Actuator control of edgewise vibrations in wind turbine blades

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

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Actuator control of edgewise vibrations in wind turbine blades A. Staino a, B. Basu a,n, S.R.K. Nielsen b a b

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

a r t i c l e i n f o

abstract

Article history: Received 16 March 2011 Received in revised form 29 October 2011 Accepted 4 November 2011 Handling Editor: D.J. Wagg Available online 6 December 2011

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 edgewise vibrations. The control is based on a pair of actuators/active tendons mounted 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 timevarying 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-megawatt wind turbines is increasing, the blades are becoming more flexible and hence are subjected to vibrations induced by

n

Corresponding author. Fax: þ 353 1 6773072. E-mail addresses: [email protected] (A. Staino), [email protected] (B. Basu), [email protected] (S.R.K. Nielsen).

0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.11.003

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external wind loading and tower interactions. Because of the impact on the mechanical components and the fatigue induced in the blades, uncontrolled vibrations could sometimes cause structural/mechanical damage, leading to a significant 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 of significant concern in wind turbines, as this mode is lightly damped and can lead to violent vibrations [3,4]. In fact, under certain conditions the first edgewise mode may exhibit a very low or even negative damping, i.e. the total damping due to structural properties and aerodynamic effects can be less than zero. This corresponds to the case in which the aerodynamic forces supply energy to the vibrating system causing large amplitude vibrations [5]. As a result, the blades and the tower may 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 of the 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 the structure to wind-induced oscillations. Passive control techniques have been investigated for structural control of both onshore and offshore wind turbines [1,10]. A semi-active method based on tuned mass dampers is described in [2] for the control of flapwise vibrations in wind turbine blades. Active control strategies have also been the focus of attention very recently. Studies on the use of synthetic jet actuators [11], microtabs and trailing edge flaps [12,13] have been considered by the researchers. Use of active strut elements based on resonant controllers [14] inspired by the concept of tuned mass dampers has been proposed for active control of vibrations in wind turbines. Individual pitch control for reducing loads on the wind turbine structure has also been investigated [15,16], even though this solution requires careful design of the control 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 to suppress 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 a Lagrangian–Eulerian approach, based on energy considerations. The use of active control devices (two actuators/active tendons generating an edgewise control force), located inside each blade, is considered in order to suppress vibrations and mitigate their damaging effects. To this end, the model is formulated by introducing controllable forces acting on the blades that can be varied according to a prescribed control law. The effect of centrifugal stiffening and gravity has been considered. Quasi static aerodynamic wind loading conditions are also modeled and time-series of the loads are computed by applying the corrected blade element momentum method [18]. The wind loading contains harmonic components due to vertical wind shear and fluctuating components due to turbulence. Multi-blade coordinate transformation is used on a reduced order model for modal analysis and control design [19]. Linear quadratic regulators are designed and tested based on numerical simulations, which show that a significant reduction in blade response can be achieved by means of the proposed active control system. Two competing LQ controllers, one synthesized based on the time varying system directly and the other on the Coleman transformed non-rotating system, are investigated and compared. Furthermore, the vibration 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 several components, including structural elements like tower, rotor (consisting of nacelle and blades) and other mechanical and 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 getting accurate models entails studying the dynamics of many degrees of freedom (DOFs), leading to a high dimensional set of equations. Because we are interested in studying the edgewise dynamics of rotor vibrations in a wind turbine, here we formulate a mathematical model that takes into account only the relevant states or degrees of freedom, representing the edgewise vibration responses and the associated coupling of the blade with the tower/nacelle motion [21]. A schematic representation of a three-bladed HAWT is shown in Fig. 1. The blades are modeled as Bernoulli–Euler cantilever beams of length ‘L’, with variable bending stiffness EI(x) and variable mass per unit length mðxÞ along the length. The blades rotate at a constant rotational speed O rad s  1 and the azimuthal angle Cj ðtÞ of blade ‘j’ at the time instant ‘t’ is given by

Cj ðtÞ ¼ C1 ðtÞ þ ðj1Þ

2p , 3

C1 ðtÞ ¼ Ot;

j ¼ 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 the tower and the mass of the nacelle. The variables u~ j ðx,tÞ, j ¼1,2,3 and u~ 4 ðtÞ denote the edgewise blade and nacelle displacements, respectively. The generalized (or modal) stiffness of the tower is represented by k4.

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

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Fig. 1. Wind turbine model for edgewise vibration.

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 used in the numerical simulations to assess the performances of the controller. The controller however will be designed on a reduced order model. In the generalized representation of the wind turbine, each mode of vibration is associated to the corresponding modeshape Fi ðxÞ, for which an appropriate function approximation can be computed from the eigenanalysis of the blade structural data. The system is therefore described by 3N þ1 generalized coordinates, that provides an ~ accurate description of the flexible blade behaviour. Let qðtÞ be the vector of the generalized coordinates of the system defined as 2~ 3 q 11 ðtÞ 6 q~ ðtÞ 7 6 12 7 6 7 6 ^ 7 6 7 6 q~ ðtÞ 7 6 1N 7 6 7 3N þ 1 ~ 7 ~ ¼6 qðtÞ 6 q 21 ðtÞ 7 2 R 6 ^ 7 6 7 6 7 6 q~ ji ðtÞ 7 6 7 6 ^ 7 4 5 q~ 4 ðtÞ

(2)

The degree of freedom q~ ji ðtÞ, j ¼ 1; 2,3, i ¼ 1, . . . ,N denotes the i-th edgewise mode for the blade ‘j’. The variable q~ 4 ðtÞ ¼ u~ 4 ðtÞ represents the motion of the nacelle in the rotor plane. The total edgewise displacement along the blade is given by u~ j ðx,tÞ ¼

N X

Fi ðxÞq~ ji ðtÞ

(3)

i¼1

~ q~_ ðtÞÞ of a dynamical system is a function of the generalized coordinates In classical mechanics, the Lagrangian LðqðtÞ, and their time derivatives, representing the difference between the kinetic and the potential energies ~ ~ ~ q~_ ðtÞÞVðqðtÞÞ q~_ ðtÞÞ ¼ T ðqðtÞ, LðqðtÞ,

(4)

where T and V denote the kinetic and potential energy, respectively. The equations of motion of the system are given by ! ~ ~ d @LðqðtÞ, q~_ ðtÞÞ @LðqðtÞ, q~_ ðtÞÞ  ¼ Q~ ext,i ðtÞ (5) dt @q~ i ðtÞ @q~_ i ðtÞ Eq. (5) is known as the Euler–Lagrange equations; Q~ ext,i ðtÞ denotes the i-th component of Q~ ext ðtÞ which is the vector of generalized non-conservative (i.e. dissipative or external) loads (forces/torques) transferred to the system. For simplicity in notation, we will omit the dependency on time of generalized coordinates and loads, as well as the dependency of L, T and V on q~ and q~_ .

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2.1.1. Kinetic energy Let v~ b,j ðx,tÞ denote the total velocity of blade ‘j’ at a distance ‘x’ from the hub in the edgewise direction, including the nacelle motion that causes the blade displacement (Fig. 1). The use of this variable in the formulation ensures inclusion of the coupling between the blade and the nacelle. The square of the total velocity for blade ‘j’ is !2 !2 N N X X v~ 2 ¼ q~_ sinðC ÞO F q~ þ q~_ cosðC Þ þ Ox þ F q~_ (6) 4

b,j

j

i ji

j

4

i

i¼1

ji

i¼1

The total kinetic energy of the system (i.e. the three blades and the tower) is given by 2 !2 !2 3 3 Z L N N X X 1X 1 _ _ _ 4 mðxÞ q~ 4 sinðCj ÞO Fi q~ ji þ q~ 4 cosðCj Þ þ Ox þ Fi q~ ji 5 dx þ M0 q~_ 24 T ¼ 2j¼1 0 2 i¼1 i¼1

(7)

2.1.2. Potential energy The total potential (strain) energy of the system V is obtained by considering the potential energy of the cantilever blade in bending, the contribution from centrifugal stiffening [22], the contribution given from the component of the gravity along the blade axis (Fig. 4), and the potential energy of the nacelle. The centrifugal force on blade acting at the point ‘x’ from the hub is Z L F c ðxÞ ¼ O2 mðxÞx dx (8) x

where ‘x’ is the distance from ‘x’ to the current element considered. Similarly, for the j-th blade, the effect of the component of the gravitational force acting along the blade at a distance ‘x’ from the blade root is Z L Z L F g,j ðxÞ ¼  mðxÞg cosðCj Þ dx ¼ g cosðCj Þ mðxÞ dx (9) x

x

The total potential strain energy can be modeled as V¼

3 1 1X k4 q~ 24 þ 2 2j¼1

N X N X

! ðK e,ik þ K w,ik cosðCj Þ þ K g,ik Þq~ ji q~ jk

(10)

i¼1k¼1

where K e,ik ¼

K g0,ik ¼

Z

L 0

Z

L

EIðxÞ½Fi 00 Fk 00  dx,

0

Z

L x



K g,ik ¼

Z

L 0

F c ðxÞ½Fi 0 Fk 0  dx ¼ O2 K g0,ik

mðxÞx dx ½Fi 0 Fk 0  dx K w,ik ¼ g

Z 0

L

Z

L x



mðxÞ dx ½Fi 0 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 the relevant axis and Fi 0 ðxÞ, Fi 00 ðxÞ, 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 control scheme, the active vibration control is implemented by means of two linear actuators (or active tendons) located inside the blade (Fig. 2). The actuators/tendons are mounted on a frame supported from the nacelle. Vector analysis of the equilibrium of forces transmitted to the blade results in a net control force acting on the blade tip in the edgewise direction. For the j-th blade, the net force from the actuators/tendons is proportional to the force Tj(t) and the sine of the angle W. In the mathematical framework used in this study, the active control force is modeled as an external force acting on each blade tip and is given by f j ðtÞ ¼ 2T j ðtÞ sinðWÞ. The reaction forces are transmitted along the supporting structure finally to the nacelle. The support structure for applying the control forces has to satisfy the requirement of transferring the force to the hub. This has to be accomplished ideally by avoiding the generation of a reaction force in the edgewise direction of the blade or practically by eliminating the possibility of any reaction force in the close to medium spatial proximity of the tip. This design condition can be achieved by introducing active elements in the support structure (such as active braces or active tendons) as is typically used in large engineering structures for protection against wind or earthquake loads. Such a system with active tendon control 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. The active elements (active tendon in this case) produce forces which are external to the support structure and hence nullify the forces in the edgewise direction (e.g. the net edgewise load at joints A or B is identically zero).

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

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

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

2.3. Generalized loads In the edgewise model formulation, wind, gravity loadings and active control forces have been considered. Wind excitation is modeled as an external modal load applied to the blade in the edgewise direction, while a variable control force f j , j ¼ 1; 2,3 is applied to each blade in order to mitigate vibrations. ~ done by external active control forces on the blades and nacelle is given by The total virtual work dW ~ ¼ dW

3 X

f j ðduj ðL,tÞ þ dq~ 4 cosðCj ÞÞ

j¼1

3 X

f j dq~ 4 cosðCj Þ ¼

j¼1

3 X N X

f j dq~ ji

(12)

j¼1i¼1

and the corresponding generalized force vector F~ is ~ ~ ¼ dW FðtÞ dq~

(13)

The generalized controlled force on the blade ‘j’ for the i-th mode, then, corresponds to the control force fj, while the resulting generalized control force on the nacelle is zero. ~ L done by the external wind load is The virtual work dW ! 3 N X X ~ L¼ dW Qj Fi dq~ ji þ dq~ 4 cosðCj Þ (14) j¼1

i¼1

where Qj ¼

Z

L 0

pj ðx,tÞ dx,

j ¼ 1, . . . ,3

(15)

with pj ðx,tÞ representing the variable wind load intensity along the blade length in the edgewise direction. ~ L with respect to the generalized coordinates, the generalized loads result in Differentiating the virtual work dW ~ L dW Q~ load ðtÞ ¼ dq~

(16)

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

Therefore, the generalized aerodynamic load on the blade ‘j’ for the i-th mode is computed as Z L pj ðx,tÞFi ðxÞ dx Q ji ¼

(17)

0

and the generalized load on the nacelle corresponds to Q4 ¼

3 Z X j¼1

L

pj ðx,tÞ dx cosðCj Þ

0

(18)

Furthermore, the load due to gravity has also been considered. The component of the gravity force acting in the edgewise direction on an element of length dx of the blade ‘j’ (Fig. 4) is dF g,j ¼ mðxÞ dx g sinðCj Þ

(19)

~ g , is obtained as The total virtual work done due to the gravity, dW "Z !# Z 3 N 3 X N L X X X ~ dW g ¼ mðxÞg sinðCj Þ dx Fi dq~ ji þ dq~ 4 cosðCj Þ ¼ g dq~ ji j¼1

since

P3

j¼1

0

i¼1

j¼1i¼1

L 0

mðxÞFi dx sinðCj Þ

sinðCj Þ cosðCj Þ ¼ 0. Differentiating with respect to the generalized displacements vector Z L ~ g dW , Q g,ji ¼ g mðxÞFi dx sinðCj Þ, Q g,4 ¼ 0 Q~ g ðtÞ ¼ dq~ 0

(20)

(21)

where Q g,ji and Q g,4 are the components of Q~ g and represent the generalized gravitational load on the blade ‘j’ for the i-th mode and on the nacelle, respectively. For the considered system the total generalized load in the Euler–Lagrange formulation is given by Q~ ext ðtÞ ¼ F~ þ Q~ load þ Q~ g

(22)

2.4. Euler–Lagrange equations The Euler–Lagrangian equation (5) for the system considered is ! 8 > d @T @T @V > > þ ¼ f j þQ ji þ Q g,ji ,  > > @q~ ji @q~ ji < dt @q~_ ji ! > d @T @T @V > > > þ ¼ Q4  > : dt @q~_ @q~ 4 @q~ 4 4

j ¼ 1; 2,3, i ¼ 1, . . . ,N (23)

By introducing the quantities m1i ¼

Z 0

L

mðxÞFi dx, m2i ¼

Z

L 0

mðxÞF2i dx, m4 ¼ 3

Z 0

L

mðxÞ dx þM0

(24)

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

1239

RL and assuming orthogonal modeshapes such that mf,ik ¼ 0 mðxÞFi Fk dx ¼ 0, the equations of motion for the considered wind turbine edgewise vibration model with 3N þ1 degrees of freedom can be written as ~ q~_ þ KðtÞ ~ ~ q~ ¼ F~ þ Q~ ~ q~€ þ CðtÞ MðtÞ load þ Q g

(25)

~ K ~ C, ~ are provided in Appendix A. Details on the system matrices M, 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 states required for implementing the control and hence to decrease the computational cost associated with the calculation of the control 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 3 2 3 2 q1 ðtÞ q~ 11 ðtÞ 7 6 7 6~ 6 q2 ðtÞ 7 6 q 21 ðtÞ 7 7 7¼6 (26) qðtÞ ¼ 6 6 q3 ðtÞ 7 6 q~ ðtÞ 7 4 5 4 31 5 q4 ðtÞ q~ 4 ðtÞ The equations of motion of the wind turbine are reduced to MðtÞq€ þCðtÞq_ þ KðtÞq ¼ F þQ load þQ g

(27)

The reduced system matrices in (27) are derived from the matrices of the model with higher modes (Appendix A) by considering the fundamental mode of vibration only. By defining the following quantities in order to simplify notations: m2 ¼ m21 ,

m1 ¼ m11 ,

K e ¼ K e,11 ,

K g,0 ¼ K g0;11

the matrices of the reduced order model can be written as 2 0 m2 6 0 m2 6 MðtÞ ¼ 6 6 0 0 4 m1 cosðC1 Þ m1 cosðC2 Þ 2 6 6 CðtÞ ¼ 6 6 4 2 6 6 KðtÞ ¼ 6 6 4

K w ¼ K w,11 (28)

0 0 m2 m1 cosðC3 Þ

m1 cosðC1 Þ

3

0

3

3

7 m1 cosðC2 Þ 7 7 m1 cosðC3 Þ 7 5 m4

cb

0

0

0

cb

0

7 07 7 07 5 c4

0

0

cb

2Om1 sinðC1 Þ

2Om1 sinðC2 Þ

2Om1 sinðC3 Þ

k2 þK w cosðC1 Þ

0

0

0

0

k2 þ K w cosðC2 Þ

0

0

k2 þ K w cosðC3 Þ

7 07 7 07 5 k4

0 2

O m1 cosðC1 Þ

2

O m1 cosðC2 Þ

2

O m1 cosðC3 Þ

(29)

where k2 ¼ K e þ O2 K g,0 O2 m2 , 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 the blade, K g ¼ O2 K g,0 is the geometrical stiffness and Kw is the stiffness arising out of gravity effects. The term Ke can be expressed as K e ¼ o2b m2

(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 higher modes and can be written as 2 3 2 3 2 3 2 3 2 3 Q g,1 Q g,11 Q1 Q 11 f1 6 7 6 7 6Q 7 6Q 7 6f 7 6 Q 2 7 6 Q 21 7 6 g,2 7 6 g,21 7 6 27 7¼6 7, Q g ðtÞ ¼ 6 7¼6 7 FðtÞ ¼ 6 7, Q load ðtÞ ¼ 6 (31) 6 7 6 7 6 7 6 7 4f3 5 4 Q 3 5 4 Q 31 5 4 Q g,3 5 4 Q g,31 5 Q g,4 Q g,4 Q4 Q4 0

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The total generalized load in the reduced order formulation is therefore Q ext ðtÞ ¼ F þ Q load þ Q g

(32)

3. Coleman transformed equation Wind turbine equations of motion in the edgewise direction (27) contain periodic term, that depends on the azimuthal angle of blades. Classical time-invariant analysis and control techniques are not suitable for dealing with the periodic behaviour of a HAWT system. For this reason, the time-varying description (27) needs to be re-formulated as a time invariant model by introducing the multi-blade coordinate transformation, also referred as Coleman transformation. MBC is a mathematical tool for three-bladed rotors that allows for mapping the dynamic variables described in local blade coordinates into a non-rotating reference frame. Assuming that the rotor is isotropic, i.e. all blades are identical, identically pitched and symmetrically mounted on the hub, the main idea is to refer the motions of individual blades in the same coordinate system as the structure supporting the rotor [22]. In this way, the periodic terms in the governing equations are eliminated. The rotating frame degrees of freedom vector qðtÞ is expressed as a function of the non-rotating frame degrees of freedom vector qnr ðtÞ by 2 3 1 cosðC1 Þ sinðC1 Þ 0 6 1 cosðC Þ sinðC Þ 0 7 2 2 6 7 qðtÞ ¼ PðtÞqnr ðtÞ, P ¼ 6 (33) 7 4 1 cosðC3 Þ sinðC3 Þ 0 5 0

0

0

1

where qnr ðtÞ 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 q4 ðtÞ. From (33), transformation in multi-blade coordinates is obtained as 2 3 1 1 1 0 3 3 3 62 7 6 3 cosðC1 Þ 23 cosðC2 Þ 23 cosðC3 Þ 0 7 7 (34) qnr ðtÞ ¼ P1 ðtÞqðtÞ, P1 ¼ 6 62 7 4 3 sinðC1 Þ 23 sinðC2 Þ 23 sinðC3 Þ 0 5 0 0 0 1 By using Eqs. (33), (34) in (27), and omitting time dependency for notational simplicity, equations of motion in Coleman domain are given by ! 2 d ½Pqnr  d½Pqnr  1 þ KPqnr ¼ P1 Q ext P M þC (35) dt dt 2 2

d½Pqnr  _ _ _ ¼ Pq nr þ Pq nr ¼ q, dt _ ðtÞ ¼ O 8j, we get Since C j

d ½Pqnr  € __ € € ¼ Pq nr þ 2P q nr þ Pq nr ¼ q dt 2

2

P_ ¼ OP1 ,

0 60 6 P1 ¼ 6 40 2

_ P1 þ O2 P2 , P€ ¼ O

3

sinðC1 Þ

cosðC1 Þ

0

sinðC2 Þ

cosðC2 Þ

sinðC3 Þ

cosðC3 Þ

07 7 7 05

0

0

0

(37)

0 3

0

cosðC1 Þ

sinðC1 Þ

0

60 6 P2 ¼ 6 40

cosðC2 Þ

sinðC2 Þ

cosðC3 Þ

sinðC3 Þ

07 7 7 05

0

0

0

(36)

(38)

0

Eq. (35) can be rewritten as 1 € __ _ € _ Q ext P1 ðMðPq nr þ2P q nr þPq nr Þ þ CðPqnr þ Pq nr Þ þ KPqnr Þ ¼ P

(39)

which leads to the edgewise model in the Coleman domain (i.e. in the non-rotating frame) as Mc q€ nr þCc q_ nr þKc qnr ¼ Q cext with Mc ¼ P1 MP Cc ¼ 2OP1 MP1 þP1 CP

(40)

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

1241

_ P1 MP1 þ O2 P1 MP2 þ OP1 CP1 þP1 KP Kc ¼ O Q cext ¼ P1 Q ext

(41)

_ ¼ 0), the wind turbine Finally, substituting Eqs. (33)–(39) in (41) and assuming constant rotor angular speed (i.e. O model in the Coleman domain is defined by the following matrices: 2 3 0 0 0 m2 6 0 m 0 m1 7 6 7 2 7 Mc ¼ 6 6 0 7 0 m 0 2 4 5 2 0 m1 0 3 m4 2

cb 60 6 Cc ¼ 6 60 4 0 2

0

0

cb

2Om2

2Om2

cb

0

0

3

0 7 7 7 0 7 5 2 3 c4

Kw 2

0

k2 þ K2w O2 m2

Ocb

Ocb

k2  K2w O2 m2

0

0

k2

6 6 Kw 6 Kc ¼ 6 6 0 4 0

0

0

3

7 0 7 7 7 0 7 5 2 3 k4

(42)

It should be noted that, by applying the Coleman transformation to the HAWT edgewise model proposed, skew-symmetric damping 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 based on the Blade Element Momentum (BEM) theory have been adopted [18]. These models allow to obtain a detailed quantitative description of the wind turbine rotor behaviour, which is based on the aerodynamic properties of the blade section airfoils, the geometrical characteristics of the rotor, as well as the wind speed and the rotational velocity of the blades. 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 the hub (Fig. 5), and it has chord length c ¼ cðrÞ and width dr. The rotor has radius L and angular velocity O. Assuming no radial dependency for the annular sections, i.e. no aerodynamic interactions between different elements, and assuming that the forces on the blade elements depend only on the lift and drag characteristics of the airfoil shape of the blades, the BEM theory provides a method to estimate the axial and tangential induction factors, a and a0 , respectively. Once these parameters are known, local loads on each segment can be determined. The total forces acting on the blade can then be computed by performing numerical integration along the blade span. The rotational sampled turbulence spectra is nonhomogenous in nature. However, for simplicity an isotropic, homogenous turbulence has been assumed over the rotor field, corresponding to the turbulence represented at the hub height for illustration of the application of the controllers in the paper. In order to describe the BEM algorithm for calculating quasi-static aerodynamic wind loads, the following quantities are defined: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ~ V rel ðr,tÞ ¼ ðV o ðr,tÞð1aÞ þ wðtÞÞ þ O2 r 2 ð1 þ a0 Þ2

fðr,tÞ ¼ tan1



~ ð1aÞV o ðr,tÞ þ wðtÞ ð1 þ a0 ÞOr

aðr,tÞ ¼ fðr,tÞbðtÞkðrÞ



(43)

where V rel and Vo denote the relative and the instantaneous wind speed, respectively, f is the flow angle, a the ~ instantaneous 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 steady wind field impacting on the rotor. The local lift and drag forces can be respectively computed as pL ðr,tÞ ¼

1 rV 2rel ðr,tÞcðrÞC l ðaÞ 2

pD ðr,tÞ ¼

1 rV 2rel ðr,tÞcðrÞC d ðaÞ 2

(44)

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

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

where r is the density of air and C l ðaÞ, C d ðaÞ represent the lift and drag coefficients, respectively, whose values depend on the local angle of attack. Finally, the aerodynamic forces normal to and tangential to the rotor plane (corresponding to the aerodynamic loads in the flapwise and edgewise direction, respectively) can be obtained by projecting the lift and the drag along the normal and the tangential planes, as shown in Fig. 6. Therefore, the local flapwise and edgewise loads are given by pN ðr,tÞ ¼ pL ðr,tÞ cosðfÞ þ pD ðr,tÞ sinðfÞ

(45a)

pT ðr,tÞ ¼ pL ðr,tÞ sinðfÞpD ðr,tÞ cosðfÞ

(45b)

As suggested in [18], in order to improve the accuracy of the model, Prandtl’s tip loss factor and Glauert correction have been applied. The former corrects the assumption, used in the classical blade element momentum theory, of an infinite number of blades, while the latter has been applied in order to compute the induced velocities more accurately when the induction 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 Appendix B. Once the local loads on the blade elements have been calculated, by integrating (45b) along the blade length and considering 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 be approximately assumed as a constant wind speed linearly varying with height.

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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 described above. Since the model based only on the first mode captures the essential dynamics of the system, the design of the controller is based on the 4 degrees of freedom model formulated in Section 2. LQ algorithm is an optimal control strategy for linear systems in the state-space domain that guarantees closed-loop stability and robustness. Consider the infinitehorizon LQ control design framework, given an n-th order stabilizable linear system in the form _ ¼ AxðtÞ þ BuðtÞ, xðtÞ A 2 Rnn ,

t Z 0,

xð0Þ ¼ x0

B 2 Rnm

(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 gain G 2 Rmn such that the static, full-state feedback control law uðtÞ ¼ G xðtÞ satisfies the following criteria: %

%

 The closed-loop system is asymptotically stable and R  the quadratic cost functional JðGÞ ¼ 12 0þ 1 ½xðtÞT R1 xðtÞ þ uðtÞT R2 uðtÞ dt is minimized. R1 2 Rnn and R2 2 Rmm are the weighting matrices such that R1 Z0 and R2 40; the former penalizes the distance of system states from the equilibrium, while the latter penalizes the control input, in the minimization process. The main aim of the LQ regulator is to drive the states of the system from x0 towards the equilibrium by using a minimum amount of energy, according to a dynamics that can be influenced by choosing appropriate weighting matrices R1 and R2 . If the couple ðA,BÞ in (46) is controllable (or at least stabilizable, i.e. non-controllable modes are stable), the controller G can be obtained in a closed form by solving a continuous-time algebraic Riccati equation. A detailed introduction to the optimal LQ control theory can be found in [23]. For the HAWT edgewise model considered in this work, state vector xðtÞ is assumed to be 2 3 2 3 q1 x1 6 7 6q 7 6 x2 7 6 2 7 6 7 6 7 6 x3 7 6 q3 7 6 7 6 7 6 7 6 7 6 x4 7 6 q4 7 7 6 7 xðtÞ ¼ 6 (47) 6 x5 7 ¼ 6 q_ 1 7 6 7 6 7 6 7 6_ 7 6 x6 7 6 q 2 7 6 7 6 7 6 x7 7 6 q_ 7 4 5 4 35 q_ 4 x8 %

The strategy described so far refers to LTI time-invariant systems, i.e. linear systems for which A and B matrices are not depending 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 MBC transformation, 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 q_ nr , X1 ,X2 2 R4 , we get ( _1¼ X X2 (48) _ 2 ¼ M1 Kc X1 M1 Cc X2 þM1 P1 ðF þQ X þQ Þ c

c

c

load

g

So for the case considered in this paper, the dynamic matrix Ac 2 R88 of the system in the non-rotating frame is given by " # O44 I44 Ac ¼ (49) M1 M1 c Kc c Cc States in the rotating and non-rotating frame are related through the following transformation:   P O44 xðtÞ ¼ Pxnr ðtÞ, P ¼ _ P P The control vector uðtÞ corresponding to the controlled forces applied to the blades is defined as 2 3 f1 6 7 uðtÞ ¼ 4 f 2 5 2 R3 f3

(50)

(51)

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

and is related to the control vector unr ðtÞ through the following expression: 2 3 1 cosðC1 Þ sinðC1 Þ 6 1 cosðC Þ sinðC Þ 7 33 uðtÞ ¼ Pu unr ðtÞ, Pu ¼ 4 2 2 5 2 R 1 cosðC3 Þ sinðC3 Þ Therefore, the control matrix Bc 2 R83 of the system in the non-rotating frame is 2 3 1 0 0 " # 60 1 07 O43 6 7 Bc ¼ U¼6 7 1 M1 P UP 40 0 15 u c 0

0

(52)

(53)

0

Since the controllability matrix of the system associated with the pair ðAc ,Bc Þ has full rank, the resulting LTI model in the 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 unr ðtÞ ¼ Gnr xnr ðtÞ

(54)

%

with Gnr minimizing the cost functional Z Z 1 þ1 1 þ1 JðGnr Þ ¼ ½xðtÞT R1 xðtÞ þuðtÞT R2 uðtÞ dt ¼ ½xnr ðtÞT Rc1 xnr ðtÞ þunr ðtÞT Rc2 unr ðtÞ dt 2 0 2 0 %

(55)

T

where Rc1 ¼ P R1 P and Rc2 ¼ PTu R2 Pu . Using Eqs. (50), (52), the control action for the HAWT in the rotating frame is 1

uðtÞ ¼ Pu unr ðtÞ ¼ Pu Gnr xnr ðtÞ ¼ Pu Gnr P xðtÞ |fflfflfflfflfflffl{zfflfflfflfflfflffl} %

%

(56)

G ðCj Þ %

The resulting feedback gain implemented is therefore, periodic since it depends on the azimuthal angle of blades, 1 through the transformation matrices Pu and P . 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 rotating frame, without performing the MBC transformation. In this case, by considering (29) at the initial time instant t ¼0, a nominal model is derived. The design of the control law is then based on the nominal model, and the time-varying dynamics 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 this paper. In particular, the control based on the 4DOF model is here applied to a 7DOF wind turbine model, which includes two vibration modes for each blade in the edgewise direction. Specifications of the NREL offshore 5-MW baseline wind turbine [24] have been considered for model building and control system simulation testing. The details of the 5-MW wind turbine 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 mass of 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 the considered blade are shown in Fig. 8. These have been computed from blade structural data (distributed mass and stiffness) by using Modes [25] which performs eigen-analysis to compute mode shapes and frequencies. Since the mode

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Table 1 Properties of NREL 5-MW baseline HAWT [24]. NREL 5-MW baseline wind turbine properties Basic description

Max. rated power Rotor orientation, configuration Rotor diameter Hub height Cut-in, rated, cut-out wind speed Cut-in, rated rotor speed

5000 kW Upwind, 3 blades 126 m 90 m 3 m s  1, 11.4 m s  1, 25 m s  1 6.9 rpm, 12.1 rpm

Blade (LM 61.5 P2)

Length Overall (integrated) mass Second mass moment of inertia 1-st edgewise mode natural frequency 2-nd edgewise mode Structural-damping ratio (all modes)

61.5 m 17 740 kg 11 776 kg m 1.08 Hz 4.05 Hz 0.48%

Hub þnacelle

Hub diameter Hub mass Nacelle mass

3m 56 780 kg 240 000 kg

Tower

Height above ground Overall (integrated) mass 1-st fore-aft mode natural frequency Structural-damping ratio (all modes)

87.6 m 347 460 kg 0.32 Hz 1%

2

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

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

F1 ðxÞ ¼ 0:6952x 6 þ 2:3760x 5 3:5772x 4 þ 2:5337x 3 þ 0:3627x 2 F2 ðxÞ ¼ 1:9678x 6 3:1110x 5 þ12:3693x 4 5:0703x 3 1:2202x 2

(57)

with x ¼ x=L and Fj ð1Þ 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 algorithm described in Appendix B. The computation is carried out using airfoil-data tables containing lift and drag curves for the aerofoil considered, as provided in [24]. The wind passing through the rotor-swept area is modeled as a constant mean wind velocity at the hub in addition to a linear wind shear in the vertical direction producing a periodic loading variation. The period corresponds to the rotor angular velocity O. A mean wind speed value of V o ¼ 12 m s1 at the hub has been used for

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Fig. 9. Edgewise aerodynamic loads on blades and nacelle using BEM algorithm ðO ¼ 1:2671 rad s1 Þ.

simulation. For the wind shear, a maximum change DV o ¼ 2 m s1 in wind speed has been assumed in the vertical direction from the hub to the blade tip. Therefore, the resulting wind speed V oj ðr,tÞ (43) for the j-th blade is given by r V oj ðr,tÞ ¼ V o þ DV o cosðCj Þ (58) L Furthermore, the rated value O ¼ 12:1 rpm has been chosen for rotor angular velocity. Turbulence corresponding to the hub height (90 m) is also included in the model. A 1-D fully coherent turbulence has been generated compatible with Kaimal spectra with parameters as in [21]. The intensity of the turbulence has been assumed to be 10%. The time series of turbulence has been generated following the digital simulation algorithm with random phases as proposed by [26]. A screenshot of the edgewise aerodynamic loads acting on blades and nacelle subjected to a steady wind with homogenous isotropic turbulence is shown 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 the generalized 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 numerical study carried out in this paper, the weight R1 has been set to the identity matrix, that is same relative importance is assigned to the regulation error of each state variable. Different controllers have been synthesized by varying the weighing matrix 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 and I 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, better performances are achieved (Fig. 10). The active control system achieves a significant reduction of the blade tip displacement. The maximum value for uncontrolled response exceeds 1.13 m, whereas for the controlled one the maximum deflection ranges between 0.009 m and 0.76 m for the different regulators considered. Furthermore, a significant overall reduction is obtained in the vibrational response, as the root mean square (RMS) value of the displacement is shown to be remarkably smaller in the controlled cases. For example, the results obtained by setting g ¼ g1 ¼ 1010 and g ¼ g2 ¼ 1011 are shown in Fig. 11. The maximum edgewise displacement for the first blade is reduced from 1.13 m to 0.39 m and 0.13 m, respectively. Furthermore, the RMS displacement for the controlled response exhibits a reduction of 56% (g ¼ g1 ) and 85% ðg ¼ g2 Þ compared the uncontrolled one. Finally, the active control system provides a lower peak-to-peak excursion and the average deflection of the blade is also 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 suppressing peaks in the uncontrolled response due to the rotational effect. In fact, a substantial reduction is achieved in the response corresponding to the peak in the spectrum around 0.2 Hz. This is associated with the rotational speed of the blades and is also the generalized load frequency. The controller is also effective in eliminating the peak corresponding to the natural frequency 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 the edgewise displacements of blade 2 and blade 3. The LQ controller provides a significant reduction of the low-frequency components which mainly dominate the response, that is the contribution in vibration due to the rotational frequency and the contribution around the natural frequency of the blade. Since no active control is operated on the nacelle, negligible

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

reduction is observed for the degree of freedom q~ 4 (Fig. 13), corresponding to the nacelle response. However, in the present study the nacelle displacement is not a major issue since loads in the edgewise direction do not induce large magnitude 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 control effort. 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 g ¼ g1 , assuming W ¼ 201, the maximum magnitude of the force required for achieving the performances in Fig. 11 is found to be about 50 kN (Fig. 14). This means that mounting actuators

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

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

capable of exerting a force of about 5 tons (  28% of the blade weight), achieves for the case considered a reduction of 65% in the maximum 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 force requirement) by varying the control parameter g. The angle W between the actuators is set to 201. In each simulation, the 7DOF 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 the edgewise 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 the rotating frame (assuming t ¼0 for the controller design) have been computed. From (29), it can be observed that the timevarying nature of the edgewise model is due to the variation of the azimuthal angle Cj ðtÞ of the blades over time.

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1249

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

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 static gain is then designed based on this LTI model, which represents the HAWT with the first blade in the vertical upright position. In this case, the dynamic matrix A0 2 R88 is given by " # O44 I44 A0 ¼ (59) M1 M1 0 K0 0 C0 where M0 ¼ M9C

1

¼0

,

K0 ¼ K9C

1

¼0

,

C0 ¼ C9C

1

¼0

(60)

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Fig. 14. Control force on blade 1 (Coleman based LQ controller). (a) W ¼ 151, (b) W ¼ 201.

The control matrix B0 2 R83 is " B0 ¼

#

O43 , M1 0 U0

2

3

1

0

0

60 6 U0 ¼ 6 40

1 0

07 7 7 15

0

0

0

(61)

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

g

Max displacement (m)

RMS displacement (m)

Peak force (kN)

RMS force (kN)

Uncontrolled LQ-MBC 1 LQ-MBC 2 LQ-MBC 3 LQ-MBC 4 LQ-MBC 5 LQ-MBC 6

– 10  9 10  10 10  11 10  12 10  13 10  14

1.133 0.766 0.398 0.137 0.043 0.015 0.009

0.494 0.391 0.201 0.073 0.023 0.006 0.004

– 41.8 55.7 70.5 79.1 82.3 83.4

– 14.7 25.9 34.7 38.3 39.5 39.9

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

The LQ control law based on the assumed nominal representation of the system is then u0 ðtÞ ¼ G0 xðtÞ %

(62)

where G0 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 marginal improvement 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 the edgewise model, the control parameter g has been set to 10  11 and a value of 201 has been considered for the structural angle W. Numerical results (Fig. 16) show that the frequency content of blade 1 response and the corresponding input force %

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

required for implementing the control are comparable to the ones obtained by applying the LQ controller in the nonrotating frame. An algorithm based on direct velocity feedback control has also been applied in this paper in order to suppress edgewise vibrations and to compare with the results from LQ controller. In this case, each blade is subjected to a control force which is proportional to the velocity of the edgewise vibration. A model-free control approach is therefore used, since the synthesis of the controller is not based on the mathematical model of the wind turbine but the control gains are simply

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

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

tuned through an iterative trial and error procedure. The edgewise displacement of the first blade using DVF is shown in Fig. 17. It can be seen (Fig. 18) that even though the DVF control provides a good reduction of the peaks corresponding to the rotational speed and the natural frequency of the blade, it is not particularly effective in suppressing the average edgewise deflection of the blade in comparison to the LQ controller. Indeed, the static deflection obtained by applying DVF is comparable 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 the one computed for the LQ controller (Fig. 19). In particular, for the performances shown in Fig. 17, the maximum force required by 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 are 92.28 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 has been proposed. An innovative hardware configuration, adopting linear actuators/active tendons mounted inside the

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

blades, has been considered in order to implement an active control strategy for reducing the edgewise vibrations experienced by the turbines. The control force can be appropriately applied by following a proposed MBC transformed LQ control 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–Lagrangian approach. The control problem has been addressed first by transforming the dynamic model into an equivalent timeinvariant representation using the multi-blade coordinate transformation. This allows for expressing the generalized coordinates into a non-rotating reference frame and hence eliminates the dependency of the equations of motion upon the azimuthal angles of the blades. The effectiveness of the proposed control scheme has been investigated by numerically applying 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 flexible multi-modal turbine rotor model. The results indicate that the active control system is successfully able to reduce the blade responses. It has been shown that promising performances can be achieved in suppressing edgewise vibrations due to steady state wind loads and under wind loads in turbulent condition. Depending on the tuning of the regulator parameters, different levels of vibration reduction can be attained. Quantitative analysis of the case considered has shown that, in comparison to the uncontrolled response, the active control system can provide a reduction of 65% in the maximum 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 RMS displacement and 85% reduction in the RMS acceleration). Also, for the case considered, the maximum peak to peak excursion 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 be successfully used for suppressing edgewise vibrations. Basic LQ and DVF controllers have been used as illustrative applications for establishing the validity of the proposed approach. Further investigations are needed for testing the behaviour of the controlled system under uncertainty in the structural parameters (such as mass and stiffness) and constraints on control forces.

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

Appendix A. System matrices of the multi-modal flexible HAWT model ~ of the wind turbine model considering N modes of vibration for each blade is defined as The mass matrix M 2 6 6 ~ ¼6 M 6 4

M2

0

0

M2

0

0

0

M2

T M1 c

1

M1 Tc

0

2

T M1 c

M1 c 1

3

3

M1 c2 7 7 ð3N þ 1Þð3N þ 1Þ 7 M1 c 3 7 2 R 5 m4

(A.1)

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

with

2

m21

6 6 0 M2 ¼ 6 6 ^ 4 0

0

...

m22

...

^

&

0

0

3

0

2

7 ^ 7 7 2 RNN , ^ 7 5

m2N

3

6 m cosðC Þ 7 6 12 j 7 7 2 RN1 M1 c j ¼ 6 6 7 ^ 4 5 m1N cosðCj Þ

The damping matrix, including structural damping, is given by 2 0 0 6 0 0 6 C~ ¼ C~ struct þ 6 6 0 0 4 2OM 1 Tc 2OM 1 Tc 1

with

m11 cosðCj Þ

1255

3 0 07 7 7 07 5 0

0 0 0 T 2OM 1 c

2

3

(A.2)

(A.3)

2

C~ struct 2 Rð3N þ 1Þð3N þ 1Þ ,

The stiffness matrix is

2 6 6 6 ~ K¼6 6 4

with

2 6 6 6 K2 ¼ 6 6 4

M 1 cj

3 m11 sinðCj Þ 6 m sinðC Þ 7 6 12 j 7 7 2 RN1 ¼6 6 7 ^ 4 5 m1N sinðCj Þ

K2 þ Kw c1

0

0

0

K2 þ Kw c2

0

0

K2 þKw c3

0 2

2

T

O M1 c

1

2

T

O M1 c

2

0

T

O M1 c

3

3

7 07 7 2 Rð3N þ 1Þð3N þ 1Þ 07 7 5 k4

K e,11 þ O2 ðK g0;11 m21 Þ

O2 K g0;12

...

O2 K g0;1N

O2 K g0;21

K e,22 þ O2 ðK g0;22 m22 Þ

...

^

^

&

^ 2

2

O K g0,N1 2

O K g0,N2

K w,11 cosðCj Þ

6K 6 w,21 cosðCj Þ K w cj ¼ 6 6 ^ 4 K w,N1 cosðCj Þ

(A.4)

...

^ 2

K e,NN þ O ðK g0,NN m2N Þ

K w,12 cosðCj Þ

...

K w,1N cosðCj Þ

K w,22 cosðCj Þ ^

... &

^ ^

K w,N2 cosðCj Þ

...

K w,NN cosðCj Þ

(A.5)

3 7 7 7 NN 7 2 R 7 5

3 7 7 7 2 RNN 7 5

(A.6)

In (A.6), assuming orthogonal modes of vibration implies K e,ik ¼ 0, 8 iak. Furthermore, (11) implies K g0,ik ¼ K g0,ki

K w,ik ¼ K w,ki

This condition ensures symmetry for the blade stiffness matrices K2 and Kw cj . Appendix B. Blade element momentum method Algorithm 1. Wind loading computation (BEM method) 1: Initialize aðkÞ , a0 ðkÞ, ac, B, g (e.g. að0Þ ¼ 0, a0 ð0Þ ¼ 0, ac ¼ 0.2, B¼ 3, g ¼ 0:01)   ~ ð1aðkÞ ÞV o þ w 2: Compute the flow angle f ¼ tan1 ð1 þ a0 ðkÞÞOr 3: Compute the angle of attack a ¼ fbk 2 B Lr 4: Compute Prandtl’s tip loss correction F ¼ cos1 ðef Þ, f ¼ 2 r sinðfÞ p 5: Retrieve lift and drag coefficients C l ðaÞ and C d ðaÞ from airfoil data table for the calculated a 6: Compute normal load coefficient C N ðaÞ ¼ C l ðaÞ cosðfÞ þ C d ðaÞ sinðfÞ 7: Compute tangential load coefficient C T ðaÞ ¼ C l ðaÞ sinðfÞC d ðaÞ cosðfÞ cðrÞB 8: Compute solidity sðrÞ ¼ 2pr 9: if a r ac then 10:

aðk þ 1Þ ¼

1 1þ

11: else

2

4F sin ðfÞ sC N

(A.7)

1256

12:

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

aðk þ 1Þ ¼

1 ½2þ Kð12ac Þ 2

13: endif 14: a0 ðk þ 1Þ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4F sin ðfÞ ðKð12ac Þþ 2Þ2 þ 4ðKa2c 1Þ, K ¼ sC N

1

4F sinðfÞcosðfÞ sC T 15: if ðaðk þ 1Þ aðkÞ 4 gÞ ða0 ðkþ 1Þa0 ðkÞ4 gÞ 16: k ¼ k þ 1, GOTO ¼)2 17: else 1 þ

18: Obtain the axial and tangential induction factors a ¼ aðkÞ , a0 ¼ a0 ðkÞ 19: endif qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ 2 þ O2 r 2 ð1 þ a0 Þ2 20: Compute V rel ¼ ðð1aÞV o þ wÞ 1 1 21: Compute pN ¼ rV 2rel cC N ðaÞ, pT ¼ rV 2rel cC T ðaÞ 2 2

References [1] P.J. Murtagh, A. Ghosh, B. Basu, B.M. Broderick, Passive control of wind turbine vibrations including blade/tower interaction and rotationally sampled turbulence, Wind Energy 11 (4) (2008) 305–317. [2] J. Arrigan, V. Pakrashi, B. Basu, S. Nagarajaiah, Control of flapwise vibrations in wind turbine blades using semi-active tuned mass dampers, Structural Control and Health Monitoring (2010), 10.1002/stc.404. [3] M.H. Hansen, Aeroelastic instability problems for wind turbines, Wind Energy 10 (6) (2007) 551–577. [4] K. Thomsen, J.T. Petersen, E. Nim, S. Øye, B. Petersen, A method for determination of damping for edgewise blade vibrations, Wind Energy 3 (4) (2000) 233–246. [5] V.A. Riziotis, S.G. Voutsinas, E.S. Politis, P.K. Chaviaropoulos, Aeroelastic stability of wind turbines: the problem, the methods and the issues, Wind Energy 7 (4) (2004) 373–392. [6] A. Dixit, S. Suryanarayanan, Towards pitch-scheduled drive train damping in variable-speed, horizontal-axis large wind turbines, in: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, 2005, pp. 1295–1300. [7] E.A. Bossanyi, The design of closed loop controllers for wind turbines, Wind Energy 3 (3) (2000) 149–163. [8] P.K. Chaviaropoulos, E.S. Politis, D.J. Lekou, N.N. Sørensen, M.H. Hansen, B.H. Bulder, D. Winkelaar, C. Lindenburg, D.A. Saravanos, T.P. Philippidis, C. Galiotis, M.O.L. Hansen, T. Kossivas, Enhancing the damping of wind turbine rotor blades, the dampblade project, Wind Energy 9 (1–2) (2006) 163–177. [9] H. Namik, K. Stol, Individual blade pitch control of floating offshore wind turbines, Wind Energy 13 (1) (2010) 74–85. [10] M.A. Lackner, M.A. Rotea, Passive structural control of offshore wind turbines, Wind Energy 14 (3) (2011) 373–388. [11] V. Maldonado, J. Farnsworth, W. Gressick, M. Amitay, Active control of flow separation and structural vibrations of wind turbine blades, Wind Energy 13 (2–3) (2010) 221–237. [12] S.J. Johnson, J.P. Baker, C.P. van Dam, D. Berg, An overview of active load control techniques for wind turbines with an emphasis on microtabs, Wind Energy 13 (2–3) (2010) 239–253. [13] M.A. Lackner, G. van Kuik, A comparison of smart rotor control approaches using trailing edge flaps and individual pitch control, Wind Energy 13 (2–3) (2010) 117–134. [14] M.N. Svendsen, S. Krenk, J. Høgsberg, Resonant vibration control of rotating beams, Journal of Sound and Vibration 330 (9) (2011) 1877–1890. [15] E.A. Bossanyi, Further load reductions with individual pitch control, Wind Energy 8 (4) (2005) 481–485. [16] T.J. Larsen, H.A. Madsen, K. Thomsen, Active load reduction using individual pitch, based on local blade flow measurements, Wind Energy 8 (1) (2005) 67–80. [17] E.A. Bossanyi, Wind turbine control for load reduction, Wind Energy 6 (3) (2003) 229–244. [18] M.O.L. Hansen, Aerodynamics of Wind Turbines, Earthscan, 2008. [19] X. Wei, M. Verhaegen, T. Van Engelen, Sensor fault detection and isolation for wind turbines based on subspace identification and Kalman filter techniques, International Journal of Adaptive Control and Signal Processing 24 (2010) 687–707. [20] S.-C. Choi, J.-S. Park, J.-H. Kim, Vibration control of pre-twisted rotating composite thin-walled beams with piezoelectric fiber composites, Journal of Sound and Vibration 300 (1–2) (2007) 176–196. [21] P. Murtagh, B. Basu, B. Broderick, Along-wind response of a wind turbine tower with blade coupling subjected to rotationally sampled wind loading, Engineering Structures 27 (8) (2005) 1209–1219. [22] M.H. Hansen, Improved modal dynamics of wind turbines to avoid stall-induced vibrations, Wind Energy 6 (2) (2003) 179–195. [23] H. Kwakernaak, R. Sivan, Linear Optimal Control Systems, John Wiley & Sons Inc., New York, NY, USA, 1972. [24] J.M. Jonkman, S. Butterfield, W. Musial, G. Scott, Definition of a 5-MW Reference Wind Turbine for Offshore System Development, National Renewable Energy Laboratory, Technical Report NREL/TP-500-38060, Golden, Colorado, 2009. [25] M. Buhl, NWTC Design Codes (Modes by Marshall Buhl), May 2005, /http://wind.nrel.gov/designcodes/preprocessors/modes/S. [26] M. Shinozuka, C.M. Jan, Digital simulation of random processes and its applications, Journal of Sound and Vibration 25 (1) (1972) 111–128.