Hydrodynamic Responses of a Novel Modular Floating Structure System With Multi-Direction Expansion

A modular floating structure (MFS) is usually composed of a series of modules expanded through connectors, which can provide a space on the sea for the utilization of both marine energy and ocean resources [1–3]. Different from the traditional very large floating structure (VLFS), the MFS is convenient for production, transportation, installation and maintenance [4,5], and the large bending moment caused by the excessive size of the continuous structure can be effectively reduced [6,7]. At present, the research on the MFS system mainly focuses on the hydrodynamic and hydroelastic responses [8]. Loukogeorgaki et al. [9] analyzed the hydroelastic responses of a modular floating breakwater system through both numerical simulation and physical model test, and the deformation characteristics of the MFS system have been investigated. Ren et al. [10] proposed a seven-module MFS system, and the outermost connector effect on the dynamic response characteristics has been clarified. Wang et al. [11] conducted a hydrodynamic analysis for an MFS system sheltered by a floating seawall, and useful suggestions for improving the habitability of the MFS system have been pointed out.

The integration of wave energy converters (WEC) into floating structures has become a cost-effective way to both utilize marine renewable energy and improve floating structures’ motion performance [12–15]. Zhang et al. [16] installed a WEC for an MFS system to utilize the modules’ relative motion for wave energy production. Nguyen et al. [17] attached two WEC auxiliary plates on the outside of a VLFS for using the movement of the auxiliary plate to convert wave energy. Ren et al. [18] coupled the WEC devices with the outermost modules of an MFS system, and it indicates that the outermost modules can effectively reduce the wave loads acting on the inner modules, as well as provide considerable wave power.

For the research of MFS systems, the connector and the mooring system are the two key factors [19,20]. Due to the limited size of each module, structural deformation mainly occurs at flexible connectors. Therefore, the MFS can be viewed as a type of rigid-module-flexible-connector (RMFC) system. Michailides et al. [21] studied the mechanical characteristics of different flexible connectors for an MFS system, which provided a helpful reference for the optimal design of connectors. In addition, there will be a large relative vertical displacement among adjacent modules of MFS systems due to potential large vertical loads difference among modules [22]. Considering the buoyancy of the tension leg platform is much greater than its gravity [23,24], Ren et al. [22] developed research on an MFS system with tension leg platforms, and it indicates that the tension leg MFS system can effectively solve the issue of modules’ large vertical displacement difference.

In summary, previous studies on MFS systems mainly focused on rectangular or semi-submersible modules in unidirectional series [25–27]. However, there is very limited research on MFS systems with different module shapes, especially for multi-directional expansions. Therefore, a novel MFS system with multi-directional expansion has been proposed in this work, which involves inner hexagonal tension leg platform (TLP) modules and outermost floating artificial reef modules coupled with WECs. Floating artificial reef modules can not only improve the stability of the MFS system by effectively attenuating wave loads, but also improve the marine ecological environment by providing a habitat for marine organisms [28,29]. The multi-directional expansion characteristic can attenuate wave loads from different wave directions, which can significantly reduce responses for inner modules. In addition, it is different from the previous relative pitch or heave wave energy generation of modular floating structures integrated with WECs, and the novel MFS system can utilize the relative yaw motion between adjacent connected modules to convert wave energy.

Considering the randomness of wave directions, the present work particularly studied the effect of incident wave directions on dynamic responses of the MFS system under typical sea cases. Both the hydrodynamic interaction effect and the mechanical coupling effect have been considered. The effects of power take-off (PTO) parameters, incident wave directions, and wave periods on the wave power generation performance of WECs have been further investigated. In addition, the safety of the proposed MFS system has been effectively verified under extreme sea conditions.

A novel modular floating structure (MFS) system has been proposed in this work, which includes four inner hexagonal TLP modules and three outermost floating artificial reef modules coupled with the function of WECs. The proposed MFS system symmetrically expands in three directions, which can utilize the floating artificial reef modules to attenuate waves in multiple directions, as shown in Fig. 1. Main design parameters of the proposed MFS system are listed in Table 1. For the proposed MFS system, each module is marked as Mi (i = 1–7), each connector is marked as C_i (i = 1–6), and each WEC is marked as WECi (i = 1–6), as shown in Fig. 1. The main modules of the MFS system are described as follows:

1. Hexagonal TLP module:

   Each hexagonal floating structure is moored to seabed by four tension legs, and modular expands through flexible ball connectors. Tension legs are symmetrically distributed at four bottom corners of each hexagonal floating structure, and fenders are installed at the bottom edge of adjacent modules to monitor the possible collision.

2. Floating artificial reef module:

   The floating artificial reef adopts the hollow box structure without a top cover, which has the advantages of simple configuration, convenient production, and self-balance (Buoyancy equals Gravity). It can play a role as a breakwater and also provide a habitat for marine organisms [30,31]. Environmentally friendly marine concrete is suggested as the material.

   Under wave loads, relative motions between adjacent modules of the MFS system can happen due to flexible ball connectors. Hence, combining WECs function with ball connectors can effectively convert wave energy into electricity by utilizing relative motions between adjacent modules. In Fig. 1, C1, C2, and C3 are defined as inside connectors, while C4, C5, and C6 are defined as outside connectors. Corresponding WEC locations are defined by the same principle as the connectors. To further improve the connector performance, two connector types have been proposed as follows, which are shown in Fig. 2:

   1. A ball connector with additional rotational linear dampers (K_p). The relative rotational motions between two adjacent connected modules can be limited and utilized for wave power production by the additional linear damper effects of the WEC’s PTO system.

   2. A ball connector with both additional linear rotational dampers (K_p) and linear rotational springs (K_s). Compared with connector A, connector B can further limit the relative rotational motions between two adjacent connected modules due to the effect of additional rotational springs, which is specially designed for outermost floating artificial reef modules.

The effects of the key connector parameters, the wave direction, and the wave period on dynamic responses of the proposed MFS system have been investigated. The safety of the MFS system under typical extreme sea cases has been further checked.

To investigate the effect of the connector stiffness (K_s) on dynamic responses of the MFS system, a typical regular sea condition (incident wave direction β = 0 deg, wave period T = 8 s, and wave height H = 2 m) has been selected for the hydrodynamic analysis. First, all connectors have been set as Type A (with K_p = 1 × 10⁸ Nms/rad and K_s = 0). The main dynamic responses of the MFS system are shown in Fig. 4. For each hexagonal TLP module, motion amplitudes in both roll and pitch are approximately 0, while the yaw motion is considerably large (in Fig. 4(a)). It indicates that tension legs can effectively limit the roll and pitch motion of hexagonal modules, but the yaw motion is still noticeable. As a result, the connector bending moment in yaw is much greater than those in both roll and pitch (in Fig. 4(b)). In addition, compared with inside connectors, outside connectors suffer larger bending moments in both roll and pitch. That is mainly because the outside floating artificial reef modules are without the constraints of tension legs.

It should be noticed that bottom fenders monitored the collision between the outside floating artificial reef module and its adjacent hexagonal TLP module during the time-domain analysis. The collision time of the MFS system is shown in Fig. 5, and the time history of the fender force is shown in Fig. 6. It can be clearly seen that the large yaw motion of the outside floating artificial reef module is the main reason for the collision. Therefore, outside connectors have been further set as Type B to improve the stability of floating artificial reef modules by adding springs in the yaw. For the optimization of the outside connectors, K_s, dynamic responses of the floating artificial reef module with different K_s have been further analyzed under the same sea condition.

The time history of the M5 yaw and the C4 bending moment Mz is shown in Fig. 7, and there is no observed bottom impact during simulations with outside connectors of Type B. As shown in Fig. 7(a), the M5 still occurs in slow drift motion for K_s = 0.5 × 10⁷ Nm/rad. When the Ks increases to 1.0 × 10⁷ Nm/rad, the yaw responses of the M5 tend to be stable with smaller amplitude. As shown in Fig. 7(b), the bending moment Mz increases as the K_s increases. Therefore, to avoid the bottom collision, it is suggested to select 1 × 10⁷ Nm/rad as the preliminary optimal stiffness without significantly increasing the connector force.

Based on the analysis of Sec. 3.1, it may be possible that utilizing the relative rotational motions among adjacent modules of the proposed MFS system drives WECs to produce considerable electricity power. To optimize the power generation performance of both inside WECs (between TLP modules) and outside WECs (between the floating artificial reef module and the TLP module), the effects of the corresponding connector damping (K_p) have been further investigated.

First, the inside WECs’ performance (between TLP modules) has been analyzed with different inside connector K_p under a typical regular sea condition (β = 0 deg, T = 8 s, and H = 2 m). The main dynamic responses of the MFS system are shown in Fig. 8, including the yaw of different inside hexagonal TLP modules, the bending moment Mz of inside connectors, and the average output power of inside WECs. It can be found that the corresponding responses of M2, C1, and WEC1 are the largest for the incident wave direction of 0 deg. In Fig. 8(a), the yaw amplitudes of the inside hexagonal TLP modules can be effectively reduced with the increase of K_p, especially for M2. However, the corresponding bending moment Mz of C1 increases from 20 MNm to 38 MNm (in Fig. 8(b)). In Fig. 8(c), the average output power of each inside WEC first increases as the K_p increases for the K_p less than 4 × 10⁹ Nms/rad, but it gradually decreases for the K_p larger than 4 × 10⁹ Nms/rad. The total average output power curve of inside WECs with a series of the K_p is shown in Fig. 8(d), and the maximum output power of inside WECs (about 220 KW) is obtained with the K_p of 4 × 10⁹ Nms/rad. Therefore, it is suggested to select 4 × 10⁹ Nms/rad as the preliminary optimal K_p for inside WECs.

Then, the effect of the K_p on outside WECs’ performance (between the floating artificial reef module and the TLP module) has also been further investigated under the same sea condition (β = 0 deg, T = 8 s, and H = 2 m). In Fig. 9(a), the relative yaw velocity ω_ref between M2 and M5 is the largest for the incident wave direction of 0 deg. Therefore, it is reasonable that the corresponding average output power of the WEC4 (between M2 and M5) is the largest among outside WECs (Fig. 9(b)). In Fig. 9(a), it can be observed that the ω_ref decreases with the increase in the K_p, so the ω_ref can be effectively adjusted by the K_p. In Fig. 9(b), with the increase in the K_p, the average output power of each outside WEC first increases for the K_p less than 2 × 10⁸ Nms/rad, but it decreases for the K_p larger than 2 × 10⁸ Nms/rad. The total average output power curve of outside WECs with different K_p is shown in Fig. 9(c). It rapidly rises to nearly 115 KW as the K_p increases to 2 × 10⁸ Nms/rad, and then falls sharply as the Kp increases larger than 2 × 10⁸ Nms/rad. Thus, it is recommended to select 2 × 10⁸ Nms/rad as the optimal K_p for outside WECs. In Fig. 9(d), it can be seen that the power generation contribution of outside WECs due to the relative yaw is the most significant among the three relative rotations.

Furthermore, the preliminary optimal design parameters of corresponding connectors have been listed in Table 2, which are applied for the following research.

Considering the randomness of incident wave directions, the main dynamic responses of the MFS system have been further investigated under a typical regular sea condition (T = 8 s, H = 2 m) with different incident wave directions (from 0 deg to 90 deg). The main motion responses of each hexagonal TLP module under different incident wave directions are compared in Fig. 10. It can be seen that the motion responses of each module are greatly affected by the change in the wave direction. The maximum responses of the yaw, the sway, and the heave all appear with the incident wave direction of 0 deg or 60 deg, while the motion responses of each module are almost all relatively small with the incident wave direction of 90 deg. Therefore, when the floating artificial reef is the up-wave or the down-wave module, it can play a good shelter role for inner TLP modules. In Fig. 10(c), the heave responses of each module under different incident wave directions are negligible due to the good performance of the tension leg system.

The main connection forces of each connector under different incident wave directions are compared in Fig. 11. Compared with inside connectors, Fx, Fy, and Mz of outside connectors are significantly small for all incident wave directions (in Figs. 11(a), 11(b), and 11(d)). The reason is that the mass and constraints of the hexagonal module and the floating artificial reef module are different. In Fig. 11(c), the Fz responses of both inside and outside connectors are quite similar, which is due to the small relative heave motion of adjacent modules. It should be noted that the Fz obtains a huge value for β = 90 deg, which is attributed to significant relative heave responses between M1 and M2. It also illustrates that the MFS system with the multi-direction expansion is sensitive to wave directions. In addition, it should be noticed that the maximum of Fy, Fz, and Mz all occur at β = 90 deg, indicating that the connector loads tend to increase when the floating artificial reef is the up-wave module.

The power generation performance of each WEC under different incident wave directions is shown in Fig. 12. It can be seen that the WEC performance varies with respect to wave directions. In Fig. 12(a), the maximum average output power of most WECs appears in the incident wave direction of 90 deg, which indicates the good power generation performance of the WEC at β = 90 deg. Figure 12(b) depicts the trend of the output power of WECs with different incident wave angles. It can be seen that the MFS system can obtain the maximum energy production (about 600 KW) at β = 90 deg.

According to the work in Sec. 3.3, dynamic responses of the MFS system are relatively good at β = 90 deg, while those seem unfavorable at β = 0 deg. Therefore, the wave period effect on dynamic responses of the MFS system has been further investigated with the incident wave direction of both 0 deg and 90 deg (H = 2 m, T = 4 s∼20 s). At β = 0 deg, both the surge and the yaw of each module are relatively significant, so Fig. 13 shows the corresponding trends with different wave periods. In Fig. 13(a), the surge curves for all modules almost coincide, which gradually increase as the wave period increases. In Fig. 13(b), all curves first rise. The peaks of both M1 and M2 occur at the wave period of about 8 s, while the peaks of both M3 and M4 occur at the wave period of 12 s. That is because of the difference in the modules’ relationship between the wavelength and the relative structure dimension at β = 0 deg.

At β = 90 deg, both the sway and the yaw responses of each module are relatively significant, so Fig. 14 presents their trends with different wave periods. Similar to β = 0 deg, the sway curves of each module are with the same trend. It gradually rises as the wave period increases (Fig. 14(a)). In Fig. 14(b), the yaw responses of both M1 and M2 are almost 0 for any wave period, which is due to both the symmetric wave loads and the good shielding effect of the floating artificial reef module. The yaw curves of both M3 and M4 coincide well. They rise steeply when T < 10 s and fall steadily when T > 10 s. It indicates that the wave period of 10 s is the most unfavorable for the MFS system at β = 90 deg, which is also due to the relationship between the structure dimension (at β = 90 deg) and the wavelength.

The trends of connector forces with different wave periods have been further analyzed under both β = 0 deg and β = 90 deg, as shown in Fig. 15. It can be observed that the Fz response is the smallest among the three connector forces. With the increase in the wave period, both the Fx and the Fy rise until reaching their corresponding peaks, and then they gradually fall. It should be mentioned that the Fy peak of the C1 under β = 0 deg and β = 90 deg appears at wave periods of 6 s and 8 s, respectively. It is mainly attributed to different relationships between the wave period and the structure dimension under different wave directions.

The wave period effect on WECs’ performance has further been investigated with β = 0 deg and H = 2 m. The average output power information for both inside and outside WECs is shown in Fig. 16. In Fig. 16(a), each WEC performance is sensitive to the wave period. All curves first rise with the increase in the wave period, then fall, and finally remain stable. A similar trend can also be seen in Fig. 16(b). In Fig. 16(a), the maximum outside WEC power (about 200 KW) occurs at about 6 s for the WEC5, while the maximum inside WEC power (about 150 KW) occurs at about 10 s for the WEC2. In Fig. 16(b), it can be seen that the outside WEC tends to produce more power for the wave period less than 8 s, while the inside WECs display better performance for the wave period from 8 s to 14 s. In addition, it is noticeable that the curve of average output power for all WECs appears two peaks. One is due to the main contribution from outside WECs with a wave period of 6 s, and the other is due to the main contribution from inside WECs with a wave period of 10 s. Therefore, the two ideal wave periods for WECs of the MFS system are about 6 s and 10 s.

To validate the safety of the MFS system under extreme sea conditions, extreme responses of the MFS system have been further analyzed. The JONSWAP spectrum (λ = 3.3, H_s = 4 m, and T_p = 10 s) has been applied for simulating extreme sea conditions with the incident wave direction of both 0 deg and 90 deg. The main extreme response characteristics of the MFS system are listed in Table 3. For β = 0 deg, both the surge and the yaw of M2 are the largest (4.144 m and 3.80 deg, respectively), while the sway of M4 is the largest (2.249 m). The extreme Fx of the C2 is the largest connector force (14.685 MN), which is much larger than the extreme Fy (11.387 MN) and the extreme Fz (1.452 MN). These phenomena are consistent with the previous results in Fig. 15. In addition, WECs of the MFS system can produce considerable energy power under extreme sea condition, and the transient maximum output power can be up to about 9900 KW.

For β = 90 deg, the surge, the sway, and the yaw responses of M3 and M4 are significantly affected by the structural symmetry, which is similar to the conclusion obtained earlier. The extreme connector forces tend to appear in the C1 and the C3. The extreme Fy of the C1 is the largest (13.198 MN) due to the incident wave direction. In addition, both the maximum out power and the average out power are much larger (by about 50%) than those for β = 0 deg.

By comparing the extreme responses of the MFS system with the two incident wave directions, it can be concluded that the response of the modules’ motions, the tension leg forces, and the output energy power is much better for the β = 90 deg. It indicates that the MFS system is more suitable for survival (under extreme sea conditions) with the floating artificial reef as the up-wave module (β = 90 deg). In addition, there is no modules’ bottom impact (no fender force observed) during the analysis, so the safety of the MFS system has been preliminarily verified under typical extreme sea conditions.

This work has proposed a novel MFS system with multi-directional expansion, which consists of inner hexagonal TLP modules, outermost floating artificial reef modules and WECs. Both the hydrodynamic interaction effect and the mechanical coupling effect have been considered. Effects of incident wave directions, wave periods, and key connector parameters on dynamic responses of the MFS system have been emphatically studied. The main results are summarized as follows:

1. The incident wave direction exerts a significant influence on the dynamic responses of the MFS system. When the outermost floating artificial reef is the up-wave module, the motion responses of each hexagonal TLP module and the wave power generation can be significantly improved, although the connector forces increase to a certain extent.

2. For dynamic responses of the MFS system with different wave periods, motion responses seem sensitive to long-wave periods. The maximum connector forces tend to appear for the wave period 6–8 s, which is due to the relationship between the module dimension and the wavelength.

3. The relative yaw motion between adjacent modules is more significant, compared with the roll and pitch motions. It indicates that the relative yaw motion between adjacent modules can drive the WECs to produce more considerable wave power. The optimal outside WEC damping and inside WEC damping are suggested to be 2 × 10⁸ Nms/rad and 4 × 10⁹ Nms/rad, respectively. The corresponding optimal wave period for WECs is about 6 s and 10 s, respectively.

4. The safety of the MFS system has been further verified under typical extreme sea conditions. As the incident wave direction is 90 deg, extreme responses of the MFS system are much better than those with the incident wave direction of 0 deg, especially for the tension leg force and wave power production. It can provide an instructive reference for the installation layout of the MFS system.

Many challenges still remain for the MFS system, such as the validation of model tests, the MFS system analysis with more modules, the attracting effect of the floating artificial reef, and the long-term fatigue damage analysis of main structures. These challenges should be studied in future work.

This research was supported by the Natural Science Foundation of Hainan Province (Grant Nos. 520RC552 and ZDYF2021GXJS034), the National Natural Science Foundation of China (Grant No. 52161041), Foundation of State Key Laboratory of Coastal and Offshore Engineering (Grant No. LP2119). The financial supports are greatly acknowledged.

There are no conflicts of interest.

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

H =

wave height

T =

wave period

K_p =

damping coefficient

K_s =

elastic stiffness

Fx =

horizontal force of connector

Fy =

vertical force of connector

Fz =

shear force of connector

Mz =

yaw bending moment of connector

JONSWAP =

Joint North Sea Wave Project

β =

incident wave direction