# Shape Optimization of Oscillating Buoy Wave Energy Converter Based on the Mean Annual Power Prediction Model

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Geometric Model

#### 2.2. Mean Annual Power

**n**is the direction vector, ${A}_{m}$ is the added mass, ${B}_{C}$ is the radiation damping. The complex amplitude of the exciting force acting on the buoy can be expressed as:

#### 2.3. Prediction Model Method

#### 2.3.1. Response Surface Method

#### 2.3.2. Radial Basis Functions Neural Network

#### 2.3.3. Elliptical Basis Functions Neural Network

#### 2.3.4. Error Analysis

^{2}is used to evaluate the consistency between the estimated value of the prediction model and the real value. It can be expressed as:

^{2}is closer to 1, the model accuracy is higher. It is generally considered that when the R

^{2}is above 0.9, the accuracy of the prediction model can meet the engineering needs [30].

#### 2.4. Optimization Algorithms and Processes

## 3. Building and Testing Prediction Models

#### 3.1. Sample Library

#### 3.1.1. Determining the Design Space

_{PTO}are selected as the optimization parameters.

- (1)
- The diameter of buoy is preferably 5–10% of the main wavelength [37]. Therefore, the value range of the buoy diameter can be expressed as follows:$$0.6\mathrm{m}2R6\mathrm{m}$$
- (2)
- In order to ensure the rationality of the buoy shape, the draft of the buoy is normalized, which is expressed as D = H/R. The value range of D can be expressed as follows:$$0.5<D<1$$
- (3)
- The traditional design method for the PTO system damping is based on the spectral peak frequency resonance, which leads to low energy capturing efficiency [20]. Therefore, the global search method is used to find the optimal PTO system damping which matches the wave resources. The value range of C
_{PTO}can be expressed as follows:$$50\mathrm{KNs}/\mathrm{m}{C}_{PTO}300\mathrm{KNs}/\mathrm{m}$$

#### 3.1.2. Hydrodynamic Calculation Verification

#### 3.1.3. Calculating the Sample Points

#### 3.2. Training and Testing of the Prediction Model

^{2}of the training set is calculated. The R

^{2}of RSM prediction model is 0.99269. The R

^{2}of BFNN prediction model is 0.99913. The R

^{2}of EBFNN prediction model is 0.99963. The test results show that the obtained model could be used to predict the mean annual power.

## 4. Optimization Results and Discussion

^{2}. The accuracy of the prediction model established by the EBFNN method is the best, while the accuracy of the prediction model established by the RSM method is the worst.

_{PTO}. In Figure 11a, when the D is same, the $\overline{\mathrm{P}}$/S first increases and then decreases with the increase of the buoy radius. The $\overline{\mathrm{P}}$/S reaches peak when the radius is around 1.33 m. In Figure 11b, when the R is the same, the $\overline{\mathrm{P}}$/S decreases with the increase of the D. According to the change trend in Figure 11, it can be judged that the optimal parameters of the buoy are about when the R is 1.33 m and the draft is 0.665 m.

_{PTO}on the capture power of WEC is analyzed, as shown in Figure 12. In Figure 12a, when D is the same, the optimal C

_{PTO}increases with the increase of R, and the smaller D, the faster the optimal C

_{PTO}increases with the radius. In Figure 12b, when the R is the same, the optimal C

_{PTO}decreases with the increase of the D, and the larger the radius, the faster the optimal C

_{PTO}decreases with the increase of D. Since the wave energy is mainly distributed on the sea surface, wave force decays rapidly with increasing water depth. In Figure 12c, the $\overline{\mathrm{P}}$/S first increases and then decreases with the increase of C

_{PTO}. In the design space, each set of shape parameters has an optimal C

_{PTO}. According to the change trend in Figure 12, it can be judged that the optimal C

_{PTO}is 50 KNs/m.

## 5. Conclusions

- (1)
- The comparison shows that the prediction model established by the RSM method has the worst accuracy, the prediction model trained by the RBFNN method has better accuracy, and the prediction model trained by the EBFNN method has the best accuracy. The mean annual power prediction model trained by the EBFNN method can more accurately reflect the mapping relationship between the input and output. According to the shape parameters of the buoy, the mean annual power can be accurately predicted.
- (2)
- Taking the wave statistics data of the Chengshantou sea area near Weihai City, Shandong Province, China, as an example, the method of combining MIGA and the mean annual prediction model is adopted to obtain a high-performance design scheme, which provides a reference for engineering design. In the optimization process, the mean annual power prediction model replaces the simulation calculation, which can reduce a lot of workload (i.e., repeated modeling, simulation, and calculation). Compared with optimization design based on simulation results, this method can save considerable time and cost, effectively shorten the optimization design cycle, and improve the optimization efficiency. This optimization method can also be extended to the optimal design of other sea areas or types of WECs. In the future, we will continue to explore WEC array optimization methods. In order to promote the development of WEC commercialization, WEC array optimization methods will be investigated in the future.
- (3)
- The three optimization parameters have a significant impact on the energy capture performance. When the buoy shape is determined, the $\overline{\mathrm{P}}$/S first increases and then decreases with the increase of damping, and there is an optimal C
_{PTO}. The optimal C_{PTO}is significantly affected by buoy radius and draft, which is positively correlated with the buoy radius and negatively correlated with the buoy draft. When C_{PTO}is the optimal damping, the $\overline{\mathrm{P}}$/S increases first and then decreases with the increase of radius and the $\overline{\mathrm{P}}$/S decreases with the increase of draft.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Subpopulation/Island generation principle in MIGA [32].

**Figure 5.**The simulation results are verified with the results published by Mei [38].

**Figure 8.**Mean annual power prediction model. (

**a**) RSM prediction model. (

**b**) RBFNN prediction model. (

**c**) EBFNN prediction model.

**Figure 9.**Prediction models error analysis. (

**a**) RSM prediction model error analysis. (

**b**) RBFNN prediction model error analysis. (

**c**) EBFNN prediction model error analysis.

**Figure 10.**Optimization results based on three prediction models. (

**a**) RSM optimization results. (

**b**) RBFNN optimization results. (

**c**) EBFNN optimization results.

**Figure 11.**$\overline{\mathrm{P}}$/S with buoy-shape parameters under optimal C

_{PTO}. (

**a**) $\overline{\mathrm{P}}$/S with different R. (

**b**) $\overline{\mathrm{P}}$/S with different D.

**Figure 12.**Optimal C

_{PTO}with buoy shape parameters. (

**a**) Optimal C

_{PTO}with different R. (

**b**) Optimal C

_{PTO}with different D. (

**c**) $\overline{\mathrm{P}}$/S with different C

_{PTO}.

**Table 1.**The average probability of occurrences of significant wave height and average wave period from 2011 to 2020.

H_{s} (m) | T_{av} (s) | Total | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

2.5 | 3.5 | 4.5 | 5.5 | 6.5 | 7.5 | 8.5 | 9.5 | 10.5 | 11.5 | ||

0.25 | 3.25 | 14.00 | 7.37 | 2.64 | 0.77 | 0.16 | 0.07 | 0.03 | 0.01 | 0.00 | 28.30% |

0.75 | 0.10 | 15.56 | 18.28 | 5.19 | 1.63 | 0.67 | 0.32 | 0.19 | 0.04 | 0.02 | 42.00% |

1.25 | 0.00 | 0.54 | 10.88 | 3.37 | 0.46 | 0.14 | 0.08 | 0.02 | 0.02 | 0.02 | 15.53% |

1.75 | 0.00 | 0.00 | 1.64 | 5.24 | 0.31 | 0.07 | 0.00 | 0.00 | 0.00 | 0.01 | 7.27% |

2.25 | 0.00 | 0.00 | 0.02 | 2.76 | 0.89 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 | 3.70% |

2.75 | 0.00 | 0.00 | 0.00 | 0.39 | 1.62 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 | 2.04% |

3.25 | 0.00 | 0.00 | 0.00 | 0.01 | 0.79 | 0.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.88% |

3.75 | 0.00 | 0.00 | 0.00 | 0.00 | 0.11 | 0.11 | 0.01 | 0.00 | 0.00 | 0.00 | 0.23% |

4.25 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.05 | 0.00 | 0.00 | 0.00 | 0.00 | 0.05% |

4.75 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00% |

Total | 3.35% | 30.10% | 38.19% | 19.60% | 6.58% | 1.34% | 0.48% | 0.24% | 0.07% | 0.05% | 100% |

Run# | R | D | C_{PTO} | Run# | R | D | C_{PTO} |
---|---|---|---|---|---|---|---|

1 | 0.52 | 0.68 | 240 | 33 | 2.89 | 0.78 | 240 |

2 | 2.35 | 0.94 | 90 | 34 | 3 | 0.58 | 210 |

3 | 0.41 | 0.88 | 110 | 35 | 2.24 | 0.62 | 280 |

4 | 1.16 | 0.76 | 160 | 36 | 0.52 | 0.82 | 270 |

5 | 2.78 | 0.6 | 130 | 37 | 1.7 | 0.6 | 80 |

6 | 2.89 | 0.7 | 220 | 38 | 1.27 | 0.96 | 260 |

7 | 2.68 | 0.5 | 230 | 39 | 1.49 | 0.52 | 200 |

8 | 2.03 | 0.74 | 70 | 40 | 0.95 | 0.54 | 290 |

9 | 1.7 | 0.98 | 260 | 41 | 2.57 | 0.98 | 230 |

10 | 1.81 | 0.56 | 100 | 42 | 1.16 | 0.7 | 160 |

11 | 1.49 | 0.96 | 150 | 43 | 1.81 | 0.86 | 190 |

12 | 1.06 | 0.54 | 180 | 44 | 0.41 | 0.64 | 220 |

13 | 0.3 | 0.66 | 140 | 45 | 2.68 | 0.66 | 90 |

14 | 1.27 | 0.58 | 280 | 46 | 2.78 | 0.88 | 150 |

15 | 0.62 | 0.92 | 210 | 47 | 0.73 | 0.9 | 170 |

16 | 2.24 | 0.64 | 290 | 48 | 0.62 | 0.56 | 110 |

17 | 1.38 | 0.9 | 50 | 49 | 1.06 | 0.74 | 50 |

18 | 0.95 | 0.72 | 60 | 50 | 2.03 | 0.84 | 300 |

19 | 2.57 | 0.86 | 270 | 51 | 2.46 | 0.94 | 60 |

20 | 2.14 | 0.82 | 170 | 52 | 0.3 | 0.76 | 120 |

21 | 0.73 | 0.84 | 300 | 53 | 2.73 | 0.8 | 275 |

22 | 0.84 | 0.52 | 80 | 54 | 1.11 | 0.65 | 300 |

23 | 3 | 0.8 | 120 | 55 | 1.65 | 0.75 | 175 |

24 | 2.46 | 1 | 190 | 56 | 0.57 | 0.55 | 150 |

25 | 1.6 | 0.78 | 250 | 57 | 2.19 | 0.5 | 225 |

26 | 1.92 | 0.62 | 200 | 58 | 0.3 | 0.85 | 200 |

27 | 0.84 | 0.92 | 70 | 59 | 2.46 | 0.95 | 100 |

28 | 1.38 | 0.72 | 250 | 60 | 1.92 | 0.6 | 50 |

29 | 1.6 | 1 | 130 | 61 | 3 | 0.7 | 125 |

30 | 1.92 | 0.8 | 100 | 62 | 0.84 | 0.9 | 75 |

31 | 2.35 | 0.5 | 140 | 63 | 1.38 | 1 | 250 |

32 | 2.14 | 0.68 | 180 |

Prediction Model | R (m) | D (m/m) | C_{PTO}(KNs/m) | $\overline{\mathrm{P}}$/S (W/m ^{2}) | Simulation Result | Error |
---|---|---|---|---|---|---|

RSM | 0.91 | 0.5 | 50 | 120.18 | 109.80 | 9.45% |

RBFNN | 1.56 | 0.5 | 50 | 119.10 | 126.08 | 5.54% |

EBFNN | 1.34 | 0.5 | 50 | 131.46 | 131.63 | 0.13% |

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**MDPI and ACS Style**

Liu, T.; Liu, Y.; Huang, S.; Xue, G.
Shape Optimization of Oscillating Buoy Wave Energy Converter Based on the Mean Annual Power Prediction Model. *Energies* **2022**, *15*, 7470.
https://doi.org/10.3390/en15207470

**AMA Style**

Liu T, Liu Y, Huang S, Xue G.
Shape Optimization of Oscillating Buoy Wave Energy Converter Based on the Mean Annual Power Prediction Model. *Energies*. 2022; 15(20):7470.
https://doi.org/10.3390/en15207470

**Chicago/Turabian Style**

Liu, Tiesheng, Yanjun Liu, Shuting Huang, and Gang Xue.
2022. "Shape Optimization of Oscillating Buoy Wave Energy Converter Based on the Mean Annual Power Prediction Model" *Energies* 15, no. 20: 7470.
https://doi.org/10.3390/en15207470