Hybrid Power Transmission Line Model: Exact Observability via Sensors at Discontinuous Joints with Uniformly Observable Approximations

We introduce a novel power transmission line model featuring both continuous and discontinuous joints, where sensors are strategically placed at the discontinuous joints to capture real-world voltage discontinuities. This unique sensor topology sets it apart from conventional models. The system is governed by wave equations derived from the telegrapher’s equations, representing voltage propagation in an LC transmission line with distinct wave speeds and segment lengths. Voltage continuity is enforced at even-numbered joints, while the discontinuous joints at odd-numbered segments allow for precise sensor placement, creating a hybrid framework. In contrast to traditional models, where sensors at continuous joints may compromise exact observability, we employ the multiplier method to achieve exact observability with minimal observation time, utilizing energy identities and hidden regularity estimates. Furthermore, we develop a Finite Difference approximation that preserves observability uniform as the discretization parameter approaches zero. The proof hinges on discrete multipliers, closely mimicking the continuous PDE system. This work offers practical insights into controlling voltage propagation, providing more robust and efficient strategies for monitoring and stabilizing power transmission lines.