# Wave Energy Extraction by Flexible Floaters

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

#### 2.1. Diffraction Potential Solution

#### 2.2. Radiation Potential Solution

#### 2.3. Structural Response and the Haskind–Hanaoka Formula

#### 2.4. Wave Power Extraction and Theoretical Maximum Efficiency

## 3. Results and Discussion

#### 3.1. Effects of the PTO

#### 3.2. Effects of the Ridge Height

#### 3.3. Effects of the Plate Stiffness

## 4. Comparison with Preliminary Demonstrator Data

#### 4.1. Description of the Demonstrator Device

#### 4.2. Demonstrator Results

#### 4.3. Comparison with Mathematical Model

## 5. Power Extraction in Irregular Waves

## 6. Conclusions

- The effect of the plate elasticity is to increase the number of the resonant frequencies with respect to a rigid plate, while wave power extraction and the bandwidth of the capture factor become larger. The same result has been obtained both in monochromatic and irregular waves.
- The PTO distribution plays a significant role, and it is seen that, by increasing the number of PTO devices, modal optimisation occurs and the overall efficiency of the system improves.
- We also investigated the effect of the ridge height below the plate. Analytical results showed that if a bottom structure is needed, the floater can be properly designed to maximise power extraction, despite reduced incident wave transmission. This aspect has potentially strong implications for the design of nearshore structures for coastal protection.
- We analysed the plate response to irregular waves described by a JONSWAP spectrum. We showed that the presence of a broad wave frequency range reduces the maximum resonant peaks of the system. However, away from resonance, the efficiency can be larger than that of the monochromatic case and the benefit of irregular waves is significant.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LCOE | Levelised cost of energy |

PTO | Power take-off |

WEC | Wave energy converter |

## Appendix A

## References

- Babarit, A. Ocean Wave Energy Conversion; Elsevier: London, UK, 2018. [Google Scholar]
- Renzi, E. Hydroelectromechanical modelling of a piezoelectric wave energy converter. Proc. R. Soc. A-Math. Phys.
**2016**, 472, 20160715. [Google Scholar] [CrossRef] [Green Version] - Buriani, F.; Renzi, E. Hydrodynamics of a Flexible Piezoelectric Wave Energy Harvester Moored on a Breakwater. In Proceedings of the 12th European Wave and Tidal Energy Conference (EWTEC 2017), Cork, Ireland, 27 August–1 September 2017. [Google Scholar]
- Zheng, S.; Meylan, M.H.; Zhu, X.; Greaves, D.; Iglesias, G. Hydroelastic interaction between water waves and an array of circular floating porous elastic plates. J. Fluid Mech.
**2020**, 900, A20. [Google Scholar] [CrossRef] - Zheng, S.; Meylan, M.H.; Fan, L.; Greaves, D.; Iglesias, G. Wave scattering by a floating porous elastic plate of arbitrary shape: A semi-analytical study. J. Fluids Struct.
**2020**, 92, 102827. [Google Scholar] [CrossRef] - Meylan, M.H.; Bennetts, L.G.; Peter, M.A. Water-wave scattering and energy dissipation by a floating porous elastic plate in three dimensions. Wave Motion
**2017**, 70, 240–250. [Google Scholar] [CrossRef] [Green Version] - Selvan, S.A.; Behera, H. Wave energy dissipation by a floating circular flexible porous membrane in single and two-layer fluids. Ocean Eng.
**2020**, 206, 107374. [Google Scholar] [CrossRef] - Karmakar, D.; Sahoo, T. Gravity wave interaction with floating membrane due to abrupt change in water depth. Ocean Eng.
**2008**, 35, 598–615. [Google Scholar] [CrossRef] - Wei, Y.; Barradas-Berglind, J.J.; Yua, Z.; van Rooij, M.; Prins, W.A.; Jayawardhana, B.; Vakis, A.I. Frequency-domain hydrodynamic modelling of dense and sparse arrays of wave energy converters. Renew. Energ.
**2019**, 135, 775–788. [Google Scholar] [CrossRef] - Wei, Y.; Bechlenberg, A.; van Rooij, M.; Jayawardhana, B.; Vakis, A.I. Modelling of a wave energy converter array with a nonlinear power take-off system in the frequency domain. Appl. Ocean Res.
**2019**, 90, 101824. [Google Scholar] [CrossRef] - Reddy, J.N. Theory and Analysis of Elastic Plates and Shells; CRC Press, Taylor & Francis Group: London, UK, 2007. [Google Scholar]
- Newman, J.N. Wave effects on deformable bodies. Appl. Ocean Res.
**1994**, 16, 47–59. [Google Scholar] [CrossRef] - Wu, C.; Watanabe, E.; Utsunomiya, T. An eigenfunction expansion-matching method for analyzing the wave-induced responses of an elastic floating plate. Appl. Ocean Res.
**1995**, 17, 301–310. [Google Scholar] [CrossRef] - Drimer, N.; Agnon, Y.; Stiassnie, M. A simplified analytical model for a floating breakwater in water of finite depth. Appl. Ocean Res.
**1992**, 14, 33–41. [Google Scholar] [CrossRef] - Mei, C.C.; Stiassnie, M.A.; Yue, D.K.-P. Theory and Application of Ocean Surface Waves; World Scientific: Singapore, 2017. [Google Scholar]
- Linton, C.M.; McIver, P. Mathematical Techniques for Wave/Structure Interactions; Chapman & Hall/CRC: London, UK, 2017. [Google Scholar]
- Michele, S.; Sammarco, P.; d’Errico, M. Theory of the synchronous motion of an array of floating flap gates oscillating wave surge converter. Proc. R. Soc. Lond. A
**2016**, 472, 20160174. [Google Scholar] [CrossRef] - Michele, S.; Renzi, E. A second-order theory for an array of curved wave energy converters in open sea. J. Fluids Struct.
**2019**, 88, 315–330. [Google Scholar] [CrossRef] - Sammarco, P.; Michele, S.; d’Errico, M. Flap gate farm: From Venice lagoon defense to resonating wave energy production. Part 1: Natural modes. Appl. Ocean Res.
**2013**, 43, 206–213. [Google Scholar] [CrossRef] - Michele, S.; Renzi, E.; Perez-Collazo, C.; Greaves, D.; Iglesias, G. Power extraction in regular and random waves from an OWC in hybrid wind-wave energy systems. Ocean Eng.
**2019**, 191, 106519. [Google Scholar] [CrossRef] - Michele, S.; Sammarco, P.; d’Errico, M.; Renzi, E.; Abdolali, A.; Bellotti, G.; Dias, F. Flap gate farm: From Venice lagoon defense to resonating wave energy production. Part 2: Synchronous response to incident waves in open sea. Appl. Ocean Res.
**2015**, 52, 43–61. [Google Scholar] [CrossRef] [Green Version] - Michele, S.; Sammarco, P.; d’Errico, M. The optimal design of a flap gate array in front of a straight vertical wall: Resonance of the natural modes and enhancement of the exciting torque. Ocean Eng.
**2016**, 118, 152–164. [Google Scholar] [CrossRef] - Michele, S.; Sammarco, P.; d’Errico, M. Weakly nonlinear theory for oscillating wave surge converters in a channel. J. Fluid Mech.
**2018**, 834, 55–91. [Google Scholar] [CrossRef] [Green Version] - Michele, S.; Renzi, E.; Sammarco, P. Weakly nonlinear theory for a gate-type curved array in waves. J. Fluid Mech.
**2019**, 869, 238–263. [Google Scholar] [CrossRef] [Green Version] - Manresa, G.M. Analysis and Comparison of Wave Energy Extraction in the Ocean Grazer’s Wave Tank Experimental Setup; University of Groningen: Groningen, The Netherlands, 2017. [Google Scholar]
- Brenes Casasola, J.; Muñoz Arias, M.; Barradas-Berglind, J.J.; Prins, W.A.; Vakis, A.I.; Jayawardhana, B. Energy Extraction Analysis of the Ocean Grazer WEC via Digital Particle Image Velocimetry; University of Groningen: Groningen, The Netherlands, 2017. [Google Scholar]
- Bögels, M. Validating Floater Blanket Models for the Ocean Grazer; University of Groningen: Groningen, The Netherlands, 2017. [Google Scholar]
- Dias, F.; Renzi, E.; Gallagher, S.; Sarkar, D.; Wei, Y.; Abadie, T.; Cummins, C.; Rafiee, A. Analytical and computational modelling for wave energy systems: The example of oscillating wave surge converters. Acta Mech. Sin.
**2017**, 33, 647–662. [Google Scholar] [CrossRef] [Green Version] - Parau, E.; Dias, F. Nonlinear effects in the response of a floating ice plate to a moving load. J. Fluid Mech.
**2002**, 460, 281–305. [Google Scholar] [CrossRef] - Wang, P.; Cheng, Z. Nonlinear Hydroelastic Waves beneath a Floating Ice Sheet in a Fluid of Finite Depth. Abstr. Appl. Anal.
**2013**, 2013, 108026. [Google Scholar] [CrossRef] - Goda, Y. Random Seas and Design of Maritime Structures; World Scientific: Singapore, 2000. [Google Scholar]

**Figure 1.**Artist’s sketch of a system of floater blanket wave energy converters (WECs) connected to offshore wind turbines. Source: www.oceangrazer.com.

**Figure 2.**Side view of the floating plate WEC. The power take-off (PTO) mechanisms are located at points ${x}_{i}$, $i=1,\dots ,M$.

**Figure 4.**Behaviour of the Capture Factor versus frequency of the incident waves and PTO-Coefficient. (

**a**) PTO at the ends ${x}_{i}=\pm L$; (

**b**) Equally spaced PTO every 5 m.

**Figure 5.**Behaviour of the Capture Factor versus frequency of the incident waves and PTO-Coefficient. (

**a**) Ridge height $c=2$ m; (

**b**) $c=4$ m.

**Figure 6.**Behaviour of the Capture Factor versus frequency of the incident waves and PTO-Coefficient. (

**a**) Flexible plate with stiffness factor $EI=6.9\times {10}^{3}$ kg m${}^{3}$s${}^{-2}$; (

**b**) The case of a rigid plate.

**Figure 7.**The floater blanket, made by a two-layer red silicone sheet, inside the wave tank. The absorbing beach is visible at the end of the tank.

**Figure 9.**Top view of the pumping system. The high-performance polyethylene cables transfer the motion from the flexible floater to the pistons inside the cylinders.

**Figure 10.**Performance (power per single piston in mW) of the experimental flexible floater device. Left panel: configurations S1 and S3; Right panel: configurations S2 and S4. The incident wave is coming from the right. Connecting lines are for graphical illustration purposes.

**Figure 11.**Extracted power by the single pistons for the mathematical and experimental models. Connecting lines are for graphical illustration purposes.

**Figure 12.**Behaviour of the Capture Factor in irregular sea waves versus peak frequency of the incident JONSWAP spectrum and PTO-Coefficient. (

**a**) Softened flexible plate with stiffness factor $EI\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}6.9\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$ kg m${}^{3}$s${}^{-2}$; (

**b**) The case of a rigid plate.

Set | Floater | Wave Height | Wave Period |
---|---|---|---|

S1 | Continuous | $0.06$ m | $1.36$ s |

S2 | Continuous | $0.08$ m | $1.62$ s |

S3 | Discontinuous | $0.06$ m | $1.36$ s |

S4 | Discontinuous | $0.08$ m | $1.62$ s |

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**MDPI and ACS Style**

Michele, S.; Buriani, F.; Renzi, E.; van Rooij, M.; Jayawardhana, B.; Vakis, A.I.
Wave Energy Extraction by Flexible Floaters. *Energies* **2020**, *13*, 6167.
https://doi.org/10.3390/en13236167

**AMA Style**

Michele S, Buriani F, Renzi E, van Rooij M, Jayawardhana B, Vakis AI.
Wave Energy Extraction by Flexible Floaters. *Energies*. 2020; 13(23):6167.
https://doi.org/10.3390/en13236167

**Chicago/Turabian Style**

Michele, Simone, Federica Buriani, Emiliano Renzi, Marijn van Rooij, Bayu Jayawardhana, and Antonis I. Vakis.
2020. "Wave Energy Extraction by Flexible Floaters" *Energies* 13, no. 23: 6167.
https://doi.org/10.3390/en13236167