Optimal Operation of Transient Gas Transport Networks Kai - PowerPoint PPT Presentation

Optimal Operation of Transient Gas Transport Networks Kai Hoppmann-Baum Combinatorial Optimization @ Work 2020

Gas Network Control Problem General Description ◮ Optimizing the short-term transient control of large real-world gas transport networks ◮ “Navigation system” for gas network operators Source: Open Grid Europe Problem Given Goal ◮ Network topology ◮ Control each element such that the network is operated “best” ◮ Initial network state ◮ Good control here means: ◮ Short-term supply/demand and pressure Satisfy all supplies and demands while forecast, e.g., 12–24 hours changing network control as little as possible 1

Example Gas Grid 2

Example Gas Grid - Network Stations 3

Two Main Sources of Complexity ∂ t + ∂ ( ρ v ) ∂ρ = 0 ∂ x ∂ x + ∂ ( ρ v 2 ) ∂ ( ρ v ) + ∂ p + λ a | v | v ρ + gs a ρ = 0 2 D a ∂ t ∂ x Combinatorics of Network Stations Transient Gas Flow in Pipelines Isothermal Euler Equations 4

KOMPASS - Basic Algorithmic Framework 1. Replace network stations by simplified graph representation 5

KOMPASS - Basic Algorithmic Framework 1. Replace network stations by simplified graph representation 2. Further network simplifications: 5

KOMPASS - Basic Algorithmic Framework 1. Replace network stations by simplified graph representation 2. Further network simplifications: ◮ Merge pipes (parallel, sequential) 5

KOMPASS - Basic Algorithmic Framework 1. Replace network stations by simplified graph representation 2. Further network simplifications: ◮ Merge pipes (parallel, sequential) ◮ Remove distribution network parts 5

KOMPASS - Basic Algorithmic Framework 1. Replace network stations by simplified graph representation 2. Further network simplifications: ◮ Merge pipes (parallel, sequential) ◮ Remove distribution network parts ◮ . . . 5

KOMPASS - Basic Algorithmic Framework 1. Replace network stations by simplified graph representation 2. Further network simplifications: ◮ Merge pipes (parallel, sequential) ◮ Remove distribution network parts ◮ . . . 3. Solve transient operation problem using linearized gas flow equations (Netmodel Algorithm) 5

KOMPASS - Basic Algorithmic Framework 1. Replace network stations by simplified graph representation 2. Further network simplifications: ◮ Merge pipes (parallel, sequential) ◮ Remove distribution network parts ◮ . . . 3. Solve transient operation problem using linearized gas flow equations (Netmodel Algorithm) 4. Retrieve pressure value and flow value time-series for the boundaries nodes of network stations 5

KOMPASS - Basic Algorithmic Framework 1. Replace network stations by simplified graph representation 2. Further network simplifications: ◮ Merge pipes (parallel, sequential) ◮ Remove distribution network parts ◮ . . . 3. Solve transient operation problem using linearized gas flow equations (Netmodel Algorithm) 4. Retrieve pressure value and flow value time-series for the boundaries nodes of network stations 5. Solve transient operation problem for original network stations (Station Model Algorithm) 5

KOMPASS - Basic Algorithmic Framework 1. Replace network stations by simplified graph representation 2. Further network simplifications: ◮ Merge pipes (parallel, sequential) ◮ Remove distribution network parts ◮ . . . 3. Solve transient operation problem using linearized gas flow equations (Netmodel Algorithm) 4. Retrieve pressure value and flow value time-series for the boundaries nodes of network stations 5. Solve transient operation problem for original network stations (Station Model Algorithm) 6. Retrieve control suggestions for dispatchers, i.e., operation modes, target values,. . . 5

Outside Network Stations - Pipelines Gasflow in a pipe ( u , v ) between timesteps t i and t i +1 can be described by p u , t i +1 − p u , t i + p v , t i +1 − p v , t i + R s T z ∆ t ( q v , t i +1 − q u , t i +1 ) = 0 2 2 L A � | q u , t i | q u , t i + | q v , t i | q v , t i � λ R s T z L 4 A 2 D p u , t i p v , t i g s L + 2 R s T z ( p u , t i + p v , t i ) + p v , t i − p u , t i = 0 6

Outside Network Stations - Pipelines Gasflow in a pipe ( u , v ) between timesteps t i and t i +1 can be described by p u , t i +1 − p u , t i + p v , t i +1 − p v , t i + R s T z ∆ t ( q v , t i +1 − q u , t i +1 ) = 0 2 2 L A � | q u , t i | q u , t i + | q v , t i | q v , t i � λ R s T z L 4 A 2 D p u , t i p v , t i g s L + 2 R s T z ( p u , t i + p v , t i ) + p v , t i − p u , t i = 0 7

Outside Network Stations - Pipelines Gasflow in a pipe ( u , v ) between timesteps t i and t i +1 can be described by p u , t i +1 − p u , t i + p v , t i +1 − p v , t i + R s T z ∆ t ( q v , t i +1 − q u , t i +1 ) = 0 2 2 L A � � λ L Fixing absolute velocity: | v u , 0 | q u , t i + | v v , 0 | q v , t i 4 A D g s L + 2 R s T z ( p u , t i + p v , t i ) + p v , t i − p u , t i = 0 8

Simplifying Network Stations ◮ Network stations are bounded by fence nodes 9

Simplifying Network Stations ◮ Network stations are bounded by fence nodes 9

Simplifying Network Stations ◮ Network stations are bounded by fence nodes ◮ Elements between fence nodes are removed 9

Simplifying Network Stations ◮ Network stations are bounded by fence nodes ◮ Elements between fence nodes are removed ◮ Fence nodes with similar “behaviour” are grouped into fence groups 9

Simplifying Network Stations ◮ Network stations are bounded by fence nodes ◮ Elements between fence nodes are removed ◮ Fence nodes with similar “behaviour” are grouped into fence groups ◮ Nodes in a fence group are merged into a single nodes 9

Simplifying Network Stations ◮ Network stations are bounded by fence nodes ◮ Elements between fence nodes are removed ◮ Fence nodes with similar “behaviour” are grouped into fence groups ◮ Nodes in a fence group are merged into a single nodes 9

Simplifying Network Stations ◮ Network stations are bounded by fence nodes ◮ Elements between fence nodes are removed ◮ Fence nodes with similar “behaviour” are grouped into fence groups ◮ Nodes in a fence group are merged into a single nodes ◮ Auxiliary nodes (for modelling purposes) may be introduced 9

Simplifying Network Stations ◮ Network stations are bounded by fence nodes ◮ Elements between fence nodes are removed ◮ Fence nodes with similar “behaviour” are grouped into fence groups ◮ Nodes in a fence group are merged into a single nodes ◮ Auxiliary nodes (for modelling purposes) may be introduced 9

Simplifying Network Stations ◮ Network stations are bounded by fence nodes ◮ Elements between fence nodes are removed ◮ Fence nodes with similar “behaviour” are grouped into fence groups ◮ Nodes in a fence group are merged into a single nodes ◮ Auxiliary nodes (for modelling purposes) may be introduced ◮ Auxiliary links represent the capabilities of a network station 9

Flow Directions and Simple States For each network station ( V , A ) we are given ◮ Flow directions F ⊆ P ( V ) × P ( V ) with f = ( f + , f − ) ∈ F ◮ Simple states S ⊆ P ( F ) × P ( A ) × P ( A ) with s = ( s f , s on a , s off a ) ∈ S 10

Example I 11

Example II 12

Flow Directions and Simple States For each network station ( V , A ) we are given ◮ Flow directions F ⊆ P ( V ) × P ( V i ) (example: ( f + , f − )) ◮ Simple states S ⊆ P ( F ) × P ( A ) × P ( A ) (example: ( s f , s on a , s off a )) ◮ x f , t ∈ { 0 , 1 } for flow direction f ∈ F and time step t ∈ T ◮ x s , t ∈ { 0 , 1 } for simple state s ∈ S and time step t ∈ T ◮ x a , t ∈ { 0 , 1 } for artificial arc a ∈ A and time step t ∈ T � f ∈F x f , t = 1 ∀ t ∈ T � f ∈ s f x f , t ≥ x s , t ∀ s ∈ S , ∀ t ∈ T � s ∈S x s , t = 1 ∀ t ∈ T ∀ s ∈ S , ∀ a ∈ s on x s , t ≤ x a , t a , ∀ t ∈ T ∀ s ∈ S , ∀ a ∈ s off 1 − x s , t ≥ x a , t a , ∀ t ∈ T ... additional flow direction related constraints ... 13

Shortcuts For a shortcut a = ( u , v ) and each t ∈ T : Not Active ( x a , t = 0): ◮ Decoupled pressure values ◮ No flow allowed Active ( x a , t = 1): ◮ Coupled pressure values ◮ Bidirectional flow up to an amount of q a (Big-M). p u , t − p v , t ≤ (1 − x a , t )( p v − p u ) p u , t − p v , t ≥ (1 − x a , t )( p v − p u ) q → a , t ≤ x a , t q a q ← a , t ≤ x a , t q a . 14

Regulating Arcs For a regulating arc a = ( u , v ) and each t ∈ T : Not Active ( x a , t = 0): ◮ Decoupled pressure values ◮ No flow allowed Active ( x a , t = 1): ◮ Pressure at u not smaller than pressure at v ◮ Unidirectional flow up to an amount of q a (Big-M). p u , t − p v , t ≥ (1 − x a , t )( p v − p u ) q → a , t ≤ x a , t q a . 15

Compressing Arcs Not Active ( x a , t = 0): ◮ No machine assigned ◮ Decoupled pressure values ◮ No flow allowed Active ( x a , t = 1): ◮ Assign machines to compressing arc ◮ Pressure at v not smaller than pressure at u ◮ Pressure at v at most r a times greater than p u , 0 ◮ Flow limited by sum of max flows of assigned machines ◮ Respect approximated power bound equation 16