Stability of the Theta method for systems with multiple time-delayed variables

The paper focuses on the numerical stability and accuracy of implicit time-domain integration (TDI) methods when applied for the solution of a power system model impacted by time delays. Such a model is generally formulated as a set of delay differential algebraic equations (DDAEs) in non index-1 Hessenberg form. In particular, the paper shows that numerically stable ordinary differential equation (ODE) methods, such as the trapezoidal and the Theta method, can become numerically unstable when applied to a power system that includes a significant number of delayed variables. Analysis reveals how numerical stability is primarily governed by the number of delayed variables and the strength of their Jacobian coefficients, whereas the influence of the delay magnitudes themselves is found to be comparatively minor. Numerical stability is examined through a scalar test delay differential equation, as well as through a matrix pencil approach that accounts for the DDAEs of any given dynamic power system model. Simulation results are presented in a case study based on the IEEE 39-bus system, as well as the real-world scale model of the All-Island Irish Transmission System (AIITS). • Standard numerical methods can be destabilized when applied to power system models impacted by multiple time-delayed variables. • A systematic analysis of the phenomenon can be carried out through a matrix-pencil-based framework. • Findings can be employed to check numerical stability and select an appropriate time step size that meets prescribed simulation accuracy criteria.