Identifying attractors in a ferroresonant circuit: a simulated perturbation approach

In a previous paper (S.K. Chakravarthy, Nonlinear oscillations due to spurious energisation of transformers, Proc IEE, EPA, Vol. 145, No. 6, Nov. 1998, 585–592) we have demonstrated the nature of oscillations that can occur in a spuriously energised transformer. The system, in that case, was a third order nonlinear (non-autonomous) ode from which bifurcation equations were determined by the method of multiple scales. The computational difficulty became awesome when analysing oscillations due to spurious energisation of power transformers. The order of the system increased from a third order to a fourth order system in this case. We further demonstrated that nonlinear interaction in power system involves dynamic phenomena of widely different speeds. Using the ‘brute force’ of simulation to search the phase space for the attractors of the system is not feasible. In this paper, we hence study higher order nonlinear systems by an approach similar to the method of perturbations. We propose to use numerical simulation as the basic tool for assessing the effect of perturbations and have coined the term ‘simulated perturbation’. In this approach, the starting point is the conservative autonomous system for which all possible oscillatory modes can be determined. We then introduce other variables in the equations one parameter at a time and then, study the changes in the nature of the oscillations. As we shall show, the method allows one to study a complex nonlinear ode with relative ease and yet obtain a feel for the factors behind the behaviour observed.