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Statistical assessment of a model for global illuminance on inclined surfaces from horizontal global illuminance
Energy Conversion and Management 43 (2002) 693–708 www.elsevier.com/locate/enconman
Statistical assessment of a model for global illuminance on inclined surfaces from horizontal global illuminance Enrique Ruiz a, Alfonso Soler b,*, Luis Robledo c a
Facultad de Ciencias, Departamento de Fısica Aplicada, Universidad Aut onoma de Madrid, Cantoblanco, Madrid, Spain b Departamento de Fısica, Escuela T ecnica Superior de Arquitectura, Universidad Polit ecnica de Madrid, Avda. Juan de Herrera 4, 28040 Madrid, Spain c Departamento de Sistemas Inteligentes Aplicados, E.U. de Inform atica, Universidad Polit ecnica 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 irradiance on inclined planes, which only requires the horizontal global irradiance and the sun elevation and azimuth as input parameters. From now on, this model will be referred to as the Olmo model. Statistical assessment of this model can be considered as important, taking into account that available models for estimation of global irradiance or illuminance on an inclined surface require information of global, and direct or diffuse irradiance or illuminance on a horizontal surface. The version of the Olmo model for global illuminance is tested in the present work using mean 15 min values of global illuminance obtained with 20 sensors of different slopes (zenith angles) and azimuths. The sensors were placed on a spherical dome located at one of the corners of the roof of the experimental site and ground shielded by black mat painted honeycomb material. Assuming a value of the honeycomb albedo of 0% values of the obtained RMSD go from about 8% for surface slopes of 12° to about 30% for a vertical surface facing east. For a north facing vertical surface, receiving mostly diffuse illuminance, a value of about 52% is obtained for the RMSD. Assuming a value of the albedo of 5%, too high for our experimental set up, similar results are obtained. In general the model over estimates global illuminance on inclined surfaces in Madrid, for experimental global illuminance values higher than about 60 klux. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Global illuminance modeling; Inclined surfaces; All sky conditions
*
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 Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 1 ) 0 0 0 6 3 - 2
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1. Introduction There are a number of models available to estimate global irradiance or illuminance on inclined surfaces from irradiance or illuminance on a horizontal surface, such as Refs. [1–4]. These models require information of the global and the direct or the diffuse irradiance or illuminance on a horizontal surface. Usually, the models are tested with data obtained for vertical surfaces, because the departure from horizontal surface values is the largest in this case. An example is the anisotropic model by Perez [3], regarded as one of the most reliable to estimate both irradiance and illuminance on inclined surfaces, as confirmed, among others, by Robledo and Soler [5–8], Utrillas and 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 the simplest version [5–10]. In this context, the recent publication by Olmo et al. [11] of a simple model to estimate global irradiance on inclined planes, which only requires the horizontal global irradiance and the sun elevation and azimuth as input parameters can be considered a priori as a daring proposal. From now on, in the present work, we refer to the Olmo et al. model as the Olmo model. Obviously, if such a model worked properly, allowing for accurate estimation of global radiation on inclined planes, 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 vertical surfaces facing south, east, north and west [12]. Two main conclusions were drawn from the results obtained. The first was that, in general, the model is not accurate enough to predict values of global irradiance on vertical surfaces. The second was that the model gives rather large errors for the surface facing north and, in general, when the vertical plane does not ‘‘see’’ the sun disk or when, for nearly overcast/overcast skies, no direct radiation is received on both vertical and horizontal planes [12]. The general equations of a global irradiance model for inclined surfaces are of the same form as the equations for a global illuminance model, as an illuminance model just relates to the visible part of the whole broadband solar spectrum [3]. If, in the Olmo model, global irradiance is changed to global illuminance, the only extra changes that need to be made to obtain the equivalent illuminance model is to substitute for the clearness index and ground albedo for global solar radiation by the clearness index and ground albedo for the visible part of the spectrum. The version of the Olmo model for global illuminance on inclined surfaces is tested in the present work using mean 15 min values of global illuminance obtained with 20 sensors of different elevations and azimuths placed on a spherical dome and ground shielded by black mat painted honeycomb material.
2. Experimental data and statistical methods The experimental illuminance data consist of two data sets. The first data set consists of mean hourly values of global illuminance measured on a horizontal surface and on vertical surfaces facing north, east, south and west during the period June 1994–July 1995. The second data set includes 15 min mean hourly values of global illuminance measured on a horizontal surface and on 16 surfaces of different orientations and slopes during the period January 2000–June 2000. All
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the data were obtained with LICOR illuminance sensors in the International Daylight Measurement Programme (IDMP) station placed at the unobstructed roof of the Escuela Tecnica Superior de Arquitectura at Madrid (40°2500 N, 3°410 W). The sensors were calibrated every six months and always a few days before the beginning of an experiment. The sensors used for the period January 2000–June 2000 were fixed on a semi-spherical dome placed at one of the corners of the roof in 16 of the 145 available positions, distributed according to a pattern suggested by the CIE [13]. All the sensors were used with artificial horizons made of mat black painted honeycomb material, so that they could only ‘‘see’’ the honeycomb material and the open air. Shielding the four vertical sensors from the ground is relatively easy, but in our case, shielding the 16 sensors corresponding to the second experimental set up imposed some limits in the azimuths of the sensors. We are not aware of published research performed using simultaneous continuous measurements with such a number of fixed illuminance sensors. A Lambda 9 Perkin–Elmer spectrophotometer for a 4° incidence was used to measure the spectral reflectance of the black mat paint used. A mean value of 3.4% was measured for 400 < k < 700 nm; Dk ¼ 10 nm: Thus, the ground reflected irradiance may be considered as very low, taking into account that the honeycomb had a depth of 16 cm. Experimental data were not used for solar elevations lower than 5°. Solar elevation and azimuth are taken at the middle of the corresponding period. P The accuracy of the model was determined by using, as statistical estimators, the MBD ¼ ðyi xi Þ=N and the RMSD ¼ P f ðyi xi Þ2 =N g1=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 inclined surfaces The Olmo model was developed to estimate global irradiance on inclined surfaces using data obtained at Granada [11]. The measurement system consisted in a pyranometer mounted on a device 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 underlying uncolored concrete was measured as no shielding from the concrete was provided. The data consisted of 114 clear sky experiments distributed over all the year. To take the effect of ground reflected radiation into account the authors proposed a multiplying factor. Although only clear sky data were used, the model was proposed for all sky conditions. The authors have remarked the general applicability of their model, which could be used with instantaneous values, as well as averaged measurements. The model was tested with the Skyscan’834 data set [14] that contains slope irradiance measurements, although only for a surface oriented to the South with an elevation angle of 44°. The system used to obtain the Skyscan’834 data set was shielded from the ground as in the case of the experiments 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 a horizontal surface GH with the following equation:
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GW ¼ GH expðkt ðW2 W2H ÞÞ
ð1Þ
where W, expressed in radians, is called the scattering [15] or incidence angle [16], that is the angle between the normal to the inclined surface and the sun–earth vector, taken from the center of the sun disk. kt is the global to extraterrestrial horizontal irradiance value, which takes into account the influence of general atmospheric conditions and clouds. WH is the angular distance, expressed in radians, between the normal direction to the horizontal plane and the sun’s position taken from the 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° #S Þ þ cosð90° #Þ cosð90° #S Þ cosðaS aÞ
ð2Þ
where # represents the zenith angle and a the azimuth, and the subscript S refers to the sun position. The azimuth is zero for south, 90° for east, 180° for north and 270° for west. The zenith angle 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] give the following expression for cos W: cos W ¼ sinð#Þ sinð#S Þ þ cosð#Þ cosð#S Þ 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 solar elevation, giving a model independent of the azimuth. Olmo et al. may have used the correct expression, Eq. (2), for their calculations. To allow for ground reflected radiation, Olmo et al. include a factor that considers the effect of anisotropic 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 other expressions as the Temps and Coulson [17] formula that depends on three geometrical factors. They propose [11]: Fc ¼ 1 þ q sin2 ðW=2Þ
ð4Þ
where q is the albedo of the underlying surface, which they take as q ¼ 0:35 (35%) for the uncoloured concrete floor present in their experiment. In this way, the mathematical expression for their model, including ground reflected radiation, reads: ð5Þ GW ¼ GH exp kt W2 W2H Fc The general equations of a global irradiance model for inclined surfaces have the same form as the equations for a global illuminance model, as an illuminance model just relates to the visible part of the whole broadband solar spectrum. If, in the Olmo model, global irradiance is changed to global illuminance, the only extra changes that need to be made to obtain the equivalent illuminance model are to substitute for the clearness index and albedo for global solar radiation by the clearness index and albedo for the visible part of the spectrum. Thus, the version of the Olmo model for global illuminance can be written as:
E. Ruiz et al. / Energy Conversion and Management 43 (2002) 693–708
EW ¼ EH expðkv ðW2 W2H ÞÞFc
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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 extraterrestrial illuminance, taking this last one as 133.8 klux [18]. The global illuminance on a horizontal surface can 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 for the 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 experimental section. However, before presenting the results we briefly consider the equivalence between 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 for different periods. The irradiance data were obtained for the period August 92–July 93, and the illuminance data for the period June 1994–July 1995, as already indicated in the experimental data section. 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) versus the measured global illuminance for a south facing vertical surface. Only data for kt < 0:85 and kv < 0:85 were used in Fig. 1(a) and (b) respectively, and these limits for the cloudiness index are kept throughout the paper. The similarity between Fig. 1(a) and 1(b) is evident. The same conclusion is obtained when the corresponding figures for north facing vertical surfaces are compared, 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 global illuminance model given by Eq. 6 and the number of data available for each sensor, are indicated in Table 1. Plots for the global illuminance values calculated with Eq. (6) versus the measured global illuminance have been obtained for each of the 20 sensors with q ¼ 0 in Eq. (6), and some of 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 for q ¼ 0, we observe that for vertical surfaces (h ¼ 90°), the RMSD ranges from about 23% for the south facing surface to about 30% and 33% for east and west facing surfaces, respectively, and to
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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.
about 52% for the north facing surface. This increase of the RMSD when going from south to north facing surfaces is similar to the one noted for the global irradiance model, more properly called Olmo model [12]. This increase is perhaps related to the increase when going from south to north in the number of cases when only diffuse illuminance/irradiance is received on the vertical surface and diffuse plus direct illuminance/irradiance is received on the vertical surface. For the surface facing north, which receives basically diffuse radiation the model shows a rather bad performance. In this respect, we note that the Skyscan’834 data set used to test the Olmo model
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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.
was obtained for surface facing south and this could favor lower RMSD values than for a north facing surface. For h ¼ 84°, the RMSD ranges from about 20% for a ¼ 0° to about 26% for a ¼ 264°. For h ¼ 72°, the values of the RMSD are about 23–25% for different sensor azimuth angles, and for h ¼ 60°, the RMSD values are about 23–25%. The RMSD values show an important decrease when we consider the sensors with h ¼ 36°, and RMSD values of about 18% are obtained in this case. For h ¼ 24°, the RMSD value obtained is about 14% for the only sensor available. Finally,
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Table 1 Values of the MBD and the RMSD in % for the 20 inclined surfaces, specified by sensor zenith angle h and azimuth a and values of q ¼ 0% and 5% # (°)
a (°)
Number of data
90 90 90 90 84 84 84 84 72 72 72 60 60 60 36 36 24 12 12 12
0 90 180 270 0 264 300 324 288 300 336 0 270 300 270 300 330 0 120 140
4248 4649 4649 4272 4050 4132 4038 4047 4041 3616 3510 4152 3508 3989 3683 3712 4035 4190 4068 4130
q ¼ 0%
q ¼ 5%
MBD (%)
RMSD (%)
MBD (%)
RMSD (%)
5.74 9.83 18.22 1.17 2.01 6.96 3.88 6.50 3.32 7.16 3.37 11.06 0.49 6.88 2.34 7.14 6.39 2.38 0.5 3.06
23.28 29.90 52.13 33.52 20.15 26.45 22.88 24.24 25.56 24.83 23.90 25.18 23.66 23.47 17.58 18.10 13.53 6.65 8.52 8.74
7.01 8.74 15.88 2.36 2.84 5.93 4.74 7.34 4.18 7.92 4.05 11.70 0.39 7.58 3.61 7.99 7.26 3.38 1.82 1.78
23.46 29.53 52.10 33.64 20.06 6.33 23.07 24.38 25.78 25.01 23.79 25.31 23.60 23.60 17.93 18.52 14.02 7.18 8.78 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°.
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Fig. 4. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for a surface of slope 24° and azimuth 300°.
Fig. 5. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for a surface of slope 36° and azimuth 300°.
for h ¼ 12° and different sensor azimuths the RMSD values are about 7–9%. It is also observed in Table 1 that if a value q ¼ 5% (0.05) is assumed, well above the albedo expected for the black mat honeycomb material, RMSD values close to those for q ¼ 0 are obtained. These results for the RMSD are not very suggestive in relation to the use of the model.
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Fig. 6. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for a surface of slope 60° and azimuth 300°.
Fig. 7. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for a surface of slope 72° and azimuth 288°.
For the Skyscan’834 data set, a RMSD of 9.3% is obtained by Olmo et al. when the solar irradiance model given by Eq. (5) is used to estimate global irradiance values for a surface with h ¼ 44°, a ¼ 0 (facing south). For with h ¼ 36°, a ¼ 300°, a value of the RMSD of 18.3% is obtained in our case. The explanation for this difference may be related to the fact that the number
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Fig. 8. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for a surface of slope 84° and azimuth 324°.
Fig. 9. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with a slope of 12°.
of measured values in the present work is larger than the number in the Skyscan’834 data set. The model 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 global illuminance for the 20 sensors, grouped in relation with their zenith angles have also been
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Fig. 10. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with a slope of 36°.
Fig. 11. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with a slope of 60°.
obtained. Thus, and to give an example, Fig. 9 is for the sensors with slope h ¼ 12° and azimuths a 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 is for h ¼ 84°, and Fig. 14 is for h ¼ 90°. The MBD and RMSD obtained for each value of h, and q ¼ 0 are given in Table 2. It is observed that RMSD values higher than 24% are obtained for h > 60° and that maximum MBD values of about 6% result. In general, the model over predicts
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Fig. 12. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with a slope of 72°.
Fig. 13. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with a slope of 84°.
for measured global illuminance values higher than about 60 klux. Values of the MBD and the RMSD for q ¼ 0:05 are similar to those for q ¼ 0. The MBD values show that the model overestimates, 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 that the model clearly over estimates for global illuminance values higher than about 60 klux.
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Fig. 14. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with 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% # (°) 90 84 72 60 36 24 12
q ¼ 0%
q ¼ 5%
MBD (%)
RMSD (%)
MBD (%)
RMSD (%)
1.77 1.95 4.55 6.80 5.09 6.39 0.12
32.53 24.53 24.90 24.57 18.13 13.53 7.87
0.45 2.89 5.31 7.51 6.12 7.26 1.30
32.84 24.57 24.97 24.67 18.53 14.02 8.04
Fig. 15. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the 20 sensors.
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5. Conclusion We have statistically assessed the illuminance version of the Olmo model (Eq. 6) using mean 15 min values of global illuminance obtained with 20 sensors of different slopes and azimuths. Assuming a zero albedo, values of the RMSD obtained go from about 8% for surface slopes of 12° to about 30% for high surface slopes and surfaces facing from east to south to west. For a north facing vertical surface receiving mostly diffuse illuminance, a value of 52.13% is obtained for the RMSD. Similar results are obtained for an albedo of 0.05 (5%), clearly above the value expected from the experimental set up. The high values of the RMSD obtained for medium and high surface 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 measuring site. The model clearly over estimates global illuminance for measured values higher than about 60 klux.
Acknowledgements The present work has been made possible by financial support from the Ministerio de Ciencia y Tecnologa through the project PB 98-0736. References [1] Hay JE, McKay DC. Estimating solar irradiance on inclined surfaces: a review and assessment of methodologies. Int J Solar Energy 1985;3:203–40. [2] Klucher TM. Evaluation of models to predict insolation on tilted surfaces. Solar Energy 1979;23(2):11–114. [3] Perez R, Ineichen P, Seals F, Michalsky J, Stewart R. Modeling daylight availability and irradiance components from direct and global irradiance. Solar Energy 1990;44:271–89. [4] Skartveit A, Olseth JA. Modeling slope irradiance at high latitudes. Solar Energy 1986;36:333–44. [5] Robledo L, Soler A. Point source Perez illuminance model. Dependence of coefficients on surface orientation at Madrid. Lighting Res Technol 1996;28:141–8. [6] Robledo L, Soler A. Modeling daylight on inclined surfaces. Renewable Energy 1997;11:149–52. [7] Robledo L, Soler A. Modeling irradiance on inclined planes with an anisotropic model. Energy 1998;23:193–201. [8] Robledo L, Soler A. Dependence on surface orientation of coefficients in the circumsolar simplified Perez illuminance model for vertical planes at Madrid. Energy Convers Mgmt 1998;39:1585–95. [9] Utrillas PM, Martinez Lozano JA. Performance evaluation of several versions of the Perez tilted diffuse irradiance model. Solar Energy 1994;53:155–62. [10] Li DHW, Lam JC. Evaluation of slope irradiance models against measured Hong Kong data. Building Environ 2000;44:501–6. [11] Olmo FJ, Vida J, Foyo I, Castro Diez Y, Alados-Arboledas L. Prediction of global irradiance on inclined surfaces from horizontal global irradiance. Energy 1999;24:689–704. [12] Ruiz E, Soler A, Robledo L. Assessment of the Olmo model for vertical surfaces facing north, east, south, west. Energy, submitted for publication. [13] Littlefair PJ. A comparison of sky luminance models with measured data for Garston, United Kingdom. Solar Energy 1995;53:315–22. [14] Brunger AP, Hooper FC. Measurements of short wave sky radiance in an urban atmosphere. Solar Energy 1991;47:137–42. [15] CIE Technical Report 110, Spatial distribution of daylight-luminance distributions of various reference skies, 1994.
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[16] Iqbal M. An introduction to solar radiation. New York: Academic Press; 1983. [17] Temps RC, Coulson KL. Solar radiation incident upon slopes of different orientations. Solar Energy 1997;19:179– 84. [18] Kittler R, Darula S, Perez R. A set of standard skies characterizing daylight conditions for computer and energy conscious design. Final Report. Available from the Institute of Construction and Architecture, Slovak Academy of Sciences, Bratislava 84220, Slovakia, 1998. [19] Robledo L, Soler A. Luminous efficacy of global solar radiation for clear skies. Energy Convers Mgmt 2000;41:1769–79. [20] Robledo L, Soler A, Ruiz E. Luminous efficacy of global solar radiation for intermediate and overcast skies. Theor Appl Climatol, in press. [21] Ruiz E, Soler A, Robledo L. Assessment of Muneers luminous efficacy models in Madrid and a proposal of new simple models based on his approach. J Solar Energy Engng (ASME), in press.
Statistical assessment of a model for global illuminance on inclined surfaces from horizontal global illuminance Enrique Ruiz a, Alfonso Soler b,*, Luis Robledo c a
Facultad de Ciencias, Departamento de Fısica Aplicada, Universidad Aut onoma de Madrid, Cantoblanco, Madrid, Spain b Departamento de Fısica, Escuela T ecnica Superior de Arquitectura, Universidad Polit ecnica de Madrid, Avda. Juan de Herrera 4, 28040 Madrid, Spain c Departamento de Sistemas Inteligentes Aplicados, E.U. de Inform atica, Universidad Polit ecnica 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 irradiance on inclined planes, which only requires the horizontal global irradiance and the sun elevation and azimuth as input parameters. From now on, this model will be referred to as the Olmo model. Statistical assessment of this model can be considered as important, taking into account that available models for estimation of global irradiance or illuminance on an inclined surface require information of global, and direct or diffuse irradiance or illuminance on a horizontal surface. The version of the Olmo model for global illuminance is tested in the present work using mean 15 min values of global illuminance obtained with 20 sensors of different slopes (zenith angles) and azimuths. The sensors were placed on a spherical dome located at one of the corners of the roof of the experimental site and ground shielded by black mat painted honeycomb material. Assuming a value of the honeycomb albedo of 0% values of the obtained RMSD go from about 8% for surface slopes of 12° to about 30% for a vertical surface facing east. For a north facing vertical surface, receiving mostly diffuse illuminance, a value of about 52% is obtained for the RMSD. Assuming a value of the albedo of 5%, too high for our experimental set up, similar results are obtained. In general the model over estimates global illuminance on inclined surfaces in Madrid, for experimental global illuminance values higher than about 60 klux. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Global illuminance modeling; Inclined surfaces; All sky conditions
*
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 Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 1 ) 0 0 0 6 3 - 2
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1. Introduction There are a number of models available to estimate global irradiance or illuminance on inclined surfaces from irradiance or illuminance on a horizontal surface, such as Refs. [1–4]. These models require information of the global and the direct or the diffuse irradiance or illuminance on a horizontal surface. Usually, the models are tested with data obtained for vertical surfaces, because the departure from horizontal surface values is the largest in this case. An example is the anisotropic model by Perez [3], regarded as one of the most reliable to estimate both irradiance and illuminance on inclined surfaces, as confirmed, among others, by Robledo and Soler [5–8], Utrillas and 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 the simplest version [5–10]. In this context, the recent publication by Olmo et al. [11] of a simple model to estimate global irradiance on inclined planes, which only requires the horizontal global irradiance and the sun elevation and azimuth as input parameters can be considered a priori as a daring proposal. From now on, in the present work, we refer to the Olmo et al. model as the Olmo model. Obviously, if such a model worked properly, allowing for accurate estimation of global radiation on inclined planes, 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 vertical surfaces facing south, east, north and west [12]. Two main conclusions were drawn from the results obtained. The first was that, in general, the model is not accurate enough to predict values of global irradiance on vertical surfaces. The second was that the model gives rather large errors for the surface facing north and, in general, when the vertical plane does not ‘‘see’’ the sun disk or when, for nearly overcast/overcast skies, no direct radiation is received on both vertical and horizontal planes [12]. The general equations of a global irradiance model for inclined surfaces are of the same form as the equations for a global illuminance model, as an illuminance model just relates to the visible part of the whole broadband solar spectrum [3]. If, in the Olmo model, global irradiance is changed to global illuminance, the only extra changes that need to be made to obtain the equivalent illuminance model is to substitute for the clearness index and ground albedo for global solar radiation by the clearness index and ground albedo for the visible part of the spectrum. The version of the Olmo model for global illuminance on inclined surfaces is tested in the present work using mean 15 min values of global illuminance obtained with 20 sensors of different elevations and azimuths placed on a spherical dome and ground shielded by black mat painted honeycomb material.
2. Experimental data and statistical methods The experimental illuminance data consist of two data sets. The first data set consists of mean hourly values of global illuminance measured on a horizontal surface and on vertical surfaces facing north, east, south and west during the period June 1994–July 1995. The second data set includes 15 min mean hourly values of global illuminance measured on a horizontal surface and on 16 surfaces of different orientations and slopes during the period January 2000–June 2000. All
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the data were obtained with LICOR illuminance sensors in the International Daylight Measurement Programme (IDMP) station placed at the unobstructed roof of the Escuela Tecnica Superior de Arquitectura at Madrid (40°2500 N, 3°410 W). The sensors were calibrated every six months and always a few days before the beginning of an experiment. The sensors used for the period January 2000–June 2000 were fixed on a semi-spherical dome placed at one of the corners of the roof in 16 of the 145 available positions, distributed according to a pattern suggested by the CIE [13]. All the sensors were used with artificial horizons made of mat black painted honeycomb material, so that they could only ‘‘see’’ the honeycomb material and the open air. Shielding the four vertical sensors from the ground is relatively easy, but in our case, shielding the 16 sensors corresponding to the second experimental set up imposed some limits in the azimuths of the sensors. We are not aware of published research performed using simultaneous continuous measurements with such a number of fixed illuminance sensors. A Lambda 9 Perkin–Elmer spectrophotometer for a 4° incidence was used to measure the spectral reflectance of the black mat paint used. A mean value of 3.4% was measured for 400 < k < 700 nm; Dk ¼ 10 nm: Thus, the ground reflected irradiance may be considered as very low, taking into account that the honeycomb had a depth of 16 cm. Experimental data were not used for solar elevations lower than 5°. Solar elevation and azimuth are taken at the middle of the corresponding period. P The accuracy of the model was determined by using, as statistical estimators, the MBD ¼ ðyi xi Þ=N and the RMSD ¼ P f ðyi xi Þ2 =N g1=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 inclined surfaces The Olmo model was developed to estimate global irradiance on inclined surfaces using data obtained at Granada [11]. The measurement system consisted in a pyranometer mounted on a device 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 underlying uncolored concrete was measured as no shielding from the concrete was provided. The data consisted of 114 clear sky experiments distributed over all the year. To take the effect of ground reflected radiation into account the authors proposed a multiplying factor. Although only clear sky data were used, the model was proposed for all sky conditions. The authors have remarked the general applicability of their model, which could be used with instantaneous values, as well as averaged measurements. The model was tested with the Skyscan’834 data set [14] that contains slope irradiance measurements, although only for a surface oriented to the South with an elevation angle of 44°. The system used to obtain the Skyscan’834 data set was shielded from the ground as in the case of the experiments 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 a horizontal surface GH with the following equation:
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GW ¼ GH expðkt ðW2 W2H ÞÞ
ð1Þ
where W, expressed in radians, is called the scattering [15] or incidence angle [16], that is the angle between the normal to the inclined surface and the sun–earth vector, taken from the center of the sun disk. kt is the global to extraterrestrial horizontal irradiance value, which takes into account the influence of general atmospheric conditions and clouds. WH is the angular distance, expressed in radians, between the normal direction to the horizontal plane and the sun’s position taken from the 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° #S Þ þ cosð90° #Þ cosð90° #S Þ cosðaS aÞ
ð2Þ
where # represents the zenith angle and a the azimuth, and the subscript S refers to the sun position. The azimuth is zero for south, 90° for east, 180° for north and 270° for west. The zenith angle 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] give the following expression for cos W: cos W ¼ sinð#Þ sinð#S Þ þ cosð#Þ cosð#S Þ 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 solar elevation, giving a model independent of the azimuth. Olmo et al. may have used the correct expression, Eq. (2), for their calculations. To allow for ground reflected radiation, Olmo et al. include a factor that considers the effect of anisotropic 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 other expressions as the Temps and Coulson [17] formula that depends on three geometrical factors. They propose [11]: Fc ¼ 1 þ q sin2 ðW=2Þ
ð4Þ
where q is the albedo of the underlying surface, which they take as q ¼ 0:35 (35%) for the uncoloured concrete floor present in their experiment. In this way, the mathematical expression for their model, including ground reflected radiation, reads: ð5Þ GW ¼ GH exp kt W2 W2H Fc The general equations of a global irradiance model for inclined surfaces have the same form as the equations for a global illuminance model, as an illuminance model just relates to the visible part of the whole broadband solar spectrum. If, in the Olmo model, global irradiance is changed to global illuminance, the only extra changes that need to be made to obtain the equivalent illuminance model are to substitute for the clearness index and albedo for global solar radiation by the clearness index and albedo for the visible part of the spectrum. Thus, the version of the Olmo model for global illuminance can be written as:
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EW ¼ EH expðkv ðW2 W2H ÞÞFc
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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 extraterrestrial illuminance, taking this last one as 133.8 klux [18]. The global illuminance on a horizontal surface can 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 for the 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 experimental section. However, before presenting the results we briefly consider the equivalence between 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 for different periods. The irradiance data were obtained for the period August 92–July 93, and the illuminance data for the period June 1994–July 1995, as already indicated in the experimental data section. 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) versus the measured global illuminance for a south facing vertical surface. Only data for kt < 0:85 and kv < 0:85 were used in Fig. 1(a) and (b) respectively, and these limits for the cloudiness index are kept throughout the paper. The similarity between Fig. 1(a) and 1(b) is evident. The same conclusion is obtained when the corresponding figures for north facing vertical surfaces are compared, 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 global illuminance model given by Eq. 6 and the number of data available for each sensor, are indicated in Table 1. Plots for the global illuminance values calculated with Eq. (6) versus the measured global illuminance have been obtained for each of the 20 sensors with q ¼ 0 in Eq. (6), and some of 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 for q ¼ 0, we observe that for vertical surfaces (h ¼ 90°), the RMSD ranges from about 23% for the south facing surface to about 30% and 33% for east and west facing surfaces, respectively, and to
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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.
about 52% for the north facing surface. This increase of the RMSD when going from south to north facing surfaces is similar to the one noted for the global irradiance model, more properly called Olmo model [12]. This increase is perhaps related to the increase when going from south to north in the number of cases when only diffuse illuminance/irradiance is received on the vertical surface and diffuse plus direct illuminance/irradiance is received on the vertical surface. For the surface facing north, which receives basically diffuse radiation the model shows a rather bad performance. In this respect, we note that the Skyscan’834 data set used to test the Olmo model
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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.
was obtained for surface facing south and this could favor lower RMSD values than for a north facing surface. For h ¼ 84°, the RMSD ranges from about 20% for a ¼ 0° to about 26% for a ¼ 264°. For h ¼ 72°, the values of the RMSD are about 23–25% for different sensor azimuth angles, and for h ¼ 60°, the RMSD values are about 23–25%. The RMSD values show an important decrease when we consider the sensors with h ¼ 36°, and RMSD values of about 18% are obtained in this case. For h ¼ 24°, the RMSD value obtained is about 14% for the only sensor available. Finally,
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Table 1 Values of the MBD and the RMSD in % for the 20 inclined surfaces, specified by sensor zenith angle h and azimuth a and values of q ¼ 0% and 5% # (°)
a (°)
Number of data
90 90 90 90 84 84 84 84 72 72 72 60 60 60 36 36 24 12 12 12
0 90 180 270 0 264 300 324 288 300 336 0 270 300 270 300 330 0 120 140
4248 4649 4649 4272 4050 4132 4038 4047 4041 3616 3510 4152 3508 3989 3683 3712 4035 4190 4068 4130
q ¼ 0%
q ¼ 5%
MBD (%)
RMSD (%)
MBD (%)
RMSD (%)
5.74 9.83 18.22 1.17 2.01 6.96 3.88 6.50 3.32 7.16 3.37 11.06 0.49 6.88 2.34 7.14 6.39 2.38 0.5 3.06
23.28 29.90 52.13 33.52 20.15 26.45 22.88 24.24 25.56 24.83 23.90 25.18 23.66 23.47 17.58 18.10 13.53 6.65 8.52 8.74
7.01 8.74 15.88 2.36 2.84 5.93 4.74 7.34 4.18 7.92 4.05 11.70 0.39 7.58 3.61 7.99 7.26 3.38 1.82 1.78
23.46 29.53 52.10 33.64 20.06 6.33 23.07 24.38 25.78 25.01 23.79 25.31 23.60 23.60 17.93 18.52 14.02 7.18 8.78 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°.
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Fig. 4. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for a surface of slope 24° and azimuth 300°.
Fig. 5. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for a surface of slope 36° and azimuth 300°.
for h ¼ 12° and different sensor azimuths the RMSD values are about 7–9%. It is also observed in Table 1 that if a value q ¼ 5% (0.05) is assumed, well above the albedo expected for the black mat honeycomb material, RMSD values close to those for q ¼ 0 are obtained. These results for the RMSD are not very suggestive in relation to the use of the model.
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Fig. 6. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for a surface of slope 60° and azimuth 300°.
Fig. 7. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for a surface of slope 72° and azimuth 288°.
For the Skyscan’834 data set, a RMSD of 9.3% is obtained by Olmo et al. when the solar irradiance model given by Eq. (5) is used to estimate global irradiance values for a surface with h ¼ 44°, a ¼ 0 (facing south). For with h ¼ 36°, a ¼ 300°, a value of the RMSD of 18.3% is obtained in our case. The explanation for this difference may be related to the fact that the number
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Fig. 8. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for a surface of slope 84° and azimuth 324°.
Fig. 9. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with a slope of 12°.
of measured values in the present work is larger than the number in the Skyscan’834 data set. The model 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 global illuminance for the 20 sensors, grouped in relation with their zenith angles have also been
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Fig. 10. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with a slope of 36°.
Fig. 11. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with a slope of 60°.
obtained. Thus, and to give an example, Fig. 9 is for the sensors with slope h ¼ 12° and azimuths a 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 is for h ¼ 84°, and Fig. 14 is for h ¼ 90°. The MBD and RMSD obtained for each value of h, and q ¼ 0 are given in Table 2. It is observed that RMSD values higher than 24% are obtained for h > 60° and that maximum MBD values of about 6% result. In general, the model over predicts
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Fig. 12. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with a slope of 72°.
Fig. 13. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with a slope of 84°.
for measured global illuminance values higher than about 60 klux. Values of the MBD and the RMSD for q ¼ 0:05 are similar to those for q ¼ 0. The MBD values show that the model overestimates, 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 that the model clearly over estimates for global illuminance values higher than about 60 klux.
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Fig. 14. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the sensors with 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% # (°) 90 84 72 60 36 24 12
q ¼ 0%
q ¼ 5%
MBD (%)
RMSD (%)
MBD (%)
RMSD (%)
1.77 1.95 4.55 6.80 5.09 6.39 0.12
32.53 24.53 24.90 24.57 18.13 13.53 7.87
0.45 2.89 5.31 7.51 6.12 7.26 1.30
32.84 24.57 24.97 24.67 18.53 14.02 8.04
Fig. 15. Global illuminance calculated with Eq. (6) with q ¼ 0 versus measured global illuminance for the 20 sensors.
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5. Conclusion We have statistically assessed the illuminance version of the Olmo model (Eq. 6) using mean 15 min values of global illuminance obtained with 20 sensors of different slopes and azimuths. Assuming a zero albedo, values of the RMSD obtained go from about 8% for surface slopes of 12° to about 30% for high surface slopes and surfaces facing from east to south to west. For a north facing vertical surface receiving mostly diffuse illuminance, a value of 52.13% is obtained for the RMSD. Similar results are obtained for an albedo of 0.05 (5%), clearly above the value expected from the experimental set up. The high values of the RMSD obtained for medium and high surface 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 measuring site. The model clearly over estimates global illuminance for measured values higher than about 60 klux.
Acknowledgements The present work has been made possible by financial support from the Ministerio de Ciencia y Tecnologa through the project PB 98-0736. References [1] Hay JE, McKay DC. Estimating solar irradiance on inclined surfaces: a review and assessment of methodologies. Int J Solar Energy 1985;3:203–40. [2] Klucher TM. Evaluation of models to predict insolation on tilted surfaces. Solar Energy 1979;23(2):11–114. [3] Perez R, Ineichen P, Seals F, Michalsky J, Stewart R. Modeling daylight availability and irradiance components from direct and global irradiance. Solar Energy 1990;44:271–89. [4] Skartveit A, Olseth JA. Modeling slope irradiance at high latitudes. Solar Energy 1986;36:333–44. [5] Robledo L, Soler A. Point source Perez illuminance model. Dependence of coefficients on surface orientation at Madrid. Lighting Res Technol 1996;28:141–8. [6] Robledo L, Soler A. Modeling daylight on inclined surfaces. Renewable Energy 1997;11:149–52. [7] Robledo L, Soler A. Modeling irradiance on inclined planes with an anisotropic model. Energy 1998;23:193–201. [8] Robledo L, Soler A. Dependence on surface orientation of coefficients in the circumsolar simplified Perez illuminance model for vertical planes at Madrid. Energy Convers Mgmt 1998;39:1585–95. [9] Utrillas PM, Martinez Lozano JA. Performance evaluation of several versions of the Perez tilted diffuse irradiance model. Solar Energy 1994;53:155–62. [10] Li DHW, Lam JC. Evaluation of slope irradiance models against measured Hong Kong data. Building Environ 2000;44:501–6. [11] Olmo FJ, Vida J, Foyo I, Castro Diez Y, Alados-Arboledas L. Prediction of global irradiance on inclined surfaces from horizontal global irradiance. Energy 1999;24:689–704. [12] Ruiz E, Soler A, Robledo L. Assessment of the Olmo model for vertical surfaces facing north, east, south, west. Energy, submitted for publication. [13] Littlefair PJ. A comparison of sky luminance models with measured data for Garston, United Kingdom. Solar Energy 1995;53:315–22. [14] Brunger AP, Hooper FC. Measurements of short wave sky radiance in an urban atmosphere. Solar Energy 1991;47:137–42. [15] CIE Technical Report 110, Spatial distribution of daylight-luminance distributions of various reference skies, 1994.
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