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Renewable Energy 35 (2010) 2874e2880

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Renewable Energy

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A statistical investigation of wind characteristics and wind energy potential basedon the Weibull and Rayleigh models in Rwanda

Bonfils Safari*, Jimmy GasoreDepartment of Physics, National University of Rwanda, P.O. Box 117, Huye, South Province, Rwanda

a r t i c l e i n f o

Article history:Received 9 July 2009Accepted 28 April 2010Available online 7 June 2010

Keywords:Wind speedWind energyWeibull distributionRayleigh distributionWind rosesRwanda

* Corresponding author. Tel.: þ250 0850 86 69.E-mail address: bsafari@nur.ac.rw (B. Safari).

0960-1481/$ e see front matter � 2010 Elsevier Ltd.doi:10.1016/j.renene.2010.04.032

a b s t r a c t

A wind energy system converts the kinetic energy of the wind into mechanical or electrical energy thatcan be harnessed for practical uses and transform the economy of rural areas where access to water andelectricity is very restricted and industry is almost nonexistent in most of the developing countries likeRwanda. Assessing wind power potential for a location is an imperative requirement before makinga decision for the installation of windmills or a wind electric generator and evaluating plans for relatingprojects. The aim of the present study was to evaluate the potential of wind resource in Rwanda and toconstitute a database for the users of the wind power. A time series of hourly daily measured wind speedand wind direction for the period between 1974 and 1993 on five main Rwandan meteorological stationswas provided by the National Meteorology Department. Statistical methods applying Weibull and Ray-leigh distribution were presented to evaluate the wind speed characteristics and the wind powerpotential at a height of 10 m above ground level using hourly monthly average data. Those characteristicswere extrapolated for higher levels in altitude. The results give a global picture of the distribution of thewind potential in different locations of Rwanda.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Rwanda is a small hilly and mountainous, landlocked country inthe Great Lakes region of Africa. Bordered by the DemocraticRepublic of the Congo (DRC), Burundi, Tanzania and Uganda, it islocated at 2�:00 Latitude South and 30�:00 Longitude East. Totalland area is about 24,950 km2, and inland lakes cover about1390 km2. Rwanda’s population of more than 9.1 million (17%urban) is growing at an annual rate of 2.6% [20,21]. The electricpower generation capacity is very low and access to energy is verylimited. ELECTROGAZ, the only and state-owned energy companyin Rwanda, serves urban areas close to the national grid. At presentonly 4.3% of the population has access to electricity with 23.4% ofthe population in urban area and less than 1% in rural areas [19,20].Rwanda’s land area covered by forest is 20% of the total. Energyconsumed in the rural areas, where the majority of Rwandans live,is 85% of the total energy.Wood-fuel constitutes 90% of rural energyconsumption. Domestic energy demand has increased drasticallydue to population growth and the increase in economic activitiesduring the last ten years [20]. As for water, ELECTROGAZ is the main

All rights reserved.

water supplier in the urban areas of the country. In rural area,a major part of water is supplied by installations constructed byinternational non-governmental organizations and run by thecommunities. 64% of the population has access to drinkable water,but in the city of Kigali, ELECTROGAZ can only meet 40% of theneeds [19e21].

Whilst Rwanda has enough renewable energy potential tosustain its energy needs and support economic development inrural areas, harnessing of these resources has to date been limited.It is therefore important to exploit those resources in view to findsolution to energy shortage and environmental degradation thecountry is experiencing.

In the past, in Rwanda, wind energy has been unexplored and inliterature, no study has been conducted in order to explore thatresource. This present study discusses a branch of a compositeanalysis whose objective is to investigate the potential of windenergy resource in Rwanda. Many studies have been conducted onelectricity generation and water pumping by direct mechanicalmeans from wind energy conversion systems. They have proventhat they are technically and economically efficient [3e13]. Tech-nologies behind for wind energy conversion systems have beendeveloped worldwide [14e18].

Anyproject of installing awind system requires the knowledge oftwo elements: the location and the wind. The location will depend

Fig. 1. Geographical locations of the five meteorological stations in Rwanda.

B. Safari, J. Gasore / Renewable Energy 35 (2010) 2874e2880 2875

on the strength of the wind and its annual and seasonal steadinessobtained through a study of a long-term record. Furthermore thewind characteristics are fundamental for the estimation of thewindpower of the location. Our study is essentially related to the analysisof long term monthly average wind data recorded at five meteoro-logical stations in Rwanda shown in Fig. 1.

2. Data and methods

2.1. Data

Time-series of measured hourly daily wind speed data for theperiod between 1974 and 1993 on five main Rwandan meteoro-logical stations have been provided by the Rwandan NationalDepartment of Meteorology. They were averaged on a monthlybasis and stored as monthly hourly values for each year. Fig. 1 andTable 1 show the geographic location and geographic coordinates ofthe stations. A two-dimensional cubic spline interpolation usingMatlab 6.5.1 was performed and applied to the monthly hourlywind speed, and the results were then plotted in an orthogonalview three dimensional graphics as presented in Fig. 2. Sixteendirectional Wind Rose presenting frequencies of direction of eachwind speed for each station was plotted with the use of WRPLOTView 5.9�1998e2008 Lakes Environmental Software.

2.2. Methods

In literature, many studies base their statistical analysis of windcharacteristics and wind energy potential on the assumption thatthe Weibull distribution approximates wind speed [1, 3-16]. Thereason is because of the easy estimation of its parameter toapproximate the empirical distribution of wind observations [3-

Table 1Geographical coordinates and elevations of the stations.

Observatory 4 l h (m)

Butare 02�360S 29�440E 1760Kigali 01�580S 30�080E 1490Ruhengeri 01�300S 29�360E 1878Gisenyi 01�400S 29�150E 1554Ruhengeri 01�300S 29�360E 1878

4: Latitude, l: longitude, h: elevation.

16]. In a recent study [3], two other distributions, the gamma andthe square-root normal distributions were compared to theWeibulldistribution. It was found that all three models give a goodapproximation irrespective of the orographic environment for kvalues significantly higher than 2. The Weibull distribution wasrecommended, due to the difficulty of parameter determination forthe other models. Very often, for k values less or equal to 2, theWeibull and the Rayleigh distributions are used.

2.2.1. The Weibull probability density functionA random variable V, here the wind speed, has a Weibull

distribution if its probability density function is defined by [1]:�f ðvÞ ¼ f ðv; k; cÞ ¼ k

c

�vc

�k�1e�ðv=cÞk ; v > 0; k; c > 0

f ðvÞ ¼ 0; v � 0; (1)

where k is a so-called shape parameter (a dimensionless number)and c is scale parameter (m/s).

If k¼ 2, then we have a special case of the Weibull distributioncalled the Rayleigh distribution [15,16] whose distribution density is:

f ðvÞ ¼ f ðv; cÞ ¼ 2vc2e�ðv=cÞ2 : (2)

With such a distribution, the expected value of a probabilityvariable is:

m ¼ cffiffiffip

p2

: (3)

In this situation, the scale parameter c is in proportion to theaverage.

The Weibull cumulative distribution function is given by:

Fðv; k; cÞ ¼ 1� e�ðv=cÞk (4)

2.2.2. Wind speed variation with altitude above the Earth’ surfaceIn general, wind speed measurements are made at a standard

altitude such as 10 m above the Earth’ surface. For projectsinvolving wind conversion system, it is required to estimate windspeeds at various elevation. The wind speed increase with height.When record of wind speed exists at different height for a station,the commonly power law can be used to obtain the extrapolatedvalues of wind speed at different heights [2,9]:

v ¼ va

�zza

�a

(5)

where v0 is the wind speed measured at anemometer height z0, v isthe wind speed to be calculated at the height z, a is the power lawexponent depending on the surface roughness and obtainedempirically.

Another technique uses theWeibull probability density functionto obtain the extrapolated values of wind speed at different heights.Parameters ofWeibull distribution functions kz and cz for altitudes zabove the anemometer level are obtained using the followingrelations [3,8]:

kz ¼ ka½1� 0:088lnðza=10Þ�=½1� 0:088lnðz=10Þ�; (6)

cz ¼ caðz=zaÞn; (7)

where ka and ca are, respectively, the shape parameter and the scaleparameter at the anemometer height za and the exponent n is givenby the relation:

n ¼ ½0:37� 0:088lnca�=½1� 0:088ðlnza=10Þ�: (8)

Fig. 2. Hourly monthly average wind patterns for five stations in Rwanda.

Fig. 3. Wind roses for five studied stations in Rwanda.

B. Safari, J. Gasore / Renewable Energy 35 (2010) 2874e28802876

Fig. 4. Weibull and Rayleigh windedensity distribution for five stations in Rwanda.

B. Safari, J. Gasore / Renewable Energy 35 (2010) 2874e2880 2877

2.2.3. Estimation of parameters k and c with the maximumlikelihood method

Various methods have been developed for estimating theparameters of the Weibull probability distribution function. Themost commonly used have been the method of moments [3,5,6],the Maximum Likelihood method [4] the least square method[3,6,7] and Chi-square method [6]. However, the Maximum Likeli-hood method has proved to be the most efficient [4,5] in deter-mining the parameters of Weibull probability distribution function.Therefore that method has been used in the present study.

Assume ðv1;v2;.; vnÞ is a random sample with a probabilitydensity function, here the Weibull function, of the form given by(1). The likelihood function of the random sample ðv1;v2;.; vnÞdenoted by Lðk; c; v1; v2;.; vnÞ is the joint density of the variablesinvolved, that is:

Table 2Comparison of statistical errors RMSE and MBE (%) for Weibull and RayleighDistributions.

Station Weibull Distribution Rayleigh Distribution

RMSE MBE (%) RMSE MBE (%)

Butare 0.04 0.43 0.05 1.00Kigali 0.04 1.20 0.04 1.22Ruhengeri 0.07 1.21 0.10 �2.74Gisenyi 0.02 �0.10 0.03 �0.06Kamembe 0.04 �0.09 0.05 �0.34

Lðk; c; v1; v2;.; vnÞ ¼i¼1

f ðk; c; viÞ (9)

Yn

Then, we have:

lnL ¼Xni¼1

ln½f ðviÞ�

¼ n½lnk� klnc� þ ðk� 1ÞXni¼1

lnðviÞ � c�kXni¼1

ðviÞk (10)

For n independent data ðv1;v2;.; vnÞ of variable V, the maximumof the function Eq. (9) is determined by solving the followingsystem of equation:(

vln Lvk ¼ 0

vln Lvc ¼ 0

(11)

Solutions of Eq. (11) must satisfy the following system ofequations:8>>>><>>>>:bc ¼ �

1nPn

i¼1 vi

�1bk ðaÞ

nbk�nln�bc�þPn

i¼1 lnðviÞ�Pn

i¼1

�vibc�bk ln�vibc�¼ 0 ðbÞ

(12)

By eliminating bc from the system of Eq. (12), we obtain thefollowing equationwhich gives the value of bk fromwhich the valueof bc can be obtained by Eq. (12(a)):

Butare

0

10

20

30

40

50

0 1 2 3 4 5 6 7 8 9Wind speed (m/s)

)%(

ytisnedytilibabor

P

Kigali

0

10

20

30

40

0 1 2 3 4 5 6 7 8 9Wind speed (m/s)

)%(ytisned

ytilibaborP

Ruhengeri

0

20

40

60

80

100

0 1 2 3 4 5 6 7Wind speed (m/s)

)%(

ytisnedytilibaborP

Gisenyi

05

1015

202530

0 1 2 3 4 5 6 7 8 9 10 11 12 13Wind speed (m/s)

)%(

ytisnedytilibaborP

Kamembe

0

10

20

30

40

50

60

0 1 2 3 4 5 6Wind speed (m/s)

ytisnedytilibaborP

10m30m60m100m

Fig. 5. Weibull windedensity distribution at different altitudes for five stations in Rwanda.

B. Safari, J. Gasore / Renewable Energy 35 (2010) 2874e28802878

b "Pni¼1 vi

bk lnðviÞ 1Xn #

k ¼ Pn

i¼1 vibk �

ni¼1

lnðviÞ

�1

(13)

Eq. (12) is solved with an iterative method starting with thevalue bk0 given by [4]:bk0 ¼ �

vffiffiffiffiffiffiffivar

p �1:086; (14)

where v and var are, respectively, the sample mean and variance ofthe series.

0.51

1.52

2.53

3.54

4.55

5.5

10 30 45 60 90 100

]s/m[

deepsdni

wnae

M

ButareKigaliRuhengeriGisenyiKamembe

z (m)

Fig. 6. Computed mean wind speeds hviz calculated for the whole measurementperiod at heights za¼ 10 m, z¼ 30 m, 45 m, 60 m, 90 m, 100 m.

2.2.4. Evaluation of mean wind energy densityThe most important wind characteristic is the wind energy

density. Assume A is a cross-section through which the wind ofspeed v flows perpendicularly. The available wind power is definedas the flow of kinetic energy which is obtained by the relation [3,4]:

PðvÞ ¼�12v2�vA ¼ 1

2rv3A ½W� (15)

where r is the air density which depend on pressure (altitude),temperature and humidity. It is assumed to be constant since itsvariation does not affect significantly wind resource calculation[3,4]. The commonly used value is r ¼ 1:225 kgm�3 correspond-ing to standard conditions (sea level, 15 �C).

The power density distribution gives the distribution of windenergy at different wind speeds. It is obtained by multiplying thewind power density with the probability of each wind speed asfollows:

PðvÞA

f ðv; k; cÞ ¼ 12rv3f ðv; k; cÞ

hWm�3s

i(16)

By integrating Eq. (15) for the period of study we obtain themean wind power density:

P ¼ 12r

Z N

0v3f ðv; k; cÞdv ¼ 1

2r G

�1þ 3

k

� hWm�2

i(17)

The wind speed vmec at which the power density distribution isa maximum is called wind Speed of Maximum Energy Carrier. It

Table 3Parameters and fundamental statistical data of the Weibull distribution describing the distribution of monthly average wind speeds at za anemometer height and at five otherlevels for the period of study.

Station / Butare

Altitude Y n k c hvi cv s vm vM vmec P

za 0.29 2.42 2.36 2.09 0.44 0.92 1.89 2.03 3.03 9.0830 2.67 3.26 2.90 0.40 1.17 2.74 2.84 4.02 22.4845 2.78 3.68 3.27 0.39 1.27 3.13 3.22 4.46 31.4860 2.87 4.00 3.57 0.38 1.35 3.44 3.52 4.81 40.0090 3.00 4.51 4.02 0.36 1.46 3.94 3.99 5.35 56.13100 3.03 4.65 4.15 0.36 1.50 4.07 4.12 5.50 61.31

Kigaliza 0.30 1.95 2.34 2.07 0.54 1.11 1.62 1.94 3.36 10.7330 2.16 3.24 2.86 0.49 1.40 2.42 2.73 4.39 25.6245 2.24 3.65 3.23 0.47 1.52 2.80 3.10 4.84 35.4460 2.31 3.97 3.52 0.46 1.61 3.11 3.39 5.20 44.6590 2.41 4.47 3.97 0.44 1.75 3.59 3.84 5.75 61.92100 2.44 4.62 4.09 0.44 1.79 3.72 3.97 5.90 67.43

Ruhengeriza 0.38 1.02 0.88 0.88 1.01 0.89 0.02 0.61 2.69 2.6030 1.09 1.34 1.29 0.91 1.18 0.14 0.96 3.45 6.3945 1.14 1.56 1.49 0.88 1.31 0.25 1.13 3.79 8.9660 1.17 1.74 1.65 0.85 1.41 0.25 1.13 4.06 11.4290 1.23 2.03 1.90 0.82 1.56 0.51 1.51 4.47 16.11100 1.24 2.12 1.97 0.81 1.60 0.56 1.57 4.59 17.63

Gisenyiza 0.27 1.55 3.05 2.75 0.66 1.81 1.57 2.41 5.21 32.8730 1.72 4.12 3.67 0.60 2.20 2.48 3.33 6.45 68.5645 1.79 4.60 4.09 0.58 2.37 2.91 3.74 7.00 90.2860 1.84 4.97 4.41 0.56 2.49 2.91 3.74 7.41 109.8990 1.92 5.55 4.92 0.54 2.67 3.79 4.59 8.04 145.23100 1.95 5.71 5.06 0.54 2.71 3.94 4.73 8.21 156.21

Kamembeza 0.33 1.12 1.49 1.43 0.89 1.28 0.21 1.08 3.72 8.2630 1.24 2.16 2.01 0.81 1.63 0.58 1.61 4.67 18.6545 1.29 2.47 2.28 0.78 1.78 0.78 1.86 5.09 25.3260 1.33 2.72 2.50 0.76 1.90 0.78 1.86 5.42 31.5390 1.39 3.12 2.84 0.73 2.07 1.25 2.39 5.91 43.05100 1.41 3.23 2.94 0.72 2.12 1.34 2.49 6.05 46.71

B. Safari, J. Gasore / Renewable Energy 35 (2010) 2874e2880 2879

corresponds to the mode of the power density distribution and isgiven by [3]:

vmec ¼ cð1þ 2=kÞ1=khm s�1

i(18)

3. Results and discussion

From the observation of Fig. 2, the windy season coincides withthe rainy season in most parts of the country (OctobereDecemberand JanuaryeApril) except the western part of the country where

020406080

100120140160180

10 30 45 60 90 100

m/w[

ytisnedrewop

naeM

2 ]

ButareKigaliRuhengeriGisenyiKamembe

z(m)

Fig. 7. Mean power density calculated for the whole measurement period at heightsza¼ 10 m, z¼ 30 m, 45 m, 60 m, 90 m, 100 m.

the windy season coincides with the dry season (JulyeSeptember).The results obtained revealed that the wind energy potential inRwanda is relatively high (�4.5 ms�1 at 60 m) in the Western partof the country during the dry season while it is fairly high(�4.5 ms�1 at 60 m) in the Center and the Southern part of thecountry during the rainy season.

The charts of the wind roses observed in Fig. 3 demonstratethat most of the wind flows from South and Southeast, except forthe station of Ruhengeri where Northeast calm winds arepredominant. This station is on the eastern border side of thehigh-elevated volcanic mountains located in the Northeast part ofthe country.

The distribution of original time series of average monthlywind speeds is well approximated by the Weibull probabilitydensity function. This is shown by Fig. 4 and Table 2. The RMSEand MBE (%) are, respectively, the root mean square and the meanbiased error computed from the Weibull and Rayleigh probabilitydensity function relatively to the empirical probabilitydistribution.

In Figs. 5, 6, 7 and Table 3 we present the results obtained fromextrapolated wind characteristics. In general, good values of meansof monthly wind and power density can be found at 60 m. Thestation of Gisenyi is found to have the greatest means of monthlywind and power density at 60 m altitudeðhvi ¼ 4:49 ms�1; P ¼ 109:89 Wm�2Þ. The observed windenergy at that station can serve as a basis for electrical conversionpurpose. The stations of Kigali and Butare are found to have

B. Safari, J. Gasore / Renewable Energy 35 (2010) 2874e28802880

relatively good values of means of monthly wind and power densityat 60 m altitude (hvi ¼ 3:57 ms�1; P ¼ 40:00 Wm�2 for Butareandhvi ¼ 3:52 ms�1; P ¼ 44:65 Wm�2 for Kigali). The station ofRuhengeri appears to be marginal with very low values of means ofmonthly wind and power density ðhvi ¼ 1:65 ms�1;

P ¼ 11:42 Wm�2Þ. This is due to the fact that the station ofRuhengeri is located at the lowest altitude (1878 m) in the vicinityof the eastern slope of the high elevated north-western Karisimbivolcanic mountain (4507 m).

In the present study we discussed a branch of a compositeanalysis whose objective was to investigate the potential of windenergy resource in Rwanda. The results obtained are satisfying.Meanwhile, further investigations are to be done based on a moredetailed and systematic analysis of wind speed patterns. Exami-nation of wind energy distribution on a seasonal, monthly, dailyand hourly basis from continuous wind speed data for a period of atleast one year should be done and results should be compared tothose obtained in the present study in order to obtain a morereliable long term average for the use of energy conversion systems.

4. Conclusion

In the present study, wind energy potential in Rwanda wasinvestigated using statistical methods to analyze the time series ofhourly monthly average wind speed in the period between 1974 and1993 measured on five main Rwandan meteorological stations.Observeddistributionofmeasuredmonthlyaveragewindspeedswasapproximated by the Weibull and Rayleigh probability density func-tions. TheWeibullprobabilitydensitydistributionwas found tobe thebestfittingmodel for the empiricaldistribution. The scaleparameter cand the shapeparameter kweredefined foreach site and this allowedgenerating a model giving the distribution of monthly average windspeed andwind characteristics at different levels above anemometerlevel. The results give a global picture of the distribution of the windpotential in different locations of Rwanda.

Analysis of wind rose shows that on monthly average the windflows from South and Southeast in Rwanda. On average, the windyseason coincides with the rainy season in most parts of the country(OctobereDecember and JanuaryeApril) except the western part ofthe country where the windy season coincides with the dry season(JulyeSeptember). On the annual average, the wind speeds atGisenyi, Kigali and Butare are fairly high. The available wind energycan therefore be harnessed to generate electricity especially duringthe corresponding windy season.

Meanwhile, it is recommended that before any decision, bypolicy makers, on harvesting wind energy in Rwanda, furtherinvestigations should be done based on a more detailed andsystematic analysis of wind speed patterns. Examination of windenergy distribution on a seasonal, monthly, daily and hourly basisfrom continuous wind speed data for a period of at least one yearshould be done and results should be compared to those obtainedin the present study in order to obtain a more reliable long term

average. The results from such investigations can further be used inthe design and estimation of performance of the types of windenergy conversion systems to be used in Rwanda.

Acknowledgements

The author is grateful to the National Meteorological Service ofthe Ministry of Infrastructure, for providing relevant informationfor this article. The present study has been supported by theResearch Commission of the National University of Rwanda (NUR)through a partnership with the Swedish International Agency SIDA/SAREC.

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