Harmonic Solution of Higher-Dimensional Second Order Kuramoto Oscillator Network

Obtaining the solution of Kuramoto oscillator model is an effective way to understand the mechanism behind the dynamical behaviors it represents. However, it is not an easy work to get, even the approximation solution for the second order Kuramoto oscillator network because of its complex coupling interaction and strong nonlinearity. Hence, this paper first establishes a new polynomial mathematical model for the second order Kuramoto oscillator network, which is partly decoupled in a polynomial fashion. Then, a quadratic harmonic balance method (HBM) is developed to get the approximate solution for quadratic differential algebraic equation (QDAE) system. An equivalent quadratic technique is introduced to convert the polynomial Kuramoto oscillator model into a QDAE system aiming to obtain its harmonic solution. Finally, the suggested harmonic solution is utilized to explain the morphological characteristics of the stable motion of oscillators, and the relative motions for the oscillators in Kuramoto network is concluded by a set of nonlinear state output equations. It is also found that the distance between the maximum phase angle and the controlling unstable equilibrium point (UEP) can be used as a stability margin for the oscillator network. A single-machine infinite bus (SMIB) power system, an IEEE 3-machine 9-bus power system and an IEEE 10-machine 39-bus power system are employed as different dimension second order Kuramoto oscillator network cases to demonstrate the validation and accuracy of the proposed method and analysis.