# Power Curve Modeling of Wind Turbines through Clustering-Based Outlier Elimination +
## 4 Case Study  
The proposed procedure outline in the previous section is now to be demonstrated with the case study of Hankyung wind farm in the Republic of Korea, which is consisted of three 1.5 MW and five 3.0 MW turbines. Data from one of the 1.5 MW turbines are taken over the period of 2018-2020 as shown in Figure 2a. The turbine is manufactured by VESTAS and its cut-in and cut-off speeds are to be 4 m/s and 25 m/s, respectively. Unlike outliers of Type II and III, the cause of Type I outliers is relatively clear and it is intuitively obvious to take these outliers out of the dataset. Removed in [Step 1] are 59 data points that no power is generated at the wind speed higher than the cut-in speed of 4.0 m/s in this study. Potentially influential outliers are now to be screened using clustering algorithms. It is widely recognized that density-based clustering techniques are more efficient than partition-based techniques when it comes to arbitrary shaped clusters or anomaly detection [17]. As an unsupervised learning method, the DBSCAN algorithm is first employed with epsilon = 0.15 and minPts = 5, which are chosen based on testing different combinations of parameters, in (2-1) of [Step 2]. Based on these two parameters, data points are categorized as core point, border point, or outlier, and those corresponding to outlier may be removed [15; 18]. Figure 4 depicts the results from DBSCAN, where black dots represent data points classified as outlier. It can be observed that there are still a multitude of points classified as border point which seems influential but irrelevant in modeling the power curve.  
The centroid-based algorithm of K-means clustering is then applied to identify those remaining influential outliers in (2-2) of [Step 2], and it is noted that the Mahalanobis distance instead of the usual Euclidian distance is used to reflect the correlation between wind speed and power output [10]. Figure 5 depicts the formation of clusters resulting from K-means clustering. It can be observed that clusters 3, 6, and 8 are corresponding to Type I and III outliers which may lead to a poor goodness-of-fit. The resultant data points from [Step 2] depicted in Figure 6 are to be used for modeling power curve of the 1.5 MW wind turbine. The procedure is also applied to data from the 3.0 MW wind turbine in the site, of which result is depicted in Figure 7.  
Figure 4: Outliers from DBSCAN Clustering.  
Several parametric models are frequently adopted to fit the power curve, and the goodness-of-fit of polynomial, Gompertz, logistics, and Weibull functions, are to be tested in [Step 3] with the performance metrics of RMSE, MAE, and \(R^{2}\). The results of goodness-of-fit analysis are summarized in Table 1 with 4 parameters for different parametric models to ensure the same degrees of freedom. It is indicated that the data from the 1.5 MW turbine are well fitted with the Weibull function by observing the absolute measure of \(R^{2}\) and the relative measure of MAE while the polynomial function fits the data better with respect to the measure of RMSE. On the other hand, the data for the 3.0 MW turbine are best fitted with Gompertz function regardless of measures. Figure 8 depicts different parametric functions fitted with the refined data shown in Figures 6 and 7.  
Figure 5: Results from K-Means Clustering.  
Figure 6: Refined Data for Power Curve Modeling of 1.5 MW Turbine.  
Figure 8: Fitted Functions of Power Curve with Refined Data. (**a**) 1.5 MW Wind Turbine; (**b**) 3.0 MW Wind Turbine.  
Figure 7: Refined Data for Power Curve Modeling of 3.0 MW Turbine.