Swing equation in power systems: approximate analytical solution for the dynamics of the system inertia

The system inertia, H(t) and the rotor angle, δ(t) play an important role in the stability study of a power grid, with the latter dictating the active power being delivered to the grid. Since δ(t) is a changing parameter within the power system, its nonlinear dynamic behaviour can prove to be very challenging to be studied. In the present work, we establish an ordinary differential equation (ODE) which gives the temporal variation of H(t) as a function of δ(t). We show that the solution of the ODE is bounded under certain conditions on the mechanical power input, P_m and the electrical power output P_max of the generator. Furthermore, in this preliminary work, we also prove that the solution grows unbounded when δ(t) is close to certain points. A numerical experiment is conducted to solve the swing equation together with the derived ODE, using Runge-Kutta 4th order method, to verify the derivations. The current preliminary results confirm that H(t) can be evaluated based on δ(t), hence confirming the significance of the rotor angle in a power system.