# Parametric Analysis Towards the Design of Micro-Scale Wind Turbines: A Machine Learning Approach
## 3 Results and Discussion
### Normal Force  
The same eleven models were used to predict the desired output, but this time, it is the normal force midspan of the blade. This value is explicitly chosen because it could be used to calculate the mean flapwise bending moment at the blade root, which is an important structural consideration. The dataset is split again into 70% for training and 30% for testing the models, but with choosing the blade normal force as the output to be predicted. Table 3 shows the results of the different ML models in predicting the normal force. For the normal force prediction, the Decision Trees Regressor has exceeded the accuracy of the Random Forest Regressor slightly by 0.004%. Cross-validation by repeating the train-test sets splitting many times could change the results. This was performed many times, and the results came close each time, where Random Forest and Decision Trees exchanged roles at the top performance. The maximum accuracy of Decision Trees in this \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**ML Model** & **R2** & **RMSE** & **Max Error** & **MAE** & **Training Time (s)** \\ \hline Decision Tree & 0.984 & 0.159 & 2.177 & 0.040 & 0.000 \\ Regression & 0.984 & 0.160 & 2.003 & 0.061 & 0.328 \\ Gradient Boosting & 0.892 & 0.418 & 4.819 & 0.237 & 0.125 \\ Regression & 0.862 & 0.473 & 5.182 & 0.263 & 0.125 \\ Neural Network & 0.856 & 0.483 & 5.437 & 0.273 & 0.000 \\ Nonlinear Regression & 0.818 & 0.543 & 6.159 & 0.333 & 0.016 \\ Ridge Regression & 0.817 & 0.544 & 6.160 & 0.334 & 0.000 \\ Linear Regression & 0.816 & 0.545 & 6.212 & 0.334 & 0.000 \\ Elastic Net Regression & 0.815 & 0.547 & 5.793 & 0.259 & 2.094 \\ Support Vector & 0.814 & 0.548 & 6.240 & 0.336 & 0.000 \\ Regression & 0.793 & 0.580 & 3.174 & 0.384 & 0.172 \\ Regression & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Machine learning models results for the prediction of normal force. Figure 7: Random Forest Regressor results for power coefficient: (**a**) the actual vs. predicted values, and (**b**) residuals distribution. run is 98.7%, which is also very satisfactory in predicting dynamic loads on the blade, which can be transferred to dynamic loads on the rotor hub and thus the supporting tower. Another key observation is that the linear models (Linear, Ridge, Elastic Net, and Lasso) could predict the normal force with an accuracy of around 75%. While this accuracy is not high as compared to the ensemble methods, it could be used in preliminary structural design stages because of its simplicity, and since the structural loads are usually multiplied by a factor of safety to account for errors in calculations. The linear regression model output is a first-order equation where the output can be calculated based on the linear relationship with different features. The output equation of the Linear Regression model for predicting the normal force is shown in Equation (8). \[ \begin{array}{l}F_{N}=2.97-2.4\times 10^{-2}\theta+32.6\times 10^{-2}V_{w}-0.75 \times 10^{-2}\beta-\\ 91.33\times 10^{-2}\ T.R.-22.09\times 10^{-2}\ C_{max,P}-20.23\times 10^{-2}\ C_{ max,pos}-\\ 15.51\times 10^{-2}T_{max,P}\end{array}  \tag{8}\]  
where \(\theta\) is the twist angle, \(V_{w}\) is the wind velocity, \(\beta\) is the pitch angle, \(T.R.\) is the taper ratio, \(C_{max,P}\) is the maximum camber percent, \(C_{max,pos}\) is the maximum camber position, and \(T_{max,P}\) is the maximum thickness percent. However, linear models can be used only for preliminary design stages when the loads are multiplied by a factor of safety. However, for more advanced prediction of the loads, ensemble models are the most recommended. The results of Decision Tree predictions for the normal force midspan of the blade are shown in Figure 8, where the actual output is plotted against the predicted value, and the density distribution of the residuals is shown. The predicted values are mostly aligned with the 45\({}^{\circ}\) line, representing the accuracy of prediction of the Decision Trees regressor.