Robust energy-efficient train speed profile optimization in a scenario-based position–time–speed network

doi:10.1007/s42524-021-0173-1

Front. Eng

2021, Vol. 8

Issue (4) : 595-614 https://doi.org/10.1007/s42524-021-0173-1

RESEARCH ARTICLE

Robust energy-efficient train speed profile optimization in a scenario-based position–time–speed network

Yu CHENG, Jiateng YIN(

), Lixing YANG

State Key Laboratory of Rail Traffic Control & Safety, Beijing Jiaotong University, Beijing 100044, China

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Abstract

Train speed profile optimization is an efficient approach to reducing energy consumption in urban rail transit systems. Different from most existing studies that assume deterministic parameters as model inputs, this paper proposes a robust energy-efficient train speed profile optimization approach by considering the uncertainty of train modeling parameters. Specifically, we first construct a scenario-based position–time–speed (PTS) network by considering resistance parameters as discrete scenario-based random variables. Then, a percentile reliability model is proposed to generate a robust train speed profile, by which the scenario-based energy consumption is less than the model objective value at $α$ confidence level. To solve the model efficiently, we present several algorithms to eliminate the infeasible nodes and arcs in the PTS network and propose a model reformulation strategy to transform the original model into an equivalent linear programming model. Lastly, on the basis of our field test data collected in Beijing metro Yizhuang line, a series of experiments are conducted to verify the effectiveness of the model and analyze the influences of parameter uncertainties on the generated train speed profile.

Keywords robust train speed profile percentile reliability model scenario-based position–time–speed network mixed-integer programming

Corresponding Author(s): Jiateng YIN

Just Accepted Date: 09 September 2021 Online First Date: 12 October 2021 Issue Date: 01 November 2021

Cite this article:

Yu CHENG,Jiateng YIN,Lixing YANG. Robust energy-efficient train speed profile optimization in a scenario-based position–time–speed network[J]. Front. Eng, 2021, 8(4): 595-614.

URL:

https://academic.hep.com.cn/fem/EN/10.1007/s42524-021-0173-1
https://academic.hep.com.cn/fem/EN/Y2021/V8/I4/595

Tab.1 Differences between recent publications and our work on the train speed profile optimization problem

Notations	Detailed definition
Parameters
$F$	Force provided by the train
$R$	Resistance force of the train
$a$	Acceleration/Deceleration of the train
$S$	Distance between two stations
$T$	Planned travel time of the train between two stations
$V$	A large value of the speed
$Δ s$ , $Δ t$ , $Δ v$	Length of each discrete segment for position, time, and speed dimensions, respectively
$H 1$ , $H 2$ , $H 3$	Total number of discrete segments for position, time, and speed dimensions, respectively
$(s, t, v)$ , $(s ′, t ′, v ′)$	Nodes in the PTS network
$N$	Set of nodes in the PTS network
$(s, t, v, s ′, t ′, v ′)$	Arc in the PTS network
$A$	Set of arcs in the PTS network
$F s, t, v, s ′, t ′, v ′$	Force provided by the train moving on arc $(s, t, v, s ′, t ′, v ′) ∈ A$
$e s, t, v, s ′, t ′, v ′$	Traction energy consumption on arc $(s, t, v, s ′, t ′, v ′) ∈ A$
$R B s, t, v, s ′, t ′, v ′$	Basic resistance of the train moving on arc $(s, t, v, s ′, t ′, v ′) ∈ A$
$R G s, t, v, s ′, t ′, v ′$	Resistance caused by the gradient of the railway track
$m$	Weight of the train, including onboard passengers
$a s, t, v, s ′, t ′, v ′$	Acceleration/Deceleration of the train on arc $(s, t, v, s ′, t ′, v ′) ∈ A$
$β 1$ , $β 2$ , $β 3$	Parameters of Davis’ formula about the basic resistance of the train
$g$	Acceleration parameter of gravity
$v ¯ s, t, v, s ′, t ′, v ′$	Average speed of the train on arc $(s, t, v, s ′, t ′, v ′) ∈ A$
$θ s$	Gradient at position $s$
$V max$	Maximum speed the train can achieve
$a max$ , $d max$	Maximum acceleration and deceleration the train can reach, respectively
$N *$	Node set of the reduced PTS network
$A *$	Arc set of the reduced PTS network
$a^s, t, v$ , $d^s, t, v$	Maximum acceleration and deceleration the train can reach at node $(s, t, v)$ , respectively
$v^s, t, v$ , $v ⌣ s, t, v$	Maximum and minimum speeds of the follower nodes for node $(s, t, v)$ , respectively
$V^s$	Speed limit at position $s$
$ω$	Random scenario
$P ω$	Probability of scenario $ω$
$Ω$	Set of scenarios
$β 1 ω$ , $β 2 ω$ , $β 3 ω$	Parameters of Davis’ formula about the basic resistance of the train under scenario $ω$
$e s, t, v, s ′, t ′, v ′ ω$	Traction energy consumption on arc $(s, t, v, s ′, t ′, v ′) ∈ A$ under scenario $ω$
$a^s, t, v ω$ , $d^s, t, v ω$	Maximum acceleration and deceleration the train can reach at node $(s, t, v)$ under scenario $ω$ , respectively
$M$	A sufficiently large number
$α$	Confidence level of the PRM
$γ$	Upper limit of the comfort level
Variables
$x s, t, v, s ′, t ′, v ′$	Binary decision variable; it takes 1 if arc $(s, t, v, s ′, t ′, v ′) ∈ A$ is selected; otherwise, it takes 0
$φ$	Critical value for the total energy consumption of a speed profile, introduced by PRM
$L$ , $y ω$	Introduced variables for the linearization of PRM

Tab.2 Relevant parameters and variables used in this paper

Fig.1 Speed profiles over an interstation.

Fig.2 Example of a PTS network.

Fig.3 Projection region of the feasible speed profile in each coordinate plane.

Algorithm 1 Algorithm for identifying the feasible nodes in the PTS network
Input:	$V max$ , $a max$ , $d max$ , $S$ , $T$ , $V$ , $Δ s$ , $Δ t$ , $Δ v$
Output:	Initialize: $I N$ // Feasible node indicator, 1 if feasible, and 0 otherwise
1:	$I N ← 0$
2:	$H 1 ← ? S / Δ s ?$ , $H 2 ← ? T / Δ t ?$ , $H 3 ← ? V / Δ v ?$
3:	$t 1 ← ? (V max / a max) / Δ t ?$ , $t 2 ← H 2 − ? (V max / d max) / Δ t ?$
4:	$s 1 ← ? (V max) 2 / (2 a max) / Δ s ?$ , $s 2 ← H 1 − ? (V max) 2 / (2 d max) / Δ s ?$
5:	for $k = 1 : H 2$ do // Generate a feasible region boundary in Speed–Time plane ( $S T$ )
6:	if $1 ≤ k < t 1 − 1$ then
7:	? $S T k ← a max (k Δ t) / Δ v$
8:	?else if $t 1 ≤ k ≤ t 2$ then
9:	?? $S T k ← V max / Δ v$
10:	?else
11:	?? $S T k ← d max ((H 2 − k) Δ t) / Δ v$
12:	for $h = 1 : H 1$ // Generate a feasible region boundary in Speed–Position plane ( $S P$ )
13:	?if $1 ≤ h < s 1 − 1$ then
14:	?? $S P h ← 2 a max h Δ s / Δ v$
15:	?else if $s 1 ≤ h ≤ s 2$ then
16:	?? $S P h ← V max / Δ v$
17:	?else
18:	?? $S P h ← 2 d max (H 1 − h) Δ s / Δ v$
19:	for $l = 1 : H 1$ do // Generate a feasible region boundary in Time–Position plane ( $T P$ )
20:	?if $1 ≤ l < s 1 − 1$ then
21:	?? $T P l ← 2 (l Δ s) / a max / Δ t$
22:	?else
23:	?? $T P l ← (l Δ s / V max + V max / (2 a max)) / Δ t$
24:	for $s = 1 : H 1$ do // Check the feasibility of the network nodes
25:	?for $t = 1 : H 2$ do
26:	??for $v = 1 : H 3$ do
27:	???if $v ≤ S P s & v ≤ S T t & t ≥ T P s$ then
28:	???? $I N s, t, v ← 1$

Tab.3 Algorithm for identifying the feasible nodes in the PTS network

Algorithm 2 DP algorithm for generating feasible arcs in the PTS network
Input:	$I N$ , $V^s$ , $a max$ , $d max$ , $β 1$ , $β 2$ , $β 3$ , $S$ , $T$ , $V$ , $Δ s$ , $Δ t$ , $Δ v$ , $m$ , $g$
Output:	$A $ , $N $ , $a s, t, v, s ′, t ′, v ′$ , $? e s, t, v, s ′, t ′, v ′$ , $(s, t, v, s ′, t ′, v ′) ∈ A *$
1:	Initialize: $A * ← Null$ , $C N ← {(0, 0, 0)}$ , $F N ← Null$ , $? O N ← Null$ , $D N ← Null$
1:	// $C N$ : Current node set, $F N$ : Feasible follower node set, $O N$ : Set of origin node for arcs in $A $ , $D N$ : Set of destination node for arcs in $A $
2:	$H 1 ← ? S / Δ s ?$ , $H 2 ← ? T / Δ t ?$ , $H 3 ← ? V / Δ v ?$
3:	for $k = 1 : H 2 − 1$ do // Search for feasible arcs
4:	?for $(s 0, t 0, v 0) ∈ C N$ do
5:	?? $a^s 0, t 0, v 0 ← (m a max − R B s 0, t 0, v 0 − R G s 0, t 0, v 0) / m$
6:	?? $v^s 0, t 0, v 0 ← min (v 0 + a^Δ t, V^s)$ , $H v^← ? v^/ Δ v ?$
7:	??if $k = 1$ then // When the current node is the origin node, acceleration only
8:	??? $v ⌣ 0, 0, 0 ← Δ v$ , $H ⌣ v ← 1$
9:	??else
10:	??? $d^s 0, t 0, v 0 ← (m d max + R B s 0, t 0, v 0 + R G s 0, t 0, v 0) / m$
11:	??? $v ⌣ s 0, t 0, v 0 ← max (v 0 − d^Δ t, Δ v)$ , $H ⌣ v ← ⌣ v / Δ v$
12:	for $v = H ⌣ v : H v^$ do
13:	?? $s ← s 0 + ? ((v 0 + v) Δ v / 2) Δ t / Δ s ?$
14:	??if $I N s, k, v = 1 & (v 0 + v) Δ v ≤ V^s$ then
15:	?? $a s 0, t 0, v 0, s, k, v ← (v − v 0) Δ v / Δ t$
16:	?? $e s 0, t 0, v 0, s, k, v ←$ Eqs. (2)–(6)
17:	?? $A * ← A * ∪ {(s 0, t 0, v 0, s, k, v)}$
18:	?? $F N ← F N ∪ {(s, k, v)}$
19:	?? $C N ← F N$ , $F N ← Null$
20:	for $k = H 2$ do // Search for the feasible arcs ended with the destination node $(H 1, H 2, 0)$
21:	?for $(s 0, t 0, v 0) ∈ C N$ do
22:	??? $d^s 0, t 0, v 0 ← (m d max + R B s 0, t 0, v 0 + R G s 0, t 0, v 0) / m$
23:	??? $s ← s 0 + ? (v 0 Δ v / 2) Δ t / Δ s ?$
24:	??? $a s 0, t 0, v 0, H 1, H 2, 0 ← v 0 Δ v / Δ t$
25:	?if $s = H 1 & a s 0, t 0, v 0, H 1, H 2, 0 ≥ – d^s 0, t 0, v 0$ then
26:	?? $e s 0, t 0, v 0, s, k, v ←$ Eqs. (2)–(6)
27:	?? $A * ← A * ∪ {(s 0, t 0, v 0, H 1, H 2, 0)}$
28:	?? $O N ← {(s 0, t 0, v 0) \| (s 0, t 0, v 0, s, t, v) ∈ A *}$
29:	?? $D N ← {(s, t, v) \| (s 0, t 0, v 0, s, t, v) ∈ A *}$
30:	for $(s, t, v) ∈ D N * \ {(H 1, H 2, 0)}$ do // Remove the disconnected arcs from $A *$
31:	?if $(s ′, t ′, v ′) ∉ O N$ then
32:	?? $A * ← A * \ {(s, t, v, s ′, t ′, v ′)}$
33:	?? $O N ← {(s 0, t 0, v 0) \| (s 0, t 0, v 0, s, t, v) ∈ A *}$
34:	?? $D N ← {(s, t, v) \| (s 0, t 0, v 0, s, t, v) ∈ A *}$
35:	?? $N * ← D N ∪ {(0, 0, 0)}$

Tab.4 DP algorithm for generating feasible arcs in the PTS network

Fig.4 Example of the process of generating a feasible arc set.

Variable/Constraints	Total number at most
Variable/Constraints	PRM	LPRM
Binary variable $X$	$\| A \|$	$\| A \|$
Continuous variable $φ$	1	0
Binary variable $Y$	0	$\| Ω \|$
Continuous variable $L$	0	1
Constraint (33)	1	0
Flow balance constraint (34)	$\| N \|$	$\| N \|$
Comfort constraint (35)	$\| N \|$	$\| N \|$
Constraint (39)	0	$\| Ω \|$
Constraint (40)	0	1

Tab.5 Numbers of variables and constraints in PRM and LPRM, respectively

Fig.5 Values of speed limit and track gradient between Rongjingdongjie and Wanyuanjie stations on the Yizhuang line.

Scenario $ω$	$β 1$	$β 2$	$β 3$
Scenario 1	$2.0 × a 1$	$2.0 × a 2$	$2.0 × a 3$
Scenario 2	$1.8 × a 1$	$1.8 × a 2$	$1.8 × a 3$
Scenario 3	$1.6 × a 1$	$1.6 × a 2$	$1.6 × a 3$
Scenario 4	$1.5 × a 1$	$1.5 × a 2$	$1.5 × a 3$
Scenario 5	$1.4 × a 1$	$1.4 × a 2$	$1.4 × a 3$
Scenario 6	$1.2 × a 1$	$1.2 × a 2$	$1.2 × a 3$
Scenario 7	$1.0 × a 1$	$1.0 × a 2$	$1.0 × a 3$
Scenario 8	$0.8 × a 1$	$0.8 × a 2$	$0.8 × a 3$
Scenario 9	$0.6 × a 1$	$0.6 × a 2$	$0.6 × a 3$
Scenario 10	$0.4 × a 1$	$0.4 × a 2$	$0.4 × a 3$

Tab.6 List of resistance parameter values for different scenarios

Fig.6 Optimal speed profiles in different scenarios.

$α$	Energy consumption (J)					Objective value (J)	Probability
$α$	Scenario 1	Scenario 3	Scenario 6	Scenario 8	Scenario 10	Objective value (J)	Probability
0.2	$1.32 × 108$	$1.13 × 108$	$9.34 × 107$	$7.39 × 107$	$5.66 × 107$	$5.66 × 107$	0.2
0.4	$1.29 × 108$	$1.09 × 108$	$8.95 × 107$	$7.01 × 107$	$6.11 × 107$	$7.01 × 107$	0.4
0.6	$1.24 × 108$	$1.04 × 108$	$8.56 × 107$	$7.63 × 107$	$6.98 × 107$	$8.56 × 107$	0.6
0.8	$1.23 × 108$	$1.02 × 108$	$8.91 × 107$	$7.86 × 107$	$6.98 × 107$	$1.02 × 108$	0.8
1.0	$1.21 × 108$	$1.05 × 108$	$9.52 × 107$	$8.46 × 107$	$7.40 × 107$	$1.21 × 108$	1.0

Tab.7 Results of the conducted experiments on different scenarios

$α$	Objective value (J)		EEC (J)		CT (s)
$α$	5 scenarios	10 scenarios	5 scenarios	10 scenarios	5 scenarios	10 scenarios
1.0	$1.21 × 108$	$1.21 × 108$	$9.62 × 107$	$9.67 × 107$	1.9	4.7
0.9	$1.21 × 108$	$1.11 × 108$	$9.62 × 107$	$9.41 × 107$	2.0	2.7
0.8	$1.02 × 108$	$1.02 × 108$	$9.26 × 107$	$9.26 × 107$	2.3	3.1
0.7	$1.02 × 108$	$9.85 × 107$	$9.26 × 107$	$9.26 × 107$	2.3	2.7
0.6	$8.56 × 107$	$9.40 × 107$	$9.18 × 107$	$9.21 × 107$	2.4	3.2
0.5	$8.56 × 107$	$8.56 × 107$	$9.18 × 107$	$9.19 × 107$	2.3	2.7
0.4	$7.01 × 107$	$7.78 × 107$	$9.17 × 107$	$9.16 × 107$	2.3	2.6
0.3	$7.01 × 107$	$5.66 × 107$	$9.17 × 107$	$9.51 × 107$	2.2	3.0
0.2	$5.66 × 107$	$5.66 × 107$	$9.38 × 107$	$9.51 × 107$	2.2	3.2
0.1	$5.66 × 107$	$5.66 × 107$	$9.38 × 107$	$9.51 × 107$	2.1	3.1

Tab.8 Results of experiments on two sets of scenarios with different confidence levels

Fig.7 Optimal speed profiles with different confidence levels.

Test index	Network discretization				EEC (J)	CT (s)	Network scale
Test index	$Δ s$	$Δ t$	$Δ v$	$Δ a$	EEC (J)	CT (s)	Node	Arc
1	5	5	2	0.40	$1.05 × 108$	0.6	274	644
2	2	4	1	0.25	$1.02 × 108$	1.1	1115	4358
3	0.5	2	0.5	0.25	$9.82 × 107$	437.1	37186	233695
4	10	10	2	0.20	$9.82 × 107$	0.5	105	239
5	2.5	5	1	0.20	$9.62 × 107$	2.1	1830	9477
6	5	10	1	0.10	$9.30 × 107$	0.7	483	2191
7	1.25	5	0.5	0.10	$9.46 × 107$	66.0	10406	112996
8	2.5	10	0.5	0.05	$9.26 × 107$	2.3	2064	18646
9	2	10	0.4	0.04	$9.23 × 107$	15.7	3443	39791

Tab.9 Results of the experiments on different network discretizations

Running time (s)	EEC (J)	Network scale		CT (s)
Running time (s)	EEC (J)	Node	Arc	CT (s)
86	$1.18 × 108$	2990	12722	5
88	$1.02 × 108$	16871	96834	59
90	$9.82 × 107$	37186	233695	417
92	$9.65 × 107$	62313	410596	1147
94	$9.28 × 107$	91409	620717	4449
96	$9.21 × 107$	123919	859515	10986

Tab.10 Results of the experiments on different total running times

Fig.8 Optimal speed profiles with different total running times.

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