A Comparison of the Capture Width and Interaction Factors of WEC Arrays That Are Co-Located with Semi-Submersible-, Spar- and Barge-Supported Floating Offshore Wind Turbines

Abstract

This research paper explores an approach to enhancing the economic viability of the heaving wave energy converters (WECs) of both cylinder-shaped and torus-shaped devices, by integrating them with four established, floating offshore wind turbines (FOWTs). Specifically, the approach focused on the wave power performance matrix. This integration of WECs and FOWTs not only offers the potential for shared construction and maintenance costs but also presents synergistic advantages in terms of power generation and platform stability. The study began by conducting a comprehensive review of the current State-of-the-Art in co-locating different types of WECs with various foundation platforms for FOWTs, taking into consideration the semi-submersible, spar and barge platforms commonly employed in the offshore wind industry. The research took a unified approach to investigate more and new WEC arrays, totaling 20 configurations across four distinct FOWTs. The scope of this study’s assumption primarily focused on the hydrodynamic wave power performance matrix, without the inclusion of aerodynamic loads. It then compared their outcomes to determine which array demonstrated superior wave energy under the key metrics of total absorbed power, capture width, and interaction factor. Additionally, the investigation could serve to reinforce the ongoing research and development efforts in the allocation of renewable energy resources.

1. Introduction

The global pursuit of sustainable energy solutions has led to the exploration and development of offshore renewable energy (ORE) sources as essential components of the future energy landscape, in addition to fossil fuels [1,2]. With a growing recognition of their potential contributions to meeting energy demands and mitigating climate change, ORE technologies have gained prominence and support in international energy policies, for example, the European Union (EU) has set the target of achieving 188 GW and 460 GW of installed capacity for ocean (wave and tidal) and offshore wind energies by 2050, respectively [3]. Wave energy, a subset of ocean energy, holds immense untapped potential, with a theoretical annual resource of 32,000 TWh [4]. Despite this potential, wave energy conversion technology has economic disadvantages due to the high construction and maintenance costs, resulting in a significant gap between its levelized cost of electricity (LCOE) and other more established renewable sources, such as wind and solar [5,6]. To bridge this gap and facilitate the commercialization of wave energy, innovative approaches are being sought to optimize resource utilization, enhance energy generation, and reduce costs.

An emerging solution for addressing the economic viability of wave energy converters (WECs) lies in their integration with existing offshore structures, particularly with breakwaters [7,8,9] and well-established offshore wind turbines [10,11,12]. Such integration not only offers the advantage of shared construction and maintenance costs but also presents synergistic benefits in terms of power generation and platform stability [13]. Furthermore, the integration of wave and wind energy systems contributes to the reduction in power variability and downtime, thereby enhancing the overall reliability and performance of offshore energy farms [14,15].

While both offshore wind and ocean energy present promising avenues for clean energy generation, they present unique challenges in the respective stages of technological development. Offshore wind, having reached a higher level of maturity with established commercial installations and workforce engagement, demonstrates a higher degree of readiness for large-scale deployment [16]. In contrast, ocean energy—which encompasses diverse forms such as tidal streams, ocean thermal energy, and waves—is still in the relatively early stages of development because of the technological and economic barriers that hinder its wider adoption [17]. The concept of hybrid offshore wind and wave energy systems that involve the integration of both energy sources at a single location has emerged as a promising innovation [18]. This approach leverages the abundant availability of wind and wave resources in the same marine environment, facilitating a streamlined grid connection and an optimized power supply [19,20]. By co-locating WECs with offshore wind turbines, the spatial efficiency is improved and the potential for mitigating platform motion is realized [21]. A co-located wave–wind system also offers a smoother power supply, less energy curtailment, and higher farm-to-grid efficiency than a wind farm [22]. Moreover, the incorporation of wave energy technologies into established offshore structures contributes to the United Nations’ Sustainable Development Goals of reduced environmental impact, improved energy yield, and cost-effective operation and maintenance [12].

This research paper first reviews the State-of-the-Art in the integration of WECs with floating offshore wind turbines (FOWTs) by examining the performance of the combined platform. This study followed a unified approach to evaluate more and new WEC arrays, totaling 20 configurations across four different FOWTs, and compared their wave power generation using the capture width, absorbed power, and interaction factor. The performance of multiple configurations of the integrated systems under regular and irregular waves were studied.

2. State-of-the-Art

Numerous studies have considered the integration of WECs and FOWTs. The floating foundation commonly used for FOWTs is a column stabilized unit, such as a semi-submersible, tension leg platform and spar, or a flat-bottom floating structure, such as a pontoon platform or barge. This section shall review the State-of-the-Art of the various concepts that consider the integration of WECs with semi-submersible, spar, and barge platforms being used as foundations for FOWTs.

2.1. Integration of WECs with Semi-Submersibles

A semi-submersible (SS) is commonly proposed as a foundation for FOWTs due to its ability to provide greater stability and buoyancy and its ability to operate in rough sea conditions. Semi-submersibles can be of two different designs, i.e., with bracing or without bracing, in which the SSs used for the FOWTs usually comprise three outer columns and one central column that houses the wind turbine. Shi et al. [23] considered a braceless SS FOWT (semi-submersible floating offshore wind turbine), where a torus-shaped WEC was attached to the central column. In their study, an optimization technique was used to maximize the wave energy power production of the WEC. Their findings revealed that an optimal WEC design features a smaller inner radius and a larger draft, resulting in an enhanced heave response amplitude. Wang et al. [24] also considered four torus-shaped WECs of different geometries, which were attached to the central column of an SS. They investigated the impact of the geometries on the dynamic response and power performance of the WECs. A comparative analysis of four WEC shapes revealed that the concave shape yielded the best dynamic response and power generation due to the greater generation of added mass and radiation damping, which effectively mitigated the dynamic responses of the braceless SS FOWT, while enhancing its power output. Their study examined multiple WECs that were attached to the outer and central columns of the SS; larger hydrodynamic responses were observed with the increase in the number of WECs [25]. These responses were also influenced by the layout of the WECs, for instance, the stability of the SS was further improved by the attachment of WECs at the outer columns [26]. The heave and pitch motions of the integrated SS were reduced, in contrast to the standalone SS FOWT [27].

Ghafari et al. [28] considered the multiple Wavestar-type heaving point absorber WECs attached along the three pontoons of an SS FOWT, where they investigated the hydrodynamic interactions between the FOWT and the WEC arrays. The results indicated that, while each WEC of the FOWT–Wavestar system’s power performance is generally lower than a standalone Wavestar WEC, it absorbs higher power than the standalone WEC in wave periods greater than 8 s. Ghafari et al. [29] also considered the attachment of Wavestar WECs on the three outer columns of the SS FOWT. They found that increasing the number of WECs from one to four could significantly reduce the platform heave and surge motions while increasing the absorption power. A similar concept of a hybrid FOWT–Wavestar system considered in China found that the SS FOWT’s response increased with the increase in the number of WECs [30], whereas the hydrodynamic interaction exhibited a destructive effect for power absorption between 5 to 7 s. However, a constructive interference effect existed for a wave period between 8 to 12 s, thereby increasing the power absorption. The total power captured by the WECs remained consistent for wave directions between 0° and 180°, despite the platform’s asymmetrical shape.

Semi-submersibles with bracing have also been considered, in which the bracing can provide greater structural integrity to the FOWT. Chen et al. [31] considered a cylindrical-shaped WEC mounted under the center column of an SS FOWT. Their results indicated that the wave power generation from the WEC integrated with a SS FOWT could be significantly enhanced by 50% when compared to a standalone WEC counterpart. Chen et al. [32] then extended their work by integrating three heaving cylinder WECs on each side of the SS. They used a coupled aero-hydro-servo-elastic-mooring numerical model to study the impact of different power take-off (PTO) control strategies. The results revealed that the PTO bang–bang control could effectively mitigate the platform dynamic loads and fatigue behaviors. The presence of different WEC arrays reduced the platform heave motion in the vicinity of the platform resonance frequency and its effect on the mooring loads was also similar to that on the platform motions [33].

Zhang et al. [34] considered a similar SS FOWT design, integrated with a heaving point absorber WEC, where they aimed to optimize the size of the WEC. Their findings indicated that WECs, with a radius of 4 m emerged as the optimal choice, displaying the highest power efficiency for the integrated system. Zhang et al. [35] investigated the combined system of SS FOWTs with two different WECs of torus-shaped and Wavestar-type. Their study showed that the dynamic response, stability, and power generation were better than those from its standalone FOWT and the other two combined systems, i.e., the spar FOWT with a torus-shaped WEC [36] and the SS FOWT with a flap-type WEC [37]. The research by [38] proposed a toolkit comprising an OpenFAST-AQWA framework, with a bridging code for the full couplings of the SS FOWT (DeepCWind) and WEC (Wavestar) combined system. Their findings showed the effective simulation of the complex interactions within this hybrid system, allowing for design optimization, the system’s stability improvements, and an optimized mooring force.

2.2. Integration of WECs with Spar Foundations

A spar is a type of offshore floating platform that consists of a vertical cylindrical hull moored to the seabed. The upper portion of the structure could be used to support the FOWT, whereas the lower portion of the hull is submerged to a significantly low center of gravity, such that the structure could be unconditionally stable. Skene et al. [39] considered an integrated spar FOWT with a single cylinder-shaped heaving WEC. Their research highlighted that an optimal relationship between the spar and WEC occurs when the mass of the spar and WEC are equal. However, this poses challenges due to the increased spar motion when a large WEC is used, posing practical difficulties for system implementation.

Multiple Wavestar-type WECs—in sets of three, six, nine, and twelve—arranged around a spar FOWT were considered by Ghafari et al. [40]. They highlighted the importance of WEC placement and orientation with respect to the wave direction in optimizing power production, where placing a WEC behind the spar structure (where it is shielded from wave direction) leads to reduced power generation. In addition, maximum power generation and capture width ratios were observed at wave periods of around 5 s and 6 s due to the WECs’ natural frequency.

A spar FOWT supported by tensioned tendons similar to the tension leg platform (TLP) was considered by Kim et al. [41], where four Wavestar WECs were deployed surrounding the spar structure. They found that the heaving response of the single spar FOWT was amplified by around three times under second-order sum-frequency wave loads, compared with its counterpart integrated with WECs. This is because the integrated platform’s motion was damped by the PTO damping of the WECs. Additionally, WEC heaving is slightly larger under second-order wave loads compared to the first-order wave loads. The dynamic response of a spar TLP FOWT combined with a torus-shaped WEC has also been studied by using both a numerical simulation and a scale model test [42]. Another torus-shaped, WEC-integrated concept—which was with three different types of monopile: long spar, short spar, and bottom-fixed spar—has been studied and has provided insights into the different dynamic responses of the three concepts [43]. Furthermore, alongside the prevalent hybrid concept of WECs with horizontal-axis FOWTs, a torus WEC attached to a vertical-axis FOWT spar was assessed by Cheng et al. [44].

The research by [45] developed a numerical framework that performed the aero-servo-hydro-mooring couplings in the hybrid system of a spar-type FOWT with an annular WEC. The findings showed that the integration enhanced platform stability in pitch and provided a substantial contribution of wind–wave energy, with the increased heaving motion of WEC as well. Another study, ref. [46], introduced a novel self-protection mechanism, where a submerged horizontal perforated plate was connected to the oscillating water column (OWC) to reduce wave loads on a system with a monopile foundation wind turbine, while sustaining effective energy capture. Another monopile foundation wind turbine was investigated in [47], with an attached, annular-shaped OWC, revealing its potential as a robust solution for effective wave energy absorption under any wave direction and for stabilizing wave forces on the system.

2.3. Integration of WECs with Floating Platform Foundation

A conventional floating platform, such as a barge or pontoon structure, can be used to house an FOWT. Such a structure is relatively easy to construct and deploy, has a wide workability area, and is relatively stable compared to the column-stabilized structure. To the best of the authors’ knowledge, all the WECs that were co-located with the FOWT barge only considered the OWC WEC. Aboutalebi et al. [48] considered an FOWT housed on a floating barge with four OWC WECs mounted at each corner of the platform. They investigated the effectiveness of the four OWC WECs in mitigating the platform motion and compared it with the counterpart without WECs. The study demonstrated an effective reduction in the pitch motion for wave period ranges between 6.4 s to 12.25 s. However, the study revealed that the performance advantage diminished for wave periods extending from 12.25 s to 20 s, in which the barge without OWCs showed a better performance. Li et al. [49] also introduced a comprehensive investigation of a multi-purpose floating barge platform, equipped with a wind turbine, an OWC WEC, and fish cages integrated within the barge moonpool. The study confirmed the technical feasibility of the proposed multi-purpose platform design. The elasticity of the support structure was found to have a negligible impact on the overall coupled dynamics of the platform. In addition, certain strong, nonlinear hydrodynamic phenomena, such as slamming and green water, were acknowledged, suggesting the need for more accurate analyses in advanced design phases. These findings have collectively contributed to an understanding of the multi-purpose offshore platform’s behavior, design optimization, and its performance under various conditions.

3. Problem Definition

While numerous research papers have investigated the performance of WEC–FOWT integrated systems, a notable observation persists that these investigations have primarily concentrated on a single type of FOWT. This paper endeavors to merge the knowledge dispersed across different FOWT structures in one place, with a diverse range of WEC array configurations, which can be comparable with each other as well. By offering a consolidated overview, it presents a perspective on the wave power generation in patterns of total absorbed power, capture width, and interaction factor across the spectrum of co-located systems. This distilled insight can facilitate the identification of optimal combinations, resulting in enhanced energy yield, efficiency, cost-effectiveness, and design considerations in future resource allocation.

This study considered four FOWT platforms: spar [50]; semi-submersible without braces, denoted as SS [51]; semi-submersible with braces, denoted as SS-B [52]; and barge [53]. For numerical modelling, these platforms’ design parameters, such as principal dimensions, geometry, displacement, draft, and center of mass, were adapted and are shown in Figure 1 and

Table 1. Parameters for foundations of various FOWTs.

		Spar	Semi-Submersible Without Braces (SS)	Semi-Submersible with Braces (SS-B)	Barge
Draft (m)	$T_{spar}$	$120$	–	–	–
	$T_{pontoon}$	–	6	–	–
	$T_{column}$	–	24	–	–
	${\left(T_{column}\right)}_{bottom}$	–	–	6	–
	${\left(T_{column}\right)}_{upper }$	–	–	14	–
	$T_{barge}$	–	–	–	10
Length (m)	$L_{pontoon}$	–	42.9	–	–
	${\left(L_{column}\right)}_{upper}$	–	–	50	–
	$L_{barge}$	–	–	–	60
	$L_{moonpool}$	–	–	–	20
Diameter (m)	${\varnothing }_{spar}$	6.5	–	–	–
	${\varnothing }_{column}$	–	6.5	–	–
	${\left({\varnothing }_{column}\right)}_{inner}$	–	–	6.5	–
	${\left({\varnothing }_{column}\right)}_{upper}$	–	–	12	–
	${\left({\varnothing }_{column}\right)}_{bottom}$	–	–	24	–
Displacement (kg)		$7.47\times {10}^6$	$1.06\times {10}^7$	$1.35\times {10}^7$	$3.22\times {10}^7$
CoG below SWL (m)		$89.915$	$24.36$	$13.46$	$4.86$
$I_{xx}$ (kgm2)		$4.23\times {10}^9$	$1.05\times {10}^{10}$	$6.83\times {10}^9$	$9.66\times {10}^9$
$I_{yy}$(kgm2)		$4.23\times {10}^9$	$1.05\times {10}^{10}$	$6.83\times {10}^9$	$9.66\times {10}^9$
$I_{zz}$ (kgm2)		$1.64\times {10}^8$	$8.24\times {10}^9$	$1.23\times {10}^{10}$	$1.93\times {10}^{10}$

. The wave directions, $\theta$, are shown in Figure 1a for the spar foundation and are applicable to the rest of the FOWT foundations.

The heaving point absorber WECs in both cylindrical and toroidal shapes and integrated with each foundation of the FOWT given in Figure 1. The linear PTO system was employed to convert the mechanical motion of the WEC into electricity. The configurations of the arrays of WECs integrated with the spar, SS, SS-B, and Barge are presented in

Table 2. Parameters of integrated spar and WEC array configuration.

Spar—WEC Array Configuration	${\mathit{h}}_{\mathit{w}\mathit{e}\mathit{c}}$ (m)	${\mathit{\varnothing }}_{\mathit{w}\mathit{e}\mathit{c}}$ (m)	${\mathit{N}}_{\mathit{w}\mathit{e}\mathit{c}}$	${\mathit{B}}_{\mathit{P}\mathit{T}\mathit{O}\left(\mathbf{m}\mathbf{i}\mathbf{n}\right)}$ (Ns/m)	${\mathit{s}}_{\mathit{p}}$ (m)	${\left({\mathit{A}}_{\mathit{w}\mathit{p}}\right)}_{\mathit{T}}$ (m2)
Spar I	3	3.00	8	2359	2	56.55
Spar II		3.46	6	3492	2
Spar III		4.24	4	5174	2
Spar IV		11.66 (o) 8.00 (i)	1	29,084	0.75

,

Table 3. Parameters of integrated SS and WEC array configuration.

SS and WEC Configuration	${\mathit{h}}_{\mathit{w}\mathit{e}\mathit{c}}$ (m)	${\mathit{\varnothing }}_{\mathit{w}\mathit{e}\mathit{c}}$ (m)	${\mathit{N}}_{\mathit{w}\mathit{e}\mathit{c}}$	${\mathit{B}}_{\mathit{p}\mathit{t}\mathit{o}\left(\mathit{m}\mathit{i}\mathit{n}\right)}$ (Ns/m)	${\mathit{s}}_{\mathit{p}}$ (m)	${\left({\mathit{A}}_{\mathit{w}\mathit{p}}\right)}_{\mathit{T}}$ (m2)
SS-I	3	3.00	8 × 4	2359	2	56.55 × 4
SS-II		3.46	6 × 4	3492	2	56.55 × 4
SS-III		4.24	4 × 4	5174	2	56.55 × 4
SS-IV		11.66 (o) 8.00 (i)	1 × 4	29,084	0.75	56.55 × 4
SS-V		11.66 (o) 8.00 (i)	1 × 3	29,084	0.75	56.55 × 3
SS-VI		11.66 (o) 8.00 (i)	1	29,084	0.75	56.55

,

Table 4. Parameters of integrated SS-B and WEC array configuration.

SS-B and WEC Array Configuration	${\mathit{h}}_{\mathit{w}\mathit{e}\mathit{c}}$ (m)	Centre Column			Side Column			${\mathit{s}}_{\mathit{p}}$(m)	${\mathit{A}}_{\mathit{w}\mathit{p}}$ (m2)
		${\mathit{\varnothing }}_{\mathit{w}\mathit{e}\mathit{c}}$ (m)	${\mathit{N}}_{\mathit{w}\mathit{e}\mathit{c}}$	${\mathit{B}}_{\mathit{p}\mathit{t}\mathit{o}\left(\mathit{m}\mathit{i}\mathit{n}\right)}$ (Ns/m)	${\mathit{\varnothing }}_{\mathit{w}\mathit{e}\mathit{c}}$ (m)	${\mathit{N}}_{\mathit{w}\mathit{e}\mathit{c}}$	${\mathit{B}}_{\mathit{p}\mathit{t}\mathit{o}\left(\mathit{m}\mathit{i}\mathit{n}\right)}$ (Ns/m)
SS-B-I	3	2.83	9	971	2.83	9	1983	2	56.55 × 4
SS-B-II		3.46	6	3492	3.79	5	4888	2	56.55 × 4
SS-B-III		4.90	3	7184	4.90	3	6957	2	56.55 × 4
SS-B-IV		11.66 (o) 8.00 (i)	1	29,084	-	-	-	0.75	56.55
SS-B-V		-	-	-	4.90	3	6957	2	56.55 × 3
SS-B-VI		4.90	3	7184	-	-	-	2	56.55

, and

Table 5. Parameters of integrated Barge and WEC array configuration.

Barge—WEC Array Configuration	${\mathit{h}}_{\mathit{w}\mathit{e}\mathit{c}}$ (m)	${\mathit{\varnothing }}_{\mathit{w}\mathit{e}\mathit{c}}$ (m)	${\mathit{N}}_{\mathit{w}\mathit{e}\mathit{c}}$	${\mathit{B}}_{\mathit{p}\mathit{t}\mathit{o}\left(\mathit{m}\mathit{i}\mathit{n}\right)}$ (Ns/m)	${\mathit{s}}_{\mathit{p}}$ (m)	Moonpool Size	${\mathit{A}}_{\mathit{w}\mathit{p}}$ (m2)
Barge-I	3	2.45	12	606	2	20 m × 20 m	56.55 × 5
Barge-II		3.00	8	2359
Barge-III		4.24	4	5174
Barge-IV		4.24	4	5174		30 m × 30 m

, respectively. The parameters of the WECs for the spar, SS, SS-B, and barge floating foundations are summarized in Table 2, Table 3, Table 4, and Table 5, respectively. The investigation employed numerical simulations using ANSYS AQWA’s diffraction analysis. This approach enabled a comprehensive analysis of the hydrodynamic interactions between the heaving WECs and the FOWTs under regular and irregular wave conditions. The simulations encompassed a wave height, $H$, of 2 m; a water depth of 100 m; and wave directions, $\theta$, ranging from 0° to 180°, at intervals of 45°, since all four FOWTs were symmetric about their center line. The wave frequency range considered was from 0.05 to 2.5 rad/s, with an interval of 0.05 rad/s.

Figure 2 shows the four foundations considered for the FOWT, i.e., spar, semi-submersible without braces (SS), semi-submersible with braces (SS-B) and barge. The number, $N_{wec}$, and diameter, ${\varnothing }_{wec}$, of the WECs in an array were mainly set in such a way that the waterplane area, ($A_{wp}$), was equal to 56.55 ${\mathrm{m}}^2$. E.g., an array of eight WECs, each WEC with ${\varnothing }_{wec}=3 \mathrm{m}$, was deployed at each column of the SS-I, whereas an array of six WECs, each WEC with $\varnothing =3.46 \mathrm{m}$, was deployed at each column in the SS-II in Figure 3, where both arrays had a $A_{wp}=56.55 {\mathrm{m}}^2$. All the $A_{wp}$ of the WEC arrays deployed for each configuration in Figure 2, Figure 3, Figure 4 and Figure 5 had the same waterplane area. The total waterplane area, ${\left(A_{wp}\right)}_T$, for each configuration is given in Table 2, Table 3, Table 4 and Table 5. Note that the waterplane area is an important parameter, which decides the heaving hydrostatic stiffness. The two semi-submersibles considered had WECs at their center and three side columns (a total of four footprints of $A_{wp}$), the spar had WECs at one column (one footprint of $A_{wp}$), and the barge had WECs at its four sides on the outer side and inside the moonpool (five footprints of $A_{wp}$). The capture width on the energy extraction efficiency, absorbed power, and interaction factor was investigated and compared across different platforms and configurations.

4. Methodology

The study adopted the ANSYS AQWA software (2021 R2), an acknowledged numerical tool among researchers in the wave energy industry for the numerical modelling of hydrodynamic simulations, by employing the boundary element method (BEM). The consistent application of the software across numerous publications underscores its validation through both experimental and numerical findings, demonstrating its methodological suitability. Some of the publications that have used ANSYS AQWA in their research of hydrodynamic performance and multi-bodies’ interactions between WEC arrays and FOWTs are attributed to Shi et al. [23], Wang et al. [24], Wang et al. [54], Tian et al. [25], Ghafari et al. [28], Zhang et al. [30], Chen et al. [32], Ghafari et al. [29], and Kim et al. [41]. In Section 5, validations of the numerical model employed are presented to verify this research study.

The mathematical framework for assessing the performance of the integrated WECs with FOWT foundations was based on the principle of three-dimensional potential flow theory, which assumes the fluid to be inviscid and incompressible, and its motion irrotational. The viscous effect is assumed to have been neglected in this research due to two main reasons: the need for computational time efficiency and the dominance of the structure’s inertia. Firstly, the primary focus of the research was to investigate the hydrodynamic power performance of WECs when co-located with FOWTs. Parametric comparison studies of various WEC array configurations on different platforms had to be conducted, involving changing variables, data, and simulations. Therefore, the consideration of the boundary element method for solving potential flow problems could have led to considerable time savings to achieve the research objectives. Secondly, previous research undertaken by Bhinder et al. [55] examined the effect of viscous damping on WECs and found that, in the case where the Keulegan–Carpenter (KC) number was low (an indication that the inertia forces dominate over the viscous forces), the difference in power generation with and without viscous damping was found to be negligible (4%). In addition, Journée and Massie [56] also highlighted that, for low values of the KC number (KC < 3), which was the case for the WECs studied in this paper, the viscosity could be neglected. Thus, in such a case, the flow does not travel far enough relative to the cylinder diameter to generate much of a boundary layer, indicating that the flow could be assumed to be inviscid, and that potential flow theory was applicable.

The investigation in this study was grounded upon the frequency domain and the steady-state condition, without the turbulent and transient wind. Although we considered an assumption without aerodynamic effects, we acknowledge the impact that aerodynamic forces can have on platform motion. Study [57] showed that optimal PTO controls on WECs can reduce pitch motion by counteracting aerodynamic loads through increased damping, particularly under below-rated wind conditions, thereby lessening the influence of aerodynamic forces on the system’s movement. Additionally, aligning WECs with wave direction can generate counteracting hydrodynamic forces, further mitigating the aerodynamic effects under low wind; however, these forces remain significant, especially in transient high-wind conditions, where they dominate pitching motion [58]. Active control strategies, such as pitch control and additional damping mechanisms, are crucial for stabilizing pitch and yaw in turbulent winds [59]. Semi-submersibles, for instance, exhibit increases in the heave motion of up to 15% under turbulent winds [60], while heave plates, although effective in damping motion, experience limits as the heave motion rises by approximately 10–15%, when winds exceed rated conditions [61]. Spar platforms, known for their stability, nonetheless have shown up to a 10% increase in heave under high-wind conditions due to aerodynamic interactions [62]. These findings illustrate the differences that can be expected in platform motion without the inclusion of aerodynamic forces, stating the scope of this study’s assumption in primarily focusing on hydrodynamic wave energy generation.

There have been numerous research studies that have aimed to highlight the importance of the performance of the floating system through the influence of various wave energy spectra, hydrodynamic aspects, and key findings. All were executed by excluding the aerodynamic aspects due to the consideration of a steady state in frequency domain analyses. To name a few, some of these research publications have been from Wang et al. [24], Wang et al. [54], Tian et al. [25], Ghafari et al. [28], Zhang et al. [30], Chen et al. [31], Zhang et al. [34], Skene et al. [39], Ghafari et al. [63], and Kim et al. [41].

4.1. Governing Equation of Motion

The governing equation—Laplace’s equation—was formulated in the frequency domain to account for wave-induced responses. The methodology employed the boundary element method (BEM) to solve the Laplace equation (Equation (1)) for the velocity potential $\Phi \left(x,y,z,t\right)$.

$$\Phi \left(x,y,z,t\right)=Re\left\{\phi \left(x,y,z\right)e^{-i\omega t}\right\}$$

(1)

The geometry of the floating structure was discretized into surface elements, and the boundary integral equation was derived using a free surface Green’s function to relate the velocity potential values of these elements. The BEM efficiently accounted for the complex geometries by focusing on the wetted surface of the floating bodies. The boundary integral equation employed a linear system of equations, where the unknowns were the velocity potential values at the surface discretized node points. An iterative solver, such as the generalized minimal residual method, was used to solve the linear system.

In the irregular wave analysis, the frequency domain facilitated the analysis of resonant phenomena and the determination of the steady-state responses of the combined system by allowing the decomposition of the wave time series into distinct frequency components, whereby Fourier transforms were applied to express the velocity potential as a sum of complex amplitudes corresponding to different wave frequencies.

The hydrodynamic model of the system in the frequency domain can be expressed with the equation of motion,

$$\left\{-{\omega }^2\left[A\left(\omega \right)+M\right]+i\omega \left[B\left(\omega \right)+B_{PTO}\right]+C\right\} \widehat{x}={\widehat{F}}_e$$

(2)

where $A\left(\omega \right)$ is the frequency-dependent added mass; $M$ is the body inertia; $B\left(\omega \right)$ is the frequency-dependent radiation damping; $B_{PTO}$ is the linear power take-off damping control; $C$ is the hydrostatic stiffness; $\widehat{x}$ is the displacement; and ${\widehat{F}}_e$ is the wave excitation force.

All six degrees of freedom and their interactions for the hydrodynamic coefficients of the floaters were considered in the diffraction analysis. By looking at only the heaving mode of the combined system, Equation (2) becomes,

$$\left\{-{\omega }^2 \left[\begin{array}{cc}{A}_{33}\left(\omega \right)+M_{33}& A_{39}\left(\omega \right)\\ A_{93}\left(\omega \right)& A_{99}\left(\omega \right)+M_{99}\end{array}\right]+i\omega \left[\begin{array}{cc}{B}_{33}\left(\omega \right)+B_{PTO}& B_{39}\left(\omega \right)-B_{PTO}\\ B_{93}\left(\omega \right)-B_{PTO}& B_{99}\left(\omega \right)+B_{PTO}\end{array}\right]+\left[\begin{array}{cc}{C}_{33}& 0\\ 0& C_{99}\end{array}\right]\right\} \left\{\begin{array}{c}{\widehat{x}}_3\\ {\widehat{x}}_9\end{array}\right\}=\left\{\begin{array}{c}{\widehat{F}}_{3, e}\\ {\widehat{F}}_{9, e}\end{array}\right\}$$

(3)

The subscripts 3 and 9 in Equation (3) refer to the heaving motion of the WEC and FOWT, respectively.

4.2. Power Take-Off Damping

Based on linear theory [64], Equation (4), for the optimum PTO damping coefficient ($B_{PTO}$) is expressed as,

$$B_{PTO}=\sqrt{{\displaystyle \frac{{[K_{33}-{\omega }^2\left(M+A_{33}\left(\omega \right)\right]}^2}{{\omega }^2}}+{B_{33}\left(\omega \right)}^2}$$

(4)

where $B_{PTO}$ is the optimum PTO damping; $\omega$ is the wave frequency; $M$ is the body mass; $K_{33}$ is the heaving hydrostatic stiffness; $A_{33}$ is the added mass; and $B_{33}$ is the radiation damping in heave. Using this equation, an optimum PTO damping was achieved at each frequency and a minimum value was considered for each configuration.

4.3. Power Generation—Regular Wave

The mean power absorption of a WEC corresponds to the mean power consumed by the PTO linear damper over a wave period. The mean absorbed power over a regular wave period is,

$$P_a={\displaystyle \frac{1}{T}}{\int }_0^T{B}_{PTO}.u^{2}dt={\displaystyle \frac{1}{2}}{B}_{PTO}.{\omega }^2.{\left|\widehat{x}\right|}^2$$

(5)

where $\left|\widehat{x}\right|=\left|{\widehat{x}}_3\right|-\left|{\widehat{x}}_9\right|$ $=$ the relative heave response between the WEC and FOWT; $P_a$ is the mean absorbed power of a WEC in a regular wave; $u$ is the heaving velocity of the WEC; and $T$ is the wave period.

The total absorbed power is given by,

$${\left(P_a\right)}_T=\sum _{n=1}^{{\left(N_{wec}\right)}_T}{\left(P_a\right)}_n$$

(6)

where ${\left(P_a\right)}_n$ is the absorbed of the $n^{\mathrm{t}\mathrm{h}}$ WEC and ${\left(N_{wec}\right)}_T$ is the total number of WECs in the FOWT.

The average absorbed power, $\overline{P_a}$, generated by the total number of WECs, ${\left(N_{wec}\right)}_T$ was then given as,

$$\overline{P_a}={\displaystyle \frac{{\left(P_a\right)}_T}{{\left(N_{wec}\right)}_T}}={\displaystyle \frac{\sum _{n=1}^{{\left(N_{wec}\right)}_T}{\left(P_a\right)}_n}{{\left(N_{wec}\right)}_T}}$$

(7)

To capture the performance of the interaction between the array of the WECs and the FOWTs, the $q$-factor was expressed as,

$$q={\displaystyle \frac{\overline{P_a}}{\overline{P_0}}}$$

(8)

where ${\overline{P}}_0$ is the average power of the total number of WECs deployed in the open sea, i.e., without the presence of an FOWT. It is to be noted that a $q$ value of greater than 1.0 indicated a constructive interaction and that one smaller than 1.0 indicated a destructive interaction.

The average capture width, ($\overline{CW}$), was the ratio of the mean absorbed power, $\overline{P_a}$, to the wave power resource, expressing the WECs’ power generation efficiency, given as

$$\overline{CW}={\displaystyle \frac{\overline{P_a}}{P_{resource}}}$$

(9)

The power generation efficiency can also be expressed by the average capture width ratio, ($\overline{CWR}$), defined as the ratio of the average capture width, ($\overline{CW}$), to the diameter of the WEC, (${\varnothing }_{wec}$) [65].

$$\overline{CWR}={\displaystyle \frac{\overline{P_a}}{{\left(P_{resource}\right)}_{reg}. {\varnothing }_{wec}}}\times 100\%={\displaystyle \frac{\overline{CW}}{{\varnothing }_{wec}}}\times 100\%$$

(10)

The wave power resource, ${\left(P_{resource}\right)}_{reg}$, in Equations (9) and (10) was given as in [66], as follows,

$${\left(P_{resource }\right)}_{reg}={\displaystyle \frac{\rho g^2}{32\pi }}{H}^{2}T$$

(11)

where $H$ is the wave height and $T$ is the wave period.

4.4. Power Generation—Irregular Wave

The irregular wave was represented by the Bretschneider (the ISSC) wave spectrum, which is a two-parameter Pierson Moskowitz spectrum, given as,

$$S_{\omega }\left(\omega \right)=5{\pi }^4{\displaystyle \frac{H_s^2}{T_p^4}} {\displaystyle \frac{1}{{\omega }^5}} \exp \left(-{\displaystyle \frac{20{\pi }^4}{T_p^4}} {\displaystyle \frac{1}{{\omega }^4}}\right)$$

(12)

where $H_s$ is the significant wave height, $T_p$ the peak wave period, and $\omega$ the wave frequency.

To obtain the mean absorbed power, the power response spectrum, $S_R\left(\omega \right)$, was first computed using Equation (13) [67],

$$S_R\left(\omega \right)=P_a^2 · S_{\omega }\left(\omega \right)$$

(13)

With the power response spectrum, the mean absorbed power ($P_{irr}$) of a WEC in an irregular wave can then be obtained as in Cruz [68],

$$P_{irr}=2\sqrt{\int S_R\left(\omega \right)·d\omega }$$

(14)

5. Validation of the Numerical Model

5.1. Validation of Hydrodynamics

To validate the numerical model of ANSYS AQWA for simulating the hydrodynamic interaction of multiple bodies placed with close spacing between the WECs and the structure, we conducted a numerical simulation involving five WECs with a breakwater. The research undertaken by Konispoliatis [69] assessed a theoretical hydrodynamic evaluation for this configuration. Our study compared the results from this publication with those from our replicated numerical model using ANSYS AQWA. The parameters of our study are depicted in Figure 6. It featured a five-number, cylindrical-shaped WEC array, spaced equidistantly at 18 m and situated parallel to a 110-m-long breakwater. A constant power take-off damping of 446.9 kg/s was applied. Given that the breakwater was fixed to the shallow water seabed and interacted with five closely spaced WECs at a distance of 1 m, the hydrodynamic interaction complexity was deemed high, necessitating a stable numerical method.

The heaving RAOs of the No. 1 and No. 3 WEC placed in front of a fixed breakwater and that of a standalone WEC were determined and compared, as depicted in Figure 7. The interaction behavior between the WECs and the breakwater was observed, when compared to the standalone WEC without the breakwater structure. It was noted that the waves reflected from the breakwater significantly increased the heave motion of the WEC compared to the standalone WEC, particularly within the lower frequency range of 0.3 to 1.5 rad/s. It was also observed that the fluctuation of the heaving RAOs occurred within that frequency range in our method. This phenomenon is likely attributed to the interaction between the closely placed breakwater and the WEC, in which the wave diffraction could be captured by the ANSYS AQWA, whereas it may not have been accurately predicted by the theoretical solution. Overall, the findings revealed good agreement, with a consistent trend in the WECs’ heaving RAO curves in the theoretical results and our present numerical results, indicating the reliability of the numerical model employed in this study, as long as the minimum mesh size is taken as $\mathsf{\lambda }/10$, where $\mathsf{\lambda }$ is the wavelength.

5.2. Validation of Absorbed Power

In order to validate the numerical model in ANSYS AQWA, the results of the mean absorbed power, $P_a$, generated by a point-absorber WEC integrated with a semi-submersible FOWT were first validated using the results published by Zhang et al. [34]. Zhang et al. considered a cylindrical-shaped WEC with a radius of 4 m and a draft of 3 m, and used ANSYS AQWA in assessing the performance of the WEC. A single heaving WEC was placed in the middle of the semi-submersible (known as OC4 in the paper) platform layout, and a linear PTO was considered above the top of the WEC, with a PTO damping control assumed to be attached at the FOWT.

The hydrodynamic analyses for both the combined WEC–FOWT model and the single WEC model were carried out in the frequency domain. The added mass, radiation damping, and response amplitude operators (RAOs) were computed and, subsequently, the mean absorbed power, $P_a$, of a heaving WEC was calculated. Figure 8 shows the comparison of the $P_a$, computed from the present method with its counterpart from [34]. The $P_a$ for an isolated WEC in the open sea (denoted as a single floater) and the $P_a$ for a WEC integrated with the semi-submersible (denoted as combined) are presented here. Overall, the results were found to agree with the corresponding counterparts in the publication. Therefore, the numerical model, the simulation parameters, and the methodology considered in this paper are deemed to be verified.

6. Results and Discussions

6.1. Regular Wave

6.1.1. Resonant Frequency

The resonant periods of the cylindrical WECs for different diameters, ${\varnothing }_{\mathrm{w}\mathrm{e}\mathrm{c}}$, are summarized in

Table 6. Heave resonant period for WECs under different diameters and drafts.

		Heaving Resonant Period (s)
$\mathbf{Diameter} {\mathit{\varnothing }}_{\mathit{w}\mathit{e}\mathit{c}}$ (m)	$\mathbf{Draft} {\mathit{T}}_{\mathit{w}\mathit{e}\mathit{c}}=$	2 m	3 m	4 m	5 m
		3.37	3.92	4.40	4.84
2.83		3.44	3.98	4.46	4.89
3.00		3.47	4.01	4.49	4.91
3.46		3.56	4.09	4.55	4.98
4.24		3.71	4.21	4.67	5.08
4.90		3.82	4.32	4.76	5.17

. An approximation for the heaving resonant period of a freely floating object with a circular cross-section can be deduced from Equation (15) [70],

$$T_N \approx 2\pi \sqrt{{\displaystyle \frac{\left(T_{wec}+\frac{{\varnothing }_{wec}}{3}\right)}{g}}}$$

(15)

where $T_N$ is the heaving resonant period, and $T_{wec}$ and ${\varnothing }_{wec}$ are the draft and diameter of the WEC, respectively.

Table 6 shows that $T_N$ increased with the increase in ${\varnothing }_{wec}$ and $T_{wec}$. The change in draft had a significant effect on the WEC’s heaving resonance, which can possibly be adjusted to the desired resonant period range for higher power absorption. The determination of the optimal dimension for the wave-absorbing body of a WEC was correlated to the encountering wave period. In the results, which are presented in the subsequent sections, it was found that the highest power absorption was concentrated on the corresponding WEC’s resonant period region.

6.1.2. Total Absorbed Power

With the numerical model having been validated in Section 5.1, taking the minimum mesh size as $\mathsf{\lambda }$/10, the performance of the WECs was assessed in ANSYS AQWA. The total absorbed power, ${\left(P_a\right)}_T$, given in Equation (6) was used to quantify the wave power extraction from the WECs. Contour map plots were used to effectively compare the ${\left(P_a\right)}_T$ generated from the various FOWT–WEC configurations, as presented in Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13. It is to be noted that the hybrid layouts, i.e., SS-V, SS-VI, SS-B-V, and SS-B-VI, were eliminated in the figures as their WECs were only deployed to either the center column or side column. Figure 9, Figure 10, Figure 11, Figure 12, and Figure 13 show the total absorbed power, ${\left(P_a\right)}_T$, of the WECs for wave direction, $\theta =0°, 45°, 90°, 135°$, and $180°$, respectively. The FOWTs were subjected to regular wave conditions, with wave periods, $T$, ranging from 3 s to 7 s. The total absorbed power was expected to be different in each configuration due to the different number of WECs, $N_{wec}$, and diameters, ${\varnothing }_{wec}$. As stated in Section 3, the footprint of the waterplane area, $A_{wp}$, for the WEC array in each FOWT foundation was assumed to be the same. This allowed the observation and comparison of the wave power performance, as well as the determination of the favourable and unfavourable wave periods from the contour map plots.

It was found that the head seas, beam seas, and following seas (i.e., 0°, 90°, and 180°, respectively) have a maximum ${\left(P_a\right)}_T$ of around 995 kW, while oblique seas (i.e., 45° and 135°) have a maximum ${\left(P_a\right)}_T$ of around 1265 kW, both were seen in the Barge-II layout. For the two semi-submersible configurations, it was observed that SS-II and SS-B-II generally performed better in all wave directions. A summary of the performance of the integrated spar, SS, SS-B, and barge with WEC arrays is presented in the subsequent sections.

• Integrated Spar with WECs

Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 show that the configurations for the torus-shaped WEC with spar (i.e., Spar-IV) generated the least total power. Among the integrated spar and cylindrical-shaped WEC arrays, Spar-I ${(N}_{wec}=8, {\varnothing }_{wec}=3 \mathrm{m})$ and Spar-II ${(N}_{wec}=6, {\varnothing }_{wec}=3.46 \mathrm{m})$ exhibited a similarity in power generation output for all wave directions and performed slightly better than Spar-III ($N_{wec}=4, {\varnothing }_{wec}=4.24 \mathrm{m})$ in the oblique sea. In general, Spar-I, II, and III were not significantly different in their power generation as the WECs’ co-location footprint of the structure was limited.

• Integrated semi-submersible without braces (SS) with WECs

The integrated SS and WEC array produced the highest total absorbed power, ${\left(P_a\right)}_T$, under oblique seas (i.e., 45° and 135°) for SS-I (with $N_{wec}=32, \varnothing =3 \mathrm{m})$ and SS-II ($N_{wec}=24, {\varnothing }_{wec}=3.46 \mathrm{m})$. For SS-III ($N_{wec}=16$,${\varnothing }_{wec}=4.24 \mathrm{m}$), it was more effective in generating energy under the head sea (i.e., 0°) and following sea (i.e., 180°) conditions. SS-IV (i.e., with torus-shaped WECs) absorbed less total power than SS-I, II, and III in all wave directions. This also agreed with the results from the integrated Spar, where the configurations with cylindrical-shaped WECs were more favourable than configurations with the torus-shaped WECs in terms of total power absorption.

• Integrated semi-submersible with braces (SS-B) and WECs

The integrated SS-B with WEC arrays generated the highest ${\left(P_a\right)}_T$ under all wave directions for SS-B-II ($N_{wec}=21$, ${\varnothing }_{wec}=3.79 \mathrm{m}$). The power generated from SS-B-II was most effective under the 45° wave direction because the WEC array was exposed to more wavefront when waves arrived from the oblique sea. It was also observed that SS-B-III ($N_{wec}=12$, ${\varnothing }_{wec}=4.9 \mathrm{m}$) performed better than SS-B-I ${(N}_{wec}=36$, ${\varnothing }_{wec}=2.83 \mathrm{m}$) for wave periods greater than 4.2 s. Moreover, SS-B-IV (i.e., with a single torus-shaped WEC) generated more total power than Spar-IV (i.e., with a single torus-shaped WEC) under 0° and 180° wave directions, but the power generation output for SS-B-IV and Spar-IV were not significantly different under oblique seas (i.e., 45° and 135°).

• Integrated Barge with WECs

To demonstrate the energy generation under resonance, the ${\left(P_a\right)}_T$ under wave periods ranging from $T=$ 7 s to 11 s for wave direction $\theta =0°$ (headsea) and $\theta =45°$ (oblique wave) are presented in Figure 14. It was observed that Barge-II ($N_{wec}=40$, ${\varnothing }_{wec}=3 \mathrm{m}$) and Barge-III ($N_{wec}=20$, ${\varnothing }_{wec}=4.24 \mathrm{m}$) performed much better in energy generation compared to their counterparts at wave period $T=7.85 \mathrm{s}$. This is because this wave period fell in the moonpool resonant region of the barge. As the moonpool size increased, i.e., Barge-IV, the moonpool resonant period increased correspondingly, thereby resulting in a drop in power generation in the range of periods considered in Figure 14. It can also be seen that the ${\left(P_a\right)}_T$ for all the configurations dropped below 9 kW when the wave period, $T$, was greater than 11 s.

In general, the waterplane area of the WEC has to be sufficiently big to harness a significant portion of the wave energy resources, while remaining small enough to fully engage the heaving stroke in normal wave periods. This can be seen from the single torus WEC (which had a notably bigger waterplane), which generated lesser total power than its counterpart of cylindrical WECs, despite both configurations having the same total $A_{wp}$ and volume.

Similarly, for the barge–WEC configurations, Barge-II ($N_{wec}=40$, ${\varnothing }_{wec}=3 \mathrm{m}$), in the oblique sea, produced the highest ${\left(P_a\right)}_T$, of around 12,265 kW at $T=$ 4 s. Barge-III ($N_{wec}=20$, ${\varnothing }_{wec}=4.24 \mathrm{m}$) produced more power than Barge-I ($N_{wec}=60$, ${\varnothing }_{wec}=2.45 \mathrm{m}$) from wave period $T=$ 4.1 s onwards. When the moonpool size was increased in Barge-IV (and by keeping the same WEC array configuration as Barge-III), there was a change in the resonant period at 5 s in the head sea, resulting in a drop in absorbed power.

6.1.3. Average Absorbed Power

To evaluate the efficiency of each combined layout, the average absorbed power, ${\overline{P}}_a$, was determined, as shown in Figure 15, Figure 16 and Figure 17. These included the hybrid layouts for the two semi-submersibles considered, i.e., SS-V, SS-VI, SS-B-V, and SS-B-VI, with the WEC array deployed to either the center column or side column.

Although the total power in torus-shaped WEC arrays was the least among all other configurations, as presented in Section 6.1.2, they exhibited the highest average power (i.e., Spar-IV, SS-IV, SS-V, SS-VI, and SS-B-IV) as they were exposed to more incoming waves, with less of a shielding effect. It was observed that SS-VI (with a torus-shaped WEC deployed only at the center column) had a higher ${\overline{P}}_a$ than SS-IV (with torus-shaped WECs deployed at both the center and side columns) and SS-V (with torus-shaped WECs at the side columns), except in 0°, where the center WEC was shielded by the side column.

Among the single torus WECs deployed at the center column of the two semi-submersibles (i.e., SS-VI ($\theta =$ 0°) and SS-B-IV ($\theta =$ 0°), with both WECs at each centre column and being shielded by the side column from the head sea), SS-B-IV, with the bigger side column, exhibited significantly more power generation due to the constructive interference of the wave diffraction between the columns. This also implied that the wave diffraction from the larger side column made SS-B more suitable for torus-shaped WECs to be deployed at the centre column. This was also evidenced by the energy generation of SS-B-IV, which produced more power than Spar-IV (both with single torus-shaped WECs).

By comparing SS-B-IV (with a torus-shaped WEC at the center column) and SS-B-VI (with a cylindrical-shaped WEC at the center column), the power generation efficiency in SS-B-IV was significantly more favorable than SS-B-VI. This showed that the SS-B’s center column was more suitable to coexist with single torus-shaped WECs, rather than with multiple cylinder-shaped WECs. It was also observed that the power efficiency in SS-B-III could be improved by co-locating the center column with torus-shaped WECs instead of cylindrical-shaped WECs. To confirm this effect, an additional hydrodynamic analysis was performed. As shown in Figure 18, the outcome proved that the power efficiency became significantly higher, especially in 0° and 180°, when SS-B-III’s center column’s WECs were replaced with a single torus WEC.

6.1.4. Average Capture Width Ratio, ($\overline{CWR}$)

Based on the available wave power resource at different wave periods, the $\overline{CW}$ and $\overline{CWR}$ were determined using Equations (9) and (10) for all the configurations. This analysis included all the configurations considered in Figure 2, Figure 3, Figure 4 and Figure 5 for regular wave periods spanning from $T=4 \mathrm{s}$ to 8 s and for wave height $H=2 \mathrm{m}$.

The $\overline{CWR}$ for $T=$ 4 s, 5 s, 6 s, 7 s, and 8 s are presented in Figure 19. The highest $\overline{CWR}$ value observed within a single wave direction was selected from among all the wave directions considered. This simplification was employed to obtain the most favourable and optimal data for comparative purposes. Generally, the $\overline{CWR}s$ observed in wave directions of 0°, 90°, and 180° may exhibit similarities, as do the values observed at 45° and 135°, owing to the axisymmetric nature of the configuration. The $\overline{CWR}s$ in most of the configurations exhibited significantly high values at their resonant period of $T=$ 4 s, except for the torus-shaped WEC arrays (i.e., Spar-IV, SS-IV to VI, and SS-B-IV). In the subsequent wave periods of $T=$ 6 s and 7 s, the torus-shaped WEC arrays displayed a higher rate of $\overline{CWR}$ than their counterparts because of the higher effective average power.

At $T=$ 8 s, it was observed that resonance occurred in the Barge-III moonpool. Enlarging the moonpool size for Barge-IV (from Barge-III) generally had a better effect on the $\overline{CWR}$, except for $T=$ 5 s and 8 s. For the comparison of $\overline{CWR}$ in the two types of semi-submersibles (for both I and III), which have equivalent WEC array configurations, SS performed moderately better than SS-B. Among all the layouts considered (except for the torus types, which had the least total absorbed power), Barge-III’s $\overline{CWR}$ generally showed better performance for wave period $T\ge$ 4 s, in addition to its attributes of being more spacious to co-locate greater WEC footprints and of generating additional energy amplification from the moonpool resonance.

6.1.5. Interaction q-Factor

The hydrodynamic interaction between the WEC arrays and the FOWT foundations was examined to determine the effectiveness in energy generation when co-locating the WEC arrays with the floating foundation. This investigation included scenarios of head seas (0°) and oblique seas (45°) under regular wave conditions. The performance of the integrated systems was compared with their counterparts when the WEC arrays were deployed in the open sea (standalone system). The interaction factor, given as the q-factor in Equation (8), was used where values greater than 1.0 denoted constructive interference, whereas values less than 1.0 denoted destructive interference. It is to be noted here that the torus-shaped WEC layouts were not subjected to an interaction factor analysis due to their inherent design as their donut-shaped section may not have been suitable to function as a standalone WEC. Instead, it was designed to be attached to the floating foundation’s columns.

Figure 20 and Figure 21 show the interaction factor of the various configurations under wave directions $\theta =0°$ and $45°$. The white cell depicted in both figures indicates values that are below the threshold, assumed as 1.05, in which destructive interference occurred; therefore, they were excluded from the figures. With the exception of the barge configurations, all the other configurations displayed constructive interferences within the wave periods ranging from approximately 3.1 s to 6.5 s. Notably, higher q-factors were primarily concentrated within the wave period range of $T=$ 3.5 s to 5.0 s.

Constructive interference was observed in the SS-B configurations, with longer constructive wave period ranges, extending up to approximately $T=$ 8 s. The magnitudes were also moderately amplified, especially in SS-B-VI, where the WECs were exclusively positioned at the central column. It was observed that the hydrodynamic interaction of the WECs with the SS-B generally performed more favorably compared to that with SS-I to III due to the larger column diameter of SS-B (approximately twice the size of SS). This preference aligned consistently with the findings for the total absorbed power generation, discussed in Section 6.1.2.

Within the barge configurations, the moonpool effect prominently influenced the constructive q-factor, particularly in the wave periods of $T=$ 5 s and 8 s. Consequently, a supplementary investigation was conducted to assess the impact of the moonpool resonance, employing the Barge-III configuration with and without WEC arrays positioned within the moonpool. The outcome of this study is presented in Figure 22. The results revealed that when only WECs were placed in the moonpool, constructive interference was observed solely at the wave periods $T=$ 3.5 s, 5 s, and 8 s. Conversely, when WEC arrays were exclusively positioned around the exterior of the barge, constructive interference extended from $T=$ 3.4 s to 7.0 s. Notably, these findings highlighted that the WECs located outside the barge also exhibited an effective performance independently.

6.2. Irregular Wave—Total Absorbed Power

The irregular wave was modelled using the Bretschneider wave spectrum, in which the significant wave height was $H_s=$ 2 m, with the peak wave period, $T_p=$ 4 s, 6 s, 8 s, and 10 s. It is noteworthy that the same wave height was consistently maintained for the comparisons between the power generation in regular and irregular wave conditions to highlight the intensity in their differences.

The Bretschneider wave spectrum for the four different peak periods are shown in Figure 23. The total power generation in irregular waves was attained using Equation (12). To streamline the post-processing, a single configuration was selected from each group of FOWT structures under head sea conditions, where the results were then compared with the counterparts obtained from the regular waves, as presented in Figure 24.

The analysis revealed that the absorbed power in the irregular waves exhibited a decrease in magnitude when compared to the power obtained from the regular waves. It should also be noted that, even when taking an irregular wave spectrum with a peak period of 10 s, which encompasses a broader spectral integration range than the currently considered 4 s spectrum, the power generation in irregular waves remained lower than the power observed in regular wave conditions. This was due to the wave energy being spread out in the irregular wave spectrum and agreed with the results presented by Tay [71,72], and Tay and Venugopal [73,74]. It was observed that the greater the magnitude of the absorbed power, the more pronounced the disparity between the regular and irregular wave conditions, which was particularly evident in the peak periods. For instance, in regular wave conditions, the absorbed power for SS-III reached around 910 kW, whereas, in its irregular counterpart, it attained a value of 817 kW. Therefore, the results presented in the irregular wave analysis were less conservative than their counterparts for the regular wave analysis. This implies that it is important to carry out irregular wave analyses in order to more accurately predict the performance of the WECs under actual sea state conditions.

7. Conclusions

In this study, we investigated the hydrodynamic and power performance of various layout configurations of heaving WEC arrays, when co-located with four different types of FOWTs. The simulations were conducted in the frequency domain, using regular wave conditions. For the simplicity of computational time and to capture the power performance of all configurations, we opted to employ a single constant minimum B_PTO, whose frequency mostly fell in the vicinity of its respective WEC’s resonance. It is worth noting that one may explore a varying optimal B_PTO as its operating sea state to enhance power generation.

The objective of this analysis was to assess and compare the performance of each configuration against key metrics, such as absorbed power, capture width, and interaction factor, with the expectation of determining the type of WEC layout and FOWT combination that showed a superior performance under specific conditions. Furthermore, we validated the numerical model used in our analyses by comparing the simulated absorbed power generation as well as the heaving RAOs with two different publications. These findings indicated a good agreement between the simulated results obtained by the numerical model and the established data during the verification process. The main findings included the following:

• Energy capture tends to decrease as WEC arrays increase in complexity, meaning that using the maximum number of WECs, such as in Type I, is not as efficient as its equivalent $(A_{wp}$), such as Type II or III, for maintaining a stable and high capture width.
• Among the tested FOWT foundations, the SS-B demonstrated superior performance across multiple wave directions, particularly at oblique angles (45°). The SS-B foundation’s geometry contributed to a higher interaction factor, maximizing power absorption. This finding highlights the critical role of platform design in enhancing WEC effectiveness.
• The type and layout of WECs (e.g., cylindrical versus torus-shaped) play a pivotal role, with cylindrical configurations generally showing better performance across various sea states compared to torus-shaped arrays. The configurations with torus-shaped WECs generated the least total power.
• The highest power absorption fell in the WECs’ resonant period region, which was directly related to the WECs’ draft and diameters. The configurations tuned to the vicinity of the natural resonant frequencies of the WECs yielded optimal energy generation. This supports the need for tailored designs and controls that align with the anticipated wave frequencies at deployment sites.
• Among the spar configurations, Spar-I, II, and III were not significantly different in their power generation, as the WECs’ co-location footprint of the structure was limited.
• The OC4 semi-submersible-WEC layout generated the highest total power for SS-B-II under all wave directions. SS-B-II’s power was the most effective under the 45° wave direction.
• For two semi-submersible configurations (both I and III), it was observed that SS-II and SS-B-II generally performed better in all wave directions.
• At wave period 7.85 s, Barge-II and III performed much better in power generation than all other configurations because of the power enhancement at the moonpool resonance.
• Although the highest total power in the torus-shaped WEC arrays was the least among all other configurations, they exhibited the highest average power (Spar-IV, SS-IV to VI, and SS-B-IV) as they were open to most wave directions, with less of a shielding effect.
• Among the single torus WECs deployed at the center columns of SS-VI and SS-B-IV, the latter, with the bigger side column, exhibited significantly more power generation due to the constructive interference of the wave diffraction between the columns. This also implies that the wave diffraction from the larger side column made SS-B more suitable for torus-shaped WECs to be deployed at the center columns. SS-B-IV also produced more power than Spar-IV (with a single torus WEC).
• Among all the layouts, except torus types, which had the least total absorbed power, Barge-III’s CWR generally showed better performance at wave periods above 4 s.
• When the WEC arrays were exclusively positioned around the exterior of the barge, the constructive q-factor extended from 3.4 s to 7.0 s, with an average q-factor of around 1.5. When combined with the moonpool-based WECs, the overall barge configuration derived additional resonance benefits.
• The results indicating a lower absorbed power in irregular wave conditions compared to regular waves suggested that regular wave conditions are likely to overestimate power generation.

This study was significant in its power performance assessment of the WEC arrays when co-located with the various foundations of FOWT. For a front-end analysis approach, future work should address the limitations of the present model, such as the effect of viscosity and the coupling of aerodynamic loads from the wind turbine, to achieve more reliable outcomes. Additionally, a time domain analysis should be included to assess the performance of WEC arrays accurately when co-located with FOWT foundations and moorings.