# Methods to Apply a 3-Parameter Logistic Model to Wind Turbine Data
## 1 Introduction  
Wind turbine performance is usually described by means of the relationship between output power and wind speed. The manufacturers provide these curves, or alternatively a set of points representing pairs of wind speeds and powers, after testing the wind turbine according to the IEC (International Electrotechnical Commission) Standard procedure (IEC 61400-12) [1]. The process to obtain such curves is based on the definition of bins of 0.5 m/s, from where pairs of data are taken (wind speed, output power). The Standard states that at least 180 min of total data must be taken and, also, at least 30 min of data per bin must be included in the test. Besides, turbulence is a very important circumstance to be taken into account when obtaining the power curve because the output power of a wind turbine is also influenced by it. Moreover, some specific corrections must be included in the test in order to reach the proper power curve as explained in [2; 3]. These corrections are not considered in this research and may be considered as a future work. The information described in the previous paragraph is enough to analyze the behavior of a wind turbine under normal operating conditions and the results provided can be considered valid when estimating or assessing its performance in an aggregate way. However, in the case of installed wind turbines, the working conditions may be very different from the expected ones: weather [4; 5; 6], terrain conditions [7; 8; 9] and aging [10; 11; 12; 13] may affect their performance. In the latter case, only real data can provide an accurate information of the wind turbine behavior. In a real case, discarding wrong or out of range data is critical. Errors in the measurement process or even in the wind turbine performance cannot be considered when trying to model or simulate its real behavior. Therefore, real data filtering before analysis is needed [14; 15; 16]. Filtered data can be analyzed in order to model or simulate wind turbine behavior [17]. As there is a wide variability in the real data [18], and the objective is to have accurate information of the installed wind turbine, a possible option is to use the simplest model for the power curve with valid results. Linear [19], quadratic [20] and cubic [21] models are discarded due to their inaccuracy. Splines [22] are very complex due to the high number of parameters and the lack of a unique expression due to the piecewise definition. Another option might be to use the sigmoid functions ('S' shape functions) that combine a low number of parameters with a single expression which makes them suitable to be implemented in direct calculations and in compounded expressions. Among them, the logistic-type functions have been successful in wind power curve modeling. Logistic functions were introduced as models of population growth and nowadays are used in statistics, medicine, linguistics, agriculture, economics or sociology [23; 24; 25; 26]. Lately, they were introduced to model wind turbine power curves with success due to their simplicity, diversity and versatility. Among them, the so called 3 PLE (Three parameter logistic exponential) was chosen as the most suitable one because it only needs three parameters and its error level is very low [27; 28]. In spite of all the above, there is not just a single way to apply a specific model to wind turbine data analysis. From the filtered data to the model itself there are several options to proceed. The simplest method to apply the model may be to define a number of intervals for the wind speed, to identify each interval by the mean value and to assign the mean output power as the corresponding value for each wind speed identifier. With these pairs of values, i.e., points in a graph, the model can be defined as the function that best represents the points, by using an optimization process for an objective function which is the error made [29; 30; 31].