# Optimization of Buoy Shape for Wave Energy Converter Based on Particle Swarm Algorithm
## 3 Results  
In order to visualize the variation of power capture during the optimization of buoy shapes, irregular waves were calculated in the time domain to analyze the shape evolution and, finally, the hydrodynamic performance and energy capture capacity of the different buoys. The model parameters are shown in Table 2.  
In order to simulate the hydrodynamic performance and shape optimization process of the buoy in real sea conditions, in this section, the improved JONSWAP wave spectrum proposed by [PERSON] [44] is selected for numerical simulation, which can be used to generate irregular wave data according to the following wave spectrum equation. According to the theory of linear superimposition, irregular waves can be considered as the superposition of a large number of regular table lines with different periods, phases, and amplitudes. \[S_{f}=\beta_{J}H_{S}^{2}T_{P}^{-4}f^{-5}\exp\left[-\frac{5}{4}(T_{P}f)^{-4} \right]\cdot\gamma^{\exp\left[-(f/f_{P}-1)^{2}/2\sigma^{2}\right]} \tag{11}\]  
\[\beta_{J}=\frac{0.06238}{0.230+0.0336\gamma-0.185(1.9+\gamma)}\cdot\left[1.904 -0.019151\ln\gamma\right] \tag{12}\]  
\[T_{P}=\frac{T_{H_{S}}}{1-0.132(\gamma+0.2)^{-0.559}} \tag{13}\]  
where \(H_{S}\) is the significant wave height, \(\gamma\) is the peak factor of the spectrum, \(f\) is the frequency, \(f_{P}\) is the peak frequency, \(\sigma\) is the peak type parameter, \(\sigma=0.07(\omega\leq\omega_{P})\), \(\sigma=0.09(\omega>\omega_{P})\), \(\omega_{P}\) is the circular frequency at the spectral peak, and \(T_{P}\) is the peak period, which is equal to \(1/f_{P}\). Offshore wind farms are situated in the Yellow and Bohai Seas of China. [PERSON] et al [45], analyzed the wave heights and periods of the Chinese Yellow and Bohai Seas, so the most probable sea state was chosen for this paper. The spectral density function curves generated by the wave conditions are shown in Figure 9.  
The above wave spectrum was simulated in WEC-Sim in the time domain; the WEC-Sim model is shown in Figure 10. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Value [m]** \\ \hline \(H\) & 10 \\ \(D\) & 6.6 \\ \(h_{1}\) & 1.2 \\ \(h_{2}\) & 3.7 \\ \hline \end{tabular}
\end{table} 
Table 2: System parameters of WEC. Figure 8: The simulation results compared with the results published by [PERSON] et al. [43]. As shown in Figure 11, when \(C\) took 0, the objective function converged in 16 steps and the fitness value converged to 13.422 kW; when \(C\) took \(5.0\times 10^{5}\) Nms/rad, the objective function converged in 12 steps and the fitness value converged to 12.132 kW; when \(C\) took \(1.0\times 10^{6}\) Nms/rad, the objective function converged in 17 steps and the fitness converged to 12.390 kW. At the beginning of the iteration, the value of \(P_{in}\) was small and, as the iteration progressed, the motion response of the buoy gradually increased and the energy of the entire swing arm system was efficiently captured. Obviously, the fitness value of \(C\) was greater than that of the other two cases, and the power generation system worked in a more efficient energy capture range. As shown in Figure 12, (a) is the motion response of the buoy in pitching degrees of freedom and (b) is the motion response of the system in pitching degrees of freedom. The maximum swing angle of the buoy was obtained when \(C\) took 0, the pitch angle reached 34 degrees, and, under the same damping coefficient, the swing angle of the system showed the same movement trend as the swing angle of the buoy, and the maximum swing angle of the system reached 9 degrees. In general, the motion response of the buoy affected the motion response of the system, and the more violently the buoy moved, the more violently the system swung. Briefly, as shown in Tables 3-5, data from six representative iterations out of 20 were selected and had distinct characteristics. Figure 10: WEC-Sim model. Figure 9: JS spectral density function curve. Figure 11: Iterative curve graph.