Solar Energy 75 (2003) 447–454

www.elsevier.com/locate/solener

Fuzzy inference systems applied to the dailyultraviolet radiation evaluation (295–385 nm) from daily

global radiation

Lourdes Ramirez Santigosa a, Jesus Polo Martinez a,*, Llanos Mora Lopez b,Mariano Sidrach de Cardona Ortin c, Julian Blanco Galvez a

a Dpto. de Energ�ııas Renovables, CIEMAT, Av. Complutense, 22 28040 Madrid, Spainb Dpto. de lenguajes y CC de la Computaci�oon, Universidad de M�aalaga, Spain

c Dpto. de F�ıısica Aplicada II, Universidad de M�aalaga, Spain

Received 26 March 2003; received in revised form 19 September 2003; accepted 19 September 2003

Abstract

This work presents the results of applying different automatic learning techniques to the calculation of daily ul-

traviolet radiation from daily global radiation on a horizontal surface. Using the data from three Spanish locations, a

zonal study was made, which was finally combined in models for general application. Using the corresponding at-

mospheric transparency index, three models based on multivariate linear regression, non-linear regression and gener-

ation of fuzzy inference systems, respectively, were designed. The results obtained verify the good behavior of the fuzzy

inference system calculated.

� 2003 Elsevier Ltd. All rights reserved.

1. Introduction

One of the limitations of using solar technologies for

photochemical decomposition of organic compounds is

the lack of information on the availability of ultraviolet

radiation (295–385 nm) at different sites (Riordan,

1990). This article reports the results of a study on the

possibility of determining daily ultraviolet radiation

from daily horizontal global radiation sets, which al-

though scant, are more frequent than the ultraviolet

radiation measurement sets. In related work, such as

that of (Baker-Blocker et al., 1984), with data measured

at the South Pole, that of (Al-Aruri et al., 1988) in

Kuwait or that of (Elhadidy et al., 1990) in Saudi Ara-

bia, the ratio of daily ultraviolet to the corresponding

* Corresponding author. Tel.: +34-91-3466043; fax: +34-91-

3466037.

E-mail addresses: lourdes.ramirez@ciemat.es (L. Ramirez

Santigosa), jesus.polo@ciemat.es (J. Polo Martinez).

0038-092X/$ - see front matter � 2003 Elsevier Ltd. All rights reserv

doi:10.1016/j.solener.2003.09.014

global radiation has been found to vary over a range

between 2.7% and 6%. Later, (Ambach et al., 1991),

demonstrated that this ratio increases when global ra-

diation decreases, that is, with overcast skies. It is pos-

sible to attempt to diminish the strong dependence of

these ratios on place of measurement by studying a

transformed variable instead of measured variables. For

this purpose, the index of atmospheric transparency

were calculated for the two variables and their ratios

were studied. The results of the ratios found in the

bibliography do not allow a clear conclusion to be ar-

rived at, since, although in (Pedr�oos et al., 1997) good fits

for a third order polynomial function are found, in

(Mart�ıınez-Lozano et al., 1994) the results of this fit are

not satisfactory for daily values. Using data from three

Spanish locations, in this work, the possibility of

studying new general models is proposed. Different au-

tomatic learning techniques were applied for this: mul-

tivariate linear regression, non-linear models and fuzzy

inference systems. Finally, the results of the different

models are analyzed and compared.

ed.

Fig. 1. Dataset from the PSA (Wh/m2).

448 L. Ramirez Santigosa et al. / Solar Energy 75 (2003) 447–454

2. Description of data used

Simultaneous daily global and ultraviolet radiation

data from three different locations in Spain (Almer�ııa,Cordoba and Madrid) were available for this work. The

ultraviolet radiation was measured in all the three cases

with a model TUVR Eppley pyranometer, having a

spectral response range of 295 to 385 nm.

• The data from Almer�ııa (PSA) were measured at the

Plataforma Solar (Latitude: 37�100; Longitude:

2�360), in 1994, 1995, 1996 and 1997. 1079 valid data

remained available after filtering.

• The data for Cordoba (Latitude: 37�850; Longitude:4�800) were provided by the Department of Applied

Physics of the Polytechnic University and correspond

to the same years as above (1994 through 1997). The

data were valid for a total of 1200 observations.

• The data from Madrid were recorded by the CIE-

MAT Department of Renewable Energies Photovol-

taic Laboratory for 1994 and represented a total of

293 data. Although this series is considerably shorter,

it was decided to use it in the study since the basic ob-

jective is to study the behavior of the models in differ-

ent locations.

In all cases, the relationships of the index of atmo-

spheric transparency were also studied. The index of

atmospheric transparency is defined as the ratio between

the radiation received on the surface of the earth and the

corresponding extraterrestrial radiation. Thus, it holds

that:

KTGD ¼ GD

GD0

and KTUD ¼ UD

UD0

Where: KTGD and KTUD represent the index of trans-

parency corresponding to the daily global and the daily

ultraviolet radiation, respectively; GD and UD repre-

sent the daily global and daily ultraviolet radiation; GD0

and UD0 the global extraterrestrial radiation corre-

sponding to the global and ultraviolet radiation, re-

spectively. The calculation of the corresponding

extraterrestrial radiation is done based on a series of

algorithms that depend on the day of the year and the

latitude of the location under study (Iqbal, 1983). For

ultraviolet radiation in the 295–385-nm range, the value

of the integral of the solar constant for this range (ISCUV)

should be considered. This constant is very sensitive to

small variations in the upper limit of the integral, and in

the related bibliography values found vary between

104.4 Wh/m2 (Blanco and Malato, 1996) and 78 Wh/m2

(Mart�ıınez-Lozano et al., 1994). In this work, the values

for ISCUV using the spectral distribution proposed by

(Frohlich and Brusa, 1981) of 82.14 Wh/m2 and by

(CIE, 1989) of 80.87 Wh/m2 were studied and finally a

value of 89.33 Wh/m2, which is the AM0 proposed by

the ASTM (ASTM, 1987) was selected. The use of index

of atmospheric transparency has two basic advantages.

From a statistical point of view, the radiation time series

does not comply with the basic hypotheses of the re-

gression model, since observations are not independent

(Brandt, 1983), while use of the corresponding index of

transparency is equivalent to transforming the input

variables and eliminating seasonality. From the point of

view of physics, when the dividend is normalized by a

factor that includes the effects of day and latitude of

location, it is assumed that many of the local effects have

been eliminated, making it possible, therefore, to find

more general relationships than if the untransformed

variable is used. For these reasons, in the models de-

veloped here it was decided to use the index of trans-

parency as an input variable.

3. Fit by location

In the studies by location presented below, the results

of the linear fit of radiation and index of transparency,

as well as the third order polynomial fit of the latter are

shown. These results will enable results to be compared

with those of similar studies in other locations.

3.1. Almer�ııa

The four figures shown below show the results of the

fits made to the data from Almer�ııa.A measured dataset is shown in Fig. 1 and the cor-

responding XY graph is given in Fig. 2, where the av-

erage ultraviolet radiation values are observed to be

4.8% of the global radiation. Although, obviously, the

ordinate at origin does not make physical sense, it is

used so that the fit is not conditioned. Their goodness of

fit with an R2 of 0.979 may also be observed. The linear

L. Ramirez Santigosa et al. / Solar Energy 75 (2003) 447–454 449

fit of the index of transparency for the two variables

studied are shown in Figs. 3 and 4 for a third order

polynomial fit (Pedr�oos et al., 1997). It may be seen how

the function resulting from the polynomial fit is better in

all the ranges of KT, but R2 only goes from 0.914 to

0.915.

Fig. 3. Linear fit of index of transparency (PSA).

Fig. 4. Polynomial fit of index of transparency (PSA).

Fig. 2. Linear fit of the PSA data (Wh/m2).

Fig. 5. Dataset from Cordoba (Wh/m2).

Fig. 6. Linear fit of data from Cordoba (Wh/m2).

3.2. Cordoba

The results of fit, in which the same order obtained

by (Pedr�oos et al., 1997) is found, are shown in the Figs. 5

and 6. When comparing these results with those ob-

tained for the data from Almer�ııa, notice a decrease in

the slope of the line, indicating an average ultraviolet

radiation of 3.9% of the global value with regard to the

linear regression of radiation (considerably less than the

4.8% found in Almer�ııa).Figs. 7 and 8 show the results of fit of the index of

transparency. As in the case of Almer�ııa, the R2 values

are lower and do not significantly increase with poly-

nomial linear fit.

3.3. Madrid

Figs. 9–12 demonstrate that the slopes of linear fit are

slightly lower than those estimated for Almer�ııa, and

higher than for Cordoba. On the other hand, the R2 for

Fig. 7. Linear fit of index of transparency (Cordoba).

Fig. 9. Dataset from Madrid (Wh/m2).

Fig. 8. Polynomial fit of index of transparency (Cordoba).

Fig. 10. Linear fit of data from Madrid (Wh/m2).

Fig. 11. Linear fit of index of transparency (Madrid).

Fig. 12. Polynomial fit of index of transparency (Madrid).

450 L. Ramirez Santigosa et al. / Solar Energy 75 (2003) 447–454

fit of transparency index are outstandingly low, with

values of 0.7859 for linear fit and 0.8105 for the poly-

nomial. These values, considerably lower than for the

other two locations (always above 0.91) would indicate

that there are other factors modulating the ultraviolet

radiation, although as the data are for only one year and

there are no data for other locations available, no rele-

vant conclusions can be arrived at in this regard.

Since the purpose of this work is to study models that

are valid for several locations, the results obtained from

L. Ramirez Santigosa et al. / Solar Energy 75 (2003) 447–454 451

the above fit with linear functions for the index of

transparency are summarized in the Table 1.

From the above expressions it may be deduced that

the coefficients of fit, while different, are of the same

order in the three locations. This enables more general

models, such as those below, to be proposed.

4. Design of general models

The purpose of this work is to obtain a general model

for the Iberian Peninsula. It is obvious that for this

purpose it would be advisable to have measurements

from some location at a high latitude or at around 42�,but at present, none with simultaneous global and ul-

traviolet radiation series are available. With the data

available from the three locations studied, 2572 obser-

vations organized at random are available of which 2000

are used in the learning phase of the models and 572 are

set aside for testing. Fig. 13 gives the data available

according to each location.

4.1. Multivariate linear regression model (MLR)

In this section, the results of developing a multivar-

iate linear model, in which the transparency index of

ultraviolet radiation represents the dependent variable,

are presented. As seen in the fit by location, the trans-

Table 1

Summary of linear fits for the three Spanish locations

Location Linear fit R2 Expression

Almer�ııa KTUD(KTGD) 0.9141 y ¼ 0:0501þ 0:6515xC�oordoba KTUD(KTGD) 0.9406 y ¼ 0:0509þ 0:5479xMadrid KTUD(KTGD) 0.7859 y ¼ 0:0214þ 0:5965x

Fig. 13. Data available from the three Spanish locations.

parency index for global radiation is able to explain

from 78% to 94% of the population, depending on the

location. In addition to this variable, the declination

(expressed in radians) is considered an independent

variable to study its significance in a general model.

Table 2 shows the statistical parameters of the multi-

variate linear regression model.

In the table above, declination is a sufficiently sig-

nificant variable, justifying its inclusion in a general

model, which would then be:

KTUD ¼ 0:0455þ 0:037 � dþ 0:6001 �KTGD

where KTUD represents the atmospheric transparency

index for daily ultraviolet radiation, d represents the

declination in radians and KTGD the index of atmo-

spheric transparency of the daily global radiation.

4.2. Multivariate non-linear regression model (MNLR)

In this section, the results of fitting the learning data

from the three locations to the non-linear model are

presented. The R2 is 0.8669 (Table 3), slightly higher

than for the MLR model, due to better fit in all the KT

ranges. The expression obtained is

KTUD ¼ 1:7802 �KT3GD � 2:2761 �KT2

GD þ 1:4416 �KTGD

þ 0:031 � d� 0:0352

Table 2

Statistical results from the multivariate linear regression model

Coefficients Std. error

of coeffi-

cients

t(1997) p-level

Intercept 0.045479 0.003650 12.46061 0.000000

d 0.037 0.004 10.08606 0.000000

KTGD 0.600115 0.006179 97.11473 0.000000

R ¼ 0:9273; R2 ¼ 0:8599; R2Fit ¼ 0:8598; F ð2; 1997Þ ¼ 6129:8;

p < 0:0000; std. error of estimation: 0.04255.

Table 3

Statistical results from the multivariate non-linear regression

model

Coefficients Std. error

of coeffi-

cients

t(1995) p-level

Intercept )0.035156 0.011333 )3.10201 0.0019

d 0.03078 0.00359 8.55292 0.0000

KTGD 1.441610 0.095407 15.11000 0.0000

KT2GD )2.276120 0.234991 )9.68599 0.0000

KT3GD 1.780260 0.175503 10.14370 0.0000

R ¼ 0:9313; R2 ¼ 0:8672; R2Fit ¼ 0:8669; F ð4; 1995Þ ¼ 3258:5;

p < 0:0000; std. error of estimation: 0.04143.

Fig. 15. Membership functions for the KTGD INDEX variable.

Fig. 16. Fuzzy inference system rules.

452 L. Ramirez Santigosa et al. / Solar Energy 75 (2003) 447–454

4.3. Fuzzy inference system model (FISM)

A fuzzy inference system is a system that enables the

behavior of one variable to be modeled based on others,

using fuzzy logic. The process of generating a fuzzy in-

ference system consists of a series of stages that can be

synthesized as: fuzzification of the input variable; ap-

plication of the rules; clustering of consequences; de-

fuzzification of the output variable.

There are two types of fuzzy inference systems for

modeling the behavior of a set of variables. The one

most used is the Mamdani method (Mamdani and

Assilian, 1975), which is supported by the work of

Zadeh (Zadeh, 1973) on the decision process in complex

systems. In this type of fuzzy inference system, the

output variable clusters in fuzzy sets with their corre-

sponding characteristic membership functions. The

other method, the Sugeno method (Sugeno and Ya-

sukawa, 1993), is similar to the Mamdani in almost all

respects, but the main difference is that the membership

functions of the output variable are only equal to a

linear function or a constant. When the output of each

rule is a constant, the only difference from the Mamdani

method is the fact that the output membership function

is a ‘‘single peak/value’’ and the implication (multipli-

cation) and clustering (inclusion of all the values)

methods are fixed. One way to visualize a first order

system when the membership functions are linear is to

imagine that each rule defines the location of a ‘‘moving

peak’’. That is, the single output peak can move linearly

in the output space. Furthermore, this tends to make the

output system notation very compact and efficient.

From the point view of using automatic learning tech-

niques, the Sugeno method adapts better since these

techniques can be used to generate membership func-

tions that make the fuzzy system model the data better.

Fig. 14. Membership functions for the DECLINATION vari-

able (in radians· 1000). Fig. 17. Output surface.

Table 4

Summary of FISUV model results

Rules VarDep b1 b2 b3

1. (Dec.¼WIN) & (KTGD ¼LOW) KTUD )0.01033 0.5991 )0.03562. (Dec.¼WIN) & (KTGD ¼MEDIUM) KTUD 0.02293 0.4273 0.2184

3. (Dec.¼WIN) & (KTGD ¼HIGH) KTUD 0.01333 0.0517 0.4989

4. (Dec.¼ SUM) & (KTGD ¼LOW) KTUD 0.01036 0.5509 0.0163

5. (Dec.¼ SUM) & (KTGD ¼MEDIUM) KTUD )0.00726 0.6536 0.0813

6. (Dec.¼ SUM) & (KTGD ¼HIGH) KTUD 0.00361 1.6380 )0.6831

Fig. 18. Distances from the estimated series to the measured

series versus KTUD ranges.

L. Ramirez Santigosa et al. / Solar Energy 75 (2003) 447–454 453

Using a set of input/output data, a fuzzy inference sys-

tem is designed in which the parameters of the mem-

bership functions are trained using back propagation in

combination with the least squares method. This enables

the system to ‘‘learn’’ from the data it is modeling. A

fuzzy system in which the input variables are the clarity

index of daily global radiation and the output variable is

the clarity index of the ultraviolet radiation is generated

from the training data. The characteristics of the model

generated are shown below. Fig. 14 shows the mem-

bership functions for declination, in which two situa-

tions are differentiated during the year (WINTER AND

SUMMER). Following, Fig. 15 shows the membership

functions corresponding to the KTGD variable. Al-

though in the model design some input membership

functions are assigned, the end functions were obtained

by self-adaptive techniques. In Fig. 16, a specific case of

values for each one of the variables and application of

the rules to obtain the final result are given as examples.

Finally, Fig. 17 shows the system output surface.

In the Table 4, the rules that govern the system are

given, as well as the linear output functions, in which b1

represents the value of the coefficient of the declination

(in radians), b2 is the value of the coefficient of the KTGD

index and b3 is the value of the coefficient of the inde-

pendent term.

5. Comparison of results

Now that the three general models have been de-

signed, in this section the results of their application are

compared to the test data. To compare the results, two

procedures are followed. In the first place, the following

test statistic is studied.

Dn ¼ jFnðxÞ � F ðxÞj

where Dn is defined in the Kolmogorov–Smirnov Test as

the critical distance at which the distributions of two

series may be found to consider them the same. FnðxÞ isthe sample empirical distribution function and F ðxÞ is

the theoretical distribution of the population that we

want to test. The critical value calculated for n ¼ 572 is

0.068. Fig. 18 shows how only the series generated with

the FISUV model behaves as a series measured in all

ranges of frequencies.

Finally, the values of the mean square error of the

series generated by the different models are compared,

observing that here too it is the FISUV model that

provides the best results.

MSE MLR ¼ 0:0431 MSE MNLR ¼ 0:0425

MSE FISUV ¼ 0:0418

6. Conclusions

The main conclusions from the work presented may

be summarized in the following points:

• The possibility of estimating a general model for the

Iberian Peninsula to calculate daily ultraviolet solar

radiation from daily global radiation was sought.

• Three models applying different techniques were

studied and in the end it was demonstrated that the

one based on the generation of fuzzy inference sys-

tems is the one that behaves the best.

• The FISUV model fulfills the Kolmogorov–Smirnov

Test, which enables us to state that the estimated se-

ries has the same distribution as the measured series.

454 L. Ramirez Santigosa et al. / Solar Energy 75 (2003) 447–454

• It is demonstrated that the fuzzy logic techniques are

applicable to the study of variables related to solar

radiation.

• It would be necessary to use the data from other loca-

tions to contrast the functioning of the system and if

necessary, improve it, adding these new observations.

Acknowledgements

The authors would like to thank Deborah Fuldauer

for helping in the translation of the paper. In addition,

the authors wish to thank Faustino Chenlo and Gerardo

Pedr�oos for supplying the Madrid and C�oordoba data,

respectively.

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