The surge-heave-pitch coupling motions of the Φ-type vertical axis wind turbine supported by the truss Spar floating foundation ＊

Ying Guo, Li-qin Liu, Yan Li, Chang-shui Xiao, You-gang Tang

State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China

Abstract: The motions of the floating vertical axis wind turbine (VAWT) with consideration of the coupling between the aerodynamics and the hydrodynamics are calculated in this paper. The surge-heave-pitch coupling nonlinear equations of the floating VAWT are established. The aerodynamic loads are obtained by applying the double-multiple-stream tube theory with consideration of the dynamic stall and the floating foundation motion. The motion performances of a 5 MW Φ-Darrieus type floating VAWT are studied with its foundation of the Spar type with heave plates. It is shown that the amplitude of the heave motion of the floating VAWT is small when the heave plates are equipped. The effects of the wind and wave loads on the floating VAWT motions are assessed. The results show that the mean values of the surge and the pitch of the floating VAWT are mainly related to the wind loads;the standard deviations of the surge, the heave, and the pitch are mainly related to the wave loads. In the regular wave cases, the frequencies of the surge, the heave and the pitch are dominated by the wave frequencies, and the components of the 2P responses(caused by the aerodynamic loads) of the pitch are small. The 2P responses of the pitch are more significant in the irregular wave cases as compared with those in the regular wave cases.

Key words: Floating VAWT, truss Spar floating foundation, surge-heave-pitch motions, aerodynamics, regular waves, irregular waves

Introduction

Compared with the fixed offshore wind turbine,the benefit of the floating offshore wind turbine(FOWT) is pronounced in the aspects of economy,convenience of installation and total capacity,especially in the deep-water zone[1]. In order to further increase the power generated by the offshore wind turbine, the offshore wind farms should be located in the open sea area beyond the littoral zone for even higher wind speed where the water depth is normally larger than 50m.The floating foundations, including normally the Spar type, the tension leg platform (TLP)type, the semi-submersible type and the barge type,are used to support the offshore wind turbines.According to the direction of the rotation axis in space,the wind turbines are divided into two types, the horizontal-axis wind turbine (HAWT) and the vertical-axis wind turbine (VAWT). Compared with the HAWT, the VAWTs enjoy the advantages of a lower centre of gravity (with smaller over-turning moment), the insensitivity to the wind direction changes (allowing for a large rotor size), and so on[2].Therefore, the VAWTs are in a superior position in the development of the large-scale offshore wind power.

Some codes for the preliminary design of the floating VAWT were developed, such as the FloVAWT developed by Cranfield University[3], the enhanced Hawc2 developed by Technical University of Denmark[4], the Simo-Riflex-DMS and the Simo-Riflex-AC developed by Norwegian University of Science and Technology[5-6], and a rigid-flexible coupling code developed by Sandia National Labs[7].To calculate the aerodynamic loads, the double multiple stream tube theory was used in the FloVAWT and the Simo-Reflex-DMS, and the actuator cylinder (AC)theory was used in the Hawc2 and the Simo-Reflex-AC, the free wake vortex method[8]was used in the code of Sandia National Labs.

Vita[4]analyzed the feasibility of a 5 MW Darrieus type wind turbine supported by a rotating Spar type foundation (the Deepwind concept) by applying the enhanced Hawc2. Blusseau and Patel[9]analyzed the effect of the gyroscopic force on the motions of a V-type VAWT supported by a semisubmersible floating foundation in the frequency domain, and the aerodynamic loads were calculated by applying the double-multiple-stream tube theory.Borg and Collu[10]compared the motion responses of a 5 MWΦ-type Darrieus type wind turbine supported by different floating foundations, using the code FloVAWT. Cheng et al.[11]analyzed the dynamic response of theΦ-type VAWT mounted on three different floating support foundations (the Spar type,the submersible type, and the TLP type) by applying the code Simo-Riflex-DMS. They found that the TLP was not a good supporting structure for the VAWT as the influences of the 2P (twice-per-revolution)aerodynamic loads were significant. Compared with the floating HAWT, the floating VAWT is not well studied, and it is desirable to gain more insight into the dynamics of the floating VAWT with consideration of the coupled effect of the aerodynamics, the hydrodynamics and the mooring.

In this paper, a novel Spar type floating foundation with heave plates is used to support a 5MW VAWT, and it has a shorter length than the existing Spar type floating foundation (such as the Spar type foundation in Refs. [11-14]) and it has a lighter weight than the existing semi-submersible type floating foundation (such as the semi-submersible type floating foundation in Refs. [2, 5]), which were all used to support the same wind turbine. A computing code for the aerodynamic forces is developed with consideration of the dynamic stall and the motions of the floating foundation. The coupling equations of the surge, heave and pitch motions of the floating VAWT are established, and the motion performances are evaluated.

1. The dynamical model of the floating VAWT

1.1 The dynamical equations

The shape of the floating VAWT system studied in this paper is shown in Fig. 1, where the wind turbine is of theΦ-Darrieus type and the floating foundation is of the Spar type. The blades drive the tower to rotate under the wind force, and the mechanical energy is converted into the electrical energy by the transmission mechanisms and the power generation devices.

The foundation is composed of an upper buoyancy tank, an upper mechanical tank, truss structures, heave plates, and a bottom ballast tank. The buoyancy tank is used to provide the buoyancy and the restoring force. The upper mechanical tank is used to arrange the transmission mechanisms and the power generation devices, and it can also be used to adjust the ballast. The truss structures are composed of six vertical poles. The heave plates increase the heave damping and reduce the heave motion of the floating VAWT. The bottom ballast tank is used to adjust the center of gravity of the floating VAWT. The upper buoyancy tank and the upper mechanical tank are cylindrical, and the cross sections of the heave plates and the bottom ballast tank are shaped as a regular hexagon.

Fig. 1 The floating VAWT system

Fig. 2 The coordinate definitions of the floating VAWT

The floating VAWT is dealt with as a rigid body oscillating in the Cartesian coordinate systemo-xyz.The origin of the coordinate systemois located at the center of gravity of the floating VAWT,1ξ,3ξand5ξare the displacements of the surge, the heave and the pitch of the floating VAWT, respectively. The positive directions of the surge and heave motions are along the positive directions of thexandzaxes,respectively, the positive direction of the pitch is from theyaxis into thexozplane according to the right-hand rule. Thexozplane is shown in Fig. 2,where Fig. 2(a) is the plane in calm water and Fig. 2(b)is the plane in waves.

For the moored floating VAWT, the coupled motion equations of the surge-heave-pitch are as follows

whereMis the mass matrix of the floating VAWT,A∞is the added mass at the infinite frequency, andh(t-τ) is the retardation function accounting for the fluid memory effect.X,X˙ andX˙ represent the displacement, the velocity and the acceleration of the floating VAWT at the center of gravity, respectively,Kis the restoring force matrix,including the hydrostatic restoring force, the restoring force of the mooring system, and the nonlinear hydrostatic restoring force.F(t,X,X˙) is the exciting force, including the Froud-Krylov forceFfk(t), the diffraction forceFd(t), the wind loadsFw(X,X˙,t)(including the aerodynamic loadsFwa(X,X˙,t) and the wind loadscalculated by the wind pressure theory), and the hydrodynamic viscous forceFv1=whereDis the matrix of the viscous coefficient. Then we have

For the Spar type floating foundation, the nonlinear coupling between the heave and pitch motions is significant. For the nonlinear heave-pitch coupling motion equations of the deep-sea truss Spar platform[15-16], the restoring force matrix can be written as follows

The fairleads of the mooring lines are near the centre of gravity of the floating VAWT, the mooring restoring moment in the pitch is insignificant and therefore is not considered here. The mooring restoring force stiffnessis computed by the quasi-static catenary model[17].

1.2 Wind loads

If the rotor is rotating, the aerodynamic loads on the blades are calculated based on the doublemultiple-stream tube theory[18]. The blades are dealt with as rigid bodies, and the dynamic stall is considered. The influences of the velocities of the floating foundation (in the surge, pitch and heave motions) are also considered. The blades are divided into the stream tubes normal to the central axis along the tower. Moreover, the stream tubes are divided into the upwind and downwind sectors. The vector relations of the wind turbine are shown in Fig. 2.

The steady wind is considered here, as a function of the height above the water level, and for a local heightzi, the wind speed can be estimated as follows according to the IEC61400-3[19]

whereV∞is the average wind velocity at a reference heightzref, the distance from the centre of the blades to the water level is taken as the reference height in this paper, and with the powera=0.14.

The wind speed and the speeds caused by the motions of the floating foundation are decomposed into components in two directions: parallel and perpendicular to the tower. Assume that the velocity parallel to the direction of the stream tube affects the induced velocities and that the velocity perpendicular to the direction of the stream tube only affects the local relative velocities. The local position and the inflow velocity of each stream tube are obtained by considering the motions of the floating foundation in the following parts.

For the upwind sector, the distance between the local position of the blade and the centre of gravity of the floating VAWT can be written as

whereris the local radius of the wind turbine.θis the azimuthal angle of the rotor, and denotes the rotation angle of the rotor in the space. Figure 2 is the sketch ofθ=0, and the direction of the wind turbine is along thex-axis.H0is the distance from each stream tube to the bottom of the wind blade.L1is the distance from the bottom of the wind blade to the water level when the structure is in the calm water andHgis the distance from the centre of gravity of the floating VAWT to the water level when the structure is in the calm water, as shown in Fig. 2(a).

The angle between the local position of the blade and the tower is denoted byγ, and it can be written as

then the local wind velocity of the blade is given by

whereHis the half-height of the rotor. Then, the resultant velocities parallel to and vertical to the direction of the stream tube for the local position can be written as:

Similarly, the resultant velocities parallel to and vertical to the direction of the stream tube for the downwind sector are obtained as follows:

whereVeis the local balancing induced velocity, andis the local wind velocity of the downwind, they are given by

whereu＜1 is the interference factor of the upwind sector.

The induced velocitiesVfor the upwind sector andV′ for the downwind sector are given by the following formulae[18]:

whereu'is the secondary interference factor.

For the rotor of the upwind sector, the azimuthal angleθis in the range of [-π/2,π/2], and the local relative velocity of the rotor can be written as follows

whereδis the angle between the normal direction of the blade and the equatorial plane, as shown in Fig.2(a), andωis the rotation velocity of the blade.

The local attack angleαis

The azimuthal angle of the upwind sector of the rotor is divided into several angular tubes uniformly,and the angle of each tube is Δθ. Assuming that the induced velocity of each angular tube is a constant,then the interference factor of each angular tube is a constant, and it can be expressed as whereRis the equator radius of the blade,nis the number of blades, andcis the chord length.η=r/R, andfup(θ) is given by

whereCNandCTare the normal force and tangential force coefficients, respectively, their expressions are as follows:

whereCLis the lift coefficient andCDis the resistance coefficient. These coefficients are obtained by interpolation of the experimental data[18]with consideration of the local Reynolds number and the local attack angleα, and then they are corrected by the Gormont-Berg method[18]with consideration of the dynamic stall. The Reynolds number is given byRet=cW/ν∞, where the coefficient of the kinematic viscosity isν=1.77× 10-5m2/s (when the air tem-

∞perature is 15° centigrade).

Given the initial value ofu()θ, the realu()θcan be obtained by iteration of the loop from Eq. (14)to Eq. (19). Then, the tangential force coefficientCTand the normal force coefficientCNof the blade at each position can be obtained.

The normal forceFNand the tangential forceFTof the blade at differentθare further calculated by the following formulae[18]

whereSis the swept area of the rotor,ζ=z/His the local height of the stream tube based on the equator plane of the rotor.

The torque is calculated according to the rotation centre of the blade, and the overall torque of the blade at eachθis integrated by the following formula

The upwind power coefficient of the wind turbine is

Similarly, the normal forceand the tangential forceFT′ in the downwind sector can also be calculated, and the downwind sector power coefficientCp2can be obtained. The power coefficient for one cycle can be obtained as follows

The lift force and the thrust force on the floating VAWT can be obtained by decomposing the normal and tangential forces. By calculating the moment of the thrust force about the center of gravity of the floating VAWT, the aerodynamic moment causing the pitch motion of the floating VAWT can be obtained.Then, the aerodynamic forces along the direction of the heave and surge motions of the floating VAWT and the moment along the direction of the pitch motions of the floating VAWT are obtained. By substituting these external forces (moment) into the motion Eqs. (1)-(3) of the floating VAWT, the motion responses in the time domain can be obtained.

1.2.2 Wind loadson the tower and the blades The wind pressure and the wind heeling moment on the tower and the blades (assuming that the wind turbine is in parking) are calculated in accordance with the rules of the API[20], as follows:

whereCs=0.5 is the shape coefficient of the windbearing component.Chis the height coefficient of the wind-bearing component, and Table 1 shows the values ofChin different height ranges.is the frontal projected area of the wind-bearing component in the wind direction.is the distance from the wind force acting points to the rotation center of the floating VAWT.

In the calculation, the structure is divided into several elements along the structure length, then the values ofCsandChare chosen for each element to calculate the corresponding pressure and moment on each element.

Table 1 The height coefficient of wind-bearing component

1.3 Hydrodynamics of the floating foundation

The hydrodynamic loads are calculated by using the potential theory and the Morison's equation[21].The potential theory is used to calculate the wave loads and the hydrodynamic parameters of the largescale structure (including the upper buoyancy tank,the upper mechanical tank, the heaving plates, and the bottom ballast tank), and the frequency dependent added mass and the radiation damping are applied in the time domain by the convolution calculation. The first-order wave loads and the second-order mean drift forces are considered in this paper.

The Morison equation is used to calculate the inertial load and the viscous drag load of the slender structure (truss structures,/λ＜0.2, whereis the diam eter of the st ructure andλis t he w ave length).Thetransversehydrodynamicforceperunit length can be expressed as follows

whereCaandCdare the added mass and quadratic drag coefficients,uwis the transverse wave particle velocity andubis the local transverse body velocity.

The heave plates are used to reduce the heave motions of the floating VAWT, and their viscous damping is very important. For the Spar floating foundation studied in this paper, the viscous forces on the main hull and the heave plates, and the quadratic viscous drag term of Eq. (29) are incorporated and represented by(see Eq. (2)), where the viscous coefficientDcan be obtained by the CFD method, the model tests or the empirical estimation(5%-8% of the critical damping). In this paper, the viscous coefficientsDare obtained by the model tests of the free decay motions[22].

Fig. 3 Computing process

The hydrodynamic calculation is carried out by Sesam/Wadam[23](the wind turbine is in parking). The transfer functions of the wave force (moment) and the added mass (added moment of inertia) are calculated by Wadam, and the results are substituted into Eq. (1)to make the time domain calculation.

1.4 The code of the coupling motions of the floating VAWT

A coupled simulation code in the MATLAB platform is developed for the time domain simulation of the floating VAWT, and the detailed computing process is shown in Fig. 3.

Fig. 4 Validation of the aerodynamics computations (θ is the azimuth angle, V∞ is the wind speed at equator)

2. Validation of the aerodynamic load calculation

The aerodynamics computation is validated in this section. The coupling between the aerodynamic loads and motions of the floating VAWT is not considered here.

For the upwind sector, the tangential force can be further expressed as follows

whereis the coefficient of the tangential force,which can be expressed as follows

Taking the Sandia 17 m wind turbine(NACA0015 airfoil, Sandia type) as an example, the values ofCDandCLare obtained by interpolation[18],and the tangential force coefficient of the upwind sector and the power of the wind turbine are calculated and compared with the results of Paraschivoiu[24], as shown in Fig. 4. Figure 4 shows that the results of this paper agree well with the results of Paraschivoiu[24]and the aerodynamic computing code of this paper is verified.

3. Case study

3.1 The parameter of a 5MW floating VAWT

The motion performance of a 5 MW floating VAWT is studied in this section. The wind turbine was proposed by Vita[3], with the suggested main parameters given in Table 2. The hydrodynamics of a truss Spar floating foundation were analyzed by Liu et al.[25], the dimensional parameters of the floating foundation are shown in Table 3. The floating VAWT is moored by six catenary mooring lines which are divided into three groups, two lines for each group.The angle between each group of mooring lines is 120°. The specific parameters of the mooring lines are shown in Table 4.

Table 2 Parameters of the wind turbine

3.2 Load cases

The wave direction is aligned with that of the wind, and the water depth is 200 m. A series of load cases are used to assess the motion performances of the floating VAWT, as follows:

For the load case 1, the free decay motions with the wind turbine in parking (rotation speed of the wind urbine is 0 rpm) are simulated, and the calculation parameters are shown in Table 5.

Table 3 Parameters of the floating foundation

Table 4 Parameters of the mooring line

Table 5 Free decay simulations (LC1)

Table 6 Wind loads cases (LC2)

Table 7 Regular wave and steady wind cases (LC3)

For the load case 2, only the wind loads (including the wind pressureon the tower, and the aerodynamic loads(X,,t) on the blades) are considered to assess the impact of the wind loads on the motions of the floating VAWT. The rated rotation speed 5.26 rpm is considered, the critical wind speed is the cut-out wind speed 25 m/s, and other calculation parameters are shown in Table 6.

For the load case 3, the regular wave and the wind loads are considered to assess the responses of the floating VAWT. ForV∞≤25 m/s , we have the wind pressureon the tower and the aerodynamic loadson the blades. ForV∞＞25 m/s,we have the wind pressureon the tower and the blades. The rated rotation speed 5.26 rpm is considered in the cases LC3.1 and 3.2, and the wind turbine is in parking in the case LC3.3. Three special wind speeds are considered, that are the rated wind speed 14 m/s, the cut-out wind speed 25 m/s, and the extreme wind speed 40m/s. Other calculation parameters are shown in Table 7.

Table 8 Irregular wave and steady wind cases (LC4)

For the load case 4, the irregular wave and wind loads (with the wind pressureon the tower, and the aerodynamic loads(X,X˙,t) on the blades)are considered to assess the responses of the floating VAWT. The random wave is decomposed into a series of regular waves, and the JONSWAP sea spectrum is applied, and the random wave loads can be calculated according to the transfer functions of the wave force(moment) obtained by Wadam. The rated rotation speed 5.26 rpm is considered, the critical wind speed is the cut-out wind speed 25 m/s, and the significant wave height and the peak spectral period of each wave are specified in Table 8.

3.3 Results and discussions

3.3.1 Results of LC1

In the free decay simulations, the wind turbine is in parking and the wind loads are not considered. The time histories of the free decay and the natural periods obtained are shown in Fig. 5, Table 9.

Table 9 Natural periods of the floating VAWT

Fig. 5 Free decay curves of the floating VAWT (t is time,X1, X3, X5 is the motion of surge, heave and pitch,respectively)

It is shown in Table 9 that the natural periods of the surge, the heave, and the pitch of the floating VAWT are all not in the range of the wave periods where the main wave energy is concentrated.

3.3.2Results of LC2

The results in the case LC2 are shown in Figs. 6,7. In Fig. 6, the solid lines denote the mean displacements of the surge, heave and pitch motions of the floating VAWT, and the error bars denote the corresponding standard deviation. Figure 7 shows the amplitude spectra of the pitch of the floating VAWT and the spectra of the corresponding aerodynamic pitch moment (evaluated at the center of gravity of the floating VAWT) in the case LC2.3.

Fig. 6 Mean displacements and standard deviations of the floating VAWT (cases LC2) (Vw is the wind speed)

Figure 6 shows that the standard deviations of the surge, heave and pitch motions of the floating VAWT in the caase LC2 are all negligible, the mean displacements of the heave are small as well. So, the aerodynamics mainly influence the mean displacements of the surge and pitch motions of the floating VAWT.

Figure 7 shows that the aerodynamic pitch moments are dominated by a 2P frequency (twice of the rotational velocity of the rotor, 1.104 rad/s). The 2P response arises from the characteristics of the aerodynamic loads on the two-blade VAWT with the resultant aerodynamic forces and torque varying twice per revolution. For the pitch motion of the floating VAWT, the response frequency is dominated by the frequency of the aerodynamic load. There is a component of the pitch natural frequency, while its response amplitude is less than 50% of the 2P response amplitude.

Fig. 7 Spectra of pitch of floating VAWT and pitch moment caused by aerodynamic (case LC2.3) (f is the frequency, Sp5 is the spectra of pitch motion and SpM5 is the spectra of aerodynamic pitch moment)

Fig. 8 Surge responses of the floating VAWT (LC3) (Sp1 is the spectra of surge motion)

3.3.3Results of LC3

The time histories of the surge, heave and pitch motions of the floating VAWT are calculated and the corresponding response spectra can be obtained by using the fast Fourier transform (FFT) of the time history responses, as shown in Figs. 8-10. The standard deviations and the mean values of the time histories are evaluated, as shown in Table 10.

Fig. 9 Heave responses of the floating VAWT (LC3) (Sp3 is the spectra of heave motion)

Fig. 10 Pitch responses of the floating VAWT (LC3)

Table 10 shows that, the max standard deviations of the surge, heave and pitch motions of the floating VAWT are 3.82 m, 0.99 m and 2.58°, respectively, the mean values of the heave are all small, and the max mean values of the surge and pitch motions in the three cases are 9.46 m, 7.76°, respectively. Compared with the case LC3.2, the pitch standard deviation of the floating VAWT in the case LC3.3 increases by approximately 139%, and the corresponding mean value decreases by approximately 40%.

Table 10 The standard deviations and mean values (LC3)

Fig. 11 The pitch responses of the floating VAWT (of LC4.3,LC4.6)

The surge, heave and pitch motion frequencies of the floating VAWT are dominated by the wave frequencies in the three cases as shown in Figs. 8(b)-10(b). For the pitch motion frequencies, there are components of the rotor rotational frequency (1.104 rad/s,2P response) in the cases LC3.1, LC3.2, there are also the pitch natural frequency (0.21 rad/s) response in the three cases, however, these components are not significant. The 2P pitch response is not present in the case LC3.3, as the rotors are in parking and the aerodynamic forces are not considered. The 2P response is very small for the surge and is not present for the heave. This is because the aerodynamic loads in the surge direction of the floating VAWT are much smaller as compared with the wave loads, and there is no aerodynamic load in the heave direction of the floating VAWT.

Fig. 12 Response statistics of the floating VAWT (cases LC4)

3.3.4Results of LC4

The responses of the floating VAWT are evaluated with consideration of the irregular wave and steady wind loads in the case LC4. The time histories and the spectra of the pitch in the cases LC4.3, LC4.6 are presented in Fig. 11. The max values, the mean values and the standard deviations of the surge, heave and pitch responses of the floating VAWT are calculated and the results are shown in Fig. 12.

Figure 11 shows that the pitch motions increase with the increase of the wave height and the wind speed. Comparing Fig. 11(b) with Fig.10(b), it is shown that the 2P responses caused by the aerody-namic loads are more significant in the cases of LC4.The wave-frequency response and the 2P responses dominate the pitch motion of the floating VAWT,while the pitch resonant responses are very small when the wave frequencies are far away from the natural frequency of the pitch of the floating VAWT.On account of the effect of the heave plates, the heave responses are small and the max value of the heave is 1.52m in the case LC4. The standard deviations of the surge and pitch motions are small, which shows that the floating VAWT motion performs well under the wave and wind conditions.

Fig. 13 Comparison between the cases LC2, LC4

3.3.5Comparison between the cases LC2, LC4

In this section we discuss the influence of the wave and wind loads on the motions of the floating VAWT by comparing the results in the cases LC2,LC4, as shown in Fig. 13, where the solid lines denote the mean displacements of the floating VAWT, and the error bars denote the corresponding standard deviation.

Figure 13 shows that the mean displacements of the surge, the heave and the pitch in the cases LC4 are almost the same as those in the cases LC2. However,the standard deviations in the cases LC4 are much larger than those in the cases LC2, especially for the surge and pitch motions of the floating VAWT.Therefore, the mean displacements of the floating VAWT motions are mainly related to the wind loads,and their standard deviations are mainly related to the wave loads.

4. Conclusions

The motion responses of the floating VAWT are studied with consideration of the coupling between the aerodynamics and the hydrodynamics. The surge-heave-pitch coupled nonlinear motion equations of the floating VAWT are established. An in-house code is developed to solve the equations numerically.The motion performances of a 5 MW floating VAWT are studied, with the turbine being of theΦ-Darrieus type and the floating foundation of the Spar type with heave plates. The main conclusions are as follows:

(1) The aerodynamic load calculations of the Darrieus wind turbine are validated, and the aerodynamic forces on the blades are calculated with consideration of the surge, the heave and the pitch of the floating foundation.

(2) The natural periods of the surge, heave, and pitch motions of the 5 MW floating VAWT are all away from the periods of the usual waves and the aerodynamic loads, and the floating VAWT performs well under the wave and wind conditions.

(3) The surge, heave and pitch motions of the floating VAWT are dominated by the wave frequencies, and there are components of 2P responses for the pitch motions. Compared with the regular wave &steady wind cases, the 2P responses are more significant in the irregular wave and steady wind cases.

(4) The amplitudes of the heave motions of the floating VAWT are small when the heave plates are equipped. The code in this paper could be used for the preliminary design of the floating VAWT with higher efficiency (3 600 s simulation requires about 40 min by a desktop computer with a CPU of i7-4790 and the RAM of 8G).

The quasi-static catenary model is used to calculate the mooring forces, a more accurate method, such as the finite element method should be considered.The total floating VAWT is dealt with as a rigid body,and the effect of the flexibility of the blades and the tower should be further studied.

Acknowledgements

This work was supported by the Natural Science Foundation of Tianjin (Grant No. 16JCYBJC21200),the Fund of the State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University (Grant No.1501).