# Performance Analysis of a Floating Wind–Wave Power Generation Platform Based on the Frequency Domain Model

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## Abstract

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## 1. Introduction

## 2. Theory Background

#### 2.1. Frequency-Domain Model for Multiple Floating Bodies with Constraints

**E**is defined to describe this coupling term:

**M**, hydrostatic stiffness matrix

**C**, viscous damping matrix

**Bv**, damping coefficient matrix of the PTO

**Bpto**, and stiffness matrix of the PTO

**Kpto**are all diagonal and can be expressed as follows:

**E**can be rewritten as follows:

#### 2.2. Assessment of Point-Absorber Power Generation

## 3. Layout of Wind–Wave Platform and Hydrodynamic Analysis

#### 3.1. Layout of Floating Wind–Wave Power Generation Platform

#### 3.2. Multi-Body Hydrodynamic Analysis

## 4. Optimization of Power Generation in Frequency Domain

#### 4.1. Verification of Motions of Multiple Floating Bodies with Constraints

#### 4.2. Optimization of PTO

^{5}N/m. The maximum CWR values in Figure 11 were statistically analyzed, as shown in Table 3. It can be observed that the CWR of the point absorber decreases as the PTO stiffness increases in all four models. The optimal PTO stiffness values are found within the range of 0–10

^{4}N/m, with the maximum CWR occurring at a stiffness value of 0, corresponding to wave frequencies near the natural frequency of the point absorber. Furthermore, comparing Figure 11b and Figure 11f, it is noted that when a floating platform is considered, the CWR exhibits a secondary peak near a wave frequency of 1.7 rad/s. The stiffness values corresponding to this secondary peak are different from those at the primary peak. Figure 12 illustrates the relative motion RAO comparison between the two models at the frequency corresponding to the secondary peak. It is observed that the RAO at the secondary peak frequency has increased, and the spectral width is wider compared to the case with a stiffness value of 0. This reflects that considering the floating platform alters the system’s motion characteristics and alters the original power generation mode of the point absorber. Therefore, to determine the optimal PTO parameters, a PTO stiffness of 10

^{−6}N/m is selected for further research.

^{5}Ns/m was selected, and two typical wave angles at −180° and −120° were compared in terms of the total power for each model, as shown in Figure 18. Subsequently, a comparison of the q factor for each model was conducted, as shown in Figure 19. The black dashed line represents a q factor of 1. It can be observed that at wave frequencies lower than 1 rad/s, the average performance of point absorbers in different arrays is close to that of a single point absorber. Additionally, the q factors for all three array arrangements are very similar. As the frequency increases, all three array arrangements display multiple peaks at the same frequency. The maximum peak of the q factor can reach up to 70 and occurs at high frequencies where the total power tends towards zero. This reflects that even though the motion responses of each model are small at high frequencies, the hydrodynamic resonance generated by the array of point absorbers on the floating platform significantly improves their average power performance at certain specific wave frequencies. Therefore, when conducting practical engineering design, it is advisable to select sea conditions that are close to these peak frequencies, which will also be the subject of future research.

#### 4.3. The Expected Power in the South China Sea

^{5}N/m, showing significant differences from the result in Table 1. Although the PA-WP model exhibits the best performance in terms of power generation, its expected power is only slightly improved compared to SPA and even occasionally falls below the performance of SPA. This suggests that while the SPA-WP solution may have better peak performance, its performance stability across all wave frequencies is relatively poor. Similarly, when comparing the two layout options for the three point absorbers (TPA-WP and TPA-WP2), it is evident that the second layout option yields significantly higher expected power than the first. The first layout option also exhibits performance close to that of TPA, indicating that it is more affected by the wave angle. Finally, the model with six point absorbers (SIXPA-WP) demonstrates more stable performance, suggesting that the placement scheme with six point absorbers is better suited for the sea conditions in the South China Sea.

## 5. Conclusions

- (1)
- The hydrodynamic coefficients in the heave DOF of the point absorbers are significantly influenced by the floating platform. Regardless of the arrangement of point absorbers on the platform, both added mass and radiation damping exhibit varying degrees of increase. This leads to higher RAO peak values compared to the model without a floating platform. The hydrodynamic coefficients of the floating platform, however, are minimally affected by the point absorbers.
- (2)
- Regardless of the arrangement of point absorbers, the optimal PTO stiffness that maximizes the CWR tends towards zero. Only when the floating foundation is considered does the CWR exhibit a secondary peak, with the corresponding PTO stiffness of around 3 × 10
^{5}–4 × 10^{5}N/m. This suggests that increasing stiffness can change the resonance frequency of the hybrid system to some extent. - (3)
- When considering the optimal PTO damping coefficient, the CWR values of the point absorbers all exhibit a peak near their natural frequencies. When a three-point-absorber array is arranged on the floating platform, the peak shifts towards the natural frequency of the platform, and the optimal PTO stiffness significantly increases. It can be inferred that array arrangements can alter the working conditions and adaptability range of point absorbers.
- (4)
- When considering different arrangements of point-absorber arrays on the floating platform, it is observed that when considering the floating platform, the point-absorber array’s maximum total power generation is minimally affected by the wave angle but offers little improvement over individual point absorbers. When considering a floating platform, the maximum power generation of each model is improved compared to that for individual point absorbers. The arrangement of a single point absorber on the floating platform results in the highest increase in maximum power generation, but it is significantly affected by the angle of waves. On the other hand, both arrangements of three point absorbers exhibit more stable performance. Similarly, the arrangement of six point absorbers is also significantly influenced by the wave angle. Additionally, when considering whether the average performance of point-absorber arrays on a floating platform is superior to that of a single point absorber on the same platform, it was observed that due to the hydrodynamic resonance generated by the array arrangements, multiple peaks occurred in the q factor at the same frequency.
- (5)
- The expected power performance of point absorbers in different arrangements in the South China Sea differs significantly from their performance in the maximum power analysis. While arranging a single point absorber on a floating foundation yields the best peak performance, its stability across all wave frequencies is poor, even dropping below that of a single point absorber. The more point absorbers are arranged in an array, the more stable their performance becomes, demonstrating better adaptability.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$A(\omega )$ | Added mass matrix [-] | ${\omega}_{p}$ | Peak frequency [rad/s] |

$B(\omega )$ | Radiation damping matrix [-] | ${\stackrel{\u2322}{x}}_{Relative}$ | The relative displacement between the platform and the point absorber [m] |

Bpto | Damping coefficient matrix of PTO [-] | $\gamma $ | Peak factor [-] |

Bv | Viscous damping matrix [-] | Abbreviations | |

C | Hydrostatic stiffness matrix [-] | ||

d | Water depth [m] | CFD | Computational fluid dynamics |

D | Capture width [m] | CWR | Capture Width Ratio |

E | Constraint matrix [-] | DOF | Degree of freedom |

${\stackrel{\u2322}{f}}_{pto}$ | The force applied to the PTO [N] | PSD | Power Spectral Density |

H | Wave height [m] | PTO | Power Take-Off |

${H}_{s}$ | Significant wave height [m] | RAO | Response Amplitude Operator |

Kpto | Stiffness matrix of PTO [-] | SIXPA-WP | Six point absorbers combined with floating wind turbine platform |

${k}_{0}$ | Wave number [-] | SPA | Single point absorber |

M | Mass matrix [-] | SPA-WP | Single point absorber combined with floating wind turbine platform |

${P}_{ave(N)}$ | The average power generation of the Nth point absorber [W] | TPA | Three point absorbers |

P | The output power of the point absorber [W] | TPA-WP | Three point absorbers combined with floating wind turbine platform |

${P}_{w}$ | Incident power of the wave per unit width [W] | TPA-WP2 | The second placement scheme for three point absorbers combined with floating wind turbine platform |

${W}_{Expected}$ | The expected power [W] | WECs | Wave energy converters |

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**Figure 3.**The wind–wave power generation platform comprising the combination of the OC4 platform and point absorbers.

**Figure 7.**Surface contours under different layouts ($H=2\mathrm{m},\omega =0.66\text{rad}/\mathrm{s}$).

**Figure 10.**Comparison of RAO between present frequency-domain model and time-domain model in ANSYS-AQWA.

**Figure 11.**Contour diagram of the CWR of the point absorber for varying PTO stiffness values and wave periods under different models.

**Figure 13.**Contour diagram of the CWR of the point absorber for varying PTO damping coefficients and wave periods under different models.

**Figure 16.**Contour diagram of the total power generation of the point absorber for varying PTO damping coefficients and wave periods under different models.

**Figure 17.**The variation in the maximum total power generation of point absorbers in different models.

Item | Value | Unit |
---|---|---|

Diameter of base columns | 24 | m |

Diameter of upper columns | 12 | m |

Length of base columns | 6 | m |

Length of upper columns | 26 | m |

Column center to center | 50 | m |

Operating draft | 20 | m |

Bracing diameter | 1.6 | m |

Distance between COG and SWL | 13.46 | m |

Displacement tonnage | 1.3473 × 10^{4} | t |

Item | Value | Unit |
---|---|---|

Radius | 4 | m |

Length | 5 | m |

Operating draft | 3 | m |

Distance between COG and SWL | 1 | m |

Displacement tonnage | 1.5457 × 10^{2} | t |

Model | Number of Point Absorbers | Maximum CWR | Optimal PTO Stiffness (N/m) | Optimal Wave Frequency (rad/s) | |
---|---|---|---|---|---|

SPA | 0.4122 | 0 | 1.35 | ||

SPA-WP | 0.6873 | 0 | 1.29 | ||

TPA | PA1 | 0.2993 | 0 | 1.29 | |

PA2 and PA3 | 0.4831 | 0 | 1.44 | ||

TPA-WP | PA1 | 0.2887 | 0 | 1.29 | |

PA2 and PA3 | 0.4969 | 0 | 1.38 |

Model | Number of Point Absorbers | Maximum CWR | Optimal PTO Damping Coefficient (Ns/m) | Optimal Wave Frequency (rad/s) | |
---|---|---|---|---|---|

SPA | 0.4193 | 5.80 × 10^{4} | 1.35 | ||

SPA-WP | 0.7633 | 8.50 × 10^{4} | 1.29 | ||

TPA | PA1 | 0.3044 | 6.50 × 10^{4} | 1.26 | |

PA2 and PA3 | 0.4787 | 4.50 × 10^{4} | 1.41 | ||

TPA-WP | PA1 | 0.3634 | 3.07 × 10^{5} | 0.87 | |

PA2 and PA3 | 0.5017 | 2.19 × 10^{5} | 1.05 |

Model | Maximum Total Power Generation (W) | Optimal PTO Damping Coefficient (Ns/m) | Optimal Wave Frequency (rad/s) |
---|---|---|---|

SPA | 6.2417 × 10^{4} | 7.60 × 10^{4} | 1.29 |

SPA-WP | 1.1666 × 10^{5} | 8.50 × 10^{4} | 1.29 |

TPA | 1.9231 × 10^{5} | 2.51 × 10^{5} | 0.99 |

TPA-WP | 2.2623 × 10^{5} | 2.39 × 10^{5} | 0.99 |

TPA-WP2 | 2.7278 × 10^{5} | 8.40 × 10^{4} | 1.26 |

SIXPA-WP | 3.8422 × 10^{5} | 2.38 × 10^{5} | 0.99 |

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**MDPI and ACS Style**

Chen, M.; Deng, J.; Yang, Y.; Zhou, H.; Tao, T.; Liu, S.; Sun, L.; Hua, L.
Performance Analysis of a Floating Wind–Wave Power Generation Platform Based on the Frequency Domain Model. *J. Mar. Sci. Eng.* **2024**, *12*, 206.
https://doi.org/10.3390/jmse12020206

**AMA Style**

Chen M, Deng J, Yang Y, Zhou H, Tao T, Liu S, Sun L, Hua L.
Performance Analysis of a Floating Wind–Wave Power Generation Platform Based on the Frequency Domain Model. *Journal of Marine Science and Engineering*. 2024; 12(2):206.
https://doi.org/10.3390/jmse12020206

**Chicago/Turabian Style**

Chen, Mingsheng, Jiang Deng, Yi Yang, Hao Zhou, Tao Tao, Shi Liu, Liang Sun, and Lin Hua.
2024. "Performance Analysis of a Floating Wind–Wave Power Generation Platform Based on the Frequency Domain Model" *Journal of Marine Science and Engineering* 12, no. 2: 206.
https://doi.org/10.3390/jmse12020206