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Wind speed and power density analysis based on
Weibull and Rayleigh distribution functions
Year: 2012
Asif Jalal
2012-MS-MED-24
Supervisor
Prof. Dr. Nasir Hayat
Department of Mechanical Engineering
University of Engineering & Technology Lahore-Pakistan
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Wind speed and power density analysis based on Weibull
and Rayleigh distribution functions
This thesis is submitted to Department of Mechanical Engineering, University of Engineering &Technology, Lahore, for the partial fulfillment of the requirement for the award of degree in
Master of Science in
Mechanical Design Engineering
Approval on __________________
Department of Mechanical Engineering
University of Engineering & Technology Lahore-Pakistan
External Examiner
Dean
Faculty of Mechanical Engineering
Internal Examiner
Professor Dr. Nasir Hayat
Chairman
Department of Mechanical Engineering
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In the Name of ALLAH, the Most Gracious and the Most Merciful.
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This thesis is dedicated to my parents, my
brothers and especially, my sister.
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Acknowledgements
Firstly, I would like to express my gratitude, praise and acknowledgement to Almighty ALLAH,
who is the only Creator and Owner of the universe, I am nothing merely the kindness of ALLAH.
Through, His kindness of My Lord, I am able to achieve my all goals in my life.
Secondly, I would like to thanks my dedicated supervisor Professor Dr. Nasir Hayat. His
motivation, direction and guidance really helped me to achieve my ultimate task. I shall be very
thankful to him for this act of kindness.
And, I would not restrain myself to express acknowledgement and honest appreciation to Mr.
Nadeem Faisal Deputy Director PMD (CPDC Karachi) Pakistanfor providing me precious
and confidential department data and technical help in this research work.
Lastly, I would yield this opportunity to thank respected Chairman and Dean of the faculty along
with all Post Graduate Research Committee and faculty members for their insightful comments
and observations.
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Abstract
Large scale industrial growth and depleted hydro power sources in the country urges the
researchers to transcend the focus towards the renewable energy sources because they are clean,
abundant and environment friendly as compared to orthodox energy sources. Among the
renewable energy sources, the wind energy is most rapidly growing than others. The key objective
of this paper is to study and investigate the wind characteristics and wind potential at the site of
Turbat. For this purpose, the measured 3-hourly time series data was taken from the Pakistan
Meteorological department (CPDC, Karachi) for the period of 21 months (Jan 2012- Sep13). The
obtained data was statistically evaluated with the help of Weibull and Rayleigh functions. The
average wind speed is more than 4 m/s for every month. The average values of most probable wind
speed and wind speed carrying maximum energies are 3.83 m/s and 7.732 m/s respectively. The
overall variation in the values of standard deviation is from 1.699 to 3.306. The values of Weibull
shape parameter are in the range of 1.508 to 3.010. Similarly, the range of Scale parameter is from
4.674 m/s to 7.429 m/s. The monthly mean value of wind power and energy densities of the
selected site are 140.145 W/m2and 101.775 kWh/m2respectively. The mean power potential of
the site is then compared with the power calculated through Weibull and Rayleigh functions. It
was exposed that the Weibull distribution describe the actual data better than the Rayleigh
function. This statement was further enriched by the assessment of performance of these twodistribution with the RMSE, 2, R2tests.
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Table of ContentsAcknowledgements------------------------------------------------------------------------------------------------------------- 5
Abstract---------------------------------------------------------------------------------------------------------------------------- 6
Chapter # 1 ----------------------------------------------------------------------------------------------------------------------- 12
INTRODUCTION: ------------------------------------------------------------------------------------------------------------ 12
Chapter # 2 ----------------------------------------------------------------------------------------------------------------------- 16
LITERATURE REVIEW --------------------------------------------------------------------------------------------------- 16
Chapter # 3 ----------------------------------------------------------------------------------------------------------------------- 19
BASIC THEORY IN WIND ENERGY --------------------------------------------------------------------------------- 19
3.1 Introduction of Wind energy --------------------------------------------------------------------------------------- 19
3.2 Brief history of Wind energy --------------------------------------------------------------------------------------- 20
3.3 Wind Resource Assessment----------------------------------------------------------------------------------------- 20
3.4 Extrapolation of wind speed at different height------------------------------------------------------------- 22
3.5 Presentation of Data -------------------------------------------------------------------------------------------------- 22
3.5.1 Data collection---------------------------------------------------------------------------------------------------- 22
3.5.2 Organization of Data-------------------------------------------------------------------------------------------- 22
3.5.3 Tabulation of data ----------------------------------------------------------------------------------------------- 23
3.5.4 Graphical representation -------------------------------------------------------------------------------------- 23
3.6 Measure of central tendency of the data----------------------------------------------------------------------- 23
3.6.1 Mean ----------------------------------------------------------------------------------------------------------------- 23
3.6.2 Median-------------------------------------------------------------------------------------------------------------- 24
3.6.3 Mode ----------------------------------------------------------------------------------------------------------------- 24
3.7 Measure of Dispersion----------------------------------------------------------------------------------------------- 24
3.7.1 Range ---------------------------------------------------------------------------------------------------------------- 24
3.7.2 Mean deviation--------------------------------------------------------------------------------------------------- 24
3.7.3 Standard deviation---------------------------------------------------------------------------------------------- 25
3.8 Wind data parameter estimation--------------------------------------------------------------------------------- 25
Chapter # 4----------------------------------------------------------------------------------------------------------------------- 27
DATA COLLECTION AND ANALYSIS------------------------------------------------------------------------------ 27
4.1 Site description and data collection ------------------------------------------------------------------------------ 27
4.2 Statistical analysis of the data------------------------------------------------------------------------------------- 28
4.3 Frequency distribution Table -------------------------------------------------------------------------------------- 30
4.4 Statistical Distributions ---------------------------------------------------------------------------------------------- 30
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4.4.1 Weibull distribution --------------------------------------------------------------------------------------------- 32
4.4.2 Rayleigh distribution------------------------------------------------------------------------------------------- 34
4.5 Evaluation of Weibull and Rayleigh distributions---------------------------------------------------------- 35
4.6 Rated wind speed for wind turbine------------------------------------------------------------------------------ 36
4.7 Cut in velocity ----------------------------------------------------------------------------------------------------------- 36
4.8 Rated wind speed of Turbine generator ------------------------------------------------------------------------ 36
4.9 Full load------------------------------------------------------------------------------------------------------------------ 36
4.10 Cut out Velocity------------------------------------------------------------------------------------------------------ 37
4.11 Wind power density calculation --------------------------------------------------------------------------------- 37
4.11.1 Swept area of Blade -------------------------------------------------------------------------------------------- 37
4.11.2 Density of air ----------------------------------------------------------------------------------------------------- 38
4.11.3 Wind power density using weibull distribution------------------------------------------------------- 38
4.11.4 Wind power density using Rayleigh distribution ---------------------------------------------------- 39
4.11.5 Error in the calculation of power density--------------------------------------------------------------- 39
4.12 Wind energy density calculation-------------------------------------------------------------------------------- 39
4.13 Analysis type ----------------------------------------------------------------------------------------------------------- 39
Chapter # 5 ----------------------------------------------------------------------------------------------------------------------- 41
RESULTS & DISCUSSIONS ---------------------------------------------------------------------------------------------- 41
5.1 Measurement of average wind speed ---------------------------------------------------------------------------- 41
5.2 Measurement of standard deviation ----------------------------------------------------------------------------- 42
5.3 Monthly values of Vmpand Vmax.E--------------------------------------------------------------------------------- 43
5.4 Coefficient of Variation ---------------------------------------------------------------------------------------------- 44
5.5 Weibull parameters estimations---------------------------------------------------------------------------------- 44
5.6 Frequency distribution table --------------------------------------------------------------------------------------- 47
5.7 Comparison of Actual, Weibull and Rayleigh probabilities---------------------------------------------- 49
5.7.1 Monthly comparison -------------------------------------------------------------------------------------------- 49
5.7.2 Yearly comparison----------------------------------------------------------------------------------------------- 54
5.8 Performance estimation of Statistical models----------------------------------------------------------------- 56
5.8.1 Monthly R2value ------------------------------------------------------------------------------------------------- 57
5.8.2 RMSE and Chi-square test----------------------------------------------------------------------------------- 59
5.9 Calculation of Wind power density------------------------------------------------------------------------------ 59
5.10 Calculation of Wind energy density ---------------------------------------------------------------------------- 62
5.11 Error estimation during power density calculations ------------------------------------------------------ 64
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5.11.1 Weibull distribution ------------------------------------------------------------------------------------------- 64
5.11.2 Rayleigh distribution ------------------------------------------------------------------------------------------ 65
5.11.3 Comparison of error estimation using Weibull function and actual wind data------------ 66
CONCLUSIONS --------------------------------------------------------------------------------------------------------------- 67
REFERENCES ----------------------------------------------------------------------------------------------------------------- 69
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List of Figures
Figure 1.1: Total installed capacity of the world wind energy ever since 1997 to 2014 ....... 13
Figure 1.2: Wind map of Pakistan showing wind potential of all provinces [5] ................... 15
Figure 4.1: Location of Turbat in Balochistan, Pakistan........................................................ 28
Figure 4.2: Swept area of the blade of wind turbine............................................................... 38
Figure 5.1: Average wind speed per month for year 2012-13................................................. 41
Figure 5.2: Monthly values of standard deviation................................................................... 42
Figure 5.3: Monthly values of Coefficient of Variation for year 2012-13 .............................. 44
Figure 5.4: Monthly values of shape parameter for year 2012-13......................................... 46
Figure 5.5: Monthly variation in the values of scale parameter for year 2012-13 ................ 46
Figure 5.6: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Feb 2012.. 49
Figure 5.7: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Jun 2012.. 50
Figure 5.8: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Sep 2012.. 50
Figure 5.9: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Nov 2012. 51
Figure 5.10: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Jan 2013 51
Figure 5.11: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Jun 2013 52
Figure 5.12: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Aug 2013 53
Figure 5.13: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Sep 2013 53
Figure 5.14: Comparison of Actual, Weibull and Rayleigh PDF, CDF for year 2012......... 54
Figure 5.15: Comparison of Actual, Weibull and Rayleigh PDF, CDF for year 2013......... 55
Figure 5.16: Comparison of Actual, Weibull and Rayleigh PDF, CDF for year 2012-13 .... 56
Figure 5.17: Monthly values of R2 for Weibull and Rayleigh distribution, year 2012......... 57
Figure 5.18: Monthly values of R2 for Weibull and Rayleigh distribution, year 2013......... 58Figure 5.19: Average monthly wind power density of the Turbat site.................................. 61
Figure 5.20: Average monthly wind power density for Turbat site....................................... 61
Figure 5.21: Monthly wind energy density calculated with mean speed, Weibull and
Rayleigh functions for year 2012............................................................................................... 62
Figure 5.22: Monthly wind energy density calculated with mean speed, Weibull and
Rayleigh functions for year 2013............................................................................................... 63
Figure 5.23: Comparison of average error estimated for 2 years, evaluated by Weibull and
Rayleigh distributions w.r.t actual observed wind data.......................................................... 65
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List of Tables
Table 1.1: Wind resources assessment of Pakistan [4] ............................................................ 14
Table 5.1: Monthly values of Vmpand Vmax.E for the year 2012-13 ........................................ 43Table 5.2: Monthly values for Shape and Scale parameters of Weibull distribution........... 45
Table 5.3: Distribution of frequencies calculated from wind speed data for year 2012 ....... 47
Table 5.4: Distribution of frequencies calculated from wind speed data for year 2013 ....... 48
Table 5.5: Evaluation of Statistical models used ...................................................................... 59
Table 5.6: Monthly comparison of power densities calculated from actual wind data,
Weibull and Rayleigh models .................................................................................................... 60
Table 5.7: Yearly average values of wind power and energy density .................................... 63
Table 5.8: Classes of wind power density at 10 m and 50 m height [33] ............................... 64
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Chapter # 1
INTRODUCTION:
Pakistan is an emerging country having population of above 180 million but the average energy
per capita of the year is about 450 kWh; whereas, the Worlds average value of energy
consumption per year is 2730 kWh [1]. Almost, 37% population of remote areas are still awaiting
to be connected with the national grid [1].Billions of dollars are being spent on the import of oil
for thermal power houses; resultantly, worse economy, air pollution, acid rain emission and abrupt
climate change. Enormous industrialization is taking place, Hydro Power /water resource are
depleting and oil and gas resources are, if local than scarce in availability and else, wounding the
budget of whole economy. Hence, depletion, pollution and energy shortage is further scorching to
our deficit economy badly, which requires severe attention and ultimate solution. Therefore, to
meet the emergent electricity demand of the country and to save the environment as well as
economy, we need Renewable energy sources in addition with existing power resources as these
are free, environment friendly and enriched in Pakistan.
In renewable energy sources, wind is most speedily growing source in the world. Likewise, other
renewable energy source, it is clean, abundantly available, reliable and does not affect the
environment. Wind energy added more consideration and importance worldwide after the oil crisis
in 1973 and 1979 [2]. Currently, based upon the global wind energy statistics, the total wind energy
capability of the world at the end of 2014 is 371,559 MW with the added capacity of 52,654 MW
in 2014 only, having annual growth rate of 16.4% [3]. This wind energy capacity of the world, at
the end of 2013 and 2012, was 319,036 MW and 282,810 MW respectively [3]. In Fig. 1.1the
total mounted aptitude of the wind energy of the world has been shown since year 1997 to 2014
[3].
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Figure 1.1: Total installed capacity of the world wind energy ever since 1997 to 2014
In the Word, China is leading with total wind energy installed capacity of 114,763 MW by the
addition of 23,350 MW in the year 2014 only, having growth rate of 25.7% [3]. USA came 2ndin
the list having total installed capacity of the wind 66,754 MW at the end of year 2014. Germany,
Spain and India are at the number 3 rd, 4thand 5thspot having total installed capacity of 40,468
MW, 22,986.5 MW and 22,465 MW respectively by the end of year 2014 [3].
In Europe, total wind energy capacity has grown from 119 GW in 2013 to 132 GW in 2014 [3].
Germany is leading in Europe having total install capacity of 40,468 MW followed by Spain, UK,
France and Italy with installed capacity of 22,986 MW, 12,440 MW, 9,296 MW and 8,662 MW
respectively [3]. In Asia, china is at the top of wind installed capacity followed by India, Turkey
(most wind form is on its Asian part), Japan and South Korea having wind installed capacities of
22,465 MW, 3,763 MW, 2,788 MW and 609 MW respectively [3]. In Pakistan, work is in progress
on different projects regarding wind energy and currently has total wind installed capacity of 256
MW by the end of 2014, with the addition of 156 MW in the subsequent year [3]. Pakistan is at
the 43thnumber in the list of world wind installed capacity [3].
Wind energy has been recognized as a one of the significant and latent source of energy for
Pakistan. The country has a total estimated gross wind power capacity of around 346,000 MW [4].
Currently, the total installed power generation capacity (containing hydel, thermal and nuclear
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sources) of Pakistan is 22,800 MW while the actual power generation is around 9,000 MW due to
many aspects [4]. The demand varies between 16,500 MW (in summer) to 10,000 MW (in winter)
[4]. Hence this shortfall of electricity can easily be encountered by the installation of more wind
power projects.
Table 1.1, reveals that Pakistan has over 9% of total land which is appropriate for utmost utility-
scale wind turbine application [4]. The total wind generation capacity of this area is above 349,000
MW [4]. This is worthwhile to mention here that about 3.5% of the area has a wind potential of
class 4 or greater and has a potential of 132,990 MW, which is the prerequisite of price effective
wind power generation [4]. So, the demand of the country can easily be encountered by installing
more wind energy projects.
Table 1.1: Wind resources assessment of Pakistan [4]
Pakistan Meteorological department (PMD) with the collaboration of National Renewable energy
laboratory (NREL) and U.S Agency for international development (USAID) has developed a
mesoscale map of Pakistan as shown in Fig. 1.2.This map shows that there is an immense wind
potential available in areas of southern Sind, north Balochistan, central KPK, Gilgit-Baltistan and
different areas of Punjab and Azad Kashmir.
Furthermore, PMD conducted a survey (Phase-I) alongside the coastal belt of Sind province having
total area of 9300 sq. km and pointed out that this area has exploitable electric potential of about
11,000 MW [5]. The wind corridor of Gharo-katibandar, spreading 60 km along the coastal of
Sind, alone has a wind potential of 60,000 MW [6]. Now, in Phase-II, survey is being done on the
northern area of Pakistan.
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Figure 1.2: Wind map of Pakistan showing wind potential of all provinces [5]
Keeping in the view of highlighted most wind potential regions in Pakistan, this article is hereby
purposed to deep spot the site of Turbat, Balochistan as having good wind power potential and
energy density. Data was collected from Pakistan meteorological department (PMD), analyzed and
results were then statistically equated using Weibull and Rayleigh distribution functions,
accordingly. In order to get precise results, monthly and yearly analysis was done on the data.
Most probable wind speed and maximum carrying energy by wind are also calculated here, which
are required for the design of wind turbine blade profile for a particular site. Both, Weibull and
Rayleigh functions were then evaluated with the help of performance tests, to determine which
distribution function better described the actual data. After selecting the suitable distribution, the
Wind power density and energy density has been calculated. And based upon these results, it is
decided that in which class this site falls in wind power classification table. The error which
occurred during the calculation of wind power density has also been calculated here.
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Chapter # 2
LITERATURE REVIEW
Many researchers have calculated the wind power potential at different sites of the World. In
Pakistan alongside PMD, many scholars calculated the wind power potential at different locations
of the country. Wind power density has been calculated at the place of Hawksbay Karachi, Gharo,
Kati-Bandar, Jiwani, Quetta, Babaurband and Jhimpir (mostly beside the coastal belt of Pakistan).
In [1]the wind power potential is calculated at Hawksbay Karachi Sind, Pakistan. 2 years (2009-
10) data was taken and power densities were calculated using Weibull and Rayleigh distributions.
Mean wind speed and power density on yearly basis at the site was found to be 5.9 m/s and 197
W/m2 at the height of 80m. The site was found appropriate for wind energy projects. In [2]the
wind power density is calculated at the site, located in Gharo Sind, Pakistan. Five years data was
taken (2003-07), measured at 30 m height and investigated with the help of Weibull and Rayleigh
function. The average wind speed at the site was above 5 m/s. The wind power and energy density
of the site are 260 W/m2 and 2300 kWh/m2 respectively [2]. These power generations then
evaluated through the wind turbine of dissimilar manufacturers and found that this site has suitable
wind power potential and appropriate for wind generation projects [2]. In [4] the wind power
potential of three provinces of the country was analyzed and Jiwani (a site of Balochistan) was
taken as a case study and its specific power density was assessed. At the end, a practical scheme
is proposed for the integration of wind power output of the turbine to the national grid.
Kharo SF et al. [7]investigated the power density of wind for the site Babaurband Sind, Pakistan
at 4 different altitudes and then assessed the wind power likely to be generated over commercial
wind turbines. At 80m height, the yearly-mean wind speed was found to be 6.712 m/s while wind
power density was 310 W/m2. The values of power densities were higher from April to August.
To illustrate the feasibility of wind turbine installed, economic analysis was done and cost per
kWh was found to be 0.0263 US$/kWh. The site was found to be suitable for wind energy projects.
Calculation of wind power potential outside the Pakistan includes; S.H. Pishgar-Komleh et al. [8]
computed power potential of the wind using Weibull and Rayleigh functions at the site of
Firouzkooh region of Iran. 10 years (2001-10) data, based upon 3-h period, was taken and
analyzed. The average value of wind power was found to be 203 and 248 Wm-2 year-1based upon
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average and root-mean cube speed methodologies [8]. The site was found suitable for wind turbine
installation, because it falls in class 4 of wind power classification. Results shows that Weibull and
Rayleigh models close-fitting the actual data well with R2value of 0.97. E.K Akpinar et al. [9]
calculated power potential of wind at Keban-Elazig, a site in the Turkey. 5 years (1998-2002) data
was taken and analysed through the Weibull and Rayleigh distribution functions. Weibull
distribution better describe the observed wind data than the Rayleigh distribution at the nominated
site. The yearly value of average power density was found to be 15.603 W/m2, so this is not a
suitable site for grid connection usage. This power density of this rank can be utilized for non-grid
application like battery charging and water pumping [9]. L. Fyrippis et al. [10]inspected the wind
power prospective of Koronos village, a location present in the Naxos Island, Greece. Wind
characteristics were statistically analysed with the help of Weibull and Rayleigh distribution
functions. It was exposed that the Weibull distribution tailored the actual data finer than the
Rayleigh. The yearly values of mean wind speed and power density at the designated site was 7.4
m/s and 420 W/m2respectively,so this site comes underneath Class 7 of the wind classification
i.e. this is an excellent site for wind energy projects. B. Safari, J. Gasore [11] statistically
investigated the wind power potential with the help of Weibull and Rayleigh models in Rwanda at
five different stations. 20 years (1974-93) data was taken from the meteorological station of
Rwanda. At the selected site, Weibull probability density function was found to be more suited for
the empirical distribution of the data.
In South Africa, Ayodele TR et al. [12]examined physical characteristics and power potential of
the wind in the coastal region at ten different sites. 1 year data was taken based on the 10-min
average wind speed at 20m and 60m height. Performance tests like Correlation coefficient and
RMSE test were applied firstly to check that which distribution can better describe the data at the
site. The daily variation, wind speed characteristics, peak periods of electricity and the turbulence
intensity were found at the selected site. Besides these, Optimum and most probable wind speed
and shear exponential were also calculated here. Data was analyzed using Weibull distribution
because of its better fitting than the Rayleigh and lognormal distribution and results showed that
Napier site has excellent wind potential while Calvinia site has low wind potential having values
of 694 W/m2 and 216.29 W/m2respectively.
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Likewise, many scholars have also calculated the characteristics and power potential of the wind
at different places of many countries of the world similar to, determination of wind energy potential
in Bishkek, Kyrgyzstan presented in [13], statistical analysis was done using Weibull and Rayleigh
models to determine the wind power density at the site of Malaysia in [14], Wind speed
characteristics and power potential of wind was determined in Akure, South west Nigeria and
presented in [15], Characteristics and power potential of wind was examined in Osmaniye, Turkey
[16].
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Chapter # 3
BASIC THEORY IN WIND ENERGY
3.1 Introduction of Wind energy
The movement of atmospheric air, sensed as wind by us, touches every single corner of the earth
in this planet. There might be some places that never experienced any rain but nobody ever
envisioned a place where wind lost. The movement of air is just because of the solar heating of the
earth and winds carrying a portion of energy in solar radiation, which is firstly absorbed by earth
and then transferred in various forms back to the atmosphere. As, the surface of the earth is not
similar because of the presence of the land, seas, desert and forest; therefore, the portion of energy
that is fascinated varies both in space and time. This generates, difference in atmospheric
temperature, pressure and density, which in sequence produces forces that transport air from one
place to other. As water and land absorbs radiations in a different way just like valley and
mountains do, this difference causes the air to move.
Throughout the year, as compare to Polar Regions, tropical sections of the earth receive additional
solar energy to radiate the earth. Because, neither tropic regions always remain warm from year
to year nor do the poles get cold, there is an altercation of thermal energy through latitudes where
wind serve as the medium of convection. Likewise: uneven heating, earths rotation also plays an
important part in the movement of air. Rotation of earth generates Coriolis forces which move the
air particles toward the Northern and Southern hemisphere. So, localized wind systems get
established over the, mountains, valleys, deserts and seas.
However, unlike sunshine (i.e. comparatively uniform in a particular region over a specified period
of time) winds are inconsistent, changing in space and time, varying from one place to other,
gusting in fluctuating strengths over different period of time and commencing seconds to years.
Winds transmit the kinetic energy of the atmosphere, which is convertible into electricity with the
help of wind turbines and conveyed through wires in desired areas far-off from the point of
generation. Wind generation capacity can be harnessed on the large scale and after connecting it
with national grid, it can contribute to encounter the load and energy necessities of the country.
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3.2 Brief history of Wind energy
The usage of wind energy can be sketched back to ancient civilization, thousands of years ago. It
had been exposed from ancient human antiquities that wind energy was discovered and used at
different sites of the earth. Before the invention of steam engine, wind was the only source ofpower for ships at that time. These wind mills were introduced to the western world in 1100 AD.
By the start of 13th century these windmills extensively used in the Europe. Near the
commencement of 14thcentury, Dutch had become leading in the designing of windmills.
Wind energy had been in usage ever since the earlier civilization for the grinding of grains,
pumping water from well and to command the sail boats. Before the industrialization of Europe,
wind mill were used for different purposes like irrigation, pumping of drainage and processing of
other merchandises like Spices, paints and tobacco etc. There were many types of the wind mill
operational by the middle if the century in Holland. These wind mills had a rotating shaft
accommodated at the mainpost. Wind mills got severe competition and hurried to decline after the
invention and advancement in the steam engine and the growth of the electricity distribution
systems in the 19thcentury.
The motivation behind the development of wind energy was due to increased prices of oil and
apprehension over restricted fossil fuel resources during the oil embargo of 1973. The trend of
declining of fuel prices abruptly reversed and many countries started thinking to utilize the
renewable energy sources like wind, solar, biomass and wave, because these sources have very
low CO2 emission throughout their life cycle and have a potential to limit the abrupt climate
change.
3.3 Wind Resource Assessment
Equatorial regions of earth collect more solar heat than Polar Regions, which origins huge scale
convection current in the stratosphere and as a result wind generates. It has been estimated by
meteorologists that about 1% of the arriving solar energy at earth is rehabilitated into wind energy.
Solar energy which reaches earth in ten day has more energy contents than the worlds entire fossil
fuels assets (including Coal, oil and gas), hence wind resource is extremely larger than we expect.
1% of daily incoming energy is equal to the daily consumption of the present world. It is inspiring
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that the global wind reserve is very large and is widely distributed. As the development is made,
technically and frugally, wind energy can serve as a remedy to the world energy crisis.
Even though wind resource approximations have significance, more detailed calculations are
required to enumerate the resources in a specific area. Winds are sporadic source of energy but
they characterize a reliable energy source year by year. Now days, large number of studies
regarding grid systems incorporation reveals that the irregularity of wind energy is no more barrier
for its large scale use. Further, most of the wind turbines are being use for the generation of
electricity in the worldwide either parallel to grid systems or in remote locations. The purpose of
installing wind turbine is to diminish the fossil fuel consumption and to reduce the total electricity
generation price.
First step in the assessment of wind resource is to identify the preliminary area. In this way
relatively large region is separated for appropriate wind resource area grounded on the statistics
like airport data of wind, topography, highlighted trees and supplementary facts.
Second step is to evaluate the wind resource and this relates to wind assessment plans to illustrate
the wind potential in a distinct area where wind power improvement is being deliberated. The
utmost common objectives of this extent of measurement are to:
Enumerate, whether adequate wind resource occurs inside the area to rationalize extra site
precise inquiries;
Relate areas to discriminate comparative progress prospective. Wind resources are
characterized by wind power classes and each class symbolizes a range of yearly average
wind power densities and corresponding mean wind speed;
Achieve demonstrative data for approximating the performance and the financial
sustainability of particular wind turbine;
Display of potential wind turbine scheduling locations by matrix investigation of wind, site
and wind turbine.
Third stage is regarding micrositing at the minimum scale of wind resource evaluation. The chief
objective of these three stages is to compute the small size inconsistency of the wind resource
above the territory of curiosity. Eventually, micro siting is being used to position one or additional
wind turbines on a portion of land to exploit the total energy yield of the wind plant.
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3.4 Extrapolation of wind speed at different height
Wind speed changes with the change of height, so there is need of an equation which can predict
the wind speed at one height to other height; because, it is not necessary that wind speed data is
always measured from wind turbine hub height. To adjust the data with the required wind turbine
hub height, we use power law method as shown in equation [17];
=
(1)
Where V represents the wind speed at a required height; Z represents the wind speed at
reference height; Zr and is the surface roughness factor varies from 0.128 to 0.160 for a
homogenous surface [18]. The typical value of surface roughness factor is 0.14 which is widely
accepted for low roughness surfaces and well unprotected sites [12].
3.5 Presentation of Data
Statistical procedures and investigation are predominantly concerned with the collection,
organization and explanation of arithmetical data which form the sample or subset.
3.5.1 Data collection
There are two types of data available in statistics as hereunder:
1. Original data is called primary data; because, it has not gone through any kind of statistical
treatment;
2. The data which has gone through any kind of statistical treatment like tabularized,
categorized or presented in some other form at least once is called secondary data.
3.5.2 Organization of Data
There are three methods which are mostly used in the statistics to organize the data which are [35]:
1. Classification;
2. Tabulation;
3. Graphical representation.
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In order to transport the data in appropriate and easy form, primary data is summarized and
grouped together into classes, this process is called classification. Two ends of the class are called
the class interval. Class boundaries are achieved by the accumulation of the upper limit of one
class interval and the lower limit of next upper class and divided by two.
3.5.3 Tabulation of data
An organized arrangement of data for statistical information characterized in rows and columns is
called the tabular form of the data. These table are further classified into two types such as
(1) General purpose, constructed for reference
(2) Particular purpose used to analyze or support in investigating data
3.5.4 Graphical representation
This is a pictorial representation of data between different variables likewise the graph between
mean wind speed and variance of the observed data. There may be different kinds of graphs
dependent upon the nature of the data used and the purpose of the graph. Similarly, in frequency
distribution, a graphical demonstration of frequency gives brief and clear information.
3.6 Measure of central tendency of the data
Central tendency measure and predicts the central region of data. Central tendency of the data can
be enumerated with the following methods;
3.6.1 Mean
The mean or average value of the wind speed data can be calculated by adding all the values of
velocities and then dividing the total number of sample values
Vavg= +2+3++ (2)Vavg=
Vini= (3)
Here, n represents the total observing wind speed data available.
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3.6.2 Median
The median of the set of data values, is the value which inhabits the middle position when data has
been organized into ascending or descending order. If, in any case two middle values are accessible
then median will be the average of these two.
Med = [V(n+1) /2] when n is odd (4)
Med = [V n/2, V (n+2) / 2] whenn is even (5)
Here, n is againrepresents the total number of observing wind speed data.
3.6.3 Mode
The mode of the data is the value which occur the most number of time. Mode is not essentially a
single value and actually may not present some time. In case of frequency distribution, the valuewhich is maximum is called the mode of the distribution. If the frequency curve has one peak it
has one mode while, curve having two maxima, have two modes.
3.7 Measure of Dispersion
The measure of central tendency of any data just describe the central point of the data but in order
to calculate; how positions are spacious or scattered on both side of the centre; we use measure of
dispersion of the data. Following methods are used to measure the dispersion:
3.7.1 Range
The simplest method to measure the dispersion of the data is to calculate the range, which is a
difference between the lowermost and uppermost values in the distribution. Range is easy to
calculate and to measure the dispersion but it does not give the measurement of dispersion
comparative to centre value.
3.7.2 Mean deviation
The mean deviation basically calculate the ranges of the magnitude of the deviance from central
value like from mean, median and mode. The formula to calculate the mean deviation is as
Md = = (6)
Mean deviation is premeditated from arithmetic mean.
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3.7.3 Standard deviation
In order to get more information about the results of data set, the scattering or inconsistency of
the data can be examined with the help of Variance or standard deviation. The formula to
calculate the variance of the data set is;
2=
2= (7)The division with n-1 is referred to as the number of degrees of freedom. If, we just take the
positive square root of the variance we will get the standard deviation of the data, so the formula
to calculate the standard deviation of the data becomes
Standard deviation = = 2So equation (8) in case of standard deviation becomes
= 2= /2
(8)
Now, the coefficient of variation can also be defined as
Coefficient of variation = COV =
100So, COV can also be written as
COV = 100 (9)Coefficient of variation gives the amount of deviation present in the hourly time sequence data or
the amount of turbulence present in the wind data. It has no unit because standard deviation and
mean of data both have same dimensions of variable used.
3.8 Wind data parameter estimation
Besides, calculating the average wind speed and standard deviation of the data, the other two
parameters, which are also very important for the design of wind turbine are most probable wind
speed and wind speed carrying maximum energy. Here most probable wind speed Vmpcorrespond
to the peak of the probability density function, whereas the wind speed carrying maximum energy
(represented as Vmax.E ) is used to estimate the design of wind turbine or rated wind speed [2].
These wind speeds can be calculated as [17, 19];
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Vmp= c /
(10)
Vmax.E = c +2 /
(11)
For the design of wind turbine, it is proposed that rated wind speed must be very much near to thewind speed transporting maximum energy because wind speed has maximum efficiency at rated
wind speed [20].
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Chapter # 4
DATA COLLECTION AND ANALYSIS
4.1 Site description and data collection
Pakistan has immense wind power potential available in different regions of the country but most
of the sites which have studies yet lies along the coastal belt of the Sind and Balochistan. Besides
the coastal belt, very few areas have been studied to assess the wind potential of the site. Turbat is
one of those sites which despite having the wind energy capacity has not yet harnessed for wind
potential. So, keeping in view of the above, a site of Turbat is selected to determine the wind
potential of the site and to determine, either this site is feasible for wind energy projects or not.
Turbat is a city situated in southern Balochistan, which is a largest province (Area-wise) of
Pakistan. Turbat is positioned on the left bank of Kech River and has total population of around
about 90,000. The climate of Turbat is relatively cool and windy. Turbat metrology station is
located at 250 59 N latitude and 630 04 E longitude at a height of 155m (508feet) above sea level.
The site has high population and many limitations (like hilly areas) to supply energy, therefore
searching for other alternatives renewable energy sources like wind or solar is essential for the
survival of mankind. Now a days, due to the more concern of world towards the global warming
and green-house effect of the gases by the exhausts of power plants, alternate energy should be
clean, environmental friendly and renewable in nature.
The selected site is suitable for the installation of wind projects because:
Surface is not smooth due to the presence of the hilly areas so, water or thermal projects
are not suitable at the site while wind energy projects can be utilized for small scale
purposes i.e. for single house only.
Unavailability of infrastructure for water or thermal power houses.
Site is situated 155 m above the ground level and as height increases, capacity of power
generation also increases from wind turbine.
Site has good observed wind speed.
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Figure 4.1: Location of Turbat in Balochistan, Pakistan
The data was taken from Pakistan Meteorological department (PMD) Karachi centre (CPDC
Karachi), consisting of 21 months (Jan 2012 - Sep 13) and is based upon the 3-hourly time series
data measured using anemometer at a height of 8.8 m above the ground level. Figure. 4.1.Shows
the location of Turbat and nearby regions of the selected site in Baluchistan.
So as to estimate the wind potential of a specific site, it is very imperative to determine the
following characteristics such as mean wind speed, the availability of wind speed for particular
time period, persistency of wind gusting, the most probable wind speed, the wind carrying
maximum energy and wind power potential of the site [2]. The estimation of wind power density
describes the amount of energy available for each unit of time and swept area of the blade is
accessible at the nominated site for transformation of electricity with the help of a wind turbine.
4.2 Statistical analysis of the data
The most important and critical aspect in the assessment of wind resources is the dispersal of wind
speed. Wind speed data has wide ranges of wind and in order to determine the key parameters
from the data, we have to apply some statistical analysis on the data. In statistical analysis, we
apply statistical distributions functions.
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Wind speed is measure in the form of the integer and each value is observed many times during
the month or year. As in previous section, it has been explained in detail that average value of the
data set can be calculated with the help of equation (2)or (3). Besides measuring the mean value
of data, we are also fascinated in the unpredictability of the set of numbers and we want to find the
abnormality present in each number with respect to the mean value and then locate the average of
these abnormalities.
In order to develop the frequency curves, we use continuous mathematical functions; which are
convenient to use instead of using the discrete values. So, the probability values p (Vi) become a
density function which is f (V) and is called as Probability density function (PDF). If velocity V
of the wind is a continuous random variable having range (0,) then it must satisfyf (V) 0 , 0 < V < 1 (12)
An additional useful probability measure of the data is Cumulative density function (CDF). In case
of continuous function, let F (V) represents the cumulative density function for the variable V,
which is a continuous variable having range
0 < V < The CDF can be defined as:
F (V) = (13)Here, speed is considered as per continuous random variable and so F (V) has the properties of
CDF are:
F (0) = 0 (14)
F (= 1 (15)It is not necessarily that the value of F (0) will be zero in the case of discrete random variablebecause there are cool spells with generally zero wind speed measure and included in F (0).
However, in the case of continuous random variable F (0) is the integral with the sign of integration
limits.
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4.3 Frequency distribution Table
The number of occurrence of an event in each class is entitled the frequency of the class. If we
convert this frequency in tabular form, then this frequency is called the frequency table.
In order to describe and analyze the population and climatological data, the important and
compulsory step is to draw the frequency distribution of the data. This is done by approximating
the features of the population frequency distribution of the sample.
Frequency distribution are of two types:
1.
Discrete distribution
2. Continuous distribution
Discrete distribution has a function of discrete random variable i.e. one that varies in stage, for
probability distribution.
While Continuous distribution has a function of continuous random variable i.e. temperature,
pressure, wind speediness, for probability density.
Frequently, for our convenience, a discrete random variable might be treated as continuous random
variable while for some special purposes continuous random variable can be transformed to
discrete random variable. More consideration is now giving to better fit the frequency distributions
to climatological data.
4.4 Statistical Distributions
There are many distributions which can be utilize to describe and analyze the wind data such as:
Weibull distribution
Rayleigh distribution
Chi-square distribution
Normal distribution
Log-normal distribution
Gamma distribution
Among all these distributions, we will discuss here some continuous frequency distributions only,
which are associated to our work. The selected distributions are Weibull, Rayleigh, normal and
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Weibull distribution is most important, common and well acknowledged distribution for the
calculation of wind power and energy all over the world because of its adaptability. This is the
only distribution which cannot be derived from any further distribution like other continuous
distributions. By keeping the shape parameter (k) of Weibull distribution as k = 3.6, k = 2, k = 1,
Weibull distribution changes into normal, Rayleigh and exponential distribution respectively. So,
the Weibull distribution is only the essential and unique continuous distribution from others.
Various statistical models are being used to describe and analyze the wind data in which some are
normal, lognormal, Weibull and Rayleigh distributions functions [21, 22]. Among all these
distributions functions, Weibull and Rayleigh distributions functions are generally applied and
accepted worldwide because both these distributions provide finer approximations to observed
wind speed data [13].
4.4.1 Weibull distribution
The Weibull distribution (entitled afterward the Swedish physicist Weibull, who practical it while
reviewing material in stiffness and fatigue in the 1930s) provides a close estimate to the probability
laws of several natural occurrences. Previously, it has been used to characterize the wind speed
distribution for application in wind capacity studies. In existing years most attention has been
concentrated on this method for wind energy solicitation not only due to its greater tractability and
easiness but also because it can provide a good fitting to investigational data.
The two parameter Weibull distribution function is most suitable, widely recognized and
recommended by many researches for analyzing the wind speed data [23, 24, 25]. The variation in
wind speed is categorized by probability density function and Cumulative density function in
Weibull distribution [2]. Probability density function (PDF) describes the probability of wind at a
specified velocity of wind i.e. what is the probability of wind speed having magnitude 3 m/s, can
be describe by the Probability density function while the cumulative distribution function (CDF)
of wind velocity V gives the probability of wind velocity equal to, lower than or inside the specifiedrange of the wind speed i.e. what is probability of wind blowing between 1 m/s and 3 m/s, is
describe by cumulative density function. The Probability density function (PDF) in case of Weibull
distribution is described and calculated as [2];
f (V) =
exp [ ] (18)
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Here V represents the wind speed, K is shape parameter for weibull having no unit, C denotes the
Weibull scale parameter possessing similar unit as wind speed and f (V) indicates probability
density function of perceiving wind speed.
The corresponding cumulative density function (CDF) is described as [2];
F (V) = 1- exp [ ] (19)
Here F (V) represents the cumulative density function for wind speed (V) .
4.4.1.1 Estimation of Weibull shape and Scale parameter
There are numerous methods which are being used to compute the Weibull shape and scale
parameters like:
Maximum likelihood method;
Empirical method;
Modified maximum likelihood method;
Energy pattern factor method.
There is a lot of search going on to determine that which method can better describe the value of
Weibull shape and scale parameters. The value of shape and scale parameters mostly depend upon
the sites specification, duration of the data and the dispersion of the data. Here in this study theseparameters are calculated using Empirical method.
In empirical method, these parameters are calculated with the help of average wind speed and
variance or standard deviation. These parameters are calculated here using equations (3)& (7).
The formulas for the calculation of k and c are [2];
k = .86
(20)
c = +( ) (21)
Here ( ) is the Gamma function and describes as [26, 27];
( ) = dt (22)
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Similarly, scale parameter for weibull distribution can also be evaluated by using the equation
[28];
c =.7
.84 +.86 .7 23)Using the weibull shape parameter (k) and scale parameter (c), mean wind speed of the data can
be determine or verified as
Vm= c 1 (24)
Here Vmdenotes the mean wind speed and k is a shape while c is a scale parameter.
4.4.1.2 Behavior of calm wind speed
Weibull distribution better describe the data when wind speed is more than 1 knot or 0.5 m/s. It
has been experimentally proved that Weibull distribution does not give better fit for wind speed
having magnitude less than 0.5 m/s, so in order to apply Weibull distribution to data, wind speed
data must be greater than zero.
So, observation of calm cannot be involved in the computation of F (V) while applying Weibull
distribution to any observed data of wind speed. To include the consequence of zero wind speed,
values like Vavgand
Vavg3are to be multiplied with the factor of (1-f) where f characterizes that
portion of time when zero wind speed has been logged.
4.4.2 Rayleigh distribution
Besides having many advantages, Weibull distribution also has certain constraints too. The most
vital restrain in the application of weibull distribution is that it cannot describe the probability of
very small or zero wind speed [29]. For this purpose, additional distribution for example Rayleigh
is used and verified here, which is being extensively used by the researchers to determine power
potential of the wind at the particular site. In this research work, two specified distributions arebeing used and results of both are compared, so that we could find out which distribution is better
to predict the wind speed data.
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In Rayleigh distribution function, we assumed that the shape parameter (k) has fixed value such as
K = 2 [19, 30], so in this distribution, expressions to calculate the probability density function
(PDF) and cumulative density function (CDF) are [2];
f (V) = 2 exp 4 2 (25)
F (V) = 1 - exp 4 2 (26)
Here the shape parameter is taken to be 2 while scale parameter can be calculated by equations
(21)or (23).
4.4.2.1 Facts of Rayleigh distribution
It is very hard to make comprehensive generalization concerning the capability of Rayleigh
distribution function to fit the data, although curves display very good amplitude and extended
width for higher wind speeds. The effectiveness of this distribution is to predict the power output
of the site. Both distribution may yield acceptable results but the results of Weibull distribution
are more accurate while Rayleigh distribution is easier to use. When the mean wind speed is larger,
graphs of data give flatter shape while peak becomes sharp when average speed of wind is less
than 2.0 m/s.
4.5 Evaluation of Weibull and Rayleigh distributions
So as to compute the accuracy of the Weibull and Rayleigh models, the root mean square error
(RMSE) test, the Chi-square (2) test then the correlation coefficient or coefficient of
determination (R2) can be used [12].
These parameters can be premeditated as follow [12];
RMSE = 2= /2
27)
2 = 28)R2 =
29)
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4.10 Cut out Velocity
The speed of the wind at which wind turbine shuts dow to avaoid destruction due to high intensity
of wind is called cut out velocity. The range of this high velocity might be 20.0 m/s to 28.0 m/s.
This is also called as furling velocity.
4.11 Wind power density calculation
The power density of wind is an estimator that displays the ability of the site regarding resources
of wind. Wind has immense power that flows by speed Vthrough a swept area of wind turbine
Aincreases as a cube of its velocity and can be described as [31];
Power =2 2
Power = 2 (AV) 2So we have
P (V) =23 (31)
PD=P
=23 (32)
Here P (V) denotes power the wind in (W) and P (V) / A denotes the wind density for each swept
area A (Wm-2) and expressesdensity of the air in (kg/m3) at this particular site.
So, equation (31) reveals that in order to get a maximum power from wind, it requires:
Higher wind speed as power output from wind turbine is directly proportional to cubic
power of mean wind speed so, a small deviation in wind speed can consequence in a huge
alteration in the wind power output.
Longer length of blades in order to attain a large swept area of blades of the wind turbine
Higher value of air density
4.11.1 Swept area of Blade
Swept area of the as shown in Fig. 4.2.The swept area of the wind turbine blade can be calculated
by this equation:
A = 2 2= l (l+ 2r) (33)
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First of all, monthly analysis was done and all the values of parameters were calculated like
Average wind speed, Variance of the data, most feasible wind speed, wind speed bearing
maximum energy, Weibull shape and scale parameters. After calculating these parameters,
frequency distribution table was drawn. In frequency distribution table comparison between the
actual, Weibull and Rayleigh probabilities is made. Then after selecting the suitable distribution
to better fit the data, Wind power potential and wind energy density has been calculated.
Similarly, analysis was done on the 1 yearly basis and all these parameters again calculated and
then again analysis was done 2-years basis and all these parameters are again calculated.
At the end, comparison has been made between the values calculated monthly, 1 yearly and 2
yearly basis.
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Chapter # 5
RESULTS & DISCUSSIONS
In this research work, analysis and assessment of both; wind speed data and power potential of
wind is executed. Wind speed characteristics such as mean, availability of wind, wind duration
and standard deviations are determined to compute the power potential and energy density of the
wind at the selected site.
5.1 Measurement of average wind speed
InFig. 5.1.The monthly mean wind speeds (Vavg) have been calculated at the site of Turbat for
the year 2012 and 2013. It could be seen that the average value of wind speed is more than 4 m/s
in each month. The average wind speed remains high from the start of the year to the mid of the
year i.e. January to June and then its value decreases. The range of average wind speeds is between
4.065 ms-1and 6.422 ms-1. The maximum monthly mean wind speeds are 6.422 ms -1and 5.610
ms-1for the months of Feb 2012 and Mar 2012, respectively. Similarly, minimum speeds are 4.065
ms-1and 4.180 ms-1for the months of Oct 2012 and Sep 2013, respectively.
Figure 5.1: Average wind speed per month for year 2012-13
0
1
2
3
4
5
6
7
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
AverageWindspeed(m/s)
Months
Year 2012 Year 2013
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From Fig. 5.1.It could be seen that the average wind speed has higher values for the months of
June, July and August, which are among the warmer months. The energy demand in these months
also increases due to hot weather in Pakistan. In contrast, average wind speed has lowest values
for the months of October, November and December, which are among the coldest months.
5.2 Measurement of standard deviation
The monthly variation in the values of standard deviation has been shown in Fig. 5.2
Figure 5.2: Monthly values of standard deviation
The value of standard deviation for 2012 is maximum for the month of Feb 2012 having value of
3.306 while minimum value is 1.683 for the month of Aug 2012. Similarly, the maximum and
minimum values of standard deviation for the year 2013 are 3.152 and 2.146 for the months of
July 2013 and Feb 2013 respectively. The range of standard deviation is 1.607 having overall
minimum and maximum values as 1.699 and 3.306 for the months of January 2012 and February
2012 respectively.
For better results, analysis was done on month basis and then average value of all the months was
taken. Hence the overall average value of Mean velocity is Vavg = 4.83 ms-1and standard deviation
is = 2.54.
0
0.5
1
1.5
2
2.5
3
3.5
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Stan.Deviation()
Months
Year 2012 Year 2013
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5.3 Monthly values of Vmpand Vmax.E
Beside average wind speed and standard deviation, the other two crucial considerations for the
evaluation of wind power potential are maximum feasible wind speed (Vmp) and wind speed
bearing maximum energy (Vmax.E), bothare correspondingly calculated here. Inside Table. 5.1.
Monthly values of both these velocities are shown. It can be seen that the maximum values of Vmp
for year 2012 are 5.244 m/s and 4.916 m/s for the months of February and July respectively
whereas the minimum value is 3.220 m/s for the month of Nov 2012. Similarly, for the year 2013,
the maximum values of Vmp are 4.002 m/s and 3.974 m/s for the months of June and March
respectively while the minimum value is 2.481 m/s for the month of Aug 2013.
The maximum values of Vmax.E for the year 2012 are 10.086 m/s and 9.009 m/s for the months of
February and March respectively; while, the minimum value is 5.842 m/s for the month of Aug
2012. Similarly, for the year 2013, the maximum values of Vmax.E are 8.928 m/s and 8.759 m/s for
the months of August and June respectively while the minimum value is 6.588 m/s for the month
of Feb 2013.
Table 5.1: Monthly values of Vmpand Vmax.E for the year 2012-13
Month 2012 2013
Vmp Vmax.E Vmp Vmax.E
January 4.588 6.214 2.876 7.702February 5.244 10.086 3.505 6.588
March 4.434 9.009 3.974 7.412
April 4.332 8.015 3.464 6.772
May 4.711 7.612 3.592 7.694
June 4.603 8.666 4.002 8.759
July 4.916 7.959 3.218 8.061
August 4.051 5.842 2.481 8.928
September 4.194 7.055 2.659 7.561
October 3.499 6.138 - -
November 3.220 7.682 - -
December 3.245 7.842 - -
Yearly 4.253 3.265 7.677 7.807
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The average value of all the months for Vmpand Vmax.E are 3.83 m/s and 7.732 m/s respectively.
5.4 Coefficient of Variation
Coefficient of determination is a measure of the dispersion of data in any probability distribution.
It describes the amount of variation or the turbulence present in the hourly time series data. In Fig.
5.3. The monthly values of coefficient of variation has been shown. COV is calculated with the
help of Equation (9). The graph of COV shows that starting six months of the year have lower
values of COV which shows that there is less amount of turbulence or dispersion is available, so
these months are most suitable than the last six months for the wind energy application. COV is
some time expressed as a percentage value.
Figure 5.3: Monthly values of Coefficient of Variation for year 2012-13
5.5 Weibull parameters estimations
Two parameters of Weibull distribution i.e. k and c are calculated here using equation (20)and
(21)respectively. Values of Weibull shape and scale parameters for each month are presented in
Table. 5.2., for the year 2012 and 2013. As it could be seen that Weibull shape parameter has
maximum value of 3.010 for the month of Jan 2012 and minimum value of 1.784 for the month of
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
CoefficientofVariation(COV)
Months
Year 2012 Year 2013
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Dec 2012. Similarly for year 2013, the maximum value of k is 2.105 for the month of Mar 2013
while has minimum value of 1.508 for the month of Jul 2013.
Table 5.2: Monthly values for Shape and Scale parameters of Weibull distribution
Month Parameter 2012 2013January k
c3.0105.247
1.6954.865
February kc
2.0577.249
2.0934.782
March KC
1.9786.329
2.1055.397
April kc
2.1195.856
2.0324.834
May kc
2.3935.906
1.9125.291
June kc
2.0906.285
1.8875.972
July kc
2.3886.169
1.5085.102
August kc
2.7384.782
1.7515.218
September kc
2.3005.375
1.6504.674
October kc
2.2144.590
--
November kc
1.7965.064
--
December kc
1.7845.145
--
Yearly kc
2.4965.666
1.8195.137
The monthly deviation in the values of Shape parameter has been shown in Fig. 5.4.
The scale parameter for the year 2012 has topmost value of 7.249 m/s for the month of Feb 2012
and has minimum value of 4.590 m/s for the month of Oct 2012.
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Figure 5.4: Monthly values of shape parameter for year 2012-13
Similarly for the year 2013, the topmost value of scale parameter is 5.291 m/s and minimum value
is 4.674 m/s for the month of Sep 2013. The value of scale parameter varies from 4.674 m/s to
7.249 m/s and has a range of 2.575.
In Fig. 5.5monthly variation of scale parameter has been represented.
Figure 5.5: Monthly variation in the values of scale parameter for year 2012-13
0
0.5
1
1.5
2
2.5
3
3.5
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Shapeparamete
r(k)
Months
Year 2012 Year 2013
0
1
2
3
4
5
6
7
8
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
S
caleparameter(c)
Months
Year 2012 Year 2013
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These parameters control the sketch of curve formed by Weibull and Rayleigh distributions.Shape
parameter defines the range and determine the shape of plot while Weibull scale parameter
determines the profile of the distribution just as a transformation of the Abscissa scale.
The value of shape factor c shows the probability of wind speed flowing, so greater value of cindicates the higher value for the possibility of wind speed. In contrast, the maximum value of k
displays the greater possibility for frequent and uniform wind blowing across the side.
5.6 Frequency distribution table
In order to derive best results using statistical analysis, it is better to arrange the actual time series
data in the form of frequency distribution. For year 2012, time series data has been organized in
the form of frequency distribution and given here in Table. 5.3.
Table 5.3: Distribution of frequencies calculated from wind speed data for year 2012
i Bins Vavg fi Actual Weibull Rayleigh
1 0-1 0.5 0 0.000 0.000 0.000
2 1-2 1.5 70 0.033 0.106 0.109
3 2-3 2.5 285 0.135 0.140 0.1404 3-4 3.5 369 0.175 0.154 0.151
5 4-5 4.5 256 0.122 0.147 0.143
6 5-6 5.5 425 0.204 0.126 0.122
7 6-7 6.5 213 0.101 0.098 0.0958 7-8 7.5 163 0.078 0.069 0.068
9 8-9 8.5 121 0.058 0.045 0.046
10 9-10 9.5 68 0.032 0.027 0.028
11 10-11 10.5 83 0.039 0.015 0.01612 11-12 11.5 13 0.006 0.007 0.008
13 12-13 12.5 13 0.006 0.004 0.004
14 13-14 13.5 6 0.003 0.002 0.002
15 14-15 14.5 4 0.002 0.001 0.001
16 15-16 15.5 8 0.004 0.000 0.000
17 16-17 16.5 0 0.000 0.000 0.000
18 17-18 17.5 0 0.000 0.000 0.00019 18-19 18.5 1 0.001 0.000 0.000
Wind speed frequency inside specific ranges (bins) has been indexed in column IV while the
column V represents the actual probability density distribution computed from actual data of wind
speed. Similarly, column VI and VII represents the probabilities calculated using Weibull and
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Rayleigh distributions respectively. The Weibull probability distribution function and cumulative
distribution functions are being calculated here by equations (18) and (19) respectively. In the
same way probability density function and cumulative density function for Rayleigh distribution
are calculated with the help of equations (25)and (26)respectively.
Likewise, for year 2013 frequency distribution table has been shown in Table 5.4.
Table 5.4: Distribution of frequencies calculated from wind speed data for year 2013
i Bins Vavg fi Actual Weibull Rayleigh
1 0-1 0.5 0 0.000 0.000 0.000
2 1-2 1.5 139 0.090 0.137 0.130
3 2-3 2.5 258 0.167 0.155 0.161
4 3-4 3.5 260 0.168 0.151 0.1655 4-5 4.5 235 0.152 0.131 0.147
6 5-6 5.5 228 0.148 0.105 0.116
7 6-7 6.5 119 0.077 0.078 0.083
8 7-8 7.5 85 0.055 0.055 0.054
9 8-9 8.5 73 0.047 0.036 0.032
10 9-10 9.5 58 0.038 0.013 0.009
11 10-11 10.5 58 0.038 0.013 0.009
12 11-12 11.5 12 0.008 0.007 0.004
13 12-13 12.5 10 0.006 0.004 0.00214 13-14 13.5 5 0.003 0.002 0.001
15 14-15 14.5 0 0.000 0.000 0.000
16 15-16 15.5 4 0.003 0.000 0.000
Similarly, comparison of Actual, Weibull and Rayleigh probabilities has been made in Table. 5.4.
For year 2013. It can be seen in the table that probabilities of wind calculated using Weibull
distribution are much closer to the probabilities computed from actual frequencies of the wind than
the Rayleigh distribution. So, Weibull distribution better describing the actual data than the
Rayleigh distribution.
As, Weibull distribution displays better fit upon measured data of wind speed at the particular site
therefore Weibull distribution should be applied to evaluate the power potential of wind at the
candidate site instead of using Rayleigh distribution.
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5.7 Comparison of Actual, Weibull and Rayleigh probabilities
The comparison between the Actual, Weibull and Rayleigh distribution has been made using the
frequency distribution given in Table 5.3 and Table 5.4. With the intention of check that which
distribution better fit the actual data. First of all, monthly analysis was done among these three
probabilities, then yearly comparison was also made to check the better distribution between
Weibull and Rayleigh.
5.7.1 Monthly comparison
First of all, monthly comparison between the Actual probability density function has been made
with Weibull and Rayleigh probability density function. Some months graphs have been shown
as a sample here. One month is selected from each Quarter of the month i.e. each one month in
three month (Quarter). So, total of the 8 graphs of monthly analysis have been shown.
Figure 5.6: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Feb 2012
In Fig. 5.6.actual, Weibull and Rayleigh probability density functions have been shown for the
month of Feb, 2012. Both distributions describing the actual data well but Weibull distribution
seems to describe the actual data better than Rayleigh distribution having R2value of 0.817 while
for Rayleigh is 0.813.
In Fig. 5.7.probabilities are compared for the month of June 2012. Both distributions describing
the actual data well having R2value of near about 90%.
0
0.05
0.1
0.15
0.2
0.25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Probabilitydensityfunction
Wind speed (m/s)
Actual Probability
Weibull Dist.
Rayleigh Dist.
February, 2012
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Figure 5.9: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Nov 2012
In Fig. 5.10.the comparison of probabilities is done for the first month of 2013 i.e. January. Again
here Rayleigh distribution describe the data better than the Weibull having R2of very close to one
i.e. 96.6 %
Figure 5.10: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Jan 2013
0
0.05
0.1
0.15
0.2
0.25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
ProbabilityDensity
Function
Wind speed (m/s)
Actual Probability
Weibull Dist.
Rayleigh Dist.
November 2012
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
ProbabilityDensityFunctio
n
Wind speed (m/s)
Actual probability
Weibull dist.
Rayleigh Dist.
January, 2013
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Similarly, for the month of June, comparison has also made between the probabilities calculated
from Actual data, Weibull and Rayleigh models as presented in Fig. 5.11.Both distribution
describing the Actual data well.
Figure 5.11: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Jun 2013
For the month of August 2013, comparison of the probabilities calculated from actual wind speed
data, weibull and Rayleigh distribution functions has been made and been represented in the Fig.
5.12.
It could be seen from the figure that, for this particular month, although both distribution
representing the actual probability well but Rayleigh distribution seems to donate the actual
distribution better than the Weibull distribution.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Probab
ilityDensityFunction
Wind speed (m/s)
Actual probability
Weibull dist.
Rayleigh Dist.
June, 2013
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Figure 5.12: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Aug 2013
Likewise, for the month of September, 2013 Rayleigh distribution seems to donate the actual
data better than the distribution as shown in Fig. 5.13.
Figure 5.13: Comparison of Actual, Weibull and Rayleigh Pdf for the month of Sep 2013
0
0.05
0.1
0.15
0.2
0.25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
ProbabilityDensityFunction
Wind speed (m/s)
Actual Probability
Weibull Dist.
Rayleigh Dist.
August 2013
0
0.05
0.1
0.15
0.2
0.25
1 2 3 4 5 6 7 8 9 10 11 12 13
ProbabilityDensityFunction
Wind speed (m/s)
Actual Probability
Weibull Dist.
Rayleigh Dist.
September 2013
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5.7.2 Yearly comparison
As monthly comparison has been made between the probabilities calculated with the help of Actual
data, Weibull distribution and Rayleigh distribution, now comparison between these three
probabilities has been made on yearly basis. Along with the comparison of PDF, comparison has
also been made between the cumulative density function, calculated with the help of Weibull and
Rayleigh functions. These probabilities are calculated using the frequency distribution table as
shown in Table 5.3. & 5.4. For year 2012, comparison has been shown in Fig. 5.14.
Figure 5.14: Comparison of Actual, Weibull and Rayleigh PDF, CDF for year 2012
From Fig. 5.14.it is clear that for the year 2012, in case of probability density function, Weibull
is representing the data well with R2value of 91%. Similarly, cumulative density function is also
better describe by the Weibull distribution.
Comparison of PDF and CDF has also made for the year 2013 as shown in Fig. 5.15.
0
0.2
0.4
0.6
0.8
1
1.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
CumulativeDensityFu
nction
ProbablityDensityFu
nction
Wind speed (m/s)
Actual Probability
Weibull Pdf
Rayleigh Pdf
Weibull Cdf
Rayleigh Cdf
Jan 2012 - Dec 12
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Figure 5.16: Comparison of Actual, Weibull and Rayleigh PDF, CDF for year 2012-13
So, at the site of Turbat both Weibull and Rayleigh distributions matched the perceived data quite
well. The deviation within the distributions depends upon the value of shape factor k. In
Rayleigh distribution k has fixed value i.e. 2, consequently in Weibull distribution the calculated
value of k is also approaches to 2 i.e. 2.059 therefore the results of both distributions are almost
similar. Whereas the difference between these two distributions is very minor but Weibull seems
to denote the actual data better than the Rayleigh. It is also clear from the above figures that there
are very less chances for the wind speed to exceed the limit of 20 m/s at the site.
5.8 Performance estimation of Statistical models
With the intention of estimating the proficiency of the two distributions taken into account here,
error analysis is also done on the data. Following tests are performed:
The coefficient of determination (R2) or Correlation test,
The root mean square error (RMSE) test,
the Chi-square (2) test
These tests are used to check the accuracy of the distributions in this study.
0
0.2
0.4
0.6
0.8
1
1.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Cu
ulativedensity
Function
ProbabilityDensityFun
ction
Wind speed (m/s)
Actual Probability
Weibull PDF
Rayleigh PDF
Weibull Cdf
Rayleigh Cdf
Jan 2012 - Sep 13
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5.8.1 Monthly R2value
Monthly R2values have been evaluated for the Weibull and Rayleigh distribution, in order to check
that which distribution better describe the data. In Fig 5.17. Monthly values of R2 have been
representing for the year 2012. It is clear that although both distribution have almost similar values
but Weibull distribution has comparatively higher values of months than the Rayleigh.
Figure 5.17: Monthly values of R2 for Weibull and Rayleigh distribution, year 2012
Similarly, for the year, 2013, R2values are also calculated for each month as shown in Fig. 5.18.
Here again, Weibull has higher values of R2test than Rayleigh distribution. So, theses graphs show
that Weibull distribution is better describing the observed wind data at the site of Turbat.
0
0.2
0.4
0.6
0.8
1
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
R2
Months
Weibull Dist. Rayleigh Dist.
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Figure 5.18: Monthly values of R2
for Weibull and Rayleigh distribution, year 2013
Average value of R2for all month calculate for both distributions are
Monthly average R2value in case of Weibull distribution = 0.874
Monthly average R2value in case of Rayleigh distribution = 0.852
Similarly, yearly R2values for Weibull and Rayleigh distributions are
Year 2012:
Weibull distribution = 0.944Rayleigh distribution = 0.951
Year 2013:
Weibull distribution = 0.918
Rayleigh distribution = 0.905
Year 2012 & 2013:
Weibull distribution = 0.946
Rayleigh distribution = 0.942
So, after viewing the statistics given above, R2test give better result as the number of observation
increases as monthly R2values are mostly below the 90% while the yearly values are above the
90% showing good correspondence of both distributions with actual wind data. As Weibull
distribution has higher values of R2test, so Weibull distribution better describe the actual data.
0
0.2
0.4
0.6
0.8
1
1.2
Jan Feb Mar Apr May Jun Jul Aug Sep
R2
Months
Weibull Dist. Rayleigh Dist.
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5.8.2 RMSE and Chi-square test
Monthly values or RMSE and Chi-square are also calculated just like R2values. After calculating
the monthly values, average values of all months has been evaluated for both RMSE and Chi-
square tests.
Table 5.5: Evaluation of Statistical models used
Index Weibull Rayleigh
RMSE 0.0040 0.0043
2 2.019 10-6 2.347 10-6
From Table. 5.5.it is clear that Weibull has lower values for both RMSE and Chi-square tests.
Any distribution model having values of both tests i.e. RMSE and Chi-Square tests closer to zero
while the value of R2near to unity; is considered to be the better for the estimation of real wind
speed data, so Weibull distribution better describe the perceived data than the Rayleigh one
because it has higher value of R2and lower values for both RMSE and 2tests.
5.9 Calculation of Wind power density
The monthly values of power density for wind are shown in Table. 5.6.Average values of power
density are relatively high for the months from Feb to Jul than the months from Aug to Dec. The
power consumption of the country in these months (i.e. Feb-July) is also comparatively higher
than the other months (i.e. AugDec). The highest value of average wind power density is for the
month of Feb 2012 i.e. 306.090 W/m2 and the minimum value of average wind power density is
for the month of Oct 2012 i.e. 75.100 W/m2.
The power density at the sel