June 1998 • NREL/TP-560-24833

Improving the Accuracy of Using 

Pyranometers to Measure the Clear 

Sky Global Solar Irradiance 

Ibrahim Reda 

National Renewable Energy Laboratory 

1617 Cole Boulevard 

Golden, CO 80401 

National Renewable Energy Laboratory 

1617 Cole Boulevard 

Golden, Colorado 80401-3393 

A national laboratory of the U.S. Department of Energy 

Managed by Midwest Research Institute 

for the U.S. Department of Energy 

under contract No. DE-AC36-83CH10093 
June 1998 • NREL/TP-560-24833

Improving the Accuracy of Using 

Pyranometers to Measure the Clear 

Sky Global Solar Irradiance 

Ibrahim Reda 

National Renewable Energy Laboratory 

1617 Cole Boulevard 

Golden, CO 80401 

National Renewable Energy Laboratory 

1617 Cole Boulevard 

Golden, Colorado 80401-3393 

A national laboratory of the U.S. Department of Energy 

Managed by Midwest Research Institute 

for the U.S. Department of Energy 

under contract No. DE-AC36-83CH10093 

Prepared under Task No. 56002010 

June 1998
NOTICE 

This report was prepared as an account of work sponsored by an agency of the United States 

government. Neither the United States government nor any agency thereof, nor any of their employees, 

makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, 

completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents 

that its use would not infringe privately owned rights. Reference herein to any specific commercial 

product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily 

constitute or imply its endorsement, recommendation, or favoring by the United States government or any 

agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect 

those of the United States government or any agency thereof. 

 
ABSTRACT 

Pyranometer users have customarily applied one responsivity value when calculating the global 

solar irradiance. Usually, the responsivity value is reported by either the manufacturer or a 

calibration facility. Many pyranometer calibrations, made both at NREL and elsewhere, have 

shown that the responsivity of a pyranometer changes with the change in solar zenith and 

azimuth angles. Depending on how well the pyranometer sensor is radiometrically leveled, these 

changes can exceed AS% of the reported responsivity, which means that errors in the calculated 

global solar irradiance can exceed &5% from the nominal values. This paper describes a method 

to decrease the errors resulting fiom the change of the solar zenith angle under clear sky 

conditions. Two responsivity functions, morning and afternoon, were used instead of one 

responsivity value. The two functions have been chosen because of asymmetry of the morning 

and afternoon cosine responses demonstrated by some pyranometers. 
Acknowledgments 

I wish to thank Cede Warner for making this publication possible; my colleagues, Daryl 

Myers and Tom Stoffel, for their continuous radiometry contributions and teachings; and my 

colleague, Steve Wilcox, for developing the RCC sohare that made this paper possible. 

Additionally, I would like to thank NREL staff members who provided help with this 

publication, specifically Bev Kay and Rene Howard. 
Contents 

Chapter 

2.0Procedure ................................................................. 5 

3.0 Results for EPLAB PSP, Serial Number 25825F3 ................................ 10 

4.0Conclusions ..............................................................17 

5.0References ...............................................................18 

Tables 

Table 1.1 Bin versus Responsivity for EPPLEY PSP 25825F3 from BORCAL9702 Report ... 1 

Table 3.1.1 Data Points Used to Solve the Equations ................................. 10 

Table 3.2.1 Morning and Afternoon Coefficients for 25825F3 .......................... 11 

Table 3.4.1 Percentage Errors Resulting from Using RS,,,, and Functions RS(0) for 25825F3. 12 

Figures 

Figure 1.1 Responsivity versus zenith angle for 25825F3, fiom BORCAL9702. ............ .2 

Figure 1.2 An example of a pyranometer with a radiometrically unleveled sensor. ........... 4 

Figure 3.3.1 Measured and fitted responsivities versus zenith angle for 25825F3, from 

BORCAL9702 ......................................................... 13 

Figure 3.3.2 Measured and fitted responsivities versus time for 25825F3, from BORCAL9702.14 

Figure 3.4.1 Percentage errors resulting from using responsivity of zenith angle range, 45 " -5 5 ", 

and responsivity as a function of zenith angle versus zenith angle for 25825F3 ....... 15 

Figure 3 A.2 Percentage errors resulting fiom using responsivity of zenith angle range, 45 "-55 ", 

and responsivity as a function of zenith angle versus time for 25825F3. ............ 16 
1.0 Introduction 

Pyranometers are used to measure the total hemispheric, broadband, shortwave solar irradiance, 

also called global irradiance. NREL calibrates pyranometers using the component summation 

technique [I]. In this technique, global irradiance equals the normal incident component plus the 

diffuse sky irradiance, where the normal incident component is the beam irradiance (direct 

normal) times the cosine of the solar zenith angle. The procedure NREL uses for such 

calibrations is called Broadband Outdoor Radiometer Calibration (BORCAL). At every 

calibration event, BORCAL can calibrate and characterize up to 90 radiometers under natural, 

clear sky conditions. Different types of radiometers can be accommodated, such as the EPPLEY 

PSP (Precision Spectral Pyranometer) and NIP (Normal Incidence Pyrheliometer), KIPP & 

ZONEN and LICOR radiometers. The software used to acquire the data, calculate responsivities, 

and print the final BORCAL report is called Radiometer Calibration and Characterization (RCC) 

software [2]. An example of a reported responsivity from an RCC BORCAL report for a 

pyranometer is shown in Table 1.1 and Figure 1.1. The reported responsivities were calculated 

from data collected on June 26 and 30 and July 1,2, 3 and 4, 1997. 

Where: 

Table 1.1 Bin versus Responsivity for EPPLEY PSP 25825F3 from BORCAL9702 Report 

- Bin is a zenith angle range of 9", the 0" to 90" range is divided into 10 bins 

- RS is the responsivity of the pyranometer in pv/W/m2, at each bin. 

RS (pV/W/mA2) 

9.681 

9.669 

9.639 

9.583 

9.533 

9.501 

9.546 

9.477 

Bin 

Bin Number 

1 

2 

3 

4 

5 

Bin Range (") 

0 to 9 

9 to 18 

18 to 27 

27 to 36 

36 to 45 

9.422 

9.530 

10 81 to 90 

45 to 55 

45 to 54 

7 

8 

9 

54 to 63 

63 to 72 

72 to 81 

Pyranometer users have customarily applied the responsivity for the range from 45" to 55" to 

calculate the global solar irradiance. 

Where: 

- G, is the ith calculated global solar irradiance in w/mZ 

- Vi is the i" output voltage of the pyranometer in microvolts (pV) 

- RS,,,, is the responsivity at solar zenith angle range from 45" to 55" in pV/W/rn2. 

From Table 1.1 and Figure 1.1, one can note that the responsivity of the pyranometer changes 

with zenith angle. Thus, using one responsivity, RS,,,,, can introduce errors into the global 

irradiance calculation. 

The change of responsivity versus zenith angle is called the cosine response. The cosine response 

is unique for each pyranometer. It depends on how well the sensor is leveled during assembly, 

the resolution of the spirit bubble level in the base of the pyranometer, and the surface roughness 

of the sensor. Figure 1.2 shows an example of a pyranometer with an unleveled sensor. In this 

case, if one responsivity value were used, the resulting errors are from -5% to 1.7% in global 

irradiance measurements. 

This method can reduce these errors in global solar irradiance measurements, using 

pyranometers, by using a fimction of responsivity versus zenith angle. Because of the asymmetry 

between the morning and afternoon cosine response for a pyranometer [2], two functions, RS,(B) 

and RS,(B), were chosen to fit the morning and the afternoon responsivity versus zenith angle. 

Then, the global solar irradiance is calculated in the morning, using the following equation: 

Where: 

- Gi is the ith global solar irradiance reading in W/rnZ 

- Vi is the i" output voltage of the pyranometer in microvolts (pV) 

- RS,(Bi) is the pyranometer's responsivity calculated by substituting the ith zenith angle, 

8, in the morning function RSa,(0). 

In the afternoon, the global solar irradiance is calculated using equation 1.2 and substituting 

Rsam(0i) by RSP(Bj)* 

In order to calculate the coefficients of each fitting function, a number of data points are chosen, 

at a 5" zenith angle increment. The data points are then substituted in the function. This will 

form a set of (n+l) equations with (n+l) unknown coefficients. Matrices are then used to solve 

the (n+l) equations. 

2.0 Procedure 

2.1 Because of the cosine behavior of the responsivity versus zenith angle, the following two 

cosine functions were chosen: 

Where : 

- RS,(B) and RS,(O) are the morning and afternoon responsivity functions of a 

pyranometer, respectively 

- n and m are the number of moming and afternoon data points used to solve 

the (n+ l)m and (m+ l)fi order equations, respectively 

- a, and pj are the i" moming and jfi afternoon coefficients, respectively 

- ei,,, and are the iLh morning and jfi afternoon zenith angles, respectively. 

2.2 Determine the data points used to solve equations 2.1.1 and 2.1.2. The data points can be 

chosen from an RCC BORCAL data file (responsivity versus zenith angle file) using the 

following procedure: 

2.2.1 Divide the 0" to 90" zenith angle range into 5 " increments (i-e., 0°, So, lo0, ....., 90") 

2.2.2 Calculate the morning mean responsivity at each increment, RS,,,, 

Where : 

- RS,, is the g" moming responsivity in the range, &, e.g. 4.7" to 5.3" 

- f is the increment 

- h is the number of data points in the range, 4, 
2.2.3 Calculate the standard deviation of RS, SD, 

2.2.4 Reject all outlier responsivities, RSKjeCt, 

2.2.5 Recalculate RS,,. 

2.2.6 Repeat steps 2.2.2 through 2.2.5 until all outliers are rejected. 

2.2.7 Repeat steps 2.2.2 through 2.2.6 to calculate the mean afternoon responsivity at each 

increment, RSS,. 

2.2.8 Include all'the peaks and valleys, if the responsivity versus zenith angle curve at zenith 

angles higher than 60" is the same shape as Figure 1.1. 

2.2.9 For the boundary conditions, include the data point at solar noon of the day the calibration 

was performed, for both the morning and afternoon data points. Also include the two data points 

for the largest zenith angles for the morning and afternoon data. 

2.3 Substituting the chosen data points from step 2.2 in equations 2.1.1 and 2.1.2, one can 

calculate all the coefficients, 4's and pj's. 

2.3.1 The following matrix equations were used to solve the n+l equations, for the morning 

function, and the m+l equations, for the afternoon hction[3]: 
Where: 

- matrix [RS,] is constructed in the form: 

where, RSo,am to RS,,,, are the responsivities chosen, using step 2.2 

- matrix [A] is constructed in the form: 

where, %to a,,, are the unknown coefficients of the morning function RS,(0). 

- matrix [cosO,] is constructed in the form: 

where, O,,, to On,,,, are the morning zenith angles chosen from step 2.2. 

The same procedure used to construct the morning matrices can be followed to construct the 

afternoon matrices. 
2.3.2 The following steps describe how to calculate the coefficients of the morning function, 

RS,(0). The same steps are followed to calculate the coefficients of the afternoon function, 

RSpm(0) PI* 

2.3 -2.1 Equation 2.3.1 can be rearranged to be, 

[RYm] *[cos~,]-'=[&[A] 

Where: 

- [cos0,]-' is the inverse (reciprocal) of matrix [cosO,] 

- [I] is the unit matrix. 

2.3.2.2 Calculate [cos~,]'~ using the following steps: 

- Construct the matrix of cofactors for matrix [cos~,], [MI, 

where, mii is a cofactor that equals the determinant value of the matrix formed by deleting row i 

and col& j of matrix [cos~,]. 

- Construct the adjoint of matrix [case,], Adj.[cos0,], - 

"00 Mlo %o 

m~l "11 mn~ 

A@. [cos~,]= . . ... 
- Calculate the determinant value of matrix [cosO,], A, 

Where: 

- co&,,,'s are the first raw of matrix [cosO,], from equation 2.3.5 

- qj's are the cofactors of the first raw of matrix [MI, fiom equation 2.3.7. 

- Calculate the matrix [cos8,]-', 

4 

Matrix [cosO,]" will be in the following form: 

[cos earn] -1=1 . 

where, q,, is the element in the i" raw of the j" column of the matrix. 

2.3.2.3 Substituting equations 2.3.3 and 2.3.11 in equation 2.3.6, matrix [A] will yield the 

following: 
2.3.2.4 From equation 2.3.12, the morning function coefficients are: 

n n 

Similarly, the afternoon function coefficients are: 

m m 

3.0 Results for EPLAB PSP, Serial Number 25825F3 

3.1 Table 3.1.1 shows the chosen data points used for solving equations 2.3.1 and 2.3.2. Note 

that zenith angles 57.5 " and 61.5 " were chosen because data points around 60" were missing. 

Table 3.1.1 Data Points Used Solve the Enuations 

Zenith Responsivity (pV/W/m2) 

Angle(") Mornine I Afternoon 

11 Responsivity at solar noon 

11 Maximum morning responsivity 

11 Minimum morning responsivity 

I Maximum afternoon responsivity 

11 Minimum afternoon responsivity 

1) Morning responsivity at greatest zenith angle 

Afternoon responsivity at greatest zenith angle 
3.2 Table 3.2.1 shows the resultant morning and afternoon coefficients. Because the numbers are 

so large, it is important to use double precision numbers in the data processing. Labview software 

was used for this paper. 

Table 3.2.1 Morning and Afternoon Coefficients for 25825F3 I 

Morning Afternoon 

Coefficients 

0 322.0557 - 177040,642 1 

Morning Afternoon 

Coefficients 

9 -10051108841 6218266505.218 

3.3 For pyranometer PSP 25825F3, Figure 3.3.1 shows the responsivity versus zenith angle, 

measured through BORCAL9702, and the responsivity versus zenith angle using the two fitting 

responsivity functions, RS,(0) and RS,(B). Figure 3 -3.2 shows the responsivities versus the 

true solar time. 

3.4 Figure 3.4.1 shows the percentage error in the global solar irradiance measurements, if one 

responsivity (e.g., responsivity at zenith angle range from 45" to 55 ") is used, and the percentage 

error if the two fitting responsivity functions, RS,(8) and RS,(B), are used. Figure 3 -4.2 shows 

the errors versus the true solar time. The percentage errors are calculated as follows: 

Where: 

- E,,, is the error of the i" data point using one responsivity, RS,,,, 

- Gi,,, is the i" reference global solar irradiance 

- Vi is the i" output voltage from the pyranometer, in microvolts 

- RS,,,, is the responsivity of the pyranometer at 45" to 55" zenith angle bin. 
Where: 

- Ei,,,R,(,, is the error of the ith data point using the morning responsivity function 

- RS,(Bi) is the morning responsivity calculated by substituting 0 by Bi in the 

morning responsivity function, RS,(0), where Bi is the ih zenith angle. 

The errors resulting from using the afternoon responsivity function, Ej,,,Rs(8,, are calculated using 

equation 3.4.2 by substituting Vi and RS,(0,), by V, and RS,(Bj). 

Table 3 -4.1 shows the maximum and minimum percentage errors at zenith angle ranges 16.5" to 

45" and 55" to 82". 

Table 3.4.1 Percentage Errors Resulting from Using RS,,-,, and Functions RS(0) for 25825F3 

Zenith Angle 

Range 

. Using RS4,,, Using RS,(0) and RS,(0) 

Maximum Error Minimum Error Maximum Error Minimum Error 




4.0 Conclusions 

4.1. Figure 3.4.1 and Table 3.4.1 show that the values of percentage errors resulting fiom using 

RS,,,, are larger than the percentage errors that result fiom using the two responsivity functions, 

RS,(0) and RSp,(B). The table and figure also show that the errors are increasing outside the 

zenith angle range from 45" to 55" when RS,,, is used, but the errors stayed about the same 

using the two responsivity functions, RS,(0) and RS,(B). 

4.2. Using the two responsivity functions will improve the accuracy of measuring the global solar 

irradiance, in the example of 25825F3, by at least 68% in certain zenith angle ranges. 

Where: 

- 1.58 is the maximum percentage error that results from using RS,,,, at zenith angle 

range 16.5 " to 45 " from Table 3.4.1 

- 0.5 is the maximum percentage error that results fiom using the two morning and 

afternoon fimctions from Table 3.4.1. 

The improvement can be much better than 68% for pyranometers with sensors that are not 

radiometrically leveled. The uncertainty of these pyranometers can reach *(5 to lo)% if RS,,,, is 

used to calculate the global solar irradiance from sunrise to sunset. 

4.3 The two responsivity functions, RS,(8) and RSpm(0), are only valid between and at the 

boundary conditions. For this reason, it is important to calibrate the pyranometer from sunrise to 

sunset in order to cover the zenith angle range from 0" to 90". If zenith angle at solar noon does 

not reach 0" on the day of calibration or if calibration were stopped before 90" is reached, 

extreme care should be taken in estimating data points outside the boundary conditions, to cover 

the range 0" to 90 O. These estimated data points can then be included in the data set (see Table 

3.1. I), used to calculate the morning and afternoon coefficients of the functions RS,(0) and 

RSprn(0). 

4.4 This paper describes the first step toward improving the uncertainty of measuring the clear 

sky global solar irradiance. The second step is to develop two responsivity functions, RS& 4) 

and RS,(B, $), where 4 is the solar azimuth angle, to include the azimuthal response of a 

pyranometer. This was not discussed in this paper because of the complexity of the functions and 

the lack of calibration data from different days of the year. More calibrations from different days 

of the year are recommended to describe the zenith and azimuthal responses more accurately for 

a pyranometer. 
5.0 References 

1. Myers, D.R., Emery, K. and Stoffel, T.L., 1984, Uncertainty Estimates for Global Solar 

Irradiance Measurements Used to Evaluate PVDevice Performance, Solar Cells, 27, 1987, 

p. 456. 

2. National Renewable Energy Laboratory (NREL), NREL BORCAL reports, Golden, CO: 

NREL. 

3. ONeil, P.V., 1995, Advanced Engineering Mathematics, fourth edition, Boston, MA: PWS 

Publishing Company, pp. 3 16-3 97.