In electric power grids, electromechanical oscillations are continuously presented during both normal and disturbed conditions. Because a power grid is essentially a nonlinear multi-oscillator system, its oscillations have inherent nonlinearities and are different from linear system oscillations. Formulation and characterization of nonlinear oscillations will lead to a new paradigm for online angular stability analysis and control for large-scale power grids. This presentation will introduce a new methodology, called nonlinear modal decoupling (NMD), which inversely constructs a set of decoupled oscillators for a given high-dimensional multi-oscillator system like a power grid. Each of the decoupled oscillators is a fictitious 2nd-order nonlinear system that is virtually islanded from the others and corresponds only to a single oscillation mode of the original nonlinear system. By such a decoupling transformation, analysis on the dynamics and stability related to each mode can be performed on the corresponding oscillator, which will be easier than on the original complex system. This new methodology can easily be applied to online stability analysis for large-scale power grids by means of wide-area measurements.