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,