# Sliding Mode Control of a Nonlinear Wave Energy Converter Model

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## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Diffraction Force

#### 2.2. Radiation Force

#### 2.3. Froude-Krylov Forces

## 3. Control Design

## 4. Simulation

## 5. Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Wave Period (s) | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|

Wave frequency ${\omega}_{0}$ (rad/s) | 2 | 1.57 | 1.26 | 1.05 | 0.89 | 0.78 | 0.7 |

Wavenumber $\chi \left({\omega}_{0}\right)$ | 0.42 | 0.25 | 0.16 | 0.11 | 0.082 | 0.063 | 0.05 |

Radiation damping $c\left({\omega}_{0}\right)\text{}\left(\mathrm{N}/\left(\mathrm{m}/\mathrm{s}\right)\right)$ | 16.19 | 16.81 | 14.35 | 11.21 | 8.51 | 6.64 | 5.02 |

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**Figure 2.**Heaving spherical point absorber at the still water level (SWL) on the left (

**a**); to the right (

**b**) is shown the displacement from resting position ${z}_{d}\left(t\right)$ and the wave elevation $\eta \left(t\right)$.

**Figure 5.**Relative displacement ${z}_{d}\left(t\right)-\eta \left(t\right)$ of the WEC when implementing SMC, CCC, and NL−CCC at $T=6\text{}\mathrm{s}$.

**Figure 6.**Absorbed energy (

**a**) and control force (

**b**) when implementing SMC, NL−CCC and CCC at T = 6 s.

**Figure 7.**Absorbed Power for the different control strategies with varying wave periods and constant steepness of regular waves.

**Figure 8.**Phase−plane plot of sliding surface $s\left(t\right)$ with SMC coefficient of ${A}_{c}=10\text{}kN$ on the left (

**a**), and ${A}_{c}=100\text{}MN$ on the right (

**b**).

**Figure 9.**(

**a**) Displacement error between reference and WEC displacement, (

**b**) Control Force, and (

**c**) Energy Absorption of the Exact Model and the Perturbed Model.

Parameter | Values |
---|---|

Buoy radius $R$ | $2.5\text{}\mathrm{m}$ |

Buoy mass $m$ | $\mathrm{32,725}\text{}\mathrm{kg}$ |

Buoy draft ${h}_{d}$ | $2.5\text{}\mathrm{m}$ |

Added mass ${m}_{a}\left({\omega}_{\infty}\right)$ | $\mathrm{14,019}\text{}\mathrm{kg}$ |

Radiation damping $c\left({\omega}_{0}\right)$ | $\mathrm{11,208}\text{}\mathrm{N}/\left(\mathrm{m}/\mathrm{s}\right)$ |

Water density $\rho $ | $1000\text{}\mathrm{kg}/{\mathrm{m}}^{3}$ |

Gravity constant $g$ | $9.81\text{}\mathrm{m}/{\mathrm{s}}^{2}$ |

Wave amplitude $A\left({\omega}_{0}\right)$ | $0.5\text{}\mathrm{m}$ |

Wave frequency ${\omega}_{0}$ | $1.05\text{}\mathrm{rad}/\mathrm{s}$ |

Wavenumber $\chi \left({\omega}_{0}\right)$ | $0.112$ |

SMC convergence rate $w$ | $8$ |

SMC coefficient $\varphi $ | $1000$ |

SMC coefficient ${A}_{c}$ | $10\text{}k\mathrm{N}$ |

Wave Period (s). | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|

Wavelength (m) | 15 | 25 | 39 | 56 | 77 | 100 | 127 |

Wave Amplitude (m) | 0.13 | 0.22 | 0.35 | 0.50 | 0.69 | 0.90 | 1.14 |

SMC Reference Amplitude (m) | 0.6 | 1.1 | 1.8 | 2.19 | 2.14 | 2.05 | 1.9 |

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**MDPI and ACS Style**

Demonte Gonzalez, T.; Parker, G.G.; Anderlini, E.; Weaver, W.W.
Sliding Mode Control of a Nonlinear Wave Energy Converter Model. *J. Mar. Sci. Eng.* **2021**, *9*, 951.
https://doi.org/10.3390/jmse9090951

**AMA Style**

Demonte Gonzalez T, Parker GG, Anderlini E, Weaver WW.
Sliding Mode Control of a Nonlinear Wave Energy Converter Model. *Journal of Marine Science and Engineering*. 2021; 9(9):951.
https://doi.org/10.3390/jmse9090951

**Chicago/Turabian Style**

Demonte Gonzalez, Tania, Gordon G. Parker, Enrico Anderlini, and Wayne W. Weaver.
2021. "Sliding Mode Control of a Nonlinear Wave Energy Converter Model" *Journal of Marine Science and Engineering* 9, no. 9: 951.
https://doi.org/10.3390/jmse9090951