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Electrical Modelling of a DC Railway System with Multiple Trains^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Case Study

## 3. Model Description

#### 3.1. Vehicle Model

#### 3.2. Traction Calculation

- The drag force $Q(v)$ in $kN$ is represented by the Davis equation:$$Q(v)=a+bv+c{v}^{2}.$$

- The maximum tractive force $F(v)$ in $kN$ is given by:$$\{\begin{array}{ll}F(v)=310\text{\hspace{0.17em}},& v\le 10\text{}\mathrm{m}/\mathrm{s}\\ F(v)=310-(10v-100),& 10v\le 22.2\text{}\mathrm{m}/\mathrm{s}\end{array}\text{}$$
- The maximum electrical braking force $B(v)$ in $kN$ is given by:$$\{\begin{array}{ll}B(v)=260\text{\hspace{0.17em}},& v\le 15\text{}\mathrm{m}/\mathrm{s}\\ B(v)=260-(18v-270),& v15\text{}\mathrm{m}/\mathrm{s}\end{array}\text{}$$

#### 3.3. Electrical Modelling

## 4. Simulation Results

## 5. Consideration for Braking Energy

## 6. Model Validation

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**(

**a**) The tractive and drag force of a single train; (

**b**) Speed profile and tractive or braking force of a train for the entire journey on a single railway track.

**Figure 10.**The electrical resistance difference between Train 2 and the train behind or the previous passenger station.

**Figure 11.**The electrical resistance difference between Train 2 and the train ahead or the next virtual station.

**Figure 13.**The substations’ voltages of a single railway track: (

**a**) Substation 1; (

**b**) Substation 2; and (

**c**) Substation 3.

**Figure 16.**The voltages at the trains’ locations on a single railway track: (

**a**) Train 1; (

**b**) Train 2; (

**c**) Train 3; (

**d**) Train 4; (

**e**) Train 5; and (

**f**) Train 6.

**Figure 20.**The impact of changing the headway on: (

**a**) The total energy dissipated through the onboard braking resistors; and (

**b**) The receptivity of the line to accept the available regenerative energy.

**Figure 23.**The total electrical resistance of the single railway track by totalling the electrical resistance of the changing electric network configuration of: (

**a**) Section 1; and (

**b**) Section 2.

Symbol | Quantity | Value |
---|---|---|

$m$ | train mass | $27,215.5\text{}\mathrm{kg}$ |

$a$ | Davis equation constant coefficient | $2.965\text{}\mathrm{N}$ |

$b$ | Davis equation linear term coefficient | $0.23\text{}\mathrm{Ns}/\mathrm{m}$ |

$c$ | Davis equation quadratic term coefficient | $0.005{\text{}\mathrm{Ns}}^{2}{/\mathrm{m}}^{2}$ |

${V}_{s}$ | substation dc voltage | $600\text{}\mathrm{V}$ |

${R}_{s}$ | substation inner resistance | $20\text{}\mathrm{m}\mathsf{\Omega}$ |

${R}_{d}$ | rail electrical resistance | $15\text{}\mathrm{m}\mathsf{\Omega}/\mathrm{km}$ |

${V}_{\mathrm{max}}$ | Voltage threshold | $740\text{}\mathrm{V}$ |

$\sum {\mathit{E}}_{\mathit{s}}}\text{}(\mathbf{kWh})\text{$ | $\sum {\mathit{E}}_{\mathit{r}}\text{}}(\mathbf{kWh})\text{$ | $\sum {\mathit{E}}_{\mathit{t}}}\text{}(\mathbf{kWh})\text{$ | $\sum {\mathit{E}}_{\mathit{l}\mathit{i}\mathit{n}\mathit{e}\text{\hspace{0.17em}}\mathit{l}\mathit{o}\mathit{s}\mathit{s}\mathit{e}\mathit{s}}}\text{}(\mathbf{kWh})\text{$ | $\sum {\mathit{E}}_{\mathit{s}\mathit{u}\mathit{b}\mathit{s}\mathit{t}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\text{\hspace{0.17em}}\mathit{l}\mathit{o}\mathit{s}\mathit{s}\mathit{e}\mathit{s}}\text{}}(\mathbf{kWh})\text{$ | $\sum {\mathit{E}}_{\mathit{c}\mathit{b}}\text{}}(\mathbf{kWh})\text{$ |
---|---|---|---|---|---|

857.94 | 207.5 | 893.06 | 82.44 | 50.76 | 47.69 |

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## Share and Cite

**MDPI and ACS Style**

Alnuman, H.; Gladwin, D.; Foster, M.
Electrical Modelling of a DC Railway System with Multiple Trains. *Energies* **2018**, *11*, 3211.
https://doi.org/10.3390/en11113211

**AMA Style**

Alnuman H, Gladwin D, Foster M.
Electrical Modelling of a DC Railway System with Multiple Trains. *Energies*. 2018; 11(11):3211.
https://doi.org/10.3390/en11113211

**Chicago/Turabian Style**

Alnuman, Hammad, Daniel Gladwin, and Martin Foster.
2018. "Electrical Modelling of a DC Railway System with Multiple Trains" *Energies* 11, no. 11: 3211.
https://doi.org/10.3390/en11113211