Electric vehicles such as trains must match their electric power supply and demand, such as by using a composite energy storage system composed of lithium batteries and supercapacitors. In this paper, a predictive control strategy based on a Markov model is proposed for a composite energy storage system in an urban rail train. The model predicts the state of the train and a dynamic programming algorithm is employed to solve the optimization problem in a forecast time domain. Real-time online control of power allocation in the composite energy storage system can be achieved. Using standard train operating conditions for simulation, we found that the proposed control strategy achieves a suitable match between power supply and demand when the train is running. Compared with traditional predictive control systems, energy efficiency 10.5% higher. This system provides good stability and robustness, satisfactory speed tracking performance and control comfort, and significant suppression of disturbances, making it feasible for practical applications.

Urbanization reflects the level of development of a civilized society. Social and economic development can lead to the expansion of urban scale which, in turn, will result in gradual increases in the urban population, the scale of population movement and the frequency of population mobility [

Generally, energy storage systems can be added to urban rail trains to solve these energy issues. A vehicle energy storage system can store regenerative braking energy. Then, when the train is accelerating, the energy can be released to power the train [

Aiming to resolve the issues caused by the separate use of batteries and supercapacitors, this paper adopts a composite energy storage system comprised of these two types of power supplies. This not only improves efficiency but also extends range. Also, the efficiency of a composite energy storage system mainly depends on its topology and its power distribution under operating conditions [

While there are many studies on the application of composite energy storage systems to electric vehicles, there are few on urban rail trains. There has been extensive research on the energy distribution strategies of composite energy electric vehicles [

Real-time optimization based on model predictive control does not require prior knowledge of the future driving characteristics of the vehicle and is not limited by specific cycle conditions. The computation task is small and easy to implement on the basis of a guaranteed sub-optimal power distribution [

As shown in

According to the architecture of the composite energy storage system shown in

where

To build a model, the whole urban rail train can be treated as a particle point and longitudinal dynamics can be applied to attain the demand dynamics model:

where

The energy consumed during train running is:

During the charging and discharging processes, complex chemical reactions are generated inside the lithium battery, which can cause it to exhibit a high degree of nonlinearity and strong coupling, thus making the accurate modelling and control of lithium batteries challenging [

From the KVL law of Kirchhoff, the circuit can be analyzed to acquire the load power of the lithium battery during operation:

where

So, the state of charge (SOC) of the lithium battery can be attained as:

where

The working principles of supercapacitors and lithium batteries are different; e.g., supercapacitors do not involve complex chemical reactions during the operating process. Currently, supercapacitor models are generally categorized into classical models, trapezoidal models, three-branch models, and impedance-based models according to their electrical characteristics. In this paper, the supercapacitor group model applies the classic RC circuit model, as shown in

From the Kirchhoff principle [

where

With

Thus, the SOC of the supercapacitor can be obtained as:

where

In the MPC algorithm, a model describing the dynamic performance of an object is required, whose job is to forecast the future dynamics of the system. For instance, the output at time

Based on the above reasons, it can be concluded that the state of charge trajectories of the battery and supercapacitor can directly affect the final control effect of the predicted-energy control system [

To achieve optimal control of the energy in the composite energy storage system, it is necessary to optimize the power distribution between the lithium battery and the supercapacitor. In a specified operating interval under the conditions of known train speed, acceleration and running resistance, the total power required for train operation can be solved with

where

Based on

In the process of designing the model predictive controller, there is a certain functional relationship between the required power and the charging states of the lithium battery and supercapacitor, and there are mutual constraints within a certain range. As a result, the charging state of the lithium battery

From our analysis, the power solution and predictive control of the composite energy storage system can be approximated as a nonlinear and time-discrete system, with the following formulas obtained:

where

In this study, it is assumed that the global optimization objective function of urban rail trains running in a certain section is:

Specifically, during the operation of urban rail trains, the performance of the composite energy storage system is affected by the performance of the lithium batteries and supercapacitors. Under certain constraints, it is necessary to ensure that the power required by the vehicle reaches the ideal state in order to ensure stable, safe and energy-efficient train operation. The constraints can be summarized in

In summary, the energy control problem related to the dynamic planning of a composite energy storage system for urban rail transit can be investigated through the following steps:

(1) The system can be divided into several phases and the state variables can be discretized within the allowable range of variation.

(2) The required power and power allocation factor can be computed in the system interval to determine the objective function.

(3) In the process of going from the initial state

(4) Based on the initial value of the state variable, the optimal control sequence can be sought in the forward direction for the entire-cycle condition.

When the train runs at certain intervals, it is crucial to construct a predictive model for each stage of the travel process using information such as the current vehicle speed and acceleration to predict the speed and acceleration within a finite period. This prediction can be applied to forecast the operational state of the train in the time domain and calculate the power demand of the train for use in the later energy optimization control problem [

The train speed and acceleration state are adopted in the Markov model to forecast the train’s running state. By selecting the specified operating interval collection data as the observation sample, the nearest-neighbour method can be selected to discretize the sample vehicle speed and acceleration information into a limited number of series.

The collected train speed and acceleration samples are analyzed and summarized at a certain interval between the two stations, and the maximum likelihood estimation method can be adopted for the control system. The probability that the acceleration corresponding to the discrete velocity point

where

As a result, a one-step state transition probability matrix

From the current train running speed

According to the prediction and control model of the train’s composite energy storage system, it can be expected that the future prediction time-domain disturbance is known. Consequently, its control system can only attain the local optimal solution in the prediction time-domain at each control moment. Regarding the condition of the known global disturbance, the global optimal power allocation control strategy can be realized via the dynamic programming algorithm within a certain operating interval, thus obtaining the reference trajectory of the lithium battery and the supercapacitor SOC under ideal cases.

To acquire the desired reference trajectory, it is essential to guarantee that the lithium battery and supercapacitor SOCs operate within a certain range. The objective function of each period is constructed within the operating interval as shown in

where

Considering the features of the energy storage system and train operation, and to optimize the performance of the control system, it is necessary to constrain each part of the composite system according to

If the optimal control variables attained in the first step can be applied to the composite system, the next cycle is started and the process can be repeated. Based on the dynamic programming solution, the composite optimization problem can be solved to accomplish optimal energy control and consumption for the system.

This paper used the 4M2T grouping method for a B-type urban rail train. The partial parameters are: whole-vehicle mass = 180 t, traction motor power = 180–300 kW, and DC-DC conversion efficiency = 0.987. The motor controller has a conversion efficiency of 0.975 and the power supply voltage is 1500 V DC. Also, the maximum running speed is 80 km/h, the maximum starting acceleration (0–35 km/h) is 0.95 m/s^{2}, and the average acceleration (0–60 km/h) is ≥ 0.5 m/s^{2}. Besides, the deceleration is 1.0 m/s^{2}, the emergency braking deceleration is 1.2 m/s^{2}, the DC-DC conversion efficiency is 0.987, and the motor controller conversion efficiency is 0.992. The entire vehicle’s traction and resistance can be computed and the train’s running speed and acceleration can be measured based on real-time acquisition and calculation. Assuming that the predicted time domain is 10 and the control time domain is 6, the composite energy storage system and predictive control energy controller were simulated in the MATLAB/Simulink environment to better confirm the optimization effect of the energy control system control strategy according to the predictive control model [

In the simulation, experiments were conducted in a part of the city with the upper and lower steep slopes of Line 2. The effects of the energy control strategy are compared and analyzed from the dynamic programming and model prediction control. The simulation findings are shown in

From the above, the dynamic programming-based energy control allocation strategy is globally optimal according to the specific cycle conditions. Meanwhile, the results of the model predictive control strategy are similar, being suboptimal but easier to implement online.

A Markov model was proposed to predict the speed and acceleration states of a train in a future time domain, thereby precisely predicting the required power. The energy control strategy based on model predictive control can provide real-time performance, providing a solid foundation for the design of real-time energy control strategies.

Based on a Markov model of predictive control, a control strategy for predicting the energy of an urban rail composite energy storage system was designed. By selecting characteristic operation intervals for simulation testing, the effects of dynamic planning and model-based predictive control were compared. Dynamic programming is a globally optimal offline static method for analyzing future working conditions and highlights the effectiveness and real-time performance of model predictive energy control strategies.

We thank the team members for their hard work, the scientific research platform provided by the University, and the strong support of government funding.