Abstract
The chapter focuses on the numerical solution of parametrized unsteady Eulerian flow of compressible real gas in pipeline distribution networks. Such problems can lead to large systems of nonlinear equations that are computationally expensive to solve by themselves, more so if parameter studies are conducted and the system has to be solved repeatedly. The stiffness of the problem adds even more complexity to the solution of these systems. Therefore, we discuss the application of model order reduction methods in order to reduce the computational costs. In particular, we apply two-sided projection via proper orthogonal decomposition with the discrete empirical interpolation method to exemplary realistic gas networks of different size. Boundary conditions are represented as inflow and outflow elements, where either pressure or mass flux is given. On the other hand, neither thermal effects nor more involved network components such as valves or regulators are considered. The numerical condition of the reduced system and the accuracy of its solutions are compared to the full-size formulation for a variety of inflow and outflow transients and parameter realizations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
We always identify flux conditions with demand, i.e., with outflux, and pressure boundary conditions with supply, i.e., with influx. Without further modification, this (somehow arbitrary) identification can be relaxed such that demand can also be modeled as pressure conditions and supply as mass or volumetric fluxes.
- 2.
The direction given to the edges serves the sole purpose of topology definition and is independent of the direction of the flux within the pipe that results from the laws of continuum mechanics.
References
Alamian, R., Behbahani-Nejad, M., Ghanbarzadeh, A.: A state space model for transient flow simulation in natural gas pipelines. J. Nat. Gas Sci. Eng. 9, 51–59 (2012)
Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010). doi:10.1137/090766498
Grundel, S., Hornung, N., Klaassen, B., Benner, P., Clees, T.: Computing surrogates for gas network simulation using model order reduction. In: Surrogate-Based Modeling and Optimization, pp. 189–212. Springer, New York (2013)
Grundel, S., Jansen, L., Hornung, N., Clees, T., Tischendorf, C., Benner, P.: Model order reduction of differential algebraic equations arising from the simulation of gas transport networks. In: Schöps, S., Bartel, A., Günther, M., ter Maten, E.J.W., Müller, P.C. (eds.) Progress in Differential-Algebraic Equations, Differential-Algebraic Equations Forum, pp. 183–205. Springer, Berlin/Heidelberg (2014). doi:10.1007/978-3-662-44926-4_9
Haasdonk, B.: Convergence rates of the POD–greedy method. ESAIM: M2AN 47(3), 859–873 (2013). doi:10.1051/m2an/2012045. http://dx.doi.org/10.1051/m2an/2012045
Herty, M., Seaïd, M.: Simulation of transient gas flow at pipe-to-pipe intersections. Int. J. Numer. Methods Fluids 56(5), 485–506 (2008). doi:10.1002/fld.1531. http://dx.doi.org/10.1002/fld.1531
Higham, D., Trefethen, L.: Stiffness of ODEs. BIT Numer. Math. 33(2), 285–303 (1993). doi:10.1007/BF01989751
Hinze, M., Kunkel, M.: Discrete empirical interpolation in POD model order reduction of drift-diffusion equations in electrical networks. In: Scientific Computing in Electrical Engineering SCEE 2010, pp. 423–431. Springer (2012)
Kiuchi, T.: An implicit method for transient gas flows in pipe networks. Int. J. Heat Fluid Flow 15(5), 378–383 (1994)
Osiadacz, A.: Simulation of transient gas flows in networks. Int. J. Numer. Methods Fluids 4(1), 13–24 (1984). doi:10.1002/fld.1650040103. http://dx.doi.org/10.1002/fld.1650040103
Schmidt, M., Steinbach, M.C., Willert, B.M.: High detail stationary optimization models for gas networks. Optimization and Engineering, 16(1), 131–164 (2015). doi:10.1007/s11081-014-9246-x
Swamee, P.K., Jain, A.K.: Explicit equations for pipe-flow problems. J. Hydraul. Div. 102(5), 657–664 (1976)
Volkwein, S.: Optimal control of a phase-field model using proper orthogonal decomposition. ZAMM – J. Appl. Math. Mech./ Z. Angew. Math. Mech. 81(2), 83–97 (2001). doi:10.1002/1521-4001(200102)81:2¡83::AID-ZAMM83¿3.0.CO;2-R
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Grundel, S., Hornung, N., Roggendorf, S. (2016). Numerical Aspects of Model Order Reduction for Gas Transportation Networks. In: Koziel, S., Leifsson, L., Yang, XS. (eds) Simulation-Driven Modeling and Optimization. Springer Proceedings in Mathematics & Statistics, vol 153. Springer, Cham. https://doi.org/10.1007/978-3-319-27517-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-27517-8_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-27515-4
Online ISBN: 978-3-319-27517-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)