The use of resonant depolarization has been suggested for precise beam energy measurements (better than 100 keV) in the $e^{+}e^{-}$ Future Circular Collider (FCC-$e^{+}e^{-}$) for Z and WW physics at 45 and 80 GeV beam energy respectively. Longitudinal beam polarization would benefit the Z peak physics program; however it is not essential and therefore it will be not investigated here. In this paper the possibility of self-polarized leptons is considered. Preliminary results of simulations in presence of quadrupole misalignments and beam position monitors (BPMs) errors for a simplified FCC-$e^{+}e^{-}$ ring are presented.