# Parametric Analysis Towards the Design of Micro-Scale Wind Turbines: A Machine Learning Approach
## 2 Methodology
### Exploratory Data Analysis  
After the data acquisition phase, the dataset is processed to check the relationship among the different features, clean the data, and check if any transformations are needed. The original dataset after removing all rows with null values at any cell contained 4703 entries, with 20 columns representing the different design parameters. For instance, the chord at each of the 5 blade sections was given individually and so was the twist angle along the blade span. Since the changes in chord length and twist angles are linear, the columns were reduced by making the following changes:  
1. The five values of chord length along the blade span have been replaced by a single value representing the blade taper ratio, following Equation (1). 2. The five columns of twist angle at each section have been reduced to a single column representing the twist angle at the blade tip. \[T.R.=\frac{C_{hub}-C_{tip}}{b}\] (1) where _T.R._ is the taper ratio, \(C_{hub}\) and \(C_{tip}\) are the chord at the blade hub and tip, respectively, and \(b\) is the blade span. The dataset has been also checked for anomalies and outliers. Originally, there were few outliers above the maximum, especially in the tangential force output. This could be due to some errors in the simulation or in the data acquisition. The outliers have been checked by measuring the quantiles of the torque output at 99%, 97%, and 95%. After checking the quantiles, it was enough to remove the highest 1% of the values to remove the outliers. Figure 3 shows a box plot for the tangential force before and after removing the outliers. Figure 2: Airfoils used for the simulations as shown in the Airfoil Design Module. After removing the outliers, the dataset has been reduced to 4415 entries, and column reduction yielded into 9 columns. Where the output parameters are 4 columns, leaving five features to be studied as inputs for each prediction model. The relationship between the different features has been studied for each prediction model for each output measure. It was observed that wind velocity had the highest effect on the blade loads compared to the blade design parameters. On the other hand, blade design parameters had the highest effect on the coefficient of power with minimal effect of the wind velocity, which is expected because the coefficient of power is a design output. Figure 4 shows the correlation heatmap between the numerical features and the outputs. Despite the low direct correlation between features and the desired outputs, there was an indirect effect for each feature on the relationship between outputs and each other. For instance, the correlation between the taper ratio and the blade normal force is \(-0.0053\)  
Figure 4: Pearson correlation between features and outputs. Figure 3: Boxplot of tangential force midspan of the blade: (**a**) before removing the outliers, and (**b**) after removing the outliers. which indicates that the taper angle could be removed to predict the normal force. However, further analyses have shown there is an indirect effect such that the blade taper ratio affects how the blade's normal force is related to the rotor torque. This is shown in Figure 5 where blade normal force is plotted against the rotor torque with different taper angles. It is clear that the highest taper angle has changed the slope of the relation between the rotor torque and the blade's normal force. This could be explained by the reduced blade projected area when the taper ratio increases. There is less area facing the wind, such that the normal force induced by the wind is reduced. This is also clear in the Kernel Density Estimator (KDE) plots that are shown on the upper and right axes of the figure. There is a peak in the blade's normal force caused by the highest taper ratio. For that reason, it was decided not to remove any feature while predicting all four outputs, to avoid missing any hidden correlations between different features.