Direct beam radiation: Projection onto tilted surfaces

Solar Ener.ey Vol. 40. No. 3. pp, 237-24.7. 1988

0038-0°2X/88 $ 3 0 0 + .(30 Copyright ~ 1988 Pergamon Journals Ltd.

Printed in the U S . A .

DIRECT BEAM RADIATION: PROJECTION ONTO TILTED SURFACES F. VIGNOLA* and D. K. MCDANIELS* Physics Department, University of Oregon, Eugene, OR 97403. U.S.A. Abstract--The projection of direct beam radiation onto tilted surfaces is studied on a daily and monthly basis. Direct tilted values, calculated from hourly beam data. are compared to estimates made by multiplying daily and monthly average direct beam values by the average value of the cosine of the incident angle. The direct horizontal component calculated by this geometrical approximation is underestimated by 10 to 20% for locations in the Pacific Northwest. The reason for this discrepancy is that the average beam intensity changes over the day and thus gives different weights to the cosine of the incident angle. This weighting has been neglected in the geometrical approximation. Previously we introduced a clear day model to weight the cosine over the day. In this article, the fit to the data has been improved by introducing a dependency on beam intensity into the clear day model weighting procedure. This modified clear day model, called the atmospheric weighting model, removes the systematic deviations between the data and values obtained using the clear day model. Direct values calculated using the atmospheric weighting model are compared to data on various tilts and at several orientations. Sensitivity of the atmospheric weighting model to changes in the parameters is examined and the model is shown to be applicable over a wide range of climates with the parameters given in this article.

1. INTRODUCTION

aged direct tilted values is described. A comparison

Knowledge of the incident solar radiation is required

is then made between a simple geometrical model of the beam ratio factor and the data. This comparison establishes a need for a more refined model of the

for the design of solar energy systems and passive solar buildings. Solar radiation data are seldom available at the tilts and orientations desired; incident energy usually has to be calculated from global insolation values using methods developed by Liu and Jordan[l, 2] and Page[3]. The first step in this process is to separate the global radiation into diffuse

beam ratio factor and brings out several characteristics useful in the development of the model. A model of the average instantaneous beam values, called the atmospheric weighting model (AWM), is developed by modifying a clear day model. The A W M model is then used to calculate a beam ratio factor that is

and direct components using an empirical correlation procedure[ 1 , 3 - 1 0 ] . The horizontal direct and diffuse components are then transformed into their corre-

compared to Pacific Northwest data. Finally, the sensitivity of the predicted beam ratio factor to changes in the A W M model's parameters is examined.

sponding tilted components using geometrically based models[2-3, 7, I l - 18]. Each step in the above process contributes to uncertainty and bias in the calculated incident energy. The goals of this article are to develop a model that improves the estimation of direct radiation on tilted surfaces from daily and monthly averaged data, and to determine the uncertainty associated with the calculations. Unlike other articles on this subject that study the ratio of direct tilted to direct horizontal radiation, we examine the beam ratio factor, the ratio

2. THE GEOMETRICAL MODEL Since many concepts in the standard approach carry over to the method we use, it is worthwhile to briefly review the standard methodology. The relationship between the direct tilted component and the direct horizontal component is known exactly on an instantaneous basis.

br = bo" [cos 0r/COS 00]

(1)

where br is the instantaneous direct component on a tilted surface, bo is the instantaneous direct horizontal component, Or is the angle of incidence to the tilted surface, and 00 is the angle of incidence to the horizontal surface. A relationship between the daily di-

of direct radiation on tilted surfaces to direct normal b e a m t radiation. This allows us to study the transformation of direct radiation in a more sensitive manher. This article is organized as follows. First, the standard treatment of this problem is discussed and the basis for our approach is established. Next, the method used to obtain the daily and monthly aver-

rect components is obtained by integrating eqn (1) over the day,

*ISES member, tThroughout this article, the terms "direct normal radiation" and "beam radiation" will be used interchangeably while "direct component on a surface" will be referred to as "direct radiation."

Br = Bo" Rb,

(2)

Rb = J,,, (bo/Bo) • (cos 0r/cos 0o) 0oJ.

(2a)

where ['~"

237

238

F. VtGNOLAand D. K. McDANIELS

Br is the daily direct tilted value, Bo is the daily direct horizontal value, to is the hour angle, and tot, and toss are, respectively, the sunrise hour angle and the sunset hour angle on the tilted surface. The ratio factor, labeled R0, is the term model in most studies, Early models of the ratio factor were developed by Page[3] and Liu and Jordan[2]. Page developed a numerical model for the bo/Bo ratio on a monthly average basis that was then used to evaluate the integral for the ratio factor. With this method, Rb was determined for a limited number of latitudes and orientations. Liu and Jordan[2] suggested that a good approximation for the ratio factor could be obtained by assuming the normal beam component (bN) is constant over the day. The Liu and Jordan model (LJ) is widely used because it involves only the geometrical cosine terms, which makes the ratio factor easy to calculate for any location at any tilt. The LJ model, as with other models of Rb, is exact for equatorial facing surfaces at equinox because the functional dependence on the hour angle (¢o) cancels for the ratio of the cosines when the declination is zero. When this occurs, the ratio of the cosines can be removed from the integral and the integral of bo/Bo over the day is equal to unity. During the late 1970s, the diversity of solar applications increased and a need developed for solar radiation values on nonequatorial facing surfaces. For instance, passive solar designers wanted to know the solar intensity on all surfaces of a building. Klein[ 1 1] presented a generalized version of the LJ model that enabled estimation of the direct component on any surface at any orientation. Later, Collares-Pereira and Rabl[12, 13], using results from their hourly correlation study[7], inserted an analytic model for the instantaneous direct horizontal component (bo) into the integral for the ratio factor. This represented a significant improvement because it took into account the average change in the direct horizontal component over the day and enabled calculations to be made for any surface at any orientation. This model was further developed and tested by Klein and Theilacker[14]. In the above studies, models of Rb were only tested as part of the overall transformation process. More recent studies[15-17] begin to specifically test the direct ratio factor of calculating R b from hourly direct horizontal values obtained by subtracting diffuse from global data. The sensitivity of these tests suffer because the ratio of the cosines associated with the ratio factor diminishes the differences between models, Furthermore, uncertainties are introduced into the diffuse data by shade-ring corrections. In this article, as in a preliminary article[ 18] on this subject, the discrepancies between models are enhanced by taking a different approach; the ratio of direct tilted values to normal beam values is examined. The beam ratio factor contains only one cosine, which eliminates the cancellations found with the direct ratio factor that involves the ratio of two cosines. The precision of this study is further enhanced by the use of high qual-

ity beam data, which greatly reduces the uncertainty associated with the use of global and diffuse data. Since the direct ratio factor and the beam ratio factor are related, the development of the standard approach described previously is similar to the development of our approach that is described in detail next. A simple geometrical relationship exists between direct radiation on a tilted surface and beam radiation on an instantaneous basis; the direct tilted component is equal to the beam intensity (b~.) times the cosine of the incident angle. br

=

bu" COS

(3)

0/-

This equation is equivalent to eqn equal to b0/cos 00. When eqn (3) the day, the difference between proach and our approach becomes

(1) because bx is is integrated over the standard apmore apparent.

Br = BN" RNr

(4)

where R~.r = '

(b,~/Bu)"cos 0r0m.

fl °',, '

(4a)

The beam ratio factor, Rut, relates the daily beam radiation, B.v, with the daily direct component on a tilted surface. This method of calculating Br is more sensitive to the model of the instantaneous beam component because only one cosine term is involved and the cancellation of effects associated with the rario of cosines is avoided. The T in the beam ratio factor refers to the tilt of the surface as measured from the horizontal. The direct ratio factor is related to the beam ratio factor since both methods calculate the same Br. Mathematically, Rb equals R,~r (beam to tilted surface) divided by R,vo (beam to horizontal surface). A simple approximation for the beam ratio factor is obtained by assuming that the beam radiation is constant over the day. This is the same assumption used by LJ. In this case, (~,,,

R,vr ~

/f~,, cos OrOto bto -,~,,, ----~,,

(5)

where tos is the sunset hour angle on a horizontal surface. Note the different limits of integration. The beam values are averaged over the entire period that the sun is above the horizon and therefore the limits of both integrals should be to,. However, during part of the year the sun sets behind the tilted surface, leading to negative contributions from the cosine. Since the contribution during this period is actually zero, this time period is eliminated from the integral of the cosine. The model of RNr given in eqn (5) assumes that the cos Or is weighted by a constant and will be referred to as the geometrical model. In our analysis of the beam ratio factor, RNr will

Direct beam radiation be treated as though it has two components. The geometrical aspect of the ratio factor is contained in the cosine term, and the simple geometrical model given in eqn (5) will be used to represent the contribution of the cosine term to RNr. The other component of the ratio factor is generated by the dependence on the instantaneous beam radiation term that is subject to modeling. The difference between the data and the predictions of the geometrical model will be thought of as representing the contribution of the beam radiation term to Rut.

239

method, which uses the unweighted average of the cosine of the incident angle. For example, when the clear day model is used to project hourly beam values onto a horizontal surface the resulting estimate of the daily direct horizontal component is improved by up to 100 KJ/m"/day over the standard method. Even though the improvement is small, the clear day method was used to ensure the accuracy of the direct tilted values. The measured daily beam and direct tilted values were normalized by dividing each by the incident extraterrestrial radiation. For beam radiation

3. NEED FOR THE ATMOSPHERIC WEIGHTING MODEL Beam data employed in this study were gathered at eight sites in the University of Oregon (UO) Solar Monitoring Network[19] and at the Solar Energy Meteorological Research and Training Site[20] in Corvallis, operated by Oregon State University. All sites were equipped with an Eppley Normal Incident Pyrheliometer (NIP) and a microprocessor-based data acquisition system. The absolute accuracy of the data is estimated t° be -+2%' These sites enc°mp ass a wide range of climates from the verdant Willamette Valley to the high desert plateau of eastern Oregon and Idaho. In all, 12,500 days of hourly beam data were used. A major effort was made to ensure the quality of the solar radiation data gathered by the UO Solar Monitoring Lab. For example, log records that detail the alignment of the pyrheliometer were used in conjunction with chart records to eliminate any beam data affected by misalignment of the NIP. Complete details of the data analysis procedures can be found in our data book[19], Daily beam values were obtained by summing hourly beam data. Daily direct tilted values were calculated by projecting hourly beam data onto the various tilted surfaces and summing over the day. The hourly beam data were projected onto tilted surfaces by multiplying the hourly beam values by the integral of the cosine of the incident angle times a clear model over the hour. In mathematical terms, f b~- = b~," C D M ' c o s

/f

OrOto

CDM 0to

where CDM is the clear day model, b~- is the hourly direct tilted value, b~, is the hourly beam value and the integrals are over the hour or while the sun was incident on the surface or above the horizon. Because the instantaneous beam values calculated with the clear day model only vary significantly during the morning and evening hours, hourly direct tilted values calculated using this method were affected only during those periods. As a check, values obtained by this method were compared to values obtained by projecting 5-rain beam data onto the tilted surface. Less than 1% variation between the two data sets was observed for daily totals[9]. The clear day method does represent a slight improvement over the standard

r.

~' O0t

where KBN is the daily beam value divided by the extraterrestrial radiation, lo is the average solar flux incident outside the earth's atmosphere (1370 W / m "~ was used), and r is the earth-sun distance factor. For direct radiation on tilted surfaces, ,5

KR,=Br/(lo'r'f~ I

cos 0r0to)

(7)

where KBr is the daily direct component on a tilted surface by the appropriate extraterrestrial radiation. The T is the tilt of the surface. For example. Kso is the normalized direct horizontal component. Normalization of the data is useful because it removes large seasonal variations in the data that puts all the data on the same footing throughout the year. This greatly facilitates the analysis and modeling of the beam ratio factor. We take advantage of this by introducing a normalized beam ratio factor, R~r.

KBr/Ks.v=-R,'vr f '°' =

RNr"

/l"

Oto

.,- . . . . .

COS 0rbto.

(8)

~,

The usefulness of .normalizing the beam ratio factor can best be demonstrated by substituting the geometrical model of the Rur given in eqn (5) into eqn (8). The integrals cancel and the geometrical model of R~,r = 1 over the whole year. The advantage of defining R~r this way is now obvious considering that Rut varies by a factor of l0 over the year. In general, models of Rj~-rgive values close to unity. Only when the cosine term is weighted by a constant, which is the case in the geometrical model, does R,~r = I. Later in this section, the need for proper weighting will be demonstrated by showing that the ratio of the actual normalized data is significantly different from unity. The procedure developed above for analysis of the daily beam ratio factor can also be used fo r the monthly average beam ratio factor. Monthly average values of I(sr and/~B,v were found by averaging daily normalized values using a moving average approach[9] in order to make more complete use of available data.

240

F. VIGNOLAand D. K. McDANIELS

The moving average approach calculates monthly avI'O t I I 1 I ~ 1 I l I / 1 erage values by averaging data over 30-day intervals starting at the beginning of the data set and p r o c e e d - ! i MONTHLI[3AIA ing to each subsequent 30-day interval. If more than 0.8 80% of the days in the interval have data, then ~(st and /(8,,: are calculated. The whole data set is then 0.6 reanalyzed five more times; each time the start day o is shifted forward 5 days. The moving average apI-"= preach results in a more representative data set be0.4 cause it makes fuller use of the available data. It is easy to look for any difference between the actual data and the geometrical model prediction. This 0.2 model, which assumes that the beam radiation is constant over the day, predicts that Ksr = KB,v (R.~,.r = 0.0 !). Plots of KB0, the normalized direct horizontal 0.0 0.2 0.4 0.6 0.8 1.0 component, as a function of Ks,v on a daily and KSN monthly average basis are shown in Figs. l and 2. The accuracy of the geometrical model is determined Fig. 2. Plot of/(s0 verses/(aN for monthly averaged data from all sites. by how close the normalized data points lie to the line KBo = Kn:v. The measured points fall about 15% above the geometrical model predictions and can be Several important inferences can immediately be approximated by the line Kn0 = I. 16. KB:~. Some dedrawn from Figs. 1 to 4. First, both the daily and the viation from a completely linear behavior is expected monthly Kno/Kn,v ratios deviate significantly from since outside the atmosphere both K~o and KB:v are unity. This deviation results from weighting of the equal to unity and a linear model would not approach cosine term by the instantaneous beam radiation. In that point, the geometrical model, the instantaneous beam raDeviations from the geometrical model can be diation was assumed constant and the values of the better studied by plotting the ratio of KBT/KB:¢ as a cosine were weighted equally over the day. This refunction of Ks,v. From the above discussion, it is clear suited in the prediction that the normalized beam rathat the data should center around a horizontal line tie factor would be unity. The data shown in Figs. 1 with value 1.0 if the assumption underlying the geeto 4 imply that the cosine factor in R~,o must be metrical model is satisfactory. Plots of Keo/K~,v for weighted in some fashion. A procedure that neglects daily and monthly averaged data from all sites are this weighting will result in an unacceptable error. shown in Figs. 3 and 4, respectively. As expected It should further be noted that while both the daily from Figs. 1 and 2, the data falls above the prediction and monthly KBo/KB,~ ratios average about 1.t6 z of the geometrical model. The spread in the data is 0.04, their dependence on beam intensity are strika little deceptive as the data covers all sites over the ingly different. Another useful observation is that the entire year. Over 12,000 points are represented in the daily plot and over 2000 points are represented in the .. I I I I I I I I I monthly averaged plot. The density of the data points ~.6 L.. ,REGRESSION FIT is greatest around the regression curve which is shown ~o:~ii:.:I - - - AWM -as a solid line in the figures. ~.4 ~ . ~ : % ~ : :. . . . ,~=,~,.,=,~,g~.:-...~:.::::....

/q

-

1.0

~'x~.~

i~."

DALLY DATA

~o= ""

0.6

~ /

0.4;

~ / '~B~

"--I

~

~

~

I'OI ~=g:; ~ o-~" :: ~'~:'g.':::'~:?":'":' " .~:,7".. :" ""

~"':

•~

E~.:i o.o

SIMPLE __ CLEAR DAY

/

~'~'.~'"'"'~:~,~.~,.%L~,~;>..

0.8

o Q3

-

• I o.2

~, ~GEOMETRICAL MOOEL

/

t--

DALLY DATA

l 0.4

1

~

o.6

i

I o.a

i I.o

KBN 0.2

0.0, O.O

0.2

0.4

0.6

0.8

1.0

KBN

Fig. 1. Plot of/(8o verses K~¢ for daily data from all sites.

Fig. 3. Plot of K~o/Kt~,~verses Ka:¢for daily data from all sites. The dots (.) represent one or two data points. The open circles (O) represent more than two data points. The line at K~o/K~ = 1 is the geometrical model predictions. The line at K~o/Ks~ = 1.16 is the CDM prediction. The solid curve through the data is the regression fit to the data. The dashed curve is the AWM prediction.

Direct beam radiation r.el__

i

I

l

i

I

I

i

I

REGRESSIONFIT - - - awu

-

[-

-

...... DR MOOEL

4

I..: I ..~- ....

.~.-~

SIMPLE /CLEAR DAY MODEL

/ ~

........

i~ i.z o ........ L

~ L GEOMETRC I AL MODEL

I.O

o.B

I ~o

I 02

I

MONTHLYDATA I I I I I 1 0.4 o.6 o,e

~.o

RaN

Fig. 4. Plot of K,o/K~,Nverses/(,N for monthly data from all sites. The dots (-) represent one or two data points. The open circles (O) represent more than two data points. The line at IKBo//('8,,= I is the geometrical model predlction, The line at /~~,.//~B.v = 1.16 is the CDM prediction. The solid cui've through the data is the regression fit to the data. The dashed curve is the AWM prediction. The dotted curve is the prediction derived from a model of Collares-Pereira and Rabl[12].

variance from the regression curves decreases significantly with increasing beam intensity. This implies that any model of the beam ratio factor will be most constrained by clear day data. Correlation procedures can also be used to study the deviations from the geometrical model estimates of the normalized beam ratio factor. When Kno is correlated against a quartic polynomial in Key, the resuiting regression curve describes the data well. The residuals about the regression fit can be used to further characterize the difference between the model and the data. The averaged residuals between Ks0 and the regression fit are plotted against time of year in Fig. 5. Here, the residuals appear normally distrib-

L

I

0041

1

i

I

i

uted about the regression curve and therefore the data are well described by this correlation. This is not the case for surfaces tilted away from the horizontal. Figure 6 shows a plot of the averaged residuals from a correlation betwen Kngo and a quartic polynomial in Ks,,, for the data from all sites. A systematic variation with time of year is found, showing that the correlation does not accurately describe the normalized data on tilted surfaces. The above discussion demonstrates that the geometrical model is unsatisfactory for use with Pacific Northwest data. Our goal is to develop an improved model, given the constraints imposed by the data as shown in Figs. 1 to 6. The geometrical model is based on the assumption that the beam intensity is constant o v e r the day. Clearly, some model that appropriately simulates the average beam intensity over the day will more accurately determine the beam ratio factor. Several studies[3, 12-14, 17] of the direct horizontal ratio factor used models of the monthly averaged instantaneous direct horizontal radiation to weight the ratio of cosines in the calculation of Rb. As noted earlier, by equals bo/cos 00, therefore models of instantaneous beam radiation can be derived from models of instantaneous direct horizontal radiation. Collares-Pereira and Rabl[12] suggested that their model of the instantaneous direct horizontal radiation could be used to calculate the monthly average RuT. The dotted line in Fig. 4 shows the predicted R:,,0 using their model. Why the predictions using their model differ so much from the Pacific Northwest data is unknown, however, this difference shows that the beam ratio factor is very. sensitive to the particular instantaneous beam model used. Another approach taken i n a few recent papers[17, 18] is to simulate bu with a clear day model (CDM). There are important advantages to this approach. The CDM closely simulates the beam radiation on clear days when the data imposed rigid constraints on the beam ratio factor. Clear day models

/

I

/

Kso vs f(Ks.)

I

°'°21

~.'-•_.:. "" • " . . .

1

.

o.oo ~_-=~'~.-.---'.-".~'~;.~.-

": ~

...................

t

-"--.... "" :""" : ' " " ""

~cr D

---

I 1. 1 ." :%.":-'.-"" :"-.

l~

uJ

,,.,

,-, ODD

....

UJ

• .

•

%7~-~ . . . . . . . . . . . . .

•

..~ ..........

"

J- : :

_

"" -O.O2 <~

I

", : ' " ":",'.,,'-- RESIDUALSOAILY-

J 0.02

~

.t.

0.04 -

J

DAILY RESIDUALS

I-

241

"#

•

.:,

%,

<>-O.O2 :.:." "

•

::~..:

.-".... .:" -..%

I0~04

IOiO~

t

JAN

MAR

I

I

I

MAY dUN SEP DAY OF YEAR

I

NOV

Fig. 5. Averaged residuals from the daily correlation of Kao and Kay,plotted against time of year for all sites,

.

KB9 0 I

JAN

MAR

1

vs

f(KBN) I

I

MAY dUN SEP DAY OF YEAR

I

NOV

Fig. 6. Averaged residuals from the daily correlation of/(,,,, and Ka.vplotted against time of year for all sites.

242

F. VIG"/OLAand D. K. MCDANIELS

also have a prescribed method for adjustment to different climatic regions. Furthermore, it is easier to make a regional test of a CDM than it is to test models of long term average instantaneous beam radiation, The CDM of Dogniaux[21] was used in our study because it closely simulates b,~. on clear days in the Pacific Northwest. The clear day approach has one major shortcoming which can seen from Figs. 3 and 4 where R.~,,o predicted using a simple clear day model is shown. Although the predicted ratio of 1.16 is in agreement with the average for the data, the predicted ratio is independent of beam intensity. Figures 3 and 4 show that the beam ratio factors are dependent on beam intensity. In an earlier study[ 18], we showed that the use of a CDM to estimate the beam ratio factor resuits in an improvement of about 15% over predictions of the geometrical model. However systematic deviations of up to 5% still exist between R~,oderived using a CDM and the ratio of normalized data. Since these deviations are about a third of the gain achieved by using the CDM, there is still room for further iraprovement in the calculation of the beam ratio factor.

simple function of KB.v.* The data from our Kimberly, Idaho, site was used for this procedure because Kimberly is representative of the our high desert sites that have the highest values of beam intensity. We wanted to be certain to fit the clear day data that provide the most stringent constraints and hoped that the data from other sites that did not have days or months with such high beam intensities would match the Kimberly fit at lower values of Once a fit to the Kimberly data was obtained, that parametefization was used for all sites. A detailed description of the atmospheric weighting model is given in Appendix 1. Improved estimates of the direct component on any surface can now be made by using the beam ratio factor derived with the AWM in which the water vapor and turbidity parameters are functionally dependent on beam intensity. Weighting the cosine factor in eqn (4a) with the AWM results in more accurate estimates of the beam ratio factor. Formally, we have

The systematic errors in the CDM's estimates apply to any model that does not incorporate a dependence on beam intensity. In order to improve on the CDM estimates, we have devised a method to incorporate the dependence on beam intensity into the clear day model.

The normalized beam ratio factor predicted from the AWM can be obtained by replacing R v~. in eqn

4. THE ATMOSPHERIC WEIGHTINGMODEL In the previous section, the beam ratio factor obtained by using a clear day model was described. While considerable improvement was made over the geometrical model given in eqn (5), the resulting R,,~r is independent of beam intensity, whereas the ratio of the normalized data changes significantly with beam intensity. In order to provide for the observed dependence on beam intensity, the clear day model was modified by making the water vapor and turbidity parameters functionally dependent on KB,,. This modifled CDM is called the atmospheric weighting model (AWM). The functional dependence of water vapor and turbidity parameters on beam intensity was derived as follows. First, clear day model parameters were determined for clear days. Monthly average water vapor coefficients from Boise, Idaho, were taken from Machta[22] and the turbidity coefficients were adjusted so that the clear day model matched the typical clear day solar noon transmission values for our sites in eastern Oregon and Idaho[ 19]. These turbidity values are quite similar to those given for Idaho Falls, Idaho[22]. This set of water vapor and turbidity coefficients was used for the entire region. The functional dependence of these parameters on was derived

Ksx

by adjusting the parameters so that R~,o would match the average ratio of the normalized data over a small range of KB,v. The adjusted values were then fit by

KBN.

R,vr=

/f I~'

fi""

AWM • cos 0.r0oJ

AWM Oto.

(9)

(8) with the beam ratio factor given in eqn (9). Thus, (~,,

R,'~r= J,,, •

AWM - cos 0r0m

&o to,

COS 0r00J" ,,

AWM &o .

(I0)

,

Because R.~,.rderived using the geometrical model is equal to unity, the differences between calculated

R.'vr

using eqn (10) and unity is the AWM model's contribution to the normalized beam ratio factor. The AWM estimates of R~,o are shown as the dashed lines in Figs. 3 and 4. As expected, the AWM estimates closely match the regression fit to the data and accurately duplicate the dependence on beam intensity. In the preceding discussions we have weighted the cosine factor, which projects the beam radiation onto the tilted surface, in three different ways. If all values of the cosine are weighted equally over the day, as is the case for geometrical model, then is unity. Weighting with the simple CDM gives a better es-

R.'~.r

timate of the data, but R~,r still lacks dependence on beam intensity, KBu. With the AWM, the proper dependence of on is obtained. Another way to test the AWM is to compare the

R,~r KnN

predicted direct tilted values, Ksr, with the data. A *The functional dependence of water vapor and turbidity on Katederived for the atmospheric weighting model is only intended to be used to match the functional dependence of the beam ratio factor on beam intensity. It should be noted that once these parameters become functions of Ks.,,, there no longer is a direct connection with measured watervapor and turbidity values.

Direct beam radiation simple way to do this is to correlate Ksr with Ks,v" R~T. If the AWM correctly weights the cosine factor, the regression coefficient (RC) should be unity and the AWM modeled R~-r would, on the average, correctly describe the ratio of Ksr/Ks,v. Results from a one-parameter correlation study are presented for each site in Table 1. The first column is the regression coefficient, whichh indicates the bias of the model. The bias is within --.L/2% for all sites except Portland. The Portland site is in a large urban environment that is likely to result in a much more turbid atmosphere than the rest of the sites. Increasing turbidity would increase the calculated value of R~0, which could account for the bias at the Portland site. The second column contains the standard deviation (or) of the data from predicted values. This standard deviation is about the same as the standard deviation from the correlation between Kso and a quartic polynomial in Ks,v, which implies that the AWM model describes the normalized data as well as a high order polynomial regression fit to the data. The accuracy to which the AWM reproduces the dependence on beam intensity is demonstrated in Fig. 7 where the averaged residuals from the correlation between Kno and Ks.v" R,'vo are plotted against Ks.v for Eugene, Oregon. The advantage of the A W M over the polynomial regression fit is that the dependence on latitude, tilt, beam intensity, and time of year are an integral part of the model. The A W M model can also be reliably used to estimate the value of direct radiation on tilted surfaces. Table 2 presents results from correlations between Knr and KsN" R.~.r for various equatorial facing tilted surfaces in Eugene, Oregon. The regression coefficient is the same within uncertainty for all tilts. The standard deviation increases with tilt because during part of the year the sun can be above the horizon and behind the tilted surface. During this time period, which increases with tilt, the instantaneous beam intensity contributes only to B,v. The scatter in the data increases because the percentage of BN obtained when the sun is behind the tilted surface varies, Standard deviations for the tilted surface correlations are significantly less than those from the correlation between Ksr and a quartic polynomial in Ks,~.. The reason for this improvement is that variations

Table 1. Correlation of Kno with Kau" R.~.0 Site

RC"

cr

Bums, OR Coeur D'Alene, ID Corvallis, OR Eugene, OR Hermiston, OR Hood River, OR Kimberly. ID Portland, OR Whitehorse Ranch, OR

0.999 1.005 0.999 1.004 0.998 1.000 1.000 1.016 1.003

0.020 0.015 0.016 0.015 0.016 0.015 0.018 0.014 0.018

"Regression coefficient,

243 t

I

J

t

0.04 -

t

I

I

t

t

DAILY RESIDUALS

KBo vs K.~.R~o ~ 0.02 r~ ~ a: ~ 0.o0 ~ ~: "-0.02 <>

-

-

"t--'~'~-~:" . .

".."---°~'.-.-'--t'- ............... -.

.

-

-0.04t 0.0

I 0.2

I

I I I I I I 0.4 0.6 0.8 KSN

1.0

Fig. 7. Averaged differences between Ks. and KBv'R'vo plotted against KBv for Eugene, Oregon.

produced by the atmospheric weighting model match the seasonal variations inherent in the data. The averaged residuals from the daily correlation between Asgo and Ks.v'R,~,,,~ are plotted against time of year for Eugene, Oregon in Fig. 8. Note that the averaged deviations from the A W M estimates are normally distributed, which is not the case for those in Fig. 6 that come from a quartic correlation in Ksu. The same methods used to develop the daily AWM can be used to generate the monthly AWM. The only difference between the daily and the monthly averaged AWM is the functional dependence of the water vapor and turbidity parameters on Ks v. Details of the functional dependence are given in Appendix I. R~, calculated using the monthly averaged A W M model is shown in Fig. 4. As was the case for daily data, the AWM prediction closely matches the polynomial regression fit to the data. The monthly averaged AWM model can also be tested using a one-parameter correlation b e t w e e n / ( s t and /('s,,~'R.~,.r. Results from this correlation are presented in Table 3. Close agreement between the data and the AWM predictions are found. The standard deviation for the monthly average correlation is only one quarter of that found from the daily correlation (see Tables 1 and 2). So far, tests of the A W M model have been limited to equatorial facing surfaces, but there is no reason why this method cannot be extended to surfaces not facing the equator. In fact, the most stringent test of the model should occur for surfaces facing the pole Table 2. Correlation of Ksr with KnN"R'~T Tilt

RC'

o-

0° 30" 60° 90°

1.004 1.004 1.004 1.005

0.015 0.017 0.020 0.026

aRegression coefficient.

244

F. VIGNOLAand D K. ~|CDANIELS 1

I • •

0.04

.:

t

I t i DAILYRESIDUALS •. K090 ~

mation of the direct horizontal c o m p o n e n t to various tilted s u r f a c e s This standard approach was not used by us because it lacks sensitivity to the transformation p r o c e s s For example, for equatorial-facing surfaces during the equinoxes, the standard approach gives the same answer independent of the model used. In order t o model the cosine weighting factor in the transformation integral more accurately, we have studied the transformation of the b e a m radiation to tilted surfaces. The physical process modeled in both cases is the same due to the simple mathematical relationship between instantaneous direct horizontal and beam radiation. However, the transformation process can be investigated with m u c h greater sensitivity by examining the b e a m ratio factor since it contains only one cosine term whereas the direct ratio factor in-

I

KBN'Rt~90 -

~ 0.02 • . .• o . : . . ." "..:'. :':,... • _ "' ...":'" ". .... : .":-: O'O0'~'~-'m-~"r:";':;:r"'" ;" " ' ; ' " ~ 2 " r ~ ' ~ ' " t.9 .... : '::'" • ° • """ ":"° • " < n,

~

-

. " . . . . -.- _.. . .: : . . . . . - .

•; " ~ ' - - "" -O.OZ" "" " • : " ." •. -O.04• I I • I', I JAN MAR MAY JUN SEP

•

I NOV

DAY OF YEAR

Fig. 8. Averaged differences between Ks~o and KB," R,',.~ plotted against time of year for Eugene, Oregon.

because these surfaces only have sunlight during the morning and evening hours when the attenuation of beam radiation by the atmosphere is changing the fastest. As can be seen from Fig. 9, R~9o for a vertical north-facing surface estimated using the A W M agrees very well with the monthly average data from Eugene, Oregon. Notice that these results increase with beam intensity and are only one-third to one-half of the value predicted by the geometrical model of the normalized beam ratio factor, 5. DISCUSSION The goal of this article has been to develop an improved model for accurately estimating direct radiation on tilted surfaces from daily and monthly averaged data. The simplest approach, which we call the geometrical model, transforms direct normal beam radiation to the direct tilted c o m p o n e n t using an unweighted cosine factor. W e have demonstrated with precise data from our monitoring sites in the Pacific Northwest that this procedure can easily lead to large errors. Analysis of these systematic errors led us to model the transformation process by weighting the cosine factor with a modified clear day model. Previous investigations have studied the transfor-

volves the ratio of two cosines that markedly reduces the ability to distinguish between various models of the beam intensity. One result from the improved sensitivity of our approach is the ability to clearly distinguish between daily and monthly average b e a m ratio factors. This study was also greatly aided by the use of the normalized b e a m ratio factor. This significantly reduced the seasonal variations in the data and greatly simplified comparison between measured data and model predictions. Good agreement between measured data and predicted values was obtained with our new atmospheric weighting model that matched the data's variation with beam intensity. This is best seen in Figs. 3 and 4 where R~0 calculated with the A W M matches the average variation of K B o / K n N with KtJ,v, while the ratio calculated with the simple C D M leads to systematic errors of up to 5% because it lacks dependence on beam intensity. Although the A W M model can be used to accurately predict direct values on north and south facing surfaces, the model has more difficulty when used to

t.O

x

Tilted

Site

0°

RC'

~

RC"

o-

Bums, OR Coeur D'Alene, I D Corvallis, OR

0.995 1.002 0.994

0.006 0.005 0.004

0.993 0.998 0.991

0.0tO 0.007 0.007

Eugene, OR Hermiston, OR Hood River, OR Kimberly, ID Portland, OR Whitehorse Ranch, OR

0.999 1.005 1.003 0.999 1.010 1.002

0.004 0.004 0.004 0.005 0.005 0.005

1.000 1.010 1.002 1.001 1.008 1.003

0.007 0.007

'Regression coefficients,

0.006

0.008 0.006 0.006

I

I

t~ t I I I \ "GEOMETRICALMODEL

/

0.6o

no

Tilted 90 °

J

~ 0.8 _z _ F-

Table 3. Monthly average comparison

t

-

• ".:'-::-

C

AWM

-

-"

.__.

~ 0.4 -

-

~.,~,d" •

~

MONTHLY DATA -

,,_m O.2 -

NORTH FACING -VERTICAL SURFACE_

-

O.(

I .0

I 0.2

I

1 0.4

I

I 0.6

I

I 0.8

I h0

KBN

Fig. 9. Monthly average ratio of geoo/l~B.,~ versus/~sr¢ for a north-facing vertical surface for Eugene, Oregon, from the middle of May through the middle of July. The dashed line is the prediction using the atmospheric weighting model.

Direct beam radiation predict direct values for east and west facing surfaces if, on the average, the surfaces receive different amounts of direct radition. Several studies[14-17] have indicated that direct radiation on an east-facing surface differs from direct radiation on a west-facing surface. This difference varies from site to site and with time of year. Data from the Pacific Northwest also contains this asymmetry. Eugene, Oregon summertime values ofl~ego/Ig,s,v for monthly averaged direct radiation on an east-facing vertical surface are plotted along with similar values for a west-facing surface in Fig. 10. The normalized beam ratio factor estimated using the A W M fits the average of the eastand west-facing data very well, but it does not match either the east- or west-facing data well. It is impossible for any model using only daily or monthly average data to predict an asymmetry between morning or evening hours (unless it is built into the model in some arbitrary fashion.) For daily Eugene data during the summer, the difference between direct radiation on vertical west- and east-facing surfaces varies considerably with the degree of cloudiness, Knu. On clear days, the direct radiation incident on a west-facing surface equals the direct radiation incident on an east-facing surface, while on partially cloudy days when KBu ~ 0.35, the direct radiation on a west-facing vertical surface averages nearly twice that on an east-facing vertical surface. This asymmetry varies with time of year and is probably site specific. This presents a considerable problem when trying to accurately estimate the solar radiation on the east- and west-facing surfaces of buildings using daily or monthly averaged data and demonstrates the usefulness of hourly data. Serious errors can result if local water vapor and

I.~ . I Z

O

1.2--

_ ~- I.O "' z~ -'~ 0.8 -~n

--

I~ 0.4

I

I

l

MOmHL¥ ~rA _

-

~OMETR~CAL --

o o~

• WF'ST FACING

--

%

EASTFACING ,, AVERA~

--

'o OooO~

0.6 --

I

MOOF. L

, ~

°°0

o.o

I

-

_

""

I

• :. ." . ,,mmo~ • .:." ""--]" .,_" " '',~" " ..,,.~,~a.~

"' I~

I

o~

oo0%0 \ A W M

~° o ° ° oo •

-

o°

I

I

o.2

I

I

0.4

I

I

(~s

I

I

o.a

[

Lo

KeN Fig. 10.. Monthly average ratio of/(e~//~B,~ versus/¢aN for east- and west-facing vertical surfaces for Eugene, Oregon. from the middle of May through the middle of July. The solid circles (e) indicate data for the west-facing surface, and the open circles (O) indicate data for the east-facing surface. The open triangles (A) show the average of the east- and west-facing data. The dashed line is the prediction using the atmospheric weighting model,

245

turbidity values are directly substituted into the AWM because the AWM is a modified clear day model. The functional dependence of the water vapor and turbidity parameters on beam intensity was determined for a specific set of water vapor and turbidity values. If these atmospheric values were changed, the functional dependence on beam intensity would also change. Another reason local values cannot be used is that water vapor and turbidity also correlate with beam intensity and the effects of this correlation were incorporated into the AWM. In general, on clear days the beam intensity decreases as water vapor and turbidity increase. The water vapor and turbidity parameters of the AWM also increase when the beam intensity decreases (eqns. (5.1)-(8.1) in Appendix 1). Because the parameters of the A W M for high values of Ksu were fit to a range of clear days, the correlation betwen atmospheric values and beam intensity is reflected in the parameterization of the AWM. Therefore, the AWM should be used with atmospheric values given in Appendix 1. The sensitivity of the normalized beam ratio factor calculated with the A W M changes in the water vapor and turbidity parameters demonstrates that much of the influence of local differences in atmospheric values have already been incorporated into the model's dependence on Ksv. R'vr estimated using the A W M is relatively insensitive to change in water vapor. Doubling the value of the water vapor parameter changes R',~oby at most 0.6%, whereas the chan~e~ in R,~9ois about 1.0%. On the other hand, there is an appreciable sensitivity to the turbidity parameter. Doubling the value of the turbidity parameter increases R~,0 by almost 6% and R~,9o by 10 to 1 i%. These changes are extremely large, considering that the AWM is about a 15% change from the geometrical model. At first glance, the above results seem to imply that the A W M is extremely sensitive to turbidity values and local turbidity values should be used at each location. Actually, just the opposite is true. The above results when combined with the correlation results given in Table 3 demonstrate that the AWM. with the Pacific Northwest climate parameters, is applicable over a wide range of climate conditions. From the previous paragraph, it is seen that the AWM-derived beam ratio factor is insensitive to chan~es in the water vapor parameters and extremely sensitive to changes in the turbidity parameter. Thus, one would expect that at least two sets of turbidity parameters would be needed to fit the Pacific Northwest data because the turbidity in the urban areas is about twice that in the high desert region. However, the correlation results given in Table 3 show that one set of water vapor and turbidity parameters can be used to fit all the Pacific Northwest data. The reason why only one set of parameters is needed is that a dependence on beam intensity is incorporated into the turbidity parameter. This dependence on beam intensity automatically increases the turbidity as the beam intensity decreases. Therefore, the parameters chosen

246

F. VIG~OLAand D. K. McDANIELS

Table 4. Daily R',T versus latitude. Averaged from May thru July. Ks.~ = 0.5 Latitude

Tilt 0 °

Tilt 60 °

0° 15° 30° 45 ° 60°

1.161 1.160 1. 166 1.182 1.222

1.275 1.244 1.241 1.264 1.333

for the Pacific Northwest should work well for a wide variety of climatic regions, It may be possible to develop an approximate beam ratio model that can be applied over a wide range of latitudes because the normalized beam ratio factor predicted by the A W M model does not vary much with latitude. As shown in Table 4, the normalized beam ratio factor calculated with the A W M model changes by less than 2% from the 45 ° latitude results for all latitudes less than 50 ° . Earlier, we showed that the normalized b e a m ratio factor automatically adjusts for local climate variations and thus R,VT can be treated as a function of time of year, beam intensity, and tilt of the surface. Therefore, it should be possible to develop standard tables of the normalized beam ratio factor that could be applied worldwide. The universality of the normalized beam ratio factor arises because normalization of the beam ratio factor separated the geometrical aspect from the transformation calculation and because climatic variation was incorporated into the atmospheric weighting model when the parameters were made dependent on beam intensity. The generalized applicability of the A W M will be discussed in more detail in the future paper.

Acknowledgements--We gratefully acknowledge the contributions of Pat Ryan and Susan Ota who helped with maintaining the University of Oregon Solar Monitoring Network and helped with the data analysis. We would also like to acknowledge the Eugene Water and Electric Board. Pacific Power, and Washington Water Power for financial support of the network without which this work would not be possible. REFERENCES 1. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse, and total solar radiation. Solar Energy4, 1-19(1960). 2. B. Y. H. Liu and R. C. Jordan, A rational procedure for predicting the long-term average performance of flatplate solar-energy collectors with design data for the U.S., its outlying possessions and Canada. Solar Energy 7, 53-74 (1963). 3. J. K. Page, The estimation of monthly mean values of daily total short wave radiation on vertical and inclined surfaces from sunshine records for latitudes 40°N-40°S. Proc. UN Conference on New Sources of Energy. paper no. 35/5/98, 378-390 (1961).

4. N. K. D. Choudhury, Solar radiation at New Dehli. Solar Energy 7, 44 (1963). 5. D. W. Ruth and R. E. Chant, The relationship of diffuse radiation to total radiation in Canada. Solar Energy 18, 153-154 (1976). 6. S. E. Tuller, The relationship betwen diffuse, total, and extraterrestrial solar radiation. Solar Energy 18, 259-263 (1976). 7. M. Collares-Pereira and A. Rabl, The average distribution of solar radiation-Correlations between diffuse and hemispherical and daily and hourly insolation values. Solar Energy 22, 155-164 (1979). 8. D. G. Erbs, S. A. Klein and J. A. Duffle, Estimation of the diffuse radiation fraction for hourly, daily, and monthly averaged global radiation. Solar Energy 28, 293-302 (1982). 9. F. Vignola and D. K. McDaniels, Correlation between diffuse and global insolation for the Pacific Northwest. Solar Energy 32, 161-168 (1984). 10. F. Vignola and D. K. McDaniels. Diffuse-global correlation: seasonal variations. Solar Energy 33, 397402 (1984). I I. S. A. Klein. Calculation of monthly average insolation on tilted surfaces. Solar Energy 19, 325-329 (1977). 12. Manuel Collares-Pereira and Ari Rabl, Derivation of • method for predicting long term average energy deliver?. of solar collectors. Solar Energy 23, 223-233 (1979). 13. Manuel Collares-Pereira and Ari Rabl. Simple procedure for predicting long term average performance of nonconcentrating and concentrating solar collectors. Solar Energy 23, 235-253 (1979). 14. S. A. Klein and J. C. Theilacker, An algorithm for calculating monthly-average radiation on inclined surfaces. J. Solar Energy Eng. 103, 29-33 (1981). 15. J. R. Simonson, The use of weighted/~ factors in calculating monthly average insolation on tilted surfaces. Solar Energy 27, 445-447 (1981). 16. U. v. Desnica, B. G. Petrovic and D. Desnica, Calculation of monthly average daily insolation on tilted, variously oriented surfaces using analytically weighted ,q~ factors. Solar Energy 37, 81-90 (1986). 17. Christian Gueymard. Mean daily averages of beam radiation received by tilted surfaces as affected by the atmosphere. Solar Energy 37, 261-278 (1986). 18. F. Vignola and D. K. McDaniels, Transformation of direct solar radiation to tilted surfaces. Proc. 1984 Annual Meeting American Solar Energy Socie~,, Inc.. Anaheim, CA, 651-655 (1984). 19. Pacific Northwest Solar Radiation Data. University of Oregon Solar Monitoring Laboratory, Eugene, OR (April 1, 1987). 20. C. R. Nagaraja Rao, T. Y. Lee, W. Bradley and G. Dorsch, Solar Radiation and Related Meteorological Data for Corvallis, Oregon, 1981. Department of Atmospheric Sciences, Oregon State University, Corvallis. OR (April 1982). 21. R. Dogniaux, Programme general de calcul des eclairements solaires energetiques et lumineux des surfaces orienteees et inclinees ciels clairs, couverts et variables. L'Institut Royal Mrtrorologique de Belgique, Avenue Circulaire, 3, 1180 Bruxelles (1985). 22. Lester Machta, Workbook for Approximate Calibration of Solar Radiation Sensors. Air Resources Laboratodes. Silver Spring, MD (August 1978). 23. F. Vignola and D. K. McDaniels, Beam-global correlations in the Pacific Northwest. Solar Energy 36, 409-4t8 (1986).

APPENDIX 1 The atmospheric weighting model is described in detail below. The clear day model is from Dogniaux[21] and was

chosen because it fit the clear day data in the Pacific Northwest. The parameters of the AWM given below are depen-

Direct beam radiation

247

Table 1.1. Water vapor and turbidity Month

Jan

Water Turb

0.86 38

Feb

Mar

Apt

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

0.76 46

0.73 55

0.82 60

1.05 66

1.43 66

1.44 68

1.44 67

1.24 66

1.00 58

0.97 40

0.78 36

dent on this clear day model because the functional dependence of the parameters on Kn, was derived using this clear day model. Any good clear day model can be used, but the functional dependence on Knu has to be determined for each model. The formula given below is called the normalized beam ratio factor and it multiplies Kn.~. to give Ksr. f~i/ 2tos cos Or exp --[mzSRTL]OoJ

R'~-c =

f.i;

( I. 1)

cos 0r0tO

exp--[mz~RTi.]Oto

The formulas for water vapor and turbidity are as follows: w = 0 . 9 3 . water "f(K~N) [3a = 0.00263. turb "f(Ks~)

(5.1)

(6.1)

where water is the ~ater vapor content of the atmosphere in centimeters and mrb is the turbidity factor for the atmosphere. The monthly averaged values used for all the Pacific Northwest sites are given in Table 1.1. Values for the daily ratio are obtained by interpolation. For the daily atmospheric weighting model

f(KB,.) = - 2 . 3 1 + 13.2-K~,/~?-12.5'K~,.

(7.1)

All the terms except the atmospheric weighting terms in the exponential have been defined in the body of the article, The first term in the exponential, m:, is the air mass term. the attenuation due to Rayleigh scattering is ~R. and the

For values of K s , < 0.1. flKn~.) is set equal to fl0.1 ) because the above formula goes negative for very small values of beam intensity. For the monthly averaged atmospheric weighting model.

Linke turbidity factor is TL. A complete discussion of the clear day model is given in reference[21]. The atmospheric weighting model differs from the clear day model only in the Linke turbidity factor in which the water vapor and the turbidity parameters are made functions of Kn,.. The air mass term is defined as follows:

fll~n,.) = 1.69 - 3.46.K,~u + 2.53. R'~,.

(8.1)

It is not necessary to have beam data in order to use the AWM. Beam values for use in the calculation can be estimated from global data by using a beam-global correlation[23] or from direct horizontal values using one of the following equations. For daily calculations.

m: = (1.0 - 0.1 • alt)/{cos 0,, + 0.15[(90 ° - 0,,) + 3.885] -t -"~}

(2.1)

where ah is the altitude of the site in kilometers and 0,, is the sun's zenith angle given in degrees. The attenuation due to Rayleigh scattering is given by

~ = I/(0.9.m. + 9.4).

(9.1)

If Keo is less than 0.1 then Knx = 0.9. KBo. For monthly averaged calculations. R'nu = - 0.003 + 0.845. K',o + 0.065. R'~.

(10.1)

(3.1)

The Linke turbidity factor is given by

TL = [(90 ° -- 0o) + 85]/(39.5e-" + 47.4) + 0. l + (16 + 0.22w)[3A

Knx ~ 0.020 -,- 0.687. K~,, + 0.253. K~-,.

(4.1)

where w is related to the water vapor content of the atmosphere and 13a is related to the turbidity of the atmosphere. Equations (2. I) to (4.1) are from Dogniaux[21].

The accuracy of these estimates of Knu is sufficient for use in the A W M because R',r is a slowly v a ~ i n g function of beam intensity. The integrals in eqn (1.1) can also be evaluated using summation techniques. R',-r can be estimated to better than 1% of dividing the period between sunrise and sunset into approximately 20 intervals and s u m m i n g over the day the product of the interval (in radians) and the function evaluated at the middle of each interval.