Predicting the performance of a floating wind energy converter in a realistic sea

Renewable Energy 101 (2017) 637e646

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Predicting the performance of a floating wind energy converter in a realistic sea Yingguang Wang a, b, c, *, Lifu Wang d a

State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, 200240, PR China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, 200240, PR China c School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, PR China d Department of Electrical Automation, Shanghai Maritime University, Shanghai, 201306, PR China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 7 April 2016 Received in revised form 22 August 2016 Accepted 14 September 2016

In this paper, the performances of a floating wind energy converter (the National Renewable Energy Laboratory 5 MW wind turbine installed on the ITI energy barge) in a realistic, multi-directional random sea are rigorously investigated. The wind loads acting on the floating wind energy converter are also fully considered in the numerical simulation process. Meanwhile, in order to improve the simulation efficiency, a new state space model (the FDI-SS model) is utilized to approximate the convolution integral term when solving the motion equation of the floating wind energy converter. For comparison purpose, the simulation results when the convolution integral term in the motion equation is approximated by a commonly used state space model based on the time domain (TD) realization theory are also included. The simulation results in this paper are systematically analyzed and compared, and the accuracy and efficiency of the new FDI-SS model are verified. Moreover, the simulation results in this article demonstrate the great necessity of using a realistic, multi-directional random sea state when calculating the generated electrical power and the dynamic responses of a floating wind energy converter. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Wind energy Realistic sea Wind loads State-space model Numerical simulation

1. Introduction This paper investigates the methods for predicting the performances of a floating wind energy converter, i. e., a wind turbine installed on a floating platform moored to a sea bed. In the worldwide offshore wind energy industry, the performance analysis of a floating wind energy converter is typically carried out by solving the converter motion equation with a complicated convolution integral term representing the hydrodynamic memory effects (see e. g., the publications: Jonkman [1-3], Jonkman and Buhl [4], Jonkman and Matha [5], Robertson and Jonkman [6], Wang et al. [7], Xia [8], Xia and Wang [9], etc.). Calculating the convolution integral term is difficult, time consuming and requiring a large amount of memory on a computing machine. Moreover, all the aforementioned research publications have only investigated the performances of floating wind energy converters in an ideal, unidirectional random sea. The wave spectra applied in these

* Corresponding author. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, 200240, PR China. E-mail address: [email protected] (Y. Wang). http://dx.doi.org/10.1016/j.renene.2016.09.025 0960-1481/© 2016 Elsevier Ltd. All rights reserved.

research publications are all uni-directional spectra, i.e., assuming that wave energy is traveling in one direction. In reality, however, wind-generated wave energy does not necessarily propagate in the same direction as the wind; instead, the energy usually spreads over various directions. Therefore, although the analyses in the above-mentioned publications are likely performed correctly and are useful for many applications (for example, when comparing to tank test data for the purposes of model validation), the calculation results in these publications can hardly be used for predicting the floating wind energy converter real world performances, let alone the simulation methods in these publications are also not timeefficient. To the best of our knowledge, in the existing literature, there is only one paper (Duarte et al. [10]) that has investigated the motion responses of a floating wind energy converter in a realistic, multidirectional random sea. However, in the study of Duarte et al. [10] the wind loads acting on the floating wind energy converter have not been included in the simulation process. Therefore, the predicting results in Duarte et al. [10] cannot be deemed reliable. As pointed out by Duarte et al. in the conclusion of their paper: “…… a preliminary study was performed on the OC4 semisubmersible platform. The comparison between the unidirectional and

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Y. Wang, L. Wang / Renewable Energy 101 (2017) 637e646

multidirectional sea state without wind loads showed a significant increase in the platform sway and roll motion. These findings should motivate further studies to carefully assess the impact of the multidirectional loads on the platform's ultimate loads and fatigue life.” Motivated by the aforementioned facts, in this article, the performances of a floating wind energy converter (the National Renewable Energy Laboratory 5 MW wind turbine installed on the ITI energy barge) in a realistic, multi-directional random sea will be rigorously investigated. The wind loads acting on the floating wind energy converter will be fully considered in the numerical simulation process. Meanwhile, in order to improve the simulation efficiency, a new state space model (the FDI-SS model) will be utilized to approximate the convolution term when solving the motion equation of the floating wind energy converter. The simulation results will be systematically analyzed and compared, and some valuable conclusions will finally be pointed out.

_ tÞdt þ CxðtÞ Kðt  tÞxð

0

¼ Pwave ðtÞ þ Pwind ðtÞ þ Pothers ðtÞ

(1)

where MRB is the rigid body inertia matrix, Að∞Þ is the constant infinite-frequency added mass matrix, xðtÞ is a vector of linear or angular displacements and C is a matrix containing the hydrostatic restoring forces or moments coefficients. Meanwhile, in Eq. (1) KðtÞ is a matrix of retardation functions. Due to the motion of the floating wind energy converter, waves will be generated in the free surface. These waves will persist at all subsequent times and affect the motion of the floating wind energy converter. This is known as the hydrodynamic memory effects, and they are captured in Eq. (1) _ tÞ and the by the convolution integral term which is a function of xð retardation functions KðtÞ. Calculating the convolution integral term in Eq. (1) is difficult, time consuming and requiring a large amount of memory on a computing machine. In this paper, in order to improve the computational efficiency, a new state space model (the FDI-SS model) will be utilized to fit a parametric model to this convolution integral term. This will be further explained in the next section. On the right hand side of Eq. (1) Pwave ðtÞ denote wave loads (forces or moments), Pwind ðtÞ denote wind loads, and Pothers ðtÞ denote other loads due to hydrodynamic viscous effects and mooring lines forces. The frequency domain counterpart of Eq. (1) is written



  u2 ½MRB þ AðuÞ þ juBðuÞ þ C XðjuÞ ¼ Pwave ðjuÞ þ Pwind ðjuÞ þ Pothers ðjuÞ

Z∞

BðuÞ ¼

1

u

Z∞

KðtÞsinðutÞdt

(3)

0

KðtÞcosðutÞdt

(4)

0

Eq. (4) is rewritten using the inverse Fourier transform

KðtÞ ¼

2

Z∞

p

BðuÞcosðutÞdu

(5)

0

Fourier Transforming KðtÞ leads to

Z∞

KðtÞejut dt ¼ BðuÞ þ ju½AðuÞ  Að∞Þ

(6)

0

The vector-form time domain motion equations of a floating wind energy converter subjected to wind, wave and other loads can be expressed as:

Zt

AðuÞ ¼ Að∞Þ 

KðjuÞ ¼

2. The motion equations of the floating wind energy converter

€ ðtÞ þ ½MRB þ Að∞Þx

and those of the frequency domain equation (2):

3. Identification of the hydrodynamic memory effects As explained in Section 2, calculating the convolution integral term in Eq. (1) is difficult, time consuming and requiring a large amount of memory on a computing machine. In this paper, in order to improve the computational efficiency, a state space model will be developed to fit a parametric model to this convolution integral term as follows:

m¼

Zt 0

 0 0_ _ _ tÞdtz z ¼ A z þ0 B x Kðt  tÞxð m¼Cz

We can notice that in order to derive the state-space system, the matrices A0 , B0 and C0 must be first calculated. An approach based on the time domain (TD) realization theory can be used to perform the identification of the state-space system. In the following the theoretical background of this approach will be elucidated. We recall that the matrices AðuÞ and BðuÞ of a floating wind energy converter can be calculated using a hydrodynamic boundary element method computer code. After BðuÞ is calculated, the inverse Fourier transform in Eq. (5) can then be used to obtain the retardation functions KðtÞ. However, the inverse Fourier transform process in Eq. (5) will inevitably lead to additional errors. For numerical implementation, the cosine transformation described in Eq. (5) can be carried out by using a trapezoidal integration rule as follows:

K ij ðtÞ ¼

1 X Du kmax Du  2Bij ðkDuÞcosðkDutÞ þ B ð0Þ p k¼1 p ij

þ Bij ðkmax Þcosðkmax DutÞ (2)

where AðuÞ denotes the added mass matrix, BðuÞ denotes the damping matrix and j denotes the imaginary unit. In the practice of ocean engineering, it is routine work to compute the matrices AðuÞ and BðuÞ by a 3D hydrodynamic boundary element method computer code. We will rely on a result from Ogilvie [11] to obtain the relationship between the parameters of the time domain equation (1)

(7)



(8)

where kmax is the number of the frequency vector entries calculated using a boundary element method computer code. Du is a step size of the angular frequency. Once the retardation functions (impulseresponse functions) are obtained by Eq. (8), an identification scheme based on the Hankel Singular Value Decomposition (SVD) can subsequently be applied. This method was proposed by Ref. [12] and is available in the MATLAB function imp2ss. However, because errors have been introduced in the process of calculating KðtÞ as explained previously, the identified state-space system

Y. Wang, L. Wang / Renewable Energy 101 (2017) 637e646

based on the KðtÞ and the SVD scheme will also become inaccurate. In order to improve the computational accuracy, in this paper, we will utilize a new state space model (the FDI-SS model) recently developed by the first author of this article in Wang [13]. However, in Wang [13] the advantages of the new FDI-SS model was only tested against a floating wind turbine operating in an ideal, unidirectional random sea. In this paper, going a step further, the accuracy and efficiency of the new FDI-SS model will be validated by applying it in predicting the dynamic responses and performances of a floating wind energy converter operating in a realistic, multidirectional random sea. In the following, the theoretical background of the new FDI-SS model will be thoroughly illustrated. We recall that the matrices AðuÞ and BðuÞ of a floating wind energy converter can be calculated using a hydrodynamic boundary element method computer code. Then the retardation matrix KðjuÞ can be calculated by utilizing Eq. (6). For each entry of the retardation matrix (Kðjul Þ) we can fit a relative degree one transfer function as follows: m m1 þ / þ b s þ b 1 0 ~ qÞ ¼ Pðs; qÞ ¼ bm s þ bm1 s Kðs; n n1 Q ðs; qÞ s þ an1 s þ / þ a1 s þ a0

(9)

In Eq. (9) the parameters vector q is defined to be:

q ¼ ½bm ; bm1 /b1 ; b0 ; an1 ; /a1 ; a0 T

(10)

The essence of our new FDI-SS model is to decide the parame~ qÞ can best fit Kðju Þ in the least squares ters vector q so that Kðs; l sense:

q* ¼ argmin q

2 X X Pðjul ; qÞ 2 Kðjul Þ  ¼ argmin frl ðqÞg Q ðjul ; qÞ q l

z_ ¼ A0 z þ B0 x_ m ¼ C0 z

639

(16)

4. Calculation examples 4.1. The chosen floating wind energy converter In the following we will show our calculation results regarding a specific floating wind energy converter, i. e. the National Renewable Energy Laboratory 5 MW wind turbine installed on the ITI energy barge moored to the sea bed (see Fig. 1). The National Renewable Energy Laboratory 5 MW wind turbine is an upwind, 3-bladed turbine with a rotor diameter of 126 m. The hub diameter is 3 m, and the hub height is 90 m from the still water surface. The masses of the rotor, nacelle and tower are 110,000 kg, 240,000 kg and 347,500 kg respectively (Jonkman, [1]). The ITI energy barge is a box-typed platform with a length of 40 m, a breadth of 40 m and a depth of 10 m. It has a water displacement of 6000m3 at a draft of 4 m. The mass of the barge (including ballast) is 5,452,000 kg and the center of gravity of the barge is 0.282 m below the still water surface. The platform has a rolling inertia about the center of gravity (CG) of 726, 900, 000kgm2, a pitching inertia about CG of 726, 900, 000kgm2 and a yawing inertia about CG of 1,453, 900, 000 kgm2. The barge is mounted to the sea floor by 8 anchor lines. The diameter of each anchor line is 0.0809 m, and the mass density of each anchor line is 130.4 kg/m. The ITI energy barge is installed in a sea area with a water depth of 150 m. 4.2. The long-crested sea cases versus the short-crested sea cases

l

¼ argmin SðqÞ q

(11) We proceed repeatedly utilizing a damped GausseNewton alð0Þ gorithm after making an initial guess of q as follows:



qðiþ1Þ ¼ qðiÞ  li JTr Jr

1

  ðiÞ ðiÞ ¼ q þ li DðiÞ JTr r q

(12)

In order to investigate the performances of the abovementioned floating wind energy converter, we first numerically integrate Eq. (1) with the convolution integral term using the National Renewable Energy Laboratory fully coupled Aero-HydroServo-Elastic simulation tool FAST. A specific load case for an operational condition is chosen, and the numerical details of this operational condition are as follows: The 10-min average wind speed at the top of the tower is 11.2 m/s, and the turbulence wind intensity is 0.15. A Kaimal power spectrum is applied to describe

where li is determined by finding its value that minimizes S, and the Jacobian matrix entries are calculated as:

ðJr Þlk ¼

  ðiÞ vrl q

(13)

vqk

The iteration continues until the solution has converged, i.e., the norm of the gradient vector ( VSðqðiÞ Þ ) is less than a specific tolerance value (e.g. 0.01). ~ qÞ have been After all the degree one transfer functions Kðs; obtained, the hydrodynamic memory effects m can be approxi~ qÞ, such that: mated by a matrix HðsÞ containing Kðs;

m ¼ HðsÞx_

(14)

with

HðsÞ ¼ C0 ðsI  A0 Þ

1 0

B

(15)

where I is an identity matrix. Consequently, the corresponding state-space model is in the form

Fig. 1. An ADAMS model of the 5 MW floating wind energy converter mounted on the ITI energy barge.

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Y. Wang, L. Wang / Renewable Energy 101 (2017) 637e646

the turbulence wind random field over the rotor plane of turbine. The aforementioned information regarding the turbulence wind is used for calculating the wind loads Pwind ðtÞ in Eq. (1) before numerically integrating Eq. (1) using FAST. The widely used JONSWAP wave spectrum is adopted in the process of simulation of the uni-directional random waves (long-crested random waves). The significant wave height (Hs) for the specific sea state is 5 m, the spectral peak period (Tp) is 12.4s and the spectral peakedness factor (g) is 2.0. The mathematical expression of the JONSWAP wave spectrum S(u) is written as:

"

SðuÞ ¼ aHS2



u5 u exp  1:25 up u4 p

4 #





2 2 2 gexp ðuup Þ =ð2up s Þ

(17) where u is the wave angular frequency (rad/s) and up ¼ 2p=Tp is spectral peak angular frequency. Meanwhile, in the above equation:

a¼ s¼

0:0624 0:230 þ 0:0336g  0:185=ð1:9 þ gÞ 

0:07 0:09

for u  up for u > up

(18)

(19)

Fig. 2 shows the JONSWAP wave spectrum with Hs ¼ 5m, Tp ¼ 12.4s and g ¼ 2.0. However, the above-mentioned JONSWAP wave spectrum is an uni-directional spectrum, i.e. assuming that wave energy is traveling in one direction. The sea state described by this unidirectional spectrum is an ideal, uni-directional random sea (also called a long-crested random sea). In reality, however, windgenerated wave energy does not necessarily propagate in the same direction as the wind; instead, the energy usually spreads over various directions. Thus, for an accurate description of random seas, it is necessary to clarify the spreading status of energy. The wave spectrum to model a sea state with waves coming from many different directions is called a directional spectrum, and the sea state thus obtained is called a multi-directional random sea (also called a short-crested random sea). A directional spectrum S(u; q) can be obtained by multiplying an uni-directional wave spectrum S(u) by a spreading function D(q), i.e.

Sðu; qÞ ¼ SðuÞDðqÞ

(20)

In this study, we choose a cos-2s type spreading function (see,

e.g., Wang and Xia [14]) as follows:

 Gðs þ 1Þ q cos2s DðqÞ ¼ pffiffiffi 2 2 pGðs þ 1=2Þ

(21)

Fig. 3 shows a directional JONSWAP wave spectrum S(u; q) with Hs ¼ 5m, Tp ¼ 12.4s, g ¼ 2.0. and a cos-2s type spreading function (s ¼ 15). We show in Fig. 4 a simulated multi-directional random sea surface on a square of 128 m by 128 m for a sea with the directional JONSWAP wave spectrum S(u; q) as shown in Fig. 3. We can obviously notice that waves are coming from different directions. We next show our calculated dynamic responses of the aforementioned floating wind energy converter operating in an ideal long-crested random sea versus operating in a realistic shortcrested random sea. Fig. 5 shows the calculation results of the radiation sway forces. In Fig. 5 the black curve represents the 630s time series of the calculated radiation sway forces when the floating wind energy converter is operating in a realistic, short-crested random sea with a directional JONSWAP wave spectrum S(u; q) as shown in Fig. 3. In Fig. 5 the red curve represents the 630s time series of the calculated radiation sway forces when the floating wind energy converter is operating in an ideal, long-crested random sea with a 2D JONSWAP wave spectrum S(u) as shown in Fig. 2. We notice that the black curve is oscillating within a radiation sway force interval of about [-400000 N 300000 N]. The red curve is oscillating within a radiation sway force interval of about [-100000 N 50000 N]. Obviously, the ideal, long-crested sea case significantly underestimates the predicted radiation sway forces. These calculation results demonstrate the great necessity of using a realistic, short-crested random sea state when calculating the dynamic responses of a floating wind energy converter. Fig. 6 shows our calculation results of the radiation rolling moments. In Fig. 6 the black curve represents the 630s time series of the calculated radiation rolling moments when the floating wind energy converter is operating in a realistic, short-crested random sea with a directional JONSWAP wave spectrum S(u; q) as shown in Fig. 3. In Fig. 6 the red curve represents the 630s time series of the calculated radiation rolling moments when the floating wind energy converter is operating in an ideal, long-crested random sea with a 2D JONSWAP wave spectrum S(u) as shown in Fig. 2. We notice that the black curve is oscillating within a radiation rolling moment interval of about [-20000000 N-m 20000000 N-m]. The red curve is oscillating within a radiation rolling moment interval of

Fig. 2. The JONSWAP wave spectrum with Hs ¼ 5m, Tp ¼ 12.4s and g ¼ 2.0.

Y. Wang, L. Wang / Renewable Energy 101 (2017) 637e646

641

Fig. 3. A directional JONSWAP wave spectrum with Hs ¼ 5m, Tp ¼ 12.4s, g ¼ 2.0 and a cos-2s type spreading function (s¼15).

The short-crested sea case The long-crested sea case

Fig. 4. Simulated sea surfaces on a square of 128 [m] by 128 [m] based on the wave spectrum as shown in Fig. 3.

Rolling moments (N-m)

20000000

10000000

0

-10000000

300000

-20000000 0

200000

100

200

300

400

500

600

Radiation sway forces (N)

Time (s)

100000

Fig. 6. Radiation rolling moments time series of the floating wind energy converter.

0 -100000

3

The short-crested sea case The long-crested sea case

-200000

2 The short-crested sea case The long-crested sea case

-400000 0

100

200

300 Time (s)

400

500

600

Fig. 5. Radiation sway forces time series of the floating wind energy converter.

about [-13000000 N-m 13000000 N-m]. We can obviously find that the ideal, long-crested sea case underestimates the predicted radiation rolling moments. These calculation results also demonstrate the necessity of using a realistic, short-crested random sea state when calculating the dynamic responses of a floating wind energy converter. Fig. 7 shows our calculation results of the rolling angles of the aforementioned floating wind energy converter. In Fig. 7 the black curve represents the 630s time series of the calculated rolling angles when the floating wind energy converter is operating in a realistic, short-crested random sea with a directional JONSWAP wave spectrum S(u; q) as shown in Fig. 3. In Fig. 7 the red curve

Rolling angle (degree)

-300000

1

0

-1

-2 0

100

200 300 Time (s)

400

500

600

Fig. 7. Rolling angles time series of the floating wind energy converter.

represents the 630s time series of the calculated rolling angles when the floating wind energy converter is operating in an ideal,

642

Y. Wang, L. Wang / Renewable Energy 101 (2017) 637e646

long-crested random sea with a 2D JONSWAP wave spectrum S(u) as shown in Fig. 2. We notice that the black curve is oscillating within a rolling angle interval of about [-2 2.8 ]. The red curve is oscillating within a rolling angle interval of about [-1 1 ]. Obviously, the ideal, long-crested sea case significantly underestimates the predicted rolling angles. These calculation results also demonstrate the great necessity of using a realistic, short-crested random sea state when calculating the dynamic responses of a floating wind energy converter. However, it takes more than 408 s to numerically integrate Eq. (1) with the convolution integral term on a desktop computer (Dell Vostro 230, Intel (R) Core (TM) 2 Duo CPU E7500 @2.93 GHz) for obtaining the black curve in Fig. 5 (or in Fig. 6, or in Fig. 7). This is not very time-efficient. More importantly, according to the design standards [15-16] of the International Electrotechnical Commission (IEC), in order to predict the long-term extreme dynamic responses of a wind turbine, a large number of 600s time series corresponding to different environmental states have to be simulated. This will require tremendous computing time if each of the 600s time series is obtained by numerically integrating Eq. (1) with the convolution integral term. Therefore, in order to improve the calculation efficiency, we have tried to fit a state space model to approximate the convolution integral term in Eq. (1) using two different methods, and our calculation results are summarized in the next sub-section. 4.3. The new FDI-SS model versus the state space model based on the realization theory The two different methods we will use to fit a state space model are our new FDI-SS model and the commonly used model based on the time domain (TD) realization theory. In Fig. 8 the green line represents an example of the roll-roll entry of the fitted retardation ~ matrix KðtÞ by using the commonly used model based on the realization theory. The blue line in Fig. 8 represents the roll-roll entry of the original retardation matrix obtained using Eq. (8). In Fig. 9 the green line represents an example of the sway-sway ~ entry of the fitted retardation matrix KðsÞ by using our new FDI-SS model as detailed in Section 3. The blue line in Fig. 9 represents the sway-sway entry of the original retardation matrix obtained using Eq. (6). We next show the performances and dynamic responses of the aforementioned floating wind energy converter when the hydrodynamic memory effects are obtained through the state space

models. For comparison purpose, the calculation results when the hydrodynamic memory effects are obtained through the convolution integral are also included. In Fig. 10a the red line represents the radiation rolling moments of the floating wind energy converter when the hydrodynamic memory effects are obtained through the state space model based on the time domain (TD) realization theory. It takes about 303 s to numerically integrate Eq. (1) with the TD realization theory state space model on a desktop computer (Dell Vostro 230, Intel (R) Core (TM) 2 Duo CPU E7500 @2.93 GHz) for obtaining the red curve in Fig. 10a. The black line in Fig. 10a represents the radiation rolling moments of the floating wind energy converter when the hydrodynamic memory effects are obtained through the convolution integral in Eq. (1). During the aforementioned calculations the floating wind energy converter is supposed to be operating in a realistic sea state with a directional JONSWAP wave spectrum S(u; q) as shown in Fig. 3. It can be noticed that the red line does not fit the black line very well. In order to demonstrate the differences between the red line and the black line more clearly, we have made a zoom-in plot of the radiation rolling moments in the time range of 200se300s and shown it in Fig. 10b. It can be obviously found that the red line systematically underestimates the radiation rolling moments of the floating wind energy converter. This fact motivates us to utilize our new FDI-SS model to represent the hydrodynamic memory effects when simulating the dynamic responses of the floating wind energy converter, and our simulation results are summarized in Fig. 11a and Fig. 11b. In Fig. 11a the red line represents the radiation rolling moments of the floating wind energy converter when the hydrodynamic memory effects are obtained through our new FDI-SS state space model. The black line in Fig. 11a represents the radiation rolling moments of the floating wind energy converter when the hydrodynamic memory effects are obtained through the convolution integral in Eq. (1). During the aforementioned calculations the floating wind energy converter is supposed to be operating in a realistic sea state with a directional JONSWAP wave spectrum S(u; q) as shown in Fig. 3. It can be found that the red line fits the black line fairly well. In order to demonstrate the good fitness between the red line and the black line more clearly, we have made a zoom-in plot of the radiation rolling moments in the time range of 200se300s and shown it in Fig. 11b. It can be obviously found that the red line can predict the radiation rolling moments of the floating wind energy converter fairly accurately. Meanwhile, it

Fig. 8. Example of the roll-roll entry of the retardation matrix.

Y. Wang, L. Wang / Renewable Energy 101 (2017) 637e646

643

Fig. 9. Example of the sway-sway entry of the retardation matrix.

Memory effects through convolution integral Memory effects through TD realization model

30000000

Memory effects through convolution integral Memory effects through FDI-SS model

a

a 20000000

Rolling moments (N-m)

Rolling moments (N-m)

20000000

10000000

0

0

-10000000

-10000000

-20000000

-20000000

0

30000000

b

100

200

300 Time (s)

400

500

0

600

Memory effects through convolution integral Memory effects through TD realization model

b

20000000

100

200 300 Time (s)

400

500

600

Memory effects through convolution integral Memory effects through FDI-SS model

20000000 Rolling moments (N-m)

Rolling moments (N-m)

10000000

10000000

0

-10000000

10000000

0

-10000000 -20000000

-20000000 200

Time (s)

300

Fig. 10. a. The time series of the radiation roll moments of the floating wind energy converter.b. A zoom-in plot of the time series of the radiation roll moments of the floating wind energy converter.

200

Time (s)

300

Fig. 11. a. The time series of the radiation roll moments of the floating wind energy converter. b. A zoom-in plot of the time series of the radiation roll moments of the floating wind energy converter.

Y. Wang, L. Wang / Renewable Energy 101 (2017) 637e646

400000 Tower base fore-aft bending moments (KN-m)

takes only about 306 s to numerically integrate Eq. (1) with the FDISS state space model on a desktop computer (Dell Vostro 230, Intel (R) Core (TM) 2 Duo CPU E7500 @2.93 GHz) for obtaining the red curve in Fig. 11a. We notice that the time spent on the numerical integration (306 s) is approximately equal to the time spent on numerically integrating Eq. (1) with the TD realization theory state space model, but is much less than the time spent on numerically integrating Eq. (1) with the convolution integral term (i. e. 408 s). The accuracy and efficiency of our new FDI-SS model in the prediction of the dynamic responses of floating wind energy converters in a realistic multi-directional random sea can thus be verified. In the following, we show several examples regarding the comparisons of the tower loads, platform motions, etc. of the floating wind energy converter. In Fig. 12 the red line represents 300s time series of the tower base fore-aft bending moments of the floating wind energy converter when the hydrodynamic memory effects are obtained through the state space model based on the time domain (TD) realization theory. The black line in Fig. 12 represents the tower base fore-aft bending moments of the floating wind energy converter when the hydrodynamic memory effects are obtained through the convolution integral in Eq. (1). During the aforementioned calculations the floating wind energy converter is supposed to be operating in a realistic sea state with a directional JONSWAP wave spectrum S(u; q) as shown in Fig. 3. It can be noticed that the red line does not fit the black line very well. In Fig. 13 the red line represents 300s time series of the tower base fore-aft bending moments of the floating wind energy converter when the hydrodynamic memory effects are obtained through our new FDI-SS state space model. The black line in Fig. 13 represents the tower base fore-aft bending moments of the floating wind energy converter when the hydrodynamic memory effects are obtained through the convolution integral in Eq. (1). During the aforementioned calculations the floating wind energy converter is supposed to be operating in a realistic sea state with a directional JONSWAP wave spectrum S(u; q) as shown in Fig. 3. It can be noticed that the red line fits the black line very well. In Fig. 14 the red line represents the floating wind energy converter sway distance when the hydrodynamic memory effects are obtained through the state space model based on the time domain (TD) realization theory. The black line in Fig. 14 represents the floating wind energy converter sway distance when the hydrodynamic memory effects are obtained through the convolution integral in Eq. (1). During the aforementioned calculations the floating

Memory effects through convolution integral Memory effects through FDI-SS model

300000 200000 100000 0 -100000 -200000 -300000 -400000 200

Time (s)

300

400

Fig. 13. The time series of the tower base bending moments of the floating wind energy converter.

Memory effects through convolution integral Memory effects through TD realization model

6 5 4 3 Platform sway distance (m)

644

2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 0

100

200

300 400 Time (s)

500

600

700

Fig. 14. The time series of the platform sway distance of the floating wind energy converter.

Memory effects through convolution integral Memory effects through FDI-SS model 4 3 Platform sway distance (m)

Tower base fore-aft bending moments (KN-m)

400000 300000 200000 100000 0

2 1 0 -1 -2

-100000

-3

-200000

-4

-300000

Memory effects through convolution integral Memory effects through TD realization model

-5 -100

-400000 200

Time (s)

300

400

Fig. 12. The time series of the tower base bending moments of the floating wind energy converter.

0

100

200 300 Time (s)

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Fig. 15. The time series of the platform sway distance of the floating wind energy converter.

Y. Wang, L. Wang / Renewable Energy 101 (2017) 637e646

Memory effects through convolution integral Memory effects through TD realization model

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Fig. 16. The time series of the generated electrical power of the floating wind energy converter.

wind energy converter is supposed to be operating in a realistic sea state with a directional JONSWAP wave spectrum S(u; q) as shown in Fig. 3. It can be noticed that the red line fit the black line poorly. In Fig. 15 the red line represents the floating wind energy converter sway distance when the hydrodynamic memory effects are obtained through our new FDI-SS state space model. The black line in Fig. 15 represents the floating wind energy converter sway distance when the hydrodynamic memory effects are obtained through the convolution integral in Eq. (1). During the aforementioned calculations the floating wind energy converter is supposed to be operating in a realistic sea state with a directional JONSWAP wave spectrum S(u; q) as shown in Fig. 3. It can be noticed that the red line fit the black line fairly well. The accuracy of our new FDI-SS model in the prediction of the dynamic and motion responses of floating wind energy converters in a realistic multi-directional random sea can thus be verified. These calculation examples show that our new FDI-SS model is superior for predicting global wind turbine performance of a realistic wind turbine system. Finally, in order to further demonstrate the suitableness of our new FDI-SS model in the prediction of the performances of floating

Memory effects through convolution integral Memory effects through FDI-SS model

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Fig. 17. The time series of the generated electrical power of the floating wind energy converter.

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wind energy converters, we have carried out further calculations regarding the generated electrical power of the aforementioned floating wind energy converter, and our calculation results are summarized in Fig. 16 and Fig. 17. In Fig. 16 the red line represents the generated electrical power of the aforementioned floating wind energy converter when the hydrodynamic memory effects are obtained through the state space model based on the time domain (TD) realization theory. The green line in Fig. 16 represents the generated electrical power of the aforementioned floating wind energy converter when the hydrodynamic memory effects are obtained through the convolution integral in Eq. (1). During the above calculations the floating wind energy converter is supposed to be operating in a realistic sea state with a directional JONSWAP wave spectrum S(u; q) as shown in Fig. 3. It can be noticed that the red line deviates a lot from the green line. In Fig. 17 the red line represents the generated electrical power of the aforementioned floating wind energy converter when the hydrodynamic memory effects are obtained through our new FDISS state space model. The green line in Fig. 17 represents the generated electrical power of the aforementioned floating wind energy converter when the hydrodynamic memory effects are obtained through the convolution integral in Eq. (1). During the above calculations the floating wind energy converter is supposed to be operating in a realistic sea state with a directional JONSWAP wave spectrum S(u; q) as shown in Fig. 3. It can be noticed that the red line fit the green line almost perfectly. Meanwhile, it takes only about 306 s to numerically integrate Eq. (1) with the FDI-SS state space model on a desktop computer (Dell Vostro 230, Intel (R) Core (TM) 2 Duo CPU E7500 @2.93 GHz) for obtaining the red curve in Fig. 17. This is much less than the time spent on numerically integrating Eq. (1) with the convolution integral term (i. e. 408 s) for obtaining the green curve in Fig. 17. These calculation results once again substantiate the accuracy and efficiency of utilizing our new FDI-SS state space model in the prediction of the performances of floating wind energy converters in a realistic multi-directional random sea. 5. Conclusions In this study we have rigorously investigated the performances of a floating wind energy converter (the National Renewable Energy Laboratory 5 MW wind turbine installed on the ITI energy barge) in a realistic, multi-directional random sea. The wind loads acting on the floating wind energy converter have also been fully considered in our numerical simulation process. Our simulation results in this paper have demonstrated the great necessity of using a realistic, multi-directional random sea state when calculating the generated electrical power and the dynamic responses of the floating wind energy converter. Furthermore, in order to improve the simulation efficiency, a new state space model (the FDI-SS model) has been utilized to approximate the convolution integral term when solving the motion equation of the floating wind energy converter. For comparison purpose, the simulation results when the convolution integral term in the motion equation is approximated by a commonly used state space model based on the time domain realization theory have also been included. The simulation results in this paper have been systematically analyzed and compared, and the accuracy and efficiency of our new FDI-SS model have been verified. The research findings in this paper demonstrate that our new FDI-SS model can be utilized as a valuable tool for engineers in their design of floating wind turbines. This will help us exploit more wind power, which is considered as the most promising renewable energy in the 21st century and has become a pillar of the energy systems in many countries [17-18].

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Acknowledgment The work reported in this paper has been supported by the State Key Laboratory of Ocean Engineering of China (Grant No. GKZD010038). Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.renene.2016.09.025. References [1] J.M. Jonkman, Dynamics Modeling and Loads Analysis of an Offshore Floating Wind Turbine, Technical Report, NREL/TP-500e41958, 2007. [2] J.M. Jonkman, Dynamics of offshore floating wind turbinesdmodel development and verification, Wind Energ 12 (2009) 459e492. [3] J.M. Jonkman, Definition of the Floating System for Phase IV of OC3, Technical Report, NREL/TP-500e47535, 2010. [4] J.M. Jonkman, M.L. Buhl Jr., Loads Analysis of a Floating Offshore Wind Turbine Using Fully Coupled Simulation, NREL/CP-500e41714, 2007. [5] J.M. Jonkman, D. Matha, Dynamics of offshore floating wind turbinesdanalysis of three concepts, Wind Energy 14 (2011) 557e569. [6] A.N. Robertson, J.M. Jonkman, Loads Analysis of Several Offshore Floating Wind Turbine Concepts, NREL/CP-5000e50539, 2011. [7] Y.G. Wang, Y.Q. Xia, X.J. Liu, Establishing robust short-term distributions of load extremes of offshore wind turbines, Renew. Energy 57 (2013) 606e619.

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