An alternative to the physical limits approach is to rely on Eq. (1.1), the theoretical relation between the three components. This approach has important advantages: it is objective and can be performed a posteriori. It can also be implemented directly, or more simply using irradiance values normalized to extraterrestrial beam (Io) values. These normalized values are referred to as clearness indices, Kn = B/Io for direct beam, Kt = G/ [Io cos(z)] for global total hemispherical, and Kd = D/[Io cos(z)] for diffuse sky clearness. The closure equation, Eq. (1.1), then becomes
Kt = Kn + Kd. (1.7)
With a large collection of historical data, the site-specific relation between any two of the clearness indices with any other can be developed, and the physical limits boundaries greatly reduced to an envelope of acceptable data, bounded by limiting curves, rather than zero and some upper limit. Figure 1.12 shows a schematic relationship between Kt and Kn for a site, with analytical boundaries defined by doubleexponential Gompertz functions (Maxwell 1993; Younes et al. 2005).
The equation of the Gompertz curves (Parton and Innes 1972) is:
Y = A ■ Wc wDX (1.8)
where choice of A, W, C, and D, along with judicious “shifting” left and right along the X-axis, result in the proper “S” shaped boundaries around the data. Acceptable data then falls ‘within’ the analytic boundary curves. A library of curves can be build up for sites, times, and air mass conditions. An important point to keep in mind regarding either a direct computation of the closure condition or the clearness index approach is that with the known uncertainties in measured data, a tolerance, or acceptable deviation from perfect closure is needed. Typically, with measurement data uncertainties of 3% to 5% in total global and direct beam data, tolerances
Fig. 1.12 Clearness index relation between Kt and Kn showing schematic envelope of analytical (Gompertz) functions (solid curved lines) which can be assigned for acceptable data
of ±5% in the balance are allowed. This means tolerances of about 0.02 to 0.03 in the clearness-index approach.
Note again that if calibrations and measurements are performed with bias (type B) errors inherent in each of the instruments for measuring different components, the closure test can be “passed” even though the measurements themselves still contain errors, perhaps several times as large as the tolerances.
Other approaches to establishing envelopes around physical relations between measured data components (such as diffuse and direct to total global ratio) have also been developed (Younes et al. 2005).