# Classical Flutter and Active Control of Wind Turbine Blade Based on Piezoelectric Actuation

Authors: Tingrui Liu
Journal: Shock and Vibration
 (2015)

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## Abstract

The aim of this article is to analyze classical flutter and active control of single-cell thin-walled composite wind turbine blade beam based on piezoelectric actuation. Effects of piezoelectric actuation for classical flutter suppression on wind turbine blade beam subjected to combined transverse shear deformation, warping restraint effect, and secondary warping are investigated. The extended Hamilton’s principle is used to set up the equations of motion, and the Galerkin method is applied to reduce the aeroelastic coupled equations into a state-space form. Active control is developed to enhance the vibrational behavior and dynamic response to classical aerodynamic excitation and stabilize structures that might be damaged in the absence of control. Active optimal control scheme based on linear quadratic Gaussian (LQG) controller is implemented. The research provides a way for rare study of classical flutter suppression and active control of wind turbine blade based on piezoelectric actuation.


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## Body

## 1. Introduction

Although instability of stall nonlinear flutter has generally been an important issue in flutter research [1], classical flutter was observed involving a high-speed rotating blade in pitch excitation process. Simultaneously, with the advent of large wind turbine fitted with relatively slender blades, classical flutter may become a more important design consideration. In addition, innovative blade designs involving the use of aeroelastic tailoring and structural tailoring, wherein the blade twists as it bends under the action of aerodynamic loads to shed load resulting from wind turbulence, may increase the blade’s proclivity for flutter [2].Most of the literature of classical flutter focused on the helicopter blades, noncoupling wing sections, and fan rotor blades [3–5]. As for active control, most of the literature focused on beam structures rather than the blade sectional shapes. Kapuria and Yasin [6] study the active vibration suppression of hybrid composite and fiber metal laminate plates integrated with piezoelectric fiber reinforced composite sensors and actuators. Phung-Van et al. [7] present an effective formulation based on higher-order shear deformation theory to investigate free vibration and dynamic control of piezoelectric composite plates integrated with sensors and actuators. Supersonic flutter control of a three-layered sandwich curved panel of rectangular plan form with an adaptive electrorheological fluid core layer is investigated by Hasheminejad and Motaaleghi [8].Park and Kim [9] study the active twist rotor blade incorporating single crystal macrofiber composite actuators and analyze the aeroelasticity. Although the rotor blade properties dynamically represent a real rotor blade, the analytical objects, are the torsional behavior of helicopter blades or the vibration behavior of airfoils in which only the torsional motions are involved. Simultaneously, the objects of the existing literature mostly concentrate on the frequency research rather than time response analysis.In this work the classical flutter and flutter suppression of composite blade beam are investigated for single-cell thin-walled structure with piezoelectric patch embedded. The validity of the piezoelectric actuation is tested and illustrated by time domain response analysis rather than frequency research. The analysis is applied to a laminated host structure of the circumferentially asymmetric stiffness (CAS) that produces bending-twist-transverse shear coupling. The governing system can be derived by the extended Hamilton principle. The spanwise distributed PZT-4 sensor/actuator pair is embedded into the orthotropic host. The net voltage output from sensor is fed to a controller for the purpose of actuation. For piezoelectric actuation, active feedback control law and linear quadratic Gaussian controller are implemented. The purpose of present study is to investigate the validity of piezoelectric actuation under extreme conditions especially in critical region.


## 2. Analytical Model and Equations of Motion

Consider the thin-walled structure in which piezoelectric patch is embedded as Figure1. The length L of the blade is along z direction. The origin of the rotating axis system is located at the rigid root in which the blade beam is mounted. Ω is the rotating speed; c is the chord length; h is the thickness of section; rn is the radius of curvature of the middle surface; ϕ is the twist angle of section; U is the wind velocity. It is assumed that c≪L, h≪c, and h≪rn. The equation of the middle line of the closed section is as follows [2]:(1a)y=eexp⁡0.051-cos⁡θφ2+0.05sin⁡θφ2+exp⁡-0.051-cos⁡θφ2-0.05sin⁡θφ2·cos⁡θφ,(1b)z=eexp⁡0.051-cos⁡θφ2+0.05sin⁡θφ2-exp⁡-0.051-cos⁡θφ2-0.05sin⁡θφ2·sin⁡θφ,where e is the 1/4 chord length and θφ changes from 0~2π.Figure 1
Coordinate system and aerodynamics and structure.The configuration of the piezoelectric actuator is assumed to be distributed over the entire blade span. The characteristics of the host composite structure and piezoelectric structure are depicted as (a) transverse shear, warping restraint effect, and secondary warping as depicted in [10]; (b) the master structure consists of 6-layer CAS configuration; the actuator is 1 piezoelectric layer, with the height bka and width ska along the circumferential s, spanwise z, and transverse n directions; the width of piezoelectric layer ska is relatively small, generally less than or equal to 1/6 chord length; (c) the piezoelectric elements may be employed concurrently for sensing and actuation and relate the voltage produced by the piezoelectric layer to the strain in the host structure; (d) the feedback control is achieved through the action of piezoelectrically induced vertical transverse shear motion at the blade tip.To simplify the analysis, a structural tailoring technology for symmetric thin-walled structure is applied [10]. Consider the cantilever vibration first, ignore rotation about y, and retain rotation about x according to the requirements of piezoelectric actuation, the displacements u, v, w, θx, and ϕ are displayed in Figure 1, and the system kinetic energy T and potential energy V are, respectively, expressed as(2a)T=12∫0L∮∑k=1N∫hkρku˙-yϕ˙2+v˙+xϕ˙2+W˙+yθ˙x-Fwϕ˙′-dxdsθ˙x+aϕ˙′2dndsdz,where the primary warping function Fw is depicted as(3)Fw=∫0srns-2Aeβds,where Ae denotes the section area bounded by the midline and β denotes the length of the contour midline. The radius of curvature of the middle surface is expressed as(4)rn=xdyds-ydxds,2bV=12·∫0L∮∑k=1N∫hkρkσzzkw′+yθx′-Fwϕ′′-ndxdsθx′+aϕ′′+σszku′dxds+v′+θxdyds+2Aeβϕ′+σnzku′dyds-v′+θxdxdsdndsdz,where σijk denotes the stress component and represents the three-dimensional constitutive equations for the actuator layers with strain components εij. It can be deduced as(5)σssσzzσsz=C11C120C12C11000C11-C122εssεzzεsz-e31ξ3RnRsRze31ξ3RnRsRz0,where Cij means elastic coefficient and the last terms identify the actuation stresses induced by the applied electric field. In addition, R is a spatial function expressed by Heaviside distribution [10]. The composite parameters and piezoelectric properties can be found in Table 1.Table 1
Composite parameters and piezoelectric properties.
Composite items
Values
Piezoelectric items
Values
Maximum exterior width
24.21 × 10−3 m
Thicknessta
1.9 × 10−4 m
Maximum exterior height
2.643 × 10−3 m
Piezoelectric coefficiente31
−2.05 × 102 Pa V
Blade densityρm
1672 kg/m3
Density
7.65 × 102 kg s2/m4
Ply layers
6
Electrical permittivityξ3p
1.2 × 10−8 F/m
Ply thickness
127 × 10−6 m
Elastic coefficientC11
1.39 × 1011 Pa
Ply layers
6
G
12/v12
3.5 GPa/0.34
Elastic coefficientC12
7.778 × 1010 Pa
E
11/E22
25.8 GPa/8.7 GPa
Piezoelectric widthska
c/6 mBased on the extended Hamilton principle, the governing system of cantilever vibration can be derived as follows:(6)∫t1t2δT-δV+δWdt=0,where δW is the virtual work of the nonconservative forces depicted as [10](7)δW=∫0LFuδu0+Fvδv0+Mφδϕ-Mxaδθx′+Bwaϕ′′dz,where Mxa and Bwa are the piezoelectrically induced terms expressed as(8)Mxa=∮ξ3bkae31Rs,z·y1-A12A11+dxdsB12A11-nkads,Bwa=∮ξ3bkae31Rs,z·Fw1-A12A11-aB12A11-nkads,where (9)nka=12c-bka-18h,a=-ysdyds-xsdxdsand ξ3 is the applied electric field on which the piezoelectrically induced moment depends related with patch area Ap. It is computed as(10)ξ3ξ3pAp=∬e31ysθ˙xz2-θ˙xz1dsdtand Aij and Bij denote local stretching and stretching-bending coupling rigidity quantities, respectively.It is complicated to directly deduce the equations of motion of the rotating section with piezoelectric fiber rein forced composite actuators. Hence another simplified method is applied here. Based on (6), the equations of motion of cantilever vibration can be obtained first. With the rotating centrifugal motion [11] and the CAS structural tailoring technology being considered [10], the equations of rotating motion with vertical bending-twist-transverse shear coupling can be deduced as(11a)b1v¨-a1v′′-a2ϕ′′′-a1θx′-12mΩ2v′′L2-z2+v′-2z=Fcos⁡ψ,(11b)b2ϕ¨-b3ϕ¨′′+a3ϕ′′′′-a5ϕ′′+a2v′′′+a4θx′′+Bwa′′-12mΩ2KA2L2-z2ϕ′′+mΩ2KA2zϕ′+mΩ2Km22-Km12ϕ=M+Fecos⁡ψ-φ,(11c)b0θ¨x-a6θx′′+a2-a0ϕ′′+a1v′+a1θx+Mxa′=0,where the related composite structural parameters aj and bj are displayed in appendix, and the classical aerodynamics of lift F and moment M can be expressed as [2](12)F=-ρacV02πα+v′,M=-ρac2V02πeα+v′,where(13)Ω=λUL,Km12=1m∬ρmx2dydx,Km22=1m∬ρmy2dydx,Km2=Km12+Km22,A0=∬dydx,A0KA2=∬y2+x2dydx.


## 3. Solution Methodology

### 3.1. Application of Galerkin Method

To analyze the aeroelastic system given by (11a)–(11c), the Galerkin method is used so as to reduce the aeroelastic coupled equations into the state-space form [11]. Firstly the representation of displacement functions is as follows:(14)Vz,t=VTzqvt,ϕz,t=ΦTzqφt,(15)θxz,t=STzqθt,where test functions are required to satisfy the kinematics and force boundary conditions of the cantilever blade and can be written as(16)VTz=v1,v2,v3,v4,…,vN,ΦTz=φ1,φ2,φ3,φ4,…,φN,STz=θ1,θ2,θ3,θ4,…,θN,vjz=cosh⁡βjz-cos⁡βjz+αjsinh⁡βjz-sin⁡βjz,φjz=2sin⁡γjz,θjz=2sin⁡γjz,γj=πj-12,βj=j-12π,αj=-cos⁡βj+cosh⁡βjsin⁡βj+sinh⁡βj.Secondly substituting (15) into (11a)–(11c), with Galerkin method applied, gives 3N matrix equations as follows:(17)MPNX¨+KPNX=QPN,where the state variable X=[qvN×1T,qφN×1T,qθN×1T]T, and the related coefficient matrices are(18)MPN=MPN1MPN2MPN3,MPN1=∫0Lb1VTzvjdz00,MPN2=0∫0Lb2ΦTz-b3ΦTz′′φjdz0,MPN3=00∫0Lb0STzθjdz,KPN=KPN1KPN2KPN3,KPN1=∫Qv+πρacV02VTz′cos⁡ψvjdz∫a2VTz′′′+πρaV02VTz′ecc+cos⁡ψφjdz∫a1VTz′θjdz,KPN2=-∫a2ΦTz′′′+πρacV02ΦTzcos⁡ψvjdz∫Qφ-πρaV02ΦTzecc+cos⁡ψφjdz∫a2-a0ΦTz′′θjdz,KPN3=-∫a1STz′vjdz∫PBwaSTz′′z=L+a4STz′′φjdz∫Qθθjdz,QPN=-∫πρacV02ψcos⁡ψvjdz-∫πρaV02ψecc+cos⁡ψφjdz0,where(19)Qv=-a1VTz′′-mΩ22L2-z2VTz′′+mΩ2zVTz′,Qφ=a3ΦTz′′′′-a5ΦTz′′+mΩ2kA2zΦTz′+mΩ2km22-km12·zΦTz-mΩ22kA2L2-z2ΦTz′′,Qθ=-a6STz′′+a1STz+STz′z=L·4e31bkataξ3taSTLqθ·∮y1-A12A11+dxdsB12A11-dydsnkads,PBwa=4e31bkataξ3taSTLqθ·∮Fw1-A12A11-aB12A11+ankads.Equation (17) is the linear decoupling equation governing the motion of the aeroelastic system, which is a matrix equation with 3N subequation structure.


### 3.2. Active Control

In active flutter suppression, it is essential to properly select actuators and properly use control method and mathematical algorithm. The feedback control is achieved through the piezoelectrically induced vertical transverse shear motion at the blade tip, considered in conjunction with the implementation of a combined feedback control law, when external voltage of opposite sign is applied in the upper and bottom piezoactuator layer. The applied electric fieldξ3 on which the piezoelectrically induced moment depends may be expressed through a prescribed linear functional relationship with the kinematical response quantities characterizing the blade’s response. The piezoelectrically induced bending moment Mxa intervenes solely in the boundary conditions prescribed at the blade tip and plays the role of a boundary moment control due to special distribution of piezoactuators [12]. A feedback control law is implemented with Mxa at the blade tip expressed as(20)MxaL,t=kvθ˙xL,t+kaθ¨xL,t,where kv and ka denote the velocity and acceleration feedback gains, respectively. So for active control with feedback law, system equation (17) is rewritten as(21)MPNX¨+CPNX˙+KPNX=QPN,where the new coefficient matrices are as follows:(22)MPN3=00∫0Lb0STz+KaSTz′z=Lθjdz,CPN=00000000∫KvSTz′z=Lθjdz.In order to determine the time domain response of the dynamic system of (21), a Runge-Kutta time-marching approach is applied. Using the method presented in [1], (21) will be expressed in state-space form. In general, upon defining the state vector Y=[XT,X˙T]T and adjoining the identity equation X˙=X˙, (21) can be converted to the state-space expression:(23)Y˙=AY+B,YO=CY+D,where 6N×6N state matrix A and 6N×1 matrix B are given by(24)A=0IE-MPN-1KPN-MPN-1CPN,B=0MPN-1QPN.Herein IE is the 3N unitary matrix, C is 6N unitary matrix, and D is 6N×1 zero matrix.In order to minimize settling time and required control energy, optimal control techniques are necessary for controller design. Since these sensitive structures often experience time-varying loads, their safe and effective design requires accurate response [13]. The linear quadratic regulator (LQR) is often used to analyze vibration control of rotating composite beam. An advantage of the quadratic optimal control method is that it provides a systematic way of computing the state feedback control gain matrix [14]. In [13], the Galerkin method, along with either instantaneous or classical LQR methods, is used to analyze vibration control of a rotating composite pretwisted single-celled box beam, exhibiting transverse shear flexibility and restrained warping.Here in order to suppress the too large initial vibration amplitude in LQR process and decrease the influence of measurement noise (might be produced by unsteady aerodynamics), the linear quadratic Gaussian (LQG) controller is used [4, 15]. It is assumed that the measurement noise and disturbance signals (process noise) are stochastic with known statistical properties and are hidden in system equation (23). The dynamic equation of LQG controller can be represented as(25)X˙C=ACXC+BCu,YC=CCXC,where(26)AC=A-BKR-KFC+KFDKR,BC=KF=XFCTR1-1,CC=-KR=-R+DTQD-1BTXR+DTQC,where Q and R are the weighting matrices of system outputs and control inputs, respectively; KF and KR are the gain matrices of the Kalman state estimator and the gain matrix of the optimal regulator, respectively; XF and XR are the positive finite solution of algebraic Riccati equations as follows:(27)AXF+XFAT-XFCTR1-1CXF+Q1=0,ATXR+XRA-CTQD+XRBR+DTQD-1·BTXR+DTQC+CTQC=0,where Q1 and R1 are the intensity matrices of the gust input and the Gaussian white noise process of measurement, respectively. For the LQG control law, the gain matrix KR can be determined firstly by choosing the weighting matrices as Q being 6N unitary matrix and R=0.001. Then the gain matrix KF of the Kalman state estimator is determined via Q1=1 and R1 being 6N unitary matrix.The LQG control problem is to find the optimal controlu(t) which minimizes(28)J=ElimT→∝⁡1T∫0TYTQY+uTRudtand the required solution to the LQG problem is then found by replacing Y by XC, to give the expression(29)ut=-KRXCt.


## 3.1. Application of Galerkin Method

To analyze the aeroelastic system given by (11a)–(11c), the Galerkin method is used so as to reduce the aeroelastic coupled equations into the state-space form [11]. Firstly the representation of displacement functions is as follows:(14)Vz,t=VTzqvt,ϕz,t=ΦTzqφt,(15)θxz,t=STzqθt,where test functions are required to satisfy the kinematics and force boundary conditions of the cantilever blade and can be written as(16)VTz=v1,v2,v3,v4,…,vN,ΦTz=φ1,φ2,φ3,φ4,…,φN,STz=θ1,θ2,θ3,θ4,…,θN,vjz=cosh⁡βjz-cos⁡βjz+αjsinh⁡βjz-sin⁡βjz,φjz=2sin⁡γjz,θjz=2sin⁡γjz,γj=πj-12,βj=j-12π,αj=-cos⁡βj+cosh⁡βjsin⁡βj+sinh⁡βj.Secondly substituting (15) into (11a)–(11c), with Galerkin method applied, gives 3N matrix equations as follows:(17)MPNX¨+KPNX=QPN,where the state variable X=[qvN×1T,qφN×1T,qθN×1T]T, and the related coefficient matrices are(18)MPN=MPN1MPN2MPN3,MPN1=∫0Lb1VTzvjdz00,MPN2=0∫0Lb2ΦTz-b3ΦTz′′φjdz0,MPN3=00∫0Lb0STzθjdz,KPN=KPN1KPN2KPN3,KPN1=∫Qv+πρacV02VTz′cos⁡ψvjdz∫a2VTz′′′+πρaV02VTz′ecc+cos⁡ψφjdz∫a1VTz′θjdz,KPN2=-∫a2ΦTz′′′+πρacV02ΦTzcos⁡ψvjdz∫Qφ-πρaV02ΦTzecc+cos⁡ψφjdz∫a2-a0ΦTz′′θjdz,KPN3=-∫a1STz′vjdz∫PBwaSTz′′z=L+a4STz′′φjdz∫Qθθjdz,QPN=-∫πρacV02ψcos⁡ψvjdz-∫πρaV02ψecc+cos⁡ψφjdz0,where(19)Qv=-a1VTz′′-mΩ22L2-z2VTz′′+mΩ2zVTz′,Qφ=a3ΦTz′′′′-a5ΦTz′′+mΩ2kA2zΦTz′+mΩ2km22-km12·zΦTz-mΩ22kA2L2-z2ΦTz′′,Qθ=-a6STz′′+a1STz+STz′z=L·4e31bkataξ3taSTLqθ·∮y1-A12A11+dxdsB12A11-dydsnkads,PBwa=4e31bkataξ3taSTLqθ·∮Fw1-A12A11-aB12A11+ankads.Equation (17) is the linear decoupling equation governing the motion of the aeroelastic system, which is a matrix equation with 3N subequation structure.


## 3.2. Active Control

In active flutter suppression, it is essential to properly select actuators and properly use control method and mathematical algorithm. The feedback control is achieved through the piezoelectrically induced vertical transverse shear motion at the blade tip, considered in conjunction with the implementation of a combined feedback control law, when external voltage of opposite sign is applied in the upper and bottom piezoactuator layer. The applied electric fieldξ3 on which the piezoelectrically induced moment depends may be expressed through a prescribed linear functional relationship with the kinematical response quantities characterizing the blade’s response. The piezoelectrically induced bending moment Mxa intervenes solely in the boundary conditions prescribed at the blade tip and plays the role of a boundary moment control due to special distribution of piezoactuators [12]. A feedback control law is implemented with Mxa at the blade tip expressed as(20)MxaL,t=kvθ˙xL,t+kaθ¨xL,t,where kv and ka denote the velocity and acceleration feedback gains, respectively. So for active control with feedback law, system equation (17) is rewritten as(21)MPNX¨+CPNX˙+KPNX=QPN,where the new coefficient matrices are as follows:(22)MPN3=00∫0Lb0STz+KaSTz′z=Lθjdz,CPN=00000000∫KvSTz′z=Lθjdz.In order to determine the time domain response of the dynamic system of (21), a Runge-Kutta time-marching approach is applied. Using the method presented in [1], (21) will be expressed in state-space form. In general, upon defining the state vector Y=[XT,X˙T]T and adjoining the identity equation X˙=X˙, (21) can be converted to the state-space expression:(23)Y˙=AY+B,YO=CY+D,where 6N×6N state matrix A and 6N×1 matrix B are given by(24)A=0IE-MPN-1KPN-MPN-1CPN,B=0MPN-1QPN.Herein IE is the 3N unitary matrix, C is 6N unitary matrix, and D is 6N×1 zero matrix.In order to minimize settling time and required control energy, optimal control techniques are necessary for controller design. Since these sensitive structures often experience time-varying loads, their safe and effective design requires accurate response [13]. The linear quadratic regulator (LQR) is often used to analyze vibration control of rotating composite beam. An advantage of the quadratic optimal control method is that it provides a systematic way of computing the state feedback control gain matrix [14]. In [13], the Galerkin method, along with either instantaneous or classical LQR methods, is used to analyze vibration control of a rotating composite pretwisted single-celled box beam, exhibiting transverse shear flexibility and restrained warping.Here in order to suppress the too large initial vibration amplitude in LQR process and decrease the influence of measurement noise (might be produced by unsteady aerodynamics), the linear quadratic Gaussian (LQG) controller is used [4, 15]. It is assumed that the measurement noise and disturbance signals (process noise) are stochastic with known statistical properties and are hidden in system equation (23). The dynamic equation of LQG controller can be represented as(25)X˙C=ACXC+BCu,YC=CCXC,where(26)AC=A-BKR-KFC+KFDKR,BC=KF=XFCTR1-1,CC=-KR=-R+DTQD-1BTXR+DTQC,where Q and R are the weighting matrices of system outputs and control inputs, respectively; KF and KR are the gain matrices of the Kalman state estimator and the gain matrix of the optimal regulator, respectively; XF and XR are the positive finite solution of algebraic Riccati equations as follows:(27)AXF+XFAT-XFCTR1-1CXF+Q1=0,ATXR+XRA-CTQD+XRBR+DTQD-1·BTXR+DTQC+CTQC=0,where Q1 and R1 are the intensity matrices of the gust input and the Gaussian white noise process of measurement, respectively. For the LQG control law, the gain matrix KR can be determined firstly by choosing the weighting matrices as Q being 6N unitary matrix and R=0.001. Then the gain matrix KF of the Kalman state estimator is determined via Q1=1 and R1 being 6N unitary matrix.The LQG control problem is to find the optimal controlu(t) which minimizes(28)J=ElimT→∝⁡1T∫0TYTQY+uTRudtand the required solution to the LQG problem is then found by replacing Y by XC, to give the expression(29)ut=-KRXCt.


## 4. Results and Discussion

To testify the validity of active flutter suppression, numerical results of time responses at blade tip for LQG controllers are presented, compared with related results of other cases. Generally, rotating blades have specific rotating speed, ply angle, and tip speed ratio for operation. Here basic testing parameters are ply angleθp=30°; tip speed ratio of wind turbine λ=3; blade length L=1.5 m.In general, the critical wind speed of a 2D section can be determined by v-g method [4] by eigenvalue analysis. However, the object of present research is a 3D entity with bending-twist-transverse shear coupling. After decoupling the number of subequations in (17) reaches 3N (here N≥5), with the number of eigenvalues being 6N. V-g method has lost the utility. However, we can use another approximate method by the eigenvalue analysis. For different wind speeds, the maximum value of all the eigenvalues of each wind speed is determined, and then the critical wind speed is determined from the fluctuation of the largest eigenvalues. Figure 2(a) displays maximum real parts of eigenvalues versus velocities from 20~45 m/s. It can be obviously demonstrated that the critical wind speed is U=34.69 m/s.Figure 2
Maximum real parts of eigenvalues versus velocities (a) and the responses for bothU=34.69 m/s and U=34.695 m/s under situation of without piezoelectric materials (b).
(a)
Maximum real parts of eigenvalues versus velocities
(b)
The responses of vertical bending(v), twist (ϕ), and transverse shear (θx) motions for both U=34.69 m/s and U=34.695 m/sFor flutter analysis, an approximate critical state range based onU=34.69 m/s~34.695 m/s is considered. When U<34.69 m/s, the system is convergent with smaller amplitude. When U>34.695 m/s, the system presents the divergent state with a large amplitude.In general, the damping ratio increases constantly as the feedback control gain increases. However feedback control gain cannot be increased infinitely because the applied voltage must be limited for the sake of breakdown voltage of actuators [16]. According to the requirements of blade structure and actual control hardware here, the fixed feedback gains of kv=1 and ka=1 are applied.Figure2(b) shows the time responses of vertical bending (v), twist (ϕ), and transverse shear (θx) motions under situation of without piezoelectric materials. Two critical wind speed values are considered at the same time; one is U=34.69 m/s (in blue mark) and the other one is U=34.695 m/s (in green mark).From the whole response trend in the case ofU=34.69 m/s, the three displacements seem to be convergent. In fact for the vertical bending displacement v, within 2 s time, the amplitude of the vibration quickly exceeds the length of the blade L, so actually the vertical bending (v) motion has been in the state of divergence, so do twist (ϕ) and transverse shear (θx) motions. The instability also can be testified by Imag.-Real plot of eigenvalues of homogeneous equation system in Figure 3(a), where some closed-loop poles lie in the right-half plane, resulting in an unstable system.Figure 3
Imag.-Real plot of homogeneous equation system forU=34.69 m/s defined in state-space.
(a)
Without piezoelectric materials
(b)
With gains ofkv=1 and ka=1, but without optimal controller
(c)
With LQG controllerAs for another case ofU=34.695 m/s in Figure 2(b), the three displacements rapidly diverge from the start with larger amplitude. These two cases in Figure 2(b) both can be regarded as the unstable states.Take the case ofU=34.69 m/s; for example, analysis and discussion for active control and flutter suppression will be carried out in the following in Figures 3~5.Figure3 shows Imag.-Real plot of homogeneous equation system (only concerning the characteristic matrices of A and AC) for U=34.69 m/s defined in state-space: (a) without piezoelectric materials; (b) with gains of kv=1 and ka=1, but without optimal controller; (c) with LQG controller. Figures 3(a)~3(b) show the system instability. Some closed-loop poles lie in the right-half plane, and others are in the left-half plane. It can be seen from the Imag.-Real plot of Figure 3(c) that the homogeneous equation system obtained by control of LQG controller is stable, with all the closed-loop poles being in the left-half plane.Figure4 shows comparisons of the three displacements responses for U=34.69 m/s with fixed feedback gains of kv=1 and ka=1, which are characterized by the state without optimal controller (in blue mark) and LQG controller (in red mark).Figure 4
The responses of the three displacements under condition of fixed feedback gains ofkv=1 and ka=1, which are characterized by the state without optimal controller (in blue mark) and LQG controller (in red mark) for U=34.69 m/s.
(a)
(b)
(c)Figure 5
The time responses of the three displacements for wind velocity ofU=38 m/s, which are characterized by the state without piezoelectric materials and actuation by LQG controller.
(a)
(b)
(c)It can be seen that piezoelectric actuation based on fixed feedback gains ofkv=1 and ka=1 and without optimal controller (blue mark in Figure 4) can greatly suppress the flutter, compared with the results in Figure 2(b). However, the results of these three displacements still maintain a larger vibration amplitude, and what is more, combined with the eigenvalue analysis in Figure 3(b), the system (with blue mark in Figure 4) might be conditionally stable. In practice, conditionally stable systems are not desirable. If system parameter assumes a value corresponding to unstable operation, the system may break down or may become nonlinear due to a saturation nonlinearity that may exist. Therefore, further active control based on theoretical algorithm, such as LQG algorithm, is necessary.As far as the trend of response is concerned, the three displacements based on LQG controller in Figure4 are rapidly convergent from the start. Figure 4 (red mark) also shows that the flutter amplitudes of all the three displacements decrease rapidly with the change of time and tend to be steady within 2 s. It shows obvious effect of LQG flutter suppression on aeroelastic instability.In contradistinction with the trends without feedback control occurring in Figure2(b) for U=34.69 m/s, the results displayed in Figure 4 reveal that LQG algorithm and active feedback control law have positive effects on classical flutter suppression. In order to verify the LQG law of universal, a further validation of LQG controller concerns the case of rapidly divergent state characterized by wind velocity of U=38 m/s without piezoelectric materials in Figure 5 (blue mark). From this figure it becomes apparent that when the wind speed is greater than the critical value, with increase of time (within 1 s), the magnitudes of the vibration amplitudes of the three displacements increase rapidly, even to unbelievable extent.The effect of LQG controller on flutter suppression is illustrated in Figure5 (green mark). The results show an expected conclusion, namely, that, for wind speed beyond the critical value, even in the adverse circumstances, the effect of LQG controller on flutter suppression is prominent. As a matter of fact, all the three responses under LQG controller are always stabilized to a constant value with smaller deviation when the time is extended beyond 2 s; meantime the three velocities at the blade tip are stabilized to zero within 2 s (see Figure 6), which verify the control performance for the state variables of (29).Figure 6
The three velocities at the blade tip.It should be stated that the purpose of present study is to investigate the validity of piezoelectric actuation under extreme conditions forU⩾34.69 m/s as mentioned in Figures 2(b)~6. Of course, in general, the LQG algorithm is effective. A further validation of LQG controller concerns that the cases of low velocity for U=10 m/s and U=5 m/s are displayed in Figure 7. The results show the same conclusion as Figure 5, namely, that, for wind speed in low area, the effect of LQG controller on flutter suppression is obvious. Simultaneously all the three responses under LQG controller are always stabilized to constant values when the time is extended beyond 3 s.Figure 7
The time responses of the three displacements for wind velocities ofU=10 m/s (a) and U=5 m/s (b), which are characterized by the state without piezoelectric materials and actuation by LQG controller.
(a)
U
=
10 m/s
(b)
U
=
5 m/sIn addition, the cost of piezoelectric materials is high with the mass of the system increasing as more actuator is used, with greater length and thickness being applied. Hence, adequate flutter suppression performance with LQG controller might be obtained through the proper arrangement and distributed size of sensor/actuator pair.


## 5. Conclusions

The purpose of present research is to investigate the validity of piezoelectric actuation under extreme conditions. An analytical study devoted to the mathematical modeling of single-cell thin-walled composite wind turbine blade beam is presented in the paper. Simultaneously classical flutter and active control based on piezoelectric actuation are investigated. The validity of the piezoelectric actuation is tested and illustrated by time domain response analysis. Since the related research on wind turbine is scarce, only numerical simulation technology is investigated here. The numerical illustrations reveal validation of the solution methodology and control algorithm used in the paper. Some concluding remarks can be drawn from the results as follows:(1)
Flutter suppression for classical flutter blade with vertical bending-twist-transverse shear coupling is investigated and discussed. Galerkin method is used to solve the coupling aeroelastic equations.(2)
Piezoelectric actuation is realized by active controller with feedback gains. It is obviously demonstrated that the LQG controller is robust and the effect on flutter suppression is apparent.(3)
It should be stated that the performance of LQG regulation depends on the choice of weighting matricesQ, R, and Q1, R1 which have no analytic solution. Hence in some sense, the LQG controller in fact is man-made.



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*Source: 292368-2015-07-29.xml*