The Energy-Efficient Operation Problem of a Freight Train Considering Long-Distance Steep Downhill Sections
Abstract
:1. Introduction
1.1. Previous Work for the Train Energy-Efficient Operation
1.1.1. Classic Methods
1.1.2. Intelligent Methods
1.2. Main Contribution
1.3. Main Structure of this Paper
2. Freight Train Model and Problem Formulation
2.1. Real Characteristics of a Freight Train
2.2. Train Model
2.3. Problem Formulation
- (1)
- In most electric trains, the regenerative braking energy can be fed back to the traction power supply system if other trains along the railway line could utilize it. Currently, freight trains are not equipped with on-board energy storage devices. Hence, the regeneration efficiency is decided by the availability of other trains. In the realistic traction power supply system, the regeneration efficiency is a variable. The railway operation scheme (including the timetable and number of freight trains that are running along the railway line) should be investigated to obtain the accurate regeneration efficiency. Based on the operation data in months or years, we could calculate the regeneration efficiency. In China, the railway electrification system is 2 × 25 kV alternating current (AC) single phase with a frequency of 50 Hz. Two phases of the three-phase AC power from the public power grid are used to supply for the electric trains. The phases at different feeders of the substations will be switched between A–C, C–B, B–A, to maintain the overall balance of the public power grid (see Figure 5). Since the double-track railway including up rail and down rail is very common in China, the traction network of up rail and down rail connects in parallel. We can optimize the timetable to make the up train and down train run on the same LDSD section. For example, for a fully loaded train that is running on the up rail of the LDSD section, if electrical braking is applied, the electricity generated by the up train will be fed back to the traction network. At the same time, if the down train with empty wagons is going uphill with power on the LDSD section or the fully loaded up train is powering on the normal gradient (see Figure 5), of the energy generated by up train will be absorbed by the down train.
- (2)
- In this paper, our aim is to calculate the optimal speed trajectory considering the LDSD section. Our work in this paper is the operation optimization for a single train. The regeneration efficiency is introduced into the optimal control problem to evaluate the ratio of coasting and regenerative braking. To simplify the optimal control problem in this paper, the regenerative efficiency is considered as a constant.
3. Solution
3.1. Necessary Condition of the Train Energy-Efficient Operation
- Full power (FP): , when .
- Partial power (PP): , when .
- Coasting (C): when .
- Partial electrical braking (PEB): , , when .
- Full electrical braking (FEB): , , when .
- Full braking (FB): , when .
3.2. Transformation of the Time Constraint
3.3. Solution to the Optimal Control Problem with the Speed Constraint
3.4. Analysis of the Special Optimal Controls
3.4.1. Partial Power
3.4.2. Full Braking
3.5. Uniqueness Analysis of the Optimal Control at the Terminal Position of the LDSD Section
4. Linkage of the Speed Holding Section
4.1. Direction of the Linkage
4.2. Sufficient Conditions of State Variable Inequality Constraint
- and denote the left-hand side and the right-hand side of the train position .
- .
- .
- .
- or .
- (1)
- While , , jumps.
- (2)
- While , , jumps.
4.3. Linkage Case
4.3.1. Case 1
- (1)
- In Figure 9, the optimal control of the freight train switches from PP to C before entering the LDSD section. Based on the linkage direction in Section 4.1, the intersection of forward profile (C) and backward profile 1 (FEB) is .
- (2)
- Entering the LDSD section, the speed of the freight train keeps increasing even if FEB is applied. The adjoint variable () is still decreasing. While , . Although the jump condition of is satisfied, the jump will not happen yet due to the optimal controls. At the next moment, , FB is applied. We have .
- (3)
- Once FB is applied, pneumatic braking is released until . Then , jumps at . We have , then FEB is applied. And FEB is kept until and . Then FB is applied again.
- (4)
- While the freight train is running at the terminal position of the LDSD section, we have . For , jumps from to . Then FP is applied while the freight train is leaving the LDSD. Finally, the freight train is accelerated to .
4.3.2. Case 2
- (1)
- In Figure 10, the optimal control of the freight train switches from PP to C before entering the LDSD section. Based on the linkage direction in Section 4.1, the intersection of forward profile (C) and backward profile 1 (FEB) is .
- (2)
- While , . At the next moment, , FB is applied. We have .
- (3)
- Once FB is applied, pneumatic braking is released until . Then , jumps at . We have , FEB is applied. And FEB is kept until and . Then FB is applied again.
- (4)
- While the freight train is running at the terminal position of the LDSD section, we have . For , jumps from to . Then FP is applied while the freight train is leaving the LDSD. Finally, the freight train will coast to . And the optimal control between PEB section and PP section is C.
5. Relationship between Journey Time and Energy Consumption
- (1)
- Strategy 1: FEB+FB+FEB, the pneumatic braking release speed is equal to .
- (2)
- Strategy 2: C+FB+C, the pneumatic braking release speed is equal to .
6. Numerical Algorithm
- (1)
- Calculate the shortest time () from 0 to X, obtain the flat-out running trajectory.
- If , then go to step (2).
- If , optimization terminates, the flat-out running trajectory is returned.
- (2)
- Initializing , then solve by combining (39) and (45). Find the speed holding interval with PP and PEB, exclude the speed holding intervals where the freight train can’t keep constant speed with FP. If and , the solution is (42) and (47).
- (3)
- Generate the slope partition table which consists of the initial and terminal position of the speed holding interval. Suppose the set () represents the slope partition table. We have , where, is the number of the speed holding interval.
- (4)
- Linkage of the speed holding intervals.
- Linkage of the adjacent speed holding intervals.Suppose , and , . Connect and . If the connection of the two speed holding intervals fails, then jump to the linkage of the non-adjacent speed holding intervals.
- Linkage of the non-adjacent speed holding intervals (see Figure 13).Suppose , . The last speed holding interval that connects with successfully is (). is equal to . Connect and . If the connection fails, repeat until the connection succeeds in the domain of , i.e., .
- (5)
- The speed profile and optimal controls during the whole trip are recorded. Calculate the train operation time from 0 to . We have .
- (6)
- Evaluate the terminal condition:If (56) is satisfied, optimization terminates. Or, if , increase ; if , decrease .
- (7)
- Repeat steps 2–6 until the terminal condition is satisfied.
- (8)
- Return the optimal trajectory, optimal controls and energy consumption.
- (9)
- Optimization terminates.
7. Case Study
7.1. Train Operation Simulation under Different Journey Time
7.2. Comparison between The Proposed Algorithm and Other Methodologies
7.2.1. Fuzzy Predictive Control
7.2.2. Field Operation Data
8. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Parameter | Value | Unit |
---|---|---|
Train marshalling | 1 locomotive + 100 wagons | - |
Locomotive mass | 200 | t |
Locomotive length | 38 | m |
Maximum velocity | 120 | km/h |
Maximum traction force | 777 | kN |
Locomotive unit basic resistance | N/t | |
Energy efficiency ratio of the traction system | 0.9 | - |
Energy efficiency ratio of the electrical braking system | 0.9 | - |
Regenerative coefficient | 0.9 | - |
Mass per one wagon | 100 | t |
Length per one wagon | 12.2 | m |
Wagon unit basic resistance | N/t | |
Auxiliary reservoir air-filled time | 130 | s |
Origin Location (km) | Terminal Location (km) | Value (km/h) |
---|---|---|
0 | 2.9 | 35 |
2.9 | 67 | 75 |
67 | 67.720 | 70 |
67.720 | 70 | 35 |
Journey Time(s) | (km/h) | Energy Consumption (kWh) | Regenerative Energy(kWh) | Total Energy Consumption (kWh) |
---|---|---|---|---|
5704.20 | 25.5 | 1701.85 | 5583.39 | −3323.20 |
4914.49 | 40 | 1784.06 | 5608.62 | −3263.70 |
4779.73 | 45 | 1821.44 | 5627.82 | −3243.60 |
4682.57 | 50 | 1873.84 | 5667.86 | −3227.23 |
Strategies | Journey Time (min) | (km/h) | Energy Consumption (kWh) | Regenerative Energy (kWh) | Total Energy Consumption (kWh) |
---|---|---|---|---|---|
Optimal algorithm | 95 | 25.5 | 1701.85 | 5583.39 | −3323.20 |
FPC | 95 | - | 1821.50 | 5492.60 | −3121.84 |
Field operation | 95 | - | 1906.50 | 5231.21 | −2801.59 |
Strategies | FPC (%) | Field Operation (%) |
---|---|---|
Optimal algorithm | 6.5 | 18.6 |
FPC | - | 11.4 |
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Lin, X.; Wang, Q.; Wang, P.; Sun, P.; Feng, X. The Energy-Efficient Operation Problem of a Freight Train Considering Long-Distance Steep Downhill Sections. Energies 2017, 10, 794. https://doi.org/10.3390/en10060794
Lin X, Wang Q, Wang P, Sun P, Feng X. The Energy-Efficient Operation Problem of a Freight Train Considering Long-Distance Steep Downhill Sections. Energies. 2017; 10(6):794. https://doi.org/10.3390/en10060794
Chicago/Turabian StyleLin, Xuan, Qingyuan Wang, Pengling Wang, Pengfei Sun, and Xiaoyun Feng. 2017. "The Energy-Efficient Operation Problem of a Freight Train Considering Long-Distance Steep Downhill Sections" Energies 10, no. 6: 794. https://doi.org/10.3390/en10060794