In a previous work (Cucumo M. et al., 2001), the authors applied the first method of Perez to calculate the hourly incident solar radiation on the vertical south-, west-, north — and east-facing surfaces, obtaining mean deviations for all surfaces ranging from 0% to 6,5% and mean square deviations from 18% to 23%. The Perez method for the calculation of global solar radiation is similar to that for the calculation of global illuminance, the structure of the equation used being similar to that of eq. 1.

In this paper the authors attempt to modify the first Perez model to obtain better predictions of illuminance on vertical surfaces. The subject of a future paper will be the analysis and attempted improvement of the second Perez model.

The high mean deviations and mean square deviations for west, north and east-facing surfaces could be justified by the fact that, for calculation of illuminance on vertical surfaces, as many as 5 correlations are used: the Erbs resolution correlation, the correlation of the effective illuminance of direct radiation, the correlation of the effective illuminance of diffuse radiation and the two correlations linking parameters F1 and F2 to the independent variables of the Perez model; whereas only 3 correlations are used for the calculation of global radiation.

Initially an attempt was made to find new correlations for the parameters F1 and F2, still following the Perez methodology, in such a way as to minimise the deviations between the calculated and experimental values. The results of these calculations are shown in table 3:

Table 3 — Mean deviations and mean square deviations between the experimental and calculated hourly illuminance using the first (Perez 1) method — with refit values of Fi and F2. South West North East 8 -11.5 -43.2 -57.5 -27.2 RMS 25.2 74.8 80.4 54.8

Table 3 compared with table 1, indicates an improvement of predictions on all the surfaces. The deviations are however still too great.

South Wall hour —■—Exper. —■—calc.

West Wall hour Exper. calc.

A closer analysis of calculated and experimental values revealed that the percentage errors are more contained during the hours when the surface is exposed to the sun, that is when direct light falls on it, whereas they are greater when the surface is only lit by diffuse light. This is clear in the graphs of Fig. 1, in which the calculated and experimental illuminance trend values for the four vertical surfaces during one day are shown.

л 100.000 "S’ 80.0 ш 60.000 40.000 20.0 0 0 2 4 6 8 10 12 14 16 18 20 22 24 hour Exper. —■—calc.

Exper. calc.

East Wall 0 2 4 6 8 10 12 14 16 18 20 22 24 hour

North Wall

Fig. 1 — Experimental and calculated hourly illuminance E for the four vertical surfaces on 30 June 2001.

The same observation can be deduced, indirectly, from table 1, observing that the errors relative to the south-facing surface, which is lit by the sun for the most hours, are lower compared to those of the other surfaces.

To point out this characteristic of the Perez model more quantitatively, the error calculations were repeated, separating the hours in which the surfaces are lit by the sun (Rb>0) from the hours in which the surfaces are not lit by the sun (Rb<0). The results of the calculation are shown in tables 4 and 5.

Table 4 — Mean deviations and mean square deviations between the experimental hourly illuminance and those calculated using the first Perez method for hourly data with Rb<0. South West North East 8 -43.6 -89.8 -74.0 -62.8 RMS 56.2 108.3 89.0 81.6

	South	West	North	East
8	-16.8	-14.6	-23.1	-10.2
RMS	26.0	35.0	35.0	28.4

Table 5 — Mean deviations and mean square deviations between the experimental hourly illuminance and those calculated using the first Perez method for hourly data with Rb>0.

Given the lack of ability of the first Perez method, both in the original and in the modified version with the refit of parameters F1 and F2, to predict well the experimental data of Arcavacata, the authors have developed the following simplified calculation method: the diffuse illuminance from the horizon is neglected, since its weight is always negligible, and, the following equation for the calculation of illuminance on a surface orientated in any direction:

E = Eb0Rb + Ed0(1 — F1) f+ E^FR + (Eb0 + Ed0)pf4)

Parameter Fi, which takes circumsolar radiation into account, was obtained, retaining it to be dependent on the same independent variables used in the Perez model (sky clearness index є, Д sky illuminance, zenith angle z), using the hourly experimental illuminance data of the single south-facing surface, during the hours in which that surface is lit by direct light (Rb>0). In fact, the south-facing surface is almost always lit by direct light, at the latitude of Arcavacata.

E

1+cosP 2

1 — cos p 2

5)

+ (Eb0 + Ed0) P

Edo(1- Fi)

For the hours when the surfaces are not lit by the sun (Rb<0), their illuminance is calculated with the relation:

The parameter F1′ was obtained, as a function of the variables used in the Perez model, considering the hourly experimental illuminance of all the vertical surfaces. The quantity (1-F1) is the diffuse illuminance reduction factor on the horizontal surface with the aim of evaluating the diffuse light striking the inclined surface in the absence of direct light. This reduction is due to the lower value of the mean sky luminance affecting the non-south­facing walls.

The coefficients f11, f12, f13, f’11, f’12 and f’13, on which the parameters F1 and F1 depend, were obtained with the squared minima method and are shown in the appendix.

Table 6 shows the mean deviations and the correlations of variations between the experimental and calculated data using this latter method.

Table 6 — Mean deviations and mean square deviations between the experimental and calculated hourly illuminance using the method proposed by the authors. South West North East 8 -5,7 -11,7 -7,6 -2,4 RMS 19,8 33,2 26,7 26,1

An examination of table 6 indicates that the deviations are notably improved compared to the values obtained using the first Perez method, see table 1. The mean deviations are contained between -2,4% and -11,7%, the correlations of variations between 20% and 33%.

The illuminance predictions could be further improved by obtaining the fit of parameter F{ for each vertical surface; that is in agreement with the fact that each surface is exposed to a portion of the sky having its own mean luminance.

The authors propose to continue this work examining the incidence on the deviations of the use of the Erbs resolution correlation (using the experimental values of direct and diffuse radiation resolution is avoided), experimentally testing, for the Arcavacata area, the effective illuminance correlations proposed by Perez for calculating direct and diffuse illuminance, and in the future also testing the sky illuminance correlation proposed by Perez for the use of the second model.

3. CONCLUSIONS

Some correlations and calculation methods of natural incident illuminance on variously orientated surfaces have been tested, using experimental hourly global illuminance data of a horizontal surface and four vertical surfaces facing south, west, north and east, recorded at Arcavacata di Rende for a period of 22 months. More than 23,000 data were considered overall.

Illuminance on the horizontal surface proved to be well-predicted by the correlation of effective illuminance proposed by Perez, whereas, for the variously orientated vertical surfaces, notable differences were observed between the values calculated using the Perez methods and the experimental data.

In particular the Perez method was analysed, based on the composition of the illuminance as the sum of the direct illuminance, diffuse circumsolar illuminance, diffuse isotropic illuminance, and diffuse illuminance from the horizon. This method appears to be inadequate for predicting the experimental data of Arcavacata.

The authors have proposed a simplified method, in which illuminance due to the illuminance from the horizon is eliminated and two correlations are used, the first to be used in those hours when the surfaces are lit by the sun and the second to be applied when those surfaces are lit only by diffuse light. This method, which will be further perfected, is able to reduce notably the deviations with respect to the original Perez method.

This argument is particularly complex and requires further examination and development.

APPENDIX Erbs Correlation

-16,638k3 + 12,336k4

k < 0,22 0,22 < k < 0,80 k > 0,80

Dh

H

h

1,0-0,09k =■ 0,9511-0,1604k + 4,388 k2

l0,

165

Erbs resolution Correlation (Erbs et al., 1982) of the global radiation on the horizontal plane Hh in the diffuse Dh and direct Bh components:

where k is the hourly clearness index, defined as the ratio of the hourly global energy Hh incident to the ground on the horizontal plane and hourly energy Hh, ex incident on the horizontal plane outside the atmosphere.

Perez Correlations

Illuminance on the horizontal plane

The effective illuminance of the global radiation on the horizontal plane is calculable with the equation:

— = a; + b;w + q cosz + djn Д G0

where w is the precipitation water content in the atmosphere; z is the zenith angle of the sun, Д sky illuminance; ai, b, c; and di are the coefficients (Perez et al., 1986) calculable as a function of the sky clearness index e.

Sky illuminance is defined in this way

Д = m-^ Iq

where m is the relative air mass, Ido diffuse radiation on the horizontal plane and Io normal extra-atmospheric radiation.

The clearness index e is defined in this way

I + bo

— sen^ + 5,535.1Q-6z3

e = !d0_____________________

1 + 5,535 ■ 10_6z3

where Ib0 is direct radiation on the horizontal plane, a is the solar height, the quantity Ib0/sena is direct normal radiation to the ground, and the zenith angle z is expressed in sexagesimal degrees.

Precipitation water w is calculable with the relation

w = e(0-07Td -0,075)

Where, Td is the dew temperature in °C, calculable with correlations in literature, as a function of the temperature and of the relative humidity of the outdoor air (ASHRAE HANDBOOK, 2001).

Illuminance on inclined surfaces

The effective illuminance of the diffuse radiation is calculable with the equation

= a; + b;w + q cosz + djn Д

‘dO

where Ed0 and I d0 are respectively the diffuse illuminance and the diffuse radiation on the horizontal plane; while the effective illuminance of the direct radiation can be calculated using the equation

Eb0 = a| + b|w + c|e(5,73z 5) + d|A

‘bO

the coefficients ai, Ь;, c; and di, different for the calculation of global, diffuse and direct illuminance, are a function of the sky clearness index e (Perez et al., 1986).

The instantaneous incident illuminance on an inclined surface however orientated can be calculated using the equation:

E = EboRb + Edo(1 — Fi) (+ e^F, і + E^senp + (Ebo + Edo) p (

where Rb is the inclination factor of the direct radiation, p is the inclination of the surface on the horizontal plane, p is the coefficient of reflection from the ground, F1 and F2 are coefficients respectively linked to circumsolar illuminance and to that of the horizon; a and b are quantities defined as:

a = max [o, cosi] b = max [cos 85°, cos z]

being the incidence angle of the solar rays on the inclined surface and z the zenith angle. The quantities F1 and F2 are calculated by means of the equations:

F2 = max o, ff2i + f22^ + f23z V!

2 [ і21 22 23 18o jj

being the illuminance coefficients fn, f12, f13, f21, f22, f23 obtainable from table A1 as a function of parameter e.

Є	fn	f12	f13	f21	f22	f23
1,000 — 1,065	0,011	0,570	-0,081	-0,095	0,158	-0,018
1,065 — 1,230	0,429	0,363	-0,307	0,050	0,008	-0,065
1,230 — 1,500	0,809	-0,054	-0,442	-0,181	-0,169	-0,092
1,500 — 1,950	1,014	-0,252	-0,531	0,275	-0,350	-0,096
1,950 — 2,800	1,282	-0,420	-0,689	0,380	-0,559	-0,114
2,800 — 4,500	1,426	-0,653	-0,779	0,425	-0,785	-0,097
4,500 — 6,200	1,485	-0,210	-0,784	0,411	-0,629	-0,082
6,200 — over	1,170	-0,300	-0,615	0,518	-1,892	-0,055

Table A1 — Illuminance coefficients of the Perez correlation for illuminance calculation.

Є fn f12 f13 1,000 — 1,065 -0,0346 1,5049 -0,2358 1,065 — 1,230 0,8060 -0,9129 -0,2185 1,230 — 1,500 1,0254 -1,7199 -0,0405 1,500 — 1,950 1,1750 -2,5873 0,1214 1,950 — 2,800 0,8914 -0,7927 -0,0899 2,800 — 4,500 0,5667 2,2778 -0,4713 4,500 — 6,200 0,3390 7,0715 -1,1760 6,200 — over 4,3546 0,3028 -3,3401

Table A3 — Illuminance coefficients for the calculation of coefficient Fi in eq. (5).

Є f’11 f’12 f 13 1,000 — 1,065 0,1247 1,2897 -0,0690 1,065 — 1,230 -0,1006 1,9303 -0,00966 1,230 — 1,500 -0,1256 2,0265 0,0170 1,500 — 1,950 0,7355 -0,2616 -0,0232 1,950 — 2,800 0,9867 -0,9543 -0,0387 2,800 — 4,500 1,1789 -1,4358 -0,1143 4,500 — 6,200 1,9573 -6,2762 -0,2830 6,200 — over 9,7606 -2,0360 -7,5381

Table A2 — Illuminance coefficients for the calculation of coefficient F1 in eq. (4)