Journal of Advanced Transportation

Volume 2018, Article ID 7308058, 16 pages

https://doi.org/10.1155/2018/7308058

## Optimal Operation of High-Speed Trains Using Hybrid Model Predictive Control

School of Information Science and Engineering, Central South University, Changsha 410083, China

Correspondence should be addressed to Zhiwu Huang; nc.ude.usc@wzh

Received 1 August 2017; Revised 8 January 2018; Accepted 11 April 2018; Published 15 May 2018

Academic Editor: Paola Pellegrini

Copyright © 2018 Yingze Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The high-speed train operation process is highly nonlinear and has multiple constraints and objectives, which lead to a requirement for the automatic train operation (ATO) system. In this paper, a hybrid model predictive control (MPC) framework is proposed for the controller design of the ATO system. Firstly, a piecewise linear system with state and input constraints is constructed through piecewise linearization of the high-speed train’s nonlinear dynamics. Secondly, the piecewise linear system is transformed into a mixed logical dynamical (MLD) system by introducing the auxiliary binary variables. For the transformed MLD system, a hybrid MPC controller is designed to realize the precise control under hard constraints. To reduce the online computation complexity, the explicit control law is computed offline by employing the mixed-integer linear programming (MILP) technique. Simulation results validate the effectiveness of the proposed method.

#### 1. Introduction

With rapid development of high-speed railway, the operation safety, punctuality, and energy consumption of train have received more and more attentions. Nowadays, in China, the high-speed railway generally adopts distributed traction power to improve speed and traction efficiency, where locomotives and wagons are marshalled together to form a high-speed train [1]. For distributed traction power trains, there are some critical issues that need to be solved, such as vehicle traction and brake force allocation, operation safety, punctuality, and energy consumption [2].

To achieve the safety and multiobjective optimization, automatic train control system is developed for high-speed railway, and it is one of the key on-board equipment with high safety assurance [3]. The automatic train operation (ATO) system is the essential component that plays a key role in the train operation process [4]. The ATO system controls the train traction/brake force to follow the reference speed. Moreover, ATO system can reduce the energy consumption and improve riding comfort. Now, it is still a challenge to develop an efficient modeling and control method for ATO system [5].

Currently, the researches of automatic train control are mainly based on single-point model and multipoint model [6]. For single-point model, the whole high-speed train is simplified as a single rigid mass point, which ignores the coupling characteristics among the vehicles and the allocation of the distributed traction/brake force. The dynamic properties of single-point model are simple and easy to design. However, high-speed train adopts distributed traction power mode; thus the single-point model is obviously not able to describe the coupling characteristics and hydrodynamic dispersion characteristics of the train. Thus multipoint model is favoured to study the automatic train control strategy.

Yang and Sun [7, 8] analyzed the coupling characteristics among vehicles of high-speed train and established train multipoint dynamical model. For both push-pull driving and distributed driving train types, the hybrid automatic train controller was designed and was compared by controller and controller, respectively. Lin et al. [9] proposed a control strategy based on improved sliding mode control to overcome the effect of the disturbance in the process of train operation, which had good robustness and effectively restrained the high frequency chattering phenomenon. Thus the work reduced the impact of the frequent switching of control input and increased riding comfort. To simplify the controller design, Song et al. [10] utilized the geometric topology to reduce multiple positions to one position for multipoint model. In their work, to address the nonlinear saturation constraints of traction and brake forces, the robust adaptive controller was designed with low computing load, which had robustness for train disturbance and uncertainty of model parameters.

Based on the multipoint model, Zhuan and Xia [11] have achieved constructive results on the automatic train control strategy of the heavy-duty train in South Africa. Specifically, Zhuan and Xia [12] put forward three kinds of offline open-loop optimal operation strategies, whose goals are to minimize the workshop bonding force, ensure the safety of train driving, and reduce the maintenance costs of the coupler buffer. Then, Chou and Xia [13] proposed a closed-loop cruising linear quadratic regulator controller to minimize the running cost of electric air brake system in heavy-haul train, in which the control objective includes the speed tracking, coupling force, and energy consumption. In order to overcome the communication limitation, the vehicle barrier was introduced and the controller was reconstructed based on the current orbital slope. In addition, Zhuan and Xia [14] adopted the output feedback adjustment method to regulate the speed of the heavy train and verified the application conditions of their proposed method.

As a summary, Scheepmaker et al. [15] presented an extensive review on energy-efficient train control and the related topic of energy-efficient train timetabling, from the first simplest model of a train running on a level track to the advanced models and algorithms of the last decade dealing with varying gradients and speed limits and including regenerative braking. And there have appeared various theories and technologies to realize the efficient and safety train operation, such as the passivity-based cruise control [16], the robust adaptive control [17], and the iterative learning and fault detection approaches [18, 19].

Li et al. [20] considered the optimal guaranteed cost of cruise control for high-speed train movement. The sufficient condition for the existence of guaranteed cost cruise control law is given in terms of linear matrix inequalities. And a convex optimization problem is formulated to determine the optimal guaranteed cost that minimizes the performance upper bound of the cruise control law. Ye and Liu [21] propose a novel approach to solve the complex optimal train control problems in closed loop by introducing some simplifications. The operation sequence consists of maximum traction, speed holding, coasting, and maximum braking on each subsection of the track or a constant force is applied on each subsection. Zhao et al. [22] proposed a new cluster consensus technique to design the distributed control law, by which the trains can track the desired speeds asymptotically and the distance between the neighboring cars can be kept in the ideal range.

However, with the improvement of the requirement for ATO control performance, the above methods lack the ability to address the nonlinear, multiconstraint, and multiobjective problem based on the multipoint model, which leads to their limitations in practical applications. Especially for online operation process of high-speed trains, it raises the nonlinear resistance force which makes many classical control methods based on the linear resistance force model or the single equilibrium point linearizable model difficult to be implemented. And it is necessary to consider the safe speed, the saturation characteristics of traction and braking units, and the work conditions such as maximization of bonding forces, operating punctuality, energy-efficiency, and reduction of coupling wear forces. The model predictive control has the advantages of fully considering the input and state constraints of the system and the multiobjective optimization problems in the design of the controller, which is more suitable for the design of ATO controller [23, 24].

In this paper, an optimal automatic train control strategy is proposed for high-speed trains based on model predictive control. Due to the high computation complexity of nonlinear train model, the common method is to linearize the nonlinear system model at the equilibrium points [25]. However, if the nonlinear system has a wide range of operating condition, there are many deviations on entire running process for singular linearized model with one equilibrium point. Thus, the piecewise linear systems are utilized to approximate the original system, which can describe complex nonlinear systems accurately [26]. In this paper, the nonlinear running resistance is fitted by multiple line segments to construct the piecewise linear system model of train.

In the traditional controller design based on piecewise linear model, the linear submodel will be determined according to the initial state of receding horizon; then the predicted control is solved based on the determined sublinear model, which is fixed in the entire receding horizon [27]. However, for each actual control step, the system state changes and the corresponding submodel switch may also occur. The fixed submodel is difficult to approximate the original nonlinear model, especially near the model switching point. Therefore, the traditional model prediction control just according to current linear submodel can result in a large prediction error.

In order to solve the above problems, the logic variables are introduced to describe the switch of different linear submodels in this paper. By using this method, the piecewise linear system can be transformed into a hybrid logic dynamical system [28, 29]. For the hybrid logic dynamical system, the model switch can be described by changing the logical variables. Then the hybrid model predictive control (MPC) can be utilized to generate the control inputs based on the constructed mixed logic dynamical system.

The hybrid MPC controller uses the mixed logic dynamical model to predict the future evolution of the system at each time step, and a certain performance index is optimized under operating constraints with respect to a sequence of future input moves. The first of such optimal solution is the control action applied to the plant. For the hybrid MPC controller, the optimal problem that needs to be solved is a mixed integer linear program (MILP) due to the hybrid system description. Because the mixed logic dynamical model ensures that the submodel can switch on the entire receding horizon. Thus, more accurate approximation can be obtained by hybrid MPC and the prediction error is reduced.

Compared to the existing work, the proposed hybrid MPC makes a superior trade-off between the model complexity and control performance. The proposed controller can obtain the optimal input sequence without excessive simplifying assumption, while the piecewise linear model reduces the computation complexity under the sufficient precision compared to the classical MPC method. The main contributions in this paper can be summarized as follows.

(1) The nonlinear multiple mass point dynamical model of high-speed trains is constructed. In the model, the complex nonlinear characteristics of running resistance are piecewise linearized and the piecewise linear description is transformed to mixed logic dynamical system.

(2) Based on the MLD system model, a hybrid MPC framework is introduced and the MLD model is used as the prediction model. Based on the MLD prediction model, the control problem can be formulated as a unified mixed integer linear programming (MILP) problem, even the piecewise linear characteristics of train model.

(3) For the formulated MILP problem containing logical variables and continuous variables, the branch and bound method () is utilized to transform the original MILP problem into linear programming (LP) by relaxing the logic variable constraints.

The remainder of the paper is organized as follows. In Section 2, we establish the piecewise linear model of the high-speed train and transform the piecewise linear model to MLD model. In Section 3, a mixed logic dynamical system is developed. In Section 4, a hybrid MPC controller is designed by analyzing the performance indices and constraints of high-speed trains. Then, the simulation results are provided to validate the effectiveness of the controller in Section 5. Finally, the major conclusion of this paper is given in Section 6.

#### 2. The Model of High-Speed Train

In this section, we establish the nonlinear multiple mass-point dynamic model of high-speed trains by analyzing their dynamical characteristics. Because the nonlinear model is complex for designing the system controller, the nonlinear curve of the running resistance is piecewise linearized and the piecewise linear model of the train is constructed.

##### 2.1. The Multiple Mass-Point Model of Trains

Figure 1 presents the multiple mass-point model structure for the simplified electric multiple power units. The total number of locomotives and wagons is denoted by . These vehicles are connected by the vehicle hook couplers. The mechanical properties of the couplers are usually described as the “elastic-damping” system. Thus, the hook coupler force between vehicle and vehicle is as follows:where is the offset value of the relative equilibrium position of the hook coupler between vehicle and vehicle , is the natural length of the coupler and is the half length of vehicle , and is the displacement of vehicle . Since the length of each vehicle is essentially the same and constant (denoted by ) and the hook coupler has the same type, the balance position between two vehicles can be denoted by . Thus . And the hook coupler force (1) can be transferred to