The light speed anisotropy, i.e., the variation of the light speed with respect to the direction in an "absolute" reference frame, is a profound issue in physics. The one-way experiment, performed at the GRAAL facility of the European Synchrotron Radiation Facility (ESRF) in Grenoble, reported results on the light speed anisotropy by Compton scattering of laser photons on high-energy electrons. So far, most articles concerned with the GRAAL data have established only the upper bounds on the anisotropy parameters based on available theories. We use a new theory of the Lorentz invariance violation to analyse the available GRAAL data and obtain the stringent upper limit of the order 2.4×10-14 on the Lorentz violation parameters. In the meantime, we also can reproduce the allowed light speed anisotropy appearing in the azimuthal distribution of the GRAAL experimental data, and find that the best-fit parameters are compatible with the competitive upper bounds.