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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.

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.

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., Comment: Submitted to the Bulletin of Irkutsk State University - series Mathematics

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., Comment: 9 pages, 12 figures; accepted to IEEE Transactions on Industrial Informatics

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., Comment: 6 pages; accepted to 38th China Control Conference 2019

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 $\varepsilon$., Comment: 24 pages

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.

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., Comment: submitted to Applied Spectroscopy Journal

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 $\partial \Omega$ 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 \Omega, $ $\Omega \subset {\mathbb R}^{m+1}.$ 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 $\Omega$ and their ranges in $E,$ which satisfy certain system of homogeneous boundary conditions. The abstract function $f: \Omega \subset {\mathbb R}^{m+1} \rightarrow E $ is assumed to be given. It is requested to find the solution $u: \Omega \subset {\mathbb R}^{m+1} \rightarrow E_{\partial},$ which satisfy certain condition on boundary $\partial \Omega.$ 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.

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., Comment: arXiv admin note: text overlap with arXiv:1507.06484

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., Comment: 6 pages, 4 figures, 4 tables. Submitted to PSSC

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.

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, Comment: Submitted to APNUM Journal

One of the most common problems of scientific applications is computation of the derivative of a function specified by possibly noisy or imprecise experimental data. Application of conventional techniques for numerically calculating derivatives will amplify the noise making the result useless. We address this typical ill-posed problem by application of perturbation method to linear first kind equations $Ax=f$ with bounded operator $A.$ We assume that we know the operator $\tilde{A}$ and source function $\tilde{f}$ only such as $||\tilde{A} - A||\leq \delta_1,$ $||\tilde{f}-f||< \delta_2.$ The regularizing equation $\tilde{A}x + B(\alpha)x = \tilde{f}$ possesses the unique solution. Here $\alpha \in S,$ $S$ is assumed to be an open space in $\mathbb{R}^n,$ $0 \in \overline{S},$ $\alpha= \alpha(\delta).$ As result of proposed theory, we suggest a novel algorithm providing accurate results even in the presence of a large amount of noise.

A novel hybrid data-driven approach is developed for forecasting power system parameters with the goal of increasing the efficiency of short-term forecasting studies for non-stationary time-series. The proposed approach is based on mode decomposition and a feature analysis of initial retrospective data using the Hilbert-Huang transform and machine learning algorithms. The random forests and gradient boosting trees learning techniques were examined. The decision tree techniques were used to rank the importance of variables employed in the forecasting models. The Mean Decrease Gini index is employed as an impurity function. The resulting hybrid forecasting models employ the radial basis function neural network and support vector regression. Apart from introduction and references the paper is organized as follows. The section 2 presents the background and the review of several approaches for short-term forecasting of power system parameters. In the third section a hybrid machine learning-based algorithm using Hilbert-Huang transform is developed for short-term forecasting of power system parameters. Fourth section describes the decision tree learning algorithms used for the issue of variables importance. Finally in section six the experimental results in the following electric power problems are presented: active power flow forecasting, electricity price forecasting and for the wind speed and direction forecasting.

The numerical method for solution of the weakly regular scalar Volterra integral equation of the 1st kind is proposed. The kernels of such equations have jump discontinuities on the continuous curves which starts at the origin. The mid-rectangular quadrature rule is employed. The accuracy of proposed numerical method is $\mathcal{O}(1/N).$

The sufficient conditions are obtained for existence of the main solution of the nonlinear Volterra integral equation of the second kind on the semi-axis and on a finite interval. The method for computation of this boundary interval is designed. Beyond such integral the solution has the blow-up. The efficiency of proposed technique is demonstrated on concrete examples.

In this text matrix Volterra integral equation of the first kind is addressed. It is assumed that kernels of the equation have jump discontinuities on non-intersecting curves. Such equations appear in the theory of evolving dynamic systems. Differentiation of such equations with jump discontinue kernels yields the new class of the Volterra integral equations with functionally perturbed argument. The algorithm for construction of the logarithmic power asymptotics of the desired continuous solutions is proposed. The theorem of existance of the parametric families of solutions is proved. Finally the sufficient conditions for existence and uniqueness of continuous solution are derived., Comment: submitted to Russian Mathematics Journal

The sufficient conditions for existence and uniqueness of continuous solutions of the Volterra operator equations of the first kind with piecewise continuous kernel are derived. The asymptotic approximation of the parametric family of solutions are constructed in case of non-unique solution. The algorithm for the solution's improvement is proposed using the successive approximations method., Comment: 17 pages

Sufficient conditions for existence and uniqueness of the solution of the Volterra integral equations of the first kind with piecewise continuous kernels are derived in framework of Sobolev-Schwartz distribution theory. The asymptotic approximation of the parametric family of generalized solutions is constructed. The method for the solution's regular part refinement is proposed using the successive approximations method.

The Volterra integral equations of the first kind with piecewise smooth kernel are considered. Such equations appear in the theory of optimal control of the evolving systems. The existence theorems are proved. The method for constructing approximations of parametric families of solutions of such equations is suggested. The parametric family of solutions is constructed in terms of a logarithmic-power asymptotics., Comment: 11 pages, bibl 14, Submitted to Studia Informatica Universalis

The main solutions in sense of Kantorovich of nonlinear Volterra operator-integral equations are constructed. Convergence of the successive approximations is established through studies of majorant integral and majorant algebraic equations. Estimates are derived for the solutions and for the intervals on the right margin of which the solution has blow-up or solution start branching.