statistical assessment of a model for global illuminance on inclined surfaces from horizontal global...

Statistical assessment of a model for global illuminanceon inclined surfaces from horizontal global illuminance

Enrique Ruiz a, Alfonso Soler b,*, Luis Robledo c

a Facultad de Ciencias, Departamento de F�ıısica Aplicada, Universidad Aut�oonoma de Madrid, Cantoblanco, Madrid, Spainb Departamento de F�ıısica, Escuela T�eecnica Superior de Arquitectura, Universidad Polit�eecnica de Madrid,

Avda. Juan de Herrera 4, 28040 Madrid, Spainc Departamento de Sistemas Inteligentes Aplicados, E.U. de Inform�aatica, Universidad Polit�eecnica de Madrid,

Ctra. de Valencia Km 7, 28031 Madrid, Spain

Received 3 November 2000; accepted 19 March 2001

Abstract

Olmo et al. [Energy 24 (1999) 689] have recently proposed a simple model to estimate global irradianceon inclined planes, which only requires the horizontal global irradiance and the sun elevation and azimuthas input parameters. From now on, this model will be referred to as the Olmo model. Statistical assessmentof this model can be considered as important, taking into account that available models for estimation ofglobal irradiance or illuminance on an inclined surface require information of global, and direct or diffuseirradiance or illuminance on a horizontal surface. The version of the Olmo model for global illuminance istested in the present work using mean 15 min values of global illuminance obtained with 20 sensors ofdifferent slopes (zenith angles) and azimuths. The sensors were placed on a spherical dome located at one ofthe corners of the roof of the experimental site and ground shielded by black mat painted honeycombmaterial. Assuming a value of the honeycomb albedo of 0% values of the obtained RMSD go from about8% for surface slopes of 12� to about 30% for a vertical surface facing east. For a north facing verticalsurface, receiving mostly diffuse illuminance, a value of about 52% is obtained for the RMSD. Assuming avalue of the albedo of 5%, too high for our experimental set up, similar results are obtained. In general themodel over estimates global illuminance on inclined surfaces in Madrid, for experimental global illumi-nance values higher than about 60 klux. � 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Global illuminance modeling; Inclined surfaces; All sky conditions

Energy Conversion and Management 43 (2002) 693–708www.elsevier.com/locate/enconman

* Corresponding author. Tel.: +34-9-1336-6569; fax: +34-9-1336-6554.

E-mail address: [email protected] (A. Soler).

0196-8904/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0196-8904(01)00063-2

1. Introduction

There are a number of models available to estimate global irradiance or illuminance on inclinedsurfaces from irradiance or illuminance on a horizontal surface, such as Refs. [1–4]. These modelsrequire information of the global and the direct or the diffuse irradiance or illuminance on ahorizontal surface. Usually, the models are tested with data obtained for vertical surfaces, becausethe departure from horizontal surface values is the largest in this case. An example is the an-isotropic model by Perez [3], regarded as one of the most reliable to estimate both irradiance andilluminance on inclined surfaces, as confirmed, among others, by Robledo and Soler [5–8], Utrillasand Martinez Lozano [9] and Li and Lam [10] for vertical surfaces. However, if this model is used,a large number of site and orientation dependent coefficients have to be determined, even for thesimplest version [5–10].

In this context, the recent publication by Olmo et al. [11] of a simple model to estimate globalirradiance on inclined planes, which only requires the horizontal global irradiance and the sunelevation and azimuth as input parameters can be considered a priori as a daring proposal. Fromnow on, in the present work, we refer to the Olmo et al. model as the Olmo model. Obviously, ifsuch a model worked properly, allowing for accurate estimation of global radiation on inclinedplanes, its availability would imply a significant advance in this field of research.

We recently performed an assessment of the Olmo model using irradiance data for verticalsurfaces facing south, east, north and west [12]. Two main conclusions were drawn from theresults obtained. The first was that, in general, the model is not accurate enough to predict valuesof global irradiance on vertical surfaces. The second was that the model gives rather large errorsfor the surface facing north and, in general, when the vertical plane does not ‘‘see’’ the sun disk orwhen, for nearly overcast/overcast skies, no direct radiation is received on both vertical andhorizontal planes [12].

The general equations of a global irradiance model for inclined surfaces are of the same form asthe equations for a global illuminance model, as an illuminance model just relates to the visiblepart of the whole broadband solar spectrum [3]. If, in the Olmo model, global irradiance ischanged to global illuminance, the only extra changes that need to be made to obtain theequivalent illuminance model is to substitute for the clearness index and ground albedo for globalsolar radiation by the clearness index and ground albedo for the visible part of the spectrum. Theversion of the Olmo model for global illuminance on inclined surfaces is tested in the present workusing mean 15 min values of global illuminance obtained with 20 sensors of different elevationsand azimuths placed on a spherical dome and ground shielded by black mat painted honeycombmaterial.

2. Experimental data and statistical methods

The experimental illuminance data consist of two data sets. The first data set consists of meanhourly values of global illuminance measured on a horizontal surface and on vertical surfacesfacing north, east, south and west during the period June 1994–July 1995. The second data setincludes 15 min mean hourly values of global illuminance measured on a horizontal surface andon 16 surfaces of different orientations and slopes during the period January 2000–June 2000. All

694 E. Ruiz et al. / Energy Conversion and Management 43 (2002) 693–708

the data were obtained with LICOR illuminance sensors in the International Daylight Mea-surement Programme (IDMP) station placed at the unobstructed roof of the Escuela T�eecnicaSuperior de Arquitectura at Madrid (40�2500 N, 3�410 W). The sensors were calibrated every sixmonths and always a few days before the beginning of an experiment. The sensors used for theperiod January 2000–June 2000 were fixed on a semi-spherical dome placed at one of the cornersof the roof in 16 of the 145 available positions, distributed according to a pattern suggested by theCIE [13]. All the sensors were used with artificial horizons made of mat black painted honeycombmaterial, so that they could only ‘‘see’’ the honeycomb material and the open air. Shielding thefour vertical sensors from the ground is relatively easy, but in our case, shielding the 16 sensorscorresponding to the second experimental set up imposed some limits in the azimuths of thesensors. We are not aware of published research performed using simultaneous continuousmeasurements with such a number of fixed illuminance sensors. A Lambda 9 Perkin–Elmerspectrophotometer for a 4� incidence was used to measure the spectral reflectance of the black matpaint used. A mean value of 3.4% was measured for 400 < k < 700 nm;Dk ¼ 10 nm: Thus, theground reflected irradiance may be considered as very low, taking into account that the honey-comb had a depth of 16 cm.

Experimental data were not used for solar elevations lower than 5�. Solar elevation and azi-muth are taken at the middle of the corresponding period. The accuracy of the model wasdetermined by using, as statistical estimators, the MBD ¼

Pðyi � xiÞ=N and the RMSD ¼

fP

ðyi � xiÞ2=Ng1=2, where yi is the predicted ith value, xi the ith measured value and N the number

of values.

3. The Olmo model and the corresponding model for the estimation of global illuminance on inclinedsurfaces

The Olmo model was developed to estimate global irradiance on inclined surfaces using dataobtained at Granada [11]. The measurement system consisted in a pyranometer mounted on adevice with the ability to vary both the elevation (with 15� intervals) and the azimuth (with 45�intervals) of the inclined surface. Unavoidably ground reflected radiation from the underlyinguncolored concrete was measured as no shielding from the concrete was provided. The dataconsisted of 114 clear sky experiments distributed over all the year. To take the effect of groundreflected radiation into account the authors proposed a multiplying factor. Although only clearsky data were used, the model was proposed for all sky conditions. The authors have remarkedthe general applicability of their model, which could be used with instantaneous values, as well asaveraged measurements.

The model was tested with the Skyscan’834 data set [14] that contains slope irradiance mea-surements, although only for a surface oriented to the South with an elevation angle of 44�. Thesystem used to obtain the Skyscan’834 data set was shielded from the ground as in the case of theexperiments performed to obtain the data used in the present paper.

In the case of no reflection from the ground, the Olmo model estimates, for all sky conditions,the global irradiance incident on an inclined surface GW from the global irradiance incident on ahorizontal surface GH with the following equation:

E. Ruiz et al. / Energy Conversion and Management 43 (2002) 693–708 695

GW ¼ GH expð�ktðW2 � W2HÞÞ ð1Þ

where W, expressed in radians, is called the scattering [15] or incidence angle [16], that is the anglebetween the normal to the inclined surface and the sun–earth vector, taken from the center of thesun disk. kt is the global to extraterrestrial horizontal irradiance value, which takes into accountthe influence of general atmospheric conditions and clouds. WH is the angular distance, expressedin radians, between the normal direction to the horizontal plane and the sun’s position taken fromthe center of the sun’s disk, that is WH reduces to the solar zenith angle #S.

The scattering angle W is evaluated in Ref. [16] and other works from:

cos W ¼ sinð90�� #Þ sinð90�� #SÞ þ cosð90�� #Þ cosð90�� #SÞ cosðaS � aÞ ð2Þ

where # represents the zenith angle and a the azimuth, and the subscript S refers to the sunposition. The azimuth is zero for south, 90� for east, 180� for north and 270� for west. The zenithangle for an inclined surface is the slope of the surface, so that for a horizontal surface, # ¼ 0,sin# ¼ 0 and the scattering angle W is just the solar zenith angle #S. However, Olmo et al. [11] givethe following expression for cos W:

cos W ¼ sinð#Þ sinð#SÞ þ cosð#Þ cosð#SÞ cosðaS � aÞ ð3Þ

Eq. (3) is not the correct expression for the scattering angle. If, as in our case, we use Eq. (3)from Olmo et al., for vertical surfaces facing east, north, south and west, # ¼ 90�, cos# ¼ 0,sin# ¼ 1 and cos W ¼ sin#S, that is the scattering or incidence angle should be equal to the solarelevation, giving a model independent of the azimuth. Olmo et al. may have used the correctexpression, Eq. (2), for their calculations.

To allow for ground reflected radiation, Olmo et al. include a factor that considers the effect ofanisotropic radiation [11]. In the Olmo model, the mathematical expression for the factor Fc

depends only on one geometrical parameter, which is of course more convenient than using otherexpressions as the Temps and Coulson [17] formula that depends on three geometrical factors.

They propose [11]:

Fc ¼ 1 þ q sin2ðW=2Þ ð4Þ

where q is the albedo of the underlying surface, which they take as q ¼ 0:35 (35%) for the un-coloured concrete floor present in their experiment. In this way, the mathematical expression fortheir model, including ground reflected radiation, reads:

GW ¼ GH exp�� kt W2

�� W2

H

��Fc ð5Þ

The general equations of a global irradiance model for inclined surfaces have the same form asthe equations for a global illuminance model, as an illuminance model just relates to the visiblepart of the whole broadband solar spectrum. If, in the Olmo model, global irradiance is changedto global illuminance, the only extra changes that need to be made to obtain the equivalent il-luminance model are to substitute for the clearness index and albedo for global solar radiation bythe clearness index and albedo for the visible part of the spectrum. Thus, the version of the Olmomodel for global illuminance can be written as:


EW ¼ EH expð�kvðW2 � W2HÞÞFc ð6Þ

where EW and EH are the global illuminance on an inclined surface and on a horizontal surface,respectively, and kv is the luminous clearness index defined as the ratio of global to extraterrestrialilluminance, taking this last one as 133.8 klux [18]. The global illuminance on a horizontal surfacecan be measured or estimated with models for the luminous efficacy of global irradiance [19–21].Obviously, W, WH and Fc have the same meaning as in Eq. (5), Fc now being the surface albedo forthe visible radiation.

4. Performance assessment of the global illuminance model using data for 20 inclined surfaces

The model given by Eq. (6) has been tested using the available data sets indicated in the ex-perimental section. However, before presenting the results we briefly consider the equivalencebetween prediction of global irradiance with Eq. (5) and prediction of global irradiance with Eq.(6). We note that the irradiance and the illuminance data mentioned next were obtained fordifferent periods. The irradiance data were obtained for the period August 92–July 93, and theilluminance data for the period June 1994–July 1995, as already indicated in the experimental datasection.

In Fig. 1(a), we can see the global irradiance calculated for q ¼ 0 with the model given by Eq.(5) plotted versus the measured global irradiance for a south facing vertical surface [12]. In Fig.1(b), we can see the global illuminance calculated for q ¼ 0 with the model given by Eq. (6) versusthe measured global illuminance for a south facing vertical surface. Only data for kt < 0:85 andkv < 0:85 were used in Fig. 1(a) and (b) respectively, and these limits for the cloudiness index arekept throughout the paper. The similarity between Fig. 1(a) and 1(b) is evident. The same con-clusion is obtained when the corresponding figures for north facing vertical surfaces are com-pared, as shown in Fig. 2(a) and (b), for global irradiance and global illuminance respectively.

4.1. Performance assessment for 20 inclined surfaces, as specified by their zenith angle and azimuth

The zenith angle (the same as the slope) h and azimuth a of the 20 sensors used to test the globalilluminance model given by Eq. 6 and the number of data available for each sensor, are indicatedin Table 1. Plots for the global illuminance values calculated with Eq. (6) versus the measuredglobal illuminance have been obtained for each of the 20 sensors with q ¼ 0 in Eq. (6), and someof the corresponding plots are given in this work. Fig. 3 is for a sensor with h ¼ 12� and a ¼ 0�;Fig. 4 is for a sensor with h ¼ 24� and a ¼ 300�; Fig. 5 is for a sensor with h ¼ 36� and a ¼ 300�;Fig. 6 is for a sensor with h ¼ 60� and a ¼ 300�; Fig. 7 is for a sensor with h ¼ 72� and a ¼ 288�;Fig. 8 is for a sensor with h ¼ 84� and a ¼ 324�.

The values for the MBD and the RMSD obtained for the 20 sensors are given in % in Table 1.For q ¼ 0, we observe in Table 1 that a few MBD values are very large. Also, in Table 1 and forq ¼ 0, we observe that for vertical surfaces (h ¼ 90�), the RMSD ranges from about 23% for thesouth facing surface to about 30% and 33% for east and west facing surfaces, respectively, and to


about 52% for the north facing surface. This increase of the RMSD when going from south tonorth facing surfaces is similar to the one noted for the global irradiance model, more properlycalled Olmo model [12]. This increase is perhaps related to the increase when going from south tonorth in the number of cases when only diffuse illuminance/irradiance is received on the verticalsurface and diffuse plus direct illuminance/irradiance is received on the vertical surface. For thesurface facing north, which receives basically diffuse radiation the model shows a rather badperformance. In this respect, we note that the Skyscan’834 data set used to test the Olmo model

Fig. 1. (a) Global irradiance calculated with Eq. (5) for q ¼ 0 versus measured global irradiance for a vertical surface

facing south and (b) Global illuminance calculated with Eq. (6) for q ¼ 0 versus measured global illuminance for a

vertical surface facing south.


was obtained for surface facing south and this could favor lower RMSD values than for a northfacing surface.

For h ¼ 84�, the RMSD ranges from about 20% for a ¼ 0� to about 26% for a ¼ 264�. Forh ¼ 72�, the values of the RMSD are about 23–25% for different sensor azimuth angles, and forh ¼ 60�, the RMSD values are about 23–25%. The RMSD values show an important decreasewhen we consider the sensors with h ¼ 36�, and RMSD values of about 18% are obtained in thiscase. For h ¼ 24�, the RMSD value obtained is about 14% for the only sensor available. Finally,

Fig. 2. (a) Global irradiance calculated with Eq. (5) for q ¼ 0 versus measured global irradiance for a vertical surface

facing north and (b) Global illuminance calculated with Eq. (6) for q ¼ 0 versus measured global illuminance for a

vertical surface facing north.


Table 1

Values of the MBD and the RMSD in % for the 20 inclined surfaces, specified by sensor zenith angle h and azimuth aand values of q ¼ 0% and 5%

# (�) a (�) Number of data q ¼ 0% q ¼ 5%

MBD (%) RMSD (%) MBD (%) RMSD (%)

90 0 4248 5.74 23.28 7.01 23.46

90 90 4649 �9.83 29.90 �8.74 29.53

90 180 4649 �18.22 52.13 �15.88 52.10

90 270 4272 1.17 33.52 2.36 33.64

84 0 4050 2.01 20.15 2.84 20.06

84 264 4132 �6.96 26.45 �5.93 6.33

84 300 4038 3.88 22.88 4.74 23.07

84 324 4047 6.50 24.24 7.34 24.38

72 288 4041 3.32 25.56 4.18 25.78

72 300 3616 7.16 24.83 7.92 25.01

72 336 3510 3.37 23.90 4.05 23.79

60 0 4152 11.06 25.18 11.70 25.31

60 270 3508 �0.49 23.66 0.39 23.60

60 300 3989 6.88 23.47 7.58 23.60

36 270 3683 2.34 17.58 3.61 17.93

36 300 3712 7.14 18.10 7.99 18.52

24 330 4035 6.39 13.53 7.26 14.02

12 0 4190 2.38 6.65 3.38 7.18

12 120 4068 0.5 8.52 1.82 8.78

12 140 4130 �3.06 8.74 �1.78 8.38

Fig. 3. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for a surface of slope

12� and azimuth 0�.


for h ¼ 12� and different sensor azimuths the RMSD values are about 7–9%. It is also observed inTable 1 that if a value q ¼ 5% (0.05) is assumed, well above the albedo expected for the black mathoneycomb material, RMSD values close to those for q ¼ 0 are obtained. These results for theRMSD are not very suggestive in relation to the use of the model.






For the Skyscan’834 data set, a RMSD of 9.3% is obtained by Olmo et al. when the solarirradiance model given by Eq. (5) is used to estimate global irradiance values for a surface withh ¼ 44�, a ¼ 0 (facing south). For with h ¼ 36�, a ¼ 300�, a value of the RMSD of 18.3% isobtained in our case. The explanation for this difference may be related to the fact that the number






of measured values in the present work is larger than the number in the Skyscan’834 data set. Themodel over predicts for values of the global illuminance higher than about 60 klux.

4.2. Performance assessment for 20 inclined surfaces, as a function of their zenith angle

Graphs for the global illuminance calculated with Eq. (6) for q ¼ 0 versus the measured globalilluminance for the 20 sensors, grouped in relation with their zenith angles have also been



Fig. 9. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with a

slope of 12�.


obtained. Thus, and to give an example, Fig. 9 is for the sensors with slope h ¼ 12� and azimuthsa ¼ 0�, 120� and 140�, Fig. 10 is for h ¼ 36�, Fig. 11 is for h ¼ 60�, Fig. 12 is for h ¼ 72�, Fig. 13 isfor h ¼ 84�, and Fig. 14 is for h ¼ 90�. The MBD and RMSD obtained for each value of h, andq ¼ 0 are given in Table 2. It is observed that RMSD values higher than 24% are obtained forh > 60� and that maximum MBD values of about 6% result. In general, the model over predicts

Fig. 10. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with

a slope of 36�.


a slope of 60�.


for measured global illuminance values higher than about 60 klux. Values of the MBD and theRMSD for q ¼ 0:05 are similar to those for q ¼ 0. The MBD values show that the modeloverestimates, and the RMSD values are not favourable to the model tested.

Finally, Fig. 15 shows all the illuminance data calculated with Eq. (6) for q ¼ 0 with h ¼ 12�,24�, 36�, 60�, 72� and 84� versus the measured global illuminance values. Again it is observed thatthe model clearly over estimates for global illuminance values higher than about 60 klux.


a slope of 72�.


a slope of 84�.



a slope of 90�.

Table 2

Values of the MBD and the RMSE in % for all the data corresponding to each of the values of the zenith angle h and

values of q ¼ 0% and 5%

# (�) q ¼ 0% q ¼ 5%

MBD (%) RMSD (%) MBD (%) RMSD (%)

90 1.77 32.53 �0.45 32.84

84 1.95 24.53 2.89 24.57

72 4.55 24.90 5.31 24.97

60 6.80 24.57 7.51 24.67

36 5.09 18.13 6.12 18.53

24 6.39 13.53 7.26 14.02

12 0.12 7.87 1.30 8.04

Fig. 15. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the 20 sensors.


5. Conclusion

We have statistically assessed the illuminance version of the Olmo model (Eq. 6) using mean 15min values of global illuminance obtained with 20 sensors of different slopes and azimuths. As-suming a zero albedo, values of the RMSD obtained go from about 8% for surface slopes of 12� toabout 30% for high surface slopes and surfaces facing from east to south to west. For a northfacing vertical surface receiving mostly diffuse illuminance, a value of 52.13% is obtained for theRMSD. Similar results are obtained for an albedo of 0.05 (5%), clearly above the value expectedfrom the experimental set up. The high values of the RMSD obtained for medium and highsurface slopes do not make the model suitable for illuminance estimation.

Values of global illuminance up to 110 klux are obtained on a horizontal surface at the mea-suring site. The model clearly over estimates global illuminance for measured values higher thanabout 60 klux.

Acknowledgements

The present work has been made possible by financial support from the Ministerio de Ciencia yTecnologa through the project PB 98-0736.

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