# Mooring Analysis of a Floating OWC Wave Energy Converter

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Campaign

#### 2.2. Numerical Simulation Setup

#### 2.2.1. Geometry and Numerical Domain

_{min}= 4.8 s, λ

_{min}≈ 36 m at full scale). A similar convergence study was conducted and documented in [24] for the hydrodynamic diffraction analysis of a tension leg platform wind turbine (TLPWT). Finally, the FOWC model was meshed with a maximum element size of 1.5 m, equating to a total of 9047 elements, 5144 of which were diffracting. Rigid body decay tests and regular wave interactions were considered during the convergence study validations, which confirmed that the 1.5 m size is sufficient for the element size of the panel mesh. The resulting element size and simulation parameters were transferred for use with the FOWC model during mooring analysis. Figure 2 depicts the definition of the axis, degrees of freedom, and meshed body of the FOWC.

#### 2.2.2. Additional Viscous Damping

_{ij}represents the damping matrix, M

_{ij}the mass matrix, Ma

_{ij}the additional mass matrix, TN

_{ij}the undamped natural period of the structure, and ζ the non-dimensional damping coefficient. Equation (2) represents the exponential curve that can be drawn through the succeeding peaks of a damped oscillation, from which the damping coefficient can be extracted [26]. This damping coefficient is representative of the entire system damping; hence, the radiation damping coefficient should be subtracted to isolate the viscous damping coefficient.

_{o}represents the initial peak amplitude and ${\omega}_{N}$ the undamped natural frequency derived from the damped angular frequency (${\omega}_{N}$) in Equation (3) [26].

_{ij}) to be applied as an external viscous correction for the numerical model.

_{ij}represents the hydrostatic stiffness matrix.

#### 2.2.3. Moonpool Damping Lid

## 3. Results and Discussion

#### 3.1. Numerical Model Validations

_{s}= 1.584 m and T

_{p}= 9.012 s and an elevated sea state of H

_{s}= 3.384 m and T

_{p}= 14.598 s. These parameters were used to generate respective Joint North Sea Wave Project (JONSWAP) spectrums in the numerical simulations, with a factor of 3.3 for the non-dimensional peak shape parameter (γ). A time history of 36 min was simulated to perform analysis in both time and frequency domains. Table 6 presents a comparison between experimental and numerical results for irregular wave conditions for wave spectral parameters and significant motion amplitudes. In the operational sea state (see Figure 9), the motions were predicted relatively well for frequencies matching those within the irregular wave train; however, there was a significant response in heave and pitch motions at frequencies outside the wave field. Although the response was also apparent in the experimental results, it was numerically overpredicted. As the peaks of these responses directly aligned with the natural frequency peaks evident in the spectral analysis of the decay tests, the overprediction is likely related to the understated viscous correction. For the elevated condition (see Figure 10), relative agreement in the heave, pitch, and surge motions was obtained with overpredictions in the heave and pitch motions and underprediction of the surge displacement. It should be noted that the considerable differences in the free surface elevation will lead to error propagation when comparing the motion of the device.

#### 3.2. Mooring Analysis

_{max}= 11.16 m) and a peak period of 12 s. The drift force was calculated through numerical analysis using the near-field solution, as the solver cannot facilitate the far-field solution when incorporating an external lid [23]. The characteristics of the mooring spread are listed in Table 7. The moorings were incorporated into the numerical model as non-linear catenary cables, and the simulations were run with a dynamic analysis method, considering the hydrodynamic drag effect of the cables. The drag coefficients were adopted as those specified in DNV-OS-E301 [37].

#### 3.2.1. Effect of Wave Height and Incident Wave Direction

_{s}= 9 m and T

_{p}= 12 s. Further analysis of the data indicated that all three failures occurred within the first hour of the storm, at wave heights of 13.5 m, 14.6 m, and 11.3 m, respectively. For this reason, data corresponding to Case 5, and the stated angles, were excluded from the following discussions as the motions and tensions after this failure point cannot be assumed accurate.

_{s}= 2.5 m, T

_{p}= 8 s) compared to those of Case 3 (H

_{s}= 4 m, T

_{p}= 12 s), particularly evident in the sway motions at a 135° wave angle. This suggests that the peak wave period of the irregular sea state has a controlling influence on the horizontal restoring forces of the mooring system, and this should be considered in conjunction with the wave heights during catenary mooring optimisations. This is specifically crucial in the design of devices in proximity. Although the static method followed in the mooring design assumed a maximum offset of 15% water depth at an H

_{s}of 6 m, the significant horizontal motions were observed to exceed this limit for some cases.

#### 3.2.2. Effect of Mooring Line Failure

## 4. Conclusions and Recommendations

- The lack of viscosity led to overpredictions of the heave and pitch motions around the resonant frequencies. With the addition of an external viscous damping factor, derived experimentally, the solver was able to replicate the free decay tests with considerable accuracy in both magnitude and the period of oscillation.
- The validation of the device motions in regular sea states and RAOs showed a close correlation for shorter wave periods. For longer wave periods, the accuracy of the experimental results was found to be likely affected by reflected waves in the testing facility. In irregular sea states, good agreeance was observed for the motions corresponding to the wave frequencies. In all cases, the numerical results slightly overpredicted the response around the resonant frequencies, suggesting that the additional viscous damping could be increased.
- In the catenary mooring study, wave direction was observed to have a minimal effect on heave and pitch motions and a more prominent effect on sway, roll, and yaw motions. The surge motion was consistent across all directions, excluding beam seas (90°), which showed less response.
- For all cases tested, the highest tensions were experienced in the mooring line/s on the forward side of the device, relative to the incoming waves. For devices located in areas with a predominant swell direction, mooring design should consider a heavier chain on the forward lines and a lighter chain on the rear, as a cost reduction strategy. Future design iterations should also consider an increased number of incident wave angles, in conjunction with coupled loading from wind and current forces.
- The mooring line tensions were observed to exceed the minimum breaking limit for a sea state of H
_{s}= 9 m and T_{p}= 12 s. Following the simulated loss of Line 4 under a 135° incident wave, the device showed good recovery for operational to moderate sea states but experienced a catastrophic loss of positive stability in H_{s}= 6 m and T_{p}= 12 s. This indicates that a heavier mooring line or alternative configuration must be considered if the expected wave conditions are within this scale. - Larger horizontal motions were evident in the moderate sea state when compared to the rough sea state, despite the increase in significant wave height. It is suggested that this is the consequence of the shorter wave period in the moderate sea state. To determine the validity of this assumption, a moderate sea state with a longer period should be analysed and compared with the original results.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Australian Maritime College’s Model Test Basin (AMC MTB) experimental setup (left), with the main dimensions of the floating oscillating water column (FOWC) model shown on the right side.

**Figure 2.**(

**a**) FOWC axis and degree of freedom definition, blue depicting the submerged volume. (

**b**) panel mesh on the FOWC device.

**Figure 3.**Time series of experimental free decay tests: (

**a**) heave-free decay test and (

**b**) pitch-free decay test.

**Figure 4.**Interrogation of experimental free decay tests: (

**a**) heave component during heave-free decay test and (

**b**) pitch component during pitch-free decay test.

**Figure 5.**(

**a**) Heave response amplitude operator (RAO) and (

**b**) pitch response amplitude operator (RAO).

**Figure 6.**(

**a**) Heave displacement during heave decay test, (

**b**) pitch rotation during heave decay test, (

**c**) pitch rotation during pitch decay test, (

**d**) heave displacement during pitch decay test, (

**e**) heave spectral response, and (

**f**) pitch spectral response.

**Figure 7.**(

**a**) Time series of surge displacement of free decay tests and (

**b**) the corresponding spectral density plot.

**Figure 8.**Experimental and numerical results for regular waves (H = 1.404 m for all cases; T = 7.2–14.4 s).

**Figure 9.**Experimental and numerical results for an operational, irregular sea state (H

_{s}= 1.584 m, T

_{p}= 9.012 s): (

**a**) wave spectral density, (

**b**) heave spectral density, (

**c**) pitch spectral density, and (

**d**) surge spectral density.

**Figure 10.**Experimental and numerical results for an elevated, irregular sea state (H

_{s}= 3.384 m, T

_{p}= 14.598 s): (

**a**) wave spectral density, (

**b**) heave spectral density, (

**c**) pitch spectral density, and (

**d**) surge spectral density.

**Figure 11.**(

**a**) Plan view of the FOWC with mooring arrangement and incident wave directions, (

**b**) three-dimensional representation of the mooring arrangement.

**Figure 13.**Mean tensions (kN) in mooring lines for Cases 1–5 over a 3 h storm duration and under varied incident wave directions and sea states.

**Figure 14.**Significant values of the moored FOWC device motions in (

**a**) surge, (

**b**) sway, (

**c**) heave, (

**d**) roll, (

**e**) pitch, and (

**f**) yaw for all test cases (data withheld for Case 5 at incident waves angles of 90°, 135°, and 180°).

**Figure 15.**Mooring line tensions pre- and post-failure of Line 4 at 1800 s for (

**a**) Case 1, (

**b**) Case 2, (

**c**) Case 3, and (

**d**) Case 4.

**Figure 16.**Motions of the moored FOWC device in (

**a**) surge, (

**b**) roll, (

**c**) sway, (

**d**) pitch, (

**e**) heave, and (

**f**) yaw following the simulated failure of Line 4 at 1800 s.

Description | Model Scale (1:36) | Designed Full Scale | Units | |
---|---|---|---|---|

FOWC structure | Mass displacement | 27.0695 | 1.263 × 10^{6} | Kg |

LCG (from heave plate aft edge) | 0.420 | 15.12 | M | |

VCG (from keel) | 0.246 | 8.856 | M | |

Draft | 0.400 | 14.4 | M | |

Mass Moments of Inertia (Ixx, Iyy, Izz) | 3.719, 3.683, 4.881 | 0.225 × 10^{9}, 0.223 × 10^{9}, 0.295 × 10^{9} | kg·m^{2} | |

Soft moorings | Spring stiffness | 647.8 | 839632 | N/m |

Pre-tension | 29.43 | 1.37 × 10^{6} | N | |

AMC MTB | Water depth | 0.893 | 32.15 | m |

Regular wave conditions | Wave height | 0.039 | 1.404 | m |

Wave periods | 0.8–5 | 4.8–30 | s | |

Irregular wave conditions | Significant wave heights | 0.044, 0.094 | 1.584, 3.384 | m |

Peak periods | 1.502, 2.433 | 9.012, 14.598 | s |

Description | Value | Units |
---|---|---|

Lid damping factor | 0.0655 | - |

Gap for external lid | 14.4 | m |

Degree of Freedom | T_{N} (s) | ζ (%) | External Viscous Force/Moment | |
---|---|---|---|---|

Experimental | Numerical | |||

Heave | 23.742 | 22.71 | 2.99 | 164,240.03 N/m/s |

Pitch | 23.477 | 22.71 | 3.19 | 7,911,707.55 N·m/rad/s |

T_{N} (s) | ζ (%) | External Viscous Force (N/m/s) | Mooring Pre-Tension (N) | |
---|---|---|---|---|

Experimental | Numerical | |||

124.29 | 124.7 | 9.0 | 60153.9 | 2.51 × 10^{6} |

**Table 5.**Comparison between experimental and numerical results for heave, pitch, and surge response in regular waves for H = 1.404 m.

Condition | Heave (m) | Pitch (Degrees) | Surge (m) | |||
---|---|---|---|---|---|---|

Experimental | AQWA | Experimental | AQWA | Experimental | AQWA | |

Run 1 (T = 7.2 s) | 0.214 | 0.213 | 0.601 | 0.623 | 0.204 | 0.328 |

Run 2 (T = 8.4 s) | 0.278 | 0.256 | 0.500 | 0.428 | 0.257 | 0.408 |

Run 3 (T = 10.2 s) | 0.283 | 0.308 | 0.482 | 0.128 | 0.296 | 0.494 |

Run 4 (T = 14.4 s) | 0.704 | 1.073 | 2.930 | 4.333 | 0.708 | 0.539 |

Condition | H_{s} (m) | T_{p} (s) | Heave (m) | Pitch (Degrees) | Surge (m) | |
---|---|---|---|---|---|---|

Operational | Experiment | 1.584 | 9.012 | 0.54 | 1.848 | 3.204 |

AQWA | 1.602 | 8.814 | 0.73 | 3.139 | 3.197 | |

Elevated | Experiment | 3.384 | 14.598 | 2.700 | 11.311 | 6.768 |

AQWA | 3.349 | 14.79 | 3.056 | 11.953 | 3.29 |

Description | Value | Units | Description | Value | Units |
---|---|---|---|---|---|

Chain grade | R3 studless | - | Added mass coefficient | 1 | - |

Chain diameter | 76 | mm | Number of mooring lines | 4 | - |

Mass/unit length in air | 126 | kg/m | Fairlead locations (x,y,z) | +/– 15.1, +/– 15.1, 0.0 | m |

Submerged mass/unit length | 100.5 | kg/m | Anchor locations (x,y,z) | +/– 491, +/– 491, –100 | m |

Minimum breaking load (MBL) | 4.8843 × 10^{6} | N | No. of cable elements | 100 | - |

Stiffness (EA) | 6.3 × 10^{8} | N | Line length | 715 | m |

Transverse drag coefficient | 2.4 | - | Safety factor (SF) | 2.0 | - |

Longitudinal drag coefficient | 1.15 | - |

Case No. | Description | Input Parameters | Spectral Analysis | Wave Directions (Degrees) | |||
---|---|---|---|---|---|---|---|

H_{s} (m) | T_{p} (s) | H_{s} (m) | T_{p} (s) | H_{max} (m) | |||

1 | Operational | 1.25 | 8.00 | 1.25 | 8.19 | 2.38 | 0, 45, 90, 135, 180 |

2 | Moderate | 2.50 | 8.00 | 2.50 | 8.19 | 4.75 | |

3 | Rough | 4.00 | 12.00 | 4.00 | 12.47 | 7.55 | |

4 | Very rough | 6.00 | 12.00 | 6.01 | 12.47 | 11.2 | |

5 | Survivable | 9.00 | 12.00 | 9.02 | 12.47 | 16.74 |

**Table 9.**Maximum tension (kN) in the mooring lines over a 3 h storm duration and under varied incident wave directions and sea states.

Wave Direction (Degrees) | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 |
---|---|---|---|---|---|

Max. Tension/Line No. | Max. Tension/Line No. | Max. Tension/Line No. | Max. Tension/Line No. | Max. Tension/Line No. | |

0 | 219.7/Line 2 | 320.1/Line 1 | 340.76/Line 2 | 621.6/Line 2 | 3411/Line 1 |

45 | 222.1/Line 3 | 301.4/Line 1 | 370.2/Line 1 | 517.7/Line 1 | 2933/Line 1 |

90 | 227.9/Line 1 | 367.9/Line 1 | 373.2/Line 1 | 2056/Line 1 | 10,450/Line 2 |

135 | 242.7/Line 4 | 518.5/Line 4 | 387.7/Line 4 | 1480/Line 1 | 20,511/Line 2 |

180 | 221.7/Line 1 | 329.7/Line 3 | 398.8/Line 1 | 750.5/Line 3 | 4995/Line 3 |

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## Share and Cite

**MDPI and ACS Style**

Pols, A.; Gubesch, E.; Abdussamie, N.; Penesis, I.; Chin, C. Mooring Analysis of a Floating OWC Wave Energy Converter. *J. Mar. Sci. Eng.* **2021**, *9*, 228.
https://doi.org/10.3390/jmse9020228

**AMA Style**

Pols A, Gubesch E, Abdussamie N, Penesis I, Chin C. Mooring Analysis of a Floating OWC Wave Energy Converter. *Journal of Marine Science and Engineering*. 2021; 9(2):228.
https://doi.org/10.3390/jmse9020228

**Chicago/Turabian Style**

Pols, Alana, Eric Gubesch, Nagi Abdussamie, Irene Penesis, and Christopher Chin. 2021. "Mooring Analysis of a Floating OWC Wave Energy Converter" *Journal of Marine Science and Engineering* 9, no. 2: 228.
https://doi.org/10.3390/jmse9020228