Mathematical analysis of wind turbines dynamics under control limits: boundedness, existence, uniqueness, and multi time scale simulations

In this study, a mathematical analysis of a wind turbine dynamics is presented. The model represented by control blocks and transfer functions taken from recognized papers and studies, is translated to a system of nonlinear differential algebraic equations. For easier computational and numerical study, we prove the existence of a unique terminal voltage solution, which eliminates the algebraic constraint. Our study provides rigorous proofs of boundedness, existence, and uniqueness for the initial value problem of the system, allowing for the assurance that convergent numerical solutions converge to a unique solution for a given initial condition. This allows scholars to have a free simulator that will aid in dynamical studies of wind turbines without the need for software and Simulink limitations. A safe region within grid parameter space (R and X) is defined, in which existence and uniqueness are guaranteed. We presented time scale analysis and simulations to show that the system can be studied in smaller sizes. Lastly, we introduce cases of two and three time scales.