Measure valued solutions of sub-linear diffusion equations with a drift term

In this paper we study nonnegative, measure valued solutions of the initial value problem for one-dimensional drift-diffusion equations when the nonlinear diffusion is governed by an increasing $C^1$ function $\beta$ with $\lim_{r\to +\infty} \beta(r)<+\infty$. By using tools of optimal transport, we will show that this kind of problems is well posed in the class of nonnegative Borel measures with finite mass $m$ and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the so called $L^2$-Wasserstein distance. Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass ${m}_{\rm c}$, which can be explicitely characterized in terms of $\beta$ and of the drift term. If the initial mass is less then ${m}_{\rm c}$, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure. Conversely, when the total mass $m$ of the solutions is greater than the critical one, the steady state has a singular part in which the exceeding mass ${m} - {m}_{\rm c}$ is accumulated.Comment: 30 page

Tracking power system events with accuracy-based PMU adaptive reporting rate

Fast dynamics and transient events are becoming more and more frequent in power systems, due to the high penetration of renewable energy sources and the consequent lack of inertia. In this scenario, Phasor Measurement Units (PMUs) are expected to track the monitored quantities. Such functionality is related not only to the PMU accuracy (as per the IEC/IEEE 60255-118-1 standard) but also to the PMU reporting rate (RR). High RRs allow tracking fast dynamics, but produce many redundant measurement data in normal conditions. In view of an effective tradeoff, the present paper proposes an adaptive RR mechanism based on a real-time selection of the measurements, with the target of preserving the information content while reducing the data rate. The proposed method has been tested considering real-world datasets and applied to four different PMU algorithms. The results prove the method effectiveness in reducing the average data throughput as well as its scalability at PMU concentrator or storage level

Supercapacitor Sizing for Fast Power Dips in a Hybrid Supercapacitor—PEM Fuel Cell System

Polymer electrolyte membrane fuel cell (PEM FC) operation is likely to be characterized by voltage dips on timescales shorter than 1 s, arising from temporary flooding of gas channels or porous layers, particularly when the FC is operated at high humidity levels. If supercapacitors are employed in hybrid systems, they can make up for the temporary lack of energy produced by the FC. However, the steep slopes of the voltage dips affect the energy that can be actually delivered by the supercapacitor because of its series impedance, and this should be taken into account when sizing it. This paper presents a simplified approach for sizing the supercapacitor, based on some observed peculiar features of the FC dips, which allow a simple but accurate model for the evaluation of the supercapacitor response to such dips. The validity of such an approach is supported by simulation and experimental results performed on a single PEM FC and on a supercapacitor

Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section

This paper focuses on the study of existence and uniqueness of distributional and classical solutions to the Cauchy Boltzmann problem for the soft potential case assuming $S^{n-1}$ integrability of the angular part of the collision kernel (Grad cut-off assumption). For this purpose we revisit the Kaniel--Shinbrot iteration technique to present an elementary proof of existence and uniqueness results that includes large data near a local Maxwellian regime with possibly infinite initial mass. We study the propagation of regularity using a recent estimate for the positive collision operator given in [3], by E. Carneiro and the authors, that permits to study such propagation without additional conditions on the collision kernel. Finally, an $L^{p}$-stability result (with $1\leq p\leq\infty$) is presented assuming the aforementioned condition.Comment: 19 page

Adaptive Polynomial Harmonic Distortion Compensation in Current and Voltage Transformers Through Iteratively Updated QR Factorization

Measuring current and voltage harmonics has paramount importance for improving the power quality of distribution grids. However, the achieved accuracy strongly depends on the adopted instrument transformer (IT). This article proposes an adaptive technique that enables an effective compensation of both the filtering behavior and the harmonic distortion (HD) introduced by current and voltage transformers (VTs), namely the strongest nonlinear effect at low-order harmonics. The approach is based on a flexible, linear in the parameters polynomial modeling of HD in the frequency domain. Model complexity can be different from one harmonic to the other, and it is selected through an automatic iterative process to suit the nonlinear behavior at each specific harmonic order, while avoiding overfitting. In particular, the number of parameters is increased by progressively updating the QR factorization of the regressor matrix trough Householder reflections until a convergence condition is reached. Experimental tests performed on an inductive VT and current transformer (CT) highlight the effectiveness of the approach

Self-similarity and power-like tails in nonconservative kinetic models

In this paper, we discuss the large--time behavior of solution of a simple kinetic model of Boltzmann--Maxwell type, such that the temperature is time decreasing and/or time increasing. We show that, under the combined effects of the nonlinearity and of the time--monotonicity of the temperature, the kinetic model has non trivial quasi-stationary states with power law tails. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution. The same idea is applied to investigate the large-time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and/or decreasing of the mean wealth. In this last case, the large-time behavior of the solution shows a Pareto power law tail. Numerical results confirm the previous analysis

Frequency-domain nonlinear modeling approaches for power systems components - A comparison

Harmonic simulations play a key role in studying and predicting the impact of nonlinear devices on the power quality level of distribution grids. A frequency-domain approach allows higher computational efficiency, which has key importance as long as complex networks have to be studied. However, this requires proper frequency-domain behavioral models able to represent the nonlinear voltage-current relationship characterizing these devices. The Frequency Transfer Matrix (FTM) method is one of the most widespread frequency domain modeling approaches for power system applications. However, others suitable techniques have been developed in the last years, in particular the X-parameters approach, which comes from radiofrequency and microwave applications, and the simplified Volterra models under quasi-sinusoidal conditions, that have been specifically tailored for power system devices. In this paper FTM, X-parameters and simplified Volterra approaches are compared in representing the nonlinear voltage-current relationship of a bridge rectifier feeding an ohmic-capacitive dc load. Results show that the X-parameters model reaches good accuracy, which is slightly better than that achieved by the FTM and simplified Volterra models, but with a considerably larger set of coefficients. Simplified Volterra models under quasi-sinusoidal conditions allows an effective trade-off between accuracy and complexity

On a kinetic model for a simple market economy

In this paper, we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading. We show that the kinetic model admits non trivial quasi-stationary states with power law tails of Pareto type. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution of wealth among individuals. For this equation the stationary state can be easily derived and shows a Pareto power law tail. Numerical results confirm the previous analysis

The role of pre-existing discontinuities in the development of extensional faults: an analog modeling perspective

Several mountainous regions are currently affected by syn- or post-orogenic active extension. We investigate how a newly-formed normal fault interacts with structures inherited from a previous contractional phase. To this end, we use analog models that adopt an innovative technique for performing a precut that mimics such inherited structures into a clay layer; this clay layer is laid on top of a master fault simulated by two rigid blocks sliding along an inclined plane. We carry out six experiments with variously oriented precuts and compare the results with those obtained in a reference isotropic experiment. All other conditions are identical for all seven realizations. Fault evolution is monitored by taking closely-spaced snapshots analyzed through the Digital Image Correlation method. We find that the upward propagation of the normal fault can be either accelerated or decelerated depending on the presence of a precut and its orientation. Such precuts can also promote or inhibit the formation of bending-moment faults. These interactions between master fault and precut also affect the shape of the fault-related syncline-anticline pair

A Low-Cost Approach to the Skin Effect Compensation in Cylindrical Shunts

In this paper the development of a new design solution for high-current shunt resistors is presented, which allows achieving very good accuracy while requiring a simple and low-cost manufacturing process. It is based on a solid cylinder having the voltage measurement circuit which runs through two holes drilled in the cylinder itself. Starting from the well-known expression of the current density in a cylindrical conductor, the frequency response of the shunt is obtained in closed form as a function of the geometric parameters. In turn, the positions of the voltage measurement terminals are chosen by optimizing the frequency response function over a specified range. A shunt prototype has been manufactured and its measurement performance has been evaluated. The experimental results confirm the validity of the approach and highlight the significant improvement with respect to the single-hole cylindrical shunt which has been recently proposed by the authors. The obtained measurement accuracy is noticeable when compared with the ease of manufacturing