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Applied Energy 87 (2010) 3748–3758
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Applied Energy
journal homepage: www.elsevier .com/ locate/apenergy
Stochastic prediction of hourly global solar radiation for Patra, Greece
S. Kaplanis *, E. KaplaniMechanical Engineering Dept., T.E.I. of Patras, Meg. Alexandrou 1, Patra, Greece
a r t i c l e i n f o
Article history:Received 18 February 2010Received in revised form 18 April 2010Accepted 7 June 2010Available online 14 July 2010
Keywords:Stochastic simulationHourly and daily solar radiation prediction
0306-2619/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.apenergy.2010.06.006
* Corresponding author. Tel.: +30 2610 325102.E-mail address: [email protected] (S. Kaplanis).
a b s t r a c t
This paper describes the stochastic prediction of the hourly profile of the intensity of the global solar radi-ation, I(h; nj) for any day nj at a site. The prediction model requires one, two, or three morning measure-ments of the global solar radiation in a day nj, makes use of a rich data bank of past years recorded data,and provides I(h; nj) values for the rest hours of the day. The model is validated by comparing the I(h; nj)profiles generated for Patra, Greece, with the solar radiation measurements recorded for Winter, Autumnand Spring days, when solar radiation fluctuations often appear to be strong, while also comparing withthe predicted by the METEONORM package I(h; nj) profile. Conclusions are deducted for the predictivepower of the model. The proposed model, which is developed in MATLAB for the purpose of this research,provides I(h; nj) profile predictions very close to the measured values and offers itself as a promising toolfor a predictive on-line daily load management.
� 2010 Elsevier Ltd. All rights reserved.
1. Introduction
The prediction of the global solar radiation, I(h; nj), at any hour,h, in a day, nj, at a site is a necessary process not only for meteoro-logical requirements, but also when investigating the energy per-formance and the sizing of solar energy plants. This was arguedin [1–3]. Several approaches and models were developed in orderto predict daily and hourly solar radiation mean expected values.These models include statistical approaches, outlined in [4–6], sto-chastic approaches in [7–9], neural networks [10–13] and fuzzy lo-gic techniques [14]. However, the outputs predicted by the abovetechniques were not extensively compared to the measured hourlyglobal solar radiation profile around the year. Such a comparison isdeemed necessary when building a reliable solar radiation predic-tion model for the on-line management of intelligent solar build-ings. This is carried out in this paper, where I(h; nj) prediction iscompared to the corresponding measured values and also to theI(h; nj) profile provided by the METEONORM package [15].
The I(h; nj) prediction is certainly an essential part of the design,necessary to build a dynamic model for a stand alone PV plant per-formance. The I(h; nj) fluctuations affect significantly the sizing ofPV generators to meet the loads effectively. The effect is really crit-ical, if sizing is to be based on Winter or even Spring conditions andthe PV system is a stand alone one, as discussed in [1,3]. Conse-quently, I(h; nj) profile prediction may be used for a dynamic on-line load management, which is important for an effective PV plantsizing or for the design of intelligent energy buildings.
ll rights reserved.
This paper presents an effective improvement of the stochasticapproach proposed by the authors in [9] in order to predict I(h; nj).The prediction of I(h; nj) values in a day was based on only onemorning measurement with reference to the mean expected valuescalculated on the basis of a theoretical model proposed by theauthors. The prediction was achieved through a solar radiation sto-chastic generator, which took into account the degree of deviationof the morning solar radiation measurement from the correspond-ing mean expected value. Although that model showed a good pre-dictive power on hourly global solar radiation when compared tothe measured values, it was considered that prediction could bemore successful if it were based on more than one morning mea-surements. According to the approach proposed hereto, the predic-tion of I(h; nj) values is based on up to three morning global solarradiation measurements. In addition, the proposed model is mostlyflexible and uses as a reference either the theoretically producedIm,exp(h; nj) values or the hourly data of the average global solarradiation, Iav(h; nj) profile, from a databank of I(h; nj) measure-ments, as recorded over the past years [16]. This paper carriesout an investigation into the predictive power of the proposedmodel and gives a comparison of the daily prediction profiles gen-erated with one, two and three morning measurements, againstthe measured profiles and the METEONORM package predictionprofiles.
Figs. 1 and 2 show the average global solar radiation and themeasured Imeas(h; nj) values for the 17th January and 16th March,respectively, over the years 1995–2000. Higher dispersion of datais evident during Spring time. Due to such strong fluctuations ob-served, the global solar radiation H(nj) in a day nj, may lie far awayfrom both the average value, Hav(nj), as calculated by the recorded
Nomenclature
H(nj) the daily global solar radiation at horizontal in a day nj
Hav(nj) the average of the daily global solar radiation, H(nj),over a number of years
Hmeas(nj) the measured daily global solar radiation in a day nj
Hm,exp(nj) the expected mean global solar radiation in a day nj
Hpr(nj) the predicted by this model daily global solar radiation,in a day nj
I(h; nj) the global solar radiation intensity at horizontal, at hourh, in a day nj
Iav(h; nj) the average of the measured global solar radiationintensity values at horizontal, at hour h, in a day nj, overa number of years
Im,exp(h; nj) the mean expected global solar radiation intensity athorizontal, at hour h, in a day nj
Imeas(h; nj) the measured global solar radiation intensity at hori-zontal at hour h, in a day nj
Ipr(h; nj) the predicted, by this model, global solar radiationintensity at horizontal, at hour h, in a day nj
h the solar hournj the number of the day. Start of numbering is the 1st of
Januarys.d. the standard deviationrI(h; nj) the standard deviation of I(h; nj), at hour h, for a day nj
rHpr the standard deviation of the Hpr(nj)
S. Kaplanis, E. Kaplani / Applied Energy 87 (2010) 3748–3758 3749
data over the years, and, also, the mean expected daily solar radi-ation Hm,exp(nj), as determined by the model in [9].
2. The model to predict I(h; nj) values from morningmeasurements
For the needs of this project, a databank of recorded I(h; nj) val-ues over the years 1995–2000 for Patra, Greece, is used togetherwith their averages and standard deviation, rI(h; nj), values. Themodel presented by the authors in [9], proposes the use of a meanexpected hourly global solar radiation values Im,exp(h; nj) extractedfrom a theoretical model developed. These values lie very close tothe average hourly data, Iav(h; nj). On the contrary, this paper pro-poses the use of the average values as a reference line for the pre-diction in order to remove the dependency on the theoreticalmodel, in cases, it may not fit the climate characteristics of the site.However, in a case that there are no past years data provided for
Fig. 1. Measured horizontal global solar radiation intensity Imeas(h; nj), for the 17th JaIav(h; nj).
the site, then the theoretical model might be used instead. The lat-ter provides the mean expected, Im,exp(h; nj), values which may beused instead of the average, Iav(h; nj). To develop a dynamic systemfor on-line load management with a cost effectively sized PV sys-tem, a model was developed to predict as close as possible theintensity of the global solar radiation hourly profile, I(h; nj).
The model is based on the assumption that the difference be-tween the measured Imeas(h1; nj) value at a morning hour h1 fromthe Iav(h1; nj) value for the same hour h1 as recorded over the years,follows a Gaussian probability density function, (p.d.f.), [2].
Therefore, the following expression may hold:
t1 ¼Imeasðh1; njÞ � Iavðh1; njÞ
rIðh1 ; njÞð1Þ
where t1 is a measure of the distance between Imeas(h1; nj) from theaverage Iav(h1; nj), in standard deviation units.
nuary, for Patra, Greece, and for the years 1995–2000, and average hourly profile
Fig. 2. Measured horizontal global solar radiation intensity Imeas(h; nj), for the 16th March, for Patra, Greece, for the years 1995–2000, and average hourly profile Iav(h; nj).
3750 S. Kaplanis, E. Kaplani / Applied Energy 87 (2010) 3748–3758
2.1. Mode I: I(h; nj) prediction based on one morning measurement
The solar radiation at hour, h, in a day, nj, is predicted by theexpression:
Iprðh; njÞ ¼ Iavðh; njÞ þ R � rIðh; njÞ ð2Þ
h, takes values from sunrise to sunset, and R is a random numberGaussianly distributed, (0, 1), with a mean equal to 0 and a standarddeviation equal to 1. Accepted values of R, according to this model,should lie within the interval [t1 ± 1], where t1 is determined for theprevious hour h1 by Eq. (1). The predicted value Ipr(h; nj) should bepositive and less than Iext(h; nj) or less than Iav(h; nj) + 3 � rI(h; nj),which is necessary to cut off the Gaussian tail for high values abovethe average.
In order to eliminate cases that might result in biased deviationof the predicted values from the measured profile, Imeas(h; nj), twobias selection rules were introduced.
1. A random subject variation which limits the sampling group inthe region of [t1 ± 1] for solar radiation fluctuations within twoconsecutive hours. Thus, any trip of I(h; nj) values to muchhigher or lower values is avoided, as from past years data anal-ysis it is noticed that such cases may occur very rarely within anhour.
2. The Ipr(h; nj) value should be positive. Hence, negative valuesare zeroed if they occur as a result of the sampling process. Arandom bias selection rule was introduced for cases whereIpr(h; nj) was higher than 3 � rI(h; nj), as it is considered unlikelyfor values higher than Iav(h; nj) + 3 � rI(h; nj), to occur within anhour.
Using these bias selection rules the predicted Ipr(h; nj) profilesare deployed in general much closer to the Imeas(h; nj) than the
Im,exp(h; nj) ones (see Figs. 4–7), or than the Iav(h; nj) ones (see Figs.4–13).
2.2. Mode II: I(h; nj) prediction based on two morning measurements
The Mode II of the proposed model considers for the I(h; nj) pre-diction two morning solar radiation measurements. In addition toMode I, it takes into account the rate of change of the difference[Imeas(h; nj) � Iav(h; nj)], during the period from h1 to h2. Conclu-sively, it includes two stochastic terms, one term as in Eq. (2),which stands for the stochastic fluctuations at hour h3, and a sec-ond term to stand for the rate of change of the I(h; nj), within thetime interval [h1, h2].
Similarly, as in Mode I, the parameter t2 is determined by thefollowing expression:
t2 ¼Imeasðh2; njÞ � Iavðh2; njÞ
rIðh2 ; njÞð3Þ
Finally, the I(h3; nj) value for the next time interval, h3, is pre-dicted by an improved expression:
Iprðh3; njÞ ¼ Iavðh3; njÞ þ R � rIðh3 ; njÞ þ14
� t2 � rIðh2 ; njÞ � t1 � rIðh1 ; njÞ
� �� R1 ð4Þ
rI(h2; nj) and rI(h3; nj) are the s.d. of the measured I(h; nj) values athours h2 and h3, respectively, in the day nj, as obtained from thedatabank.
Furthermore, the model proceeds to predict the Ipr(h4; nj) value.This is based on Ipr(h3; nj), which is the previous hour, h3, predictedvalue and the measured Imeas(h2; nj) one. Further on, it may predictIpr(h5; nj), based on the previously predicted values Ipr(h3; nj) andIpr(h4; nj), and so on. The flowchart of this model is given in Fig. 3.
Fig. 4. Predicted Ipr(h; nj) values by Modes I, II and III, for the 17th January 1996, compared to the measured values Imeas(h; nj), the average Iav(h; nj), the mean expectedIm,exp(h; nj), and the METEONORM package predicted results.
Calculate t1
Calculate t2
Calculate Ipr
(Ipr-Iavg) >3*sd ?
Ipr<0 ?Ipr=0
Generate Gaussian random number R using bias rules
Generate Gaussian random number R1 using bias rules
YES
YES
NO
NO
BEGIN Predict Ipr
END Predict Ipr
START
Enter measurement at 1st hour (Imeas1)
Enter measurement at 2nd hour (Imeas2)
For firsthour+2 to lasthour-1
Predict Ipr based on Imeas1, Imeas2
Imeas1=Imeas2
Imeas2=Ipr
Plot results
END
Enter month, day
Read Database with Iavg and sddata for specified date
Store result
Fig. 3. Flowchart of Ipr(h; nj) prediction model for Mode II.
S. Kaplanis, E. Kaplani / Applied Energy 87 (2010) 3748–3758 3751
Fig. 6. Predicted Ipr(h; nj) values by Modes I, II and III, for the 16th March 1995, compared to the measured values Imeas(h; nj), the average Iav(h; nj), the mean expectedIm,exp(h; nj), and the METEONORM package predicted results.
Fig. 5. Predicted Ipr(h; nj) values by Modes I, II and III, for the 17th January 2000, compared to the measured values Imeas(h; nj), the average Iav(h; nj), the mean expectedIm,exp(h; nj), and the METEONORM package predicted results.
3752 S. Kaplanis, E. Kaplani / Applied Energy 87 (2010) 3748–3758
Going back to Eq. (4), R is a random number which may takevalues within t2 ± 1. This implies that from hour to hour, one
may not expect weather variations larger than ±1 � rI of the p.d.f.of the hourly data for I(h; nj). R1 is randomly distributed according
Fig. 8. Predicted Ipr(h; nj) values by Modes I, II and III, for the 16th February 1997, compared to the measured values Imeas(h; nj), the average Iav(h; nj), and the METEONORMpackage predicted results.
Fig. 7. Predicted Ipr(h; nj) values by Modes I, II and III, for the 16th March 2000, compared to the measured values Imeas(h; nj), the average Iav(h; nj), the mean expectedIm,exp(h; nj), and the METEONORM package predicted results.
S. Kaplanis, E. Kaplani / Applied Energy 87 (2010) 3748–3758 3753
to a Gaussian p.d.f. (0, 1). The term (t2 � rI(h2; nj) � t1 � rI(h1; nj)) � R1stands for the contribution to the I(h; nj) prediction by its rate of
change during the 2 previous hours. This contribution is estimatedby the relative positions of the measured Imeas(h1; nj) and
Fig. 10. Predicted Ipr(h; nj) values by Modes I, II and III, for the 15th May 1996, compared to the measured values Imeas(h; nj), the average Iav(h; nj), and the METEONORMpackage predicted results.
Fig. 9. Predicted Ipr(h; nj) values by Modes I, II and III, for the 15th April 1998, compared to the measured values Imeas(h; nj), the average Iav(h; nj), and the METEONORMpackage predicted results.
3754 S. Kaplanis, E. Kaplani / Applied Energy 87 (2010) 3748–3758
Imeas(h2; nj), with reference to the average values Iav(h1; nj) andIav(h2; nj), respectively.
Formula (4) expresses the superposition principle composed bytwo processes. The first one is the short term stochastic behaviour
Fig. 12. Predicted Ipr(h; nj) values by Modes I, II and III, for the 14th November 1996, compared to the measured values Imeas(h; nj), the average Iav(h; nj), and the METEONORMpackage predicted results.
Fig. 11. Predicted Ipr(h; nj) values by Modes I, II and III, for the 15th October 1997, compared to the measured values Imeas(h; nj), the average Iav(h; nj), and the METEONORMpackage predicted results.
S. Kaplanis, E. Kaplani / Applied Energy 87 (2010) 3748–3758 3755
which provides expected hourly fluctuations based on the past his-tory of the stored data for the hour h of a day nj. The second one
represents the present trend of the hourly I(h; nj) measurementsduring the interval h1–h2. This trend is weighted over a Gaussian
Fig. 13. Predicted Ipr(h; nj) values by Modes I, II and III, for the 10th December 1995, compared to the measured values Imeas(h; nj), the average Iav(h; nj), and the METEONORMpackage predicted results.
3756 S. Kaplanis, E. Kaplani / Applied Energy 87 (2010) 3748–3758
p.d.f., (0, 1), which underlines that the two processes, short andlong term, are independent to each other, and this ensemble repre-sents the real phenomenon. Note that the bias selection rules de-scribed in Mode I are also enforced here.
2.3. Mode III: I(h; nj) prediction based on three morningmeasurements
The Mode III of the proposed model takes into considerationthree morning solar global radiation measurements for the predic-tion of I(h; nj). According to the concept presented earlier, the pre-diction of I(h4; nj) at hour h4 is based on the following formula,which is more advanced than the previous two; see Eqs. (2) and(4).
Iprðh4; njÞ ¼ Iavðh4; njÞ þR �rIðh4 ; njÞ þ14� t3 �rIðh3 ; njÞ � t2 �rIðh2 ; njÞ
� �� R1
þ 19� t3 �rIðh3 ; njÞ �2 � t2 �rIðh2 ; njÞ þ t1 �rIðh1 ; njÞ
� �� R2
ð5Þ
In Eq. (5) appears a third and fourth term in comparison toMode I. The third term in Eq. (5) gives a measure of the contribu-tion of the rate of change of [Imeas(h; nj)-Iav(h; nj)], during the twohours, [h2, h3], prior to the hour, h4, of the prediction. The fourthterm gives the rate of change of the above difference during thethree previous hours. Thus, it provides the contribution to theI(h; nj) prediction of the second derivative of [Imeas(h; nj) �Iav(h; nj)], with respect to h. R1 and R2 are random numbersGaussianly distributed with a mean equal to zero and standarddeviation equal to 1. In order to avoid any bias entering theI(h; nj) prediction, the bias selection rules discussed in Mode I arealso used here.
3. Results
For the validation of the model, the three modes outlined abovewere executed for several cases. Results of the I(h; nj) predictionprofiles are presented for the 17th January, 16th February, 16thMarch, 15th April, 15th May, 15th October, 14th November, and10th December. The results are compared to the correspondingmeasured values, and to the METEONORM package predicted onesfor those dates. These cases in Winter, Autumn and Spring, are cho-sen to show the prediction power of the proposed modes, when thesolar radiation profile usually deviated significantly from the aver-age Iav(h; nj) values.
Figs. 4 and 5 show the predicted I(h; nj) profiles, using Modes I,II and III of the model, for the 17th January, for Patra, Greece for theyears 1996 and 2000, respectively. In this example, the first mea-surement of the global solar radiation was taken at 8 o’clock solartime. Mode I predicts Ipr(h; nj) for hour 9 and then, taking this as areference, predicts Ipr(h; nj) at hour 10, and so on for the rest hoursof the day. Mode II takes into account the measured Imeas(h; nj) val-ues at hours 8 and 9, and then predicts using Eq. (4), the Ipr(h; nj) athour 10. Then, it takes the measured Imeas(h; nj) at hour 9 and thepredicted Ipr(h; nj) at hour 10 as a reference, in order to predict theIpr(h; nj) at hour 11 and continues, taking the predicted Ipr(h; nj) athours 10 and 11 to predict Ipr(h; nj) at hour 12, and so on for therest hours of the day.
Mode III takes into account the measured Imeas(h; nj) at hours 8,9 and 10, and predicts Ipr(h; nj) at hour 11. Then, it takes into ac-count the measured Imeas(h; nj) at hours 9 and 10 and the predictedIpr(h; nj) at hour 11, in order to predict the Ipr(h; nj) at hour 12. Itcontinues, the same way the prediction procedure, taking the mea-sured Imeas(h; nj) at hour 10 and the predicted Ipr(h; nj) at hours 11and 12 as a reference, in order to predict Ipr(h; nj) at hour 13. Then,
S. Kaplanis, E. Kaplani / Applied Energy 87 (2010) 3748–3758 3757
takes the predicted Ipr(h; nj) at hours 11, 12 and 13 as a reference,in order to predict Ipr(h; nj) at hour 14, and so on for the rest hoursof the day.
Modes II and III of this model, were compared against Mode I,on the basis of the predicted I(h; nj) values against the recordedI(h; nj) values. Especially, for the year 2000, as may be observedin Fig. 5, there are large deviations of the global solar radiationfor the 17th January, in contrast to the average values. Both, ModesII and III, predict the daily I(h; nj) profile with a high degree ofeffectiveness. On the other hand, Fig. 4 shows a case where solarradiation profile exhibits a peculiar shape far from the mathemat-ical functions described in [4]. In this case, the solar radiationshows a high peak or equivalently an extreme fluctuation fromnoon onwards. The improvement in the predicted Ipr(h; nj) profilecompared to the measured data, is evident for Mode II and ModeIII, with Mode III performing better.
The same conclusion is drawn when studying the predictionprofiles of Modes I, II, and III, for the 16th March for years 1995and 2000. In this case, large deviations from the average data areobserved, as seen in Figs. 6 and 7, respectively. Here, due to earliersunrise time, hour 7 might be taken as the first hour of measure-ments. Although, the solar radiation profiles measured in March,and especially for years 1995 and 2000, show extremely large fluc-tuations from the average Iavg(h; nj), Mode III gave fairly good pre-dictions. Thus, in cases of high fluctuations, taking 3 morningmeasurements may highly improve the prediction. It is clear thatthe 3 morning measurements might start from hour 8, to 9, to10, instead of hour 7, to 8, to 9. In this case, one might obtain, asit is evident, better prediction.
Figs. 8–13 show the predicted I(h; nj) values by the three modesof the proposed model for the 16th February, 15th April 15th May,15th October, 14th November and 10th December. These are com-pared to the measured Imeas(h; nj) values for the above dates and tothe corresponding values as predicted by the METEONORMpackage.
In cases where the measured Imeas(h; nj) values follow a patternwhich does not lie systematically far from the mean expected val-ues (as these may be determined as in [2,4]), then all three modesof the model provide good estimates. In all cases it is clear thatMode III of the proposed model gives the best prediction. Specifi-cally, in cases when fluctuations are kept within ±1 � rI(h1; nj)and the Imeas(h; nj) values are close to the Im,exp(h; nj) ones, all threemodes outlined above provide very close predictions for I(h; nj).
The effect of the random values, R, R1 and R2, in Eqs. (2), (4), and(5), introduced by the stochastic parts of the model, was investi-gated running the three modes for a number of times for January
Table 1Daily global solar radiation, Hpr(nj), predicted values, in W h/m2, obtained by 10 program exmean expected Hm,exp(nj) for January and March for Patra, Greece.
Program execution Hpr(nj) for 17th January 1995 (W h/m2)
Mode I Mode II M
1st 2468 2661 272nd 2555 2614 273rd 2632 2471 264th 2586 2684 285th 2523 2613 276th 2538 2675 297th 2533 2595 268th 2557 2747 269th 2606 2715 2810th 2533 2739 27
Average: 2553 W h/m2 2651 27rHpr(nj) = 46 W h/m2 82 1Hmeas(nj): 2928 W h/m2
Hm,exp(nj): 2613 W h/m2
and March as indicative months of high fluctuations, while keepingthe same initial conditions in the several program executions. Theresults of the predicted daily Hpr(nj) are shown in Table 1. In fact,there is a dependence, as expected, of Ipr(h; nj) on the R, R1, R2 val-ues. However, this dependence does not affect significantly thepredicted solar radiation daily profile Hpr(nj), as shown in Table 1for the various program executions. The s.d. of the average of thepredicted daily solar radiation, Hpr(nj), increases from Mode I toMode II to Mode III, as it was expected. It is important to note thatthose solar radiation profiles obtained by these stochastic predic-tion modes have equal probability to occur. Hence, they are to beaccepted as profiles of equal probability. A further study showsthat the sum of the predicted Ipr(h; nj) profiles over a day, providesthe Hpr(nj) values, shown in Table 1, which lie within the s.d. oftheir average value.
The entire analysis of I(h,nj) prediction and the repeatability testvalues shown in Table 1 give the following statistical results. ForJanuary it is easily estimated that Hpr(nj) predicted values have rel-ative deviation from the measured values 12.8 ± 1.6% for Mode I,9.5 ± 2.8% for Mode II and 5.9 ± 3.6% for Mode III, while the relativedeviation of the Hm,exp(nj) from the Hmeas(nj) is 10.8%. Similarly, forMarch the predicted H(nj) values show the following relative devi-ations: 8.8 ± 5% for Mode I, 0.2 ± 6% for Mode II, and 1.9 ± 9.3% forMode III, while the relative deviation of the Hm,exp(nj) from theHmeas(nj) is 1.1%. This proves that Modes II and III give better pre-dictions than Mode I.
4. Discussion
The prediction model outlined in this paper is based on theassumption that the relative position of Imeas(h; nj) with respectto Im,exp(h; nj) or with respect to Iav(h; nj) may not change morethan ±1 � rI per hour, for the mild Mediterranean climates. Gener-ally, Mode I provides good estimates of I(h; nj). In order to comparethe three modes, cases were taken for the representative days ofJanuary and March, when strong solar radiation fluctuations oc-curred during the day. In these cases, the I(h; nj) prediction byMode I differs significantly from the measured values. The compar-ison was also extended for the representative days of February,April, May, October, November and December.
A comparison of the results shows that predicted Ipr(h; nj) val-ues by Mode II lie generally closer to the measured values thanfor Mode I, especially in cases where the level of Imeas(h; nj) valuesduring morning hours lies far from the mean expected Im,exp(h; nj)values, and the pattern of the differences [Imeas(h; nj) � Im,exp(h; nj)]undergoes fluctuations within ±1 � rI(h; nj). This improvement is
ecutions, in comparison to the measured daily global solar radiation Hmeas(nj) and the
Hpr(nj) for 16th March 1999 (W h/m2)
ode III Mode I Mode II Mode III
48 2666 2432 268613 2844 2628 247866 2934 2507 311409 2801 2632 301898 2863 2774 256728 2887 2760 265244 2665 2297 229007 2741 2730 252293 3071 2562 266235 2966 2730 2634
54 2843 2605 266204 130 157 243
Hmeas(nj): 2612 W h/m2
Hm,exp(nj): 2640 W h/m2
3758 S. Kaplanis, E. Kaplani / Applied Energy 87 (2010) 3748–3758
brought in by the third factor in Eq. (4). In general, Mode II pro-vides to a good degree the shape or the I(h; nj) profile in the major-ity of the cases examined. It is only the high peaked profile shownin Fig. 4 and the case in Fig. 6 where morning fluctuations are largeand far away from the average values, which are not predicted at agood estimate by this mode.
Mode III of this model gives much better results compared tothe other modes, and provides good profiles even in cases whereI(h; nj) shows higher degree of fluctuations, as shown for themonths November, December, January, February and March inFigs. 4–8 and Figs. 12 and 13. The improvement is attributed tothe fourth term in Eq. (5), which takes into account the secondderivative of the difference [Imeas(hi; nj) � Iavg(hi; nj)] for the previ-ous hours.
For Summer time, the effect of fluctuations, at least for the mildclimates, is diminished and, hence, all three modes provide verygood results.
Furthermore, it is important to underline that Hpr(nj), as pro-vided by Modes II and III, differs from the measured H(nj) withinabout 5% in most cases. This is very important, as the averageHav(nj) or the mean expected Hm,exp(nj) values for Winter or Springmay differ considerably from the measured H(nj) values, in manycases. This highlights the power of this model for on-line manage-ment of PV & Loads systems, in contrast to the PV sizing method-ology based on Hav(nj), or Hm,exp(nj).
It must be noted that the prediction model developed may useeither the mean expected Im,exp(h; nj) values or their averagesIav(h; nj) as taken from the data bank. In cases where reliable datafrom the past years are available, the average Iav(h; nj) values maybe used. Otherwise, the mean expected values obtained from themodel in [9] may be used instead.
For cases when past data are limited, then in order to build theHav(nj) we may take I(h; nj) values from the days at the neighbour-hood of nj, that is nj + 1 and nj � 1.
The model proposed here can be also used to predict hourlyprofiles for any period of the day, the prediction based on the pre-vious measured values. This makes the program adjustable, flexi-ble, and very useful for the daily management of the PV systembased on the power delivered and the loads demand.
5. Conclusions
A program was developed in MATLAB to simulate solar radia-tion fluctuations and implement a stochastic model generatingthe Ipr(h; nj) hourly profile of the global solar radiation to occurin a day, based on corresponding morning measurements. The pre-dicted profiles were compared to the measured values and ModesII and III gave predictions closer to the measurements than Mode I.Especially for months where high fluctuations occurred, Mode IIIgave the best results. The proposed model was found to providereliable results for the I(h; nj) profile, for any execution of theprogram.
The predicted results for the hourly global solar radiation forWinter, Autumn and Spring seasons were also compared to the re-sults provided by the METEONORM package, and in all cases theproposed model, and especially Modes II and III, gave significantlybetter predictions, even for cases where I(h; nj) deviates far fromthe average values.
It was understood from the analysis and the prediction resultsfor the entire year, that the predictive power of the model in-creased from Mode I, to Mode II, and further to Mode III. This isattributed to its characteristic by which it takes into account boththe stochastic fluctuations based on the past records for each day(see Mode I) and the contribution of the trend of the evolution ofI(h; nj) during the first early morning hours (see Modes II and III).
As it is evident from the analysis of the results the model is adynamic one and may respond from the early morning hours toprovide the I(h; nj) values for the rest hours of the day with a verygood predictive power. The analysis shows that the prediction ofthe daily solar radiation is much more pragmatic, i.e. close to themeasured daily solar radiation, than the average Hav(nj) or themean expected Hm,exp(nj) are.
Finally, the proposed model is not only suitable for Patra,Greece, but is also reliable for any climate were the fluctuation pro-file of the solar radiation varies as much as ±1 � rI(h; nj) per hour.
Acknowledgements
The Project was co-funded by the European Social Fund and theNational Resources – (EPEAEK II) – Programme ARCHIMIDIS I. Thedata of the period 1995–2000 were provided by the Hellenic MeteoOrganization (E.M.Y.).
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