### 1. Introduction

^{nd}-order sum-frequency wave loads. This leads to an increase in the fatigue of mooring lines (Kim, 1991). Therefore, it is necessary to identify the motion response characteristics of a combined wind-wave energy platform precisely based on the TLP platform. In particular, understanding the motion characteristics of TLP platform under 2

^{nd}-order wave loads provides the basic knowledge required for designing combined energy platforms in the future; it makes it possible to secure the diversity of platforms to be designed.

^{nd}-order wave loads to identify the precise motion response of a combined wind-wave energy platform based on the TLP structure. A platform that combined MIT-NREL TLP (Matha et al., 2010), a TLP-type offshore wind turbine model disclosed by NREL, with Wavestar-type WECs was set as a numerical model. As mentioned above, this is to present a new type of combined wind-wave energy platform to which a movable body-type WEC is attached while utilizing the excellent motion response performance of TLP on waves and the advantages of installing multiple WECs. In particular, the wind load acting on the wind turbine was excluded, and focus was given to the analysis of the motion by the wave load to identify the motion response characteristics of the structure that combined TLP with hemispherical WECs under 2

^{nd}-order wave loads. The motion characteristics of the combined platform were compared according to the application of the 2

^{nd}-order sum-frequency wave load and the attachment of WECs. The power take-off (PTO) model was applied to WECs.

^{nd}–order wave loads and the nonlinear Froude-Krylov force that changes every time step due to body motion to the platform and apply the PTO damping force to the WECs, a user-subroutine was developed and numerical analysis was performed in connection with the AQWA program.

### 2. Numerical Analysis Model

^{nd}-order wave loads.

### 3. Numerical Analysis Method

^{nd}-order wave force that cannot be considered in the AQWA-NAUT program to the platform in time domain analysis. The time-series data of the 2

^{nd}-order wave force acting on FOWT were calculated by substituting the quadratic transfer function (QTF) of the external force calculated in AQWA-LINE into AQWA-DRIFT. They were applied to the combined energy platform at each time step to calculate its motion response. In addition, the Coulomb damping equation (Kim et al., 2021) was substituted by setting WEC PTO as a hydraulic cylinder model. In the section below, specific calculation formulas were described.

### 3.1 Hydrodynamic Force Calculation Process

^{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).

*ϕ*is the incident wave potential;

_{O}*ϕ*is the diffracted wave potential, and

_{S}*S*is the wetted surface area of the structure.

_{b}*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).

^{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*is the radiated wave potential and

_{R}*ϕ*is the radiated wave potential of the body according to each motion mode.

_{j}*∊*is the unit amplitude of the body motion and

_{j}*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.

### 3.2 Second-order Hydrodynamic Coefficient Calculation

^{nd}-order wave force of the combined platform (

^{nd}-order transfer function calculated in AQWA-LINE into AQWA-DRIFT through Eqs. (6) and (7).

##### (7)

^{nd}-order sum-frequency wave load generated between the

*j*

^{-th}and

*k*

^{-th}wave components.

^{nd}-order difference-frequency wave load.

*a*and

_{j}*a*are the amplitudes of the

_{k}*j*

^{-th}and

*k*

^{-th}wave components.

*ω*and

_{j}*ω*are the frequencies of the

_{k}*j*

^{-th}and

*k*

^{-th}wave component.

*α*and

_{j}*α*are the phase angles of the

_{k}*j*

^{-th}and

*k*

^{-th}wave components, respectively.

### 3.3 Equation of Motion of the Platform

*m*and

_{F}*m*are the masses of FOWT and WEC, respectively.

_{W}*m*

_{F}_{,}

*and*

_{a}*m*

_{W}_{,}

*are the added masses of FOWT and WEC.*

_{a}^{nd}-order wave forces, and

*F*

_{F}_{,}

*and*

_{R}*F*

_{W}_{,}

*denote the radiation-damping forces.*

_{R}*F*

_{F}_{,}

*and*

_{STIFF}*F*

_{W}_{,}

*are the restoring forces acting on each structure, and*

_{STIFF}*F*

_{W}_{,}

*is the mooring load acting on FOWT due to mooring line motion. Finally,*

_{STIFF}*F*

_{F}_{,}

*and*

_{CONS}*F*

_{W}_{,}

*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.*

_{CONS}*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.

*k*, which is the restoring force by the hydrostatic force acting on the floating body, and

_{hydrostatic}*k*, 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).

_{mooring}*x*(

_{F}*t*) and

*x*(

_{W}*t*) are the displacements of FOWT and WEC at time

*t*.

*K*] is the linear stiffness matrix between FOWT and mooring lines, and [

^{L}*P̃*] is the position matrix between FOWT and mooring lines. [

*K*] is the rotational stiffness matrix between FOWT and mooring lines, and [

^{θ}*R̃*] is the rotation matrix between FOWT and mooring lines. In addition, [

*C̃*] and [

*D̃*] are the translational and rotational motion matrices of FOWT.

*F*

_{F}_{,}

*and*

_{CONS}*F*

_{W}_{,}

*, 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).*

_{CONS}*F*and

_{j}*F*are the resultant forces acting on FOWT and WEC, excluding the constraint forces. [

_{k}*U*] and [

_{j}*U*] indicate the six-degree-of-freedom motions of FOWT and WEC, respectively.

_{k}*H*is the boundary condition matrix for the constraints.

*R*and

_{j}*R*are the local coordinate system vectors at the joint of FOWT and WEC, respectively.

_{k}*E*is the unit position coordinate matrix at the joint. G is a matrix for defining hinge constraints (ANSYS, 2016).

*G*is the Coulomb damping force gradient according to the velocity of WEC. Δ

_{pto}*p*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.

### 4. Numerical Analysis Results

### 4.1 Verification of the Motion Analysis of the Single Platform Model (FOWT-Only)

### 4.2 Analysis of the Motion Responses of the Combined Energy Platform

^{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.

^{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.

^{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.

^{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.

^{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.

### 4.3 FOWT and WEC Motion Analysis According to the Presence or Absence of WEC PTO

^{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.

^{nd}-order wave load condition than the 1

^{st}-order wave load condition, regardless of the presence or absence of PTO.

### 4.4 Platform Mooring Line Tension Analysis

^{st}-order wave load, and the influence of the 2

^{nd}-order wave loads is insignificant.

^{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.

^{nd}-order sum-frequency wave load.

### 5. Conclusion

^{nd}-order wave loads to identify the precise motion response of a combined wind-wave energy platform based on the TLP structure. The motion response characteristics of single and combined platforms were compared to analyze the change in the motion response of the platform due to the attachment of Wavestar-type WECs. ANSYS AQWA, a hydrodynamic program, was used. A user subroutine was developed for time-domain analysis of the effect of 2

^{nd}-order wave loads and WEC PTO damping force acting on the combined energy platform.

^{nd}-order wave loads, the heave response was found to be large. In particular, the magnitude of the heave response of the single platform increased by approximately three times under the 2

^{nd}-order sum-frequency wave load. This is because of the influence of the slowly decaying second-order pressure field as the incident wave is reflected from the cylinder-shaped floating body, which was the motion characteristic of the TLP structure. The motion response of the combined platform model, however, was barely affected by the 2

^{nd}-order sum-frequency wave load because of the influence of the WEC PTO force. The PTO force serves as a heave damper of the combined platform, decreasing the WEC motion in all frequency ranges. The motion response of WECs was slightly larger when the 2

^{nd}-order wave load was applied than when the 1

^{st}-order wave load was applied. 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 platform was dominantly affected by the heave of the platform, and the mooring line tension of the single platform increased rapidly under the influence of the 2

^{nd}-order sum-frequency wave load.

^{nd}-order wave loads. Through the identification of such motion response characteristics, it is possible to control the excessive mooring line response of the TLP-type wind turbine platforms due to the 2

^{nd}-order wave loads and to seek more stable energy extraction. Based on this, the motion response characteristics of combined platforms will be analyzed under realistic ocean environmental conditions by adding the aerodynamic load acting on the wind turbine.