# Eco-Driving in Railway Lines Considering the Uncertainty Associated with Climatological Conditions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulation

#### 2.1. Simulation Model

- ❖
- Rolling stock.
- i.
- Physical parameters: mass of the train, the mass of the load, length, maximum speed, rotatory inertia, and adhesion traction.
- ii.
- Traction effort curve, braking effort curve, running resistance coefficients, energetic and power systems and auxiliary systems consumptions.

- ❖
- Railway line. Includes Kilometric Points of relevant information as stations (K.P.), grades, track curvatures, speed limits, tunnels, and grade transitions considering the length of the train and the electric neutral zones.

- a is the acceleration of the train.
- F
_{m}is the motor force. - R is the running resistance of the train defined by the Davis formula (Equation (7)).
- F
_{g}is the force due to the railway grades. - F
_{r}is the force due to track curvature. - m
_{eq}is the equivalent mass.

- m
_{0}is the empty train mass. - m
_{l}is the load mass. - λ is the dimensionless rotating mass factor.

- g is the gravity acceleration.
- θ is the slope angle.

- K
_{r}is an empirical constant defined by track gauge. - r is the radius of track curvature.

_{m}) is bounded by a maximum value of the electrical traction effort curve and a maximum electrical braking effort curve both dependent on the train speed. The simulation model considers at every moment the speed limits established along the rail track by the railway administration. Speed limits are defined by stretches along the railway where the train must drive respecting the maximum speed limit considering the total length of the train. When a train passes through a neutral zone, auxiliary systems cannot be fed from a catenary, and motors apply at least a constant braking effort to maintain the charge of the batteries that feed auxiliary systems along neutral zones.

- a is a constant related to the mechanical resistance to the motion.
- b is a constant related to the resistance due to the air inlet in the train.
- c is a constant related to the aerodynamic resistance.
- k is the tunnel factor.
- v
_{train}is the velocity of the train.

^{3}).

_{pantograph}is expressed as shown in Equation (8) while the train energy consumption estimated at the electrical substation (E

_{substation}) is defined by means of Equation (9):

- P
_{pantograph}is the electrical power consumed by the train measured at the pantograph. - P
_{substation}is the estimation of the power consumed at the substation.

- P
_{mec}is the mechanical power. - P
_{aux}is the power consumed by the auxiliary systems. - η
_{T}is the electrical chain efficiency for the traction case. - η
_{B}is the electrical chain efficiency for the braking case. - V is the nominal line voltage.
- cos φ is the power factor.
- r(s) is the electrical line resistance.
- nz
_{start}and nz_{end}are the initial and final position of every neutral electrical zone.

#### 2.2. Climatological Model

- c is the aerodynamic coefficient of the running resistance model (Equation (7)).
- ρ
_{ISA}is the density of the air under standard conditions [87]. - A is the cross-sectional area.
- F
_{d}is the drag force.

- ρ is the density.
- p is the pressure.
- R is the ideal gas constant.
- T is the temperature.

- v
_{wind}is the velocity of the wind. - v
_{rel}is the relative train speed of the train, graphically defined in Figure 1.

## 3. Fuzzy Climatological Model

_{ρ}, the expression is:

## 4. Eco-Driving Optimisation with Fuzzy Parameters

_{m}). Then, the problem and proposed algorithm for the problem resolution will be presented: Genetic Algorithm with Fuzzy Parameters (GA-F).

#### 4.1. Efficient Driving Commands

_{m}) [24]. The matrix is the result of the optimisation process and means the division of the railway line where the driver is required to apply certain driving commands. The C

_{m}is defined as a matrix formed with the values of the points where every stretch end and the value of the regulation speed without braking are applied in every section. The end (Kilometric Point) of the divisions forms the first column, and the regulation speed value without braking forms the second column. Thus, the initial point of a division is the end of the previous section except for the first division, where the initial point is the journey starting position. Furthermore, the end of the last division will be the coasting section’s starting point until the braking curve is reached up to the station. The numbers of rows of the matrix plus 1 are the number of the sections defined.

_{m}) obtained as a solution to the problem will be efficient, safe, and comfortable driving.

#### 4.2. Genetic Algorithm with Fuzzy Parameters: GA-F

- w
_{e}and w_{t}are weighting factors. - ${E}_{flat-out}$ is the energy consumption when the flat-out driving is applied.
- t
_{obj}is the target running time.

_{m}as the genome of the individuals. Additionally, crossover and mutation operators are applied to generate and offspring of individuals from the elite group at each iteration. The fuzzy optimisation algorithm is the procedure for the calculation of the driving commands when the fuzzy parameters are included. The GA-F process is graphically described in Figure 6 and it is explained as follows:

- First random generation of individuals (C
_{m}) equal to the population size N_{pop}. - Simulation of command matrix of the population. This step generates a running time ${\widehat{T}}_{r}$ and an energy consumption $\widehat{E}$ associated to each individual.
- Evaluation of each individual by the fitness function (Equation (29)).
- Sorting of each individual in increasing order by means of their fitness value.
- Elite group selection. The first individuals N
_{e}survive in the next iteration population and the rest are eliminated. - Offspring generation. Mutation and crossover operators are applied to the elite group to generate new individuals until the population reaches its size N
_{pop}. In this step, a number of mutations N_{m}and a number of crossovers N_{c}are applied. - The process is repeated from step 2 until the iteration number (it) is equal to the maximum numbers of iterations defined (it
_{max}), giving the best solution to the fittest in the last population.

_{p}, the upper limit of the α-cuts of the fuzzy running time (equivalent to the upper limit of the α-cuts of the fuzzy running time) is necessary for the evaluation of the punctuality constraint. For this upper limit to be obtained, given the fuzzy numbers associated with the climatological parameters, it has to be analysed whether the running time is an increasing or a decreasing function in each of them.

- (1)
- $0\xb0<\widehat{\varphi}<180\xb0$ then ${F}_{1}$ is monotonically decreasing with the angle $\varphi $.
- (2)
- $180\xb0<\widehat{\varphi}<360\xb0$ then ${F}_{1}$ is monotonically increasing with the angle $\varphi $.

- (1)
- $0\xb0<\widehat{\varphi}<90\xb0$ or $270\xb0<\widehat{\varphi}<360\xb0$ then ${F}_{1}$ is monotonically increasing with the velocity of the wind.
- (2)
- $90\xb0<\widehat{\varphi}<270\xb0$ then ${F}_{1}$ is monotonically decreasing with the velocity of the wind.

_{2}(Equation (31)) is monotonous with all climatological parameters as F

_{1}which permits the application of α-cut calculations. Moreover, the dependence of ${F}_{2}$ with these parameters is the same as F

_{1}because the greater the running resistance, the greater the traction effort and energy consumption.

_{p}. The parameter n

_{p}is the imposed punctuality constraint. The algorithm will search for the solution in the form of a Command Matrix with a fuzzy running time associated. Its upper α-cut gives the objective running time as a result, considering punctuality requirements and minimising the energy consumption.

## 5. Case Study

#### 5.1. Introduction

#### 5.2. Fuzzy Climatological Parameters

#### 5.3. Results

#### 5.3.1. Flat-Out Driving

#### 5.3.2. Eco-Driving Design without Climatological Parameters

#### 5.3.3. Eco-Driving with Fuzzy Climatological Parameters Applying GA-F

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Initial K.P.–Final K.P. | Angle of the Rail Track (^{o}) |
---|---|

0–45 | 65 |

45–135 | 48 |

135–200 | 59 |

200–245 | 57 |

245–275 | 41 |

275–335 | 114 |

335–435 | 82 |

435–495 | 117 |

495–515 | 161 |

515–580 | 62 |

580–621 | 145 |

Fuzzy Temperature | |||
---|---|---|---|

Initial K.P. | Summer (°C) | ||

Lower Limit | Core | Upper Limit | |

0 | 28.25 | 34.31 | 40.37 |

95 | 27.86 | 33.30 | 38.74 |

170 | 24.14 | 32.38 | 40.61 |

240 | 25.10 | 33.11 | 41.12 |

330 | 25.21 | 32.07 | 38.93 |

400 | 26.89 | 33.63 | 40.37 |

475 | 27.03 | 31.39 | 35.75 |

Initial K.P. | Winter (°C) | ||

Lower Limit | Core | Upper Limit | |

0 | 5.46 | 11.77 | 18.09 |

95 | 4.90 | 11.35 | 17.80 |

170 | 5.00 | 11.47 | 17.93 |

240 | 3.54 | 11.71 | 19.87 |

330 | 3.85 | 11.22 | 18.59 |

400 | 1.10 | 10.69 | 20.27 |

475 | 9.55 | 15.35 | 21.14 |

Fuzzy Pressure | |||
---|---|---|---|

Initial K.P. | Pressure (mbar) | ||

Lower Limit | Core | Upper Limit | |

0 | 927.91 | 941.17 | 954.44 |

45 | 929.15 | 942.42 | 955.68 |

95 | 916.55 | 929.58 | 942.60 |

170 | 931.78 | 945.25 | 958.72 |

240 | 968.48 | 983.10 | 997.72 |

330 | 956.92 | 971.09 | 985.25 |

400 | 976.49 | 991.22 | 1005.94 |

475 | 990.38 | 1004.74 | 1019.11 |

500 | 995.28 | 1009.65 | 1024.01 |

Fuzzy Intensity of the Wind/Summer and Winter (m/s) | |||
---|---|---|---|

Initial K.P.–Final K.P. | Lower Limit | Core | Upper Limit |

0–45 | 0.6 | 3.2 | 5.8 |

45–200 | 1.4 | 4 | 6.6 |

200–240 | 0 | 2.8 | 5.6 |

240–330 | 0 | 3.6 | 7.9 |

330–400 | 0 | 3.9 | 8.8 |

400–475 | 0 | 2.5 | 5.4 |

475–500 | 1.6 | 4.5 | 7.4 |

500–550 | 0.4 | 3.3 | 6.2 |

550–580 | 0 | 3.7 | 7.4 |

580–621 | 0 | 3 | 6.7 |

Fuzzy Angle of Incidence of the Wind (°)/Summer | ||||
---|---|---|---|---|

Initial K.P.–Final K.P. | Lower Limit | Core | Upper Limit | Type of Wind |

0–45 | 215 | 230 | 245 | Unfavourable |

45–135 | 55 | 70 | 85 | Favourable |

135–200 | 59 | 70 | 85 | |

200–240 | 25 | 40 | 55 | |

240–330 | 295 | 310 | 325 | Unfavourable |

330–335 | 245 | 260 | 275 | |

335–400 | 245 | 260 | 262 | |

400–475 | 245 | 260 | 275 | |

475–495 | 103 | 110 | 125 | Favourable |

495–500 | 95 | 110 | 125 | |

500–515 | 125 | 140 | 155 | |

515–550 | 125 | 140 | 152 | |

550–580 | 62 | 70 | 85 | |

580–621 | 145 | 160 | 175 | |

Fuzzy Angle of Incidence of the Wind (^{o})/Winter | ||||

Initial K.P.–Final K.P. | Lower Limit | Core | Upper Limit | Type of Wind |

0–45 | 5 | 20 | 35 | Unfavourable |

45–135 | 55 | 70 | 85 | Favourable |

135–200 | 59 | 70 | 85 | |

200–240 | 25 | 40 | 55 | |

240–330 | 295 | 310 | 325 | Unfavourable |

330–335 | 245 | 260 | 275 | |

335–400 | 245 | 260 | 262 | |

400–475 | 245 | 260 | 275 | |

475–495 | 103 | 110 | 125 | Favourable |

495–500 | 95 | 110 | 125 | |

500–515 | 265 | 280 | 295 | |

515–550 | 265 | 280 | 295 | |

550–580 | 62 | 70 | 85 | |

580–621 | 145 | 160 | 175 |

Scenario | Energy Consumption (MWh) | Running Time (hh:mm:ss) |
---|---|---|

Initial Case | 10.804 | 2:18:49 |

Winter | 10.603 | 2:18:48 |

Summer | 10.061 | 2:18:40 |

Command Matrix | ||
---|---|---|

Initial K.P. | Final K.P. | Speed (km/h) |

0 | 271.08 | 244.73 |

271.08 | 422.03 | 230.16 |

422.03 | 621 | 240.17 |

Energy Consumption | Flat-Out (MWh) | Eco-Driving (MWh) | Saving |
---|---|---|---|

Initial Case | 10.804 | 7.589 | 29.76% |

**Table 9.**Fuzzy results obtained with the eco-driving design applying GA-F with np = 0.5 in winter and in summer.

Season | Winter | Summer | ||||
---|---|---|---|---|---|---|

Value | α | Energy (MWh) | Running Time (hh:mm:ss) | α | Energy (MWh) | Running Time (hh:mm:ss) |

Lower | 0 | 6.989281 | 2:44:06 | 0 | 6.550131 | 2:44:13 |

0.5 | 7.220638 | 2:44:25 | 0.5 | 6.800661 | 2:44:31 | |

Core | 1 | 7.451263 | 2:44:43 | 1 | 7.055194 | 2:44:46 |

Upper | 0.5 | 7.710178 | 2:45:00 | 0.5 | 7.335098 | 2:45:00 |

0 | 7.957147 | 2:45:14 | 0 | 7.602826 | 2:45:12 |

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## Share and Cite

**MDPI and ACS Style**

Blanco-Castillo, M.; Fernández-Rodríguez, A.; Fernández-Cardador, A.; Cucala, A.P.
Eco-Driving in Railway Lines Considering the Uncertainty Associated with Climatological Conditions. *Sustainability* **2022**, *14*, 8645.
https://doi.org/10.3390/su14148645

**AMA Style**

Blanco-Castillo M, Fernández-Rodríguez A, Fernández-Cardador A, Cucala AP.
Eco-Driving in Railway Lines Considering the Uncertainty Associated with Climatological Conditions. *Sustainability*. 2022; 14(14):8645.
https://doi.org/10.3390/su14148645

**Chicago/Turabian Style**

Blanco-Castillo, Manuel, Adrián Fernández-Rodríguez, Antonio Fernández-Cardador, and Asunción P. Cucala.
2022. "Eco-Driving in Railway Lines Considering the Uncertainty Associated with Climatological Conditions" *Sustainability* 14, no. 14: 8645.
https://doi.org/10.3390/su14148645