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- A joint model of probabilistic/robust constraints for gas transport management in stationary networks (2017)
- We present a novel mathematical algorithm to assist gas network operators in managing uncertainty, while increasing reliability of transmission and supply. As a result, we solve an optimization problem with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise in the presence of uncertain parameters having partially stochastic and partially non-stochastic character. The application that drives this new approach is a stationary network with uncertain demand (which are stochastic due to the possibility of fitting statistical distributions based on historical measurements) and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of attainable measurements). We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is allowed (shaping a 'maximal' uncertainty set) around nominal roughness coefficients, such that random demands in a stationary gas network can be satisfied at given high probability level for no matter which realization of true roughness coefficients within the uncertainty set. One ends up with a constraint, which is probabilistic with respect to the load of gas and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions. The numerical solution of a corresponding optimization problem is illustrated. The results might assist the network operator with the implementation of cost-intensive roughness measurements.

- MIP-Based Instantaneous Control of Mixed-Integer PDE-Constrained Gas Transport Problems (2017)
- We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics that are described by the system of partial differential Euler equations. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.

- Two-Stage Stochastic Semidefinite Programming: Theory, Algorithms, and Application to AC Power Flow under Uncertainty (2017)
- In real life decision problems, one almost always is confronted with uncertainty and risk. For practical optimization problems this is manifested by unknown parameters within the input data, or, an inexact knowledge about the system description itself. In case the uncertain problem data is governed by a known probability distribution, stochastic programming offers a variety of models hedging against uncertainty and risk. Most widely employed are two-stage models, who admit a recourse structure: The first-stage decisions are taken before the random event occurs. After its outcome, a recourse (second-stage) action is made, often but not always understood as some "compensation". In the present thesis, the optimization problems that involve parameters which are not known with certainty are semidefinite programming problems. The constraint sets of these optimization problems are given by intersections of the cone of symmetric, positive semidefinite matrices with either affine or more general equations. Objective functions, formally, may be fairly general, although they often are linear as in the present thesis. We consider risk neutral and risk averse two-stage stochastic semidefinite programs with continuous and mixed-integer recourse, respectively. For these stochastic optimization problems we analyze their structure, derive solution methods relying on decomposition, and finally apply our results to unit commitment in alternating current (AC) power systems. Furthermore, deterministic unit commitment in AC power transmission systems is addressed. Beside traditional unit commitment constraints, the physics of power flow are included. To gain globally optimal solutions a recent semidefinite programming (SDP) approach is used which leads to large-scale semidefinite programs with discrete variables on top. As even the SDP relaxation of these programs is too large for being handled in an all-at-once manner by general SDP solvers, it requires an efficient and reliable method to tackle them. To this end, an algorithm based on Benders decomposition is proposed. With power demand (load) and in-feed from renewables serving as sources of uncertainty, two-stage stochastic programs are set up heading for unit commitment schedules which are both cost-effective and robust with respect to data perturbations. The impact of different, risk neutral and risk averse, stochastic criteria on the shapes of the optimal stochastic solutions will be examined. To tackle the resulting two-stage programs, we propose to approximate AC power flow by semidefinite relaxations. This leads to two-stage stochastic mixed-integer semidefinite programs having a special structure. To solve the latter, the L-shaped method and dual decomposition have been applied and compared.

- Unit commitment under uncertainty in AC transmission systems via risk averse semidefinite stochastic programs (2017)
- This paper addresses unit commitment under uncertainty of load and power infeed from renewables in alternating current (AC) power systems. Beside traditional unit-commitment constraints, the physics of power flow are included. To gain globally optimal solutions a recent semidefinite programming approach is used, which leads us to risk averse two-stage stochastic mixed integer semidefinite programs for which a decomposition algorithm is presented.

- Deciding Robust Feasibility and Infeasibility Using a Set Containment Approach: An Application to Stationary Passive Gas Network Operations (2017)
- In this paper we study feasibility and infeasibility of nonlinear two-stage fully adjustable robust feasibility problems with an empty first stage. This is equivalent to deciding set containment of a projection of the feasible region and the uncertainty set. For answering this question, two very general approaches using methods from polynomial optimization are presented --- one for showing feasibility and one for showing infeasibility. The developed methods are approximated through sum of squares polynomials and solved using semidefinite programs. Deciding robust feasibility and infeasibility is important for gas network operations, which is a \nonconvex quadratic problem with absolute values functions. Concerning the gas network problem, different topologies are considered. It is shown that a tree structured network can be decided exactly using linear programming. Furthermore, a method is presented to reduce a tree network with one additional arc to a single cycle network. In this case, removing the absolute values and solving the problem can be decided with linearly many polynomial optimization problems. Lastly, the effectivity of the methods is tested on a variety of small cyclic networks. For instances where robust feasibility or infeasibility can be decided, level~2 or level~3 of the Lasserre relaxation hierarchy is typically sufficient.

- Probability of Feasible Loads in Passive Gas Networks with up to Three Cycles (2017)
- Gas networks are of growing importance for the economy and offer interesting mathematical problems at the same time. The classical linear network ﬂow allows for approximate models that more and more have come to their limits. This has raised interest in nonlinear but, for simplicity, still steady-state models. The present paper aims at mobilizing techniques from symbolic computation and reparametrization of multivariate integrals to enable validation of stochastic nominations following Gaussian distributions in passive gas networks with more than one cycle.

- A Multilevel Model of the European Entry-Exit Gas Market (2017)
- In entry-exit gas markets as they are currently implemented in Europe, network constraints do not affect market interaction beyond the technical capacities determined by the TSO that restrict the quantities individual firms can trade at the market. It is an up to now unanswered question to what extent existing network capacity remains unused in an entry-exit design and to what extent feasible adjustments of the market design could alleviate inefficiencies. In this paper, we offer a four-level modeling framework that is capable of analyzing these issues and provide some first results on the model structure. In order to decouple gas trading from network congestion management, the TSO is required to determine technical capacities and corresponding booking fees at every entry and exit node up front. Firms book those capacities, which gives them the right to charge or discharge an amount of gas at a certain node up to this capacity in every period. Beyond these technical capacities and the resulting bookings, gas trade is unaffected by network constraints. The technical capacities have to ensure that transportation of traded quantities is always feasible. We assume that the TSO is regulated and determines technical capacities, fees, and transportation cost under a welfare objective. As a first step we moreover assume perfect competition among gas traders and show that the booking and nomination decisions can be analyzed in a single level. We prove that this aggregated model has a unique solution. We moreover show that the TSO's decisions can be subsumed in one level as well. If so, the model boils down to a mixed-integer nonlinear bilevel problem with robust aspects. In addition, we provide a first-best benchmark that allows to assess welfare losses that occur in an entry-exit system. Finally we discuss and provide guidance on how to include several important aspects into the approach, such as network and production capacity investment, uncertain data, market power, and intra-day trading.

- A System to Evaluate Gas Network Capacities: Concepts and Implementation (2017)
- Since 2005, the gas market in the European Union is liberalized and the trading of natural gas is decoupled from its transport. The transport is done by so-called transmissions system operators (TSOs). The market model established by the European Union views the gas transmission network as a black box, providing shippers (gas traders and consumers) the opportunity to transport gas from any entry to any exit. TSOs are required to offer maximum independent capacities at each entry and exit such that the resulting gas flows can be realized by the network without compromising security of supply. Therefore, evaluating the available transport capacities is extremely important to the TSOs. This paper gives an overview of the toolset for evaluating gas network capacities that has been developed within the ForNe project, a joint research project of seven research partners initiated by Open Grid Europe, Germany's biggest TSO. While most of the relevant mathematics is described in the book "Evaluating Gas Network Capacities", this article sketches the system as a whole, describes some developments that have taken place recently, and gives some details about the current implementation.

- Adaptive Refinement Strategies for the Simulation of Gas Flow in Networks using a Model Hierarchy (2017)
- A model hierarchy that is based on the one-dimensional isothermal Euler equations of fluid dynamics is used for the simulation and optimisation of gas flow through a pipeline network. Adaptive refinement strategies have the aim of bringing the simulation error below a prescribed tolerance while keeping the computational costs low. While spatial and temporal stepsize adaptivity is well studied in the literature, model adaptivity is a new field of research. The problem of finding an optimal refinement strategy that combines these three types of adaptivity is a generalisation of the unbounded knapsack problem. A refinement strategy that is currently used in gas flow simulation software is compared to two novel greedy-like strategies. Both a theoretical experiment and a realistic gas flow simulation show that the novel strategies significantly outperform the current refinement strategy with respect to the computational cost incurred.