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Modeling of atmospheric wind speed sequence using a lognormal continuous stochastic equation
Abstract
a b s t r a c t In this paper, we have presented a spectral and a multifractal analysis performed on 412 time series of wind speed data each of duration of 350 s and sampled at 20 Hz. The average spectrum for the wind speed displays a scaling behavior, in the inertial range, over two decades, with b ¼ 1:68 close to the Kolmogorov value À 5/3. A multifractal analysis has been motivated by the presence of scaling invariance in data set. Then we have considered their scaling properties in the framework of fully developed turbulence and multifractal cascades. The results obtained for wind speed confirm that the exponent scaling function z V ðqÞ is nonlinear and concave. This exponent characterizes the scaling functions in the inertial range indicating that the wind speed is intermittent and multifractal. Moreover the theoretical quadratic relation for lognormal multifractals is well fitted. We investigate the consequence for wind energy production: we generate stochastic simulations of a multifractal random walk, and using a power curve derived from experimental data, we generate the associated power time series. We show that, due to the saturation of the power curve for large speed values, when the input time series (turbulent wind speed) is multifractal, the output can be almost monofractal.
... The obtained relationship among wind speed, pressure and temperature in [1] must not be expressed by certain equation, because the method he used is based on statistic. Calif and Schmit modeled wind speed sequence using lognormal continuous stochastic equation [2], they can not get results expressed by certain formula, but by numerical results, since the method they used is based on probability. Zhang et al used Poisson-Gumbel distribution to wind speed calculation for the southeast coastland of China [3], Chu et al used regression method for climate prediction [4]. ...
... The combination of Boyle's Law and Charles' Law for the point is: pV = RT, (2)(3)(4)(5) where p = p(r, θ, z), V = V(r, θ, z), and T = T(r, θ, z) are pressure, volume, and temperature respectively. R is a constant. ...
... Expanding p(r + ∆r) into Taylor expansion and keeps two terms, i.e., Where a θ is the acceleration component in θ −direction，and p(θ) = p(r, θ, z) emphasizes on that only θ is varied (r, z keep unchange) for simple. Expanding p(θ + ∆θ) into Taylor expansion and keeps two terms, i.e., p(θ + ∆θ) = p(θ) + ∆θ ப୮(θ) பθ , (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) Substituting (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) into (2)(3)(4)(5)(6)(7)(8)(9)(10)(11) and neglecting the smaller terms, we have For the case of the box in UCM within a horizontal plane, the components of velocity u ୰ and u θ are related each other, which can be obtained from vector triangle (vector addition of u θ , u ୰ and u θା∆θ ), of a box in circular motion, i.e., u + u ୰ = u ା∆ , (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) Where ܝ ܚ = u ୰ ܍ ܚ , ܝ ી = u θ ܍ ી , ܝ ીା∆ી = u θା∆θ ܍ ીା∆ી . ...
This paper uses a point model of cylindrical box wrapped by zero-weighted membrane to derive a wind
speed equation of circular cyclone by method of section based on Boyle’s law, Charles’ law and Newton’s
laws. The obtained equation is a non-linear partial differential equation (PDE) with two unknown
functions, seldom seen in literatures. The obtained vertical motion is upward which has been confirmed
by a recorded video; and it also shows that the mass of air approaches to zero at isothermal layer where
the temperature keeps unchanged for any height in the layer.
... For these reasons, the study of wake development behind wind turbine blades is important to minimize the effects of wake interaction and increase the performance of the wind turbine [4]. Moreover, in recent studies the intermittent turbulent character of atmospheric wind speed [5] is transferred to the power output from wind farms [6,7]. In other words, the power output of wind farm possesses intermittent and multifractal properties [5,6]. ...
... Moreover, in recent studies the intermittent turbulent character of atmospheric wind speed [5] is transferred to the power output from wind farms [6,7]. In other words, the power output of wind farm possesses intermittent and multifractal properties [5,6]. Calif and Schmitt proposed methodologies to describe the intermittency of wind power fluctuations at all intensities [6]. ...
... Although the tip local Reynolds number is 5 10 c tip Re ≈ for the upstream turbine, a full-scale wind turbine has a much higher value (at least one order of magnitude higher). It should be noted that the tip local Reynolds number has significant effects on the functional characteristics of wind turbines. ...
A Large Eddy Simulation (LES) has been employed in order to study the flow field in a single-wind turbine and in two in-line wind turbines. The present study focuses on the flow around a horizontal axis wind turbine in a virtual wind tunnel. The anisotropic residual stress tensor is driven by the Smagorinsky model. The results are consistent with experimental data presented in literature. Streamwise velocity is increased and cross stream velocity is decreased as wake moves in downstream direction. A faster rate of wake recovery is seen for the two in-line setup. The results reveal that turbulence intensity is increased by increasing the downstream distance and two in-line turbines show greater intensity. Wind turbine performance can be affected by the turbulent structures. If this phenomenon occurs, information about turbulent structures would be useful in order to investigate the effect of turbulent structures on wind turbine performance. As a result, we aim to reveal the effect of turbulent structures, by using the technique in this study, and to investigate the performance of wind turbines in different conditions. Furthermore, the effects of blade rotation direction are studied in this paper. It is concluded that wind turbine efficiency is increased by 4%.
... wind [4]. On the other hand, the amount of energy produced by wind power is a function of the wind speed. ...
... It could provide useful information to decision-making to maximize the rentability of wind power production, e.g., information about the future error probability distribution to know the risk of bidding certain power output. Zhang et al. [31] classifies the probabilistic approaches in three categories according to the uncertainty representation: probabilistic (e.g., [4,32,33]), risk index (e.g., [34,35]), and space time scenario forecasts [36][37][38]. ...
This article presents a comparison of wind speed forecasting techniques, starting with the Auto-regressive Integrated Moving Average, followed by Artificial Intelligence-based techniques. The objective of this article is to compare these methods and provide readers with an idea of what method(s) to apply to solve their forecasting needs. The Artificial Intelligence-based techniques included in the comparison are Nearest Neighbors (the original method, and a version tuned by Differential Evolution), Fuzzy Forecasting, Artificial Neural Networks (designed and tuned by Genetic Algorithms), and Genetic Programming. These techniques were tested against twenty wind speed time series, obtained from Russian and Mexican weather stations, predicting the wind speed for 10 days, one day at a time. The results show that Nearest Neighbors using Differential Evolution outperforms the other methods. An idea this article delivers to the reader is: what part of the history of the time series to use as input to a forecaster? This question is answered by the reconstruction of phase space. Reconstruction methods approximate the phase space from the available data, yielding m (the system’s dimension) and τ (the sub-sampling constant), which can be used to determine the input for the different forecasting methods.
... The standard procedure as described by Moriarty and Hansen (2005) linearly interpolates the on-grid values to the required inter-grid positions. However, as discussed in Section 3.1 this distorts the wind statistics, which are important for turbine load and fatigue analysis as well as power output predictions Calif and Schmitt, 2012;Milan et al., 2013;Calif et al., 2013). This deficit can be overcome by applying wind speed increment interpolation instead. ...
... Based on these 45,500 wind speed values a normal distribution was fitted to the results via the built-in Matlab function fitdist(). Although real wind is known to be not a Gaussian process (Boettcher et al., 2007;Calif and Schmitt, 2012), for the used synthetic wind speed data set the normal distribution was found to be a very good fit. If a more complex wind model is used, or experimental data is available, a different distribution might be a better choice. ...
Advancing towards ‘better’ wind turbine designs engineers face two central challenges:
first, current aerodynamic models (based on Blade Element Momentum theory) are
inherently limited to comparatively simple designs of flat rotors with straight blades.
However, such designs present only a subset of possible designs. Better concepts
could be coning rotors, swept or kinked blades, or blade tip modifications. To be
able to extend future turbine optimization to these new concepts a different kind of
aerodynamic model is needed. Second, it is difficult to include long term loads (life time
extreme and fatigue loads) directly into the wind turbine design optimization. This is
because with current methods the assessment of long term loads is computationally
very expensive – often too expensive for optimization. This denies the optimizer the
possibility to fully explore the effects of design changes on important life time loads,
and one might settle with a sub-optimal design.
In this dissertation we present work addressing these two challenges, looking at
wing aerodynamics in general and focusing on wind turbine loads in particular. We
adopt a Lagrangian vortex model to analyze bird wings. Equipped with distinct tip
feathers, these wings present very complex lifting surfaces with winglets, stacked in
sweep and dihedral. Very good agreement between experimental and numerical results
is found, and thus we confirm that a vortex model is actually capable of analyzing
complex new wing and rotor blade geometries.
Next stochastic methods are derived to deal with the time and space coupled
unsteady aerodynamic equations. In contrast to deterministic models, which repeatedly
analyze the loads for different input samples to eventually estimate life time load
statistics, the new stochastic models provide a continuous process to assess life
time loads in a stochastic context – starting from a stochastic wind field input
through to a stochastic solution for the load output. Hence, these new models allow
obtaining life time loads much faster than from the deterministic approach, which
will eventually make life time loads accessible to a future stochastic wind turbine
optimization algorithm. While common stochastic techniques are concerned with
random parameters or boundary conditions (constant in time), a stochastic treatment
of turbulent wind inflow requires a technique capable to handle a random field. The
step from a random parameter to a random field is not trivial, and hence the new
stochastic methods are introduced in three stages.
iv
First the bird wing model from above is simplified to a one element wing/ blade
model, and the previously deterministic solution is substituted with a stochastic
solution for a one-point wind speed time series (a random process). Second, the wind
inflow is extended to an n-point correlated random wind field and the aerodynamic
model is extended accordingly. To complete this step a new kind of wind model
is introduced, requiring significantly fewer random variables than previous models.
Finally, the stochastic method is applied to wind turbine aerodynamics (for now based
on Blade Element Momentum theory) to analyze rotor thrust, torque, and power.
Throughout all these steps the stochastic results are compared to result statistics
obtained via Monte Carlo analysis from unsteady reference models solved in the
conventional deterministic framework. Thus it is verified that the stochastic results
actually reproduce the deterministic benchmark. Moreover, a considerable speed-up
of the calculations is found (for example by a factor 20 for calculating blade thrust
load probability distributions).
Results from this research provide a means to much more quickly analyze life
time loads and an aerodynamic model to be used a new wind turbine optimization
framework, capable of analyzing new geometries, and actually optimizing wind turbine
blades with life time loads in mind. However, to limit the scope of this work, we only
present the aerodynamic models here and will not proceed to turbine optimization
itself, which is left for future work.
... As one kind of the rapidly growing renewable energy sources, wind energy has been recognized as an attractive alternative to conventional fossil fuels due to several advantages, including renewability and pollution-free environment [2]. However, wind power is recognized as a stochastic process [3] because of the intermittent and multi-scale characteristics of wind speed fluctuation [4,5]. With the increasing penetration of wind power in electric grids, this presents a number of challenges to power system operation, both technically and economically [6]. ...
... Finally, a feature selection process is used to capture the useful features of wind speed fluctuations and determine the optimal inputs of the prediction models. (4) In order to avoid the over-fitting limitation and reduce the influence of outliers, an improved ELM named WRELM is employed as a basic predictor for building the prediction model by using these selected features. ...
With the growing penetration of wind power into electric grids, improving wind speed prediction accuracy has become particularly valuable for the exploitation of wind power. In this paper, a novel hybrid strategy based on a three-phase signal decomposition (TPSD) technique, feature extraction (FE) and weighted regularized extreme learning machine (WRELM) is developed for multi-step ahead wind speed prediction. The TPSD including seasonal separation algorithm (SSA), fast ensemble empirical mode decomposition (FEEMD) and variational mode decomposition (VMD) is proposed for the first time to handle the complex and irregular natures of wind speed comprehensively. The FE process is used to capture the useful features of wind speed fluctuations and determine the optimal inputs for a prediction model. The WRELM is employed as a basic predictor for building the prediction model by these selected features. Four real wind speed prediction cases are utilized to evaluate the proposed model, and experimental results verify the effectiveness of the proposed model compared with the benchmark models.
... Let us begin with known results about the power spectrum of solar and wind power. The power spectra computed from high frequency time series (with sample rate 1 Hz) of solar irradiance, wind velocity and wind power exhibit a power-law behaviour with an exponent ∼ 5/3 (Kolmogorov exponent [2,35]) in the frequency domain 0.001 < f < 0.1 Hz, indicating that they are turbulent-like sources [35,36,37]. This is reconfirmed here in Figs. ...
Wind and solar power are known to be highly influenced by weather events and may ramp up or down abruptly. Such events in the power production influence not only the availability of energy, but also the stability of the entire power grid. By analysing significant amounts of data from several regions around the world with resolutions of seconds to minutes, we provide strong evidence that renewable wind and solar sources exhibit multiple types of variability and nonlinearity in the time scale of {\it seconds} and characterise their stochastic properties. In contrast to previous findings, we show that only the jumpy characteristic of renewable sources decreases when increasing the spatial size over which the renewable energies are harvested. Otherwise, the strong non-Gaussian, intermittent behaviour in the cumulative power of the total field survives even for a country-wide distribution of the systems. The strong fluctuating behaviour of renewable wind and solar sources can be well characterised by Kolmogorov-like power spectra and exponential probability density functions. Using the estimated potential shape of power time series, we quantify the jumpy or diffusive dynamic of the power. Finally we propose a time delayed feedback technique as a control algorithm to suppress the observed short term non-Gaussian statistics in spatially strong correlated and intermittent renewable sources.
... Wind power plays a leading role and could provide nearly 35% of that contribution (Center, B. P, 2020). Wind power, for all its benefits, has the undesirable effect of introducing variability into the energy portfolio due to the non-stationary nature and nonlinear character of wind (Calif and Schmitt, 2012). To appreciate the large variability in wind data, please see the visualization of wind speed and wind direction in Figure 4.4 of (Ding, 2019, Page 106). ...
The forecast of wind speed and the power produced from wind farms has been a challenge for a long time and continues to be so. This work introduces a method that we label as Wavelet Decomposition-Neural Networks (WDNN) that combines wavelet decomposition principles and deep learning. By merging the strengths of signal processing and machine learning, this approach aims to address the aforementioned challenge. Treating wind speed and power as signals, the wavelet decomposition part of the model transforms these inputs, as appropriate, into a set of features that the neural network part of the model can ingest to output accurate forecasts. WDNN is unconstrained by the shape, layout, or number of turbines in a wind farm. We test our WDNN methods using three large datasets, with multiple years of data and hundreds of turbines, and compare it against other state-of-the-art methods. It’s very short-term forecast, like 1-h ahead, can outperform some deep learning models by as much as 30%. This shows that wavelet decomposition and neural network are a potent combination for advancing the quality of short-term wind forecasting.
... There are many distribution functions proposed in the literature for evaluating wind data around the world such as Weibull (Wan et al., 2021), Rayleigh (Serban et al., 2020), Gamma (Lo Brano et al., 2011) Burr (Soukissian, 2013), Normal, truncated normal (Carta, 2009), Loglogistic (Wu et al., 2013), and lognormal (Calif & Schmitt., 2012). The main function of the WAsP software is the Weibull distribution. ...
This paper presents a site suitability analysis for a 20 MW wind farm project in western Algeria’s highlands. The aim is to improve the quality of the electricity grid’s service and increase Algeria’s renewable energy utilization. The wind potential of three regions, Mecheria, El Kheiter, and Naâma, was evaluated using the Weibull function and the wind atlas analysis and application program (WAsP) with a ten-year database (2011-2021) at 10 m hub height. The assessment encompassed a comprehensive analysis of various wind resource parameters, including mean wind speed, prevailing direction, and power densities. In comparison to other sites, the Mecheria region has the best wind potential, with a mean annual wind speed of 6.31 m/s, a power density of 283 W/m2, and Weibull parameters A = 7.1 m/s and k = 2.02. These promising results prompted us to design a wind farm in this region using Power Wind 90/2000 kW turbine technology facing the predominant wind directions of the site, producing 103.91 GWh of total annual gross energy produced (gross AEP) and 103.75 GWh of total annual net energy produced (net AEP). Finally, it appears that the wind resources in the selected region are well-suited for electricity generation, offering a promising opportunity to reduce the country's dependence on fossil fuels.
... estimated over a sequence of length N of the considered signal x(t) defined as in [19] and described by equation Eq.(4): ...
The characterization of irradiance variability needs tools to describe and quantify variability at different time scales in order to optimally integrate PV onto electrical grids. Recently in the literature, a metric called nominal variability defines the intradaily variability by the ramp rate’s variance. Here we will concentrate on the quantification of this parameter at different short time scales for tropical measurement sites which particularly exhibit high irradiance variability due to complex microclimatic context. By analogy with Taylor law performed on several complex processes, an analysis of temporal fluctuations scaling properties is proposed. The results showed that the process of intradaily variability obeys Taylor’s power law for every short time scales and several insolation conditions. The Taylor power law for simulated PV power output has been verified for very short time scale (30s sampled data) and short time scale (10 min sampled data). The exponent λ presents values between 0.5 and 0.8. Consequently, the results showed a consistency of Taylor power law for simulated PV power output. These results are a statistical perspective in solar energy area and introduce intradaily variability PV power output which are key properties of this characterization, enabling its high penetration.
... Although wind is a source of clean energy, wind power generation is characterized by fluctuations because of the changeable nature of wind speed [22]. Given that the outputs of offshore wind farms are considerably influenced by wind speed, identifying locations where wind speeds are high and steady is crucial. ...
Renewable energy is produced using renewable natural resources, including wind power. The Taiwan government aims to have renewable energy account for 20% of its total power supply by 2025, in which offshore wind power plays an important role. This paper explores the application of index insurance to renewable energy for offshore wind power in Taiwan. We employ autoregressive integrated moving average models to forecast power generation on a monthly and annual basis for the Changhua Demonstration Offshore Wind Farm. These predictions are based on an analysis of 39 years of hourly wind speed data (1980–2018) from the Modern-Era Retrospective analysis for Research and Applications, Version 2, of the National Aeronautics and Space Administration. The data analysis and forecasting models describe the methodology used to design the insurance contract and its index for predicting offshore wind power generation. We apply our forecasting results to insurance contract pricing.
... Wind speed and direction cannot be predicted precisely because of wind's stochastic nature [2]. The natural behavior of wind at a prospective site must be observed and evaluated to identify wind characteristics. ...
Choosing the right wind site and estimating the extracted energy of the wind turbines are essential to successfully establishing a wind farm in a specific wind site. In this paper, a method for estimating the extracted energy of the wind farms using several mathematical models is proposed. The estimating method, which was based on five wind turbines, Q1, Q2, Q3, Q4, and Q5 and three wind distribution models, gamma, Weibull, and Rayleigh, was used to suggest suitable specifications of a wind turbine for a specific wind site and maximize the extracted energy of the proposed wind farm. An optimization problem, developed for this purpose, was solved using the whale optimization algorithm (WOA). The suggested method was tested using several potential wind sites in Jordan. The proposed wind farms at these sites achieved the maximum extracted energy, maximum capacity factor (CF), and minimum levelized cost of energy (LCoE) based on the solution of the developed optimization problem. The developed model with Q3 and the Rayleigh distribution function was validated with real measurement data from several wind farms in Jordan. Error analysis showed that the difference between the measured and estimated energy was less than 20%. The study validated the provided model, which can now be utilized routinely for the assessment of wind energy potential at a specific wind site.
... Particularly, in some specific areas, the effect of fractional Gaussian noise imposed on complex systems is more complicated than classical noise, for example, in the turbulence propagation model of pollutants, it is better to use fractional Brownian noises to describe the character of turbulent [4]. It's worth pointing out that the fractional Gaussian noise is an incremental process of the fractional Brownian motion (fBm) [1,20], which has been widely used in many scientific fields. So far, the hotspot of stochastic systems has moved from the classical Gaussian noise to the fBm. ...
This paper investigates the problem of disturbance-observer-based sliding mode control for stabilization of stochastic systems driven by fractional Brownian motion (fBm). By proposing a novel disturbance observer, an integral-type sliding surface is put forward with the estimated disturbance error confined within a certain value. Meanwhile, by virtue of fractional infinitesimal operator and linear matrix inequality, a sufficient criterion is derived to guarantee the asymptotic stability of obtained sliding mode dynamics. Further, an observer-based sliding mode controller is designed to ensure finite-time reachability of state trajectories onto the predefined sliding surface. Lastly, an illustrative example is utilized to verify the reliability and applicability of the proposed control strategy.
... Nevertheless, explicit determination of the multipoint PDF (2) is rather difficult if the field u(x) exhibits non-Gaussian properties and one typically has to resort to approximations, e.g., the assumption of a Markov process in scale which reduces equation (2) to a product of three-point quantities [35,36,49]. We introduce here an explicit construction of the joint multipoint PDF (2) with multifractal scaling that belongs to the class of the Kolmogorov-Oboukhov (K62) model of turbulence [50,51] (see also [52][53][54][55] for further references on the use of such models in turbulence). The method is based on an ensemble of fractional Ornstein-Uhlenbeck processes which have been modified by the introduction of fluctuating length scales. ...
Complex systems often involve random fluctuations for which self-similar properties in space and time play an important role. Fractional Brownian motions, characterized by a single scaling exponent, the Hurst exponent H , provide a convenient tool to construct synthetic signals that capture the statistical properties of many processes in the physical sciences and beyond. However, in certain strongly interacting systems, e.g., turbulent flows, stock market indices, or cardiac interbeats, multiscale interactions lead to significant deviations from self-similarity and may therefore require a more elaborate description. In the context of turbulence, the Kolmogorov–Oboukhov model (K62) describes anomalous scaling, albeit explicit constructions of a turbulent signal by this model are not available yet. Here, we derive an explicit formula for the joint multipoint probability density function of a multifractal field. To this end, we consider a scale mixture of fractional Ornstein–Uhlenbeck processes and introduce a fluctuating length scale in the corresponding covariance function. In deriving the complete statistical properties of the field, we are able to systematically model synthetic multifractal phenomena. We conclude by giving a brief outlook on potential applications which range from specific tailoring or stochastic interpolation of wind fields to the modeling of financial data or non-Gaussian features in geophysical or geospatial settings.
... The second method is the numerical method. Based on numerical simulation, the continuous stochastic equation [17], thermal equation [18], kinematic equation and state equation [19] are used to fit atmospheric numerical value, to achieve the purpose of analyzing meteorology and temperature. The third method is model analysis [20][21][22]. ...
In recent years, the global temperature is continuously rising and has the trend of accelerating. The frequent occurrence of extremely high temperatures and heat waves has caused widespread concern from all walks of life. How to fully understand the change law of temperature becomes very important. In view of the temperature change in Xi’an, this paper introduces a new method called visibility graph to establish the temperature network in Xi’an. On this basis, firstly, this paper studies the relationship between temperature fluctuation and network degree. We find that short-term fluctuations do not cause long-term effects. Then, through the study of network degree distribution, it is revealed that the temperature network conforms to the law of power-law distribution. In addition, this paper also completes the community detection of temperature network, and finds that some communities have fewer nodes (between June and August), which means that the correlation between summer temperature and other seasons in Xi’an is low, and it is easy to form extreme weather. To sum up, the research in this paper provides a new theoretical method and research ideas for mining and mastering the variation law of temperature in Xi’an.
... This framework corresponds to a cascade of energy as proposed by Richardson in the 1920s (Richardson, 1922): the energy is injected at large scales, is cascading in the inertial range and is dissipated at small scales, smaller than the Kolmogorov scale, which is of the order of millimeters. The Reynolds number in the atmosphere is in the order of 7 10 to 9 10 , and the 5 / 3 slope has been found from atmospheric observations by many authors (Calif & Schmitt, 2012;Schmitt & Huang, 2016;Sreenivasan & Antonia, 1997). ...
Turbulence or turbulence‐like phenomena are ubiquitous in nature, often showing a power‐law behavior of the fluctuations in either spatial or temporal domains. This power‐law behavior is due to interactions among different scales of motion, and to the absence of characteristic scale. In this work, we consider the multiscale dynamics of China France Oceanography SATellite (CFOSAT) data as atmospheric and oceanic quantities influenced by turbulence. Fourier power spectra were estimated for the data provided by the CFOSAT via the Wiener‐Khinchine theorem to extract multiscale information for both wind speed (WS) and significant wave height (Hs). The WS data were collected from December 18, 2018 to August 31, 2020, and the Hs data from July 29, 2019 to August 31, 2020. Fourier power spectra for both WS and Hs exhibit power‐law features in the ranges of 100–3,000 km with a scaling exponent β varying from 5/3 to 3. The global distributions and seasonal variations of β for both WS and Hs have also been considered. The results show that due to the energetic convective activities in the low‐latitude zones, the scaling exponents β in these regions are closer to the value of 5/3. Concerning the seasonal variations, for most regions, the scaling exponents in winter are larger than those in summer for WS. The seasonal variations of β in low‐latitudes are stronger than those in the mid‐latitudes. Our preliminary results enrich the fundamental knowledge of ocean surface processes and also provide a benchmark for either oceanic or atmospheric models.
... As output, the component gives a list containing the time-frequency that the wind is coming from for each direction and the average wind velocity. However, a logarithmic normal distribution is considered to better describe the probability distribution of the wind speed frequency curves [82,83]. Therefore, both normal and lognormal distributions were used ( Figure 3) when analyzing the wind speed statistical distribution, and the best fit was selected. ...
Predicting building air change rates is a challenge for designers seeking to deal with
natural ventilation, a more and more popular passive strategy. Among the methods available for
this task, computational fluid dynamics (CFD) appears the most compelling, in ascending use.
However, CFD simulations require a range of settings and skills that inhibit its wide application.
With the primary goal of providing a pragmatic CFD application to promote wind-driven ventilation
assessments at the design phase, this paper presents a study that investigates natural ventilation
integrating 3D parametric modeling and CFD. From pre- to post-processing, the workflow addresses
all simulation steps: geometry and weather definition, including incident wind directions, a model
set up, control, results’ edition, and visualization. Both indoor air velocities and air change rates
(ACH) were calculated within the procedure, which used a test house and air measurements as
a reference. The study explores alternatives in the 3D design platform’s frame to display and
compute ACH and parametrically generate surfaces where air velocities are computed. The paper
also discusses the effectiveness of the reference building’s natural ventilation by analyzing the CFD
outputs. The proposed approach assists the practical use of CFD by designers, providing detailed
information about the numerical model, as well as enabling the means to generate the cases, visualize,
and post-process the results.
... Many distribution functions have been proposed in the literature to assess the wind speed data of hundreds sites around the world. Weibull [25][26][27], Rayleigh [28,29], Gamma [30], Burr [31], Loglogistic [32], Normal [33], truncated normal [33], and lognormal [34] probability density functions are the commonly-used functions to model wind speed measurements [35]. These functions have different degrees of complexity, fitness, and accuracy. ...
Accurate estimation of wind speed distributions is a challenging task in wind power planning and operation. The selection of convenient functions for describing wind speed distribution is a crucial requisite. In this paper, remarkable bi-parameter Weibull function is presented to estimate the wind energy potential. Weibull parameters based on different six estimation methods, namely graphical, method of moment, energy pattern factor, mean standard deviation, power density methods, and genetic algorithm are evaluated. Besides, the goodness of fit of the estimation methods is investigated via mean absolute error, root mean square error, normalized mean absolute error, Chi-square error, and regression coefficient. To plainly identify the best matching estimation method, Net Fitness test is also presented. Catalca in the Marmara region in Istanbul, Republic of Turkey, is selected to be the underlying site. The experimental results show the effectiveness of the estimation methods in modeling wind distribution but with relatively small differences in terms of performance. However, the genetic algorithm and energy pattern factor accomplish the best and worst matching estimation methods, respectively.
... The traditional tool to capture this aspect is the power spectrum obtained by Fourier analysis [16]. Several studies have examined the power spectral density both for wind speed [17,18] and power time series [19,20], also uncovering how aggregation on different spatial scales impacts power fluctuations [21,10]. Other classical methods range from structure functions to estimators for the Hurst exponent based on scale dependent variance calculation on the respective time series [22]. ...
Wind farms can be regarded as complex systems that are, on the one hand, coupled to the nonlinear, stochastic characteristics of weather and, on the other hand, strongly influenced by supervisory control mechanisms. One crucial problem in this context today is the predictability of wind energy as an intermittent renewable resource with additional non-stationary nature. In this context, we analyze the power time series measured in an offshore wind farm for a total period of one year with a time resolution of 10 min. Applying detrended fluctuation analysis, we characterize the autocorrelation of power time series and find a Hurst exponent in the persistent regime with cross-over behavior. To enrich the modeling perspective of complex large wind energy systems, we develop a stochastic reduced-form model ofpower time series. The observed transitions between two dominating power generation phases are reflected by a bistable deterministic component, while correlated stochastic fluctuations account for the identified persistence. The model succeeds to qualitatively reproduce several empirical characteristics such as the autocorrelation function and the bimodal probability density function.
... The accurate prediction of wind speeds is an important component of effectively utilizing wind energy to generate electricity. However, due to the stochasticity and intermittence of wind speed [1][2][3][4], it is difficult to accurately predict the wind speed estimation errors and meet the actual requirements of wind power generation. Therefore, the accurate prediction of wind speed is of great significance. ...
Due to the ever-increasing environmental pollution becoming progressively more serious, wind power has been widely used around the world in recent years. However, because of their randomness and intermittence, the accurate prediction of wind speeds is difficult. To address this problem, this article proposes a hybrid system for short-wind-speed prediction. The system combines the autoregressive differential moving average (ARIMA) model with a three-layer feedforward neural network. An ARIMA model was employed to predict linear patterns in series, while a feedforward neural network was used to predict the nonlinear patterns in series. To improve accuracy of the predictions, the neural network models were trained by using two methods: first-order transition rules and fuzzy first-order transition rules. The Levenberg–Marquardt (LM) algorithm was applied to update the weight and deviation of each layer of neural network. The dominance matrix method was employed to calculate the weight of the hybrid system, which was used to establish the linear hybrid system. To evaluate the performance, three statistical indices were used: the mean square error (MSE), the root mean square error (RMSE) and the mean absolute percentage error (MAPE). A set of Lorenz-63 simulated values and two datasets collected from different wind fields in Qilian County, Qinghai Province, China, were utilized as to perform a comparative study. The results show the following: (a) compared with the neural network trained by first-order transition rules, the prediction accuracy of the neural network trained by the fuzzy first-order transition rules was higher; (b) the proposed hybrid system attains superior performance compared with a single model; and (c) the proposed hybrid system balances the forecast accuracy and convergence speed simultaneously during forecasting. Therefore, it was feasible to apply the hybrid model to the prediction of real time-series.
... The finescale intermittency of turbulent velocity fluctuations was found to be prevalent in the atmospheric boundary layer (Schertzer et al. 1997;Boettcher et al. 2003;Vindel and Yagüe 2011;Wächter et al. 2012;Liu et al. 2019). To improve the modeling of turbulent flows (Friedrich and Peinke 1997;Baïle et al. 2011;Calif and Schmitt 2012) and obtain a better understanding of particle dispersal such as air pollutants and seeds (Wei et al. 2018;Duman et al. 2016) and the propagation of light and sound wave (Sreenivasan and Antonia 1997) in the atmospheric boundary layer, a better knowledge of the finescale intermittency of atmospheric turbulence is still needed. ...
The intermittency of atmospheric turbulence plays an important role in the understanding of particle dispersal in the atmospheric boundary layer and in the statistical simulation of high-frequency wind speed in various applications. There are two kinds of intermittency, namely the magnitude intermittency (MI) related to non-Gaussianity and the less studied clusterization intermittency (CI) related to long-term correlation. In this paper, we use a 20 Hz ultrasonic data set lasting for one month to study CI of turbulent velocity fluctuations at different scales. Basing on the analysis of return time distribution of telegraphic approximation series, we propose to use the shape parameter of the Weibull distribution to measure CI. Observations of this parameter show that contrary to MI, CI tends to weaken as the scale increases. Besides, significant diurnal variations, showing that CI tends to strengthen at daytime (under unstable conditions) and weaken at nighttime (under stable conditions), are found at different observation heights. In the convective boundary layer, the mixed-layer similarity is found to scale CI exponent better than the Monin-Obukhov similarity. At night, CI is found to vary more slightly with height in the regime with large mean wind speeds than in the regime with small mean wind speeds, according to the Hockey-Stick theory.
... Although wind power is a clean energy source, wind power is characterized by fluctuations due to the non-stationary nature of wind speed [21]. Hence, wind farm output is significantly influenced by wind speed, and it is essential to identify the locations with high and steady wind speed. ...
Wind power variations at two heights (the surface level and turbine hub level) were investigated at 20 locations in the shelf seas of India using hourly fifth generation European Centre for Medium-Range Weather Forecasts (ECMWF) atmospheric reanalyses of the global climate (ERA5) data covering the last 40 years (1979 to 2018). The interannual and seasonal variability in wind power was studied. The wind power density, the exceedance probability of power density and the exploitable wind resources were examined. In the Indian shelf seas, the annual mean wind power density at 10 m above mean sea level varies from 82 to 353 W/m2. Wind power density at 110.8 m is 20% to 40% higher than at 10 m above mean sea level. The study shows that the shelf seas have an abundance of wind power, with wind speeds over 3 m/s during 90% of the time at locations 1 to 3, 12 and 13, with a high occurrence of exploitable wind energy above 0.7 × 103 kWh/m2. Among the locations studied, the most power-rich area was location 12, where during ~62% of the time power was greater than 200 W/m2. A significant change (~10–35%) in inter-annual wind power density was detected at a few locations, and these variations were associated with Indian summer monsoon and El Niño–Southern Oscillation events. Trend analysis suggests a decreasing trend in the annual mean wind power density for most of the locations in the Indian shelf seas over the last 40 years. Wind power has considerable directional distribution, and at different locations the annual wind power from the dominant direction is 10% to 79% of the total available power from all directions.
... 15 The traditional tool to capture this aspect is the power spectrum obtained by Fourier analysis. 16 Several studies have examined the power spectral density for both wind speed 17,18 and power time series, 19,20 also uncovering how aggregation on different spatial scales impacts power fluctuations. 10,21 Other classical methods range from structure functions to estimators for the Hurst exponent based on the scale dependent variance calculation on the respective time series. ...
Wind farms can be regarded as complex systems that are, on the one hand, coupled to the nonlinear, stochastic characteristics of weather and, on the other hand, strongly influenced by supervisory control mechanisms. One crucial problem in this context today is the predictability of wind energy as an intermittent renewable resource with additional non-stationary nature. In this context, we analyze the power time series measured in an offshore wind farm for a total period of one year with a time resolution of 10 min. Applying detrended fluctuation analysis, we characterize the autocorrelation of power time series and find a Hurst exponent in the persistent regime with crossover behavior. To enrich the modeling perspective of complex large wind energy systems, we develop a stochastic reduced-form model of power time series. The observed transitions between two dominating power generation phases are reflected by a bistable deterministic component, while correlated stochastic fluctuations account for the identified persistence. The model succeeds to qualitatively reproduce several empirical characteristics such as the autocorrelation function and the bimodal probability density function.
... However, the volatility and uncertainty of wind speed give a fundamental challenge to power system operations. Because the basic characteristics of the wind is its intermittency and random fluctuations [3,4], the integration of wind power into power systems puts forward a series of challenges. The most effective way to resolve the challenges is to improve the prediction accuracy of wind speed and power forecasting [5][6][7]. ...
Most regression techniques assume that the noise characteristics are subject to single noise distribution whereas the wind speed prediction is difficult to model by the single noise distribution because the noise of wind speed is complicated due to its intermittency and random fluctuations. Therefore, we will present the ν -support vector regression model of Gauss-Laplace mixture heteroscedastic noise (GLM-SVR) and Gauss-Laplace mixture homoscedastic noise (GLMH-SVR) for complex noise. The augmented Lagrange multiplier method is introduced to solve models GLM-SVR and GLMH-SVR. The proposed model is applied to short-term wind speed forecasting using historical data to predict future wind speed at a certain time. The experimental results show that the proposed technique outperforms the single noise technique and obtains good performance.
... The Weibull distribution was also considered, but the logarithmic normal distribution captured better the frequency of the measured wind speeds ( Figure 6). The log-normal distribution is generally considered a suitable distribution to describe the probability distribution of wind speed data [30][31][32][33]. ...
Wind pressure coefficients (Cp) are important values for building engineering applications, such as calculation of wind loads or wind-induced air infiltration and especially for tall buildings that are more susceptible to wind forces. Wind pressure coefficients are influenced by a plethora of parameters, such as building geometry, position on the façade, exposure or sheltering degree, and wind direction. On-site measurements have been performed on a twin medium-rise building complex. Differential pressure measurements have been employed in order to determine the wind pressure coefficients at various positions along the windward façades of the twin buildings. The measurements show that one building provides substantial wind shelter to its twin and the microclimatic effect is captured by the measured wind pressure coefficients. They also showed that the wind pressure coefficients vary significantly spatially along the windward façades of the medium-rise buildings. Furthermore, the pressure measurements showed that the wind pressure coefficients fluctuate significantly during the measuring period. The use of the fluctuating Cp values by means of probability distribution function (pdf) for the calculation of air infiltration has been evaluated. The results indicate that the air flows deriving using fluctuating Cp values are more accurate than the ones calculated by the conventional method of using mean Cp values.
... The most popular SC-based models are neural network (NN), neuro-fuzzy system (NF), support vector regression (SVR), least square support vector regression (LSSVR), and M5 regression tree (M5RT) models. Wind power production is mainly affected by wind speed fluctuations [20][21][22]. Thus, SC-based models overcome the shortcomings of statistical models in handling the nonlinearity of the data (e.g., wind speed) [23,24]. ...
Accurate predictions of wind speed and wind energy are essential in renewable energy planning and management. This study was carried out to test the accuracy of two different neuro fuzzy techniques (neuro fuzzy system with grid partition (NF-GP) and neuro fuzzy system with substractive clustering (NF-SC)), and two heuristic regression methods (least square support vector regression (LSSVR) and M5 regression tree (M5RT)) in the prediction of hourly wind speed and wind power using a cross-validation method. Fourfold cross-validation was employed by dividing the data into four equal subsets. LSSVR’s performance was superior to that of the M5RT, NF-SC, and NF-GP models for all datasets in wind speed prediction. The overall average root-mean-square errors (RMSE) of the M5RT, NF-GP, and NF-SC models decreased by 11.71%, 1.68%, and 2.94%, respectively, using the LSSVR model. The applicability of the four different models was also investigated in the prediction of one-hour-ahead wind power. The results showed that NF-GP’s performance was superior to that of LSSVR, NF-SC, and M5RT. The overall average RMSEs of LSSVR, NF-SC, and M5RT decreased by 5.52%, 1.30%, and 15.6%, respectively, using NF-GP.
... As the value of the WP penetration increases, power systems have lower inertia and the power fluctuation from a wind turbine (WT) can reduce the power quality, sometimes leading to severe stability problems. Since WP is a stochastic process [3] highly dependent on turbulent fluctuations in wind speed [4,5], many studies have been carried out to integrate WP successfully without causing stability problems. To improve the stability when WP is integrated into a power system with high penetration, different methods for grid frequency regulation have been studied. ...
This paper proposes a method for the coordinated control of a wind turbine and an energy storage system (ESS). Because wind power (WP) is highly dependent on wind speed, which is variable, severe stability problems can be caused in power systems, especially when the WP has a high penetration level. To solve this problem, many power generation corporations or grid operators have begun using ESSs. An ESS has very quick response and good performance for reducing the impact of WP fluctuation; however, its installation cost is high. Therefore, it is important to design the control algorithm by considering both the ESS capacity and WP fluctuation. Thus, we propose a control algorithm to mitigate the WP fluctuation by using the coordinated control between the wind turbine and the ESS by considering the ESS capacity and the WP fluctuation. Using de-loaded control, according to the WP fluctuation and ESS capacity, we can expand the ESS lifespan and improve grid reliability by avoiding the extreme value of state of charge (SoC) (i.e., 0 or 1 pu). The effectiveness of the proposed method was validated via MATLAB/Simulink by considering a small power system that includes both a wind turbine generator and conventional generators that react to system frequency deviation. We found that the proposed method has better performance in SoC management, thereby improving the frequency regulation by mitigating the impact of the WP fluctuation on the small power system.
... Multifractality theory is widely used to quantitatively delineate the nonlinear evolution of a complicated system and the multiscale characteristics of physical quantities. In tropical Atmospheric Boundary Layer (thereafter ABL), it is known that wind speed and solar radiation which are responsible of transport and production of atmospheric pollutants present multifractal properties (Calif and Schmitt, 2012;Calif et al., 2013b). It is the reason why we have decided to investigate the possible multifractal behavior of atmospheric pollutants. ...
A good knowledge of the intermittency of atmospheric pollutants is crucial for air pollution management. We consider here particulate matter PM10 and ground-level ozone O3 time series in Guadeloupe archipelago which experiments a tropical and humid climate in the Caribbean zone. The aim of this paper is to study their scaling statistics in the framework of fully developed turbulence and Kolmogorovs theory. Firstly, we estimate their Fourier power spectra and consider their scaling properties in the physical space. The power spectra computed follows a power law behavior for both considered pollutants. Thereafter we study the scaling behavior of PM10
and O3 time series. Contrary to numerous studies where the multifractal detrended fluctuation analysis is frequently applied, here, the classical structure function analysis is used to extract the scaling exponent or multifractal spectrum ζ(q); this function provides a full characterization of a process at all intensities and all scales. The obtained results show that PM10 and O3 possess intermittent and multifractal properties. The singularity spectrum MS(α) also confirms both pollutants multifractal features. The originality of this work comes from a statistical modeling performed on ζ(q) and MS(α) by a lognormal model to compute the intermittency parameter μ . By contrast with PM10 which mainly depends on puffs of Saharan dust (synoptic-scale), O3 is more intermittent due to variability of its local precursors. The results presented in this paper can help to better understand the mechanisms governing the dynamics of PM10 and O3 in Caribbean islands context.
... Many laboratory and geophysical turbulence studies have shown that the pdfs of velocity increments, pdf [δu], are scaledependent and change steadily within the inertial subrange. Specifically, these distributions have shown to exhibit strong non-Gaussian behavior at small increments, then become more Gaussian as separation increases [15,56,[61][62][63][64][65]. A few years ago, Barndorff-Nielsen et al. [56] demonstrated that the normal inverse Gaussian (NIG) distribution has the inherent ability to capture such scale-dependent traits in a parsimonious manner. ...
Utilizing synthetically generated random variates and laboratory measurements, we document the inherent limitations of the conventional structure function approach in limited sample size settings. We demonstrate that an alternative approach, based on the principle of maximum likelihood, can provide nearly unbiased structure function estimates with far less uncertainty under such unfavorable conditions. The superiority of this approach over the conventional approach does not diminish even in the case of strongly correlated samples. Two entirely different types of probability distributions, which have been reported in the turbulence literature, are found to be compatible with the proposed approach.
... On the other hand, several other authors reported the inertial range breaking at much shorter time-scales, ranging between a few seconds to minutes (e.g. [8][9][10][11][12][13]). Most of these latter works found a transition to a β % 1 − 1.3 regime after the break. ...
The equivalency between the power law behavior of Multiscale Entropy (MSE) and of power spectra opens a promising path for interpretation of complex time-series, which is explored here for the first time for atmospheric fields. Additionally, the present manuscript represents a new independent empirical validation of such relationship, the first one for the atmosphere. The MSE-fractal relationship is verified for synthetic fractal time-series covering the full range of exponents typically observed in the atmosphere. It is also verified for near-surface wind observations from anemometers and CFSR re-analysis product. The results show a ubiquitous β ≈ 5/3 behavior inside the inertial range. A scaling break emerges at scales around a few seconds, with a tendency towards 1/f noise. The presence, extension and fractal exponent of this intermediate range are dependent on the particular surface forcing and atmospheric conditions. MSE shows an identical picture which is consistent with the turbulent energy cascade model: viscous dissipation at the small-scale end of the inertial range works as an information sink, while at the larger (energy-containing) scales the multiple forcings in the boundary layer act as widespread information sources. Another scaling transition occurs at scales around 1–10 days, with an abrupt flattening of the spectrum. MSE shows that this transition corresponds to a maximum of the new information introduced, occurring at the time-scales of the synoptic features that dominate weather patterns. At larger scales, a scaling regime with flatter slopes emerges extending to scales larger than 1 year. MSE analysis shows that the amount of new information created decreases with increasing scale in this low-frequency regime. Additionally, in this region the energy injection is concentrated in two large energy peaks: daily and yearly time-scales. The results demonstrate that the superposition of these periodic signals does not destroy the underlying scaling behavior, with both periodic and fractal terms playing an important role in the observed wind time-series.
In this study, the multiscale dynamics of 38 oceanic and atmospheric pCO2 time series from fixed Eulerian buoys recorded with 3 h resolution are considered, and their multifractal properties are demonstrated. The difference between these time series, the sea surface temperature data and the sea surface salinity data were also studied. These series possess multiscale turbulent-like fluctuations and display scaling properties from 3 h to the annual scale. Scaling exponents are estimated through Fourier analysis, and their average quantities were considered globally for all parameters, as well as for different ecosystems such as coastal shelf, coral reefs and open ocean. Sea surface temperature is the only parameter for which a spectral slope close to 5/3 is found, corresponding to a passive scalar in homogeneous and isotropic turbulence. The other parameters had smaller spectral slopes, from 1.22 to 1.45. By using empirical mode decomposition of the time series, together with generalized Hilbert spectral analysis, the intermittency of the time series was considered in the multifractal framework. Concave moment functions were estimated, and Hurst indices H and intermittency parameters μ were determined in the framework of a lognormal multifractal fit. We obtained mean values of H=0.26 and 0.21, respectively, for oceanic and atmospheric pCO2 time series and μ=0.08 for both. It is the first time that atmospheric and oceanic pCO2 and their difference ΔpCO2 are studied using such an intermittent turbulence framework. The ΔpCO2 time series was shown to possess a power-law scaling with an exponent of β=1.36±0.19.
Many wind speed forecasting approaches have been proposed in literature. In this paper a new statistical approach for jointly predicting wind speed, wind direction and air pressure is introduced. The wind direction and the air pressure are important to extend the forecasting accuracy of wind speed forecasts. A good forecast for the wind direction helps to bring the turbine into the predominant wind direction. We combine a multivariate seasonal time varying threshold autoregressive model with interactions (TVARX) with a threshold seasonal autoregressive conditional heteroscedastic (TARCHX) model. The model includes periodicity, conditional heteroscedasticity, interactions of different dependent variables and a complex autoregressive structure with non-linear impacts. In contrast to ordinary likelihood estimation approaches, we apply a high-dimensional shrinkage technique instead of a distributional assumption for the dependent variables. The iteratively re-weighted least absolute shrinkage and selection operator (LASSO) method allows to capture conditional heteroscedasticity and a comparatively fast computing time. The proposed approach yields accurate predictions of wind speed, wind direction and air pressure for a short-term period. Prediction intervals up to twenty-four hours are presented.
In this paper, we are concerned with a stochastic semilinear differential equations driven by both Brownian motion and fractional Brownian motion. Firstly, we establish an inequality for the distance between finite‐dimensional distributions of a random process at two different moments. Then, using the properties of stochastic integrals, fixed point theorems, and based on this inequality, we establish the existence and uniqueness of Besicovich almost automorphic solutions in finite‐dimensional distributions for this type of semilinear equation. Finally, we provide an example to demonstrate the effectiveness of our results. Our results are new to stochastic differential equations driven by Brownian motion or stochastic differential equations driven by fractional Brownian motion.
The rapid increase in world's installed wind energy capacity may have masked the power loss caused by declining global surface wind speed (termed ‘global stilling’), particularly for China with huge wind energy investments. Here, we estimated the potential impact of global stilling on wind energy production in China, using data from 1226 meteorological stations between 1971 and 2015. We show that surface wind speeds have on average declined at a rate of 5.5% decade⁻¹, while the corresponding wind power density dropped by 24.5% decade⁻¹. Although the decline in wind speeds has slowed after 1991, the decline in wind power density did not show this slowdown. This was attributed to the absence of slowdown in the declining trend of strong winds. Compared with the wind speed levels in the 2000s, the mean capacity factor would have dropped from 20.7% in 2001 to 14.9% in 2015, with the largest absolute change in the “Three North” regions. Furthermore, wind electricity generation would have declined by 40 TWh/a using the installed capacity of 2015, representing a decrease of 16%. The correlation analysis indicates that the decreased wind speed is most likely caused by changes in the large-scale atmospheric circulation rather than increased surface roughness.
A real-time correction method is proposed to accurately and comprehensively collect mine air volume data using wind velocity sensors; ventilation network calculation is realized without relying upon wind resistance measurements. For single-roadway wind velocity correction, an improved Kalman filter is employed to identify the error and adapt to dynamic changes in the ventilation system, improving the error updating process and thus the convergence rate. For node air mass flow balance correction, three types of sensor redundancy layouts are summarized with corresponding correction methods. The principle of the proposed real-time network calculation algorithm is discussed, the calculation method for each sensor layout is provided, and the robustness is analyzed accordingly. Simulation experiment results demonstrate significant improvements in the accuracy of the data obtained using the proposed data correction and network calculation algorithm. Thus, the designed real-time calculation algorithm can be practically applied to realize the accurate collection of ventilation data.
This article aims to analyze the daily fluctuations of the time series of wind speed in some municipalities in the State of Bahia, Brazil from January 2009 to December 2018 with the approach of sliding windows. The analysis will be performed, mainly, with the method known in the literature as Detrended Fluctuation Analysis (DFA) able to identify and measure autocorrelation in non-stationary time series on different time scales (Peng et al., 1994). To meet the proposed objective, we chose five cities of Bahia, by methodological option: Barreiras, Feira de Santana, Guanambi, Salvador and Vitória da Conquista. The results indicated a persistent and statistically significant behavior at the level of 5% for all the studied cities and period. The description with sliding windows (w=365) found a predominance of relative variation above 15%, kurtosis and positive asymmetry. Our findings can be used as one more proposal to model wind speed data and contribute to research related to climatological data.
This article is concerned with exponential mean square stabilization of stochastic systems driven by fractional Brownian motion subject to state-delay and uncertainties by sliding mode control. By applying the proposed method, the states of the system reach the sliding surface in finite time. Then, some sufficient conditions are given in terms of linear matrix inequalities (LMIs) to guarantee the mean-square exponential stability of the sliding motion. The LMI conditions for mean-square exponential stability of the sliding mode dynamics are derived by constructing a novel Lyapunov functional. Finally, a simulation example is presented which corroborates the accuracy of the results.
The major difficulty in studying the response of multi-degrees-of-freedom (MDOF) nonlinear dynamical systems driven by fractional Gaussian noise (fGn) is that the system response is not Markov process diffusion and thus the diffusion process theory cannot be applied. Although the stochastic averaging method (SAM) for quasi Hamiltonian systems driven by fGn has been developed, the response of the averaged systems still needs to be predicted by using Monte Carlo simulation. Later, noticing that fGn has rather flat power spectral density (PSD) in certain frequency band, the SAM for MDOF quasi-integrable and nonresonant Hamiltonian system driven by wideband random process has been applied to investigate the response of quasi-integrable and nonresonant Hamiltonian systems driven by fGn. The analytical solution for the response of an example was obtained and well agrees with Monte Carlo simulation. In the present paper, the SAM for quasi-integrable and resonant Hamiltonian systems is applied to investigate the response of quasi-integrable and resonant Hamiltonian system driven by fGn. Due to the resonance, the theoretical procedure and calculation will be more complicated than the nonresonant case. For an example, some analytical solutions for stationary probability density function (PDF) and stationary statistics are obtained. The Monte Carlo simulation results of original system confirmed the effectiveness of the analytical solutions under certain condition.
The load frequency control (LFC) aims to keep the frequency fluctuation of the power grids within certain specified limits, under various load disturbances. However, with the increased usage of renewable energy sources (RESs) in smart grids, it is essential to regulate the conventional power plants, based on renewable energy penetration levels. Moreover, with the decentralized nature of the control operation in smart grids, the communication network between the control center and actuator faces the challenge of random communication delays and packet drops in the form of cyberattacks. In this article, the conventional thermal power plant operations within an LFC have been modified using energy storage elements with an emphasis on maximizing the RES utilization while tackling the problems associated with cyber-physical systems, such as packet drops and random time delays. A filtered proportional–integral–derivative (PID) controller is tuned in the LFC using the particle swarm optimization (PSO) algorithm, including random time delays and cyberattacks modeled as random packet drops. The tuned PID control performance in the LFC scheme is tested with synthetic stochastic as well as real profiles of RES and load demands. The numerical analysis has been conducted on two-area LFC model with Monte Carlo simulations of stochastic demand and generation profiles.
Note to Practitioners
—We are moving toward more renewable energy-based cyber-physical power grids, and it is becoming increasingly important to understand the limits of the balance between more renewable energy integration versus changing the prime-mover in thermal power generation units to meet uncertain load demands. This article proposes a new load frequency control (LFC) scheme employing a nonlinear dead-zone element between the control signals and the actuators (prime movers), which allows the utilization of the available renewable energy sources (RESs) to meet the load and then send a set-point change command in the thermal power generators if the RES does not meet the load. The robustness of the smart grid stability is also verified with the denial-of-service (DoS)-type cyberattack, which is represented in the form of high probability of packet drops and stochastic time delays in the communication channels between the control center and the generation units. It is also essential to understand how different stochastic profiles may affect the stability and performance of the tuned LFC loops, under random time delays and packet dropouts. These are quantified using the uncertainty bounds of grid frequency, its rate of change, control inputs, and power exchange between the two areas that are analyzed using Monte Carlo simulations with different types of nonstationary load and RES profiles and also using real data to show the effectiveness of the LFC scheme with communication constraints and the resulting imperfections.
In this dissertation, different data driven (machine learning) models are employed in the field of hydrological related problems (runoff, sediment, evaporation, drought etc.,), whereas the conventional approaches are cumbersome and complex in view of computational analysis. A good physical understanding of the hydrologic process being modelled can help in selecting the input vector and designing a more efficient network.
The reliability of quasi integrable and non-resonant Hamiltonian system under fractional Gaussian noise (fGn) excitation is studied. Noting rather flat fGn power spectral density (PSD) in most part of frequency band, the fGn is innovatively regarded as a wide-band process. Then, the stochastic averaging method for quasi integrable Hamiltonian systems under wide-band noise excitation is applied to reduce 2n-dimensional original system into n-dimensional averaged Itô stochastic differential equations (SDEs). Reliability function and mean first passage time are obtained by solving the associated backward Kolmogorov equation and Pontryagin equation. The validity of the proposed procedure is tested by applying it to an example and comparing the numerical results with those from Monte Carlo simulation.
The intermittency of wind turbine power is an important issue limiting the massive development of this renewable energy. To address this issue, we consider the theoretical framework of multifractal energy cascades, which is a classical framework for describing and characterizing the fluctuations in the turbulent wind input. The multi-scaling statistics of the input turbulent wind are inherited by the wind power produced, and these multi-scaling statistics correspond to a memory in the process. There is memory coming from the fact that the Hurst scaling exponent is smaller than 1/2, and memory coming from the scale invariant cascade process generating intermittency. This memory can be exploited for prediction purposes. Here we test an approach based on an analogy of the power scaling properties with a fractional brownian motion. This is illustrated on real exploitation systems.
The prevalent multifractal characteristics of turbulent velocity fluctuations in the atmosphere are important for estimating various wind effects in wind engineering. Here, the multifractal characteristics of turbulent velocity fluctuations, including the small-scale multiscaling, the long-tail distributions and the intermittency, are thoroughly investigated by using a high-frequency dataset of three-dimensional velocities (100 Hz) collected at three levels during one month. To reduce uncertainties in the estimate of multiscaling exponents, a new method, the sequential extended self-similarity, is proposed. Based on this method, we obtain the multiscaling exponents of qth-order moments of velocity increments as a function of q, that is the so-called multifractal spectrum. The multifractal random walk (MRW) model is then shown to describe the various multifractal spectra of turbulent velocity fluctuations. With the help of this model, we find a connection between the small-scale multiscaling and the long-tail distributions, which is generally observed in our dataset, again validating the MRW model. A non-linear multifractal spectrum is commonly considered to be related to the intermittency of turbulent velocity fluctuations at small scales and its curvature is usually used as a quantification of intermittency intensity. However, we suggest that models capturing the non-linear multifractal spectrum may fail to represent the long-tail distribution, which is a more direct quantification of intermittency. Finally, qualitative variations of validated indicators with specific boundary-layer parameters are investigated. Results show that the intermittency of turbulent velocity fluctuations is more relevant to the friction velocity, compared with the average wind speed, the average temperature, and the surface-layer stability.
In this paper, a class of linear stochastic systems driven by fractional Brownian motion are investigated. The fractional infinitesimal operator and stability criterion based on the Lyapunov approach for the systems with fractional stochastic noise are employed, which are different from the results of classical stochastic systems. Firstly, the robust H ∞ filtering problem is studied, and the stochastic stability and H ∞ performance of the filtering error system are guaranteed by the feasibility of linear matrix inequalities. Secondly, robust H ∞ control problem is investigated, and the closed-loop system driven by a fractional Brownian motion is stochastically stable and has H ∞ performance if some linear matrix inequalities are feasible under the designed controller. Finally, two numerical examples show the effectiveness and correctness of the proposed methods.
This paper addresses the robust observer design problem for uncertain time-delay fractional Ito stochastic systems. A sliding surface is proposed and it is shown that state estimates converge towards it and remain there for the subsequent time. Additionally, by constructing a novel Lyapunov functional, a sufficient condition for the stability of the sliding motion of the estimated states is given in the form of Linear Matrix Inequalities (LMIs). It is demonstrated that the state estimates are stabilizable in probability provided that the LMI is feasible. Moreover, a finite-time sliding mode control law based on the estimated states is proposed. Simulation examples are given to show the validity and effectiveness of the results. © 2018
The stationary response of multidegree-of-freedom (MDOF) strongly nonlinear system to fractional Gaussian noise (FGN) with Hurst index 1/2 < H < 1 is studied. First, the system is modeled as FGN-excited and -dissipated Hamiltonian system. Based on the integrability and resonance of the associated Hamiltonian system, the system is divided into five classes: partially integrable and resonant, partially integrable and nonresonant, completely integrable and resonant, completely integrable and nonresonant, and nonintegrable. Then, the averaged fractional stochastic differential equations (SDEs) for five classes of quasi-Hamiltonian systems with lower dimension and involving only slowly varying processes are derived. Finally, the approximate stationary probability densities and other statistics of two example systems are obtained by numerical simulation of the averaged fractional SDEs to illustrate the application and compared with those from original systems to show the advantages of the proposed procedure.
Many wind speed forecasting approaches have been proposed in literature. In this paper a new statistical approach for jointly predicting wind speed, wind direction and air pressure is introduced. The wind direction and the air pressure are important to extend the forecasting accuracy of wind speed forecasts. A good forecast for the wind direction helps to bring the turbine into the predominant wind direction. We combine a multivariate seasonal time varying threshold autoregressive model with interactions (TVARX) with a threshold seasonal autoregressive conditional heteroscedastic (TARCHX) model. The model includes periodicity, conditional heteroscedasticity, interactions of different dependent variables and a complex autoregressive structure with non-linear impacts. In contrast to ordinary likelihood estimation approaches, we apply a high-dimensional shrinkage technique instead of a distributional assumption for the dependent variables. The iteratively re-weighted least absolute shrinkage and selection operator (LASSO) method allows to capture conditional heteroscedasticity and a comparatively fast computing time. The proposed approach yields accurate predictions of wind speed, wind direction and air pressure for a short-term period. Prediction intervals up to twenty-four hours are presented.
Wind energy, inserted into the power grid by wind turbines, is strongly influenced by the turbulent fluctuations of wind speed in the atmospheric layer. Here we investigate the power production of a wind farm and show that due to the presence of large-scale and long-time correlation in wind velocity, turbines interact with each other. This interaction can result in phase locking in pairs of turbines. We show that there are time intervals during which some pairs of turbines are temporally phase locked. This intermediate phase locking leads to the statistical effect that the short-time fluctuations of the cumulative power output of the wind farm become non-Gaussian, i.e., intermittent power production occurs. Contrary to phase-locked states, there are some time intervals where all turbines are phase unlocking and consequently the probability density function of the temporal increment of cumulative power production of the wind farm has almost Gaussian distribution. The phase-locked states, which can be distinct from phase-unlocked states by their dynamical features, are evaluated by reconstructed stochastic differential equations.
Turbulent intermittency plays a fundamental role in fields ranging from combustion physics and chemical engineering to meteorology. There is a rather general agreement that multifractals are being very successful at quantifying this intermittency. However, we argue that cascade processes are the appropriate and necessary physical models to achieve dynamical modeling of turbulent intermittency. We first review some recent developments and point out new directions which overcome either completely or partially the limitations of current cascade models which are static, discrete in scale, acausal, purely phenomenological and lacking in universal features. We review the debate about universality classes for multifractal processes. Using both turbulent velocity and temperature data, we show that the latter are very well fitted by the (strong) universality, and that the recent (weak, log-Poisson) alternative is untenable for both strong and weak events. Using a continuous, space-time anisotropic framework, we then show how to produce a causal stochastic model of intermittent fields and use it to study the predictability of these fields. Finally, by returning to the origins of the turbulent "shell models" and restoring a large number of degrees of freedom (the Scaling Gyroscope Cascade, SGC models) we partially close the gap between the cascades and the dynamical Navier–Stokes equations. Furthermore, we point out that beyond a close agreement between universal parameters of the different modeling approaches and the empirical estimates in turbulence, there is a rather common structure involving both a "renormalized viscosity" and a "renormalized forcing". We conclude that this gives credence to the possibility of deriving analytical/renormalized models of intermittency built on this structure.
Depuis quelques décennies, l'énergie éolienne connaît une croissance considérable. Cependant cette énergie est dépendante de la vitesse du vent variant en intensité sur des échelles de temps qui incluent l'année, la journée à quelques secondes. Il est donc fondamental de bien parvenir à comprendre et décrire le caractère non-linéaire et stochastique de ces fluctuations dans la production électrique issue des éoliennes.
L'objectif de ce travail est de caract\'eriser les fluctuations d'une série temporelle de production éolienne. Dans un premier temps, nous vérifions l'utilisation de la loi de Taylor, relation de puissance entre l'écart-type et la moyenne. Cette relation fut observée en écologie, en finance, dans les sciences du vivant et pour des données de traffic internet. De récents travaux fournissent des hypothèses d'explication quant à l'origine de cette loi \cite{Agata2010,Kendall2011}. L'exposant alpha caractérise le type de dynamique du processus considéré et varie entre 1/2 et 1. Dans notre cas d'étude, l'estimation de l'exposant alpha est proche de 1. Quand alpha=1, les processus considérés sont à invariance d'échelle \cite{eisler2008}.
Pour mettre en évidence les propriétés d'invariance d'échelle de notre série temporelle, nous effectuons une analyse multifractale pour estimer la fonction exposant d' échelle l'aide des moments d'ordre q de l'incrément temporel des données de production éolienne , telle que \cite{Schertzer1997}. La fonction est concave et non-linéaire: plus elle est concave, plus la série analysée sera intermittente. Nous montrons que la série temporelle de la production éolienne considérée est intermittente et possède des propriiétés multifractales. De plus le modèle de cascades aléatoires log-normal se révèle pertinent pour décrire ces fluctuations.
Measurements are presented of the velocity structure function on the axis of a turbulent jet at Reynolds numbers Rλ [less-than-or-equal] 852 and in a turbulent duct flow at Rλ = 515. Moments of the structure function up to the eighteenth order were calculated, primarily with a view to establish accurately the dependence on the order of the inertial range power-law exponent and to draw conclusions about the distribution of energy transfer in the inertial range. Adequate definition of the probability density of the structure function was achieved only for moments of order n [less-than-or-equal] 10. It is shown, however, that, although the values of moments of n > 10 diverges from their true values, the dependence of the moment of the structure function on the separation r is still given to a fair accuracy for moments up to n [approximate] 18. The results demonstrate that the inertial-range power-law exponent is closely approximated by a quadratic dependence on the power which for lower-order moments (n [less, similar] 12) would be consistent with a lognormal distribution. Higher-order moments diverge, however, from a lognormal distribution, which gives weight to Mandelbrot's (1971) conjecture that ‘Kolmogorov's third hypothesis’ is untenable in the strict sense. The intermittency parameter μ, appearing in the power-law exponent, has been determined from sixth-order moments [left angle bracket](δμ)6[right angle bracket] [similar] r2−μ to be μ = 0.2 ± 0.05. This value coincides with that determined from non-centred dissipation correlations measured in identical conditions.
Cambridge Core - Nonlinear Science and Fluid Dynamics - Turbulence - by Uriel Frisch
A frequent challenge for wind energy applications is to grasp the impacts of turbulent wind fields properly. The wind turbine power performance curve, usually determined applying the IEC standard 61400-12, provides a functional relation between the measured wind speed u and the corresponding power output P of the turbine. But it is not sufficient to describe the actual dynamics of the power output on small time scales. Recently, we introduced an alternative method to estimate the power curve of a wind turbine based on high-frequency data. The crucial point of this method is that we first reconstruct the wind turbine's entire response dynamics that, then, yields the power curve by a fixed-point analysis. To reconstruct the power conversion dynamics of the wind turbine, we apply a dynamical systems approach, assuming that the variable P(t) follows a diffusion process. Its evolution in time is given by a Langevin equation consisting of a deterministic relaxation term and a stochastic noise term. The coefficients defininig these terms can directly be extracted from the given data sets. On the basis of simulated data, we verified the method and showed that it is more accurate than the standard method. For measured data we reconstructed the deterministic dynamics of a representative wind turbine and estimated the corresponding power characteristics to demonstrate the application of the method. The split of the dynamics into a deterministic and a stochastic part allows to distinguish the actual behaviour of the wind turbine, i.e. relaxation and control effects, from the turbulent effects of the wind. We gain an understanding of something like the turbulent dynamics of wind energy conversion and obtain a method to describe the actual fluctuations in power output.
A summary of experimental results on structure functions obtained using extended self-similarity in various flow configurations (jet, grid, mixing layer, duct flow, cylinder) at Reynolds numbers ranging between 30 and 5000 is presented.
We study wind turbulence with the help of universal multifractals, using atmospheric high resolution time series. We empirically determine the three universal indices (H, C1, and α) which are sufficient to characterize the statistics of turbulence. The first, H, which characterizes the conservation of the field, is theoretically and empirically known to be ≈1/3, while C1 corresponds to the inhomogeneity of the mean field (C1=0 for homogeneous fields, and C1>0 for inhomogeneous and intermittent fields). The most important index is the Lévy index α corresponding to the degree of multifractality (0≤α≤2, α=0 for a monofractal). The two latter indices are directly obtained by applying the double trace moment technique (DTM) on the turbulent field. Analyzing various atmospheric velocity measurements we obtain: α≈1.45±0.1 and C1≈0.25±0.1. These results show that atmospheric turbulence has nearly the same multifractal behavior everywhere in the boundary layer, corresponding to unconditionally hard multifractal (α≥1) processes. This describes the entire hierarchy of singularities of the Navier-Stokes equations.
The existence of universal power laws at low wavenumbers (K) in the energy spectrum (Eu) of the turbulent longitudinal velocity (u) is examined theoretically and experimentally for the near-neutral atmospheric surface layer. Newly derived power-law solutions to Tchen's approximate integral spectral budget equation are tested for strong- and weak-interaction cases between the mean flow and turbulent vorticity fields. To verify whether these solutions reproduce the measured Eu at low wavenumbers, velocity measurements were collected in the dynamic sublayer of the atmosphere at three sites and in the inner region of a laboratory open channel. The atmospheric surface layer measurements were carried out using triaxial sonic anemometers over tall corn, short grass, and smooth desert-like sandy soil. The open channel measurements were performed using a two-dimensional boundary-layer probe above a smooth stainless steel bed. Comparisons between the proposed analytical solution for Eu, the dimensional analysis by Kader and Yaglom, and the measured Haar wavelet Eu spectra are presented. It is shown that when strong interaction between the mean flow and turbulent vorticity field occurs, wavelet spectra measurements, predictions by the analytical solution, and predictions by the dimensional analysis of Kader-Yaglom (KY) are all in good agreement and confirm the existence of a -1 power law in Eu(= Cuuu2
* K-1, where Cuu is a constant and u* is the friction velocity). The normalized upper wavenumber limit of the -1 power law (Kz = 1, where z is the height above the zero-plane displacement) is estimated using two separate approaches and compared to the open channel and atmospheric surface-layer measurements. It is demonstrated that the measured upper wavenumber limit is consistent with Tchen's budget but not with the KY assumptions. The constraints as to whether the mean flow and turbulent vorticity strongly interact are considered using a proposed analysis by Panchev. It is demonstrated that the arguments by Panchev cannot be consistent with surface-layer turbulence. Using dimensional analysis and Heisenberg's turbulent viscosity model, new constraints are proposed. The new constraints agree with the open channel and atmospheric surface-layer measurements, Townsend's inactive eddy motion hypothesis, and the Perry et al. analysis.
Regions interesting for the installation of windmills possess large mean wind velocity, but this is also associated to large Reynolds numbers and huge velocity fluctuations. Such fully developed turbulence is generally characterized by intermittent fluctuations on a wide range of scales. There are many gusts (large fluctuations of the wind velocity at small scales), producing extreme loading on wind turbine structure. Here we consider the dynamics of gusts: the recurrent times of large fluctuations for a fixed time increment, and also “inverse times” corresponding to the times needed to have wind fluctuations larger than a fixed threshold. We discuss theoretically these two different quantities, and provide an example of data analysis using wind data recorded at 10 Hz. The objective here is to provide theoretical relations that will help for the interpretation of available data, and for their extrapolation.
Basic problems arising in the growing use of the wind energy are discussed. We mainly focus on problems which are related to the turbulent character of the wind. The meaning of large and meso-scale atmospheric turbulence for the energy production is discussed. We show for the wind turbulence on small scales how statistics with anomalous probabilities arise, which can be set into the context of wind gusts. Furthermore, we discuss a new method to extract a stochastic process from given data which enables a new and more profound statistical description of the complexity of turbulence and turbulent wind fields.
The structure df atmospheric surface layer turbulence low wavenumbers was analyzed using 56 Hz triaxial velocity and temperature measurements above a uniform dry lake bed, A key feature of this experiment was the, small roughness length of the surface that resulted in a small roughness Reynolds number. Under near-neutral atmospheric stability conditions, a -1 power law was observed in both measured velocity and temperature spectra which is consistent with previously proposed dimensional analysis for rough and smooth turbulent boundary layer flows. The wavenumber at which the -1 power law terminates and the -5/3 power law commences was derived as a function of the Kolmgorov and von Karman constants, Good agreement between the predicted and the measured transition. wavenumber from -1 to -5/3, was holed for fully rough-flow conditions. However, this was ndt the case fdr other roughness conditions. The similarity theory constants for the neutral case were determined and they compared favorably with other laboratory and field studies. For unstable atmospheric conditions, directional dimensional analysis was used to predict the slopes of the power spectra of temperature and velocity. It was demonstrated that for moderately unstable conditions, the temperature and vertical velocity power spectra exhibited a -1 power law, but the longitudinal velocity exhibited a -2 power law. The agreement between predicted and measured power laws was within experimental errors. Some differences between the constants determined from this experiment and other experiments are also discussed.
We introduce a class of multifractal processes, referred to as multifractal random walks (MRWs). To our knowledge, it is the first multifractal process with continuous dilation invariance properties and stationary increments. MRWs are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio. The MRWs are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments. We briefly explain how, in the same way, one can build stationary multifractal processes or positive random measures.
We define a large class of continuous time multifractal random measures and processes with arbitrary log infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal multifractal random walk [J.F. Muzy, J. Delour, and E. Bacry, Eur. J. Phys. B 17, 537 (2000), E. Bacry, J. Delour, and J.F. Muzy, Phys. Rev. E 64, 026103 (2001)] and the log-Poisson "product of cylindrical pulses" [J. Barral and B.B. Mandelbrot, Cowles Foundation Discussion Paper No. 1287, 2001 (unpublished)]. Our construction is based on some "continuous stochastic multiplication" [as introduced in F. Schmitt and D. Marsan, Eur. J. Phys. B. 20, 3 (2001)] from coarse to fine scales that can be seen as a continuous interpolation of discrete multiplicative cascades. We prove the stochastic convergence of the defined processes and study their main statistical properties. The question of genericity (universality) of limit multifractal processes is addressed within this new framework. We finally provide a method for numerical simulations and discuss some specific examples.
this paper is to build a multifractal process X(t), referred to as a Multifractal Random Walk (MRW), with stationary increments and such that Eq. (1) holds for all l
Atmospheric wind speeds and their fluctuations at different locations (onshore and offshore) are examined. One of the most striking features is the marked intermittency of probability density functions (PDF) of velocity differences -- no matter what location is considered. The shape of these PDFs is found to be robust over a wide range of scales which seems to contradict the mathematical concept of stability where a Gaussian distribution should be the limiting one. Motivated by the instationarity of atmospheric winds it is shown that the intermittent distributions can be understood as a superposition of different subsets of isotropic turbulence. Thus we suggest a simple stochastic model to reproduce the measured statistics of wind speed fluctuations.
Multiplicative cascades have been introduced in turbulence to generate random or deterministic fields having intermittent values and long-range power-law correlations. Generally this is done using discrete construction rules leading to discrete cascades. Here a causal log-normal stochastic process is introduced; its multifractal properties are demonstrated together with other properties such as the composition rule for scale dependence and stochastic differential equations for time and scale evolutions. This multifractal stochastic process is continuous in scale ratio and in time. It has a simple generating equation and can be used to generate sequentially time series of any length.
§1. We shall denote by u α ( P ) = u α ( x 1 , x 2 , x 3 , t ), α = 1, 2, 3, the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 , x 2 , x 3 . In considering the turbulence it is natural to assume the components of the velocity u α ( P ) at every point P = ( x 1 , x 2 , x 3 , t ) of the considered domain G of the four-dimensional space ( x 1 , x 2 , x 3 , t ) are random variables in the sense of the theory of probabilities (cf. for this approach to the problem Millionshtchikov (1939) Denoting by Ᾱ the mathematical expectation of the random variable A we suppose that ῡ ² α and (d u α /d x β ) ² ― are finite and bounded in every bounded subdomain of the domain G .
Two limits of Newtonian mechanics were worked out by Kolmogorov. On one side it was shown that in a generic integrable Hamiltonian system, regular quasi-periodic motion persists when a small perturbation is applied. This result, known as Kolmogorov-Arnold-Moser (KAM) theorem, gives mathematical bounds for integrability and perturbations. On the other side it was proven that almost all numbers on the interval between zero and one are uncomputable, have positive Kolmogorov complexity and, therefore, can be considered as random. In the case of nonlinear dynamics with exponential (i.e. Lyapunov) instability this randomness, hidden in the initial conditions, rapidly explodes with time, leading to unpredictable chaotic dynamics in a perfectly deterministic system. Fundamental mathematical theorems were obtained in these two limits, but the generic situation corresponds to the intermediate regime between them. This intermediate regime, which still lacks a rigorous description, has been mainly investigated by physicists with the help of theoretical estimates and numerical simulations. In this contribution we outline the main achievements in this area with reference to specific examples of both low-dimensional and high-dimensional dynamical systems. We shall also discuss the successes and limitations of numerical methods and the modern trends in physical applications, including quantum computations.
A general hierarchical statistical framework for the characterization of wind turbulence is proposed. This framework should ease the understanding of different existing statistical approaches and enable a clear path for refined methods of characterization. Low-order statistical descriptions were extended by higher-order statistics with respect to one-point and two-point statistics. In particular, we showed that proper analysis leads to a superstatistics approach for the probability of velocity fluctuations. To demonstrate the importance of our considerations, we analysed wind time series of the research platform FINO 1. On one side, we showed how different statistical aspects can be reproduced quite accurately, whereas on the other, the necessity of more profound approaches was worked out. The analysis of the measured data provides some deep insights into the nature of wind turbulence. Most interestingly, for conditioned data subsets, we found higher-order statistical behavior well known for idealized turbulence. Finally, we give an outlook on how to achieve a general n-point statistical description. Copyright © 2011 John Wiley & Sons, Ltd.
In this paper we present the results of a four month measurement campaign of the wind velocity at the wind energy production site of Petit Canal in Guadeloupe (FWI). Data acquisitions are performed with a sampling rate of 1 Hz. This allows an analysis of the statistical properties of the turbulence for time scale smaller than one hour. The statistical properties of the wind speed fluctuations are identified for time scales ranging from 600s to 3600s. A method to classify the wind speed fluctuations characteristics in different classes, involving different statistical properties, is proposed. The isotropy of the turbulence characteristics is also investigated. The results are finally discussed from the point of view of the short time range wind forecast.
This paper explores the wind stochastic field from a new viewpoint of stochastic Fourier spectrum (SFS). The basic random parameters of the wind stochastic field, the roughness length z0 and the mean wind velocity at 10m height U10, as well as their probability density functions (PDF), are obtained. It provides opportunities to use probability density evolution method (PDEM), which had been proved to be of high accuracy and efficiency, in computing the dynamic response and reliability of tall buildings subject to the wind loading. Principals and corresponding numerical solving algorithm of the PDEM are first presented. Then, the adopted model of the wind stochastic field is described briefly. The simulation method of the fluctuating wind velocity based on the SFS is introduced. Finally, as an example of the application of the PDEM, a 20-storey frame subject to wind loading is investigated in detail. The responses, including the mean value and the standard deviation, and the reliabilities of the frame are evaluated by the PDEM. The results demonstrate that the PDEM is applicable and efficient in the dynamic response and reliability analysis of wind-excited tall building.
A method for simulating a stationary Gaussian process on a fine rectangular grid in [0, 1] ⊂ℝ is described. It is assumed that the process is stationary with respect to translations of ℝ, but the method does not require the process to be isotropic. As with some other approaches to this simulation problem, our procedure uses discrete Fourier methods and exploits the efficiency of the fast Fourier transform. However, the introduction of a novel feature leads to a procedure that is exact in principle when it can be applied. It is established that sufficient conditions for it to be possible to apply the procedure are (1) the covariance function is summable on ℝ, and (2) a certain spectral density on the d-dimensional torus, which is determined by the covariance function on ℝ, is strictly positive. The procedure can cope with more than 50,000 grid points in many cases, even on a relatively modest computer. An approximate procedure is also proposed to cover cases where it is not feasible to apply the procedure in its exact form.
The purpose of this paper is to establish a spectral theory for certain types of random fields and random generalized fields (multidimensional random distributions) in the Euclidean n-space similar to the well-known spectral theory for stationary random processes.Let D denote the Schwartz space of all complex-valued functions defined on whose carrier is compact. Following Ito [6] and Gelfand [7] we shall call the random linear functional on D satisfying (1.4) the random generalized field. We can identify a continuous random field on with a random generalized field (1.5) and therefore we can consider the ordinary random fields as special cases of random generalized fields.We shall only deal with the first moment and the second moment of the random generalized field and shall call them a mean value functional and a covariance functional of this fie...
Project development. Initial site selection, project feasibility assessment including the measure-correlate-predict technique for estimating energy yields from wind farm sites, micro-siting of turbines, the importance of public consultation and an overview of the preparation of environmental impact assessments.Visual and landscape assessment. Wind farm design and mitigation measures, assessment of visual impact and the use of Zones of Visual Impact (ZVI), wire-frame representations and photomontages.Noise. Terminology and basic concepts, sources of noise from a wind turbine, measurement and prediction of wind farm noise.Electro-magnetic interference. Impact of wind farms on various types of communication signals, modelling and prediction of electro-magnetic interference from wind turbines.Ecological assessment. Impact on birds.Financing wind farm developments. Project appraisal using discounted cash flow techniques, project finance and support mechanisms for wind energy development.
The hypotheses concerning the local structure of turbulence at high Reynolds number, developed in the years 1939-41 by myself and Oboukhov (Kolmogorov 1941 a,b,c; Oboukhov 1941 a,b) were based physically on Richardson's idea of the existence in the turbulent flow of vortices on all possible scales l < r < L between the L and the l and of a certain uniform mechanism of energy transfer from the coarser-scaled vortices to the finer.
The wind farm power collection system.Earthing (grounding) of wind farms against power system faults and transient overvoltages.Wind turbine lightning protection systems.Embedded (dispersed) wind generation. Electrical distribution networks and the impact of dispersed generation, the per-unit system, power flows and voltages in simple radial distribution networks, connection of embedded wind generation, power system studies.Power (voltage) quality. Voltage flicker, harmonics from variable speed wind turbines, measurement and assessment of power quality of grid connected wind turbines.Electrical protection. Wind farm and generator protection, islanding and self-excitation of induction generators, protection of the interface of the wind farm with the distribution network.Economic aspects of dispersed generation. Losses in the distribution network, reactive power charges and voltage control, connection charges “deep” and “shallow”, use of system charges, impact of wind energy on central generators.
The multiscaling statistics of atmospheric surface-layer winds at low wavenumbers above farmland and in the lee of a mountain range were examined using a hot-wire and lightweight cup anemometer. It was found that the horizontal velocity spectra could be broken into high and low-wavenumber regimes according to the parameters given by this analysis. The low-wavenumber end of the spectrum possessed a spectral slope parameter that varied between values of 0.8 and 1.35 at the farmland site during the period of the experiment, and the high-wavenumber end – corresponding to the inertial range – possessed a spectral slope slightly greater than -5/3. The larger values for this parameter for the low-wavenumber end appeared to coincide with unstable conditions. In the lee of the mountain range, the low-wavenumber spectral slope parameter was larger still, at 1.45. The low-wavenumber signals over farmland were much less intermittent than inertial-range signals, but in the lee of the mountain range the intermittency increased. From this analysis, it was shown that the statistical properties of the recorded wind signal could be reproduced using a bounded random multiplicative cascade. The model was successfully used to simulate the wind velocity field directly, rather than simulating the energy dissipation field. Since the spectral slope parameter for low wavenumbers appeared to be a function of atmospheric stability, the method presented is a simple way of generating wind signals characteristic of a variety of atmospheric conditions.
Literature on turbulence modeling is rich in empirical, semi-empirical and theoretical spectral equations whose parameters assume deterministic values. Starting from a critical review of the state of the art, this paper proposes a unified model of atmospheric turbulence especially suited to determine the 3-D gust-excited response of structures. Unlike classical models, all parameters are assigned through first and second order statistical moments derived from a wide set of selected experimental measurements. A general discussion is also provided about model errors and other sources of randomness. Due to these properties the model proposed is suitable for carrying out reliability analyses which take into account the propagation of the uncertainties.
The behaviour of spectra and cospectra of turbulence in the surface layer is described within the framework of similarity theory using wind and temperature fluctuation data obtained in the 1968 AFCRL Kansas experiments. With appropriate normalization, the spectra and cospectra are each reduced to a family of curves which spread out according to z/L at low frequencies but converge to a single universal curve in the inertial subrange. The paper compares these results with data obtained by other investigators over both land and water.
Spectral constants for velocity and temperature are determined and the variability in the recent estimates of the constants is discussed. The high-frequency behaviour is consistent with local isotropy. In the inertial subrange, where the spectra fall as n−5/3, the cospectra fall faster: uω and ωθ as n−7/3, and uθ, on the average, as n−5/2. The 4/3 ratio between the transverse and longitudinal spectral levels is observed at wavelengths of the order of the height above ground under unstable conditions and at wavelengths of the order of L/10 under stable conditions. This lower isotropic limit is shown to be governed by the combined effects of shear and buoyancy on small-scale eddies.
Hilbert-Huang transform is a method that has been introduced recently to decompose nonlinear, nonstationary time series into a sum of different modes, each one having a characteristic frequency. Here we show the first successful application of this approach to homogeneous turbulence time series. We associate each mode to dissipation, inertial range and integral scales. We then generalize this approach in order to characterize the scaling intermittency of turbulence in the inertial range, in an amplitude-frequency space. The new method is first validated using fractional Brownian motion simulations. We then obtain a 2D amplitude-frequency representation of the pdf of turbulent fluctuations with a scaling trend, and we show how multifractal exponents can be retrieved using this approach. We also find that the log-Poisson distribution fits the velocity amplitude pdf better than the lognormal distribution.
The inertial-range scaling laws of fully developed turbulence are described in terms of scalings of a sequence of moment ratios of the energy dissipation field εl coarse-grained at inertial-range scale l. These moment ratios ε(p)l=/(p=0, 1, 2,...,) form a hierarchy of structures. The most singular structures ε(∞)l are assumed to be filaments, and it is argued that ε(∞)l~l-2/3. Furthermore, a universal relation between scalings of successive structures is postulated, which leads to a prediction of the entire set of the scaling exponents: ~ltaup, taup=-2/3p+2[1-( 2) / 3 )p] and ~lzetap, zetap=p/9+2[1-(2/3)p/3].
Stochastic modelling of a wind turbine's power output with special respect to turbulent dynamics An amplitude–frequency study of turbulent scaling intermittency using Hilbert spectral analysis
- J Gottschall
- J Peinke
Gottschall, J., Peinke, J., 2007. Stochastic modelling of a wind turbine's power output with special respect to turbulent dynamics. Journal of Physics: Conference Series 75, 012045. Huang, Y., Schmitt, F.G., Lu, Z., Liu, Y., 2008. An amplitude–frequency study of turbulent scaling intermittency using Hilbert spectral analysis. Europhysics Letters 84, 40010.
A Fit Course in Turbulence Progress in the statistical theory of turbulence
- H Tennekes
- J L Lumley
Tennekes, H., Lumley, J.L., 1972. A Fit Course in Turbulence. The MIT Press. von Karman, T., 1948. Progress in the statistical theory of turbulence. Proceedings of the National Academy of Sciences of the United States of America 34, 530–539.
The Fourier Transform and Its Applications, third ed. McGraw-Hill Science Wind Energy Handbook
- R Bracewell
- T Burton
- D Sharpe
- N Jenkins
- E Bossanyi
Bracewell, R., 1999. The Fourier Transform and Its Applications, third ed. McGraw-Hill Science. Burton, T., Sharpe, D., Jenkins, N., Bossanyi, E., 2001. Wind Energy Handbook. John Wiley & Sons, Chichester. (p. 12).