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A novel approach to estimate the clear day global radiation
RenewableEnergy Vol. 1, No. 1, pp. 119-123, 1991 Printed in Great Britain.
0960 1481/91 $3.00+.00 Pergamon Press plc
D A T A BANK A novel approach to estimate the clear day global radiation A . BAIG, P . AKHTER a n d
A. MUFTI
National Institute o f Silicon Technology, No. 25, H-9, Islamabad, Pakistan
(Received 4 December 1989 ; accepted 18 December 1989) Abstract--A model based on the modified version of Gaussian distribution function has been proposed to estimate the clear day global radiation. The model fits very well on the recorded data which is further exploited to develop a relationship between the ratio of hourly to daily global radiation and day length (time between sunrise to sunset). This has helped us to calculate the distribution of the broad-band global radiation on any clear day of the year.
1. INTRODUCTION The exact knowledge of solar radiation and its pattern on hourly basis is becoming more and more important due to the development of new technologies and application of solar energy. Its importance has been highlighted in many papers [1-3]. Also because of unavailability of recorded radiation data at many places, a number of models have been developed to estimate the daily sums; to draw curves of ratios of hourly to daily radiation for major parts of the day and then to extend these towards sunrise and sunset [4-8]. In such an attempt Jain [8] has proposed a Gaussian function to fit the experimental daily radiation data using the following equation :
1
r, = ax/2rt exp (-- (t-- 12)2/2er2).
(1)
Here r, is the ratio of hourly to daily global radiation, t is the local time (in hours) and a is the S.D. obtained by matching the experimental recorded value of r, at t = 12 (noon time) in eq. (1). However the fitting is rather poor and equation gives mismatch in major parts of the day except very near to the noon time. This mismatch is quite large near sunrise and sunset and is also clear in the author's own work. We have introduced a correcting factor which gives much better fit to the experimental data throughout the day. Furthermore we have also developed an approach to estimate the hourly global radiation on any day o f the year from sunrise to sunset. The model gives good fit to the experimental data.
Daily global radiation, Hg (kWH/m2), available on any day has been calculated as :
Hg = tA ~ (io) i. )
(2)
Here ig (kW/m 2) is the (measured) value of global irradiance in mid of the j t h interval of time, tA, which in our case is taken equal to 0.5 h each and summation is from sunrise to sunset. Values of hourly global radiation, Ig (kWH/mZ), have been taken on half hourly basis. It has been assumed that the value of ia does not vary significantly within _+ 15 min and the instantaneous value of ig in the middle of the time interval (each equal to half an hour) represents the hourly global radiation within that interval of time. The ratio of hourly to daily global radiation has been calculated using the following equation :
r, = Ig/Hg.
(3)
We have applied Jain's approach [8] to calculate rt for our experimental data. The results of only six selected months are shown (as dotted curves) in Fig. 1. The theoretical values so obtained are almost always less than the recorded values for the main part o f the day. The mismatch is much wider during early mornings and late afternoons when the model gives values much higher than the recorded data. This is because eq. (1) decreases asymptotically with the time and becomes zero only at infinity whereas practically there is no radiation before sunrise or after sunset.
3. O U R P R O P O S E D M O D E L 2. ANALYSIS OF E X P E R I M E N T A L DATA The radiation recording station at the National Institute of Silicon Technology, Islamabad (33.73°N, 73.08°E and at elevation of 150.80 m) is routinely used to record irradiance components [9]. The global radiation data recorded on a half hourly basis on every 15th or nearest clear day of each month from September, 1986 to March, 1989 has been analysed first on Jain's model [8] i.e. using eq. (1) and then on our proposed model. A number of parameters used in our model are defined as under.
We have introduced a correction factor to Jain's model [8], so that the new proposed empirical relationship is, r, --
1
~ {exp ( -- (t -- 12) 2/20"2) 2a~/2n +cos(180°(t-12)/(So--1))}
(4)
where (i) 0" is again the constant whose value is obtained by using noon time (t = 12) recorded value of r, in eq. (4) which 119
Data Bank
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.
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9 10 11 iz
Fig. 1. Plot of the ratio of hourly to daily global radiation, r,, against the local time.
gives, 1
(5)
rt= 1 2 ~
This value, in principle, is the same as obtained from eq. (1). (ii) So (hours) is the daylength, i.e. the time between sunrise to sunset and is obtained from the following two standard equations [10] 6 = 23.45 sin {360°(n + 284)/365}
(6)
2 S0 = 15 arccos ( - tan ~btan 6).
(7)
Here 6 represents declination angle, n is day of the year (n = 1 for 1 January) and ~b stands for latitude of the site.
Once the values of a and So are known, we can use eq. (4) to trace the theoretical curves which are shown by solid lines in Fig. 1. The results show a reasonably good fit to the experimental data points throughout the day, i.e. from sunrise to sunset• However occasionally, the measured data shows a slight misfit on certain points. This is primarily because of the asymmetry of the observed data due to changes in the atmospheric turbidity during the day. We believe that such slight and occasional misfit can be improved if one uses data averaged over many years instead of single day data. 4. E S T I M A T I O N OF CLEAR DAY GLOBAL RADIATION To estimate r t (ratio of hourly to daily global radiation) at any time of the day from sunrise to sunset we need the
Data Bank
! 21
9.
3.6 3-4
3.2
~
•
T A
•
2.8
2.6 2.4
e • •
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8.
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6.
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1'2 '
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13 '
2
14 " 15
'S_'(Hrs)
Hg(Meosurod ) (kWH/rn z )
Fig. 2. Variation in the constant e with day length.
Fig. 4. Comparison of estimated and experimental values of daily global radiation.
value of a for the respective day ; then eq. (4) can be used to find out the theoretical values of r, at any time of the day. For clear days the value of a, as obtained and discussed in Section 3, are plotted in Fig. 2 against the daylength So. The plot shows practically a linear behaviour and solid line is the linear regression of all the data points. Empirically we can write : (8)
a = ASo+B
where A = 0.21 +0.02 and B = 0.26_+0.18 are the slope and y-intercept respectively. Similar behaviour of a as function of So has also been reported earlier by Jain [8] for recorded data for Trieste, Italy. Figure 2 shows that most of the points lie quite close to the line. However, some data points such as M or N show some deviation from the line or theoretical values. The practical data points which show deviation of 5% or more are about 25% and those points which show deviation more than 10% are only 10% of our total data points recorded so far. Once t7 is known from eq. (8) then using eq. (4) one can theoretically find out the values of r, at any time of the day (as has been explained earlier in Section 3). Obviously the plots of r, (calculated) against the local time will give extremely good fit to practical data for all those days for
(a)
which the value of a lies on or close to the line in Fig. 2. In other words the practical value of a, in such cases, will give good fit to eq. (8). Such plots are already shown in Fig. 1 and discussed. However for the days such as M or N as shown in Fig. 2 where measured values deviate from eq. (8) the fit may not be as good. This has been shown in Fig. 3 for the 2 days (M and N) that show maximum deviation from theoretical values. The figure shows that even in these cases of maximum discrepancies between experimental and theoretical calculated values, the fit is very good for mornings and late afternoons. The misfit is only in the middle of the day and is equal to I0% at noon time. The frequency of such days is rather smaller and is only 10% of our total data points as discussed earlier. One has to realise that the values o f r t have little utilization unless exploited further to calculate other, practically more important parameters such as the aggregated values o f solar radiation of a given day or values of hourly global radiation at any time of the day. These values for any clear day now can easily be calculated by using the estimated values of r, in eq. (3) provided we know the hourly solar radiation value at any one time (say noon time) of that day. We have used the measured noon time hourly radiation values and estimated daily global radiation, Ho. These estimated values have been
(b}
MAY f5,1988
[0.18
JANUARY 14,1989 (Point N inFig.2)
r.o.16
(Point M in Fig.2)
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6
8
10
12 t (Hrs)
14
16
18
20
4
6
8
10
12 1(Hrs)
14
16
18
Fig. 3. Plot o f r, vs local time for 2 days which show maximum misfit between theory and experiment (for details see text).
20
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0-0 . . . . . . . . . . . . :, , , , 0400 0600 0800 1000 1200 1400 1600 1800 2000 TIME(Hrs)
TIME (Hrs}
Fig. 5. Hourly global radiation on selected days.
plotted against the practically measured values in Fig. 4 which shows that most of the estimated values are very close to the measured values except for a few points such as M or N which are in fact the points already shown in Fig. 2 and discussed. This explains that the errors in the estimated values of Hg have mainly come from the estimation of the values. We have gone one step further and used the estimated values of H a and r, in eq. (3) to calculate the hourly global radiation Ig at any time of the clear day. Such estimated values for selected days (same as in Fig. 1) have been plotted in Fig. 5 along with the measured values. These plots also show very good fit to the experimental data over the year and build further confidence in the validity of this approach.
R E F E R E N C E S
1. Peter J. Robinson and Howard L. Hill, Towards a policy for climate impacts. Bull. Am. Meteorol. Soc. 68, 769772 (1987). 2. S. R a h m a n and B. H. Chowdhury, Simulation of photovoltaic power systems and their performance prediction. IEEE Transactions on Ener#y Conversion 3, 440446 (1988). 3. S. X. Cheng and X. S. Ge, Preliminary research in the measurement of solar radiation by transient technique. Solar Energy 30, 391-395 (1983)• 4. A. Whillier, The determination o f hourly values of total solar radiation from daily summation. Archs Meteorol. Geophys. Bioklimatol., Ser. B. 7, 197-204 (1956)•
Data Bank 5. H. C. Hottel and A. Whillier, Evaluation of flat plate solar collector performance. Transactions of the Conference on the use of solar energy : the scientific basis. Vol. II(1), Section A, pp. 74-104 (1955). 6. B. Y. H. Liu and R. C. Jordan, The interrelation and characteristic determination of direct, diffuse and total solar radiation. Solar Energy 4, 1-19 (1960). 7. K.K. Gopinathan, Diurnal variation of the hourly hemispherical insolation. Solar Wind Technol. 5, 661--665 (1988). 8. P. C. Jain, Comparison of techniques for the estimation
123
of daily global irradiation and a new technique for the estimation of global irradiation. Solar Wind Technol. I, 123-134 (1984). 9. P. Akhter, A. Baig, I. Mazhar, M. Usman and A. Mufti, Measurement of solar irradiance at Islamabad. International symposium-workshop on Silicon technology development and its role in the sun-belt countries, 1418 June 1987, Islamabad, pp. CN-1-9 (1987). 10. J. Duffle and W. A. Bechman (Editors) Solar Engineering of Thermal Processes, Chap. 2, John Wiley & Sons Inc., New York (1980).
0960 1481/91 $3.00+.00 Pergamon Press plc
D A T A BANK A novel approach to estimate the clear day global radiation A . BAIG, P . AKHTER a n d
A. MUFTI
National Institute o f Silicon Technology, No. 25, H-9, Islamabad, Pakistan
(Received 4 December 1989 ; accepted 18 December 1989) Abstract--A model based on the modified version of Gaussian distribution function has been proposed to estimate the clear day global radiation. The model fits very well on the recorded data which is further exploited to develop a relationship between the ratio of hourly to daily global radiation and day length (time between sunrise to sunset). This has helped us to calculate the distribution of the broad-band global radiation on any clear day of the year.
1. INTRODUCTION The exact knowledge of solar radiation and its pattern on hourly basis is becoming more and more important due to the development of new technologies and application of solar energy. Its importance has been highlighted in many papers [1-3]. Also because of unavailability of recorded radiation data at many places, a number of models have been developed to estimate the daily sums; to draw curves of ratios of hourly to daily radiation for major parts of the day and then to extend these towards sunrise and sunset [4-8]. In such an attempt Jain [8] has proposed a Gaussian function to fit the experimental daily radiation data using the following equation :
1
r, = ax/2rt exp (-- (t-- 12)2/2er2).
(1)
Here r, is the ratio of hourly to daily global radiation, t is the local time (in hours) and a is the S.D. obtained by matching the experimental recorded value of r, at t = 12 (noon time) in eq. (1). However the fitting is rather poor and equation gives mismatch in major parts of the day except very near to the noon time. This mismatch is quite large near sunrise and sunset and is also clear in the author's own work. We have introduced a correcting factor which gives much better fit to the experimental data throughout the day. Furthermore we have also developed an approach to estimate the hourly global radiation on any day o f the year from sunrise to sunset. The model gives good fit to the experimental data.
Daily global radiation, Hg (kWH/m2), available on any day has been calculated as :
Hg = tA ~ (io) i. )
(2)
Here ig (kW/m 2) is the (measured) value of global irradiance in mid of the j t h interval of time, tA, which in our case is taken equal to 0.5 h each and summation is from sunrise to sunset. Values of hourly global radiation, Ig (kWH/mZ), have been taken on half hourly basis. It has been assumed that the value of ia does not vary significantly within _+ 15 min and the instantaneous value of ig in the middle of the time interval (each equal to half an hour) represents the hourly global radiation within that interval of time. The ratio of hourly to daily global radiation has been calculated using the following equation :
r, = Ig/Hg.
(3)
We have applied Jain's approach [8] to calculate rt for our experimental data. The results of only six selected months are shown (as dotted curves) in Fig. 1. The theoretical values so obtained are almost always less than the recorded values for the main part o f the day. The mismatch is much wider during early mornings and late afternoons when the model gives values much higher than the recorded data. This is because eq. (1) decreases asymptotically with the time and becomes zero only at infinity whereas practically there is no radiation before sunrise or after sunset.
3. O U R P R O P O S E D M O D E L 2. ANALYSIS OF E X P E R I M E N T A L DATA The radiation recording station at the National Institute of Silicon Technology, Islamabad (33.73°N, 73.08°E and at elevation of 150.80 m) is routinely used to record irradiance components [9]. The global radiation data recorded on a half hourly basis on every 15th or nearest clear day of each month from September, 1986 to March, 1989 has been analysed first on Jain's model [8] i.e. using eq. (1) and then on our proposed model. A number of parameters used in our model are defined as under.
We have introduced a correction factor to Jain's model [8], so that the new proposed empirical relationship is, r, --
1
~ {exp ( -- (t -- 12) 2/20"2) 2a~/2n +cos(180°(t-12)/(So--1))}
(4)
where (i) 0" is again the constant whose value is obtained by using noon time (t = 12) recorded value of r, in eq. (4) which 119
Data Bank
120 (a)
(b)
FEBRUARY 15,1988
•
• 0.18
APRIL13,1987
- 0-18
. . . . Join [8] -0.16 o E X work / % o T h .i .14 s
T2'swork f ,o.1o •
0.O8
"~•
• 0.o6
~e
~
- 0.06
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.
.
.
.
.
0.04
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.
,
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 t (Hours)
(d) AUCKJST17,1988
0.18 0.16
. . . . Join[8]
• 0.18 ,0.16
. . . . Join [8] This work • E x p t . /
0.14
This work
~
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 t (Hours)
(c) JUNE14,1987
"~
0-02
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5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 t (Hours}
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(e) OCTOBER14,1987 . . . . Join [8] • ?is .work
5-;
- 0.18
@ _~ "
• 0-16
- 0.18
. . . . Join [8] ~ This work • Expl. ~"
~~1"~4
/
'~
.0.O6
DECEMBER15,1987
0.04
0?\
1/
"¢
.0-12 I1~ ~i--"- 0-10 0.08 \~_ .0.o6 - 0~14
,
.
5
6
7
8
9
10
~,\
"~,,,
• 0.o2
5 6 7 8
11 12 13 14 15 16 17 18 19 t(Hours)
.
.
.
mll
1314 1516 17 1'81~ t (Hours)
9 10 11 iz
Fig. 1. Plot of the ratio of hourly to daily global radiation, r,, against the local time.
gives, 1
(5)
rt= 1 2 ~
This value, in principle, is the same as obtained from eq. (1). (ii) So (hours) is the daylength, i.e. the time between sunrise to sunset and is obtained from the following two standard equations [10] 6 = 23.45 sin {360°(n + 284)/365}
(6)
2 S0 = 15 arccos ( - tan ~btan 6).
(7)
Here 6 represents declination angle, n is day of the year (n = 1 for 1 January) and ~b stands for latitude of the site.
Once the values of a and So are known, we can use eq. (4) to trace the theoretical curves which are shown by solid lines in Fig. 1. The results show a reasonably good fit to the experimental data points throughout the day, i.e. from sunrise to sunset• However occasionally, the measured data shows a slight misfit on certain points. This is primarily because of the asymmetry of the observed data due to changes in the atmospheric turbidity during the day. We believe that such slight and occasional misfit can be improved if one uses data averaged over many years instead of single day data. 4. E S T I M A T I O N OF CLEAR DAY GLOBAL RADIATION To estimate r t (ratio of hourly to daily global radiation) at any time of the day from sunrise to sunset we need the
Data Bank
! 21
9.
3.6 3-4
3.2
~
•
T A
•
2.8
2.6 2.4
e • •
3.0 b
8.
oM
.E_
S
2-2
3
iN
2.0
6.
' 10 '
1'1 '
1'2 '
Dcty length
13 '
2
14 " 15
'S_'(Hrs)
Hg(Meosurod ) (kWH/rn z )
Fig. 2. Variation in the constant e with day length.
Fig. 4. Comparison of estimated and experimental values of daily global radiation.
value of a for the respective day ; then eq. (4) can be used to find out the theoretical values of r, at any time of the day. For clear days the value of a, as obtained and discussed in Section 3, are plotted in Fig. 2 against the daylength So. The plot shows practically a linear behaviour and solid line is the linear regression of all the data points. Empirically we can write : (8)
a = ASo+B
where A = 0.21 +0.02 and B = 0.26_+0.18 are the slope and y-intercept respectively. Similar behaviour of a as function of So has also been reported earlier by Jain [8] for recorded data for Trieste, Italy. Figure 2 shows that most of the points lie quite close to the line. However, some data points such as M or N show some deviation from the line or theoretical values. The practical data points which show deviation of 5% or more are about 25% and those points which show deviation more than 10% are only 10% of our total data points recorded so far. Once t7 is known from eq. (8) then using eq. (4) one can theoretically find out the values of r, at any time of the day (as has been explained earlier in Section 3). Obviously the plots of r, (calculated) against the local time will give extremely good fit to practical data for all those days for
(a)
which the value of a lies on or close to the line in Fig. 2. In other words the practical value of a, in such cases, will give good fit to eq. (8). Such plots are already shown in Fig. 1 and discussed. However for the days such as M or N as shown in Fig. 2 where measured values deviate from eq. (8) the fit may not be as good. This has been shown in Fig. 3 for the 2 days (M and N) that show maximum deviation from theoretical values. The figure shows that even in these cases of maximum discrepancies between experimental and theoretical calculated values, the fit is very good for mornings and late afternoons. The misfit is only in the middle of the day and is equal to I0% at noon time. The frequency of such days is rather smaller and is only 10% of our total data points as discussed earlier. One has to realise that the values o f r t have little utilization unless exploited further to calculate other, practically more important parameters such as the aggregated values o f solar radiation of a given day or values of hourly global radiation at any time of the day. These values for any clear day now can easily be calculated by using the estimated values of r, in eq. (3) provided we know the hourly solar radiation value at any one time (say noon time) of that day. We have used the measured noon time hourly radiation values and estimated daily global radiation, Ho. These estimated values have been
(b}
MAY f5,1988
[0.18
JANUARY 14,1989 (Point N inFig.2)
r.o.16
(Point M in Fig.2)
Using 0" votues from • Experimental dota
r
"0.14 r
Using 0" volues from • Experimental dora
e_°_
./
!0.18
012 040
\
.0.06
o.o2.. ,\. 4
6
8
10
12 t (Hrs)
14
16
18
20
4
6
8
10
12 1(Hrs)
14
16
18
Fig. 3. Plot o f r, vs local time for 2 days which show maximum misfit between theory and experiment (for details see text).
20
Data Bank
122 (a) 10-
1.0
FEBRUARY15,1988 Th~swork
08
•
(b) APRIL13,1987 _ This work
~
"~•
Expt.
0.6-
0-4
~
/
0.2-
00 . . . . . 0400 0600 08O0 1000
. 0.20-0
120(] 1400
1600
18(](]
.
040(3 0600
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1000
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(C) JUNE 14,1987 This work Expt
•
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~,,=
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0600 0800 1000 1200 1400 1600 1800 2000
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. 0600 080(3 1000
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1.0-
OCTOBER14,1987 - - lhis work •
1200
. 1400
.
. 1600
. 1800
2(]0(3
TIME (Hrs)
TIME (Hrs)
0.8-
1600
J<
~" 0-4.
1.0 ~
1400
TIME {Hrs)
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(f)
DECEMBER15,1987 __ This work • Expt
06
~'~ 0.6-
0.4.~ 0,2-
T.,~,
0.0 0400 0600 0800
1000
0.2-
120(} 1400
16(]0 1800 '2000
0-0 . . . . . . . . . . . . :, , , , 0400 0600 0800 1000 1200 1400 1600 1800 2000 TIME(Hrs)
TIME (Hrs}
Fig. 5. Hourly global radiation on selected days.
plotted against the practically measured values in Fig. 4 which shows that most of the estimated values are very close to the measured values except for a few points such as M or N which are in fact the points already shown in Fig. 2 and discussed. This explains that the errors in the estimated values of Hg have mainly come from the estimation of the values. We have gone one step further and used the estimated values of H a and r, in eq. (3) to calculate the hourly global radiation Ig at any time of the clear day. Such estimated values for selected days (same as in Fig. 1) have been plotted in Fig. 5 along with the measured values. These plots also show very good fit to the experimental data over the year and build further confidence in the validity of this approach.
R E F E R E N C E S
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