Assume the case that the measured efficiency ^me is equal to the modeled efficiency ^mo during the whole test, we will get a straight line as presented in Figure

1. This line we call identity line. In the real case (Figure 2 and Figure 3) ^me and ■qmo are different. Measured and modeled efficiency can be determined with their uncertainties for a chosen confidence limit. A confidence limit of 95% is chosen here. Combining the uncertainties of ^me and ^mo we get an uncertainty ellipse for each measuring point. The uncertainty ellipse shows, with a probability of 95%, where the single measuring point can be located. If the uncertainty ellipse has an intersection with the identity line, we assume for this measuring point that the measured efficiency is equal to the modeled efficiency. To prove the value and the confidence limit of the estimated uncertainties, we tested for which percentage of the pairs (measured / modeled efficiency) the uncertainty ellipse (with 95% confidence limit) covers the identity line with:

Л measured = ‘Л modeled (see figure 1).

The intersections of the ellipse were calculated using equation (6).

SHAPE * MERGEFORMAT

SHAPE * MERGEFORMAT

i)MeSP [2] : intersection of the identity line with the elipse

VMe [2]; ЛмОо [2] : pair of the modeled and measured eficieny

UriMfe[i] ; Um [2]: measured and modeled uncertainties of the eficiencies

The modeled uncertainties for the i-th data point were calculated with:

UX1[;]….. UX6[;] : uncertainity of the variables;

UX1[;]….. UX6[;]: uncertainity of the coeficients

The uncertainty of the coefficients is taken from the vector C as discussed above for the WLS method. For the LS method we take the uncertainty of the coefficients from the result of the Excel™ spread sheet program.

We calculated the measured uncertainty for the i-th data point with equation (8).

Um [2]: Uncertainty of the collector power

Ug [2] : Uncertainty of the global radiation Ua [2] : Uncertainty of the collector area

Standard transducer uncertainties as defined by EN 12975 [ 1 ] are used for the calculation of equations (7) and (8).

For about 4,5% with the LS method and for about 7% with the WLS method the uncertainty ellipse does not cover the identity line (expected value is 5%). In the examples discussed here, we tested 134 of the efficiency pairs from one collector test. To get the 95% confidence limit result with both methods, we had to increase the uncertainty of the pyranometer from 10 W/m2 to 10 W/m2 + 1%. This indicates that the estimated uncertainty assumes the same value, with the same confidence limit (95%), with which the uncertainty was calculated using the WLS and the LS methods. The outliner points for the Ls and the WLS methods are indicated in Figure 2 and Figure 3.