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Decomposing global solar radiation into its direct and
diffuse components
John Boland
n
, Jing Huang, Barbara Ridley
School of Mathematics and Statistics, Barbara Hardy Institute, University of South Australia, Mawson Lakes Boulevard, Mawson Lakes, SA 5095, Australia
article info
Article history:
Received 3 December 2012
Received in revised form
5 August 2013
Accepted 11 August 2013
Keywords:
BRL model
Direct normal radiation
Diffuse radiation
Statistical modelling
abstract
To assess the viability of proposed solar installations, knowledge of global solar radiation is not sufficient.
For stationary photovoltaic plant, we require global radiation series, but also the contemporaneous
diffuse radiation series. Alternatively, for concentrated solar thermal, we need global and direct normal
solar radiation. In this paper, we investigate whether one can simply use a model for predicting diffuse
radiation using multiple predictions derived by our research team, the Boland–Ridley–Lauret (BRL)
model, to give delineations of both diffuse and direct or if we need to use another model for direct or
develop a new direct normal statistical model.
&2013 Elsevier Ltd. All rights reserved.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749
2. Historical development of the diffuse fraction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750
3. Logistic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752
4. Comparison with other models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752
5. Boland–Ridley–Lauret (BRL) model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755
7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756
1. Introduction
The evaluation of the performance of a solar collector such as a
solar hot water heater or photovoltaic cell requires knowledge of the
amount of solar radiation incident upon it. Solar radiation measure-
ments are typically only for global radiation on a horizontal surface.
They may be on various time scales, by minute, hour or day.
Additionally, one can infer daily totals from satellite images. These
global values comprise two components, the direct and the diffuse.
DNI, “the direct normal irradiance, is the energy of the direct solar
beam falling on a unit area perpendicular to the beam at the Earth's
surface. To obtain the global irradiance the additional irradiance
reflected from the clouds and the clear sky must be included”[1].
This additional irradiance is the diffuse component.
For various applications, one needs knowledge of diffuse solar
radiation and for others, one needs to have measured or estimated
values of direct solar radiation. For flat plate collectors and house
energy analysis, we require global and diffuse radiation series but
for concentrated solar thermal, we need global and direct solar
radiation. If only global radiation on a horizontal surface is
available through measured data or inferred from satellite images,
one will need some type of model to estimate either the diffuse or
direct from the global values. When research first began on this
topic, the solar collectors in use were all flat plate, and so attention
was focused on developing diffuse radiation models.
There is an added reason for computing values of the diffuse
radiation. Typically solar collectors are not mounted on a hori-
zontal surface but tilted at some angle to it. Thus it is necessary to
calculate values of total solar radiation on a tilted surface given
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/rser
Renewable and Sustainable Energy Reviews
1364-0321/$- see front matter &2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.rser.2013.08.023
n
Corresponding author. Tel.: þ61 8 83025781.
E-mail address: john.boland@unisa.edu.au (J. Boland).
Renewable and Sustainable Energy Reviews 28 (2013) 749–756
values for a horizontal surface. It is not possible to merely employ
trigonometric relationships to calculate the solar radiation on a
tilted collector. This is because the diffuse radiation is anisotropic
over the sky dome and the “radiative configuration factor from the
sky to the tilted solar collector is not only a function of the
collector orientation, but is also sensitive to the assumed distribu-
tion of the diffuse solar radiation across the sky”[2]. There are two
different approaches to calculating the diffuse radiation on a tilted
surface; using analytic models, for example the Brunger approach
[2] or empirical models such as the BRL model [3]. Each rely on
knowledge of the diffuse radiation on a horizontal surface. The
diffuse component is not generally measured. Consequently, a
method must be derived to estimate the diffuse radiation on a
horizontal surface based on the measured global radiation on that
surface.
Numerous researchers have studied this problem and have
been successful to varying degrees. Liu and Jordan [4] developed a
relationship between daily diffuse and global radiation which has
also been used to predict hourly diffuse values. The predictor
typically used in studies is not precisely the global radiation but
the “hourly clearness index k
t
, the ratio of hourly global horizontal
radiation to hourly extraterrestrial radiation”[5]. Orgill and
Hollands [6] and Erbs et al. [7] correlate the hourly diffuse
radiation with k
t
, but Iqbal [8] extended the work of Bugler [9]
to develop a model with two predictors, k
t
and the solar altitude.
Reindl et al. [5] use stepwise regression to “reduce a set of 28
potential predictor variables down to four significant predictors:
the clearness index, solar altitude, ambient temperature and
relative humidity.”They further reduced the model to two
predictor variables, k
t
and the solar altitude, because the other
two variables are not always readily available. Another possible
reason was that some combinations of predictors may produce
unreasonable values of the diffuse fraction, eg. greater than 1.0 [5].
Skartveit et al. [10] developed a model which in addition to using
clearness index and solar altitude as predictors, also added a
variability index. This is meant to add the influence of scattered
clouds on the sky dome. As well, Gonzales and Calbo [11] stress
the importance of including the altitude and the variability of the
clearness index in any predictions of the diffuse fraction. Aguiar
[12] fitted an exponential model to Mediterranean daily data using
only the clearness index and found a consistency of fit amongst
locations of similar climate.
Boland et al. [13] presented the use of a decaying logistic
function to estimate the diffuse fraction from knowledge of the
clearness index. Subsequently, the lead author of that paper
combined with other researchers to provide a theoretical basis
for selecting that form of the model [14]. This concept was further
developed by adding more predictor variables to enhance the fit,
resulting in the Boland–Ridley–Lauret (BRL) model [3]. The mod-
elling effort in these three studies can be classified as from a
frequentist approach to statistical modelling. This refers to the
classical least squares estimation procedure that was used to
perform the parameter estimation. In related work [15,16], the
problem was undertaken using an alternative statistical starting
proposition, Bayesian model building and parameter estimation. It
was reassuring that using two separate modelling approaches, the
same predictor variables were found to be significant and the
parameter estimates proved to be very similar.
In recent years, there has been increasing interest in both
concentrating solar thermal (CSP) and concentrating solar photo-
voltaic (CPV) installations, and as a consequence, an increasing
interest in reliable estimation of direct normal radiation. So, we
now have the situation where for some applications, we need to
estimate diffuse radiation from global radiation, and for others,
direct normal radiation (DNI) from global radiation. As testimony
to this, Perez–Higueras et al. [17] have developed a simplified
model to predict direct normal from global. Additionally, the latest
version of Meteonorm software [18] includes two models in this
area, one statistically based model, the BRL model [3] for estimat-
ing diffuse from global, and one physically based model, the Perez
model [19], to estimate DNI from global.
The question that comes immediately to mind is whether we
need a plethora of models, specifically do we need a “best”model
for estimating diffuse from global and a “best”model for estimat-
ing DNI from global? Or, can one model suffice, wherein estima-
tion of the diffuse from global is performed, for instance, and then
the DNI is calculated from the other two components? In this
paper, we will provide evidence that using the BRL model [3] to
estimate diffuse solar fraction, and from it calculate DNI performs
as well as any present model specifically designed to estimate the
DNI from knowledge of the global. The implication is that we do
not need another complex model to model direct solar radiation,
because the direct solar radiation coming from the modelling of
diffuse solar radiation is sufficient.
The paper is organized as follows. Section 2 describes the
development of the logistic function model of hourly direct
normal solar radiation with multiple predictors. Comparison of
the logistic function model with other models and error analysis is
given in Section 3. How direct normal solar radiation is calculated
from the BRL model for modelling diffuse solar fraction with
multiple predictors and comparison of this procedure with other
models is described in Section 4. The final section is devoted to
conclusions.
2. Historical development of the diffuse fraction model
The original approach to diffuse fraction estimation from the
clearness index relied on a basic assumption that there are three
separate regions in the scatterplot –Fig. 1,reflecting differing
processes. Lanini [20] discusses this with using the Reindl model
[5] to as an example. The model is given below:
d¼η
1
þγ
1
k
t
þδ
1
sin α0rk
t
r0:3dr1:0
d¼η
2
þγ
2
k
t
þδ
2
sin α0:3ok
t
o0:78 0:1rdr0:97
d¼η
3
þγ
3
k
t
þδ
3
sin αk
t
Z0:78 0:1rd
Lanini shows that for an example data set, the diffuse fraction
varies in the middle sub-interval of 0:3rk
t
r0:78, with solar
altitude αbut has very little variation in the end ranges k
t
o0:3
and k
t
40:78. This is then used as the justification for breaking the
interval for k
t
into three segments and using separate models in
each sub-interval. Reindl may well have been guided by earlier
work where a similar splitting was done. Many of the earlier
approaches, such as that of Orgill and Hollands [6], used piecewise
Fig. 1. Diffuse fraction versus clearness index for Adelaide.
J. Boland et al. / Renewable and Sustainable Energy Reviews 28 (2013) 749–756750
linear models for the sections. There are, in our opinion, a number
of problems with this splitting.
Various authors use different end points for the sub-intervals.
Orgill and Hollands [6] use 0.35, 0.75 while Erbs [7] uses
0.22, 0.8.
The argument used above assumes that there are only two
significant predictors. Reindl uses only two as they are the ones
from his set of 28 that are most easily obtained for any location.
This covering of the spread of data in the middle sub-interval
by using the solar altitude works to an extent, but as we will
see later, the use of other equally available predictor variables
covers more of that spread and also the spread in the two sub-
intervals at the ends.
For the Reindl model per se, there are discontinuities at the
boundaries of the sub-intervals.
It must be stated that the third reason was not necessarily
apparent to any of the early modellers. The adoption of a single
model formulation for the whole interval for k
t
enables the
alteration of the model to suit climate change projections,
much more easily than have fixed sub-intervals.
We now encapsulate the steps in the development of the BRL
model. When we first began looking at the problem in preparation
for the first version of the model [13], it seemed from a purely
curve fitting perspective, that a type of decay function should be
appropriate. This led to the idea to first construct the variation of
the diffuse fraction as a moving average through the k
t
interval.
Such an exercise is depicted in Fig. 2. From this, it was relatively
straight forward to select a decaying logistic function as the
appropriate one. The next iteration was done in a much more
systematic manner [14]. It was decided to transform the data to a
form that is amenable to standard linear regression techniques.
Note that for linear regression
y
i
¼β
0
þβ
1
x
i
þϵ
i
ð1Þ
the assumption is that the x
i
are known, and the y
i
are random
variables that are independent and identically distributed (iid).
This means that the transformation should be of a type to result in
a homogeneous band of variation in the dependent variable as the
independent variable increases, as happens after the transforma-
tion –see Fig. 3. The data was now in a form suitable for modelling
with a line of best fit–see Fig. 4, whereupon the data and line
were back transformed to give the fit to the original data –see
Fig. 5. This seemed a more mathematical approach to the problem
than a simple moving average, and from it we felt we were
justified in selecting a decaying logistic function to use. The final
step in the development for the single predictor model was to
perform the activity above for several locations. The parameter
estimates were sufficiently similar to set in train the idea of
combining data sets from the various locations and constructing
a model that may be used for any location –see for example the
model applied to another location in Fig. 6.
The final step in the process resulted in the BRL model
[3,15,16 ]. What was added was four other predictor variables to
cover much more of the spread of the data. These are apparent
solar time AST, solar altitude angle α, daily clearness index K
t
and
persistence ψ. The first three are self-explanatory in terms of what
they are. The solar altitude had been employed in various other
models. The inclusion of AST reflects the fact that the atmosphere
is generally more turbid in afternoon than morning. See [3] for
more details. The result of adding these extra predictors is shown
in Fig. 7,fitting the spread of the data much better.
Fig. 2. Diffuse fraction versus clearness index for Adelaide with moving average
superimposed.
Fig. 3. Diffuse fraction versus clearness index for Adelaide transformed.
Fig. 4. Diffuse fraction versus clearness index for Adelaide transformed with line of
best fit.
Fig. 5. Diffuse fraction versus clearness index for Adelaide back transformed
with fit.
J. Boland et al. / Renewable and Sustainable Energy Reviews 28 (2013) 749–756 751
3. Logistic model
Boland et al. [13,14] found that using a decaying logistic
function is a good way to model diffuse solar radiation and also
it is widely used in ecology and species growth representations
[21,22]. To use a logistic function for modelling direct normal solar
radiation, it is growth with respect to clearness index.
According to Banks [23] and Jeffrey [24] using standard
integration techniques, Thornley et al. [25] obtains a modified
logistic function as
G
t
¼NM
NþðMNÞe
r
0
t
ð2Þ
Here, G
t
represents population number, and Nand Mhave
biological meaning for populations with a strong interaction
among individuals that controls their reproduction. If NoM, this
represents logistic growth which is the situation needed here and
if N4M, there is logistic decay. r
0
is the maximum possible rate of
population growth. If r
0
is low this means a slow rate of growth,
otherwise it will be fast. When applying this structure to model-
ling direct solar radiation, G
t
is replaced by the direct normal solar
radiation I
DN
,r
0
is replaced by β
1
and tis replaced by the clearness
index k
t
. Ridley et al. [3] also suggest using four other important
parameters for modelling the diffuse fraction which are apparent
solar time AST, solar altitude angle α, daily clearness index K
t
and
persistence ψ. These are adopted here as well. The multiple
predictor logistic model is
I
DN
¼NM
NþðMNÞe
β
1
k
t
β
2
AST
β
3
α
β
4
K
t
β
5
ψ
ð3Þ
The data chosen to build the model is multiple location data [3]
which is aggregated data from seven locations worldwide
(Adelaide, Darwin, Bracknell, Lisbon, Macau, Maputo and Uccle
from year 2001 to 2005, there are 7338 hourly diffuse radiation
data points). The method of ordinary least squares in Solver
(an optimization tool in EXCEL) is used to obtain all the parameter
estimates, Eq. (3) becomes
I
DN
¼0:006 4:38
0:006þð4:380:006Þe
7:75k
t
1:185AST1:05
α
0:004K
t
þ0:003
ψ
ð4Þ
Eq. (4) is applied to four individual locations; Adelaide (from
year 2003 to 2004, 4741 hourly diffuse radiation data points),
Darwin (from year 2001 to 2005, 2597 hourly diffuse radiation
data points), Lisbon (for year 1980, 3422 hourly diffuse radiation
data points) and Mt Gambier (from year 1973 to 1977, 14,058
hourly diffuse radiation data points) to test the efficacy of the
modelling of hourly direct normal solar radiation. The data
obtained from the Bureau of Meteorology, Australia. Fig. 8 shows
that the logistic model gives a good fit for the direct normal data
for the southern hemisphere location of Adelaide. In the northern
hemisphere location of Lisbon there is also a good fit as shown in
Fig. 9.
4. Comparison with other models
There are many models used to predict direct normal solar
radiation, but one of the most recognized is the Perez model. So,
we will use it to compare with the logistic model.
The Perez model [19] is a four dimensional coefficient matrix
model based on the Maxwell's model [26] for estimating direct
normal solar radiation. The basic idea of the Perez model is to use
a coefficient function XðK
′
t
;Z;W;ΔK
′
t
Þto improve the estimate
values I
disc
from Maxwell's model, as given in
I
DN
¼I
disc
XðK
′
t
;Z;W;ΔK
′
t
Þð5Þ
Here, K
′
t
is a zenith angle dependent expression of the clear-
ness, Zis solar zenith angle, Wis atmospheric perceptible water
and ΔK
′
t
is the stability index.
Fig. 10 shows the Perez model against the actual data in
Adelaide and it is not performing well for higher clearness index
values. The Perez model seems to have limitations for predicting
the higher values of direct normal. The same problem also appears
in the other three locations. Thus feature is a common problem for
use of models for either diffuse or direct radiation that have been
developed for Northern Hemisphere locations, when they are
applied to Southern Hemisphere sites. It does also seem to be an
issue for Lisbon –see Fig. 11.
Fig. 7. The BRL diffuse fraction model fit to Adelaide data.
Fig. 8. The logistic model fit for direct normal data in Adelaide.
Fig. 6. Diffuse fraction versus clearness index for Geelong with fit.
J. Boland et al. / Renewable and Sustainable Energy Reviews 28 (2013) 749–756752
For the purpose of formal error analysis of the proposed models,
the following measures are considered: median absolute percentage
error (MeAPE), mean bias error (MBE), normalised root mean square
error (NRMSE) and Kolmogorov–Smirnov test integral (KSI). MeAPE
captures the size of the errors, while MBE is used to determine
whether any particular model is more biased than another. NRMSE is a
measure of overall model quality related to regression fit. What this
meansisthatishowfarthedatadeviatesfromthemodel.Whatis
more informative is in essence how far the regression line is from the
line Y¼X,wherethey's are the predicted values from the model, and
x's are the data values. Interestingly, Willmott and Matsuura [27]
produce convincing arguments as to why the mean absolute error
(MAE) is a superior error measure to the RMSE. They argue that the
RMSE is a function of three characteristics of a set of errors.
It varies with the variability within the distribution of error
magnitudes and with the square root of the number of errors
(n
1=2
), as well as with the average-error magnitude (MAE).
KSI is a new model validation measure based on the
Kolmogorov–Smirnov test [28] which has the advantage of being
nonparametric. The KSI measure was proposed by Espinar et al.
[29] to assess the similarity of the cumulative distribution func-
tions (CDFs) of actual and modelled data over the whole range of
observed values.
Definitions of all the measures are as follows:
MeAPE ¼MEDIAN ^
y
i
y
i
y
i
100
MBE ¼1
n∑
n
i¼1
ð^
y
i
y
i
Þ
NRMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
∑
n
i¼1
ð^
y
i
y
i
Þ
2
n
s
y
where ^
y
i
are predicted values, y
i
are measured values and yare
average of measured values.
KSIð%Þ¼100 R
x
max
x
min
D
n
dx
α
critical
where x
max
and x
min
are the extreme values of the independent
variable, and α
critical
is calculated as α
critical
¼V
c
ðx
max
x
min
Þ. The
critical value V
c
depends on population size Nand is calculated for
a 99% level of confidence as V
c
¼1:63=ffiffiffiffi
N
p;NZ35. D
n
are the
differences between the cumulative distribution functions (CDFs)
for each interval. The higher the KSI value, the worse the fitof
model to data.
Table 1 shows that the logistic model is better than the Perez
model in all four error analyses at all locations, except MBE in
Lisbon. This is further illustrated in Figs. 12 and 13 for the KSI
which show observed and modelled CDFs, as well as differences
between them over the whole range of the data. Clearly, the
logistic model obtains estimates closer to the measured values and
lower values of D
n
. Thus, the logistic model appears at least as
accurate as the Perez model for predicting hourly direct normal
solar radiation. Therefore, there appears to be no advantage of
using arguably the best performing Direct Normal model in the
literature over using the multiple predictor direct normal model
developed here.
Fig. 10. The Perez model fit for direct normal data in Adelaide.
Table 1
Results of error analysis of two models in four locations.
Error measure Adelaide Darwin Lisbon Mt Gambier
Logistic model
MeAPE 9.20% 8.36% 10.92% 25.47%
MBE 0.030 0.040 0.122 0.094
NRMSE 14.20% 13.35% 16.27% 28.44%
KSI 22.27% 13.01% 26.11% 20.36%
Perez model
MeAPE 20.94% 13.94% 18.18% 27.60%
MBE 0.145 0.231 0.087 0.139
NRMSE 24.81% 17.73% 22.35% 33.74%
KSI 80.04% 60.43% 47.70% 30.82%
Fig. 11. The Perez model fit for direct normal data in Lisbon.
Fig. 9. The logistic model fit for direct normal data in Lisbon.
J. Boland et al. / Renewable and Sustainable Energy Reviews 28 (2013) 749–756 753
5. Boland–Ridley–Lauret (BRL) model
The performance of the direct normal radiation model derived
in Section 2 is compared with the direct normal radiation
estimated through using the BRL model [3]. The purpose of this
comparison is to ascertain whether using a model already
reported in the literature, the BRL diffuse fraction model [3],is
sufficient for estimating DNI from global through the chain global-
diffuse-DNI. In Section 3, it was shown that the model in Eq. (4)
performs at least as good as the best performing model in the
literature. If the BRL model performs just as well, then there is no
need for this new approach.
To obtain the direct normal solar radiation from the BRL model,
the following steps will be applied.
First, use the BRL model given by Ridley et al. [3]
d¼1
1þe
5:38 þ6:63k
t
þ0:006AST0:007
α
þ1:75K
t
þ1:31
ψ
ð6Þ
to obtain the diffuse fraction, d.
Second, using the following equation, we can calculate the
direct normal solar radiation.
I
DN
¼I
G
ðdI
G
Þ
sin ðαÞð7Þ
Here, I
G
is global solar radiation, αis solar altitude angle and I
DN
is direct normal solar radiation. Using Eq. (7), we can obtain the
BRL model fit to the direct normal data in Adelaide which is shown
in Fig. 13.
Comparing Figs. 8 and 14 shows that the BRL model results
seem to cover the data more than the logistic model, but it is hard
to see the difference between these two figures. So the same error
analysis as before has also been used for the predicted values of
the BRL model.
Table 2 shows that for all error measures the BRL model
performs as well as the newly derived model and thus at least
as well as or better than the Perez model. Since it performs slightly
better than the logistic model in MeAPE and NRMSE we could say
that the BRL model, in a ‘local’sense, is better than the logistic
Fig. 12. Plot of the logistic model of CDF for the measured and predicted data sets
(left) and the differences D
n
between those (right) at Adelaide. The dotted line
marks the critical value V
c
.
Fig. 13. Plot of the Perez model of CDF for the measured and predicted data sets
(left) and the differences D
n
between those (right) at Adelaide. The dotted line
marks the critical value V
c
.
Fig. 14. The BRL model fit for direct normal data in Adelaide.
J. Boland et al. / Renewable and Sustainable Energy Reviews 28 (2013) 749–756754
model. However, for error measures MBE and KSI, the logistic
model is slightly better than the BRL model, except in Mt Gambier.
It is an illustration that, in a ‘global’sense, the logistic model
developed here has better predictive ability, but not for all
locations. Figs. 12 and 15 also show that instead of slightly
different ‘global’and ‘local’performance, the BRL model obtains
better predicted values for most of the ranges (in the middle
ranges from 0.5 to 3) and the logistic model is a little bit better at
the ends of the range (from 0 to 0.5 and 3 to 4). Therefore, based
on their ‘global’,‘local’performance, D
n
and the value of difference
of D
n
, it is concluded that the BRL model is a slightly better than
the logistic model.
6. Discussion
The development of models for estimating diffuse solar radiation
began in 1960. Liu and Jordan [4] used the diffuse solar radiation
depending on different degrees of cloudiness or ranges of clearness
index for 98 locations across United States and Canada. Numerous
models have been presented for estimating diffuse radiation since
then, such as Orgill and Hollands [6] who used a correlation
function to estimate hourly diffuse radiation on a horizontal surface
and Erbs et al. [7] who used a curvilinear function to establish a
relationship between the hourly diffuse fraction and the hourly
clearness index k
t
. For the purpose of reducing the standard error of
the correlation function, Reindl et al. [5] proposed 28 potential
predictor variables for estimating diffuse fraction data and utilized
stepwise regression reducing the 28–4significant predictors: the
clearness index, solar altitude, ambient temperature and relative
humidity. In 1992, Perez et al. [19] developed a direct normal
radiation model which is a four dimensional coefficient matrix
model based on the Maxwell's model [26]. Until now, the Perez
model has been recognized as one of the most accurate models for
estimating direct normal radiation. In 2001, Boland et al. [13]
developed a logistic function to estimate diffuse fraction which is
unlike previous methods that used either piecewise linear or simple
nonlinear functions. To further validate the logistic model, Boland
et al. [14] outlined the theoretical development of the logistic
function, for estimating diffuse solar radiation. Then, Ridley et al.
[3] developed the Boland–Ridley–Lauret (BRL) model to improve
the accuracy of the logistic function by adding more predictors
which was then verified further by using a Bayesian approach to
arrive at the same model structure [15,16].
Recently, in the literature, most papers about diffuse fraction
study either test many models [30,38] or add their own correla-
tions [32,33]. For example, Kudish and Evseev [31] evaluated four
different correction models: Drummond [34], LeBaron et al. [35],
Battles et al. [36] and Muneer and Zhang [37]. Through error
analysis and scoring systems, such as the coefficient of determina-
tion of R
2
, RMSE, MBE, Percentage average deviation (PAD),
deviation (SD), t-statistic, accuracy score (AS) and Kudish and
Rahima (K&R) [38], Kudish and Evseev [31] concluded that overall
the Muneer and Zhang is the best model among these four
different correlation models for hourly diffuse radiation data at
Beer Sheva, Israel. Since their model requires extra variables to be
measured, we cannot test their model on our data, so we choose
normalized error measures in order to make comparisons. When
using the same error measures to compare the Muneer and
Zhang's model with the BRL model [3], we found that the two
models have similar accuracy when modelling hourly diffuse
radiation. For example, the coefficient of determination of R
2
from
the Muneer and Zhang's model is 0.9301 and the normalized MBE
is 1.4%, whereas the BRL model the measures are 0.9628 and
3.7% respectively for hourly diffuse fraction Adelaide, Australia
data. Since the evaluation is performed on separate sites, no direct
comparison can be made but the results are similar in nature.
Dervishi and Mahdavi [30] also assessed eight models, such as
Erbs [7], Reindl [5], Orgill and Hollands [6], Lam and Li [39],
Skartveit and Olseth [40], Louche et al. [41], Maxwell [26] and
Vignola and McDaniels [42], for estimating diffuse fraction by
using radiation data at Vienna, Austria. They found that three
models, Erbs, Reindl and Orgill and Hollands performed better in
obtaining estimates of diffuse fraction. Since Ridley et al. [3]
showed that the BRL model performed better than the Reindl
model at many locations mentioned in their paper, so the results
show that the BRL model performs at least as well as one of their
best performing ones. One should note in addition that the BRL
model, because of its structure, is easier to implement than many
of the competing models.
a
b
Fig. 15. Plot of the BRL model of CDF for the measured and predicted data sets (left)
and the differences D
n
between those (right) at Adelaide. The dotted line marks the
critical value V
c
.
Table 2
Results of error analysis of the BRL model in four locations.
Error measure Adelaide Darwin Lisbon Mt Gambier
Logistic model
MeAPE 9.20% 8.36% 10.92% 25.47%
MBE 0.030 0.040 0.122 0.094
NRMSE 14.20% 13.35% 16.27% 28.44%
KSI 22.27% 13.01% 26.11% 20.36%
BRL model
MeAPE 8.87% 8.12% 9.65% 24.90%
MBE 0.068 0.049 0.139 0.045
NRMSE 14.49% 12.71% 16.16% 27.98%
KSI 24.72% 13.39% 29.21% 10.64%
J. Boland et al. / Renewable and Sustainable Energy Reviews 28 (2013) 749–756 755
Instead of evaluating existing models, Li et al. [32] use a
combination of different predictors, clearness index, relative
sunshine duration, ambient temperature and relative humidity
to estimate diffuse radiation. They compared with four other
models in the literature and found that their model can perform
well for estimating the monthly average daily diffuse radiation.
However, they were testing their model on a longer time scale,
daily, than the hourly BRL model, and so no direct comparison can
be made.
From the previous studies [3,13–16], it has been shown that the
multiple predictors logistic function is a suitable model for diffuse
radiation. Thus, using the same method to estimate direct radia-
tion followed naturally. This new direct radiation model was
compared with the model that has been regarded as the industry
standard, the Perez model [19]. It performed better overall than
the Perez model. The next step was to compare this newly
designed statistical direct model with the sequence of estimating
the diffuse fraction with the BRL model, and subsequently calcu-
lating the direct irradiance. There proved to be no advantage in
using the newly derived direct model, thus showing that there is
no need to go further than using the chain of global to diffuse to
direct using the BRL model.
7. Conclusion
This paper focused first on the development of models for
diffuse solar radiation and then moved to discuss how to best
obtain estimated hourly direct normal solar radiation. First, a
logistic model for direct normal solar radiation using multiple
location data was constructed. Then, the use of the logistic and
Perez models in four different locations was compared. The results
of four error analyses show that the logistic model performed
arguably better than the Perez model. Afterwards, the BRL model
was used to obtain hourly diffuse radiation and from that direct
normal solar radiation. The predicted values from that chain of
steps was compared with results from the logistic model. Con-
sidering all locations and error analyses, the results show that the
BRL model is well equipped not to estimate the diffuse solar
irradiance from the global solar, but also to go from there to
estimate the direct normal irradiance.
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