Optimal model switching for gas flow in pipe networks

Fabian Rüffler; Volker Mehrmann; Falk M. Hante; Fabian Rüffler; Volker Mehrmann; Falk M. Hante

doi:10.3934/nhm.2018029

Networks and Heterogeneous Media

2018, Volume 13, Issue 4: 641-661. doi: 10.3934/nhm.2018029

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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

Received: 01 January 2018 Revised: 01 August 2018
Primary: 35L50, 76N25, 93C30; Secondary: 35R02, 49J20

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.
- Gas networks,
- model adaptivity,
- switching systems,
- strongly continuous semigroups,
- optimization
Citation: Fabian Rüffler, Volker Mehrmann, Falk M. Hante. Optimal model switching for gas flow in pipe networks[J]. Networks and Heterogeneous Media, 2018, 13(4): 641-661. doi: 10.3934/nhm.2018029

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Abstract

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.

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]	Gradient descent approach to optimal mode scheduling in hybrid dynamical systems. Journal of Optimization Theory and Applications (2008) 136: 167-186.
[3]	Coupling conditions for gas networks governed by the isothermal Euler equations. Networks and Heterogeneous Media (2006) 1: 295-314.
[4]	Gas flow in pipeline networks. Networks and Heterogeneous Media (2006) 1: 41-56.
[5]	MPEC problem formulations and solution strategies with chemical engineering applications. Computers & Chemical Engineering (2008) 32: 2903-2913.
[6]	Flows in networks with delay in the vertices. Mathematische Nachrichten (2012) 285: 1603-1615.
[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]	Gas pipeline models revisited: model hierarchies, nonisothermal models, and simulations of networks. SIAM Journal on Multiscale Modeling and Simulation (2011) 9: 601-623.
[10]	J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, 2016. doi: 10.1002/9781119121534
[11]	Optimal control of a class of hybrid systems. IEEE Transactions on Automatic Control (2001) 46: 398-415.
[12]	G. Cerbe, Grundlagen der Gastechnik, Hanser, 2016. doi: 10.3139/9783446449664
[13]	Cascading of fluctuations in interdependent energy infrastructures: Gas-grid coupling. Applied Energy (2015) 160: 541-551.
[14]	P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Courier Corporation, 2007.
[15]	Classical solutions and feedback stabilization for the gas flow in a sequence of pipes. Networks and Heterogeneous Media (2010) 5: 691-709.
[16]	Adaptive refinement strategies for the simulation of gas flow in networks using a model hierarchy. Electronic Transactions Numerical Analysis (2018) 48: 97-113.
[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]	Adjoint-based error control for the simulation and optimization of gas and water supply networks. Journal of Applied Mathematics and Computing (2015) 259: 1003-1018.
[19]	Transition-time optimization for switched-mode dynamical systems. IEEE Transactions on Automatic Control (2006) 51: 110-115.
[20]	Maximal controllability for boundary control problems. Applied Mathematics & Optimization (2010) 62: 205-227.
[21]	Vertex control of flows in networks. Networks and Heterogeneous Media (2008) 3: 709-722.
[22]	Stationary states in gas networks. Networks and Heterogeneous Media (2015) 10: 295-320.
[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]	Modeling and simulation of a gas distribution pipeline network. Applied Mathematical Modelling (2009) 33: 1584-1600.
[26]	Adjoint calculus for optimization of gas networks. Networks and Heterogeneous Media (2007) 2: 733-750.
[27]	Optimal switching between autonomous subsystems. Journal of the Franklin Institute (2014) 351: 2675-2690.
[28]	Second-order switching time optimization for nonlinear time-varying dynamic systems. IEEE Transactions on Automatic Control (2011) 56: 1953-1957.
[29]	Transient analysis of isothermal gas flow in pipeline networks. Chemical Engineering Journal (2000) 76: 169-177.
[30]	Spectral properties and asymptotic periodicity of flows in networks. Mathematische Zeitschrift (2005) 249: 139-162.
[31]	Mild solution and constrained local controllability of semilinear boundary control systems. Journal of Dynamical and Control Systems (2017) 23: 735-751.
[32]	C. B. Laney, Computational Gasdynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9780511605604
[33]	Control parametrization enhancing technique for optimal discrete-valued control problems. Automatica (1999) 35: 1401-1407.
[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]	A mixed integer approach for time-dependent gas network optimization. Optimization Methods and Software (2010) 25: 625-644.
[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]	Hybrid systems of differential-algebraic equations - Analysis and numerical solution. Journal of Process Control (2009) 19: 1218-1228.
[39]	E. S. Menon, Gas pipeline Hydraulics, CRC Press, 2005.
[40]	Pipe networks: Coupling constants in a junction for the isentropic Euler equations. Energy Procedia (2015) 64: 140-149.
[41]	D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, 2014. doi: 10.1007/978-3-319-04621-1
[42]	Simulation of transient gas flows in networks. International Journal for Numerical Methods in Fluids (1984) 4: 13-24.
[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]	Validation of nominations in gas network optimization: Models, methods, and solutions. Optimization Methods and Software (2015) 30: 15-53.
[45]	Optimal switching for hybrid semilinear evolutions. Nonlinear Analysis and Hybrid Systems (2016) 22: 215-227.
[46]	Optimality Conditions for Switching Operator Differential Equations. Proceedings in Applied Mathematics and Mechanics (2017) 17: 777-778.
[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]	Flows in networks with dynamic ramification nodes. Journal of Evolution Equations (2005) 5: 441-463.
[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]	Switched-mode systems: Gradient-descent algorithms with Armijo step sizes. Discrete Event Dynamic Systems: Theory and Applications (2015) 25: 571-599.
[52]	Optimal control of switched autonomous systems. Proceedings of the 41st IEEE Conference on Decision and Control (2002) 4: 4401-4406.
[53]	Optimal control of switched systems based on parameterization of the switching instants. IEEE Transactions on Automatic Control (2004) 49: 2-16.
[54]	Optimal control of hybrid switched systems: A brief survey. Discrete Event Dynamic Systems: Theory and Applications (2015) 25: 345-364.

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