Hydrodynamic Performance and Motion Prediction Before Twin-Barge Float-Over Installation of Offshore Wind Turbines

Abstract

In recent years, the twin-barge float-over method has been widely used in offshore installations. This paper conducts numerical simulation and experimental research on the twin-barge float-over installation of offshore wind turbines (TBFOI-OWTs), focusing primarily on seakeeping performance, and also explores the influence of the gap distance on the hydrodynamic behavior of TBFOI-OWTs. Model tests are conducted in the ocean basin at Tsinghua Shenzhen International Graduate School. A physical model with a scale ratio of 1:50 is designed and fabricated, comprising two barges, a truss carriage frame, two small wind turbines, and a spread catenary mooring system. A series of model tests, including free decay tests, regular wave tests, and random wave tests, are carried out to investigate the hydrodynamics of TBFOI-OWTs. The experimental results and the numerical results are in good agreement, thereby validating the accuracy of the numerical simulation method. The motion RAOs of TBFOI-OWTs are small, demonstrating their good seakeeping performance. Compared with the regular wave situation, the surge and sway motions in random waves have greater ranges and amplitudes. This reveals that the mooring analysis cannot depend on regular waves only, and more importantly, that the random nature of realistic waves is less favorable for float-over installations. The responses in random waves are primarily controlled by motions’ natural frequencies and incident wave frequency. It is also revealed that the distance between two barges has a significant influence on the motion RAOs in beam seas. Within a certain range of incident wave periods (10.00 s < T < 15.00 s), increasing the gap distance reduces the sway RAO and roll RAO due to the energy dissipated by the damping pool of the barge gap. For installation safety within an operating window, it is meaningful but challenging to have accurate predictions of the forthcoming motions. For this, this study employs the Whale Optimization Algorithm (WOA) to optimize the Long Short-Term Memory (LSTM) neural network. Both the stepwise iterative model and the direct multi-step model of LSTM achieve a high accuracy of predicted heave motions. This study, to some extent, affirms the feasibility of float-over installation in the offshore wind power industry and provides a useful scheme for short-term predictions of motions.

1. Introduction

Currently, offshore wind turbines are tending towards a larger capacity and larger dimensions. Against this backdrop, the float-over method has emerged as a potential solution for the installation of giant offshore wind turbines, due to its unique advantages, such as high efficiency and cost-effectiveness, large lifting capacity, and broad applicability in different waters. The float-over method not only effectively addresses the complexity and challenges of installing large platforms and wind turbines, but also significantly enhances the installation efficiency while ensuring safety [1], providing strong support for the development of renewable energy.

The float-over installation method can be classified in detail according to different standards [2]. In terms of the type of barge, float-over installations can be categorized into single-barge float-over, twin-barge float-over, and dynamically positioned barge float-over. The twin-barge float-over installation can further be divided into twin-hull float-over installation, rigid connection twin-barge float-over installation, and hinged connection twin-barge float-over installation. In November 2006, Technip successfully installed a 3400-ton module on the Kikeh Spar platform at a water depth of 1320 m in waters off the eastern coast of Malaysia using rigid connection twin-barge float-over technology, demonstrating its strong technical capabilities and installation competence [3].

Numerical simulations and model tests are two important methods for investigating the hydrodynamic performance of marine engineering structures. In the numerical sense, references [4,5,6,7] proposed a variety of modeling approaches for float-over installation, aiming at simulating the actual float-over process and improving computational accuracy. Chu et al. [8] considered mooring lines, fender systems, ballast systems, and mating devices when simulating the dynamic loads of mooring lines and the impacts between the barge fenders and the concrete gravity substructure during the float-over installation process in the time domain. A number of researchers [9,10,11,12,13] have extensively investigated the hydrodynamic performance of transportation barges during float-over installation using various commercial software packages (such as WAMIT, SESAM, ANSYS AQWA, and ORCAFLEX) from both time-domain and frequency-domain perspectives. Du et al. [14] developed a three-dimensional potential-flow-theory-based program for time-domain calculations of multibody coupled motions, focusing on the float-over installation process of a semi-submersible platform. They compared the time-domain responses under mutual interaction and non-interaction conditions, revealing the shielding effect of the semi-submersible platform on the transport barge. Choi et al. [15] employed a high-order boundary element method (BEM) to analyze the hydrodynamic interactions of floating multibody systems. They addressed the radiation and diffraction problems of two rectangular barges. Xu et al. [16] conducted hydrodynamic performance studies on a twin-barge system during a float-over installation process based on three-dimensional potential flow theory. Their focus was on hydrodynamic parameters and free surface elevation at various distances, and the mutual influence mechanisms of multiple floats. Wang [17] introduced a state-space model to replace time-domain convolution integrals for twin-barge float-over installation systems, thereby improving the efficiency of time-domain simulations to some extent. Li [18] investigated the hydrodynamic characteristics of multiple floats arranged in parallel under various configurations, emphasizing the effects of different wave parameters. Li et al. [19] utilized WAMIT software to numerically simulate the hydrodynamic characteristics of a twin-barge system during float-over operations. They studied the influence of the distance between two barges on hydrodynamic properties. Zhang et al. [20] proposed a twin-barge lifting system for the decommissioning of large offshore platforms. As their focus was on the transfer of upper modules to transport barges, they conducted numerical simulations using ANSYS AQWA to compute hydrodynamic parameters such as added mass, radiation damping, and wave forces for the three-barge system. It was concluded that the system is sensitive to environmental conditions.

In the experimental sense, physical model tests can help researchers gain insights into the dynamic behavior and loading conditions of marine structures under complex environmental factors such as wind, waves, and currents. Chu et al. [8] employed model tests to validate the numerical simulation results, including mooring line tensions, barge motions, and environmental conditions throughout the float-over installation process. Koo et al. [21] divided the float-over installation process of a spar platform into three stages: load transfer at 0%, 30%, and 80%. They considered the forces acting on the mooring lines, fender systems, and leg mating units (LMUs). Their experimental results were found to align closely with numerical simulations. Kurian et al. [22] carried out physical model tests on the barge float-over installation of a jacket platform, with a focus on the motion response of the barge. Their experimental findings were in good agreement with the results derived from diffraction theory using WAMIT software. Magee et al. [23] simulated the ballast of the barge by altering its height relative to the jacket through water discharge. They utilized a combined lifting and ballast system to mitigate high dynamic impact loads during the mating process. Xiong et al. [24] conducted 1:30-scale physical model tests on a large float-over barge, exploring various water depths and wave directions through white noise wave tests. The experiments highlighted significant shallow water effects and examined the influence of water depth on barge motion responses and mooring line tensions. Dessi et al. [25,26] conducted physical model tests on the twin-barge float-over installation process. They examined both rigid connections [25] and flexible connections [26]. References [27,28,29,30,31,32,33] conducted model tests focusing on different load transfer stages during various float-over installation processes, with particular attention paid to barge motions, mooring line tensions, and impact loads on the docking system. Tao et al. [34] introduced an innovative remote rapid ballast system based on intelligent algorithms, capable of continuously adjusting the draft of installation barges. This system considered deck support units (DSUs), LMUs, and fender devices, facilitating continuous load transfer. The experiment revealed two noteworthy phenomena: the load transfer rate of the upper module was closely related to the trim angle of the transport barge, and the separation between the upper module’s tip and the receiving cone of the DSU was asynchronous. He et al. [35] developed a time-domain method to simulate the nonlinear motions during the mating process of the float-over installation, accurately modeling the forces on the fendering system. Their findings were further validated through 1:36-scale physical model tests, confirming the accuracy of the numerical simulations. Li et al. [36] executed 1:40-scale physical model tests on the twin-barge float-over installation of a 10,000-ton upper module. They measured and collected six degrees of freedom motion data from both barges, analyzing the tension in the mooring lines in comparison with numerical simulation results.

Currently, no dedicated twin-barge systems exist for the float-over installation of offshore wind turbines, and research on their hydrodynamic performance is limited. Contrastingly to previous research on the float-over installation, the floater in the present paper has a relatively taller center of gravity because of the multi-turbines on deck. Due to the importance of safety in wind turbine installations, it is also essential to study how the distance between two barges influences the floater’s hydrodynamic performance. At the same time, some researchers have made valuable attempts in the field of ocean meteorological data forecasting [37], but research works on the motion prediction of float-over installations in the offshore wind industry are limited. In order to ensure successful installation and reduce operational risks, it is important to carry out TBFOI-OWT motion prediction.

The organization of this paper is as follows. Section 2 introduces the concept and outlines the key parameters of the full-scale TBFOI-OWT. Section 3 covers the scaled physical model along with the sensor arrangement. Section 4 and Section 5 present the numerical simulation methodology and describe the load cases. Section 6 provides an overview of the preliminary calibration tests. In Section 7, the test results of the main load cases are thoroughly discussed, including statistical analysis in the time domain and spectral analysis in the frequency domain. Section 8 builds a WOA-LSTM model to predict the motions of TBFOI-OWT. Finally, Section 9 summarizes the key findings and offers suggestions for future research.

2. Prototype Description

Figure 1 illustrates the conceptual diagram of TBFOI-OWTs, which primarily consists of two identical barges; a truss (for rigidly connecting the two barges); a deck support frame (DSF, used to support the upper modules and transferring loads); a truss carriage frame (for securing two wind turbines); cantilever beams (for securing the tower); connectors (for joining the tower to the foundation); and two identical 16 MW wind turbines. Notably, the DSF has a substantial weight, and the barges are equipped with significant ballast, which effectively lowers the overall center of gravity of the model, ensuring the stability of TBFOI-OWTs. The origin O of the coordinate system Oxyz is positioned at the midpoint of the water surface between the midsections of the two hulls. The positive x-axis extends along the twin barge’s length towards the bow, the positive y-axis points towards the port side along the twin barge’s width, and the positive z-axis is oriented vertically upwards. The additional parameters of TBFOI-OWTs are presented in

Table 1. Parameters of TBFOI-OWTs.

Parameters	Values
Length overall of a single barge (m)	171
Molded breadth of a single barge (m)	35
Molded depth of a single barge (m)	9
Draft (m)	4.66
Water depth (m) 1	250
Mass of a single barge (t)	11,161.21
Displacement of twin barge (t)	39,141.29
Center of gravity of twin barge (m)	(0.21, 0, 15.97)
Roll moment of inertia about the CG of twin barge (kg·m2)	7.98 × 1010
Pitch moment of inertia about the CG of twin barge (kg·m2)	1.11 × 1011
Yaw moment of inertia about the CG of twin barge (kg·m2)	6.34 × 1010

¹ Water depth of 250 m is assumed for deep-water spar projects like Hywind Tampen.

.

To achieve the accuracy of installation, it is necessary to position the twin-barge system through the mooring system. Designed for a water depth of 250 m, the mooring employs an eight-chain catenary system using studless chain R4. The mean drift forces in beam seas are greater than those in head seas. Thus, larger diameter chain links are selected for the mooring in the y direction. The arrangement is illustrated in Figure 2, and the specific parameters of the mooring system are detailed in

Table 2. Parameters of mooring system.

Parameters	Line 1,2,3,4	Line 5,6,7,8
Type	Studless chain R4	Studless chain R4
Diameter (mm)	157	180
Equivalent diameter (mm)	283	324
Break load (kN)	21,234	26,278
Mass density (kg/m)	493	648
Axis stiffness (kN)	1.96 × 106	2.55 × 106
Length (m)	775	880
Mooring radius (m)	786	862
Depth of fairleads below SWL (m)	0.66	0.66
Distance between fairleads and axis z (m)	89	54

.

3. Physical Model Description

3.1. Ocean Engineering Basin Description

The model tests are conducted in the Ocean Engineering Basin at the Tsinghua Shenzhen International Graduate School. The basin has a length of 25 m (in the wave direction), a width of 8 m, and an effective water depth of 5 m, as shown in Figure 3. The wave maker is capable of producing both regular and random waves with heights up to 0.75 m and wave periods ranging from 0.4 to 3.5 s. To minimize wave reflection, a wave absorber was installed at the rear of the basin. The excellent performance of the laboratory has been verified in several model tests [38,39,40].

3.2. Physical Model and Test Set-Up Description

Considering the water depth and wave-making capacity of the wave maker, the physical model of the TBFOI-OWT is constructed at a 1:50 scale, following Froude similarity (

Table 3. Froude scaling.

	Scale Factor (Prototype/Model)
Linear dimension	λ
Linear velocity	λ1/2
Linear acceleration	1
Time or period	λ1/2
Angle	1
Mass 2	γλ3
Displacement volume	γλ3
Force	γλ3
Moment	γλ4
Moment of inertia	γλ5

² γ is the ratio of seawater density to fresh water density, taken as 1.025 in this paper.

) on the condition that the gravitational and inertia forces are the main external loads. The physical model is divided into four parts: the twin barge, the topside module, the tower and wind turbines, and the mooring system. Underwater tension sensors are placed in the fairlead holes to measure the mooring tension, while accelerometers are installed at the top of the tower to measure the acceleration at the top of the tower. Markers are placed on the twin barge, and the Qualisys 3D motion capture system is used to record the six degrees-of-freedom (6-DOF) motions of TBFOI-OWTs, as shown in Figure 4 and

Table 4. Sensors used in the model tests.

	Type	Capacity	Resolution
Wave probe	JM5900	±500 mm	0.02 mm
Accelerometer	AI050	10 g	0.0001 g
Underwater tension sensor	LA1	300 N	0.05 N
6-DOF motion capture system	Qualisys 700+	-	0.1 mm

.

3.2.1. Twin-Barge Model

For the twin-barge model, considering the large size of the model and the low center of gravity, fiberglass material was used to make the model. The model is fabricated according to the prototype dimensions and geometric similarity. A clay model is created first, and then layers of resin and fiber cloth are applied to the clay mold, with a total of 10 layers. After the outer shell of the ship model is accomplished, it is demolded, cleaned, weighed, balanced, and tested for watertightness. The twin-barge model is shown in Figure 5, and the specific parameters are shown in

Table 5. Parameters of twin-barge model.

	Full Scale	1:50 Scale (Target)	1:50 Scale (Achieved)	Deviation
Length overall (m)	171	3.42	3.42	0
Molded breadth (m)	35	0.70	0.70	0
Molded depth (m)	9	0.18	0.18	0
Mass of black barge (kg)	11,161,210	87.11	87.39	+0.31%
Mass of yellow barge (kg)	11,161,210	87.11	87.76	+0.75%
Center of gravity of black barge (m)	6.40	0.13	0.13	−1.93%
Center of gravity of yellow barge (m)	6.40	0.13	0.13	−2.00%

.

3.2.2. Topside Module Model

Although the geometric dimensions of the prototype have been determined, directly scaling down the prototype based on geometric similarity presents challenges. For example, the diameter and wall thickness of the truss carriage frame members are relatively small, making it impossible to fabricate them through welding. Additionally, the truss carriage frame, cantilever beams, and connectors are complex and difficult to construct. Therefore, it is necessary to simplify and modify the topside module to ensure that the model and prototype adhere to the similarity. Since this part of the structure is located above the water, its shape has almost no influence on the overall hydrodynamic performance of the structure.

The truss carriage frame is fabricated using 22 × 18 mm aluminum alloy tubes, with the main external dimensions meeting the geometric similarity of the prototype. The deck support frame is replaced by 4 mm thick steel plates, with the main dimensions in the length and width directions matching the geometric similarity of the prototype. The cantilever beams are made of 3 mm thick steel plates, with a central circular hole of a 60 mm diameter for securing the tower. The differences between the model’s mass and center of gravity and the target values would be compensated by adding ballast. The concept and physical images of the upper module are shown in Figure 6, and the specific parameters are listed in

Table 6. Parameters of topside module model.

	Size (mm)	Material	Mass (kg)	Center of Gravity (m)
Truss carriage frame (OD × ID)	22 × 18	Aluminum alloy	12.721	0.406
Deck support frame	1324 × 952 × 4	Steel	39.578	0.002
Cantilever beams	1260 × 310 × 3	Steel	9.132	0.028
Ballast (L × B × H)	678 × 318 × 8 722 × 362 × 2	Steel	17.613	0.223

and

Table 7. Parameters and error of topside module model.

Parameters	Full Scale	1:50 Scale (Target)	1:50 Scale (Achieved)	Deviation
Mass (kg)	11,490,581.70	89.68	88.18	−1.68%
Center of gravity (m)	5.42	0.11	0.11	+1.85%
Roll moment of inertia about the CG (kg·m2)	3.24 × 109	10.11	10.65	+5.31%
Pitch moment of inertia about the CG (kg·m2)	5.92 × 109	18.47	18.13	−1.85%
Yaw moment of inertia about the CG (kg·m2)	6.86 × 109	21.43	21.05	−1.77%

. Note that the center of gravity in the table has the deck as the reference. It can be seen from Table 7 that the mass properties of the model made are quite close to the target values.

3.2.3. Tower and Wind Turbines Model

Since this experimental study focuses on the motion of TBFOI-OWTs in waves, the wind turbine model mainly satisfies the similarity relationship in terms of the mass and center of gravity. The target total weight of the tower and wind turbine model is 7.02 kg. The model is adjusted to meet this target by using a connection flange and adding ballast.

For ease of fabrication, the tower is constructed using aluminum alloy tubes, with the specifications selected from the available standards that meet the design requirements. During the design, the height of the tower remains unchanged while its specifications (diameter and wall thickness) are adjusted, along with the ballast, to ensure that the overall mass and center of gravity height of the tower match the theoretical values of the model. The main body of the tower is made of 6061 aluminum alloy tubes, with an outer diameter of 60 mm, a wall thickness of 10 mm, a length of 2760 mm, a weight of 11.71 kg, and a center of gravity height of 1380 mm. Lead wire, with a mass of 1.17 kg, is uniformly wrapped around the tower 941 mm above the base for additional ballast. Once the model is assembled, further adjustments can be made to ensure that the weight and center of gravity closely match the target values. The wind turbine and tower model are shown in Figure 7, and specific parameters are listed in

Table 8. Parameters of tower and wind turbine model.

Parameters	Full Scale	1:50 Scale (Target)	1:50 Scale (Achieved)	Deviation
Mass of wind turbine1 (kg)	900,000	7.02	7.00	−0.30%
Mass of wind turbine2 (kg)	900,000	7.02	6.98	−0.60%
Center of gravity of wind turbine (m)	143	2.86	2.86	0
Mass of tower1 (kg)	1,650,000	12.88	12.92	+0.30%
Mass of tower2 (kg)	1,650,000	12.88	12.94	0.05%
Center of gravity of tower (m)	67	1.34	1.34	0
Overall mass (kg)	2,550,000	19.90	19.92	+0.10%
Overall center of gravity (m)	93.824	1.87	1.87	−0.11%

. Note that the center of gravity in the table also has the deck as the reference. The two towers are fixed to the truss carriage frame in Figure 6.

3.2.4. Mooring System Model

In this experiment, a catenary mooring system is employed. However, due to limited basin dimensions, a truncated mooring system is used [41]. Since the prototype mooring system provides distinct restoring stiffnesses in the x and y directions, it is necessary to design two sets of truncated mooring systems. The head sea truncated mooring system must ensure that the restoring forces in terms of surge, heave, and pitch match those of the prototype mooring system. Similarly, the beam sea truncated mooring system must ensure that the restoring forces in terms of sway, heave, and roll are consistent with those of the prototype mooring system.

To accommodate both head and beam seas during the experiment, four identical truncated mooring chains are symmetrically arranged. The specific parameters for the head sea and beam sea truncated mooring systems are detailed in

Table 9. Parameters of head sea truncated mooring system.

Parameters	Upper	Lower
Link diameter (mm)	303.46	506.49
Equivalent diameter (mm)	546.29	911.77
Length (m)	335.21	186.46
Mass density in air (kg/m)	1839.93	5125.39
Mass density in water (kg/m)	1624.02	4523.95
Equivalent stiffness (N)	1.24 × 1010	3.46 × 1010

and

Table 10. Parameters of beam sea truncated mooring system.

Parameters	Upper	Lower
Link diameter (mm)	261.17	499.01
Equivalent diameter (mm)	477.14	898.30
Length (m)	369.71	126.09
Mass density in air (kg/m)	1362.77	4975.15
Mass density in water (kg/m)	1202.85	4931.33
Equivalent stiffness (N)	9.21 × 109	3.36 × 1010

. Figure 8 illustrates the comparison between the restoring forces of the truncated mooring system and the original mooring system in full scale. As shown in this figure, within an offset range of 0 to 10 m, the two restoring forces match well, with a maximum error of less than 1%. The mooring arrangement is depicted in Figure 9.

Based on the truncated mooring system parameters and the Froude similarity, the mass properties of the model mooring system are calculated. Mooring chains with mass properties close to the target are procured, and combinations of two mooring chains are employed to make the weight of the mooring chain model approach the desired value. Figure 10 illustrates the measurement of the mooring chain’s length and weight, while

Table 11. Parameters of mooring system model.

	Parameters	1:50 Scale (Target)	1:50 Scale (Achieved)	Deviation
Head sea	Mass density of upper (kg/m)	0.72	0.72	0.84%
	Mass density of lower (kg/m)	2.00	2.07	3.65%
Beam sea	Mass density of upper (kg/m)	0.53	0.53	0.38%
	Mass density of lower (kg/m)	1.94	1.90	−2.01%

presents the discrepancies between the actual values of the mooring chain model and the target values.

4. Numerical Simulation Methodology

The numerical simulation methodology for TBFOI-OWTs is illustrated in Figure 11. The hydrodynamic analysis is conducted using WAMIT [42], which includes the evaluation of the hydrostatic stiffness, damping, added mass, and wave forces. In addition, the second-order mean drift forces that affect the mooring performance are also calculated. To enhance the computational accuracy, a higher-order panel model is employed when solving the hydrodynamics of TBFOI-OWTs in WAMIT. Since two barges are symmetrical in layout, only a half-model needs to be established for analysis, significantly reducing the computational resources and time. The higher-order panel model for the twin barge is shown in Figure 12.

In this study, the wave loads acting on the floating foundation are calculated using the potential flow theory while the wind load is ignored. The floating foundation is assumed to be a 6-DOF rigid body, and the wave force response amplitude operator (RAO), added mass coefficients, radiation damping coefficients, and hydrostatic coefficients of the “vessel” are computed based on the wetted surface model using the WAMIT higher-order panel method. As shown in Figure 11, these frequency-domain quantities are exported to ORCAFLEX [43] for the time-domain calculation. ORCAFLEX is a nonlinear time-domain finite element software developed by Orcina, primarily used for the static and dynamic response analysis of marine structures.

In ORCAFLEX, the mooring chains are simulated using a line model, with the initial positions and mechanical parameters of each mooring chain specified in Table 2. Both the bending and torsional stiffness of the mooring chains are considered negligible, with only axial stiffness taken into account, and the mooring chains are not allowed to have compression. The hydrodynamic forces on the mooring chains are calculated using the Morrison equation, with C_m set to 1.0 and C_d set to 2.4 [44]. The dynamic response of the mooring chains is computed using the finite element method. The time-domain simulation model is illustrated in Figure 13.

5. Load Cases

To investigate the hydrodynamic performance of TBFOI-OWTs, a series of load cases (LCs) are considered, including free decay tests, regular wave tests, and random wave tests. Finally, the influence of the gap distance on the motion RAOs is explored. The regular waves are defined by the wave height H and wave period T. The random waves are represented by the JONSWAP spectra, defined by H_s, T_p, and γ, which represent the significant wave height, spectral peak period, and peak enhancement factor.

For LC1, as shown in

Table 12. Load case 1 (LC1): free decay tests.

Number	Description	DOF
LC1	Unmoored state	Heave, roll, and pitch
	Head sea moored state	Surge, sway, heave, roll, pitch, and yaw
	Beam sea moored state	Surge, sway, heave, roll, pitch, and yaw

, in the unmoored state, the model only exhibits restoring forces in three degrees of freedom: heave, roll, and pitch. Therefore, during the free decay tests, only these three degrees of freedom are considered. By contrast, in the head sea or beam sea moored state, restoring forces are present in all six degrees of freedom, necessitating the consideration of all six degrees during the free decay tests.

For LC2, as shown in

Table 13. Load case 1 (LC2): motion RAO tests.

Number	Description	Full Scale		Model Tests
		H/Hs (m)	T (s)	H/Hs (m)	T (s)
LC2	Regular wave	2.50	5.00–20.00	0.05	0.71–2.83
	White noise wave	2.50	5.00–20.00	0.05	0.71–2.83
	White noise wave	1.25	5.00–20.00	0.025	0.71–2.83

, a series of model tests are conducted under regular wave conditions, primarily to assess the model’s motion response under waves of different periods and heights, including both head seas and beam seas. In the head sea moored state, the focus is on the model’s motions in surge, heave, and pitch, while in the beam sea moored state, attention is directed toward the model’s motions in sway, heave, and roll. Additionally, white noise wave tests are performed to validate the motion RAOs obtained from the series of regular waves and to compare them with the numerical simulation results.

For LC3–LC6, as shown in

Table 14. Load case 3 and load case 4 (LC3 and LC4): regular wave tests.

Number	Description	Full Scale		Model Tests
		H (m)	T (s)	H (m)	T (s)
LC3	Head sea	4.00	6.30, 7.40, 10.47, 16.20	0.08	0.89, 1.05, 1.48, 2.29
LC4	Beam sea	4.00	6.30, 7.40, 16.20	0.08	0.89, 1.05, 2.29

and

Table 15. Load case 5 and load case 6 (LC5 and LC6): random wave tests.

Number	Description	Full Scale		Model Tests		$\mathit{\gamma }$
		Hs (m)	Tp (s)	Hs (m)	Tp (s)
LC5	Head sea	2.50	7.00, 8.20, 11.60, 18.00	0.05	0.99, 1.16, 1.64, 2.55	1.90, 1.00, 1.00, 1.00
LC6	Beam sea	2.50	7.00, 8.20, 18.00	0.05	0.99, 1.16, 2.55	1.90, 1.00, 1.00

, tests are conducted in both head seas and beam seas under regular and random conditions, including typical wind waves, swell waves, and waves with characteristic wavelengths. In ocean engineering, natural waves typically consist of both wind waves and swells. The periods of wind waves are usually determined based on the SNAME guidelines [45]. In addition to wind waves and swells, certain wave periods such as those corresponding to the characteristic length of the floaters can also induce significant responses. SNAME recommends conducting both deterministic analyses under regular waves and stochastic analyses under random waves, and provides conversion relationships among the relevant parameters. Regular wave periods in Table 14 correspond to 0.9 times the peak wave periods in Table 15, i.e., T_max = 0.9 T_p, and regular wave heights in Table 14 correspond to 1.6 times more significant wave heights in Table 15, i.e., H_max = 1.6 H_s. The focus of tests is on the 6-DOF motions of TBFOI-OWTs. Each regular wave scenario is repeated twice, with each trial lasting 5 min in model tests, and the statistical values are averaged over both trials. During the random wave experiments, six trials are conducted, with each random wave test lasting 15 min and of distinct random seeds. In data post processing, transient responses were removed, allowing for a comprehensive assessment of the dynamic responses under random sea states. Thus, the six 15 min random wave tests after scaling correspond to three random sea states whose durations are all 3 h.

For LC7, as shown in

Table 16. Load case 7 (LC7): different distances.

Number	Description	Full Scale		Model Tests
		Hs (m)	T (s)	Hs (m)	T (s)
LC7	8 m, 12 m, 16 m	2.50	5.00–20.00	0.05	0.71–2.83

, motion RAO tests are primarily conducted to analyze the influence of varying gap distances under white noise waves. The tests under white noise waves consider two scenarios: the head sea and beam sea states. Each white noise scenario lasts for 15 min. After the transient response is removed, the steady motions of the last 13 min are analyzed.

6. Preliminary Calibration Experiment

6.1. Validation of Wave Fields

The accuracy of the wave field is crucial for the experimental results, and three wave probes are used to measure the wave elevations. For the regular wave with H = 50.00 mm and T = 2.26 s (corresponding to a prototype H of 2.50 m and T of 16.00 s in LC2), the comparison between the measured and expected wave elevations is shown in Figure 14. For the random wave following the JONSWAP spectra with H_s = 50.00 mm and T_p = 2.55 s (corresponding to a prototype H_s of 2.50 m and T_p of 18.00 s in LC5), the comparison between the measured and expected wave elevations is shown in Figure 15. For the white noise wave with H_s = H_rms of 50 mm, with period ranging from 0.71 s to 2.83 s (corresponding to a prototype H_s of 2.50 m, period ranging from 5.00 s to 20.00 s in LC2), the comparison between the measured and expected wave elevations is shown in Figure 16. It can be seen that all waves generated in basins agree well with the target requirements.

6.2. Validation of Mooring Lines’ Tension

In still water, the moored model is laterally displaced in the horizontal direction by 40 mm, 80 mm, 120 mm, 160 mm, and 200 mm (corresponding to prototype scales of 0 m, 2 m, 4 m, 6 m, 8 m, and 10 m, respectively). The motion capture system is used to precisely control the distance of the model. After the model is pushed to these specified positions, it is held stationary for the stable tension in the mooring chain to be measured and subsequently averaged over a specified duration.

The mooring chain tension is shown in Figure 17. It can be observed that the mooring chain tension of the fabricated truncated mooring system aligns well with the target values, with a maximum error of no more than 5%, indicating that the truncated mooring system serves as an appropriate alternative to the prototype mooring system. This deviation might be attributed to errors of mooring chain fabrication and anchor locations.

6.3. Free Decay Tests

During the free decay tests, an initial displacement is applied to the model in the degrees of freedom of interest. The Qualisys motion capture system records the model’s trajectory, allowing for the determination of the natural period from the free decay motion time histories. Three states are considered in the free decay tests: unmoored, head sea moored, and beam sea moored.

In the unmoored state, heave, roll, and pitch are considered during the tests, with three appropriate initial displacements incrementally applied for each degree. In the moored states (both head sea and beam sea), restoring forces are present in all six degrees of freedom. Similarly to the unmoored state, three initial displacements are also considered for each degree, incrementally applied. The model’s natural frequency is determined using the Fourier transform of the time histories of the measured motion. Figure 18 illustrates the surge free decay time history curve with an initial displacement of 6.18 m in the head sea moored state and the roll free decay time history curve with an initial displacement of 3.02 deg in the beam sea moored state, showing good agreement between the numerical simulation results using ORCAFLEX and the experimental outcomes.

Figure 19 presents the time histories and power spectra of the surge free decay tests of TBFOI-OWTs in the head sea moored state. Surge1, Surge2, and Surge3 represent three tests conducted with different initial displacements. All three spectral curves have an identical peak at the natural frequency of the surge, i.e., f_surge = 0.18 Hz or T_surge =5.65 s. At the same time, we can calculate the damping ratios of three initial displacements as 0.75%, 1.11%, and 4.17%, respectively, according to the surge free decay time histories. Note that the damping levels not only depend on amplitudes, but also on the model scale [46]. Similarly, Figure 20 illustrates the time histories and power spectra of the pitch free decay tests in the head sea moored state. The time histories do not exhibit an obvious exponential decay. Such behavior is attributed to the coupling of pitch with heave and roll motions, as evidenced by the multiple peaks visible in the power spectra. This coupling occurs because of the geometric asymmetry of the bow and stern, which also causes heave motion to unavoidably take place.

Figure 21 displays the time histories and power spectra of the sway free decay tests of TBFOI-OWTs in the beam sea moored state. The power spectra contain two distinct peaks, corresponding to f_sway and f_yaw. This coupling also occurs because of the asymmetry of the bow and stern, as well as the uneven action on the floater. Likewise, Figure 22 presents the time histories and power spectra for the roll free decay tests of TBFOI-OWTs in the beam sea moored state, where the power spectra show a single peak corresponding to f_roll = 0.10 Hz or T_roll = 10.48 s.

Table 17. Natural periods obtained from free decay model tests and their deviations against ORCAFLEX simulations (T_simulation) (Unit: s).

State	Freedom	Model Test				Tsimulation	Deviation
		T1	T2	T3	Taverage
Unmoored state	Heave	5.59	5.60	5.68	5.62	5.64	−0.35%
	Roll	10.40	10.38	10.41	10.40	10.21	1.86%
	Pitch	6.14	6.07	5.82	6.01	6.05	−0.66%
Head sea moored state	Surge	77.54	78.01	78.76	78.10	76.20	2.49%
	Heave	5.66	5.66	5.66	5.66	5.65	0.19%
	Pitch	6.06	6.08	6.07	6.07	6.08	−0.12%
Beam sea moored state	Sway	61.46	60.66	60.87	61.00	58.51	4.24%
	Heave	5.81	5.81	5.81	5.81	5.65	2.80%
	Roll	10.39	10.40	10.39	10.39	10.48	−0.83%

presents the natural periods of TBFOI-OWT unmoored, head sea moored, and beam sea moored tests. T₁, T₂, and T₃ correspond to different initial displacements in model tests, while T_average represents the average value and T_simulation represents the natural periods obtained from the numerical results via ORCAFLEX.

7. Results and Discussion

7.1. Motion RAO Tests

The model tests under a series of regular waves primarily evaluate the motion RAOs of the model under varying periods and wave heights. During these tests, it is essential to ensure that the interval between two trials is sufficiently long for the water surface to calm down. When the model is in head seas, the wave propagation direction aligns with the model’s x-axis in Figure 1. Hence, in a moored state, the focus is primarily on the model’s motions in surge, heave, and pitch motions. Conversely, when the model is subjected to beam seas, the wave propagation direction aligns with the y-axis, concentrating on the model’s motions in sway, heave, and roll motions.

The model’s motion RAOs can not only be obtained from the series of regular waves, but also from tests under white noise waves of a finite bandwidth. Assuming the entire system is linear, the model’s motion RAOs can be derived from the motion response spectra and the input wave spectra according to Equation (1).

$$S_{response}(f)=S_{wave}(f){\left|RAO\right|}^2$$

(1)

The tests under white noise waves also consider two states: head seas and beam seas. Numerically, RAOs can be obtained using frequency-domain potential flow software like WAMIT and ORCAWAVE, or the time-domain module in ORCAFLEX. The corresponding numerical results, along with the experimental results from the series of regular waves and white noise waves, are plotted on a single graph. Figure 23 illustrates the surge RAO, heave RAO, and pitch RAO in head seas, revealing that the motion RAOs obtained from all software are very close to each other, as are those derived from the series of regular waves and white noise waves. The numerical simulation results for surge and pitch RAOs align well with the experimental results, while some discrepancies are observed for the heave RAO under long-period waves. Figure 24 presents the sway RAO, heave RAO, and roll RAO in beam seas. The data indicate that the motion RAOs obtained from all software closely match each other, and those derived from the series of regular waves and the white noise waves are also in close agreement. The numerical simulation results for roll RAO correspond well with the experimental results, whereas there are some deviations for the sway and heave RAOs.

The deviation in the heave RAO arises in that the numerical RAOs calculated using the potential flow theory are derived from only the wetted surface of TBFOI-OWTs, which does not account for the varying surface effect and water impact. But, in real experiments, when the waves propagate toward the model, they exert slamming from time to time on the deck support framework connecting the two barges (represented by a steel plate in the tests in Figure 4). Such slamming amplifies the heave motion and increases the transient behavior of heave. Consequently, the experimental heave RAO is greater than that obtained from the numerical simulations.

7.2. Regular Wave Tests for H = 4.00 m

LC3 and LC4, respectively, correspond to head seas and beam seas of severity IV on the China scale [47], i.e., a maximum H_s of 2.50 m in random waves or an equivalent wave height H_max of 4 m in regular wave trains. Such severity is the highest installation scenario examined in this study. The head sea state includes four wave periods, while the beam sea state consists of three periods (Table 14). Figure 25 presents a comparison of the numerical simulation results via ORCAFLEX and the experimental results in head seas for T = 16.20 s. The graph shows good consistency between the two outcomes, with identical periods and slightly greater amplitudes in experiments.

Figure 26 displays the maximum, average, and minimum values for surge, heave, and pitch responses in head seas with H = 4.00 m and T = 6.30 s, T = 7.40 s, T = 10.47 s, and T = 16.20 s. For example, in Figure 26a, the surge motion at T = 7.40 s oscillates at the mean offset of −4.21 m, with the oscillation range only from −4.38 m (Min) to −4.21 m (Max). Thereby, the data indicate that the greatest surge takes place at T = 7.40 s, while the greatest heave of 2.32 m occurs at T = 16.20 s, and the largest pitch of 1.90 deg is at T = 10.47 s. Figure 27 presents the maximum, average, and minimum values for sway, heave, and roll responses of TBFOI-OWTs in beam seas with H = 4.00 m at periods T = 6.30 s, T = 7.40 s, and T = 16.20 s. From this figure, it can be observed that the greatest sway of 6.65 m is at T = 7.40 s, while the greatest heave of 2.19 m and roll of 2.32 deg are highest at T = 16.20 s.

7.3. Random Wave Tests

LC5 and LC6 correspond to random waves in head seas and beam seas, respectively. The head sea state includes four T_p, while the beam sea state encompasses three T_p.

Figure 28 compares the time histories and power spectra of surge, heave, and pitch responses in head seas with H_s = 2.50 m and T_p = 18.00 s. The graph demonstrates good consistency between the numerical simulation results and the experimental results. It is worthwhile to point out that though the random seeds adopted in numerical and experimental studies differ, the resultant spectra are in good agreement, especially for moderate and long waves.

Table 18. Comparisons of response statistics in head seas: regular waves versus random waves.

	Load Case	Max	Mean	Min
Surge (m)	Hs = 2.50 m, Tp = 7.00 s	2.79	−0.57	−4.43
	H = 4.00 m, T = 6.30 s	−2.23	−2.45	−2.66
	Hs = 2.50 m, Tp = 8.20 s	3.05	−0.56	−4.42
	H = 4.00 m, T = 7.40 s	−4.01	−4.21	−4.38
	Hs = 2.50 m, Tp = 11.60 s	2.81	−0.35	−3.70
	H = 4.00 m, T = 10.47 s	−1.01	−1.91	−2.80
	Hs = 2.50 m, Tp = 18.00 s	2.16	−0.09	−2.62
	H = 4.00 m, T = 16.20 s	2.34	0.00	−2.35
Heave (m)	Hs = 2.50 m, Tp = 7.00 s	0.38	−0.01	−0.39
	H = 4.00 m, T = 6.30 s	0.25	−0.17	−0.57
	Hs = 2.50 m, Tp = 8.20 s	0.60	0.00	−0.52
	H = 4.00 m, T = 7.40 s	0.45	0.03	−0.38
	Hs = 2.50 m, Tp = 11.60 s	1.48	0.01	−1.24
	H = 4.00 m, T = 10.47 s	1.21	0.06	−0.99
	Hs = 2.50 m, Tp = 18.00 s	2.13	0.01	−2.01
	H = 4.00 m, T = 16.20 s	2.32	0.05	−2.18
Pitch (deg)	Hs = 2.50 m, Tp = 7.00 s	0.63	0.01	−0.62
	H = 4.00 m, T = 6.30 s	0.36	−0.05	−0.44
	Hs = 2.50 m, Tp = 8.20 s	0.90	0.00	−0.87
	H = 4.00 m, T = 7.40 s	0.76	0.09	−0.61
	Hs = 2.50 m, Tp = 11.60 s	1.82	0.00	−1.71
	H = 4.00 m, T = 10.47 s	1.90	0.02	−1.72
	Hs = 2.50 m, Tp = 18.00 s	1.86	0.00	−1.85
	H = 4.00 m, T = 16.20 s	1.83	0.00	−1.79

illustrates the comparisons of statistical values between regular waves and random waves in head seas. The data reveal that the greatest surge of 4.43 m occurs at T_p = 7.00 s, while the greatest heave of 2.13 m is at T_p = 11.60 s, and the greatest of 1.86 deg pitch is at T_p = 18.00 s.

Unlike the regular wave situation in the last section, the surge motions in random waves have greater ranges and amplitudes. For example, [−4.43 m 2.79 m] is for T_p = 7.00 s in random waves, but its counterpart is [−2.66 m −2.23 m] for T = 6.30 s in regular waves. All these demonstrate that the mooring analysis cannot depend on regular waves only, and more importantly, that the random nature of realistic waves is less favorable to float-over installations. On the other hand, in long waves of T = 16.20 s and T_p = 18.00 s, the two ranges for surge are close to each other, as swell dominates the wave train. This again reveals that for float-over installation in wind-dominated seas, one should be more cautious of the influence of stochastic waves.

The response spectra shown in Figure 29 exhibit distinct wave frequency peaks and spectral peaks corresponding to the natural periods of TBFOI-OWTs, indicating that the motions are primarily dominated by both f_wave and natural frequencies of motions. It can also be deduced that in scale IV sea states, as T_p increases, the wave frequency motions increase. But, for float-over installations, the unfavorable level of a sea state still depends on the aforementioned total response (wave frequency and natural frequency inclusive) and its extreme value.

Table 19. Comparisons of response statistics in beam seas: regular waves versus random waves.

	Load Case	Max	Mean	Min
Sway (m)	Hs = 2.50 m, Tp = 7.00 s	7.85	1.13	−4.89
	H = 4.00 m, T = 6.30 s	5.14	3.77	2.16
	Hs = 2.50 m, Tp = 8.20 s	7.05	1.01	−4.88
	H = 4.00 m, T = 7.40 s	6.65	6.19	5.60
	Hs = 2.50 m, Tp = 18.00 s	3.18	0.13	−2.83
	H = 4.00 m, T = 16.20 s	2.50	0.01	−2.48
Heave (m)	Hs = 2.50 m, Tp = 7.00 s	0.61	−0.04	−0.67
	H = 4.00 m, T = 6.30 s	0.69	−0.09	−0.89
	Hs = 2.50 m, Tp = 8.20 s	0.81	0.00	−0.71
	H = 4.00 m, T = 7.40 s	0.14	−0.04	−0.21
	Hs = 2.50 m, Tp = 18.00 s	2.17	0.01	−2.15
	H = 4.00 m, T = 16.20 s	2.19	0.05	−2.13
Roll (deg)	Hs = 2.50 m, Tp = 7.00 s	1.75	0.09	−1.40
	H = 4.00 m, T = 6.30 s	0.80	0.20	−0.33
	Hs = 2.50 m, Tp = 8.20 s	3.00	0.06	−2.86
	H = 4.00 m, T = 7.40 s	1.34	0.34	−0.77
	Hs = 2.50 m, Tp = 18.00 s	3.75	0.00	−3.62
	H = 4.00 m, T = 16.20 s	2.32	0.00	−2.26

and Figure 30 present similar information on beam seas with H_s = 2.50 m at T_p = 7.00 s, T_p = 8.20 s, and T_p = 18.00 s. It can be perceived that the greatest sway of 7.85 m is at T_p = 7.00 s, while the greatest heave of 2.17 m and roll of 3.75 deg are at T_p = 18.00 s. The response spectra again display clear wave frequency peaks alongside those corresponding to TBFOI-OWTs’ natural periods, reaffirming that the motions of TBFOI-OWTs are primarily subjected to both f_wave and natural frequencies of motions. Moreover, the statistics once again reflect that the sway and roll motions have greater ranges and amplitudes in random waves than in regular waves. A conservative estimation of the safety during float-over installation should rely on random analysis.

Based on the above results, it can be deduced that the sway and roll motions of the twin-barge system in beam seas are significantly larger than the surge and pitch motions in head seas. Therefore, during actual float-over installation operations, it is advisable to avoid exposing the twin-barge system to beam seas whenever possible. In head seas, the heave and pitch motions under long-period waves are considerably greater than those under short-period waves. As such, random waves with H_s = 2.50 m and T_p = 18.00 s in head seas are identified as the most unfavorable scenario. Special attention should thus be paid to the presence of long-period swell during the installation process.

7.4. Influence of the Gap Distance Between Two Barges

The gap distance is a critical factor influencing the hydrodynamic performance of TBFOI-OWTs. Because in the numerical analysis of a gap, the artificial damping has to be applied and the level of damping is quite subjective [48], it is necessary to quantitatively investigate the gap’s influence preferably through experimental study. Figure 31 and Figure 32 present the derived motion RAOs in model tests for each degree of freedom under both moored states.

From Figure 31, it is evident that as the incident wave period increases, the surge RAO shows an increasing trend, with very close surge RAOs observed across varying gap distances. The heave RAO also increases with longer incident wave periods, while exhibiting a small peak at short periods. Additionally, as the distance between the two barges increases, the period corresponding to this peak slightly increases. When T exceeds 10.00 s, the heave RAOs across varying distances merge. For the pitch RAO, a small peak is observed at short periods, with the period also slightly increasing with a greater gap distance. This is because a bigger distance would trigger a larger added mass of water. Again, when T exceeds 10.00 s, the pitch RAOs for the three gaps become consistent with each other. When the model is in head seas, waves can propagate through the gap between the two barges, resulting in minimal energy dissipated by the gap. Therefore, the consistency of RAOs across different gaps can be explained.

Differently from the head seas, as shown in Figure 32, the sway RAO decreases with increasing gap distance within the period range of 11.00–17.00 s, while for other ranges of T, the differences in sway RAOs across various distances are very mild. With increasing T, all roll RAOs increase first and then decrease. For T between 10.00 s and 15.00 s, they decrease with increasing gaps, while for other ranges, the roll RAOs across varying distance are more or less the same. When T approaches the roll natural period (10.48 s in Table 17), resonance occurs. The gap in this situation forms a damping pool [49], where oscillatory water motions are trapped and dissipated. As the gap distance increases, the volume and surface area of this damping pool expand, which enhances its ability to dissipate the energy associated with wave-induced fluid motions within the gap. This increased damping effect helps suppress excessive roll responses by attenuating the resonant energy through viscous dissipation and wave reflection inside the enlarged gap. In contrast, when the incident wave period deviates significantly from the roll natural period, resonance is less likely to occur, and the influence of the gap size becomes less pronounced. Under such non-resonant conditions, the hydrodynamic responses across different gap configurations exhibit similar trends and magnitudes.

For T in the range of 5.00–11.00 s, it can be seen that the heave RAO increases with a greater gap, because the larger the distance between the two barges, the greater the area exposed to slamming. For other ranges, the heave RAOs across three gaps remain largely consistent.

8. Motion Prediction

In the float-over installation process, the motion characteristics of the TBFOI-OWT play a decisive role in the precise alignment of the upper and lower connectors. Accurate motion prediction is crucial for ensuring successful installation and reducing operational risks and efforts. Therefore, this section focuses on the short-time motion prediction of the TBFOI-OWT using the Long Short-Term Memory (LSTM) neural network as the core methodology.

8.1. Introduction to LSTM

LSTM improves upon the traditional recurrent neural network (RNN) by introducing specialized memory cells to store long-term information and designing a precise gating mechanism to control information updating and forgetting. This gating structure consists of an input gate, a forget gate, and an output gate, which work together to selectively update, delete, or output information. As a result, LSTM optimizes the flow of information, enhances the ability to learn from long sequential data, and overcomes the vanishing and exploding gradient problems encountered in standard RNN when processing long time series like seakeeping motions.

Figure 33 illustrates the information propagation mechanism of the LSTM model over time steps, revealing its core structure and information flow in sequence data processing. In the figure, the blue rectangles represent LSTM cells, which process input data at the current time step while maintaining long-term memory to capture dependencies within the sequence. The red circles denote input data, which are fed into the network at each time step. The green circles represent output information, which is computed by the LSTM unit at the current time step and passed to the next time step to sustain the dynamic information flow in the time series. The long-term memory state, illustrated by the purple arrows, is responsible for storing and managing historical information, enabling the model to effectively learn and retain important long-term dependencies without losing information due to the vanishing gradient problem. The brown arrows indicate the data propagation direction across time steps, demonstrating how LSTM unfolds over time, where the current state is influenced not only by the present input, but also by past information.

Figure 34 illustrates the detailed structure of an LSTM cell, which primarily consists of the forget gate, input gate, output gate, and cell state. The forget gate dynamically regulates the retention and discarding of information, ensuring the model’s ability to efficiently capture long-term dependencies. The input gate controls the extent to which new information is integrated into the cell state, determining which new data should be incorporated to update long-term memory. The cell state serves as the primary memory component, storing and transmitting crucial information across long sequences. The output gate determines which parts of the cell state should be output to the next time step or utilized for the current time step’s prediction, based on the current cell state and input information. Equations (2)–(7) show the calculation relationship in the LSTM cell.

$$f_t=\sigma (W_f•\left[h_{t-1},x_t\right]+b_f)$$

(2)

where f_t is the output of the forget gate, $\sigma$ is the sigmoid activation function, W_f is the weight value, h_t−1 is the output of the previous step, x_t is the input for the current step, and b_f is the bias amount.

$$i_t=\sigma (W_i\left[h_{t-1},x_t\right]+b_i)$$

(3)

where i_t is the output of the sigmoid activation function in the input gate, W_i is the weight value, and b_i is the bias amount.

$$\widetilde{c_t}=\tanh (W_c\left[h_{t-1},x_t\right]+b_c)$$

(4)

where $\widetilde{c_t}$ is the output of the tanh function activation function in the input gate, W_c is the weight value, and b_c is the bias amount.

$$c_t=f_t•c_{t-1}+i_t•\widetilde{c_t}$$

(5)

where c_t is the cell state of the current step and c_t−1 is the cell state of the previous step.

$$o_t=\sigma (W_o[h_{t-1},x_t]+b_o)$$

(6)

where o_t is the output of the sigmoid activation function in the output gate, W_o is the weight value, and b_o is the bias amount.

$$h_t=o_t•\tanh (c_t)$$

(7)

where h_t is the output of the output gate.

8.2. Introduction to WOA

The Whale Optimization Algorithm (WOA) is a nature-inspired heuristic optimization algorithm stemming from the hunting behavior of humpback whales. It consists of four key phases: population initialization, encircling prey, bubble-net feeding (including the shrinking encircling mechanism and spiral updating strategy), and the search phase.

In the population initialization phase, the algorithm randomly generates an initial population to ensure the solution’s diversity. During the encircling prey phase, whale individuals dynamically adjust their positions to gradually approach the optimal solution, in order to form the cooperative encirclement behavior of whales. The bubble-net feeding phase further refines the hunting strategy according to Equation (8): the shrinking encircling mechanism contracts the search space globally, while the spiral updating strategy enhances the local search capability by imitating the spiral motion of whales around their prey. In the search phase, whales perform random movements to enhance global exploration, to ensure that the algorithm avoids local optima and improves the optimization performance. The computational process is illustrated in Figure 35.

$$X(t+1)=\left\{\begin{array}{cc}{X}^{*}(t)-A•D,& rand<0.5\\ X^{*}(t)+D^{*}•e^{bl}•\cos (2\pi l),& rand\ge 0.5\end{array}\right.$$

(8)

where X(t + 1) is the updated location of the whale population, X*(t) is the current optimal solution position, A is the coefficient vector, D is the enclosing step size, D* is the distance between a whale individual and the optimal whale individual, b is the shape parameter of the spiral line, and l is a random number in the range of −1 to 1.

8.3. Model Construction

In MATLAB2023b, an LSTM neural network model can be constructed with key components, including an input layer, an LSTM layer, a ReLU activation layer, a fully connected layer, and a regression layer, as shown in Figure 36. The input layer serves as the entry point of the network to receive external input data. The LSTM layer is the core of the network, responsible for processing sequential data. The activation layer introduces nonlinearity to enhance the model’s expressive capacity and to learn complex features. The fully connected layer maps the features extracted by the LSTM layer to the final output space. Depending on the task, the output layer can be a regression layer for continuous value predictions or a classification layer for probability distribution outputs.

In the present study, the dataset of the LSTM neural network selects the heave motion data in head seas, with H_s = 2.50 m and T_p = 18.00 s collected in the experiment, as shown in Figure 37. Before training the LSTM neural network, data preprocessing is required. In the experiment, each random wave condition underwent six repeated trials, each lasting 15 min with different seeds. After removing the transient response phase, six segments of 13 min experimental data were obtained. Then, the maximum and minimum normalization processing was carried out.

Due to the high sampling frequency, the large dataset may impose significant computational demands. Therefore, before forming the dataset, downsampling is applied to reduce the data volume while preserving the motion waveform, resulting in 6 × 8000 data points. The time interval between the two data points is 0.57 s at the prototype scale. The last portion of each segment is designated as the test set, while the remaining data are used for training. After training a model on the combined training set, predictions are made on each of the six segments individually, and the results are compared with the true motions.

To compare the different prediction approaches, two forecasting models are established in this study. The first model is a stepwise iterative model, where every 100 data points are used to predict one future point. The predicted value is then fed back as input for the next prediction step, iteratively forecasting a total of 20 points for validation. The division of the training and test sets for the stepwise iterative model is shown in

Table 20. Training set of stepwise iterative model.

Input	Output
X1, X2, X3……, X100	X101
X2, X3, X4……, X101	X102
……	……
X7780, X7781, X7782……, X7879	X7880

and

Table 21. Test set of stepwise iterative model.

Input	Output	True Value
X7881, X7882, X7883……, X7980	Y1	X7981
X7882, X7883, X7884……, X7980, Y1	Y2	X7982
……	……	……
Y7900, Y7901, Y7902…X7980, Y1, …, Y19	Y20	X8000

. The second model is a direct multi-step model, where every 100 data points are used to predict 20 future points. In this approach, the last 100 historical data points are directly used to predict 20 points for validation. The division of the training and test sets for the direct multi-step model is presented in

Table 22. Training set of direct multi-step model.

Input	Output
X1, X2, X3……, X100	X101-X120
X2, X3, X4……, X101	X102-X121
……	……
X7761, X7762, X7763……, X7860	X7861-X7880

and

Table 23. Test set of direct multi-step model.

Input	Output	True Value
X7881, X7882, X7883……, X7980	Y1-Y20	X7981-Y8000

. It is worth noting that the following four tables take the delay of 100 steps as an example.

8.4. Prediction Results

Based on the fundamental principles of the WOA, this algorithm is used in MATLAB to optimize the LSTM model. The optimization focuses on tuning the time delay step size and the number of LSTM units. The evaluation metric is the mean Root Mean Square Error (RMSE) of the six prediction results, which serves as the fitness function for WOA to determine the optimal whale position within the population.

The optimization results for the stepwise iterative model indicate that the optimal time delay step size is 84 instead of aforementioned 100, and the number of LSTM units is 62. For the direct multi-step model, they are, respectively, 82 and 93. Figure 38 illustrates the six prediction results of both models after WOA optimization.

Table 24. Evaluation of the prediction results of stepwise iterative model after WOA optimization.

	1	2	3	4	5	6	Mean
R2	0.9023	0.9985	0.9135	0.9171	0.9442	0.9275	0.9339
MAE	0.1019	0.0208	0.1330	0.1958	0.1356	0.1087	0.1160
RMSE	0.1173	0.0238	0.1601	0.2263	0.1609	0.1449	0.1389

and

Table 25. Evaluation of the prediction results of direct multi-step model after WOA optimization.

	1	2	3	4	5	6	Mean
R2	0.9693	0.9912	0.9373	0.9804	0.9490	0.9168	0.9573
MAE	0.0562	0.0493	0.1066	0.0926	0.1348	0.1166	0.0927
RMSE	0.0657	0.0568	0.1363	0.1100	0.1539	0.1553	0.1130

show the accuracy evaluation of six prediction results after WOA optimization in terms of the R-squared (R²), Mean Absolute Error (MAE), and RMSE.

Table 26. Evaluation of the prediction results of two models before and after WOA optimization.

	Stepwise Iterative Model		Direct Multi-Step Model
	Before Optimization	After Optimization	Before Optimization	After Optimization
R2	0.8148	0.9339	0.9435	0.9573
MAE	0.1870	0.1160	0.1045	0.0927
RMSE	0.2290	0.1389	0.1276	0.1130

further shows the accuracy evaluation before and after WOA optimization.

For the stepwise iterative model, it can be seen from Figure 38 that Sequence 2 exhibits the best prediction performance, with an R² value close to 1, a Mean Absolute Error (MAE) of 0.0208, and a Root Mean Square Error (RMSE) of 0.0238. The prediction performance of the other five sequences is similar, with the R² values ranging from 0.90 to 0.95, MAE values between 0.10 and 0.20, and RMSE values between 0.11 and 0.24. The LSTM neural network model, after WOA optimization, demonstrates a significant improvement in predictive capability. As shown in Table 26, thanks to WOA, R² rises from 0.8148 to 0.9339.

For the direct multi-step model, Sequence 2 also achieves the best performance among all six series, with an R² value close to 1, an MAE of 0.0493, and an RMSE of 0.0568. The prediction performance of the other five sequences is also comparable, with R² values ranging from 0.91 to 0.98, MAE values between 0.05 and 0.14, and RMSE values between 0.05 and 0.16. Meanwhile, the pitch motion is also predicted with such a direct model. Once again, the application of the WOA improves the model’s performance, as evidenced by the increase in R² from 0.7761 to 0.8477 and the reductions in the MAE and RMSE from 0.1101 to 0.0934 and 0.1438 to 0.1390, respectively, thereby verifying the generalization capability of the WOA-LSTM model in pitch motion prediction. Overall, the direct multi-step model to some extent outperforms the stepwise iterative model. Both models demonstrate their capacity to successfully predict short upcoming motions that are valuable to float-over installation. In particular, if the predicted motion exceeds the upper threshold value prescribed beforehand, the installation crew may take measures to reduce risks.

9. Conclusions

Based on model tests and numerical simulations, this study investigates the hydrodynamic performance of TBFOI-OWTs under different wave conditions and employs WOA-LSTM to predict the motion during installation. It covers a description of the prototype parameters of the floater, the fabrication of various parts of the physical model, the scheme of numerical simulation, the comparison between the numerical and experimental dynamic responses, and the methodology and results of WOA-LSTM prediction. The following conclusions can be drawn.

The results of static, decay, regular, white noise, and random wave tests all reveal that there are good consistencies between numerical simulation and the experimental approach. Therefore, the adopted numerical simulation method is validated.

The TBFOI-OWT has excellent seakeeping performance. Within common wave period ranges, the surge RAO is less than 1.5 m/m, the sway RAO is less than 1.6 m/m, the heave RAO is under 1.2 m/m, the roll RAO is below 4.0 deg/m, and the pitch RAO is under 1.2 deg/m.

Compared with the regular wave situation, the surge and sway motions in random waves have greater ranges and amplitudes. Such discrepancies demonstrate that the mooring analysis cannot depend on regular waves only, and more importantly, it should be pointed out that the random nature of realistic waves is less favorable to float-over installations. The power spectral analysis suggests that the motions of TBFOI-OWTs are primarily determined by both the wave frequency and natural frequencies of motions. Given the significant wave height, as the spectral peak period increases, the wave frequency motions increase. But for float-over installations, the unfavorable level of a sea state still depends on the aforementioned total response (wave frequency and natural frequency inclusive) and its extreme value.

In head seas, the influence of the gap on the motion RAOs is small, while it becomes significant in beam seas. Within a certain range of beam wave periods (10.00 s < T < 15.00 s), increasing gap distance helps to reduce sway RAO and roll RAO due to the energy dissipation by the damping pool of the barge gap. Instead, for 5.00 s < T < 10.00 s, the larger the gap distance, the larger heave motion, due to gap water slamming on the DSF plate. We recognize that this plate is an imperfection of the model fabrication in the experimental study.

Using LSTM, both the stepwise iterative model and the direct multi-part model can accurately predict the motion of TBFOI-OWTs. After WOA optimization, the prediction accuracy of the stepwise iterative model noticeably improves, while the prediction accuracy of the direct multi-step model slightly improves. Even so, the performance of the direct multi-step model is better than that of the stepwise iterative model, as illustrated in six episodes of motions. Both models pave a path to successfully predict the short upcoming motions that is of engineering significance and interest for float-over installations.

Nevertheless, this study has some limitations, and in-depth research can be carried out in future. It would be beneficial to place wave probes between twin barges and carry out CFD-based validations to investigate the influence of gaps on the wave elevation for acquiring valuable insights into the complicated hydrodynamics of TBFOI-OWTs. Additionally, with rapid progress in artificial intelligence, more powerful algorithms can be explored to further improve the accuracy and length of predicted motions. Fortunately, recent years have seen more and more published works on this issue.