# Analysis and Detection of Erosion in Wind Turbine Blades

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Wind Turbine Blades

#### 2.1. Aerodynamic Analysis

^{3}. In particular, the work considers the wind conditions of La Ventosa, Oaxaca, Mexico in the Isthmus of Tehuantepec for the simulations, because this geographical zone is located between the Pacific Ocean and the Gulf of Mexico, which is a perfect location for a wind farm [16]. The wind farms are placed near the Pacific Ocean, within a range of 20 and 60 km. In this zone, wind conditions are considered as good (class 4) and excellent (class 7). With a wind speed of 8 m/s, the power generated was 1.38 kW, and the power coefficient was ${C}_{p}=0.4197$ with a simulation of 1.4 s.

#### 2.2. Modal Analysis

#### 2.3. Numerical Analysis

#### 2.4. Experimental Modal Analysis

## 3. Erosion Detection with Machine Learning

#### 3.1. Related Work

#### 3.2. Data Set

#### 3.3. Feature Extraction

- Power: ${P}_{\xi}=\frac{1}{T}{\mathsf{\Sigma}}_{-\infty}^{\infty}{\left|\xi \left(t\right)\right|}^{2}$;
- First difference: ${\delta}_{\xi}=\frac{1}{T-1}{\mathsf{\Sigma}}_{t-1}^{T-1}|\xi (t+1)-\xi \left(t\right)|$;
- Normalized first difference: ${\overline{\delta}}_{\xi}=\frac{{\delta}_{\xi}}{{\sigma}_{\xi}}$;
- Second difference: ${\gamma}_{\xi}=\frac{1}{T-2}{\mathsf{\Sigma}}_{t-1}^{T-2}|\xi (t+2)-\xi \left(t\right)|$;
- Normalized second difference: ${\overline{\gamma}}_{\xi}=\frac{{\gamma}_{\xi}}{{\sigma}_{\xi}}$.

#### Classification and Regression Problems

#### 3.4. Auto Machine Learning with H2O-DAI

#### 3.4.1. Classification Results

#### 3.4.2. Regression Results

## 4. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Wind turbine blade design. (

**b**) Area of erosion damage. (

**c**) Blue line represents the clean Airfoil FX 63_137, while the dark line is the eroded blade considering case $B{I}_{1}$.

**Figure 3.**(

**a**) Relationship between the drag coefficient (${C}_{D}$) and the lift coefficient (${C}_{L}$). (

**b**) Lift coefficient ${C}_{L}$ Vs angle of attack $\alpha $. (

**c**) Power coefficient ${C}_{p}$ for $B{I}_{1}$ and the clean wind turbine blade. (

**d**) Power output ${P}_{G}$ for $B{I}_{1}$ and the clean blade.

Parameter | Value | Variables | Units |
---|---|---|---|

Rated power | 1000 | ${P}_{\mathrm{nom}}$ | W |

Number of blades | 3 | B | [-] |

Rated speed | 8 | ${u}_{\mathrm{nom}}$ | m/s |

Speed of rotation | 295.63 | $\mathsf{\Omega}$ | rpm |

Gearbox efficiency | 0.95 | ${\eta}_{\mathrm{gearbox}}$ | [-] |

Generator efficiency | 0.95 | ${\eta}_{\mathrm{gen}}$ | [-] |

Wind density | 1.185 | $\rho $ | kg/m^{3} |

Dynamic Viscosity | 1.78 × 10${}^{-5}$ | $\mu $ | Pa × s |

(a) | ||
---|---|---|

Position | Airfoil | Chord |

0 | Circular Airfoil | 0.07 |

0.077 | Circular Airfoil | 0.07 |

0.077 | Circular Airfoil | 0.07 |

0.228 | E216 | 0.14 |

0.46 | E216 | 0.12 |

0.693 | E216 | 0.11 |

0.926 | E216 | 0.1 |

1.158 | FX 63-137 | 0.08 |

1.391 | FX 63-137 | 0.06 |

1.6323 | FX 63-137 | 0.023 |

(b) | ||

Property | Value | Units |

Density | 2000 | (Kg/m^{3}) |

Orthotropic elasticity | ||

Young’s Module in x | 50,000 | MPa |

Young’s Module in y | 8000 | MPa |

Young’s Module in z | 8000 | MPa |

Poisson’s ratio $xy$ | 0.3 | - |

Poisson’s ratio $yz$ | 0.4 | - |

Poisson’s ratio $xz$ | 0.3 | - |

Stiffness module $xy$ | 5000 | MPa |

Stiffness module $yz$ | 3846.2 | MPa |

Stiffness module $xz$ | 5000 | MPa |

Mode | ${\mathit{B}\mathit{I}}_{1}$ | ${\mathit{B}\mathit{I}}_{2}$ | ${\mathit{B}\mathit{I}}_{3}$ | Clean |
---|---|---|---|---|

(Hz) | ||||

1 | 3.8954 | 3.2521 | 3.9509 | 3.9887 |

2 | 15.631 | 12.874 | 14.903 | 14.317 |

3 | 23.60 | 22.268 | 23.411 | 22.469 |

4 | 42.867 | 35.699 | 38.514 | 38.207 |

**Table 4.**Comparison of the leading edge in three different erosion’s depth for the drag coefficient ${C}_{d}$ and the lift coefficient ${C}_{l}$. Last two columns show the percentage difference relative to the clean blade (${C}_{d}$(%) and ${C}_{l}$(%)).

Blade | $\mathit{\alpha}$ | ${\mathit{C}}_{\mathit{d}}$ | ${\mathit{C}}_{\mathit{l}}$ | ${\mathit{C}}_{\mathit{d}}$(%) | ${\mathit{C}}_{\mathit{l}}$(%) |
---|---|---|---|---|---|

Clean | 10 | $0.036$ | $1.66$ | - | - |

$B{I}_{1}$ | 10 | $0.036$ | $1.64$ | $0\%$ | $1.3\%$ |

$B{I}_{2}$ | 10 | $0.03$ | $1.46$ | $16.7\%$ | $13\%$ |

$B{I}_{3}$ | 10 | $0.028$ | $1.44$ | $22.3\%$ | $14\%$ |

**Table 5.**Modal analysis of the simulated (FEM) and physical (experimental) blade under different erosion conditions; values given in Hz.

Mode | FEM | Experimental |
---|---|---|

Level of Erosion $B{I}_{1}$ | ||

1 | 3.89 | 3.45 |

2 | 15.6 | 14.6 |

3 | 23.6 | 29.3 |

Level of Erosion $B{I}_{2}$ | ||

1 | 3.25 | 3.5 |

2 | 12.8 | 14.9 |

3 | 22.2 | 29.2 |

Level of Erosion $B{I}_{3}$ | ||

1 | 3.95 | 3.49 |

2 | 14.9 | 14.8 |

3 | 23.4 | 29.0 |

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

Time (s) | 60 |

Timesteps | 100 |

Point per direction | 20 |

Simulation Parameter | Value |

Rotor Radius (m) | 30 |

Hub Height (m) | 60 |

Mean Wind Speed (m/s) | 13 |

Measurement Height (m) | 10 |

Turbulence Intesity (%) | 10 |

Roughness Length (m) | 1.00 × ${10}^{-2}$ |

Bottom | Top | Both | Error | |
---|---|---|---|---|

Bottom | 99% | 1% | 0 | 1% |

Top | 0 | 100% | 0 | 0 |

Both | 0 | 1% | 99% | 1% |

**Table 8.**Average confusion matrix using both the power signal and the acceleration signal for feature extraction.

Bottom | Top | Both | Error | |
---|---|---|---|---|

Bottom | 100% | 0 | 0 | 0% |

Top | 0 | 100% | 0 | 0 |

Both | 0 | 0 | 100% | 0 |

**Table 9.**Regression results for H2O DAI estimating the percentage of erosion showing the average and standard deviation.

Power | Acceleration | Both | |
---|---|---|---|

MAE | 0.002 (0.0007) | 0.001 (0.0007) | 0.0009 (0.0001) |

${R}^{2}$ | 0.98 (0.008) | 0.98 (0.012) | 0.99 (.0004) |

RMSPE | 2.8 (0.4) | 1.9 (1.14) | 0.97(0.14) |

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**MDPI and ACS Style**

Enríquez Zárate, J.; Gómez López, M.d.l.Á.; Carmona Troyo, J.A.; Trujillo, L.
Analysis and Detection of Erosion in Wind Turbine Blades. *Math. Comput. Appl.* **2022**, *27*, 5.
https://doi.org/10.3390/mca27010005

**AMA Style**

Enríquez Zárate J, Gómez López MdlÁ, Carmona Troyo JA, Trujillo L.
Analysis and Detection of Erosion in Wind Turbine Blades. *Mathematical and Computational Applications*. 2022; 27(1):5.
https://doi.org/10.3390/mca27010005

**Chicago/Turabian Style**

Enríquez Zárate, Josué, María de los Ángeles Gómez López, Javier Alberto Carmona Troyo, and Leonardo Trujillo.
2022. "Analysis and Detection of Erosion in Wind Turbine Blades" *Mathematical and Computational Applications* 27, no. 1: 5.
https://doi.org/10.3390/mca27010005