# Effect of the Dynamic Froude–Krylov Force on Energy Extraction from a Point Absorber Wave Energy Converter with an Hourglass-Shaped Buoy

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## Abstract

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## 1. Introduction

## 2. Hourglass Buoy Model

#### 2.1. Froude–Krylov Force

#### 2.2. Radiation Force

## 3. Contribution of Froude–Krylov Forces to the Free Response

## 4. Control Design

## 5. Spherical Buoy Model

## 6. Results

#### 6.1. Effect of the Nonlinear FK Force

#### 6.2. Hourglass and Spherical Buoy Comparison

#### 6.3. Sensitivity Analysis

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Simulated steady state results comparing the three noninertial terms of Equation (11): ${F}_{b}={F}_{fk,s}+{F}_{g}$, ${F}_{fk,d}$, $b\dot{\zeta}$ and their sum (upper plot) and the buoy displacement and wave elevation (lower plot) for a deep water wave with $T=18$ s and the buoy parameters shown in Table 1.

**Figure 3.**Simulated steady state results of a model without ${F}_{fk,d}$. The upper plot is ${F}_{b}={F}_{fk,s}+{F}_{g}$ force and the lower plot is the buoy displacement.

**Figure 4.**Control architecture illustrating its access to wave elevation, buoy states, and reference states.

**Figure 5.**Simulink model of the closed loop system. The plant model for both the hourglass and spherical buoys was implemented using a MATLAB Function block and an integrator.

**Figure 6.**Closed–loop hourglass buoy response comparing the effect of including ${F}_{fk,d}$ in the control law.

**Figure 7.**Closed–loop response and performance comparing the hourglass and spherical buoys. The hourglass buoy height and the spherical buoy diameter are both 5 m.

**Figure 8.**Closed–loop response and performance comparing the hourglass and spherical buoys. The hourglass buoy’s height is 5 m, and the spherical buoy diameter is 7.5 m so as to yield a similar energy extraction performance as the hourglass buoy.

**Figure 9.**Comparison of the controller performance when using the exact model parameters and perturbed values.

Feature | Symbol | Value | Units |
---|---|---|---|

Height | $2h$ | 5 | m |

Cone Angle | $\alpha $ | 60 | degree |

Draft | ${h}_{0}$ | $2.5$ | m |

Mass | m | $\mathrm{50,376}$ | kg |

Density | ${\rho}_{b}$ | 1000 | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

Added Mass | ${m}_{a}$ | $\mathrm{59,250}$ | kg |

Damping | b | $\mathrm{20,000}$ | N/(m/s) |

Feature | Symbol | Case 1 | Case 2 | Units |
---|---|---|---|---|

Radius | R | $2.5$ | $3.75$ | m |

Draft | ${h}_{d}$ | $2.5$ | $3.75$ | m |

Mass | m | 32,725 | 110,446 | kg |

Density | ${\rho}_{b}$ | 500 | 500 | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

Added Mass | ${m}_{a}$ | 14,019 | 47,492 | kg |

Damping | b | 11,208 | 18,665 | N/(m/s) |

First State Gain | Displacement Error (m) |
---|---|

10 | $0.130$ |

20 | $0.098$ |

50 | $0.064$ |

100 | $0.015$ |

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**MDPI and ACS Style**

Yassin, H.; Demonte Gonzalez, T.; Parker, G.; Wilson, D. Effect of the Dynamic Froude–Krylov Force on Energy Extraction from a Point Absorber Wave Energy Converter with an Hourglass-Shaped Buoy. *Appl. Sci.* **2023**, *13*, 4316.
https://doi.org/10.3390/app13074316

**AMA Style**

Yassin H, Demonte Gonzalez T, Parker G, Wilson D. Effect of the Dynamic Froude–Krylov Force on Energy Extraction from a Point Absorber Wave Energy Converter with an Hourglass-Shaped Buoy. *Applied Sciences*. 2023; 13(7):4316.
https://doi.org/10.3390/app13074316

**Chicago/Turabian Style**

Yassin, Houssein, Tania Demonte Gonzalez, Gordon Parker, and David Wilson. 2023. "Effect of the Dynamic Froude–Krylov Force on Energy Extraction from a Point Absorber Wave Energy Converter with an Hourglass-Shaped Buoy" *Applied Sciences* 13, no. 7: 4316.
https://doi.org/10.3390/app13074316