• Previous Article
    Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface
  • NHM Home
  • This Issue
  • Next Article
    Fluvial to torrential phase transition in open canals
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.

[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.

[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.

[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.

[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.

[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.

[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. 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.

[10]

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

[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.

[12]

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

[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.

[14]

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

[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.

[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.

[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. 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.

[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.

[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.

[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.

[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.

[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-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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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.

[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.

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. 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.

[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.

[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.

[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.

[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.

[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. 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.

[10]

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

[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.

[12]

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

[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.

[14]

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

[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.

[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.

[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. 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.

[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.

[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.

[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.

[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.

[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-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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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.

[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.

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
[1]

Samir EL Mourchid. On a hypercylicity criterion for strongly continuous semigroups. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 271-275. doi: 10.3934/dcds.2005.13.271

[2]

Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069

[3]

Tomáš Gedeon. Attractors in continuous –time switching networks. Communications on Pure & Applied Analysis, 2003, 2 (2) : 187-209. doi: 10.3934/cpaa.2003.2.187

[4]

Michael Herty, Veronika Sachers. Adjoint calculus for optimization of gas networks. Networks & Heterogeneous Media, 2007, 2 (4) : 733-750. doi: 10.3934/nhm.2007.2.733

[5]

Michael Herty. Modeling, simulation and optimization of gas networks with compressors. Networks & Heterogeneous Media, 2007, 2 (1) : 81-97. doi: 10.3934/nhm.2007.2.81

[6]

Fritz Colonius, Guilherme Mazanti. Decay rates for stabilization of linear continuous-time systems with random switching. Mathematical Control & Related Fields, 2018, 8 (0) : 1-20. doi: 10.3934/mcrf.2019002

[7]

Mapundi K. Banda, Michael Herty, Axel Klar. Gas flow in pipeline networks. Networks & Heterogeneous Media, 2006, 1 (1) : 41-56. doi: 10.3934/nhm.2006.1.41

[8]

Martin Gugat, Falk M. Hante, Markus Hirsch-Dick, Günter Leugering. Stationary states in gas networks. Networks & Heterogeneous Media, 2015, 10 (2) : 295-320. doi: 10.3934/nhm.2015.10.295

[9]

Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963

[10]

Daniel M. N. Maia, Elbert E. N. Macau, Tiago Pereira, Serhiy Yanchuk. Synchronization in networks with strongly delayed couplings. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3461-3482. doi: 10.3934/dcdsb.2018234

[11]

Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447

[12]

Sebastián Donoso. Enveloping semigroups of systems of order d. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2729-2740. doi: 10.3934/dcds.2014.34.2729

[13]

Radu C. Cascaval, Ciro D'Apice, Maria Pia D'Arienzo, Rosanna Manzo. Flow optimization in vascular networks. Mathematical Biosciences & Engineering, 2017, 14 (3) : 607-624. doi: 10.3934/mbe.2017035

[14]

A. Zeblah, Y. Massim, S. Hadjeri, A. Benaissa, H. Hamdaoui. Optimization for series-parallel continuous power systems with buffers under reliability constraints using ant colony. Journal of Industrial & Management Optimization, 2006, 2 (4) : 467-479. doi: 10.3934/jimo.2006.2.467

[15]

Ian D. Morris. Ergodic optimization for generic continuous functions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 383-388. doi: 10.3934/dcds.2010.27.383

[16]

Xingwu Chen, Weinian Zhang. Normal forms of planar switching systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6715-6736. doi: 10.3934/dcds.2016092

[17]

David Cowan. A billiard model for a gas of particles with rotation. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 101-109. doi: 10.3934/dcds.2008.22.101

[18]

Ciprian Preda, Petre Preda, Adriana Petre. On the asymptotic behavior of an exponentially bounded, strongly continuous cocycle over a semiflow. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1637-1645. doi: 10.3934/cpaa.2009.8.1637

[19]

Giuseppe Da Prato. Transition semigroups corresponding to Lipschitz dissipative systems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 177-192. doi: 10.3934/dcds.2004.10.177

[20]

Giuseppe Buttazzo, Filippo Santambrogio. Asymptotical compliance optimization for connected networks. Networks & Heterogeneous Media, 2007, 2 (4) : 761-777. doi: 10.3934/nhm.2007.2.761

2017 Impact Factor: 1.187

Metrics

  • PDF downloads (12)
  • HTML views (28)
  • Cited by (0)

Other articles
by authors

[Back to Top]