Applied Energy 92 (2012) 809–814
Contents lists available at SciVerse ScienceDirect
Applied Energy
journal homepage: www.elsevier.com/locate/apenergy
Theoretical derivation of wind power probability distribution function
and applications
_
Abdüsselam Altunkaynak, Tarkan Erdik, Ismail
Dabanlı ⇑, Zekai Sß en
_
Istanbul Technical University, Faculty of Civil Engineering, Department of Hydraulics, Maslak 34469, Istanbul,
Turkey
a r t i c l e
i n f o
Article history:
Received 7 June 2011
Received in revised form 3 August 2011
Accepted 19 August 2011
Available online 17 September 2011
Keywords:
Perturbation
Power
Statistical parameter
Weibull distribution
Wind
a b s t r a c t
The instantaneous wind power contained in the air current is directly proportional with the cube of the
wind speed. In practice, there is a record of wind speeds in the form of a time series. It is, therefore, necessary to develop a formulation that takes into consideration the statistical parameters of such a time series. The purpose of this paper is to derive the general wind power formulation in terms of the statistical
parameters by using the perturbation theory, which leads to a general formulation of the wind power
expectation and other statistical parameter expressions such as the standard deviation and the coefficient
of variation. The formulation is very general and can be applied specifically for any wind speed probability distribution function. Its application to two-parameter Weibull probability distribution of wind
speeds is presented in full detail. It is concluded that provided wind speed is distributed according to
a Weibull distribution, the wind power could be derived based on wind speed data. It is possible to determine wind power at any desired risk level, however, in practical studies most often 5% or 10% risk levels
are preferred and the necessary simple procedure is presented for this purpose in this paper.
Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Industrializations and economic development require further
energy sources especially as clean alternatives that are friendly with
the environment and especially help to attenuate the climate change
effects globally. Since, present atmospheric pollution is due greatly
to as fossil fuel usages, the clean and environmentally friendly
power resources are becoming increasingly important in domestic
and industrial energy productions. Among these clean resources,
wind power has become popular due to recent reduction in wind
turbine costs and rising fuel oil prices. Fuel consumption for many
industrial and other uses gives emissions to the atmospheric composition, especially in the form of carbon dioxide, and consequently
nuisances such as the global warming, atmospheric pollution,
greenhouse effect and ozone layer depletion become observable in
an increasing manner [1]. Although there is an international agreement after the famous Rio gathering for the reduction of atmospheric loads due to fuel consumption in 1992, and industrially
rich countries abide by the regulations drawn since then, they have
more or less completed their developments, but unfortunately many
developing countries did not sign such an agreement because they
want to reach the industrial level of rich countries. It is, therefore,
not surprising that today industrialized countries are trying to shift
towards the clean and environmentally friendly energy resources.
⇑ Corresponding author. Tel.: +90 2122853728; fax: +90 2122853710.
_ Dabanlı).
E-mail address: [email protected] (I.
0306-2619/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.apenergy.2011.08.038
Wind energy plays a major role in such activities. It has been
studied more frequently in the scientific area in recent years. This
is partly due to increased interest in alternative and less expensive
energy resources. It is strongly emphasized in this study that development of wind generation depends on the following factors [2]:
(a) Increasing public consciousness on carbon emissions, climate change and environmental issues.
(b) Consciousness about consumption of oil and gas reserves
and increasing production capacity of oil.
(c) Convenience costs of wind generation due to the development in relevant technologies.
In order to assess the variability of a wind energy conversion
system at a particular location, it is necessary to know various statistical characteristics of wind speed and wind power behavior.
After determining the wind power potential for a relevant site, a
suitable wind turbine can be erected for energy production [3].
Proper turbine design, sizing and storage facilities require knowledge of the wind velocity and wind power level variability, preferably at a given risk level.
Many wind power studies are based on average wind speed values at different sites, however, wind power estimations should be
based on elaborated wind speed statistics including the standard
deviation and the coefficient of skewness due to actual skewed distribution of the wind speed record. Determination of wind speed
characteristic is possible through wind speed probability distribu-
810
A. Altunkaynak et al. / Applied Energy 92 (2012) 809–814
Nomenclature
a
b
E ()
f ()
i
k
n
P
PD
Pi
PDF
Sp
standard deviation
V
wind speed (m/s)
V
average wind speed (m/s)
V0
perturbation term (m/s)
W-PDF Weibull wind speed probability distribution function
WW-PDF two component mixture Weibull PDF
scale parameter (m/s)
dimensionless shape parameter
energy
probability density function
rank of data
order term
data number
wind power (W/m2)
power density
probability of exceedence
probability distribution function
Greek letters
C
Gamma function
q
air density (kg/m3)
c
skewness coefficient
tion function (PDF) as Weibull, Gamma or any other suitable distribution. Recent studies indicate that the Weibull PDF seems to be
the best fitting model for the empirical wind distribution [3,4].
In practice, several PDF’s have been proposed in order to estimate more accurate wind speed characteristics. Majority of the
studies have focused on the typical two-parameter Weibull wind
speed PDF (W-PDF) [2]. Recent studies also tried to establish new
methodologies such as two-component mixture Weibull PDF
(WW-PDF), involving five parameters and the power-density (PD)
[5–7]. Beside these methods, moment, empirical, graphical, maximum likelihood, modified maximum likelihood and energy pattern
factor methods are used for determining wind speed characteristics. If wind speed distribution matches well with Weibull PDF,
these six methods are applicable; but if not, the maximum likelihood method performs best [4]. In this study, Weibull parameters
of the wind speed values are estimated by the maximum likelihood
method.
The purpose of this study is to derive the most general wind
power statistics by perturbation method with its application to a
two-parameter Weibull wind speed PDF. One can observe after
the derivation of wind power statistics that the wind power PDF
also abides by the Weibull PDF but with different parameters,
however, there exist analytical relationships between wind speed
and power Weibull PDF parameters.
2. Wind power perturbation
The theoretical derivation of wind power initiates from the consideration of the kinetic energy definition in physics, and for any
given instantaneous wind speed, V, the instantaneous wind power,
P, expression becomes as [8].
P¼
1
qV 3
2
ð1Þ
and therefore, in most of the analytical derivations the independence of q from V3 is taken into consideration for the simplification
of formulation derivation.
Perturbation methodology has been used extensively in many
research studies concerning turbulent flow in channels [15]. Since
the wind also presents a turbulent flow in the atmosphere, the
instantaneous wind speed, V, can be thought of consisting two
components namely, average wind speed, V, and the perturbation
term V0 as,
V ¼ V þ V0
ð3Þ
The perturbation term has the expected value, which is equal to
zero, i.e., E(V0 ) = 0 and its variance is equal to the variance of the
instantaneous wind speed record, that is VarðVÞ ¼ ðV 02 Þ. The substitution of Eq. (3) into Eq. (2) leads after sophisticated manipulations
to,
EðPÞ ¼
i
1 h 3
q E ðVÞ þ 3EðVÞEðV 02 Þ þ EðV 03 Þ
2
ð4Þ
Unfortunately, in all the wind power researches, this expression
has been approximated by ignoring the last term in the big bracket
and it is written in an approximate form as,
EðPÞ ¼
i
1 h 3
q E ðVÞ þ 3EðVÞEðV 02 Þ
2
ð5Þ
This expression is valid exactly for symmetrical PDF’s such as
the Gaussian distribution but the wind speed is never symmetrically distributed in nature. Most often the wind speed has skewed
distribution according to the Weibull, Gamma, Chi-squared, logarithmic-normal distribution, etc. In this paper, refined wind power
expectation in Eq. (4) will be developed for the Weibull distribution in the following section.
The general stochastic definition of variance of a random variable such as the wind energy is defined as,
VarðPÞ ¼ EðP2 Þ E2 ðPÞ
ð6Þ
where q is the standard density of air. Many researchers have assumed that the air density is independent of the wind speed cubed
and constant as for the standard atmosphere being equal to
1.293 kg/m3 In such a case the statistical expectation of the wind
power is,
The second term on the right hand side is the square of expectation given in Eq. (4) and it is necessary to calculate E(P2). For this
purpose, it is necessary first to substitute Eq. (3) into Eq. (1) and
then taking the square of both sides and finally the expectation
of both sides which leads to,
1
EðPÞ ¼ qEðV 3 Þ
2
EðP2 Þ ¼
ð2Þ
This formulation has been used by many researchers [9–14].
The dependence of q on V3 has been considered yielding to about
5% difference from the previous assumption [8]. However, in practice most often such an error is assumed practically insignificant,
1 2
q E V 6 þ 6V 5 V 0 þ 15V 4 V 02 þ 20V 3 V 03 þ 15V 2 V 04
4
þ 6VV 05 þ V 06
ð7Þ
Since, by definition EðV 0 ¼ 0Þ, this last expression takes the following explicit form,
A. Altunkaynak et al. / Applied Energy 92 (2012) 809–814
1
EðP2 Þ ¼ q2 E6 ðVÞ þ 15E4 ðVÞEðV 02 Þ þ 20E3 ðVÞEðV 03 Þ
4
þ 15E2 ðVÞEðV 04 Þ þ 6EðVÞEðV 05 Þ þ EðV 06 Þ
ð8Þ
1
E2 ðPÞ ¼ q2 E6 ðVÞ þ E2 ðV 03 Þ þ 6E4 ðVÞEðV 02 Þ þ 2E3 ðVÞEðV 03 Þ
4
þ 6EðVÞEðV 02 ÞEðV 03 Þ þ 9E2 ðVÞE2 ðV 02 Þ
ð9Þ
E2 ðV 03 Þ 2E3 ðVÞEðV 03 Þ 6EðVÞEðV 02 ÞEðV 03 Þ þ EðV 06 Þ
ð10Þ
This is the general variance equation after perturbation of the
wind speed value that can be applied specifically to any PDF.
3. Weibull PDF and wind power
The two parameter Weibull W-PDF is employed in most of the
wind speed assessments [10]. If mean wind speed and standard
deviation are available, this method is more useful and simple than
graphical method [16,17]. It has been shown that W- PDF of wind
speed is a useful practical tool for estimation of future power in
many climatic regions. It has been shown that the W-PDF wind
speed is a useful practical tool for estimation of future power from
windmills in Denmark [18]. A review of the relevant statistical
methods for estimation of Weibull parameters is given with
emphasis on efficiency [19]. The wind speed is treated as a random
variable which abides with a two-parameter W-PDF. The PDF of
such a distribution is given mathematically as,
FðVÞ ¼
" #
b1
b
a V
V
exp b a
a
ð12Þ
in which, C = Gamma pdf. Hence, the expected value (k = 1) and the
variance from Eq. (6) of a W-PDF can be obtained analytically as,
Substitution of this expression together with (5) into (6) leads
after algebraic manipulations to,
1
VarðPÞ ¼ q2 9E4 ðVÞEðV 02 Þ þ 15E2 ðVÞEðV 04 Þ 9E2 ðVÞE2 ðV 02 Þ
4
where a is a scale parameter with the same dimension as V, and b is
a dimensionless shape parameter. In many applications of the WPDF, it is necessary to use the statistical moments. In general, k-th
order statistical moment of the W-PDF wind speed, V, is given as,
k
EðV k Þ ¼ ak C 1 þ
b
and
811
ð11Þ
1
EðVÞ ¼ aC 1 þ
b
ð13Þ
and
2
1
C2 1 þ
VarðVÞ ¼ a2 C 1 þ
b
b
ð14Þ
In order to obtain an analytical expression for the W-PDF wind
speed, wind power expectation according to Eq. (4), it is necessary
to evaluate various terms in the big brackets. The first term is ready
from Eq. (14) but other two perturbation terms can be obtained as
2
1
C2 1 þ
EðV 02 Þ ¼ a2 C 1 þ
b
b
ð15Þ
and
EðV 03 Þ ¼ EðV VÞ3 ¼ EðV 3 Þ 3EðV 2 ÞEðVÞ þ 2E3 ðVÞ
ð16Þ
Under the light of Eq. (12) this expression takes the following
form,
3
1
2
1
3C 1 þ C 1 þ
þ 2 C3 1 þ
EðV 03 Þ ¼ a3 C 1 þ
b
b
b
b
ð17Þ
On the other hand, in order to find an explicit expression for the
wind power variance according to Eq. (10), it is necessary to evaluate some other perturbation term higher order moment. The
fourth order perturbation term expectation becomes,
EðV 04 Þ ¼ EðV VÞ3 ¼ EðV 4 Þ 4EðVÞEðV 3 Þ þ 6E2 ðVÞEðE2 Þ
Fig. 1. Location map.
4E3 ðVÞEðEÞ þ E4 ðVÞ
ð18Þ
812
A. Altunkaynak et al. / Applied Energy 92 (2012) 809–814
Table 1
Statistical description of the wind speed data.
Station name
Wind speed (m/s)
Kahta/Adiyaman
Merzifon/Amasya
Karaburun/Istanbul
Min
Mean
Max
Std. dev.
Median
Mode
0.021
0.016
0.010
7.57
6.88
7.39
33.29
25.04
26.56
4.13
3.41
4.38
6.74
6.54
6.62
10.20
12.00
7.86
However, for the W-PDF after the substitution of the necessary
moments from Eq. (12), one can obtain,
4
3
1
2
4C 1 þ
þ 6C 1 þ
EðV 04 Þ ¼ a4 C 1 þ
C 1þ
b
b
b
b
1
1
2
4
C 1þ
3C 1 þ
ð19Þ
b
b
The general fifth moment expression for the perturbation velocity can be expressed as,
EðV 05 Þ ¼ EðV VÞ5 ¼ EðV 5 Þ 5EðVÞEðV 4 Þ þ 10E2 ðVÞEðE3 Þ
10E3 ðVÞEðE2 Þ þ 5E4 ðVÞEðVÞ E5 ðVÞ
ð20Þ
Its specific form can be obtained from Eq. (12) one as,
5
4
1
3
5C 1 þ
þ 10C 1 þ
EðV 05 Þ ¼ a5 C 1 þ
C 1þ
b
b
b
b
1
2 3
1
1
2
5
10C 1 þ
þ 4C 1 þ
ð21Þ
C 1þ
C 1þ
b
b
b
b
Furthermore, the sixth perturbation moment by stochastic definition can be expanded into the following form,
EðV 06 Þ ¼ EðV VÞ6 ¼ EðV 6 Þ 6EðVÞEðV 5 Þ þ 15E2 ðVÞEðE4 Þ
20E3 ðVÞEðE3 Þ þ 15E4 ðVÞEðV 2 Þ 6E5 ðEÞEðEÞ þ E6 ðVÞ ð22Þ
Consideration each term from of Eq. (12) lead to,
6
5
1
4
EðV 06 Þ ¼ a5 C 1 þ
C 1þ
6C 1 þ
þ 15C 1 þ
b
b
b
b
1
3 3
1
2
2
20C 1 þ
þ 15C 1 þ
C 1þ
C 1þ
b
b
b
b
1
1
4
6
5C 1 þ
ð23Þ
C 1þ
b
b
The substitution of all these necessary terms into Eq. (10) gives
after the necessary manipulations to desired variance of wind
power as,
VarðPÞ ¼
1 2 6
6
3
q a C 1 þ C2 1 þ
4
b
b
ð24Þ
The standard deviation Sp, of wind power is equal to the square
root of this final equation.
Sp ¼
1 3
qa
2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6
3
C2 1 þ
C 1þ
b
b
ð25Þ
Substitution of the necessary expectation terms into equation
(4) is performed and wind power expectation of a W-PDF can be
obtained analytically as,
EðPÞ ¼
1 3
3
qa C 1 þ
2
b
ð26Þ
It is also possible to evaluate the dimensionless skewness coefficient (c) of the wind energy is given below as,
c¼
EðP 3 Þ 3EðP 2 ÞEðPÞ þ 2E3 ðPÞ
S3P
Fig. 2. Weibull distribution function fitted to wind speed data of stations.
ð27Þ
The substitution of necessary terms into this expressions leads
after the necessary manipulations to,
h i
C 1 þ 9b 3C 1 þ 6b C 1 þ 3b þ 2C2 1 þ 3b
c¼
hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
i3
C 1 þ 6b C2 1 þ 3b
ð28Þ
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A. Altunkaynak et al. / Applied Energy 92 (2012) 809–814
Table 2
Statistic parameters of wind power perturbation theory.
Station name
Adiyaman/Kahta
Amasya/Merzifon
Istanbul/Karaburun
Wind speed (m/s) Parameters of the W-pdf
W-pdf and wind power
a (scale)
Perturbation
8.56
7.77
8.29
b (shape)
1.94
2.12
1.74
Mean (kg/s3)
Standard deviation (kg/s3)
Dimensionless skewness
502.00
348.35
508.99
689.44
438.24
792.23
6.83
5.14
9.62
4. Study area and data
Wind speeds are measured as 10-min interval averages, between the years 2000 and 2002, in both Merzifon/Amasya and Kah_
ta/Adiyaman and as one hour averages in Karaburun/Istanbul.
Of
these locations, Karaburun, a district of Istanbul province is in
the northwestern Turkey (see Fig. 1) with geographic coordinates
of 41.3380 N latitude and 28.6770 E longitude. Merzifon, a town
and a district in Amasya Province in the central Black Sea region
of Turkey, covers an area of 970 km2, with the geographic coordinates of 40.8750 N latitude and 35.4600 E longitude. Kahta, in Adiyaman province, is bounded on the east and southeast by the
Euphrates River and on the northeast by Gerger tributary; it is almost 2500 m above mean sea level, with the geographic coordinates of 37.7830 N latitude and 38.6170 E longitude.
Whilst Karaburun/Istanbul wind speed gaging station covers
365 days, Merzifon/Amasya and Adiyaman Kahta stations cover
633 and 539 days, respectively. The statistical description of the
data used is provided in Table 1.
Two-parameter W-PDF, with scale and shape parameters, has
been the most popular one due to its ability to fit most accurately
the variety of wind speed data measured at different geographical
locations in the world. While the Weibull scale parameter controls
the abscissa scale of data distribution and shape parameter describes the width of the PDF. The larger the shape parameter, the
narrower is the distribution and the higher is its peak.
5. Application
The application of the methodology developed in the Sections 2
and 3 is applied to three wind speed measurement locations. This
methodology is in the form of a general formulation including statistical parameters such as expectation, standard deviation and
dimensionless skewness coefficient. The same methodology can
be applied to any region provided that wind speed matches WPDF. In Fig. 2a–c, good fits are observed for wind speed data by
W-PDF’s. The proper fits between the theoretical W-PDF and
empirical histograms are checked with Kolmogorov-Smirnov
goodness-of-the-fit test, which tries to determine closeness of distribution function. It is calculated that both theoretical W-PDF and
empirical histograms comply with each other at least at 5% significance level [20].
The parameters of W-PDF’s are presented in Table 2 with scale,
a, and shape, b, parameters. The calculation of wind power statistics parameters at the Adıyaman/Kahta station, according to the
mean, (Eq. (26)) standard deviation (Eq. (24)) and dimensionless
skewness (Eq. (28)) are 502 kg/s3, 689.44 kg/s3 and 6.89, respectively. The same parameters for Amasya/Merzifon station are
348.35 kg/s3, 438.24 kg/s3 and 5.14. For Istanbul/Karaburun station, they are 508.99 kg/s3, 792.23 kg/s3 and 9.62, respectively.
The wind power statistics parameters are presented in Table 2. It
is obvious that Istanbul/Karaburun station has the highest value
than the other stations. The same station is found to produce wind
power better than other two stations when evaluated in terms of
dimensionless skewness statistic parameter. Since wind time
Fig. 3. Wind energy cumulative distribution function.
814
A. Altunkaynak et al. / Applied Energy 92 (2012) 809–814
7. Conclusion
Fig. 4. Wind power-risk level relationships.
series record has random variability above and below the mean value, in order to make more feasible investment, the perturbation
theory should be applied for wind energy production in any optimum wind turbine design.
Wind power stochastic characteristics play a significant role in
planning, design, and operation of the wind turbines. Theoretically,
these characteristics are functions of recorded wind speed behaviors. It is, therefore, necessary to develop analytical relationships
between wind speed statistics and wind power characteristics. This
study presents derivation of wind power stochastic characteristics
in terms of the expected values of the means, standard deviation
and the dimensionless skewness. The perturbation method was
used to derive first general expressions for the wind power statistics from the wind speed records, and then Weibull probability distribution function (PDF) was employed for the presented wind
power formulation. After the derivation of wind power statistics,
their comparisons with the corresponding characteristics of wind
speed PDF are exposed that if the wind speed PDF abides by the
Weibull PDF than the wind power also abides with the WeibullPDF, but with different statistical parameters that can be expressed
in terms of the wind speed statistical characteristics. In the present
study, derived general formulation is applied to three potential
sites in Turkey homely at locations of Adıyaman/Kahta, Amasya/
Merzifon and Istanbul/Karaburun.
Acknowledgment
6. Wind energy distribution
As explained before the PDF of wind power accords with the
Weibull distribution, and therefore only the right hand tail is significant for wind energy generation risk calculations. Prior to any
risk calculation this section provides conformation between the
practically calculated wind power calculations and their theoretical Weibull PDF matching. For this purpose, two-step procedure
is applied as follows.
(1) According to Eq. (1) wind energy amounts are calculated for
each time interval wind speed records and their empirical
scatter is obtained by attaching the probability of exceedence, Pi, according to empirical probability calculation
under the light of the following classical formulation [21],
Pi ¼
i
nþ1
where i is the i-th rank of data in ascending order and n is the data
number.
(2) The theoretical Weibull cumulative PDF is obtained after the
best fitting to empirical data scatter.
The two cases, namely, empirical and theoretical wind power
solutions are shown in Fig. 3 for Kahta/Adiyaman, Merzifon/Amasya and Karaburun/Istanbul stations. On the other hand, Fig. 4
shows the relationship between the risks level and wind power
amounts on the horizontal logarithmic axis. The reason for logarithmic scale is to provide uniform scatter of very high and low
wind power amounts. Otherwise, it is not possible to distinguish
clearly between these values. Herein, the risk level is defined as
the exceedence probability for a given wind power or conversely,
given the exceedence probability then the corresponding wind
power is the risk level. It is obvious from this figure that as the risk
level increases in all stations wind power generation amounts decrease. There is an inverse and non-linear relationship between the
two variables. In practical risk level ranges (between 5% and 10%)
Kahta/Adıyaman and Amasya/Merzifon stations have almost the
same wind energy generation possibilities, but at Karaburun/Istanbul station at the same risk levels about 350 kW less energy can be
generated.
The authors would like to express their gratitude to Dr. Ahmet
Duran SAHIN for sharing us wind speed database.
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