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CO@Work 2020 Gas Networks Introduction Topics are: 1. European Regulations 2. Gas Network Basics 3. Network Design 4. Gas Network Capacity 5. Gas Network Control September 2020 Online http://co-at-work.zib.de The Energy Team (ZIB, TU

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SLIDE 1

September 2020 Online http://co-at-work.zib.de

CO@Work 2020

Gas Networks Introduction

Topics are:

  • 1. European Regulations
  • 2. Gas Network Basics
  • 3. Network Design
  • 4. Gas Network Capacity
  • 5. Gas Network Control
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SLIDE 2

The Energy Team (ZIB, TU Berlin, MODAL GasLab)

Thorsten Koch 2

Tom Christine Jan Milena Janina Charlie Inci Nils Mark Carsten Lovis Felix Kai Jaap Thai Ying

… and me

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SLIDE 3

A pipeline

↓

𝛽 π‘Ÿ π‘Ÿ = π‘ž!"#

$

βˆ’ π‘ž%&

$

𝛽 depending on dimension and inclination of the pipeline, the friction in the

pipeline, gas temperature, gas composition, outside temperature, and more.

Thorsten Koch 20

50 bar 20 bar pipeline

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SLIDE 4

The network

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Given a graph 𝐻 = (π‘Š, 𝐡) with pressure π‘ž!, and flow π‘Ÿ", for 𝑣 ∈ π‘Š, 𝑏 ∈ 𝐡 and 𝜌! = π‘ž!

#.

Exists 𝒓, 𝝆 subject to 1

"∈%!(!)

π‘Ÿ" ( 1

"∈%"(!)

π‘Ÿ" = 𝑒! for all 𝑣 ∈ π‘Š 𝛽" ∣ π‘Ÿ" ∣ π‘Ÿ" = 𝜌! βˆ’ 𝛾"𝜌) for all 𝑏 = 𝑣, 𝑀 ∈ 𝐡 𝜌! ≀ 𝜌! ≀ 𝜌! for all 𝑣 ∈ π‘Š π‘Ÿ" ≀ π‘Ÿ" ≀ π‘Ÿ" for all 𝑏 ∈ 𝐡

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SLIDE 5

Line Theorem with bounded variables

Theorem (Maugis, 1977, Collins at al, 1978, Humpola, K., et al, 2013) Let 𝑒 ∈ ℝ* be a balanced demand and Ξ¦" strictly increasing function. Then the solution space of exists π‘Ÿ, 𝜌 subject to is either empty or fulfills the conditions: 1. The flow is unique 2. The squared pressure component 𝜌 has the form πœŒβˆ— + πœƒ, πœƒ ≀ πœƒ ≀ Μ… πœƒ } for some πœŒβˆ—, πœƒ, Μ… πœƒ .

Thorsten Koch 22

1

"∈%!(!)

π‘Ÿ" ( 1

"∈%"(!)

π‘Ÿ" = 𝑒! for all 𝑣 ∈ π‘Š Ξ¦"(π‘Ÿ") = 𝜌! βˆ’ 𝜌) for all 𝑏 = 𝑣, 𝑀 ∈ 𝐡 𝜌! ≀ 𝜌! ≀ 𝜌! for all 𝑣 ∈ π‘Š π‘Ÿ" ≀ π‘Ÿ" ≀ π‘Ÿ" for all 𝑏 ∈ 𝐡

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SLIDE 6

Line Theorem: The Line

β–· Flow is unique and Ξ¦'is strictly increasing. β–· 𝜌" βˆ’ 𝜌( are uniquely determined for all arcs. β–· Shifting all values by a constant is feasible. β–· Constant shift is the only possible source of difference.

Consider two feasible squared pressure vectors: 𝜌) und 𝜌)). Both are shifted in such a way that they coincide in the value at 𝑣. The difference is constant which implies 𝜌"

) βˆ’ 𝜌( ) = 𝜚' π‘Ÿ' = 𝜌" )) βˆ’ 𝜌( )).

Since 𝜌"

) = 𝜌" )), we also have 𝜌( ) = 𝜌( )) .

Thorsten Koch 23

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SLIDE 7

Convex Slack Reformulation

The solution space of the problem

min 1

!∈*

Ξ”- + 1

"∈.

Ξ”" subject to

is non-empty and is convex.

Thorsten Koch 24

1

"∈%!(!)

π‘Ÿ" ( 1

"∈%"(!)

π‘Ÿ" = 𝑒! for all 𝑣 ∈ π‘Š Ξ¦"(π‘Ÿ") = 𝜌! βˆ’ 𝜌) for all 𝑏 = 𝑣, 𝑀 ∈ 𝐡 𝜌! βˆ’ Ξ”- ≀ 𝜌! for all 𝑣 ∈ π‘Š 𝜌! + Ξ”! β‰₯ 𝜌! for all 𝑣 ∈ π‘Š Ξ”! β‰₯ 0 for all 𝑣 ∈ π‘Š π‘Ÿ" βˆ’ Ξ”" ≀ π‘Ÿ" for all 𝑏 ∈ 𝐡 π‘Ÿ" + Ξ”" β‰₯ π‘Ÿ" for all 𝑏 ∈ 𝐡 Ξ”" β‰₯ 0 for all 𝑏 ∈ 𝐡

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SLIDE 8

Element Function Symbol

Valve Switch

  • n/off

Regulator Decrease pressure Compressor Increase pressure Active elements in gas networks

Thorsten Koch 25

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SLIDE 9

Controllable Networks

This is an example of just a compressor station

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SLIDE 10

Compressor machines

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Compressor performance depends on input pressure, output pressure, flow, temperature, composition, compressor power.

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SLIDE 11

Compressor stations

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How precise can we model a compressor? Let us assume we assume the gas temperature 10Β°C too low. This gives about 3% more power to the compressor station. This might be enough to get another 1500 MW gas through.

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SLIDE 12

3 Questions

β–Ά Design a new network

(or extend an existing one)

β–Ά Determine the capacity

  • f a given network

β–Ά Control a network to achieve

maximum efficiency

Thorsten Koch 29

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SLIDE 13

Thorsten Koch

CO@Work Online, September 2020

Network Design

(Just a teaser)

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SLIDE 14

Network Construction Steiner Tree

The Steiner tree problem in graphs (STP) Given an undirected connected graph 𝐻 = (π‘Š, 𝐹), costs 𝑑: 𝐹 β†’ β„š! and a set π‘ˆ βŠ‚ π‘Š of terminals, find a minimum weight tree 𝑇 βŠ‚ 𝐻 which spans π‘ˆ. The STP is one of the classical 21 NP-hard problems.

Thorsten Koch 31

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SLIDE 15

Direct Cut Integer Programming Formulation for SAP

min 𝒅𝑼𝒛 subject to 𝑧 πœ€"

#

β‰₯ 1, for all 𝑋 βŠ‚ π‘Š, 𝑠 ∈ 𝑋, π‘Š \ 𝑋 ∩ π‘ˆ β‰  βˆ… 𝑧 πœ€$

% 2

= 0, if 𝑀 = 𝑠; = 1, if 𝑀 ∈ π‘ˆ \ 𝑠; ≀ 1, if 𝑀 ∈ 𝑂 for all 𝑀 ∈ π‘Š 𝑧 πœ€$

% ≀ 𝑧 πœ€$ # ,

for all 𝑀 ∈ 𝑂; 𝑧 πœ€$

% β‰₯ 𝑧&,

for all 𝑏 ∈ πœ€$

#, 𝑀 ∈ 𝑂;

0 ≀ 𝑧& ≀ 1, for all 𝑏 ∈ 𝐡; 𝑧& ∈ 0,1 , for all 𝑏 ∈ 𝐡, where 𝑂 = π‘Š βˆ– π‘ˆ, πœ€'

# ≔ { 𝑣, 𝑀 ∈ 𝐡|𝑣 ∈ π‘Œ, 𝑀 ∈ π‘Š βˆ– X}, πœ€' % ≔ πœ€(βˆ–' #

for π‘Œ βŠ‚ V.

Thorsten Koch 32

See, e.g., Koch, Martin, Solving Steiner tree problems in graphs to optimality, Networks (1998) Polzin, Algorithms for the Steiner problem in networks, Uni Saarland, 2004, Rehfeldt, Koch, Combining NP-Hard Reduction Techniques and Strong Heuristics in an Exact Algorithm for the Maximum-Weight Connected Subgraph Problem, SIAMOPT (2019), Shinano, Rehfeldt, Koch, Building Optimal Steiner Trees on Supercomputers by Using up to 43,000 Cores, CPAIOR 2019, LNCS 11494, and the references there in.

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SLIDE 16

SCIP-Jack is a solver for Steiner Tree Problems in Graphs

It is part of the SCIP Optimization Suite and can solve:

β–· Steiner Tree Problem in Graphs (STP) β–· Steiner Arborescence Problems in Graphs (SAP) β–· Rectilinear Steiner Minimum Tree (RSMTP) β–· Node-weighted Steiner Tree (NWSTP) β–· Prize-collecting Steiner Tree (PCSTP) β–· Rooted Prize-collecting Steiner Tree (RPCSTP) β–· Maximum-weight Connected Subgraph (MWCSP) β–· Degree-constrained Steiner Tree (DCSTP) β–· Group Steiner Tree (GSTP) β–· Hop-constrained directed Steiner Tree (HCDSTP)

http://scipopt.org

Thorsten Koch 33

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SLIDE 17

Please continue with the lecture on Gas Network Cacapcity

Thorsten Koch 34

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SLIDE 18

Thorsten Koch

CO@Work Online, September 2020

Gas Network Capacity

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SLIDE 19

What does β€œCapacity of the Network” mean?

The Technical Capacity is defined as the maximum flow bounds at the entry- and exit-nodes, such that any possible balanced demand scenario within these bounds can be fulfilled by the network.

Thorsten Koch 36

1000 1000 1000 1000 1000 1000

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SLIDE 20

What does β€œCapacity of the Network” mean?

The Technical Capacity is defined as the maximum flow bounds at the entry- and exit-nodes, such that any possible balanced demand scenario within these bounds can be fulfilled by the network. Now adding a connection…

Thorsten Koch 37

1000 1000 1000 1000 1000 1000 500 1000 1000

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SLIDE 21

What does β€œCapacity of the Network” mean?

The Technical Capacity is defined as the maximum flow bounds at the entry- and exit-nodes, such that any possible balanced demand scenario within these bounds can be fulfilled by the network. Now adding a connection … does not necessarily increase it.

Thorsten Koch 38

1000 1000 1000 1000 1000 1000 500 1000 1000 500 500 500 500

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SLIDE 22

What does β€œCapacity of the Network” mean?

The new connection also adds restrictions due to pressure coupling. Not all splits of the inflow between the two entries are easily possible anymore. See also Braess’s paradox.

Thorsten Koch 39

80 bar 60 bar 50 bar 34 bar 70 bar 42 bar

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SLIDE 23

Transient Models

Transient models describe the network state over time. Advantages

β–· This is quite realistic (depending on the time step size)

Disadvantages

β–· Can only be computed over a finite time horizon β–· Requires a forecast of the in- and outflow over time β–· Requires a start state, which is not known for planning β–· Deviations between the predicted and the physical network

state grow over time If we want to decide the feasibility of a future demand scenario should we test against:

β–· A worst case start state? Far too pessimistic β–· All possible start states? Infinitely many β–· A suitable start state? Likely overly optimistic

Thorsten Koch 40

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SLIDE 24

Stationary Models

Stationary models describe a (timeless) equilibrium network state. Advantages

β–· Stable situation (by definition)

modelling an β€œaverage network” state

β–· No start state needed, no time horizon to consider β–· Ensures that the situation is sustainable

(we cannot paint ourselves easily into a corner)

β–· Much less data requirements, simpler physics

Disadvantages

β–· Using pipes as gas storage (linepack) cannot be modelled β–· Transition between states cannot be modelled β–· Too pessimistic, especially regarding short-term peak situations

Often better suited for medium and long-term planning.

Thorsten Koch 41

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SLIDE 25

Demand Scenarios

Thorsten Koch 42

Demand Scenario

𝑒 specifies for each 𝑣 ∈ π‘Š the amount of flow that enters 𝑒! β‰₯ 0 or leaves 𝑒! ≀ 0 the network at 𝑣. The scenario 𝑒 is balanced, i.e., we have 1

!∈*

𝑒! = 0. The task is to decide, whether or not the scenario d can be realized in the network by controlling the active elements. Given a directed graph 𝐻 = (π‘Š, 𝐡) that models a gas network. The arcs represent the elements of the network. We distinguish between passive network elements (pipes and resistors), whose behavior cannot be influenced, and active network elements (valves, control valves, and compressors), which allow to control the network. The active and passive elements are collected in the arc sets 𝐡"/01)2 and 𝐡3"441)2, respectively.

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SLIDE 26

Are all contractually legitimate scenarios technically feasible?

Thorsten Koch 43

Contractually legitimate Contractually illegitimate

A possible approach:

  • 1. Experts derive scenarios

that are on the border of feasibility.

  • 2. These scenarios are

checked using stationary models.

  • 3. If they are feasible, it is

concluded that also all

  • ther scenarios are

feasible.

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SLIDE 27

Assumptions on the Set of all feasible Scenarios

Thorsten Koch 44

Non-smooth Non-convex Non-compact

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SLIDE 28

Assumptions on the Scenarios

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β–· Span the set β–· Convexity β–· Sweep the border β–· Monotonicity

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SLIDE 29

Decisions regarding the Momentum-Equation

Thorsten Koch 46

Momentum-Equation Analytically Discretisation Approximation Friction model HP-PC Colebrook Hofer Nikuradse Smooth Approximation

For medium temperature For medium pressure

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SLIDE 30

Assumptions on the Border

Thorsten Koch 47

The border is not unique

β–· different Models/Methods β–· different Parameters β–· different Implementations

The border is the frontier between technical feasible and technical infeasible.

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SLIDE 31

Assumptions on the Border

Thorsten Koch 48

Vertraglich zulΓ€ssig und realistisch Contractual legitimate and reasonable What we want is not the contractual legitimate area, but the contractual legitimate and reasonable area. This area is not well defined.

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SLIDE 32

Shit!

Thorsten Koch 49

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SLIDE 33

Scenario Generation

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We are able to generate scenarios and test their technical feasibility (with some limitations). Generation of scenarios:

β–· Manual generation of expert scenarios

  • limited number,

+ high quality

β–· Automatic generation of extreme value scenarios

  • difficult to distinguish between contractual legitimate and reasonable,

+ high number

β–· Automatic generation of stochastic scenarios

  • seldom extreme,

+ reasonable, + high number, + probable distribution

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SLIDE 34

Manual Expert Scenarios

Advantages:

β–Ά High Quality β–Ά Very well understood

Problems:

β–Ά Risk of being too much tuned towards a particular model,

parameter setting, or implementation β€žlet us see how far we can drive thisβ€œ Disadvantages:

β–Ά Small number β–Ά Complete coverage of the feasible set is hard to ensure β–Ά Only extreme value scenarios, i.e., the probability for the

scenarios to happen in reality is nearly zero

Thorsten Koch 51

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SLIDE 35

Automatic Extreme Value Scenarios

Advantages:

β–Ά High number β–Ά Can be generated with different methods

Problems:

β–Ά Hard to detect contractual legitimate but unreasonable

Disadvantages:

β–Ά Only the border of the feasible set is covered β–Ά Only extreme value scenarios, i.e. the probability for the

scenarios to happen in reality is nearly zero

Thorsten Koch 52

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SLIDE 36

Stochastic Scenarios

Advantages:

β–Ά High number β–Ά Can be generated with different methods β–Ά Realistic β–Ά Coverage of the inner part of the feasible set β–Ά Can be attributed with probabilities

Problems:

β–Ά Methods for generation are involved and need much data β–Ά Refer to the past

Disadvantages:

β–Ά Extreme values occur seldom since they are not probable

to happen

Thorsten Koch 53

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SLIDE 37

Combination

β–Ά If all three methods are combined, there are nearly no

disadvantages left.

β–Ά This may lead to a huge number of scenarios of which

many are possibly similar.

β–Ά This can be countered by methods for scenario reduction. β–Ά Testing the technical feasibility of the scenarios has

necessarily to happen automatically.

β–Ά This can be troublesome regarding extremal value

scenarios.

β–Ά Now it is possible to automatically validate scenarios. β–Ά This can be extended to compute capacities. β–Ά It is possible to extend these concepts to answer more

sophisticated questions, e.g. looking at probabilities for buy back happening when overbooking.

Thorsten Koch 54

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SLIDE 38

Using Optimization rather than Simulation

Thorsten Koch 55

Simulation:

β–Ά allows very accurate gas

physics models

β–Ά Relies on human

experience to decide feasibility

β–Ά Therefore cannot

determine infeasibility Optimization:

β–Ά Works on simplified models

  • f gas physics

β–Ά Automatically finds settings

for active elements

β–Ά Eventually can prove

infeasibility of a scenario Beware: different solution spaces due to different modeling

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SLIDE 39

Forschungskooperation Netzoptimierung

Thorsten Koch 56

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SLIDE 40

Nova: Nomination Validation

Goal of the project: Build an automatic system that given a network and a set of supplies and demands at the border points computes settings for the active elements

  • f the network such that the resulting stationary

scenario is feasible.

  • Prof. Dr. T. Koch ZIB Mathematics to Gas-Industry

57

Thorsten Koch 57

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SLIDE 41
  • Prof. Dr. T. Koch ZIB Mathematics to Gas-Industry

S t a r t

  • f

p r

  • j

e c t 09/2008

Nova Progress

58

07/2016 Thorsten Koch 58

xxxxxxxxxxxxxxxxxxxxxxxxx
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SLIDE 42

Simplified Subnetwork

Thorsten Koch

09/2008 02/10

Element Count Pipes 136 Compressor groups 3 Resistors 8 Control valves 7 Valves 1 Entries 26 Exits 14

59

Start of project First running code and data: 30 min

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SLIDE 43

Simplified Subnetwork

Thorsten Koch

09/2008 02/10 05/10

60

Start of project First running code and data: 30 min nonlinear preprocess: 15 sec

Element Count Pipes 136 Compressor groups 3 Resistors 8 Control valves 7 Valves 1 Entries 26 Exits 14

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SLIDE 44

H-Gas Network (north)

Thorsten Koch

09/2008 02/10

Element Count Pipes 470 Compressor groups 6 Resistors 8 Control valves 23 Valves 34 Entries 26 Exits 87

05/10 10/10

61

Start of project First running code and data: 30 min nonlinear preprocess: 15 sec novel MIP relaxation: 3 min

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SLIDE 45

H-Gas Network (north)

Thorsten Koch

09/2008 02/10 05/10 10/10 01/11

62

Start of project First running code and data: 30 min nonlinear preprocess: 15 sec novel MIP relaxation: 3 min subnet operation modes: 1 min

Element Count Pipes 470 Compressor groups 6 Resistors 8 Control valves 23 Valves 34 Entries 26 Exits 87

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SLIDE 46

Entire L-Gas Network

Thorsten Koch

09/2008 02/10

Element Count Pipes 3,638 Compressor groups 12 Resistors 26 Control valves 121 Valves 308 Entries 9 Exits 860

05/10 10/10 01/11 03/12

63

Start of project First running code and data: 30 min nonlinear preprocess: 15 sec novel MIP relaxation: 3 min subnet operation modes: 1 min first results L-gas network: 2 h

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SLIDE 47

Entire L-Gas Network

Thorsten Koch

09/2008 02/10 05/10 10/10 01/11 03/12 02/13

64

Start of project First running code and data: 30 min nonlinear preprocess: 15 sec novel MIP relaxation: 3 min subnet operation modes: 1 min first results L-gas network: 2 h novel PADM algorithm: 20 min

Element Count Pipes 3,638 Compressor groups 12 Resistors 26 Control valves 121 Valves 308 Entries 9 Exits 860

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SLIDE 48

H-Gas Network (south)

Thorsten Koch

09/2008 02/10

Element Count Pipes 1,218 Compressor groups 34 Resistors 47 Control valves 115 Valves 473 Entries 17 Exits 286

05/10 10/10 01/11 03/12 02/13 06/13

65

Start of project First running code and data: 30 min nonlinear preprocess: 15 sec novel MIP relaxation: 3 min subnet operation modes: 1 min first results L-gas network: 2 h novel PADM algorithm: 20 min first results for south H-gas: 9 h

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SLIDE 49

Entire H-Gas Network

Thorsten Koch

Start of project 09/2008 First running code and data: 30 min 02/10

Element Count Pipes 1,747 Compressor groups 41 Resistors 85 Control valves 145 Valves 545 Entries 45 Exits 429

Nonlinear preprocess: 15 sec 05/10 novel MIP relaxation: 3 min 10/10 subnet operation modes: 1 min 01/11 first results L-gas network: 2 h 03/12 novel PADM algorithm: 20 min 02/13 first results for south H-gas: 9 h 12/14 tailored heuristics: 57 min 06/13

66

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SLIDE 50

The Research Cooperation Network Optimization ForNe ran for 6 years and involved more than 30 people from around 10 universities and institutes along with more than 10 employees from Germany’s largest gas network system

  • perator OGE.

Γ§ Here are the results.

Thorsten Koch 67

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SLIDE 51

Computational outcome

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Number of infeasible scenarios Number of scenarios reaching iteration limit Feasible scenarios

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SLIDE 52

Time spend

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Running time in h Number of computed scenarios

Feasible Infeasible Iteration limit

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SLIDE 53

Additional difficulties

β–Ά Technology is changing all the time,

new systems mix with existing ones

β–Ά Often hard to determine actual system limits

(many assumptions are necessary for optimization)

β–Ά Optimization drives errors to cluster in one direction β–Ά For larger models/scenarios hard to check against reality

(also very hard to find data errors/inconsistencies)

β–Ά Marketization makes it difficult to predict behavior of

participants, since the objectives of the individual players are unknown, and the outcome is result of a game (negotiation).

Thorsten Koch 70

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SLIDE 54

Some insights

Thorsten Koch 71

Relevant real-world questions can be tackled efficiently with mathematical optimization and algorithmic intelligence.

However,

β–Ά a substantial effort is needed to succeed, β–Ά the setup cost is high compared to pure research, β–Ά close cooperation with practitioners is indispensable, β–Ά different disciplines need to collaborate, β–Ά access to and curation of data is essential.

Good luck!

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SLIDE 55

Thorsten Koch 72

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SLIDE 56

Thank you very much

Thorsten Koch 73

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SLIDE 57

Please continue with the lecture on Gas Network Control

Thorsten Koch 74