Models of Diffuse Solar Fraction

John Boland and Barbara Ridley

1 Introduction

This chapter continues the work of Boland and Scott (1999) and Boland, Scott and Luther (2001) who developed models for some Australian locations using the clearness index and time of day as predictors. More recently, Boland and Ridley (2007) have presented the theoretical basis for a generic model for diffuse radia­tion, and additionally, a methodology for identifying possibly spurious values of measured diffuse. There is strong motivation for undertaking this study, wherein a number of Australian locations have been included. Spencer (1982) adapted Orgill and Hollands (1977) model and tested it on a number of Australian data sets for the reason that most of the work in the field has been performed using higher latitude North American and European data sets.

The evaluation of the performance of a solar collector such as a solar hot water heater or photovoltaic cell requires knowledge of the amount of solar radiation in­cident upon it. Solar radiation measurements are typically only for global radiation on a horizontal surface. They may be on various time scales, by minute, hour or day. Additionally, one can infer global radiation from satellite images. We have used in­ferred daily totals of global radiation. Presently, there is some satellite inferred data available at the three hour time scale, and it is expected that this will become more widespread in the future. At present we will only assume daily data available for a wide range of locations.

These global values comprise two components, the direct and the diffuse. “Idn, the direct normal irradiance, is the energy of the direct solar beam falling on a unit area perpendicular to the beam at the Earth’s surface. To obtain the global irradiance

John Boland

University of South Australia, Mawson Lakes, e-mail: john. boland@unisa. edu. au Barbara Ridley

University of South Australia, Mawson Lakes, e-mail: barbara. ridley@unisa. edu. au the additional irradiance reflected from the clouds and the clear sky must be in­cluded” (Lunde 1979, p. 69). This additional irradiance is the diffuse component.

Typically solar collectors are not mounted on a horizontal surface but tilted at some angle to it. Thus it is necessary to calculate values of total solar radiation on a tilted surface given values for a horizontal surface. It is not possible to merely employ trigonometric relationships to calculate the solar radiation on a tilted col­lector. This is because the diffuse radiation is anistropic over the sky dome and the “radiative configuration factor from the sky to the tilted solar collector is not only a function of the collector orientation, but is also sensitive to the assumed distri­bution of the diffuse solar radiation across the sky” (Brunger 1989). There are two different approaches to calculating the diffuse radiation on a tilted surface; using an­alytic models (Brunger 1989) or empirical models such as that of Perez et al. (1990). Each relies on knowledge of the diffuse radiation on a horizontal surface. The dif­fuse component is not generally measured. Consequently, it is very useful to have a method to estimate the diffuse radiation on a horizontal surface based on the mea­sured global solar radiation on that surface.

Numerous researchers have studied this problem and have been successful to varying degrees. Liu and Jordan (1960) developed a relationship between daily dif­fuse and global radiation which has also been used to predict hourly diffuse val­ues. The predictor typically used in studies is not precisely the global radiation but the “hourly clearness index kt, the ratio of hourly global horizontal radiation to hourly extraterrestrial radiation” (Reindl et al. 1990). Orgill and Hollands (1977) and Erbs (1982) correlate the hourly diffuse radiation with kt, but Iqbal (1980) ex­tended the work of Bugler (1977) to develop a model with two predictors, kt and the solar altitude. Skartveit and Olseth (1987) also use these two predictors in their correlations. Reindl et al. (1990) use stepwise regression to “reduce a set of 28 po­tential predictor variables down to four significant predictors: the clearness index, solar altitude, ambient temperature and relative humidity.” They further reduced the model to two predictor variables, kt and the solar altitude, because the other two variables are not always readily available. Another possible reason was that some combinations of predictors may produce unreasonable values of the diffuse fraction, eg. greater than 1.0 (Reindl et al. 1990). Skartveit et al. (1998) developed a model which in addition to using clearness index and solar altitude as predictors, have added a variability index. This is meant to add the influence of scattered clouds on the sky dome.

As well, Gonzalez and Calbo (1999) stress the importance of including the al­titude and the variability of the clearness index in any predictions of the diffuse fraction. Aguiar (1998) fitted an exponential model to Mediterranean daily data us­ing only the clearness index and found a consistency of fit amongst locations of similar climate.

Boland et al. (2001) developed a validated model for Australian conditions, using a logistic function instead of piecewise linear or simple nonlinear func­tions. Recently, Jacovides et al. (2006) have verified that this model performs well for locations in Cyprus. Their analysis includes using moving average tech­niques to demonstrate the form of the relationship, which corresponds well to
a logistic relationship. Suehrcke and McCormick (1988) and McCormick and Suehrcke (1991) present some significant work on modelling diffuse radiation, in­cluding pointing out that “instantaneous diffuse fraction correlation differs markedly from the correlations obtained for integrated diffuse fractions”. However, in most in­stances, it is integrated values that are normally available for modelling purposes, and indeed it is integrated values that are used in performance estimation software. Thus, we are responding to this specific need in providing understanding of the modelling issues on an hourly time scale.

We have made significant advances in both the physically inspired and formal justification of the use of the logistic function. In the mathematical development of the model utilising advanced non-parametric statistical methods, we have also constructed a method of identifying values that are likely to be erroneous. The method, using quadratic programming, will be described. Using this method, we can eliminate outliers in diffuse radiation values, the data most prone to errors in measurement. Additionally, this is a first step in identifying the means for de­veloping a generic model for estimating diffuse from global and other predictors. Examples for both Australian and locations in other parts of the world will be presented.

Updated: August 4, 2015 — 5:08 am