| Synchronization has been widely studied in the fields of physics,biology,and chemistry.One of the most classic models for studying synchronization is the coupled system model proposed by Kuramoto.Due to the wide application of the Kuramoto model,there is a large amount of literature studying this model.More and more scholars have begun to pay attention to the application of complex network theory in power systems.However,most of the existing work is carried out under conditions of complete connection or symmetric coupling but little research on asymmetric coupling.This thesis will study the first-order asymmetric Kuramoto model related to the power system.We first introduce the swing equation and the Kuramoto model.When some parametric conditions are fulfilled,the second-order Kuramoto model can be approximated by a first-order Kuramoto model by singular perturbation theory.However,the Kuramoto models considered in these settings should be asymmetric,so it is interesting to analyze the Kuramoto model with asymmetric coupling.In literature,an important approach for synchronization is the energy estimate and the existing works rely on the symmetric coupling.In this thesis,we will use energy method to study the asymmetric Kuramoto model;the main work is as follows: Firstly,we discuss the bounded synchronization of the system when the coupling is asymmetric.By constructing the energy function and using the symmetrization method,we overcome the difficulties caused by the asymmetry.We find some sufficient conditions leading to the bounded synchronization.Of course,this approach can be applied for the case of symmetric coupling.Secondly,we analyze the bounded synchronization for the asymmetric coupled model with frustration and sufficient conditions for bounded synchronization are derived.Finally,we give some numerical simulations for the first-order Kuramoto models with different conditions,which show that the system can achieve synchronization when the conditions are fulfilled. |
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