arXiv subjects

Simone Göttlich

Publications and source records attributed to Simone Göttlich.

At least 55 records · Page 3Linked to original sources

arXiv2020

Second-order traffic flow models on networks

This paper deals with the Aw-Rascle-Zhang model for traffic flow on uni-directional road networks. For the conservation of the mass and the generalized momentum, we construct weak solutions for Riemann problems at the junctions. We particularly focus on a novel approximation to the homogenized pressure by introducing an additional equation for the propagation of a reference pressure. The resulting system of coupled conservation laws is then solved using an appropriate numerical scheme of Godunov type. Numerical simulations show that the proposed approximation is able to approximate the homogenized pressure sufficiently well. The difference of the new approach compared with the Lighthill-Whitham-Richards model is also illustrated.

math.NA↗

arXiv2020

Modelling arterial travel time distribution using copulas

The estimation of travel time distribution (TTD) is critical for reliable route guidance and provides theoretical bases and technical support for advanced traffic management and control. The state-of-the art procedure for estimating arterial TTD commonly assumes that the path travel time follows a certain distribution without considering segment correlation. However, this approach is usually unrealistic as travel times on successive segments may be dependent. In this study, copula functions are used to model arterial TTD as copulas are able to incorporate for segment correlation. First, segment correlation is empirically investigated using day-to-day GPS data provided by BMW Group for one major urban arterial in Munich, Germany. Segment TTDs are estimated using a finite Gaussian Mixture Model (GMM). Next, several copula models are introduced, namely Gaussian, Student-t, Clayton, and Gumbel, to model the dependent structure between segment TTDs. The parameters of each copula model are obtained by Maximum Log Likelihood Estimation. Then, path TTDs comprised of consecutive segment TTDs are estimated based on the copula models. The scalability of the model is evaluated by investigating the performance for an increasing number of aggregated links. The best fitting copula is determined in terms of goodness-of-fit test. The results demonstrate the advantage of the proposed copula model for an increasing number of aggregated segments, compared to the convolution without incorporating segment correlations.

stat.AP↗

arXiv2020

Properties of the LWR model with time delay

In this article, we investigate theoretical and numerical properties of the first-order Lighthill-Whitham-Richards (LWR) traffic flow model with time delay. Since standard results from the literature are not directly applicable to the delayed model, we mainly focus on the numerical analysis of the proposed finite difference discretization. The simulation results also show that the delay model is able to capture Stop & Go waves.

math.NA↗

arXiv2020

Modeling and simulation of sector-coupled networks: A gas-power benchmark

In this contribution, we aim at presenting a gas-to-power benchmark problem that can be used for the simulation of electricity and gas networks in a time-dependent environment. Based on realistic data from the IEEE database and the GasLib suite, we describe the full set up of the underlying equations and motivate the choice of parameters. The simulation results demonstrate the applicability of the proposed approach and also allow for a clear visualization of gas-power conversion.

eess.SY↗

arXiv2020

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

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.

math.OC↗

arXiv2019

The global classical solution to compressible Euler system with velocity alignment

In this paper, the compressible Euler system with velocity alignment and damping is considered, where the influence matrix of velocity alignment is not positive definite. Sound speed is used to reformulate the system into symmetric hyperbolic type. The global existence and uniqueness of smooth solution for small initial data is provided.

math.AP↗

arXiv2019

Modeling random traffic accidents by conservation laws

We introduce a stochastic traffic flow model to describe random traffic accidents on a single road. The model is a piecewise deterministic process incorporating traffic accidents and is based on a scalar conservation law with space-dependent flux function. Using a Lax-Friedrichs discretization, we show that the total variation is bounded in finite time and provide a theoretical framework to embed the stochastic process. Additionally, a solution algorithm is introduced to also investigate the model numerically.

math.PR↗

arXiv2019

Car Path Tracking in Traffic Flow Networks with Bounded Buffers at Junctions

This article deals with the modeling for an individual car path through a road network, where the dynamics is driven by a coupled system of ordinary and partial differential equations. The network is characterized by bounded buffers at junctions that allow for the interpretation of roundabouts or on-ramps while the traffic dynamics is based on first-order macroscopic equations of Lighthill-Whitham-Richards (LWR) type. Trajectories for single drivers are then influenced by the surrounding traffic and can be tracked by appropriate numerical algorithms. The computational experiments show how the modeling framework can be used as navigation device.

math.NA↗

arXiv2019

The Mean Field Kinetic Equation for Interacting Particle Systems with Non-Lipschitz Force

In this paper, we prove the global existence of the weak solution to the mean field kinetic equation derived from the $N$-particle Newtonian system. For $L^1\cap L^\infty$ initial data, the solvability of the mean field kinetic equation can be obtained by using uniform estimates and compactness arguments while the difficulties arising from the non-local non-linear interaction are tackled appropriately using the Aubin-Lions compact embedding theorem.

math.AP↗

arXiv2019

Uncertainty quantification with risk measures in production planning

This paper is concerned with a simulation study for a stochastic production network model, where the capacities of machines may change randomly. We introduce performance measures motivated by risk measures from finance leading to a simulation based optimization framework for the production planning. The same measures are used to investigate the scenario when capacities are related to workers that are randomly not available. This corresponds to the study of a workforce planning problem in an uncertain environment.

math.OC↗

arXiv2019

Feedback Stabilization for a coupled PDE-ODE Production System

We consider an interlinked production model consisting of conservation laws (PDE) coupled to ordinary differential equations (ODE). Our focus is the analysis of control laws for the coupled system and corresponding stabilization questions of equilibrium dynamics in the presence of disturbances. These investigations are carried out using an appropriate Lyapunov function on the theoretical and numerical level. The discrete $L^2-$stabilization technique allows to derive a mixed feedback law that is able to ensure exponential stability also in bottleneck situations. All results are accompanied by computational examples.

math.OC↗

arXiv2019

Optimal control of compressor stations in a coupled gas-to-power network

We introduce a tool for simulation and optimization of gas pipeline networks coupled to power grids by gas-to-power plants. The model under consideration consists of the isentropic Euler equations to describe the gas flow coupled to the AC powerflow equations. A compressor station is installed to control the gas pressure such that certain bounds are satisfied. A numerical case study is presented that showcases effects of fast changes in power demand on gas pipelines and necessary operator actions.

math.OC↗

arXiv2019

Well-posedness of a non-local model for material flow on conveyor belts

In this paper, we focus on finite volume approximation schemes to solve a non-local material flow model in two space dimensions. Based on the numerical discretisation with dimensional splitting, we prove the convergence of the approximate solutions, where the main difficulty arises in the treatment of the discontinuity occurring in the flux function. In particular, we compare a Roe-type scheme to the well-established Lax-Friedrichs method and provide a numerical study highlighting the benefits of the Roe discretisation. Besides, we also prove the L1-Lipschitz continuous dependence on the initial datum, ensuring the uniqueness of the solution.

math.AP↗

arXiv2019

A Production Model with History Based Random Machine Failures

In this paper, we introduce a time-continuous production model that enables random machine failures, where the failure probability depends historically on the production itself. This bidirectional relationship between historical failure probabilities and production is mathematically modeled by the theory of piecewise deterministic Markov processes (PDMPs). On this way, the system is rewritten into a Markovian system such that classical results can be applied. In addition, we present a suitable solution, taken from machine reliability theory, to connect past production and the failure rate. Finally, we investigate the behavior of the presented model numerically in examples by considering sample means of relevant quantities and relative frequencies of number of repairs.

math.PR↗

arXiv2019

Optimal inflow control penalizing undersupply in transport systems with uncertain demands

We are concerned with optimal control strategies subject to uncertain demands. An Ornstein-Uhlenbeck process describes the uncertain demand. The transport within the supply system is modeled by the linear advection equation. We consider different approaches to control the produced amount at a given time to meet the stochastic demand in an optimal way. In particular, we introduce an undersupply penalty and analyze its effect on the optimal output in a numerical simulation study.

math.OC↗

arXiv2019

Validation of Flow Models as New Simulation Approach for Parcel Handling in Bulk Mode

Flow Models based on hyperbolic partial differential equations (conservation laws) are a well-established approach for the material flow simulation. Until now, they have been exclusively used for the simulation of cylindrical cargo. This paper investigates the application of flow models on cubical cargo as a new simulation approach for parcel logistics. Selected flow model parameters are adapted to cover this new situation. The simulation results of this macroscopic model are compared with the microscopic Discrete Element Method (DEM) where the cargo is approximated by superquadrics. An experimental setup especially designed for the validation of the considered flow model for cubical cargo bulk flow is presented. Results of the flow model are analyzed and validated against the results of experiment and DEM for the test setting.

math.NA↗

arXiv2018

Modeling and simulation of gas networks coupled to power grids

In this paper, a mathematical framework for the coupling of gas networks to electric grids is presented to describe in particular the transition from gas to power. The dynamics of the gas flow are given by the isentropic Euler equations, while the power flow equations are used to model the power grid. We derive pressure laws for the gas flow that allow for the well-posedness of the coupling and a rigorous treatment of solutions. For simulation purposes, we apply appropriate numerical methods and show in a experimental study how gas-to-power might influence the dynamics of the gas and power network, respectively.

math.NA↗

arXiv2018

Continuous approximation of $M_t/M_t/1$ distributions with application to production

A single queueing system with time-dependent exponentially distributed arrival processes and exponential machine processes (Kendall notation $M_t/M_t/1$) is analyzed. Modeling the time evolution for the discrete queue-length distribution by a continuous drift-diffusion process a Smoluchowski equation on the half space is derived approximating the forward Kolmogorov equations. The approximate model is analyzed and validated, showing excellent agreement for the probabilities of all queue lengths and for all queuing utilizations, including ones that are very small and some that are significantly larger than one. Having an excellent approximation for the probability of an empty queue generates an approximation of the expected outflow of the queueing system. Comparisons to several well-established approximation from the literature show significant improvements in several numerical examples.

math.PR↗