Optimal Control of Compressor Stations in a Coupled Gas-to-Power Network

Fokken, Eike; Göttlich, Simone; Kolb, Oliver

doi:10.1007/978-3-030-32157-4_5

Cited by 3 publications

(15 citation statements)

References 16 publications

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“…The nonlinear gas transport can be described by the so‐called isentropic Euler equations . They can be obtained from Equation () by putting in the parameters of GtP_a 4.3 of Table 2:

\begin{align} {((), \begin{matrix} subarray \\ array ρ^{e} \\ array q^{e} \end{matrix})}_{t} & + {((), \begin{matrix} subarray \\ array q^{e} \\ array p (ρ^{e}) + \frac{{(q^{e})}^{2}}{ρ^{e}} \end{matrix})}_{x} = ((), \begin{matrix} subarray \\ array 0 \\ array g (ρ^{e}, q^{e}) \end{matrix}), \end{align}

\begin{align} ρ^{e} false(x, 0 false) & = ρ_{0}^{e} false(x false), q^{e} false(x, 0 false) = q_{0}^{e} false(x false) . \end{align}

Thereby, q 1 e is denoted by

ρ^{e}

and describes the density, q 2 e is identified with q and gives the flow, g a given source term chosen as in Reference 63, and p e is the pressure on edge e , which is specified by the pressure law

p^{e} (ρ^{e}) = d^{2} \cdot ρ^{β}

for all edges e (with

β = 1

and

d = 340 \frac{m}{s}

in t...…”

Section: Numerical Resultsmentioning

confidence: 99%

“…To ensure the well‐posedness of the system, we also need coupling conditions at the nodes. As in Reference 63, we use pressure equality and mass conservation at all nodes at all times (Kirchhoff‐type‐coupling), that is, for all

v \in 𝒱

and for all t ∈ [0, T ], we have

\begin{align} p_{i n}^{v} false(t false) & = p_{o u t}^{v} false(t false), \\ \sum_{e \in δ^{prefix-} false(v false)} q^{e} false(t false) & = \sum_{\overset{e}{true} \in δ^{+} false(v false)} q^{\overset{e}{true}} false(t false) . \end{align}

To couple the gas network to the power system, we adapt the deterministic coupled gas‐to‐power‐system described in Reference 63, and extend it by an uncertain power demand given by the OUP (). The adapted setting is sketched in Figure 5.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…It can be compensated by a compressor station. The modeling of the compressor station is taken from Reference 63: the compressor is modeled as edge e 1 with time‐independent in‐ and outgoing pressure

p^{v_{0}}

and

p^{v_{1}}

, and flux values

q^{v_{0}}

, and

q^{v_{1}}

through the nodes v 0 and v 1 . We assume that the compressor is run via an external power source only linked to the scenario in Figure 5 via the cost component C 1 in () accounting for operator costs of the compressor.…”

Section: Numerical Resultsmentioning

confidence: 99%

See 2 more Smart Citations

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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“…The nonlinear gas transport can be described by the so‐called isentropic Euler equations . They can be obtained from Equation () by putting in the parameters of GtP_a 4.3 of Table 2:

\begin{align} {((), \begin{matrix} subarray \\ array ρ^{e} \\ array q^{e} \end{matrix})}_{t} & + {((), \begin{matrix} subarray \\ array q^{e} \\ array p (ρ^{e}) + \frac{{(q^{e})}^{2}}{ρ^{e}} \end{matrix})}_{x} = ((), \begin{matrix} subarray \\ array 0 \\ array g (ρ^{e}, q^{e}) \end{matrix}), \end{align}

\begin{align} ρ^{e} false(x, 0 false) & = ρ_{0}^{e} false(x false), q^{e} false(x, 0 false) = q_{0}^{e} false(x false) . \end{align}

Thereby, q 1 e is denoted by

ρ^{e}

and describes the density, q 2 e is identified with q and gives the flow, g a given source term chosen as in Reference 63, and p e is the pressure on edge e , which is specified by the pressure law

p^{e} (ρ^{e}) = d^{2} \cdot ρ^{β}

for all edges e (with

β = 1

and

d = 340 \frac{m}{s}

in t...…”

Section: Numerical Resultsmentioning

confidence: 99%

v \in 𝒱

and for all t ∈ [0, T ], we have

\begin{align} p_{i n}^{v} false(t false) & = p_{o u t}^{v} false(t false), \\ \sum_{e \in δ^{prefix-} false(v false)} q^{e} false(t false) & = \sum_{\overset{e}{true} \in δ^{+} false(v false)} q^{\overset{e}{true}} false(t false) . \end{align}

Section: Numerical Resultsmentioning

confidence: 99%

p^{v_{0}}

and

p^{v_{1}}

, and flux values

q^{v_{0}}

, and

q^{v_{1}}

Section: Numerical Resultsmentioning

confidence: 99%

See 1 more Smart Citation

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

Göttlich

Kolb

Lux

2020

Optim Control Appl Methods

Self Cite

View full text Add to dashboard Cite

show abstract

“…To ensure the well-posedness of the system, we also need coupling conditions at the nodes. As in [17], we use pressure equality and mass conservation at all nodes at all times (Kirchhoff-typecoupling), i.e. for all v ∈ V and for all t ∈ [0, T ], we have…”

Section: Nonlinear System Of Hyperbolic Balance Laws: Gas-to-power Symentioning

confidence: 99%

“…where ∆t is the discretization step. The first passage time density obtained by the iterative procedure(17) converges to the true first passage time density as the step size tends to zero, i.e. lim ∆t→0 |g(t k ) −g(t k )| = 0 for all k ∈ {1, · · · , N } To use the iteratively approximated first passage time density to obtain the risk level corresponding to S(t), we apply Algorithm 3.1.…”

mentioning

confidence: 99%

Optimal Inflow Control Penalizing Undersupply in Transport Systems with Uncertain Demands

Göttlich¹,

Korn²,

Lux³

2019

Progress in Industrial Mathematics at ECMI 2018

Self Cite

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In this paper, 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 (PDEs) 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.

show abstract

Network and Storage

Schwarz

Lacalandra²,

Schewe

et al. 2020

AIRO Springer Series

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Optimal Control of Compressor Stations in a Coupled Gas-to-Power Network

Cited by 3 publications

References 16 publications

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

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

Optimal Inflow Control Penalizing Undersupply in Transport Systems with Uncertain Demands

Network and Storage

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