For polycrystalline materials, the experimental determination of residual stresses neglects the so-called 2 nd order fluctuations arising from e.g. plastic or thermal incompatibilities from grain to grain. This constitutes a serious limitation of the classical measurements methods, since these 2 nd residual stresses are known to have a major influence on the mechanical behavior of metallic alloys, especially if these are strongly textured. In the present paper, a new methodology for the treatment of the measured data is described and compared to classical ones. In order to do so, the simulation of a tensile test is performed using a self-consistent elasto-plastic model, in order to constitute a virtual experimental data set. The 1 st and 2 nd order stresses are extracted from the simulation for various macroscopic stress levels. Two approaches (the classical sin 2 ψ method and a method based on the simultaneous analysis of several X-ray diffraction peaks) are then used to quantify the 1 st order stresses from these " experimental " data. It is clearly shown that the method based on multi-peak analysis allows to minimize the error made by neglecting the so-called 2 nd order stresses and leads to a better quantitative estimation of the 1 st order stresses. * This article is dedicated to Prof. José Gracio, with whom I had the pleasure of working several times and discussing about the links between microstructure and mechanical properties of engineering materials. Every moment of exchange gave me the opportunity to discover his availability, creativity and enthusiasm for life and research. For all of us, he has been an example and a source of inspiration and renewed motivation. Surely, he would have actively participated to discussions about residual stresses.