# Dynamics of Heaving Buoy Wave Energy Converters with a Stiffness Reactive Controller

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## Abstract

**:**

## 1. Introduction

## 2. Modelling

#### 2.1. Governing Equations

#### 2.2. Energy Frequency

#### 2.3. Available and Absorbed Power

## 3. Stiffness Reactive Controller

#### 3.1. CVT Controller

#### 3.2. Inertia Effect of Cvt Mechanism

#### 3.3. Controller Operating Bandwidth

#### 3.4. PTO Damping

#### 3.5. Experimental Validation

## 4. Discussion and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

WEC | Wave energy converter |

SDOF | Single-degree-of-freedom |

PTO | Power take-off system |

CVT | Continuous variable transmission |

## Appendix A

## Appendix B

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**Figure 3.**Controller bandwidth: $\Delta $ and ${k}_{eff}$ at different excitation frequencies $\omega $.

**Figure 4.**Effect of the PTO damping b in irregular waves on (

**a**) the average absorbed power ${P}_{b}$ in RMS, (

**b**) the capture width $\lambda $ in RMS.

**Figure 5.**The experimental setup: (

**a**) 3D schematic showing the exciting and controller mechanism, (

**b**) photograph of the controller mechanism (aligned vertically), showing the CVT.

Parameters | ${\mathit{m}}_{\mathit{b}}$ | ${\mathit{R}}_{\mathit{b}}$ | ${\mathit{K}}_{\mathit{ext}}$ | ${\mathit{I}}_{\mathit{P}}$ | r |
---|---|---|---|---|---|

[kg] | [m] | [kN/m] | [kg.m${}^{2}$] | [mm] | |

Value | 8000 | 0.4 | 8 | 2.63 | 75 |

Parameters | ${\mathit{m}}_{\mathit{b}}$ | ${\mathit{k}}_{\mathit{es}}$ | ${\mathit{k}}_{\mathit{ext}}$ | ${\mathit{I}}_{\mathit{P}}$ | r |
---|---|---|---|---|---|

[kg] | [N/m] | [N/m] | [kg.m${}^{2}$] | [mm] | |

Value | 4.75 | 200 | 330 | 0.0019 | 20 |

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**MDPI and ACS Style**

Sakr, A.H.; Metwalli, S.M.; Anis, Y.H.
Dynamics of Heaving Buoy Wave Energy Converters with a Stiffness Reactive Controller. *Energies* **2021**, *14*, 44.
https://doi.org/10.3390/en14010044

**AMA Style**

Sakr AH, Metwalli SM, Anis YH.
Dynamics of Heaving Buoy Wave Energy Converters with a Stiffness Reactive Controller. *Energies*. 2021; 14(1):44.
https://doi.org/10.3390/en14010044

**Chicago/Turabian Style**

Sakr, Ahmed H., Sayed M. Metwalli, and Yasser H. Anis.
2021. "Dynamics of Heaving Buoy Wave Energy Converters with a Stiffness Reactive Controller" *Energies* 14, no. 1: 44.
https://doi.org/10.3390/en14010044