Energy 24 (1999) 689–704
www.elsevier.com/locate/energy
Prediction of global irradiance on inclined surfaces from
horizontal global irradiance
F.J. Olmo, J. Vida, I. Foyo, Y. Castro-Diez, L. Alados-Arboledas*
Dpto. Fı́sica Aplicada, Universidad de Granada, Campus Fuentenueva, s/n 18071, Granada, Spain
Received 23 June 1998
Abstract
Knowledge of the radiation components incoming at a surface is required in energy balance studies,
technological applications such as renewable energy and in local and large-scale climate studies. Experimental data of global irradiance on inclined planes recorded at Granada (Spain, 37.08°N, 3.57°W) have
been used in order to study the pattern of the angular distribution of global irradiance. We have modelled
the global irradiance angular distribution, employing horizontal global irradiance as the only radiometric
input, and geometric information. We have obtained good results (root mean square deviation about 5%),
except for surfaces affected by artificial horizon effects, which are not allowed for in this new model. The
Skyscan’834 data set has also been used in order to test the model under completely different conditions
from those in Granada, with respect to the amount of cloud, local peculiarities, experimental design and
instrumentation. The results prove the validity of our model, even when compared with the Perez et al.
model. The model offers a reliable tool for use when solar radiance data are scarce or limited to global
horizontal irradiance.  1999 Elsevier Science Ltd. All rights reserved.
1. Introduction
Knowledge of short-wave global irradiance incoming on inclined surfaces is often necessary
in order to study the surface energy balance, the local and large-scale climate, or to design technological applications such as renewable energy conversion systems, either in urban or in rural
zones. Historically, at many national meteorological stations, global irradiance has been measured
only on horizontal surfaces and rarely on inclined ones. Thus different estimation methods have
* Corresponding author. Tel.: ⫹ 34-958-244024; fax: ⫹ 34-958-243214; e-mail: [email protected]
0360-5442/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 6 0 - 5 4 4 2 ( 9 9 ) 0 0 0 2 5 - 0
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F.J. Olmo et al. / Energy 24 (1999) 689–704
been developed, such as Refs. [1–6]. Nevertheless, calculations using these kinds of models are
not simple, since they require information of the direct or diffuse irradiance. This information
can be obtained from different techniques that involve horizontal diffuse irradiance measurements,
as in Refs. [7–10]. A detailed analysis of the contributions received on an arbitrarily oriented
surface suggests that the parameters involved in these models must depend on the atmospheric
state, properties of the adjacent surfaces, topography and geometrical factors.
Different authors have tested the results of inclined surfaces models against experimental data
[1,6,11–15]. Feuermann and Zemel [6] have shown that a combined application of the Perez et
al. model [3] with empirical correlations to estimate the direct component from horizontal global
irradiance, can yield accurate predictions for locations where only global horizontal data are available.
There has been growing interest in recent decades about the use of satellite data to obtain solar
irradiance at a surface. This method provides the possibility of continuous and global monitoring
of this radiation flux [16–19]. Although this technique seems powerful, the use of remote sensing
data in the case of complex topography terrains requires additional efforts [20,21]. The algorithms
that deal with the influence of complex topography in the solar irradiance field require the consideration of solar irradiance impinging on non-horizontal surfaces. This information must be
acquired with the unique knowledge of horizontal solar global irradiance.
In this paper, we present a simple model to estimate global irradiance on inclined surfaces,
under all weather conditions, which only requires the horizontal global irradiance, and the sun’s
elevation and azimuth as input parameters. The added value of this model lies in its applicability
to sites where only horizontal global irradiance is measured, as is the case at most conventional
meteorological stations, or satellite-derived global irradiance data. In addition, this method allows
for the estimation of the global irradiance distribution by means of the horizontal global irradiance
in a way that may on occasions be more practical than the methods of Feuermann and Zemel [6]
or Perez et al. [3]. These latter methods require a previous estimation of direct and diffuse
irradiance from the horizontal global irradiance.
This work represents a previous stage in the study of topographic effects on the solar irradiance
field, with the goal of mapping solar radiation on local and large scales. At present we only intend
to test the validity of the inclined global irradiance model that we propose in contrast to other
type of models—such as the one used in [20] or [21]. These authors use the Erbs et al. model
[22], which can present errors of about 37% and 22% when splitting global into diffuse and direct
irradiance, respectively [23].
2. Data base
Measurements of incoming global irradiance on surfaces with different slopes and orientations
have been carried out at Granada (Spain). The measurement system consists of a pyranometer
Kipp-Zonen CM-5, mounted on a device with the ability to vary both the elevation (0, the horizontal, to 90°, at 15° intervals) and azimuth (0, the south meridian, to 360°, at 45° intervals) of the
inclined surface. This system has been installed on the terrace of the Science Faculty of the
University of Granada (650 m a.s.l.) [24,25]. The stability of the radiometer calibration factor
has been periodically tested against a reference pyranometer used as substandard and not exposed
to the sun except during the intercomparison trials.
F.J. Olmo et al. / Energy 24 (1999) 689–704
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Each sky scan includes 49 instantaneous measurements, acquired during an interval of 45 min,
that we standardise for the solar height changes during the experiment (refer to 2D polar plot in
Fig. 1). The data base consists of a total of 114 clear sky experiments distributed over the year
(all months) and carried out with the same sequence of measurements at different times of the
day distributed uniformly around solar noon. Thus, a complete range of solar azimuth (from
247.5° to 104°) and solar height combinations (from 10° to 76°) are included in our data base.
In order to carry out a test of the model, trying to avoid the limitation of our data set (clear
skies only) and possible local dependencies from Granada in our results, we have used the Skyscan’834 data set [26,27]. The measurement system consists of an Eppley Precision Spectral
Pyranometer (PSP) installed on the Mechanical Engineering Building rooftop (University of
Toronto, 43.7°N, 79.4°W, 111 m a.s.l.). The pyranometer was calibrated by the National Atmospheric Radiation Centre in July 1982. This is a well-known data set, widely used and referenced
in this type of works. Skyscan’834 covers a full year of measurements in a completely different
environment to that of our trials in Granada. The main limitation is that the Skyscan’834 data
set only contains slope irradiance measurements for the case of surfaces oriented to the south with
an elevation angle of 44°. On the other hand, the system has the advantage that the pyranometer is
shielded from the radiation reflected from the ground [26]. We have characterised the different
atmospheric conditions by means of the clearness index, kt, defined as the global horizontal
irradiance to extraterrestrial horizontal irradiance ratio. kt ranges from 0.4 to 0.8 in Granada and
from 0 to 1 in the Skyscan’834 data set. In the latter, we observe certain values for which an
increase of the diffuse fraction comes together with an increase of kt. This fact is associated with
partly cloudy conditions with some clouds located near the sun. Under these circumstances, cloud
borders reflect the direct irradiance during periods when the sun is unshaded by the surrounding
clouds, thus enhancing both global and diffuse irradiance, and these situations are associated with
changing sky conditions [28,29].
3. Model formulation
In order to represent graphically the measurements of global irradiance obtained, we have used
two types of diagram: polar and three dimensional. Fig. 1 presents a three-dimensional and polar
global irradiance diagram for one of our experiments. In the figure caption, specific features of
this experimental series are explained. In short, this diagram facilitates the visual information of
each experience. Both diagrams have been obtained from experimental values, using the kriging
interpolation method [30].
The complete Granada data set has been analysed by means of this type of diagram. In this
sense, Fig. 1 can be considered as an example of the general performance of the irradiance distribution. These diagrams show the existence of a dependence between global irradiance values and
the angular distance subtended by the normal to the inclined plane and the sun (␺). To evidence
this correlation we have restricted our analysis to the solar zenith plane. This is the plane that
contains the zenith and the sun’s position (Fig. 2). Fig. 3 shows the global irradiance values in
the solar zenith plane (G␺sz) versus the angular distance to the sun’s position (␺sz), which, for
this plane, is the difference between the surface zenith angle and the sun’s zenith angle. In this
way, we eliminate the azimuth dependence. In this figure, we have plotted only the values that
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Fig. 1. (a) Three-dimensional; (b) projection on a horizontal plane; (c) detailed polar representation of the global
irradiance for one scan. Sun elevation, 74°; sun azimuth, 341°.
F.J. Olmo et al. / Energy 24 (1999) 689–704
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Fig. 2. Sky dome showing the solar zenith plane geometry: ␺sz, angular distance, in the zenith plane, of the normal
to the considered surface (n⬘) with respect to the sun’s position; ␺, angular distance, for an arbitrary plane, of the
normal to the considered surface (n) with respect to the sun’s position; ␪ and ␪s, zenith angles for the normal (n⬘) and
the sun, respectively; ␣ and ␣s, azimuth angles for the normal (n⬘) and the sun, respectively.
correspond to the intervals 60–80 and 20–30° of solar elevation in the Granada data set. It is well
known that direct irradiance on an inclined plane can be written as direct irradiance on a horizontal
surface multiplied by a geometrical factor [31]. This geometrical factor is a ratio of cosines. In
order to investigate if a similar ratio could be applied when dealing with global irradiance let us
refer to Fig. 3. In this figure, in which Gn is the global irradiance in the surface normal to the
sun beam, we investigate the ratio G␺zs/Gn, which suggests a dependence in terms of angular
distance to the sun’s position (␺sz). An additional analysis shown in Fig. 4 indicates that this ratio
depends also on the clearness index, kt (global to extraterrestrial horizontal irradiance ratio). This
is especially true for inclined planes far from the sun’s position (greater ␺sz). In our case, we
have taken the global irradiance at normal incidence to the sun, Gn, as maximum (Fig. 3). Then
we have modelled the global irradiance distribution in the solar zenith plane in terms of this
maximum value and an exponential function that includes the clearness index, kt, and the angular
distance to the sun’s position. The exponential function is adopted by convenience, as it will be
explained later, even though we could also model the global irradiance in the solar zenith plane
as a function of cosines with similar behaviours. In this procedure, we have only used 8% of the
data set, i.e. data lying exactly in the solar zenith plane.
Thus, we propose the following expression:
G␺zs ⫽ Gn exp( ⫺ kt␺2zs)
(1)
where ␺zs is expressed in radians. The kt value takes into account the influence of sky conditions,
turbidity and clouds, as a modulating function in the solar zenith plane.
In previous works [32,33], the effects of turbidity on the global solar irradiance measurements
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Fig. 3. Global irradiance on solar zenith plane vs. angular distance to the sun’s position for two extreme solar elevation intervals.
for inclined and oriented surfaces have been studied. It has been shown that the influence of
turbidity on global solar irradiance has great importance for surfaces normal to the sun.
Moreover, the symmetries observed in Fig. 2 suggest the possibility of extending geometrically
this procedure to the entire hemisphere (by means of the experimental function adopted). For this
purpose the angular distance ␺ can be evaluated as follows:
cos ␺ ⫽ sin ␪ sin ␪s ⫹ cos ␪ cos ␪s cos(␣s ⫺ ␣)
(2)
where ␪ represents the zenith angle and ␣ the azimuth. The subscript s refers to the sun’s position.
We should point out that the angular distance ␺ is the so-called scattering angle.
The scheme developed provides a good representation of the global irradiance distribution on
the complete hemisphere. Nevertheless, the global irradiance on a surface normal to the sun is not
measured in most radiometric networks. Therefore, considering that horizontal global irradiance is
the usually available term, we have modified the proposed scheme in order to use as input the
horizontal global irradiance.
For horizontal global irradiance our model reads:
GH ⫽ Gn exp( ⫺ kt␺2H)
(3)
where ␺H denotes the angular distance between the normal direction to the horizontal plane and
the sun’s position, that is, ␺H reduces to the solar zenith angle ␪s.
F.J. Olmo et al. / Energy 24 (1999) 689–704
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Fig. 4. Normalised global irradiance on solar zenith plane vs. quadratic angular distance to the sun’s position, for
different clearness index intervals.
From Eq. (3) and extending Eq. (1) to the whole hemisphere, we can obtain:
G␺ ⫽ GH exp( ⫺ kt(␺2 ⫺ ␺2H))
(4)
where ␺ and ␺H are expressed in radians.
This simple equation enables us to calculate the global irradiance distribution using as inputs
the horizontal global irradiance and the solar position.
4. Performance assessment
At first, we tested the model against our Granada data base. Considering that the model development has been carried out using only data in the solar zenith plane, which represents about 8%
of the data base, this part of the data base has not been used in the testing of the model.
Fig. 5 shows the scatter plot of calculated (Eq. (4)) versus measured global irradiance for cases
with the sun’s elevations in the range 60–80° at Granada. We must take into account that this
subset includes data for different ranges of clearness index and the fact that we represent experimental instantaneous values.
Nevertheless, it must be pointed out that in the present formulation the model does not allow
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Fig. 5. Global irradiance: measured vs. calculated values from Eq. (4); 60–80° sun elevation range.
for the effect of ground reflected radiation. In this sense, it could be worthy to include a factor
that considers the effect of anisotropic reflections. The multiplying factor that we propose is a
modified version of that proposed by Temps and Coulson [34]:
Fc ⫽ 1 ⫹ ␳ sin2(␺/2)
(5)
where ␳ is the albedo of the underlying surface. In our case, for uncoloured concrete, ␳ ⫽ 0.35
[31]. Finally, our model reads:
G␺ ⫽ GH exp( ⫺ kt(␺2 ⫺ ␺2H))Fc
(6)
Taking into account the expression for Fc, the correction is stronger for surfaces at 180° azimuth
angle from the sun, that present the greater contribution of ground reflected radiation. The use of
the anisotropic reflection factor (Eq. (6)) provides an improvement over the results shown in Fig.
5, as we can see in Fig. 6.
Fig. 7 shows the scatter plot of measured versus calculated global irradiance for the whole data
base (excluding data in the solar zenith plane). In Table 1 we show the model’s statistical results
for six different solar elevation ranges, where a is the slope, b the intercept and r the correlation
coefficient of the experimental versus calculated values [35]. The correlation coefficient gives an
evaluation of the experimental data variance explained by the model, while the other two provide
information about the tendency to over- or under-estimate in a particular range. Moreover, the
model performance was evaluated using the root mean square deviation (RMSD) and the mean
F.J. Olmo et al. / Energy 24 (1999) 689–704
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Fig. 6. Global irradiance: measured vs. calculated values from Eq. (4) taking into account the effect of ground anisotropic reflection (Fc); 60–80° sun elevation range.
bias deviation (MBD) [35]. These statistics allow for the detection of differences between the
experimental data and the model estimates and the existence of systematic over- or under-estimation tendencies, respectively. The slopes and the intercept variability can be explained because
of the influence of almost vertical surfaces. These surfaces present anisotropic ground reflectance
effects. This fact explains the great accumulation of values in the lowest part of the plot in Figs.
6 and 7. We must point out that the terrace of the Science Faculty presents a horizon obstructed
by buildings, which reaches in the worse case elevations of 20° (over the terrace level), especially
in the north and west directions. Furthermore, most of the surrounding buildings are painted white.
Because of this situation, we find great changes for some surfaces when the solar elevation is
lower than 40°. In these cases, the reflection from the buildings around is very important, and
we have not introduced these effects on the model. The departure of the slope from the ideal
value in the 10–20° category can be explained by this fact.
We must remember that the global irradiance values correspond to instantaneous measurements.
On the other hand, although the solar azimuth angle changes during the experiment, we use in
Eq. (4) a mean value for all the experimental points in each scan. Moreover, as already mentioned,
our experimental measurements recorded in each scan correspond to an interval of 45 min. During
this time, the global irradiance variation can reach 20% for mean solar elevations. These circumstances can explain the RMSD between model and experimental data, which increases when the
solar elevation decreases. Nevertheless, as suggested by Table 1 and Figs. 6 and 7, the model
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Fig. 7. Global irradiance: measured vs. calculated values for our data set, taking into account the effect of ground
anisotropic reflection (Fc).
Table 1
Statistical model results for different solar elevations in Granada data set; where a is the slope, b the intercept, r the
correlation coefficient of the experimental vs. calculated values, MBD the mean bias deviation and RMSD the root
mean square deviation
Solar elevation
a
b
(Wm−2)
r
MBD
(%)
10–20°
20–30°
30–40°
40–50°
50–60°
60–80°
0.888
1.009
1.104
1.041
1.081
0.998
19.5
⫺ 1.0
⫺ 20.9
⫺ 22.5
⫺ 53.0
9.3
0.887
0.936
0.969
0.963
0.972
0.982
⫺ 1.2
0.6
5.3
0.7
⫺ 2.1
1.3
38.0
26.9
17.8
16.0
13.0
8.1
10.1
0.966
0.2
17.8
Complete data set 1.027
RMSD
(%)
F.J. Olmo et al. / Energy 24 (1999) 689–704
699
provides a good estimation for solar elevations greater than 40°. The relatively marked overestimation in 30–40° may be explained as a result of the shorter number of cases in this category,
which corresponds to afternoon series.
As mentioned above, we have also carried out a second study involving the Skyscan’834 data
set [26,27] to gain general applicability of the model. We have also considered the Perez model
[3] in parallel testing with our model, in order to have a comparison with one of the most reliable
existing methodologies.
It is important to point out that our model can be used with either instantaneous measurements
of global horizontal irradiance or any kind of averaged values. In contrast, the Perez model needs
hourly values. If we don’t know the direct and diffuse horizontal irradiance values, they have to
be calculated by means of the model described in [36]. In order for the Perez method to calculate
the direct and diffuse irradiance on a horizontal surface, knowledge of the hourly global irradiance
in the previous, next and actual hour of the considered period is required. As the horizontal global
irradiance has to be the only radiometric input, several previous steps had to be carried out. Using
the Perez model to obtain the direct irradiance and the diffuse irradiance on a horizontal surface
[36], we had to compute both magnitudes employing Skyscan’834, even though both magnitudes
were available in the data set. We also had to carry out the proper averaging process in the
Skyscan’834 data set that the Perez model demands. After this it was possible to compute the
direct irradiance projection and the diffuse irradiance on a tilted surface, the latter by means of
the method described in [3], to get the global irradiance on a tilted surface by adding the other
two and having the same initial conditions for both models, that is, global horizontal irradiance
as the only input.
In Figs. 8 and 9, we can observe the performance of the Perez model and our model for G␺
hourly values, respectively. We have not used the factor Fc when testing the model against the
Skyscan’834 data set, because of the very well shielded pyranometer as mentioned above. As the
data set covers a full year period, a variety of solar elevation angles as well as cloudy and turbidity
conditions are included. Just one thing has to be pointed out before we analyse the statistics for
both models. The Skyscan’834 data set is composed of 5845 instantaneous measurements for the
sloping global irradiance, but they turn into 1029 after the averaging process.
Having this in mind, in Table 2 we show the statistical results of the regression analyses for
both models. As can be observed in Table 2, together with Figs. 8 and 9, our model offers an
over-estimation of 22.6 W/m2 (4.8% of the averaged measured slope irradiance) in contrast to an
under-estimation of the Perez model of only 3.9%. The RMSD follows a similar behaviour, being
1% greater for our model than for the Perez model. But considering the statistic as a whole, our
model gives good results as well, with a slope close to 1 and a general performance that, together
with its simple formulation and input requirements, makes it quite useful and reliable.
If we test our model against the instantaneous measurements, we find similar results (Fig. 10).
Considering the whole data set (all types of skies), the results show a general over-estimation of
5.2%, a correlation coefficient greater than 0.99 and a slope close to 1. Thus, we can conclude
that our model offers a reliable tool for use when solar radiation data are scarce or limited to
global horizontal irradiance. Additionally, a fast and simple estimation is often necessary and
preferred. Compared with the Perez model, our model gives a similar performance but has the
advantage of its simpler formulation.
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Fig. 8. The Perez et al. model for the estimation of global slope irradiance vs. Skyscan’834 hourly values for all
weather conditions.
5. Conclusions
As our study shows, it is possible to obtain the global irradiance angular distribution from
knowledge of the global horizontal irradiance and the astronomic parameters, by means of a model
that has been proposed for our latitude. This model depends on local atmospheric conditions by
means of kt, and avoids the classical partitioning of the global solar irradiance into direct and
diffuse components. This fact gives a general applicability character to the model, which could
be used in sites where only horizontal global irradiance is measured.
This model has been developed taking into account the exponential pattern shown by the global
irradiance angular distribution and its symmetry using polar diagrams. Our model takes into
account the ground reflection effects by means of an anisotropic factor, although the effects of
an artificial horizon are not taken into account. This provides a good estimation for instantaneous
global irradiance values on inclined surfaces.
In order to establish its applicability, the model had to be validated against other experimental
data sets. To this end, the Skyscan’834 data set has been used. This is a well-known data base,
which includes information of slope irradiance under different types of cloud cover and turbidity
conditions. The results corroborate our predictions and show that when used with either instantaneous or averaged measurements, the new model performs well.
Owing to the appropriate performance of the model for different solar elevations and inclined
F.J. Olmo et al. / Energy 24 (1999) 689–704
701
Fig. 9. The new model for the estimation of the global slope irradiance vs. Skyscan’834 hourly values for all
weather conditions.
Table 2
Statistical results obtained from testing the Perez model and the new model against Skyscan’834 hourly and instantaneous values
Model
a
b
(W/m2)
r
MBD
(%)
RMSD
(%)
Hourly values
Perez model
New model
0.975
1.01
⫺ 6.0
19.5
0.994
0.993
⫺ 3.9
4.8
8.3
9.3
Instantaneous values
New model
1.00
22.7
0.993
5.2
10.1
surfaces we deem that it could be a good tool for the study of hourly and daily values of solar
irradiance on inclined surfaces, using the horizontal global irradiance as unique input. The estimations provided by the model can be used for the estimation of the energy balance, in technological applications or in local and large-scale climate studies. On the other hand, this work represents
a previous stage in the study of topographic effects on the solar irradiance field, with the goal of
mapping solar radiation on local and large scales.
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F.J. Olmo et al. / Energy 24 (1999) 689–704
Fig. 10. The new model for the estimation of the global slope irradiance vs. Skyscan’834 instantaneous values for
all weather conditions.
Acknowledgements
This work was supported by La Dirección General de Ciencia y Tecnologı́a from the Education
and Research Spanish Ministry through the project No. CLI98-0912-C02-01. We would like to
thank sincerely Dr. Alfred Brunger for lending us the Skyscan’834 data set and for the permission
to use it. We are also very grateful to Dr. Richard Perez for sending us the Perez model code.
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