The three-dimensional potential flow theory was used for the frequency-domain hydrodynamic analysis. The radiation and diffraction potentials could be obtained by solving the integral equation with flow field boundary conditions. Based on the velocity potential of flow, the first-order hydrodynamic force could be expressed as (ANSYS and AQWA, 2018) F j e − i ω t = [ − i ω ρ ∫ S 0 φ ( X → ) n j d S ] e − i ω t where ω is the wave frequency, ρ is the fluid density, S 0 is the mean wetted surface of body, φ is the velocity potential, X → = ( X , Y , Z ) is the location of a point on the submerged body surface, n j is the unit normal vector of the body surface pointing outwards, and j = 1 ∼ 6 represent 6-DOF.

The added mass and radiation damping of floating structure could be obtained by solving following equation (ANSYS and AQWA, 2018): A j k ( ω ) + i B j k ( ω ) ω = F r j k = − i ρ ω ∫ S 0 φ r k ( X → ) n j d S where A j k and B j k are respectively added mass and radiation damping coefficients in frequency domain, F r j k is the radiation force induced by k-th unit amplitude motion, and φ r k is the radiation potential.

The motions of FOWT are significantly influenced by the second-order wave force. In this article, the second-order wave loads were mainly dominated by the mean drift force and difference frequency force, which were respectively calculated by the far-/near-field methods and full QTF method (ANSYS and AQWA, 2018).

The FOWT is under the coupled effects of aerodynamic, hydrodynamic, and mooring loads simultaneously. Therefore, it is necessary to carry out the time-domain coupled analysis of FOWT. The hydrodynamic parameters have been obtained by the frequency-domain analysis.

The aerodynamic loads on wind turbine rotor could be obtained by blade element momentum (BEM) theory, which discretizes the blade into blade elements and calculates the aerodynamic loads on each element with the theorem of momentum. The aerodynamic thrust and torque on each blade element could be expressed as (Jonkman and Buhl, 2005) d F = 1 2 ρ a v r 2 ( C L ⁡ sin ⁡ ϕ − C D ⁡ cos ⁡ ϕ ) c d r d Q = 1 2 ρ a v r 2 ( C L ⁡ cos ⁡ ϕ + C D ⁡ sin ⁡ ϕ ) c r d r where F is the thrust, Q is the torque, ρ a is the air density, v r is the relative wind speed, C L and C D are respectively lift and drag coefficients, ϕ is the inflow angel, c is the chord length of blade element, r is the radius of blade element, and d r is the length of blade element. After the calculation of aerodynamic loads on blade elements, the total aerodynamic loads on a blade could be obtained by integrating radially.

The tension of mooring line could be calculated by the lumped-mass method to account for the dynamic effects, which discretizes the mooring line into Morison elements. The mass and applied/internal forces of each element are assembled on one point. The dynamic governing equation of mooring line element could be expressed as (ANSYS and AQWA, 2018) ∂ T → ∂ S e + ∂ V → ∂ S e + w → + F → h = m ∂ 2 R → ∂ t 2 ∂ M → ∂ S e + ∂ R → ∂ S e × V → = − q → where T → , M → , and V → are respectively tension force, bending moment, and shear force at the first node of the element; S e is the length of the element; w → and F → h are element weight and external hydrodynamic loads per unit length, respectively; m is the structural mass per unit length; q → is the distributed moment per unit length; and R → is the position vector of the first node of mooring line element.

The coupled motion responses of FOWT could be solved by the Cummins equation, expressed as follows (Cummins, 1962; ANSYS and AQWA, 2018): ( M s + M a ) X ¨ ( t ) + ∫ 0 t X ⋅ ( τ ) R ( t − τ ) d τ + C X ⋅ ( t ) + KX ( t ) = F e ( t ) where M s and M a are respectively structural mass and added mass matrixes, R is the velocity impulse function, C is the damping matrix including radiational and viscous components, K is the hydrostatic restoring stiffness matrix, and F e is the exciting force, which includes aerodynamic loads, wave loads (first order, mean drift, and difference frequency wave forces), and mooring forces. M a and R could be calculated as follows (ANSYS and AQWA, 2018): M a = A ( ∞ ) R ( t ) = 2 π ∫ 0 ∞ B ( ω ) s i n ( ω t ) ω d ω

The flowchart of time-domain coupled analysis is given in Figure 6 to better introduce this methodology.

Accounting for the elasticity and deflection, the blade and tower are modeled as beam model and discretized into elements. The dynamics could be governed by the following equation: R I ( r , r ¨ , t ) + R D ( r , r ˙ , t ) + R S ( r , t ) = R E ( r , r ˙ , t ) where R I is the inertia force, R D is the damping force, R S is the internal structural reaction force, R E is the external force, and r , r ˙ , and r ¨ are respectively the structural displacement, velocity, and acceleration. Obtaining the elastic and mass parameters of each element, the structural dynamics under the influences of external loads could be solved.

To verify the quality of hydrodynamic mesh, the far-field and near-field method were employed to calculate the transfer functions of mean drift force in surge DOF, as shown in Figure 7. The far-field method solves the mean drift force based on the change rate of linear and angular momentum, which has relatively high precision. The near-field method solves the mean drift force based on the potential flow theory and the calculation precision relies on the quality of hydrodynamic mesh. Therefore, the quality of mesh could be checked according to the coincidence of the mean drift forces obtained by these two methods. It was showed in Figure 7 that the results achieved good coincidence, which indicated that the quality of mesh was good enough to obtain accurate hydrodynamic parameters.

To investigate the motion responses of FOWT under two types of mooring system, the time-domain coupled analysis was carried out. The environmental conditions are given in Table 3 (Jonkman and Musial, 2010). The wind and wave incident is along the positive direction of X-axis. The surge, heave, and pitch were the most important DOFs to the mooring system design of FOWT. Therefore, the motion responses of these three DOFs and corresponding power spectral density (PSD) are calculated and presented in Figure 8 and Figure 9. The total duration of numerical simulation was 10,000 s; for the sake of analysis, only 1,000 s of the motion responses are shown in figures.

According to Figure 8, the motion responses of innovative type in surge and pitch DOFs were significantly reduced, compared with the conventional type. The heave motion of innovative type was also reduced. It could be confirmed that the innovative mooring system could effectively reduce the motion responses of the FOWT, especially in the surge and pitch DOFs. Moreover, according to Figure 5 in Section Mooring System, the horizontal static restoring stiffnesses of two mooring systems were basically equal with 0° global pitch angle. Therefore, it could be inferred that the global pitch motions could improve the horizontal motion inhibiting ability of innovative mooring system. Also, it was elaborately explained in the Section Restoring Forces and Moments

According to Figure 9, it was obvious that the motion responses of the FOWT were dominated by the low-frequency component induced by the difference frequency wave forces. The PSD of motion responses under the effects of innovative mooring system were reduced significantly in these three DOFs, especially the low-frequency component, compared with the conventional type. It also proved the better motion inhibiting performance of the innovative mooring system. In addition, it was noticeable that the frequencies corresponding to the peak values of PSD were basically equal between the innovative and conventional types. It indicated that the innovative mooring system would not change the natural frequencies to cause new problems while reducing the motion responses of FOWT.

Furthermore, to quantify the comparison of motion responses, the results were processed statistically, as shown in Table 4. The mean, maximum values, and standard deviations of motion responses under the effects of innovative mooring system were all reduced significantly. Especially in the surge DOF, the mean and maximum values of motion were respectively reduced 37.97% and 38.57%, and in the pitch DOF, the mean and maximum values of motion were respectively reduced 17.87% and 20.64%, compared with the conventional type. It could be confirmed that the innovative mooring system would improve the motion performance of FOWT.

To verify the influences of innovative mooring system on the structural loads of FOWT, the bending moment and shearing force at the tower base are calculated and presented in Figure 10. The corresponding PSD are also given in Figure 11.

The bending moment and shearing force at tower base were the most essential structural load components, which were easy to cause problems of structural strength. According to Figures 10 and 11, compared with the conventional type, the bending moment and shearing force at tower base were reduced under the effects of innovative mooring system, especially the low-frequency component, which was consistent with the motion responses. Furthermore, the statistics of structural loads are also given in Table 5 to quantify the comparison. According to the statistics, for the innovative type, the mean values of bending moment and shearing force were respectively reduced 6.45% and 7.81%, compared with the conventional type. The maximum values were respectively reduced 6.14% and 8.18%. In addition, for the innovative type, the standard deviations of these two structural loads were also reduced and it was beneficial to the reduction of fatigue damage.

As verified earlier, the innovative mooring system had significant improvement on motion inhibiting performance in surge and pitch DOFs, compared with the conventional type. To better analyze and compare the performances of these two mooring systems, the restoring force and moment in surge and pitch DOFs are calculated and given in Figure 12. The corresponding statistics are also given in Table 6. It should be noted that the negative signs in figures referred to the direction of force and moment in the reference coordinate system.

According to the figures, it was noticeable that the restoring moment of innovative type in pitch DOF was nearly a magnitude greater than the conventional type, which was consistent with the calculation of static restoring stiffness. The statistics showed that the mean and maximum values of restoring moment provided by the innovative mooring system were respectively increased 401.13% and 498.15%, compared with the conventional type. However, according to Figure 12 and Table 6, the horizontal restoring forces generated by innovative and conventional mooring systems in surge DOF were basically equal. As presented in Section Motion Responses, the surge motions of innovative type were significantly reduced. It meant that the restoring stiffness in surge DOF was significantly increased, which was not consistent with the horizontal static restoring stiffness in Section Mooring System. As mentioned before, it might have resulted from the coupling effects of surge and pitch motions.

For further analysis, the PSD of horizontal restoring force are presented in Figure 13. It was obvious that the horizontal restoring forces of these two mooring systems were both dominated by the low-frequency component, which was consistent with the natural frequency of surge motion (0.069 rad/s) presented in Figure 9. However, the PSD of innovative type had a peak at the natural frequency of pitch motion (about 0.119 rad/s). It indicated that the horizontal restoring force generated by the innovative mooring system was not only induced by the horizontal motion but also influenced by the pitch motion, which was consistent with the improvement of horizontal restoring stiffness presented in Section Motion Responses. It was caused by the larger vertical distance between two groups of fairleads, which made the innovative mooring system more sensitive to the pitch motion. Moreover, for the innovative mooring type, the wave-frequency component of horizontal restoring force was more significant than the conventional type. This was because the wave-frequency component of pitch motion was more significant than the horizontal motion, as shown in Figure 9, and it was enlarged by the innovative mooring system. Based on the analysis earlier, it could be confirmed that the innovative mooring system could provide greater horizontal restoring stiffness taking advantage of the pitch motion.