**2. Mathematical Model**

As shown in Figures 1 and 2, to prevent the damage of the typhoon waves, the turbine and the floating platform are submerged to a depth of less than 60 m. Therefore, the influence of the wave impact is almost negligible. In general, the dynamic response of a large-surfaced structure subjected to wave impact force, which is non-uniform and transient, is generally in the coupled translational–rotational (pitching, rolling and yawing) motion. The good conditions for ocean current power generation are high flow rate and stable flow direction, so the site is often a considerable distance from the shore: less affected by the coast and less likely to produce breaking waves. The impact of breaking waves is not considered in this manuscript.

When ocean currents flow through the blades of the ocean turbine, the turbine rotates and drives the power generator to generate electricity. Meanwhile, the turbine unit is subjected to the force of the ocean current; to fix the turbine unit, it is pulled by the floating platform connected by rope B. The floating platform provides buoyancy and is anchored to the deep seabed with lightweight, high-strength PE ropes. In addition, the ocean current turbine is connected to pontoon 4 via rope D, and the balance between the current generator and the pontoon is reached so that the depth of the turbine when the current is not affected can be determined by the length *LC* of rope C. On one side of the floating platform, rope B is used to pull the ocean current generator, and the other side of the floating platform is pulled down and anchored on the deep seabed. The buoyancy of the floating platform can be adjusted to be smaller than that of static balance so that, when ropes A and B are pulled, the floating platform has negative buoyancy and pontoon 3 has positive buoyancy, and rope C is used to connect the floating platform and pontoon 3 to achieve a balance of positive and negative buoyancy. In this way, the depth of the floating platform can be calculated by the length *LC* of rope C.

**Figure 1.** Coordinates of the current energy system composed of submarined ocean turbine, pontoons, floating platform, traction ropes and mooring foundation in the static state under steady ocean current.

**Figure 2.** Coordinates of the current energy system composed of submarined ocean turbine, pontoon, floating platform, traction rope and mooring foundation in the dynamic state under steady ocean current and wave.

Lin and Chen [26] showed that a PE rope can be assumed to be a straight line under a certain amount of ocean current drag force because the force deformation of the PE rope is negligible. The linear elastic model presented by Lin and Chen [26] is used to analyze the motion equation of the overall mooring system.

Several assumptions are made based on these facts about ocean current energy converters (OCEC):


According to these assumptions, the motion of the mooring system is translational. The coupled translational–rotational motion of a system subjected to non-uniform and impulsive force from the wave current will be discussed in future research. The coupled, linear, ordinary differential equations of the system are derived based on the assumptions. Due to the wave fluctuation, the buoyance forces of the pontoons stimulate the mooring system to vibrate. The coupled vibration motion of the system includes horizontal and vertical oscillations.

The global displacements (*xi*, *yi*) for the *i*-th element shown in Figures 1 and 2 are the sum of two parts: (1) the static one subjected to the steady current and (2) the dynamic one subjected to the wave, as follows:

$$\mathbf{x}\_{i} = \mathbf{x}\_{i\text{s}} + \mathbf{x}\_{\text{id}\prime} \quad \mathbf{y}\_{i} = \mathbf{y}\_{\text{is}} + \mathbf{y}\_{\text{id}\prime} \text{ \'n = 1\text{, 2, 3, 4}} \tag{1}$$

where *x* and *y* are the vertical and horizontal displacements, respectively. Because of the pontoon buoyancy and the short length of rope between the turbine and the pontoon, the horizontal dynamic displacements of the turbine and pontoon 4 are almost the same: *y*2*<sup>d</sup>* ≈ *y*4*d*. In a similar way, the horizontal dynamic displacements of the floating platform and pontoon 3 are almost the same: *y*1*<sup>d</sup>* ≈ *y*3*d*. In addition, the total tensions of ropes A, B, C and D are also composed of two parts: (1) the static one and (2) the dynamic one, as follows:

$$T\_i = T\_{is} + T\_{id\prime} \text{ } i = \text{A} \text{ } \text{B} \text{ } \text{C} \text{ } \text{D} \tag{2}$$

(3)
