Network Models for Supply Chains and Gas Pipelines
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M. Herty and A. Klar
in cooperation with
S. Göttlich and M. Banda
Contents
Introduction Network models for supply chains Network models for gas networks Numerical results and optimization Outlook
Introduction
Supply Chain:
Gas Pipeline
Networks:
Networks
Tasks:
Determine dynamics on the arcs
Define „correct“ coupling conditions
Supply Chain Modelling
See Armbruster, Degond, Ringhofer et al.
Basic equations:
: density of parts
: maximum processing capacity
L/T: processing velocity
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
Consecutive Processors
Supply Chain Network Modelling
Theoretical Investigations
Definition:
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)
Comparison with ADR: N-curve from ADR is obtained from
Junctions
Dispersing Junction:
Junctions
Merging Junction:
Numerical Results (Example 1, see ADR)
Density: Queues 1,2,3:
Inflow:
Example 2 (Optimization of distribution rates):
Example 3 (Optimization of processing velocities):
Results (queues):
Example 4, Braun, Frankfurt
Results (Optimization of processing velocity of processor 5):
Comparison of CPU times:
Modelling of Gas Networks
Isothermal Euler equations with friction
or without friction
Conditions
Gas Networks
Simplifying assumptions:
Discuss Riemann problems at the vertices
Consecutive pipelines
Theorem:
Remark (Demand and Supply functions):
1-waves and 2-waves for given left state
Demand function
Supply function
Remark (Construction of the solution ):
General networks
Remark:
Similar to the above, solutions can be constructed, see example 1.
However: Corresponding maximization problem can have no solution.
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.
Example 1:
Coupling conditions:
Remark: Existence, uniqueness?
Example 1 (Construction of a solution fulfilling the constraints):
Numerical Results (Example 1, with friction):
Numerical Results (Example 2)
Pressure increase on the two vertical pipes 2 and 4
Numerical Results (Example 2):
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,