With wide applications of power electronic devices in modern power systems, simulation using traditional\ud electromechanical and electromagnetic tools suffers low speed and imprecision. Multi-rate methods can enhance\ud efficiency of simulation by decreasing the scale of systems in small time-steps. However, the existing traditional\ud methods for multi-rate simulation suffer the problems of instability and simulation errors. These have hindered the\ud application of multi-rate simulation in power industry. Therefore theoretical evaluation on different multi-rate simulation\ud methods is crucial to understand the feasibility and limitation of the methods, and to contribute to overcome the\ud drawbacks of the traditional methods. In this paper, the multi-rate simulation performance based on two traditional\ud technologies and a Modified Thevenin Interface are evaluated to provide an overall feasibility of multi-rate algorithms\ud in the power simulation. The Modified Thevenin Interface is proposed to overcome the drawbacks in synchronization.\ud Three theorems are proposed and proved for theoretically analyzing the stability of the simulation methods. Error\ud analyses of the multi-rate methods are performed to identify the relationships between errors and simulation\ud conditions. Besides, the accuracy and efficiency performance in a practical project of VSC-MTDC shows the feasibility\ud and necessity by using multi-rate simulation. Through the theoretical analysis, the issues of stability and accuracy of\ud multi-rate simulation for the DC grids have been better understood, based on which an improved simulation algorithm\ud has been proposed to overcome these issues. Long-term system dynamics of large-scale systems containing DC grids\ud and fast transients of HVDC converters can be investigated simultaneously with high speed and sufficient accuracy.