# Wave Energy Converter’s Slack and Stiff Connection: Study of Absorbed Power in Irregular Waves

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. WEC Model

#### 2.1.1. Slack Connection

#### 2.1.2. Stiff Connection

#### 2.2. Hydrodynamic Properties of Buoys

#### 2.3. Damping Force

#### 2.4. Radiation Force

- Damping $B\left(\omega \right)$ tends to zero when $\omega $ tends to zero. The difference $A\left(0\right)-A\left(\omega \right)$ becomes finite, resulting in ${lim}_{\omega \to 0}K\left(j\omega \right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$.
- Damping $B\left(\omega \right)$ tends to zero when $\omega $ tends to infinity. Therefore, ${lim}_{\omega \to \infty}K\left(j\omega \right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$.
- Passivity of the system. It ensures that there is no own energy generated by the system, the energy is only stored or dissipated, supplied by the excitation force of the wave. The influence of passivity in linear and nonlinear control systems is shown in [42].

#### 2.4.1. Transfer Function in the Frequency Domain

- Defining the initial weights for the fitting of TF by the rational function;
- Calculation of the hydrodynamic coefficients using the least square method;
- Improving the fit by choosing an appropriate weight vector, corresponding to a minimal chosen error;
- A passivity check; roots with a positive real part are identified, and passivity reinforcement is performed: the sign of negative real parts for the unstable roots is flipped according to the convolution properties described above.

#### 2.4.2. Rational Approximation by Vector Fitting

- Calculation of poles, using the initial given set: a default starting set of poles can be provided as starting poles, in an iterative process these poles are improved;
- Calculation of residues, made by the least square method;
- A passivity check, where the new poles are ensured to be stable and passivity is reinforced; if needed, the sign of the unstable poles’ real parts are inverted.

## 3. Accuracy of the Model

#### Comparison of the Two Approximations

## 4. Results

#### 4.1. Irregular Waves

#### 4.2. Absorbed Power

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The two WEC concepts and the C-GEN linear generator: (

**a**) a schematic illustration of the slack connection; (

**b**) a schematic illustration of the stiff connection; (

**c**) an image of the C-GEN linear generator during construction at Quartz Elec, Rugby, UK. The translator with the embedded magnets aligns with the white vertical guide in the middle of the device, the stator coils are potted in blue epoxy and are placed between the translator teeth.

**Figure 2.**Photos of the C-GEN linear generator modular parts: (

**a**) modules of the translator, (

**b**) stator coils during production and (

**c**) the finished stator blade of the coils in epoxy.

**Figure 3.**Illustration of the different forces and their directions for buoy and translator, where origins of the coordinate system are the center of mass.

**Figure 4.**(

**a**) Heave RAO of the buoy with a radius of 4 m and four-leg platform for different wave heading angles; (

**b**) top view of the buoy and columns’ arrangement relative to the platform, and wave heading angles are indicated.

**Figure 5.**The added mass ${A}_{33}$ and the radiation damping ${B}_{33}$ of the buoy with a radius of 4 m for two WEC concepts: slack concept (dashed line) and stiff concept (solid line).

**Figure 7.**Approximation of transfer function for the buoy with a radius of 2 m: (

**a**) TFFD (${k}_{c}\approx $ 1, ${\u03f5}_{r}$ = 0.05) and RAVF (${k}_{c}\approx $ 1, ${\u03f5}_{r}$ = 0.08) methods for the fitting order of 4; (

**b**) TFFD (${k}_{r}\approx $ 1, ${\u03f5}_{r}$ = 0.02) and RAVF (${k}_{r}\approx $ 1, ${\u03f5}_{r}$ = 0.06), for the fitting order of 8. The passivity condition was fulfilled.

**Figure 8.**Approximation of transfer function using RAVF for the 2 m radius buoy. The fitting order is 16, the passivity condition was fulfilled.

**Figure 9.**Comparison of RAO, made for the buoy with a radius of 2 m, calculated with: (

**a**) the fitting order of 4; (

**b**) the fitting order of 8. The passivity condition was fulfilled.

**Figure 11.**Absorbed power for the slack connection case, calculated for March 2018 wave data. The colour bar shows the average absorbed power in kW.

**Figure 12.**Absorbed power for the stiff connection case for March 2018. The colour bar shows the average absorbed power in kW.

**Figure 13.**Absorbed power, calculated by ${c}_{damp}$, R- and $RC$-loads for stiff connection. The buoy with a radius of 1 m was used.

**Figure 14.**Translator dynamics relative to the wave elevation for the stiff connection. The damping force is calculated by different approaches for the buoy with a radius of 1 m.

**Figure 15.**The maximum damping coefficient, calculated by different approaches for the buoy with a radius of 1 m.

Parameters | Values |
---|---|

Mass of Translator | 5600 kg |

Number of stages | 4 |

Translator length | 2 m |

Stroke length | 3 m |

Pole width | 0.083 m |

Number of coils | 18 |

Number of turns | 258 |

Number of poles | 24 |

Peak velocity | 1 m/s |

Phase Voltage, rms | 240 V |

Windings phase resistance | 13.35 $\mathsf{\Omega}$ |

Windings phase inductance | 0.21 H |

Flux density of airgap | 0.45 T |

Radius, m | Draft, m | Height, m | Mass, kg |
---|---|---|---|

1 | 2.4 | 3.5 | 2200 |

2 | 0.7 | 2 | 3000 |

3 | 0.3 | 1 | 4000 |

4 | 0.2 | 1 | 6000 |

**Table 3.**Comparison of the two approaches for the buoy with a radius of 2 m for different approximation orders N.

N | ${\mathit{k}}_{\mathit{r}}$ | ${\mathit{\u03f5}}_{\mathit{r}}$ | Time, s | |||
---|---|---|---|---|---|---|

TFFD | RAVF | TFFD | RAVF | TFFD | RAVF | |

3 | 0.9959 | 0.7908 | 0.7273 | 7.5642 | 0.0935 | 2.1992 |

4 | 0.9997 | 0.9989 | 0.0446 | 0.0848 | 0.0211 | 0.7260 |

5 | 0.9996 | 0.7925 | 0.0005 | 26.9476 | 0.0153 | 0.8855 |

6 | 0.9999 | 0.9999 | 0.0268 | 0.0828 | 0.1129 | 0.6632 |

7 | 0.9999 | 0.6079 | 0.0221 | 51.5098 | 0.0771 | 0.7957 |

8 | 0.9999 | 0.9999 | 0.0159 | 0.0607 | 182.18 | 0.6523 |

9 | − | 0.9999 | − | 0.0686 | − | 0.6288 |

10 | − | 0.9878 | − | 1.5843 | − | 1.7884 |

11 | − | 0.7368 | − | 7.5926 | − | 0.8021 |

12 | − | 0.8741 | − | 2.6359 | − | 0.7282 |

13 | − | 0.9995 | − | 0.4739 | − | 0.6668 |

14 | − | 0.9017 | − | 4.5169 | − | 0.7067 |

15 | − | 0.7720 | − | 0.5195 | − | 0.7279 |

16 | − | 0.9999 | − | 0.0532 | − | 3.5091 |

17 | − | 0.9976 | − | 0.6541 | − | 1.4792 |

18 | − | 0.9999 | − | 0.0362 | − | 0.7475 |

19 | − | 0.4588 | − | 20.67 | − | 0.7801 |

20 | − | 0.9999 | − | 0.0321 | − | 1.9464 |

21 | − | 0.9992 | − | 0.4968 | − | 0.8284 |

22 | − | 0.9999 | − | 0.0402 | − | 0.7147 |

23 | − | 0.9999 | − | 0.0349 | − | 0.6947 |

24 | − | 0.9999 | − | 0.0487 | − | 0.7139 |

25 | − | 0.9999 | − | 0.0315 | − | 0.7282 |

**Table 4.**Comparison of the two approaches for the buoy with a radius of 4 m for the different order of approximation N.

N | ${\mathit{k}}_{\mathit{r}}$ | ${\mathit{\u03f5}}_{\mathit{r}}$ | Time, s | |||
---|---|---|---|---|---|---|

TFFD | RAVF | TFFD | RAVF | TFFD | RAVF | |

3 | 0.9992 | 0.9982 | 0.0708 | 1.1462 | 0.0865 | 2.3621 |

4 | 0.1964 | 0.9987 | 7$\xb7{10}^{64}$ | 1.2971 | 0.0194 | 0.8734 |

5 | 0.1964 | 0.9991 | 4$\xb7{10}^{65}$ | 0.8850 | 0.0114 | 0.8990 |

6 | 0.1964 | 0.9991 | 10${}^{54}$ | 1.2095 | 0.0276 | 0.7253 |

7 | 0.1964 | 0.9991 | 2.5$\xb7{10}^{42}$ | 1.1208 | 0.0674 | 0.7466 |

8 | − | 0.9998 | − | 0.9605 | − | 0.6875 |

9 | − | 0.9997 | − | 1.1580 | − | 0.7518 |

10 | − | 0.9999 | − | 0.7258 | − | 0.7378 |

11 | − | 0.9998 | − | 0.8217 | − | 0.7028 |

12 | − | 0.9998 | − | 0.8386 | − | 0.7286 |

13 | − | 0.9999 | − | 0.7882 | − | 0.7305 |

14 | − | 0.9997 | − | 1.0562 | − | 0.7556 |

15 | − | 0.9991 | − | 0.8054 | − | 0.8148 |

Concept | ${\mathit{r}}_{\mathit{b}}$, m | ${\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}}^{{\mathit{C}}_{\mathit{d}\mathit{a}\mathit{m}\mathit{p}}}$, kW | ${\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathit{R}}$, kW | ${\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathit{R}\mathit{C}}$, kW | ${\mathit{C}}_{\mathit{w}}^{{\mathit{C}}_{\mathit{d}\mathit{a}\mathit{m}\mathit{p}}}$, % | ${\mathit{C}}_{\mathit{w}}^{\mathit{R}}$, % | ${\mathit{C}}_{\mathit{w}}^{\mathit{R}\mathit{C}}$, % |
---|---|---|---|---|---|---|---|

1 | 2.12 | 1.92 | 1.95 | 10.49 | 9.49 | 9.62 | |

Slack | 2 | 7.57 | 3.61 | 3.80 | 18.71 | 8.93 | 9.39 |

connection | 3 | 10.91 | 3.83 | 4.08 | 18.98 | 6.32 | 6.72 |

4 | 12.89 | 3.96 | 4.23 | 15.94 | 4.90 | 5.24 | |

1 | 2.22 | 1.97 | 2.00 | 10.96 | 9.75 | 9.89 | |

Stiff | 2 | 8.22 | 3.66 | 3.87 | 20.33 | 9.06 | 9.58 |

connection | 3 | 20.63 | 3.86 | 4.13 | 34.01 | 6.36 | 6.80 |

4 | 32.04 | 3.76 | 4.03 | 39.62 | 4.65 | 4.98 |

Concept | ${\mathit{r}}_{\mathit{b}}$, m | ${\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}}^{{\mathit{C}}_{\mathit{d}\mathit{a}\mathit{m}\mathit{p}}}$, kW | ${\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathit{R}}$, kW | ${\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathit{R}\mathit{C}}$, kW |
---|---|---|---|---|

1 | 72.58 | 45.08 | 47.49 | |

Slack | 2 | 336.81 | 73.50 | 76.85 |

connection | 3 | 588.60 | 84.95 | 88.04 |

4 | 1032.91 | 84.27 | 85.80 | |

1 | 78.45 | 46.92 | 46.81 | |

Stiff | 2 | 310.11 | 65.93 | 68.96 |

connection | 3 | 1058.40 | 64.83 | 69.68 |

4 | 1373.41 | 77.74 | 80.90 |

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**MDPI and ACS Style**

Potapenko, T.; Burchell, J.; Eriksson, S.; Temiz, I.
Wave Energy Converter’s Slack and Stiff Connection: Study of Absorbed Power in Irregular Waves. *Energies* **2021**, *14*, 7892.
https://doi.org/10.3390/en14237892

**AMA Style**

Potapenko T, Burchell J, Eriksson S, Temiz I.
Wave Energy Converter’s Slack and Stiff Connection: Study of Absorbed Power in Irregular Waves. *Energies*. 2021; 14(23):7892.
https://doi.org/10.3390/en14237892

**Chicago/Turabian Style**

Potapenko, Tatiana, Joseph Burchell, Sandra Eriksson, and Irina Temiz.
2021. "Wave Energy Converter’s Slack and Stiff Connection: Study of Absorbed Power in Irregular Waves" *Energies* 14, no. 23: 7892.
https://doi.org/10.3390/en14237892