Hydro-Electric Open Channel

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Description

Motivation

The so-called Saint-Venant equations are largely used in the hydraulic domain to model the dynamics of an open channel flow. These equations consist of two nonlinear hyperbolic PDEs. In the considered benchmark, under mild simplifying assumptions detailed in , the St Venant PDE equations describing the height variation $h$ of the river as a function of the inflow $q_i$ and outflow $q_o$ variations, at location $x$ ($x\in [0L]$, $L\in {\mathbb{ℝ}}_{+}$), obtained around some flow and height linearisation point, can be formulated as follows:

$h(x,s)={\mathbf{G}}_{\mathbf{𝐢}}(x,s)q_i(s)-{\mathbf{G}}_{\mathbf{𝐨}}(x,s)q_o(s)=\mathbf{𝐇}(x,s)u(s).$

The ${\mathbf{𝐆}}_i$ and ${\mathbf{𝐆}}_o$ functions are irrational and read

${\mathbf{G}}_{\mathbf{𝐢}}(x,s)=\frac{{\lambda }_1(s)e^{{\lambda }_2(s)L+{\lambda }_1(s)x}-{\lambda }_2(s)e^{{\lambda }_1(s)L+{\lambda }_2(s)x}}{B_{0}s(e^{{\lambda }_1(s)L}-e^{{\lambda }_2(s)L})}$

and

${\mathbf{G}}_{\mathbf{𝐨}}(x,s)=\frac{{\lambda }_1(s)e^{{\lambda }_1(s)x}-{\lambda }_2(s)e^{{\lambda }_2(s)x}}{B_{0}s(e^{{\lambda }_1(s)L}-e^{{\lambda }_2(s)L})}$

Considered data

The benchmark contains the above irrational model description together with the numerical data as used in the reference paper, given as Matlab handle functions.

Origin

Collaboration between ONERA and EDF. The data from the simplified hydro electric open channel model come from V. Dalmas and G. Robert while and the treatment performed jointly with P. Vuillemin, and C. Poussot-Vassal.

Data

The HydroelectricChannel.zip (67Ko) repository contains three files:

• The dataONERA_FlexibleAircraft.mat data file, with
  • W : the frequency values in rad/s (real $1\times 421$ vector).
  • H : transfer function matrix evaluation at different output measurements points of the aircraft (complex $92\times 1\times 421$ matrix).

• The dataONERA_FlexibleAircraft_withMOR.mat data file, with 3 ROMs obtained with the MOR toolbox using the Loewner method
  • Hr1 : linear rational ROM with varying dimensions (state-space models in Matlab form $n_y\times n_u$).
  • Hr2 : linear rational ROM with varying dimensions, with stability post enforcement and input/output normalisation to catch all transfer whatever the amplitude (state-space models in Matlab form $n_y\times n_u$).

• The startONERA_FlexibleAircraft.m script file, used to loads and plots the data for illustration.

The transfer function matrix represents the transfer from the gust input to the 92 measurements gathering from

• 1--44: the local aerodynamic lift on the aerodynamic strips.
• 45--88: the local aerodynamic pitch moment on the aerodynamic strips.
• 89--92: the four generalized coordinates derivative (heave and pitch derivatives) and the first two flexible modes.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

The MORwiki Community, Hydro-Electric Open Channel. MORwiki - Model Order Reduction Wiki, 2018. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Hydro-Electric_Open_Channel

@inproceedings{DalmasECC:2016,
  author    = {V. Dalmas and G. Robert and C. Poussot-Vassal and I. {Pontes Duff}  and C. Seren},
  title     = {From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity},
  booktitle = {Proceedings of the 15th European Control Conference},
  address   = {Aalborg, Denmark},
  month     = {July},
  year      = {2016},
  pages     = {1982-1987},
}