Frequency Probability Density Research Articles

Impact of continuous stochastic and spatially distributed perturbations on power system frequency stability

With the increase in variable inverter-based renewable energy sources, power systems become increasingly subjected to stochastic sources. Considering the decline in rotational inertia, the consequence of the continuous spatially distributed perturbations is stochastic frequency dynamics. In this paper, the stochastic dynamics of the power system dynamics, subjected to spatially distributed disturbances, are investigated using a stochastic differential equation and the Fokker-Planck equation. This research shows that the reactance distance between perturbation sources, like wind farms, and the location of system inertia (synchronous generators/condensers), reduces the variance of the system’s rate of change of frequency probability density function. Thus, for a power system under continuous and spatially distributed stochastic feed-in, greater reactance distances decrease the probability of significant frequency dynamics.

Deadbands, Droop, and Inertia Impact on Power System Frequency Distribution

Power system inertia is falling as more energy is supplied by renewable generators, and there are concerns about the frequency controls required to guarantee satisfactory system performance. The majority of research into the negative effect of low inertia has focused on poor dynamic response following major disturbances, when the transient frequency dip can become unacceptable. However, another important practical concern—keeping average frequency deviations within acceptable limits—was mainly out of the sight of the research community. In this manuscript, we present a method for finding the frequency probability density function (PDF) for a given power system. We pass from an initial stochastic dynamic model to deterministic equations for the frequency PDF, which are analyzed to uncover key system parameters influencing frequency deviations. We show that system inertia has little effect on the frequency PDF, making virtual inertia services insufficient for keeping frequency close to nominal under ambient load fluctuations. We establish that aggregate system droop and deadband width are the only parameters that have major influence on the average frequency deviations, suggesting that energy storage might be an excellent solution for tight frequency regulation. We also show that changing the governor deadband width does not significantly affect generator movement.

Bayesian modal identification with particle filtering for sediment property inversion

Sequential Bayesian filtering methods have been previously used in dispersion curve tracking for long range sound propagation in the ocean. Modal frequency probability density functions were extracted for sound speed inversion. Here, we calculate modal arrival time densities, instead, and employ them for inversion for sediment sound speed and thickness and water column depth. Bayesian mode identification is performed to this end. We investigate two methods for describing the statistical errors in power spectra, which we use in the arrival time density calculation using normal modes. We then link these densities to the parameters of interest. The approaches are tested with synthetic data as well as data collected in the Gulf of Mexico. [Work supported by ONR.]

Hurricane wind fields needed to assess risk to offshore wind farms

In their paper in PNAS, Rose et al. (1) applied a statistical model to estimate hurricane wind losses to wind turbines over a 20-y typical wind farm lifetime. They combined a county annual landfall frequency probability density function with a generalized extreme value (GEV) fit of maximum wind speeds to model the expected 20-y losses attributable to hurricane activity at four hypothetical offshore wind farm sites.

Study of ExacTrac X-ray 6D IGRT setup uncertainty for marker-based prostate IMRT treatment.

Novalis Tx ExacTrac X‐ray system has the 6D adjustment ability for patient setup. Limited studies exist about the setup uncertainty with ExacTrac X‐ray system for IMRT prostate treatment with fiducial markers implanted. The purpose of this study is to investigate the marker‐based prostate IMRT treatment setup uncertainty using ExacTrac 6D IGRT ability for patient setup. Forty‐three patients with prostate cancers and markers implanted have been treated on the Novalis Tx machine. The ExacTrac X‐ray system has been used for the patient pretreatment setup and intratreatment verification. In total, the shifts data for 1261 fractions and 3504 correction times (the numbers of X‐ray images were taken from tube 1 and tube 2) have been analyzed. The setup uncertainty has been separated into uncertainties in 6D. Marker matching uncertainty was also analyzed. Correction frequency probability density function was plotted, and the radiation dose for imaging was calculated. The minimum, average, and maximum translation shifts were: −5.12±3.89 mm, 0.20±2.21 mm, and 6.07±4.44 mm, respectively, in the lateral direction; −6.80±3.21 mm, −1.09±2.21 mm, and 3.12±2.62 mm, respectively, in the longitudinal direction; and −7.33±3.46 mm, −0.93±2.70 mm, and 5.93±4.85 mm, respectively, in the vertical direction. The minimum, average, and maximum rotation shifts were: −1.23° ± 1.95°, −0.25° ± 1.30°, and −2.38° ± 2.91°, respectively, along lateral direction; −0.67° ± 0.91°, −0.10° ± 0.61°, and −1.51° ± 2.04°, respectively, along longitudinal direction; and −0.75° ± 1.01°, −0.02° ± 0.50°, and −0.82° ± 1.13°, respectively, along vertical direction. On average, each patient had three correction times during one fraction treatment. The radiation dose is about 3 mSv per fraction. With the ExacTrac 6D X‐ray system, the prostate IMRT treatment with marker implanted can achieve less than 2 mm setup uncertainty in translations, and less than 0.25° in rotations as overall interfraction mean error. The imaging dose is less than kV (CBCT) for setup verification.PACS number: 87.55.km, 87.55.Qr, 87.56.bd

Luminescence-dated aeolian deposits of late Quaternary age in the southern Tibetan Plateau and their implications for landscape history

Aeolian deposits are widely distributed in the interior of the Tibetan Plateau, and their chronology is poorly known. It is not yet clear whether they accumulated only after the last deglaciation, or over a longer time. We applied quartz OSL dating to aeolian samples from the Lhasa area with OSL ages ranging from 2.9 ± 0.2 to at least 118 ± 11 ka. The probability density frequency (PDF) distribution of 24 ages reveals age clusters at about 3, 8, 16–21, 33, and 79–83 ka, indicating enhanced sediment accumulation then. The results show that aeolian deposition occurred throughout most of the last 100 ka. This implies that: 1) an ice sheet covering the whole Tibetan Plateau during the last glacial maximum (LGM) could not have existed; and 2) erosion during the last deglaciation was not as strong as previously proposed, such that not all pre-Holocene loess was removed. The age distribution shown in the PDF indicates that aeolian accumulation is episodic. Sand-formation events revealed by age clusters at 3, 8, and 16–21 ka imply roughly synchronous environmental responses to corresponding global-scale arid events.

Impulse response of a linear diffusion analogy model as a flood frequency probability density function

The impulse response of a linear convective-diffusion analogy (LD) model used for flow routing in open channels is proposed as a probability distribution for flood frequency analysis. The flood frequency model has two parameters, which are derived using the methods of moments and maximum likelihood. Also derived are errors in quantiles for these parameter estimation methods. The distribution shows that the two methods are equivalent in terms of producing mean values—the important property in case of unknown true distribution function. The flood frequency model is tested using annual peak discharges for the gauging sections of 39 Polish rivers where the average value of the ratio of the coefficient of skewness to the coefficient of variation equals about 2.52, a value closer to the ratio of the LD model than to the gamma or the lognormal model. The likelihood ratio indicates the preference of the LD over the lognormal for 27 out of 39 cases. It is found that the proposed flood frequency model represents flood frequency characteristics well (measured by the moment ratio) when the LD flood routing model is likely to be the best of all linear flow routing models.

PDF calculations of turbulent nonpremixed flames with local extinction

A joint velocity–composition–turbulence frequency probability density function (PDF) model is used to calculate a series of piloted-jet flames of methane. The ingredients of the present model include the simplified Langevin model for velocity, a stochastic model of turbulence frequency, the Euclidean minimum spanning tree (EMST) mixing model, and the 16-species augmented reduced mechanism for methane. The solutions are obtained using a particle/mesh algorithm, and the chemistry mechanism is implemented via the in situ adaptive tabulation (ISAT) algorithm. These flames exhibit an increasing amount of local extinction with increasing jet velocity, and are good cases to test the capabilities of turbulent combustion models to account for local extinction in turbulent nonpremixed flames. The calculation results are compared extensively with the experimental data, and demonstrate the ability of the PDF model to represent, quantitatively, the processes of local extinction and reignition that occur in these flames.

Probability characteristics of the frequency spectra of natural vibrations of turbocompressor blades

Blades of the same state of the turbocompressor of gas turbine units always have scatter associated with their natural vibration frequencies. Establishing a law governing the probability density of turbocompressor blades is necessary to analyze resonance stresses in blades. The probability density of the natural vibration frequencies of a single lot of blades obeys a normal law. The object of this work is to propose a form of natural vibration frequency probability density function for the blades of a GT-750-6 turbocompressor on the basis of a physical model of the natural vibration frequencies and previous analysis of similar data.

AM and FM quantum noise in semiconductor lasers - Part I: Theoretical analysis

AM and FM quantum noise properties of semiconductor lasers have been studied theoretically. Theoretical formulations for the AM noise spectrum, photon number probability density, FM noise spectrum, instantaneous frequency probability density, and power spectrum are presented here as functions of semiconductor laser material, structural, and pumping parameters. Two theoretical approaches are employed: one is based on the quantum mechanical Langevin equation, and the other on the density matrix equation. Starting from the quantum mechanical Langevin equation, three different formulations, that is, the rate equation, Fokker-Planck equation, and van der Pol equation, are derived. The parameters which represent stimulated emission, spontaneous emission, and refractive-index dispersion are obtained by using the Kane function interpolated to Halperin-Lax bandtail and the Stern's improved matrix element. The above four different theoretical formulations are related to each other, and the applicability for each method is discussed.