# Wave-to-Wire Model Development and Validation for Two OWC Type Wave Energy Converters

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## Abstract

**:**

## 1. Introduction

## 2. Wave-to-Wire Models

#### 2.1. Hydrodynamics

#### 2.2. Thermodynamics

^{4}Pa. In those conditions, the error introduced by the linearisation of the isentropic relationship between density and pressure does not exceed 0.7%. Thus, for simplicity, Equation (4) is also used in this work.

#### 2.3. Tupperwave Non-Return Valves

#### 2.4. Turbine Model

#### 2.5. Generator Model, Bypass Valve and Control Law

#### 2.6. Numerical Integration

## 3. Turbine-Generator Systems Dimensioning

#### 3.1. Turbines

- Identification of the sea states for which the devices are the most productive over the year.
- Assessment of optimal damping coefficients for the most productive sea states.
- Assessment of turbine diameter and rotational speed to achieve optimal damping.
- Verification that the damping achieved by the chosen turbine is close to optimal.

#### 3.2. Generator

## 4. Numerical Model Validation

#### 4.1. Objective and Method

#### 4.2. Hardware-in-the-Loop Results

## 5. Wave-to-Wire Models Results

#### 5.1. Along the Power Conversion Chain

#### 5.2. Yearly Power Performance Comparison

- Annual electrical power production.
- Electrical power fluctuation.
- Use of the security system (bypass valve).

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

OWC | Oscillating Water Column |

HP | High Pressure |

LP | Low Pressure |

## References

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**Figure 1.**Two-dimensional schematic of the full scale conventional oscillating water column (OWC) and Tupperwave devices.

**Figure 5.**Generator efficiency curve as a function of the load based on [22].

**Figure 6.**Schematics of the radial-inflow twin-rotor turbine modelled in the conventional OWC device and corresponding unidirectional radial-inflow turbine modelled in the Tupperwave device [5]. (

**a**) Twin-rotor turbine. (

**b**) Single rotor turbine.

**Figure 7.**Total-to-static efficiencies for the twin-rotor and unidirectional turbines as functions of the flow coefficient.

**Figure 9.**Evolution of pneumatic power available to the turbine with turbine damping coefficient for the conventional OWC.

**Figure 10.**Evolution of pneumatic power available to the turbine with turbine damping coefficient for the Tupperwave device.

**Figure 11.**Relationship between pressure head $\Delta {P}_{t}$ and volumetric flow rate ${q}_{t}$ for the twin-rotor turbine at various rotational speeds $\mathsf{\Omega}$ compared to the optimal fixed damping relationships.

**Figure 12.**Relationship between pressure head $\Delta {P}_{t}$ and volumetric flow rate ${q}_{t}$ for the unidirectional turbine at various rotational speeds $\mathsf{\Omega}$ compared to the optimal fixed damping relationships.

**Figure 13.**Rotary test rig of the Lir National Ocean Test Facilities (Lir-NOTF), used to emulate the turbine-generator systems of the devices.

**Figure 14.**Hardware-in-the-Loop flow chart with scaling laws applied to the turbine speed $\mathsf{\Omega}$ and output torque ${T}_{turb}$.

**Figure 15.**Time series of electrical power ${P}_{e}$ and generator rotational speed ${\mathsf{\Omega}}_{gen}$ obtained for the Tupperwave device with HIL and fully numerical model in sea state $\{{H}_{s}=2\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$; ${T}_{p}=8\phantom{\rule{0.166667em}{0ex}}\mathrm{s}\}$.

**Figure 16.**Time series of electrical power ${P}_{e}$ and generator rotational speed ${\mathsf{\Omega}}_{gen}$ obtained for the conventional OWC with HIL and fully numerical model in sea state $\{{H}_{s}=2\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$; ${T}_{p}=8\phantom{\rule{0.166667em}{0ex}}\mathrm{s}\}$.

**Figure 17.**Power applied by the internal water surface (IWS) on the air in the OWC chamber (or absorbed wave power) in a conventional OWC and Tupperwave device (case 2) in sea state (${H}_{s}=3\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$; ${T}_{p}=9\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$). Solid lines: Time series; dash lines: Average values.

**Figure 18.**Pneumatic power available to the turbines in a conventional OWC and Tupperwave device (case 2) in sea state (${H}_{s}=3\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$; ${T}_{p}=9\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$). Solid lines: Time series; dash lines: Average values.

**Figure 19.**Percentage of occurrence of turbine working points in sea state (${H}_{s}=3\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$; ${T}_{p}=9\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$).

**Figure 20.**Electrical power produced by the conventional OWC and Tupperwave device (case 2) in sea state (${H}_{s}=3\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$; ${T}_{p}=9\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$). Solid lines: Time series; dash lines: Average values.

**Figure 21.**Electrical power produced by the conventional OWC and Tupperwave device (case 2) in sea state (${H}_{s}=3\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$; ${T}_{p}=9\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$). A flywheel of inertia 36.3 N·m${}^{2}$ was added to the Tupperwave power-take off (PTO) so that the resulting PTO inertia is the same as in the conventional OWC. Solid lines: Time series; dash lines: Average values.

**Figure 22.**Average power along the power conversion chain of the conventional OWC and Tupperwave device in sea state (${H}_{s}=3\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$; ${T}_{p}=9\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$).

**Figure 23.**Electrical energy production on all sea states of the EMEC wave energy test site over a year by the Tupperwave (case 2). Quasi-identical figure is obtained for the conventional OWC device.

Unidirectional | Twin-Rotor | |
---|---|---|

${\mathsf{\Phi}}_{opt}$ | 0.053 | 0.07 |

${\mathsf{\Omega}}_{d}$ (rpm) | 4000 | 1000 |

$\overline{{q}_{t}}$ $({\mathrm{m}}^{3}\xb7{\mathrm{s}}^{-1})$ | 2.8 | 9.8 |

D (m) | 0.5 | 1.10 |

Tupperwave | Conventional OWC | ||
---|---|---|---|

Turbine | Type | Unidirectional radial inflow turbine | Self-rectifying radial inflow twin-rotor turbine |

Diameter (m) | 0.50 | 1.10 | |

Inertia (kg·m${}^{2}$) | 1.7 | 38 | |

Max. efficiency (%) | 86.6 | 73.9 | |

Gearbox | Gearing Ratio | 4 | 1 |

Generator | Rated power (kW) | 100 | |

Inertia ($\mathrm{kg}\xb7{\mathrm{m}}^{2}$) | 3.6 | ||

Design speed (rpm) | 1000 | ||

Max. speed (rpm) | 2000 | ||

Min. speed (rpm) | 400 |

**Table 3.**Pearson correlation coefficients between Hardware-in-the-Loop (HIL) and fully numerical results for the time series of electrical power ${P}_{e}$ and generator rotational speed ${\mathsf{\Omega}}_{gen}$ in three common sea states of the EMEC wave climate.

Tupperwave | Conventional OWC | |||
---|---|---|---|---|

${\mathit{P}}_{\mathit{e}}$ | ${\mathsf{\Omega}}_{\mathit{gen}}$ | ${\mathit{P}}_{\mathit{e}}$ | ${\mathsf{\Omega}}_{\mathit{gen}}$ | |

${H}_{s}$ = 1.5 m; ${T}_{p}$ = 7.5 s | 0.954 | 0.956 | 0.953 | 0.990 |

${H}_{s}$ = 2 m; ${T}_{p}$ = 8 s | 0.978 | 0.958 | 0.959 | 0.990 |

${H}_{s}$ = 3 m; ${T}_{p}$ = 9 s | 0.974 | 0.961 | 0.944 | 0.985 |

Tupperwave Valves | ${\mathit{p}}_{0}$ (Pa) | $\mathit{\alpha}{\mathit{A}}_{\mathit{v}}$ $\left({\mathbf{m}}^{2}\right)$ |
---|---|---|

Case 1 | 1700 | 0.286 |

Case 2 | 150 | 0.286 |

Case 3 | 150 | 1.3 |

Tupperwave | Conventional OWC | ||||
---|---|---|---|---|---|

Valve characteristics | case 1 | case 2 | case 3 | case 2 | - |

Flywheel inertia (N·m${}^{2}$) | - | - | - | 36.3 | - |

Annual electrical production (MWh) | 70.6 | 97.6 | 119.5 | 99.3 | 99.9 |

Average power fluctuation (%) | 64.9 | 55.0 | 55.1 | 23.2 | 70.2 |

Annual pneumatic energy dissipated in bypass valves (MWh) | 0.009 | 0.051 | 0.619 | 0.012 | 0.582 |

Bypass valve opening per year | 44 | 287 | 3209 | 65 | 8537 |

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## Share and Cite

**MDPI and ACS Style**

Benreguig, P.; Kelly, J.; Pakrashi, V.; Murphy, J. Wave-to-Wire Model Development and Validation for Two OWC Type Wave Energy Converters. *Energies* **2019**, *12*, 3977.
https://doi.org/10.3390/en12203977

**AMA Style**

Benreguig P, Kelly J, Pakrashi V, Murphy J. Wave-to-Wire Model Development and Validation for Two OWC Type Wave Energy Converters. *Energies*. 2019; 12(20):3977.
https://doi.org/10.3390/en12203977

**Chicago/Turabian Style**

Benreguig, Pierre, James Kelly, Vikram Pakrashi, and Jimmy Murphy. 2019. "Wave-to-Wire Model Development and Validation for Two OWC Type Wave Energy Converters" *Energies* 12, no. 20: 3977.
https://doi.org/10.3390/en12203977