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The best Weibull distribution methods for the assessment of wind energy potential at different altitudes in desired locations are statistically diagnosed in this study. Seven different methods, namely graphical method (GM), method of moments (MOM), standard deviation method (STDM), maximum likelihood method (MLM), power density method (PDM), modified maximum likelihood method (MMLM) and equivalent energy method (EEM) were used to estimate the Weibull parameters and six statistical tools, namely relative percentage of error, root mean square error (RMSE), mean percentage of error, mean absolute percentage of error, chisquare error and analysis of variance were used to precisely rank the methods. The statistical fittings of the measured and calculated wind speed data are assessed for justifying the performance of the methods. The capacity factor and total energy generated by a small model wind turbine is calculated by numerical integration using Trapezoidal sums and Simpson's rules. The results show that MOM and MLM are the most efficient methods for determining the value of
Energy and environment are the twin major crises in the world [
Research is ongoing worldwide on the Weibull distribution to find the most reliable methods for wind energy estimation. The main question is how precisely the values of the Weibull shape factor “
Seguro and Lambart [
This study summarizes the results of the 10min time series wind speed data measured at 20 m and 30 m height in three windy sites, namely Kuakata, Kutubdia and Sitakunda, located in Bangladesh.
To investigate the feasibility of the wind energy resource at any site, there are basically two ways at present to evaluate wind power. The first and the most accurate method to calculate wind power potential is based on measured values that are recorded at meteorological stations. The second method to assess wind power potential is by using probability distribution functions, namely the Rayleigh distribution, Chisquared distribution, Normal distribution, Binomial distribution, Poisson distribution and Weibull distribution. In this study, the authors used only the Weibull distribution for wind power assessment as presented and discussed below.
The Weibull probability density function is a twoparameter function characterized by a dimensionless shape parameter (
Cumulative distribution function is the integration of the Weibull density function. It is the cumulative of relative frequency of each velocity interval [
All these distributions are used to determine the probability of occurrence. The nature of the occurrence affects the shape of the probability curve, and in the case of the wind regime, the cumulative curve probability nature mostly fits to the Weibull Function. Several methods to estimate Weibull factors are found in the literature. Some of these methods are:
Graphical method (GM);
Method of moments (MOM);
Standard deviation method (STDM);
Maximum likelihood method (MLM);
Power density method (PDM);
Modified maximum likelihood method (MMLM);
Equivalent energy method (EEM).
The graph is constructed in such a way that the cumulative Weibull distribution becomes a straight line, with the shape factor k as its slope. Taking the logarithm of both sides, the expression of
The above equation represents a relationship between ln(
The MOM is another technique commonly used in the field of parameter estimation. If the numbers
After a few manipulations:
This formula can easily be handled by pocket calculators in energy output calculations. The accuracy of the approximation is within 0.5% for 1.6 <
When we divide m_{2} by the square of m_{1}, we get an expression which is a function of the shape factor
On taking the square roots of the equation, we have the coefficient of variation (
In this case, this method can be used as an alternative to the MLM. The value of
After some calculation we can find:
The Weibull scale factor can be calculated by:
In the STDM, the Weibull factors can be obtained as follows:
By determining the mean wind speed
One can find next an expression for σ in terms of
Maximum likelihood estimation has been the most widely used method for estimating the parameters of the Weibull distribution. The commonly used procedure of MLM proposed by Cohe [
Thus,
On taking the logarithms of both sides of
Differentiating
From
When
Substituting
Substituting
Therefore,
To obtain the shape factor and scale factor through this method, firstly the energy pattern factor is computed. The energy pattern factor usage is for turbine aerodynamic design. The energy pattern factor is related to the averaged data of wind speed and is defined as a ratio between mean of cubic wind speed to cube of mean wind speed. The energy pattern factor
Once the energy pattern factor is calculated by using the above equation, the Weibull shape factor and scale factor can be estimated from the following formulas:
The MMLM can only be considered if the available data of wind speed are already in the shape of the Weibull distribution. The solution of the equations in the MLM requires some numerical iteration by the NewtonRaphson method [
Consider a random sample of
This observation random is also related with the Weibull parameters
The first hypothesis says that “The energy density is a parameter that helps in the determination of parameters of the Weibull distribution for applications in wind energy”. The related factor part deterministic must meet the following conditions: (a) be variable with random value expected value equal to 0:
Substituting
The estimate of the parameter
The
Wind energy is indirect solar energy because it is generated by the temperature difference between the equator and the poles which drives the thermal system by solar radiation. It is known that wind speed varies with altitude, however, wind blows relatively slowly at low altitude and wind speed then increases with altitude. Different relationships are found in the literature to calculate wind speed at any height [
The mean energy density over a period of time,
To find the best method for the analysis, some statistical parameters were used to analyze the efficiency of the above mentioned methods. The following tests were used to achieve this goal:
zRelative percentage of error (RPE)
Root mean square error (RMSE)
Mean percentage error (MPE)
Chisquare error
Analysis of variance or efficiency of the method
In this statistical analysis, data from three wind monitoring stations were used to diagnose the best method of the Weibull distribution. The most important results of this analysis, based on hourly, monthly, seasonal and annual figures, are presented.
A sample data frequency distribution and cumulative frequency distribution has been presented in
The hourly variation of wind speed at the selected altitude has been presented in
The hourly mean
From the sample frequency distribution (
In the statistical analysis, seven methods were used to determine the shape parameter
From
The procedure mentioned in this study is not only applicable in case study sits, it can be applied in any climatic conditions at any site in any countries in the world. For example, Mohammadi and Mostafaeipour [
The monthly mean available power is analyzed using both MOM and MLM methods in
Therefore, the six months from April to September show more potential wind generated power than other months in the year. In this research work, the Weibull parameters at heights of 30 m and 20 m were determined.
The mean velocity, shape factor, scale factor,
The energy estimation of wind regimes by the Weibull based approach has been presented in
In this work, statistical diagnosis of the best Weibull distribution methods for wind data analysis is presented. By using the available wind data, the values of shape factor
The authors declare no conflict of interest.
Total power, W/m^{2}
Area, m^{2}
Practically extractable power, W/m^{2}
Dimensionless shape parameter
Scale parameter (m/s)
Weibull probability density function
Cumulative distribution function
Constant
Likelihood function
Coefficient of variation
Random sample of wind speed central to bin
Number of sample or bin
Weibull frequency for wind speed ranging within bin
Probability for wind speed ≥ 0
Observed frequency of the wind speed
Mean of the cubic wind speed, m/s
Error of the approximation
Reference height, m
Weibull shape factor at desired height
Weibull scale factor at desired height, m/s
Wind speed at desired height, m/s
Power at desired height, W/m^{2}
Total number of observations
Mean of
Analysis of variance
Gamma function
Standard deviation of wind speed, m/s
Air density, kg/m^{3}
Unknown parameter for maximum likelihood function
Power law coefficient
Chisquare error
Hourly mean wind speed at 20 m and 30 m height for the selected sites.
Hourly mean
Site name, location for wind speed measuring at 20 m and 30 m height above surface.
StationI  Kuakata  21°54.76′  90°08.24′  3 m  [ 
StationII  Kutubdia  21°54.71′  91°52.43′  0–4 m  [ 
StationIII  Sitakunda  22°35.68′  91°42.52′  9 m  [ 
Monthly mean wind speed (m/s) data for selected wind stations at 20 m and 30 m height.
 

StationI  H = 30 m  744  672  744  720  744  720  744  744  720  744  720  744  8760  
3.11  3.57  3.54  4.98  4.97  5.07  5.74  5.97  4.95  2.57  2.98  3.22  4.22  
5.08  5.79  5.52  8.26  8.28  7.55  9.46  9.29  9.52  3.5  5.77  7.45  7.12  
1.66  2.25  2.44  2.52  2.32  2.48  3.23  2.34  1.97  1.5  1.31  1.26  2.11  
σ (m/s)  0.78  0.86  0.82  1.54  1.68  1.56  1.61  2.13  2.14  0.54  0.76  1.39  1.32  
25.08  24.09  23.16  30.92  33.8  30.77  28.05  35.68  43.23  21.01  25.5  43.17  30.37  
 
H = 20 m  744  672  744  720  744  720  744  744  720  744  720  744  8760  
2.15  2.62  2.65  3.76  3.81  3.82  4.44  4.75  3.81  1.76  2.12  2.55  3.19  
3.65  4.14  4.29  6.03  6.25  5.58  8.15  8.21  7.33  2.77  4.9  7.2  5.71  
1.22  1.68  1.78  1.88  1.69  1.77  2.71  1.72  1.53  0.99  0.9  0.97  1.57  
σ (m/s)  0.52  0.69  0.66  1.1  1.24  1.17  1.37  1.83  1.73  0.45  0.65  1.4  1.07  
24.19  26.34  24.91  29.26  32.55  30.63  30.86  38.53  45.41  25.57  30.66  54.9  32.82  
 
StationII  H = 30 m  744  672  744  720  744  720  744  744  720  744  720  744  8760  
3.11  2.8  3.15  3.73  4.24  4.53  5.42  4.74  3.39  2.25  2.96  2.65  3.58  
5.26  5.69  4.81  6.36  6.95  7.35  7.58  7.58  7.48  3.31  4.15  4.37  5.91  
1.64  1.29  1.67  2.22  2.35  2.06  2.82  1.99  1.29  1.25  1.7  1.73  1.83  
σ (m/s)  0.95  1.33  0.69  1.16  1.39  1.52  1.17  1.56  1.74  0.59  0.5  0.64  1.1  
30.55  47.5  21.9  31.1  32.78  33.55  21.59  32.91  51.33  26.22  16.89  24.15  30.87  
 
H = 20 m  744  672  744  720  744  720  744  744  720  744  720  744  8760  
1.69  1.79  1.9  2.73  3.3  3.81  4.63  3.95  2.58  1.26  1.45  1.11  2.52  
3.51  4.47  3.7  5.1  6.31  6.31  7.26  6.79  7.13  2.41  2.68  2.42  4.84  
0.7  0.64  0.76  1.49  1.39  1.38  2.24  1.61  0.87  0.56  0.59  0.5  1.06  
σ (m/s)  0.71  1.25  0.61  1.08  1.42  1.41  1.18  1.45  1.84  0.46  0.43  0.44  1.02  
42.01  69.83  32.11  39.56  43.03  37.01  25.49  36.71  71.32  36.51  29.66  39.64  41.91  
 
StationIII  H = 30 m  744  672  744  720  744  720  744  744  720  744  720  744  8760  
2.72  3.17  3.36  4.35  4.46  4.81  5.36  5.15  10.09  2.25  2.7  2.32  4.23  
4.16  6.21  5.14  7.36  7.55  8.16  7.59  13.76  39.2  3.48  3.69  3.67  9.16  
1.16  1.24  1.96  2.09  2.63  2.14  2.91  2.31  1.49  1.23  1.86  1.11  1.84  
σ (m/s)  0.84  1.43  0.82  1.42  1.4  1.79  1.14  2.58  13.07  0.48  0.43  0.71  2.18  
30.88  45.11  24.4  32.64  31.39  37.21  21.27  50.1  129.53  21.33  15.93  30.6  39.2  
 
H = 20 m  744  672  744  720  744  720  744  744  720  744  720  744  8760  
1.96  2.47  2.64  3.72  3.84  4.26  4.79  4.87  5.97  1.43  1.82  1.66  3.29  
3.36  5.61  4.35  6.62  6.86  7.34  6.89  14.35  18.58  2.6  2.67  2.7  6.83  
0.66  0.79  1.33  1.65  2.01  1.56  2.31  1.73  1.28  0.76  0.93  0.86  1.32  
σ (m/s)  0.73  1.45  0.82  1.42  1.42  1.79  1.09  2.88  5.33  0.42  0.36  0.52  1.52  
37.24  58.7  31.06  38.17  36.98  42.02  22.76  59.14  89.28  29.37  19.78  31.33  41.32 
Frequency distribution and cumulative frequency distribution for the selected sites in June.


 

0–1  0.403  0.403  0.269  0.269  0.134  0.134 
1–2  2.151  2.554  1.882  2.151  0.806  0.94 
2–3  5.242  7.796  3.629  5.78  5.645  6.585 
3–4  9.005  16.801  14.247  20.027  17.339  23.924 
4–5  17.742  34.543  19.086  39.113  24.059  47.983 
5–6  20.43  54.973  24.328  63.441  18.414  66.397 
6–7  18.011  72.984  18.28  81.721  17.742  84.139 
7–8  12.634  85.618  11.559  93.28  8.199  92.338 
8–9  8.602  94.22  5.242  98.522  3.898  96.236 
9–10  4.032  98.252  1.344  99.866  2.016  98.252 
10–11  1.613  99.865  0  99.866  1.344  99.596 
11–12  0.134  100  0.134  100  0.269  99.865 
12–13          0.134  100 
The Weibull distribution analysis for StationI at 20 m height. GM: graphical method; MOM: method of moments; STDM: standard deviation method; MLM: maximum likelihood method; PDM: power density method; MMLM: Modified maximum likelihood method; and EEM: equivalent energy method.

 

GM  3.63  3.55  0.0785  0.0500  0.8283  4.2571  0.1432  0.9577 
MOM  3.58  3.54  −0.0523  0.0408  −0.0394  0.0394  0.0001  0.9999 
STDM  3.99  4.46  26.6736  0.9219  27.9826  27.982  2.2593  0.1542 
MLM  3.5  3.55  0.0000  0.0000  −0.0438  0.3700  0.0005  0.9998 
PDM  3.11  3.53  −0.7845  0.1581  −0.6649  0.6648  0.0285  0.9917 
MMLM  3.54  3.55  0.0784  0.0500  0.0723  0.1885  0.0002  0.9999 
EEM  3.37  3.54  −0.2876  0.0957  0.1507  2.4687  0.0451  0.9866 
Note: Dimensionless shape factor (
The Weibull distribution analysis for StationI at 30 m height.

 

GM  2.71  4.79  0.8880  0.1930  0.6250  1.7040  0.0350  0.9900 
MOM  3.87  4.68  −0.5530  0.1530  −0.5420  0.5890  0.0023  0.9990 
STDM  2.64  4.78  1.7370  0.2710  2.3310  3.7180  0.2380  0.9320 
MLM  3.78  4.68  0.0000  0.0000  −0.0560  0.3440  0.0010  0.9997 
PDM  2.58  4.66  −1.2040  0.2250  −1.1870  1.2390  0.0150  0.9960 
MMLM  3.18  4.67  −0.8090  0.1850  −0.9210  0.9540  0.0064  0.9980 
EEM  3.26  4.73  0.4930  0.1440  0.6290  1.6220  0.0290  0.9920 
The Weibull distribution analysis for StationII at 20 m height.

 

GM  3.11  2.70  −3.8742  0.3120  −4.1546  4.5644  0.1336  0.9787 
MOM  2.82  2.81  0.0662  0.0408  0.0422  0.0422  0.0002  0.9999 
STDM  3.4  3.74  34.2053  0.9278  40.0035  40.003  2.8453  0.3654 
MLM  2.84  2.9  3.0463  0.2768  6.7823  7.0891  0.3063  0.9475 
PDM  2.65  2.82  0.1324  0.0577  0.1170  0.2672  0.0003  0.9999 
MMLM  2.84  2.85  1.5894  0.2000  3.4823  3.6671  0.0795  0.9865 
EEM  2.88  2.76  −1.8212  0.2140  −1.9377  2.1966  0.0337  0.9945 
The Weibull distribution analysis for StationII at 30 m height.

 

GM  4.17  3.88  −1.8150  0.2550  −1.5198  2.7444  0.0610  0.9790 
MOM  3.99  3.94  −0.5820  0.1440  −0.5930  0.6490  0.0074  0.9974 
STDM  4.47  4.83  22.6440  0.9004  23.8630  23.863  1.9020  0.1780 
MLM  3.92  3.96  −0.0930  0.0580  −0.0920  0.5220  0.0020  0.9990 
PDM  3.25  3.99  −0.2560  0.0960  −0.2560  0.2560  0.0004  0.9998 
MMLM  3.59  3.98  −6.5160  0.4830  −8.3160  8.5270  2.3420  0.2280 
EEM  4.23  4.39  11.1940  0.6330  11.7710  11.771  0.5180  0.7970 
The Weibull distribution analysis for StationIII at 20 m height.

 

GM  3.04  3.66  0.0254  0.0289  −0.8825  4.4901  0.1568  0.979 
MOM  3.14  3.62  −0.2536  0.0913  −0.1711  0.1711  0.0031  0.9996 
STDM  3.43  4.95  35.1255  1.0743  34.5353  34.535  5.7001  0.0331 
MLM  3.13  3.65  0.1775  0.0764  0.0826  0.4453  0.0011  0.9998 
PDM  2.77  3.65  −0.0507  0.0408  −0.0311  0.0311  0.0001  0.9999 
MMLM  3.14  3.64  −1.1666  0.1958  −0.8205  1.4034  0.0924  0.9877 
EEM  2.91  3.66  0.0000  0.0000  −0.449  2.2702  0.041  0.9945 
The Weibull distribution analysis for StationIII at 30m height.

 

GM  2.1  4.71  −0.5910  0.1580  −0.5250  2.8630  0.0640  0.9950 
MOM  3.76  4.45  0.0790  0.0580  0.1320  0.1650  0.0010  0.9999 
STDM  2.3  3.73  −21.540  0.9540  −11.410  13.920  20.680  0.3470 
MLM  3.66  4.5  −0.4730  0.1414  −0.3095  0.5340  0.0140  0.9990 
PDM  2.32  4.74  1.4980  0.2520  −0.0730  3.4630  0.3080  0.9740 
MMLM  2.99  4.62  0.1580  0.0820  −0.4510  1.9760  0.0580  0.9950 
EEM  3.03  4.09  −11.845  0.7080  −6.0510  7.4310  5.9690  0.5632 
Ranking of the methods by statistical test results.


 

GM  Sixth  Sixth  Fifth  Fourth  Sixth  Fourth   
MOM  First  Second  First  Third  Third  First  The First choice 
STDM  Seventh  Seventh  Seventh  Seventh  Seventh  Seventh   
MLM  Third  First  Sixth  Second  Second  Second  The second choice 
PDM  Fourth  Fourth  Second  First  First  Fifth  The third choice 
MMLM  Second  Third  Fourth  Sixth  Fifth  Third   
EEM  Fifth  Fifth  Third  Fifth  Fourth  Sixth   
Monthly mean wind speed, power law coefficient (η), the Weibull shape factor (

 

 
January  0.262  2.15  4.33  2.35  3.11  4.42  3.40  3.55  4.63  3.89 
February  0.25  2.62  4.18  2.89  3.57  4.41  3.91  4.05  4.62  4.44 
March  0.251  2.65  4.12  2.91  3.54  4.41  3.87  4.02  4.62  4.4 
April  0.219  3.76  3.86  4.16  4.98  3.58  5.53  5.57  3.75  6.18 
May  0.219  3.81  3.45  4.25  4.97  3.30  5.55  5.58  3.46  6.21 
June  0.218  3.82  3.95  4.23  5.07  3.84  5.63  5.69  4.02  6.29 
July  0.207  4.44  3.41  4.93  5.74  3.82  6.34  6.39  4.00  7.05 
August  0.203  4.75  2.91  5.34  5.97  3.17  6.69  6.66  3.32  7.42 
September  0.218  3.81  2.4  4.31  4.95  2.52  5.60  5.56  2.64  6.26 
October  0.28  1.76  4.36  1.94  2.57  5.65  2.79  2.98  5.92  3.22 
November  0.266  2.12  3.06  2.34  2.98  3.70  3.26  3.37  3.87  3.73 
December  0.256  2.55  1.99  2.89  3.22  2.49  3.63  3.68  2.61  4.14 
Monthly mean power based on measured data and calculated data by statistical methods.

 

January  18.75  4.50  3.41  21.84  1.7578  0.9904  4.42  3.4  21.67  1.7088  0.9724 
February  27.87  4.71  3.90  32.63  2.1817  4.41  3.91  33.24  2.3173  
March  26.71  4.92  3.86  31.40  2.1656  4.41  3.87  31.96  2.2913  
April  73.11  3.58  5.53  97.57  4.9457  3.58  5.53  97.57  4.9457  
May  71.47  3.25  5.55  101.46  5.4763  3.3  5.55  101.04  5.4378  
June  75.87  3.6  5.63  102.85  5.1942  3.84  5.63  101.21  5.0339  
July  110.11  3.98  6.34  143.45  5.7741  3.82  6.34  144.7  5.8813  
August  123.88  3.06  6.68  181.11  7.5651  3.17  6.69  179.36  7.4485  
September  70.61  2.48  5.58  118.12  6.8927  2.52  5.6  117.25  6.8293  
October  10.05  5.47  2.78  11.70  1.2845  5.65  2.79  11.8  1.3229  
November  16.21  4.42  3.27  19.38  1.7804  3.7  3.26  19.82  1.9000  
December  20.81  2.48  3.63  32.46  3.4132  2.49  3.63  32.37  3.4000 
Seasonal mean wind speed in (m/s), the Weibull shape factor (), scale factor (m/s) and power density (W/m^{2}) at 30 m and 50 m height.

 

 
Winter  0.24  3.19  3.5  3.55  26.21  4.22  3.78  4.7  59.1  4.76  3.96  5.27  83.6 
Spring  0.24  3.01  4.05  3.32  20.2  4.03  4.13  4.4  48.2  4.55  4.33  5.01  68.7 
Summer  0.21  4.02  3.6  4.47  48.64  5.26  3.65  5.8  108.24  5.89  3.83  6.52  157.2 
Autumn  0.23  3.44  3.22  3.86  34.1  4.5  3.78  5.1  72.3  5.07  3.96  5.63  100.67 
Wind power classification.


 

ClassI  Poor  ≤4.4  ≤5.1  ≤5.6  
ClassII  Marginal  ≤5.1  ≤6.0  ≤6.0  
ClassIII  Moderate  ≤5.6  ≤6.5  ≤7.0  
ClassIV  Good  ≤6.0  ≤7.0  ≤7.5  
ClassV  Very good  ≤6.4  ≤7.5  ≤8.0  
ClassVI  Excellent  ≤7.0  ≤8.2  ≤8.8  
ClassVII  Excellent  ≤9.4  ≤11.0  ≤11.9 
The most frequent wind velocity (
Winter  4.31  5.24  0.0586  129.408 
Spring  4.15  4.88  0.0492  105.082 
Summer  5.35  6.58  0.1148  253.531 
Autumn  4.64  5.63  0.0728  160.667 
 
Yearly mean  4.31  5.24  0.0586  513.412 
Energy generated by the wind turbine and the capacity factor by numerical integration. The turbine rated power is 20 kW or 0.02 MW, rated speed is 8 m/s and hub height at 30 m.
 

 
Winter  0.1238  5.4689  0.1238  5.4689 
Spring  0.0920  3.9283  0.0920  3.9283 
Summer  0.3010  13.2937  0.3010  13.2938 
Autumn  0.1698  7.5005  0.1698  7.5005 
 
Yearly mean  0.1238  21.6973  0.1238  21.6974 