Increasing nonlinearity in today’s grid challenges the conventional small-signal (modal) analysis (SSA) tools. For instance, the interactions among modes, which are not captured by SSA, may play significant roles in a stressed power system. Consequently, alternative nonlinear modal analysis tools, notably Normal Form (NF) and Modal Series (MS) methods are being explored. However, they are computation-intensive due to numerous polynomial coefficients required. This paper proposes a fast NF technique for power system modal interaction investigation, which uses characteristics of system modes to carefully select relevant terms to be considered in the analysis. The Coefficients related to these terms are selectively computed and the resulting approximate model is computationally reduced compared to the one in which all the coefficients are computed. This leads to a very rapid nonlinear modal analysis of the power systems. The reduced model is used to study interactions of modes in a two-area power system where the tested scenarios give same results as the full model, with about 70% reduction in computation time.