Robust optimal discrete arc sizing for tree-shaped potential networks

Robinius, Martin; Schewe, Lars; Schmidt, Martin; Stolten, Detlef; Thürauf, Johannes; Welder, Lara

doi:10.1007/s10589-019-00085-x

Cited by 21 publications

(38 citation statements)

References 29 publications

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“…When only considering the sub-problem of verifying feasibility over trees, we have managed to improve this complexity to O(|V | 2 ). However, let us also remark that the main goal of [37] is different to ours and our setting is only contained there as a special case. Thus, it is reasonable that our tailored solution approach outperforms the one in [37].…”

Section: Since We Are Considering Trees We Havementioning

confidence: 95%

“…We finish this section with some remarks on the tree case. In [37], the authors consider the problem of robust discrete arc sizing for potential-based trees. This problem consists in determining optimal diameters for the pipes in a tree network such that a certain set of nominations is feasible.…”

Section: Since We Are Considering Trees We Havementioning

confidence: 99%

“…Further assuming that every pipe can take only a single diameter, namely the one in place in the existing network, we can verify the feasibility of the booking q b using the approach of robust discrete arc sizing. In Lemma 4.5 of [37] it is shown that this leads to a time complexity of O(|V | 4 ). When only considering the sub-problem of verifying feasibility over trees, we have managed to improve this complexity to O(|V | 2 ).…”

Section: Since We Are Considering Trees We Havementioning

confidence: 99%

“…of the sets of feasible nominations and bookings are established in [40]. Finally, an algorithm for solving the problem of robust discrete arc sizing is presented in [37]. Verifying the feasibility of bookings on a tree can be seen as a special case of this problem.…”

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: 95%

Section: Since We Are Considering Trees We Havementioning

confidence: 99%

Section: Since We Are Considering Trees We Havementioning

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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“…We introduce auxiliary variable ∈ R | pi | to model the pressure drop in Equations (38c), (38d). A closer look at (27) shows that the linear objective function can be written as c T − v + w using the coefficient vector…”

Section: Precomputation Of the Right-hand Side Of Problem (35) Using mentioning

confidence: 99%

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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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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Robust optimal discrete arc sizing for tree-shaped potential networks

Cited by 21 publications

References 29 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

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

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