the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Estimating vertical wind power density using tower observation and empirical models over varied desert steppe terrain in northern China

### Shaohui Zhou

### Yuanjian Yang

### Zhiqiu Gao

### Xingya Xi

### Zexia Duan

### Yubin Li

A complex and varied terrain has a great impact on the
distribution of wind energy resources, resulting in uncertainty in
accurately assessing wind energy resources. In this study, three wind speed
distributions of kernel, Weibull, and Rayleigh type for estimating average
wind power density were first compared by using meteorological tower data
from 2018 to 2020 under varied desert steppe terrain contexts in northern
China. Then three key parameters of scale factor (*c*) and shape factor (*k*) from the Weibull model and surface roughness (*z*_{0}) were investigated for estimating wind energy resource. The results show that the Weibull distribution is the most suitable wind speed distribution over that terrain. The scale factor (*c*) in the Weibull distribution model increases with an increase in height, exhibiting an obvious form of power function, while there were two different forms for the relationship between the shape factor (*k*) and height: i.e., the reciprocal of the quadratic function and the logarithmic function, respectively. The estimated roughness length
(*z*_{0}) varied with the withering period, the growing period, and the lush period, which can be represented by the estimated median value in each
period. The maximum and minimum values of surface roughness length over the
whole period are 0.15 and 0.12 m, respectively. The power-law model and
the logarithmic model are used to estimate the average power density values
at six specific heights, which show greater differences in autumn and
winter, and smaller differences in spring and summer. The gradient of the
increase in average power density values with height is largest in autumn
and winter, and smallest in spring and summer. Our findings suggest that
dynamic changes in three key parameters (*c*, *k*, and *z*_{0}) should be accurately considered for estimating wind energy resources under varied desert steppe terrain contexts.

Wind energy is a renewable, environmentally friendly, and popular alternative source of clean energy (Islam et al., 2013; Gabbasa et al., 2013), and as a source of power it has great potential (Chaurasiya et al., 2019). In 2020, 93 GW of new installations brought the global cumulative wind power capacity up to 743 GW. In the onshore market, 86.9 GW was installed, an increase of 59 % compared to 2019. China and the United States remain the world's largest markets for new onshore installations (Joyce and Feng, 2021). To use this kind of nonpolluting energy, a lot of research has been conducted through a variety of different methods to develop an accurate and reliable wind energy evaluation model.

The probability density function (PDF) of wind speed can effectively characterize wind speed. Therefore, the wind speed PDF is of great significance in selecting wind turbine sites, in wind farm design, in generator design, in determining the dominant wind direction, and in evaluating the management and operation of wind conversion systems (Masseran, 2015; Li and Shi, 2010). Wind displays large differences with various topographies, landforms, and meteorological conditions. The magnitude and direction of the wind speed exhibit significant differences when wind flows over rough ground or obstacles in a complex terrain. In addition, the surface topography and roughness of the area around the location of the wind measurement tower will affect the predicted wind resources (Kim and Lim, 2017). Therefore, the wind speed PDF and roughness are important input factors in the estimation of the power density of wind energy.

Different distribution functions have different fitting effects on the actual wind speed values in distinct study areas. According to previous studies (Lo Brano et al., 2011; Celik, 2004; Masseran et al., 2012), seven wind PDFs have been widely used to fit the actual wind speed values: i.e., Weibull, Rayleigh, lognormal, gamma, inverse Gaussian, Pearson type V, and Burr. These models exhibited different advantages and disadvantages for estimating wind probability density. For instance, Celik (2004) used the Weibull and Rayleigh models to perform a statistical analysis of wind energy density in southern Turkey and found that the Weibull model not only fits the measured monthly probability density distribution better but also provides a better power density estimation compared to the Rayleigh model. Masseran et al. (2012) used nine different wind speed PDF models to describe wind speed conditions in different regions of Malaysia and found that gamma, Weibull, and inverse gamma models fit the wind speed data better. Chang (2011a) used six different PDFs, namely, Weibull, a mixture of gamma and Weibull, a mixture of normal, a mixture of normal and Weibull, a mixture of Weibull, and principle of maximum entropy distributions. They were tested on the wind data of three wind farms in Taiwan and it was found that, when the current wind speed distribution is unimodal, the fitting effects of these six PDFs are not significantly different. When the wind speed distribution is bimodal, the other five PDFs are better than Weibull at describing wind characteristics. In addition, many other PDFs have been invented to provide more accurate results for the estimation of wind power density in a specific area (Masseran, 2015; Carta et al., 2009; Jaramillo and Borja, 2004).

Among the aforementioned types of wind speed PDFs, the Weibull and Rayleigh distributions remain the more
traditional and widely applicable typical wind speed distribution forms. The
key issue in the study of the Weibull distribution is how to accurately
determine the values of Weibull scale factor *c* and shape factor *k* (Azad et al.,
2014; Kaplan, 2017). Generally, six different methods, i.e., graphical
method (Basu et al., 2009), empirical method (Costa Rocha et al., 2012;
Kaoga et al., 2014), maximum likelihood method (Andrade et al., 2014; Azad
et al., 2014), energy trend method (Chang, 2011b; Akdağ and Dinler,
2009), energy pattern method (Andrade et al., 2014), and the moment method
(Azad et al., 2014; Kaplan, 2017; Costa Rocha et al., 2012), have been
employed to calculate the *c* and *k* of the Weibull distribution model. But these
methods perform differently in different regions. For instance, Kaplan (2017) found that the energy pattern method and the moment method were the
best methods between 2009 and 2013 in the Hatay and Osmaniye regions. When
the time series of wind data is provided, the maximum likelihood method is
more robust and accurate than other methods (Seguro and Lambert, 2000;
George, 2014). In addition, there is a strong time dependence and a high
change dependence for the changes in shape factor *k* and scale factor *c* (Lun and Lam, 2000; Justus and Mikhail, 1976): e.g., the scale factor *c* has a power-law functional relationship with height and the shape factor *k* has a reciprocal logarithmic functional relationship with height. Therefore, we
can explore its general laws by studying the seasonal changes and height
changes in shape and scale parameters in a specific area.

Roughness length plays a key role in estimating wind energy resources. For
example, Laporte (2010) pointed out that the roughness estimation error can
cause 5–10 % of the wind energy resource estimation error. Current wind energy resource assessment is based on measured wind data at a height of 60–80 m from the ground, but the actual height of the hub may be greater than these heights. Therefore, we need to combine the surface
roughness length and the known wind speed value of the measured height to
extrapolate the wind speed value at the height of the hub (Nayyar and Ali,
2020). Theoretically, the surface roughness length *z*_{0} is the height at
which the average wind decreases to zero with height. The value *z*_{0} varies with the
underlying surface (Davenport et al., 2000; Duan et al., 2021). Currently,
three approaches (the analysis method, the Charnock method, and the
statistical method) are widely applied to estimate the surface
roughness length of offshore wind energy (Golbazi and Archer, 2019). Among
them, the statistical method is convenient, as it needs only three layers of
wind speed data. After comparing the average value and median value of
roughness *z*_{0}, it is found that the median value is an order of magnitude
closer to the roughness length calculated from the other two methods.
Therefore, when using the field measurement method to statistically
determine the surface roughness length, attention should be paid to using
the median value instead of the average value; otherwise huge errors will be
generated when the wind speed is extrapolated to the height of the hub,
which will have a major impact on the evaluation of wind energy resources.

As an important production base of wind power energy in northern China,
Inner Mongolia is under the influence of the westerly wind all year round.
The types of underlying surfaces of wind power towers in China are complex
and diverse, including offshore, mountainous, urban outskirts, and
grasslands. In Inner Mongolia, especially the desert grassland, the terrain
is open, the vegetation is low and sparse, and its wind resources are very
rich. Thus, taking the Ningyuanbailiutu site as an example, in-depth data
mining was carried out on the four heights of 10, 30, 50, and 70 m for
the meteorological element data of a 100 m wind tower from the autumn of
2018 to the summer of 2020 in Damaoqi, Baotou City, Inner Mongolia, China.
The following three steps are used to study the three important key
parameters that affect the evaluation of wind energy resources: the surface
roughness length *z*_{0}, the scale factor *c*, and the shape factor *k* in the Weibull distribution function. First, we need to determine the uniqueness and importance of the Weibull distribution function in the wind speed time series data in the Damaoqi area. This is reflected in the advances and shortcomings of the kernel distribution model, the Rayleigh distribution
model, the Weibull distribution function, and the frequency distribution
model using actual wind speed, which are used to calculate the monthly,
seasonal, and all-time average power densities. Second, by studying the
monthly and seasonal changes in the surface roughness length and the changes
in different incoming flow directions, we will gain a comprehensive
understanding of the roughness of the site area in Inner Mongolia. Finally,
by using two different models, namely, the power-law model with scale
parameter *c* and the logarithmic model with roughness information, the average wind power densities at six specific heights (75, 80, 85, 90, 95,
and 100 m) per month, per season, and throughout the period are calculated.
In this way, we discuss the application significance of the two models for
wind energy development, and provide a scientific reference to further our
understanding of the wind energy resources in the region.

## 2.1 Study site and data

In this study, long-term in situ measurement was conducted in Damaoqi,
Baotou City, Inner Mongolia (42^{∘}04^{′}25.738^{′′} N, 110^{∘}29^{′}2.778^{′′} E; 1376 m above sea level) from 1 September 2018 to 31 August
2020 (Fig. 1). The observation wind tower is located at the northern foot
of Daqing Mountain in the central area of the Inner Mongolia Plateau. Wind
speed and wind direction (010C cup anemometers and 020C wind vanes, Metone,
USA), atmospheric pressure (CS106 Campbell, USA), air temperature, and
humidity (HC2-S3, Rotronic, Switzerland) were measured at four levels (i.e., 10, 30, 50, and 70 m) of the tower, which is surrounded by typical desert
grassland. The site is characterized by a middle temperate zone and
semi-arid continental climate. During the study period, the daily air
temperature at the 2 m height ranged between −27.3 and 33.9^{∘}, with an average value of 6.3^{∘} (Fig. 2a).
Surface-level air pressure has an inverse relation with air temperature,
with an average value of 862.9 hPa (Fig. 2b). In addition, the daily
average relative humidity at the 2 m height maintains a level of 41.02 %
and fluctuates back and forth. The average wind speed at the 70 m height is
7.6 m s^{−1}. The daily averaged wind speed in spring and autumn occasionally exceeds the level of 10.0 m s^{−1}, indicating that the site has sufficient wind
resources in these two seasons (Fig. 2c). The predominant wind direction
was southwesterly and northwesterly during the whole observation period
(Figs. 2d and 3).

## 2.2 Methods

### 2.2.1 Kernel, Weibull, and Rayleigh distributions

The kernel density estimator is the estimated PDF of a random variable. For any real values of *v*, the formula for the
kernel density estimator is given by:

where *v*_{1}, *v*_{2}, …, *v*_{i} are random wind samples from an
unknown distribution, *n* is the sample size, *K*(⋅) is the kernel
smoothing function, and *h* is the bandwidth.

The PDF of the Weibull distribution is given by:

The Rayleigh model is a special and simplified case of the Weibull model. It
is obtained when the shape factor *k* of the Weibull model is assumed to be
equal to 2.

The maximum likelihood estimation method is a mathematical expression
recognized as a likelihood function of the wind speed data in a time series
format. In this method, many numerical iterations can be required to
determine the *k* and *c* parameters of the Weibull function. The parameter
estimation formula of the maximum likelihood method is as follows:

The average value and standard deviation of the wind speed can be obtained from the following formulas, respectively:

Alternatively, the mean wind speed can be determined from:

if the PDF is known.

If Eq. (7) is solved together with Eq. (2) making the substitution
of $\mathit{\xi}=(v/c{)}^{k}$ for *v*, the following is obtained for the mean wind
speed:

Note that the gamma function has the properties of $\mathrm{\Gamma}\left(x\right)={\int}_{\mathrm{0}}^{\mathrm{\infty}}{\mathit{\xi}}^{x-\mathrm{1}}\mathrm{exp}(-\mathit{\xi})\mathrm{d}\mathit{\xi}$ and $\mathrm{\Gamma}(\mathrm{1}+x)=x\mathrm{\Gamma}\left(x\right)$.

### 2.2.2 Power density

The mean power density for the kernel smoothing function becomes:

The mean power density for the Weibull function becomes:

The mean power density for the Rayleigh model is found to be:

where *ρ* is the air density.

### 2.2.3 Weibull parameters

The relationship between scale factor *c* and height can be expressed as
follows:

Here *c*_{10} represents the scale factor at 10 m height, *z* represents the
height, and *α* represents the power exponent parameter to be
estimated.

The relationship between scale factor *k* and height can be expressed as
follows:

where *a*, *b*^{′}, and *d* are unknown parameters to be fitted to the quadratic function.

In addition, as shown in Fig. 6c, Justus and Mikhail (1976) gave
the following formula for the shape factor *k* with height:

where *k*_{10} is the shape factor at a reference height of 10 m. At a
reference height of 10 m, *b*=*b*_{10} is just some constant, whose value can be determined by a least squares fit of relation (14) to the data.

### 2.2.4 Surface roughness length

When the wind speed at three or more heights is measured, the roughness length calculated by the least square regression (Archer and Jacobson, 2003; Archer, 2005; Golbazi and Archer, 2019) is:

where *z*_{R} is the reference height, *z*_{i} is the height of the
other three layers, *N*=4 representing four vertical layers, and *U*_{i} is the wind speed corresponding to the height of the other three layers. In
most cases, it is a purely mathematical statistical method, and therefore this simple
mathematical method does not require a physical explanation for roughness
estimation.

In addition, the aforementioned method is obtained from the logarithmic wind speed profile, which is a typical form of wind speed profile under neutral stratification. A calculation of the wind speed at other altitudes under the reference altitude can be obtained from the following formula (Golbazi and Archer, 2019; Archer and Jacobson, 2003):

where *z*_{0} is the estimated surface roughness length, assuming that the
friction speed near the ground does not change with height.

## 3.1 Comparisons of kernel, Weibull, and Rayleigh models

The monthly, seasonal, and annual average wind speed values and standard
deviations calculated using Eqs. (5) and (6) for the available time
series data are shown in Table 1. It can be seen from Table 1 that the
highest average wind speeds occurred in May and December 2019 and in May
2020, and the lowest average wind speeds occurred in February and August
2019. Over about 2 years, it was found that the average wind speed in the
spring of 2019 and 2020 was higher, and the average wind speed in the summer
of 2019 and 2020 was lower. During the entire study period, the
average wind speed values at 10, 30, 50, and 70 m were 6.0, 6.8, 7.2, and 7.6 m s^{−1}, respectively, which also shows that the wind
speed value increases with an increase in altitude.

Figure 4 shows the frequency density histogram of the wind speed at 70 m for
about 2 years and the probability density curves of the Weibull, kernel,
and Rayleigh distributions. First of all, it is obvious from the frequency
histogram that the wind speed at 70 m fluctuated drastically in the autumn
of 2018, spring of 2019, and summer of 2020. This conclusion can also be
confirmed from the data in Table 1. The shape factor *k* values of these three
specific seasons are 2.18, 2.13, and 2.11, respectively, which are slightly
higher than the shape factor *k* value of the Rayleigh distribution. In
combination with Table 1, it is also found that the higher the value of the
scale factor *c*, the smoother the three specific probability distribution
curves. By contrast, as shown in Fig. 4d, its three specific probability
density curves are very sharp.

Although the kernel distribution also has specific parameters to control its
probability density curve, it does not have the general form of wind speed
distribution. Moreover, the *k* value of the Weibull distribution is
∼ 2. To select the specific wind speed distribution form
suitable for the Ningyuanbailiutu site, therefore, the model prediction
accuracies of the Weibull distribution and the Rayleigh distribution for
average wind power need to be compared further.

The mean power densities calculated from the measured probability density
distributions and those obtained from the models are shown in Fig. 5. The
mean power density shows significant monthly and seasonal variation. The
minimum average power density appeared in August 2019 and was only 214.9 W m^{−2}. In addition, smaller mean wind power densities appeared in July and September 2019 and January, July, and August 2020, which were generally lower than 350.0 W m^{−2}. Generally, the maximum value of monthly mean wind power density reached 862.4 W m^{−2} in May 2019, and the seasonal mean wind power density peaked in spring 2020.

The differences between the kernel distribution, Weibull distribution, and
Rayleigh distribution are explored when calculating the average wind power
density and the frequency distribution using the original wind speed data.
The 2-year mean absolute percentage error (MAPE) values in calculating the
mean power density using the kernel, Weibull, and Rayleigh functions are
1.17 %, 1.05 %, and 4.20 %, respectively. The RMSEs of the kernel distribution, Weibull distribution, and Rayleigh
distribution are 45.8, 60.5, and 875.3 W m^{−2},
respectively. The Weibull and kernel models return smaller error values in
calculating the mean power density compared to the Rayleigh model. The mean
power density is estimated by the Rayleigh model to have a very large
absolute error value of 83.1 W m^{−2} in December 2019. On the other hand, the highest absolute error value occurs in May 2019 with 21.3 W m^{−2} for the Weibull model.

Analysis of residual error and average percentage error suggests that the average wind power density estimated by the Weibull distribution with specific parameter control is very similar to the kernel distribution, which is closest to the original wind frequency distribution (Fig. 5c). The lower limit of the 95 % prediction interval is each predicted value minus 1.96 standard deviations, and the upper limit is each predicted value plus 1.96 standard deviations (Fig. 5b–d). This suggests that the interval applicability of the three specific distribution models is good.

## 3.2 Vertical characteristics of Weibull parameters

Figure 6 shows the characteristic variation of the scale factor *c* and the
shape factor *k* with height estimated from the Weibull distribution for
original wind speed data during the study period, exhibiting power
exponential and quadratic function variations, respectively.

Table 2 gives in detail the values of *α*, *a*, *b*^{′}, *d*, and *b*_{10}
obtained by the least squares fitting method for each month, each season,
and all time periods, as well as the corresponding RMSEs obtained from the formula.
When using the power exponent formula (12) to fit the relationship between
the scale factor *c* and the height, the RMSE has the smallest values in
January 2019 and July and August 2020. However, in December 2019, January
2020, and February 2020 it has the largest values. This shows that formula (12) has a better fitting effect in the winter of 2018, and a poor fitting effect in the winter of 2019. Justus and Mikhail (1976) found that the mean value of *α* was 0.23. In the present study, the mean value of *a* for each month over the whole 2 years is 0.117, and the corresponding standard deviation is 0.016.

Figure 6b and c indicate that the two different formula forms have a good
fitting relationship for shape factor *k* and height. The RMSEs of Table 2 also suggest that the effect of the quadratic function fitting is better than the logarithmic reciprocal function of Justus and Mikhail (1976). The RMSE of
the quadratic function fitted to all data for 2 whole years is 0.0078, but
the RMSE of the logarithmic reciprocal function is 0.0214, which is close to
a multiple of 1 : 3. Both of these two types of formula are basically applicable only to heights below 100 m. In addition, from a comparison of Fig. 6b and c, it can be seen that there will be different trends in the
change in the *k* value with height, and the increasing or decreasing speed of the *k* value in the form of a quadratic function will be higher than that
found by Justus and Mikhail (1976) when the height is greater than 70 m.
This different trend will lead to large errors in estimating wind energy
resources above 70 m.

## 3.3 Spatial–temporal variations in surface roughness length

The shape of the wind profile is greatly affected by the surface roughness in the direction of the incoming flow. Thus, surface roughness is a key element in wind energy resource evaluation and forecasting models. In calculating aerodynamic roughness, especially in practical applications, the least squares approximation of the logarithmic profile equation to the measured wind speed profile method has been widely used, referred to as the “logarithmic profile method”.

After calculating the 15 min continuous wind speed data using the above
method, quality control of the data is carried out. In this study, we eliminated wind speeds greater than or equal to 6.0 m s^{−1} at 50 m, and the estimated abnormal roughness data are infinitely large or infinitely small. Figure 7 shows that both the average and median monthly roughness lengths in January, February, and March 2019 are significantly less than those in August, September, and October 2019. The largest value of median roughness
was close to 0.19 m in October 2019, and the maximum value of average
roughness was approximately equal to 0.27 m. In June 2020, the median and
average roughness values reached 0.18 and 0.25 m, respectively. The
minimum value of median roughness was about 0.10 m in January 2019, and the
smallest value of average roughness value was about 0.20 m in January 2020.

In addition, the median and average roughness length were lowest at about 0.12 and 0.22 m in the winter of 2018 and 2019, while the highest were about 0.15 and 0.25 m in the autumn of 2019. It is notable that the roughness length steadily increases from winter to autumn. In short, this suggests that the grassland vegetation in the site area has an obvious wilting period, growing season, and lush period. Compared with the average roughness length, the representative roughness length of the area fitted the median value more closely.

According to the Davenport land type roughness classification (Davenport et
al., 2000) and the summary of roughness length over the wind tower sites and the
corresponding land types (Li et al., 2021), in the case of land types with less
vegetation and cropland, the roughness length is generally estimated to be a
slightly rough open area of about 0.10 m. The area we studied belongs to the
grassland vegetation type, and the roughness estimate should be around 0.13 m, and will not be classified as rough; that is, the roughness length is as high as 0.25 m. In addition, in a study (Golbazi and Archer, 2019) on the estimation of sea surface roughness length in coastal waters, it is mentioned that the statistical method uses a single constant value of
*z*_{0} in the representative area, and the median value can be worth
recommending.

Figure 8 shows the variation in the estimated roughness length in 12
different incoming wind directions. When the wind direction is
120 or 240^{∘}, the estimated roughness length is
highest, and the median value and average value are about 0.23 and 0.30 m,
respectively. Secondly, when the wind direction is 30 or
300^{∘}, the estimated roughness length is lowest, and the median
value and average value are about 0.08 and 0.18 m, respectively.
Therefore, between the highest and lowest estimated roughness lengths, there
is a specific trend of increasing or decreasing. This phenomenon can be
explained in conjunction with Figs. 1 and 3. There is a hillside to
the west of the wind tower. Therefore, when the incoming wind direction is
120 or 240^{∘}, it is on the windward side or leeward
side, respectively, of the wind-measuring tower. In this way, there will be
a pressure difference, which will increase friction loss and increase the
estimate of the effective roughness length. When the incoming wind direction
is 30 or 330^{∘}, it is found that the wind passing
through the wind measurement tower will not be greatly affected by the
terrain. The terrain is relatively flat, and the estimated roughness length
is close to the normal value of 0.10 m. In addition, in the plot of
roughness length estimation with wind direction, there are obviously more
data points in the wind directions from 180 to 330^{∘} than in the other wind directions. The 240^{∘} wind direction has the
most data points, which also shows that the site has a southwesterly wind
blowing all year round.

## 3.4 Extrapolation of the average wind power density

With the scale factor *c* changing with height in the form of a power function, and shape factor *k* changing with height in the form of a quadratic function, the scale factor *c* and the shape factor *k* at 75, 80, 85, 90, 95, and 100 m are calculated. Then the average wind power density (Fig. 9b) is calculated for each month, each season, and the whole time period from formula (10). On the other hand, when studying the roughness length parameter in the previous section, we assume that the roughness length
calculated from the four-layer height is dynamic. Then through the
logarithmic form of formula (16), we can calculate the wind speed values
at 75, 80, 85, 90, 95, and 100 m every 15 min. Finally, the
“reference average power density” (Fig. 9a) at six specific heights can
also be obtained.

Both the power-law model and the logarithmic model can estimate the average
wind power density of six specific heights, and as seen in Fig. 9 the values estimated by the two methods show greater differences in
autumn and winter, and smaller differences in spring and summer. In
addition, the two different models both show that the average power density
values are largest in spring and smallest in summer. Although the average
power density values increased with height over the whole experiment, the
gradient of the increase in average power density values with height is
largest in autumn and winter, and smallest in spring and summer. Figure 9c
shows that relative to the power-law model, the average power density of the
logarithmic model extrapolated at 70–100 m is smaller in the winter of 2018
and in July and August of 2020, while it is larger in other study periods.
Generally speaking, the difference between the estimated average power
density values is very small. However, the data and methods used in the
estimation of the two models are different. The result of this estimation
gives us important guidance for studying two Weibull parameters, namely, the
scale factor *c* and the shape factor *k*, and the surface roughness length
parameter.

## 4.1 Applicability of Weibull and Rayleigh models

There are various statistical distribution functions for describing and analyzing wind data, including normal, lognormal, Rayleigh, and Weibull probability distributions (Fagbenle et al., 2011; Ozerdem and Turkeli, 2003). It has been found that the Weibull and Rayleigh distributions are the most accurate and adequate in wind analysis as well as in interpreting the actual wind speed data and in predicting the characteristics of the prevailing wind profile. A kernel distribution is a nonparametric representation of the PDF of a random variable. A kernel distribution is defined by a smoothing function and a bandwidth value, which control the smoothness of the resulting density curve (Kafadar et al., 1999). In fact, some scholars have used the probability density distribution of wind speed to compare the advantages and disadvantages of the Weibull distribution and the Rayleigh distribution (Celik, 2004; Pishgar-Komleh et al., 2015). In our present work, the kernel function exhibits the feature of the smooth function, and is also closer to the actual frequency distribution (Fig. 4), which can be used to fit the original wind speed data. Therefore, the kernel function is employed as a medium for comparing the pros and cons of the Weibull and Rayleigh functions in the desert steppe area.

Celik (2003) employed the Weibull function to analyze wind power density in
six different regions around the world, and the average percentage errors
obtained were relatively low. The reason may be that the scale factor *c*
representing the average wind and the shape factor *k* are relatively small,
the original wind speed is relatively stable and fluctuates little, and also
the Weibull function has strong applicability. Celik (2004, 2011) used the
Weibull function and Rayleigh function to calculate wind power density in Turkey, showing the average annual relative errors of < 8 % and 37 %, respectively. Pishgar-Komleh et al. (2015), on the other hand, reported annual average errors of 55.00 % for both the Weibull function and the Rayleigh function. These values vary largely from our present results, due
to the different applicability of the two specific distribution functions in
different regions. Therefore, when developing new wind farms, it is
extremely important and necessary to compare the applicability of various
wind speed distribution functions in the local area. In general, we found that the
Weibull distribution is applicable for depicting the wind speed distribution
at the Ningyuanbailiutu site in northern China.

In addition, although the mean wind power density calculated in this study is in good agreement with the actual grid-connected average power density (Fig. 5a), there is a significant difference in these two values. This is because the wind turbines are not always connected to the grid, due to failures, or to other wind turbines outside the range of the wind measurement tower. As a result, the wind measured by a single wind tower will underestimate the wind speed of other wind turbines. The limitation of this study is that data from only a single wind tower were derived. For future wind energy density estimations, it is worth collecting data from more wind towers to obtain a more realistic wind resource distribution in the study area.

## 4.2 The complexity of Weibull parameters and surface roughness

Figures 6, 7, 8, and Table 2 show the spatiotemporal
variability of scale factor *c*, shape factor *k*, and surface roughness
*z*_{0}, which can be attributed to the following three aspects: (a) the type
of surface land and meteorological conditions (Golbazi and Archer, 2019);
(b) the uncertainty of Weibull parameters calculated using maximum likelihood
method (Mohammadi et al., 2016) and uncertainty of the roughness length
calculated using statistical mathematical methods (Kim and Lim, 2017); (c) the limitations of extrapolating high-level Weibull parameters (Justus and Mikhail, 1976). The shape factor *k* varies with height and exhibits not only the form of the reciprocal of the logarithmic function but also the form of the quadratic function. From the RMSE in Table 2, it can be seen that the
quadratic function is the most suitable for this study area. Therefore, to
use the Weibull function to evaluate the high-level wind speed distribution
in a specific research area in the future, we should consider high-density
observations with more fitting methods so as to obtain the best functional form of the Weibull parameter varying with height. Table 3 reviews scale factor *c*,
shape factor *k*, surface roughness *z*_{0}, and yearly mean absolute percentage
error over different topographies, showing obvious regional differences, due
to various climate and topography contexts with different methods. For
example, Pishgar-Komleh et al. (2015) used a constant surface roughness
value of 0.14 to extrapolate wind speed, ignoring the dynamic changes in
surface roughness throughout the year. The calculated annual mean absolute
percentage error is much higher than the value calculated by the dynamic
surface roughness in this study. Therefore, we should take dynamic roughness
into account based on a reliable and accurate topographic map, rather than
assuming surface roughness as a constant.

The differences and uncertainties between the logarithmic and power-law
models can also be seen in Fig. 9. Its uncertainty is manifested in the
absence of verification of actual high-level wind speeds. The reason for the
difference is that the shape factor *k* in the winter of 2018 and in July and
August of 2020 shows a decreasing trend with height, and the average wind
energy density is inversely proportional to the shape factor *k*, according to formula (10). Therefore, the limitation of this paper is that the
extrapolated results need to be further confirmed by future encrypted
observations of high-level wind speed data. In addition, the encouraging
news is that this gives us two reliable options for future extrapolation of
high-level wind energy density: In the case of non-Weibull winds, only the
logarithmic model can be considered, and in the case of Weibull winds, both
the logarithmic model and the power-law model are good choices.

The present work investigated the scale factor *c* and the shape factor *k* that affect the Weibull distribution of wind speed, by directly estimating the energy potential of the wind speed resource at four different heights, and the surface roughness length parameter that directly affects the shape and law of the wind profile. The main conclusions are:

The 2-year mean absolute percentage error values in calculating the mean
power density using the kernel, Weibull, and Rayleigh functions are
1.17 %, 1.05 %, and 4.20 %, respectively. The Weibull wind speed
distribution model is the most suitable wind speed distribution model for
the Ningyuanbailiutu site. The scale factor *c* increases with an increase in
height, showing an obvious form of power function. The shape factor *k*
increases or decreases with height and has two different forms, which are
the reciprocal of the quadratic function and the logarithmic function. For
further determination of the changes in form factor with height, it will be necessary in future to set up intensive observations for heights
above 70 m and below 100 m.

When estimating the surface roughness length, the median value is selected
as the representative value of the surface roughness length. This is based
not only on recognition of actual previous research, but also on
confirmation of actual grassland vegetation types. Although the
statistically calculated *z*_{0} does not have a proper physical explanation,
it gives the most accurate wind speed estimate at the required height. The
estimated roughness length varies with the seasons of the grassland
vegetation at the site. The estimated roughness lengths of the wilting
period, growing season, and lush period are about 0.12, 0.13, and 0.15 m, respectively. The estimated surface roughness length will be affected by
the windward and leeward sides. When the wind flows across the hillside,
there will be a pressure difference, which will increase the friction loss
and increase the estimated effective roughness length. The prevailing wind
direction at this site is 240^{∘}, which happens to be the direction
of the windward side of the site. The estimated roughness length is about
0.23 m. Finally, the power-law model and the logarithmic model were employed
to estimate the average power density values at 75, 80, 8, 90, 95, and 100 m. The two models show greater differences in autumn and winter,
and smaller differences in spring and summer. The gradient of the increase
in average power density values with height is largest in autumn and winter,
and smallest in spring and summer.

In general, against a carbon-neutral background, the determination of the
potential for economical and clean wind energy resources is an important
scientific issue in the development of renewable energy worldwide. Our
research has determined the possible relationship between Weibull natural
wind mesoscale parameter *c* and shape factor *k* with height under the conditions
of a desert steppe terrain in northern China; this has great potential in
wind power generation, but there is a lack of comprehensive investigations
into key parameters for estimating wind power density from tower data. In
the present study, we gained an enhanced understanding of the seasonal
changes in the surface roughness of the desert grassland and the changes in
the incoming wind direction. Our findings also have important implications
for the assessment of wind energy resources for the establishment of new
wind farms in areas experiencing varied desert steppe terrains throughout
the world.

The model and all implementation and analysis codes in this paper are based on MatLab and Python, and are available from the author (20191203039@nuist.edu.cn) upon request.

The data in this paper come from an observation wind tower and are available upon request from the author (20191203039@nuist.edu.cn).

YY and ZG were responsible for the conceptualization, supervision and funding acquisition. SZ developed the software and prepared the original draft. SZ and YY developed the methodology and carried out the formal analysis. XX and SZ validated the data. ZG, YY, XX, ZD, and YL reviewed and edited the text. SZ was responsible for visualization. All authors have read and agreed on the published version of the paper.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We are very grateful to two reviewers for their careful review and valuable comments, which led to a substantial improvement of the paper.

This research has been supported by the National Natural Science Foundation of China (grant no. 41875013).

This paper was edited by Simone Lolli and reviewed by two anonymous referees.

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*c*and shape factor

*k*with height under the conditions of a desert steppe terrain in northern China, which has great potential in wind power generation. We have gained an enhanced understanding of the seasonal changes in the surface roughness of the desert grassland and the changes in the incoming wind direction.