A simple hourly all-sky solar radiation model based on meteorological parameters

Solar Energy Vol. 32. No. 2. pp. 195-.-~4, 1984

O0.';g-.O92X]g4 $3.00 § .00 Pergamon Pre~r lad.

Printed in Grea! Britain.

A SIMPLE HOURLY ALL-SKY SOLAR RADIATION MODEL BASED ON METEOROLOGICAL PARAMETERS J. E. SftER~Y and C. G. Jusrust School of GeophysicalSciences, Georgia Institute of Technology,Atlanta, GA 30332. U.S.A.

(Receired 24 May 1982; accepted 20 April 1983) Abstract--An hourly solar radiation model for cloudy skies, based on meteorologicaldata. was developed and tested. As a means of comparison,the SOLMET regressionand Watt models were also tested. The present model was examined for individualcloud types using measured solar radiation to judge the effectivenessof the model in the presence of particular clouds. It is then possible to determine an approximate amount of thin cirrus for each reported cirrus layer by multiplying the reported fractional amount of the cirrus layer (N~) by the percent of reported thin cirrus in the sky (Rci,), which can be expressed as:

INTRODUCTION

Atmospheric parameters which deplete solar radiation vary considerably in the magnitude of their effect. According to Watt[l] the five clear-sky, solar-radiationdepleting atmospheric parameters and their' relative effect on solar radiation are ozone (0.5-3.0%), upperlayer aerosol (1.9-11%), dry air (i1-13%), water vapor (3.5-14%), and lower layer aerosol (0.1-26%). If all-sky conditions are considered, this list is increased to include clouds. Of all the radiation-depleting parameters, clouds have the greatest potential attenuating effect on solar radiation. Clouds, although well-reported by a nationwide network of surface observations, are highly variable in amount and optical properties. The amount of cloudiness can change very rapidly, making calculations of solar radiation estimates very difficult. In order to calculate the solar radiation in the presence of clouds, a cloudy-sky model is developed here using the Suckling and ttay cloud-sky model[2] as a basis, with several important modifications. The required input to the present model for cloudy skies includes the results of the Georgia Tech (GT) clear sky model (Sherry and Justus,[3]), the 3-hourly National Weather Service (NWS) observations, the mixing height as reported twice daily by the NWS, and the continuous measurement of sunshine duration.

Rr = Nci,lN, i.

(3)

Since only one cloud observation is available for every three hours and because this observation is taken approximately on the hour, the model applies the 3-hourly NWS observation to the hour ending on the hour of the 3-hourly observation as well as to the following hour. Since cloud reports are applied in this manner and because cloud amounts can vary so quickly, a case could develop whereby no clouds are reported in the 3-hourly NWS observation but are indicated by a fractional sunshine (F,,) reading of less than i.0. If no clouds are reported in the 3-hourly NWS observation and the F,, is less than or equal to 0.9, then the climatologically most prevalent monthly cloud type is assigned to the first cloud layer and the fractional amount of the first cloud layer (NI) can be expressed as: Na = i . 0 - F . .

(4)

The determination of sunshine duration should employ the best available data. A problem arises, however, when instrument measurements are used for low solar elevations. Sunshine duration is defined as the length of time the direct beam exceeds 200 W/m:'. When the solar elevation is low and the atmosphere is hazy, the direct beam may still be present but may not exceed the 200 W/m2 threshold defining "sunshine". During such a period, the sunshine duration would be measured as zero and model calculations for the direct beam would erroneously calculate a value of zero. In order to allow for direct beam radiation which is far below the 200 W/m2 threshold at low solar elevations, the present model replaces measured fractional sunshine for solar elevation angles of less than 11.5 degrees (i.e. air masses greater than 5.0) with a value calculated from the 3hourly NWS reported opaque cloud cover. Thus, for solar elevations less than 11.5 degrees fractional sun-

Cloudy sky model For cloudy skies, the 3-hourly NWS cloud report is first interpreted to determine the simple sum of the fractional (0-1) amounts of cirrus (No-0 and non-cirrus (Nctay) clouds in the four reported cloud layers. Then the fractional amount of thin cirrus (Ncl,) can be calculated using the expression: (l) where Noraq is the reported fractional amount of opaque cloud cover. The fractional amount of opaque cirrus (N~o) is then: N~io= N , i - N~i,. (2)

tlSESR Member. IO~

196

J.E. SHERRYand C. G. Jusrus

shine (F,,) would be expressed as: F+, = 1.0- No,+,

(5)

Before proceeding further, it is now necessary to discuss two methods for modifying the amount of reported cloud (N~) in each of the four layers. The first involves making a correction for changes in opaque cloud amount. As was already mentioned, the amount of cloud can vary rapidly and some method of modifying the 3-hourly cloud observation would seem desirable. In order to modify each of the opaque cloud layers as well as the opaque portion of the cirrus layers, a method has been developed here which uses the amount of reported opaque cloud cover (Nop~q) and the fractional sunshine (F,). The ratio (R) of ! . 0 - F,~ and Nop~,ais an indicator of how the total amount of opaque clouds has changed from the NWS report. Then assuming that each opaque layer and the opaque portion of cirrus layers changes in an amount relative to the total change in opaque cloudiness and that all reported cloud types remain present with no new types developing, a relationship based on the ratio R = ( I . 0 - F,~,)lNor, a,:+,

(6)

can be used to modify each cloud layer. The product (N0 of R and the reported amount of opaque cloud in each layer (N0 should then be more representative of the changing opaque cover. In order to allow for thin cirrus, N~ is expressed in two forms: Non-cirrus layers: N~ = N~R.

(7)

Ni = Nil(! - R~i,)R + R~it]

(8)

Cirrus layers:

where Rc~, is the fractional per cent of reported thin cirrus to the total reported cirrus. This method does not alter the amount of thin clouds as reported in the cirrus layers, and care should be taken that the sum of the modified cloud layers is not made to exceed 1.0. To avoid such an occurrence, if the sum of the four modified layers exceeds 1.0, then the amount of thin cirrus should be decreased so that the sum of the four cloud layers equals 1.0. A second modification to cloud amount is not dependent on the opacity of clouds. This second modification, as developed by Suckling and ttay[2], modifies the amount of clouds in each layer (N0 to include clouds obstructed from view by lower clouds. The resulting cloud amounts (NCOR0 are expressed by Suckling and Hay[2] as:

" 0.1

NL4

m 0.9

NCOR 4 = 1.0

~3

" 0.1

NL3

= 0.8

NCOR3= 0.5 N2

= 0.3

NL2

~ 0.5

NCOR 2 u 0 . 6

1,: 1

= 0.5

NL1

= 0.0

NCOR 1 - 0 . 5

( C ~

Actual

tenth

Estimated rro~ the

•

of

tenth surface

cloud of

visible

cloud

from

obstructed

the

surface

from view

Fig. I. Example of the correction of cloud layer amounts for layers obstructed by lower cloud. modified cloud report can be seen in Fig. 1, where the shaded clouds are the unobstructed clouds and the unshaded clouds are obstructed from view at the surface. The Haurwitz[4] cloud transmissivity study, whose results are often employed in solar radiation models (Suckling and Hay,[5]; Davies,J6]), is used in this model. The Haurwitz study determined cloud transmissivities at the Blue Hill Observatory for the period 1938--45. In order to make this determination, only conditions of complete overcast, as indicated by hourly cloud observations, were considered. Haurwitz assumed that, for overcast conditions, the relation between measured hourly total horizontal global insolation, (l~ho) and air mass could be expressed as: l, h o = ( a l m ) e - ( ~ )

(I0)

where a and b are constants determined by the method of least squares, l~ho is in kJ/m 2, m is air mass (sec z), and z is the solar zenith angle. For the eight cloud types indicated by an (*) in Table !, there was a sufficient number of overcast observations (Table 2) to determine the coefficients a and b (eqn 10). The average air-massdependent cloud transmissivities (To), for the eight cloud types already mentioned, can then be obtained by taking the ratio of IRho (eqn 10) and the modeled clear sky horizontal global insolation l~hc T+ = 1,~d t, hc.

( I I)

The value of hourly total lgh+ (kJ/m2), for clear sky conditions, used by Haurwitz can be expressed as: I~hc = (3951.6/m)

NCORI = NJ(I.0- N0,

Nr

e -tO'O59 m)

(12)

(9)

where N is the simple sum of the unobstructed lower layers (NO and i is the reported cloud layer from 1 to 4. An example of this procedure for a four-layer F,,-

It was unfortunate that other atmospheric optical parameters (i.e. such as water vapor and aerosol amounts) were not taken into consideration at the time of the Haurwitz study. The omission of other con-

A simple hourly all-sky solar radiation model

197

Table 1. Haurwitz [4] cloud transmissioncoefficients Cloud Type 0

a(kJ/m 2)

b

None

1

*Fog

645.3

0.028

2

*Stratus

997.2

0.159

3

*Stratocumulus

1453.9

0.104

1453.9

0.104

645.3

0.028

4

Cumulus

5

Cumulonimbus

6

*Altostratus

1634.1

0.063

7

*Altocumulus

2199.8

0.112

8

*Cirrus

3444.2

0.079

9

*Cirrostratus

3649.5

0.148

i0

Stratus Fractus

997.2

0.159

ii

Cumulus Fractus

1453.9

0.104

12

Cumulonimbus Mama

645.3

0.028

469.3

-0.167

13

*Nimbostratus

14

Altocumulus Castellanus

2199.8

0.112

15

Cirrocumulus

3649.5

0.148

16

Obscuring phenomena other than fog (including precipitation)

645.3

0.028

*Indicates cloud transmlsslvitfes derived by Haurwitz. assigned subjectively.

siderations has forever linked the determination of cloud transmissivity with eqns.(10) and (12), for it cannot otherwise be determined what effect local aerosol and water vapor amounts may have had on the resulting values of a and b. Thus, it is suggested that if the Haurwitz cloud transmissivities are used, the calculation of those cloud transmissivities should be done using eqns (10) and (12) and no substitute for eqn (12) should be employed. Because the Haurwitz study was unable to assign cloud transmissivities to all cloud types, an alternate approach had to be taken for several cloud types. These cloud types include cirrus (which has already been discussed), stratus fractus, cumulus fractus, altocumulus castellanus, cumulus, cumulonimbus, cumulonimbus mama and obscuring phenomena other than fog. As can be seen in Table !, values for these undetermined cloud transmissivities have been subjectively assigned based on the most representative of the eight Haurwitz cloud type transmissivities. In trying to apply Haurwitz cloud transmissivities to a four-layer insolation cloud model, special care should be taken. Because the method employed by Haurwitz to obtain cloud transmissivities used overcast conditions, the resulting transmissivities represent average cloud conditions above the overcast layer (i.e. there may or may not be clouds present above the overcast). Since the

All others are

four-layer cloud insolation model for partly cloudy skies has a mechanism which uses both the observed cloud amounts for each of four layers and an estimate of cloud amount in each of the layers which is obscured by lower cloud layers, it is necessary to increase the Haurwitz cloud transmissivities of the lower cloud layers for partly cloudy conditions. In order not to account for higher clouds twice, when definite multiple cloud layers are present, the Haurwitz cloud transmissivities for lower clouds (i.e. stratus, stratocumulus, fog, cumulus, cumulonimbus, stratus fractus, cumulus fractus and nimbostratus) should be modified to increase their transmissivity. This investigator found that for overcast conditions, the average standard deviation for the lower clouds was about 0.14 and when this amount of 0.14 is used in the present cloud sky model to increase nonovercast lower level cloud transmissivities, good results are obtained. For the case of cumulonimbus and cumulonimbus mama, a value of 0.21 was used. When conditions are overcast, Haurwitz cloud transmissivities are used unmodified. The treatment of cirrus in the present cloudy-sky model is considered for both its effect on diffuse and direct solar radiation; a global transrnissivity will not apply for this case. It was found, through trial and error, that the values in Table 3 for cloud transmissivities for cirrus, when applied to the determination of diffuse and

198

J. E. SHERRYand C. G. JUSTUS Table 2. Number of observations for the Haurwitz cloud transmission study Air

Mass

Ci

Cs

Ac

As

Sc

St

Ns

Fog

i.i 1.2 1.3 1.4 1.5

80 66 36 28 58

77 51 37 47 42

96 70 68 44 84

85 91 96 51 77

340 263 220 147 241

60 45 41 27 52

23 22 --20

170 i01 iii 89 154

1.6 1.7 1.8 1.9 2.0

21 21 22 26 29

35 42 24 29 18

50 44 37 39 44

52 43 50 62 52

156 128 139 171 183

29 19 19 24 25

21 -14 17 26

75 39 75 117 108

2.1 2.2 2.3 2.4 2.5

14 19 22 17 12

21 20 24 38 --

35 40 34 45 17

35 51 37 69 32

i01 125 iii 154 49

10 21 ii 14 ii

-18 i0 29 13

67 70 63 81 41

2.6 2.7 2.8 2.9 3.0

ii 13 17 -25 18 . . . . 18 I0

19 27 43 20 21

30 36 47 16 19

73 72 103 49 63

13 17 15 13

-15 15 --

42 29 86 32 57

3.1 3.2 3.3 3.4 3.5

15 -. . . . . . . . -ll 12 --

16 27 ii 22 20

16 20 18 14 22

51 44 34 45 32

12 -ii -. . . . . . . . . . . .

46 63 25 32 26

3.6 3.7 3.8 3.9 4.0

I0 14 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

61 13 19 20 20

-. . . . . . . .

48 17 17 15 19

4.1 4.2 4.3 4.4 4.6 4.7 4.8 4.9 5.0

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

1

.

. .

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

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

. . . . . . . .

I0 .

.

i0 . . .

. . .

direct insolation in the present cloudy-sky model, produce favorable results. At this stage, enough information is known to begin calculating the various components of solar radiation for cloudy skies. The direct normal solar radiation for all cloudiness conditions received at the surface can be expressed as: Re,, Nk~,(i 0 - 0 6)

I0

.

18 . 16 17 12 16 14 13 i0

.

. . . .

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

15 . . . . .

.

.

. . . . . . . . .

.

.

. . . .

.

.

.

.

ii

the sky (D,) can be expressed as: D,, = (!.0 - NcioR - Nc~,- NI)Iah,,

(14)

where Ia,c is the clear-sky diffuse as calculated in the Georgia Tech clear-sky model. The horizontal diffuse component for the cirrus portion of the sky is given by:

(13)

4

Dci = ~ . 0.6(1 - Rcit)

where N](r is the fractional amount of cirrus in the layer (i) and N](ci~is equal to zero for non-cirrus layers, I,~,r is the clear-sky direct normal calculated using the Georgia Tech clear-sky model (Sherry and Justus[3]), and 0.6 is the transmissivity coefficient for the direct normal component of cirrus clouds. The determination of the horizontal diffuse component in the cloudy sky model for all cloudiness conditions (l,~h,,) is somewhat more involved. The sky is first divided into three categories, which are the cloudless, the cirrus and the remaining cloudy portions of the sky. The horizontal diffuse component-for the clear-sky portion of

14 i0 i0

Nj(ci)RIghc + 1.4 Rci,Ni(ciJdhc,

(15) where N~(c~is the corrected amount of cirrus in the layer (i), (N,ci~ equals zero for non-cirrus layers), l~,c is the clear sky horizontal global as calculated using the G T model and 0.6 and !.4 are the diffuse transmissivity coefficients for opaque and thin cirrus clouds (Table 3). For the remaining portion of the cloudy sky diffuse component (Dc~y) is expressed by Suckling and Hay[2] as"

Dday = Nddy IRhc~ ~,, i=!

(16)

A simple hourly all-sky solar radiation model

199

Table 3. Transmissivity (%) for all types of cirrus clouds Opaque

Thin Direct Normal

60% of Clear Sky Direct Normal

0% of Clear Sky Direct Normal

Horizontal Diffuse

140% of Clear Sky Horizontal Diffuse

60% of Clear Sky Horizontal Global

where ,/,, is the individual layer transmissivities for the cloudy portions of the sky and is expressed by Suckling and Hay[2] as:

ttay[2l as: D,,,, = (I,,,,,, + De, + Dr + Dc,d~,)A~(Ar

+ A~,(N~oR + Nci,)), if, = 1.0- (1.0- T~,~) NCORJN[,

(17)

where Tc.~ is the individual cloud type transmissivities, as derived by Haurwitz (1948) (eqn 11), including the Haurwitz global transmissivities for cirrus and NCOR~ for cirrus layers (NCOR,,~) should be expressed as:

(18)

NCOR,~,~ = NCORi - NI.

The reason that cirrus is treated in this section in the manner expressed above is that the portion of cirrus which is obstructed from view by lower-layer clouds has a contributing effect on solar radiation in the cloudy portion of the sky, and since only the diffuse component of solar radiation passes through this portion of the sky, it is more correct to use the Haurwitz cirrus cloud transmissivities which were derived using global solar radiation data. An additional component of the horizontal diffuse radiation is the multiple reflection term (D,,,). whose first-order component is expressed by Suckling and

L~J

Kx:~ 9

l,~h, = D , + D.:i + Dc,dy+ Din.

(20)

The horizontal global solar radiation received at the surface for all cloudiness conditions is then simply the sum of the horizontal component of l,~h, from eqn (13), and Idho from eqn (20), and can be expressed as: l~,h,~= I~.0 cos(z)+ I.,.. where z is the solar zenith angle.

Y* 1 0 3 .

[,163.|78

'%,

x

590 GT DIFFUSE MODEL (KJIMZ) SL is the slope o f the least squares best f i t l i n e Y is the y-intercept of the least squares best fit line E is the ~v~ error between the y-ordinate and the least squares best fit line ~MS between the x and y - 173.3 AVG between the x and y - 255.6 ~U~/AVG (%) - 67.8

Fig. 2. Measured vs Georgia Tech horizontal diffuse for fog. SE Vol. 32, No. 2--D

(19)

where A~ is the albedo of the ground which according to Suckling and Hay[2] is equal to 0.2, except in cases of snow cover when it is set equal to 0.8. Addyand A~ are the albedos of non-cirrus and cirrus clouds and, according to Watt[I], are equal to 0.2 and 0.6-0.8, respectively. The total ditIuse component o[ solar radiation received at the surface (l,,ha) for all cloudiness conditions is then the sum of these four previously determined components.

e:L- . 7 6 0

FOG

Nc,dy

(21)

2OO

J. E. SHERRYand C. G. JusTus

RESULTS The results of the Georgia Tech (GT) cloudy-sky model were tested in three ways. The first was to compare the results of the GT diffuse, cloudy-sky model vs measured diffuse solar radiation for conditions when only one cloud type was present. Sufficient data were available for nine cloud types, the results of which are shown in Figs. 2-10 and listed in Table 4. Secondly, the GT direct normal cloudy-sky model results were compared to measured direct normal solar radiation for conditions when only cirrus or cirrostratus clouds were present. These results are shown in Figs. 12 and 13 and listed in Table 4. Thirdly, the results of the GT cloudysky model for diffuse, direct and global solar radiation were compared with the results of the SOLMET regression and Watt models (Sherry,[7]) (Table 5).

CONCLUSION

In conclusion, it is shown that the GT cloudy-sky model gives good results. When individual cloud types were examined (Figs. 2 and 3, Table 4), it was found that both the GT cloudy-sky diffuse and direct normal models compared well with measured solar radiation. Also, when all-cloud conditions were examined, it was found that the GT cloudy-sky model gave significantly better results than the less complicated Watt model (Table 5). The SOLMET regression model results, although better than the Watt model results, were not as good as those of the Georgia Tech model (Table 5). Overall, the Georgia Tech cloudy-sky model gave better results than the other models tested.

Table 4. GeorgiaTech diffuse horizontal and direct normal RMSIAVG(%) for various cloud types (Apt-Dec 1979) at GeorgiaTech Horizontal Diffuse Cloud

~

,Type

Observations

AVG

(kj/m2-hr)

(kJ(m2-hr)

P~IS/AVG

(%)

Fog

46

173.3

255.6

67,8

Sf

79

193.9

284.7

68.1

St

67

226.1

254.8

88.7

Sc

260

249.4

444.4

56.1

Cu

252

212.6

732.1

29.0

Cb

16

182.2

470.9

38.7

Ac

107

76.1

191.7

39.7

Ci

203

112.7

236.7

47.6

Cs

143

220.4

250.5

88.0

Rain

280

181.6

153.8

118.1

Direct Normal C1

203

294.5

1636.6

18.0

Cs

143

285.8

801.5

35.7

Table 5. Hourly model RMS/AVG(%) for all sky conditions(Atlanta, Georgia Tech, Apr-Dec 1979) Diffuse Horizontal

Direct Normal

Global Horizontal

Watt

78.1

47.0

47.1

SOLMET Regression !

50.3

43.7

19.2

Georgia Tech

43.2

24.4

17.3

Model

IThe SOLMET regression model uses the Randall regression model for direct normal and the ARL regression model for horizontal global (Nashville coefficients).

A simple hourly all-sky solar radiation model

ST;~TU$

F/~L~[TU$

5L-

i

u

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I

26.2

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201

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[.|E/l.~t|

SL- . g B | I

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?ooo

T- 3 1 . 5 I

[,2"6.gS$

x x

x

x

15oo

1509

x

x

t,_ t.,.

xxx~ x

x

ff ~ x x x x x

1000

I O0O

x

xx,k

x

xx ~

~x ~ x xx~

x

x xx x

xX'~

x ~x 0

xX

GT

x

xX

~oo

0

?.5~o

x

xx

~x x

)~ x x

L~J

~

xx

Jooo DIFFUSE

,s~o

2o~o

MODEL ( K J / H Z )

SL is the slope of the least squares b e s t fit line Y is the y - l n t e r c e p t of the least squares best fit llne E is the ~v~ e r r o r b e t w e e n the y - o r d i n a t e and the least squares b e s t fit llne

SL iS the slope of the least squares best fit llne Y is the y - ~ n t e r c e p t of the least squares best fit line E is the ~-~ error b e t w e e n the y - o r d l n a t e and the least squares b e s t fit line

~MS b e t w e e n x a n d y - 193.9 A V G between x and y - 284.7 ~MS/AVG (~) - 68.1

~MS b e t w e e n x and y - 249.4 A V G b e t w e e n x a n d y - 444.4 K M S / A V G (%) 56.1

Fig. 3. Measured vs Georgia Tech horizontal diffuse for stratus fractus clouds.

IR~TUS

SI-

:~500

I

I

Fig. 5. Measured vs Georgia Tech horizontal diffuse for stratocumulus clouds.

u

I.E5

i

1].7

[-227'.

996

I

E000

X

1500

X x

/a_ u-

x

x

x

I000 x

x x

x

--

500

K

xx

xx

x x

x

xx

D;

D

2000

G'I DIFFUSE M O D E L

2"5O0

(KJIMZ)

S L is t h e s l o p e O f t h e l e a s t s q u a r e s b e s t f i t l i n e Y is t h e y - i n t e r c e p t of the least squares best fit line E is t h e ~ M S e r r o r b e t w e e n t h e y - o r d i n a t e and the least squares best fit line ~MS between AVG between ~MS/AVG (~)

the the

x and x and

y - 226.1 y - 254.8 88.7

Fig. 4. Measured vs Georgia Tech horizontal diffuse for stratus clouds.

202

J.E. SHERRYand C. G. Jusrus

(;UMULU$

5L= .796

~00

|

I

u

E-ITLB2'I

86.1

1

I

E0,33

:7

x

x

1500 x

x

xx

9 ~x~l

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x

~ xxx

f~t

x

xx

1000 Jk

~x

x

x x

x

L>300 GT

DIFFUSE

HODEL

(KJ/M~)

S L is t h e s l o p e o f t h e l e a s t s q u a r e s b e s t Y is t h e y - i n t e r c e p t of the least squares E is t h e ~ M S e r r o r b e t w e e n t h e y - o r d i n a t e squares best fit line ~M5 between AVG between ~MS/AVG (%)

x and x and

fit line best fit line and the least

y - 212.6 y - 732.1 ~ 29.0

Fig. 6. Measured vs Georgia Tech horizontal diffuse for cumulus clouds.

L~O0

SL- .991

[UMULONIMBUS =

T- 35.& I

'E-19|.1~67

SL= .9S9

I~LTOEUMULU$

u

B.9!

E- 75.387

L~O0

=

^

s

l( X

ISO0

1500

N u_ ZL =

Lx

X

I000

IDO0

X X

,,'R, (z: x

,.=,

~o

xK

N

x

:iz

x ~x x

x

5oo

x

x

x~

x x 9 xx D

?500

E50O

0

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GT O1FFOSE , O = , .

(KJ,,~

SL iS the slope of the least squares b e s t fit line Y i s the y - i n t e r c e p t of the least s q u a r e s best fit line E is the ~,~ error b e t w e e n the y - o r d i n a t e and the least squares best fit line

S L is the slope of the least squares b e s t fit line Y is the y - l n t e r c e p t of the least s q u a r e s best fit line E is the ~MS e r r o r b e t w e e n the y - o r d i n a t e and the least squares best fit line

~MS b e t w e e n x and y - 182.2 AVG b e t w e e n x and y - 470.9 ~v~/AVG (%} 38.7

~ 3 between x and y 76.1 A V G between x and y - 191.7 ~v3/AVG (%) 39.7

Fig. 7. Measured vs Georgia Tech horizontal diffuse for cumulonimbus clouds.

Fig. 8. Measured vs Georgia Tech horizontal diffuse for altocumulus clouds.

A simple hourly all-sky solar radiation model

SL- .EZi~

203

SL= . 9 B ~

V - - 2 . IH L~500

v

1,50"3 9

x

X

w x

x x

10'33

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x u/

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~r D~FFUSE MOOEL(~J~.z~ S L is the s l o p e o f the l e a s t s q u a r e s b e s t f i t l i n e Y is the y - i n t e r c e p t of t h e l e a s t s q u a r e s b e s t f i t l i n e E is the F ~ S e r r o r b e t w e e n the y - o r d i n a t e a n d t h e least s q u a r e s b e s t fit l l n e

S L is t h e s l o p e o f t h e l e a s t s q u a r e s b e s t fit l i n e Y is t h e y - i n t e r c e p t O f t h e l e a s t s q u a r e s b e s t f i t l l n e E is the ~ v ~ e r r o r b e t w e e n t h e y - o r d i n a t e a n d t h e l e a s t s q u a r e s b e s t fit l i n e

~t~ between x and y - 112.7 A V G b e t w e e n x a n d y - 236.7 P~MS/AVG (%) - 47.6

~ between x and y - 181.6 AVG between x and y - 153.8 ~ M S / A V G (%) - 118.1

Fig. 9. Measured vs Georgia Tech horizontal diffuse for cirrus clouds.

I;l;t~O~la;tlU~

St-

I

.726

I

u I

&9.11

Fig, ll. Measured vs Georgia Tech horizontal diffuse for rain.

~-I~B.I3g

SL-

lll~'ds

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l

,

,

,

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l

,

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,

i

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2000 v

v

1500

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9

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x

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x

x

x

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x

o

x

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x

x x

m

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,~

iL~ DIFFUSE

MODEL

2dm

500

~500

(KJIMZ)

S L is the s l o p e of t h e l e a s t s q u a r e s b e s t fit l l n e Y is the y - i n t e r c e p t o f t h e l e a s t s q u a r e s b e s t f i t l i n e E is the ~MS e r r o r b e t w e e n t h e y - o r d i n a t e a n d the l e a s t squares best fit line

S L is t h e s ] o p e of the l e a s t s q u a r e s b e s t fit l i n e Y is t h e y - l n t e r c e p t o f the l e a s t s q u a r e s b e s t fit l i n e E is t h e ~MS e r r o r b e t w e e n t h e y - o r d l n a t e a n d the l e a s t s q u a r e s b e s t fit l i n e

~ M S b e t w e e n x a n d y - 220.4 AVG between x and y - 250.5 ~ M S / A V G {%) B8.0

~v~ b e t w e e n x a n d y - 294.5 AVG between x and y- 1636.6 ~S/AVG (%} 18.0

Fig. 10. Measured vs Georgia Tech horizontal diffuse for cirrostratus clouds.

Fig. 12. Measured vs Georgia Tech direct normal for cirrus clouds.

204

J. E. SHERRYand C. G. Jusrus c]~csl~lus 4900

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i

J

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S L is t h e s l o p e of t h e least s q u a r e s b e s t f i t l i n e Y is t h e y - i n t e r c e p t of t h e l e a s t s q u a r e s b e s t flt l l n e E is t h e p v~ e r r o r b e t w e e n the y - o r d i n a t e a n d the l e a s t s q u a r e s b e s t fit l l n e ~ b e t w e e n x a n d y - 285.8 A V G b e t w e e n x a n d y - 801.5 ~Lv~/AVG (%) 35.7

Fig. 13. Measured vs Georgia Tech direct normal for cirrostratus clouds. REFERENCES 5. P. W. Suckling and J. E. Hay, Modeling direct, diffuse and total solar radiation for cloudless days. Atmosphere 14, 298-308 I. A. D. Watt, On the nature and distribution of solar radiation. (1976). HCP/T2552--01, U.S. Department of Energy, Washington, 6. J. A. Davies, Models for estimating incoming solar irradiance, D.C., U.S.G.P.O. (1978). DSS/OSU79--00163, Hamilton, Ontario: Department Geo2. P. W. Suckling and J. E. Hay, A cloud layer-sunshine model graphy, McMaster University (1980). for estimating direct, diffuse and total solar radiation. 7. J. E. Sherry, A simpIe hourly solar insolation model based on Atmosphere 15, 194-207 (1977). meteorological parameters and its application to solar radia3. J. E. Sherry and C. G. Justus, A simple clear-sky solar tion resource assessment in the southeastern United States. radiation model based on meteorological parameters. Solar Masters Dissertation, Atlanta, Georgia: Georgia Institute Energy 30, 425--431 (1983). of Technology (1980. 4. B. Haurwitz, Insolation in relation to cloud type. J. Meteor. 5, 110-113 (1948).