A comparative study of correlation functions for estimation of monthly ...

2729 Vol. 5

Indian Journal of Science and Technology

No. 5 (May 2012)

ISSN: 0974- 6846

A comparative study of correlation functions for estimation of monthly mean daily global solar radiation for Jaipur, Rajasthan (India) V.K. Marwal1, R.C. Punia1, N. Sengar2, S. Mahawar1 and P. Dashora1 Department of Physics, University of Rajasthan Jaipur-302 004, India 2 Department of Pure and Applied Physics, University of Kota, Kota-324 005, India [email protected], [email protected], [email protected], [email protected], [email protected] 1

Abstract Six empirical correlations have been employed to predict monthly mean global solar radiation on a horizontal surface from relative duration of bright sunshine hours for Jaipur (26.92° N, 75.87° E). These correlations are Angstrom-Prescott linear correlation and modified functions such as quadratic, cubic, exponential, logarithmic and power function of relative duration of sunshine hours. Predicted values of monthly mean global solar radiation were compared with observed values using statistical parameters coefficient of determination R2, mean bias error MBE and root mean square error RMSE. The cubic correlation has been found to be the best while logarithmic correlation is proved worst for Jaipur. The proposed power correlation shows the values of statistical parameters close to cubic correlation and has a simpler form. Keywords: Global solar radiation, Sunshine hours, Empirical correlation, Statistical parameters. Introduction is suitable region for installation of solar energy devices. Knowledge of solar radiation is necessary to decide Intensity of global solar radiation in ranging between 6the design and installation parameters of any solar 6.4 KWH/m2/day in about half of Rajasthan, but literature energy systems at particular place. Since solar radiation survey reveals that very few studies have been done measuring instruments have high cost, high maintenance about solar radiation correlations for this region except for and calibration requirement, only scant solar radiation Jodhpur (26.30oN,73.02oE) which is situated in west data is available in India. However, information about region of Rajasthan (Modi & Sukhatme 1979). Jaipur solar radiation can also be achieved using empirical (26.92° N, 75.87° E) is the capital city of Rajasthan and correlations between solar radiation and other having high annual average of mean daily global solar meteorological and geographical parameters. Several radiation (19.42 MJ/m2/day) which is comparable to such attempts have been made earlier e.g. Kasten and annual average of mean daily global solar radiation at Czeplak (1980) have correlated monthly mean global Jodhpur (19.97 MJ/m2/day). solar radiation with clouds amount. Zhou et al. (2005) Jaipur is also an appropriate place for implementation have expressed monthly mean global solar radiation as a of solar energy devices. The aim/object of the present function of latitude and altitude. Others (Hawas & Munar, paper is to validate and compare various correlations to 1983; Garg & Garg, 1985; Holouani et al.,1993; Almorox predict monthly mean daily global solar radiation on a & Hontoria, 2004; Samual, 1991) have proposed Table 1. Various correlations used in this study correlations of monthly mean global solar radiation with Correlations Regression equations relative duration of sunshine hours for different places H  S  around the world. Such relations have several constants Linear  a  b  Ho  So  which need to be evaluated for specific meteorological and geographical conditions. 2 H S  S  In prediction of monthly mean global solar radiation, Quadratic  a  b   c  the most widely used correlation is a linear relationship Ho  So   So  between the ratio of mean daily global solar radiation to 2 3 H S  S  S  the corresponding value on a clear sky day and the ratio Cubic  a  b   c   d   of mean daily sunshine duration to the maximum possible Ho  So   So   So  sunshine duration proposed by Angstrom (1924). This   S  H correlation was modified by Prescott (1940), who Exponential  a. exp b   eliminated the problem of computing clear sky radiation Ho   So   by replacing clear sky radiation with extraterrestrial H S  radiation. Logarithmic  a  b. log   In India also various researchers developed and Ho  So  investigated correlations in monthly mean global solar b radiation and relative duration of sunshine (Modi & H  S  Power  a   Sukhatme, 1979; Garg & Garg, 1985). Due to high solar Ho  So  *proposed by Authors potential and large number of clear sky days, Rajasthan Research article Indian Society for Education and Environment (iSee)

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V.K.Marwal et al. Indian J.Sci.Technol.

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Indian Journal of Science and Technology horizontal surface using relative duration of sunshine at Jaipur located in north-east region of Rajasthan. These correlations would also be applicable for a number of states receiving high solar potential, in underdeveloped and developing countries having same meteorological and geographical conditions. Table 2. Various statistical parameters Statistical parameter MBE

n

 H

ic 

n

RMSE

H im 

i 1

 H

ic 

H im 

2

i 1

n n

 H

ic 

Hac Him  Ham 

i 1

R2

n

n

i 1

i 1

2 2  Hic  Hac   Him  Ham 

Data In this study data have been supplied by India Meteorological Department (IMD) from 1987-2002 for monthly mean of daily global solar radiation and monthly mean of daily sunshine duration for Jaipur. Methodology Several correlations have been found in literature to estimate global solar radiation. Some of them were selected and developed to predict monthly mean daily global solar radiation at horizontal surface at Jaipur. The correlations used in this study are shown in Table 1. Among these correlations, the linear, quadratic, cubic, exponential and logarithmic correlation have been used earlier also for different places (Akinoglu & Ecevit, 1990; Almorox & Hontoria, 2004), while power correlation has been proposed by us. In above correlations a, b, c and d are regression coefficients which have been found using least square method by fitting available data. H is monthly mean daily global solar radiation on a horizontal surface, Ho is the monthly mean daily extraterrestrial global solar radiation on same surface, S is monthly mean daily sunshine duration and So is monthly mean daily maximum possible sunshine duration. The extraterrestrial solar radiation was computed using the following equation:

Ho 

24 ' I scs sin sin  cos cos sins1 

Where

360 n   I ' sc  Isc  1  0 . 033    2  365   2 Here Іsc is the solar constant (1367 W/m ) and n is st number of the day (n=1 for 1 Jan.). In equation (1)  is Research article Indian Society for Education and Environment (iSee)

ISSN: 0974- 6846

the latitude for place  is the solar declination angle and ωs is the sunset hour angle at the place. The solar declination angle and sunset hour angle computed from following equation:

 360 284  n   23.45 sin ...........(3) 365  

s  tan  tan  ............(4)

Formula

1 n

No. 5 (May 2012)

Performance of various correlations Performance of the various correlations were evaluated in terms of coefficient of determination (R2), mean bias error(MBE) and root mean square error(RMSE).These statistical parameters are computed by equations shown in Table 2. Where Him is the ith measured value of monthly mean global radiation on a horizontal surface while Hic is the ith computed value monthly mean global radiation on the same surface, Ham average of all measured values of monthly mean global radiation, Hac is the average of all computed values of monthly mean global radiation and n is the number observations. The goodness of fitting of a correlation was judged by coefficient of determination R2. For a better correlation R2 should have a higher value (Ideal value is 1 or 100%). Mean Bias Error MBE provides information on the long-term performance of correlation. A positive MBE represents an over estimation whereas a negative MBE shows an under estimation. A low value of MBE is desired for a good correlation (ideal value is 0). Root mean square error RMSE provides information on the short-term performance of correlation. The lower value of RMSE represent better estimation (Ideal value is 0). Results and discussion Regression coefficients appeared in correlation equations (as shown in Table1) have been found for Jaipur using least square method by fitting available data of monthly mean daily global radiation and bright sunshine hours at Jaipur. Computed values of monthly mean daily global solar radiation using empirical correlations have been compared with observed values as shown in Fig.1-6. Each correlation with their computed 2 regression coefficients and statistical parameters (R , MBE and RMSE) has been shown in Table 3. On comparing the obtained results for various correlations, we can say that all the correlations give good results. The 2 coefficient of determination (R ) is more than 80% for each correlation, so all the correlations fit the data adequately. Thus, these can be used for the prediction of radiation values at Jaipur. Table 3 clearly shows that the cubic correlation has 2 maximum value of R (85.51%) and minimum value of errors likes MBE and RMSE, so cubic correlation gives the best estimation of the global solar radiation at Jaipur. On the other hand, the logarithmic correlation has 2 minimum value of R (80.05) and maximum value MBE and RMSE. Thus, it can be seen that logarithmic correlation seems to be the worst correlation among all

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Fig.3.Comparison between observed and computed global solar radiation using third degree correlation

Research article Indian Society for Education and Environment (iSee)

ISSN: 0974- 6846

Fig.4. Comparison between observed and computed global solar radiation using exponential correlation

Fig.1.Comparison between observed and computed global solar radiation using linear correlation

Fig.2.Comparison between observed and computed global solar radiation using quadratic correlation

No. 5 (May 2012)

Fig.5.Comparison between observed and computed global solar radiation using logarithmic correlation

Fig.6.Comparison between observed and computed global solar radiation using power correlation

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V.K.Marwal et al. Indian J.Sci.Technol.

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Indian Journal of Science and Technology

Correlation Linear Quadratic

Table 3. Correlations with their computed regression coefficients. and statistical parameters R2 MBE Regression Equation (MJ/m2/day) H  S   0.2687  0.4885   0.8050 -0.0753 Ho  So 

H S   S   0.0861  1.1584   0.5555  Ho S o    So  2

Cubic

No. 5 (May 2012)

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RMSE (MJ/m2/day) 1.3073

2

H S  S  S   0.3462  0.4222   2.329   1.634  Ho  So   So   So 

0.8423

-0.0396

1.1997

0.8551

-0.0363

1.1425

3

Exponential

 H  S   0.3158. exp 0.9251  Ho  So  

0.8006

-0.1520

1.4223

Logarithmic

H  S   0.7166  0.2705. log   Ho  So 

0.8005

-0.8368

1.6393

0.8517

-0.1086

1.2390

Power

H S   0.7399  Ho  So 

0.5201

these six correlations. A careful analysis of Table 3 clearly shows that the power and the quadratic correlations would also predict mean monthly global radiation with good confidence level since the values of statistical parameters are close to those for cubic correlation and are better than other correlations. Power correlation has only two regression coefficients while the cubic correlation has four regression coefficients. Thus, power correlation has a much simpler form as compared to the cubic correlation and it is much easy for calculation. Conclusions and scope for future work Six empirical correlations based on relative duration of sunshine hours have been employed to predict monthly mean daily global solar radiation on horizontal surface for Jaipur. Regression coefficients have been calculated and statistical parameters (R2, MBE and RMSE) have been computed. It has been found that among these correlations cubic correlation is the best while logarithmic correlation has proved to be worst for Jaipur, Rajasthan (India). The proposed power correlation shows the values of statistical parameters close to cubic and has a simple form. Thus due to simplicity, power correlation is also suggested along with cubic correlation for calculation of mean global radiation at Jaipur. Further, in future, these correlations would be compared for different places having good solar potential such as Jodhpur, and other areas of Rajasthan and Gujarat, and improvements in power correlation would be reported. Acknowledgement We are thankful to India Meteorological Department (IMD), Pune for providing meteorological data. References 1. Angstrom A (1924) Solar and terrestrial radiation. Quart. J. Roy. Met. Soc. pp: 50121-501126. Research article Indian Society for Education and Environment (iSee)

2. Prescott JA (1940) Evaporation from a water surface in relation to solar radiation. Trans. Roy. Soc. Aust. pp: 64114-61148. 3. Almorox J and Hontoria C (2004) Global solar radiation estimation using sunshine duration in Spain. Energy Conver. Mgmt. 45, 1529-1535. 4. Zhou Jin, Yeheng Wu and Gang Yan (2005) General formula for estimation of monthly average daily global solar radiation in China. Energy Conver. Mgmt. pp: 46257-46268. 5. Hawas MM and Munar T (1983) Correlation between global solar radiation and sunshine data for India. Solar Energy. 30(3),289-290. 6. Holouani N,Nguyen and CTVo-Ngoc D (1993) Calculation of monthly average global solar radiation on horizontal surface using daily hours of bright sunshine. Solar Energy. 50(3), 247-258. 7. Kasten F and Czeplak G (1980) Solar and terrestrial radiation dependent on the amount and type of cloud. Solar Energy. 24, 177-189. 8. Garg HP and Garg SN (1985) Correlation of monthly average daily global, diffuse and beam radiation with bright sunshine hours. Energy Conver. Mgmt. 25(4), 409-417. 9. Modi VS and ukhatme SP (1979) Estimation of daily total and diffuse isolation in India from weather data. Solar Energy. 22, 407-411. 10. Akinoglu BG and Ecevit A (1990) Construction of a quadratic model using modified Angstrom coefficients to estimate global solar radiation. Solar Energy. 45(2), 85-92. 11. Ertekin C and Yaldiz O (2000) Comparison of some existing models for estimating global solar radiation for Antalya (Turkey). Energy Conver. Mgmt. 41(4), 311330. 12. Samual TDMA (1991) Estimation of global solar radiation for Srilanka. Solar Energy. 47(5), 333-337.

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