Monthly averaged-hourly solar diffuse radiation model for the UK

Monthly-averaged daily global irradiation data are now easily available from NASA website. Using established models it is then possible to decompose the daily to averaged-hourly global irradiation. The missing link so far has been hourly averaged diffuse irradiation. In this article data was pooled from 10 UK locations to obtain a regression model to complete the above missing link. It was presently shown that the averaged-data regressions are distinctly different from previously available hour-by-hour regressions. Practical application : The present work has explored a correlation between averaged-hourly diffuse and global irradiation using the diffuse ratio–clearness index envelope. Results show that a strong regression relationship is thus obtainable. The present work has therefore the potential for other similar research that may be followed up in a likewise manner.


Introduction
Solar radiation data are essential for the design of building services. These data are needed for obtaining solar energy transmission through glazing and also through opaque fabric such as walls and roof.
In response to the above demand posed by industry, CIBSE provides, and is in the process of extending the work related to production of computed data sets of: a) Hourly time series of global and diffuse illuminance with the view to obtain frequency of any given level of illuminance b) Sol-air temperature tables for horizontal and vertical surfaces. These are produced for days that belong to 97.5 th percentile when daily irradiation totals are arranged in an ascending order. c) Monthly-average irradiation for horizontal, vertical and sloped surfaces.
In the preparation of successive editions of CIBSE Guides A [1, 2, 3, 4, 5, 6], Guide J [7] and weather data sets a long-term series of hourly global and diffuse irradiation were required. Not always and also not for all locations (countries which cannot afford the measurement equipment and techniques involved) are measurements of the latter component available though.
Furthermore, as of year 2000, the UK Meteorological Office has stopped recording diffuse irradiation at all but two locations -Camborne (50.21N) and Lerwick (60.15N). Therefore, for all of the other major and minor conurbations in the UK the latter component has to be estimated from models that are based on historical records.
The new CIBSE Guide A, to be published in the year 2014, will contain solar radiation tables for 14 UK locations. The above tables will contain information on global and diffuse illuminance, irradiation totals for surfaces of various aspects and slopes, and sol-air temperatures for walls and roof. To produce these tables it is necessary to obtain horizontal-diffuse irradiation, provided global energy data are available. For the above-mentioned 14 locations within the UK it is not always possible to acquire measured diffuse irradiation data as these are seldom available. Hence, it is necessary to develop a mathematical model, built from measured data that is available for some of the above locations.
Another constraint in the present CIBSE Guide A production was that in view of costs, only averagedhourly data were provided by the UK Meteorological Office. The models that were previously developed [10] were not for averaged-data. Rather, they were for use for hour-by-hour records. Hence the previous work [10] was found to be redundant for the present production of CIBSE Guide A.
This article detailed reviews the prior work that has been undertaken in this respect and then progresses to report newer, averaged-hourly diffuse irradiation models. The use and justification of such models is also presented.

The need for diffuse irradiation models
In addition to their use in other disciplines such as agriculture, there are a large number of engineering applications where, beam and diffuse irradiation hourly or sub-hourly are required. The two components are used to estimate, among other things, slope radiation. A brief list of those applications may be drawn thus: • Solar transmission through building fenestration

• Daylight transmission through building fenestration
• Sol-air temperature estimation for opaque building fabric • Solar water heating design and product assessment • Solar PV design and product assessment For above mentioned items (a) and (b), in Section 1, an hour-by-hour diffuse irradiation model is required and previous research has made such work available for the UK [8,9].
In Section 3 some of the above work is reviewed.
For generation of tables related to item (c) in Section 1, new work had to be undertaken to develop monthly-averaged hourly regressions and that is the main subject of this article.
A typical scatter plot for diffuse ratio (ratio of hourly diffuse -to global irradiation) -clearness index (ratio of global -to extraterrestrial irradiation) relationship is shown in Figure 1. The diffuse ratio is represented by k and the clearness index by kt.
Insert Figure 1 This plot, obtained from hour-by-hour data provided by the UK-Meteorological Office for the period 1981-1983 displays a convex shape. The regressed curve reported by Muneer and Saluja [10] is: In the following section 4 an attempt shall be made on developing � -� relationship based on monthlyaveraged hourly data. A justification shall also be stated for such work.

Previous work
As mentioned in Section 1 this work attempts to present a new model that relates averaged-diffuse irradiation to its global counterpart. All previous work has been related to regressions that involve hourby-hour energy quantities. In the following paragraphs a brief review of the older work is carried out as the basic mathematical formulation is unchanged, i.e. diffuse ratio is regressed against clearness index.
Liu and Jordan [11] were pioneers in correlating the relationship between diffuse and global radiation on a horizontal surface, however, the original correlation of Liu and Jordan was developed for daily-not hourly values.
Very many research teams have since developed hourly regressions that primarily relate k, the diffuse ratio (diffuse to global irradiation) to kt, the clearness index (global to extra-terrestrial irradiation ratio) (see Table 1). long-wave radiation which acts as a surrogate for cloud-cover information on the regional scale. An objective evaluation showed that the diffuse solar radiation is better reproduced by neural network synthetic series than by a correlation model.

. Monthly-averaged, � − ���� relationship
The previous section reviewed the k-kt relationship that was based on hour-by-hour data. It was also shown that during the past forty years such regressions have been presented for very many regions of the world. It is however, interesting to note that there is a dearth of such knowledge for diffuse ratioclearness index regressions that are based on averaged data.
The need to produce k � − k t � relationship presently stemmed from three factors: is provided in Table 2. This information was downloaded from the above-mentioned NASA website. Note that the NASA reported irradiation data were compared against averaged measured data for the period 1981-1983 (three complete years) (see Figure 3). The statistics within the latter figure shows that there is a close concordance between the satellite-based NASA irradiation-and the UK Meteorological Office measured data set.
Insert Table 2 Insert Figure 3 It is therefore possible to construct a three-step computational chain that links with the NASA data that now exists in public domain to obtain all manner of solar energy calculations that require hourly horizontal and slope, global and diffuse irradiation. Figure 4 shows the above mentioned computational chain.
Insert Figure 4

Analysis and discussion
The data used for this study were obtained from UK Meteorological Office. Ten locations were chosen which are detailed in Table 3 and Figure 5. Data consisted of hourly global and diffuse irradiation values for several years for each location, covering the most of the range of latitude for the country. The location names have been arranged in increasing order of latitude.
Insert Table 3 Insert Figure 5 Monthly-averaged hourly values were calculated for the global and diffuse radiation considering the data period for each location.
For each of them, the diffuse ratio (k) and the clearness index (kt) for every hour in each month were calculated. The following conditions were used in each case to remove erroneous recorded data.
The monthly-averaged clearness index was then regressed against the monthly-averaged diffuse ratio for each location. Figure 6 shows one such scatter plot for Bracknell. Furthermore, for each increment at bandwidth of clearness index of 0.05 width, the corresponding values of diffuse ratio shown in Figure   7 were again averaged.
Insert Figures 6 and 7 The data from all ten locations were found to have a similar degree of scatter as depicted in Figure 5. Figures 8 and 9 respectively display the all-location scatter plot and bin-wise averaged plot.
Insert Figures 8 and 9 In solar radiation studies, it is common to encounter data that lie unusually far from the bulk of the data population. These data are called outliers and in literature there are standard statistical tests to ascertain if those data are indeed outliers [25].
In this study though, none of the data were found to be outliers.
A point worth mentioning here is that in almost all cases of the data from 10 locations that is under examination, there is an increasing trend of diffuse ratio for the top end of clearness index. This is a well-known phenomenon that is associated with simultaneous occurrence of two astronomical / weather-related conditions: • Low solar altitude angle, and • Sun shining strongly through a broken cloud The result is that a high clearness index is obtained from a high beam irradiation augmented with skydiffuse and cloud-reflected radiation as is also a high value of diffuse ratio. The above phenomenon is depicted in Figure 10.
Insert Figure 10 Table 4 sheds further light on those points that belong to the class of data under discussion.
Insert Table 4 A single regression curve for UK was obtained by pooling all data include in Figure 9. That regression model is given in the following equation:

Conclusions
It was presently shown that monthly-averaged hourly data for solar radiation are much more economical to obtain than hour-by-hour records. Furthermore, through NASA website one may freely obtain daily solar data which may easily be decomposed into hourly global irradiation time-series. The missing link so far has been averaged-hourly diffuse irradiation.
This work fills that gap. A single regression equation has presently been obtained for the UK as a whole.
Application of this equation has enabled production of Vertical Surface irradiation data for 24 key UK locations that form an important part of CIBSE Guide A (2014) [6].      Decompose into monthly-averaged hourly global irradiation [10,23,24] Split into monthly-averaged hourly diffuse and beam radiation using presently proposed regression for averaged data.
Step I Step II Step III