Modeling zenith luminance on Madrid partly cloudy skies from diffuse illuminance on a horizontal surface and solar altitude

Energy Conversion and Management 45 (2004) 2591–2601www.elsevier.com/locate/enconman

Modeling zenith luminance on Madrid partly cloudyskies from diffuse illuminance on a horizontal

surface and solar altitude

Alfonso Soler a,b,*, Kannam K. Gopinathan a,1

a Departamento de F�ısica, Escuela T�ecnica Superior de Arquitectura, Universidad Polit�ecnica de Madrid,

Avda. Juan de Herrera 4, 28040 Madrid, Spainb Facultad de Ciencias Ambientales, INEF, Universidad Polit�ecnica de Madrid, Avda. Mart�ın Fierro s/n,

28040 Madrid, Spain

Received 28 June 2003; accepted 27 September 2003

Available online 27 April 2004

Abstract

The present work deals with zenith luminance estimation for Madrid partly cloudy skies (cloud cover

from 1 to 7 oktas) from solar altitude and diffuse illuminance on a horizontal surface. Two different ap-

proaches are followed, as in our recent work for cloudless skies. The first approach has been to compare the

experimental zenith luminance values with the corresponding values estimated using the equations for the

five standards for partly cloudy skies recently proposed by Kittler et al. The dependence of zenith lumi-

nance on solar altitude is similar for both the best fit of the experimental data with a 5th degree polynomial

and the best fit of the estimated values obtained with the mentioned sky standards. The zenith luminance vs.solar altitude curve for sky standard IV.3 lies closest to the best fit of the experimental data. Next, a

multiple linear correlation based on the equation for sky standard IV.3 is developed, and excellent

agreement is found with the best fit of experimental data. Three empirical coefficients and the two constants

given in the equation for sky standard IV.3 are needed in this approach, and the equation of the model is

not a simple one.

The second approach has been to try different fits of the experimental data, looking for a simple equation

to be used as a model. As in our recent investigation for the cloudless skies case, it is observed that a simple

inverse equation relating the ratio between zenith luminance and diffuse illuminance to the cosine of thesolar altitude, containing only two empirically determined coefficients, gives an adequate fit to the experi-

mental data. From this equation, Lz can be estimated if diffuse illuminance on a horizontal surface and solar

* Corresponding author. Address: Facultad de Ciencias Ambientales, INEF, Universidad Polit�ecnica de Madrid,

Avda. Mart�ın Fierro s/n, 28040 Madrid, Spain. Tel.: +34-91-3366569; fax: +34-91-3366554.

E-mail address: [email protected] (A. Soler).1 Permanent address: Dean of Science, The National University of Lesotho, Roma, Lesotho (Southern Africa).

0196-8904/$ - see front matter � 2003 Published by Elsevier Ltd.

doi:10.1016/j.enconman.2003.09.034

mail to: [email protected]

2592 A. Soler, K.K. Gopinathan / Energy Conversion and Management 45 (2004) 2591–2601

altitude are known with approximately the same values for the statistical estimators as if the multiple linear

correlation obtained with the first approach is used. As a consequence, the proposed model is given by the

simple inverse equation.

� 2003 Published by Elsevier Ltd.

Keywords: Zenith luminance modeling; Partly cloudy skies; Sky standards; Diffuse illuminance on a horizontal surface

1. Introduction

Since 1991, continuous measurements of global and diffuse illuminance on horizontal surfaces,global illuminance on vertical surfaces, the direct normal or beam illuminance, the zenith lumi-nance and the sky luminance distribution are being obtained in different stations in the world inthe framework of the International Daylight Measurement Program (IDMP) [1].

The Spanish station located at the flat roof of the Escuela T�ecnica Superior de Arquitectura deMadrid (40�250N, 3�410W) joined the IDMP in 1993, and a number of papers have been producedrelating illuminance measurements and daylighting [2–26]. Concerning luminance measurements,the dependence of zenith luminance Lz on solar altitude a and the visible turbidity factor has beenreported for cloudless skies [27], and the dependence of Lz on a for overcast skies has been in-vestigated [28]. A seasonal analysis of zenith luminance data for all sky conditions and the de-pendence on the clearness index were reported [29]. Recently, Lz was modeled for cloudless skiesfrom diffuse illuminance on a horizontal surface and solar altitude [30], and the month–hourdistribution of zenith luminance and diffuse illuminance were studied [31]. The present work isdevoted to modeling Lz for partly cloudy skies. Two different approaches are followed, as in ourwork for cloudless skies. The first approach has consisted of comparing the experimental zenithluminance values with the corresponding values estimated with the equations for the five stan-dards for partly cloudy skies recently proposed by Kittler et al. [32] in order to know if anequation of the type proposed for the sky standards can be used to model the experimental data.In the second approach, different types of fits are tried for the experimental data in the process oflooking for a simple equation to be recommended for Lz modeling on Madrid partly cloudy skies.

2. Experimental data

The experimental data are mean values of Lz obtained each 15 min using a sampling time of 5 sin the period July 1997–June 1998. The sensor used is a LICOR illuminance sensor fitted with ablack collimator tube designed with a view angle of 10.5� following a standard procedure [33]. Thesensor was calibrated as both illuminance (each six months) and luminance sensor. A referenceilluminance sensor, circulated among participants in the IDMP had a calibration constant within0.2% of the one measured in Madrid. In the present work, the experimental Lz data for partlycloudy skies were used. Partly cloudy skies are defined here as those with a cloud cover of 1–7oktas. Visual observations routinely performed by trained observers of the Instituto Nacional deMeteorolog�ıa at the airport of Madrid, combined with visual observations and analysis of thesunshine fraction data routinely obtained at the station, were used.

A. Soler, K.K. Gopinathan / Energy Conversion and Management 45 (2004) 2591–2601 2593

Mean 15 min values of the diffuse illuminance Dv on a horizontal surface obtained at the stationfor the same period were also measured with a LICOR sensor, and have been used after correctionfor sky anisotropy [34]. For partly cloudy skies, a total of 8648 data were available for Lz. Thenumber of data sets with values for both Lz and Dv were 4769 due to different experimentalproblems. Only data sets with values for both Lz and Dv have been used in the present work.

The statistical estimators used are the determination coefficient r2, the MBE ¼P

iðyi � xiÞ=N ,and the RMSE ¼ ½

Piðyi � xiÞ2=N �1=2, where yi is the ith predicted value, xi is the ith measured

value and N is the number of values. As usual, the MBE and RMSE are given as percentage of themean of all the mean 15 min values.

3. Modeling zenith luminance on partly cloudy skies using the equations for sky standards

In Ref. [32] and papers cited therein, a new set of 15 standard skies has been developed. Thetask was undertaken partly because of the need for linking the whole spectrum of skies betweenthe already standardised CIE overcast and CIE clear skies to cover the real conditions. In thisrespect, more than a hundred selected cases, scanned in Berkeley, Tokyo and Sidney with com-mercial sky scanners, were analysed, tested and compared. Our interest in the present work is tofind the most suitable model to be proposed as the best fit for the partly cloudy skies dataavailable for Lz in Madrid, and in this respect, it seems logical to test if the equations used in Ref.[32] to characterise partly cloudy skies can be used for best fitting our experimental data.

The probable zenith luminance Lz under any sky condition is given in Ref. [32] as:

Lz ¼ ðDv=EvÞ ðBðsin aÞC=ðcos aÞDÞh

þ Eðsin aÞi

ð1Þ

where B, C, D and E are parameters with specific values for each standard, Dv is the diffuse il-luminance on a horizontal surface, a is the solar altitude and Ev is the extraterrestrial illuminancecomputed using a luminous solar constant of 133.9 Klux. The expression for Lz=Dv given by Eq.(1) was developed as an approximation to the actual ratio, as expressed using the gradationfunction and the indicatrix function [32]. While the indicatrix function takes into account the lawsfor diffusion of solar radiation by air molecules and aerosols (Rayleigh and Mie scattering), thegradation function is actually expressed as an exponential function of the inverse of the sine of thealtitude of the point of the sky considered [35].

The experimental values of Lz obtained for partly cloudy skies when Dv values are also availablehave been plotted against a in Fig. 1, together with the best fit of the data obtained with a 5thdegree polynomial. The corresponding best fits obtained with the coefficients B, C, D and E givenin Ref. [32] for the partly cloudy sky standards have been obtained and are also plotted in Fig. 1,assuming that the range of Dv values at the measuring site can be used for any of the five stan-dards. It can be noticed from the above figure that the graphs for the five standards and the onefor the best fit have similar shapes. However, one may have expected the best polynomial fit to belocated between the graphs for the standards. That is not the case. In fact, the five graphs for thestandards, although well within the region of the experimental data, are above the best fit of theexperimental values.

0 10 20 30 40 50 60 700.1

1

10

III-4

III-3

IV-2

IV-3

III-2

Measured

Zeni

th lu

min

ance

(kcd

/m 2 )

Solar altitude (degrees)

Fig. 1. Experimental values of Lz against a for partly cloudy skies when Dv is also available, best fit with a 5th degree

polynomial, and corresponding graphs from Eq. (1), for standard skies III.2, III.3, III.4, IV.2 and IV.3.


The graph for sky standard IV.3 in Fig. 1, named in Ref. [32] as partly cloudy with a brightercircumsolar effect, is the closest to the best fit with a 5th degree polynomial. In Fig. 2, one canverify the disagreement between the mean experimental values of Lz for partly cloudy skies cal-culated at every 5� intervals of a and the corresponding values predicted with sky standard IV.3.An attempt was, thus, made to fit best the experimental data with a multiple linear correlationrelated to Eq. (1) as formulated for sky standard IV.3:

Lz ¼ aþ bðDv=EvÞ½ðsin aÞC=ðcos aÞD� þ eðDv=EvÞ½sin a� ð2Þ
where a, b and e are empirical constants, and C and D have the same values given in Ref. [31] forsky standard IV.3. The values obtained for the constants in Eq. (2) are: a ¼ 0:0842, b ¼ 20:1238,e ¼ 17:0162, with r2 ¼ 0:8180. The mean values of Lz calculated at every 5� intervals of a with Eq.(2) for the given values of a, b and e are plotted in Fig. 3 together with the mean experimentalvalues of Lz, also calculated at every 5� intervals of a, and a good correspondence is observed. TheLz values estimated with the multiple linear correlation given by Eq. (2) are plotted against theexperimental data in Fig. 4.
4. Modeling the zenith luminance on partly cloudy skies from a simple equation

Although Eq. (2) can be used to model zenith luminance for partly cloudy skies, this equation isfar from being simple if we consider that a, b, e, C and D are needed to estimate Lz. Attempts weremade to obtain a simple, yet accurate model with a smaller number of constants. Experimental

0 10 20 30 40 50 60 700.1

1

10 IV-3

Measured

Mea

n ze

nith

lum

inan

ce (k

cd/m

2 )


Fig. 2. Mean experimental values of Lz for partly cloudy skies against a at every 5� intervals of a, calculated for

standard sky IV.3, and with experimental data when both Lz and Dv are available.

0 10 20 30 40 50 60 700.1

1

10

Eq. (2)

Measured

Mea

n ze

nith

lum

inan

ce (k

cd/m

2 )


Fig. 3. Mean experimental values of Lz against a for partly cloudy skies plotted at every 5� intervals of a, as calculatedwith Eq. (2), and with experimental data when both Lz and Dv are available.


0 5 10 15 20 250

5

10

15

20

25

Estim

ated

illu

min

ace

L Z (kc

d/m

2 )

Measured illuminance LZ (kcd/m 2)

Fig. 4. Values of Lz for partly cloudy skies estimated with Eq. (2) against measured values.


values of Lz=Dv were fitted against different simple functions of a. The highest coefficient ofcorrelation is obtained for Lz=Dv from a 5th degree polynomial in cos a:

Lz=Dv ¼ 1:3181� 5:2892 cos aþ 10:83284 cos2 a

� 11:40389 cos3 aþ 5:43567 cos4 a� 0:76134 cos5 a; r2 ¼ 0:7135 ð3Þ

The experimental values of Lz=Dv are plotted from Eq. (3) against cos a in Fig. 5. Obviously, thereis no interest in using Eq. (3) to estimate Lz, among other reasons because six constants areneeded, and it will be difficult to compare polynomials obtained at different locations just bysimply comparing the values of the empirically determined constants. Thus, values of Lz=Dv werefitted against cos a using different types of simple relations: power, exponential, logarithmic etc.On the basis of both the r value and the number of empirically determined constants, the fol-lowing inverse relation for Lz=Dv from cos a is chosen as the best model for local data when theprocedure outlined in this section is used:

Lz=Dv ¼ �0:0183þ ð0:1433= cos aÞ; r2 ¼ 0:709 ð4Þ

The value of r2 for Eq. (4) is almost the same as that obtained for Eq. (3). In agreement withthat observation, it is seen in Fig. 6 that the curves for Eqs. (3) and (4) are rather close. In Fig. 7,mean experimental values of Lz calculated at every 5� intervals of a have been plotted against solaraltitude together with the corresponding mean values obtained from Eq. (4), and it is observedthat the agreement between the experimental and the estimated values is rather good.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

L Z / D

V (kc

d·m

-2/k

lx)

Cos α

Fig. 5. Experimental values of Lz=Dv for partly cloudy skies against cos a and best fit with the 5th degree polynomial

given by Eq. (3).

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

Eq.(4)

Eq.(3)

L Z / D

V (kc

d·m-2

/klx

)

cos α

Fig. 6. Representation of the 5th degree polynomial for Lz=Dv from cos a given by Eq. (3) and the inverse relation for

Lz=Dv from cos a given by Eq. (4).


0 10 20 30 40 50 60 700.1

1

10

Eq.(4)

Measured

Mea

n ze

nith

lum

inan

ce (k

cd/m

2 )


Fig. 7. Mean values of Lz for partly cloudy skies calculated at every 5� intervals of a with Eq. (4), and with experimental

data when both Lz and Dv values are available.

0 5 10 15 20 250

5

10

15

20

25

Estim

ated

LZ (

kcd/

m 2 )

Measured LZ (kcd/m 2)

Fig. 8. Values of Lz for partly cloudy skies estimated with Eq. (4) against measured values.


Table 1

Values for r, the %MBE and the %RMSE when Lz is estimated with Eqs. (2) and (4)

r %MBE %RMSE

Lz estimated with Eq. (2) 0.9659 0.136 28.976

Lz estimated with Eq. (4) 0.9644 )1.712 29.315


Finally, values of Lz estimated with Eq. (4) and the measured values are compared in Fig. 8, andthe graph obtained is similar to the corresponding one in Fig. 4.

5. Statistical evaluation of models

The %MBE and %RMSE obtained when Lz values are estimated with the models given by Eqs.(2) and (4) are shown in Table 1, and it is observed that the statistical indicators have similarvalues for both equations, although the model given by Eq. (2) slightly over estimates and themodel given by Eq. (4) slightly under estimates Lz. The advantage of Eq. (4) over Eq. (2) is clearlythat only two constants are needed to fit the data. From this consideration, one can take Eq. (4) asthe best local model.

6. Conclusions

In the present work, two different approaches have been followed to model Lz for partly cloudyskies from solar altitude and diffuse illuminance on a horizontal surface. The first approach hasbeen to compare the experimental zenith luminance values with the corresponding values esti-mated with the equations for the five standards for partly cloudy skies recently proposed byKittler et al. [32] in order to know if an equation of the type proposed for the sky standards can beused to model the experimental data. A multiple linear correlation, given by Eq. (2), based on theequation for sky standard IV.3 has been developed, and an excellent agreement is found with thebest fit of the experimental data. However, three empirical coefficients and the two constants givenin the equation for sky standard IV.3 are needed in this approach, and the equation for the modelis not a simple one.

The second approach has been to try different fits of the experimental data, looking for a simpleequation to be used for this purpose, and it is observed that Eq. (4), a simple inverse equationrelating the ratio between zenith luminance and diffuse illuminance to the cosine of the solaraltitude, gives statistical indicators similar to those obtained with Eq. (2).

As mentioned elsewhere, Eqs. (1) and (2) are developed as an approximation to the actual ratioLz=Dv expressed using the gradation function and the indicatrix function, which takes into ac-count the laws for the diffusion of solar radiation by air molecules and aerosols. Thus, in someway, Eqs. (1) and (2) are related to the physical mechanisms involved in the complex absorptionprocesses of solar radiation taking place in the atmosphere. Eq. (4) obtained for the proposedmodel is just a best fit of the data but contains only two empirically determined coefficients, al-lowing easy use and comparison between similar equations obtained at different sites. As aconsequence, the proposed model is given by Eq. (4) in spite of its apparent lack of physicalsignificance.


Acknowledgements

The present work was made possible by the Spanish Government through research grant PB98-0763. Professor K.K. Gopinathan thanks the Spanish Ministry of Science and Technology forfinancial help for his sabbatical.

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Documents

Modeling zenith luminance on Madrid partly cloudy skies from diffuse illuminance on a horizontal surface and solar altitude