The unique nature of solar radiation regressions

Historically speaking, a large number of research teams from around the world have produces k-kt regressions that were based on an hour-by-hour, daily, monthly or annual data. Examples that may be cited here, are Liu and Jordan (daily, and monthly-averaged daily) (Liu & Jordan 1960), Erbs et al. (hourly, daily and monthly-averaged daily) (Erbs et al. 1982), Hawas and Muneer (hour-by-hour, daily, monthly- and annual-averaged daily) (Hawas & Muneer 1984; Muneer & Hawas 1984; Muneer et al. 1984) and Stanhill (monthly- and annual-averaged daily) (Stanhill 1966).

Presently, Figs. 2 and 3 show the unique nature of hour-by-hour (Fig. 2: k − k_t plot) and monthly-averaged hourly regressions (Fig. 3: IEq1_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ \overline{k}-{\overline{k}}_t \] \end{document}). An important point to note is that while Fig. 2 shows a convex profile, Fig. 3 demonstrates a concave behaviour. The latter two figures are based on data from common UK locations.

While there are established models for data of Fig. 2 there are no regressions available in literature for averaged-hourly data such as those shown in Fig. 3. The object of this article is to present the latter type of regressions.

Presently developed monthly-averaged hourly IEq2_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ \overline{k}-{\overline{k}}_t \] \end{document} regressions

Fourteen worldwide locations were chosen for this study, details of which are shown in Table 2.

Data consisted of hourly global and diffuse irradiation values for several years for each location, covering most of the range of latitude for the country. The location names have been arranged in an increasing order of latitude.

Monthly-averaged hourly values were calculated for the global and diffuse radiation considering the data period for each location. For each of them, the monthly-averaged hourly diffuse ratio IEq3_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ \left(\overline{k}\right) \] \end{document} and the corresponding clearness index IEq4_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ \left({\overline{k}}_t\right) \] \end{document} were calculated from sunrise to sunset. The following conditions were used in each case to remove erroneously recorded data.

The monthly-averaged clearness index was then regressed against the monthly-averaged diffuse ratio for each location. Figure 4 shows one such scatter plot for Chennai and Lisbon. Furthermore, for each increment at bandwidth of clearness index of 0.05 width, the corresponding values of averaged diffuse ratio shown in Fig. 5 were obtained, shown here for pooled data from two Indian locations.

Figures 6 and 7 respectively show the regressions for locations in a narrower range of latitudes (20-42° N) and worldwide sites with a more diverse range of latitudes (13-58°N).

Note that Fig. 6 shows the potential for a single regression model. Figure 7 on the other hand indicates the existence of different sub-models and these shall now be explored further.

Figures 8, 9 and 10 respectively present regressions models that were obtained by pooling data from locations with a latitude range of 13-20° N, 20-42° N and 50-58° N.

Table 3 presents regressions equations and coefficient of determination (R²) for each location.

Three points are worthy of note from Figs. 8, 9 and 10:

1. A strong correlation is observed between IEq16_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ \overline{k} \] \end{document} and IEq17_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ {\overline{k}}_t \] \end{document} with the respective coefficient of determination of 0.87, 0.80 and 0.83 (corresponding values of coefficient of correlation are 0.93, 0.89 and 0.91),
2. In each case the shape of the regressed curve is concave, contrary to the convex profile for hour-by-hour regressions reported by research teams from around the world, and
3. it is not possible to produce a single regressed curve for worldwide locations.

The latter point is reinforced via Fig. 11.