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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

* Corresponding author

Received  January 2018 Revised  August 2018 Published  November 2018

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.

Citation: Fabian Rüffler, Volker Mehrmann, Falk M. Hante. Optimal model switching for gas flow in pipe networks. Networks & 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. Google Scholar

[2]

H. AxelssonY. WardiM. 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.  Google Scholar

[3]

M. K. BandaM. 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.  Google Scholar

[4]

M. K. BandaM. Herty and A. Klar, Gas flow in pipeline networks, Networks and Heterogeneous Media, 1 (2006), 41-56.  doi: 10.3934/nhm.2006.1.41.  Google Scholar

[5]

B. BaumruckerJ. 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.  Google Scholar

[6]

F. BayazitB. Dorn and A. Rhandi, Flows in networks with delay in the vertices, Mathematische Nachrichten, 285 (2012), 1603-1615.  doi: 10.1002/mana.201100163.  Google Scholar

[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.  Google Scholar

[8]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, vol. 20, Oxford University Press on Demand, 2000.  Google Scholar

[9]

J. BrouwerI. 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.  Google Scholar

[10]

J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, 2016. doi: 10.1002/9781119121534.  Google Scholar

[11]

C. G. CassandrasD. 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.  Google Scholar

[12]

G. Cerbe, Grundlagen der Gastechnik, Hanser, 2016. doi: 10.3139/9783446449664.  Google Scholar

[13]

M. ChertkovS. 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.  Google Scholar

[14]

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Courier Corporation, 2007. Google Scholar

[15]

M. DickM. 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.  Google Scholar

[16]

P. DomschkeA. DuaJ. J. StolwijkJ. 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.  Google Scholar

[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. Google Scholar

[18]

P. DomschkeO. 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.  Google Scholar

[19]

M. EgerstedtY. 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.  Google Scholar

[20]

K.-J. EngelM. K. FijavžB. KlössR. Nagel and E. Sikolya, Maximal controllability for boundary control problems, Applied Mathematics & Optimization, 62 (2010), 205-227.  doi: 10.1007/s00245-010-9101-1.  Google Scholar

[21]

K.-J. EngelM. K. FijavzR. 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.  Google Scholar

[22]

M. GugatF. M. HanteM. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[25]

A. Herrán-GonzálezJ. De La CruzB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

C. B. Laney, Computational Gasdynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9780511605604.  Google Scholar

[33]

H. W. J. LeeK. L. TeoV. 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.  Google Scholar

[34]

R. J. Le, Veque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. doi: 10.1007/978-3-0348-8629-1.  Google Scholar

[35]

R. J. Le, Veque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[36]

D. MahlkeA. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[39]

E. S. Menon, Gas pipeline Hydraulics, CRC Press, 2005. Google Scholar

[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.  Google Scholar

[41]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, 2014. doi: 10.1007/978-3-319-04621-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

M. E. PfetschA. FügenschuhB. GeißlerN. GeißlerR. GollmerB. HillerJ. HumpolaT. KochT. LehmannA. MartinA. MorsiJ. RövekampL. ScheweM. SchmidtR. SchultzR. SchwarzJ. SchweigerC. StanglM. C. SteinbachS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[48]

E. Sikolya, Semigroups for Flows in Networks, PhD thesis, Eberhard-Karls-Universität Tübingen, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[51]

Y. WardiM. 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.  Google Scholar

[52]

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.  Google Scholar

[53]

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.  Google Scholar

[54]

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.  Google Scholar

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. Google Scholar

[2]

H. AxelssonY. WardiM. 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.  Google Scholar

[3]

M. K. BandaM. 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.  Google Scholar

[4]

M. K. BandaM. Herty and A. Klar, Gas flow in pipeline networks, Networks and Heterogeneous Media, 1 (2006), 41-56.  doi: 10.3934/nhm.2006.1.41.  Google Scholar

[5]

B. BaumruckerJ. 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.  Google Scholar

[6]

F. BayazitB. Dorn and A. Rhandi, Flows in networks with delay in the vertices, Mathematische Nachrichten, 285 (2012), 1603-1615.  doi: 10.1002/mana.201100163.  Google Scholar

[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.  Google Scholar

[8]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, vol. 20, Oxford University Press on Demand, 2000.  Google Scholar

[9]

J. BrouwerI. 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.  Google Scholar

[10]

J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, 2016. doi: 10.1002/9781119121534.  Google Scholar

[11]

C. G. CassandrasD. 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.  Google Scholar

[12]

G. Cerbe, Grundlagen der Gastechnik, Hanser, 2016. doi: 10.3139/9783446449664.  Google Scholar

[13]

M. ChertkovS. 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.  Google Scholar

[14]

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Courier Corporation, 2007. Google Scholar

[15]

M. DickM. 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.  Google Scholar

[16]

P. DomschkeA. DuaJ. J. StolwijkJ. 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.  Google Scholar

[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. Google Scholar

[18]

P. DomschkeO. 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.  Google Scholar

[19]

M. EgerstedtY. 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.  Google Scholar

[20]

K.-J. EngelM. K. FijavžB. KlössR. Nagel and E. Sikolya, Maximal controllability for boundary control problems, Applied Mathematics & Optimization, 62 (2010), 205-227.  doi: 10.1007/s00245-010-9101-1.  Google Scholar

[21]

K.-J. EngelM. K. FijavzR. 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.  Google Scholar

[22]

M. GugatF. M. HanteM. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[25]

A. Herrán-GonzálezJ. De La CruzB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

C. B. Laney, Computational Gasdynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9780511605604.  Google Scholar

[33]

H. W. J. LeeK. L. TeoV. 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.  Google Scholar

[34]

R. J. Le, Veque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. doi: 10.1007/978-3-0348-8629-1.  Google Scholar

[35]

R. J. Le, Veque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[36]

D. MahlkeA. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[39]

E. S. Menon, Gas pipeline Hydraulics, CRC Press, 2005. Google Scholar

[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.  Google Scholar

[41]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, 2014. doi: 10.1007/978-3-319-04621-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

M. E. PfetschA. FügenschuhB. GeißlerN. GeißlerR. GollmerB. HillerJ. HumpolaT. KochT. LehmannA. MartinA. MorsiJ. RövekampL. ScheweM. SchmidtR. SchultzR. SchwarzJ. SchweigerC. StanglM. C. SteinbachS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[48]

E. Sikolya, Semigroups for Flows in Networks, PhD thesis, Eberhard-Karls-Universität Tübingen, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[51]

Y. WardiM. 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.  Google Scholar

[52]

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.  Google Scholar

[53]

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.  Google Scholar

[54]

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Figure 1.  A gas network with a supply node $N_1$ and two costumer nodes $N_2$ and $N_3$
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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