# A Shape Optimization Method of a Specified Point Absorber Wave Energy Converter for the South China Sea

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis

_{wave}denotes the wave exciting force, and m is the mass of buoy. K

_{PTO}and R

_{PTO}represent the damping and elastic properties of the PTO system, respectively. K is on behalf of the spring characteristics of the hydrostatic force. In this paper, K

_{PTO}is ignored since the PTO is deemed as a pure damping system. The PA is considered to be a two-body WEC when the reference is a submerged body but a one-body WEC when the reference is the sea bed, and this paper concentrates on the latter case.

#### 2.1. Mathematical Model

#### 2.2. Objective Function

## 3. Geometry Optimization Methodology

_{av}), the South China Sea’s wave spectrum is obtained. Secondly, given the range of the buoy geometry parameters, the candidate buoy library is obtained by the Taguchi method. Then, the hydrodynamic performance of the candidate buoy is analyzed, along with the optimal PTO damping coefficient of each buoy being calculated to obtain the absorption power spectrum. On the basis of obtaining these two spectra, the objective function is applied to evaluate the performance so as to obtain the approximate range of the optimal buoy parameters. Finally, the RSM is used for local optimization to obtain the optimal buoy parameter configuration.

## 4. Wave Spectrum

_{av}) are presented in Table 1. SWH represents the mean wave height of the top one-third of the waves. The background color scale illustrates the occurrence probability level, showing that the most probable SWH values are below 4.5 m and the majority of SWH values fall in the 0.5–2.5 m interval. Most of the wave periods are distributed within the range of 3.5–9.5 s. The wave power level, P, per unit width in a wave can be calculated as follows [46]:

^{2}/(rad/s).

## 5. Absorption Power Spectrum

#### 5.1. Geometry Library Generation Based on Taguchi Design

^{3}= 125 tests are required to establish a database. In order to improve the optimization efficiency, Taguchi design is applied in this study. As shown in Table 4, an L

_{25}(5

^{3}) orthogonal experiment table is applied and a geometry library which contains 25 buoys is generated.

#### 5.2. Absorption Power Spectrum Calculation

#### 5.2.1. Hydrodynamic Parameters Calculation

#### 5.2.2. Optimal PTO Damping Determination

## 6. Parameter Optimization Based on RSM

## 7. Results and Discussion

#### 7.1. Optimal Geometry Configuration Analysis

#### 7.2. Performance Characteristic Analysis

#### 7.2.1. Energy Absorption Efficiency

#### 7.2.2. Resonance Frequency

#### 7.2.3. Absorption Bandwidth

_{PTO}should be greater than the radiation damping. According to Equation (16), the R

_{PTO}proposed in this paper is not less than the radiation damping, therefore all candidate buoys have a wider absorption bandwidth than the bandwidth of the wave spectrum (0.42 rad/s).

#### 7.2.4. Maximum Absorbed Power

#### 7.2.5. Effects of Adjusting Geometrical Parameters

#### 7.2.6. Effects of Adjusting PTO Damping

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Joint probability distribution of significant wave height (SWH) and wave average period (T

_{av}).

Wave Average Period (s) | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1.5 | 3.5 | 4.5 | 5.5 | 6.5 | 7.5 | 8.5 | 9.5 | 10.5 | 11.5 | 12.5 | 13.5 | 14.5 | ||

significant wave height (SWH, m) | 10.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

9.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

8.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

7.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

6.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 9 | 2 | 0 | 0 | 0 | |

5.5 | 0 | 0 | 0 | 0 | 0 | 0 | 28 | 54 | 24 | 1 | 0 | 0 | 0 | |

4.5 | 0 | 0 | 0 | 0 | 0 | 50 | 373 | 240 | 37 | 7 | 0 | 0 | 0 | |

3.5 | 0 | 0 | 0 | 0 | 163 | 1317 | 950 | 436 | 86 | 0 | 0 | 0 | 0 | |

2.5 | 0 | 0 | 0 | 874 | 4743 | 2891 | 1459 | 546 | 54 | 0 | 0 | 0 | 0 | |

1.5 | 0 | 19 | 3720 | 9300 | 5011 | 2925 | 1131 | 149 | 7 | 1 | 0 | 0 | 0 | |

0.5 | 531 | 4859 | 11,299 | 6525 | 3159 | 997 | 189 | 38 | 3 | 2 | 0 | 0 | 0 |

Wave Average Period (s) | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1.5 | 3.5 | 4.5 | 5.5 | 6.5 | 7.5 | 8.5 | 9.5 | 10.5 | 11.5 | 12.5 | 13.5 | 14.5 | ||

significant wave height (SWH, m) | 10.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

9.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

8.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

7.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

6.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 196 0.0% | 1959 0.3% | 476 0.1% | 0 | 0 | 0 | |

5.5 | 0 | 0 | 0 | 0 | 0 | 0 | 3533 0.6% | 7616 1.2% | 3741 0.6% | 170 0.0% | 0 | 0 | 0 | |

4.5 | 0 | 0 | 0 | 0 | 0 | 3727 0.6% | 31,510 5.1% | 22,660 3.7% | 3861 0.6% | 800 0.1% | 0 | 0 | 0 | |

3.5 | 0 | 0 | 0 | 0 | 6370 1% | 59,386 9.6% | 48,549 7.8% | 24,902 4% | 5429 0.9% | 0 | 0 | 0 | 0 | |

2.5 | 0 | 0 | 0 | 14,745 2.4% | 94,569 15.2% | 66,511 10.7% | 38,042 6.1% | 15,911 2.6% | 1739 0.6% | 0 | 0 | 0 | 0 | |

1.5 | 0 | 74 0.0% | 18,486 3% | 56,485 9.1% | 35,969 5.8% | 24,226 3.9% | 10,616 1.7% | 1563 0.3% | 81 0.0% | 12 0.0% | 0 | 0 | 0 | |

0.5 | 98 0.0% | 2087 0.3% | 6239 1% | 4403 0.7% | 2519 0.4% | 917 0.1% | 197 0.0% | 44 0.0% | 3 0.0% | 2 0.0% | 0 | 0 | 0 |

Geometrical Parameter | Minimum | Maximum |
---|---|---|

Base radius (r) | 3 (m) | 12 (m) |

Cone angle ($\theta $) | 40° | 120° |

Draft ($d$) | 0.5r | 2.5r |

Buoy ID | Base Radius (m) | Cone Angle (°) | Draft Ratio (d/r) | CoG (m) | CoB (m) |
---|---|---|---|---|---|

1 | 3 | 40 | 0.5 | −0.9 | −0.7 |

2 | 3 | 60 | 1 | −1.8 | −1.28 |

3 | 3 | 80 | 1.5 | −2.7 | −1.7 |

4 | 3 | 100 | 2 | −3.6 | −2 |

5 | 3 | 120 | 2.5 | −4.5 | −2.26 |

6 | 6 | 40 | 1 | −3.6 | −2.7 |

7 | 6 | 60 | 1.5 | −5.4 | −3.62 |

8 | 6 | 80 | 2 | −7.2 | −4.3 |

9 | 6 | 100 | 2.5 | −9 | −4.8 |

10 | 6 | 120 | 0.5 | −1.8 | −1.2 |

11 | 8 | 40 | 1.5 | −7.2 | −5.2 |

12 | 8 | 60 | 2 | −9.6 | −6.1 |

13 | 8 | 80 | 2.5 | −12 | −6.9 |

14 | 8 | 100 | 0.5 | −2.4 | −1.7 |

15 | 8 | 120 | 1 | −4.8 | −2.8 |

16 | 10 | 40 | 2 | −12 | −8.3 |

17 | 10 | 60 | 2.5 | −15 | −9.2 |

18 | 10 | 80 | 0.5 | −3 | −2.2 |

19 | 10 | 100 | 1 | −6 | −3.8 |

20 | 10 | 120 | 1.5 | −9 | −4.9 |

21 | 12 | 40 | 2.5 | −18 | −12 |

22 | 12 | 60 | 0.5 | −3.6 | −2.8 |

23 | 12 | 80 | 1 | −7.2 | −4.9 |

24 | 12 | 100 | 1.5 | −10.8 | −6.4 |

25 | 12 | 120 | 2 | −14.4 | −7.5 |

Base Radius | Cone Angle | Draft | |
---|---|---|---|

Maximum | 7282 | 6683 | 7264 |

Minimum | 4591 | 5484 | 5310 |

Range | 2691 | 1237 | 1954 |

Percentage | 46% | 21% | 33% |

Rank | 1 | 3 | 2 |

Base Radius | Cone Angle | Draft | |
---|---|---|---|

Maximum (rad/s) | 0.7820 | 0.7560 | 0.7335 |

Minimum (rad/s) | 0.5263 | 0.5364 | 0.5735 |

Range (rad/s) | 0.2557 | 0.2196 | 0.1600 |

Percentage | 40% | 35% | 25% |

Rank | 1 | 2 | 3 |

Base Radius | Cone Angle | Draft | |
---|---|---|---|

Maximum (rad/s) | 0.8919 | 0.6856 | 0.8039 |

Minimum (rad/s) | 0.5064 | 0.5762 | 0.5439 |

Range (rad/s) | 0.3855 | 0.1096 | 0.2600 |

Percentage | 51% | 15% | 34% |

Rank | 1 | 3 | 2 |

Base Radius | Cone Angle | Draft | |
---|---|---|---|

Maximum (kW) | 1254 | 1169 | 951 |

Minimum (kW) | 201 | 423 | 297 |

Range (kW) | 1053 | 746 | 654 |

Percentage | 43% | 30% | 27% |

Rank | 1 | 2 | 3 |

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## Share and Cite

**MDPI and ACS Style**

Wen, Y.; Wang, W.; Liu, H.; Mao, L.; Mi, H.; Wang, W.; Zhang, G.
A Shape Optimization Method of a Specified Point Absorber Wave Energy Converter for the South China Sea. *Energies* **2018**, *11*, 2645.
https://doi.org/10.3390/en11102645

**AMA Style**

Wen Y, Wang W, Liu H, Mao L, Mi H, Wang W, Zhang G.
A Shape Optimization Method of a Specified Point Absorber Wave Energy Converter for the South China Sea. *Energies*. 2018; 11(10):2645.
https://doi.org/10.3390/en11102645

**Chicago/Turabian Style**

Wen, Yadong, Weijun Wang, Hua Liu, Longbo Mao, Hongju Mi, Wenqiang Wang, and Guoping Zhang.
2018. "A Shape Optimization Method of a Specified Point Absorber Wave Energy Converter for the South China Sea" *Energies* 11, no. 10: 2645.
https://doi.org/10.3390/en11102645