The Fokker-Planck Equation for Power System Stability Probability Density Function Evolution

2013 ◽  
Vol 28 (3) ◽  
pp. 2994-3001 ◽  
Author(s):  
Keyou Wang ◽  
Mariesa L. Crow
Keyword(s):  
Probability Density Function ◽  
Power System ◽  
Probability Density ◽  
Density Function ◽  
Planck Equation ◽  
Power System Stability ◽  
System Stability ◽  
Fokker Planck Equation ◽  
Fokker Planck

Nonstationary Response Envelope Probability Densities of Nonlinear Oscillators

2006 ◽  
Vol 74 (2) ◽  
pp. 315-324 ◽  
Author(s):  
P. D. Spanos ◽  
A. Sofi ◽  
M. Di Paola
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Planck Equation ◽  
Nonlinear Oscillators ◽  
Basis Functions ◽  
Time Dependent ◽  
Fokker Planck Equation ◽  
Response Envelope ◽  
Fokker Planck

The nonstationary random response of a class of lightly damped nonlinear oscillators subjected to Gaussian white noise is considered. An approximate analytical method for determining the response envelope statistics is presented. Within the framework of stochastic averaging, the procedure relies on the Markovian modeling of the response envelope process through the definition of an equivalent linear system with response-dependent parameters. An approximate solution of the associated Fokker-Planck equation is derived by resorting to a Galerkin scheme. Specifically, the nonstationary probability density function of the response envelope is expressed as the sum of a time-dependent Rayleigh distribution and of a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. These functions are the eigenfunctions of the boundary-value problem associated with the Fokker-Planck equation governing the evolution of the probability density function of the response envelope of a linear oscillator. The selected basis functions possess some notable properties that yield substantial computational advantages. Applications to the Van der Pol and Duffing oscillators are presented. Appropriate comparisons to the data obtained by digital simulation show that the method, being nonperturbative in nature, yields reliable results even for large values of the nonlinearity parameter.

scholarly journals Investigation of Phase-Locked Loop Statistics via Numerical Implementation of the Fokker–Planck Equation

2020 ◽  
Vol 10 (7) ◽  
pp. 2625
Author(s):  
Dah-Jing Jwo
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Additive White Gaussian Noise ◽  
Phase Error ◽  
Planck Equation ◽  
Phase Locked Loop ◽  
Comprehensive Treatment ◽  
Error Statistics ◽  
Fokker Planck

The goal of this paper is to explore the effect of various parameters on the information geometric structure of the phase-locked loop (PLL) statistics, both transient and stationary. Comprehensive treatment on the behavior of PLL statistics will be given. The behavior of the phase-error statistics of the first-order PLL, in the presence of additive white Gaussian noise (WGN) is investigated through solving the differential equations known as the Fokker–Planck (FP) equation using the implicit Crank–Nicolson finite-difference method. The PLL is one of the most commonly used circuits in electrical engineering. A full knowledge of probability density functions (PDFs) of the phase-error statistics becomes essential in understanding the PLLs. Several illustrative examples are presented to provide profound insights on understanding the PLL statistics both qualitatively and quantitatively. Results covered include the transient and stationary statistics for the nonmodulo-2π probability density function, modulo-2π probability density function, and cycle slipping density function, of the phase error. Various numerical settings of PLL parameters are involved, including the detuning factor and signal-to-noise ratio (SNR). The results presented in this paper elucidate the link between various parameters and the information geometry of the phase-error statistics and form a basis for future investigation on PLL designs.

scholarly journals On universal features of the turbulent cascade in terms of non-equilibrium thermodynamics

2018 ◽  
Vol 848 ◽  
pp. 117-153 ◽  
Author(s):  
Nico Reinke ◽  
André Fuchs ◽  
Daniel Nickelsen ◽  
Joachim Peinke
Keyword(s):  
Probability Density ◽  
Entropy Production ◽  
Turbulent Flows ◽  
Planck Equation ◽  
Small Scale ◽  
Equilibrium Thermodynamics ◽  
Fokker Planck Equation ◽  
Velocity Increments ◽  
Turbulent Cascade ◽  
Fokker Planck

Features of the turbulent cascade are investigated for various datasets from three different turbulent flows, namely free jets as well as wake flows of a regular grid and a cylinder. The analysis is focused on the question as to whether fully developed turbulent flows show universal small-scale features. Two approaches are used to answer this question. First, two-point statistics, namely structure functions of longitudinal velocity increments, and, second, joint multiscale statistics of these velocity increments are analysed. The joint multiscale characterisation encompasses the whole cascade in one joint probability density function. On the basis of the datasets, evidence of the Markov property for the turbulent cascade is shown, which corresponds to a three-point closure that reduces the joint multiscale statistics to simple conditional probability density functions (cPDFs). The cPDFs are described by the Fokker–Planck equation in scale and its Kramers–Moyal coefficients (KMCs). The KMCs are obtained by a self-consistent optimisation procedure from the measured data and result in a Fokker–Planck equation for each dataset. Knowledge of these stochastic cascade equations enables one to make use of the concepts of non-equilibrium thermodynamics and thus to determine the entropy production along individual cascade trajectories. In addition to this new concept, it is shown that the local entropy production is nearly perfectly balanced for all datasets by the integral fluctuation theorem (IFT). Thus, the validity of the IFT can be taken as a new law of the turbulent cascade and at the same time independently confirms that the physics of the turbulent cascade is a memoryless Markov process in scale. The IFT is taken as a new tool to prove the optimal functional form of the Fokker–Planck equations and subsequently to investigate the question of universality of small-scale turbulence in the datasets. The results of our analysis show that the turbulent cascade contains universal and non-universal features. We identify small-scale intermittency as a universality breaking feature. We conclude that specific turbulent flows have their own particular multiscale cascades, in other words, their own stochastic fingerprints.

A nonlocal theory for stress in bound, Brownian suspensions of slender, rigid fibres

1995 ◽  
Vol 296 ◽  
pp. 271-324 ◽  
Author(s):  
Richard L. Schiek ◽  
Eric S. G. Shaqfeh
Keyword(s):  
Brownian Motion ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Fluid Velocity ◽  
Nonlocal Theory ◽  
Velocity Shear ◽  
Slender Body Theory ◽  
Nonlocal Stress ◽  
Fokker Planck

A nonlocal theory for stress in bound suspensions of rigid, slender fibres is developed and used to predict the rheology of dilute, rigid polymer suspensions when confined to capillaries or fine porous media. Because the theory is nonlocal, we describe transport in a fibre suspension where the velocity and concentration fields change rapidly on the fibre's characteristic length. Such rapid changes occur in a rigidly bound domain because suspended particles are sterically excluded from configurations near the boundaries. A rigorous no-flux condition resulting from the presence of solid boundaries around the suspension is included in our nonlocal stress theory and naturally gives rise to concentration gradients that scale on the length of the particle. Brownian motion of the rigid fibres is included within the nonlocal stress through a Fokker–Planck description of the fibres’ probability density function where gradients of this function are proportional to Brownian forces and torques exerted on the suspended fibres. This governing Fokker–Planck probability density equation couples the fluid flow and the nonlocal stress resulting in a nonlinear set of integral-differential equations for fluid stress, fluid velocity and fibre probability density. Using the method of averaged equations (Hinch 1977) and slender-body theory (Batchelor 1970), the system of equations is solved for a dilute suspension of rigid fibres experiencing flow and strong Brownian motion while confined to a gap of the same order in size as the fibre's intrinsic length. The full solution of this problem, as the fluid in the gap undergoes either simple shear or pressure-driven flow, is solved self-consistently yielding average fluid velocity, shear and normal stress profiles within the gap as well as the probability density function for the fibres’ position and orientation. From these results we calculate concentration profiles, effective viscosities and slip velocities and compare them to experimental data.

scholarly journals Linking Nonlinearity and Non-Gaussianity of Planetary Wave Behavior by the Fokker–Planck Equation

2005 ◽  
Vol 62 (7) ◽  
pp. 2098-2117 ◽  
Author(s):  
Judith Berner
Keyword(s):  
Stochastic Model ◽  
Probability Density ◽  
Planetary Wave ◽  
Multiplicative Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
State Dependent ◽  
Nonlinear Stochastic Model ◽  
Non Gaussian ◽  
Fokker Planck

Abstract To link prominent nonlinearities in the dynamics of 500-hPa geopotential heights to non-Gaussian features in their probability density, a nonlinear stochastic model of atmospheric planetary wave behavior is developed. An analysis of geopotential heights generated by extended integrations of a GCM suggests that a stochastic model and its associated Fokker–Planck equation call for a nonlinear drift and multiplicative noise. All calculations are carried out in the reduced phase space spanned by the leading EOFs. It is demonstrated that this nonlinear stochastic model of planetary wave behavior captures the non-Gaussian features in the probability density function of atmospheric states to a remarkable degree. Moreover, it not only predicts global temporal characteristics, but also the nonlinear, state-dependent divergence of state trajectories. In the context of this empirical modeling, it is discussed on which time scale a stochastic model is expected to approximate the behavior of a continuous deterministic process. The reduced model is then used to determine the importance of the nonlinearities in the drift and the role of the multiplicative noise. While the nonlinearities in the drift are crucial for a good representation of planetary wave behavior, multiplicative (i.e., state dependent) noise is not absolutely essential. It is found that a major contributor to the stochastic component is the Branstator–Kushnir oscillation, which acts as a fluctuating force for physical processes with even longer time scales, like those that project on the Arctic Oscillation pattern. In this model, the oscillation is represented by strongly correlated noise.

Analysis of the Chang–Cooper discretization scheme for a class of Fokker–Planck equations

2015 ◽  
Vol 23 (3) ◽  
Author(s):  
Masoumeh Mohammadi ◽  
Alfio Borzì
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Numerical Experiments ◽  
Parabolic Type ◽  
Total Probability ◽  
Discretization Scheme ◽  
Finite Differencing ◽  
Fokker Planck Equations ◽  
Fokker Planck

AbstractThe Chang-Cooper discretization scheme for a class of Fokker-Planck equations is investigated. These equations of parabolic type govern the time evolution of the probability density function of stochastic processes, such that positivity of the density function and conservativeness of the total probability is guaranteed. It is shown that the Chang-Cooper scheme combined with backward first- and second-order finite differencing in time provides stable and accurate solutions that are conservative and positive. These properties are theoretically proven and validated by numerical experiments.

The fBm-driven Ornstein-Uhlenbeck process: Probability density function and anomalous diffusion

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Caibin Zeng ◽  
YangQuan Chen ◽  
Qigui Yang
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Anomalous Diffusion ◽  
Planck Equation ◽  
Viscosity Coefficient ◽  
Infinite Variance ◽  
Nonlinear Stochastic Differential Equations ◽  
The Fourier Transform ◽  
Fractional Ito Formula

AbstractThis paper deals with the Ornstein-Uhlenbeck (O-U) process driven by the fractional Brownian motion (fBm). Based on the fractional Itô formula, we present the corresponding fBm-driven Fokker-Planck equation for the nonlinear stochastic differential equations driven by an fBm. We then apply it to establish the evolution of the probability density function (PDF) of the fBm-driven O-U process. We further obtain the closed form of such PDF by combining the Fourier transform and the method of characteristics. Interestingly, the obtained PDF has an infinite variance which is significantly different from the classical O-U process. We reveal that the fBm-driven O-U process can describe the heavy-tailedness or anomalous diffusion. Moreover, the speed of the sub-diffusion is inversely proportional to the viscosity coefficient, while is proportional to the Hurst parameter. Finally, we carry out numerical simulations to verify the above findings.

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