Active Vibration Control of Wind Turbine Using Virtual TMD Algorithm Based on Aerodynamic-Structure-Servo Coupling Model

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Introduction
With the increasingly severe energy crisis and environmental pollution, renewable energies, especially wind energy, have attracted wide attention and become the key to the energy transition.Te global installed wind power capacity has reached 906 GW by the end of 2022, and the increment is 77.6 GW within the year 2022 [1].Te increment of the global installed wind power capacity will be expected to reach 143 GW by the end of 2030, which is 13% higher than previous forecasts [1].As the worldwide demand for wind power continues to increase, wind turbine hubs become taller and rotor diameters become larger to capture more wind energy and reduce the levelized cost of energy [2].However, this trend leads to a reduction in wind turbine natural frequencies and structural damping ratios, resulting in a signifcant increase in wind-induced vibrations.Excessive wind-induced vibrations not only afect the normal operation of wind turbines but also cause damage to wind turbines or even collapse.Dueñas-Osorio and Basu [3] indicated that the malfunctioning of acceleration-sensitive equipment in wind turbines might increase the annual failure rate during normal operating conditions.Colwell and Basu [4] showed that wind-induced vibrations may reduce the fatigue life of wind turbines.Chou and Tu [5] observed that wind turbines can cause structural damage and collapse under strong wind loads.From 2010 to 2022, there were 145 wind turbine accidents caused by structural failures worldwide, with a considerable proportion caused by windinduced vibrations [6].Tus, it is obvious that the vibration of wind turbines will not only adversely afect the production of electrical energy but also afect the structural safety.Terefore, it is necessary to reduce the vibration of wind turbines.
Diferent control strategies have been investigated to reduce the vibration of wind turbine towers and blades [7,8].Generally, there are three strategies for reducing the vibration of wind turbines: passive, semiactive, and active control.Passive control has been extensively studied on wind turbines over the past decades due to its simplicity of installation, convenient maintenance, low cost, and good stability.Studies have shown that a tuned mass damper system (TMD) can efectively suppress the vibration of wind turbine towers [9][10][11][12][13][14]. Zuo et al. [15] proposed multiple tuned mass dampers (MTMDs) to reduce the vibration of ofshore wind turbine towers under parked conditions.Te results showed that MTMDs can efectively control vibrations from the fundamental and higher modes of ofshore wind turbine tower under the multihazards of the wind, wave, and earthquake.Many scholars have also proposed tuned mass-damper-inerter (TMDI) systems to suppress the vibration of wind turbines [16][17][18].Teir results have shown that the TMDI can reduce the structural response with a smaller mass ratio and stroke than traditional TMD.Chen and Georgakis [19] proposed a tuned rolling ball damper to control the bidirectional vibration of wind turbine.Te experimental results demonstrated that this damper can efectively reduce the vibration of the wind turbine.Sun and Jahangiri [20] utilized a three-dimensional pendulum-tuned damper (3d-PTMD) to reduce the bidirectional vibration of wind turbine under the multihazards of the wind, wave, and earthquake.Teir fndings indicated that the 3d-PTMD can suppress the vibration of the wind turbine with a smaller stroke and a better control efect than dual linear passive TMDs.Mensah and Dueñas-Osorio [21] found that tuned liquid column dampers (TLCDs) can suppress the vibration of wind turbines, which can efectively reduce the vulnerability and improve the reliability of wind turbine towers.Although the passive control can efectively suppress the vibration of wind turbines under certain conditions, it may lose its efectiveness or even aggravate the structural vibration due to the complex environments and the timevarying characteristics of wind turbines.It is well known that wind turbine is a time-varying system.Ghassempour et al. [13] found that the optimal tuning frequency of TMD should vary with the wind speed to achieve the best vibration reduction efect.Consequently, the passive control based on conventional design may not be that appropriate for reducing wind-induced vibrations of wind turbines under various conditions.
Compared with passive control, the semiactive and active control can adjust their parameters in real time.Terefore, they are more suitable for wind turbine systems, especially under operational conditions.Hence, scholars have studied the semiactive and active control of wind turbines.For the semiactive control method, Chen et al. [22] developed a magnetorheological (MR) damper using a semiactive fuzzy control strategy, which can efectively suppress the in-plane vibration of wind turbine blades under extreme wind loads.Sun [23] and Hemmati and Oterkus [24] investigated the vibration reduction efect of the semiactive variable stifness TMD on ofshore monopile wind turbines subjected to the wind, wave, and earthquake.Te authors found that the semiactive TMD adjusted in real time not only had more efectiveness in controlling the structural response but also had a smaller stroke than passive TMD.For the active control of wind turbines, ATMDs have been commonly used [8].Diferent control algorithms have been proposed such as H infnity [25,26], static state feedback [27][28][29], and adaptive sliding-mode [30].In their studies, Lackner and Rotea [25] and Brodersen et al. [27] investigated the responses of ofshore monopile wind turbine without dampers and with ATMD and TMD.Trough frequency domain and time-domain analysis, it was shown that the vibration reduction efect of the ATMD was better than that of the TMD with the same mass ratio.Ten, Li and Gao [26] proposed the generalized H infnity optimal approach to derive the control force for the ATMD.Teir fndings indicated that active control is more efective in reducing the primary fatigue load than passive control.Fitzgerald et al. [28,29] explored the efectiveness of ATMD based on optimal LQR control in mitigating the responses of wind turbines.It was found that the active control strategy used achieved greater response reductions than the passive TMD.Hu et al. [30] proposed an adaptive sliding-mode controller to reduce the load of wind turbines.Te fnding indicated that the proposed control law improves the overall performance of the wind turbine under the consideration of the system uncertainties.Te above fndings show that semiactive and active control can adjust their own parameters in real time according to the time-varying characteristics of the structure, which can signifcantly reduce the vibration of wind turbines and have good robustness.
Although much research has been conducted to reduce the vibration of wind turbines using diferent control strategies, limitations in these strategies still exist.For operating wind turbines, the tower sufers the periodic (3P, 6P, etc.) loadings caused by the rotor rotation, which is diferent from the traditional high-rise structure [31].Te noticeable 3P frequency can be observed in the response of the wind turbine tower [32].However, the principle of the vibration mitigation in the most aforementioned literature is to achieve vibration control by changing the transfer function of the structure.Te response of the wind turbine at the natural frequency can be signifcantly reduced, while the control efect of the response at far away from the natural 2 Structural Control and Health Monitoring frequency caused by the periodic loadings is not obvious.Perhaps the LQR controller for ATMD [28,29] can efectively reduce the responses of the wind turbine, but there are still several limitations existing.Firstly, it is challenging to control two widely separated frequencies simultaneously using the LQR controller due to the limited efective bandwidth of the controller.Secondly, the LQR controller requires full states of wind turbines, and the obtainment of these states is a challenge for the measurement.
On the other hand, the wind turbine is a complicated dynamic system including the coupling efects among the aerodynamics, tower-nacelle-blade, and servo system.To accurately evaluate the system behavior, diferent models have been developed by scholars over the decades.Finite element models have been commonly adopted to study the wind-induced responses of the tower in structural analysis [33,34].Ten, Jonkman and Marshall [35] developed a timedomain simulation tool called FAST in 2006, which utilizes Kane's equations to analyze the dynamic responses of wind turbines in depth.Te accurate prediction of the wind turbine response is a prerequisite for vibration control research.Zhang [36] developed a reduced-order coupled model for wind turbines with 13 degrees of freedom using the Euler-Lagrangian together with nonlinear aerodynamics, which served as a basis for extensive research on passive and active vibration control.Zuo et al. [8] emphasized the importance of considering the infuences of the coupling efects and the interactions among the components of wind turbines in vibration control research.Terefore, a fully coupled wind turbine model should be adopted in vibration control research to study the efectiveness of the proposed control method more accurately.
To address the 3P vibration problem of wind turbines with limited state measurement information, an active control method based on the virtual TMD is proposed and validated with an aerodynamic-structure-servo coupling model in this paper.Te virtual TMD algorithm has a wide efective bandwidth and only requires the acceleration information on the top of the tower or the relative displacement of the active TMD.Firstly, an aero-structureservo coupling model (ASSC model) will be established in MATLAB/Simulink for dynamic analysis of the wind turbine.Secondly, the accuracy of the ASSC model will be verifed by comparative analysis with the wind turbine analysis package FAST (fatigue, aerodynamics, structures, and turbulence).Tirdly, an ATMD active controller is designed by the virtual TMD algorithm to suppress the vibration in the fore-aft (FA) direction of the wind turbine tower.Finally, compared with passive TMD and the traditional LQR algorithm, the established model is utilized to evaluate the vibration reduction efect of the virtual TMD algorithm on the wind turbine under diferent conditions.

The Holistic Coupled Model
In this section, a holistic coupled model of an aero-structureservo system is established, including the aerodynamic module, structural module, and servo system module, which is named as the ASSC model in the paper.Te aerodynamic module is mainly adopted to determine the aerodynamic loading of the structure by using wind feld information and the responses of the wind turbine.Te structural module is used to obtain the responses of the wind turbine according to aerodynamic loadings and commands of the servo system.Te servo system module is generally utilized to control the rotating speed and pitch angle of the blade based on the wind speed and dynamic response characteristics.
where T and V are the system kinetic and potential energy, q(t) is the generalized coordinates vector, Q i (t) is the generalized force corresponding to the ith component of q(t), and dot (•) represents the frst derivative with respect to time.
To simplify the complex physical model of the wind turbine, the following assumptions are given as follows: (1) A wind turbine consists of blades, hub, nacelle, and tower.Te blade and tower are modeled as elastic cantilevers, the hub and nacelle are assumed to be the lumped mass, and a rigid connection is assumed among the rotor, nacelle, and tower [35,37,38].(2) It is assumed that the bottom of the tower is fxed to the ground.Te nacelle is fxed on the top of the tower, while there is a certain distance from the top of the tower to the center of mass of the nacelle [37].
Te rotor is composed of three blades, which are connected with a nacelle through the hub.Te nacelle yaw is neglected in the model [12].(3) Te response of the tower and blades is dominated by the contribution of the fundamental mode, neglecting the infuence of the higher modes [20,28].

Coordinate System of the Structure Model.
Te wind turbine is a rotating system composed of multiple components.In order to determine the distribution of load, displacement, velocity, and other vectors in time and space when establishing equations, it is necessary to build multiple coordinate systems, which is convenient for the conversion among wind turbine components.Te simplifed model is illustrated in Figure 1(a), which is described by four coordinate systems, as shown in Figure 1(b).First, the global coordinate system C 0 is established at the bottom of the tower.Te tower-top coordinate system C 1 is nonrotating and located at the center of the tower top.Te hub coordinate system C 2 is fxed on the rotating shaft.Te blade section coordinate system C 3 is aligned with one of the blades.As the rotor rotates, the angle between the y b -z b and y h -z h planes is the azimuth angle.

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According to Reference [39], the transformation from tower-top system C 1 to global system C 0 is given as follows: where h is the height of the tower.Te transformation from hub system C 2 to tower-top system C 1 is given as follows: where θ s represents the tilted angle of the shaft, x hub is the distance between the hub center and z-axis, and z hub is the vertical distance between the hub center and the tower-top base plate.Te transformation from blade system C 3 to hub system C 2 is given as follows: where the azimuth angle φ is the angle between y h and y b , R hub and r denote the radius of hub and the distance from blade cross-section to the blade root, respectively.Te motion of the wind turbine can be simplifed by 8 degrees of freedom.Figure 2 illustrates the generalized coordinates of the tower-top base plate and blades.In Figure 2, parameters q ed1 ∼ q ed3 and q fp1 ∼ q fp3 denote the edgewise and fapwise tip displacement of the three blades, respectively.Parameters q fa and q ss denote the tower-top base plate displacement in the fore-aft (FA) and side-side (SS) directions.Te detailed information is provided in Appendix A. Hence, the 8 degrees of freedom of a wind turbine can be defned by q � q 1 • • • q 8   Τ � q ed1 q ed2 q ed3 q fp1 q fp2 q fp3 q fa q ss   Τ .
(5)   Structural Control and Health Monitoring By defning the blade rotating speed Ω(t), the azimuth angle of the ith blade can be given as follows: 2.1.2.Tower Modeling.As described above, the absolute tower displacement can be expressed as follows: where ϕ fa (z) and ϕ ss (z) are the fundamental mode shape of the tower in the fore-aft and side-side directions, respectively.
In terms of equation ( 7), the kinetic energy of the tower is where M(z) is the mass density per unit length of the tower.
In terms of Reference [37], the potential energy of the tower associated with the distributed stifness and gravity is obtained by where EI t (z) and w(z, t) denote the distributed stifness and the axial defection at the height z of the tower, respectively.g is the acceleration of gravity.w(z, t) can be expressed as follows: where z ′ denotes the dummy variable along the tower height.

Nacelle and Hub
Modeling.With the small defection assumption of the tower and blades [37], the tower-top rotations θ fa and θ ss in both the fore-aft and side-side directions can be approximated: Considering the additional displacement due to the tower-top rotations, the absolute displacement of the nacelle can be expressed as follows: where z nac is the vertical distance between the nacelle gravity center and the tower-top base plate, x nac is the distance between the nacelle gravity center and z-axis.Same as above, the absolute displacement of the hub can be expressed as follows: Te total kinetic energy of the nacelle and hub is Blade 1 Blade 2 Blade 3

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Te total potential energy of the nacelle and hub is where M n and M h denote the mass of the nacelle and hub, respectively.
2.1.4.Blade Modeling.As shown in Figure 2, parameters u ie (r, t) and u if (r, t) denote the edgewise and fapwise defections of blade cross-section at a distance r from the blade root, following as where ϕ 1e (r) and ϕ 1f (r) represent the edgewise and fapwise fundamental mode shape, respectively.At a distance r from the blade root, the displacement of blade can be expressed as follows: Ten, the displacement of blade in the global coordinate system C 0 can be given as follows: where θ → is the tower-top rotation due to the bending of the tower, TH ��→ denotes the vector between the origins of coordinate system C 1 and C 2 , and HB ��→ represents the vector between the origins of coordinate system C 2 and C 3 .Tey can be expressed as follows: Te kinetic energy of the blades can be expressed as follows: where m(r) is the mass density per unit length of the blades, and R denotes the distance from the blade tip to the root.Te potential energy is obtained by considering the strain energy of the blades due to the bending, the additional strain energy owing to the gravity, and the centrifugal stifening efect.In terms of Reference [39], the total potential energy of blades can be obtained by where EI eg (r) and EI fp (r) denote the edgewise bending stifness and the fapwise bending stifness, respectively.r ′ represents the dummy variable along the blade length.

Te Wind Turbulence Model.
Wind speed can be expressed as the combination of the mean and fuctuating components, v(z, t) � v(z) +  v(z, t), the exponential wind profle is adopted to calculate the mean speed: where v Hub represents the mean wind speed at the height of the hub, z Hub and α 0 denote the height of the hub and wind shear exponent, respectively.

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Te fuctuating component is computed using the IEC Kaimal spectral model [32].Te spatial correlation between two points in space is determined by the coherent function in IEC61400-1 [40].
Based on the above theory, a three-dimensional wind feld profle covering the domain of the rotor disk and tower is generated using the TurbSim [41] program.Te wind feld is sampled in real time, and bilinear interpolation is used to obtain wind speeds on the blades and tower using the developed MATLAB code.

Aerodynamic Loading.
Te aerodynamic loading of the wind turbine consists of the rotating blades and the tower.Te aerodynamic loading of the tower is calculated by quasi-steady theory [42].Te aerodynamic loading per unit length of tower can be expressed as follows: where ρ is the air density, and C dt and d t (z) denote the drag coefcient and the outside diameter of the tower section, respectively.Under F t (z, t), the virtual work done by external wind load is Te aerodynamic loading acting on the rotating blades can be calculated by applying the blade element momentum theory [31], eddy current theory [31], and computational fuid dynamics [43].In the paper, the blade element momentum (BEM) theory is used to estimate the aerodynamic loading, neglecting unsteady indicial variation and the dynamics of vorticity shed.Te BEM is a combination of momentum theory and blade element theory.It is assumed that there is no radial interaction, which means that the aerodynamic force at one element cannot be infuenced by the others.It is assumed that the forces on a blade element can be calculated by means of two-dimensional airfoil characteristics under an angle of attack, ignoring three-dimensional efects.Considering the aeroelasticity of the wind turbine, the blade velocities are returned to the BEM theory so that the aerodynamic loads are estimated based on the relative velocities.Terefore, the time series of the aerodynamic loading can be calculated from the blade characteristics, blade velocity, and wind speed.Te blade characteristics consist of the number of blades, pretwist, chord distribution, airfoils, and pitch angle.Figure 3 illustrates the blade element of wind turbines for BEM analysis.
Figure 4 shows all the velocities and forces relative to the blade chord line at radius r from the blade root.From Figure 4, the resultant relative velocity at the blade is where vb represents the wind speed in the blade coordinate system, _ u bxi and _ u byi denote the velocity components of the blade cross-section at a distance r from the blade root in the blade coordinate system.a and a ′ are the axial and tangential induction factors, respectively.In general, a and a ′ are unknown and obtained by iterations.
Te fow angle ψ to the plane of rotation can be calculated by Te angle of attack α is given by where β is the pitch angle, and c is the twist angle which is determined by the airfoil.
In terms of the attack angle, the lift and drag coefcients C l and C d can be looked up by the airfoil data which is obtained by experiments [38].On a span-wise length of each blade, the lift force is normal to the direction of the resultant relative velocity, and the drag force which is parallel to the resultant relative velocity can be expressed as follows: where c is the chord length.Te normal forces P N and the tangential forces P T can be computed as follows: Te aerodynamic loadings acting on the rotating blades can be calculated by equations ( 25)-( 29) via BEM.In order to obtain better results, it is necessary to apply two corrections to the BEM.Both Prandtl's tip loss factor and Glauert correction are considered in this study, and the Structural Control and Health Monitoring related theories can be found in literature [39].Ten the virtual work done by P N and P T can be computed by: Hence, the virtual work done by aerodynamic loadings can be expressed as follows: In terms of the principles of work and energy, the generalized force can be determined as follows: Substituting expression of the equation ( 31) into equation (32), the generalized force can be provided by 2.3.Te Servo System.Te servo system of the wind turbine includes a generator-torque controller and a full-span rotorcollective blade-pitch controller [38].Te fowchart of the whole servo control system is shown in Figure 5.According to Reference [38], the dynamic model of the drivetrain can be obtained as follows: Te generator torque versus generator speed response curve is adopted by the generator-torque controller.With proportionalintegral (PI) control, the control law of pitch angle is In terms of the kinetic energy and potential energy of wind turbines, the generalized force, the servo system, and the Euler-Lagrange equation, the motion of wind turbine can be computed as follows: where the structural damping of the blade and tower adopts Rayleigh damping, which is directly added to the C matrix.Te detailed information on matrices is provided in Appendix B.
Rotor plane Based on the aerodynamic module, structure module, and servo system module, the holistic coupled ASSC model is established on MATLAB/Simulink, which can be seen in Figure 6.Also, Table 1 shows the detailed information of the developed ASSC model.

Verification and Analysis of the ASSC Model
First, the natural frequencies of the 5 MW onshore wind turbine structure are compared by using the ASSC model and FAST.Ten, the dynamic responses of wind turbine under the ASSC model and FAST model are analyzed, and the accuracy and efectiveness of the ASSC model are evaluated.Te vibration at the tower top in the fore-aft direction is the largest for the tower, and its control is focused.Consequently, it mainly verifes and analyzes the foreaft response of the wind turbine.As shown in Table 2, the 5 MW onshore wind turbine given by NREL is summarized.

Comparison of Natural
Frequencies.Te natural frequencies of the wind turbine in FAST are determined by the linearization module at the initial moment.Te frequencies of the ASSC model are also obtained at the same time by the eigenvalue analysis of equation (36).Ten, the results of the ASSC model and FAST are compared in Table 3.It can be seen from the table that the ASSC model can predict the natural frequencies of wind turbine well, with an error of less than 3%.

Comparison of Responses in Turbulent Winds.
Te dynamic responses of the ASSC and FAST models under turbulent winds are analyzed and compared.Tree typical conditions are considered in this paper: Case 1: the mean wind speed is 30 m/s at parked condition; Case 2: the mean wind speed is 8 m/s at operational condition; and Case 3: the mean wind speed is 16 m/s at operational condition.Te turbulence intensities (TI) for these conditions are 0.18.
Te time histories and power spectra under these conditions are shown in Figures 7-9.It can be seen that the structural responses of the ASSC model and FAST are in good agreement both in the time and frequency domain.Te tower response is dominated by the fundamental frequency (0.322 Hz) at parked condition, as shown in Figure 7. Figures 8 and 9 show the tower top responses with the wind speed of 8 m/s and 16 m/s at operational conditions, respectively.Te following observations can be found.Te tower fundamental frequency, 3P frequency, and 6P frequency can be observed in tower top FA responses.Note that the 3P frequency is three times the rotor rotation frequency.At operational condition, the tower displacement is dominated by the tower fundamental frequency, while the acceleration is dominated by the 3P frequency.Compared with the parked condition, the tower fundamental frequencies at wind speeds of 8 m/s and 16 m/s have changed to 0.337 Hz and 0.347 Hz, respectively.It is indicated in Figures 8(c) and 8(d) that the 3P frequency is not constant and exists in a wide frequency band, which is due to the fact that the rotor speed is not constant.However, as shown in Figures 9(c) and 9(d), the 3P frequency is constant at 0.61 Hz, which is due to the fact that the blade-pitch angle changes in real time to keep the rotational speed constant.
Clearly, during normal operational conditions, the aerodynamic loads and structural characteristics of wind turbines will change at diferent wind speeds due to the existence of the servo control system.As a result, it can be concluded that the developed ASSC model has good precision and accuracy at both parked and operational conditions, which can be used to calculate the structural responses of the wind turbine.

Wind Turbine-ATMD Control Model.
From Figure 10, it is seen that ATMD is installed at the top of the tower to reduce tower vibration in the fore-aft direction.Compared with the 8 degrees of freedom of the uncontrolled model, the displacement and speed of ATMD need to be considered.Terefore, the wind turbine-ATMD controlled model has nine degrees of freedom.On the basis of the developed ASSC model, the motion equation of the wind turbine-ATMD controlled model needs to be rededuced.Te new equation of motion is as follows: where M 1 , K 1 , C 1 , and Q 1 are the derived mass, stifness, damping, and load matrices after adding ATMD, the detailed information on matrices are provided in Appendix C, Structural Control and Health Monitoring respectively.x ATMD (t) is the relative displacement of ATMD device to the top of the tower.U is the generalized active control force vector of ATMD, expressed as follows: where the control infuence matrix, B is given by where u a is the active control force of ATMD.
Te new equation of motion is rewritten into a statespace equation and the new state variable is Te new state space equation is where,

Active Control Algorithm Based on the Virtual TMD
System.Te proposed active control algorithm based on the virtual TMD system is presented in this subsection.As shown in Section 3, the vibration of a wind turbine tower is dominated by the tower fundamental frequency and 3P frequency at operational condition.Terefore, a control scheme with a wide efective bandwidth is required to achieve the vibration reduction of wind turbine.When the TMD mass ratio is large enough (e.g., 15%∼30%), it is expected to meet the requirements of wide-band vibration reduction.With this inspiration, the performance and mechanism of vibration reduction for the TMD with a mass ratio of 20% are analyzed frstly.Ten, the control force of the TMD with a large mass ratio is applied to wind turbines by an ATMD with a mass ratio of 2% to achieve the same reduction efect as the TMD with a large mass ratio.Finally, the parameters of virtual TMD are optimized to improve the control efect of ATMD.

Control Efect of TMD with Large Mass Ratio and
Damping Ratio. Figure 11 shows the power spectral density (PSD) function of tower top acceleration at operational condition and frequency response functions under uncontrolled and three controlled cases.Tese controlled conditions are: Case 1, the mass ratio of the TMD is 2%, the damping ratio is 9%, and the tuning frequency is the fundamental frequency; Case 2, the mass ratio of the TMD is 2%, the damping ratio is 9%, and the tuning frequency is the 3P frequency; Case 3, the mass ratio of the TMD is 20%, the damping ratio is 25%, and the tuning frequency is the 3P frequency.It is seen that the tower-top acceleration response is dominated by the 3P frequency.Te fundamental tower frequency is also observed in tower top acceleration.For Case 1, the response amplitude at the fundamental frequency is signifcantly reduced, and that at 3P almost does not change, compared with those of the uncontrolled case.For Case 2, although the tuning frequency of TMD changes  to the 3P frequency, the amplitude at the 3P frequency does not change obviously compared with that of the uncontrolled case.It indicates that the response amplitude at the frequency far away from the natural frequency cannot be reduced by the traditional passive control like TMD.However, for Case 3, the amplitudes both at the fundamental frequency and 3P frequency have been signifcantly reduced by TMD with a large mass and damping ratio.

Te Virtual TMD Algorithm for the ATMD Control.
From the above, it indicates that TMD with a large mass ratio and damping ratio can signifcantly reduce the response of wind turbine.Although this type of TMD cannot be realized in the wind turbine, the concept of the virtual TMD with a large mass ratio and damping ratio can be used here.Te active control force of the ATMD with 2% mass ratio can be obtained by the virtual TMD with a large mass ratio and damping ratio.Ten, the ATMD can achieve the same control efect as the virtual TMD system.On this basis, an active control algorithm based on virtual TMD is proposed to mitigate tower vibration in the fore-aft direction.
In the proposed control algorithm, the motion equation of the virtual TMD system can be given as follows: where € x s and € x vTMD represent the acceleration of the structure and the virtual TMD, respectively.k vTMD and c vTMD denote the stifness and damping of the virtual TMD, respectively.
Ten, the control force of the virtual TMD system can be expressed as follows: Te ATMD system can be used to simulate the control force of the virtual TMD system.Ten, the motion of the ATMD device can be expressed as follows: where k TMD and c TMD denote the stifness and damping of the TMD, respectively.Also, € x TMD represents the acceleration of the TMD.
Te ATMD feedback control force is determined by the virtual TMD system.Te active control force of ATMD can be obtained by equations ( 42)- (45).Ten, the responses of the wind turbine at the 3P frequency can be signifcantly reduced.Figure 12 illustrates the diagram of Simulink for the virtual TMD algorithm.Te algorithm has good robustness and broadband characteristics due to the large mass ratio of the 12 Structural Control and Health Monitoring virtual TMD.Furthermore, unlike the LQR control algorithm, the proposed control algorithm does not require the full states.Te acquisition of sole requirement is to acquire either the acceleration information on the top of the tower or the relative displacement of the active TMD.Tis information can be conveniently obtained.Once the acceleration information is available, the acceleration € x TMD of the active TMD can be determined by solving equations ( 42)- (44).Integrating the acceleration € x TMD allows determination of the displacement x TMD and velocity _ x TMD of the active TMD.Ten the active control force of the ATMD can be calculated by substituting x TMD , _ x TMD and € x TMD into equation (45).Similarly, knowing the displacement of the active TMD enables obtaining the active control force of the ATMD by solving equations ( 42)- (45).Considering the impact of noise on the signal, we recommend obtaining the relative displacement information of the active TMD. Figure 13 illustrates the comparison of results between the pure virtual passive TMD and the ATMD using the virtual TMD algorithm under the same parameters.It can be seen from Figures 13(a) and 13(b) that the acceleration responses of the wind turbine and the control force of the two are almost the same.It indicates that the ATMD utilizing the virtual TMD algorithm can accurately replicates the control force of the pure virtual passive TMD and achieve the same control efect.Structural Control and Health Monitoring 4.2.3.Optimal Design for the Virtual TMD Algorithm.In this paper, the proposed control algorithm is adopted to realize the vibration control of the wind turbines at parked and operational conditions.At parked conditions, the optimal frequency and damping ratio of the virtual TMD system can be obtained in the literature [44].For operational conditions, a numerical search is adopted to obtain the optimal design.Te acceleration RMS is set as the objective function.Te optimal objective is to minimize the RMS of the tower-top acceleration.
Te frequency ratio is defned as the ratio of the tuned frequency to the 3P frequency.With a given mass ratio, the value of the objective function corresponding to diferent frequency and damping ratio values are calculated.Figure 14 shows the value of the objective function against frequency and damping ratio, where the mass ratio is 30%.In Figure 14

14
Structural Control and Health Monitoring the frequency and damping ratio that makes the minimum value of the objective function is considered as the optimal design.Based on the optimal design, the efectiveness of ATMD will be evaluated in the following section under the constraints of stroke limitation and control force.

Vibration Control under Parked and Operational
Conditions.Te efectiveness of the proposed virtual TMD algorithm in controlling the along-wind vibration of the wind turbine is evaluated in this subsection.Te TMD and traditional LQR algorithms [29] are adopted for comparison.
In terms of the literature [44], the tuning frequency of TMD is set as the tower fundamental frequency.Also, the TMD mass is selected as percentage of the total mass.Tree diferent mass ratios are selected, 1%, 2%, and 5%, as in similar studies [10,13].Te vibration control efect of wind turbines will be assessed under parked conditions and operational conditions: v Hub � 16 m/s at parked maintenance condition (PLC1); v Hub � 30 m/s at normal parked condition (PLC2); v Hub � 8 m/s at operational condition (NLC1); and v Hub � 16 m/s at operational condition (NLC2).Te turbulence intensities for these conditions are 0.18. 10 wind feld samples are generated by TurbSim [41] for each condition, and the tower top accelerations of the wind turbine are calculated by equations ( 36) and (41).

Parked Conditions
(1) PLC1.Under this condition, Figure 15 illustrates the tower top accelerations under diferent control conditions in wind feld sample 1.It can be seen from Figure 15(a) that TMD, LQR, and virtual TMD have good vibration control efects.Among them, TMD, LQR, and virtual TMD can reduce the RMS response by 40.56%, 62.77%, and 54.54%, and the maximum response by 29.22%, 55.71%, and 35.92%, respectively.From Figure 15(b), it is seen that the LQR and virtual TMD outperform TMD in mitigating the response at the tower fundamental frequency.Te peak value of the TMD controlled wind turbine at the tower fundamental frequency is about 1/9 of the uncontrolled wind turbine, while the peak value of the LQR and virtual TMD is approximately 1/4 of the TMD. Figure 16 shows the stroke and control force of the dampers.It is seen that the stroke of LQR and virtual TMD is within 0.4 m, and the maximum control force is 13 kN.
(2) PLC2.Under this condition, Figure 17 illustrates the tower-top accelerations under diferent control conditions in wind feld sample 1.It can be seen from Figure 17(a) that TMD, LQR, and virtual TMD have good vibration control efects.Among them, TMD, LQR, and virtual TMD can reduce the RMS response by 28.96%, 56.66%, and 44.97%, and the maximum response by 19.46%, 49.76%, and 39.75%, respectively.From Figure 17(b), it is seen that LQR and virtual TMD outperform TMD in mitigating the response at the tower fundamental frequency.Te peak value of the TMD controlled wind turbine at the tower fundamental frequency is about 1/8 of the uncontrolled wind turbine, while the peak value of the LQR and virtual TMD is approximately 1/3 of the TMD. Figure 18 shows the stroke and control force of the dampers.It is seen that the stroke of LQR and virtual TMD is within 2.0 m, and the maximum control force is 60 kN. Figure 19 is the mean square response of the wind turbine acceleration in 10 wind feld samples under PLC1 and PLC2 conditions.It is demonstrated that the LQR has a better efect than the virtual TMD in mitigating windinduced vibration at parked conditions.However, the LQR algorithm, as the optimal feedback control, needs to obtain the full states of the wind turbine.On the contrary, only the acceleration information on the top of the tower or the relative displacement of the active TMD is required for the virtual TMD algorithm.Tis not only makes the proposed virtual TMD algorithm easier to implement but also reduces the cost of installing sensors.

Operational Conditions
(1) NLC1.Under this condition, Figure 20 illustrates the tower-top accelerations under diferent control conditions in wind feld sample 1.It can be seen from Figure 20(a) that Structural Control and Health Monitoring the LQR and virtual TMD still have a good vibration reduction control efect, while TMD has less control efect.Among them, TMD, LQR, and virtual TMD can reduce the RMS response by 5.96%, 33.39%, and 45.11%, and the maximum response by 7.14%, 46.20%, and 54.17%, respectively.It is found that the virtual TMD outperforms LQR in mitigating the tower-top acceleration.Te RMS response reduction can be improved by 11.72% using the   Structural Control and Health Monitoring virtual TMD.From Figure 20(b), it is seen that TMD cannot reduce the tower fundamental frequency and 3P frequency components.Te reason is that the tower-top acceleration response at operational condition is dominated by the 3P frequency.In this case, the tower fundamental frequency is not the optimal tuning frequency of TMD.However, LQR and virtual TMD have a good control efect and can signifcantly reduce the tower fundamental frequency and 3P frequency components.Figure 21 shows the stroke and control force of the dampers.It can be seen that the virtual TMD requires a smaller stroke and control force than the LQR.Also, the stroke of the virtual TMD is within 0.5 m, and the maximum control force is 30 kN.
(2) NLC2.Under this condition, Figure 22 illustrates the tower top accelerations under diferent control conditions in wind feld sample 1.It can be seen from Figure 22(a) that the virtual TMD still has a good vibration reduction control efect, while LQR has less control efect and TMD basically has no control efect.Among them, TMD, LQR, and virtual TMD can reduce the RMS response by 2.41%, 14.16%, and 32.39%, and the maximum response by 2.38%, 15.06%, and 27.53%, respectively.It is found that the virtual TMD outperforms LQR in mitigating the tower-top acceleration.Te RMS response reduction can be improved by 18.03% using the virtual TMD algorithm.From Figure 22(b), it is seen that TMD cannot reduce the tower fundamental frequency and 3P frequency components.LQR can signifcantly reduce the tower fundamental frequency component and cannot reduce the 3P frequency component.Under the NLC2 condition, the 3P frequency is constant at 0.61 Hz, which is not within the efective frequency band of the LQR algorithm.However, the virtual TMD algorithm has a wider band and can signifcantly reduce the tower fundamental frequency and 3P frequency components.Figure 23 shows the stroke and control force of the dampers.can be seen that the virtual TMD requires smaller stroke and control force than the LQR.Also, the stroke of the virtual TMD is within 2.0 m, and the maximum control force is 60 kN.
Figure 24 shows the mean square responses of the wind turbine acceleration in 10 wind feld samples under NLC1 and NLC2 conditions.It is demonstrated that the virtual TMD has a better efect than the LQR in mitigating windinduced vibration at operational conditions.
According to the results of 10 wind fled samples under the four conditions, the mean square and maximum control Structural Control and Health Monitoring efects of the accelerations for TMD, LQR, and virtual TMD are obtained, as shown in Table 4.It is seen that the proposed virtual TMD algorithm has a signifcant control efect both at parked and operational conditions.However, the LQR algorithm and TMD have a less control efect at operational conditions.In order to further verify the control efect of the proposed virtual TMD algorithm, 10 wind feld samples with wind speed of 5-30 m/s are generated by TurbSim [41].Te    Ten, the tower-top accelerations of the uncontrolled, LQR, and virtual TMD wind turbine are calculated.Figure 25 illustrates the mean square values of tower top acceleration under diferent control conditions at diferent wind speeds.
From Figure 25, it is seen that the virtual TMD can signifcantly suppress wind-induced vibrations at diferent wind speeds, whose control efect can reach more than 30%.
It is illustrated that the control efect of LQR is far less than the virtual TMD at operational conditions, especially near the rated wind speed.Figures 26(a)-26(c) illustrate the maximum stroke, control forces, and output power under diferent control conditions at diferent wind speeds, respectively.It can be observed that the stroke of the virtual TMD is within 2 m, the control force demand is within 85 kN, and the output power demand is within 80 kW at

20
Structural Control and Health Monitoring diferent wind speeds.And the stroke of the LQR is within 3 m, the control force demand is within 115 kN, and the output power demand is within 80 kW at diferent wind speeds.

Conclusions
Based on the holistic coupled dynamic analysis model of wind turbines, this paper proposes a virtual TMD algorithm for ATMD control to reduce the along-wind vibration of the tower.Also, the optimum design of the virtual TMD algorithm has been obtained via a numerical search approach.Te performance of this algorithm is compared with passive TMD and the traditional LQR algorithm.According to the results, the following conclusions can be drawn: (1) Wind turbines exhibit signifcant time-varying behavior.Te along-wind acceleration response of the wind turbine tower is dominated by the natural frequency at parked conditions.However, under operational conditions, the along-wind acceleration response is dominated by the 3P frequency, and the 3P frequency varies with the wind speed.(2) TMD tuned to the fundamental frequency of the tower has a good control efect at parked conditions, while has a little poor control efect at operational conditions.Tis suggests that the traditional design approach based on the fundamental frequency of the tower may not be suitable for the wind turbines due to their time-varying characteristics.(3) Te proposed virtual TMD algorithm for ATMD demonstrates excellent adaptability and can significantly reduce vibration in various operating conditions of wind turbines.With the constraints of 80 kW output power and a 2 m stroke limitation, the proposed virtual TMD algorithm has a control efect of more than 30% under various wind speeds.(4) In comparison with the traditional LQR algorithm, the proposed virtual TMD algorithm has a wider efective bandwidth and only requires the tower-top acceleration information or the relative displacement of the active TMD.Te proposed virtual TMD algorithm can improve the reduction efect of the wind turbine response at operational conditions.In addition, the virtual TMD algorithm experiences smaller stroke and control force than the traditional LQR algorithm, which is benefcial to the practical application.
In general, the virtual TMD algorithm outperforms passive TMD and the traditional LQR algorithm in mitigating the vibration of the wind turbine, which has better robustness and can adapt to various conditions.However, it is also worth noting that this paper only focuses on the along-wind vibration control of the wind turbine tower.In addition, references [14,45] have demonstrated that the performance of the control system can be infuenced by soil-structure interaction and wind load uncertainty.Based on this study, future research will be conducted to investigate the bidirectional vibration control of wind turbines, considering soil-structure interaction and the uncertainty of wind loads.

Appendix Description of Relevant Symbols
A comprehensive description of all relevant symbols is provided in Table 5.

B. Uncontrolled System Matrices
Mass, stifness, and damping matrix of uncontrolled system: where

Data Availability
Te datasets generated and/or analyzed during the present study are available from the corresponding authors on reasonable request.

2. 1 .
Structure Model.Te structural model of the wind turbine is established by using the Lagrangian equation.Te equation is as follows:

Figure 1 :
Figure 1: Simplifed wind turbine model and coordinate systems.(a) Simplifed model and global coordinate.(b) Local coordinate systems.

Figure 4 :
Figure 4: Velocity and force distribution on blade cross-section.

Figure 6 :
Figure 6: Diagram of the ASSC model.

Figure 11 :Figure 12 :
Figure 11: Te power spectral density of acceleration and frequency response functions under diferent control cases.

Figure 13 :
Figure 13: Te tower-top acceleration responses and damper control forces between the pure virtual passive TMD and the ATMD using virtual TMD algorithm.(a) Time histories of acceleration.(b) Time histories of control force.

Figure 14 :
Figure 14: Te value of the objective function under diferent frequency and damping ratio values.(a) 3D view.(b) 2D view.

Figure 18 :Figure 19 :
Figure 18: Stroke and control force comparison between TMD, LQR, and virtual TMD for 2% mass ratio (PLC2, wind feld sample 1).(a) Time histories of stroke.(b) Time histories of control force.

Figure 21 :
Figure 21: Stroke and control force comparison between TMD, LQR, and virtual TMD for 2% mass ratio (NLC1, wind feld sample 1).(a) Time histories of stroke.(b) Time histories of control force.

Figure 23 :
Figure 23: Stroke and control force comparison between TMD, LQR, and virtual TMD for 2% mass ratio (NLC2, wind feld sample 1).(a) Time histories of stroke.(b) Time histories of control force.

Figure 25 :
Figure 25: RMS response of tower-top acceleration under diferent wind speeds.(a) RMS response of tower-top acceleration.(b) RMS reduction.

Figure 26 :
Figure 26: LQR and virtual TMD stroke displacement, control force, and output power under diferent wind speeds.(a) Maximum stroke displacement.(b) Maximum control force.(c) Maximum output power.

Table 1 :
Te detail information of the developed ASSC model.

Table 2 :
Parameters of the NREL 5-MW baseline wind turbine.

Table 3 :
Frequencies comparison between the ASSC model and FAST. ,

Table 4 :
Responses reduction comparison between TMD, LQR, and virtual TMD at park and operational conditions.