Algorithmic results for potential‐based flows: Easy and hard cases

Groß, Martin; Pfetsch, Marc E.; Schewe, Lars; Schmidt, Martin; Skutella, Martin

doi:10.1002/net.21865

Cited by 24 publications

(18 citation statements)

References 40 publications

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“…It is sufficient to replace the corresponding quantities by using Corollary 19 and (14). As a final remark, it is also interesting to point out that the results of this section can easily be extended to series-parallel networks, since they can be reduced to trees [19] in our potential-based setting.…”

Section: Since We Are Considering Trees We Havementioning

confidence: 93%

“…This potential function links the flow on an arc with the potentials on its endpoints. It can be defined as follows; see, e.g., [19].…”

Section: Main Definitions and Notationmentioning

confidence: 99%

“…We refer to [19], where this example is discussed in depth. (ii) Water transport networks: Hydraulic heads in water networks can also be interpreted as potentials.…”

Section: Main Definitions and Notationmentioning

confidence: 99%

“…Furthermore, we choose a rather abstract physics model by considering a steadystate potential-based flow model. These models have been introduced in [5] and can, besides gas flow, also be applied to model water or power transport networks [19,39]. They are governed by Kirchhoff's first law and a special variant of the second law [24].…”

Section: Introductionmentioning

confidence: 99%

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Bookings in the European gas market: characterisation of feasibility and computational complexity results

Plein

Schmidt

2019

Optim Eng

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As a consequence of the liberalisation of the European gas market in the last decades, gas trading and transport have been decoupled. At the core of this decoupling are so-called bookings and nominations. Bookings are special capacity right contracts that guarantee that a specified amount of gas can be supplied or withdrawn at certain entry or exit nodes of the network. These supplies and withdrawals are nominated at the day-ahead. The special property of bookings then is that they need to be feasible, i.e., every nomination that complies with the given bookings can be transported. While checking the feasibility of a nomination can typically be done by solving a mixed-integer nonlinear feasibility problem, the verification of feasibility of a set of bookings is much harder. The reason is the robust nature of feasibility of bookingsnamely that for a set of bookings to be feasible, all compliant nominations, i.e., infinitely many, need to be checked for feasibility. In this paper, we consider the question of how to verify the feasibility of given bookings for a number of special cases. For our physics model we impose a steady-state potential-based flow model and disregard controllable network elements. For this case we derive a characterisation of feasible bookings, which is then used to show that the problem is in coNP for the general case but can be solved in polynomial time for linear potential-based flow models. Moreover, we present a dynamic programming approach for deciding the feasibility of a booking in tree-shaped networks even for nonlinear flow models. It turns out that the hardness of the problem mainly depends on the combination of the chosen physics model as well as the specific network structure under consideration. Thus, we give an overview over all settings for which the hardness of the problem is known and finally present a list of open problems.Date: June 5, 2019. 2010 Mathematics Subject Classification. 90B10, 90C30, 90C35, 90C39, 90C90.

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Section: Since We Are Considering Trees We Havementioning

confidence: 93%

“…This potential function links the flow on an arc with the potentials on its endpoints. It can be defined as follows; see, e.g., [19].…”

Section: Main Definitions and Notationmentioning

confidence: 99%

“…We refer to [19], where this example is discussed in depth. (ii) Water transport networks: Hydraulic heads in water networks can also be interpreted as potentials.…”

Section: Main Definitions and Notationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bookings in the European gas market: characterisation of feasibility and computational complexity results

Plein

Schmidt

2019

Optim Eng

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“…The nonlinear model used in this study for minimizing the pipeline costs is developed and explained in detail in Robinius et al [17]. The pressure drop is considered by using the Weymouth equation [18,19,[24][25][26] for the approximated relationship between mass flow, diameter, and gas pressures. As the pipeline investment costs are mainly dependent on the pipeline diameters, the overall system costs are optimized by selecting the diameters.…”

Section: Pipeline Design Modelingmentioning

confidence: 99%

Modeling hydrogen networks for future energy systems: A comparison of linear and nonlinear approaches

Reuß

Welder

Thürauf

et al. 2019

International Journal of Hydrogen Energy

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Common energy system models that integrate hydrogen transport in pipelines typically simplify fluid flow models and reduce the network size in order to achieve solutions quickly. This contribution analyzes two different types of pipeline network topologies (namely, star and tree networks) and two different fluid flow models (linear and nonlinear) for a given hydrogen capacity scenario of electrical reconversion in Germany to analyze the impact of these simplifications. For each network topology, robust demand and supply scenarios are generated. The results show that a simplified topology, as well as the consideration of detailed fluid flow, could heavily influence the total pipeline investment costs. For the given capacity scenario, an overall cost reduction of the pipeline costs of 37% is observed for the star network with linear cost compared to the tree network with nonlinear fluid flow. The impact of these improvements regarding the total electricity reconversion costs has led to a cost reduction of 1.4%, which is fairly small. Therefore, the integration of nonlinearities into energy system optimization models is not recommended due to their high computational burden. However, the applied method for generating robust demand and supply scenarios improved the credibility and robustness of the network topology, while the simplified fluid flow consideration can lead to infeasibilities. Thus, we suggest the utilization of the nonlinear model for postprocessing to prove the feasibility of the results and strengthen their credibility, while retaining the computational performance of linear modeling.

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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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Algorithmic results for potential‐based flows: Easy and hard cases

Cited by 24 publications

References 40 publications

Bookings in the European gas market: characterisation of feasibility and computational complexity results

Bookings in the European gas market: characterisation of feasibility and computational complexity results

Modeling hydrogen networks for future energy systems: A comparison of linear and nonlinear approaches

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

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