Comparison of elastic vs. inelastic penstock model using OpenModelica

Abstract

The possibility for modelling and simulating hydropower systems as accurately as possible take an important role in order to develop a control structure and to make efficient analysis tools for testing a designed controller for stability and performance in different operating regimes. Both the simulation time for such models as well as the accuracy are important.A high head hydropower system is considered for this study. The pipe with the main part of the height drop is known as pressure shaft or penstock, and it can be modeled with two levels of accuracy which have been compared in this studying. A simple model with one nonlinear ODE considers inelastic walls of the penstock and incompressible water. A more realistic model for large pressure variations assumes a penstock with elastic walls and compressible water column in the penstock. This more detailed model of a penstock is described with two nonlinear PDEs which have been solved using the Kurganov-Petrova scheme.Comparing results from these two models it can be concluded that the simple ODE model shows by and large the same results as the PDE model with just slightly smoothed dynamics. Obviously, the simulation time for the inelastic penstock model is considerably smaller. Both models show reasonable results and can be further used for control synthesis and analysis. In cases where the time consumption is most important, the simple ODE model for the penstock is preferred. On the other hand, for more accurate studies the elastic/compressible model for the penstock or even for other waterway units, such as conduit, is more useful.The modeling part for both cases was done in OpenModelica using our own hydropower library, where all models for different units of the hydropower system have been developed and collected.

Nonlinear observer for hydropower system

“…A typical structure for a high head hydropower plant is depicted in Fig. 1, Vytvytskyi and Lie (2017). For simulations in this paper, data from the Sundsbarm hydropower plant in Telemark, Norway is used with data provided in Vytvytskyi and Lie (2017).…”

“…In this library, different waterway components of the hydropower system are described by both mass and momentum balance, and include compressible/incompressible water or elastic/inelastic pipe walls. An overview of the mathematical models and methods used in this library is given in Vytvytskyi and Lie (2017); Splavska et al (2017).…”

“…It is of interest to see how the reduced order nonlinear observer behave when the synthetic measurements are based on a more detailed/realistic model while the simple model is used for the design of the observer. The detailed model used for the synthetic measurements is similar to the model presented above, but includes water compressibility and pipe elasticity in the penstock, see Vytvytskyi and Lie (2017) for more information about this model. Moreover, the Darcy friction factor varies with the flow rate for all pipe units in this detailed model.…”

OpenHPL for Modelling the Trollheim Hydropower Plant

“…Our work on modelling the waterway and the Francis turbine for high head hydropower systems using the multiphysics tool OpenModelica [20], has been reported in [21][22][23]. Unit models have been developed for our in-house open-source Modelica [24] library-OpenHPL (Open Hydro Power Library is developed by the first author within his PhD study).…”

“…The electricity produced by the generator is then transferred through power lines to consumers. More detailed description of the hydropower system has been presented in our previous work [21,22]. A typical structure of a high head hydropower plant is depicted in Figure 1 [21].…”