~ Pergamon Renewable Eneryy, Vol. 11, No. 2, pp. 149 152, 1997

© 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain

P I I : S 0 9 6 0 - 1 4 8 1 (97)00003-7 0960-1481/97 $17.00 + 0.00

M O D E L L I N G D A Y L I G H T ON INCLINED S U R F A C E S F O R A P P L I C A T I O N S TO

D A Y L I G H T CONSCIOUS A R C H I T E C T U R E

LUIS ROBLEDO Departamento de Sistemas Inteligentes Aplicados, E.U. Informfitica,

Universidad Polit6cnica de Madrid, Ctra de Valencia km 7, 28031 Madrid, Spain

and

ALFONSO SOLER Departamento de Fisica e Instalaciones Aplicadas, E.T.S. de Arquitectura,

Universidad Polit6nica de Madrid, Avda Juan de Herrera 4, 28040, Madrid, Spain

(Received 20 November 1996 ; accepted 6 December 1996)

Abs t rac t - - In the present work the circumsolar and point source versions of the Perez model have been evaluated for south facing vertical surfaces at Madrid. Different data sets have been used. The coefficients F,j in the model have been determined using joint experimental data obtained for vertical planes facing N, E, S and W, and also with data for just the south vertical planes. Different values of the half angle ~ corresponding to the circumsolar region have been used and the most simple version of the model is recommended. © 1997 Elsevier Science Ltd.

INTRODUCTION

One way energy can be saved in buildings is through better use of daylight. Design of windows, use of innovative daylighting systems and prediction of the savings from new forms of electric lighting control are important in relation to this aim. However, to quantify the energy effects of daylight or to simply estimate daylight illuminances in different parts of a room, knowledge of illuminances received at external surfaces with different slopes is necessary. One way to estimate these values is to convert available or estimated radiation data into external horizontal illuminances using a luminous efficacy model, and then to estimate illuminances at inclined surfaces from illuminances at horizontal surfaces. In recent years illuminance measurements have increased world wide, mainly through the International Daylight Measuring Programme (IDMP), launched in 1991 by the Com- mission Internationale de l 'Eclairage (Report no. IAE-SHCP-17E-2, International Energy Agency, 1994). Thus, direct modelling of illuminances on inclined surfaces from illu- minances on horizontal surfaces is now becoming a more realistic approach. There are

149

150 L. ROBLEDO and A. SOLER

different models to predict the illuminance on an inclined surface. The chosen model must be easy to use and should predict illuminances with the highest possible accuracy. The model proposed by Perez et al. [1], mostly tested for solar irradiances instead of illuminances, is usually considered as one of the more accurate. However, in its original formulation the model is rather complex and difficult to use. A simplified version of the model is available [2], which considers the circumsolar region characterized by a half angle ~ and an infini- tesimal horizon band with an elevation of 0 °. The most easy to use being the point source version, which considers circumsolar radiation as coming from a point at the centre of the sun's disk. To use any of the versions one must know a set of coefficients obtained from illuminance data from horizontal and inclined surfaces.

In the present work we have statistically assessed with different data sets for different values of a, the simplified version for vertical south facing surfaces at Madrid, Spain, where continuous measurements of illuminances are routinely obtained in a General Class Station in the framework of the IDMP.

MODELS

In general, global illuminance on an inclined plane is obtained from its direct and diffuse components. Direct illuminance can be calculated from the difference between global and diffuse illuminance on a horizontal surface, or from measurements of direct normal illuminance.

The hourly diffuse illuminance on an inclined surface with a slope fl is obtained in the simplified Perez model from the following equation :

E~ = Eh[0.5(1 --El )(1 + cos fl) + (a/b)F~ + F2sinfl] (1)

where Eh is the horizontal illuminance, and F1 and F2 are coefficients which respectively express the degree of anisotropy of the circumsolar and the horizon regions. These coefficients show a dependence on the parameters that define the sky conditions :

(a) The zenithal angle, Z. (b) The clearness index e defined through :

= (Eh + En)/Eh. (2)

A modified clearness index, ~', has been proposed :

e' = [(Eh +En)/(Eh +kZ3)]/[1 + k Z 3] (3)

En being the direct normal illuminance.

(c) The sky's brightness A defined by :

A = Ehm/Eo (4)

where Eo is the mean extraterrestrial normal illuminance and m the optical air mass.

The model considers a set of categories for e or e', and for each of them Fl and F2 are given as :

F l = F I I + F 1 2 , A + F I 3 , Z

f 2 = f21 +F22,A-F-F23,Z. (5)

Modelling daylight on inclined surfaces 151

The coefficients F~ corresponding to each category can be obtained by fitting the measured data.

In the point source version (~ = 0 °) a and b are given as

a = max(0,cos0) ; b = max(0.087,cosZ) (6)

0 being the incidence angle of the sun on the surface and Z the zenithal angle. In the circumsolar version of the simplified Perez model (~ # 0 °) the corresponding

expressions for a and b are the same as in the first version of the model [1]. They are more complex than those given in eq. (6) and the utilization of the model results more arduous. There is a general agreement that the circumsolar version is more accurate than the point source version. In the available literature, coefficients F, i in eq. (1) are calculated using joint data for four orientations, usually N, E, S and W, and/~ = 90 °.

In the present work the circumsolar and point source versions have been evaluated for exclusively south facing vertical surfaces as follows :

• M 1. Point source version with coefficients obtained by Perez using data for all orien- tations and some U.S. stations [3].

• M2. Point source version with coefficients obtained for Madrid with data for all orientations.

• M3. Point source version with coefficients developed for Madrid with data for a south orientaUon.

• M4. Point source version with two sets of coefficients obtained for Madrid for a south orientaUon, but separating the data for a/b v ~ 0 and the data for a/b = 0. (Cor- responding, respectively, to the vertical plane, respectively, seeing or not the centre of the sun.) A detailed justification of this modification of the point source version is given in Robledo and Soler [4].

• M5-M9. Circumsolar simplified Perez model with coefficients obtained for a south orientation and ~ taking, respectively, values 7.5 °, 15 °, 20 °, 25 ° and 35 °.

• M 10. Circumsolar version for ~ = 25 ° with coefficients obtained using jointly obtained data for the four vertical planes facing N, E, S and W.

M E T H O D S

Experimental data consist of mean hourly values of diffuse and global illuminance on a horizontal surface, and global illuminance on vertical surfaces facing N, E, S and W. The data used, obtained in the flat roof of the Escuela T6chica Superior de Arquitectura in Madrid (40.3 '~ N, 4.4 ° W), belong to the period August 1992-September 1993. A complete description of the experiment is given in Ref. [4].

To statistically assess the model 's validity two different estimators have been calculated, the root mean square deviation (RMSD) and the mean bias deviation (MBD).

RESULTS

In Table 1 we give the relative values MBD and R M S D for M1 M10, for global illuminance.

Relating to the RMSD it is clear that the models which predict best are M3 M9, for which Fij values have been calculated using only data for the south facing surface. Of these,

152 L. ROBLEDO and A. SOLER

Table 1. Statistical performance of models for the south orientation at Madrid

Global illuminance

Model M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 RMSD 3.00 3.16 1.85 1.76 1.84 1.73 1.67 1.64 1.67 2.42 MBD -2.28 1.23 0.05 0.01 0.01 0.01 0.01 0.01 0.01 1.02

Average global illuminance 32.44 klux

M6 M9 (15 ° ~< ~ ~< 35 °) give a similar RMSD. However, M3 and M4, both point source (M4 obtained by separating the data for the surface seeing or not the sun), predict almost as well as M6-M9, and are much easier to use. Thus, when looking for an equilibrium between prediction accuracy and easy use, the most simple version of the Perez model, the point source, especially M4, can be recommended. It has to be emphasized that both M3 and M4 are developed using only data for the south plane.

CONCLUSIONS

The circumsolar and point source versions of the Perez model have been tested for Madrid. For the south facing surface, prediction accuracy is similar for both versions when the coefficients Fa used are determined only with data for this orientation. As a consequence, if we look for both prediction accuracy and easy use, the point source version can be recommended.

REFERENCES

1. Perez, R., Stewart, R., Arbogast, C., Seals, R. and Scott, J., An anisotropic hourly diffuse radiation model for sloping surfaces: Description, performance validation, site dependency evaluation. Solar Energy, 1986, 36, 487-497.

2. Perez, R., Seals, R., Ineichen, P., Stewart, R. and Menicucci, D., A new simplified version of the Perez diffuse irradiance model for tilted surfaces. Solar Energy, 1987, 39, 221-231.

3. Perez, R., Ineichen, P., Seals, R., Michalsky, J. and Stewart, R., Modeling daylight availability and irradiance components from direct and global irradiance. Solar Energy, 1990, 44, 271-289.

4. Robledo, L. and Soler, A., Point-source simplified illuminance model for vertical surfaces at Madrid: Dependence of model coefficients on surface orientation. Int. J. Lighting Res. Technol., 1996, 28(3) 141-148.