Nonlinear Systems of Volterra Equations with Piecewise Smooth Kernels: Numerical Solution and Application for Power Systems Operation
Abstract
:1. Introduction
2. Problem Statement
3. Description of the Iterative Approximate Method
The Convergence Theorem
- Equation (8) has a unique solution for , i.e., there exists ;
- where ;
- .
4. Discretization of the System of Linear Integral Equations
4.1. Problem Formulation
4.2. Piecewise Constant Approximation
4.3. Polynomial Collocation Method
4.3.1. Description of the Problem
4.3.2. Collocation
5. Numerical Results
5.1. System of Linear Integral Equations
5.2. Nonlinear Equations
5.3. Nonlinear Systems of Equations
6. Storage System Analysis in Microgrid Using System of Volterra Equations
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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m | |||||
---|---|---|---|---|---|
2 | 1.0 | 2.0 | |||
3 | 1.0 | 2.0 | |||
5 | 1.0 | 2.0 | |||
8 | 1.0 | 2.0 | |||
12 | 1.0 | 2.0 | |||
15 | 1.0 | 2.0 |
m | |||||||
---|---|---|---|---|---|---|---|
2 | 0.6(6) | 1.3(3) | 2.0 | ||||
3 | 0.6(6) | 1.3(3) | 2.0 | ||||
5 | 0.6(6) | 1.3(3) | 2.0 | ||||
8 | 0.6(6) | 1.3(3) | 2.0 | ||||
12 | 0.6(6) | 1.3(3) | 2.0 | ||||
15 | 0.6(6) | 1.3(3) | 2.0 |
Piece-Wise Constant Approximation | |||||
h | |||||
m | 5 | 5 | 5 | 5 | 5 |
0.0286877 | 0.0152708 | 0.00730057 | 0.0044031 | 0.00386043 | |
Piece-Wise Linear Approximation | |||||
h | |||||
m | 6 | 9 | 9 | 9 | 10 |
0.001496942 | 0.0008320731 | 0.0004604894 | 0.0002356853 | 0.0001186141 |
m | ||||
---|---|---|---|---|
1 | 3 | |||
10 | 3 | |||
20 | 3 |
m | ||||
---|---|---|---|---|
1 | 5 | |||
5 | 5 | |||
10 | 5 | |||
20 | 5 |
m | ||||
---|---|---|---|---|
1 | 10 | |||
5 | 10 | |||
10 | 10 | |||
20 | 10 |
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Sidorov, D.; Tynda, A.; Muftahov, I.; Dreglea, A.; Liu, F. Nonlinear Systems of Volterra Equations with Piecewise Smooth Kernels: Numerical Solution and Application for Power Systems Operation. Mathematics 2020, 8, 1257. https://doi.org/10.3390/math8081257
Sidorov D, Tynda A, Muftahov I, Dreglea A, Liu F. Nonlinear Systems of Volterra Equations with Piecewise Smooth Kernels: Numerical Solution and Application for Power Systems Operation. Mathematics. 2020; 8(8):1257. https://doi.org/10.3390/math8081257
Chicago/Turabian StyleSidorov, Denis, Aleksandr Tynda, Ildar Muftahov, Aliona Dreglea, and Fang Liu. 2020. "Nonlinear Systems of Volterra Equations with Piecewise Smooth Kernels: Numerical Solution and Application for Power Systems Operation" Mathematics 8, no. 8: 1257. https://doi.org/10.3390/math8081257