# Hydrodynamic Interactions and Enhanced Energy Harnessing amongst Many WEC Units in Large-Size Wave Parks

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## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Linear Potential Theory and the BEM

#### 2.2. WEC Hydrodynamics and Interaction Effects

#### 2.3. WEC Dynamics

## 3. Numerical Model

#### 3.1. The Bottom-Fixed Heaving Point Absorber WEC

#### 3.2. Drag Elements

#### 3.3. PTO System

#### 3.4. Power Performance Calculation

## 4. Study Cases

#### 4.1. Single WEC and Park Simulations

## 5. Results and Discussion

#### 5.1. Validation of the Single WEC Unit

#### 5.2. Effects of Wave Periods and Wave Heights on Power Performance

#### 5.3. Effects of Water Depth on Power Performance

#### 5.4. Effects of the Array layout on Power Performance

#### 5.5. Effects of Wave Directions on Power Performance

#### 5.6. Effects of WEC Distances on Power Performance

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Thorpe, T.W. A Brief Review of Wave Energy; A Report Produced for the UK Department of Energy, No ETSU R120; The UK Department of Trade and Industry: London, UK, 1999.
- Clément, A.; McCullen, P.; Falcão, A.; Fiorentino, A.; Gardner, F.; Hammarlund, K.; Lemonis, G.; Lewis, T.; Nielsen, K.; Petroncini, S.; et al. Wave energy in Europe: Current status and perspectives. Renew. Sustain. Energy Rev.
**2002**, 6, 405–431. [Google Scholar] [CrossRef] - Falnes, J. A review of wave-energy extraction. Mar. Struct.
**2007**, 20, 185–201. [Google Scholar] [CrossRef] - Falcao, A.F.d.O. Wave energy utilization: A review of the technologies. Renew. Sustain. Energy Rev.
**2010**, 14, 899–918. [Google Scholar] [CrossRef] - Budal, K. Theory for absorption of wave power by a system of interacting bodies. J. Ship Res.
**1977**, 21, 248–254. [Google Scholar] [CrossRef] - Falnes, J. Radiation impedance matrix and optimum power absorption for interacting oscillators in surface waves. Appl. Ocean Res.
**1980**, 2, 75–80. [Google Scholar] [CrossRef] - Yang, S.H.; Ringsberg, J.W.; Johnson, E. Wave energy converters in array configurations—Influence of interaction effects on the power performance and fatigue of mooring lines. Ocean Eng.
**2020**, 211, 107294. [Google Scholar] [CrossRef] - Li, X.; Xiao, Q.; Zhou, Y.; Ning, D.; Incecik, A.; Nicoll, R.; McDonald, A.; Campbell, D. Coupled CFD-MBD numerical modeling of a mechanically coupled WEC array. Ocean Eng.
**2022**, 256, 111541. [Google Scholar] [CrossRef] - Krivtsov, V.; Linfoot, B. Basin Testing of Wave Energy Converters in Trondheim: Investigation of Mooring Loads and Implications for Wider Research. J. Mar. Sci. Eng.
**2014**, 2, 326–335. [Google Scholar] [CrossRef] - Babarit, A. Impact of long separating distances on the energy production of two interacting wave energy converters. Ocean Eng.
**2010**, 37, 718–729. [Google Scholar] [CrossRef] - Stallard, T.; Stansby, P.K.; Williamson, A.J. An experimental study of closely spaced point absorber arrays. In Proceedings of the ISOPE International Ocean and Polar Engineering Conference, Vancouver, BC, Canada, 6–11 July 2008; p. ISOPE-I. [Google Scholar]
- Göteman, M. Wave energy parks with point-absorbers of different dimensions. J. Fluids Struct.
**2017**, 74, 142–157. [Google Scholar] [CrossRef] - Vasiliki, L.; Areti, L.; Eva, L. Hydrodynamic Interaction Effects and Performance of an Offshore Wave Farm. In Proceedings of the Twenty-Seventh 2017 International Ocean and Polar Engineering Conference, San Francisco, CA, USA, 25–30 June 2017; pp. 207–214. [Google Scholar]
- Weller, S.; Stallard, T.; Stansby, P. Experimental measurements of irregular wave interaction factors in closely spaced arrays. IET Renew. Power Gener.
**2010**, 4, 628–637. [Google Scholar] [CrossRef] - Lee, H.; Poguluri, S.; Bae, Y. Performance Analysis of Multiple Wave Energy Converters Placed on a Floating Platform in the Frequency Domain. Energies
**2018**, 11, 406. [Google Scholar] [CrossRef] - Hu, J.; Zhou, B.; Vogel, C.; Liu, P.; Willden, R.; Sun, K.; Zang, J.; Geng, J.; Jin, P.; Cui, L.; et al. Optimal design and performance analysis of a hybrid system combing a floating wind platform and wave energy converters. Appl. Energy
**2020**, 269, 114998. [Google Scholar] [CrossRef] - Ghafari, H.R.; Ghassemi, H.; Abbasi, A.; Vakilabadi, K.A.; Yazdi, H.; He, G. Novel concept of hybrid wavestar- floating offshore wind turbine system with rectilinear arrays of WECs. Ocean Eng.
**2022**, 262, 112253. [Google Scholar] [CrossRef] - Jin, P.; Zheng, Z.; Zhou, Z.; Zhou, B.; Wang, L.; Yang, Y.; Liu, Y. Optimization and evaluation of a semi-submersible wind turbine and oscillating body wave energy converters hybrid system. Energy
**2023**, 282, 128889. [Google Scholar] [CrossRef] - Zhou, B.; Hu, J.; Jin, P.; Sun, K.; Li, Y.; Ning, D. Power performance and motion response of a floating wind platform and multiple heaving wave energy converters hybrid system. Energy
**2023**, 265, 126314. [Google Scholar] [CrossRef] - Zhu, K.; Shi, H.; Zheng, S.; Michele, S.; Cao, F. Hydrodynamic analysis of hybrid system with wind turbine and wave energy converter. Appl. Energy
**2023**, 350, 121745. [Google Scholar] [CrossRef] - Göteman, M.; Engström, J.; Eriksson, M.; Isberg, J. Optimizing wave energy parks with over 1000 interacting point-absorbers using an approximate analytical method. Int. J. Mar. Energy
**2015**, 10, 113–126. [Google Scholar] [CrossRef] - Göteman, M.; Engström, J.; Eriksson, M.; Isberg, J. Fast Modeling of Large Wave Energy Farms Using Interaction Distance Cut-Off. Energies
**2015**, 8, 13741–13757. [Google Scholar] [CrossRef] - Stratigaki, V.; Troch, P.; Stallard, T.; Forehand, D.; Kofoed, J.P.; Folley, M.; Benoit, M.; Babarit, A.; Kirkegaard, J. Wave basin experiments with large wave energy converter arrays to study interactions between the converters and effects on other users in the sea and the coastal area. Energies
**2014**, 7, 701–734. [Google Scholar] [CrossRef] - Rahm, M.; Svensson, O.; Boström, C.; Waters, R.; Leijon, M. Experimental results from the operation of aggregated wave energy converters. IET Renew. Power Gener.
**2012**, 6, 149–160. [Google Scholar] [CrossRef] - Da Fonseca, F.C.; Gomes, R.; Henriques, J.; Gato, L.; Falcão, A. Model testing of an oscillating water column spar-buoy wave energy converter isolated and in array: Motions and mooring forces. Energy
**2016**, 112, 1207–1218. [Google Scholar] [CrossRef] - Wolgamot, H.A.; Taylor, P.H.; Taylor, R.E.; Van Den Bremer, T.; Raby, A.; Whittaker, C. Experimental observation of a near-motion-trapped mode: Free motion in heave with negligible radiation. J. Fluid Mech.
**2016**, 786, R5. [Google Scholar] [CrossRef] - Mercadé Ruiz, P.; Ferri, F.; Kofoed, J.P. Experimental validation of a wave energy converter array hydrodynamics tool. Sustainability
**2017**, 9, 115. [Google Scholar] [CrossRef] - Sinha, A.; Karmakar, D.; Soares, C.G. Performance of optimally tuned arrays of heaving point absorbers. Renew. Energy
**2016**, 92, 517–531. [Google Scholar] [CrossRef] - Tay, Z.Y.; Venugopal, V. Hydrodynamic interactions of oscillating wave surge converters in an array under random sea state. Ocean Eng.
**2017**, 145, 382–394. [Google Scholar] [CrossRef] - Shao, X.; Ringsberg, J.W.; Yao, H.D.; Li, Z.; Johnson, E.; Fredriksson, G. A comparison of two wave energy converters’ power performance and mooring fatigue characteristics—One WEC vs many WECs in a wave park with interaction effects. J. Ocean Eng. Sci.
**2023**, 8, 446–460. [Google Scholar] [CrossRef] - Newman, J.N. Marine Hydrodynamics; The MIT Press: Cambridge, MA, USA, 2018. [Google Scholar]
- Faltinsen, O. Sea Loads on Ships and Offshore Structures; Cambridge University Press: Cambridge, UK, 1993; Volume 1. [Google Scholar]
- HydroD. 2024. Available online: https://www.dnv.com/services/hydrodynamic-analysis-and-stability-analysis-software-hydrod-14492 (accessed on 7 April 2024).
- SIMA. SIMA Documenattion. 2024. Available online: https://sima.sintef.no/doc/4.4.0/sima/index.html (accessed on 7 April 2024).

**Figure 6.**Power output of the $4\times 4$ array layout (d = 150 m) for the regular wave conditions RL 1 to 11 and RH 1 to 11; see Table 3 for details. (

**b**) Interaction factors for the cases presented in (

**a**); see the text for details.

**Figure 7.**Illustration of an abnormal heave motion response of a WEC near resonance. Under regular waves, the wave height is 4 m, and the wave period is 10 s.

**Figure 8.**Power output of the $4\times 4$ array layout (d = 150 m) for the water depths 46 m and 55 m and the irregular sea states IRL 1 to 3. The percentages are the interaction factors indicating the additional power generation due to interaction effects.

**Figure 9.**Power output of the $4\times 4$ and $8\times 2$ array layouts (d = 150 m) for the irregular sea states IRL 1 to 3. The percentages are the interaction factors indicating the additional power generation or power reduction due to interaction effects.

**Figure 10.**Power output of individual WEC units in the $4\times 4$ (

**left**) and $8\times 2$ (

**right**) array layouts (d = 150 m) for the IRL 1 sea state.

**Figure 11.**Power output of individual WEC units in the $4\times 4$ (

**left**) and $8\times 2$ (

**right**) array layouts (d = 150 m) for the IRL 3 sea state.

**Figure 12.**Power output of the $4\times 4$ array layout (d = 150 m) for different wave directions and the irregular sea states IRL 1 to 3. The percentages are the interaction factors indicating the additional power generation due to interaction effects.

**Figure 13.**(

**a**) Power output of the $4\times 4$ array layout for different WEC distances for regular wave conditions RL 1 to 11. (

**b**) Interaction factors for different WEC distances under regular wave conditions RL 1 to 11.

Property | Value |
---|---|

Mass [kg] | $60.0\times {10}^{3}$ |

Draft [m] | 5.421 |

Diameter [m] | 9 |

Volume [m^{3}] | 200 |

Centre of gravity, ${\mathrm{COG}}_{\mathrm{w}}\left[\mathrm{m}\right]$ ^{1} | −0.314 |

Roll inertia relative to ${\mathrm{COG}}_{\mathrm{w}},{I}_{\mathrm{xx}}\left(\right)open="["\; close="]">\mathrm{kg}\times {\mathrm{m}}^{2}$ | $2.5\times {10}^{6}$ |

Pitch inertia relative to ${\mathrm{COG}}_{\mathrm{w}},{I}_{\mathrm{vv}}\left(\right)open="["\; close="]">\mathrm{kg}\times {\mathrm{m}}^{2}$ | $2.5\times {10}^{6}$ |

Yaw inertia relative to ${\mathrm{COG}}_{\mathrm{w}},{I}_{\mathrm{zz}}\left(\right)open="["\; close="]">\mathrm{kg}\times {\mathrm{m}}^{2}$ | $5.0\times {10}^{5}$ |

Water depth [m] | 46.0 or 55.0 |

Pre-tension [N] | $1.4\times {10}^{6}$ |

^{1}The origin of the reference Cartesian coordinate system (x, y, z) is placed on the plane of the water surface at the geometric centre of the WEC buoy when it is in its unloaded neutral position.

Drag Element No. | Radius R [m] | Height H [m] |
---|---|---|

D1 | 2.5 | 2.00 |

D2 | 4.0 | 2.00 |

D3 | 4.5 | 2.00 |

D4 | 4.0 | 2.00 |

D5 | 2.5 | 2.00 |

D6 | 1.3 | 0.42 |

Sea State | Wave Height [m] | Wave Period [s] | Useful Damping [kNs/m] | Direction | |
---|---|---|---|---|---|

Regular Low | RL 1 | 1 | 5 | 137 | 0° |

RL 2 | 1 | 6 | 115 | 0° | |

RL 3 | 1 | 7 | 94 | 0° | |

RL 4 | 1 | 8 | 80 | 0° | |

RL 5 | 1 | 9 | 78 | 0° | |

RL 6 | 1 | 10 | 88 | 0° | |

RL 7 | 1 | 11 | 107 | 0° | |

RL 8 | 1 | 12 | 129 | 0° | |

RL 9 | 1 | 13 | 152 | 0° | |

RL 10 | 1 | 14 | 176 | 0° | |

RL 11 | 1 | 15 | 199 | 0° | |

Regular High | RH 1 | 4 | 5 | 144 | 0° |

RH 2 | 4 | 6 | 124 | 0° | |

RH 3 | 4 | 7 | 107 | 0° | |

RH 4 | 4 | 8 | 163 | 0° | |

RH 5 | 4 | 9 | 227 | 0° | |

RH 6 | 4 | 10 | 287 | 0° | |

RH 7 | 4 | 11 | 343 | 0° | |

RH 8 | 4 | 12 | 395 | 0° | |

RH 9 | 4 | 13 | 445 | 0° | |

RH 10 | 4 | 14 | 494 | 0° | |

RH 11 | 4 | 15 | 541 | 0° | |

Irregular Low | IRL 1 | 1.75 | 7.5 | 90 | 0° or 4.7° |

IRL 2 | 1.75 | 9.5 | 86 | 0° or 9.4° | |

IRL 3 | 1.75 | 11.5 | 118 | 0° or 14° |

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**MDPI and ACS Style**

Shao, X.; Ringsberg, J.W.; Yao, H.-D.; Gowda, U.R.S.L.; Khedkar, H.N.; Todalshaug, J.H.
Hydrodynamic Interactions and Enhanced Energy Harnessing amongst Many WEC Units in Large-Size Wave Parks. *J. Mar. Sci. Eng.* **2024**, *12*, 730.
https://doi.org/10.3390/jmse12050730

**AMA Style**

Shao X, Ringsberg JW, Yao H-D, Gowda URSL, Khedkar HN, Todalshaug JH.
Hydrodynamic Interactions and Enhanced Energy Harnessing amongst Many WEC Units in Large-Size Wave Parks. *Journal of Marine Science and Engineering*. 2024; 12(5):730.
https://doi.org/10.3390/jmse12050730

**Chicago/Turabian Style**

Shao, Xinyuan, Jonas W. Ringsberg, Hua-Dong Yao, Uday Rajdeep Sakleshpur Lokesh Gowda, Hrishikesh Nitin Khedkar, and Jørgen Hals Todalshaug.
2024. "Hydrodynamic Interactions and Enhanced Energy Harnessing amongst Many WEC Units in Large-Size Wave Parks" *Journal of Marine Science and Engineering* 12, no. 5: 730.
https://doi.org/10.3390/jmse12050730