Influence of Heave Plate on the Dynamic Response of a 10 MW Semisubmersible Floating Platform

Abstract

Based on the 10 MW OO-Star semi-submersible floating platform, this study proposes internal and external heave plates to enhance its stability and explores their influence on the platform’s hydrodynamic characteristics. The platform’s structural behavior is analyzed in both frequency and time domains using numerical simulation methods. The study investigates the effects of the porosity and number of holes (with an equal porosity) of the inner heave plate and the opening angle (with the equal area) of the external heave plate on the platform’s hydrodynamic characteristics, ultimately obtaining the optimal arrangement for the inner and external heave plates. Results indicate that the best scheme involves a 10% porosity with 16 holes, which reduces the heave amplitude by 5.7% compared to the original structure. Additionally, reducing the opening angle of the external heave plate increases the added mass and natural period in the heave and pitch directions of the platform. At an opening angle of 140°, the added mass in the heave direction can increase by 25.2% compared to the original structure. Overall, the internal and external heave plates effectively reduce the heave and pitch amplitude of the platform under severe sea conditions.

1. Introduction

Wind energy, as a renewable and clean source of energy, is supported by development policies in an increasing number of countries [1]. China’s offshore wind resources are located near economically developed regions with high power demand, making the development of offshore wind power an effective solution to reduce transmission costs. The infrastructure of offshore wind turbines is transitioning from fixed foundations to floating foundations due to the gradual expansion of offshore wind power into deeper waters. And the deep sea contains a vast amount of stable wind energy resources [2], underscoring the immense potential held by FOWT in the forthcoming era [3]. Figure 1 illustrates the global forecast for floating offshore wind platforms, showing rapid growth in North America, East Asia, and Europe, which collectively hold the majority of the global market share. As more countries plan to develop their first floating offshore wind farms, continuous growth is expected, with a significant leap in global installed capacity projected by 2030. The fully coupled time-domain simulation of three semi-submersible floating wind turbines in different water depths was studied by [4]. Different methods are used to simulate the structural responses of vertical axial wind turbine in ultimate limit state design evaluation [5]. The dynamic responses of the STLP wind turbine during the operation phase were studied by [6], with emphasis on the effect of second-order wave loads, wind–wave misalignment, and water depth. The study reveals that research on offshore floating wind power is advancing rapidly worldwide [7]. Moreover, the design of platform structures is increasingly critical to meeting the stability requirements of future developments, presenting new challenges to motion stability.

The complexity of design increases with larger turbines installed in deeper waters, resulting in different types of foundations [8]. Currently, there are four main types of floating wind turbine platforms, as shown in Figure 2: semi-submersible, spar, tension-leg, and barge. Among them, the semi-submersible platform is suitable for a wide range of water depths, offers flexible installation, and has a short operational window for offshore operations. However, the stability required for normal operation of the floating wind turbine relies on the larger waterline surface area of the semi-submersible platform, resulting in a larger platform size and increased construction costs. Nonlinear wave loads, which are much smaller than direct (linear) wave excitation, often trigger large surge and pitch motions in semi-submersible FOWTs due to resonance effects [9,10]. To improve the hydrodynamic performance of the floating platform, it is necessary to increase the inherent period of pitch and heave, which should be much greater than the peak period of waves to avoid resonance [11]. Some researchers have suggested that the inherent heave period can be increased by adding a heave plate, which also increases the added mass and improves the natural period in the heave direction, keeping it away from the concentrated frequency band of wave energy [12].

Scholars at home and abroad have carried out a series of studies on heave plates to effectively improve the hydrodynamic performance of floating platforms. Li et al. [13] compared single circular perforated and double circular solid heave plates to the original single solid heave plate of the OC4-DeepCwind semi-submersible FWT. Compared to the single solid plate, both heave and pitch motions increased by up to 40% with the perforated or double plates. However, the perforated plate showed reduced heave motion under turbulent wind, and pitch motion decreased by up to 15% under turbulent wind or irregular wave conditions. Huang et al. [14] mounted a heave plate with bionic fractal structures on the bottom of the spar-type FOWT. In their study, the aero-hydro-mooring dynamic method of the FOWT is established to develop a reliable numerical solution model through the DFBI module using computational fluid dynamics software STARCCM+. Zhang et al. [15] conducted experimental investigations of the hydrodynamic performance of the heave plates via the forced oscillation method. The results showed that the added mass and damping coefficients are independent of oscillation frequencies, while the KC number significantly influences the hydrodynamic coefficients. Yu et al. [16] investigated ship collision responses of a semi-submersible floating offshore wind turbine, the OO-Star [17] floater with the DTU 10 MW blades, using the nonlinear finite element code USFOS. In recent years, numerous scholars have validated the effectiveness of heave plates in suppressing motion through experimental models, numerical simulations, and secondary development techniques [18,19,20,21].

While heave plates are known for their performance in motion suppression, existing research primarily focuses on variations in shape characteristics, positioning, and quantity. However, not enough research has been conducted on how to design the opening size of heave plates without increasing material usage based on existing dimensions. Furthermore, the impacts of internal and external heave plates on floating platforms can differ significantly. Designing open chambers at the bottom of columns for internal heave plates is another important research direction. This paper uses Sesam software version 2021 to study the effects of internal and external heave plates on the hydrodynamic performance of the OO-Star semi-submersible platform. The first section outlines the research background and objectives. The second section describes the primary formulas involved in hydrodynamic simulations. The third section details the model parameters, sizes of internal and external heave plates, and calculation conditions. For external heave plates, the same surface area is used with varying opening sizes; for internal heave plates, the same total opening area is kept but with different numbers of openings. At the same time, we validated the reliability of the numerical simulations presented in this paper. The fourth section verifies the hydrodynamic performance of the model under frequency-domain conditions. Time-domain analysis is performed under various conditions to study the hydrodynamic performance of the optimal design under the influence of DTU 10 MW wind turbine thrust, with statistical analysis and comparison across six environmental conditions. The fifth section provides a categorical summary of the research findings.

Figure 1. New floating wind installations worldwide. Data Source: GWEC [22].

Figure 1. New floating wind installations worldwide. Data Source: GWEC [22].

Figure 2. Classification of floating wind turbine platforms.

Figure 2. Classification of floating wind turbine platforms.

2. Analysis and Calculation Theory

2.1. Wind Load

Natural wind speed always fluctuates above and below its average value, which is a random process. This paper employs the NPD wind spectrum, a gust wind spectrum developed by the International Organization for Standardization in ISO 19902:2020 [23], to describe the fluctuations in wind speed. The corresponding variance spectrum captures the magnitude of the fluctuating wind speed, and the fluctuating wind speed of the NPD wind spectrum can be expressed as:

$$S(f,z)={\displaystyle \frac{320{(\frac{U_0}{10})}^2{(\frac{z}{10})}^{0.45}}{{(1+{\tilde{f}}^n)}^{\frac{5}{3n}}}}$$

(1)

$$\tilde{f}=172f{({\displaystyle \frac{z}{10}})}^{{\displaystyle \frac{2}{3}}}{({\displaystyle \frac{U_0}{10}})}^{-0.75}$$

(2)

As shown in Equation (1): $S(f,z)$ is the spectral density when the frequency is $f$; $f$ is frequency, with a value range of $0.00167 \mathrm{Hz}\leqq f\leqq 0.5 \mathrm{Hz}$; $U_0$ is 1 h sustained wind speed at height $z(z=10)$ above MSL; $n$ is a parameter, and $n=0.468$.

2.2. Wave Load

In the hydrodynamic calculations of this study, the JONSWAP spectrum from irregular wave spectra is utilized. The JONSWAP spectrum is widely used to describe North Sea conditions and is based on the Pierson–Moskowitz (PM) spectrum, with further inclusion of a sharper peak enhancement factor to better capture the energy distribution in real marine environments. By employing the JONSWAP spectrum, this paper can more accurately simulate the dynamic response of floating platforms under complex sea states, facilitating the assessment of platform stability and performance under irregular wave conditions:

$$S_{JS}(f)=0.3125H_S^2{T}_P({\displaystyle \frac{f}{f_P}})\exp (-1.25{({\displaystyle \frac{f}{f_P}})}^{-4})\times (1-0.287\ln \gamma ){\gamma }^{\exp \left[-{\displaystyle \frac{(f-f_P)}{2{\sigma }^2{f}_P^2}}\right]}$$

(3)

As shown in Equation (3): $T_P$ is wave period; $H_S$ is significant wave height; $f$ is frequency; when $f\le 1/T_P$, $\sigma =0.07$ and $f>1/T_P$, $\sigma =0.09$; variable $\gamma$ is JONSWAP peak parameter:

$$\gamma =\left\{\begin{array}{l}5,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{T}_P/\sqrt{H_S}\le 3.6 \\ \exp (5.75-1.15T_P/\sqrt{H_S}), 3.6<T_P/\sqrt{H_S}<5.0\\ 1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{T}_P/\sqrt{H_S}>5.0\end{array}\right.$$

(4)

2.3. Frequency Domain Characteristics

In Sesam, when solving for the frequency domain of a floating body, the hull is treated as a rigid body. The formula used is:

$$M_{ij}{a}_j=F_i$$

(5)

As shown in Equation (5): $a_j(j=1,2,\cdots ,6)$ represents the acceleration of the floating body in six degrees of freedom; $M_{ij}(i,j=1,2,\cdots ,6)$ is the generalized added mass matrix; $F_i$ is the wave force acting on the floating body.

Assuming potential flow theory, $F_i$ can be divided into incident force $F_i^{fk}$, radiation force $F_i^r$, diffraction force $F_i^d$, and hydrostatic restoring force $F_i^s$.

$$F_i^{fk}=\rho i\omega {\displaystyle \underset{s_0}{\iint }{\varphi }_0{n}_i}\mathrm{d}s$$

(6)

$$F_i^r=-{\lambda }_{ij}{\ddot{x}}_j-{\mu }_{ij}{\dot{x}}_j$$

(7)

$$F_i^d=\rho i\omega {\displaystyle \underset{s_0}{\iint }{\varphi }_7{n}_i}\mathrm{d}s$$

(8)

$$F_i^s=-C_{ij}{x}_j$$

(9)

As shown in Equations (6)–(9): ${\lambda }_{ij}$ is the added mass coefficient; ${\mu }_{ij}$ is the damping coefficient; ${\varphi }_0$ and ${\varphi }_7$ are the incident and diffraction potentials, respectively; $C_{ij}$ is the hydrostatic restoring force coefficient matrix.

By substituting Equations (6)–(9) into Equation (5), one finds:

$$M_{ij}{a}_j=F_i^{fk}+F_i^s+F_i^r+F_i^d$$

(10)

In hydrodynamic frequency domain calculations, the motion equation of a floating body under external loads is derived using Newton’s second law and the angular momentum theorem [24]. Simplifying Equation (10):

$$\left(M+m\right)\ddot{x}+Q\dot{x}+Kx=F_{\mathrm{w}}$$

(11)

As shown in Equation (11): $M$ is generalized mass matrix; $K$ is the hydrostatic stiffness matrix; $F_{\mathrm{w}}$ is wave excitation force; $m$ and $Q$ are hydrodynamic added mass matrix and damping coefficient matrix.

Under the action of linear, small-amplitude waves with frequency $\omega$, the floating wind turbine platform exhibits harmonic motion. The first-order wave excitation force can be simplified as:

$$F_w=f_w{e}^{-i\omega t}$$

(12)

$$\mathrm{x}=\overline{\mathrm{x}}{e}^{-i\omega t}$$

(13)

In Equations (12)–(13): $f_w$ denotes the amplitude of the first-order wave force on the FOWT under the influence of incident waves in frequency domain $\omega$ with unit wave amplitude; $\overline{\mathrm{x}}$ represents the response amplitude of the FOWT under the incident wave in frequency domain $\omega$ with unit wave amplitude.

Substituting Equations (12) and (13) into Equation (11) yields [25]:

$$\left[-{\omega }^2(M+m)-\mathrm{i}\omega Q+K\right]\overline{x}=f_{\mathrm{w}}$$

(14)

By solving the algebraic equations of the complex coefficient, when the frequency $\omega$ is obtained, the motion response $\overline{x}$ of the floating body under the first-order wave force $f_{\mathrm{w}}$ can be obtained.

2.4. Equation of Motion

The time-domain motion of the floating body can be obtained by superimposing the wave-frequency and low-frequency motions of the floating body. During the calculation, the motion of the floating body is assumed to be linear, so the motion in different directions and the corresponding velocity potentials can be solved via superposition. The radiation potential of the floating body can be expressed as:

$${\mathrm{\Phi }}_i\left(t\right)={\nu }_i\left(t\right){\phi }_j+{\displaystyle \underset{-\infty }{\overset{t}{\int }}{\nu }_i\left(\tau \right)}{\chi }_j\left(t-\tau \right)\mathrm{d}\tau$$

(15)

As shown in Equation (15): ${\mathrm{\Phi }}_i\left(t\right)$ is the velocity potential of the floating body in the $i(i=1,2,\cdots ,6)$ mode; ${\nu }_i\left(t\right)$ is the velocity equation of the floating body; ${\phi }_j$ is the velocity potential generated by the unit impulse motion of the floating body; ${\chi }_j$ denotes the velocity potential at time $\tau$ after a unit impulse motion of the floating body.

A function is defined as follows:

$$m_{ij}=-{\displaystyle {\int }_S\rho {\phi }_j{n}_i}\mathrm{d}S$$

(16)

$$h_{ij}(t)={\displaystyle {\int }_S\rho \frac{\partial {\chi }_j(t)}{\partial t}}{n}_i\mathrm{d}S$$

(17)

Taking the floating wind turbine as a complete system, the equation of motion of the system under wind, wave, and current load can be obtained from Newton’s law of motion:

$$(M+m)\ddot{x}(t)+{\displaystyle {\int }_0^{t}h(t-\tau )}\dot{x}(\tau )\mathrm{d}\tau +Df(\dot{x})+K(x)x=q(t,x,\dot{x})$$

(18)

As shown in the aforementioned Equation (18): let $M$ and $m$ represent generalized mass matrix and the hydrodynamic added mass matrix; $D$ and $K$ denote the system’s damping coefficient matrix and hydrostatic stiffness matrix; $h(t-\tau )$ is the system’s retardation function, with $\tau$ as the retardation time. Retardation function, representing the hydrodynamic memory effect, is the influence of the structure’s past movements on its current state of motion. The values $x$, $\dot{x}$ and $\ddot{x}$ represent the displacement vector, velocity vector, and acceleration vector in the platform’s six degrees of freedom, respectively; $q$ represents the external loads, which includes the environmental forces on the buoy and the nonlinear restoring forces from mooring tensions [26].

The expressions of the generalized mass matrix $M$ and the hydrostatic stiffness matrix $K$ are shown as follows:

$$M=\left[\begin{array}{cccccc}{m}_0& 0& 0& 0& 0& 0\\ 0& m_0& 0& 0& 0& 0\\ 0& 0& m_0& 0& 0& 0\\ 0& 0& 0& I_{xx}& 0& 0\\ 0& 0& 0& 0& I_{yy}& 0\\ 0& 0& 0& 0& 0& I_{zz}\end{array}\right]\hspace{1em}K=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& K_{33}& K_{34}& K_{35}& 0\\ 0& 0& K_{43}& K_{44}& K_{45}& K_{46}\\ 0& 0& K_{53}& K_{54}& K_{55}& K_{56}\\ 0& 0& 0& 0& 0& 0\end{array}\right]$$

(19)

$$K_{33}=A\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{K}_{34}=K_{43}={\displaystyle \underset{A}{\int }y\mathrm{d}A}$$

(20)

$$K_{35}=K_{53}=-{\displaystyle \underset{A}{\int }A\mathrm{d}A}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{K}_{44}={\displaystyle \underset{A}{\int }{y}^2\mathrm{d}A}+z_{gb}V$$

(21)

$$K_{45}=K_{54}=-{\displaystyle \underset{A}{\int }xy\mathrm{d}A}\hspace{1em}\hspace{1em}{K}_{46}=-x_{gb}V$$

(22)

$$K_{55}={\displaystyle \underset{A}{\int }{x}^2\mathrm{d}A}+z_{gb}V\hspace{1em}\hspace{1em}\hspace{1em}{K}_{56}=-y_{gb}V$$

(23)

As shown in Equations (20)–(23): $m_0$ is the mass of the platform; $I_{xx}$, $I_{yy}$, and $I_{zz}$ are the moments of inertia of the platform around the $X$, $Y$, and $Z$ axes, respectively; $A$ is the waterline area; $x$, $y$, and $z$ are the coordinates in the platform’s coordinate system; $V$ is the displaced volume; $x_{gb}$, $y_{gb}$, and $z_{gb}$ are the coordinates of the buoyancy center relative to the center of gravity.

3. Floating Platform Model with Heave Plate

3.1. Parameters of FOWT and Simulation Verification

The present analysis is based on the 10 MW OO-Star semi-submersible floating platform developed by [17], whose structural parameters are illustrated in Figure 3 and Figure 4, and domestic and foreign scholars have conducted multi-angle and multi-directional research on this platform. Li et al. [27] used OpenFAST to conduct the aero-hydro-servo-elastic coupled simulations under selected environmental conditions. The surge motion and fairlead tension were compared in both time-domain and frequency-domain for different mooring configurations. Yang et al. [28] examined the complicated interaction effects between the wind and tidal turbines of a 10 MW-class semi-submersible floating wind–current integrated system (IFES) that was composed of the DTU 10 MW wind turbine, OO-Star platform, and indefinite 550 kW tidal turbines. A 0.5 m-thick heave plate was placed at the bottom of the platform, and its internal and external structural designs were adjusted to investigate changes in the hydrodynamic characteristics of the platform. Other platform parameters are listed in

Table 1. Parameters of the semi-submersible floating platform.

Property	Parameter
Total platform mass [kg]	2.17 × 107
Centroid position (x,y,z) [m]	(0, 0, −15.23)
Center of buoyancy position (x,y,z) [m]	(0, 0, −14.24)
Moment of inertia of roll Ixx [kg·m2]	9.43 × 109
Moment of inertia of pitch Iyy [kg·m2]	9.43 × 109
Moment of inertia of yaw Izz [kg·m2]	1.63 × 1010
Drainage volume [m3]	2.35 × 104
Draught [m]	22
Number of lines	3
Pre-tension [N]	1.67 × 106
Equivalent mass per length in air [kg/m]	375.38
Equivalent weight per length in water [N/m]	3200.6
Extensional stiffness EA [N]	1.506 × 109
Physical chain diameter [m]	0.137

. The platform has a total design water depth of 130 m, a mass of 2.17 × 10⁷ kg, and a design draft of 22 m.

Table 2. The results of FAST and Sesam were compared.

Parameter	FAST	Sesam	Deviation
Heave	20.92	21.67	3.56%
Pitch	31.65	29.92	5.45%

compares the results obtained using FAST from the literature [17] with those calculated by Sesam in this study. The results indicate that our calculations can accurately simulate the motion response of the FOWT. Furthermore, this allows for a more in-depth comparative analysis of the internal and external heave plates discussed later in the text.

3.2. Structural Design of Heave Plate

The study begins by analyzing the porosity of the internal heave plate to determine its optimal value, followed by a comparative analysis of different hole numbers under the same porosity to obtain the design scheme with the optimal number of openings. The external heave plate was compared under different opening angles but with the same heave plate area. To ensure the effectiveness of the numerical simulation comparison experiment of the internal and external heave plates, the overall center of gravity, draught, and moment of inertia of the ballast water control platform were kept consistent with the parameters of the original platform. Chen et al. [29] who proposed a porous viscoelastic model for a deformable floating platform, which was relevant to this wave–body interaction problem.

Figure 4 shows the design scheme of the inner and outer heave plates, including the ballast tanks located at the bottom of the three side columns as cabins connected with the outside world in the blue section of the upper left corner. The upper right corner illustrates the layout of the internal heave plate with different hole numbers under different porosities and equal porosity. The circular array was arranged evenly relative to the center of the circle. In the design of the external heave plate, to eliminate the influence of the heave plate area, a comparative experiment was carried out on the basis of a 20-degree interval ranging from 120 degrees to 220 degrees. The blue part shown in the lower left corner of Figure 3 represents the changing area of the external heave plate structure, and the layout diagram of the external heave plate under different tension angles was shown in the lower right corner.

Figure 4. Design scheme of inner and external heave plate. (a) Inner heave plate; (b) external heave plate.

Figure 4. Design scheme of inner and external heave plate. (a) Inner heave plate; (b) external heave plate.

3.3. Parameters of Environment

The time-domain calculations utilize the NPD wind spectrum and the JONSWAP wave height spectrum. To analyze and compare the impact of different wind conditions and irregular waves on the platform’s motion response, and to enhance the generalizability of the conclusions, six typical calculation conditions from [30] were considered, as shown in

Table 3. Calculation conditions.

Condition	Effective Wave Height [m]	Peak Period [s]	Wind Speed [m/s]
EC1	2.34	9.56	7
EC2	3.3	9.71	8.5
EC3	4.14	10.04	11.4
EC4	5.16	10.29	14.7
EC5	6.18	10.51	17.8
EC6	6.99	11.12	21.3

. The wind speed for the three operating conditions corresponds to the rated wind speed of the DTU 10 MW wind turbine [17,31] while the corresponding rated thrust is 1500 kN.

4. Analysis of Numerical Simulation Results

4.1. Frequency Domain Analysis

Sesam software was used to calculate the response amplitude operator (RAO) and added mass of the semi-submersible floating platform in surge, heave, and pitch degrees of freedom. The internal and external heave plate designs were compared and analyzed.

The theoretical foundation of Sesam for designing floating structures relies on robust finite element methods and the radiation-diffraction theory in hydrodynamics. Its HydroD module provides stability analysis and hydrodynamic calculations in both frequency and time domains. The Sima module offers advanced numerical tools for coupled simulations of floating offshore wind turbines. The air–liquid–servo-elastic aerodynamic method enhances the coupling analysis of hydrodynamic loads and motions of the body (Simo module) with structural responses of the mooring system (Riflex module), completing nonlinear time-domain dynamic simulations.

In hydrodynamic analysis, to fully account for the influence of viscous effects, we’ve set a damping coefficient of 0.09 based on practical engineering experience. By incorporating this into the calculated inertia mass, added mass, restoring stiffness, and viscous damping into the equations from section two, we can accurately derive the motion amplitude and acceleration of the floating body. In addition, the integration of aerodynamics and structural dynamics provides a comprehensive simulation approach to reasonably simulate wind viscous effects in Sesam, thereby enhancing the precision and reliability of structural design and wind load evaluations.

(1) Frequency domain results of internal heave plate

Figure 5 and Figure 6 show that the structure without perforation exhibits larger amplitudes in the heave and pitch directions, while the amplitudes were smaller, at 10% and 20% porosity levels. Introducing porosity within the internal heave plate can reduce the amplitude in the heave direction; however, the differences in the heave direction among various porosity levels were minimal. This indicates that altering porosity does not effectively reduce the heave amplitude. In the heave direction, the added mass of the 10% porosity after wave periods exceeding 12 s was larger. In the heave and pitch directions, the added mass of the 30% porosity after wave periods exceeding 20 s was larger. Considering the RAO and added mass in the inherent period, a 10% porosity was determined as the optimal design, which was further studied to determine the number of holes.

Under the assumption that the porosity of the internal heave plate was 10%, four different numbers of holes, namely 8, 16, 24, and 32, were selected for analysis. The RAOs of surge, heave, and pitch degrees of freedom were presented in Figure 7, indicating almost no difference in amplitude between surge and heave directions, but with the pitching amplitude of 8 and 16 holes being approximately 4.8% better than that of the other two. The added mass for each degree of freedom was shown in Figure 8. In the heave and pitch directions, the amplitude of 8 and 16 holes was greater than 24 and 32 holes by about 1.6%. As the frequency domain calculation does not distinguish between the advantages and disadvantages of 8 and 16 holes, a further time-domain analysis was conducted for these two options.

(2) Frequency domain results of external heave plate

The frequency domain plot of the external heave plate is shown in Figure 9, revealing that the response amplitude operators of different opening angles exhibit almost no difference in surge motion. In the heave direction, the 220-degree opening angle has an amplitude that was 2.5% better than that of the 120-degree opening angle. In the pitch direction, the natural period corresponding to the 120- and 140-degree opening angles was slightly shifted. Added mass in surge motion shows no significant differences across all opening angles. However, the scheme with a 120-degree opening angle has significantly higher added mass in both heave and pitch directions compared to other opening angles, with the pitch amplitude being 33.9% greater than that of the 220-degree opening angle. This is illustrated in Figure 10.

4.2. Time-Domain Analysis

Time-domain analysis was conducted using the Sima module in Sesam, where the DTU 10 MW wind turbine thrust was coupled with the platform’s motion response, as shown in Figure 11. The flow and wind direction were along the positive X-axis, with the computational domain length set at 1000 m and the platform located at the center of the coordinate system. To guarantee computation accuracy and duration, the total simulation time was set to 2000 s, with an iteration time step of 0.01 s and a time increment of 0.1 s. For each operating condition, 20,000 data points were collected and statistically analyzed for surge, heave, and pitch degrees of freedom, as shown in Figure 12, which illustrates the six-degree-of-freedom motion of the platform in surge, sway, heave, roll, pitch, and yaw directions.

(1) Motion response of internal heave plate

Time-domain analysis was further conducted for the 8-hole and 16-hole options, as shown in Figure 13. The time history curves between 1500 s and 2000 s under EC3 operating conditions were selected. It was found that there was no significant difference between the designs with 8 holes and 16 holes. Due to the longer natural period of pitch compared to heave, the periodic fluctuation frequency in the time history curve was significantly smaller. As the time-domain curves were unable to effectively compare the differences between the two options, statistical analysis was performed on the data for each operating condition.

The time history curve data for the six operating conditions were analyzed, and the results were compiled and summarized in

Table 4. Statistics of the calculation results of different opening numbers.

EC 1	Number of Holes	8 Holes			16 Holes
	DOF 2	Surge [m]	Heave [m]	Pitch [deg]	Surge [m]	Heave [m]	Pitch [deg]
EC1	MV 3	0.115	−0.017	0.343	0.115	−0.017	0.343
	SD 4	0.397	0.138	0.361	0.397	0.138	0.365
	AMP 5	2.270	1.174	2.087	2.267	1.177	2.085
EC2	MV	0.072	−0.021	0.479	0.073	−0.021	0.479
	SD	1.166	0.194	0.841	1.163	0.194	0.836
	AMP	6.399	1.587	4.295	6.246	1.598	4.213
EC3	MV	0.182	−0.021	0.789	0.174	−0.022	0.788
	SD	1.198	0.260	1.038	1.238	0.260	1.050
	AMP	6.850	1.995	5.417	7.061	1.979	5.590
EC4	MV	0.057	−0.023	0.499	0.052	−0.023	0.498
	SD	1.434	0.359	1.469	1.460	0.358	1.458
	AMP	8.658	2.723	9.333	8.697	2.714	9.157
EC5	MV	−0.062	−0.026	0.459	−0.020	−0.026	0.460
	SD	1.790	0.711	2.798	1.720	0.695	2.665
	AMP	11.440	5.336	14.894	10.379	5.171	13.553
EC6	MV	−0.068	−0.030	0.412	−0.005	−0.036	0.402
	SD	2.040	0.593	2.570	2.163	0.565	2.466
	AMP	11.024	4.769	13.111	11.272	4.786	13.553

¹ EC: Environment condition; ² DOF: degrees of freedom; ³ MV: mean value; ⁴ SD: standard deviation; ⁵ AMP: amplitude.

. In the surge direction, the 16-hole option exhibited smaller amplitudes under EC1–2 conditions, while the 8-hole design performed better under more severe sea state variations. As wave height increased, the amplitude difference between the two options in the heave direction gradually expanded but remained insignificant. In the pitch direction, the amplitude of the 16-hole option was slightly larger than that of the 8-hole design under EC3 and EC6 conditions, but the difference did not exceed 3.37%, with the amplitude of the remaining conditions being better than that of the 8-hole option. Under the EC3 condition corresponding to the rated wind speed, the maximum thrust of the wind turbine resulted in a maximum mean value for overall pitch motion, which was about 65% greater than that of neighboring operating conditions, such as EC2 and EC4. Comparing the mean and standard deviation of the three degrees of freedom, the 16-hole design exhibited lower deviation from the initial position and less fluctuation amplitude, indicating better stability than the 8-hole option.

(2) Motion response of external heave plate

Time-domain analysis was performed for the external heave plate with opening angles of 120, 140, and 220 degrees over a period of 2000 s. Figure 14 shows the time-domain plot after 1500 s under EC3 operating conditions. Due to the larger natural period in pitch direction for the 120- and 140-degree options, a significant phase difference appeared in the time-domain curve, gradually increasing with simulation time, which confirmed the frequency-domain calculation results. The natural period in the heave direction was smaller than that in pitch, resulting in more frequent periodic fluctuation in the time-domain curve. Under the influence of wind turbine thrust, the mean value of the time history curve in pitch direction significantly increased, with greater amplitude variation compared to surge and heave directions.

Based on the statistical analysis of the time history data for the external heave plate provided in

Table 5. Statistics of the calculation results of different opening numbers.

EC	Angle	120-Degree			140-Degree			220-Degree
	DOF	Surge [m]	Heave [m]	Pitch [deg]	Surge [m]	Heave [m]	Pitch [deg]	Surge [m]	Heave [m]	Pitch [deg]
EC1	MV	0.123	−0.017	0.411	0.129	−0.017	0.409	0.132	−0.017	0.404
	SD	0.417	0.128	0.722	0.445	0.129	0.579	0.373	0.140	0.534
	AMP	2.493	0.989	3.580	2.595	0.961	2.659	2.262	1.037	2.391
EC2	MV	0.210	−0.017	0.576	0.194	−0.018	0.575	0.185	−0.018	0.571
	SD	0.490	0.190	0.551	0.548	0.194	0.831	0.621	0.199	0.838
	AMP	2.851	1.539	3.257	3.231	1.487	4.713	3.540	1.632	4.006
EC3	MV	0.238	−0.021	0.948	0.248	−0.019	0.952	0.265	−0.020	0.943
	SD	1.104	0.278	1.422	0.889	0.264	1.666	0.950	0.261	1.375
	AMP	6.801	2.042	6.776	5.413	2.244	7.173	5.910	2.098	6.598
EC4	MV	0.090	−0.025	0.600	0.076	−0.020	0.621	0.113	−0.022	0.592
	SD	1.485	0.337	2.303	1.197	0.332	2.084	1.380	0.382	1.276
	AMP	8.939	2.721	12.846	7.579	2.836	10.383	6.999	3.068	6.509
EC5	MV	0.054	−0.029	0.521	0.034	−0.025	0.533	−0.169	−0.041	0.493
	SD	1.835	0.494	2.724	1.642	0.423	1.812	2.654	0.578	3.222
	AMP	12.832	3.534	17.144	8.609	3.097	9.019	14.803	4.326	18.523
EC6	MV	−0.114	−0.044	0.491	−0.067	−0.035	0.480	−0.179	−0.048	0.444
	SD	2.489	0.663	3.769	2.246	0.667	2.769	2.819	0.668	3.478
	AMP	13.614	4.392	19.233	13.171	4.480	16.083	15.652	4.079	19.285

, it was found that under severe sea-state conditions, the 140-degree opening angle exhibited amplitudes that were 3.2% better than those of other opening angles in the surge direction. In the heave direction, the 120-degree opening angle had relatively smaller amplitudes under EC1–4 conditions, while the average value of the 140-degree opening angle performed the best across all sea states. In the pitch direction, the amplitude of the 220-degree opening angle was smaller under EC1–4 conditions, and the 140-degree opening angle had a 16.4% better performance than other opening angles under EC5–6 conditions. Comparing the mean and standard deviation of the three degrees of freedom, the 140-degree opening angle showed a smaller deviation from the initial position in the heave direction. The 140-degree opening angle had better damping performance, combining the advantages of the diameter of the heave plate and extension length of its edges, which could also show better hydrodynamic performance under more severe operating conditions.

5. Conclusions

This study was based on the OO-Star semi-submersible platform prototype and investigates the impact of inner and outer heave plates on the platform’s hydrodynamic performance. The results provide a reference for the design of heave plates for similar semi-submersible platforms. The main conclusions of this study are as follows:

(1) The installation of inner and outer heave plates effectively reduces the heave and pitch amplitudes of the platform, with a 10% porosity resulting in optimal stability. Frequency domain calculations show that an inner heave plate with a 10% porosity and 16 holes can reduce heave amplitude by approximately 5.7% compared to the original structure. In time-domain calculations, the 16-hole inner heave plate design exhibits better overall performance, with less deviation from the initial position and lower fluctuation amplitude under most operating conditions.

(2) For outer heave plate installation, reducing the opening angle can increase the added mass and natural period in the heave and pitch directions, respectively, under equal area conditions. A 140-degree opening angle for the outer heave plate combines the advantages of a smaller diameter and a longer extended edge, exhibiting superior stability under most operating conditions. Compared to designs with larger opening angles, the 140-degree opening angle can increase the heave-based added mass by 25.2% while reducing heave and pitch amplitudes, fluctuation amplitude, and deviation from the initial position under severe sea state conditions.

(3) Changes in the presence and size of inner and outer heave plates have a small effect on the response amplitude operator (RAO) in the pitch direction but significantly affect the added mass in heave and pitch directions. Adjusting the outer heave plate has a more significant impact on the hydrodynamic performance of the platform.