arXiv subjects

Denis Sidorov

Publications and source records attributed to Denis Sidorov.

At least 19 recordsLinked to original sources

arXiv2025

Magnetically Insulated Diode: Existence of Solutions and Complex Bifurcation. I

In order to avoid the electron oscillation of the cathode and enhance the work efficiency of a vacuum diode, an approach for analyzing the solutions and complex bifurcation has been proposed and used to determine the optimal trajectory of electron motion of the vacuum diode. This work is focusing on the stationary self-consistent problem of magnetic insulation in a space-charge-limited vacuum diode, modeled by a singularly perturbed 1.5-dimensional Vlasov-Maxwell system. We focus on the insulated regime, characterized by the reflection of electrons back toward the cathode at a point $x^{*}.$ The analysis proceeds in two primary stages. First, the original Vlasov-Maxwell system is reduced to a nonlinear singular system of ordinary differential equations governing the electric and magnetic field potentials. Subsequently, this system is further reduced to a novel nonlinear singular ODE for an effective potential $\theta(x).$ The existence of non-negative solutions to this final equation is established on the interval $[0, x^{*})$, where $\theta(x)>0$. This is achieved by reformulating the associated initial value problem into a system of coupled nonlinear Fredholm integral equations and proving the existence of fixed points for the corresponding operators. The most significant and previously unexplored case occurs when $\theta(x)<0$ on the interval $(x^{*}, 1]$, which corresponds to the fully insulated diode. For this regime, we present a novel numerical analysis of complex solution bifurcations, examining their dependence on system parameters and boundary conditions. Bifurcation diagrams illustrating the solution $\theta(x)$ as a function of the free boundary $x^{*}$ is constructed, and the insulated diode spacing is determined.

math.AP↗

arXiv2025

Boundary value problem of magnetically insulated diode: existence of solutions and complex bifurcation

The paper focuses on the stationary self-consistent problem of magnetic insulation for a vacuum diode with space-charge limitation, described by a singularly perturbed Vlasov-Maxwell system of dimension 1.5. The case of insulated diode when the electrons are deflected back towards the cathode at the point $x^{*}$ is considered. First, the initial VM system is reduced to the nonlinear singular limit system of ODEs for the potentials of electric and magnetic fields. The second step deals with the limit system's reduction to the new nonlinear singular ODE equation for effective potential $\theta(x)$. The existence of non-negative solutions is proved for the last equation on the interval $[0, x^{*})$ where $\theta(x)>0$. The most interesting and unexplored case is when $\theta(x)<0$ on the interval $(x^{*}, 1]$ and corresponds to the case of an insulated diode. For the first time, a numerical analysis of complex bifurcation of solutions in insulated diode is considered for $\theta(x)<0$ depending on parameters and boundary conditions. Bifurcation diagrams of the dependence of solution $\theta(x)$ on a free point (free boundary) $x^{*}$ were constructed. Insulated diode spacing is found.

math.AP↗

arXiv2025

Numerical solution of locally loaded Volterra integral equations

Volterra's integral equations with local and nonlocal loads represent the novel class of integral equations that have attracted considerable attention in recent years. These equations are a generalisation of the classic Volterra integral equations, which were first introduced by Vito Volterra in the late 19th century. The loaded Volterra integral equations are characterised by the presence of a load which complicates the process of their theoretical and numerical study. Sometimes these equation are called the equations with ``frozen'' argument. The present work is devoted to the study of Volterra equations with locally loaded integral operators. The existence and uniquness theorems are proved. Among the main contributions is the collocation method for approximate solution of such equations based on the piecewise linear approximation. To confirm the convergence of the method, a number of numerical results for solving model problems are provided.

math.NA↗

arXiv2024

Solving Nonlinear Energy Supply and Demand System Using Physics-Informed Neural Networks

Nonlinear differential equations and systems play a crucial role in modeling systems where time-dependent factors exhibit nonlinear characteristics. Due to their nonlinear nature, solving such systems often presents significant difficulties and challenges. In this study, we propose a method utilizing Physics-Informed Neural Networks (PINNs) to solve the nonlinear energy supply-demand (ESD) system. We design a neural network with four outputs, where each output approximates a function that corresponds to one of the unknown functions in the nonlinear system of differential equations describing the four-dimensional ESD problem. The neural network model is then trained and the parameters are identified, optimized to achieve a more accurate solution. The solutions obtained from the neural network for this problem are equivalent when we compare and evaluate them against the Runge-Kutta numerical method of order 4/5 (RK45). However, the method utilizing neural networks is considered a modern and promising approach, as it effectively exploits the superior computational power of advanced computer systems, especially in solving complex problems. Another advantage is that the neural network model, after being trained, can solve the nonlinear system of differential equations across a continuous domain. In other words, neural networks are not only trained to approximate the solution functions for the nonlinear ESD system but can also represent the complex dynamic relationships between the system's components. However, this approach requires significant time and computational power due to the need for model training.

cs.LG↗

arXiv2024

DC-DC Converters Optimization in Case of Large Variation in the Load

The method for controlling a DC-DC converter is proposed to ensures the high quality control at large fluctuations in load currents by using differential gain control coefficients and second derivative control. Various implementations of balancing the currents of a multiphase DC-DC converter are discussed, with a focus on achieving accurate current regulation without introducing additional delay in the control system. Stochastic particle swarm optimization method is used to find optimal values of the PID controller parameters. An automatic constraint-handling in optimization are also discussed as relevant techniques in the field.

eess.SY↗

arXiv2024

A note on spectral theory of integral-functional Volterra operators

A concise overview of the spectral theory of integral-functional operators is provided. In the context of analysis, a technique is described for deriving solutions to equations involving operators in a closed form. A constructive theorem has been established, outlining a procedure for determining the eigenvalues and eigenfunctions of these operators. Based on this foundation, an analytical approach for generating solutions to a Volterra-type integro-functional inhomogeneous equation is proposed.

math.DS↗

arXiv2023

Volterra black-box models identification methods: direct collocation vs least squares

The Volterra integral-functional series is the classic approach for nonlinear black box dynamical systems modeling. It is widely employed in many domains including radiophysics, aerodynamics, electronic and electrical engineering and many other. Identifying the time-varying functional parameters, also known as Volterra kernels, poses a difficulty due to the curse of dimensionality. This refers to the exponential growth in the number of model parameters as the complexity of the input-output response increases. The least squares method (LSM) is widely acknowledged as the standard approach for tackling the issue of identifying parameters. Unfortunately, the LSM suffers with many drawbacks such as the sensitivity to outliers causing biased estimation, multicollinearity, overfitting and inefficiency with large datasets. This paper presents alternative approach based on direct estimation of the Volterra kernels using the collocation method. Two model examples are studied. It is found that the collocation method presents a promising alternative for optimization, surpassing the traditional least squares method when it comes to the Volterra kernels identification including the case when input and output signals suffer from considerable measurement errors.

math.NA↗

arXiv2019

Control of accuracy on Taylor-collocation method for load leveling problem

High penetration of renewable energy sources coupled with the decentralization of transport and heating loads in future power systems will result in even more complex unit commitment problem solution using energy storage system scheduling for efficient load leveling. This paper employees an adaptive approach to load leveling problem using the Volterra integral dynamical models. The problem is formulated as the solution of the Volterra integral equation of the first kind which is attacked using Taylor-collocation numerical method which has the second-order accuracy and enjoys self-regularization properties, which is associated with confidence levels of system demand. Also, the CESTAC method is applied to find the optimal approximation, optimal error and optimal step of the collocation method. This adaptive approach is suitable for energy storage optimization in real-time. The efficiency of the proposed methodology is demonstrated on the Single Electricity Market of the Island of Ireland.

math.NA↗

arXiv2019

A Dynamic Analysis of Energy Storage with Renewable and Diesel Generation using Volterra Equations

Energy storage systems will play a key role in the power system of the twenty first century considering the large penetrations of variable renewable energy, growth in transport electrification and decentralisation of heating loads. Therefore reliable real time methods to optimise energy storage, demand response and generation are vital for power system operations. This paper presents a concise review of battery energy storage and an example of battery modelling for renewable energy applications and second details an adaptive approach to solve this load levelling problem with storage. A dynamic evolutionary model based on the first kind Volterra integral equation is used in both cases. A direct regularised numerical method is employed to find the least-cost dispatch of the battery in terms of integral equation solution. Validation on real data shows that the proposed evolutionary Volterra model effectively generalises conventional discrete integral model taking into account both state of health and the availability of generation/storage.

math.NA↗

arXiv2019

Energy balancing using charge/discharge storages control and load forecasts in a renewable-energy-based grids

Renewable-energy-based grids development needs new methods to maintain the balance between the load and generation using the efficient energy storages models. Most of the available energy storages models do not take into account such important features as the nonlinear dependence of efficiency on lifetime and changes in capacity over time horizon, the distribution of load between several independent storages. In order to solve these problems the Volterra integral dynamical models are employed. Such models allow to determine the alternating power function for given/forecasted load and generation datasets. In order to efficiently solve this problem, the load forecasting models were proposed using deep learning and support vector regression models. Forecasting models use various features including average daily temperature, load values with time shift and moving averages. Effectiveness of the proposed energy balancing method using the state-of-the-art forecasting models is demonstrated on the real datasets of Germany's electric grid.

eess.SP↗

arXiv2018

Control of accuracy on Taylor-collocation method to solve the weakly regular Volterra integral equations of the first kind by using the CESTAC method

Finding the optimal parameters and functions of iterative methods is among the main problems of the Numerical Analysis. For this aim, a technique of the stochastic arithmetic (SA) is used to control of accuracy on Taylor-collocation method for solving first kind weakly regular integral equations (IEs). Thus, the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method is applied and instead of usual mathematical softwares the CADNA (Control of Accuracy and Debugging for Numerical Applications) library is used. Also, the convergence theorem of presented method is illustrated. In order to apply the CESTAC method we will prove a theorem that it will be our licence to use the new termination criterion instead of traditional absolute error. By using this theorem we can show that number of common significant digits (NCSDs) between two successive approximations are almost equal to NCSDs between exact and numerical solution. Finally, some examples are solved by using the Taylor-collocation method based on the CESTAC method. Several tables of numerical solutions based on the both arithmetics are presented. Comparison between number of iterations are demonstrated by using the floating point arithmetic (FPA) for different values of $\varepsilon$.

math.NA↗

arXiv2018

Basins of attraction of nonlinear systems' equilibrium points: stability, branching and blow-up

This paper presents a nonlinear dynamical model which consists the system of differential and operator equations. Here differential equation contains a nonlinear operator acting in Banach space, a nonlinear operator equation with respect to two elements from different Banach spaces. This system is assumed to enjoy the stationary state (rest points or equilibrium). The Cauchy problem with the initial condition with respect to one of the desired functions is formulated. The second function controls the corresponding nonlinear dynamic process, the initial conditions are not set. The sufficient conditions of the global classical solution's existence and stabilization at infinity to the rest point are formulated. It is demonstrated that a solution can be constructed by the method of successive approximations under the suitable sufficient conditions. If the conditions of the main theorem are not satisfied, then several solutions may exist. Some of solutions can blow-up in a finite time, while others stabilize to a rest point. The special case of considered dynamical models are nonlinear differential-algebraic equation (DAE) have successfully modeled various phenomena in circuit analysis, power systems, chemical process simulations and many other nonlinear processes. Three examples illustrate the constructed theory and the main theorem. Generalization on the non-autonomous dynamical systems concludes the article.

math.DS↗

arXiv2017

Discrete Spectrum Reconstruction using Integral Approximation Algorithm

An inverse problem in spectroscopy is considered. The objective is to restore the discrete spectrum from observed spectrum data, taking into account the spectrometer's line spread function. The problem is reduced to solution of a system of linear-nonlinear equations (SLNE) with respect to intensities and frequencies of the discrete spectral lines. The SLNE is linear with respect to lines' intensities and nonlinear with respect to the lines' frequencies. The integral approximation algorithm is proposed for the solution of this SLNE. The algorithm combines solution of linear integral equations with solution of a system of linear algebraic equations and avoids nonlinear equations. Numerical examples of the application of the technique, both to synthetic and experimental spectra, demonstrate the efficacy of the proposed approach in enabling an effective enhancement of the spectrometer's resolution.

math.NA↗

arXiv2016

Nonclassic boundary value problems in the theory of irregular systems of equations with partial derivatives

The linear PDE ${\mathbf B} {\mathbf L} (\frac{\partial}{\partial x}) u ={\mathbf L}_1(\frac{\partial}{\partial x})u +f(x)$ with nonclassic conditions on boundary $\partial Ω$ is considered. Here ${\mathbf B}$ is linear noninvertible bounded operator acting from linear space $E$ into $E,$ $x=(t,x_1,\dots, x_m) \in Ω, $ $Ω\subset {\mathbb R}^{m+1}.$ It is assumed that ${\mathbf B}$ enjoys the skeleton decomposition ${\mathbf B}={\mathbf A}_1 {\mathbf A}_2,$ ${\mathbf A}_2 \in {\mathcal L}(E\rightarrow E_1),$ ${\mathbf A}_1 \in {\mathcal L}(E_1\rightarrow E)$ where $E_1$ is linear normed space. Differential operators ${\mathbf L}, \, {\mathbf L}_1$ are partial differential operators. In the concrete cases the domains of definition of operators ${\mathbf L}, {\mathbf L}_1$ consist of linear manifolds $E_{\partial}$ of sufficiently smooth abstract functions $u(x)$ with domain in $Ω$ and their ranges in $E,$ which satisfy certain system of homogeneous boundary conditions. The abstract function $f: Ω\subset {\mathbb R}^{m+1} \rightarrow E $ is assumed to be given. It is requested to find the solution $u: Ω\subset {\mathbb R}^{m+1} \rightarrow E_{\partial},$ which satisfy certain condition on boundary $\partial Ω.$ The concept of a skeleton chains is introduced as sequence of linear operators ${\mathbf B}_i \in {\mathcal L}(E_i \rightarrow E_i), \, i=1,2,\dots, p,$ where $E_i$ are linear spaces corresponding to the skeleton decomposition of operator ${\mathbf B}.$ It is assumed that irreversible operator ${\mathbf B}$ generates skeleton chain of the finite length $p.$ The problem is reduced to a regular split system with respect to higher order derivative terms with certain initial and boundary conditions.

math.AP↗

arXiv2016

Application of Volterra Equations to Solve Unit Commitment Problem of Optimised Energy Storage and Generation

Development of reliable methods for optimised energy storage and generation is one of the most imminent challenges in moder power systems. In this paper an adaptive approach to load leveling problem using novel dynamic models based on the Volterra integral equations of the first kind with piecewise continuous kernels. These integral equations efficiently solve such inverse problem taking into account both the time dependent efficiencies and the availability of generation/storage of each energy storage technology. In this analysis a direct numerical method is employed to find the least-cost dispatch of available storages. The proposed collocation type numerical method has second order accuracy and enjoys self-regularization properties, which is associated with confidence levels of system demand. This adaptive approach is suitable for energy storage optimisation in real time. The efficiency of the proposed methodology is demonstrated on the Single Electricity Market of Republic of Ireland and Sakhalin island in the Russian Far East.

eess.SY↗

arXiv2016

Ensemble Methods of Classification for Power Systems Security Assessment

One of the most promising approaches for complex technical systems analysis employs ensemble methods of classification. Ensemble methods enable to build a reliable decision rules for feature space classification in the presence of many possible states of the system. In this paper, novel techniques based on decision trees are used for evaluation of the reliability of the regime of electric power systems. We proposed hybrid approach based on random forests models and boosting models. Such techniques can be applied to predict the interaction of increasing renewable power, storage devices and swiching of smart loads from intelligent domestic appliances, heaters and air-conditioning units and electric vehicles with grid for enhanced decision making. The ensemble classification methods were tested on the modified 118-bus IEEE power system showing that proposed technique can be employed to examine whether the power system is secured under steady-state operating conditions.

cs.AI↗

arXiv2015

Generalized quadrature for solving singular integral equations of Abel type in application to infrared tomography

We propose the generalized quadrature methods for numerical solution of singular integral equation of Abel type. We overcome the singularity using the analytical calculation of the singular integral expression. The problem of solution of singular integral equation is reduced to nonsingular system of linear algebraic equations without shift meshes techniques employment. We also propose generalized quadrature method for solution of Abel equation using the singular integral. Relaxed errors bounds are derived. In order to improve the accuracy we use Tikhonov regularization method. We demonstrate the efficiency of proposed techniques on infrared tomography problem. Numerical experiments show that it make sense to apply regularization in case of highly noisy sources only. That is due to the fact that singular integral equations enjoy selfregularization property.

math.NA↗

arXiv2015

Numerical solution of Volterra integral equations of the first kind with discontinuous kernels

We propose the numerical methods for solution of the weakly regular linear and nonlinear evolutionary (Volterra) integral equation of the first kind. The kernels of such equations have jump discontinuities along the continuous curves (endogenous delays) which starts at the origin. In order to linearize these equations we use the modified Newton-Kantorovich iterative process. Then for linear equations we propose two direct quadrature methods based on the piecewise constant and piecewise linear approximation of the exact solution. The accuracy of proposed numerical methods is $\mathcal{O}(1/N)$ and $\mathcal{O}(1/N^2)$ respectively. We also suggest a certain iterative numerical scheme enjoying the regularization properties. Furthermore, we adduce generalized numerical method for nonlinear equations. We employ the midpoint quadrature rule in all the cases. In conclusion we include several numerical examples in order to demonstrate the efficiency of proposed numerical methods

math.NA↗