Solar Energy Vol. 51, No. 2, pp. 121-138, 1993
0038-092X/93 $6.00 + .00
Copyright © 1993 Pergamon Press Ltd.
Printed in the U.S.A.
CRITICAL ANALYSIS AND PERFORMANCE ASSESSMENT
OF CLEAR SKY SOLAR IRRADIANCE MODELS USING
THEORETICAL AND MEASURED DATA
CHRISTIANGUEYMARD*
Florida Solar Energy Center, 300 State Road 401, Cape Canaveral, FL 32920, U.S.A.
Abstract--Eleven clear sky irradiance models have been selected for this analysis. All predict the beam,
diffuse, and global radiation on a horizontal surface. Three types of analyses are made in order to test the
validity of the models, their limitations, and their performance for standard or real conditions. First, a
detailed analysis of the main equations of the models is performed. It is shown that atmospheric effects are
not always correctly modeled. The modeling of water vapor absorption, and more importantly of aerosol
extinction mostlyconditionsthe overallmodel accuracy. Second,the performanceof each model is statistically
evaluated by comparison with a benchmark constituted by the predictions of three sophisticated spectral
codes. Third, real life performance is evaluated by comparison with a large number of measured data from
seven sites around the world, encompassing a wide range of atmospheric conditions. The more physical
models are found to be generally of higher accuracy and greater flexibilitythan empirical models.
1. INTRODUCTION
The modeling of the clear sky irradiance components
of solar radiation is necessary in many applications of
solar energy (systems design and simulation, control
process of the accuracy of radiometers, data quality
control, gaps filling process, etc.), as well as in routine
engineering practice (e.g., the peak cooling load of
buildings is determined for a hot, cloudless, summer
day.) A number of models of varying complexity have
been proposed in the literature, spanning from simple
empirical formulae to highly sophisticated spectral
codes. For most engineering applications, the latter
tools are not necessary, so that solar radiation is generally computed by means of either empirical or physical models, using a nonspectral (broadband) approach. In the acceptance used here, physical means
that no equation is obtained by statistical means from
observed irradiance data. On the contrary, a physical
model tries to interpret the physical extinction processes as close as possible, except that broadband
transmittances are derived instead of spectral ones.
Thus, this approach departs significantly from both
theoretical (spectral) and empirical models.
In the present contribution, 11 physical or empirical
models are analyzed and compared. In section 3, their
individual transmittance formulations are analyzed
(whenever possible) in order to detect their limitations,
using a large range for each of the input parameters.
In section 4, the predictions of the models are compared
to reference data obtained from sophisticated spectral
computations. In section 5, the model predictions are
compared to carefully measured data in a variety of
climates for cloudless-sky conditions.
Only models aimed at the computation of clear sky
irradiances on a horizontal surface are considered here.
It is possible to derive the corresponding irradiances
on tilted surfaces by means of a clear sky conversion
* ISES member.
algorithm (such as[l]). A literature survey showed that
very few extensive assessments of clear-sky radiation
models have been reported. In most cases, only a very
limited number of models (generally one to three) have
been tested against one set of measured (or more rarely,
theoretical) data (see, e.g., [2-6]). For the general case
of continuously changing cloudiness, some irradiance--or more exactly, hourly irradiation--models
have been compared to a large measured data set during
the International Energy Agency (IEA) Task IX Study
on Solar Radiation Models[7,8]. The clear sky part
of two of these "all-skies" models have been selected
for further analysis here. The present study will therefore complement both the few existing assessment
studies for clear-sky conditions and the IEA findings,
particularly giving some insights on the clear-sky part
of all-skies models. Because cloud effects generally induce most of the variability and uncertainty in radiation calculations, years of hourly data are necessary to
constitute a valid reference data set whenever a general
assessment covering all types of sky conditions is
needed, as was the case in the lEA study. A different
approach is used here because clear-sky irradiances are
far less variable with time than cloudy-sky irradiances.
Compared to the IEA study, less emphasis is placed
on the statistical analysis of a large number of data
pairs of model predictions and measured data. Conversely, more emphasis will be placed on the internal
consistency of the models, in order to find their limitations and propose possible improvements. Such improvements could in turn result in more accurate allskies models when their cloud transmittance part is
added.
2. MODEL SELECTION AND DESCRIPTION
The main criteria for the selection of irradiance
models were that: (a) they had to be able to evaluate
all the components of solar radiation on a horizontal
121
122
C. GUEYMARD
plane (beam, diffuse, and global); (b) they were to be
valid in any climate; and (c) all the necessary equations
had to be available in a closed form and use commonly
available input parameters. Criterion (a) eliminated
several global-only irradiance models, criterion (b)
eliminated all models that were not fully programmable
on a computer and those using mostly empirical coefficients tuned to specific locations, while criterion (c)
eliminated the models using special input data, such
as satellite imagery. Eleven models were thus selected,
and their description follows.
Hansen [ 12 ]. The absorptance due to water vapor aw
is also taken from[ 12]. The resulting beam irradiance
at normal incidence Ebn is finally obtained as:
Eb, = Eo.( ToTR - aw) TA
where Eo, is the extra-terrestrial irradiance at normal
incidence.
The diffuse irradiance for a perfectly absorbing
ground (i.e., with an albedo of 0.0) is obtained as:
Ea
=
2.1 M A C m o d e l
Eo.[0.5To(
1 -
TR)
+ wABA(ToTR - a w ) ( 1 - TA)]
It results from the continuing effort of researchers
at McMaster University (hence its name, as found in
the literature). Its first version appeared in[9 ], while
its latest versions are described in[8,10]. This model
is representative of the physical category, as it is based
on a preliminary derivation of the broadband transmittance (or absorptance), corresponding to each extinction process in the atmosphere.
One important particular case is the way that the
aerosol transmittance TA is handled. Following the
suggestion of Houghton [ 11 ], TA is given as a simple
power function of the absolute (pressure corrected)
optical air mass corresponding to Rayleigh scattering
(3)
(4)
where WAis the aerosol single-scattering albedo (held
constant at 0.75) and BA is the forward aerosol scatterance, given as:
BA = 0.93 -- 0.21 In mR.
(5)
Atmospheric backscatter takes into account the
multiple reflection process between the ground and
sky. It is calculated as a function of their respective
albedos. The clear sky albedo is evaluated by MAC as:
Ps = 0.0685 + WA(1 -- T ] ) ( I -- B])
(6)
mR :
TA = k "~
(1)
where k is an aerosol transparency coefficient. Unfortunately, k is not a true turbidity index (to the contrary
of A,ngstr6m's/3 and Schiiepp's B) that can be related
to the aerosol spectral optical thickness. (It will be
shown in section 3.2.6 that, for fixed aerosol characteristics, its value is dependent on air mass.) It is thus
particularly difficult to estimate its appropriate value
for a particular period (month, day, hour, etc.) or for
design applications, even if/3 or B is known. In fact,
an average k has to be obtained from an educated guess
or by trial and error [ 8 ]. Thus, for the 15 stations from
7 countries around the world that formed the data set
of the IEA validation part of the Task IX study, k was
assigned a constant value for each station, ranging from
0.87 for Kew (near London, Great Britain) to 1.0 for
four stations in Australia [ 8 ]. (Note that this latter value
would mean that no aerosol extinction takes place,
which is unrealistic.)
The other transmittance functions considered in
MAC are for Rayleigh scattering TR and ozone absorption To. Different formulations have been proposed over time for TR in the MAC model. A derivation
of TR using a preliminary integration of the spectral
Rayleigh scattering optical thickness is given in[7]:
2.2 Josefsson m o d e l
This model (hereafter JOS) has been submitted by
its author to the lEA Task IX committee as an unpublished manuscript, but its equations are listed in [ 8 ].
Basically a variant of MAC, its equation for beam irradiance also uses eqn ( 1) to obtain TA, but the beam
transmittance derivation is different than MAC's:
Eb, = Eo,,(ToTRTAaTAs - aw)
(7)
T o = 0.95545
(8)
with
TR = 0.9768 -- 0.0874mR + 0.010607552m~
-- 8.46205 X 10-4m~ + 3.57246 X 10-Sm,~
-- 6.0176 X 10-Vm~
Ta,~= 1 - ( 1
TR = 1/[(1 + exp(--2.12182
+ 0.791532 In mR - 0.024761 ln2m.~)].
where B~ = 0.8236 and T~ = k TM.
In Canada, the all-skies version of MAC has been
used by the Atmospheric Environment Service (AES)
to generate hourly radiation data for 100 meteorological stations[13]. It is planned to reprocess the files
using an expanded version of MAC and make these
data available on CD-ROM format for engineering and
research purposes[14].
(2)
The ozone transmittance To is obtained from an
absorptance parameterization proposed by Lacis and
--¢OA)(I -- TA)
TAs = 1 --w~(l -- TA).
(9)
(10)
(11)
k, aw, wA, and T~ are the same as in the MAC model.
The diffuse irradiance for a zero albedo is:
Clear sky solar irradiance models
Ed = Eon[O.5ToTxaTAs( 1 -- TR)
+ BA(ToTRTA,,- aw)(1 - TA~)] (12)
where BA is a stepwise function of h. The clear sky
albedo is calculated as in MAC, except that B] is now
0.8832.
From the lEA validation study [ 8 ], it appears that
JOS performs marginally better than MAC when compared to routinely measured hourly irradiation data of
mixed cloudiness. (There are some differences in the
way that the two models evaluate the transfer of radiation through the cloud cover.) Unfortunately, no
detailed statistics have been provided to evaluate the
performance of MAC and JOS for clear sky conditions
only. The results in sections 4 and 5 will allow to compare their accuracy to that of clear-sky-only models.
2.3 CPCR2 model
This two-band model[15 ] is based on a more rigorous approach than the other models considered here,
as it separates the solar spectrum into two bands,
avoiding most transmittance overlapping problems
inherent to the typical one-band approach.
A preliminary version of this model (including a
cloud-layer submodel to treat the cloudy sky cases)
had been submitted to the lEA Task IX committee,
but unlike MAC and JOS, it has not been evaluated.
However, it has been compared to rigorous spectral
codes [ 15 ], suggesting that it could perform well in a
variety of atmospheric conditions under clear skies.
The beam and global irradiances in the band 0.290.70 tzm were also shown to compare well to one of
these codes. All the governing equations may be found
in[15].
2.4 lqbal's parameterization models A, B, and C
These three physical models are fully described
in [ 16 ]. Their predictions have also been compared to
measured data in France[3 ], while the original models
from which they closely derive have been evaluated
using data from Greece [ 2 ].
2.4.1 ModelA. Model A (hereafter IQA) is based
on the earliest version of MAC[9]. However, an important modification regarding the aerosol extinction
transmittance has been introduced by Iqbal. Equation
( 1) is replaced by a function of mR,/3, and ~,ngstrrm's
turbidity exponent, cc
TA = (0.12445a -- 0.0162) + (1.003 -- 0.125a)
× exp[--mR/3(1.089a + 0.5123)].
(13)
There are also a few minor departures from MAC.
As is the case with the two other Iqbal models, the
amount of precipitable water w is pressure and temperature corrected. The aerosol forward scatterance is
given in a look-up table only. As this is only a secondorder parameter, the corresponding equation in the
latest MAC version has been used here for ease of
computation.
123
2.4.2 Model B. Model B (hereafter IQB) offers a
practical improvement over Hoyt's model[ 17 ]. Essentially, the look-up tables of the latter have been replaced by exponential fits, thus simplifying computerized calculations. The original Hoyt's model has been
used as a part of the methodology used to rehabilitate
the historic U.S. radiation data[ 18 ].
2.4.3 Model C. Model C (hereafter IQC) is essentially identical to Bird and Hulstrom's model [ 19 ]. Besides a change of coefficient to conform to an updated
value of the solar constant (1367 W / m 2 instead of
1353), the only difference between Model C and Bird's
model lies in the treatment of water vapor and aerosol
paths. As mentioned above, a pressure and temperature
correction is used for w in IQC, whereas no correction
is used in Bird and Hulstrom's model. Moreover, IQC
uses the absolute (pressure corrected) air mass in the
computation of TA, while Bird and Huistrom's model
does not. Both models rely on the aerosol optical
thicknesses 6AXat 0.38 and 0.5 #m in order to compute
TA. As these parameters are available only at a limited
number of sites (see for example the data in[20]),
they have been replaced here using ~mgstrrm's equation:
5AX= /3X-5.
(14)
Bird and Hulstrom's broadband optical thickness /)a
then becomes:
tSA = (0.2758 × 0.38-~ + 0.35 × 0.5-")/3.
(15)
2.5 ASHRAE model
Thanks to its very simple formulation, the ASHRAE model has gained acceptance among the engineering community. In the present study, its computerized version [ 21 ] is used, and the treatment of diffuse
radiation follows the recommendation in [ 22 ].
The main difficulty in using the ASHRAE model
is the proper selection of the so-called clearness number
C, that has been introduced to make beam and diffuse
irradiances conform to nonstandard atmospheric conditions. C, should mainly reflect the local atmospheric
turbidity and precipitable water, but no means to derive
it is provided, except a small map of the U.S.A., giving
contours of the summer and winter C, values ranging
from 0.9 to 1.1.
The internal limitations and inconsistencies of this
model in its different variants have been discussed in
detail elsewhere (e.g., [22,23]). Though previous tests
have shown that its performance was far from that of
other more detailed models for conditions of mixed
cloudiness[ 13,24 ], it is still supported by the ASHRAE
and widely used, so that it has been included here for
reference purpose.
2.6 Powell's model
This modified ASHRAE clear sky model (hereafter
POW) was introduced by Powell [6]to alleviate the
problem of determining C,. This coefficient is thus
124
C. GUEYMARD
simply removed, and a pressure corrected air mass is
used, on the ground that elevated stations have a clearer
(high C.) atmosphere. This apparently helps decrease
the monthly mean bias error and root mean square
errors at many U.S. stations most of the time [6,25 ].
POW has also been shown to perform well (if using
somewhat corrected coefficients) in Italy [ 26 ].
2.7 Mtichler & lqbal's model
This model (hereafter M&I) has been proposed as
an alternative to the various versions of the ASHRAE
model [ 27 ]. It consists of a simple parameterization of
IQC. The aerosol transmittance is calculated as a function of visibility V which is observed at airports. As
the present study necessitated to rely on a turbidity
coefficient to allow comparisons, Vwas obtained from
a and/3 by solving equation 10 in [27]:
V = 1 4 7 . 9 4 4 - 1740.523[/3X
- (/32X2 - 0.17/3X + 0.011758) °5]
(16)
where X = 0.55-".
The simpler version of this model considers a fixed
value for the precipitable water in the atmosphere ( w
= 1.5 cm). A correction formula has been proposed
by its authors when this condition does not apply and
better accuracy is needed:
F ( w ) = (1.0223 - 0.0149w)m~ a7.
(17)
No formal water vapor transmittance equation is given,
but it is possible to obtain it indirectly, if it is considered
that it must be equal to the ratio of the beam irradiances
calculated for a fixed w and for w = 0 cm. Using eqn
(17) for both these cases, it is easily shown that:
Tw = F ( w ) / F ( O ) = (1 - 0.01457w)m~ 27.
(39.5 e x p ( - w ) + 47.4)]
(20)
where w is the precipitable water in cm.
No specific water vapor transmittance equation is
provided, but it has been derived for the present work
in a similar manner than for M&I, using eqn (19), as
well as eqn (20) with/3 = 0. The aerosol transmittance
is also derived here from eqn (20), using a fixed value
of precipitable water ( l cm) and the ratio of eqn (19)
determined for a fixed/3 and/3 = 0, respectively.
The diffuse irradiance equation is empirically obtained as:
Ed = 0.5Fl(F2Eon
-
Ee.)sin h
(21)
where F~ and Fz are polynomial functions ofh and TL
(see [ 28 ] for details.).
The EEC model represents the typical case of an
empirical model (particularly in the case of diffuse irradiance). Considerable effort was devoted to its development, but no independent performance statistics
seem to have been published to date.
2.9 PSI model
The Parameterized Solar Irradiance (hereafter PSI)
model is a simplified, one-band version of CPCR2
which has been presented recently[29]. It gives the
beam and global irradiances as an expansion of solar
elevation, with coefficients dependent on precipitable
water, Angstrrm's/3 turbidity coefficient, zonal (or regional) ground albedo, and station's altitude. (A discussion on the sources of data for the zonal albedo and
its conceptual difference from the local albedo can be
found in[29] .) PSI has been shown to closely duplicate
CPCR2 predictions [ 29 ], but it has not yet been tested
against independent data.
(18)
Though this model was proposed as an alternative to
ASHRAE, no comparative statistics of their respective
performance have been published.
2.8 EEC model
The clear sky irradiance model developed as a joint
effort from participants in the European countries
(members of the EEC, hence its name) has been described in [ 28 ]. It is based on a simplistic, though semiphysical, formulation for the beam irradiance, using
the Linke turbidity factor Tt:
Eb. = Eo,, exp[-mgTL/(O.9mR + 9.4)].
TL = 0.1 + (16 + 0.22w)/3 + [(h + 85)/
(19)
As TL is not a pure turbidity coefficient (it varies
significantly with precipitable water and air mass for
a constant aerosol loading), it has been replaced for
the present analysis by a function of w and /3 using
Dogniaux's empirical equation, also given in[28 ]:
3. DETAILED ANALYSIS OF THE MODELS
Some important results that have been discovered
during this study will be highlighted here. This critical
analysis of the individual equations used by the models
is essential for a better understanding of their limitations and their performance statistics that will be presented later on.
3.1 Input data
Besides solar elevation h and the extraterrestrial irradiance E0n that are considered here as external parameters, more or less input data are needed by the
different models among the following list: p (station
pressure), T (station temperature), Uo (ozone reduced
path), w (precipitable water), a and /3 (Angstr6m's
turbidity parameters), wA (aerosol single-scattering albedo), TL (Linke's turbidity coefficient), k (aerosol
transparency coefficient), fax (aerosol optical thickness
at fixed wavelengths), Cn (clearness number), V(visibility), and pg (average zonal albedo). Table 1 lists
Clear sky solar irradiance models
125
Table 1. Input data requested by the various models investigated here
Model
p
ASHRAE
CPCR2
EEC
IQA
IQB
IQC
JOS
MAC
M&I
POW
PSI
•
•
•
•
•
•
•
•
•
•
T
Uo
w
a
:1
to,4
o~
•
•
•
•
•
•
•
•
•
•
A
F
F
F
•
•
•
•
•
•
•
F
•
•
A
TL
6Ax
k
V
D
•
•
•
F
•
A
F
F
F
A
A
•
•
•
•
•
6".
D
D
• = Requested by the model and used in present study.
A = Alternate input used in present study.
D = Requested by the model but derived from alternate input(s) for present study.
F = Requested by the model but fixed or default value used in present study if no measured data available.
See text for nomenclature.
the different inputs requested by each model, and how
they have been handled in the present work.
Note that if station pressure is not available, it is
possible to estimate it from the station's altitude and
latitude, using the empirical equation proposed in [ 29 ].
All the results presented in this section pertain to a
sea-level site, so that p = Po (where Po is the standard
sea-level pressure, 101.325 k Pa) and the absolute and
relative air masses are equal: ms = m ( p / p o ) = m .
3.2 B e a m irradiance
3.2.1 Optical masses. Most of the models use Kasten's equation [ 30 ] to obtain the optical masses from
h. A S H R A E and M&I use the simple sine function
(m = 1/ sin h), while M A C uses Rogers's equation [ 31 ].
CPCR2 [ 15 ] used Kasten's equation for the Rayleigh
optical air mass and other equations for the water vapor, ozone, and aerosol optical masses. Now that more
recent data are available [ 32 ], the air mass values tabulated by Kasten appear too low at very low solar elevations. Therefore, these equations have been revised
using least-square fits of these data. The optical air
mass is now parameterized in C P C R 2 as:
m=
1/[sinh+al(90-h)(a2+h)
~3
(22)
where h is in degrees, a~ = 1.76759 × 10 -3, a2 =
4.37515, and a3 = - 1.21563. For h = 0, this equation
predicts that m = 37.81, compared to m = 38.17 from
the tabulated data i n [ 3 2 ] , and m = 36.51 from Kasten's equation. Unlike Kasten's equation, the form of
eqn (22) allows that m be strictly equal to 1.0 for h =
90 °. (After the completion of this work, the author's
attention has been drawn by a reviewer to a recent
publication co-authored by Kasten[33 ]. It provides
revised coeffcients for Kasten's air mass equation [ 30 ],
resulting from updated atmospheric profiles and a corrected numerical technique. The new equation predicts
m = 37.92 for h = 0 and rn = 0.9997 for h = 90 ° .
However, no specific coefficients are given either for
the water-vapor mass or the ozone mass.)
For the revised version of CPCR2, the water vapor
mass is obtained from data in [ 32 ] still with eqn (22)
but with different coefficients: al = 4.29452 × 10 -4,
a2 = 2.24849, and a3 = -1.25290. Finally, the ozone
mass is obtained in a similar manner, with a~ = 1.07489
× 10 -2, a2 = 6.62667, and a3 = -1.38802.
Differences between all these equations are only
significant below about h = 5-10 °. Figure 1 shows a
plot of the different optical mass equations used in the
models tested here, for solar elevations below 10 °. Calculations have been extended to a solar depression of
- 1o. (Solar depressions of a few degrees may be encountered at elevated sites.) A truly physical parameterization of an optical mass should predict an everincreasing value for decreasing solar elevations or
depressions. Rogers equation and the 1/sin h equation
are symetrical about the zero elevation value, so that
they do not follow this rule. It is also apparent from
Fig. 1 that a large range of values are obtained for h =
0, depending on the type of optical mass. For example,
the extremes predicted by eqn (22) are 14.3 for the
ozone mass and 71.4 for the water vapor mass. The
use of separate equations based on recent data for the
different optical paths, such as eqn (22), is recommended in the general case where calculations at low
solar elevations are considered. (Although the accuracy
of calculations at low solar altitudes may seem of marginal interest in usual solar energy applications, it may
be an issue in mountaineous areas or for some theoretical studies. Because the future use of a model is
never known, it is this author's perception that all
equations be as physical as possible, covering the whole
range of possible conditions, or at least that their limitations be duly stated.)
The absolute air mass is used to compute the aerosol
transmittance for IQA, IQB, IQC, MAC, M&I, and
JOS, rather than m in Bird and Hulstrom's model and
CPCR2. MAC, JOS, and M&I also use mR for the calculation of water vapor absorption. The other models
use m instead, but IQA, IQB, and IQC use a pressure
and temperature correction on w. (According to [ 34 ],
126
C. GUEYMARD
I
75
\
65"
I
I
\
....
ASHRAE / M&I
.....
MAC / POW
....
Kasten
- -
CPCP,2 Air
CPCR2 Water
--
55
or)
09
<
--
CPCR2 Ozone
\
45
-..I
o
In
O
\\
35
25
15
5
•
-1
0
1
SOLAR
I
I
I
2
3
4
ELEVATION
(°)
Fig. 1. Optical mass predictions at low solar elevations.
no temperature correction would be necessary on w;
the justification of the pressure correction remains debatable to the eyes of the present author.) MAC uses
mR again in the calculation of ozone absorption.
It is this author's recommendation that mR be used
only for the calculation of the transmittances by air
molecules (Rayleigh) and uniformly mixed gases because the concentration of these constituents in the
atmosphere is pressure dependent. The altitude dependence of water vapor, aerosols, and ozone follows
different patterns, so that their respective (nonpressure
corrected) optical masses should be used along with
the corresponding reduced paths (w,/3, and Up) observed directly above the site. As discussed in [ 15 ], the
aerosol optical mass may be taken equal to the water
vapor optical mass.
3.2.2 Rayleigh scattering. Though Rayleigh scattering is considered as a well described process, its
broadband transmittance T~ is not always correctly
formulated. For instance, anomalous beam irradiance
values are obtained with JOS for h < 2 °, with IQA for
h < 4 °, and with IQC for h < 3 ° when a Rayleigh
atmosphere is considered (Fig. 2). With JOS and IQA,
negative or excessively high irradiances are also ob-
tained in normal atmospheric conditions at low solar
elevations. For MAC, the equation giving TR has
changed in each successive publication. For instance,
two different equations are recommended in the latest
references[8,10]. When the typo in[8]is corrected,
both equations appear to give nearly identical results.
Equation (2), as recommended in[10], has been preferred here because it is documented in a refereed paper[7]. CPCR2 and MAC used similar techniques to
derive TR from a physical approach and their results
are in close agreement at any air mass. EEC predicts
about 8% lower than CPCR2 and 4% below MAC at
m = 10, but otherwise gives similar results. (For EEC,
the Rayleigh transmittance is simply obtained from
eqn (19) with Tz = 0. This simplification may explain
in part the apparent lower transmittance of this model,
because in reality it incorporates the ozone transmittance as well [ 35 ] .)
3.2.3 Ozone absorption. The ozone transmittance
results of IQA, IQC, and MAC agree to those of CPCR2
within _+ 1% as long as muo is not larger than about 5
cm (Fig. 3). (Note that surprisingly for MAC, an unusually complex transmittance equation is used, while
a fixed value of the ozone reduced path, Up = 0.35 cm
.... ....i,"'" 'i,i
127
Clear sky solar irradiance models
RAYLEIGH SCATTERING
1.0
,
. . . .
,
0.9-
0.8.
LU
O
Z
<
}F-
0.7
.~
0.6-
Z
nI.-
0.5-- -
- Iqbal A
- - - - Iqbal B
.....
Iqbal C
......
MAC
........ Josefsson
0.4-
. . . . .
0.3-
" % " '-'- ~- . . . . ._
',,,
" "";"
i
1
It
a ~
i
i
*l
s
I
0.2
'
'
'
'
I
. . . .
I
5
0
'
'
'
10
'
I
'
'
'
15
AIR
'
I
. . . .
2O
MASS
Fig. 2. Rayleigh transmittance predicted by different models.
OZONE ABSORPTION
1.0
ILl
O
Z
<
.~
0.9
O3
Z
,<
nI--
-
-
CPCR2
-
Iqbal
A
-
Iqbal
B
. . . . .
Iqbal
C
........
Josefsson
--
--
/ MAC
0.8
0
2
4
6
m * u z (cm)
Fig. 3. Ozone transmittance predicted by different models.
j
l
,
,
I
25
30
128
C. GUEYMARD
MIXED GASES ABSORPTION
i
1.00
i
I
I
i
i
I
1
I
l
.
t
i
I
I
l
I
I
I
. . . .
I
I
\
Ll.I
0.95
£0
Z
,,~
tr
I--
0.90
O
Z
<
- -
CPCR2
-- -- - Iqbal B
.....
Iqbal C
0.85
I
. . . .
I
5
. . . .
10
AIR
15
20
MASS
Fig. 4. Uniformly mixed gases transmittance predicted by different models.
is recommended--see, e.g., [ 8 ] - - b u t this limitation has
been overridden here to improve MAC's accuracy.)
JOS uses a constant ozone transmittance (0.95545),
corresponding to m u o .~ 1.35 (Fig. 3). As Uo has a
representative latitudinal and yearly average of 0.35
cm[36], this condition corresponds to m ~. 3.9 or a
solar elevation of about 15°. Therefore, JOS will tend
to underestimate the ozone transmittance whenever h
,~
1
L
~
l
i
> 15 ° (i.e., in the majority of cases, except at very high
latitudes).
Because of their common root, IQA and MAC give
exactly identical results as long as a sea-level calculation
is performed.
3.2.4 M i x e d gases absorption. Only CPCR2, IQB,
and IQC take the mixed gases absorption into consideration, though it is of the same order of magnitude
WATER VAPOR ABSORPTION
t i i i i i i i i i
,.
~
~ . ~ _ .
_____._
CPCR2
IqbaIA/MAC/JOS
--.-
Iqbal C
EEC
M&I
0.9-
i
i
Iqbal B
'
' ~"
\\
I:
u
oo
Z
<
n."
I-
i
.-.......
~x
IJJ
i
-.
"'-:x
0.8-
p = 1013 mb
""
"-.
°
0.7
0
5
10
m,w
15
(cm)
Fig. 5. Water vapor transmittance predicted by different models.
20
Clear sky solar irradiance models
as the other absorption processes (by ozone and water
vapor). CPCR2 and IQC give almost identical results,
while comparatively to CPCR2 and IQC, IQB tends
to underpredict significantly at high air masses
(Fig. 4).
3.2.5 W a t e r v a p o r a b s o r p t i o n . The prediction of
water vapor absorption is generally in close agreement
from model to model, at least for m w < 1 cm. For the
models using the absorptance approach (IQA, IQB,
JOS, and MAC), the corresponding transmittances
were obtained as T w = 1 - a w . For m w = 20 cm, the
transmittance predicted by IQC is about 4% over
CPCR2, IQA, JOS, and MAC, and about 14% over
IQB (Fig. 5 ). EEC and M&I have a pronounced erratic
behavior, due to their nonphysical derivation. (For
these models, the water vapor transmittance equation
has been derived for this study as explained above, but
is a function of m and w separately; the individual
values necessary to obtain Fig. 5 result from averaging
many calculations made with different (m, w) pairs
with the same r n w value, so that the resulting transmittance may not be considered as a unique function
of m w as with the other models.)
3.2.6 A e r o s o l e x t i n c t i o n . For clear sky conditions,
aerosols are generally the main source of extinction in
the atmosphere. Their opposite effect on the beam and
diffuse irradiances is documented elsewhere (e.g.[ 15 ]).
Surprisingly, as Fig. 6 demonstrates, most models agree
well on the prediction of TA for low to relatively high
turbidities (/3 = 0 - 0.47) and medium to high solar
129
elevations (h > 40 ° ). However, this agreement deteriorates as solar elevation decreases and m/3 increases.
For instance, IQA asymptotically reaches a minimum
TA of 0.1456 if t~ = 1.3 (value that has been used
throughout in this section), while the other models
predict far lower values if/3 > 0.2 (Fig. 7).
For/3 -- 0.8 and w from 1 to at least 4 cm, it appears
that EEC predicts a slightly negative global irradiance
for h < 6 °. For k = 0.7 and w = 4 cm, JOS predicts
negative beam irradiances for h lower than about 12°.
It may therefore be concluded that eqn (7) is not physically sound and eqn (3) should be preferred.
The relationship between/3 and k has been investigated using the aerosol transmittance predicted by
CPCR2, for a large range of/3 (0 - 0.47). As CPCR2
does not consider a multiplicative aerosol transmittance such as eqn (3) with MAC, a modified broadband
(0.29-2.7 #m) transmittance had to be defined as the
ratio of the values of Eb, respectively predicted for a
given/3 and for/3 = 0. For each TA thus generated, k
was then obtained by solving eqn ( 1). This procedure
was done for different solar elevations and turbidities,
and the results appear in Table 2. If the predictions of
CPCR2 may be considered accurate, it appeals that k
is not only a function of turbidity, but also of air mass
(k essentially decreases with m for a fixed/3), thus is
not appropriate for accurate radiation modeling.
Finally, it may be noted from Figs. 6 and 7 that
IQA, IQB, and M&I incorrectly predict a transmittance
value below 1.0 for/3 = 0.
AEROSOL EXTINCTION
1.0
I,,,,~
J , , , l ~ , , , , , , , , I , , , t , , , , , I , ~ , , *
h =45
J ~ , l , , l l l l l l
I
°
0.9
- -
I
uJ
o
z
,<
p = 1013 mb
CPCR2
Iqbal A
Iqbal B
Iqbal C
M~chler
EEC
.....
--....
0.7
% ,
oo
z
0.6
%
nI-
\
0.5
"
0.4
0.3 - ~
0.0
0.1
0.2
0.3
0.4
0.5
BETA
Fig. 6. Aerosol transmittance predicted by different models for a solar elevation of 45°.
130
C. GUEYMARD
AERO~DL ~ C T I O N
1.0
0.8
I . . . . . . . . .
lt
I
I . . . . . .
, , , I
'
. . . . . . . . .
I . . . . . . . .
. . . . . . . .
I
h = 5 °
---
CPCR2
~
~
\
cz= 1.3
p = 1013 mb
~
Iqbal A
Iqbal B
Iqbal C
....
---
LU
0
Z
<
II
M~.chler
EEC
....
0.6
\
\
z
,,~
tr
I-.-
0.4
\
\
~
0.2
0.0
,
"--L
~
0,0
.
0.1
0.3
0.2
.
.
.
0.4
.
0.5
BETA
Fig. 7. Aerosol transmittance predicted by different models for a solar elevation of 5 °.
3.3 Diffuse irradiance
Unlike beam radiation, diffuse radiation cannot be
modeled in terms of individual transmittances. Therefore, no such detailed analysis, as done in section 3.2,
can be done in this case.
The diffuse irradiance values predicted by the 11
models were only compared for a few atmospheric
conditions. As clear-sky diffuse radiation is mostly dependent on solar elevation and turbidity, plots of the
variation of the predicted irradiance with h have been
made for different /3 values. Two such plots corresponding to somewhat extreme turbidity conditions
(B = 0 and 0.47) are displayed in Fig. 8.
It appears that large differences between the predicted irradiances occur, at least on a relative scale. It
is also evident that, relatively to the bulk of other models, A S H R A E predicts a very low diffuse irradiance for
a turbid sky (/~ = 0.47), even if an unusually small
value of the clearness number ( C , = 0.8) is used. At
h = 60 °, the value of C, needed to replicate the CPCR2
prediction would be 0.367. To replicate the prediction
of beam radiation in the same conditions, C, would
have to be equal to 0.443. As was discussed in[23],
the A S H R A E model has been empirically tuned to fit
conditions o f " v e r y clear skies" (i.e., with low turbidity)
for an important engineering purpose (peak cooling
load calculations). Therefore, its use in turbid conditions is not recommended. Because of the simplistic
approach used in the A S H R A E model, C, appears as
an empirical, fine tuning coefficient. If accurate calculations were to be done with this model, C, should
be derived as a function of turbidity, precipitable water,
ozone amount, pressure, zonal albedo, and solar elevation. As the derivation of such an intricate function
Table 2. Equivalent values of the aerosol transparency coefficient k for fixed turbidity conditions
and different solar elevations
B
0.0
0.05
0.1
0.2
0.3
0.4
0.47
0°
5°
15°
30 °
45 °
90 °
1.000
1.000
1.000
1.000
1.000
1.000
0.929
0.941
0.932
0.927
0.925
0.923
0.883
0.895
0.874
0.897
0.857
0.854
0.813
0.823
0.781
0.754
0.744
0.735
-0.768
0.708
0.668
0.652
0.638
-0.722
0.649
0.598
0.576
0.558
-0.694
0.614
0.556
0.531
0.510
131
Clear sky solar irradiance models
120
I=,,al,,,=l,,,,I
.... I .... t .... I .... I .... I .... I
600
[ntnl[
....
I ....
- CPCR2
....
MAC
-- -- - JOS
.....
Iqbal A
Iqbal B
....
Iqbal C
.....
Machler
.....
EEC
........
ASHRAE
......... POW
......
PSI
LU 8O
o
z
t~
60
LU
m
,p
-
,~.;,..
¢_
-~-.,~_.-.. .. .. .. .. .. .. .. .. .. ....
500
ILl
O
Z
oc= 1.3
w--lcm
p = 1013 mb
pg = 0.2
13=0
20
(A)
k--1
C n = 1.1
Uo= 0.35 ¢ n
....
0
I ....
10
I ....
20
I ....
3O
I ....
40
] ....
50
I ....
50
I ....
70
I ....
50
9O
SOLAR ELEVATION
I,,,,I
....
I
. ..... " - "
400
300
I,U
,P
l,
a
....
n"
_¢
03
40
Ill,,I,,,,I
13 = 0 . 4 7
k=0.S2
C n = 0.8
(b)
100
I ....
200
B
a
100
')
"o_
e,::..,
0
10
....
20
_ .......................
, ....
30
, ....
40
, ....
50
, ....
60
, ....
70
,
....
80
90
SOLAR ELEVATION
Fig. 8. Diffuse irradiance predicted by all the selected models for specified conditions. (a)/~ = 0.0;
(b) ~ = 0.47.
is beyond the scope of this study, no attempt has been
made here to adjust this coefficient to each particular
case.
Because POW does not consider C, or any parameter related to the atmospheric conditions (except
pressure), it predicts even lower diffuse irradiances at
high turbidities than ASHRAE. Conversely, M&I
seems to predict excessively high diffuse irradiances at
very low turbidities. MAC and JOS have been used
with an optimized value of k, as obtained from Table
2 (e.g., k = 0.52 for/3 = 0.47). This special treatment
allows MAC and JOS predictions to stay in close
agreement with those of the other physical models.
4.
THEORETICAL
VALIDATION
Results from three sophisticated spectral codes are
used here to test the performance of the 11 models
described above, with the same methodology as in [ 15 ].
These spectral codes are based on such close representations of the radiative transfer process that their prediction accuracy--though not perfect--is certainly
better than that of broadband models. They will be
used as reference theoretical calculations to validate
the simpler broadband models described above.
• Results of the spherical harmonics code of Braslau
& Dave (hereafter B&D) are available for h = 10 °,
30 ° , 60 ° , 90 ° , and six basic atmospheres (code
named by these authors A, B, C, CI, D, and DI )
ranging from a Rayleigh atmosphere (no absorption
nor aerosol extinction) to a very turbid sky with absorbing aerosols[ 37](see Table 4 of [15]for details).
• S O L T R A N is a special version of the L O W T R A N
3 transmittance code [ 38 ] that has already been used
to test broadband irradiance models[ 19 ]. Its published results pertain to two typical atmospheres
(MidLatitude Summer, MLS, and SubArctic Winter,
SAW), whose precipitable water values are 1.42 cm
and 0.42 cm, respectively. Each of these atmospheres
is combined with a 23 km visibility aerosol model
( a = 0.94 and/3 = 0.133 according to spectral data
i n [ 3 8 ] a n d a 5 km model ( a = 0.94 and/~ = 0.47).
Results are available for h = 5, 10, 15, 30, 50,
and 90 ° .
* BRITE is a rigorous spectral code using a MonteCarlo method, that has been used extensively to derive spectral data sets[39]. Spectral tabulations for
a MLS atmosphere (w = 1.42 cm) and two aerosol
loadings are available[39]. They have been integrated for this study, using a rectangular rule. The
25 km visibility atmosphere may be described by a
= 1.07 a n d / 3 = 0.118, according to data i n [ 4 0 ] ,
while a = 1.07 and/~ = 0.045 for the 150 km visibility
atmosphere. BRITE's results are available for h =
10, 15, 20, 30, 41.8, 53, and 90 °.
It is important to note that all these rigorous calculations predict the true beam irradiance, without any
contribution from the parasitic circumsolar (diffuse)
irradiance. Therefore, these predictions are not exactly
comparable to field measurements with pyrheliometers
having a nonnegligible field-of-view.
These data sets from rigorous calculations constitute
a very discriminating benchmark of 61 values of beam
irradiance and 38 values of global and diffuse irradiance
to test simpler broadband models. For further reference, this benchmark data set appears in Table 3.
The statistical results of this comparison are given
in terms of the mean bias error (MBE) and root mean
square error ( R M S E ) , statistics that are standard in
assessing the performance of solar radiation models
(see, e.g., [3,6,8,24,25,41,42 ]). These statistics appear
in Tables 4, 5, and 6, respectively for the beam, diffuse,
and global irradiances.
132
C. GUEYMARD
Table 3. Reference irradiances (W/m 2) predicted by rigorous spectral codes for specified solar elevations
h
5
10
Beam
B&D
A
20
917.1
688.4
420.3
421.7
95.2
96.0
B
C
Ct
D
Dl
BRITE
MLS/150
MLS/25
SOLTRAN
MLS/23
MLS/5
SAW/23
SAW/5
Diffuse
B&D
A
15
102.8
7.0
120.0
9.0
C
C1
D
DI
BRITE
MLS/150
MLS/25
Global
B&D
A
475.4
292.6
605.0
423.4
258.9
42.4
289.0
50.5
386.3
97.2
423.3
112.0
693.2
521.8
128.2
109.8
804.5
655.5
53
879.5
750.6
58.7
83.5
70.6
84.7
132.6
307.7
M A C a n d JOS have b e e n used with a c o n s t a n t "effective" k c o r r e s p o n d i n g to the value of/~ given above
for each typical a t m o s p h e r e (using Table 2, h = 45 ° ).
This o p t i m i z a t i o n has been d o n e in order to prevent
487.0
460.3
90
1194.3
1038.2
937.1
938.0
678.0
679.9
1210.3
1059.9
969.7
970.6
732.4
734.3
969.7
869.7
756.6
833.5
530.6
881.9
568.5
804.3
461.1
77.9
69.7
150.1
141.2
348.2
312.3
99.3
157.2
107.6
171.7
620.9
525.3
511.5
505.3
471.3
445.5
215.3
193.1
60
922.0
806.1
617.7
270.3
662.5
297.8
195.4
151.6
135.1
131.7
108.0
97.4
C
CI
D
DI
BRITE
MLS/150
MLS/25
50
61.4
55.1
117.4
110.6
244.7
217.9
45.6
59.0
B
41.8
1118.9
940.3
788.1
789.3
453.1
455.1
36.1
32.1
62.1
58.5
91.5
80.7
B
30
120.6
205.4
1112.2
968.8
961.7
953.5
935.4
901.1
685.6
657.6
82.8
74.1
158.9
149.6
376.2
337.9
843.9
815.5
1293.1
1134.0
1128.6
1120.2
1108.6
1072.2
1090.6
1075.1
excessive errors due to the uncertainty o n k i n h e r e n t
to these models, as in the previous section. Because
this special fine t u n i n g t r e a t m e n t is unlikely from the
n o r m a l user, the corresponding results in Tables 4 - 6
Table 4. Percent mean bias errors (MBE) and root mean square errors (RMSE) for the beam irradiance Ebn predicted by
different models for typical atmospheres
Reference model
Mean Ebn (W/m 2)
N
Model
ASHRAE
CPCR2
EEC
IQA
IQB
IQC
JOS
MAC
M&I
POW
PSI
B&D
763.6
24
SOLTRAN
373.2
23
BRITE
690.6
14
All
599.7
61
MBE
RMSE
MBE
RMSE
MBE
RMSE
MBE
RMSE
(-15.7)
-0.5
-9.4
-0.6
-3.9
4.6
(-14.8)
(-8.4)
0.3
-4.3
-6.8
(16.3)
2.2
11.0
3. l
8.0
5.4
(18.7)
(10.2)
12.3
24.9
9.8
(28.7)
0.4
-2.8
0.4
- 12.6
2.0
(-17.5)
(-9.9)
8.6
47.2
-1.6
(41.3)
0.9
10.8
5. l
13.7
3.5
(20.8)
(11.5)
10.6
62.8
7.0
(14.2)
0.6
1.5
- 1.4
-7.0
0.7
(-9.7)
(-4.6)
6.0
4.2
-1.5
(16.4)
0.9
3.5
2.3
7.5
0.8
(10.5)
(5.4)
6.2
32.0
3.1
(2.6)
0.0
-5.0
-0.6
-6.7
2.9
(-14.1)
(-7.8)
3.8
10.0
-4.2
(22.4)
1.8
9.9
3.4
9.2
4.5
(17.9)
(9.7)
11.2
35.8
8.5
Parentheses indicate optimized results after fine tuning of the model (see text).
Clear sky solar irradiance models
133
Table 5. Percent mean bias errors (MBE) and root mean square errors (RMSE) for diffuse irradiance Ea predicted by
different models for typical atmospheres
Reference model
Mean Ed (W/m 2)
N
B&D
143.6
24
BRITE
107.4
13
All
130.9
37
Model
MBE
RMSE
MBE
RMSE
MBE
RMSE
ASHRAE
CPCR2
EEC
|QA
IQB
1QC
JOS
MAC
M&I
POW
PSI
(-35.2)
-0.4
2.4
-9.3
12.7
-17.4
(10.6)
(26.7)
-1.4
-39.0
7.8
(57.2)
2.9
16.7
14. I
23.9
25.0
(20.6)
(37.7)
24.3
72.1
22.7
(-38.0)
7.9
-3.0
2.3
2.2
3.9
(-2.1)
(16.3)
11.7
-35.6
14.6
(42.6)
8.9
8.5
6.2
16.5
8.8
(9.9)
(17.3)
15.3
40.5
21.0
(-36.0)
2.0
0.8
-5.9
9.7
-11.2
(7.0)
(23.7)
2.4
-38.0
9.8
(54.6)
5.0
15.3
12.9
22.6
22.5
( 18.9)
(34.4)
22.7
66.7
22.5
Parentheses indicate optimized results after fine tuning of the model (see text).
appear inside brackets, indicating that nonoptimized
prediction errors would be larger. The performance of
M A C is not excellent for the prediction of the beam
and diffuse irradiances (Tables 4-5 ). It even compares
poorly to IQA in these cases, showing that eqn (13) is
significantly superior to eqn ( l ). However, some cancellation effects allow M A C with the optimized k to
obtain the lowest R M S E for global radiation
(Table 6).
The negative beam irradiance values predicted by
JOS for high w and/~ values and solar elevations up
to 15 ° were zeroed. In any case, JOS's performance is
surprisingly disappointing, even though the best estimate o f k was most probably provided. This is certainly
related to the unphysical form of eqn (7), as already
discussed.
A S H R A E was used with its December set of coefficients whenever ~ < 0.05 or w < 2 cm. Otherwise,
the July set of coefficients was used. Clearness numbers
were adjusted to vary between 0.8 for the most turbid
atmosphere to 1.2 for the Rayleigh atmosphere. Results
in parentheses also indicate that higher errors would
have been possibly obtained with less optimized Cn
values.
IQC performs generally well, specially against
BRITE. This could be expected, as its most important
equations are fitted to BRITE's results.
As a whole, CPCR2 appears unsurpassed, most
probably because of the extra flexibility and accuracy
provided by its detailed two-band parameterizations.
Other good performers are IQA and IQC for beam
radiation (with an overall R M S E below 5% of the
theoretical m e a n ) , IQA and EEC for diffuse radiation
( R M S E < 20%), and, finally, IQA, IQC, MAC, and
PSI for global radiation ( R M S E < 5%).
5. EXPERIMENTAL VALIDATION
With regularly calibrated instruments, frequent inspections and detailed data quality control, it is now
Table 6. Percent mean bias errors (MBE) and root mean square errors (RMSE) for the global irradiance E = Eb,, sin h +
Ea predicted by different models for typical atmospheres
Reference model
Mean Ed (W/m 2)
N
B&D
691.2
24
BRITE
543.8
13
All
639.4
37
Model
M BE
RMSE
MBE
RMSE
MBE
RMSE
ASHRAE
CPCR2
EEC
IQA
IQB
IQC
JOS
MAC
M&I
POW
PSI
(-8.6)
-0.7
-7.4
-2.2
-2.0
-1.0
(-7.9)
(-0.1)
(12.0)
1.6
9.4
3.3
7.3
3.0
(9.8)
(1.2)
5.8
12.6
5.2
(2.5)
1.8
-0.7
- 1.2
-5.3
1.2
(-5.7)
(l. 1)
5.9
0.5
1.6
(8.5)
1.9
3.2
4.4
9.0
2.0
(6.0)
(1.7)
6.2
5.7
2.1
(-5.3)
0.0
-5.4
- 1.9
-2.9
-0.4
(-7.2)
(0.2)
1.7
-7.3
-2.1
(11.3)
1.7
8.3
3.6
7.8
2.8
(9.1)
(1.3)
6.0
11.4
4.7
-0.1
- 10.6
-3.7
Parentheses indicate optimized results after fine tuning of the model (see text).
134
C. GUEYMARD
estimated that the overall field accuracy (including
systematic and random errors) is about + 3-5% for
pyrheliometers and ___7-10% for pyranometers [ 43 ].
This uncertainty for first-class pyranometers is somewhat higher than what is generally quoted (___5%) in
older references, because recent national and international assessment studies have shown that some instrumental problems such as cosine response and azimuthal sensitivity had greater influence than previously
estimated. Although no uncertainty analysis was available for the experimental data sets presented below,
all were apparently obtained with the highest care possible, so that the above mentioned uncertainties should
be valid here.
The performance of the different models in real
conditions has been tested against different data sets
of measured beam and global irradiance. The data sets
include measurements from seven sites around the
world, covering a large range of atmospheric conditions, solar elevations, and station altitudes (Table 7).
Table Mountain is a rather dry and unpolluted elevated site in California that has been used in the past
by the Smithsonian Institution to record the solar irradiance and make observations of the possible variations of the solar constant[44]. Data of beam irradiance only have been extracted from spectral and filtered measurements [45,46 ].
For Montreal, the experimental data were obtained
by the present author in 1982-1983. The setup was
installed on the roof of Ecole Polytechnique's building
(located on University of Montreal's campus in a hilly
residential part of the city). It mainly consisted of Eppley sensors (a pyrheliometer with three filters and two
pyranometers), regularly calibrated by the AES. Some
of the necessary meteorological observations were obtained from nearby Dorval airport. Uccle is a residential
suburb of Brussels. The broadband data used
here [47 ]are interesting because they are matched with
concomittent spectral measurements of beam and
global irradiance. Linke turbidity data are available
and the corresponding/5 values have been obtained
from eqn (20).
Data from Weissfluhjoch [48,49]constitute an exceptional subset because of its altitude and the great
care taken to record these detailed solar and meteorological measurements. Beam irradiance was measured with an active cavity pyrheliometer, an instrument known to be more accurate than ordinary thermopile ones (a field accuracy of _+ 2% is attainable.)
Very low turbidities were encountered most of the time
due to the altitude. Their observed range was 0.0120.093. Because of the mountaineous environment,
some masking of sky diffuse radiation and some parasitic ground reflected radiation may occur, which may
slightly affect the diffuse and, to a lesser extent, the
global radiation results.
The data subset for Bangalore and Nandi has already been used to validate Hoyt's model, from which
IQB comes [ 5 ].
Data from Carpentras[50]are in fact hourly irradiations. This data subset has been added because irradiance models are often used to generate hourly solar
data for energy simulation studies, etc. In that particular case, only data from complete hours were retained.
lrradiance calculations were performed at each hourly
period mid-point.
For each station, all the individual data were carefully screened to remove those where clouds could have
been present. The diffuse irradiance was obtained by
difference between the global and beam horizontal
components. The different input parameters were generally measured on site (or close to it) but time interpolations were often necessary. The ozone reduced
thickness was measured in India[ 5 ], otherwise calculated from Van Heuklon's formula[36]. Because the
assessment of the physical consistency maintained in
the derivation of the models was the primary objective
of this study and because turbidity plays a major role
in the accuracy of predicting beam and diffuse solar
radiation, only high quality and detailed data from stations for which /3 was available (or traceable from
spectral or filtered data) were used in this study.
The coefficient a was not experimentally determined, except in India where a constant value of 2.0
has been used [ 5 ]. In Montreal, Carpentras, and Uccle,
a constant a = 1.3 was used. Inference data have shown
that ct was sharply decreasing with wavelength below
about 0.4 #m (thus reversing the normal trend and
translating into a negative a value in that part of the
spectrum) at the very low turbidities typical of elevated
sites such as Weisstluhjoch[51]. Therefore, its value
in the visible band was set to 0.0 for this latter station,
and 1.0 for Table Mountain. The infrared value was
maintained at 1.3, and the resulting broadband average
a was obtained as 0.65 and 1.15, respectively.
Constant (station-wise) values of k for MAC and
JOS were obtained from an educated guess, using data
in Table 2 and[8]as guidelines. A fixed clearness
Table 7. Characteristicsof the experimental sites
Fixed
parameters
Coordinates
Station
Table Mountain
Montreal
Uccle
Weissfluhjoch
Carpentras
Bangalore
Nandi
Country
U.S.A.
Canada
Belgium
Switzerland
France
India
India
Latitude
Longitude
Altitude
(m ASL)
34.37N
45.50N
50.80N
46.82N
44.05N
12.97N
13.37N
117.68W
73.60W
4.35E
9.83E
5.03E
77.58E
77.68E
2286
183
105
2667
99
950
1479
(k)
(C,)
Source of
data
(Reference)
0.93
0.91
0.87
0.94
0.95
0.88
0.88
1.10
1.00
0.90
1.15
1.00
1.00
1.10
[45]-[46]
Gueymard
[47]
[48]-[49]
[50]
[5]
[5]
MBE
-48.2
-1.7
12.9
23.7
21.8
23.7
-42.2
-30.8
22.1
-48.3
18.3
Model
ASHRAE
CPCR2
EEC
IQA
IQB
IQC
JOS
MAC
M&I
POW
PSI
MBE
180.4
68
66.4
10.4
19.7
28.5
34.4
27.7
61.4
56.4
22.2
66.5
23.1
-3.3
-0.1
7.6
1.7
-6.7
0.8
-3.0
-1.2
-0.9
-5.8
-0.8
6.3
1.4
7.7
2.1
6.8
1.6
3.9
3.0
2.0
6.7
2.3
25.5
- 1.9
10.8
4.6
2.7
-6.7
10.5
14.5
3.3
28.0
6.3
MBE
26.8
5.0
11.5
6.5
6.2
8.6
15.2
18.2
5.5
29.2
8.2
RMSE
2.6
9
790.6
Bangalore
19.0
2.4
12.7
6.9
1.0
0.7
1.4
4.4
5.7
11.7
7.6
MBE
20.0
4.5
13.2
7.8
3.9
4.2
5.7
7.2
6.8
13.1
8.5
RMSE
1.1
6
897.4
Nandi
15.1
2.2
6.1
0.9
-1.6
-0.5
7.1
12.5
7.2
28.2
2.7
MBE
25.7
5.1
7.6
5.0
6.7
4.6
20.7
23.0
8.6
35.0
6.0
RMSE
5.7
63
607.3
Uccle
-12.0
-2.5
7.1
3.0
-10.3
0.8
-9.9
-8.2
-4.4
-13.2
-7.7
MBE
18.7
4.8
9.5
22.9
12.0
5.6
13.3
12.3
7.6
16.4
8.4
RMSE
1.3
169
868.0
Weissfluhjoch
3.4
-0.9
4.2
-2.8
-7.0
-2.9
5.9
8.5
2.9
4.1
0.3
MBE
-98.8
8.2
-50.5
-5.7
-48.5
8.8
-61.7
-51.0
13.0
-98.0
-24.0
MBE
228.6
7
89.9
20.4
45.6
17.9
44.6
19.5
60.5
52.2
22.1
89.2
25.2
RMSE
Bangalore
-59.0
2.3
-41.7
-2.4
- 39.6
0.1
-7.9
4.8
8.4
-48.8
- 15.1
MBE
144.5
6
Nandi
63.5
21.8
43.7
19.3
45.9
22.8
25.5
24.5
24.2
53.8
26.4
RMSE
-28.8
-1.1
- 11.0
8.1
3.4
9.4
-24.1
-9.9
9.1
-42.2
6.6
MBE
144.5
63
Uccle
46.0
12.5
17.5
15.9
16.0
15.1
41.9
35.5
15.1
55.5
15.4
RMSE
- 11.9
6.3
-40.4
24.8
- 8.0
0.1
32.7
51.0
76.7
31.0
35.8
MBE
53.8
169
38.8
18.1
48.1
36.4
17.7
17.6
49.3
63.9
17.6
53.6
44.8
RMSE
Weissfluhjoch
- 18.2
13.1
1.1
24.5
10.1
21.6
- 15.8
-4.8
31.4
- 17.6
20.2
MBE
85.6
132
41.3
29.6
24.5
36.9
19.4
27.0
42.2
39.4
27.0
41.0
42.2
RMSE
Carpentras
12.4
5.6
8.3
5.8
9.3
6.2
15.2
16.4
6.6
12.5
8.5
RMSE
4.1
132
704.3
Carpentras
Table 9. Performance statistics (MBE and RMSE) for diffuse radiation, expressed in percent of the corrected mean irradiance
RMSE
45.4
4.6
5.0
5.0
8.2
5.5
38.4
41.3
9.0
45.2
5.5
RMSE
0.7
10
6.7
95
RMSE
994.7
Montreal
31.3
0.5
0.4
-0.9
-5.6
-2.6
21.2
26.3
7.5
31.1
2.1
ASHRAE
CPCR2
EEC
IQA
IQB
IQC
JOS
MAC
M&I
POW
PSI
Mean corrected
Et,n (W/m 2)
N
MBE
Table Mountain
602.3
Montreal
Model
Mean corrected
Ebn (W/m 2)
Mean circumsolar
correction (%)
N
Table 8. Performance statistics (MBE and RMSE) for beam radiation, expressed in percent of the corrected mean irradiance
-31.1
4.3
-9.4
19.3
5.1
14.4
- 16.4
-3.1
32.4
-24.7
17.8
MBE
All
99.4
445
3.3
-0.9
5.3
0.7
-7.2
- 1.0
1.8
4.7
1.2
4.2
-2.2
MBE
64.7
19.5
30.1
31.1
31.9
26.4
58.4
54.9
26.4
67.7
32.1
RMSE
23.6
5.0
8.8
16.4
10.3
5.7
19.5
20.4
7.6
23.8
7.9
RMSE
3.3
484
738.8
All
136
C. GUEYMARD
o.
I
I
I
d~
8
Ill
~11
II
(o
I
I
O
i.
oo
ud
I
8
oo
¢,
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~5
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ud
III
number for each site was also guessed for each site in
order to use ASHRAE. The resulting estimates of k
and Cn appear in Table 7. For CPCR2, the aerosol
single-scattering albedo was determined from Table l
in [ 15 ], using the values corresponding to rural clear
aerosols for Weissfluhjoch, rural average for Bangalore
and Nandi, rural/urban clear for Carpentras and Uccle,
and urban average for Montreal.
The pyrheliometric data include a nonnegligible
circumsolar contribution because of the relatively large
field-of-view of the instrument (5.7 ° ). As the physical
models do not make provision for such a contribution
(that may become important at large air mass/turbidity
conditions), the measured beam data were individually
corrected to obtain the beam irradiance alone, using
eqns (27-28) in[l]. This has been performed before
obtaining the indirectly observed diffuse component.
This helped to significantly reduce the error statistics
(MBE and RMSE) for all models, except EEC. (This
is understandable because a part of this model has been
fitted to observed data that were not corrected for the
circumsolar contribution.) The mean percentage of
circumsolar contribution contained in the beam radiation ranged from a low 0.7% in Table Mountain to
a high 6.7% in Montreal, where high turbidities were
recorded (the maximum/~ was 0.49).
The overall performance of the models has been
evaluated with the same standard statistics as in section
4. For each model, the resulting MBEs and RMSEs
were expressed as a percentage of the observed mean
irradiance, corrected for the circumsolar contribution
in the case of beam and diffuse irradiance. The results
for each model and each site appear in Table 8 for
beam radiation, Table 9 for diffuse radiation, and Table
l0 for global radiation. The total data set includes
nearly 500 individual measured values and cover a
broad range of cloudless atmospheric conditions, so
that the results may be considered as highly significant.
The statistics of Tables 8-10 tend to confirm the
main results of section 4, so that no detailed discussion
will be necessary. However, something should be noted
about POW and ASHRAE. Although more than onethird of the whole data set comes from elevated stations,
POW does not perform better than ASHRAE. This
finding does not agree with the results in [6,25 ], that
were based solely on data of global radiation from the
U.S. network, the quality of which is known to have
suffered a lot of problems in the past.
The errors on diffuse radiation appear large on a
relative scale but correspond to 20-30 W / m 2 on an
absolute scale, i.e., the same magnitude as the errors
on beam and global radiation. The relative errors on
global radiation are lower than for the beam radiation,
as could be expected. CPCR2, EEC, IQC, and PSI are
the most accurate, with RMS errors within--or close
of--the instrumental random error (e.g., about 5% for
global radiation).
6. CONCLUSIONS
The analysis and tests presented herein show that:
• Among the 11 selected irradiance models, CPCR2,
Clear sky solar irradiance models
•
•
•
•
•
•
IQC, EEC, and PSI (in this order) perform best
against carefully recorded experimental data. They
all have an R M S error below 6% for global radiation,
and below 9% for beam radiation. Therefore, it appears possible to obtain the irradiance components
with an accuracy comparable to routine measurements given the necessary input parameters. Particularly, good estimates of turbidity are necessary to
obtain accurate predictions of beam and diffuse radiation.
These same four models, and also IQA, perform best
against theoretical (reference) data from rigorous
spectral codes. This data set may be considered as a
benchmark test for clear sky irradiance models.
Many models suffer from various types of limitations
(e.g., uncertainty on input data for MAC, JOS, and
A S H R A E ) or even flaws (leading to anomalous itradiance values for JOS, IQA, and IQC in certain
conditions). The source of these problems has been
traced back either to the derivation of some transmittance functions [e.g., eqns ( I ) and (7)], that
should be revised, or to the use of " c u s t o m " parameters (k and Cn) that are difficult to relate to actual
atmospheric conditions.
P O W is generally not an improvement over ASHR A E if reasonable clearness numbers may be obtained for the latter. In any case, both models are
not recommended because their too simplistic formulations lead to large errors on the beam and diffuse
components. M&I offers a consistent, though sometimes small, i m p r o v e m e n t over these two models.
Validation studies such as this one should take care
of the circumsolar contribution from diffuse radiation that augments the apparent beam irradiance in
a nonnegiigible way. This problem deserves more
attention.
The two physical models that performed best against
hourly data of mixed cloudiness during the lEA Task
IX study are not r e c o m m e n d e d for clear sky predictions if beam or diffuse irradiances are needed ( M A C
and JOS), or low solar elevations are encountered ( J O S ) .
Two-band modeling seems to offer extra-accuracy
over more conventional one-band models. The best
performers of each kind ( C P C R 2 and IQC) may
predict radiation with an accuracy comparable to
more sophisticated spectral codes, but with far less
computational effort.
Acknowledgments--The author wishes to thank Mr. Robert
Couderc, Dr. Alain Heimo, and Dr. Peter Valko who were
highly instrumental in providing some of the measured data
sets that have been used here. Thanks also to Dr. Ross
McCluney who revised the manuscript.
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