Phase change material (PCM) has received significant attention due to its great potential for thermal energy storage. However, the major undesirable property of PCM is related to its low thermal conductivity. In this work, the electrohydrodynamic (EHD) enhancement of PCM melting in circular-elliptical annuli is investigated numerically by using the lattice Boltzmann method (LBM). The key motivation for our choice of the elliptical shape is due to the fact that the more curved elliptical surface corresponds to stronger charge injection strength, which may lead to stronger flow field, and the consequent increase of heat transfer rate. The influences of several non-dimensional parameters, including electric Rayleigh number T, thermal Rayleigh number (Ra) and the aspect ratio (AR) of the inner ellipse are investigated in detail. Based on the numerical results, it is found that the radial electro-convective flow induced by the external electric field makes a significant contribution to the enhancement of melting heat transfer, and specially, the maximum time saving in some cases is more than 85%. Moreover, we observe that when the Coulomb force is dominant over the buoyancy force, no matter the inner elliptical tube is oriented horizontally or vertically, the total melting times in these two cases are nearly the same, and the melting performance obtained for the circular electrode is usually better than the other cases. However, when the flow regime is dominated by the buoyancy force, the use of a slender vertical-oriented elliptical electrode instead of the circular one is more efficient.