– Forschungskooperation Netzoptimierung

Optimization of Gas Networks

Martin Grötschel

2012-06-05

The PartnersZuse-Institut BerlinDiskrete Methoden / OptimierungDr. Thorsten Koch (Projektkoordinator)

Friedrich-Alexander Universität Erlangen-NürnbergEconomics · Discrete Optimization · MathematicsProf. Dr. Alexander Martin

Leibniz-Universität HannoverInstitut für Angewandte MathematikProf. Dr. Marc Steinbach

Universität Duisburg-EssenFachbereich MathematikProf. Dr. Rüdiger Schultz

Technische Universität BraunschweigInstitut für Mathematische OptimierungProf. Dr. Marc Pfetsch

Humboldt-Universität zu BerlinInstitut für MathematikProf. Dr. Werner Römisch

Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)Dr. Rene Henrion

Open Grid EuropeNetzplanung und -steuerung / NetzoptimierungDr. Klaus Spreckelsen (Projektleiter)

BMWiProjektträger Jülich, Geschäftsbereich EnergietechnologienChristoph Jessen (Projektbetreuer)

Martin Grötschel Optimization of Gas Networks 2 / 31

The Team

T. Berthold, J. Bödecker, M. Ebbers, A. Emgrunt, A. Fügenschuh, G. Gamrath, B. Geißler, N. Geißler, R. Gollmer, U. Gotzes,

M. Grötschel, C. Hayn, S. Heinz, R. Henrion, B. Hiller, L. Huke, J. Humpola, J. Hülsewig, I. Joormann, W. Knieschweski,

T. Koch, V. Kühl, H. Leövey, F. Malow, D. Mahlke, A. Martin, R. Mirkov, A. Morsi, G. Möhlen, A. Möller, F. Nowosatko,

D. Oucherif, M. Pfetsch, W. Römisch, L. Sax, L. Schewe, M. Schmidt, R. Schultz, R. Schwarz, J. Schweiger, K. Spreckelsen,

C. Stangl, M. Steckhan, M. Steinbach, A. Steinkamp, J. Szabó, H. Temming, I. Wagner-Specht, B. Willert, S. Vigerske, A. Zelmer

Martin Grötschel Optimization of Gas Networks 3 / 31

The Entry/Exit Model

. the entry/exit model was introduced as ameans to liberalize the gas market(directive 2003/55/EC, GasNZV)

. in the entry/exit model network users booktransmission capacity at entries and exitsseparately

. transmission cost may depend on entry/exit,but not on transportation path

. customers later nominate the actual inflowand outflow within their booked capacities⇒ nomination of all customers

. transmission system operator has to ensurethat each nomination within the bookedcapacities can technically be realized

• •

••

•

entries

exits

booked capacities

10 5

520

10

nomination 1

5 5

00

10

nomination 2

8 3

52

4

Martin Grötschel Optimization of Gas Networks 4 / 31

The Entry/Exit Model

. the entry/exit model was introduced as ameans to liberalize the gas market(directive 2003/55/EC, GasNZV)

. in the entry/exit model network users booktransmission capacity at entries and exitsseparately

. transmission cost may depend on entry/exit,but not on transportation path

. customers later nominate the actual inflowand outflow within their booked capacities⇒ nomination of all customers

. transmission system operator has to ensurethat each nomination within the bookedcapacities can technically be realized

• •

••

•

entries

exits

booked capacities

10 5

520

10

nomination 1

5 5

00

10

nomination 2

8 3

52

4

Martin Grötschel Optimization of Gas Networks 4 / 31

The Entry/Exit Model

. the entry/exit model was introduced as ameans to liberalize the gas market(directive 2003/55/EC, GasNZV)

. in the entry/exit model network users booktransmission capacity at entries and exitsseparately

. transmission cost may depend on entry/exit,but not on transportation path

. customers later nominate the actual inflowand outflow within their booked capacities⇒ nomination of all customers

. transmission system operator has to ensurethat each nomination within the bookedcapacities can technically be realized

• •

••

•

entries

exits

booked capacities

10 5

520

10

nomination 1

5 5

00

10

nomination 2

8 3

52

4

Martin Grötschel Optimization of Gas Networks 4 / 31

The Entry/Exit Model

. the entry/exit model was introduced as ameans to liberalize the gas market(directive 2003/55/EC, GasNZV)

. in the entry/exit model network users booktransmission capacity at entries and exitsseparately

. transmission cost may depend on entry/exit,but not on transportation path

. customers later nominate the actual inflowand outflow within their booked capacities⇒ nomination of all customers

. transmission system operator has to ensurethat each nomination within the bookedcapacities can technically be realized

• •

••

•

entries

exits

booked capacities

10 5

520

10

nomination 1

5 5

00

10

nomination 2

8 3

52

4

Martin Grötschel Optimization of Gas Networks 4 / 31

The Entry/Exit Model

. the entry/exit model was introduced as ameans to liberalize the gas market(directive 2003/55/EC, GasNZV)

. in the entry/exit model network users booktransmission capacity at entries and exitsseparately

. transmission cost may depend on entry/exit,but not on transportation path

. customers later nominate the actual inflowand outflow within their booked capacities⇒ nomination of all customers

. transmission system operator has to ensurethat each nomination within the bookedcapacities can technically be realized

• •

••

•

entries

exits

booked capacities

10 5

520

10

nomination 1

5 5

00

10

nomination 2

8 3

52

4

Martin Grötschel Optimization of Gas Networks 4 / 31

The Entry/Exit Model

. the entry/exit model was introduced as ameans to liberalize the gas market(directive 2003/55/EC, GasNZV)

. in the entry/exit model network users booktransmission capacity at entries and exitsseparately

. transmission cost may depend on entry/exit,but not on transportation path

. customers later nominate the actual inflowand outflow within their booked capacities⇒ nomination of all customers

. transmission system operator has to ensurethat each nomination within the bookedcapacities can technically be realized

• •

••

•

entries

exits

booked capacities

10 5

520

10

nomination 1

5 5

00

10

nomination 2

8 3

52

4

Martin Grötschel Optimization of Gas Networks 4 / 31

The Entry/Exit Model

. the entry/exit model was introduced as ameans to liberalize the gas market(directive 2003/55/EC, GasNZV)

. in the entry/exit model network users booktransmission capacity at entries and exitsseparately

. transmission cost may depend on entry/exit,but not on transportation path

. customers later nominate the actual inflowand outflow within their booked capacities⇒ nomination of all customers

. transmission system operator has to ensurethat each nomination within the bookedcapacities can technically be realized

• •

••

•

entries

exits

booked capacities

10 5

520

10

nomination 1

5 5

00

10

nomination 2

8 3

52

4

Martin Grötschel Optimization of Gas Networks 4 / 31

The Entry/Exit Model

. the entry/exit model was introduced as ameans to liberalize the gas market(directive 2003/55/EC, GasNZV)

. in the entry/exit model network users booktransmission capacity at entries and exitsseparately

. transmission cost may depend on entry/exit,but not on transportation path

. customers later nominate the actual inflowand outflow within their booked capacities⇒ nomination of all customers

. transmission system operator has to ensurethat each nomination within the bookedcapacities can technically be realized

• •

••

•

entries

exits

booked capacities

10 5

520

10

nomination 1

5 5

00

10

nomination 2

8 3

52

4

Martin Grötschel Optimization of Gas Networks 4 / 31

Essential Subtask: Validating Nominations

Given: . a detailed description of a gas network. a nomination specifying amounts of gas flow

at entries and exits

Task: Find

1. settings for the active devices(valves, control valves, compressors)

2. values for the physical parameters of thenetwork

that comply with. gas physics. legal and technical limitations

human experience

simulation tool

Issue: How to decide whether a nomination is technically feasible?

Martin Grötschel Optimization of Gas Networks 5 / 31

Essential Subtask: Validating Nominations

Given: . a detailed description of a gas network. a nomination specifying amounts of gas flow

at entries and exits

Task: Find

1. settings for the active devices(valves, control valves, compressors)

2. values for the physical parameters of thenetwork

that comply with. gas physics. legal and technical limitations

human experience

simulation tool

Issue: How to decide whether a nomination is technically feasible?

Martin Grötschel Optimization of Gas Networks 5 / 31

Essential Subtask: Validating Nominations

Given: . a detailed description of a gas network. a nomination specifying amounts of gas flow

at entries and exits

Task: Find1. settings for the active devices

(valves, control valves, compressors)

2. values for the physical parameters of thenetwork

that comply with. gas physics. legal and technical limitations

human experience

simulation tool

Issue: How to decide whether a nomination is technically feasible?

Martin Grötschel Optimization of Gas Networks 5 / 31

Essential Subtask: Validating Nominations

Given: . a detailed description of a gas network. a nomination specifying amounts of gas flow

at entries and exits

Task: Find1. settings for the active devices

(valves, control valves, compressors)2. values for the physical parameters of the

network

that comply with. gas physics. legal and technical limitations

human experience

simulation tool

Issue: How to decide whether a nomination is technically feasible?

Martin Grötschel Optimization of Gas Networks 5 / 31

Essential Subtask: Validating Nominations

Given: . a detailed description of a gas network. a nomination specifying amounts of gas flow

at entries and exits

Task: Find1. settings for the active devices

(valves, control valves, compressors)2. values for the physical parameters of the

networkthat comply with. gas physics. legal and technical limitations

human experience

simulation tool

Issue: How to decide whether a nomination is technically feasible?

Martin Grötschel Optimization of Gas Networks 5 / 31

Essential Subtask: Validating Nominations

Given: . a detailed description of a gas network. a nomination specifying amounts of gas flow

at entries and exits

Task: Find1. settings for the active devices

(valves, control valves, compressors)2. values for the physical parameters of the

networkthat comply with. gas physics. legal and technical limitations

human experience

simulation tool

Issue: How to decide whether a nomination is technically feasible?

Martin Grötschel Optimization of Gas Networks 5 / 31

Essential Subtask: Validating Nominations

Given: . a detailed description of a gas network. a nomination specifying amounts of gas flow

at entries and exits

Task: Find1. settings for the active devices

(valves, control valves, compressors)2. values for the physical parameters of the

networkthat comply with. gas physics. legal and technical limitations

human experience

simulation tool

Issue: How to decide whether a nomination is technically feasible?

Martin Grötschel Optimization of Gas Networks 5 / 31

Essential Subtask: Validating Nominations

Given: . a detailed description of a gas network. a nomination specifying amounts of gas flow

at entries and exits

Task: Find1. settings for the active devices

(valves, control valves, compressors)2. values for the physical parameters of the

networkthat comply with. gas physics. legal and technical limitations

human experience

simulation tool

Issue: How to decide whether a nomination is technically feasible?

Martin Grötschel Optimization of Gas Networks 5 / 31

Using Optimization Rather Than SimulationSimulation. allows very accurate gas physics

models. relies on human experience to

decide feasibility. is thus inappropriate to determine

infeasibility of a nomination

Optimization. works on simplified models of gas

physics. automatically finds settings for

active devices. eventually proves infeasibility of an

infeasible nomination

Beware: different solution spaces due to different modeling

simulation A

simulation B

optimization A

optimization B

Martin Grötschel Optimization of Gas Networks 6 / 31

Using Optimization Rather Than SimulationSimulation. allows very accurate gas physics

models. relies on human experience to

decide feasibility. is thus inappropriate to determine

infeasibility of a nomination

Optimization. works on simplified models of gas

physics. automatically finds settings for

active devices. eventually proves infeasibility of an

infeasible nomination

Beware: different solution spaces due to different modeling

simulation A

simulation B

optimization A

optimization B

Martin Grötschel Optimization of Gas Networks 6 / 31

Using Optimization Rather Than SimulationSimulation. allows very accurate gas physics

models. relies on human experience to

decide feasibility. is thus inappropriate to determine

infeasibility of a nomination

Optimization. works on simplified models of gas

physics. automatically finds settings for

active devices. eventually proves infeasibility of an

infeasible nomination

Beware: different solution spaces due to different modeling

simulation A

simulation B

optimization A

optimization B

Martin Grötschel Optimization of Gas Networks 6 / 31

Transient Models

Transient models describe the network state evolution over time.

. similar to reality

but. can only be computed over a finite time horizon. require a forecast of the in- and outflow over time. require a start state, which is not known for planning. deviations between predicted / physical network state grow over time

Deciding feasibility of a future nomination requires to test it against

. a worst case start state?

definitely far too pessimistic

. all likely start states?

infinitely many

. a suitable start state?

might be overly optimistic

Martin Grötschel Optimization of Gas Networks 7 / 31

Transient Models

Transient models describe the network state evolution over time.

. similar to reality

but. can only be computed over a finite time horizon. require a forecast of the in- and outflow over time. require a start state, which is not known for planning. deviations between predicted / physical network state grow over time

Deciding feasibility of a future nomination requires to test it against

. a worst case start state?

definitely far too pessimistic

. all likely start states?

infinitely many

. a suitable start state?

might be overly optimistic

Martin Grötschel Optimization of Gas Networks 7 / 31

Transient Models

Transient models describe the network state evolution over time.

. similar to reality

but. can only be computed over a finite time horizon. require a forecast of the in- and outflow over time. require a start state, which is not known for planning. deviations between predicted / physical network state grow over time

Deciding feasibility of a future nomination requires to test it against

. a worst case start state?

definitely far too pessimistic

. all likely start states?

infinitely many

. a suitable start state?

might be overly optimistic

Martin Grötschel Optimization of Gas Networks 7 / 31

Transient Models

Transient models describe the network state evolution over time.

. similar to reality

but. can only be computed over a finite time horizon. require a forecast of the in- and outflow over time. require a start state, which is not known for planning. deviations between predicted / physical network state grow over time

Deciding feasibility of a future nomination requires to test it against. a worst case start state?

definitely far too pessimistic

. all likely start states?

infinitely many

. a suitable start state?

might be overly optimistic

Martin Grötschel Optimization of Gas Networks 7 / 31

Transient Models

Transient models describe the network state evolution over time.

. similar to reality

but. can only be computed over a finite time horizon. require a forecast of the in- and outflow over time. require a start state, which is not known for planning. deviations between predicted / physical network state grow over time

Deciding feasibility of a future nomination requires to test it against. a worst case start state? definitely far too pessimistic. all likely start states?

infinitely many

. a suitable start state?

might be overly optimistic

Martin Grötschel Optimization of Gas Networks 7 / 31

Transient Models

Transient models describe the network state evolution over time.

. similar to reality

but. can only be computed over a finite time horizon. require a forecast of the in- and outflow over time. require a start state, which is not known for planning. deviations between predicted / physical network state grow over time

Deciding feasibility of a future nomination requires to test it against. a worst case start state? definitely far too pessimistic. all likely start states? infinitely many. a suitable start state?

might be overly optimistic

Martin Grötschel Optimization of Gas Networks 7 / 31

Transient Models

Transient models describe the network state evolution over time.

. similar to reality

but. can only be computed over a finite time horizon. require a forecast of the in- and outflow over time. require a start state, which is not known for planning. deviations between predicted / physical network state grow over time

Deciding feasibility of a future nomination requires to test it against. a worst case start state? definitely far too pessimistic. all likely start states? infinitely many. a suitable start state? might be overly optimistic

Martin Grötschel Optimization of Gas Networks 7 / 31

Stationary Models

Stationary models describe a (timeless) equilibrium network state.

. stable situation (by definition) modeling an “average network state”

. no start state needed, no time horizon

. ensures that the situation is sustainable

. much less data requirements, simpler physics

But. using pipes as gas storage (linepack) cannot be modelled. transition between nominations cannot be modelled. too pessimistic especially regarding short-term peak situations

Nevertheless, the better choice for medium and long-term planning.

Martin Grötschel Optimization of Gas Networks 8 / 31

Stationary Models

Stationary models describe a (timeless) equilibrium network state.

. stable situation (by definition) modeling an “average network state”

. no start state needed, no time horizon

. ensures that the situation is sustainable

. much less data requirements, simpler physics

But. using pipes as gas storage (linepack) cannot be modelled. transition between nominations cannot be modelled. too pessimistic especially regarding short-term peak situations

Nevertheless, the better choice for medium and long-term planning.

Martin Grötschel Optimization of Gas Networks 8 / 31

Stationary Models

Stationary models describe a (timeless) equilibrium network state.

. stable situation (by definition) modeling an “average network state”

. no start state needed, no time horizon

. ensures that the situation is sustainable

. much less data requirements, simpler physics

But. using pipes as gas storage (linepack) cannot be modelled. transition between nominations cannot be modelled. too pessimistic especially regarding short-term peak situations

Nevertheless, the better choice for medium and long-term planning.

Martin Grötschel Optimization of Gas Networks 8 / 31

Stationary Models

Stationary models describe a (timeless) equilibrium network state.

. stable situation (by definition) modeling an “average network state”

. no start state needed, no time horizon

. ensures that the situation is sustainable

. much less data requirements, simpler physics

But. using pipes as gas storage (linepack) cannot be modelled. transition between nominations cannot be modelled. too pessimistic especially regarding short-term peak situations

Nevertheless, the better choice for medium and long-term planning.

Martin Grötschel Optimization of Gas Networks 8 / 31

Gas Network: H-Nord

. 32 entries, 142 exits

. 498 pipes,9 resistors,33 valves,26 control valves,7 compressor stations

. 32 cycles

Martin Grötschel Optimization of Gas Networks 9 / 31

Network and Variables

Mathematical model description:Network: directed Graph G = (V ,E ) with vertices V and edges EVariables: . pressure at node i ∈ V : pi

. mass flow, volumetric flow rate on edge e ∈ E : qe , Qe

. decision for active element ea ∈ Ea ⊂ E : xea

. temperature at node i ∈ V : Ti

. velocity on edge e ∈ E : ve

. fuel, power for compressor eCS ∈ ECS ⊂ E : beCS , PeCS

. density at node i ∈ V : ρi

. real gas factor of gas at node i ∈ V : zi

. calorific value of gas at node i ∈ V : B̂i

. speed of compressor eCS ∈ ECS : neCS

. adiabatic head of compressor eCS ∈ ECS : Had ,eCS

. adiabatic efficiency of compressor eCS ∈ ECS : ηad ,eCS

Martin Grötschel Optimization of Gas Networks 10 / 31

Hierarchical Modeling

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

PDE WeymouthHPPCs

Weymouth

Accurate Approximate

QuadraticModel

PolyhedralModel

IdealCompressor

AccurateModel

ApproximateModel

AccurateModel

ApproximateModel

Martin Grötschel Optimization of Gas Networks 11 / 31

Euler Differential Equation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

PDE WeymouthHPPCs

Weymouth

Euler-differential equation&

equation of state for real gases:

∂ρ

∂t+∂ (ρv)

∂x= 0

∂ (ρv)

∂t+∂(ρv2)

∂x+∂p∂x

+ gρ∂h∂x

+ λ(q)|v |v2D

ρ = 0

Aρcp

(∂T∂t

+ v∂T∂x

)− A

(1 +

Tz∂z∂T

)∂p∂t

−AvTz∂z∂T

∂p∂x

+ Avgρdhdx

+ QE = 0

ρ− ρ0z0T0

p0· pz(p,T )T

= 0

Martin Grötschel Optimization of Gas Networks 12 / 31

Weymouth Equation of HPPCs Model

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

PDE WeymouthHPPCs

Weymouth

Weymouth-equation

based on HPPCs-model:

p2j =

(p2i − Λ · φ (q)

eS − 1S

)e−S

Λ =

(4π

)2 Lp0z (pm,Tm) Tm

D5ρ0z0T0

S = 2Lgdhdx

ρ0z0T0

p0z (pm,Tm) Tm

Martin Grötschel Optimization of Gas Networks 13 / 31

Weymouth Equation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

PDE WeymouthHPPCs

Weymouth

Simplified Weymouth-equation:

Friction coefficient according to Nikuradze

λ =

(2 log10

(Dk

)+ 1.138

)−2

yields

p2j =

(p2i − Λ|qe |qe

eS − 1S

)e−S

Martin Grötschel Optimization of Gas Networks 14 / 31

Flow-dependent Resistor

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

Accurate Approximate

Exact flow dependent resistor equation:

Pressure decrease:

pi − pj =8ρ0p0

π2z0T0

ξTi

D4|qe |qez(pi ,Ti )

pi

Temperature decrease due to Joule-Thomson effect:

Te,out = Te,in + µJT (pj − pi )

Martin Grötschel Optimization of Gas Networks 15 / 31

Resistor Approximation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

Accurate Approximate

Approximation

of flow dependent resistors:

ε(p2i − p2

j)

=8ρ0p0

π2z0T0

ξTD4 |qe |qe

with

minε∈R

u∫l

pv∫l

(p2v − pv · pw

1 + ζe · pv− ε · (p2

v − p2w )

)2dpw dpv

+

u∫l

pw∫l

(−

p2w − pv · pw

1 + ζe · pw− ε · (p2

v − p2w )

)2dpv dpw

12

Martin Grötschel Optimization of Gas Networks 16 / 31

Valves and Control Valves

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

Valve: open or closed

One decision variable xe ∈ {0, 1}:

closed: xe = 0 =⇒ qe = 0open: xe = 1 =⇒ pi = pj

Control Valve: active, bypassed, or closed

Two decision variables xbypasse and xactive

e :

xbypasse + xactive

e ≤ 1

closed: xbypasse = xactive

e = 0 =⇒ qe = 0

bypass: xbypasse = 1 =⇒ pi = pj

active: xactivee = 1 =⇒ ∆ ≤ pi − pj ≤ ∆

Martin Grötschel Optimization of Gas Networks 17 / 31

Valves and Control Valves

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

Valve: open or closed

One decision variable xe ∈ {0, 1}:

closed: xe = 0 =⇒ qe = 0open: xe = 1 =⇒ pi = pj

Control Valve: active, bypassed, or closed

Two decision variables xbypasse and xactive

e :

xbypasse + xactive

e ≤ 1

closed: xbypasse = xactive

e = 0 =⇒ qe = 0

bypass: xbypasse = 1 =⇒ pi = pj

active: xactivee = 1 =⇒ ∆ ≤ pi − pj ≤ ∆

Martin Grötschel Optimization of Gas Networks 17 / 31

Valves and Control Valves

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

Valve: open or closed

One decision variable xe ∈ {0, 1}:

closed: xe = 0 =⇒ qe = 0open: xe = 1 =⇒ pi = pj

Control Valve: active, bypassed, or closed

Two decision variables xbypasse and xactive

e :

xbypasse + xactive

e ≤ 1

closed: xbypasse = xactive

e = 0 =⇒ qe = 0

bypass: xbypasse = 1 =⇒ pi = pj

active: xactivee = 1 =⇒ ∆ ≤ pi − pj ≤ ∆

Martin Grötschel Optimization of Gas Networks 17 / 31

Valves and Control Valves

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

Valve: open or closed

One decision variable xe ∈ {0, 1}:

closed: xe = 0 =⇒ qe = 0open: xe = 1 =⇒ pi = pj

Control Valve: active, bypassed, or closed

Two decision variables xbypasse and xactive

e :

xbypasse + xactive

e ≤ 1

closed: xbypasse = xactive

e = 0 =⇒ qe = 0

bypass: xbypasse = 1 =⇒ pi = pj

active: xactivee = 1 =⇒ ∆ ≤ pi − pj ≤ ∆

Martin Grötschel Optimization of Gas Networks 17 / 31

Quadratic Compressor Model

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

QuadraticModel

PolyhedralModel

IdealCompressor

Quadratic approximation

of turbo-compressor:

Had (Q, n) = a1 + a2n + a3n2 +(a4 + a5n + a6n2)Q

+(a7 + a8n + a9n2)Q2

η (Q, n) = b1 + b2n + b3n2 +(b4 + b5n + b6n2)Q

+(b7 + b8n + b9n2)Q2

Had ≤ s1 + s2Q + s3Q2

Had ≥ c1 + c2Q + c3Q2

P = HadQ/η

Martin Grötschel Optimization of Gas Networks 18 / 31

Polyhedral Compressor Model

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

QuadraticModel

PolyhedralModel

IdealCompressor

Polyhedral approximation based on:

Had =ziRsTiκ

κ− 1

((pj

pi

)κ−1κ

− 1

)

Q =p0z(pi ,T )T3.6z0T0

qpi

pi

pj

q

= pi

1

( κ−1ziRsTiκ

Had + 1)κκ−1

3.6z0T0p0z(pi ,T )T Q

Martin Grötschel Optimization of Gas Networks 19 / 31

Idealized Compressor

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

QuadraticModel

PolyhedralModel

IdealCompressor

Idealized compressor:

Pe =κ

κ− 1ρ0RTiz(pi ,Ti )

ηadm

((pj

pi

)κ−1κ

− 1

)qe

Martin Grötschel Optimization of Gas Networks 20 / 31

Compressor Station

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

AccurateModel

ApproximateModel

Compressor Station:

. union of single compressor machines

. compressor station can operate in differentconfigurations

. configuration: selected compressormachines operateI parallellyI sequentiallyI parallelly and sequentially

Martin Grötschel Optimization of Gas Networks 21 / 31

Compressor Station Approximation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

AccurateModel

ApproximateModel

1. The feasible operating range of a compressormachine is mainly described by the characteristicdiagram:

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

Q

Had

M1

Martin Grötschel Optimization of Gas Networks 22 / 31

Compressor Station Approximation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

AccurateModel

ApproximateModel

1. The feasible operating range of a compressormachine is mainly described by the characteristicdiagram:

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

Q

Had

M1

Martin Grötschel Optimization of Gas Networks 22 / 31

Compressor Station Approximation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

AccurateModel

ApproximateModel

2. Approximation of the feasible operating range ofa configuration by convex approximation incombination with Fourier-Motzkin-Elimination:

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

Q

Had

M1

M3 M1 ‖ M3

Martin Grötschel Optimization of Gas Networks 22 / 31

Compressor Station Approximation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

AccurateModel

ApproximateModel

2. Approximation of the feasible operating range ofa configuration by convex approximation incombination with Fourier-Motzkin-Elimination:

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

Q

Had

M1M3

M1 ‖ M3

I parallel operation of machines M1 ‖ M3

Martin Grötschel Optimization of Gas Networks 22 / 31

Compressor Station Approximation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

AccurateModel

ApproximateModel

2. Approximation of the feasible operating range ofa configuration by convex approximation incombination with Fourier-Motzkin-Elimination:

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

Q

Had

M1M3 M1 ‖ M3

I parallel operation of machines M1 ‖ M3

Martin Grötschel Optimization of Gas Networks 22 / 31

Compressor Station Approximation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

AccurateModel

ApproximateModel

2. Approximation of the feasible operating range ofa configuration by convex approximation incombination with Fourier-Motzkin-Elimination:

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

Q

Had

M1M3 M1 ‖ M3

I parallel operation of machines M1 ‖ M3

Martin Grötschel Optimization of Gas Networks 22 / 31

Compressor Station Approximation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

AccurateModel

ApproximateModel

2. Approximation of the feasible operating range ofa configuration by convex approximation incombination with Fourier-Motzkin-Elimination:

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

Q

Had

M1M3

M1 � M3

I sequential operation of machines M1 � M3

Martin Grötschel Optimization of Gas Networks 22 / 31

Compressor Station Approximation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

AccurateModel

ApproximateModel

2. Approximation of the feasible operating range ofa configuration by convex approximation incombination with Fourier-Motzkin-Elimination:

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

Q

Had

M1M3

M1 � M3

I sequential operation of machines M1 � M3

Martin Grötschel Optimization of Gas Networks 22 / 31

Compressor Station Approximation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

AccurateModel

ApproximateModel

2. Approximation of the feasible operating range ofa configuration by convex approximation incombination with Fourier-Motzkin-Elimination:

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

Q

Had

M1M3

M1 � M3

I sequential operation of machines M1 � M3

Martin Grötschel Optimization of Gas Networks 22 / 31

Compressor Station Approximation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

AccurateModel

ApproximateModel

3. Convex hull of the union of all configurationsyields approximation of the feasible operatingrange of a compressor station

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

10

20

30

40

50

60

70

Q

Had

parallelserial

1. Machines

2. Configurations3. Approximation of Compressor Station

Martin Grötschel Optimization of Gas Networks 22 / 31

Compressor Station Approximation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

AccurateModel

ApproximateModel

3. Convex hull of the union of all configurationsyields approximation of the feasible operatingrange of a compressor station

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

10

20

30

40

50

60

70

Q

Had

parallelserial

1. Machines2. Configurations

3. Approximation of Compressor Station

Martin Grötschel Optimization of Gas Networks 22 / 31

Compressor Station Approximation

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

AccurateModel

ApproximateModel

3. Convex hull of the union of all configurationsyields approximation of the feasible operatingrange of a compressor station

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

10

20

30

40

50

60

70

Q

Had

parallelserial

1. Machines2. Configurations3. Approximation of Compressor Station

Martin Grötschel Optimization of Gas Networks 22 / 31

Subnetwork Operation Modes

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

April 20, 2011Zuse Institute BerlinDep. Optimization

Schematischer Stationsaufbau (neu)

PORZ_VS-S(11p/s, 12p/s)

PORZ_VS-H(5, 6)

PORZ_VS-H1(5, 6)

VA_219 VA_221

VA_193 VA_192 VA_198

VA_200

VA_194 VA_195 VA_199

VA_205

PI_798 VA_198O0003 80060012008

VA_192I

PI_799 VA_199O0004

VA_193O

PI_863

HEROS-H-S-FI

4

800600120010

. each operation mode is described by abinary vector giving the state of each valve

. we use the convex hull of these binaryvectors to include the operation modes inour model

Martin Grötschel Optimization of Gas Networks 23 / 31

Subnetwork Operation Modes

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

April 20, 2011Zuse Institute BerlinDep. Optimization

Schematischer Stationsaufbau (neu)

PORZ_VS-S(11p/s, 12p/s)

PORZ_VS-H(5, 6)

PORZ_VS-H1(5, 6)

VA_219 VA_221

VA_193 VA_192 VA_198

VA_200

VA_194 VA_195 VA_199

VA_205

PI_798 VA_198O0003 80060012008

VA_192I

PI_799 VA_199O0004

VA_193O

PI_863

HEROS-H-S-FI

4

800600120010

April 20, 2011Zuse Institute BerlinDep. Optimization

2a. Verdichten von Stolberg (M11 o. M12)

5

PORZ_VS-S(11p/s, 12p/s)

PORZ_VS-H(5, 6)

PORZ_VS-H1(5, 6)

VA_219 VA_221

VA_193 VA_192 VA_198

VA_200

VA_194 VA_195 VA_199

VA_205

PI_798 VA_198O0003 80060012008

VA_192I

PI_799 VA_199O0004

VA_193O

PI_863

HEROS-H-S-FI

800600120010

. each operation mode is described by abinary vector giving the state of each valve

. we use the convex hull of these binaryvectors to include the operation modes inour model

Martin Grötschel Optimization of Gas Networks 23 / 31

Subnetwork Operation Modes

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

April 20, 2011Zuse Institute BerlinDep. Optimization

Schematischer Stationsaufbau (neu)

PORZ_VS-S(11p/s, 12p/s)

PORZ_VS-H(5, 6)

PORZ_VS-H1(5, 6)

VA_219 VA_221

VA_193 VA_192 VA_198

VA_200

VA_194 VA_195 VA_199

VA_205

PI_798 VA_198O0003 80060012008

VA_192I

PI_799 VA_199O0004

VA_193O

PI_863

HEROS-H-S-FI

4

800600120010

April 20, 2011Zuse Institute BerlinDep. Optimization

2b. Verdichten von Stolberg (M5 o. M6)

6

PORZ_VS-S(11p/s, 12p/s)

PORZ_VS-H(5, 6)

PORZ_VS-H1(5, 6)

VA_219 VA_221

VA_193 VA_192 VA_198

VA_200

VA_194 VA_195 VA_199

VA_205

PI_798 VA_198O0003 80060012008

VA_192I

PI_799 VA_199O0004

VA_193O

PI_863

HEROS-H-S-FI

800600120010

. each operation mode is described by abinary vector giving the state of each valve

. we use the convex hull of these binaryvectors to include the operation modes inour model

Martin Grötschel Optimization of Gas Networks 23 / 31

Subnetwork Operation Modes

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

April 20, 2011Zuse Institute BerlinDep. Optimization

Schematischer Stationsaufbau (neu)

PORZ_VS-S(11p/s, 12p/s)

PORZ_VS-H(5, 6)

PORZ_VS-H1(5, 6)

VA_219 VA_221

VA_193 VA_192 VA_198

VA_200

VA_194 VA_195 VA_199

VA_205

PI_798 VA_198O0003 80060012008

VA_192I

PI_799 VA_199O0004

VA_193O

PI_863

HEROS-H-S-FI

4

800600120010

April 20, 2011Zuse Institute BerlinDep. Optimization

3a. Verdichten von Paffrath nach Stolberg und nach Scheidt, oder verdichtet aus Paffrath und unverdichtet aus Scheidt nach Stolberg

7

PORZ_VS-S(11p/s, 12p/s)

PORZ_VS-H(5, 6)

PORZ_VS-H1(5, 6)

VA_219 VA_221

VA_193 VA_192 VA_198

VA_200

VA_194 VA_195 VA_199

VA_205

PI_798 VA_198O0003 80060012008

VA_192I

PI_799 VA_199O0004

VA_193O

PI_863

HEROS-H-S-FI

800600120010

. each operation mode is described by abinary vector giving the state of each valve

. we use the convex hull of these binaryvectors to include the operation modes inour model

Martin Grötschel Optimization of Gas Networks 23 / 31

Subnetwork Operation Modes

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

April 20, 2011Zuse Institute BerlinDep. Optimization

Schematischer Stationsaufbau (neu)

PORZ_VS-S(11p/s, 12p/s)

PORZ_VS-H(5, 6)

PORZ_VS-H1(5, 6)

VA_219 VA_221

VA_193 VA_192 VA_198

VA_200

VA_194 VA_195 VA_199

VA_205

PI_798 VA_198O0003 80060012008

VA_192I

PI_799 VA_199O0004

VA_193O

PI_863

HEROS-H-S-FI

4

800600120010

April 20, 2011Zuse Institute BerlinDep. Optimization

3a. Verdichten von Paffrath nach Stolberg und nach Scheidt, oder verdichtet aus Paffrath und unverdichtet aus Scheidt nach Stolberg

7

PORZ_VS-S(11p/s, 12p/s)

PORZ_VS-H(5, 6)

PORZ_VS-H1(5, 6)

VA_219 VA_221

VA_193 VA_192 VA_198

VA_200

VA_194 VA_195 VA_199

VA_205

PI_798 VA_198O0003 80060012008

VA_192I

PI_799 VA_199O0004

VA_193O

PI_863

HEROS-H-S-FI

800600120010

. each operation mode is described by abinary vector giving the state of each valve

. we use the convex hull of these binaryvectors to include the operation modes inour model

Martin Grötschel Optimization of Gas Networks 23 / 31

Subnetwork Operation Modes

Pipeline

Resistor

Valve

Control Valve

Compressor

Compressor Station

Operation Modes

April 20, 2011Zuse Institute BerlinDep. Optimization

Schematischer Stationsaufbau (neu)

PORZ_VS-S(11p/s, 12p/s)

PORZ_VS-H(5, 6)

PORZ_VS-H1(5, 6)

VA_219 VA_221

VA_193 VA_192 VA_198

VA_200

VA_194 VA_195 VA_199

VA_205

PI_798 VA_198O0003 80060012008

VA_192I

PI_799 VA_199O0004

VA_193O

PI_863

HEROS-H-S-FI

4

800600120010

April 20, 2011Zuse Institute BerlinDep. Optimization

3a. Verdichten von Paffrath nach Stolberg und nach Scheidt, oder verdichtet aus Paffrath und unverdichtet aus Scheidt nach Stolberg

7

PORZ_VS-S(11p/s, 12p/s)

PORZ_VS-H(5, 6)

PORZ_VS-H1(5, 6)

VA_219 VA_221

VA_193 VA_192 VA_198

VA_200

VA_194 VA_195 VA_199

VA_205

PI_798 VA_198O0003 80060012008

VA_192I

PI_799 VA_199O0004

VA_193O

PI_863

HEROS-H-S-FI

800600120010

. each operation mode is described by abinary vector giving the state of each valve

. we use the convex hull of these binaryvectors to include the operation modes inour model

Martin Grötschel Optimization of Gas Networks 23 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Visualization of Solutionsutilization

entries

exits

Martin Grötschel Optimization of Gas Networks 24 / 31

Automatic Testing of Many Nominations

There are mathematically sound methods to reduce a large set ofnominations to a much smaller representative set.

1 500 000 nominations ca. 4 000 representative nominations

Martin Grötschel Optimization of Gas Networks 25 / 31

Future Tasks

As usual:Citius,Altius,Fortius

bigger networks,faster computations,higher precision

. we need to deal with bigger networks as the market areas increase

. calculation times have to be reduced

. we have to incorporate more detailed physics

. we should be able to handle multi-scale networks.

Martin Grötschel Optimization of Gas Networks 26 / 31

Future Tasks

As usual:Citius,Altius,Fortius

bigger networks,faster computations,higher precision

. we need to deal with bigger networks as the market areas increase

. calculation times have to be reduced

. we have to incorporate more detailed physics

. we should be able to handle multi-scale networks.

Martin Grötschel Optimization of Gas Networks 26 / 31

Future Tasks

As usual:Citius,Altius,Fortius

bigger networks,faster computations,higher precision

. we need to deal with bigger networks as the market areas increase

. calculation times have to be reduced

. we have to incorporate more detailed physics

. we should be able to handle multi-scale networks.

Martin Grötschel Optimization of Gas Networks 26 / 31

Future Tasks

As usual:Citius,Altius,Fortius

bigger networks,faster computations,higher precision

. we need to deal with bigger networks as the market areas increase

. calculation times have to be reduced

. we have to incorporate more detailed physics

. we should be able to handle multi-scale networks.

Martin Grötschel Optimization of Gas Networks 26 / 31

Future Tasks

As usual:Citius,Altius,Fortius

bigger networks,faster computations,higher precision

. we need to deal with bigger networks as the market areas increase

. calculation times have to be reduced

. we have to incorporate more detailed physics

. we should be able to handle multi-scale networks.

Martin Grötschel Optimization of Gas Networks 26 / 31

Future Tasks

As usual:Citius,Altius,Fortius

bigger networks,faster computations,higher precision

. we need to deal with bigger networks as the market areas increase

. calculation times have to be reduced

. we have to incorporate more detailed physics

. we should be able to handle multi-scale networks.

Martin Grötschel Optimization of Gas Networks 26 / 31

Bigger Networks

H-Süd. 47 entries, 265 exits. 1136 pipes,

45 resistors,224 valves,78 control valves,29 compressor stations

. 175 cycles

Martin Grötschel Optimization of Gas Networks 27 / 31

Bigger Networks

L-Gas. 12 entries, 1001 exits. 3623 pipes,

26 resistors,300 valves,118 control valves,12 compressor stations

. 259 cycles

Martin Grötschel Optimization of Gas Networks 28 / 31

Some Insights

But. a substantial effort is needed to succeed,

. the setup cost is high compared to pure research,

. close cooperation with practitioners is necessary,

. different disciplines have to collaborate.

Martin Grötschel Optimization of Gas Networks 29 / 31

Some Insights

Relevant real-world questions can be tackled efficiently bymathematical optimization.

But. a substantial effort is needed to succeed,

. the setup cost is high compared to pure research,

. close cooperation with practitioners is necessary,

. different disciplines have to collaborate.

Martin Grötschel Optimization of Gas Networks 29 / 31

Some Insights

Relevant real-world questions can be tackled efficiently bymathematical optimization.

But. a substantial effort is needed to succeed,

. the setup cost is high compared to pure research,

. close cooperation with practitioners is necessary,

. different disciplines have to collaborate.

Martin Grötschel Optimization of Gas Networks 29 / 31

Some Insights

Relevant real-world questions can be tackled efficiently bymathematical optimization.

But. a substantial effort is needed to succeed,

. the setup cost is high compared to pure research,

. close cooperation with practitioners is necessary,

. different disciplines have to collaborate.

Martin Grötschel Optimization of Gas Networks 29 / 31

Some Insights

Relevant real-world questions can be tackled efficiently bymathematical optimization.

But. a substantial effort is needed to succeed,

. the setup cost is high compared to pure research,

. close cooperation with practitioners is necessary,

. different disciplines have to collaborate.

Martin Grötschel Optimization of Gas Networks 29 / 31

Some Insights

Relevant real-world questions can be tackled efficiently bymathematical optimization.

But. a substantial effort is needed to succeed,

. the setup cost is high compared to pure research,

. close cooperation with practitioners is necessary,

. different disciplines have to collaborate.

Martin Grötschel Optimization of Gas Networks 29 / 31

Future Challenges

How can we incorporate transient effects intostationary optimization models?

. To be able toI compute probabilities for interruptible capacities,I take storage into account when computing capacities,I compute capacities less pessimistically while still ensuring

security of supply,

. we need toI analyse the gap between transient and stationary models,I understand transitions between successive nominations better,I make improvements on the stochastic treatment,I develop more powerful optimization algorithms.

Research on all levels –from basic theory to practicalapplication– is needed to face future challenges!

Martin Grötschel Optimization of Gas Networks 30 / 31

Future Challenges

How can we incorporate transient effects intostationary optimization models?

. To be able toI compute probabilities for interruptible capacities,I take storage into account when computing capacities,I compute capacities less pessimistically while still ensuring

security of supply,

. we need toI analyse the gap between transient and stationary models,I understand transitions between successive nominations better,I make improvements on the stochastic treatment,I develop more powerful optimization algorithms.

Research on all levels –from basic theory to practicalapplication– is needed to face future challenges!

Martin Grötschel Optimization of Gas Networks 30 / 31

Future Challenges

How can we incorporate transient effects intostationary optimization models?

. To be able toI compute probabilities for interruptible capacities,I take storage into account when computing capacities,I compute capacities less pessimistically while still ensuring

security of supply,

. we need toI analyse the gap between transient and stationary models,I understand transitions between successive nominations better,I make improvements on the stochastic treatment,I develop more powerful optimization algorithms.

Research on all levels –from basic theory to practicalapplication– is needed to face future challenges!

Martin Grötschel Optimization of Gas Networks 30 / 31

Future Challenges

How can we incorporate transient effects intostationary optimization models?

. To be able toI compute probabilities for interruptible capacities,I take storage into account when computing capacities,I compute capacities less pessimistically while still ensuring

security of supply,

. we need toI analyse the gap between transient and stationary models,I understand transitions between successive nominations better,I make improvements on the stochastic treatment,I develop more powerful optimization algorithms.

Research on all levels –from basic theory to practicalapplication– is needed to face future challenges!

Martin Grötschel Optimization of Gas Networks 30 / 31

Thank you very much!

Martin Grötschel Optimization of Gas Networks 31 / 31