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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.

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.

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

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.

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

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.