# Wind Turbine Power Curves Based on the Weibull Cumulative Distribution Function
## 1 Introduction  
The development of wind energy has been increasing all over the world during the last few decades, and mainly since the end of last century. Countries like Germany, Denmark, the United States and Spain, first, and others such as India or China, some years later, began to promote wind energy power plants as a part of their electrical power systems. The use of such a clean energy resource can be an incentive for multiple countries to get a high level of energy independence. Additionally, the reduction of greenhouse gas emissions encourages many of them to use renewables and, in particular, wind [1] and solar (photovoltaic) [2] energy. This is especially relevant because of its undoubted ecological aspect. This development goes hand in hand with its corresponding research, and a part of this research is the adequate use of wind turbine (WT) power curves, i.e., curves relating wind speed at hub height and wind power delivered by the WT. The WT power curves are generally provided by WT manufacturers as graphs where their points can be identified by pairs of the mentioned values. They are calibrated under certain requirements given by the International Electrotechnical Commission [3]. Figure 1 presented in [3] allows researchers to check the variability and uncertainty of such curves. In most calculation procedures, it is not easy to deal with such a degree of uncertainty, and more deterministic functions are needed. The WT power curves are used in different calculation algorithms, especially when the users have a certain knowledge about wind speed, and the corresponding power value is needed. An example of this is load flow analysis in those electrical power systems where wind farms (WF) account for an important part of the load. More specifically, in probabilistic load flow (PLF) analysis, with the help of Monte Carlo (MC) simulations, such operations must be done repeatedly. In the literature, there are several proposals for WT power curve models. They have frequently been represented by means of different groups of functions, either piece-wise or continuous [4; 5]. Among continuous ones, logistic functions [6; 7; 8] have been very popular. Continuous representations of power curves seem to be advantageous when they must be included in algorithms with a very high number of operations, where power values have to be deduced from wind speeds. For instance, when MC simulations are needed in PLF analysis, the use of continuous functions can simplify operations [9; 10]. Logistic functions have proven to be very accurate according to simulations validated with the help of power curves given by several manufacturers [7]. Weibull distributions have often been used for representing the behavior of wind speed in a cumulative way. So far, the Weibull distribution is used in applications such as synthetic generation of wind speed time series [11] and prediction of wind power/speed time series [12]. The fact that wind speed in a given location can be cumulatively modeled by means of such a distribution has been generally accepted. The shape of those curves is of interest in this work due to its similarity with WT power curves. This observation induces researchers to think about the possibility of fitting Weibull functions, by means of finding their appropriate parameters, to WT power curves. To the best of the authors' knowledge, this is the first attempt to accommodate the power curve feature into a form of Weibull CDF expressions. Summarizing, it can be said that logistic approaches achieve a high degree of accuracy, although they require several parameters and a certain degree of complexity during the fitting process. A further step consists of simplifying such a process, i.e., of achieving functions that provide a similar degree of accuracy, but easier to deal with. This is the case of functions such as Weibull ones, simpler than the logistic functions, continuous and very accurate, as well. As logistic functions offer very good results, they have been used for the validation process.