## Abstract

The paper discusses some of the recent developments in vibration control strategies for wind turbines, and in this context proposes a new dual control strategy based on the combination and modification of two recently proposed control schemes. Emerging trends in the vibration control of both onshore and offshore wind turbines are presented. Passive, active and semi-active structural vibration control algorithms have been reviewed. Of the existing controllers, two control schemes, active pitch control and active tendon control, have been discussed in detail. The proposed new control scheme is a merger of active tendon control with passive pitch control, and is designed using a Pareto-optimal problem formulation. This combination of controllers is the cornerstone of a dual strategy with the feature of decoupling vibration control from optimal power control as one of its main advantages, in addition to reducing the burden on the pitch demand. This dual control strategy will bring in major benefits to the design of modern wind turbines and is expected to play a significant role in the advancement of offshore wind turbine technologies.

## 1. Introduction

The past decade has witnessed an unprecedented development in the wind energy sector, and this growth is expected to continue over the next decade, particularly in the offshore area. A hallmark of this development is the design and deployment of multi-megawatt machines with higher ratings and larger rotor diameters. Along with the larger machines come challenges such as increased flexibility of blades/towers and associated vibration issues, which can not only cause fatigue damage with increased operating and maintenance costs but also compromise the power output of the turbines [1].

To curb the negative impacts of vibration and fatigue damage, researchers in recent years have introduced the concept of pitch control in wind turbines [2,3] as a means to control aerodynamic loads. The use of pitch control in general marks a major advancement towards control of vibration, though it brings in associated difficulties related to careful design consideration to avoid interference with torque control [4]. As often unrecognized in the field of wind energy, load control and vibration control are two faces of the same coin. Hence, vibration control strategies could be considered as an alternative effective means to achieve the identical aim of control of structural vibrations and fatigue-related damage in the same way that pitch control would do by load control. In fact, vibration control strategies would not have the drawback of interference with torque control or stresses generated by pitching the blades. Although structural vibration control has been an active area of research for the past two decades, developing structural control strategies and applying those to wind turbines is a relatively new field of research.

The primary modes of vibration of wind turbine blades are in-plane (primarily edgewise) and out-of-plane (primarily flapwise). These two modes of vibration are coupled, as the blades are structurally pre-twisted in general. Besides, owing to the coupling of the aerodynamic loads in the in-plane and out-of-plane directions, the responses (in-plane and out-of-plane) of the blades are coupled too. This coupling gives rise to nonlinearity due to aero-elasticity, as the loads are dependent on the response of the flexible blades. Of the two modes, the edgewise mode has very low or almost no aerodynamic damping, whereas the flapwise mode is aerodynamically damped in nature. Thus, the edgewise modes are of concern, as they may induce instabilities, while the flapwise modes contribute to fatigue damage. In addition, the flexibility of the towers may couple the motion of the tower to that of the blades in both the side-to-side and fore–aft directions, and is a source for generating undesirable blade/tower vibrations [5–7]. The importance of dynamic coupling of the wind turbine tower and blades has been highlighted by Murtagh *et al.* [8] among others. They derived a model of a wind turbine including blade (flapwise)–tower interaction. On comparing the results from their model with the results from a model ignoring the coupling, an increase in the blade tip displacement up to 256% was observed. The blade–tower interaction may also become critical for the edgewise (or side-to-side) modes, which are inherently lightly damped [9,10]. Recognizing the importance of controlling vibrations in wind turbines, several types of vibration controllers have been proposed by researchers in recent years.

Passive control techniques have been investigated for structural control of both onshore and offshore wind turbines [11–13]. One of the issues associated with the use of passive devices for wind turbine structural control is that they may become ineffective as a result of environmental/operational changes or in the presence of parametric variations. The use of traditional passive tuned mass dampers (TMDs) in wind turbines also needs careful design. In fact, if the primary structure is very large (e.g. a wind turbine blade), the TMD will inevitably have a small mass ratio. As a consequence, accurate tuning of the natural frequency of the damper to the natural frequency of the primary structure may become difficult, and small deviations from the optimal tuning may result in poor control performance. Further, tuning of the damper to the rotational frequency of the turbine (which is generally associated with harmonic components that mainly dominate the blade response) is not feasible, as it will lead to instabilities in the TMD–blade coupled system [14].

More recently, there has been a shift in vibration control strategies from passive to active and semi-active. Use of active strut elements based on resonant controllers [15] inspired by the concept of TMDs has been proposed for active control of vibrations in wind turbines. Investigations on the use of synthetic jet actuators [16], microtabs and trailing edge flaps [17,18] have also been considered by researchers. Active control strategies have been proposed by Staino *et al.* [19] and Fitzgerald *et al.* [14]. The control hardware takes advantage of the fact that the structure of the blade is hollow in nature. This allows controllers to be installed inside the blade without affecting the external aerodynamic characteristic of the blades and hence does not interfere with aerodynamic performance. By exploiting the space available in the blade, Staino *et al.* [19] have proposed an active tendon controller and Fitzgerald *et al.* [14] have proposed an active tuned mass damper (ATMD) for control of the vibration of blades. Control of vibration for variable-speed rotors has also been proposed by Staino & Basu [20]. A variant of the ATMD has been proposed by Fitzgerald & Basu [21] to reduce the requirement of the actuator control force in an ATMD. The controller, termed a cable-connected active tuned mass damper (CCATMD), consists of a classical ATMD connected to the tip of the blade by a cable.

Semi-active control of wind turbines has been explored by Arrigan *et al*. [22,23] to suppress the vibrations by using semi-active TMDs. The control algorithm is based on a short-time Fourier transform and retuning the damper to track the dominant/natural frequencies of the structural system. Control of both edgewise and flapwise vibrations has been investigated.

Recently, the issue of structural and mechanical vibrations in wind turbines due to the occurrence of electrical grid faults has been analysed by Basu *et al.* [24] and Staino *et al.* [25]. The study [24,25] indicated that the effect of electrical grid faults may propagate through the mechanical subsystem of the turbine and cause major structural vibrations. The application of custom power devices to counteract the effect of grid fault-induced vibration is considered in the study. Numerical results show that flexible alternating current transmission system (FACTS) devices and unified power quality conditioner (UPQC) in particular are successful in mitigating vibrations due to electrical faults, and they can be conveniently applied to stabilize the generator shaft speed, drive train oscillations, edgewise blade vibrations and tower responses.

Offshore wind energy has gained increasing attention owing to increased electricity generation of offshore power plants with respect to their onshore counterparts and also other advantages such as reduced visual impacts, less turbulence and lower noise constraints. For floating offshore wind turbines, it has been proposed to modify the blade pitch angle and the generator torque to improve the damping of problematic motions and loads on floating platforms [26–28]. Recently, passive control of floating wind turbine nacelle and spar vibrations using multiple TMDs has been investigated by Dinh & Basu [29]. In an overview of floating offshore wind turbines [30], the spar-type floating offshore wind turbine (S-FOWT) was shown to be the most suitable concept for deep-water areas because of its lowered centre of mass, small water plane area and deep draft. The use of single and multiple TMDs for passive control of edgewise vibrations of nacelle/tower and spar of S-FOWTs was validated by Dinh *et al.* [31].

In this paper, first a multi-modal flexible wind turbine model based on an Euler–Lagrange formulation is presented. Subsequently, a brief review of an active pitch controller is presented. A recently proposed active tendon controller for vibration control of wind turbine blades has also been discussed and some results have been presented. Following the discussion on the pitch and active tendon controllers, a new controller combining active tendon and passive pitch control has been proposed and investigated. This dual control strategy merges the benefits of the two controllers while eliminating some of the difficulties associated with active pitch control. One of the major benefits of the proposed improved control strategy (active tendon–passive pitch) is that it decouples vibration control from optimal power tracking. A Pareto-optimal problem is formulated for the design of the proposed controller, and the results show the encouraging potential of the controller in the future, particularly for offshore deployments.

## 2. Euler–lagrange wind turbine model

A flexible multi-body representation is adopted to describe the dynamic behaviour of a three-bladed horizontal-axis wind turbine (HAWT). The proposed structural model focuses on the vibrational response of the rotor blades and of the supporting tower. The model formulated consists of three rotating cantilever beams, representing the turbine blades, with variable bending stiffness and variable mass along the length. The blades rotate at *Ω*(*t*) rad s^{−1} about the rotor hub’s horizontal axis. A schematic representation of the proposed wind turbine structural model is shown in figure 1. The formulation considers the blade in-plane and out-of-plane vibration modes and the tower side-to-side and fore–aft motions. The inclusion of the degrees of freedom (d.f.) associated with the tower into the Lagrangian formulation allows one to capture the dynamic coupling between the blades and the tower [11]. The blade–tower interaction has been found to have a significant impact on the dynamic response of the turbine [8], as it may lead to amplification of the oscillations associated with the blade response. The proposed vibration model also describes the centrifugal stiffening effect induced in the blades from the rotation and the contribution arising from gravity. The coupling between the in-plane and out-of-plane modes due to the structural pre-twist of the blades is also considered. Furthermore, the effects of variation in the rotor speed are incorporated into the formulation.

The wind turbine rotor blades are represented as rotating pre-twisted cantilever beams of length *L*_{b} with variable mass, stiffness and thickness per unit length. The cross section of the blade at distance *x* from the root has structural pre-twist *ϑ*(*x*), chord length *c*(*x*) and mass per unit length *μ*_{b}(*x*). The pre-twist is measured with respect to the free end of the beam, i.e. *ϑ*(*L*_{b})=0°. The motions of the blades are described by a modal approximation in the co-rotating frame (*x*,*y*,*z*), where the *x*-axis is the blade axis and the *z*-axis coincides with the *Z*-axis of the ground-fixed frame (*X*,*Y*,*Z*).

Each mode of vibration is associated with the corresponding mode shape, for which an appropriate function approximation can be computed from the eigen-analysis of the blade structural data. The coupled mode shapes have been computed using a finite-element approach. Using the Rayleigh–Ritz method [32], the deformation variable *u*_{j1}(*x*,*t*) associated with the in-plane displacement of blade *j* is approximated by *N*_{1} vibration modes as
2.1
where *q*_{ji1}(*t*) are the generalized in-plane displacements and *Φ*_{i1}(*x*) the corresponding in-plane mode shape functions such that *Φ*_{i1}(*L*_{b})=1 for all *i*. Similarly, the blade response in the out-of-plane direction *u*_{j2}(*x*,*t*) is represented by *N*_{2} degrees of freedom
2.2
where *q*_{jk2}(*t*) is the *k*th out-of-plane vibration mode of blade *j* and *Φ*_{k2}(*x*) is the *k*th out-of-plane mode shape with *Φ*_{k2}(*L*_{b})=1 for all *k*.

The azimuthal angle *Ψ*_{j}(*t*) of blade *j* with respect to the *Z*-axis is given by
2.3
where *Ψ*_{1}(*t*) is the azimuthal angle of the first blade,
2.4

The wind turbine nacelle is modelled as a rigid discrete mass *M*_{nac} located at the tower top. The supporting tower is modelled as a flexible Euler–Bernoulli beam of length *L*_{T} and variable mass *μ*_{T}(*z*) along its length. The tower side-to-side and fore–aft motions, denoted by *v*_{1}(*z*,*t*) and *v*_{2}(*z*,*t*), respectively, are modelled as
2.5
where *Φ*_{h3}(*z*) and *Φ*_{l4}(*z*) are the side-to-side and fore–aft tower mode shape functions, respectively, and *q*_{4h1}(*t*) and *q*_{4l2}(*t*) the corresponding generalized coordinates. The tower mode shape functions are such that *Φ*_{h3}(*L*_{T})=*Φ*_{l4}(*L*_{T})=1 for all *h*,*l*. The parameters *M*_{1} and *M*_{2} denote the number of assumed modes for *v*_{1}(*z*,*t*) and *v*_{2}(*z*,*t*), respectively.

The equations of motion for the considered wind turbine vibration model with *N* degrees of freedom can be written as
2.6
where is the mass matrix of the system, is the damping matrix (including structural damping), is the stiffness matrix and represents the vector of generalized non-conservative (i.e. dissipative or external) loads (forces/torques) transferred to the system. The aerodynamic loads associated with the wind passing through the rotor area and the load arising due to gravitational effects have been considered as generalized loads. A detailed description of the formulation of the Lagrangian for a three-bladed HAWT system can be found in [19]. The time-dependent entries in the system matrices arise from the formulation of the blade dynamics in a rotating frame of reference and represent mass, damping and stiffness contributions. Details of the wind turbine system matrices obtained using the proposed Euler–Lagrange approach are provided in [33].

From the time-varying system matrices in [33], it can be seen that the formulation with the blade–tower interaction gives rise to negative damping terms for the edgewise dynamics. This is consistent with the study carried out by Thomsen *et al*. [6], who reported that total damping of the blade vibrations in the edgewise direction may become negative, with a detrimental impact on the structural performance of the blade. It is important to note that the occurrence of negative damping arises from the coupling between the tower and the blades. In fact, the side-to-side motion of the tower may result in coupling with the in-plane motion of the blade, leading to an amplification of the oscillations associated with the time-varying dynamics. This is reflected in time-varying stiffness and time-varying damping, with the possibility of negative damping. The omission of the blade–tower interaction in the model would not allow one to capture this phenomenon and therefore it could result in an underestimation of the blade response.

If variations of the rotor speed are taken into account (i.e. ), the stiffness matrix of the system can be decomposed as 2.7 where describes the contribution due to the elastic properties of the blades and the tower, the contribution from centrifugal stiffening and the contribution arising from gravitational effects. The entries in are proportional to and they account for the impact of rotor speed changes on the stiffness of the blades through the blade–tower interaction.

In this study, the aerodynamic load due to the wind and the effect of gravity are modelled as external loads on the wind turbine structure and are denoted by {*Q*_{L}(*t*)} and {*Q*_{W}(*t*)}, respectively. Further, an additional generalized load term associated with the rotor acceleration is derived from the Lagrangian formulation due to the blade–tower interaction. The total generalized external load vector ({*Q*_{ext}(*t*)}) on the right-hand side of (2.6) is expressed as
2.8

## 3. Combined active tendon and pitch control

### (a) Conventional active pitch control

Pitch control is the most widely used method to regulate the mechanical power generated in wind turbines [34]. In fact, commercially available wind turbines are nowadays equipped with pitch control systems to enhance the efficiency of wind energy conversion and to improve the safety of the plant in case of high wind speeds or emergency situations [35]. Active pitch control is achieved by rotating each blade about its axis by means of hydraulic or electric actuators appropriately located inside the wind turbine structure, depending upon the configuration of the active pitch control system (collective pitch control or individual pitch control). Below rated wind speed, there is generally no need to vary the pitch angle, since the wind turbine should produce as much power as possible, though some optimization of energy capture below rated wind speed is also possible [36]. At high wind speeds, pitch control can be used to prevent excessive mechanical power production and to limit the power generated to the rated level. In classical designs, PI (proportional and integral) and PID (proportional, integral and derivative) controllers are used for wind turbine control applications [36]. The control strategy is designed to provide the demanded pitch angle *β*^{⋆} as a function of the power error, defined as the difference between the rated power and the actual power being generated. The main control loop to limit the power in a fixed-speed pitch-regulated wind turbine is illustrated in figure 2. When the measured power is below the rated value, the power error is negative and the demanded pitch is set at the fine pitch limit (e.g. *β*^{⋆}=0), to maximize the aerodynamic efficiency of the rotor. For positive values of the power error, *β*^{⋆} is appropriately controlled to meet the prescribed power requirements.

Power regulation is operated by pitching the blades in order to decrease the angle of attack (and hence the lift coefficient) along the length of the blade. As a consequence, pitch control also has an important effect on structural loads [4] and strongly affects the aerodynamic loads generated by the rotor. By assuming local quasi-static two-dimensional flow conditions around aerofoil sections and ignoring three-dimensional phenomena at the blade tip, the instantaneous local angle of attack *α*(*r*,*t*) [19] at a distance *r* from the blade root can be expressed as
3.1
where *β* is the pitch angle of the blade and *ϕ*(*r*,*t*) is the instantaneous local flow angle. From (3.1), it can be seen that the angle of attack can be reduced by pitching the leading edge of the blades up against the wind, i.e. by increasing *β*(*t*). This, in turn, has an effect on the aerodynamic loads on the blades *p*_{N}(*r*,*t*) and *p*_{T}(*r*,*t*), in the normal and tangential directions, respectively (with respect to the rotor plane). The aerodynamic loads are modelled as
3.2a
and
3.2b
where *ρ* is the density of air, *V* _{rel} is the relative wind velocity, *C*_{l}(*α*) is the lift coefficient and *C*_{d}(*α*) is the drag coefficient associated with the blade aerofoil.

Equations (3.1) and (3.2*a*,*b*) show that the loads on the wind turbine structure can be alleviated by appropriately modifying the pitch angle *β*. However, careful design of the control algorithm is required in order to avoid alleviation of loads using the pitch controller from interfering with power generation. In fact, small changes in pitch angle can have a dramatic effect on the power output [34]. The application of active pitch control for wind turbine load reduction is presented in [2,4,36]. Active pitch control interacts strongly with the turbine dynamics, in particular with the tower dynamics, and under certain circumstances can lead to instability of the active system if the gains of the controller are not designed properly. For large wind turbines, a significant amount of force is required and intense mechanical stresses are experienced at the blade root in order to operate fast and effective pitch control. Further, as mentioned above, major reduction of the structural loads using active pitch control only is not possible without compromising on the mechanical power generated by the rotor. The approach suggested in this paper aims at complementing the application of pitch control with active tendon control. In this way, it is possible to reduce the burden on active pitch control for load reduction by decoupling the mitigation of structural loads from the power regulation issue.

### (b) Offline pitch control

The conventional active pitch control proposed by Bossanyi [2,4] now forms an integral part of any variable-speed wind turbine machine. Though the primary action of the pitch controller is to track the optimum tip speed ratio for maximum power generation (in what is called region II control [37,38]) and regulating the speed of the turbine rotor at the rated speed (in the zone termed region III control [37,38]), pitch control can also be used for load control. The aerodynamic load control of wind turbines is achieved by pitching the angle of the blades based on appropriate control algorithms. This has the effect of ‘dampening’ (reducing) blade vibrations and can also incorporate effective damping in towers by using an active state feedback using the pitch controller of the blades. Two types of pitch control are possible: collective and individual [2,36]. In the case of collective pitch control, all the blades are pitched at the same angle using a feedback control algorithm, whereas for individual pitch controllers the algorithms are designed to control the pitch angle of each individual blade separately and can be more effective than collective control. It has been shown [2,4] that pitch controllers can effectively control fatigue and in that way prolong the life of the wind turbine blades. Even though the advent of pitch controllers has been a landmark in the advancement of wind turbine control, including the design of machines with higher capacity, they suffer from certain inherent drawbacks, particularly for future offshore developments, as discussed previously. Hence, it would be ideal if the pitch demand for a wind turbine, located offshore in particular, could be significantly reduced or if at the least controllers could be designed such that the role of the controllers in load control (vibration control) and power control could be decoupled from each other.

By taking advantage of the active tendon controller conceptualized and examined in [19,20], it is proposed to improve it further by combining it with a pitch controller, thereby achieving the objective of decoupling vibration control from power control in a wind turbine. Active tendon control can control vibrations with tendons deployed inside the hollow structure of the blades. It has the advantage of not interfering with the external aerodynamic surface and indeed having no major impact on the power generation capability of the turbine. When used with pitch control of the blades, a reduction in the force demand of the actuator of the active tendon is also possible (owing to the inherent aerodynamic load reduction by virtue of pitch angle), which is an added benefit. This arrangement will relieve the burden on the pitch controller, with the primary responsibility of the pitch controller being to track optimal power till saturation, and even in those situations the level of vibration will be significantly diminished due to the use of active tendon control. Vibration reduction (fatigue load control) and mechanical stress alleviation can be performed by active tendon control. The apparent conflict between load/fatigue control and generating maximum possible power when exclusively pitch control is used will be eliminated by this dual (active tendon–pitch) control strategy.

As the actuator dynamics of the pitch controller is of additional concern, the pitch control algorithm in the proposed dual control strategy is designed to avoid a real-time active control strategy. By contrast, an offline approach is proposed. This is based on a pre-calculated design decision table (based on, for example, wind speeds, blade–tower–drive train responses, power production) rather than a state feedback/PID controller in real time. The use of an offline control scheme avoids the requirement for fast changing actuator dynamics, which not only reduces the actuator demand and problems of actuator dynamics but also eliminates dynamic stresses generated due to pitching. Essentially, the pitch will need to be changed only infrequently at relatively long intervals of time (fast changing dynamics and stability will not be an issue) and the rate of pitching can be slow (quasi-static) to avoid dynamic stresses . An illustration of such dual schemes (active tendon–offline pitch) follows.

### (c) Active tendon control

The application of active devices as in [19] is now considered for suppression of rotor vibrations. The active tendon control strategy works by providing a cable anchorage to the structure, with the cable force or the displacement of the cable support controlled by an actuator. Staino *et al.* [19] proposed the use of active tendon control for suppressing edgewise vibrations, but the control system arrangement could be equally applicable for controlling flapwise displacements as well (by rotating the cable/tendon arrangement inside the blade by 90°) as shown in the following sections. In fact, two sets of orthogonal cable arrangements can be used to facilitate independent control in two orthogonal directions. An appropriate control algorithm can then be used to actuate the tendon controller. Active tendon control to mitigate flapwise blade vibrations has been investigated in [39,40]. The hollow nature of wind turbine blades makes them suitable for the installation of the control devices. The control devices can mitigate the dynamic response without affecting the aerodynamic performance of the structure. In particular, for practical advantages (e.g. ease of installation inside the blades in comparison with other power devices), the use of active tendons is proposed as a part of the dual strategy and numerically analysed for vibration control. The tendons are mounted on a frame supported from the nacelle (figure 3). Vector analysis of the equilibrium of forces transmitted to the blade results in a net control force acting on the blade tip in the flapwise direction. For the *j*th blade, the net force from the actuators/tendons is proportional to the force *T*_{j}(*t*) and the sine of the angle *φ*_{0}, as illustrated in figure 3. In the mathematical framework used in this study, the active control force is modelled as an external force acting on each blade tip and is given by .

The active elements are drawn in thin lines, while the support structure (e.g. a truss or a frame) is shown in bold. The introduction of active elements in the support structure allows transfer of the control force to the hub without the generation of a reaction force in the flapwise direction of the blade. In fact, the active elements produce forces that are external to the support structure and hence nullify the forces in the flapwise direction (e.g. the net flapwise load at joints A or B is identically zero). Hence, mechanistically, this eliminates any flapwise reaction forces for the support structure if the algebraic condition shown in figure 3 is met. The forces *T*_{0},*T*_{1},… are static prestressing forces in the cable. The control force determines the value of Δ*T*_{0}(*t*) and the other active tendon forces (such as Δ*T*_{1}(*t*)) can be calculated subsequently based on geometry and force equilibrium. The net control force acting on the blade tip in the flapwise direction for the *j*th blade is given by
3.3
In the mathematical framework adopted in this study, the active control forces have been modelled as external modal loads applied to each blade. The principle of virtual work is used again in order to include the effect of the controller into the Lagrangian formulation.

A numerical simulation is carried out to illustrate the capabilities of the active tendon controller described above. The numerical model presented in §2 has been simulated using the specifications of the NREL 5 MW baseline reference wind turbine [41]. It should be noted that, although offshore dynamics are not included in this study, the control principles illustrated in what follows are equally valid for both onshore and offshore systems. In fact, the numerical results presented in the paper can readily be extended to offshore wind turbines. A linear quadratic (LQ) controller with design parameter *γ*=10^{−9} as in [19] has been synthesized based on a reduced-order 4 d.f. model of the wind turbine. The parameter *γ* is associated with the weight *R* assigned to the control input {*u*(*t*)} in the LQ cost function
3.4
where {*x*} is the reduced state of the wind turbine and *R*=*γ*[*I*], with [*I*] the 3×3 identity matrix. The controller that minimizes the cost functional (3.4) is obtained by solving the LQ control problem as in [19]. It will be shown later that, as the value of *γ* is decreased, allowing larger values in the control effort, better vibration control performances are achieved. Of course, a smaller value for *γ* entails a higher force requirement from the active tendons. The fundamental modes of vibration of the blades (in-plane and out-of-plane) and of the tower (side-to-side and fore–aft) have been considered for validation of the controller, leading to an 8 d.f. system. The mode shapes and frequencies have been computed by using ‘Modes’ [42]. Aerodynamic loads on the blades and on the nacelle have been simulated using the modified blade element momentum (BEM) theory as described in [19,33]. The wind exciting the structure is represented as a steady wind flow, including vertical wind shear effects due to the rotation of the blades. The wind loading scenario therefore corresponds to a steady wind at the rated speed of 11.4 m s^{−1}, with a maximum change of 1 m s^{−1} in the vertical direction from the hub to the blade tip to simulate the wind shear effect. The rotor speed has been set to the rated value of 12.1 r.p.m. The time history of the blade displacements (in-plane and out-of-plane) and of the tower top displacements (side-to-side and fore–aft) in the controlled (solid line, blue) and in the uncontrolled (dashed line, red) case are shown in figure 4.

It should be noted that the blade oscillations (figure 4) are mainly caused by the load arising from gravitational effects at the rotational speed *Ω*=12.1 r.p.m. It can be observed that by using active tendon control a substantial improvement of the out-of-plane blade response is achieved. The maximum blade tip out-of-plane displacement in steady operation is reduced from 5.2 to 2.8 m, and a reduction of the peak-to-peak oscillation is also attained. Because of the structural coupling induced by the pre-twist of the blade, a reduction of the in-plane blade response is also obtained. As no active control is operated on the nacelle, no significant improvement of the nacelle fore–aft motion is observed, although the side-to-side motion exhibits some reduction in the controlled case.

### (d) Dual strategy with active tendon and offline pitch control

A numerical study to assess the performance of the combined active tendons–pitch controller has been carried out. The proposed wind turbine model subjected to a steady wind flow at the rated speed of 11.4 m s^{−1} has been considered. For the offline pitch controller, the desired collective blade pitch demand is sent as a command to the pitch actuator model. The blade pitch actuator allows a minimum blade pitch setting of 0°, a maximum blade pitch setting of 90° and a maximum blade pitch rate of blade pitch setting of 8° s^{−1}. For the active tendon controller, four different LQ configurations have been synthesized, corresponding to different values of the design parameter *γ*. In solving the LQ control problem for the active tendons, by appropriately assigning the design parameter *γ*, it is possible to relatively prioritize the suppression of the out-of-plane blade response and the force usage to implement the control.

The results of the application of the combined active tendons–offline pitch controller are illustrated in figure 5.

Figure 5*c* shows that the active tendon controller has no impact on the power production, whereas the pitch control has the effect of reducing vibrations (figures 5*a* and 6) and has a significant impact on reducing the power production.

#### (i) Wind speed variation

The results shown in figure 5 correspond to the application of the combined controller in the case of constant wind speed. A scenario with increased wind speed excitation has been simulated to test the performance of the proposed controller in the presence of wind speed variations. This analysis is of particular interest because power regulation using pitch control is required when the wind turbine is operating above the rated wind speed conditions. The simulation of the application of the combined control under varying wind speed is illustrated in figure 6. Initially, the turbine is operating at rated wind speed and the active tendon controller with *γ*=10^{−9} is employed to reduce the blade displacement. Pitch control is disabled in the initial phase of the simulation, as active tendon control is effective in controlling vibrations and reduction in aerodynamic loads is not necessary. If the tendon controller had not been effective, the blade out-of-plane displacements could have gone up to about 5 m. At an instant of time *t*=35 s, the wind speed is linearly increased to 12.5 m s^{−1} over a 15 s period. It can be observed that, while the blade tip response does not experience a significant increase due to the active tendon controller, the mechanical power generated by the turbine exhibits a strong increase up to more than 7 MW. At *t*=90 s, pitch control with *β*^{⋆}=4.5° is activated and correspondingly the power is restored to its rated value after a short transient. The activation of the blade pitch also leads to a reduced load on the blade, and as a consequence the vibration of the blade is further reduced. This also implies a smaller control force requirement for the active tendon controller as shown in figure 6*d*.

Figure 6 also shows that active tendon and collective pitch control are almost uncoupled in terms of impact on the dynamic response of the blade.

#### (ii) Pareto-optimal formulation for dual control strategy

As discussed previously, the objectives of reduction in vibration in wind turbines and maximizing the power generated are conflicting in nature. This, as expected, has also been confirmed by figure 5, where one can observe the price of reduced power generation (figure 5*c*), to be paid for reducing vibration (figure 5*a*,*b*) using pitch control. The use of an active tendon controller can decouple the two objectives to some extent but may be constrained by limits on the actuation capacity as well. Also, there are constraints or limits on feasible pitch angles. Based on these considerations, it is possible to cast a Pareto-optimal optimization problem to solve for the design variables, i.e. control force *F* and pitch angle *β*, in the dual active tendon–pitch control strategy. The objective function is formulated by assigning weights to the two conflicting objectives of vibration reduction and power generation maximization as
3.5
such that (2.6) is satisfied. The term denotes the relative reduction of the controlled out-of-plane blade peak displacement with respect to the uncontrolled blade peak displacement and is defined as
3.6
where and are the blade out-of-plane peak response in the controlled and uncontrolled cases, respectively. The term *r*_{P}(*β*) represents the relative power loss and it is computed as
3.7
where *P*_{ctr} and *P*_{unctr} denote the mechanical power generated by the turbine in the controlled and uncontrolled cases, respectively. The term λ is a scalar, and λ and 1−λ are the weights used for the vibration reduction ratio and power loss ratio, respectively. To solve this Pareto-optimal optimization problem with only two variables, it is possible to carry out a numerical analysis with a discrete search technique by sampling the values over the possible ranges of *F* and *β*. However, even in that case, the total computation time may be significant. To avoid the computationally intensive approach, a simpler technique is followed by exploiting the fact that the power loss is independent of the tendon tension *F* (figure 5*c*) and hence *r*_{P}(*β*) is a function only of *β*. Further, it can be observed from figure 5 that, to achieve the maximum possible vibration reduction or to minimize the vibrational response, the maximum value of should be used (since *F* has no impact on power production). Hence, the first term in (3.5) is a function of *β* as *F* is known. Thus, the first and second terms in the objective function in (3.5) can be expressed as
3.8
where *g*_{1} and *g*_{2} are functions of *β*. Using figure 5*a*,*c*, the functions *g*_{1} and *g*_{2} can be regressed by polynomial functions for ease of representation and subsequent optimization calculations. Hence, the minimal value of the objective function is obtained by satisfying the condition
3.9
which leads to the following equation:
3.10
For a given value of λ, the solution of (3.10) gives the optimal value of *β*. The optimal value of the tendon force is given as . Using these two values, the desired minimization of the objective function is attained with a specified weight λ. For the purpose of numerical illustration, we consider the variation of *β*^{⋆} as a function of λ with constraints and (figure 7).

The simulated scenario corresponds to rated wind speed conditions. By selecting λ=0.6, the corresponding optimal pitch angle is *β*^{⋆}=3.2° and the corresponding force requirement is kN. Using the proposed optimal controller, 62.74% reduction of the out-of-plane blade peak response is achieved and the power loss with respect to the uncontrolled case is about 14.81%. If a different tuning of the pitch control is selected, either an increased power loss or an increased blade vibration is obtained, as illustrated in table 1.

## 4. Conclusion

A review of different structural vibration control strategies has been carried out in this paper. Of the different controllers proposed recently, specific attention has been focused on two promising control strategies: active pitch control and the recently proposed active tendon control. Details on these controllers, including some key results, have been presented. A new dual control strategy by combining active tendon control with passive pitch control has been formulated and presented. This new controller, which is an improved version of the active tendon controller, merges the benefits of the two different control schemes while eliminating some of the drawbacks of each of the individual control schemes. The dual control strategy avoids the actuator dynamics of the active pitch control and the generation of dynamic stresses due to pitching, as it uses a passive pitch algorithm. Also, the force demand on the active tendon is reduced, as the pitch control has the effect of reducing the aerodynamic loads. Further, the proposed controller decouples the vibration control from the control for optimal power tracking, as the active tendon control does not interfere with the power production. A Pareto-optimal optimization formulation has been cast and solved for the design of the proposed dual controller. The results indicate that, for example, using weights of 0.6 and 0.4 for the vibration reduction and power production criteria, respectively, achieves a vibration reduction of about 63% with a power production loss of approximately 15%. If the pitch controller alone was used to achieve the same level of vibration reduction, a power loss of over 50% would have occurred, or if the active tendon control was solely used instead an increase in approximately 40% of the tendon force would have been required (though without any power loss). The proposed dual strategy shows encouraging prospects of applications in future with the significant positive impact it may have on the growth of offshore wind energy.

## Footnotes

One contribution of 17 to a theme issue ‘New perspectives in offshore wind energy’.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.