  SAXO+ is a proposed upgrade to SAXO, the AO system of the SPHERE instrument
on the ESO Very Large Telescope. It will improve the capabilities of the
instrument for the detection and characterization of young giant planets. It
includes a second stage adaptive optics system composed of a dedicated
near-infrared wavefront sensor and a deformable mirror. This second stage will
remove the residual wavefront errors left by the current primary AO loop
(SAXO). This paper focuses on the numerical simulations of the second stage
(SAXO+) and concludes on the impact of the main AO parameters used to build the
design strategy. Using an end-to-end AO simulation tool (COMPASS), we
investigate the impact of several parameters on the performance of the AO
system. We measure the performance in minimizing the star residuals in the
coronagraphic image. The parameters that we study are : the second stage
frequency, the photon flux on each WFS, the first stage gain and the DM number
of actuators of the second stage. We show that the performance is improved by a
factor 10 with respect to the current AO system (SAXO). The optimal second
stage frequency is between 1 and 2 kHz under good observing conditions. In a
red star case, the best SAXO+ performance is achieved with a low first stage
gain of 0.05, which reduces the first stage rejection.
