Control of Accuracy on Taylor-Collocation Method for Load Leveling Problem

Abstract

High penetration of renewable energy sources coupled with decentralization of transport and heating loads in future power systems will result 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 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 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.

Error Estimation of the Homotopy Perturbation Method to Solve Second Kind Volterra Integral Equations with Piecewise Smooth Kernels: Application of the CADNA Library

“…Usually, in order to find the numerical results we need to apply the usual packages like Mathematica and Matlab. Here, instead of them we introduce the CADNA library and the CESTAC method to validate the numerical results [1,55,56,62].…”

“…The problem of finding approximate solution for linear Volterra IEs is one of the oldest problems in the applied mathematics researches. Specially, this problem with discontinuous kernel has many applications in the load leveling problems, energy storage with renewable and diesel generation, charge/discharge storages control and others [1][2][3].…”

“…There are various methods for solving linear and nonlinear problems [4][5][6][7][8][9][10] specially the Volterra IEs with discontinuous kernel. Muftahov et al in [11] applied the Lavrentiev regularization and direct quadrature method, Sidorov in [12] used the successive approximations and Noeiaghdam et al studied the Taylor-collocation method for solving Volterra IEs with discontinuous kernel [1,13]. Also, the nonlinear system of Volterra IE with applications was studied in [14,15].…”

Integro-Differential Equation for the Non-Equilibrium Thermal Response of Glass-Forming Materials: Analytical Solutions

“…Viscoelasticity is one Symmetry 2021, 13, 256 2 of 17 of the fields of physics where the Volterra equations are often used [25]. The applications of Volterra equations in renewable energy is an example of their use in industry [30][31][32]. Volterra integro-differential equations are usually difficult to solve analytically; therefore, approximate solutions and numerical methods are often used [33].…”

A Novel Algorithm to Evaluate Definite Integrals by the Gauss-Legendre Integration Rule Based on the Stochastic Arithmetic: Application in the Model of Osmosis System

“…Eberhart et al [40] presented a high performance numerical validation using the SA. In recent years, the CESTAC method was applied to validate the results of several numerical methods such as estimating the value of interpolation polynomials [41], improper and definite integrals [42,43], dynamical control on Gauss-Laguerre integration rule [44], evaluating the fuzzy definite integrals [45], numerical validation of the Sinccollocation method to solve linear integral equations [46], finding optimal iteration of the power and inverse iteration methods, solving linear systems, fuzzy Newton-Cotes integration rules [47], finding the optimal parameter of the homotopy analysis method for solving integral equations [48], solving fuzzy integral equations [49], validating Taylorcollocation method for solving Volterra integral equations with discontinuous kernels [50] and load leveling problem [51].…”