Using Artificial Intelligence to Predict the Aerodynamic Properties of Wind Turbine Profiles

Abstract

This study describes the use of artificial intelligence to predict the aerodynamic properties of wind turbine profiles. The goal was to determine the lift coefficient for an airfoil using its geometry as input. Calculations based on XFoil were taken as a target for the predictions. The lift coefficient for a single case scenario was set as a value to find by training an algorithm. Airfoil geometry data were collected from the UIUC Airfoil Data Site. Geometries in the coordinate format were converted to PARSEC parameters, which became a direct feature for the random forest regression algorithm. The training dataset included 60% of the base dataset records. The rest of the dataset was used to test the model. Five different datasets were tested. The results calculated for the test part of the base dataset were compared with the actual values of the lift coefficients. The developed prediction model obtained a coefficient of determination ranging from 0.83 to 0.87, which is a good prognosis for further research.

1. Introduction

The use of wind energy has increased significantly since 2000, reaching 906 GW of installed capacity in 2022 [1]. As a result, electricity production from wind power has increased significantly and accounted for approximately 7.5% of global electricity production in 2022 [2]. According to data from the IRENA laboratory, global wind energy generation capacity has increased approximately 98 times over the last two decades [3]. In the case of onshore wind turbines, the installed capacity increased from 22 GW in 2001 to 842 GW in 2022. Meanwhile, the installed capacity of offshore wind farms increased from zero in 2002 to 64 GW in 2022 [1].

Newly installed wind turbines are mainly three-blade horizontal axis units (HAWT), with a highest efficiency of approximately 50% [4]. It should be added that in the coming years a very rapid increase in new capacity installed in both onshore and offshore wind farms is expected [1,5].

Despite these optimistic forecasts of an increase in the number of new installations, wind farms are performing worse than producers expected. Results from existing wind farms indicate that capacity factors are often overestimated by 10% to 30%. The average realized value for Europe in 2004–2009 was less than 21%, which reduced expected profits by over 60% and resulted in a 40% lower than expected reduction in CO₂ emissions [6].

One of the key reasons for this is the underestimation of the deterioration of the aerodynamic properties of turbine blades over time, caused by changes in their roughness due to erosion and contamination with foreign bodies, peeling of the coating, icing, as well as the wind energy deficit in the aerodynamic cross-section behind the wind turbines.

The paper in [7] showed that wind turbine blades can accumulate significant leading edge roughness, which adversely affects aerodynamic parameters. The results showed that increasing roughness decreased lift and increased drag force. Similarly, in the work in [8], it was shown that the periodic change in the roughness of the airfoils was caused by the accumulation of dead insects on the leading edge, which caused a drop in electricity production by as much as 25%, despite good wind conditions. This was caused by accelerated transition and separation of boundary layers compared to clean airfoils.

In the work in [9,10], field experiments were carried out to investigate the accumulation of ice on 50 m long turbine blades and the power losses in the electricity production of multi-megawatt wind turbines caused by icing. It was shown that, despite strong winds, icy wind turbines rotated much slower and even shut down frequently. As a consequence, the power loss caused by icing was reduced by up to 80%.

The review paper in [11] summarized and comprehensively analyzed typical types of damage to wind turbine blades, such as trailing edge cracking, lightning strike, leading edge corrosion contamination, icing, and delamination, as well as the mechanisms of their formation. Their work can serve as a guide to analyzing changes in the shape and roughness of aerodynamic profiles during the operation of wind turbines. Based on the information provided by the review paper, in the work in [12], the shapes of icy and eroded airfoil profiles were recreated and used for calculations and estimation of their impact on the generation of aerodynamic forces. It was shown that in the case of ice-covered, dirty, and eroded turbine blades, the power production can decrease by approximately 50%, 16%, and 12.5%, respectively.

The above literature review draws attention to a very important aspect related to permanent or temporary change in the shape of profiles during the use of a wind turbine and the need to take into account their impact on the generation of electricity.

In this context, it is still important to study the aerodynamic properties of the airfoils that make up turbine blades, especially in the context of reducing the impact of roughness changes on their operation, as well as the possibility of developing new aerodynamic shapes with specific properties. Moreover, this may also be a very interesting and important research topic regarding the use of artificial intelligence algorithms.

In recent years, many articles have been published related to the use of artificial intelligence to research turbines and wind farms, especially in the context of the creation of aerodynamic forces by airfoils. In this context, the authors of various works have used various machine learning methods, such as linear regression, support vector, radial basis function, K-nearest neighbors, decision tree, gradient boosting tree, random forest, AdaBoost, mutli-gene genetic programming, and neural networks. In most cases, the use of so many methods was motivated by checking their efficiency and suitability for specific research problems related to the performance of aerodynamic shapes.

In the research presented in [13], an investigation of automatic programming to predict the aerodynamic coefficients and power efficiency of the AH 93-W-145 wind turbine blade at different Reynolds numbers and angles of attack was conducted. Their results showed that the multi gene GP method achieved the highest accuracy. Meanwhile, the article [14] focused on an intensive comparative study on the approximation performance of three artificial intelligence methods: an artificial neural network, radial basis function, and support vector regression. Finally, the support vector regression model was selected and combined with the Monte Carlo simulation method for uncertainty analysis of a wind turbine airfoil. It was shown that this algorithm could capture the uncertainty propagation from the surface roughness to the airfoil aerodynamic performance.

In the work in [15], the Elman neural network method was used to estimate the aerodynamic coefficients of a delta wing. Meanwhile, in the research reported in the article [16], a neural network was used to predict the aerodynamic forces acting on a NACA 2415 airfoil. The results showed a good estimate of the lift force and a slightly poorer estimate of the drag force.

Especially important in the context of the current study is the investigation presented in the paper [17], where five machine learning algorithms were studied in the context of lift and drag force prediction: random forest, gradient boosting regression, decision tree regressor, AdaBoost algorithm, and linear regression. It was shown that the random forest method was superior in its predictive performance. Similarly, the article in [18] examined a Darrieus-type wind turbine with standard and modified S1046 airfoils and found that of four different machine learning models, decision tree, linear regression, K-nearest neighbors, and random forest, the last performed the best.

The above literature review shows that the prediction of airfoil aerodynamic properties is most promising when performed using random forest regression and that in most cases a graphical representation of airfoil shapes was used as input. A possible alternative to the graphical representation of an airfoil may be the use of dedicated numerical parameterization, which unambiguously describes its shape and can be applied to a wide range of airfoils of various types. One example of such parameterization is PARSEC, which defines 11 parameters describing the shape of an aerodynamic profile [19,20].

The main novelty of the current research is the use of PARSEC parameterization together with the random forest regression algorithm to determine the ability to predict the lift force of the aerodynamic shape described by this parameterization. Although the research focused only on the lift force for a given attack angle, this indicates a potential direction in developing a methodology for proposing new aerodynamic shapes with the desired aerodynamic characteristics.

2. Materials and Methods

2.1. Data Collection

In the case of the current research, the investigated aerodynamic properties were limited to lift force coefficient $C_L$. In order to predict the value of $C_L$ of wind turbine blades using the selected artificial intelligence method, it was necessary to appropriately prepare a set of training and test data. The dataset had to be fully labeled, because a supervised learning algorithm was used. The created prediction learning algorithm used specific geometric properties of wind turbine blades.

The airfoil geometries used in the research were downloaded from the UIUC Airfoil Data database [21], which contains source files for hundreds of different types of airfoils. The

Table 1. Airfoil types used in the current study and downloaded from the UIUC airfoil database [21].

Archer	Wortmann
Drela	Gottingen
Andrew	Hollom Quabeck
NASA/AMES/Hicks	I.S.A.
RUTAN CANARD	NACA/Munk
Rotol	ARA-D Marske
Verbitsky	NACA
CLARK	RAE
Davis	RAF
Eppler	NASA/Langley
Eiffel	Selig

shows the airfoil types used in the current study.

Because these files contained geometric coordinates of the airfoils saved in several different formats, it was necessary to adapt them to the target format used in the research.

This standardization was necessary to be able to use files describing the geometry of individual airfoils in an automated way to calculate lift and drag in the XFOIL 6.94 software [22,23] and to parameterize them using the PARSEC method (described below). Finally, the file format contained normalized Cartesian coordinates describing subsequent points of the profile contour, starting from point (1,0), then passing through point (0,0) and ending at the starting point (1,0). An exemplary airfoil shape is shown in the Figure 1.

Figure 2 shows all the airfoils used in the current study, which numbered 688. All airfoils were scaled to have an aerodynamic chord of one. The graphic shows the outline of the airfoils, plotted on top of each other to visualize the geometric diversity that was accounted for by the machine learning algorithm. The airfoils were characterized by various thicknesses, varying degrees of asymmetricity, and curvature. This resulted in PARSEC parameters (see below) having a wide range of values.

2.2. PARSEC Parametrization

To use learning algorithms effectively, it is necessary to reduce the full set of airfoil shape coordinates into a smaller set of parameters that convey the required information about this shape. For this purpose, PARSEC parameterization was adopted in the current research [20,24]. It defines 11 parameters to describe the shape of the aerodynamic profile. A description of the individual PARSEC parameters is presented in Figure 3.

As can be seen in Figure 3 the PARSEC parameterization consisted of values representing the dimensions of the leading edge radius $R_{le}$, the upper crest point $Y_{up}$, the position of upper crest $X_{up}$, the upper crest curvature $Y_{{XX}_{up}}$, the lower crest point $Y_{lo}$, the position of lower crest $X_{lo}$, the lower crest curvature $Y_{{XX}_{lo}}$, the trailing edge direction angle ${\alpha }_{TE}$, the trailing edge wedge angle ${\beta }_{TE}$, the trailing edge thickness $T_{TE}$, and the trailing edge offset $T_{off}$. The PARSEC parameters take into account the airfoil geometry in the two-dimensional plane (shown as XY in Figure 3). It assumes all parameters are constant in the width direction.

It should be added that PARSEC parameterization allows the description of a wide range of airfoils belonging to different types. Additionally, it allows examining the influence of individual parameters that describe a specific geometric feature of the profile, such as its thickness or selected curvatures.

Table 2. Selected statistics of PARSEC parameters of all airfoil profiles used in the study, including minimum, maximum, mean, and median values.

PARSEC	Min	MAX	Median	Mean
$R_{le}$	0.00379	0.17550	0.01348	0.01705
$Y_{up}$	0.17309	0.45714	0.30431	0.30511
$X_{up}$	0.02429	0.18732	0.09080	0.09225
$Y_{{XX}_{up}}$	−4.63757	−0.16621	−0.89782	−1.03904
$Y_{lo}$	0.02566	0.49657	0.28917	0.24503
$X_{lo}$	−0.17842	0.00777	−0.03199	−0.03801
$Y_{{XX}_{lo}}$	0.05867	40.24080	0.82125	2.25580
${\alpha }_{TE}$	−0.32172	0.41028	−0.00609	−0.00847
${\beta }_{TE}$	−0.09119	0.56564	0.19309	0.20546
$T_{TE}$	−0.00361	0.03663	0.00039	0.00160
$T_{off}$	−0.01515	0.04547	0.00098	0.00148

shows the selected statistics of the PARSEC parameters of all airfoil profiles used in the study, including the minimum, maximum, mean, and median values. For many of the parameters considered, a significant dispersion of their values can be observed, which results from the geometric diversity of the selected profiles. As was mentioned earlier, the entire set of studied profiles numbered 688, of which 60% constituted the training set for the machine learning algorithm used.

2.3. Lift Calculation

In the current work, the full-inverse panel method, based on a complex-mapping formulation and mixed-inverse method, implemented in XFOIL software, was used to determine the lift coefficients for a selected set of aerodynamic airfoils. XFOIL is an program for the design and analysis of subsonic isolated airfoils for a wide range of Reynolds numbers. It is an opensource software and is distributed under the GNU general public license. It can be used for viscous and inviscid analysis, airfoil design, and inverse design of airfoils [22].

The inviscid formulation was based on an inviscid linear-vorticity panel method with Karman–Tsien compressibility correction. In the case of viscous flows, source distributions were superimposed on the airfoil and wake to model a viscous layer on the potential flow. Consequently, a two-equation lagged dissipation integral method was used [23].

In XFOIL, laminar and turbulent flows are treated with the en—type amplification formulation to determine the transition point in the boundary layer. Finally, the boundary layer and transition equations are solved simultaneously with the inviscid flow field by a global Newton method.

In the current research, calculations were performed for the Reynolds number $Re=8·{10}^6$ and the angle of attack $\alpha =8^{\circ }$. The choice of this angle of attack and Reynolds number was dictated by the values of these quantities for wind turbines with a rated power of several MW. For example, for the reference wind turbine developed in [25], with a rated power of 5 MW, for rated conditions, the Reynolds number ranged from 7.14 to 9.83 million, while the angle of attack ranged from 5 to 10 degrees.

In the case of the aerodynamic profiles, the Reynolds number was defined as follows:

$$Re=\frac{Uc}{\nu }$$

(1)

where U is the average free-stream velocity, c is a chord line of an airfoil, and $\nu$ is kinematic viscosity. Figure 4 shows exemplary calculation results showing the distribution of the pressure coefficient $C_p$ for the FX 84-W-127 airfoil.

2.4. Random Forest Regression

The current research used a machine learning method based on the random forest regression (RFR) algorithm. This is one of the more popular methods and, as shown in the introduction, it gives good results in the case of aerodynamic problems.

The created computational algorithm was based on the Python scikit-learn library [26]. The RFR algorithm ran in parallel on multiple decision trees without interaction between them and generated a forecast as the average of all trees. As mentioned earlier, the current research used 688 aerodynamic profiles, which were divided into a training set and a test set for the RFR algorithm. Each aerodynamic profile was represented by 11 PARSEC parameters, detailing its shape and the value of the lift coefficient $C_L$. A total of 12 parameters constituted a single input record of the considered database. PARSEC parameters were used as features, which were independent variables that constituted the input to the machine learning algorithm. On the other hand, the calculated lift coefficient was the target value predicted by the algorithm.

The default settings for the RFR algorithm were used in the calculations. Only one parameter was passed to the scikit-learn function. That parameter was the random_state, and it controlled the randomness of the bootstrapping of the samples used when building the decision trees. It was also responsible for the sampling of the features to consider when looking for the best split at each node.

3. Results and Discussion

The random forest regressor method solves regression tasks by minimizing the mean squared error formulated as

$$MSE=\frac{1}{N}\sum _{i=1}^N{\left(C_{L,i}^{RFR}-C_{L,i}^{XFOIL}\right)}^2$$

(2)

where N is the number of data points, $C_{L,i}^{RFR}$ is the value of $C_L$ predicted by the RFR algorithm, and $C_{L,i}^{XFOIL}$ is the actual value of $C_L$ (calculated by the XFOIL software).

In order to check the predictive ability of the used RFR model, the considered dataset was divided into two subsets, the first for training purposes, the second for testing. The training set had 60% and the test set had 40% of the records of the base dataset.

Table 3. Comparison of the real values and values predicted by the RFR algorithm of the lift force coefficient $C_L$ for five exemplary airfoils.

	No. 1	No. 2	No. 3	No. 4	No. 5	No. 6
real $C_L$	1.2545	1.0559	1.1098	1.0941	1.2309	1.2209
predicted $C_L$	1.2543	1.0556	1.1091	1.0933	1.2319	1.2198
Abs Error	0.0002	0.0003	0.0007	0.0008	0.0010	0.0011
Rel Error	0.0001	0.0003	0.0007	0.0007	0.0008	0.0009

presents sample results of the comparison of the actual values and the values predicted by the RFR algorithm of the lift coefficient $C_L$ of airfoils described by the PARSEC parameters presented in Table 2. Randomly selected sample results from Table 3 indicate that the method used has the potential to produce satisfactory outcomes. It should be noted that both the relative error, marked in the table as Rel Error, varied in a range from 0.0001 to 0.0009, while the absolute error Abs Error varied in a range from 0.0002 to 0.0011. Both of these readings were at a satisfactory level.

In this work, the considered predictive RFR algorithm was used to run five different cases with airfoils randomly assigned to the training and testing datasets, assuming 60% and 40% to the corresponding sets. The prediction results of the RFR model for each case are presented in Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. In the following figures, the results are arranged in ascending order according to the values of the lift coefficients for the considered airfoils. Attention should be paid to the significant range of lift force coefficients, from approximately 0.6 to almost 2.2, and the non-linear character of the curve. This means that the profiles considered in the study were characterized by a considerably large variety of shapes and, consequently, their usage. It can be seen that the prediction algorithm performed best in the linear part of the curve and worst in the area of low $C_L$ values, which is particularly visible in Figure 5.

Summarizing the results from Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, it can be seen that the choice of aerodynamic profiles for the training set had a certain impact on the quality of the results obtained. For example, comparing Figure 6 and Figure 7 with Figure 5 and Figure 8, one can notice a difference in the quality of predicting the $C_L$ value within large values of this coefficient. This may mean that in order to reduce the sensitivity of the model to the selection of elements of the training set, the absolute number of airfoils in the training set should be further increased.

In order to check the predictive power of the model, the coefficient of determination $R^2$, was defined as

$$R^2=\frac{{\sum }_{i=1}^N{\left(C_{L,i}^{XFOIL}-\overline{C_{L,i}^{RFR}}\right)}^2}{{\sum }_{i=1}^N{\left(C_{L,i}^{RFR}-\overline{C_{L,i}^{RFR}}\right)}^2}$$

(3)

where $\overline{C_{L,i}^{RFR}}$ is the arithmetic mean of the predicted values of $C_L$ by the RFR algorithm (for explanation of the other symbols see Equation (2)).

The values of $R^2$ calculated using Equation (3) for each considered case are presented in

Table 4. Coefficients of determination $R^2$ for the considered calculations with randomly assigned airfoils to the training and test sets.

Test No.	${\mathit{R}}^2$	Corresponding Figure
1	0.85	Figure 5
2	0.84	Figure 6
3	0.87	Figure 7
4	0.85	Figure 8
5	0.83	Figure 9

. It can be seen that the value of the $R^2$ coefficient was approximately 0.85. The obtained $R^2$ values agreed with the values reported in [17], which ranged from 0.944 to 0.863, depending on the ratio of the size of the training set and the test set (a train/test ratio ranging from 0.2 to 0.8 was tested). It should be added that, in the work [17], the aerodynamic airfoils were represented graphically and not as a set of parameters as in the current work, and that the train/test ratio in the current work was 1.5.

On this basis, it can be concluded that the RFR machine learning method coped satisfactorily with predicting the lift force, even though the training set consisted of airfoils with very different geometries and was represented by 11 independent PARSEC parameters.

It should be emphasized that the methodology used was relatively simple, and yet it produced promising results. Figure 10 shows that about 60% of predictions had a relative error below 5%, and more than 85% had a relative error below 10%. In Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, it can be seen that the results with the worst predictability occurred quite sporadically and are randomly distributed in the middle part of the presented curves. As was mentioned before, the few cases with higher relative errors may have resulted from the large diversity of the geometries of the considered airfoils, which, despite a relatively large dataset, could have led to a situation where some characteristic geometric features were not represented an appropriate number of times. With this in mind, it can be concluded that the dataset should be selected so that each PARSEC parameter is represented an appropriate number of times in an appropriately wide range.

To find which of the PARSEC parameters could potentially have the greatest impact on the quality of prediction, selected statistics from 5% of profiles with the largest relative prediction error and 5% of profiles with the smallest prediction error were compared. These values of the statistics are listed in

Table 5. Selected PARSEC parameter statistics for the 5% of airfoils for which the worst predictions were obtained.

PARSEC	Min	Max	Median	Mean
$R_{le}$	0.0110	0.0814	0.0146	0.0245
$Y_{up}$	0.1909	0.3848	0.2774	0.2868
$X_{up}$	0.0567	0.1784	0.1148	0.1142
$Y_{{XX}_{up}}$	−3.6830	−0.5035	−1.4239	−1.5649
$Y_{lo}$	0.1067	0.4805	0.3706	0.3491
$X_{lo}$	−0.1784	−0.0081	−0.0433	−0.0569
$Y_{{XX}_{lo}}$	0.2393	2.0044	1.0944	1.0723
${\alpha }_{TE}$	−0.2365	0.3767	0.0258	0.0476
${\beta }_{TE}$	0.0456	0.5587	0.2631	0.2828
$T_{TE}$	−0.0012	0.0366	0.0011	0.0053
$T_{off}$	−0.0145	0.0455	0.0009	0.0040

and

Table 6. Selected PARSEC parameter statistics for the 5% of airfoils for which the best predictions were obtained.

PARSEC	Min	Max	Median	Mean
$R_{le}$	0.0064	0.0331	0.0130	0.0164
$Y_{up}$	0.2224	0.4039	0.3336	0.3191
$X_{up}$	0.0576	0.1373	0.0914	0.0970
$Y_{{XX}_{up}}$	−1.9614	−0.2533	−0.9504	−1.0343
$Y_{lo}$	0.0451	0.4947	0.1688	0.2095
$X_{lo}$	−0.0758	0.0010	−0.0326	−0.0355
$Y_{{XX}_{lo}}$	0.1497	8.7170	0.6225	2.2716
${\alpha }_{TE}$	−0.1228	0.1315	−0.0449	−0.0386
${\beta }_{TE}$	0.0365	0.3628	0.1677	0.1864
$T_{TE}$	−0.0008	0.0067	0.0001	0.0007
$T_{off}$	0.0002	0.0094	0.0007	0.0016

, respectively.

Table 7. Influence of PARSEC parameters on the prediction quality calculated using Equation (4).

PARSEC	${\mathit{S}}_{\mathit{rel}}^{\mathit{min}}$	${\mathit{S}}_{\mathit{rel}}^{\mathit{max}}$	${\mathit{S}}_{\mathit{rel}}^{\mathit{median}}$	${\mathit{S}}_{\mathit{rel}}^{\mathit{mean}}$
$R_{le}$	0.424	0.593	0.113	0.332
$Y_{up}$	0.165	0.050	0.203	0.113
$X_{up}$	0.016	0.230	0.204	0.151
$Y_{{XX}_{up}}$	0.467	0.497	0.333	0.339
$Y_{lo}$	0.577	0.030	0.545	0.400
$X_{lo}$	0.575	1.122	0.248	0.376
$Y_{{XX}_{lo}}$	0.375	3.349	0.431	1.118
${\alpha }_{TE}$	0.481	0.651	2.737	1.810
${\beta }_{TE}$	0.199	0.351	0.363	0.341
$T_{TE}$	0.298	0.817	0.933	0.871
$T_{off}$	1.011	0.794	0.256	0.615

shows the relative comparison of the selected statistics of the 5% worst and 5% best predictions calculated according to the following formulas:

$$\begin{array}{cc}\hfill S_{rel}^{min}& =\frac{P_{best}^{min,i}-P_{worst}^{min,i}}{P_{worst}^{min,i}}\hfill \\ \hfill S_{rel}^{max}& =\frac{P_{best}^{max,i}-P_{worst}^{max,i}}{P_{worst}^{max,i}}\hfill \\ \hfill S_{rel}^{median}& =\frac{P_{best}^{median,i}-P_{worst}^{median,i}}{P_{worst}^{median,i}}\hfill \\ \hfill S_{rel}^{mean}& =\frac{P_{best}^{mean,i}-P_{worst}^{mean,i}}{P_{worst}^{mean,i}}\hfill \end{array}$$

(4)

where $P_{worst}^{<·>,i}=(R_{le},Y_{up},X_{up},Y_{{XX}_{up}},Y_{lo},X_{lo},Y_{{XX}_{lo}},{\alpha }_{TE},{\beta }_{TE},T_{TE},T_{off})$ is the value of individual PARSEC parameters from Table 5 for a given statistic: min, max, median, mean, and $P_{best}^{<·>,i}$ is the value of the corresponding PARSEC parameter from Table 6. Note that Equation (4) is in the form of a relative error formula, which can be interpreted that the largest values correspond to the greatest impact of a given PARSEC parameter on the prediction accuracy. A particularly influential PARSEC parameter for the prediction quality is one for which all the values of $S_{rel}^{min},S_{rel}^{max},S_{rel}^{median},S_{rel}^{mean}$ are simultaneously relatively large. One such parameters is ${\alpha }_{TE}$, which is responsible for the thickness of the trailing edge of the airfoil, see Figure 3.

On the contrary, the PARSEC parameters having the least impact on the quality of predictability by the RFR algorithm are those for which all values $S_{rel}^{min},S_{rel}^{max},S_{rel}^{median},S_{rel}^{mean}$ are simultaneously relatively small. Analyzing the data collected in Table 7, such parameters include the pairs of $Y_{up}$ and $X_{up}$, and $Y_{lo}$ and $X_{lo}$, which correspond to the location of the curvature of the lower and upper crest of the airfoil, see Figure 3.

4. Conclusions

This work used a machine learning algorithm based on the random forest regression method to predict the lift force of aerodynamic airfoils. In the study, 688 airfoils belonging to different families and characterized by very diverse geometries were used as a dataset. The PARSEC parameterization was used to describe the shape of the airfoils, resulting in 11 independent parameters that were used as input to the machine learning algorithm.

A series of calculations were performed for the selected Reynolds number and angle of attack. Despite very diverse geometries, the prediction results turned out to be satisfactory, giving $R^2$ in the range between 0.83 and 0.87. This means that the random forest regression method can be used for these types of calculations and predictions.

It can be concluded that the presented research could be successfully extended to a wide range of Reynolds numbers and angles of attack, and used to develop a methodology based on the reverse engineering principle, in which one could potentially search for a wind turbine or a wing airfoil shape that gives specific aerodynamic properties.

The future direction of research related to the presented methodology should include calculations for a wide range of angle of attacks (above aerodynamic stall), both for lift and drag forces, and for a range of Reynolds numbers taking into account the flow regime of the planned application.

Possible applications of the developed method could concern both the optimization of the aerodynamic profiles of wind turbines as well as the profiles of wings or propellers of flying vehicles. In the case of wind turbines, care should be taken to ensure that the set of profiles contains a broad representation of the families of aerodynamic profiles used for wind turbine blades.

The main limitation of the presented research is the presentation of the operation of the random forest regression algorithm only for one selected Reynolds number and angle of attack. Therefore, there is no guarantee that the algorithm would provide consistent and appropriate results if the training set were expanded to include a wide range of Reynolds numbers and angles of attack. Nevertheless, based on the results obtained in another work [17], where a graphical representation of profiles was used, which seems more challenging than using PARSEC parameters, satisfactory predictions were obtained for a wide range of angles of attack. Therefore, this can be treated as a good prognosis for the methodology presented in the current research.

Another limitation is related more to the use of PARSEC parameterization, as there is no guarantee that all possible airfoil shapes, both existing and proposed, can be described using PARSEC.