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

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  • Renewable Energy 32 (2007) 14141425

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

    r 2006 Elsevier Ltd. All rights reserved.

    ARTICLE IN PRESS

    www.elsevier.com/locate/renene

    0960-1481/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.

    doi:10.1016/j.renene.2006.06.014

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

    E-mail address: kaplanis@teipat.gr (S. Kaplanis).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 [113]. 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) andReceived 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 uctuating 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

    19952000 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.A model to predict expected mean and stochastichourly global solar radiation I(h;nj) values

    S. Kaplanis, E. Kaplani

  • ARTICLE IN PRESS

    Rg the radius of the Earth

    x(yz) the distance the solar beam travels in the atmosphere when suns zenith

    angle is yzx the mean daily distance the solar beam travels in the atmospherensimrea

    1.2.

    efflatprpr

    wh

    bostuhoj the number of a day. Start of numbering is the 1st January.

    INomenclature

    H(nj) the daily global solar radiation for the day njHatm the height of the atmosphereHext(nj) the extra-terrestrial radiation during the day njh the solar hourhss the solar hour at sunsetI(h;nj) the global solar radiation at hour h in a day njIMET(h;nj) the predicted, by METEONORM, global solar radiation at hour h in a

    day njIm,pr(h;nj) the predicted mean global solar radiation at hour h in a day njImes(h;nj) the measured global solar radiation at hour h in a day njpr(h;nj) the predicted global solar radiation at hour h in a day nj

    S. Kaplanis, E. Kaplani / Renewable Energy 32 (2007) 14141425 1415ulating the statistical nature of I(h;nj), is a challenge. A model to predict I(h;nj) close tol values would be useful in problems such as

    effective and reliable sizing of the solar power systems i.e. PV generators [14].management of solar energy sources; e.g. output of the PV systems, as affected by themeteo conditions, in relation to the power loads to be met.

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

    Ih; nj a b cos2ph=24, (1)ere, a and b are constants, which depend on the day nj and the site j.Eq. (1) shows some similarities to the CollaresPereira and Rabl model [11]. However,th models differ in the way a and b parameters are estimated. The I(h;nj) model [15], asdied in detail, overestimates somehow I(h;nj) at the early morning and late afternoonurs, while it underestimates I(h;nj) around the solar noon hours. Although, the predicted

    m

    d the suns declination angleyz the suns zenith anglem(nj) the solar beam attenuation coefcient in a day njsI the standard deviation of I(h;nj)j the latitudeo the hour angleoss the hour angle at sunset

  • (h;nj) values fall, in general, within the range of the standard deviation of the measured,Imes(h;nj) uctuations, 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 uctuations, 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 19952000 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

    emnj xWzemnjxWz;o0, (2)

    where yz is the zenith angle and m(nj) the solar beam attenuation coefcient, determined in

    ARTICLE IN PRESSS. Kaplanis, E. Kaplani / Renewable Energy 32 (2007) 141414251416this approach by

    mnj lnHnjHextnj

    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 tted in afunction, as in Eq. (4), below. The tting results provide the daily global solar radiationFig. 1. The six climatic zones of Greece based on the mean monthly solar radiation.

  • ARTICLE IN PRESSH(nj) where,

    Hnj C1 C2 cos 2p=364nj C3

    , (4)

    C1C3 are given in Table 1, for the 6 climatic zones of Greece. Note that, the correlationcoefcient 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 and

    x(yz;o 0) is this distance at solar noon (o 0). Notice that

    xyz Rg cosyz R2g cos

    2yz R2 R2gq

    (5)

    and

    R Rg Hatm, (6)

    where Rg is the earths radius, Rg 6.35 103 km, and Hatm is the height of theatmosphere, Hatm 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.

    cosyz cosj cosd coso sinj sind. (7)

    For o 0 or equivalently h 12, i.e. at solar noon, Eq. (7) givescosyz;0 cosj cosd sinj sind cosj d (8)

    oryz;0 j d. (9)

    Table 1

    Constants C1C3 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

    S. Kaplanis, E. Kaplani / Renewable Energy 32 (2007) 14141425 1417Notice that, at sunset yz 901. Therefore, Eq. (5) gives for sunsetxyz 90 xoss

    R2 R2g

    q. (10)

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

    xm Pn

    i1xyziN

    . (11)

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

  • wh

    ARTICLE IN PRESSvalues

    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 38.251 andL 21.731). The Imes(h;nj) values were recorded during the period 19952000. 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 uctuations, 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, uctuations are really strong. Therefore, prediction ofstatistically uctuating 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 ShapiroWilk statistic

    for normal distribution. In almost all cases, 19 out of the 20, the Imes(h;nj) values3.satz) is determined by Eqs. (5) and (7). From Eqs. (14) and (15), one may obtain A and Bich are nj and j dependent.

    Comparison of the mean expected values, predicted by this model, with the measuredx(yFor more accurate predictions, xm should be weighted over the hourly global solarintensity. That provides for xm the following formula:

    xm Pn

    i1 xyzIh; nj

    iPni1Ih; nji

    . (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

    Ih; nj A Bemnj xh cos 2ph=24

    emnjxh12

    . (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. Ih; nj 0 at h hswith hss 12 oss=15 and oss a cos tanj tand . 14

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

    Hnj 2AZ oss0

    do 2BZ oss0

    emnjxyz

    emnjxyz;o0cos2ph=24dh. (15)

    ,

    S. Kaplanis, E. Kaplani / Renewable Energy 32 (2007) 141414251418ised the above test for normal distribution. For summer months, the standard

  • ARTICLE IN PRESS0100200300400500600700800900

    0 10 122 144 166 188 20 22 24hour

    Sola

    r Rad

    iatio

    n (W

    h/m2 ) 1995

    19961997199819992000ModelAvg 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 19952000, for Patra, Greece.

    S. Kaplanis, E. Kaplani / Renewable Energy 32 (2007) 14141425 1419deviation of the hourly means of solar radiation, take values which for morning andlate afternoon hours lie around s/I 58%, while for hours around the solar noons/I lies within the range of 23%. 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 the

    measured data (19952000); 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 uctuations, for the 17th January and 17thJuly. The METEONORM results show the presence of expected strong uctuations inI(h;nj), especially, for winter months; see Fig. 4. Obviously, in summer period, uctuationsare neither strong nor frequent; see Fig. 6 for comparison. To smooth the IMET(h;nj)values, one should take the hourly means for72 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.

    0

    200

    400

    600

    800

    1000

    1200 199519961997199819992000ModelAvg Measured

    0 10 122 144 166 188 20 22 24hour

    Sola

    r Rad

    iatio

    n (W

    h/m2 )

    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 19952000, for Patra, Greece.

  • ARTICLE IN PRESS

    0

    100

    200

    300

    400

    500

    600

    Model Avg Measured METEONORM

    0 10 122 144 166 188 20 22 24hour

    Sola

    r Rad

    iatio

    n (W

    h/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 15.01METEONORM 16.01METEONORM 17.01METEONORM 18.01METEONORM 19.01Avg METEONORMAvg MeasuredModel

    0

    100

    200

    300

    400

    500

    600

    0 10 122 144 166 188 20 22 24hour

    Sola

    r Rad

    iatio

    n (W

    h/m2 )

    Fig. 5. A comparison between the predicted IMET(h;nj) values by METEONORM for the days 15th19th 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 24hour

    Sola

    r Rad

    iatio

    n (W

    h/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.

    S. Kaplanis, E. Kaplani / Renewable Energy 32 (2007) 141414251420

  • 4.

    shob

    (a(b

    eahothme

    2.

    3.

    determine the l0sI0 interval of the normal distribution in which the I(h2;nj) value wouldl

    ARTICLE IN PRESSvalues of |l|X3, the permitted range for l 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 l0sI0, as in Eq. (17).

    0 0this model, values within the range l71, with l now forced to an integer value. For0r

    ie. For instance, for nj 17 and a morning hour h2, sI0 25%Im,pr(h2;17). l0 isandomly selected through Gaussian sampling and is permitted to take, according toestimate of the Ipr(h2;nj), taking into account the value of l determined in step 4. In thismeasured Imes(h1;17). The result is compared to sI, as in Eq. (16). Let this deviation dIbe lsI.

    Imesh1; 17 Im;prh1; 17sI

    dIsI

    l. (16)

    5. An attempt is made to predict Ipr(h2;nj), at hour h2 h1+1. The model gives an

    step, the model samples from a Gaussian probability density function in order to4.

    example, at a morning hour h and nj 17, sI 25% Im,pr(h;17).Let us start with hour h1. The program predicts Im,pr(h1;17) and subtracts it from the100% 25% at morning and evening hours, and 12% around the solar noon, forwinter months; and similarly 8% and 3%, respectively, for summer months. Fordeveloped with MATLAB program for the purpose of this research project.Let the solar intensity measurement at hour h1 obtained be Imes(h1;nj) with a standarddeviation sI.sI is pre-determined for the morning period [hsr, hsr+s), afternoon period (hsss, hss],and hours around the solar noon [hsr+s, hsss], with s equal to 3 for winter and 4 forsummer. Let the probability density function be a normal distribution with (s/I)1.rly I(h;nj) measurement, at hour h1. The model to predict I(h;nj) values for the remainingurs, as described in this paper, introduces a stochasti...

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