The inclusion of higher-order terms in small-signal (modal) analysis has been an intensive research topic in nonlinear power system analysis. Inclusion of second-order terms with the method of normal forms (MNF) has been well developed and investigated, overcoming the linear conventional small-signal methods used in the power system control and stability analysis. However, application of the MNF has not yet been extended to include third-order terms in a mathematically accurate form to account for nonlinear dynamic stability and dynamic modal interactions. Due to the emergence of larger networks and long transmission line with high impedance, modern grids exhibit predominant nonlinear oscillations and existing tools have to be upgraded to cope with this new situation. In this paper, first, fundamentals of normal form theory along with a review of existing tools based on this theory is presented. Second, a new formulation of MNF based on a third-order transformation of the system’s dynamic approximation is proposed and nonlinear indexes are proposed to make possible to give information on the contribution of nonlinearities to the system stability and on the presence of significant third-order modal interactions. The induced benefits of the proposed method are compared to those afforded by existing MNFs. Finally, the proposed method is applied to a standard test system, the IEEE 2-area 4-generator system, and results given by the conventional linear small signal and existing MNFs are compared to the proposed approach. The applicability of the proposed MNF to larger networks with more complex models has been evaluated on the New England–New York 16-machine 5-area system.