# Analysing the Near-Field Effects and the Power Production of Near-Shore WEC Array Using a New Wave-to-Wire Model

^{*}

## Abstract

**:**

^{®}OSWEC developed by AW Energy Ltd. The investigation in this study provides a proof of concept of the proposed W2W model in investigating potential commercial WEC projects.

## 1. Introduction

- coupling between the BEM solver NEMOH and the mild-slope wave propagation model MILDwave,
- development of an iterative technique to model WEC Farms composed of clustered WEC arrays,
- development of a realistic time-domain Power Take-off (PTO) module.

^{®}technology [6]. WaveRoller is an OSWEC (Oscillating Surging Wave Energy Converter) that has been successfully deployed over various generations and independently certified by the ratings agency Lloyd’s register [7]. The WEC farm investigated in this study is to be located in the Baie d’Audierne near Pouldreuzic in Bretagne, France at a latitude of 47.93${}^{\xb0}$ N and a longitude of 4${}^{\xb0}$44${}^{\prime}$ W. The project location is shown in Figure 1 on a map of the western part of the Finistère peninsula of Bretagne.

- a wave climate representative of that observed at the installation site,
- a realistic sloping bathymetry,
- a WEC with approximate dimensions to the WEC technology that is to be deployed,
- a hydraulic PTO system simulating that of the proposed WEC,
- a WEC farm layout that seeks to maximize power absorption over a limited coastal length.

^{®}technology. The PTO system is reproduced in WEC-Sim [12] as a simplified, yet accurate, hydraulic time-domain simulation which has been introduced in [1] and detailed in [11]. The WEC farm absolute value of the total free surface elevation $\left|\eta \right|$ is determined via an iterative the method first developed in [1] and detailed in Section 2.8.

#### 1.1. Study Location and Geographical Context

#### 1.2. Site Bathymetry and Approximation

#### 1.3. Analysis of the Wave Climate at the Investigation Site

#### 1.4. WEC Farm and Clustered WEC Array Layout

## 2. Wave-to-Wire Model Methodology

#### 2.1. Modelled Scenarios

#### 2.2. NEMOH BEM Model Parameters

#### 2.3. MILDwave Wave Propagation Model Parameters

#### 2.4. Coupling of NEMOH to MILDwave

#### 2.5. Simulating Irregular Sea States

#### 2.6. Modelled OSWECs

#### 2.7. Hydraulic PTO System and Derivation of the Optimal Coefficients for Irregular Waves

^{®}environment and then simulated in WEC-Sim, an open source purpose-built WEC dynamics simulator developed jointly by Sandia Laboratories and the National Renewable Energy Laboratory in the USA [12,34]. A schematic of the hydraulic system parameters is presented in Figure 8.

#### 2.8. Calculating the Total Wave Field in the WEC Farm

## 3. Calculating the Power Output of a WEC Farm Composed of Multiple WEC Arrays

- the wave field inside each WEC array is computed in NEMOH using Equation (2),
- the power of each WEC in the array is calculated in WEC-Sim using the amplitudes output by NEMOH and summed for the $\mathcal{M}$ WECs,
- the average perturbed 1st order wave field of the W2W model is computed at the WEC array perimeter,
- the power of the WEC array is multiplied by the wave field computed in the previous step,
- the power of the WEC farm is then the sum of the power of all constituent WEC arrays.

## 4. Results for a 2-Array 10 OSWEC Farm

#### 4.1. The 10-OSWEC Farm Wave Field for a Regular Wave at $\beta $ = 0${}^{\xb0}$ Incidence

#### 4.2. The 10-OSWEC Farm Wave Field for a Regular Wave at $\beta $ = 24${}^{\xb0}$ Incidence

#### 4.3. The 10-OSWEC Farm Wave Field for an Irregular Wave at $\beta $ = 0${}^{\xb0}$ Incidence

#### 4.4. The 10-OSWEC Farm $\eta $ for an Irregular Wave at $\beta $ = 24${}^{\xb0}$ Incidence

## 5. Results for a 10 Array 50 OSWEC Farm

#### 5.1. The 50-OSWEC Farm Wave Field for the Site Winter Climate

#### 5.2. The 50-OSWEC Farm Wave Field for the Site Summer Climate

#### 5.3. The 50-OSWEC Farm Wave Field for the Autumn Wave Climate

#### 5.4. The Power Output of a 10 Array 50 OSWEC Farm for the Seasonal Wave Climate

#### 5.4.1. Absolute Power Output of the 50-WEC Farm

#### 5.4.2. Relative Power Output of the 50-WEC Farm

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BEM | Boundary Element Method |

CANDHIS | Centre d’Archivage National des Données de Houle In Situ |

DoF | Degree of Freedom |

OSWEC | Oscillating Surge Wave Energy Converter |

PTO | Power Take-Off |

RAO | Response Amplitude Operator |

WEC | Wave Energy Converter |

WSI | Wave-Strucure Interaction |

W2W | Wave-to-wire |

$A(\omega )$ | added moment of inertia (kg·m${}^{2}$) |

$\beta $ | angle of incidence of the incoming wave to the x-axis (${}^{\xb0}$) |

${d}_{x}$, ${d}_{y}$ | WEC–WEC separation distances in the x and y direction (m) |

$B(\omega )$ | hydrodynamic damping (kg/s${}^{2}$) |

${B}_{PTO,l}$ | power-take-off linear damping coefficient (kg/s${}^{2}$) |

${B}_{PTO,h}$ | power-take-off hydraulic damping equivalent coefficient (kg/s${}^{2}$) |

${D}_{m}$ | variable motor displacement (rev/s) |

${K}_{PTO}$ | power take-off linear stiffness coefficient ($\frac{N}{m}$) |

$\mathcal{M}$ | number of bodies in the WEC array |

$\mathcal{N}$ | number of WEC arrays in a WEC farm |

$\left|\eta \right|$ | absolute value of the complex free surface elevation $\eta $ (m) |

${f}_{PTO,h}$ | PTO system-force for hydraulic PTO system |

${p}_{ij}$ | perturbed wave of order j for array i (-) |

${P}_{l}$ | mechanical power produced by the WEC with a linear PTO system |

${P}_{h}$ | mechanical power produced by the WEC with a hydraulic PTO system |

${P}_{isolated}$ | total power output of a WEC farm as if it were composed of isolated WEC arrays (kW) |

${P}_{array}$ | total power output of a WEC array including the intra-array effects (kW) |

${P}_{farm}$ | total power output of an WEC farm including array effects (kW) |

q | q-value, defined as ratio of power of the $\mathcal{M}$-WEC array to the power produced by the sum of $\mathcal{M}$ isolated WECs |

${s}_{c}$ | piston area (m${}^{2}$) |

${T}_{r}$ | resonance or natural period of an oscillating body (s) |

${\mathcal{T}}_{PTO,l}$ | PTO-torque for linear PTO system |

${\mathcal{T}}_{PTO,h}$ | PTO-torque for hydraulic PTO system |

${Z}_{i}$ | complex amplitude of heave displacement |

$z\left(t\right)$ | heave displacement in time domain (m) |

$\lambda $ | wavelength (m) |

$\mathsf{\Theta}$ | complex amplitude of pitch angular displacement |

$\theta \left(t\right)$ | pitch angular displacement in time domain (rad) |

$\zeta $ | wave amplitude (m) |

$\omega $ | wave angular frequency (rad/s) |

‘array effects’ = the hydrodynamic effects of WECs in an array that produce | |

a perturbation in the incident wave field | |

‘intra-array’ referring to effects between WECs inside an array | |

‘inter-array’ referring to effects between disparate WEC arrays inside a WEC farm | |

‘near-field’ referring to wave field modification effects in the general location of the WECs inside an array | |

‘far-field’ referring to wave field modification effects outside the immediate area of the WEC array(s) | |

‘perturbed wave’ = radiated + diffracted wave |

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**Figure 1.**Map locating the investigation domain at the proposed WATTMOR project site in the Baie d’Audierne and the CANDHIS buoy 05605 (Belle-île) which serves as the source of the wave data off the coast of Bretagne, France.

**Figure 2.**Detailed map showing the project area (red box) and the W2W model domain investigated in this study (yellow box). Point water depths are shown in meters [6].

**Figure 3.**Map showing the detailed bathymetric survey of the project area (red box) with the present investigation domain (yellow box) overlaid [6].

**Figure 6.**Schematic showing the coupling methodology domain with the MILDwave Empty Basin on the left and the coupled NEMOH-MILDwave perturbed wave run. SL indicates the sponge layer as in Section 2.3 and ${\lambda}_{max}$ is the simulated wavelength. (

**A**) shows the Empty Basin set-up; (

**B**) the coupled model run set-up.

**Figure 7.**Pitching OSWEC (right) schematic. The wavy line indicates the undisturbed free surface elevation z = 0 [11].

**Figure 8.**Hydraulic PTO system working principle of a generic OSWEC [11].

**Figure 9.**Iterative procedure for determining the perturbed wave field for a regular wave input. Incident wave $\lambda $ is coming from the left [1].

**Figure 11.**Coupled $\left|\eta \right|$ for H = 2.0 T = 10.0 s and $\beta $ = 0${}^{\xb0}$ regular wave for a 10 WEC 2-Array farm over a sloping bathymetry shown in Figure 10.

**Figure 12.**Coupled absolute total wave amplitude $\left|\eta \right|$ for H = 2.0 m, T = 10.0 s and $\beta $ = −24${}^{\xb0}$ for a 10 WEC 2-array farm over the sloping bathymetry shown in Figure 10.

**Figure 13.**Coupled total wave amplitude $\left|\eta \right|$ for ${H}_{s}$ = 2.0 m, ${T}_{p}$ = 10.0 s and $\beta $ = 0${}^{\xb0}$ regular wave for a 10 WEC 2-Array farm over a sloping bathymetry shown in Figure 10.

**Figure 14.**Coupled $\left|\eta \right|$ for H = 2.0 T = 10.0 s and $\beta $ = −24${}^{\xb0}$ regular wave for a 10 WEC 2-Array farm over a sloping bathymetry shown in Figure 10.

**Figure 15.**Coupled total $\left|\eta \right|$ for the mean winter wave ${H}_{m0}$ = 2.55 m, ${T}_{p}$ = 11.71 s, and $\beta $ = −20${}^{\xb0}$ for a 50 WEC 10-Array farm.

**Figure 16.**Coupled total $\left|\eta \right|$ for the mean summer wave ${H}_{m0}$ = 1.20 m, ${T}_{p}$ = 8.71 s, and $\beta $ = −30${}^{\xb0}$ for a 50 WEC 10-Array farm.

**Figure 17.**Coupled total $\left|\eta \right|$ for the mean autumn wave ${H}_{m0}$ = 1.80 m, ${T}_{p}$ = 10.54 s, and $\beta $ = −22${}^{\xb0}$ for a 50 WEC 10-Array farm.

**Table 1.**Summary statistics of the wave climate at the Belle île measurement buoy located at 47${}^{\xb0}$17${}^{\prime}$ N and 3${}^{\xb0}$17${}^{\prime}$ W for a 9-year period Oct. 2010–Mar. 2019.

Winter | Spring | Summer | Autumn | Year | |
---|---|---|---|---|---|

${H}_{s}$ (m) | 2.55 | 1.75 | 1.20 | 1.80 | 1.87 |

${T}_{p}$ (s) | 11.71 | 10.45 | 8.71 | 10.54 | 10.34 |

${\theta}_{m}\phantom{\rule{3.33333pt}{0ex}}{(}^{\xb0})$ | 261.74 | 263.32 | 270.12 | 263.16 | 264.48 |

Wave Height H (m) | Wave Period T (s) | Wave Incidence Angle $\mathit{\beta}\phantom{\rule{3.33333pt}{0ex}}{\mathbf{(}}^{\xb0}\mathbf{)}$ |
---|---|---|

2.0 | 10.0 | 0 |

2.0 | 10.0 | 20 |

Simulated Case | Winter | Summer | Autumn |
---|---|---|---|

${H}_{m0}$ (m) | 2.55 | 1.20 | 1.80 |

${T}_{p}$ (s) | 11.71 | 8.71 | 10.54 |

$\beta $ (${}^{\xb0}$) | −20.0 | −30.0 | −22.0 |

**Table 4.**Optimal hydraulic damping coefficients ${B}_{PTO,h,irr}$ for a single OSWEC (${10}^{6}\times $ m${}^{2}$·kg/s).

wave peak period | ${T}_{P}$ (s) | 8.71 | 10.54 | 11.71 |

hydraulic PTO damping coefficient | ${B}_{PTO,h,irr}$ | 198.7 | 145.6 | 121 |

**Table 5.**Power output in kW for the 50 OSWEC farm for an irregular wave of ${H}_{m0}$ = 2.55 m and ${T}_{p}$ = 11.71 s. Wave incidence angle $\beta $ = −20${}^{\xb0}$. Wave climate is based on the 9-year site winter mean.

Array i Array vi | Array ii Array vii | Array iii Array viii | Array iv Array ix | Array v Array x | Total Power ${\mathbf{P}}_{\mathit{farm}}$ per Row [kW] | Total Power ${\mathbf{P}}_{\mathit{farm}}$ [kW] |
---|---|---|---|---|---|---|

743.13 | 774.77 | 772.83 | 744.26 | 733.79 | 3768.78 | |

771.63 | 620.77 | 546.93 | 585.96 | 567.38 | 3092.67 | 6861.45 |

**Table 6.**Power output in kW for the 50 OSWEC farm for an irregular wave of ${H}_{m0}$ = 1.7 m and ${T}_{p}$ = 8.71 s. Wave incidence angle $\beta $ = −30${}^{\xb0}$. Wave climate is based on the 9-year site summer mean.

Array i Array vi | Array ii Array vii | Array iii Array viii | Array iv Array ix | Array v Array x | Total Power ${\mathbf{P}}_{\mathit{farm}}$ per Row [kW] | Total Power ${\mathbf{P}}_{\mathit{farm}}$ [kW] |
---|---|---|---|---|---|---|

607.14 | 629.09 | 615.59 | 621.04 | 603.41 | 3076.27 | |

595.74 | 595.26 | 489.20 | 490.51 | 484.22 | 2654.93 | 5731.20 |

**Table 7.**Power output in kW for the 50 WEC farm for an irregular wave of ${H}_{m0}$ = 1.80 m and ${T}_{p}$ = 10.54 s. Wave incidence angle $\beta $ = −22${}^{\xb0}$. Wave climate is based on the 9-year site autumn mean.

Array i Array vi | Array ii Array vii | Array iii Array viii | Array iv Array ix | Array v Array x | Total Power ${\mathbf{P}}_{\mathit{farm}}$ per Row [kW] | Total Power ${\mathbf{P}}_{\mathit{farm}}$ [kW] |
---|---|---|---|---|---|---|

686.76 | 699.57 | 705.83 | 708.32 | 698.83 | 3499.30 | |

699.34 | 592.99 | 498.44 | 561.49 | 546.08 | 2898.34 | 6397.64 |

**Table 8.**WEC farm q-values for the 50 WEC farm for an irregular wave of ${H}_{m0}$ = 2.55 m and ${T}_{p}$ = 11.71 s. Wave incidence angle $\beta $ = −20${}^{\xb0}$. Wave climate is based on the 9-year mean site winter mean.

Array i Array vi | Array ii Array vii | Array iii Array viii | Array iv Array ix | Array v Array x | Average q-Value per Row | Average q-Value Farm |
---|---|---|---|---|---|---|

0.93 | 0.97 | 0.97 | 0.93 | 0.92 | 0.95 | |

0.97 | 0.78 | 0.69 | 0.74 | 0.72 | 0.78 | 0.86 |

**Table 9.**WEC farm q-values for the 50 WEC farm for an irregular wave of ${H}_{m0}$ = 1.7 m and ${T}_{p}$ = 8.71 s. Wave incidence angle $\beta $ = −30${}^{\xb0}$. Wave climate is based on the 9-year site summer mean.

Array i Array vi | Array ii Array vii | Array iii Array viii | Array iv Array ix | Array v Array x | Average q-Value per Row | Average q-Value Farm |
---|---|---|---|---|---|---|

0.99 | 1.03 | 1.00 | 1.01 | 0.98 | 1.00 | |

0.97 | 0.97 | 0.80 | 0.80 | 0.79 | 0.87 | 0.93 |

**Table 10.**WEC farm q-values for the 50 WEC farm for an irregular wave of ${H}_{m0}$ = 1.80 m and ${T}_{p}$ = 10.54 s. Wave incidence angle $\beta $ = −22${}^{\xb0}$. Wave climate is based on the 9-year site autumn mean.

Array i Array vi | Array ii Array vii | Array iii Array viii | Array iv Array ix | Array v Array x | Average q-Value per Row | Average q-Value Farm |
---|---|---|---|---|---|---|

0.91 | 0.93 | 0.94 | 0.94 | 0.93 | 0.93 | |

0.93 | 0.79 | 0.66 | 0.75 | 0.73 | 0.77 | 0.85 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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*, 1137.
https://doi.org/10.3390/w11061137

**AMA Style**

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(6):1137.
https://doi.org/10.3390/w11061137

**Chicago/Turabian Style**

Balitsky, Philip, Nicolas Quartier, Vasiliki Stratigaki, Gael Verao Fernandez, Panagiotis Vasarmidis, and Peter Troch. 2019. "Analysing the Near-Field Effects and the Power Production of Near-Shore WEC Array Using a New Wave-to-Wire Model" *Water* 11, no. 6: 1137.
https://doi.org/10.3390/w11061137