# A Real-Time Detection System for the Onset of Parametric Resonance in Wave Energy Converters

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Parametric Resonance in WECs

#### 1.2. Suppression Control Methods for Parametric Resonance in WECs

#### 1.3. Objectives and Outline of the Paper

## 2. A Real-Time Detection System for Early Warning of Parametric Resonance in WECs

#### 2.1. The Linear Time-Varying Model

#### 2.2. Real-Time Parameter Identification

#### 2.3. Detecting Instability

## 3. Test Case

- Correctly warning when parametric resonance occurs (correct positive) and not giving a false warning when parametric resonance does not occur (correct negative).
- How early the system detects the onset of parametric resonance and sends a warning.

#### 3.1. The Device

#### 3.2. Input Waves

#### 3.3. Numerical Model

- ${F}_{I,i}$ is the inertia, detailed in Section 3.3.1.
- ${F}_{R,i}$ is the hydrostatic restoring force, detailed in Section 3.3.2.
- ${F}_{D,i}$ is the hydrodynamic damping force, detailed in Section 3.3.3.
- ${F}_{E,i}$ is the wave excitation force, detailed in Section 3.3.4.

#### 3.3.1. Inertia

#### 3.3.2. Hydrostatic Restoring Force/Moment

#### 3.3.3. Hydrodynamic Damping

#### 3.3.4. Wave Excitation

#### 3.3.5. Model Parameters

#### 3.4. Simulation Details and the Implementation of an Early Warning Detection System

#### 3.4.1. Simulation Details

#### 3.4.2. Recursive Least Squares Implementation

#### 3.4.3. Detection Syste Implementation

## 4. Results

#### 4.1. Monochromatic Waves

#### 4.1.1. Post Process Identification of Parametric Resonance

#### 4.1.2. Performance of the Early Warning Detection System

#### 4.2. Polychromatic

#### 4.2.1. Post Process Identification of Parametric Resonance

#### 4.2.2. Performance of the Early Warning Detection System

#### 4.3. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1. | Note: to limit the scale, the maximum amplitude is clipped at 40°, since for any simulation in which the amplitude exceeded this value, the pitch displacement grew to 90° and the simulation was terminated. |

**Figure 1.**Example of the input waves: (

**a**) monochromatic, (

**b**) polychromatic, (

**c**) power spectral density (PSD) of the polychomatic waves.

**Figure 2.**The excitation force coefficients (

**a**) ${H}_{E,3}$, (

**b**) ${\varphi}_{E,3}$, (

**c**) ${H}_{E,5}$ and (

**d**) ${\varphi}_{E,5}$.

**Figure 3.**The online adaptation of ${\theta}_{i,k}$ using the recursive least squares algorithm. For monochromatic waves with: (

**a**) a normalised wave frequency of 1.6 and varying wave height; (

**b**) a normalised wave amplitude of 0.01 and varying wave frequency; and (

**c**) a normalised wave frequency of 3, a normalised wave height of 0.1 and varying initial values.

**Figure 4.**The amplitude and frequency range spanned by the monochromatic waves test cases. The white region denotes where parametric resonance was observed. Note the inclusion of grid lines in this figure to highlight the frequency and amplitude resolution, with a result simulated at every grid intersection. For reference, as discussed in Section 3.1, the wave height is normalised against $\overline{GM}$ and the wave frequency by ${\omega}_{n,5}$.

**Figure 5.**The pitch displacement time series and resulting PSD for the points (A) A, (B) B and (C) C in Figure 4.

**Figure 6.**The performance of the early warning detection system in monochromatic waves; correct positives (white), correct negatives (grey), false positives (dark grey) and false negatives (black).

**Figure 7.**The pitch displacement time series (

**top**), the magnitudes of the eigenvalues of the matrix $\mathbf{A}$ (

**middle**) and the PSD of the pitch displacement (

**bottom**), for the points (A) A, (B) B and (C) C in Figure 6.

**Figure 8.**The pitch displacement time series (

**top**), the magnitudes of the eigenvalues of the matrix $\mathbf{A}$ (

**middle**) and the PSD of the pitch displacement (

**bottom**), for the points (D) D, (E) E and (F) F in Figure 6.

**Figure 9.**The performance of the early warning detection system in monochromatic waves. (

**a**) The amplitude of the pitch displacement when the warning system detected parametric resonance. (

**b**) The maximum amplitude of the pitch displacement when parametric resonance occurred (clipped at 40°, above which simulations always crashed). (

**c**) The time between the when the warning system detected parametric resonance and when the maximum pitch displacement occurred.

**Figure 10.**The energy increase in the pitch displacement from the start to the end of the simulation as a function of the normalised wave frequency and height. The contour lines have a resolution of 0.25.

**Figure 11.**The amplitude and frequency range spanned by the polychromatic waves’ test cases. The region in which parametric resonance was observed is shown in white.

**Figure 12.**The pitch displacement time series and resulting PSD for the points (A) A, (B) B and (C) C in Figure 11.

**Figure 13.**The pitch displacement time series and resulting PSD for the points (D) D, (E) E and (F) F in Figure 11.

**Figure 14.**The performance of the early warning detection system in polychromatic waves; correct positives (white), correct negatives (grey), false positives (dark grey) and false negatives (black).

**Figure 15.**The time series for the pitch displacement (

**top**), the magnitudes of the eigenvalues of the matrix $\mathbf{A}$ (

**middle**) and the PSD of the pitch displacement (

**bottom**), for the points (A) A, (B) B and (C) C in Figure 14.

**Figure 16.**The time series for the pitch displacement (

**top**), the magnitudes of the eigenvalues of the matrix $\mathbf{A}$ (

**middle**) and the PSD of the pitch displacement (

**bottom**), for the points (D) D, (E) E and (F) F in Figure 14.

**Figure 17.**The performance of the early warning detection system in polychromatic waves. (

**a**) The amplitude of the pitch displacement when the warning system detected parametric resonance. (

**b**) The maximum amplitude of the pitch displacement when parametric resonance occurred (clipped at 40°, above which simulations always crashed). (

**c**) The time between the when the warning system detected parametric resonance and when the maximum pitch displacement occurred.

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

${A}_{C}$ | 1087 m^{2} | $\rho $ | 1000 kg/m^{3} | M | 2.15 × 10^{8} kg |

${L}_{D}$ | 198.1 m | g | 9.81 m/s^{2} | ${m}_{3}$ | $1.37\times {10}^{7}$ kg |

$\overline{GM}$ | 10.1 m | ${C}_{3}$ | $1.19\times {10}^{6}$ kg/s | ${I}_{5}$ | $1.12\times {10}^{12}$ kgm |

${L}_{MS}$ | 109.1 m | ${C}_{5}$ | $7.54\times {10}^{9}$ kgm/s | ${m}_{5}$ | $7.26\times {10}^{11}$ kg |

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**MDPI and ACS Style**

Davidson, J.; Kalmár-Nagy, T.
A Real-Time Detection System for the Onset of Parametric Resonance in Wave Energy Converters. *J. Mar. Sci. Eng.* **2020**, *8*, 819.
https://doi.org/10.3390/jmse8100819

**AMA Style**

Davidson J, Kalmár-Nagy T.
A Real-Time Detection System for the Onset of Parametric Resonance in Wave Energy Converters. *Journal of Marine Science and Engineering*. 2020; 8(10):819.
https://doi.org/10.3390/jmse8100819

**Chicago/Turabian Style**

Davidson, Josh, and Tamás Kalmár-Nagy.
2020. "A Real-Time Detection System for the Onset of Parametric Resonance in Wave Energy Converters" *Journal of Marine Science and Engineering* 8, no. 10: 819.
https://doi.org/10.3390/jmse8100819