Analysis of Backward Euler Method in Presence of Saturation Nonlinearity and Applications in Power Systems Simulation

Dynamic simulation of a power system involves the solutions of many nonlinear differential-algebraic equations that are computationally expensive. While quasi steady state approximation methods are computationally efficient, they cannot capture many power system phenomena such as controller-induced instabilities. Recently, Backward Euler Method (BEM) has been used to produce a coarser approximation of the ground truth (obtained from Trapezoidal method) at a lower computational effort. However, no fundamental analysis exists in the literature for understanding the properties of BEM in presence of saturation nonlinearity in a dynamical system. This paper mathematically investigates the properties of BEM when applied to a 1 -dimensional and a 2 - dimensional system with saturation nonlinearity. Our analyses show that besides hyper-stability, unlike in a linear time-invariant system, BEM can also suffer from hyperinstability in a system with saturation. Based on the mathematical analyses, qualitative recommendations are presented for adaptively varying the stepsizes of BEM such that the solution can resemble the ground truth in an averaged sense at a significantly lower computational cost. BEM with adaptive stepsize variation is applied to simulate (i) a single-generator system (with saturation nonlinearity in the governor's dynamics) feeding a standalone load and (ii) a 6-bus system with a synchronous generator and inverter-based resources having saturation nonlinearity. It is shown that by adaptively varying the stepsizes based on the presented recommendations, BEM can produce the same end result as in the ground truth while consuming significantly less cpu time.