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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.

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

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

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 $\mathcal{O}(1/N).$