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CALIBRATION OF SOLAR POWER METER
BY LINEAR REGRESSION METHOD
Presented By
Tanisha Gaur &
Devendra Singh
Authored By
Tanisha Gaur, Devendra Singh, Anil Kumar and Prasant Baredar
Department of EnergyMaulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Introduction
• Solar power meters are used to measure solar radiation intensity coming on to the earth surface.
• Calibration is a stastical technique of enhancing the accuracy by reducing the error in the instrument’s reading.
Department of EnergyMaulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Calibration Technique Used
• Linear regression defines relationships between variables, usually under an assumption of normally distributed errors.
• Linear regression uses the fact that there is a statistically significant correlation between two variables to allow you to make predictions about one variable based on your knowledge of the other.
Department of EnergyMaulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Experimentation Setup• The solar radiation data was collected (global and
diffused radiation) by Solar power meters for one day on 11/05/2013, from 11:00 AM to 05:00 PM through both TM 206 and TM 207, simultaneously at both horizontal and inclined surfaces in MANIT Bhopal.
• The angle of inclination was set on 23o as latitude (23.2500o N) of the Bhopal. The angle of inclination of solar energy systems is set up according to the latitude of the place. Then data collected was analyzed and a relationship was established between them.
Department of EnergyMaulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Instruments Used
• Solar power meter model TM 206 made by TENMARS is considered as a reference instrument and calibration is done for model TM 207.
• In this analysis solar power meter TM 207 needs to be calibrated through linear regression method as TM 207 gives the values of solar radiation intensity on a earth surface more than 1000W/m2, which is not practically possible in partial cloudy weather conditions.
Department of Energy Maulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Tenmars TM 206
Department of EnergyMaulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Tenmars TM 207
Department of Energy Maulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Linear Regression Equation
• Relationship between global radiations of both the solar power meters on horizontal surface are expressed by means of a linear equation of the form Y = mX + c.
• Normally, it agrees to reserve “Y” for the variable, which is to be predicted in terms of other.
Department of Energy Maulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
The Coefficient Of Correlation
• Coefficient of correlation “r2” which measures the degree of association between the two values of related variables given in the data set.
Where, “X” will be values of TM 206
“Y” will be values of TM 207.
Department of Energy Maulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Calculations of Coefficient of Correlationn Tm 207
(Y)TM 206
(X)XY X2 Y2
1 967 864 835488 746496 9350892 1038 895 929010 801025 10774443 1036 925 958300 855625 10732964 1053 938 987714 879844 11088095 1015 914 925680 831744 10302256 914 788 720232 620944 8353967 832 710 590720 504100 6922248 927 802 743454 643204 8593299 742 601 445942 361201 550564
10 587 464 272368 215296 34456911 225 246 55350 60516 5062512 246 140 34440 19600 60516
SUM ∑Y= 9582 ∑X= 8285 ∑XY= 7498698
∑X2 =6539595
∑Y2 =8618086
• Now, the value of coefficient of correlation is calculated mathematically, by putting values of “X” and “Y”, in the mentioned equation.
• Value of r2 is not perfectly equal to “1”,that means there is some error between the two values of solar power meter TM 206 and TM 207, and thus the need of calibration of TM 207, arises.
Department of EnergyMaulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Error in TM 207
Global radiations of both the solar power meters
100 200 300 400 500 600 700 800 900 10000
200
400
600
800
1000
1200
f(x) = 1.08416135515408 x + 17.161375442304R² = 0.978988662174831
Series1
Linear (Series1)
Value of 206
Val
ue
of 2
07
Diffused radiations of both the solar power meters
70 80 90 100 110 120 130 140 150 160 1700
20
40
60
80
100
120
140
160
180
200
f(x) = 1.00923724519624 x + 28.5095035792701R² = 0.693139014758766
Series1Linear (Series1)
Value of 206
Val
ue
of 2
07
Department of EnergyMaulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Calibration Procedure• Now for this experimental setup, linear
equation Y=mX+c, can be solved for calculating “m” and “c” using formulaes:
Department of EnergyMaulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Calculations for Regression Constantsn Tm 207 (Y) TM 206 (X) XY X2
1 967 864 835488 746496
2 1038 895 929010 801025
3 1036 925 958300 855625
4 1053 938 987714 879844
5 1015 914 925680 831744
6 914 788 720232 620944
7 832 710 590720 504100
8 927 802 743454 643204
9 742 601 445942 361201
10 587 464 272368 215296
11 225 246 55350 60516
12 246 140 34440 19600
SUM ∑Y= 9582 ∑X= 8285 ∑XY= 7498698∑X2 =
6539595
Regression constantsSurface Global
radiation“M”
Global radiation
“C”
Diffused radiation
“M”
Diffused radiation
“C”
Horizontal 1.0776 5.737 1.2203 54.473
Inclined 1.0842 28.51 1.0092 17.161
Average value
m = 1.0809
c = 17.1235
m = 1.11475
c = 35.817
Department of EnergyMaulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Cont….
• Using above equations , Average values of “m” and “c” are calculated.
• Now, the modified calibrated values of TM 207 are calculated by putting values of average “m” & “c” in the following equation.
• Where, “X” represents modified calibrated values of TM 207.
Department of Energy Maulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Cont….• To prove the mathematical relationship of
closeness between two values, between the original and modified values of TM 207, the coefficient of correlation is again calculate.
• Now the value of “r2” is obtained “1”, that is perfect closeness between them.
Department of Energy Maulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Minimized Error in TM 207
Global radiations of both the solar power meters
100 200 300 400 500 600 700 800 900 10000
200
400
600
800
1000
1200
f(x) = 1.0809 x + 35.8155000000007R² = 1
Series1yLinear (y)
Modified calibrated value of 207
Val
ue
of 2
07
Diffused radiations of both the solar power meters
40 60 80 100 120 140 160 180 200 2200
50
100
150
200
250
300
f(x) = 1.11475 x + 17.1235000000001R² = 1
yLinear (y)
Modified calibrated value of 207
Val
ue
of 2
07
Department of EnergyMaulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
Conclusion
• Calibration of Solar power meter modal TM 207 can done using linear regression method.
• Calibration factor are calculated for both global and diffused radiation. The obtained values are: global (m = 35.817 and c = 1.0809) and diffused (m = 17.1235 and c = 1.11475).
• This study is useful for the manufacturing company TENMARS, and similar method can be adopted for calibration of other doubtful instruments. Department of Energy
Maulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
References
[1] Fourth edition, Modern Elementary Statistics, by John E. Freund professor of mathematics, Arisona state University.
[2] http://www.tenmars.com.
[3] http://www.biddle.com/documents/bcg_comp
chapter3.pdf.
[4] http://people.duke.edu/~rnau/regintro.htm.
[5] Karmel, P.H. and Polasek, M. 1986; Applied Statistics for Economists, Fourth Edition, Chapter 8,Khosla Publishing House, Delhi.
Department of EnergyMaulana Azad National Institute of Technology, Bhopal (M.P.) INDIA
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