A new neural network model for evaluating the performance of various hourlyslope irradiation models: Implementation for the region of Athens
E.D. Mehleri
*
, P.L. Zervas, H. Sarimveis, J.A. Palyvos, N.C. Markatos
National Technical University of Athens, School of Chemical Engineering, Zographou Campus, Athens 15780, Greece
a r t i c l e i n f o
Article history:
Received 27 May 2009Accepted 30 September 2009Available online 26 November 2009
Keywords:
Hourly slope irradiation modelsIsotropic modelsAnisotropic modelsRadial basis function (RBF)
a b s t r a c t
The present study is divided into two parts. The ﬁrst part deals with the comparison of various hourlyslope irradiation models, found in the literature, and the selection of the most accurate for the region of Athens. In the second part the prediction of global solar irradiance on inclined surfaces is performed,based on neural network techniques.The models tested are classiﬁed as isotropic (Liu and Jordan, Koronakis, Jimenez and Castro, Badescu,Tian) and anisotropic (Bugler, Temps and Coulson, Klucher, Ma and Iqbal, Reindl) based on the treatmentof diffuse irradiance. For the aforementioned models, a qualitative comparison, based on diagrams, wascarried out, and several statistical indices were calculated (coefﬁcient of determination
R
2
, mean biaserror MBE, relative mean bias error MBE/A(%), root mean square error RMSE, relative root mean squareerror RMSE/A(%),statistical index tstat), in order to select the optimal.The isotropic models of ‘‘Tian’’ and ‘‘Badescu’’ show the best accordance to the recorded values. Theanisotropic model of ‘‘Ma&Iqbal’’ and the pseudoisotropic model of ‘‘Jimenez&Castro’’, show poorperformance compared to other models. Finally, a neural network model is developed, which predictsthe global solar irradiance on a tilted surface, using as input data the total solar irradiance on a horizontalsurface, the extraterrestrial radiation, the solar zenith angle and the solar incidence angle on a tiltedplane. The comparison with the aforementioned models has shown that the neural network model,predicts more realistically the total solar irradiance on a tilted surface, as it performs better in regionswhere the other models show underestimation or overestimation in their calculations.
2009 Elsevier Ltd. All rights reserved.
1. Introduction
Solarenergyisasustainable,safeandabundantenergyresourceand therefore there are no restrictions of time and space for itsexploitation [1]. Concerning the exploitation of solar energy, it isdivided into three basic application categories: passive solarsystems, active solar systems and photovoltaic systems (Fig. 1).Passive and active solar systems exploit the thermal energy of global solar irradiance, while photovoltaic systems convert globalsolar irradiance to electrical power through the photovoltaicphenomenon.Estimating global solar irradiance on tilted surfaces is necessaryas the majority of solar systems are inclined according to the siteof installationandutilization.Moreover,beamanddiffusecomponentsof global solar irradiance on inclined surfaces are essential in orderto calculate theelectricpowerofphotovoltaicsystems, design solarthermal systems and to evaluate their longterm average performance.Despite the fact that manymeteorological stationsmeasureglobal and diffuse irradiation received on horizontal surfaces thedata on inclined surfaces are not available and are estimated withseveral models [2], using the components of global solar irradianceon horizontal surfaces. It must be noted that the knowledge of thecomponents of global solar irradiance on horizontal surfaces isessential for the prediction of global solar irradiance on tiltedsurfaces,asitisdifﬁculttodevelopasimplemodelconvertingsolarirradiance received by the horizontal plane to that arriving to aninclined area for two main reasons [3]:
Global solar irradiance reaching a tilted surface includes irradiance reﬂected from the surroundings.
The view angle of a tilted surface cuts out a limited solid angleof the sky. This sky irradiation not only depends on the tiltangle,ontheazimuthofthecollectorandonthesolarelevationand azimuth but also depends on the cloud conditions.
Abbreviations:
ANN, Artiﬁcial neural network; MBE, Mean bias error; MBE/A(%),Relative mean bias error (%); RBF, Radial basis function; RMSE, Root mean squareerror; RMSE/A (%), Relative root mean square error (%); tstat, t

statistical.
*
Corresponding author. Tel.:
þ
30 210 7723235; fax:
þ
30 210 7723228.
Email address:
emehleri@chemeng.ntua.gr (E.D. Mehleri).
Contents lists available at ScienceDirect
Renewable Energy
journal homepage: www.elsevier.com/locate/renene
09601481/$ – see front matter
2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.renene.2009.11.005
Renewable Energy 35 (2010) 1357–1362
The models for predicting global solar irradiance on tiltedsurfaces are classiﬁed as isotropic and anisotropic.The isotropic models [4–8], predict the diffuse irradiance ona tilted surface, assuming the uniformity of diffuse sky irradianceover the sky dome. However, his theory is not correct [9] andtherefore additional models are developed, known as anisotropicmodels. The anisotropic models assume the anisotropy of thediffuse sky irradiance in the circumsolar region (sky near the solardisk) and an isotropically diffuse component for the rest of the skydome. Besides the anisotropic models that have been tested in thispaper are: Ma & Iqbal [10], Bugler [11], Temps & Coulson [12],
Klucher [13], Reindl [14].
This paper consists of two parts. In the ﬁrst part, several modelsfor predicting global solar irradiance on inclined surfaces are presented and evaluated by the computation of several statisticalindices. In the second part, a new empirical model for predictingglobal solar irradiance on inclined surfaces is developed, based onNeural Network Techniques [15].
2. Presentation of the models
In this section, the procedure for predicting global solar irradiance on inclined surfaces is analytically presented.
2.1. Theory and models
Global solar irradiance incident on a tilted plane (
It
), consists of three components: beam (
It
b
) which derives from the solar disc,diffuse (
It
d
) which derives fromthe sky dome and groundreﬂected(
It
r
) which derives from the surrounding area:
It
¼
It
b
þ
It
d
þ
It
r
(1)
The amount of beam irradiance incident on a tilted surfacedepends on the angle of solar incidence. Diffuse and groundreﬂected irradiance on a tilted surface, do not depend on theorientation of the tilted surface and they do not come from thewhole sky dome or the surrounding area. Hence, the diffuse irradiance incident on the tilted surface comes from the partof the skydome that ‘‘sees’’ this surface.The beam and groundreﬂected components are calculated byusing simple algorithms. The nature of the diffuse component ismore complicated and needs more attention.The equations used for the prediction of the beam and groundreﬂected components are presented below:
It
b
¼
I
b
$
R
b
(2)
R
b
¼
cos
q
=
cos
q
z
(3)cos
q
z
¼
sin
4
$
sin
d
$
cos
4
$
cos
d
$
cos
u
(4)cos
q
¼
sin
d
$
sin
4
$
cos
b
sin
d
$
cos
4
$
sin
b
$
cos
g
þ
cos
d
$
cos
4
$
cos
b
$
cos
u
þ
cos
d
$
sin
4
$
sin
b
$
cos
g
$
cos
u
þ
cos
d
$
sin
b
$
sin
g
$
sin
u
(5)
where
It
b
is the hourly beam irradiance on inclined surfaces,
q
z
isthe solar zenith angle,
q
is the solar incidence angle on a tiltedplane,
4
is the latitude of the location,
d
is the solar declination,
b
is the tilt angle,
u
is the solar hour angle and
g
is the azimuthalangle. The groundreﬂected component, on the other hand, isgiven by
It
r
¼
1
=
2
$
r
g
$
I
$
ð
1
cos
b
Þ
(6)
where
r
g
isthegroundreﬂectivityand
I
istheglobalsolarirradianceon horizontal surfaces.
2.2. Diffuse irradiance models
Table1,presentsthemodelsforpredictingthediffuseirradiancecomponent of global solar irradiance on a tilted surface, chosen forthis study.
Fig. 1.
Basic applications of global solar irradiance.
Nomenclature
A
i
Ratio of the hourly beam solar irradiance ona horizontal surface to the extraterrestrial solarradiation (
I
b
/
I
oh
) (dimensionless)
b
Tilt angle (
)
f
Horizon brightening
ﬃﬃﬃﬃﬃﬃﬃﬃ
I
b
=
I
p
(dimensionless)
F
ð
1
ð
I
d
=
I
Þ
2
Þ
(dimensionless)
I
Globalsolarirradianceonahorizontalsurface(
kW
/
m
2
)
Irec
Recorded values of hourly global solar irradiance ona horizontal surface (
kW
/
m
2
)
I
b
Hourly beam irradiance on a horizontal surface(
kW
/
m
2
)
I
brec
Recorded values of hourly beam irradiance ona horizontal surface (
kW
/
m
2
)
I
d
Hourly diffuse irradiance on a horizontal surface(
kW
/
m
2
)
I
drec
Recorded values of hourly diffuse irradiance ona horizontal surface (
kW
/
m
2
)
I
oh
Hourly extraterrestrial radiation (
kW
/
m
2
)
It
Hourly global solar irradiance on an inclined surface(
kW
/
m
2
)
It
b
Hourlybeamirradianceonaninclinedsurface(
kW
/
m
2
)
It
d
Hourly diffuse irradiance on an inclined surface(
kW
/
m
2
)
It
r
Hourly groundreﬂected irradiance on an inclinedsurface (
kW
/
m
2
)
It
rec
Recorded values of hourlyglobal solar irradiance on aninclined surface (
kW
/
m
2
)
k
t
Clearness index (dimensionless)
r
g
Ground reﬂectivity (dimensionless)
R
2
Coefﬁcient of determination (dimensionless)
R
b
cos
q
/
cos
q
z
(dimensionless)
g
Solar azimuth angle (
)
d
Solar declination (
)
q
Solar incidence angle on a tilted surface (
)
q
z
Solar zenith angle (
)
4
Latitude of the site (
)
u
Solar hour angle (
)
E.D. Mehleri et al. / Renewable Energy 35 (2010) 1357–1362
1358
3. Formulation of the database
The data used for the comparison of the models mentionedabove and the derivation of the empirical model for predictingglobal solar irradiance on inclined surfaces were recorded at thestation of the National Technical University of Athens (37
58
0
26
00
B,23
47
0
16
00
A, 219 m above mean sea level) [16]. These data are tenminutevaluesofglobal,beamanddiffuseirradianceonahorizontalsurfaceandoninclinedsurfaces(32
)respectively, fortheperiodof one year (January 2004–December 2004), transformed into 8760hourly values. Then, after extracting the zero values of global solarirradiance on tilted surfaces, the ﬁnal database consists of 4283values, as presented in Table 2.
4. Statistical methods of comparison
The evaluation of the models presented in Table 1, as well as of the RBF neural network model, was carried out based on thecomputation of several statistical indices [17–19].Coefﬁcient of Determination:
R
2
¼
P
ni
¼
1
ð
It
i
Itmean
Þ
$
ð
Itrec
i
Itrecmean
Þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ P
ni
¼
1
ð
It
i
Itmean
Þ
2
#v uut
$
P
ni
¼
1
ð
Itrec
i
Itmean
Þ
2
#
Root mean square error:
RMSE
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃX
ni
¼
1
ð
It
i
Itrec
i
Þ
2
=
n
v uut
Mean bias error:
MBE
¼
X
ni
¼
1
ð
It
i
Itrec
i
Þ
=
n
tstatistical:
t
stat
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð
n
1
Þ
$
MBE
2
RMSE
2
MBE
2
s
Relative mean bias error:
MBE
=
A
ð
%
Þ ¼
MBE
P
ni
¼
1
It
i
n
100
Relative root mean square error:
RMSE
=
A
ð
%
Þ ¼
RMSE
P
ni
¼
1
It
i
n
100
Mean error:
Mean Error
¼
P
ni
¼
1
ð
It
i
Itrec
i
Þ
=
Itrec
i
n
100
where
It
i
is the
ith
predicted value of Global Solar Irradiance oninclined surfaces,
Itrec
i
is the
ith
measured value,
Itmean
is thepredicted mean value,
Itrecmean
is the measured mean value,
n
isthe number of data analyzed.
5. An introduction to artiﬁcial neural networks
Neurons are the basic elements of the human brain. Their basicfunction is to provide us with the ability to apply our previousexperiences to our actions [20]. Artiﬁcial Neural Networks (ANNs)are computing algorithms that mimic the four basic functions of these biological neurons. These functions receive inputs from otherneurons or sources, combine them, perform operations on theresult, and output the ﬁnal result. What makesANNs exciting is the
Table 1
Details and mathematical relationships of diffuse models selected for current study.Model Year Type
a
Equation Ref.
b
Liu & Jordan 1962 Iso
It
d
¼
Id
$
ðð
1
þ
cos
b
Þ
=
2
Þ
[4]Temps & Coulson 1977 Ani
It
d
¼
Id
$
cos
2
ð
b
=
2
Þ
$
½
1
þ
sin
3
ð
b
=
2
Þ
$
½ð
1
þ
cos
2
q
Þ
$
sin
3
q
z
[12]Bugler 1977 Ani
It
d
¼ ð
1
þ
cos
b
=
2
Þ
$
ð
Id
0
:
05
$
It
b
$
cos
q
z
Þ þ ð
0
:
05
$
It
b
$
cos
q
Þ
[11]Klucher 1979 Ani
It
d
¼
Id
$
ðð
1
þ
cos
b
Þ
=
2
Þ
$
ð
1
þ
F
$
sin
3
ð
b
=
2
ÞÞ
$
ð
1
þ
F
$
cos
2
b
$
sin
3
q
z
Þ
F
¼
1
ð
Id
=
I
2
Þ
[13]Ma&Iqbal 1983 Ani
It
d
¼
Id
$
½ð
k
t
$
R
b
þ ð
1
k
t
Þ
$
ð
cos
2
ð
b
=
2
ÞÞ
k
t
¼
I
=
I
oh
R
b
¼
cos
q
=
cos
q
z
[10] Jimenez & Castro 1986 Pseudoiso
It
d
¼ ð
0
:
5
Þ
$
ð
0
:
2
Þ
$
I
$
ð
1
þ
cos
b
Þ
It
b
¼
0
:
8
$
I
$
R
b
[6]Koronakis 1986 PseudoIso
It
d
¼ ð
1
=
3
Þ
$
Id
$
ð
2
þ
cos
b
Þ
[5]Reindl 1990 Ani
It
d
¼
Id
$
½ð
1
A
i
Þ
$
ðð
1
þ
cos
b
Þ
=
2
Þ
$
ð
1
þ
f
$
ð
sin
3
ð
b
=
2
ÞÞ þ ð
A
i
$
R
b
Þ
f
¼
ﬃﬃﬃﬃﬃﬃﬃﬃ
Ib
=
I
p
A
i
¼
Ib
=
I
oh
R
b
¼
cos
q
=
cos
q
z
i
[14]Tian 2001 Iso
It
d
¼
Id
$
ð
1
b
=
180
Þ
[8]Badescu 2002 Iso
It
d
¼
Id
$
ðð
3
þ
cos
ð
2
b
ÞÞ
=
4
Þ
[7]
a
Iso: Isotropic, Ani: Anisotropic.
b
Ref: Reference.
Table 2
The database.No of Data Ib
rec
Id
rec
I
rec
It
rec
1 0.00055176 0.020282 0.020833 0.0141672 0.19003 0.066301 0.25633 0.36833– – – – –– – – – –40 0.0010649 0.041602 0.042667 0.03916741 0.00025615 0.0022438 0.0025 0.0025– – – – –4283 0.00038376 0.0019496 0.0023333 0.0016667In kW/m
2
.
E.D. Mehleri et al. / Renewable Energy 35 (2010) 1357–1362
1359
fact that, once a network has been set up, it can learn in a selforganizing way that emulates brain functions such as patternrecognition, classiﬁcation, and optimization. An ANN is characterized by its architecture, training or learning algorithm, and activation function. The architecture describes the connection betweenthe neurons. It consists of an input layer, an output layer and,generally, one or more hidden layers inbetween [21]. The layers inthese networks are interconnected by communication links thatare associated with weights which dictate the effect on the informationpassing through them.The activation function, on the otherhand, relates the output of a neuron to its input, based on theneuron’s input activity level [22].
5.1. RBF network implementation
The radial basis function network consists of an input layer, anoutput layer and usually but not necessarily, one hidden layer inwhich the activation function is Gaussian [23], as presented inFig.2.Thisarchitectureexhibitsimportantadvantagescomparedtoother neural network architectures, such as the simple networktopology, the local rather than global approximation they achieve,and the fast algorithms that have been developed for training suchnetworks. The learning process of an RBF network involves usingthe input–output data to determine the number of the hiddennodes, the centres of the hidden nodes, and the output weights, sothat the error between the predicted and the recorded values isminimal.
6. Development of the RBF neural network model
In the present work the RBF neural network is structured sothat it can predict the global solar irradiance on inclined surfaces,
It
, as an output variable, using as independent input parameters:the global solar irradiance on a horizontal surface, the extraterrestrial radiation, the solar zenith angle and the solar incidenceangle on a tilted plane. Thus the input vector
x
is deﬁned as
x
¼
[
I
,
I
oh
,
q
z
,
q
z
]
T
.The neural network output provides the global solar irradianceon inclined surfaces
I
t
as an output and it is calculated as theweighted sum of the responses of the hidden layers:
I
t
¼
X
L j
¼
1
w
j
z
j
ð
x
Þ
where
z
j
ð
x
Þ ¼
f
ðk
x
x
j
k
22
Þ
In the above equations
z
j
is the response of the
jth
node,
f
is theradial basis function,
x
j
is the centre of the
jth
node,
L
is the totalnumber of hidden nodes and
w
j
is the weight corresponding to theresponse of the
jth
node.An RBF training procedure, aims at the determination of thenumber of nodes in the hidden layer, the hidden node centres andtheoutputweights,inordertominimizethedeviationbetweenthemeasured and predicted values of the output variables over the setof the available database.The training method used in this work is based on the fuzzypartition of the input space, which is produced by deﬁninga number of triangular fuzzy sets in the domain of each inputvariable [24]. The centres of these fuzzy sets form a multidimensional grid on the input space. A rigorous selection algorithmchooses the most appropriate vertices on the grid, which are thenused as the hidden node centres in the resulting RBF networkmodel. The idea behind the selection algorithm is to place thecentres in the multidimensional input space, so that the distancebetween any two centre locations is guaranteed to be greater thana lower limit, which is deﬁned by the length of the edges on thegrid. At the same time, the algorithm ensures that for any inputexample in the training set there is at least one selected hiddennode that is close enough, according to an appropriately deﬁneddistance criterion. The socalled ‘‘fuzzymeans’’ training methoddoes not need the number of centres to be ﬁxed before theexecution of the method. Due to the fact that it is a onepassalgorithm,itisextremelyfast,eveninthecaseofalargedatabaseof input–output training data. One additional advantage is that thetraining algorithm used needs only one tuning parameter, namelythe number of fuzzy sets that are utilized to partition each inputdimension.The training procedure was used several times by altering ateach run the fuzzy partition of the input space (number of fuzzysets deﬁned in each input dimension), which is in fact the onlydesign parameter that must be deﬁned by the user when utilizingthe fuzzymeans algorithm.Four neural network models were developed by altering thefuzzy partition of the input space. The results were graduallyimproved up to the point where ten fuzzy sets were used topartition the domain of each input variable. A further increaseresultstotheovertrainingphenomenon,wheretheperformanceof the produced model is not improved, although the model increasesin size.The RBF neural network model technique was validated bydeveloping neural network models using 100 random partitions of the data into training and validation sets (75% and 25% of the data,respectively). For each random partition, four neural networkmodels were developed by altering the fuzzy partition of the inputspace. The results were gradually improved up to the point whereten fuzzy sets were used to partition the domain of each inputvariable. A further increase results to the over training phenomenon, where the performance of the produced model is not
Fig. 2.
Standard topology of an RBF neural network.
Table 3
Statistical indices corresponding to RBF neural network models.No of fuzzy sets
R
2
RMSE Mean Error RMSE/A (%)7 0.9397 0.0713 8.4082 19.25658 0.9426 0.0688 7.9939 18.43919 0.9521 0.0628 7.8760 17.503110
0.9600 0.0580 7.6969 15.357
E.D. Mehleri et al. / Renewable Energy 35 (2010) 1357–1362
1360