A Cooperative Train Control Model for Energy Saving - IEEE Xplore

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Mar 27, 2015 - Abstract—Increasing attention is being paid to energy efficiency in subway systems to reduce operational cost and carbon emis- sions.
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IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 16, NO. 2, APRIL 2015

A Cooperative Train Control Model for Energy Saving Shuai Su, Tao Tang, and Clive Roberts

Abstract—Increasing attention is being paid to energy efficiency in subway systems to reduce operational cost and carbon emissions. Optimization of the driving strategy and efficient utilization of regenerative energy are two effective methods to reduce the energy consumption for electric subway systems. Based on a common scenario that an accelerating train can reuse the regenerative energy from a braking train on the opposite track, this paper proposes a cooperative train control model to minimize the practical energy consumption, i.e., the difference between traction energy and the reused regenerative energy. First, we design a numerical algorithm to calculate the optimal driving strategy with the given trip time, in which the variable traction force, braking force, speed limits, and gradients are considered. Then, a cooperative train control model is formulated to adjust the departure time of the accelerating train for reducing the practical energy consumption during the trip by efficiently using the regenerative energy of the braking train. Furthermore, a bisection method is presented to solve the optimal departure time for an accelerating train. Finally, the optimal driving strategy is obtained for the accelerating train with the optimal departure time. Case studies based on the Yizhuang Line, Beijing Subway, China, are presented to illustrate the effectiveness of the proposed approach on energy saving. Index Terms—Cooperative train control, energy-efficient operation, optimal driving strategy, regenerative braking.

I. I NTRODUCTION

S

UBWAY systems play an important role in local transportation within cities and urban areas, providing frequent, fast, safe, and comfortable journeys to a large number of passengers. Subways have become a necessary part of public transportation for reducing traffic congestion and are regarded as a “green” transportation mode compared with buses and private cars. However, subway systems, particularly in big cities, also consume a great amount of energy. For example, 17 subway lines are currently put into service in Beijing, China. The total distance is around 465 km, and the energy consumption is over 500 000 MW · h per year. Energy efficiency in subway systems is more and more important because of environmental concerns Manuscript received October 14, 2013; revised February 9, 2014, March 21, 2014, and May 20, 2014; accepted June 17, 2014. Date of publication July 15, 2014; date of current version March 27, 2015. This work was supported in part by the Beijing Laboratory of Urban Rail Transit, by the Beijing Key Laboratory of Urban Rail Transit Automation and Control, by the Fundamental Research Funds for the Central Universities under Grant 2014YJS029, and by projects funded by the Beijing Municipal Science and Technology Commission under Grants D131100004113002 and D131100004013001. The Associate Editor for this paper was F.-Y. Wang. S. Su and T. Tang are with the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China (e-mail: [email protected]). C. Roberts is with the School of Electronic, Electrical and Computer Engineering, University of Birmingham, Birmingham B15 2TT, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TITS.2014.2334061

and rising energy price. Therefore, many measures have been taken to reduce the energy consumption and the operational cost, such as improvements in energy conversion, reduction of the aerodynamic resistance, energy-efficient operation, and reuse of regenerative braking energy. The main use of energy in subway systems is to tow the train during the acceleration phase. However, the kinetic energy of trains in dc railways can be converted into electric energy when the train is braking. This regenerative energy can be transmitted backward to the overhead catenary or the third rail and reused by other trains. Hence, the practical energy consumption should be the difference between the energy consumption during the trip and the reused energy from regenerative braking. Most traditional studies focus on minimizing the energy consumption during the trip, in which an optimal train control model was formulated, and the optimal driving strategy was solved by analytical methods, numerical methods, or evolutionary algorithms. However, this work [1]–[3] does not introduce regenerative braking or consider the dynamic coordination between trains when regenerative braking is applied [4]. On the other hand, some work is attempted to maximize the use of regenerative energy by optimizing the timetable [5]–[7], without considering a detailed energy consumption model during the trip. The optimization of energy during the trip and maximum use of regenerative energy are two closely dependent parts of improving energy consumption, which should be combined to reduce the practical energy consumption. To improve the energy efficiency of subway systems, this paper proposes a cooperative train control model to minimize the practical energy consumption by slightly adjusting the timetable. Moreover, a numerical algorithm is designed to calculate the optimal departure time and the energy-efficient driving strategy for trains. The rest of this paper is organized as follows. In Section II some important work in the area of energy-efficient driving strategy and regenerative braking is reviewed. In Section III a cooperative train control model is proposed and a numerical algorithm is designed to solve the optimal departure time based on the solution to the optimal train control problem. In Section IV some numerical examples are presented based on the infrastructure and operational data from the Yizhuang Line, Beijing Subway, China, which illustrates that the proposed approach can produce useful results. II. L ITERATURE R EVIEW Previous work has focused on minimizing the energy consumption of towing trains by solving the energy-efficient driving strategy between successive stations. Studies of this

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SU et al.: COOPERATIVE TRAIN CONTROL MODEL FOR ENERGY SAVING

problem can date back to the 1960s. Ishikawa [8] formulated an optimal train control model based on the assumption that the train runs on a flat track with constant gradient and traction efficiency. By using the Pontryagin maximum principle, the energy-efficient driving strategy is proved to consist of maximum acceleration, cruising, coasting, and maximum braking. Although this research has some restrictive assumptions, this theoretical result has laid the foundation of modern train control theory. As an extension of the optimal train control theory, Howlett et al. [1], Howlett [9], and Milroy [10] considered variable gradients, variable speed limits, and traction efficiency in the optimal train control model and gave a complete analysis on the switching points of the optimal driving strategy. Liu and Golovitcher [2] analyzed the possible switching of optimal control sequences by using the Pontryagin maximum principle and gave an analytical method to solve the switching points among different control phases. In addition, Howlett et al. [11] also investigated this optimal control problem for freight trains, in which the trip contains more than one steep slope. The route is first divided into several small parts such that each part contains only one steep slope. Then, the precise switching strategy for each part was solved by using a local optimal principle. In 2000, Khmelnitsky [4] presented a complete study on the optimal train control problem, in which regenerative braking, variable gradients, variable traction efficiency, and arbitrary speed limits were all considered. In addition, Miyatake and Matsuda [12] proposed an energy-saving model that integrated the application of the energy storage devices and then used sequential quadratic programming to solve the optimal control sequences for trains. In 2010, Miyatake and Ko [13] made a comparison among the dynamic programming (DP), gradient method, and sequential quadratic programming methods, in which sequential quadratic programming is emphasized because it can solve the speed profiles and the optimal state of charge profiles of energy storage with less computation time. Franke et al. [14], [15] proposed a DP method to obtain the energy-efficient control sequences for trains. The given work is based on the assumption that the motor can provide continuous traction and braking force, while only certain discrete throttle settings are possible for some trains, and each setting determines a constant rate of power supply. The Scheduling and Control Group at the University of South Australia undertook some work on the discrete control problem [16], [17]. For example, Cheng and Howlett [18]–[20] proposed a discrete optimal control model and presented an analytic method to find an idealized strategy by solving the optimal duration of the control sequences. Howlett [16] analyzed both the continuous and discrete control problems and noted that any sequence of positive measurable control can be closely approximated by a sequence of power–coast pairs. Howlett finally used the Kuhn–Tucker equation to find the key equations that determine the optimal switching times for the discrete control problem. Recent research around the energy-efficient operation in railways has extended to timetable optimization [5], [21]– [23]. For example, Pena et al. [5] proposed a timetable design method to reduce the energy consumption from substations by maximizing the use of regenerative energy. In addition to the analytical and numerical methods, there is also some work

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on evolutionary algorithms. For example, Chang and Sim [24] found optimal coast control sequences by using the genetic algorithm with considering punctuality, riding comfort, and energy consumption. Ke et al. [25] proposed a combinatorial optimization model and designed an ant colony optimization algorithm to find the optimal speed profile. Rémy [26] used the genetic algorithm to minimize travel time, delays, and energy consumption. Rémy also integrated the concept of Pareto optimization into the selection process. III. M ODEL F ORMULATION AND S OLUTION A. Symbols The parameters and variables used in this paper are listed here. Decision Variables T trip time; t1 departure time; u traction or braking force from the train. Parameters v train speed; x train position; V¯ speed limits; v0i initial speed for the ith section; vtii final speed for the ith section; S trip distance; m train mass; Ep practical energy consumption; E energy consumption during the trip; Er reused regenerative energy; g gradient resistance; w running resistance. Intermediate Variables Ei energy consumption for the ith section; Ti trip time for the ith section. B. Model Formulation Trains in subway systems are generally powered by electricity, which has several advantages over diesel and combustion trains, i.e., high efficiency in energy transmission, less noise, and less local emission. In addition, electric machines are able to convert the kinetic energy to electric energy when trains are braking, which is known as regenerative braking. The regenerative energy can be used to supply onboard auxiliary systems or be fed back to the third rail and reused by other trains within the power supply network [27]. In other words, the energy for towing the train is mainly provided by the electric substation in the power supply network, and part of the towing energy may come from the regenerative braking of other trains. As shown in Fig. 1, train A accelerates to achieve a high travel speed from the station. Area “1” denotes the energy consumption during the accelerating process. Regenerative braking will be enabled if there is another train (Train B) braking near the same station. In this case, a percentage of the regenerative energy (area “2”) can be reused by the accelerating train. Hence, if we assume that a train is departing from the station when

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t, respectively, then the reused regenerative energy during the coordination process between trains can be calculated as T Er =

  min Ega (s, t), Egb (s, t) dt

(6)

t=0

where Ega (s, t) and Egb (s, t) can be further described as  a 2 − 12 mvt2 Eg (s, t) = 12 mvt+Δt  1 b 2 . Eg (s, t) = ϕ(s) 2 mvt2 − 12 mvt+Δt

Fig. 1. Coordination between trains at stations.

another train in the same power supply network is braking, we can formulate the following objective to minimize the practical energy consumption: min Ep = E − Er

(1)

in which the energy consumption during the trip can be calculated as T

T max {u(t), 0} v(t) dt =

E= 0

u(t) + |u(t)| v(t) dt. (2) 2

0

The control variable u is the traction or braking force applied by the train, which satisfies the maximum traction and braking force constraints, i.e., umin ≤ u ≤ umax .

(3)

State variable v denotes the train speed. The initial speed, final speed, and the maximum speed constraints are given as v0 = vT = 0, v ≤ V¯ (x).

(4)

In addition, the train movement can be described by the following differential equations:  dx dt = v(t) (5) m dv dt = u(t) − w(v) − g(x). Considering regenerative energy Er , the amount of the reused regenerative energy relies on the efficient and cooperative operation between accelerating trains and braking trains, since the voltage level of the power supply system will be increased when regenerative braking is enabled, and the regenerative energy is fed back to the third rail. If no trains can use the regenerative energy at this time, the overvoltage protection system will then automatically consume the regenerative energy at the braking resistance. In addition, the distance between the accelerating and braking trains should not be too large, so as to avoid the loss of regenerative energy during transmission. If Ega (s, t) and Egb (s, t) are used to denote the potential use of the regenerative energy for the accelerating trains and the available regenerative energy from the braking trains at time

(7)

Here, ϕ(s) is a parameter between 0 and 1, whose value depends on the distance between trains and the transmission efficiency. Generally, a smaller distance between trains results in a large value of ϕ(s), and hence, a higher percentage of the regenerative energy is used due to the reduced losses. s in (6) and (7) means that the positions of the cooperative trains are close enough to utilize the regenerative energy, not implying that trains are in the same position. When the cooperative operation between trains is enabled, both Ega (s, t) and Egb (s, t) are positive. If not, ϕ(s) and Egb (s, t) will be zero. It is noted that the accelerating distance of the subway vehicles is approximately 200 m, which is close to the substation. Furthermore, the accelerating processes for trains are similar to each other. Hence, the average energy losses due to the energy transfer between trains and substations are assumed to be constant in this paper. The mechanical energy consumption divided by the average traction efficiency is used to calculate the electrical energy consumption during the trip [28], [29]. Additionally, the cooperative control of two trains at stations happens between close trains, and the energy losses due to the energy transfer between trains are assumed to be fixed [30], [31]. IV. S OLUTION The solution to the cooperative problem is divided into two parts, i.e., the optimal train control model and the cooperative model. The optimal train control model aims to solve the optimal driving strategy between stations with a given trip time, which can also be used to calculate the energy consumption during the trip. Based on the result of the optimal train control model, the bisection method is applied to the cooperative model to obtain the optimal departure time that minimizes the trains’ practical energy consumption, which is the difference between the energy consumption during the trip and the reused regenerative energy. A. Optimal Train Control Model Based on (2)–(5), the optimal train control model is ⎧ T u(t)+|u(t)| ⎪ ⎪ v(t) dt ⎪ min E = 2 ⎪ ⎪ 0 ⎪ ⎨ s.t. m dv dt = u(t) − w(v) − g(x) dx ⎪ ⎪ ⎪ dt = v(t) ⎪ ⎪ ¯ v ⎪ 0 = vT = 0, v ≤ V (x) ⎩ umin ≤ u ≤ umax .

(8)

SU et al.: COOPERATIVE TRAIN CONTROL MODEL FOR ENERGY SAVING

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TABLE I O PTIMAL D RIVING R EGIMES

Fig. 3. Decreasing character of the E-T function.

sequences of the rest of the journey will be covered by CO and MB phases. The details for obtaining the speed sequences of a given section is described in Algorithm 1. Algorithm 1: Optimal speed sequences for a given section

Fig. 2. limit.

Optimal driving strategy on a section with constant gradient and speed

By applying the Pontryagin maximum principle [32], the optimal control sequences {uo } should maximize the Hamiltonian function with respect to the control variable u, i.e., H = p1 v(t) + p2 (u(t) − w(v) − g(x)) −

u(t) + |u(t)| v(t). 2 (9)

As a consequence, there are five optimal driving regimes in the following sequence (see Table I). Thomas explained the optimal driving strategy as follows [27]. After a stop at a station, acceleration with maximum traction force must be applied. Then, depending on the available running time reserve, the coasting phase immediately follows, or the maximum speed is reached, and therefore, cruising is applied. Finally, the train has to brake with maximum operational braking to stop at the next station (see Fig. 2). On steep descents, coasting may precede cruising at the speed limit. Many papers consider the optimal switching points of the energy-efficient driving strategy [2], [4], [11], [33] to minimize the energy consumption during the trip. Previous work [21], [34] has proposed an algorithm to solve the optimal train control problem, which first presents a numerical algorithm to calculate the optimal driving strategy with the energy constraint for a section and then solves the energy-efficient driving strategy for the whole trip by distributing the energy units to sections. For a section with constant gradient and speed limit, the minimum energy consumption is uniquely determined by the trip time, and vice versa (see Fig. 3). Hence, the optimal driving strategy can be calculated with either the known trip time or the known energy consumption. With the given energy consumption, we can first generate the speed sequences for MA. The CR speed sequences will be calculated with the remaining energy when the train speed has reached the speed limit. The speed

Step 1: Initialize the initial speed v0j and the energy consumption Ej for section j; Step 2: Divide the section into nj pieces such that the distance of each piece is Δx; Step 3: Generate the speed sequences for the MA phase; 2 j 2 − vij = vij = v0j , while Ej > 0, vij < V , do vi+1 E=E− 2Δx(F (vij )/m − r(vij )/m − g(xi )), F (vij )Δx; Step 4: If the speed vij has reached the speed limit, then partial braking or partial power is applied to keep cruising, and the speed sequences are calculated as j = vij ; vi+1 Step 5: Generate the speed sequences for CO phase as 2 j 2 − vij = 2Δx(−r(vij )/m − g(xi )); vi+1 Step 6: If MB phase exists, we first calculate the braking   2 j 2 − vij = speed sequences {vij } as vkj = vtj , vi+1   2Δx(−B(vkj )/m − r(vkj )/m − g(xk )), And then  let vij = min(vij , vij ); Step 7: Return the optimal speed sequences vij and the trip

nj time of this section Tj = i=0 (Δx/vij ). Δx is a small distance unit, which can be 1 m in the algorithm. For dealing with variable gradients and speed limits, the trip is partitioned into several sections such that each section has a constant gradient and speed limit. The speed sequence of each section can be generated by Algorithm 1. Then, the energy unit (a small amount of energy that is defined as 0.03 kW · h in this paper) will be attempted to distribute to different sections for achieving the corresponding time reductions. After a comparison among these time reductions, the energy unit will be finally distributed to the section that can achieve the maximum time reduction. This distribution process will be repeated to shorten the primary trip time until the practical trip

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Fig. 5.

Fig. 4. Flowchart for calculating the optimal driving strategy for trains between stations.

time is delivered, after which the driving strategy will also be obtained. The flowchart of the algorithm is described in Fig. 4 [21], [23].

Optimal departure time.

slice. With a given departure time, we then can calculate the speed sequence of the departure train and obtain the energy consumption needed for each time slice during the acceleration process as well as the total energy consumption E for the trip. Finally, the reused regenerative energy of each time slice can be calculated and summed to be Er according to (6). Specifically, Fig. 6 shows the specific cooperative processes for different departure times with the given braking speed sequences. Trains need more energy to accelerate at high speed. As a result, the traction energy increases with respect to T . On the contrary, trains have a large amount of kinetic energy at high speed and less kinetic energy at low speed. Thus, the available regenerative energy is a decreasing function. For the cooperative state (a), Ega (s, t) > Egb (s, t). According to (6), we have

B. Cooperative Model As shown in Fig. 1, there is a common scenario in subway systems that one train stops at the station and will depart to the next station in a few seconds, when another train is braking in the same power network. Then, the regenerative energy from the braking train can be used by the departing train; hence, the practical energy consumption can be reduced. In subway systems, the distance between stations is usually short, and therefore, frequent accelerating and braking can contribute to efficient utilization of the regenerative energy. Two important elements of the practical operational energy consumption are the energy consumption during the trip and the reused amount of the regenerative braking energy. The former element is related to the trip time and driving strategy between stations, and the latter element depends on the cooperative control between the accelerating and braking trains. To satisfy the requirement of punctuality, accelerating trains need to arrive at the next station on time. Hence, a delayed departure time implies a shorter running time between stations, which could increase the energy consumption during the journey (see Fig. 3). As shown in Fig. 5, the energy consumption between stations increases when the train has a departure delay. However, later departure may create an opportunity for making better use of the regenerative energy. Consequently, we formulate a cooperative model to deal with this tradeoff problem to minimize the practical energy consumption by solving the optimal departure time for the accelerating train. For the calculation of the practical energy consumption, we first obtain the speed sequences of the braking train and the available regenerative energy at each time

T Er =

min



Ega (s, t), Egb (s, t)



t1 Egb (s, t)dt. (10)

dt = t0

t=0

It is obtained that Er will be an increasing function with respect to t1 during process (a). Since Egb (s, t) > 0 during the interval [t0 , t1 ]. For process (b), Er can be calculated as t2

t1 Ega (s, t)dt

Er = t0

Egb (s, t)dt

+

(11)

t2

from which we can obtain dEr dt1 b dt1 b = Ega (s, t2 ) + Eg (s, t1 ) − Egb (s, t2 ) = E (s, t1 ). dt2 dt2 dt2 g (12) It is obvious that t2 will increase with t1 ; thus, dt1 /dt2 > 0. Moreover, Egb (s, t1 ) is positive. Hence, dEr /dt2 > 0, and the amount of reused regenerative energy will keep increasing. When the cooperative process reaches state (c), Er should be presented as in t3 Egb (s, t)dt

Er =

(13)

t0

where t0 is a constant, and Er is a decrease function with respect to t3 . In other words, the amount of regenerative energy will decrease with a later departure time in this situation.

SU et al.: COOPERATIVE TRAIN CONTROL MODEL FOR ENERGY SAVING

Fig. 6.

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Coordination between trains at stations.

Hence, the amount of reused regenerative energy shows an increasing, increasing, and then decreasing trend from states (a)–(c). Based on the continuity of the function of Er with respect to departure time, we conclude that the amount of reused regenerative energy is a unimodal function with respect to t1 , i.e., ⎧ dE ∗ r ⎪ ⎨ dt1 > 0, when t < t dEr ∗ (14) dt1 = 0, when t = t ⎪ ⎩ dEr < 0, when t > t∗ . dt1 In addition, E increases with a later departure time, i.e., dE >0 dt1 dEp dE dEr = − . dt1 dt1 dt1

(15) (16)

From (14)–(16), we analyze function Ep as the following two cases. • If dE/dt1 > dEr /dt1 , i.e., dEp /dt1 > 0, it implies that it will cost more practical energy consumption with a later departure time. In other words, trains should leave the station as early as possible to reduce the practical energy consumption. • If dE/dt1 < dEr /dt1 , we can obtain ⎧ dEp ∗∗ ⎪ ⎨ dt1 < 0, when t < t dEp ∗∗ (17) dt = 0, when t = t ⎪ ⎩ dE1p ∗∗ dt1 > 0, when t > t . In this case, the practical energy consumption Ep will decrease first and then increase with the departure time postponed. In conclusion, there must be one inflexion at most for the practical energy consumption function. Hence, we solve the optimal departure time by using the bisection method [35], [36] in this paper, which is shown in Fig. 7. First, the earliest and latest departure times are initialized. The gradient of the practical energy consumption for these two departure times is then calculated. If both of the gradients are positive, it implies that the practical energy consumption will increase with

Fig. 7. Flowchart of the bisection method.

a later departure time, and the earliest departure time is the optimal solution. Otherwise, the extremum will be obtained by shrinking the range of the best solution with the bisection method until the defined procedure is completed. The possible solution to the practical application of the proposed algorithm is also presented. Note that most subway systems are equipped with the communication-based train control system, which can implement communication among trains, track equipment, and the control center. The train position and speed are known more accurate than the traditional signaling system, which lays the foundation for cooperation among trains and makes it possible to efficiently manage the railway traffic. The sequence diagram of the cooperation process in subway systems is described in Fig. 8. First, sensors on trains obtain the train information (such as train position, and train speed), and vehicle onboard controllers (VOBCs) send this information to the zone controller (ZC) and the Operation Control Center (OCC) through the data communication system (DCS), after which the ZC will generate the moving authority for braking trains. Then, based on the possible departure time, the OCC calculates the optimal departure time to achieve a good cooperation between trains for the departure train. Finally, the OCC

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TABLE II I NFRASTRUCTURE DATA B ETWEEN J IUGONG S TATION AND Y IZHUANGQIAO S TATION , S YMBOL “−” IN THE S ECOND C OLUMN M EANS T HERE IS A D OWNHILL G RADIENT

Fig. 8. Sequence diagram of the cooperation process.

Fig. 10.

Fig. 9. Maximum traction force, maximum braking force, and running resistance for vehicles in the Yizhuang Line.

feeds back the departure information to trains through the DCS, and VOBCs on trains can calculate the driving strategy for the following journey with moving authority. V. E XAMPLES Some examples based on the Yizhuang Line, Beijing Subway are presented in this section to illustrate the effectiveness of the proposed approach. The maximum traction force, maximum electrical braking force, and running resistance of vehicles can be calculated with the data in Fig. 9. The braking force is 260 kN, and air braking will become effective when electric braking is not enough [28]. The average traction efficiency is assumed to be 70% in the case study [28], [29] with the efficiency of the reused regenerative energy being around 40% [30], [31].

Example A.

Example A: The first example presents a cooperative control for two trains running between Jiugong Station and Yizhuangqiao Station. The infrastructure data between stations are shown in Table II. Train A has stopped at Jiugong Station to allow passengers to alight and board. The original arrival time is 27 35 and the earliest departure time is 27 58 in off-peak hours. (The shortest dwell time mainly includes the door-open time, passengers’ boarding and alighting time, and the door-closing time.) In addition, the latest departure time is 28 20, and the train should arrive at Yizhuangqiao Station at 30 30 to follow the punctuality according to the planned timetable. Train B is running in the interval between Jiugong and Yizhuangqiao and starts to brake at 28 05 for stopping at Jiugong Station at the opposite track (see Fig. 10). By applying the proposed algorithm, the optimal departure time is solved as 28 07, and the convergence process of the algorithm is presented in Fig. 11. The details of the energy consumption for different departure schedules is also presented in Table III. Without considering the cooperation between trains, the departing train follows the earliest departure time, and the energy consumption calculated from the optimal driving strategy is 13.600 kW · h. No regenerative energy is reused. Hence, the practical energy consumption without changing the departure time is 13.600 kW · h. However, by adjusting the departure time with the cooperative model, the energy consumption during the trip is calculated to be

SU et al.: COOPERATIVE TRAIN CONTROL MODEL FOR ENERGY SAVING

Fig. 11.

Convergence of the bisection method for the off-peak-hours example. TABLE III C OMPARISON ON THE E NERGY C ONSUMPTION FOR D IFFERENT D EPARTURE T IMES

Fig. 12. Optimal driving strategy for the departure train.

14.229 kW · h, and the regenerative energy is 0.956 kW · h. Hence, the practical energy consumption calculated from the cooperative model is 13.273 kW · h (see Table III), which reduces the practical energy consumption by 2.4%. In addition, the optimal driving strategy obtained from the cooperative model is shown in Fig. 12.

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Fig. 13. Convergence of the bisection method for the peak-hours example. TABLE IV C OMPARISON ON E NERGY C ONSUMPTION D URING 8:00 A . M .–9:00 A . M .

There is another scenario that the train arrives at the same station in peak hours or a delay happens, resulting in that the earliest departure time becomes 28 08, and the state of the braking train does not change. Then, the cooperative approach is applied to solve the optimal departure time, which is shown in Fig. 13. The result implies that the earlier the train departs from the station, the less energy consumption it will cost, since the accelerating train has missed the best opportunity to cooperate with the braking train, and the practical energy consumption mainly depends on the energy consumption during the trip. Example B: To efficiently utilize the regenerative energy and then reduce the energy consumption, the proposed approach achieves the coordination between trains by changing the departure time, which is scheduled by the timetable. Hence, in Example B, we apply the proposed approach to modify the timetable for a fixed time period from 8:00 A . M . to 9:00 A . M . This time period is one of the peak hours with more passengers alighting and boarding the Yizhuang Line. Generally, 11 trains will be put into service to meet the passenger demand with an approximately cyclic timetable. The signaling system is capable of recording the operational data, such as train mass, speed, position, traction force, and braking force. Hence, the energy consumption during trips and reused regenerative energy are first calculated for the original timetable. As shown in Table IV, the energy consumption in practice for the 8:00 A . M .–9:00 A . M . period (case 1) is 3451.43 kW · h, in which the regenerative energy accounts for 38.90 kW · h. Then, we apply the optimal train control approach to optimize the driving strategy for trains during the time period of 8:00 A . M .–9:00 A . M ., and the practical energy consumption can be reduced by 10.87%, because the energy consumption during the trip is efficiently reduced. It should be noted that a small difference occurs to

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R EFERENCES

Fig. 14. Modification of timetable for train coordinations. The circles denote the generated cooperative operation for trains.

the amount of regenerative energy due to the change of control sequences when the optimal driving strategy is applied (see case 2 in Table IV). However, the optimal control model does not contribute to efficiently use the regenerative energy. Finally, we apply the cooperative approach for the same time period and calculate the corresponding energy consumption (case 3 in Table IV). For the cooperative approach, the energy consumption shows a gentle rise since some trip times are shortened, going with costing more energy. A great increase in the utilization of the regenerative energy is achieved from 40.41 kW · h to 86.73 kW · h. Specifically, many acceleration–braking train pairs are efficiently coordinated, which is shown in Fig. 14. Hence, the practical energy consumption is ultimately reduced by 11.34%, and the proposed cooperative approach provides a better performance on energy saving, which achieves more energy reduction by 0.47% than the optimal train control model. The simulation is performed on a personal laptop with processor speed of 2.4 GHz and memory size of 3 GB. The average computation time of the proposed algorithm is about 0.4 s, which can meet the requirement of real-time control.

VI. C ONCLUSION The contribution of this paper is to propose a cooperative train control model for multitrains based on the energy-efficient driving strategy for a single train. A numerical algorithm has been designed to adjust the departure time of trains for better usage of regenerative energy. The proposed approach can efficiently reduce the practical energy consumption for subway systems and does not influence the punctuality of trains. Some examples based on the operation data of the Yizhuang Line, Beijing Subway show that the cooperative model can save 2.4% of energy for one trip and 0.47% for one peak hour operation compared with the optimal train control approach, which illustrates the effectiveness of the proposed model. In addition, the computation time is short enough to apply the algorithm to the real-time control system.

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Shuai Su received the B.S. degree from the School of Sciences, Beijing Jiaotong University, Beijing, China, in 2010. He is currently working toward the Ph.D. degree with the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University. His research interests include energy-efficient operation and control in railway systems, such as timetable optimization, optimal driving strategy, and rescheduling.

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Tao Tang received the Ph.D. degree from Chinese Academy of Sciences, Beijing, China, in 1991. He is the Academic Pacesetter with National Key Subject Traffic Information Engineering and Control and the Director of the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing. He is also a Specialist with the National Development and Reform Commission and the Beijing Urban Traffic Construction Committee. His research interests include both highspeed and urban railway train control systems and intelligent control theory.

Clive Roberts received the B.Eng. degree (Hons) in electronics from University of Wales, Cardiff, U.K., in 1996. In 1997 he joined University of Birmingham, Birmingham, U.K., as a Research Fellow and began working on projects looking at railway condition monitoring of railway infrastructure assets. In 2007 he was promoted to Senior Research Fellow and took on the role of the Director of Research for the Birmingham Center for Railway Research and Education. In 2010 he became a Professor of railway systems. Over the last 14 years, he has developed a broad portfolio of research aimed at improving the performance of railway systems. He leads the University’s contribution in a number of large EPSRC, European Commission, and industryfunded projects. He works extensively with the railway industry in Britain and overseas.