# Fractional Order Modeling of Thermal Circuits for an Integrated Energy System Based on Natural Transformation

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## Abstract

**:**

## 1. Introduction

- (1)
- Compared with the electricity circuit model, the thermal circuit model is studied.
- (2)
- The fractional-order model of the thermal circuit is proposed.
- (3)
- The natural transform method is used to solve the thermal circuit model.
- (4)
- The influence of different fractional orders is analyzed.

## 2. Thermal Circuit Analysis of Heating Network

#### 2.1. Classical Mathematical Modeling

_{t}is thermal resistance; L

_{t}is thermal inductance; C

_{t}is heat capacity; and G

_{t}is thermal conductivity.

_{0}is radial direction thermal diffusion coefficient; S is the cross-sectional area of the heating pipeline.

#### 2.2. Fractional-Order Mathematical Model

## 3. Solution Method for Solving Model Based on Natural Transformation

#### 3.1. Natural Transformation

#### 3.2. Model Solving

#### 3.3. Comparison with Existing Thermal Circuit Models

## 4. Main Results and Discussion

#### 4.1. Parameters

#### 4.2. Analysis and Simulation Results

_{0}= 40[(exp(t)/(1 + exp(t)) + 1) is assumed to step function. It can be seen that the results of proposed method are very close to those from original partial differential equation. The results in this paper are slightly smaller than the solution of the original partial differential equation. It is mainly due to the fact that the first four terms of the polynomial solution are taken. The relative error is less than 0.02% for i = 3, which can meet the calculation requirements.

#### 4.3. Discussion Based on the Results

## 5. Conclusions

- (1)
- Compared with the electric circuit model, the thermal circuit model has similar forms.
- (2)
- The proposed method reduces the computational complexity of the thermal circuit model. Compared with other models, the thermal circuit model in this paper can meet the accuracy requirements and the results can dynamically display the changes of pipeline temperature with time and position.
- (3)
- Fractional-order values have an effect on the results of heat transfer. The higher fractional-order causes a lower temperature. Different fractional-order values can be used to correspond to different conditions.
- (4)
- The model in this paper is the thermal circuit analysis theory, which is expected to contribute to the dynamic modeling of integrated energy system in the future. However, different fractional-order values correspond to different models under different conditions, which is the future research direction.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Abbreviations | |

IES | integrated energy system |

CHP | combined heat and power |

PDE | partial differential equation |

Parameters and Variables | |

A\B\C | refers to constant coefficient of partial differential equation |

c | refers to the specific heat capacity |

C_{t} | refers to heat capacity |

h | refers to heat power (W) |

G_{t} | refers to thermal conductivity |

l | refers to the length of pipeline |

L_{t} | refers to thermal inductance |

m | refers to the mass flow (kg·s^{−1}) |

R_{t} | refers to thermal resistance |

S | refers to the cross-sectional area of the heating pipeline (m^{2}) |

T | refers to the temperature field (°C) |

t | refers to time |

x | refers to the position along the pipe |

u\s | refer to the natural transform variables |

ρ | refers to the density (kg·m^{−3}) |

λ | refers to the heat-loss factor of the pipeline (W·mK^{−1}) |

γ_{0} | refers to radial direction thermal diffusion coefficient (m^{2}·s^{−1}) |

β | refers to fractional-order |

$\Gamma $ | refers to the gamma function |

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**Figure 1.**Heating pipeline model: (

**a**) Schematic diagram; (

**b**) Distributed-parameter thermal circuit model.

λ (W·mK^{−1}) | S/(m^{2)} | L (m) | m (Kg·s^{−1}) | γ_{0} (m^{2}·s^{−1}) |
---|---|---|---|---|

0.6 | 0.01 | 200 | 3 | 0.16 |

t (min) | Numerical Results of PDE | Numerical Results of Proposed Method | |||
---|---|---|---|---|---|

T (°C) | i = 3 | i = 4 | |||

T (°C) | Relative Errors | T (°C) | Relative Errors | ||

33 | 58.87636 | 58.87039 | 0.000101 | 58.87042 | 0.000101 |

34 | 58.85114 | 58.8448 | 0.000108 | 58.84483 | 0.000107 |

35 | 58.82592 | 58.8192 | 0.000114 | 58.81924 | 0.000114 |

36 | 58.80071 | 58.79361 | 0.000121 | 58.79364 | 0.00012 |

37 | 58.77552 | 58.76801 | 0.000128 | 58.76805 | 0.000127 |

38 | 58.75034 | 58.74242 | 0.000135 | 58.74246 | 0.000134 |

39 | 58.72516 | 58.71682 | 0.000142 | 58.71687 | 0.000141 |

40 | 58.70000 | 58.6912 | 0.000149 | 58.69128 | 0.000149 |

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**MDPI and ACS Style**

Li, M.; Ye, J.
Fractional Order Modeling of Thermal Circuits for an Integrated Energy System Based on Natural Transformation. *Electronics* **2022**, *11*, 914.
https://doi.org/10.3390/electronics11060914

**AMA Style**

Li M, Ye J.
Fractional Order Modeling of Thermal Circuits for an Integrated Energy System Based on Natural Transformation. *Electronics*. 2022; 11(6):914.
https://doi.org/10.3390/electronics11060914

**Chicago/Turabian Style**

Li, Ming, and Jin Ye.
2022. "Fractional Order Modeling of Thermal Circuits for an Integrated Energy System Based on Natural Transformation" *Electronics* 11, no. 6: 914.
https://doi.org/10.3390/electronics11060914