A 2^nd-order transfer function was obtained using AQWA-LINE to calculate the 2^nd-order wave force acting on the platform. The boundary integral equation to solve the diffraction and radiation problems for deriving the 1^st-order hydrodynamic coefficient is as follows. In addition, the boundary conditions of the platforme are given by Eq. (2).

Diffraction problem boundary integral equation where ϕ_O is the incident wave potential; ϕ_S is the diffracted wave potential, and S_b is the wetted surface area of the structure. G indicates the Green function and n is the normal vector perpendicular to the surface of the body. The Green function used is expressed as Eq. (3).

Radiation problem boundary integral equation

The integral boundary equation to obtain the 1^st-order radiation force coefficient is given by Eq. (4). In this case, the boundary conditions of the floating body are given by Eq. (5). ϕ_R is the radiated wave potential and ϕ_j is the radiated wave potential of the body according to each motion mode. ∊_j is the unit amplitude of the body motion and j is the motion mode. j = 1 and 3 indicate the translational directions of the x-axis and z-axis, while j = 4 and 6 indicate rotational directions. In addition, when there are m floating bodies, including FOWT and multiple WECs, the radiated wave potential in consideration of the hydrodynamic interaction between each floating body can be expressed as a coupling term as in Eq. (5) through linear superposition.

The 2^nd-order wave force of the combined platform ( FEX2) was calculated in the time domain by substituting the 2^nd-order transfer function calculated in AQWA-LINE into AQWA-DRIFT through Eqs. (6) and (7). where Pjk+ and Qjk+ are the real and imaginary parts of the 2^nd-order sum-frequency wave load generated between the j^-th and k^-th wave components. Pjk− and Qjk− are the real and imaginary parts of the 2^nd-order difference-frequency wave load. a_j and a_k are the amplitudes of the j^-th and k^-th wave components. ω_j and ω_k are the frequencies of the j^-th and k^-th wave component. α_j and α_k are the phase angles of the j^-th and k^-th wave components, respectively.

To calculate the equation of motion of the combined energy platform in the time domain, the input values obtained through the user subroutine were substituted into AQWA-NAUT program. Eqs. (8) and (9) are the equations of motion that include all loads acting on FOWT and WECs. where m_F and m_W are the masses of FOWT and WEC, respectively. m_F,a and m_W,a are the added masses of FOWT and WEC. FF,FK1 and FW,FK1 indicate the Froude-Krylov forces, FF,DIFF1 and FW,DIFF1 are the diffraction forces of FOWT and WEC, respectively. In addition, FF,EX2 and FW,EX2 indicate the 2^nd-order wave forces, and F_F,R and F_W,R denote the radiation-damping forces. F_F,STIFF and F_W,STIFF are the restoring forces acting on each structure, and F_W,STIFF is the mooring load acting on FOWT due to mooring line motion. Finally, F_F,CONS and F_W,CONS are constraint forces acting on FOWT and WEC, respectively, due to the motion of FOWT and WEC. These constraint forces are acting at the hinge joint where FOWT and WEC are connected.

In addition, the Froude–Krylov force, which can be expressed as the incident wave force, is calculated for the wetted surface of the floating body that changes at each time step through the incident wave potential and Bernoulli equation, as shown in Eq. (10). The diffraction force was calculated considering the mean wetted surface area of the floating body through the diffracted wave potential and Bernoulli equation, as shown in Eq. (11).

In the time domain, the radiation-damping force can be calculated as a convolution integral, as shown in Eq. (12). ẋ_F(τ) and ẋ_W(τ) are the velocities of FOWT and WEC at time τ, respectively. The retardation function R(t) can be calculated using the radiation damping coefficient B(ω) as shown in Eq. (13). The radiation damping force can be calculated in the same way for both FOWT and WEC, which was calculated by applying each radiation damping coefficient.

The restoring force of FOWT consists of k_hydrostatic, which is the restoring force by the hydrostatic force acting on the floating body, and k_mooring, which is the mooring restoring force caused by the mooring lines, as shown in Eq. (14). The restoring force of WEC includes the restoring force by a hydrostatic force and the PTO force (Eq. 21), as shown in Eq. (15). x_F (t) and x_W (t) are the displacements of FOWT and WEC at time t.

The translational and rotational forces acting on FOWT due to the motion of mooring lines are given by Eqs. (16) and (17) (Ran, 2000). [K^L] is the linear stiffness matrix between FOWT and mooring lines, and [P̃] is the position matrix between FOWT and mooring lines. [K^θ] is the rotational stiffness matrix between FOWT and mooring lines, and [R̃] is the rotation matrix between FOWT and mooring lines. In addition, [C̃] and [D̃] are the translational and rotational motion matrices of FOWT.

FOWT and WEC are constrained by a hinge joint, as mentioned above. This causes F_F,CONS and F_W,CONS, which are constraint forces acting on FOWT and WEC, affecting the motion of FOWT and WEC, respectively. The constraint forces occur at the joint of FOWT and WEC, can be calculated using Eq. (18) (Ansys, 2016). F_j and F_k are the resultant forces acting on FOWT and WEC, excluding the constraint forces. [U_j] and [U_k] indicate the six-degree-of-freedom motions of FOWT and WEC, respectively. [KjjKjkKkjKkk] is the stiffness matrix between the two floating bodies. H is the boundary condition matrix for the constraints. R_j and R_k are the local coordinate system vectors at the joint of FOWT and WEC, respectively. E is the unit position coordinate matrix at the joint. G is a matrix for defining hinge constraints (ANSYS, 2016).

The PTO force acting on WEC was implemented through the Coulomb damping force equation (Kim et al., 2021). G_pto is the Coulomb damping force gradient according to the velocity of WEC. Δp and Sc are the pressure difference and cross-sectional area of the hydraulic cylinder of the PTO system, respectively. where ż is the vertical velocity of WEC.

Before motion analysis of the combined energy platform model and the single platform model (FOWT-Only), the motion response of the single platform model to regular waves in the time domain was compared with the calculation results of FAST (Jonkman and Buhl, 2005), an open-source program provided by NREL, as shown in Fig. 4. The comparison was performed under the following conditions, and the validity of the calculation was verified: the representative condition of the target sea state (a wave height of 4 m and a wave period of 8 seconds) and the long wave condition (a wave height of 6 m and a wave period of 15 seconds) were considered in this study. In addition, the response of the platform for surge, heave, and pitch was compared while the wind load acting on the wind turbine was not considered, and the turbine and tower were assumed to be rigid bodies. The results were in good agreement with the FAST calculation results. The heave response, however, is slightly different from the FAST results. This result appears to be because the buoyancy of the spokes, which was not considered in the FAST model, was included in the TLP structures of the combined and single platform models. Therefore, the initial tension of mooring lines to offset this effect was increased by approximately 10% compared to the FAST model. In other words, the initial tension difference corresponding to the offset of the buoyancy of the appendage for the same calculation conditions partially affected the calculation results. In addition, each response is slightly different from the FAST results under the conditions of a wave height of 4 m and a wave period of 8 seconds. This result was attributed to the external force acting on the platform being slightly different from the nonlinear Froude-Krylov force. The nonlinear restoring force by the floating body motion was also included in this study. Nevertheless, there is no significant difference between the two calculation results, indicating that the analysis method used in this study is reliable.

Calculations were performed according to each load condition and WEC attachment to examine the effect of 2^nd-order wave loads on the motion response of the combined energy platform. The wind load acting on the wind turbine was excluded in the analysis to precisely identify the influence of 2^nd-order wave loads on the platform motion response. In this study, the following three load conditions were used: 1^st-order wave load (LC1), 2^nd-order difference-frequency wave load in addition to the 1^st-order wave load (LC2), and both 2^nd-order difference-frequency and sum-frequency wave loads in addition to the 1^st-order wave load (LC3). Detailed load conditions are described in Table 4. These load conditions were applied to the combined and single platform models. Mooring line tension changes and characteristics of the motion response were mainly compared. The environmental conditions were selected by referring to a previous study that performed TLP analysis with specifications similar to this numerical model. The environmental conditions used were the JONSWAP spectrum with a significant wave height of 4 m, a peak period of 7.5 seconds, and a gamma of 2 (Bae and Kim, 2013). In addition, all incident waves were introduced in the x-axis direction.

The time-series data of motion response of the combined and single platform models were calculated in the time domain. They were converted to motion response spectra and compared in Fig. 5. Fig. 5(a) and 5(b) show the results of the surge responses of the platforms. For both the combined and single platforms, a large motion response occurred near the natural frequency (0.11 rad/s), while the response in the peak frequency of the incident wave (7.5 s = 0.83 rad/s) was negligible. When only the 1^st-order wave load (LC1) was applied to the combined platform, the surge response was larger than that of the single platform. This is because the horizontal force acting on the entire platform is larger than the single platform under the influence of WECs. The surge response of the single platform tended to be larger when LC2 was applied than when LC1 was applied. The response of the single platform in the natural frequency range decreased when LC3 was applied compared to when LC2 was applied. In the case of the combined platform, however, the magnitude of the surge response in the natural frequency range hardly changed regardless of the presence or absence of the 2^nd-order wave load. This is because the WEC PTO of the combined platform offsets the effect of the 2^nd-order sum-frequency load.

Fig. 5(c) and 5(d) compare the heave responses of the platforms. Similar to the surge response, the heave response was large in the natural frequency range (2.5 rad/s). This indicates that a large response occurred at the natural frequency because of the characteristics of the TLP platform. The magnitude of the response was negligible in the peak frequency range of the incident wave (0.83 rad/s) because it was far from the natural frequency of the platform. For both the combined and single platforms, the magnitude of the heave response did not change significantly when LC1 and LC2 were applied. When LC3 was applied to the single platform, however, the magnitude of the heave response was increased by approximately three times. This result was attributed to a phenomenon in which the load increases under the influence of the slowly decaying second-order pressure field as the incident wave is reflected from the cylinder-shaped floating body, and it has already been reported in several studies (Min and Koo, 2022; Koo and Kim, 2007; Kim, 1993, Kim and Yue, 1989). The motion characteristics of TLP structures respond to 2^nd-order sum-frequency wave loads. In the case of the combined platform model with WECs, however, the effect of the 2^nd-order sum-frequency wave load can hardly be observed. This appears to be due to the influence of WEC PTO, and the analysis of this phenomenon is described below.

Similar to surge and heave responses, the pitch responses of the platforms (Fig. 5(e) and 5(f)) were observed in the pitch natural frequency of the floating body (1.7 rad/s) and the peak frequency of the incident wave (0.83 rad/s). The magnitude of the pitch response at the peak frequency, however, is similar to that at the natural frequency. This appears to be because the peak frequency of the incident wave is relatively close to the natural frequency, and a relatively large pitch occurs in the incident wave spectrum. No significant change was observed in the pitch response under the application of the 2^nd-order wave loads for single and combined platforms. The pitch response, however, slightly increased in the combined platform compared to the single platform. This result was attributed to the motion response (pitch) of WECs in the peak frequency (0.83 rad/s), as mentioned in section 4.3, affecting the pitch increase of the combined platform. WEC attachment also appears to slightly increase the pitch of the combined platform in the pitch natural frequency (1.7 rad/s) of the floating body.

Therefore, regarding the surge response of the TLP-type platform, the motion of the single platform model tends to increase under the influence of 2^nd-order wave loads. In particular, when the 2^nd-order sum-frequency wave load was applied, the heave response in the natural frequency range tended to increase dramatically. The heave of the combined platform with WECs was not affected significantly by the 2^nd-order wave loads. In the case of pitch, the influence of the 2^nd-order wave loads was negligible for both models, but the response of the combined platform was slightly larger than the single platform.

The heave of the single platform model was significantly increased by the 2^nd-order sum-frequency wave load, but its effect on the combined platform was insignificant. This reason can be explained by the influence of the PTO acting on WEC. Fig. 6 compares the heave response spectra of the combined platform when the WEC PTO force was excluded (w/o PTO) and added (w/ PTO). For the combined platform under the application of LC1, the heave of the platform was significantly reduced when the PTO force was added (Fig. 6(a)). The application of the PTO force also reduced the heave of the combined platform significantly under the application of LC3 (Fig. 6(b)). The PTO force made the heave of the combined platform smaller than that of the single platform when LC3 was applied. Hence, the PTO force in the combined platform serves as a motion damper that attenuates the heave of the platform.

Fig. 7 compares the rotational motion response spectra of WECs according to the application of the PTO force. WECs operate one-dimensional rotational motion because they are connected to the platform through arms, and the arms are connected to the platform using hinge joints. As shown in Fig. 2, the direction of the motion of WEC2 is perpendicular to the incident wave, while WEC3 is located in front of the incident wave direction. The motion response of WEC2 is similar to that of WEC4 because the arrangement of WECs is symmetrical with respect to FOWT. The response of WEC3, which is expected to be most significantly affected by the incident wave, was analyzed. The WEC showed responses at the surge natural frequency of FOWT (0.11 rad/s), half the peak frequency of the incident wave (0.4 rad/s), and the peak frequency of the incident wave (0.83 rad/s). The motion response of WECs is significantly large when the PTO force is not applied, but the application of the PTO force rapidly decreases the responses in all frequency ranges. Therefore, the PTO force sharply decreases the overall motion response of WECs, and only the response in the peak frequency range of the incident wave occurs (Figs. 7(c) and 7(d)). The rotational response of WECs was slightly larger when LC3 was applied than when LC1 was applied. It appears that the WEC motion response occurred in the surge natural frequency range of the FOWT platform (0.11 rad/s) because WECs are attached to the platform and are thus partially affected by the surge of the platform.

The response that occurs at half the peak frequency of the incident wave is affected by the nonlinear Froude–Krylov force and the nonlinear restoring force of the floating body, and the steepness of the incident wave (Yang et al., 2020). This phenomenon can be explained by the WEC response being reduced rapidly at half the peak frequency of the incident wave if calculation is performed by fixing the wetted surface of the floating body (linear calculation).

In summary, a comparison of the motion response of the combined platform and WECs according to the presence or absence of PTO confirmed that PTO serves as a heave motion damper of the combined platform and reduces the WEC motion rapidly in all frequency ranges. In addition, as can be seen from the condition without PTO, the WEC motion is affected by the surge natural frequency of FOWT (0.11 rad/s), but FOWT is unaffected by the natural frequency of hinged WECs (1.26 rad/s). This indicates that the influence of WECs is limited because their size and weight are very small compared to those of FOWT. Moreover, the motion response is larger under the 2^nd-order wave load condition than the 1^st-order wave load condition, regardless of the presence or absence of PTO.

From the mooring line layout in Fig. 1(b), two mooring lines are vertically attached to each of the four spokes. Figs. 8 and 9 compare the time-series data and spectrum for mooring line tension according to the floating body motion. Because the arrangement of mooring lines is symmetrical with respect to FOWT, as with the arrangement of WECs, the results of mooring lines 3 and 4 will be similar to those of mooring lines 7 and 8. The tension of mooring lines 5 and 6, which are expected to be significantly affected by the incident wave, was analyzed. Mooring lines 5 and 6, located in front of the incident wave direction, showed a similar tendency to that of the pitch response spectrum of the platform. An additional response was also observed in the natural frequency range of the heave response spectrum of the platform (2.5 rad/s). This suggests that mooring lines 5 and 6 are affected most significantly by the pitch of the platform and by the heave of the platform to some extent because of their location. Therefore, the mooring line tension in front of the incident wave direction is affected significantly by the 1^st-order wave load, and the influence of the 2^nd-order wave loads is insignificant.

Fig. 9 compares the tension time-series data and spectrum for mooring lines 3 and 4, which were installed on the side of the floating body. They generally showed a similar tendency to that of the heave response spectrum of the platform for both the combined and single platforms. This suggests that the mooring lines are affected by the heave of the platform and hardly affected by the surge or pitch of the platform, which can be expected. Therefore, the influence of the 2^nd-order sum-frequency wave load (LC3), which had a significant impact on the increase in heave of the single platform, acted similarly on mooring lines 3 and 4, increasing the mooring line tension by approximately three times (Fig. 9(a)). This also applies to mooring lines 7 and 8, which are symmetrically installed.

In summary, the mooring line tension in front of the incident wave direction was dominantly affected by the pitch of the platform and partially affected by the heave of the platform. The mooring line tension on the side of the floating body was dominantly affected by the heave of the platform, and the mooring line tension of the single platform (FOWT-Only) increased rapidly under the influence of the 2^nd-order sum-frequency wave load.