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Network Models for Supply Chains and Gas Pipelines

Network Models for Supply Chains and Gas Pipelines. M. Herty and A. Klar in cooperation with S. Göttlich and M. Banda. Hier Partnerlogo einfügen An linker oberer Ecke dieses Rechtecks ausrichten, Rechteck anschließend löschen.

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Network Models for Supply Chains and Gas Pipelines

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  1. Network Models for Supply Chains and Gas Pipelines M. Herty and A. Klar in cooperation with S. Göttlich and M. Banda Hier Partnerlogo einfügen An linker oberer Ecke dieses Rechtecks ausrichten, Rechteck anschließend löschen. Wenn kein Partnerlogo verwendet wird, Rechteck und Gliederungslinie löschen.

  2. Contents • Introduction • Network models for supply chains • Network models for gas networks • Numerical results and optimization • Outlook

  3. Introduction Supply Chain: Gas Pipeline Networks:

  4. Networks • Tasks: • Determine dynamics on the arcs • Define „correct“ coupling conditions

  5. Supply Chain Modelling • See Armbruster, Degond, Ringhofer et al. Basic equations: : density of parts : maximum processing capacity L/T: processing velocity

  6. Model • Idea: • Each processor is described by one arc • Use above equations to describe dynamics of the processor. • Add equation for the queues in front of the processor • Advantage: • Standard treatment of equations (constant maximal processing rate) • Straightforward definitions for complicated networks, junctions Start with simple structure:consecutive processors

  7. Consecutive Processors

  8. Supply Chain Network Modelling

  9. Theoretical InvestigationsDefinition:

  10. Theorem: Proof: Explicit solutions of Riemann problems, Front Tracking, Bounds for the number of interactions of discontinuities, see Holden, Piccoli et al. Remark 1: Possible increase of total variation due to influence of queues Remark 2: Not a weak solution across the junction in the usual network sense (queues)

  11. Comparison with ADR: N-curve from ADR is obtained from

  12. Junctions Dispersing Junction:

  13. Junctions Merging Junction:

  14. Numerical Results (Example 1, see ADR) Inflow: Density: Queues 1,2,3:

  15. Example 2 (Optimization of distribution rates):

  16. Example 3 (Optimization of processing velocities):

  17. Results (queues):

  18. Example 4, Braun, Frankfurt

  19. Results (Optimization of processing velocity of processor 5):

  20. Comparison of CPU times:

  21. Modelling of Gas Networks Isothermal Euler equations with friction or without friction

  22. Conditions

  23. Gas Networks Simplifying assumptions: Discuss Riemann problems at the vertices

  24. Consecutive pipelines Theorem:

  25. Remark (Demand and Supply functions): 1-waves and 2-waves for given left state Demand function Supply function

  26. Remark (Construction of the solution ):

  27. General networks Remark: Similar to the above, solutions can be constructed, see example 1. However: Corresponding maximization problem can have no solution.

  28. Discussion • Remark: The solution is not a weak solution in the usual network sense. The second moment is not conserved • Remark: In contrast to traffic networks the distribution of flow for a dispersing junction can not be chosen, but is implicitly given by the equality of pressure. • Remark: For real world applications the pressure at the vertex is reduced by so called minor losses. This is modelled by a pressure drop factor depending on geometry, flow and density at the intersection.

  29. Example 1: Coupling conditions: Remark: Existence, uniqueness?

  30. Example 1 (Construction of a solution fulfilling the constraints):

  31. Numerical Results (Example 1, with friction):

  32. Numerical Results (Example 2) Pressure increase on the two vertical pipes 2 and 4

  33. Numerical Results (Example 2):

  34. Outlook • Simplified problems: • ODE on networks • Mixed Integer Problems (MIP) derived from PDE, see traffic networks • Optimization problems: • Supply Chains: Improve optimization procedures (Adjoint calculus etc.) • Gas networks:pressure distribution corrected by compressors discrete optimization,

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