Solar Enerav. Vol. 53. No. 2. DD. 177-185. 1994 Copyright 0 1994 Ei&er Science Ltd Printed in the USA. All rights reserved

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MEASUREMENTS AND MODELS FOR TOTAL SOLAR IRRADIANCE ON INCLINED SURFACE

IN ATHENS, GREECE

H. D. KAMBEZIDIS,* B. E. PSILOGLOU,* and C. GUEYMARD** *Atmospheric Research Team, Institute of Meteorology and Physics of the Atmospheric Environment, National Observatory of Athens, P.O. Box 20048, GR- 118 10 Athens, Greece, * *Florida Solar Energy

Center, 300 State Road 40 I, Cape Canaveral, FL 32920, U.S.A.

Abstract-This article presents a comparative assessment of tilted irradiation models, using hourly mea- surements of total solar irradiation on a surface tilted 50 degrees and oriented south in Athens. Detailed measurements on inclined surfaces are carried out at the National Observatory of Athens and are unique in Greece. Twelve sky diffuse submodels are used with four albedo submodels to estimate the global irradiation on the tilted surface from data on the horizontal plane. Root mean square errors (rmse) and mean bias errors (mbe) are used to determine the intrinsic performance of each diffuse tilt/albedo submodel combination. GUEYMARD, HAY, REINDL, and SKARTVEIT-OLSETH diffuse tilt submodels are found to have the best overall performances, in conjunction with either one of three albedo submodels (constant albedo. seasonally varying albedo, and anisotropic al&do). The PEREZ model’s performance was below expectations. probably due to the particular atmospheric environment of Athens. The anisotropic and seasonally varying albedo submodels do not improve the performance of the four better diffuse tilt models (compared to their performance using an albedo fixed at 0.2) for the moderately tilted surface investigated in this article.

1. INTRODU(TIION

Global and diffuse solar components on a horizontal surface compose the basic instrumentation for any solar radiation measuring station. Nevertheless, there is in- creasing demand for knowledge of solar irradiance fluxes or hourly irradiation on inclined surfaces with arbitrary azimuthal orientation for building design and energy analysis purposes. There are only a few stations worldwide which measure solar components on in- clined surfaces. Therefore, the development of various algorithms that predict the diffuse solar irradiance on inclined surfaces is necessary.

In Greece, three stations have relatively long-term records of diffuse and global solar radiation on a hor- izontal plane: Athens, Kythnos Island, and Rhodes Is- land; there are also nine others that measure only the global component on a horizontal plane. The radiation center in Athens belongs to the National Observatory of Athens (NOA ), whereas most of the remainder be- longs to the Greek Public Power Corporation.

NOA started measuring total solar irradiance on inclined surfaces with various azim;lthal orientations on January I, 1988. These records, together with those of diffuse solar component on inclined planes facing south (since February 1989), are still unique in Greece and among a few of this kind worldwide.

The purpose of this article is to test 12 of the most known models for sky diffuse radiation on inclined surfaces. Each of them has been associated with four different albedo submodels: a constant reference mean value of 0.2, as frequently used in the literature since Liu and Jordan ( 1963); an albedo expression depen- dent on season and the latitude of the location; an anisotropic albedo model dependent on the solar zenith

angle; and, finally, another anisotropic albedo model dependent on the sun/surface geometry. The latter two anisotropic albedo models have provisions for hourly variations of the ground albedo adjacent to the inclined plane.

The predictions of the sky diffuse models with each type of albedo are compared with the measured NOA data. Most of these models have been tested against real data in other countries ( Abdelrahman and Elha- didy, 1986; Harrison and Coombes, 1989: Hay and McKay, 1988; Louche et al., 1988; Newland, 1989; Reddy and Attalage, 1988; Reindl et al., 1990; San- tamouris et al., 1990; Stoffel et al., 1987; Utrillas et al., 1992). In Greece, such results have not yet been presented because of lack of measurements on inclined surfaces. The only studies that have appeared in the literature are related to the prediction of global solar radiation on inclined surfaces on a daily, monthly, or annual basis using the trivial albedo of 0.2 ( Katsoulis, 199 1; Koronakis, 1986; Lalas et al., 1982; Pissimanis et al., 1987). Therefore, the comparison between the aforementioned twelve sky diffuse submodels (the iso- tropic and 11 anisotropic) using the four albedo sub- models with measurements of total solar irradiance on an inclined surface facing south is done for the first time in Greece.

2. DATA COLLECTION

NOA’s radiometric station is located on top of Pnyx (lat. 37.98” N, long. 23.75” E, alt. 107 m a.m.s.l.), a hill 1.9 km from downtown Athens. All NOA solar radiation components are recorded as hourly irradia- tions in MJ m-*. For the present study, 17 months of hourly data (beginning in January 1990) of global ir-

177

178 H. D. KAMBEZIDIS, B. E. PSILOGLOU, and c. G[!EYMAKt>

radiation on a surface facing south at a tilt of 50” were H, is the solar zenith angle and % is the incidence angle used. No data for other slopes or azimuths and no of the solar rays onto the tilted surface. albedo data are available for this period, but such data BUGLER’s model ( Bugler, 1977 ) :

will become available for future assessment studies. The inputs to the 12 selected models are the global ffd(B) = [fId(o) - O.O5H,,(B)lcos %,]RA and diffuse hourly solar irradiations on the horizontal surface measured by two Eppley PSP pyranometers + 0.05Hh( /3 kos rl (I) (one of them being equipped with a standard Eppley- type shade ring). The total and diffuse components on where @ = 0 denotes the horizontal plane.

the tilted surface are measured with CM5 Kipp and GUEYMARD’s model (Gueymard, 1984, 1986.

Zonen pyranometers (a shade ring is also used to elim- 1988):

inate beam radiation). All these pyranometers are reg- ularly calibrated against reference instruments to en- H‘/(e) = Hd(O)]( 1 ~ ~~‘,v)R, + hZ,b,l (2)

sure consistent measurements. Before using the data, some necessary quality con- where Rd,, and Rd, are expressions describing the effect

trol tests were performed. The two first tests conform of radiance patterns from the clear and overcast skies,

to the criteria set forth by the Daylight Research Group respectively (Gueymard, 1988). N,, is a nebulosity

of the Commission of the European Communities, in weighting term. Because hourly cloud observations

which NOA participates. The other tests are intended were not available, iv’,>, has been estimated from radia-

to eliminate spurious data as well as inaccurate mea- tion data using an approximate function that was orig-

surements resulting from the cosine response error of inally proposed (Gueymard. 1988) for situations in

the pyranometers. These tests consist of which no coincident cloud opacity information is

1. Rejecting all diffuse horizontal values that are available:

greater than 1.1 times the corresponding global horizontal ones; N,, = max[min(Y. I). O] (3a)

2. Rejecting all global horizontal hourly values which are greater than 1.2 times the solar constant and where

diffuse values greater than 0.8 times the solar con- stant (i.e., greater than 5.915 and 3.937 MJ rnm2, Y = 6.6667[rr,(O)/H,(O)] ~ 1.4167 (3b)

respectively) : 3. Rejecting all global horizontal values equal to or

if ffd(0)/H,(O) I 0.227: otherwise

less than 0.0007 MJ me2: 4. Rejecting all values for a solar elevation, h, less than

Y = 1.2121 [Hd(0)/ff,(O)] - 0.1758 (3c)

5 degrees; 5. Rejecting all values if the direct component [i.e.,

where If,(O) is the total (or global) irradiation on the

(global - diffuse)/sin h] exceeds the extraterrestrial horizontal (MJ mm’).

solar irradiation; and HAY’s model (Hay, 1979; Hay and McKay. 1988 ):

6. Applying a shade ring correction to the observed diffuse horizontal radiation values according to the HI?(P) = ffd(O)[&& + (1 - b)Rdil (‘+a)

method described by Littlefair ( 1989 ). Thus 5228 data points out of a total of 1 I,4 14 passed

with

the quality control tests. Nearly all the rejected data points (6 157 of 6 186) were eliminated by test 4. Among

Kh = min(H,(0)/HO, 1) (4b)

the 29 other rejected data points, 26 were eliminated by test 1 and 3 by test 3.

where H0 is the extraterrestrial irradiation on a hori- zontal surface for a solar constant of 4.92 1 MJ mm2.

HAY-WILLMOTT’s model (Willmott, 1982 ): 3. SKY DIFFUSE MODELS

Twelve sky diffuse submodels have been selected f-r,(@) = H,(O)[K,/cos %i i- c;{ 1 - (&/COS H;)}]

as the most promising for our study, based on published (5,

results for other locations. These models are listed in alphabetical order and described succinctly in this sec- where C; , Kh , and K, are expressions given by Willmott

tion. Hd( fi) and Hb( p), respectively, denote the diffuse (1982).

irradiation and the beam irradiation on a plane inclined ISOTROPIC model (see, for example, Liu and Jor-

0 degrees from the horizontal, using the SI unit (MJ dan, 1963):

me2) throughout. Hb(p) is obtained easily from the measured global and diffuse irradiations on the hori- fr,(?) = ffd(O)Rdr (6)

zontal (Iqbal, 1983). In what follows, Rdi = ( 1 + cos /3)/2 is the isotropic sky configuration factor for the KLUCHER’s model ( Kiucher. 1979 ): considered tilted surface, and Rb = max( cos %/cos %,, 0) is the conversion factor for beam radiation, where &(,(l) = fd(O)&,M,lv? (7)

Total solar irradiance 179

where MI and M2 are expressions given by Klucher where P, and Pz are expressions given by Temps and (1979). Coulson (1977).

MUNEER’s model (Muneer, 1990):

H&I) = F&(O) T (8a)

for surfaces in shade and sunlit surfaces under overcast sky, and

WILLMOTT’s model (Willmott, 1982): The expression for the sky diffuse component is the same as the one used in Hay-Willmott’s model. eqn ( 5 ) The only difference between the two is that in the present model, the reflected radiation (eqn 14) is mul- tiplied by a coefficient K, ( Willmott, 1982 ).

Hd(@) = &(O)]T( 1 - F) + t;‘&l (8b)

for surfaces under nonovercast sky, where

T= Rd,+N,N2 (8~)

N, = 0.00263 - 0.7 120F - 0.6883 F* (8d)

iVz = sin @ - ,B cos /3 - K sin*(/3/2) (8e)

4. REFLECTED AND TOTAL RADIATION

To calculate the total irradiation on a surface with slope 0, the following relationship was used:

H,(P) = Hb(P) + Hd(P) + Hr(P) (13)

where H,(p) is the total irradiation on the tilted plane. H,(P) is defined as

F = Hb( 0)cos O,/HO ( 80

The coefficients of the quadratic fit in eqn (8d) have been empirically obtained for Geneva, Switzerland. Other coefficients are also available for Easthampstead, Great Britain, and Eindhoven, Netherlands (Muneer, 1991).

H,(P) = p&H,(O) ( 14)

where px is the ground albedo and R, = ( 1 - cos p)/ 2 is the configuration factor between the ground and the receiver plane.

PEREZ’s model (Perez et al., 1990):

Hd(P) = Hd(O)]&( 1 - F;)

+ F;(cos B/cos 0,) + F; sin fl] (9)

where F; and Fi are expressions for the circumsolar and horizon brightening effects as empirically deter- mined by Perez et a/. ( 1990).

RElNDL’s model (Reindl et a/., 1990):

Four possible expressions for the albedo are con- sidered in this study: 1. Isotropic constant model. This is the most simple

and commonly used albedo estimate, with a con- stant value of 0.2:

P‘0 = 0.2 (15)

Hd(8)

= Hd(O)]( 1 - &)&r{l f .fSin3(P/2) + &%)I (10)

where Kh is defined as in eqn (4b). SKARTVEIT-OLSETH’s model (Skartveit and

Olseth, 1986):

F&(P) = I-I,(O)]&& + B cos P

+(1 -Kh-B)&-S(‘-Qt,~i)I (11)

where B = max { (0.3 - 2Kb), 0 } , and the function S( wi , 0;) evaluates the diffuse irradiation screened by obstacles on the horizon, w, and 0, being the solid and incidence angle for the ith sector of the actual horizon according to Skartveit and Olseth ( 1986). In our case the obstacles on the horizon are almost nonexistent so that this term has been neglected.

TEMPS-COULSON’s model (Temps and Coulson, 1977):

F&(P) = &(0)&F,& (12)

2. Isotropic seasonal model. This model is a function of latitude and month and is fully described else- where (Gueymard, 1993). It was originally devel- oped to estimate the monthly zonal albedo in North America. Zonal albedo refers to the average albedo over a large land area. Therefore, the zonal albedo is the space average of a myriad of individual local albedos. Although strictly speaking this model should not be used for the evaluation of local albedo, it has already been used in such an extrapolated way (lneichen et al., 1990) and has shown an ac- curacy comparable to the constant trivial 0.2 value in predicting the local albedo of six midlatitude ra- diation stations. The zonal albedo model has been included in this study to test its ability to withstand such an extrapolation in another climatic area. It may also be seen as a better estimate than a constant a priori value when no albedo measurement is available (which is generally the case), or when dif- ferent potential sites for solar energy system instal- lations are compared, or when developing maps of incident solar irradiation on inclined planes at the country or continental scale level (for example, constant a priori albedo values of 0.15 and 0.2, re- spectively, have been used by Olseth and Skartveit, 1985, for a radiation atlas of Norway, and Palz, 1984, for a European atlas, whereas seasonal and site-specific albedo values have been used in Canada by McKay and Morris, 1985).

180 H. D. KAMBEZIDIS, B. E. PSILOGLO~J. and (‘. GUI-YMARII

3.

4.

The seasonal model for north latitudes between 20” and 60” is

where 4 is latitude in degrees and monthly values of coefficients a, are provided (Gueymard, 1993 ). For the latitude of Athens, pg2 varies between 0. I4 1 in June and 0.288 in February, for a yearly average of 0.18. Although existing coefficients a, are not site specific, they could be recalculated for various en- vironments from the ad hoc basic information if available (e.g., maps of land use, or maps of seasonal zonal albedo obtained from satellites). Further- more, daily albedo values can be obtained easily from their monthly counterparts by interpolation. Although these refinements have not been consid- ered for this study, they will be investigated in future work. Ciimuto~ogical anisotropic model. Developed by Nkemdirim (1972) and Arnfield ( 1975), this model expresses the albedo as a function of the solar zenith angle:

PK3 = u exp( hf9,) (17)

where u = 0.244 and h = 0.00526 deg.’ for morning hours, and a = 0.2 I2 and h = 0.00891 deg-’ for afternoon hours, for any day of the year. The cor- responding typical values of px, are 0.264 (morning) and 0.242 (afternoon) at low zenith angle ( Oz = l5”, the extreme value at the summer’s solstice for Athens’s latitude), and 0.382 (morning) and 0.452 (afternoon) at high zenith angle (fl, = 85”). This model therefore predicts a large hourly albedo fluc- tuation and an average albedo significantly larger than p,, and pn2. Semiphysical anisotropic model. It has been shown by lneichen ( 1983) that the apparent ground albedo may be significantly dependent on the beam radia- tion incident on it, Hh( 0). Therefore, it makes sense to consider separate albedos for the beam and diffuse components on the horizontal plane. In this respect, a model developed by Gueymard ( 1988) has been selected because it distinguishes between the reflec- tance for beam and diffuse radiation. The resulting average albedo for global radiation is

where

pa = .ii,~& + ~ci ( 1 - h) (18a)

Ph = Pn + WA .f,,)

X IcosaIexp(-1.77 - 1.53h’- 3.61h’*) (18b)

is the reflectance for beam radiation, and

Pd = Pn + o.ow.fk +./$) (18~)

is the reflectance for diffuse radiation, pn is the ground albedo for purely isotropic reflection,,f$, and .fa, are coefficients respectively describing the back- ward and forward increase of reflectance, F( f;,,. ,/i, ) is a function defined in Gueymard ( 1988). ,fl;, is the shadow factor for beam radiation. ki, = ff,,(O)/ ff,( 0), N is the solar azimuth relative to the receiving surface (or “wall-solar azimuth”). and h ’ = 0.0 117, The values of the parameters ,f;,,, . ,/A ,, and tL\ used

in this work are 0.5. 0.5, and I .O, respectively. These coefficients are for pasture land. no beam shading, with reference to the guidelines provided (Gueymard. I988 ). (The area around the measuring station is quite dry in summer, with a few trees: “pasture land” is the closest available description of this type of vegetation.) Accordingly, ~1~ has been estimated to bc constant at 0. I6 so that p,, becomes 0.183.

When modeled by eqn ( l8), the albedo of a par- ticular ground surface is a function of time of day (through h and cy) and of atmospheric conditions (through li,). For example. on a typical early morning or late afternoon around the winter solstice, h and (1 would be about 5” and 55”. respectively, at Athens’s latitude. At such a low solar elevation. ki, is about 0.5 for clear conditions (low turbidity. I cm precipitable water) according to a two-band solar irradiance model (Gueymard, 1989). This would yield pt> := 0.205 and

QY4 = 0,194. For typical noon conditions (cloudless but turbid sky) around the summer solstice, one would similarly obtain pi, .= 0. I64 and pK4 = 0.170. For over- cast conditions at any time. l\,, = 0. so that ,J~. = ~1~ =- 0.183. .Thus it appears that. for the extreme conditions examined, eqn ( I8 ) predicts relatively low variations of albedo from one hour to the other (i.e., nearly isotropic reflection. compared to the predictions of eqn I7 ).

To apply eqn (11), one-minute values of I!(,(@). Hd( @). and ff,(b) were added 60 times to derive the hourly integral, II,(B). This method is more accurate than computing only once an hour using preintegrated hourly values of the same input data.

Because the calculation of the irradiations by most of the aforementioned sky and albedo models need the knowledge of the solar position in the sky, the SUNAE algorithm (Walraven. 1978) was used with all modi- fications suggested by Wilkinson ( 1981). Muir ( 1983). Kambezidis and Papanikolaou ( 1990). and Kambc- zidis and Tsangrassoulis ( 1993 ).

5. S’l’.4’L‘ISl’ICAl. RESCJLI’S AND DISCUSSIOPU

The accuracy of the 48 diIIerent models ( 12 sky diffuse submodels times 4 albedo submodels) was as- sessed using two widely used statistics: rmse and mbe. The following expressions for the rmse, in MJ rn-‘. and mbe, in percent of a,,,( 8) (the average tilted global irradiation during the whole period, measured to be 1.53 MJ mm’), were used:

rmse = { 2 [H,(p) - ff,,,(@)]s/lV j Ii2 (19)

Total solar irradiance 181

Azimuth: 0 degrees

MEA&ED HO”:LY TOTA; IRRADIAAON (MJ,

(a)

b i i ii 121 MEASURED GLOBAL ON A TILTED SURFACE (MJ;mB)

(b)

$,” Location: NOA

e p$$: &Qp-~~&9~ , , /I’

2 Albed;: 0.2 I

/

s4 Tilt: 50 degrees 6 Azimuth: 0 degrees /

.^

Fig. I. Estimated versus measured hourly total irradiations on a 50” tilted surface facing south in Athens (Period: January I, 1990-May 5, 199 1). Best performing (a) and worst performing (b) sky diffuse submodels

using albedo submodel I, eqn ( 15 ).

where N is the number of data points (5228), and Ht,,( /I) is the measured total solar irradiation on a sur- face with tilt /I = 50’ and zero azimuth. The mbe and rmse have been computed for the whole dataset as well as for each daily period. Equation ( 13) was used to derive the hourly total solar irradiations incorporating

I

I

/’ , / 9’

Location: NOA Period: 19/i/90-20/5/91 Model: HAY Albedo: A*EXP(B’Z) Tilt: 50 degrees Azimuth: 0 degrees ,

I

MEASURED GLOBAL ON A TILTED SURFACE (M

(a)

each sky diffuse submodel and each albedo submodel separately.

Figures 1 through 4 show examples of the predicted versus the measured hourly total irradiations using each one of the aforementioned albedo submodels. Each figure consists of two scatter graphs, diagram (a) show- ing the best-performing model and diagram (b) the worst-performing model. For each figure, the selection process was based on the minimum absolute mbe to pick the best model, and on the maximum absolute mbe to pick the worst. The dashed diagonal line rep- resents the ideal match between the estimated and measured values. Comparing these scatter graphs. it

Location: NOA /

Period: 19 l/90-20/5/91 G

I /

Model: WIL MOTT I

Albedo: A*EXP(B’Z) I

/ Tilt: 50 degrees

, I

Azimuth: 0 degrees I

0 hm~wRlED CLOBA~L ON A T~ILTED S&ACE (M

W

Fig. 2. AS in Fig. 1, but using albedo submodel 2, eqn ( 16)

H. D. KAMBEZIDIS, B. E. PSILOGLOU, and C. GutwARn

Location: NOA Period: 19

c l/90-20/5/91

,’ , Model: IS0 ROPIC , Albedo: F(latitude)

/ I Tilt: 50 degrees / I Azimuth: 0 degrees /

..’ .

0 MEASUI&D GLOB:L ON A GLTED &FACE (h&2) MEASURED GLOBAL ON A TILTED SURFACE (M

(a) (b)

Fig. 3. As in Fig. I, but using albedo submodel 3, eqn ( I7 ).

appears that the albedo submodel introduces only minimal difference. This can be explained by the rel- atively small tilt (50” ) and the low albedo of the en- vironment. When this assessment exercise is repeated for vertical surfaces, we will expect to observe more influence of the reflected irradiation because the con- figuration factor, R,, for a vertical surface is 2.8 times more than for a surface tilted 50”.

The differences introduced by the albedo submodel are only detectable when comparing the rmse and mbe (Table 1). All models are reacting similarly to a change in the albedo submodel. More scatter is observed on the rmse (which can vary by 0.01 to 0.03 MJ m-‘. depending on the model) than on the mbe (which gen-

Location: NOA Period: 19

IG Model: REI l/90-20/5/91 DL

Albedo: F(pb,pd) Tilt: 50 degrees Azimuth: 0 degrees

,’ / , / I

I

I

1

I

4 0

MEAS”:ED .OU:LY TOTA: IRRADIAAON (h&Z)

(a)

erally shows a 3%, decrease) when switching from al- bedo submodel 3 to submodel 4; this could be expected because of the difference in albedo prediction between these submodels already noted. The effect on the global tilted radiation would be noticeably larger for a vertical plane.

For further reference. Table 2 displays the monthly mean global and diffuse horizontal as well as total in- clined irradiation for the study period. If used in con- junction with the results of Table I. it is possible to obtain the mbe on an absolute scale.

Considering now the effect of sky diffuse modeling. a general observation is that all models exhibit com- parable and relatively low scatter, except Bugler and

Location: NOA / Period: 19

CM l/90-20/5/91

I / Model: WIL OTT / Albedo: F(pb.pd) Tilt: 50 degrees / Azimuth: 0 degrees I ,

I I-T-~

MEAS&ED HOURLY TOW? IRRADIAT~~ON (MJ,

(W

n2)

Fig. 4. As in Fig. 1, but using albedo submodel 4. eqn ( 18 )

Total solar irradiance 183

Table I. Root mean square errors (rmse) in MJ me2 and mean bias errors (mbe) in percent of the measure average tilted irradiation for the

12 sky diffuse submodels and four albedo submodels of this work

Model rmse mbe

Bugler

Gueymard

Hay

Hay-Willmott

Isotropic

Klucher

Muneer

Perez

Reindl

Skartveit-Olseth

Temps-Coulson

Willmott

0.16 -4.15 0.15 -5.05 0.14 -2.81 0.16 -5.84 0.12 I.17 0.1 I 0.85 0.13 3.10 0.1 I 0.08 0.1 I 0.23 0.1 I -0.08 0.12 2.16 0.1 I -0.86 0.14 -4.19 0.14 -4.50 0.12 -2.26 0.14 ~5.28 0.13 -2.63 0.13 ~2.94 0.12 -0.70 0.13 ~3.72 0.12 2.49 0.1 I 2.18 0.14 4.43 0.1 I 1.40 0.12 2.19 0.12 1.88 0.14 4.12 0.12 1.10 0.13 3.52 0.13 3.22 0.15 5.45 0.12 2.44 0.1 I 1.08 0.1 I 0.77 0.12 3.00 0.1 I PO.02 0.1 I -0.09 0.1 I -0.40 0.12 1.83 0.11 -1.18 0.16 5.39 0.15 5.08 0.17 1.32 0.15 4.29 0.14 -5.13 0.14 -5.26 0.13 -3.81 0.15 -5.89

The first row of numbers refers to the sky diffuse submodel run with albedo submodel 1 [eqn (I 5)]. the second with albedo submodel 2, etc.

Temps-Coulson, which have less precision. The mea- surement period ( 17 months) is long enough to smooth out all deficiencies of a good model due to particular atmospheric conditions. Most models appear to over- estimate slightly at high irradiations. This may be due to an overestimation of the reflected radiation or an overestimation of the horizon radiance, a small part of which may be blocked out by the few existing low- rise obstructions.

Table 1 gives the rmse and mbe for the 12 models as estimated from the hourly values. From this it is seen that the mean rmse is in the range of 0.11 to 0.17 MJ me2 for all sky diffuse/albedo submodel combi- nations. The largest mean rms errors are obtained with Bugler, Hay-Willmott, Perez, Temps-Coulson, and Willmott. All these models perform apparently worse than the Isotropic model. From the rmse standpoint, it appears that the best performance (i.e., a mean rmse of 0.11 MJ m-*) is equally obtained with Gueymard (albedo submodels 2 and 4), Hay (albedo submodels 1, 2, and 4), Klucher (albedo submodels 2 and 4), Reindl (albedo submodels 1, 2, and 4)) and Skartveit- Olseth (albedo submodels 1, 2, and 4). The lowest absolute mbe (5 1% ) is obtained with Gueymard (al- bedo submodels 2 and 4), Hay (albedo submodels I, 2, and 4), Isotropic (albedo submodel 3), Reindl (al- bedo submodels 2 and 4), and Skartveit-Olseth (alhedo submodels I and 2). Thus the only good result obtained with albedo submodel 3 is when coupled with Isotropic. It is known that the latter tends to underpredict diffuse radiation on a plane facing the equator, compared to anisotropic models. It may therefore be concluded that the good overall result of Isotropic/albedo submodel 3 is merely caused by cancellation of errors between the diffuse and reflected components. This, in turn, suggests that neither Isotropic nor albedo submodel 3 can be recommended for our site, if used indepen- dently.

From these results, it may be concluded that at least four sky diffuse submodels (Gueymard, Hay, Reindl, and Skartveit-Olseth) perform equally when combined with either one of three albedo submodels ( 1, 2. and 4) and therefore may be used to obtain accurate irra- diation estimates in Athens.

An unexpected result of this study is that Perez does not perform as well as the Isotropic model and exhibits more bias and more random error than the group of

Table 2. Mean monthly global horizontal, H,(O), diffuse horizontal, HAO), and global tilted, H,(50) irradiations

as calculated from the corresponding measured hourly values at NOA (MJ m-*)

Year

1990

1991

1990-1991

Month H,(O) Hd(O) H,(50)

I 0.98 0.34 1.70 2 I .08 0.40 1.59 3 I.58 0.41 1.95 4 I.61 0.59 1.59 5 1.64 0.63 1.37 6 1.87 0.53 1.44 7 1.85 0.54 I .47 8 1.83 0.54 1.69 9 1.54 0.46 1.74

10 1.31 0.38 I.84 11 0.92 0.38 I .49 I2 0.68 0.34 I.13

1 0.78 0.31 1.35 2 0.91 0.46 1.28 3 I .23 0.58 I .44 4 1.44 0.65 1.39 5 1.68 0.60 1.42

All 1.35 0.48 1.53 -

184 H. D. KAMBEZIDIS, B. E. PSILOC;LOU, and C’. GUEYMARI)

best-performing models. This is surprising because previous assessment studies generally concluded that Perez was performing better in most cases. For different reasons (the care taken at all stages of the measurement and quality control processes; the large number ofdata points, which eliminates the possibility of fortuitous error accumulation with Perez and error cancellation with other models; and finally the estimation of re- flected radiation by different methods, thus excluding a possible counterperformance due to systematic errors in the albedo estimation), it is most certain that the present findings reflect more modeling noise than in- strumental noise.

It is possible that the Perez model is more sensitive to the combination of the available inputs (global and diffuse horizontal irradiations, rather than global and direct beam, which should be used according to Perez et al., 1990) than the other models. (The sensitivity of diffuse tilt models to the various possible input com- binations has been demonstrated by Gueymard, 1988.) A more likely explanation is that the empirical coef- ficients derived from different data sets in Perez do not match the specificity of Athens’s climate. The NOA site is surrounded by Athens city (except for an area with a radius of 400 m approximately around the city, where there are pine trees and walking places for the Athenians). The climate of Athens is Mediterranean, with predominantly clear-sky conditions. Air pollution is often important, with frequent occurrence of pho- tochemical smog. Those conditions have a definite in- fluence on the sky radiance patterns and may generate significantly different distributions of diffuse radiation than were considered during the development of the Perez model. This conclusion is substantiated by the fact that a similar slight counterperformance of this model has been found at another Mediterranean site in Cyprus (Santamouris et al.. 1990) as well as at the Belgian Building Research Institute in Belgium and at Catania, Italy (Balaras, 1993). Furthermore, the IEA Task IX study (Hay and McKay, 1988). which used data from 25 stations around the world, showed that although the Perez model was the overall best per- former, it was not always so, depending on tilt value. orientation, and climate. For example. Gueymard’s model (and, in very few instances, Hay’s model) ob- tained equivalent or better results than Perez for various tilt/azimuth combinations at Albany and Golden (U.S.A.), Bergen (Norway), Bracknell (U.K.), Geneva and Locarno (Switzerland), Griffith (Australia), lspra (Italy), Vaerlose (Denmark), Valentia (Ireland), and Vancouver (Canada). On the other hand, the IEA Task

IX study showed that Skartveit-Olseth did not perform as well as Gueymard, Hay, or Perez, contrary to the present findings (no result for Reindl was available because it was not yet published when the IEA work was performed).

Because of these slight disagreements between re- sults from similar studies, further work (using more tilt/azimuth combinations and possibly ground albedo as well as reflected radiation measurements) will be necessary to obtain more discrimination between the

various sky diffuse models. The present work suggests

that for polluted environments and climates similar to Athens, it would be interesting to investigate whether the empirical set of coefficients in the Perez model might relate to some atmospheric condition, such as turbidity caused by natural aerosols and pollutants.

Finally, the problem of selecting the right albedo submodel appears to be of secondary importance as long as the reflected radiation is relatively low. which typically occurs when the tilt angle of the surface is low to moderate and the local albedo is low and nearly isotropic. For these conditions, there is no significant modeling performance gain to be expected when switching from a constant albedo of 0.2 to more so- phisticated anisotropic or seasonally varying albedo submodels. Data for vertical surfaces and different re- flecting surface types will be necessary to discriminate better between the different modeling options for the local albedo.

.~lclino~,/cd~,~?m~-Dr. Richard Perez was particularly instru- mental in revising our computer version of his model.

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