# Optimization of a Lightweight Floating Offshore Wind Turbine with Water Ballast Motion Mitigation Technology

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^{2}

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## Abstract

**:**

## 1. Introduction

#### Proposed Design and Solution Method

## 2. Materials and Methods

#### 2.1. Genetic Algorithm and Constraint Handling

#### 2.1.1. Input Variables

- r, the outer radius of the platform;
- w, the outer width of the platform;
- d, the draft of the platform;
- ${h}_{p}$, the displacement limit which is a bound on the travel of the TMD;
- f, the freeboard of the platform;
- a, the aspect ratio, which is the ratio between the inner length along the radius and the inner width of the platform.

#### 2.1.2. Constraints

- The hull is initially unstable: $g}_{1}=\frac{-GM}{{N}_{h}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}16.44$, where $GM$ is the metacentric height of the hull, and the baseline value of $GM=16.44\phantom{\rule{0.222222em}{0ex}}\mathrm{m}$ is from an initial system design. This accounts for metacentric heights less than zero, which are obviously infeasible.
- The ballast water does not fit in the ballast chamber: $g}_{2}=\frac{{y}_{TMD}-{y}_{vac}}{{N}_{h}{y}_{TMD}$, where ${y}_{TMD}$ is the travel limit of the TMD and ${y}_{vac}$ is the height of the vacant space in the ballast tank above the ballast water. If the required ballast mass with the TMD at the limit of its travel interferes with the top of the chamber, this constraint is non-zero.
- Negative ballast mass required: $g}_{3}=\frac{-{m}_{b}}{{N}_{h}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}6.85\times {10}^{6}$, where ${m}_{b}$ is the ballast mass in the hull and $6.85\times {10}^{6}\phantom{\rule{0.222222em}{0ex}}\mathrm{kg}$ is the ballast mass required from an initial system design. This accounts for situations where the buoyancy of the hull requires a negative ballast mass to reach the specified draft.
- Linear hydrostatics violated: $g}_{4}=\frac{-{f}_{min}}{{N}_{h}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}3.79$, where ${f}_{min}$ is the minimum freeboard under the rated thrust. This constraint becomes non-zero when the deck is just exposed to the waterline.
- Towout draft too large: $g}_{5}=\frac{{d}_{tow}-10}{{N}_{h}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}10$, where ${d}_{tow}$ is the towout draft (the draft without ballast) and 10 m is the maximum draft allowable. This constraint ensures the hull does not sit too deep in the port.
- Ballast chamber does not fit: $g}_{6}=\frac{{L}_{bal}-{L}_{avl}}{{N}_{h}{L}_{avl}$, where ${L}_{bal}$ is the length of the ballast chamber and ${L}_{avl}$ is the available space inside the hull along the radius for the ballast water. This accounts for situations where the combination of the aspect ratio and width is incompatible with the space available.

- The horizontal RNA acceleration limit is exceeded: $g}_{7}=\frac{{a}_{RNA,x}-2.5}{{N}_{f}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}2.5$, where ${a}_{RNA,x}$ is the horizontal acceleration of the RNA and $2.5\mathrm{m}/{\mathrm{s}}^{2}$ is a typical value set by a turbine OEM.
- The vertical RNA acceleration limit is exceeded: $g}_{8}=\frac{{a}_{RNA,z}-2.0}{{N}_{f}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}2.0$, where ${a}_{RNA,z}$ is the vertical acceleration of the RNA and $2.0\mathrm{m}/{\mathrm{s}}^{2}$ is a typical value set by a turbine OEM.
- The pitch angle limit is exceeded: $g}_{9}=\frac{{\theta}_{p}-10}{{N}_{f}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}10$, where ${\theta}_{p}$ is the pitch angle of the tower and ${10}^{\xb0}$ is a typical value set by a turbine OEM.
- The TMD travel limit is exceeded: $g}_{10}={y}_{tmd$, where ${y}_{tmd}$ is the maximum travel of the TMD. This constraint accounts for designs where there are no damper configurations (one period and varied damping ratios) that keep the TMD within the limits for all design load cases. See the section on the FDF for details on how the period and damping ratios were chosen.

#### 2.1.3. Objective

#### 2.2. Hydrostatic Function

#### 2.3. Frequency Domain Function

#### 2.3.1. Response Surface Model

#### 2.3.2. Controller Scheduling

#### 2.3.3. Design Load Case Down-Selection

#### 2.4. Metric Space Calculation

#### Mechanical System Costs

## 3. Results

#### 3.1. Optimized Platform Summary

#### 3.2. Turbine Specifications

#### 3.3. Wind and Wave Conditions

- UMaine PhOG designation: E01
- NOAA buoy designation: station 44032;
- Deployment location: 43${}^{\xb0}$42${}^{\prime}$94${}^{\u2033}$ N, 69${}^{\xb0}$21${}^{\prime}$32${}^{\u2033}$ W;
- Data range used: 9 July 2001–12 September 2014;
- Data types used: significant wave height, peak wave period, wind speed and direction, and current speed and direction.

- UMaine PhOG designation: E02
- NOAA buoy designation: N/A;
- Deployment location: 43${}^{\xb0}$42${}^{\prime}$39${}^{\u2033}$ N, 69${}^{\xb0}$19${}^{\prime}$18${}^{\u2033}$ W;
- Data range used: 11 August 2010–7 October 2011 and 14 November 2014–17 September 2015;
- Data types used: significant wave height and mean wave direction.

- UMaine PhOG designation: N/A
- NOAA buoy designation: station 44007;
- Deployment location: 43${}^{\xb0}$31${}^{\prime}$30${}^{\u2033}$ N, 70${}^{\xb0}$8${}^{\prime}$26${}^{\u2033}$ W;
- Data range used: 1 January 2007–20 June 2017;
- Data types used: wave spectral parameters.

#### 3.4. Genetic Algorithm Specifications and Convergence

#### 3.5. Optimized Platform Design

#### 3.5.1. Hydrostatic Specifications

#### 3.5.2. Frequency Domain Inputs and Dynamic Performance

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

RNA | Rotor and nacelle assembly |

LCOE | Levelized cost of energy |

DOE | Department of Energy |

ARPA-E | Advanced Research Projects Agency—Energy |

ATLANTIS | Aerodynamic Turbines Lighter and Afloat with Nautical Technologies and |

Integrated Servo-control | |

TMD | Tuned mass damper |

GA | Genetic algorithm |

HDF | Hydrostatic function |

FDF | Frequency domain function |

RSM | Response surface model |

NREL | National Renewable Energy Laboratory |

DTU | Technical University of Denmark |

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**Figure 8.**A diagram of the FDF model [17].

**Figure 10.**A graph showing a surface mesh of the platform below the waterline. Due to the symmetry in the two planes, only one quarter of the platform was generated.

**Figure 11.**A graph comparing the ${X}_{1}$ values in terms of period from WAMIT with the polynomial fit.

**Figure 18.**Surface plot of LCOE vs. radius and width with constraint values overlayed on the surface.

Variable | Lower Limit | Upper Limit |
---|---|---|

r (m) | 32.5 | 45 |

w (m) | 8 | 21 |

d (m) | 7.5 | 15 |

${h}_{p}$ (m) | 3 | 7 |

f (m) | 3 | 15 |

a | 1 | 2 |

Matlab Variable ^{1} | Description |
---|---|

$r,w,d,f,{h}_{p},a$ | Optimizer variables as described in Table 1 |

h | Height, $f+d$ |

t | Nominal wall thickness, 0.3 m |

${r}_{ts}$ | Outer radius of tower support, 5 m |

${h}_{s}$ | Height of support above deck $15-f$ |

${n}_{wall}$ | Number of additional walls for damage stability, 0 |

${L}_{bal}$ | Length of ballast tank, $a\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}(w-2t)$ |

${r}_{p}$ | Radius of TMD element |

${A}_{0}$ | Water plane area, $2wr+w(2r-w)$ |

${V}_{0}$ | Volume below waterline, ${A}_{0}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}d$ |

${F}_{b}$ | Buoyant force, $g{V}_{0}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\rho}_{ocean}$ |

${I}_{wp}$ | Waterplane area moment of inertia, $(2r-w){w}^{3}/12+w{\left(2r\right)}^{3}/12$ |

$BM$ | Distance between center of buoyancy and metacentric height, ${\mathit{I}}_{\mathit{wp}}/{\mathit{V}}_{\mathit{0}}$ |

$KB$ | Distance between keel and center of buoyancy, $d/2$ |

${TMD}_{lim,TMD}$ | Limit of TMD travel, ${h}_{p}-0.5$ |

${TMD}_{lim,h20}$ | Limit of travel of water, $TM{D}_{lim,h20}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\pi {r}_{p}^{2}/\left((w-2t){L}_{bal}\right)$ |

^{1}The variables under this heading are identically named to the variables in the MATLAB funcion, except where subscripts shown here are represented by underscores in the code.

Matlab Variable | Description |
---|---|

${L}_{wz}$ | Distance from the system CG to the waterline |

${I}_{s}$ | Mass moment of inertia in the pitch DOF about the center of gravity |

${K}_{11}$ | Mooring stiffness in the surge direction |

${K}_{33}$ | Heave stiffness |

${z}_{cg,tower}$ | Tower z center of gravity |

${M}_{tower}$ | Mass of the tower |

${z}_{cg,hull}$ | Distance from CG of dry hull to system CG |

${M}_{hull}$ | Mass of the hull without ballast |

${z}_{cg,RNA}$ | RNA z center of gravity |

${M}_{RNA}$ | Mass of the RNA |

$M{p}_{total}$ | Total ballast mass |

$M{p}_{xcg}$ | Ballast x center of gravity |

$M{p}_{zcg}$ | Ballast z center of gravity |

${L}_{tbz}$ | Distance from the system CG to the hull and tower interface |

${h}_{tank}$ | Inner height of the ballast tank |

${w}_{tank}$ | Inner width of the ballast tank |

${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | |||||
---|---|---|---|---|---|---|

DLC | ${\mathbf{\zeta}}_{\mathbf{1}}$ | ${\mathbf{\zeta}}_{\mathbf{2}}$ | ${\mathbf{\zeta}}_{\mathbf{3}}$ | ${\mathbf{\zeta}}_{\mathbf{1}}$ | ${\mathbf{\zeta}}_{\mathbf{2}}$ | ${\mathbf{\zeta}}_{\mathbf{3}}$ |

$DL{C}_{1}$ | 2.0 | 3.0 | 4.0 | 6.0 | 5.5 | 5.1 |

$DL{C}_{2}$ | 5.5 | 4.0 | 4.5 | 5.5 | 4.0 | 6.0 |

${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | |||||
---|---|---|---|---|---|---|

DLC | ${\mathbf{\zeta}}_{\mathbf{1}}$ | ${\mathbf{\zeta}}_{\mathbf{2}}$ | ${\mathbf{\zeta}}_{\mathbf{3}}$ | ${\mathbf{\zeta}}_{\mathbf{1}}$ | ${\mathbf{\zeta}}_{\mathbf{2}}$ | ${\mathbf{\zeta}}_{\mathbf{3}}$ |

$DL{C}_{1}$ | 1.0 | 2.0 | 3.0 | 1.0 | 2.0 | 3.0 |

$DL{C}_{2}$ | 1.0 | 2.0 | 3.0 | 1.0 | 2.0 | 3.0 |

${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | |||||
---|---|---|---|---|---|---|

DLC | ${\mathbf{\zeta}}_{\mathbf{1}}$ | ${\mathbf{\zeta}}_{\mathbf{2}}$ | ${\mathbf{\zeta}}_{\mathbf{3}}$ | ${\mathbf{\zeta}}_{\mathbf{1}}$ | ${\mathbf{\zeta}}_{\mathbf{2}}$ | ${\mathbf{\zeta}}_{\mathbf{3}}$ |

$DL{C}_{1}$ | 1.0 | 2.0 | 3.0 | 1.0 | 2.0 | 3.0 |

$DL{C}_{2}$ | 1.0 | 2.0 | 3.0 | 1.0 | 2.0 | 3.0 |

${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | |||||
---|---|---|---|---|---|---|

DLC | ${\mathbf{\zeta}}_{\mathbf{1}}$ | ${\mathbf{\zeta}}_{\mathbf{2}}$ | ${\mathbf{\zeta}}_{\mathbf{3}}$ | ${\mathbf{\zeta}}_{\mathbf{1}}$ | ${\mathbf{\zeta}}_{\mathbf{2}}$ | ${\mathbf{\zeta}}_{\mathbf{3}}$ |

$DL{C}_{1}$ | 8.0 | 9.0 | 10.5 | 8.0 | 9.0 | 10.5 |

$DL{C}_{2}$ | 8.0 | 9.0 | 10.5 | 8.0 | 9.0 | 10.5 |

DLC | ${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ |
---|---|---|

$DL{C}_{1}$ | ${\zeta}_{1}$ | ${\zeta}_{3}$ |

$DL{C}_{2}$ | ${\zeta}_{2}$ | ${\zeta}_{2}$ |

Condition | DLC |
---|---|

Power production, normal sea state | 1.1 |

Power production, extreme sea state | 1.6 |

Parked, 50 years of wind and waves | 6.1 |

DLC | Wind Bins (m/s) |
---|---|

1.1 | 10, 24 |

1.6 | 10, 12, 14, 16, 18, 20, 22, 24 |

6.1 | 50 years of wind and waves |

Material | ${\mathit{f}}_{\mathit{t}}$ | UMaine Adjusted ${\mathit{f}}_{\mathit{t}}$ |
---|---|---|

Aluminum alloys | 4.0 | - |

Brass (70Cu30Zn, annealed) | 1.1 | - |

CFRP laminate (carbon fiber reinforced polymer) | 80.0 | - |

Copper alloys | 1.5 | - |

GFRP laminate (glass-fiber, reinforced plastic, or fiberglass) | 4.0 | - |

Lead alloys | 0.6 | - |

Nickel alloys | 3.0 | - |

Pre-stressed concrete | 0.3 | 0.13 |

Titanium alloys | 22.5 | - |

Steel of reference to calculate ${f}_{t}$ factors | 1.0 | - |

Component | ${\mathit{f}}_{\mathit{m}}$ | ${\mathit{f}}_{\mathit{i}}$ |
---|---|---|

Rotor | 3.87 | 0.10 |

Hub | 11.00 | 0.10 |

Nacelle | 9.49 | 0.10 |

Tower | 1.69 | 0.10 |

Floating platform | 2.00 | 0.13 |

Mooring system | 0.14 | 0.52 |

Anchor system | 6.70 | 3.48 |

Item | Actual Mass (kg) | Equivalent Steel Mass (kg) | Percentage of Equivalent Mass (%) |
---|---|---|---|

Rotor | 194,126 | 3,859,200 | 18.5 |

Hub | 190,000 | 2,299,000 | 11.0 |

Nacelle | 607,275 | 6,431,000 | 30.9 |

Tower | 1,262,967 | 3,523,700 | 16.9 |

Floating Platform | 7,905,400 | 3,216,700 | 15.4 |

Mooring System | 140,040 | 232,470 | 1.12 |

Anchor System | 114,000 | 1,274,520 | 6.12 |

Feature | Value |
---|---|

Generator | |

Rated power (MW) | 15 |

Power control strategy | Variable speed, collective pitch |

Rotor diameter (m) | 240 |

Hub height (m) | 150 |

Cut-in wind speed (m/s) | 3 |

Rated wind speed (m/s) | 10.59 |

Cut-out wind speed (m/s) | 25 |

Range of rotational speed (RPM) | 5–7.56 |

Blade | |

Maximum tip speed (m/s) | 95 |

Swept area (${\mathrm{m}}^{2}$) | 45,000 |

Turbine component masses | |

Nacelle (t) | 507.3 |

Hub (t) | 190.0 |

Yaw bearing (t) | 100.0 |

Blade x3 (t) | 194.1 |

Total (t) | 991.4 |

Wind Speed (m/s) | Power (MW) | ${\mathbf{C}}_{\mathbf{P}}$ | Thrust (MN) | ${\mathbf{C}}_{\mathbf{T}}$ |
---|---|---|---|---|

3 | 0.07 | 0.10 | 0.59 | 0.82 |

4 | 3.71 | 0.36 | 0.74 | 0.81 |

5 | 2.72 | 0.44 | 0.95 | 0.82 |

6 | 1.19 | 0.48 | 1.21 | 0.83 |

7 | 4.34 | 0.49 | 1.46 | 0.81 |

8 | 6.48 | 0.49 | 1.79 | 0.80 |

9 | 9.23 | 0.49 | 2.15 | 0.80 |

10.59 ^{1} | 15.0 | 0.49 | 2.73 | 0.77 |

11 | 15.0 | 0.44 | 2.38 | 0.61 |

12 | 15.0 | 0.34 | 2.05 | 0.43 |

13 | 15.0 | 0.26 | 1.86 | 0.32 |

14 | 15.0 | 0.21 | 1.72 | 0.25 |

15 | 15.0 | 0.17 | 1.62 | 0.20 |

16 | 15.0 | 0.15 | 1.54 | 0.17 |

17 | 15.0 | 0.12 | 1.47 | 0.14 |

18 | 15.0 | 0.10 | 1.41 | 0.12 |

19 | 15.0 | 0.09 | 1.36 | 0.16 |

20 | 15.0 | 0.07 | 1.31 | 0.09 |

21 | 15.0 | 0.06 | 1.28 | 0.08 |

22 | 15.0 | 0.05 | 1.25 | 0.07 |

23 | 15.0 | 0.05 | 1.21 | 0.06 |

24 | 15.0 | 0.04 | 1.19 | 0.05 |

25 | 15.0 | 0.04 | 1.17 | 0.05 |

^{1}Rated wind speed.

Wind Design Parameters | Value |
---|---|

Annual average wind speed at 100 m (m/s) [28] | 8.75 |

Extreme 10-min average 1-year wind speed at 4 m (m/s) [27] | 18.4 |

Extreme 10-min average 10-year wind speed at 4 m (m/s) [27] | 21.8 |

Extreme 10-min average 50-year wind speed at 4 m (m/s) [27] | 24.1 |

Extreme 10-min average 500-year wind speed at 4 m (m/s) [27] | 26.7 |

Normal wind shear power law exponent per ABS [19] | 0.14 |

Extreme wind shear power law exponent per ABS [19] | 0.26 |

Metocean and Site Design Parameters | Value |

1-year significant wave height (m) [27] | 6.4 |

10-year significant wave height (m) [27] | 8.5 |

50-year significant wave height (m) [27] | 9.8 |

500-year significant wave height (m) [27] | 11.5 |

Mean peak period associated with 1-year sig wave height (s) [27] | 11.7 |

Mean peak period associated with 10-year sig wave height (s) [27] | 13.3 |

Mean peak period associated with 50-year sig wave height (s) [27] | 14.2 |

Mean peak period associated with 500-year sig wave height (s) [27] | 15.0 |

1-year individual extreme wave height (m) [27] | 11.9 |

10-year individual extreme wave height (m) [27] | 15.8 |

50-year individual extreme wave height (m) [27] | 18.2 |

500-year individual extreme wave height (m) [27] | 23.0 |

Extreme 1-year sea current at depths 2 m/10 m/30 m/62 m (cm/s) [24] | 77/63/48/45 |

Extreme 1-year sea current at depths 2 m/10 m/30 m/62 m (cm/s) [24] | 89/79/67/67 |

Extreme 50-year sea current at depths 2 m/10 m/30 m/62 m (cm/s) [24] | 105/88/81/88 |

Extreme 500-year sea current at depths 2 m/10 m/30 m/62 m (cm/s) [24] | 127/99/104/129 |

DLC | Wind Speed (m/s) | ${\mathbf{H}}_{\mathbf{s}}$ (m) | ${\mathbf{T}}_{\mathbf{p}}$ (s) | $\mathbf{f}\mathbf{l}$ | Current Speed (m/s) |
---|---|---|---|---|---|

1.1 | 10 | 1.03 | 7.12 | 1.5 | 0.158 |

1.1 | 24 | 3.07 | 9.01 | 1.8 | 0.307 |

1.6 | 10 | 8.1 | 12.8 | 2.75 | 0.158 |

1.6 | 12 | 8.5 | 13.1 | 2.75 | 0.163 |

1.6 | 14 | 8.5 | 13.1 | 2.75 | 0.174 |

1.6 | 16 | 9.8 | 14.1 | 2.75 | 0.190 |

1.6 | 18 | 9.8 | 14.1 | 2.75 | 0.211 |

1.6 | 20 | 9.8 | 14.1 | 2.75 | 0.238 |

1.6 | 22 | 9.8 | 14.1 | 2.75 | 0.270 |

1.6 | 24 | 9.8 | 14.1 | 2.75 | 0.307 |

6.1 | 58.7 | 10.7 | 14.2 | 2.75 | 1.05 |

Parameter | Value |
---|---|

Maximum generations | 100 |

Population number | 120 |

Number of genes | 6 |

Elite parameter | 1 |

Best parameter | 1 |

Probability of crossover | 0.9 |

Probability of SBX crossover | 0.5 |

Crossover strength parameter | 1 |

Probability of mutation | 0.02 |

Probability of PBM operation | 0.5 |

Mutation strength parameter | 100 |

Maximum allowable niching distance | 0.1 |

Individuals checked during niching parameter | 0.25 |

Drop parameter | 0.5 |

Dyn parameter | 0.001 |

Variable | Optimizer Run 1 | Optimizer Run 2 | Percent Difference |
---|---|---|---|

r (m) | 37.58 | 37.89 | 0.83 |

w (m) | 15.53 | 14.86 | 4.37 |

d (m) | 12.50 | 12.33 | 1.37 |

${h}_{p}$ (m) | 6.33 | 6.79 | 6.98 |

f (m) | 6.14 | 6.65 | 7.92 |

a | 1.90 | 1.99 | 4.51 |

Variable | Converged Value | Standard Deviation |
---|---|---|

r (m) | 37.58 | 0.535 |

w (m) | 15.53 | 0.507 |

d (m) | 12.50 | 0.295 |

${h}_{p}$ (m) | 6.33 | 0.117 |

f (m) | 6.14 | 1.69 |

a | 1.90 | 0.069 |

Variable | Optimized | Baseline | Percent Change |
---|---|---|---|

r (m) | 37.58 | 43.50 | −13.61 |

w (m) | 15.53 | 11.00 | 41.18 |

d (m) | 12.50 | 10.50 | 19.05 |

${h}_{p}$ (m) | 6.33 | * | * |

f (m) | 6.14 | 8.00 | −20.88 |

a | 1.90 | * | * |

Parameter | Optimized | Baseline | Percent Change |
---|---|---|---|

Hull displacement $\left({\mathrm{m}}^{3}\right)$ | 26,170 | 18,827 | 39.00 |

Waterplane area $\left({\mathrm{m}}^{2}\right)$ | 2093 | 1790 | 16.93 |

Hull concrete mass (t) | 7084 | 9382 | −24.59 |

Ballast mass, fluid (t) | 15,850 | 6853 | 131.3 |

TMD equipment steel mass (t) | 821.9 | * | * |

Vertical COG from SWL (m) | 6.701 | 10.82 | −38.07 |

Vertical COB from SWL (m) | −6.251 | −5.25 | 19.07 |

Roll inertia about COG $\left(\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}\right)$ | $3.399\times {10}^{10}$ | $2.924\times {10}^{10}$ | 16.24 |

Pitch inertia about COG $\left(\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}\right)$ | $3.410\times {10}^{10}$ | $2.924\times {10}^{10}$ | 16.62 |

Yaw inertia about COG $\left(\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}\right)$ | $1.464\times {10}^{10}$ | $1.027\times {10}^{10}$ | 42.55 |

KG $\left(\mathrm{m}\right)$ | 19.20 | 21.32 | −9.94 |

KB $\left(\mathrm{m}\right)$ | 6.25 | 5.25 | 19.05 |

BM $\left(\mathrm{m}\right)$ | 21.70 | 32.51 | −33.25 |

GM $\left(\mathrm{m}\right)$ | 8.75 | 16.44 | −46.78 |

Heave natural period $\left(\mathrm{s}\right)$ | 11.38 | 9.81 | 16.00 |

Pitch natural period $\left(\mathrm{s}\right)$ | 27.15 | 21.61 | 25.64 |

TMD Position | Pitch Stiffness (Nm/rad) | Percent Change vs. Resting |
---|---|---|

Up limit | $1.84\times {10}^{9}$ | −17.61 |

Resting | $2.23\times {10}^{9}$ | 0 |

Down limit | $2.63\times {10}^{9}$ | 17.61 |

Matlab Variable | Value |
---|---|

${L}_{wz}$ (m) | −6.701 |

${I}_{s}\phantom{\rule{0.277778em}{0ex}}(\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3})$ | 3.410 $\times {10}^{10}$ |

${K}_{11}\phantom{\rule{0.277778em}{0ex}}\left(\mathrm{N}/\mathrm{m}\right)$ | 6.360 $\times {10}^{4}$ |

${K}_{33}\phantom{\rule{0.277778em}{0ex}}\left(\mathrm{N}/\mathrm{m}\right)$ | 2.104 $\times {10}^{7}$ |

${z}_{cg,tower}$ (m) | 49.31 |

${M}_{tower}$ (kg) | 1,263,000 |

${z}_{cg,hull}$ (m) | −9.636 |

${M}_{hull}$ (kg) | 7.084 $\times {10}^{6}$ |

${z}_{cg,RNA}$ (m) | 142.2 |

${M}_{RNA}$ (kg) | 9.914 $\times {10}^{5}$ |

$M{p}_{total}$ (kg) | 1.585 $\times {10}^{7}$ |

$M{p}_{xcg}$ (kg) | 23.08 |

$M{p}_{zcg}$ (kg) | −8.093 |

${L}_{tbz}$ (m) | 8.299 |

DLC/Wind Speed | $\mathbf{\u0131}$ | ${\mathbf{r}}_{6}\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{m}/{\mathbf{s}}^{2}\right)$ | ${\mathbf{r}}_{7}\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{m}/{\mathbf{s}}^{2}\right)$ | ${\mathbf{r}}_{8}\phantom{\rule{0.277778em}{0ex}}{(}^{\xb0})$ | $\mathbf{Twbsmt}\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{kN}\phantom{\rule{0.166667em}{0ex}}\mathbf{\xb7}\phantom{\rule{0.166667em}{0ex}}\mathbf{m}\right)$ | ${\mathbf{r}}_{9}\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{m}\right)$ |
---|---|---|---|---|---|---|

DLC 1.1/10 m/s | 3 | 0.390 | 0.175 | 7.139 | 4.46 × ${10}^{5}$ | 0.127 |

DLC 1.1/24 m/s | 1 | 0.731 | 0.640 | 3.285 | 1.96 × ${10}^{5}$ | 1.146 |

DLC 1.6/10 m/s | 0.7 | 1.313 | 1.630 | 8.570 | 6.12 × ${10}^{5}$ | 5.081 |

DLC 1.6/12 m/s | 0.7 | 1.262 | 1.673 | 8.227 | 5.77 × ${10}^{5}$ | 5.359 |

DLC 1.6/14 m/s | 0.7 | 1.504 | 1.680 | 7.332 | 5.34 × ${10}^{5}$ | 5.359 |

DLC 1.6/16 m/s | 0.9 | 1.561 | 1.847 | 5.151 | 4.15 × ${10}^{5}$ | 5.339 |

DLC 1.6/18 m/s | 0.9 | 1.640 | 1.846 | 4.477 | 4.01 × ${10}^{5}$ | 5.339 |

DLC 1.6/20 m/s | 0.9 | 1.538 | 1.846 | 4.234 | 3.82 × ${10}^{5}$ | 5.339 |

DLC 1.6/22 m/s | 0.9 | 1.684 | 1.848 | 4.326 | 3.81 × ${10}^{5}$ | 5.339 |

DLC 1.6/24 m/s | 0.9 | 1.698 | 1.847 | 4.320 | 3.57 × ${10}^{5}$ | 5.339 |

DLC 6.1/58.7 m/s | 0.9 | 1.415 | 1.999 | −0.252 | 7.99 × ${10}^{4}$ | 5.822 |

Property | Maximum Value | DLC/Wind Speed |
---|---|---|

Horizontal RNA Acceleration $\left(\mathrm{m}/{\mathrm{s}}^{2}\right)$ | 1.698 | DLC 1.6/24 m/s |

Vertical RNA Acceleration $\left(\mathrm{m}/{\mathrm{s}}^{2}\right)$ | 1.999 | DLC 6.1/58.7 m/s |

Platform Pitch ${(}^{\xb0})$ | 8.570 | DLC 1.6/10 m/s |

Tower Base Moment $\left(\mathrm{kN}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\right)$ | 6.12 × ${10}^{5}$ | DLC 1.6/10 m/s |

TMD Displacement (m) | 5.822 | DLC 6.1/58.7 m/s |

**Table 27.**Comparison of frequency domain model and OpenFAST [17].

Motion | % Diff. Freq. Dom. and OpenFAST |
---|---|

RNA horizontal acceleration | 12.2 |

RNA vertical acceleration | 1.5 |

Platform pitch | 7.6 |

Heave | 4.4 |

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## Share and Cite

**MDPI and ACS Style**

Ramsay, W.; Goupee, A.; Allen, C.; Viselli, A.; Kimball, R.
Optimization of a Lightweight Floating Offshore Wind Turbine with Water Ballast Motion Mitigation Technology. *Wind* **2022**, *2*, 535-570.
https://doi.org/10.3390/wind2030029

**AMA Style**

Ramsay W, Goupee A, Allen C, Viselli A, Kimball R.
Optimization of a Lightweight Floating Offshore Wind Turbine with Water Ballast Motion Mitigation Technology. *Wind*. 2022; 2(3):535-570.
https://doi.org/10.3390/wind2030029

**Chicago/Turabian Style**

Ramsay, William, Andrew Goupee, Christopher Allen, Anthony Viselli, and Richard Kimball.
2022. "Optimization of a Lightweight Floating Offshore Wind Turbine with Water Ballast Motion Mitigation Technology" *Wind* 2, no. 3: 535-570.
https://doi.org/10.3390/wind2030029