Today, the use of wind energy by wind turbines has grown significantly, and this development is due to the production of required energy and tourism attraction of wind turbines. But according to the standard technologies used in this industry, the operating costs are very high. For this reason, before the construction of wind farms, potential measurements should be done along with economic analysis. Therefore, in this research, a statistical analysis of wind farms has been done. In the statistical analysis, Weibull and Rayleigh distribution functions were used to predict the wind speed of the studied area. MATLAB software is used to model prediction functions. Among the important results of wind speed prediction by the Weibull distribution function, it can be mentioned that the wind speed is variable between 0.6 and 7 m/s in the studied area. The total power density and wind energy in the 10 years are equivalent to 28 W/m2 and 810/0534 kWh/m2 at the height of 10 m calculated.

Topics
Electrical energy, Power plants, Wind energy, Wind turbines, Atmospheric thermodynamics, MATLAB, Statistical models, Probability theory, Descriptive statistics, Gaussian processes

In technical energy literature, there are different definitions for renewable energy that comes from natural processes that are constantly revived, renewed, and exploited. Although this definition is debatable in terms of the time horizon of resource renewal, it has been accepted as a general definition so far. Since 2011, renewable resources have had growth at an annual rate of about 1.2%. This growth rate for energy sources such as wind and sun has more growth.1,2 Most of it has been in the member countries of the Organization for Economic Cooperation and Development (OECD) and due to the development of executive programs in this field in some countries similar to Denmark and Germany.3,4 Some of the most important reasons mentioned for using non-fossil energy that the possible increase in the use of fossil energy resources as a result of the increase in population, decrease in available reserves, increase in environmental pollution, and emission of greenhouse gases.5,6 In Iran, the capacity of installing turbines to produce wind-electric energy until 2014 is equal to 51.42 MW and 32 turbines.7 This amount will reach 121 MW by the middle of 1312. In general, the capacity of these turbines ranges reported from 131 to that 991 kW, which is the highest frequency belonging to 991 kW turbines. It should be noted that the share of electricity–wind energy produced in 2014 is equal to 14.1% of the electricity that was produced in the country.8,9 Related information to wind speed for different regions of the country in different months of the year is reported by the Meteorological Department. The purpose of this research is to examine the general condition of the wind and its characteristics, including direction, speed, and stability, in Markazi province, Saveh city. In addition, the determination of wind potential and the feasibility of energy extraction with the help of statistical methods in this field.

In meteorology, the wind is considered a vector that is like any other vector has direction and magnitude. The size of this vector expresses the speed of the wind. Wind speed is the distance that the wind travels in a unit of time and with units, such as knots (one knot equals 1/152 km/h), m/s (1 m/s is equivalent to 3.9 km/h), and km/h (1 km/h equivalent to 1.222 m/s).10,11

Several statistical distributions (Weibull, Rayleigh, Johnson, Pearson, Chi-2, Lognormal, Inverse Gaussian) to determine the wind energy potential that there are many researchers of the Weibull distribution for various reasons.12,13 For the superiority of Weibull distribution in wind energy, potential analysis use the following reasons:14,15

  • The distribution of wind speed has a skewness that is well-defined in this distribution.

  • In the areas where the wind power is maximum, wind speed data have the highest average and the lowest parameter value.

There are also some advantages to using this distribution:

  • Weibull distribution for wind speed data has less error in comparison to the square root of the normal distribution.

  • By changing the position of the height of the anemometer, it is still possible to estimate the distribution parameters Weibull.

  • Ease of doing calculations compared to other distributions, such as distribution bivariate normal, because this distribution has only two parameters.

Several types of research regarding the description of wind behavior have been done, which are based on meteorological statistics. In a study, wind speed and direction were simulated through the Weibull distribution and this is the result according to the wind histogram; the normal distribution is not a suitable distribution for wind speed data, whereas the Weibull distribution is the most extensive model that can be used in the field of wind speed.16,17 Kaplan presented a calculation of Weibull distribution parameters at low wind speed and performance analysis.18 Yang et al. presented a bivariate distribution mixture of wind speed and air density for wind energy assessment.19 In 2023, presented a Weibull distribution analysis of wind power-generating potential on the southwest coast of Aceh, Indonesia.20 Khanh et al. presented a determination of profitable wind farm generating capacity based on Weibull distribution of wind speed in the competitive electricity market.21 

According to the research background, more studies on the feasibility of wind energy are needed in different cities in Iran. In this study, the information related to wind speed for different regions of Iran in different months of the year is reported by the meteorological department. The purpose of this research is to investigate the general condition of wind, its characteristics in the wind direction, and its speed and stability in Markazi province Saveh city. In addition, determination of wind potential and exploitation feasibility of energy. According to the previous research, for the first time, MATLAB software was used for this purpose.

Saveh is one of the northern cities of the Arak province of Iran. According to the census of 1331, the population of this city was equal to 50 650 people. Saveh is located at a longitude of 36° and 15 min and a latitude of 35° and 63 min and the height is 1205 m above sea level. Figure 1 shows the geographical characteristics of the Saveh.

FIG. 1.
Geographical characteristics of the Saveh.
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Geographical characteristics of the Saveh.

FIG. 1.
Geographical characteristics of the Saveh.
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Geographical characteristics of the Saveh.

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One of the important criteria in the construction of places for the exploitation of wind energy, such as wind power plants, is the measurement of the annual average speed and the wind power density.22,23 For this purpose and to perform statistical analyses, consecutive 3 h data, including wind speed over the years 2018 to 2020, were collected in Saveh city of Markazi province. It should be noted that the raw data of wind speed, which are entered in terms of (Knots), were converted to meters per second with a factor of 1.605 because the unit of meters per second was used in most of the previous sources and research in this field.

The collected data, including wind speed and direction, were arranged in such a way that the wind direction–speed frequency table, which indicates the frequency of the occurrence of wind with a certain speed, was drawn for each month and finally all the years during 11 years. Many types of research have shown that the wind speed at higher altitudes is obtained from Eq. (1), which is an empirical formula24 
(1)
V is the wind speed at height H, and Vmes is the wind speed at height Hmes (the base height is 10 m). α indicates the roughness coefficient of the surface, which is related to surface complications, time, wind speed, and temperature. Its value in this research is assumed to be between 0.128 and 0.160. The average value for this coefficient is 0.14 (for surfaces with low roughness), and for flat lands, water, or frozen surfaces, the lowest value is equal to 0.10 and for forest lands covered by trees is more than 0.25. As mentioned, the value of the roughness coefficient is usually the middle limit of this interval and is considered equal to 1/143 that in this study also used this value.
The meaning of wind energy potential is the ability of wind to produce electrical energy with the increase in wind energy potential. The amount of electricity production has increased, and as a result, the efficiency of the wind energy conversion system has increased. Studies have shown that a 10% change in wind speed changes the power generated from the turbine by 25%. Wind speed data in wind energy potentiometric calculations are continuous random variables that a certain interval can include all possible values. But because data collection from meteorological stations is done at 3 h intervals has taken, the distribution function of such probability will be discrete, and in this situation, it is necessary to use a continuous distribution function to distribute the frequency of wind speeds. For this purpose, previous studies, mostly from Weibel and Rayleigh probability distribution function. The Rayleigh distribution is much simpler compared to the Weibull distribution due to its one-parameter nature. The density value of this distribution function is calculated from Eq. (2) for different values of wind speed (V),24 
(2)
f(V) is the probability of occurrence of different values of wind speed, c (c > 1) is the distribution scale parameter in terms of (m/s), and V wind speed [for V < 0 the amount of f(V) = 0], which is in m/s. Weibull distribution probability density function [Eq. (3)] has two parameters, including the shape parameter (k) and the scale parameter (c). The equation of the probability density function of this distribution shows that the Rayleigh distribution is a special case of this distribution, taking into account the shape parameter (k) equal to 2,
(3)

V, c, k (k > 0) are, respectively, wind speed, scale parameter (average), and dimensionless parameter known as shape parameter (variance or skewness). It is necessary to explain that the probability density function of the mentioned distribution has been reported that the Weibull distribution is more flexible than the Rayleigh distribution with velocities close to zero on representing the fraction of time that the desired wind speed prevails in that area.

As mentioned, Weibull distribution parameters (c, k) are effective in the shape of the graph and skewness. As shown in Fig. 2, the distribution graph becomes wider as the value of (c) increases for fixed values of (k). It is for this reason that parameter c is called the scale parameter, whose unit is m/s (equal to the unit of wind speed). Also, it can be deduced from the graphs that by increasing the value of (k) for certain values of (c), the maximum value of the probability density function increases. Therefore, parameter (k) is known as the shape parameter. The cumulative distribution function (CDF) of the Weibull distribution can be calculated using
(4)
fV is a cumulative distribution function for observed values of the speed of the wind. Considering k = 2, the cumulative distribution function of Rayleigh distribution is obtained, which is in the form of
(5)
FIG. 2.
The Weibull distribution diagram with different (k) and (c) parameters.
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The Weibull distribution diagram with different (k) and (c) parameters.

FIG. 2.
The Weibull distribution diagram with different (k) and (c) parameters.
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The Weibull distribution diagram with different (k) and (c) parameters.

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The maximum value of the probability density function (PDF) is called the most probable point or in statistical terms the mode. By deriving the probability density function of the Weibull distribution [Eq. (3)] and then setting it equal to zero according to
(6)
By factoring the expression inside the bracket and setting it equal to zero with
(7)
By solving the above equation, Eq. (8) is obtained to calculate the most probable speed (view) for the probability density function of the Weibull distribution (Vmp)
(8)

There are different methods for calculating shape and scale parameters in Weibull and Rayleigh functions. The “least square fitting” method is one of the most common methods that is obtained by using the regression equation and the linear equation between the wind speed values and the probability of each occurrence. The “maximum likelihood estimate” method (MLE) is also another method of estimating distribution parameters. When an operation is performed on a set of data, a statistical model is obtained, then maximum accuracy can provide an estimation of model parameters. The general mode of the method (MLE) in the case of a specific set of data consists of assigning values to the model parameters, as a result of a distribution that is produced that gives the highest probability to the observed data.

Weibull and Rayleigh distribution parameters can also be easily calculated by having random wind speed data and using the maximum accuracy estimation method. The parameters calculated for the probability distributions with the help of this method are the most probable state according to the information available from the random data, which can be considered the correct estimation of the parameters. With the help of MATLAB software and using the (wblfit or data) command, the distribution parameters were calculated using the MLE method and for data at heights of 11, 31, and 41 m. The output of this command is c and k parameters, respectively.

As mentioned before, wind power is the result of kinetic energy per unit of wind mass (⁠V22⁠) and the mass flow rate of wind that flows between the blades of the turbine (ρVA). Obviously, the higher the wind speed in the region, the higher the potential of wind energy to produce electricity. Other factors, such as the size of the turbine, also affect the output power of the turbine. The increase in the size of the turbine along with the increase in the length of the blades increases the contact surface of the wind with the turbine, and as a result, the rotation speed of the turbine blades also increases. According to the description above, the power of the turbine by considering the two parameters of wind speed and its contact surface with the blade can be calculated from
(9)
where P̄ is the average wind power capacity (W), ρ is the air density (kg/m3), A is the indicator of the swept surface (π in Rotor radius – rπ) (m3), and V is the wind speed (m/s). According to Eq. (9), wind power is proportional to the average cube of wind speed whose value can be estimated from the equation (third-order torque of the Weibull distribution function). It should be noted that the average cube of wind speed is different from the average cube of speed. Equations (10) and (11) provide, respectively, the value of V3̄ according to Weibull distribution parameters (distribution Riley) and its third root (Vrmc) according to
(10)
(11)
(12)
where Γ is the famous Gamma function
(13)
According to Eq. (9), the wind power density (WPD), which is the wind power per area swept by the wind turbine [P(V)/A], is obtained from Eq. (14). In the above equation, air density (kg/m3) in the studied area is calculated from25,26
(14)
where P is the average air pressure (Pa), T is the average air temperature (°K), and R is the gas constant (278 J kg−1°−K−1). For the region in this study, the air density is 1.08 kg/m3 and was used in this study. Air density is a function of pressure and temperature; which changes are not considered very important in wind energy calculations. Air density decreases with the increase of water vapor in it, which can be corrected in this research. According to what was discussed in this section, considering the constant values of Eq. (14), the final power calculation equation for a wind turbine can be rewritten as
(15)
where D is the diameter of the rotor or the diameter of the surface of the turbine that is affected by the wind.

1. Calculation of wind energy density

In the next step, after calculating the wind energy density (WED) Eq. (14), the amount of wind energy density can be calculated for a region and in a period (T in hours) with the help of Eq. (16) (Eskin et al., 2008). T is considered for a monthly period (221 or 244 h) or an annual period
(16)

2. Other specifications of wind speed

The most probable wind speed or the most likely wind speed VEmp was introduced in the previous sections. The nominal wind speed or the wind speed that produces the maximum electrical energy is the wind speed that has the potential to produce the most energy throughout the year. This speed with VMaxe or VEmp is shown is considered one of the important parameters in the design of wind turbines and it is obtained from
(17)
Each turbine has other characteristics, such as minimum speed (Vi) and maximum speed (V0). (Vi) is the lowest value of the wind speed is that it is not able to generate electricity, and in fact, it is the wind speed to start producing power by the turbine. (V0) shows the maximum value of the wind speed, which decreases to the maximum power at speeds higher than the generated power. In fact, at high speeds, the turbine is designed to prevent damage in such a way that the blades are placed parallel to the direction of the wind and the wind prevents the proper rotation of the blades (stops), and as a result, the amount of electricity produced and the efficiency of the turbine decreases. The minimum speed and maximum speed in most turbines are considered to be 3 and 25 m/s, respectively. The speed limit is 3 m/h/s because the minimum speed of many commercial turbines in the country is important. The maximum speed of the turbines also varies between 21 and 25 m/s, and its value cannot exceed 25 m/s. Having information on the minimum and maximum speed can be obtained by using Eq. (18) to determine the probability of wind speed between two values of (V1 and V2),
(18)
The researchers have shown that the economic performance of wind turbines requires at least 4000 h of operation per year. As seen in the above equation, the nominal power can be calculated based on the nominal speed. According to the description above, the importance of proper turbine speed in the range Vi to V0 is essential for the correct calculation of turbine power. There are several methods for calculating the hours of wind existence in the area that is often time-consuming. With the help of Eq. (20), hours can estimate the presence of wind in a place in a simpler way,
(19)
In this regard, WE is equal to the amount of wind (h in year), fi is the abundance values of wind speed categories, N is the length of the statistical period of the study item (year), and (t) is the time interval between wind data collection (h).

Standard deviation is one of the most common criteria for statistical data dispersion. In this study, the standard deviation expresses the number of changes in wind speed that shows the stability of the wind in the study location. Therefore, the lower the value of this parameter is toward zero, the less dispersion around the mean is less and the data are more homogeneous; therefore, sudden and strong wind speeds occur less.

Since wind speeds are considered random events, it is expected that a probability distribution can be fitted to them, which is necessary to fit the data well. In most of the studies conducted, the distribution of the wind speed is measured by the Weibull distribution without checking the fit and validating the assumption. Undoubtedly, it is impossible to give a definite opinion on this matter without using suitable tools. Therefore, in this research, to provide a decision on the reliability of the model error, the goodness of fit test for the Weibull and Rayleigh distributions is implemented. In practice, investment in wind projects cannot be implemented based on decisions with a low degree of certainty. In this study, the Lilliefors tests and curve fitting tests were used in the MATLAB software environment, which will be explained further about these methods. Also in this research, validation of the curve fitting method using parameters statistics, such as coefficient of explanation (R2), the sum of squared error (SSE), and the smallest root mean square error (RMSE), was done as follows:
(20)
(21)
(22)

Xi is the observed value, Xî is the predicted amount, and X̄ is equal to the average of the observed values. To show the direction of the wind in the region the 19-direction-wind charts were drawn in different months for 11 years. The purpose of drawing such diagrams is to choose the most suitable turbine according to the wind conditions of the region. For this reason, the data of the prevailing wind direction in the region for a height of 11 m from the ground surface has been used, which was obtained from the weather station.

Because the power of wind energy per unit area (wind energy density) is also the direction of these winds, one of the important and determining factors in distinguishing the potential of wind energy in an area and the most suitable place to install the turbine. In addition, it is necessary to know the speed and direction of the wind on an hourly basis to find the frequency of the wind speed and ultimately the energy production potential. This research, uses the data obtained from the meteorological station of Saveh city in Markazi province for wind speed for 11 years (2010–2019AD) for potential measurement. Also, several models, such as Weibull and Rayleigh, were used to express the distribution of wind speed. More detailed results of these calculations are described below.

To understand the general condition of the wind in the studied station at a height of 11 m, the graph of the average wind speed for the year 2010 to 2019 was drawn (Fig. 3). These charts show the change in wind speed in different years. Generally, the wind speed varies between 1.9 and 2 m/s. The average wind speed during the statistical period is about 39.2 m/s. Another point that emerges from Fig. 3 is that the lowest wind speeds belong in the hot months of the year. On the contrary, the highest wind speed occurs in February and March, which can be considered one of the coldest months of the year.

FIG. 3.
The average wind speed in different months at the height of 10 m during the years 2010–2019.
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The average wind speed in different months at the height of 10 m during the years 2010–2019.

FIG. 3.
The average wind speed in different months at the height of 10 m during the years 2010–2019.
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The average wind speed in different months at the height of 10 m during the years 2010–2019.

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Table I shows the standard deviation of the data. As mentioned in Sec. III, the standard deviation of the wind speed data indicates the stability of the wind in a region. The monthly and annual values of the standard deviation of wind speeds at 10 and 40 m are given in the table. According to this table, in August at both altitudes, the value of the standard deviation is the lowest, which indicates the homogeneity of the wind speed data and the absence of sudden and extreme speeds in this month. Also, according to the average standard deviation in all months of the year, wind stability decreases with increasing altitude. According to Eq. (19) were calculated the probability of wind blowing with a speed between 3 and 25 m/s (presumed minimum and maximum speeds for most commercial turbines in the country) at a height of 10 and 40 m for the desired station is 36.4 and 43.6 percent of the total the hours of wind existence (14.1096 h/year), respectively. With these calculations, it can be claimed that the economic performance of the wind turbine in the study area is about 399 and 477.94 h/year, respectively. It is necessary to explain that the total number of wind hours in the region was calculated by using Eq. (20), which the parameters ∑fi,N,t equal to 3654, 10, 3, respectively, and thus the total hours of the existence of wind in the subject station comment was estimated as 1096.2 h/year. The daily wind speed changes are shown in Fig. 4. A similar trend is evident for the average wind speed at different hours during the studied years. The trend of changes for the year 2000 is different only with a slightly higher level of wind speed in this year, which can be due to the special weather conditions of the year. Also, it is possible to determine the hours of the day graph when there is the lowest and highest wind speed on the graph.

TABLE I.

The monthly and annual standard deviation of wind speeds at 10 and 40 m above the ground for Saveh station.

MonthWind speed at a high of 10 m (m/s)Wind speed at a high of 40 m (m/s)
January 3.5672 4.3312 
February 3.8977 4.7325 
March 3.2483 3.9441 
April 2.8330 3.4398 
May 1.9978 2.4258 
June 1.2935 1.5706 
July 1.1997 1.4567 
August 0.9430 1.1450 
September 1.2238 1.4859 
October 1.5396 1.8694 
November 2.6809 3.2552 
December 3.5417 4.3003 
MonthWind speed at a high of 10 m (m/s)Wind speed at a high of 40 m (m/s)
January 3.5672 4.3312 
February 3.8977 4.7325 
March 3.2483 3.9441 
April 2.8330 3.4398 
May 1.9978 2.4258 
June 1.2935 1.5706 
July 1.1997 1.4567 
August 0.9430 1.1450 
September 1.2238 1.4859 
October 1.5396 1.8694 
November 2.6809 3.2552 
December 3.5417 4.3003 
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FIG. 4.
The average daily changes in wind speed in 2010–2019 at Saveh.
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The average daily changes in wind speed in 2010–2019 at Saveh.

FIG. 4.
The average daily changes in wind speed in 2010–2019 at Saveh.
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The average daily changes in wind speed in 2010–2019 at Saveh.

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The lowest wind level between 12:00 PM and 6:00 AM and the highest wind speed from 9 AM to 3 PM was observed.

The distribution of wind speed in the hours of the day and night for the statistical period studied is shown in Fig. 5. This diagram is drawn with the aim of better analysis for the optimal exploitation of wind in energy production. As can be seen, the maximum wind speed during the day (9–12 AM) is equal to 46 and 4.2 m/s, and its lowest value is at midnight (3 and 6 AM). This graph is in the form of a polynomial graph that in the noon and afternoon hours reaches its peak. This result can also be logically justified by considering the temperature difference. Therefore, there are always stormy and turbulent winds in most parts of the world during the day and the wind situation is calmer at night. With this description, the potential of wind energy production is more during the day and in contrast to most of the electricity consumed at night, which is considered one of the advantages of this phenomenon.

FIG. 5.
The average wind speed distribution during the day and night hours for the statistical period.
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The average wind speed distribution during the day and night hours for the statistical period.

FIG. 5.
The average wind speed distribution during the day and night hours for the statistical period.
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The average wind speed distribution during the day and night hours for the statistical period.

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Weibull and Rayleigh distribution parameters were calculated by using the maximum likelihood estimation method in MATLAB software. So, the results for the years 2010 to 2019 are shown in Table II. According to the table, the lowest and the highest value of the parameter (k) was calculated as 1.1 and 2.094 in 2018 and 2015, respectively. The average value of the parameter (k) for the entire statistical period is equal to 1.48. The value of (c) was also a maximum of 4.5 and a minimum of 2.76 m/s and with an average of 3.23 m/s. Most likely wind speed values (Vmp) and nominal wind speed (VEmp) are presented in Table II. The most probable wind speed varies between 0.475 and 1.976. The average nominal speed in the studied station is about 7.81 m/s. This figure means that if wind energy is to be used in the station, it is better to advise the wind turbine manufacturers that the maximum load of the wind turbine output can be obtained at this speed. Because the high speeds are often observed in the data collection station in this research within the nominal speed range. The value of the predicted speed by years is shown in the last two columns of Table II.

TABLE II.

Wind energy components at Saveh station at a height of 10 m for the years 2010–2019.

YearK (-)C (m/s)Vmp (m/s)VEmp (m/s)(%) P (3 < v < 25)V̄ measuredV̄ predicted by WeibullV̄ predicted by Railly
2010 1.510 4.525 1.687 10.376 58.4117 4.084 4.225 4.039 
2011 1.475 3.169 1.126 9.052 39.7504 2.877 3.035 2.850 
2012 1.455 3.388 1.187 7.305 43.261 3.103 3.230 3.039 
2013 1.421 3.618 1.360 8.045 46.465 3.299 3.462 3.228 
2014 1.246 3.233 0.740 8.213 33.414 2.945 3.263 2.889 
2015 1.111 2.762 0.511 9.797 40.074 2.513 3.048 2.497 
2016 1.301 2.628 0.475 8.018 30.479 2.458 2.681 2.389 
2017 1.377 2.637 0.848 6.810 30.297 2.463 2.637 2.399 
2018 2.094 3.080 1.657 5.546 38.814 2.904 2.756 2.780 
2019 1.845 3.287 1.976 5.242 42.964 2.994 2.977 2.936 
YearK (-)C (m/s)Vmp (m/s)VEmp (m/s)(%) P (3 < v < 25)V̄ measuredV̄ predicted by WeibullV̄ predicted by Railly
2010 1.510 4.525 1.687 10.376 58.4117 4.084 4.225 4.039 
2011 1.475 3.169 1.126 9.052 39.7504 2.877 3.035 2.850 
2012 1.455 3.388 1.187 7.305 43.261 3.103 3.230 3.039 
2013 1.421 3.618 1.360 8.045 46.465 3.299 3.462 3.228 
2014 1.246 3.233 0.740 8.213 33.414 2.945 3.263 2.889 
2015 1.111 2.762 0.511 9.797 40.074 2.513 3.048 2.497 
2016 1.301 2.628 0.475 8.018 30.479 2.458 2.681 2.389 
2017 1.377 2.637 0.848 6.810 30.297 2.463 2.637 2.399 
2018 2.094 3.080 1.657 5.546 38.814 2.904 2.756 2.780 
2019 1.845 3.287 1.976 5.242 42.964 2.994 2.977 2.936 
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During this research, the empirical probability of discrete values of wind speed was replaced by Weibull and Rayleigh distribution. During this research, the empirical probability of discrete values of wind speed was replaced by Rayleigh and Weibull distribution. The correctness of this action can be evaluated with various goodness tests. The tests like chi-square, Anderson–Darling, Kolmogorov–Smirnov, and Li force test are among these. All these tests are used to determine whether the sample taken from the society follows a specific distribution or not. By using these tests, it is possible to measure the correctness of replacing the experimental probability of the data with the desired distribution. In this study, the goodness of fit test of the following assumptions was proposed:

  • The observed wind speeds in the check stations have a Weibull distribution.

  • The observed wind speeds in the check stations do have not a Weibull distribution.

This test is done by dividing the data into two groups and estimating the observed and expected values for each and finally calculating the assumption test statistic. All these tests can be used in MATLAB software, and in this research, the Li force test was used according to the target distribution and population size. From the command h = lillietest (x, alpha, distr) for this purpose used, where (h) is the parameter of test assumptions. The test calculates the true value of h = 1 if the opposite hypothesis is accepted at the 5% level and h = 0 if the test accepts the null hypothesis at the 5% level. (X) represents the data under test, and here the frequency of wind speed during different years of 2000–2009 is the result of measurement of the meteorological station and alpha is the significance level of test 5% and 1%. The argument of the input variable (dister) specifies the desired statistical distribution, which in this research was the Weibull and Rayleigh distribution. Using the equation between the limit value distribution and the Weibull log(W) = EW, the test for the Weibull distribution can be performed using the lillietest command. By using this test and calculating the h parameter, it was concluded that the observed wind speeds in the investigated station are consistent with the Weibull distribution.

h = lillietest (observed wind speed data), 0.05 eV

h = 0

There is another way to test the goodness of fit of the data, which is called curve fitting. The purpose of this test is to determine the appropriate fit of wind speed data with Weibull and Rayleigh distribution. For this purpose, (cftool) was used in MATLAB software. The output of using this tool for the frequency data of wind speed at heights of 10 and 40 m from the ground for the years studied in this research is described in Table III. Also, Figs. 6 and 7 show the fitting diagram of each height of 10 and 40 m for wind speed data. From Table III, it is obvious that the best results for Weibull distribution were obtained at a height of 10 m, which has the highest coefficient of explanation (R2), the lowest amount sum of squared error (SSE), and the least root means square error (RMSE). This conclusion is similar to the results from the study of twenty-year wind speed data from 2000 to 2019 of ten synoptic stations in Saudi Arabia, which was carried out in 1994, it showed that the wind speed data have a very good agreement with the Weibull distribution. In 1983, to check the matching of wind speed data on the Indian, Atlantic, and Pacific oceans with the Weibull distribution, it was determined that despite the matching of this distribution with the wind data, the amount of this matching has changed that the highest matching occurs in the areas with the lowest percentage of calm winds. In the present research, according to Figs. 6 and 7, the matching of two distributions with the measured data is more at higher speeds.

TABLE III.

The validation of Rayleigh and Weibull statistical distribution at 10 and 40 m height.

Statistical parametersR2AdjustedR2SSERMSE
The height of the turbine         
from the ground level (m) 10 40 10 40 10 40 10 40 
Weibull 0.989 0.968 0.989 0.967 0.00271 0.0069 0.0069 0.0106 
Rayleigh 0.548 0.449 0.548 0.449 0.0486 0.0466 0.0441 0.0432 
Statistical parametersR2AdjustedR2SSERMSE
The height of the turbine         
from the ground level (m) 10 40 10 40 10 40 10 40 
Weibull 0.989 0.968 0.989 0.967 0.00271 0.0069 0.0069 0.0106 
Rayleigh 0.548 0.449 0.548 0.449 0.0486 0.0466 0.0441 0.0432 
View Large
FIG. 6.
The fitting of wind speed frequency data was measured with Rayleigh and Weibull distribution at a height of 10 m from the ground.
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The fitting of wind speed frequency data was measured with Rayleigh and Weibull distribution at a height of 10 m from the ground.

FIG. 6.
The fitting of wind speed frequency data was measured with Rayleigh and Weibull distribution at a height of 10 m from the ground.
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The fitting of wind speed frequency data was measured with Rayleigh and Weibull distribution at a height of 10 m from the ground.

Close modal
FIG. 7.
The fitting of wind speed frequency data was measured with Rayleigh and Weibull distribution at a height of 40 m from the ground.
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The fitting of wind speed frequency data was measured with Rayleigh and Weibull distribution at a height of 40 m from the ground.

FIG. 7.
The fitting of wind speed frequency data was measured with Rayleigh and Weibull distribution at a height of 40 m from the ground.
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The fitting of wind speed frequency data was measured with Rayleigh and Weibull distribution at a height of 40 m from the ground.

Close modal

By Eq. (4) in which the cumulative wind frequency for Weibull and Rayleigh distribution is calculated, the probability of cumulative occurrence of different wind speeds between 2000 and 2009 was obtained and compared through the graph with the cumulative occurrence probability obtained from the measured data. Figure 8, which is drawn only for the height of 10 m from the ground, shows that at speeds higher than 10 m/s, the Rayleigh distribution is in perfect agreement with the cumulative frequency of wind and the cumulative frequency of occurrence of different wind speeds is about 100%. On the other hand, at speeds less than 10 m/s, the Weibull distribution has the highest agreement with cumulative frequencies of wind speeds.

FIG. 8.
The comparison of a cumulative probability distribution of measured data with values calculated by Rayleigh and Weibull distribution at 10 m height.
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The comparison of a cumulative probability distribution of measured data with values calculated by Rayleigh and Weibull distribution at 10 m height.

FIG. 8.
The comparison of a cumulative probability distribution of measured data with values calculated by Rayleigh and Weibull distribution at 10 m height.
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The comparison of a cumulative probability distribution of measured data with values calculated by Rayleigh and Weibull distribution at 10 m height.

Close modal

The power of wind energy per unit area (wind energy density) is very important in determining the location of the construction site and also the potential measurement of wind energy. This factor as well as the wind power density for the studied station were calculated as 657.67 (kWh/m2) and 79 (W/m2), respectively, for the time series of different years (2010–2019) at a height of 10 m from the ground. The more detailed results of this investigation according to wind speed levels are given in Table IV. Because the frequency of wind speed was well estimated by both Weibull and Rayleigh distributions, wind power and energy were also well estimated in this section. Moreover, in this section, only its value for different wind speeds at the height of 10, 30, and 40 m is reported. The energy resulting from the wind speed for some time is the swept area under the surface of the power curve multiplied by the number of days for the time. Table V shows that with the increase in height, the energy and power that can be extracted increases; therefore, a height of 40 m above the ground level is always recommended for wind turbine manufacturers. In the grading of the areas based on the wind power available at a height of 10 m above the ground, the places with wind power equal to 200–250 W/m2 are mentioned as suitable for installing wind power plants. Therefore, according to the more suitable density of wind power at a height of 40 m and above, as well as other calculated components, for Saveh station, this station does not have a relatively good potential for using wind energy. It is also carefully observed in the table that for the approximate speeds of 7–14 m/s, the power density and subsequently the wind energy density reaches their maximum value, while at speeds of 0 and 25 m/s, and both parameters decrease to zero.

TABLE IV.

The power density and wind energy in Saveh area.

Wind speedWind power density (W m2 year−1)The wind energy density (KW h m2 year−1)
(m/s)10 (m)30 (m)40 (m)10 (m)30 (m)40 (m)
0 0 0 0 0 0 0 
1 0.1259 0.1106 0.0955 0.4116 0.3338 0.2649 
2 0.8891 0.7509 0.8692 4.5107 3.4117 3.9448 
3 1.8474 1.8839 1.7122 11.4260 10.6928 9.5338 
4 2.6483 3.1220 3.1031 18.1587 20.1933 19.7209 
5 3.9717 4.2130 4.2130 29.2824 29.5549 29.0796 
6 5.8416 5.5877 5.6197 45.6344 41.5459 41.1629 
7 5.6266 7.5548 7.7576 45.4535 58.8732 59.6718 
8 5.5992 7.0388 8.0227 46.2271 56.4228 63.7515 
9 6.5718 7.7590 9.1599 55.2187 63.5367 74.6568 
10 6.3547 9.6085 7.8347 54.0504 80.1814 64.8522 
11 6.6878 8.6571 10.2312 57.4293 73.1564 85.9664 
12 6.3842 8.9404 8.6849 55.2058 76.3008 73.6830 
13 5.8442 7.4697 11.6917 50.7887 64.1616 100.2023 
14 2.8386 9.3294 7.3013 24.7164 80.6512 62.8906 
15 5.9852 7.9825 8.9803 52.2869 69.3135 77.7406 
16 2.4213 3.0274 9.0823 21.1755 26.3241 78.9508 
17 3.6303 7.2626 3.6313 31.7927 63.3239 31.6096 
18 1.7237 8.6211 7.7590 5.1041 75.3758 67.7079 
19 1.0136 2.0278 7.0975 8.8844 17.7396 62.0545 
20 0 4.7304 3.5478 0 41.4266 31.0444 
21 1.3668 2.7380 4.1070 11.9990 23.9914 35.9673 
22 0 1.5740 3.1481 0 13.7960 27.5845 
23 0 0 1.7986 0 0 15.7641 
24 2.0430 2.0435 0 17.9159 17.9159 0 
25 0 0 2.3097 0 0 20.2500 
Total years 79 122 138 657.6723 1008.2332 138.0550 
Wind speedWind power density (W m2 year−1)The wind energy density (KW h m2 year−1)
(m/s)10 (m)30 (m)40 (m)10 (m)30 (m)40 (m)
0 0 0 0 0 0 0 
1 0.1259 0.1106 0.0955 0.4116 0.3338 0.2649 
2 0.8891 0.7509 0.8692 4.5107 3.4117 3.9448 
3 1.8474 1.8839 1.7122 11.4260 10.6928 9.5338 
4 2.6483 3.1220 3.1031 18.1587 20.1933 19.7209 
5 3.9717 4.2130 4.2130 29.2824 29.5549 29.0796 
6 5.8416 5.5877 5.6197 45.6344 41.5459 41.1629 
7 5.6266 7.5548 7.7576 45.4535 58.8732 59.6718 
8 5.5992 7.0388 8.0227 46.2271 56.4228 63.7515 
9 6.5718 7.7590 9.1599 55.2187 63.5367 74.6568 
10 6.3547 9.6085 7.8347 54.0504 80.1814 64.8522 
11 6.6878 8.6571 10.2312 57.4293 73.1564 85.9664 
12 6.3842 8.9404 8.6849 55.2058 76.3008 73.6830 
13 5.8442 7.4697 11.6917 50.7887 64.1616 100.2023 
14 2.8386 9.3294 7.3013 24.7164 80.6512 62.8906 
15 5.9852 7.9825 8.9803 52.2869 69.3135 77.7406 
16 2.4213 3.0274 9.0823 21.1755 26.3241 78.9508 
17 3.6303 7.2626 3.6313 31.7927 63.3239 31.6096 
18 1.7237 8.6211 7.7590 5.1041 75.3758 67.7079 
19 1.0136 2.0278 7.0975 8.8844 17.7396 62.0545 
20 0 4.7304 3.5478 0 41.4266 31.0444 
21 1.3668 2.7380 4.1070 11.9990 23.9914 35.9673 
22 0 1.5740 3.1481 0 13.7960 27.5845 
23 0 0 1.7986 0 0 15.7641 
24 2.0430 2.0435 0 17.9159 17.9159 0 
25 0 0 2.3097 0 0 20.2500 
Total years 79 122 138 657.6723 1008.2332 138.0550 
View Large
TABLE V.

The classification of wind power based on PNL.

Category numberWind power at the height of 10 m (W m−2)Wind speed at the height of 10 m (m s−1)Wind power at the height of 30 m (W m−2)Wind speed at the height of 30 m (m s−1)Wind power at the height of 50 m (W m−2)Wind speed at the height of 50 m (m s−1)
1 100≥ 4.4≥ 160≥ 5.1≥ 200≥ 5.6≥ 
2 150≥ 1.5≥ 240≥ 6.0≥ 330≥ 6.4≥ 
3 200≥ 5.6≥ 320≥ 6.5≥ 400≥ 7.0≥ 
4 250≥ 6.0≥ 400≥ 7≥ 500≥ 7.5≥ 
5 300≥ 6.4≥ 480≥ 7.5≥ 600≥ 8.0≥ 
6 400≥ 7.0≥ 640≥ 8.2≥ 800≥ 8.8≥ 
7 1000≥ 9.4≥ 1600≥ 11.0≥ 2000≥ 11.9≥ 
Category numberWind power at the height of 10 m (W m−2)Wind speed at the height of 10 m (m s−1)Wind power at the height of 30 m (W m−2)Wind speed at the height of 30 m (m s−1)Wind power at the height of 50 m (W m−2)Wind speed at the height of 50 m (m s−1)
1 100≥ 4.4≥ 160≥ 5.1≥ 200≥ 5.6≥ 
2 150≥ 1.5≥ 240≥ 6.0≥ 330≥ 6.4≥ 
3 200≥ 5.6≥ 320≥ 6.5≥ 400≥ 7.0≥ 
4 250≥ 6.0≥ 400≥ 7≥ 500≥ 7.5≥ 
5 300≥ 6.4≥ 480≥ 7.5≥ 600≥ 8.0≥ 
6 400≥ 7.0≥ 640≥ 8.2≥ 800≥ 8.8≥ 
7 1000≥ 9.4≥ 1600≥ 11.0≥ 2000≥ 11.9≥ 
View Large

The last case of this research is the pattern of 19-directional wind in different months over 10 years in which the percentage related to the prevailing wind and its direction is specified (Fig. 9).

FIG. 9.
The wind direction in 2010–2019 at Saveh.
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The wind direction in 2010–2019 at Saveh.

FIG. 9.
The wind direction in 2010–2019 at Saveh.
View largeDownload slide

The wind direction in 2010–2019 at Saveh.

Close modal

In this study, the use of wind turbines with lower capacity in smaller dimensions and for villages and small communities is recommended. The comparison of turbines with small dimensions in different models available in the market in terms of production power and economic performance is suggested in future research. Based on this diagram shows 22%, 18%, and 17% of the winds in the region with the highest frequency in the east, south, and west directions, respectively.

In this study, the wind energy potential extracted in the region during the years 2010–2019 was analyzed. For this purpose, the distribution function of Weibull and Rayleigh is used to estimate wind speeds with the help of data obtained from the meteorological station. Also, the correctness of doing this was measured by goodness of fitting methods and the results of curve fitting showed that the wind speed data are consistent with the Weibull distribution. The results of the wind speed analysis showed that the wind speed during the investigated period was variable in the range of 0.6–7 m/s. The average wind speed during the statistical period was also reported as 96.2 m/s. Also, the lowest wind speeds in the hot months of the year were observed. The analysis of monthly and annual standard deviation showed that the wind speed in August has the lowest value, which indicates the homogeneity of wind speed data and the absence of sudden and extreme speeds in this month. Checking the wind speed during the day and night hours showed that the lowest level of wind is between 06:00 AM and 09:00 AM and the highest wind speed is between 09:00 AM and 15:00. Average parameter (k and c) for the entire statistical period equals 1.48 and 3.23 m/s, respectively. The average nominal speed to obtain maximum wind energy in the studied station was estimated at 7.81 m/s. The total power density and wind energy in the 10 years were calculated as 79 W/m2 and 657.6723-kW h m2 at a height of 10 m. Also, these values had an upward trend with increasing the height of the turbine. By using the PNL classification, Euclid city was placed in the first class in terms of extractable wind power, and from this point of view, the production of electricity in large dimensions is not justifiable. The prevailing wind direction, respectively, from east and west, had the largest share among the 16 directions. According to the said results, the following suggestions are provided:

According to the data analysis method in this research, it is recommended to use other possible distributions, such as log-logistic, beta, bar, Pareto, and Chule normal, to accurately determine the appropriate statistical distribution to describe the wind situation in different regions. It is effective to use different fit improvement methods to determine the best statistical model of wind speed frequency, and today, various software programs with different accuracies perform these tests. Due to the large size of the area, it is recommended to forecast the wind speed in other neighboring stations and compare the results with each other. Wind speed forecasting for future periods after the construction of the wind energy site is also suggested with the help of modeling methods. A study on the economic justification of the wind turbine site construction plan for energy extraction considering the purpose of using wind energy in any of the industrial, agricultural, residential, etc., is recommended in the continuation of this research. Keeping in mind the practical power of wind turbines, including their efficiency, should be considered in choosing wind turbines. To complete future studies and increase the accuracy of decisions, it is necessary to draw and analyze the power-continuity, and speed-interval curves. Also, if more accurate data and fewer time intervals (in terms of minutes) are used, the results of the study will be much more accurate. For this reason, it is suggested that the government, by allocating funds, provides the possibility of recording such information to make the best use of this renewable source of energy.

The authors have no conflicts to disclose

Mohammed. I. Alghamdi: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Resources (equal); Software (equal). Oriza Candra: Conceptualization (equal); Data curation (equal); Investigation (equal); Project administration (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Arif Sari: Conceptualization (equal); Data curation (equal); Software (equal); Validation (equal); Visualization (equal). Iskandar Muda: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Mujtaba Zuhair Ali: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal). Karrar Shareef Mohsen: Conceptualization (equal); Data curation (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Reza Morovati: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Behnam Bagheri: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.

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