Modelling Solar Radiation

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OCTOBER 1999 3105S A N T A M O U R I S E T A L .

1999 American Meteorological Society

Modeling the Global Solar Radiation on the Earths Surface Using AtmosphericDeterministic and Intelligent Data-Driven Techniques

M. SANTAMOURIS AND G. MIHALAKAKOU

Laboratory of Meteorology, Division of Applied Physics, Department of Physics, University of Athens, Athens, Greece

B. PSILOGLOU

Laboratory of Meteorology, Division of Applied Physics, Department of Physics, University of Athens, andInstitute of Meteorology and Physics of the Atmospheric Environment, National Observatory of Athens, Athens, Greece

G. EFTAXIAS

Laboratory of Meteorology, Division of Applied Physics, Department of Physics, University of Athens, Athens, Greece

D. N. ASIMAKOPOULOS

Laboratory of Meteorology, Division of Applied Physics, Department of Physics, University of Athens, andInstitute of Meteorology and Physics of the Atmospheric Environment, National Observatory of Athens, Athens, Greece

(Manuscript received 6 July 1998, in final form 13 January 1999)

ABSTRACT

Three methods for analyzing and modeling the global shortwave radiation reaching the earths surface arepresented in this study. Solar radiation is a very important input for many aspects of climatology, hydrology,atmospheric sciences, and energy applications. The estimation methods consist of an atmospheric deterministicmodel and two data-driven intelligent methods.

The deterministic method is a broadband atmospheric model, developed for predicting the global and diffusesolar radiation incident on the earths surface. The intelligent data-driven methods are a new neural networkapproach in which the hourly values of global radiation for several years are calculated and a new fuzzy logic

method based on fuzzy sets theory. The two data-driven models, calculating the global solar radiation on ahorizontal surface, are based on measured data of several meteorological parameters such as the air temperature,the relative humidity, and the sunshine duration.

The three methods are tested and compared using various sets of solar radiation measurements. The comparisonof the three methods showed that the proposed intelligent techniques can be successfully used for the estimationof global solar radiation during the warm period of the year, while during the cold period the atmosphericdeterministic model gives better estimations.

1. Introduction

Solar radiation incident on the earths surface is afundamental input for many aspects of climatology, hy-drology, biology, and architecture. In addition, it is animportant parameter in solar energy applications, in

electricity generation, and in daylighting. In locationswhere radiation measurements are sparse, theoretical es-timates of the available solar energy can be used topredict it from other existing data. Therefore, various

Corresponding author address: Dr. G. Mihalakakou, University ofAthens, Department of Physics, Division of Applied Physics, Lab-oratory of Meteorology, University Campus, Bldg. PHYS-V, Athens15784, Greece.E-mail: [email protected]

algorithms have been developed for the prediction ofthe available solar irradiance from other existing data,which usually consist of the standard climatological pa-rameters that are measured extensively such as air tem-perature, relative humidity, sunshine duration, and

cloudiness.In principle, the amount of solar radiation reachingthe earths surface could be calculated by subtractingfrom the extraterrestrial radiation, which is known withsufficient accuracy, the radiation losses in the atmo-sphere, which are caused by several processes such asabsorption and scattering (Iqbal 1983; Pisimanis et al.1987). This is indeed the case for the clear-sky directcomponent of the radiation for which several models,both rigorous and simple, exist and provide adequateestimates.


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3106 VOLUME 12J O U R N A L O F C L I M A T E

The majority of models, which calculate clear-skysolar radiation components on a horizontal surface, con-sider a one-band calculation (Atwater and Ball 1978;Hoyt 1978; Bird and Hulstrom 1981; Davies and McKay1982; Sherry and Justus 1983) or a two-band calculation(Lacis and Hansen 1974; Paulin 1980; Gueymard 1989).The most sophisticated models are those of the spectraltype (Braslau and Dave 1973; Bird et al. 1983; Kneizyset al. 1988). These models are useful for applicationswith spectrally varying optical characteristics, but theyare computationally complicated and often require veryspecific input data that are rarely available. Conversely,very simple one-band radiation models, such as ASH-RAE (1976) and its variations (Powell 1984; Machlerand Iqbal 1985) are widely used by engineers becauseof their computational convenience, but they are limitedwith respect to climatic variety (Gueymard 1986).

The effect of cloudiness has been included in variousmodels calculating solar radiation (Angstrom 1924; Bar-

baro et al. 1979; Collares-Pereira and Rabl 1979; Kleinand Theilacker 1981; Erbs et al. 1982; Dogniaux 1984;Page 1986; Pisimanis et al. 1987). A broadband at-mospheric model designed for predicting the global anddiffuse solar radiation incident on the earths surfaceunder clear- or cloudy-sky cases has been developedand used in the present study. The atmospheric trans-mittance of each atmospheric parameter contributing tosolar depletion, such as water vapor ozone, uniformlymixed gases, molecules, and aerosols, is calculated us-ing parameterized expressions resulting from integratedspectral transmittance functions. The beam and diffuseradiation are obtained as a function of the specific at-mospheric transmittances. The model is validated using

extensive sets of measurements, and a close agreementbetween the calculated and the measured values of glob-al and diffuse solar radiation is obser ved. The validationof the model is performed for the city of Athens, whichis a large-sized near-coastal area. For the case of Athens,the part of the model predicting the spectral and broad-band aerosol transmittance has taken into account thepollution problems and the high concentrations of sea-salt particles observed in a coastal or near-coastal en-vironment like Athens. Although many accurate at-mospheric models have been proposed and tested withsufficient accuracy, the present model is selected as itis designed to fit with the specific climatological dataof Athens, where the comparison will be performed.

Furthermore, a neural network approach and a fuzzylogic method are used in this study to estimate the globalsolar radiation. Neural networks and fuzzy logic tech-niques belong to the class of data-driven approachesinstead of model-driven approaches (Chakraborty et al.1992). In the data-driven models the analysis dependsonly on the available data, with little rationalizationabout possible interactions. Relationships between var-iables, models, laws, and predictions are constructedafter building a machine that simulates the considereddata. Both neural networks and fuzzy systems have been

shown to have the capability of modeling complex non-linear processes to arbitrary degrees of accuracy.

The main objective of the present study is the pre-sentation and comparison of three models, one deter-ministic atmospheric model and two intelligent data-driven models, for the estimation of the global short-wave radiation using as inputs several meteorologicalparameters. The atmospheric model is an analytical ap-proach, based on parameterized expressions, which re-quires as inputs several climatological parameters suchas air temperature, relative humidity, sunshine duration,cloudiness, surface albedo, etc. The model is able togive sufficiently accurate estimations provided that allthe required input parameters are available. Taking intoaccount that the climatological measurements networkin developed countries is still in progress and that thereare locations where measured data are rather sparse, thedesign of data-driven approaches could be very effec-tive. Intelligent data-driven approaches such as neural

networks and fuzzy logic methods present several ad-vantages over conventional, deterministic analyticalmodels. Besides simplicity, another major advantage isthat they do not require any assumption to be madeabout the underlying function or model to be used. Allthey need are the historical data of the target and thoserelevant input factors for training the data-driven sys-tem. Once the system is well trained and the error be-tween the target and the method estimations has con-verged to an acceptable level, it is ready for use. Variousauthors have already designed intelligent data-driventechniques for several energy applications (von Altrocket al. 1994; Dash et al. 1995; Mihalakakou et al. 1998).

However, the results of these methods have never

been tested using accurate deterministic models. Thus,there is an uncertainty as regards the applicability of themethods, their field of applicability, and their advan-tages and disadvantages. The present study aims at in-vestigating the accuracy of two intelligent data-drivenmethods, a neural network and a fuzzy logic technique,by comparing their results primarily with testing sets ofmeasured data and secondarily with the outputs of ananalytical and accurate atmospheric model. Finally, thepresent paper proposes specific information on the ap-plicability of each model.

The paper is organized as follows: the three modelsused in the present study are presented in the first sectionof the article, each one in a separate paragraph, while

in the second section a comparison of the three modelsresults can be found. Finally, the conclusions are givenin the last section.

2. Modeling the global solar radiation

a. The atmospheric deterministic model

A broadband atmospheric model is used in the presentstudy. The proposed model is developed for calculatingthe beam, diffuse, and global solar radiation incident on


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TABLE 1. The values of the atmospheric models parameters for watervapor, O3, CO2, CO, N2O, CH4, O2 and aerosol total extinction.

Gas A B C D

WaterOzoneCO2CON2OCH4O2Aerosol total extinction

3.0140.25540.07210.00620.03260.01920.00030.2579

119.36107.26

377.89243.67107.413166.095476.934

0.04001

0.6440.2040.58550.42460.55010.42210.4892

2.8451

5.8140.4713.17091.72220.90930.71860.12610.2748

the earths surface, under clear- or cloudy-sky cases.The revised Neckel and Labs (1981, 1984) extraterres-trial solar spectrum was used in the above model. The-oretical studies of the solar radiation absorption andscattering caused by the principal atmospheric constit-uents have permitted the development of correspondingtransmission functions. The atmospheric transmittanceof each atmospheric component contributing to solarradiation depletion, such as water vapor (Psiloglou etal. 1994); atmospheric ozone (Psiloglou et al. 1996);uniformly mixed gases such as CO, CO 2 , CH 4 , N 2O,and O 2 (Psiloglou et al. 1995); and molecules and aero-sols (Psiloglou et al. 1997), was calculated using pa-rameterized expressions resulting from integrated spec-tral transmittance functions. The beam and diffuse ra-diation components were obtained as a function of thespecific atmospheric transmittances.

1) CLEAR-SKY RADIATION MODEL

The beam (Ib) under clear-sky conditions on a hori-zontal surface can be expressed as

beam: Ib Io coszTwTRTmg TA(ext), (1)TO3

where Io is the extraterrestrial solar radiation; z is thezenith angle; and the T terms are the broadband trans-mission functions for water vapor (Tw), uniformly mixedgases (Tmg), ozone absorption ( ), Rayleigh scatteringTO3(TR ), and aerosol total extinction due to scattering andabsorption (TA(ext)).

The diffuse solar radiation (Id) under clear-sky con-ditions and on a horizontal surface is regarded as thesum of a portion of beam solar radiation single scattered

from the atmospheric constituents (Id1 ), and of a mul-tiple-scattering component (Id2) that is caused by a sin-gle reflection of the (Ib) and (Id1) components at theearths surface followed by backscattering atmosphericconstituents. Thus, the diffuse radiation can be modeledas follows:

I I I ,diffuse: d d1 d2

where

I I cosT T T T (1 T T )/2,d1 o z w mg 03 A(abs) A(sct) R

I (I I )[a a /(1 a a )],d2 b d1 g s g s

where TA(abs) is the aerosol broadband transmission func-tion due only to absorption attenuation, TA(sct) is the aero-sol broadband transmission function due only to scat-tering attenuation, ag is the ground surface albedo, andas is the albedo of the cloudless sky.

The global solar radiation (It) for clear sky conditionscan be expressed as follows:

It Ib Id (Ib Id1)/(1 agas ). (2)

The atmospheric albedo (as ) for clear-sky conditionscan be approximated using the following form:

as ar aa,

where ar represents the albedo due to molecular Ray-leigh scattering and aa is the atmospheric aerosol albedodue to aerosol scattering.

The transmission functions for water vapor and ozoneabsorption can be expressed by the following equation(Psiloglou et al. 1994, 1996):

AMUiT 1 ,i C[(1 BMU) DMU]i i

where A, B, C, and D are parameters, given in Table 1,for water vapor, O 3, CO 2, CO, N 2O, CH 4, and O 2 . HereM is the relative optical air mass and Ui is the absorberamount in a vertical column.

The broadband transmission function due to uniform-ly mixed gases total absorption is calculated by thefollowing equation:

Tmg ,T T T T T CO CO N O CH O2 2 4 2

where , , , , and are the transmit-T T T T T CO CO N O CH O2 2 4 2tances due to absorption of CO 2, CO, N 2O, CH 4, and

O 2, respectively.The transmittance corresponding to Rayleigh scatter-

ing is calculated from the following expression:

TR exp[0.1128M0.8346(0.9341 M0.9868

0.9391 M)].

The absorption and scattering aerosol broadbandtransmittance functions, TA(abs) and TA(sct), are calculatedas follows:

3 5 2T 1.0 1.405 10 M 9.013 10 MA(abs)

6 3 2.2 10 M

T T /TA(sct) A(ext) A(abs)

1.6364 1M [cos 0.50572(96.07995 ) ] ,z z

optical air mass.

2) CLOUD-SKY RADIATION MODEL

The beam (Icb) and the diffuse (Icd ) solar radiationunder cloudy-sky conditions are represented by the fol-lowing forms:


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FIG. 1. Temporal variation of the estimated with the atmosphericmodel and of the measured global solar radiation values for themonthly mean day of Jan and of Jul 1995.

beam: I I f(n/N);cb b

diffuse: I I I , (3)cd cd1 cd2

where

I

I f(n/N)

0.33[1

f(n/N)](I

I ),cd1 d1 b d1

I (I I )[(a a )/(1 a a )],cd2 cb cd1 g s g cs

where n/N is the relative insolation (the ratio of the realsunshine duration, n, to the maximum possible numberof sunshine duration, N), and acs is the albedo of thecloudy sky.

The global solar radiation, (Ict ), for cloudy sky on ahorizontal surface is represented as follows:

Ict Icb Icd (Icb Icd1 )/(1 agacs ), (4)

where acs ar aa ac and ac is the albedo of theclouds.

The inputs to the above model were the air temper-ature, the relative humidity, the air pressure, the totalozone amount in a vertical column, the sunshine du-ration, and the surface albedo.

The accuracy of the model has been verified by com-parisons of the theoretical results with the correspondingdetailed radiation data measured at two stations withslightly different characteristics [National Observatoryof Athens (NOA) and Penteli] in the Athens basin,where global and diffuse radiation measurements areavailable, for a period of 34 months for NOA and 23for Penteli. The NOA (altitude: 107 m) station is locatedon a small hill near the center of Athens, while thePenteli station (altitude: 500 m) is situated in a relatively

less populated area in the northern part of Athens. Theclear-sky part of the model was tested for 70 individualclear days with 2-min intervals, while the wholemodel was checked with monthly mean days andhourly mean values.

Close agreement between the predicted from the mod-el and the measured values of global and diffuse radi-ation is observed, which verifies the accuracy of theproposed expressions for the solar radiation expressions.For the NOA station the calculated values of root-mean-square errors between the measured and estimated glob-al solar radiation values for the monthly mean day variedbetween 2.6% and 5.9%. Similarly, for the Penteli sta-

tion the calculated root-mean-square errors fluctuatedbetween 1.5% and 5.2%.Figure 1, as an example, shows the temporal variation

of that estimated with the atmospheric model and of thatmeasured at the NOA station global solar radiation val-ues for the monthly mean day of January and July 1995.

It can be seen from this figure there is a good agree-ment between measured and estimated values. The root-mean-square error between the measured and the modelestimated values was found equal to 4.46% for themonth of January and 2.69% for the month of July.

b. The neural network approach

1) NEURAL NETWORK ARCHITECTURE

Artificial neural networks are computing systems con-taining many simple nonlinear computing units or nodesinterconnected by links:

w n a

p F (5)

a F(wp).

A neuron with a single input and no bias is shown inEq. (5). The scalar input p is transmitted through a con-

nection that multiplies its strength by the scalar weightw to form the product wp, again a scalar. The weightedinput wp is the only argument of the transfer functionF, which produces the scalar output a (Demuth andBeale 1994):

w n a

p F (6)b

a F(wp b).

The neuron in Eq. (6) has a scalar bias b. The bias canbe viewed as simply being added to the product wp.The transfer function net input n, again a scalar, is the

sum of the weighted input wp and the bias b. The F isa transfer function, typically a step function, a linear ora sigmoid function, that takes the argument n and pro-duces the output a.

In a feed-forward network, the units can be parti-tioned into layers, with links from each unit in the kthlayer being directed to each unit in the (k 1)th layer.Inputs from the environment enter the first layer, andoutputs from the network are manifested in the last layer.A dn1 network is a three-layer feed-forward networkwith dinputs, n units in the intermediate hidden layer,


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FIG. 2. Architecture of the neural network used in this study.

and one unit in the output layer (Weigend et al. 1990;Chakraborty et al. 1992). A weight is associated with

each link, and the network learns or is trained by mod-ifying these weights. A multilayer feed-forward neuralnetwork can be observed in Fig. 2. The network consistsof three layers: an input layer, an output layer, and anintermediate or hidden layer. The neurons in the inputlayer act only as buffers for distributing the input signalsto the neurons in the hidden layer. The dotted lines inFig. 2 mean that there are more neurons in each layerthan are represented in this figure.

In nonlinear estimation problems the artificial neuralnetworks provide an implementation for the real-timeestimation of parameters and the reconstruction of sig-nals corrupted by random noise and distorted by para-sitic components. Therefore, for estimation problems a

suitable neural network model is constructed, which,when subject to an input parameter u(k), produces anoutput (k), which estimates the output y(k) of the sys-tem in the sense that the specified cost function of theerrors e(k) y(k)(k) is minimal. The cost function(E) is defined as follows (Cichocki and Unbehauen1993):

12E |e(k)| k 0, 1, 2, . . . . (7)

2

For the nonlinear system estimation the models cantake the general form (Korenberg and Paarmann 1991)

(k) F[y (k 1), . . . , y (k na), u(k 1),

. . . , u(k nb )], (8)where u(k) and y(k) are, respectively, the multidimen-sional system input and output; F is the multidimen-sional system function; and () is the estimated vectorof y(k).

The estimation problem can be separated into threesuccessive steps or subproblems:

R model building or neural network architecture,R the learning or training procedure, andR the testing or diagnostic checking.

In the present study a multiple network based on aback-propagation learning procedure is designed for es-timating the global solar radiation. The selected neuralnetwork architecture consists of one hidden layer of 15log-sigmoid neurons followed by an output layer of onelinear neuron. Linear neurons are those that have a lineartransfer function, while the sigmoid neurons use a sig-moid transfer function. Back-propagation networks usethe log-sigmoid (logsig) or the tan-sigmoid (tansig)transfer function.

Several learning techniques exist for optimization ofneural networks (Rumelhart and McClelland 1986). Inthe present neural network approach learning isachieved using the back-propagation algorithm of Ru-melhart et al. (1986). Mathematically, back propagationis the gradient descent of the mean-square error as afunction of the weights (Weibel et al. 1995). If the mean-square error exceeds some small predetermined value,a new epoch (cycle of presentations of all training

inputs) is started after termination of the current one.One of the main parameters of the back-propagationalgorithm is the learning rate. The learning rate specifiesthe size of changes that are made in the weights andbiases at each epoch. A learning rate of 0.2 was selected,while the number of epochs varied between 3000 and4000 in all cases.

2) RESULTS AND DISCUSSION

Global solar radiation measured on a horizontal sur-face at the NOA has been simulated using the neuralnetwork approach. For the global solar radiation esti-mation the measurements of three meteorological pa-

rameters were used: air temperature, relative humidity,and sunshine duration.The NOA Institute is situated on a hill at the center

of Athens (37.967N, 23.717E, altitude: 107 m). Con-tinuous observations of standard meteorological param-eters have been performed at this location, the closesurroundings of which have remained unaltered since1864.

Integrated hourly, daily, and monthly values of globalsolar radiation in MJ m2 are measured at the obser-vatory with KippZonen and Eppley actinometers andpyranometers, respectively. Sunshine hours are mea-sured with a CampbellStokes heliograph.

Hourly values of air temperature, relative humidity,

and sunshine duration as well as hourly integrated val-ues of global solar radiation for 12 yr (198495) andfor various months of the year were used for trainingand testing the network. Analytically, 11 years (198494) were used for training the neural network and oneyear (1995) for testing the training data. The nighttimevalues of global solar radiation, which probably are zerovalues, are omitted from the training and testing sets,and therefore they are not used in the training and testingprocesses.

The network was trained over a certain part of the


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FIG. 3. (a) Comparison of the measured with the neural networkestimated global solar radiation values for two years from the trainingset of data for the month of Jul. (b) Comparison of the measuredwith the neural network predicted total solar radiation values for two

years from the training set of data for the month of Jan.

FIG. 4. Testing of the neural network results using the actual radia-tion values of the year 1995 for the month of (a) Jul and (b) Jan.

climatic data, and once training was completed, the net-work was tested over the remaining data.

The input parameters of the neural network modelwere the following:

R air temperature measurements in C,R relative humidity measurements (percent),R sunshine duration measurements in hours, andR calculated extraterrestrial radiation values in kJ m2

h1.

Analytically, the extraterrestrial irradiation (I0) wascalculated using the equation of Iqbal (1983): I0 ISCE0(sin sin cos cos cosi), where ISC is thesolar constant, E0 is the eccentricity correction factor, is the solar declination, is the geographic latitude,and i is the hour angle. The output was the global solarradiation values. Training is performed using hourly val-ues of the input climatic parameters for the estimationof integrated hourly global solar radiation values for 11years (198494) and for various months of the year. Aslearning occurs the mean-square error decreases.Resultsfrom trial runs indicated that adding more hidden layersor nodes did not significantly improve the networksprediction capabilities; rather this only slowed the con-

vergence. Calculations have been performed for variousmonths of the year, and the following two time periodswere selected for the presentation of results.

R The cold period of the year, which consists of themonths of December, January, February, and March.The month of January was regarded as representativeof the cold period for the presentation of results.

R The warm period of the year, which consists of themonths of June, July, August, and September. Ac-cordingly, the month of July was considered to be the

representative month of the warm period for the pre-sentation of results.

Figures 3a and 3b show the comparison of the mea-sured integrated hourly global solar radiation valueswith the neural network estimated ones for two yearsfrom the training set of data (1987 and 1989) and forthe months of July and January, respectively. As can beseen from these figures, there is a good agreement be-tween measured and estimated values.

For most cases, the radiation differences are less than0.25 MJ m2 , while the root-mean-square error betweenthe measured and the estimated values was found to be

equal to 0.16 MJ m2

for the month of July and 0.19MJ m2 for the month of January.The accuracy of the neural network estimations was

tested by comparing the measurements of the testing setof data, which consists of the radiation values of theyear 1995, with the estimated results of the neural net-work approach. Figures 4a and 4b show the comparisonbetween the estimated hourly values for the year 1995and the measured values of the testing set of data (1995)for July and January, respectively. The mean-square er-rors were found to be equal to 0.22 MJ m2 for Julyand 0.20 MJ m2 for January. The present results arequite encouraging for developing a feed-forward back-propagation neural network approach able to simulate

and predict the future values of global solar radiationtime series by extracting knowledge from their past val-ues.

Figures 5a and 5b show the temporal variation of theestimated and measured global solar radiation valuesfor two randomly selected days of the warm period (2July 1992 and 15 July 1995). Accordingly for the coldperiod, Figs. 5c and 5d present the temporal variationof the estimated and measured radiation for two ran-domly selected days (7 January 1993 and 12 January1995). In these figures, the continual line indicates the


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FIG. 5. Temporal variation of the estimated with the neural network and of the measured globalsolar radiation values for (a) 2 Jul 1992, (b) 15 Jul 1995, (c) 7 Jan 1993, and (d) 12 Jan 1995.

measured global solar radiation values, while the crosssymbols indicate the model estimations. As shown, thereis a good agreement between the estimated and the mea-sured data. Similar performance was seen for the whole

set of data.

c. The fuzzy logic method

Fuzzy set theory provides a means for representinguncertainties. A real system could be very complicated,but as humans learn more and more about it, its com-plexity decreases (Zadehb 1975). As complexity de-creases, the precision afforded by computational meth-ods becomes more useful in modeling the system. Forsystems with little complexity, hence little uncertainty,closed-form mathematical expressions provide precisedescriptions of the system. For systems that are a littlemore complex, but for which significant data exist, mod-

el-free methods, such as artificial neural networks, pro-vide a powerful and robust means to reduce some un-certainty through the process of learning. Finally, forthe most complex systems where few numerical dataexist and where only ambiguous or imprecise infor-mation may be available, fuzzy methods provide a wayto understand the system behavior by allowing us tointerpolate approximately between observed input andoutput situations.

Fuzzy logic starts with the concept of a fuzzy set. Afuzzy set is a set without a crisp, clearly defined bound-

ary. It can contain elements with only a partial degreeof membership.

A fuzzy logic method for modeling nonlinear func-tions of arbitrary complexity is based on the following

processes (Ross 1995).R The classification of the systems variables in cate-

gories or classes where the value of each parameterparticipates in each of the above class with a certaindegree of membership. The degree of membership ofa variables value in each class is defined by the mem-bership function. A membership function is a curvethat defines how each point in the input space ismapped to a membership value (or degree of mem-bership) between 0 and 1. The membership functionembodies all fuzziness for a particular fuzzy set, andits description is the essence of a fuzzy property oroperation (Dubois and Prade 1980).

R The fuzzification is the process of making a crisp

quantity fuzzy and creating the fuzzy sets. It involvesalso the development of the membership functions.During this process the membership functions wereassigned to fuzzy variables. This assignment processcan be intuitive or it can be based on some algorithmicor logical operations. There are several methods fordeveloping membership functions such as intuition,inference, neural networks, genetic algorithms, fuzzystatistics, etc.

R The formation and application of several conditionalrules when observing some complex process.


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FIG. 6. Comparison of the measured with the fuzzy logic methodestimated global solar radiation values for two years from the trainingset of data for the month of (a) Jul and (b) Jan.

FIG. 7. (a) Comparison of the measured with the fuzzy logic methodestimated total daily global solar radiation for the testing year (1995)for (a) Jul and (b) Jan.

R The defuzzification process includes the conversionof a fuzzy quantity to a precise quantity, just as fuzz-ification is the conversion of a precise quantity to afuzzy quantity. The output of a fuzzy process can bethe logical union of two or more fuzzy membershipfunctions.

In classification, the most important issue is decidingwhat criteria to classify against. For time series appli-cations the process of pattern recognition is extensivelyused for the classification of the system parameters. Pat-tern recognition can be defined as a process of identi-fying structure in data by comparisons to known struc-

ture (Fukunaga 1972; Bezdek 1981). The purpose of thepattern recognition is to assign each input to one ofpossible pattern classes (or data clusters). Presumably,different input observations should be assigned to thesame class if they have similar features and to differentclasses if they have dissimilar features.

The data used to design a pattern recognition systemare usually divided into the following two categoriesmuch like the categorization used in neural networks:

R the training data andR the testing data.

The training data are used to establish the algorithmicparameters of the pattern recognition system, while the

testing data are used to test the overall performance ofthe pattern recognition system.

In the present study, for the estimation of the globalsolar radiation on a horizontal surface, the same inputparameters as in the neural network approach were used.Moreover, the same sets of measured data of the Instituteof Meteorology and Physics of the Atmospheric Envi-ronment, National Observatory of Athens, and for thesame time period as in the neural network model wereused for training and testing the system.

The training data were classified as follows:

R air temperature data in four classes,R relative humidity data in three classes, andR sunshine duration in eight classes.

The global solar radiation data were classified in 13classes. For all classes the trigonal symmetric mem-bership function was used.

Figures 6a and 6b show the comparison of the mea-sured integrated hourly solar radiation values with thefuzzy logic method estimated ones for two years fromthe training set of data (1987 and 1989) and for themonths of July and January, respectively. A very good

agreement is observed from the comparison. For mostcases the radiation differences are less than 0.20 MJm2 , while the root-mean-square error between the mea-sured and the estimated values was found equal to 0.16MJ m2 for July and 0.21 MJ m2 for January.

In order to check the accuracy of the method esti-mations, the results were tested using the measurementsof the testing set of data which consists of the radiationvalues of the year 1995. Figures 7a and 7b show thecomparison between the estimated and measured totaldaily values of global solar radiation for the months ofJuly and January, respectively. As shown from thesefigures, there is a relatively good agreement betweenthe measured and the estimated values. The root-mean-

square errors were 0.26 MJ m2

for January and 0.22MJ m2 for July.

3. Comparison of the three models

The global solar radiation values estimated from eachone of the three models were compared with the cor-responding measured values at the station of the Na-tional Observatory of Athens. The comparison was per-formed for the solar radiation hourly values of the year1995.


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FIG. 8. Temporal variation of the relative difference (%) between the measured and the estimated from the three models global solarradiation values for the monthly mean day of Jul 1995.

When a model has been fitted it is checked whetheror not the model provides an adequate description ofthe data. This is usually done by working out the re-siduals, which are defined as measured values minusestimated values. The visual inspection of a plot of theresiduals themselves is an indispensable first step in thechecking process.

Figures 8 and 9 show the temporal variation of therelative difference (%RD) between measured and esti-mated values from the three models global solar ra-diation values for the monthly mean day of July and forthe monthly mean day of January, respectively:

(%RD) [(Rmeas Rest )/Rmeas ]100,

where Rmeas and Rest are the measured and estimated fromthe three models global-solar radiation values, respec-tively.

For the month of July, which in Athens consists usu-ally of clear days, and which represents the warm periodof the year, there is a close agreement between the at-mospheric model estimations and the measured datawith the relative difference ranging from 5.5% to2.2%. Respectively, a relatively good agreement hasbeen observed between the data-driven models esti-

mations and the measured data. The performance of theneural network approach was quite satisfactory and therelative difference varied between 4.7% and 5.3%. Asfor the fuzzy logic method, the relative differencesranged from 5.7% to 6.6%. The performance of theatmospheric model is trivially better because it is moreanalytical and takes into account many more involvedparameters than the two data-driven models. However,using too few inputs can result in inadequate modeling,whereas too many inputs can excessively complicate themodel. The data-driven procedure provides a good per-formance for the warm period of the year despite theunavailability of an analytical theoretical model under-

lying the observed phenomena.In the three models the higher and lower values ofthe relative difference are observed early in the morning(0600 or 0700 LT) or in the late afternoon (1700 or1800 LT), when the values of the global solar radiationare relatively small. During the day the values of therelative difference varied between 3% and 3%.

Therefore, for the warm period of the year the twodata-driven models provide a satisfactory performance,which, compared with the performance of the deter-ministic atmospheric model, is quite similar.


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FIG. 9. Temporal variation of the relative difference (%) between the measured and the estimated from the three models global solarradiation values for the monthly mean day of Jan 1995.

The cold period of the year in Athens is representedby January, which is a month with a great number of

cloudy days. The relative difference between the at-mospheric model estimations and the measurement fluc-tuated between 10.5% and 3.2%. The lower value(10.5%) is observed in the morning, while during theday the relative difference varied between 0% and 3%.

Respectively, the neural network model estimationsgave a relative difference ranging from 6.7% to14.3%, while in the fuzzy logic method the observedrelative differences varied between 13.2% and 14.3%.As in the month of July, the two data-driven modelsgave their higher and lower values of the relative dif-ference in the morning and in the afternoon with theobserved low values of the global solar radiation. Dur-ing the day the values of the relative difference varied

between 4% and 4%.In January, the atmospheric model gives better esti-mates of the global solar radiation than the two data-driven models. This is observed especially during thecloudy days; it can be explained mainly by the fact thatthe atmospheric model consists of several formulationscalculating separately the beam, diffuse, and global solarradiation for a clear and for a cloudy day, and it takesinto account a large number of involved parameters. Onthe contrary, the two data-driven models cannot use somany inputs, as that would imply slower training and

slower convergence. Moreover, January in Athens is amonth with a great number of cloudy days and generally

with weather phenomena such as cloud coverage, rain-fall, and storms, and the data-driven models cannot al-ways simulate successfully the days with these variousweather phenomena because their results depend strong-ly on the training data.

4. Summary and conclusions

The hourly values of global solar radiation are esti-mated in the present study using the following threemodels.

R A deterministic atmospheric model.R A new neural network system based on back-propa-

gation techniques, designed and trained to model theglobal solar radiation. Remarkable success has beenachieved in training the networks to learn the hourlyradiation values. After training the network the resultswere tested over another number of data not used inthe training procedure, and it was found that the neuralnetwork predicted values perform well on the testingset of measurements.

R A new fuzzy logic method based on fuzzy sets formodeling nonlinear functions. The fuzzy logic methodcontains the classification of the systems variables in


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classes, the fuzzification, the formation and applica-tion of conditional rules, and the defuzzification pro-cess. The same sets of data as in the neural networkapproach were used for training and testing the sys-tem. The trained values were compared with the cor-responding actual values and they were found to bein close agreement.

The measured data and the results of the three modelswere compared, and this comparison led to the followingobservations.

R For the warm period of the year, which in Athensconsists mainly of clear and sunny days, the threemodels can give accurate estimations. The atmospher-ic model provides a trivially better performance,which is caused by the fact that it requires a largernumber of input parameters. However, the perfor-mance of the two data-driven models can be char-acterized as very satisfactory for the summer period.

Therefore, taking into account the two major advan-tages of the data-driven models, which are the sim-plicity and the fact that they do not require any as-sumption to be made about the underlying functionor model to be used, the proposed data-driven modelscan be successfully used for the global solar radiationestimation in Athens.

R During the cold period of the year, which usually con-sists of a great number of cloudy days and variousweather phenomena, the atmospheric model is able togive quite better estimations than the two data-drivenmodels. This can be explained by the fact that theatmospheric model consists of various formulationssimulating separately the global solar radiation under

several weather conditions. On the other hand, theproposed data-driven models cannot work efficientlywhen a large number of inputs are involved. More-over, the results of the data-driven models dependstrongly on the training sets of data, and it is notpossible to make long-term estimations on chaotictime series such as the global solar radiation data dur-ing the cold period of the year, which is characterizedby the high frequency of different weather phenom-ena.

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Modelling Solar Radiation