The automated generation of geometry-only digital twins of Overhead Line Equipment (OLE) system in existing railways from point clouds is an unsolved problem. Currently, this process is highly reliant upon manual inputs, needing 10 times more labour hours than scanning the physical asset. The resulting modelling cost counteracts the expected benefits of the digital twin. We tackle this challenge using a novel model-driven method that exploits the highly regulated and standardised nature of railways. It starts by restricting the search for OLE elements relative to point clusters of the railway masts. The resulting point clusters of the OLE elements are then converged with various parametric models of different catenary configurations to verify the presence of OLE elements and to find the best possible fit. The method outputs a geometry-only digital twin of the OLE system in Industry Foundation Classes (IFC) format. The method was tested on an 18 km railway point cloud and achieves overall detection rates of 93.2% F1 score for OLE cables and 98.1% F1 score for other OLE elements. The accuracy of the generated model is evaluated using distance-based metrics between the ground truth model and the automated model. The average modelling distance is 3.82 cm Root Mean Square Error (RMSE) for all 18 km data.