555 research outputs found
Consequences of uncertain friction for the transport of natural gas through passive networks of pipelines
Assuming a pipe-wise constant structure of the friction coefficient in
the modeling of natural gas transport through a passive network of pipes via
semilinear systems of balance laws with associated linear coupling and
boundary conditions, uncertainty in this parameter is quantified by a Markov
chain Monte Carlo method. Here, information on the prior distribution is
obtained from practitioners. The results are applied to the problem of
validating technical feasibility under random exit demand in gas transport
networks. In particular, the impact of quantified uncertainty to the
probability level of technical feasible exit demand situations is studied by
two example networks of small and medium size. The gas transport of the
network is modeled by stationary solutions that are steady states of the time
dependent semilinear problems
Consequences of uncertain friction for the transport of natural gas through passive networks of pipelines
Assuming a pipe-wise constant structure of the friction coefficient in the modeling of natural gas transport through a passive network of pipes via semilinear systems of balance laws with associated linear coupling and boundary conditions, uncertainty in this parameter is quantified by a Markov chain Monte Carlo method. Here, information on the prior distribution is obtained from practitioners. The results are applied to the problem of validating technical feasibility under random exit demand in gas transport networks. In particular, the impact of quantified uncertainty to the probability level of technical feasible exit demand situations is studied by two example networks of small and medium size. The gas transport of the network is modeled by stationary solutions that are steady states of the time dependent semilinear problems
On probabilistic capacity maximization in a stationary gas network
The question for the capacity of a given gas network, i.e., determining the maximal amount of gas that can be transported by a given network, appears as an essential question that network operators and political administrations are regularly faced with. In that context we present a novel mathematical approach to assist gas network operators in managing uncertainty with respect to the demand and in exposing free network capacities while increasing reliability of transmission and supply. The approach is based on the rigorous examination of optimization problems with nonlinear probabilistic constraints. As consequence we deal with solving an optimization problem with joint probabilistic constraints over an infinite system of random inequalities. We will show that the inequality system can be reduced to a finite one in the situation of considering a tree network topology. A detailed study of the problem of maximizing free booked capacities in a stationary gas network is presented that comes up with an algebraic model involving Kirchhoff's first and second laws. The focus will be on both the theoretical and numerical side. We are going to validate a kind of rank two constraint qualification implying the differentiability of the considered capacity problem. At the numerical side we are going to solve the problem using a projected gradient decent method, where the function and gradient evaluations of the probabilistic constraints are performed by the approach of spheric-radial decomposition applied for multivariate Gaussian random variables and more general distributions
On probabilistic capacity maximization in a stationary gas network
The question for the capacity of a given gas network, i.e., determining
the maximal amount of gas that can be transported by a given network, appears
as an essential question that network operators and political administrations
are regularly faced with. In that context we present a novel the demand and
in exposing free network capacities while increasing reliability of
transmission and supply. The approach is based on the rigorous examination of
optimization problems with nonlinear probabilistic constraints. As
consequence we deal with solving an optimization problem with joint
probabilistic constraints over an infinite system of random inequalities. We
will show that the inequality system can be reduced to a finite one in the
situation of considering a tree network topology. A detailed study of the
problem of maximizing free booked capacities in a stationary gas network is
presented that comes up with an algebraic model involving Kirchhoffs first
and second laws. The focus will be on both the theoretical and numerical
side. We are going to validate a kind of rank two constraint qualification
implying the differentiability of the considered capacity problem. At the
numerical side we are going to solve the problem using a projected gradient
decent method, where the function and gradient evaluations of the
probabilistic constraints are performed by the approach of spheric-radial
decomposition applied for multivariate Gaussian random variables and more
general distributions
Joint model of probabilistic-robust (probust) constraints with application to gas network optimization
Optimization problems under uncertain conditions abound in many real-life applications. While solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually applied in case solutions are sought that are feasible for all realizations of uncertainties within some predefined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we introduce a class of joint probabilistic/robust constraints. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for finding their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound with high probability
Joint Model of Probabilistic-Robust (Probust) Constraints Applied to Gas Network Optimization
Optimization problems under uncertain conditions abound in many real-life applications. While solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually applied in case solutions are sought that are feasible for all realizations of uncertainties within some predefined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we deal with a class of joint probabilistic/robust constraints. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for finding their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound with high probability
On convex lower-level black-box constraints in bilevel optimization with an application to gas market models with chance constraints
Bilevel optimization is an increasingly important tool to model hierarchical decision making. However, the ability of modeling such settings makes bilevel problems hard to solve in theory and practice. In this paper, we add on the general difficulty of this class of problems by further incorporating convex black-box constraints in the lower level. For this setup, we develop a cutting-plane algorithm that computes approximate bilevel-feasible points. We apply this method to a bilevel model of the European gas market in which we use a joint chance constraint to model uncertain loads. Since the chance constraint is not available in closed form, this fits into the black-box setting studied before. For the applied model, we use further problem-specific insights to derive bounds on the objective value of the bilevel problem. By doing so, we are able to show that we solve the application problem to approximate global optimality. In our numerical case study we are thus able to evaluate the welfare sensitivity in dependence of the achieved safety level of uncertain load coverage
Solving joint chance constrained problems using regularization and Benders' decomposition
In this paper we investigate stochastic programms with joint chance constraints. We consider discrete scenario set and reformulate the problem by adding auxiliary variables. Since the resulting problem has a difficult feasible set, we regularize it. To decrease the dependence on the scenario number, we propose a numerical method by iteratively solving a master problem while adding Benders cuts. We find the solution of the slave problem (generating the Benders cuts) in a closed form and propose a heuristic method to decrease the number of cuts. We perform a numerical study by increasing the number of scenarios and compare our solution with a solution obtained by solving the same problem with continuous distribution
Deciding Feasibility of a Booking in the European Gas Market on a Cycle is in P for the Case of Passive Networks
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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