Solar Energy, Vok 19, pp. 217-218. Pergamon Press 1977. Printed in Great Britain

TECHNICAL NOTE

A distributed lag model to predict incoming solar radiation

SUSAN S. HAMLENt and WILLIAM A. HAMLEN~ State University of New York at Buffalo, Buffalo, NY 14214, U.S.A.

(Received IMarch 1976; in revised form 8 July 1976)

INTRODUCTION This paper presents a brief description and test of a simple model which can be used to estimate the amount of incoming short wave solar radiation. Such a model has obvious uses in many applications including the construction of atmospheric diffusion models and research in solar energy. This particular solar radiation model emanates from two recent studies.

In the first study an atmospheric diffusion model was developed by Hamlen[1] to predict the variation in the pollution potential using only the airways surface observation (A.S.O.) data which are collected at most airports. A theoretical estimation of direct and diffuse solar radiation is used within the atmospheric diffusion model. Subsequent comparisons of the predictions of these theoretical estimations with measured levels of solar radiation have shown that a distributed lag relationship is useful in constructing a reasonably accurate model of solar radiation.

In a second study Hamlen and Hamlen [2] have developed and tested a new method of estimating distributed lag functions which significantly reduces the problem of multicoUinearity and thus yields estimates of the distributed lag coefficients which are less biased by the particular sample data used. This method is described below and used in constructing the distributed lag model.

TIlE MODEL

Tverskoi ([3], p. 181) presents a small set of data which gives the direct and diffuse solar radiation as a function of the solar altitude h(.) and the level of atmospheric turbidity, Tb. Various functional forms were examined and the following equations gave the best linear regression fits on the transformed variables.

In S = 1.5781 - 0.0509Tb +0.0025 In h(.) (1)

In D = 0.3981 + 0.0658Tb +0.0157 In h(.) (2)

where S, is the direct short wave radiation (cal. cm 2. min-~) and D, is the diffused short wave radiation ('~.

Equations (1) and (2) illustrate the expected relationships that both direct and diffused radiation increase with increasing solar altitude but while increased atmospheric turbidity reduces the direct solar radiation, it increases the diffused solar radiation. The net effect of increased turbidity is to reduce the total short wave radiation. An index of turbidity can be estimated by the following equation

Tb = 1+ W+R. (3)

The three components of (3) are the turbidity of an ideal atmosphere (=1), the humidity turbidity factor, W and the residual turbidity factor, R.

K. Ya Kondratyev ([4], p. 294) has given an equation for estimating the humidity turbidity factor as

W = 0.5ev -'3 (4)

where ev, is the water vapor pressure (mm).

The vapor pressure can be estimated with the use of A.S.O. data

ev = 6714× 10-Tp(tc - t~). r/(1 - r) (5)

where p, is the atmospheric pressure (ram), to, is the temperature (centigrade), tw, is the wet bulb temperature and r, is the relative humidity as a decimal.

Hamlen[1] has derived an estimate of the residual turbidity factor as

R = (935.031 x 103)/V (6)

where V, is the visibility (cm) recorded at the airport. The potential amount of incoming solar radiation is reduced

whenever cloud cover is present. A simple reduction equation is given by Sutton (1953) as

Sh = (D + S)[0.235 + 0.765(1 - coy)] (7)

where Sh, is the short wave radiation which reaches the earth's surface and coy, is the cloud cover as a fraction between 0 (clear sky) and 1 (complete clouds).

From eqns (1)-(7) an estimate of the short wave radiation reaching the earth's surface can be made using the following meteorological measurements defined above: V, p, to, tw, r and coy. The solar altitude, h(.), is a basic astronomical measurement which can be easily calculated for any given time, date and location.

A simple linear regression between Sh of (7) and actual measure solar radiation gave a correlation coefficient of 0.75 and the estimated equation

Y, = -0.0083 + 1.42Sh,

where Y,, is the measured level of solar radiation (cal. cm 2. min-').

Empirical testing of the model has supported the expectation that a distributed lag model of the following form can be used to more accurately predict solar radiation

Y, =c,+#oSh,+13,Sh,_,+.. , +/3NSh,_N+ ~, (8)

where N, is the number of lags. The improved prediction accuracy of the distributed lag model

can be explained by three aspects. First, the A.S.O. data consist of point estimates while the measured solar radiation is recorded throughout each hour. This would imply that a distributed lag model be used to construct a "best" weighted average of the results of the A.S.O. data over the present and past few hours. The greater weights would be expected of the present observations. Second, the A.S.O. data provides micrometeorological conditions which normally precede the large-scale modifications that determine the actual amount of solar radiation reaching the earth's surface during any given hour. Finally, incoming solar radiation has a cyclical component which should be included in a distributed lag model. The interest here is not to separate these three

217

218 Technical Note

components but to derive a single model for prediction purposes. Estimation of the above simple successive lag equation by

linear regression produced a strong multicollinearity effect. This effect, caused by the dependence between the independent variables, produces large variances in the distributed lag coefficients and thus the coefficients are sample-oriented and cannot be trusted for general use.

A well-known technique was proposed by Almon[5] to avoid this problem. Almon assumes that there exists an unknown distributed lag function/3(Z) such that Z = 0 gives/3o in (8), Z = 1 gives/31, etc. An approximation of this true but unknown function is made using a polynomial of degree r or

#(Z) ~ Bo + B,Z + B J 2 +. • • + B,Z" (9)

where Z, is the lag in unit increments. Substituting (9) into (8) for various lags yields

Y , = a + B o X,_~ + B j ~ X , , ' i i +tx, r < N (10)

The coefficients a, Bo, Bj, (j = 1 . . . . . r) are estimated by linear regression. The results of the B's are then substituted into (9) to determine the /3's. Unfortunately, these new transformation variables are also highly collinear and the problem of mul- ticollinearity is not eliminated. Alternatively, Hamlen and Hamlen[2] have modified the Almon technique by assuming that the unknown distributed lag function can be approximated by a harmonic function. Such functions have an orthogonality property which makes the new transformation variables almost completely independent while maintaining all of the positive attributes of polynomial functions. In this modification the following function is substituted for (9)

L

13(Z) =/~ + ~, fiAt sin (360/(N + 1)./'. Z)] j = l

+[Bj cos (3601(N + 1). j . Z)]} (11)

where/~, is a constant term equal to the mean of the series, N, is the number of lags and L = NI2 when N is even and (N + 1)/2 when N is odd.

As in the polynomial method, (11) is substituted into (8) and the following equation analogous to (9) is formed.

Y , = a + B X, , +~_~ Aj X, , s in (3601(N+l) . i . j ) L i=O J j ~ l iffi

+ X,_~ cos (360](N + 1)- i . + IX,. (12)

The coefficients a, /~, Aj, Bj (j = 1 . . . . , L) are estimated by linear regression and substituted into (11) to obtain the distributed lag coefficients. In order to obtain a smooth distributed lag function with the harmonic method, only the first few harmonics in (12) need be estimated. In most cases, the first few harmonics account for the largest percentage of the variation in the dependent variable.

THE DISTRIBUTED LAG FUNCTION OF SOLAR RADIATION The measured levels of solar radiation were obtained from

Argonne National Laboratories for August 1970. Thirteen hours per day between 6 a.m. and 7 p.m. were available. While the solar radiation was measured at the Argonne National Laboratories the meteorological observations used in the hourly calculations of (7) were recorded at Midway Airport. The distance between these locations, while not great (about 15 miles), would be expected to lower the accuracy of the test. A lag of 6 hr was used in order to include the cyclical component which would be included in any model used primarily to predict solar radiation.

In order to obtain a smoother distributed lag function only the first two harmonics were estimated. Thus the intercept term,/~,

A,, A2, B, and B2 were estimated from (11). At the 95 per cent level only the intercept, B and B2 were significantly different from zero. The correlation coefficient was an encouraging 0.93 and the total relationship highly significant at the 99 per cent level.

The resulting distributed lag equation is

Y,=-O.279+O.811Sh,+O.603Sh, ,+0.135Sh, 2-0.239Sh,_3

-0.239Sh,_, + 0.635Sh,_5 + 0.603Sh,_6. (13)

This equation can be used to provide estimates of incoming solar radiation for cities which have airports and thus the A.S.O. data but which do not have recorded measurements of solar radiation. In as much as air pollution affects visibility at the airport, the effect of air pollution on incoming solar radiation is also included within the estimation.

Finally, in order to compare this model with other simple methods of using A.S.O. data as well as to evaluate the decrease in accuracy due to the distance between Argonne and Midway the following three models were tested by substituting for Shj, j = t . . . . . t - N , in (8)

Sh' = n/N, R = 0.61 (14)

where n/N, is the ratio between actual and possible hours of sunshine ([6], p. 167), R, is the correlation coefficient between the model and actual measurements of solar radiation.

Sh"= temperature at the surface, R = 0.017 (17)

and

Sh" = QA(0.18 + 0.55nlN)(l - a) - ~Ta'(0.56 - 0.092v'es) ×(O.lO+0.90nlN), R =0.62 (16)

where (16) is given as (12) in [6] and QA = Angot's value, e,, is the saturation vapor pressure at the dew point temperature (mm) and a, is the surface albedo.

The values used for (16) were available as hourly estimates from the model described in [1]. These were chosen in order to remain true to the original purpose of this study which is to develop a solar radiation model that uses only A.S.O. data. However, the restriction to hourly data might explain why the correlation coefficient for (16) was lower than obtained by Ojo[6].

Acknowledgements--This research was partially supported by a Grant from the Research Foundation of State University of New York and by a summer internship from the Erie County Environmental Management Council of New York. In addition, Marve Wesely of Argonne National Laboratories is gratfuUy acknowledged for providing the solar radiation measurements. A revision was completed under time provided by a Baldy Summer Fellowship, SUNYAB.

REFERENCES

1. William A. Hamlen, A model to predict the pollution potential with the use of airways surface observation data. Atmos. Environ. (forthcoming).

2. Susan Hamlen and William Hamlen, Harmonic Substitution in the Almon Polynomial Technique. State University of New York at Buffalo, School of Management Paper Series, No. 243 (1976).

3. P. N. Tverskoi, Physics of the Atmosphere. Israel Program for Scientific Translations, translated from Russian, Jerusalem, Isreal (1%5).

4. K. Ya. Kondratyev, Radiation in the Atmosphere. Academic Press, New York (1%9).

5. S. Almon, The distributed lag between capital appropriations and expenditures. Econometrica 33, 178-196 (1%5).

6. Oyediran Ojo, Solar radiation, net radiation and temperature in Argonne, Chicago.

7. O. G. Sutton, Micrometeorology. McGraw-Hill, New York (1953).