A Discrimination Method Between Transformer Inrush and Fault Current Based on Chromatic Monitoring

or successful transformer differential protection employment, correct discrimination between inrush current and fault current is essential. It is one of the main focuses of research and one of the main challenges for transformer protection. In this paper, a discrimination method based on utilizing chromatic monitoring of the box dimension algorithm outcome curve for transformer differential current in time-domain analysis is proposed. The x-L chromatic mapping is employed for general detection of fault cases, while the x-y chromatic mapping result is used for distinguishing inrush current from the fault current cycles. The preliminary results show that the proposed method can effectively provide correct discrimination of the current type within quite a short time and thus help in providing efficient decision-making supportive protection tool.

Fractal behaviours of networks induced on infinite tree structures by random walks

Tree graphs such as Cayley trees provide a stage to support the self-organization of fractal networks by the flow of walkers from the root vertex to the outermost shell of the tree graph. This network model is a typical example that demonstrates the ability of a random process on a network to generate fractality. However, the finite scale of the tree structure assumed in the model restricts the size of fractal networks. In this study, we removed the restriction on the size of the trees by introducing a lifetime τ (number of steps of random walks) of walkers. As a result, we successfully induced a size-independent fractal structure on a tree graph without a boundary. Our numerical results show that the mean number of offspring d b of the original tree structure determines the value of the fractal box dimension db through the relation d b — 1 = (n b — 1) -θ . The lifetime τ controls the presence or absence of small-world and scale-free properties. The ideal fractal behaviour can be maintained by selecting an appropriate value of τ. The numerical results contribute to the development of a systematic method for generating fractal small-world and scale-free networks while controlling the value of the fractal box dimension. Unlike other models that use recursive rules to generate self-similar structures, this model specifically produces small-world fractal networks with scale-free properties.

Health monitoring of Bridges based on multifractal theory

The multifractal theory is applied in an analysis of bridge disturbance signals with the aim of investigating their nonlinear characteristics, and then the recognisable fault features are extracted from them. By calculating the box dimension and correlation dimension of the bridge disturbance signal, the dimensional characteristics of the disturbance data are analysed to distinguish the health-state of the bridge. Finally, taking the bridge disturbance data as an example, and by using the multifractal spectrum analysis of the disturbance data, it is concluded that the multifractal method can accurately identify the fault state and realise the bridge health monitoring.

RELATIONSHIP BETWEEN UPPER BOX DIMENSION OF CONTINUOUS FUNCTIONS AND ORDERS OF WEYL FRACTIONAL INTEGRAL

In this paper, we mainly investigate relationship between fractal dimension of continuous functions and orders of Weyl fractional integrals. If a continuous function defined on a closed interval is of bounded variation, its Weyl fractional integral must still be a continuous function with bounded variation. Thus, both its Weyl fractional integral and itself have Box dimension one. If a continuous function satisfies Hölder condition, we give estimation of fractal dimension of its Weyl fractional integral. If a Hölder continuous function is equal to 0 on [Formula: see text], a better estimation of fractal dimension can be obtained. When a function is continuous on [Formula: see text] and its Weyl fractional integral is well defined, a general estimation of upper Box dimension of Weyl fractional integral of the function has been given which is strictly less than two. In the end, it has been proved that upper Box dimension of Weyl fractional integrals of continuous functions is no more than upper Box dimension of original functions.

Description of Fracture Network of Hydraulic Fracturing Vertical Wells in Unconventional Reservoirs

Based on the distribution of complex fractures after volume fracturing in unconventional reservoirs, the fractal theory is used to describe the distribution of volume fracture network in unconventional reservoirs. The method for calculating the fractal parameters of the fracture network is given. The box dimension method is used to analyze a fracturing core, and the fractal dimension is calculated. The fractal index of fracture network in fracturing vertical wells are also firstly calculated by introducing an analysis method. On this basis, the conventional dual-media model and the fractal dual-media model are compared, and the distribution of reservoir permeability and porosity are analyzed. The results show that the fractal porosity/permeability can be used to describe the reservoir physical properties more accurately. At the same time, the flow rate calculating by conventional dual-media model and the fractal dual-media model were calculated and compared. The comparative analysis found that the flow rate calculated by the conventional dual-media model was relatively high in the early stage, but the flow rate was not much different in the later stage. The research results provide certain guiding significance for the description of fracture network of volume fracturing vertical well in unconventional reservoirs.

Study on Mesoscopic Damage Evolution Characteristics of Single Joint Sandstone Based on Micro-CT Image and Fractal Theory

The different directions of joints in rock will lead to great differences in damage evolution characteristics. This study utilizes DIP (digital image processing) technology for characterizing the mesostructure of sandstone and combines DIP technology with RFPA2D. The mesoscale fracture mechanics behavior of 7 groups of jointed sandstones with various dip angles was numerically studied, and its reliability was verified through theoretical analysis. According to digital image storage principle and box dimension theory, the box dimension algorithm of rock mesoscale fracture is written in MATLAB, the calculation method of fractal dimension of mesoscale fracture was proposed, and the corresponding relationship between mesoscale fractal dimension and fracture damage degree was established. Studies have shown that compressive strength as well as elastic modulus of sandstone leads to a U-shaped change when joint dip increases. There are a total of six final failure modes of joint samples with different inclination angles. Failure mode and damage degree can be quantified by D (fractal dimension) and ω (mesoscale fracture damage degree), respectively. The larger the ω, the more serious the damage, and the greater the D, the more complex the failure mode. Accumulative AE energy increases exponentially with the increase of loading step, and the growth process can be divided into gentle period, acceleration period, and surge period. The mesoscale fracture damage calculation based on the fractal dimension can be utilized for quantitatively evaluating the spatial distribution characteristics of mesoscale fracture, which provides a new way to study the law of rock damage evolution.

Dimensions of Fractional Brownian Images

This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$$\alpha $$ α fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.

Micro-morphologies of SBS modifier at mortar transition zone in asphalt mixture with thin sections and fluorescence analysis

The microstructure control of modified asphalt, especially the micro-dispersion of the SBS modifier in the mortar transition zone, is a critical technology for the performance design of modified asphalt. To characterize the micro-dispersive morphology of SBS modifiers, thin-section preparation techniques that can be used to analyze the original microstructure of the asphalt mixture were proposed and introduced in this study. Flexible resin is filled into the mixture at vacuum conditions to ensure accepted sample conditions for preparing thin sections of asphalt mixture. The morphology parameters, including SBS area ratio, box dimension, SBS average particle area and its coefficient of variation, area-weighted average axis ratio, and coefficient of variation, were plotted from fluorescence images to characterize the micro-morphological distribution of the SBS modifier in detail. Results have shown that the area ratio increased with the increase in SBS content, while the box dimension was reduced and the distribution uniformity of the particles decreased. The superfluous SBS modifier in the binder at a too high adding ratio will decrease the value of the box dimension. Lower modification temperature worsened the SBS modifier in the mixture, resulting in a wide range of particle size, higher axis ratio, and higher area ratio. The micro-morphologies of SBS in the asphalt mixture phase varied a lot from the asphalt binder phase. The additional materials of mineral filler and fine aggregate, together with the other heating processes, will significantly influence the swelling state and particle size of the SBS modifier.