WLS is a weighted regression method, as the name says. The method is used for collector coefficients and uncertainties calculations. Theoretical background of this method is given by Press [ 3 ]. Compared to LS it has the following advantages:

— the collector coefficients are determined also using the measurement uncertainties of the transducers

— the measurement uncertainty of the transducer can be different in different measurement ranges

— the statistic distribution of the measurement uncertainties doesn’t have to be normally distributed

— measurement uncertainties may be weakly correlated with each other

Equation (2) shows the individual error squares as function of the collector coefficients a1…a6 and the measurement variables Xi…. Хє. Like in the least square method (LS), first this „squares of the deviation between model and measurement” (2) have to be determined also by the WLS. To do this, the variables for the collector model have to be calculated with the data from the measurement

(1) . The initial coefficients could be determined by the LS regression.

In the next step, the uncertainties of all determined error-values Uerror[i] have to be calculated for every 5 min mean value (3). The measurement uncertainty of every variable U^me; Ux1… Ux6 has to be determined with help of the measurement uncertainties of the transducers to calculate the combined uncertainty Uerror (3).

After the squared quotients of the error — and Uerror-values have been determined, the new weighted evaluation function can be implemented (4).

The collector coefficients are varied iteratively by a spreadsheet program (e. g. EXCEL™) in a way that the sum of x2 is minimized. If the uncertainty of the measurements Uerror is very large for one 5 min mean value, the weight or participation of this data point in the regression process is reduced due to the decreased contribution of the respective error value to the overall value of %2 (see equation 4). That works because the %2-value of this data point is lower.

7me[i]: measured efficiency; ^mo[i] = ^: modeled or calculated efficiency;

Gd : beam radiation [w/m2 ] ; Gb : diffuse radiation [w/m2 ] ; Gb = G — Gd Tm : average collector temperature = Tm = (Tin + Tout) / 2 [°Cj; AT = Tm — Ta Ta : ambient temperature; в : incidence angle

Coeficients a1….. a6 obtained from the regression :

1: rj0 = a1 : zero loss efficiency[-],

1 : V0 • b0 = a2 : (b0 : factor to determine the incident angle modifier of the beam irradiance [-]),

2 : r/0 ■ IAMGd = a3 : (IAMGd : incident angle modifier for diffuse radiation [-] )

4: k1 = a4 : heat loss coeficient[W/(m2K)], negative

5: k2 = a5 : heat loss coeficient[W/(m2K)], negative 6: k3 = a6 : coeficient for the thermal capacity [kJ/(m2K)]

(errO[i] )2 =( rime[i] — Vmof )2=(^ V me[i] “Z (Xk[l] ‘ ak )2 j

„2 = у (erroti])2

& (Uerro [i])2

N : quantity of 5 min mean values

Calculation of the collector coefficients uncertainties with WLS-method:

For the determination of the uncertainties of collector coefficients in the framework WLS method the method described by [ 3 ] is applied here. The application of this method for the SST is described by article [ 6 ]; here it is used for the QDT and calculated the uncertainties for the collector coefficients that were determined by the WLS regression. With equation (5) the collector coefficients uncertainties are calculated using the variables Xk(i) from a data set of one collector test and the uncertainties Uerror[i] (3) of each “5 min mean value” from the same data set. The author [ 5 ] shows that C = (AT ■ A) _1 of eqn. (5) yields a 6×6 matrix with squared uncertainties of the free parameters as diagonal elements and the co-variances between the parameters as off-diagonal elements.

n = number of measurement points