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Modelling irradiance on inclined planes with an anisotropic model
Pergamon
PII: S0360-5442(97)00083-2
Energy Vol. 23, No. 3, pp. 193–201, 1998 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0360-5442/98 $19.00 + 0.00
MODELLING IRRADIANCE ON INCLINED PLANES WITH AN ANISOTROPIC MODEL LUIS ROBLEDO†‡ and ALFONSO SOLER§ † Departamento de Sistemas Inteligentes Aplicados, E. U. Informatica. Universidad Polite´cnica de Madrid. Ctra. de Valencia km 7. 28031 Madrid, Spain; §Departamento de Fisica e Instalaciones Aplicadas, E. T. S. de Arquitectura. Universidad Polite´cnica de Madrid. Avda. Juan de Herrera, 4. 28040 Madrid, Spain
(Received 19 March 1997)
Abstract—The circumsolar ( ␣ = 25°) and point-source versions of the simplified Perez model were developed to estimate irradiances on inclined planes from values for horizontal planes through the determination of sets of empirical coefficients. This model has been evaluated by us at Madrid. Mean hourly values of irradiances on vertical surfaces facing north, east, south and west were used. A modification in the structure of the model was made by analyzing separately the data obtained when the surface "sees" or does not "see" the circumsolar disk (for the point-source version, the centre of the disk). The validity of the different versions is assessed statistically using experimental and estimated values for the irradiance. For both the point-source and circumsolar versions, with data for all orientations, the model performs better with local coefficients than with those given by Perez. The accuracy was increased when local coefficients were calculated independently for each orientation. When the surface does not "see" the circumsolar disk (the centre of the disk in the case of the point-source version), the specified data separation produces an equation that is simpler and easier to use than the corresponding equation in the original model while yielding some increase in accuracy. 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
In the last two decades, different models have been developed to predict the diffuse solar irradiance on inclined surfaces with different orientations. Of all the anisotropic models, that by Perez [1], which is used to predict diffuse irradiance/illuminance on an inclined surface from diffuse irradiance/illuminance on a horizontal surface, has received special attention because of its good performance [2]. In its original formulation, the Perez model is somewhat difficult to use. The original model has been greatly simplified [3,4], to obtain circumsolar and point-source versions. Apparently the original formulation does not perform better than the simplified model [5]. Some authors have assessed both the original and the simplified versions [5–8] and have shown that empirical coefficients derived from experimental data are location-dependent. Robledo and Soler [7,8], using illuminance data obtained at Madrid for vertical surfaces facing north, east, south and west (in 1992–1993), have shown that the coefficients for the two versions depend on surface orientation. In the point-source version [7], the data were separated into sets for the surface "seeing" or not "seeing" the centre of the sun disk. The function which includes the coefficients of the model was then optimised independently for each set. In this way, an increasingly simplified model is obtained when the surface does not "see" the centre of the circumsolar disk, and the error in the estimation of the experimental values is reduced. The present work is based on 1994–1995 measurements and represents a more detailed study of the two simplified versions. The results obtained for irradiance confirm the conclusions of previous illuminance work. A modification of the circumsolar version is proposed. 2. MODEL DESCRIPTION
The main difficulty concerning the study of diffuse radiation is its anisotropic nature. The most important anisotropic effects for solar radiation passing through the atmosphere are forward scattering ‡Author for correspondence. Fax: 34 1 3367522. 193
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by aerosols and multiple scattering and retroscattering near the horizon. In the simplified Perez model the sky dome consists of one isotropic and two anisotropic zones; the circumsolar region is characterized by the half angle ␣ and the horizon band is characterized by the angle . Radiation from each of these regions is assumed to be constant in the areas where they are defined. Another basic characteristic of the model is parametrization of the sky conditions, which was first done in [1,9] using the solar zenith angle Z, horizontal diffuse radiation Dh and clearness index ⑀ = (Dh + I)/Dh as variables. I is the direct normal irradiance. With the objective of reducing the dependence of the second parameter on the sun’s position (zenith angle Z), the new quantity ⌬ = Dhm/IO
(1)
with m = optical air mass and IO = extraterrestrial irradiance. For the same reason the clearness index was redefined as
⑀⬘ = ( ⑀ + kZ3 )/(1 + kZ3 ),
(2)
where k = 1.041 and Z is in radians. Thus, the sky conditions are now given in terms of the solar zenith angle Z, the brightness index ⌬ and the clearness index ⑀⬘. If L, L1 and L2 are radiances corresponding, respectively, to the isotropic, circumsolar and horizon zones, the original model gives the diffuse irradiance D on an inclined plane with slope  as follows: D = Dh
0.5(1 + cos ) + a(f1 − 1) + b(f2 − 1) 1 + c(f1 − 1) + d(f2 − 1)
(3)
where Dh is the diffuse irradiance on a horizontal plane, f1 = L1 /L, f2 = L2 /L, and a, b, c and d are quantities which depend on the solar zenith angle Z, half angle of the circumsolar disk ␣ and solar incident angle on the inclined plane [9]. The coefficients f1 and f2 in Eq. (3) are ⱖ 1. For f1 = f2 = 1, Eq. (3) reduces to the equation for the isotropic model. The non-linearity of Eq. (3) makes the evaluation of f1 and f2 difficult by a least-mean-squares fit. For this reason, Perez [3] proposed a reformulation of Eq. (3) as D = Dh[0.5(1 + cos )(1 − F1 − F2 ) + (a/c)F1 + (b/d)F2 ],
(4)
with F1 and F2 representing the respective normalized contributions of the circumsolar and horizon zones to the total diffuse irradiance received on a horizontal plane. In Eq. (4), F1 and F2 range from 0 to 1. If F1 = F2 = 0, Eq. (4) reduces to the equation for the isotropic model. The quantities a, b, c and d are the same as those in Eq. (3). Other simplifications were introduced by Perez [3] into the geometry of the model. The first of these is based on the horizon zone being infinitesimally thin at 0° elevation (linear horizon). With this supposition, 1 − F1 − F2 ⯝ 1 − F1, b/d = sin, and Eq. (4) becomes now D = Dh[0.5(1 + cos )(1 − F1 ) + (a/c)F1 ) + F2sin ].
(5)
A major simplification involves the assumption that all circumsolar radiation comes from a point placed at the centre of the circumsolar disk (point source). Then a = max(0,cos ),c = max(0.087, cosZ),
(6)
where 0.087 = cos85° is introduced because values corresponding to low solar elevations ( ␣ ⬍ 5°) are not considered and discontinuities for a/c = ⬁ are avoided. Equation (5) establishes a linear dependence of D on the coefficients F1 and F2. When assuming a discrete set of categories or bins for ⑀⬘, the Fi may be considered to be linear functions of ⌬ and Z for each of the bins in the form
Modelling irradiance on inclined planes with an anisotropic model
195
Fi = Fi1 + Fi2⌬ + Fi3Z,i = 1,2.
(7)
Thus, from Eqs. (5) and (7) and for each of the categories of ⑀⬘, a set of 6 coefficients Fij is obtained by a least-squares fit to the experimental data for D and Dh. As stated Robledo and Soler [7], when evaluating the point-source version for illuminance measured on vertical surfaces facing N, E, S and W at Madrid, found that when the model was used to fit data for all orientations, it performed better with local coefficients than with those proposed by Perez, and that the model accuracy increased further when it was used with local coefficients derived independently for each orientation. In fact, different behaviour of the model was found for each orientation. For example, for the north-facing surface, which receives almost no direct radiation, atypical and artificial coefficients were obtained for some of the bins because, when the function given by Eq. (5) was optimized by a least-mean-squares fit, the resulting coefficients Fi (i.e. F1 ) lacked physical meaning due to the absence of a significant circumsolar contribution. In [7], a slight modification was introduced in the utilization of the point-source version. Thus, the data obtained were fitted separately when the surface "sees" the centre of the circumsolar disk (a/c⫽0, ⬍ /2, cos ⬎ 0), and when the surface does not "see" the centre of the disk (a/c = 0, ⬎ /2, cos ⬍ 0). For vertical planes and a/c⫽ 0, using Eq. (7), Eq. (5) becomes D = Dh[0.5 + F21 + (a/c − 0.5)F11 + (a/c − 0.5)⌬F12 + (a/c − 0.5)ZF13 + F22⌬ + F23Z].
(8)
For a/c = 0, Eq. (5) shows a linear dependence of D /Dh on ⌬ and Z. The diffuse irradiance on a vertical plane becomes now [7] D /Dh = (0.5 + F)
(9)
with F = a1 + a2⌬ + a3Z representing the normalized contribution of the anisotropic (circumsolar and horizon) to the total diffuse irradiance on a horizontal surface. When F = 0, Eq. (9) reduces to the isotropic model. Once D is obtained, the global irradiance on tilted planes can be predicted if the direct irradiance on the horizontal plane is known. The direct irradiance on a vertical plane is obtained from its value for a horizontal plane. 3. EXPERIMENTAL DATA AND STATISTICAL METHODS
The experimental data consist of mean hourly values of diffuse and global irradiance on a horizontal surface and global irradiance on vertical surfaces facing north, east, south and west, obtained at Madrid (40.3°N, 4.4°W) for the period from June 1994 to July 1995. All vertical sensors are used with artificial horizons made of matt black painted honeycomb material. Thus, the ground-reflected irradiance is considered as negligible. Experimental data were not used when Z ⬎ 85° or when negative values of the direct horizontal or diffuse vertical irradiances were obtained. The accuracy of the model was determined by using as statistical estimators the MBD = ⌺i(yi − xi )/N and the RMSD = [⌺i(yi − xi )2 /N]1/2 where yi is the ith predicted value, xi the ith measured value and N the number of values. Experimental values of diffuse irradiance on vertical surfaces have been obtained as the difference between the experimental values of global and direct irradiances. 4. RESULTS AND DISCUSSION
4.1. Point-Source version The coefficients for the point-source version in Table 1 refer to the simplified Perez model using irradiance data for all orientations. If irradiance data are used for each orientation, new tables are obtained with coefficients that for the E, S and W planes follow the same type of variation with ⑀⬘ as the data in Table 1. However, if the coefficients for the last two bins of the north plane given in Table 2 are considered (especially for bin 8), corresponding to clear skies, rather atypical values result. The
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Table 1. Irradiance coefficients determined for Madrid, from experimental data for all of the vertical planes in the point-source version
⑀⬘bin 1 2 3 4 5 6 7 8
Upper limit 1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
Cases (%) 15.3 6.1 6.0 6.3 11.6 16.7 17.3 20.6
F11 0.057 0.366 0.784 0.838 1.025 0.921 0.636 − 0.044
F12 0.307 0.263 − 0.560 − 0.149 − 0.619 0.189 0.243 4.844
− − − − − − − −
F13 0.057 0.257 0.338 0.490 0.554 0.644 0.477 0.391
F21 − 0.003 0.169 0.254 0.211 0.442 0.486 0.299 0.279
F22 0.075 − 0.365 − 0.564 − 0.177 − 0.936 − 1.402 − 1.483 − 0.884
F23 − 0.046 − 0.047 − 0.028 − 0.071 − 0.077 0.003 0.167 0.049
Circumsolar brightening coefficient = F1 = F11 + F12⌬ + F13Z Horizon brightening coefficient = F2 = F21 + F22⌬ + F23Z
Table 2. Irradiance coefficients determined for Madrid, from experimental data for the north plane in the point-source version.
⑀⬘bin 1 2 3 4 5 6 7 8
Upper limit 1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
Cases (%) 16.1 4.9 5.3 6.3 10.5 19.3 13.6 24.0
F11 0.264 1.576 1.108 2.057 3.147 3.052 4.716 − 93.47
F12 0.692 0.522 − 0.044 − 0.631 − 0.243 0.216 7.878 − 27.41
F13 − 0.266 − 1.155 − 0.662 − 1.228 − 2.087 − 2.186 − 4.335 95.99
F21 0.116 0.797 0.515 1.044 1.579 1.453 2.522 − 46.22
F22 − 0.062 − 0.304 − 0.506 − 1.029 − 0.946 − 0.886 1.643 − 19.65
F23 0.070 − 0.487 − 0.221 − 0.485 − 0.872 − 0.769 − 1.861 48.33
Circumsolar brightening coefficient = F1 = F11 + F12⌬ + F13Z Horizon brightening coefficient = F2 = F21 + F22⌬ + F23Z
reason is that for N orientation there is almost no direct irradiance. For clear skies, the Fij are difficult to justify. We note that for a vertical surface, Eq. (5) becomes D = Dh[0.5 + (a/c − 0.5)F1 + F2 ].
(10)
For bin 8, a/c = 0 for almost all data. Examination of Table 2 shows that experimentally F1 ⯝ 2F2, so that Eq. (10) leads to D = 0.5Dh.
(11)
Eq. (11) corresponds to the isotropic radiance distribution, which may be adequate for overcast but not for clear skies. Under these conditions the 6 coefficients Fij appearing in the last bin do not have any meaning. The fact that there is almost no direct irradiance makes utilization of the Perez model difficult, because the mathematically justified coefficients obtained for the last bins are difficult to understand in physical terms. This situation leads us to consider separating data for a/c⫽0 from data for a/c = 0. In the first of these cases Eq. (5) is used to calculate the diffuse irradiance, while for the second case Eq. (9) is the most advisable. With the specified modification, comparisons of the results for different orientations now show a uniform variation of the ci and Fij with intervals of ⑀⬘, not only in the S, E and W orientation, but also in the N orientation as is shown in Table 3. For all orientations and a/c = 0 F has values close to zero for small values of ⑀⬘ (overcast skies) but increases as expected on physical grounds with ⑀⬘, because the anisotropy increases when the skies become clearer. 4.2. Circumsolar version In Table 4 coefficients are given for the circumsolar version of the simplified Perez model using data for all orientations. In Tables 4 and 1 no appreciable difference is observed relating the way the Fij vary with the different bins considered for ⑀⬘. For the model coefficients obtained using data for each specific orientation, a similar pattern of variation with ⑀⬘ as that for the point source version results. Coefficients for the N plane given in Table
Modelling irradiance on inclined planes with an anisotropic model
197
Table 3. Irradiance coefficients determined for Madrid, from experimental data for the north plane in the point − source version; (a) a/c = 0, (b) a/c⫽0. (a)
⑀ bin⬘ 1 2 3 4 5 6 7 8
Upper limit
Cases (%)
1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
15.0 5.0 5.1 6.0 9.4 16.6 14.0 27.0
Upper limit
Cases (%)
1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
9.5 4.5 7.1 8.7 19.0 39.3 11.9 —
a1 − 0.012 − 0.003 0.034 0.006 0.032 0.096 0.174 0.511
− − − − − − − −
a2 0.418 0.551 0.497 0.600 0.637 0.762 2.319 5.951
a3 0.061 0.089 0.105 0.097 0.153 0.288 0.292 0.340
(b)
⑀ bin⬘ 1 2 3 4 5 6 7 8
F11 0.593 0.689 − 0.532 0.963 2.134 0.409 0.143 —
F12 0.568 0.432 − 0.157 − 0.496 0.009 1.344 3.805 —
F13 − 0.474 − 0.533 0.492 − 0.526 − 1.466 − 0.519 − 0.699 —
F21 0.111 0.568 − 0.191 0.554 1.094 0.706 0.628 —
F22 0.204 − 0.276 − 0.312 − 0.963 − 1.437 − 1.582 − 1.985 —
F23 − 0.107 − 0.309 0.306 − 0.104 − 0.371 − 0.066 0.033 —
Anisotropic brightening coefficient = F = a1 + a2⌬ + a3Z. Circumsolar brightening coefficient = F1 = F11 + F12⌬ + F13Z. Horizon brightening coefficient = F2 = F21 + F22⌬ + F23Z. Table 4. Irradiance coefficients determined for Madrid, from experimental data for all the vertical planes in the circumsolar version.
⑀⬘ bin 1 2 3 4 5 6 7 8
Upper limit
Cases (%)
1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
15.3 6.1 6.0 6.3 11.6 16.7 17.3 24.8
F11 − 0.114 0.213 0.617 0.670 0.995 0.873 0.673 − 0.065
F12 0.562 0.318 − 0.608 − 0.225 − 1.123 0.143 0.183 5.540
F13 0.052 − 0.123 − 0.151 − 0.294 − 0.386 − 0.568 − 0.491 − 0.430
F21 − 0.041 0.110 0.191 0.149 0.377 0.425 0.290 0.276
F22 0.081 − 0.350 − 0.611 − 0.231 − 0.959 − 1.298 − 1.450 − 0.799
F23 − 0.012 − 0.003 0.045 − 0.001 − 0.014 0.040 0.167 0.044
Circumsolar brightening coefficient = F1 = F11 + F12⌬ + F13Z. Horizon brightening coefficient = F2 = F21 + F22⌬ + F23Z.
5 are again anomalous for very clear skies. For this reason the data have also been separated in two groups (a/c⫽0 and a/c = 0) as for the point-source version. However for the circumsolar version the separation is less radical because now we shall have cases (included in a/c⫽0) for which the circumsolar disk is only partially seen from the considered surface. Using this procedure, the coefficients have been calculated for the N, E and W orientation but not for S, because there are almost no cases with a/c = 0 for this orientation. For the N, E and W planes, the coefficients follow a similar variation with ⑀⬘, as is shown in Table 6 for the N plane. 5. VALIDATION OF THE MODEL
To calculate the diffuse and global irradiances on vertical surfaces facing N, E, S and W, different sets of coefficients have been used as follows: (a) The coefficients of Perez [4] for the point-source version. (b) Local coefficients calculated for the point-source version using data obtained at Madrid for all four orientations (Madrid 1). (c) Coefficients calculated for each specific orientation with corre-
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Table 5. Irradiance coefficients determined for Madrid, from experimental data for the north plane in the circumsolar version.
⑀⬘ bin 1 2 3 4 5 6 7 8
Upper limit
Cases (%)
1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
16.2 4.9 5.3 6.3 10.5 19.3 13.6 24.0
F11 − 0.255 0.741 0.546 1.686 2.217 3.077 4.801 3.774
F12 1.289 0.910 − 0.075 − 1.250 − 0.642 − 0.864 − 1.783 21.228
F13 0.035 − 0.586 − 0.152 − 0.749 − 1.267 − 1.965 − 3.297 − 4.000
F21 − 0.131 0.369 0.244 0.795 1.064 1.368 2.393 2.400
F22 0.200 − 0.106 − 0.508 − 1.073 − 1.027 − 1.045 − 2.309 3.991
F23 0.075 − 0.202 0.007 − 0.281 − 0.472 − 0.673 − 1.328 − 1.631
Circumsolar brightening coefficient = F1 = F11 + F12⌬ + F13Z. Horizon brightening coefficient = F2 = F21 + F22⌬ + F23Z. Table 6. Irradiance coefficients determined for Madrid, from experimental data for the north plane in the circumsolar version; (a) a/c = 0, (b) a/c⫽0 (a)
⑀⬘ bin 1 2 3 4 5 6 7 8
Upper limit
Cases (%)
1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
23.6 5.6 6.0 6.8 8.5 13.6 11.3 24.6
Upper limit
Cases (%)
1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
8.4 4.2 4.7 5.8 12.6 25.1 15.9 23.4
a1 0.111 − 0.008 − 0.032 − 0.039 0.077 − 0.201 − 0.132 0.474
− − − − − − − −
a2 0.495 0.477 0.449 0.363 0.892 0.108 0.562 7.658
a3 0.054 0.070 0.071 0.052 0.081 0.214 0.313 0.510
(b)
⑀⬘ bin 1 2 3 4 5 6 7 8
F11 − 0.032 0.610 0.243 1.469 2.221 2.302 4.019 3.361
F12 0.978 1.015 − 0.140 − 1.332 − 1.035 − 0.030 0.238 0.441
F13 − 0.078 − 0.524 0.058 − 0.613 − 1.233 − 1.599 − 2.980 − 1.496
F21 0.074 0.336 0.080 0.631 0.958 1.084 2.101 2.209
F22 0.265 − 0.181 − 0.460 − 1.037 − 0.978 − 1.299 − 2.215 − 5.302
F23 0.021 − 0.160 0.125 − 0.155 − 0.393 − 0.413 − 1.114 − 0.528
Anisotropic brightening coefficient = F = a1 + a2⌬ + a3Z. Circumsolar brightening coefficient = F1 = F11 + F12⌬ + F13Z. Horizon brightening coefficient = F2 = F21 + F22⌬ + F23Z.
sponding data obtained at Madrid, using the point-source version (Madrid 2). (d) Coefficients for each orientation determined independently with Madrid data for a/c⫽0 and a/c = 0, using the point-source version (Madrid 3). (e) Coefficients given by Perez [3] for the circumsolar version. (f) Local coefficients calculated with the circumsolar version using data obtained at Madrid for all four orientations (Madrid 4). (g) Coefficients calculated for each orientation with data obtained at Madrid for that orientation, using the circumsolar version (Madrid 5). (h) Coefficients for each orientation determined independently with Madrid data for a/c⫽0 and a/c = 0, using the circumsolar version (Madrid 6). Table 7 gives the RMSD and MBD for each of the models. If we compare the results for both versions and the same type of coefficients, in general, a somewhat better prediction is obtained when the circumsolar version is used. However if we compare the RMSD for any of the point source models with the RMSD for any of the circumsolar models, we observe that the point source models with specific coefficients (Madrid 2 and Madrid 3) give lower RMSDs than the circumsolar models with non specific coefficients (Perez, [3] and Madrid 4). Relating the MBD all the models with local coefficients obtained using jointly data for all orientations underestimate the irradiance, while when specific coefficients for each orientation are used the MBD
Modelling irradiance on inclined planes with an anisotropic model
199
Table 7. Model performance. Model
(a) Perez [4] (0°) (b) Madrid 1 (0°) (c) Madrid 2 (0°) (d) Madrid 3 (0°) (e) Perez [3] (25°) (f) Madrid 4 (25°) (g) Madrid 5 (25°) (h) Madrid 6 (25°)
North 90°
18.5 11.4 10.2 9.8 15.2 10.6 8.7 8.4
East 90° South 90° RMSD (W/m2 ) 24.5 15.8 12.9 12.4 21.4 15.3 12.9 11.8
West 90°
All planes
22.0 16.4 14.9 14.9 21.2 16.0 14.9 −
21.7 16.1 14.4 14.0 18.5 15.3 13.3 13.0
21.8 15.0 13.2 12.9 19.2 14.4 12.6 11.2
MBD (W/m2 ) (a) Perez [4] (0°) (b) Madrid 1 (0°) (c) Madrid 2 (0°) (d) Madrid 3 (0°) (e) Perez [3] (25°) (f) Madrid 4 (25°) (g) Madrid 5 (25°) (h) Madrid 6 (25°)
− 12.7 − 1.9 0.1 0.0 − 10.6 − 1.9 0.1 0.0
− 4.5 − 1.5 − 0.1 0.0 − 3.1 − 1.5 2.4 0.0
− 3.6 − 2.9 − 0.2 0.0 − 1.2 − 2.9 0.0 −
− 1.2 0.7 0.0 0.1 − 0.4 0.7 0.0 0.0
− 5.5 − 1.4 − 0.1 0.0 − 3.9 − 1.4 0.6 0.0
Average global irradiance (W/m2 )
71.3
199.4
274.5
203.3
185.6
Average diffuse irradiance (W/m2 )
64.9
74.1
83.0
73.6
73.7
No. of Events
3219
3177
3008
3173
12577
is practically zero for both, point-source version and circumsolar version. The effect of separating the data for a/c⫽0 from the data for a/c = 0 has been assessed for all the models studied in the present work. Tables 8 and 9 show the RMSD and MBD for all models. For the point-source version, Table 8 shows than for both cases (a/c⫽0 and a/c = 0) the best estimation is obtained with the model Madrid 3 (0°). When we consider the models Madrid 2 (0°) and Madrid 3 (0°) the largest difference in the RMSDs for a/c⫽0 is obtained for the north orientation, but for the south facing plane there is no difference in the RMSDs. An explanation of this result is that for the north orientation there are many data for a/c = 0, and when these are suppressed prediction accuracy increases, while for the south orientation as there are few data for a/c = 0, when these are suppressed there is little influence in accuracy. Globally, predictions using one or other method are not too different. Relating the MBD, in both cases values obtained are close to zero. For the circumsolar version, Table 9 shows the RMSDs and the MBDs obtained with Madrid 5 (25°) and Madrid 6 (25°) for N, E and W orientations. For the S plane a distinction between a/c = 0 and a/c⫽0 can not be made because there are almost no data for a/c = 0. The prediction accuracy is better for the model Madrid 6 (25°) with data separation for both a/c = 0 and a/c⫽0. We may summarize the results obtained when data separation between the vertical plane "seeing" or not "seeing" the circumsolar disk, by the remark that the RMSD values are somewhat reduced. Furthermore, when a/c = 0, the easier-to-use Eq. (9) results. 6. CONCLUSIONS
From the results given in Tables 7–9, some conclusions can be obtained as follows. For the pointsource and circumsolar versions, when local coefficients are used with data for all orientations (Madrid 1 and 2) the model performs better than with the coefficients given by Perez [3,4]. Although this may be expected, because the Perez model is an empirical model, this fact has been emphasized in the present work due to the universal validity usually attributed to the Perez model. For both versions the model’s accuracy is increased when local coefficients calculated independently for each orientation are
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L. Robledo and A. Soler Table 8. Performance of the point-source version when (a) a/c = 0 and (b) a/c⫽0.
(a) Model
(a) (b) (c) (d)
Perez [4] Madrid 1 Madrid 2 Madrid 3
North 90°
(0°) (0°) (0°) (0°)
(a) Perez [4] (0°) (b) Madrid 1 (0°) (c) Madrid 2 (0°) (d) Madrid 3 (0°) Average global irradiance (W/m2 ) Average diffuse irradiance (W/m2 ) No. of Events (b) Model
(a) (b) (c) (d)
Perez [4] (0°) MadriD 1 (0°) Madrid 2 (0°) Madrid 3 (0°)
(a) Perez [4] (0°) (b) Madrid 1 (0°) (c) Madrid 2 (0°) (d) Madrid 3 (0°) Average global irradiance (W/m2 ) Average diffuse irradiance (W/m2 ) No. of Events
East 90° South 90° RMSD (W/m2 )
West 90°
All plane
18.1 10.8 10.1 10.0
16.5 11.1 9.0 8.8 MBD (W/m2 )
17.8 10.4 8.0 7.4
15.9 12.8 10.7 10.0
17.1 11.4 9.8 9.5
− 13.0 − 0.9 0.5 0.0 65.0
− 9.5 1.8 0.8 0.0 60.0
− 13.5 − 5.4 0.2 − 0.1 50.7
− 7.8 3.9 − 0.7 0.0 57.3
− 10.8 0.7 0.3 0.0 60.8
65.0
60.0
50.7
57.3
60.8
2840
1718
472
1641
6671
West 90°
All plane
North 90°
East 90° South 90° RMSD (W/m2 )
21.1 14.7 11.0 8.1
31.6 20.1 16.3 15.7 MBD (W/m2 )
22.7 17.3 15.9 15.9
26.5 19.0 17.6 17.2
24.5 17.2 15.2 14.9
− 11.0 − 9.2 − 3.3 0.0 118.0
1.2 − 5.5 − 1.1 0.0 363.6
− 1.8 − 2.5 − 0.2 0.0 316.1
5.9 − 2.7 0.8 0.1 359.6
0.3 − 3.3 − 0.3 0.0 326.4
64.0
90.7
88.9
91.1
88.3
379
1459
2536
1532
5906
used (Madrid 2 and Madrid 5). Another approach relates the Madrid 3 (0°) and 6 (25°)models. For both, the data have been separated into two groups, depending on the vertical surface "seeing" or "not seeing" the centre of the circumsolar disk (point-source model, Madrid 3) or part of this disk (circumsolar model, Madrid 6). Besides a decrease of the RMSD values, this approach offers a more adequate physical interpretation of the model’s coefficients. It also allows the use of a precise and simple model [Eq. (9)] when calculating the irradiance received on a vertical surface without the possibility of "seeing" the centre of the circumsolar disk in the point source version or part of the circumsolar disk in the circumsolar version. Acknowledgements—This research was funded in part by the Universidad Polite´cnica de Madrid through an Accio´n Concertada. REFERENCES
1. Perez, R., Scott, J. and Stewart, R., Proceedings of ASES. Minneapolis, MN, 1983, p. 883. 2. Hay, J. E. and McKay, D. C., IEA Solar Heating and Cooling Program, Task IX Final Report. Atmospheric Environment Service, Downsview, Ontario, Canada, 1987. 3. Perez, R. and Seals, R., Solar Energy, 1987, 39, 221. 4. Perez, R., Ineichen, P., Seals, R., Michalsky, J. and Stewart, R., Solar Energy, 1990, 44, 271. 5. Utrillas, M. P. and Martı´nez-Lozano, J. A., Solar Energy, 1994, 53, 155. 6. Kambezidis, H. D., Psiloglou, B. E. and Gueymard, C., Solar Energy, 1994, 53, 177. 7. Robledo, L. and Soler, A., Lighting Res. Technol., 1996, 28, 141. 8. Robledo L. and Soler, A., Submitted for publication to Energy Conversion and Management. 9. Perez, R., Stewart, R., Arbogast, C., Seals, R. and Scott, J., Solar Energy, 1986, 36, 487.
Modelling irradiance on inclined planes with an anisotropic model
201
Table 9. Performance of the circumsolar version when (a) a/c = 0 and (b) a/c⫽0. (a) Model
North 90°
East 90° South 90° RMSD (W/m2 )
West 90°
All plane
(g) Madrid 5 (25°) (h) Madrid 6 (25°)
8.0 7.5
8.2 6.0 MBD (W/m2 )
— —
7.6 6.8
7.9 6.9
(g) Madrid 5 (25°) (h) Madrid 6 (25°) Average global irradiance (W/m2 ) Average diffuse irradiance (W/m2 ) No. of events (b)
1.1 0.0 60.4
− 0.9 0.0 48.8
— — —
1.9 0.0 45.7
0.7 0.0 53.0
60.4
48.8
—
45.7
53.0
1631
1092
—
1031
3754
West 90°
All plane
Model
North 90°
East 90° South 90° RMSD (W/m2 )
(g) Madrid 5 (25°) (h) Madrid 6 (25°)
9.4 9.2
14.8 13.9 MBD (W/m2 )
— —
15.3 15.1
13.7 13.3
(g) Madrid 5 (25°) (h) Madrid 6 (25°) Average global irradiance (W/m2 ) Average diffuse irradiance (W/m2 ) No. of Events
− 1.0 0.0 82.4
− 2.2 0.0 278.3
— — —
− 0.9 0.0 279.1
− 1.4 0.0 225.1
69.5
87.3
—
87.1
82.4
1588
2085
—
2142
5815
NOMENCLATURE
D = Mean hourly diffuse irradiance on a plane with slope  (W/m2 ) Dh = Mean hourly horizontal diffuse irradiance (W/m2 ) I = Direct normal irradiance (W/m2 ) IO = Mean extraterrestrial irradiance (W/m2 ) L = Radiance corresponding to the isotropic zone L1 = Radiance corresponding to the circumsolar disk L2 = Radiance corresponding to the horizon band F = Anisotropic brightening coefficient (dimensionless) F1 = Circumsolar brightening coefficient (dimensionless) F2 = Horizon brightening coefficient (dimensionless) Fij = Perez simplified model coefficients (Fi1 and Fi2 dimensionless, Fi3 in degrees−1 ) Z = Zenith angle (degrees) a = Solid angle occupied by the circumsolar disk, weighted by its average incidence on the slope (sr) b = Solid angle occupied by the horizon band, weighted by its average incidence on the slope (sr)
c = Solid angle occupied by the circumsolar disk, weighted by its average incidence on the horizontal (sr) d = Solid angle occupied by the horizon band, weighted by its average incidence on the horizontal (sr) ci = Coefficients used to calculate values of F (c1 and c2 are dimensionless and c2 is in degrees−1 ) f1 = Original circumsolar brightening coefficient f2 = Original horizon brightening coefficient m = Optical air mass (dimensionless) GREEK LETTERS
␣ = Circumsolar disk half angle (degrees)  = Surface inclination (degrees) ⌬ = Brightness index (dimensionless) ⑀ = Sky clearness index (dimensionless) ⑀⬘ = Modified sky clearness index (dimensionless) = Solar incidence angle on vertical surface (degrees) = Angular width of horizon band (degrees)
PII: S0360-5442(97)00083-2
Energy Vol. 23, No. 3, pp. 193–201, 1998 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0360-5442/98 $19.00 + 0.00
MODELLING IRRADIANCE ON INCLINED PLANES WITH AN ANISOTROPIC MODEL LUIS ROBLEDO†‡ and ALFONSO SOLER§ † Departamento de Sistemas Inteligentes Aplicados, E. U. Informatica. Universidad Polite´cnica de Madrid. Ctra. de Valencia km 7. 28031 Madrid, Spain; §Departamento de Fisica e Instalaciones Aplicadas, E. T. S. de Arquitectura. Universidad Polite´cnica de Madrid. Avda. Juan de Herrera, 4. 28040 Madrid, Spain
(Received 19 March 1997)
Abstract—The circumsolar ( ␣ = 25°) and point-source versions of the simplified Perez model were developed to estimate irradiances on inclined planes from values for horizontal planes through the determination of sets of empirical coefficients. This model has been evaluated by us at Madrid. Mean hourly values of irradiances on vertical surfaces facing north, east, south and west were used. A modification in the structure of the model was made by analyzing separately the data obtained when the surface "sees" or does not "see" the circumsolar disk (for the point-source version, the centre of the disk). The validity of the different versions is assessed statistically using experimental and estimated values for the irradiance. For both the point-source and circumsolar versions, with data for all orientations, the model performs better with local coefficients than with those given by Perez. The accuracy was increased when local coefficients were calculated independently for each orientation. When the surface does not "see" the circumsolar disk (the centre of the disk in the case of the point-source version), the specified data separation produces an equation that is simpler and easier to use than the corresponding equation in the original model while yielding some increase in accuracy. 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
In the last two decades, different models have been developed to predict the diffuse solar irradiance on inclined surfaces with different orientations. Of all the anisotropic models, that by Perez [1], which is used to predict diffuse irradiance/illuminance on an inclined surface from diffuse irradiance/illuminance on a horizontal surface, has received special attention because of its good performance [2]. In its original formulation, the Perez model is somewhat difficult to use. The original model has been greatly simplified [3,4], to obtain circumsolar and point-source versions. Apparently the original formulation does not perform better than the simplified model [5]. Some authors have assessed both the original and the simplified versions [5–8] and have shown that empirical coefficients derived from experimental data are location-dependent. Robledo and Soler [7,8], using illuminance data obtained at Madrid for vertical surfaces facing north, east, south and west (in 1992–1993), have shown that the coefficients for the two versions depend on surface orientation. In the point-source version [7], the data were separated into sets for the surface "seeing" or not "seeing" the centre of the sun disk. The function which includes the coefficients of the model was then optimised independently for each set. In this way, an increasingly simplified model is obtained when the surface does not "see" the centre of the circumsolar disk, and the error in the estimation of the experimental values is reduced. The present work is based on 1994–1995 measurements and represents a more detailed study of the two simplified versions. The results obtained for irradiance confirm the conclusions of previous illuminance work. A modification of the circumsolar version is proposed. 2. MODEL DESCRIPTION
The main difficulty concerning the study of diffuse radiation is its anisotropic nature. The most important anisotropic effects for solar radiation passing through the atmosphere are forward scattering ‡Author for correspondence. Fax: 34 1 3367522. 193
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L. Robledo and A. Soler
by aerosols and multiple scattering and retroscattering near the horizon. In the simplified Perez model the sky dome consists of one isotropic and two anisotropic zones; the circumsolar region is characterized by the half angle ␣ and the horizon band is characterized by the angle . Radiation from each of these regions is assumed to be constant in the areas where they are defined. Another basic characteristic of the model is parametrization of the sky conditions, which was first done in [1,9] using the solar zenith angle Z, horizontal diffuse radiation Dh and clearness index ⑀ = (Dh + I)/Dh as variables. I is the direct normal irradiance. With the objective of reducing the dependence of the second parameter on the sun’s position (zenith angle Z), the new quantity ⌬ = Dhm/IO
(1)
with m = optical air mass and IO = extraterrestrial irradiance. For the same reason the clearness index was redefined as
⑀⬘ = ( ⑀ + kZ3 )/(1 + kZ3 ),
(2)
where k = 1.041 and Z is in radians. Thus, the sky conditions are now given in terms of the solar zenith angle Z, the brightness index ⌬ and the clearness index ⑀⬘. If L, L1 and L2 are radiances corresponding, respectively, to the isotropic, circumsolar and horizon zones, the original model gives the diffuse irradiance D on an inclined plane with slope  as follows: D = Dh
0.5(1 + cos ) + a(f1 − 1) + b(f2 − 1) 1 + c(f1 − 1) + d(f2 − 1)
(3)
where Dh is the diffuse irradiance on a horizontal plane, f1 = L1 /L, f2 = L2 /L, and a, b, c and d are quantities which depend on the solar zenith angle Z, half angle of the circumsolar disk ␣ and solar incident angle on the inclined plane [9]. The coefficients f1 and f2 in Eq. (3) are ⱖ 1. For f1 = f2 = 1, Eq. (3) reduces to the equation for the isotropic model. The non-linearity of Eq. (3) makes the evaluation of f1 and f2 difficult by a least-mean-squares fit. For this reason, Perez [3] proposed a reformulation of Eq. (3) as D = Dh[0.5(1 + cos )(1 − F1 − F2 ) + (a/c)F1 + (b/d)F2 ],
(4)
with F1 and F2 representing the respective normalized contributions of the circumsolar and horizon zones to the total diffuse irradiance received on a horizontal plane. In Eq. (4), F1 and F2 range from 0 to 1. If F1 = F2 = 0, Eq. (4) reduces to the equation for the isotropic model. The quantities a, b, c and d are the same as those in Eq. (3). Other simplifications were introduced by Perez [3] into the geometry of the model. The first of these is based on the horizon zone being infinitesimally thin at 0° elevation (linear horizon). With this supposition, 1 − F1 − F2 ⯝ 1 − F1, b/d = sin, and Eq. (4) becomes now D = Dh[0.5(1 + cos )(1 − F1 ) + (a/c)F1 ) + F2sin ].
(5)
A major simplification involves the assumption that all circumsolar radiation comes from a point placed at the centre of the circumsolar disk (point source). Then a = max(0,cos ),c = max(0.087, cosZ),
(6)
where 0.087 = cos85° is introduced because values corresponding to low solar elevations ( ␣ ⬍ 5°) are not considered and discontinuities for a/c = ⬁ are avoided. Equation (5) establishes a linear dependence of D on the coefficients F1 and F2. When assuming a discrete set of categories or bins for ⑀⬘, the Fi may be considered to be linear functions of ⌬ and Z for each of the bins in the form
Modelling irradiance on inclined planes with an anisotropic model
195
Fi = Fi1 + Fi2⌬ + Fi3Z,i = 1,2.
(7)
Thus, from Eqs. (5) and (7) and for each of the categories of ⑀⬘, a set of 6 coefficients Fij is obtained by a least-squares fit to the experimental data for D and Dh. As stated Robledo and Soler [7], when evaluating the point-source version for illuminance measured on vertical surfaces facing N, E, S and W at Madrid, found that when the model was used to fit data for all orientations, it performed better with local coefficients than with those proposed by Perez, and that the model accuracy increased further when it was used with local coefficients derived independently for each orientation. In fact, different behaviour of the model was found for each orientation. For example, for the north-facing surface, which receives almost no direct radiation, atypical and artificial coefficients were obtained for some of the bins because, when the function given by Eq. (5) was optimized by a least-mean-squares fit, the resulting coefficients Fi (i.e. F1 ) lacked physical meaning due to the absence of a significant circumsolar contribution. In [7], a slight modification was introduced in the utilization of the point-source version. Thus, the data obtained were fitted separately when the surface "sees" the centre of the circumsolar disk (a/c⫽0, ⬍ /2, cos ⬎ 0), and when the surface does not "see" the centre of the disk (a/c = 0, ⬎ /2, cos ⬍ 0). For vertical planes and a/c⫽ 0, using Eq. (7), Eq. (5) becomes D = Dh[0.5 + F21 + (a/c − 0.5)F11 + (a/c − 0.5)⌬F12 + (a/c − 0.5)ZF13 + F22⌬ + F23Z].
(8)
For a/c = 0, Eq. (5) shows a linear dependence of D /Dh on ⌬ and Z. The diffuse irradiance on a vertical plane becomes now [7] D /Dh = (0.5 + F)
(9)
with F = a1 + a2⌬ + a3Z representing the normalized contribution of the anisotropic (circumsolar and horizon) to the total diffuse irradiance on a horizontal surface. When F = 0, Eq. (9) reduces to the isotropic model. Once D is obtained, the global irradiance on tilted planes can be predicted if the direct irradiance on the horizontal plane is known. The direct irradiance on a vertical plane is obtained from its value for a horizontal plane. 3. EXPERIMENTAL DATA AND STATISTICAL METHODS
The experimental data consist of mean hourly values of diffuse and global irradiance on a horizontal surface and global irradiance on vertical surfaces facing north, east, south and west, obtained at Madrid (40.3°N, 4.4°W) for the period from June 1994 to July 1995. All vertical sensors are used with artificial horizons made of matt black painted honeycomb material. Thus, the ground-reflected irradiance is considered as negligible. Experimental data were not used when Z ⬎ 85° or when negative values of the direct horizontal or diffuse vertical irradiances were obtained. The accuracy of the model was determined by using as statistical estimators the MBD = ⌺i(yi − xi )/N and the RMSD = [⌺i(yi − xi )2 /N]1/2 where yi is the ith predicted value, xi the ith measured value and N the number of values. Experimental values of diffuse irradiance on vertical surfaces have been obtained as the difference between the experimental values of global and direct irradiances. 4. RESULTS AND DISCUSSION
4.1. Point-Source version The coefficients for the point-source version in Table 1 refer to the simplified Perez model using irradiance data for all orientations. If irradiance data are used for each orientation, new tables are obtained with coefficients that for the E, S and W planes follow the same type of variation with ⑀⬘ as the data in Table 1. However, if the coefficients for the last two bins of the north plane given in Table 2 are considered (especially for bin 8), corresponding to clear skies, rather atypical values result. The
196
L. Robledo and A. Soler
Table 1. Irradiance coefficients determined for Madrid, from experimental data for all of the vertical planes in the point-source version
⑀⬘bin 1 2 3 4 5 6 7 8
Upper limit 1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
Cases (%) 15.3 6.1 6.0 6.3 11.6 16.7 17.3 20.6
F11 0.057 0.366 0.784 0.838 1.025 0.921 0.636 − 0.044
F12 0.307 0.263 − 0.560 − 0.149 − 0.619 0.189 0.243 4.844
− − − − − − − −
F13 0.057 0.257 0.338 0.490 0.554 0.644 0.477 0.391
F21 − 0.003 0.169 0.254 0.211 0.442 0.486 0.299 0.279
F22 0.075 − 0.365 − 0.564 − 0.177 − 0.936 − 1.402 − 1.483 − 0.884
F23 − 0.046 − 0.047 − 0.028 − 0.071 − 0.077 0.003 0.167 0.049
Circumsolar brightening coefficient = F1 = F11 + F12⌬ + F13Z Horizon brightening coefficient = F2 = F21 + F22⌬ + F23Z
Table 2. Irradiance coefficients determined for Madrid, from experimental data for the north plane in the point-source version.
⑀⬘bin 1 2 3 4 5 6 7 8
Upper limit 1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
Cases (%) 16.1 4.9 5.3 6.3 10.5 19.3 13.6 24.0
F11 0.264 1.576 1.108 2.057 3.147 3.052 4.716 − 93.47
F12 0.692 0.522 − 0.044 − 0.631 − 0.243 0.216 7.878 − 27.41
F13 − 0.266 − 1.155 − 0.662 − 1.228 − 2.087 − 2.186 − 4.335 95.99
F21 0.116 0.797 0.515 1.044 1.579 1.453 2.522 − 46.22
F22 − 0.062 − 0.304 − 0.506 − 1.029 − 0.946 − 0.886 1.643 − 19.65
F23 0.070 − 0.487 − 0.221 − 0.485 − 0.872 − 0.769 − 1.861 48.33
Circumsolar brightening coefficient = F1 = F11 + F12⌬ + F13Z Horizon brightening coefficient = F2 = F21 + F22⌬ + F23Z
reason is that for N orientation there is almost no direct irradiance. For clear skies, the Fij are difficult to justify. We note that for a vertical surface, Eq. (5) becomes D = Dh[0.5 + (a/c − 0.5)F1 + F2 ].
(10)
For bin 8, a/c = 0 for almost all data. Examination of Table 2 shows that experimentally F1 ⯝ 2F2, so that Eq. (10) leads to D = 0.5Dh.
(11)
Eq. (11) corresponds to the isotropic radiance distribution, which may be adequate for overcast but not for clear skies. Under these conditions the 6 coefficients Fij appearing in the last bin do not have any meaning. The fact that there is almost no direct irradiance makes utilization of the Perez model difficult, because the mathematically justified coefficients obtained for the last bins are difficult to understand in physical terms. This situation leads us to consider separating data for a/c⫽0 from data for a/c = 0. In the first of these cases Eq. (5) is used to calculate the diffuse irradiance, while for the second case Eq. (9) is the most advisable. With the specified modification, comparisons of the results for different orientations now show a uniform variation of the ci and Fij with intervals of ⑀⬘, not only in the S, E and W orientation, but also in the N orientation as is shown in Table 3. For all orientations and a/c = 0 F has values close to zero for small values of ⑀⬘ (overcast skies) but increases as expected on physical grounds with ⑀⬘, because the anisotropy increases when the skies become clearer. 4.2. Circumsolar version In Table 4 coefficients are given for the circumsolar version of the simplified Perez model using data for all orientations. In Tables 4 and 1 no appreciable difference is observed relating the way the Fij vary with the different bins considered for ⑀⬘. For the model coefficients obtained using data for each specific orientation, a similar pattern of variation with ⑀⬘ as that for the point source version results. Coefficients for the N plane given in Table
Modelling irradiance on inclined planes with an anisotropic model
197
Table 3. Irradiance coefficients determined for Madrid, from experimental data for the north plane in the point − source version; (a) a/c = 0, (b) a/c⫽0. (a)
⑀ bin⬘ 1 2 3 4 5 6 7 8
Upper limit
Cases (%)
1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
15.0 5.0 5.1 6.0 9.4 16.6 14.0 27.0
Upper limit
Cases (%)
1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
9.5 4.5 7.1 8.7 19.0 39.3 11.9 —
a1 − 0.012 − 0.003 0.034 0.006 0.032 0.096 0.174 0.511
− − − − − − − −
a2 0.418 0.551 0.497 0.600 0.637 0.762 2.319 5.951
a3 0.061 0.089 0.105 0.097 0.153 0.288 0.292 0.340
(b)
⑀ bin⬘ 1 2 3 4 5 6 7 8
F11 0.593 0.689 − 0.532 0.963 2.134 0.409 0.143 —
F12 0.568 0.432 − 0.157 − 0.496 0.009 1.344 3.805 —
F13 − 0.474 − 0.533 0.492 − 0.526 − 1.466 − 0.519 − 0.699 —
F21 0.111 0.568 − 0.191 0.554 1.094 0.706 0.628 —
F22 0.204 − 0.276 − 0.312 − 0.963 − 1.437 − 1.582 − 1.985 —
F23 − 0.107 − 0.309 0.306 − 0.104 − 0.371 − 0.066 0.033 —
Anisotropic brightening coefficient = F = a1 + a2⌬ + a3Z. Circumsolar brightening coefficient = F1 = F11 + F12⌬ + F13Z. Horizon brightening coefficient = F2 = F21 + F22⌬ + F23Z. Table 4. Irradiance coefficients determined for Madrid, from experimental data for all the vertical planes in the circumsolar version.
⑀⬘ bin 1 2 3 4 5 6 7 8
Upper limit
Cases (%)
1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
15.3 6.1 6.0 6.3 11.6 16.7 17.3 24.8
F11 − 0.114 0.213 0.617 0.670 0.995 0.873 0.673 − 0.065
F12 0.562 0.318 − 0.608 − 0.225 − 1.123 0.143 0.183 5.540
F13 0.052 − 0.123 − 0.151 − 0.294 − 0.386 − 0.568 − 0.491 − 0.430
F21 − 0.041 0.110 0.191 0.149 0.377 0.425 0.290 0.276
F22 0.081 − 0.350 − 0.611 − 0.231 − 0.959 − 1.298 − 1.450 − 0.799
F23 − 0.012 − 0.003 0.045 − 0.001 − 0.014 0.040 0.167 0.044
Circumsolar brightening coefficient = F1 = F11 + F12⌬ + F13Z. Horizon brightening coefficient = F2 = F21 + F22⌬ + F23Z.
5 are again anomalous for very clear skies. For this reason the data have also been separated in two groups (a/c⫽0 and a/c = 0) as for the point-source version. However for the circumsolar version the separation is less radical because now we shall have cases (included in a/c⫽0) for which the circumsolar disk is only partially seen from the considered surface. Using this procedure, the coefficients have been calculated for the N, E and W orientation but not for S, because there are almost no cases with a/c = 0 for this orientation. For the N, E and W planes, the coefficients follow a similar variation with ⑀⬘, as is shown in Table 6 for the N plane. 5. VALIDATION OF THE MODEL
To calculate the diffuse and global irradiances on vertical surfaces facing N, E, S and W, different sets of coefficients have been used as follows: (a) The coefficients of Perez [4] for the point-source version. (b) Local coefficients calculated for the point-source version using data obtained at Madrid for all four orientations (Madrid 1). (c) Coefficients calculated for each specific orientation with corre-
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L. Robledo and A. Soler
Table 5. Irradiance coefficients determined for Madrid, from experimental data for the north plane in the circumsolar version.
⑀⬘ bin 1 2 3 4 5 6 7 8
Upper limit
Cases (%)
1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
16.2 4.9 5.3 6.3 10.5 19.3 13.6 24.0
F11 − 0.255 0.741 0.546 1.686 2.217 3.077 4.801 3.774
F12 1.289 0.910 − 0.075 − 1.250 − 0.642 − 0.864 − 1.783 21.228
F13 0.035 − 0.586 − 0.152 − 0.749 − 1.267 − 1.965 − 3.297 − 4.000
F21 − 0.131 0.369 0.244 0.795 1.064 1.368 2.393 2.400
F22 0.200 − 0.106 − 0.508 − 1.073 − 1.027 − 1.045 − 2.309 3.991
F23 0.075 − 0.202 0.007 − 0.281 − 0.472 − 0.673 − 1.328 − 1.631
Circumsolar brightening coefficient = F1 = F11 + F12⌬ + F13Z. Horizon brightening coefficient = F2 = F21 + F22⌬ + F23Z. Table 6. Irradiance coefficients determined for Madrid, from experimental data for the north plane in the circumsolar version; (a) a/c = 0, (b) a/c⫽0 (a)
⑀⬘ bin 1 2 3 4 5 6 7 8
Upper limit
Cases (%)
1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
23.6 5.6 6.0 6.8 8.5 13.6 11.3 24.6
Upper limit
Cases (%)
1.065 1.230 1.500 1.950 2.800 4.500 6.200 —
8.4 4.2 4.7 5.8 12.6 25.1 15.9 23.4
a1 0.111 − 0.008 − 0.032 − 0.039 0.077 − 0.201 − 0.132 0.474
− − − − − − − −
a2 0.495 0.477 0.449 0.363 0.892 0.108 0.562 7.658
a3 0.054 0.070 0.071 0.052 0.081 0.214 0.313 0.510
(b)
⑀⬘ bin 1 2 3 4 5 6 7 8
F11 − 0.032 0.610 0.243 1.469 2.221 2.302 4.019 3.361
F12 0.978 1.015 − 0.140 − 1.332 − 1.035 − 0.030 0.238 0.441
F13 − 0.078 − 0.524 0.058 − 0.613 − 1.233 − 1.599 − 2.980 − 1.496
F21 0.074 0.336 0.080 0.631 0.958 1.084 2.101 2.209
F22 0.265 − 0.181 − 0.460 − 1.037 − 0.978 − 1.299 − 2.215 − 5.302
F23 0.021 − 0.160 0.125 − 0.155 − 0.393 − 0.413 − 1.114 − 0.528
Anisotropic brightening coefficient = F = a1 + a2⌬ + a3Z. Circumsolar brightening coefficient = F1 = F11 + F12⌬ + F13Z. Horizon brightening coefficient = F2 = F21 + F22⌬ + F23Z.
sponding data obtained at Madrid, using the point-source version (Madrid 2). (d) Coefficients for each orientation determined independently with Madrid data for a/c⫽0 and a/c = 0, using the point-source version (Madrid 3). (e) Coefficients given by Perez [3] for the circumsolar version. (f) Local coefficients calculated with the circumsolar version using data obtained at Madrid for all four orientations (Madrid 4). (g) Coefficients calculated for each orientation with data obtained at Madrid for that orientation, using the circumsolar version (Madrid 5). (h) Coefficients for each orientation determined independently with Madrid data for a/c⫽0 and a/c = 0, using the circumsolar version (Madrid 6). Table 7 gives the RMSD and MBD for each of the models. If we compare the results for both versions and the same type of coefficients, in general, a somewhat better prediction is obtained when the circumsolar version is used. However if we compare the RMSD for any of the point source models with the RMSD for any of the circumsolar models, we observe that the point source models with specific coefficients (Madrid 2 and Madrid 3) give lower RMSDs than the circumsolar models with non specific coefficients (Perez, [3] and Madrid 4). Relating the MBD all the models with local coefficients obtained using jointly data for all orientations underestimate the irradiance, while when specific coefficients for each orientation are used the MBD
Modelling irradiance on inclined planes with an anisotropic model
199
Table 7. Model performance. Model
(a) Perez [4] (0°) (b) Madrid 1 (0°) (c) Madrid 2 (0°) (d) Madrid 3 (0°) (e) Perez [3] (25°) (f) Madrid 4 (25°) (g) Madrid 5 (25°) (h) Madrid 6 (25°)
North 90°
18.5 11.4 10.2 9.8 15.2 10.6 8.7 8.4
East 90° South 90° RMSD (W/m2 ) 24.5 15.8 12.9 12.4 21.4 15.3 12.9 11.8
West 90°
All planes
22.0 16.4 14.9 14.9 21.2 16.0 14.9 −
21.7 16.1 14.4 14.0 18.5 15.3 13.3 13.0
21.8 15.0 13.2 12.9 19.2 14.4 12.6 11.2
MBD (W/m2 ) (a) Perez [4] (0°) (b) Madrid 1 (0°) (c) Madrid 2 (0°) (d) Madrid 3 (0°) (e) Perez [3] (25°) (f) Madrid 4 (25°) (g) Madrid 5 (25°) (h) Madrid 6 (25°)
− 12.7 − 1.9 0.1 0.0 − 10.6 − 1.9 0.1 0.0
− 4.5 − 1.5 − 0.1 0.0 − 3.1 − 1.5 2.4 0.0
− 3.6 − 2.9 − 0.2 0.0 − 1.2 − 2.9 0.0 −
− 1.2 0.7 0.0 0.1 − 0.4 0.7 0.0 0.0
− 5.5 − 1.4 − 0.1 0.0 − 3.9 − 1.4 0.6 0.0
Average global irradiance (W/m2 )
71.3
199.4
274.5
203.3
185.6
Average diffuse irradiance (W/m2 )
64.9
74.1
83.0
73.6
73.7
No. of Events
3219
3177
3008
3173
12577
is practically zero for both, point-source version and circumsolar version. The effect of separating the data for a/c⫽0 from the data for a/c = 0 has been assessed for all the models studied in the present work. Tables 8 and 9 show the RMSD and MBD for all models. For the point-source version, Table 8 shows than for both cases (a/c⫽0 and a/c = 0) the best estimation is obtained with the model Madrid 3 (0°). When we consider the models Madrid 2 (0°) and Madrid 3 (0°) the largest difference in the RMSDs for a/c⫽0 is obtained for the north orientation, but for the south facing plane there is no difference in the RMSDs. An explanation of this result is that for the north orientation there are many data for a/c = 0, and when these are suppressed prediction accuracy increases, while for the south orientation as there are few data for a/c = 0, when these are suppressed there is little influence in accuracy. Globally, predictions using one or other method are not too different. Relating the MBD, in both cases values obtained are close to zero. For the circumsolar version, Table 9 shows the RMSDs and the MBDs obtained with Madrid 5 (25°) and Madrid 6 (25°) for N, E and W orientations. For the S plane a distinction between a/c = 0 and a/c⫽0 can not be made because there are almost no data for a/c = 0. The prediction accuracy is better for the model Madrid 6 (25°) with data separation for both a/c = 0 and a/c⫽0. We may summarize the results obtained when data separation between the vertical plane "seeing" or not "seeing" the circumsolar disk, by the remark that the RMSD values are somewhat reduced. Furthermore, when a/c = 0, the easier-to-use Eq. (9) results. 6. CONCLUSIONS
From the results given in Tables 7–9, some conclusions can be obtained as follows. For the pointsource and circumsolar versions, when local coefficients are used with data for all orientations (Madrid 1 and 2) the model performs better than with the coefficients given by Perez [3,4]. Although this may be expected, because the Perez model is an empirical model, this fact has been emphasized in the present work due to the universal validity usually attributed to the Perez model. For both versions the model’s accuracy is increased when local coefficients calculated independently for each orientation are
200
L. Robledo and A. Soler Table 8. Performance of the point-source version when (a) a/c = 0 and (b) a/c⫽0.
(a) Model
(a) (b) (c) (d)
Perez [4] Madrid 1 Madrid 2 Madrid 3
North 90°
(0°) (0°) (0°) (0°)
(a) Perez [4] (0°) (b) Madrid 1 (0°) (c) Madrid 2 (0°) (d) Madrid 3 (0°) Average global irradiance (W/m2 ) Average diffuse irradiance (W/m2 ) No. of Events (b) Model
(a) (b) (c) (d)
Perez [4] (0°) MadriD 1 (0°) Madrid 2 (0°) Madrid 3 (0°)
(a) Perez [4] (0°) (b) Madrid 1 (0°) (c) Madrid 2 (0°) (d) Madrid 3 (0°) Average global irradiance (W/m2 ) Average diffuse irradiance (W/m2 ) No. of Events
East 90° South 90° RMSD (W/m2 )
West 90°
All plane
18.1 10.8 10.1 10.0
16.5 11.1 9.0 8.8 MBD (W/m2 )
17.8 10.4 8.0 7.4
15.9 12.8 10.7 10.0
17.1 11.4 9.8 9.5
− 13.0 − 0.9 0.5 0.0 65.0
− 9.5 1.8 0.8 0.0 60.0
− 13.5 − 5.4 0.2 − 0.1 50.7
− 7.8 3.9 − 0.7 0.0 57.3
− 10.8 0.7 0.3 0.0 60.8
65.0
60.0
50.7
57.3
60.8
2840
1718
472
1641
6671
West 90°
All plane
North 90°
East 90° South 90° RMSD (W/m2 )
21.1 14.7 11.0 8.1
31.6 20.1 16.3 15.7 MBD (W/m2 )
22.7 17.3 15.9 15.9
26.5 19.0 17.6 17.2
24.5 17.2 15.2 14.9
− 11.0 − 9.2 − 3.3 0.0 118.0
1.2 − 5.5 − 1.1 0.0 363.6
− 1.8 − 2.5 − 0.2 0.0 316.1
5.9 − 2.7 0.8 0.1 359.6
0.3 − 3.3 − 0.3 0.0 326.4
64.0
90.7
88.9
91.1
88.3
379
1459
2536
1532
5906
used (Madrid 2 and Madrid 5). Another approach relates the Madrid 3 (0°) and 6 (25°)models. For both, the data have been separated into two groups, depending on the vertical surface "seeing" or "not seeing" the centre of the circumsolar disk (point-source model, Madrid 3) or part of this disk (circumsolar model, Madrid 6). Besides a decrease of the RMSD values, this approach offers a more adequate physical interpretation of the model’s coefficients. It also allows the use of a precise and simple model [Eq. (9)] when calculating the irradiance received on a vertical surface without the possibility of "seeing" the centre of the circumsolar disk in the point source version or part of the circumsolar disk in the circumsolar version. Acknowledgements—This research was funded in part by the Universidad Polite´cnica de Madrid through an Accio´n Concertada. REFERENCES
1. Perez, R., Scott, J. and Stewart, R., Proceedings of ASES. Minneapolis, MN, 1983, p. 883. 2. Hay, J. E. and McKay, D. C., IEA Solar Heating and Cooling Program, Task IX Final Report. Atmospheric Environment Service, Downsview, Ontario, Canada, 1987. 3. Perez, R. and Seals, R., Solar Energy, 1987, 39, 221. 4. Perez, R., Ineichen, P., Seals, R., Michalsky, J. and Stewart, R., Solar Energy, 1990, 44, 271. 5. Utrillas, M. P. and Martı´nez-Lozano, J. A., Solar Energy, 1994, 53, 155. 6. Kambezidis, H. D., Psiloglou, B. E. and Gueymard, C., Solar Energy, 1994, 53, 177. 7. Robledo, L. and Soler, A., Lighting Res. Technol., 1996, 28, 141. 8. Robledo L. and Soler, A., Submitted for publication to Energy Conversion and Management. 9. Perez, R., Stewart, R., Arbogast, C., Seals, R. and Scott, J., Solar Energy, 1986, 36, 487.
Modelling irradiance on inclined planes with an anisotropic model
201
Table 9. Performance of the circumsolar version when (a) a/c = 0 and (b) a/c⫽0. (a) Model
North 90°
East 90° South 90° RMSD (W/m2 )
West 90°
All plane
(g) Madrid 5 (25°) (h) Madrid 6 (25°)
8.0 7.5
8.2 6.0 MBD (W/m2 )
— —
7.6 6.8
7.9 6.9
(g) Madrid 5 (25°) (h) Madrid 6 (25°) Average global irradiance (W/m2 ) Average diffuse irradiance (W/m2 ) No. of events (b)
1.1 0.0 60.4
− 0.9 0.0 48.8
— — —
1.9 0.0 45.7
0.7 0.0 53.0
60.4
48.8
—
45.7
53.0
1631
1092
—
1031
3754
West 90°
All plane
Model
North 90°
East 90° South 90° RMSD (W/m2 )
(g) Madrid 5 (25°) (h) Madrid 6 (25°)
9.4 9.2
14.8 13.9 MBD (W/m2 )
— —
15.3 15.1
13.7 13.3
(g) Madrid 5 (25°) (h) Madrid 6 (25°) Average global irradiance (W/m2 ) Average diffuse irradiance (W/m2 ) No. of Events
− 1.0 0.0 82.4
− 2.2 0.0 278.3
— — —
− 0.9 0.0 279.1
− 1.4 0.0 225.1
69.5
87.3
—
87.1
82.4
1588
2085
—
2142
5815
NOMENCLATURE
D = Mean hourly diffuse irradiance on a plane with slope  (W/m2 ) Dh = Mean hourly horizontal diffuse irradiance (W/m2 ) I = Direct normal irradiance (W/m2 ) IO = Mean extraterrestrial irradiance (W/m2 ) L = Radiance corresponding to the isotropic zone L1 = Radiance corresponding to the circumsolar disk L2 = Radiance corresponding to the horizon band F = Anisotropic brightening coefficient (dimensionless) F1 = Circumsolar brightening coefficient (dimensionless) F2 = Horizon brightening coefficient (dimensionless) Fij = Perez simplified model coefficients (Fi1 and Fi2 dimensionless, Fi3 in degrees−1 ) Z = Zenith angle (degrees) a = Solid angle occupied by the circumsolar disk, weighted by its average incidence on the slope (sr) b = Solid angle occupied by the horizon band, weighted by its average incidence on the slope (sr)
c = Solid angle occupied by the circumsolar disk, weighted by its average incidence on the horizontal (sr) d = Solid angle occupied by the horizon band, weighted by its average incidence on the horizontal (sr) ci = Coefficients used to calculate values of F (c1 and c2 are dimensionless and c2 is in degrees−1 ) f1 = Original circumsolar brightening coefficient f2 = Original horizon brightening coefficient m = Optical air mass (dimensionless) GREEK LETTERS
␣ = Circumsolar disk half angle (degrees)  = Surface inclination (degrees) ⌬ = Brightness index (dimensionless) ⑀ = Sky clearness index (dimensionless) ⑀⬘ = Modified sky clearness index (dimensionless) = Solar incidence angle on vertical surface (degrees) = Angular width of horizon band (degrees)