Journal of Applied Science and Engineering, Vol. 17, No. 3, pp. 283-292 (2014)
DOI: 10.6180/jase.2014.17.3.09
Study on Wind Characteristics Using Bimodal Mixture
Weibull Distribution for Three Wind Sites in Taiwan
Feng-Jiao Liu1, Hong-Hsi Ko2, Shyi-Shiun Kuo 2, Ying-Hsin Liang2 and Tian-Pau Chang2*
1
Department of Electrical and Information Technology, Nankai University of Technology,
Nantou, Taiwan 542, R.O.C.
2
Department of Multimedia Animation and Applications, Nankai University of Technology,
Nantou, Taiwan 542, R.O.C.
Abstract
Some wind speed distributions in Taiwan have been found deviating from conventional
Weibull distribution. In this paper the mixture Weibull distribution was adopted to analyze the wind
data observed at three wind sites having different climatic environments. The Kolmogorov-Smirnov
test and wind potential energy were considered as indicators to show how the mixture Weibull function
characterizes wind speed distribution. Relevant mathematical expressions are derived originally for
wind energy assessment. The results show that the mixture Weibull function performs quite better than
a conventional Weibull function particularly for a region where the wind speed distribution reveals
two humps on it. The similar result is obtained also when wind power density is considered. The
maximum errors of cumulative distribution function between observation data and mixture Weibull
function are always below the critical value of 95% confidence level in Kolmogorov-Smirnov test.
The relative percentage errors of wind potential energy between time-series data and theoretical values
from mixture Weibull function never exceed 0.1%. It is found that the distribution pattern of wind
speed would affect a lot to the electrical energy generated by an ideal turbine.
Key Words: Wind Characteristics, Wind Speed, Wind Power, Weibull Distribution, Mixture
Weibull Distribution
1. Introduction
Accurately analyzing wind characteristics for a particular region is the first step for wind resource utilization. A number of mathematical functions have been
published to model the frequency distribution of wind
speed [1,2]. Typical two-parameter Weibull function
presents a series of advantages with respect to other
probability density functions (pdf) [3-15]. However recent researches have shown that the conventional Weibull function cannot model all the wind regimes observed in nature. Akpinar and Akpinar [16] proposed a
theoretical approach using the concept of maximum entropy principle (MEP) to determine wind speed distribu*Corresponding author. E-mail: [email protected]
tion and concluded that the MEP type distribution describes wind speed and wind power density more accurately than the typical Weibull distribution. Shamilov et
al. [17] first applied the MinMaxEnt distribution to wind
energy field to model wind data; similar conclusions
were obtained that the MinMaxEnt distribution shows
better flexibility than the known Weibull distribution in
estimating wind speed distribution and power density. In
some cases there are two peaks in the wind speed distribution; where a bimodal probability density function, i.e.
the two-component mixture Weibull distribution (Weibull-Weibull), is found to be more suitable in describing
the data [1,18,19]. As stated by Carta and Ramirez [20]
even the data simply reveals unimodal distribution, the
mixture Weibull distribution is applicable.
The application of mixture Weibull function to Tai-
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wan has never been found in literature so far. Taiwan
situates at a special climatic environment; when northeast monsoon prevails in winter and spring it experiences high wind especially in western offshore, while in
summer and autumn wind speed is commonly low. Wind
resource here reveals an obviously seasonal discrepancy
that in some degree would cause a challenge to wind energy utilization. In this study the mixture Weibull function is applied to wind speed data, which are observed
each 10 minutes from 2006 to 2007 by three wind turbines, conducted by the Ministry of Economic Affairs,
having different weather conditions. Turbine specifications are shown in Table 1. Station Dayuan is located at
the northwestern plain of Taiwan having strong wind in
winter months; station Hengchun is at the southern peninsula experiencing more stable weather conditions over
the year; Chungtun is at a small island in Taiwan Strait
experiencing the highest wind in winter and spring, with
different turbine specifications from other two stations.
The mixture Weibull function consists of five parameters needed to be resolved; two shape parameters, two
scale parameters and a weight parameter which represents the proportion of mixing of the component distributions. Several numerical methods can be found in
literature for estimating the parameters such as the moment method, least-square method, maximum likelihood
method, etc. In this paper the maximum likelihood method is adopted for its popularity and feasibility [21,22].
The Kolmogorov-Smirnov test [12,23] as well as the relative percentage error of wind potential energy between
time-series data and theoretical calculation will be considered as indicators to show how the mixture Weibull
function is superior to conventional Weibull function. A
computer source code finished using MATLAB toolbox
software could be available through email request via
[email protected].
2. Mixture Weibull Function
The probability density function of conventional Weibull distribution and its cumulative distribution function
(cdf) are expressed respectively as:
(1)
(2)
where, v is the wind speed, a is the dimensionless shape
parameter, b is the scale parameter with the same unit as
v.
The probability density function of two-component
mixture Weibull distribution is expressed as [24,25]:
(3)
Its cumulative distribution function is given by:
(4)
where 0 £ f £ 1 is a weight parameter; which represents
the proportion of component distributions being mixed.
a1, a2 > 0 are the shape parameters; b1, b2 > 0 are the
scale parameters. The most probable speed with the
given probability function is calculated by [26]:
(5)
The wind speed carrying maximum energy is:
(6)
For a wind turbine designed with the cut-in speed vi
and cut-off speed vo, the percentage of time that the turbine might operate, named availability factor, equivalent
to the occurrence probability of wind speed between vi
and vo can be calculated by:
Table 1. Specifications of wind turbine used in this study
Stations
Dayuan &
Hengchun
Chungtun
Manufacturer
Rated power (kW)
Cut-in speed (m/s)
Rated speed (m/s)
Cut-off speed (m/s)
Hub height (m)
Rotor diameter (m)
GE (USA)
1500
4.0
12
25
64.7
70.5
Enercon (Germany)
600
2.5
12.5
25
46
43.7
Study on Wind Characteristics Using Bimodal Mixture Weibull Distribution for Three Wind Sites in Taiwan
285
(7)
(13)
3. Wind Potential Energy
Wind potential energy for a specified time period T
based on actual time-series data is calculated by:
(8)
5. Parameter Estimation
where, r is the air density; A is the blade sweep area of
wind turbine; v 3 is the mean of wind speed cubes. The
similar energy based on the theoretical mixture Weibull
function can be obtained by:
(9)
where, G( ) is the Gamma function given as:
The maximum likelihood method was adopted to
compute the five parameters of the mixture Weibull distribution (f, a1, b1, a2, b2) by numerically maximizing
the distribution’s log-likelihood (the joint probability
density function) shown as below. Maximizing the logarithm of likelihood function gives the same result as
maximizing the likelihood function, since logarithm is a
monotonic function.
(10)
4. Electrical Energy Generated by an Ideal
Wind Turbine
For an idealized wind turbine, it starts to operate at
cut-in speed then its output power P1 increases with wind
speed until the rated speed vr. For wind speed between
the rated and cut-off speed the output power remains a
constant value, i.e. the rated power P2. The turbine will
be shut down while wind speed exceeds cut-off speed.
Where
(11)
(12)
The output energy of an ideal turbine Eid can be calculated by the following numerical integration:
(14)
where vj is the wind speed in time step j; n is the number
of data points. The maximum likelihood estimators can
be obtained through differentiating with respect to the
parameters that we want to solve. In this paper some restrictions have been set to these parameters for rapid
converging to a tolerance level during iterative searching. Where mixing parameter begins with the initial
value of 0.5; shape and scale parameters begin with 2
and 10, respectively. Additionally shape parameters are
limited between 0.2 and 10, while scale parameters are
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Feng-Jiao Liu et al.
between 1 and 30, which are commonly encountered in
wind regimes. The maximum number of iterations could
be adjusted if needed.
6. Result Discussions
Figures 1-3 show the hourly mean wind speed throughout the year for station Dayuan, Hengchun and
Chungtun respectively; the time-series data has been fitted using a second-order polynomial. As seen it reveals
high wind near winter and most of the low speeds occur
in summer as marked with vertical dashed lines.
Figures 4-6 show the corresponding histograms of
wind speed with bin size of 1 m/s; both the Weibull and
mixture Weibull (Weibull-Weibull) probability density
function curves are compared together. There are two
Figure 1. Hourly mean wind speed at Dayuan over the year.
Figure 3. Hourly mean wind speed at Chungtun over the
year.
Figure 2. Hourly mean wind speed at Hengchun over the
year.
Figure 4. Wind speed probability and cumulative distribution
function for Dayuan.
Study on Wind Characteristics Using Bimodal Mixture Weibull Distribution for Three Wind Sites in Taiwan
287
Figure 5. Wind speed probability and cumulative distribution
function for Hengchun.
Figure 6. Wind speed probability and cumulative distribution
function for Chungtun.
significant humps on it for the stations of Dayuan and
Chungtun; but it is not significant for station Hengchun
due to its more stable wind speed distribution over the
year.
Tables 2-4 summarize the yearly wind characteristics for the three stations. The most probable speed for
specific hump is available in the Tables. The mixture
Weibull function performs better in describing wind speed
distribution than the conventional Weibull function. Being an example, as in Figure 4, Dayuan, if conventional
Weibull function is considered, the occurrence probability of wind speed between 11 and 16 m/s is underestimated, however the probability between 4 and 10 m/s is
seriously overestimated. The curve of cumulative distribution function (cdf) calculated using mixture Weibull
function matches very well with the observed one for all
the stations.
To examine whether a theoretical probability density
function is suitable to characterize observation data or
not, the Kolmogorov-Smirnov test is considered, which
Table 2. Yearly wind characteristics for Dayuan*
Parameters
Weibull
Weibull-Weibull
Probability density function
a = 1.944
b = 9.115 (m/s)
Most probable speed (m/s)
Vmp = 6.287
Speed carrying max energy (m/s)
Vme = 13.116
f = 0.6375
a1 = 4.166
a2 = 2.445
b1 = 11.824 (m/s)
b2 = 3.880 (m/s)
Vmp1 = 11.070
Vmp2 = 3.1290
Vme1 = 12.990
Vme2 = 4.9540
2.0636 ´ 107
2.0649 ´ 107
0.062
1.5895 ´ 107
0.7540
0.0113
Potential energy by theoretical function (kWh)
Potential energy by time-series data (kWh)
Relative percentage error of potential energy (%)
Output energy of ideal turbine (kWh)
Availability factor
Max error of cumulative distribution function
2.1748 ´ 107
2.0649 ´ 107
5.319
1.4104 ´ 107
0.8170
0.0866
* Note: mean speed = 8.09 (m/s); standard deviation = 4.31 (m/s); skewness = 0.067; kurtosis = 1.813.
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Feng-Jiao Liu et al.
Table 3. Yearly wind characteristics for Hengchun*
Parameters
Weibull
Weibull-Weibull
Probability density function
a = 2.150
b = 8.539 (m/s)
Most probable speed (m/s)
Vmp = 6.383
Speed carrying max energy (m/s)
Vme = 11.594
f = 0.0468
a1 = 9.980
a2 = 2.263
b1 = 15.949 (m/s)
b2 = 8.109 (m/s)
Vmp1 = 15.782
Vmp2 = 6.2670
Vme1 = 16.243
Vme2 = 10.726
1.6184 ´ 107
1.6176 ´ 107
0.045
1.2346 ´ 107
0.8260
0.0138
Potential energy by theoretical function (kWh)
Potential energy by time-series data (kWh)
Relative percentage error of potential energy (%)
Output energy of ideal turbine (kWh)
Availability factor
Max error of cumulative distribution function
1.6148 ´ 107
1.6176 ´ 107
0.177
1.2539 ´ 107
0.8220
0.0154
* Note: mean speed = 7.55 (m/s); standard deviation = 3.72 (m/s); skewness = 0.547; kurtosis = 2.889.
Table 4. Yearly wind characteristics for Chungtun*
Parameters
Weibull
Weibull-Weibull
Probability density function
a = 1.689
b = 11.231 (m/s)
Most probable speed (m/s)
Vmp = 6.606
Speed carrying max energy (m/s)
Vme = 17.836
f = 0.4426
a1 = 4.151
a2 = 2.354
b1 = 17.334 (m/s)
b2 = 6.149 (m/s)
Vmp1 = 16.221
Vmp2 = 4.862
Vme1 = 19.057
Vme2 = 7.984
1.8139 ´ 107
1.8152 ´ 107
0.072
0.7375 ´ 107
0.9320
0.0139
Potential energy by theoretical function (kWh)
Potential energy by time-series data (kWh)
Relative percentage error of potential energy (%)
Output energy of ideal turbine (kWh)
Availability factor
Max error of cumulative distribution function
1.8736 ´ 107
1.8152 ´ 107
3.219
0.7311 ´ 107
0.9030
0.0775
* Note: mean speed = 9.997(m/s); standard deviation = 6.143(m/s); skewness = 0.540; kurtosis = 2.115.
is defined as the max error between the two cumulative
distribution functions. As listed in the Tables 2-4, the
cdf max errors obtained by mixture Weibull function are
quite smaller than those by conventional Weibull function and never exceed the critical value of 95% confidence level in Kolmogorov-Smirnov test (i.e., 0.0145).
On the other hand, the relative percentage errors of wind
potential energy for the mixture Weibull function never
exceed 0.1%, which are less than those for the conventional one. Where the relative percentage error mentioned above is defined as the difference between the
wind energy calculated from actual time-series data and
that from theoretical function divided by the wind energy
from actual time-series data.
Figures 7-9 demonstrate the wind power frequency
with respect to wind speed. Once again as in the case of
wind speed histogram, the mixture Weibull function matches with observed data better than the conventional
Weibull function. Otherwise wind speeds carrying maximum energy are higher than the most probable speeds,
independent of station, since wind energy is proportional
to the cube of speed.
Study on Wind Characteristics Using Bimodal Mixture Weibull Distribution for Three Wind Sites in Taiwan
Figure 7. Wind power probability for Dayuan.
Figure 8. Wind power probability for Hengchun.
It is worth to note that, as in Table 4, even though station Chungtun’s availability factor reveals a high value
as 0.932, an ideal turbine in Chungtun extracts only
40.6% (= 0.7375/1.8152) of wind potential energy calculated by observed time-series data; whereas the similar
quantity reaches more than 76.3% for other two stations,
as in Tables 2 and 3. There are basically two reasons that
can be attributed to this phenomenon, as illustrated in
289
Figure 9. Wind power probability for Chungtun.
Eq. (13), i.e. (1) the operation speeds of wind turbine and
(2) the Weibull parameters of wind speed distribution.
Through computation, it is found that an ideal wind turbine in Chungtun will lose 7.6% and 9.0% of output energy, respectively in Weibull and mixture Weibull calculation, if the ideal turbine operates with the speeds of 4
m/s (cut-in), 12 m/s (rated) and 25 m/s (cut-off) instead
of 2.5, 12.5 and 25 m/s. Contrarily an ideal turbine in
Dayuan will increase only 6.2% and 6.6% of output
energy, respectively in Weibull and mixture Weibull
calculation, if the ideal turbine operates with the speeds
of 2.5 m/s (cut-in), 12.5 m/s (rated) and 25 m/s (cut-off)
instead of 4, 12 and 25 m/s; the similar values are 5.0%
and 5.2% for the ideal turbine in Hengchun. As a result,
it could be concluded that the main reason causing the
lower energy output of an ideal turbine in Chungtun is
the second one, i.e. the distribution pattern of wind speed
observed.
7. Conclusions
Wind characteristics of three wind sites in Taiwan
have been analyzed according to bimodal mixture Weibull distribution. Mathematical expressions for assessing wind resource are proposed. The performance comparisons between the mixture and conventional Weibull
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Feng-Jiao Liu et al.
functions are made through Kolmogorov-Smirnov test
and relative percentage error of wind energy. The conclusions are summarized as follows:
(a) Mixture Weibull function provides more accurate assessment of wind energy than conventional Weibull
function especially while wind speed distribution reveals two humps on it, which is widely found in wind
regimes in the world.
(b) Max errors of cumulative distribution function obtained by mixture Weibull function lie below the significant level of 95% in Kolmogorov-Smirnov test.
(c) Relative errors of wind energy calculated by mixture
Weibull function are all below 0.1%.
(d) Distribution pattern of wind speed for a particular
area must be seriously assessed while selecting wind
farm or designing wind energy conversion system.
v
vi
vj
vme
vmp
vo
vr
v3
wind speed, m/s
cut-in speed of turbine, m/s
observed wind speed in stage j, m/s
wind speed carrying maximum energy, m/s
most probable wind speed, m/s
cut-off speed of turbine, m/s
rated speed of turbine, m/s
mean of wind speed cubes, m3/s3
Greek Letters
a
shape parameter of Weibull function, dimensionless
b
scale parameter of Weibull function, m/s
r
air density, kg/m3
f
weight parameter of mixture Weibull function, dimensionless
G( ) Gamma function
Acknowledgements
References
The authors would deeply appreciate the Central
Weather Bureau and Ministry of Economic Affairs for
providing observation data and deeply thank Dr. Wu CF
and Dr. Huang MW, researchers of the Institute of Earth
Sciences, Academia Sinica, Taiwan, for their useful
comments. This study is supported partly by the National
Science Council under the contract NSC100-2221-E252-014.
Nomenclature
sweep area of turbine blade, m2
availability factor, dimensionless
wind potential energy by time-series data, kWh
output energy of ideal turbine, kWh
wind potential energy by mixture Weibull function, kWh
f ( ) Weibull pdf
fww( ) mixture Weibull pdf
F( ) cumulative Weibull function
Fww( ) cumulative mixture Weibull function
L( ) likelihood function
LL( ) log-likelihood function
n
sample size, dimensionless
P
output power of wind turbine, W
T
time period, hr
A
AF
Ea
Eid
Eww
[1] Carta, J. A., Ramirez, P. and Velazquez, S., “A Review
of Wind Speed Probability Distributions Used in Wind
Energy Analysis Case Studies in the Canary Islands,”
Renewable and Sustainable Energy Reviews, Vol. 13,
pp. 933-955 (2009). doi: 10.1016/j.rser.2008.05.005
[2] Celik, A. N., Makkawi, A. and Muneer, T., “Critical
Evaluation of Wind Speed Frequency Distribution
Functions,” J. Renewable and Sustainable Energy,
Vol. 013102, pp. 1-16 (2010). doi: 10.1063/1.329
4127
[3] Ucar, A. and Balo, F., “Investigation of Wind Characteristics and Assessment of Wind-Generation Potentiality in Uludag-Bursa, Turkey,” Applied Energy, Vol.
86, pp. 333-339 (2009). doi: 10.1016/j.apenergy.
2008.05.001
[4] Akdag, S. A. and Dinler, A., “A New Method to Estimate Weibull Parameters for Wind Energy Applications,” Energy Conversion and Management, Vol. 50,
pp. 1761-1766 (2009). doi: 10.1016/j.enconman.
2009.03.020
[5] Jowder, F. A. L., “Wind Power Analysis and Site
Matching of Wind Turbine Generators in Kingdom of
BAHRAIN,” Applied Energy, Vol. 86, pp. 538-545
(2009). doi: 10.1016/j.apenergy.2008.08.006
[6] Raichle, B. W. and Carson, W. R., “Wind Resource As-
Study on Wind Characteristics Using Bimodal Mixture Weibull Distribution for Three Wind Sites in Taiwan
sessment of the Southern Appalachian Ridges in the
Southeastern United States,” Renewable and Sustainable Energy Reviews, Vol. 13, pp. 1104-1110 (2009).
doi: 10.1016/j.rser.2007.12.005
[7] Kwon, S. D., “Uncertainty Analysis of Wind Energy
Potential Assessment,” Applied Energy, Vol. 87, pp.
856-865 (2010). doi: 10.1016/j.apenergy.2009.08.038
[8] Zhou, W., Yang, H. X. and Fang, Z. H., “Wind Power
Potential and Characteristic Analysis of the Pearl
River Delta Region, China,” Renewable Energy, Vol.
31, pp. 739-753 (2006). doi: 10.1016/j.renene.2005.
05.006
[9] Seguro, J. V. and Lambert, T. W., “Modern Estimation
of the Parameters of the Weibull Wind Speed Distribution for Wind Energy Analysis,” J Wind Eng Indus
Aerod, Vol. 85, pp. 75-84 (2000). doi: 10.1016/
S0167-6105(99)00122-1
[10] Akpinar, E. K. and Akpinar, S., “Determination of the
Wind Energy Potential for Maden-Elazig, Turkey,”
Energy Conversion and Management, Vol. 45, pp.
2901-2914 (2004). doi: 10.1016/j.enconman.2003.
12.016
[11] Celik, A. N., “A Statistical Analysis of Wind Power
Density Based on the Weibull and Rayleigh Models at
the Southern Region of Turkey,” Renew Energy, Vol.
29, pp. 593-604 (2003). doi: 10.1016/j.renene.2003.
07.002
[12] Sulaiman, M. Y., Akaak, A. M., Wahab, M. A.,
Zakaria, A., Sulaiman, Z. A. and Suradi, J., “Wind
Characteristics of Oman,” Energy, Vol. 27, pp. 35-46
(2002). doi: 10.1016/S0360-5442(01)00055-X
[13] Yang, H. X., Lu, L. and Burnett, J., “Weather Data and
Probability Analysis of Hybrid Photovoltaic-Wind
Power Generation Systems in Hong Kong,” Renewable Energy, Vol. 28, pp. 1813-1824 (2003). doi:
10.1016/S0960-1481(03)00015-6
[14] Chang, T. J., Wu, Y. T., Hsu, H. Y., Chu, C. R. and
Liao, C. M., “Assessment of Wind Characteristics and
Wind Turbine Characteristics in Taiwan,” Renewable
Energy, Vol. 28, pp. 851-871 (2003). doi: 10.1016/
S0960-1481(02)00184-2
[15] Dorvlo, A. S. S., “Estimating Wind Speed Distribution,” Energy Conversion and Management, Vol. 43,
pp. 2311-2318 (2002). doi: 10.1016/S0196-8904(01)
00182-0
291
[16] Akpinar, S. and Akpinar, E. K., “Wind Energy Analysis Based on Maximum Entropy Principle (MEP)Type Distribution Function,” Energy Conversion and
Management, Vol. 48, pp. 1140-1149 (2007). doi:
10.1016/j.enconman.2006.10.004
[17] Shamilov, A., Kantar, Y. M. and Usta, I., “Use of
MinMaxEnt Distributions Defined on Basis of MaxEnt Method in Wind Power Study,” Energy Conversion and Management, Vol. 49, pp. 660-677 (2008).
doi: 10.1016/j.enconman.2007.07.045
[18] Carta, J. A. and Mentado, D., “A Continuous Bivariate
Model for Wind Power Density and Wind Turbine Energy Output Estimations,” Energy Conversion and
Management, Vol. 48, pp. 420-432 (2007). doi: 10.
1016/j.enconman.2006.06.019
[19] Jeromel, M., Malacic, V. and Rakovec, J., “Weibull
Distribution of Bora and Sirocco Winds in the Northern Adriatic Sea,” Geophysical, Vol. 26, pp. 85-100
(2009).
[20] Carta, J. A. and Ramirez, P., “Analysis of Two-Component Mixture Weibull Statistics for Estimation of
Wind Speed Distributions,” Renewable Energy, Vol.
32, pp. 518-531 (2007). doi: 10.1016/j.renene.2006.
05.005
[21] Razali, A. M. and Salih, A. A., “Combing Two Weibull
Distributions Using a Mixing Parameter,” European
Journal of Scientific Research, Vol. 31, pp. 296-305
(2009).
[22] Jiang, S. Y. and Kececioglu, D., “Maximum Likelihood Estimation from Censored-Data Mixed Weibull
Distribution,” IEEE Transactions on Reliability, Vol.
41, pp. 248-255 (1992). doi: 10.1109/24.257791
[23] Chang, T. P., “Performance Comparison of Six Numerical Methods in Estimating Weibull Parameters for
Wind Energy Application,” Applied Energy, Vol. 88,
pp. 272-282 (2011). doi: 10.1016/j.apenergy.2010.
06.018
[24] Carta, J. A. and Ramirez, P., “Use of Finite Mixture
Distribution Models in the Analysis of Wind Energy in
the Canarian Archipelago,” Energy Conversion and
Management, Vol. 48, pp. 281-291 (2007). doi: 10.
1016/j.enconman.2006.04.004
[25] Jaramillo, O. A. and Borja, M. A., “Wind Speed Analysis in La Ventosa, Mexico, a Bimodal Probability
Distribution Case,” Renewable Energy, Vol. 29, pp.
292
Feng-Jiao Liu et al.
1613-1630 (2004). doi: 10.1016/j.renene.2004.02.
001
[26] Jamil, M., Parsa, S. and Majidi, M., “Wind Power Statistics and an Evaluation of Wind Energy Density,” Renewable Energy, Vol. 6, pp. 623-628 (1995). doi:
10.1016/0960-1481(95)00041-H
Manuscript Received: Nov. 22, 2013
Accepted: Aug. 14, 2014