energies
Article Energy Optimization for Train Operation Based on an Improved Ant Colony Optimization Methodology
Youneng Huang 1,2,*, Chen Yang 1 and Shaofeng Gong 1
1 School of Electronics and Information Engineering, Beijing Jiaotong University, Haidian District, Beijing 100044, China; [email protected] (C.Y.); [email protected] (S.G.) 2 National Engineering Research Center of Rail Transportation Operation and Control System, Beijing Jiaotong University, Haidian District, Beijing 100044, China * Correspondence: [email protected]; Tel.: +86-10-5168-8532
Academic Editor: Enrico Sciubba Received: 4 May 2016; Accepted: 2 August 2016; Published: 9 August 2016
Abstract: More and more lines are using the Communication Based Train Control (CBTC) systems in urban rail transit. Trains are operated by tracking a pre-determined target speed curve in the CBTC system, so one of the most effective ways of reducing energy consumption is to fully understand the optimum curves that should prevail under varying operating conditions. Additionally, target speed curves need to be calculated with optimum real-time performance in order to cope with changed interstation planning running time. Therefore, this paper proposes a fast and effective algorithm for optimization, based on a two-stage method to find the optimal curve using a max-min ant colony optimization system, using approximate calculations of a discrete combination optimization model. The first stage unequally discretizes the line based on static gradient and speed limit in low-density and it could conduct a comprehensive search for viable energy saving target speed curves. The second stage unequally discretizes the line based on first stage discretion results, it makes full use of first-stage optimization information as pheromone, quickly optimizing the results to satisfy real-time demands. The algorithm is improved through consideration of the experience of train drivers. Finally, the paper presents some examples based on the operation data of Beijing Changping Subway Line, which is using CBTC system. The simulation results show that the proposed approach presents good energy-efficient and real-time performance.
Keywords: CBTC; ant colony optimization; discrete combination; optimization of energy-savings
1. Introduction With growing concerns about environmental problems, the huge energy consumption of urban rail transit systems has attracted much attention. The energy consumption of train traction makes up nearly half of a subway system’s total energy consumption. In an urban rail transit, more and more lines are using the Communication Based Train Control (CBTC) systems. Based on these systems, trains are controlled by tracking a target speed curve, and this allows the control center operators to change in real-time the planned interstation running time according to whether the train arrived at a station early or late. This requires that the system have the ability of computing real-time target distance curves online. Therefore, optimizing the energy-expenditure of those speed curves is regarded as one of the most effective ways to realize energy-efficient train operation. However, energy-saving target speed curves must be calculated with real-time performance data, as the distances between two adjacent stations can be relatively short, causing frequent acceleration and deceleration switches. Thus, this paper aims to design a fast and efficient real-time algorithm for optimization of energy-saving train speed curves. Many studies dating as far back as the1960s have previously focused on energy-efficient train operation, For example, Ishikawa et al. carried out the first study focusing on energy-efficient train
Energies 2016, 9, 626; doi:10.3390/en9080626 www.mdpi.com/journal/energies Energies 2016, 9, 626 2 of 18 operation strategies [1], followed by Howlett et al. [2,3] who confirmed the fundamental optimality of a maximum acceleration-coasting, minimum-braking, train control strategy. In addition, they designed the “Metromiser”, which can be used to advise drivers when to coast and brake, so that the train arrives on time and consumes as little energy as possible [4]. In studies on energy-efficient train operations, off-line optimization has attracted more attention. In 1994, Wang studied optimized control methods for energy-efficient operation, and proposed corresponding operator sequences and optimization strategies [5]. González-Gil et al. gave an insightful overview on the potential of urban rail systems to reduce their energy consumption [6]. Ding et al. discussed an optimized model and the corresponding heuristic algorithm for locomotives to work under a fixed running time regime between given stations in a practical operating environment [7–9]. In 2009, Fu et al. studied the control strategies of a train when disturbed and obtained target speed profiles using a genetic algorithm (GA) based on an optimized model [10]. Also, Howlett et al. proposed a method for calculation of critical switching points for a globally optimal strategy on a track with steep gradients [11]. In 2011, Ke et al. proposed MMAS to search for the optimal speed codes of each section and train acceleration was controlled by a fuzzy-PID gain scheduler to meet the determined speed commands [12]. Yong et al. presented the two-level optimization model based on a Genetic Algorithm for minimal energy consumption of trains. Then, Domínguez et al. proposed a Multi Objective Particle Swarm Optimization (MOPSO) algorithm to obtain the consumption or time Pareto front based on the simulation of a train with a real Automatic Train Operation (ATO) system [13,14]. Besides, Huang et al. optimized both trip time and driving strategy for multiple interstation segments by using a multi-population genetic algorithm (MPGA) [15]. The works of Roberts et al. have assessed Enhanced Brute Force (EBF), Ant Colony Optimization (ACO), and GA searching methods for calculating the most appropriate train target speed series to optimize the train operation and suggested that both GA and ACO are suitable. In addition, Li et al. have explored the merits of optimizing the speed curve using ACO for train operation [16,17]. The real-time optimization of energy-efficient train operation needs to be considered in further studies. From 2009 to 2011, Ke et al. discretized the optimization problem of train energy-efficient operation and made a combinatorial optimization using an ant colony algorithm, based on linear computing discrete combination optimization [18,19]. Wong et al. presented an application of GA to search for the appropriate coasting points and investigates the possible improvement on fitness of genes. Single and multiple coasting point control with simple GA are developed to attain the solutions and their corresponding train movement is examined [20]. In 2010, Miyatake studied the problems of energy-consumption minimization with energy storage equipment, and discussed three methods: the gradient method, the dynamic programming method and the quadratic programming method, and real-time control methods are mentioned in research [21]. Jin et al. discussed the optimization problem of energy-efficient train operation with variable gradients, and proposed a simulation optimization model which combined local optimization with global optimization strategies. Then, they created an optimized speed profile online, using this model, to guide the train to energy-efficient operation [22]. Su et al. proposed an integrated energy-efficient train operation algorithm combined with optimal timetabling [23,24]. Sicre et al. proposed a GA with fuzzy parameters to calculate a new efficient driving mode in real-time to be manually executed on high speed trains when significant delays arise [25]. Furthermore, Yin et al. considered multiple train operation objectives and proposed two intelligent train operation algorithms in order to minimize the energy consumption of train operation online [26]. Khmelnitsky used the Pontryagin maximum principle and proved the optimization problem, under which the optimal driving strategy consisted of acceleration, cruising, coasting and maximum braking; he derived a numerical solution based on a successive model, and the method can be added in real-time feedback design in many different ways [27]. Gu et al. proposed a novel multiple-model-based switching optimization framework to reduce energy consumption while guaranteeing the punctuality during train’s real-time tracking operation [28]. Yang et al.’s survey of energy-efficient train operation assessed the existing literature, including the use of several optimization algorithms, and the MMAS mentioned in the research is more suitable to study discretization intervals and online optimization problem [29]. Energies 2016, 9, 626 3 of 18
In conclusion, train energy-efficient operation can be researched from several perspectives such as driving strategy, timetable, real time, comfortableness and so on. Different research contents correspondEnergies 2016 to, 9 different, 626 research methods. This paper researches train energy-efficient3 operation of 18 mainly from a real-time perspective. A method is therefore needed that achieves energy-efficient operationIn together conclusion, with train good energy real-time‐efficient optimization-performance. operation can be researched Consequently, from several perspectives this paper such proposes as driving strategy, timetable, real time, comfortableness and so on. Different research contents an improved ant colony algorithm, based on a two-stage optimization, to search for the optimal target correspond to different research methods. This paper researches train energy‐efficient operation speed profile capable of fast operation as well as achieving enhanced energy efficiency. mainly from a real‐time perspective. A method is therefore needed that achieves energy‐efficient operation together with good real‐time optimization‐performance. Consequently, this paper 2. Problem Statement proposes an improved ant colony algorithm, based on a two‐stage optimization, to search for the optimalGenerally target speaking, speed profile the capable core of problem fast operation of energy-efficient as well as achieving train enhanced operation energy is efficiency. to find the energy-optimal speed-curve for trains to track. To do so, we first discretize a subway segment between2. Problem two adjacent Statement stations into n intervals i.e.,: Generally speaking, the core problem of energy‐efficient train operation is to find the energy‐ optimal speed‐curve for trains to track.X PTo t Xdo1 ,so,X2 we, X 3first,..., discretizeXnu a subway segment between two (1) adjacent stations into intervals i.e.,: Similarly, the target speed corresponding to the distance point is: ∈ , , ,…, (1)
Similarly, the target speed correspondingV P tV1, toV2 the, V3 distance,..., Vn upoint is: (2)
∈ , , ,…, (2) where Xi is the train discrete point location and Vi is the tracking target speed code(in km/h). A target where is the train discrete point location and is the tracking target speed code(in km/h). speed sequence builds up a target speed curve. A target speed sequence builds up a target speed curve. As shown in Figure1, the object this paper researches is the train target speed curve, which As shown in Figure 1, the object this paper researches is the train target speed curve, which needsneeds to be to calculated be calculated off-line off‐line in in some some literatures. literatures. The The train train reaches reaches this this target target speed speed by traction, by traction, coastingcoasting or braking. or braking. Different Different operation operation mode mode is related is related to line to conditions line conditions and running and running time constraints. time The trainconstraints. tracks The the train target tracks speed the target curve speed by cycles curve of by traction-braking-coasting cycles of traction‐braking‐coasting around around the objective the speed,objective but that speed, problem but that is the problem optimization is the optimization of ATO control, of ATO which control, is different which is withdifferent the with problem the this paperproblem researches. this paper The optimization researches. The of optimization ATO control of is ATO also control one of is the also hotspots one of the in trainhotspots energy-efficient in train operationenergy research‐efficient [operation30]. research [30].
Train Operation Curve Speed Target Speed Curve Vm Vm-1 … Vm-2 … … … … … …
V5 … V4 … V3 … V2 … V1 … V0 X1 X2 X3 … Xn-2 Xn-1 Xn Distance FigureFigure 1. 1.Target Target speed speed curvecurve and and train train operation operation curve. curve.
Therefore, the objective function of the energy‐efficient train operation model is: Therefore, the objective function of the energy-efficient train operation model is: (3) br gr Fai “ M ˆ ai ` Fi ` Fi (4) (3)
br gr (5) Fci “ F ` F (4) i i
Ei “ F ai ˆ Si ` Fci ˆ D i (6) (5) n Etotal “ min Ei (6) i“ ÿ1 Energies 2016, 9, 626 4 of 18
br where M is the total mass of the train; ai is acceleration of the train in the ith interval; Fi is the gr basic resistance of the train in the ith interval; Fi is the gradient resistance of the train in the ith interval; Fai is traction force in acceleration process of the train in the ith interval, Fci is traction force in cruising process of the train in the ith interval, Si is acceleration distance of the train of the ith interval; Di is cruising distance of the train in the ith interval; Ei is tracking energy consumption in the ith interval; Etotal is the total energy that the train consumes after completing all discrete-points speed code selection (in kWh); and n is the number of discrete points after discretization of the line. Under normal circumstances, the running time of the train on the segments will strictly conform to the requirements of the schedule. The driving strategies will attempt to achieve the train operation and running times specified in the schedule are the same [31]. The driving strategy will also ensure that the actual operation time of the train is within the allowable difference between the two, even under extreme circumstances. Therefore, in the process of the algorithm design, the operation time needs to be constrained: Vi ´ Vi´1 Di Ti “ ` (7) ai Vi n T “ Ti (8) i“ ÿ1 T ´ Trunning ď ∆t (9) ˇ ˇ where Ti is the actual running time of theˇ train in theˇith interval; ai is acceleration of the train in the ith interval; Vi is the tracking target speed code in the ith interval; Di is cruising distance of the train in the ith interval; T is the actual total running time of the train (in seconds); Trunning is the running time according to the timetable scheduled requirements (in seconds); ∆t is the permissible value of the train running time, which is a positive value, n is the number of discrete points after discretization of the line. The ant ACO is modeled on ants’ foraging process in the natural world. Generally, the ants’ foraging space describes the problem search space and the ant colony can be regarded as a set of effective solutions to the search space, and ant paths are used to represent feasible solutions. Therefore each ant can independently search the problem space for feasible solutions. The pheromone will be left behind during the foraging process. Meanwhile, the pheromone in the path will evaporate as time goes by. Therefore, the ant will choose the right path through perceiving the pheromone concentration, and then by continuous iteration obtain the optimal solution. Finally, the ant colony will centralize to the optimal path, which is the optimal solution to the problem. The correspondence between actual ant phenomena and our model are shown in Table1.
Table 1. The corresponding elements table between ant colony foraging phenomenon and ant colony optimization algorithm.
Ant Colony Foraging Phenomenon Ant Colony Optimization Algorithm Ant colony A set of effective solutions to the search space Foraging space Problem search space Pheromone Pheromone concentration Path from nest to food Effective solution Found the shortest path The optimal solution to the problem
By summarizing the above fundamental elements of the ACO algorithm, we can summarize its main features as follows:
‚ It possesses positive feedback, and heuristic hunting characters ‚ It uses distributed control but not centralized control ‚ It is has strong robustness ‚ Each individual can perceive only local information, but not global information Energies 2016, 9, 626 5 of 18
Energies 2016, 9, 626 5 of 18 MAX-MIN Ant System (MMAS) is one of the most effective algorithms in ACO. In addition, focusingMAX on‐MIN the optimization Ant System of(MMAS) the energy-efficient is one of the train most operation, effective comparedalgorithms with in ACO. other optimizationIn addition, focusingalgorithms, on MMAS the optimization has the following of the three energy advantages:‐efficient train operation, compared with other optimization algorithms, MMAS has the following three advantages: ‚ It has good real-time performance of the energy-efficient train operation It has good real‐time performance of the energy‐efficient train operation ‚ The setting of heuristic pheromone can effectively improve the convergence speed of the algorithm The setting of heuristic pheromone can effectively improve the convergence speed of the algorithm. ‚ The method can avoid getting the local optical solution and has better search capabilities of The method can avoid getting the local optical solution and has better search capabilities of global information global information To the best of our knowledge, there is little research in energy energy-efficient‐efficient train operation that uses the two-stagetwo‐stage MMAS optimization algorithm. By By considering the the characteristics of the two-stagetwo‐stage MMAS algorithm, and the present research conditions, using this algorithm is expected to expand on prevailing ideas and methods in the fieldfield ofof energy-efficientenergy‐efficient traintrain operation.operation.
3. Improved Improved Ant Ant Colony Colony Algorithm Algorithm Modeling Modeling The ant colony optimization algorithm is motivated by simulating the process of ants foraging to solve discrete combination-optimizationcombination‐optimization problems. The advantage of the ant colony optimization algorithm is that it is superior to other algorithms [32], [32], as it is more effective in terms of computing speed and convenience. MMAS MMAS is is a a type type of improved ACO algorithm that that aims to solve a discrete optimization problem. It generally contains two important parts: path construction and pheromone updating. Therefore, Therefore, the the problem problem of of energy energy-efficient‐efficient train train operation operation using using MMAS MMAS is divided is divided into intotwo partstwo parts as respectively as respectively described described in the in following the following two twosections. sections.
3.1. Partition of the Algorithm Discretization in the ACO algorithm affects the size of the problem: a reasonable discretization can reduce the solution-spacesolution‐space energy-savingenergy‐saving train-operationtrain‐operation optimization, and accelerate the convergence speed. ACO algorithms usually use equal discretization when solving the discrete combination optimization problem.problem. Figure Figure2 shows 2 shows the equal the discretization: equal discretization: the train canthe choose train thecan corresponding choose the correspondingtarget speed of target every speed segment. of every A target segment. speed A sequence target speed builds sequence up a target builds speed up a curve. target speed curve.
Speed Limit Speed Target Speed Curve Vm Vm-1 … Vm-2 … … … … …
V5 … V4 … V3 … V2 … V1 … V0 X1 X2 X3 … Xn-2 Xn-1 Xn Distance
Figure 2. Equal discretization.
In reality, the line information and traffic operation characteristics must be taken into In reality, the line information and traffic operation characteristics must be taken into consideration. consideration. Therefore it discretizes the line into segments based on the variety of static gradient Therefore it discretizes the line into n segments based on the variety of static gradient and speed limit. and speed limit. As shown in Figure 3, the speed limit and gradient in each discretized segment are As shown in Figure3, the speed limit and gradient in each discretized segment are constant, so that constant, so that the algorithm can optimize conveniently. the algorithm can optimize conveniently.
Energies 2016, 9, 626 6 of 18 Energies 2016, 9, 626 6 of 18
Speed Limit Gradient Speed Target Speed Curve Vm Vm-1 … Vm-2 … … … … … …
V5 … V4 … V3 … V2 … V1 … V0 X1 X2 X3 … Xn-2 Xn-1 Xn Distance
Figure 3. Unequal discretization. Figure 3. Unequal discretization. 3.2. Pheromone 3.2. PheromoneThe pheromone plays a guiding role for the algorithm allowing it to find the optimal speed profile. It reflects the effect of the a priori factors and the certainty factors and so is critical in the The pheromone plays a guiding role for the algorithm allowing it to find the optimal speed profile. search process described herein. In this paper, the pheromone setting studies the driving experience It reflectsof the train effect drivers. of the The a principle priori factors is as follows: and the taking certainty the real factorsdriving experience and so is into critical account, in the the search train process describedspeed herein. tends In to this accelerate paper, when the downhill pheromone and tends setting to decelerate studies the when driving uphill. This experience principle of affects train drivers. The principlethe probability is as follows: when searching taking thethe next real path driving in solution experience space, can into change account, the searching the train direction, speed tends to accelerateaccelerate when downhill the convergence and tends rate of to the decelerate algorithm, whenand improve uphill. the This performance principle of affectsthe algorithm. the probability Based on the two aspects of the improvement, a target speed curve optimization method based on when searchingMMAS is theproposed next in path this paper. in solution space, can change the searching direction, accelerate the convergence rate of the algorithm, and improve the performance of the algorithm. Based on the two aspects of3.3. the Path improvement, Construction a target speed curve optimization method based on MMAS is proposed in this paper.Path construction uses random proportion rules to choose the next step: the random proportion rule of the MMAS algorithm is basically consistent with the earliest ant system (AS) algorithm, and 3.3. Paththe Construction specific proportion random rule is defined as follows:
1 ∙ , ∙ , Path construction uses random proportion rules to choose the next step:∈ the random proportion , ∑ 1 ∙ , ∙ , (10) rule of the MMAS algorithm is basically ∈ consistent with the earliest ant system (AS) algorithm, and the 0 specific proportion random rule is defined as follows: where , is the selection probability of the tracking target speed when the th ant is in the th discrete point to the 1th discrete point; is a constant value to regulate the proportion of p1´γq¨ϕpVi,Vi`1q`γ¨ρpVi,Vi`1q 0 1 , Vj`1 P I pheromone trails and the heuristic information,rp1´ withγq¨ϕ pVi,Vj`1q`; γ¨ρpV i,V j`1qis the heuristic information Pn pVi, Vi`1q “ Vj`1P I (10) values for the tracking target $speed ; , being the pheromone trail values of the tracking 0ř other target speed when the &th ant is in the th discrete point to the 1th discrete point; is the collection of all the feasible tracking target speeds in the 1th discrete point; n is the grade of the ant. where Pn pVi, Vi`1q is the selection% probability of the tracking target speed Vi`1 when the nth ant is in the ith discrete3.4. Pheromone point Updating to the i ` 1th discrete point; γ is a constant value to regulate the proportion of pheromone trailsIn the and MMAS the algorithm, heuristic the information, pheromone updating with 0 ď ruleγ hasď 1 evolved; ϕ pVi, significantlyVi`1q is the compared heuristic with information values fortraditional the tracking ACO. target The MMAS speed algorithmVi`1; ρ p Vfocusesi, Vi` 1onq being the larger the pheromoneimprovements trail of the values pheromone of the tracking updating, which enhances algorithm performance. The specific pheromone updating rule is defined target speed Vi`1 when the nth ant is in the ith discrete point to the i ` 1th discrete point; I is the as follows: collection of all the feasible tracking target speeds in the i ` 1th discrete point; n is the grade of the ant.
3.4. Pheromone Updating In the MMAS algorithm, the pheromone updating rule has evolved significantly compared with traditional ACO. The MMAS algorithm focuses on the larger improvements of the pheromone updating, which enhances algorithm performance. The specific pheromone updating rule is defined as follows:
λ ¨ ∆ρn pVi, Vi`1q ` p1 ´ δq ¨ ρn pVi, Vi`1q if ant is iteration ´ best ρ pVi, Vi`1q “ $ p1 ´ λq ¨ ∆ρn pVi, Vi`1q ` p1 ´ δq ¨ ρn pVi, Vi`1q i f ant is so ´ f ar ´ best (11) &’ p1 ´ δq ¨ ρn pVi, Vi`1q other
%’ Energies 2016, 9, 626 7 of 18
Energies 2016, 9, 626 7 of 18 ∙∆ , 1 ∙ , , 1 ∙∆ , 1 ∙ , (11) E τ 1 ∙ , sbest if ant is iteration ´ best Eibest ∆ρ pV , V q “ (12) n i i`1 $ ´1 ¯ i f ant is so ´ f ar ´ best &’ 0 other ∆ , (12) where ρ pVi, Vi`1q being the pheromone%’1 trail values of the tracking target speed Vi`1 when the nth ant is in the ith discrete point to the i `0 1th discrete point; ∆ρn pVi, Vi`1q is the released pheromone trail values of the tracking target speed V when the nth ant is in the ith discrete point to the i ` 1th where , being the pheromone traili` 1values of the tracking target speed when the th ant discrete point; λ is the weighting factor of the iteration-best solutions, with 0 ă λ ă 1; 1 ´ λ is the is in the th discrete point to the 1th discrete point; ∆ , is the released pheromone trail weighting factors of the best-so-far solutions; δ is the pheromone evaporation rate, with 0 ă δ ď 1; values of the tracking target speed when the th ant is in the th discrete point to the 1th discreteτ is a constant point; which is the can weighting be set to factor determine of the whether iteration a‐best pheromone solutions, is with initialized 0 1 again;; 1 Eibest isis the the weightingminimum factors energy of consumption the best‐so‐ infar this solutions; iteration (inis the kWh); pheromoneEsbest is the evaporation minimum rate, energy with consumption 0 1; so far (in kWh). The process of line discretization is described as follows: is a constant which can be set to determine whether a pheromone is initialized again; is the minimum energy consumption in this iteration (in kWh); is the minimum energy consumption Step 1. Import line data containing the gradients and speed limits, and initialize the so far (in kWh). The process of line discretization is described as follows: related parameters; Step 1. Import line data containing the gradients and speed limits, and initialize the Step 2. Set the discretization density, and then discretize lines and running target speeds; related parameters; StepStep 2. 3. SetCalculate the discretization the equivalent density, value and based then ondiscretize the gradient lines data;and running target speeds; StepStep 3. 4. CalculateCalculate the the equivalent energy-consumption value based and on runningthe gradient time, data; including the train starting phase, Step 4. Calculatetrain running the energy phase‐consumption and train braking and running phase; time, including the train starting phase, Step 5. trainStore running the data phase and and create train the braking energy consumptionphase; and running time lookup table; StepStep 5. 6. StoreOutput the data the lookup and create table. the energy consumption and running time lookup table; Step 6. Output the lookup table. The above provides a concrete model for the discrete optimization and the MMAS algorithm, whichThe lays above the foundationsprovides a concrete for the model design for of the train discrete energy-saving optimization speed and curve the MMAS optimization algorithm, based which on a laystwo-stage the foundations optimization for the algorithm. design of Thetrain first energy stage,‐saving i.e., lowspeed density curve optimization discretization, based is fully on a optimized two‐stage optimizationto search for algorithm. the best solution; The first the stage, second-stage, i.e., low density i.e., high discretization, density discretization, is fully optimized will to refer search to the for first the beststage solution; information the second and quickly‐stage, i.e., search high fordensity the optimaldiscretization, solution. will The refer algorithm to the first design stage information is described and in quicklythe following search section. for the optimal solution. The algorithm design is described in the following section.
4.4. Overall Overall Design Design of of Improved Improved ACO ACO InIn terms terms of of energy energy-efficient‐efficient train train operation [13], [13], the the train train needs to to accelerate at at the the beginning andand brake brake on arrival. When When the the train train is is running running between between stations, stations, the the train train uses uses a a combination of of accelerating,accelerating, cruising, cruising, coasting coasting and and braking. braking. To To achieve achieve this, this, the the algorithm algorithm is is designed designed to to use use the the optimizationoptimization results results from from the the first first phase phase as as a reference a reference for for second second stage stage optimization. optimization. This This means means that twothat stages two stages of optimization of optimization design design are needed are needed and both and use both the use MMAS the MMAS algorithm. algorithm. In the first In the stage, first westage, divide we dividethe subway the subway line to achieve line to achieve low‐density low-density discretization; discretization; in the second in the second stage, we stage, derive we a derive high densitya high densitydiscretization discretization to search to search for the for more the more viable viable path. path. The The second second stage stage of of high high-density‐density discretizationdiscretization therefore therefore cuts cuts the the line line into into many many small small lengths lengths with with the the aim aim of of searching the optimal path as accurately and and fast as possible. The The results results of of the the first first stage stage will already have converged to to thethe vicinity vicinity of of the the optimal path and the the high high-density‐density discrete optimization simulation simulation of of the the second second stagestage takes takes full full advantage advantage of of that that to to initialize initialize the the optimization optimization of of pheromone pheromone values. values. The The second second stage stage thereforetherefore finds finds a better path path in in the the shortest shortest time, time, improving improving the the efficiency efficiency of of the the overall overall optimization optimization andand improving improving real real-time‐time performance. The overall architecture is further illustrated by Figure4 4..
FigureFigure 4. 4. OverallOverall architecture architecture diagram diagram of of the the train train energy energy-efficient‐efficient operation operation optimization optimization simulation. simulation.
Energies 2016, 9, 626 8 of 18 Energies 2016, 9, 626 8 of 18
4.1.4.1. Preparation Preparation of of Optimized Optimized Stage Stage TheThe main main purposes purposes in in preparing preparing the the optimization optimization stage stage are are to to discretize discretize running running lines lines and and target target speedspeed for for the train,train, form form different different operating operating modes modes and and the the corresponding corresponding energy energy consumptions consumptions under underdifferent different target target speeds, speeds, and establish and establish a look-up a look table‐up table of running of running times. times. The The look-up look‐up table table allows allows the theoptimization optimization stage stage to to search search different different operation operation modes modes and and targettarget speeds in in each each discrete discrete section, section, inin order order to to prepare prepare for for efficient efficient optimization. optimization.
4.2.4.2. Optimization Optimization of of the the First First Stage Stage TheThe main main functions functions of of the the optimized optimized first first stage stage are are to to search search every every possible possible running running path path of of the the traintrain constructed constructed in in the the prepared prepared optimization optimization phase, phase, being being as as comprehensive comprehensive as as possible, possible, find find the the qualifyingqualifying paths, paths, then then gradually gradually converge converge to to the the optimal optimal path path and and finally finally save save all all the the pheromone pheromone informationinformation for for the the path. path. In In addition, addition, the the look look-up‐up table table of of pheromones pheromones for for optimization optimization of of the the second second stagestage is is also also built. built. ThenThen the the ACO ACO needs needs to to set set heuristic heuristic pheromone pheromone in in the the initialization initialization phase. phase. The The strategies strategies for for heuristicheuristic pheromone pheromone in in this this design design are are set set by by referring referring to to the the experience experience of of train train drivers. drivers. The The main main settingsetting principles principles are are described described as: as: take take the the real real driving driving experience experience into into account, account, the the train train speed speed tends tends toto accelerate accelerate when downhill andand tends tends to to decelerate decelerate when when uphill. uphill. The The detailed detailed process process of the of first-stagethe first‐ stageoptimization optimization is shown is shown in Figure in Figure5. 5.
start
set the MMAS corresponding parameters and operation time limit
set the gradient and heuristic information
iteratiom=0
Y output optimal operation curve with pheromone information, iteraton>set value corresponding operation mode and speed code information of each N section, then stored
Iteration ++
according to each section pheromone determine ant path
calculate the ant energy consumption and running time
ants run time > operation time limit value Y
N
according to the Iteration_best_ant update each segment pheromones
Iteration‐best‐ant energy Y N update the global_best_ant energy consumption, running time and speed code information, operation mode information update pheromone(according to global‐best‐ant) store the pheromone information of global‐best‐ant optimization curve FigureFigure 5. 5. FirstFirst stage stage optimization optimization flow flow chart. chart. Energies 2016, 9, 626 9 of 18 Energies 2016, 9, 626 9 of 18 4.3. Optimization of the Second Stage 4.3. Optimization of the Second Stage The center operators would change in real-time the planned interstation running time according to whenThe the center train operators arrived at would the station change early in real or‐ late,time so the we planned need to interstation further optimize running the time target according speed curveto when in the the second train arrived stage. Second-stageat the station optimizationearly or late, so mainly we need considers to further two optimize factors: the target speed curve in the second stage. Second‐stage optimization mainly considers two factors: ‚ Whether the route searching is comprehensive Whether the route searching is comprehensive ‚ Whether the optimization time is fast enough Whether the optimization time is fast enough WeWe mainly mainly consider consider the the first first factor factor during during the the first first stage stage and and provide provide reference reference information information for for optimizationoptimization of of the the second second stage. stage. The The second second factor factor will will be mainly be mainly considered considered during during the second the second stage. Thestage. result The of result the low-density of the low first-stage‐density optimizationfirst‐stage optimization will have already will providedhave already an approximation provided an ofapproximation what is optimal. of what The pheromoneis optimal. The look-up pheromone table formed look‐up after table the formed end of after optimization the end of is optimization used as the valueis used for as the the pheromone value for the initialization. pheromone initialization. This improves This the improves pheromone the pheromone distribution distribution when searching when pathssearching during paths optimization during optimization of the second of stage. the second Note that stage. the Note optimization that the of optimization the second stage of the involves second high-densitystage involves discretization high‐density and discretization we have a and better we search have a direction better search at the direction beginning at ofthe optimization. beginning of Therefore,optimization. according Therefore, to this according design, the to optimizedthis design, speed the optimized of the second speed stage of will the besecond further stage improved, will be andfurther the optimizedimproved, timeand the will optimized be significantly time will reduced. be significantly The procedure reduced. of the The second procedure stage of is the shown second in Figurestage 6is. Theshown set strategyin Figure for 6. heuristicThe set strategy pheromone for inheuristic this optimization pheromone stage in this is achieved optimization in the stage same is wayachieved as the in first the stage, same andway we as the also first add stage, the train and drivers’ we also driving add the experience. train drivers’ driving experience. FigureFigure 6. 6.Second Second stage stage optimization optimization flow flow chart. chart. 4.4.4.4. Overall Overall Design Design TheThe overall overall design design process process ofof thethe traintrain energy-savingenergy‐saving optimization simulation is shown in in Figure Figure 77.. Energies 2016, 9, 626 10 of 18 Energies 2016, 9, 626 10 of 18 Figure 7. The overall design process. Figure 7. The overall design process. 5. Simulations Based on Beijing Subway Changping Line 5. Simulations Based on Beijing Subway Changping Line 5.1. Basic Data To illustrate the effectiveness of the proposed model and algorithm, a case based on the Beijing subway Changping Changping line line is provided is provided as an as example. an example. The chosen The chosen line covers line a covers length of a length3800 m, of consisting 3800 m, ofconsisting two stations of two and stations one interstation. and one interstation. The practical The running practical time running is 246 s and time the is 246energy s and consumption the energy forconsumption this interstation for this trip interstation is 21.62 kWh. trip As is shown 21.62 kWh. in Tables As shown2 and 3 in respectively, Tables2 and the3 respectively, subway line thedata derivedsubway from line data Beijing derived subway from Changping Beijing subway line contains Changping static line gradients, contains and static speed gradients, limits. and The speed train speed code is partitioned from 40 km/h to 100 km/h every 5 km/h, as shown in Table 4. In the case study, the train formation form is three motor cars (M1, M2 and M3) and three trailer cars (T1, T2 and T3), the total mass of the train is 199 t, as shown in Table 5. Table 6 shows the parameters of MMAS. Energies 2016, 9, 626 11 of 18 limits. The train speed code is partitioned from 40 km/h to 100 km/h every 5 km/h, as shown in Table4. In the case study, the train formation form is three motor cars (M1, M2 and M3) and three trailer cars (T1, T2 and T3), the total mass of the train is 199 t, as shown in Table5. Table6 shows the parameters of MMAS. Table 2. The slope data of the simulation cases. Starting Position (m) Final Position (m) Slope (Micrometer Degrees) 0 398 0 398 598 ´23.835 598 798 5.6 798 1878 ´3 1878 2184 ´7.99 2184 2386 2.971 2386 2739 14.873 2739 3453 3 3453 3698 ´12.449 3698 3800 0 Table 3. The speed limit data of the simulation cases. Starting Position (m) Final Position (m) Speed Limit Value (km/h) 0 2092 100 2092 2739 86 2739 2949 100 2949 3719 84 3719 3800 100 Table 4. The partition of speed code data. Speed Code V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12 V13 Speed value 40 45 50 55 60 65 70 75 80 85 90 95 100 Table 5. Weight of the vehicles. Vehicle T1 (t) M1 (t) T2 (t) M2 (t) M3 (t) T3 (t) Total (t) Weight 33 35 28 35 35 33 199 Table 6. Parameters of MMAS. Parameters Names Parameters Values Iteration 160 The number of ants 30 The maximum of pheromone 9.99 The minimum of pheromone 0.01 The pheromone evaporation rate 0.05 The weighting factor of the iteration-best solutions 2.3 The weighting factors of the best-so-far solutions 2.1 The proportion of pheromone trails and the heuristic information 0.7 5.2. The First Stage Simulation Result By using the method proposed above, the speed profile is optimized based on the constraints of the running time. Energies 2016, 9, 626 12 of 18 Figure8 shows the first stage convergence analysis from the algorithm. In the problem solution space, with the increasing of iterations, the so far best energy and the iteration average energy decrease gradually and remain stable at about the 60th and 110th generation, respectively. After the first stage Energiesoptimization, 2016, 9, 626 a nearly globally optimal solution has been obtained. 12 of 18 Energies 2016, 9, 626 12 of 18 Figure 8. The first stage optimization iteration convergence. Figure 8. The first first stage optimization iteration convergence. Figure 9 illustrates that the algorithm obtains the first stage optimal speed profile. According to the gradientFigure 99 and illustratesillustrates the speed thatthat limit thethe algorithm algorithmdata, the interval obtainsobtains thehasthe firstfirstbeen stagestage discretized optimaloptimal into speedspeed 13 sections. profile.profile. According AccordingEach section toto hasthe gradient a constant and gradient the speed and limit speed data, limit the value. interval The train has been gains discretized traction when into starting 13 sections. and brakesEach section when approachinghas a constant a gradient station. In and the speed other limit sections, value. the The train train train uses gains gains different traction traction operation when starting modes. and brakes when approaching a station. In the other sections, the train uses different operation modes. Figure 9. The V‐S curve of the first stage optimization. Figure 9. TheThe V V-S‐S curve of the first first stage optimization. 5.3. The Second Stage Simulation Result 5.3. The Second Stage Simulation Result In this case, according to the practical train operation in the Changping Line, the planning runningIn thisthis time case, case, has according accordingbeen adjusted to to the the practicalto practical251 s. train Figure train operation 10 operation shows in the the in Changping secondthe Changping stage Line, optimization the Line, planning the planningiteration running convergence.runningtime has time been Based adjustedhas been on totheadjusted 251 first s. Figurestage to 251 optimization 10 s. shows Figure the 10 results, second shows MMAS stagethe second optimization searches stage for optimization iteration the optimal convergence. iterationsolution nearconvergence.Based the on first the firstBasedstage stage optimalon the optimization first solution. stage results,optimizationThe so MMASfar the results, searchesbest energyMMAS for theand searches optimal the iteration for solution the optimal average near solution theenergy first decreasenear the firstgradually stage optimaland remain solution. stable The at aboutso far thethe 5thbest and energy 60th and generation, the iteration respectively, average energywhich reflectsdecrease fast gradually convergence. and remain stable at about the 5th and 60th generation, respectively, which reflects fast convergence. Energies 2016, 9, 626 13 of 18 stage optimal solution. The so far the best energy and the iteration average energy decrease gradually andEnergies remain 2016, 9 stable, 626 at about the 5th and 60th generation, respectively, which reflects fast convergence.13 of 18 Energies 2016, 9, 626 13 of 18 Figure 10. The second stage optimization iteration convergence. FigureFigure 10. TheThe second second stage stage optimization optimization iteration convergence. Figure 11 describes the second stage optimal speed profile. Based on the first stage discretization, Figure 11 describes the second second stage stage optimal optimal speed speed profile. profile. Based Based on on the the first first stage stage discretization, discretization, the second stage discretizes the line into 21 sections. Compared with the first stage optimal speed thethe second stage discretizes the line into 21 sections. Compared Compared with with the the first first stage optimal speed profile, the second stage optimal speed profile has some differences in operation modes, but the profile,profile, thethe second second stage stage optimal optimal speed speed profile profile has has some some differences differences in operation in operation modes, modes, but the but general the general tendency is approximately the same. generaltendency tendency is approximately is approximately the same. the same. Figure 11. The V‐S curve of the second stage optimization. Figure 11. The V‐S curve of the second stage optimization. Figure 11. The V-S curve of the second stage optimization. TheThe simulationsimulation resultsresults don’tdon’t havehave tootoo manymany coastingcoasting sectionssections asas describeddescribed inin thethe literature,literature, becausebecauseThe the simulationthe datadata isis derived resultsderived don’t fromfrom have practicalpractical too many operationoperation coasting basedbased sections onon the asthe described BeijingBeijing subwaysubway in the literature, ChangpingChangping because line,line, andtheand data not not only isonly derived to to meet meet from the the practicalrequirements requirements operation of of running running based ontime time the interstation, interstation, Beijing subway but but also also Changping to to meet meet the line, the requirements requirements and not only oftoof the meetthe higherhigher the requirements technicaltechnical speed.speed. of running time interstation, but also to meet the requirements of the higher technicalTableTable speed. 77 showsshows thethe comparisoncomparison betweenbetween thethe resultresult ofof thethe firstfirst stagestage andand thethe secondsecond stage.stage. TheThe runningrunningTable timestimes7 shows areare 242.71242.71 the comparison ss andand 248.05248.05 between s,s, respectively,respectively, the result whichwhich of both theboth first satisfysatisfy stage thethe operation andoperation the second requirementsrequirements stage. ofTheof the the running timetable. timetable. times The The are second second 242.71 stage sstage and is is 248.05 online online s, optimization, respectively,optimization, which and and the the both algorithm algorithm satisfy the optimization optimization operation requirements time time is is 19.58 19.58 s,ofs, whilewhile the timetable. thethe dwelldwell The timetime second isis 3030 s.s. stage TheThe optimizationoptimization is online optimization, timetime isis smallersmaller and than thethan algorithm thethe dwelldwell optimization time,time, soso thethe energyenergy time is‐‐ efficientefficient speedspeed profileprofile forfor thethe nextnext interstationinterstation cancan bebe quicklyquickly obtainedobtained duringduring thethe dwelldwell time.time. ItIt indicatesindicates thatthat thethe algorithmalgorithm hashas thethe capabilitycapability ofof fastfast optimizationoptimization ofof thethe speedspeed profile.profile. Energies 2016, 9, 626 14 of 18 19.58 s, while the dwell time is 30 s. The optimization time is smaller than the dwell time, so the energy-efficient speed profile for the next interstation can be quickly obtained during the dwell time. It indicates that the algorithm has the capability of fast optimization of the speed profile. Energies 2016, 9, 626 14 of 18 Table 7. Comparison between the result of the first stage and the second stage. Table 7. Comparison between the result of the first stage and the second stage. Planning Running Running Energy Computing OptimizationOptimization StagePlanning Running Running Energy Consumption Computing Stage TimeTime (s) (s) TimeTime (s) (s) Consumption(kWh) (kWh) TimeTime (s) (s) The firstThe stage first stage 246 246242.71 242.71 18.8718.87 39.6739.67 The secondThe second stage stage 251 251248.05 248.05 18.6118.61 19.5819.58 InIn TableTable8 8,, the the simulation simulation results results and and practical practical data data are are compared compared to to verify verify the the effectiveness effectiveness of of thethe optimizationoptimization methodmethod usedused herein.herein. TheThe resultsresults illustrateillustrate thatthat thethe algorithmalgorithm cancan quicklyquickly obtainobtain thethe energy-efficientenergy‐efficient speedspeed profileprofile under under the the constraint constraint of of running running time. time. SignificantSignificant energyenergy savingsaving effectseffects areare achieved: achieved: thethe energyenergy savingsaving raterate is is 13.92%. 13.92%. TableTable 8. 8. ComparisonComparison between between simulation simulation results results and and practical practical data. data. Planning Running Running Energy Consumption Computing Planning Running Running Energy Computing ResultsResults TimeTime (s) (s) Time (s) (s) Consumption(kWh) (kWh) TimeTime (s) (s) PracticalPractical data data246 246 245.00245.00 21.6221.62 – – SimulationSimulation result result 251 251 248.05248.05 18.6118.61 19.58 19.58 6.6. Influence Influence of Different Factors on Simulation 6.1.6.1. ImprovementImprovement BasedBased onon PartitionPartition TheThe discretizationdiscretization ofof MMASMMAS influencesinfluences thethe convergenceconvergence raterate andand thethe energy-efficientenergy‐efficient raterate veryvery much.much. This paper paper discretized discretized the the line line based based on on gradients gradients and and speed speed limits. limits. To Toverify verify the theeffect effect of our of ouralgorithm algorithm a comparison a comparison between between equal equal discretization discretization algorithms algorithms (as shown (as shownin Figure in 12) Figure and unequal12) and unequaldiscretization discretization algorithms algorithms (as shown (as in shown Figure in9) Figureis made.9) The is made. difference The difference between the between two algorithms the two algorithmsis simply the is simplydiscretization the discretization method, other method, parameters other parameters remaining the remaining same in the order same to inensure order the to ensureeffectiveness the effectiveness of the comparison. of the comparison. FigureFigure 12.12. TheThe V-SV‐S curvecurve basedbased onon equal equal discretization discretization algorithms. algorithms. As shown in Figure 13, the equal discretization and unequal discretization so far best energy decrease gradually and keep stable at about 110th and 60th generation respectively. Unequal discretization can achieves better convergence and energy‐efficient performance. Energies 2016, 9, 626 15 of 18 As shown in Figure 13, the equal discretization and unequal discretization so far best energy decrease gradually and keep stable at about 110th and 60th generation respectively. UnequalEnergies 2016 discretization, 9, 626 can achieves better convergence and energy-efficient performance. 15 of 18 FigureFigure 13. 13.Equal Equal discretization discretization convergence convergence and and unequal unequal discretization discretization convergence. convergence. As shown in Table9 9,, while while the the practical practical running running time time is is 246 246 s, s, the the optimal optimal running running times times of of the the two algorithmsalgorithms are are 243.25 243.25 s and s and 242.71 242.71 s, which s, which both satisfyboth satisfy the operation the operation requirements requirements of the timetable. of the Thetimetable. energy The consumption energy consumption of the unequal of discretizationthe unequal discretization algorithm is 18.87 algorithm kWh, which is 18.87 is smaller kWh, which than the is energysmaller consumption than the energy of the consumption equal discretization of the equal algorithm. discretization Also, the algorithm. computation Also, time the of computation the unequal discretizationtime of the unequal algorithm discretization is much smaller algorithm than that is much of the smaller equal discretization than that of algorithm.the equal Thisdiscretization indicates thatalgorithm. the unequal This discretizationindicates that algorithm the unequal achieves discretization better real-time algorithm performance. achieves Therefore, better real unequal‐time discretizationperformance. Therefore, of the line isunequal shown discretization to be needed. of the line is shown to be needed. Table 9. Comparison between equal discretization and unequalunequal discretization.discretization. Planning Running Running Energy Consumption Computing Discretization Method Planning Running Running Energy Computing Discretization Method TimeTime (s) (s) TimeTime (s) (s) Consumption(kWh) (kWh) TimeTime (s) (s) Equal discretization 246 243.25 20.57 81.20 UnequalEqual discretization discretization 246 246 243.25242.71 20.5718.87 81.2039.67 Unequal discretization 246 242.71 18.87 39.67 6.2. Improvement Based on Driving Experience 6.2. Improvement Based on Driving Experience In this study heuristic pheromone values were set as shown in Table 10, which were derived In this study heuristic pheromone values were set as shown in Table 10, which were derived through study of driver experience: the train speed tends to accelerate when downhill and tends to through study of driver experience: the train speed tends to accelerate when downhill and tends to decelerate when uphill. decelerate when uphill. Table 10. Heuristic pheromone setting based on driving experience. Table 10. Heuristic pheromone setting based on driving experience. Heuristic Pheromone Weighting Value Gradient Value AccelerateHeuristic PheromoneCruise WeightingDecelerate Value Gradient Value 0.2 0.5 0.8 >0 Accelerate Cruise Decelerate =0 0.3 0.7 0.3 >0 <0 0.20.8 0.5 0.50.2 0.8 =0 0.3 0.7 0.3 <0 0.8 0.5 0.2 The heuristic pheromone influences the probability of searching‐ants search finding a path in the solution space and changes the search direction of the ants. In order to verify the effects of these heuristicThe heuristicpheromone pheromone settings, influencesa comparison the probabilitybetween two of heuristic searching-ants pheromone search settings finding ahas path been in themade, solution other heuristic space and parameters changes the remaining search direction the unchanged of the ants. in order In order to ensure to verify the effectiveness the effects of of these the comparison. The algorithm in this paper obtained the optimal speed profile with constant value heuristic pheromone setting (Figure 14) and driver experience (Figure 9). As shown in Figure 15, the constant value heuristic pheromone setting and with driving experience heuristic pheromone setting have different so far best energy convergence, decrease gradually and remain stable at about 80th and 60th generation respectively. Energies 2016, 9, 626 16 of 18 heuristic pheromone settings, a comparison between two heuristic pheromone settings has been made, other heuristic parameters remaining the unchanged in order to ensure the effectiveness of the comparison. The algorithm in this paper obtained the optimal speed profile with constant value heuristic pheromone setting (Figure 14) and driver experience (Figure9). As shown in Figure 15, the constant value heuristic pheromone setting and with driving experience heuristic pheromone setting have different so far best energy convergence, decrease gradually and remain stable at about 80th and 60th generationEnergies 2016, respectively. 9, 626 16 of 18 Energies 2016, 9, 626 16 of 18 Figure 14. The V‐S curve based on constant value pheromone setting. Figure 14. The V V-S‐S curve based on constant valuevalue pheromonepheromone setting.setting. Figure 15. Constant value convergence and driving experience convergence. Figure 15. Constant value convergence and driving experience convergence. Table 11 shows the energy consumption and computing time comparison between different Table 11 shows the energy consumption and computing time comparison between different heuristicTable pheromone 11 shows the settings. energy The consumption pheromone and settings computing based on time driving comparison experience between achieve different better heuristic pheromone settings. The pheromone settings based on driving experience achieve better heuristicreal‐time pheromone and energy settings.‐efficient The performance. pheromone Therefore, settings based our improved on driving method experience is shown achieve to better have real‐time and energy‐efficient performance. Therefore, our improved method is shown to have real-timesignificant and potential energy-efficient for improving performance. the computation Therefore, rate and our so improved aid in incorporating method is optimized shown to target have significant potential for improving the computation rate and so aid in incorporating optimized target significantspeed curves potential into real for‐time improving rail systems. the computation rate and so aid in incorporating optimized target speed curves into real‐time rail systems. speed curves into real-time rail systems. Table 11. Comparison between different heuristic pheromone settings. Table 11. Comparison between different heuristic pheromone settings. Heuristic Pheromone Energy Consumption (kWh) Computing Time (s) HeuristicConstant Pheromone value Energy Consumption19.54 (kWh) Computing67.53 Time (s) DrivingConstant experience value 19.5418.87 67.5339.67 Driving experience 18.87 39.67 7. Conclusions 7. Conclusions In CBTC systems, the center operators would change in real‐time the planned running time of interstationIn CBTC segments systems, according the center to operators whether wouldthe train change arrived in at real the‐time station the early planned or late, running so it is time a very of interstationimportant problemsegments to according consider to the whether real‐time the trainproperties arrived in at thethe stationprocess early of train or late, energy so it ‐isefficient a very importantoperation. Aimingproblem at to the consider problem the that real is abstracted‐time properties from a practicalin the process subway of line, train this energy paper ‐proposesefficient operation. Aiming at the problem that is abstracted from a practical subway line, this paper proposes Energies 2016, 9, 626 17 of 18 Table 11. Comparison between different heuristic pheromone settings. Heuristic Pheromone Energy Consumption (kWh) Computing Time (s) Constant value 19.54 67.53 Driving experience 18.87 39.67 7. Conclusions In CBTC systems, the center operators would change in real-time the planned running time of interstation segments according to whether the train arrived at the station early or late, so it is a very important problem to consider the real-time properties in the process of train energy-efficient operation. Aiming at the problem that is abstracted from a practical subway line, this paper proposes a discrete-combined model based on linear approximation calculations to optimize train energy-efficient operation. According to the static gradient and the speed limit, the line is unequally discretized in low density in the first stage and in high density in the second stage, and each stage has different running time restrictions. Based on the operation data from the Beijing subway Changping line, the simulation results shows that the energy consumption is reduced by 13.92% after two stage MMAS optimization. The second stage optimization time is 19.58 s, which shows good energy-efficient and real-time performance compared with the method used in previous researches. With the decrease of train headway in the CBTC system, research on the process of multi-train energy saving operation also needs to be considered in the real-time running time changes of each train, and this will be a topic in our future research. Acknowledgments: This work was funded by Beijing Municipal Science and Technology Plan Projects under Grant D151100005815001 and Z161100001016008, the Shenhua Group scientific research funding projects under Grant 20140269. The authors would also like to thank the reviewers for their corrections and helpful suggestions. Author Contributions: Youneng Huang mainly proposed the problem that is abstracted from practical subway lines and a discrete-combined model based on linear approximation calculations to optimize train energy-efficient operation. Chen Yang and Shaofeng Gong built up the simulation model and helped to program. All of the authors did the simulation analysis, experiment and results discussions, as well as contributed to the paper writing work. Conflicts of Interest: The authors declare no conflict of interest. References 1. Ishikawa, K. Application of optimization theory for bounded state variable problems to the operation of trains. Bull. JSME Nagoya Univ. 1968, 11, 857–865. [CrossRef] 2. Howlett, P. Optimal strategies for the control of a train. Automatica 1996, 32, 519–532. [CrossRef] 3. Cheng, J.X.; Howlett, P. A note on the calculation of optimal strategies for the minimization of fuel consumption in the control of trains. IEEE Trans. Autom. Control 1993, 38, 1730–1734. 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