Solar Energy 75 (2003) 43–51

S tatistical comparison of models for estimating the monthlyaverage daily diffuse radiation at a subtropical African site

*M. BashahuDepartment of Physics and Technology, Institute of Applied Pedagogy, University of Burundi, P.O. Box 5223, Bujumbura, Burundi

Received 2 October 2002; received in revised form 14 April 2003; accepted 5 May 2003

Associate Editor: Richard Perez

Abstract

Nine correlations have been developed in this paper to estimate the monthly average diffuse radiation for Dakar, Senegal.A 16-year period data on the global (H ) and diffuse (H ) radiation, together with data on the bright sunshine hours (N), thed

fraction of the sky’s cloud cover (Ne /8), the water vapour pressure in the air (e) and the ambient temperature (T ) have beenused for that purpose. A model inter-comparison based on the MBE, RMSE andt statistical tests has shown that estimates inany of the obtained correlations are not significantly different from their measured counterparts, thus all the nine models arerecommended for the aforesaid location. Three of them should be particularly selected for their simplicity, universal

¯ ¯ ¯ ¯ ¯applicability and high accuracy. Those are simple linear correlations betweenK and N /N , Ne /8 or K . Even presentingd d t

adequate performance, the remaining correlations are either simple but less accurate, or multiple or nonlinear regressionsneeding one or two input variables. 2003 Elsevier Ltd. All rights reserved.

excluded, the simplest way to obtain values of that1 . Introductionquantity is to estimate them. The existing models forestimating that quantity discern one from another the

¯ ¯ ¯~estimation ofI , I , H , I andH (the annexed nomencla-The solar global radiation on a horizontal surface is the d d d d d

ture gives the meaning of those quantities).minimum information generally needed as an input param-In this work, we only deal with and our objectives areeter in any project for building a solar energy system.

two fold. First to develop different relatively simpleNevertheless, the diffuse radiation is frequently necessary¯candidate statistical correlations for determiningH overin various applications such as agriculture, architecture, d

Dakar (Senegal, Africa) and thus over neighbouring sta-climatology and illumination. Particularly, one alwaystions or over locations which have the same climate.needs separate values of the direct and diffuse radiation on

horizontal surfaces when calculating the insolation on Second to compare those correlations. There are at leastinclined surfaces, analyzing the energy balance in the three underlying reasons to these goals. The first one is theatmosphere or describing photobiological processes. Fur- belief shared with other workers (Iqbal, 1983; Deris, 1983;thermore, measurements of the diffuse radiation are per- Erbs et al., 1982) that statistical determination of diffuseformed in very few stations whereas those of the global radiation should be limited to restricted regions withradiation are made in many locations. When direct mea- similar climates in order to achieve a reasonable degree ofsurement of the diffuse radiation at a given location is accuracy. The second reason is that work on estimation of

H is too scare for the subtropical African region. The thirdd

one is that data on the required physical quantities of this*Fax: 1257-22-32-88. study were available for the aforesaid site.

0038-092X/03/$ – see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0038-092X(03)00213-5

44 M. Bashahu / Solar Energy 75 (2003) 43–51

Nomenclature

* A dash over a given physical quantity means the monthly average of that quantity.e Water vapour pressure in the air (in mb)v

H Daily global radiation incident on a horizontal surfaceH Daily diffuse radiation incident on a horizontal surfaced

H Extraterrestrial daily radiation incident on a horizontal surfaceo

H9 Daily global radiation emerging from the atmosphere before striking the ground9H Daily diffuse radiation emerging from the atmosphere before striking the groundd

h Solar elevation at solar noonn

I Hourly global radiation incident on a horizontal surfaceI Hourly diffuse radiation incident on a horizontal surfaced~I Global irradiance incident on a horizontal surface~I Beam irradiance incident on a horizontal surfaceb~I Diffuse irradiance incident on a horizontal surfaced

L Longitude of the stationK Diffuse fraction,H /Hd d

K Clearness index,H /Ht o

N Daily number of bright sunshine hoursN Day length in hoursd

Ne /8 Fraction, in eighths, of the sky covered by cloudsN Modified day length, in hours, which excludes the fraction during which the sun is less than 58j

above the horizonT Ambient temperature in Kt Student’s statistical variableW Atmospheric water vapour content (in cm)w Solar hour anglez Height of the station above sea leveld Solar declination

¯d Characteristic declination, i.e. the declination on whichH is identical to its monthly averageHc o o

F Latitude of the stationr Ground albedor Clear-sky albedoa

r Cloud albedoc

AcronymCERER Centre for study and research on renewable energyMBE Mean bias errorRMSE Root mean square errorWMO World Meteorological Organization

2 . Background and selected correlations linear relationships of this kind for the indicated regions:(i) Page (1961)on the basis of data from stations spread in

¯ ¯2 .1. Correlations of H with H the latitude range of 408S–409N; (ii) Iqbal (1979) ford

Canadian data; (iii)Tuller (1970) for data from four2 .1.1. Linear relationships Canadian stations; (iv) de Brichambaut and Ben-Kaddour

A linear correlation of the following form should be (Sfeir and Guarraccino, 1981) for stations in France andfound for any station, set of stations or region: Morocco, respectively; (v)Deris (1983)for 19 station in

Europe and 14 in Turkey, respectively.¯ ¯K 5 a 1 b K (2.1)d 1 1 t

2 .1.2. Nonlinear relationshipsThis expression may depend on the extent of the consid- Liu and Jordan have developed a third degree polyno-

¯ ¯ered region and its climate. By way of illustration, let us mials correlation ofK vs. K (Liu and Jordan, 1960) basedd t

state that the following workers have obtained different on global radiation from several stations with a wide range

M. Bashahu / Solar Energy 75 (2003) 43–51 45

¯of K values (0.3–0.7). On the basis of data from different precipitable water, the atmospheric turbidity, the solart

countries (India, Australia, Israel and Canada), Klein and elevation andI. Examining the residual differences be-Duffie have shown that the Liu and Jordan relationship is tween measured diffuse fractions andK values determinedd

not universal and that individual correlations are therefore from the best one-parameter (K ) correlations for sixt

required for different climatic and geographic regions Pacific Northwest stations,Vignola and McDaniels (1984)(Iqbal, 1983). Third degree polynomials relationships of have been led to the hypothesis that the atmospheric¯ ¯K vs. K have been also developed, respectively by Ruth factors contributing to the transmission values, i.e. aird t

and Chant for Canadian stations, and Ohlam, Desplanches mass, water vapour and turbidity, are the same onesand Antoine for French stations (Sleir and Guarraccino, responsible for the observed seasonal variations.1981; Ruth and Chant, 1976). Using data from four US Similarly, in terms of monthly averages, models for

¯stations,Erbs et al. (1982)have obtained seasonal depen- estimatingH show that this quantity may be expressed asd¯dent correlations of the previous kind. All those features a function ofH and one or several amongst the following

allow us to believe that a relationship of the following type climatological parameters and optical properties of theshould be found for any station, set of stations or region: earth’s atmosphere: the average daily bright sunshine hours

]¯ ¯N, cloud coverNe, water vapour contentW, atmospheric2 3¯ ¯ ¯ ¯K 5 a 1 b K c K 1 d K (2.2) ¯ ¯d 2 2 t 2 t 2 t ¯turbidity T , solar elevationh, ground albedor, cloudat

¯ ¯albedor and clear sky albedor .c aNevertheless, a fourth degree polynomials correlation of¯ ¯K vs. K has been developed by Collares-Pereira (Sfeird t

and Guarraccino, 1981) for US stations. Furthermore, ¯ ¯ ¯2 .2.1. H vs. H and NdCollares-Pereira and Rabl (1979)have used pyrheliometer When the sky is completely covered by clouds, theand pyranometer data from the US to obtain a cosine number of bright sunshine hoursN is zero, and the global

¯ ¯relationship of K vs. K , whose coefficients vary withd t radiation is of diffuse nature. That is why correlationsseason.Mosalam et al. (2001)have compared monthly ¯ ¯ ¯betweenK and the relative sunshine durationN /N ared daverage diffuse radiation data measured at El-Menia likely.(Egypt) with those calculated by the means of Page’s and 1. The simplest correlations of this kind ore linearCollares-Pereira’s correlations. Bruno (Sfeir and Guarrac- relationships of the following form:

¯ ¯cino, 1981) has developed a sine relationship ofK vs. Kd t

¯based on data from Hamburg (Germany).Coppolino N¯ ]K 5 a 1 b (2.4)(1981) has proposed the following correlation which d 4 4 Ndincludes the solar elevation at solar noon,h :n

¯ ¯]] Two input measured variables, i.e.H andN are necessarya b3 3¯ ¯H 5 k9(K ) usinh u (2.3)d t n in those correlations. Iqbal (Ma and Iqbal, 1984) hasdeveloped one of such correlations on the basis of datawhere the empiric constantsa , b and k9 have to be3 3from three Canadian stations.determined for the considered station, set of stations or

¯ ¯ ¯ ¯2. ReplacingK by H /H in Eq. (2.4), andH by theregion. d d

following expression proposed by Prescott and others (Maand Iqbal, 1984,Ref. No. 16):¯ ¯2 .2. Correlations of H with H, optical properties of thed

earth’ s atmosphere and climatological quantities¯ ¯H N] ]5 a91 b9 (2.49)¯ ¯H NIn terms of irradiances or hourly or daily radiations, the 0 d

existing models for estimating the diffuse radiation showone obtains the following second degree polynomialsthat this quantity depends upon various other quantitiescorrelation:besides the global radiation. As a matter of fact,Buckius

and King (1978)have expressedI in terms of climated 2¯ ¯H Ntype, air mass and surface reflexivity.Hollands and Crha ] ]5 a 1 b (2.5)S D5 5¯ ¯(1987) have presented a revised model forK vs. K in H Nd t 0 d

which they include the ground albedo as a parameter. As¯The only input parameter In this correlation isN; e.g. Iqbalindicated by Skartveit and Olseth (1987),Braslau and

(Ma and Iqba, 1984) has proposed a relationship of thisDave have shown that the division ofI into its beamI andb

kind through regression analysis based on data from thediffuse I components depends strongly upon five vari-d

aforesaid Canadian stations.ables, i.e. the surface albedo, the atmospheric gases, the3. Coppolino has demonstrated that the following non-atmospheric turbidity, the solar elevation and the amount

¯linear correlation betweenH . the fractional sunshineof clouds.From an extensive data base, Garrisson (Skar- d

duration and the solar elevation at solar noon is alsotveit and Olseth, 1987,Ref. No. 2) has displayed thepossible:dependence ofI upon the surface albedo, the atmosphericd

46 M. Bashahu / Solar Energy 75 (2003) 43–51

a 3 . Methodology6N ]] b6¯ ]H 5 k0 (sinh ) (2.6)S Dd nNd 3 .1. Description of the data

The empiric constantsk9, a and b of Eq. (2.6) may be6 6 The global and diffuse radiation data of this study referdetermined from regression analysis for a considered to a 16-year period (1967–1982) of continuous recordsstation, set of stations or region. performed in an open area at theCERER, Dakar-Hann

Senegal (05148439N; L5278269W; z510 m). They havebeen taken from available reports in terms of hourly and

¯ ¯2 .2.2. H vs. H and other quantitiesd daily values. The accuracy of both types of data was not1. Considering radiation before and after multiple reflec- stated in those reports. However, according to these

tions between the earth and cloud cover, Hay (Ma and reports, the calibration of the two Kipp and Zon pyranome-Iqbal, 1984,Ref. Nos. 13 and 14) has proposed a calcula- ters used to obtain such data was maintained by the WMO.

¯ ¯tion procedure forH which includes the knowledge ofH,d Furthermore, the diffuse data were certified to have been¯c, a andN, together with the average modified day length preliminarly corrected for the shadow band effects.

N . which is given byj In terms of reports of daily records, the climatologicaldata have been kindly provided to us by theDepartment of

1 cos 852 sin I sind21 Climatology located at Dakar-Yoff, near Dakar Airport.¯ ] ]]]]]N 5 cos (2.69)F Gj 7.5 cosI cosd The related quantities are: the bright sunshine hours (N).c

the relative cloud cover(Ne /8), the water vapour pressurethe values of the characteristic declinationd being tabu- in the air (e in mb) and the ambient temperature (T in 8C).c v

¯lated (Iqbal, 1983). The monthly average atmospheric water vapour content (W2. Following the fundamental idea of the beginning of in cm) has boon calculated by means of the following

Section 2.2.1, it is obvious that when the sky is completely Leckner’s expression (Harrison, 1992):covered by clouds, the daily relative cloud coverNe /8

W5 0.227r (3.1)W(whenNe is given in eighths) is equal to one, the fractionalsunshine durationN /N is zero and all the global radiationd where the water vapour densityr expressed in terms ofWis of diffuse nature. It has been shown elsewhere (Bashahu the ambient temperatureT (in K) and the water vapour

¯et al., 1984; Bashahu and Nkundabakura, 1994) that Ne /8 pressure (e ) as follows:v¯ ¯is practically the complement to one ofN /N . Correlationsd¯ ¯ 217ebetweenK and Ne /S or (12Ne /8) are therefore likely. vd ]]r 5 (3.2)WThe simplest are linear relationships of this kind: T

¯In the classical relationships used to computeH andH ,o oNe 22~¯ ] the new solar constantI 5 1367 W m has been em-K 5 a 1 b (2.7) od 7 7 ]]8ployed. In its turn, the quantity sinh has been calculatedn

with w equal to zero (solar noon) andd equal tod ascNe¯ defined in the annexed nomenclature and given in theIqbalS D]K 5 a 1 b 12 (2.8)d 8 8 8 (1983) tables.

Further order degree polynomial relationships may also3 .2. Correlations obtainedobviously develop.

¯3. Following the work of Garg and Garg to estimateHFrom the aforesaid selected models (Section 2.3) and¯ ¯ ¯with N /N and W, and that of Hussain (1984)whichd

data (Section 3.1), the following correlations (Table 1)¯extended the method to findH for stations near sea leveldhave been obtained as the best least square fits to estimateof North and Central India, a general correlation of thethe monthly average diffuse radiation for Dakar site.following type should be proposed for any station, set of

stations or region:3 .3. Comparison methods

¯ ¯H Nd ¯] ]5 a 1 b 1 c W (2.9) The root mean square error (RMSE), the mean bias9 9 9¯ ¯H No d error (MBE) and thet-statistic have been used to evaluatethe accuracy of the correlations described above and tocompare them.2 .3. Selected correlations

For their relative simplicity of handling, the correlations 3 .3.1. RMSE and MBE(2.1)–(2.9) have been selected for investigation in this The following expressions define the two first statisticalwork. tests:

M. Bashahu / Solar Energy 75 (2003) 43–51 47

T able 1¯Correlations for estimating monthly average diffuse radiationH at Dakard

Selected Expression Eq. Range of the Correlationcorrelation no variables coefficient

¯ ¯ ¯2.1 K 51.1321.29K (3.3.1) 0.51#K # 0.68 20.93d t t2¯ ¯ ¯ ¯2.2 K 537.4368261.3179K 10.2856K (3.3.2) 0.51#K # 0.68 10.87d t t t

3¯2 0.8451K t ]]21.173 1.801¯ ¯ ¯2.3 H 5 1.281(K ) usin h u (3.3.3) 0.51#K # 0.68 10.92d t n t]]

0.798# usin h u# 0.9996n

¯ ¯N N¯ ] ]2.4 K 51.05920.984 (3.3.4) 0.60# #0.80 20.95d ¯ ¯N Nd d

2¯ ¯ ¯ ¯H N N Nd] ] ] ]2.5 5 2 7.07441 31.4386 215.5906 (3.3.5) 0.60# #0.80 10.93S D¯ ¯ ¯ ¯H N N No d d d

21.242¯ ¯N N]] 1.975¯ ] ]2.6 H 5 1.539 usin h u (3.3.6) 0.60# #0.80 10.98S Dd n¯ ¯N Nd d

]]0.798# usin h u# 0.9996n

¯ ¯Ne Ne¯ ] ]2.7 K 50.1210.52 (3.3.7) 0.28# #0.65 10.88d 8 8

¯ ¯Ne Ne¯ S]D ]2.8 K 50.6720.59 (3.3.8) 0.35# 12 #0.72 20.94d 8 8¯ ¯ ¯H N Nd ¯] ] ]2.9 5 0.22712 0.1302 10.0198W (3.3.9) 0.60# #0.80 10.91¯ ¯ ¯H N No d d

¯2.91#W # 4.96

n line of perfect estimation), and on the other hand, a small1 / 21 2](i) RMSE5 O d (3.4) RMSE and a large SINE (a consistently small over orF Gin i51 underestimation). Thus, although these two tests generallyprovide a reasonable procedure to compare models, theywheredo not objectively indicate whether a model’s estimates are

d 5Y 2 Y (3.5)i i,calc i,meas significantly different from their measured counterparts.Those are the reasons why an additional indicator, the

Y being theith calculated value,Y the ith mea-i,calc i,meas t-statistic, is generally required. As shown elsewheresured value andn the total number of observations. ˇ ˇ(Togrul and Togrul, 2002), this statistic not only allows

n models to be compared and at the same time indicates1](ii) MBE 5 O d (3.6)i whether or not a model’s estimates are statistically signifi-n i51

cant at a particular confidence level, hut also gives moreThe RMSE test provides information on the short-term reliable and explanatory results when used in addition with

performance of a given correlation. It allows a term by RMSE and MBE.term comparison of the actual deviation between thecalculated and the measured values. The smaller the value,

3 .3.2. t-Statisticthe better the model’s performance. Nevertheless, a fewAs defined by Student in one of the tests for meanlarge errors in the sum can produce a significant increase

values (Honerkamp, 1999),the random variablet within RMSE.n 2 1 degrees of freedom may be written here as follows:The MBE test provides information on the long-term

performance of a correlation. A low MBE is desired. An1positive value stands for the average amount of overesti-

]O dinmation in the calculated value and vice versa, However, i51]]t 5 (3.7)1 / 2overestimation of an individual observation can cancel S /n

underestimation in a separate observation.It is possible to have, on one hand, a large RMSE value where S is the standard deviation of the differencesd ,i

and at the same time a small lINE (a large scatter about the between estimated and measured values, and is given by:

48M

.B

ashahu/

SolarE

nergy75 (2003) 43–51

T able 2Monthly mean values of the principal quantities used in this work for Dakar, 1967–1982

Month J F M A M J J A S O N D

22 21H (J cm day ) 2969.91 3269.65 3595.61 3806.60 3853.52 3833.62 3824.17 3793.55 3646.27 3351.74 3025.50 2864.02o22 21H (MJ m day ) 5.08 6.04 6.77 6.95 6.86 6.37 5.00 5.38 5.40 5.67 5.15 4.6622 21H (MJ m day ) 1.40 1.66 1.82 2.09 2.44 2.57 2.64 2.65 2.30 1.96 1.63 1.63d

K 0.62 0.67 0.68 0.66 0.62 0.60 0.53 0.51 0.54 0.61 0.61 0.59t

K 0.28 0.27 0.27 0.33 0.36 0.40 0.47 0.49 0.42 0.35 0.32 0.35d

¯ ¯H /H 0.17 0.18 0.18 0.20 0.23 0.24 0.25 0.25 0.23 0.21 0.19 0.20d o

N (hours) 8.49 9.14 9.32 9.90 9.58 8.75 7.63 7.52 7.62 8.72 8.44 7.57

N (h) 11.23 11.52 11.92 12.34 12.69 12.86 12.78 12.48 12.07 11.65 11.31 11.14d

¯ ¯N /N 0.76 0.79 0.78 0.80 0.75 0.68 0.60 0.60 0.03 0.73 0.75 0.68d

Ne /8 0.44 0.36 0.31 0.28 0.37 0.50 0.65 0.65 0.63 0.50 0.44 0.50¯12Ne /8 0.56 0.64 0.69 0.72 0.63 0.50 0.35 0.35 0.37 0.50 0.56 0.50

c (deg) 220.84 213.32 22.40 9.46 18.78 23.04 21.11 13.28 1.97 29.84 219.02 223.12]sin h 0.8135 0.8826 0.9557 0.9958 0.9975 0.9895 0.9938 0.9996 0.9754 0.9095 0.8316 0.7898n

T (K) 294.28 293.59 293.96 294.58 295.49 293.48 299.99 300.34 300.50 300.54 298.63 295.90

e (mb) 17.36 17.89 18.86 20.01 21.61 25.57 27.65 29.44 30.27 28.92 24.09 18.85v

W (cm) 2.91 3.00 3.16 3.35 3.60 4.22 4.54 4.83 4.96 4.74 3.97 3.14

M. Bashahu / Solar Energy 75 (2003) 43–51 49

T able 3 statistical tables, the criticalt value, i.e.t at a level ofa / 2MBE, RMSE andt values for the nine correlations of this work significance and (n 2 1) degrees of freedom. For the

model’s estimates to be judged statistically significant atEq. no. MBE RMSE tthe (12a) confidence level, the calculatedt value must be

3.3.1 20.005 0.026 0.645 less than the critical value.3.3.2 20.089 2.669 0.1113.3.3 20.002 0.139 0.0503.3.4 0.003 0.024 0.4193.3.5 0.001 0.060 0.050

4 . Results and discussion3.3.6 0.013 0.165 0.2603.3.7 0.007 0.035 0.709

As computed from classical formulae and/or from the3.3.8 0.000 0.035 0.0133.3.9 0.000 0.015 0.020 fundamental data stated in Section 3.1, the monthly mean

values of the daily quantities used in this work arepresented inTable 2. The correlations ofTable 1 havebeen obtained from those results which allow us to noten 2

the following features;O dn i1 i512 2 (i) Even if its mean fraction of cloud cover is variable,]] ]]S 5 3O d 21 2 4 (3.8)in 2 1 ni51 Dakar location is in average characterized by a very¯ ¯ ¯sunny sky since itsK andN /N values are rather hight dUsing Eqs. (3.4) and (3.5) in the expression (3.8), we

(ranges of 0.51–0.68 and 0.60–0.80, respectively).have:

(ii) Higher values of the average diffuse radiation at2 2 Dakar, which correspond to higher values of then[(RMSE) 2 (MBE) ]2 ]]]]]]S 5 (3.9) relative cloud cover and to lower values of the relativen 2 1

sunshine duration occur during the rainy (and farming)season (June–September).Substituting forS in Eq. (3.7) gives:

¯(iii)At the opposite, lower values ofK occur during thed2 1 / 2 Harmattan season (November–May).(n 21)(MBE)

]]]]]t 5 (3.10)F G2 2 Table 3summarizes the results of the MBE, RMSE and(RMSE) 2 (MBE)t values as computed for the nine correlations obtained.

The smaller the value oft, the better the performance. From those results and from the correlations themselves,To determine whether a model’s estimates are statistically the following features may be drawn out by way of modelsignificant, one simply has to determine, from standard intercomparison elements:

T able 4Characteristics of the obtained models

Eq. Type of correlation Measured input variables Additional Evaluation of the natureno. parameter of estimation

Kind Number

¯ ¯ ¯3.3.1 Simple correlationK vs. K H 1 – Overall small underestimationd t

Very small scatter about the LPE¯ ¯ ¯3.3.2 3rd degree polynomials correl.K vs. K H 1 – Overall small underestimationd t

Large scatter about the LPE]]¯ ¯ ¯3.3.3 Nonlinear correlationK vs. K H 1 sin h Overall slight underestimationd t n

Mean scatter about the LPE¯ ¯ ¯ ¯ ¯3.3.4 Simple linear correlationK vs. N /N H, N 2 – Overall slight overestimationd d

Very small scatter about the LPE¯ ¯ ¯ ¯ ¯3.3.5 2nd degree polynomials correl.H /H vs. N /N N 1 – Overall slight overestimationd o d

Mean scatter about the LPE]]¯ ¯ ¯ ¯3.3.6 Nonlinear correlationH vs. N /N N 1 sin h Overall mean overestimationd d n

Mean scatter about the LPE¯ ¯ ¯ ¯3.3.7 Simple linear correlationK vs. Ne /8 H, Ne /8 2 – Overall slight overestimationd

Small scatter about the LPE¯ ¯ ¯ ¯3.3.8 Simple linear correlationK vs. 12Ne /8 H, Ne /8 2 – Overall very slight overestimationd

Very small scatter about the LPE¯ ¯ ¯ ¯ ¯ ¯ ¯3.3.9 Double linear correlationH /H vs. N /N andW N, W 2 – Overall very slight overestimationd o d

Very small scatter about the LPE

LPE, line of perfect estimation.

50 M. Bashahu / Solar Energy 75 (2003) 43–51

(i) Since all the obtainedt values are less thant 5 4.318, A cknowledgementsc

i.e. the critical t value given in standard tables(Honerkamp, 1999) at the levela 5 0.005 of signifi- The author would like to thank the responsible people ofcance (or atg 50.995 confidence level) and 11 the CERER, Dakar-Hann and of the Department ofdegrees of freedom (n 5 12), all the nine correlations Climatology, Dakar-Yoff, for having kindly provided thehave statistical significance. This is in a good agree- fundamental data of this work.ment with the high correlation coefficients observedfor any of them (Table 1).

(ii) Eqs. (3.3.8) and (3.3.9) give the best results. In adecreasing order of performance, they are followed by R eferencesEqs. (3.3.3), (3.3.5), (3.3.2), (3.3.6), (3.3.4), (3.3.1)and (3.3.7). B ashahu, M., Terrissol, M., Laplaze, D., 1984. Estimation du

` ´ ´ ´ `rayonnement solaire a l’aide de donnees meteotologiques, IVe(iii) Nevertheless, a more complete comparison of the´Seminaire sur l’Energie Solaire, 10–21 Septembre, Triesteobtained models may also take into account their

(Italie), pp. 537—547.evaluation in terms of characteristics such as theB ashahu, M., Nkundabakura, P., 1994. Analysis of daily globalfollowing: the degree of complexity, number and kind

irradiation data for five sites in Rwanda and one in Senegal.of measured input variables, possible additional inputRenewable Energy 4 (4), 425–435.

parameters. Those elements are summarized inTableB uckius, R.O., King, R., 1978. Diffuse solar radiation on a

4. horizontal surface for a clear sky. Solar Energy 21, 503–509.(iv) For their simplicity, practical universal applicability A rchives of the Centre d’Etudes et de Recherches sue les Energies

¯(simple linear correlations relatingH /H to sunshine Renouvelables (CERER), Dakar-Hann, Senegal.d

duration or cloud cover data) and high accuracy (high C ollares-Pereira, M., Rabl, A., 1979. The average distribution ofcorrelation coefficients and low MBE, RMSE andt solar radiation correlations between diffuse and hemispherical

and between daily and hourly insolation values. Solar Energyvalues), correlations (3.3.8), (3.3.4) and (3.3.1) should¯ 22 (2), 155–164.be the first to be recommended for stimulatingH atd

C oppolino, S., 1981. Extensive applicability of a new model forthe Dakar site.estimating diffuse solar radiation from clearness index and(v) Eq. (3.3.7) is also simple, but less accurate.minimum air mass. Renewable Energy 1 (2), 293–297.(vi) Model (3.3.9) presents a very high performance, but,

A rchives of the Direction de la Climatologie, Dakar-Yoff, Senegal.as a double linear regression, it needs two measured

D eris, N., 1983. Statistical determination of diffuse radiation in¯ ¯ ¯input variables (N /N and W ).d Europe and Turkey. In: Najat Veziroglu, T. (Ed.). Alternative(vii) Even accurate, the remaining models are nonlinear Energy Sources III, Solar Energy 1, Vol. 1. Hemisphere

relationships. Publishing, Washington, New York, London, pp. 47–58.E rbs, D.G., Klein, S.A., Duffie, J.A., 1982. Estimation of the

diffuse radiation fraction for hourly, daily and monthly-averageglobal radiation. Solar Energy 28 (4), 293–302.

H arrisson, J.D., Solar Energy 48(3), (1992).5 . ConclusionsH ollands, K.G.T., Crha, S.J., 1987. An improved model for

diffuse radiation: correction for backscattering. Solar Energy 38Based on a 16-year set of data on the global and diffuse(4), 233–236.

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data on climatological quantities, nine correlations have Numerical Methods, Data Analysis. VCH, New York,been developed to estimate the monthly average diffuse Weinheim, Cambridge.radiation for or Dakar. Those correlations have been H ussain, M., 1984. Estimation of global and diffuse irradiationcompared in terms of MBE, RMSE andt values. For any from sunshine duration and atmospheric water vapour contest.of those models, estimates are not significantly different Solar Energy 33 (2), 2217–2220.

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