U-Pu-Zr nuclear fuels experience a redistribution of constituents and a number of phase transformations when subjected to the thermal gradient present in nuclear reactors. This redistribution and phase separation leads to several undesirable fuel performance issues. In an effort to better understand how different alloys compositions are affected by this thermal gradient, we utilize the recently introduced Hybrid Potts-phase Field Method to study the U-Pu-Zr system. The recently introduced Hybrid method couples microstructural and compositional evolutions of a system so that the two phenomena can be studied together rather than separately, as is frequently done. However, simulation of the U-Pu-Zr system required several adaptations to the modeling framework. First the model was adapted to incorporate a thermodynamic database for free energy calculations, as well as thermal diffusion (the Soret effect). These abilities were tested in the Al-Si system. Second, the modeling framework was expanded to simulate three component systems such that ternary U-Pu-Zr alloys could be studied.Simulations capture constituent redistribution and the appropriate phase transformations as compared to experimentally irradiated a U-16Pu-23Zr (at%) nuclear fuel. Additional simulations analyze constituent redistribution over the entire spectrum of U-Pu-Zr compositions. Analysis of these simulation results indicate alloys that are likely to experience minimal constituent redistribution and fewer phase boundaries, such that their fuel performance should be improved. The outcomes of the work include a coupled microstructural-compositional modeling framework for ternary alloys and suggestions of U-Pu-Zr alloys that could lead to improved fuel performance.