# Machine Learning-Based Approach to Wind Turbine Wake Prediction under Yawed Conditions

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## Abstract

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## 1. Introduction

#### Background

## 2. Literature Review

#### 2.1. Wind Turbine Wake Aerodynamics

#### 2.2. Wake in Yawed Wind Turbines

#### 2.3. Analytical Models for Yawed Wind Turbines

#### 2.3.1. Jiménez et al. [52] Wake Model for Yawed Conditions, 2010

#### 2.3.2. Bastankhah and Porté-Agel [58] Wake Model for Yawed Conditions, 2016

- Advantages
- -
- A cost-effective analytical approach for computational prediction of wake characteristics in the far wake [62].

- Limitations
- -
- The estimation of two empirical parameters is necessary to figure out the initiation of the far wake zone. However, obtaining universal values for these parameters is challenging, as their forecasts heavily rely on computer simulations or experiments. Consequently, the practicality of the wake model is significantly constrained [61].
- -
- Wake is significantly impacted by turbulence intensity, which is not sufficiently taken into consideration in this model [62].

#### 2.3.3. Qian and Ishihara [9] Wake Model for Yawed Conditions, 2018

- Advantages
- -
- In contrast to the Bastankhah and Porté-Agel model, this model incorporates input parameters that are expressed as functions of ambient turbulence intensity and thrust coefficient. This consideration is believed to improve the practicality of the model [61].

- Limitations
- -
- The model has a tendency to underestimate the maximum velocity deficit in scenarios involving yaw angles of 10${}^{\circ}$ and 20${}^{\circ}$ [63].
- -
- The underestimation of maximum velocity deficit is especially evident in the instances of yaw angles $\gamma ={20}^{\circ}$ and $\gamma ={30}^{\circ}$ [61].
- -
- More validation studies are necessary to support the efficacy of this model [61].

#### 2.3.4. General Limitations of Existing Analytical Models for Yawed Wind Turbines

#### 2.4. Role of Machine Learning in Wind Turbine Wake Modeling

#### 2.5. Original Contributions and Objectives of the Study

- To develop a data-driven symbolic regression model aimed at accurately capturing vital aerodynamic parameters, including velocity deficit at hub height, velocity deficit for a yawed wake center, and wake deflection.
- To move beyond traditional modeling assumptions, such as actuator disc models and Gaussian velocity deficit estimates, in an effort to achieve a more faithful representation of the intricate physics involved in wind turbine operations.
- To employ the actuator line method as the computational foundation of this research, recognizing its merits in better representing complex flow dynamics compared to traditional actuator disc models.
- To make use of symbolic regression’s natural ability for interpretability, with the aspiration of revealing not just empirical relationships but also the underlying physical principles that govern aerodynamic behaviors.
- To conduct a thorough parametric study, covering a meaningful range of yaw angles and thrust coefficients, with the intent of validating the model’s efficacy and broadening its range of applicability.

## 3. Methodology

#### 3.1. WindSE

#### 3.1.1. Unsteady Solver in WindSE

#### 3.1.2. Actuator Line Method in WindSE

#### 3.2. Symbolic Regression

#### 3.3. Data Generation

#### 3.3.1. NREL 5-MW Wind Turbine

#### 3.3.2. Case Setup

#### 3.4. Procedure: Yawed Wake Model Development through Symbolic Regression

#### 3.4.1. Objective

#### 3.4.2. Input Parameters

- Yaw angle, $\gamma $, measured in radians;
- Downstream distance normalized to the rotor diameter, represented as $x/D$;
- Yaw-specific thrust coefficient, defined as ${C}_{{T}_{\gamma}}={C}_{{T}_{\gamma =0}}\times cos\left(\gamma \right)$;
- Ambient turbulence intensity, ${I}_{a}$, capturing the environmental dynamics the turbine operates within.

#### 3.4.3. Symbolic Regression

- Automated Input Selection: The algorithm autonomously sifts through potential input variables, zeroing in on those of paramount importance for creating mathematical representations of $y/D$, $\Delta U$ at hub height, and $\Delta U$ of yawed wake.
- Optimization Technique: The optimization leverages the prowess of Simulated Annealing (SA). SA is prized in computational research for its unparalleled ability to perform exhaustive searches within vast solution spaces, ensuring that the optimal solution is approached.
- Balancing Complexity and Accuracy: To strike the delicate balance between a model’s complexity and its representational accuracy, we have intertwined Pareto Simulated Annealing within our approach. This method is a specialized variant of Multi-Objective Combinatorial Optimization (MOCO) and ensures our model remains both versatile and true to the data it represents.

#### 3.4.4. Mathematical Operators in Symbolic Regression

- Multiplication;
- Division;
- Exponentiation;
- Square root function;
- Trigonometric functions.

#### 3.4.5. Model Training and Validation

- ${R}^{2}$ (Coefficient of Determination): A measure that illustrates how well the model predictions approximate the real data points. An ${R}^{2}$ value of 1 indicates perfect predictions, while values closer to 0 indicate a model that fails to capture the underlying data trend.
- RMSE (Root Mean Square Error): It provides a quantifiable measure of how far off our model predictions are from the actual values. Lower RMSE values signify that the model predictions are close to the true values, while higher values suggest potential issues with the model or underlying data.

## 4. Results and Discussion

#### 4.1. Velocity Deficit at Hub Height ($\Delta {U}_{\mathit{hub}}/{U}_{\infty}$)

#### 4.2. Maximum Velocity Deficit of Yawed Wake ($\Delta {U}_{\mathit{yawed}}/{U}_{\infty}$)

#### 4.3. Wake Deflection $y/D$

## 5. Conclusions and Future Work

#### 5.1. Summary of Findings

- The symbolic regression model demonstrated the ability to characterize aerodynamic parameters, notably velocity deficit and wake deflection.
- By moving beyond actuator disc models and Gaussian velocity deficit assumptions, the study approached a more nuanced depiction of the physics involved in wind turbine operations.
- The actuator line model served as the computational foundation, highlighting its potential in representing complex flow dynamics.
- Symbolic regression’s inherent interpretability facilitated insights into the underlying physical principles that govern wind turbine aerodynamics.
- The extensive parametric study encompassed a diverse range of yaw angles and thrust coefficients, reinforcing the model’s adaptability and potential applicability.

#### 5.2. Future Directions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic representation of a wind turbine yawed by the angle $\gamma $, illustrating the resulting wake deflection characterized by the wake deflection ${y}_{d}$. Adapted from [9].

**Figure 2.**Correlation between CFD results and symbolic regression predictions for the velocity deficit at hub height $\Delta {U}_{\mathrm{hub}}/{U}_{\infty}$.

**Figure 3.**Comparison of CFD outcomes with the symbolic regression predictions for yaw-induced maximum velocity deficit $\Delta {U}_{\mathrm{yawed}}/{U}_{\infty}$.

**Figure 4.**Comparison of the CFD results and the proposed symbolic regression model for wake deflection $y/D$.

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**MDPI and ACS Style**

Gajendran, M.K.; Kabir, I.F.S.A.; Vadivelu, S.; Ng, E.Y.K.
Machine Learning-Based Approach to Wind Turbine Wake Prediction under Yawed Conditions. *J. Mar. Sci. Eng.* **2023**, *11*, 2111.
https://doi.org/10.3390/jmse11112111

**AMA Style**

Gajendran MK, Kabir IFSA, Vadivelu S, Ng EYK.
Machine Learning-Based Approach to Wind Turbine Wake Prediction under Yawed Conditions. *Journal of Marine Science and Engineering*. 2023; 11(11):2111.
https://doi.org/10.3390/jmse11112111

**Chicago/Turabian Style**

Gajendran, Mohan Kumar, Ijaz Fazil Syed Ahmed Kabir, Sudhakar Vadivelu, and E. Y. K. Ng.
2023. "Machine Learning-Based Approach to Wind Turbine Wake Prediction under Yawed Conditions" *Journal of Marine Science and Engineering* 11, no. 11: 2111.
https://doi.org/10.3390/jmse11112111