Wind Turbine Power Curves Based on the Weibull Cumulative Distribution Function

applied sciences Article Wind Turbine Power Curves Based on the Weibull Cumulative Distribution Function Neeraj Bokde1,*,† , Andrés Feijóo 2,*,† and Daniel Villanueva 2,† 1 Department of Electronics and Communication Engineering, Visvesvaraya National Institute of Technology, Nagpur 440010, India 2 Departamento de Enxeñería Eléctrica-Universidade de Vigo, Campus de Lagoas-Marcosende, 36310 Vigo, Spain; [email protected] * Correspondence: [email protected] (N.B.); [email protected] (A.F.); Tel.: +91-90-2841-5974 (N.B.) † These authors contributed equally to this work. Received: 6 September 2018; Accepted: 26 September 2018; Published: 28 September 2018 Abstract: The representation of a wind turbine power curve by means of the cumulative distribution function of a Weibull distribution is investigated in this paper, after having observed the similarity between such a function and real WT power curves. The behavior of wind speed is generally accepted to be described by means of Weibull distributions, and this fact allows researchers to know the frequency of the different wind speeds. However, the proposal of this work consists of using these functions in a different way. The goal is to use Weibull functions for representing wind speed against wind power, and due to this, it must be clear that the interpretation is quite different. This way, the resulting functions cannot be considered as Weibull distributions, but only as Weibull functions used for the modeling of WT power curves. A comparison with simulations carried out by assuming logistic functions as power curves is presented. The reason for using logistic functions for this validation is that they are very good approximations, while the reasons for proposing the use of Weibull functions are that they are continuous, simpler than logistic functions and offer similar results. Additionally, an explanation about a software package has been discussed, which makes it easy to obtain Weibull functions for fitting WT power curves. Keywords: wind turbine; power curve; curve fitting; Weibull CDF 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]. Figure1 presented in [ 3] allows researchers to check the variability and uncertainty of such curves. Appl. Sci. 2018, 8, 1757; doi:10.3390/app8101757 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 1757 2 of 18 In most calculation procedures, it is not easy to deal with such a degree of uncertainty, and more deterministic functions are needed. ) ) p − 1 ( g o l − b ( + g ⋅ x o m l = y m = k = y b = −k ⋅ log(C) x = log(v) Figure 1. Representation of Equation (5) in the form of a straight line. 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 [6–8] 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. Finally, there is another reason for using Weibull functions when modeling WT power curves, and that has to do with WF power curves. The consideration that a WF power curve consist of the sum of its WT power curves is very simple. In general, due to some aerodynamic processes such as the wake effect, obtaining a WF power curve from their WT power curves generally requires more complex Appl. Sci. 2018, 8, 1757 3 of 18 operations [13,14]. However, these operations can be simplified by using Weibull functions, as will be explained. In the rest of the paper, the procedure is explained for obtaining WT power curves based on the Weibull CDF. The symbols used in the following sections are shown in Table1. Table 1. Symbols used in the following sections. Symbols Meaning F(v) CDF of wind speed series P a function of wind speed Pmax maximum WT power Pmin minimum WT power Pi calculated power with proposed function s Pi specified power (values given by the manufacturers) k shape parameter C scale parameter v hub height CT the thrustcoefficient Kw the Waledecay constant d the distance between WTs D the rotor diameter of WTs x wake coefficient xw the mean value of coefficient of mWTs 2. Weibull CDF Model The CDF of the Weibull distribution [15,16] is expressed as: −( v )k F(v) = 1 − e C (1) where F(v) represents the CDF of the wind speed series, k is the shape parameter and C is the scale parameter. Following the proposed analogy, the power generated by a WT, P, as a function of the wind speed at hub height, v, can be expressed as: −( v )k P = Pmax · 1 − e C (2) where Pmax is the maximum WT power. The parameters of the Weibull distribution can be obtained from the data given by the manufacturer, by interpreting the power values as frequencies. The procedure for wind speed series against frequencies can be read in [17] and is briefly explained in the following lines. With the double logarithmic action on both sides of Equation (2), it can be operated to be written as: log(− log(1 − P)) = k · log(v) − k · log(C) (3) In order to simplify the process, the values of power, P, can be normalized, i.e., the process is presented in per unit values (p.u.). The range of values of P, [Pmin, Pmax], can be transformed into the interval [0, 1], according to Equation (4). P − P p = min (4) Pmax − Pmin Appl. Sci. 2018, 8, 1757 4 of 18 In the case of wind power curves, Pmin can be assumed to equal zero, because it is the minimum value for the wind, since dealing with negative values does not make any sense. Eventually, Equations (4) and (3) are further modified as shown in Equation (5). log(− log(1 − p)) = k · log(v) − k · log(C) (5) Equation (5) can be represented in the form of a straight line, as illustrated in Figure1: y = m · x + b (6) where y = log(− log(1 − p)), x = log(v), m = k and b = −k · log(C). By solving Equation (5), the parameters k and C can be calculated. With the known values of k and C, the relationship between p and v is given by Equation (7). −( v )k p = 1 − e C (7) which is equal to Equation (1) with the only difference that F(v) has been changed to p.

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