Boundary Identification for Traction Energy Conservation Capability of Urban Rail Timetables

energies

Article Boundary Identiﬁcation for Traction Energy Conservation Capability of Urban Rail Timetables: A Case Study of the Beijing Batong Line

Jiang Liu 1,*, Tian-tian Li 2, Bai-gen Cai 3 and Jiao Zhang 4 1 School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China 2 Beijing Engineering Research Center of EMC and GNSS Technology for Rail Transportation, Beijing 100044, China; [email protected] 3 School of Computer and Information Technology, Beijing Jiaotong University, Beijing 100044, China; [email protected] 4 Vehicle and Equipment Technology Department, Beijing Subway Operation Technology Centre, Beijing 100044, China; [email protected] * Correspondence: [email protected]

Received: 14 February 2020; Accepted: 20 April 2020; Published: 24 April 2020

Abstract: Energy conservation is attracting more attention to achieve a reduced lifecycle system cost level while enabling environmentally friendly characteristics. Conventional research mainly concentrates on energy-saving speed profiles, where the energy level evaluation of the timetable is usually considered separately. This paper integrates the train driving control optimization and the timetable characteristics by analyzing the achievable tractive energy conservation performance and the corresponding boundaries. A calculation method for energy efficient driving control solution is proposed based on the Bacterial Foraging Optimization (BFO) strategy, which is utilized to carry out batch processing with timetable. A boundary identification solution is proposed to detect the range of energy conservation capability by considering the relationships with average interstation speed and the passenger volume condition. A case study is presented using practical data of Beijing Metro Batong Line and two timetable schemes. The results illustrate that the proposed optimized energy efficient driving control approach is capable of saving tractive energy in comparison with the conventional traction calculation-based train operation solution. With the proposed boundary identification method, the capability space of the energy conservation profiles with respect to the energy reduction and energy saving rate is revealed. Moreover, analyses and discussions on effects from different passenger load conditions are given to both the weekday and weekend timetables. Results of this paper may assist the decision making of rail operators and engineers by enhancing the cost effectiveness and energy efficiency.

Keywords: energy conservation; urban transit; train driving control; timetable; bacterial foraging optimization; boundary identiﬁcation; case study

1. Introduction In recent years, by providing safe, eﬃcient and convenient transport services to a large number of passengers in a short period of time, metro systems are playing a more important role in modern transportation systems all over the world [1]. It has been a common sense that urban rail transit systems have great advantages in enhancing the energy utilization eﬃciency over private cars and other public transportation modes, which enables a greener choice with higher environmental friendliness capabilities [2,3]. However, according to the fast development of the urban rail transit networks in many cities, the extended energy consumption reduction solution is becoming a big concern for

Energies 2020, 13, 2111; doi:10.3390/en13082111 www.mdpi.com/journal/energies Energies 2020, 13, 2111 2 of 25

bothEnergies the 2020 research, 13, x FOR and PEER rail REVIEW operation fields. Energy efficient operation solutions are attracting2 of more 25 attention to optimize the system cost in the whole lifecycle, reduce carbon emissions and realize public environmentalattracting more responsibilities. attention to optimize the system cost in the whole lifecycle, reduce carbon emissions andAmong realize public the di ffenvironmentalerent factors ofresponsibilities. energy consumption in the urban rail transit trains, including the trainAmong braking, the different lighting, factors aeration of energy and air consumption condition, the in the train urban traction rail transit plays trains, the most including significant the role,train which braking, reaches lighting, about aeration 52% of and the air total condition, energy the consumption train traction of theplays train the systemmost significant in Beijing role, (see Figurewhich1)[ reaches4]. A about collection 52% of ofthe research total energy works consumptio publishedn of in the the train 1990s system begun in Beijing to focus (see on Figure optimal 1) train[4]. A control collection methods of research [5–8]. works In recent published years, in great the e1990sfforts begun have beento focus made on optimal to develop train advanced control energy-emethodsffi cient[5–8]. traction In recent methods, years, great which efforts calculate have and been suggest made to energy-e developffi cientadvanced driving energy-efficient strategies that aretraction feasible methods, with the which permitted calculate trip and time suggest schedule ener andgy-efficient track conditions, driving strategies and do notthat requireare feasible extra railwaywith the infrastructure permitted investments.trip time schedule Many existingand trac worksk conditions, in the literature and do concentrate not require on extra the calculation railway ofinfrastructure energy-efficient investments. train driving Many control existing strategies works toin etheffectively literature reduce concentrate tractive on energy the calculation consumption of underenergy-efficient specific constraints, train driving including control the strategies traveling to time, effectively running reduce distance tractive within energy adjacent consumption rail stations, trainunder traction specific/braking constraints, force including characteristics, the traveling train parameters time, running and distance the passenger within adjacent distribution rail stations, features, etc.train The traction/braking most typical solution force characteristics, for solving the train condition-constrained parameters and the tractive passenger energy distribution saving problem features, is anetc. intelligent The most optimization typical solution approach, for solving such the as genetic condition-constrained algorithm [9], particle tractive swarm energy optimization saving problem [10 ], simulatedis an intelligent annealing optimization [11], differential approach, evolution such [as12 ],genetic artificial algorithm bee colony [9], method particle [ 13swarm], etc. optimization Some of these methods[10], simulated have been annealing validated [11], with differ fieldential experiences evolution in[12], design artificial and b optimizationee colony method of the [13], train’s etc. driving Some of these methods have been validated with field experiences in design and optimization of the train’s curves for automatic train operation [14,15]. In addition, efforts also have been made to the analysis driving curves for automatic train operation [14,15]. In addition, efforts also have been made to the of the correlation between the energy efficient control and specific issues. Zhang et al. [16] proposed analysis of the correlation between the energy efficient control and specific issues. Zhang et al. [16] the Comprehensive Evaluation Index (CEI) to analyze integrated optimization strategies of urban proposed the Comprehensive Evaluation Index (CEI) to analyze integrated optimization strategies of rail transport by calculating the traction energy cost and the technical operation time. Lin et al. [17] urban rail transport by calculating the traction energy cost and the technical operation time. Lin et al. realized the optimization of the train dwell time with a multiple train operation model for an energy [17] realized the optimization of the train dwell time with a multiple train operation model for an conservation target. Tian et al. [18] proposed a multi-train traction power network modelling method to energy conservation target. Tian et al. [18] proposed a multi-train traction power network modelling determine the system energy flow of the rail system with regenerating braking trains. Hamid et al. [19] method to determine the system energy flow of the rail system with regenerating braking trains. exploredHamid et how al. errors[19] explored in train how position errors data in atrainffect po thesition overall data energy affect consumptionthe overall energy under consumption a single train trajectoryunder a single optimization train trajectory scheme. optimization Gomes et al.scheme. [20] evaluated Gomes et theal. [20] potentials evaluated of thethe energypotentials effi ofciency the fromenergy “fixed efficiency block” tofrom the “fixed “moving block” block” to the type “mov usinging the block” empirical type datausing and the the empirical statistical data approaches. and the Zhoustatistical et al. [approaches.21] take into Zhou account et al. multiple [21] take issues into account in the energymultiple minimization issues in the ofenergy the traction minimization system, includingof the traction train headway,system, including interstation train runtime,headway, coasting interstation on downhillruntime, coasting slopes, and on downhill the variations slopes, of trainand massthe variations due to the of change train mass of on-board due to the passengers. change of Yangon-board et al. passengers. [22] further Ya investigatedng et al. [22] the further effect frominvestigated multi-phase the effect speed from limits mult in optimizingi-phase speed the limits total in energy optimizing consumption the total ofenergy a metro consumption line using theof optimizationa metro line techniques.using the optimization techniques.

Figure 1. Constitution of energy consumption for the whole urban rail transit system and the rail Figure 1. Constitution of energy consumption for the whole urban rail transit system and the rail vehicle system in the Beijing Metro system. vehicle system in the Beijing Metro system. Different from the abovementioned solutions for optimized driving control, the timetable, which Different from the abovementioned solutions for optimized driving control, the timetable, which describes the temporal-spatial constraints to the planned trains by defining the running time between describes the temporal-spatial constraints to the planned trains by defining the running time between two stations, has a direct and significant influence on the energy-efficiency of train operations. In recent two stations, has a direct and significant influence on the energy-efficiency of train operations. In years, the topic of energy-efficient timetabling has drawn the attention of researchers [23]. Lv et al. [24] recent years, the topic of energy-efficient timetabling has drawn the attention of researchers [23]. Lv et al. [24] proposed an energy-efficient timetable optimization method using a Mixed-Integer Non- Linear Programming (MINLP) model. Feng et al. [25] developed a simulation- based approach to

Energies 2020, 13, 2111 3 of 25 proposed an energy-efficient timetable optimization method using a Mixed-Integer Non-Linear Programming (MINLP) model. Feng et al. [25] developed a simulation- based approach to optimize both the timetable and control strategy considering the time margin that is caused by dynamically uncertain passenger demands in off-peak hours. Gupta et al. [26] proposed a novel two-step linear optimization model for energy-efficient metro timetables and realized a significant reduction in the energy consumption (the worst case being 19.27%). Canca et al. [27] formulated the timetable design problem as a passenger load-dependent optimization problem and presented an effective solution using the sequential MILP algorithm. Scheepmaker et al. [28] designed a method to solve the energy efficient operation problem taking into account the desired robustness and the optimal distribution of the running time supplements. Nitisiri et al. [29] developed a parallel multi-objective evolutionary algorithm with hybrid sampling and learning-based mutation to solve the train scheduling problem. To further investigate the rail timetable rescheduling problem, Yang et al. [30] introduced the Deep Deterministic Policy Gradient (DDPG) algorithm over a conventional Q-learning strategy to realize the energy-aimed rescheduling by adjusting the cruising speed and dwelling time continuously. It can be found that energy consumption level of a given timetable is actually an inherent attribute. As a secondary timetable quality indicator, it can be adjusted through a fine-tuning operation to the basic timetable solution with certain time allowances. Therefore, the quantitative evaluation of the traction energy conservation capability would be a decisive issue to determine the direction and the concrete strategy to adjust the timetable in the fine-tuning process. Given a real-world timetable, the individual driving control strategy of each planned train can be deduced with the practical field conditions and parameters, where an appropriate traction energy consumption reduction solution can be directly utilized. However, the knowledge about energy conservation results obtained from the train-level operation deduction is not sufficient to enable the fine-tuning of a candidate timetable scheme. It is of great value to illustrate the correlation between the achievable energy conservation capability and the specific energy consumption influencing factors, e.g., the interstation speed, train operation strategy, passenger load rate, train formation and train routing scheme. There is a lack of the correlation analysis for the achievable traction energy conservation capability with a specific rail timetable scheme. Furthermore, a rapid development of advanced swarm intelligence optimization technology in recent years enables great opportunities to solve the energy conservation problems in rail transport. The Bacterial Foraging Optimization (BFO) algorithm is a new swarm intelligence optimization technique that possesses a series of advantages including insensitivity to initial values and parameter selection, strong robustness, simplicity, ease of implementation, parallel processing and global search [31]. BFO has been applied in a wide range of optimization problems such as the energy forecasting [32], expert energy management considering the uncertainty [33], the imbalanced data classification [34], and robotic cell scheduling [35]. The advantages of BFO against conventional swarm intelligence approaches greatly encourages us to solve the problems in enhancing the train driving control profiles and evaluate the energy saving space of the train scheduling schemes. In 2017, a National Key R&D Program was launched to evaluate and demonstrate novel energy- efficiency enhancing techniques in Batong Line of Beijing Metro. With a rail operation organization perspective, the traction energy conservation capability of the real-world timetables is evaluated. This paper provides a case study of Beijing Batong Line. The contribution of this paper is to describe the correlation between the timetable-based tractive energy conservation capability and typical influencing factors. An optimization-based solution of energy efficient train driving control is introduced, and the correlation between energy-efficient-driving-enabled energy saving capability and some timetable-related influencing factors (i.e. running speed and passenger load rate) covering all the planned trains is revealed through data fitting models and boundary identification. The test results can give the metro operators and decision makers the overview reference information to understand the optimization space of specific timetable schemes, which would help them to adjust and update the operation plans with the energy-efficiency improving purpose. Energies 2020, 13, 2111 4 of 25

EnergiesThe 2020 rest, 13 of, x thisFOR PEER paper REVIEW is organized as follows: In Section2, the energy e fficient control method4 of 25 and the corresponding calculating solution are presented. In Section3, the principle and procedures of the2. Section correlation 4 shows boundary a case study identification in Beijing isBatong described Line using by using two thetypical strategies timetable introduced schemes in in practical Section2 . Sectionoperation.4 shows A short a case discussion study in and Beijing the planned Batong Linefuture using works two are typical provided timetable in the final schemes Section in practical 5. operation. A short discussion and the planned future works are provided in the final Section5. 2. Energy-efficient Train Driving Control Method 2. Energy-eIn orderfficient to evaluate Train Driving the possible Control energy Method consumption level of the train traction system in field operationIn order according to evaluate to the the time possible schedule, energy the consumptionkey issue is to level estimate of the the train train traction operation system profiles in field in operationpractical operations, according towhich the time will help schedule, us to find the keyout the issue train is todriving estimate control the characteristics train operation (see profiles Figure in practical2) and analyze operations, the possible which will energy help consumption us to find out by the the train trac drivingtion system control quantitatively. characteristics (see Figure2) and analyze the possible energy consumption by the traction system quantitatively.

Distance

Station B

Braking

Coasting Speed limit Speed

Cruising

Motoring Current time Current

Station A Running speed Time

Figure 2. Typical train movement modes and the relationship with the train plan. Figure 2. Typical train movement modes and the relationship with the train plan. The train kinematic modeling is the first step to derive the train operation profiles. As shown in The train kinematic modeling is the first step to derive the train operation profiles. As shown in Figure2, based on the general constraint from the time schedule, the practical moving operation of the Figure 2, based on the general constraint from the time schedule, the practical moving operation of train usually can be modeled using four typical movement modes. The driving status and the force the train usually can be modeled using four typical movement modes. The driving status and the conditionforce condition under specificunder specific movement movement mode canmode be can determined be determined accordingly. accordingly. Based Based on Newton’s on Newton’s second lawsecond of motion, law of themotion, force the for force a moving for a moving train can train be written can be written as the following as the following form: form:    dvt dv  M dt ⋅==t F(vt) W −(vt) −Rg ·M Fv()−tt Wv ()− Rg (1)  dst = dtv  dt t (1) ds  t = v where M is the train mass, st and νt denote dt thet time-varying running distance and the speed of the train at time t, F(νt) and W(νt) indicate the traction force and resistance with respect to the running where M is the train mass, st and νt denote the time-varying running distance and the speed of the speed, Rg is the force due to the gradient. When a train is operating in a specific movement mode, train at time t, F(νt) and W(νt) indicate the traction force and resistance with respect to the running the force analysis can be performed accordingly based on the kinematic model in Equation (1), which speed, Rg is the force due to the gradient. When a train is operating in a specific movement mode, the can be summarized as follows: force analysis can be performed accordingly based on the kinematic model in Equation (1), which can be summarized as follows: 1) Motoring The forward traction power is applied to accelerate the train against the resistance force, which raises the train speed until it reaches the required level to enable the following cruising operation.

Energies 2020, 13, 2111 5 of 25

(1) Motoring The forward traction power is applied to accelerate the train against the resistance force, which raises the train speed until it reaches the required level to enable the following cruising operation. Therefore, all the elements in Equation (1), including the traction force F(νt), resistance W(νt) and the gradient force Rg, are involved in the description of the kinematics. (2) Cruising It is expected that the train keeps a constant running speed along the track. Therefore, traction force is provided as the sum of the resistance and gradient force, which means F(νt) = W(νt) + Rg. Due to the physical limitations of the traction system, a constant speed target cannot always be realized under time-varying moving conditions, and thus a fluctuation range (e.g., 2 km/h) for the cruising ± speed is suggested in field implementation. (3) Coasting The forward traction power is switched off in the coasting mode. The train will decelerate with the . resistance and the gradient force, which means M νt = W(νt) Rg. Under the constraints of section · − − length sAB and the operation time TB from the timetable, the sufficient utilization of the coasting mode is one major approach to reduce the tractive energy since the movement of the train relies more on the inertia effect without consuming traction power. (4) Braking

The backward braking force B(νt) will replace the forward traction force to decelerate the train, . which means the Equation (1) will be written as M νt = B(νt) W(νt) Rg. · − − It can be found that there would be different integration solutions of the above listed operation modes. As shown in Figure2, the black “speed-distance” curve represents an alternative which indicates a lower ratio of the coasting mode, where the temporal-spatial constraints by the timetable also can be fulfilled. In order to evaluate the energy conservation space and identify the boundaries, it is required to find specified energy efficient driving control solutions for all the planned trains in the timetable. Therefore, a calculation approach that is sufficiently effective in optimization and also computationally inexpensive is important for batch processing of a large amount of train plans.

2.1. BFO Algorithm The Bacterial Foraging Optimization algorithm was proposed in [36] based on the foraging behavior of E. coli bacteria. The natural principle of the survival of the fittest illustrates that the natural selection will favor the bacteria with successful foraging strategies. The BFO algorithm was developed to realize the optimization capability by eliminating the bacteria with the poor fitness or reshaping them into good ones after a number of generations [37]. By simulating basic movements of the bacteria, including the swimming and tumbling, the foraging behavior of the E. coli can be utilized to establish the BFO architecture. The computational intelligence technique used in BFO is not affected by non-linearity and volume of the population. In the last few years, the BFO algorithm has been successfully applied to solve real-world engineering optimization problems [38]. It has been pointed out that the BFO algorithm can reduce the global convergence, computational burden, time and also handle a number of objective functions [39]. Comparisons with the conventional intelligent evolutionary computation algorithms, such as the Genetic Algorithm (GA) and Particle Swarm Optimization (PSO), have shown that BFO can realize a better optimization performance in many practical problems and real applications [40–42]. Considering the advantages of BFO over the conventional measures, it is adopted in this paper to evaluate the energy-efficient capability of the timetables. The standard BFO architecture consists of four typical operations as follows [43]: Energies 2020, 13, 2111 6 of 25

(1) Chemotaxis The swimming and tumbling movements are concerned in researching the behavior of bacteria. The swimming movement indicates moving within a pre-defined direction and tumbling means different directions can be selected during their entire lifetime. In the chemotaxis process, bacteria will perceive the level of nutrient content in the surrounding environment and thus guide the chemotaxis behavior. The generated unit length by tumbling will be used to define the direction of movement, which can be written as:

i i ∆(i) θ (j + 1, k, m) = θ (j, k, m) + C(i, j) p (2) ∆T(i)∆(i) where θi(j, k, m) indicates the population of bacteria, the location of the i-th bacterium at the j-th chemotactic step, in the k-th reproduction step and m-th elimination and dispersal event, C(i,j) is the step size in a random direction, and ∆ depicts a vector in the random direction whose elements lie in ( 1,1). − (2) Swarming When E. coli bacteria find a nutrient rich environment, bacterial interactions with each other will encourage the accumulative effect at the high bacterial density area. By considering the swarming effect, the cost function J(i, j, k, m), which is also known as the nutrient function, will be denoted by:

i JSW(i, j, k, m) = J(i, j, k, m) + JCC(θ (j, k, m)) (3)

i where JCC(θ (j, k, m)) is the cost function value to be added to the actual cost to be minimized to present a time varying cost function. (3) Reproduction In the reproduction process, the selected healthiest bacteria will live and the least healthy will die. According to the fitness value indicating whether an area is suitable for the bacteria survival, the bacteria in the place where the nutrients are rich will self-replicate, and they will be placed in the same location to keep a constant bacteria population. (4) Elimination and dispersal Gradual or sudden environmental changes for bacteria may happen due to different reasons. In order to prevent the premature or the local optimal solutions in the chemotaxis, the elimination and dispersal are necessary in the bacterial foraging. As the population-level long-term behavior, the elimination and dispersal events have possible positive effect on assisting the chemotaxis by placing the bacteria near the nutritious areas. The detailed theoretical derivations of the procedures for searching the global optimal solution can be found in [36,44].

2.2. Calculation of Energy-efficient Driving Control Solution The track section between two rail stations is the fundamental unit for calculating the energy consumption level. It is usually divided into several sub-sections in generating the train operation profiles when considering the variation of track gradient and speed limit. Different from a detailed modeling to the traction control that is useful in analyzing the microscopic behaviors of the traction system [45], all the types of driving control phases are considered during the whole trip in a section. For a sub-section, the suggested driving control solution consists of accelerating, cruising, coasting and braking according to the abovementioned kinematic model. In order to find an energy efficient n o train driving control solution, a state sequence is defined as vINT,n, vCRU,n, vBRK,n, vEND,n for the nth sub-section, where vINT,n demotes the initial train speed in a sub-section, vCRU,n represents the target cruising speed, vBRK,n is the initial speed in the braking phase which is lower than vCRU,n and Energies 2020, 13, 2111 7 of 25

represents the existence of the coasting operation, vEND,n is the final speed at the end of the sub-section. The values of the components in the state sequence are based on the speed limit of each sub-section and its neighborhoods. It means that the value space of a state sequence has to be collaboratively determined by the probable driving phases of its neighborhood sub-section(s), and that would be complicated under the long section cases with multiple changing points for the speed limit. The value of this state sequence directly determines the train operation profile, which provides a quantitative approach of carrying out the fine searching with the energy saving purpose. By using the BFO algorithm, the energy efficient traction solution can be derived by an optimal train state sequences for all the sections. With the definition of the state sequence, it can be found that the multiple state sequences are taken as the variables need to be optimized for a planned train. The number of the variables is determined by the number of rail sections. With the derived results of state sequence, the speed values enable us to determine the control points for every control phase and draw the speed curves. The fitness of the bacterium is defined as the consumed traction energy by using the corresponding state sequence-based train speed profile. The amount of traction energy can be written as: η Z XΩ E = F(vt) vtdτ = e (t) (4) TRA · TRA = 0 t 1 where η denotes the total journey time according to the derived state sequence, which should not exceed the planned time cost, eTRA,t is the traction energy consumption in a time interval, and Ω represents the total number of time steps within the operation period of η. It can be found that the energy model depicted by Equation (4) only considers the traction and cruising operations. To derive the energy-conservation-oriented operation profiles and calculate consumed tractive energy, the BFO-enabled procedures can be described in Figure3. It has to be noticed that a simplified tractive energy calculation model is adopted in the presented solution compared to those consider detailed locomotive traction and train dynamics as [46,47]. Utilization of a detailed modeling solution will enhance the resolution of evaluation by integrating more realistic behaviors of the trains. Considering the fact that rail sections in the target line are not very long, we can use a relatively simple energy calculation model to guarantee the calculation efficiency of a timetable-based batch processing method. The model resolution enhancement for the energy consumption computation will be considered in our future research on integrated design of rail dispatching and driving control. Figure3 illustrates the framework of the energy e fficient driving control solution calculation method using the BFO strategy. As shown in Figure3, the presented method mainly consists of two parts, including the offline database collection and online optimization. In the offline database collection process, information about the train, rail track and the timetable schemes is required to enable the following online calculation, and the correctness and accuracy of the database are significant to the creditability of the BFO-based results. Based on that, the online optimization will be carried out by performing the speed profile prediction and energy-saving-oriented optimization. By utilizing a population of the bacteria, which indicate a group of train state sequences for each section, the BFO operation will be carried out according to the abovementioned procedures. With the optimization result, the optimal energy efficient speed profile is derived and that is able to enable the analysis of the energy conservation characteristics. It has to be noticed that the calculation of energy efficient traction solutions as Figure3 and the following boundary evaluation is based on the following assumptions according to the real-world operation properties:

(1) We consider the temporal constraints from the timetable by utilizing the indicated arrival and departure times at each station. The speed proﬁle of each planned train is determined by BFO-based calculation under the safety and running time limitations. Energies 2020, 13, x FOR PEER REVIEW 8 of 25

4) The passenger load condition is considered as a constant, where several average passenger load values have been evaluated in this research according to the statistical result in real operations.

To solve the energy-saving optimization model minETRA subject to the motion equations in (1), some constraints have been considered as follows: ≤≤ 0 sst AB (5)

== vvs(0) (t ) 0 (6) Energies 2020, 13, 2111 8 of 25 ≤≤ 0()vstt vlim () s (7) (2) To simplify the solution procedure, a similar assumption as [48] for the train’s driving phase  ≤ sequence in each rail section has been made,||vat wheremax four sequential phases are involved as shown(8) in Figure2, including motoring, cruising, coasting and braking. −≤Δ* (3) Some parameters in the energy consumption||tTAB model AB and Ta proﬁle calculation, such as train’s weight,(9) acceleration and braking characteristics, and the resistance, are considered as the constant values where s denotes the section length between the two adjacent stations A and B, vs() indicates accordingAB to practical situations. t (4)theThe running passenger speed load at the condition locationis s consideredt , vslim ()t as is athe constant, speed limit, where a severalmax represents average the passenger maximum load values have been evaluated in this research according* to the statistical result in real operations. value of the acceleration or deceleration, tAB and TAB denote the derived and the scheduled trip Δ times, and Ta is the acceptable limit to the trip time error.

FigureFigure 3. 3.Flowchart Flowchart of of thethe energy-eenergy-efficientﬃcient trac tractiontion solution solution calculation calculation method. method.

To solve the energy-saving optimization model minETRA subject to the motion equations in (1), some constraints have been considered as follows:

0 st s (5) ≤ ≤ AB

v(0) = v(st) = 0 (6)

0 v(st) v (st) (7) ≤ ≤ lim . vt amax (8) ≤

t T∗ ∆Ta (9) AB − AB ≤ Energies 2020, 13, 2111 9 of 25

where sAB denotes the section length between the two adjacent stations A and B, v(st) indicates the running speed at the location st, vlim(st) is the speed limit, amax represents the maximum value of the acceleration or deceleration, tAB and TAB∗ denote the derived and the scheduled trip times, and ∆Ta is the acceptable limit to the trip time error. Energies 2020, 13, x FOR PEER REVIEW 9 of 25

3. Boundary3. Boundary Identiﬁcation Identification Method Method for for Timetable Timetable

3.1. Calculation3.1. Calculation of Energy of Energy Conservation Conservation Profiles Profiles forfor MultipleMultiple Planned Planned Trains Trains WithWith the presentedthe presented BFO-based BFO-based calculation calculation solutionsolution for for rail rail sections, sections, it is it possible is possible to evaluate to evaluate the the energy-consumptionenergy-consumption features features of of all all the the planned planned trains in in the the timetable. timetable. This This cancan be regarded be regarded as an as an extensionextension of the of trainthe train identity identity dimension. dimension. ByBy integratingintegrating time time schedule schedule data data for formultiple multiple trains trains that that are plannedare planned with with the the similar similar movement movement in in the the samesame transit line, line, the the same same parameters parameters are adopted are adopted to to performperform the BFO-based the BFO-based calculation. calculation. Some Some differences differences among among different different numbered numbered trains trains in the in timetable the will betimetable considered will be in considered the calculation, in the calculation, including in stationcluding stopstation schedule stop schedu plans,le plans, running running speeds speeds within within the sections and the passenger flow distribution conditions. These factors are also regarded as the sections and the passenger flow distribution conditions. These factors are also regarded as the the independent variables in deriving the energy conservation profiles and identifying independentcorresponding variables boundaries in deriving of the the traction energy energy conservation conservation profiles space and with identifying respect to correspondingthe given boundariestimetable. of the traction energy conservation space with respect to the given timetable. FigureFigure4 shows 4 shows the the flowchart flowchart of of the the energy energy conservation conservation profile profile calculatio calculationn method. method. The Thedata data of of train,train, track track and timeand time schedules schedules of multiple of multiple planned planned trains trains are are taken taken as as the the inputs. inputs. With With a a specific specific input and parameterinput and parameter configuration, configuration, the unique the unique energy energy efficient efficient operation operation solution solution for eachfor each train train plan plan can be obtained,can be with obtained, which with the which optimized the optimized train movement train movement can becan described be described by by speed-distance speed-distance curve and speed-timeand speed-time curve respectively. curve respectively. For the Fo evaluationr the evaluation purpose, purpose, reference reference train train operation operation profilesprofiles are also computedalso computed in parallel. in parallel. The fully The consistent fully consistent assumptions assumptions and constraints and constraints for derivingfor deriving the the BFO-enabled BFO- enabled solution as stated in Section 2.2 are adopted in the calculation of reference profiles. The solution as stated in Section 2.2 are adopted in the calculation of reference profiles. The coasting phase coasting phase is also involved in the derived profiles according to the practice of driving control in is alsothe involved target line. in A the fixed derived deviation profiles limit accordingto the cruisi tong thespeed practice and initial of driving speed of control the braking in the phase target is line. A fixedadopted deviation to derive limit the to distribution the cruising of speed the driving and initial phases. speed The major of the difference braking phasebetween is the adopted optimized to derive the distributionand the reference of the solution driving is phases.the utilization The major of the diiterativefference optimization between the logic. optimized The reference and thesolution reference solutiononly is uses the utilizationthe basic train of the traction iterative calculation optimization to generate logic. the The driving reference control solution profiles only without uses the an basic trainoptimization traction calculation process. to generate the driving control profiles without an optimization process.

FigureFigure 4. Flowchart4. Flowchart of of the the energy energy conservation profile proﬁle generation generation method. method.

Energies 2020, 13, 2111 10 of 25

When we use ETRA,α,β to represent the BFO-derived tractive energy consumption in the β-th section for the α-th train plan, the reduced energy consumption ξα,β against the reference proﬁle is:

Ref ΩX XΩ ξ = ERef E = eRef (t) e (t) (10) α,β TRA,α,β − TRA,α,β TRA,α,β − TRA,α,β t = 1 t = 1 where the superscript ‘Ref’ corresponds to the reference solution-based result. Based on that, the energy saving rate for each train at all the sections can be obtained by (11):

P Ref P Ω eRef (t) Ω e (t) ξα,β t = 1 TRA,α,β t = 1 TRA,α,β µ = 100% = − 100% (11) α,β Ref P Ref E × Ω eRef (t) × TRA,α,β t = 1 TRA,α,β

Thus, the saved energy ξα,β and energy saving rate µα,β can be applied to generate energy conservation profiles. These profiles can realize the description of the correlation with independent variables, especially the interstation speed which fully reflects the dynamic characteristic of the trains under the timetable scheme. The acquirable energy conservation profiles are summarized as follows: (1) ASS-REN-PLF profile It describes the relationship between Average Section Speed (ASS) and Reduced Energy (REN) against the reference solution, under a specific Passenger Load Factor (PLF). The profile is represented n o by a series feature points PARnP(ξα,β, vα,β φα,β) , where vα,β indicates the average speed in the β-th section for the α-th train plan, and φα,β represents the amount of passengers per train. (2) ASS-RER-PLF profile Similar to the ASS-REN-PLF profile, the ASS-RER-PLF profile describes the relationship between the average interstation speed and the Reduced Energy Rate (RER) against a reference solution. Thus, n o a feature point in this profile can be represented by PARrP(µα,β, vα,β φα,β) .

3.2. Boundary Identification Method With the knowledge of the energy conservation profiles, the curve fitting approach is capable of establishing the numerical models of different factors. Besides that, it would be necessary to identify the upper and lower boundaries with the obtained feature point sets. The boundaries, especially the upper bound, bring the users the reference information to determine the way to evaluate the timetable schemes. In some existing applications, several strategies have been developed to realize an analytical determination of the boundaries of a given data set, e.g., the simple polynomial-based boundary fitting and the adaptive spline-based smooth boundary fitting. However, the existing examples aim to reach an analytical form of the “smooth” boundary by tolerating the residual error between the fitted values and the truth, which may not be acceptable for the capability evaluation case concerned in this paper. In order to derive tight boundaries for the acquired energy conservation profiles, a specific boundary identification strategy is proposed in this research using a piecewise detection strategy. Taking the ASS-REN-PLF profile as an example, the piecewise detection is carried out in a series of speed intervals with a fixed piece width δS. That means the whole value space of the interstation speed is divided into a number of pieces, and boundary identification will be carried out piecewisely. The value of δS illustrates the resolution of the derived boundaries. The principle of identifying the upper and lower boundaries are described in Figure5. Energies 2020, 13, 2111 11 of 25 Energies 2020, 13, x FOR PEER REVIEW 11 of 25

[vq + 4] δS δS [vq + 5] UB(vq + 4)UB(vq + 5) Boundary relaxation … … … … Reduced energy (kw·h) energy Reduced (kw·h) energy Reduced

~ ~ LB(vq + 4)LB(vq + 5) Speed (km/h) Speed

(a) (b)

FigureFigure 5. 5.Principle Principle ofof boundary boundary identi identiﬁcationfication method method andand the boundary the boundary relaxation relaxation strategy. strategy. (a) (a) Boundary identiﬁcation;Boundary (b) Boundary identification; relaxation (b) Boundary strategy. relaxation strategy. n o As shown in Figure5, in the j-th speed section, there would be multiple feature points PARnP,i As shown in Figure 5, in the j-th speed section, there would be multiple feature points {}P j ARnP,i j describing the relationship between the average interstation speed and the reduced traction energy. describing the relationship between the average interstation speed and the reduced traction energy. By extracting the maximum and minimum values, the boundaries will be: By extracting the maximum and minimum values, the boundaries will be:

Nj Nnj o u (v ) = P j = y B,j j ==max ARnP,i , 1, , ul (12)(12) uvB,jj()i = max1 {} P ARnP, ij , j 1,,··· y ul i=1 j

Nj n o N lB,j(vj) = min j PARnP,i , j = 1, , yul (13) ==i = 1 j ··· (13) lvB,jj() min{} P ARnP, i , j 1,, y ul i=1 j where uB,j(vj) and lB,j(vj) are the upper and lower bounds, vj represents the speed value of the boundaries in the j-th sections, N indicates the number of the feature points fall in the j-th sections, where uvB, j ()j and lvB, j ()j arej the upper and lower bounds, v j represents the speed value of the and yul denotes the total number of the divided sections. boundaries in the j-th sections, N j indicates the number of the feature points fall in the j-th sections, It can be found that a small value for δS is strongly suggested since a higher resolution makesand y itul possible denotes the to total track number the characteristics of the divided sections. of the boundaries precisely. To avoid possible acute or frequent changes in the derivedδ boundaries, a further boundary relaxation strategy It can be found that a small value for S is strongly suggested since a higher resolution makes is proposed to refine the original results by enhancing their smoothness. For a certain speed it possible to track the characteristics of the boundaries precisely. To avoid possible acute or frequent piece, a sliding window with a fixed width (2h + 1) is adopted to find a candidate boundary changes in the derived boundaries, a further boundary relaxation strategy is proposed to refine the ensemble and extract the representative characteristics of the boundaries. Thus, the upper original results by enhancing their smoothness. For a certain speed piece, a sliding window with a or lower bound with respect to a specific vj would not be determined only by the feature fixed width (2ℎ + 1) is adopted to find a candidate boundary nensemble and extracto the pointsrepresentative in the q- characteristicsth piece. Instead, of the boundaries. a set of candidate Thus, the boundariesupper or lower(uB, boundc(vc), l B,withc(vc ))respectare to used a to identify the refined boundary values, where c [j h 1, j + h]. As seen in the specific v j would not be determined onlyn by the feature ∈ points − in −the q-th piece. Instead, a set of o example in Figure5, the candidate sets uB,q+1 vq+1 , lB,q+1 vq+1 , , uB,q+7 vq+7 , lB,q+7 vq+7 n o ··· candidate boundaries {(uvlvB,cc ( ), B, cc ( ))} are used to identify the refined boundary values, where and uB q+ vq+ , lB q+ vq+ , , uB q+ vq+ , lB q+ vq+ are collected to generate the refined , 2 2 , 2 2 ··· , 8 8 , 8 8 results∈[−ℎ−1,+ℎ] for the (q + 4).-th As and theseen( q +in 5)-ththe speed example section. in Therefore, Figure the5, finalthe outputcandidate of the boundariessets are: {(, ( ), , ( )), ⋯ , (, ( ), , ( ))} and j + h n o {(, ( ), , ( )), ⋯ , (, ( ), , ( ))} are collected to generate the UBj(vj) = max uB,c(vc) , c [h + 1, yul h] (14) ( + 4) c = j(h + 5) ∈ − refined results for the -th and the − -th speed section. Therefore, the final output of the boundaries are: j + h n o LBj(vj) = minjh+ lB,c(vc) , c [h + 1, yul h] (15) c = j h ∈ − UB (vuvchyh )=∈+− max− {} ( ) , [ 1, ] (14) jj=− B, cc ul The procedures of the proposed boundarycjh identification method are summarized in Table1.

jh+ LB (vlvchyh )=∈+− min{} ( ) , [ 1, ] (15) jjcjh=− B, cc ul

The procedures of the proposed boundary identification method are summarized in Table 1.

Energies 2020, 13, 2111 12 of 25

Table 1. Procedures of the boundary identiﬁcation method.

Boundary Identification of Energy Conservation Profiles for a Given Timetable. Step 1: Collect the energy conservation profiles with different feature point sets for a given timetable, including n o n o PARnP(ξα,β, vα,β φα,β) and PARrP(µα,β, vα,β φα,β) . Step 2: Divide the speed value space into several pieces, such that the length of each piece is δS or δT. Step 3: Determine the piecewise upper and lower bounds for each energy conservation profile.

Nj n o uB,j(vj) = max PPro,i , Pro ARnP, ARrP i = 1 j ∈ { } N j n o lB,j(vj) = min PPro,i , Pro ARnP, ARrP i = 1 j ∈ { } Step 4: Initialize the reﬁned boundaries according to the width of the sliding window. UBj(vj) = u j(vj), j 1, , h, y h + 1, , y B, ∈ ··· ul − ··· ul LB (v ) = l (v ), j 1, , h, y h + 1, , y j j B,j j ∈ ··· ul − ··· ul Step 5: Calculate the reﬁned boundaries using the boundary relaxation strategy. j+h n o UBj(vj) = max uB,c(vc) , c [h + 1, yul h] c = j h ∈ − − j+h n o LBj(vj) = min lB,c(vc) , c [h + 1, yul h] c = j h ∈ − −

With the above procedures, all the proﬁles can be examined and the corresponding boundaries can be obtained. The derived data sets will be given to the users to evaluate or even enhance the time schedule by using the quantitative results of the traction energy conservation capability.

4. Case Study In this section, case studies for Beijing Metro Batong Line with both the weekday and weekend timetables are employed to illustrate the effectiveness of the presented energy efficient train traction solution and the boundary identification method for energy conservation capability evaluation.

4.1. Data Preparation The Beijing Metro Batong Line is 18.96 km long with 13 stations. The southern extension section with 2 additional stations is not considered in this study. This line connects the central city of Beijing andEnergies the Tongzhou2020, 13, x FOR district, PEER REVIEW which is the sub-center of this city (see Figure6). 13 of 25

FigureFigure 6. 6.Beijing Beijing MetroMetro networknetwork and Batong Line.

Table2 showsTable the basic 2. Characteristics information of ofthe trains trains servicingoperating in Beijing Batong Metro Line. Batong The resistance Line. coe ﬃcients are deﬁned as the technical document to identify the resistance force as the following formula: Passenger capacity (AW1) 248 (3M3T) Passenger capacity (AW2) 2 1428 (3M3T) W(vt) = w1 + w2vt + w3vt (16) Train weight (Mc) 36 t Train weight (T) 29 t Average acceleration (0~40 km/h) ≥0.84 m/s2 Average acceleration (0~80 km/h) ≥0.53 m/s2 Braking deceleration (electric brake adhesion coefficient ≤ 0.16) 0.833 m/s2 Maximum running speed 80 km/h Motor efficiency 92% Resistance coefficient w1 9.8(1.65 Wm + 0.78 Wt) Resistance coefficient w2 9.8(0.0247 Wm + 0.0028 Wt) Resistance coefficient w3 9.8(0.0028 + 0.0078(Nmt − 1))

By using true train parameters and track profiles of Batong Line, the following calculation and analyzes are performed to reflect the practically achievable energy conservation space of the given timetable schemes.

(a) (b)

Figure 7. Statistical results of daily passenger volume in weekdays and weekend days. (a) Results in weekdays; (b) Results in weekend days.

Based on the prepared data sets, the energy efficient train driving control profile calculation and boundary identification are performed and the results are given in the following sections.

4.2. Energy Efficiency Improved by BFO-based Solution

Energies 2020, 13, x FOR PEER REVIEW 13 of 25 Energies 2020, 13, 2111 13 of 25

North Transfer Line 1 where Wm and Wt are total weights of the motor unit and trailer unit, and Nmt is theSiHui total (SH) unit number. SiHuiDong (SHD) Table 2. Characteristics of the trains operating in Beijing Metro Batong Line. GaoBeiDian (GBD) Passenger capacity (AW1) 248 (3M3T)ChuanMeiDaXue (CMDX) Passenger capacity (AW2) 1428 (3M3T)ShuangQiao (SQ) Train weight (Mc) 36GuanZhuang t (GZ) BaLiQiao (BLQ) Train weight (T)Batong Line 29 t TongZhouBeiYuan (TZBY) Average acceleration (0~40 km/h) GuoYuan0.84 m/s (2GY) JiuKeShu≥ (JKS) 2 Average acceleration (0~80 km/h) LiYuan0.53 (LY m) /s LinHeLi≥ (LHL) Braking deceleration (electric brake adhesion coeﬃcient 0.16) 0.833 m/s2 ≤ TuQiao (TQ) Maximum running speed 80 km/h Figure 6. Beijing Metro network and Batong Line. Motor eﬃciency 92%

Table 2. ResistanceCharacteristics coeffi ofcient the wtrains1 operating in Beijing Metro9.8(1.65 BatongW mLine+ 0.78. Wt) Resistance coefficient w 9.8(0.0247 W + 0.0028 W ) Passenger capacity (AW2 1) 248 (3M3T)m t ResistancePassenger coe capacityfficient (wAW3 2) 9.8(0.00281428+ 0.0078((3M3T)N mt 1)) − Train weight (Mc) 36 t According to different passengerTrain weight flow (T distribution) features, two timetable29 schemes t are adopted 2 for weekday and weekend.Average A acceleration whole weekday (0~40 km/h) timetable (5:00~24:00) of the≥0.84 Batong m/s Line contains 2 438 planned trains, whichAverage indicates acceleration 219 trains (0~80 km/h) in each direction. For the weekend≥0.53 m/s version, there are Braking deceleration (electric brake adhesion coefficient ≤ 0.16) 0.833 m/s2 380 planned trains, where 190 trains are scheduled in each direction. From peak hour train graphs and Maximum running speed 80 km/h the statistics of planned trains, it can be found that there are obvious differences in the passenger flow Motor efficiency 92% prediction and train management in weekdays and weekend days from the operator. Resistance coefficient w1 9.8(1.65 Wm + 0.78 Wt) As one major factor that would influence the traction energy consumption level, the passenger Resistance coefficient w2 9.8(0.0247 Wm + 0.0028 Wt) load of Batong Line is also investigated in this case study. Statistical report about the daily passenger Resistance coefficient w3 9.8(0.0028 + 0.0078(Nmt − 1)) volume are collected from the operator when the two train graphs are utilized in practical operation. FigureBy7 showsusing true the reported train parameters results of and passenger track profile volumes of in Batong both weekdays Line, the and following weekend calculation days during and aanalyzes period over are 4performed months. It to is reflect obvious the that practically the passenger achievable load in energy weekend conservation days is much space lower of the than given that intimetable the weekdays, schemes. and that is consistent with the train operation plan arrangements.

(a) (b)

FigureFigure 7.7. StatisticalStatistical resultsresults ofof dailydaily passengerpassenger volumevolume inin weekdaysweekdays andand weekendweekend days.days. ((aa)) ResultsResults inin weekdays; (b) Results in weekendweekdays; days. (b) Results in weekend days.

By using true train parameters and track profiles of Batong Line, the following calculation and Based on the prepared data sets, the energy efficient train driving control profile calculation and analyzes are performed to reflect the practically achievable energy conservation space of the given boundary identification are performed and the results are given in the following sections. timetable schemes. 4.2. EnergyBased onEfficiency the prepared Improved data by sets,BFO the-based energy Solution efficient train driving control profile calculation and boundary identification are performed and the results are given in the following sections.

Energies 2020, 13, 2111 14 of 25 Energies 2020, 13, x FOR PEER REVIEW 14 of 25

4.2.Energies EnergyWe 2020 firstly E,ffi 13ciency, x FORinvestigate Improved PEER REVIEW the by performance BFO-based Solution of the proposed BFO-based energy calculation solution14 of 25 for energy efficient train driving control solutions. A typical passenger load value is selected within theWe statisticalWe firstly firstly investigate results investigate to theenable the performance performance the optimization of theof the proposed calculation. proposed BFO-based BFO-based For the energyderivation energy calculation calculation of energy solution solutionefficient for energydrivingfor energy effi controlcient efficient train solutions drivingtrain driving under control thecontrol solutions. given solutions. timetable A typical A information,typical passenger passenger an load average load value value load is selected is condition selected within withinof 738 the statisticalpassengersthe statistical results per results totrain enable isto adopted enable the optimization thein the optimization evaluation calculation. calculation.of the For weekday the For derivation timetable,the derivation of and energy theof eenergytypicalfficient efficientaverage driving controlvaluedriving solutions of control525 passengers undersolutions the per givenunder train timetablethe is utilized given information,timetable for the weekend information, an average timetable an load average case. condition Results load ofcondition can 738 be passengers found of 738 in perthepassengers train following is adopted per Figure train in 8is the andadopted evaluation Figure in 9. the ofevaluation the weekday of the timetable, weekday timetable, and the typical and the average typical valueaverage of 525value passengers of 525 passengers per train isper utilized train is forutilized the weekendfor the weekend timetable timetable case. Resultscase. Results can becan found be found in thein followingthe following Figures Figure8 and 89 .and Figure 9.

Figure 8. BFO-based optimized speed-distance profiles of all planned trains listed in the weekday timetable. Direction: TuQiao (TQ) – SiHui (SH). Figure 8. BFO-based optimized speed-distance proﬁles of all planned trains listed in the weekday Figure 8. BFO-based optimized speed-distance profiles of all planned trains listed in the weekday timetable.timetable. Direction: Direction: TuQiao TuQiao (TQ) (TQ) – – SiHui SiHui (SH). (SH).

Figure 9. BFO-based optimized speed-distance profiles of all planned trains listed in the weekend Figure 9. BFO-based optimized speed-distance profiles of all planned trains listed in the weekend timetable.timetable. Direction: Direction: SiHui SiHui (SH) (SH) – – TuQiao TuQiao (TQ). (TQ). Figure 9. BFO-based optimized speed-distance profiles of all planned trains listed in the weekend Figure8 shows the derived train speed profiles in the TQ-SH direction by using the presented Figuretimetable. 8 shows Direction: the SiHui derived (SH) train – TuQiao speed (TQ). profiles in the TQ-SH direction by using the presented BFO-based energy efficient driving control solution with the weekday timetable. Speed-distance BFO-based energy efficient driving control solution with the weekday timetable. Speed-distance curvescurves coveringFigure covering 8 shows all all 438 438 the planned planned derived train train train in in speed both both the profilesthe SH-TQ SH-TQ in andtheand TQ-SH TQ-SHTQ-SH direction directions by have using been been the calculated, calculated,presented fromfromBFO-based which which it can energyit can be easilybe efficient easily found founddriving that that the control the energy energy solution effi cientefficient with operation operationthe weekday solutions solutions timetable. of di offf erentdifferent Speed-distance trains trains by BFO by iterations,BFOcurves iterations, covering even in even theall 438 same in plannedthe track same section, train track in section, vary both obviously the vary SH-TQ obviously with and eachTQ-SH with other. eachdirections Similarly, other. have Similarly, all been the 390 calculated,all plannedthe 390 from which it can be easily found that the energy efficient operation solutions of different trains by BFO iterations, even in the same track section, vary obviously with each other. Similarly, all the 390

Energies 2020, 13, 2111 15 of 25 trains in both directions with the weekend timetable have been evaluated, and Figure9 depicts the optimized speed profiles in the SH-TQ direction. At the same time, the reference driving control solutions for all trains in the weekday and weekend timetables are also calculated and compared with the optimized results. Table3 summaries the energy conservation performance of the BFO-derived operation speed profiles over the calculated references, where Np represents the adopted average passenger load condition per train, ∆ESec indicates the saved energy consumption per section against the reference solution-based results, and ‘∆ESec rate’ denotes the energy saving rate against the referencing results. To illustrate the performance differences for the trains, the train-level indices are also computed and summarized as shown in Table4, where ∆ETrain is used to represent the reduced energy consumption for a planned train during the whole trip, and thus ‘∆ETrain rate’ indicates the train-level energy saving rate considering all the 12 sections. Statistical results of energy consumption level for the reference profiles are also given in these tables, where ∆ESec-Ref and ∆ETrain-Ref denote the tractive energy per section and the train-level energy consumption by the reference solution.

Table 3. Performance of the BFO-based energy eﬃcient proﬁles over the reference solutions.

Weekday Timetable (N = 738) Weekend Timetable (N = 525) Performance Indices p p SH-TQ Direction TQ-SH Direction SH-TQ Direction TQ-SH Direction Average E (kW h) 21.5846 21.8804 19.9229 19.9256 Sec-Ref · Maximum E (kW h) 33.2176 37.0205 31.5019 31.0593 Sec-Ref · Average ∆E (kW h) 1.6663 1.7049 1.5641 1.5655 Sec · Maximum ∆E (kW h) 2.4116 2.6649 2.3994 2.5312 Sec · Standard Deviation of 0.0797 0.3150 0.2612 0.2897 ∆E (kW h) Sec · Average ∆ESec rate (%) 7.92 8.03 8.16 8.20

Maximum ∆ESec rate (%) 14.96 14.00 14.03 15.07 Standard Deviation of 1.61 1.71 1.66 1.79 ∆ESec Rate (%)

Table 4. Train-level performance comparison results over the reference solutions.

Weekday Timetable (N = 738) Weekend Timetable (N = 525) Performance Indices p p SH-TQ Direction TQ-SH Direction SH-TQ Direction TQ-SH Direction Average E (kW h) 259.0147 262.5644 239.0748 239.1067 Train-Ref · Average E (kW h) 277.7405 285.7818 252.6382 255.3068 Train-Ref · Average ∆E (kW h) 19.7782 20.1960 18.7305 18.6987 Train · Maximum ∆E (kW h) 21.7280 22.9194 19.9803 20.3491 Train · Standard deviation of 1.3532 1.5066 0.8207 0.9583 ∆E (kW h) Train · Average ∆ETrain rate (%) 7.64 7.69 7.84 7.82

Maximum ∆ETrain rate (%) 9.14 8.63 8.94 8.78 Standard deviation of 0.32 0.34 0.28 0.32 ∆ETrain rate (%)

From these results, it can be found that the proposed energy efficient control solution calculation method using the BFO strategy is capable of reducing the tractive energy consumption in all sections against the conventional traction computation results. For the comparison about the amount of energy reduction at both the section-level and train-level, it can be found that the weekday timetable holds an advanced energy saving capability than the weekend timetable. This situation may be resulted by the difference in the passenger load rate in weekdays and weekend days. A relatively higher passenger load in the weekdays will increase the total mass of the trains and raise the used energy by the train traction system. With the same energy efficient driving control solution derived by the BFO technique, Energies 2020, 13, x FOR PEER REVIEW 16 of 25 Energies 2020, 13, 2111 16 of 25 passenger volume case for the weekend days. However, the weekend timetable realizes a better energy saving rate, which takes advantage of the smaller absolute value of the section-level and train- the absolute value of the energy reduction would be greater than the lower passenger volume case for level energy consumption. the weekend days. However, the weekend timetable realizes a better energy saving rate, which takes Overall, the proposed method for calculating the energy efficient driving control solution based advantage of the smaller absolute value of the section-level and train-level energy consumption. on BFO is able to save the tractive energy consumption under a given time schedule scheme. With Overall, the proposed method for calculating the energy efficient driving control solution based the same configuration, quantitative evaluation of the energy saving performance against the on BFO is able to save the tractive energy consumption under a given time schedule scheme. With the referencing solutions can be performed, and that makes it possible of realizing additional analyses same configuration, quantitative evaluation of the energy saving performance against the referencing on the different influencing factors for the timetables and detecting the boundaries of specific profiles. solutions can be performed, and that makes it possible of realizing additional analyses on the different The following sections will address the issue of boundary identification according to the proposed influencing factors for the timetables and detecting the boundaries of specific profiles. The following solution. sections will address the issue of boundary identification according to the proposed solution.

4.3.4.3. Boundary IdentificationIdentification ofof Speed-orientedSpeed-oriented EnergyEnergy ConservationConservation ProfilesProfiles TheThe energyenergy conservation conservation profiles profiles for for two two timetables timetables are establishedare established by collecting by collecting the feature the feature point sets.point In sets. this In section, this section, speed-oriented speed-oriented profiles profiles are firstly are concerned firstly concerned to illustrate to illustrate the relationship the relationship between thebetween energy the effi energycient performance efficient performance and the average and sectionthe average speed section according speed to the according train schedule to the plans.train Thisschedule section plans. illustrates This section the boundary illustrates identification the boundary results identification for the ASS-REN-PLF results for the andASS-REN-PLF ASS-RER-PLF and ASS-RER-PLF profiles under the weekday timetable. The piece width for boundary detection is set profiles under the weekday timetable. The piece width for boundary detection is set as δS = 0.1 km/h, andas a window = 0.1 km/h width, and of ha window= 5 is taken width in of the ℎ boundary = 5 is taken relaxation. in the boundary In order to relaxation. depict the In diff ordererences to duedepict to thethe passengerdifferences volume due to condition, the passenger two typical volume values condition, of the average two typical passenger values volume of the per average train, passenger volume per train, including Np = 635 (minimum) and Np = 773 (maximum), are used to including Np = 635 (minimum) and Np = 773 (maximum), are used to carry out the presented boundary detectioncarry out andthe presented relaxation boundary strategies. detection Representative and relaxa resultstion are strategies. given in Representative the following figures. results Figureare given 10 in the following figures. Figure 10 shows the boundaries in the SH-TQ plans under the condition Np shows the boundaries in the SH-TQ plans under the condition Np = 635. The results from the TQ-SH = 635. The results from the TQ-SH direction is given in Figure 11 with Np = 773. Similarly, boundaries direction is given in Figure 11 with Np = 773. Similarly, boundaries under the given weekend timetable under the given weekend timetable in both directions are shown in Figure 12 and Figure 13, where in both directions are shown in Figures 12 and 13, where the minimum (Np = 414) and maximum the minimum (Np = 414) and maximum (Np = 593) passenger volume values from the statistical report (Np = 593) passenger volume values from the statistical report in those weekend days are adopted. in those weekend days are adopted.

2.5 15 Feature points Feature points Lower bound Lower bound 2 Upper bound Upper bound 10 1.5

1 5

0.5

0 0 30 35 40 45 50 55 60 65 30 35 40 45 50 55 60 65 Average speed per section (km/h) Average speed per section (km/h) (a) (b)

FigureFigure 10. 10. Boundaries of of ASS-REN-PLF ASS-REN-PLF and and ASS-RER-PLF ASS-RER-PLF profiles profiles (weekday, (weekday, SH-TQ, SH-TQ, NpN = p635).= 635). (a) (a) ASS-REN-PLF profileASS-REN-PLF result; (b) profile ASS-RER-PLF result; ( profileb) ASS-RER-PLF result. profile result.

Energies 2020, 13, x FOR PEER REVIEW 17 of 25 EnergiesEnergies2020 2020, ,13 13,, 2111 x FOR PEER REVIEW 17 ofof 2525 Energies 2020, 13, x FOR PEER REVIEW 17 of 25

2.5 15 Feature points 2.5 15 Feature points FeatureLower bound points 2.5 15 FeatureLower bound points 2 FeatureLowerUpper bound points FeatureLowerUpper bound points 2 LowerUpper boundbound 10 LowerUpper boundbound 1.52 Upper bound 10 Upper bound 1.5 10 1.51 1 5 5 0.51 5 0.5 0.50 0 030 35 40 45 50 55 60 65 030 35 40 45 50 55 60 65 030 35Average 40 speed 45 per section 50 (km/h) 55 60 65 030 35Average 40 speed 45 per section 50 (km/h) 55 60 65 30 35Average 40 speed 45 per 50section (km/h) 55 60 65 30 35Average 40 speed 45 per 50section (km/h) 55 60 65 Average speed(a )per section (km/h) Average speed (b per) section (km/h) (a) (b) Figure 12. Boundaries(a) of ASS-REN-PLF and ASS-RER-PLF profiles (weekend, (b) SH-TQ, Np = 414). (a) pN FigureFigure 12. 12. BoundariesBoundariesASS-REN-PLF of ASS-REN-PLF profile and and result; ASS-RER-PLF ASS-RER-PLF (b) ASS-RER-PLF profiles profiles profile (weekend, (weekend, result. SH-TQ, SH-TQ, N =p 414).= 414). (a) (aFigure) ASS-REN-PLF 12. Boundaries profileASS-REN-PLF of result; ASS-REN-PLF (b) ASS-RER-PLFprofile and result; ASS-RER-PLF ( profileb) ASS-RER-PLF result. profiles profile (weekend, result. SH-TQ, Np = 414). (a) ASS-REN-PLF profile result; (b) ASS-RER-PLF profile result.

3 15 3 Feature points 15 Feature points 2.53 FeatureLower bound points 15 FeatureLower bound points 2.5 FeatureLowerUpper bound points FeatureLowerUpper bound points 2.52 LowerUpper boundbound 10 LowerUpper boundbound 2 Upper bound 10 Upper bound 1.52 10 1.5 1.51 5 1 5 0.51 5 0.5 0.50 0 030 35 40 45 50 55 60 030 35 40 45 50 55 60 030 35Average 40 speed 45per section 50 (km/h) 55 60 030 35Average 40 speed 45per section 50 (km/h) 55 60 30 35Average 40 speed 45per section 50 (km/h) 55 60 30 35Average 40 speed 45per section 50 (km/h) 55 60 Average speed(a )per section (km/h) Average speed (b per) section (km/h) (a) (b) FigureFigure 13. 13.Boundaries Boundaries(a) ofof ASS-REN-PLF and and ASS-RER-PLF ASS-RER-PLF profiles profiles (weekend, (weekend, (b) TQ-SH, TQ-SH, NpN =p 593).= 593). (a) (aFigure) ASS-REN-PLF 13. Boundaries profileASS-REN-PLF of result; ASS-REN-PLF (b) ASS-RER-PLFprofile and result; ASS-RER-PLF ( profileb) ASS-RER-PLF result. profiles profile (weekend, result. TQ-SH, Np = 593). (a) Figure 13. BoundariesASS-REN-PLF of ASS-REN-PLF profile and result; ASS-RER-PLF (b) ASS-RER-PLF profiles profile (weekend, result. TQ-SH, Np = 593). (a) ASS-REN-PLF profile result; (b) ASS-RER-PLF profile result. FromFrom the the derived derived results results of of the the boundaries, boundaries, it canit ca ben be found found that that there there is obvious is obvious diff erencedifference in the in generalthe generalFrom relationships the relationships derived with results with the reduced ofthe the reduced boundaries, energy energy consumption it consumption can be found and theand that energy the there energy savingis obvious saving rate. differencerate. In both In both the in From the derived results of the boundaries, it can be found that there is obvious difference in SH-TQthe general and TQ-SHrelationships directions, with the the reduced reduced energy energy consumption consumption tends and tothe increase energy with saving a larger rate. sectionIn both thethe generalSH-TQ relationshipsand TQ-SH directions, with the reduced the reduced energy energy consumption consumption and thetends energy to increase saving withrate. aIn larger both speedthe SH-TQ level. and An enhanced TQ-SH directions, upper bound the revealsreduced that energy the potential consumption energy tends conservation to increase capability with a oflarger the thesection SH-TQ speed and level.TQ-SH An directions, enhanced the upper reduced bound energy reveals consumption that the tendspotential to increase energy withconservation a larger timetablesection speed using thelevel. energy An eenhancedfficient driving upper control bound solution reveals would that bethe proportional potential energy to the corresponding conservation sectioncapability speed of the level. timetable An enhanced using the upperenergy bound efficien ret drivingveals that control the solutionpotential would energy be conservationproportional averagecapabilityto the corresponding interstation of the timetable speed. average using However, interstation the energy the boundaries speed. efficien Hot drivingwever, of energy controlthe saving boundaries solution rate-related wouldof energy profilesbe proportionalsaving behave rate- capabilityto the corresponding of the timetable average using interstation the energy speed.efficien Hot drivingwever, controlthe boundaries solution wouldof energy be proportional saving rate- to the corresponding average interstation speed. However, the boundaries of energy saving rate-

Energies 2020, 13, x FOR PEER REVIEW 18 of 25 Energies 2020, 13, 2111 18 of 25 related profiles behave differently. With the growth of average interstation speed, the achievable energy saving rate by BFO optimized train driving control solution tends to decrease gradually. diBasedfferently. on the With investigation the growth to of the average boundaries, interstation it can speed,be concluded the achievable that an energyincreased saving average rate speed by BFO in optimizedthe rail sections traindriving is surely control suggested. solution At the tends same todecrease time, one gradually. has to be Basedclearly on aware the investigation of the constraint to the in boundaries,the achieved it energy can be concludedsaving efficiency. that an increasedWhen tuni averageng the speedrunning in thespeed rail configuration sections is surely of a suggested. timetable Atscheme, the same multiple time,one factors has toand be constraint clearly aware conditions of the constraint have to inbe theconsidered achieved with energy the saving energy-saving- efficiency. Whenoriented tuning recommendations. the running speed configuration of a timetable scheme, multiple factors and constraint conditions have to be considered with the energy-saving-oriented recommendations. 4.4. Analysis on Passenger Load Factor 4.4. Analysis on Passenger Load Factor From the boundary identification results mainly consider the effect from average running speed, someFrom preliminary the boundary knowledge identification results results have mainlybeen acqu considerired to the assist effect fromthe eval averageuation running of a specific speed, sometimetable preliminary scheme. knowledge In order to results further have discuss been on acquired the correlation to assist the among evaluation the energy of a specific efficient timetable driving scheme.control, the In ordertrain tospeed further parameter discuss and on the the correlation probable passenger among the load energy conditions, efficient drivingmore passenger control, thevolume train samples speed parameter participate and in the the probable analysis passenger to demons loadtrate conditions, the effect from more the passenger passenger volume flow features. samples participateSix additional in the passenger analysis volume to demonstrate values are the sampled effect from from the the passenger statistics as flow shown features. in Figure Six additional8 to enrich passengerthe data set volume for both values the two are sampledtimetables, from which the statisticsmeans {635, as shown692, 718, in 732, Figure 747,8 to 773} enrich for the the weekday data set fortimetable both the and two {414, timetables, 446, 480, which 511, 545, means 593} {635, for the 692, weekend 718, 732, timetable. 747, 773} forThe the statistical weekday results timetable of energy and {414,consumption 446, 480, 511,reduction 545, 593} and for the the energy weekend saving timetable. rate, including The statistical the average results and of the energy maximum consumption values, reductionare calculated. and the Figure energy 14 saving and Figure rate, including 16 summariz the averagee the andenergy the maximumreduction values,results arefrom calculated. the two Figurestimetables. 14 and Statistical 15 summarize results theof the energy energy reduction saving results rates are from compared the two timetables. in Figure 15 Statistical and Figure results 17 ofrespectively. the energy These saving results rates illustrate are compared that there in Figures are not 16 great and differences 17 respectively. in both These the averagely results illustrate reduced thatenergy there and are energy not great saving differences rate by in a both varied the pa averagelyssenger reducedvolume, energyexcept andfor energythe SH-TQ saving minimum rate by a varied passenger volume, except for the SH-TQ minimum passenger load case (N = 635) with passenger load case (Np = 635) with the weekday timetable. As for the statistical presults of the themaximum weekday values, timetable. the differences As for the statistical between resultstwo dire ofctions the maximum become values,significant. the di Aff erenceslager maximum between twovalue directions of energy become reduction significant. can be Aachieved lager maximum for planned value TQ-SH of energy trains, reduction especially can for be the achieved weekday for plannedtimetable. TQ-SH However, trains, the especially maximum for energy the weekday saving timetable. rates behave However, differently. the maximum The weekday energy results saving do ratesnot indicate behave dia ffunifiederently. trend The weekdayin two directions results do with not a indicate varied apassenger unified trend volume, in two while directions the weekend with a variedtimetable passenger results volume,in a lowe whiler maximum the weekend value in timetable TQ-SH. results in a lower maximum value in TQ-SH.

2 3.5 D1(SiHui-TuQiao) D2(TuQiao-SiHui) D1(SiHui-TuQiao) D2(TuQiao-SiHui) 3 1.5 2.5

2 1 1.5

1 0.5 0.5

0 0 635 692 718 732 747 773 635 692 718 732 747 773 Number of passengers per train Number of passengers per train (a) (b)

FigureFigure 14.14. Statistical results results of of average average energy energy reductio reductionn per per section section from from the the weekday weekday timetable. timetable. (a) (a) Comparison ofComparison average results; of average (b) Comparison results; (b) ofComparison maximum results.of maximum results.

Energies 2020, 13, x FOR PEER REVIEW 19 of 25 Energies 2020, 13, x FOR PEER REVIEW 19 of 25 Energies 2020, 13, 2111 19 of 25 Energies 2020, 13, x FOR PEER REVIEW 19 of 25 10 20 10 D1(SiHui-TuQiao) D2(TuQiao-SiHui) 20 D1(SiHui-TuQiao) D2(TuQiao-SiHui) D1(SiHui-TuQiao) D2(TuQiao-SiHui) D1(SiHui-TuQiao) D2(TuQiao-SiHui) 108 20 15 8 D1(SiHui-TuQiao) D2(TuQiao-SiHui) 15 D1(SiHui-TuQiao) D2(TuQiao-SiHui) 68 15 6 10 46 10 4 10 5 2 4 5 2 5 02 0 0 635 692 718 732 747 773 0 635 692 718 732 747 773 635Number 692 of 718 passengers 732 per train 747 773 635Number 692 of 718 passengers 732 per train 747 773 0 Number of passengers per train 0 Number of passengers per train 635 692 718 732 747 773 635 692 718 732 747 773 Number of passengers(a) per train Number of passengers(b) per train (a) (b) Figure 15. Statistical( aresults) of average energy saving ra te per section from the(b) weekday timetable. Figure 15. Statistical(a) Comparison results of of average average energy results; saving (b) Comparison rate per section of maximum from the results. weekday timetable. FigureFigure 15. 15.Statistical Statistical(a) Comparison results results ofof of averageaverage average energyenergy results; savingsaving (b) Comparison raterate per section of maximum from from the the results. weekday weekday timetable. timetable. (a) Comparison(a of) Comparison average results; of average (b) Comparison results; (b of) Comparison maximum results. of maximum results. 2 3 2 D1(SiHui-TuQiao) D2(TuQiao-SiHui) 3 D1(SiHui-TuQiao) D2(TuQiao-SiHui) D1(SiHui-TuQiao) D2(TuQiao-SiHui) 2.5 D1(SiHui-TuQiao) D2(TuQiao-SiHui) 3 1.52 2.5 1.5 D1(SiHui-TuQiao) D2(TuQiao-SiHui) 2 D1(SiHui-TuQiao) D2(TuQiao-SiHui) 2.5 2 1.51 1.5 2 1 1.5 1 1.5 0.51 1 0.5 0.5 1 0.5 0.50 0 0.5 0 414 446 480 511 545 593 0 414 446 480 511 545 593 414Number 446 of 480 passengers 511 per 545train 593 414Number 446 of 480 passengers 511 per train 545 593 0 Number of passengers per train 0 Number of passengers per train 414 446 480 511 545 593 414 446 480 511 545 593 Number of passengers(a) per train Number of passengers(b) per train (a) (b) Figure 16. Statistical (resultsa) of average energy reductio n per section from the ( bweekend) timetable. (a) Figure 16. Statistical results of average energy reduction per section from the weekend timetable. Figure 16. StatisticalComparison results of of average average energy results; reductio (b) Comparisonn per section of maximum from the results.weekend timetable. (a) (aFigure) Comparison 16. Statistical ofComparison average results results; ofof averageaverage (b) Comparison energyresults; reductio (b) ofComparison maximumn per section results.of maximum from the results.weekend timetable. (a) Comparison of average results; (b) Comparison of maximum results. 10 20 10 D1(SiHui-TuQiao) D2(TuQiao-SiHui) 20 D1(SiHui-TuQiao) D2(TuQiao-SiHui) D1(SiHui-TuQiao) D2(TuQiao-SiHui) D1(SiHui-TuQiao) D2(TuQiao-SiHui) 108 20 15 8 D1(SiHui-TuQiao) D2(TuQiao-SiHui) 15 D1(SiHui-TuQiao) D2(TuQiao-SiHui) 68 15 6 10 46 10 4 10 5 2 4 5 2 5 02 0 0 414 446 480 511 545 593 0 414 446 480 511 545 593 414Number 446 of 480 passengers 511 per train 545 593 414Number 446 of 480 passengers 511 per train 545 593 0 Number of passengers per train 0 Number of passengers per train 414 446 480 511 545 593 414 446 480 511 545 593 Number of passengers(a) per train Number of passengers(b) per train (a) (b) FigureFigure 17. 17.Statistical Statistical( results aresults) ofof average average energyenergy saving ratera te per section from from the the(b) weekend weekend timetable. timetable. (aFigure) Comparison 17. Statistical(a of) Comparison average results results; of of average average (b) Comparison energy results; saving (b of) Comparison maximum rate per section results. of maximum from the results. weekend timetable. Figure 17. Statistical(a) Comparison results of of average average energy results; saving (b) Comparison rate per section of maximum from the results. weekend timetable. ToTo deepdeep analyzeanalyze(a) Comparison thethe eeffectsffects of fromfrom average specificspecific results; passengerpass (b)enger Comparison volumevolume of conditionsmaximumconditions results. onon thethe derivedderived profileprofile boundaries,boundaries,To deep thethe analyze weekdayweekday the andeffectsand weekendweekend from specific timetable-derivedtimetable- passderivedenger volume upperupper bounds,conditionsbounds, whichwhich on the describedescribe derived extremeextreme profile capabilitycapabilityboundaries,To deep spacespace the analyze ofofweekday thethe givengiventhe effectsand timetablestimetables weekend from usingusingspecific timetable- thethe pass optimizedderivedenger volume drivingupper conditions bounds, control solution,solution, which on the describe areare derived summarized extreme profile andandcapabilityboundaries, compared space the (see ofweekday the Figure Figures given and 18 timetables18 toweekend– 21Figure) to using demonstrate 21)timetable- to the demonstrate optimizedderived the energy driving upperthe energy conservation bounds,control conservation solution, which characteristics describe are characteristics summarized extreme with multiplewithandcapability comparedmultiple factors. space factors. (see of the Figure given 18 timetables to Figure using 21) to the demonstrate optimized driving the energy control conservation solution, are characteristics summarized withand comparedmultiple factors. (see Figure 18 to Figure 21) to demonstrate the energy conservation characteristics with multiple factors.

Energies 2020, 13, x FOR PEER REVIEW 20 of 25 Energies 2020, 13, x FOR PEER REVIEW 20 of 25

EnergiesEnergies2020 2020, ,13 13,, 2111 x FOR PEER REVIEW 20 ofof 2525 (kw·h) rate (%) Sec Sec (kw·h) rate (%) Sec Sec (kw·h) rate (%) Sec Sec Upper bound of ΔE Upper bound of Upper bound of ΔE Upper bound of Upper bound of ΔE Upper bound of Upper bound of ΔE Upper bound of

Upper bound of ΔE Upper bound of (a) (b) Upper bound of ΔE Upper bound of (a) (b) Figure 18. Comparison of upper boundaries with different Np values (weekday, SH-TQ). (a) ASS- Figure 18. ComparisonREN-PLF of upper boundariesboundaries; with (b) ASS-RER-PLF different Np values boundaries. (weekday, SH-TQ). (a) ASS- (a) REN-PLF boundaries; (b ) ASS-RER-PLF boundaries. (b)

FigureFigure 18. 18. ComparisonComparison of ofupper upper boundaries boundaries with withdifferent diﬀ Nerentp valuesNp (weekday,values (weekday, SH-TQ). (a SH-TQ).) ASS- (a) ASS-REN-PLF boundaries;REN-PLF (b) ASS-RER-PLF boundaries; (b boundaries.) ASS-RER-PLF boundaries. (kw·h)

rate (%) Sec Sec (kw·h) rate (%) Sec Sec (kw·h) rate (%) Sec Sec Upper bound of ΔE Upper bound of Upper bound of ΔE Upper bound of Upper bound of ΔE Upper bound of Upper bound of ΔE Upper bound of

Upper bound of ΔE Upper bound of (a) (b) Upper bound of ΔE Upper bound of (a) (b) Figure 19. Comparison of upper boundaries with different Np values (weekday, TQ-SH). (a) ASS- FigureFigure 19. 19. ComparisonComparisonREN-PLF of ofupper upper boundariesboundaries; boundaries with (b) ASS-RER-PLF withdifferent diﬀ Nerentp values boundaries.Np (weekday,values (weekday, TQ-SH). (a TQ-SH).) ASS- (a) ASS-REN-PLF boundaries;(a) REN-PLF ( b ) ASS-RER-PLF boundaries; (b boundaries. ) ASS-RER-PLF boundaries. (b)

Figure 19. Comparison of upper boundaries with different Np values (weekday, TQ-SH). (a) ASS- REN-PLF boundaries; (b) ASS-RER-PLF boundaries. (kw·h) rate (%) Sec Sec (kw·h) rate (%) Sec Sec (kw·h) rate (%) Sec Sec Upper bound of ΔE Upper bound of Upper bound of ΔE Upper bound of Upper bound of ΔE Upper bound of Upper bound of ΔE Upper bound of

Upper bound of ΔE Upper bound of (a) (b) Upper bound of ΔE Upper bound of (a) (b) Figure 20. Comparison of upper boundaries with diﬀerent N values (weekend, SH-TQ). Figure 20. Comparison of upper boundaries with different Np valuesp (weekend, SH-TQ). (a) ASS- (a)Figure ASS-REN-PLF 20. Comparison boundaries;REN-PLF of upper (b) ASS-RER-PLF boundariesboundaries; with (b boundaries.) ASS-RER-PLF different Np values boundaries. (weekend, SH-TQ). (a) ASS- (a) REN-PLF boundaries; (b ) ASS-RER-PLF boundaries. (b)

Figure 20. Comparison of upper boundaries with different Np values (weekend, SH-TQ). (a) ASS- REN-PLF boundaries; (b) ASS-RER-PLF boundaries.

EnergiesEnergies2020 2020, 13, 13, 2111, x FOR PEER REVIEW 21 ofof 2525 (kw·h) Sec rate (%) Sec Upper bound of ΔE Upper bound of Upper bound of ΔE Upper bound of

(a) (b)

FigureFigure 21. 21. ComparisonComparison of ofupper upper boundaries boundaries with withdifferent diﬀ erentNp valuesNp (weekend,values (weekend, TQ-SH). (a TQ-SH).) ASS- (a) ASS-REN-PLF boundaries;REN-PLF (b) ASS-RER-PLF boundaries; (b boundaries.) ASS-RER-PLF boundaries.

FiguresFigure 1818 and and Figure 19 depict 19 depict the passengerthe passenger volume-based volume-based upper upper bounds bounds covering covering the the energyenergy conservationconservation profilesprofiles for for both both the the SH-TQ SH-TQ and and TQ-SH TQ-SH weekday weekday timetable, timetable, in which in which the different the diff erentcolors colorsof the boundary of the boundary curves represent curves represent the adopted the pass adoptedenger passengervolume samples volume in samplesthe energy in conservation the energy conservationcapability evaluation. capability Similarly, evaluation. Figure Similarly, 20 and Figures Figure 2120 showand 21 the show computed the computed upper bounds upper from bounds the fromweekend the weekend timetable timetable in both directions. in both directions. From the Fromfigures, the it figures,is easy to it isfind easy out to the find similarly out the consistent similarly consistenttrend toward trend both toward the energy both the reduction energy reduction and energy and sa energyving rate saving when rate varied when passenger varied passenger load rates load are ratesassumed. are assumed. An increasing An increasing upper boun upperd can bound be derived can be derived as a larger as a aver largerage average interstation interstation speed no speed matter no matterwhether whether the passenger the passenger load is load high is or high low. or At low. the At same the sametime, time,the upper the upper bounds bounds fall gradually fall gradually with withthe increased the increased average average running running speed speed per per section. section. However, However, it itis is difficult difficult to to find find out the didifferencesfferences duedue toto thethe passenger passenger loadload condition condition directlydirectly fromfrom the the neighboring neighboring upper upper bound bound curves. curves. ToTo realizerealize numericalnumerical comparisons,comparisons, thethe linearlinear curvecurve fittingfitting method method is is applied applied to to obtain obtain the the analytic analytic forms forms of of the the boundaries,boundaries, and and thus thus the the fitting fitting slopes slopes of of the the boundaries boundaries are are computed computed and and summarized. summarized. TheThe fittingfitting slopeslope ofof anan upperupper boundbound curvecurve indicatesindicates thethe slopeslope valuevalue ofof aafitted fitted straight straightline line thatthat representsrepresents thethe approximatedapproximated linear linear model model of of the the boundary boundary profile. profile. Tables Table5 5and and6 summarize Table 6 summarize the comparison the comparison results ofresults fitting of slopes fitting of slopes the ASS-REN-PLF of the ASS-REN-PLF and ASS-RER-PLF and ASS-RER-PLF upper upper boundaries. boundaries. FromFrom thethe results in the the tables, tables, similar similar fitting fitting results results can can be befound found in the in same the same direction direction with withthese thesetwo involved two involved timetable timetable schemes, schemes, except except for forthe the results results of of the the ASS-RER-PLF ASS-RER-PLF profile profile in thethe SH-TQSH-TQ direction.direction. DiDifferencesfferences stillstill existexist duedue toto thethe adopted adopted passenger passenger volumevolume value,value, which which indicate indicate thatthat the the passengerpassenger load load condition condition should should be be taken taken into into consideration consideration in in fully fully exploring exploring the the potentialpotential of of energyenergy conservationconservation forfor aa time time schedule schedule scheme. scheme. ToTo showshow thethe details details of of the the comparison comparison results, results, the the derived derived fittingfitting slopes slopes of ofboth boththe theASS-REN-PLF ASS-REN-PLF andand ASS-RER-PLFASS-RER-PLF profileprofile boundariesboundaries areare givengiven inin FigureFigure 2222,, wherewhere the the symbol symbol ‘D1’ ‘D1’ and and ‘D2’ ‘D2’ represent represent the the SH-TQ SH-TQ and and TQ-SH TQ-SH directions directions respectively. respectively.

TableTable 5. 5.Comparison Comparison of of ﬁtting fitting slopes slopes of of ASS-REN-PLF ASS-REN-PLF upper upper boundaries. boundaries.

Weekday Timetable WeekendWeekend Timetable Timetable PassengerPassenger PassengerPassenger Volume Np SH-TQSH-TQ TQ-SHTQ-SH Volume Np SH-TQSH-TQ TQ-SHTQ-SH Volume Np Volume Np DirectionDirection DirectionDirection DirectionDirection DirectionDirection 635635 0.0366 0.0366 0.04800.0480 414414 0.03100.0310 0.04490.0449 692692 0.0337 0.0337 0.04880.0488 446446 0.03000.0300 0.04670.0467 718718 0.0368 0.0368 0.04820.0482 480480 0.03120.0312 0.04600.0460 732732 0.0345 0.0345 0.04930.0493 511511 0.03230.0323 0.04810.0481 747747 0.0370 0.0370 0.04940.0494 545545 0.03260.0326 0.04750.0475 773773 0.0342 0.0342 0.04890.0489 593593 0.03240.0324 0.04720.0472

Energies 2020, 13, 2111 22 of 25

Energies 2020, 13, x FOR PEER REVIEW 22 of 25 Table 6. Comparison of ﬁtting slopes of ASS-RER-PLF upper boundaries.

Table 6. ComparisonWeekday Timetable of fitting slopes of ASS-RER-PLF upperWeekend boundaries. Timetable Passenger Passenger Volume Np SH-TQWeekday TimetableTQ-SH Volume Np SH-TQWeekend TQ-SHTimetable Passenger Passenger DirectionSH-TQ DirectionTQ-SH DirectionSH-TQ DirectionTQ-SH Volume Np Volume Np 635 Direction0.1138 Direction0.2085 414 0.2148Direction 0.2045Direction − − − − 635692 −0.11380.1667 −0.20850.1996 446414 0.2211−0.2148 0.2267−0.2045 − − − − 692718 −0.16670.1663 −0.19960.2204 480446 0.2161−0.2211 0.2022−0.2267 − − − − 718732 −0.16630.1869 −0.22040.2265 511480 0.2075−0.2161 0.1952−0.2022 − − − − 747 0.1524 0.2233 545 0.2127 0.1857 732 −−0.1869 −−0.2265 511 − −0.2075 − −0.1952 773 0.1706 0.1997 593 0.2193 0.1996 747 −−0.1524 −−0.2233 545 − −0.2127 − −0.1857 773 −0.1706 −0.1997 593 −0.2193 −0.1996

(a) (b)

FigureFigure 22. 22Comparison. Comparison of of ﬁtting fitting slopesslopes corresponding to to the the derived derived boundaries boundaries with with different diﬀerent passengerpassenger volume volume values. values. (a) Weekend (a) Weekend timetable timetable results; results; (b) Weekday (b) Weekday timetable timetable results. results.

ItIt isis clearclear thatthat thethe fittingfitting slopesslopes varyvary withwith thethe operationoperation directiondirection exceptexcept forfor upperupper boundsbounds ofof thethe energyenergy savingsaving rate-relatedrate-related fittingfitting resultsresults basedbased onon thethe weekendweekend timetable.timetable. TheThe slopesslopes ofof thethe weekdayweekday timetable-basedtimetable-based ASS-REN-PLFASS-REN-PLF upperupper boundsbounds inin D2D2 andand thethe weekendweekend timetable-basedtimetable-based ASS-REN-PLFASS-REN-PLF upperupper bound-basedbound-based fittingfitting resultsresults inin bothboth D1D1 andand D2D2 dodo notnot changechange greatlygreatly withwith aa variedvaried passengerpassenger volume.volume. Under the weekday weekday D1 D1 case case with with respect respect to to the the ASS ASS-REN-PLF-REN-PLF upper upper bounds, bounds, the the volume volume Np N= p747= 747realizes realizes a maximum a maximum fitting fitting slope, slope, which which indicates indicates a shaper a shaperincrease increase of the upper of the bound upper to bound saved totractive saved energy tractive consumption energy consumption with a raising with average a raising section average speed section over speed other overNp-derived other Nresults.p-derived The results.situation The for situation the ASS-RER for the-PLF ASS-RER-PLF boundary fitting boundary is much fitting different. is much The di volumefferent. N Thep = 446 volume for a Nweekendp = 446 forachieves a weekend the maximum achieves thefitting maximum slope in fittingboth the slope D1 inand both D2 thedirections D1 and against D2 directions other N againstp conditions. other ForNp conditions.the weekday For timetable the weekday-based comparison, timetable-based a maximum comparison, fitting a slope maximum is activated fitting by slope the same is activated condition by theNp = same732 in condition both the twoNp directions.= 732 in both According the two to directions. the changing According trend with to thethe changingpassenger trendload condition, with the passengerwe can draw load the condition, conclusion we that can the draw identified the conclusion boundaries that with the identified respect to boundaries both the ASS with-REN respect-PLF toand both ASS the-RER ASS-REN-PLF-PLF profiles and vary ASS-RER-PLF with different profiles passenger vary load with conditions different passenger, while the load changing conditions, rates whileof the thederived changing boundaries, rates of theincluding derived the boundaries, growing rate including for ASS the-REN growing-PLF rateand fordecrease ASS-REN-PLF rate for ASS and- decreaseRER-PLF, rate do fornot ASS-RER-PLF,change monotonously. do not change Therefore, monotonously. to better utilize Therefore, capability to better of the utilize proposed capability BFO- ofbased the proposed energy efficient BFO-based train energy driving effi controlcient train method, driving the control timetable method, scheme the timetable is suggested scheme to be is suggestedestablished to orbe established updated by or matchingupdated by the matching identified the identifiedASS-REN ASS-REN-PLF-PLF and ASS and-RER ASS-RER-PLF-PLF profile profileboundaries, boundaries, which is which significant is significant to accomplish to accomplish the energy the conservation energy conservation purpose from purpose the operation from the operationorganization organization perspective. perspective.

5.5. ConclusionsConclusions andand FutureFuture WorkWork TheThe mainmain contributioncontribution ofof thisthis paperpaper isis toto givegive aa casecase studystudy forfor boundaryboundary identiﬁcationidentification ofof thethe energyenergy conservation conservation capability capability by by using using the the optimized optimized traintrain driving driving control control method.method. An energy-energy- efficient driving control solution using the Bacterial Foraging Optimization algorithm. Based on that, the connection between the energy saving space of the timetable and the specific influencing factor is described with the derived energy conservation train speed profiles. The proposed energy efficient

Energies 2020, 13, 2111 23 of 25 efficient driving control solution using the Bacterial Foraging Optimization algorithm. Based on that, the connection between the energy saving space of the timetable and the specific influencing factor is described with the derived energy conservation train speed profiles. The proposed energy efficient optimization method and the boundary identification solution are implemented with the data of the Beijing Metro Batong Line. Results of the case study show that the developed approach can realize the quantitative evaluation of achievable space and boundaries for the energy saving capability with a given timetable. A maximum average energy saving rate of 8.20% per section can be realized with the involved timetables by using the BFO-based solution, and an extreme reduction rate of 15.07% is achieved for the weekend timetable. A series of tractive energy saving capability boundaries have been obtained in the case study, which reveal the relationship with the average speed per section and the passenger load condition. The results could give important implications to transit operators, management departments and engineers that the energy conservation operations should be tightly connected with the practical situation and dynamic requirement of the rail transit system. It should be noticed that this research only concerns the train’s tractive energy consumption in the capability evaluation and boundary identification. In our future work, the regenerative braking factor will be involved in the evaluation of energy conservation profile boundaries with the energy efficient control solution comprehensively under the given timetable schemes. Several influencing factors, including the adopted regenerative braking strategy, capability of the energy storage system and the cooperation of the trains’ movement, will be jointly analyzed with the timetable parameters. Additionally, the effects from the advanced system design, e.g., the lightweight trains with the novel materials, the energy efficient metro infrastructures, and the effective energy storage equipment, are also planned to be investigated to carry out more general analyses on the energy conservation topic. According to the practical operation of the urban rail transit system, this study will be performed continuously for Batong Line and more other metro lines based on the field experiences.

Author Contributions: All the authors contributed to the work of this paper. J.L. contributed significantly to the development of the algorithms, analysis and manuscript preparation. T.-t.L. helped to the programming and data analysis. B.-g.C. contributed to the conception of the study and coordinated the overall research. J.Z. supported the data acquisition and contributed to the result analysis with constructive discussions. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Key R&D Program of China (No. 2017YFB1201105), National Natural Science Foundation of China (No. 61873023), Beijing Natural Science Foundation (No. L191014), and Beijing Nova Program of Science and Technology (No. Z191100001119066). Conflicts of Interest: The authors declare no conflict of interest.

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