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Study on Energy-Saving Train Trajectory Optimization Based on Coasting Control in Metro Lines

Study on Energy-Saving Train Trajectory Optimization Based on Coasting Control in Metro Lines Hindawi Journal of Advanced Transportation Volume 2023, Article ID 1217352, 12 pages https://doi.org/10.1155/2023/1217352 Research Article Study on Energy-Saving Train Trajectory Optimization Based on Coasting Control in Metro Lines 1,2 1 3 3 Bo Jin , Song Yang, Qingyuan Wang, and Xiaoyun Feng Zhejiang Scientifc Research Institute of Transport, Hangzhou 310023, China Institute of Intelligent Transportation Systems, College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China School of Electrical Engineering, Southwest Jiaotong University, Chengdu 611756, China Correspondence should be addressed to Bo Jin; kimbojin@163.com Received 4 November 2022; Revised 3 January 2023; Accepted 17 January 2023; Published 22 March 2023 Academic Editor: Luca Pugi Copyright © 2023 Bo Jin et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. With increasing energy consumption in urban rail transit systems, researchers have paid signifcant attention to energy-saving train control. In this paper, we propose an efective train trajectory optimization method to reduce the energy consumption based on coasting control, in which coasting control regimes are added to balance running time and energy consumption. For better determining the starting points of coasting control regimes, the whole train running process is divided into several subintervals. Ten, aiming to achieve energy efciency, coasting regimes are added to the subintervals with high energy-saving efects, in which more energy consumption can be reduced with the same running time addition. Based on this, a coasting control method is proposed to generate energy-saving trajectories considering train dynamics, safety, and punctuality. In addition, the proposed method can solve the multisection energy-saving train trajectory optimization problem to obtain optimal running time schemes and related trajectories. Finally, numerical examples based on one of the Beijing metro lines are implemented to verify the efectiveness of the proposed method. Te results show that, for the single-section train control problem, the proposed coasting control algorithm can achieve signifcant energy-saving efects compared to the practical trajectory and calculate energy-saving trajectory in shorter computation times compared to the dynamic programming method. Meanwhile, for the multisection train control problem, energy consumption can be further reduced by optimizing trajectories and running times integratedly. Energy-efcient train control mainly focuses on re- 1. Introduction ducing energy consumption by optimizing train trajectory Urban rail transit (URT) systems are developing rapidly in (or called speed profle and driving strategy). In recent years, recent years to meet the increasing passenger demands. many works have been devoted to design algorithms for Meanwhile, URT systems consume a huge amount of energy, generating energy-saving train trajectories while satisfying especially in big cities (e.g., Beijing, New York, and Tokyo). operational constraints. Te frst research on the optimal With rising energy prices and environmental issues, energy cost train control problem was carried out in 1968, in which is becoming a grand challenge. Terefore, energy-saving Pontryagain’s maximum principle (PMP) was used to solve strategies are being implemented to reduce energy consump- the problem for level tracks [3]. Based on the PMP, the tion. Tese strategies mainly include [1] applying the energy- optimal train control regimes (i.e., maximum acceleration efcient rolling stock; demand-driven train timetabling aiming (MA), cruising (CR), coasting (CO), and maximum braking to reduce the number of train services; energy-efcient train (MB)) for the energy-efcient operation were proposed. In timetabling; and energy-efcient train control. In this paper, we addition, many researchers have applied the PMP to solve focus on the energy-efcient train control. More details about the optimal train control problem considering varying speed other strategies can be found in works [1, 2]. restrictions and gradients [4–7]. Especially, the scheduling 2 Journal of Advanced Transportation a good solution and cause a long computation time. In study and control group at the University of South Australia presented a systematic review of the optimal train control [23], coasting control was applied to calculate energy-saving trajectories, considering the utilization of regenerative theory from the viewpoint of PMP [8, 9]. Methods in studies [3–9] belong to the indirect methods, in which the optimal braking energy. solutions are obtained with complicated computational Diferent from the existing coasting control methods processes and large computation times. [20–23], this paper proposes a novel searching method to With the application of automatic train operation (ATO) determine the points of coasting regimes. First, we formulate systems in railway systems, especially in URT systems, the a distance-based model to describe the train dynamics. recommended trajectory for the next running process Considering the operational constraints and the energy- saving objective, an optimal train control model is built. should be determined before train departure. Tus, direct methods are applied to calculate optimal train trajectories Ten, fay-out trajectory and subinterval are introduced. Te former clarifes the boundary of trajectory optimization, into with shorter computation times. In direct methods, the control actions (i.e., traction/braking force or acceleration) which coasting regimes can be added for energy con- sumption reduction with running time addition. Meanwhile, and/or state variables (i.e., position, speed, and time) are discretized to transform the optimal train control problems the trajectory is divided into several subintervals for better into mathematical programming problems [10]. By splitting determining the starting points of coasting regimes. Finally, the train running process to build a discrete-position model a coasting control algorithm is designed to calculate the and linearization with piecewise approximation, the original energy-saving train trajectory meeting the constraint of pre- optimal train control problem was rebuilt into a mixed given running time. Coasting regimes are added into the integer linear programming model, which can be solved subintervals with high energy consumption reduction ef- ciency, which means that more energy consumption can be efectively by existing solvers [11, 12]. Dynamic pro- gramming (DP) algorithm has been widely applied in the reduced with the same running time addition. Specially, the proposed coasting control algorithm can be applied to the optimal train control problem [13–17]. It was necessary to transform the optimal problem into multiple decision energy-saving train control problem for multisection, and then the energy-saving train trajectory and running time processes in the DP algorithm, which can be realized by state space discretization [13]. Specially, for the energy-efcient schemes can be generated simultaneously. train control problem, the cost of state transition was set as Te rest of this paper is organized as follows. In Section energy consumption and decision actions were set as 2, we describe the energy-saving train control problem. traction and braking forces. By backwards calculating the Ten, we propose the solution methodology for the optimal optimal policies of each state space point and forwards control problem based on coasting control in Section 3. In searching, a set of optimal control actions and the optimal Section 4, we give two numerical examples, i.e., a single- section case and a multisection case based on one of the trajectory with minimum energy consumption can be ob- tained. In study [14], the performance of the DP algorithm, Beijing metro lines, to demonstrate the efectiveness and efciency of the proposed approaches. We conclude this genetic algorithm, and ant colony optimization algorithm was contrasted and compared. It was found that the DP paper in Section 5. algorithm can obtain the best solution with more compu- tational resources compared to the other two algorithms. 2. Problem Formulation Te pseudospectral method was also employed in the op- timal train control problem, in which the continuous-time 2.1. Defnition of Symbols. For a better understanding of our optimal control problem was transcribed into a discrete paper, we defne the necessary notations and parameters for nonlinear programming problem [18, 19]. the energy-efcient train control problem in Table 1. On the other hand, some works focus on determining the optimal conversion points of control regimes to formulate the energy-efcient train trajectories. Train trajectory op- 2.2. Train Dynamics Modeling. By considering the train timization based on coasting control is a popular method- traction, braking force, running basic resistance, and line ology to enhance the energy efciency, in which energy resistance, the dynamics model of train motion can be consumption is reduced by adjusting coasting regimes. formulated as follows [24]. Generally, in coasting control, the starting points of coasting regimes are determined by genetic algorithms [20–23]. A dv F − B − W − W 0 l v � , genetic algorithm was proposed to search for the points of dx M(1 + c) coasting regimes where the number of coasting regimes was (1) dt 1 predetermined [20]. To deal with complex operation situ- � . ations, a hierarchical genetic algorithm was introduced to dx v integrate the determination of the number of coasting re- Te train traction force F can be expressed as follows: gimes and points of coasting regimes [21]. Te simulation results showed that coasting control was an economical F � μ F (v), (2) f m approach to balance running time and energy consumption, which can achieve a good performance in energy-saving. where F is the maximum traction force, which is de- However, genetic algorithms may fail to converge onto pendent on the train characteristics and speed; μ is the f Journal of Advanced Transportation 3 Table 1: Notations and parameters. Based on equation (7), v can be calculated as follows: k+1 􏽱��������� Index Description (8) v � v + 2a ∆x, k+1 k i Index of station j Index of subinterval where a is the train acceleration in subsection [x , x ], k k k+1 k Index of discretizing point which can be calculated as follows: Parameters Description μ F − μ B − W − W f,k m,k b,k m,k 0,k l,k M Train mass (t) a � , (9) M(1 + c) c Rotary mass coefcient η Efciency of train motor where F , B , W , and W are the train traction force, k k 0,k l,k V Speed restriction (m/s) braking force, basic resistance, and line resistance in sub- S Position of station i (m) section [x , x ], respectively. Finally, the train running T Running time from station i to station i (s) k k+1 i,i times of discretizing points are denoted as 􏼈t , . . . , t 􏼉, and W Train running basic resistance (kN) 1 K+1 W Line resistance (kN) satisfy: 2∆x F Maximum train traction force (kN) t � + t . (10) k+1 k B Maximum train braking force (kN) v + v k k+1 State variables Description x Train running position (m) 2.3. Energy-Saving Train Control Model for Single Section. t Train running time (s) In this section, based on the distance-based modelling, we v Train running speed (m/s) introduce the energy-saving train control model for a single- F Train traction force (kN) B Train braking force (kN) section[S , S ]. Te train running process from station i to i i+1 station i + 1 is analyzed, which is divided into K subsection. Decision variables Description First, the train traction energy consumption is introduced. μ Train traction force adjustment coefcient At each subsection, the train traction energy consumption μ Train braking force adjustment coefcient can be calculated as follows: ∆x traction force adjustment coefcient, μ ∈ [0, 1]. Similarly, E � F . (11) k k the train braking force B can be expressed as follows: Ten, the cost function of energy-saving train control B � μ B (v), (3) b m problem for the single-section can be described as follows: where B is the maximum braking force, which is dependent on the train characteristics and speed; μ is the braking force J � 􏽘 E . (12) b i,i+1 adjustment coefcient, μ ∈ [0, 1]. Based on the adjustment k�1 coefcients μ and μ , the corresponding train control re- f b For the train running process from station i to station gimes can be described as shown in Table 2. i + 1, the following constraints should be considered. Te Te train running basic resistance can be formulated train speed at station i and station i + 1 are equal to zero: based on the Davis equation. v � v � 0. (13) W (v) � M􏼐a + a v + a v 􏼑, (4) 0 1 2 3 K+1 1 To keep the safe operation, the train speed must be less where a , a , and a are nonnegative coefcients. Moreover, 1 2 3 than the speed restrictions: the line resistance, caused by track slope, can be calculated as follows: 0 ≤ v ≤ V . (14) k k W (x) � Mg sin(θ(x)), (5) To keep the punctual operation, the pre-given running where θ(x) is the track slope at position x. time should be satisfed: In addition, we adopt distance-based modelling to t − t � T . (15) generate the energy-efcient train control strategy. Te K+1 1 i,i+1 whole section [S , S ] is divided into K subsections, i i+1 Ten, the energy-saving train control problem for the K∆x � S − S , as shown in Figure 1. Te postions of i+1 i single-section [S , S ] can be stated as follows. i i+1 discretizing points are denoted as 􏼈x , . . . , x 􏼉, and satisfy: 1 K+1 x � x +∆x. (6) k+1 k ⎧ ⎪ min J � 􏽘 E , i,i+1 k Meanwhile, the train running speed of discretizing k�1 (16) points are denoted as 􏼈v , . . . , v 􏼉. Te relationship be- ⎪ 1 K+1 ⎪ s.t. μ , μ ∈ [0, 1], f,k b,k tween v and v can be described as follows: ⎩ k+1 k Equations (6) to (10), and (13) to (15). 2 2 v − v � 2a ∆x. (7) k+1 k k 4 Journal of Advanced Transportation Table 2: Description of control regimes based on μ and μ . Equations (6–10), (14), (18), and (19). f b Te energy-efcient train trajectory from station i to Regime Description station i + I can be obtained by solving the optimal train MA μ � 1 and μ � 0 f b control problem 21. Meanwhile, the energy-efcient dis- CR μ ∈ (0, 1) and μ � 0 or μ � 0 and μ ∈ (0, 1) f b f b ∗ ∗ tribution of running time T , . . . , T can be 􏽮 􏽯 i,i+1 i+I−1,i+I CO μ � 0 and μ � 0 f b generated, which can be calculated as follows: MB μ � 0 and μ � 1 f b ∗ ∗ ∗ ′ T � t − t , ∀i ∈ {i, . . . , i + I − 1}. (21) ′ ′ i ,i +1 k k ′ ′ i +1 i a (μ , μ ) k f,k b,k 3. Coasting Control Algorithm k+1 Typically, the pre-given running times are not those asso- ciated with the minimum running times (fat-out trajecto- ries), but they include running time supplements to be able to recover delays when necessary or fulfl running at a lower speed with less energy consumption [25]. Coasting control is x x x x x x x x x an economical approach to balance running time and energy 1 2 3 k k+1 K–1 K K+1 S S consumption in train operation [21]. In this section, we i i+1 introduce a coasting control algorithm to calculate the Figure 1: Te illustration of the section discretizing. energy-saving trajectory. First, the calculation of fat-out trajectory is introduced to generate the minimum running By solving the abovementioned optimal control problem time. Ten, the subinterval is proposed to divide the whole ∗ ∗ 17, optimal control actions μ , . . . , μ and 􏽮 􏽯 running process into sections, in which coasting regimes can f,1 f,K ∗ ∗ 􏽮μ , . . . , μ 􏽯 can be obtained to generate the energy- be added. Finally, coasting points are determined based on b,1 b,K efcient train trajectory. the principle that adding coasting regimes to subintervals with high energy-saving efects. 2.4. Energy-Saving Train Control Model for Multisection. Based on the energy-saving train control model for a single- 3.1. Flat-Out Trajectory. Under fat-out running, a train is section, the model for multisection is built in this section. travelling close to speed restrictions. As shown in Figure 3, Considering the train running process from station i to when the train speed is less than the speed restriction, MA station i + I, I ≥ 2, the whole section [S , S ] is also divided i i+I regime is applied to speed up; when the speed is close to the into K subsections, K∆x � S − S , as shown in Figure 2. A i+I i speed restriction, CR regime is applied to keep the train set 􏼈κ , . . . , κ 􏼉 is introduced to represent the indexes of the i i+I running at high-speed; MB regime is applied for low-speed discretizing points that overlap with the station positions restrictions and stopping. Tus, there is no coasting regime 􏼈S , . . . , S 􏼉, which can be described as follows: i i+I in the trajectory. Tis kind of trajectory is defned as the fat- out trajectory [23], which can be calculated as shown in x � S , ∀i ∈ {i, . . . , i + I}. (17) Algorithm 1 and Figure 4. In the fat-out trajectory calcu- κ i lation algorithm, train control regimes are determined based Considering the middle stations in the multisection on the relationship between the train running speed and running process, the constraint 14 that limits the train speed speed restriction. Under the constraints of safety and train at stations should be rewritten as follows: characteristics, the algorithm keeps the train running speed as close to or within speed restrictions. v � 0, ∀k ∈ 􏼈κ , . . . , κ 􏼉. (18) k i i+I Based on the fat-out trajectory calculation algorithm, the minimum running time can be calculated as follows: Meanwhile, the punctuality constraint 16 should be flat flat rewritten as follows: (22) T � t − t , i,i K+1 flat t − t � T , (19) where T is the minimum running time from station i to K+1 1 i,i+I ′ i,i ′ flat station i ; t is the train running time at the ending point K+1 where only the total running time from station i to station ′ (station i ) of the fat-out trajectory. i + I is limited, the running times of each section are un- limited, which can be adjusted in the optimization process. 3.2. Defnition of Subinterval. Based on the fat-out trajec- Ten, the energy-saving train optimal control problem tory, coasting regimes can be added into the control se- for the multisection [S , S ] can be stated as follows. i i+I quence, as shown in Figure 3. For better determining the starting points of coasting regimes, the whole running ⎧ ⎨ min J � 􏽘 E , s.t. μ , μ ∈ [0, 1]. (20) process is divided into several subintervals. Te subinterval i,i+I k f,k b,k k�1 means the train running process starts with an accelerating regime and ends with a braking regime. Meanwhile, there Journal of Advanced Transportation 5 v v Subinterval Subinterval x x x x x x x 1 2 k k' K K+1 S S S S i i+1 i+I–1 i+I Figure 2: Te illustration of multisection discretizing. MA CR MB CR MB can only be one accelerating phase and one braking phase in Flat-out trajectory a subinterval, which makes the accelerating phase and its CO regime subsequent adjacent braking phase form a subinterval. As Speed restriction shown in Figure 3, the frst subinterval begins with a MA Figure 3: Te illustration of fat-out trajectory and coasting re- (accelerating) regime and ends with an MB (braking) re- gimes addition. gime, and the MA and MB regimes are adjacent. Four important points (x , x , x , and x ) are introduced to a c d b j j j j describe the subinterval j, as shown in Figure 5. Start (1) x : the beginning position of the accelerating regime (2) x : the beginning position of the coasting regime Section discretizing (3) x : the ending position of the coasting regime Set k=1, v =0 and v =0 1 K+1 (4) x : the ending position of the braking regime. v < V No k k+1 Without coasting regime addition, x is equal to the ending position of the accelerating regime, and x is equal v == V d No k k+1 j Yes to the starting position of the braking regime. As the du- Adopt MA regime, set μ =1 ration of the coasting regime increases, the beginning po- f,k Yes and μ =0 to calculate v b,k k+1 sition of the coasting regime x moves to x , and the ending c a j j position of the coasting regime x moves to x . Specially, d b Adopt CR regime, set μ =0 j j f,k when x � x or x � x , the duration of the coasting and μ =0 to calculate v d b c a b,k k+1 j j j j regime cannot increase, as shown in Figure 6. For two Set k= k + 1 adjacent subintervals, if x � x in the frst subinterval and d b j j Adopt MB regime, set v =V k+1 k+1 x � x in the second subinterval, then these two c,j+1 a,j+1 and reverse search the starting adjacent subintervals can be merged as a new subinterval, as point of MB regime shown in Figure 6. Specially, the beginning position of the accelerating regime and the beginning position of the coasting regime of the new subinterval are equal to those of Yes k < K subinterval j, the ending position of the coasting regime and No the ending position of the braking regime of the new subinterval are equal to those of subinterval j + 1. End Figure 4: Te illustration of the fat-out trajectory calculation algorithm. 3.3. Coasting Points Determination. Due to the line char- acteristics (like speed restrictions and track slope), the fat- out trajectory consists of several subintervals, as shown in the running time change value after adding the coasting Figure 3. We propose a coasting control algorithm to dis- regime of subinterval j. In addition, the larger ρ means that tribute the running time supplements to subintervals. Te the more energy consumption can be reduced in subinterval distribution criterion distributes the running time supple- j with the same running time supplement. Considering the ments to subintervals where energy consumption can be whole running process from station i to station i , the limited reduced more signifcantly. To evaluate the efciency of running time supplement should be distributed to the energy consumption reduction, indicator ρ is introduced: subinterval with the maximum ρ for energy saving. Based on this, the coasting control algorithm is proposed: 􏽐 E − E 􏼁 ∆E k k k�a j Specially, for the multisection running process, the (23) ρ � − � − , ′ proposed coasting control algorithm is efective. Due to the ∆T t − t j b b j j multisection running process, the speed limit constraints of where ρ is the energy-saving efect of subinterval j; ∆E � middle stations need to be considered additionally. First, the j j ′ multisection running process will be divided into several 􏽐 (E − E ) is the energy consumption change value after k k k�a adding the coasting regime of subinterval j;∆T � t − t is subintervals with the same number of sections, as shown in j b b j j 6 Journal of Advanced Transportation (1) Divide the whole running section into K subsections. Set k � 1, v � 0, and V � 0. 1 K+1 (2) while k ≤ K do (3) if v < V (MA regime) then k k+1 (4) Set μ � 1 and μ � 0 to calculate v . f,k b,k k+1 (5) else if v � V (CR regime) then k k+1 (6) Set v � Vk + 1 to calculate μ and μ . k+1 f,k b,k (7) else if v > V (MB regime) then k k+1 ′ ′ (8) Set v � V and k � k + 1. Ten, reverse search the starting point of MB regime (k ): k+1 k+1 (9) repeat ′ ′ ′ ′ ′ (10) Set μ ′ � 0 and μ ′ � 1 to calculate v ′ . Since , set k � k − 1. f,k b,k k −1 (11) until v � v k k ′ ′ ′ ′ ′ (12) Set μ � μ , μ � μ , and v � v , for κ ∈ 􏽮k , k + 1, . . . , k􏽯 f,κ f,κ b,κ b,κ κ κ (13) end if (14) k � k + 1 (15) end while ALGORITHM 1:Flat-out trajectory calculation. Subinterval j Subinterval j+1 Subinterval j x x x x a c d b j j j j x = x x = x d b c a j j j+1 j+1 Figure 5: Te illustration of the subinterval. New Subinterval Figure 2. Ten, each section will be divided into several subintervals based on the defnition of the subinterval. Tus, the running time supplement can be distributed for the multisection running process as described in the Algo- rithm 2, to generate energy-saving running time schemes and related trajectories. 4. Numerical Examples Figure 6: Te illustration of the subinterval merging. In this section, we present two numerical examples to demonstrate the performance of the proposed energy- ⎧ ⎪ 200 0 ≤ v ≤ 15.28, efcient coasting control algorithm. Te frst example sol- ves the energy-saving train control problem along the single- F (v) � section, in which only the energy-saving train trajectories are 15.28 < v < ≤ 22.22, optimized. Te second one involves the energy-saving train 2 (24) control problem along the multisection, in which both the energy-saving train trajectories and running time scheme B (v) � 159.6 0 ≤ v ≤ 22.22, are optimized. All examples are based on the data of one of the W (v) � 216 􏼐4.5024 + 0.1089v + 0.0108v 􏼑. Beijing metro lines. Te speed restrictions and track 0 gradient between station 1 and station 5 are shown in Figure 7. Te parameters of the running train are listed in Examples are tested under the MATLAB environment Table 3. Te maximum train traction force, the maximum on a personal computer with Intel Core i5 2.30 GHz CPU braking force, and running basic resistance are given as and 8 GB RAM. follows: Journal of Advanced Transportation 7 (1) Divide the whole running section into K subsections. Calculate the fat-out trajectory based on the Algorithm 1. Initialize the sup flat running time supplement T � T − T . Divide the whole running section into J subintervals based on the defnition of the ′ i,i ′ i,i i,i subinterval. sup (2) while T ≥ 0 do i,i (3) for j � 1 to J do (4) Move the beginning position of coasting regime x with step ∆x backward temporarily, c � c − 1. Calculate the ending c j j ′ ′ ′ position of coasting regime x with μ � μ � 0, for k ∈ c , . . . , d . Calculate ρ based on the temporarily modifed ′ ′ 􏽮 􏽯 d f,k b,k j j j trajectory. (5) end for sup sup ′ ′ ′ ′ (6) Determine the subinterval j with maximum ρ , ρ ≥ ρ , for j ∈ {1, . . . , J}. Update c ′ � c ′, d ′ � d ′, and T � T −∆T ′. j j j j j j j ′ ′ j i,i i,i (7) for j � 1 to J − 1 do (8) if x � x and x � x then d b c,j+1 a,j+1 j j (9) Merge the subinterval j and subinterval j + 1. Update J � J − 1. (10) end if (11) end for (12) end while ALGORITHM 2: Coating control algorithm. (4) T-DP: Optimal trajectory calculated based on dy- namic programming, more details can be seen in Appendix. Specially, the DP algorithm is introduced to compare with the coasting control algorithm, aiming to demonstrate its efect. ∆x is set to be 1 m in the coasting control algo- rithm, and then the running processes of four sections are 0 0 divided into 990, 1225, 1257, and 776 subsections, re- 0 990 2215 3472 4248 S S spectively. Meanwhile, ∆x is also set to be 1 m in the DP S S 1 3 4 5 algorithm, and ∆v is set to be 0.02 m/s. For four diferent Position (m) types of trajectories, the trajectories and control commands Figure 7: Te illustration of speed restriction and gradient are shown in Figure 8, and the performance is shown in changing. Table 4. As shown in Figure 8, T-fat is keeping close to the speed restrictions in the running process, in which there is no Table 3: Train parameters. coasting regime. Te T-fat corresponds to the maximum Parameters Value energy consumption and the minimum running time in each Train mass, M 216 (t) section, as shown in Table 4. For T-Pra, accelerating regimes are Rotary mass coefcient, c 0.08 applied to reach a high speed at the beginning of the running Efciency of train motor, η 0.85 process, then braking regimes are applied for the low-speed restriction and train stops. Compared to T-CC and T-DP, we can observe that fewer coasting regimes are applied in T-Pra, as 4.1. Example 1: Scenarios of Single-Section Running. In this shown in Figure 8. For T-CC and T-DP, MA regimes are example, the train trajectories of sections [S , S ], [S , S ], 1 2 2 3 applied at the beginning of the running process and MB re- [S , S ], and [S , S ] were analyzed. Four diferent types of 3 4 4 5 gimes are applied at the stopping process. Tis kind of strategy trajectories are compared to verify the performance of the avoids the train from staying in low-speed phases and wasting proposed coasting control algorithm for single-section train running time supplements. Considering the running time control. More details about these four trajectories are as constraint, there will be more time for train coasting to reduce follows: energy consumption. Terefore, as shown in Table 4, the (1) T-Pra: Practical trajectory obtained from the comparison results of single-section running with the same equipped ATO systems running times show that T-CC can achieve 26.16%, 37.51%, (2) T-fat: Flat-out trajectory with a minimum running 12.12% and 35.31%, energy-saving for sections [S , S ], [S , S ], 1 2 2 3 time and maximum energy consumption, which can [S , S ], and [S , S ], respectively, in comparison to T-Pra. 3 4 4 5 be calculated based on the Algorithm 1; Meanwhile, T-DP can achieve 26.46%, 37.35%, 12.40%, and 36.19% energy-saving for four sections, respectively. Te little (3) T-CC: Optimal trajectory calculated based on the deviations in energy-saving performance between T-CC and proposed coasting control algorithm (Algorithm 2); Speed Restriction (m/s) Height (m) 8 Journal of Advanced Transportation Table 4: Te comparison of diferent trajectories and running time distribution schemes from S to S . 1 5 [S , S ] [S , S ] [S , S ] [S , S ] [S , S ] (total) 1 2 2 3 3 4 4 5 1 5 Section T J T J T J T J T J 1,2 1,2 2,3 2,3 3,3 3,4 4,5 4,5 1,5 1,5 index (s) (kWh) (s) (kWh) (s) (kWh) (s) (kWh) (s) (kWh) T-fat 80.97 20.38 103.51 17.63 91.59 22.24 71.72 15.94 347.79 76.19 T-Pra 88.40 13.15 109.80 12.45 100.90 14.44 79.80 9.09 378.90 49.13 T-DP 88.31 9.67 109.72 7.80 100.90 12.65 79.78 5.80 378.71 35.92 T-CC 88.31 9.71 109.77 7.78 100.82 12.69 79.78 5.88 378.68 36.06 T-CC with optimal running times 89.46 9.12 110.17 7.58 101.03 12.59 78.22 6.53 378.88 35.83 20 20 15 15 10 10 1 1 5 5 0 u 0 u 0 -1 0 -1 0 200 400 600 800 990 0 200 400 600 800 1000 1225 Position (m) Position (m) T-Pra T-CC T-Pra T-CC T-Flat Speed restriction T-Flat Speed restriction T-DP T-DP (a) (b) 20 20 15 15 10 10 1 1 5 5 0 u 0 0 0 -1 -1 0 200 400 600 800 1000 1257 0 200 400 600 776 Position (m) Position (m) T-Pra T-Pra T-CC T-CC T-Flat T-Flat Speed restriction Speed restriction T-DP T-DP (c) (d) Figure 8: Te illustration of trajectories and control commands μ for four sections. (a) Section [S , S ]. (b) Section [S , S ]. (c) Section 1 2 2 3 [S , S ]. (d) Section [S , S ]. 3 4 4 5 T-DP might come from the speed discretization in the DP In addition, to verify the feasibility of the coasting algorithm and the small diferences in running times. control algorithm, the optimal trajectories with diferent In terms of computation time, the average computation running times in sections [S , S ] are analyzed. T-CC with 1 2 times of T-CC and T-DP for four sections are 1.1 s and diferent running times is shown in Figure 9, and the cor- 261.9 s, respectively. Tis means that, for single-section train responding running time supplements of each subinterval control, applying the coasting control algorithm can achieve and energy consumption are shown in Table 5. First, the the similar energy-saving efect as the DP algorithm with less whole running process from S to S is divided into two 1 2 computation time. Specially, the computation time of the subintervals due to the low-speed restriction, as shown in coasting control algorith m can reach a 10 ms level when Figure 9. As the running time increases, running time running in a C environment. supplements are added into the subintervals with coasting Speed (m/s) Speed (m/s) Speed (m/s) Speed (m/s) Journal of Advanced Transportation 9 Table 5: Te comparison of T-CC with diferent running times from S to S . 1 2 Pre-given Running time supplement Running time supplement Energy consumption (kWh) running time (s) in subinterval 1 (s) in subinterval 2 (s) 80.97 0 0 20.38 81.97 (+1) 0.96 0.03 16.16 82.97 (+2) 1.91 0.07 14.34 85.97 (+5) 4.86 0.08 11.23 90.97 (+10) 9.90 0.08 8.47 100.97 (+20) 20.318 (merged subinterval) 5.72 110.97 (+30) 29.672 (merged subinterval) 4.58 Subinterval 1 2 0 200 400 600 800 990 Position (m) Speed restriction 85.97 80.97 90.97 81.97 100.97 82.97 110.97 Figure 9: Te illustration of T-CC with diferent running times from S to S . 1 2 15 15 10 10 5 5 0 0 0 0.05 0.1 0 10 20 Runtime supplement (s) Runtime supplement (s) (a) (b) Figure 10: Te illustration of energy-saving efects ρ for the subinterval 1 (a) and 2 (b) duration addition. In addition, the energy-saving efects ρ of 4.2. Example 2: Scenarios of Multisection Running. In this the subintervals 1 and 2, as shown in Figure 10, guide the example, we optimize the train trajectories and related to distribution of running time supplements. Running time running time schemes for the running process from S to S , 1 2 supplement is added to the subinterval with the larger ρ for based on the proposed coasting control algorithm. T-CC more energy consumption reduction. Specially, the maxi- with optimal running times is compared with the other mum running time supplement of the subinterval 2 is 0.08 s. trajectories with pre-given running times, to verify the When the running time supplement of subinterval reaches performance of the coasting control algorithm for the the maximum one, there is no room for coasting regime multisection train control. T-CC with optimal running times addition, like the cases 85.97 s and 90.97 s. Since, when the represents the optimal trajectories calculated based on the running time supplement is large enough, two subintervals optimal model 21, in which trajectories and running times are merged into one subinterval, as in the cases 100.97 s and for the multisection running process are optimized inte- 110.97 s in Figure 9. gratedly. T-CC with optimal running times and T-CC with ρ (kWh/s) Speed (m/s) ρ (kWh/s) 2 10 Journal of Advanced Transportation 0 990 2215 3472 4248 Position (m) T-CC with pre-given runtimes T-CC with optimal runtimes Speed restriction Figure 11: Te illustration of T-CC with optimal running times and T-CC with pre-given running times from S to S . 1 5 v v k,m k+1,m k k+1 v v k,m –1 k+1,m –1 k k+1 x x x x x x x x x x 1 2 3 4 k k+1 K–2 K–1 K K+1 Trajectory Speed restriction Grid point Figure 12: Te illustration of the speed-distance network. pre-given running times are compared in Figure 11, and the constraints and energy-saving objective, we developed train running times and energy consumption of each section distance-based train trajectory optimization models for the for T-CC with optimal running times are shown in Table 4. single-section and multi-section operations. A coasting As shown in Table 4, by comparing T-CC with pre-given control algorithm was proposed to generate the energy- running times and T-CC with optimal running times, we can efcient trajectories, in which the coasting control regime observe that the running times of each section are diferent points were determined according to the energy-saving in these two plans. Due to the change in running times, there efect. are also diferences in the trajectories. As shown in Figure 11, Numerical examples based on one of the Beijing metro with running time addition, more coasting regimes are lines were implemented in two diferent cases, i.e., single- added into the trajectories in sections [S , S ], [S , S ], and section and multisection operation, to demonstrate the 1 2 2 3 [S , S ] of T-CC with optimal running times. Meanwhile, the performance of the proposed coasting control algorithm. 3 4 energy consumption in these three sections can be reduced Te computational results showed that, by applying the in comparison to those of T-CC with pre-given running coasting control algorithm, the energy consumption of times. On the other hand, with running time reduction in the single-section operation can be reduced efectively by sections [S , S ], the energy consumption of T-CC with around 12.12% to 37.35% in comparison to the practical 4 5 optimal running times in this section is larger than it of trajectories obtained from equipped ATO systems. Mean- T-CC with the pre-given running times. In terms of the while, the coasting control algorithm was compared with the DP algorithm; the former can achieve a similar energy- whole running process, for T-CC with optimal running times, the total energy consumption can be reduced from saving performance in shorter computation times. For the 36.06 kWh to 35.83 kWh compared to T-CC with pre-given multisection operation, the proposed coasting control al- running times. gorithm can generate energy-saving running time schemes and related trajectories by optimizing the whole running process integratedly. 5. Conclusion Our future research will focus on the online train control In this paper, we studied the optimal train control problem problem to deal with the dynamic situations, like temporary to reduce energy consumption. Combining the operational speed restrictions. Tis paper only deals with the ofine train Speed (m/s) Journal of Advanced Transportation 11 trajectory optimization problem. However, the train tra- References jectories will be adjusted in real-time operation to keep safe [1] G. M. Scheepmaker, R. M. Goverde, and L. G. Kroon, “Review and punctual operations. of energy-efcient train control and timetabling,” European Journal of Operational Research, vol. 257, no. 2, pp. 355–376, Appendix [2] A. 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Study on Energy-Saving Train Trajectory Optimization Based on Coasting Control in Metro Lines

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Hindawi Journal of Advanced Transportation Volume 2023, Article ID 1217352, 12 pages https://doi.org/10.1155/2023/1217352 Research Article Study on Energy-Saving Train Trajectory Optimization Based on Coasting Control in Metro Lines 1,2 1 3 3 Bo Jin , Song Yang, Qingyuan Wang, and Xiaoyun Feng Zhejiang Scientifc Research Institute of Transport, Hangzhou 310023, China Institute of Intelligent Transportation Systems, College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China School of Electrical Engineering, Southwest Jiaotong University, Chengdu 611756, China Correspondence should be addressed to Bo Jin; kimbojin@163.com Received 4 November 2022; Revised 3 January 2023; Accepted 17 January 2023; Published 22 March 2023 Academic Editor: Luca Pugi Copyright © 2023 Bo Jin et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. With increasing energy consumption in urban rail transit systems, researchers have paid signifcant attention to energy-saving train control. In this paper, we propose an efective train trajectory optimization method to reduce the energy consumption based on coasting control, in which coasting control regimes are added to balance running time and energy consumption. For better determining the starting points of coasting control regimes, the whole train running process is divided into several subintervals. Ten, aiming to achieve energy efciency, coasting regimes are added to the subintervals with high energy-saving efects, in which more energy consumption can be reduced with the same running time addition. Based on this, a coasting control method is proposed to generate energy-saving trajectories considering train dynamics, safety, and punctuality. In addition, the proposed method can solve the multisection energy-saving train trajectory optimization problem to obtain optimal running time schemes and related trajectories. Finally, numerical examples based on one of the Beijing metro lines are implemented to verify the efectiveness of the proposed method. Te results show that, for the single-section train control problem, the proposed coasting control algorithm can achieve signifcant energy-saving efects compared to the practical trajectory and calculate energy-saving trajectory in shorter computation times compared to the dynamic programming method. Meanwhile, for the multisection train control problem, energy consumption can be further reduced by optimizing trajectories and running times integratedly. Energy-efcient train control mainly focuses on re- 1. Introduction ducing energy consumption by optimizing train trajectory Urban rail transit (URT) systems are developing rapidly in (or called speed profle and driving strategy). In recent years, recent years to meet the increasing passenger demands. many works have been devoted to design algorithms for Meanwhile, URT systems consume a huge amount of energy, generating energy-saving train trajectories while satisfying especially in big cities (e.g., Beijing, New York, and Tokyo). operational constraints. Te frst research on the optimal With rising energy prices and environmental issues, energy cost train control problem was carried out in 1968, in which is becoming a grand challenge. Terefore, energy-saving Pontryagain’s maximum principle (PMP) was used to solve strategies are being implemented to reduce energy consump- the problem for level tracks [3]. Based on the PMP, the tion. Tese strategies mainly include [1] applying the energy- optimal train control regimes (i.e., maximum acceleration efcient rolling stock; demand-driven train timetabling aiming (MA), cruising (CR), coasting (CO), and maximum braking to reduce the number of train services; energy-efcient train (MB)) for the energy-efcient operation were proposed. In timetabling; and energy-efcient train control. In this paper, we addition, many researchers have applied the PMP to solve focus on the energy-efcient train control. More details about the optimal train control problem considering varying speed other strategies can be found in works [1, 2]. restrictions and gradients [4–7]. Especially, the scheduling 2 Journal of Advanced Transportation a good solution and cause a long computation time. In study and control group at the University of South Australia presented a systematic review of the optimal train control [23], coasting control was applied to calculate energy-saving trajectories, considering the utilization of regenerative theory from the viewpoint of PMP [8, 9]. Methods in studies [3–9] belong to the indirect methods, in which the optimal braking energy. solutions are obtained with complicated computational Diferent from the existing coasting control methods processes and large computation times. [20–23], this paper proposes a novel searching method to With the application of automatic train operation (ATO) determine the points of coasting regimes. First, we formulate systems in railway systems, especially in URT systems, the a distance-based model to describe the train dynamics. recommended trajectory for the next running process Considering the operational constraints and the energy- saving objective, an optimal train control model is built. should be determined before train departure. Tus, direct methods are applied to calculate optimal train trajectories Ten, fay-out trajectory and subinterval are introduced. Te former clarifes the boundary of trajectory optimization, into with shorter computation times. In direct methods, the control actions (i.e., traction/braking force or acceleration) which coasting regimes can be added for energy con- sumption reduction with running time addition. Meanwhile, and/or state variables (i.e., position, speed, and time) are discretized to transform the optimal train control problems the trajectory is divided into several subintervals for better into mathematical programming problems [10]. By splitting determining the starting points of coasting regimes. Finally, the train running process to build a discrete-position model a coasting control algorithm is designed to calculate the and linearization with piecewise approximation, the original energy-saving train trajectory meeting the constraint of pre- optimal train control problem was rebuilt into a mixed given running time. Coasting regimes are added into the integer linear programming model, which can be solved subintervals with high energy consumption reduction ef- ciency, which means that more energy consumption can be efectively by existing solvers [11, 12]. Dynamic pro- gramming (DP) algorithm has been widely applied in the reduced with the same running time addition. Specially, the proposed coasting control algorithm can be applied to the optimal train control problem [13–17]. It was necessary to transform the optimal problem into multiple decision energy-saving train control problem for multisection, and then the energy-saving train trajectory and running time processes in the DP algorithm, which can be realized by state space discretization [13]. Specially, for the energy-efcient schemes can be generated simultaneously. train control problem, the cost of state transition was set as Te rest of this paper is organized as follows. In Section energy consumption and decision actions were set as 2, we describe the energy-saving train control problem. traction and braking forces. By backwards calculating the Ten, we propose the solution methodology for the optimal optimal policies of each state space point and forwards control problem based on coasting control in Section 3. In searching, a set of optimal control actions and the optimal Section 4, we give two numerical examples, i.e., a single- section case and a multisection case based on one of the trajectory with minimum energy consumption can be ob- tained. In study [14], the performance of the DP algorithm, Beijing metro lines, to demonstrate the efectiveness and efciency of the proposed approaches. We conclude this genetic algorithm, and ant colony optimization algorithm was contrasted and compared. It was found that the DP paper in Section 5. algorithm can obtain the best solution with more compu- tational resources compared to the other two algorithms. 2. Problem Formulation Te pseudospectral method was also employed in the op- timal train control problem, in which the continuous-time 2.1. Defnition of Symbols. For a better understanding of our optimal control problem was transcribed into a discrete paper, we defne the necessary notations and parameters for nonlinear programming problem [18, 19]. the energy-efcient train control problem in Table 1. On the other hand, some works focus on determining the optimal conversion points of control regimes to formulate the energy-efcient train trajectories. Train trajectory op- 2.2. Train Dynamics Modeling. By considering the train timization based on coasting control is a popular method- traction, braking force, running basic resistance, and line ology to enhance the energy efciency, in which energy resistance, the dynamics model of train motion can be consumption is reduced by adjusting coasting regimes. formulated as follows [24]. Generally, in coasting control, the starting points of coasting regimes are determined by genetic algorithms [20–23]. A dv F − B − W − W 0 l v � , genetic algorithm was proposed to search for the points of dx M(1 + c) coasting regimes where the number of coasting regimes was (1) dt 1 predetermined [20]. To deal with complex operation situ- � . ations, a hierarchical genetic algorithm was introduced to dx v integrate the determination of the number of coasting re- Te train traction force F can be expressed as follows: gimes and points of coasting regimes [21]. Te simulation results showed that coasting control was an economical F � μ F (v), (2) f m approach to balance running time and energy consumption, which can achieve a good performance in energy-saving. where F is the maximum traction force, which is de- However, genetic algorithms may fail to converge onto pendent on the train characteristics and speed; μ is the f Journal of Advanced Transportation 3 Table 1: Notations and parameters. Based on equation (7), v can be calculated as follows: k+1 􏽱��������� Index Description (8) v � v + 2a ∆x, k+1 k i Index of station j Index of subinterval where a is the train acceleration in subsection [x , x ], k k k+1 k Index of discretizing point which can be calculated as follows: Parameters Description μ F − μ B − W − W f,k m,k b,k m,k 0,k l,k M Train mass (t) a � , (9) M(1 + c) c Rotary mass coefcient η Efciency of train motor where F , B , W , and W are the train traction force, k k 0,k l,k V Speed restriction (m/s) braking force, basic resistance, and line resistance in sub- S Position of station i (m) section [x , x ], respectively. Finally, the train running T Running time from station i to station i (s) k k+1 i,i times of discretizing points are denoted as 􏼈t , . . . , t 􏼉, and W Train running basic resistance (kN) 1 K+1 W Line resistance (kN) satisfy: 2∆x F Maximum train traction force (kN) t � + t . (10) k+1 k B Maximum train braking force (kN) v + v k k+1 State variables Description x Train running position (m) 2.3. Energy-Saving Train Control Model for Single Section. t Train running time (s) In this section, based on the distance-based modelling, we v Train running speed (m/s) introduce the energy-saving train control model for a single- F Train traction force (kN) B Train braking force (kN) section[S , S ]. Te train running process from station i to i i+1 station i + 1 is analyzed, which is divided into K subsection. Decision variables Description First, the train traction energy consumption is introduced. μ Train traction force adjustment coefcient At each subsection, the train traction energy consumption μ Train braking force adjustment coefcient can be calculated as follows: ∆x traction force adjustment coefcient, μ ∈ [0, 1]. Similarly, E � F . (11) k k the train braking force B can be expressed as follows: Ten, the cost function of energy-saving train control B � μ B (v), (3) b m problem for the single-section can be described as follows: where B is the maximum braking force, which is dependent on the train characteristics and speed; μ is the braking force J � 􏽘 E . (12) b i,i+1 adjustment coefcient, μ ∈ [0, 1]. Based on the adjustment k�1 coefcients μ and μ , the corresponding train control re- f b For the train running process from station i to station gimes can be described as shown in Table 2. i + 1, the following constraints should be considered. Te Te train running basic resistance can be formulated train speed at station i and station i + 1 are equal to zero: based on the Davis equation. v � v � 0. (13) W (v) � M􏼐a + a v + a v 􏼑, (4) 0 1 2 3 K+1 1 To keep the safe operation, the train speed must be less where a , a , and a are nonnegative coefcients. Moreover, 1 2 3 than the speed restrictions: the line resistance, caused by track slope, can be calculated as follows: 0 ≤ v ≤ V . (14) k k W (x) � Mg sin(θ(x)), (5) To keep the punctual operation, the pre-given running where θ(x) is the track slope at position x. time should be satisfed: In addition, we adopt distance-based modelling to t − t � T . (15) generate the energy-efcient train control strategy. Te K+1 1 i,i+1 whole section [S , S ] is divided into K subsections, i i+1 Ten, the energy-saving train control problem for the K∆x � S − S , as shown in Figure 1. Te postions of i+1 i single-section [S , S ] can be stated as follows. i i+1 discretizing points are denoted as 􏼈x , . . . , x 􏼉, and satisfy: 1 K+1 x � x +∆x. (6) k+1 k ⎧ ⎪ min J � 􏽘 E , i,i+1 k Meanwhile, the train running speed of discretizing k�1 (16) points are denoted as 􏼈v , . . . , v 􏼉. Te relationship be- ⎪ 1 K+1 ⎪ s.t. μ , μ ∈ [0, 1], f,k b,k tween v and v can be described as follows: ⎩ k+1 k Equations (6) to (10), and (13) to (15). 2 2 v − v � 2a ∆x. (7) k+1 k k 4 Journal of Advanced Transportation Table 2: Description of control regimes based on μ and μ . Equations (6–10), (14), (18), and (19). f b Te energy-efcient train trajectory from station i to Regime Description station i + I can be obtained by solving the optimal train MA μ � 1 and μ � 0 f b control problem 21. Meanwhile, the energy-efcient dis- CR μ ∈ (0, 1) and μ � 0 or μ � 0 and μ ∈ (0, 1) f b f b ∗ ∗ tribution of running time T , . . . , T can be 􏽮 􏽯 i,i+1 i+I−1,i+I CO μ � 0 and μ � 0 f b generated, which can be calculated as follows: MB μ � 0 and μ � 1 f b ∗ ∗ ∗ ′ T � t − t , ∀i ∈ {i, . . . , i + I − 1}. (21) ′ ′ i ,i +1 k k ′ ′ i +1 i a (μ , μ ) k f,k b,k 3. Coasting Control Algorithm k+1 Typically, the pre-given running times are not those asso- ciated with the minimum running times (fat-out trajecto- ries), but they include running time supplements to be able to recover delays when necessary or fulfl running at a lower speed with less energy consumption [25]. Coasting control is x x x x x x x x x an economical approach to balance running time and energy 1 2 3 k k+1 K–1 K K+1 S S consumption in train operation [21]. In this section, we i i+1 introduce a coasting control algorithm to calculate the Figure 1: Te illustration of the section discretizing. energy-saving trajectory. First, the calculation of fat-out trajectory is introduced to generate the minimum running By solving the abovementioned optimal control problem time. Ten, the subinterval is proposed to divide the whole ∗ ∗ 17, optimal control actions μ , . . . , μ and 􏽮 􏽯 running process into sections, in which coasting regimes can f,1 f,K ∗ ∗ 􏽮μ , . . . , μ 􏽯 can be obtained to generate the energy- be added. Finally, coasting points are determined based on b,1 b,K efcient train trajectory. the principle that adding coasting regimes to subintervals with high energy-saving efects. 2.4. Energy-Saving Train Control Model for Multisection. Based on the energy-saving train control model for a single- 3.1. Flat-Out Trajectory. Under fat-out running, a train is section, the model for multisection is built in this section. travelling close to speed restrictions. As shown in Figure 3, Considering the train running process from station i to when the train speed is less than the speed restriction, MA station i + I, I ≥ 2, the whole section [S , S ] is also divided i i+I regime is applied to speed up; when the speed is close to the into K subsections, K∆x � S − S , as shown in Figure 2. A i+I i speed restriction, CR regime is applied to keep the train set 􏼈κ , . . . , κ 􏼉 is introduced to represent the indexes of the i i+I running at high-speed; MB regime is applied for low-speed discretizing points that overlap with the station positions restrictions and stopping. Tus, there is no coasting regime 􏼈S , . . . , S 􏼉, which can be described as follows: i i+I in the trajectory. Tis kind of trajectory is defned as the fat- out trajectory [23], which can be calculated as shown in x � S , ∀i ∈ {i, . . . , i + I}. (17) Algorithm 1 and Figure 4. In the fat-out trajectory calcu- κ i lation algorithm, train control regimes are determined based Considering the middle stations in the multisection on the relationship between the train running speed and running process, the constraint 14 that limits the train speed speed restriction. Under the constraints of safety and train at stations should be rewritten as follows: characteristics, the algorithm keeps the train running speed as close to or within speed restrictions. v � 0, ∀k ∈ 􏼈κ , . . . , κ 􏼉. (18) k i i+I Based on the fat-out trajectory calculation algorithm, the minimum running time can be calculated as follows: Meanwhile, the punctuality constraint 16 should be flat flat rewritten as follows: (22) T � t − t , i,i K+1 flat t − t � T , (19) where T is the minimum running time from station i to K+1 1 i,i+I ′ i,i ′ flat station i ; t is the train running time at the ending point K+1 where only the total running time from station i to station ′ (station i ) of the fat-out trajectory. i + I is limited, the running times of each section are un- limited, which can be adjusted in the optimization process. 3.2. Defnition of Subinterval. Based on the fat-out trajec- Ten, the energy-saving train optimal control problem tory, coasting regimes can be added into the control se- for the multisection [S , S ] can be stated as follows. i i+I quence, as shown in Figure 3. For better determining the starting points of coasting regimes, the whole running ⎧ ⎨ min J � 􏽘 E , s.t. μ , μ ∈ [0, 1]. (20) process is divided into several subintervals. Te subinterval i,i+I k f,k b,k k�1 means the train running process starts with an accelerating regime and ends with a braking regime. Meanwhile, there Journal of Advanced Transportation 5 v v Subinterval Subinterval x x x x x x x 1 2 k k' K K+1 S S S S i i+1 i+I–1 i+I Figure 2: Te illustration of multisection discretizing. MA CR MB CR MB can only be one accelerating phase and one braking phase in Flat-out trajectory a subinterval, which makes the accelerating phase and its CO regime subsequent adjacent braking phase form a subinterval. As Speed restriction shown in Figure 3, the frst subinterval begins with a MA Figure 3: Te illustration of fat-out trajectory and coasting re- (accelerating) regime and ends with an MB (braking) re- gimes addition. gime, and the MA and MB regimes are adjacent. Four important points (x , x , x , and x ) are introduced to a c d b j j j j describe the subinterval j, as shown in Figure 5. Start (1) x : the beginning position of the accelerating regime (2) x : the beginning position of the coasting regime Section discretizing (3) x : the ending position of the coasting regime Set k=1, v =0 and v =0 1 K+1 (4) x : the ending position of the braking regime. v < V No k k+1 Without coasting regime addition, x is equal to the ending position of the accelerating regime, and x is equal v == V d No k k+1 j Yes to the starting position of the braking regime. As the du- Adopt MA regime, set μ =1 ration of the coasting regime increases, the beginning po- f,k Yes and μ =0 to calculate v b,k k+1 sition of the coasting regime x moves to x , and the ending c a j j position of the coasting regime x moves to x . Specially, d b Adopt CR regime, set μ =0 j j f,k when x � x or x � x , the duration of the coasting and μ =0 to calculate v d b c a b,k k+1 j j j j regime cannot increase, as shown in Figure 6. For two Set k= k + 1 adjacent subintervals, if x � x in the frst subinterval and d b j j Adopt MB regime, set v =V k+1 k+1 x � x in the second subinterval, then these two c,j+1 a,j+1 and reverse search the starting adjacent subintervals can be merged as a new subinterval, as point of MB regime shown in Figure 6. Specially, the beginning position of the accelerating regime and the beginning position of the coasting regime of the new subinterval are equal to those of Yes k < K subinterval j, the ending position of the coasting regime and No the ending position of the braking regime of the new subinterval are equal to those of subinterval j + 1. End Figure 4: Te illustration of the fat-out trajectory calculation algorithm. 3.3. Coasting Points Determination. Due to the line char- acteristics (like speed restrictions and track slope), the fat- out trajectory consists of several subintervals, as shown in the running time change value after adding the coasting Figure 3. We propose a coasting control algorithm to dis- regime of subinterval j. In addition, the larger ρ means that tribute the running time supplements to subintervals. Te the more energy consumption can be reduced in subinterval distribution criterion distributes the running time supple- j with the same running time supplement. Considering the ments to subintervals where energy consumption can be whole running process from station i to station i , the limited reduced more signifcantly. To evaluate the efciency of running time supplement should be distributed to the energy consumption reduction, indicator ρ is introduced: subinterval with the maximum ρ for energy saving. Based on this, the coasting control algorithm is proposed: 􏽐 E − E 􏼁 ∆E k k k�a j Specially, for the multisection running process, the (23) ρ � − � − , ′ proposed coasting control algorithm is efective. Due to the ∆T t − t j b b j j multisection running process, the speed limit constraints of where ρ is the energy-saving efect of subinterval j; ∆E � middle stations need to be considered additionally. First, the j j ′ multisection running process will be divided into several 􏽐 (E − E ) is the energy consumption change value after k k k�a adding the coasting regime of subinterval j;∆T � t − t is subintervals with the same number of sections, as shown in j b b j j 6 Journal of Advanced Transportation (1) Divide the whole running section into K subsections. Set k � 1, v � 0, and V � 0. 1 K+1 (2) while k ≤ K do (3) if v < V (MA regime) then k k+1 (4) Set μ � 1 and μ � 0 to calculate v . f,k b,k k+1 (5) else if v � V (CR regime) then k k+1 (6) Set v � Vk + 1 to calculate μ and μ . k+1 f,k b,k (7) else if v > V (MB regime) then k k+1 ′ ′ (8) Set v � V and k � k + 1. Ten, reverse search the starting point of MB regime (k ): k+1 k+1 (9) repeat ′ ′ ′ ′ ′ (10) Set μ ′ � 0 and μ ′ � 1 to calculate v ′ . Since , set k � k − 1. f,k b,k k −1 (11) until v � v k k ′ ′ ′ ′ ′ (12) Set μ � μ , μ � μ , and v � v , for κ ∈ 􏽮k , k + 1, . . . , k􏽯 f,κ f,κ b,κ b,κ κ κ (13) end if (14) k � k + 1 (15) end while ALGORITHM 1:Flat-out trajectory calculation. Subinterval j Subinterval j+1 Subinterval j x x x x a c d b j j j j x = x x = x d b c a j j j+1 j+1 Figure 5: Te illustration of the subinterval. New Subinterval Figure 2. Ten, each section will be divided into several subintervals based on the defnition of the subinterval. Tus, the running time supplement can be distributed for the multisection running process as described in the Algo- rithm 2, to generate energy-saving running time schemes and related trajectories. 4. Numerical Examples Figure 6: Te illustration of the subinterval merging. In this section, we present two numerical examples to demonstrate the performance of the proposed energy- ⎧ ⎪ 200 0 ≤ v ≤ 15.28, efcient coasting control algorithm. Te frst example sol- ves the energy-saving train control problem along the single- F (v) � section, in which only the energy-saving train trajectories are 15.28 < v < ≤ 22.22, optimized. Te second one involves the energy-saving train 2 (24) control problem along the multisection, in which both the energy-saving train trajectories and running time scheme B (v) � 159.6 0 ≤ v ≤ 22.22, are optimized. All examples are based on the data of one of the W (v) � 216 􏼐4.5024 + 0.1089v + 0.0108v 􏼑. Beijing metro lines. Te speed restrictions and track 0 gradient between station 1 and station 5 are shown in Figure 7. Te parameters of the running train are listed in Examples are tested under the MATLAB environment Table 3. Te maximum train traction force, the maximum on a personal computer with Intel Core i5 2.30 GHz CPU braking force, and running basic resistance are given as and 8 GB RAM. follows: Journal of Advanced Transportation 7 (1) Divide the whole running section into K subsections. Calculate the fat-out trajectory based on the Algorithm 1. Initialize the sup flat running time supplement T � T − T . Divide the whole running section into J subintervals based on the defnition of the ′ i,i ′ i,i i,i subinterval. sup (2) while T ≥ 0 do i,i (3) for j � 1 to J do (4) Move the beginning position of coasting regime x with step ∆x backward temporarily, c � c − 1. Calculate the ending c j j ′ ′ ′ position of coasting regime x with μ � μ � 0, for k ∈ c , . . . , d . Calculate ρ based on the temporarily modifed ′ ′ 􏽮 􏽯 d f,k b,k j j j trajectory. (5) end for sup sup ′ ′ ′ ′ (6) Determine the subinterval j with maximum ρ , ρ ≥ ρ , for j ∈ {1, . . . , J}. Update c ′ � c ′, d ′ � d ′, and T � T −∆T ′. j j j j j j j ′ ′ j i,i i,i (7) for j � 1 to J − 1 do (8) if x � x and x � x then d b c,j+1 a,j+1 j j (9) Merge the subinterval j and subinterval j + 1. Update J � J − 1. (10) end if (11) end for (12) end while ALGORITHM 2: Coating control algorithm. (4) T-DP: Optimal trajectory calculated based on dy- namic programming, more details can be seen in Appendix. Specially, the DP algorithm is introduced to compare with the coasting control algorithm, aiming to demonstrate its efect. ∆x is set to be 1 m in the coasting control algo- rithm, and then the running processes of four sections are 0 0 divided into 990, 1225, 1257, and 776 subsections, re- 0 990 2215 3472 4248 S S spectively. Meanwhile, ∆x is also set to be 1 m in the DP S S 1 3 4 5 algorithm, and ∆v is set to be 0.02 m/s. For four diferent Position (m) types of trajectories, the trajectories and control commands Figure 7: Te illustration of speed restriction and gradient are shown in Figure 8, and the performance is shown in changing. Table 4. As shown in Figure 8, T-fat is keeping close to the speed restrictions in the running process, in which there is no Table 3: Train parameters. coasting regime. Te T-fat corresponds to the maximum Parameters Value energy consumption and the minimum running time in each Train mass, M 216 (t) section, as shown in Table 4. For T-Pra, accelerating regimes are Rotary mass coefcient, c 0.08 applied to reach a high speed at the beginning of the running Efciency of train motor, η 0.85 process, then braking regimes are applied for the low-speed restriction and train stops. Compared to T-CC and T-DP, we can observe that fewer coasting regimes are applied in T-Pra, as 4.1. Example 1: Scenarios of Single-Section Running. In this shown in Figure 8. For T-CC and T-DP, MA regimes are example, the train trajectories of sections [S , S ], [S , S ], 1 2 2 3 applied at the beginning of the running process and MB re- [S , S ], and [S , S ] were analyzed. Four diferent types of 3 4 4 5 gimes are applied at the stopping process. Tis kind of strategy trajectories are compared to verify the performance of the avoids the train from staying in low-speed phases and wasting proposed coasting control algorithm for single-section train running time supplements. Considering the running time control. More details about these four trajectories are as constraint, there will be more time for train coasting to reduce follows: energy consumption. Terefore, as shown in Table 4, the (1) T-Pra: Practical trajectory obtained from the comparison results of single-section running with the same equipped ATO systems running times show that T-CC can achieve 26.16%, 37.51%, (2) T-fat: Flat-out trajectory with a minimum running 12.12% and 35.31%, energy-saving for sections [S , S ], [S , S ], 1 2 2 3 time and maximum energy consumption, which can [S , S ], and [S , S ], respectively, in comparison to T-Pra. 3 4 4 5 be calculated based on the Algorithm 1; Meanwhile, T-DP can achieve 26.46%, 37.35%, 12.40%, and 36.19% energy-saving for four sections, respectively. Te little (3) T-CC: Optimal trajectory calculated based on the deviations in energy-saving performance between T-CC and proposed coasting control algorithm (Algorithm 2); Speed Restriction (m/s) Height (m) 8 Journal of Advanced Transportation Table 4: Te comparison of diferent trajectories and running time distribution schemes from S to S . 1 5 [S , S ] [S , S ] [S , S ] [S , S ] [S , S ] (total) 1 2 2 3 3 4 4 5 1 5 Section T J T J T J T J T J 1,2 1,2 2,3 2,3 3,3 3,4 4,5 4,5 1,5 1,5 index (s) (kWh) (s) (kWh) (s) (kWh) (s) (kWh) (s) (kWh) T-fat 80.97 20.38 103.51 17.63 91.59 22.24 71.72 15.94 347.79 76.19 T-Pra 88.40 13.15 109.80 12.45 100.90 14.44 79.80 9.09 378.90 49.13 T-DP 88.31 9.67 109.72 7.80 100.90 12.65 79.78 5.80 378.71 35.92 T-CC 88.31 9.71 109.77 7.78 100.82 12.69 79.78 5.88 378.68 36.06 T-CC with optimal running times 89.46 9.12 110.17 7.58 101.03 12.59 78.22 6.53 378.88 35.83 20 20 15 15 10 10 1 1 5 5 0 u 0 u 0 -1 0 -1 0 200 400 600 800 990 0 200 400 600 800 1000 1225 Position (m) Position (m) T-Pra T-CC T-Pra T-CC T-Flat Speed restriction T-Flat Speed restriction T-DP T-DP (a) (b) 20 20 15 15 10 10 1 1 5 5 0 u 0 0 0 -1 -1 0 200 400 600 800 1000 1257 0 200 400 600 776 Position (m) Position (m) T-Pra T-Pra T-CC T-CC T-Flat T-Flat Speed restriction Speed restriction T-DP T-DP (c) (d) Figure 8: Te illustration of trajectories and control commands μ for four sections. (a) Section [S , S ]. (b) Section [S , S ]. (c) Section 1 2 2 3 [S , S ]. (d) Section [S , S ]. 3 4 4 5 T-DP might come from the speed discretization in the DP In addition, to verify the feasibility of the coasting algorithm and the small diferences in running times. control algorithm, the optimal trajectories with diferent In terms of computation time, the average computation running times in sections [S , S ] are analyzed. T-CC with 1 2 times of T-CC and T-DP for four sections are 1.1 s and diferent running times is shown in Figure 9, and the cor- 261.9 s, respectively. Tis means that, for single-section train responding running time supplements of each subinterval control, applying the coasting control algorithm can achieve and energy consumption are shown in Table 5. First, the the similar energy-saving efect as the DP algorithm with less whole running process from S to S is divided into two 1 2 computation time. Specially, the computation time of the subintervals due to the low-speed restriction, as shown in coasting control algorith m can reach a 10 ms level when Figure 9. As the running time increases, running time running in a C environment. supplements are added into the subintervals with coasting Speed (m/s) Speed (m/s) Speed (m/s) Speed (m/s) Journal of Advanced Transportation 9 Table 5: Te comparison of T-CC with diferent running times from S to S . 1 2 Pre-given Running time supplement Running time supplement Energy consumption (kWh) running time (s) in subinterval 1 (s) in subinterval 2 (s) 80.97 0 0 20.38 81.97 (+1) 0.96 0.03 16.16 82.97 (+2) 1.91 0.07 14.34 85.97 (+5) 4.86 0.08 11.23 90.97 (+10) 9.90 0.08 8.47 100.97 (+20) 20.318 (merged subinterval) 5.72 110.97 (+30) 29.672 (merged subinterval) 4.58 Subinterval 1 2 0 200 400 600 800 990 Position (m) Speed restriction 85.97 80.97 90.97 81.97 100.97 82.97 110.97 Figure 9: Te illustration of T-CC with diferent running times from S to S . 1 2 15 15 10 10 5 5 0 0 0 0.05 0.1 0 10 20 Runtime supplement (s) Runtime supplement (s) (a) (b) Figure 10: Te illustration of energy-saving efects ρ for the subinterval 1 (a) and 2 (b) duration addition. In addition, the energy-saving efects ρ of 4.2. Example 2: Scenarios of Multisection Running. In this the subintervals 1 and 2, as shown in Figure 10, guide the example, we optimize the train trajectories and related to distribution of running time supplements. Running time running time schemes for the running process from S to S , 1 2 supplement is added to the subinterval with the larger ρ for based on the proposed coasting control algorithm. T-CC more energy consumption reduction. Specially, the maxi- with optimal running times is compared with the other mum running time supplement of the subinterval 2 is 0.08 s. trajectories with pre-given running times, to verify the When the running time supplement of subinterval reaches performance of the coasting control algorithm for the the maximum one, there is no room for coasting regime multisection train control. T-CC with optimal running times addition, like the cases 85.97 s and 90.97 s. Since, when the represents the optimal trajectories calculated based on the running time supplement is large enough, two subintervals optimal model 21, in which trajectories and running times are merged into one subinterval, as in the cases 100.97 s and for the multisection running process are optimized inte- 110.97 s in Figure 9. gratedly. T-CC with optimal running times and T-CC with ρ (kWh/s) Speed (m/s) ρ (kWh/s) 2 10 Journal of Advanced Transportation 0 990 2215 3472 4248 Position (m) T-CC with pre-given runtimes T-CC with optimal runtimes Speed restriction Figure 11: Te illustration of T-CC with optimal running times and T-CC with pre-given running times from S to S . 1 5 v v k,m k+1,m k k+1 v v k,m –1 k+1,m –1 k k+1 x x x x x x x x x x 1 2 3 4 k k+1 K–2 K–1 K K+1 Trajectory Speed restriction Grid point Figure 12: Te illustration of the speed-distance network. pre-given running times are compared in Figure 11, and the constraints and energy-saving objective, we developed train running times and energy consumption of each section distance-based train trajectory optimization models for the for T-CC with optimal running times are shown in Table 4. single-section and multi-section operations. A coasting As shown in Table 4, by comparing T-CC with pre-given control algorithm was proposed to generate the energy- running times and T-CC with optimal running times, we can efcient trajectories, in which the coasting control regime observe that the running times of each section are diferent points were determined according to the energy-saving in these two plans. Due to the change in running times, there efect. are also diferences in the trajectories. As shown in Figure 11, Numerical examples based on one of the Beijing metro with running time addition, more coasting regimes are lines were implemented in two diferent cases, i.e., single- added into the trajectories in sections [S , S ], [S , S ], and section and multisection operation, to demonstrate the 1 2 2 3 [S , S ] of T-CC with optimal running times. Meanwhile, the performance of the proposed coasting control algorithm. 3 4 energy consumption in these three sections can be reduced Te computational results showed that, by applying the in comparison to those of T-CC with pre-given running coasting control algorithm, the energy consumption of times. On the other hand, with running time reduction in the single-section operation can be reduced efectively by sections [S , S ], the energy consumption of T-CC with around 12.12% to 37.35% in comparison to the practical 4 5 optimal running times in this section is larger than it of trajectories obtained from equipped ATO systems. Mean- T-CC with the pre-given running times. In terms of the while, the coasting control algorithm was compared with the DP algorithm; the former can achieve a similar energy- whole running process, for T-CC with optimal running times, the total energy consumption can be reduced from saving performance in shorter computation times. For the 36.06 kWh to 35.83 kWh compared to T-CC with pre-given multisection operation, the proposed coasting control al- running times. gorithm can generate energy-saving running time schemes and related trajectories by optimizing the whole running process integratedly. 5. Conclusion Our future research will focus on the online train control In this paper, we studied the optimal train control problem problem to deal with the dynamic situations, like temporary to reduce energy consumption. Combining the operational speed restrictions. Tis paper only deals with the ofine train Speed (m/s) Journal of Advanced Transportation 11 trajectory optimization problem. However, the train tra- References jectories will be adjusted in real-time operation to keep safe [1] G. M. Scheepmaker, R. M. Goverde, and L. G. Kroon, “Review and punctual operations. of energy-efcient train control and timetabling,” European Journal of Operational Research, vol. 257, no. 2, pp. 355–376, Appendix [2] A. 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Published: Mar 22, 2023

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