# INFLUENCES OF THE ESTIMATING METHODS ON WEIBULL PARAMETER FOR

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```					INFLUENCES OF THE ESTIMATING METHODS ON WEIBULL PARAMETER FOR
WIND SPEED DISTRIBUTION IN JAPAN

Takeshi KAMIO*, Makoto IIDA* and Chuichi ARAKAWA*
*Department of Mechanical Engneering, The University of Tokyo
7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656, JAPAN
e-mail: kami0tch@cfdl.t.u-tokyo.ac.jp

ABSTRACT: The Weibull distribution is useful to represent the wind speed frequency distribution and estimate the
amount of the wind resource. This paper shows the differences between the several methods of estimating the two
parameters in the Weibull distribution function using the wind data that were measured at more than 300 locations to
clarity the wind characteristics in a complex terrain. The turbulence intensity was also analyzed to discuss the
relationship between the error of the estimated Weibull parameters and the turbulence intensity and it is found that the
errors of the estimated Weibull parameters might be lager in the higher turbulent category. Some methods, which are
based on the least square method, were examined to estimate the Weibull parameters and it is found that the nonlinear
least square method and the weighted linear least square are more appropriate to estimate the Weibull parameters
from the experimental wind speed frequency distribution than the ordinary linear least square method. The errors of
the estimated Weibull parameters are small even if the turbulent intensity was higher.

Key Words: Wind Speed Frequency Distribution, Weibull Distribution, Nonlinear Least Square, Turbulence Intensity

1.   Introduction                                                        the Weibull parameters could be useful to estimate the
When planning the installation of the wind energy, the              electrical output of the wind turbines.
assessment of the wind resource is important. Though it has                    The function of Weibull distribution is described as
been noted the complexity of wind structure in complex                   below:
terrain, it is not enough to investigate it. This paper
kU k −1       U k 
mentions the analysis of the wind speed distribution, the                  f (U ) =         exp −   
Weibull distribution and the methods of parameter
ck          c 
        
estimation of the Weibull function.
Wind speed distribution, which can be derived from                  where k is the shape parameter and c is the scale
the experimental wind data for a year or longer, is important            parameter. And the cumulative version of Weibull
parameters in the wind resource assessment and it is related             distribution is described like below.
to the amount of the wind energy in the site and the fatigue
loads for the wind turbine systems. When analyzing a                                         U k 
character of the distribution, Weibull distribution is                    F (U ) = 1 − exp −   
generally used and the two parameters of its function are                                   c 
        
derived from the wind data. This is based on the
well-known fact that Weibull distribution fit to wind speed                    Reyleigh distribution is also famous for the
distribution well and its parameters can represent the                   function which fits wind speed distribution and it is a
characters of the wind. Saito[7] suggested that the value of             special case of Weibull distribution with k=2.0.
0.18                                                                          1.20                          Wind speed distribution
Wind speed distribution
Weibull distribution                                                            Weibull distribution

0.16
1.00
0.14

0.12                                                                          0.80

Distribution
Distribution

0.10
0.60
0.08

0.06                                                                          0.40

0.04
0.20
0.02

0.00                                                                          0.00
0.0     5.0         10.0      15.0             20.0                        0.0           5.0         10.0      15.0            20.0
Wind speed[m/s]                                                                  Wind speed[m/s]
Figure 1 The Weibull distribution                                    Figure 2 The cumulative Weibull distribution

When calculating the Weibull parameters from the                            a phenomenon and it can give an estimation of the
experimental wind data, several methods, such as moment                                   parameters. When using least square method, the sum of
methods, maximum likelihood and linear least square                                       the squares of the deviations S which is defined as below,
fitting, are available and generally the linear least square                              should be minimized.
method might be chosen. This is because the linear least
square could be mathematically easier than others.                                                  n
S = ∑ wi [ y i − g ( xi )]
2                    2
Then, there is the problem that some examples exist
that the Weibull parameters which were calculated by linear                                         i

least square method were not precise and the actually
obtained energy were much smaller than expected. The                                               In the equation, xi is the wind speed, yi is the
main reason might be that the turbulence included in the                                  probability of the wind speed rank, so (xi, yi) mean the
wind and the turbulence caused to a peculiar figure of wind                               data plot, wi is a weight value of the plot and n is a
speed distribution. So the problem can occur evenly in the                                number of the data plot. Though the weight values wi
regions where the wind has the large turbulence component,                                should be based on the reliability of each data plot, it is
and Japan could be included in the regions.                                               often that the value for every data plot is set to 1.
Then, the authors take notice of another reason, which                               In this paper, the analysis was undertaken with
is the error in the methods of estimating the Weibull                                     the four estimating methods which are based on the least
parameters. For example, linear least square could include                                square method and they are the linear least square
the large error in the process of the calculation. So, the old                            method, nonlinear least square method, which is after all
methods should be taken place by the other methods.                                       divided into two methods, and the weighted linear least
In this paper, the several estimating methods are                           square method.
examined and the error data for each method are analyzed.                                          In the following sections, the details of the four
The experimental wind dat used in this paper are gathered                                 estimating methods are described.
from more than 300 locations in Japan, and this data set
could cover almost all the wind characters in Japan.                                       2.2 Linear least square method
The linear least square method (LLS) is a special
2.            Estimating the Weibull parameter for wind                                   case for the least square method with a formula which
speed distribution                                                          consists of some linear functions and it is easy to use.
And in the more special case that the formula is line, the
2.1 General explain for Least square method                                        linear least square method is much easier. The Weibull
Least square method is used to calculate the                               distribution function is a non-linear function, but the
parameter(s) in a formula when modeling an experiment of                                  cumulative Weibull distribution function is transformed
to a linear function like below:                                 and the value wi is defined as follow.

ln[− ln{ − F (U )}] = k ln U − k ln c
1                                                        wi =
[(1 − Fi ) ln(1 − Fi )]2
∑ [(1 − F )ln(1 − F )]
n
2
j          j
Here, the linear least square method can be                     j =1

used to estimate the Weibull parameters for the wind speed
distribution. If the values of the weight of every data plot             Originally, the weight function is described with
are set to 1, the value S which should be minimized is           statistical median and the formula is changed by the
described as below.                                              authors, and it is described only with the values of date
sets.
n
S = ∑ [yi − g i ]
2
3.      Turbulence in the wind
i                                                                The turbulence intensity represents the variation of

yi = ln[− ln{ − F (U i )}]
the wind speed within a period. For the wind, the period
1
might be a minute or ten minutes and it might be longer
than the period which is used when we discuss the
g i = k ln U i − k ln c
ordinary fluid machines. The authors assume the
turbulence might cause the irregular figure of the wind
2.3 Non-linear least square method                              speed distribution and investigated the error of the
Non-linear least square method is available for           Weibull parameters according to the turbulence intensity.
estimating the parameters in a nonlinear function. By using              In the wind energy systems, the turbulence intensity
the nonlinear least square method, the Weibull parameters        is the important parameter which enables to evaluate the
can be estimated with the original function of the Weibull       fatigue load for the structure, for example, the wind
distribution. In this study, the Levenberg-Marquardt method      turbine. To investigate the relationship between the
is used to find the minimum of the sum of squares. Here,         turbulent intensity and the precision of the estimated
when estimating the parameters, both the Weibull                 Weibull parameters, every wind data is divided to the
distribution function and the cumulative function are            turbulence category in IEC61400-1 Ed.3[9], in which the
available as a formula which should be fit to the data plots.    wind turbine class is determined with the mean speed and
So both the nonlinear least square method with the               the turbulence intensity of the wind. The category is
cumulative Weibull distribution function (NLS-CWDF)              written like Table 1, in which the expected value Iref of the
and the nonlinear least square method with the                   turbulence intensity is used defined for the categories. Iref
non-cumulative Weibull distribution function (NLS-WDF)           is defined as below.
were examined and the results of the two estimations were
compared.
Table 1 IEC61400-1 Ed.3
2.4 Weighted linear least square method                                               Wind Turbine Classes
Weighted linear least square method (WLLS) is
modified case for linear least square method and it has a                Class            I        II        III      S
different weight value for each data set in the formula of the        Vref[m/s]          50       42.5   37.5
Values
value S. The authors chose a weight function proposed by              Vav[m/s]           10       8.5        7.5
specified
Wen-Liang Fung[8], which is the weight function for the                           A               0.16
by the
estimation of Weibull parameter with linear least square              Iref        B               0.14
designer
method. The weight function is based on the theory that the                       C               0.12
weight value which equal to 1 for the plot (Ui, F(Ui)) should
be transform the value for the plot (lnUi, ln[-ln(1-F(Ui))])
10.0                                                   1.0                                       10.0                                               1.0

LLS                                                                                            NLS-CWDF
8.0                                                   0.8                                        8.0                                               0.8

Computational averaged
Computational averaged

wind speed[m/s]
wind speed[m/s]

6.0                                                   0.6                                        6.0                                               0.6

Variance
Variance
4.0                                                   0.4                                        4.0                                               0.4

2.0                                                   0.2                                        2.0                                               0.2

0.0                                                   0.0                                        0.0                                               0.0
0.0       2.0      4.0      6.0     8.0     10.0                                              0.0       2.0      4.0      6.0     8.0     10.0
Experimental averaged wind speed[m/s]                                                         Experimental averaged wind speed[m/s]

10.0                                                   1.0                                       10.0                                               1.0

WLLS                                                                                           NLS-WDF
8.0                                                   0.8                                        8.0                                               0.8

Computational averaged
Computational averaged

wind speed[m/s]
wind speed[m/s]

6.0                                                   0.6                                        6.0                                               0.6

Variance
Variance

4.0                                                   0.4                                        4.0                                               0.4

2.0                                                   0.2                                        2.0                                               0.2

0.0                                                   0.0                                        0.0                                               0.0
0.0       2.0      4.0      6.0     8.0     10.0                                              0.0       2.0      4.0      6.0     8.0     10.0
Experimental averaged wind speed[m/s]                                                         Experimental averaged wind speed[m/s]

Figure 3 The comparison of the averaged wind speed between the observed and the analyzed data
for the four methods

4.                      Wind data                                                                the NEDO field test project and the wind data which
The wind data used in this study come from more                          have the high turbulence intensity came from many sites.
than 300 locations all over Japan. Those observations were                                       Seeing in another point of view, the Japanese sites which
undertaken as the wind energy field test project by NEDO                                         are appropriate for the wind energy are frequently
(New Energy and Industry Technology Development                                                  categorized in higher turbulent wind climates and the
Organization). In the observations, the ten minutes                                              standards and the requirements for the higher turbulent
averaged wind speed and wind direction were measured for                                         wind climates is important.
a year at each site. The authors analyzed all data, in which
mainly the wind speed frequency distribution and the
turbulence intensity, and investigated the estimating method                                         Table 2 Number of data in turbulent category
of the Weibull parameters.
The number of the data in each turbulent category is                                                    turbulent category    number of data
showed in Table 2. The observations in the some locations                                                                           Iref <0.12               39
were undertaken for 2 heights and the total number of the                                                                        0.12< Iref <0.14            45
data counts 440.                                                                                                                 0.14< Iref <0.16            78
The table shows that the number of the data in the                                                          0.16< Iref              278
highest turbulent category (Iref>0.16) is largest. The reason
total                440
for this might be that the analysis of the wind data in the
higher turbulent regions in Japan is one of the purposes of
y=x
LLS
10.0             NLS-CWDF
WLLS
LLS-WDF

8.0
Computational averaged
wind speed[m/s]

6.0

4.0

2.0

0.0
0.0      2.0     4.0     6.0     8.0    10.0
Experimental averaged wind speed[m/s]
Figure 4 The Comparison between the errors of the four method in the case of the high turbulent

5.            Analysis                                               differences between the computational and experimental
To investigate the differences of the precision of     wind speeds are larger or not in the higher wind speed
estimations by the methods and for the turbulence intensity,         rank.
the wind data were analyzed and compared the estimations                     Figure 3 shows the differences between the results
for the methods and for the turbulence intensity. In the             of the four methods. Figure 5, 6, 7 and 8 shows the
comparison, the subject is the relationship between the              differences between the turbulent categories for each
averaged wind speed of the original observed data and the            method. Figure 5 shows the result for the normal linear
analyzed data.                                                       least square method, Figure 6 for the nonlinear least
The results of the analysis are shown in Figure 3-8.   square method with the cumulative function of the
Each figure shows the relationships of the averaged wind             Weibull distirbution, Figure 7 for the weighted linear
speeds between the observed data and the analyzed data.              least square and Figure 8 for the nonlinear least square
One plot represents a relation between the experiment and            with the normal function of the Weibull distribution.
the observed data from one location data. If the plot is                     Then Figure 4 shows the results from the nine
nearer to the line y=x (y is computational data and x is             locations’ wind data that were classified into the highest
experimental data), the estimation is better. It is important        turbulent categories and it shows the large differences
whether the plot is over or under the line y=x, because it           between the linear least square method and the other
means the analyzed data shows the overestimated or                   methods.
underestimated wind energy.
The diagrams also indicate the variance of the
computational averaged wind speeds for each experimental
averaged wind speed rank. This shows whether the
10.0                                                   1.0                                       10.0                                               1.0

Iref<0.12                                                                                      0.12<Iref<0.14
8.0                                                   0.8                                        8.0                                               0.8

Computational averaged
Computational averaged

wind speed[m/s]
wind speed[m/s]

6.0                                                   0.6                                        6.0                                               0.6

Variance
Variance
4.0                                                   0.4                                        4.0                                               0.4

2.0                                                   0.2                                        2.0                                               0.2

0.0                                                   0.0                                        0.0                                               0.0
0.0       2.0      4.0      6.0     8.0     10.0                                              0.0       2.0      4.0      6.0     8.0     10.0
Experimental averaged wind speed[m/s]                                                         Experimental averaged wind speed[m/s]

10.0                                                   1.0                                       10.0                                               1.0

0.14<Iref<0.16                                                                                 0.16<Iref
8.0                                                   0.8                                        8.0                                               0.8

Computational averaged
Computational averaged

wind speed[m/s]
wind speed[m/s]

6.0                                                   0.6                                        6.0                                               0.6

Variance
Variance

4.0                                                   0.4                                        4.0                                               0.4

2.0                                                   0.2                                        2.0                                               0.2

0.0                                                   0.0                                        0.0                                               0.0
0.0       2.0      4.0      6.0     8.0     10.0                                              0.0       2.0      4.0      6.0     8.0     10.0
Experimental averaged wind speed[m/s]                                                         Experimental averaged wind speed[m/s]

Figure 5 The comparison of the averaged wind speed between the observed and the analyzed data
for the normal linear least square method (LLS)

6.                      Result of the analysis for the four methods                              wider than NLS-CWDF.
Figure 3 shows the data analyzed by the four methods.                                            Finally, in the result of the nonlinear least square
First, many computational averaged wind speed in the                                             method with the non-cumulative Weibull distribution
diagram for the linear least square method (LLS) are larger                                      function (NLS-WDF), the computational averaged wind
than the experimental averaged wind speed. The                                                   speeds considerably differ from the experimental data
differences are lager in the higher wind speed. That means                                       and some are larger than the experimental data and
that the averaged wind speed might be overestimated by the                                       others are smaller. These results are worse than LLS.
linear analysis and the amount of the wind energy might be                                                               From these result, it is found that the most
also overestimated.                                                                              appropriate method for the estimation of the Weibull
Second, the diagram for the nonlinear least square                       parameters might be NLS-CWDF. Then WLLS might
method with the cumulative Weibull distribution function                                         be also suitable for the estimation considering the
(NLS-CWDF) indicates that every computational averaged                                           process of the method is easier than NLS-CWDF,
wind speed by NLS-CWDF is nearly equal to the                                                    though it is slightly inferior to NLS-CWDF in the
experimental averaged wind speed and the data plots gather                                       precision of the result.
on the line y=x. So, NLS-CWDF enables to analyze the                                                                     These results are more clearly shown in Figure 4,
probability of the wind speed more precisely than LLS.                                           in which the data plots of the methods are plotted in the
Next, the result of the weighted linear least square                     same diagram and it is well shown that the differences
(WLLS) shows those estimations are much better than LLS                                          between the four methods, especially the differences
and the plots also gather on the line y=x, but spread slightly                                   between the linear least square method and the others.
10.0                                                   1.0                                       10.0                                               1.0

Iref<0.12                                                                                      0.12<Iref<0.14
8.0                                                   0.8                                        8.0                                               0.8

Computational averaged
Computational averaged

wind speed[m/s]
wind speed[m/s]

6.0                                                   0.6                                        6.0                                               0.6

Variance
Variance
4.0                                                   0.4                                        4.0                                               0.4

2.0                                                   0.2                                        2.0                                               0.2

0.0                                                   0.0                                        0.0                                               0.0
0.0       2.0      4.0      6.0     8.0     10.0                                              0.0       2.0      4.0      6.0     8.0     10.0
Experimental averaged wind speed[m/s]                                                         Experimental averaged wind speed[m/s]

10.0                                                   1.0                                       10.0                                               1.0

0.14<Iref<0.16                                                                                 0.16<Iref
8.0                                                   0.8                                        8.0                                               0.8

Computational averaged
Computational averaged

wind speed[m/s]
wind speed[m/s]

6.0                                                   0.6                                        6.0                                               0.6

Variance
Variance

4.0                                                   0.4                                        4.0                                               0.4

2.0                                                   0.2                                        2.0                                               0.2

0.0                                                   0.0                                        0.0                                               0.0
0.0       2.0      4.0      6.0     8.0     10.0                                              0.0       2.0      4.0      6.0     8.0     10.0
Experimental averaged wind speed[m/s]                                                         Experimental averaged wind speed[m/s]

Figure 6 The comparison of the averaged wind speed between the observed and the analyzed data
for the nonlinear least square method with cumulative Weibull distribution function (NLS-CWDF)

7.                      Result          of   the     estimations       of   Weibull              the development of the wind energy in the higher
paramater for turbulence intensity                                       turbulent wind climates regions would be important and
The differences between the four methods are shown                       the analyzing methods which can be useful to analyze
in Figure 3 and 4, and especially Figure 4 shows the                                             the higher turbulent wind data might be demanded. For
differences in the highest turbulent category. So it is found                                    the analysis of the wind speed frequency distribution,
that there might be the differences of the errors between the                                    LLS might be not useful enough to analyze the high
turbulence categories. In this section, there are the results                                    turbulent wind and other methods are expected to
that shows the differences of the error between the                                              replace LLS.
turbulence categories for each method.
7.2 NLS-WCDF
7.1 LLS                                                                                                           Figure 6 shows the result of NLS-CWDF. It is
Figure 5 shows LLS. In the higher turbulent                             found that the errors of the averaged wind speed are
categories, the computational averaged wind speeds are                                           considerable small. Though there are some plots away
different from the experimental averaged wind speeds and                                         from the line y=x in the highest turbulent category, those
it is remarkable in the highest category. In the lower                                           errors are much smaller than the errors in the estimation
turbulent categories, the errors are smaller but .                                               by LLS.
This result indicates that LLS is useful enough to
estimate the Weibull parameter for the lower turbulent wind
and it is not useful for the higher turbulent wind. In future,
10.0                                                   1.0                                       10.0                                               1.0

Iref<0.12                                                                                      0.12<Iref<0.14
8.0                                                   0.8                                        8.0                                               0.8

Computational averaged
Computational averaged

wind speed[m/s]
wind speed[m/s]

6.0                                                   0.6                                        6.0                                               0.6

Variance
Variance
4.0                                                   0.4                                        4.0                                               0.4

2.0                                                   0.2                                        2.0                                               0.2

0.0                                                   0.0                                        0.0                                               0.0
0.0       2.0      4.0      6.0     8.0     10.0                                              0.0       2.0      4.0      6.0     8.0     10.0
Experimental averaged wind speed[m/s]                                                         Experimental averaged wind speed[m/s]

10.0                                                   1.0                                       10.0                                               1.0

0.14<Iref<0.16                                                                                 0.16<Iref
8.0                                                   0.8                                        8.0                                               0.8

Computational averaged
Computational averaged

wind speed[m/s]
wind speed[m/s]

6.0                                                   0.6                                        6.0                                               0.6

Variance
Variance

4.0                                                   0.4                                        4.0                                               0.4

2.0                                                   0.2                                        2.0                                               0.2

0.0                                                   0.0                                        0.0                                               0.0
0.0       2.0      4.0      6.0     8.0     10.0                                              0.0       2.0      4.0      6.0     8.0     10.0
Experimental averaged wind speed[m/s]                                                         Experimental averaged wind speed[m/s]

Figure 7 The comparison of the averaged wind speed between the observed and the analyzed data
for the weighted linear least square method (WLLS)

7.3 WLLS                                                                                  there are some results of NLS-WDF that the errors are
Figure 7 shows the result of WLLS. The errors of the                    large in the comparison of the experimental and
averaged wind speed are small and the figures looks like                                         computational averaged wind speed but the Weibull
the figures of NLS-WCDF. Though the errors are slightly                                          distributions fit well the experimental wind speed
larger than the errors of NLS-WCDF, the process of the                                           frequency distribution in the figure.
estimation by the WLLS is much easier than NLS and it is
considerable to use WLLS.                                                                        8.                      Conlusion
In this paper, the four methods that are based on the
7.4 NLS-CDF                                                                               least square method were examined for the purpose of
Figure 8 shows the result of NLS-WDF. About this                        the better estimation of the Weibull parameters from the
method, the errors are much larger than the other methods                                        experimental wind data. As a result, NLS-CWDF gives
for all the categories. In this case, the errors might be not                                    most exact estimations and weighted linear least square
caused by the turbulent or by the problem of NLS-WDF.                                            method also gives the better estimations, though the
Then it is also thought that there is a problem in the                  process of WLLS is easier.
comparison of the result. In this study, the comparison is                                                               Then the influences of the turbulent intensity on the
based on the averaged wind speed and the experimental                                            estimations ware investigated and it is showed that there
data and computational data are compared. The other                                              are the larger estimating errors in the higher turbulent
parameters which are appropriate to judge the precision of                                       categories, though it could be given the exact
the estimated Weibull parameters should be chosen. In fact,                                      estimations with NLS-CWDF.
10.0                                                   1.0                                       10.0                                               1.0

Iref<0.12                                                                                      0.12<Iref<0.14
8.0                                                   0.8                                        8.0                                               0.8

Computational averaged
Computational averaged

wind speed[m/s]
wind speed[m/s]

6.0                                                   0.6                                        6.0                                               0.6

Variance
Variance
4.0                                                   0.4                                        4.0                                               0.4

2.0                                                   0.2                                        2.0                                               0.2

0.0                                                   0.0                                        0.0                                               0.0
0.0       2.0      4.0      6.0     8.0     10.0                                              0.0       2.0      4.0      6.0     8.0     10.0
Experimental averaged wind speed[m/s]                                                         Experimental averaged wind speed[m/s]

10.0                                                   1.0                                       10.0                                               1.0

0.14<Iref<0.16                                                                                 0.16<Iref
8.0                                                   0.8                                        8.0                                               0.8

Computational averaged
Computational averaged

wind speed[m/s]
wind speed[m/s]

6.0                                                   0.6                                        6.0                                               0.6

Variance
Variance

4.0                                                   0.4                                        4.0                                               0.4

2.0                                                   0.2                                        2.0                                               0.2

0.0                                                   0.0                                        0.0                                               0.0
0.0       2.0      4.0      6.0     8.0     10.0                                              0.0       2.0      4.0      6.0     8.0     10.0
Experimental averaged wind speed[m/s]                                                         Experimental averaged wind speed[m/s]
Figure 8 The comparison of the averaged wind speed between the observed and the analyzed data
for the nonlinear least square method with non-cumulative Weibull distribution function (NLS-WDF)

Acknowledgement                                                                                   pp.2311-2318.
This study used the field test data by NEDO project. We                                           5. AL Hasan, M., Nigmatullin, RR., Identification of the
would like to thank to NEDO for their great support.                                              generalized Weibull distribution in wind speed data by
the Eigen-coordinates method, Renewable Energy,
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