Estimation of global illuminance on inclined surfaces for clear skies

Estimation of global illuminance on inclinedsurfaces for clear skies

Luis Robledo a,*, Alfonso Soler b

a Departamento de Sistemas Inteligentes Aplicados, E.U. Inform�aatica, Universidad Polit�eecnica de Madrid,

Ctra. de Valencia km 7, 28031 Madrid, Spainb Departamento de F�ıısica e Instalaciones Aplicadas, E.T.S.A.M. Universidad Polit�eecnica de Madrid,

Avda. Juan de Herrera 4, 28040 Madrid, Spain

Received 6 August 2002; accepted 11 December 2002

Abstract

In the present work, we have studied the luminous efficacy of global solar radiation incident on vertical

surfaces for clear skies (cloudless and rather clean skies) and mean hourly values of global solar radiation

(i.e. averaged over each hour). Luminous efficacy models similar to those previously obtained for a hori-

zontal surface have been developed using solar elevation as the independent variable. Thus, the hypothesis

that the slope of the surface does not influence the general formulation of the model has been assumed.

However, the final formulation of the models clearly depends on the insolation conditions of the inclined

surface and, more specifically, on whether direct radiation is or is not incident on it. As a consequence, twodifferent types of models have been proposed to take both insolation conditions into account. On the other

hand, global luminous efficacy models have also been obtained as the best fits of the experimental data.

Global illuminance has been estimated for the vertical surfaces considered, facing north, east, south and

west, with the models obtained as the best fits of the experimental data and the other two types of models

mentioned in the preceding paragraph. Comparison of the performance of models for global illuminance

estimation has validated the initial hypothesis that the slope of the surface does not influence the general

formulation of the model. The performances of the known Perez model and the proposed models are also

compared.� 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Global illuminance; Luminous efficacy; Vertical surfaces; Clear skies

Energy Conversion and Management 44 (2003) 2455–2469www.elsevier.com/locate/enconman

*Corresponding author. Tel.: +34-91-336-7854; fax: +34-91-336-7522.

E-mail address: [email protected] (L. Robledo).

0196-8904/03/$ - see front matter � 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0196-8904(03)00003-7

mail to: [email protected]

1. Introduction

The global illuminance on an inclined surface can be obtained as the sum of the direct anddiffuse illuminances. The direct illuminance is usually calculated by multiplying its value on ahorizontal surface by a slope dependent geometrical factor. The diffuse illuminance can be esti-mated from its horizontal value using different available models, i.e. Refs. [1–3]. However, globalilluminance can also be obtained from luminous efficacy models. If we consider that the luminousefficacy of global solar radiation is defined as the ratio between global illuminance and globalirradiance, then by multiplying the values obtained from the luminous efficacy model and thosefor the global irradiance, the global illuminance values can be estimated. Exploration of thismethodology is of great interest because of the small amount of stations, apparently about 50 allover the world, where global illuminance is measured on a routine basis, as compared to the largenumber of stations where global solar irradiance is monitored.A few papers are available concerning luminous efficacy models for inclined surfaces. In Ref.

[4], a luminous efficacy model for global solar irradiance on a horizontal surface has been used topredict the global illuminance on vertical surfaces. In Ref. [5], an approach to estimate the verticaloutdoor illuminance from computed vertical luminous efficacy based on measured horizontalsolar irradiance and illuminance data has been presented. Other works [6,7] have considered anapparent linear relation between global illuminance and global irradiance on inclined surfaces,thus taking a constant value for the corresponding luminous efficacy. This last approach can beconsidered rather acceptable from a practical point of view because it is a good approximation toreality, especially if, as in Ref. [7], different constant values are considered for clear skies (cloudlessand rather clean skies) and skies with different degrees of cloudiness. However, and from a morerigorous perspective, to consider the constancy of the luminous efficacy for global solar radiationdoes not produce a fully satisfactory explanation of the dependence of the efficacy on some of theimplied independent variables, such as the solar elevation. Also, the constant value assumption isless accurate the more the inclined surface is exposed to direct solar radiation and, particularly, inthe limiting case of south-facing surfaces.The present authors have developed luminous efficacy models for diffuse irradiance on vertical

surfaces for clear skies [8] and for all sky conditions [9] of the same type as those they had pre-viously reported for horizontal surfaces [10]. Basically, the same hypothesis used in Ref. [8], thatthe slope of the inclined surface does not influence the general formulation of the model, has beenassumed here, although specific considerations concerning the different conditions for the fourplanes have been taken into account. The present work is devoted to modeling the luminousefficacy of global solar radiation on vertical planes facing north, east, south and west in order topredict the global illuminance on these vertical planes.

2. Experimental data and sky characterization

The experimental data used are mean hourly values of global illuminance and irradiance (i.e.averaged over each hour) on vertical surfaces facing north, south, east and west and global anddiffuse irradiance on a horizontal surface. Measurements were performed at the flat roof of theTechnical School of Architecture in Madrid (40.4� N, 3.70�W). The LICOR illuminance sensors

2456 L. Robledo, A. Soler / Energy Conversion and Management 44 (2003) 2455–2469

used were calibrated each six months by representatives of the manufacturer following the ap-proved procedure. Irradiance values were obtained with Kipp & Zonen CM6B pyranometers thatwere calibrated once a year at the Instituto Nacional de Meteorolog�ııa in Madrid. Furthermore,the calibration of the illuminance sensors was tested against a reference standard circulated by theCommision Internationale de l�Eclairage (CIE) through stations in the International DaylightMeasurement Program.The data were obtained for the period June 1994–November 1995. The data for the period June

1994–May 1995 have been used to develop the models, while those for the period June 1995–November 1995 have been used to assess them statistically. The measurements with verticalsensors are routinely performed with artificial horizons made of matte black painted honeycombmaterial, so that the ground reflected solar radiation could be considered negligible.To establish what we considered as clear sky, we use the clearness index e0 and the brightness

index D, as defined by Perez et al. [1], e0 ¼ ½ðEdh þ IÞ=Edh þ kZ3�=ð1þ kZ3Þ, where Edh is the diffuseirradiance on a horizontal surface, I the direct normal irradiance calculated from the values ofglobal and diffuse horizontal irradiance, Z the solar zenith angle and k a constant equal to 1.041for Z in radians, and D ¼ Edhm=Io, m being the optical air mass and Io the extraterrestrial irra-diance. We have considered clear skies when e0 > 5:0 and D < 0:12, as evaluated from the solarradiation data.The accuracy of the models was determined using as statistical indicators, the coefficient of

correlation r, the mean bias error ðMBEÞ ¼ Rðyi � xiÞ=N and the root mean square errorðRMSEÞ ¼ ½Rðyi � xiÞ2=N �1=2, xi and yi being the ith predicted and the ith measured values, re-spectively, and N the number of values. Both the MBE and the RMSE are expressed as per-centages of the corresponding mean values.

3. Methodology

In the last years, we have performed a complete study of the luminous efficacy of global, directand diffuse solar radiation on horizontal surfaces, developing luminous efficacy models [10–14].The method used consisted in obtaining equations to model both the illuminance and irradianceand deducing from the ratio of the developed models the corresponding luminous efficacy model.The luminous efficacy models obtained in this way are mathematically coherent with the corres-ponding illuminance and irradiance models, contrary to the usual polynomial models obtainedjust by fitting the experimental luminous efficacy data. It is difficult to interpret polynomialmodels or to extrapolate them to locations different from those used to develop them, but this isnot the case when the mentioned method for luminous efficacy modeling is used.It is difficult to implement this method when dealing with inclined surfaces because illuminance

and irradiance modeling for inclined surfaces is much more complex than for horizontal surfaces.When modeling the luminous efficacy of diffuse solar radiation for vertical surfaces [8], we in-troduced the hypothesis, based on the observation of available graphs, that the slope of the in-clined surface influenced both the diffuse illuminance and diffuse irradiance in the same way, atleast qualitatively. So that, if we consider that each of these quantities ‘‘experiences’’ the verti-cality of the surface in the same way, we can suppose that their ratio, that is the luminous efficacy,

L. Robledo, A. Soler / Energy Conversion and Management 44 (2003) 2455–2469 2457

is not going to exhibit the effect of the verticality in relation to the form of the luminous efficacymodel but only in relation to the model coefficients.However, it is not easy to use the corresponding hypothesis to model the luminous efficacy of

the global solar irradiance for vertical surfaces. The reason is that considering global solar ra-diation as the sum of its direct and diffuse components, while for clear skies a horizontal surface isalways receiving direct radiation, its main component in this case, the same is not certain for avertical surface because it will receive more or less direct insolation depending on the orientation.While the south facing vertical plane receives direct radiation most of the time, the north facingvertical plane receives little direct radiation, and an intermediate situation is obtained for the eastand west facing vertical planes.From the preceding arguments, the south facing vertical surface is the most similar to a hori-

zontal surface, and our hypothesis that verticality does not change the form of the equation forthe model of the luminous efficacy of global solar radiation for a horizontal surface would be validin this case. In this respect, one can observe that graph (a) of Fig. 1 for the luminous efficacy ofglobal solar radiation for clear skies, Kgclv vs. solar altitude a, is very similar for the south facingsurface to the corresponding graph for a horizontal surface [13]. In the mentioned graph (a) ofFig. 1, one can see that Kgclv increases with solar elevation for low values of a up to a ¼ 30� or 35�,seems to get stabilized in an approximately constant value for a up to about 60� and then shows acertain tendency to diminish for higher values of a. This is more clearly observed in graph (b) inFig. 1 where mean Kgclv values, calculated at every 2.5� interval of solar altitude, have beenrepresented vs. the mean value of a for each interval. We can observe in the mentioned graph (b)that there is a zone of maximum luminous efficacy at about the interval 35� < a < 55�, as for ahorizontal surface [13]. However, for the horizontal surface, the effect is somewhat more pro-nounced than for the south facing surface. This is a logical result because for high values of solaraltitude, we have already associated the tendency for Kgclv to diminish for increasing values of awith the amount of direct solar radiation incident on the surface, which, for clear skies, depends

Fig. 1. Clear sky global luminous efficacy plotted against solar altitude for the south vertical plane: (a) all measured

data; (b) mean measured data calculated at every 2.5� interval of a.


on the atmospheric composition and geometrical factors, such as solar altitude and the incidenceangle of the sun on the surface, h. In relation with h, a vertical surface will receive, for high valuesof a, a less proportion of direct radiation than a horizontal one, resulting in a smaller curvatureeffect in the Kgclv vs. a graph. Thus, in the present case, if in a first approximation, we do notconsider this effect and assume that Kgclv reaches a constant value for a certain solar elevation,there is no risk of making important errors. However, and to be rigorous, when establishing amathematical model for the luminous efficacy of global solar radiation for the vertical surface, wemust take the effect into account. In Ref. [13], for clear skies and a horizontal surface, themaximum in the luminous efficacy of global solar radiation vs. solar elevation graph was justifiedwith an exponential term in the equation of the model, and in Ref. [14], the exponential term wasrelated to the proportion of direct radiation incident on the surface. So, the luminous efficacymodel for global radiation on a horizontal surface Kgclh given by:

Kgclh ¼ Aðsin aÞBe�Ca ð1Þwhere a is in degrees, as for all other equations in the paper, and A, B and C are empirical co-efficients, can be adapted, in principle, to a vertical south facing surface by simply optimizing thefunction given in Eq. (1), using the corresponding experimental data. Coefficient C in Eq. (1) willhave, of course, less influence for the vertical than for the horizontal surface, and if, from apractical point of view, we do not take it into account, Eq. (1) will be reduced to:

Kgclv ¼ Aðsin aÞB ð2ÞIn preceding works, where the study of the luminous efficacy of direct and global solar radi-

ation on a horizontal surface for clear skies was undertaken [11,13], we found that from a merelystatistical point of view, when the model given by Eq. (2) was used for the prediction of illumi-nance, it gave deviations not very different from those offered by Eq. (1). However, the modelgiven by Eq. (2) did not justify, from a mathematical point of view, the descent of the luminousefficacy with solar elevation for values of a higher than about 60�. In the case of the south facingvertical plane, one can expect that the difference between the statistical deviations produced whenusing the two models to estimate the global illuminance will be still less.Contrary to the south facing plane, the north facing plane can only receive a small amount of

direct solar radiation, and so, for clear skies, the luminous efficacy model for diffuse solar radi-ation on vertical surfaces [8] with an equation of the same form as Eq. (2) can be assumed, inprinciple, as valid for this plane. In fact, the graph for Kgclv vs. a in Fig. 2, obtained from data forthe north facing plane, is practically coincident with the one obtained in Ref. [8] for the sameplane. Also, we note that both are similar to the graph for Kdclh vs. a, Kdclh being the luminousefficacy for diffuse solar radiation on a horizontal surface [10].For east and west facing surfaces, we get an intermediate situation between the north and south

facing ones in relation with the possibility of receiving direct radiation. In Fig. 3, we represent theluminous efficacy for global radiation on the west facing plane as a function of solar altitude. Asimilar dependence with a is obtained for the east facing surface. There is obviously a geometricalsymmetry between the east and west facing vertical surfaces. For the same value of a, h will takethe same values when the surfaces receive direct radiation. Often, if clear skies are available in themorning, they persist in the evening, and in general, we can consider that the amounts of solarenergy incident on both planes will be similar, with small differences depending on changes in


atmospheric composition. In Fig. 4, we represent the Kgclv values obtained with all the data forboth surfaces vs. a. It can be observed that there is a clear similarity between the graph in Fig. 4and the one shown in Fig. 3 for the west facing surface. Taking this into account, the data for eastand west facing surfaces have been grouped in one set to obtain the luminous efficacy models thatwill allow estimation of the global illuminance received on them. Considering the specific char-acteristics of these surfaces, it is not possible to suggest a priori which of the models, as given by

Fig. 2. Clear sky global luminous efficacy plotted against solar altitude for the north vertical plane.

Fig. 3. Clear sky global luminous efficacy plotted against solar altitude for the west vertical plane.


Eqs. (1) or (2), will be the most convenient for the grouped data. Evidently, what seems mostreasonable is to separate the data in two sets, for the sunlit or shadowed surfaces, that is receivingor not receiving direct radiation. Graphs (a) and (b) in Fig. 5 show the values for Kgclv vs. a for thetwo mentioned data sets on the west facing surface. A clear difference between both graphs can beobserved. When the surface is not seeing the sun, Fig. 5(a), we can see a cloud of points with asimilar tendency to that obtained for the north plane (Fig. 2), with Kgclv diminishing for increasingvalues of a. On the contrary, when the surface receives direct solar radiation, Fig. 5(b), the cloud

Fig. 4. Clear sky global luminous efficacy plotted against solar altitude for the east west vertical planes.

Fig. 5. Clear sky global luminous efficacy against solar altitude for the west vertical plane: (a) shadowed surface;

(b) sunlit surface.


of points shows a similar tendency to that obtained for the south facing plane. In Fig. 6(a) and (b),the graphs corresponding to those shown in Fig. 5 are given for the grouped data obtained for theeast and west facing planes, and a great coincidence is observed between both figures. Apparently,no other report on this behavior is available in the literature.In the present work and in order to check the influence on the statistics of including or not

including the exponential term in Eq. (1), the models given by Eqs. (1) and (2) have been obtainedfor the north, south and east west facing surfaces. In this last case, also by separating the data intwo groups, the models have been obtained for the sunlit or shadowed surface.A comparison is also undertaken between the statistical performance of the proposed models

and that of the models obtained by best fitting the Kgclv values vs. a, using a stepwise procedure.The statistical performance of the proposed models is also compared with that of the well knownand much more complex model by Perez.

4. Models and statistical assessment

4.1. Models obtained without data separation

For the rest of the paper, the models given by Eqs. (1) and (2) will be referred to as models 1and 2, respectively. To obtain the coefficients of these models for each vertical surface, we opti-mize Eqs. (1) and (2) using regression analysis with all the luminous efficacy values. In Tables 1and 2, one can see the coefficients for each model and plane and the correspondent coefficients ofcorrelation. In Table 2, one can observe that for the south facing vertical plane, the model co-efficients considerably differ from those for the other orientations. On the other hand, the coef-ficient of correlation for the south facing vertical plane is rather lower than those for the other

Fig. 6. Clear sky global luminous efficacy against solar altitude for the east west vertical planes: (a) shadowed surface;

(b) sunlit surface.


surfaces. This is in agreement with the fact that model 2 does not include an exponential term asdoes model 1.

4.2. Models obtained by using stepwise regressions

Using a stepwise regression for Kgclv vs. a for all the vertical planes, introducing as independentvariables a, ln a, sin a, lnðsin aÞ and powers of a up to the fifth degree, the following models areobtained:

North : Kgclv ¼ 128:76� 0:512a; r ¼ 0:602 ð3Þ

East–west : Kgclv ¼ 136:21� 0:467a; r ¼ 0:531 ð4Þ

South : Kgclv ¼ 132:75ðsin aÞ0:130e�0:0022a r ¼ 0:508 ð5Þ

Eq. (5), obtained with the stepwise procedure, corresponds to that for model 1, and for the restof the planes, simple luminous efficacy models of the linear type result. As a consequence, ourhypothesis concerning the validity of adapting the luminous efficacy models for a horizontalsurface to vertical surfaces is correct for the south facing surface. Statistical evaluation of all themodels with the data used for testing them will allow us to verify if this success can be extended tothe other orientations.

4.3. Statistical evaluation of models

In Table 3, one can see the %RMSE and the %MBE obtained when the global illuminance isestimated for the four vertical planes with models 1 and 2 using the empirical constants given inTables 1 and 2, and when the global illuminance is estimated with the models obtained by stepwiseregression for all the surfaces, given by Eqs. (3)–(5).We can see in Table 3 that the global illuminance for the south facing surface is estimated with

models 1 and 2 with an RMSE of about 5% and an MBE of about 4%. The estimation is slightly

Table 1

Coefficients for the model given by Eq. (1)

Plane A (lm/W) B C r

North 152.47 0.104 )0.0067 0.598

East–west 137.24 0.002 )0.0040 0.528

South 132.75 0.130 )0.0022 0.508

Table 2

Coefficients for the model given by Eq. (2)

Plane A (lm/W) B r

North 100.15 )0.199 0.581

East–west 106.29 )0.233 0.504

South 115.85 0.037 0.391


more favorable with model 2, in spite of the fact that we are proposing model 1 as the mostsuitable model for this surface. However, our proposal is still valid because, although in relationwith statistical results and from a practical point of view, model 2 can be used to estimate theglobal illuminance on a south facing vertical plane, it does not interpret in the same way as model1 the experimental data in the graph for Kgclv vs. a. In Fig. 7, the curves for models 1 and 2,obtained with all the experimental data for the south plane, Fig. 1(a), have been superimposed onthe mean experimental values for this plane, Fig. 1(b), and one can see that model 2 does notjustify the dependence of Kgclv on a neither for about a > 60� or a < 20�, but model 1 does justifythe observed dependence.For the north facing plane, model 2, considered as the most adequate for this orientation taking

into account the practical absence of direct radiation for a north facing surface, offers RMSEvalues 10.8% and 18.6% lower than those obtained for model 1 and Eq. (3), respectively. Also, theMBE obtained for model 2 is 23.3% and 57.4% lower than those for models 1 and Eq. (3), res-

Table 3

Statistical assessment of models without separating data

Plane Model 1 Model 2 Stepwise Average (klx)

RMSE MBE RMSE MBE RMSE MBE

North 9.3 )3.0 8.3 )2.3 10.2 )5.4 6.45

East 9.6 )2.9 9.0 )3.0 9.6 )2.5 26.43

South 5.1 )4.2 4.8 )4.0 5.1 )4.2 41.86

West 11.4 )1.9 11.6 )2.4 11.5 )1.5 27.74

Fig. 7. Clear sky global luminous efficacy plotted against solar altitude for the south vertical plane: mean measured

data calculated at every 2.5� interval of a and curves given by models 1 and 2.


pectively. However, models 1 and 2 show RMSE values lower than 10%, and for the model givenby Eq. (3) the RMSE is only slightly above this value.Taking into account that there is more equilibrium between the direct and diffuse radiation

incident on them, the east and west facing surfaces are the most conflicting ones regardingmodeling. For the east facing plane, model 2 works the best, although there are not large dif-ferences with the other two. For the west facing plane, the three models show similar statistics. Inorder to perform a better comparison between these last two models and taking into account thearguments given in the methodology section of the present work, the data for east and west facingsurfaces have been separated into two groups, for the surface receiving or not receiving directradiation, and models have been obtained for each of them.We can also mention that if the luminous efficacy model developed for a horizontal surface [13]

is used to obtain the global luminous efficacy on the south facing vertical plane, RMSE and MBEvalues of 11.1% and 8.0% are, respectively, obtained. This is pointing to the fact that the surfaceslope has a quantitative influence on the model, and in fact, the model coefficients are different forhorizontal and vertical surfaces. However, for the north facing plane, the model coefficients andstatistical results are practically coincident if instead of using the method developed here, Eq. (1)with the coefficients in Table 2, the luminous efficacy model developed in Ref. [8] for clear skies isused.

4.4. Models obtained for east and west facing surfaces by separating data in relation with insolation

conditions

In the graphs of Fig. 6, luminous efficacy values for global radiation and clear skies were givenfor the east and west facing shadowed or sunlit surfaces. Best fits of the experimental clouds ofpoints in the mentioned graphs result in the following models for the east and west facing surfaces:(a) Shadowed surfaces:

Model 1 : Kgclv ¼ 169:40ðsin aÞ0:083e�0:0091a; r ¼ 0:666 ð6Þ

Model 2 : Kgclv ¼ 102:89ðsin aÞ�0:245; r ¼ 0:618 ð7Þ

Stepwise regression : Kgclv ¼ 147:58� 0:719a; r ¼ 0:677 ð8Þ

(b) Sunlit surfaces:

Model 1 : Kgclv ¼ 135:70ðsin aÞ0:138e�0:0028a; r ¼ 0:422 ð9Þ

Model 2 : Kgclv ¼ 115:31ðsin aÞ�0:059; r ¼ 0:396 ð10Þ

Stepwise regression : Kgclv ¼ 135:70ðsin aÞ0:138e�0:0028a; r ¼ 0:422 ð11Þ

One can see that for shadowed surfaces, the model obtained using a stepwise regression offers alinear relation, while when the surfaces are sunlit, the best fit obtained with this method is co-incident with model 1. This was expected, taking into account that in this case, the east and westfacing planes will behave similarly to the south facing or the horizontal surfaces.


When the preceding models are used to estimate the global illuminance on vertical surfacesfacing east and west with the data reserved for the statistical evaluation, the results presented inTable 4 are obtained. In Table 4, one can observe that when the surface is shadowed, global il-luminance is obtained with more precision with model 2 than with model 1, model 2 giving a valueof the RMSE 6.7% lower for the east facing plane and 17.5% lower for the west facing plane thanmodel 1. The linear model obtained for the shadowed surface using stepwise regression offers verysimilar results to those obtained with model 1 and worse than those given by model 2, confirmingthe validity of the models proposed in the present work.When the surface is sunlit, the RMSE values obtained for model 1 are, for both surfaces,

somewhat lower than those offered by model 2, 7% lower for the east facing plane and 7.5% lowerfor the west facing plane. In this case, we have already observed that the stepwise regressionanalysis takes us back to model 1.It can be pointed out that all the models work better for the sunlit surface than for the

shadowed one. This observation can also be obtained from Table 3, where one can see that formodels 1 and 2, the RMSE values are clearly lower for the south facing surface, which receives thesun beam most of the time, than for the rest of the surfaces.We can still widen our considerations regarding the statistical analysis of data for the east and

west facing planes if we compare the results in Table 3 with those reported in Table 4, where wealso report the composite RMSE and MBE obtained using the specific models for shadowed orsunlit surfaces. When we compare the results in both tables, we observe that the prediction ac-curacy of global illuminance for east and west facing surfaces is better when specific models areused for each of the insolation conditions.

4.5. A comparison with the Perez model

We now refer to the statistical comparison between the developed models and a well-knownavailable model used to predict mean hourly illuminance on a vertical surface, the Perez model [1].In Table 5, we report the values of the %RMSE and %MBE obtained when this model is appliedto global illuminance estimation on vertical planes facing north, east, south and west for clearskies, using the 48 needed coefficients, specifically developed for Madrid [3].

Table 4

Statistical assessment of models separating data for shadowed and sunlit surface

Plane Model 1 Model 2 Stepwise Average (klx)

RMSE MBE RMSE MBE RMSE MBE

(a) East

Shadowed 10.5 )0.1 9.8 0.5 10.4 0.5 6.63

Sunlit 4.0 )3.6 4.3 )3.9 4.0 )3.6 46.67

All (composite) 9.0 )1.8 7.7 )1.6 8.0 )1.5 26.43

(b) West

Shadowed 9.7 )5.1 7.9 )4.5 9.3 )4.6 6.42

Sunlit 4.9 )3.7 5.3 )4.2 4.9 )3.7 50.49

All (composite) 7.8 )4.4 6.6 )4.4 7.5 )4.2 27.74


From the results in Tables 5 and 3, one can infer that the Perez model works appreciably betterthan the ones proposed in the present work for the south facing vertical plane. For the east facingsurface and from the results in Table 4, the Perez model shows similar results to those obtained forthe developed models and works clearly worse than them for the other two orientations, especiallyfor the north plane. However, this comparison is vitiated because, in fact, the Perez model esti-mates the diffuse illuminance on the vertical plane, and then the direct illuminance is added toobtain the global illuminance, the direct illuminance being obtained from horizontal measure-ments by using a geometrical factor. This additive method implies that the RMSE and MBE inklux obtained when estimating diffuse illuminance are the same as those for global illuminance.When the RMSE and MBE for global illuminance are divided by its mean value, the %RMSE and%MBE finally given for global illuminance estimation are much smaller than those for diffuseilluminance. An exception will be, of course, the north facing plane because, in this case, thedifference between global and diffuse illuminance will be small. Obviously, with the procedureadopted by Perez, the larger the direct radiation incident on the surface, the larger is the reductionof the relative value (%) of the statistical indicators. Even so, the %MBE for the west facing planeand the %RMSE, for the east and west facing planes, which receive important amounts of directradiation, are clearly larger when using the Perez model than if the proposed models are used.

5. Conclusions

In the present work, a study of the luminous efficacy of global solar radiation for clear skies onvertical surfaces facing north, east, south and west has been undertaken, and from the developedmodels, the global illuminance on vertical surfaces has been estimated. In Ref. [8] and in order tomodel the luminous efficacy of diffuse irradiance on vertical surfaces, we formulated the plausiblehypothesis, supported by the observation of different graphs, that the slope of the inclined surfacedoes not influence the general formulation of the model, although the model coefficients willdepend on surface slope. To translate this hypothesis into the modeling of global radiation forclear skies has required some considerations because of the existence of the direct component. Ahorizontal surface always receives direct radiation for clear skies, but this is not the case for avertical surface. From this perspective, we have argued that only the vertical surface facing southkeeps a certain parallelism with the horizontal surface because it receives direct radiation most ofthe time, although obviously with an inclination different from that of the horizontal surface. Thisamounts to a geometrical factor that will affect the model coefficients but not the model structure.Thus, for the south facing vertical surface, one can adapt the type of luminous efficacy modeldeveloped for global solar radiation on a horizontal surface [13], as given by Eq. (1).

Table 5

%RMSE and %MBE values obtained for vertical global illuminance by using the Perez illuminance model

Plane North East South West

RMSE 19.7 9.5 3.3 10.2

MBE 1.4 0.8 )0.2 5.4


For the north facing surface, we have the opposite situation because it receives very little directradiation. Thus, the line of thinking we are introducing in this work suggests that for the northfacing vertical surface, one can adapt the type of luminous efficacy model developed for diffusesolar radiation on a horizontal surface [8], as given by Eq. (2).For east and west facing vertical planes, we get intermediate situations between those for the

two above-mentioned planes, and it is difficult to decide on using one model or the other, that isEqs. (1) or (2). On the other hand, in a previous work [13], we observed that when used to estimateglobal illuminance, the model given by Eq. (2) offered values of the statistical estimators not verydifferent from those obtained with model 1 or with polynomial models. However, Eq. (2) did notjustify the shape of the variation of the luminous efficacy of global solar radiation with solarelevation, and the exponential factor which appears in the model given in Eq. (1) was introducedto model this shape.Taking the preceding comments into account, we have developed both types of models in the

present work, as given by Eqs. (1) and (2), for the vertical surfaces facing north, south and east–west and compared their statistical performance when used to estimate the global illuminancereceived on these planes. Table 3 shows that for the north facing plane, model 2 works better thanmodel 1, as we had foreseen. For the south facing surface, model 1 does not work better thanmodel 2 as we expected, but as observed in Fig. 7, it gives a more adequate fit to the observeddependence of the luminous efficacy on solar elevation for low and high values of a.Relating east and west facing surfaces, models 1 and 2 and a third model obtained by a stepwise

regression analysis have been developed, without reaching a clear conclusion concerning which ofthem is the most adequate. This is due, without any doubt, to the fact that these two planes arereceiving important amounts of both direct and diffuse radiation. Taking this into account, thedata have been separated for further analysis into two groups for sunlit or shadowed surfaces, andnew models have been developed for each of the two situations. These new models have beenstatistically evaluated in Table 4 for east and west facing planes, and the following conclusionshave been obtained:

(a) Model 2 works better than model 1 when the surfaces are shadowed. When the surfaces aresunlit, model 1 offers better statistics than model 2, although with a smaller difference than inthe preceding case.

(b) Both models 1 and 2 are more accurate when the surface receives direct radiation than when itdoes not see the sun beam. This can also be observed in Table 3 if the results for the north andsouth facing planes are compared.

(c) From the results in Tables 4 and 5, the use of specific models for the situations when the sur-face receives or does not receive the sun beam is recommended.

Together with models 1 and 2, functional models obtained by fitting the data with a stepwiseprocedure have been developed. These models do not perform better than models 1 and 2 buthave been useful to reaffirm the validity of the proposed models.We conclude by stating that the values of the statistical estimators obtained when global il-

luminance is calculated for north, east, south and west facing vertical surfaces can be termed asexcellent. The %RMSE and the %MBE values are always lower than 10% and 6%, respectively,although the models always underestimate global illuminance.


Acknowledgement

The present work was performed with financial assistance from the Spanish Governmentthrough grant PB98-0736.

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Estimation of global illuminance on inclined surfaces for clear skies