Comparison of diffuse/global ratios calculated from one-minute, hourly and daily solar radiation data

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  • Solar Energy Vol. 32, No. 3. pp. 417~t23. 1984 0038-092X/84 $3.00 + .00 Printed in Great Britain, Pergamon Press Lid.

    COMPARISON OF DIFFUSE/GLOBAL RATIOS CALCULATED FROM ONE-MINUTE, HOURLY AND

    DAILY SOLAR RADIATION DATAf

    P. J. SMIETANA, JR., R. G. FLOCCHINI, R. L. KENNEDY and J. L. HATFIELD Biosearch, Inc., 2980 Kerner Boulevard, San Rafael, CA 94901, U.S.A.

    (Received 9 October 1981; revision received 30 December 1982; accepted 21 April 1983)

    Abstract--One-minute values of direct, diffuse and global radiation have been continuously collected at Davis, California (38.5N, 121.1W) since 1 January, 1979. These datasets are quality controlled to insure the most accurate and reliable data possible. Analysis of one-minute data has provided an opportunity to evaluate some of the bias that may be inherent in statistical representation of solar radiation data. A simple mean and standard deviation do not adequately describe the variation in the data and we show that a more representative treatment includes the box and whisker analysis. In this the mean, median, first and third quartiles, and the maximum and minimum ranges are presented. It is possible to compute the variability between days more completely with this technique while the means may be very close. This has application to evaluation of solar collectors as a better method of evaluating theire efficiency. This is applied to diffuse/global ratios which show a seasonal dependence although some clear winter days have ratios close to clear summer values; however, the first and third quartile and median distinctly separate these days. Analysis of solar radiation data should be conducted with caution as shown by these results.

    A simple model is proposed to compute hourly global values from the integrated daily total. Comparisons of calculated with measured hourly values indicated less than a 10 per cent error between 0700 to 1600 with the maximum value being slightly underestimated. This procedure allows one to evaluate solar collectors with only daily values and presents a method for thoroughly evaluating our solar resources.

    I. INTRODUCTION

    Throughout the world solar flux measurements have been made in order to provide scientists and engineers with local solar data. These measurements are usually integrated hourly, daily or monthly values and from these data inferences pertaining to longer and sometimes shorter periods are made [I-14]. Many models have been developed to predict global, diffuse and direct solar radiation on a horizontal and inclined surface with various solar and meteorological components as in- dependent parameters [I-14]. These models have used simple linear regressions [9-14], multiple regressions [6, 7] and statistical Markovian [4, 7] techniques to arrive at general empirical relationships which have a variability of predictive use. However, it has been necessary to use mean hourly, daily and monthly values due to an insufficient number of reliable measurements. The datasets may have suffered from instruments which were not recalibrated at regular intervals, varying samp- ling time schedules and insufficient quality control. The latter involves, for example, making global and diffuse solar radiation measurements but not making simul- taneously corresponding direct or beam solar radiation measurements. Solar shortwave and longwave radiation measurements and meteorological parameters at the University of California at Davis, as part of a Solar Energy Research and Meteorological Training Site (SEMRTS) grant, have been recorded on a minute inter- val since January 1979. Davis is one of eight sites recording similar one-minute measurements and then archiving these measurements at the Solar Energy Research Institute (SERI) in Golden, Colorado [21]. These sites are located throughout the United States in

    fContribution from the California Agricultural Experiment Station. Research supported by the Department of Energy, Grant DE-FG03-79ET20187.

    areas which have different solar radiation and atmos- pheric conditions thus providing an extensive geo- graphical database. A quality control procedure, which is applied to each one-minute measurement, and an in- strument calibration procedure were established for the eight sites thus providing a reliable data base. A com- plete description of the instrumentation, data collection procedures and data quality control at Davis is given in Hatfield et a1.[15]. Tables of reduced one-minute measurements to hourly and daily integrated values and to monthly summaries for 1979 and 1980 are given in Hatfield et aL[16].

    The purpose of this paper is to provide a statistical comparison of one-minute values and ratios with hourly and daily integrated means and ratios. The results from the comparisons should provide a reliability or confidence level in the use of earlier recorded hourly or daily values. A method of generating hourly or smaller time divisions from a given integrated daily value is also explored. The extent of cloudiness on a day, which can be obtained from the presented statistics, is discussed.

    The amount of solar radiation approximates a sine curve [19] during the day suggesting that simple measurements throughout the day and calculation of an average will not adequately provide a complete and accurate description of the radiation flux for the day. This implies that averaging measurements taken within an hour will also not be valid since they would be biased toward the end of the hour in the morning and toward the beginning of the hour in the afternoon. The extent of error is analyzed and methods of presenting statistical analysis is part of the objectives of this paper.

    2. EXPERIMENTAL PROCEDURES

    Solar radiation and meteorological data are collected on a one-minute basis at Davis, California (38.5N, 121.1VO. Data collected include direct, diffuse and glo-

    417

  • 418

    bal radiation with Eppley Normal Incidence Pyr- heliometer and Precision Spectral Pyranometers, res- pectively. Diffuse radiation is measured by means of a shading disk which continually provides a blockage of the direct beam from the pyranometer.

    The hourly and daily values are computed by per- forming a trapazoidal numerical integration on the ori- ginal one-minute measurements. This method was chosen because of the sinusoidal nature of the data. The general trapazoidal integral equation is

    Area=(Xo+X~o)12+(X ,+X2+. . . + X~9) (1)

    where Xo and X,o are the Values for the starting minutes of two consecutive hours and Xt-Xs9 are the intervening 59 rain. The daily integral is calculated by summing of the individual hourly integrals. The programs for these conversions are given in Hatfield et al. [15].

    Statistical calculations were performed using BMDP (1979) computer programs (17). Box and whisker diagrams (Tukey [20]) are used to graphically present a compact statistical summary of one-minute measure- ments.

    a. RESELTS AND mSCUSSION

    3.1 Statistical comparisons Daily global radiation values (calculated from the one-

    minute measurements) for 1980 are illustrated in Fig. 1. The envelope of this curve follows the predicted extra- terrestrial radiation curve but shows considerable day-to- day variation caused by cloudiness, changing air mass and aerosols. Tables 1-3 present a summary of yearly per cent sunshine statistics for years 1979-1981. These values were computed using the daily total number of minutes of sunshine as measured by a Campbell-Stokes Sunshine Recorder and dividing by the number of minutes between sunrise and sunset. In Table 1 it should be noted that the median is approximately 18 percentage points above the mean which indicates that Davis has predominantly sunny days. Table 2 presents the frequency of percent sunshine values in range intervals of 5 per cent. This table shows that approximately 10 per cent of the days have less than 5 per cent sunshine, approximately 20 per cent of the days have between 5 to 70 per cent sunshine and the remaining 70 per cent have

    %

    19::-:0 JI_IL I All BAT

    Fig, 1. Daily global shortwave radiation for 1980 from one- minute measurements and predicted extraterrestrial solar flux on

    a horizontal surface.

    P. J. SMIETANA, JR. et al.

    Table 1. Yearly per cent sunshine statistics for Davis, California

    1979 1980 1981

    Maximum 99.00 99.50 98.10

    3rd Quartile 95.30 94.40 92.90

    Median 88.80 88.20 83.10

    Mean 70.30 70.60 65.90

    Ist Quartile 51.10 52.90 38.00

    Minimum .00 .00 .00 i

    Table 2. Yearly frequency distribution of daily per cent sunshine values in five per cent range intervals for Davis, California

    r

    Interval Frequency Number Percent 1979 1980 1981

    I 0 - 5 31 32 a3 2 5 - I0 5 5 I0 3 10 - 15 4 5 3 4 15 - 20 6 3 4 5 2O - 25 4 6 7 6 2 5 - 3 O 3 3 6

    7 3 0 - 3 5 3 8 7

    8 35 - 40 6 3 6 9 40 - 45 3 10 4 I0 45 - 5O 7 6 9 11 50 - 55 8 5 4 12 55 - 60 4 8 7 13 60 - 65 5 10 2 14 65 - 70 5 3 8 15 70 - 75 17 9 13 16 75 - 80 13 18 19 17 80 - 85 17 17 30 18 85 - 90 16 34 27 19 90 - 95 62 82 85 20 95 -100 81 7O 45

    Number of Days 300 337 339

    greater than 70 per cent sunshine with it being heavly skewed to the 90 per cent value. Figure 2 pictorially illustrates the 1980 daily per cent sunshine variability and similar figures were obtained for 1979 and 1981. As shown in Fig. 2 cloudy days (less than 70 per cent sunshine) exist predominantly in the winter months, but they also occur in the summer months.

    In Table 3 the mean per cent sunshine values were determined for each hour of the day. It can be seen that the values cluster about the yearly mean; however, the skewed data in Table 2 suggest that the median value would have been a better measurement statistic. These

    Table 3. Hourly mean per cent sunshine values determined from Campbell-Stokes sunshine recorder data for Davis, California 197~ ~9 6~ 6~ 70 7~ 72 73 7~ 76 7~ 7~ rO 64 63

    1~aO 7J 61 71 77 rB 78 T8 79 79 ~9 79 r3 68 ~8

    1981 69 ~9 66 72 7~ 7~ 75 76 76 76 7~ 68 59 56

  • Comparison of diffuse/global ratios 419

    i F:

    . . . . . . I '1" ' ! j ;i

    19: . : :0 . h J l i~tn I)~t~

    Fig. 2. Daily per cent sunshine values for 1980 determined from Campbell-Stokes sunshine recorder data.

    4-,

    L

    00,0

    ~e.O

    ~ T T ~

    T J - L

    T i T

    - J- T

    J U L 1 , 1 9 7 9 H o u r

    Fig. 4. Direct solar radiation on a horizontal surface for ], July 1980 from one-minute measurements.

    means also suggest little variability since the bimodal distribution has been averaged out. They could however be used as lower bound estimates of per cent sunshine or direct radiation.

    As an example of the daily variability the global values for July, 1980 are shown in Fig. 3 where the data are from one-minute measurements. The mean values for each day have approximately the same value typical for July; however, the maximum and minimum ranges exhi- bit large diurnal variations with some variation between days for the maximum values. The major part of the daily variation is explained by the diurnal cycle as illus- trated in the plot of hourly mean, maximum and mini- mum values for the direct component of global solar radiation which were calculated from one-minute measurements for 1 July, 1980 (Fig. 4). It is shown that maximum and minimum values should be reported along with mean values when the effect of the diurnal cycle is present. This also applies to monthly mean values since the beginning and ending predicted extraterrestrial solar radiation values for a month vary from as little as 290 KJ/M 2 (June) to as much as 7200 K JIM 2 (March).

    Since all solar radiation measurements fluctuate due to a variety of changes, e.g. intermittent clouds, diurnal and seasonal sun elevation changes, these could cause the ratios of any radiation measurements to also fluctuate with any of these temporal variations. This suggests that traditional methods of analysis may not be adequate to describe the daily variations in relationships between parameters. One of the ratios most frequently used to describe the solar radiation components is the diffuse/global ratio. During the winter with numerous cloudy and foggy days, ratios near 1.0 are evident (Fig. 5)

    and during the summer with typical clear days, ratios near 0.15 are evident. However, it should be noted that on clear winter days in Davis ratios near 0.15 are also present. Ratios greater than 1.0 are due to inclusion of measurements at low sun elevation.

    Ratios of diffuse to global solar radiation remove the diurnal variation in the data and thus calculating averages and standard deviations would seem applicable descrip- tors. However, a simple mean and standard deviation may not be adequate to completely describe the variation in these data. It has been found that calculating quartiles and plotting them along with means, medians, maximums and minimums provides a more meaningful description of data calculated from diffuse/global ratios [20]. This representation utilizes the box and whisker diagram which not only presents the maximum and minimum range bars (whiskers), but one can quickly see the clus- tering of measurements about the mean and median by the size of the box determined by the 1st and 3rd quartiles. The relative positions of the mean and median are also not static since the mean is sensitive to ranges during the day indicating its value alone could be quite misleading. As an illustration of this technique, Box and Whisker diagrams are presented for January and June in 1980 (Figs. 6.1 and 6.2) where one-minute data with the sun elevation less than 5 degrees were excluded. A complete set of these graphs for all months in 1980 is given in Hatfield et al. [15]. The winter months tend to produce more days with ratios near one, but on clear winter days the ratios are nearly the same as those during the summer months; however, on these clear days the mean and median still exhibit a wide separation as compared to clear days in the summer. Another pre-

    T

    lllliilliiiiili ! J U L Y 198~3 Julian Bays

    Fig. 3. Daily global shortwave radiation for July 1980 from one-minute measurements.

    JULIAN DAY

    Fig. 5. Daily diffuse/global ratios for 1980 from one-minute measurements.

  • 420 P.J. SMIETANA, JR. et al.

    D '~I IIII T'i , I JAl l 19::':0 DAYS

    Fig. 6. I.

    30 31

    - , - - - _

    J l J I l I '7:~0 BA'/S

    Fig. 6.2. Fig. 6. Box and Whisker diagrams of diffuse/global radiation for January and June of 1980 calculated from one-minute measure-

    ments.

    dominant feature is that during the summer months the minimum ratios are close to the mean and median values as shown for June (Fig. 6.2).

    Even though two days (e.g. 2 January and 10 January or 20 January and 30 January) have similar maximums and minimums their 50 percent clustering, means and

    c, .d t~

    D u .

    il;,

    i~urt 7 . 1

    l -

    \ oo

    (x

    i ,

    I i : re>

    JQHI.IQRY 1 6 , 1 9 8 0 HOUR

    0.0

    ~4

    F i ~ r e 7.2

    Fig, 7. Time of day plots using one-minute measurements on 16, January 1980: (1)Diffuse/global ratios calculated from one- minute measurements and hourly integrated values. Curves labeled 1-3 are the same hourly ratios plotted at the beginning, midpoint and end of the hour respectively; (2) Global and diffuse

    solar radiation and diffuse/global ratios.

    medians indicate quite different atmospheric conditions (Fig. 6.1). In addition by using only the mean and extremes, days with the same 50 per cent clustering and median value could be interpreted as having different relative amounts of diffuse and global radiation because of different extremes, but basically they would be the same (Fig. 6.2 for 8 June and 25 June). These differences are due to large amounts of diffuse relative to the global early in the morning.

    A plot of diffuse/global ratios for a very foggy 16 January, 1980 is shown in Fig. 7.1. In the morning the values cluster near 1.0 until the fog begins to burn off at approximately 1330. At this time the ratios become smaller and the magnitude of the fluctuations increase due to intermittent cloudiness. This variability is shown in the measured values of global and diffuse solar radia- tion and diffuse/global ratios for each minute of the day (Fig. 7.2). Near sunset the fog begins to reform and the fluctuations decrease.

    Even though the diffuse/global ratio removes the diurnal variation, Figs. 7 and 8 show a foggy and clear sky diurnal variation respectively which is primarily due to the increase in optical air mass in the morning and evening over midday values. A correlation between diffuse/global ratios and air mass values can be seen by comparing time of day plots of diffuse/global ratios with a time of day plot of air mass for a Rayleigh atmosphere by Boer [18]. The plot of air mass is symmetric with respect to solar noon; however, the Davis data indicates an asymmetric air mass distribution with respect to solar noon. The slightly smaller evening increase is due to a higher inversion layer thus even though the total number of photochemical particulates increased during the day the concentration is less than in the early morning hours. This effect was not always present and analysis indicated that local atmospheric heating did not occur when clouds or fog were present.

    The same diffuse/global ratios calculated from hourly integrated diffuse and global values are plotted at the beginning, midpoint and end of the hour respectively (Fig. 8.1). The plot at the midpoint gives the best ap- proximation for the one-minute ratios and very different results are obtained for the plots at the beginning and the end of the hour. This is important since the hourly integrals do not contain time explicitly. It should also be noticed how hourly ratios remove the extensive one- minute ratio fluctuations. This is important in the

    Z $

    t

    o< . . . . . . . I ~ ~2 o

    r -

    \ , " " " ] C ,

    JULY :31 ~1979 HOURS

    Fig. 8. Plot of clear sky one-minute diffuse/global ratios for 31 July 1980 and residual sum of squares (R.S.S. 104) determined

    from linear regression analysis.

  • Comparison of diffuse/global ratios

    presentation of data which is representative of the true value. , .s - .

    A series of linear regressions were performed on clear sky diffuse/global ratios for 31 July, 1979 (Fig. 8) over

    ,-i

    periods of one hour. Given the brief time period, no ,-~ attempt was made to fit a polynomial, or to linearize the > dependent variable. The variance attributable to the "~ o curvature of the diurnal function relative to a one-hour period was found to be small compared to that attribut- ~, able to the linear term and error. All hour subsets con-

    e .

    tained sixty observations, beginning with the first com- plete hour after sunrise, and ending with the last com- plete hour before sunset. Figure 8 contains the residual sum of squares which has a diurnal curvature similar to = o the plot of ratios. Using linear analysis on hourly data, which has a curvature due to the time of day, a diurnal variance results and thus requires that any statistical analysis be used with caution. This caution also extends to the diffuse/global ratios.

    It was also found that the sun elevation at which the diffuse/global ratio reaches its clear sky value of ap- t .s proximately 0.15 is seasonally dependent which is con- sistant with the plots of air mass [18]. The air mass diurnal 0.l variability introduced into diffuse/global ratios must be ,~ taken into consideration when performing statistical cal-

    t . e culations. Calculation of standard deviations for these ,v ratios must be used with caution as described for the diurnal measurements. ~,

    The calculation of daily mean ratios by averaging individual hourly and minute ratios are given in Figs. 9.1 ~= o.s and 9.2. The daily ratios were calculated by dividing the >, integrated daily total diffuse value by the integrated daily total global value. Values of each ratio range from 0.1 to '~ 1.0 with predominant some clustering in the 0.1--0.3 range and around 1.0. The ratios calculated from daily total values are greater than those from the hourly in- tegrated values and the one-minute measurements (Fig. 9). These data suggest that care be exercised in the analysis of ratios of diffuse to global radiation because of the biased results from the traditional use of the mean t.s and standard deviation. Detailed one-minute data provide a unique opportunity to evaluate the variability in radia- tion measurements and statistical analysis of the data. With a large data base of daily total global or diffuse values a limited evaluation of solar collectors or tilted t.o surface models can be made.

    3.2 Estimation of hourly global from daily total global radiation

    A simple model to determine hourly integrated values from integrated total daily global solar radiation is presented and compared to measured values. Hirschman [19] showed that if the maximum global irradiance (Imp) at solar noon and if the time of sunrise and sunset are known then a sinusoidal equation approximates global irradiance (I0 at any time t after sunrise (eqn 2).

    [t = Imax sin (~rt/n) (2)

    where: n = day length in hours (or minutes); t = hours (or minutes) after sunrise

    1 . 0

    . ~ ,~

    :7: O.O~ I .O / ' ' ' ' ' 0 Ic.. '

    Figure 9.1

    i . . . . / ,

    / /

    /' / "

    118 . . . .

    One-Minute Measurements

    421

    o l

    : >

    " o

    ~J

    { : n o

    e -

    r ~

    Figure 9.2

    i i

    , / /

    ;2 ~. ",

    , , C ' : f

    ~ l . e l / . . . . . ! . . . . I . . . . e .e e . s I . e I . 5

    Hourly Integrated Values

    e.S

    Figure 9.3

    . . . . t . . . . I . . . . / /

    . / . /

    / / :

    /.:,/'?',

    ~; ' : , , ,'.~:i,".

    . . . . e l . s e.e I . . . . ~ . 0 t . t .S

    One-Minute Measurements

    Fig. 9. Scatter p.ot of daily diffuse/global radiation ratios cal- culated for 1980 from: (l) Hourly integrated values (Y-axis) and one-minute measurements (X-axis); (2) Daily (Y-axis) and hourly integrated values (X-axis; (3) Dally integrated values (Y-axis)

    and one-minute measurements (X-axis).

  • 422 P. J. SMIETANA, Jr. et al.

    This equation can be integrated to calculate the total daily flux

    (3) /daily = f I . ,~ sin (Trt/n) dt

    = Im,x 2nHr.

    If however the total integrated daily flux is measured then the maximum irradiance /max can be determined from eqn (3).

    However, instead of integrating eqn (3) for an entire day it can be integrated to obtain eqn (4) for any time interval [h, t2] between sunrise and sunset.

    I2 - l , = (,/Tr) Im~x /COS (Trtl/n) - COS ('rrt2/n)] (4)

    After replacing Imp, in eqn (4) by eqn (3) and substituting the measured daily flux for Ida/y, flux values were cal- culated and compared with actual measurements. A representative plot of calculated and measured global shortwave radiation for days with greater than 90 (5 August, 1980 (218)), approximately 50 (April 5, 1980 (112)) and approximately 0 (7 January, 1980 (7)) per cent sunshine is given in Fig. 10. The sine curve can be seen to crudely approximate the sinusoidal global radiation curve indicating that a decomposition of a total daily integrated value into smaller time increments is possible. The maximum irradiance is underestimated for clear days and is closely approximated on cloudy days. Days with intermittent cloudiness are underestimated and smoothed; however, the areas under each curve are the same. This suggests that this technique could be used for days without minimal atmospheric fluctuation or could be utilized on cloudy days if a rough approximation of each hour is needed.

    The mean per cent difference between measured and calculated hourly global values were determined for years 1979-1980 by using the above method of decom- posing daily global integrated solar radiation values into hourly integrated values. Table 4 shows that the best agreement occurs between 0700 and 1600 which is when the sun elevation is above 10 degrees and the Ith hour represents the next 60 rain; e.g. hour 11 represents lh00 to 11:59. The worst agreement is during low sun ele-

    I - i L t ~ \

    / / / \

    HE~:m~F[D

    C ~LCmt~TE~

    1 9 : - - : 0 H O U R

    ,~o0,

    Ld

    C_i

    . J ,::= ,ooo.

    --J

    o.

    Fig. 10. Comparison of calculated hourly vs measured hourly integrated values for :(1) 5 August, 1980 (218) greater than 90 per cent sunshine; (2) 21 April, 1980 (112) approximately 50 per cent sunshine; (3) 7 January, 1980 (7) approximately 0 per cent sun-

    shine.

    Table 4. Hourly mean per cent differences between measured (Gin) and calculated (Go) global solar radiation per cent difference =(Gm-Gc)*IOO/G~ where the per cent for the lth hour represents the next 60 minutes. For example, hour 11 represents

    11:00-11:59

    . o . r o r t h e oax

    yea r Mean 5 7 8 9 10 11 12 $3 1'~ 15 16 17 IB

    ~979 -7 -2~ .1'~ -'~ 3 5 6 B 8 1 . 3 . 1 6 -23 -'~5

    1980 -7 .27 . 1 8 .7 .Z ] ~ 7 6 2 . ~ -18 . 2 ~ . ~

    19a~ -9 -3o -19 -9 ~ 5 6 7 5 1 .7 - z2 -30 -56

    19az - i 0 . . . . . . . 7 0 5 7 8 6 la -3 . i ~ -a6 - - -

    Table 5. Per cent of values in each per cent rangegroup forTable4

    Year Total (-I00,-50) (-50,-25) (-25,0) (0,25) (25,50) (50,100)

    1979 ~090 12$ 10% 28% 42% 4% 4%

    198o " o 7 8 I I ~ II% 28% ~2% 5$ 2%

    1981 4082 125 10% 29% 43% q% 2%

    1982 2035 125 11% 29~ 44% 3% I%

    vation as expected since the sine curve approximation does not have a component which takes increased air mass into consideration. During the midday hours the approximation is not as good as during the ascent and descent regions which correlates with the under- estimation of maximum irradiance. Table 5 shows that over 70 per cent of the per cent differences were in the -25 to + 25 range which includes the range of hours normally considered for solar energy utilization. The results in Table 4 therefore suggest that less than a 10 per cent error would be obtained in using this decomposition procedure. We feel that the utilization of this approach provides a method for the evaluation of solar systems which require hourly input but where only daily totals are available.

    CONCLUSIONS

    Simple mean and standard deviations for solar radia- tion are not adequate to describe the variability in the data. This is particularly evident in diffuse/global ratios which, when calculated from one-minute values, exhibit large variations during a day for partly cloudy skies but small variations on extremely clear or foggy days. The use of a box and whisker diagram provides a better representation of the variability contained in a data as well as the bias which is introduced by an examination of only integrated hourly values. This bias extends to in- tegrated daily values which underestimate the true ratios of diffuse/global ratios. The one-minute data provide a unique opportunity to evaluate the response of alternate energy systems without the inherent bias of the hourly or daily values.

    A simple model was constructed to estimate the hourly values from an integrated daily total and the daylength. This approach provides a method for prediction of hourly values. In general the maximum value is under- estimated and on partly cloudy days the reconstructed values are smoothed through the large varying values. A

  • Comparison of diffuse/global ratios 423

    comparison to measured hourly values indicated less than a ten per cent error be tween 0700 and 1600 with the best agreement taking place during the mid-morning and mid-af ternoon hours which is when the sun elevation gradient is largest. Therefore , these techniques provide a method of obtaining hourly values f rom measured daily global solar radiation data to within 10 per cent. These techniques should be incorporated into all programs which need an unbiased evaluat ion of passive, act ive and photovol taic system.

    REFERENCES 1. D. V. Hoyt, A model for the calculation of solar global

    insolation. Solar Energy 21,.27-35 (1978). 2. J.W. Bugler, The determination of hourly insolation on an

    inclined plane using a diffuse irradiance model based on hourly measured global horizontal insolation. Solar Energy 19(5), 477--491 (1977).

    3. B. Goldberg, W. H. Klein and R. D. McCartney, A com- parison of some simple models to predict solar irradiance on a horizontal surface. Solar Energy 23, 81--83 (1979).

    4. C. Mustacchi, V. Cena and M. Rocchi, Stochastic simulation of hourly global radiation sequences. Solar Energy 23, 47-51 (1979).

    5. V. M. Puri, Estimation of half-hour solar radiation values from hourly values. Solar Energy 21, 409--414 (1978).

    6. M. Iqbal, Prediction of hourly diffuse solar radiation from measured hourly global radiation on a horizontal surface. Solar Energy 24(5), 491-503 (1980).

    7. J. E. Hay, Calculation of monthly mean solar radiation for horizontal and inclined surfaces. Solar Energy 23(4), 301-307 (1979).

    8. M. Iqbal, Correlation of average diffuse and beam radiation with hours of bright sunshine. Solar Energy 23, 169-173 (1979).

    9. B. Y. H, Liu, and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse and total radiation. Solar Energy 4(3), 1-19 (1960).

    10. B. D. Katsoulia and C. E. Papachristopoulos, Analyis of solar radiation measurementts at Athens observatorry and estimates of solar radiation in Greece. Solar Energy 21, 217-226 (1978).

    11. T. N. Goh, Statistical study of solar radiation information in an equatorial region (Singapore). Solar Energy 22, 105-111 (1979).

    12. V. Modi and S. P. Sukhatme, Estimation of daily total and diffuse insolation in India from weather data. Solar Energy 22, 407--411 (1979).

    13. M. Collares-Pereira and A. Rabl, The average distribution of solar radiation--Correlations between diffuse and hemis- pherical and between daily and hourly insolation values. Solar Energy 22(2), 155-164 (1979).

    14. R. D. Sears, R. G. Flocchini and J. L. Hatfield, Correlations of total, diffuse and direct solar radiation with the percentage of possible sunshine for Davis, California. Solar Energy 27(4), 357-360 (1981).

    15. J. L. Hatfield, P. J. Smietana, Jr., J. J., Carroll, R. G. Flocchini, R. H. Hamilton, Solar energy-implementation of a research and training site at Davis, California. Land, Air and Water Resources Papers Series, Nos 10005 (1981).

    16. J. L. Hatfield, J. J. Carroll, R. G. Flocchini, P. J. Smietana, Jr,, R. H. Hamilton, R. L. Kennedy, Tables of hourly, daily and monthly solar radiation data for Davis, California-1979 and 1980; Land, Air and Water Resources Papers Series, Nos. 10002 and 10004 respectively (1981).

    17. BMDP Biomedical Computer Programs P-Series, University of California Press (1979).

    18. K. W. Boer, The solar spectrum at typical clear weather days. Solar Energy, 19(5), 525-538 (1977).

    19. J. R. Hirschman, The cosine function as a mathematical expression for the processes of solar energy, Solar Energy 16(2), 117-124 (1974).

    20. J. W. Tukey, Exploratory Data Analysis, Addison Wesley (1977).

    21. A list of the eight Solar Energy and Meteorological Research Training Sites and the specifics and availability of the archived one-minute data can be obtained from the Solar Energy Research Institute in Golden, Colorado.

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