Physics-Informed Neural Networks for Real-Time State Estimation in High-Altitude Hydropower Cascades

Abstract

Real-time state estimation of hydraulic transients in high-head hydropower cascades is a prerequisite for closed-loop governor control and for the fast frequency reserves that mountainous power systems increasingly rely on. Classical solvers for the water hammer equations, most notably the method of characteristics, are constrained by the Courant–Friedrichs–Lewy condition, which couples the temporal and spatial resolutions and forces the reconstruction of an entire trajectory even when only a handful of pointwise state values are required. This work develops a physics-informed neural network framework for the one-dimensional unsteady pipe flow system that governs penstock dynamics, and analyses it both theoretically and empirically. On the theoretical side we exploit the symmetric hyperbolic structure of the nondimensionalised system together with the monotonicity of the quadratic friction term to derive an a posteriori error estimate in which the growth of the bound is at most proportional to the square root of the horizon rather than exponential, and we combine this estimate with a quasi-Monte Carlo quadrature bound and a tanh approximation result to obtain a total error guarantee expressed entirely in terms of computable training residuals. The analysis identifies exact imposition of boundary data and characteristic-aligned domain decomposition as the two architectural choices that control the constants, and both are adopted in the method. On the empirical side we construct a reference solution for a 1200 m, 300 m gross head penstock using a method of characteristics solver operating at unit Courant number, verified to machine precision against the Joukowsky and Michaud–Allievi limits and shown to be grid independent to eight significant figures. Against this reference the proposed model attains a relative L₂ error of 1.14% in piezometric head and 1.32% in discharge, degrades to 1.42% under 5% measurement noise and to 1.41% with two measurement channels, and recovers the acoustic wave speed to within 0.4% in the inverse setting. A parametric formulation amortises training across the closure-time family and supports online assimilation of supervisory telemetry in 6.4 ms. We also correct a claim that recurs in this literature: measured against a competently implemented reference solver the wall-clock advantage of the surrogate on full-field reconstruction is modest, and the operationally decisive advantages are instead constant-time pointwise evaluation, amortisation across operating scenarios, and end-to-end differentiability for control.