Numeric solution of Volterra integral equations of the first kind with discontinuous kernels

Muftahov, Ildar; Tynda, Aleksandr; Sidorov, Denis

doi:10.1016/j.cam.2016.09.003

Cited by 37 publications

(22 citation statements)

References 13 publications

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“…Here we would like to mention some other results, at least. In [55] the modified Newton-Kantorovich method combined with collocation were applied non linear and nonlinear VIE with piecewise smooth kernels. Such VIE were introduced in [56] and asymptotic approximations to parametric families of solutions were constructed and the existence of continuous solutions was proved.…”

Section: Discussionmentioning

confidence: 99%

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Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

Cardone

Conte

D’Ambrosio

et al. 2018

Axioms

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We present a collection of recent results on the numerical approximation of Volterra integral equations and integro-differential equations by means of collocation type methods, which are able to provide better balances between accuracy and stability demanding. We consider both exact and discretized one-step and multistep collocation methods, and illustrate main convergence results, making some comparisons in terms of accuracy and efficiency. Some numerical experiments complete the paper.

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Section: Discussionmentioning

confidence: 99%

“…We can summarize them in the following theorem. Moreover, if hypotheses of Theorem 8 are fulfilled, Q and Q i defined in (55) have order 2m + r and 2m + r − 1 respectively, then the order of the discretized method (56) and (57), at the mesh points, is 2m + r − 1.…”

Section: Theoremmentioning

confidence: 99%

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

Cardone

Conte

D’Ambrosio

et al. 2018

Axioms

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“…The theory and regularised numerical methods to relieve the ill-posedness of the problem are employed in Sec. III as suggested in papers [33], [34]. Proposed approach to dynamical analysis of energy storage is based on the solid mathematical theory [31] including the following existence and uniqness theorem.…”

Section: Volterra Modelmentioning

confidence: 99%

A Dynamic Analysis of Energy Storage With Renewable and Diesel Generation Using Volterra Equations

Sidorov

Muftahov

Tomin

et al. 2020

IEEE Trans. Ind. Inf.

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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.

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“…Such models can be employed to simulate the degradation processes in storage systems of MGs using retrospective time series of generation and load for specific location. Numerical results of proposed integral model were derived using the collocation numerical method proposed in [36,37] for determination APF and SoC will be shown on real datasets below.…”

Section: The Relationship Between Local and Centralized Levelsmentioning

confidence: 99%

Toward Zero-Emission Hybrid AC/DC Power Systems with Renewable Energy Sources and Storages: A Case Study from Lake Baikal Region

et al. 2020

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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.

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Numeric solution of Volterra integral equations of the first kind with discontinuous kernels

Cited by 37 publications

References 13 publications

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

A Dynamic Analysis of Energy Storage With Renewable and Diesel Generation Using Volterra Equations

Toward Zero-Emission Hybrid AC/DC Power Systems with Renewable Energy Sources and Storages: A Case Study from Lake Baikal Region

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