# Wind Turbine Power Curve Modelling with Logistic Functions Based on Quantile Regression
## 1 Introduction  
The wind turbine power curve (WTPC) is defined as the relationship between electrical power output and hub height wind speed of a wind turbine [1], and it is important for energy assessment, wind power forecasting and condition monitoring [2]. As mentioned in [3], the manufacturer provides a design power curve to describe the characteristics of wind turbine power generation. However, affected by the variability of the local environment and the adjustment of wind turbine internal parameters, the design power curve is unable to meet the requirements of wind farm operators. To enhance the fitting accuracy, many published literatures used supervisory control and data acquisition (SCADA) data to establish data-driven WTPC models, which are generally divided into parametric methods and nonparametric methods [4]. Parametric methods are based on solving mathematical expressions, including determination of expressions and parameter estimation. According to [5], linearized segmented model has been widely used in practical production. In [4, 6], polynomial regression (PR) with different orders was used for WTPC modelling, and the results show that 6 th-order and 9 th-order PR have better fitting accuracy. In addition to PR, exponential functions,hyperbolic tangent functions and power coefficient methods have all been applied for WTPC modelling [7; 8]. In recent years, logistic functions (LF) have been utilized for WTPC fitting because of their continuity and good nonlinear mapping ability [9; 10; 11; 12]. Reference [9] made comparative studies of LF with three to six model parameters, and the results show that 5-parameter logistic function (5 PL) generally has the best fitting effect. Nonparametric methods do not impose any prespecified mathematical expressions, and the modelling process is entirely based on the observed data. Reference [13] proved that cubic spline interpolation (CSI) can fit smooth and accurate power curves. Reference [14] divided the raw dataset into 10 phases according to K-means clustering, and for each phase, smoothing spline was utilized as the fitting function. With the development of statistics and computer science, machine learning techniques such as neural networks (NN) [15], adaptive neural-fuzzy inference systems (ANIFS) [16] and K-Nearest Neighbors (KNN) [17] have been gradually applied for WTPC modelling. Most of the aforementioned methods belong to deterministic WTPC models, which only describe the relationship between wind speed and power average but cannot reflect the uncertainty of wind power. Accordingly, probabilistic WTPC models were developed to reveal the variation and uncertainty of the power generation process, and improve the reliability for WTPC based applications, such as energy assessment and condition monitoring [18]. According to [19; 20; 21; 22; 23], the Gaussian process (GP) is the widely used method for probabilistic WTPC modelling, which can qualify the uncertainty of power generation via the predicted confidence intervals (CI). To reduce the computation cost, [20] used Cholesky decomposition to solve the inverse matrix in GP. Reference [21] proposed a heteroscedastic GP to enhance the interval predictions. Reference [22] combined GP with LF and proposed a semi-parametric model for probabilistic WTPC modelling. Reference [23] indicated that adding pitch angle and rotor to the model inputs of GP can improve the fitting accuracy. In [24], a multivariate WTPC was established by using support vector machine (SVM) with Gaussian kernel. Reference [25] combined SVM with pointwise CI and simultaneous CI to estimate the uncertainty of wind power. The Monte Carlo algorithm [18], Copula function [26] and relevance vector machine (RVM) [27] were also applied for probabilistic WTPC modelling. The key challenges of most probabilistic WTPC models lies in the assumption that the wind power follows a specific prior distribution during the training process.