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Page 1: A model to predict expected mean and stochastic hourly global solar radiation I(h;nj) values

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Renewable Energy 32 (2007) 1414–1425

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A model to predict expected mean and stochastichourly global solar radiation I(h;nj) values

S. Kaplanis�, E. Kaplani

Renewable Energy Laboratory, T.E.I. of Patra, Meg. Alexandrou 1, Patra 26334, Greece

Received 26 July 2005; accepted 27 June 2006

Available online 20 September 2006

Abstract

This paper describes an improved approach for (a) the estimation of the mean expected hourly

global solar radiation I(h;nj), for any hour h of a day nj of the year, at any site, and (b) the estimation

of stochastically fluctuating I(h;nj) values, based on only one morning measurement of a day.

Predicted mean expected values are compared, on one hand with recorded data for the period of

1995–2000 and, on the other, with results obtained by the METEONORM package, for the region of

Patra, Greece. The stochastically predicted values for the 17th January and 17th July are compared

with the recorded data and the corresponding values predicted by the METEONORM package. The

proposed model provides I(h;nj) predictions very close to the measurements and offers itself as a

promising tool both for the on-line daily management of solar power sources and loads, and for a

cost effective PV sizing approach.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Hourly solar radiation; Prediction; On-line management; Statistical simulation; Dynamic PV-sizing

1. Introduction

The prediction of the global solar radiation, I(h;nj), on an hourly, h, basis, for any day,nj, was the target of many attempts [1–13]. Review papers, such as [1,2], outline themethodologies to obtain mean expected I(h;nj) values. A reliable methodology to predictI(h;nj), based on the least available data, taking into account morning measurement(s) and

see front matter r 2006 Elsevier Ltd. All rights reserved.

.renene.2006.06.014

nding author. Tel.: +30 2610 369015; fax: +30 2610 314366.

dress: [email protected] (S. Kaplanis).

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Nomenclature

H(nj) the daily global solar radiation for the day nj

Hatm the height of the atmosphereHext(nj) the extra-terrestrial radiation during the day nj

h the solar hourhss the solar hour at sunsetI(h;nj) the global solar radiation at hour h in a day nj

IMET(h;nj) the predicted, by METEONORM, global solar radiation at hour h in aday nj

Im,pr(h;nj) the predicted mean global solar radiation at hour h in a day nj

Imes(h;nj) the measured global solar radiation at hour h in a day nj

Ipr(h;nj) the predicted global solar radiation at hour h in a day nj

nj the number of a day. Start of numbering is the 1st January.Rg the radius of the Earthx(yz) the distance the solar beam travels in the atmosphere when sun’s zenith

angle is yz

xm the mean daily distance the solar beam travels in the atmosphered the sun’s declination angleyz the sun’s zenith anglem(nj) the solar beam attenuation coefficient in a day nj

sI the standard deviation of I(h;nj)j the latitudeo the hour angleoss the hour angle at sunset

S. Kaplanis, E. Kaplani / Renewable Energy 32 (2007) 1414–1425 1415

simulating the statistical nature of I(h;nj), is a challenge. A model to predict I(h;nj) close toreal values would be useful in problems such as

1.

effective and reliable sizing of the solar power systems i.e. PV generators [14]. 2. management of solar energy sources; e.g. output of the PV systems, as affected by the

meteo conditions, in relation to the power loads to be met.

The above issues drive the research activities towards the development of an improvedeffective methodology to predict the I(h;nj), for any day nj of the year at any site withlatitude j, for a dynamic sizing of solar energy systems. One of those methodologies topredict the mean expected hourly global solar radiation, as proposed by the authors [15],provided a simple approach model based on the function

Iðh; njÞ ¼ aþ b cosð2ph=24Þ, (1)

where, a and b are constants, which depend on the day nj and the site j.Eq. (1) shows some similarities to the Collares–Pereira and Rabl model [11]. However,

both models differ in the way a and b parameters are estimated. The I(h;nj) model [15], asstudied in detail, overestimates somehow I(h;nj) at the early morning and late afternoonhours, while it underestimates I(h;nj) around the solar noon hours. Although, the predicted

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(h;nj) values fall, in general, within the range of the standard deviation of the measured,Imes(h;nj) fluctuations, a more accurate and dynamic model had to be developed, for theneeds of the on-line system management and cost-effective PV-sizing. Such a model shouldhave inbuilt statistical fluctuations, as is the case with the METEONORM package [13],but with a more effective prediction power. A comparison of the predicted results betweenthe METEONORM approach and the one to be proposed in this paper, as compared tomeasured Imes(h;nj) values, during the period 1995–2000 for the region of Patra, Greece,will be presented, hereafter.

2. Model description

Due to the above-mentioned drawback of the simple model of Eq. (1), a correctionfactor is introduced, which takes into account the difference in the air mass, that the globalsolar radiation penetrates during the daytime hours. This new factor, normalized at solarnoon, takes the form

e�mðnj ÞxðWzÞ�e�mðnjÞxðWz;o¼0Þ, (2)

where yz is the zenith angle and m(nj) the solar beam attenuation coefficient, determined inthis approach by

mðnjÞ ¼ � lnHðnjÞ

HextðnjÞ

� ��xm. (3)

Hext is the extra-terrestrial radiation during the day nj [18]. H(nj) is the daily global solarradiation at horizontal. For the case of Greece, it was obtained from a database ofmonthly radiation data [16]. For instance, the 12 monthly global solar energy, E(nj),values, given in the databank for the 6 climatic zones of Greece; see Fig. 1, are fitted in afunction, as in Eq. (4), below. The fitting results provide the daily global solar radiation

Fig. 1. The six climatic zones of Greece based on the mean monthly solar radiation.

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Table 1

Constants C1–C3 to estimate H(nj), from Eq. (4), for the six climatic zones of Greece

Zone 1 2 3 4 5 6

C1 16.607 15.862 15.140 14.892 14.283 13.742

C2 9.731 9.305 10.326 9.650 9.290 8.950

C3 9.367 9.398 9.405 9.394 28.265 78.535

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H(nj) where,

HðnjÞ ¼ C1 þ C2 cos ð2p=364Þnj þ C3

� �, (4)

C1–C3 are given in Table 1, for the 6 climatic zones of Greece. Note that, the correlationcoefficient for all cases is higher than 0.996. Corresponding values for any region may beobtained the same way.

Furthermore, x(yz) is the distance the solar beam travels within the atmosphere andx(yz;o ¼ 0) is this distance at solar noon (o ¼ 0). Notice that

xðyzÞ ¼ �Rg cosðyzÞ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

g cos2ðyzÞ þ ðR

2 � R2gÞ

q(5)

and

R ¼ Rg þHatm, (6)

where Rg is the earth’s radius, Rg ¼ 6.35� 103 km, and Hatm is the height of theatmosphere, Hatm ¼ 2.5 km for the calculations. The results provided by the program didnot change essentially with changing Hatm values. Also, cos(yz) is given by Eq. (7), where dis the solar declination and o the hour angle.

cosðyzÞ ¼ cosðjÞ cosðdÞ cosðoÞ þ sinðjÞ sinðdÞ. (7)

For o ¼ 0 or equivalently h ¼ 12, i.e. at solar noon, Eq. (7) gives

cosðyz;0Þ ¼ cosðjÞ cosðdÞ þ sinðjÞ sinðdÞ ¼ cosðj� dÞ (8)

oryz;0 ¼ j� d. (9)

Notice that, at sunset yz ¼ 901. Therefore, Eq. (5) gives for sunset

xðyz ¼ 90�Þ ¼ xðossÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � R2

g

q. (10)

xm is the mean daily distance the solar beam travels in the atmosphere. It is determined bythe formula

xm ¼

Pni¼1xðyzÞi

N. (11)

i is associated to the hour interval h; also, yz and, therefore, x(yz) are determined byEqs. (7) and (5) respectively.

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For more accurate predictions, xm should be weighted over the hourly global solarintensity. That provides for xm the following formula:

xm ¼

Pni¼1 xðyzÞIðh; njÞ� �

iPni¼1Iðh; njÞi

. (12)

This xm notion is in favour of winter times.Finally, the proposed model, which gives better I(h;nj) prediction results than Eq. (1), for

the mean expected hourly global solar radiation, takes the form

Iðh; njÞ ¼ Aþ Be�mðnj ÞxðhÞ cos 2ph=24

� �e�mðnjÞxðh¼12Þ

. (13)

A and B, in Eq. (13), are determined the same way as in [15], using the boundary conditionshighlighted below. However, the A and B values of Eq. (13), do not take the same values asa and b obtained by the model of Eq. (1).The two boundary conditions are

1.

Iðh; njÞ ¼ 0 at h ¼ hs

with hss ¼ 12þ oss=15� and oss ¼ a cos � tanðjÞ tanðdÞð Þ. ð14Þ

2.

Integration of Eq. (13) over a day, from osr to oss, provides H(nj) at the left side ofEq. (13). Hence, it gives

HðnjÞ ¼ 2A

Z oss

0

doþ 2B

Z oss

0

eð�mðnjÞxðyzÞÞ

eð�mðnjÞxðyz;o¼0ÞÞcosð2ph=24Þdh. (15)

x(yz) is determined by Eqs. (5) and (7). From Eqs. (14) and (15), one may obtain A and B,which are nj and j dependent.

3. Comparison of the mean expected values, predicted by this model, with the measured

values

Predictions of mean expected Im,pr(h;nj) values by this model, (see Eq. (13)), werecompared against the measured Imes(h;nj) values for Patra, Greece (where j ¼ 38.251 andL ¼ 21.731). The Imes(h;nj) values were recorded during the period 1995–2000. Thepredicted mean values, Im,pr(h;nj), are shown in Figs. 2 and 3 for the 17th January and 17thJuly, for Patra, Greece, along with Imes(h;nj).For the validation of the results of this method, several dates were selected along the

year to compare the predicted values and validate the model.The recorded Imes(h;nj) values show statistical fluctuations, whose means are fairly close

to the predicted values, both for January and July (see Figs. 2 and 3). On the other hand,for winter months, e.g. January, fluctuations are really strong. Therefore, prediction ofstatistically fluctuating I(h;nj) values, which may simulate closely the measured data, is arequirement both for the sizing projects and the on-line management.The measured data for Patra city in Greece were tested with the Shapiro–Wilk statistic

for normal distribution. In almost all cases, 19 out of the 20, the Imes(h;nj) valuessatisfied the above test for normal distribution. For summer months, the standard

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0

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1996

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Avg Measured

0 10 122 144 166 188 20 22 24

hour

Sola

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Fig. 3. A comparison of the predicted mean Im,pr(h;198) and the measured Imes(h;198) values for the 17th July,

during the years 1995–2000, for Patra, Greece.

0

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0 10 122 144 166 188 20 22 24

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Avg Measured

Fig. 2. A comparison of the predicted mean Im,pr(h;17) and the measured Imes(h;17) values for the 17th January,

during the years 1995–2000, for Patra, Greece.

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deviation of the hourly means of solar radiation, take values which for morning andlate afternoon hours lie around s/I ¼ 5–8%, while for hours around the solar noons/I lies within the range of 2–3%. On the contrary, for winter months s/I is around25% for morning and late afternoon periods, while s/I is about 12% around the solarnoon.

Figs. 4 and 6 show, respectively, the means of Imes(h;nj), as determined from themeasured data (1995–2000); the Im,pr(h;nj) values, as predicted by the model developed inthis research, and the IMET(h;nj) values, provided by the METEONORM package, withthe inbuilt random generator to provide for fluctuations, for the 17th January and 17thJuly. The METEONORM results show the presence of expected strong fluctuations inI(h;nj), especially, for winter months; see Fig. 4. Obviously, in summer period, fluctuationsare neither strong nor frequent; see Fig. 6 for comparison. To smooth the IMET(h;nj)values, one should take the hourly means for 72 days around the 17th January, as seen inFig. 5. The smoothed METEONORM predicted results are compared, in Fig. 5, with themean predicted values by this method and the mean measured values for the 17th Januaryfor Patra, Greece.

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0

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200

300

400

500

600

Model Avg Measured METEONORM

0 10 122 144 166 188 20 22 24hour

Sola

r R

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(Wh/

m2 )

Fig. 4. Values of the means for Imes(h;17), Im,pr(h;17) and IMET(h;17) for the 17th January, for Patra, Greece.

METEONORM 15.01

METEONORM 16.01

METEONORM 17.01

METEONORM 18.01

METEONORM 19.01

Avg METEONORM

Avg Measured

Model

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0 10 122 144 166 188 20 22 24

hour

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(Wh/

m2 )

Fig. 5. A comparison between the predicted IMET(h;nj) values by METEONORM for the days 15th–19th January

and, consequently, their average values on one hand, and the predicted corresponding values Im,pr(h;nj) by this

model, with reference to the average measured Imes(h;nj) values.

0

200

400

600

800

1000

1200

Model Avg Measured METEONORM

0 10 122 144 166 188 20 22 24

hour

Sola

r R

adia

tion

(Wh/

m2 )

Fig. 6. Values of the means for Imes(h;198), Im,pr(h;198) and IMET(h;198) for the 17th July, for Patra, Greece.

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4. Predicted I(h;nj) values by the stohastic approach of this model

Fluctuations of solar radiation data, like those recorded in Figs. 2 and 3, as well as thoseshown by the METEONORM results, in Fig. 5, affect the sizing process and the resultsobtained. However, the METEONORM package results:

(a)

follow a pattern based on an inbuilt random generator which is not accessed. (b) have the same profile for I(h;nj), without taking into account, as a reference, a morning

I(h;nj) value. Therefore, the prediction is not really a dynamic one.

For an effective load management, on a daily basis, a more accurate and reliable

prediction methodology for I(h;nj), with an inbuilt stochastic contribution for fluctuations,had to be established. The prediction methodology, proposed, takes into account a firstearly I(h;nj) measurement, at hour h1. The model to predict I(h;nj) values for the remaininghours, as described in this paper, introduces a stochastic factor, which takes into accountthe previous hour I(h�1;nj) value. The steps followed to predict I(h;nj) based on a morningmeasurement, Imes(h;nj), are outlined below.

1.

Let Im,pr(h;nj) predicted by Eq. (13). These values are easily generated with routinesdeveloped with MATLAB program for the purpose of this research project.

2.

Let the solar intensity measurement at hour h1 obtained be Imes(h1;nj) with a standarddeviation sI.

3.

sI is pre-determined for the morning period [hsr, hsr+s), afternoon period (hss�s, hss],and hours around the solar noon [hsr+s, hss�s], with s equal to 3 for winter and 4 forsummer. Let the probability density function be a normal distribution with (s/I)100% ¼ 25% at morning and evening hours, and 12% around the solar noon, forwinter months; and similarly 8% and 3%, respectively, for summer months. Forexample, at a morning hour h and nj ¼ 17, sI ¼ 25% Im,pr(h;17).

4.

Let us start with hour h1. The program predicts Im,pr(h1;17) and subtracts it from themeasured Imes(h1;17). The result is compared to sI, as in Eq. (16). Let this deviation dI

be lsI.

Imesðh1; 17Þ � Im;prðh1; 17Þ

sI

¼dI

sI

¼ l. (16)

5.

An attempt is made to predict Ipr(h2;nj), at hour h2 ¼ h1+1. The model gives anestimate of the Ipr(h2;nj), taking into account the value of l determined in step 4. In thisstep, the model samples from a Gaussian probability density function in order todetermine the l0sI

0 interval of the normal distribution in which the I(h2;nj) value wouldlie. For instance, for nj ¼ 17 and a morning hour h2, sI

0 ¼ 25%Im,pr(h2;17). l0 israndomly selected through Gaussian sampling and is permitted to take, according tothis model, values within the range l71, with l now forced to an integer value. Forvalues of |l|X3, the permitted range for l0 would be either [�4, �3] or [3, 4] accordingto the sign of l for all day long. It is important to note that if l is as extreme as this, theIpr(h;nj) lies in the same sI region, for all day long, without jumping to other sI intervals.

6.

The predicted value at hour h2, Ipr(h2;nj), is determined based on the mean expectedIm,pr(h2;17) and the new deviation value l0sI

0, as in Eq. (17).

Iprðh2; njÞ ¼ Im;prðh2; njÞ � l0s0I . (17)

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hour

Sola

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Patra, 17 January

modeldata 2000predicted (8h)

Fig. 7. Mean predicted hourly global solar radiation values, Im,pr(h;17); the measured ones, Imes(h;17) for the 17th

January 2000 and the predicted Ipr(h;17) values, based on a single morning measurement at 8 h, for Patra, Greece.

S. Kaplanis, E. Kaplani / Renewable Energy 32 (2007) 1414–14251422

7.

The model, then, compares the predicted Ipr(h2;nj) value with the mean expectedIm,pr(h2;nj). It repeats steps 4–6, for the hour h3 and so on until the hour hss�1.

Fig. 7 shows an example with the mean expected Im,pr(h;17) values, as determined by theproposed model, see Eq. (13); the Imes(h;17) values, as obtained from the measured data in2000 for the 17th January (nj ¼ 17), and the Ipr(h;17) values, as predicted by the proposedmodel with the inbuilt stochastic generator, taking into account an initial measured valueImes(8;17) at 8 h, for the year 2000 . The corresponding values for the 17th July (nj ¼ 198),based on this model and triggered by an initial measurement taken at 7 h, are shown inFig. 8.The effect of the random sampling, i.e. the sequence of random numbers produced by

the inbuilt random generator, on the predicted values Ipr(h;nj), as estimated by thepreviously described stochastic model, is shown in Figs. 9 and 10. Predicted Ipr(h;17)values, for the whole day, starting with a measured value Imes(8;17), as provided by the2000 data, is displayed in Fig. 9. The corresponding graphs for the 17th July andImes(7;198), are shown in Fig. 10. Due to the small standard deviation values in summermonths, the effect of this random value is expected to be very small. This may be realizedthrough the closeness of the values in series Ipr�1 to �4, in Fig. 10. A much larger effect isseen during the winter months (see curves in Fig. 9), which is, however, within the limits ofone standard deviation.Mean hourly predicted values Im,pr(h;nj) and stochastically fluctuating ones, provided by

this model based on only one-morning measurement, are shown against measured I(h;nj)data for the years 1998–2000 in Fig. 11. It is evident from the comparison betweenpredicted and real measured data that they lie very close to each other at the correspondinghours, especially for the years 1999 and 2000. For the year 1998 the prediction is not verygood at noon hours due to an unexpected and really unusual large and rapid drop of the

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h/m

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Ipr-1Ipr-2Ipr-3Ipr-4

Fig. 9. Four runs (series Ipr�1 to �4) of the daily Ipr(h;17) values based on the measured Imes(8;17) value for the

17th January and the 8 h from the data records of 2000.

0 2 4 6 8 10 12 14 16 18 20 22 240

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m2 )

Patra, 17 July

modeldata 2000predicted (7h)

Fig. 8. Mean predicted hourly global solar radiation values, Im,pr(h;198); the measured ones, Imes(h;198) for the

17th July 2000 and the predicted Ipr(h;198) values, based on a single morning measurement at 7 h, for Patra,

Greece.

0

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hour

Sola

r R

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h/m

2 )

Ipr-1

Ipr-2

Ipr-3

Ipr-4

Fig. 10. Four runs (series Ipr�1 to �4) of the daily Ipr(h;198) values based on the measured Imes(7;198) value for

the 17th July and 7 h from the data records of 2000.

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data 2000

model

predicted (8h;2000)

data 1999

predicted (8h;1999)

data 1998

predicted (8h;1998)

Fig. 11. Mean predicted hourly global solar radiation Im,pr(h;17), and the measured Imes(h;17) values for the 17th

January and the years 1998, 1999, 2000 along with the predicted Ipr(h;17) values based on one morning

measurement.

S. Kaplanis, E. Kaplani / Renewable Energy 32 (2007) 1414–14251424

I(h;nj) around the solar noon. Drops of this magnitude are not considered in this modelwhich is built for rather mild changing weather conditions.

5. Conclusion

The proposed model predicts mean expected Im,pr(h;nj) values for any hour h of a day nj,with a very good accuracy, as the comparisons with the measured Imes(h;nj) values, within aperiod of years (1995–2000), have shown. The predicted Im,pr(h;nj) values lie closer to themeasured means, than the results obtained from Eq. (1) [15]. A program was developed inMATLAB to provide and plot the I(h;nj) solar hourly estimations. The model was checkedwith repeated runs and different sequences of random numbers, as required for theprediction of I(h;nj). The results were within the limits of the standard deviation andintroduce no instability effect, for the daily on-line management and effective PV-sizing.The dynamic part of the proposed model presents a very good behaviour, providing

predicted values Ipr(h;nj), based only on one morning measurement of I(h1;nj). The modelpermits I(h;nj) to take stochastically based values inside the domain Ipr(h;nj)7sI whenpassing from the h1 interval to the h2 one.A similar behaviour, providing random fluctuations, is shown in the predicted results by

METEONORM. However, its inbuilt dynamic predictive power provides the same I(h;nj)profile and does not incorporate real daily variations, as is the case with the proposedmodel of this study, whose predictions are based on a real morning measurement, I(h1;nj).It is important to note that examples of a worse case scenario given by the highly deviant

data of year 2000 from the measured means of 1995–2000, were used to show theprediction power of this methodology.The I(h;nj) fluctuations do affect the sizing of PV generators to meet the loads and

consequently, the load management based on a daily/hourly basis. The effect is reallycritical, if sizing is based on winter conditions and the PV system is a stand alone one [17].The model proposed in this paper, which predicts global solar radiation during a day

based on one morning measurement, has applications in predictive load managementwhere the daily solar radiation profile needs to be determined in advance. The validity ofthe model has been examined for the climate of Greece and also holds for similar climates.Further work is currently being undertaken to gain further reliability in the proposed

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model when applied in northern climates, in which weather conditions change rapidlyduring the day. In this case the prediction of I(h;nj) takes into account two rather than onemorning measurements, at hours h1 and h2, and the I(h;nj) rate of change over h will be theprime criterion.

Acknowledgements

The project for the dynamic prediction of I(h;nj) was funded by the Greek Ministry ofEducation and specifically the Archimides programme. The data of the period 1995–2000were provided by the Hellenic Meteo Organization (HMO).

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