Modeling Wind-Speed Statistics beyond the Weibull Distribution

Abstract

While it is well known that the Weibull distribution is a good model for wind-speed measurements and can be explained through simple statistical arguments, how such a model holds for shorter time periods is still an open question. In this paper, we present a systematic investigation of the accuracy of the Weibull distribution to wind-speed measurements, in comparison with other possible “cousin” distributions. In particular, we show that the Gaussian distribution enables one to predict wind-speed histograms with higher accuracy than the Weibull distribution. Two other good candidates are the Nakagami and the Rice distributions, which can be interpreted as particular cases of the Weibull distribution for particular choices of the shape and scale parameters. These findings hold not only when predicting next-point values of the wind speed but also when predicting the wind energy values. Finally, we discuss such findings in the context of wind power forecasting and monitoring for power-grid assessment.

1. Introduction: What Is the Model Underlying Wind-Speed Distributions?

Historically, sets of wind-speed measurements have been modeled as a two-parameter distribution, the so-called Weibull distribution [1], since it follows from simple statistical assumptions, as we briefly explain below. However, although this model has been found to be the most appropriate [2], being used in the large majority of studies [3], recent studies have pointed towards other alternative models, such as the inverse Weibull distribution and the Rayleigh distribution, which show stronger accuracy to fit wind-speed distributions in some cases [4,5,6]. For example, in Ref. [2] the authors develop a system of wind-speed distributions able to retrieve the main statistical features of wind-speed regimes, combining statistical models with machine-learning implementations. Modeling distributions of stochastic variables have also been addressed in other contexts, namely in finance to assess the non-stationary evolution of volume-price in the stock market [7].

An accurate model for the distribution of wind-speed measurements is important in the present conjuncture of energy transfer strategies from conventional to renewable energy sources [8,9], since the dynamical features, including the high fluctuations and turbulent character of the wind speed, are reflected in the power output of wind turbines [10,11,12] and, consequently, in the economical viability of renewable energy strategies in many countries [13,14,15,16,17]. Models for the distributions of wind-speed measurements are fundamental to estimate the power output of wind turbines [18] and in the application of mathematical and artificial-intelligence methods to power-grid modeling and monitoring [19]. In particular, wind-speed fluctuations have been modeled with auto-regressive integrated moving average (ARIMA) models [20] to reproduce heavy-tail features of power output. Moreover, approaches derived from statistical physics out of equilibrium were proposed to derive deterministic and stochastic contributions in wind-speed time-series, handling wind speed as a stochastic process [21,22,23], which can then be used to assess the power performance of wind turbines [11,24] and to model short-time fluctuations [25] as well as 10-min average wind power curves [12], wind turbine vibrations [26], torque [27], and fatigue loads [28].

The purpose of this paper is to present a systematic investigation of a family of two-parameter distributions of wind speeds measured over different time-periods, spanning from 10-min to one-month windows. As an overall outcome, we show that while the Weibull distribution is a good model for large time windows, the simple Gaussian distribution as well as particular variants of the Weibull distribution, namely the Nakagami and the Rician distributions, show better goodness-of-fit for short time windows. We begin in Section 2 by presenting the main models and methods and describing the data from FINO-1. In Section 3, we present the main results, which are then discussed in Section 4, which also presents the main conclusions of the paper.

2. Models, Methods, and Data

2.1. Wind Data Measured at FINO-1 Tower at the North Sea

Our analysis is carried out with data from the FINO-1 platform, located at the North Sea, within Alpha Ventus, a wind farm with 12 wind turbines used for research in wind energy production and wind energy systems; see Figure 1. FINO stands for Forschungsplattformen in Nord- und Ostsee and comprises three research platforms, funded by the Federal Ministry for Economic Affairs and Energy (BMWi), with the aim of reducing wind energy costs and providing more reliable operation procedures for wind turbines.

The datasets comprise 1 Hz wind-speed measurements collected at the Offshore Research Platfrom FINO-1 located in the North Sea, approximately 45 km north of the German island of Borkum (N 54°${00}^{\prime }$${53.5}^{″}$, E 6°${35}^{\prime }$${15.15}^{″}$). As shown in Figure 1 (left), FINO-1 is located in the immediate vicinity of the wind farm Alpha Ventus. The dataset spans three years and is collected at different heights of the tour—cf. Figure 1 (right). Here we only consider the anemometer at 100 m.

While it is typical to use 10-min or 15-min aggregated data of wind speeds when monitoring and analysis wind turbine functioning, there are several reasons to consider higher sampling frequency [11]. First, turbulent fluctuations with time scales of less than 10 min are part of the local wind structure. Second, the main shaft of the turbine eolian responds to gusts within time-frames of one second. Third, it is known that inhomogeneous and intermittent effects in turbulent systems, such as the atmosphere, manifest at several time-scales, down to the order of seconds, and therefore are reflected in the dynamics of wind power generation.

We detected that 1% of the total data points were NaNs (Not a Number). This fraction was discarded from the analysis. Furthermore, we analyze the data in different time-frames of length T, namely $T=10,30$, and 60 min and also 1 h, 3 h, 6 h, 1 day, and 1 month. We also discarded time windows with more than 40% of NaNs. Figure 2 illustrates both typical time series within this time-frame of length T (right), together with its wind-speed distribution (left). In the latter plots we also indicate the best Weibull and Gaussian distribution. It should be noted that, whereas for large time-frames, e.g., 1 month, the Weibull fits better than the Gaussian, for shorter time scales, of 10 min, Gaussian models seems to surpass Weibull. Below we address this point in more detail, after introducing our models and metrics.

2.2. The Weibull Distribution as a Model for Wind-Speed Measurements and Some of Its “Cousins”

The Weibull distribution is a common model for the distribution of measurements of wind speed v, defined as

$$f_W\left(v\right)=\frac{k}{\lambda }{\left(\frac{v}{\lambda }\right)}^{k-1}exp\left(-{\left(\frac{v}{\lambda }\right)}^k\right),$$

(1)

where $\lambda$ and k are the scale and shape parameters, respectively. This was introduced by Waloddi Weibull in 1951 [29,30].

As shown in [31], a Weibull wind-speed distribution follows from a Gaussian distribution of each one of the velocity components in the two-dimensional plane, assuming that both components are completely uncorrelated and have zero mean and the same standard deviation, $\sigma$. For that particular case, the Weibull distribution yields $k=2$ and $\lambda =\sqrt{2}\sigma$. Indeed, as reported in the literature [32,33], when fitting wind-speed measurements with a Weibull distribution (1) the shape parameter k takes values typically around 2, and therefore such assumptions are acceptable.

Having introduced the Weibull distribution, we can now describe two of its “cousins”, namely, the Nakagami and the Rician distributions.

The Rician distribution [34] has a probability density function defined as

$$f_R\left(v\right)=\frac{x}{{\sigma }_R^2}exp\left(-\frac{v^2+{\nu }^2}{2{\sigma }_R^2}\right)I\left(\frac{v\nu }{{\sigma }_R^2}\right),$$

(2)

where $I\left(x\right)$ represents the modified Bessel function of the first kind of x and $\sigma$ and $\nu$ represent the spread parameter and the distance between the reference point and center of the distribution, respectively. Notice that, while the Bessel function is an odd function, $I(-x)=-I\left(x\right)$, here the argument x is always positive since $\sigma ,v,\nu >0$. For $\nu =0$, the Rician distribution reduces to a Weibull distribution with $k=2$ and $\lambda =\sqrt{2}{\sigma }_R$.

The Nakagami distribution is defined by [35]

$$f_N\left(v\right)=\frac{2}{\Gamma \left(m\right)}{\left(\frac{m}{\Omega }\right)}^m{v}^{2m-1}exp\left(-\frac{m}{\Omega }{v}^2\right),$$

(3)

where $\Omega$ and m represent the spread and the shape parameters, respectively. For the particular case of $m=1$, the Nakagami distribution reduces to a Weibull distribution with $\lambda =\sqrt{\Omega }$ and $k=2/{\lambda }^2$. Since in the case of wind-speed measurements $k\sim 2$, we conclude that Nakagami distribution is a good model for wind-speed data series with a scale parameter of $\lambda \sim 1$.

Together with these two “cousin” distributions of the Weibull model, we also consider the usual Gaussian distribution, defined as

$$f_G\left(v\right)=\frac{1}{\sigma \sqrt{2\pi }}exp\left(-\frac{1}{2}{\left(\frac{v-\mu }{\sigma }\right)}^2\right),$$

(4)

where $\mu$ is the mean and $\sigma$ is the standard deviation. As we will see, each one of these three variants is sometimes better than the Weibull distribution as a model for s.

Finally, for consistency, we will also consider other distributions, not usually assumed in wind-data modeling, namely the Gamma distribution, the inverse-Gamma distribution, and the log-normal distribution, none of which are reducible to a Weibull distribution. The Gamma distribution is defined by

$$f_G\left(v\right)=\frac{1}{\Gamma \left(k\right){\theta }^k}{v}^{k-1}exp\left(\frac{-v}{\theta }\right),$$

(5)

where $\theta$ and k are the scale and shape parameters, respectively. The inverse-Gamma distribution is defined by

$$f_I\left(v\right)=\frac{{\beta }^{\alpha }}{\Gamma \left(\alpha \right)}{v}^{-\alpha -1}exp\left(\frac{-\beta }{v}\right),$$

(6)

where $\beta$ and $\alpha$ are the shape and the scale parameters, respectively. The log-normal distribution is defined by:

$$f_L\left(v\right)=\frac{1}{v\sigma \sqrt{2\pi }}exp\left(\frac{{\left(lnv-\mu \right)}^2}{-2{\sigma }^2}\right),$$

(7)

where $\mu$ represents the mean and $\sigma$ the standard deviation of the logarithm of the wind speed v.

Figure 3 illustrates each one of these distributions for two representative time windows, one with $T=10$ min and another with $T=1$ month. Note that the fit here is done by solving the parameter values of each distribution above to match the same mean and variance of the empirical wind-speed measurements in that window.

2.3. Performance Measures: Evaluating the Fitness of Each Model

We evaluate the performance of each model with respect to each of three questions:

(I)

How well does each model enable one to predict the next value of the wind speed?

(II)

How well does each distribution model fit the empirical histogram of measurements?

(III)

How well does each model enable one to predict the energy associated with the wind speed, i.e., the square of the wind speed?

To evaluate the prediction power of the next value of the wind speed, we define a quantity called Next-Point-Probability (NPP), defined as the probability of observing the empirical data point immediately following the time window where the probability distribution was calculated. When NPP is close to one, the predictive power is high, whereas a probability close to zero shows a low predictive power. For clarity, we consider henceforth the logarithm of the NPP, $\mathcal{L}=log\left(NPP\right)$. It is important to note that this quantity serves the purpose of an out-of-sample metric of a distribution’s fit quality and also of a short-term propagator.

To fit each distribution to the empirical histogram of each time window, we consider the Kullback–Leibler divergence (KLD), which is a non-negative and non-commutative measurement, fulfilling a maximum likelihood estimation scheme [36]. It is also called Relative Entropy and is defined as

$$D_{KL}(f_X\left|\right|{\rho }_{\mathrm{emp}})=\sum _{i\in \mathrm{bins}/\mathrm{data}}{f}_X\left(i\right)log\left(\frac{f_X\left(i\right)}{{\rho }_{\mathrm{emp}}\left(i\right)}\right),$$

(8)

where ${\rho }_{\mathrm{emp}}$ is the empirical distribution and $f_X$ is one of the models above. The sum runs over the number of bins for which the empirical histogram is defined or over the data points in each time window in case a kernel-based estimator is chosen, depending on how it is implemented. Empirical distributions with two local maxima were discarded, which represented less than $2\%$ of the total number of time windows. For the inverse-Gamma fits, infinite values could be obtained when the likelihood was lower than computational resolution, though this occurs for only $0.1\%$ of the total number of time windows, and therefore we discarded these windows for this model.

Finally, to estimate the prediction power of each model for predicting the energy from wind-speed data, we evaluated the square velocity relative deviation, namely

$$\Delta =〈\frac{v_{\mathrm{emp}}^2-v_{\mathrm{fit}}^2}{v_{\mathrm{emp}}^2}〉,$$

(9)

where the average runs over the time window. Our own routine was implemented. It should be highlighted that the fitting was performed on the empirical measurements (not their square), and from that, $v_{\mathrm{fit}}^2$ was estimated. Empirical distributions with two local maxima were discarded, which cover $2\%$ of the total number of time windows.

3. Results: Turbulence Features, Wind-Speed Statistics, and Prediction of Speed and Energy

We present our results by showing first how the average speed and respective standard deviation change with the size T of the time windows. Those two quantities are shown in Figure 4a and Figure 4b, respectively. We observe an almost constant average speed, $\mu$, together with a standard deviation, $\sigma$, which increases five times when increasing the size of the time windows from $T=10$ min to $T=1$ month. Consequently, the intermittency of wind-speed measurements, which is defined as the quotient between the standard deviation and the mean, namely $I=\frac{\sigma }{\mu }$, increases linearly with the standard deviations; see Figure 4c.

Since the intermittency increases monotonically with the time-window size T, one would expect that the fitness or prediction power, assessed with the metrics introduced above ($\mathcal{L},D_{KL}$, and $\Delta$), should decrease with T. However, as we will see next, this is not always the case.

To address each one of the questions introduced above, in Section 2.3 we compute the respective metrics, namely $\mathcal{L}$ for question I, $D_{KL}$ for question II, and $\Delta$ for question III. The results are shown in Figure 5.

For the NPP, measured by $\mathcal{L}$, the lower curves in Figure 5a are observed for the inverse-Gamma and log-normal. These two are therefore the worst models for next-point probability of the wind speed. As for all the other models, the performance in NPP is very similar and decreases monotonically with T. Interestingly, here the Gaussian model is ∼2% better than the Weibull one, being surpassed only by the Nakagami model, and sometimes the Rice model. Note that for windows of 1 month, all models perform equally well, with the exception of the log-normal model.

In Figure 5b we show how well the different distribution models fit the empirical wind-speed histogram, using $D_{KL}$. Again, the log-normal and inverse-Gamma models perform poorly. Notably, for windows of 1 day or less, the Gauss, Gamma, Nakagami, and Rice distributions perform similarly, representing an improvement relative to the canonically used Weibull distribution (inset of Figure 5b).

When assessing the error in energy prediction, again we observe that the log-normal and inverse-Gamma distributions perform worse than the other distributions. We see that the Gauss, Gamma, Nakagami, and Rice distributions perform better than the Weibull here as well for smaller time windows. We observe in Figure 5c a performance of a few percent or less ($\Delta \lesssim 0.03$) for all models, except log-normal and inverse-Gamma distribution, and for all window sizes T.

Finally, we assess the error in energy prediction. In Figure 5c, one observes a performance of a few percent or less, $\Delta \lesssim 0.03$, for all models, except log-normal and inverse-Gamma distribution, and for all window sizes T. Here one observes that the Weibull model performs robustly, with an error of only $\Delta \lesssim 0.01$. However, it does not surpass the performance of both Gauss and Nakagami. The Rice model is only worse than these latter three models for time windows of one month. Surprisingly, of all the models, the Gauss distribution model is the one retrieving the least amount of errors in energy predictions, independent of the window size. For better comparison, we provide in

Table 1. All values of the curves in Figure 5.

	Time(s)	Gauss	Weibull	Gamma	InvGam	Nakagami	Rice	LogN
	600	0.077	0.107	0.080	1.875	0.078	0.077	0.769
		(±0.001)	(±0.001)	(±0.001)	(±0.004)	(±0.001)	(±0.001)	(±0.002)
	1800	0.0910	0.119	0.094	1.636	0.092	0.091	0.733
		(±0.001)	(±0.001)	(±0.001)	(±0.006)	(±0.001)	(±0.001)	(±0.003)
	3600	0.099	0.126	0.101	1.487	0.099	0.098	0.721
		(±0.002)	(±0.002)	(±0.002)	(±0.008)	(±0.002)	(±0.002)	(±0.004)
$D_{KL}$	10,800	0.112	0.135	0.114	1.22	0.111	0.111	0.694
		(±0.003)	(±0.003)	(±0.003)	(±0.01)	(±0.003)	(±0.003)	(±0.006)
	21,600	0.126	0.145	0.129	1.03	0.124	0.124	0.668
		(±0.005)	(±0.005)	(±0.005)	(±0.02)	(±0.005)	(±0.005)	(±0.008)
	86,400	0.151	0.151	0.16	0.67	0.149	0.146	0.59
		(±0.009)	(±0.008)	(±0.01)	(±0.03)	(±0.009)	(±0.009)	(±0.02)
	2,592,000	0.06	0.039	0.07	0.47	0.046	0.041	0.40
		(±0.015)	(±0.006)	(±0.01)	(±0.07)	(±0.007)	(±0.006)	(±0.04)
	600	−0.96	−1.01	−0.97	−2.485	−0.96	−0.97	−5.245
		(±0.01)	(±0.01)	(±0.01)	(±0.004)	(±0.01)	(±0.01)	(±0.007)
	1800	−1.20	−1.23	−1.20	−2.464	−1.19	−1.19	−5.11
		(±0.02)	(±0.02)	(±0.01)	(±0.007)	(± 0.01)	(±0.01)	(±0.01)
	3600	−1.41	−1.44	−1.42	−2.47	−1.41	−1.41	−5.06
		(±0.02)	(±0.02)	(±0.02)	(±0.01)	(±0.02)	(±0.02)	(±0.02)
$\mathcal{L}$	10,800	−1.88	−1.88	−1.88	−2.53	−1.87	−1.88	−4.98
		(±0.04)	(±0.04)	(±0.05)	(±0.02)	(±0.05)	(±0.05)	(±0.04)
	21,600	−2.24	−2.22	−2.24	−2.61	−2.23	−2.23	−4.90
		(±0.07)	(±0.07)	(±0.07)	(±0.03)	(±0.06)	(±0.07)	(±0.06)
	86,400	−2.7	−2.7	−2.7	−2.90	−2.7	−2.7	−4.7
		(±0.1)	(±0.1)	(±0.1)	(±0.09)	(±0.1)	(±0.1)	(±0.2)
	2,592,000	−3.1	−3.0	−2.9	−3.0	−3.0	−3.0	−4
		(±0.6)	(±0.4)	(±0.4)	(±0.4)	(±0.5)	(±0.5)	(±1)
	600	0.0002	0.0067	0.0002	0.563	0.0002	0.0002	0.0901
		(±0.0001)	(±0.0001)	(±0.0001)	(±0.001)	(±0.0001)	(±0.0001)	(±0.0001)
	1800	0.0004	0.0061	0.0006	0.496	0.0004	0.0004	0.095
		(±0.0001)	(±0.0001)	(±0.0001)	(±0.001)	(±0.0001)	(±0.0001)	(±0.001)
	3600	0.0005	0.0059	0.0012	0.449	0.0007	0.0006	0.100
		(±0.0001)	(±0.0001)	(±0.0001)	(±0.002)	(±0.0001)	(±0.0001)	(±0.001)
$\Delta$	10,800	0.0010	0.0058	0.0034	0.361	0.0017	0.0011	0.110
		(±0.0001)	(±0.0001)	(±0.0001)	(±0.004)	(±0.0001)	(±0.0001)	(±0.002)
	21,600	0.0014	0.006	0.006	0.296	0.003	0.0016	0.120
		(±0.0001)	(±0.001)	(±0.001)	(±0.005)	(±0.001)	(±0.0001)	(±0.003)
	86,400	0.0021	0.007	0.014	0.172	0.007	0.004	0.140
		(±0.0001)	(±0.001)	(±0.002)	(±0.006)	(±0.001)	(±0.001)	(±0.007)
	2,592,000	0.01	0.01	0.03	0.031	0.01	0.02	0.3
		(±0.01)	(±0.01)	(±0.01)	(±0.009)	(±0.01)	(±0.01)	(±0.2)

all values of the curves plotted in Figure 5, for a better inspection of the values of all performance values.

4. Discussion and Conclusions

In this paper, we presented a systematic assessment of different distribution models for wind-speed measurements. The aim was to compare the performance of these models with the standard Weibull and Gauss distributions.

We provided quantitative evidence that, similarly to other studies, the Weibull distribution is not always the distribution that best fits wind speeds [4,5,6] and also not for predicting wind speeds and wind energy (square of the speed). For larger time-frames, namely within 1 day and 1 month, Weibull retrieves the best fit of the wind-speed distribution, supporting the standard assumption that—for large enough samples—wind-speed measurements follow a Weibull distribution.

However, we have provided evidence that, for all time windows, the Weibull model is not necessarily the best model for retrieving the prediction of the average wind speed and average wind energy. Moreover, while the Rice and Nakagami models, being related to the Weibull distribution, can retrieve more accurate predictions, it is surprising that Gauss performs equally well, and in some cases better than any other model.

Our findings may be of importance in the context of wind power forecasting and monitoring for power-grid assessment. Commonly, the Gauss model is used when addressing wind power and other observables in wind energy systems. Our results show that for 10-min or larger windows, Gauss models provide reasonable performance, and in some cases it is even the best model. However, one should note that, differently from what was shown in Figure 4, at time-scales smaller than minutes towards a few seconds or sub-seconds the intermittency may be higher than observed for 10-min windows.

For future research, one can consider studying the process in scale, seeing not only how standard deviation scales with the time-window scale, but also performing an analysis of correlations and Hurst exponents. Such analysis may shed further light on the limitations (or lack of them) of assigning a particular statistical distribution to wind speed.