We grew graphene epitaxially on 6H-SiC(0001) substrate by the simulated annealing method. The mechanisms that govern the growth process were investigated by testing two empirical potentials, namely, the widely used Tersoff potential [J. Tersoff, Phys. Rev. B 39, 5566 (1989)] and its more refined version published years later by Erhart and Albe [Phys. Rev. B 71, 035211 (2005)]. Upon contrasting the results obtained by these two potentials, we found that the potential proposed by Erhart and Albe is generally more physical and realistic, since the annealing temperature at which the graphene structure just coming into view at approximately 1200 K is unambiguously predicted and close to the experimentally observed pit formation at 1298 K within which the graphene nucleates. We evaluated the reasonableness of our layers of graphene by calculating carbon-carbon (i) average bond-length, (ii) binding energy, and (iii) pair correlation function. Also, we compared with related experiments the various distance of separation parameters between the overlaid layers of graphene and substrate surface.