Power quality state estimation (PQSE) is a set of techniques with the objective to estimate diverse power quality phenomena in electric power systems; transient state estimation (TSE) being one such technique. Its objective is to estimate node voltage waveforms in a network after it has been subject to an electromagnetic transient. This work focuses on TSE using the numerical integrator substitution (NIS) method to model the system transients. The objective of this work is to further extend TSE with NIS, to include non-linear and distributed parameter elements. The intended application for TSE is to serve as an analysis tool for post-disturbance root cause analysis. The technique can be considered under development but in the future must be capable, considering certain requirements are met for measurements and system modelling, to help in the identification of the cause(s) for failure or malfunction when electromagnetic transients are recorded. The use of TSE removes the need of multiple simulation runs that are currently required in order to find the cause of disturbance (each simulation with a different system configuration trying to match measured waveforms). The contributions of this work include: analysis of numerical oscillation in TSE with NIS using single-phase circuits, the application of TSE with NIS to non-linear branches using the piecewise linear method, the development of an extended observability analysis capable of identifying unobservable islands, the application of virtual measurements to linear and lumped circuits, and the application of TSE with NIS to transmission lines modelled with distributed parameters.