DFG Research CenterMATHEON

mathematics forkey technologieswww.matheon.de

Project B20

Optimization of gas transport

Martin Grötschel René Henrion Thorsten Koch Werner RömischThomas Arnold Timo Berthold Stefan Heinz Stefan Vigerske

Domain of Expertise: Energy and utilities

Background

⊲ political regulations (e.g., Gasnetzzugangs-verordnung) led to a strict separation of gastrading and gas transport in Germany

⊲ these newly imposed political requirements in-fluence the technical processes of gas transport

⊲ as a result, the already complex task of plan-ning and operating gas networks becomes evenmore challenging

⊲ however, suitable algorithms or software arecurrently not available to solve today’s gastransport optimization problems

Project Goal

Integration of aspects from Mixed IntegerProgramming, Nonlinear Programming,Constraint Programming, and StochasticProgramming into a general purpose solver.

SCIPHeuristic

actcons

divingcoef

diving

cross

over

dins

feaspump

fixand

infer

fracdiving

guided

diving

intdiving

int

shifting

linesearch

diving

local

branching

mutation

objpscost

diving

octane oneopt

pscost

diving

rens

rins

rootsol

diving

rounding

shifting simple

rounding

veclen

diving

Variable

· · ·

Branch

allfull

strong

full

strong

in

ference

leastinf

mostinf

pscostrandom

relps

cost

Conflict

Constraint

Handler

and

bound

disjunc.

count

sols

indi

cator

integral

knap

sack

linear logicor

or

setppc

sos1

sos2

var

bound

xor

Cutpool

LP

clp

cpx msk

none

qso

spx

xprs

Dialog

default

Display

default

Node

selector

bfs

dfs

estimate

hybrid

estim

restart

dfs

Event

default

Presolver

bound

shift

dualfix

implics

intto

binaryprobing

trivial

Impli

cations

Tree

Reader

ccg

cip

cnf

fix

lp

mpsopb

ppm

rlp

sol

sos

zpl

Pricer

Separator

clique

cmir

flow

cover

gomoryimplied

bounds

intobj

mcf

redcost

strong

cg

zero

half

Relaxer

Propa

gator

pseudo

obj

root

redcost

Mathematical Aspects of Gas Transport

Mixed Integer Programming

Network configuration and design

qu,v = 0 ∨qmin

u,v ≤ qu,v ≤ qmax

u,v

pu = pv

⇒ Combinatorial decisions

Nonlinear Programming

Pressure loss in a pipeline

p2v =

(

p2u − Λ|qu,v|qu,veS − 1

S

)

e−S

⇒ Nonlinear nonconvex constraints

Constraint Programming

Operating map of a compressorstation can be well represented bya union of nested polyhedra

⇒ Disjunction constraints

Stochastic Programming

Demand of gas underliesuncertainties, e.g., weather

⇒ Chance constraints

Gasflow on Exitvs. Date

June February July

Achievements

⊲ ForNE solves topology planning problems on “H-Nord” with SCIP⊲ SCIP supports quadratic and cumulative scheduling constraints⊲ LNS heuristics extended to MINLP and general CIPs⊲ new MINLP heuristic “Undercover”

100

101

102

103

104

0

20

40

60

80

SCIP 2.0.1

SCIP 1.2.0

BARON 9.0.7

LindoGlobal 6.1.1Couenne 0.3

time limit: 1 hour

time factor w.r.t. fastest solver

%in

stan

ces

solv

ed

Performance on 104 MIQQP benchmark instances(from MINLPLib and testsets of J.N. Hooker, H. Mittelmann, J.P. Vielma, IBM/CMU)

Open Grid Europe’s “H-Nord”network (594 nodes, 624 edges)

Future Plans

⊲ solve larger topology planning instances⊲ support general MINLPs in SCIP⊲ transfer of more MIP/CP technology to MINLP/CIP⊲ handling of disjunction/conjunction constraints⊲ support for chance constraints

q10 20 30 40 50

p out

1,2

1,4

1,6

1,8

2,0

20) eS_MITB_VSOperating map of compressor(nested union of convex sets)

Probability density function ofnormal distribution