  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.
