Study on the Dynamic Response of Mooring System of Multiple Fish Cages under the Combined Effects of Waves and Currents

Abstract

Deep-sea aquaculture can alleviate the spatial and environmental pressure of near-shore aquaculture and produce higher quality aquatic products, which is the main development direction of global aquaculture. The coastline of China is relatively flat, with aquaculture operations typically operating in sea areas with water depths of approximately 30–50 m. However, with frequent typhoons and poor sea conditions, the design of mooring system has always been a difficult problem. This paper investigated the multiple cages, considering two layouts of 1 × 4 and 2 × 2, and proposed three different mooring system design schemes. The mooring line tension of the mooring systems under the self-storage condition was compared, and it was observed whether the mooring line accumulation and the contact between the mooring line and the steel structure occurred on the leeward side. Additionally, flexible net models were compared with rigid net models to evaluate the impact of net deformation on cage movement and mooring line tension. Finally, based on the optimal mooring design, the dynamic response of the mooring system under irregular wave conditions was analyzed and studied, providing important reference for the safety and economic design of the mooring system of multiple fish cages.

1. Introduction

With the saturation of offshore development, the ecological pressure of marine aquaculture is increasing rapidly. Meanwhile, the demand for high-quality fish is also increasing [1,2,3]. It has gradually become a trend to develop offshore marine aquaculture and build marine ranches [4,5]. Developing deep-sea aquaculture can produce cleaner and healthier high-quality aquatic products while reducing the occupation of nearshore aquaculture space, thereby alleviating pressure on coastal environments. Major countries in the world also attach great importance to aquaculture, and it has become a hot industry competing for development throughout the world.

The current research of fish cages focuses on the structure strength analysis, the hydrodynamic characteristics of net and mooring system design [6,7]. Fredriksson et al. [8] obtained the material properties of the high-density polyethylene floating coil through tensile tests, carried out the local strength check of the floating collar under the tension of mooring lines, and verified the accuracy of the numerical method through model experiments. Then, the stress of the floating collar and the critical load leading to the local failure of the structure were evaluated by a numerical method under typical working conditions. Liu et al. [9] carried out elastoplastic mechanical tests and finite element analysis on the guardrails of gravity cages. They investigated areas prone to structural failure, analyzed the frequency of loads leading to resonance fatigue failure, and optimized the section shape of the guardrail to enhance horizontal bending performance.

In the aspect of net hydrodynamics, the complete net system of offshore fish cages is usually regarded as a combination of limited net units. Based on this idea, Aarsnes and Rudi [10,11] systematically carried out flow load tests on single netting units, and established a mathematical mapping relationship between the drag coefficient and lift coefficient with net solidity ratio and the angle of attack of the incoming flow; this has been widely recognized in industry and academia. Zhan et al. [12]. further investigated the influence of Reynolds number and net shape on the hydrodynamic coefficients of nets through drag testing, and proposed a new mathematical model. Zhao et al. [13,14] carried out a systematic net unit test, revealing not only the effects of solidity ratio, angle of attack, and Reynolds number but also the influence of net material and structure on hydrodynamic coefficients.

In terms of the mooring system of floating fish cage, it mainly includes a single point mooring system and a multi-point mooring system [15]. DeCew et al. [16]. conducted numerical and experimental studies on the performance of small submerged aquaculture cages under single-point mooring subjected to current. Huang and Pan [17] assessed the risk of rope failure in single point mooring systems based on the long-term environmental load and presented the failure probability during the service life. Hou [18] carried out a reliability analysis on the fatigue damage in the mooring system of the deep-sea fish cage. Shainee et al. [19] studied the submerged characteristics of a self-submersible single-point mooring cage system under the action of wave and current. Fredriksson et al. [20] used numerical models and field-measured data to study the tension of a mooring system in a large fishing ground without waves. Huang et al. [21] used numerical models to analyze the influence of uniform wave and current on mooring line tension and net deformation of multi-point mooring cage system. Liang et al. [22] designed a new type of cage floating collar system, compared it with traditional designs, and conducted a comparative analysis of four common multi-point mooring systems. For multi-cage systems, Cifuentes et al. [23] employed the Morison model to calculate the dynamic response of the cages in wave and current fields and compared the results with experimental data. The wake effects due to the fish net were considered in the analysis. Selvan. et al. [24] investigated the surface wave scattering of multi-cage systems, elucidating the impact of the cages on far-field waves. Xu et al. [25] conducted numerical simulations of the dynamic response of multiple cages under combined wave and current conditions and studied the loads on the mooring lines. Zhao et al. [26] conducted both numerical and experimental studies on the dynamic response of various multi-cage arrangement configurations and evaluated the mooring loads and current velocity distributions.

This paper primarily investigates the performance of mooring systems in different layouts of combined cages. It compares and analyzes the dynamic response of mooring systems under self-storage sea conditions for various cage design schemes, considering both flexible and rigid net models. Furthermore, under the optimal design scheme, the study analyzes the motion response of multiple fish cages and the load characteristics of the cables between different cages.

2. Hydrodynamic Response Analysis Method of Floating Cages

2.1. Motion Response Equation of Cage

For floating cages, the motion equation can be written as follows:

$$\left[m\right]\left\{\ddot{x}\right\}+\left[c\right]\left\{\dot{x}\right\}+\left[k\right]\left\{x\right\}=\left\{G\right\}+\left\{B\right\}+\left\{F\right\}$$

(1)

where [m], [c], and [k] are, respectively, the mass matrix, damping matrix, and stiffness matrix of the entire system of the cage. The external forces acting on the structure include gravity{G}, buoyancy force {B}, and hydrodynamic load {F}. $x$, $\dot{x}$, $\ddot{x}$ represent the cage movement displacement, speed, and acceleration, respectively. For the partially submerged floating frame structure, the influence of instantaneous wave surface is considered in calculating the buoyancy force {B}. In addition, this study focuses on the performance of the mooring system, treating the floating structure as a rigid body to improve calculation efficiency. The numerical approach for hydrodynamic analysis of the fish cages is shown in Figure 1.

2.2. Current and Wave Load

The floating structure of the cage is all slender rods. Therefore, the current and wave load subjected to the floating collar, the net system, and the mooring line are calculated by the Morison equation:

$$\{\begin{array}{l}\left\{F\right\}={\displaystyle \sum \left\{f_{Di}\right\}}+{\displaystyle \sum \left\{f_{Ii}\right\}}\hfill \\ \{f_{Di}\}=C_{Di}{\displaystyle \frac{1}{2}}\rho D_i{L}_i\left|\left\{u_i^n\right\}+\left\{v_i^n\right\}-\left\{{\dot{x}}_i^n\right\}\right|(\{u_i^n\}+\{v_i^n\}-\{{\dot{x}}_i^n\})\hfill \\ \{f_{Ii}\}=\rho C_{Mi}{\displaystyle \frac{\pi D_i^2{L}_i}{4}}\{{\dot{u}}_i^n\}-\rho (C_{Mi}-1){\displaystyle \frac{\pi D_i^2{L}_i}{4}}\{{\ddot{x}}_i^n\}\hfill \end{array}$$

(2)

where, the hydrodynamic load $F$ includes drag force $f_{Di}$ and inertia force $f_{Ii}$. $L_i$ is the element length and $\rho$ is the fluid density. $u_i$ and ${\dot{u}}_i$ are the water particle velocity and acceleration caused by the wave, respectively. $v_i$ is the current velocity. $D_i$ is the hydrodynamic diameter of the slender structure. ${C_M}_i$ and ${C_D}_i$ are the inertia and drag force coefficient, respectively. Since the net is a slender structure with a very small diameter, the inertial force is negligible. Therefore, only the drag force is considered. For the partially submerged floating body, e.g., the floating collar and the pontoon, its load is adjusted by the immersion ratio and applied to the center of the submerged part. The immersion ratio $p_w$ can be written in the following form:

$$p_w={\displaystyle \frac{h_w}{h}}$$

(3)

where, $h$ is the total height of the float and $h_w$ represents the height of the float below the instantaneous wave surface.

2.3. Equivalent Group Method of Netting

Due to the large number of real-scale netting units, the equivalent simplification of full-scale fishing nets is needed to improve the computational efficiency. For the simplified model, it is necessary to ensure that the hydrodynamic force, tensile stiffness, and mass of the net remain the same as in the original as the following [27,28]:

$$\begin{array}{l}{A}_{model}={\displaystyle \sum A_{net}}\\ A_{section\_model}={\displaystyle \sum A_{section\_net}}\\ M_{\mathrm{mod}el}={\displaystyle \sum M_{net}}\end{array}$$

(4)

where, $A_{net}$ and $A_{section\_net}$ are the projected area and cross-sectional area of the real net, and $M_{net}$ is the mass of the real scale net.

2.4. Mooring Load

The mooring lines of cages studied in this paper adopt catenary form. The dynamic responses of cages and mooring lines are calculated by the fully coupled time-domain method. In the calculation process, the effect of gravity, drag force, and axial elastic tension of the mooring lines are taken into account.

For the mooring line element, the differential equation of motion can be written as the following form under the action of external hydrodynamic load and internal inertial load:

$${\displaystyle \frac{\partial \left\{T\right\}}{\partial s}}+\{f_m\}+\{w\}={\rho }_m{A}_m{\displaystyle \frac{{\partial }^2\left\{R\right\}}{\partial t^2}}$$

(5)

where, ${\rho }_m$ is the density of the mooring line, $A_m$ is the equivalent cross-sectional area of the mooring line, $\left\{R\right\}$ is the position vector of the mooring line unit. $f_m$ and $w$ are the hydrodynamic load and gravity of the mooring line unit on the unit length, respectively. $\left\{T\right\}$ is the axial tension vector on the mooring line unit, and $s$ represents the axial vector of the mooring line unit. The dynamic response of the mooring system is calculated using the lumped mass method in Orcaflex11.2. It is a commercial software developed by Orcina Ltd. (Daltongate, Ulverston, UK) and is widely used in the numerical simulation of mooring line systems and aquacultural structures. As shown in Figure 2, the line elements are converted into concentrated mass points connected by springs for dynamic response calculations. The fluid loads on the line elements are also distributed to the concentrated mass points at both ends.

2.5. Analysis of Short-Term Sea State Extremum Response of Mooring System

As for the load of the mooring system in multiple cages under irregular waves, there are obvious differences in the load extremums of the mooring system under different random phase conditions due to the phase randomness of the wave components. Therefore, it is necessary to analyze and study multiple different samples. It is generally believed that the load extremum of the mooring system between different samples follows the Gumbel distribution, and its probability function can be written as follows:

$$F(x)=e^{-e^{-{\displaystyle \frac{x-\mu }{\beta }}}}$$

(6)

where, $\mu$ and $\beta$ are parameters of the Gumbel distribution.

3. Numerical Calculation Model

3.1. Single Cage Model

The numerical model of a single cage is shown in Figure 3, which includes the floating collars, the pontoons, the net system, and the mooring system. The pontoons are mainly used to provide sufficient reserve buoyancy. The specific size parameters are shown in

Table 1. Cage main size parameters.

Floating collar	Length × width × height (m)	41.8 × 41.8 × 1.75
	Pontoon size (m)	3.65/4.6 × 1.3 × 1.1
	Floating collar diameter (m)	0.7/0.6
Netting system	Length × width × depth (m)	36 × 36× 10
	net solidity	0.129

. The drag coefficient C_D of the floating collars is 1.2 and that of the pontoons is 2.0. The inertia coefficient C_M is 2.0. The hydrodynamic coefficient of the net is selected according to the screen model [6], and 1.5 times the marine growth coefficient is taken into account. The main body of the cage is modeled by Orcaflex11.2, and the two situations, rigid net and flexible net, are analyzed.

3.2. Layout and Mooring Scheme of Multiple Fish Cages

This study mainly considers two arrangement layouts and three deployment schemes, as shown in Figure 4, Figure 5 and Figure 6. For the first layout scheme, there are four mooring lines on the outer cage, and there are two chains between each pair of cages through cables, totaling 14 mooring lines. All chains are R3S grade stud-link chains with a diameter of 84 mm. For the second layout scheme, there are similarly four mooring lines on each outer cage, of which two have 45° arrangement. There are two mooring lines between each pair of cages connected through cables, a total of 14 mooring lines. These chains are also R3S grade stud-link chains with a diameter of 84 mm. For the third scheme, each cage has four mooring lines, totaling 16 chains, all using R3S grade stud-link mooring lines with a diameter of 84 mm. In addition, the adjacent cages are connected by cables. In all three schemes, the drag coefficient of the mooring lines are 2.6, the spacing of the cages is 50 m, and the cables are polyester cables with a diameter of 100 mm. Detailed parameters of the mooring lines are shown in

Table 2. Cable parameter.

Scheme	Chain Categories	Chain Length (m)	Mooring Radius (m)	Steel Grade
Scheme 1	A	558	550	R3S
	B	508	500	R3S
Scheme 2	A	558	550	R3S
	B	508	500	R3S
	C	508	500	R3S
Scheme 3	A	448	440	R3S

. The safety factor and breaking tension for the mooring lines are determined according to the DNVGL-OS-E302 [30].

3.3. Environmental Parameters

This paper analyzes the performance of multiple cages with different mooring schemes under extreme regular wave conditions and studies the dynamic response of the optimized scheme under irregular waves. The parameters of the extreme regular wave state are shown in

Table 3. Mooring system check calculation working condition table.

Condition	Direction (°)	Depth (m)	Wave Period (s)	Wave Height (m)	Flow Velocity (m/s)
Case1	0	32.5	12.5	8.4	Surface 1.5 Seafloor 1.0
Case2	45
Case3	90

, and those of the irregular wave state are shown in

Table 4. Irregular wave conditions.

Depth (m)	Significant Wave Height (m)	Spectral Peak Period (s)	Flow Velocity (m/s)
32.5	4.7	8.5	Surface 1.5 Seafloor 1.0

. The flow field is linear shear flow, and the velocity at the surface and seabed is 1.5 m/s and 1.0 m/s, respectively. Three different wave directions are taken into account, in which 0° are propagated forward along the x-axis and 90° are propagated forward along the y-axis. The operating water depth of the cage is 32.5 m.

4. Result Analysis and Discussion

The maximum tension of each mooring line in the different layout and mooring system design schemes under three wave directions was calculated, and the rigid net model and flexible net model were compared in various working conditions. The motion response characteristics of the cages were analyzed according to the optimal mooring design scheme.

4.1. Analysis of Load Characteristics of Mooring System

4.1.1. Scheme 1: 1 × 4 Layout and Mooring System of Multiple Cages

In this scheme, the direction of the most dangerous wave-current is 0°, and the environmental load is mainly carried by the No. 1 and No. 2 mooring lines. Under the two conditions of rigid and flexible net models, the maximum mooring line tension is 447.90 tons and 438.68 tons, respectively, as shown in Figure 7. The deformation of the net garment is shown in Figure 8. When the net system is deformed, the force of the net becomes smaller, resulting in the tension of the mooring system becoming smaller, but the influence is not significant. In the direction of 0° wave-current incidence, the strain decreases by 2.06%.

The distribution of hydrodynamic load is relatively uniform at 45° and 90° wave-current direction, and the maximum load of a single mooring line decreases obviously. The maximum mooring system load of the rigid net model is 314.83 tons and 232.23 tons, respectively. The maximum mooring system load of the flexible net model is 307.50 tons and 224.63 tons, respectively, as shown in Figure 9 and Figure 10.

According to the grade and diameter of the mooring lines, the breaking tension of the mooring line can be determined as 667.8 tons. Under the condition of different inflow direction of wave, the rigid net and flexible net model are considered, and the checking of results of the mooring system are shown in

Table 5. Results of mooring line verification in Scheme 1 (rigid net model).

	Degree	0°	45°	90°
Category
Maximum cable tension (tons)		447.90	314.83	232.23
Safety factor		1.49	2.12	2.88
Pile and touch		No	No	No

and

Table 6. Results of mooring line verification in Scheme 1 (flexible net model).

	Degree	0°	45°	90°
Category
Maximum cable tension (tons)		438.68	307.50	224.63
Safety factor		1.52	2.17	2.97
Pile and touch		No	No	No

. After calculation, it can be seen that under the arrangement of Scheme 1, in the sea state of 0° incident regular wave, the safety factor of the mooring lines does not meet the requirement that the safety factor of the self-storage condition in the specification be greater than 1.67. Through observation of the motion response process, there is no accumulation of the mooring lines or contact between the mooring lines and cage under the three working conditions. The space attitude of the mooring system under the maximum displacement state is shown in Figure 11.

4.1.2. Scheme 2: 1 × 4 Layout and Inclined Mooring System

In Scheme 2, the diagonal arrangement of No. 3 and No. 4 mooring line shares a large part of the load in the direction of 0° wave-current. In this case, the load of No. 1 and No. 2 mooring lines are significantly lower than that of Scheme 1. Without considering the deformation of the net, the maximum load of No. 1 and No. 2 mooring lines is 349.07 tons. It is 340.81 tons when considering net deformation, as shown in Figure 12.

At 45° wave-current direction, No. 4 mooring line is collinear with the wave-current direction and carries the maximum load. Under the conditions of rigid and flexible net models, the maximum loads on the mooring system are 324.81 tons and 313.39 tons, respectively. In the 90° wave-current direction, the No. 5 and No. 6 mooring lines are parallel to the wave direction and carry large loads. The oblique arrangement of part of the mooring lines leads to the low lateral carrying capacity of the mooring system, and the load of the mooring system is greatly increased compared with Scheme 1, as shown in Figure 13 and Figure 14. In Scheme 2, there is also no accumulation of mooring lines or contact between cages. The space posture of the mooring system under the condition of maximum displacement of cages is shown in Figure 15.

The mooring line check results of Scheme 2 are shown in

Table 7. Results of mooring line verification in Scheme 2 (rigid net model).

	Degree	0°	45°	90°
Category
Maximum cable tension (tons)		349.07	324.81	380.06
Safety factor		1.91	2.06	1.75
Pile and touch		No	No	No

and

Table 8. Results of mooring line verification in Scheme 2 (flexible net model).

	Degree	0°	45°	90°
Category
Maximum cable tension (tons)		340.81	313.39	370.25
Safety factor		1.96	2.13	1.8
Piling and touching		No	No	No

. In Scheme 1 and Scheme 2, four cages are arranged in a straight line, resulting in a large hydrodynamic load of cages in the 0° wave-current direction. Therefore, the maximum tension of No. 1 and No. 2 mooring lines in Scheme 1 are much larger than that of others, which increases the cost. Compared with Scheme 1, Scheme 2 changes the arrangement angle of No. 3 and No. 4 mooring lines. Therefore, No. 3 and No. 4 mooring lines share the load of No. 1 and No. 2 mooring lines in the direction of 0° wave-current, and carry greater load in the direction of 45° wave-current. Due to the reduction of the lateral carrying capacity of the mooring line, the load increases under the condition of 90° incidence, but it still meets the requirement of the safety factor of 1.67. Compared with Scheme 1, the mooring line tension in Scheme 2 is more average under each working condition, which can reduce the construction cost of the mooring system to a certain extent.

4.1.3. Scheme 3: 2 × 2 Layout and Mooring System of the Tandem Cage

For Scheme 3, the wave directions of 0° and 90° are symmetrical along the diagonal of the whole system, so only the wave directions of 0° and 45° are calculated under this scheme, and the stress of No. 1–4 mooring lines is observed, as shown in Figure 16 and Figure 17. In the 0° wave-current incidence, mooring line 1 and 2 carry the main hydrodynamic loads. The maximum mooring line tension of the rigid and flexible net models is 328.45 tons and 319.86 tons, respectively, so the safety factor exceeds 1.67 and meets the safety design requirements, as shown in

Table 9. Results of mooring line verification in Scheme 3 (rigid net model).

	Degree	0°	45°
Category
Maximum cable tension (tons)		328.45	235.76
Safety factor		2.03	2.83
Pile and touch		No	No

and

Table 10. Results of mooring line verification in Scheme 3 (flexible net model).

	Degree	0°	45°
Category
Maximum cable tension (tons)		319.86	224.47
Safety factor		2.09	2.98
Pile and touch		No	No

. In the working condition of 45° wave-current incidence, since the hydrodynamic load is shared by eight mooring lines, the load is relatively small compared with the working condition of 0° wave injection. In the third scheme, there is also no accumulation of mooring lines or contact between cages.

Compared with the first two schemes, the four cages in Scheme 3 are arranged in a square manner, the hydrodynamic load is relatively small, and the mooring lines are arranged in a symmetrical manner, so the maximum tension distribution of the mooring lines is uniform, and the total length of the mooring lines is short, which is conducive to reducing the cost of the mooring system.

4.1.4. Comparative Analysis of Performance of Different Mooring System Design Schemes

In the first scheme, the hydrodynamic load of the cage under the 0° wave-current direction is only carried by the two mooring lines against the wave, resulting in the safety factor of the mooring line being only 1.49, which does not meet the specification requirements. Scheme 2 optimizes the arrangement angle of part of the mooring lines according to Scheme 1, resulting in a more balanced maximum tension of each line under different wave-current directions, thereby increasing the safety factor. However, since the four cages in Scheme 2 are arranged in a linear manner, the hydrodynamic load at 90° is larger, while Scheme 3 improves the cage arrangement to 2 × 2, reducing the hydrodynamic load carried by a single mooring line, making it more economical than Scheme 2. Meanwhile, no accumulation of mooring lines and contact friction between the cage structure are observed. In summary, Scheme 3 is the best in terms of safety and economy.

4.2. Analysis of Cage Motion Response and Cable Bearing Characteristics between Cages under Regular Wave Conditions

For Scheme 3, this paper further studied the motion response characteristics of the cages under the 0° and 45° wave-current directions. The flexible net model was used to analyze the motion response of the cages, as shown in Figure 18 and Figure 19.

In the direction of 0° wave flow, the surge amplitude of the two cages is relatively close, and the displacement difference is far less than the distance between the two cages, thus, there is no collision risk of the cages. Due to the constraints of the mooring line, the heave amplitude of the No. 1 cage is smaller than that of the No. 2 cage.

In the 45° wave-current direction, the horizontal resilience of No. 4 cage on the leeward of the wave is the least, so the surge amplitude is the largest. The horizontal motion amplitudes of No. 1, 2, and 4 cages are not large, and there is no collision between cages. As for heave amplitude, the No. 1 cage on the wave side has four mooring lines to provide the restoring force, so the heave amplitude is the smallest. Only two of the four mooring lines on the No. 2 cage can provide the restoring force, and the four mooring lines on the No. 4 cage are in the leeside, so the heave amplitude of the No. 4 cage is the largest.

Figure 20 shows the maximum tension of the cables between the cages. The cable numbers are shown in Figure 6. At 0° of wave-current incidence, the motion of the cage is mainly surge and heave, so the tension of the cables 1 and 2, which mainly carry the horizontal resilience of the cage, is larger, while the tension of the cables 3 and 4 is smaller. At 45° of wave-current incidence, the mean surge of 1, 2, and 4 cages are close, and the maximum tension of the 1 to 4 cables are also close.

4.3. Analysis of Cage Motion Response and Cable Loading Characteristics between Cages under Irregular Wave Conditions

In the real marine environment, the wave environment of the aquaculture cages is generally irregular waves. In order to save calculation efficiency, regular wave conditions are adopted to compare the safety and economy of different mooring schemes. According to the third optimal scheme, the dynamic response characteristics of the cage and the load characteristics of the mooring system under the irregular wave condition are studied. For example, under the condition of 0° wave-current incidence, the time history and power spectrum analysis results of surge and heave are shown in Figure 21 and Figure 22. Combined with the time history and its power spectrum, it can be seen that under the action of irregular waves, the surge energy of the cage is mainly concentrated in the low frequency part, while the heave response is more significantly affected by wave frequency. Meanwhile, due to the influence of the Morison force squared term, the power spectrum has the common characteristics of the wave frequency and wave frequency multiplier. The surge amplitude of cages is smaller than the cage spacing, indicating that there is essentially no risk of collision.

The load time history of the cage mooring system and connecting cable with the corresponding power spectrum are shown in Figure 23 and Figure 24. Among them, there is little difference between the peak load of the mooring line and that of the connecting cable under the same sea state. The energy of the mooring line tension is mainly concentrated in the low-frequency part, the energy of the cable tension is mainly concentrated in the vicinity of the spectral peak period, and the low-frequency movement has little influence on the cable tension.

Aiming at the load of the mooring system under short-term sea conditions, several random seeds were selected, respectively. Gumbel distribution fitting was performed on the extreme value of the mooring line load of each sea state sample under the conditions of 0° wave-current incidence and 45° wave-current incidence, and the extreme value response analysis was performed on the mooring line load corresponding to 0.9 exceedance probability (P90). The results are shown in Figure 25, Figure 26, and

Table 11. Estimation of short-term extreme load of mooring system at different wave directions.

Working Condition	Mooring Line Load (P90)
0° wave-current incidence	288.1 tons
45° wave-current incidence	224.5 tons

, respectively.

In summary, it is evident that compared to the 0° wave-current incidence condition, the mooring system experiences a lower maximum load and is relatively safer under the 45° wave-current incidence condition in the same sea state.

5. Conclusions

This paper proposes three mooring schemes according to the working environment and hydrodynamic characteristics of floating multiple cages. Their safety under self-storage conditions was numerically simulated and evaluated, and the numerical results of the rigid net model and the flexible net model compared, focusing on the maximum load of the mooring system. The possibility of mooring line accumulation and collisions between mooring lines and the fish cages is also considered. The optimal mooring scheme was selected, and its motion response, mooring line tension, and cable load characteristics between cages studied. The main conclusions are as follows:

• Based on the three different arrangement configurations mentioned above, the mooring line tension of the fish cages is significantly higher with a 1 × 4 arrangement compared to a 2 × 2 arrangement. Under 0° wave-current incidence, the maximum tension increases by 37.14%, and under 45° wave-current incidence conditions, the maximum tension increases by 46.12%. From a safety perspective for the cages, it is recommended to adopt the 2 × 2 arrangement configuration.
• The difference between the mooring line tension calculated by the flexible and rigid net model is not more than 5% under different wave-current incidences. Therefore, for this kind of fish cages, considering the modeling workload and calculation efficiency in practical application, the rigid net model can be used to check the safety of the mooring system.
• Under irregular wave conditions, the load of the mooring system of the cage is mainly controlled by the low-frequency motion, and the load of the connecting cable is affected by the wave-frequency motion. Both surge and heave motions have a low-frequency component, which is due to there being a coupling among surge motion, mooring line tension, and heave motion. In the same irregular wave condition, for the 2 × 2 arrangement of the combined cage form, the wave-current incident along 0° incidence is more dangerous.