Dynamic analysis of an offshore wind turbine under random wind and wave excitation with soil-structure interaction and blade tower coupling

Soil Dynamics and Earthquake Engineering 125 (2019) 105699

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Dynamic analysis of an offshore wind turbine under random wind and wave excitation with soil-structure interaction and blade tower coupling

T

Arundhuti Banerjeea, Tanusree Chakrabortya,∗, Vasant Matsagara, Martin Achmusb a b

Department of Civil Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India Institute for Geotechnical Engineering, Leibniz University Hannover, Appelstr. 9A, D-30167, Hannover, Germany

ARTICLE INFO

ABSTRACT

Keywords: Blade tower coupling Equivalent spring dashpot MDOF Offshore wind turbine Pierson-Moskowitz spectrum Rotationally sampled wind turbulence Von karman spectrum

This study investigates the dynamic response of a 5 MW offshore wind turbine with monopile foundation subjected to wind and wave actions. The work includes dynamic interaction between the monopile and the underlying soil subjected to realistic offshore random wind and wave loading modeled using Von karman spectrum and Pierson Moskowitz spectrum respectively. The study also incorporates the effect of blade tower coupling in the analysis. The offshore wind turbine tower is modeled herein as a multi-degree of freedom system (MDOF) and it comprises of a rotor blade system, a nacelle, and a flexible tower. The mass of the rotor, blade, and nacelle are lumped at the top of the tower for simplicity. Separately, the effect of the rotation of blades has also been incorporated in the work. The rotational effect of the blades is taken into account considering shape filters using von Karman spectrum. The soil-structure interaction effect at the foundation level is modeled using equivalent spring-dashpot model for embedded foundations. The results are studied in time as well as frequency domain for both wind and wave loading. It has been observed that soil structure interaction effect greatly alters the response of the offshore wind turbine structure not only in the parked condition but also in operational conditions when blade tower coupling is also included. The effect of blade tower coupling and SSI on the response of the structure are observed more coherently in the case of wave induced loading.

1. Introduction The wind energy industry has developed and grown remarkably in the past years. Being renewable and immeasurably spread across the planet, it is an important source of energy when meeting with the world's growing need for energy demand. Compared to wind turbines onshore, the offshore wind turbines are dynamically sensitive structures placed in adverse environmental conditions due to the harsh offshore climate. However, as the winds are stronger and more stable at sea than onshore, offshore wind turbines are the more preferred options for generating renewable energy. Wind turbines have increased tremendously in both size and performance in the recent years. In order to minimize the cost of installation, the weight of individual components is reduced, making the structure more flexible and susceptible to dynamic excitation even at low frequencies. In a conventional design analysis, the masses of the nacelle and rotor blades are lumped at the top of the tower, and as long as the fundamental frequencies of the tower and blades are different, a dynamic analysis is carried out. While the simplicity of this technique is attractive, it results in a conservative

design approach. It is extremely important to conduct dynamic analysis on offshore wind turbine structures taking into consideration the realistic random loading due to wind and waves, the complexities of extreme loading conditions and actual slenderness ratio of the wind turbine structure. Wind turbine structures can be analyzed using two approaches. First one is a combined modal and multi-body dynamics formulation to simulate the dynamic behavior of the turbine in the time domain. The structural dynamics of the wind turbine are represented using a finitedegree-of-freedom modal model. This approach is less time consuming than the second approach which is the finite element (FE) approach which are used in commercial codes in the industry. However, for a more detailed analysis that could consider nonlinearities and stress analysis through different components of the system, FE method has to be considered. Current well known commercial computational softwares are capable of conducting aero-servo-geotechnical analysis by coupling geotechnical and aerodynamic loads on the wind turbine structure [1]; Quallen and Tao [2]; Shi et al. [3]; Abhinav and Saha [4]. It uses comprehensive simulation codes that account for the coupled

Corresponding author. E-mail addresses: [email protected] (A. Banerjee), [email protected] (T. Chakraborty), [email protected] (V. Matsagar), [email protected] (M. Achmus). ∗

https://doi.org/10.1016/j.soildyn.2019.05.038 Received 27 July 2017; Received in revised form 22 May 2019; Accepted 23 May 2019 0267-7261/ © 2019 Elsevier Ltd. All rights reserved.

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dynamics of aerodynamics, elasticity, hydrodynamics, and foundation dynamics of the support structure in the time domain in a coupled simulation environment. In the present study, the first approach has been used to analyze the wind turbine structure. A general stochastic analysis method for offshore wind turbine towers using a lumped mass model approach subjected to wave action and strong motion earthquakes have been reported in the literature by Refs. [5,6]. In a study by Shinozuka et al. [7]; the response of a fixed offshore tower using a lumped mass model subjected to ocean waves was investigated. The response of a model articulated tower subjected to random wind force and to the combined effect of the wind, wave, and current forces were investigated by Datta and Jain [8]. They observed that the displacement response of the tower due to wind force varies linearly with the mean wind velocity. The random wind force causes more displacement as compared to the random wave forces. The current velocity also has a significant influence on the displacement of the tower. A solution of the equation of motion for the wind turbine structures in which drag nonlinearity can be uncoupled in a suitable form as was suggested by Dao and Penzien [9] using a frequency domain analysis method. Further, a stochastic method of analysis for offshore wind turbine towers subjected to both random sea wave loading and strong motion earthquake loading was investigated by Penzien et al. [10] using an analytical model. In this analysis, the Pierson-Moskowitz wave height spectrum was used along with linear wave theory to define a stationary random sea state as caused by wind-generated surface waves. The finite element method was used by Chung and Yoo [11] to consider the rotational effect of the blades to obtain the dynamic properties of a rotating cantilever. An experimental work was carried out on the same research objective by Lee et al. [12]. Very few research is available on blade tower coupled effect on a wind turbine structure which plays a major role in contributing to the natural frequency of the system. Harrison et al. [13] stated that the motion of the tower is strongly associated with the motion of the blades, which ultimately alters the response of the tower. Naguleswaran [14] simplified rotating blades to cantilever beam for analyzing their dynamic characteristics. Lee et al. [12] carried out experimental studies on the vibration characteristics of rotating blades. Baumgart [15] analyzed the dynamic response of blades of an onshore wind turbine under the wind loads using a combination of finite element method (FEM) and practical running characteristics of blades. Chen et al. [16] analyzed wind-induced response characteristics of an onshore wind turbine tower with the blade-tower coupling effect and observed 300% increase in displacement of the tower when blade tower coupling is considered. Several commercial codes are available in the industry for analysing the offshore structures with blade coupling effects [Karimirad and Moan [1]; Quallen and Tao [2] and Xiong et al. [17]]. Apart from the blade tower coupling effect, soil structure interaction also plays a pivotal role in analyzing the structures. In reality, the support condition of an offshore wind turbine structure is not completely fixed. The literature present on the dynamic interaction of wind turbines taking into account the SSI effect has been somewhat sparse. The presence of flexible soil underneath the foundation of a structure increases the damping capacity of the foundation and reduces the structure's natural frequency as was shown by Veletsos and Verbic [18] highlighting the importance of soil structure interaction. The role of soil in the dynamics of onshore wind turbines has been investigated by Adhikari and Bhattacharya [19] through experimental work. Experimental results show that the natural frequencies and the damping factors of the wind turbine tower change significantly with the type of soil/foundation revalidating the work of Veletsos and Verbic [18]. Hence, it is essential to model the interactions between tower and soil by using a soil-structure interaction model. The accuracy of a wind turbine simulation is dependent on how realistically the foundation of the turbine represents the true soil-structure interaction. In general, several methods for analyzing the soil-pile interaction are available, among these are

analytical linear, viscoelastic continuum models, linear Winkler type medium models, and finite-element (FE) models [Winkler [20]; Novak [21–23], Nogami [24]; Kellezi [25]; Spyrakos and Xu [26]; Lesny and Wieman [27]; Zaaijer [28]; Achmus [29]; Zhang [30]; Adhikari and Bhattacharya [19]; Harte et al. [31]; Shi et al. [3]; Abhinav and Saha [4]]. Usually, commercial codes are adopted to investigate the complex dynamic behavior of the SSI, but the simulation time is much intensive. To capture the most important features of SSI, an effective approach is to use spring-damper systems for representing the soil-foundation interaction (SFI). In the present study, an equivalent spring-dashpot system introduced by Gazetas [32] has been used to incorporate the effect of soil structure interaction for an offshore wind turbine structure. The present study deals with analyzing an offshore wind turbine structure modeled as a multi degree of freedom system (MDOF) for wind and wave induced loading considering the effect of rotating blades and soil structure interaction using realistic offshore wind and wave loading modeled using von Karman spectra and Pierson-Moskowitz Spectra, respectively. Cases for offshore wind turbine blades lumped at the nacelle as well as the effect of rotating blades - both are simulated, to incorporate blade tower coupling and soil structure interaction on the response of the structure. The rotational effect of the blades generates turbulence that has been modeled using shape filter method described later in the section. The soil structure interaction effect has been modeled using frequency independent springs and dashpots at the base of the structure. The outline of the paper is as follows. In Section 2, the theoretical formulation considering soil-structure interaction is discussed followed by Section 3, which discusses the method of calculation of wind and wave loading in the offshore environment. In Section 4, the description of the offshore wind turbine model for numerical analysis is given. Based on the analytical approaches described, a numerical analysis has been performed in section 5. 2. Theoretical formulation 2.1. Wind turbine structural system Offshore structures can be discretized by lumping masses at selected nodal points. A schematic model is shown in Fig. 1. Let q(t) be the vector of the generalized coordinates of the offshore wind turbine structure system, then the equation of motion is defined as:

M {q¨} + C {q} + K {q} = {0}

(1)

where M is the diagonal mass matrix, C is the damping matrix and K is the stiffness matrix and the displacement vector q is given as:

{q} = [x1 1x2

2 x3 3 ........x n n ]

(2)

where x is the translational degree of freedom and θ is the rotational degree of freedom. A common modeling and analyzing method of the soil-structure interaction is using the stiffness springs to model the soil. Usually, the foundation soil has a finite stiffness. In order to represent the finite stiffness and damping of the foundation soil, a set of springs can be applied in one or several supporting points of the structure. Herein, to include the soil structure interaction effect, spring and dashpot coefficients for circular rigid footing by Gazetas [32] have been used. As seen in Fig. 1, the lateral vibration of wind turbines is controlled by three foundation springs: Kx (transverse spring), Kv (axial spring) and Kr (rotational spring). These frequency independent coefficients give the translational, axial and rotational stiffness and damping at the node. These coefficients, in turn, depend on the soil and foundation properties. The stiffness and damping coefficients of the soil-foundation system considering a soil with the shear modulus of elasticity (G), Poisson's ratio (ν) and shear wave velocity (Vs) and radius of foundation (R) are [Gazetas [32]]: 2

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stiffness coefficients of soil medium and Cx, Cv and Cr represent the horizontal, vertical and rocking viscous damping coefficients for radiation damping of the soil-foundation system. These depend on the soil properties and the geometry of the foundation. The rigid foundationsoil interaction is modeled by the equivalent spring-damper-mass model system. The equation of motion of a wind turbine structure including soil-structure interaction is given as [Novak [33]]: N i= 1

N

m i + m f x¨ f +

i= 1

N

m i x¨i +

m i h i ¨ + Cx u f + Cr

i= 1

(5)

+ Kh x f = 0 N i= 1

N

m i h i x¨ f +

i= 1

{m i h i2 + Ii} + If

N

+

i= 1

m i h i x i + Cr (6)

+ Kr = 0

where m is the mass of the tower at each node, mf is the mass of the foundation, I is the mass moment of inertia at each node of the structure, If is the mass moment of inertia of the foundation with xf and θ as its translation and rotation. The term Kx represents the horizontal stiffness matrix, Cx is the horizontal damping matrix, and h is the height of the superstructure. The matrices for the equation of motion considering SSI effect is given as:

Fig. 1. Physical representation of soil structure interaction model using springs and dashpots.

[M ] =

m1 0 0 0 0 0

0 m2 0 0 0 0

m1

m2

0 0 0 0 0

0 0 0

0 0 0 0 mn 0

0 0

mn

m1 h1 m2 h2

0 0 0 0 0 mn

1

mn

1

mn 1 hn

m1 m2

mn 1 mn mf + N i=1

mn hn

1

m1 h1 m2 h2

mn 1 hn mn hn

N i=1

mi

mi hi

If +

1

N mi hi i=1 N (I + i=1 i

mi

h i2)

(7)

k1 + k2 k2 k2 k2 + k3 0 k3 0 0 [K ] = 0 0 0 0 0

0

0 k3

0 0

0 0

+ kn kn 0

0 0 0 kn kn 0

0

0

0

kN

1

0 0 0 0 0 Kx 0

0 0 0 0 0 0 1 ( K v R2 3

+ Kr ) (8)

c1 + c2 c2 c2 c2 + c3 0 k3 0 0 [C ] = 0 0 0 0 0

0

0 c3

0 0

0 0

+ cn cn 0

0 0 0 cn cn 0

0

0

0

cn

1

0 0 0 0 0 cx 0

0 0 0 0 0 0 1 ( Cv R2 3

+ Cr ) (9)

where k1-kn represent stiffness of the elements in the superstructure and c1 to cn represent the damping of the elements in the superstructure. 3. Loads

Fig. 2. (a) Wave induced loading, and (b) Fast Fourier Transform (FFT) of wave force.

Kx =

8GR 2

Cx =

4.6GR2 (2 ) Vs

Kv =

4GR 2

Cr =

Kr =

3.1. Wave load

8GR3 3(1

)

0.4GR4 0.4GR4 Cr = (1 ) Vs (1 ) Vs

The wave force acting on a cylinder submerged in water is given by Morison et al. [34]. The Morison's equation takes into account the effect of two force components, i) an inertia force in phase with the local flow acceleration, and ii) a drag force proportional to the square of the instantaneous flow velocity. Hence, the total force per unit height, F(t) acting on the cylinder is given as Morison et al. [34].:

(3) (4)

where Kx, Kv and Kr represent the horizontal, vertical and rocking 3

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Fig. 3. Variation of normalized autocorrelation function Kv with rotational speed of shaft ψ, (b) with variation in blade length, r for a particular rotational speed shaft value of 1.57 rad/sec, and (c) fixed point spectrum Su(f) and the rotationally sampled spectrum Su(r, f) for different length of blades.

F (t ) =

w CM

4

D 2u +

1 2

w DCD A

u u = FD + FI

where CD is the drag coefficient, CI is the inertia coefficient, D is the projected diameter, u is the velocity of water. When the motion of the structure is considered, the drag force and the inertia force is reduced by a factor proportional to the acceleration of the structure. Hence, the modified Morrison's equation taking into account the relative motion of the structure becomes:

F (t ) =

1 CD A (u 2

x) u

x + (CI

1) V x¨ + CI V u¨

pm g 5

S ( )=2

(10)

pm

= 0.0081,

2

pm

p

exp

=

5 , 4

4

pm

4 p

=

(12)

0.74 pm

g Vm

(13)

where ω is the angular frequency = 2π/T, ωp is the spectral peak angular frequency, η(t) is the local free surface elevation with respect to the M.S.L., g is the acceleration due to gravity, Vm is the mean wind speed at the reference height of 19.5 m above M.S.L. The derivation of the force spectral density for the case of a fully submerged section was given by Borgman [36] on the basis of a linearization of the nonlinear drag component. The Morison equation considering the relative velocity given by Equation (10) is thus approximated as:

(11)

where u is the horizontal velocity of water particles, u¨ is the water particle acceleration, x is the structural velocity andx¨ is the structural acceleration. The first term on the right-hand side of equation (10) represents the form drag, the second term is the inertial force due to local and convective accelerations of the fluid around the body, and the third term is the inertial force due to the motion of the body as it would be in a fluid at rest condition in drag and inertial forces resulting from the motion of the fluid around the body. The presence of the square term in the structural velocity imparts nonlinearity to Equation (10).

F (z , t ) =

sinh kd

CI i cosh(kz ) + CD

8

cosh(kz )

x (z ,

t)

(t ) (14)

By taking the Fourier transform of equation (13), the force spectrum is obtained as a function of the wave energy spectral density given as:

SFF ( , z ) =

3.1.1. Power spectral density function (PSDF) of wave force In the present work a Pierson-Moskowitz wave spectrum [Pierson and Moskowitz [35]] has been adopted for dynamic analysis of the structure given as:

2

sinh kd S ( )

[CI

cosh(kz )]2 + CD

8

2

cosh(kz )

x (z ,

t) (15)

where the expression between braces in Equation (14) is the force transfer function, CD is the drag coefficient, CI is the inertia coefficient, 4

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Fig. 4. (a) Drag force with and without considering blade rotation, (b) Fast Fourier Transform (FFT) of fixed as well as rotationally sampled time history.

Fig. 5. FFT of rotationally sampled and fixed point time history of wind drag force at different rotating shaft values.

σx is the standard deviation of the water-particle velocity component in x direction, k is the wave number and z is the elevation from the bottom most point. Fig. 2(a) and (b) presents the wave induced loading in time domain as well as in frequency domain (Fast Fourier Transform (FFT)) applied in the present study.

3.2. Along-wind forcing The total wind force on any structural member is a summation of the mean and the fluctuating components. The total drag force in along wind direction, F(t), experienced by a slender structure is expressed as 5

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Fig. 6. Frequency spectrum of the dynamic loads and the rotor and blade passing frequency of 5 MW wind turbine with an operational interval of 6.9–12.1 rpm.

[Holmes [37]]:

AV (t)2 =

1 CD A [Vm + v (t)]2 2

Kv ( ) = (16)

F

2 u

( ) 1 + (70.8 ( ) ) 200

=

(21)

Lu Vm

(22)

K v (r , ) =

2

1/3

s 2

2 u 1 3

()

1.34L u

2 r sin

( )

K1/3

s s s + K2/3 1.34L u 2{1.34L u} 1.34L u

2

2

s

s = (Vm )2 + 4r 2 sin2

2 5/6

Kv ( )cos(2 f )d 0

K1/3

The power spectrum density of the wind speed turbulence seen by a point on the rotating blade at distance r from the hub is given as:

(17)

(23)

2

(24)

Fig. 3(a) shows the plot of fixed point as well as rotating wind turbulence for different values of rotating speed of the blade ψ. Fig. 3(b) shows how the normalized autocorrelation function varies with time at a distance r = 0 m (case which is identical to the fixed point autocorrelation function), 20 m, 32.5 m and 40 m. These curves display pronounced peaks after each full revolution. The rotationally sampled spectrum is obtained by applying the discrete Fourier transform (DFT) to the sampled autocorrelation function given in Equation (22). Fig. 3(c) shows the rotating wind sampled spectra as well fixed point spectra obtained after DFT in Matlab Programming for different lengths of blades.

(18)

where z is the elevation (m), Sv (z , f ) is the power spectral density function of the fluctuating wind velocity as a function of elevation and frequency f in Hz, σu is the turbulence standard deviation, Lu is the turbulence length and Vm is the mean wind speed. The terms Sv (z , f ) and the autocorrelation function, K v (f ) , are given by the Fourier transform as [Burlibas and Ceanga [40]]:

Sv (z , f ) = 4

2

()

= 1.34

Lu Vm

Lu Vm

(20)

1/3

2 u 1 3

2

K v (f ) =

In order to make the prediction of the wind speed experienced by a wind turbine as accurate as possible, wind shear and tower shadow were used for modeling the turbulence component. There are at least three methods for modeling the local effects of the wind speed. In this paper, the focus was set on the spatial/rotational filter method from von Karman [Burton [38]]. Connell [39] reported that a rotating blade is subjected to a typical fluctuating wind velocity spectrum, known as a rotationally sampled spectrum. The rotationally sampled spectrum deduction is based on the fixed point (without rotation) von Karman model, given by the following expression [Burton [38]]:

fSv (z, f )

) df

Using the von Karman spectrum model for the fixed point turbulence, the autocorrelation function K v (f ) is given as:

where CD is the coefficient of drag, ρ is the density of air, A is the surface area of the body, and V(t) is the total velocity of the wind flow. In realistic approach, the wind velocity V(t), may be decomposed into a mean component Vm and a fluctuating component v'(t). By neglecting the higher powers of gust component in equation (15), the total drag force would be composed of a mean component Fm and a fluctuating component F (t ) given as:

1 CD A (Vm )2 + CD AVm v (t ) = Fm + F (t ) 2

Sv (f )cos(2 f 0

(19) 6

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Fig. 7. Mode shapes of MDOF system.

For modeling the wind speed turbulence in time series, two filters are introduced after the fixed point wind speed -one that takes care of the averaging effect (the Spatial Filter), and the other one that takes into account the effect of the shear, the tower shadow and effects introduced by the rotating blades (the rotational sampling filter). For the spatial filter, the following transfer function is used given as [Burlibas and Ceanga (2013) [40]]:

H1 (s) =

For the rotational sample filter, the following transfer functions are used: 1. A third order subsystem, dependent on parameters m1 and m2 mentioned above, with the transfer function given as [Burlibas (2013)]:

H2 (s ) =

4Tf (m1 Tf2 s + 1) (Tf2 s + 1)(m2 Tf2 s + 1)

(25)

(m2 Tf2 s + 1)2 (1/2 )2 s + 1 2.9 4 (m1 Tf2 s + 1)(0.12m2 Tf2 s + 1)(0.01s + 1)

(26)

2. A combination of four resonant filters, with the transfer function given by the following expression as [Burlibas and Ceanga [40]]:

where Tf = Lu/Vm is the time constant of the shaping filter, Tf2 = 70.8 Tf , m1 and m2 are filter parameters whose values are 0.4 and 0.25, respectively. 7

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Table 1 Summary of results. Response

Displacement (m)

Wind Fixed 0.543 V40 1.961 V70 1.266 V100 1.198 V200 0.990 Wave Fixed 0.321 V40 2.112 V70 1.240 V100 1.209 V200 0.856 Combined Wind-Wave Fixed 0.766 V40 2.686 V70 2.017 V100 1.331 V200 1.075

4

(F k )2 pk

Hr (s ) = k= 1

( )s + ( )s + 1 2

1 2

2csis k pk

2

2

2 2csiik kF s 2

Acceleration (m/sec2)

Rotation (Radian)

1.238 1.321 1.835 2.053 2.393

−0.0091 −0.0213 −0.0158 −0.0164 −0.0154

3.464 11.988 7.734 7.511 5.471

−0.0084 −0.0329 −0.0189 −0.0226 −0.0157

3.654 9.008 8.197 7.287 5.450

−0.0137 −0.0325 −0.0303 −0.0253 −0.0216

Blade Coupled 1.346 3.918 3.024 2.455 1.828 Blade Coupled 0.637 4.357 2.760 1.751 1.382 Blade Coupled 1.473 4.070 3.110 3.034 1.695

3. Two auxiliary resonant filters given by the transfer functions as [Burlibas and Ceanga [40]]:

( ) s + 1.8 ( ) s + 1.35

Hral (s ) =

( ) + 1.2 ( ) s + 1.6

1 2

2

2

1 2

2

1.5 2 2 s 2 1 2 2

csi1 F 3.6 2

2

( ) s+( )

s+

1.5 F 3.6 2

3.6

1.5 csi1 F s 2

2

1.5 F s 2

F 2 3.6 F 2

(28)

+ (1.4F1)2

+ (1.4 F1) 2

Rotation (Radian)

1.555 2.597 1.839 1.642 2.577

−0.0211 −0.0417 −0.0362 −0.0326 −0.0277

3.694 14.873 7.082 6.619 6.004

−0.012 −0.050 −0.036 −0.027 −0.024

3.560 8.616 6.831 7.943 5.447

−0.024 −0.046 −0.041 −0.043 −0.029

The structural model considered herein is an offshore wind turbine structure from the study by Colwell and Basu [41] as shown in Fig. 1 and has been validated by his work. In the present work, a 5 MW (megawatt) three-bladed offshore wind turbine with a hollow tubular tower is considered with a base radius of 2.15 m, the thickness of 18 mm and top radius of 1.75 m with a thickness of 10 mm. The tower is of height 104 m. The blade length has been taken as 63 m. The system under present investigation is an offshore wind turbine structure under parked condition. The wind and wave load used in the present study are as per North Sea conditions. The modulus of elasticity of steel is taken as 2 × 1011 N/m2. The structural damping ratio is assumed to be 0.01. A suitable discretization of the exposed structure is carried out for load application, at each node a random force is specified which corresponds to the effect of random waves (for the node at the mean sea level) or wind excitation acting on the corresponding area of exposure. Among the seven nodes, one node at the bed level, three nodes along the splash zones (e.g., high water level, mean water level and low water level), one at the platform level, one at the midlength of the tower and the last one at the tower top have been considered. The soil condition for the North Sea has been taken as having shear wave velocity (Vs) = 100 m/ sec and density of soil (ρ) = 1700 kg/m3. However for the parametric study, the analyses are carried out for four different soil conditions in terms of shear wave velocity - (i) V40 for 40 m/sec (very soft soil), (ii) V70 for 70 m/sec (soft soil), (iii) V100 for 100 m/sec (medium dense) and, (iv) V200 for 200 m/sec (very dense soil). These shear wave velocities of soil represent the actual scenario of soil site conditions. An example of a very soft soil is a loose sandy soil fill which amplifies the structural response by increasing its time period. In the present work, the soil used is sandy soil. The vibration frequencies and vibration modes of the wind turbine are calculated by carrying out an eigenvalue analysis. Fig. 7(a)–7(e) show the mode shapes of the structure obtained from the present study. The modal participation factors for the fundamental vibration mode is 91%. The soil structure interaction analysis is carried out by considering first five mode shapes that are extracted through modal analysis. From Fig. 7(a)–7(e), it is evident that as the shear wave velocity of the soil decreases the fundamental modal frequency of the system drops.

(27)

where F = ψ/2 and parameters csisk, csiik and pk are given in the appendix.

Hral (s ) =

Acceleration (m/sec2)

4. Numerical model

+ (pk )2 + (F k )2

Displacement (m)

(29)

where F1 = F/2 and the parameter csi1 is given in the appendix. By multiplying all the mentioned transfer functions from equations (24)–(28) one obtains a shaping filter which represents the combination of two different shaping filters - the fixed point shaping filter, given by equation (24) and the rotational sampled filter, given by the multiplication of the remaining transfer functions. To obtain the rotating speed turbulence, a white noise signal is filtered with the fixed point shape filter. After obtaining this, the refined signal is again passed through the rotating shape filter. Fig. 4(a) shows the fixed as well rotationally sampled wind time history obtained using this method. Fig. 4(b) shows the Fast Fourier Transform (FFT) of the time history and distinct peaks can be seen at the respective rotor speed of 1.57 rad/sec and 3.14 rad/sec. Fig. 5 presents the FFT of rotationally sampled wind speed turbulence obtained through the shape filters for different values of blade shaft frequencies to give an outlook about the frequency spectrum of these loads. Fig. 6 presents the frequency spectrum of fixed point as well as rotationally sampled wind turbulence and the wave load in terms of drag force applied in the present study. Light excitation due to excessive or mass imbalances at the nacelle occurs at the rotor frequency (often termed 1P) which lies in the range 0.115–0.2 Hz and the corresponding blade passing frequency (3P), arising from the individual blades passing by the tower for a three-bladed turbine lies in the range 0.345–0.6 Hz. The wind turbine should be designed such that the fundamental modal period lies in between the 1P and 3P range, and for stiff soil conditions. However, for soft soil, the fundamental modal period slips down into the 1P range (0.717 Hz for V40 soil) which may lead to undesirable dynamic interaction caused by resonance.

5. Numerical study In this section, the dynamic analysis of the structure is discussed. Firstly, time history analysis of the offshore wind turbine structure 8

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Fig. 8. Time history of (a) displacement, (b) acceleration, (c) rotation at tower top at node 8 for wind loading only.

under wind and wave loading with soil structure interaction is discussed. The comparisons are with respect to the fixed base structure. The effect of blade coupling is also shown here. Secondly, the effect of blade tower coupling is taken into consideration for (i) the fixed base, and (ii) V40 soil condition. The structure is subjected to wind and wave loading time histories simulated at two discrete points along the tower, which are at node 8 and node 4, respectively. The mean wind velocity taken to calculate the forces at the top of the tower has been taken as 25 m/s. The density of air, the coefficient of drag and the roughness length for the water surface are taken as 1.2 kg/m3, 1.2 and 0.002, respectively. The modal damping ratio is taken as 5% and 20% for tower bending and flap blade bending i.e., in the first mode, respectively. A high damping ratio for the blade-bending mode is assumed due to aerodynamic damping. The random wave elevation spectra have been simulated using the Pierson-Moskowitz spectra. The following parameter values have been used in the wave spectra simulation: Cd = 1.2, ρ0 = 1.2 kg/m3 (at sea

level), Vm = 25 m/s (mean wind speed 19 m above sea level). The time histories for wave loading at node 4 and wind loading at node 8 of 120 s duration is shown in Figs. 2(a) and 6(a) respectively. The effect of blade tower coupling effect is considered for the study at the top of the tower. To analyze the influence of blade-tower coupling effects on the tower, the wind-induced response of the tower was calculated in the time domain in two cases. In the first case, the mass of the blades and the hub altogether being added at the top of the tower as a lumped mass, in the other case, blade-tower coupling vibration is included. The three rotating blades are assumed to be identical uniform cantilever beams of the rectangular hollow cross section. The properties of the blades are taken from the work by Murtagh and Basu (2005) [42]. Each blade is 30 m in length with a hub radius of 3 m. The width of the blade is 2.8 m, with a depth of 0.8 m and a thickness of 0.01 m. Modal damping ratios of 1% of the critical are assumed for the first three modes of the blade. The mass, stiffness and damping matrices are developed and incorporated in the total model. The effect of the rotation of the blades is 9

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Fig. 9. Fast Fourier transform of displacement and acceleration time history for wind load at node 8.

excitation. Introducing SSI effect in structures reduces the stiffness of the system and decreases its natural frequency thereby exposing it to low-frequency excitation of wind. It is clearly seen in Fig. 9(a) from the FFT of displacement response of the system with V40 soil that the peak response occurs at a lower frequency as compared to the fixed base response. Structural response under wind loading is a superposition of two components, referred to as the background response and the resonant response. Background response refers to the quasi-static response to the fluctuating portion of the wind load that would occur if the natural frequency is extremely high. On the other hand, the resonant response occurs at the natural frequency of the system. It is observed in the present study that the background response for the structure with SSI effect (V40 soil condition) is higher compared to the fixed base condition. While considering SSI effect it should be noted that the first natural frequency is for the soil and the second frequency is the fundamental frequency of the structure. Hence, it can be seen that the resonance for V40 soil condition occurs at 0.717 Hz instead of 0.313 Hz. At higher frequencies, the response dampens out for a structure with SSI effect. It is observed from Fig. 9(a) that the major peaks occur around 0–2 Hz and at around 3P depending on soil stiffness. It is seen from Fig. 9(b) that the second mode also plays a major role in acceleration response of the structure with SSI effect (V40) as a distinct peak can be observed at a higher frequency other than the fundamental frequency.

taken into consideration in the response of the structure. The blade rotational frequency is taken as 1.57 rad/sec. Here in the results and discussion, time history till 90 s is shown for better clarity of the plots. The summary of the results for all the cases for has been provided in Table 1. 5.1. Results and discussions 5.1.1. Wind loading with and without SSI effect The random wind load time history applied at node 7 (top of the tower) without blade interaction effect is presented in Fig. 4(a). Fig. 8(a)–8(d) show the displacement, acceleration and the rotation time history response for the offshore wind turbine structure considering fixed base as well as soil structure interaction parameters. Total displacement at the top includes horizontal foundation displacement, lateral displacement due to foundation rotation and the relative displacement of the nacelle as it incorporates the rotational effect due to the foundation [Harte et al. (2012) [31]]. As expected, there is a large increase in response when the shear wave velocity of the soil drops, caused by the rotation of the foundation increasing the displacement of the nacelle as observed from Fig. 8(a). It is seen that the total displacement time history is maximum for V40 soil. It shows an amplification of 295% as compared to the fixed base condition in Fig. 8(a). For V70 soil, the total displacement response is amplified by 116%. For V100 and V200 soil, this amplification is 48.69% and 16.99% respectively. Similarly, for acceleration, it is observed that the response is increased by 6.70% as shown in Fig. 8(b) for the softest soil strata V40. However, for V70, V100, and V200 soil condition, it is seen that the acceleration response increases by 48.22%, 65.83%, and 93.2% respectively. Hence, it can be concluded that as the soil becomes harder (stiffer), its acceleration response increases. Acceleration response of V40 soil condition increases as compared to fixed base conditions due to the kinematic interaction between the soil base and structure. No significant difference between the shear and moment in the foundation and tower base was found considering SSI. Hence, the results were not presented. The foundation rotation plays a significant role in determining the response of the tower, Fig. 8(c) shows the time history of rotation of the tower. It is seen that introducing SSI effect, increases the rotation by 97.6% when soil with shear wave velocity (V40) is introduced. For soil V70, V100 and V200 this increase is 71.5%, 54.5%, 31.3% respectively. Results in terms of displacement and acceleration response are presented in the frequency domain and are shown in Fig. 9 (a) and 9(b). For structures with foundations, the soil-structure interaction depends primarily on the stiffness of the structure and on the nature of external

5.1.2. Wave loading The random wave load time history applied at node 4 has been presented in Fig. 2(a). Time history response in displacement and acceleration are presented in Fig. 10(a)–10(c). Fig. 10(a) shows that the displacement response is amplified by 557% for V40 soil condition considering fixed base condition as the reference. Since, wave loading is of higher magnitude and lies close to the ground level as compared to the wind load (which is of lower magnitude), it causes higher rotation of the tower. This high amplification is pertaining to the fact that wave load causes more rotation of the foundation due to its magnitude and as it is an inertial load. Hence, it causes higher acceleration of the structure too. Similarly, V70 and V100 causes an amplification of 286% and 276% respectively in the displacement response. For V200, this amplification in displacement is reduced to 167%. Similarly, for acceleration, as seen in Fig. 10(b), there is an amplification by 246% for V40 soil. The response is increased by 123% and 116.8% for V70 and V100 soil respectively. For V200 soil, this amplification is 57.9%. This amplification in displacement, as well as acceleration response, is quite higher for wave loading as compared to wind loading even if the total response is lesser for wave loading. Fig. 10(c) shows the tower top rotation time history under wave 10

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Fig. 10. Time history of (a) displacement, (b) acceleration, (c) rotation at tower top at node 8 for wave loading only.

loading and it is seen that there is an increase of 291% for V40 soil condition as compared to the fixed base case. This increase in the rotation is higher as compared to wind loading. Hence, it can be concluded that wave loading has a higher contribution in tower top rotation response of the structure when SSI effect is considered as compared to wind loading. This increase is 125% for V70 soil condition, 169% for V100 soil condition and 86% for V200 soil condition. It is seen that this increase is higher as that compared to wind loading. Fig. 11(a) and (b) show the displacement and acceleration response of the structure in the frequency domain for wave loading. It is seen that major peaks of displacement response (resonant response) of the structure with SSI fall within 1.5 Hz frequency range where as for the fixed base structure, it is up to 2 Hz. Wave loading affects the second mode of the fixed base structure which was not included in the case of wind loading. However, for acceleration response, it is seen that the major peaks or the zone of influence are up to 3 Hz frequency range. It is seen that the V40 soil strata have a higher response in all the

frequency ranges except the second frequency of the fixed base structure, compared to wind loading where it dominated the background response only. It also shows that the peak response does not dampen out after 2 Hz as compared to wind loading case where the response would dampen out. 5.1.3. Combined wind and wave loading For combined wind and wave loading, the displacement and acceleration response is presented in Fig. 12(a)–12(c). Fig. 12(a) shows that the displacement response is amplified by 263.3% for the V40 soil condition. This amplification in displacement response is recorded for V70, V100 and V200 soil which is 97.89%, 66.01%, and 21.80% respectively. In a similar way, for acceleration response, Fig. 12(b) shows that the amplification is 132.6% for V40 soil. Further, for V70, V100 and V200 soil, this amplification in acceleration is 83.01%, 74.4%, and 21.14%. The rotation at the tower top is highest for V40 soil condition, which is 0.0125 radian as seen from Fig. 12(c). It is higher than fixed 11

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Fig. 11. Fast Fourier transform of displacement and acceleration time history for wave load at node 8.

coupling effect and it can be concluded that blade coupling effect significantly increases the base shear response and base moment of the structure when rotation of blades is considered in fixed base conditions. Fig. 15(a)–15(f) presents the time history response of the structure considering blade-coupling effect under wave induced loading. Fig. 15(a) and (b) present the results for displacement response of the structure with and without considering SSI and blade coupling effect. It is seen from Fig. 15(a), for a fixed base structure when blade tower coupling is considered, it is seen that the displacement response is increased by 93%. For soil condition V40, the displacement response increases by 106%. When compared to a fixed base structure without SSI effect [Fig. 15(a)], this amplification is significantly huge. Hence, it can be concluded that wave induced loading together with SSI and blade tower coupling significantly alters the response of the structure. Hence, if these effects are not considered in the structural design of an offshore wind turbine structure, it can significantly underestimate the response of the structure. Fig. 15(c) presents the acceleration response of the fixed base structure with and without considering blade tower coupling. As seen from the figure, the acceleration response increases by 7%. Fig. 15(d) presents the acceleration response for V40 soil condition and it can be concluded from the figure that blade tower coupling increases the response by 100%. When compared with a fixed base structure, the acceleration response increases by 329% when SSI and blade coupling effect both is considered. Hence, it can be concluded that the soil structure effect plays a very vital role and therefore while considering the blade coupling effect, SSI should also be considered. It can also be concluded that while considering the blade coupling effect or in other words, under operating conditions, it is seen that displacement, as well as acceleration response, is amplified significantly when SSI and blade coupling effect is also considered. Fig. 15(e) and (f) present the base shear and base moment of the fixed base structure considering blade-coupling effect. There is no significant effect on the base shear and base moment response of the structure under wave loading alone when blade coupling effect is considered. It is seen that Fig. 16(a)–16(f) present the response of the structure for combined wind and wave loading. Under combined wind and wave loading, it is seen that the displacement response of the fixed base structure increases by 85% with blade tower coupling effect as presented in Fig. 16(a). Fig. 16(b) presents the displacement response when V40 soil is considered. In this case, the displacement response increases by 51%. However, when compared to the fixed base response, including SSI effect and blade tower coupling this amplification in displacement response is as high as 583%. For acceleration response as

base condition by 145.96%. This percentage increase is lesser than wave-induced loading due to the aerodynamic damping induced by wind loading. Hence, it can be concluded that the highest contribution in displacement response is due to wind-induced loading but for acceleration and rotation at tower top, the highest contribution happens due to wave-induced effect combined with SSI effect. Hence, while analyzing the offshore structures it is extremely necessary to analyze the structure for realistic wave loading. Fig. 13(a) and (b) present the displacement and acceleration response of the structure in the frequency domain and it is seen that the majority of the peaks fall in the range of 1.5 Hz for displacement and 3 Hz for the case of acceleration same as that of wave loading. Hence, wave loading plays a significant role in the response of the structure. 5.1.4. Blade tower coupling effect In order to consider the effect of blade rotation on the response of the structure including soil structure interaction, the offshore wind turbine structure is analyzed for blade coupling under wind excitation as shown in Fig. 4(a) for the fixed base condition as well as considering SSI effect for V40 soil condition only. The natural frequency of the fixed base structure without SSI and blade coupling is 0.31 Hz. Blade coupling effect alters the natural frequency of the structure and reduces it to 0.213 Hz which is 104% reduction in natural frequency. Similarly, for V40 soil condition, the natural frequency of the structure was 0.171 Hz. However, incorporating blade effect reduces the natural frequency by 75% (0.043 Hz). Fig. 14(a)–14(f) presents the time history response of the structure considering blade-coupling effect under wind induced loading. It is seen that the displacement response is amplified by 76% as shown in Fig. 14(a) for fixed base structure and when SSI effect is considered this amplification goes up to 29% as presented in Fig. 14(b). When compared to a fixed base structure without SSI effect, this amplification is significantly high (262%) when SSI and blade coupling effect is considered. Fig. 14(c) shows the acceleration response comparison of the structure without considering SSI effect. It is seen that the acceleration response increases by 20%. However, considering SSI effect (V40), this amplification is high by 106%. Fig. 14(d) shows the acceleration response of the structure with and without considering blade coupling for V40 soil. It is seen from the figure that blade tower coupling increases the acceleration response by 32.4% when the structure is in V40 soil. However, when compared to the fixed base structure, this amplification in acceleration response is 109%. Fig. 14(e) and (f) present the base shear and base moment time history of the fixed base structure with and without considering blade

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Fig. 12. Time history of (a) displacement, (b) acceleration, (c) rotation at tower top at node 8 for combined wind loading and wave loading.

shown in Fig. 16(c) and (d), there is an amplification of 210% for the fixed base structure when SSI and blade coupling effect is also included. Hence, it is again concluded that the response of the structure considering blade tower coupling but without SSI effect gives unsafe results and hence it is very important to consider both blade coupling as well as SSI effect while analyzing offshore wind turbine structure. It is seen from the results that blade tower coupling along with SSI effect amplifies the response of the structure significantly. Fig. 16(e) and (f) present the base shear and base moment plots for the fixed base structure considering blade tower coupling. Under combined effect of wind and wave loading it is seen there is no much significant increase in base shear and base moment response of the strucure.

6. Conclusions In this study, an MDOF model of an offshore wind turbine has been considered, which is subjected to random wind and wave loading simulated by using Von Karman spectrum and Pierson-Moskowitz spectrum, respectively and analyzed for SSI as well blade coupling effects. The behavior of the soil which imparts flexibility to the structure has been simulated using Gazeta's equivalent spring-dashpot model for four different soil conditions varying from soft to stiff soils. The modeling approach is used to estimate the tower tip displacement as well as the acceleration for a particular value of blade rotational frequency, 1.57 rad/sec to investigate the effect of blade coupling along with the SSI effect for offshore wind turbine structure. The following conclusions can be drawn from the present study:

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Fig. 13. Fast Fourier transform of displacement and acceleration time history for combined wind and wave load at node 8.

Fig. 14. Time history of response under wind loading considering rotation of blades.

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Fig. 15. Time history of response under wave loading considering rotation of blades.

1. Under wind loading, the total displacement of the nacelle at tower top's response increases as shear wave velocity decreases, especially for lower shear wave velocity values, i. e, for softer soil strata, significant peaks of displacement response occur up to 1.5 Hz frequency. The acceleration response is higher for the stiffer structures and highest for the fixed base structure. 2. Background response for the structure under wind loading including SSI effect is higher as compared to the fixed base condition. 3. Under wave loading, the response of the offshore wind turbine structure considering SSI effect has a higher response in all the frequency ranges except for the second frequency of the fixed base structure, as compared to wind loading where it dominated the background response only. Significant peaks under acceleration response fall in the range from 0 to 3 Hz when SSI effect is included in

the fixed base model. 4. Wave loading has a higher contribution in tower top rotation response of the structure when SSI effect is considered as compared to wind loading. Wave loading significantly dominates the acceleration response of the structure. 5. Under combined wind and wave loading, wind loading governs the displacement response of the structure. However, acceleration, as well as tower top rotation, is significantly governed by wave-induced loading. 6. Blade coupling effect along with soil structure interaction not only alters the natural frequency of the system significantly, but it also amplifies the response of the offshore wind turbine structure significantly, especially for wave-induced loading.

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Fig. 16. Time history of response under combined wind and wave loading considering rotation of blades.

Appendix Rotationally sampled shaping filter parameters obtained through the optimization procedure: cs1 = 0.8347; csisk = [0.24 0.1103 0.2879 0.8229]; csiik = [0.1141 0.0982 0.0701 0.5841]; pk = [0.4462 0.8594 0.6610 1.0493]. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.soildyn.2019.05.038.

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