# Design and Research of Slope-Pendulum Wave Energy Conversion Device

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## Abstract

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## 1. Introduction

## 2. Mathematical

#### 2.1. S-PWEC

#### 2.2. Motion Response

- $\theta $, $\dot{\theta}$, and $\ddot{\theta}$ are the angular displacement, angular velocity, and angular acceleration of the device moving at the pivot point, respectively.
- $I$ and ${A}_{55}\left(\infty \right)$ are the moment of inertia of the device and the additional moment of inertia at infinite frequency, respectively.
- ${{\displaystyle \int}}_{0}^{t}{K}_{R}\left(t-\xi \right)\dot{\theta}\left(t\right)d\xi $ is the convolution term of the radiation moment, which represents the memory effect of the fluid, where $\xi $ is a small time step.$${K}_{R}=\frac{2}{\pi}{{\displaystyle \int}}_{0}^{+\infty}B\left(\omega \right)\mathrm{cos}\left(\omega t\right)dt$$

- ${M}_{b}$ and ${M}_{G}$ are the buoyancy moment and the gravitational moment, respectively, which constitute the recovery moment of the floating plate, which is ${K}_{S}\theta $.
- ${M}_{ex}$ is the excitation moment; the excitation moment of the regular wave can be given by the following equation:$${M}_{ex}\left(t\right)=Re\left\{{R}_{f}\left(t\right)\frac{H}{2}{\tilde{M}}_{ex}\left(\omega ,\theta \right){e}^{i\omega t}\right\}$$$${R}_{f}\left(t\right)=\left(\right)open="\{">\begin{array}{cc}\frac{1}{2}\begin{array}{c}\left(1+\mathrm{cos}\left(\pi +\frac{\pi t}{{t}_{r}}\right)\right)\end{array}& \frac{t}{{t}_{r}}1\\ 1& \frac{t}{{t}_{r}}\ge 1\end{array}$$

- ${M}_{vis}$ is the secondary viscous drag moment; in practice, the wave energy converter device in the process of action with the wave will induce vortex detachment, resulting in a viscous effect [18].$${M}_{vis}=\left(-{C}_{D}\left|\mathrm{V}-{V}_{0}\right|\left(\mathrm{V}-{V}_{0}\right)\right)\times {L}_{vis}$$
- ${M}_{PTO}$ is the damping torque of the hydraulic power take-off.$${M}_{PTO}=\Delta {p}_{piston}{A}_{piston}\times {L}_{PTO}$$

## 3. Results and Discussion

#### 3.1. Initial Condition Setting

#### 3.1.1. Analysis of Sea State Conditions

#### 3.1.2. Single WEC Model

#### 3.2. Motion Response

#### 3.3. Power Generation Efficiency of Single WEC

#### 3.4. Power Generation Efficiency of Array WEC

## 4. Conclusions and Future Work

- The slope-pendulum wave energy conversion device adds a slope structure at the bottom to reflect through waves. The innovation of this structure can effectively enhance the ability of the floating plate to capture wave energy and improve the overall efficiency at a smaller cost.
- It can also cope with extreme sea conditions with large wave heights and has a certain “self-locking” function to keep the floating plate working normally, and the power generation efficiency of the wave energy conversion device is effectively improved under regular and irregular wave sea conditions.
- The array floating plate has a longitudinal optimal arrangement distance, the spacing is too large or too small, and it is not conducive to the array floating plate power generation. By experimental calculation, when the arrangement distance is 25 m, the whole wave energy conversion system electrical energy output power reaches the maximum.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Property | Floating Plate | Slope |
---|---|---|

Mass (kg) | 802.36 | 5065.41 |

${X}_{CG}\left(\mathrm{m}\right)$ | −2.24 | −2.651 |

${Y}_{CG}\left(\mathrm{m}\right)$ | 0 | 0 |

${Z}_{CG}\left(\mathrm{m}\right)$ | 0.22 | −2.7 |

${I}_{xx}\left(\mathrm{kg}{\mathrm{m}}^{2}\right)$ | 1379.1 | 60,802 |

${I}_{yy}\left(\mathrm{kg}{\mathrm{m}}^{2}\right)$ | 4211.8 | 42,574 |

${I}_{zz}\left(\mathrm{kg}{\mathrm{m}}^{2}\right)$ | 5439.5 | 17,467 |

Description | Value |
---|---|

Wave height (m) | H |

Wave period (s) | T |

Simulink solver | ode45 |

Maximum time step (s) | T/200 |

Wave ramp time (s) | 5 T |

Convolution integral time (s) | 25 |

Simulation end time (s) | 25 T |

Description | T(s) | H(m) |
---|---|---|

Value | 2–12 | 0.5–3 |

Average Power GenerationEfficiency (%) | WEC | |
---|---|---|

NS-PWEC | S-PWEC | |

Regular wave | 10.32 | 23.56 |

Irregular wave | 17.26 | 27.32 |

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**MDPI and ACS Style**

Wan, Z.; Li, Z.; Zhang, D.; Zheng, H.
Design and Research of Slope-Pendulum Wave Energy Conversion Device. *J. Mar. Sci. Eng.* **2022**, *10*, 1572.
https://doi.org/10.3390/jmse10111572

**AMA Style**

Wan Z, Li Z, Zhang D, Zheng H.
Design and Research of Slope-Pendulum Wave Energy Conversion Device. *Journal of Marine Science and Engineering*. 2022; 10(11):1572.
https://doi.org/10.3390/jmse10111572

**Chicago/Turabian Style**

Wan, Zhanhong, Ze Li, Dahai Zhang, and Honghao Zheng.
2022. "Design and Research of Slope-Pendulum Wave Energy Conversion Device" *Journal of Marine Science and Engineering* 10, no. 11: 1572.
https://doi.org/10.3390/jmse10111572