*3.1. Initial Conditions*

For the proposed model, it is necessary to define the initial condition for all segments, as listed in Table 1.


**Table 1.** Initial conditions for proposed model.

All angle values in Table 1 are in radian. The initial angles are set for every segment to represent an untensioned mooring line. This method can be applied to a general mooring system as well. The parameters for the turbine, buoy, and mooring line are shown in Table 2; the turbine diameter was set to be the same as that of the SIMEC Atlantis Energy tidal turbine AR2000 [26] and the weight was half of its weight. The definitions of *l*1, *l*<sup>2</sup> and *l*<sup>3</sup> are shown in Figure 2.

**Table 2.** The parameters of the turbine, buoy and mooring line.


#### *3.2. Sea States*

Table 3 lists the sea states investigated in the simulations to obtain the thrust and torque on the turbine. Steep and swell waves are investigated, comparing how wave excitation on the buoy affects the load on the turbine. The assumed hypothetical site for the generic turbine was chosen off the north east coast of the Orkney islands, Scotland. This site provided a flow speed of approximately 2.5 [m/s] and an average depth of 50 [m] [28].


**Table 3.** Sea states.

A 3 min window of 0.1 [s] time step was simulated for each sea states.

#### **4. Results**

Firstly, the results with and without wave excitation on the buoy are compared in the same sea states. In sea state 1, a three-step approximate wave–current interaction model was applied in this simulation because its steepness was larger than 0.02 [29]. However, the hub height dropped to approximately 22.5 [m] during the operation compared with the original height, which was set at 30 [m] from the seabed. Therefore, the hub height for the rigid supported turbine was set equal to that of the mooring supported one. Figure 7 exhibits the result in sea state 1. These results indicate that the mooring supported turbine can more effectively reduce the peak thrust on the turbine when compared with the rigid supported turbine. Furthermore, the results from considering wave excitation on the buoy indicate a different waveform without wave excitation. The mean values of the three curves are close: 1.019 [MN] for the rigid supported turbine, 1.019 [MN] for that without wave excitation on the buoy, and 1.016 [MN] for that with wave excitation on the buoy. However, the standard deviation of the thrust on the mooring supported turbine considering wave excitation on the buoy is 28.23 [kN] and that without wave excitation is 16.90 [kN], which are 69.69% and 41.73% of that on the rigid structure value of 40.50 [kN].

The results for torque indicate a different trend compared with that of the thrust. Torques on the mooring-supported turbine with and without wave excitation on the buoy both have more fluctuations than that of the rigid supported turbine. However, the mean values are similar, that is, 364.91 [kN·m], 364.66 [kN·m], and 361.98 [kN·m] for rigid, without, and with, wave excitation on the buoy, respectively. The respective standard deviations are 18.19 [kN·m], 87.14 [kN·m], and 83.41 [kN·m]. This is because of modifying the relative velocity in the vertical direction, which is discussed later in this section.

Figure 8 exhibits the result for the turbines operating in sea state 2. It shows a favorable performance in the reduction of peak loading both in thrust and torque and the wave excitation on the buoy provides a positive effect in load reduction. The reason is that the buoy will oscillate more regularly in the wave with long wave periods. In this sea state, the mean values of thrust are 1.260 [MN], 1.265 [MN] and 1.265 [MN] for rigid supported turbine, mooring turbine without and with wave excitation on a buoy, respectively. The standard deviations are 47.35 [kN], 11.59 [kN] (24.5% of 47.35 [kN]) and 6.56 [kN] (13.9% of 47.35 [kN]) separately. The average torque values are 694.47 [kN·m], 699.92 [kN·m] and 699.06 [kN·m] and the standard deviations are 58.19 [kN·m], 24.98 [kN·m] and 20.86 [kN·m].

**Figure 7.** Thrust and torque in sea state 2.

**Figure 8.** Thrust and torque in sea state 2.

In sea state 3, the performance of the system is different from all the sea states discussed so far as shown in Figure 9. The peak loading of thrust as a result of wave excitation on the buoy is larger when compared with the rigid supported turbine. The mean values of the thrust are 1.045 [MN], 1.044 [MN], and 1.040 [MN] for rigid supported turbine, mooring supported turbine without and with wave excitation on the buoy. The respective standard deviations are 5.82 [kN], 6.28 [kN] and 13.43 [kN]. The respective average torque values are 396.44 [kN·m], 394.84 [kN·m], and 390.25 [kN·m] and the standard deviations are 169.08 [kN·m], 20.73 [kN·m], and 35.48 [kN·m].

**Figure 9.** Thrust and torque in sea state 3.

The results for harsh winter conditions (the extreme storm sea state in the North Sea) are presented in Figure 10. The peak loading reduction in the thrust is similar to that in sea state 2. The mean values are 930.03 [kN], 897.42 [kN], and 894.36 [kN], respectively. The respective standard deviations are 390.80 [kN], 86.67 [kN] (22.2%), and 37.57 [kN] (9.6%). However, the torque has negative values, which indicate that the directions of the torque have reversed. In reality, this means that the turbine will be stalled due to displacement being significantly large enough for the velocities on blade elements to change (see Figure 11).

**Figure 10.** Thrust and torque in harsh winter.

**Figure 11.** Inflow velocity vectors on blade sections in harsh winter.
