# Power Curve Modeling of Wind Turbines through Clustering-Based Outlier Elimination +
## 3 Proposed Procedure  
Having long been under investigation in the community of data analysis, outlier detection is a method to identify data points that are significantly different from the majority in a dataset. Outliers may affect the accuracy of data analysis and models, and there exists a variety of different methods for outlier detection. The choice of method depends on the characteristics of the data and the research question at hand. Encountered with excessive inherent variations in wind speed and potential difficulties with data collection of enough time resolution, power curve modeling experiences the lack of detailed information regarding the operation of wind turbines. Under such circumstances, clustering algorithms can be effectively employed. Clustering algorithms are a type of unsupervised machine learning technique used to group similar data points together into clusters. The main goal of clustering algorithms is to find natural groupings in the data without any prior knowledge of the labels or categories that the data points belong to. The most popular applications of clustering include customer segmentation, image classification, and recommendation systems. It can also be used for anomaly or outlier detection by identifying data points that do not belong to any of the existing clusters. These data points can be considered as outliers or anomalies and may indicate unusual behavior, errors, or other anomalies in the data. There are many different types of clustering algorithms, but some of the most commonly used ones include partition- and density-based clustering. The most popular partition- and density-based methods are DBSCAN and K-means, respectively. As a density-based clustering algorithm, DBSCAN can handle arbitrary-shaped clusters and does not require a predefined number of clusters. It groups data points that are close to each other and separates outliers that are far from any cluster by defining a neighborhood around each data point and looking for areas of high density. It can be observed from Figure 2 that data points far from the usual power curves are sparsely distributed and they may most probably be seen as manifest outliers. Thus, the density-based DBSCAN can be effective to rule out these points for power curve modeling. There are two key parameters in DBSCAN: epsilon, which defines the radius of the neighborhood around each data point, and minPts, which sets the minimum number of data points required to form a dense region. Unless the shape of remaining data points after DBSCAN clustering may closely coincide with the usual power curve, the power curve modeling can hardly be effective and it is not intuitive to apply the density-based clustering to remaining data points that are already densely distributed. Instead, partition-based methods such as K-means clustering can be effective to further group densely populated data points. K-means is a centroid-based clustering algorithm that partitions data into a fixed number of K clusters on their distances from a set of K centroids. It works by iteratively assigning each point to the nearest centroid and then updating the centroid based on the mean of the points assigned to it. This process continues until the centroids no longer move significantly or a maximum number of iterations is reached. It is not unusual to use the Euclidian distance for proximity measure [15], but it is pointed out in [16] that Mahalanobis distance is more appropriate for correlated data and it is defined as  
\[d(x,y)=\sqrt{(x-y)^{T}C(x-y)} \tag{1}\]  
where the Mahalanobis distance between \(x\) and \(y\) is denoted by \(d(x,y)\), and \(C\) is the covariance matrix of \(x\) and \(y\). It is noted that the Mahalanobis distance is more appropriate than the usual Euclidian one in clustering for power curve modeling. The effectiveness of power curve modeling depends upon the detection and elimination of outliers, and it is proposed to employ clustering algorithms of DBSCAN and K-means in a sequential manner.