Wind and Wave-Induced Vibration Reduction Control for Floating Offshore Wind Turbine Using Delayed Signals

Abstract

Active vibration control is a critical issue of the wind turbine in the field of marine energy. First, based on a three-degree-of-freedom wind turbine, a state space model subject to wind and wave loads is obtained. Then, a delayed state feedback control scheme is illustrated to reduce the vibration of platform pitch angle and tower top foreaft displacement, where the control channel includes time-delay state signals. The designed controller’s existence conditions are investigated. The simulation results show that the delayed feedback $H_{\infty }$ controller can significantly suppress wind- and wave-induced vibration of the wind turbine. Furthermore, it presents potential advantages over the delay-free feedback $H_{\infty }$ controller and the classic linear quadratic regulator in two aspects: vibration control performance and control cost.

1. Introduction

With the urgent global demand for clean renewable energy and the acceleration of industrialization, offshore wind power has attracted increasing attention [1,2,3]. However, several challenges still exist [4,5,6,7]. It is known that the complexity of the marine environment will aggravate the platform and tower oscillation to increase the risk of potential damage to floating wind turbine (FWT) systems [8,9,10,11,12,13]. For instance, wind, waves, and ocean currents can impact normal FWT operation, potentially leading to structural damage and economic losses. Thus, during wind turbine operations, it is essential to assess and control vibration responses properly.

For the wind turbine, the general method for reducing vibration amplitudes is to utilize structural control techniques, such as adjusting the pitch angle of blades in [14]. Moreover, an additional spring–mass device attached to the main structure is also commonly used, known as a tuned mass damper (TMD) [15,16,17,18,19]. For instance, in [20,21], a TMD is installed on an FWT to verify the feasibility of suppressing vibration. In [22], a variety of parameter optimization schemes are used to further investigate the inhibition effect of the FWT’s performance under proper TMD parameters. By combining the TMD with an inerter (TMDI), in [23], the TMDI submodule is developed to mitigate vibrations in a FWT subject to external disturbance. From the above research, it is clear that TMDs have been widely utilized in FWTs, consistently proving their effectiveness in improving FWTs’ vibration responses.

Due to the problems of aging and the service life of mechanical devices, the control performance of passive devices is generally limited. To overcome the above problems and ensure the structure safety, a feasible solution is to apply an active control method to the mechanical device. For example, in [24], a turbine cabin is equipped with a stroke-limited mixed mass damper and combined with a linear quadratic regulator (LQR) to reduce the internal oscillation amplitudes. In [25], a static output feedback method is developed to design the control law, and on this basis, an advanced generalized $H_{\infty }$ controller is developed to control the operation of an FWT. By considering the system uncertainties caused by inaccurate damping and natural frequency estimation, in [26], an adaptive sliding mode control idea was introduced to deal with these problems regarding parameter perturbations. In [27], a reinforcement learning idea is applied to control the wind turbine, which effectively enhances system robustness. In [28], based on a proportional integral control strategy, the vibration mitigation effect is verified by using the TMDI. In addition, the auto-disturbance rejection control algorithm and the model predictive control strategy are developed to study the control performance of FWTs, respectively [29,30]. These active control methods significantly suppress vibration characteristics in wind turbine systems. Regrettably, these active controllers tend to ignore the delayed information between state signals. Naturally, one would like to know that if delayed signals are considered in the design of the controller, can it be determined that the FWT‘s control performance will be better while the necessary control cost decreases? Therefore, exploring an active control method that minimizes wind–wave-induced vibration in wind turbines and takes advantage of this delayed information is the motivation of this paper.

In practical control systems, time delays may cause some negative effects on the system stability [31,32]. However, if an appropriate time delay is introduced into an unstable system’s feedback channel, the control system will most likely stabilize, proving that time delays possess beneficial impacts on certain systems [33,34,35,36]. In this situation, there are few results on delayed feedback schemes applied to control FWT systems. Inspired by this, a delayed feedback control idea is introduced to design a controller for an FWT subject to disturbance. The vibration attenuation control effects of the wind turbine with delayed state feedback controllers are investigated. In this paper, the main contributions can be further summarized as follows:

• A state space model of WFT subject to wind and wave disturbance is developed based on the Euler–Lagrange equation.
• By considering delayed information between state signals, a delayed full-state feedback $H_{\infty }$ control scheme is proposed for the control system and the designed controller’ existence conditions are derived.
• Compared to a delayed state feedback $H_{\infty }$ controller and the linear quadratic regulator, the former performs better than the latter in terms of vibration responses of the FWT.

The rest of this paper is structured as follows. Section 2 gives the modeling of the FWT subject to external disturbance. Section 3 details the design process of the delayed feedback controller. Section 4 provides simulation results to show the feasibility of the proposed control scheme. Section 5 outlines a brief conclusion.

2. Problem Formulation

Figure 1a shows a barge-type FWT subject to wind and wave disturbance. The schematic diagram of a simplified turbine model is exhibited in Figure 1b, where the symbols p, t, and T are the subsystem platform, tower, and TMD, respectively; ${\theta }_p$, ${\theta }_t$, and $x_T$ are the platform pitch angle (PtfmPitch), tower top foreaft displacement (TTFD), and TMD displacement, respectively; $m_p$, $m_t$, and $m_T$ are the masses of the subsystems; $k_p$, $k_t$, and $k_T$ are the corresponding stiffness coefficients; $c_p$, $c_t$, and $c_T$ are the equivalent damping coefficients of the subsystems; $R_p$ is the distance from the center of the barge mass to the tower’s rotational hinge; $R_t$ is the distance from the tower’s rotational hinge to the center of tower mass; $R_T$ is the distance from the center of TMD mass to the tower hinge; and g represents the acceleration of gravity.

To give a dynamic model of the FWT, it is assumed that the tower is a cantilever beam, fixed at the bottom and connected to the barge with a linear spring and damper, and the hydrodynamic effect and mooring lines are also represented by a linear damper and spring, respectively [22]. Then, for a non-conservative system with multiple degrees of freedom, one can establish the Euler–Lagrange equations, as follows:

$$\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial {\dot{q}}_i}\right)-\frac{\partial \mathcal{L}}{\partial q_i}={\mathcal{Q}}_{i}i=1,2,3,\dots ,n$$

(1)

where ${\mathcal{Q}}_i$ represents the generalized non-potential force, $\mathcal{L}=\mathcal{T}-\mathcal{V}$, with $\mathcal{T}$ and $\mathcal{V}$ being the total kinetic energy and potential energy, respectively.

Note

$$\left\{\begin{array}{c}\mathcal{T}=(I_t{\dot{\theta }}_t^2+I_p{\dot{\theta }}_p^2+m_T{\dot{x}}_T^2)/2\hfill \\ \mathcal{V}=[k_t{\left({\theta }_t-{\theta }_p\right)}^2+k_p{\theta }_p^2]/2+m_{t}g{R}_t\cos {\theta }_t\hfill \\ +m_{T}g\left(R_T\cos {\theta }_t-(x_T-R_T\sin {\theta }_t)\tan {\theta }_t\right)\hfill \\ +k_T{\left[(x_T-R_T\sin {\theta }_t)/\cos {\theta }_t\right]}^2/2-m_{p}g{R}_p\cos {\theta }_p\hfill \end{array}\right.$$

(2)

By substituting (2) into (1) and using the small-angle approximation [37], one yields

$$\left\{\begin{array}{c}{I}_p{\ddot{\theta }}_p=-c_p{\dot{\theta }}_p-k_p{\theta }_p+c_t({\dot{\theta }}_t-{\dot{\theta }}_p)+w_{wave}\left(t\right)\hfill \\ +k_t({\theta }_t-{\theta }_p)-m_{p}g{R}_p{\theta }_p\hfill \\ I_t{\ddot{\theta }}_t=-c_t({\dot{\theta }}_t-{\dot{\theta }}_p)-R_{T}u\left(t\right)-c_T(R_T{\dot{\theta }}_t-{\dot{x}}_T)R_T\hfill \\ -k_t({\theta }_t-{\theta }_p)+k_T(x_T-R_T{\theta }_t)R_T\hfill \\ +m_{t}g{R}_T{\theta }_t-m_{T}g(R_T{\theta }_t-x_T)+w_{wind}\left(t\right)\hfill \\ m_T{\ddot{x}}_T=c_T(R_T{\dot{\theta }}_t-{\dot{x}}_T)-k_T(x_T-R_T{\theta }_t)+m_{T}g{\theta }_t+u\left(t\right)\hfill \end{array}\right.$$

(3)

where $I_p$ and $I_t$ represent the subsystem’s moments of inertia, respectively; $w_{wind}\left(t\right)$ and $w_{wave}\left(t\right)$ are the wind and wave disturbance forces, respectively; and $u\left(t\right)$ is the required control force.

Let $\tilde{z}={\left[{\tilde{z}}_1{\tilde{z}}_2{\tilde{z}}_3{\tilde{z}}_4{\tilde{z}}_5{\tilde{z}}_6\right]}^T$ and $\xi ={\left[w_{wave}{w}_{wind}\right]}^T$, where ${\tilde{z}}_1={\theta }_p$, ${\tilde{z}}_2={\theta }_t$, ${\tilde{z}}_3=x_T$, ${\tilde{z}}_4={\dot{\theta }}_p$, ${\tilde{z}}_5={\dot{\theta }}_t$, and ${\tilde{z}}_6={\dot{x}}_T$. Let

$$\begin{array}{cc}\hfill & a_{41}=-(k_p+k_t+m_{p}g{R}_p)/I_p,a_{44}=-(c_p+c_t)/I_p\hfill \\ \hfill & a_{52}=-(-m_{t}g{R}_t+k_t+k_T{R}_T^2+m_{T}g{R}_T)/I_t\hfill \\ \hfill & a_{53}=(k_T{R}_T+m_{T}g)/I_t\hfill \\ \hfill & a_{55}=-(c_t+c_T{R}_T^2)/I_t\hfill \\ \hfill & a_{62}=(k_T{R}_T+m_{T}g)/m_T\hfill \end{array}$$

Then, from (3), one obtains the dynamic model of FWT as follows:

$$\dot{\tilde{z}}\left(t\right)=A\tilde{z}\left(t\right)+Bu\left(t\right)+D_w\xi \left(t\right),\tilde{z}\left(0\right)={\tilde{z}}_0$$

(4)

where the system matrices $A,B$, and $D_w$ are given as

$$\left\{\begin{array}{c}A=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ a_{41}& k_t/I_p& 0& a_{44}& c_t/I_p& 0\\ k_t/I_t& a_{52}& a_{53}& c_t/I_t& a_{55}& c_T{R}_T/I_t\\ 0& a_{62}& k_T/m_T& 0& c_T{R}_T/m_T& -c_T/m_T\end{array}\right]\hfill \\ B={\left[\begin{array}{cccccc}0& 0& 0& 0& -R_T/I_t& 1/m_T\end{array}\right]}^T\hfill \\ D_w={\left[\begin{array}{cccccc}0& 0& 0& 1/I_p& 0& 0\\ 0& 0& 0& 0& 1/I_t& 0\end{array}\right]}^T\hfill \end{array}\right.$$

and ${\tilde{z}}_0$ represents the initial state.

The control output equation is provided as

$$\eta \left(t\right)=C_{\eta }\tilde{z}\left(t\right)+E_{\eta }\xi \left(t\right)$$

(5)

where $C_{\eta }$ and $E_{\eta }$ are known matrices with proper dimensions.

To reduce external disturbance-induced vibration of the control system (4), a delayed feedback control law is denoted as

$$u\left(t\right)=K_h\tilde{z}(t-h)$$

(6)

where $K_h$ is a $1\times 6$ gain matrix to be determined, and $h>0$ is a time-delay signal to be designed.

The aim of this paper is to design a delayed feedback controller (6) such that (i) the resulting closed-loop system with $\xi \left(t\right)\equiv 0$ of the floating wind turbine system (4) is asymptotically stable; and (ii) the $H_{\infty }$ performance index with a prescribed attenuation level $\gamma >0$ satisfies the subsequent inequality:

$${\int }_0^{\infty }{\eta }^T\left(t\right)\eta \left(t\right)dt\le {\gamma }^2{\int }_0^{\infty }{\xi }^T\left(t\right)\xi \left(t\right)dt$$

(7)

Before obtaining the main proposition to design the proposed controller, Lemma 1 is needed.

Lemma 1

([38]). Let y be a differentiable function: [$c_1$, $c_2$]→${\mathbb{R}}^{n\times n}$. For any symmetric positive definite constant matrix R ∈ ${\mathbb{R}}^{n\times n}$, and matrices $\mathcal{X}={\left[{\mathcal{X}}_1{\mathcal{X}}_2{\mathcal{X}}_3\right]}^T$ and $\mathcal{Y}={\left[{\mathcal{Y}}_1{\mathcal{Y}}_2{\mathcal{Y}}_3\right]}^T$ with ${\mathcal{X}}_i,{\mathcal{Y}}_i\in {\mathcal{R}}^{n\times n},i=1,2,3$, the following inequality holds:

$$-{\int }_{c_1}^{c_2}{\dot{y}}^T\left(z\right)\tilde{R}\dot{y}\left(z\right)dz\le {\zeta }^T(\mathcal{S}+\mathcal{H})\zeta$$

(8)

where $\zeta ={\left[y^T\left(c_2\right)y^T\left(c_1\right)\frac{1}{c_2-c_1}{\int }_{c_1}^{c_2}{y}^T\left(z\right)dz\right]}^T$, and

$$\begin{array}{cc}\hfill & \mathcal{S}=(c_2-c_1)(\mathcal{X}{\tilde{R}}^{-1}{\mathcal{X}}^T+\mathcal{Y}{\tilde{R}}^{-1}{\mathcal{Y}}^T/3)\hfill \\ \hfill & \mathcal{H}=\left[\begin{array}{ccc}{\mathcal{X}}_1+{\mathcal{X}}_1^T+{\mathcal{Y}}_1+{\mathcal{Y}}_1^T& -{\mathcal{X}}_1^T+{\mathcal{X}}_2+{\mathcal{Y}}_1^T+{\mathcal{Y}}_2& {\mathcal{X}}_3+{\mathcal{Y}}_3-2{\mathcal{Y}}_1^T\\ *& -{\mathcal{X}}_2-{\mathcal{X}}_2^T+{\mathcal{Y}}_2+{\mathcal{Y}}_2^T& -{\mathcal{X}}_3-2{\mathcal{Y}}_2^T+{\mathcal{Y}}_3\\ *& *& -2{\mathcal{Y}}_3-2{\mathcal{Y}}_3^T\end{array}\right]\hfill \end{array}$$

3. Design of Delayed Feedback Controller

In this section, the delayed feedback control scheme is adopted to reduce the vibration of the wind turbine. The existence conditions of the designed controller are investigated, and the corresponding solution algorithm of the gain matrix $K_h$ is presented.

From (4) and (6), the closed-loop control system can be expressed as follows:

$$\dot{\tilde{z}}\left(t\right)=A\tilde{z}\left(t\right)+B{K}_h\overline{z}(t-h)+D_w\xi \left(t\right),\tilde{z}\left(0\right)={\tilde{z}}_0$$

(9)

Let $\tilde{z}\left(t\right)=e_1\alpha \left(t\right),\tilde{z}(t-h)=e_2\alpha \left(t\right)$, and ${\int }_{t-h}^t\tilde{z}\left(v\right)dv/h=e_3\alpha \left(t\right)$, where $\alpha \left(t\right)={[{\tilde{z}}^T\left(t\right){\tilde{z}}^T(t-h){\int }_{t-h}^t{\tilde{z}}^T\left(v\right)dv/h]}^T$, $e_1=\left[I00\right],e_2=\left[0I0\right]$, and $e_3=\left[00I\right]$. Then, the closed-loop control system (9) can be further written as

$$\dot{\tilde{z}}\left(t\right)=(A{e}_1+B{K}_h{e}_2)\alpha \left(t\right)+D_w\xi \left(t\right),\tilde{z}\left(0\right)={\tilde{z}}_0$$

(10)

To obtain this paper’s primary results, the following proposition offers a stability criterion for the closed-loop control system (10).

Proposition 1.

For given $\gamma >0$, $h>0$, the closed-loop control system (10) with $\xi \left(t\right)=0$ is asymptotically stable, and the $H_{\infty }$ performance index (7) is guaranteed, if there exist $6\times 6$ matrices $\mathcal{P}>0$, $\mathcal{Q}>0$, $\mathcal{R}>0$, ${\mathcal{X}}_1$, ${\mathcal{X}}_2$, ${\mathcal{X}}_3$, ${\mathcal{Y}}_1$, ${\mathcal{Y}}_2$, ${\mathcal{Y}}_3$, and $1\times 6$ matrix $K_h$ such that

$$\begin{array}{c}\left[\begin{array}{cccccc}{\mathrm{\Xi }}_{11}& e_1^T\mathcal{P}{D}_w& {\mathrm{\Omega }}_{12}& e_1^T{C}_{\eta }^T& {\mathrm{\Xi }}_{13}& {\mathrm{\Xi }}_{14}\\ *& -{\gamma }^{2}I& h{D}_w^T\mathcal{R}& E_{\eta }^T& 0& 0\\ *& *& -\mathcal{R}& 0& 0& 0\\ *& *& *& -I& 0& 0\\ *& *& *& *& -\mathcal{R}& 0\\ *& *& *& *& *& -3\mathcal{R}\end{array}\right]<0\hfill \end{array}$$

(11)

where

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill {\mathrm{\Xi }}_{11}& =e_1^T(\mathcal{P}A+A^T\mathcal{P}+\mathcal{Q})e_1+e_1^T(\mathcal{P}B{K}_h+K_h^T{B}^T\mathcal{P})e_2\hfill \\ \hfill & -e_2^T\mathcal{Q}{e}_2+h{e}_1^T{\mathrm{\Psi }}_{11}{e}_1+h{e}_1^T({\mathrm{\Psi }}_{12}+{\mathrm{\Psi }}_{12}^T)e_2\hfill \\ \hfill & +h{e}_1^T({\mathrm{\Psi }}_{13}+{\mathrm{\Psi }}_{13}^T)e_3+h{e}_2^T{\mathrm{\Psi }}_{22}{e}_2\hfill \\ \hfill & +h{e}_2^T({\mathrm{\Psi }}_{23}+{\mathrm{\Psi }}_{23}^T)e_3+h{e}_3^T{\mathrm{\Psi }}_{33}{e}_3\hfill \\ \hfill {\mathrm{\Xi }}_{12}& =h\mathcal{R}(e_1^T{A}^T+e_2^T{K}_h^T{B}^T)\hfill \\ \hfill {\mathrm{\Xi }}_{13}& =h{e}_1^T{\mathcal{X}}_1^T+h{e}_2^T{\mathcal{X}}_2^T+h{e}_3^T{\mathcal{X}}_3^T\hfill \\ \hfill {\mathrm{\Xi }}_{14}& =h{e}_1^T{\mathcal{Y}}_1^T+h{e}_2^T{\mathcal{Y}}_2^T+h{e}_3^T{\mathcal{Y}}_3^T\hfill \end{array}\hfill \end{array}\right.$$

(12)

and

$$\begin{array}{cc}\hfill & {\mathrm{\Psi }}_{11}={\mathcal{X}}_1+{\mathcal{X}}_1^T+{\mathcal{Y}}_1+{\mathcal{Y}}_1^T,{\mathrm{\Psi }}_{12}=-{\mathcal{X}}_1^T+{\mathcal{X}}_2+{\mathcal{Y}}_1^T+{\mathcal{Y}}_2\hfill \\ \hfill & {\mathrm{\Psi }}_{13}={\mathcal{X}}_3+Y_3-2{\mathcal{Y}}_1^T,{\mathrm{\Psi }}_{22}=-{\mathcal{X}}_2-{\mathcal{X}}_2^T+{\mathcal{Y}}_2+{\mathcal{Y}}_2^T\hfill \\ \hfill & {\mathrm{\Psi }}_{23}=-{\mathcal{X}}_3-2{\mathcal{Y}}_2^T+{\mathcal{Y}}_3,{\mathrm{\Psi }}_{33}=-2{\mathcal{Y}}_3-2{\mathcal{Y}}_3^T\hfill \end{array}$$

Proof. 

A Lyapunov–Krasovskii functional candidate is chosen as

$$\mathcal{V}\left(t\right)={\tilde{z}}^T\left(t\right)\mathcal{P}\tilde{z}\left(t\right)+{\int }_{t-h}^t{\tilde{z}}^T\left(v\right)\mathcal{Q}\tilde{z}\left(v\right)dv+h{\int }_{-h}^0{\int }_{t+v}^t{\dot{\tilde{z}}}^T\left(s\right)\mathcal{R}\dot{\tilde{z}}\left(s\right)dsdv$$

(13)

where $\mathcal{P}>0$, $\mathcal{Q}>0$, and $\mathcal{R}>0$.

Taking the time derivative of $\mathcal{V}\left(t\right)$ along the trajectory of (10), we have

$$\begin{array}{cc}\hfill \dot{\mathcal{V}}\left({\tilde{z}}_t\right)=& {\alpha }^T\left(t\right)\left[e_1^T(\mathcal{P}A+A^T\mathcal{P})e_1+e_1^T(\mathcal{P}B{K}_h+K_h^T{B}^T\mathcal{P})e_2\right]\alpha \left(t\right)\hfill \\ \hfill & +{\alpha }^T\left(t\right)(e_1^T\mathcal{Q}{e}_1-e_2^T\mathcal{Q}{e}_2)\alpha \left(t\right)+2{\alpha }^T\left(t\right)e_1^T\mathcal{P}{D}_w\xi \left(t\right)\hfill \\ \hfill & -h{\int }_{t-h}^t{\dot{\tilde{z}}}^T\left(v\right)\mathcal{R}\dot{\tilde{z}}\left(v\right)dv+h^2{\dot{\tilde{z}}}^T\left(t\right)\mathcal{R}\dot{\tilde{z}}\left(t\right)\hfill \end{array}$$

(14)

By Lemma 1, for real matrices ${\mathcal{X}}_i$ and ${\mathcal{Y}}_i,i=1,2,3$, one obtains the following inequality:

$$\begin{array}{cc}\hfill & -h{\int }_{t-h}^t{\dot{\overline{z}}}^T\left(v\right)\mathcal{R}\dot{\overline{z}}\left(v\right)dv\hfill \\ \hfill & \le h{\alpha }^T\left(t\right)\left[e_1^T{\mathrm{\Psi }}_{11}{e}_1+e_1^T({\mathrm{\Psi }}_{12}+{\mathrm{\Psi }}_{12}^T)e_2+e_1^T({\mathrm{\Psi }}_{13}+{\mathrm{\Psi }}_{13}^T)e_3\right]\alpha \left(t\right)\hfill \\ \hfill & +h{\alpha }^T\left(t\right)\left[e_2^T{\mathrm{\Psi }}_{22}{e}_2+e_2^T({\mathrm{\Psi }}_{23}+{\mathrm{\Psi }}_{23}^T)e_3+e_3^T{\mathrm{\Psi }}_{33}{e}_3\right]\alpha \left(t\right)\hfill \\ \hfill & +{\alpha }^T\left(t\right)\left[{\mathrm{\Xi }}_{13}^T{\mathcal{R}}^{-1}{\mathrm{\Xi }}_{13}+{\mathrm{\Xi }}_{14}^T{\left(3\mathcal{R}\right)}^{-1}{\mathrm{\Xi }}_{14}\right]\alpha \left(t\right)\hfill \end{array}$$

(15)

Denote $\beta \left(t\right)={\left[{\alpha }^T\left(t\right){\xi }^T\left(t\right)\right]}^T$. Then, one obtains

$$h^2{\dot{\tilde{z}}}^T\left(t\right)\mathcal{R}\dot{\tilde{z}}\left(t\right)={\beta }^T\left(t\right)\left[\begin{array}{c}{\mathrm{\Xi }}_{12}\\ h{D}_w^T\mathcal{R}\end{array}\right]{\mathcal{R}}^{-1}\left[\begin{array}{cc}{\mathrm{\Xi }}_{12}^T& h\mathcal{R}{D}_w\end{array}\right]\beta \left(t\right)$$

(16)

From (14)–(16), we have

$$\dot{\mathcal{V}}\left(t\right)\le {\beta }^T\left(t\right)\left\{{\mathrm{\Pi }}_0+\left[\begin{array}{c}{\mathrm{\Xi }}_{12}\\ h{D}_w^T\mathcal{R}\end{array}\right]{\mathcal{R}}^{-1}\left[\begin{array}{cc}{\mathrm{\Xi }}_{12}^T& h\mathcal{R}{D}_w\end{array}\right]\right\}\beta \left(t\right)$$

(17)

where

$$\begin{array}{c}{\mathrm{\Pi }}_0=\left[\begin{array}{cc}{\mathrm{\Xi }}_{11}+{\mathrm{\Xi }}_{13}{\mathcal{R}}^{-1}{\mathrm{\Xi }}_{13}^T+{\mathrm{\Xi }}_{14}{\left(3\mathcal{R}\right)}^{-1}{\mathrm{\Xi }}_{14}^T& e_1^T\mathcal{P}{D}_w\\ *& 0\end{array}\right]\hfill \end{array}$$

(18)

Denote

$${\mathrm{\Pi }}_1={\mathrm{\Pi }}_0+\left[\begin{array}{cc}0& 0\\ 0& -{\gamma }^{2}I\end{array}\right]$$

Then, one further obtains

$$\dot{\mathcal{V}}\left(t\right)+{\eta }^T\left(t\right)\eta \left(t\right)-{\gamma }^2{\xi }^T\left(t\right)\xi \left(t\right)\le {\alpha }^T\left(t\right)\mathrm{\Pi }\alpha \left(t\right)$$

(19)

where

$$\begin{array}{c}\hfill \mathrm{\Pi }={\mathrm{\Pi }}_1+\left[\begin{array}{c}{\mathrm{\Xi }}_{12}\\ h{D}_w^T\mathcal{R}\end{array}\right]{\mathcal{R}}^{-1}\left[\begin{array}{cc}{\mathrm{\Xi }}_{12}^T& h\mathcal{R}{D}_w\end{array}\right]+\left[\begin{array}{c}{e}_1^T{C}_{\eta }^T\\ E_{\eta }^T\end{array}\right]\left[\begin{array}{cc}{C}_{\eta }{e}_1& E_{\eta }\end{array}\right]\end{array}$$

(20)

Note that if $\mathrm{\Pi }<0$ holds, we yield $\dot{\mathcal{V}}\left(t\right)+{\eta }^T\left(t\right)\eta \left(t\right)-{\gamma }^2{\xi }^T\left(t\right)\xi \left(t\right)\le 0$. In fact, from the known condition (11), one can yield the inequality $\mathrm{\Pi }<0$ by the Schur complement, which implies that for the system (10), the $H_{\infty }$ performance index (7) is guaranteed. Moreover, the inequality (11) indicates that the following inequality holds:

$$\begin{array}{c}\left[\begin{array}{cccc}{\mathrm{\Xi }}_{11}& {\mathrm{\Xi }}_{12}& {\mathrm{\Xi }}_{13}& {\mathrm{\Xi }}_{14}\\ *& -\mathcal{R}& 0& 0\\ *& *& -\mathcal{R}& 0\\ *& *& *& -3\mathcal{R}\end{array}\right]<0\hfill \end{array}$$

(21)

Further, from (21), one obtains

$$\dot{\mathcal{V}}\left(t\right)\le {\alpha }^T\left(t\right)\left[{\mathrm{\Xi }}_{11}+{\mathrm{\Xi }}_{12}{\mathcal{R}}^{-1}{\mathrm{\Xi }}_{12}^T+{\mathrm{\Xi }}_{13}{\mathcal{R}}^{-1}{\mathrm{\Xi }}_{13}^T+{\mathrm{\Xi }}_{14}{\left(3\mathcal{R}\right)}^{-1}{\mathrm{\Xi }}_{14}^T\right]\alpha \left(t\right)<0$$

(22)

which means that the closed-loop control system (10) with $\xi \left(t\right)=0$ is asymptotically stable. This completes the proof.

To calculate the controller gain matrix $K_h$, denote

$$\begin{array}{c}\hfill \aleph =\mathrm{diag}\{{\aleph }_0,I,{\mathcal{R}}^{-1},I,{\mathcal{P}}^{-1},{\mathcal{P}}^{-1}\}\end{array}$$

where ${\aleph }_0=\mathrm{diag}\{{\mathcal{P}}^{-1},{\mathcal{P}}^{-1},I\}$. Pre- and post-multiply the matrix ℵ and its transpose on the right-hand side of (11), respectively; set $\overline{\mathcal{P}}={\mathcal{P}}^{-1}$, $\overline{\mathcal{Q}}={\mathcal{P}}^{-1}\mathcal{Q}{\mathcal{P}}^{-1}$, $\overline{\mathcal{R}}={\mathcal{P}}^{-1}\mathcal{R}{\mathcal{P}}^{-1}$, ${\overline{\mathcal{X}}}_i={\mathcal{P}}^{-1}{\mathcal{X}}_i{\mathcal{P}}^{-1}$, ${\overline{\mathcal{Y}}}_i={\mathcal{P}}^{-1}{\mathcal{Y}}_i{\mathcal{P}}^{-1}(i=1,2,3)$, ${\overline{K}}_h=K_h{\mathcal{P}}^{-1}$; and utilize $-\overline{\mathcal{P}}{\overline{\mathcal{R}}}^{-1}\overline{\mathcal{P}}\le -2\overline{\mathcal{P}}+\overline{\mathcal{R}}$. Then, one directly yields the following proposition. □

Proposition 2.

For given $\gamma >0$, $h>0$, if there exist $6\times 6$ matrices $\overline{\mathcal{P}}>0$, $\overline{\mathcal{Q}}>0$, $\overline{\mathcal{R}}>0$, ${\overline{\mathcal{X}}}_1$, ${\overline{\mathcal{X}}}_2$, ${\overline{\mathcal{X}}}_3$, ${\overline{\mathcal{Y}}}_1$, ${\overline{\mathcal{Y}}}_2$, ${\overline{\mathcal{Y}}}_3$, and $1\times 6$ matrix ${\overline{K}}_h$ such that

$$\begin{array}{c}\left[\begin{array}{cccccc}{\overline{\mathrm{\Xi }}}_{11}& e_1^T{D}_w& {\overline{\mathrm{\Xi }}}_{12}& e_1^T\overline{\mathcal{P}}{C}_{\eta }^T& {\overline{\mathrm{\Xi }}}_{13}& {\overline{\mathrm{\Xi }}}_{14}\\ *& -{\gamma }^{2}I& h{D}_w^T& E_{\eta }^T& 0& 0\\ *& *& -2\overline{\mathcal{P}}+\overline{\mathcal{R}}& 0& 0& 0\\ *& *& *& -I& 0& 0\\ *& *& *& *& -\overline{\mathcal{R}}& 0\\ *& *& *& *& *& -3\overline{\mathcal{R}}\end{array}\right]<0\hfill \end{array}$$

(23)

where

$$\begin{array}{cc}\hfill {\overline{\mathrm{\Xi }}}_{11}=& e_1^T(A\overline{\mathcal{P}}+\overline{\mathcal{P}}{A}^T+\overline{\mathcal{Q}})e_1+e_1^T(B{\overline{K}}_h+{\overline{K}}_h^T{B}^T)e_2-e_2^T\overline{\mathcal{Q}}{e}_2\hfill \\ \hfill & +h{e}_1^T{\overline{\mathrm{\Psi }}}_{11}{e}_1+h{e}_1^T({\overline{\mathrm{\Psi }}}_{12}+{\overline{\mathrm{\Psi }}}_{12}^T)e_2+h{e}_1^T({\overline{\mathrm{\Psi }}}_{13}+{\overline{\mathrm{\Psi }}}_{13}^T)e_3\hfill \\ \hfill & +h{e}_2^T{\overline{\mathrm{\Psi }}}_{22}{e}_2+h{e}_2^T({\overline{\mathrm{\Psi }}}_{23}+{\overline{\mathrm{\Psi }}}_{23}^T)e_3+h{e}_3^T{\overline{\mathrm{\Psi }}}_{33}{e}_3\hfill \\ \hfill {\overline{\mathrm{\Xi }}}_{12}=& h{e}_1^T\overline{\mathcal{P}}{A}^T+h{e}_2^T{\overline{K}}_h^T{B}^T\hfill \\ \hfill {\overline{\mathrm{\Xi }}}_{13}=& h{e}_1^T{\overline{\mathcal{X}}}_1^T+h{e}_2^T{\overline{\mathcal{X}}}_2^T+h{e}_3^T{\overline{\mathcal{X}}}_3^T\hfill \\ \hfill {\overline{\mathrm{\Xi }}}_{14}=& h{e}_1^T{\overline{\mathcal{Y}}}_1^T+h{e}_2^T{\overline{\mathcal{Y}}}_2^T+h{e}_3^T{\overline{\mathcal{Y}}}_3^T\hfill \end{array}$$

with

$$\begin{array}{cc}\hfill {\overline{\mathrm{\Psi }}}_{11}& ={\overline{\mathcal{X}}}_1+{\overline{\mathcal{X}}}_1^T+{\overline{\mathcal{Y}}}_1+{\overline{\mathcal{Y}}}_1^T,{\overline{\mathrm{\Psi }}}_{12}=-{\overline{\mathcal{X}}}_1^T+{\overline{\mathcal{X}}}_2+{\overline{\mathcal{Y}}}_1^T+{\overline{\mathcal{Y}}}_2\hfill \\ \hfill {\overline{\mathrm{\Psi }}}_{13}& ={\overline{\mathcal{X}}}_3+{\overline{\mathcal{Y}}}_3-2{\overline{\mathcal{Y}}}_1^T,{\overline{\mathrm{\Psi }}}_{22}=-{\overline{\mathcal{X}}}_2-{\overline{\mathcal{X}}}_2^T+{\overline{\mathcal{Y}}}_2+{\overline{\mathcal{Y}}}_2^T\hfill \\ \hfill {\overline{\mathrm{\Psi }}}_{23}& =-{\overline{\mathcal{X}}}_3-2{\overline{\mathcal{Y}}}_2^T+{\overline{\mathcal{Y}}}_3,{\overline{\mathrm{\Psi }}}_{33}=-2{\overline{\mathcal{Y}}}_3-2{\overline{\mathcal{Y}}}_3^T\hfill \end{array}$$

Then, the control gain matrix $K_h$ is solved by $K_h={\overline{K}}_h{\overline{\mathcal{P}}}^{-1}$.

Remark 1.

In (6), setting $h=0$ obtains a delay-free full-state feedback control law as

$$u\left(t\right)=K_0\tilde{z}\left(t\right)$$

(24)

where the gain matrix $K_0={\overline{K}}_0{\overline{\mathcal{P}}}^{-1}$, with ${\overline{K}}_0$ and $\overline{\mathcal{P}}$ satisfying the following inequality:

$$\begin{array}{c}\left[\begin{array}{ccc}A\overline{\mathcal{P}}+\overline{\mathcal{P}}{A}^T+B{\overline{K}}_0+{\overline{K}}_0^T{B}^T& D_w& \overline{\mathcal{P}}{C}_{\eta }^T\\ *& -{\gamma }^{2}I& E_{\eta }^T\\ *& *& -I\end{array}\right]<0\hfill \end{array}$$

(25)

Remark 2.

The controller in (24) uses the current full-state signals, while the one in (6) uses the delayed full-state signals. It is found that by taking delayed information into account, the designed delayed controllers have potential advantages over the delay-free controller for the active vibration control of FWT, which will be illustrated by the simulation results below.

4. Simulation Results

In this section, a delayed state feedback controller is designed for the wind turbine system subject to external disturbance, and the controlled performance of the FWT is investigated based on simulation results. The designed delayed state feedback controller is compared with the delay-free state feedback controller and the classic linear quadratic regulator [24] from two aspects: system performance and control cost.

4.1. Main Parameters of FWT System

A barge-type FWT with three blades is selected, with a rating power of 5 MW. The values of the main parameters are taken from [24], where the mass of the nacelle, platform, tower, and rotor is 240,000, 5,452,000, 347,460, and 110,000 kg, respectively. The remaining parameters are listed in

Table 1. Parameters of FWT.

Parameters	Values	Parameters	Values
$D_w$	126 m	$R_p$	0.28 m
$R_t$	64.21 m	$R_T$	90.6 m
$D_p$	40 × 10 × 10 ${\mathrm{m}}^3$	$D_n$	24 × 8 × 6 ${\mathrm{m}}^3$
$k_p$	1.8735 × ${10}^9$ N m/rad	$k_t$	1.3604 × ${10}^{10}$ N m/rad
$c_p$	5.4151 × ${10}^7$ N m s/rad	$c_t$	2.3087 × ${10}^7$ N m s/rad
$k_T$	5274 N/m	$c_T$	10,183 N/m
$I_p$	2.0577 × ${10}^9$ kg ${\mathrm{m}}^2$	$I_t$	3.1051× ${10}^9$ kg ${\mathrm{m}}^2$

, where $D_w$ is the diameter of the wheel, $D_p$ is the dimension of the platform, and $D_n$ is the dimension of the nacelle. Then, using the aforementioned parameters, the system matrices (4) can be obtained as follows:

$$\left\{\begin{array}{c}A=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ -7.5291& 6.6113& 0& -0.0375& 0.0112& 0\\ 4.3812& -4.2651& 0.0003& 0.0074& -0.0344& -0.0003\\ 0& 21.7523& -0.1318& 0& 23.0645& -0.2546\end{array}\right]\hfill \\ B={10}^{-5}\times {\left[\begin{array}{cccccc}0& 0& 0& 0& -0.003& 2.500\end{array}\right]}^T\\ D_w={10}^{-10}\times {\left[\begin{array}{cccccc}0& 0& 0& 0& 3.221& 0\\ 0& 0& 0& 4.860& 0& 0\end{array}\right]}^T\end{array}\right.$$

(26)

In the control output Equation (5), the matrices $C_{\eta }$ and $E_{\eta }$ are set as

$$\begin{array}{c}{C}_{\eta }=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\end{array}\right],E_{\eta }=\left[\begin{array}{cc}0.005& 0\\ 0& 0.005\end{array}\right]\hfill \end{array}$$

(27)

Referencing IEC61400-3, the original Stevenson meteorological station is selected as the environmental reference point. The Kaimal wind speed spectrum is selected, generated by TurbSim software v2.00 [39], in which the power index rate, the turbulence intensity, and wind speed are set to 0.14, 18%, and 12 m/s, respectively. To obtain the wave load, the JONSWAP spectrum is employed. The peak enhancement factor for the JONSWAP spectrum is 3.3, the peak spectral period is set to 11.8 s, and the significant wave height is set to 2.6 m. Figure 2 shows the curves of wind and wave force acting on the FWT, and the wave spectrum is shown in Figure 3.

To compare different controllers quantitatively, the standard deviations of the PtfmPitch, TTFD, and TMD displacement are denoted as $\sigma \left({\theta }_p\right)$, $\sigma \left({\theta }_t\right)$, and $\sigma \left(x_T\right)$, respectively. Their peak values are represented by ${\widehat{\theta }}_p$, ${\widehat{\theta }}_t$, and ${\widehat{x}}_T$, respectively, and the root mean square (RMS) values $J_u$ of the control force is computed by $J_u=\sqrt{{\int }_0^{t_f}{u}^2\left(t\right)dt/t_f}$. The standard deviation inhibition rate $\overline{\eta }$ of tower top foreaft displacement is defined as $\overline{\eta }=({\sigma }_s-{\sigma }_t)/{\sigma }_s$ [22], where ${\sigma }_s$ and ${\sigma }_t$ are the standard deviation of TTFD without TMD and with TMD, respectively.

4.2. Performance of Wind Turbine System with Different Controllers

First, the vibration characteristics of the FWT with no control and with the TMD-based passive controller (TMD-PC) are investigated. Meanwhile, the standard deviations and peak values of PtfmPitch, TTFD and TMD displacement of the FWT are computed, respectively. In fact, under the passive controller, the standard deviations of PtfmPitch and TTFD of the FWT are reduced from 4.2518 and 0.5941 of the subsystems without control to 1.8275 and 0.2564, respectively. The peak values of PtfmPitch and TTFD of the wind turbine are significantly reduced from 10.1134 and 1.5055 to 5.8882 and 0.8516 of the systems without control, respectively. It is can be found that under the passive controller, the standard deviations of PtfmPitch and TTFD of the wind turbine are reduced to about 43% and 43% of the original values, respectively, and their peak values are reduced to about 58% and 57% of their original values, respectively. The standard deviation inhibition rate is calculated to be 57%.

Set $\gamma =0.45$. Then, by solving the matrix inequality (25), one obtains the control gain matrix $K_0$ of the current state feedback controller (CSFC) (24) as

$$K_0=-{10}^7\times [1.6907-3.06230.00531.79151.91770.0148]$$

(28)

According to Proposition 2, let $h=0.25$, and the gain matrix $K_h$ of the delayed state feedback controller (DSFC) (6) can be obtained as

$$K_h={10}^7\times [0.5166-0.43070.0000-3.87493.6078-0.0002]$$

(29)

The corresponding delayed controller is denoted as DSFC25. Under the CSFC and DSFC25, the RMS values of the control force under CSFC and DSFC25 and the standard deviations and peak values of PtfmPitch, TTFD, and TMD displacement of the FWT are listed in

Table 2. The standard deviations, peak values of ${\theta }_p$, ${\theta }_t$, $x_T$ of the wind turbine system, and RMS values of control force with no control, TMD-PC, CSFC, and DSFC25.

Controllers	$\mathit{\sigma }\mathbf{\left(}{\mathit{\theta }}_{\mathit{p}}\mathbf{\right)}$	$\mathit{\sigma }\mathbf{\left(}{\mathit{\theta }}_{\mathit{t}}\mathbf{\right)}$	$\mathit{\sigma }\mathbf{\left(}{\mathit{x}}_{\mathit{T}}\mathbf{\right)}$	${\widehat{\mathit{\theta }}}_{\mathit{p}}$	${\widehat{\mathit{\theta }}}_{\mathit{t}}$	${\widehat{\mathit{x}}}_{\mathit{T}}$	${\mathit{J}}_{\mathit{u}}\mathbf{\left(}{\mathbf{10}}^{\mathbf{4}}\mathbf{\right)}$
No control	4.2518	0.5941	–	10.1134	1.5055	–	–
TMD-PC	1.8275	0.2564	4.2241	5.8882	0.8516	13.9194	–
CSFC	1.1117	0.1631	5.9557	3.8250	0.6155	21.1931	3.9723
DSFC25	1.2881	0.1800	5.2710	4.6297	0.6565	17.4487	2.6508

, where their values of the system without control and with TMC-PC are also provided for comparison purposes.

The responses of PtfmPitch, TTFD, and TMD displacement of the wind turbine system under no control, TMD-PC, CSFC, and DSFC25 are displayed in Figure 4, Figure 5 and Figure 6, respectively. The control force curves of CSFC and DSFC25 are given in Figure 7.

Table 2 and Figure 4, Figure 5, Figure 6 and Figure 7 show that the designed CSFC and DSFC25 are both effective in degrading wind- and wave-induced vibration. Specifically, in this situation, one can clearly observe that the delayed controller DSFC25 outperforms the delay-free one, i.e., CSFC, in terms of the control cost. In fact, the RMS value of control force by DSFC25 is reduced to about 67% of the one by CSFC, which shows that considering delayed signals to the designed controller may improve the system’s performance. Additionally, comparing DSFC25 with TMD-PC, the suppression responses of FWT vibration are significantly more effective with the former than with the latter, which demonstrates that DSFC25 ensures more reliable and efficient operation of the FWT and its advantages over the pure TMD-PC approach. Figure 8 and Figure 9 show the Spectral Amplitude Operators (SAOs) of PtfmPitch and TTFD by TMD-PC and the designed DSFC25, respectively. From the figures, it is not difficult to find that the significant peaks are clustered within the frequency range of 0.05 to 0.2 Hz. It indicates that the system has significant oscillations or resonances near that frequency. In particular, DSFC25 significantly reduces the peaks compared to TMD-PC, thereby ensuring the better steady-state performance of the wind turbine system.

To illustrate the superiority of the proposed controller, the designed DSFC25 is compared with a linear quadratic regulator (LQR) [24].

For system (4), the matrices Q and r of the LQR are taken as follows:

$$Q={10}^5\times \mathrm{diag}\{15,15,0.001,130,190,0.001\},r=4\times {10}^{-7}$$

(30)

Then, the gain matrix $K_{LQR}$ of the LQR can be obtained as

$$K_{LQR}=-{10}^6\times [1.8608-5.90850.01274.0205-1.08190.0323]$$

(31)

Table 3. The standard deviations, peak values of ${\theta }_p$, ${\theta }_t$, $x_T$ of the wind turbine system, and RMS values of the control force by the LQR and DOFC25.

Controller	$\mathit{\sigma }\mathbf{\left(}{\mathit{\theta }}_{\mathit{p}}\mathbf{\right)}$	$\mathit{\sigma }\mathbf{\left(}{\mathit{\theta }}_{\mathit{t}}\mathbf{\right)}$	$\mathit{\sigma }\mathbf{\left(}{\mathit{x}}_{\mathit{T}}\mathbf{\right)}$	${\widehat{\mathit{\theta }}}_{\mathit{p}}$	${\widehat{\mathit{\theta }}}_{\mathit{t}}$	${\widehat{\mathit{x}}}_{\mathit{T}}$	${\mathit{J}}_{\mathit{u}}\mathbf{\left(}{\mathbf{10}}^{\mathbf{4}}\mathbf{\right)}$
LQR	1.4217	0.2023	5.6245	4.9293	0.7245	19.5667	2.7515
DSFC25	1.2881	0.1800	5.2710	4.6297	0.6565	17.4487	2.6508

shows the standard deviations, peak values of PtfmPitch, TTFD, and TMD displacement, and the RMS values of control force under the LQR and DSFC25. The response curves of PtfmPitch, TTFD, and the TMD displacement of the wind turbine system are depicted in Figure 10, Figure 11 and Figure 12, respectively. The curves of control force with the LQR and DSFC25 are shown in Figure 13.

From Figure 10, Figure 11, Figure 12 and Figure 13 and Table 3, one yields that (i) the control cost with the LQR is slightly larger than that with the DSFC25; (ii) the standard deviations and peak values of PtfmPitch and TTFD of the wind turbine system with DSFC25 are all smaller than those with LQR; (iii) the peak value of TMD displacement of the FWT by the DSMC25 is reduced to about 96% of the value by LQR, which means that to obtain better vibration reduction effects, the larger displacement response of the TMD should be avoided. In this sense, the designed delayed control method provides a better solution to the vibration control of the FWT subject to external disturbance.

5. Conclusions

The active vibration control of a floating wind turbine system subject to external disturbance has been studied. First, a dynamic model was established by considering wind and wave loads. Then, a delayed full-state feedback controller was proposed, where the existence conditions of the designed controller were derived based on Lyapunov stability theory. Simulation results show that under the proposed delayed feedback scheme, the wind- and wave-induced vibration of the wind turbine system was significantly reduced. Additionally, when compared to the delayed feedback controller with the delay-free state feedback controller and the classic optimal controller for the wind turbine system, the former provided potential advantages over the latter from two perspectives: system performance and control cost.

This paper introduces an active control scheme for wind turbine systems aimed at suppressing vibration characteristics. However, there are critical issues regarding vibration analysis to be further investigated. Therefore, a more general dynamic model and easily implemented active control schemes need to be investigated to guarantee the safety of the wind turbine.