# Nonlinear Systems of Volterra Equations with Piecewise Smooth Kernels: Numerical Solution and Application for Power Systems Operation

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

**Theorem**

**1.**

## 3. Description of the Iterative Approximate Method

#### The Convergence Theorem

**Theorem**

**2.**

- Equation (8) has a unique solution for $m=0$, i.e., there exists ${\mathsf{{\rm Y}}}_{0}={\left[{P}^{\prime}\left({X}^{0}\right)\right]}^{-1}$;
- $\parallel \Delta {X}^{1}\parallel \u2a7d\eta ,$ where $\Delta {X}^{m}={X}^{m}-{X}^{m-1},\phantom{\rule{0.277778em}{0ex}}m=0,1,\dots .$;
- $\parallel {\mathsf{{\rm Y}}}_{0}{P}^{\u2033}\left(X\right)\parallel \u2a7dL,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}X\in {\mathsf{\Omega}}_{0}$.

**Proof.**

## 4. Discretization of the System of Linear Integral Equations

#### 4.1. Problem Formulation

#### 4.2. Piecewise Constant Approximation

**Remark**

**1.**

#### 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

**Example**

**1.**

**Example**

**2.**

#### 5.2. Nonlinear Equations

#### 5.3. Nonlinear Systems of Equations

**Remark**

**2.**

## 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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**Figure 1.**24 h efficiency changeover: blue line: constant; green line: nonlinear effect caused by temperature drop.

**Figure 2.**Alternating power functions (APF in MW) with state of charge information (SOC in %) computed by the system of Volterra integral Equation (34) for the winter period.

**Table 1.**Results for the problem (28).

m | ${\mathit{\epsilon}}_{1}$ | ${\mathit{t}}_{\mathbf{max}}^{1}$ | ${\mathit{\epsilon}}_{2}$ | ${\mathit{t}}_{\mathit{max}}^{2}$ | $\mathit{\epsilon}$ |
---|---|---|---|---|---|

2 | $9.82294\times {10}^{-3}$ | 1.0 | $6.72940\times {10}^{-2}$ | 2.0 | $6.80072\times {10}^{-2}$ |

3 | $1.60472\times {10}^{-3}$ | 1.0 | $2.35676\times {10}^{-2}$ | 2.0 | $2.36222\times {10}^{-2}$ |

5 | $6.67315\times {10}^{-6}$ | 1.0 | $3.95344\times {10}^{-4}$ | 2.0 | $3.95400\times {10}^{-4}$ |

8 | $1.72968\times {10}^{-8}$ | 1.0 | $1.80165\times {10}^{-7}$ | 2.0 | $1.80994\times {10}^{-7}$ |

12 | $1.31014\times {10}^{-8}$ | 1.0 | $4.12674\times {10}^{-9}$ | 2.0 | $1.37360\times {10}^{-8}$ |

15 | $2.79384\times {10}^{-8}$ | 1.0 | $1.75046\times {10}^{-8}$ | 2.0 | $3.29692\times {10}^{-8}$ |

**Table 2.**Results for problem (29).

m | ${\mathit{\epsilon}}_{1}$ | ${\mathit{t}}_{\mathit{max}}^{1}$ | ${\mathit{\epsilon}}_{2}$ | ${\mathit{t}}_{\mathit{max}}^{2}$ | ${\mathit{\epsilon}}_{3}$ | ${\mathit{t}}_{\mathit{max}}^{3}$ | $\mathit{\epsilon}$ |
---|---|---|---|---|---|---|---|

2 | $1.75798\times {10}^{-3}$ | 0.6(6) | $3.30114\times {10}^{-3}$ | 1.3(3) | $6.72940\times {10}^{-2}$ | 2.0 | $3.44752\times {10}^{-2}$ |

3 | $2.75534\times {10}^{-4}$ | 0.6(6) | $3.77383\times {10}^{-3}$ | 1.3(3) | $4.51700\times {10}^{-3}$ | 2.0 | $5.89245\times {10}^{-3}$ |

5 | $1.68420\times {10}^{-6}$ | 0.6(6) | $2.89243\times {10}^{-5}$ | 1.3(3) | $9.1497\times {10}^{-5}$ | 2.0 | $9.59747\times {10}^{-5}$ |

8 | $7.86707\times {10}^{-10}$ | 0.6(6) | $6.86599\times {10}^{-9}$ | 1.3(3) | $4.15579\times {10}^{-8}$ | 2.0 | $4.21286\times {10}^{-8}$ |

12 | $1.33907\times {10}^{-9}$ | 0.6(6) | $1.55675\times {10}^{-9}$ | 1.3(3) | $1.97100\times {10}^{-10}$ | 2.0 | $2.06287\times {10}^{-9}$ |

15 | $1.69313\times {10}^{-9}$ | 0.6(6) | $1.00072\times {10}^{-9}$ | 1.3(3) | $1.43989\times {10}^{-9}$ | 2.0 | $2.43750\times {10}^{-9}$ |

**Table 3.**Error analysis (30).

Piece-Wise Constant Approximation | |||||

h | $1/32$ | $1/64$ | $1/128$ | $1/256$ | $1/512$ |

m | 5 | 5 | 5 | 5 | 5 |

$\epsilon $ | 0.0286877 | 0.0152708 | 0.00730057 | 0.0044031 | 0.00386043 |

Piece-Wise Linear Approximation | |||||

h | $1/32$ | $1/64$ | $1/128$ | $1/256$ | $1/512$ |

m | 6 | 9 | 9 | 9 | 10 |

$\epsilon $ | 0.001496942 | 0.0008320731 | 0.0004604894 | 0.0002356853 | 0.0001186141 |

**Table 4.**Results for problem (31).

${\mathit{N}}_{\mathit{it}}$ | m | ${\mathit{\epsilon}}_{1}$ | ${\mathit{\epsilon}}_{2}$ | $\mathit{\epsilon}$ |
---|---|---|---|---|

1 | 3 | $0.426435$ | $0.133873$ | $0.446955$ |

10 | 3 | $5.97261\times {10}^{-5}$ | $1.91177\times {10}^{-5}$ | $6.27112\times {10}^{-5}$ |

20 | 3 | $2.81636\times {10}^{-9}$ | $9.78086\times {10}^{-10}$ | $2.98137\times {10}^{-9}$ |

${\mathit{N}}_{\mathit{i}\mathit{t}}$ | m | ${\mathit{\epsilon}}_{1}$ | ${\mathit{\epsilon}}_{2}$ | $\mathit{\epsilon}$ |
---|---|---|---|---|

1 | 5 | $0.156589$ | $0.0170903$ | $0.157519$ |

5 | 5 | $5.04072\times {10}^{-4}$ | $4.89738\times {10}^{-5}$ | $5.06445\times {10}^{-4}$ |

10 | 5 | $5.22785\times {10}^{-6}$ | $4.16019\times {10}^{-6}$ | $6.68114\times {10}^{-6}$ |

20 | 5 | $1.97551\times {10}^{-6}$ | $3.69009\times {10}^{-6}$ | $4.18562\times {10}^{-6}$ |

${\mathit{N}}_{\mathit{i}\mathit{t}}$ | m | ${\mathit{\epsilon}}_{1}$ | ${\mathit{\epsilon}}_{2}$ | $\mathit{\epsilon}$ |
---|---|---|---|---|

1 | 10 | $0.194802$ | $0.0224296$ | $0.196089$ |

5 | 10 | $3.02512\times {10}^{-3}$ | $3.21621\times {10}^{-4}$ | $3.04217\times {10}^{-3}$ |

10 | 10 | $2.28126\times {10}^{-5}$ | $2.22376\times {10}^{-6}$ | $2.29207\times {10}^{-5}$ |

20 | 10 | $4.03156\times {10}^{-9}$ | $6.04141\times {10}^{-10}$ | $4.07657\times {10}^{-9}$ |

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Sidorov, 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