The accurate description and robust simulation at relatively low cost of a size polydisperse population of fine particles in a carrier fluid is still a major challenge for many applications. For this purpose, moment methods, derived from a population balance equation, represent a very interesting strategy. However, one of the major issues of such methods is the realizability: the numerical schemes have to ensure that the moment sets stay realizable, i.e. that an underlying distribution exists. This issue is all the more crucial that some moment vectors can be at the boundary of the moment space for practical applications, corresponding to a population of particles with only one or a few sizes. It is then investigated here for the advection operator, for which it is particularly significant. Then second order realizable kinetic finite volume schemes are designed, with two strategies for the fluxes evaluation based on the work of Kah et al. (J. Comput. Phys., 231:394-422, 2012) and of Vikas et al. (J. Comput. Phys., 230:5328-5352, 2011), which are here completely revisited, analyzed and compared in a Cartesian mesh context. For a potential easiest generalization to un-structured meshes, simplified but still realizable versions of these schemes are also developed. The high accuracy of all the schemes is then numerically checked on 1D and 2D test cases, with Carte-sian meshes, and their robustness is shown, even when some moments vectors are at the boundary of the moment space.