To study the performance of a new environmentally friendly refrigerant mixture, the simulation software Aspen Plus was used to simulate the compressor discharge temperature, compression work, evaporating temperature, system cooling capacity, and COP of the zeotropic refrigerant mixture R1234yf/R116/R14 under different component ratios for a three-stage auto-cascading refrigeration system. The accuracy of the model was verified by comparing it with existing experimental data. Additionally, the simulation results were compared and analyzed. The results reveal that the compressor operating conditions and evaporating temperature were close to each other under the different compositions. The cooling capacity and COP of the system were the largest, and the performance was the best when the composition was 0.50/0.27/0.23. This study provided a theoretical basis for applying the R1234yf refrigerant in refrigeration systems.