Structural damping sensitivity affecting the flutter performance of a 10-MW offshore wind turbine

Abstract

Classical flutter of wind turbine blades is one of the most destructive instability phenomena of wind turbines especially for several-MW-scale turbines. In the present work, flutter performance of the DTU 10-MW offshore wind turbine is investigated using a 907-degree-of-freedom aero-hydro-servo-elastic wind turbine model. This model involves the couplings between tower, blades and drivetrain vibrations. Furthermore, the three-dimensional aerodynamic effects on wind turbine blade tip have also been considered through the blade element momentum theory with Bak’s stall delay model and Shen’s tip loss correction model. Numerical simulations have been carried out using data calibrated to the referential DTU 10-MW offshore wind turbine. Comparison of the aeroelastic responses between the onshore and offshore wind turbines is made. Effect of structural damping on the flutter speed of this 10-MW offshore wind turbine is investigated. Results show that the damping in the torsional mode has predominant impact on the flutter limits in comparison with that in the bending mode. Furthermore, for shallow water offshore wind turbines, hydrodynamic loads have small effects on its aeroelastic response.

Keywords

classical flutter flutter speed offshore horizontal axis wind turbines structural damping

Introduction

Classical flutter of wind turbine blade is an unfavourable aerodynamic coupling between flapwise and torsional modes of blades due to the action of aerodynamic load, inertial load and elastic load. It can result in a rapid growth of the amplitudes for both flapwise and torsional motions, and may eventually lead to structural failure when the rotor speed is beyond a certain critical value (Chen et al., 2018; Owens et al., 2013; Resor et al., 2012). Such critical rotational speed $Ω_{c r}$ is fully called ‘the critical rotational speed for flutter’ and will be simply called ‘flutter speed’ in the remainder of the present article.

Generally speaking, the flutter speed can be obtained either by eigenvalue analysis or time domain simulations (Chen et al., 2017; Hansen, 2004; Ke et al., 2015; Lobitz, 2005), and is influenced by both structural and aerodynamic parameters. Lobitz (2005) carried out a systematic research on flutter sensitivity to two principle parameters (the location of gravity centre and the ratio between the flapwise frequency and the torsional frequency) using a 35-m isolated wind turbine blade model. It was shown that the flutter speed can be increased by increasing the torsional stiffness of blade. In addition, when the gravity centre is moved towards the trailing edge, the flutter speed decreases. Furthermore, flutter is unlikely happen when the centre of mass is ahead of the shear centre (Hansen, 2007). Recently, Hafeez and El-Badawy (2018) proposed a refined single-blade wind turbine model using the ordering scheme proposed by Hodges and Ormiston (1976), and the aerodynamic loads are derived based on an improved Theodorsen’s theory. Effects of the nominal rotational speed of the rotor and shear-mass centre offset on flutter speed are investigated. Similar results have been observed with Hansen (2007). Chaviaropoulos et al. (2003) have assessed the effects of some key parameters (the torsional and bending natural frequencies, the blade mass and the reduced frequency) on flutter characteristics of wind turbine blade based on a two-dimensional (2D) section model. It was shown that the flutter instabilities occur at low reduced frequency values, and natural frequencies of the elastic system enhanced the stability, while lighter blade has a destabilising effect. According to Resor et al. (2012)’s study for the SNL 100-m isolated blade model, the flutter speed can be modestly increased by either reducing weight from trailing edge part of blade or shifting the aerodynamic centre 5% after its original location. In addition, Hansen (2002) investigated the effects of tower elastic deformations on the aeroelastic stability of a wind turbine with focus on flutter based on the eigenvalue analysis. Their study was performed on both a linear full wind turbine model and a single-blade model using quasi-steady aerodynamic model (blade element momentum (BEM) theory (Hansen, 2008)). Results showed that the single-blade model provides unconservative results since it omitted the blade-to-blade and blade-to-tower interactions. Later, such eigenvalue-based method has been extended by Hansen (2004) by coupling the BEM method with Leishman-Beddoes-type dynamic stall model (Hansen et al., 2004). Results indicated that there is only a small quantitative difference between a full wind turbine model and a single-blade model in the predicted flutter speed. Kallesøe and Kragh (2016)’s research showed that the geometric non-linearities caused by steady state deflection of the blade for NREL 5-MW Reference Wind Turbine with 63-m blades may also influence the aeroelastic stability of wind turbines. Furthermore, Lobitz (2004) assessed the influence of different aerodynamic models on flutter speed for MW-sized wind turbines. It was shown that the flutter speed obtained using quasi-steady theory is smaller than that estimated using unsteady theory. Pirrung et al. (2014) investigated the effects of different wake models (near and far wake models) on flutter speed using the commercial aeroelastic code HAWC2. Results indicated that replacing the classical BEM model with a near wake model will increase the flutter speed by roughly 4%–10%.

Above works are focused on flutter analysis of onshore wind turbines. For offshore wind turbines, the hydrodynamic effect should be considered during studying the flutter instabilities. Furthermore, previous studies omitted the contribution from structural inherent characteristics (i.e. the structural damping). In the current article, a 907-degree-of-freedom (907-DOF) aero-hydro-servo-elastic wind turbine model is used to study the flutter performance of the DTU 10-MW offshore wind turbine. And the flutter critical rotation speeds for offshore and onshore wind turbines are compared. In addition, the current article also provides insight into the influence of structural damping on the onset of flutter and quantitative measures for cases in which flutter is a design concern.

Wind turbine model

Geometrical description

Considering the wind turbine structure, two separate coordinate systems are defined. A global $(X_{1}, X_{2}, X_{3}) -coordinate$ system is placed at the bottom of the tower with an origin point O at the interface of the rigid foundation. The tilt angle and the cone angle of this wind turbine are set to zero. A local $(x_{1}, x_{2}, x_{3}) -coordinate$ system with origin $O^{'}$ is defined and with the $(x_{2}, x_{3}) -plane$ parallel to the global $(X_{2}, X_{3}) -plane$ . The azimuthal angle of $j th$ blade is defined by $Ψ_{j} (t)$ as shown in Figure 1. The $x_{3} -axis$ is oriented from the hub to the blade tip. Then, the x ₁ and $x_{2} -axes$ define the flapwise and edgewise directions, respectively. The length of the blade is L, the distance from the centre of the tower to $O^{'}$ is denoted s (i.e. hub overhang). The height of the tower from the base to the nacelle is h ₁, the height from the mean sea level (MSL) to the tower base is h ₂. The water depth from the mud line to the MSL is denoted h ₃, see Figure 1. $p_{w} (X_{3}, t)$ is the wave load, and will be elaborated in the following section.

Figure 1.

Definition of global and local coordinate systems and the degrees of freedom $q_{1} (t), \dots, q_{5} (t)$ .

The azimuthal angles $Ψ_{j} (t)$ are given as

Ψ_{j} (t) = Ω t + \frac{2 π}{3} (j - 1), j = 1, 2, 3

where $Ψ_{j} (t)$ is the $j th$ $(j = 1, 2, 3)$ blade azimuthal angle, $Ω$ is the nominal rotational speed of the rotor.

Here, the wind turbine tower and the support structure are treated as a whole, and 5 DOFs have been introduced to describe the motion of tower top. The tower top can translate in two horizontal (i.e. side-by-side and forward-and-backward) directions, described in the ground fixed $(X_{1}, X_{2}, X_{3}) -coordinate$ system by $q_{1} (t)$ and $q_{2} (t)$ , see Figure 1. The corresponding rotations of tower top in $X_{1} -$ , $X_{2} -$ and $X_{3} -directions$ are described by $q_{3} (t)$ , $q_{4} (t)$ and $q_{5} (t)$ , respectively.

Two DOFs $q_{6} (t)$ and $q_{7} (t)$ are introduced to model the drive train, as shown in Figure 2, indicating the deviations of the rotational angles at the hub and the generator from the nominal rotational angles $Ω t$ and $N Ω t$ , respectively. N denotes the gear ratio. $M_{r} (t)$ and $M_{g} (t)$ are the work-conjugated rotor torque and generator torque. J_r and J_g are the mass moment of inertia of the rotor and the generator, and k_r and k_g denote the St. Venant torsional stiffness of the rotor and generator shafts.

Figure 2.

Schematic diagram of the drive train with odd number of gear stages (Chen et al., 2018).

The DOFs of $q_{1} (t), \dots, q_{7} (t)$ are assembled in the vector $q_{0} (t)$ .

Here, each blade of wind turbine is modelled as a Bernoulli–Euler beam. Furthermore, we assumed that the elastic centre E of all blade cross sections is assumed to be placed along the $x_{3} -axis$ . For each section, a local $(y_{1}, y_{2}, y_{3}) -coordinate$ system is defined as shown in Figure 3. The $y_{3} -axis$ is parallel to the $x_{3} -axis$ , and the rotational angle between the $y_{2} -axis$ to the $x_{2} -axis$ is defined by the angle $δ (x_{3})$ . $δ (x_{3})$ is the pretwist angle of blade. A is the aerodynamic centre, S is the shear centre and G denotes the mass centre as shown in Figure 3. The position of the aerodynamic centre A is assumed to be fixed in time. The aerodynamic lift force p_l and drag force p_d acting at the aerodynamic centre are transferred to the shear centre. The gravity load is acting at the mass centre G. The relative location of the mass centre G, the elastic centre E and the aerodynamic centre A induces couplings in the equation of motion which can significantly influence on the flutter characteristics of wind turbines.

Figure 3.

Definition of velocities of an airfoil (Chen et al., 2018).

Figure 4 shows the discretization of a blade. $y_{3, i}$ indicates the position of node i of the element on the $x_{3} -axis$ . The bottom plot of Figure 4 shows the $i th$ element with nodes i and $i + 1$ , and $r_{1, i}, \dots, r_{12, i + 1}$ is the corresponding 12 DOFs of the $i th$ element. $l_{i} = x_{3, i + 1} - x_{3, i}$ indicates element length. The total DOFs of blade j are assembled in the column vector $q_{j} (t), j = 1, 2, 3$ , see equation (2). Small elastic vibrations are assumed, so that linear beam theory may be assumed with due consideration of the geometrical stiffening in the $x_{3} -direction$ , and the geometrical softening in the transverse direction. For more details about this 907-DOF wind turbine model, the reader is referred to Zhang (2015).

Figure 4.

Nodal numbering and definition of degrees of freedom of a beam element.

The total DOFs of the wind turbine system are assembled in the vectors $q (t)$ of dimension $7 + 18 n$

q (t) = [\begin{matrix} q_{0} (t) \\ q_{1} (t) \\ q_{2} (t) \\ q_{3} (t) \end{matrix}]

The kinetic energy $T (q, \dot{q})$ and the potential energy $U (q)$ of this model may be calculated as a function of $q$ and $\dot{q} (t)$ . Then, the equation of motion of this wind turbine can be obtained using Euler–Lagrange equation

\begin{array}{l} \frac{d}{d t} (\frac{\partial T (q, \dot{q})}{\partial \dot{q}}) - \frac{\partial T (q, \dot{q})}{\partial q} + \frac{\partial U (q)}{\partial q} = f (q, \dot{q}, t) \\ \Rightarrow M (t) \ddot{q} (t) + C (t) \dot{q} (t) + K (t) q (t) = f (z, t) \end{array}

Because the elastic deformations have been assumed to be linear, the mass matrix $M (t)$ , the damping matrix $C (t)$ and the stiffness matrix $K (t)$ are periodic. $f (z, t)$ on the right-hand side of equation (3) denotes the load vector which contains the self-induced aeroelastic load, pitch control load and aerodynamic load. $z (t)$ is the state vector defined as

z (t) = [\begin{matrix} q (t) \\ \dot{q} (t) \\ β (t) \end{matrix}]

The parameter $β (t)$ denotes the collective pitch angle with a time delay

\dot{β} (t) = \frac{1}{τ} (β_{0} (q_{6}, {\dot{q}}_{6}) - β (t))

The positive direction of $β (t)$ is defined in Figure 3. $β_{0} (q_{6}, {\dot{q}}_{6})$ is the pitch demand, which is modelled as proportional–integral (PI) controller with feed-back from $q_{6} (t)$ and ${\dot{q}}_{6} (t)$ (Ogata, 2010). $τ$ indicates the reaction time.

Aerodynamic and wave loads

The aerodynamic loads are obtained based on the most widely used BEM theory (Hansen, 2008). Furthermore, Shen’s three-dimensional (3D) tip loss factor and Glauert correction (Hansen, 2008; Shen et al., 2005) have also been implemented.

The turbulence velocity is considered with a turbulence model constructed based on Taylor’s hypothesis of frozen turbulence, corresponding to a frozen turbulence field that is convected into the rotor in the global X ₁ direction with a mean wind velocity V ₀. The frozen field is assumed to be a zero mean homogeneous and isotropic stochastic field, with a spatial covariance structure as specified in literature IEC61400-1 (2005). For more details about the construction of turbulence model, the reader is referred to Chen et al. (2018) and Zhang (2015).

The aerodynamic lift and drag loads $p_{l} (x_{3}, t)$ and $p_{d} (x_{3}, t)$ are given as

\begin{array}{l} p_{l} (x_{3}, t) = C_{l} (α) \frac{1}{2} ρ V^{2} (x_{3}, t) c (x_{3}) \\ p_{d} (x_{3}, t) = C_{d} (α) \frac{1}{2} ρ V^{2} (x_{3}, t) c (x_{3}) \end{array}}

where $α (x_{3}, t)$ is the angle of attack, $ρ$ is air density, $V (x_{3}, t)$ is the resulting wind velocity and $c (x_{3})$ is the chord length. $C_{l} (α)$ and $C_{d} (α)$ are the aerodynamic lift and drag coefficients obtained from static 2D wind tunnel test data. However, previous studies showed that a direct use of the static 2D airfoil characteristics shows bad agreement between measured and calculated loads and power, due to the centrifugal and Coriolis effects acting on the blades (Bak et al., 2006; Yang et al., 2014). Furthermore, both the experimental and computational fluid dynamics (CFD) results have shown significant increase in sectional maximum lift coefficients beyond the results based on 2D static measurements (Leishman, 2002). Therefore, in order to make BEM more realistic, the 2D airfoil characteristics should be modified to consider the 3D static stall delay effects (i.e. rotation effects on the boundary layer). Based on the 2D airfoil characteristics, the $C_{l} (α)$ and $C_{d} (α)$ (except for the cylinder part) used in equation (6) were 3D corrected with Bak’s model (Bak et al., 2006) (suggested by Bak et al. (2013) for DTU 10-MW horizontal axis wind turbine (HAWT)) to account for the 3D static stall delay effects.

Figure 3 shows a typical profile of a wind turbine blade. $V_{1} (x_{3}, t)$ and $V_{2} (x_{3}, t)$ are the instantaneous wind velocities in local $x_{1} -$ and $x_{2} -directions$ . These are given as

V (x_{3}, t) = \sqrt{V_{1}^{2} (x_{3}, t) + V_{2}^{2} (x_{3}, t)}

\begin{array}{l} V_{1} (x_{3}, t) = (1 - a) V_{0} + v_{1} (x_{3}, t) - {\dot{u}}_{1} (x_{3}, t) \\ V_{2} (x_{3}, t) = (1 + a^{'}) Ω x_{3} + v_{2} (x_{3}, t) - {\dot{u}}_{2} (x_{3}, t) \end{array}}

where V ₀ denotes the mean wind velocity and is assumed to be constant over the entire rotor area. $Ω$ is the rated speed of the rotor and is assumed to be constant. $v_{j} (x_{3}, t)$ indicates the turbulence velocity. ${\dot{u}}_{j} (x_{3}, t)$ denotes the velocity of blade at x ₃. a and $a^{'}$ denote the axial and tangential induction coefficients, respectively. Shen et al. (2005) carried systematic studies about some existing tip loss correction models (including the most commonly used Prandtl’s tip loss factor) and found inconsistencies in the existing correction models. They proposed an improved tip loss correction model to account for the 3D tip loss effects.

The instantaneous angle of attack

α (x_{3}, t) = φ (x_{3}, t) + β (t) + δ (x_{3})

where $φ (x_{3}, t)$ is the flow angle (see Figure 3)

tan φ (x_{3}, t) = \frac{V_{1} (x_{3}, t)}{V_{2} (x_{3}, t)}

The aerodynamic loads of blades are obtained through the BEM theory with Shen’s 3D tip loss function and Glauert correction (Hansen, 2008; Shen et al., 2005), and the 3D static stall delay effect is also incorporated.

In accordance with the IEC 61400-3 standard IEC 61400-3 (2009), sea surface elevation is modelled as a zero mean, stationary Gaussian process with the single-sided Joint North Sea Wave Observation Project (JONSWAP) spectrum. The stationary wave surface elevation process is determined by

η (X_{2}, t) = \sum_{j = 1}^{J} \sqrt{2} η_{j} cos (ω_{j} t - k_{j} X_{2} + φ_{j})

where J is the number of harmonic components, $ω_{j}$ and k_j are the angular frequency and wave number of the $j th$ harmonic component related by $ω_{j}^{2} = g k_{j}$ . $φ_{j}$ denotes samples of the random phase, which are mutually independent and uniformly distributed in $[0, 2 π]$ . $η_{j} = S_{η} (ω_{j}) Δ ω_{j}$ is the standard deviation of the $j th$ harmonic component, and $S_{η} (ω_{j})$ is the single-sided JONSWAP spectrum. Following the horizontal velocity $v (X_{3}, t)$ and acceleration $\dot{v} (X_{3}, t)$ of the water particle at the position $X_{1} = 0$

v (X_{3}, t) = \sum_{j = 1}^{J} \sqrt{2} η_{j} ω_{j} \frac{cosh (k_{j} X_{3})}{sinh (k_{j} h)} cos (ω_{j} t + φ_{j})

\dot{v} (X_{3}, t) = \sum_{j = 1}^{J} \sqrt{2} η_{j} ω_{j}^{2} \frac{cosh (k_{j} X_{3})}{sinh (k_{j} h)} sin (ω_{j} t + φ_{j})

The wave load of the tower is determined by the Morison equation (Isaacson and Sarpkaya, 1981)

p_{w} (X_{3}, t) = \frac{1}{2} ρ_{w} C_{d} D v (X_{3}, t) | v (X_{3}, t) | + \frac{π}{4} ρ_{w} C_{m} D^{2} \dot{v} (X_{3}, t)

where $ρ_{w}$ is the fluid density, C_d is the drag coefficient, C_m is the inertia coefficient (Sarpkaya, 1986) and D is the diameter of the turbine monopile. The generalised wave load corresponding to the generalised coordinate $q_{1} (t)$ can be calculated through $\int_{0}^{h_{3}} p_{w} (X_{3}, t) N_{1} (X_{3}) d X_{3}$ , $N_{1} (X_{3})$ is the modal shape of $q_{1} (t)$ .

Aeroelastic stability characteristics of DTU 10-MW offshore wind turbine

DTU 10-MW wind turbine

Here, the 907-DOF aeroelastic wind model has been used for following flutter analysis. This wind turbine model is sized and studied with application to the DTU 10-MW wind turbine (Bak et al., 2013). The parameters relevant to the monopile foundation are determined based on Jonkman and Musial (2010) and Velarde (2016). Some of the major parameters are listed in Table 1. Table 2 summarised the natural frequencies calculated through the 907-DOF wind turbine model, which are close to the reference report Bak et al. (2013). For the full database, please refer to Bak et al. (2013).

Table 1.

Principle parameters in the 907-DOF wind turbine model.

Parameter	Value	Unit
L	89.166	m
s	7	m
h ₁	89	m
h ₂	10	m
h ₃	20	m
H_s	2	m
T_p	6	s
J_r	$1.5696 \times 10^{8}$	kg m²
J_g	2328.24	kg m²
k_r	$3.48 \times 10^{9}$	N m/rad
k_g	$1.392 \times 10^{6}$	N m/rad
$Ω_{0}$	1.0053	rad/s
g	9.81	m/s²
$ρ$	1.225	kg/m³
$ρ_{w}$	$1 .0 \times 10^{3}$	kg/m³
I	0.1	–
C_d	1.2	–
C_m	2	–
D	8	m

Table 2.

Natural frequencies (rad/s) calculated in the current study and the reference Bak et al. (2013).

Item	Current study	Bak et al. (2013)
First flapwise, $ω_{f_{1}}$	4.15	3.93
Second flapwise, $ω_{f_{2}}$	11.56	10.93
Third flapwise, $ω_{f_{3}}$	23.86	22.43
First edgewise, $ω_{e_{1}}$	6.09	5.84
Second edgewise, $ω_{e_{2}}$	18.50	17.34
Third edgewise, $ω_{e_{3}}$	39.18	41.25
First torsional, $ω_{θ_{t}}$	36.06	35.75
First tower fore-aft, $ω_{f a}$	1.8	1.67
First tower side-by-side, $ω_{s s}$	1.8	1.67

Dynamic response of DTU 10-MW offshore wind turbine

Figures 5 to 9 show the aeroelastic response of the DTU 10-MW onshore and offshore wind turbines under normal working condition (i.e. mean wind velocity $V_{0} = 15 m / s$ , rated nominal rotational speed of the rotor $Ω_{0} = 1.0053 rad / s$ , turbulence intensity $I = 0.1$ ). Furthermore, the first flapwise mode damping ratio, the first edgewise damping ratio and the first flapwise torsional mode damping ratio are set to 0.005 (Bak et al., 2013; Chen et al., 2018) to generate the structural damping matrix $C_{s t}$ based on the Caughey damping model. Figure 5 shows the flapwise displacement response of blade tip and its Fourier spectrum. As seen, the resonant response component which consists of the first flapwise frequency and the first tower frequency is not predominant, while the background response component seems to have larger contribution to the total response, see Figure 5(c). For the edgewise and torsional response of blade, the supporting structure has little influence on it as shown in Figures 6(c) and 7(c). The edgewise response is mainly dominated by the first and second bending frequencies of blade, see Figure 6(c). Likewise, the torsional respond is characterised by the first and second natural frequencies of blade, see Figure 7(c). As for the tower tip, the translation response in both forward–backward and side-by-side directions are mainly dominated by its first bending mode, see Figures 8(c) and 9(c).

Figure 5.

Flapwise displacement response of blade tip and its Fourier spectrum: (a) time series $q_{f} (t)$ , (b) magnified region and (c) Fourier spectrum $q_{f} (ω)$ .

Figure 6.

Edgewise displacement response of blade tip and its Fourier spectrum: (a) time series $q_{e} (t)$ , (b) magnified region and (c) Fourier spectrum $q_{e} (ω)$ .

Figure 7.

Torsional deformation of blade tip and its Fourier spectrum: (a) time series $q_{t} (t)$ , (b) magnified region and (c) Fourier spectrum $q_{t} (ω)$ .

Figure 8.

Forward–backward displacement response of tower tip and its Fourier spectrum: (a) time series $q_{f b} (t)$ , (b) magnified region and (c) Fourier spectrum $q_{f b} (ω)$ .

Figure 9.

Side-by-side displacement response of tower tip and its Fourier spectrum: (a) time series $q_{s s} (t)$ , (b) magnified region and (c) Fourier spectrum $q_{s s} (ω)$ .

Furthermore, as shown in Figure 5(b), there is no noticeable difference between the flapwise response of both onshore and offshore wind turbine blades. It is so mainly because the wave load is relatively small in comparison with the wind load (Zuo et al., 2018), since the water depth is only $20 m$ . Similar results can be observed in edgewise, torsional deformation of blade, and the forward–backward and side-by-side response of tower, see Figures 6(b), 7(b), 8(b) and 9(b).

The influence of the structural damping on flutter limit

This wind turbine is under normal operational condition with turbulence intensity $I = 0.1$ and mean wind velocity $V_{0} = 15 m / s$ (Bak et al., 2013; Bir and Jonkman, 2007). In order to eliminate the effect from pitch controller, the pitch angle is set to 0 (i.e. $β = 0$ and $\dot{β} = 0$ ). For the following set of results, the rated speed of the rotor $Ω$ is uniformly increased until the flutter boundary is reached, and significant classical flutter oscillations have been observed. It was found that the third flapwise mode couples with the first torsional mode in flutter, which is in agreement with the result from literature Velarde (2016). Furthermore, the first flapwise mode, first edgewise mode and the first torsional mode have been chosen to generate the structural damping matrix $C_{s t}$ of this 907-DOF wind turbine model based on the Caughey’s damping model (Bak et al., 2013; Chen et al., 2018). Effects of different flapwise and torsional modal damping ratios on flutter speed are investigated in detail.

Figure 10(a) shows the flutter speed versus the flapwise mode damping ratio of blade. As seen, with the increase in flapwise mode damping ratio, the flutter speed almost remains the same value. This indicates that the flutter instability is insensitive to the structural damping of bending modes. Figure 10(b) shows the torsional mode damping ratio of blade affecting the flutter speed. Evidently, the flutter speed is found to increase monotonically for increasing torsional mode damping ratio. When the torsional mode damping ratio increased from 0.0025 to 0.01, the flutter speed increased by almost 20%, that is, torsional mode damping ratio causes the largest increase in the flutter speed. Also, the flapwise mode damping has slight influence on the results.

Figure 10.

Influence of structural damping on flutter speed: (a) flutter speed versus the flapwise mode damping ratio $ζ_{f l a p}$ and (b) flutter speed versus the torsional mode damping ratio $ζ_{t o r}$ .

Conclusion

This work presented an aero-hydro-servo-elastic wind turbine model with generator and pitch controller implemented. The model takes into account the aerodynamic load, elastic load and inertial load of the blades and hydrodynamic load of tower. Furthermore, the coupling between the rotor, drive train and the tower is also considered. Aerodynamic loading is based on the most commonly used BEM theory. Stall delay model and 3D tip loss correction model are implemented to account for the 3D effects. Numerical simulations on DTU 10-MW both offshore and onshore wind turbines shown that the hydrodynamic loads have small effects on the aeroelastic response of shallow water offshore wind turbine. Furthermore, increasing the torsional mode damping causes a more significant increase in the flutter speed. However, the structural damping of blade bending modes has slight influence on the flutter performance of wind turbine.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: The first author gratefully acknowledges the financial support from the State’s Key Project of Research and Development Plan (No. 2016YFE0127900) and the National Science Foundation of China (No. 51908209).

ORCID iDs

Xugang Hua

Bei Chen

Zili Zhang

References

Bak

Johansen

Andersen

(2006) Three-dimensional corrections of airfoil characteristics based on pressure distributions. In: Proceedings of the European wind energy conference and exhibition (EWEC), Athens, Greece, 27 February–2 March.

Bak

Zahle

Bitsche

, et al. (2013) Description of the DTU 10 MW reference wind turbine. DTU Wind Energy report-I-0092. Roskilde: Technical University of Demark.

Bir

Jonkman

(2007) Aeroelastic instabilities of large offshore and onshore wind turbines. Journal of Physics: Conference Series 75: 012069.

Chaviaropoulos

Soerensen

Hansen

, et al. (2003) Viscous and aeroelastic effects on wind turbine blades. The VISCEL project. Part II: aeroelastic stability investigations. Wind Energy 6(4): 387–403.

Chen

Hua

Zhang

, et al. (2017) Monitoring of wind turbine blades for flutter instability. Structural Monitoring and Maintenance 4(2): 115–131.

Chen

Zhang

Hua

, et al. (2018) Enhancement of flutter stability in wind turbines with a new type of passive damper of torsional rotation of blades. Journal of Wind Engineering and Industrial Aerodynamics 173: 171–179.

Hafeez

El-Badawy

(2018) Flutter limit investigation for a horizontal axis wind turbine blade. Journal of Vibration and Acoustics 140(4): 041014.

Hansen

(2002) Vibrations of a three-bladed wind turbine rotor due to classical flutter. In: ASME 2002 wind energy symposium, Reno, NV, 14–17 January, pp. 256–266. New York: American Society of Mechanical Engineers.

Hansen

(2004) Stability analysis of three-bladed turbines using an eigenvalue approach. In: Proceedings of 2004 ASME wind energy symposium, Reno, NV, 12–15 January, pp. 192–202. Reston, VA: American Institute of Aeronautics and Astronautics; New York: American Society of Mechanical Engineers.

10.

Hansen

(2007) Aeroelastic instability problems for wind turbines. Wind Energy 10(6): 551–577.

11.

Hansen

Gaunaa

Madsen

(2004) A Beddoes-Leishman type dynamic stall model in state-space and indicial formulation. Technical report, Risø-R-1354(EN). Roskilde: Risø National Laboratory.

12.

Hansen

MOL

(2008) Aerodynamics of Wind Turbines. 2nd ed. London: Earthscan Publications Ltd.

13.

Hodges

Ormiston

(1976) Stability of elastic bending and torsion of uniform cantilever rotor blades in hover with variable structural coupling. Technical report NASA TN D-8192. Moffett Field, CA: NASA’s Ames Research Center.

14.

IEC61400-1 (2005) Wind Turbine-Part 1: Design Requirements. Geneva: International Electrotechnical Commission.

15.

IEC 61400-3 (2009) Wind Turbines–Part 3: Design Requirements for Offshore Wind Turbines. Geneva: International Electrotechnical Commission.

16.

Isaacson

Sarpkaya

(1981) Mechanics of Wave Forces on Offshore Structures. New York: Van Nostrand Reinhold.

17.

Jonkman

Musial

(2010) Offshore code comparison collaboration (OC3) for IEA wind task 23 offshore wind technology and deployment. Technical report, National Renewable Energy Lab.(NREL), Golden, CO, USA, March.

18.

Kallesøe

Kragh

(2016) Field validation of the stability limit of a multi MW turbine. Journal of Physics: Conference Series 753: 042005.

19.

Wang

, et al. (2015) Aeroelastic responses of ultra large wind turbine tower-blade coupled structures with SSI effect. Advances in Structural Engineering 18(12): 2075–2087.

20.

Leishman

(2002) Challenges in modelling the unsteady aerodynamics of wind turbines. Wind Energy 5(2–3): 85–132.

21.

Lobitz

(2005) Parameter sensitivities affecting the flutter speed of a mw-sized blade. Journal of Solar Energy Engineering 127(4): 538–543.

22.

Lobitz

(2004) Aeroelastic stability predictions for a MW-sized blade. Wind Energy 7(3): 211–224.

23.

Ogata

(2010) Modern Control Engineering. 5th ed. Upper Saddle River, NJ: Pearson.

24.

Owens

Griffith

Resor

, et al. (2013) Impact of modeling approach on flutter predictions for very large wind turbine blade designs. In: Proceedings of the American helicopter society (AHS) 69th annual forum, 21–23 May 2013, Phoenix, AZ, USA.

25.

Pirrung

Madsen

Kim

(2014) The influence of trailed vorticity on flutter speed estimations. Journal of Physics: Conference Series 524: 012048.

26.

Resor

Owens

Griffith

(2012) Aeroelastic instability of very large wind turbine blades. In: Proceedings of the European wind energy conference annual event, Copenhagen, Denmark, 16–19 April.

27.

Sarpkaya

(1986) Force on a circular cylinder in viscous oscillatory flow at low Keulegan–Carpenter numbers. Journal of Fluid Mechanics 165: 61–71.

28.

Shen

Mikkelsen

Sørensen

, et al. (2005) Tip loss corrections for wind turbine computations. Wind Energy 8(4): 457–475.

29.

Velarde

(2016) Design of monopile foundations to support the DTU 10 MW offshore wind turbine. Master’s Thesis, Norwegian University of Science and Technology, Trondheim.

30.

Yang

Shen

, et al. (2014) Prediction of the wind turbine performance by using BEM with airfoil data extracted from CFD. Renewable Energy 70: 107–115.

31.

Zhang

(2015) Passive and active vibration control of renewable energy structures. PhD Thesis, Aalborg University, Aalborg.

32.

Zuo

Hao

(2018) Dynamic analyses of operating offshore wind turbines including soil-structure interaction. Engineering Structures 157: 42–62.