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December
2018, 13(4): 641-661.
doi: 10.3934/nhm.2018029
Optimal model switching for gas flow in pipe networks
| 1. | Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Lehrstuhl Angewandte Mathematik Ⅱ, Cauerstr. 11, 91058 Erlangen, Germany |
| 2. | Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany |
We consider model adaptivity for gas flow in pipeline networks. For each instant in time and for each pipe in the network a model for the gas flow is to be selected from a hierarchy of models in order to maximize a performance index that balances model accuracy and computational cost for a simulation of the entire network. This combinatorial problem involving partial differential equations is posed as an optimal switching control problem for abstract semilinear evolutions. We provide a theoretical and numerical framework for solving this problem using a two stage gradient descent approach based on switching time and mode insertion gradients. A numerical study demonstrates the practicability of the approach.
Keywords:
Gas networks,
model adaptivity,
switching systems,
strongly continuous semigroups,
optimization.
Mathematics Subject Classification:
Primary: 35L50, 76N25, 93C30; Secondary: 35R02, 49J20.
Citation:
Fabian Rüffler, Volker Mehrmann, Falk M. Hante. Optimal model switching for gas flow in pipe networks. Networks and Heterogeneous Media,
2018, 13
(4)
: 641-661.
doi: 10.3934/nhm.2018029
![]()
References:
| [1] |
M. A. Adewumi and J. Zhou,
Simulation of Transient Flow in Natural Gas Pipelines, 27th Annual Meeting of PSIG (Pipeline Simulation Interest Group), Albuquerque, NM, 1995, URL https://www.onepetro.org/conference-paper/PSIG-9508. |
| [2] |
H. Axelsson, Y. Wardi, M. Egerstedt and E. I. Verriest,
Gradient descent approach to optimal mode scheduling in hybrid dynamical systems, Journal of Optimization Theory and Applications, 136 (2008), 167-186.
doi: 10.1007/s10957-007-9305-y. |
| [3] |
M. K. Banda, M. Herty and A. Klar,
Coupling conditions for gas networks governed by the isothermal Euler equations, Networks and Heterogeneous Media, 1 (2006), 295-314.
doi: 10.3934/nhm.2006.1.295. |
| [4] |
M. K. Banda, M. Herty and A. Klar,
Gas flow in pipeline networks, Networks and Heterogeneous Media, 1 (2006), 41-56.
doi: 10.3934/nhm.2006.1.41. |
| [5] |
B. Baumrucker, J. Renfro and L. T. Biegler,
MPEC problem formulations and solution strategies with chemical engineering applications, Computers & Chemical Engineering, 32 (2008), 2903-2913.
doi: 10.1016/j.compchemeng.2008.02.010. |
| [6] |
F. Bayazit, B. Dorn and A. Rhandi,
Flows in networks with delay in the vertices, Mathematische Nachrichten, 285 (2012), 1603-1615.
doi: 10.1002/mana.201100163. |
| [7] |
L. T. Biegler,
Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes, vol. 10 of MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2010.
doi: 10.1137/1.9780898719383. |
| [8] |
A. Bressan,
Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, vol. 20, Oxford University Press on Demand, 2000. |
| [9] |
J. Brouwer, I. Gasser and M. Herty,
Gas pipeline models revisited: model hierarchies, nonisothermal models, and simulations of networks, SIAM Journal on Multiscale Modeling and Simulation, 9 (2011), 601-623.
doi: 10.1137/100813580. |
| [10] |
J. C. Butcher,
Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, 2016.
doi: 10.1002/9781119121534. |
| [11] |
C. G. Cassandras, D. L. Pepyne and Y. Wardi,
Optimal control of a class of hybrid systems, IEEE Transactions on Automatic Control, 46 (2001), 398-415.
doi: 10.1109/9.911417. |
| [12] |
G. Cerbe,
Grundlagen der Gastechnik, Hanser, 2016.
doi: 10.3139/9783446449664. |
| [13] |
M. Chertkov, S. Backhaus and V. Lebedev,
Cascading of fluctuations in interdependent energy infrastructures: Gas-grid coupling, Applied Energy, 160 (2015), 541-551.
doi: 10.1016/j.apenergy.2015.09.085. |
| [14] |
P. J. Davis and P. Rabinowitz,
Methods of Numerical Integration, Courier Corporation, 2007. |
| [15] |
M. Dick, M. Gugat and G. Leugering,
Classical solutions and feedback stabilization for the gas flow in a sequence of pipes, Networks and Heterogeneous Media, 5 (2010), 691-709.
doi: 10.3934/nhm.2010.5.691. |
| [16] |
P. Domschke, A. Dua, J. J. Stolwijk, J. Lang and V. Mehrmann,
Adaptive refinement strategies for the simulation of gas flow in networks using a model hierarchy, Electronic Transactions Numerical Analysis, 48 (2018), 97-113.
doi: 10.1553/etna_vol48s97. |
| [17] |
P. Domschke, B. Hiller, J. Lang and C. Tischendorf,
Modellierung von Gasnetzwerken: Eine Übersicht, Technical report, Technische Universität Darmstadt, 2017, URL https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/191. |
| [18] |
P. Domschke, O. Kolb and J. Lang,
Adjoint-based error control for the simulation and optimization of gas and water supply networks, Journal of Applied Mathematics and Computing, 259 (2015), 1003-1018.
doi: 10.1016/j.amc.2015.03.029. |
| [19] |
M. Egerstedt, Y. Wardi and H. Axelsson,
Transition-time optimization for switched-mode dynamical systems, IEEE Transactions on Automatic Control, 51 (2006), 110-115.
doi: 10.1109/TAC.2005.861711. |
| [20] |
K.-J. Engel, M. K. Fijavž, B. Klöss, R. Nagel and E. Sikolya,
Maximal controllability for boundary control problems, Applied Mathematics & Optimization, 62 (2010), 205-227.
doi: 10.1007/s00245-010-9101-1. |
| [21] |
K.-J. Engel, M. K. Fijavz, R. Nagel and E. Sikolya,
Vertex control of flows in networks, Networks and Heterogeneous Media, 3 (2008), 709-722.
doi: 10.3934/nhm.2008.3.709. |
| [22] |
M. Gugat, F. M. Hante, M. Hirsch-Dick and G. Leugering,
Stationary states in gas networks, Networks and Heterogeneous Media, 10 (2015), 295-320.
doi: 10.3934/nhm.2015.10.295. |
| [23] |
M. Hahn, S. Leyffer and V. M. Zavala, Mixed-Integer PDE-Constrained Optimal Control of Gas Networks,
Mathematics and Computer Science, URL https://www.mcs.anl.gov/papers/P7095-0817.pdf. |
| [24] |
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, in Industrial Mathematics and Complex Systems: Emerging Mathematical Models, Methods and Algorithms (eds. P. Manchanda, R. Lozi and A. H. Siddiqi), Springer Singapore, Singapore, 2017, 77-122.
doi: 10.1007/978-981-10-3758-0_5. |
| [25] |
A. Herrán-González, J. De La Cruz, B. De Andrés-Toro and J. Risco-Martín,
Modeling and simulation of a gas distribution pipeline network, Applied Mathematical Modelling, 33 (2009), 1584-1600.
doi: 10.1016/j.apm.2008.02.012. |
| [26] |
M. Herty and V. Sachers,
Adjoint calculus for optimization of gas networks, Networks and Heterogeneous Media, 2 (2007), 733-750.
doi: 10.3934/nhm.2007.2.733. |
| [27] |
A. Heydari and S. Balakrishnan,
Optimal switching between autonomous subsystems, Journal of the Franklin Institute, 351 (2014), 2675-2690.
doi: 10.1016/j.jfranklin.2013.12.008. |
| [28] |
E. R. Johnson and T. D. Murphey,
Second-order switching time optimization for nonlinear time-varying dynamic systems, IEEE Transactions on Automatic Control, 56 (2011), 1953-1957.
doi: 10.1109/TAC.2011.2150310. |
| [29] |
S. L. Ke and H. C. Ti,
Transient analysis of isothermal gas flow in pipeline networks, Chemical Engineering Journal, 76 (2000), 169-177.
doi: 10.1016/S1385-8947(99)00122-9. |
| [30] |
M. Kramar and E. Sikolya,
Spectral properties and asymptotic periodicity of flows in networks, Mathematische Zeitschrift, 249 (2005), 139-162.
doi: 10.1007/s00209-004-0695-3. |
| [31] |
S. Kumar and N. Tomar,
Mild solution and constrained local controllability of semilinear boundary control systems, Journal of Dynamical and Control Systems, 23 (2017), 735-751.
doi: 10.1007/s10883-016-9355-2. |
| [32] |
C. B. Laney,
Computational Gasdynamics, Cambridge University Press, 1998.
doi: 10.1017/CBO9780511605604. |
| [33] |
H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings,
Control parametrization enhancing technique for optimal discrete-valued control problems, Automatica, 35 (1999), 1401-1407.
doi: 10.1016/S0005-1098(99)00050-3. |
| [34] |
R. J. Le, Veque,
Numerical Methods for Conservation Laws, Birkhäuser, 1992.
doi: 10.1007/978-3-0348-8629-1. |
| [35] |
R. J. Le, Veque,
Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
doi: 10.1017/CBO9780511791253. |
| [36] |
D. Mahlke, A. Martin and S. Moritz,
A mixed integer approach for time-dependent gas network optimization, Optimization Methods and Software, 25 (2010), 625-644.
doi: 10.1080/10556780903270886. |
| [37] |
V. Mehrmann, M. Schmidt and J. Stolwijk, Model and Discretization Error Adaptivity within Stationary Gas Transport Optimization, to appear, Vietnam Journal of Mathematics, URL https://arXiv.org/abs/1712.02745, Preprint 11-2017, Institute of Mathematics, TU Berlin, 2017. |
| [38] |
V. Mehrmann and L. Wunderlich,
Hybrid systems of differential-algebraic equations - Analysis and numerical solution, Journal of Process Control, 19 (2009), 1218-1228.
doi: 10.1016/j.jprocont.2009.05.002. |
| [39] |
E. S. Menon,
Gas pipeline Hydraulics, CRC Press, 2005. |
| [40] |
A. Morin and G. A. Reigstad,
Pipe networks: Coupling constants in a junction for the isentropic Euler equations, Energy Procedia, 64 (2015), 140-149.
doi: 10.1016/j.egypro.2015.01.017. |
| [41] |
D. Mugnolo,
Semigroup Methods for Evolution Equations on Networks, Springer, 2014.
doi: 10.1007/978-3-319-04621-1. |
| [42] |
A. Osiadacz,
Simulation of transient gas flows in networks, International Journal for Numerical Methods in Fluids, 4 (1984), 13-24.
doi: 10.1002/fld.1650040103. |
| [43] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [44] |
M. E. Pfetsch, A. Fügenschuh, B. Geißler, N. Geißler, R. Gollmer, B. Hiller, J. Humpola, T. Koch, T. Lehmann, A. Martin, A. Morsi, J. Rövekamp, L. Schewe, M. Schmidt, R. Schultz, R. Schwarz, J. Schweiger, C. Stangl, M. C. Steinbach, S. Vigerske and B. M. Willert,
Validation of nominations in gas network optimization: Models, methods, and solutions, Optimization Methods and Software, 30 (2015), 15-53.
doi: 10.1080/10556788.2014.888426. |
| [45] |
F. Rüffler and F. M. Hante,
Optimal switching for hybrid semilinear evolutions, Nonlinear Analysis and Hybrid Systems, 22 (2016), 215-227.
doi: 10.1016/j.nahs.2016.05.001. |
| [46] |
F. Rüffler and F. M. Hante,
Optimality Conditions for Switching Operator Differential Equations, Proceedings in Applied Mathematics and Mechanics, 17 (2017), 777-778.
doi: 10.1002/pamm.201710356. |
| [47] |
S. Sager, Reformulations and Algorithms for the Optimization of Switching Decisions in Nonlinear Optimal Control, Journal of Process Control, 19 (2009), 1238-1247, URL https://mathopt.de/PUBLICATIONS/Sager2009b.pdf. |
| [48] |
E. Sikolya,
Semigroups for Flows in Networks, PhD thesis, Eberhard-Karls-Universität Tübingen, 2004. |
| [49] |
E. Sikolya,
Flows in networks with dynamic ramification nodes, Journal of Evolution Equations, 5 (2005), 441-463.
doi: 10.1007/s00028-005-0221-z. |
| [50] |
J. Smoller,
Shock Waves and Reaction-Diffusion Equations, vol. 258 of Grundlehren der mathematischen Wissenschaften, Springer, 1983.
doi: 10.1007/978-1-4612-0873-0. |
| [51] |
Y. Wardi, M. Egerstedt and M. Hale,
Switched-mode systems: Gradient-descent algorithms with Armijo step sizes, Discrete Event Dynamic Systems: Theory and Applications, 25 (2015), 571-599.
doi: 10.1007/s10626-014-0198-2. |
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X. Xu and P. J. Antsaklis,
Optimal control of switched autonomous systems, Proceedings of the 41st IEEE Conference on Decision and Control, 4 (2002), 4401-4406.
doi: 10.1109/CDC.2002.1185065. |
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X. Xu and P. J. Antsaklis,
Optimal control of switched systems based on parameterization of the switching instants, IEEE Transactions on Automatic Control, 49 (2004), 2-16.
doi: 10.1109/TAC.2003.821417. |
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F. Zhu and P. J. Antsaklis,
Optimal control of hybrid switched systems: A brief survey, Discrete Event Dynamic Systems: Theory and Applications, 25 (2015), 345-364.
doi: 10.1007/s10626-014-0187-5. |
show all references
References:
| [1] |
M. A. Adewumi and J. Zhou,
Simulation of Transient Flow in Natural Gas Pipelines, 27th Annual Meeting of PSIG (Pipeline Simulation Interest Group), Albuquerque, NM, 1995, URL https://www.onepetro.org/conference-paper/PSIG-9508. |
| [2] |
H. Axelsson, Y. Wardi, M. Egerstedt and E. I. Verriest,
Gradient descent approach to optimal mode scheduling in hybrid dynamical systems, Journal of Optimization Theory and Applications, 136 (2008), 167-186.
doi: 10.1007/s10957-007-9305-y. |
| [3] |
M. K. Banda, M. Herty and A. Klar,
Coupling conditions for gas networks governed by the isothermal Euler equations, Networks and Heterogeneous Media, 1 (2006), 295-314.
doi: 10.3934/nhm.2006.1.295. |
| [4] |
M. K. Banda, M. Herty and A. Klar,
Gas flow in pipeline networks, Networks and Heterogeneous Media, 1 (2006), 41-56.
doi: 10.3934/nhm.2006.1.41. |
| [5] |
B. Baumrucker, J. Renfro and L. T. Biegler,
MPEC problem formulations and solution strategies with chemical engineering applications, Computers & Chemical Engineering, 32 (2008), 2903-2913.
doi: 10.1016/j.compchemeng.2008.02.010. |
| [6] |
F. Bayazit, B. Dorn and A. Rhandi,
Flows in networks with delay in the vertices, Mathematische Nachrichten, 285 (2012), 1603-1615.
doi: 10.1002/mana.201100163. |
| [7] |
L. T. Biegler,
Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes, vol. 10 of MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2010.
doi: 10.1137/1.9780898719383. |
| [8] |
A. Bressan,
Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, vol. 20, Oxford University Press on Demand, 2000. |
| [9] |
J. Brouwer, I. Gasser and M. Herty,
Gas pipeline models revisited: model hierarchies, nonisothermal models, and simulations of networks, SIAM Journal on Multiscale Modeling and Simulation, 9 (2011), 601-623.
doi: 10.1137/100813580. |
| [10] |
J. C. Butcher,
Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, 2016.
doi: 10.1002/9781119121534. |
| [11] |
C. G. Cassandras, D. L. Pepyne and Y. Wardi,
Optimal control of a class of hybrid systems, IEEE Transactions on Automatic Control, 46 (2001), 398-415.
doi: 10.1109/9.911417. |
| [12] |
G. Cerbe,
Grundlagen der Gastechnik, Hanser, 2016.
doi: 10.3139/9783446449664. |
| [13] |
M. Chertkov, S. Backhaus and V. Lebedev,
Cascading of fluctuations in interdependent energy infrastructures: Gas-grid coupling, Applied Energy, 160 (2015), 541-551.
doi: 10.1016/j.apenergy.2015.09.085. |
| [14] |
P. J. Davis and P. Rabinowitz,
Methods of Numerical Integration, Courier Corporation, 2007. |
| [15] |
M. Dick, M. Gugat and G. Leugering,
Classical solutions and feedback stabilization for the gas flow in a sequence of pipes, Networks and Heterogeneous Media, 5 (2010), 691-709.
doi: 10.3934/nhm.2010.5.691. |
| [16] |
P. Domschke, A. Dua, J. J. Stolwijk, J. Lang and V. Mehrmann,
Adaptive refinement strategies for the simulation of gas flow in networks using a model hierarchy, Electronic Transactions Numerical Analysis, 48 (2018), 97-113.
doi: 10.1553/etna_vol48s97. |
| [17] |
P. Domschke, B. Hiller, J. Lang and C. Tischendorf,
Modellierung von Gasnetzwerken: Eine Übersicht, Technical report, Technische Universität Darmstadt, 2017, URL https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/191. |
| [18] |
P. Domschke, O. Kolb and J. Lang,
Adjoint-based error control for the simulation and optimization of gas and water supply networks, Journal of Applied Mathematics and Computing, 259 (2015), 1003-1018.
doi: 10.1016/j.amc.2015.03.029. |
| [19] |
M. Egerstedt, Y. Wardi and H. Axelsson,
Transition-time optimization for switched-mode dynamical systems, IEEE Transactions on Automatic Control, 51 (2006), 110-115.
doi: 10.1109/TAC.2005.861711. |
| [20] |
K.-J. Engel, M. K. Fijavž, B. Klöss, R. Nagel and E. Sikolya,
Maximal controllability for boundary control problems, Applied Mathematics & Optimization, 62 (2010), 205-227.
doi: 10.1007/s00245-010-9101-1. |
| [21] |
K.-J. Engel, M. K. Fijavz, R. Nagel and E. Sikolya,
Vertex control of flows in networks, Networks and Heterogeneous Media, 3 (2008), 709-722.
doi: 10.3934/nhm.2008.3.709. |
| [22] |
M. Gugat, F. M. Hante, M. Hirsch-Dick and G. Leugering,
Stationary states in gas networks, Networks and Heterogeneous Media, 10 (2015), 295-320.
doi: 10.3934/nhm.2015.10.295. |
| [23] |
M. Hahn, S. Leyffer and V. M. Zavala, Mixed-Integer PDE-Constrained Optimal Control of Gas Networks,
Mathematics and Computer Science, URL https://www.mcs.anl.gov/papers/P7095-0817.pdf. |
| [24] |
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, in Industrial Mathematics and Complex Systems: Emerging Mathematical Models, Methods and Algorithms (eds. P. Manchanda, R. Lozi and A. H. Siddiqi), Springer Singapore, Singapore, 2017, 77-122.
doi: 10.1007/978-981-10-3758-0_5. |
| [25] |
A. Herrán-González, J. De La Cruz, B. De Andrés-Toro and J. Risco-Martín,
Modeling and simulation of a gas distribution pipeline network, Applied Mathematical Modelling, 33 (2009), 1584-1600.
doi: 10.1016/j.apm.2008.02.012. |
| [26] |
M. Herty and V. Sachers,
Adjoint calculus for optimization of gas networks, Networks and Heterogeneous Media, 2 (2007), 733-750.
doi: 10.3934/nhm.2007.2.733. |
| [27] |
A. Heydari and S. Balakrishnan,
Optimal switching between autonomous subsystems, Journal of the Franklin Institute, 351 (2014), 2675-2690.
doi: 10.1016/j.jfranklin.2013.12.008. |
| [28] |
E. R. Johnson and T. D. Murphey,
Second-order switching time optimization for nonlinear time-varying dynamic systems, IEEE Transactions on Automatic Control, 56 (2011), 1953-1957.
doi: 10.1109/TAC.2011.2150310. |
| [29] |
S. L. Ke and H. C. Ti,
Transient analysis of isothermal gas flow in pipeline networks, Chemical Engineering Journal, 76 (2000), 169-177.
doi: 10.1016/S1385-8947(99)00122-9. |
| [30] |
M. Kramar and E. Sikolya,
Spectral properties and asymptotic periodicity of flows in networks, Mathematische Zeitschrift, 249 (2005), 139-162.
doi: 10.1007/s00209-004-0695-3. |
| [31] |
S. Kumar and N. Tomar,
Mild solution and constrained local controllability of semilinear boundary control systems, Journal of Dynamical and Control Systems, 23 (2017), 735-751.
doi: 10.1007/s10883-016-9355-2. |
| [32] |
C. B. Laney,
Computational Gasdynamics, Cambridge University Press, 1998.
doi: 10.1017/CBO9780511605604. |
| [33] |
H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings,
Control parametrization enhancing technique for optimal discrete-valued control problems, Automatica, 35 (1999), 1401-1407.
doi: 10.1016/S0005-1098(99)00050-3. |
| [34] |
R. J. Le, Veque,
Numerical Methods for Conservation Laws, Birkhäuser, 1992.
doi: 10.1007/978-3-0348-8629-1. |
| [35] |
R. J. Le, Veque,
Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
doi: 10.1017/CBO9780511791253. |
| [36] |
D. Mahlke, A. Martin and S. Moritz,
A mixed integer approach for time-dependent gas network optimization, Optimization Methods and Software, 25 (2010), 625-644.
doi: 10.1080/10556780903270886. |
| [37] |
V. Mehrmann, M. Schmidt and J. Stolwijk, Model and Discretization Error Adaptivity within Stationary Gas Transport Optimization, to appear, Vietnam Journal of Mathematics, URL https://arXiv.org/abs/1712.02745, Preprint 11-2017, Institute of Mathematics, TU Berlin, 2017. |
| [38] |
V. Mehrmann and L. Wunderlich,
Hybrid systems of differential-algebraic equations - Analysis and numerical solution, Journal of Process Control, 19 (2009), 1218-1228.
doi: 10.1016/j.jprocont.2009.05.002. |
| [39] |
E. S. Menon,
Gas pipeline Hydraulics, CRC Press, 2005. |
| [40] |
A. Morin and G. A. Reigstad,
Pipe networks: Coupling constants in a junction for the isentropic Euler equations, Energy Procedia, 64 (2015), 140-149.
doi: 10.1016/j.egypro.2015.01.017. |
| [41] |
D. Mugnolo,
Semigroup Methods for Evolution Equations on Networks, Springer, 2014.
doi: 10.1007/978-3-319-04621-1. |
| [42] |
A. Osiadacz,
Simulation of transient gas flows in networks, International Journal for Numerical Methods in Fluids, 4 (1984), 13-24.
doi: 10.1002/fld.1650040103. |
| [43] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [44] |
M. E. Pfetsch, A. Fügenschuh, B. Geißler, N. Geißler, R. Gollmer, B. Hiller, J. Humpola, T. Koch, T. Lehmann, A. Martin, A. Morsi, J. Rövekamp, L. Schewe, M. Schmidt, R. Schultz, R. Schwarz, J. Schweiger, C. Stangl, M. C. Steinbach, S. Vigerske and B. M. Willert,
Validation of nominations in gas network optimization: Models, methods, and solutions, Optimization Methods and Software, 30 (2015), 15-53.
doi: 10.1080/10556788.2014.888426. |
| [45] |
F. Rüffler and F. M. Hante,
Optimal switching for hybrid semilinear evolutions, Nonlinear Analysis and Hybrid Systems, 22 (2016), 215-227.
doi: 10.1016/j.nahs.2016.05.001. |
| [46] |
F. Rüffler and F. M. Hante,
Optimality Conditions for Switching Operator Differential Equations, Proceedings in Applied Mathematics and Mechanics, 17 (2017), 777-778.
doi: 10.1002/pamm.201710356. |
| [47] |
S. Sager, Reformulations and Algorithms for the Optimization of Switching Decisions in Nonlinear Optimal Control, Journal of Process Control, 19 (2009), 1238-1247, URL https://mathopt.de/PUBLICATIONS/Sager2009b.pdf. |
| [48] |
E. Sikolya,
Semigroups for Flows in Networks, PhD thesis, Eberhard-Karls-Universität Tübingen, 2004. |
| [49] |
E. Sikolya,
Flows in networks with dynamic ramification nodes, Journal of Evolution Equations, 5 (2005), 441-463.
doi: 10.1007/s00028-005-0221-z. |
| [50] |
J. Smoller,
Shock Waves and Reaction-Diffusion Equations, vol. 258 of Grundlehren der mathematischen Wissenschaften, Springer, 1983.
doi: 10.1007/978-1-4612-0873-0. |
| [51] |
Y. Wardi, M. Egerstedt and M. Hale,
Switched-mode systems: Gradient-descent algorithms with Armijo step sizes, Discrete Event Dynamic Systems: Theory and Applications, 25 (2015), 571-599.
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Figure 2.
Snapshot of the fully simulated solution showing density (solid, blue) and flux (dashed, red, scaled by $0.05$ ). On the outer pipes 1 to 5 we see a lot of fluctuation due to the oscillatory boundary flows. The pipes 6 to 9 of the inner circle, however, remain nearly constant
Figure 3.
(A): resulting optimized switching sequence showing, for each time step from $t_0 = 0\ \text{s}$ to $T = 1800\ \text{s}$ and each edge $e_1, \ldots, e_{10}$ , if the solution is calculated with the fine model (white) or frozen (black). (B), (C): filtered results with two different filters. (D): $L^2$ -error relative to maximum values of the solution $\bar{z}$ corresponding to freezing edges $6$ to $10$ completely
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