DFG Research CenterMATHEON

mathematics forkey technologieswww.matheon.de

Project B20

Optimization of gas transport

Martin Grötschel René Henrion Thorsten Koch Werner RömischTimo Berthold Stefan Heinz Stefan Vigerske

W e ie rstra ß -In stitu t fü r A n g e w a n d te A n a ly s is u n d S to ch a stik

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

Fuel Consumption of a Compressor

c ·

(

(

pv

pu

)

κ−1

κ

− 1

)

|qu,v| ≤ fmax

u,v

⇒ Nonlinear nonconvex constraints

Constraint Programming

Complicated legal concepts thatare difficult to model algebraically,e.g., pipelines shared betweencompanies⇒ Global constraints

Stochastic Programming

Demand of gas underliesuncertainties, e.g., weather⇒ Chance constraints

Gasflow on Exitvs. Date

June February July

Currently implemented

⊲ support for quadratic constraints (released with SCIP 1.2)⊲ support for second-order cone constraints⊲ support for nonlinear pressure loss constraints f |f | = c(p − q)

⊲ Nonlinear Relaxation-Enforced Neighbourhood Search heuristic⊲ interfaces to GAMS and ZIMPL

Performance on 80 MIQQP benchmark instances

instances taken fromMINLPLib and testsets ofJ.N. Hooker, H. Mittelmann,J.P. Vielma, and an IBM/CMUproject⋆LP solver: CPLEX 11.2,

NLP solver: IPOPT 3.8

Future plans

Further transfer of MIP/CP technology to MINLP:

⊲ primal heuristics⊲ branching rules⊲ domain store relaxation and disconnected domains⊲ conflict analysis⊲ lift and project for MIQQPs

Detecting and exploiting problem structure:

⊲ preprocessing by symbolic algebra using REDUCE⊲ upgrading of constraints⊲ detection of convexity and symmetry

Evaluation, derivation, relaxation of chance constraints