This paper proposes a new method for partitioning power grids based on the nonlinear Koopman Mode Analysis (KMA). Grid partitioning is the fundamental problem in the controlled islanding strategy. The KMA is a new technique of nonlinear modal decomposition based on properties of the point spectrum of the so-called Koopman operator. The key idea in the proposed method is to determine a set of islanded sub-grids using KMA of data on voltage angle dynamics of every bus. The method is numerically investigated with the IEEE 118-bus test system. It is shown that the proposed method provides partitions on a multiple frequency scale as well as captures the intrinsic structural properties of a grid characterized by spectral graph theory.