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Journal of Sound and Vibration

Journal of Sound and Vibration 331 (2012) 1233–1256

0022-46

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jsvi

Actuator control of edgewise vibrations in wind turbine blades

A. Staino a, B. Basu a,n, S.R.K. Nielsen b

a School of Engineering, Trinity College Dublin, Dublin 2, Irelandb Department of Civil Engineering, Aalborg University, DK-9000 Aalborg, Denmark

a r t i c l e i n f o

Article history:

Received 16 March 2011

Received in revised form

29 October 2011

Accepted 4 November 2011

Handling Editor: D.J. Waggedgewise vibrations. The control is based on a pair of actuators/active tendons mounted

Available online 6 December 2011

0X/$ - see front matter & 2011 Elsevier Ltd.

016/j.jsv.2011.11.003

esponding author. Fax: þ353 1 6773072.

ail addresses: stainoa@tcd.ie (A. Staino), basu

a b s t r a c t

Edgewise vibrations with low aerodynamic damping are of particular concern in modern

multi-megawatt wind turbines, as large amplitude cyclic oscillations may significantly

shorten the life-time of wind turbine components, and even lead to structural damages or

failures. In this paper, a new blade design with active controllers is proposed for controlling

inside each blade, allowing a variable control force to be applied in the edgewise direction.

The control forces are appropriately manipulated according to a prescribed control law.

A mathematical model of the wind turbine equipped with active controllers has been

formulated using an Euler–Lagrangian approach. The model describes the dynamics of

edgewise vibrations considering the aerodynamic properties of the blade, variable mass

and stiffness per unit length and taking into account the effect of centrifugal stiffening,

gravity and the interaction between the blades and the tower. Aerodynamic loads

corresponding to a combination of steady wind including the wind shear and the effect

of turbulence are computed by applying the modified Blade Element Momentum (BEM)

theory. Multi-Blade Coordinate (MBC) transformation is applied to an edgewise reduced

order model, leading to a linear time-invariant (LTI) representation of the dynamic model.

The LTI description obtained is used for the design of the active control algorithm. Linear

Quadratic (LQ) regulator designed for the MBC transformed system is compared with the

control synthesis performed directly on an assumed nominal representation of the time-

varying system. The LQ regulator is also compared against vibration control performance

using Direct Velocity Feedback (DVF). Numerical simulations have been carried out using

data from a 5-MW three-bladed Horizontal-Axis Wind Turbine (HAWT) model in order to

study the effectiveness of the proposed active controlled blade design in reducing edgewise

vibrations. Results show that the use of the proposed control scheme significantly

improves the response of the blade and promising performances can be achieved.

Furthermore, under the conditions considered in this study quantitative comparisons of

the LQ-based control strategies reveal that there is a marginal improvement in the

performances obtained by applying the MBC transformation on the time-varying edgewise

vibration model of the wind turbine.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Vibration control is of topical research interest for design of modern wind turbines [1,2]. As the size of multi-megawattwind turbines is increasing, the blades are becoming more flexible and hence are subjected to vibrations induced by

All rights reserved.

b@tcd.ie (B. Basu), srkn@civil.aau.dk (S.R.K. Nielsen).

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–12561234

external wind loading and tower interactions. Because of the impact on the mechanical components and the fatigueinduced in the blades, uncontrolled vibrations could sometimes cause structural/mechanical damage, leading to asignificant reduction in the operational efficiency and lifetime of the wind turbine.

The main modes of vibration for the blades are flapwise and edgewise. Out of the two, edgewise vibrations can be ofsignificant concern in wind turbines, as this mode is lightly damped and can lead to violent vibrations [3,4]. In fact, undercertain conditions the first edgewise mode may exhibit a very low or even negative damping, i.e. the total damping due tostructural properties and aerodynamic effects can be less than zero. This corresponds to the case in which the aerodynamicforces supply energy to the vibrating system causing large amplitude vibrations [5]. As a result, the blades and the towermay be subjected to unacceptably large deflections, which may potentially lead to the failure of the overall system.Furthermore, because of the strong coupling between the blade edgewise motion and the drive train torsional mode [6],oscillations induced in the drive train may also have a negative impact on the power production control system ofthe plant.

Significant research has been carried out into the control of different aspects related to large wind turbines [7–9].Different approaches have been proposed in the literature for the design of solutions for improving the response of thestructure to wind-induced oscillations. Passive control techniques have been investigated for structural control of bothonshore and offshore wind turbines [1,10]. A semi-active method based on tuned mass dampers is described in [2] for thecontrol of flapwise vibrations in wind turbine blades. Active control strategies have also been the focus of attention veryrecently. Studies on the use of synthetic jet actuators [11], microtabs and trailing edge flaps [12,13] have been consideredby the researchers. Use of active strut elements based on resonant controllers [14] inspired by the concept of tuned massdampers has been proposed for active control of vibrations in wind turbines. Individual pitch control for reducing loads onthe wind turbine structure has also been investigated [15,16], even though this solution requires careful design of thecontrol algorithm [17] in order to avoid pitch controllers from interfering with the torque control system of the plant.Thus, recent studies indicate the importance and necessity of further investigations in the area of active control systems tosuppress undesirable vibrations without compromising on the power output from the turbines.

In this paper, a mathematical model describing the dynamics of the edgewise vibrations is formulated by using aLagrangian–Eulerian approach, based on energy considerations. The use of active control devices (two actuators/activetendons generating an edgewise control force), located inside each blade, is considered in order to suppress vibrations andmitigate their damaging effects. To this end, the model is formulated by introducing controllable forces acting on theblades that can be varied according to a prescribed control law. The effect of centrifugal stiffening and gravity has beenconsidered. Quasi static aerodynamic wind loading conditions are also modeled and time-series of the loads are computedby applying the corrected blade element momentum method [18]. The wind loading contains harmonic components dueto vertical wind shear and fluctuating components due to turbulence. Multi-blade coordinate transformation is used on areduced order model for modal analysis and control design [19]. Linear quadratic regulators are designed and tested basedon numerical simulations, which show that a significant reduction in blade response can be achieved by means of theproposed active control system. Two competing LQ controllers, one synthesized based on the time varying system directlyand the other on the Coleman transformed non-rotating system, are investigated and compared. Furthermore, thevibration reduction obtained by using DVF control [20] is also considered for comparison.

2. HAWT edgewise model with controller

A modern multi-megawatt wind turbine is a highly complex mechano-electrical system consisting of severalcomponents, including structural elements like tower, rotor (consisting of nacelle and blades) and other mechanicaland electrical elements such as gears, converters, transformers, etc., as well as a high number of different sensors,actuators and controllers. It follows that the modeling of large wind turbines is also complex and challenging, and gettingaccurate models entails studying the dynamics of many degrees of freedom (DOFs), leading to a high dimensional set ofequations. Because we are interested in studying the edgewise dynamics of rotor vibrations in a wind turbine, here weformulate a mathematical model that takes into account only the relevant states or degrees of freedom, representing theedgewise vibration responses and the associated coupling of the blade with the tower/nacelle motion [21]. A schematicrepresentation of a three-bladed HAWT is shown in Fig. 1. The blades are modeled as Bernoulli–Euler cantilever beams oflength ‘L’, with variable bending stiffness EI(x) and variable mass per unit length mðxÞ along the length. The blades rotate ata constant rotational speed O rad s�1 and the azimuthal angle CjðtÞ of blade ‘j’ at the time instant ‘t’ is given by

CjðtÞ ¼C1ðtÞþðj�1Þ2p3

, C1ðtÞ ¼Ot; j¼ 1;2,3 (1)

The dynamic coupling between the blade and the tower has been included through the horizontal motion of the nacelle.The tower is modeled as a single degree of freedom system with the mass M0, which represents the modal mass of thetower and the mass of the nacelle. The variables ~ujðx,tÞ, j¼1,2,3 and ~u4ðtÞ denote the edgewise blade and nacelledisplacements, respectively. The generalized (or modal) stiffness of the tower is represented by k4.

Fig. 1. Wind turbine model for edgewise vibration.

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–1256 1235

2.1. Generalized multi-modal flexible model

A generalized flexible model of the blade with N modes of vibration is formulated. This generalized model will be usedin the numerical simulations to assess the performances of the controller. The controller however will be designed on areduced order model. In the generalized representation of the wind turbine, each mode of vibration is associated to thecorresponding modeshape FiðxÞ, for which an appropriate function approximation can be computed from the eigen-analysis of the blade structural data. The system is therefore described by 3Nþ1 generalized coordinates, that provides anaccurate description of the flexible blade behaviour. Let ~qðtÞ be the vector of the generalized coordinates of the systemdefined as

~qðtÞ ¼

~q11ðtÞ

~q12ðtÞ

^~q1NðtÞ

~q21ðtÞ

^~qjiðtÞ

^~q4ðtÞ

266666666666666664

377777777777777775

2 R3Nþ1 (2)

The degree of freedom ~qjiðtÞ, j¼ 1;2,3, i¼ 1, . . . ,N denotes the i-th edgewise mode for the blade ‘j’. The variable~q4ðtÞ ¼ ~u4ðtÞ represents the motion of the nacelle in the rotor plane. The total edgewise displacement along the blade isgiven by

~ujðx,tÞ ¼XN

i ¼ 1

FiðxÞ ~qjiðtÞ (3)

In classical mechanics, the Lagrangian Lð ~qðtÞ, _~q ðtÞÞ of a dynamical system is a function of the generalized coordinatesand their time derivatives, representing the difference between the kinetic and the potential energies

Lð ~qðtÞ, _~q ðtÞÞ ¼ T ð ~qðtÞ, _~q ðtÞÞ�Vð ~qðtÞÞ (4)

where T and V denote the kinetic and potential energy, respectively. The equations of motion of the system are given by

d

dt

@Lð ~qðtÞ, _~q ðtÞÞ@ _~q iðtÞ

!�@Lð ~qðtÞ, _~q ðtÞÞ

@ ~qiðtÞ¼ ~Q ext,iðtÞ (5)

Eq. (5) is known as the Euler–Lagrange equations; ~Q ext,iðtÞ denotes the i-th component of ~Q extðtÞ which is the vector ofgeneralized non-conservative (i.e. dissipative or external) loads (forces/torques) transferred to the system. For simplicity innotation, we will omit the dependency on time of generalized coordinates and loads, as well as the dependency of L, T andV on ~q and _~q .

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–12561236

2.1.1. Kinetic energy

Let ~vb,jðx,tÞ denote the total velocity of blade ‘j’ at a distance ‘x’ from the hub in the edgewise direction, including thenacelle motion that causes the blade displacement (Fig. 1). The use of this variable in the formulation ensures inclusion ofthe coupling between the blade and the nacelle. The square of the total velocity for blade ‘j’ is

~v2b,j ¼

_~q 4 sinðCjÞ�OXN

i ¼ 1

Fi ~qji

!2

þ _~q 4 cosðCjÞþOxþXN

i ¼ 1

Fi_~q ji

!2

(6)

The total kinetic energy of the system (i.e. the three blades and the tower) is given by

T ¼ 1

2

X3

j ¼ 1

Z L

0mðxÞ _~q 4 sinðCjÞ�O

XN

i ¼ 1

Fi ~qji

!2

þ _~q 4 cosðCjÞþOxþXN

i ¼ 1

Fi_~q ji

!224

35 dxþ

1

2M0

_~q 24 (7)

2.1.2. Potential energy

The total potential (strain) energy of the system V is obtained by considering the potential energy of the cantileverblade in bending, the contribution from centrifugal stiffening [22], the contribution given from the component of thegravity along the blade axis (Fig. 4), and the potential energy of the nacelle. The centrifugal force on blade acting at thepoint ‘x’ from the hub is

FcðxÞ ¼O2Z L

xmðxÞx dx (8)

where ‘x’ is the distance from ‘x’ to the current element considered. Similarly, for the j-th blade, the effect of thecomponent of the gravitational force acting along the blade at a distance ‘x’ from the blade root is

Fg,jðxÞ ¼�

Z L

xmðxÞg cosðCjÞ dx¼�g cosðCjÞ

Z L

xmðxÞ dx (9)

The total potential strain energy can be modeled as

V ¼ 1

2k4 ~q

24þ

1

2

X3

j ¼ 1

XN

i ¼ 1

XN

k ¼ 1

ðKe,ikþKw,ik cosðCjÞþKg,ikÞ ~qji ~qjk

!(10)

where

Ke,ik ¼

Z L

0EIðxÞ½Fi

00 Fk00� dx, Kg,ik ¼

Z L

0FcðxÞ½Fi

0 Fk0� dx¼O2Kg0,ik

Kg0,ik ¼

Z L

0

Z L

xmðxÞx dx

� �½Fi0 Fk

0� dx Kw,ik ¼�g

Z L

0

Z L

xmðxÞ dx

� �½Fi0 Fk

0� dx (11)

In (11), E is the Young’s modulus of elasticity of the material, I(x) the second moment of area of the blade about therelevant axis and Fi

0ðxÞ, Fi

00ðxÞ, respectively denote the first and the second derivative of the modeshape with respect to ‘x’.

2.2. Hardware configuration for active control scheme

The installation of active devices is required to physically operate the control of edgewise vibrations. In the controlscheme, the active vibration control is implemented by means of two linear actuators (or active tendons) located inside theblade (Fig. 2). The actuators/tendons are mounted on a frame supported from the nacelle. Vector analysis of theequilibrium of forces transmitted to the blade results in a net control force acting on the blade tip in the edgewisedirection. For the j-th blade, the net force from the actuators/tendons is proportional to the force Tj(t) and the sine of theangle W. In the mathematical framework used in this study, the active control force is modeled as an external force actingon each blade tip and is given by f jðtÞ ¼ 2TjðtÞ sinðWÞ.

The reaction forces are transmitted along the supporting structure finally to the nacelle. The support structure forapplying the control forces has to satisfy the requirement of transferring the force to the hub. This has to be accomplishedideally by avoiding the generation of a reaction force in the edgewise direction of the blade or practically by eliminatingthe possibility of any reaction force in the close to medium spatial proximity of the tip. This design condition can beachieved by introducing active elements in the support structure (such as active braces or active tendons) as is typicallyused in large engineering structures for protection against wind or earthquake loads. Such a system with active tendoncontrol is shown in Fig. 3.

The active elements are drawn with thin lines while the support structure (e.g. a truss or a frame) is shown in bold. Theactive elements (active tendon in this case) produce forces which are external to the support structure and hence nullifythe forces in the edgewise direction (e.g. the net edgewise load at joints A or B is identically zero).

Fig. 2. Actuator configuration for the proposed active control system.

Fig. 3. Implementation of active vibration blade control based on active tendons.

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–1256 1237

2.3. Generalized loads

In the edgewise model formulation, wind, gravity loadings and active control forces have been considered. Windexcitation is modeled as an external modal load applied to the blade in the edgewise direction, while a variable controlforce f j, j¼ 1;2,3 is applied to each blade in order to mitigate vibrations.

The total virtual work d ~W done by external active control forces on the blades and nacelle is given by

d ~W ¼X3

j ¼ 1

f jðdujðL,tÞþd ~q4 cosðCjÞÞ�X3

j ¼ 1

f jd ~q4 cosðCjÞ ¼X3

j ¼ 1

XN

i ¼ 1

f jd ~qji (12)

and the corresponding generalized force vector ~F is

~FðtÞ ¼d ~Wd ~q

(13)

The generalized controlled force on the blade ‘j’ for the i-th mode, then, corresponds to the control force fj, while theresulting generalized control force on the nacelle is zero.

The virtual work d ~W L done by the external wind load is

d ~W L ¼X3

j ¼ 1

Q j

XN

i ¼ 1

Fid ~qjiþd ~q4 cosðCjÞ

!(14)

where

Q j ¼

Z L

0pjðx,tÞ dx, j¼ 1, . . . ,3 (15)

with pjðx,tÞ representing the variable wind load intensity along the blade length in the edgewise direction.Differentiating the virtual work d ~W L with respect to the generalized coordinates, the generalized loads result in

~Q loadðtÞ ¼d ~W L

d ~q(16)

Fig. 4. Model for gravity loading acting on the blade.

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–12561238

Therefore, the generalized aerodynamic load on the blade ‘j’ for the i-th mode is computed as

Qji ¼

Z L

0pjðx,tÞFiðxÞ dx (17)

and the generalized load on the nacelle corresponds to

Q4 ¼X3

j ¼ 1

Z L

0pjðx,tÞ dx cosðCjÞ (18)

Furthermore, the load due to gravity has also been considered. The component of the gravity force acting in theedgewise direction on an element of length dx of the blade ‘j’ (Fig. 4) is

dFg,j ¼ mðxÞ dx g sinðCjÞ (19)

The total virtual work done due to the gravity, d ~W g , is obtained as

d ~W g ¼X3

j ¼ 1

Z L

0mðxÞg sinðCjÞ dx

XN

i ¼ 1

Fid ~qjiþd ~q4 cosðCjÞ

!" #¼X3

j ¼ 1

XN

i ¼ 1

gd ~qji

Z L

0mðxÞFi dx sinðCjÞ (20)

sinceP3

j ¼ 1 sinðCjÞ cosðCjÞ ¼ 0. Differentiating with respect to the generalized displacements vector

~Q gðtÞ ¼d ~W g

d ~q, Qg,ji ¼ g

Z L

0mðxÞFi dx sinðCjÞ, Qg,4 ¼ 0 (21)

where Qg,ji and Qg,4 are the components of ~Q g and represent the generalized gravitational load on the blade ‘j’ for the i-thmode and on the nacelle, respectively.

For the considered system the total generalized load in the Euler–Lagrange formulation is given by

~Q extðtÞ ¼~Fþ ~Q loadþ

~Q g (22)

2.4. Euler–Lagrange equations

The Euler–Lagrangian equation (5) for the system considered is

d

dt

@T@ _~q ji

!�@T@ ~qji

þ@V@ ~qji

¼ f jþQjiþQg,ji, j¼ 1;2,3, i¼ 1, . . . ,N

d

dt

@T@ _~q 4

!�@T@ ~q4þ@V@ ~q4¼Q4

8>>>>><>>>>>:

(23)

By introducing the quantities

m1i ¼

Z L

0mðxÞFi dx, m2i ¼

Z L

0mðxÞF2

i dx, m4 ¼ 3

Z L

0mðxÞ dxþM0 (24)

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–1256 1239

and assuming orthogonal modeshapes such that mf,ik ¼R L

0 mðxÞFiFk dx¼ 0, the equations of motion for the consideredwind turbine edgewise vibration model with 3Nþ1 degrees of freedom can be written as

~MðtÞ €~qþ ~CðtÞ _~qþ ~KðtÞ ~q ¼ ~Fþ ~Q loadþ~Q g (25)

Details on the system matrices ~M, ~C, ~K are provided in Appendix A.

2.4.1. Reduced order HAWT model for control synthesis

A reduced order model has been derived for the system under consideration, in order to reduce the number of statesrequired for implementing the control and hence to decrease the computational cost associated with the calculation of thecontrol law. In particular, for the design of the controller each beam is assumed to be vibrating in its fundamental mode.This leads to a reduced order model with 4DOF and the vector of the generalized coordinates of the system becomes

qðtÞ ¼

q1ðtÞ

q2ðtÞ

q3ðtÞ

q4ðtÞ

266664

377775¼

~q11ðtÞ

~q21ðtÞ

~q31ðtÞ

~q4ðtÞ

266664

377775 (26)

The equations of motion of the wind turbine are reduced to

MðtÞ €qþCðtÞ _qþKðtÞq¼ FþQ loadþQ g (27)

The reduced system matrices in (27) are derived from the matrices of the model with higher modes (Appendix A) byconsidering the fundamental mode of vibration only. By defining the following quantities in order to simplify notations:

m2 ¼m21, m1 ¼m11, Kw ¼ Kw,11

Ke ¼ Ke,11, Kg,0 ¼ Kg0;11 (28)

the matrices of the reduced order model can be written as

MðtÞ ¼

m2 0 0 m1 cosðC1Þ

0 m2 0 m1 cosðC2Þ

0 0 m2 m1 cosðC3Þ

m1 cosðC1Þ m1 cosðC2Þ m1 cosðC3Þ m4

266664

377775

CðtÞ ¼

cb 0 0 0

0 cb 0 0

0 0 cb 0

�2Om1 sinðC1Þ �2Om1 sinðC2Þ �2Om1 sinðC3Þ c4

266664

377775

KðtÞ ¼

k2þKw cosðC1Þ 0 0 0

0 k2þKw cosðC2Þ 0 0

0 0 k2þKw cosðC3Þ 0

�O2m1 cosðC1Þ �O2m1 cosðC2Þ �O

2m1 cosðC3Þ k4

266664

377775 (29)

where k2 ¼ KeþO2Kg,0�O2m2, while cb and c4 denote the structural and the aerodynamic damping associated with the

blades and the nacelle, respectively. In the reduced order formulation, Ke represents the generalized elastic stiffness of theblade, Kg ¼O2Kg,0 is the geometrical stiffness and Kw is the stiffness arising out of gravity effects. The term Ke can beexpressed as

Ke ¼o2bm2 (30)

where ob is the fundamental natural frequency of the blade and m2 is the modal mass of the blade.Similarly, the reduced generalized control force vector F, the reduced generalized aerodynamic load Q load and the

reduced generalized gravity load Q g can be derived from the corresponding quantities in the formulation including highermodes and can be written as

FðtÞ ¼

f 1

f 2

f 3

0

26664

37775, Q loadðtÞ ¼

Q1

Q2

Q3

Q4

266664

377775¼

Q11

Q21

Q31

Q4

266664

377775, Q gðtÞ ¼

Qg,1

Qg,2

Qg,3

Qg,4

266664

377775¼

Qg,11

Qg,21

Qg,31

Qg,4

266664

377775 (31)

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–12561240

The total generalized load in the reduced order formulation is therefore

Q extðtÞ ¼ FþQ loadþQ g (32)

3. Coleman transformed equation

Wind turbine equations of motion in the edgewise direction (27) contain periodic term, that depends on the azimuthalangle of blades. Classical time-invariant analysis and control techniques are not suitable for dealing with the periodicbehaviour of a HAWT system. For this reason, the time-varying description (27) needs to be re-formulated as a timeinvariant model by introducing the multi-blade coordinate transformation, also referred as Coleman transformation. MBCis a mathematical tool for three-bladed rotors that allows for mapping the dynamic variables described in local bladecoordinates into a non-rotating reference frame. Assuming that the rotor is isotropic, i.e. all blades are identical, identicallypitched and symmetrically mounted on the hub, the main idea is to refer the motions of individual blades in the samecoordinate system as the structure supporting the rotor [22]. In this way, the periodic terms in the governing equations areeliminated. The rotating frame degrees of freedom vector qðtÞ is expressed as a function of the non-rotating frame degreesof freedom vector qnrðtÞ by

qðtÞ ¼ PðtÞqnrðtÞ, P¼

1 cosðC1Þ sinðC1Þ 0

1 cosðC2Þ sinðC2Þ 0

1 cosðC3Þ sinðC3Þ 0

0 0 0 1

26664

37775 (33)

where qnrðtÞ denotes the vector of generalized coordinates in the non-rotating frame. Since the motion of tower/nacelle in(27) is described in the ground fixed frame, no transformation is required for nacelle edgewise displacement q4ðtÞ. From(33), transformation in multi-blade coordinates is obtained as

qnrðtÞ ¼ P�1ðtÞqðtÞ, P�1

¼

13

13

13 0

23 cosðC1Þ

23 cosðC2Þ

23 cosðC3Þ 0

23 sinðC1Þ

23 sinðC2Þ

23 sinðC3Þ 0

0 0 0 1

266664

377775 (34)

By using Eqs. (33), (34) in (27), and omitting time dependency for notational simplicity, equations of motion in Colemandomain are given by

P�1 Md2½Pqnr�

dt2þC

d½Pqnr�

dtþKPqnr

!¼ P�1Q ext (35)

d½Pqnr�

dt¼ _PqnrþP _qnr ¼ _q,

d2½Pqnr�

dt2¼ €Pqnrþ2 _P _qnrþP €qnr ¼ €q (36)

Since _CjðtÞ ¼O 8j, we get

_P ¼OP1, P1 ¼

0 �sinðC1Þ cosðC1Þ 0

0 �sinðC2Þ cosðC2Þ 0

0 �sinðC3Þ cosðC3Þ 0

0 0 0 0

26664

37775 (37)

€P ¼ _OP1þO2P2, P2 ¼

0 �cosðC1Þ �sinðC1Þ 0

0 �cosðC2Þ �sinðC2Þ 0

0 �cosðC3Þ �sinðC3Þ 0

0 0 0 0

26664

37775 (38)

Eq. (35) can be rewritten as

P�1ðMð €Pqnrþ2 _P _qnrþP €qnrÞþCð _PqnrþP _qnrÞþKPqnrÞ ¼ P�1Q ext (39)

which leads to the edgewise model in the Coleman domain (i.e. in the non-rotating frame) as

Mc €qnrþCc _qnrþKcqnr ¼Q cext(40)

with

Mc ¼ P�1MP

Cc ¼ 2OP�1MP1þP�1CP

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–1256 1241

Kc ¼_OP�1MP1þO

2P�1MP2þOP�1CP1þP�1KP

Q cext¼ P�1Q ext (41)

Finally, substituting Eqs. (33)–(39) in (41) and assuming constant rotor angular speed (i.e. _O ¼ 0), the wind turbinemodel in the Coleman domain is defined by the following matrices:

Mc ¼

m2 0 0 0

0 m2 0 m1

0 0 m2 0

0 m1 0 23 m4

266664

377775

Cc ¼

cb 0 0 0

0 cb 2Om2 0

0 �2Om2 cb 0

0 0 0 23 c4

266664

377775

Kc ¼

k2Kw

2 0 0

Kw k2þKw2 �O

2m2 Ocb 0

0 �Ocb k2�Kw2 �O

2m2 0

0 0 0 23 k4

2666664

3777775 (42)

It should be noted that, by applying the Coleman transformation to the HAWT edgewise model proposed, skew-symmetricdamping matrix is achieved, which confirms the property of energy conservation in the autonomous system.

4. Wind loading: blade element momentum theory

In this paper, in order to have a realistic estimate of the wind loading to which the rotor is subjected to, models basedon the Blade Element Momentum (BEM) theory have been adopted [18]. These models allow to obtain a detailedquantitative description of the wind turbine rotor behaviour, which is based on the aerodynamic properties of the bladesection airfoils, the geometrical characteristics of the rotor, as well as the wind speed and the rotational velocity of theblades. BEM analysis is carried out by combining momentum theory and blade element theory.

The blade is assumed to be discretized into N sections (elements). Each element is located at a radial distance r from thehub (Fig. 5), and it has chord length c¼ cðrÞ and width dr. The rotor has radius L and angular velocity O. Assuming no radialdependency for the annular sections, i.e. no aerodynamic interactions between different elements, and assuming that theforces on the blade elements depend only on the lift and drag characteristics of the airfoil shape of the blades, the BEMtheory provides a method to estimate the axial and tangential induction factors, a and a0, respectively. Once theseparameters are known, local loads on each segment can be determined. The total forces acting on the blade can then becomputed by performing numerical integration along the blade span. The rotational sampled turbulence spectra is non-homogenous in nature. However, for simplicity an isotropic, homogenous turbulence has been assumed over the rotorfield, corresponding to the turbulence represented at the hub height for illustration of the application of the controllers inthe paper. In order to describe the BEM algorithm for calculating quasi-static aerodynamic wind loads, the followingquantities are defined:

V relðr,tÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðVoðr,tÞð1�aÞþ ~wðtÞÞ2þO2r2ð1þa0Þ2

q

fðr,tÞ ¼ tan�1 ð1�aÞVoðr,tÞþ ~wðtÞ

ð1þa0ÞOr

� �

aðr,tÞ ¼fðr,tÞ�bðtÞ�kðrÞ (43)

where V rel and Vo denote the relative and the instantaneous wind speed, respectively, f is the flow angle, a theinstantaneous local angle of attack, b the pitch angle and k the local pre-twist of the blade (Fig. 6). The quantity ~w

represents the stochastic (turbulent) component of the wind flow on the rotor plane and has been added to the steadywind field impacting on the rotor.

The local lift and drag forces can be respectively computed as

pLðr,tÞ ¼1

2rV2

relðr,tÞcðrÞClðaÞ

pDðr,tÞ ¼1

2rV2

relðr,tÞcðrÞCdðaÞ (44)

Fig. 5. Blade model according to the BEM theory approach.

Fig. 6. Local forces and velocities in the BEM model of the blade.

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–12561242

where r is the density of air and ClðaÞ, CdðaÞ represent the lift and drag coefficients, respectively, whose values depend on thelocal angle of attack. Finally, the aerodynamic forces normal to and tangential to the rotor plane (corresponding to theaerodynamic loads in the flapwise and edgewise direction, respectively) can be obtained by projecting the lift and the dragalong the normal and the tangential planes, as shown in Fig. 6. Therefore, the local flapwise and edgewise loads are given by

pNðr,tÞ ¼ pLðr,tÞ cosðfÞþpDðr,tÞ sinðfÞ (45a)

pT ðr,tÞ ¼ pLðr,tÞ sinðfÞ�pDðr,tÞ cosðfÞ (45b)

As suggested in [18], in order to improve the accuracy of the model, Prandtl’s tip loss factor and Glauert correction havebeen applied. The former corrects the assumption, used in the classical blade element momentum theory, of an infinitenumber of blades, while the latter has been applied in order to compute the induced velocities more accurately when theinduction factor a is greater than a critical value ac.

The BEM algorithm for computing quasi-static aerodynamic wind loads for each blade element is described in AppendixB. Once the local loads on the blade elements have been calculated, by integrating (45b) along the blade length andconsidering the appropriate modeshape of the blade, the generalized edgewise load can be calculated using (17).

To account for the variation in the vertical wind shear due to the rotation of the blade, the term Vo in (43) can beapproximately assumed as a constant wind speed linearly varying with height.

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–1256 1243

5. Linear-quadratic control of edgewise vibration

In this paper, a linear quadratic approach has been used for the control of edgewise vibrations for the model describedabove. Since the model based only on the first mode captures the essential dynamics of the system, the design of thecontroller is based on the 4 degrees of freedom model formulated in Section 2. LQ algorithm is an optimal control strategyfor linear systems in the state-space domain that guarantees closed-loop stability and robustness. Consider the infinite-horizon LQ control design framework, given an n-th order stabilizable linear system in the form

_xðtÞ ¼AxðtÞþBuðtÞ, tZ0, xð0Þ ¼ x0

A 2 Rn�n, B 2 Rn�m (46)

where xðtÞ 2 Rn is the state vector and uðtÞ 2 Rm is the control input vector. The objective is to determine the matrix gainG%

2 Rm�n such that the static, full-state feedback control law uðtÞ ¼�G%xðtÞ satisfies the following criteria:

�

The closed-loop system is asymptotically stable andR � the quadratic cost functional JðGÞ ¼ 1

2þ1

0 ½xðtÞTR1xðtÞþuðtÞTR2uðtÞ� dt is minimized.

R1 2 Rn�n and R2 2 R

m�m are the weighting matrices such that R1Z0 and R240; the former penalizes the distance ofsystem states from the equilibrium, while the latter penalizes the control input, in the minimization process. The main aimof the LQ regulator is to drive the states of the system from x0 towards the equilibrium by using a minimum amount ofenergy, according to a dynamics that can be influenced by choosing appropriate weighting matrices R1 and R2. If thecouple ðA,BÞ in (46) is controllable (or at least stabilizable, i.e. non-controllable modes are stable), the controller G% can beobtained in a closed form by solving a continuous-time algebraic Riccati equation. A detailed introduction to the optimalLQ control theory can be found in [23].

For the HAWT edgewise model considered in this work, state vector xðtÞ is assumed to be

xðtÞ ¼

x1

x2

x3

x4

x5

x6

x7

x8

2666666666666664

3777777777777775¼

q1

q2

q3

q4

_q1

_q2

_q3

_q4

2666666666666664

3777777777777775

(47)

The strategy described so far refers to LTI time-invariant systems, i.e. linear systems for which A and B matrices are notdepending on time. Therefore, in order to design an appropriate LQ regulator for the HAWT edgewise model proposed in(27), it is convenient to adopt a time-invariant representation of the system, that can be obtained by applying MBCtransformation, as shown in Section 3. To this end, Eq. (40) can be easily reconstructed to a state-space formulation (46).By defining X1 ¼ qnr, X2 ¼ _qnr, X1,X2 2 R

4, we get

_X1 ¼ X2

_X2 ¼ �M�1c KcX1 �M�1

c CcX2 þM�1c P�1

ðFþQ loadþQ gÞ

((48)

So for the case considered in this paper, the dynamic matrix Ac 2 R8�8 of the system in the non-rotating frame is given by

Ac ¼O4�4 I4�4

�M�1c Kc �M�1

c Cc

" #(49)

States in the rotating and non-rotating frame are related through the following transformation:

xðtÞ ¼ PxnrðtÞ, P ¼P O4�4

_P P

� �(50)

The control vector uðtÞ corresponding to the controlled forces applied to the blades is defined as

uðtÞ ¼

f 1

f 2

f 3

264

375 2 R3 (51)

Fig. 7. Scheme for active control of edgewise vibration.

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–12561244

and is related to the control vector unrðtÞ through the following expression:

uðtÞ ¼ PuunrðtÞ, Pu ¼

1 cosðC1Þ sinðC1Þ

1 cosðC2Þ sinðC2Þ

1 cosðC3Þ sinðC3Þ

264

375 2 R3�3 (52)

Therefore, the control matrix Bc 2 R8�3 of the system in the non-rotating frame is

Bc ¼O4�3

M�1c P�1UPu

" #U¼

1 0 0

0 1 0

0 0 1

0 0 0

26664

37775 (53)

Since the controllability matrix of the system associated with the pair ðAc,BcÞ has full rank, the resulting LTI model inthe Coleman domain is fully controllable. Therefore, once system matrices have been determined, LQ controllers based on(49) and (53) can be designed. The optimal control law in the non-rotating frame is

unrðtÞ ¼�G%

nrxnrðtÞ (54)

with G%

nr minimizing the cost functional

JðGnrÞ ¼1

2

Z þ10½xðtÞTR1xðtÞþuðtÞTR2uðtÞ� dt¼

1

2

Z þ10½xnrðtÞ

TRc1xnrðtÞþunrðtÞ

TRc2unrðtÞ� dt (55)

where Rc1 ¼ P

TR1P and Rc

2 ¼ PTuR2Pu. Using Eqs. (50), (52), the control action for the HAWT in the rotating frame is

uðtÞ ¼ PuunrðtÞ ¼ �PuG%

nrxnrðtÞ ¼�PuG%

nrP�1|fflfflfflfflfflffl{zfflfflfflfflfflffl}

G%

ðCjÞ

xðtÞ (56)

The resulting feedback gain implemented is therefore, periodic since it depends on the azimuthal angle of blades,through the transformation matrices Pu and P

�1. A block scheme of the proposed control system is shown in Fig. 7.

Full-state measurements are fed back to the controller and the LQ control action in the non-rotating frame is computed.The values obtained are, then, transformed back in the original domain and applied to the HAWT edgewise model.

For the purpose of comparison, the LQ control strategy has been also directly applied to the system in the rotatingframe, without performing the MBC transformation. In this case, by considering (29) at the initial time instant t¼0, anominal model is derived. The design of the control law is then based on the nominal model, and the time-varyingdynamics is taken into account by using the feedback of the states.

6. Results

The proposed LQ strategy has been implemented in Matlab and tested on the edgewise vibration model derived in thispaper. In particular, the control based on the 4DOF model is here applied to a 7DOF wind turbine model, which includestwo vibration modes for each blade in the edgewise direction. Specifications of the NREL offshore 5-MW baseline windturbine [24] have been considered for model building and control system simulation testing. The details of the 5-MW windturbine are provided in Table 1.

The blade considered is the LM61.5 P2 (manufactured by LM Wind Power), which is 61.5 m long and it has a total massof 17 740 kg. Since the radius of the hub is 1.5 meters, the total rotor radius is L¼63 m. The first two mode shapes of theconsidered blade are shown in Fig. 8. These have been computed from blade structural data (distributed mass andstiffness) by using Modes [25] which performs eigen-analysis to compute mode shapes and frequencies. Since the mode

Table 1Properties of NREL 5-MW baseline HAWT [24].

NREL 5-MW baseline wind turbine properties

Basic description Max. rated power 5000 kW

Rotor orientation, configuration Upwind, 3 blades

Rotor diameter 126 m

Hub height 90 m

Cut-in, rated, cut-out wind speed 3 m s�1, 11.4 m s�1, 25 m s�1

Cut-in, rated rotor speed 6.9 rpm, 12.1 rpm

Blade (LM 61.5 P2) Length 61.5 m

Overall (integrated) mass 17 740 kg

Second mass moment of inertia 11 776 kg m 2

1-st edgewise mode natural frequency 1.08 Hz

2-nd edgewise mode 4.05 Hz

Structural-damping ratio (all modes) 0.48%

Hubþnacelle Hub diameter 3 m

Hub mass 56 780 kg

Nacelle mass 240 000 kg

Tower Height above ground 87.6 m

Overall (integrated) mass 347 460 kg

1-st fore-aft mode natural frequency 0.32 Hz

Structural-damping ratio (all modes) 1%

Fig. 8. Mode shapes corresponding to the first and second mode of vibration of the blade.

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–1256 1245

shapes must have zero deflection and slope at the base (blades are modeled as cantilever beams), a sixth-order polynomialwith the coefficients of order ‘0’ and ‘1’ equal to 0 represents an admissible shape function. From Modes, the followingpolynomials have been obtained:

F1ðxÞ ¼�0:6952x6þ2:3760x5

�3:5772x4þ2:5337x3

þ0:3627x2

F2ðxÞ ¼�1:9678x6�3:1110x5

þ12:3693x4�5:0703x3

�1:2202x2 (57)

with x ¼ x=L and Fjð1Þ ¼ 1.A steady wind flow with turbulence has been simulated in order to investigate the HAWT model response. A steady

wind speed is considered, resulting in a periodic load in the edgewise direction, due to the variation in vertical wind shear.The additional turbulent velocity component has been generated at the hub height using a Kaimal spectrum.

6.1. Aerodynamic load

Aerodynamic load calculation has been performed using the blade element momentum theory, according to the algorithmdescribed in Appendix B. The computation is carried out using airfoil-data tables containing lift and drag curves for the aerofoilconsidered, as provided in [24]. The wind passing through the rotor-swept area is modeled as a constant mean wind velocity atthe hub in addition to a linear wind shear in the vertical direction producing a periodic loading variation. The periodcorresponds to the rotor angular velocity O. A mean wind speed value of V o ¼ 12 m s�1 at the hub has been used for

Fig. 9. Edgewise aerodynamic loads on blades and nacelle using BEM algorithm ðO¼ 1:2671 rad s�1Þ.

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–12561246

simulation. For the wind shear, a maximum change DVo ¼ 2 m s�1 in wind speed has been assumed in the vertical directionfrom the hub to the blade tip. Therefore, the resulting wind speed Voj

ðr,tÞ (43) for the j-th blade is given by

Vojðr,tÞ ¼ V oþDVo

r

LcosðCjÞ (58)

Furthermore, the rated value O¼ 12:1 rpm has been chosen for rotor angular velocity. Turbulence corresponding to the hubheight (90 m) is also included in the model. A 1-D fully coherent turbulence has been generated compatible with Kaimalspectra with parameters as in [21]. The intensity of the turbulence has been assumed to be 10%. The time series of turbulencehas been generated following the digital simulation algorithm with random phases as proposed by [26]. A screenshot of theedgewise aerodynamic loads acting on blades and nacelle subjected to a steady wind with homogenous isotropic turbulence isshown in Fig. 9.

It may be noted that the generalized load on the nacelle is given by a combination of the generalized loads on the blades(as in Eq. (18)) weighted by cosine terms. This results in the non-zero mean load (Fig. 9) in the final expression for thegeneralized load on the nacelle.

6.2. Control of vibrations

The task of designing a LQ regulator consists of appropriately tuning the weighing matrices R1 and R2. In the numericalstudy carried out in this paper, the weight R1 has been set to the identity matrix, that is same relative importance isassigned to the regulation error of each state variable. Different controllers have been synthesized by varying the weighingmatrix R2 in the LQ cost function. The weight on the control input is assumed in the form R2 ¼ gI, where g is a scalar andI is an identity matrix of order 3�3, i.e. the control variables are equally weighted in solving the optimization problem.Numerical simulations confirm that as the value of g is reduced, allowing larger values in the control effort, betterperformances are achieved (Fig. 10). The active control system achieves a significant reduction of the blade tipdisplacement. The maximum value for uncontrolled response exceeds 1.13 m, whereas for the controlled one themaximum deflection ranges between 0.009 m and 0.76 m for the different regulators considered. Furthermore, asignificant overall reduction is obtained in the vibrational response, as the root mean square (RMS) value of thedisplacement is shown to be remarkably smaller in the controlled cases.

For example, the results obtained by setting g¼ g1 ¼ 10�10 and g¼ g2 ¼ 10�11 are shown in Fig. 11. The maximumedgewise displacement for the first blade is reduced from 1.13 m to 0.39 m and 0.13 m, respectively. Furthermore, the RMSdisplacement for the controlled response exhibits a reduction of 56% (g¼ g1) and 85% ðg¼ g2Þ compared the uncontrolledone. Finally, the active control system provides a lower peak-to-peak excursion and the average deflection of the blade isalso significantly reduced along the time history of the edgewise response.

Fourier spectrum for blade 1 tip displacement (Fig. 12) shows that the proposed LQ regulator is effective in suppressingpeaks in the uncontrolled response due to the rotational effect. In fact, a substantial reduction is achieved in the responsecorresponding to the peak in the spectrum around 0.2 Hz. This is associated with the rotational speed of the blades and isalso the generalized load frequency. The controller is also effective in eliminating the peak corresponding to the naturalfrequency of the blade (around 1.08 Hz). Similar results have been confirmed for the other two blades.

No significant contribution is observed from the second mode, around 4.05 Hz. This has been also observed in theedgewise displacements of blade 2 and blade 3. The LQ controller provides a significant reduction of the low-frequencycomponents which mainly dominate the response, that is the contribution in vibration due to the rotational frequency andthe contribution around the natural frequency of the blade. Since no active control is operated on the nacelle, negligible

Fig. 10. Performances of the active controller for different tuning of the parameter g. (a) Maximum edgewise displacement, (b) RMS value of edgewise

displacement.

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–1256 1247

reduction is observed for the degree of freedom ~q4 (Fig. 13), corresponding to the nacelle response. However, in thepresent study the nacelle displacement is not a major issue since loads in the edgewise direction do not induce largemagnitude vibrations (the amplitude of the oscillation is of the order of a few centimeters) in the nacelle.

It is obvious that a smaller value for R2 provides better performances but, at the same time, it entails a higher controleffort. Also, as the angle W for the actuators is increased, a decrease in the control force requirement is observed.

For example, by selecting the controller corresponding to g¼ g1, assuming W¼ 201, the maximum magnitude of the forcerequired for achieving the performances in Fig. 11 is found to be about 50 kN (Fig. 14). This means that mounting actuators

Fig. 11. Blade 1 edgewise displacement (Coleman based LQ controller).

Fig. 12. Blade 1 edgewise frequency spectrum (Coleman based LQ controller).

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–12561248

capable of exerting a force of about 5 tons (� 28% of the blade weight), achieves for the case considered a reduction of 65% in themaximum blade displacement, of 56% in the RMS blade tip displacement and of 85% in the RMS blade tip acceleration.

Numerical results in Table 2 show the performances of the LQ-MBC controller (in terms of blade displacement and forcerequirement) by varying the control parameter g. The angle W between the actuators is set to 201. In each simulation, the7DOF edgewise vibration model is subjected to a steady turbulent wind for 150 s.

According to the numerical study carried out in this paper, the proposed approach is highly effective in reducing theedgewise vibration in wind turbine blades.

6.3. Comparison of control strategies

For the purpose of comparison, the closed loop responses of the system using LQ regulator based on the model in therotating frame (assuming t¼0 for the controller design) have been computed. From (29), it can be observed that the time-varying nature of the edgewise model is due to the variation of the azimuthal angle CjðtÞ of the blades over time.

Fig. 13. Nacelle edgewise displacement (Coleman based LQ controller). (a) Time history, (b) frequency spectrum.

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–1256 1249

Therefore, by setting, 8t, C1 ¼ 0 (which implies C2 ¼ 2p=3,C3 ¼ 4p=3), a time invariant system is obtained. The LQ staticgain is then designed based on this LTI model, which represents the HAWT with the first blade in the vertical uprightposition. In this case, the dynamic matrix A0 2 R

8�8 is given by

A0 ¼O4�4 I4�4

�M�10 K0 �M�1

0 C0

" #(59)

where

M0 ¼M9C1 ¼ 0

, K0 ¼K9C1 ¼ 0

, C0 ¼ C9C1 ¼ 0

(60)

Fig. 14. Control force on blade 1 (Coleman based LQ controller). (a) W¼ 151, (b) W¼ 201.

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–12561250

The control matrix B0 2 R8�3 is

B0 ¼O4�3

M�10 U0

" #, U0 ¼

1 0 0

0 1 0

0 0 1

0 0 0

26664

37775 (61)

Fig. 15. Blade 1 edgewise displacement using LQ regulators.

Table 2Performances of the Coleman transform based LQ controller (blade 1).

Case g Max displacement (m) RMS displacement (m) Peak force (kN) RMS force (kN)

Uncontrolled – 1.133 0.494 – –

LQ-MBC 1 10�9 0.766 0.391 41.8 14.7

LQ-MBC 2 10�10 0.398 0.201 55.7 25.9

LQ-MBC 3 10�11 0.137 0.073 70.5 34.7

LQ-MBC 4 10�12 0.043 0.023 79.1 38.3

LQ-MBC 5 10�13 0.015 0.006 82.3 39.5

LQ-MBC 6 10�14 0.009 0.004 83.4 39.9

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–1256 1251

The LQ control law based on the assumed nominal representation of the system is then

u0ðtÞ ¼ �G%

0xðtÞ (62)

where G%

0 is determined by solving the LQ control problem assuming A0 and B0 as system matrices.Even though no significant difference is observed, the controller based on the MBC transformation indeed offers slightly

better performances. By comparing the two control strategies assuming the same tuning parameters, a marginalimprovement in the response is achieved by the controller based on the Coleman transformed equations (Fig. 15).

For the simulation of active control system based on the LQ algorithm designed on the nominal representation of theedgewise model, the control parameter g has been set to 10�11 and a value of 201 has been considered for the structuralangle W. Numerical results (Fig. 16) show that the frequency content of blade 1 response and the corresponding input force

Fig. 16. Control of blade 1 using LQ based on the nominal representation of the HAWT. (a) Frequency spectrum, (b) control force.

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–12561252

required for implementing the control are comparable to the ones obtained by applying the LQ controller in the non-rotating frame.

An algorithm based on direct velocity feedback control has also been applied in this paper in order to suppressedgewise vibrations and to compare with the results from LQ controller. In this case, each blade is subjected to a controlforce which is proportional to the velocity of the edgewise vibration. A model-free control approach is therefore used, sincethe synthesis of the controller is not based on the mathematical model of the wind turbine but the control gains are simply

Fig. 17. Blade 1 edgewise displacement (DVF control).

Fig. 18. Blade 1 edgewise frequency spectrum (DVF control).

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–1256 1253

tuned through an iterative trial and error procedure. The edgewise displacement of the first blade using DVF is shown inFig. 17.

It can be seen (Fig. 18) that even though the DVF control provides a good reduction of the peaks corresponding to therotational speed and the natural frequency of the blade, it is not particularly effective in suppressing the average edgewisedeflection of the blade in comparison to the LQ controller. Indeed, the static deflection obtained by applying DVF iscomparable to the one corresponding to the uncontrolled response.

The control gains for the DVF have been chosen such that the control effort required from the actuators is comparable to theone computed for the LQ controller (Fig. 19). In particular, for the performances shown in Fig. 17, the maximum force requiredby the LQ regulator for the first blade is 70.5 kN, while for the DFV is 74.85 kN. The maximum peak to peak excursions are92.28 kN (LQ) and 124.62 kN (DVF). Finally, the RMS values are 35.76 kN and 27.33 kN for LQ and DVF, respectively.

7. Conclusions

In this paper, a new control scheme for suppressing wind-induced edgewise vibrations in large wind turbine blades hasbeen proposed. An innovative hardware configuration, adopting linear actuators/active tendons mounted inside the

Fig. 19. Control force on blade 1 (DVF control).

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–12561254

blades, has been considered in order to implement an active control strategy for reducing the edgewise vibrationsexperienced by the turbines. The control force can be appropriately applied by following a proposed MBC transformed LQcontrol algorithm for mitigating the damaging effects of edgewise vibrations in wind turbines.

A mathematical model of the wind turbine equipped with active devices has been formulated using a Euler–Lagrangianapproach. The control problem has been addressed first by transforming the dynamic model into an equivalent time-invariant representation using the multi-blade coordinate transformation. This allows for expressing the generalizedcoordinates into a non-rotating reference frame and hence eliminates the dependency of the equations of motion upon theazimuthal angles of the blades. The effectiveness of the proposed control scheme has been investigated by numericallyapplying two optimal control strategies based on the LQ framework and one based on DVF algorithm.

Numerical simulations have been carried out using realistic data from a 5-MW offshore wind turbine and flexiblemulti-modal turbine rotor model. The results indicate that the active control system is successfully able to reduce theblade responses. It has been shown that promising performances can be achieved in suppressing edgewise vibrations dueto steady state wind loads and under wind loads in turbulent condition. Depending on the tuning of the regulatorparameters, different levels of vibration reduction can be attained. Quantitative analysis of the case considered has shownthat, in comparison to the uncontrolled response, the active control system can provide a reduction of 65% in themaximum blade displacement by applying a force of about 28% of the blade weight. By applying this amount of force,excellent performances are achieved in the suppression of the edgewise vibrations (56% reduction in the RMSdisplacement and 85% reduction in the RMS acceleration). Also, for the case considered, the maximum peak to peakexcursion of the blade tip is reduced by 74%.

The main aim of the present study was to show how the proposed active control of a wind turbine blade could besuccessfully used for suppressing edgewise vibrations. Basic LQ and DVF controllers have been used as illustrativeapplications for establishing the validity of the proposed approach. Further investigations are needed for testing thebehaviour of the controlled system under uncertainty in the structural parameters (such as mass and stiffness) andconstraints on control forces.

Acknowledgments

This research is carried out under the EU FP7 ITN project SYSWIND (Grant No. PITN-GA- 2009-238325). The SYSWINDproject is funded by the Marie Curie Actions under the Seventh Framework Programme for Research and TechnologicalDevelopment of the EU. The authors are grateful for the support.

Appendix A. System matrices of the multi-modal flexible HAWT model

The mass matrix ~M of the wind turbine model considering N modes of vibration for each blade is defined as

~M ¼

M2 0 0 M1c1

0 M2 0 M1c2

0 0 M2 M1c3

M1Tc1

M1Tc2

M1Tc3

m4

266664

377775 2 Rð3Nþ1Þ�ð3Nþ1Þ (A.1)

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–1256 1255

with

M2 ¼

m21 0 . . . 0

0 m22 . . . ^

^ ^ & ^

0 0 0 m2N

266664

377775 2 RN�N , M1cj

¼

m11 cosðCjÞ

m12 cosðCjÞ

^

m1N cosðCjÞ

266664

377775 2 RN�1 (A.2)

The damping matrix, including structural damping, is given by

~C ¼ ~Cstructþ

0 0 0 0

0 0 0 0

0 0 0 0

�2OM1Tc1�2OM1

Tc2�2OM1

Tc3

0

266664

377775 (A.3)

with

~Cstruct 2 Rð3Nþ1Þ�ð3Nþ1Þ, M1cj¼

m11 sinðCjÞ

m12 sinðCjÞ

^

m1N sinðCjÞ

266664

377775 2 RN�1 (A.4)

The stiffness matrix is

~K ¼

K2þKwc10 0 0

0 K2þKwc20 0

0 0 K2þKwc30

�O2M1Tc1�O2M1

Tc2�O2M1

Tc3

k4

2666664

3777775 2 Rð3Nþ1Þ�ð3Nþ1Þ (A.5)

with

K2 ¼

Ke,11þO2ðKg0;11�m21Þ O2Kg0;12 . . . O2Kg0;1N

O2Kg0;21 Ke,22þO2ðKg0;22�m22Þ . . . ^

^ ^ & ^

O2Kg0,N1 O2Kg0,N2 . . . Ke,NNþO2ðKg0,NN�m2NÞ

2666664

3777775 2 RN�N

Kwcj¼

Kw,11 cosðCjÞ Kw,12 cosðCjÞ . . . Kw,1N cosðCjÞ

Kw,21 cosðCjÞ Kw,22 cosðCjÞ . . . ^

^ ^ & ^

Kw,N1 cosðCjÞ Kw,N2 cosðCjÞ . . . Kw,NN cosðCjÞ

266664

377775 2 RN�N (A.6)

In (A.6), assuming orthogonal modes of vibration implies Ke,ik ¼ 0, 8 iak. Furthermore, (11) implies

Kg0,ik ¼ Kg0,ki Kw,ik ¼ Kw,ki (A.7)

This condition ensures symmetry for the blade stiffness matrices K2 and Kwcj.

Appendix B. Blade element momentum method

Algorithm 1. Wind loading computation (BEM method)

1: Initialize aðkÞ , a0ðkÞ, ac, B, g (e.g. að0Þ ¼ 0, a0ð0Þ ¼ 0, ac¼0.2, B¼3, g¼ 0:01)

2: Compute the flow angle f¼ tan�1 ð1�aðkÞÞVoþ ~w

ð1þa0ðkÞÞOr

� �3: Compute the angle of attack a¼f�b�k

4: Compute Prandtl’s tip loss correction F ¼2

p cos�1ðe�f Þ, f ¼B

2

L�r

r sinðfÞ5: Retrieve lift and drag coefficients ClðaÞ and CdðaÞ from airfoil data table for the calculated a6: Compute normal load coefficient CNðaÞ ¼ ClðaÞ cosðfÞþCdðaÞ sinðfÞ7: Compute tangential load coefficient CT ðaÞ ¼ ClðaÞ sinðfÞ�CdðaÞ cosðfÞ

8: Compute solidity sðrÞ ¼ cðrÞB

2pr9: if arac then

10: aðkþ1Þ ¼ 1

1þ4F sin2

ðfÞsCN

11: else

A. Staino et al. / Journal of Sound and Vibration 331 (2012) 1233–12561256

12: aðkþ1Þ ¼1

2½2þKð1�2acÞ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKð1�2acÞþ2Þ2þ4ðKa2

c�1Þq

�, K ¼4F sin2

ðfÞsCN

13: endif

14: a0ðkþ1Þ ¼ 1

�1þ4F sinðfÞcosðfÞ

sCT

15: if ðaðkþ1Þ�aðkÞ4gÞ ða0ðkþ1Þ�a0ðkÞ4gÞ16: k¼ kþ1, GOTO ¼)2

17: else

18: Obtain the axial and tangential induction factors a¼ aðkÞ , a0 ¼ a0ðkÞ

19: endif

20: Compute V rel ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðð1�aÞVoþ ~wÞ2þO2r2ð1þa0Þ2

q21: Compute pN ¼

1

2rV2

relcCNðaÞ, pT ¼1

2rV2

relcCT ðaÞ

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