Hydroelastic Analysis of Interconnected Offshore Floating Photovoltaic Floats Open Access

Floating photovoltaic (FPV) systems represent a new and promising technology in renewable energy. With the growing demand for environmentally friendly energy, the availability of land for energy infrastructure is becoming increasingly limited. As a result, such floating systems have emerged as a viable alternative. Research on these systems is crucial, considering their technical, economic, and environmental implications.

The use of sea areas has gained attention for the implementation of renewable energy systems, including FPV systems, which harness clean and renewable energy from marine environments. Given their large dimensions on the water surface, utility-scale FPV systems can be classified as very large floating structures (VLFS). Due to these dimensions, they experience elastic responses in addition to rigid-body motions. FPVs, as a special type of VLFS, have garnered significant research interest since the establishment of the first commercial floating facility in 2007 [1], leading to advancements in design and application to enhance economic and technical performance.

An FPV array typically consists of components such as floating platforms, mooring and anchoring systems, underwater power cables, and transformers. The floating platforms (floats) are used to support the photovoltaic (PV) modules [2]. The mooring system is designed to adapt to fluctuations in water levels while maintaining the position of the array, and the electrical components transfer the generated power to the shore. Figure 1 provides a schematic representation of an FPV array with its structural components.

Floating solar installations offer multiple benefits, including enhanced efficiency and energy production due to the water’s temperature-regulating effect, reduced maintenance costs through decreased algae growth, and minimized water evaporation, thereby preserving water for other uses. Floating platforms for FPV arrays consist of modular, prefabricated floats that are interconnected to form large sections. These platforms are assembled on land by linking rows of modular floats, which are then launched into the water sequentially. Once assembled, the entire platform is towed to its designated location using boats [3].

During the past decade, FPV technologies have advanced significantly, but they are still used mainly in freshwater bodies or sheltered coastal areas with calm waters. These locations are limited in space and cannot meet the growing demand for energy production. In contrast, the open sea offers vast space, making it possible to deploy much larger floating solar arrays without the limitations found in smaller, confined areas. For instance, Moss Maritime AS has designed an FPV system with modular, flexible floats that can adapt to the longer wavelengths of ocean waves [4]. Similarly, SolarDuck is working on designs that address the technical challenges of operating in the open sea [5]. Ocean Sun AS has developed a membrane-based concept with a circular floating structure made from high-density polyethylene (HDPE) to handle strong winds and repetitive wave impacts [6]. Figure 2 highlights two FPV systems, where the Ocean Sun [6] and the SolarDuck [7] concepts have very different features. The proprietary nature of commercial developments has led to limited information on the design specifications of such FPV concepts in the public domain.

Traditionally, FPV platforms have been designed under the assumption that their support structures behave as rigid bodies, neglecting structural deformations induced by wave action. For instance, Song et al. [8] analyzed the dynamic response of multi-connected floating solar panel systems subjected to wave loads using a rigid-body approach. Similarly, Al-Yacouby et al. [9] conducted hydrodynamic assessments of offshore FPV farms, also assuming rigid-body behavior. Recently, Tang et al. [10] proposed more flexible designs for FPV systems, highlighting the potential benefits of semi-submersible platforms with varying stiffness tensioners. Furthermore, Huang et al. [11] reviewed various FPV systems and emphasized the importance of sustainable designs and addressing environmental and structural challenges in offshore applications.

However, recent research has highlighted the importance of hydroelastic effects in FPV performance, particularly in offshore environments where wave-induced bending and plate displacements are critical considerations. To address the limitations of rigid-body assumptions, Sree et al. [12] investigated the hydroelastic response of offshore FPVs using a multi-scale numerical approach, simulating 6000 interconnected floating modular units. Building upon this, Xu and Wellens [13] employed an analytical solution up to the third order, modeling the interconnected units using the Euler–Bernoulli–von Kármán beam theory to better capture structural flexibility and dynamic response.

Despite these advancements, the application of FPVs in ocean environments remains in its early stages, presenting significant challenges such as corrosion, biofouling, and structural failure. The hydroelastic challenges faced by VLFSs under varying wave conditions can significantly impact their performance and long-term viability. Chen et al. [14] proposed a coupled constant-parameter hydrodynamic-structural model, offering a novel approach to analyzing the dynamics of multi-module floating systems with flexible connectors. This model provides valuable insights into the influence of connector flexibility on system performance and serves as a useful tool for optimizing FPV system designs in offshore conditions. Huang et al. [11] provided a comprehensive review of land-based and offshore FPV systems, discussing key projects, technological advancements, and challenges associated with photovoltaic power generation in marine environments. Their findings highlight the importance of optimizing system design for offshore applications, addressing issues such as material selection, energy efficiency, and environmental impact.

For large-scale marine floating systems comprising numerous interconnected floats, mitigating hydroelastic effects such as bending moments and plate displacements is essential. Zhang et al. [15] found that adding a single hinge to a VLFS did not reduce the maximum bending moment along its length. However, increasing the number of hinges led to a significant reduction. Additionally, when the distance between each hinge equaled half the wavelength, the vertical displacement along the structure increased substantially. Given that most VLFSs are constructed by connecting individual modules, the selection of appropriate connectors plays a crucial role in determining the overall hydroelastic response.

Further expanding on hydroelastic analysis, Tay [16] conducted a three-dimensional study of offshore floating photovoltaic systems (OFPVs), modeling the floating structure as a mat-like VLFS using the Kirchhoff-Love thin plate theory and linear potential wave theory. A hybrid boundary element method-finite element method (BEM-FEM) was employed to analyze fluid–structure interactions. The findings demonstrated the influence of structural layout and wave periods on hydroelastic response, providing valuable insights into optimal configurations for OFPV systems under varying wave conditions.

This research focuses on the hydroelastic analysis of a novel FPV concept. In this design, each float takes the form of a semi-submersible, with ropes used to connect adjacent floats to form an array [5]. Lightweight and cost-effective materials such as fibers or HDPE are considered for the floats [17]. Previous model tests demonstrated the FPV concept’s good motion performance under harsh offshore conditions, and hydrodynamic analysis for a single float has been conducted [5]. However, limited research exists on the hydroelastic behavior of rope-connected floats.

In a recent study, Leventopoulou et al. [18] investigated the hydroelasticity of this FPV concept, focusing on a double-plate configuration with a single material property and two bending stiffness levels, corresponding to fully hinged and rigid connections. A key next step is exploring the influence of realistic rope connection bending stiffness. Research on similar structures, such as prestressed cables, has shown that bending stiffness significantly affects dynamic response characteristics, including cable tensions [19,20]. Theoretical and numerical analyses have been used to calculate the bending stiffness of seven-wire steel strands and semi-parallel wire cables with simple cross sections [20]. Applying similar methodologies, the equivalent bending stiffness of realistic materials can be numerically estimated using the FEM [21].

In this study, a hydroelasticity analysis of the FPV concept is carried out considering different inter-modular connection stiffness, float layouts, and wave conditions. The scenarios of two and three interconnected plates are addressed, and the bending stiffness of the connection ropes is linearized and then implemented in a hybrid higher-order BEM-FEM code. A parametric study is conducted to study the influence of connection stiffness on the responses of the plates. Through a comparison of the displacement, bending moment, and stress distribution under various wave periods and headings, we show interesting features and structural integrity of the FPV concept.

The structure of the article is organized as follows. In Sec. 2, the problem definition and the scope of the study are defined. The methodology used in the study is presented in Sec. 3, which includes the hybrid numerical method and analytical estimates. In Sec. 4, the numerical modeling and implementation is verified through comprehensive comparisons. The case studies analyzed in the article are introduced in Sec. 5, with results and discussions presented in Sec. 6. Finally, Sec. 7 summarizes the key findings and suggests directions for future research.

This study investigates an FPV concept featuring standardized, lightweight semi-submersible floats made from circular materials. Each float has a streamlined profile and can house four standard PV panels arranged in a dual-pitch configuration to mitigate the shading effect of one panel on its neighboring panels. These floating modules are flexibly connected with ropes to form an FPV array. This configuration allows the modules to follow the wave motion in extreme seas. The conceptual design, as shown in Fig. 3, forms the basis for this research.

Figure 4 shows the analysis flow of the article. This procedure involves seven steps. First, in Step 1, the semi-submersible float is simplified as an equivalent solid rectangular plate characterized by length $L$⁠, width $B$⁠, thickness $h$⁠, Young’s modulus $E$⁠, Poisson’s ratio $ν$⁠, and mass density $ρ$⁠. The linear bending (⁠$ks$⁠) and torsional stiffness (⁠$kθ$⁠) of the connections are estimated through both analytical and numerical methods. Then, in Step 2, a numerical model of a multi-float system in the high-order hybrid BEM-FEM code is established, and the corresponding degrees-of-freedom of the corner nodes of each plate must be identified, followed by a modification of the stiffness matrix in Step 3. In Step 4, this updated code is verified against the existing literature on a plate with mooring lines to ensure the applicability of the method to FPV systems with connections. In Step 5, the case study is defined by specifying the number of plates, plate orientations, rope materials, and wave conditions to create a comprehensive framework for analyzing the effects of connection stiffness. In Step 6, numerical simulations are conducted. A mesh convergence study is first carried out to determine the optimal element size, followed by a parametric analysis of the hydroelastic responses for the single, double, and triple float configurations with varying orientations and connection stiffness. Two assumed connections with zero and infinite stiffness are considered in addition to the stiffness of two realistic materials. In Step 7, the plate responses from the parametric study are analyzed, and the influence of connection stiffness on the structural responses is highlighted.

The calculations are conducted for a circular cross-sectional rope that connects the two plates. The rope is modeled as a beam without prestress, connecting the two adjacent plate corners (Fig. 5). To obtain the bending stiffness coefficient, a finite element model of the beam is established, and different levels of axial forces are applied at one end of the beam in the $z$-direction. The resultant displacements are used to derive the linearized stiffness. A similar procedure is followed to obtain the torsional stiffness coefficient, applying different moments about the $x$-axis (Fig. 5).

One of the main inputs that identify the rope properties is the effect of the material on the connection stiffness. Two different material properties have been tested for identifying the vertical bending stiffness and rotational stiffness in the $x$-axis, which will be used to calculate the hydroelasticity effects in the hybrid higher-order BEM-FEM code. A hybrid higher-order BEM and FEM approaches [24] were employed to model the interaction between the plate and the surrounding fluid. For the numerical modeling of the floats, eight-noded quadrature plate elements were used in the FEM, with each node accounting for three unknown variables. The direct integration method [23] was applied in the BEM-FEM coupling to solve the system efficiently. The two material properties that have been used are stainless steel and HDPE, and the characteristics of the rope are presented in Sec. 5. The verification of the calculations for the bending and rotational displacement has been carried out through individual analyses based on beam theory and the FEM software. Observations from model tests [5] showed that the rope connections were subjected to tension. Furthermore, the present hydroelastic analysis only considers the vertical degrees-of-freedom (DOFs). Hence, the connection properties are derived according to the Euler–Bernoulli beam theory focusing on the bending behavior. Due to the relatively short length of the ropes, both HDPE and stainless steel exhibit behavior consistent with the beam theory. The equations used to calculate the effective bending stiffness of the rope are presented below [20]:

The global stiffness matrix now incorporates both axial and transverse DOFs. Specifically, the four DOFs considered are the horizontal displacement at node $Na$⁠, the vertical displacement at node $Na$⁠, the horizontal displacement at node $Nb$⁠, and the vertical displacement at node $Nb$⁠. To calculate the stiffness matrix, the geometric and material properties of the beam must first be determined. These properties include the horizontal and vertical projections of the end points of the beam:

To complete the transformation, the coordinates of the nodes of the beam are obtained from arrays that represent the nodal coordinates and element connectivity. For each element j, the coordinates of the start (Coord) and end nodes (Elem) are determined using the following relations:

These nodal coordinates are essential for determining the orientation of the beam and for calculating its stiffness matrix in the global coordinate system. Given that this study involves both longitudinal and transverse configurations in the vertical DOF, the stiffness matrices for horizontal and vertical connections have been modified to account for bending and torsional stiffness in different orientations. For the longitudinal connection scenario, the stiffness matrix must be modified to represent the combined effects at the nodes, as shown in Fig. 6. This figure describes the configuration for horizontal bending stiffness, with similar adjustments applied to the second rope connection located at the opposite edge of the interconnected plates.

In this section, two realistic rope connection materials—HDPE and stainless steel—are evaluated, and the analytical formulas for connection stiffness presented earlier are validated by comparison with FEM simulations performed in SAP2000 [25]. The material properties used in the FEM simulations, as well as those considered in the case study, are provided in Table 2 of Sec. 5. Figure 7 illustrates the force–displacement and bending moment–rotation curves at the tip of the connection ropes where a concentrated force or a moment is applied; refer to Eqs. (10) and (11). For validation purposes, the corresponding FEM results are also plotted in the figure. As shown in Fig. 7, the analytical solutions exhibit excellent agreement with the FEM results, demonstrating the accuracy of the method. These results are then used to derive the axial stiffness $ks$ and rotational stiffness $kθ$ of the connection ropes. The derived stiffness parameters are incorporated into the hybrid higher-order BEM-FEM code by embedding them into the stiffness matrix, enabling the analysis of the hydroelastic responses of the interconnected plates. This analysis considers a range of wave periods and examines the effects of the two different connection material properties.

The hybrid higher-order BEM-FEM method is adopted for the analysis of the coupled plate-fluid system [24]. For the purpose of verifying the accuracy of the present numerical model, a case study involving a VLFS with two vertical mooring lines positioned at the corners of the weather-facing edge, as examined by Karperaki et al. [26], is considered. For consistency, the same parameters as those in the reference study are adopted, as detailed in Table 1. In this verification case, the VLFS is modeled as a floating plate with substantially larger planar dimensions compared to its height. As such, the draft of the floating plate is neglected for simplicity. Figure 8 provides a plan view of the VLFS configuration, where the two vertical mooring lines are represented by spring constants $ks1$ and $ks2$⁠.

Figure 9 shows the vertical hydroelastic displacement amplitude along the centerline of the floating plate in its longitudinal direction. For comparison, the results reported by Karperaki et al. [26] are also included in the figure. In this analysis, two mooring line stiffness values are considered, namely, $ks1=0N/m$ and $ks2=2450N/m$⁠. The present numerical model employs a mesh configuration of $20×2$ elements. As observed in Fig. 9, the results produced by the present model match well with those by Karperaki et al. [26], thereby validating the accuracy of the present numerical approach for hydroelastic analysis of floating plates with spring connections.

The hydroelastic responses of single and connected plates under various wave excitations are next investigated. The cases to be studied consider equivalent rectangular plates with the same overall dimensions representing the floating modules that are modeled according to the Mindlin plate theory. Table 2 shows the material properties of the equivalent rectangular plate that has been used for the case study.

The case study explores three different configurations. Case 1 focuses on a mesh convergence study, assessing how the total number of elements impacts the hydroelastic simulation results to determine the required mesh density for accurate and efficient computations. Case 2 examines two floating plates connected along their longitudinal or transverse direction, as shown in Fig. 10, with wave headings of 0 deg or 90 deg. The study is further extended in case 3, which examines the effect of connection stiffness on a configuration involving three interconnected plates along either their longitudinal or transverse direction, as shown in Fig. 10. Case 3 is intended to investigate the hydroelastic behavior of a system comprising multiple connected floating modules. A numerical analysis is conducted to examine the hydroelastic responses of each configuration. For both case 2 and case 3, the two inter-modular connection materials, namely, HDPE and stainless steel, are considered. Table 3 shows the characteristics of the two materials employed in the case study. For comparison purposes, the case of zero connection stiffness and rigid connection stiffness representing two extreme connection stiffness is also taken into account. In addition to connection stiffness, torsional stiffness was also considered. Table 4 lists the axial and torsional stiffness of four sets of rope connections between floating modules.

The hydroelastic analysis of the floating modules involved a mesh convergence study for a single float to identify the most suitable mesh size for the case studies [18]. In the convergence study, four different mesh sizes were tested. Table 5 shows the computational time involved for each of the four mesh configurations. This study was performed for an operational regular wave of 5 s with a wave heading of 0 deg. In all four mesh configurations, the elements have a nearly unit aspect ratio. Figure 11 presents the hydroelastic bending moment and vertical displacement amplitudes along the centerline of the floating module. As can be seen, as the mesh configuration increased, hydroelastic displacement amplitude also increased, while bending moments decreased. The study observed negligible differences between the results produced by mesh 3 and mesh 4, indicating that for meshes finer than mesh 3, further increasing the number of elements does not significantly improve the accuracy anymore. On the other hand, mesh 4 required significantly more computational time than mesh 3. Therefore, mesh 3 was chosen as the optimal mesh for subsequent case studies, balancing the need for accuracy with practical considerations of computational efficiency and resource constraints. Note that the convergence study was carried out using matlab on a laptop with i7-1360P CPU and 32 GB installed RAM.

A sensitivity study was performed to investigate the impact of different material properties on the hydroelastic behavior of the floating structure. Since material stiffness is a key factor affecting hydroelasticity, the study focused on variations in Young’s modulus, which directly affects the flexural rigidity of the structure.

For practical implementation, the study was conducted using a floating module with predefined size dimensions, as outlined in Table 2. To systematically investigate the role of float flexural rigidity, simulations were performed for three distinct levels of Young’s modulus: $105N/m2$⁠, $5.83⋅108N/m2$⁠, and $1015N/m2$⁠. These three levels represent an artificially highly flexible float, the present design, and a nearly rigid float.

The analysis was carried out over a wave period range of 5–15 s to capture the response amplitudes under varying wave frequencies. As demonstrated in Fig. 12, hydroelastic responses, including maximum vertical displacement and bending moment amplitudes, increased with the rise in Young’s modulus. Specifically, maximum bending moment amplitudes exhibited a significant percentage difference (PD) of 58.69% between the material properties with $E3=1015N/m2$ and $E2=5.83⋅108N/m2$⁠, while the differences in float maximum vertical displacement amplitude remained small at 0.017%. This indicates that although the hydroelastic displacement responses of the current float design are virtually the same as those corresponding to a rigid float, the large discrepancies in the bending moment responses highlight the importance of conducting hydroelastic analysis. Discrepancies in the maximum vertical displacement amplitude are observed when the float Young’s modulus is reduced to $E1=105N/m2$⁠, particularly for waves shorter than 11 s.

Case 2 is next investigated. In this scenario, the role of connection stiffness is examined, with a focus on how different stiffness levels affect the hydroelastic responses for the configuration of two floating plates that are longitudinally connected to each other under a 0 deg wave heading, i.e., the head sea condition. Figure 13 shows the maximum plate displacements and bending moments across different wave periods, considering four different connection stiffness levels: stainless steel, HDPE, rigid and no connection stiffness. As can be seen, with the increase in connection stiffness, the hydroelastic effects become more pronounced. When rigid stiffness is considered for the connection, it is observed that the hydroelastic bending moment and displacement amplitudes tend to be amplified. The higher bending moment and displacement amplitudes with rigid connections indicate a more significant response of the structure to wave-induced hydroelastic actions. A soft and lightweight design results in lower hydroelastic responses. For shorter wave periods, the differences between lower connection stiffness levels and rigid connections in terms of vertical hydroelastic plate displacement amplitudes are more significant, since with rigid connections they are higher, but for periods over 13 s, these differences are found to be small. Regarding the bending moments, the effects are still higher for a more rigid connection stiffness, but the differences become more significant for longer wave periods. An interesting observation is that, between the two proposed connection materials, the hydroelastic responses are rather similar despite the large difference in the connection stiffness, indicating that the adoption of practical semi-rigid connections of different materials would not likely affect the hydroelastic responses of the proposed FPV system significantly. Overall, for all the connection stiffness levels, the general trend is that the hydroelastic bending moments and vertical displacements are higher under shorter waves. For longer waves, the results become constant after 12.5 s and remain relatively low.

In this analysis, the impact of varying degrees of connection stiffness on hydroelastic phenomena is explored for 90 deg, as shown in Fig. 14. As can be seen, higher stiffness in the connection tends to increase the hydroelastic bending moments and vertical displacements. The amplified bending moments and plate deformation under such stiffnesses reflect a stronger structural reaction to wave actions. For the cases with HDPE and stainless steel connections, the connection stiffness has a negligible effect on the hydroelastic responses. However, a substantial difference is observed when a near-rigid connection is employed. In general, higher hydroelastic responses are expected with stiffer connections, and vice versa. Since the floats are designed with flexible out-of-plane rigidity, the effect of semi-rigid connections tends to be small and close to the case without connections. Besides, the hydroelastic responses are found to be high with short waves and decrease monotonically until they converge after 14 s. It may be thus concluded that while a rigid connection does amplify the hydroelastic bending moment and vertical displacement amplitudes, the conceptual design of the floats, being inherently lightweight and flexural, are instrumental in minimizing such effects. This results in a consistent hydroelastic performance regardless of the connection stiffness levels except for fully rigid connections. Furthermore, the global configuration and wave heading also play a role in the hydroelastic responses. With a 0 deg wave heading toward longitudinally connected plates, the hydroelasticity responses are more pronounced as compared with the case with a 90 deg heading toward transversely connected plates.

Previous case studies revealed only slight differences in the maximum hydroelastic bending moment and vertical displacement amplitudes between connected floats with stainless steel and HDPE connections. To further investigate these discrepancies, the hydroelastic effects are examined by comparing the effect of two connection stiffnesses on the hydroelastic responses of the floats along their centerline in the direction of incoming waves. Figure 15 shows the results for two connected plates considering both connections at the midpoints of the plates for a wave period range of 5–15 s in longitudinal configuration. The results show that the differences between the use of two connection stiffness are rather small despite that a stiffer connection tends to lead to a marginally higher responses. For the case with transversed connected plates as shown in Fig. 16, the difference in the hydroelastic responses induced by the use of different connection stiffness is even smaller.

Figures 17 and 18 illustrate the PD in bending moment and vertical displacement amplitudes for both longitudinally and transversely connected floats. As observed, PD remains below 5% for the entire floats, which, from an engineering perspective, suggests that connection stiffness within practical variations has negligible effects on the overall hydroelastic responses of the system under wave actions.

In this section, the von Mises stress for a single float is evaluated under an operational wave period of 5 s, with a focus on HDPE as the connection stiffness material. For the double-plate configuration, the hydroelastic responses between the two floats are found to be identical, justifying the analysis of an equivalent single float. The study is conducted for a wave heading of 0 deg, which is identified as the most adverse scenario. Table 6 shows the properties that are used for the von Mises stress calculations. In this case, the second moment of area $Iy$ is 1.229 m, normalized over the section length to $0.1182m4/m$⁠. Figure 19 illustrates the equivalent rectangular float for the von Mises stress calculations.

The maximum hydroelastic bending stress amplitude, i.e., the transfer function of the hydroelastic bending stress per unit wave amplitude, $σtransfer$⁠, is evaluated to be $1.1837kN/m3$⁠, while the transfer function of the maximum shear stress per unit wave amplitude, $τtransfer$⁠, is found to be $0.4075kN/m3$⁠. Using the von Mises stress criterion, which accounts for both bending and shear stresses, the transfer function of the von Mises stress amplitude $σVM,transfer$ is evaluated to be $1.2933kN/m3$⁠. This quantity, multiplied by an extreme wave amplitude, must be lower than the yield stress of HDPE to avoid any material failure in a survival sea state. Although no site-specific condition is considered in this study, an extreme wave height of $H=14m$ can be assumed based on previous research [5]. Accordingly, the maximum von Mises stress is estimated as $σVMmax=9.05kPa$⁠. This indicates that the designed float can withstand the hydroelastic actions.

To further understand the stress distribution among the three interconnected floats, a comparative analysis is conducted, focusing on stress levels under varying connection materials and wave periods. Figure 20 illustrates a comparison of the transfer function of von Mises stress for four different connection materials: stainless steel, HDPE, rigid, and no connection stiffness, across three floats in a longitudinal orientation subjected to a 0 deg wave heading. The analysis reveals that, for shorter wave periods, the rigid connection results in higher stress levels. However, as the wave period exceeds 13 s, structures with no connection stiffness exhibit greater flexibility, allowing them to undergo larger displacements under wave-induced loading. This increased displacement occurs because the floating structure can deform more freely in the absence of rigid connector constraints. For longer wave periods, which correspond to larger wavelengths, the wave-induced loads are distributed over a larger portion of the structure, resulting in significant bending moments and shear forces. Due to the absence of rigid connectors, the structure deforms more freely, leading to higher local stresses. While the von Mises stress amplitude is generally similar across all three plates, the highest value is observed in the first floater, particularly in its middle section, where the combined effects of bending and shear are most pronounced. Longer wave periods, typically associated with larger wavelengths, can also induce resonance effects if the structure’s natural frequency coincides with the frequency of the incoming wave energy. In the absence of connection stiffness, the floating segments may experience extreme motion, further amplifying stresses in specific regions due to the relative motion between components.

Next, the hydroelastic responses of three floats connected along their longitudinal direction are examined. Figure 21 shows the maximum hydroelastic bending moment and vertical displacement amplitudes with HDPE connections with incoming regular waves of 0 deg heading. Also presented in the figure are the results corresponding to two longitudinally connected floats for the purpose of comparison. As can be seen, both maximum hydroelastic bending moment and displacement amplitudes are rather similar between the two-plate and three-plate systems despite that those for the three-plate system appear to be slightly higher for short waves. Such a discrepancy is observed to diminish when the wave is longer than 7 s. In general, these show that the inclusion of more plates does not significantly affect the overall hydroelastic responses of the system. This implies that the use of a two-plate system may be sufficient in engineering design practice for the sake of both computational efficiency and accuracy.

In the last case study, the hydroelastic responses corresponding to the configuration of three plates arranged in their transverse direction connected with vertical connection stiffness under a wave heading of 90 deg is examined and compared with its two-plate counterpart, as shown in Fig. 22. The same HDPE connections used in Sec. 6.6 are employed here. As can be seen, the same observations for the longitudinally connected plates are applicable here. It appears that the difference between the two-plate and three-plate configurations when they are transversely connected is more visible for shorter waves as compared to longitudinally connected configurations. The maximum discrepancy is found to be slightly over 10% at a long period of 5 s. For waves longer than 6.5 s, the difference becomes negligible.

This article investigates the hydroelastic behavior of a recently proposed FPV concept. We simplify the semi-submersible floats as equivalent plates and then adopt a hybrid finite element-boundary element approach combining the Mindlin plate and potential flow theories. The effect of the connection stiffness among floats is addressed by modifying the bending and torsional terms of nodal points in the stiffness matrix. After verifying the modified code against the literature, we carried out numerical simulations for the configurations of two and three interconnected plates in two different orientations. The study examines the effects of wave headings, wave periods, and connection stiffness on the hydroelastic responses to understand how these factors influence the system’s performance.

Although the observed deflections and bending moments are relatively small for the considered FPV dimensions, hydroelastic analysis remains essential for a comprehensive understanding of the system’s response. As the planar dimensions of the floats increase relative to their height, flexural motions become more significant, making the structure behave similarly to a thick plate under wave action. Furthermore, hydroelastic analysis inherently includes hydrodynamic effects while also capturing structural deformations, providing a more holistic evaluation compared to purely rigid-body hydrodynamic models. This approach is particularly relevant for optimizing the FPV system’s design, as it allows for assessing the influence of connection stiffness, float geometry, and wave-induced flexural motions. Although this study focuses on a small FPV array, consisting of a maximum of three interconnected plates, the methodology lays the foundation for future analyses of larger-scale FPV systems, where hydroelastic effects are expected to play a more dominant role.

The key findings of the study are summarized as follows. First, for a given wave heading and wave period, the double plates are sensitive to shorter wave periods than to longer waves, and this sensitivity decreases as the wave period exceeds 12.5 s. Second, for the double plates with a connection, there is an overall increase in the hydroelastic effects with rigid stiffness, but through a lightweight connection design, the effects become lower. The findings indicate that for two rope materials (HDPE and stainless steel) with similar stiffness properties, the connection stiffness has a small effect on the displacement and bending moments of the interconnected plates, but a significant difference of 41% in the plate displacement and 73% in the bending moments can be observed at the extreme stiffness levels of zero and infinity.

Overall, an increased number of interconnected plates with lower connection stiffness effectively mitigates hydroelastic effects. This trend can be useful for the design and optimization of FPV systems to ensure that they are resilient to hydroelastic stresses and can maintain structural integrity under various wave conditions.

This study is limited to the hydroelastic analysis of an FPV system with simplified connections and few plates. Future studies could address additional factors that may influence hydroelastic performance, such as marine biofouling, nonlinear environmental loads, nonlinear inter-modular connection behavior, and long-term material degradation. Incorporating these factors would provide a more comprehensive understanding of system behavior and improve the practical applicability of the findings. Furthermore, expanding the study to larger FPV arrays with mooring design would be beneficial to gain deeper insights into the interactions among interconnected floats and to develop design strategies for larger-scale deployments. Another potential area for future research is the evaluation of the failure risk of the system under various working conditions. Assessing the system’s vulnerability under different scenarios could provide crucial information for improving the reliability and safety of FPV installations.

This work has been financed by the Norwegian Ministry of Education and Research granted through the Department of Engineering Sciences, University of Agder, and by the Research Council of Norway (Solar4Sea Technical Pilot, Project Number 345637). These funding sources are gratefully acknowledged.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.