Monthly frequency distribution of daily clearness index values in Iraq

Energy Convers. Mgmt Vol. 31, No. 1, pp. 51--60, 1991 0196-8904/91 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1991 Pergamon Press ple

M O N T H L Y F R E Q U E N C Y D I S T R I B U T I O N O F D A I L Y

C L E A R N E S S I N D E X V A L U E S I N I R A Q

M. AL-RIAHI, N. AL-HAMDANI and H. AL-SAFAR Solar Energy Research Centre, Scientific Research Council, P.O. Box 13026, Jadiriyah, Baghdad, Iraq

(Received 20 April 1988; received for publication 9 January 1990)

Abstract--The monthly statistical distribution of solar radiation is a good summary of radiation data and is a particularly well studied meteorological input for solar system design tools. The aim of the present study was to prepare the monthly reference distribution of the daily values of clearness index K r from measured data of daily global solar radiation on a horizontal surface for eight stations in Iraq over 6 yr of observation. The Fischer test (F-test) is applied to the data of every single station of months characterized by the same monthly average K T values at a level of significance ~t = 0.05. The results show that, in every station, the monthly distribution of the daily K r values depends only on their monthly average value and not on the order number of the month during the year. The study also shows that the distribution function of daily Kr values collected from different locations, and referring to a given month for several years, depends only on their mean value.

Global radiation Clearness index Frequency distribution function Fractional time distribution

I N T R O D U C T I O N

The design and performance prediction of a solar energy conversion system at a particular location requires simulation models capable of estimating their long term performance, which in turn, requires accurate information about the solar energy available and climatic conditions over the whole period at that site. Such detailed information can be obtained only from direct measurements collected at the site in question.

The output of solar thermal systems is sensitive to variations in the solar energy input. In general, the relation between this output and the input of solar energy is complex, since: (i) the energy losses, which are temperature dependent, introduce a threshold under which the system does not work; (ii) the output does not only depend upon the magnitude and variation of solar radiation input, but also upon the sequence in which the variation occurs. One way in which this dependence has been accommodated lies in identifying a typical year, generally defined as a "Reference Year". In this way, the long term performance of the solar energy system can be predicted by running the computer program over the Reference Year,

For most solar systems, knowledge of the distribution of solar radiation and relative sunshine is sufficient for calculating the performance of such systems. Barbaro et al. [1, 2] have studied the monthly reference distribution of the daily relative sunshine values and constructed generalized daily sunshine distribution curves for the Italian area. Generalized statistical distributions of daily global solar radiation have been studied by numerous authors. Liu and Jordan [3] found that, for widely spread locations, the fractional time distribution could be correlated quite well with a single parameter, the monthly average clearness index K'T. They used data for 10 yr from 27 locations in the U.S.A. and Canada to develop generalized cumulative frequency distributions of KT for five values of KT between 0.30 and 0.70. However, the meteorological data and the computational facilities available to Liu and Jordan were rather limited, and Bendt et al. [4] have recently repeated the analysis of the frequency distribution, using data for 90 stations in the contiguous U.S. with approx. 20 yr of observation at each station, to develop generalized Kr distributions. The agreement with the distributions of Liu and Jordan is generally very good. Somewhat similar studies have been carried out by several researchers in different countries with notable contributions being made by Bartoli et al. [5] for Italy, Mani [6] for India and Jimrnez [7] for Spain.

For devices provided with storage units, time sequences may be studied using the auto- correlation function. In England, Brinkworth [8] reported that the sequential characteristics of the

51

52 AL-RIAHI et al.: CLEARNESS INDEX VALUES IN IRAQ

daily insolation can be represented in simple numerical terms derived from the auto-correlation functions of a straightforward stochastic model.

However, it appears from the foregoing work that there are at least three parameters of the long term radiation which characterize the actual data: average magnitude, the variance and some measure of sequential characteristics.

The first aim of the present study is to derive the monthly frequency distribution of the daily values of clearness index for every single station in Iraq. The second purpose is to verify that, for a given station, the distributions of months characterized by the same monthly averages of clearness index are consistent to be casual samples of the same statistical distribution, i.e. the statistical distribution of daily clearness index depends only on their monthly average and not on the order number of the month during the year. Furthermore, we want to test if the distribution of KT values referring to a given month for several years is a function of their ET value only, and to this purpose, we used the Fischer test [9] at a confidence level of 5%.

DATA BASE

Iraq is located roughly between 29 and 37°N lat and has regionally variable climatic characteristics. The country is divided geographically into three regions [10]. The southern region lies between 29 and 32°N, the central region between 32 and 34°N and the northern region between 34 and 37°N.

There are 27 geographical locations distributed uniformly and representing the different climatic regions in Iraq, which report all of the meteorological data to the Iraqi Meteorological Organization. However, there are only eight stations which record the daily global solar radiation on a horizontal surface. In all stations, the daily global radiation measurements were made using a Robitzsch type bimetallic actinograph. When calibrated, the accuracy of these instruments is estimated to be better than 5%.

The present statistical analysis has been performed on data of daily global solar radiation from the eight available stations, collected over a period of 6 yr. In Table l, the geographical locations and the recording period for every station is reported.

S T A T I S T I C A L T E S T I N G P R O C E D U R E

The statistical procedure here is applied to the daily clearness index values KT defined as a ratio of the experimental daily global radiation on a horizontal surface H to the corresponding extraterrestrial radiation value ETR. The ratio Kr = H / E T R is a very good index of the sky conditions of the day, independently of the season and locality, and its values always lie in the interval [0, 1], since values close to 1 identify clear days, and values close to 0 characterize overcast days.

The daily extraterrestrial radiation ETR, is deduced by the expression:

ETR = (24/~)I~[1 + 0.033 cos(360n/365)]. [cos ~b cos 6 sin Ws + (Ws2~/360)sin ~b sin 6] (1)

Table 1. Geographical location of Iraqi stations considered in this study and durations of records

Latitude Longitude Elevation Station Symbol N E (m) Records used

Nasiriya NAS 31o05 ' 46o14 ' 3 1980-1985 Kerblla KER 32037 ' 44o01 ' 29 1980-1985 Rutba RUT 33o02 , 40°17 ' 615 1980-1985 Baghdad BAG 33°14 ' 44°14 ' 34 1980-1985 Haditha HAD 34004 ' 42044 ' 108 1979-1985t Kirkuk KIR 35028 ' 44024 ' 331 1980-1985 Sulaimaniya SUL 35033 ' 45o27 ' 853 1980-1985 Mosul MOS 36°19 ' 43o09 , 223 1980-1985

f1981, 1984 missing data.

A L - R I A H I et al.: C L E A R N E S S I N D E X V A L U E S I N I R A Q

Table 2. Results o f F-test for n ine ranges o f R r , F0 is the real statistical value, F is the corresponding theoret ical va lue takes w i t h suitable degrees o f f r eedom (nt, % ) at the ~t = 0.05 level o f s ignif icance

53

Interval N A S K E R R U T B A G H A D K I R S U L M O S

0.30 <~ R T ~< 0.34

F o . . . . . 1.600 - - 0.435 F . . . . . 5.318 - - 2.455

n I . . . . . 1 - - 6 n 2 . . . . . 8 - - 28

0.35 ~< d r ~< 0.39

F 0 . . . . . 0.677 - - 1.329 F . . . . . 2.266 - - 1.917

n I . . . . . 8 - - 12 n 2 . . . . . 35 - - 51

0.40 ..<.e~ ~< 0.44

F 0 0.208 - - 0.208 - - 1.315 1.583 4.102 1.156 F 3.885 - - 2.866 - - 5.318 2.077 5.318 1,836

n I 2 - - 4 - - 1 11 1 14 n 2 12 - - 20 - - 8 46 8 57

0.45 <~ d r <~ 0.49

F o 1.104 1.355 1.748 0.019 0.944 1,290 1,929 1,754 F 1.659 2.866 2.180 5.318 2.866 1,917 2.621 2.077

n t 21 4 8 1 4 13 5 11 n 2 86 20 36 8 20 54 24 48

0,50 <~ R x ~< 0.54

F 0 0.914 1.687 0.955 1,410 1,365 0,914 0.771 F 1.608 1.836 1.836 2,334 2.077 2.459 1.659

n t 27 14 14 7 11 6 21 n 2 111 59 57 32 47 27 86

m

m

0.55 ~< d r ~< 0.59

F 0 1.770 0.451 0,487 0.940 0.742 1.322 1.237 F 2.445 1,748 2.445 1.608 1.917 1.836 2.077

n t 6 18 6 25 12 17 10 n 2 28 76 28 104 52 72 44

f

f

0.60 ~<.¢~ ~< 0.64

F 0 - - 1.611 2.182 1.701 2.154 0~651 1.296 F - - 2.180 2,334 2.077 2.334 2.866 2.445

n I - - 8 7 11 7 4 6 n 2 - - 36 32 48 32 20 28

m

m

0.65 ~< d r ~< 0.69

F0 - - 0.586 0.276 0.902 1.470 - - 0.936 F - - 1.836 3.239 2,180 2.180 - - 1.836

n a - - 19 3 8 8 - - 17 n 2 - - 72 16 36 36 - - 71

0.70 ~< KT ~< 0.74

F 0 - - - - 1.014 F - - - - 5.318

n I - - - - 1 n 2 - - - - 8

m

where I,~ is the solar constant, n is the day number of the year given for each month, $ is the latitude and ~ is the solar declination which can be approximately expressed as:

t5 = 23.45 ° sin[360(284 + n)/365]. (2)

Ws is the sunset hour angle

Ws = cos - t ( - tan ~ tan 6). (3)

For each locality, we have grouped samples putting together data of KT that refer to months whose average values of gr fall within intervals fixed on the basis of an appropriate confidence level, ~ = 0.05.

Given I ( I >1 2) independent samples (number of months) belonging to the same range of population whose average value is equal, and i ( i = 1, 2 . . . . , I ) the code-number of each month, we want to cheek if all the population have the same distribution function, that is to cheek if the samples are homogenous. Such homogenity hypothesis was verified by application of the F-test (which holds for samples of small dimension) to the daily clearness index data for every single station.


By a suitable randomization method, we divide the ith month into J~ sets of Kq elements (i = l, 2 . . . . , I; j = 1, 2 , . . . , J;), and Sij is the variance of the ith month of the j t h set; define

Fo = n2 ZiZJ(K'J - - 1)(~h- r/)2 (4) n, F.,Ej(K,j - 1)(In S~j - ~/,)2

where n I = I - 1, n 2 = Zi(J i - 1).

X j ( K i j - l)ln S 2 X, Zj~h(K,2- 1) th= Z j ( K , j - 1 ) ' r / = E , Z j ( K q - 1 )

We can check the acceptance or rejection o f a given s ta tement (null hypothesis H0), which in our case, is " the belonging o f the I samples to the same parent popula t ion" .

The hypothesis H 0 is accepted at a level o f significance ct if:

F0 -%< F~,,,,.,2

where F = . . , . . 2 is the upper ~ point o f the F-dis t r ibut ion with tl t and n 2 degrees o f f reedom at the = 0.05 level o f significance. Values of F~,.,,. 2 are tabula ted in Ref. [9].

60

0.45 ~< KT .< 0.49 0.60 .< E T ~< 0.64

40 I- I I-,8o

%

2 0 k 1 1-60

50

40

%

20

0

80

60

%

40

I60

0.55.<RT <~ 0.59

40

0

60

% 40

20

0

60

% 4 0

zoF 1 Fzo 1 I zo

0 0.5 1.0 0 0.5 1.0 KT KT

Fig. I. Distribution functions of KT values with parameter gT for Nasiriya.

0.50. < KT~< 0.54

40

20

0.65~<KT<~ 0.69

80

0.55`<KT ~ 0.59 60

40

20

0 0.5 1.0 0 0.5 K T KT

Fig. 2. Distribution functions of K T values with parameter • ~'T for Kerbella.

1.0

AL-RIAHI et al.: CLEARNESS INDEX VALUES IN IRAQ 55

BO

40

oA

20

40

VQ

20

0

60

40

oA

20

, s I I • s

0,40~R T ~ 0.44

Vo 40

o ~ ~ 0.55 -< RT ~< 0.5g

BO

I fl , I | I ! | •

0.60 <~R" T <~0,,64 "80

"60

-4o

:eli 2 800.65<~RT<~0.69

:°o I - 4 0

- 20

L. 0.70<~ KT ~0.74

-60

% . 40

20 ~ ,. 20

o o J 0 0.5 1.0 0 0.5 1.0

KT KT Fig. 3. Distribution functions of K T values with parameter

• (T for Rutba.

o.45~ RT~ 0.49 0,60~<RT~0.64

80 m

60

0,50~ RT(O-54

40

20

0.65 ~KT .~0-6 g

80

0.55 ~KT~0.59 oAI '60

L40

20

0 0.5 1.0 0 0.5 KT K T

Fig. 4. Distribution functions of K r values with parameter gT for Baghdad.

1.0


40

*/,20

0

40

*/020

0

40

%20

40

%

20

0.40~RT ~ 0.44 O.BO ERr <~O.B4

BO

0.45<~RT~-0.49 BO

40

20

0.50(RT~(

-~<0.69

BO

0.55~ R T %

B0

40

20

0 0 0.5 1.0 0 0.5 1.0

KT KT Fig. 5. Distribution functions of KT values with parameter

K T for Haditha.

40

*/,20

40

%

20

40

O/o

20

0

BO

%40

20

OL

0

0.30~<RT~0.34 o sO, T OS4 -60 , m

o/,P40

0.35~<KT~O.3g

0.55~<KT'~<0.59

50

0.40 ~KT<~0.44

40

20

0.45~<RT~<0.4 9

% "i 60

1-40

20

0.5 1.0 0 0.5 KT KT

Fig. 6. Distribution functions of K T values with parameter K r for Kirkuk.

1.0

AL-RIAHI et al.: CLEARNESS INDEX VALUES IN IRAQ 57

60

% 40

• i r • i I ' • •

0.40~KT ~< 0.44 • i , , , - i , ,

~ 400.55~KT~< 0.59

o/,

20

40

20

0

60

40

20

60

,o 40

Izo • 0.50(RT~ 0.54

0.65 ~ K'T ~< 039

" I °/° t 40

20

0 0.5 1.0 0 0.5 1.0

K T KT

Fig. 7. Distribution functions of K T values with parameter gT for Sulaimaniya.

0,30 ~KT ~ 0.34 0.40 --< KT ~ 0.44 60 -60

%

%

m

40 .40

0 I • • • I I I I I • • i

60 - 0.35~ RT~0.39 .600 .45~KT~0.49

. .

40 ~ % -40

20 12

0 0 0.5 1.0 0 0.5

KT KT

Fig. 8. Distribution functions of K T values with parameter g'T for Mosul.

1.0

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2.44

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2.45

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2.33

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3.88

5 2.

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3.88

5 5.

318

2.62

1

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T ~

0.5

4 nt

9

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2 4

0

34

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32

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12

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12

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No

vem

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Dec

emb

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318

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1.89

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2.33

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0.82

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rn

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60 AL-RIAHI eta/.: CLEARNESS INDEX VALUES IN IRAQ

R E S U L T S AND D I S C U S S I O N

The Fischer test, based on variance comparison, has been proposed to check the acceptance or rejection of a given statement. The application of the F-test is performed for the eight stations of Table 1. The /~ r values are classified into nine intervals from 0.30 to 0.74 with a step 0.05 (e.g. 0.50 ~< . ~ ~< 0.54). By analysis of the results of the F-test presented in Table 2, it can be concluded that, at a level of significance 0t = 0.05, the null hypothesis H0 may be accepted, and then the distributions of months characterized by the same monthly averages of clearness index have the same variance and then derive from the same parent population.

In Figs 1-8, we show the general empirical distribution functions of Kr for different K'T values and for each station. In Fig. 9, we show the cumulative distribution functions for different -(T values (a, b, c . . . . . i). As a comparison, we note that the curves are analogous to those mentioned in the literature. This result is very useful in setting up a correct Reference Year; in fact, it allows one to give the correct average magnitude and the correct variance to every month of the Reference Year. Moreover, we can assign the correct distribution also to months when only their monthly values are known by the correlation with other atmospheric factors.

After that, we tested whether the distribution function of daily Kr values collected from different locations referring to a given month for several years depend only on their mean value. By applying the F-test, this hypothesis has been confirmed as shown in Table 3; therefore, a reference distribution for every month of the year can be determined. An examination of Table 3 shows that, at some intervals of interest, the real values of F0 in January, February and March give results closest to those obtained by the theoretical value which the F variable takes with the suitable degrees of freedom. This could be explained by the irregularity in Kx values in those months due to the relative differences of the climate and the variability of the characteristics of global radiation in the neighbourhood of these localities. Also from this table, we deduced that, during the winter months (November-February), the lowest Kr values belong to the 0.30-0.34 range, while the highest K" T values occur in January and belong to the 0.60-0.64 range. Similarly, during the summer months (June-August), the lowest gT values belong to the 0.444).49 range, while the highest KT values occur in July and lie within the range 0.70-0.74.

R E F E R E N C E S

1. S. Barbaro, G. Cannata and S. Coppolino, Sol. Energy 31, 63 (1983). 2. S. Barbaro, G. Cannata, S. Coppolino and E. Sinagra, Sol. Energy 33, 19 (1984). 3. Y. H. Liu and R. C. Jordan, Sol. Energy 4, 1 (1960). 4. P. Bendt, M. Collares-Pereira and A. Rabl, Sol. Energy 29, I (1981). 5. B. Bartoli, S. Catalanotti, V. Cuomo, M. Francesca, C. Serio, V. Silvestrini and G. Troise, Nuovo Cim. 2C, 222 (1979). 6. A. Mani and S. Rangarajan, Solar Radiation Over India. Allied Publishers, New Delhi (1982). 7. J. I. Jimrnez, Sol. Energy 27, 271 (1981). 8. B. J. Brinkworth, Sol. Energy 19, 343 (1977). 9. H. Scheffr, The Analysis of Variance. Wiley, New York (1959).

10. I. Ahmad, N. AI-Hamdani and K. Ibrahim, Sol. Energy 31, 29 (1983).

Documents

Monthly frequency distribution of daily clearness index values in Iraq