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A numerical study of the sensitivity of the two dimensional electromagnetic ground plane cloak in the visible spectrum is performed using the finite difference time domain method. Transverse magnetic plane waves are launched at an angle towards an object placed on a ground plane and the scattering signature is suppressed by surrounding it by an all-dielectric cloak. In particular, we study the performance of the system as a function of the number of distinct components the cloak is divided into, the effects of lossy elements, the angle of incidence, as well as typical random variations of the permittivity of the building materials. Finally, we evaluate impedance matching layers that can surround the cloak and improve its effectiveness. The results provide useful information for the fabrication of practical cloaking devices.