In the moving block signalling (MBS) system where the tracking target point of the following train is moving forward with its leading train, overload of the substations occurs when a dense queue of trains starts (or restarts) in very close distance interval. This is the peak power demand problem. Several methods have been attempted in the literature to deal with this problem through changing train’s operation strategies. However, most existing approaches reduce the service quality. In this paper, two novel approaches—“Service Headway Braking” (SHB) and “Extending Stopping Distance Interval” (ESDI)—are proposed according to available and unavailable extra station dwell times, respectively. In these two methods, the restarting times of the trains are staggered and traction periods are reduced, which lead to the reduction of peak power demand and energy consumption. Energy efficient control switching points are seen as the decision parameters. Nonlinear programming method is used to model the process. Simulation results indicate that, compared with ARL, peak power demands are reduced by 40% and 20% by applying SHB and ESDI without any arrival time delay, respectively. At the same time, energy consumptions are also reduced by 77% and 50% by applying SHB and ESDI, respectively.
Moving block signalling (MBS) [
This problem could be solved by improving the infrastructure or the train operation strategies. Energy storage system (ESS) is a very important component in modern railway power supply system. Advanced control and manufacturing technology improve the stability, capacity, and weight of ESS [
There are two kinds of traditional PDR techniques based on changing operation strategies; one is called starting time delay (STD), which introduces a starting time delay to each of the following trains. Under this category, there are two specific techniques called single STD and grade STD. The difference between them is the introduced starting time delay to each of the following trains, which are the same in single STD technique but different in grade STD (showing a deceasing trend). The other one is called acceleration rate limit (ARL), which means the acceleration of the following trains is limited to a certain extent (or different extents). Under this category, there are two specific techniques called single ARL and grade ARL. The difference between them is the limited acceleration rate to each of the following trains, which are the same in ARL technique but different in grade ARL (showing a deceasing trend). In addition to the above techniques, there is also a PDR technique called coordinated PDR. It is a combination of the STD and ARL techniques with feeding the regenerated power of decelerating trains to accelerating trains in the same queue, by coordinating the movement of queued trains.
Takeuchi and his colleagues discussed these techniques in [
Although the existing techniques can reduce peak power demand at different degrees, they increase the travel time between the successive stations and decrease the service quality. And the energy consumption is increased since there are more traction periods.
In this paper, in order to reduce peak power demand and energy consumption, we first analyze the reasons of the formation of the peak power demand, and, then, two novel approaches are proposed based on the main reasons with considering the energy efficient driving strategies. One is for available extra station dwell time, named Service Headway Braking (SHB), and the other one is for unavailable extra station dwell time, named Extending Stopping Distance Interval (ESDI). Both of them are realtime adjustment methods and can be implemented before the leading train’s restarting. Therefore, there is no need to change timetable. Considering energy saving driving strategy could be seen as a kind of driving mode switching process [
Under MBS, the tracking target point of the following train moves forward continuously as the leading train travels. The instantaneous distance
The distance between two successive trains must be larger than the safety margin at any moment even if the leading train comes to a sudden halt, so we have
Based on (
The reason of the formation of the peak power demand is the restarting of the dense queue, and the reasons for the formation of the dense queue are listed as follows.
Features of moving block signaling system. (Two trains will start simultaneously if the distance interval between them is
Extra dwell time in station.
There are two kinds of extra station dwell time: one is available and the other is unavailable. In daily railway operation, there may be some exceptions, such as a passenger may be caught in the door of the train or a shortterm surge in passenger flow (i.e., passenger flow increases sharply after a football match). In these circumstances, adjusting the whole timetable is not convenient, because the circumstances only exit in a short period. In this case, the operator will arrange the train to stop a little longer and this extra station dwell time is available. However, if a train is broken in a station and we are not sure how long we need to fix it, in this situation, the extra station dwell time is unavailable.
Based on the analysis above, it is known that peak power demand could be reduced by avoiding the dense queue. In order to achieve this goal, we first analyze the relation between extra station dwell time and the number of delayed trains.
Generally speaking, each train has a required dwell time at a station. If a train stops longer than the required dwell time, we call the extra time as
Formation of dense queue.
As it is shown in Figure
When train 1 starts, the positions of following trains are
Let
Letting
then the number of delayed trains should satisfy
The power demand of the
The peak power demand of the
The total peak power demand of all delayed following trains
Because there are more traction periods, traditional PRD techniques cause energy consumption increasing. Energy conservation is the research interest in many fields [
For electric traction systems, the motion equations of train have the following forms:
Note that the basic resistance
The energy efficient operation problem is modeled as follows:
By applying Pontryagin maximum principle to solve the problem as specified in (
It is easy to prove that the Hamiltonian reaches the maximum with respect to
Full power (
Partial power (
No power and no braking (
Partial braking (
Full braking (
They could be seen as four possible driving phases, which are
acceleration with full power;
speed holding with partial power or braking;
coasting with no power and braking;
braking with full braking.
Based on the above analysis, it is seen that the energy saving strategy relies on these optimal controls. Train energy saving driving process is the process of switching from an optimal control to another, and the switching sequence is acceleration, speed holding, coasting, and braking.
Then, the optimal control switching speeds and running phases duration are treated as decision parameters. The energy and power can be expressed by these decision parameters. The power demand is used as the objective function of a nonlinear programming problem. The constraints include running time and distance. In actual train operation, maximum power and braking cannot be applied by considering the ride comfort. Instead, a service acceleration/braking rate is applied. Therefore, by solving the nonlinear programming model, a reference trajectory leading to less power and energy consumption can be obtained.
Based on the analysis above, we know the restarting of the dense queue in a small area leads to peak power demand and both of the two traditional PDR techniques are carried out after the formation of the dense queue. In this section, we propose a novel operation strategy to reduce the peak power by avoiding the formation of a dense queue based on the available extra station dwell time. In the following parts, we use
Figure
As shown in Figure
(a) Following trains’ SHB driving curves after train 1 stops for
Figure
In order to obtain
Running time of following train.



Running time of the following train arrives at station A 

400  16  82  40.26 
400  12  138  39.82 
400  8  178  39.5 
400  4  202  39.42 
400  0  210  39.38 
From Table
Based on the analysis above, the whole strategy is consisting of 6 steps, braking, waiting, traction, speed holding, coasting, and finally tracking the front train. The indexes of them are
It is worth noting that this operation strategy is energy efficient, because the duration of some steps may be 0 based on an appropriate target function. For example, if we get
In this new strategy, all the following trains brake at first; therefore, they will not stop too to cause a dense queue; at the same time, the other trains are coasting when one train reaccelerates, so the reaccelerating times of the following trains are staggered. Therefore the peak power demand is avoided.
In this section, we formulate the mathematical model of the operation process. For train
Since the traction phase is highly energy consumed and power supported, we minimize the peak power. Therefore, based on the analysis above, the problem could be seen as a constrained nonlinear programming problem as follows:
During the running process of successive following trains, the time interval among them is
The previous section shows SHB strategy; it could reduce the peak power demand if the
According to the analysis in Section
Figure
Operation strategy of ESDI.
In this new strategy, the stopping distance interval of the following trains is increased, so the density of the queue is decreased, and the peak power demand is reduced.
In this section, we formulate the mathematic model of the operation process. According to the last section, it is known that we know that the stopping intervals of the trains are extended in the new operation strategy; therefore, peak power demand can be reduced when the trains restart. Define
In order to grantee the feasibility of the new operation strategy, if leading train stops at the station, after
Since the traction phase is highly energy consumed and power supported, we minimize the peak power. Based on the analysis above, we use nonlinear programming to model the problem as follows:
where
In this section, a simulation is used to test and verify the new strategies. The length of train (
In order to choose
Measured value of coasting data.
Speed range (km/h)  Slop (m/s^{2})  Average error (m) 

37–31 

0.0227 
4140 

0.0361 
4342 

0.0364 
54–48 

0.0236 
6059 

0.0296 
6261 

0.0301 
79–75 

0.0237 
From Table
If
Figures
Comparison of the graded ARL and SHB techniques.
Arrival time of train 2 (s)  Arrival time of train 3 (s)  Peak power demand (kw/t)  Energy consumption (kw·h)  Stopping time before arrival (s)  

Train 2  Train 3  
NonPDR  289.38  338.76  25.1  15.9137  130  43.75 
Graded ARL  297.78  400.36  22.07  46.3402  130  43.75 
SHB  289.5  339.16  13.42  10.5016  65.66  0 
In order to show the peak power demand without any PDR technique and compare it with the performance of traditional PDR technique, Figures
Peak power demand profile without PDR technique.
Peak power demand profile with graded ARL technique.
Peak power demand profile with SHB technique.
Figures
The performance of applying SHB technique is shown from Figures
Table
According to Section
If the extra dwell time is 250 seconds, in graded ARL, 3 trains will be delayed (including the leading train) according to (
In ESDI, based on (
Peak power demand profile with ESDI technique.
As we see, using ESDI technique, the arrival times of the following two trains are almost the same as their times with no PDR technique. Without any PDR technique, the stopping distance interval is 190 m, so the dense queue caused by the extra station time (250 s) is 380 m. The peak power demand after 250 s within 380 m is 25.1 kw/t. In ESDI strategy, stopping distance interval is extended to 383.86 m, which is two times longer than the distance in ARL. That is to say, the restarting position of train 2 belongs to the nearby substation. Thus the power demands by train 2 and train 3 are not afforded by the same substation. Therefore, the peak power demand after 250 s within 380 m is reduced to 18.18 kw/t. Compared with nonPDR technique, the peak power demand is reduced by 18%. In ESDI, train 2 has a stopping time 153.98 s at the position of 2944.6 m. Train 3 has a stopping time 57.96 s at the position of 3328.32 m. Compared with nonPDR technique, the stopping time before arrival is increased by 18%.
According to above analysis, we can see that the new strategy reduces peak power demand by sacrificing the stopping time before arrival. However, the advantages of ESDI are still obvious, because it is more efficient on energy saving; it can reduce energy consumption by 50% compared with graded ARL technique.
From Table
Comparison the PDR techniques.
Arrival time 
Arrival time 
Peak power 
Energy 
Stopping time before arrival (s)  

Train 2  Train 3  
NonPDR  289.38  338.76  25.1  15.9137  130  43.75 
Graded ARL  297.78  400.36  22.07  46.3402  130  43.75 
SHB  289.5  339.16  13.42  10.5016  65.66  0 
ESDI  290  340  18.18  22.86  153.98  57.96 
Peak power demand reduction strategies are discussed in this paper. The reasons of peak power demand problem are analyzed deeply and two main reasons are given.
Based on the reasons and according to different situations, two new peak demand reduction techniques are proposed. One is Service Headway Braking (SHB) strategy, which is used to reduce peak power demand when the extra station dwell time is available. The other is Extending Stopping Distance Interval (ESDI) strategy, which is used to reduce peak power demand when the extra station dwell time is unavailable. Nonlinear programming approach is introduced to model the operation strategy. The simulation results show that, compared with the best traditional PDR techniques, SHB can reduce 40% of peak power demand and 77% of energy consumption without increasing the arrival time delay. ESDI can reduce 20% of peak power demand and 50% of energy consumption without increasing the arrival time delay. Therefore, SHB has a better performance than ESDI when the available and unavailable extra station dwell times are the same. The basic principle of the two strategies is to avoid the formation of the dense queue. Therefore, both of the two strategies are implemented before the formulation of the dense queue.
The authors declare that there is no conflict of interests.
This work was supported by the National High Technology Research and Development Program of China (no. 2011AA110502) and the National Natural Science Foundation of China (no. 61134001).