# Novel Machine-Learning-Based Stall Delay Correction Model for Improving Blade Element Momentum Analysis in Wind Turbine Performance Prediction

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## Abstract

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## 1. Introduction

**Figure 1.**${C}_{L}$ vs. AoA for rotating blade (RB) and non–rotating blade (NRB) at 30 percent blade length [21].

- A synopsis of the stall delay mechanism;
- A synopsis of the Blade Element Momentum theory and Inverse BEM theory;
- A brief description of existing correction models for stall delay used in BEM;
- Description of the NREL Phase VI turbine and MEXICO rotor experiments;
- Proposed new models;
- Results comparison to NREL Phase VI turbine and MEXICO rotor data, followed by discussion.

## 2. Stall Delay Mechanism

## 3. Blade Element Momentum Theory and Inverse BEM Theory

#### 3.1. Blade Element Momentum Theory

#### 3.2. Inverse Blade Element Momentum Theory

## 4. Models for Stall Delay Correction in BEM Technique

#### 4.1. Lindenburg [9]

#### 4.2. Dumitrescu and Cardos [42,43,44]

#### 4.3. Hamlaoui, Smaili and Fellouah Model [45]

## 5. Description of Experiments of NREL Phase VI Turbine and MEXICO Rotor

#### 5.1. NREL Phase VI Turbine Experiment

#### 5.2. MEXICO Experiment

## 6. Symbolic Regression

#### 6.1. Dataset

#### 6.2. Model Evaluation

## 7. New Empirical Model for Stall Delay

## 8. Results and Discussions

#### 8.1. 3D Aerodynamic Characteristics

#### 8.1.1. Angle of Attack Distribution

#### 8.1.2. Comparison of ${C}_{L}$ Prediction throughout the Blade Length

#### 8.1.3. Comparison of ${C}_{D}$ Prediction throughout the Blade Length

## 9. Conclusions and Future Works

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

c | Local chord of the blade (m) |

${C}_{1}$, ${C}_{2}$, ${C}_{3}$, ${C}_{4}$ | Constants in the proposed model |

${C}_{L}$ | Lift coefficient (dimensionless) |

${C}_{D}$ | Drag coefficient (dimensionless) |

${C}_{D}$($\alpha $=0) | Drag coefficient when AoA is zero (dimensionless) |

${f}_{L}$ | Correction term or function for lift coefficient (dimensionless) |

${f}_{D}$ | Correction term or function for drag coefficient (dimensionless) |

r | Local radius of the blade (m) |

R | Total radius of the blade (m) |

Y | Ground truth for symbolic regression model |

x | Inputs for symbolic regression model |

N | Number of observations for symbolic regression model |

${U}_{\infty}$ | Free-stream velocity (m/s) |

${U}_{rel}$ | Relative velocity (m/s) |

Greek Symbols | |

The following Greek symbols are used in this manuscript: | |

$\alpha $ | Angle of attack (rad) |

${\alpha}_{0}$ | Angle of attack when ${C}_{L}$ is zero (rad) |

$\lambda $ | Tip-speed ratio (dimensionless) |

${\lambda}_{r}$ | Local tip-speed ratio (dimensionless) |

${\lambda}_{rel}$ | Product of $\lambda $ and $\frac{{U}_{\infty}}{{U}_{rel}}$ (dimensionless) |

$\gamma $ | Constant in Dumitrescu and Cardos stall correction model |

$\Omega $ | Angular velocity (rad/s) |

Abbreviations | |

The following abbreviations are used in this manuscript: | |

2D | Two-dimensional |

3D | Three-dimensional |

AD | Actuator disk |

AI | Artificial intelligence |

AL | Actuator line |

AoA | Angle of attack |

AR | Aspect ratio |

AS | Actuator surface |

BEM | Blade element momentum |

CFD | Computational fluid dynamics |

DNW | German–Dutch wind tunnel |

DUT | Delft University of Technology |

ECN | Energy Research Centre |

GA | Genetic algorithm |

GP | Genetic programming |

HAWT | Horizontal axis wind turbine |

LES | Large eddy simulation |

MEXICO | Model rotor EXperiments In COntrolled conditions |

ML | Machine learning |

MOCO | Multi-objective combinatorial optimization |

NASA | National Aeronautics and Space Administration |

NRB | Non-rotating blade |

NREL | National Renewable Energy Laboratory |

RB | Rotating blade |

RMSE | Root mean square error |

rpm | Revolutions per minute |

SA | Simulated annealing |

URANS | Unsteady Reynolds-averaged Navier–Stokes equations |

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**Figure 2.**Stall delay phenomenon [27].

**Figure 3.**Favourable pressure gradient due to the Coriolis force in rotating flow (inspired from [29]).

**Figure 6.**Contrast of evaluated AoA throughout the blade length from BEM analysis without and with different correction models for stall delay and Inverse BEM method for NREL Phase VI turbine.

**Figure 7.**Contrast of evaluated AoAs throughout the blade length from BEM analysis without and with different correction models for stall delay and Inverse BEM method for MEXICO rotor.

**Figure 8.**Contrast of evaluated ${C}_{L}$ throughout the blade length from BEM analysis without and with different correction models for stall delay and Inverse BEM method for NREL Phase VI turbine.

**Figure 9.**Error in ${C}_{L}$ computation by each model at different radial positions with different wind speeds for NREL Phase VI turbine.

**Figure 10.**Contrast of evaluated ${C}_{L}$ throughout the blade length from BEM analysis without and with different correction models for stall delay and Inverse BEM method for MEXICO rotor.

**Figure 11.**Error in ${C}_{L}$ computation by each model at different radial positions with different wind speeds for MEXICO rotor.

**Figure 12.**Contrast of evaluated ${C}_{D}$ throughout the blade length from BEM analysis without and with different correction models for stall delay and Inverse BEM method for NREL Phase VI turbine.

**Figure 13.**Contrast of evaluated ${C}_{D}$ throughout the blade length from BEM analysis without and with different correction models for stall delay and Inverse BEM method for MEXICO rotor.

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## Share and Cite

**MDPI and ACS Style**

Syed Ahmed Kabir, I.F.; Gajendran, M.K.; Ng, E.Y.K.; Mehdizadeh, A.; Berrouk, A.S.
Novel Machine-Learning-Based Stall Delay Correction Model for Improving Blade Element Momentum Analysis in Wind Turbine Performance Prediction. *Wind* **2022**, *2*, 636-658.
https://doi.org/10.3390/wind2040034

**AMA Style**

Syed Ahmed Kabir IF, Gajendran MK, Ng EYK, Mehdizadeh A, Berrouk AS.
Novel Machine-Learning-Based Stall Delay Correction Model for Improving Blade Element Momentum Analysis in Wind Turbine Performance Prediction. *Wind*. 2022; 2(4):636-658.
https://doi.org/10.3390/wind2040034

**Chicago/Turabian Style**

Syed Ahmed Kabir, Ijaz Fazil, Mohan Kumar Gajendran, E. Y. K. Ng, Amirfarhang Mehdizadeh, and Abdallah S. Berrouk.
2022. "Novel Machine-Learning-Based Stall Delay Correction Model for Improving Blade Element Momentum Analysis in Wind Turbine Performance Prediction" *Wind* 2, no. 4: 636-658.
https://doi.org/10.3390/wind2040034