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To study problems with the Kipriyanov singular differential operator Δ B - γ with a negative parameter - γ ∈ (- 1 , 0) , an integral transformation is introduced in the work, based on the solution u = J μ of the singular Bessel equation B - γ u + u = 0 . This solution is expressed through a Bessel function of the first kind with a positive parameter μ = γ + 1 2 . An even, odd and complete FBK transform (Fourier—Bessel—Kipriyanov—Katrakhov transform) and a class of singular K -pseudodifferential operators are constructed. Theorems on the order and commutator of singular K -pseudodifferential operators are obtained. [ABSTRACT FROM AUTHOR]

Energy storage units (ESU) and distributed generation (DG) plants including those using renewable energy sources can be used to develop isolated power supply systems (IPSS) and enhance their reliability. Operation of DG plants in the IPSS requires to consider restrictions for consumers maximum load, as well as effects of abrupt load increase or load shedding on the generating plants. These problems can be resolved using ESU and intelligent technologies for DG plants control. The paper considers IPSS of an industrial facility with turbogenerator plants (TGP), wind power plant (WPP) and high-capacity ESU for which fuzzy control systems and prognostic controllers were used. The simulation was performed in the MATLAB system using the Simulink and SimPowerSystems software packages. Normal and emergency modes were studied in IPSS with TGP, WPP, and high-capacity ESU. The simulation results indicated that combined use of high-capacity ESU and prognostic controller of TGP generator rotor speed allows to ensure stability and survivability of IPSS enhancing its damping properties. The auto-prognostic controller of the TGP generator, which does not require special adjustment, provides high quality indicators of the IPSS control in all considered modes.

The wide use of distributed generation (DG) technologies in electrical energy systems (EES) requires development of new control algorithms in normal, emergency and postemergency modes. The issues of determining the static aperiodic stability (SAS) margin in EES equipped with DG plants are of particular relevance. These plants can be removed from the consumption centers, which can lead to SAS reserves depletion. The article presents the results of studies aimed at the developing SAS reserves express calculation methods in EES equipped with DG plants. An effective technique to determine stability margins was proposed based on one of the modifications of limiting modes equations. The results of determining SAS for an electrical network with distributed generation plants are presented. Additionally, the simulation of transient processes in the studied EES for various points in the space of the mode’s controlled parameters has been carried out in the Matlab system.

The paper introduces integral transformations for studying problems with a singular differential Bessel operator with a negative parameter . We construct the -Bessel transform and the corresponding class of singular -pseudodifferential operators. A theorem on the order of -pseudodifferential operators in a special class of Sobolev–Kipriyanov functions is proved. [ABSTRACT FROM AUTHOR]

Functions satisfying the Laplace equation for the Kipriyanov -operator are said to be -harmonic. The following properties of -harmonic functions are presented and proved: the Green-type integral representation of -functions, the spherical mean theorem, and the maximum principle. As a corollary, the uniqueness of the solution of the interior and exterior Dirichlet problems is proved. [ABSTRACT FROM AUTHOR]

We introduce integral transforms to study problems for the singular Bessel differential operators B−γ with negative parameter −γ < 0. A solution to the singular Bessel equation B−γu+u = 0 is a function u = 핁μ expressed in terms of the Bessel functions of the first kind with positive parameter μ = (γ + 1)/2. Based on the notion of a 핋-pseudoshift, we formulate the Levitan addition theorem. We construct the Bessel 핁-transform and the corresponding class of singular 핁-pseudodifferential operators. We prove a theorem on the order of 핁-pseudodifferential operators in the class of Sobolev–Kipriyanov functions. [ABSTRACT FROM AUTHOR]

Solutions of singular differential equations with the Bessel operator of negative order are studied. In this regard, of great interest are the solutions of the singular differential Bessel equation , which are presented in the paper as linearly independent functions and , . Some properties of the functions expressed in terms of the properties of the Bessel–Levitan -function are considered. The direct and inverse -Bessel transforms are introduced, and the -pseudoshift operator that commutes with the Bessel operator is defined. The fundamental solution of the ordinary singular differential operator is found. A representation of the fundamental solution of the Kipriyanov -operator with a singularity at the point and on the cone in the Euclidean -dimensional half-space is given. [ABSTRACT FROM AUTHOR]