Experimental verification of some clear-sky insolation models

Solar Energy. VoI. 41, No. 3, pp. 273-279. 1988

0038-092X/88 $3.00 + .00 Copyright ~ 1988 Pergamon Press plc

Printed in the U.S.A.

EXPERIMENTAL VERIFICATION OF SOME CLEAR-SKY INSOLATION MODELS A. LOUCHE,* G. SIMONNOT, and M. IQBALt Laboratoire d'Hrliornergrtique, Universite de Corse/C.N.R.S.-UA 877, Vignola--Route des Sanguinaires. 20000 Ajaccio, France and M. MERMIER I.N.R.A. Station de Bioclimatologie d'Avignon, 84140 Montfavet, France

Abstract--Clear-sky solar irradiance can be predicted when a number of essential atmospheric parameters are known. A number of parameterization methods to predict solar irradiance with various degrees of difficulty are available in the literature. In this study, three models called model A, model B and model C, with medium degree of difficulty, have been examined. In these models, the solar transmittance due to each atmospheric parameter is available in simple algebraic form. Based on these algebraic equations, the direct normal, diffuse, and global horizontal irradiance can be predicted. These models have been compared with measured data from Carpentras, a French radiometric station. At this station, several ~aily observations of the clear-sky irradiance are carried out. Corresponding instantaneous values of the AngstrOm turbidity coefficient 13 and several other necessary surface meteorological observations are also made. For diffuse irradiance, a value of 0.95 is assumed for the single-scattering albedo of the aerosols, Based on the calculation of the mean bias error and root mean square error, model C has the best correspondance with the measurements as for as direct irradiance is concerned. Model B appears to be more accurate for prediction of diffuse and global irradiance. Regression equations are provided to help the user of any one of the three models for better prediction of solar irradiance. 1. INTRODUCTION Many technological applications of solar energy are based on the availability of high intensities of direct solar radiation. High intensities of direct radiation are produced when the skies are cloudless and free of haze, here termed clear skies. Study of clear-sky insolation is also necessary for sizing refrigeration and air-conditioning systems and for protection of plants against lethal temperatures. When the air is absolutely clear and dry, the attenuation of extraterrestrial solar radiation is solely due to continuum scattering by all the atmospheric molecules, and selective absorption by gases such as CO2, 02 and O3. As the water vapor is introduced in the atmosphere, a slight relative increase in the total number of gaseous molecules has very little effect on the total molecular scattering (also called Rayleigh scattering), however, it has a substantial effect on the selective absorption of solar radiation in the infrared region. Furthermore, the gaseous water molecules have a strong tendency to coalesce and coagulate forming liquid particles. An atmosphere containing liquid water particles remains, of course, clean in a chemical sense, however, it becomes turbid in the optical sense. Unlike the continuum scattering of electromagnetic radiation by the gaseous molecules which is limited to about 1 I~m wavelength, the continuum scattering by

*Author to whom correspondence should be addressed. "tOn sabbatical leave from the University of British Columbia. 273

the water droplets depends upon their amount in the atmosphere and their average size distribution. The atmospheric pollution is derived from industrial smoke, forest fires, volcanic eruptions, and dust. This form of pollution plus water droplets plus other particles such as salt crystals and ocean spray all combined together are called aerosols. The aerosols scatter and absorb solar radiation, the energy absorbed being insignificant compared to the energy scattered. The extraterrestrial solar radiation as it enters the earth's atmosphere is partly absorbed and partly scattered, and the remaining reaches the ground in a continuous line from the sun. This radiation is called direct or beam radiation. The diffuse radiation reaching the earth has a complex origin. It is composed of: 1. The Rayleigh-scattered diffuse radiation reaching the ground after the first pass through the atmosphere. 2. The aerosol-scattered diffuse radiation reaching the ground after the first pass through the atmosphere. 3. The diffuse radiation reaching the ground after multiple reflections of the direct and diffuse radiation between the ground and the atmosphere from (1) plus (2). In many applications, calculation of the spectrally integrated (also called broadband) solar radiation reaching the ground is required. The integration procedures based on detailed knowledge of the distribution of atmospheric parameters are very lengthy.

274

A. LOUCHEet al.

Consequently, a number of researchers have developed broadband models in which the variation of the atmospheric parameters can be included through simple algebraic equations. One of the earliest parameterization model is the well-known and simple ASHRAE clear-sky formula[ 1], derived from the conditions in the United States. Result of a more recent effort in the United States is the SOLMET model[2,3], which presents regression equations with prescribed coefficients for specified locations. Reviews of these and other models appear in[4,5]. In the present study, mathematical basis for three generalized models is presented. The accuracy of the models is tested with data from a station in continental Europe, which is the main objective of this study. Such a comparison for the three models presented does not appear to be available in the literature.

spectrum. However, in an approximate but quite accurate form, 1, can be written as,

(5)

i. = eoi~¢

where the transmittance ~" has been presented in a number of forms, one of the most usual is to write it in a multiplicative form of the individual transmittances such as T = T,T0rg~w%

(6)

On the right-hand side of eqn (6), each transmittance is an integrated form of the spectral transmittance, such as (3). For example, ao

E i0,~ exp (-13h-"m,) Ah k=0

2. MATHEMATICAL BASIS OF PARAMETERIZATION

To =

(7)

is ¢

MODELS For a better appreciation of the material to follow, we present the elementary governing equations for transfer of radiation through the atmosphere. The direct solar spectral irradiance at a given instant and location can be given by

i.~ = e o i 0 . ~

(1)

where Tk :

(2)

TrhTOkTgkTwkTah

The quantities r~ are the spectral transmittances due to Rayleigh scattering, ozone absorption, mixed gas (CO2, Oz) absorption, water vapor absorption, and the aerosol attenuation (scattering and absorption), respectively.~ Each one of these transmittances can be written in terms of the Bouguer formula:

An algebraic form of (7) can be written as [7], % = (0.124455 - 0.0162) + (1.003 - 0.125e0 exp [-13m.(1.O89a + 0.5123)]

(8)

Similar algebraic expressions are available in the literature for other transmittances[4,7]. Iqbal[7] has analysed three direct and diffuse clear sky models, A, B, and C. Model A is based on the work of Paltridge and Platt[8] and others[9-12], and the transmittance for direct radiation has the form T = (TOT r - - ~ w ) Ta

(9)

The model B is from Sasamori[13] et al. and Hoyt[14,15]. The form for direct transmittance is

T =

1

~i

- -

TrTas

(10)

i=1

% = exp ( - k s m )

(3)

and the particular case of aerosol scattering as kak ~

13k a o

where a and 13 are the well-known Angstrrm turbidity parameters. Details of all the quantities (1) and (2) are available in [6,7]. The broadband direct normal irradiance can be written as

i.=eo E io. , ax

(4)

k~0

Obviously computation of (4) is quite tedious. There are about 140 values of io,,~, covering the solar tFor full description of all symbols, see the Nomenclature section at the end of the article.

Model C is from Bird and Hulstrom[16,17] and has the form for direct transmittance identical to that given in eqn (6) with a constant 0.9751 attached to it. The algebraic expressions for the individual transmittances, such as "rr, are not indentical for all the three models. There is a separate, quite similar, algorithm, however, for the computation of diffuse radiation on a horizontal surface. To avoid use of excessive space, full details of the three models are not given here. The reader is referred to [7] for this purpose. The atmospheric p. "ameters required for computation of the direct irradiance are ozone layer thickness, precipitable water thickness and a,13 of the Angstr6m turbidity formula. For the diffuse irradiance, an estimation of the ground albedo p, and the single scattering albedo ~0 of the aerosols is required. The data from the Radiometric Center at Carpentras (44°05'N; 0.5°03'E; 109 masl), France, were used

275

Clear-sky insolation models

to examine the accuracy of the three models. The data covered the period January 1984 to October 1985. The data included measurements of i, i., la, 13, precipitable water, and surface measurements of pressure and temperature. The turbidity coefficient 13 was measured (assuming ct = 1.3) with a 0.630 0.m RG 630 cutoff filter. The ozone layer thickness from [18] was used. For diffuse radiation, a constant value of 0.95 was used for tOo, the single-scattering albedo of the aerosols, and 0,2 for the ground albedo.

tO00

. . . . . . . . . l . . . . . . . . . I ' " ...... I. . . . . . . . . I . . . . . . . .

MOO_ A =900. _

": +~'~'~I~'~'~'~.,T*~--,-

DIRECTNORMAL

÷

B

O

0

+~

+~÷

--

.

÷~i,

÷

4-

i

jx,

.-..+

÷ 'e +

__

~700. +4-

3. DISCUSSIONS

Before carrying out the comparisons, the data were examined for any anomalous readings. This examination was based on the intercomparison of measurements of the direct normal irradiance by the pyrheliometer, the global horizontal by the pyranometer, and the diffuse horizontal by the shade-ring pyranometer. For example, the horizontal global radiation obtained by all the three instruments can be written as

J - (/,cos 0: + i~) = 0

(11)

where i is the horizontal global irradiance measured with a pyranometer without a shade ring, 1, is the direct normal irradiance measured with a pyrheliometer, 0: is the zenith angle, and is the horizontal diffuse irradiance measured with a shade-ring pyranometer. The data that did not agree within - 10% were discarded. The remaining data with 837 points were used for the comparison. For each of the three models, we have compared the measured data with the calculated values of the direct normal, diffuse, and global irradiance. In addition, the values of their mean bias error and the root mean square error were calculated. The relative mean bias error (MBE) was defined as

ia

(calculated-measured~/N MBE = \

measured

//

• ÷ 500.

]J

MBE =-0.03

,.I,,

I

600.

700.

""''1''''1 MODEL A

I,

800.

~ 200.

_

....

900.

~

I ....

DIFFUSE HORIZONTAL

+

--

I ....

t000

~

÷* ~ "

+

+

--~L*+

-i. ÷ ~ +

1~'''4.

"

++

4-

(13)

I ......

__

tering albedo of aerosols, ground albedo, and forward scatterance employed in this model. It seems that more accurate values of these physical parameters are needed to better examine the diffuse radiation component of model A. The comparison of model B with the measured values is plotted in Figs. 4, 5, and 6 and the values of MBE and RMSE are given in Table 1. Model B produces about as good values of the direct normal irradiance as model A. The diffuse and the global irradiance from model B result in values more accurate than those from model A. In the original model B[ 13-15], the wavelength exponent ct in the ,/~Jagstr6m turbidity formula is assumed equal to one and 13 is

'-ill2

meaNred

+

Fig. 1. Comparison of the measured with the calculated direct normal irradiance for model A.

(12)

m asured | RMSE = [ . \

+*

DIRECT NORMALIRRADIANCE (calculated) In (WIM2)

.~25o.

F

~.,w.+ ÷

~¢~¢

~600.

~300.

and the relative root mean square error (RMSE) was obtained from

÷

+

÷ -p,.~

~

+ + +

150.

~100.

For model A, the measured versus calculated values of the direct normal, diffuse, and global irradiance are presented in Figs. 1, 2, and 3. The values of MBE and RMSE are indicated on these diagrams and are also given in Table 1. Model A appears to be fairly accurate as far as the direct and the global irradiance are concerned. However, the calculated values do not correspond well with the measured values as far as the diffuse irradiance is concerned. This may be due to the arbitrary values of the single scat-

z

"50.0

HBE ....

I ....

I ....

I ....

0.3296

-

RMSE

-

I ....

0.4039 I ....

0 50.0 100. t50. 200. 250, 300. DIFFUSE HORIZONTAL IRRADIANCE (calculated) Id (W/M2)

Fig. 2. Comparison of the measured with the calculated diffuse horizontal irradiance for model A.

276

A. LoucHE etal.

1000 _

i000.

~ ........ I ......... I ......... I ......... I ......... I ...... ~T" ....... I'"'="i

~ 9 0 0 . ; EDBAL HORIZONTAL

"++

-.~.,

"=_ 16oo.

~800. ~700.

.

~

/

+ ÷¢

ij~

-

~600. ~

500.

+

: oo/2

= 600.

300.

400.

_

I

I

500.

600.

5 ~ o ....

700.

800.

900.

t000

Fig. 3. Comparison of the measured with the calcula~d global horizontal i~adiance ~ r model A.

÷~'~

:

MBE - -0.0335

"-°+

; ....

500.

GLOBALHORIZONT~IRFIAOIANCE(calculated) I (N/N2)

+

. .,.~,~;?e~.,,

+'_'~

uJ

N2oo.

200.

~

N 700. _

+ ~

÷

L ........

I .........

600.

- 0.0487

~BE

700.

I .........

800.

I .........

900.

I000

DIRECT NORMALIRRAOIANCE (calculate{I) In (N/M2) Fig. 4. Comparison of the measured with the calculated direct normal irradiance for model B.

obtained from k = 1 p.m. However, the values of 13 We have also developed the best fit curves of the at Carpentras are measured with a 0.630 i.Lm filter, measured values with the calculated values as a paa common practice in meteorological observations. rameter for direct, diffuse, and global irradiance for Therefore, it remains to be seen whether the model each of the three models. For the direct irradiance will correspond better with the observations had we Wm -2, the regression equations are the values of I3 prescribed by this model. Figures 7, 8, and 9 show the measured against the i,(measured) calculated values of the direct, diffuse, and global = 0.9418/,(calculated) + 68.545 (14) irradiance according to model C. Statistically speakmodel A ing, computation of the direct irradiance by this model is more accurate than by models A and B. It gives l,(measured) better values of the diffuse and global irradiance = 1.0129I,(calculated) + 16.934 (15) compared to model A but not compared to model B. The overall statistical comparison of the three model B models is presented in Table 1. The MBE and RMSE appear to follow each other. To sum up, model C is _ 300. more accurate for computation of direct normal ir.... I .... I .... I .... I~''''1 .... * radiance whereas model B is more accurate for comMODEL B * .= ~ + ÷ +*+ ÷÷ +" putation of diffuse and global irradiance. •,--,'~50. _DIFFUSE HORIZONTAL ÷÷

:

÷÷

+ ÷ +7. $ . . . .

Table 1. Comparison of mean bias error (MBE) and root mean square error (RMSE) of models A, B, and C

IrraOlance Direct

Model A

Model B

Model C

MBE

-0.0330

-0.0335

-0.0156

RMSE

D.0676

0.0687

0.0610

MBE

0.3296

0.1382

0.1785

RMSE

0.6039

0.2279

0.2568

MBE

0.0395

0.0068

0.0282

RMSE

0.0646

0.0531

0.0582

~ 200.

~ 150.

2

÷

.+

-

+ ÷

:

:.r.a~JE,~~ ' * " ÷

÷÷

_

4~÷

~i00.

~ E = O. t382

RMSE- 0.2279

Diffuse

Global

....

0

I ....

50.0

I ....

i00.

I ....

i50.

I ....

200.

I ....

250.

300.

DIFFUSEHORIZONTALIRAAnI~.NCE(calculateO) Id (W/M2)

Fig. 5. Comparison of the measured with the calculated diffuse horizontal irradiance for model B.

Clear-sky insolation models iO00 ........ t .........

~......... ~.........

t .........

=: ......

~30o.

I~ff;

+

÷

~¢

....

~

++ +++,~ :: " ....,. -. ~ :

NOOEL B

_ ~.OBAL HORIZONTAL

~900.

t .........

277 ,'"'l

-~

"-

....

l ....

HOO~_C

.

"

.-,'°250.~_OIFFUSE HORIZONTAL +`'.+

:

I' ' - " I " : ' ~ " ,d.

L

.."'_,'_4

+

~ +-~

I ~BO0.~ '-'

~2oo.1"

.

:

÷:,..~ +..

_

700. ~600, ,,.

~500,

_

÷

**.

÷ *

!

, " t r /

"x ~,.i~

~~

+

i

z400.

_o=

+

§5o oF_

++ ] ; .

=,

i ,~

~200.

[ ........ I .........

I .........

I .........

I .........

RN~E - 0.053~

,,o.

I .........

~=

I .........

I ........

global horizontal Jr'radiance for model B.

/,(measured) = 0.90581,(calculated) + 81.779

(16)

model C The above equations are plotted in Fig. 10 along with the tune 45 ° line, and confirm that by and large, model C is more accurate than the other two models. This is particularly so at high intensities of radiation, an aspect very important in solar technology systems using concentrators and other high temperature devices. In Fig. 10, it can also be noted that all models underpredict the direct irradiance vis-~t-vis the measurements. This is most likely due to the fact that the

-

0

i .........

, .........

i .........

,":' +~C

_

.~goo.

DIRECTNORMAL

lO0.

I ....

I ....

i5O.

I ....

200.

250.

300.

Fig. 8. Comparison of the measured with the calculated diffuse horizontal irradiance for model C.

measurements contain a portion of the circumsolar radiation. It is useful to add here that to take into account the World Radiometric Reference based new solar constant (1367 Wm -2) and its spectral distribution. Model C by Bird and Hulstrom[16] was slightly modified by Iqbal[7]. The modification pertained to changing the original constant 0.9662 to 0.9751. Our analysis has shown that there is less error with use of 0.9751 and the change seems to be justified. The regression equations for diffuse irradiance are ia(measured) = 0.7270Ja(calculated) + 7.259

(17)

model A

~ ........ i ......... I ......... I ......... I ......... [ ......... I ......... I' '~"':, ¢

_

~¢÷

, . . ~

50.0

I ....

tO00

..... ]

*~ ~+'+ 1

MODEL C

t ....

DIFFUSE HORIZONTALIRRAOIANCE(calculateO) I~ {W/N2) ¢]

tO00 .........

,-

~Eo~B~ P~SE 0.2548

0 ....

200. 300. 400. 500. 600. 700. BOO. 900. tO00 GLOBALHORIZONTALIRRAOIANCE(calculated) I ~/M2) Fig. 6. Comparison of the measured with the calculated

.,,,.-

I

~300.

E

HOOELC

i

~900.~__ GLOBAL HORIZONTAL

-- .:".'~J""

.+

~ :

_~ ~BOO.g

"¢1

- :ie~'r*

c.

,=

+

~p~.

-: :

e Boo.

~

45 4"

+÷~"

÷÷

+

50o.L

,

k

700.

++o~"

._I

~6oo.

÷**~÷

_ :~-~,,: •

,,~T ar

++

. *~-.-,-~ + +*

¢+t.,-+L÷

~o

600.

Aef " +

*

.BE--0.0i36 ~

.

÷

_

RHSE-0.04t0

÷

,.,: .... I ......... 500.

:

-. +

.~ # "

I ......... 700.

I ......... BOO.

900.

RMGE - 0.05B2

2.:..i..

I .........

200 ................ [.........[.........I.........I.........I.........i........

t000

DIRECTNORMALIRRADIkNCE (calculated} i n (W/M?.)

Fig. 7. Comparison of the measured with the calculated direct normal irradiance for model C.

200.

300.

400.

500.

600.

700.

800.

900.

iO00

GLOBALHORIZONTALIP/IAOIANCE (calculated) I O~/N2) Fig. 9. Comparison of the measured with the calculated global horizontalirradiancefor model C.

278

A. LOUCHE et al.

t000 ,

7/j,~/

........

I .........

I .........

I .........

I ....

I

},oo

_-:

!,oo.L

Ill=ll

9oo.L

-

-

-

/.-J _: /S _

r'"'~

........I.........I.........I.........I.........I.........I.........

(C':

///

7oo. _

4.000

;rf~

E

500 . . . . . . . . . .

I .........

I .........

I .........

600. 700. 800. 900. t000 DIRECT NORMALIRRAOIANCE (calculaLed) In (W/M2)

500.

Fig. 10. Comparison of the best-fit curves of direct normal irradiance for models A, B, and C. (18)

_700I,-, 500. L

(19)

/(measured) = 0.9141i(calculated) + 28.507

(20)

model A

_ 300. ' ....

I ....

I ' ' '/>

/ ....

t

._

MODELA MOOELB

~. / I

- - " ,ooa c

//2..,/

__

/.//"

~15o.L

!

""/_LJ

v

o

//:~/

/

//

/

_

f

...... I......... I........

Fig. 12. Comparison of the best-fit curve of global horizontal irradiance for models A, B, and C.

l(measured) = 1.0021(calculated) - 2.643

(21)

/(measured) = 0.9385i(calculated) + 20.224

(22)

model C

The eqns (17) to (19) are plotted in Fig. 11 and show clearly that model B is superior to the other two models. Similar equations for the global irradiance are

"-~ ~250

-

200. 300. 400. 500. 6 0 0 . 700. 800. 900. t000 GLOBAL HORIZONTALIRRADIANCE (calculatedJ I (W/M2)

model C

[ ....

/I

model B

Ia(measured) = 0.8084id(calculated) + 8.618

I ....

7

_:

~Goo.i-

model B

....

i

i.:'/

///

~200 .~.,,,,.,I ........ I.... ~ . 1 .

I .........

la(measured) = 0.84941a(calculated) + 7.782

/.,.--

/.//

F

!'°°L!

/y

NOOEL A

_

_ _ . MODELB

~800. ~

F

=soo_

_

.... ....

/

-

4. CONCLUSIONS

"

. . . .

Figure 12 contains the plots of eqns (20) to (22). Obviously, model B is the most accurate, and model C is more accurate than model A. The study of models accuracy with respect to the aerosols turbidity and water vapor is not significant. As for the models, we did not observed a deviation between low and high turbidity, and low and high water vapor. We hope the above regression equations will help the user of any one of the three models to better predict the results. The present comparisons are based on data from one station only. It will be useful to carry out similar comparisons with data from a number of other locations before casting a definite judgment about the superiority of one model over the others.

_

.

In this study, three parameterization models for direct, diffuse, and global solar irradianee have been compared with measurements from one station. Model C is most accurate for prediction of direct normal Jrradiance, and model B is most accurate for prediction of diffuse and global horizontal irradiance. In addition, regression equations are provided to help the user of any one of the three models for better prediction of solar irradiance.

0 50.0 100. t50. 200. 250. 300. DIFFUSE HORIZONTALIRRAOZANCE (calculateO) r d ~/H~ Fig. ] I . Comparison o f the best-fit curves o f diffuse horizontal irradiance for models A, B, and C.

NOMENCLATURE

Eo eccentricity correction factor of the orbit (dimensionless)

Clear-sky insolation models i /~ I~ 10~ /:,¢ ma N ct ct, etw 13 h p -r ra "ru "r~ % "to "rw ~o0

total global irradiance on a horizontal surface (W m-:) total diffuse irradiance on a horizontal surface (W m-") direct normal irradiance (W m -2) solar spectral irradiance at mean sun-earth distance over small ban centered at k(w m -z p.m-t). solar constant: 1367 W m-" air mass at actual pressure (dimensionless) total number of data points wavelength exponent in Angstr0m's turbidity equation (dimensionless) fraction of the incident energy absorbed by a particular atmospheric constituent (dimensionless) fraction of the incident energy absorbed by water vaI~or (dimensionless) AngstrOm turbidity parameter (dimensionless) wavelength (l~m); as a subscript, h indicates monochromaticity albedo of ground (or ground cover) (dimensionless) transmittance, fraction of the incident energy transmitted by the atmosphere (dimensionless) fraction of the incident energy transmitted by aerosols (dimensionless) fraction of the incident energy transmitted after scattering effects of aerosols (dimensionless) fraction of the incident energy transmitted after absorption by uniformly mixed gases (dimensionless) fraction of the incident energy transmitted after scattering by clean, dry air molecules (dimensionless) fraction of the incident energy transmitted after absorption by ozone (dimensionless) fraction of the incident energy transmitted after absorption by water vapor (dimensionless) single-scattering albedo, fraction of the incident energy scattered to total attenuation by aerosols (dimensionless)

Acknowledgments--Financial supports of the Natural Sciences and Engineering Research Council of Canada, and of the Centre National de la Recherche Scientifique of France are gratefully acknowledged. The facilities provided by the Universit~ de Corse and the sabbatical leave granted to the third author (M.I.) by the University of British Columbia are highly appreciated. We are especially grateful to Robert Coudert and his colleagues of the Radiometric Center of Carpentras for liberally providing us the necessary data. Thanks are also due to Paul Ottavi for transferring the data to the computer. REFERENCES

1. ASHRAE Applications Handbook, Chapter 58, Solar energy utilization for heating and cooling. ASHRAE, New York (1978).

279

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