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A GENERAL MODEL FOR THE PREDICTION OF HOURLYDIFFUSE SOLAR RADIATIONM. S. AUDI a & M. A. ALSAAD aa Department of Mechanical Engineering , University of Jordan , Amman, JordanPublished online: 27 Apr 2007.

To cite this article: M. S. AUDI & M. A. ALSAAD (1991) A GENERAL MODEL FOR THE PREDICTION OF HOURLY DIFFUSE SOLARRADIATION, International Journal of Solar Energy, 10:1-2, 39-49, DOI: 10.1080/01425919108941450

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Int. J. Solar Energy, 1991, Vol. 10, pp. 39-49Reprints available directly from the publisherPhotocopying permitted by license only

© 1991 Harwood Academic Publishers GmbHPrinted in the United Kingdom

A GENERAL MODEL FOR THE PREDICTIONOF HOURLY DIFFUSE SOLAR RADIATION

M. s. AUm and M. A. ALSAAD

Department of Mechanical Engineering, University of Jordan, Amman, Jordan

(Received II October 1990)

A general computer simulation model, '1', for predicting average hourly diffuse solar radiation,developed and tested for Amman, Jordan, is presented in this paper. This model is based on threesimple components with a control function 'I' that can be switched on ('I' =1) or off ('I' =0) or set athalf way position between on and off ('I' = 0.5). The three components which constitute the 'I' modelare a normal distribution, a half-sine wave, and a polynomial of the fourth degree. These areindividually tested as separate models. It is determined that none of them is alone adequatelyacceptable for the tested data. It is found that for an error of 1% or less, the polynomialmodel- although it is the best of the three - represents the data for only 18 hours of the year or 10%of the time while the 'I' model represents 53 hours or about 29% of the time. The acceptability of thethree models increases with an increase of the upper limit of error. If the error is 10% or less thepolynomial model represents the data for 92 hours or about 51% of the time of the year comparedwith 122 hours or about 68% of the time of the year representable by the 'I' model. It is also shownthat the 'I' model can represent the data for up to 92.4% of the time if the two early morning and thetwo late afternoon hours are neglected.

KEY WORDS: Solar, Radiation, Diffuse, Hourly, Model, Prediction.

INTRODUCfION

The current worldwide awareness of the cost of conventional energy and theinterest in optimizing the utilization of the conventional and the nonconventionalenergy resources forced an interest in the minute details of energy resources. Inthe area of solar energy applications, in addition to the seasonal, monthly, anddaily variation, the hour by hour variation of solar energy intensities in anydesired locality has become increasingly important.

Jordan is one of the small countries of the world which imports most of itsenergy needs. Its solar energy availability, however, is one of the highest in theworld [1,2). It is natural, therefore, that modeling research [3,4] and applications[5,6] be conducted to maximize the utilization of this source on hope to reducedependence on imported oil.

Thus, hourly or instantaneous, if possible, models are needed for thesimulation of solar systems for research and design of solar application systemsfor implementation of the research results.

Several models for the prediction of the ratio of hourly over daily solarradiation energies have been reviewed and published [7]. These models havevarying degrees of applicability and complexity.

In this paper, we tested the normal distribution model, the half-sine wavemodel, and the polynomial model on the hourly diffuse solar radiation of

39

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40 M. S. AVO! AND M. A. ALSAAD

Amman, Jordan, then developed the IJI model which is comprised of thesemodels with switching control functions.

The mathematical formulation constituting the IJI model, the procedure ofanalysis of the data, and a discussion of the results are given in the next sections,respectively.

THE IJI MODEL

The 'P model developed in this paper for the prediction of the hourly diffuse solarradiation is of the following form.

(1)

(2)

where'd is the ratio of the hourly average diffuse radiation over the daily diffuseradiation, both averaged for the period of the data in years. IJIn' IJIs r and IJIparecontrol functions. They take values of 1,0.5, or O. i and h are the month of theyear and the hour of the day, respectively. i takes the value of 1 for January, andhis 0 at noon, negative before noon and positive in the afternoon.

The f,,(i, h) function is the normal distribution relationship given by thefollowing expression.

(1 ) _h2

f,,(i, h) = a(i, h)y!2';; exp 2a2(i, h)

where a is the standard deviation which is a function of the month of the year andthe hour of the day.

The £(i, h) function is a half-sine wave relationship given by the followingexpression.

(3)

where both bo and b, are constants to be determined on the basis of the localdata.

And, the ft,(i, h) function is given by the following relationship.

ft,(i, h) = Uo +a.h + U2h2 + U3h3 +u.h· (4)

where the coefficients uo, UI, U2' U3 and U. are to be determined by using the localdata also. The degree of the polynomial was determined by testing the data forhigher degrees. The results were not improved by using higher than fourth degreepolynomials. Indeed, as it is noted below, for four of the twelve months of theyear the third degree polynomial produced better results than the fourth degree.

Finally, a combination of these models of the form given in equation (1),above, is a comprehensive representation of the data analysed in this research.

PROCEDURE OF ANALYSIS

The data used in this analysis is for five years for the city of Amman, Jordan. Itconsists of hourly diffuse radiation collected on a horizontal surface by the RoyalScientific Society.

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GENERAL MODEL 41

The analysis included the following major steps:

1. The hourly readings of each day were averaged for a month and thoseresults were averaged over the five years. Thus, for fifteen hours a day, allthe data was reduced to 180 values.

2. Regression methods were used to compute the constants of the best fit of anormal distribution, a have-sine wave, and a polynomial relationships.

3. On the basis of the results computed as described in 2, above, the 'II valueswere computed which incorporate the best of what each of the simplemodels contribute.

4. For the sake of comparison and visual inspection, the four models includingthe results of the 'II model and the data were plotted on twelve separateFigures of which only four are presented in the paper.

The statistical analysis of the data included the computation of the correlationcoefficient between the data and the computed values, R 2

, and the error ratio, er,which is the ratio of the standard error in a coefficient to the value of thecoefficient itself [9]. A value of R 2 = 1 designates a perfect correlation betweenthe data and the computed values. a value of er greater than 0.1 indicates that theconsidered coefficient does not contribute significantly to the results.

The results and the discussion of these results are given in the next section.

RESULTS AND DISCUSSION

In this section, the results of analysis of the data using the simple models, namely,the normal distribution model (NO), the half-sine wave model (SW), and thepolynomial model (PO) will be presented sequentially. The 'II model will thenfollow.

The data for four representative months of the year - January, March, June,and September - and the predicted values of rd using the NO, SW, PO and the 'IImodels are shown in Figures 1 through 4, respectively.

The Normal Distribution Model (ND)

The NO model is given in equation 2, above. The standard deviation of thenormal distribution, o, is computed by using the hourly diffuse solar radiation atnoon and the sunshine hours of the day [8]. The regression formula derived forhourly diffuse solar radiation for Amman, Jordan, is given in equation 5, below.

o, = 0.3782N, - 1.135 (5)

where N, is the average number of sunshine hours of the month i.The function fn(i, h) was computed and compared with the data. The

coefficient of correlation, R 2, for each of the twelve months is listed in Table 1,

below.By visual inspection of Figures 1 through 4, it is seen that not this nor any of

the simple models discussed, below, do adequately represent the data. In Figure1, for example, the NO curve passes through four of the fifteen data points. All ofthese points are in the late morning hours. On the other hand, Figures 3 which

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M. S. AUDI AND M. A. ALSAAD

o Data (Jon.)NOSWPOCombined

42

0.20

0.18

0.15

c0:g 0.13"00L.

Q) 0.10Ul::J-~"0 0.08~

00';:; 0.051:::J0

:r: 0.03

0.00-7 -5 -3 -1 1

Hour (noon=O)3

Figure 1 Comparison of data and prediction by all models (January).

represents the summer period shows that the ND curve passes through only twoof the fifteen data points. These points are around the noon hours.

Statistically, for an error of 1% or less, the ND model represents only 9 hoursof the 180 hours of the year, or 5% of the time. For an error of less than 10%,this model represents 61 hours or 33.8% of the time of the year. By comparisonwith the other simple model, this one is the least representative of the data.

The Half Sine Wave Model (SW)

The mathematical formulation of this model is given in equation 3. The R 2 value,the constant bo, and the coefficient b, and the error ratios of both bo and b,which are Ero and Ero, respectively, are computed and listed in Table 2.

It is seen from this table that the error ratio of the constant of the equation,Ero, is considerably greater than 0.1, one of them is as high as 6.1. Therefore, thecontribution of bo, the constant of the equation to the results is negligible. This isunderstandable since this term is related to the phase shift of the half-sine wavefrom the zero axis - the noon hour.

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GENERAL MODEL 43

o Data (March)NOSWPOCombined

0.20

0.18

0.15

co:.g0.13'tJo'-

Q) 0.10Ul:::J--

'tJ 0.08.z-o

Cl~0.05I:

:::Jo

I 0.03

0.00-7 -s -3 -1 1

Hour (noon=O)3

Figure 2 Comparison of data and prediction by all models (March).

Moreover, the least error ratios of the coefficient of the sinusoidal terms are0.045 and 0.046 for May and September, respectively. These values alsocorrespond to the highest values of R 2

• By inspecting Figure 4 one can see thatthe SW curve is very close to the data. Other Figures however confirm thestatement that this and the other simple models are alone inadequate inrepresenting the hourly diffuse solar radiation of Jordan.

Thus, for the month of September the prediction formula becomes as follows.

fs(9, h) = 0.1279 sinG: +~) (6)

The applicability of this model- although it is better than the ND model whenthe error is limited to less than 1% - is still unacceptable. Only 11 hours or about6% of the time this model is useful. When the error limit is raised to 10% or less,89 hours or about 50% of the time the correct values are predictable.

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44

0.20

0.18

0.15

co:g 0.13'tJo...<U 0.10III:::J--

'tJ 0.08~

oo~0.05c::::Jo

:I: 0.03

0.00

M. S. AUDI AND M. A. ALSAAD

D Oala (June)

NOSWPOCombined

- - --

-1 1

Hour (noon=O)J 5 7

Figure 3 Comparison of data and prediction by all models (June).

The polynomial model (PO)

The mathematical formation of this model is given in equation 4. An erroranalysis shows that, if only 1% error or less is allowable this model can predictcorrectly only 18 hours of the total of 180 hours of the year or about 10% of thetime. If the error limit is raised to 10% or less, the applicability of this modelexpands to become 92 of the 180 hours or about 51% of the time. These resultsmake this model the best of the three simple models; the ND, the SW, and thePO models.

In using a polynomial in the analysis of the data, a fourth degree was found toproduce the best statistical results. Nevertheless, for four months of theyear - May, June, July, and August - a third degree polynomial was found toproduce better results. This is based on consideration of both the correlationcoefficients and the error ratios.

Moreover, the coefficients a3 and a. of the fourth degree polynomial and thecoefficient Q3 of the third degree polynomial are all zeroes. Thus, the statisticalresults listed in Table 3 below and the corresponding error ratios are excludedfrom the table.

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GENERAL MODEL 45

0.20

0.18

0.15

c0:g 0.13

"0L-

Ql 0.101Il:1..........

:.0 0.08~

0Cl<, 0.05>,

L:10:r: 0.03

0.00-7

o Data (Sept.)NOSW

POCombined

-1 1

Hour (noon=O)3 5 7

FIglIl'e 4 Comparison of data and prediction by all models (September).

This table shows that for all the months of the year the R 2 values areacceptable, all of them are greater than 0.97. Also, the ero are all acceptable; allof them are considerably less than 0.1. Nevertheless, the other two error ratiosare not that promising. Many of them are considerably greater than 0.1. Thisindicates that this model cannot by itself adequately represent the data.

These results are also shown on the Figures 1 through 4. By inspecting Figure3, for example, it is seen that only four of the 15 points lie on the polynomialcurve, thus only four hours are adequately represented by this model for thismonth.

Table 1. R2 for the NO model computations

Jan. Feb. March April May June0.9288 0.9158 0.9114 0.8987 0.8833 0.7878

July Aug. Sept. Oct. Nov. Dec.0.7933 0.8663 0.9218 0.9189 0.9470 0.9620

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46 M. S. AUDI AND M. A. ALSAAD

Table 2. Statistical results of SW model

Month R' bo "0 b l ert,

Jan. 0..8425 -0.0400 0.63 0.1699 0.120Feb. 0.8808 -0.0320 0.62 0.1577 0.102March 0.9142 -0.0290 0.55 0.1525 0.085April 0.9672 -0.0130 0.62 0.1271 0.050May 0.9730 -0.0010 6.10 0.1081 0.045June 0.9050 0.0123 0.77 0.0863 0.089July 0.9117 0.0071 1.39 0.0936 0.086Aug. 0.9623 0.002 3.65 0.1089 0.054Sept. 0.9727 -0.014 0.52 0.1279 0.046Oct. 0.9244 -0.0260 0.55 0.1476 0.079Nov. 0.8831 -0.0370 1.79 0.1656 0.100Dec. 0.8725 -0.040 0.55 0.1694 0.106

Better results are shown by Figure 2, where it is noticed that eight of the fifteendata points fall on the curve. These points include the noon hours which are mostsignificant in solar energy applications.

As an example of the derived PO model, the following is the predictionformula for the month of March.

/p(3, h) = 0.1360 + 0.0041h - 0.OO5h2 (7)

In general, the 1% and 10% error analysis show that the PO model is best ofthe three simple models. For an acceptable error of 1% or less a total of 18 hoursor about 10% of the time are respresentable by this model. If the acceptable erroris raised to 10% or less the number of hours increase to 92 or about 51% of thetime.

The \II Model

None of the simple models considered so far can adequately represent the data.The \II model as given in equation 1, above, can predict correctly 29.4% of thedata with an error of less than 1% and 67.7% of the data with an error of 10% orless.

Table 3. Statistical results of the PO model

Month R' ao e,o a. e,. a, er

Jan. 0.9896 0.1496 0.048 0.0067 0.194 -0.007 0.057Feb. 0.9906 0.1386 0.045 0.0067 0.164 -0.005 0.080March 0.9966 0.1360 0.026 0.0041 0.146 -0.005 0.040April 0.9824 0.1152 0.057 0.0021 0.520 -0.002 0.200May 0.9912 0.1056 0.035 0.0015 0.400 -0.002 0.000June 0.9794 0.0958 0.051 0.0008 1.000 -0.001 0.000July 0.9767 0.0980 0.053 0.0014 0.640 -0.001 0.100Aug. 0.9856 0.1046 0.044 0.0009 0.700 -0.001 0.000Sept. . 0.9869 0.1143 0.050 0.0010 1.000 -0.002 0.150Oct. 0.9778 0.1292 0.069 -0.002 0.750 -0.004 0.125Nov. 0.9825 0.1460 0.062 0.0003 0.530 -0.006 0.080Dec. 0.9909 0.1505 0.045 0.000 -0.007 0.057

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GENERAL MODEL 47

Table 4. Comparison of all models

% ND SW PO 'I' '1"

I 5 6.1 10 29.4 40.25 20 28.8 35.5 57.7 78.8

10 33.8 49.4 51.1 67.6 92.4

By inspecting Figures 1 through 4, it is seen that the 'P model curve passesthrough most of the data points except the early morning and the late afternoonhours. Thus, a model 'P' formed by neglecting the two early morning hours andthe two late afternoon hours of every day improves the predictability con-siderably. In this case the predictability of the data with an error of 1% or lessrises to 40.2% and that with 10% or less rises to 92.4%.

Table 4 lists the percentage predictabilities of the data by the four models forthree levels of error: 1%,5%, and 10%.

The control functions 'Pn , 'Ps> and 'Pp , of the 'P' model are given in Tables5,6, and 7 below. Note that with the elimination of the four hours the time coverseleven hours centered at noon.

It is expected that the control functions listed in these tables remain constant,

Table S. The 'I' n values of the simulation model

h\i 2 3 4 5 6 7 8 9 10 11 12

-5 0 0 1 1 0 0 0 0 0 0 1 1-4 0.5 0.5 1 0 0 0 0 0 0 0 0.5 0-3 1 0.5 0 0 0 0.5 0.5 0 0 0 0 0-2 0 1 0 0 0 1 1 0.5 0 0 0 0-1 0 1 0 0 0 1 1 1 1 0 0 0

0 0.5 0.5 0 0 0 I I 1 0 0 1 01 0.5 0.5 0 0 0 0.5 1 1 0 0 0 0.52 0 0 0 0 0 0 0 0 0.5 0 0 03 0 0 0 0 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 0 0 0 0.5 05 0.5 1 1 0 0 0 0 0 0 0.5 0 0

Table 6. The '1', values of the simulation model

h\; 2 3 4 5 6 7 8 9 10 11 12

-5 0 0 0 0 1 1 1 1 0.5 0 0 0-4 0 0 0 0.5 0 1 1 0.5 0 0 0.5 0.5-3 0 0.5 0.5 0 0.5 0.5 0.5 1 0.5 0 1 1-2 0 0 0 0 1 0 0 0.5 1 0 0 0-1 0.5 0 0.5 0 0 0 0 0 0 1 1 0.5

0 0 0 0 1 0 0 0 0 0 0.5 0 0.5I 0 0 0 1 1 0 0 0 1 1 0 0.52 0 0 0 1 1 0.5 0 1 0.5 0 0 03 0 0.5 0.5 0.5 0 0.5 0 0.5 0.5 0 1 0.54 1 0 0.5 1 0.5 0.5 0 0.5 0.5 0.5 0.5 0.55 0 0 0 1 0 0 0.5 0.5 1 0 0 0

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48 M. S. AUDI AND M. A. ALSAAD

Table 7. The 'IIp values of the simulation model

h \i 2 3 4 5 6 7 8 9 10 11 12

-5 0 0 0 0 0 0 0 0 0.5 1 0 0-4 0.5 0.5 0 0.5 1 0 0 0.5 1 1 0 0.5-3 0 0 0.5 1 0.5 0 0 0 0.5 1 0 0-2 1 0 1 1 0 0 0 0 0 1 I 1-1 0.5 0 0.5 1 1 0 0 0 0 0 0 0.5

0 0.5 0.5 1 0 1 0 0 0 1 0.5 0 0.51 0.5 0.5 1 0 0 0.5 0 0 0 0 1 02 1 1 1 0 0 0.5 1 0 0 1 1 13 1 0.5 0.5 0.5 1 0.5 1 0.5 0.5 1 0 0.54 0 1 0.5 0 0.5 0.5 1 0.5 0.5 0.5 0 0.55 0.5 0 0 0 1 1 0.5 0.5 0 0.5 0 0

but the functions fn(i, h), fs(i, h), and t,(i, h) may improve if more data iscollected.

The significance of the qt or the qt' models is its need in computer simulationof systems using solar energy. It is fitted to the data by using the controlfunctions. These functions and the simple model relations could be checked andupdated when additional data becomes available.

CONCLUSION

This research is focused on developing a computer simulation model for the.prediction of the average hourly diffuse radiation in terms of simple models andcontrol functions. The following observations could be made.

1. None of the three simple models tested in this research could adequatelyrepresent the analyzed data. The best of the three models is the PO model;yet, it represents 51.1% of the data with an accuracy of 10% or less.

2. The qt model is comprised of a combination of the three simple models withthree control functions of values 1,0.5, or O. These are determined on thebasis of best prediction of the data by the simple models.

3. The development of the 1p model for computer simulation of the data usedin this research covers only five years of data collected in the capital cityarea of Jordan. Some improvements in the control functions and in themodel components may result when additional data becomes available.

References

1. H. EI-Mulki, "Solar and Wind Energy Potential in Jordan," Solar Energy Applications, TheRoyal Scientific Society, pp. 95-104 (1987).

2. M. A. Alsaad, "Solar Radiation Map for Jordan," Solar and Wind Technology 7, 267-275 (1990).3. M. S. Audi, "Estimation of Solar Energy Flux in Jordan," Dirasat S, No.2, 121-134 (1979).4. M. A. Alsaad, "lmproved Correlations for Predicting Global Radiation for Different Locations in

Jordan," Int. J. Solar Energy 8, 97-107 (1990).5. M. Alsaad, S. Habali, M. Hijazi and N. Rabadi, "An Inexpensive and Reliable Solar Water

Heater for Jordan," Dirasat, XII, No.1, 111-128 (1985).

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GENERAL MODEL 49

6. M. A. Alsaad, "A Sub-Atmospheric Solar Distillation Unit," Int. J. Solar Energy 5, 129-141(1987).

7. M. A. Alsaad and M. S. Audi, "Solar Radiation Characteristics in Jordan," Proceeding, FourthArab International Solar Energy Conference, Amman-Jordan, Nov. (1990).

8. P. C. Jain, "Estimation of Monthly Global and Diffuse Irradiation," Solar and Wind Technology5, No.1, 7-14 (1988).

9. M. S. Audi, "Simulation Models for Solar and Other Weather Parameters," Submitted forpublication (1990).

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