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 unstable when applied to a power system that includes a significant number of delayed variables. Numerical stability is discussed 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.