candidate function is selected in the following form:

From (

In the light of Lemma

In this section, numerical simulation results are used to validate the effectiveness and the superiority of the proposed control algorithm for the system (

In numerical simulation, to highlight the superiorities of the proposed scheme in further depth, two cases of the reference signal are implemented in keeping the rest of conditions unchanged.

The reference signal is chosen as

The reference signal is set as

The initial conditions of the state variable are set as

Time response of tracking

The curve of tracking

The tuning parameter estimations.

In order to exhibit the merits of the proposed fixed-time method, the control effect is independent of the initial conditions. Therefore, in case 2, the initial conditions of the state variable are set as

Time response of tracking

The curve of tracking

The estimations of tuning parameter

In this study, the stability analysis of a complex nonlinear drivetrain with combined harmonic excitation can be addressed by relying on nonlinear dynamics theory. First, we rebuilt the mathematical model of the wind turbine drivetrain considering the nonlinear time-varying stiffness and the nonlinear damping force. Second, the nonlinear dynamics theory is introduced to analyze the stability of the wind turbine drivetrain including bifurcation map, phase diagrams, and Lyapunov exponential spectrum under combined harmonic excitation. Then, numerical results clearly show that the greater the amplitude of the external disturbance excitation, the smaller the damping, and the larger the value of the nonlinear negative stiffness, the more unstable the wind turbine drivetrain. In order to guarantee the stability and normal functioning of the whole system, an adaptive fixed-time terminal sliding mode control approach can be implemented. Finally, in comparison with finite-time method, the applicability and superiority of the proposed scheme can be exhibited by numerical simulations, which are in good agreement with the theoretical analysis for the vibration phenomena of the wind turbine drivetrain. Meanwhile, it is noteworthy that the proposed method here can be further extended to the steady operation and design of the dual-motor driving electromechanical system. The fixed-time sliding mode control with an adaptive disturbance observer for the high-order or fractional-order dynamic system can be considered to estimate the model uncertainty directly in the future work.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This wok was supported by the National Natural Science Foundation of People’s Republic of China (Grant no. 51075326) and the Basic Ability Enhancement Project of Young Teachers in Guangxi Provincial Department of Education (Grant no. KY2015YB305).