On Stability of a Distributed Averaging PI Frequency and Active Power Controlled Differential-Algebraic Power System Model

We consider the problems of stability, frequency restoration and optimal steady-state resource allocation in a heterogeneous and structure-preserving differential-algebraic equation (DAE) power system model. Thereby, we include constant-power-controlled loads (CPCLs) and constant-power-controlled sources (CPCSs) explicitly in the analysis and network control design. This results in a power system model with mixed algebraic as well as first- and second-order differential dynamics. We show that the abovementioned control objectives can be achieved via a distributed averaging proportional integral (DAPI) control and, in particular, extend the stability proof in [1] to the resulting closed-loop DAE system

Comparison of an h- and hp-Adaptive Finite Element Solver for Chemo-Mechanically Coupled Battery Electrode Particles

Numerical investigations of mechanical stresses for phase transforming battery electrode materials on the particle scale are computationally highly demanding. The limitations are mainly induced by the strongly varying spatial and temporal scales of the underlying phase field model, which require an ultra fine mesh and time resolution, however, solely at specific stages in space and time. To overcome these numerical difficulties we present a general-purpose space and time adaptive solution algorithm based on an hp-adaptive finite element method and a variable-step, variable-order time integrator. At the example of a chemo-mechanical electrode particle model we demonstrate the computational savings gained by the hp-adaptivity. In particular, we compare the results to an h-adaptive finite element method and show the reduction of computational complexity

Simulation of the Deformation for Cycling Chemo-Mechanically Coupled Battery Active Particles with Mechanical Constraints

Next-generation lithium-ion batteries with silicon anodes have positive characteristics due to higher energy densities compared to state-of-the-art graphite anodes. However, the large volume expansion of silicon anodes can cause high mechanical stresses, especially if the battery active particle cannot expand freely. In this article, a thermodynamically consistent continuum model for coupling chemical and mechanical effects of electrode particles is extended by a change in the boundary condition for the displacement via a variational inequality. This switch represents a limited enlargement of the particle swelling or shrinking due to lithium intercalation or deintercalation in the host material, respectively. For inequality constraints as boundary condition a smaller time step size is need as well as a locally finer mesh. The combination of a primal-dual active set algorithm, interpreted as semismooth Newton method, and a spatial and temporal adaptive algorithm allows the efficient numerical investigation based on a finite element method. Using the example of silicon, the chemical and mechanical behavior of one- and two-dimensional representative geometries for a charge-discharge cycle is investigated. Furthermore, the efficiency of the adaptive algorithm is demonstrated. It turns out that the size of the gap has an significant influence on the maximal stress value and the slope of the increase. Especially in two dimension, the obstacle can cause an additional region with a lithium-poor phase

Simulation of the Deformation for Cycling Chemo-Mechanically Coupled Battery Active Particles with Mechanical Constraints

Next-generation lithium-ion batteries with silicon anodes have positive characteristics due to higher energy densities compared to state-of-the-art graphite anodes. However, the large volume expansion of silicon anodes can cause high mechanical stresses, especially if the battery active particle cannot expand freely. In this article, a thermodynamically consistent continuum model for coupling chemical and mechanical effects of electrode particles is extended by a change in the boundary condition for the displacement via a variational inequality. This switch represents a limited enlargement of the particle swelling or shrinking due to lithium intercalation or deintercalation in the host material, respectively. For inequality constraints as boundary condition a smaller time step size is need as well as a locally finer mesh. The combination of a primal-dual active set algorithm, interpreted as semismooth Newton method, and a spatial and temporal adaptive algorithm allows the efficient numerical investigation based on a finite element method. Using the example of silicon, the chemical and mechanical behavior of one- and two-dimensional representative geometries for a charge-discharge cycle is investigated. Furthermore, the efficiency of the adaptive algorithm is demonstrated. It turns out that the size of the gap has an significant influence on the maximal stress value and the slope of the increase. Especially in two dimension, the obstacle can cause an additional region with a lithium-poor phase

Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators

We discuss several adaptive mesh-refinement strategies based on (h − h/2)-error estimation. This class of adaptivemethods is particularly popular in practise since it is problem independent and requires virtually no implementational overhead. We prove that, under the saturation assumption, these adaptive algorithms are convergent. Our framework applies not only to finite element methods, but also yields a first convergence proof for adaptive boundary element schemes. For a finite element model problem, we extend the proposed adaptive scheme and prove convergence even if the saturation assumption fails to hold in general

Rotational state-changing collisions between N2+_2^+ and Rb at low energies

We present a theoretical study of rotationally elastic and inelastic collisions between molecular nitrogen ions and Rb atoms in the sub-Kelvin temperature regime prevalent in ion-atom hybrid trapping experiments. The cross sections for rotational excitation and de-excitation collisions were calculated using quantum-scattering methods on ab-initio potential energy surfaces for the energetically lowest singlet electronic channel of the system. We find that the rotationally inelastic collision rates are at least an order of magnitude smaller than the charge-exchange rates found in this system, rendering inelastic processes a minor channel under the conditions of typical hybrid trapping experiments.Comment: 6 pages, 5 figures, Computational study of rotational state changing collision

On Metric Dimension of Functigraphs

The \emph{metric dimension} of a graph $G$, denoted by $\dim(G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$ and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid v=f(u)\}$. We study how metric dimension behaves in passing from $G$ to $C(G,f)$ by first showing that $2 \le \dim(C(G, f)) \le 2n-3$, if $G$ is a connected graph of order $n \ge 3$ and $f$ is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.Comment: 10 pages, 7 figure

In situ Raman spectroscopy on silicon nanowire anodes integrated in lithium ion batteries

Rapid decay of silicon anodes during lithiation poses a significant challenge in application of silicon as an anode material in lithium ion batteries. In situ Raman spectroscopy is a powerful method to study the relationship between structural and electrochemical data during electrode cycling and to allow the observation of amorphous as well as liquid and transient species in a battery cell. Herein, we present in situ Raman spectroscopy on high capacity electrode using uncoated and carbon-coated silicon nanowires during first lithiation and delithiation cycle in an optimized lithium ion battery setup and complement the results with operando X-ray reflection diffraction measurements. During lithiation, we were able to detect a new Raman signal at 1859 cm−1 especially on uncoated silicon nanowires. The detailed in situ Raman measurement of the first lithiation/delithiation cycle allowed to differentiate between morphology changes of the electrode as well as interphase formation from electrolyte components