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Efficient numerical methods for gas network modeling and simulation. (English) Zbl 1457.76013

Summary: We study the modeling and simulation of gas pipeline networks, with a focus on fast numerical methods for the simulation of transient dynamics. The obtained mathematical model of the underlying network is represented by a system of nonlinear differential algebraic equations (DAEs). With our modeling approach, we reduce the number of algebraic constraints, which correspond to the \((2,2)\) block in our semi-explicit DAE model, to the order of junction nodes in the network, where a junction node couples at least three pipelines. We can furthermore ensure that the \((1, 1) \) block of all system matrices including the Jacobian is block lower triangular by using a specific ordering of the pipes of the network. We then exploit this structure to propose an efficient preconditioner for the fast simulation of the network. We test our numerical methods on benchmark problems of (well-)known gas networks and the numerical results show the efficiency of our methods.

MSC:

76-10 Mathematical modeling or simulation for problems pertaining to fluid mechanics
65L80 Numerical methods for differential-algebraic equations
65F08 Preconditioners for iterative methods
37M05 Simulation of dynamical systems
37N30 Dynamical systems in numerical analysis
76N15 Gas dynamics (general theory)
90B10 Deterministic network models in operations research

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[1] N. Banagaaya, S. Grundel and P. Benner, Index-aware MOR for gas transport networks with many supply inputs, in IUTAM Symposium on Model Order Reduction of Coupled Systems (eds. J. Fehr and B. Haasdonk), Springer International Publishing, Cham, 2020,191-207. · Zbl 1428.93025
[2] J. Bang-Jensen and G. Z. Gutin, Digraphs: Theory, Algorithms and Applications, Springer-Verlag, London, 2008. · Zbl 1001.05002
[3] P. Benner, S. Grundel, C. Himpe, C. Huck, T. Streubel and C. Tischendorf, Gas network benchmark models, in Differential-Algebraic Equations Forum, Springer, Berlin, Heidelberg, 2018. · Zbl 1445.76065
[4] M. Benzi; G. H. Golub; J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14, 1-137 (2005) · Zbl 1115.65034
[5] A. Bermúdez; X. López; M. E. Vázquez-Cendón, Finite volume methods for multi-component Euler equations with source terms, Comput. Fluids, 156, 113-134 (2017) · Zbl 1390.76396
[6] M. Chaczykowski, Sensitivity of pipeline gas flow model to the selection of the equation of state, Chem. Eng. Res. Des., 87, 1596-1603 (2009)
[7] R. Dembo; S. Eisenstat; T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19, 400-408 (1982) · Zbl 0478.65030
[8] H. Egger, A robust conservative mixed finite element method for isentropic compressible flow on pipe networks, SIAM J. Sci. Comput., 40 (2018), A108-A129. · Zbl 1394.76066
[9] H. Egger, T. Kugler and N. Strogies, Parameter identification in a semilinear hyperbolic system, Inverse Probl., 33 (2017), 055022. · Zbl 1515.35342
[10] A. Fügenschuh, et al., Physical and technical fundamentals of gas networks, in Evaluating Gas Network Capacities
[11] A. Fügenschuh, et al., Physical and technical fundamentals of gas networks, in Evaluating Gas Network Capacities · Zbl 1397.90092
[12] T. G. Grandón; H. Heitsch; R. Henrion, A joint model of probabilistic/robust constraints for gas transport management in stationary networks, Comput. Manag. Sci., 14, 443-460 (2017)
[13] S. Grundel, N. Hornung, B. Klaassen, P. Benner and T. Clees, Computing surrogates for gas network simulation using model order reduction, in Surrogate-Based Modeling and Optimization
[14] S. Grundel, N. Hornung and S. Roggendorf, Numerical aspects of model order reduction for gas transportation networks, in Simulation-Driven Modeling and Optimization, Springer Proceedings in Mathematics & Statistics, 153, 2016, 1-28.
[15] S. Grundel and L. Jansen, Efficient simulation of transient gas networks using IMEX integration schemes and MOR methods, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015, 4579-4584. · Zbl 1319.65077
[16] S. Grundel, L. Jansen, N. Hornung, T. Clees, C. Tischendorf and P. Benner, Model order reduction of differential algebraic equations arising from the simulation of gas transport networks, in Progress in Differential-Algebraic Equations, Differential-Algebraic Equations Forum, Springer Berlin Heidelberg, 2014,183-205. · Zbl 1332.93154
[17] M. Gugat; F. M. Hante; M. Hirsch-Dick; G. Leugering, Stationary states in gas networks, Netw. Heterog. Media, 10, 295-320 (2015) · Zbl 1396.76078
[18] F. M. Hante, G. Leugering, A. Martin, L. Schewe and M. Schmidt, Challenges in Optimal Control Problems for Gas and Fluid Flow in Networks of Pipes and Canals: From Modeling to Industrial Applications, Springer Verlag, Singapore, 2017, 77-122. · Zbl 1168.90361
[19] A. Herrán-González; J. M. D. L. Cruz; B. D. Andrés-Toro; J. L. Risco-Martín, Modeling and simulation of a gas distribution pipeline network, Appl. Math. Model., 33, 1584-1600 (2009) · Zbl 1132.76045
[20] M. Herty, Modeling, simulation and optimization of gas networks with compressors, Netw. Heterog. Media, 2, 81-97 (2007) · Zbl 1173.35080
[21] M. Herty, Coupling conditions for networked systems of Euler equations, SIAM J. Sci. Comput., 30, 1596-1612 (2008) · Zbl 1334.76131
[22] M. Herty; J. Mohring; V. Sachers, A new model for gas flow in pipe networks, Math. Methods Appl. Sci., 33, 845-855 (2010)
[23] C. Huck; L. Jansen; C. Tischendorf, A topology based discretization of PDAEs describing water transportation networks, Proc. Appl. Math. Mech., 14, 923-924 (2014) · Zbl 0618.65105
[24] C. Johnson; J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46, 1-26 (1986) · Zbl 1031.65069
[25] C. Kelley, Solving Nonlinear Equations with Newton’s Method, Society for Industrial and Applied Mathematics, Philadelphia, 2003. · Zbl 0538.76066
[26] A. Osiadacz, Simulation of transient gas flows in networks, Internat. J. Numer. Methods Fluids, 4, 13-24 (1984) · Zbl 0615.76082
[27] A. Osiadacz, Simulation and Analysis of Gas Networks, Gulf Publishing, Houston, TX, 1987. · Zbl 0664.76119
[28] A. J. Osiadacz; M. Yedroudj, A comparison of a finite element method and a finite difference method for transient simulation of a gas pipeline, Appl. Math. Model., 13, 79-85 (1989) · Zbl 1312.65104
[29] J. W. Pearson, On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems, Electron. Trans. Numer. Anal., 44, 53-72 (2015) · Zbl 1338.65078
[30] J. Pestana; A. J. Wathen, Natural preconditioning and iterative methods for saddle point systems, SIAM Rev., 57, 71-91 (2015) · Zbl 1325.65066
[31] M. Porcelli, V. Simoncini and M. Tani, Preconditioning of active-set Newton methods for PDE-constrained optimal control problems, SIAM J. Sci. Comput., 37 (2015), S472-S502.
[32] Y. Qiu, Preconditioning Optimal Flow Control Problems Using Multilevel Sequentially Semiseparable Matrix Computations, Ph.D thesis, Delft Institute of Applied Mathematics, Delft University of Technology, 2015.
[33] T. Rees, Preconditioning Iterative Methods for PDE-Constrained Optimization, Ph.D thesis, University of Oxford, 2010.
[34] S. Roggendorf, Model Order Reduction for Linearized Systems Arising from the Simulation of Gas Transportation Networks, Master’s thesis, Rheinischen Friedrich-Wilhelms-Universität Bonn, Germany, 2015. · Zbl 1031.65046
[35] Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, 2003. · Zbl 1190.65053
[36] P. Sonneveld; M. B. van Gijzen, IDR(s): A family of simple and fast algorithms for solving large nonsymmetric systems of linear equations, SIAM J. Sci. Comput., 31, 1035-1062 (2008) · Zbl 1119.93045
[37] M. C. Steinbach, On PDE solution in transient optimization of gas networks, J. Comput. Appl. Math., 203, 345-361 (2007) · Zbl 1330.65153
[38] M. Stoll and T. Breiten, A low-rank in time approach to PDE-constrained optimization, SIAM J. Sci. Comput., 37 (2015), B1-B29.
[39] W. Q. Tao; H. C. Ti, Transient analysis of gas pipeline network, Chem. Eng. J., 69, 47-52 (1998) · Zbl 0943.65100
[40] E. F. Toro; S. J. Billett, Centred TVD schemes for hyperbolic conservation laws, IMA J. Numer. Anal., 20, 47-79 (2000) · Zbl 1365.65089
[41] M. B. van Gijzen and P. Sonneveld, Algorithm 913: An elegant IDR(s) variant that efficiently exploits biorthogonality properties, ACM Trans. Math. Software, 38 (2011), 5: 1-5: 19. · Zbl 1316.65039
[42] A. J. Wathen, Preconditioning, Acta Numer., 24, 329-376 (2015) · Zbl 06822612
[43] M. Wathen, C. Greif and D. Schötzau, Preconditioners for mixed finite element discretizations of incompressible MHD equations, SIAM J. Sci. Comput., 39 (2017), A2993-A3013. · Zbl 0981.76065
[44] J. Zhou; M. A. Adewumi, Simulation of transients in natural gas pipelines using hybrid TVD schemes, Internat. J. Numer. Methods Fluids, 32, 407-437 (2000)
[45] A. Zlotnik, M. Chertkov and S. Backhaus, Optimal control of transient flow in natural gas networks, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015, 4563-4570.
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