Influence of Gaussian noise and Levy noise on the phase dynamics of the ensemble of Kuramoto-like oscillators of first and second order

The purpose of this study is to determine the stability threshold of the dynamic modes of the ensemble of phase Kuramoto-like oscillators, describing the behavior of a simple power grid model with a ring topology, under the external influence of Gaussian noise and Levy noise, to evaluate the results and determine the threshold values of noise at which the considered dynamic model is the most sensitive to noise and demonstrates a change of the steady state.  Methods. In this paper, two ensembles of Kuramoto-like phase oscillators with the same topology but different number of oscillators are investigated. The ensembles consist of second and first order phase oscillators modeling the dynamics of generators and consumers in the power grid, respectively. In this work, mode maps are computed and used, from which regions with different synchronous dynamics are selected. In the selected regions, a set of initial conditions is fixed and the ensemble under study is modeled in the presence of noise of different types and intensities. The obtained result is evaluated with the help of calculated spatio-temporal diagrams, values of the Kuramoto parameter and statistical characteristics estimated from the realizations of oscillations in time. Results. It has been shown that a power grid model consisting of Kuramoto-like phase oscillators exhibits different robustness to external noise disturbances depending on the type of noise disturbance and the steady-state dynamic regime. It was demonstrated that the frequency synchronization mode of all oscillators, independent of the initial conditions, is insensitive to the influence of white noise of high intensity, both Gaussian and Levy noise. Whereas, in the region of coexistence of synchronous and asynchronous behavior, depending on the initial conditions, a change of phase dynamics under the influence of different noise is observed. Numerical experiment has shown that the power grid model is more susceptible to Levy noise due to the noise features associated with random emissions, which in turn can be interpreted as random impulses. Conclusion. In a power grid model represented by two ensembles consisting of different numbers of Kuramoto-like phase oscillators of second and first order, different modes of frequency and phase dynamics of the oscillators are established. A numerical experiment with the influence of Gaussian noise and Levy noise is carried out for the obtained modes. It is shown that the model under study is more sensitive to Levy noise, the influence of which leads to a change of the dynamic mode due to the influence of strong random pulses.