The.main types of wind turbines-generators in the power supply system

THE PURPOSE. This paper considers the problem of relay protection functioning when the current transformer reaches the saturation mode which is provided by transient processes.METHODS. MATLAB Simulink software environment allows reproducing the method of statespace representation by using structural blocks. The model is verified by comparison the time to saturation, obtained by calculation and according to the graphical data of the model. The separation of variables method extracts and graphically displays the investigated components.RESULTS. This paper reveals that applying the requirements of IEC 61869-2:2012 standard, which determines the worst combination of series of unfavorable factors for current transformers in transient mode, can influence a serious impact on the correct operation of relay protection based on current, reactance or differential principle of action. Saturation of the current transformer can lead to both negative results: false operation of relay protection devices and their failure.CONCLUSION. According to the results of the study, it was determined that the presence of a DC component in the primary short-circuit current has the greatest effect on the protection operation. The delays in the restoration of the RMS value of the short-circuit current reached up to 0.3 seconds, which is comparable with the response time of the second protection zones for microprocessor-based relay protection devices. The DC component of the primary current and the presence of residual magnetic induction of the current transformer provides the largest content of the magnetization current, the largest angular error and also the largest content of the second harmonic component in the secondary short-circuit current.

On a nonlocal problem for a fourth-order mixed-type equation with the Hilfer operator

The present work is devoted to the study of the solvability questions for a nonlocal problem with an integrodifferential conjugation condition for a fourth-order mixed-type equation with a generalized RiemannLiouville operator. Under certain conditions on the given parameters and functions, we prove the theorems of uniqueness and existence of the solution to the problem. In the paper, given example indicates that if these conditions are violated, the formulated problem will have a nontrivial solution. To prove uniqueness and existence theorems of a solution to the problem, the method of separation of variables is used. The solution to the problem is constructed as a sum of an absolutely and uniformly converging series in eigenfunctions of the corresponding one-dimensional spectral problem. The Cauchy problem for a fractional equation with a generalized integro-differentiation operator is studied. A simple method is illustrated for finding a solution to this problem by reducing it to an integral equation equivalent in the sense of solvability. The authors of the article also establish the stability of the solution to the considered problem with respect to the nonlocal condition.

Space and Genotype-Dependent Virus Distribution during Infection Progression

The paper is devoted to a nonlocal reaction-diffusion equation describing the development of viral infection in tissue, taking into account virus distribution in the space of genotypes, the antiviral immune response, and natural genotype-dependent virus death. It is shown that infection propagates as a reaction-diffusion wave. In some particular cases, the 2D problem can be reduced to a 1D problem by separation of variables, allowing for proof of wave existence and stability. In general, this reduction provides an approximation of the 2D problem by a 1D problem. The analysis of the reduced problem allows us to determine how viral load and virulence depend on genotype distribution, the strength of the immune response, and the level of immunity.

Analytical Subdomain Model for Double-Stator Permanent Magnet Synchronous Machine with Surface-Mounted Radial Magnetization

This paper proposes an analytical subdomain model for predicting magnetic field distributions in a three-phase double-stator permanent magnet synchronous machine (DS-PMSM) during open-circuit and on-load conditions. The geometric structure of DS-PMSM is quite challenging since the stator cores are located in the outer and inner parts of the motor, while the rotor magnets are placed between these two stators. Parameters that influence the motor performance in DS-PMSM include stator outer radius, stator inner radius, magnet thickness, magnet arc, slot opening, outer and inner airgap thickness and the number of winding turns. The analytical subdomain model proposed in this paper, which can accurately predict the performances of DS-PMSM with less computational time, has an excellent advantage as a rapid design tool. The model is initially generated using the separation of variables technique in four subdomains, namely, outer airgap, outer magnet, inner magnet, and inner airgap, based on Laplace’s and Poisson’s equations in polar coordinates. The field solutions in each subdomain are derived by applying the appropriate boundary and interface conditions. Furthermore, finite element analysis (FEA) is used to validate the analytical results in fractional DS-PMSM with a different number of slots between outer and inner stators and a non-overlapping winding configuration. The electromagnetic performances that have been evaluated are the slotted airgap flux density, back-emf and output torque. The results demonstrate that the proposed analytical model is able to predict the magnetic field distributions accurately in DS-PMSM.

Retrieval of the piecewise-constant mass density in the Bergman wave equation

This investigation is concerned with the 2D acoustic scattering problem of a plane wave propagating in a non-lossy, isotropic, homogeneous fluid host and soliciting a linear, isotropic, macroscopically-homogeneous, generally-lossy, flat-plane layer in which the mass density and wavespeed are different from those of the host. The focus is on the inverse problem of the retrieval of the layer mass density. The data is the transmitted pressure field, obtained by simulation (resolution of the forward problem) in exact, explicit form via the domain integral form of the Bergman wave equation. This solution is exact and more explicit in terms of the mass-density contrast (between the host and layer) than the classical solution obtained by separation of variables. A perturbation technique enables the solution (in its form obtained by the domain integral method) to be cast as a series of powers of the mass density contrast, the first three terms of which are employed as the trial models in the treatment of the inverse problem. The aptitude of these models to retrieve the mass density contrast is demonstrated both theoretically and numerically.

MODIFIED METHOD OF SEPARATION OF VARIABLES FOR SOLVING DIFFRACTION PROBLEMS ON MULTILAYER DIELECTRIC GRATINGS

A modified method of separation of variables is proposed for solving the direct problem of diffraction of electromagnetic wave by multilayer dielectric gratings (MDG). To apply this method, it is necessary to solve a one-dimensional eigenvalue problem for a 2nd- order differential equation on a segment with piecewise constant coefficients. The accuracy of the method is verified by comparison with the results obtained by the commercially available RCWA method. It is demonstrated that the method can be applied not only to commonly used MDG elements with one line in a grating period but also to potentially promising MDG elements with several different lines in a grating period.

Integrable variable-coefficient derivative Spin-1 Gross–Pitaevskii equations and their explicit solutions

Based on the generalized dressing method, we propose a new integrable variable coefficient Spin-1 Gross–Pitaevskii equations and derive their Lax pair. Using separation of variables, we have derived explicit solutions of the equations. In order to analyze the characteristic of derived solution, the graphical wave of the solutions is plotted with the aid of Matlab.

An Improved Tunnel-Track Model in Saturated Poroelastic Soils to a Moving Point Load

To predict the mechanical response of a circular cavity/tunnel buried in saturated poroelastic soils to a moving point load, a semianalytical model is provided in this work. The soils are governed by Biot’s theory that describes the wave propagations for saturated poroelastic materials. The displacement and stress vectors for the solid skeleton and pore-water fluid are represented by scalar and vectorial potentials. The governing equations for the tunnel and surrounding soils are solved in the frequency domain with the aid of separation of variables and Fourier transformations. To check the feasibility of the present analytical model, the solution is compared with other available results calculated for the ring load case. The good agreement shows the correctness of the present model. Numerical results suggest that the mechanical response from a moving point load in a tunnel for two-phase poroelastic materials is quite different from that in single-phase elastic materials. The critical velocity of the tunnel-soil system is around the shear wave speed of soils while the second one introduced into the track-tunnel-soil system with very high value is around the critical velocity of the track structure itself.

On scalar products and form factors by Separation of Variables: the antiperiodic XXZ model

We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by Separation of Variables (SoV), and the eigenstates can be constructed in terms of Q-functions, solution of a Baxter TQ-equation, which have double periodicity compared to the periodic case. We compute in this framework the scalar products of a particular class of separate states which notably includes the eigenstates of the transfer matrix. We also compute the form factors of local spin operators, i.e. their matrix elements between two eigenstates of the transfer matrix. We show that these quantities admit determinant representations with rows and columns labelled by the roots of the Q-functions of the corresponding separate states, as in the periodic case, although the form of the determinant are here slightly different. We also propose alternative types of determinant representations written directly in terms of the transfer matrix eigenvalues.

Analytical Modeling for Water Chemistry Changes in River Bank Filtration Systems

Riverbank filtration system is considered one of the economic and sustainable solutions to river water pollution especially in tropical countries such as Malaysia. In this work, an analytical model is developed to simulate the contaminant attenuation in riverbank filtration systems by using the separation of variables method. The basic aim of the model is to understand the role of microbial activity that occurs in riverbed sediments on reducing the concentration of the contaminant in the aquifer and changing the water characteristics. Graphically, it is found that the model can simulate the infiltration process of polluted river water effectively. Also, the analytical model results, as well as experimental data, show that nitrate (18.6 and 34.1 mg-NO3/L) and sulphate (20.9 – 22.1 mg-SO4/L) can be consumed by bacteria in the first 0.5 m of the aquifer, and reduced by more than 95% for both compounds. The model is applied for the first riverbank filtration system in Malaysia. Sensitivity analysis results highlight the importance of dissolved organic matter (DOM) concentration (ranged from 1.0 to 12.4 mg/L) for RBF efficacy in which a higher concentration of DOM leads to faster consumption of pollutants.