# Influence of Power Take-Off Modelling on the Far-Field Effects of Wave Energy Converter Farms

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. MILDwave-NEMOH Coupled Model

#### 2.2. Parametric Study

#### 2.3. WEC Numerical Implementation

- A cylindrical HPA of a diameter 20 m and a height of 4 m moored to the seabed at a water depth of 30 m (Figure 1A). The HPA WEC motion is restricted to heave only, and therefore, only one degree of freedom (DoF) is considered for the simulations.
- A surface-piercing OSWEC of a width 20 m, a height of 12 m and a length of 1 m hinged to the seabed at a water depth of 10 m (Figure 1B). The OSWEC motion is restricted to pitch only, and only one DoF is considered for the simulations.

#### 2.4. PTO Numerical Implementation

#### 2.4.1. Linear PTO System

#### 2.4.2. Hydraulic PTO System

## 3. Results and Discussion

#### 3.1. Comparison between the Sponge Layer Technique in MILDwave and the MILDwave-NEMOH Coupled Model

#### 3.1.1. Single WEC

#### 3.1.2. WEC Farm

#### 3.2. PTO Modelling Techniques in the MILDwave-NEMOH Coupled Model

#### 3.2.1. Power Production

#### 3.2.2. “Far-Field” Effects

#### 3.2.3. Effect of the PTO System in the “Far-Field”

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

TRL | technology readiness level |

WEC | wave energy converter |

SPH | smoothed particle hydrodynamics |

BEM | boundary element method |

CFD | computer fluid dynamics |

DoF | degree of freedom |

PTO | power take-off |

RAO | response amplitude operator |

HPA | heaving point absorber |

OSWEC | oscillating wave surge wave energy converter |

## References

- Weber, J.; Costello, R.; Ringwood, J. WEC Technology Performance Levels (TPLs)—Metric for Successful Development of Economic WEC Technology. In Proceedings of the 10th European Wave and Tidal Energy Conference, Aalborg, Denmark, 2–5 September 2013. [Google Scholar]
- Stratigaki, V. WECANet: The First Open Pan-European Network for Marine Renewable Energy with a Focus on Wave Energy-COST Action CA17105. Water
**2019**, 11, 1249. [Google Scholar] [CrossRef] [Green Version] - Millar, D.L.; Smith, H.C.M.; Reeve, D.E. Modelling analysis of the sensitivity of shoreline change to a wave farm. Ocean. Eng.
**2007**, 34, 884–901. [Google Scholar] [CrossRef] - Delft University of Technology. SWAN (Simulating WAves Nearshore); A Third Generation Wave Model Copyright©1993–2020. 2020. Available online: http://swanmodel.sourceforge.net/features/features.htm/ (accessed on 11 November 2020).
- Smith, H.C.; Pearce, C.; Millar, D.L. Further analysis of change in nearshore wave climate due to an offshore wave farm: An enhanced case study for the Wave Hub site. Renew. Energy
**2012**, 40, 51–64. [Google Scholar] [CrossRef] - Rusu, E.; Onea, F. Study on the influence of the distance to shore for a wave energy farm operating in the central part of the Portuguese nearshore. Energy Convers. Manag.
**2016**, 114, 209–223. [Google Scholar] [CrossRef] - Onea, F.; Rusu, E. The Expected Shoreline Effect of a Marine Energy Farm Operating Close to Sardinia Island. Water
**2019**, 11, 2303. [Google Scholar] [CrossRef] [Green Version] - Raileanu, A.; Onea, F.; Rusu, E. An Overview of the Expected Shoreline Impact of the Marine Energy Farms Operating in Different Coastal Environments. J. Mar. Sci. Eng.
**2020**, 8, 228. [Google Scholar] [CrossRef] [Green Version] - Iglesias, G.; Carballo, R.; Castro, A. Development of the WaveCat wave energy converter. In Proceedings of the 31st International Conference of Coastal Engineering, Hamburg, Germany, 31 August–5 September 2008. [Google Scholar]
- Carballo, R.; Iglesias, G. Wave farm impact based on realistic wave-WEC interaction. Energy
**2013**, 51, 216–229. [Google Scholar] [CrossRef] - Fernandez, H.; Iglesias, G.; Carballo, R.; Castro, A.; Fraguela, J.; Taveira-Pinto, F.; Sanchez, M. The new wave energy converter WaveCat: Concept and laboratory tests. Mar. Struct.
**2012**, 29, 58–70. [Google Scholar] [CrossRef] - Iglesias, G.; Carballo, R. Wave farm impact: The role of farm-to-coast distance. Renew. Energy
**2014**, 69, 375–385. [Google Scholar] [CrossRef] - Abanades, J.; Greaves, D.; Iglesias, G. Coastal defence using wave farms: The role of farm-to-coast distance. Renew. Energy
**2015**, 75, 572–582. [Google Scholar] [CrossRef] [Green Version] - XBeach Open Source Community. XBeach. 2019. Available online: https://oss.deltares.nl/web/xbeach/ (accessed on 6 June 2019).
- Chang, G.; Ruehl, K.; Jones, C.; Roberts, J.; Chartrand, C. Numerical modelling of the effects of wave energy converter characteristics on nearshore wave conditions. Renew. Energy
**2016**, 89, 636–648. [Google Scholar] [CrossRef] [Green Version] - Beels, C.; Troch, P.; De Backer, G.; Vantorre, M.; De Rouck, J. Numerical implementation and sensitivity analysis of a wave energy converter in a time-dependent mild-slope equation model. Coast. Eng.
**2010**, 57, 471–492. [Google Scholar] [CrossRef] - Tuba Özkan-Haller, H.; Haller, M.C.; Cameron McNatt, J.; Porter, A.; Lenee-Bluhm, P. Analyses of Wave Scattering and Absorption Produced by WEC Arrays: Physical/Numerical Experiments and Model Assessment. In Marine Renewable Energy: Resource Characterization and Physical Effects; Springer International Publishing: Cham, Switzerland, 2017; pp. 71–97. [Google Scholar] [CrossRef]
- Stokes, C.; Conley, D. Modelling Offshore Wave farms for Coastal Process Impact Assessment: Waves, Beach Morphology, and Water Users. Energies
**2018**, 11, 2517. [Google Scholar] [CrossRef] [Green Version] - Davidson, J.; Costello, R. Efficient Nonlinear Hydrodynamic Models for Wave Energy Converter Design: A Scoping Study. J. Mar. Sci. Eng.
**2020**, 8, 35. [Google Scholar] [CrossRef] [Green Version] - Giorgi, G.; Ringwood, J.V. Articulating Parametric Nonlinearities in Computationally Efficient Hydrodynamic Models. IFAC-PapersOnLine
**2018**, 51, 56–61. [Google Scholar] [CrossRef] - Magana, M.E.; Brown, D.R.; Gaebelle, D.T.; Henriques, J.C.; Brekken, T.K. Sliding Mode Control of an Array of Three Oscillating Water Column Wave Energy Converters to Optimize Electrical Power. In Proceedings of the 13th European Wave and Tidal Energy Conference, Napoli, Italy, 1–6 September 2019. [Google Scholar]
- Gaebele, D.; Magaña, M.; Brekken, T.; Sawodny, O. State space model of an array of oscillating water column wave energy converters with inter-body hydrodynamic coupling. Ocean. Eng.
**2020**, 195, 106668. [Google Scholar] [CrossRef] - Crespo, A.; Altomare, C.; Domínguez, J.; González-Cao, J.; Gómez-Gesteira, M. Towards simulating floating offshore oscillating water column converters with Smoothed Particle Hydrodynamics. Coast. Eng.
**2017**, 126, 11–26. [Google Scholar] [CrossRef] - Verbrugghe, T.; Domínguez, J.M.; Crespo, A.J.; Altomare, C.; Stratigaki, V.; Troch, P.; Kortenhaus, A. Coupling methodology for smoothed particle hydrodynamics modelling of non-linear wave-structure interactions. Coast. Eng.
**2018**, 138, 184–198. [Google Scholar] [CrossRef] - Domínguez, J.M.; Crespo, A.J.; Hall, M.; Altomare, C.; Wu, M.; Stratigaki, V.; Troch, P.; Cappietti, L.; Gómez-Gesteira, M. SPH simulation of floating structures with moorings. Coast. Eng.
**2019**, 153, 103560. [Google Scholar] [CrossRef] - Ransley, E.; Yan, S.; Brown, S.; Musiedlak, P.H.; Windt, C.; Schmitt, P.; Davidson, J.; Ringwood, J.; Wang, J.; Wang, J. A blind comparative study of focused wave interactions with floating structures (CCP-WSI Blind Test Series 3). Int. J. Offshore Polar Eng.
**2019**, 29. [Google Scholar] [CrossRef] [Green Version] - Windt, C.; Davidson, J.; Schmitt, P.; Ringwood, J. Contribution to the CCP-WSI Blind Test Series 3: Analysis of scaling effects of moored point-absorber wave energy converters in a CFD-based numerical wave tank. In Proceedings of the 29th International Ocean and Polar Engineering Conference, Honolulu, HI, USA, 16–21 June 2019. [Google Scholar]
- Liu, Z.; Wang, Y.; Hua, X. Numerical studies and proposal of design equations on cylindrical oscillating wave surge converters under regular waves using SPH. Energy Convers. Manag.
**2020**, 203, 112242. [Google Scholar] [CrossRef] - Ransley, E.; Greaves, D.; Raby, A.; Simmonds, D.; Hann, M. Survivability of wave energy converters using CFD. Renew. Energy
**2017**, 109, 235–247. [Google Scholar] [CrossRef] [Green Version] - Devolder, B.; Stratigaki, V.; Troch, P.; Rauwoens, P. CFD simulations of floating point absorber wave energy converter arrays subjected to regular waves. Energies
**2018**, 11, 641. [Google Scholar] [CrossRef] [Green Version] - Abbasnia, A.; Soares, C.G. Fully nonlinear simulation of wave interaction with a cylindrical wave energy converter in a numerical wave tank. Ocean. Eng.
**2018**, 152, 210–222. [Google Scholar] [CrossRef] - Kim, S.; Koo, W. Numerical simulation of a latching controlled heaving-buoy-type point absorber by using a 3D numerical wave tank. In Proceedings of the 13th European Wave and Tidal Energy Conference (EWTEC 2019), Napoli, Italiy, 1–6 September 2019. [Google Scholar]
- Giorgi, G.; Gomes, R.P.F.; Bracco, G.; Mattiazzo, G. The Effect of Mooring Line Parameters in Inducing Parametric Resonance on the Spar-Buoy Oscillating Water Column Wave Energy Converter. J. Mar. Sci. Eng.
**2020**, 8, 29. [Google Scholar] [CrossRef] [Green Version] - Charrayre, F.; Peyrard, C.; Benoit, M.; Babarit, A. A Coupled Methodology for Wave-Body Interactions at the Scale of a Farm of Wave Energy Converters Including Irregular Bathymetry. In Proceedings of the ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering, San Francisco, CA, USA, 8–13 June 2014. [Google Scholar]
- Balitsky, P.; Verao Fernandez, G.; Stratigaki, V.; Troch, P. Coupling methodology for modelling the near-field and far-field effects of a Wave Energy Converter. In Proceedings of the ASME 36th International Conference on Ocean, Offshore and Arctic Engineering, Trondheim, Norway, 25–30 June 2017. [Google Scholar]
- Tomey-Bozo, N.; Babarit, J.M.A.; Troch, P.; Lewis, T.; Thomas, G. Wake Effect Assesment of a flap-type wave energy converter farm using a coupling methodology. In Proceedings of the ASME 36th International Conference on Ocean, Offshore and Arctic Engineering, Trondheim, Norway, 25–30 June 2017. [Google Scholar]
- Fernandez, G. A Numerical Study of the Far Field Effects of Wave Energy Converters in Short and Long-Crested Waves Utilizing a Coupled Model Suite. Ph.D. Thesis, Ghent University, Ghent, Belgium, 2019. [Google Scholar]
- Verbrugghe, T.; Stratigaki, V.; Troch, P.; Rabussier, R.; Kortenhaus, A. A comparison study of a generic coupling methodology for modelling wake effects of wave energy converter arrays. Energies
**2017**, 10, 1697. [Google Scholar] [CrossRef] [Green Version] - Kemper, J.; Windt, C.; Graf, K.; Ringwood, J.V. Development towards a nested hydrodynamic model for the numerical analysis of ocean wave energy systems. In Proceedings of the 13th European Wave and Tidal Energy Conference, Naples, Italy, 1–6 September 2019. [Google Scholar]
- Rijnsdorp, D.P.; Hansen, J.E.; Lowe, R.J. Simulating the wave-induced response of a submerged wave-energy converter using a non-hydrostatic wave-flow model. Coast. Eng.
**2018**, 140, 189–204. [Google Scholar] [CrossRef] - Rijnsdorp, D.P.; Hansen, J.E.; Lowe, R.J. Understanding coastal impacts by nearshore wave farms using a phase-resolving wave model. Renew. Energy
**2020**, 150, 637–648. [Google Scholar] [CrossRef] - Tomey-Bozo, N.; Babarit, A.; Murphy, J.; Stratigaki, V.; Troch, P.; Lewis, T.; Thomas, G. Wake effect assessment of a flap type wave energy converter farm under realistic environmental conditions by using a numerical coupling methodology. Coast. Eng.
**2018**. [Google Scholar] [CrossRef] [Green Version] - Fernandez, G.; Balitsky, P.; Stratigaki, V.; Troch, P. Coupling Methodology for Studying the Far Field Effects of Wave Energy Converter Arrays over a Varying Bathymetry. Energies
**2018**, 11, 2899. [Google Scholar] [CrossRef] [Green Version] - Stratigaki, V.; Troch, P.; Forehand, D. A fundamental coupling methodology for modelling near-field and far-field wave effects of floating structures and wave energy devices. Renew. Energy
**2019**, in press. [Google Scholar] [CrossRef] - Balitsky, P.; Verao Fernandez, G.; Stratigaki, V.; Troch, P. Assessment of the Power Output of a Two-Array Clustered WEC Farm Using a BEM Solver Coupling and a Wave-Propagation Model. Energies
**2018**, 11, 2907. [Google Scholar] [CrossRef] [Green Version] - Balitsky, P.; Quartier, N.; Stratigaki, V.; Verao Fernandez, G.; Vasarmidis, P.; Troch, P. Analysing the near-field effects and the power production of near-shore WEC array using a new wave-to-wire model. Water
**2019**, 11. [Google Scholar] [CrossRef] [Green Version] - Verao Fernandez, G.; Stratigaki, V.; Troch, P. Irregular Wave Validation of a Coupling Methodology for Numerical Modelling of Near and Far Field Effects of Wave Energy Converter Arrays. Energies
**2019**, 12, 538. [Google Scholar] [CrossRef] [Green Version] - Bingham, H. A hybrid Boussinesq-panel method for predicting the motion of a moored ship. Coast. Eng.
**2000**, 40, 21–38. [Google Scholar] [CrossRef] - Agamloh, E.B.; Wallace, A.K.; von Jouanne, A. Application of fluid-structure interaction simulation of an ocean wave energy extraction device. Renew. Energy
**2008**, 33, 748–757. [Google Scholar] [CrossRef] - McCallum, P.D. Numerical Methods for Modelling the Viscous Effects on the Interactions between Multiple Wave Energy Converters. Ph.D. Thesis, The University of Edinburgh, Edinburgh, UK, 2017. [Google Scholar]
- Babarit, A.; Delhommeau, G. Theoretical and numerical aspects of the open source BEM solver {NEMOH}. In Proceedings of the 11th European Wave and Tidal Energy Conference, Nantes, France, 6–11 September 2015. [Google Scholar]
- Penalba, M.; Kelly, T.; Ringwood, J. Using NEMOH for Modelling Wave Energy Converters: A Comparative Study with WAMIT. In Proceedings of the 12th European Wave and Tidal Energy Conference, Cork, Ireland, 27 August–1 September 2017. [Google Scholar]
- Troch, P. MILDwave—A Numerical Model for Propagation and Transformation of Linear Water Waves; Technical Report; Internal Report; Department of Civil Engineering, Ghent University: Ghent, Belgium, 1998. [Google Scholar]
- Troch, P.; Stratigaki, V. Phase-Resolving Wave Propagation Array Models. In Numerical Modelling of Wave Energy Converters; Folley, M., Ed.; Elsevier: Amsterdam, The Netherlands, 2016; Chapter 10; pp. 191–216. [Google Scholar] [CrossRef]
- Radder, A.C.; Dingemans, M.W. Canonical equations for almost periodic, weakly nonlinear gravity waves. Wave Motion
**1985**, 7, 473–485. [Google Scholar] [CrossRef] - Vasarmidis, P.; Stratigaki, V.; Troch, P. Accurate and Fast Generation of Irregular Short Crested Waves by Using Periodic Boundaries in a Mild-Slope Wave Model. Energies
**2019**, 12, 785. [Google Scholar] [CrossRef] [Green Version] - Stratigaki, V. Experimental Study and Numerical Modelling of Intra-Array Interactions and Extra-Array Effects of Wave Energy Converter Arrays. Ph.D. Thesis, Ghent University, Ghent, Belgium, 2014. [Google Scholar]
- European Marine Energy Centre (EMEC) Ltd. Wave Developers Database. Available online: http://www.emec.org.uk/marine-energy/wave-developers/ (accessed on 13 November 2018).
- Alves, M. Frequency-Domain Models. In Numerical Modelling of Wave Energy Converters; Folley, M., Ed.; Elsevier: Amsterdam, The Netherlands, 2016; Chapter 2; pp. 11–30. [Google Scholar] [CrossRef]
- Cargo, C. Design and Control of Hydraulic Power Take-Offs for Wave Energy Converters. Ph.D. Thesis, University of Bath, Bath, UK, 2013. [Google Scholar]

**Figure 2.**Mesh convergence study for (

**A**) the HPA WEC and (

**B**) the OSWEC for different panel discretizations.

**Figure 3.**Detail from the sponge layer technique employed in MILDwave to mimic the behaviour of the WEC. In Subplot (

**A**), the HPA WEC is represented by an ${N}_{x}$ = 5 × ${N}_{y}$ = 5 cell grid configuration. In Subplot (

**B**), the OSWEC is represented by an ${N}_{x}$ = 2 × ${N}_{y}$ = 5 grid cell configuration. Black cells indicate fully reflective cells, while the red cells are assigned a specific wave absorption coefficient.

**Figure 4.**Kd disturbance coefficient results for the MILDwave-NEMOH coupled model and the MILDwave sponge layer technique along one longitudinal cross-section at the centre of the domain for a single HPA WEC interacting with regular waves of H = 2 m and T = 8 s. The coupling region is filled in grey colour and includes the WEC’s cross-section indicated by a black vertical area. Incident waves are generated from the left to the right.

**Figure 5.**Kd disturbance coefficient results for the MILDwave-NEMOH coupled model and the MILDwave sponge layer technique along one longitudinal cross-section at the centre of the domain for a single OSWEC interacting with regular waves of H = 2 m and T = 8 s. The coupling region is filled in grey colour and includes the WEC’s cross-section indicated by a black vertical area. Incident waves are generated from the left to the right.

**Figure 6.**${K}_{d}$ disturbance coefficient results for a 21 HPA WEC farm interacting with regular waves of $H=2.0$ m, $T\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}8$ s and (

**A**) $\theta $ = 0°, (

**B**) 15° and (

**C**) 30° in the MILDwave-NEMOH coupled model. S1, S2 and S3 indicate the locations of the cross-sections. Contour levels are set at an interval of 0.05 for the ${K}_{d}$ value. The water depth is 30 m. Waves are generated along the bottom boundary.

**Figure 7.**${K}_{d}$ disturbance coefficient results along three transversal cross-sections S1, S2 and S3 as indicated in Figure 6, for a 21 HPA farm interacting with regular waves of $H=2.0$ m and $T\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}8$ s and $\theta $ = 0°, 15° and 30°, respectively.

**Figure 8.**Bar charts showing the average power output for (

**A**) a 21 HPA WEC farm (left) and (

**B**) a 21 OSWEC farm (right) with a linear optimal PTO (blue, first bar), a linear sub-optimal PTO (red, second) and a hydraulic PTO (green, third bar). The percentage difference between the linear optimal PTO and the linear sub-optimal and hydraulic PTOs, respectively, is shown at the top of the bars.

**Figure 9.**Bar chart showing the minimum ${K}_{d}$ disturbance coefficient for a 21 HPA farm with a linear optimal PTO (blue, first bar), a linear sub-optimal PTO (red, second bar) and a hydraulic PTO (green, third bar), interacting with regular waves of $H=2.0$ m, $T=8.0$ w and $\theta $ = 0°, 15° and 30°, respectively. The minimum ${K}_{d}$ is obtained along the three transversal cross-sections S1 (y = 1000 m), S2 (y = 3000 m) and S3 (y = 5000 m), as indicated in Figure 6.

**Figure 10.**Bar chart showing the minimum ${K}_{d}$ disturbance coefficient for a 21 OSWEC farm with a linear optimal PTO (blue, first bar), a linear sub-optimal PTO (red, second bar) and a hydraulic PTO (green, third bar), interacting with regular waves of $H=2.0$ m, $T=8.0$ s and $\theta $ = 0°, 15° and 30°, respectively. The minimum ${K}_{d}$ is obtained along the three transversal cross-sections S1 (y = 1000 m), S2 (y = 3000 m) and S3 (y = 5000 m), as indicated in Figure 6.

**Table 1.**${K}_{d,diff}$ (%) at $\mathrm{x}=0$ m for three different distances in the lee of the WEC (y = 100, 200 and 400 m).

${\mathit{K}}_{\mathit{d},\mathbf{diff}}$ | |||
---|---|---|---|

$T\left(s\right)$ | 8.0 | 10.0 | 12.0 |

1 HPA WEC | |||

x = 100 m | 1.96 | 1.02 | −0.10 |

x = 200 m | 1.79 | 0.88 | −0.09 |

x = 400 m | 1.52 | 0.77 | −0.05 |

1 OSWEC | |||

x = 100 m | 2.58 | 1.08 | 1.30 |

x = 200 m | 1.93 | 1.02 | 0.92 |

x = 400 m | 1.33 | 0.92 | 0.62 |

**Table 2.**${K}_{d,max,diff}$ (%) at $\mathrm{x}=0$ m for three different transversal cross-sections S1, S2 and S3 as indicated in Figure 6 for a 21 HPA WEC farm and a 21 OSWEC farm interacting with regular waves of $H=2.0$ m.

${\mathit{K}}_{\mathit{d},\mathit{max},\mathit{diff}}(\%)$ | |||||||||
---|---|---|---|---|---|---|---|---|---|

Wave Period T = 8 s | Wave Period T = 10 s | Wave Period T = 12 s | |||||||

21 HPA WEC farm | $\theta ={0}^{\xb0}$ | $\theta ={15}^{\xb0}$ | $\theta ={30}^{\xb0}$ | $\theta ={0}^{\xb0}$ | $\theta ={15}^{\xb0}$ | $\theta ={30}^{\xb0}$ | $\theta ={0}^{\xb0}$ | $\theta ={15}^{\xb0}$ | $\theta ={30}^{\xb0}$ |

S1 | 20.62 | 18.85 | 22.87 | 18.95 | 17.50 | 20.63 | 18.05 | 16.22 | 20.20 |

S2 | 9.85 | 9.74 | 11.15 | 9.05 | 8.79 | 10.33 | 8.73 | 8.75 | 9.74 |

S3 | 14.60 | 14.73 | 16.53 | 13.36 | 13.36 | 15.15 | 12.55 | 12.88 | 14.31 |

21 OSWEC farm | $\theta ={0}^{\xb0}$ | $\theta ={15}^{\xb0}$ | $\theta ={30}^{\xb0}$ | $\theta ={0}^{\xb0}$ | $\theta ={15}^{\xb0}$ | $\theta ={30}^{\xb0}$ | $\theta ={0}^{\xb0}$ | $\theta ={15}^{\xb0}$ | $\theta ={30}^{\xb0}$ |

S1 | 36.02 | 34.63 | 9.57 | 12.67 | 14.18 | 10.93 | 6.84 | 6.97 | 6.15 |

S2 | 15.87 | 16.03 | 9.36 | 9.52 | 13.65 | 8.83 | 6.11 | 7.32 | 8.16 |

S3 | 7.36 | 4.60 | 5.55 | 11.27 | 6.91 | 6.93 | 8.78 | 8.67 | 7.06 |

**Table 3.**Average power output for a farm of 21 HPA WECs for a linear optimal, a linear sub-optimal and a hydraulic PTO system.

Wave Direction $\mathit{\theta}$ (°) | Wave Height H (m) | Optimal PTO | Sub-Optimal PTO | Hydraulic PTO | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Wave Period T (s) | Wave Period T (s) | Wave Period T (s) | |||||||||

8 | 10 | 12 | 8 | 10 | 12 | 8 | 10 | 12 | |||

0 | FARM | 2.0 | 812.60 | 1151.6 | 1669.74 | 300.46 | 268.49 | 280.66 | 364.0 | 426.30 | 402.40 |

Single | 2.0 | 559.26 | 1068.35 | 1645.31 | 253.51 | 277.72 | 279.59 | 311.36 | 364.59 | 344.15 | |

q | 2.0 | 1.45 | 1.07 | 1.01 | 1.18 | 0.96 | 1.0 | 1.16 | 0.96 | 1.00 | |

15 | FARM | 2.0 | 738.94 | 1126.8 | 1679.82 | 282.80 | 266.67 | 281.52 | 343.77 | 402.54 | 379.98 |

Single | 2.0 | 559.26 | 1068.35 | 1645.31 | 254.35 | 277.72 | 279.59 | 311.36 | 364.59 | 344.15 | |

q | 2.0 | 1.32 | 1.05 | 1.02 | 1.11 | 0.96 | 1.00 | 1.10 | 0.96 | 1.0 | |

30 | FARM | 2.0 | 430.17 | 1090.1 | 1732.09 | 200.30 | 265.38 | 284.31 | 248.03 | 290.43 | 274.15 |

Single | 2.0 | 559.26 | 1068.3 | 1645.31 | 254.35 | 277.72 | 279.59 | 311.36 | 364.59 | 344.15 | |

q | 2.0 | 0.76 | 1.02 | 1.05 | 0.78 | 0.95 | 1.01 | 0.79 | 0.95 | 1.01 |

**Table 4.**Average power output for a farm of 21 OSWECs for a linear optimal, a linear sub-optimal and a hydraulic PTO systems.

Wave Direction $\mathit{\theta}$ (°) | Wave Height H (m) | Optimal PTO | Sub-Optimal PTO | Hydraulic PTO | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Wave Period T (s) | Wave Period T (s) | Wave Period T (s) | |||||||||

8 | 10 | 12 | 8 | 10 | 12 | 8 | 10 | 12 | |||

0 | FARM | 2.0 | 910.96 | 1083.01 | 1287.63 | 518.28 | 387.87 | 325.41 | 584.65 | 578.21 | 559.60 |

Single | 2.0 | 854.38 | 1222.89 | 1583.03 | 506.08 | 472.38 | 424.42 | 571.70 | 565.40 | 547.20 | |

q | 2.0 | 1.06 | 1.26 | 1.50 | 1.020 | 0.82 | 0.76 | 1.02 | 1.03 | 1.02 | |

15 | FARM | 2.0 | 900.11 | 988.85 | 1264.41 | 460.84 | 354.40 | 311.50 | 459.56 | 354.40 | 311.68 |

Single | 2.0 | 786.02 | 1131.49 | 1468.8 | 465.59 | 437.08 | 393.79 | 465.04 | 437.08 | 389.29 | |

q | 2.0 | 1.14 | 1.25 | 1.6 | 0.98 | 0.81 | 0.79 | 0.98 | 0.81 | 0.8 | |

30 | FARM | 2.0 | 502.45 | 862.12 | 1181.14 | 342.97 | 283.37 | 275.02 | 343.10 | 288.57 | 272.74 |

Single | 2.0 | 607.91 | 888.98 | 1162.84 | 360.09 | 343.40 | 311.76 | 359.66 | 344.7 | 308.20 | |

q | 2.0 | 0.82 | 1.41 | 1.94 | 0.95 | 0.82 | 0.88 | 0.95 | 0.83 | 0.88 |

**Table 5.**${K}_{d}$ disturbance coefficient for a farm of 21 HPA WECs for a linear optimal, a linear sub-optimal and a hydraulic PTO systems, at a distance of y = 1000 m (S1), y = 3000 m (S2) and y = 5000 m (S3) in the lee of the WEC farm, respectively.

Wave Direction $\mathit{\theta}$ (°) | Distance from the WEC Farm | Wave Height H (m) | Optimal PTO | Sub-Optimal PTO | Hydraulic PTO | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Wave Period T (s) | Wave Period T (s) | Wave Period T (s) | |||||||||

8 | 10 | 12 | 8 | 10 | 12 | 8 | 10 | 12 | |||

0 | y = 1000 m | 2.0 | 0.17 | 0.32 | 0.25 | 0.82 | 0.85 | 0.86 | 0.8 | 0.84 | 0.86 |

y = 2000 m | 2.0 | 0.51 | 0.51 | 0.37 | 0.81 | 0.89 | 0.91 | 0.82 | 0.88 | 0.91 | |

y = 5000 m | 2.0 | 0.84 | 0.5 | 0.47 | 0.94 | 0.92 | 0.93 | 0.95 | 0.91 | 0.93 | |

15 | y = 1000 m | 2.0 | 0.21 | 0.21 | 0.24 | 0.79 | 0.85 | 0.87 | 0.79 | 0.84 | 0.87 |

y = 3000 m | 2.0 | 0.53 | 0.45 | 0.37 | 0.81 | 0.89 | 0.91 | 0.82 | 0.88 | 0.91 | |

y = 5000 m | 2.0 | 0.86 | 0.53 | 0.47 | 0.95 | 0.92 | 0.93 | 0.96 | 0.91 | 0.93 | |

30 | y = 1000 m | 2.0 | 0.39 | 0.10 | 0.11 | 0.82 | 0.85 | 0.86 | 0.82 | 0.84 | 0.87 |

y = 3000 m | 2.0 | 0.63 | 0.42 | 0.34 | 0.86 | 0.90 | 0.91 | 0.86 | 0.89 | 0.92 | |

y = 5000 m | 2.0 | 0.94 | 0.53 | 0.45 | 0.99 | 0.92 | 0.93 | 0.99 | 0.92 | 0.94 |

**Table 6.**${K}_{d}$ disturbance coefficient for a farm of 21 OSWECs for a linear optimal, a linear sub-optimal and a hydraulic PTO systems, at a distance of y = 1000 m (S1), y = 3000 m (S2) and y = 5000 m (S3) in the lee of the WEC farm, respectively.

Wave Direction $\mathit{\theta}$ (°) | Distance from the WEC Farm | Wave Height H (m) | Optimal PTO | Sub-Optimal PTO | Hydraulic PTO | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Wave Period T (s) | Wave Period T (s) | Wave Period T (s) | |||||||||

8 | 10 | 12 | 8 | 10 | 12 | 8 | 10 | 12 | |||

0 | y = 1000 m | 2.0 | 0.22 | 0.20 | 0.24 | 0.68 | 0.78 | 0.85 | 0.69 | 0.78 | 0.85 |

y = 2000 m | 2.0 | 0.28 | 0.35 | 0.25 | 0.80 | 0.89 | 0.91 | 0.80 | 0.89 | 0.91 | |

y = 5000 m | 2.0 | 0.74 | 0.44 | 0.37 | 0.90 | 0.88 | 0.91 | 0.90 | 0.89 | 0.91 | |

15 | y = 1000 m | 2.0 | 0.42 | 0.48 | 0.11 | 0.65 | 0.86 | 0.86 | 0.66 | 0.81 | 0.86 |

y = 3000 m | 2.0 | 0.36 | 0.57 | 0.25 | 0.79 | 0.87 | 0.91 | 0.79 | 0.90 | 0.91 | |

y = 5000 m | 2.0 | 0.78 | 0.84 | 0.37 | 0.91 | 0.96 | 0.91 | 0.91 | 0.90 | 0.91 | |

30 | y = 1000 m | 2.0 | 0.39 | 0.16 | 0.05 | 0.74 | 0.79 | 0.90 | 0.74 | 0.79 | 0.89 |

y = 3000 m | 2.0 | 0.63 | 0.35 | 0.30 | 0.86 | 0.87 | 0.91 | 0.86 | 0.88 | 0.91 | |

y = 5000 m | 2.0 | 0.91 | 0.63 | 0.45 | 0.96 | 0.91 | 0.91 | 0.96 | 0.91 | 0.92 |

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**MDPI and ACS Style**

Verao Fernandez, G.; Stratigaki, V.; Quartier, N.; Troch, P.
Influence of Power Take-Off Modelling on the Far-Field Effects of Wave Energy Converter Farms. *Water* **2021**, *13*, 429.
https://doi.org/10.3390/w13040429

**AMA Style**

Verao Fernandez G, Stratigaki V, Quartier N, Troch P.
Influence of Power Take-Off Modelling on the Far-Field Effects of Wave Energy Converter Farms. *Water*. 2021; 13(4):429.
https://doi.org/10.3390/w13040429

**Chicago/Turabian Style**

Verao Fernandez, Gael, Vasiliki Stratigaki, Nicolas Quartier, and Peter Troch.
2021. "Influence of Power Take-Off Modelling on the Far-Field Effects of Wave Energy Converter Farms" *Water* 13, no. 4: 429.
https://doi.org/10.3390/w13040429