Wind Turbine Power Curves Based on the Weibull Cumulative Distribution Function

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.

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.

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 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

Table 1. Symbols used in the following sections.

Symbols	Meaning
$F\left(v\right)$	CDF of wind speed series
P	a function of wind speed
$P_{max}$	maximum WT power
$P_{min}$	minimum WT power
$P_i$	calculated power with proposed function
${P_i}^s$	specified power (values given by the manufacturers)
k	shape parameter
C	scale parameter
v	hub height
$C_T$	the thrustcoefficient
$K_w$	the Waledecay constant
d	the distance between WTs
D	the rotor diameter of WTs
$\xi$	wake coefficient
${\xi }_w$	the mean value of coefficient of mWTs

.

2. Weibull CDF Model

The CDF of the Weibull distribution [15,16] is expressed as:

$$F\left(v\right)=1-e^{-}{}^{{\left(\frac{v}{C}\right)}^k}$$

(1)

where $F\left(v\right)$ 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:

$$P=P_{max}·\left(1-e^{-}{}^{{\left(\frac{v}{C}\right)}^k}\right)$$

(2)

where $P_{max}$ 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:

$$\log (-\log (1-P\left)\right)=k·\log \left(v\right)-k·\log \left(C\right)$$

(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, $[P_{min},P_{max}]$, can be transformed into the interval $[0,1]$, according to Equation (4).

$$p=\frac{P-P_{min}}{P_{max}-P_{min}}$$

(4)

In the case of wind power curves, $P_{min}$ 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).

$$\log (-\log (1-p\left)\right)=k·\log \left(v\right)-k·\log \left(C\right)$$

(5)

Equation (5) can be represented in the form of a straight line, as illustrated in Figure 1:

$$y=m·x+b$$

(6)

where $y=\log (-\log (1-p\left)\right)$, $x=\log \left(v\right)$, $m=k$ and $b=-k·\log \left(C\right)$.

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).

$$p=1-e^{-}{}^{{\left(\frac{v}{C}\right)}^k}$$

(7)

which is equal to Equation (1) with the only difference that $F\left(v\right)$ has been changed to p.

Finally, the wind power values can be reconstructed within their actual range with the denormalization process shown in Equation (2), and this constitutes the final proposal for the power curve expression.

3. Wake Analysis

The procedure presented in Section 2 allows researchers to get a very good approximation for representing WT power curves. It is simple, and it is a given by a continuous function; this means it is easy to implement in algorithms that require operating with such data. However, the most common case of calculations consists of having to operate with WF power curves instead of WT power curves.

A well-known model for calculation of WF power curves is proposed in [13], based on the Jensen model. In [18,19], it is discussed how this model is useful in wake effect simulations in WF power curves. Li et al. [13] proposed a model for a WF power curve by accumulating all WTs in the farm. This WF power is equivalent to the total power generated by m rows, n column layout and d spacing of WTs between rows and columns in the WF, although the wind direction parameter is not considered in the model. Assuming these conditions and considering WT thrust coefficient $C_T$, the Wale decay constant $K_w$, the distance between WTs d and the rotor diameter of WTs D, the wake coefficient ($\xi$) can be calculated with: (8) as discussed in [18].

$$\xi =1-\frac{1-\sqrt{(1-C_T)}}{{(1+2k_w\frac{d}{D})}^2}$$

(8)

In the mentioned paper [13], Li et al. proposed a model in which this wake coefficient is used to estimate the total power generated in the WF. A similar operation to that given by (2) permits calculating the total wind power produced in the WF with (9).

$$P_{WF}=nm{P}_{max}·\left(1-e^{-}{}^{{\left(\frac{{\xi }_{w}v}{C}\right)}^k}\right)$$

(9)

where:

$${\xi }_w=\frac{1-{\xi }^m}{m(1-\xi )}$$

(10)

${\xi }_w$ is the mean value of the coefficient of mWTs in a column, and the mean values of the sequence are $1,\xi ,{\xi }^2,{\xi }^3,\dots ,{\xi }^{m-1}$, as shown in Equation (10). Similar to the [13] and [20] models, the power generated by any of the WTs can be calculated with Equation (2), when v is replaced by ${\xi }_{w}v$, where ${\xi }_w$ is the wake effect coefficient experienced by corresponding WT.

According to the information given in this section and in Section 2, a case study is presented in Section 4.

4. Case Study

The same WTs as in [21] have been simulated with the proposed Weibull CDF model. The performance of 4P-OP, 4P-DP, 4P-DS and 3P-DPalong with other piecewise models was already presented in [7]. In the case study proposed here, shape and scale parameters have been deduced for all WTs, with the constraint that they must satisfy Equation (5). The values obtained have been tabulated in

Table 2. Values of parameters ‘k’ and ‘C’ for several WTs using the proposed model.

Model	Vestas V80	Siemens 82	REpower 82	Nordex N90	Siemens 107	Vestas V164
k	4.6256	4.4233	4.5242	4.2424	4.5699	4.5241
C	9.3890	10.2096	9.7374	9.5649	9.8116	9.5458

. Furthermore, a comparison of errors made by means of the proposed model with those made by using the other models is expressed in terms of root mean square error (RMSE) and mean absolute error (MAE) in

Table 3. Comparison of RMSE values for several WTs.

Model	Vestas V80	Siemens 82	REpower 82	Nordex N90	Siemens 107	Vestas V164
4P-OP	25.3319	24.5692	21.2895	39.1501	42.6797	96.6218
4P-DP	56.2597	59.9937	38.1073	68.3914	85.3121	146.1193
4P-DS	55.4600	54.6059	35.5069	65.3848	80.6301	129.8385
Weibull CDF	22.0624	14.5216	18.3741	30.8761	20.1724	46.5722

and

Table 4. Comparison of MAE values for several WTs.

Model	Vestas V80	Siemens 82	REpower 82	Nordex N90	Siemens 107	Vestas V164
4P-OP	19.2696	18.1581	16.0462	27.1972	31.4132	60.5502
4P-DP	32.4614	34.7689	24.3311	38.8846	51.3634	89.9470
4P-DS	36.9029	37.5170	23.2441	41.6216	53.3197	77.3338
Weibull CDF	11.7565	7.0023	9.4974	15.1381	10.3209	22.1550

, respectively. These error measures are defined in Equations (11) and (12):

$$RMSE=\sqrt{\frac{1}{N}\sum _{i=1}^N{\left|P_i-P_i^s\right|}^2}$$

(11)

$$MAE=\frac{1}{N}\sum _{i=1}^N\left|P_i-P_i^s\right|$$

(12)

where N is the number of calculated values, $P_i$ the calculated power according to the Weibull CDF proposed function and $P_i^s$ the specified power, i.e., the value given by the manufacturer.

Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 show the curves fitted with the proposed and 4P-OP models for all power curves discussed in the case study.

5. Future Scope

The proposed approach consists of using the known Weibull CDF expression to fit a WT power curve, but keeping in mind that the fitted WT power curve expression is not a CDF equation. The methodology proposed in the article is based on a graphical method to estimate the ‘k’ and ‘C’ values of a Weibull distribution. A sound comparison of such methods can be found in [22,23,24], containing various methodologies including the maximum likelihood estimation (MLE), the method of moments (MOM) and the least squares method (LSM). These methods have been derived and simplified for Weibull distribution parameter estimations. Though these derivations cannot be directly applicable for fitting the WT power curves, a few modifications the in methodologies can make it applicable. In order to understand the scope for improvements, a trial and error method is used to estimate the optimum values of the ‘k’ and ‘C’ parameters for the mentioned WTs. Consider the example of the Vestas V80 WT. With the proposed methodology, the obtained ‘k’ and ‘C’ values are 4.6256 and 9.3890, respectively. Considering these values as a reference, all values of ‘k’ in the range from 3–5 and those of ‘C’ in the range from 8–10 at an interval of 0.01 are used to fit the WT power curves on a trail and error basis with Equation (7). The optimum parameter values for all WTs for which the minimum error arises are noted in

Table 5. Best case values of parameters ‘k’ and ‘C’ for several WTs.

Model	Vestas V80	Siemens 82	REpower 82	Nordex N90	Siemens 107	Vestas V164
k	4.57	4.13	4.08	4.36	4.54	4.53
C	9.29	10.11	9.55	9.43	9.80	9.50

, and the corresponding error values are shown in

Table 6. Comparison of the RMSE and MAE values for the best case parameters.

Model	Vestas V80	Siemens 82	REpower 82	Nordex N90	Siemens 107	Vestas V164
RMSE	18.1487	10.8834	11.2334	25.9851	19.8567	43.2761
MAE	10.66565	6.9250	6.8415	14.06718	9.2265	21.4085

.

These best case values of ‘k’ and ‘C’ ensure the possibility of the improvement in the proposed methodology.

6. Conclusions

The proposal of this paper consists of fitting WT power curves to Weibull CDF expressions. Shape and scale parameters have been deduced for a given group of WTs, and results have been compared with those obtained by means of several logistic functions proposed in the literature. This article concludes that the proposed function can be used in achieving accuracy in the wind power curve fitting process with a comparatively less complex approach. The proposed function shows comparable accuracy with three and four parameter logistic functions. Further investigation into Weibull CDFs, parameter estimation techniques and corresponding improvement in curve fitting accuracy can motivate the use of the proposed function. Further, a software package has been included in Appendix A, which makes it easy to obtain Weibull functions for fitting WT power curves.