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Numeric methods for solution of the weakly regular linear and nonlinear evolutionary (Volterra) integral equations of the first kind are proposed. The kernels of such equations have jump discontinuities along the continuous curves (endogenous delays) which start at the origin. In order to linearize these equations the modified Newton–Kantorovich iterative process is employed. Two direct quadrature methods based on the piecewise constant and piecewise linear approximation of the exact solution are proposed for linear solutions. The accuracy of proposed numerical methods is O ( 1 / N ) and O ( 1 / N 2 ) respectively. A certain iterative numerical scheme enjoying the regularization properties is suggested. Furthermore, generalized numerical method for nonlinear equations is adduced. The midpoint quadrature rule in all the cases is employed. In conclusion several numerical examples are applied in order to demonstrate the efficiency of proposed numerical methods. [ABSTRACT FROM AUTHOR]

The evolutionary integral dynamical models of storage systems are addressed. Such models are based on systems of weakly regular nonlinear Volterra integral equations with piecewise smooth kernels. These equations can have non-unique solutions that depend on free parameters. The objective of this paper was two-fold. First, the iterative numerical method based on the modified Newton–Kantorovich iterative process is proposed for a solution of the nonlinear systems of such weakly regular Volterra equations. Second, the proposed numerical method was tested both on synthetic examples and real world problems related to the dynamic analysis of microgrids with energy storage systems. [ABSTRACT FROM AUTHOR]

Tourism development in ecologically vulnerable areas like the lake Baikal region in Eastern Siberia is a challenging problem. To this end, the dynamical models of AC/DC hybrid isolated power system consisting of four power grids with renewable generation units and energy storage systems are proposed using the advanced methods based on deep reinforcement learning and integral equations. First, the wind and solar irradiance potential of several sites on the lake Baikal's banks is analyzed as well as the electric load as a function of the climatic conditions. The optimal selection of the energy storage system components is supported in online mode. The approach is justified using the retrospective meteorological datasets. Such a formulation will allow us to develop a number of valuable recommendations related to the optimal control of several autonomous AC/DC hybrid power systems with different structures, equipment composition and kind of AC or DC current. Developed approach provides the valuable information at different stages of AC/DC hybrid power systems projects development with stand-alone hybrid solar-wind power generation systems. [ABSTRACT FROM AUTHOR]