On the quantification of nomination feasibility in stationary gas networks with random load

Gotzes, Claudia; Heitsch, Holger; Henrion, René; Schultz, Rüdiger

doi:10.1007/s00186-016-0564-y

Cited by 48 publications

(84 citation statements)

References 14 publications

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“…This result was proven for pipe‐only networks without active elements in . We observe next that the result of can easily be generalized to networks with compressors, providing Assumption is satisfied. With

{scriptA}_{normalcs} \cap N = \emptyset

, it is Δ ≡ Δ ℬ and thus we can remove any

Δ_{scriptN}

from the model.Theorem Let

G = (scriptV, scriptA)

be a weakly connected digraph with nodes

V

and arcs

A

.…”

Section: Stationary Gas Network Operations With Linear Compressor Modelsmentioning

confidence: 61%

“…We next present an equivalent formulation of problem (8) where a number of variables are eliminated. This is an extension of a result in for passive networks to active gas networks with a linear compressor model. For our result, we use the following assumption:Assumption Let a gas network problem with compressors over graph

G

be given.…”

Section: Stationary Gas Network Operations With Linear Compressor Modelsmentioning

confidence: 67%

“…In , the authors show that all squared pressure variables and

| V | - 1

flow variables can be eliminated from a gas network model without compressors. This is done by expressing the flows within the network as a combination of flows on a spanning tree together with flows on the remaining cycles.…”

Section: Stationary Gas Network Operations With Linear Compressor Modelsmentioning

confidence: 99%

“…We define a function q (·) that maps nonbasic flows to flows in the whole network:

\begin{array}{l} q : ℝ^{∣ scriptN ∣} \to ℝ^{∣ scriptA ∣}, \\ q_{a} ((), q_{scriptN}) = ({, \begin{array}{l} {({\overset{true˜}{A}}_{ℬ}^{- 1} (\overset{true˜}{d} - {\overset{bold-italicA}{true}}_{scriptN} q_{scriptN}))}_{a}, & if a \in ℬ, \\ {(q_{N})}_{a}, & if a \in scriptN . \end{array}) \end{array}

Next we define a function that describes an aggregated pressure drop between the root node and the other nodes in the network. This function was first introduced in :

\overset{true˜}{g} (q, Δ) = {((), {\overset{bold-italicA}{true}}_{ℬ}^{- 1})}^{T} F_{ℬ} (q, Δ) .

We notice that each entry F a depends either on q a or on Δ a . Thus, the arguments of F ℬ can be restricted to their basis parts q ℬ and Δ ℬ .…”

Section: Stationary Gas Network Operations With Linear Compressor Modelsmentioning

confidence: 99%

“…If

q_{scriptN}

is feasible for , the remaining original variables q ℬ , π can be recovered through

q_{ℬ} = {\overset{bold-italicA}{true}}_{ℬ}^{- 1} (\overset{true˜}{d} - {\overset{bold-italicA}{true}}_{scriptN} q_{scriptN})

and

π_{v} = π_{r} - g_{v} (q_{N}, Δ_{ℬ})

for

v \in V

. The value of π r is an arbitrary given element of the interval

[{\underset{true_}{π}}_{r}^{*}, {\overset{true‾}{π}}_{r}^{*}] = [\max_{v \in V} [{\underline{π}}_{v} + g_{v} (q_{N}, Δ_{ℬ})], \min_{v \in V} [{\overset{π}{true}}_{v} + g_{v} (q_{scriptN}, Δ_{ℬ})]] .

Conversely, a vector

q_{scriptN}

that was extracted from a solution q * , π * of is feasible for . Proof The original result of is established for pipe‐only networks, that is, for

{scriptA}_{normalcs} = \emptyset

. However, the form of the pressure drop law is never exploited explicitly.…”

Section: Stationary Gas Network Operations With Linear Compressor Modelsmentioning

confidence: 99%

See 4 more Smart Citations

Decomposable robust two‐stage optimization: An application to gas network operations under uncertainty

2019

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We study gas network problems with compressors and control valves under uncertainty that can be formulated as two‐stage robust optimization problems. Uncertain data are present in the physical parameters of the pipes as well as in the overall demand. We show how to exploit the special decomposable structure of the problem to reformulate the two‐stage problem as a single‐stage robust optimization problem. The right‐hand side of the single‐stage problem can be precomputed by solving a series of optimization problems and multiple elements of the right‐hand side can be combined into one optimization task. The practical feasibility and effectiveness of our approach is demonstrated with benchmarks on several gas network instances, among them a realistic model of the Greek natural gas network. Overall, aggregation and preprocessing allow us to quickly solve large gas network instances under uncertainty for the price of slightly more conservative solutions.

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{scriptA}_{normalcs} \cap N = \emptyset

, it is Δ ≡ Δ ℬ and thus we can remove any

Δ_{scriptN}

from the model.Theorem Let

G = (scriptV, scriptA)

be a weakly connected digraph with nodes

V

and arcs

A

.…”

Section: Stationary Gas Network Operations With Linear Compressor Modelsmentioning

confidence: 61%

G

be given.…”

Section: Stationary Gas Network Operations With Linear Compressor Modelsmentioning

confidence: 67%

“…In , the authors show that all squared pressure variables and

| V | - 1

Section: Stationary Gas Network Operations With Linear Compressor Modelsmentioning

confidence: 99%

“…We define a function q (·) that maps nonbasic flows to flows in the whole network:

\begin{array}{l} q : ℝ^{∣ scriptN ∣} \to ℝ^{∣ scriptA ∣}, \\ q_{a} ((), q_{scriptN}) = ({, \begin{array}{l} {({\overset{true˜}{A}}_{ℬ}^{- 1} (\overset{true˜}{d} - {\overset{bold-italicA}{true}}_{scriptN} q_{scriptN}))}_{a}, & if a \in ℬ, \\ {(q_{N})}_{a}, & if a \in scriptN . \end{array}) \end{array}

Next we define a function that describes an aggregated pressure drop between the root node and the other nodes in the network. This function was first introduced in :

\overset{true˜}{g} (q, Δ) = {((), {\overset{bold-italicA}{true}}_{ℬ}^{- 1})}^{T} F_{ℬ} (q, Δ) .

We notice that each entry F a depends either on q a or on Δ a . Thus, the arguments of F ℬ can be restricted to their basis parts q ℬ and Δ ℬ .…”

Section: Stationary Gas Network Operations With Linear Compressor Modelsmentioning

confidence: 99%

“…If

q_{scriptN}

is feasible for , the remaining original variables q ℬ , π can be recovered through

q_{ℬ} = {\overset{bold-italicA}{true}}_{ℬ}^{- 1} (\overset{true˜}{d} - {\overset{bold-italicA}{true}}_{scriptN} q_{scriptN})

and

π_{v} = π_{r} - g_{v} (q_{N}, Δ_{ℬ})

for

v \in V

. The value of π r is an arbitrary given element of the interval

[{\underset{true_}{π}}_{r}^{*}, {\overset{true‾}{π}}_{r}^{*}] = [\max_{v \in V} [{\underline{π}}_{v} + g_{v} (q_{N}, Δ_{ℬ})], \min_{v \in V} [{\overset{π}{true}}_{v} + g_{v} (q_{scriptN}, Δ_{ℬ})]] .

Conversely, a vector

q_{scriptN}

that was extracted from a solution q * , π * of is feasible for . Proof The original result of is established for pipe‐only networks, that is, for

{scriptA}_{normalcs} = \emptyset

. However, the form of the pressure drop law is never exploited explicitly.…”

Section: Stationary Gas Network Operations With Linear Compressor Modelsmentioning

confidence: 99%

See 3 more Smart Citations

Decomposable robust two‐stage optimization: An application to gas network operations under uncertainty

2019

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show abstract

Deciding feasibility of a booking in the European gas market on a cycle is in P for the case of passive networks

2021

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We show that the feasibility of a booking in the European entry‐exit gas market can be decided in polynomial time on single‐cycle networks that are passive, i.e., do not contain controllable elements. The feasibility of a booking can be characterized by solving polynomially many nonlinear potential‐based flow models for computing so‐called potential‐difference maximizing load flow scenarios. We thus analyze the structure of these models and exploit both the cyclic graph structure as well as specific properties of potential‐based flows. This enables us to solve the decision variant of the nonlinear potential‐difference maximization by reducing it to a system of polynomials of constant dimension that is independent of the cycle's size. This system of fixed dimension can be handled with tools from real algebraic geometry to derive a polynomial‐time algorithm. The characterization in terms of potential‐difference maximizing load flow scenarios then leads to a polynomial‐time algorithm for deciding the feasibility of a booking. Our theoretical results extend the existing knowledge about the complexity of deciding the feasibility of bookings from trees to single‐cycle networks.

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Chance‐constrained optimal inflow control in hyperbolic supply systems with uncertain demand

Göttlich

Kolb

Lux

2020

Optim Control Appl Methods

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Summary In this article, we address the task of setting up an optimal production plan taking into account an uncertain demand. The energy system is represented by a system of hyperbolic partial differential equations and the uncertain demand stream is captured by an Ornstein‐Uhlenbeck process. We determine the optimal inflow depending on the producer's risk preferences. The resulting output is intended to optimally match the stochastic demand for the given risk criteria. We use uncertainty quantification for an adaptation to different levels of risk aversion. More precisely, we use two types of chance constraints to formulate the requirement of demand satisfaction at a prescribed probability level. In a numerical analysis, we analyze the chance constrained optimization problem for the Telegrapher's equation and a real‐world coupled gas‐to‐power network.

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On the quantification of nomination feasibility in stationary gas networks with random load

Cited by 48 publications

References 14 publications

Decomposable robust two‐stage optimization: An application to gas network operations under uncertainty

Decomposable robust two‐stage optimization: An application to gas network operations under uncertainty

Deciding feasibility of a booking in the European gas market on a cycle is in P for the case of passive networks

Chance‐constrained optimal inflow control in hyperbolic supply systems with uncertain demand

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