Mixed Weibull Distribution: A Case Study on Ichanda, India

Kumaravel. R1, Varun. C2 and Sarfudeen. M3Department of Wind Turbine Testing, ReGen Powertech Private Limited 403 L, 5th floor,Samson Towers, Pantheon road, Egmore, Chennai-600008, IndiaEmail: kumaravel.r@regenpowertech.com

ABSTRACTThis paper presents a case study of a new methodology to accurately characterize and predict the annual variation of windconditions applied on Ichanda site, Tamil Nadu, India showing dual behaviour. The estimate of the distribution of windconditions is necessary to quantify the available energy (power density) at a site, and to design an optimum wind farm.Wind speed frequency distribution for some sites with two distinct peaks is not represented accurately by the typical twoparameter Weibull distribution. The wind characteristics of Ichanda has been analysed by using wind data recorded bymeteorological mast installed at that location. By examining the analysis, it shows that wind speed distribution is notdemonstrated precisely by two parameter Weibull distribution. A mixed Weibull probability distribution function (PDF) isapplied to analyse wind speed frequency distribution in that region. This model can be applied to regions where the windspeed distribution presents a bimodal distribution to predict wind speed probability distribution and annual energy productionaccurately.

Keywords: Wind energy, Wind frequency distribution, Bimodal wind speed distribution, Mixed Weibull, Ichanda, Tamil Nadu

Received 09/05/2014; Accepted 04/09/2014

1. INTRODUCTIONDue to the energy demands and the shortages of fossil fuels currently in the world, the utilization ofwind resource plays a very important role in energy supply. As we know, wind speed distribution fora specified area determines the wind energy available and the performance of energy conversionsystem. Accurate wind speed modelling is critical in estimating wind energy potential for harnessingwind power effectively [1]. After obtaining the probability distribution of wind speed, the windenergy potential could be estimated accordingly. The chosen probability density function (PDF)describes the quality of wind speed assessment to measure the wind speed frequency distribution. Avariety of probability density functions have been used in literature to estimate wind energy potential,but the Weibull function is most widely adopted because of its two flexible parameters. i.e. Weibullshape parameter describes the width of data distribution, while scale parameter controls the abscissascale of a plot of data distribution. It is worth to mention that the Weibull function is not possible torepresent all the wind structures encountered in nature, particularly for a dispersive wind distributionthat might be resulted from special climatic factors [1].

In the last decades, several mixed probability functions have been proposed by researchers tomodel those complex wind distributions in the world, However relevant study concerning TamilNadu has never been found in literature.

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1Head, Department of Wind Turbine Testing2Testing Engineer, Department of Wind Turbine Testing3Testing Engineer, Department of Wind Turbine Testing

Ichanda situates at a unique geographic or climatic environment. Southwest monsoon andnortheast monsoon prevail in summer and winter seasons, respectively; wind energy potential isthus considerable here. In this paper, the bimodal Weibull function are introduced and adopted todescribe wind characterization. Their performance and validity will be compared with theconventional Weibull function.

2. WIND DATATwo main seasons are present in the windy region of Ichanda. One of them is South West monsoon.Exceptionally high wind speeds can be observed due to effect of the strong cross mountain pressurevariation. The high winds come from Indian Ocean as a northerly wind, then constricted and speedingthrough the narrow Sengottai pass in Western Ghats. Other season is North east monsoon that blowsfrom North east when the south west monsoon wind after hitting Himalayas retrieves back.

The south western summer monsoon starts from May through September. The Thar Desert andadjoining areas of the northern and central Indian subcontinent heats up during the hot summers.This leads to a low pressure area over the northern and central Indian subcontinent. To occupy thisvoid, the moisture-loaded winds from the Indian Ocean rush in to the subcontinent. These winds,enriched with moisture, blow towards the Himalayas. The Himalayas act like a high wall, blockingthe winds from passing into Central Asia, and forcing them to rise.

The southwest monsoon is generally expected to begin around the end of May and fade away bythe end of September. The moisture-loaded winds on reaching the southernmost point of the Indiansubcontinent, due to its topography, become divided into two parts: the Arabian Sea and the Bay ofBengal Branches.

Around September, with the sun fast retreating south, the northern land mass of the Indiansubcontinent starts cooling off rapidly. With this air pressure begins to rise over northern India, theIndian Ocean and its surrounding atmosphere still holds its heat. This causes cold wind to sweepdown from the Himalayas and Indo-Gangetic Plain towards the vast spans of the Indian Ocean southof the Deccan peninsula. This is known as the Northeast Monsoon or Retreating Monsoon.

Since February of 2009, wind speed data have been recorded from Ichanda site located at a radialdistance of 42.9 km from North-western direction of Tirunelveli in the state of Tamilnadu shown inFig. 2. Wind speed and wind direction were measured at 78 m above ground level. Sensors with

614 Mixed Weibull Distribution: A Case Study on Ichanda, India

Figure 1. Monsoon wind pattern in India (Source: rhapsodyinbooks.wordpress.com)

high accuracy were used. Anemometers with 3 cone shaped cups were used to measure wind speed.This type of anemometer exerts a more uniform torque throughout the revolution. Conventionalwind vanes were used to measure wind direction. The wind vane measures the azimuth angle of thewind. The wind data was recorded using a data acquisition system supplied by 50W solar panel. Thewind speed values were measured with a frequency of 1 Hz and the average wind speed wasrecorded at 10 min regular intervals.

In Ichanda during the year 2009 annual average wind speed at 78 m above ground level wasvav = 7.07 m/s with a standard deviation close to = 2.58 m/s. In Fig. 3 the monthly average windspeed at Ichanda is depicted. The maximum wind speed occurred during July due to south westmonsoon.

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Figure 2. Sengottai Pass and Ichanda Site (Source: srtm.csi.cgiar.org)

Figure 3. Monthly average wind speed at Ichanda

The frequency distribution of the wind for Ichanda is illustrated in Fig. 4. The probability ofoccurrence of wind speed presents a bimodal distribution. The left peak of the distribution ismainly associated with north east monsoon winds and the right peak with south west monsoonwinds. South west monsoon winds blow during 42 % of the time with an average wind speed of10.7 m/s.

3. THEORY OF ANALYSIS3.1. Weibull distributionIn many literatures adequate methods for describing wind speed frequency distribution are reported.The widely used conventional (two-parameter) Weibull probability density function for describingwind regimes written as [1]:

(1)

Where, v is the wind speed, k is the dimensionless shape parameter, c is the scale parameter withthe same unit as v [1].

Many methods are available for estimating Weibull shape and scale parameters such asmaximum likelihood, least squares, moments, weighted moments, linear moments and entropy.Maximum likelihood method corresponds to many well-known estimation methods in statistics. Ingeneral, for a fixed set of data and underlying statistical model, the method of maximum likelihoodselects values of the model parameters that produce a distribution that gives the observed data thegreatest probability (i.e. parameters that maximize the likelihood function) [1]. Weibull shape andscale parameters can be calculated using the maximum likelihood method as [1]

(2)

(3)

Where, vi is the wind speed in time step i and n is the number of non-zero data points [3].

( ) =

f v k c kc

v

c

v

c; , exp

k k1

=

=

=

=

kv v

v

v

n

ln( ) ln( )ikin

i

ik

i

n

ii

n

1

1

1

1

=

=c

nv

1ik

i

n k

1

1

616 Mixed Weibull Distribution: A Case Study on Ichanda, India

Figure 4. Wind speed frequency distribution

3.2. Mixed weibull probability distributionIn our analysis we considered a bimodal PDF to fit the time series measured data. This function isbased on a Mixed Weibull PDF and provides an analytical approach to estimate the wind speedfrequency distribution for Ichanda [2].

The Mixed Weibull PDF is defined by [1]

(4)

Or in an explicit manner

(5)

Where 0 w 1 is a weight parameter; which represents the proportion of componentdistributions being mixed. c1 and c2 are the scale parameters established by left and rightWeibull distribution, respectively; k1 and k2 are the shape parameters established by left andright Weibull distribution, respectively [1]. The weight component w can be obtained by usingthe following formula [2]

vav = w vav1 + (1-w) vav2 (6)Where vav is the average wind speed of time series measured data; vav1 and vav2 are the average

wind speeds of the left and right Weibull distribution respectively [2]. Figure 5 shows the wind speeddistribution from actual wind data. As seen previously, the left part of this distribution is mainly dueto north east monsoon wind and right part due to south west monsoon wind. This figure showsWeibull two parameter distribution does not fit the real wind data. The Mixed Weibull distributionPDF is an adequate statistical model for describing the wind speed frequency in Ichanda site.

The parameters obtained for Weibull PDF and Mixed Weibull PDF is shown in Table 1. Actualwind data of Ichanda site is split in to two parts; May to September which correspond to right peakof wind frequency distribution and October to April which is associated with the left peak. Separatesets of Weibull parameters are obtained using maximum likelihood method and combined both setsof distribution by applying weight parameter using equation (4).

= + G v w c k c k wF v c k w F v c k( ; , , , , ) ( ; , ) (1 ) ( ; , )1 1 2 2 1 1 2 2

=

+

=G v w c k c k w k

c

v

c

v

cw

kc

v

c

v

c( ; , , , , ) exp (1 ) expi

ki

ki

ki

k

i

n

1 1 2 21

1 1

1

1

2

2 2

1

21

1 1 2 2

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Figure 5. Wind speed distribution from actual data, Weibull PDF and Weibull & Weibull PDF

4. ANNUAL ENERGY PRODUCTION (AEP)AEP is the estimate of the total energy production of a wind turbine during a one-year period byapplying the site specific power curve to different wind speed frequency distributions, assuming100 % availability. Annual energy production for Ichanda site was calculated using both WeibullPDF and Mixed Weibull distribution and compared with the power curve data of a wind turbine of1.5 MW capacity and hub height of 85 m at an air density of 1.14 kg/m3.

AEP is calculated by

(6)

Where P (vi) and F(vi) is the power and wind frequency, respectively corresponding to windspeed vi P(vi) is taken from power curve data of the turbine under consideration. The factor 8760is the number of hours during a year. Table 2 shows the annual energy production calculated usingtwo methods discussed.

AEP calculated using Weibull PDF is only 91.801% of actual AEP of the site but applying MixedWeibull distribution to the same wind data is 97.668% of actual AEP, i.e. by using Weibull PDF weare underestimating the site.

5. CONCLUSIONThe current practice in wind industry is to calculate annual energy production using two parameterWeibull distribution. But this widely adopted method leads to highly inaccurate results in some siteswhich show bimodal behaviour. This is the case of Ichanda site which has bimodal winddistribution. With Mixed Weibull PDF, AEP is calculated close to actual, whereas using Weibulldistribution the site is underestimated close to 8.2%. Mixed Weibull distribution is ideal andflexible for sites/regions showing dual behaviour.

==

AEP P v F v( ) ( )*8760iin

i1

618 Mixed Weibull Distribution: A Case Study on Ichanda, India

Table 1. Parameters obtained for Weibull PDF and mixed Weibull PDF

Weibull parameters (78 m)Annual vav :- 7.065 [m/s]

:- 4.154 [m/s]k :- 1.780c :- 7.940 [m/s]

Mixed Weibull parameters (78 m)w :- 0.580

October-April vav1 :- 4.433 [m/s]1 :- 2.573 [m/s]k1 :- 1.805c1 :- 4.985 [m/s]

May-September vav2 :- 10.700 [m/s]2 :- 3.026 [m/s]k2 :- 3.942c2 :- 11.815 [m/s]

Table 2. Annual energy production calculated usingWeibull PDF and mixed Weibull PDF

AEP

Actual :- 5130914.867 kWhWeibull PDF :- 4710249.711 kWhMixed Weibull PDF :- 5011283.679 kWh

REFERENCES[1] Tian Pau Chang, Wind Speed and Power Density Analyses Based on Mixture Weibull and

Maximum Entropy Distributions, International Journal of Applied Science and Engineering,2010. 8, 1: 3946.

[2] O.A. Jaramillo, M.A. Borja. Wind speed analysis in La Ventosa, Mexico: a bimodalprobability distribution case. Renewable Energy 29, 2004, 16131630.

[3] Sabereh Darbandi, Mohammad Taghi Aalami, Comparison of four distributions for frequencyanalysis of wind speed, Environment and Natural Resources Research, March 2012, Vol. 2,No. 1.

[4] Ravindra Kollu, Srinivasa Rao Rayapudi, SVL Narasimham and Krishna Mohan Pakkurthi,Mixture probability distribution functions to model wind speed distributions, Kollu et al.International Journal of Energy and Environmental Engineering, 2012, 327.

[5] D. Zafirakis, El. Gavrilopoulou, K.A. Kavadias, J.K. Kaldellis. The need for the developmentof a new readjusted Weibull distribution for increased reliability of energy yield estimation.

[6] Chang, T. P. Performance comparison of six numerical methods in estimating Weibullparameters for wind energy application. Applied Energy in press, doi:10.1016/j.apenergy,2010, 06018.

[7] Akpinar, S. and Akpinar, E.K., Estimation of wind energy potential using finite mixturedistribution models, Energy Conversion and Management, 2009, 50:877884.

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