fractional integro-differential equation, state dependent delay, mild solution, stability analysis, stochastic calculus, Applied mathematics. Quantitative methods, T57-57.97

In this work, the stability results for a nonlinear mathematical model are derived, and the power system is realized by utilizing fractional calculus theory. The fixed point theorem is used to establish sufficient conditions for the existence of a mild solution and the stability of a nonlinear impul-sive fractional stochastic integro-differential equation with state-dependent delays with Mainardi’s function in a Hilbert space. Numerical simulations are provided to validate the obtained theoretical results. The proposed model supports (i) predicting the instability of synchronization between generators and the lines and (ii) stabilizing the disturbance that occurs in synchronization among generators and the lines.