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

Abstract

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.