Calculation on an hourly basis of solar diffuse irradiations from global data for horizontal surfaces in Ajaccio

Energy Conversion and Management 45 (2004) 2849–2866www.elsevier.com/locate/enconman

Calculation on an hourly basis of solar diffuse irradiationsfrom global data for horizontal surfaces in Ajaccio

G. Notton *, C. Cristofari, M. Muselli, P. Poggi

Laboratoire ‘‘Syst�emes Physiques de l’Environnement’’, Universit�e de Corse Pascal Paoli, UMR CNRS 6134,

Route des Sanguinaires, F-20000 Ajaccio, France

Received 3 November 2003; accepted 9 January 2004

Available online 5 March 2004

Abstract

Hourly global irradiations on tilted planes are required in various engineering calculations for solar

systems. In a lot of sites, at best, only global irradiations on horizontal planes are available. To calculate the

global irradiation on inclined surfaces from horizontal global irradiation, the fist step consists in deter-

mining the horizontal diffuse component. Thus, we study the variation of the diffuse component with global

irradiation on an hourly basis. Several correlations between hourly values of diffuse and global irradiations

are presented, validated and compared using solar data collected on the French Mediterranean site of

Ajaccio.� 2004 Elsevier Ltd. All rights reserved.

Keywords: Hourly solar irradiation; Horizontal diffuse component; Estimation

1. Introduction

For sizing a solar system using global solar radiation, or to estimate its productivity, manyengineers use monthly mean values of daily or hourly solar irradiation data, but in many cases, asfor instance the mathematical simulation of solar energy processes, these values are not sufficientbecause they do not provide a precise idea of the different energy phenomena that take place in theheart of the production system (inertia phenomena, shadowing masks,. . .).

Most available solar radiation data around the world are global solar radiations on a hori-zontal surface. In practice, solar collectors (flat plate thermal or photovoltaic collectors) are tilted,

* Corresponding author. Tel./fax: +33-4-95-52-41-42.

E-mail addresses: [email protected], [email protected] (G. Notton).

0196-8904/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2004.01.003

mail to: [email protected],

Nomenclature

CC correlation coefficientI hourly global solar irradiation on horizontal plane (Whm�2)Ib hourly beam solar irradiation on horizontal plane (Whm�2)Ib;n hourly normal beam solar irradiation (Whm�2)ID hourly diffuse solar irradiance on horizontal plane (Whm�2)I0 hourly extraterrestrial irradiation on horizontal plane (Whm�2)I0;n hourly normal extraterrestrial irradiation (Whm�2)K function used in Skarveith and Olsen formulaMax½Mb;?� threshold value of Mb;? for clear sky conditionMBE mean bias error (Whm�2)MBE/A ratio, MBE divided by average value of IDMb;? normal beam clearness index¼ Ib;n=I0;nMD hourly diffuse clearness index¼ ID=I0MT hourly clearness index I=I0MT;0 threshold value of MT used in Skarveith and Olsen formula (Whm�2)MT;1 threshold value of MT used in Skarveith and Olsen formula (Whm�2)N number of observationsRMBE relative mean bias errorRMSE root mean square error (Whm�2)RMSE/A ratio RMSE divided by average value of IDRRMSE relative root mean square errora coefficient of Hollands model¼ 1=sua1 coefficient of Skarveith and Olsen formulaa2 function of MT used in Maxwell modelb coefficient of Hollands model xl=2b1 coefficient of Skarveith and Olsen formulab2 function of MT used in Maxwell modelc2 function of MT used in Maxwell modeld1 variable used in Skarveith and Olsen formulaf hourly diffuse solar fraction¼ ID=Im optical thickness (or mass) of atmospheret timex parameter used in Hollands and Crha model¼MT=suxi ith measured value�x measured mean valueyi ith predicted value�y predicted mean valuea parameter used in Skarveith and Olsen formulab parameter used in Hollands and Crha model¼ 2

xl� 1þ qgxð1� 2xÞ

DMb;? function of m used in Maxwell model

2850 G. Notton et al. / Energy Conversion and Management 45 (2004) 2849–2866

c solar elevation degreeqg ground reflectancesu beam transmittance of upper layer of atmosphere in Hollands modelxl albedo of lower layer of atmosphere in Hollands model

G. Notton et al. / Energy Conversion and Management 45 (2004) 2849–2866 2851

so it is necessary to know the solar radiation incident on such tilted planes. Moreover, the par-ticularity of solar conversion systems, as for all energy systems using a phenomenological input,comes from the uncontrollable character of the energy input as a result of non-foreseeablemeteorological variations. Then, even for a ‘‘perfectly’’ known system, from a mathematical pointof view, the efficiency or the productivity of such a system is dependent on the temporal fluctu-ations of the energy input and output. Thus, the solar irradiation must be known with a time stepat most equal to an hour.

The question is ‘‘how to calculate the hourly solar irradiation on an inclined plane when onlyhourly irradiation on a horizontal plane is known?’’

To compute these hourly global solar irradiations on tilted planes, various models are availableand used [1–4], as for example the Hay model [5–7] or the Perez model [8–10], but all these modelsrequire, in the same time, the hourly global irradiations and the hourly horizontal diffuse solarirradiations.

We propose in this paper to study the determination of the hourly diffuse solar irradiation on ahorizontal plane from horizontal global irradiation, i.e. the first step of the determination of thesolar irradiation on inclined planes. To reach this goal, various models have been tested and theirperformances have been evaluated using solar data collected on the Mediterranean site of Ajaccio.

2. Description and definition of solar data

In Ajaccio (Corsica, France), a seaside Mediterranean site (Latitude 41�550N, Longitude:8�480E), we have, in the laboratory, a complete meteorological station where the direct normaland the global horizontal irradiances are, respectively, measured by an Eppley NIP pyrheliometerand a Kipp and Zonen (CM5) pyranometer. The standardization of such instruments is main-tained by the French meteorological organization. Other parameters such as temperature, pres-sure, relative humidity and wind speed and direction, are also recorded. The data are collected andrecorded every minute. About 5% of the data values are missing because of some problems withthe instruments and some defects and maintenance in the data acquisition system.

Concerning the direct irradiance, the Eppley NIP pyrheliometer is mounted on an automaticSolar Tracker, Model SMT-3 (two axis, azimuth/elevation device programmed to align directbeam instruments with the normal incidence of the sun). Every two days, the pyrheliometer iscleaned, and its alignment is verified.

The horizontal diffuse radiation ID is not measured, but is obtained from the horizontal globalradiation I and the normal beam radiation Ib;n by [11]

ID ¼ I � Ib;n sinðcÞ ð1Þ


where c is the solar altitude, also called the solar elevation. It is the angular height of the sunabove the celestial horizon of the observer and varies between 0� (sunset or sunrise) and 90�(zenith).

Hourly global I and hourly diffuse ID radiations are obtained from an integration of the cor-responding solar irradiance collected every minute.

For our study, three parameters are used

• the hourly clearness index MT ¼ I=I0, which is indicative of the clearness of the sky;• the hourly diffuse index MD ¼ ID=I0; and• the diffuse fraction f ¼ ID=I .

I0 is the hourly extraterrestrial irradiation received on a horizontal plane and is calculated usingwell known astronomical formulae [11].

We have, for this study, one year of solar data (I and Ib;n) (year 2002). The hourly values arecomputed, and the solar altitude is calculated at midhour, i.e. for the hour between t and t þ 1, c iscalculated at t þ 1=2.

The values of MT, MD and f have been computed and then checked. In the case of hourly solarradiation data, instrumental errors can be significant, especially due to the cosine response and themask effect of the surrounding mountains, which can introduce errors near sunrise and sunset, asunderlined by De Miguel et al. [12]. A quality control, suggested by De Miguel et al. [12] is appliedon the hourly data, and all data that do not meet the following conditions are not used in thisstudy:

06 ID 6 ð1:1 IÞ06 I 6 ð1:2 I0Þ06 ID 6 ð0:8 I0Þ06 Ib 6 I0

8>><>>: ð2Þ

Thus, 3925 hourly data of MT, MD and f have been used in this work.

3. Presentation of the tested models

Iqbal [11] introduced some models linking hourly diffuse irradiation to horizontal globalirradiation.

The first of this kind is the Orgill and Hollands correlation [13] based on four years of datafrom Toronto (Canada) and which divides the sky cover into three parts

f ¼ ID=I ¼ 1:0� 0:249MT for 06MT 6 0:35f ¼ 1:577� 1:84MT for 0:35 < MT 6 0:75f ¼ 0:177 for MT > 0:75

8>><>>: ð3Þ

This Eq. (3) is based on data from a high latitude site. Erbs et al. [14] reworked this correlationintroducing solar data from five US stations with latitudes between 31�N and 42�N. This Erbscorrelation is given by


f ¼ ID=I ¼ 1:0� 0:09MT for 06MT 6 0:22f ¼ 0:9511� 0:1604MT þ 4:388M2

T

�16:638M3T þ 12:336M4

T for 0:22 < MT 6 0:80f ¼ 0:165 for MT > 0:80

8>><>>: ð4Þ

Iqbal [15] noted that such correlations have reliability problems. The sky cover can be qualifiedby three conditions: cloudy, partially cloudy or very clear. This classification is important, and itwill appear several times during this study. Moreover, the validity of classification has beendemonstrated in a previous study performed on our site [16]. It seems obvious that in the twoprevious categories, the solar altitude c has an influence on the prediction of ID from I.

Iqbal [15] plotted MD versus MT for several meteorological stations, particularly for Trappes(48�460N, 02�010E) and Carpentras (44�050N, 05�030E), and noted that

• Under overcast conditions (MT < 0:35),MD increases linearly withMT and the solar altitude hasno bearing on the clearness diffuse index MD.

• After 0.35, the effects of the solar altitude begin to emerge. At the beginning of the partiallycloudy sky region, MD increases with MT and then begins to decrease as the partly cloudy skiesbecome clearer. At particular solar altitudes, a minimum value of diffuse radiation is reached,and the minimum value of MD varies with solar altitude.

• After this value, in a clear sky condition, MD increases again with MT and lower solar altitudesgive a higher percentage of diffuse radiation.

Under partially cloudy or clear skies, a solar altitude lower than 30� has a significant impact ondiffuse radiation.

Hollands [17] established a model between f and MT. This model has the originality of beingbased on a theoretical physical basis. The atmosphere is modelled as having two homogeneousand nonselectively absorbing layers: an upper layer called the ozonosphere with zero scatteringand beam transmittance su, and a lower layer with isotropic, nonselective scattering characterizedby an albedo xl. The values of su and xl have been chosen so as to make the final expression fit themeasured data (from Toronto, Canada) as closely as possible and are su ¼ 0:897 and xl ¼ 0:982.The expression of the Hollands model is

f ¼1� b�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� bÞ2 � 4ab2MTð1� aMTÞ

q� �ð2abMTÞ

ð5Þ

with a ¼ 1=su and b ¼ xl=2.This model has been revised in 1987 [18], and the effect of multiple inter-reflections between the

atmosphere and the ground (atmospheric back scattering) has been taken into account. Theground reflectance qg has been introduced as a parameter and must be evaluated for the studiedsite. The new expression is

f ¼b �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4xð1� xÞð1� q2

gx2Þ

qh ið6Þ
½2xð1� qgxÞ�


with x ¼ MT

suand b ¼ 2

xl� 1þ qgxð1� 2xÞ. The values of su and xl are different from the values

used in Eq. (5) now being su ¼ 0:855 and xl ¼ 0:985.Skarveith and Olseth [19] proposed to express the hourly diffuse fraction of the global irradi-

ation f in terms of hourly solar elevation and clearness index. They tested their model on solardata from Norway. The expression of this model is

MT 6MT;0

IDI¼ 1:00

MT;0 < MT 6 aMT;1

IDI¼ f ðMTÞ ¼ 1� ð1� d1Þ a1

ffiffiffiffiK

pþ b1K þ ð1� a1 � b1ÞK2

h iwith

K ¼ 0:5 1

�þ sinp

MT �MT;0

MT;1 �MT;0

�� 0:5

��

MT > aMT;1

IDI¼ 1� aMT;1

ð1� f ðaMT;1ÞÞMT

ð7Þ

with MT;0 ¼ 0:20, MT;1 ¼ 0:87� 0:56 expð�0:06cÞ and c expressed in degrees, a ¼ 1:09,d1 ¼ 0:15þ 0:43 expð�0:06cÞ, a1 ¼ 0:27 and b1 ¼ 0:00.

This model is perfectly in agreement with the remarks of Iqbal [15] who showed that forc < 30�, solar elevation has an important influence, and this effect disappears for upper values ofc.

De Miguel et al. [12] compared several models on the basis of solar data of Greece (1 station),Portugal (6 stations), France (3 stations) and Spain (2 stations), all these stations being situated inthe Mediterranean belt area. In a first part, they classified the MT values and found the followingresult: 8% of MT < 0:2, 75% of 0:2 < MT < 0:7 and 17% of MT > 0:7. There were no data for

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Hourly Clearness Index MT

Hou

rly

Dif

fuse

Fra

ctio

n f

(ID

/I)

Orgill and Hollands [13]

Erbs et al [14]

Hollands [17]

Hollands and Crha [18]

Climed2 [12]

Fig. 1. Illustration of various relations not taking into account the solar elevation.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


Hou

rly

Dif

fuse

Fra

ctio

n f

= I D

/I

80˚

70˚

60˚

50˚

40˚

30˚

20˚

10˚

Solar elevation in degree

Fig. 2. Illustration of Skartveit and Olseth model [19] for various solar elevations.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


Hou

rly

Dif

fuse

Cle

arne

ss I

ndex

ID

/I0

10

20

30

40

50

60

70

80

90


Fig. 3. Illustration of the Maxwell relation [20] for various solar elevations.


MT > 0:9, which is not completely in accordance with Orgill and Hollands [13]. They plotted MT

versus MD for different sun elevations (<30�, 30–45�, >45�) and noted that for MT < 0:4 (overcast


sky), the solar elevation has no effect any more, and the diffuse radiation increases linearly withglobal solar radiation, a result corroborated by Iqbal [15]. For MT > 0:4, the effect of solar ele-vation begins to appear and the minimum value of MD depends on the solar altitude. The fol-lowing region where MD increases with MT is the clear skies regions. This increasing of the diffusefraction may be due to beam radiation reflected from clouds and recorded as diffuse radiationduring periods when the sun is unobscured by the surrounding clouds. This zone is moreimportant for solar altitudes lower than 30� less important shorter for higher solar elevations. Inclear sky conditions, at lower solar elevations, the global radiation has a high diffuse fraction dueto the scattering effects of a greater air mass, and as the global radiation increases, the diffuseportion increases too, as remarked by Iqbal [15] and Skartveit and Olseth [19]. Several modelswere tested by De Miguel et al. [12]:

• the Hollands and Crha model [18] previously described;• the Maxwell model [20], which will be presented in the following;• and the CLIMED2 model elaborated by the authors [12].

The Maxwell model [20] calculates the hourly normal beam irradiation Ib;n from the hourly globalirradiation on a horizontal plane I . A threshold value of the ‘‘normal beam clearness index’’ for a

clear sky condition Mb;? ¼ Ib;nI0;n

is first computed

Max½Mb;?� ¼ 0:866� 0:122mþ 0:0121m2 � 0:000653m3 þ 0:000014m4 ð8Þ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9


Hou

rly

Dif

fuse

Fra

ctio

n f

0-5

5-10

10-15

15-20

20-25

25-30

30-35

35-40

40-45

45-50

50-55

55-60

60-65

65-70

>70


Fig. 4. The hourly diffuse fraction f versus the hourly clearness index MT.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8


Hou

rly

Dif

fuse

Cle

arne

ss I

ndex

MD

=ID

/I0

0-5

5-10

10-15

15-20

20-25

25-30

30-35

35-40

40-45

45-50

50-55

55-60

60-65

65-70

>70


Fig. 5. The hourly diffuse clearness index MD versus the hourly clearness index MT.


then a reduction of Mb;? is computed versus the air mass, m, and the hourly clearness index MT

Mb;? ¼ Max½Mb;?� � DMb;?ðm;MTÞ ð9Þ

DMb;? ¼ a2ðMTÞ þ b2ðMTÞ exp½c2ðMTÞm� ð10Þ

a2ðMTÞ ¼ �5:74þ 21:77MT � 27:49M2T þ 11:56M3

T

b2ðMTÞ ¼ 41:40� 118:50MT þ 66:05M2T þ 31:90M3

T for MT > 0:6

c2ðMTÞ ¼ �47:01þ 184:2MT � 222:0M2T þ 73:81M3

T

8><>:

a2ðMTÞ ¼ 0:512� 1:56MT þ 2:286M2T � 2:222M3

T

b2ðMTÞ ¼ 0:370þ 0:962MT for MT 6 0:6

c2ðMTÞ ¼ �0:280þ 0:932MT � 2:048M2T

8><>:

ð11Þ

where the air mass is calculated according to Kasten and Young [21]

m ¼ 1=½sin c þ 0:50572ðc þ 6:07995�Þ�1:6364� ð12Þ

At last, the hourly diffuse irradiation is calculated using Eq. (1).De Miguel et al. [12] evaluated a series of models on 11 various sites of the Mediterranean belt,

a very interesting work. They developed a model called CLIMED2 on the basis of the datacollected in these Mediterranean sites

Fig. 6. The hourly diffuse clearness index MD versus the hourly diffuse fraction f .


f ¼ 0:995� 0:081MT for MT 6 0:21f ¼ 0:724þ 2:738MT � 8:32M2

T þ 4:967M3T for 0:21 < MT 6 0:76

f ¼ 0:180 for MT > 0:76

8<: ð13Þ

It seems interesting to plot these seven models to observe their differences. The five models linkingthe solar fraction f to the clearness index MT and for which the solar elevation is not taken intoaccount are plotted in Fig. 1.

Excepted the Hollands and Crha model, the correlations are relatively similar, but differ onlyfor high values of MT.

The Skartveit and Olseth model and the Maxwell model are illustrated, respectively, in Figs. 2and 3 for various values of solar elevation c taken in equal steps from 10� to 90� by steps of 10�.

The remarks of Iqbal [15] and De Miguel et al. [12] concerning the impact of the solar elevationon the relation between diffuse radiation and global radiation are relatively taken into account inthese two models.

4. Observation of data and parameter variations

The hourly diffuse fraction f and the diffuse clearness index MD, calculated from experimentalvalues, are plotted according to the clearness index, respectively, in Figs. 4 and 5 in distinguishingthe hourly data according to their solar elevation (by steps of 5�).

We note that the repartition of MT is: 13.68% for MT < 0:2, 62.37% for 0:26MT < 0:7 and23.95% for MT P 0:7. It is very close to the distribution found by De Miguel et al. [12] for five

Fig. 7. Average value of f by interval as a function of MT.


Fig. 8. Average value of MD by interval as a function of MT.



meteorological Mediterranean stations with 8% for MT < 0:2, 75% for 0:26MT < 0:7 and 17%for MT P 0:7. Moreover, the maximum value of MT obtained during this period is 0.85, and thecorresponding value for De Miguel et al. [12] is 0.9.

If the aspect of the set of points in Fig. 4 is similar to the illustrations of the correlationspresented in Fig. 1, the influence of solar elevation is more difficult to observe. Nevertheless, wedistinguish some trends according to solar elevation except for low solar elevations, particularlyfor 0�6 c < 5�, probably for the previously explained reasons concerning the problems encoun-tered at sunset and sunrise. It seems interesting to evaluate the adequacy of each model using allsolar data on the one hand and solar data except values with 0�6 c < 5� on the other hand.

Skartveit and Olseth [19] showed that for MT < 0:2, c has no influence, which is perceptible inFig. 4. This influence appears since MT ¼ 0:35 for Iqbal [15] and since MT ¼ 0:4 for De Miguelet al. [12], but it is not clear in Fig. 4. So, we will present these data in another manner in thefollowing part of this study.

In Fig. 5, the influence of solar elevation is also difficult to observe, but the threshold of 0.2clearly appears and the Iqbal threshold [15], equal to 0.35, seems to be present because from thisvalue, the dispersion of the points is more important.

We presented in Fig. 6, the hourly diffuse fraction f versus MD. Three well distinguished areascan be observed: the first one for f < 0:4 where there is a linear relation between the two vari-ables, the second one with 0:46 f < 0:98 where it is difficult to know if a correlation exists and thethird one for f P 0:98 where for a same f , MD varies from 0.02 to 0.4.

For a best observation of the trends, for eachMT interval such as i 0:056MT < ðiþ 1Þ 0:05with i ¼ 0; 1; . . . ; 16, we computed the average value of MD and f . This average value is computedfrom a variable number of solar data, and this number is not always sufficient to have an averagevalue representative of the considered interval. Figs. 7 and 8 show the evolution of the averagevalues of, respectively, f and MD. We can see the tendencies observed in Figs. 2 and 3, corre-

Table 1

Values of the statistical coefficients for each model

Statistical coefficients MBE

(Whm�2)

RMSE

(Whm�2)

MBE/A(%)

RMSE/A(%)

CC

Orgill and Hollands [13] 8c 7.305 50.180 5.386 37.000 0.8480

c > 5� 7.908 51.743 5.516 36.095 0.8289

Erbs et al. [14] 8c )0.883 50.246 )0.652 37.048 0.8448

c > 5� )0.822 51.807 )0.573 36.140 0.8250

Hollands [17] 8c 0.598 50.741 0.441 37.414 0.8408

c > 5� 0.783 52.318 0.546 36.496 0.8204

Hollands and Crha [18] 8c 10.564 53.464 7.789 39.422 0.8322

c > 5� 11.358 55.133 7.923 38.460 0.8103

CLIMED2 [12] 8c 1.819 49.534 1.341 36.524 0.8502

c > 5� 2.069 51.073 1.443 35.628 0.8316

Skartveit and Olseth [19] 8c 13.494 50.042 9.949 36.898 0.8608

c > 5� 14.481 51.624 10.102 36.011 0.8430

Maxwell [20] 8c )4.144 51.614 )3.056 38.058 0.8420

c > 5� )4.278 53.203 )2.984 37.113 0.8229

Fig. 9. Validation of some correlations: Orgill and Hollands [13]; Erbs et al. [14]; Hollands [17] and Hollands and Crha

[18].


sponding, respectively, to the Skartveit and Olseth model [19] and to the Maxwell model [20], andwe find all the remarks underlined by Iqbal [15].

It is important to note that the aerosols represent a major source of variability in the diffuseradiation, and Corsica often receives dust from Africa.

5. Performances of the models

In order to quantify the adequacy of a model, we use statistical values as suggested by Iqbal [11]and Notton et al. [22]: the mean bias error (MBE), the root mean square error (RMSE) and thecorrelation coefficient (CC). The expressions of these parameters are

MBE ¼PN

i¼1ðyi � xiÞN

; RMSE ¼PN

i¼1ðyi � xiÞ2

N

( )1=2

;

CC ¼PN

i¼1ðyi � �yÞðxi � �xÞPNi¼1ðyi � �yÞ2

h i PNi¼1ðxi � �xÞ2

h in o1=2ð14Þ

Table 2

Values of the statistical coefficients according to the study of De Miguel et al. [12]

Athens Porto Seville CC

MBE/A(%)

RMSE/A(%)

MBE/A(%)

RMSE/A(%)

MBE/A(%)

RMSE/A(%)

Hollands and

Crha [18]

29.54 53.89 2.73 36.04 14.75 38.05 0.81–0.94

Maxwell [20] 17.71 56.09 )24.85 51.43 )7.84 48.63 0.77–0.93

CLIMED2 [12] 11.81 58.00 )2.88 36.25 6.23 38.01

Fig. 10. Validation of some correlations: De Miguel et al. [12]; Skartveit and Olseth [19]; and Maxwell [20].


where yi is the ith predicted value, xi the ith measured value, �y the predicted mean value, �x themeasured mean value and N the number of data analysed. We also use the relative values of theseparameters: the ratios of MBE and RMSE divided by the average value A of hourly diffuseirradiation, as proposed by De Miguel et al. [12]. These variables are defined by

MBE=A ¼PN

i¼1ðyi � xiÞN�x

; RMSE=A ¼PN

i¼1ðyi � xiÞ2

N

( )1=224

35,�x ð15Þ


In this study, yi and xi are the estimated and measured values of hourly diffuse irradiations on ahorizontal plane ID. As explained in the previous paragraph, it seems that some problems appearwhen the sun elevation is lower than 5�, so the statistical parameters have been computed twoways: from all the solar data and from these data excluding the irradiations for which c6 5�. Thevalues of the these parameters for each correlation are given in Table 1.

Estimated values using each model are compared with experimental ones and plotted in Figs. 9and 10. All the sets of representative points are quite similar expected for the Maxwell correlationfor which the repartition of points around the straight line, corresponding to the estimated valuesequal to the experimental ones, is more regular. Except for the Erbs model and Maxwell�s model,the other correlations overestimate the experimental ‘‘reality’’ (MBE>0 and MBE/A>0).

At the beginning of this study, we thought that using hourly solar data collected at sunset orsunrise (c < 5�) could lead to a bad performance of the models, but in Table 1, we can note thatthe value of the qualifying coefficients are often better when all the data are used to estimate theadequacy of the correlation. Nevertheless, the RMSE/A is improved when solar data with c6 5�are not taken into account in the determination of the calculation.

De Miguel et al. [12] tested the Hollands and Crha model [18], the Maxwell model [20] and theCLIMED2 model, which they elaborated from solar data collected in Athens (Greece), Porto(Portugal) and Seville (Spain) according to the CC, MBE/A and RMSE/A criteria and obtainedthe results presented in Table 2.

The results found by De Miguel et al. [12] are of the same order as the results computed for themeteorological station of Ajaccio. Only the Maxwell model [20] presents better performances forAjaccio than for the three Mediterranean cities used in the other study.

The Hollands and Crha model, which has been presented as an improved model of the Hollandmodel, leads to worse results. A recalibration of the coefficients used in this model might sub-stantially improve its performance.

Taking solar elevation into account, as in the Skartveit and Olseth model and in Maxwell�smodel, improves the performances in terms of RMSE but does not have a significant influencefrom the point of view of the other statistical parameters.

The oldest models [13,14] have correct performances compared with the more recent ones.At last, the model CLIMED2 is the most satisfying model if we do not take into account the

solar data collected for sun elevations lower than 5�.

6. Conclusion

To search a possible relation between the diffuse component and the hourly horizontal globalirradiation, two types of variation have been studied

• the variation of the hourly diffuse fraction f versus the hourly clearness index MT; and• the variation of the hourly diffuse clearness index MD as a function of the clearness index MT.

About 4000 hourly solar data have been used in this study, which confirmed the observationspreviously described by Iqbal [15], Skartveit and Olseth [19] and De Miguel et al. [12] concerningthe influence of solar elevation on the correlation between diffuse and global hourly irradiations.


Seven relations have been applied, validated and compared using statistical test parameters toquantify their accuracy.

It appears, for the meteorological station of Ajaccio-Vignola, that

• there is not a model largely better than another, whatever the statistical test chosen to deter-mine the quality of the model;

• whatever we might think, the use of correlations taking into account the influence of solar ele-vation [19,20] do not improve the determination of the hourly diffuse irradiation very much;

• not using the hourly solar data corresponding to a solar elevation less than 5� (sunset and sun-rise hours) for validation of the model does not improve the values of the statistical parametersexcept for the relative root mean square error presented as a percentage of the average value(RMSE/A);

• the ‘‘old’’ models [13,14] have satisfactory performances; and• the model showing the best performances for our solar data is the CLIMED2 model [12], which

has been elaborated on the basis of solar data collected in three Mediterranean stations.

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Documents

Calculation on an hourly basis of solar diffuse irradiations from global data for horizontal surfaces in Ajaccio