Analytical solution to swing equations in power grids

Objective: To derive a closed-form analytical solution to the swing equation describing the power system dynamics, which is a nonlinear second order differential equation. Existing challenges: No analytical solution to the swing equation has been identified, due to the complex nature of power systems. Two major approaches are pursued for stability assessments on systems: (1) computationally simple models based on physically unacceptable assumptions, and (2) digital simulations with high computational costs. Motivation: The motion of the rotor angle that the swing equation describes is a vector function. Often, a simple form of the physical laws is revealed by coordinate transformation. Methods: The study included the formulation of the swing equation in the Cartesian coordinate system, which is different from conventional approaches that describe the equation in the polar coordinate system. Based on the properties and operational conditions of electric power grids referred to in the literature, we identified the swing equation in the Cartesian coordinate system and derived an analytical solution within a validity region. Results: The estimated results from the analytical solution derived in this study agree with the results using conventional methods, which indicates the derived analytical solution is correct. Conclusion: An analytical solution to the swing equation is derived without unphysical assumptions, and the closed-form solution correctly estimates the dynamics after a fault occurs.