In this paper, the heat storage process of a latent heat thermal energy storage (LHTES) tank is studied numerically. A new type of gradient fin is added to the heat storage process in a latent heat storage tank to improve the heat transfer performance of the internal phase change material (PCM). The numerical model is verified by the experimental data. The influences of different structures on the melting rate, temperature distribution, velocity distribution, and maximum melting rate of PCM in LHTES tank are investigated, and the influence of natural convection on the energy storage system is further quantified. The results show that the melting time can be significantly reduced by 16.9% by using the four straight fins with gradient shape, and the melting time can be further reduced by 41.1% by reconstruction of the gradient fin position. The deformation in the opposite direction will slow down the melting speed and extend the melting time by more than 4.66 times. The convection criterion for judging the buoyancy convection of the melt in this paper shows that natural convection is dominant in both the upper and lower part of the melt. The closer the convection coefficient is to the overall heat transfer coefficient, the better the melting efficiency and energy efficiency will be.