International Journal of Applied Science and Engineering
2010. 8, 1: 39-46
Wind Speed and Power Density Analyses Based on Mixture
Weibull and Maximum Entropy Distributions
Tian Pau Chang *
Department of Computer Science and Information Engineering, Nankai University of Technology,
Taiwan, R.O.C.
Abstract: Wind resource is important part of the utilization of renewable energy. To effectively
estimate the wind energy potential for a given area, a variety of probability density functions (pdf)
have been available in literature. In this paper, the bimodal mixture Weibull function (WW) and
the probability function derived with maximum entropy principle (MEP) will be used and compared with the conventional Weibull function. Wind speed data measured at three wind farms
experiencing different climatic environments in Taiwan are selected as sample data to test their
performance. Judgment criterions include four kinds of statistical errors, i.e. the max error in
Kolmogorov-Smirnov test, Chi-square error, root mean square error and relative error of wind
potential energy. The results show that the proposed WW and MEP pdfs describe wind characterizations better than the conventional Weibull pdf, irrespective of wind speed and wind power
density data, particularly for a location where wind regime presents two humps on it. For wind
speed distributions, the WW pdf describes best according to the Kolmogorov-Smirnov test;
while for wind power density, the MEP pdf outperforms the others.
Keywords: wind speed; wind power density; probability density function
1. Introduction
Due to the energy demands and the shortages of fossil fuels currently in the world, the
utilization of wind resource plays a very important role in energy supply. As we know,
wind speed distribution for a specified area
determines the wind energy available and the
performance of energy conversion system.
Once the probability distribution of wind
speed is obtained, the wind energy potential
could be estimated accordingly. A variety of
probability density functions (pdf) have been
used in literature to estimate wind energy potential, but the Weibull function is most
widely adopted because of its two flexible
parameters [1-4]. i.e. Weibull shape parameter
*
describes the width of data distribution, while
scale parameter controls the abscissa scale of
a plot of data distribution. It is worth to mention that the Weibull function is not possible
to represent all the wind structures encountered in nature, particularly for a dispersive
wind distribution that might be resulted from
special climatic factors. In the last decades,
several mixed probability functions have been
proposed by researchers to model those complex wind distributions in the world, e.g. the
bimodal mixture Weibull function (WW), singly truncated normal Weibull mixture function (NW) [2, 5-9] and the probability function derived with maximum entropy principle
Corresponding author; e-mail: t118@nkut.edu.tw
Accepted for Publication: September 7, 2010
© 2010 Chaoyang University of Technology, ISSN 1727-2394
Int. J. Appl. Sci. Eng., 2010. 8, 1
39
Tian Pau Chang
(MEP) [10-15]. However relevant study concerning Taiwan has never been found in literature.
Taiwan situates at a unique geographic or
climatic environment, more than 95% of energy demand relies on the imported fossil fuels. Southwest monsoon and northeast monsoon prevail in summer and winter seasons,
respectively; wind energy potential is thus
considerable here. In this paper, the bimodal
Weibull function and the probability function
derived with maximum entropy principle are
introduced and adopted to describe wind
characterization. Their performance and validity will be compared with the conventional
Weibull function according to four kinds of
judgment criteria, i.e. the max error in the
Kolmogorov-Smirnov test, Chi-square error,
root mean square error (RMSE) as well as the
relative percent error of wind energy potential
between theoretical function and actual
time-series data. Wind speed data used are
measured per 10 minutes from 2006 to 2008
at three wind farms experiencing different
climatic environments in Taiwan. The first
wind farm, Pingtung, locates at the southern
peninsula that experiences more stable
weather condition throughout the year. The
second one, Penghu, is at a small island in the
Taiwan Strait having the highest wind in winter and spring months. The third wind farm,
Taoyuan, locates at the northwest part experiencing higher wind in winter months. The
10-minute wind data are transferred to hourly
data and have been averaged over the three
years in subsequent analyses. These observations are considered as sample data just for
testing how accurate the proposed probability
density functions describe the wind characterizations; that would be the first publication
regarding this topic in Taiwan.
2. Probability Density Functions
2.1. Weibull Distribution (W pdf)
40
Int. J. Appl. Sci. Eng., 2010. 8, 1
The conventional (two-parameter) Weibull
probability density function has widely been
used for describing wind regimes written as:
f (v; α , β ) =
α v α −1
v
( ) exp[−( )α ]
β β
β
(1)
Where, v is the wind speed, α is the dimensionless shape parameter, β is the scale
parameter with the same unit as v . The
Weibull cumulative distribution function (cdf)
is given by:
v
F (v; α , β ) = 1 − exp[−( )α ]
β
(2)
Weibull shape and scale parameters can be
calculated using the maximum likelihood
method as [1, 4]:
vα ln(vi ) ∑i =1 ln(vi ) −1
∑
i =1 i
α =[
−
]
n
α
n
v
∑i =1 i
n
β =(
n
1 n α 1/α
∑ vi )
n i =1
(3)
(4)
Where, vi is the wind speed in time step
i and n is the number of nonzero data
points.
2.2 Mixture Weibull distribution (WW pdf)
Where, vi is the wind speed in time step
i and n is the number of nonzero data
points.
2.2. Mixture Weibull distribution (WW
pdf)
The probability density function of a mixture Weibull distribution can be written as
[5-7]:
Wind Speed and Power Density Analyses Based on Mixture Weibull and Maximum Entropy Distributions
g (v; w, α1 , β1 , α 2 , β 2 ) = wf (v; α1 , β1 ) + (1 − w) f (v; α 2 , β 2 )
(5)
Its cumulative distribution function is given by:
G (v; w, α1 , β1 , α 2 , β 2 ) = wF (v; α1 , β1 ) + (1 − w) F (v; α 2 , β 2 )
Where 0 ≤ w ≤ 1 is a weight parameter;
which represents the proportion of component
distributions being mixed. α1 , α 2 > 0 are the
shape parameters; β1 , β 2 > 0 are the scale
parameters. The five parameters of the mix-
(6)
ture Weibull distribution ( w, α1 , β1 , α 2 , β 2 )
can be estimated using the maximum likelihood method that maximizes the logarithm of
likelihood function given as:
LL = ln L(vi ; w, α1 , β1 , α 2 , β 2 )
n ⎧
⎫
v
v
α v
α v
= ln Π ⎨w{ 1 ( i )α1 −1 exp[−( i )α1 ]} + (1 − w){ 2 ( i )α 2 −1 exp[−( i )α 2 ]}⎬
β1 β1
β1
β2 β2
β2
i =1 ⎩
⎭
n
⎧ α v
⎫
v
v
α v
= ∑ ln ⎨w{ 1 ( i )α1 −1 exp[−( i )α1 ]} + (1 − w){ 2 ( i )α 2 −1 exp[−( i )α 2 ]}⎬
β1
β2 β2
β2
i =1
⎩ β1 β1
⎭
2.3. Maximum Entropy Principle Distribution (MEP pdf)
For a given probability density function f (x) , its entropy is defined as [10-12]:
H = − ∫ f ( x) ln f ( x) dx
(8)
Maximizing the entropy subject to some
constraints enables one to find the most likely
probability density function if the information
available is limited. For the analysis of wind
distribution, the classic solution of the maximum entropy problem can be written by:
f (v) = exp[−∑ n = 0 λn v n ]
N
(7)
erally lies between 3 and 6 in wind researches.
The larger the N value, the greater the accuracy is. It is found by the present paper that
the accuracy of using 4 moment constraints is
quite close to that of 5 or even 6 moment constraints. For the consideration of computation
burden, the maximum entropy distribution
considered in this paper is the fourth order
(i.e. N = 4 ).
3. Wind Power Density
Wind power density is proportional to the
cube of wind speed, for a given theoretical
probability distribution f (v) , it can be calculated by the following integration:
(9)
Where λn are the Lagrange multipliers,
which can be obtained by the standard Newton method; N is the number of moment
constraints of a physical system, which gen-
P=
1
ρv 3 f (v) dv
∫
2
(10)
Where, ρ is the air density.
As for actual time-series data, wind power
density is calculated by:
Int. J. Appl. Sci. Eng., 2010. 8, 1
41
Tian Pau Chang
P=
1 3
ρv
2
(11)
I Where, v
wind speed.
3
is the mean of the cube of
5. Results and Discussions
4. Judgment Criterions
In order to check how accurate a theoretical
probability density function fits with observation data, in this paper, four types of statistical
errors are considered as judgment criterions
shown as below, basically the smaller the errors the better the fit is.
The first criterion is the max error in the
Kolmogorov-Smirnov test given by [16]:
Q = max T (v) − O(v)
(12)
Where T (v) is the cumulative distribution
function (cdf), for wind speed not exceeding
v , calculated from a theoretical function;
similarly O (v) is calculated from observation data.
The second one is the Chi-square error
given as:
χ 2 = ∑ in=1[
1
( yi − yic ) 2 ]
yic
(13)
Where, yi is the actual value at time stage
i, yic is the value computed from correlation
expression for the same stage, n is the
number of data.
The third one is the root mean square error
(RMSE) defined as:
RMSE = [
42
The fourth criterion is the relative percent
error between the wind potential energy calculated from actual time-series data and that
from theoretical probability function.
1 n
( yi − yic ) 2 ]1 / 2
∑
i =1
n
Int. J. Appl. Sci. Eng., 2010. 8, 1
(14)
To illustrate how suitable the proposed
probability functions describe wind regimes,
Figure 1-3 show the annual wind speed histograms for three stations experiencing different
weather conditions, in which the conventional
Weibull function is plotted too for comparison.
Table 1 lists the relevant parameter values
computed for different probability functions.
Various statistical errors are summarized in
Table 2. It is clear that for station Pingtung,
where only one hump is found on it, all the
three probability functions fit very well with
the observed data; the curves of cumulative
distribution functions (cdf) are quite consistent with that of observed one. Generally
smaller statistical errors are obtained as summarized in Table 2. While for stations Penghu
and Taoyuan, in which wind regimes exhibit
two significant humps on it, the WW and
MEP pdfs fit still very well with the observations having relatively smaller statistical errors; whereas the conventional Weibull function seems unsuitable for describing the distributions. For example, in Penghu, the appearance probability of wind speed 4~7 m/s
and 17~22 m/s is underestimated, however it
is overestimated for speed 8~12 m/s when
using the conventional Weibull pdf. The relative percent errors of potential energy for the
conventional Weibull pdf reach 5.4% above.
The proposed WW pdf performs best according to the Kolmogorov-Smirnov test. As for
Chi-square and RMSE errors, there is no significant discrepancy found for WW and/or
MEP pdf.
Wind Speed and Power Density Analyses Based on Mixture Weibull and Maximum Entropy Distributions
0.2
observed
W pdf
WW pdf
MEP pdf
W cdf
WW cdf
MEP cdf
observed cdf
Pingtung
0.18
0.16
0.12
0.1
0.08
1
0.8
0.06
0.6
0.04
0.4
0.02
0.2
0
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
Cumulative distribution function
Wind speed frequency
0.14
30
Wind speed (m/s)
Figure 1. Wind speed distribution for station Pingtung
0.2
observed
W pdf
WW pdf
MEP pdf
W cdf
WW cdf
MEP cdf
observed cdf
Penghu
0.18
0.16
0.12
0.1
0.08
1
0.8
0.06
0.6
0.04
0.4
0.02
0.2
0
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
Cumulative distribution function
Wind speed frequency
0.14
30
Wind speed (m/s)
Figure 2. Wind speed distribution for station Penghu
0.2
observed
W pdf
WW pdf
MEP pdf
W cdf
WW cdf
MEP cdf
observed cdf
Taoyuan
0.18
0.16
0.12
0.1
0.08
1
0.8
0.06
0.6
0.04
0.4
0.02
0.2
0
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
Cumulative distribution function
Wind speed frequency
0.14
30
Wind speed (m/s)
Figure 3. Wind speed distribution for station Taoyuan
Int. J. Appl. Sci. Eng., 2010. 8, 1
43
Tian Pau Chang
Table 1. Relevant parameters for three probability density functions
Stations
W
WW
α β (m/s) w α1 α 2 β1 (m/s) β 2 (m/s)
Ping2.150 8.539 0.0468 9.980 2.263 15.949
tung
Penghu 1.689 11.231 0.4426 4.151 2.354 17.334
Taoyuan 1.944 9.115 0.6375 4.166 2.445 11.824
λ0
λ1
MEP
λ2
λ3
λ4
8.109
5.10085163 -1.27133026 0.18858305 -0.01110984 0.00025591
6.149
3.880
5.02276188 -1.27182598 0.20579344 -0.01212952 0.00024082
4.46372824 -1.37253895 0.31984164 -0.02864190 0.00086949
Table 2. Statistical errors for three probability density functions
Wind speed
Stations
Functions
W
Pingtung WW
MEP
W
Penghu WW
MEP
W
Taoyuan WW
MEP
Max error
RMSE
0.05418
0.02218
0.02346
0.10716
0.01681
0.01998
0.10541
0.02120
0.02245
0.00628
0.00592
0.00584
0.01342
0.00759
0.00845
0.01806
0.00732
0.00820
Power density
χ
2
0.01699
0.01209
0.02203
0.14683
0.04219
0.03829
0.19421
0.05861
0.06095
Figure 4-6 show the annual wind power
distributions for the three stations. They present similar trends with those of wind speed,
i.e. both the MEP and WW pdfs perform better than the conventional Weibull function
especially for station where wind regime has
bimodal distributions and for high wind speed
ranges. Since MEP pdf is defined for zero of
wind speed, it thus can describe the appearance probability of calm winds that makes the
distribution curve may start with a non-zero
value when wind speed approaches zero. The
percent errors of potential energy calculated
by using the MEP pdf are even below 0.01%,
independent of stations.
6. Conclusions
In this paper, the performance of mixture
Weibull distribution and maximum entropy
distribution in describing wind regimes was
compared together with the conventional
Weibull function considering four kinds of
statistical errors. It is concluded that the pro-
44
Int. J. Appl. Sci. Eng., 2010. 8, 1
Max error
RMSE
0.04261
0.02829
0.02666
0.14007
0.04209
0.03237
0.23255
0.05144
0.03886
0.00976
0.00764
0.00701
0.02126
0.00891
0.00785
0.02729
0.01493
0.01064
Potential energy
χ
2
0.07022
0.03851
0.02769
0.32164
0.08362
0.07293
0.40675
0.19347
0.17393
Percent error (%)
0.1780
0.0452
1.69e-005
5.4088
0.0721
1.58e-006
5.1927
0.0623
6.36e-006
posed WW and MEP pdfs describe better
wind characterizations than the conventional
Weibull function, irrespective of wind speed
and power density data, particularly when
wind distribution has two humps on it. For
wind speed distributions, the WW pdf describes best; while for wind power density, the
MEP pdf outperforms the others. Overall both
the proposed probability functions could be a
useful alternative to the conventional Weibull
function in wind energy applications.
Acknowledgements
The author would deeply appreciate the
Taipower Company for providing observation
data and thank Dr. Wu CF and Dr. Huang MW,
researchers of the Institute of Earth Sciences,
Academia Sinica, Taiwan, for their treasure
suggestions. This work was partly supported
by the National Science Council under the
contract of NSC99-2221-E-252-011.
Wind Speed and Power Density Analyses Based on Mixture Weibull and Maximum Entropy Distributions
0.2
observed
W pdf
WW pdf
MEP pdf
W cdf
WW cdf
MEP cdf
observed cdf
Pingtung
0.18
0.16
0.12
0.1
0.08
1
0.8
0.06
0.6
0.04
0.4
0.02
0.2
0
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Cumulative distribution function
Wind power frequency
0.14
Wind speed (m/s)
Figure 4. Wind power distribution for station Pingtung
0.2
Penghu
0.18
observed
W pdf
WW pdf
MEP pdf
W cdf
WW cdf
MEP cdf
observed cdf
0.16
0.12
0.1
0.08
1
0.8
0.06
0.6
0.04
0.4
0.02
0.2
0
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Cumulative distribution function
Wind power frequency
0.14
Wind speed (m/s)
Figure 5. Wind power distribution for station Penghu
0.2
observed
W pdf
WW pdf
MEP pdf
W cdf
WW cdf
MEP cdf
observed cdf
Taoyuan
0.18
0.16
0.12
0.1
0.08
1
0.8
0.06
0.6
0.04
0.4
0.02
0.2
0
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Cumulative distribution function
Wind power frequency
0.14
Wind speed (m/s)
Figure 6. Wind power distribution for station Taoyuan
Int. J. Appl. Sci. Eng., 2010. 8, 1
45
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