Solar Radiation Data Manual
for
Flat-Plate and Concentrating Collectors


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Appendix - Methodology

This Appendix describes Version 1.1 revisions of the National Solar Radiation Data base (NSRDB) and describes the method used to calculate the monthly and yearly averages of solar radiation for flat-plate and concentrating collectors. It also describes how the solar radiation data uncertainties were determined and how the climatic information was derived.

National Solar Radiation Data Base Version 1.1

This data manual is based on the NSRDB Version 1.1, completed in March 1994; the previous Version 1.0 was completed in August 1992. Version 1.1 corrects two types of errors discovered in Version 1.0: (1) for 23 stations, the wrong time zones were used, and data values were mismatched with their time stamp by 1 or 2 hours, and (2) for 8 stations that measured solar radiation, from 1 to 3 months per station had some hourly solar radiation values that were unrealistically low.

Version 1.1 corrects time zone errors for the following stations:

Version 1.1 replaces erroneous measured solar radiation data with modeled data for the following stations:


Calculating Solar Radiation for Flat-Plate and Concentrating Collectors

The total solar radiation received by a flat-plate collector () is a combination of direct beam radiation (), diffuse (sky) radiation (), and radiation reflected from the surface in front of the collector ():

(1)         

where is the incident angle of the sun's rays to the collector. The incident angle is a function of the sun's postion in the sky and the orientation of the fixed or tracking collector. Algorithms presented by Menicucci and Fernandez (1988) were used to compute the incident angles for the various collectors. For tracking collectors, these algorithms also were used to compute collector tilt angles from the horizontal. Direct beam solar radiation hourly values from the National Solar Radiation Data Base (NSRDB) were used to determine the direct beam contribution () for each hour. Except for the first and last daylight hour, incident angles were calculated at the midpoint of the hour. For the first and last daylight hour, incident angles were calculated at the midpoint of the period during the hour when the sun was above the horizon.

The diffuse (sky) radiation, , received by the collector was calculated by an anisotropic diffuse radiation model developed by Perez et al. (1990). The model determined the diffuse (sky) radiation for the collector using hourly values (from the NSRDB) of diffuse horizontal and direct beam solar radiation. Other inputs to the model included the sun's incident angle to the collector, the collector tilt angle from the horizontal, and the sun's zenith angle. The model is an improved and refined version of their original model that was recommended by the International Energy Agency for calculating diffuse radiation for tilted surfaces (Hay and McKay 1988).

The Perez model equation for diffuse sky radiation for a tilted surface is:

         

where

          = diffuse solar horizontal radiation

          = circumsolar anisotropy coefficient, function of sky condition

          = horizon/zenith anisotropy coefficient, function of sky condition

          = tilt of the collector from the horizontal

          = 0 or the cosine of the incident angle, whichever is greater

          = 0.087 or the cosine of the solar zenith angle, whichever is greater.

The model coefficients and are organized as an array of values that are selected for use depending on the solar zenith angle, the sky's clearness, and the sky's brightness. The manner in which this is done is described by Perez et al.(1990).

      The ground-reflected radiation received by a collector is a function of the global horizontal radiation (), the tilt of the collector from the horizontal (), and the surface reflectivity or albedo ():

         

Surface albedo was adjusted depending on the presence of snow cover, as indicated by the snow depth data in the NSRDB. If there was snow on the ground, the surface albedo was set to 0.6 (albedo for snow ranges from about 0.35 for old snow to 0.95 for dry new snow). If no snow was indicated, the surface albedo was set to 0.2, a nominal value for green vegetation and some soil types.

The concentrating collectors portrayed in the manual have small fields-of-view and do not receive diffuse (sky) radiation or ground-reflected radiation. Consequently, solar radiation for these concentrating collectors is solely a function of the direct beam radiation and the sun's incident angle to the collector. Solar radiation received by the concentrating collectors simplifies to:

         

For each station location, collector type, and collector orientation, hourly values of solar radiation received by the collectors were calculated. Monthly and yearly averages were then determined for the period of 1961-1990. For a few stations, monthly and yearly averages do not include data for 1989 or 1990 or both because NSRDB data did not include those station years. Stations with less than 30 years of NSRDB data, along with their period of record, are listed below:


Estimating the Uncertainty of Solar Radiation Data

The solar radiation values presented in the manual were calculated using improved models and data. The estimated data uncertainties assigned to the calculated values show how they might compare with true values. They were determined using the uncertainty method of Abernethy and Ringhiser (1985). This root-sum-square method defines an uncertainty, , in which 95% of the time, the true value will be within plus or minus the uncertainty of the calculated value.

         

where

          = student's T distribution factor ( equals 2 for sample size greater than 30)

          = random error

          = bias error.

Random and Bias Errors. The two types of errors that contribute to uncertainties are random errors and bias errors. Random errors usually follow statistical distributions and result in values both above and below the true values. Random errors tend to cancel when individual values are used to determine an average. For example, a 30-year monthly average of solar radiation may use 10,800 hourly values (assuming 30 days per month and 12 hours of sunlight per day) to determine the average monthly solar radiation. The random error of the average is reduced by a factor of , or approximately 100. Consequently, random error sources do not contribute significantly to the uncertainty of 30-year monthly averages.

Bias errors, however, are not reduced by averaging. Bias errors, which are often referred to as fixed or systematic errors, cause values to be in error by about the same amount and direction. The reason for bias errors, as well as their magnitude and direction, may be unknown; otherwise, corrections such as changes in the calibration factor can be made. When detailed information is not known about the bias errors, reasonable estimates of the bias error magnitude can be made using procedures similar to those described here.

For the monthly averages of solar radiation, we evaluated the three major bias errors: (1) errors in direct beam radiation incident on the collector caused by errors in NSRDB direct beam radiation data, (2) errors in diffuse radiation incident on the collector caused by errors in NSRDB diffuse horizontal radiation, and (3) errors in diffuse radiation incident on the collector caused by errors in modeling the diffuse solar radiation for the collector. Climate change could also bias monthly average solar radiation values but was not considered a major source of error for this work.

The root-sum-square of the individual bias errors yields the total bias error. Because the random error is negliible, the total bias error is the same as the total uncertainty of the monthly averages. Consequently, the uncertainty, , can be expressed as:

         

where

          = errors in collector direct beam radiation caused by errors in direct beam radiation data

          = errors in collector diffuse radiation caused by errors in diffuse horizontal radiation data

          = errors in total collector radiation caused by errors in modeling the diffuse solar radiation for the collector.

The bias errors for direct beam and diffuse horizontal radiation were extracted from the NSRDB daily statistic files for each station. The NSRDB daily statistic files include, among other information, 30-year averages and their uncertainties for direct beam and diffuse horizontal radiation. An integer number represents an uncertainty range. Examples of uncertainty ranges for the monthly averages are from 6% to 9%, from 9% to 13%, and from 13% to 18% of the monthly average.

For 30-year averages, most of the stations have direct beam radiation uncertainties in the 6% to 9% range and diffuse horizontal radiation uncertainties in the 9% to 13% range. The remaining stations have direct beam radiation uncertainties in the 9% to 13% range and diffuse horizontal radiation uncertainties in the 13% to 18% range. For the purpose of extracting the bias errors from the daily statistic files, a single integer value near the midpoint of the range was used (8% for the 6% to 9% range, 11% for the 9% to 13% range, and 16% for the 13% to 18% range).

The bias error for modeling the collector radiation is attributed to the diffuse solar radiation model because the direct beam component is considered an exact solution (). An evaluation of the original Perez model by Hay and McKay (1988) provided informati on whereby we estimated the bias error to be about 5% of the total collector radiation for our applications.

The uncertainty, , can be expressed as a percentage of the total collector radiation by the following equation:

         

where

          = average monthly direct beam radiation incident on the collector

          = average monthly diffuse radiation incident on the collector

          = average monthly total radiation incident on the collector

          = percent bias uncertainty of average monthly beam radiation

          = percent bias uncertainty of average monthly diffuse horizontal radiation

          = percent bias uncertainty of the solar radiation modeling for tilted surfaces.

Uncertainty Values in Tables. Because of the large number of solar radiation values presented in the manual (546 per station), it was judged impractical with respect to space limitations to present uncertainty values for each solar radiation value. Rather, a simplifying assumption was made so that only one uncertainty value was presented for all flat-plate collectors. The assumption was that the direct beam radiation and diffuse radiation incident on the collector were of equal weight. The uncertainties of the diffuse horizontal and direct beam radiation have about the same value, so this assumption did not create large changes in calculated uncertainties for collector radiation.

Over a range of direct-beam-radiation-to-diffuse-radiation ratios (30/70 to 90/10), the assumption yielded uncertainties within 1% or 2% of that when calculated using the exact proportions of direct beam radiation and diffuse radiation (e.g., uncertainty of 8% or 10% instead of 9%). This was judged acceptable, considering that there are uncertainties associated with the uncertainty values used for the average monthly direct beam radiation, the average monthly diffuse horizontal radiation, and the solar radiation modeling for tilted surfaces. As a conservative measure, the calculated uncertainties were rounded to the next highest integer value.

For most of the stations in the data manual, uncertainties of 9% were assigned to the solar radiation data for flat-plate collectors. The few stations with higher uncertainties for direct beam and diffuse horizontal radiation were assigned uncertainties of 11%.

A separate value was assigned to the uncertainty for the direct beam radiation for concentrating collectors, which is only a function of the uncertainty () of the average monthly beam radiation. For most of the stations in the data manual, uncertainties of 8% were assigned to the solar radiation data for concentrating collectors. The few stations with higher uncertainties for direct beam radiation were assigned uncertainties of 11%.

Data values in the data manual are given to one significant figure by rounding the calculated value to the nearest tenth of a . Consequently, the data values presented are within 0.05 of the calculated values. Because of the uncertainties of the data values, there is no benefit to expressing the data values to more than one significant figure.

Deriving Climatic Data

The climatic data presented in the manual were derived using both data from the National Solar Radiation Data Base (NSRDB) and from climatic data sets provided by the National Climatic Data (NCDC), Asheville, North Carolina (704) 271-4994.

Climatic data pertaining to average temperature, average daily minimum temperature, average daily maximum temperature, average heating degree days base 18.3 ° C, and average cooling degree days base 18.3 ° C were extracted from NCDC's data tape, "1961-1990 Monthly Station Normals All Elements." This data tape includes temperature and degree day normals for about 4,775 stations in the United States and its territories. The normals are averages computed by NCDC for the period of 1961-1990.

For this data set, NCDC used procedures, when possible, to estimate missing data and to correct for other inconsistencies by using data from neighboring stations. For one of the stations in this data manual, data were not available on NCDC's data tape. For this station, in Arcata, California, the averages were computed using NSRDB data, but no attempt was made to estimate missing data or to correct for other inconsistencies.

NSRDB data were used to calculate average relative humidity and average wind speed. Record minimum and maximum temperatures were obtained primarily from NCDC's data diskette, "Comparative Climatic Data Tables 1991." This data diskette contains, among other useful parameters, record minimum and maximum temperatures for about 90% of the stations in this manual and spans periods of record back to 1948 and earlier. For the remaining 10% of the stations, record minimum and maximum temperatures are based on NSRDB data.

References
Abernethy, R.; Ringhiser, B. (1985). "The History and Statistical Development of the New ASME-SAE-AIAA-ISO Measurement and Uncertainty Methodology." 20th AIAA/SAE/ASME Joint Propulsion Conference (July 1985). AIAA-85-1403. New York: American Institute of Astronautics and Aeronautics.

Hay, J.E.; McKay, D.C. (1988). Final Report IEA Task IX - Calculation of Solar Irradiances for Inclined Surfaces: Verification of Models Which Use Hourly and Daily Data. International Energy Agency Solar Heating and Cooling Programme.

Menicucci, D.; Fernandez, J.P. (1988). User's Manual for PVFORM: A Photovoltaic System Simulation Program for Stand-Alone and Grid-Interactive Applications. SAND85-0376, Albuquerque, NM: Sandia National Laboratories.

Perez, R.; Ineichen, P.; Seals, R.; Michalsky, J.; Stewart, R. (1990). "Modeling Daylight Availability and Irradiance Components from Direct and Global Irradiance." Solar Energy, 44(5), pp. 271-289.


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