Many transposition models have been proposed in the literature to convert
solar irradiance on the horizontal plane to that on a tilted plane. The
inverse process, i.e. the conversion from tilted to horizontal is
investigated here based upon seven months of in-plane global solar irradiance
measurements recorded on the roof of the Royal Meteorological Institute of
Belgium's radiation tower in Uccle (Longitude 4.35

The usual solar radiation parameters measured on ground are the global
horizontal irradiance,

If many transposition models have been proposed in the literature (see Yang,
2016 for a review) to convert solar irradiance on the horizontal plane to
that on a tilted plane, the inverse process (i.e. converting from tilted to
horizontal) is only poorly discussed in literature. The difficulty stems from
the fact that the procedure is analytically not invertible. As an example,
single-sensor approach involving a numerical search method was proposed by
Yang et al. (2013), and evaluated using various combinations of decomposition
(i.e. models that separate direct and diffuse solar components from the
global one) and transposition models. They found their method sufficiently
accurate for small zenith angles, but reported that the conversion error
increases exponentially as zenith angle increases. More recently, Marion
(2015) presented an iterative method using a modified version of the DIRINT
decomposition model (Perez et al., 1992) in combination with the
transposition model of Perez et al. (1987) to obtain horizontal irradiance
from measurements of a single tilted sensor. Performance of his method was
found essentially the same as for the DIRINT model used in the normal
calculation mode for

The aim of the present study is to (1) evaluate the tilt to horizontal
irradiance conversion when using for model input

The paper is organized as follows: methods to perform the conversion from tilt to horizontal are presented in Sect. 2. In situ measurements are briefly described in Sect. 3. Performances of the different approaches are evaluated in Sect. 4. Final remarks and conclusions are provided in Sect. 5.

Transposition models have the general form:

Definition of incidence angle

Considering the effective global horizontal transmittance,

Given

EM approach was selected here to solve Eq. (

The present analysis is based upon seven months of global solar irradiance
measurements performed on the roof of the radiometric tower of the Royal
Meteorological Institute of Belgium (RMI) located on the Brussels Uccle
plateau (Longitude 4.35

Irradiance measurements were made with a 5 s time step and then integrated to bring them to a 10 min time step. All 10 min data have undergone a series of automated quality control procedures similar to those described in Journée and Bertrand (2011) prior to be visually inspected by a human operator.

The relative ability of the single and multi-pyranometer approaches to
predict horizontal irradiance from the tilted one was estimated by means of
two statistical error indexes: Mean Bias Error (MBE) and Root Mean Square
Error (RMSE).

Performance of the four considered transposition models
in the tilt to horizontal conversion process.
Unsuccessful conversions

Figure

With only one tilted irradiance involved in the inverse modeling approach,
the tilt angle and the surface's orientation have a major impact on the

Figure

Involving three different tilted irradiance measurements in the conversion
process always ensures solutions excepted for the PER model for which the
conversion of a bit less than one third of the data points (i.e. 30.4 %)
were unsuccessful. All models behave quite similarly in term of RMSE (overall
RMSE value of 20.0 W m

When only a single tilted sensor is used, the conversion can be carried out with a decomposition model coupled with a transposition model to solve the inverse transposition problem. In this case, there is an additional error (additional to the inverse transposition problem) in the predicted horizontal irradiance. In addition, with only one tilted irradiance involved in the inverse modeling approach, the conversion's performance is very sensitive to the tilted pyranometer angular configuration. None of the considered transposition models was found to best perform over the 3 pyranometers mounting plane configurations.

When two (or more) tilted irradiance sensors are involved, only a transposition model is required and the inverse transposition problem can be solved more accurately. In practice, there are certain sun positions for which this procedure still fails to produce a valid estimation of the global horizontal irradiance. Consequently more instruments should be used so as to overdetermine the system. Here it has been shown that three tilted pyranometers set at different orientations is sufficient to guarantee solution for three of the four considered transposition models. Indeed, because of the non bijectivity of the anisotropic transposition model of Perez et al. (1987) the conversion of a percentage of data points can be unsuccessful.

Finally, comparing the performance between the isotropic and anisotropic
approaches to the inverse transposition problem indicates that the
improvement from using the Liu and Jordan (1962) isotropic model to using
anisotropic models is not significant (e.g. a RMSE of 20.0 W m

RMI has not an open data policy, but data can be order at:

The authors declare that they have no conflict of interest.

This study was supported by the Belgian Science Policy Office (BELSPO) through the BRAIN-be research project: “Solar irradiation from the energy production of residential PV systems-SPIDER”, contract No. BR/314/PI/SPIDER. Edited by: S.-E. Gryning Reviewed by: two anonymous referees