Renewable Energy 23 (2001) 49–55www.elsevier.nl/locate/renene

A study on zenith luminance on Madridovercast skies

A. Soler a,b,*, K.K. Gopinathana,c, S.T. Clarosa

a Departamento de Fı´sica, Escuela Te´cnica Superior de Arquitectura, UPM, Avda. Juan de Herrera 4,28040 Madrid, Spain

b Facultad de Ciencias Ambientales, UPM, Avda. de Martı´n Fierro s/n, 28040 Madrid, Spainc Permanent address: Department of Physics, The National University of Lesotho, Roma, Lesotho,

South Africa

Received 28 March 2000; accepted 19 July 2000

Abstract

The zenith luminance has been measured at Madrid for skies with a cloud cover of 7–8 oktasand experimental mean 15-min values ofLz have been obtained. Among the most accurate fitsto the data, a 2nd degree polynomial forLz against tan(a) is the most simple. When only datafor a cloud cover of 8 oktas are used a similar fit is obtained, but for a smaller coefficient ofcorrelation, due to the smaller amount of data.

A linear, second degree, or power dependence forLz against sin(a) obtained by otherresearchers, does not give the best fit for our data. 2000 Elsevier Science Ltd. All rightsreserved.

1. Introduction

Characterization of the luminous climate available at different sites is of interestfor a variety of purposes, such as to quantify possible energy savings from theadequate use of photoelectric controls [1], to evaluate daylighting systems [2], toestimate illuminance levels in rooms for design purposes [3], or to evaluate day-lighting software [4].

The overcast sky conditions are the most important ones for window design anddaylighting. In the lack of local data, the CIE model for the zenith luminance and

* Corresponding author. Tel.:+34-9133-66569; fax+34-9133-66554.E-mail address:asoler@corbu.aq.upm.es (A. Soler).

0960-1481/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S0960 -1481(00 )00158-0

50 A. Soler et al. / Renewable Energy 23 (2001) 49–55

the distribution of sky luminance, are often assumed as valid. Recent research inMadrid showed that a simple, graphic daylight calculation method, based on the CIEstandard for overcast sky does not work for this city [3].

As a part of ongoing research on natural light and daylighting at the EscuelaTecnica Superior de Arquitectura de Madrid (40° 259N, 3° 419W) [2,3,5–15], andfor the first time in Spain, the present work reports on the dependence of zenithluminanceLz on solar elevationa, for nearly overcast (cloud cover of 7 oktas) andovercast skies (cloud cover of 8 oktas). Work was also performed with only datafor overcast skies. Other possible independent variables as the diffuse to extraterres-trial ratio [16] will not be taken into account for the time being. Our interest here,is just to obtain the best fit for the dependence ofLz ona, in order to compare it withthose obtained by other researchers, who have useda as the independent variable.

2. Experimental data

The experimental data used are mean 15-min values ofLz obtained for a samplingtime of 5 s in the period July 1997–June 1998. The sensor used is a LICOR illumin-ance sensor fitted with a black collimator tube, designed for a view angle of 10.5°following a standard procedure [17]. The luminance sensor was calibrated at theInstituto de Optica of the CSIC a few days before starting the experiment, and earlyin January 1998. The whole unit was fixed on top of a semi-spherical dome. TheLz sensor was one of those used in a larger experiment intended to help in thecharacterization of the luminous climate at Madrid.

In the present work only values for nearly overcast and overcast skies are used(7–8 oktas), with a total of 2717 data. Fits ofLz againsta for only 8 oktas data,with a total of 807 values, give similar results to those for 7–8 oktas, but smallercoefficients of correlation were obtained due to the fact that less values are available.Combined 7/8 oktas data are used unless specified otherwise. When available, thediffuse over extraterrestrial ratio did not show specially high values for the 7 oktasdata. Visual observations of sky cloudiness, routinely performed at the airport ofMadrid by trained observers of the Instituto Nacional de Meteorologı´a, are used.Occasional observations were sometimes undertaken at the measuring site, and foundto be in agreement with the airport reports.

3. Zenith luminance data analysis

The experimental values ofLz are plotted againsta in Fig. 1 on a logarithmicscale. The data for clear skies used in a previous work [13] and the overcast skiesdata are plotted together in Fig. 2 for comparison. Attempts were made to fit theexperimental data againsta using various forms of functional equations. Simplemodels based on exponential, polynomial, power, logarithmic and linear equationswere tried, by considering various combinations ofLz, lg(Lz), a, tan(a), sin(a), etc.,in order to obtain the best fit for the experimentally measured data. By carrying out

51A. Soler et al. / Renewable Energy 23 (2001) 49–55

Fig. 1. Experimental values of zenith luminanceLz (15-min means) against solar elevationa, for skieswith cloudiness of 7–8 oktas.

Fig. 2. Experimental values of zenith luminanceLz against solar elevationa, for cloudless skies (I)and skies with cloud cover of 7–8 oktas (s).

such an analysis, it was observed that a 2nd degree polynomial connectingLz againsttan(a) gives the best correlation in terms of the correlation coefficient. The poly-nomial obtained and plotted in Fig. 3 is:

Lz520.1581110.6664tan(a)20.5940tan2(a) with r50.902 (1)

Best fits with 5th degree polynomials forLz againsta or sin(a), were found to predictthe Lz values with almost the same accuracy.

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Fig. 3. Best fits with a second degree polynomial of:Lz against tan(a) for skies with cloud cover of 7–8 oktas; mean values ofLz calculated at every 5° interval of a, against tan(a), for skies with a cloudcover of 7–8 oktas;Lz against tan(a) for skies with cloud cover of 8 oktas.

To fully establish the dependence ofLz on tan(a), mean values ofLz were calcu-lated at every 5° interval of a, and the vales are also plotted in Fig. 3. The bestcorrelation coefficient is also obtained for a best fit with a 2nd degree polynomialrelatingLz against tan(a), clearly supporting the validity of earlier observations. Thepolynomial obtained and also plotted in Fig. 3 is:

Lz520.0803110.5460tan(a)20.6364tan2(a) with r50.999 (2)

Also in Fig. 3, the best fit obtained with 7–8 oktas data is compared with thatobtained with 8 oktas data. Both curves are similar but the coefficient of correlationobtained when only data for a cloud cover of 8 oktas is smaller than the one corre-sponding to the best fit given by Eq. (1), obtained for 7–8 oktas data. The followingpolynomial fit is obtained for a cloud cover of 8 oktas:

Lz520.169919.2425tan(a)10.1250tan2(a) with r50.827 (3)

Different authors have attempted to fit the overcast data for zenith luminance in theform of a linear dependence ofLz on sin(a) [18], a 2nd order polynomial dependenceof Lz on sin(a) [19],or a power dependence ofLz on sin(a) [20]. Mean measureddata, calculated at every 5° interval of solar elevation are plotted in Fig. 4, togetherwith the fits given in [18–20].It is seen that the equations in [18–20] under predictthe experimental data. When a dependence on sin(a) is assumed as in the abovethree models, the following equations are obtained for the best fits with local data:

Lz523.2170120,5766sin(a) with r50.848 (4)

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Fig. 4. Mean measured data ofLz at every 5° interval of a, and calculatedLz from equations in refs[18–20].

Lz51.936525.2496sin(a)125.7494sin2(a) with r50.887 (5)

Lz514.5063(sina)1.2959with r50.871 (6)

The mean values ofLz calculated at every 5° interval of a are plotted in Fig. 5,along with the values obtained from Eqs. (4)–(6). In Fig. 5, it is observed that alinear, 2nd degree or power dependence ofLz against sin(a), does not give the best

Fig. 5. Mean measured data ofLz at every 5° interval ofa, and calculatedLz from Eqs. (4)–(6).

54 A. Soler et al. / Renewable Energy 23 (2001) 49–55

fit to our data. Similar results are obtained when mean values for a sky cover of 8oktas are used.

4. Conclusions

The zenith luminance has been measured at Madrid for skies with a cloud coverof 7–8 oktas and experimental mean 15-min values ofLz are obtained. A 2nd degreepolynomial of tan(a) againstLz gives the most accurate fit to the whole data. Whenonly data for a cloud cover of 8 oktas are used a similar fit is obtained, but for asmaller coefficient of correlation, due to the smaller amount of data. Though a 5thdegree polynomial forLz againsta or sin(a) can predict theLz values with almostthe same accuracy, there is no advantage or need to have a 5th degree equation.

A linear, 2nd degree or power dependence forLz against sin(a), presented byother researchers, does not give the best fit for our data.

Acknowledgements

The present work has been performed as a part of the project PB95-0037 financedby the Spanish Government through the DGICYT. We are grateful to the InstitutoNacional de Meteorologı´a for supplying the cloud data free of charge.

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