# A Comparison of Numerical Approaches for the Design of Mooring Systems for Wave Energy Converters

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Models

#### 2.1. Quasistatic Mooring Frequency Domain

- $\left[M\right]$: Mass matrix of the floating structure
- $\left[A\left(\omega \right)\right]$: Added mass matrix
- $\left[B\left(\omega \right)\right]$: Radiation damping matrix
- $\left[{B}_{d}\right]$: Linearized drag force
- $\left[H\right]$: Hydrostatic matrix
- $\left[{K}_{m}\right]$: Linearized mooring stiffness

- ${S}_{\mathrm{SV}}\left(\mu \right)$: Slowly-varying wave drift force spectrum
- $T\left(\omega ,\omega \right)$: Drift force quadratic transfer function

#### 2.2. Quasistatic Mooring Time Domain

- ${T}_{\mu}$: Mean line tension
- ${T}_{\sigma}$: Standard deviation of line tension maxima of the set of simulations

#### 2.3. Dynamic Mooring Time Domain

#### 2.4. Reference Simulation Cases

#### 2.5. Environmental Conditions

#### 2.6. Mooring Properties

#### 2.7. Numerical Model of the Floating Wave Energy Converter

^{2}]. Its total mass is 2.434 × 10

^{6}[kg].

## 3. Results

#### 3.1. Quasistatic Frequency Domain Model Results

#### 3.2. Performance Results of Non-Linear QSTD and DynTD Models

#### Floater Dynamics

## 4. Predicted Line Tensions

#### Performance and Cost Comparison Results of Numerical Models

^{2}] and the total mooring mass, computed with the maximum suspended length resulting with each approach and applied to all lines of the mooring system, and the cost has been provided in terms of the total equivalent required mass. The equivalent required mass is the summation of the mooring mass, assuming lines made up of a single chain type, and the equivalent mass of the footprint area given a cost ratio. Assuming a cost ratio of $\frac{\left[\frac{\u20ac}{\mathrm{kg}}\right]}{\left[\frac{\u20ac}{{\mathrm{m}}^{2}}\right]}=\frac{3}{1}$, total computed costs represented in Figure 18 with both models result in increasing costs with increasing pretensions independently of the linear mass, and higher optimum linear mass with decreasing pretensions. Figure 18 derives that the QSFD approach seems applicable for early stage optimizations with mid to low linear pretensions, i.e., <1.2. Obtained cost values differ by a 10% and it can acceptably be utilized only for preliminary designs optimizations, setting tendencies of offset, footprint, and total required mass, along with design line tensions, as long as corrections in Figure 17 are accounted for.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Sensitivity analysis of the QSTD model to the iterative process of the catenary equations. Surge of the structure and the corresponding mooring horizontal force (

**a**) and relative errors of the standard deviation with different relative errors allowed in the catenary equations of both line tension and surge motion (

**b**).

**Figure 4.**Mesh representation of the boundary element method (BEM) model for the FWEC spar platform submerged part (

**a**) and its main dimensions [m] with the reference at its CoG (

**b**). Reproduced from [11] (Renewable Energy 44 (2012) 328–339), corresponding to model ‘K’.

**Figure 5.**Baseline results of the QSFD model. Performance indicators of line design tension and design offset (

**a**) and cost indicators of mooring total mass and anchor radius (

**b**).

**Figure 6.**Surge factors with respect to QSFD model of the QSTD (

**a**) and DynTD (

**b**) approaches. Accumulated bars with the weights of the mean and dynamic offsets.

**Figure 7.**Heaving (

**a**) and pitching (

**b**) std factors of the DynTD models with respect to the QSFD model.

**Figure 8.**Power spectral densities in surge (WF magnified 20 times) (

**a**), heave and pitch motions (WF reduced by a factor of 2) (

**b**) and (

**c**) comparing models’ performance with the largest and lowest non-dimensional pretension.

**Figure 9.**Normalized PDFs with the three models in surge (

**a**), heave (

**b**), pitch (

**c**), and the most loaded line tension (

**d**).

**Figure 10.**Kurtosis and skewness of surge motion (

**a**,

**b**), heave motion (

**c**,

**d**) and most loaded line tension (

**e**,

**f**) obtained with the DynTD approach.

**Figure 13.**Normalized PDFs of the most loaded line with the three approaches. High pretension (

**a**) and low pretension (

**b**).

**Figure 14.**Time series extract of buoy heaving and the corresponding tension of the most loaded line.

**Figure 15.**Line tension with large pretension (

**a**) and with low pretension (

**b**) for three models. Green: QSFD, Blue: QSTD and Orange: DynTD.

**Figure 16.**Design (

**a**) and cost (

**b**) spaces for the FWEC structure with the QSFD (red squares), QSTD (green triangles), and DynTD (blue circles) models.

**Figure 17.**Correction factors between the DynTd and the QSFD models for five linear mass values (65, 85, 105, 125, and 145) and a linear fitting of the mean values for Anchor Radius (

**a**), Suspended length (

**b**) and Design Tension (

**c**).

**Figure 18.**Cost optimization of the mooring system. Total equivalent mass is the sum of 1/3 the lease area and the total mooring mass, assuming a cost ratio of 3 [€/kg]/[€/m

^{2}].

Parameter | Return Period | Value |
---|---|---|

Significant Wave Height | 100 yrs | 10 m |

Peak Period (Tp) | 18 s | |

Current Velocity (Vc) | 50 yrs | 1.3 m/s |

**Table 2.**Mooring properties selected to be combined for simulation cases, resulting in 25 cases arisen from combining each linear mass with each non-dimensional pretension.

Mooring Linear Mass [kg/m] | Non-Dimensional Pretension [-] | Length/Mass Number |
---|---|---|

65 | 1.43 | 1 |

85 | 1.26 | 2 |

105 | 1.16 | 3 |

125 | 1.10 | 4 |

145 | 1.06 | 5 |

Degree of Freedom | Viscous Force Factor |
---|---|

Surge [N·s/m] | 1.188 × 10^{5} |

Sway [N·s/m] | 1.188 × 10^{5} |

Heave [N·s/m] | 4.469 × 10^{4} |

Roll [N·m·s] | 3.532 × 10^{9} |

Pitch [N·m·s] | 3.532 × 10^{9} |

Yaw [N·m·s] | 0 |

**Table 4.**Mean obtained kurtosis and skewness of the FWEC motions and tension of the most loaded line, obtained with the QSTD approach.

FWEC | Surge | Heave | Pitch | Mll Tension |
---|---|---|---|---|

Kurtosis (QSTD) | 2.924 | 2.671 | 2.241 | 3.700 |

Skewness (QSTD) | −0.188 | −0.039 | −0.050 | 0.665 |

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**MDPI and ACS Style**

Touzon, I.; Nava, V.; de Miguel, B.; Petuya, V. A Comparison of Numerical Approaches for the Design of Mooring Systems for Wave Energy Converters. *J. Mar. Sci. Eng.* **2020**, *8*, 523.
https://doi.org/10.3390/jmse8070523

**AMA Style**

Touzon I, Nava V, de Miguel B, Petuya V. A Comparison of Numerical Approaches for the Design of Mooring Systems for Wave Energy Converters. *Journal of Marine Science and Engineering*. 2020; 8(7):523.
https://doi.org/10.3390/jmse8070523

**Chicago/Turabian Style**

Touzon, Imanol, Vincenzo Nava, Borja de Miguel, and Victor Petuya. 2020. "A Comparison of Numerical Approaches for the Design of Mooring Systems for Wave Energy Converters" *Journal of Marine Science and Engineering* 8, no. 7: 523.
https://doi.org/10.3390/jmse8070523