Research on the Improvement of BEM Method for Ultra-Large Wind Turbine Blades Based on CFD and Artificial Intelligence Technologies

Abstract

With the development of the wind power industry, wind turbine blades are increasingly adopting ultra-large-scale designs. However, as the size of blades continues to increase, existing aerodynamic calculation methods struggle to achieve both relatively high computational accuracy and efficiency simultaneously. To tackle this challenge, this research focuses on the low accuracy issues of the traditional Blade Element Momentum theory (BEM) in predicting the aerodynamic performance of wind turbine blades. Consequently, a correction framework is proposed, to integrate the Computational Fluid Dynamics (CFD) method with the Multilayer Perceptron (MLP) neural network. In this approach, the CFD method is used to predict the airflow characteristics around the blades, and the MLP neural network is employed to model the intricate functional relationships between multiple influencing factors and key aerodynamic parameters. This process results in high-precision predictive functions for key aerodynamic parameters, which are then used to correct the traditional BEM. When this correction framework is applied to the rotor of the IEA 15 MW wind turbine, the effectiveness of MLP in predicting key aerodynamic parameters is demonstrated. The research findings suggest that this framework can enhance the accuracy of BEM aerodynamic load predictions to a level comparable to that of RANS.

1. Introduction

With the global energy paradigm transitioning towards sustainability, the imperative of sustainable development is increasingly recognized. Wind energy, valued for its cleanliness and renewability, has attracted significant attention globally. The principle of wind power generation involves converting the kinetic energy of wind into electrical energy. Harnessing wind energy offers significant advantages, including optimizing the energy structure, protecting the environment, and other benefits. The blades of wind turbines come into direct contact with the wind, and their aerodynamic characteristics are crucial for the efficiency and operational stability of the wind turbine system. Consequently, a thorough examination of the aerodynamic performance of these blades is vital for the progression of advanced wind power generation technologies.

As the scale of offshore wind turbine installations continues to expand, the mechanical loads experienced by turbine blades have increased significantly, thereby complicating the analysis of aerodynamic performance. The aerodynamics of wind turbines are characterized by a high degree of unsteadiness and complexity, due to the complex and variable nature of incoming flow conditions, as documented in the literature [1]. The increase in blade size can lead to significant interaction between aerodynamics and structural elasticity. Calculation accuracy of aerodynamic parameters is crucial for improving the efficiency and reliability of blade structural design and overall turbine configuration, with the potential to reduce the design costs of wind turbine systems.

Presently, the main simulation methods for evaluating the aerodynamic performance of wind turbines include the Computational Fluid Dynamics (CFD) method, the Vortex Wake Method (VWM), and the Blade Element Momentum theory (BEM) method. The CFD method, renowned for its high computational precision, offers a detailed depiction of the blade flow field interaction and furnishes an abundance of flow field detail, thereby facilitating a comprehensive aerodynamic performance analysis of wind turbines [2,3]. Despite its significant advantages in simulation accuracy [4], the CFD method’s high computational cost and relatively low computational efficiency [5] impede its widespread application in engineering practice. In contrast, the Vortex Wake Method (VWM) estimates the aerodynamic performance of wind turbines by calculating the vortex trajectories on the blade surface, thereby achieving relatively high computational efficiency and accuracy [6]. Nevertheless, its limitations in simulating dynamic responses, structural loads, and flexible deformations [7], as well as its significant dependence on the accuracy of experimental data [8], restrict its practicality in accurately estimating aerodynamic performance [9,10].

The Blade Element Momentum theory (BEM) method, characterized by its relatively high computational efficiency and broad applicability, has been extensively utilized in engineering practice. This method divides the blade into multiple elements along the spanwise direction, independently solving the aerodynamic parameters for each element, and then subsequently integrating these parameters to obtain the overall aerodynamic performance of the blade. However, the BEM method’s theoretical derivation, which neglects blade element interactions, the three-dimensional flow field around the blade, and assumes an infinitely long blade [11], may lead to inaccurate aerodynamic parameter calculations under complex conditions like blade stall and yaw. To address these limitations, numerous scholars have undertaken various corrections to the BEM method, including tip loss correction [12,13], dynamic stall correction [14,15], three-dimensional rotation correction [16], and angle of attack correction [17], among others. These corrections, to a certain extent, compensate for the BEM method’s deficiencies and enhance its suitability for the aerodynamic performance analysis and prediction of modern large-scale wind turbines.

Despite certain corrections, traditional Blade Element Momentum (BEM) theory still has shortcomings in elucidating the complex functional relationships that influence aerodynamic parameters from the perspective of the flow field. Moreover, the issue of error accumulation in iterative calculations remains unresolved. To address these limitations, numerous researchers have explored the integration of neural network techniques into the aerodynamic correction of the BEM method. Baisthakur et al. [18] proposed a Physics-Informed Neural Network (PINN) to improve the accuracy of the BEM model for wind turbines. By directly predicting the angle of attack at blade nodes, their approach eliminates the need for time-consuming iterative procedures inherent in conventional BEM models, thereby enhancing computational efficiency while preserving high accuracy. Abdulaziz et al. [19] utilized neural networks (NNs) to predict lift and drag coefficients, which significantly improved the precision of the BEM model. The developed BEM-NN model was subsequently incorporated into a modified BEM framework, effectively resolving convergence issues commonly encountered at high tip-speed ratios. Verma et al. [20] introduced an artificial neural network (ANN)-based method for predicting lift and drag coefficients of airfoils, given specific angles of attack and Reynolds numbers. This approach effectively mitigates the uncertainties associated with inadequate consideration of Reynolds number effects on airfoil aerodynamics during the design and performance evaluation processes.

This research introduces an innovative method that integrates Computational Fluid Dynamics (CFD) simulations with neural networks. The spatial velocity field is obtained via CFD, and the Average Azimuthal Technique (AAT) is employed to calculate the velocity induction factors at the blade. Concurrently, a neural network is trained to develop a more accurate predictive model. This integrated approach not only circumvents the error accumulation inherent in the BEM iterative process but also compensates for its inadequacies in addressing the three-dimensional characteristics of the blade flow field and aerodynamic parameters. Consequently, this method effectively elevates the computational accuracy of BEM to a level comparable to that of CFD.

This paper presents research aimed at developing an enhanced framework for the Blade Element Momentum (BEM) theory, with the goal of improving the accuracy of aerodynamic performance calculations for wind turbine generators. In Section 2, the Computational Fluid Dynamics (CFD) method and neural network techniques utilized within the enhanced framework are introduced. In Section 3, the calculation results of the improved BEM are presented and analyzed, including normal force, tangential force coefficients, thrust, and torque, to assess the efficacy of the proposed approach. Additionally, a summary of work in the paper is provided and future research directions are discussed.

2. Numerical Scheme

2.1. CFD Calculate

In this study, Computational Fluid Dynamics (CFD) is employed for numerical simulations of the (International Energy Agency) IEA 15 MW wind turbine generator, a design collaboratively developed by the National Renewable Energy Laboratory (NREL) and the Technical University of Denmark (DTU) [21], featuring a blade length of 117 m that classifies it among ultra-large wind turbine generators.

Table 1. Main parameters table of wind turbine.

Parameter	Value	Parameter	Value
Rated Power	15 MW	Tip Speed Ratio	9
Number of Blades	3	Maximum Tip Speed	95 m/s
Cut-in Wind Speed	3 m/s	Rotor Diameter	240 m
Rated Wind Speed	10.59 m/s	Blade Length	117 m
Cut-out Wind Speed	25 m/s	Hub Diameter	7.94 m
Minimum Rotational Speed	5 RPM	Root Diameter	5.2 m
Maximum Rotational Speed	7.56 RPM	Maximum Chord Length	5.77 m

delineates the key parameters of this generator.

In this study, a geometric model of the wind turbine generator is meticulously constructed using the technical documents and data published by IEA [21]. The Reynolds-Averaged Navier–Stokes (RANS) equations are chosen for the CFD solution to ensure the accuracy of aerodynamic calculation results for the blades. Furthermore, The Rotating Coordinate System (RCS) is enabled to simulate the motion of wind turbine rotor, facilitating an accurate prediction of the blades’ aerodynamic performance.

Prior to performing the Computational Fluid Dynamics (CFD) simulations, a wind turbine rotor model comprising three blades is constructed. The computational domain is defined with the following dimensions: the distance from the rotor plane to the inlet boundary is set to 5R, while the distance from the rotor plane to the outlet plane boundary is set to 10R. The radius of the entire external flow field domain is specified as 5R. A schematic illustration of the computational domain is provided in Figure 1. Given the dimensional variations across different regions and the spanwise distribution of chord lengths along the blade, a customized mesh refinement strategy is adopted to balance computational accuracy and efficiency. The grid element sizes are specified as follows; in the root region (0–30% of the blade span), the mesh size is 0.08 m; in the mid-span region (30–60%), the mesh size is 0.04 m; and in the tip region (60–100%), the mesh size is 0.02 m, the mesh schematic diagram is presented in Figure 2.

During the mesh generation process, polyhedral meshes are preferred over hexahedral meshes due to the challenges associated with generating high-quality hexahedral meshes for the complex geometry of the flow domain surrounding an ultra-large wind turbine. The use of pure hexahedral meshes typically results in a significantly higher element count, increased computational cost, and reduced mesh quality. Conversely, polyhedral meshes offer several advantages, including improved computational accuracy, a greater number of neighboring cells for more accurate gradient calculations, fewer elements at equivalent resolution, reduced computational time, and enhanced convergence performance.

To verify the accuracy of Computational Fluid Dynamics (CFD) simulations and the strategy for meshing as described above, grid independent verification work was carried out in this study. During the verification process, by gradually increasing the grid density, the thrust and torque of the wind turbine rotor were monitored in this paper to assess the stability of their numerical values. As illustrated in

Table 2. Table of wind thrust and torque changes with the number of grids.

Mesh	Number of Grids	Thrust (MN)	Torque (MNm)
M1	13 million	2.04	16.8
M2	17.43 million	2.15	18.7
M3	23.63 million	2.18	18.8

, under the specified operating conditions of an inflow wind speed of 10.59 m/s and a rotational speed of 7.56 RPM, the computational grids are categorized into three types: M1, M2, and M3. Among these, the M2 grid employs the aforementioned refinement strategy. The results indicate that when the total grid count reaches approximately 17 million, the thrust and torque experienced by the wind turbine rotor exhibit minimal variation, suggesting that these key aerodynamic parameters have reached a stable convergence. As depicted in Figure 3, the vortex structures generated by the blade evolve with increasing mesh density. When the grid count reaches 17 million (M2), the vortex structure differs significantly from that of M1, which utilizes a coarser mesh. Further increases in grid density beyond M2 result in negligible changes to the vortex structures surrounding the blade. Therefore, the mesh refinement strategy implemented in M2 effectively ensures the accuracy of the Computational Fluid Dynamics (CFD) simulations while maintaining computational efficiency. This approach substantially reduces the computational cost associated with subsequent CFD analyses.

During the simulation process, the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm was adopted in this study for pressure–velocity coupling. Proposed by Patankar and Spalding in 1972, the SIMPLE algorithm has established itself as a standard approach for incompressible flow simulations.

To ensure the accuracy and convergence of the CFD simulations and to accelerate the convergence process, this study conducted verification and comparative analyses using three turbulence models: $k-\omega$, $k-\epsilon$, and RSM. Among them, Model B exhibited the smallest relative error when compared to the data from [22]. This outcome is likely attributed to the $k-\epsilon$ model′s characteristics as a Reynolds-averaged turbulence model, which offers high versatility and robustness in computing aerodynamic parameters for ultra-large wind turbine blades. It demonstrates strong adaptability to the complex and dynamic flow field conditions around wind turbine blades. Moreover, its superior numerical stability helps suppress oscillations and prevents divergence during simulations, thereby producing more reliable and consistent aerodynamic predictions.

This study follows the official specifications of the International Energy Agency (IEA) 15 MW reference wind turbine [21]. Based on the turbine’s operational characteristics, the wind speed range is segmented into low, medium, and high intervals to comprehensively evaluate its aerodynamic performance under various environmental conditions. Within each interval, representative operating points are selected to capture wind speed fluctuations. A total of 24 operational scenarios are simulated using CFD, As shown in

Table 3. The wind speed and rotation speed of the working condition.

Working Condition	Wind Speed [m/s]	Rotor Speed [rpm]	Working Condition	Wind Speed [m/s]	Rotor Speed [rpm]
Case1	4	5	Case13	11	7.56
Case2	5	5	Case14	12	7.56
Case3	6	5	Case15	13	7.56
Case4	6.73	5	Case16	14	7.56
Case5	7.16	5.09	Case17	16	7.56
Case6	7.50	5.33	Case18	17	7.56
Case7	8.18	5.87	Case19	19	7.56
Case8	8.71	6.19	Case20	20	7.56
Case9	9.38	6.67	Case21	21	7.56
Case10	9.78	6.94	Case22	22	7.56
Case11	10.20	7.25	Case23	23	7.56
Case12	10.59	7.56	Case24	24	7.56

. The blade model, with appropriately adjusted pitch angles, is imported into commercial software for mesh generation. In the Fluent solver, varying inlet wind speeds, rotational speeds, wind inflow directions, and the coordinate axis of rotor rotation are specified. Upon completion of the CFD simulations, convergence is assessed by exporting thrust and torque reports and monitoring residuals. Finally, the normal and tangential velocity components at all flow field positions are extracted for each operating condition. To enrich and expand the training dataset for the neural network, under each operating condition, this study establishes densely distributed monitoring sections along the blade span. By extracting the flow field information within these sections, we ensure the continuity and sufficiency of the training data.

This paper conducts Computational Fluid Dynamics (CFD) calculations for the selected operating conditions to ensure the accuracy and reliability of simulation results. Subsequently, the Average Azimuthal Technique (AAT) is utilized for post-processing to extract aerodynamic parameters, including axial and tangential velocities, from key sections of the wind rotor. As shown in Equations (1) and (2), the definition of the Blade Element Momentum (BEM) method is employed in this study to derive the velocity induction factors (denoted as “a” and “b”) through CFD calculations. This comprehensive simulation and analysis approach offers a robust research framework for predicting wind turbine performance across a range of wind speed conditions.

$$V_a=V_{\infty }-v_a$$

(1)

$$a={\displaystyle \frac{v_a}{V_{\infty }}}$$

(2)

$$b=-{\displaystyle \frac{V_t}{2\Omega r}}$$

(3)

In the equations, V_∞ is the incoming wind speed. V_a is the axial velocity at the position of the wind rotor. v_a stands for the axial-induced velocity at the position of the wind wheel. a and b are the axial induction factor and tangential induction factor, respectively. V_t represents the tangential velocity at the position of the wind rotor. Ω is the rotation speed of the wind rotor. r is the spanwise position of the blade.

The CFD calculations within this research framework assume a uniform and parallel inflow for the assessment of working conditions, adhering to a standard idealization in wind turbine performance analysis. Induction factors are dimensionless and quantify the impact of the wind rotor on the inflow velocity field, directly affecting the rotor’s thrust and torque, as elaborated in Equations (4) and (5). The precise calculation of these parameters is crucial for the evaluation and prediction of wind turbine blade aerodynamics. Therefore, the employment of neural network training to refine the prediction of ‘a’ and ‘b’ remarkably enhances the accuracy of BEM-based aerodynamic computations.

$$dT=4\pi \rho V_{\infty }^{2}a\left(1-a\right)rdr$$

(4)

$$dM=4\pi \rho \Omega V_{\infty }b\left(1-a\right)r^{3}dr$$

(5)

In the equation, dT and dM are the differentials of thrust and torque at each blade element position, respectively. ρ represents the air density.

2.2. Artificial Intelligence Approaches

In the selection of parameters for neural network training that influence the wind rotor’s aerodynamic performance, to comprehensively assess the influencing factors, the following input parameters are identified based on a comprehensive literature review [23]: incoming wind speed, wind rotor rotational speed, blade chord length, twist angle, spanwise position, pre-bending, and pitch angle. After the CFD calculations, we input the abovementioned aerodynamic influence parameters and the velocity induction factors a and b, obtained from the CFD into the neural network for training.

For the purpose of fitting multivariate functional models, several approaches can be employed, including the Multilayer Perceptron (MLP), Radial Basis Function Network (RBFN), Long Short-Term Memory (LSTM), and Gated Recurrent Unit (GRU). Among these, the RBFN model is limited by the complexity of hidden layer parameter selection, while LSTM and GRU models typically involve high computational costs and are susceptible to overfitting. Traditional methods such as linear regression are inadequate for capturing nonlinear relationships among variables. In contrast, MLP exhibits several advantages, including a flexible network architecture, ease of implementation, strong generalization capability, and support for parallel computation. These attributes make it particularly well-suited for modeling high-dimensional nonlinear data, enabling the effective capture of complex variable mappings while maintaining computational efficiency.

In this study, the Multilayer Perceptron (MLP) neural network model is selected to simulate the nonlinear relationship between the input variables and the output (i.e., the velocity induction factors). The MLP is a typical feedforward neural network, which comprises an input layer, multiple hidden layers, and an output layer. The neurons in the network are connected with weighted links and undergo nonlinear processing using activation functions. In this study, the Sigmoid function is adopted as the activation function, and its specific expression and derivative are shown in Equations (7) and (8), respectively.

$$y_k=\Phi \left({\displaystyle \sum x_i{w}_{k,i}+b_{k,i}}\right)$$

(6)

$$\Phi \left(z\right)={\displaystyle \frac{1}{1+e^{-z}}}$$

(7)

$${\Phi }^{\prime }\left(z\right)=\Phi \left(z\right)\left[1-\Phi \left(z\right)\right]$$

(8)

After the input parameters are fed into individual neurons and processed through weights, biases, and activation functions (specific expressions shown in Equation (6)), the neural network ultimately outputs a predicted value. By comparing the neural network’s predicted value $y_{predicted}$ with the true value $y_{measured}$ obtained from CFD calculations, the loss function between the predicted and true values is derived (specific calculation formula shown in Equation (9)). Using gradient descent, the neural network method can automatically adjust the weights and biases at each neuron, thereby updating the parameter matrices within the network. Through these steps, the loss function value continuously decreases, eventually yielding a prediction function with a small error.

$$LOSS={\displaystyle \frac{1}{2}}\left(Outpu{t}^2-y^2\right)$$

(9)

In this paper, two neural network evaluation metrics widely used in the fields of statistics and machine learning are adopted: Mean Absolute Percentage Error (MAPE) and Coefficient of Determination (R²). The Mean Absolute Percentage Error (MAPE) is an indicator used to measure the difference between the predicted values and the actual observed values, which is obtained by calculating the average of the absolute percentages of the prediction errors. The expression of MAPE is defined as shown in Equation (10).

$$MAPE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{k=1}^n\frac{\left|{\left(y_{predicted}\right)}_k-{\left(y_{measured}\right)}_k\right|}{\left|{\left(y_{predicted}\right)}_k\right|}}\times 100\%$$

(10)

The Coefficient of Determination (R²), commonly referred to as the goodness of fit, is used as a metric for assessing a model’s explanatory power. Ranging from 0 to 1, R² quantifies the proportion of variance in the dependent variable that is predictable from the independent variables. A value closer to 1 indicates a stronger explanatory power and a higher degree of agreement between predicted and observed values. The formula for calculating R² is

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{k=1}^n{\left[{\left(y_{predicted}\right)}_k-{\left(y_{measured}\right)}_k\right]}^2}}{{\displaystyle \sum _{k=1}^n{\left[{\left(y_{measured}\right)}_k-\frac{1}{n}{\displaystyle \sum _{k=1}^n{\left(y_{measured}\right)}_k}\right]}^2}}}$$

(11)

In designing the neural network architectures, different models were developed in this paper to estimate the axial induction coefficient a and the tangential induction coefficient b, the general configuration of the mesh is illustrated in Figure 4. The architecture for factor a includes a neural network with seven hidden layers, featuring an input layer with the seven previously identified independent variables, and an output layer yielding the axial induction factor a, with neurons sequenced as 4, 8, 16, 32, 16, 4, and 2. This configuration is intended to model the complex nonlinear dynamics of the induction factor a. For factor b, a six-hidden-layer architecture is employed, utilizing the same set of input parameters, with neurons sequenced as 4, 8, 16, 32, 16, and 4. This design enhances the precision of aerodynamic performance simulation and prediction for wind turbine blades. Additionally, a learning rate of 0.001 was selected for training, with 10 samples per epoch. During the neural network training process, Dropout is applied after each hidden layer to enhance prediction accuracy by randomly deactivating neurons. In addition, the Adam optimizer with weight decay is employed to improve generalization performance. An Early Stopping mechanism is also implemented to monitor the validation loss, terminating the training process if no improvement is observed over 100 consecutive epochs.

In this study, the 9600 datasets acquired through Computational Fluid Dynamics (CFD) are considered. The dataset is partitioned into training and validation sets following standard methodologies, at an 8:2 ratio. This means 80% of the data is allocated for neural network training, and 20% is allocated for validation. The validation set is isolated from the training process and is specifically used to evaluate the model’s accuracy and generalization. Performance evaluations using the validation set enable the effective identification and mitigation of model overfitting.

2.3. Improved BEM

The traditional Blade Element Momentum (BEM) theory is predominantly employed with modification methods that take into account the rotation of trailing vortices to model the aerodynamic interactions between the airflow and turbine blades, with the influence of the blades on the airflow via induction factors. Deriving the velocity induction factors ‘a’ and ‘b’ from blade element and momentum theories, other aerodynamic parameters, including the inflow angle and angle of attack, can be calculated. Accurate computation of these parameters is essential for predicting aerodynamic forces in wind turbines. The influences of turbine operating conditions, wind speed, and blade geometry on induction factors are pivotal and must be considered. Therefore, Computational Fluid Dynamics (CFD) is utilized in this study to ascertain precise aerodynamic relationships and data, and neural networks are applied for functional fitting. The induction factors ‘a’ and ‘b’ from the Blade Element Momentum (BEM) theory are refined to the accuracy level obtained from CFD, thereby improving the precision of aerodynamic predictions.

Overall, the enhanced Blade Element Momentum (BEM) theory represents an innovative approach that integrates neural networks with Computational Fluid Dynamics (CFD) to improve prediction accuracy while preserving the computational efficiency inherent to the traditional BEM method. The initial steps of this methodology are illustrated in Figure 5. Geometric parameters of the blade, operational parameters of the turbine, and inflow wind conditions are extracted from the official documentation of the IEA 15 MW reference wind turbine to construct a high-fidelity geometric model. This model is then imported into commercial software for mesh generation. Through CFD simulations, detailed aerodynamic data―such as axial and tangential velocities at all positions within the computational domain―are obtained. Subsequently, the Average Azimuthal Technique (AAT) is employed to extract axial and tangential velocity components at blade locations. Using BEM equations, the axial induction factor (a) and tangential induction factor (b) are calculated based on the CFD-derived flow parameters. The blade geometry, turbine operating conditions, and inflow parameters serve as the independent variables for training the neural network, while the CFD-computed induction factors act as the dependent variables. The neural network optimizes its parameters by minimizing the loss function, iteratively adjusting the weights and biases of each neuron through comparison of predicted and target values to achieve convergent. Ultimately, the neural network serves as a nonlinear curve-fitting tool, forming a “black-box model” that captures the complex relationships between input parameters and induction factors. This model enables the direct prediction of induction factors from seven key input variables, yielding results that closely approximate those of CFD simulations. Consequently, the traditional iterative process in the BEM framework is replaced by a more efficient and accurate computational procedure.

The aerodynamic calculation procedure of the improved Blade Element Momentum (BEM) theory is outlined as follows:

(1) After inputting the influencing factors, the values of the induction factors a and b are obtained;

(2) The inflow angle φ is calculated;

(3) The attack angle α is calculated;

(4) The lift coefficient and drag coefficient of the blade element are obtained according to the airfoil aerodynamic characteristic curve;

(5) The normal force coefficient and tangential force coefficient of the blade element are calculated.

In this pioneering research, the neural network approach is innovatively integrated into the Blade Element Momentum (BEM) theory for the first time, enhancing the prediction accuracy of the velocity induction factor. The primary objective of this study is to rigorously evaluate the feasibility and effectiveness of this novel modification strategy. Future work will aim to further refine the BEM model by potentially incorporating advanced computational fluid dynamics (CFD) techniques, such as the large eddy simulation (LES) method, and more advanced neural network prediction methods, thereby striving for greater precision in aerodynamic modeling.

3. Results and Discussion

This study employs Early Stopping during the initial phases of neural network training with CFD data to monitor the loss function. Training is halted if the loss function fails to show a significant reduction over a set number of iterations, effectively mitigating overfitting. Figure 6 illustrates the trajectory of the loss function throughout the iteration process, indicating that the model is continuously learning and adapting to the data characteristics. Notably, the loss function’s reduction plateaus around 1400 iterations, suggesting that the model has converged on the data’s underlying patterns.

After training, Figure 7 and Figure 8 depict the R² value distributions for the axial induction factor (a) and the tangential induction factor (b). Upon examination of these figures, it is evident that the R² values for both induction factors are very close to 1, demonstrating the model’s accurate capture of the functional relationship between independent variables, namely the influencing factors, and dependent variables, specifically the flow characteristics within the blade tip region.

After the neural network model was fitted, a comprehensive evaluation of the accuracy and generalization capability of the proposed modification method was conducted by validating its performance under both training and non-training operating conditions. This two-pronged verification approach thoroughly assesses the neural network model’s performance across diverse conditions, thereby substantiating its practicality and reliability. In the course of this verification, thrust (T) and torque (M) are designated as the primary parameters for assessment. These parameters are pivotal for quantifying the wind rotor’s aerodynamic properties, with their mathematical formulations detailed in Equations (5) and (6). These equations establish a relationship between thrust, torque, the axial induction factor (a), and the tangential induction factor (b), facilitating a quantitative analysis of the wind rotor’s aerodynamic behavior. It is evident that refining the axial induction factor (a) significantly enhances the precision of thrust calculations, thereby providing a more accurate representation of the wind rotor’s flow characteristics’ influence on thrust generation. Furthermore, torque (M) is influenced not solely by the axial induction factor (a) but also by the tangential induction factor (b). By calibrating factors (a) and (b), the torque estimates are refined, enabling a more precise prediction of the wind rotor’s power output.

Figure 9 presents the thrust correction effect diagram for the Blade Element Momentum (BEM) theory across various inflow conditions, the relative error is calculated as shown in Equation (12). Observations reveal that, for both training conditions at wind speeds of 7.5 m/s and 15 m/s, and a non-training condition at 10 m/s, the thrust calculation error is reduced from 7% to approximately 3% after correcting the velocity induction factors (a and b). Significant correction effects are evident across all scenarios. This indicates a substantial reduction in discrepancies between BEM and CFD results post-correction, underscoring the efficacy of the applied correction method.

$$\mathrm{Relative}\mathrm{error}={\displaystyle \frac{\left|Y\right|}{Y_{RANS}}}$$

(12)

Figure 10 illustrates the torque correction effects for the Blade Element Momentum (BEM) theory under varying inflow conditions. It is observed that the trends in error variation are consistent between non-training and training operational conditions. Namely, after correction of the velocity induction factors (a and b), the torque calculation error is reduced from 11% to approximately 6%. Pronounced correction effects are observed across all operational conditions. Thus, the BEM exhibits a degree of generalizability in both non-training and training scenarios after the correction, indicating the robustness of the correction method applied.

Figure 11 and Figure 12 depict the distributions of the normal and tangential forces for the IEA 15 MW wind turbine at rated wind speeds. The corrected Blade Element Momentum (BEM) model demonstrates robust predictive capabilities for the spanwise distribution of both normal and tangential forces.

$$E_{r,Opt\_BEM}={\displaystyle \frac{\left|F_{n,RANS}-F_{n,Opt\_BEM}\right|}{F_{n,RANS}}}$$

(13)

$$E_{r,BEM}={\displaystyle \frac{\left|F_{n,RANS}-F_{n,BEM}\right|}{F_{n,RANS}}}$$

(14)

Table 4. Error change table of the normal force.

Wind Speed [m/s]	Name of the Errors	Type of Errors	Numerical Value
8 m/s	Er,Opt_BEM	ΔEr,max	6.87%
		ΔEr,min	0.5%
		ΔEr,mean	3.59%
	Er,BEM	ΔEr,max	36.49%
		ΔEr,min	1.92%
		ΔEr,mean	20.32%
10.59 m/s	Er,Opt_BEM	ΔEr,max	9.30%
		ΔEr,min	0.36%
		ΔEr,mean	2.65%
	Er,BEM	ΔEr,max	53.82%
		ΔEr,min	0.80%
		ΔEr,mean	18.21%

and

Table 5. Error change table of tangential force.

Wind Speed [m/s]	Name of the Errors	Type of Errors	Numerical Value
8 m/s	Er,Opt_BEM	ΔEr,max	13.3%
		ΔEr,min	0.83%
		ΔEr,mean	6.33%
	Er,BEM	ΔEr,max	112.06%
		ΔEr,min	6.4%
		ΔEr,mean	53.64%
10.59 m/s	Er,Opt_BEM	ΔEr,max	9.63%
		ΔEr,min	0.54%
		ΔEr,mean	4.55%
	Er,BEM	ΔEr,max	108.43%
		ΔEr,min	2.18%
		ΔEr,mean	53.3%

present the error variations for the normal and tangential forces following the BEM enhancement, and through Equations (13) and (14), the exact value of the error is calculated. Analysis of Table 4 and Table 5 reveals that the enhanced BEM predominantly refines estimates of the normal and tangential forces in the blade tip region, which is a critical area for the aerodynamic performance of wind turbine units.

Figure 13 clearly delineates the discrepancies in torque and thrust magnitudes exerted on the wind rotor’s blade elements following the augmentation of axial and tangential induction factors. In the figure, the values obtained from the optimized Blade Element Momentum theory (Opt_BEM) are represented in red, while the values derived from the traditional Blade Element Momentum theory (BEM) are shown in blue. A variety of symbols are employed to differentiate between distinct incoming wind velocities. It is evident that the induction factor calculations in Opt_BEM differ significantly from those in the conventional BEM approach.

As illustrated in Figure 14, the induction factors predicted by Opt_BEM show substantial deviations from those obtained using conventional BEM. In the traditional BEM method, velocity induction factors are assumed to remain nearly constant along the blade span. However, Opt_BEM―based on Prandtl’s lifting-line aerodynamic theory [24]―provides a more accurate representation of the spanwise distribution of velocity induction factors. According to Prandtl’s theory, pressure differentials near the blade tip induce spanwise flow circulation, giving rise to rotating tip vortices that distort the distributions of axial and tangential velocities. To account for this phenomenon, Prandtl introduced a tip loss factor and developed an integral model to quantify the flow field disturbances caused by tip vortices. This refinement significantly improves the accuracy of aerodynamic performance predictions for propellers. As a result, Opt_BEM is capable of capturing the spanwise variation in aerodynamic parameters with greater fidelity, leading to corrected induction factors that markedly influence the computed thrust and torque along the blade span.

The Optimized Blade Element Momentum theory (Opt_BEM) has substantially advanced the iterative refinement of velocity induction factors within the traditional Blade Element Momentum framework by integrating a sophisticated neural network prediction model. This enhancement elevates the accuracy of Opt_BEM to a level commensurate with that of the Reynolds-Averaged Navier–Stokes (RANS) approach. In contrast, Computational Fluid Dynamics (CFD) excels at precisely modeling the tip vortex structure and radial flow phenomena along the blade span, particularly in the vicinity of the blade tip. Unlike conventional tip loss correction methods such as Prandtl’s, CFD enables a more accurate prediction of the velocity induction factor’s distribution and variations under complex three-dimensional flow conditions. Consequently, this facilitates more precise computations of the differential thrust and torque values at each blade element position, thereby significantly enhancing the overall accuracy of aerodynamic calculations predicated on the Blade Element Momentum theory.

This paper introduces a Multilayer Perceptron (MLP) neural network approach. It utilizes comprehensive flow field data obtained from Computational Fluid Dynamics (CFD) simulations as training inputs, allowing the MLP network to model the intricate functional relationships between aerodynamic parameters and various influencing factors. This approach accounts for the interdependencies among influencing factors and their complex nonlinear interactions with aerodynamics. The enhanced Blade Element Momentum (BEM) framework leverages CFD data to train the neural network, resulting in a predictive model for velocity induction factors. This significantly improves the traditional iterative method for determining induction factors. The traditional assumption of independence of blade elements and the blade surface not having 3D flow in BEM was surpassed by the use of CFD methods, which incorporated centrifugal and Coriolis forces, thereby enhancing the accuracy of flow field simulations and BEM’s aerodynamic force calculations. Furthermore, the CFD application effectively addresses BEM’s limitations in modeling three-dimensional blade characteristics and complex flow in vb interactions, improving the predictive accuracy of wind turbine aerodynamic performance.

4. Conclusions

In this study, a novel comprehensive aerodynamic performance prediction method for wind turbine blades is proposed, which integrates the Computational Fluid Dynamics (CFD) method with neural networks. This approach effectively enhances the accuracy of predictions made by the Blade Element Momentum theory (BEM). When applied to the rotor of the IEA 15 MW steady parallel incoming flow, this method, following BEM correction, significantly reduces the discrepancies between the BEM aerodynamic calculation values and the CFD calculation values under both training and non-training conditions. Specifically, the relative error in thrust calculation is reduced to approximately 3%, while the relative error in torque calculation is diminished to around 6%. Furthermore, the prediction accuracy of the normal force and tangential force distributions along the blade spanwise direction is markedly improved.

Given that this is the first time neural network technology has been applied to improve Blade Element Momentum (BEM) theory, the current study primarily focuses on validating the technical feasibility of this novel approach. Future work will be focused on the adoption of more precise CFD computational techniques or the incorporation of blade elasticity and other ancillary factors into CFD simulations, with the overarching goal of enhancing the fidelity of the training set data. Concurrently, the deployment of more advanced neural network paradigms will be considered to further improve the model’s predictive accuracy and to provide additional validation and refinement from a mechanistic perspective. Moreover, for wind turbine units with ultra-large blades of greater dimensions, the integration of neural networks with data augmentation techniques can be particularly beneficial in ensuring enhanced data sufficiency.