Solar Ceils, 27 (1989) 455 - 464 455

UNCERTAINTY ESTIMATES FOR GLOBAL SOLAR IRRADIANCE MEASUREMENTS USED TO EVALUATE PV DEVICE PERFORMANCE

DARYL R. MYERS, K. A. EMERY and T. L. STOFFEL

Solar Energy Research Institute, Golden, CO 80401 (U.S.A.)

Summary

Broadband (0 .3- 3.0/~m) global solar irradiance measurements are used in the evaluation o f solar energy conversion devices. The uncertainty at. tached to such measurements is important in evaluating whether conclusions associated with the measurements are statistically valid. A standardized un- certainty analysis method, developed over the past 15 years in the arena of consensus standards and professional society organizations, is described and applied. The results of the uncertainty analysis for the instrument calibration and field data measurement process indicate that the total measurement un- certainty in pyranometry (i.e. the measurement of global solar irradiance) can approach 5%. Thus comparisons of results between laboratories using different pyranometers can have a total uncertainty of up to 10%. Statisti- cally valid conclusions on a conversion device's performance may be drawn only if such results account for known bias errors or exceed the uncertainty limits derived using this methodology.

1. Introduct ion

A pyranometer measures the combinat ion of direct beam, scattered, and ground reflected solar radiation (if tilted from the horizontal) in a 180 ° field of view. The detec tor is of ten a thermopile (group of thermocouples) embedded in an absorbing disc which heats up, when exposed to radiation, and causes the thermopile to generate a voltage (of the order of mfllivolts). The calibration factor is the conversion factor for determining the incident power density, in watts per square meter, as a function of the generated out- put voltage. This calibration factor is of ten assumed to be a constant value, independent of the incident angle of the incoming radiation, ambient tem- perature, etc. However, many pyranometer characteristics have been studied which demonstrate that this is no t the case. This paper evaluates the sources of error in the thermopile de tec tor pyranometer in the process of calibration and field measurements, including applications which involve the determina- tion of photovoltaic conversion device efficiency.

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2. Uncertainty formulation

The basic method was described by Abernethy and Ringhiser [1] and Abernethy and Benedict [2]. Each error source consists of two components, a bias (or systematic offset) component and a random (statistical) compo- nent.

The bias or offset is a fixed difference between the true value and the measured value of a parameter. This offset may have a known or unknown sign (i.e. +), as well as an unknown magnitude. No statistical technique can be used to estimate bias errors. The random component is often determined experimentally, as the standard deviation of a number of measurements which are assumed to possess the properties of a normal distribution.

The total measurement uncertainty is determined using one of two models to combine the total bias and total random components. The total bias uncertainty is derived by root-sum-squaring (RSS) the elemental bias errors together. The elemental sources of random error are RSSed together to produce the total random component of uncertainty. The total overall uncertainty is then either (1) the RSS of the total bias plus the Student 's t factor (two for n > 20) times the total random uncertainty, or (2) the sum of the total bias plus the Student 's t factor times the total random un- certainty. Model 1 is called the U9s model; model 2 is called the U99 model. Monte Carlo evaluation techniques [3] have shown the confidence intervals associated with the two methods of combining the bias components to be 95% for the RSS of total bias and total random errors, and 99% for the sum of total bias and total random errors.

The factor of two is used to scale the random, or statistical, component to include all of the measurements, where there are more than 20 degrees of freedom. The random component is usually approximated by the standard deviation of a distribution of measurements, which encompasses only 37% of the data in the distribution. This factor has been confirmed through the Monte Carlo modeling mentioned above, for n > 20.

The U9s model indicates that the true value, T, of the measurement lies in the interval M - U9s ~ T ~ M + U9s with 95% confidence, where M is the mean value of a measured parameter. It must be mentioned that the degrees of freedom are needed to characterize the statistical parameters completely, especially for small numbers of measurements, where the Student 's t factor can become much larger than the value of two suitable for n > 20.

The i bias error components are denoted as bi and the j random error components as r i. It is recommended that all bias errors (no matter how many of them there are) should be combined by the RSS technique. How- ever, for i <~ 4, bias errors can be combined by summing. It should be noted that a combination of RSS and summing is possible, especially if a known sign is attached to some bias estimates. This may result in asymmetrical bias error estimates such as +5% and --2%. The combined random component is calculated using RSS of all rj elemental estimates. Finally, the U9s or U99

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model is used to combine total bias and total random estimates into the final uncertainty using

U9s = [B 2 + (2 X R)2] °'s B = [Z~ bi2] °'s, R = [~j rj2] °'s (1)

or

U99=B + (2 XR) (2)

Either approach may result in asymmetrical uncertainty limits, as a bias estimate with known sign may be added to the above Ugs or U99 estimates, given +U9s and --Ugs or +U99 and --U99 limits which are not equal. The final limits of error are then the mean value, plus or minus the upper and lower total uncertainty limits.

Three processes are involved in the measurement of a parameter, such as global solar irradiance: calibration, acquisition, and reduction of data. In what follows, bias and random error sources are evaluated in each step of this measurement process.

3. Calibration uncer ta inty

The reference device for the calibration of pyranometers is the absolute cavity radiometer, as described by Kendall and Berdahl [4], Willson [5], Brusa and Fr6hlich [6], Fr6hlich [7], and Crommelynck [8]. The cavity radiometer is used to measure accurately the direct beam, Ib, in conjunction with a measure of the diffuse sky irradiance, Id. These parameters are used to form the reference global solar irradiance Iref according to

Iref = IbSin (a) + ]d (3)

where a is the solar elevation angle at the t ime of the measurement. Dividing the test pyranometer ' s ou tput voltage by Iref gives the calibration factor (sensitivity) for the test pyranometer .

Kendall has performed an extensive characterization of the sources of uncer ta inty listed in Table 1 for a cavity, direct-beam, radiometer.

The calibration of a reference cavity radiometer is traceable to the World Radiometric Reference (WRR) maintained by the World Meteoro- logical Organization (WMO) at the World Radiation Center, Physical Meteoro- logical Observatory, Davos, Switzerland. The WRR is defined as the weighted mean of the World Standard Group (WSG) of instruments, all similar in concept to the designs listed above [4-8] . The quoted uncertainty of WRR is 0.30%. This total uncer ta inty is separated into a bias of 0.25% and a random component of 0.12%, based upon Kendall and Berdahl's analysis of sources of error described above. The relationship of a working reference cavity radiometer to WRR is determined through International Pyrheliometric Comparisons (IPC) sponsored by WMO every 5 years. Typical

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TABLE l

Cavity radiometer uncertainty (-+%)

So urce Bias R a n d o m To tal (RSS)

Absorbance 0.12 0.05 0.15 Electronics 0.005 0.05 0,10 Aperture area 0.15 0.01 0.15 Radiative losses 0.08 0.01 0.08 Paint conductivity 0.01 0.01 0.01 Non-equivalence 0.002 0.01 0.01 Aperture heating 0.01 0.01 0.01 Reflected gains 0.01 0.002 0.01 Heat flow 0.002 0,002 0.0l Total RSS 0.247 0.074 0.25

results indicate a bias error (offset) from WRR of 0.25% and a random com- ponent of 0.12%.

The bias and random components of uncertainty related to the refer- ence radiometer are calculated as follows:

(1) the bias error = (0.00252 + 0.00252) I/2 = 0.35% (one bias contribu- tion from WRR, the other from the instrument itself),

(2) the random error = (0.00072 + 0.00122) i n = 0.12% (the 0.12% from WRR, 0.7% from the instrument itself),

(3) the total error, CAVtot, due to the reference radiometer= [(0.35) 2 + (2 × 0.12)2] In = 0.42% with respect to WRR.

4. Data acquisition uncertainty

4.1. Data logging The collection of calibration and field data both require the use of

data-logging equipment which has its own at tendant uncertainties. A data- logging system may consist of digital voltmeters and scanners with specifica- tions for accuracy, temperature coefficients, linearity, thermal electromag- netic force performance, etc. In addition, such specifications may refer to uncertainty with respect to the full scale, per cent of reading, or both. Care must be taken to ensure tha t measurement uncertainty attributed to the data logger reflects the measurement regime and performance limitations of the data logger. A typical specification might be 0.01% + 8 counts, with respect to full scale range of 64 mV, with a dynamic range of ±40 000 counts.

The dynamic range indicates 625 counts mV -1. If a pyranometer generates a 5 mV signal (typical), this is 3125 counts, and the eight count bias is 0.25% of the signal. The 0.01%, treated as a random com- ponent, gives an error of 0.0001 × 64 mV = 0.0064 mV, or 0.13% of a 5 mV signal. The uncertainty statement for such a typical data logger is bias = 0.25%, random = 0.13%, and total = 0.36% for a 5 mV signal.

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4.2. Instrument characteristics By far the greatest contributions to the uncertainty of a pyranometer

calibration or measurement are from the peculiarities of the instrument response with respect to incidence angle of the solar beam, temperature coefficients of response, etc. [9]. Figures l(a) and l(b) illustrate the variation in the response of two different pyranometers with respect to solar incidence angle during outdoor calibrations performed at SERI [10]. Figure 2 shows the variation in response with respect to temperature, as derived through multiple determinations for the same instrument, by the Solar Radiation Facility (SRF) of the National Oceanic and Atmospheric Administration (NOAA) at Boulder, CO [11].

8.40~ , , ' 2 10.20~ 4

8'30 t 1 10'10 1 32

o-oolc i ~ 8.10 f t l~%l~.,~f -1 ~ >~ 9'90

8.oo t -2 e ~ 97o~ N ~ , / ~"

z 6 o , , , , 9. o ' go go ; o 30 40 50 60 70 30 40 (a) Incidence angle (b) Incidence angle Fig. I. (a) Deviation from perfect cosine response for pyranometer 17863F3 compared with (b) deviations in cosine response for pyranometer 24034F3. Such instrument-to- instrument variation is typical. Magnitude of deviations is -+2%.

1.00

~0.9~ n- 0 / / / -e-O- Mar 84

0.96 I I I -40 -20 0 20 40

Temperature °C Fig. 2. Four determinations of temperature response for a single pyranometer, illustrating the magnitude (+0.5% to --3%) and precision (-+0.5% of such determinations).

Different models of pyranometers have been shown to possess different characteristics with respect to cosine response, temperature, etc. For instance, one unit (a PSP) may have a response to temperature that can be

460

approximated as a parabolic polynomial , and another (Kipp and Zonen CM5) as a cubic polynomial .

Other workers, particularly in the International Energy Agency (IEA) [11] have devoted much time and effor t to characterization of geometrical, thermal, linearity, t ime constant , and other sources of uncer ta inty for pyranometers . Table 2 lists estimates for various sources of error studied in the IEA and other laboratory work.

TABLE 2

Pyranometer uncertainty sources during calibration (+%)

Source Bias Random Total (RSS)

Cosine response 1.50 0.25 1.58 Azimuth response 0.50 0.25 0.71 Linearity 0.10 0.10 0.22 Temperature response 1.00 0.50 1.41 Leveling 0.10 0.10 0.22 Thermal e.m.f. 0.10 0.10 0.22 Thermal gradients 0.30 0.10 0.36 Time constants 0.10 0.10 0.22 Spectral response 0.10 0.10 0.22 Total RSS 1.91 0.70 2.32

4. 3. Fie ld data uncer ta in ty The instrument characteristics evaluated in Table 2 influence the

instrument performance in the field, as well as in the calibration process. Therefore, the uncer ta inty in the field data must account for these influences again. Similarly, the field data collection system must be analyzed for its contr ibut ion, as the calibration data logger was analyzed above. The uncer- ta inty in the reference cavity radiometer , with respect to WRR, must also be accounted for. Thus, the field data uncer ta in ty contr ibut ions can be sum- marized in Table 3, using the total uncer ta inty from each of the analyses above.

The sensitivity derived using the calibration technique described in Sec- t ion 3 is usually applied only with the sensor horizontal . If the unit is tilted, such as in a plane of array (POA) measurement, o ther sources of error come into play; namely, the incidence angles between the direct beam and the normal to the de tec tor no longer match the calibration (horizontal) situa- tion. Further , an analysis by Emery e t al. [12] has shown that an uncer- ta inty of 2 ° in each of the no r t h - sou t h and east-west directions for a pyran- ometer tilted normal to the noon-t ime sun results in errors in cosine of the incidence angle of 3% at incidence angles up to 30 ° (of f normal). Figure 3 shows the cosine error as a funct ion of incidence angle for such a POA appli- cation in the spring and winter months. This 3% error is included under inci-

T A B L E 3

Pyranometer uncertainty sources in field measurements (+%)

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Source Bias Random To tal (%)

Reference/WRR 0.35 0.12 0 .42 Cal. data logger 0.25 0 .13 0 .36 Instruments (cal.) 1 .90 0 .70 2 .36 Field data logger 0.25 0 .13 0 .36 Instruments (field) 1.90 0 .70 2 .36 Etc. ?? ?? ?? EMI ?? ?? ?? Incidence angle 3.0 0.5 3.2 etc. ?? ?? ?? Total 4.1 1.12 4.7

I

3

;;2 o 1

0

-1

0 -

-3

-4

-5 0

Fig. 3.

==== Winter oR====

~ l i n I~la

Sprmg

110 i l I I 20 30 40 50 60

Incidence angle

T h e e f fec t o f a 2 ° m i s a l i g n m e n t in t w o o r t h o g o n a ] d i rect ions (eas t -wes t and north-south) between a pyranometer sensor and the plane-of-array at incidence angles less than 30 ° , for two different seasons.

dence angle effects in Table 3. (It should be noted it accounts for the possi- bility of testing at any time of the year.)

A recent international round-robin calibration of photovoltaic reference cells and modules has been reported by Emery e t al . [12] . There, a calibra- tion value (CV) was derived from

Isc CV = - - ( 4 )

E t o t

where I=c is the spectrally corrected short-circuit current and Eto t is the total power density measured by a pyranometer.

Figure 4 is a plot of the ratio of individual CV determination vs. mean CV, (CV), as a function of solar incidence angle. The +2% variation in the CV ratios should be noted. These are representative of the precision, or random

462

Ra t io CV/(CV) 1.03 ' I ' I

1,02

1,01

1,00

0,99

0,98

0,97

0.96 t

©0

0 O0

!'i

I ' I ' I ' I ' I ' I

'~c~X 0 0 Co O0 o 0

~Z,. oo

++" i I! , ~ ,~ 0 °0000

+ AM data o PM data

I , I I I t , I I I , I a l

_~ _~ ~ ~,~' ~ ~ ~ ~

Solar Elevation Angle ( d e g r e e s )

Fig. 4. The ra t io of ind iv idual ca l ib ra t ion values CV to t h e m e a n ca l ib ra t ion value (CV) for several days o f data . T he CV values a r e w i t h r e s p e c t t o an Eto t measu red b y a pyran- ome te r . I t s h o u l d b e n o t e d t h a t t h e p r e c i s i o n i n d i c a t e d ( + - 2 % ) i s a p p r o x i m a t e l y t w i c e t h e d e r i v e d t o t a l r a n d o m error c o m p o n e n t in Tab le 3.

components of error estimated in Table 3. It should be noted how twice the estimated total random value compares with this experimental result.

Emery e t el. [12] have also analyzed the uncertainty in various types of PV device and module calibrations using these methods. Their results are summarized in Table 4. The similarity of their results for global fixed-tilt pyranometer methods vs. those in Table 3 should be noted.

T A B L E 4

S u m m a r y o f PV ca l ib ra t ion u n c e r t a i n t y ( a f t e r Emery , et al. [ 12 ])

Method Eto t Total U9s (%)

G l o b a l - t i l t P y r a n o m e t e r 4 .26 G l o b a l - t i l t Cavi ty + diff . pyran . 3 .24 G l o b a l - n o r m a l P y r a n o m e t e r 3 .69 G l o b a l - n o r m a l Cavi ty + diff . pyran. 2 .46 D i r e c t - n o r m a l Cavi ty r a d i o m e t e r 0 .72 X-25 S i m u l a t o r -+1% Refe rence cell 1 .09 SPI -S imula to r +- 1% R e f e r e n c e cell 2.97

5. Data reduction

Additional sources of error are possible through propagation of errors in data computation, or through functional relationships between measured parameters and the desired output variable. The researcher should be aware

463

of the computational accuracy of his/her analysis and try to account for errors with respect to rounding, truncation, and computations involving ratios and differences of nearly equal numbers and by tracking significant digits using well-designed algorithms.

If a secondary variable is calculated, for example the cosine of the incidence angle is calculated from time and location, the functional relation° ship should be used to derive a sensitivity factor 0i that indicates the error propagated to the secondary variable because of errors in the primary variables.

For example, if r = f(Pi,..., Pn), 0i = ar/bPi, the bias (Bp~) and random (Spi) estimates for the parameters Pt are weighted with the 0i to give precision (random) estimates Sr = [ X~n( O~Spi)~°'s and bias estimates Br = [ Z p(O~Bpt)2]°'s. Such estimates were used to arrive at part of the cosine response error terms in Table 2.

6. Conclusions

A standardized uncertainty analysis method, developed over the past 15 years in the arena of consensus standards and professional society orga- nizations, has been applied to determine the uncertainty inherent in a pyran- ometer used for global solar irradiance measurements. The results account for numerous sources of uncertainty in the calibration and field data mea- surement processes. The derived uncertainty in pyranometer field or test data is of the order of +5.0%. Statistically valid conclusions on PV device performance may be drawn only if such results account for the total uncer- tainty of global solar irradiance measurements using pyranometers.

Comparisons (say at different laboratories) of PV device performance that depend on the measurement of the global irradiance should include as part of the uncertainty analysis the appropriate combination of bias and random errors, using the uncertainty model (Ugs or U99) described previously. From the result calculated in Section 5 for typical field measurement uncer- tainty, the total uncertainty in the comparison of results from two laborato- ries (where the characterization of bias errors such as temperature and cosine effects is not carried out) would be +6.6% (U9s) or -+9.4% (U99), as outlined in Table 5.

TABLE 5

Uncertainty in comparison of global measurements at two laboratories

Source Bias R a n d o m Total

Lab. 1 4.1 1.12 4.7 Lab. 2 4.1 1.12 4.7 Total (RSS) U95 5.8 1.60 6.6% Total (Sum) U ~ 8.2 2.2 9.4

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R e f e r e n c e s

I R. B. Abernethy and B. Ringhiser, AIAA/SAE/ASME/ASES Joint Propulsion Conf., July 8-10, 1985.

2 R. B. Abernethy and R. P. Benedict, ISA Trans., 24 (1) (1985). 3 Anonymous, Measurement Uncertainty Handbook, Instrument Society of America,

(ISA), Research Triangle Park, NC, 1980. 4 J. M. Kendall and C. M. Berdahl, Appl. Opt., 9 (1970) 1082. 5 R. C. Willson, Appl. Opt., 12 (1973) 810. 6 R. W. Brusa and C. FrShlich, Scientific Discussions International Pyrheliometer

Comparisons IV, W.R.C., Davos, 1975. 7 C. Fr6hlich, Proc. Syrup. Solar Rad. Meas. Instrum., 1973, p. 61. 8 D. Crommelynck, ThJorie Instrumentale en Radiomdtrie Absolue, PubL Ser. A, No.

81, Institut Royal M4t4orologique de Belgique, Bru~els, 1973. 9 D. R. Myers, Uncertainty Analysis for Thermopile Pyranometer and Pyrheliometer

Calibrations Performed by SERI, TR-215-3294, April 1988, Solar Energy Research Institute, Golden, CO, 1988.

10 T. L. Stoffel, D. Myers, C. Wells and R. Hulstrom, SERI Calibration of Pyranometers Associated with the PVUSA Project, Solar Energy Research Institute, Golden, CO, 1988.

11 C. V. Wells and D. Myers, SERI Draft Materials for Sections 4 and 5 for International Energy Agency Task 9 Subtask C Final Rep., Pyranometry Studies, Solar Energy Research Insti tute, Golden, CO, 1988.

12 K. Emery, D. Waddington, S. Rummel, D. R. Myers, T. L. Stoffel and C. R. Ostwerwald, SERI Results from the PEP-1987 Summit Round Robin and a Compar- ison o f Photovoltaic Calibration Methods, TR-213-3472, Solar Energy Research Institute, Golden, CO, 1989.