Early‐ and late‐lumping observer designs for long hydraulic pipelines: Application to pumped‐storage power plants

Pumped‐storage power plants typically feature very long hydraulic pipelines, which can be modeled by a set of partial differential equations. The estimation of the pressure and volumetric flow along the pipes is an important task for the operation of such a plant. Therefore, this work compares different early‐ and late‐lumping–based observer designs for this system. Two late‐lumping observers, ie, a Lyapunov‐based design and an observer using the backstepping design method, are examined. The Lyapunov‐based approach uses a simple boundary correction to stabilize the estimation error dynamics. In contrast, the backstepping‐based approach allows utilizing additional in‐domain correction to obtain a faster rate of convergence. For the implementation of these distributed‐parameter observers, the spectral element method as a flexible and computationally efficient discretization method is introduced. It is shown that, compared with that of the Lyapunov‐based design, the discretization of the backstepping‐based design requires additional spatial grid points for the accurate approximation of its feedback gains. For the early‐lumping approach, the spectral element method is used to approximate the model equations by a system of differential equations. Based on this approximation, an extended Kalman filter is designed. All observer designs are validated and compared for a representative test case.