1. Introduction
The train operation system is a complex multi-objective nonlinear system, which needs to take into account multiple performance indicators such as safety, punctuality, energy saving, accurate parking and comfort [
1,
2,
3]. At the same time, the train is restricted by a variety of constraints in the operating process, and it has a lot of uncertainty [
4,
5,
6]. Therefore, how to use optimization methods to solve the train operation strategy is a hot issue in train research. At present, many intelligent optimization algorithms and their improved algorithms have been applied in train operation strategy optimization, such as genetic algorithm (GA), particle swarm optimization (PSO), simulated annealing (SA) algorithm, differential evolution (DE) algorithm, hybrid evolutionary algorithm and so on. Wang et al. [
7] use GA with global search to optimize the speed curve of automatic train operation (ATO) to obtain an accurate train control sequence, which satisfies the speed protection index, punctuality index, accurate parking index, comfort index and energy saving index. The authors in [
8], aiming at the ATO system, adopt the multi-objective optimization strategy of GA to optimize from five aspects: safety, accurate parking, punctuality, energy saving and comfort. In addition, the penalty function is added to the fitness function to improve the convergence speed of GA. Rong et al. [
9] use PSO to solve the multi-objective optimization model of train operating process. Meanwhile, to solve the problem that the basic PSO are easily trapped into the local optimum in the late evolution period, the acceleration coefficient of PSO is improved. Shangguan et al. [
10] propose a hybrid evolutionary algorithm based on DE and SA to solve the multi-objective optimization model for the speed trajectory to obtain an optimal velocity trajectory searching strategy.
The authors in [
7,
8,
9,
10] have achieved good results in the multi-objective optimization of train operation strategy through some common optimization algorithms, and one common feature of these algorithms is that the single population search method is used for the optimal solution. If the single population search strategy is extended to the multi-population search strategy, better optimization effect may be obtained. Therefore, many researchers have begun to study multi-population optimization algorithms in depth. Youneng et al. [
11] propose a multi-population genetic algorithm (MPGA) to reduce the traction energy by optimizing the train operation for multiple interstations, and this method can get a better energy-efficient driving strategy. Zhou et al. [
12] use the multi-objective multi-population ant colony optimization algorithm for continuous domain to solve the economic emission dispatch problem, and the pheromone structure of ant colony is reconstructed, which extends the original single-target method to the multi-target region. To overcome the premature convergence problem, the multi-population ant colony with different search range and speed is also proposed. Aiming at the problem of constrained optimization, Xu et al. [
13] introduce a new method combining adaptive DE with multi-population mutation operators. In the process of mutation, the external population is combined with the main population to produce the evolutionary direction towards the optimal region. In addition, the new method can adaptively adjust DE’s control parameters according to the previous statistical information. Wang et al. [
14] propose an adaptive multi-population optimization method for multi-objective optimization problem. In addition, this method combines PSO with DE to guide the exploitation of Pareto optimal solutions.
Based on the research results of literature [
11,
12,
13,
14], in this paper, a DP-GAPSO algorithm is proposed for the multi-objective optimization of train operation strategy, which can make up for the lack of a single population, a single method [
15,
16]. One population evolves by using GA, and the other population evolves by using PSO. In addition, the two branched populations adopt parallel structure to participate in evolution simultaneously [
17,
18]. Parallel structure refers to the fact that multiple tasks are not prioritized and can be carried out simultaneously to reduce the waiting time of the single task [
19]. On the one hand, it can reduce the evolution time of the whole population, and, on the other hand, it simultaneously makes the two populations produce new individuals for further operation. To make the two populations complement each other, an immigrant strategy is proposed to exchange some good individuals between two populations, which can give full play to the overall advantages of parallel structure. In addition, GA and PSO are improved, respectively called IGA and IPSO, to achieve better optimization effect. For GA, individuals are selected by adjusting the selection pressure adaptively based on the current iteration number to improve the convergence speed. In order to prevent the destruction of the best individual in each iteration, elite retention strategy (ERS) is introduced into GA, which makes an important contribution to the convergence of the algorithm. The concept of opposition-based learning (OBL) was proposed by Tizhoosh in 2005 [
20], who pointed out that the opposition solution was nearly 50% more likely to approach the optimal solution than the current solution. Therefore, GA uses the general dynamic OBL (which is a type of OBL) to generate opposition population. Then, the good individuals are selected from the current population and opposition population to form a temporary population to participate in next generation evolution, which expands the search area of the population. For PSO, the inertia weight determines the ability of the particle to inherit its previous velocity. Shi [
21] first introduced the inertia weight into PSO. He also pointed out that a larger inertia weight was beneficial to global search, while a smaller inertia weight was more beneficial to local search. Thus, linear decreasing inertia weight (LDIW) is adopted to balance the global search ability and local search ability of PSO, which can improve the convergence speed of PSO.
In addition, the train operation strategy optimization should also take into account the eco-driving design in railways. Fernández-Rodríguez et al. [
22] propose a real-time multi-objective optimization algorithm by means of fuzzy numbers to model the uncertainty of manual driving, considering the interference of delay. When a delay occurs, the system recalculates an optimal driving to reduce energy consumption. Aiming at the optimal energy-saving driving strategy of the train, Albrecht et al. [
23] adopted a fast and effective numerical algorithm to solve the problem of the local energy minimization, so as to find the best switching point. The authors in [
24] propose a robust and efficient method for designing velocity profile in the ATO equipment of the metro is proposed, which involves two objectives: running time and energy consumption. In addition, PSO is used to optimize the multi-objective model to minimize the total energy consumption. Bocharnikov et al. [
25] design a fitness function with variable weight, and find the qualitative and quantitative effects of acceleration and braking rates on train energy saving by using genetic search method. The optimal train track is determined by using the fitness function. Lu et al. [
26] propose a distance-based train track search model, which is optimized by various optimization algorithms. The results show that the ant colony algorithm achieves a good balance between stability and energy saving effect.
In order to verify that DP-GAPSO has better optimization performance for the multi-objective optimization of train operation strategy, three intervals of Rail transit line 12 and Jinpu line 1 in Dalian, China are selected for simulation test. Both MATLAB simulation and hardware-in-the-loop (HIL) simulation results show that, compared with IGA and IPSO, the multiple performance indexes obtained by DP-GAPSO have been improved to a considerable extent. Therefore, DP-GAPSO has better optimization performance.
2. Problem Description of Train Operation
Classic setup of discrete actions of train operation includes the full traction condition, constant speed condition, coasting condition and full braking condition as follows:
where the constant speed condition includes partial traction and partial braking, and the coasting condition means that the train exerts neither traction nor braking force.
A scientific research team at Beijing Jiaotong university in China has also proposed another setup of discrete actions of train operation according to the handle position of the train. Different handle positions correspond to different traction or braking forces. For example, the handle position of the train can be divided as follows:
where the maximum handle position is 4. The ratio of the current handle position to the maximum handle position is multiplied by the maximum traction or braking force to obtain the current traction or braking force.
Finally, we choose the classic setup of discrete actions of train operation () to study the train operation strategy. When the train runs in the same interval, the results obtained by using different combinations of operating conditions are also different. In this paper, we take the switching positions of the operating conditions and the corresponding operating conditions of the train as decision variables.
2.1. Safety Protection Curve
There are multiple speed limit sub-intervals in the train operating interval, and the speed of the train must be kept below the speed limit. The speed limit sub-intervals are divided into three cases: the speed constant limit sub-interval, the speed limit falling sub-interval and the speed limit rising sub-interval. These three cases are dealt with as follows:
Speed constant limit sub-interval
As shown in
Figure 1a, in the constant speed limit sub-interval, when the actual running speed of the train is greater than or equal to the speed limit, the switching point of constant speed condition is inserted into the train control sequence to make the train maintain a constant running speed under the speed limit. In addition, this constant running speed curve is the safety protection curve.
Speed limit falling sub-interval
In the speed limit falling sub-interval, it is necessary to ensure that the train enters the low speed limit area at a speed which is less than the speed limit, so the train needs to brake and slow down in advance. As shown in
Figure 1b, from the beginning point of the speed limiting section B, the reverse calculation is carried out under the full braking condition until the speed of the train equals the speed limit of the section A. By this reverse calculation method, the braking curve from the speed limiting section A to the speed limiting section B can be obtained, which is also called the safety protection curve.
Speed limit rising sub-interval
As shown in
Figure 1c,
l represents the length of the train. When the train enters the speed limit section D from the speed limit section C, it is necessary to ensure that the speed of the back for the train is below the speed limit of the section C. Therefore, the train cannot be accelerated immediately when its head leaves the speed limit section C. When the rear part of the train leaves the speed limit section C, the train accelerates under the full traction condition until the speed of the train equals the speed limit of the section D. By this method, the traction curve from the speed limiting section C to the speed limiting section D can be obtained, which is also called the safety protection curve.
The multi-segment safety protection curves can be obtained by preprocessing the whole operating interval according to the above three cases. These safety protection curves can be connected to obtain the safety protection curve of the whole operating interval. In the simulation operation, when the running speed of the train is greater than or equal to the speed of the safety protection curve, it is switched to the protection curve condition. Then, the train runs in accordance with the safety protection’s curve until the operating speed is lower than the protection curve, and the original condition of the train is restored. As shown in
Figure 2, when the train reaches point A and point C, it switches to the condition of the safety protection curve. When the train reaches point B and point D, it will return to the original operating condition.
2.2. Initialization Settings for Operating Conditions
When the intelligent optimization algorithm is used to solve the optimal control sequence of train, there will be a lot of infeasible solutions if the train operating conditions are randomly generated within the entire running interval. In order to improve the efficiency of the algorithm, the ramps in entire running interval can be divided into the following three cases according to the forced condition of the train on the ramp.
Case1:
When the train is in the coasting condition (the train exerts neither traction nor braking force), the ramp in which the train still gets the same acceleration as the driving direction is called the big downhill.
Case2:
When the train is in the full traction condition, the ramp in which the train still slows down is called the big uphill.
Case3:
The remaining ramps except for case 1 and case 2 are continuous ramps.
According to these three cases, the entire interval can be divided into multiple sub-intervals. As you can see in
Figure 3, for each sub-interval, its starting point
is the beginning position of the big uphill or the continuous ramp, its middle point
is the beginning position of the big downhill, and its end point
is the end position of the big downhill. In
Figure 3, the operating conditions within a sub-interval is set as
. The switching point of the full traction condition (1) is the starting point of the sub-interval, as shown
in
Figure 3. The setting range of the switching point
for the constant speed condition (0.5) is from the switching point
of the previous full traction condition (1) to the middle point
of the sub-interval, as shown 1 (
) in
Figure 3. The setting range of the switching point
for the coasting condition (0) is from the switching point
of the previous constant speed condition (0.5) to the end point
of the sub-interval, as shown 2 (
) in
Figure 3. The full braking condition (−1) is inserted when the train running curve hits the safety protection curve. The train control sequence of the entire interval is the combination of the operating conditions of all the sub-intervals.
3. Multi-Objective Optimization Model for the Train Operating Strategy
The train operating strategy optimization is a complex optimization problem that needs to meet multiple performance indexes such as energy saving, comfort level and punctuality at the same time [
27,
28]. The model for each performance index is as follows.
The model of energy consumption for train operation strategy optimization is
where
is the energy consumed by the urban rail train;
f and
b are the traction force and braking force;
A is the auxiliary power of the urban rail train;
is the actual running time of the urban rail train between stations;
is the conversion factor that converts electrical energy into mechanical energy during the urban rail train pulling; and
is the conversion factor that converts mechanical energy into electrical energy during the urban rail train braking.
The model of comfort index for train operation strategy optimization is
where
is passengers’ comfort level, which is generally reflected by the variation of acceleration of the train;
is the rate of change of acceleration, also called
. When
is smaller, the passenger feels more comfortable. Research shows that the value of
should be kept below 1.5 m/s
. In order to simplify the calculation, the above model is improved as follows:
where
is expressed by the sum of the absolute value of the difference in acceleration of the two adjacent time steps;
and
are the two accelerations at the
ith and
th time steps, respectively.
The model of punctuality index for train operation strategy optimization is
where
is the absolute value of the difference between the actual running time and the prescribed running time;
is the speed of the train at the
th time step;
T is the prescribed time and
is the actual running time between two stations.
Combined with the established performance indexes and multi-objective optimization theory, the multi-objective optimization model for train operating strategy is built as follows:
where
Z is the multi-objective optimization model for train operating strategy;
M is the mass of the train;
u is the control sequence of the train;
is the actual traction of the urban rail train, which is determined by
u and the actual running speed
v of the urban rail train;
is the braking force of the urban rail train, which is determined by
u and
v;
S is the distance between two given stations;
is the speed limit of train operation; and
is the actual running distance of the train between the two given stations. In general, the error between
and
S cannot exceed 30 cm, which can make passengers get on the train.
is the running resistance of train, including basic resistance (frictional resistance and air resistance), additional resistance of ramp, additional resistance of curve and additional resistance of tunnel, as follows:
where
is the running resistance of train;
is the basic resistance of train;
are empirical constants related to the train;
v is the speed of the train;
is additional resistance of ramp per unit weight;
i is the slope, and its unit is ‰; Re
3 is the additional resistance of curve;
R is the radius of the track curve;
L is the length of the train;
Lr is the length at which the train overlaps the track curve;
Re4 is additional resistance of tunnel;
Ls is the length of the tunnel.
In addition, the multi-objective optimization problem for the train operating strategy can be transformed into the single-objective problem by the linear weighting method [
29] as follows:
where
f is the target function;
,
and
are the weight coefficients of the three optimization indexes (
).
To obtain the weight of each index more accurately, linear combination weights based on entropy (LCWBE) is used, which is both subjectivity and objectivity [
30,
31]. The steps of LCWBE is as follows.
For
n given evaluation objects and
m evaluation indexes, the evaluation matrix
can be constructed as
Then, Equation (
12) is normalized to obtain
In addition, the best evaluation object is an
m-dimensional vector whose components are all 1. The experienced persons give
l weight vectors (
) of evaluation indexes. The
kth weight vector
is
, and it satisfies
is the weight vector obtained by the linear combination of (
), and
is
where
is the linear combination coefficient and it satisfies
The generalized distance
between the
ith evaluation object and the best evaluation object is
Since the combination coefficient
has uncertainty, the uncertainty can be expressed by Shannon information entropy [
32] as follows:
Before solving the linear combination weight vector
, it is necessary to determine the appropriate combination coefficient
. On the one hand, the sum of the generalized distances between all evaluation objects and the best evaluation object should be minimized as follows:
On the other hand, the uncertainty of the combination coefficient
should be eliminated as far as possible. According to Jaynes maximum entropy principle [
33], the weight coefficient
should make Shannon entropy maximum that is
Since it is a multi-objective optimization problem to solve the linear combination weight vector
, it is converted to the single objective optimization as follows:
where
is the balance coefficient between the two objectives, and it is set to 0.8.
It is verified that the above single-objective optimization problem has a unique solution, and the solution is
where
.
Train operation strategy optimization has three optimization indexes (). In this paper, we select 10 objects () and three weight vectors of optimization indexes given by the experienced persons (). After the calculation of LCWBE, the weights of the three optimization indexes are 0.5124, 0.2854, 0.2022, respectively.
6. Conclusions
In this paper, taking energy saving, comfort and punctuality as optimization indexes, the multi-objective optimization model of train operation strategy is established, and the multi-objective optimization problem is transformed into the single-objective optimization problem by using a linear weighting method. In addition, LCWBE that takes account of both subjectivity and objectivity is used to obtain the weight of each index more accurately. Aiming at the multi-objective optimization of the train operation strategy, a DP-GAPSO algorithm is proposed to obtain the optimal operation strategy, which adopts a parallel structure and double-population optimization strategy. One population evolves by using GA, and the other population evolves by using PSO. To make these two populations complement each other, an immigrant strategy is proposed to exchange some good individuals between two populations, which can give full play to the overall advantages of parallel structure.
In addition, GA and PSO are improved, respectively. For GA, its convergence rate and optimization effect are improved by adjusting the selection pressure adaptively based on the current iteration number. ERS is introduced into the GA to avoid destroying the best individual in each iteration. In order to avoid the algorithm falling into local optimum as far as possible, OBL is used to generate the opposition population. In addition, the opposite population participates in evolution along with the current population. For PSO, LDIW is proposed to better balance the global search ability and local search ability, so that better optimization effect can be obtained. Simulation results show that DP-GAPSO has achieved good results for train operation strategy optimization.
However, the individual exchange strategy such as immigrant strategy between two parallel populations is not perfect, and this strategy is the focus of the double-population algorithm. Through the analysis and summary of a large number of experimental results, a more reasonable immigrant strategy may be obtained. Therefore, the further improvement of this strategy in the future can achieve better optimization results.