# The Performance of a Spectral Wave Model at Predicting Wave Farm Impacts

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## Abstract

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## 1. Introduction

- Medium-sized arrays of 27 WECs are modeled, representing a medium-sized wave farm. Smaller arrays of 5 WECs were compared in McNatt et al. [20], and, while larger arrays could be considered in the future, an array of 27 WECs represents a medium-sized wave farm and is a stepping stone in both the development of commercial wave farms and the verification of SNL-SWAN. To model these arrays, an approach referred to as Linear Wave Interaction Theory (LWIT) was used.
- Only directionally spread sea conditions are modeled, based on literature on the range of realistic directional spreading [21]. The range of directional spreading includes highly focused storm waves, however it does not include idealized purely unidirectional seas. In previous studies, SNL-SWAN did not show good agreement with linear wave theory for unidirectional seas because without the SWAN diffraction option on, it is not capably of wave diffraction [19]. However, directional spreading is a form of wave diffraction and unidirectional seas are not representative of real sea conditions.
- A realistic beach bathymetry is used. In previous studies, the authors considered only a flat sea bottom. However, the aim of SNL-SWAN is to assess the impact of wave farms on nearshore processes. As such, the wave fields produced by each method were fed in as boundary conditions for a SWAN model that propagates the wave field to shore.

## 2. Methods

#### 2.1. Computational Methods

#### 2.1.1. SNL-SWAN

#### 2.1.2. Linear Wave Interaction Theory

#### 2.2. WEC Arrays

#### 2.3. Environment

#### 2.4. Computational Domains

- Array Domain: the domain around the WEC array located offshore. This domain is evaluated either with SNL-SWAN or with LWIT. The bathymetry is a flat bottom.
- Nearshore Domain: the domain with a beach-profile bottom extending to a depth of 0 at the shoreline. It is modeled always with SWAN.

#### 2.5. Means of Comparison

#### 2.5.1. Normalized ${H}_{s}$

#### 2.5.2. Normalized Error

#### Mean-Squared Error

#### 2.5.3. Mean-Squared Skill Score

- $MSSS=0$ would be produced by not modeling the WEC array at all.
- $MSSS<0$: SNL-SWAN produces worse results compared to not modeling the WEC array at all.
- $MSSS=1$: SNL-SWAN exactly matches LWIT results (no error).
- $0<MSSS<1$: SNL-SWAN makes some improvement to the wave field (compared to not modeling the array at all).

## 3. Results

#### 3.1. Example Results

#### 3.2. General Considerations

#### 3.2.1. Wave Breaking

#### 3.2.2. Boundary Condition Effects

#### 3.2.3. Garden Sprinkler Effect

#### 3.3. Mean-Squared Skill Score Results

#### 3.4. Visualization of Results

- How significant are the impacts of the array?
- Are the errors between SNL-SWAN and LWIT meaningful in the context of nearshore physical processes?

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BC | Boundary condition |

BEM | Boundary-element method |

${H}_{s}$ | Significant wave height |

LWIT | Linear wave interaction theory |

MSE | Mean-squared error |

MSSS | Mean-squared skill score |

RCW | Relative capture width |

SNL | Sandia National Laboratories |

SWAN | Simulating WAves Nearshore |

${T}_{p}$ | Peak wave period |

WEC | Wave energy converter |

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**Figure 1.**Result of a verification/comparison of linear wave interaction theory (LWIT) computation to a direct computation using WAMIT from McNatt [28].

**Figure 4.**WEC array configurations were considered in this study: (

**a**) two rows of WECs and (

**b**) a cluster of WECs.

**Figure 5.**The bathymetry modeled in the study consisted of a flat bottom in the region of the WEC array (WEC array centered at $x=0$), and an idealized beach profile [29] in the nearshore.

**Figure 8.**Wave fields, ${\widehat{H}}_{s}$, and error $\u03f5$ for a particular array configuration at a particular directional spreading for the three wave periods under consideration.

**Figure 9.**Longshore transects of ${\widehat{H}}_{s}$ at various cross-shore locations for the conditions given in Figure 8.

**Figure 10.**Mean-Squared Skill Score for longshore transects as a function of cross-shore location for the conditions given in Figure 8.

**Figure 11.**Mean-Squared Error ($MSE$) and the mean-squared error of Sandia National Laboratories-Simulating WAves Nearshore (SNL-SWAN) against the no-WEC condition ($MS{E}_{0}$) for ${T}_{p}=$ 6s and the conditions given in Figure 8.

**Table 1.**Mean Squared Skill Score (MSSS) model performance as given by Sutherland et al. [32].

MSSS | Performance |
---|---|

0.5–1 | Excellent |

0.2–0.5 | Good |

0.1–0.2 | Fair |

0–0.1 | Poor |

<0 | Bad |

**Table 2.**Number of cases scored as “Excellent” and “Good” in terms of the MSSS from Sutherland et al. [32] for each wave period.

T_{p} | Excellent | Good |
---|---|---|

($0.5<\mathbf{MSSS}\le 1$) | ($0.2<\mathbf{MSSS}\le 0.5$) | |

6s | 5 | 7 |

8s | 12 | 0 |

12s | 12 | 0 |

**Table 3.**Difference in wave height, $\Delta {H}_{s}={H}_{s,SNL}-{H}_{s,LWIT}$, over the nearshore domain.

${\mathit{T}}_{\mathit{p}}$ | $\mathbf{max}\left(\Delta {\mathit{H}}_{\mathit{s}}\right)$ [m] | $\mathbf{mean}\left(\Delta {\mathit{H}}_{\mathit{s}}\right)$ [m] |
---|---|---|

6s | 0.089 | 0.041 |

8s | 0.098 | 0.042 |

12s | 0.207 | 0.097 |

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**MDPI and ACS Style**

McNatt, J.C.; Porter, A.; Chartrand, C.; Roberts, J. The Performance of a Spectral Wave Model at Predicting Wave Farm Impacts. *Energies* **2020**, *13*, 5728.
https://doi.org/10.3390/en13215728

**AMA Style**

McNatt JC, Porter A, Chartrand C, Roberts J. The Performance of a Spectral Wave Model at Predicting Wave Farm Impacts. *Energies*. 2020; 13(21):5728.
https://doi.org/10.3390/en13215728

**Chicago/Turabian Style**

McNatt, J. Cameron, Aaron Porter, Christopher Chartrand, and Jesse Roberts. 2020. "The Performance of a Spectral Wave Model at Predicting Wave Farm Impacts" *Energies* 13, no. 21: 5728.
https://doi.org/10.3390/en13215728