Spatial and temporal patterns of global onshore wind speed distribution

The Weibull function is widely used due to its simplicity and ability to closely mirror the distribution of observed wind speeds. The probability density function f(x;λ,k) of a Weibull random variable x is given by,

$$\begin{eqnarray} &&f(x;\lambda ,k)=\frac{k}{\lambda }\left (\frac{x}{\lambda }\right )^{k-1}{\mathrm{e}}^{-(x/\lambda )^{k}} \end{eqnarray} \tag{ 1 }$$

where k is the Weibull shape parameter, λ is the scale parameter. Physically, the scale parameter (λ) is proportional to the mean wind speed, while the shape parameter (k) determines the width of the distribution, with narrower speed distributions for higher k values.

The mean wind speed can be written as:

$$\begin{eqnarray} &&\bar {v}=\lambda \Gamma (1+1/k) \end{eqnarray} \tag{ 2 }$$

where $\bar {v}$ is the mean of wind speed, and Γ is the gamma function.

In this study, we chose the power density (PD) method developed by Akdağ and Dinler (2009) to estimate the Weibull parameters. They found that the PD method can produce accurate estimates of Weibull parameters with less computation, and we find this to be the case here, and it might be better than graphic, maximum likelihood, and moment methods (Akdağ and Dinler 2009). For application in wind energy studies, this method is also more suitable given its focus on wind power density. In addition, the Weibull parameters can be easily estimated if only power density and mean wind speed are known.

According to the PD method by Akdağ and Dinler (2009), the Weibull k parameter can be estimated as,

$$\begin{eqnarray} &&k=1+\frac{3.6 9}{{E}_{\mathrm{pf}}^{2}} \end{eqnarray} \tag{ 3 }$$

where E_pf is the energy pattern factor. The energy pattern factor is based on the mean of wind speed cubed $(\overline{{v}^{3}})$ and the cube of mean wind speed $({\bar {v}}^{3})$ as below,

$$\begin{eqnarray} &&{E}_{\mathrm{pf}}=\frac{\Gamma (1+3/k)}{\Gamma (1+1/k)^{3}}=\frac{\overline{{v}^{3}}}{{\bar {v}}^{3}}. \end{eqnarray} \tag{ 4 }$$

With the Weibull shape parameter (k) estimated from equation (3), the scale parameter (λ) can be determined from equation (2).

Several types of wind speed datasets are available to evaluate global wind resources. These datasets are distinguished by differences in their methodology and spatial and temporal resolution (Wiser et al 2011, Zhou et al 2012). After evaluation of a number of global datasets, in this study we use 10 m wind speeds from the NCEP climate forecast system reanalysis (CFSR) (Saha et al 2010, Zhou et al 2012). While wind turbines operate at higher heights (Drechsel et al 2012), we use 10 m height wind speed because it is generally available from many datasets, allowing comparison. It is not clear, however, if the character of wind speed statistics would change with height, or if such changes would accurately be captured by reanalysis methodologies. This could be a topic of future work.

The CFSR reanalysis data has a spatial resolution of 0.3125° lat/lon and provides hourly values of wind speed for the time period from 1980 to 2009. The PD method is also useful for this large dataset because binning and solving linear least squares or iterative procedures are not necessary. Zhou et al (2012) identified some biases such as the underestimation of wind potential at wind class 5 and higher in this data set, however, and we return to this issue in the discussion.

Wind speed frequency distribution, in terms of the Weibull shape parameter (k), shows large spatial heterogeneity at the global level (figure 1(a)). The value of the shape parameter is larger than 2 in large portion of global land area (figure 1(c)). It also varies across regions¹ regarding magnitude and distribution (figure 1(c)). Regions of Africa and Australia generally show large shape parameter with a peak around values of 2.4, and Africa also shows largest regional variation (90% range of 1.9–3.0) within region. A large shape parameter indicates a narrower frequency distribution. Therefore, wind speeds in these areas, at least in this data set, are relatively stable compared to regions with a lower shape parameter. As a key parameter impacting wind energy estimation, this large spatial heterogeneity should be borne in mind when evaluating the wind energy using mean wind speed.

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The spatial variation of the Weibull scale parameter (λ) shows a pattern similar to the mean wind speed (figure 1(b)). The scale parameter is a function of the mean wind speed and Weibull k (equation (2)) and is determined largely by the mean wind speed, as Γ(1 + 1/k) shows a narrow range from 0.88 to 0.92 for shape parameter varying from 1.5 to 4 as observed in this study at the global level. The Weibull wind speed distribution is completely characterized by the mean wind speed and shape (k) parameter. The sensitivity of wind power to mean wind speed has been discussed in Zhou et al (2012). For these reasons the following analysis will focus on the Weibull shape parameter.

To understand the reliability of Weibull distribution in fitting time series wind speed data, we evaluated accuracy using three metrics: R², root mean square error (RMSE), and power density error (PDE) (%). High R² and low RMSE values indicate a good fit between the data and the Weibull distribution in terms of wind speed. A low PDE value denotes good performance of the Weibull distribution in representing power density. These metrics can be expressed as (Akdağ and Dinler 2009),

$$\begin{eqnarray} {R}^{2}&=&1-\frac{\sum _{i=1}^{N}({y}_{i}-{x}_{i})^{2}}{\sum _{i=1}^{N}({y}_{i}-\bar {y})^{2}} \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} \mathrm{RMSE}&=&\left [\frac{1}{N}\sum _{i=1}^{N}({y}_{i}-{x}_{i})^{2}\right ]^{0.5} \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} \mathrm{PDE}(\%)&=&\left \vert \frac{{\mathrm{Pd}}_{\mathrm{w}}-{\mathrm{Pd}}_{\mathrm{ts}}}{{\mathrm{Pd}}_{\mathrm{ts}}}\right \vert \end{eqnarray} \tag{ 7 }$$

where N is the number of bins (here, N is 100 for wind speed from 0.1 to 10 m s⁻¹ with bin width of 0.1 m s⁻¹), x_i and y_i are the frequencies of calculated wind speed data from Weibull distribution and reanalysis time series data, Pd_w is the power density calculated from Weibull distribution, and Pd_ts is the power density calculated from hourly reanalysis data.

According to all three metrics, Weibull distribution estimated using the PD method fits the data well (figure 2). Most areas at the global level show a R² better than 0.93 (figure 2(a)). In terms of RMSE, still, most areas show a low value indicating the high reliability of the fitted Weibull distribution (figure 2(b)). Power density errors (PDE%) are under 1% over most parts of the globe (figure 2(c)). In areas with large shape parameter values such as northeastern part of Africa, the Weibull distribution does not capture the reanalysis wind speed data quite well. This may indicate that the wind speed frequency distribution here is too complex to be fitted by the Weibull distribution. Tuller and Brett (1984) find, for example, that areas with frequent calm periods or asymmetric wind patterns are less likely to be fitted by a Weibull distribution. Alternatives for such regions include a bimodal probability distribution, proposed for La Ventosa, Mexico (Jaramillo and Borja 2004), or the general probability distribution presented by Ramírez and Carta (2006). A sector-wise fitting method might improve the estimation of a Weibull distribution in those (relatively few) areas not well fitted by a Weibull distribution, especially in the areas with a bimodal probability distribution (Achberger et al 2002).

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The performance of the Weibull distribution and associated parameters shows a relationship with mean wind speed in terms of R² (figure 2(d)). Grid cells with higher mean wind speed tend to have lower R² magnitude while RMSE and PDE are distributed more evenly over the range of wind speeds. The results indicate that the two-parameter Weibull distribution is not necessary more biased in more economically viable wind areas (large speed) compared to low wind speed areas in terms of wind power density, at least when fitted using the PD method, although the R² magnitude in terms of wind speed tends to decrease in these areas.

The wind speed distribution shows small temporal changes over 10-year periods. Compared to the base time period (1980–1989), the shape parameter is somewhat different in 1990–1999 and 2000–2009 in space (figures 3(a) and (b)). Some areas, for example, the west coast of Africa, show a consistently increasing trend, while other areas such as Eastern Europe, show an inversed pattern in 1990–1999 and 2000–2009 (figures 3(a) and (b)). At any given point in space, changes of order 0.1 of the k parameter are common, and it reaches up to 0.2 (99% range at the global level). The change of shape parameter of the Weibull distribution will have different impacts on the wind energy potential in areas with different mean wind speed, as discussed in section 3.4 on implications. When evaluated at the regional level, the k parameter shows a small difference over these three periods (figure 3(c)).

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Seasonal² changes in the wind speed distribution are larger than the decadal changes over 1980–2009 (figure 4). This is to be expected, as weather patterns vary in a consistent manner by season. While there are some consistent differences by season, these are smaller than the spatial differences. At a given point in space the k value can vary by more than 1.0 (99% range at the global level) between seasons. When evaluated at the regional level, the change of k parameter varies greatly among regions (figure 3(d)). For example, in the Former Soviet Union, the Weibull k shows a large variation across seasons while in Asia the distribution of the Weibull k is consistent across four seasons. The seasonal changes in wind shape parameter at the grid and regional level are both much smaller than the changes seen across space.

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The relatively small variations in wind statistical properties over time, as compared to much larger spatial, and often seasonal, changes, are consistent with the common industry practice of using at least one year of direct measurement data to characterize potential wind turbine sites (MEASNET 2009).

Power generation for the same mean wind speed can be very different for different Weibull shape parameters (figure 5). There are a number of factors influencing actual power production including different turbine power curves and others such as turbine height and turbine density. As an illustrative calculation, we compared wind power generated from a reference 1.5 MW GE turbine (GE 2009) for a range of mean wind speeds from 4 to 12 m s⁻¹ by varying the Weibull shape parameter. When mean wind speed is large (greater than about 6 m s⁻¹), the wind power generated with a small shape parameter (<2) is smaller than the case of the commonly assumed Rayleigh distribution (Weibull k = 2), while wind power generation with a large shape parameter (>2) is larger as compared to Rayleigh. The opposite is true when the mean wind speed is smaller than 6 m s⁻¹. This motivates us to investigate the possible bias of wind power estimation at the global level with the commonly used Rayleigh distribution (Hoogwijk et al 2004).

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Wind power is under- or over-estimated by using mean wind speed and k = 2, Rayleigh distribution (figure 6). The bias (under- or over-estimation) is calculated as the percentage difference between the energy produced using a Rayleigh distribution and a Weibull distribution with calculated k. At the global level, wind power was overestimated by an average of 6%, varying from underestimation by 20% (5% quantile) and overestimation by 30% (95% quantile). At the regional level and globally, wind energy is generally overestimated by assuming k = 2 except in Europe. The largest overestimation occurs in Africa and Australia, with an average of more than 15%, and the overestimation can even reach more than 35% (95% quantile) in Africa. The average bias is smallest in Asia and Europe, but it can still be overestimated by 10%–20% (95% quantile) in Europe and Asia. The bias in the estimation of wind power varies greatly in space because of the large spatial heterogeneity of the Weibull shape parameter. We see that bias due to an assumption of k = 2 becomes larger at smaller scales: while only 6% globally this increases to 15% in certain continental-scale regions, and up to 30–35% in smaller areas. The larger the deviation of shape parameter from 2, the higher possible bias in the estimation of wind power. Some caution is advised in these areas when analyzing wind power using a Rayleigh distribution (e.g. Hoogwijk et al 2004), although any potential error in this respect has to be weighted against the quality of the wind speed, or even reanalysis, data.

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