# mathematical model

## Modeling the energy-dynamic modes of a wind farm with a battery energy storage system (BESS)

The article presents the results of mathematical modeling of the energy-dynamic processes of a wind farm which includes a battery energy storage system (BESS). The dependence between load capacity and energy generation capabilities of the active set of a wind power plant taking into account the energy capacity of BESS has been determined. A mathematical model of the BESS has been developed. The elaborated model is compared with two other models: a black box module and a model based on equivalent circuit model.

## Analysis of magnetic fields of electrical devices based on their circuit models

The article deals with an analysis of magnetic fields of electric machines and electromagnetic devices on the basis of their circuit mathematical models. The magnetic systems of electrical devices in these models are presented in the form of planar nonlinear magnetic circuits with lumped elements. The parameters of these elements are determined on the basis of geometric dimensions taking into account the design features of the devices and the physical parameters of the environment.

## Universal mathematical model of asynchronous machine as an element microgrid in smart grid

An universal mathematical model of an induction machine (IM) has been constructed, taking into account the saturation of the main magnetic circuit of its magnetic core and the active power losses.  The proposed approach to IM modeling expands the abilities of MATLAB / Simulink environment for analysis of the electric power supply systems (microgrid in the smart grid) with the nonlinear elements and dynamic load in abnormal, asymmetric and fault regimes.

## Mathematical modeling of centrifugal machines rotors seals for the purpose of assessing their influence on dynamic characteristics

With an increase of equipment parameters, such as the pressure of the sealing medium and the speed of shaft rotation, the problems  ensuring its hermetization efficiency are rising up.  In addition to hermetization itself, the sealing system affect the overall operational safety of the equipment, especially vibratory.  Groove seals are considered as hydrostatodynamic supports capable of effectively damping rotor oscillations.  To determine the dynamic characteristics, models of grooved seals and single-disc rotors with grooved seals are examined.  The obtained analytical dependences for com

## Kinetic Regularities and Mathematical Modelling of Potassium Chloride Dissolution

The dissolution process of potassium chloride particles in the apparatus with two-blade mechanical stirrer was investigated and the mass transfer coefficient was determined. The experimental results were generalized by criterion dependence. The independence of the mass transfer coefficient from the solid particles diameter was confirmed. A countercurrent process of potassium salt dissolution in two apparatuses with a mechanical stirring was considered. A mathematical model for countercurrent dissolution was developed and the efficiency of this process was determined.

## Phase Equilibrium of Petroleum Dispersion Systems in Terms of Thermodynamics and Kinetics

The process of paraffin formation has been considered, including the peculiarities of the paraffin structure as a result of phase transitions with a decreasing temperature. Mathematical models for thermodynamic and kinetic calculations of the "solid-liquid" system phase equilibrium have been developed. To shift the "fuel oil-paraffin" balance towards the liquid, it is necessary to reduce the activity ratio of solid and liquid phases by introducing into the system a substance with a lower solubility parameter.

## Analysis of the State of Territorial Communities to Model Their Socio-Economic Development

The problems of development of united territorial communities, in particular unemployment and economic problems, are considered. Communities, in most cases, lack the resources to address economic and other issues. Therefore, it is necessary to create self-sufficient communities in which there are enough financial instruments for their own development. The mathematical model of the decision support system for the development of territorial communities using the agro-industrial sector was considered.

## Mathematical Modeling of Area Boundary of Biaxial Absolutely Elastic States of Wood

In the paper, the mathematical model for determination of the region boundary of biaxial elastic states of orthotropic materials is synthesized and the system of nonlinear algebraic equations for identification of its parameters is obtained. Using the continuous method of solution continuation concerning the best parameter and the Runge-Kutta method, the demarcation curves of absolutely elastic and non-elastic deformation regions for pine trees are depicted.

## THE ROLE OF NITROGEN OXIDES IN THE PRESENCE OF SULFUR DIOXIDE IN THE DYNAMICS OF THE CARBON DIOXIDE ABSORPTION OFCHLOROPHYLL - PRODUCING MICROALGAE

The effect of nitrogen oxides in the presence of sulfur oxide on the absorption of carbon dioxide by chlorophyll – producing microalgae  Chlorella was investigated. Experimental dependences of the dynamics of CO2 uptake by microalgae in the presence of NxOy alone and at the critical concentration of the SO2 photosynthesis inhibitor in the presence of NxOy are presented.

## Analysis and improvement of design diagrams and mathematical models of vibratory lapping machines

Problem statement. The development of energy-efficient and high-performance vibratory lapping machines demands the improvement of their design diagrams and calculation techniques. Purpose. The main objectives of this research consist in detailed analysis of existent design diagrams and mathematical models of vibratory lapping machines; designing the three-mass hanger-type structures of such machines providing circular oscillations of laps; derivation of differential equations describing the motion of their oscillatory systems.