# A Novel 2-D Point Absorber Numerical Modelling Method

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Point Absorber Wave Energy Converters

#### 2.2. Point Absorber Model Characteristics

#### 2.3. Linear Frequency-Domain Analysis

**K**and damping

**C**matrices. This last equation gives the solution in terms of displacements when the frequency-domain approach is implemented.

#### 2.4. Time-Domain Formulation

#### 2.5. First-Order Wave Loads

#### 2.6. Second-Order Wave Drift Loads

#### 2.7. Numerical Formulation

- 1.
- Definition of the free surface elevation signal $\eta \left(t\right)$ using an empirical spectrum or by importing a defined $\eta \left(t\right)$ from the real-time record;
- 2.
- Finding the points where peaks, through and zero-crossings occur;
- 3.
- Defining a signal of the instantaneous wave period;
- 4.
- Interpolation of vectors found in the previous two steps;
- 5.
- Interpolation of the reflection coefficient, which depends on the instantaneous wave period;
- 6.
- Evaluation of Equation (13);
- 7.
- Adding the found waves drift load vector to the first-order wave load horizontal component vector.

#### 2.8. Experimental Study

- Length = 76 m;
- Width = 4.6 m;
- Depth = 0.5–2.3 m;
- Waves making: variable-water-depth computer-controlled flaps wavemaker;
- Beach: variable-water-depth sloping beach (reflection coefficient typically less than 5%).

#### 2.9. Calibration and Uncertainty Analysis

## 3. Results

#### 3.1. Qualitative Assessment

#### 3.2. Validation with Experimental Data

_{ref}coefficients (Table 3), which were then used in the irregular waves numerical simulations for including the drift forces. For what concerns irregular sea, results for the surge and heave response are illustrated for a specific sea state condition (T

_{p}= 7.5 s, H

_{s}= 2 m) in Figure 20 and Figure 21, respectively. The numerical surge motion, without drift forces, is far from the measured time series (Figure 20a). Differently, when drift forces are included, the numerical solution is well in agreement with the experimental measurement (Figure 20b). Similarly, the heave motion (Figure 21) and mooring load (Figure 22) are in good agreement with experimental estimated quantities.

#### 3.3. Uncertainty Analysis Results

#### 3.4. Discussion

- The mooring line is assumed to be inelastic. The only axial mooring displacement is due to the extension of the spring component. Depending on the material used, for a real installation, the mooring component elongation might be an important factor that needs to be investigated;
- Viscous forces are neglected. Note that for the typical size of wave power devices, operating in normal sea conditions, the viscous forces usually are significantly less than the first-order wave loads. The proposed methodology is not intended to be used for modelling the device for studying survivability of sea states conditions when, eventually, viscous effects are more significant. Moreover, to precisely estimate power absorption for relatively small devices at resonance conditions, viscous forces might be an important contribution;
- For simplicity, the mooring line is assumed to be attached to the center. In practice, this is not possible. However, due to the spherical shape considered and having assumed no viscous forces, it was possible to implement such simplification;
- The PTO system is modelled as a linear damping mechanism for which a single PTO damping coefficient can be set. The model, if appropriately extended, could be suitable for analyzing different, more complex types of PTO models, such as the hydraulic, phase-controlled systems, latching control or a combination of these.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Examples of point absorbers: (

**a**) the Seabased, (

**b**) the CETO, (

**c**) the CorPower PA and (

**d**) the BOLT Lifesaver WEC.

**Figure 3.**Experimental setups used in hydrodynamic experiments of point absorbers. (

**a**,

**c**–

**f**) are devices with a surface-located floater and (

**b**,

**g**) are devices with a fully submerged floater.

**Figure 4.**Point absorber model considered. In (

**a**) the geometry and (

**b**) the mesh (200 panels), used for the numerical model, are illustrated, respectively.

**Figure 5.**A mechanical analogy of the considered system for defining the numerical approximation related to the horizontal mooring restoring force component.

**Figure 6.**Experiment setup: (

**a**) side view, (

**b**) top view and (

**c**) approximate position of the Qualysis cameras.

**Figure 7.**Photos of experiments: (

**a**) carriage, mooring line and floater location, (

**b**) low friction pulley, (

**c**) floater with reflectors for motion capturing during regular waves testing.

**Figure 11.**Radiation and excitation force impulse response functions for surge and heave directions (numerical).

**Figure 14.**System response in the irregular sea (T

_{p}= 10 s, H

_{s}= 2 m) given by the two options for estimating time-dependent wave loads (numerical).

**Figure 17.**Surge and heave response amplitude operators (RAO) validation (numerical and experimental).

**Figure 20.**Surge response numerical and experimental results for an irregular sea state (T

_{p}= 7.5 s, H

_{s}= 2.0 m). Numerical results for the simulations are without (

**a**) and with (

**b**) drift forces.

**Figure 21.**Heave response numerical and experimental results for an irregular sea state (T

_{p}= 7.5 s, H

_{s}= 2.0 m).

**Figure 22.**Mooring load numerical and experimental results for an irregular sea state (T

_{p}= 7.5 s, H

_{s}= 2.0 m).

**Figure 23.**Spectral analysis of surge motion time series (numerical and experimental) results (T

_{p}= 7.5 s, H

_{s}= 2.0 m).

Instrument | Measurement | Range | Unit |
---|---|---|---|

Motion capture system | Floater displacement | 0–500 | mm |

Standard wave probe | Free-surface elevation | 0–150 | mm |

Sonic wave probe | Free-surface elevation upstream | 0–150 | Mm |

Load cell | Mooring tension | 0–50 | N |

Laser sensor | Mooring line displacement | 0–350 | Mm |

Motor tachometer | Mooring line displacement | 0–350 | mm |

Parameter | Model Scale (1:33) | Real Scale (1:1) | Unit |
---|---|---|---|

Floater radius | $0.23$ | $7.49$ | m |

Floater mass | $21.7$ | $7.8\times {10}^{5}$ | kg |

Water depth | $2$ | $66.0$ | m |

Mooring line length | $1.69$ | $55.9$ | m |

Spring stiffness | $177.0$ | $1.93\times {10}^{5}$ | N/m |

C_{pto} damping | $0.0;\phantom{\rule{0ex}{0ex}}30.0;\phantom{\rule{0ex}{0ex}}40.0;\phantom{\rule{0ex}{0ex}}50.0$ | $0;\phantom{\rule{0ex}{0ex}}1.88\times {10}^{5};\phantom{\rule{0ex}{0ex}}2.50\times {10}^{5};\phantom{\rule{0ex}{0ex}}3.13\times {10}^{5}$ | kg/s |

Mooring pretension | $27$ | $9.70\times {10}^{5}$ | N |

**Table 3.**Reflection coefficients from regular wave tests for the calculation of drift forces during irregular sea numerical tests.

ω (rad/s) | 0.40 | 0.46 | 0.53 | 0.59 | 0.66 | 0.72 | 0.79 | 0.86 | 0.92 | 0.99 | 1.05 | 1.12 | 1.18 | 1.25 | 1.31 |

C_{refl} | 0.10 | 0.11 | 0.15 | 0.25 | 0.30 | 0.35 | 0.40 | 0.45 | 0.45 | 0.50 | 0.50 | 0.55 | 0.60 | 0.70 | 1.00 |

Units | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Waves Characteristics | Ampl. | (mm) | 30 | 30 | 60 | 30 | 30 | 60 | 30 | 30 | 60 | |

Freq. | (rad/s) | 2.3 | 6.6 | 6.6 | 2.3 | 6.6 | 6.6 | 2.3 | 6.6 | 6.6 | ||

${u}_{s-A}$ | ${u}_{s-B}$ | ${u}_{s}$ | ${u}_{e}$ | |||||||||

Heave | Ampl. (mm) | 0.763 | 1.545 | 2.39 | 0.8 | 1.105 | 1.74 | 2.52 | 4.753 | 7.482 | 10.838 | |

Freq. (rad/s) | 0.002 | 0.008 | 0.004 | 0.002 | 0.008 | 0.004 | 0.008 | 0.036 | 0.018 | |||

Surge | Ampl. (mm) | 0.66 | 0.123 | 0.31 | 0.8 | 1.037 | 0.809 | 0.858 | 4.46 | 3.481 | 3.69 | |

Freq. (rad/s) | 0.005 | 0.017 | 0.003 | 0.005 | 0.017 | 0.003 | 0.021 | 0.073 | 0.013 | |||

Direct meas. | Wave probe | Ampl. (mm) | 0.891 | 0.341 | 2.099 | 0.7 | 1.133 | 0.779 | 2.213 | 4.873 | 3.349 | 9.516 |

Freq. (rad/s) | 0.002 | 0.004 | 0.003 | 0.002 | 0.004 | 0.003 | 0.009 | 0.016 | 0.013 | |||

Load | Ampl. (N) | 0.056 | 0.119 | 0.138 | 0.015 | 0.058 | 0.12 | 0.138 | 0.25 | 0.517 | 0.595 | |

Freq. (rad/s) | 0.001 | 0.007 | 0.004 | 0.001 | 0.007 | 0.004 | 0.006 | 0.031 | 0.016 | |||

Displacement | Ampl. (mm) | 0.822 | 1.73 | 2.176 | 0.85 | 1.183 | 1.927 | 2.336 | 5.085 | 8.288 | 10.044 | |

Freq. (rad/s) | 0.002 | 0.006 | 0.003 | 0.002 | 0.006 | 0.003 | 0.007 | 0.028 | 0.011 | |||

Sim. of PTO | PTO damping | (kg/s) | 1.473 | 1.473 | 1.473 | 1.473 | NA | NA | NA | |||

Spring stiffness | (N/m) | 13.212 | 13.212 | 13.212 | 13.212 | NA | NA | NA | ||||

Mass of floater | (kg) | 0.005 | 0.005 | 0.005 | 0.005 | NA | NA | NA | ||||

Fixed param. | Radius of floater | (mm) | 0.8 | 0.8 | 0.8 | 0.8 | NA | NA | NA | |||

Length of line | (mm) | 6 | 6 | 6 | 6 | NA | NA | NA | ||||

Pretension | (N) | 0.056 | 0.119 | 0.138 | 0.015 | 0.058 | 0.12 | 0.183 | 0.25 | 0.517 | 0.786 | |

Indirect meas. | Power | (W) | 0.03 | 0.328 | 1.221 |

Test No.: | 12 | 25 | 35 | 38 | 53 | ${\mathit{u}}_{\mathit{s}-\mathit{A}}$ | |
---|---|---|---|---|---|---|---|

Heave | Ampl. (mm) | 29.527 | 28.476 | 30.847 | 30.782 | 31.258 | 0.728 |

Freq. (rad/s) | 4.585 | 4.608 | 4.588 | 4.58 | 4.594 | 0.007 | |

Surge | Ampl. (mm) | 26.168 | 23.079 | 24.98 | 26.281 | 25.295 | 0.816 |

Freq. (rad/s) | 4.643 | 4.524 | 4.589 | 4.55 | 4.592 | 0.029 | |

Wave probe | Ampl. (mm) | 33.027 | 34.506 | 34.02 | 32.872 | 33.884 | 0.438 |

Freq. (rad/s) | 4.596 | 4.609 | 4.594 | 4.592 | 4.594 | 0.004 | |

Load | Ampl. (N) | 2.156 | 2.067 | 2.248 | 2.249 | 2.278 | 0.055 |

Freq. (rad/s) | 4.588 | 4.604 | 4.589 | 4.578 | 4.594 | 0.006 | |

Displacement | Ampl. (mm) | 29.732 | 28.305 | 30.742 | 30.927 | 31.253 | 0.758 |

Freq. (rad/s) | 4.587 | 4.603 | 4.589 | 4.578 | 4.594 | 0.006 |

Quantity | Total bias (±%) | |
---|---|---|

Heave | ampl. | 21.41 |

phase | 0.55 | |

Surge | ampl. | 20.68 |

phase | 1.11 | |

Wave probe | ampl. | 10.76 |

phase | 0.25 | |

Load | ampl. | 20.36 |

phase | 0.47 | |

Displacement | ampl. | 24.27 |

phase | 0.42 | |

Power | ampl. | 64.45 |

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**MDPI and ACS Style**

Giannini, G.; Day, S.; Rosa-Santos, P.; Taveira-Pinto, F.
A Novel 2-D Point Absorber Numerical Modelling Method. *Inventions* **2021**, *6*, 75.
https://doi.org/10.3390/inventions6040075

**AMA Style**

Giannini G, Day S, Rosa-Santos P, Taveira-Pinto F.
A Novel 2-D Point Absorber Numerical Modelling Method. *Inventions*. 2021; 6(4):75.
https://doi.org/10.3390/inventions6040075

**Chicago/Turabian Style**

Giannini, Gianmaria, Sandy Day, Paulo Rosa-Santos, and Francisco Taveira-Pinto.
2021. "A Novel 2-D Point Absorber Numerical Modelling Method" *Inventions* 6, no. 4: 75.
https://doi.org/10.3390/inventions6040075