# Development of a Multi-Objective Optimization Tool for Screening Designs of Taut Synthetic Mooring Systems to Minimize Mooring Component Cost and Footprint

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## Abstract

**:**

## 1. Introduction

## 2. Implementation of the MOGA

- (1)
- A geometric feasibility constraint is implemented to avoid analyzing designs where the line lengths for a certain mooring radius yield nonsensical designs (i.e., ${g}_{1}\left(\mathit{x}\right)$).
- (2)
- Next, the FOWT platform periods are estimated so that designs which do meet minimum natural period requirements and would likely have resonance issues due to the wave loading are not analyzed (i.e., ${g}_{2}\left(\mathit{x}\right)\mathrm{and}{g}_{3}\left(\mathit{x}\right)$).
- (3)
- Designs which pass the aforementioned constraints are subjected to DLC 6.1 simulations to determine the maximum and minimum loads in the mooring system and assess constraints requiring these values (i.e., ${g}_{4}\left(\mathit{x}\right)-{g}_{6}\left(\mathit{x}\right)$).

## 3. Optimization Inputs

#### 3.1. Mooring System

#### 3.2. Description of the Turbine

#### 3.3. Design Criteria

#### 3.4. Environmental Loading

## 4. Accelerating the Simulation Process for Obtaining Design Constraint Values

#### 4.1. Extrapolating the Maximum DLC 6.1 Line Response Based on a Shorter Simulation Time

#### 4.2. Selection of OpenFAST and MoorDyn Timesteps

## 5. MOGA Mooring Optimization Results

#### Verification of Candidate Design with Full Suite of DLC 6.1 Simulations

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ABS | American Bureau of Shipping |

CDF | Cumulative distribution function |

CM | Center of mass |

DLC | Design load case |

FOS | Factor of safety |

FOWT | Floating offshore wind turbine |

GEV | Generalized extreme value fit |

IEC | International Electrotechnical Commission |

MAP++ | Mooring Analysis Program |

MBS | Minimum breaking strength |

MW | Megawatt |

QTF | Quadratic transfer function |

SLC | Survival Load Case |

SWL/MWL | Still water line/Mean water line |

UV | Ultraviolet |

${a}_{33}$ | infinite period added mass of the platform in heave |

${a}_{55}$ | infinite period added inertia of the platform in pitch |

dtM | MoorDyn timestep |

dtF | OpenFAST timestep |

${d}_{chain}$ | chain diameter |

${d}_{syn}$ | synthetic diameter |

D_{f} | depth from the mean water line (MWL) to the fairlead connection point |

${D}_{w}$ | water depth |

${F}_{f\_\mathrm{chain}}$ | chain fatigue factor |

${F}_{s\_\mathrm{syn}}$ | synthetic factor of safety for a synthetic |

${F}_{\mathrm{min}\_\mathrm{syn}}$ | minimum allowable line tension to avoid slack lines as a percentage of MBS |

H_{s} | Significant wave height |

${I}_{\mathrm{platform}}$ | platform pitch inertia |

${k}_{33}$ | platform heave stiffness |

${k}_{33\mathrm{Mooring}}$ | mooring system heave stiffness |

${k}_{55}$ | platform pitch stiffness |

${k}_{55\mathrm{Mooring}}$ | mooring system pitch stiffness |

LM | lumped masses the mooring line is discretized into using MoorDyn |

${L}_{syn}$ | length of synthetic segment (expressed as a fraction of mooring radius) |

L_{r} | total line length |

${m}_{\mathrm{platform}}$ | mass of the platform |

${\mathrm{MBS}}_{\mathrm{chain}}$ | minimum breaking strength of the chain |

${\mathrm{MBS}}_{\mathrm{syn}}$ | minimum breaking strength of the synthetic |

$R$ | mooring radius |

${R}_{f}$ | distance from the center of the platform to the fairlead connection point |

${\mathrm{T}}_{\mathrm{fairlead}\_\mathrm{max}}$ | maximum tension at the fairlead |

${T}_{n\_\mathrm{heave}}$ | platform heave natural period |

${T}_{n\_\mathrm{pitch}}$ | platform pitch natural period |

${\mathrm{T}}_{\mathrm{syn}\_\mathrm{max}}$ | maximum tension at the fairlead |

${\mathrm{T}}_{\mathrm{syn}\_\mathrm{min}}$ | minimum tension at the fairlead |

T_{p} | Peak period |

$\mathsf{\gamma}$ | Peak shape factor |

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**Figure 6.**Dry Chain Mass and Chain Load Capacity (

**left**) and Dry Synthetic Mass and Synthetic Load Capacity (

**right**).

**Figure 14.**Designs along the Pareto-Optimal Front (black) with Interpolated Designs with Constraint Violations (red).

Number of mooring lines | 3 |

Angle of mooring lines (0° aligned with positive surge axis; positive defined as counter-clockwise) | 60°, 180°, 300° |

Depth to anchors below SWL (water depth) | 55 m |

Depth to fairleads below SWL | 5.4 m |

Radius to anchors from platform centerline | Design Variable |

Radius to fairleads from platform centerline | 45.7 m |

Unstretched chain length (Leader) | 10 m |

Unstretched chain length (Anchor) | 10 m |

Unstretched synthetic length | Design Variable |

Synthetic line diameter | Design Variable |

Chain diameter | Design Variable |

Material | Cost (USD/kg) |
---|---|

Steel | 1.50 |

Synthetic | 17.00 |

Total Draft | 20.0 m |

Platform Mass, Including Ballast | 1.09 × 10^{7} kg |

Displacement | 1.17 × 10^{4} m^{3} |

Center of Mass (CM) Location Below SWL Along Platform Centerline | 11.85 m |

Platform Roll Inertia About CM | 5.23 × 10^{9} kg-m^{2} |

Platform Pitch Inertia About CM | 5.23 × 10^{9} kg-m^{2} |

Platform Yaw Inertia About CM | 8.33 × 10^{9} kg-m^{2} |

Hub Height Above SWL | 100 m |

Total Tower Top Mass | 557,000 kg |

Tower Mass | 246,000 kg |

Tower CM above SWL | 72.9 m |

Synthetic Minimum Breaking Factor of Safety | 2.18 |

Chain Minimum Breaking Factor of Safety | 6.78 |

Partial safety factor for DLC 6.1 Loads | 1.35 |

Minimum Line Tension | 2% of Synthetic MBS |

Minimum Platform Surge Period | 40 s |

Minimum Platform Heave Period | 18 s |

Minimum Platform Pitch Period | 25 s |

Wave Loading | |||
---|---|---|---|

JONSWAP Spectrum | H_{s} (m) | T_{p} (s) | γ |

10.7 | 14.2 | 2.75 | |

Mean Load due to Second-order Wave Effects | Mean Load (kN) | ||

19.9 | |||

Current Loading | |||

Mean Load due to Current | Current Velocity (m/s) | Mean Load (kN) | |

0.28 | 49 | ||

Wind Loading | |||

Mean Load due to Wind | Wind Velocity (m/s) | Mean load (kN) | |

23.8 | 290 |

**Table 6.**DLC 6.1 Results for VolturnUS 6-MW Moored with a Basin Tested 6-MW System (0 Degree Loading; Front Line).

HydroDyn SEED 1 | HydroDyn SEED 2 | Max Tension (N) | Min Tension (N) | Mean Tension (N) | STD Tension (N) |
---|---|---|---|---|---|

674,802,239 | −228,621,085 | 2.36 × 10^{6} | 2.78 × 10^{5} | 1.13 × 10^{6} | 3.44 × 10^{5} |

−2,090,187,775 | 1,455,391,302 | 2.92 × 10^{6} | 2.27 × 10^{5} | 1.13 × 10^{6} | 3.63 × 10^{5} |

−1,973,081,278 | −629,542,915 | 2.59 × 10^{6} | 2.59 × 10^{5} | 1.13 × 10^{6} | 3.67 × 10^{5} |

301,302,578 | −328,425,023 | 2.82 × 10^{6} | 1.33 × 10^{5} | 1.13 × 10^{6} | 3.88 × 10^{5} |

81,611,327 | 265,255,796 | 2.48 × 10^{6} | 2.52 × 10^{5} | 1.13 × 10^{6} | 3.38 × 10^{5} |

133,186,342 | 1,154,134,095 | 3.31 × 10^{6} | 1.75 × 10^{5} | 1.13 × 10^{6} | 3.82 × 10^{5} |

Run | Number of Peaks in 1000 s | Number of Peaks Extrapolated to 3600 s | Probability of Maximum Tension in 3600 s | Predicted Tension (N) |
---|---|---|---|---|

A | 46 | 165 | 99.39% | 2.81 × 10^{6} |

B | 45 | 162 | 99.38% | 3.97 × 10^{6} |

C | 46 | 165 | 99.39% | 3.18 × 10^{6} |

D | 44 | 158 | 99.36% | 3.67 × 10^{6} |

E | 45 | 162 | 99.38% | 2.44 × 10^{6} |

F | 51 | 183 | 99.45% | 2.17 × 10^{6} |

10 Lumped Masses | 40 Lumped Masses | 160 Lumped Masses | |||
---|---|---|---|---|---|

OpenFAST Timesteps (s) | MoorDyn Timesteps (ms) | OpenFAST Timesteps (s) | MoorDyn Timesteps (ms) | OpenFAST Timesteps (s) | MoorDyn Timesteps (ms) |

0.25 | 2.5 | 0.25 | 2.5 | 0.1 | 1.0 |

0.175 | 0.175 | 0.075 | |||

0.125 | 1.25 | 0.125 | 1.25 | 0.05 | 0.5 |

0.075 | 0.075 | 0.025 | |||

0.025 | 0.25 | 0.025 | 0.25 | 0.01 | 0.1 |

0.0025 | 0.0025 | 0.001 |

Converged Value | Value for dtM = 2.5 ms; dtF = 0.175 s; LM = 10 | Percent Difference (%) | |
---|---|---|---|

Max Fair Tension | 1273 kN | 1308 kN | 2.7 |

Max Syn Tension | 1274 kN | 1263 kN | −0.9 |

Min Syn Tension | 353 kN | 343 kN | −2.8 |

Max Anch Tension | 1271 kN | 1253 kN | −1.4 |

Max Surge Displacement | 9.89 m | 9.79 m | −1.0 |

Max Pitch Displacement (From Resting Position) * | 0.52 deg | 0.46 deg | −1.9 |

Number of mooring lines | 3 |

Angle of mooring lines (0° aligned with positive surge axis) | 60°, 180°, 300° |

Depth to anchors below SWL (water depth) | 55 m |

Depth to fairleads below SWL | 5.4 m |

Radius to anchors from platform centerline | 239 m |

Radius to fairleads from platform centerline | 45.7 m |

Unstretched chain length (Leader) | 10 m |

Unstretched chain length (Anchor) | 10 m |

Unstretched synthetic length | 167 m |

Synthetic line diameter | 121 mm |

Chain diameter | 133 mm |

Component Cost | 113,310 USD |

SEED 1 | SEED 2 | Max Tension (N) | Min Tension (N) | Mean Tension (N) | STD Tension (N) |
---|---|---|---|---|---|

674,802,239 | −228,621,085 | 1.63 × 10^{6} | 3.32 × 10^{5} | 9.46 × 10^{5} | 1.86 × 10^{5} |

−2,090,187,775 | 1,455,391,302 | 1.97 × 10^{6} | 3.29 × 10^{5} | 9.47 × 10^{5} | 1.98 × 10^{5} |

−1,973,081,278 | −629,542,915 | 1.74 × 10^{6} | 3.50 × 10^{5} | 9.45 × 10^{5} | 2.01 × 10^{5} |

301,302,578 | −328,425,023 | 1.79 × 10^{6} | 2.30 × 10^{5} | 9.45 × 10^{5} | 2.14 × 10^{5} |

81,611,327 | 265,255,796 | 1.69 × 10^{6} | 3.30 × 10^{5} | 9.43 × 10^{5} | 1.85 × 10^{5} |

133,186,342 | 1,154,134,095 | 2.16 × 10^{6} | 2.80 × 10^{5} | 9.47 × 10^{5} | 2.11 × 10^{5} |

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## Share and Cite

**MDPI and ACS Style**

West, W.; Goupee, A.; Hallowell, S.; Viselli, A.
Development of a Multi-Objective Optimization Tool for Screening Designs of Taut Synthetic Mooring Systems to Minimize Mooring Component Cost and Footprint. *Modelling* **2021**, *2*, 728-752.
https://doi.org/10.3390/modelling2040039

**AMA Style**

West W, Goupee A, Hallowell S, Viselli A.
Development of a Multi-Objective Optimization Tool for Screening Designs of Taut Synthetic Mooring Systems to Minimize Mooring Component Cost and Footprint. *Modelling*. 2021; 2(4):728-752.
https://doi.org/10.3390/modelling2040039

**Chicago/Turabian Style**

West, William, Andrew Goupee, Spencer Hallowell, and Anthony Viselli.
2021. "Development of a Multi-Objective Optimization Tool for Screening Designs of Taut Synthetic Mooring Systems to Minimize Mooring Component Cost and Footprint" *Modelling* 2, no. 4: 728-752.
https://doi.org/10.3390/modelling2040039