Co-simulation of heterogeneous systems allows for in-depth analysis of various aspects of power systems’ operation while staying within the environments of the simulation tools that are best fit to represent their respective domains. Equipped with a proprietary co-simulation platform, the paper focuses on the issue of power-conjugate coupling between parts of power grids modeled in transient stability and electromagnetic transient simulation tools. The problems of co-simulation stability and precision in presence of delays are tackled by means of designing a proper coupling interface. It is shown that two established interface methods – the V-I method and the Transmission Line Interface – are special cases of a generalized interface framework proposed in the paper. Moreover, a new interface algorithm is described by parametrizing the generalized framework. Analytical tools are also formulated to aid in the analysis of interface stability and precision via the concepts of passivity and transparency. Simulation results of benchmark systems of various complexity demonstrate the application of the developed power coupling interface.