An adaptive model for predicting of global, direct and diffuse hourly solar irradiance

Energy Conversion and Management 51 (2010) 771–782

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Energy Conversion and Management

journal homepage: www.elsevier .com/ locate /enconman

An adaptive model for predicting of global, direct and diffuse hourly solar irradiance

A. Mellit a,*,1, H. Eleuch b,1, M. Benghanem c,1, C. Elaoun d, A. Massi Pavan e

a Department of Electronics, Faculty of Sciences Engineering, LAMEL, Jijel University, Ouled-Aissa, P.O. Box 98, Jijel 18000, Algeriab Institut National des Sciences Appliquées et de Technologie, Zone Urbaine, Nord Tunis, B.P. 676, Tunisia, Cedex 1080, Tunisiac Department of Physics, Faculty of Sciences, Taibah University, P.O. Box 344, Madinah, Saudi Arabiad Laboratoire Cristal, Ecole Nationale des Sciences de l’informatique, Université de Mandouba, 210 Mandouba, Tunisiae Department of Materials and Natural Resources, University of Trieste, Via A. Valerio, 2 – 34127 Trieste, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 March 2009Accepted 30 October 2009Available online 14 December 2009

Keywords:Solar irradianceModelingPredictionNeural network

0196-8904/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.enconman.2009.10.034

* Corresponding author. Tel.: +213 551998982.E-mail addresses: [email protected], amellit@ic

1 Present address: The International Centre for TheoCostiera, 1134014 Trieste, Italy.

In this paper, an adaptive model for predicting hourly global, diffuse and direct solar irradiance isdescribed. A dataset of measured air temperature, relative humidity, direct, diffuse and global horizontalirradiance for Jeddah site (Saudi Arabia) were used in this study. Several combinations have been pro-posed, and the best performance is obtained by using sunshine duration, air temperature and relativehumidity as inputs of the developed adaptive a-model. A good agreement between measured and pre-dicted data is obtained. In fact, the correlation coefficient is more than 97% and the mean bias error is lessthan 0.8. A comparison between a Feed-Forward Neural Network (FFNN) and the adaptive proposedmodel is presented in order to demonstrate his performance.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

It is well known that solar irradiance is the basic data in manyfields, and the knowledge of the amount of solar irradiance playvery important role for many applications, such as atmosphericenergy-balance studies, analysis of the thermal load on buildings,designing, sizing, operation and economic assessment of energyand renewable energy systems and also for some environmentalimpact analysis [1–3]. Recently, there is an increasing need formore precise modeling, forecasting and prediction of solar irradi-ance that is helpful for operation control and optimization of someenergy systems.

Therefore, it is necessary to develop a more accurate method ofmodeling and predicting the hourly solar irradiance of the nexthours. If the irradiance for several hours ahead can be predicted,it is useful for the power dispatching plan.

Several models have been developed in order to estimate andgenerate the solar irradiance data, in different scales. These modelsinclude empirical models [4–6], analytical models [7–10], numeri-cal models and statistical approaches [11–17] neural networksapproaches [18–25].

Models based on statistical processes such as autoregressive(AR), moving-average (MA), autoregressive moving-average(ARMA) model, autoregressive-integrated moving-average (ARIMA)

ll rights reserved.

tp.it (A. Mellit).retical Physics (ICTP), Strada-

and Markov chain have been used widely for modeling and predic-tion of solar radiation data. However, these models need some sta-tistical transformations on the data before their using. Therefore,due to these transformations, we cannot obtain an accurately re-sults between measured solar radiation values and those estimatedby these models [3].

Artificial Neural Networks (ANNs) are powerful when applied toproblems whose solutions require knowledge that is difficult tospecify, but for which there is an abundance of examples. The neu-ral network approach does not need to know any informationregarding the process that generates the data. Nowadays, differentmodels based on the neural networks were introduced in literaturefor modeling and prediction of solar radiation data from meteoro-logical and geographical parameters [26–28]. In the neural net-works approach we need a long-term data in order to get abetter model which can be used for estimating or forecasting ofsolar radiation data [29].

In the present study, we will develop of an adaptive model(a-model) to predict the hourly global, direct and diffuse solarirradiance from sunshine duration, relative humidity and air tem-perature. The used model was developed for the times economicsseries problem [20,31]. In the same context, several examples ofthe prediction hour by hour were presented and discussed. Inaddition, a comparison between an artificial neural network andthe adaptive a-model is presented and discussed.

This paper is organized as follows: the next section provides thedataset description. A brief introduction on neural network isshown in Section 3. Section 4 deals with the presentation of themodel development, which will be used in this study for prediction

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Nomenclature

ANN artificial neural networkFFNN feed-forward neural networkMBE men bias errorRMSE root mean square errorHG global solar irradiance (W/m2)Hd direct solar irradiance (W/m2)

HD diffuse solar irradiance (W/m2)Hu relative humidity (%)r correlation coefficientS sunshine duration (h)T air temperature (�C)

700

800

900

1000Jeddah (Saudi Arabia) Latitude: 21.68 N°Longitude: 39.15 E°Elevation: 4 m

772 A. Mellit et al. / Energy Conversion and Management 51 (2010) 771–782

of hourly solar irradiance. The model implementation, results anddiscussions are given in Section 5. A comparison between the Feed-Forward Neural Network-model and the proposed adaptive modelwill be presented and discussed in Section 6.

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

100

200

300

400

500

600

Time(hours)

Hd(

W/m

2 )

Fig. 1b. Hourly diffuse irradiance, Hd (Jeddah).

2. Dataset description

The historical data used in this study are available from 1998 to2002 at the National Renewable Energy Laboratory (NREL) website.The present database includes global irradiance (HG), diffuse irradi-ance (HD), direct normal irradiance (Hd), air temperature (T) andrelative humidity (Hu). All these data have been collected each5 min since 1998 until 2002. In addition, we have calculated thesunshine duration (S) which is defined by WMO as the time inter-val when direct solar radiation exceeds 120 (W m�2) [32].

Figs. 1a–1e show the hourly evolution of the global, diffuse anddirect irradiance measured on horizontal surface, the air tempera-ture and relative humidity for 1 year at Jeddah (Saudi Arabia).

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

200

400

600

800

1000

1200

HD

(W/m

2 )

Jeddah (Saudi Arabia) Latitude: 21.68 N°Longitude: 39.15 E°Elevation: 4 m

3. Neural networks

Artificial neural networks have been used widely in many appli-cation areas. Most applications use a Feed-Forward Neural Net-work with the Back-Propagation (BP) training algorithm. Thereare numerous variants of the classical BP algorithm and othertraining algorithms [33,3]. All these training algorithms assume afixed ANN architecture and during training, they change theweights to obtain a satisfactory mapping of the data.

The main advantage of the Feed-Forward Neural Networks isthat they do not require a user-specified problem solving algo-rithm (as is the case with classic programming) but instead they‘‘learn’’ from examples, much like human beings.

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

200

400

600

800

1000

1200

1400

1600

1800

2000

Time(hours)


HG

(Wh/

m2 )

Fig. 1a. Hourly global solar irradiance, HG (Jeddah).

Time(hours)

Fig. 1c. Hourly direct irradiance, HD (Jeddah).

0 1000 2000 3000 4000 5000 6000 7000 8000 900010

15

20

25

30

35

40

45

Time(hours)

T(C

)


Fig. 1d. Hourly air temperature, T (Jeddah).

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

20

40

60

80

100

120

Time(hours)

Hu(

%)


Fig. 1e. Hourly relative humidity, Hu (Jeddah).

xp1

xp2

xpN

yp1

yp2

ypM

Netp1 Op1

Nh

Input layer

Hidden layer

Output layer

Fig. 2. Feed-Forward Neural Network.

A. Mellit et al. / Energy Conversion and Management 51 (2010) 771–782 773

Another advantage is that they possess inherent generalizationability. This means that they can identify and respond to patternswhich are similar but not identical to the ones with which theyhave been trained. On the other hand, the development of afeed-forward ANN model also poses certain problems, the mostimportant being that there is no prior guarantee that the modelwill perform well for the problem at hand.

A typical Feed-Forward Neural Network is shown in Fig. 2. Thetraining data set consists of N training patterns {(xp, tp)}, where p isthe pattern number. The input vector xp and desired output vectortp have dimensions N and M, respectively; yp is the network outputvector for the pth pattern. The thresholds are handled by augment-ing the input vector with an element xp(N + 1) and setting it equalto one [3].

4. Model development

The main objective of the present study is to propose an adap-tive suitable model for prediction of hourly solar irradiance valuesby using several combinations. The proposed adaptive model,called a-model was developed initially for finance applications inthe context of e-commerce negotiation and stock exchange previ-sions [30,31]. Therefore, in the present study we try to adapt thismodel for solar radiation prediction. This model is based on theconcept of reasoning by analogy. This approach involves a seriesof steps that generate novel inferences about an unfamiliar target

domain based on prior knowledge of a familiar domain [30].We extend this concept to this work using the formula devel-oped in the a-model. In finance applications, the parameter arepresents the fraction between the final price Pf and the startingprice PS:

a ¼ Pf

PSð1Þ

Therefore, based on parameter a, we can decide to sell or to buy.In fact, we can predict the final price Pf knowing the starting pricePS:

Pf ¼ a � PS ð2Þ

If a > 1, it is necessary to buy because the final price willincrease.If a < 1, it is necessary to sell because the final price willdecrease.

a is given by variables that influence the final price [30]:

a ¼ P2�b1S

ðVb2 Sb3P � R

b4d � T

b5B � V

b6l þ r0Þ2

ð3Þ

where Vb2 ; Sb3P ; Rb4

d ; Tb5B and Vb6

l are the variables influencing thefinal price of bourse, and r0 include the others variables with lessinfluence on the final price. The problem is to estimate the differentparameters bi of the model.

By analogy, we have set up an adaptive model (a-model) rela-tive to solar radiation by considering that there are meteorologicalvariables that influence the solar radiation data.

This is due the correlation existing between different solarparameters as indicated in literature [4,5,13,32]. Therefore, wetry to predict the solar irradiance on hour j + 1, knowing the solarirradiance, sunshine duration, air temperature and relative humid-ity measured on hour j (Fig. 3).

In fact, our problem consists here to make prediction of solarirradiance data. The First crucial step that should be done in orderto apply this model, it is to identify the variables that may predictthe solar irradiance data. The most significant parameters in ourcase are sunshine duration, air temperature and relative humidity.Using the concept of the reasoning by analogy with the relation (3),we can write:

aðjÞ ¼ yðjÞ2b5

ðSðjÞb2 HuðjÞb3 TðjÞb4 þ b1Þ2 ð4Þ

where y(j) is the output parameter of the model which will be theglobal irradiance HG, the diffuse irradiance HD, or the direct irradi-ance Hd, while the parameters S(j), Hu(j) and T(j) represent thesunshine duration, the relative humidity and the air temperaturefor the hour j respectively. The parameter a is defined as thequotient of the predicted value (yj+1) to the actual value (yj) of solarirradiance:

a ¼yjþ1

yjð5Þ

The bi coefficients are the parameters of the adaptive a-modelthat can be estimated from the history of our studied system.b1 represents the influence of all other parameters that arenot taking into account in the proposed model. We suppose herethat the other parameters are weakly influencing our system, sothat:

l ¼ b1

SðjÞb2 HuðjÞb3 TðjÞb4� 1

- Global irradiance HG(j)- Sunshine duration S(j) - Air temperature T(j)- Relative humidity Hu(j)

adaptive -model

atadtuptuOatadtupnI

- Global irradiance HG(j+1)- Diffuse irradiance HD(j+1) - Direct irradiance Hd(j+1)

Fig. 3. Input/output parameters for the adaptive a-model relative to solar radiation data.


In this case, the relation (4) yields:

LogðaðjÞÞ ¼ 2b5LogðyjÞ � 2LogðSb2j Hub3

j Tb4j þ b1Þ

¼ 2b5LogðyjÞ � 2Log Sb2j Hub3

j Tb4j 1þ b1

SSb2j Hub3

j Tb4j

! !

¼ 2b5LogðyjÞ � 2LogðSb2j Hub3

j Tb4j ð1þ lÞÞ

¼ 2b5LogðyjÞ � 2LogðSb2j Hub3

j Tb4j Þ � 2 � Lnð1þ lÞ

Then:

LogðaðjÞÞ � 2b5LogðyjÞ � 2b2LogðSjÞ � 2b3LogðHujÞ � 2b4LogðTjÞ � 2lð6Þ

5. Results and discussion

The available dataset are divided into two sets, the first one(8000 points) is used for estimating the bi coefficients and the sec-ond set (765 points) is used for testing and validating the model.The pseudo code of the developed adaptive a-model is shown inAppendix A.

Therefore, in order to predict the global solar irradiance, we getthe a parameter by using the following formula:

aHGj

j ¼H2b5

Gj

ðSb2j Hub3

j Tb4j þ b1Þ

2 ð7Þ

In this case, a parameter depends on solar irradiance, sunshineduration, relative humidity and air temperature data measured onhour j. The predicted solar irradiance for the hour j + 1 can be givenby the following expression:

HGjþ1¼ a

HGj

j � HGjð8Þ

The developed aj parameters for different components (directand diffuse) of solar radiation are given by the followingexpressions:

0 100 200 3000

200

400

600

800

1000

1200

1400

Time

Pred

icte

d, G

loba

l Hor

izon

tal I

rradi

ance

, W/m

2

Fig. 4a. Comparison between measured and predicted global irradiance usi

aHdj

j ¼H

2b05dj

ðSb02j Hu

b03j T

b04j þ b01Þ

2ð9Þ

aHDj

j ¼H

2b005Dj

ðSb002j Hu

b003j T

b004j þ b001Þ

2ð10Þ

where Hd and HD are respectively the direct and diffuse solar irradi-ance measured on hour j.

Figs. 4a–4c show a comparison between measured and pre-dicted global, diffuse and direct horizontal irradiance by usingthe sunshine duration, the air temperature and the relative humid-ity as inputs for the adaptive a-model. According to these figures, itis clearly shown that good agreements between both series areobtained. Also, we can observe that the predicted data are highlycorrelated with the measured data.

Therefore, in order to test and validate these results we havecalculated some goodness test (Root Mean Square Error ‘RMSE’,the correlation coefficient ‘r’, and Mean Bias Error, ‘MBE’) whichallow to make comparison between measured and predicted datausing the developed adaptive a-model. Figs. 5a–5c indicate thecorrelation between predicted and measured data over 765 h forJeddah site.

The results are illustrated in Table 1, according to this table itshould be noted that the MBE over the whole data set is not exceed0.8, the RMSE is arranged between 2.18 and 2.74, and the coeffi-cient of correlation r is arranged between 97.54% and 97.77%.Therefore, in point of view statistical test (goodness test), we canconclude that the predicted global solar irradiance from sunshineduration, air temperature and relative humidity show a higher cor-relation coefficient. Nevertheless, in all case, the proposed adaptivea-model is very suitable for predicting the direct, diffuse, and glo-bal solar irradiance from sunshine duration, air temperature andrelative humidity. In addition, we have plotted the cumulative dis-tribution function Fx for both measured and predicted series (seeFigs. 6a–6c), as can be seen obtained results are very closed withexperimental data.

400 500 600 700 800

(hour)

MeasuredPredicted

ng as inputs sunshine duration, air temperature and relative humidity.

0 100 200 300 400 500 600 700 8000

100

200

300

400

500

600

700

800

900

1000

Time (hour)

Pred

icte

d, D

irect

Nor

mal

Irra

dian

ce, W

/m2 Measured

Predicted

Fig. 4c. Comparison between measured and predicted direct irradiance using as inputs sunshine duration, air temperature and relative humidity.

0 100 200 300 400 500 600 700 8000

100

200

300

400

500

600

700

800

Time (hour)

Diff

use

Hor

izon

tal I

rradi

ance

, W/m

2

MeasuredPredicted

Fig. 4b. Comparison between measured and predicted diffuse irradiance using as inputs sunshine duration, air temperature and relative humidity.

0 200 400 600 800 1000 1200 14000

200

400

600

800

1000

1200

1400

Measured, Global Horizontal Irradiance, W/m2

Pred

icte

d, G

loba

l Hor

izon

tal I

rradi

ance

, W/m

2

r =97.77

Fig. 5a. The evolution of predicted global irradiation versus the measured global irradiance using as inputs sunshine duration, air temperature and relative humidity.


0 100 200 300 400 500 600 7000

100

200

300

400

500

600

700

800

Measured, Diffuse Horizontal Irradiance, W/m2

Pred

icte

d, D

iffus

e H

oriz

onta

l Irra

dian

ce, W

/m2

r=97.54

Fig. 5b. The evolution of predicted diffuse irradiation versus the measured diffuse irradiance using as inputs sunshine duration, air temperature and relative humidity.

0 100 200 300 400 500 600 700 800 9000

100

200

300

400

500

600

700

800

900

1000

Measured, Direct Normal Irradiance, W/m2

Pred

icte

d, D

irect

Nor

mal

Irra

dian

ce, W

/m2 r=97.73

Fig. 5c. The evolution of predicted direct irradiation versus the measured direct irradiance using as inputs sunshine duration, air temperature and relative humidity.

Table 1The goodness test between measured and predicted ~HG; ~Hd and ~HD using three inputsparameters S, Hu and T.

The adaptive a-models MBE RMSE r (%)

Global horizontal irradiance (HG), W/m2

aHGj

j ¼H2ð�0:0175Þ

Gj

ðS0:0036j Hu�0:0171

j T0:0146j þ0:0209Þ2

0.423 2.18 97.77

Direct normal irradiance (Hd), W/m2

aHdj

j ¼H2ð�0:0201Þ

dj

ðS0:0014j Hu�0:0281

j T0:0122j þ0:0211Þ2

0.751 2.74 97.54

Diffuse horizontal irradiance (HD), W/m2

aHDj

j ¼H2ð�0:0175Þ

Dj

ðS0:0022j Hu�0:0163

j T0:0144j þ0:0217Þ2

0.542 2.47 97.73


In the developed above models we need to use four types ofmeteorological data to predict the solar irradiance (global, directand diffuse). However, these data are not always available espe-

cially the sunshine duration, even though they exist; they not existon a long period. Therefore, we have tried to reduce the number ofinputs data on hour j to predict the solar irradiance on hour j + 1.For this purpose, new proposed configurations can be formulatedas follows:

– Predicting solar radiation from sunshine duration and air tem-perature by using the adaptive a-model, so:

0

aHGj

j ¼H2b4

Gj

ðSb2j Tb3

j þ b1Þ2 ; a

Hdj

j ¼H

2b4dj

ðSb02j T

b03j þ b01Þ

2and

aHDj

j ¼H

2b004Dj

ðSb002j T

b003j þ b001Þ

2ð11Þ

– Predicting solar irradiance from sunshine duration and relativehumidity by using the adaptive a-model, so:

0 200 400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

1.2

1.4

Cum

ulat

ive

dist

ribut

ion

func

tion

Fx

Global Horizontal Irradiance, W/m2

Fig. 6a. The cumulative distribution function between actual and predicted globalirradiance HG.

0 100 200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1

1.2

1.4

Cum

ulat

ive

dist

ribut

ion

func

tion

Fx

Diffuse Horizontal Irradiance, W/m2

Fig. 6b. The cumulative distribution function between actual and predicted diffuseirradiance HD.

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

Cum

ulat

ive

dist

ribut

ion

func

tion

Fx

Direct Normal Irradiance, W/m2

Fig. 6c. The cumulative distribution function between actual and predicted directirradiance Hd.


aHGj

j ¼H2b4

Gj

ðSb2j Hub3

j þ b1Þ2 ; a

Hdj

j ¼H

2b04dj

ðSb02j Hu

b03j þ b01Þ

2and

aHDj

j ¼H

2b004Dj

ðSb002j Hu

b003j þ b001Þ

2ð12Þ

– Predicting solar irradiance from air temperature and relativehumidity by using the adaptive a-model, so:

2b 2b04

aHGj

j ¼H 4

Gj

ðTb2j Hub3

j þ b1Þ2 ; a

Hdj

j ¼Hdj

ðTb02j Hu

b03j þ b01Þ

2and

aHDj

j ¼H

2b004Dj

ðTb002j Hu

b003j þ b001Þ

2ð13Þ

In addition, we have tried to use only one parameter as input ofthe adaptive a-model on hour j to predict the solar irradiance onhour j + 1. For this purpose, new proposed configurations can beformulated as follows:

– Predicting solar irradiance from only sunshine duration by usingthe adaptive a-model:

2b0

aHGj

j ¼H2b3

Gj

ðSb2j þ b1Þ

2 ; aHdj

j ¼H 3

dj

ðSb02j þ b01Þ

2and

aHDj

j ¼H

2b003Dj

ðSb002j þ b001Þ

2ð14Þ

– Predicting solar irradiance from only air temperature by usingthe adaptive a-model:

2b0

aHGj

j ¼H2b3

Gj

ðTb2j þ b1Þ

2 ; aHdj

j ¼H 3

dj

ðTb02j þ b01Þ

2and

aHDj

j ¼H

2b003Dj

ðTb002j þ b001Þ

2ð15Þ

– Predicting solar radiation from only relative humidity by usingthe adaptive a-model:

2b 2b0

aHGj

j ¼H 3

Gj

ðHub2j þ b1Þ

2 ; aHdj

j ¼H 3

dj

ðHub02j þ b01Þ

2and

aHDj

j ¼H

2b003Dj

ðHub002j þ b001Þ

2ð16Þ

Table 2 shows the goodness test between measured and pre-dicted solar irradiance (~HG

~Hd, ~Hd and ~HD) using two parametersas inputs of the adaptive a-model by using different combinationbetween sunshine duration S, air temperature T and relativehumidity Hu.

According to this table, good result are obtained by using thesunshine duration S and air temperature T as inputs of the a-modelsince the best correlation coefficient r, which is equal to 96.65%, theMBE is 4.75 and RMSE is 8.74 in the case of global solar irradiance.

Table 3 indicates that the goodness test between measured andpredicted ~HG; and ~HD using one parameter as inputs of the adaptivea-model. As can be seen from this table, the best configuration is ob-tained by using the sunshine duration S as input for a-model. In thiscase, the correlation HG coefficient is equal to 94.94, the MBE is5.4107% and RMSE is 8.98 in the case of global solar irradiance.

6. Comparative study between the adaptive a-model and FFNN-model

From the literature review, it has been demonstrated that theANNs have been used with success for modeling and predicting

z-1

yi+1

yi

Hui

Si

Ti

n

2

1

HG : n=17

Hd : n=12

HD : n=15

Fig. 7. The FFNN-model for predicting the solar irradiance HG, Hd and HD.

Table 3The goodness test between measured and predicted ~HG; ~Hd and ~HD using oneparameter as inputs of the a-model.


The first configuration Global horizontal irradiance (W/m2)

aHGj

j ¼H2ð0:0457Þ

Gj

ðS0:0081j �0:0515Þ2

5.4107 8.98 94.94

Direct normal irradiance (W/m2)

aHdj

j ¼H2ð0:0291Þ

dj

ðS0:0706j �0:0971Þ2

5.1043 9.18 94.80

Diffuse horizontal irradiance (W/m2)

aHDj

j ¼H2ð0:0483Þ

Dj

ðS0:0119j �0:0579Þ2

5.0857 8.15 94.93

The second configuration Global horizontal irradiance (W/m2)

aHGj

j ¼H2ð0:0459Þ

Gj

ðT0:0078j �0:0512Þ2

6.319 9.213 93.94


aHdj

j ¼H2ð0:0291Þ

dj

ðT0:0706j �0:0971Þ2

7.051 9.446 92.92


aHDj

j ¼H2ð0:0479Þ

Dj

ðT0:0187j �0:0581Þ2

6.965 9.224 93.93

The third configuration Global horizontal irradiance (W/m2)

aHGj

j ¼H2ð0:0456Þ

Gj

ðHu0:0079j �0:0520Þ2

11.475 13.62 90.89


adGj

j ¼H2ð0:0199Þ

dj

ðHu0:0710j �0:0965Þ2

12.124 16.03 90.52


aHDj

j ¼H2ð0:0479Þ

Dj

ðHu0:0214j �0:0568Þ2

12.509 16.71 90.85

Table 2The goodness test between measured and predicted ~HG; ~Hd and ~HD using twoparameters as inputs of the model (combination between T, H and S).


The first configuration Global horizontal irradiance (W/m2)

aHGj

j ¼H2ð0:01172Þ

Gj

ðS0:0338j T�0:0361

j �0:121Þ24.751 8.74 96.65


aHdj

j ¼H2ð0:01172Þ

dj

ðS0:00247j T�0:0114

j �0:03641Þ25.158 8.94 96.21


aHDj

j ¼H2ð0:08649Þ

Dj

ðS0:0620j T�0:0489

j �0:0774Þ24.984 8.87 96.45

The second configuration Global horizontal irradiance (W/m2)

aHGj

j ¼H2ð0:01171Þ

Gj

ðS0:0339j Hu�0:0359

j �0:113b1Þ25.254 9.05 95.31


aHdj

j ¼H2ð0:1171Þ

dj

ðS0:0254j Hu�0:0115

j �0:0358Þ25.685 9.17 94.85


aHDj

j ¼H2ð0:08974Þ

Dj

ðS0:0615j Hu�0:0478

j �0:0784Þ25.486 9.34 94.55

The third configuration Global horizontal irradiance (W/m2)

aHGj

j ¼H2ð0:1164Þ

Gj

ðT0:0336j Hu�0:0357

j �0:115Þ25.784 9.87 94.83


aHdj

j ¼H2ð0:1155Þ

dj

ðT0:0246j Hu�0:1008

j �0:0359Þ25.227 9.22 94.42


aHDj

j ¼H2ð0:0962Þ

Dj

ðT0:0540j Hu�0:0587

j �0:0887Þ25.659 9.54 94.65


of solar irradiation from other meteorological parameters[18,19,21,24–26,28], in this section we try to make a comparisonbetween the adapted a-model and FFNN-model in order to showhis performance for prediction solar irradiance. In the above sec-tion it has been demonstrated that the best model is the one whichhas as input the hourly temperature, humidity, sunshine durationand irradiance, for the hour j, and in his output the hourly irradi-ance at the hour j + 1. The proposed FFNN architecture is shownin Fig. 7. So:

yjþ1 ¼ f ðyj;Hj; Tj; SjÞ ð17Þ

where y can be take one parameters, Hd and HD. From the 8765available data points, 8000 points were used for training the FFNNwhile 765 data points were used for testing and validating theFFNN-models for each parameter (i.e. HG, Hd and HD). The inputsand the outputs of the FFNN are fixed previously, while the numberof hidden layers and the neurons within each layer will be adjustedduring the learning process. A soft computing program is developedby using Matlab (Ver. 7.5) environment, and the LM algorithm isused for this subject.

Traditional normalization techniques use linear or logarithmicscaling essentially. These require the designer to supply practicalestimates of maximum and minimum values of normalized vari-ables so as to improve the neural network performance. The nor-malization equation typically used is:

y ¼ ymin þx� xmin

xmax � xminðymax � yminÞ ð18Þ

where x e [xmin, xmax] and y e [ymin, ymax]. Here, x is the original datavalue and y is the corresponding normalized variable. In this studyymin = 0.1 and ymax = 0.9.

After several simulations, the best architecture is obtainedwhich consists of four neurons in the input layer, one neuron inthe output layer and one hidden layer within 17 neurons, 15 neu-rons and 12 neurons for HG, Hd and HD respectively.

Fig. 8 shows a comparison between measured and predictedsolar irradiance between by FFNN for HG, Hd and HD respectively.As can be seen, good agreement is obtained for all parameters.The developed FFNN-models can be given by the following approx-imate formula:

~y ¼XM

k¼1

2

1þ exp �PM

r¼1

PNi¼1ðw1ði; rÞxðiÞÞ þ b1ðrÞ

� �� 1

24

35

0@

1Aw2ðkÞ þ b2

ð19Þ

where w1, w2, b1 and b2 are respectively the weights and the bias ofthe networks, x, represents the inputs data which can take the sun-shine duration, air temperature, relative humidity and the irradi-ance at the hour j. M and N are the number of neurons in the

8000 8100 8200 8300 8400 8500 8600 8700 88000

100

200

300

400

500

600

700

800

900

1000

Time(hour)

Dire

ct i

rradi

ance

(Wh/

m²)

MeasuredFFNN-Predicted

8000 8100 8200 8300 8400 8500 8600 8700 88000

100

200

300

400

500

600

700

800

Time(hour)

Diff

use

irra

dian

ce (W

h/m²)


8000 8100 8200 8300 8400 8500 8600 87000

200

400

600

800

1000

1200

1400

Time(hour)

Glo

bal

irrad

ianc

e (W

h/m

²)


Fig. 8. Comparison between measured and predicted HG, Hd and HD irradiance respectively by using the developed FFNN.


hidden layer and in the input layer respectively. The experimentalvalues of w1, w2, b1 and b2 are shown in Appendix B.

Table 4 shows a comparison between FFNN-models and theadaptive a-models for predicting the solar irradiance (global, direct

and diffuse) by using the correlation coefficient, we can observethat the correlation coefficient for each simulation is arranged be-tween 98.21% and 98.53%. The comparison shows that the resultsobtained by the FFNN-modes are more accurate than the adaptive

Table 4Comparison between the FFNN-models and the adaptive a-model by using thecorrelation coefficient.

FFNN-model a-model

Correlation coefficient r (%)

Global irradiance HG (W/m2) 98.53 97.77Direct irradiance Hd (W/m2) 98.34 97.54Diffuse irradiance HD (W/m2) 98.21 97.73


a-models, however, the adapted approach is relatively easy for hisimplementation due to his simple formula while FFNN is not rela-tively easy for his implementation due to his complex architecture(sigmoid function, number neurons, number of hidden layers, etc.).

In any case, the proposed adaptive a-model can providesacceptable results compared with those predicted by the FFNN-models. In addition, the adapted model can be used for predictingmore than 1 h, for example we can calculate: yi+1 = a1yi,yi+2 = a2yi+1, . . . yi+n = anyi+n�1, subsequently we get that yi+n =a1a2 . . . anyi, therefore we can predicted n hours, however, thenumber of the predicted hours can be achieved according to oneaccurate criterion. In more, the adaptive model can be used for pre-dicting solar irradiance in different scales (min, hour and day).

7. Conclusion

In this paper, an adaptive a-model for predicting hourly global,direct and diffuse horizontal solar irradiance from sunshineduration, air temperature and relative humidity was developed.The proposed adaptive a-model has been applied for Jeddah site(Saudi Arabia), and it can predict hour by hour the solar irradiance(HG, HD and Hd) with good accuracy results. It was found that thecorrelation coefficient between measured and predicted data ismore than 97%, the mean bias error is less than 0.8 which is verysatisfactory.

Loading data H, T, HG, HD, Hd

Normalization dataWhile i 6 N,

{A [�1, Log(Si), Log(Ti), Log(Hui), Log(HGi); . . .

�1, Log(Si+1), Log(Ti+1), Log(Hui+1), Log(HGi+1); . . .



�1, Log(Si+4), Log(Ti+4), Log(Hui+4), Log(HGi+4)];B [log(HGi+1)/HGi, log(HGi+2)/HGi+1, log(HGi+3)/HGi+2, log(HGi+4)/HGi+

bi A�1 � BT

b() [b(), bi];i++}

End while�b() Mean (b()),

While j 6M,{

aj H2�bð5Þ

Gj

ðS�bð2Þj Hu

�bð3Þj T

�bð4Þj þ�bð1ÞÞ2

~HGjþ1 ajHGj

~HGjþ1ðÞ ½~HGjþ1

ðÞ; ~HGjþ1�

j++}

End while

In addition, in the case of two parameters as inputs, the bestresult is obtained by using sunshine duration and air tempera-ture (r = 96.65%), and however for one parameter as input, thebest prediction is obtained by using sunshine duration(r = 94.94%).

The advantage of the present developed adaptive model that itcan be implemented very easy, and it is flexible, i.e. we can addsome parameters or remove, especially if this parameters are notavailable, the case of sunshine duration because it is related to so-lar irradiance. Also, we can applied this model for predicting othermeteorological or climate parameters, such as temperature, rela-tive humidity, wind speed, wind direction, precipitation, pressure,rainfall.

By comparing the obtained results with the adaptive a-modelwith those obtained by using FFNN we can conclude that theadaptive a-model model is suitable for predicting the solar irradi-ance, however, the FFNN is more perform than the adaptive a-model.

In our future works, we try to adapt this model for predictingmore than one hour, for example 2, 3 and 4 h. In addition, we willcompare the proposed adaptive a-model with the most used mod-els for time series prediction such as wavelet-network, recurrentneural network and polynomial neural network.

Acknowledgements

The authors would like to thank the International Centerfor Theoretical Physics, Trieste (Italy) for providing materials andthe computers facilities for achieving the present work. Also,the first author would like to thank Prof. G. Furlan (ICTP-LAB) forhis help.

Appendix A

//{N = 1:8000, M = 8000:1:24 � 365}

//Construction of A matrix

3, log(HGi+5)/HGi+4];//Calculating of bi

//Storage of b parameters

//Calculate the mean of b()

//Calculating of a

//Predicted global radiation

//Storage predicted global radiation


Appendix B

The first FFNN-model for predicting the global irradiance(HG) 4 � 17 � 1

w1 = [�466.5180
0.1299 �1.5030 �2.1882 282.8935 �0.1623 2.0694 �2.6548 �313.9163 0.1048 2.0382 0.5977
95.3060
�0.0809 2.7453 �3.6142 285.3924 �0.3351 �1.6886 �1.1237 �347.2994 0.1304 �2.7910 �3.4668 �302.5631 �0.0712 �3.6321 �3.1648
66.2119
0.0716 3.9370 3.7949 �19.1474 0.4413 �4.0525 �2.6769 519.5431 0.2670 �1.6027 1.3791 281.2043 0.1090 0.5316 4.3682 295.5261 �0.0047 �3.6725 �0.8470 �454.8073 �0.0257 1.5313 1.9699 �353.9333 0.0487 �0.1382 2.1362 �356.8868 0.0392 �3.2643 �0.5857
281.2019
0.0312 2.8732 1.4696 347.3199 �0.1108 �3.3905 �3.0608]
w2 = [0.3604 �0.3800 �0.3383 0.0570 �0.0978 0.0803�0.3607 �0.1196 �0.2255 0.0807 0.1522 �0.0591 0.2506�0.3674 �0.0805 0.1428 0.0952]

b1 = [6.0084�1.6024 0.3351 �0.0910 0.7523 5.2452 6.1973�5.4704 1.9132 �2.6913 �4.2519 3.4586 �2.0885 �2.9658�0.0911 0.9326 4.7519]

b2 = �0.1953

The second FFNN-model for predicting the direct irradiance(Hd) 4 � 15 � 1

w1 = [�47.8739
0.2952 4.0454 3.9915 402.6107 �0.0945 3.3591 1.3505 113.5157 �0.2099 �0.5405 �2.2857 257.1350 0.2176 �1.6985 �1.5875 225.9593 �0.2011 2.2661 �3.2425 191.2485 �0.1471 �3.0096 3.2225
79.2271
0.5036 �4.1251 �3.6335 �226.7296 �0.0521 �2.7015 �2.9911 �303.5329 0.3355 �1.5346 �1.5768
298.1089
0.3421 2.9072 �2.4664 �404.8505 0.0508 2.9943 �0.0960 �8.9507 �0.1019 �4.4843 �1.5111
�290.8480
0.1113 �1.0074 4.2094 388.7716 0.0239 0.2818 3.6319 159.7174 0.1657 �0.7176 �3.5730]
w2 = [0.1246 �0.2534 �0.0221 �0.0088 0.2422 0.2030 0.4307�0.1801 0.3463 0.0813 �0.1555 �0.1554 0.2228 �0.0509�0.3473]

b1 = [�0.9002 �6.7323 1.2490 �3.4572 �0.3543 �0.57053.2068 5.4890 0.7419 �2.2441 �1.4077 2.9840 �2.7841�1.7123 2.1468]

b2 = 0.1762

The third FFNN-model for predicting the diffuse irradiance(HD)4 � 12 � 1.

w1 = [�99.4911
0.1542 5.4809 0.3272 125.0491 0.0066 �2.2068 4.4375 165.2288 �0.0046 �2.2414 �3.2262 153.0656 �0.3286 �3.5748 �3.4783 110.6842 �0.0624 3.9612 �3.2507 351.2063 �0.1463 1.2360 �3.6193 �322.6303 0.3344 �0.2299 �1.0472
253.5869
0.1600 1.8669 �0.6431 �348.1763 �0.5170 0.6237 �1.7915 �388.6074 0.0841 1.6667 2.7510
285.8377
0.0101 1.2635 3.4730 �71.2729 0.0471 3.5087 3.9945
w2 = [0.4462 0.6127 �0.3975 �0.0195 0.0081 �0.2792�0.1584 �0.0379 �0.0020 �0.1378 0.2248 0.2126]

b1 = [0.8570 �4.5261 �1.4217 2.1087 �2.2100 �0.68560.1167 �3.2122 2.8080 �2.4206 �2.8582 �6.1573]

b2 = 0.0808

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Documents

An adaptive model for predicting of global, direct and diffuse hourly solar irradiance