# A distributed lag model to predict incoming solar radiation

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• 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

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• 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+Bo X,_~ + B j~X, , ' i i +tx, r