Energy-Efficient Optimization Method of Urban Rail Train Based on Following Consistency

Abstract

Because of the short distance between stations in urban rail transit, frequent braking of urban rail trains during operation will generate a large amount of regenerative braking energy. Urban rail trains can reduce their actual traction energy consumption using regenerative braking energy. Therefore, an energy-efficient optimization method for urban rail trains is proposed. By taking the punctuality of trains as the premise, the weighted acceleration of trains is taken as the synergetic variable, the synergetic coefficient is introduced to construct the following consistency model, and its convergence is proved. By analyzing the influencing factors of the following consistency coordination time, an adaptive parameter adjustment strategy is designed to solve the latest secondary traction time and the corresponding maximum speed of the primary traction. In order to save communication resources, the event trigger function is used to construct trigger conditions, and the consistency algorithm is used to update the cooperative controller. The simulation results show that the weighted acceleration of the follower train achieves the following consistency on the premise of ensuring punctuality, and the actual traction energy consumption of the follower train is reduced by 5.73%. The proposed method provides a new strategy for the energy-efficient operation of urban rail trains.

1. Introduction

Urban rail transit has become an important mode of travel today, and as urban rail transit has grown rapidly, how to minimize the increasing energy requirements has become an increasingly important issue. Of the various end uses of energy in an urban railway system, the traction energy of the train is the most significant. Increasing attention by the researcher is therefore being focused on methods for optimizing train operations to reduce traction energy consumption [1]. Due to the characteristics of urban rail transit, the train will frequently brake during operation. Regenerative braking energy generated by braking can be fed back to the catenary for immediate use by other traction trains. Therefore, improving the utilization rate of regenerative braking energy is an important method to reduce the traction energy consumption of urban rail trains [2].

At present, many scholars at home and abroad improve the utilization of regenerative braking energy from the aspects of train schedule optimization and train coordinated control optimization. The optimization of train schedule mainly includes the optimization of train arrival and departure time, the optimization of dwell time between stations, and the allocation of redundant time between stations [3,4]. Yang et al. [5] reduced traction energy consumption by optimizing train arrival and departure time and adjusting departure intervals. Mo et al. [6] adopted an optimized comprehensive timetable to maximize the traction braking overlap time. Zhang et al. [7] optimized train energy efficiency by adjusting departure intervals. Some scholars studied the coordinated control of urban rail trains to improve the utilization rate of regenerative braking energy [8,9,10]. Xun et al. [11] introduced the traction power supply model into the optimization control algorithm of urban rail train operation, established a comprehensive model of energy-efficient driving, changed the train operation speed through the detection of traction network voltage, and realized the energy-efficient optimization control of urban rail trains. Sun et al. [8] proposed the regenerative braking energy distribution scheme and absorbed and utilized the regenerative braking energy by adjusting the running speed of adjacent trains. Bai et al. [9] proposed a train-coordinated control method based on rolling optimization to calculate the operation scheme with the lowest actual energy consumption in the next station in real time. In the literature [3,4,5,6,7,8,9,10,11], while improving the utilization rate of regenerative braking energy, without exception, it has had an impact on the train schedule and has brought trouble to residents’ travel arrangements. The urban rail train is regarded as a multi-agent, and the urban rail train cooperative control is the multi-agent cooperative control whose control basis is consistency [12]. As an important part of the consistency problem, the following consistency is highly valued by scholars. In the following consistency problem, the leader train keeps the original running state unchanged, and other trains follow the leader but change their running states under the effect of the consistency protocol [13]. In practical applications, control systems communicate through the network. Bandwidth and other network constraints bring higher challenges to multi-system cooperative control [14]. Therefore, Tabuada and other scholars proposed an event-triggered communication mode [15,16]. At present, research on multi-system consistency based on event-triggering mechanisms has achieved fruitful results [17,18]. Event triggering mechanism and consistency protocol have achieved good research results and applications in other fields, such as references [12,13,14,15,16,17,18], which provide ideas for the research of the paper.

The literature show that, at present, few articles apply the consistency theory to the research on the energy-efficient operation of urban rail trains, and the research on multi-vehicle coordinated control aiming at energy conservation is mostly at the cost of sacrificing the punctuality of trains or adding energy storage equipment. In view of this, in the process of studying the coordinated control of the energy-saving operation of urban rail trains, this paper introduces the following consistency theory and event trigger mechanism for the first time, the acceleration of each train is regarded as a consistent collaborative variable, and the collaboration coefficient is introduced to make it possible for the follower train and the leader train to achieve the following consistency. During the collaborative control process, the parameter adaptive control strategy is introduced to improve the utilization rate of regenerative braking energy of the leader train by increasing the number and time of consistent collaboration. At the same time, it ensures the punctuality of train operations between stations. The research in this paper can not only improve the utilization rate of regenerative braking energy of urban rail trains and reduce the actual traction energy consumption of urban rail trains but also expand the application field of the following consistency theory. It is of great significance to the intelligent development of urban rail transit and urban energy conservation and emission reduction.

2. Basic Knowledge and Problem Description

2.1. Algebraic Graph Theory

The undirected graph is used to describe the communication between trains in the system, which is the communication topology [14]. The communication topology can be expressed as Formula (1).

$$G=(V,E)$$

(1)

wherein G represents an undirected graph with n nodes, $V=\left\{v_1,v_2\cdots v_N\right\}$ represents the set of undirected graph nodes, and $E\subset V\times V$ represents the set of edges in undirected graphs. The adjacency matrix of the communication topology graph can be expressed as Formula (2).

$$A_n=\left. [a_{ij}\right]\in R^{n\times n}$$

(2)

wherein $a_{ij}=a_{ji}$, $a_{ii}=0$. The neighbor set of the train is defined as $N_i=\left\{v_j|v_j\in V:(v_j,v_i)\in E\right\}$. The degree matrix of a graph is expressed as $D=diag\left\{d_1,d_2,\cdots d_N\right\}$, $d_i={\displaystyle {\sum }_{j=1}^N{a}_{ij}}$. the Laplace matrix of a graph is expressed as Formula (3).

$$L={\left. [l_{ij}\right]}_{N\times N}=D-A$$

(3)

wherein $l_{ii}={\displaystyle {\sum }_{j\ne i}{a}_{ij}}$, $l_{ij}=-a_{ij}(i\ne j)$. If there is a path $\left\{(v_{i1},v_{i2}),(v_{i2},v_{i3}),\cdots (v_{ir},v_j)\right\}$, $v_i$ reaches $v_j$, then $v_i$ and $v_j$ are connected. If any two nodes in an undirected graph are connected, the graph is connected. If the graph is undirected connected, then 0 is the single-valued eigenvalue of matrix L, and the other eigenvalue is greater than 0.

2.2. Problem Description

The earlier-running urban rail trains are regarded as leaders, and the later-running trains are regarded as followers. They form a multi-agent system whose dynamic model is expressed as Formula (4).

$$\left\{\begin{array}{l}{\dot{x}}_0(t)=A{x}_0(t)\\ {\dot{x}}_i(t)=A{x}_i(t)+B\tau sat(u_i(t))\end{array}\right.$$

(4)

wherein $x_0(t)$ is the state of leaders, $x_i\left(t\right)\in {ℝ}^n$ is the states of followers, $A\in {ℝ}^{n\times n}$ and $B\in {ℝ}^{n\times p}$ are the system matrix, $\tau$ is the traction control coefficient, $u_i\left(t\right)\in {ℝ}^p$ is the control input of the follower, and $sat(\cdot )$ is a saturation function of symmetric input, as is shown in Formula (5).

$$sat(u_i)=\left\{\begin{array}{ll}{u}_{max},& u_i>u_{max}\\ u_i,& \left|u_i\right|\le u_{max}\\ -u_{max},& u_i<-u_{max}\end{array}\right.$$

(5)

In the case of adaptive adjustment of system parameters, an event-triggered consistency control strategy is proposed to reduce the update frequency of the controller, thereby reducing network consumption, avoiding network congestion, and extending network life.

The follower train achieves the purpose of energy conservation through the immediate utilization of the regenerative braking energy of the leader train [19]. Then the actual traction energy consumption E_i of train i between two stations can be expressed as Formulas (6)–(9).

$$E_i=E_t^i-E_b^{i0}$$

(6)

$$E_t^i={\displaystyle {\int }_{t\in T}{p}_t^i(t)dt={\displaystyle {\int }_{t\in T}{F}_t^i(v_i,t)\cdot v_i(t)dt}}$$

(7)

$$E_b^{i0}=\left\{\begin{array}{l}{E}_b^0 p_t^i(t)>p_b^0(t)\\ E_t^i p_t^i(t)\le p_b^0(t)\end{array}\right.$$

(8)

$$E_b^0={\displaystyle {\int }_{t\in T}{p}_b^0(t)dt={\displaystyle {\int }_{t\in T}{F}_b^0(v_0,t)\cdot v_0(t)dt}}$$

(9)

where $E_t^i$ is the total traction energy consumption of train i, $E_b^{i0}$ is the regenerative braking energy of the leader train absorbed by train i, $E_b^0$ is the regenerative braking energy generated during the braking process of the leader train, $p_t^i(t)$ is the instantaneous traction power of the train i, $p_b^0(t)$ is the leader of train i braking instantaneous power, $F_t^i(v_i,t)$ is the traction force of train i at time t, $F_b^0(v_0,t)$ is to lead the braking force of the train at time t, and $F_t^i(v_i,t)$ and $F_b^0(v_0,t)$ can be expressed as:

$$\left\{\begin{array}{l}{F}_t^i(v_i,t)=m_i\cdot a_i(t)+F_r^i(v_i,t)\\ F_b^0(v_0,t)=m_0\cdot a_0(t)-F_r^0(v_0,t)\end{array}\right.$$

(10)

wherein m_i is the quality of the train i, a_i(t) is the acceleration of train i, m₀ is the quality of the leader train, $F_r^i(v_i,t)$ is the basic running resistances of the follower train i, a₀(t) is the acceleration of the leader train, $F_r^0(v_0,t)$ is the basic running resistances of the leader train. $F_r^i(v_i,t)$ can be expressed as:

$$\left\{\begin{array}{l}{F}_r^i(v_i,t)=m_i\cdot F_{ur}^i(v_i,t)g\cdot {10}^{-3} (i=0,1,2\cdots )\\ F_{ur}^i(v_i,t)=a+b\cdot v_i(t)+c{v}_i^2(t)\end{array}\right.$$

(11)

wherein a, b, and c are empirical parameters determined experimentally, and v_i(t) is the speed of train i at time t.

The following conditions shall be met for the coordinated control of urban rail trains following consistency: ① The dwell time and running time of urban rail trains at each station are not adjustable, which are fixed constants. ② According to reference [9], the operation curve of the leader train adopts the traditional optimal energy-efficient operation curve of a single train; that is, the maximum traction force is used under traction conditions, the maximum braking force is used under braking conditions, and the coasting condition is used between traction conditions and braking conditions. ③ The leader train and the follower train are in the same power supply arm. The regenerative braking energy generated during the operation of the leader train under braking conditions can be used by the follower train under other traction conditions immediately, and the residual energy can be consumed by the trackside resistance after use. ④ The train is regarded as a single particle, and the ramp and tunnel resistance of the line are not considered temporarily.

3. Follow Consistency Cooperative Control Model

Definition 1.

For Equation (4), if the state of the agent finally satisfies$\underset{x\to \infty }{\lim }‖x_i(t)-x_0(t)‖=0$ under any initial conditions, this phenomenon is called the system obtaining the following consensus [20].

It can be seen from Equations (6)–(9) that train acceleration is the main variable affecting train energy consumption. Therefore, if the train acceleration is regarded as a synergetic variable, the energy conservation synergetic following the consistency of multiple trains can be described as Equation (12).

$$\underset{t\to \infty }{\lim }‖\mu a_i(t)-\lambda a_0(t))‖=0$$

(12)

wherein $a_0$ and $a_i$ are the acceleration of the leader train and follower train i, respectively; $\mu$ and $\lambda$ are synergy coefficients; $\mu$ is any real number; $\lambda =1$. ${\beta }_i=\mu a_i-\lambda a_0$. ${\beta }_i$ is defined as the collaborative deviation degree, whose value is the difference between the collaborative variables of the leader train and the follower train i.

In the process of energy-efficient operation of urban rail trains, the weighted accelerations of follower trains and leader trains do not need to be completely consistent. When the value of the weighted acceleration meets the requirements, the system can also be considered to have reached a consensus; therefore, when ${\beta }_i\le \epsilon$, it is said that the multi-train system has reached the $\epsilon$ synergy, where $\epsilon$ is the synergy error.

Based on the consistency protocol design method, the system adjustment coefficient p is introduced to define the event trigger function, as shown in Formula (13) [21].

$$‖e_i(t)‖\le \gamma ‖q_i(t_k^i)‖$$

(13)

If Formula (13) is valid, the system will not be triggered; otherwise, the system will update the controller. $e_i(t)=q_i(t_k^i)-q_i(t)$ is the error function of the system at time t, $t_k^i$ is the kth trigger time of train i, its trigger time is expressed as discrete ordered time series $\left\{t_0^i=0,t_1^i,\cdots ,t_k^i,t_{k+1}^i,\cdots \right\}$, $t\in \left. [t_k^i,t_{k+1}^i\right)$. $q_i(t)=b_i(x_i(t)-x_0(t))$ is the state error between agent i and the leader, $\gamma$ is the system regulating parameter, and it is required to meet $0<\gamma <1$. Based on the trigger function (13), the consistency control protocol is designed, as shown in Formula (14).

$$u_i(t)=-K{q}_i(t_k^i)=-\alpha K(x_i(t_k^i)-x_0(t_{\overline{k}}^0))$$

(14)

By taking the train acceleration as a cooperative variable, Formula (14) can be rewritten as Formula (15).

$$u_i(t)=-K{q}_i(t_k^i)=-\alpha K(\mu a_i(t_k^i)-\lambda a_0(t_{\overline{k}}^0))$$

(15)

wherein $t\in \left. [t_k^i,t_{k+1}^i\right)$ and $t_{\overline{k}}^0$ is the event triggering event sequence of leader train, $t_k^i$ is the event triggering event sequence of train i, $k$ and $\overline{k}$ are positive integers, $\alpha$ is a constant greater than 0, and $K$ is the control gain matrix. Formula (16) is then established.

$$\overline{k}\triangleq \arg \underset{l\in N,t\ge t_l^j}{\min }\left\{t-t_l^j\right\}$$

(16)

This formula indicates that in the value of natural number l, $t>t_l^j$, and $\overline{k}$ is the minimum of all values.

After the following consistency protocol is applied, the operation status of the leader train is not affected by the protocol. Therefore, only the controlled status of the follower trains is considered. The control structure of the follower train i is shown in Figure 1.

When train i meets the event triggering conditions, it receives the latest state information of the leader train and calculates the state error. Then, according to the designed consistency protocol, the following consistency control information after parameter setting is output.

Lemma  2.

Assume that for sets $R_{\zeta }=\{v,z:‖v_i-v_z‖\le u_{oi},\forall i=1,2,\cdots ,n\}$, where $v\in {ℝ}^p$ and $z\in {ℝ}^p$ are vector elements in $R_{\zeta }$, there is a dead zone nonlinear function $\varphi (v)$, then $\varphi {(v)}^{T}T(\varphi (v)+z)\le 0$, and $T\in {ℝ}^{p\times p}$ is an arbitrary positive definite diagonal matrix.

$\varphi {(v)}^{T}T(\varphi (v)+z)\le 0$ is the generalized sector condition, this equation is a deformation of sector condition $\varphi {(v)}^{T}T(\varphi (v)+v)\le 0$.

Theorem 3.

Regarding Formula (4), assuming that the system topology G is strongly connected and contains a directed spanning tree if there is a positive definite matrix $P\in {ℝ}^{n\times n}$, any diagonal positive definite matrix $T\in {ℝ}^{p\times p}$, $G\in {ℝ}^{m\times m}$, and $Z\in {ℝ}^{m\times m}$, a positive number $\eta$ and two positive scalars ${\tau }_1$ and ${\tau }_2$, $0<\gamma <1$. If Equations (17) and (18) are satisfied, where $a_{11}=(I_N\otimes PA)+(I_N\otimes A^{T}P)-2(H\otimes PBK)+{\tau }_1(I_N\otimes P)+\sigma (I_m\otimes H^T)(I_m\otimes H)$, $H=L+B$, $\sigma ={\tau }_2{\left(\frac{\gamma }{1-\gamma }\right)}^2$, then the error trajectory of the follower and the leader will be included in this ellipsoid; thus, the follower can achieve the desired consistency.

$$\left. [\begin{array}{l}{I}_n\otimes P { G}_i^T\\ \ast \eta u_{oi}^2\end{array}\right]\ge 0$$

(17)

$$\left. [\begin{array}{l}{a}_{11} I_N\otimes PB+TKH-TG I_N\otimes PBK\\ \ast -2(I_m\otimes T) TK\\ \ast \ast -{\tau }_2{I}_{Nm}\end{array}\right]<0$$

(18)

Proof of Theorem 3.

See reference [22] for the certification process. □

In order to prove that the Zeno phenomenon will not occur in the whole event-triggered control process, the paper gives Definition 2 and Theorem 2 with reference to [17].

Definition 4.

If the event trigger interval meets the conditions $\underset{k}{\inf }\left\{t_{k+1}^i-t_k^i\right\}>0$, the system can eliminate the Zeno phenomenon; thus, the system will not be infinitely triggered within the limited trigger time. Then the system does not have the Zeno phenomenon, in which the system will not be triggered infinitely in a finite time.

Theorem 5.

The system is shown in Formula (4), and its communication topology is strongly connected. Based on the event trigger control function (15) and trigger condition (13), if each train in the initial time system can obtain the initial state of other trains and each train can transmit the trigger time state information to other trains, Zeno phenomenon will not occur in all trains applying control protocol (15) and trigger condition (13).

Proof of Theorem 5.

Since $‖e_i(t)‖\le \frac{\gamma }{1-\gamma }‖q_i(t)‖$ is valid, we obtain $‖e_i(t)‖\le \frac{\gamma }{1-\gamma }‖H\widehat{x}‖$. Since ${\dot{e}}_i(t)=-{\dot{q}}_i(t)=-(H\otimes A)\overline{x}(t)-(H\otimes B)u(t)$, the result shown in Equations (19) and (20) are obtained.

$$\left|e_i(t)\right|\le {\displaystyle {\int }_{t_k^i}^t\left|(H\otimes A){\widehat{x}}_i(s)(H\otimes B)u_i(s)\right|ds}$$

(19)

$$\begin{array}{l}\left|(H\otimes A){\widehat{x}}_i(t)+(H\otimes B)u_i(t)\right|\le ‖(H\otimes A)\widehat{x}(t)+(H\otimes B)u(t)‖\\ \le ‖H\otimes A‖‖\widehat{x}(t)‖+‖H\otimes B‖‖u(t)‖\le ‖H\otimes A‖‖\widehat{x}(t)‖+‖H\otimes B‖‖HK‖‖\widehat{x}(t_k^i)‖\end{array}$$

(20)

To make ${‖\widehat{x}(t)‖}_{\infty }=\underset{t\ge 0}{\sup }‖\widehat{x}(t)‖$, Equation (19) can be converted into the following:

$$\left|e_i(t)\right|\le (t-t_k^i)\times (‖H\otimes A‖{‖\widehat{x}(t)‖}_{\infty }+‖H\otimes B‖‖HK‖‖\widehat{x}(t_k^i)‖)$$

(21)

That is,

$$(t-t_k^i)\ge \frac{\left|e_i(t)\right|}{‖H\otimes A‖{‖\widehat{x}(t)‖}_{\infty }+‖H\otimes B‖‖HK‖‖\widehat{x}(t_k^i)‖}$$

(22)

When the system is triggered, Equation (23) is valid.

$$\left|e(t)\right|>\frac{\gamma }{1-\gamma }‖H\widehat{x}‖$$

(23)

Because $0<\gamma <1$, therefore, $\left|e(t)\right|>0$ exists when the system is triggered, namely,

$$(t_{k+1}^i-t_k^i)\ge \frac{\left|e_i(t)\right|}{‖H\otimes A‖{‖\widehat{x}(t)‖}_{\infty }+‖H\otimes B‖‖HK‖‖\widehat{x}(t_k^i)‖}>0$$

(24)

Since the event trigger interval is greater than zero, the system does not have the Zeno phenomenon, and Theorem 5 is proved. □

4. Adaptive Adjustment of Control Parameters

Based on the following consistent, collaborative control model, the follower train improves the utilization rate of regenerative braking energy of the leader train by adding secondary traction conditions. Equation (25) can be obtained from the force analysis of the train.

$$\left\{\begin{array}{l}F=F_t+F_b+F_r\\ v(t)=v_0+{\displaystyle {\int }_{t_0}^{t}a(t)dt}\\ s(t)=s_0+{\displaystyle {\int }_{t_0}^{t}v(t)dt}\end{array}\right.$$

(25)

where $F$ is the resultant force on the train; $F_r$, $F_t$, and $F_b$ are train traction force, braking force, and running resistance, respectively. When the train is in traction condition, $F_b=0$ and $F_r<0$. When the train is in coasting condition, $F_b=0$, $F_t=0$, and $F_r<0$. When the train is under braking condition, $F_t=0$, $F_b<0$, $F_r<0$. $v(t)$, $a(t)$, and $s(t)$, respectively, represent the speed, acceleration, and displacement of the train at time t. $v_0$, $t_0$, and $s_0$, respectively, represent the initial speed, initial time, and initial displacement of the train.

It can be seen from Formula (15) that the weighted acceleration of the follower train is affected by the acceleration of the leader train when the following consistent, collaborative control conditions are met (the follower train starts to pull for the second time). In Equation (25), the acceleration increases, causing v to increase. Since the station spacing s remains unchanged, the train operation time t between stations decreases. If the dwell time between stations remains unchanged, the follower train will depart in advance at the next station, causing changes in the train timetable, which brings trouble to residents in travel time arrangements. At the same time, simply using secondary traction to achieve the goal of following consistent, collaborative control only improves the utilization rate of regenerative braking energy of the leader train, which does not play a role in saving actual traction energy consumption.

In view of this, the parameters that affect the actual traction energy consumption of trains were analyzed. It can be seen from Formula (15) that the event driving time of the follower train is affected by the collaboration coefficient in addition to its own collaboration variable and leader collaboration variable μ_i impact.

This paper selected $\lambda =1$ and $\epsilon =0.2$. The value of μ_i is shown in

Table 1. Value of coordination coefficient.

Leader Train Working Condition	Value of Follower Train Coordination Coefficient
Traction condition	${\mu }_i=\left\{\begin{array}{ll}1& \mathrm{When} \mathrm{train} i \mathrm{is} \mathrm{under} \mathrm{traction} \mathrm{condition}\\ -24& \mathrm{When} \mathrm{train} i \mathrm{is} \mathrm{under} \mathrm{coasting} \mathrm{condition}\\ -1& \mathrm{When} \mathrm{train} i \mathrm{is} \mathrm{under} \mathrm{braking} \mathrm{condition}\end{array}\right.$
Coasting condition	${\mu }_i=\left\{\begin{array}{ll}-0.04& \mathrm{When} \mathrm{train} i \mathrm{is} \mathrm{under} \mathrm{traction} \mathrm{condition}\\ 1& \mathrm{When} \mathrm{train} i \mathrm{is} \mathrm{under} \mathrm{coasting} \mathrm{condition}\\ 0.04& \mathrm{When} \mathrm{train} i \mathrm{is} \mathrm{under} \mathrm{braking} \mathrm{condition}\end{array}\right.$
Braking condition	${\mu }_i=\left\{\begin{array}{ll}-1& \mathrm{When} \mathrm{train} i \mathrm{is} \mathrm{under} \mathrm{traction} \mathrm{condition}\\ -1& \mathrm{When} \mathrm{train} i \mathrm{is} \mathrm{under} 1\mathrm{th} \mathrm{coasting} \mathrm{condition}\\ 24& \mathrm{When} \mathrm{train} i \mathrm{is} \mathrm{under} 2\mathrm{th} \mathrm{coasting} \mathrm{condition}\\ 1& \mathrm{When} \mathrm{train} i \mathrm{is} \mathrm{under} \mathrm{braking} \mathrm{condition}\end{array}\right.$

.

It can be seen from the train operation characteristics, Formula (15), and Table 1 that the follower train i can only carry out consistent, coordinated control with the leader train in the first coasting working condition, and the first inert time and duration are mainly affected by the first traction maximum speed v_h1. Based on the line, train parameters, and interstation operation time, the influence of v_h1 on the triggering time and participating in consistent coordination time can be obtained by changing the value of v_h1, as shown in Figure 2.

As shown in Figure 2, when $v_{h1}\ge v_{hc}$, the follower train can accept the consistent coordination of the leader train, and the time to participate in the consistent coordination decreases with the increase in $v_h$ because the trigger time of the follower train is affected by the braking time of the leader train. Therefore, when $v_{h1}\ge vhc$, the trigger time of the follower train is cft and remains unchanged with the increase in $v_{h1}$. Moreover, when $v_{h1}=v_{hc}$, the follower train takes the longest time to participate in consistency coordination. For $\forall v_{h1}$, $v_{h1}\in \left. [v_{h\min },v_h\right]$, $t_{cf}$ does not exist, which indicates that the follower train cannot participate in consistent coordination in this operation section, and the maximum traction force is used under traction conditions, the maximum braking force is used under braking conditions, and the coasting condition is used between traction conditions and braking conditions. Combined with Figure 2, we can see that $v_{h1}$ is associated with the train trigger time $t_{cf}$ and the consistency collaboration time $t_{xt}$ to obtain the parameter adaptive adjustment strategy, as shown in Formulas (26) and (27).

$$\left\{\begin{array}{l}\max t_{xt}^i=f(v_{h1}^i)\\ s.t.t_c^i\le t_{cf}^i\le t_{a1\_2}^i\\ v_h\le v_{max}\\ v_{hmin}\le v_{h1}\le v_h\end{array}\right.$$

(26)

$$\left\{\begin{array}{cc}{t}_{cf}^i=t_b^{0i}& v_{h1}\ge vhc\\ t_{xt}^i=t_{c2}^i-t_{a1\_2}^i& v_{h1}\ge vhc\end{array}\right.$$

(27)

where $t_c^i$, $t_{a1\_2}^i$, and $t_{c2}^i$ are the end time of the first traction, the start time of the second traction, and the end time of the second traction of the follower train i, and $t_b^{0i}$ is the braking time of the leader train in the corresponding driving section of the follower train i. In the process of consistency collaborative control, the follower train sets the minimum and maximum speed of primary traction and calculates the latest time of secondary traction of the follower train according to the minimum speed of primary traction. When the follower train meets the condition of consistent cooperative control, first judge whether the braking time of the leader train is greater than the starting time of the first coasting of the follower train and less than the latest time of the second traction of the follower train. If satisfied, the braking time of the leader train is the second traction time of the follower train. The second traction time of the follower train shall be regarded as the latest second traction time after the train is updated, and the maximum speed of the first traction of the follower train shall be deduced. The second traction time of the follower train is the train coordination time.

After self-adaptive adjustment of parameters, when the follower train participates in the consistency coordination control, by adjusting the maximum speed of the follower train at a time, it can not only increase the consistency coordination control time but also reduce the energy consumption of the train at a time, which truly saves the traction energy without changing the train schedule.

5. Simulation Analysis

A multi-agent system consisting of three follower trains and one leader train is established. In order to verify the effectiveness of the model, a multi-agent system simulation model was built using MATLAB. The corresponding Laplacian matrix is.

$$L=\left. [\begin{array}{cccc}3& -1& -1& -1\\ -1& 1& 0& 0\\ -1& 0& 1& 0\\ -1& 0& 0& 1\end{array}\right]$$

The degree matrix is $B=\mathrm{diag}\{3,1,1,1\}$.

By taking a line of a metro operation company as an example, the train parameters and operation parameters are shown in

Table 2. Basic parameters of trains.

Train Weight (t)	Maximum Speed Limit (km/h)	Maximum Traction (KN)	Maximum Braking Force (KN)	Maximum Traction Acceleration (m/s2)	Maximum Braking Acceleration (m/s2)	Maximum Allowable Running Time Error (s)	Basic Running Resistance Parameters
194	80	195	160	1	0.9	1	$a=2.031$ $b=0.0622$ $c=0.001807$

and

Table 3. Running distance and running time between stations.

Inter Station	1–2	2–3	3–4	4–5	5–6	6–7	7–8	8–9	9–10	10–11	11–12	12–13
Distance (m)	1850	2030	1558	1302	1395	1845	1753	2330	2105	2880	1560	2338
Running time (s)	125	135	103	88	99	123	118	156	138	185	108	159

.

The values of setting parameters $\lambda =1$, $\epsilon =0.2$, and μ_i are shown in Table 1. The consistency protocol is introduced, and the running time–speed curve of multiple trains is shown in Figure 3.

Figure 3 shows the change curve of leader train and follower train speed with time before and after the following consistency coordination. It can be seen from Figure 3 that Follower Train 1, Follower Train 2, and Follower Train 3 carried out six, seven, and three times of secondary traction, respectively; therefore, they carried out consistent, collaborative control with the leader train. It can be seen from Figure 3 that when the leader train runs for 1016 s, 1401 s, and 1995 s, more than one follower train meets the consistency and coordination conditions and conducts secondary traction to absorb and utilize the regenerative braking energy of the leader train, resulting in that the secondary traction power consumption of the follower train participating in the consistency and coordination is far greater than the regenerative braking energy of the leader trains in the corresponding time period. In order to reduce the total energy consumption of the train, no more than one follower train shall participate in the consistency coordination when the leader train is running between stations. Therefore, the coordination time of the train triggered at the same time is taken as the constraint, and the train with the longest time is taken for consistency coordination. Formula (26) is rewritten as Formula (28).

$$\left\{\begin{array}{l}\max t_{xt}^i=f{(v}_{h1}^i)\\ s.t.t_c^i\le t_{cf}^i\le t_{a1\_2}^i\\ v_h\le v_{max}\\ v_{hmin}\le v_{h1}\le v_h\\ t_{xt}^i\ge t_{xt}^j (i\ne j)\end{array}\right.$$

(28)

where $t_{xt}^i<t_{xt}^j$, $v_{h1}^i=v_h^i$, and $t_{cf}^i$ does not exist. At this time, the train operation time speed curve is shown in Figure 4.

In Figure 4, when Follower Train 1 runs to time 106 s, 276 s, 412 s, 544 s, and 1016 s; Train 2 runs to time 690 s, 853 s, 1632 s, and 1783 s; and Train 3 runs to time 1224 s, 1401 s, and 1995 s, consistency coordination is carried out, and energy-efficient optimization is carried out on the premise of ensuring the punctuality of the train.

Figure 5, Figure 6, and Figure 7, respectively, show the following change trend of follower train trigger time, consistency error, and collaboration variable under the effect of consistency protocol.

Under the action of the event trigger function, the urban rail train only communicates at the trigger time. It can be seen from Figure 5 that Follower Train 1, Train 2, and Train 3 were triggered five, four, and three times, respectively, in the operation cycle, and only one train can be triggered at the same time. As shown in Figure 4, the trigger time of the follower trains is the second traction time of the train. After the following consistency protocol is applied to the multi-train system, the weighted acceleration of the follower trains will change with the change in the leader’s weighted acceleration, as shown in Figure 7. In Figure 6, the error between the negotiation variables of the follower train and the leader train is less than 0.2. It can be seen from Figure 6 and Figure 7 that the consistency error of the follower train meets the requirement ${\epsilon }_i(t)<0.2$, so the follower train realizes coordination.

According to Formulas (6)–(11), under the above simulation conditions, the traction energy consumption and regenerative braking energy utilization of follower trains 1, 2, and 3 are shown in Figure 8, Figure 9 and Figure 10 and

Table 4. Comparison table of the regenerative energy used by the follower trains before and after the cooperation.

Follower Train	Regenerated Energy Absorbed before Collaboration (kW·h)	Actual Traction Energy Consumption before Collaboration (kW·h)	Regenerated Energy Absorbed after Collaboration (kW·h)	Actual Traction Energy Consumption after Collaboration (kW·h)	Energy- Efficient	Utilization Rate of Renewable Energy
Train1	0	81.54	19.85	75.81	5.73	23.70%
Train 2	0.01	94.55	19.64	93.01	1.54	23.45%
Train 3	0	63.61	15.75	57.15	6.46	18.81%

.

From Figure 8, Figure 9 and Figure 10, it can be seen that after the following consistency collaboration, although the traction energy consumption of the follower train increased compared with that before the collaboration, after the utilization of regenerative braking energy, the actual traction energy consumption of the follower trains decreased to varying degrees, with the largest reduction in Train 3. It can be seen from Table 4 that the total energy saving is 13.73 kW·h, and the regenerative braking energy utilization rate of the leader train is 65.96% compared with that before the following consistency coordination. The coordinated control of urban rail train tracking consistency is based on punctuality. To verify the effectiveness of the algorithm, it is also necessary to verify punctuality. After simulation, the punctuality of Follower Train 1, Train 2, and Train 3 is shown in

Table 5. Train punctuality.

		Stations 1–2	Stations 2–3	Stations 3–4	Stations 4–5	Stations 5–6	Stations 6–7	Stations 7–8	Stations 8–9	Stations 9–10	Stations 10–11	Stations 11–12	Stations 12–13
Train 1	Running time (s)	/	/	/	/	98.8	122.8	117.7	155.7	137.8	184.7	107.6	158.7
	Punctuality error (s)	/	/	/	/	0.2	0.2	0.3	0.3	0.2	0.3	0.4	0.3
Train 2	Running time (s)	124.8	134.7	102.8	87.7	98.8	122.8	117.7	155.7	137.8	/	/	/
	Punctuality error (s)	0.2	0.3	0.2	0.3	0.2	0.2	0.3	0.3	0.2	/	/	/
Train 3	Running time (s)	124.8	134.7	102.8	87.7	98.8	122.8	/	/	/	/	/	/
	Punctuality error (s)	0.2	0.3	0.2	0.3	0.2	0.2	/	/	/	/	/	/

. It can be seen from Table 5 that the punctuality errors of the three trains are all within 1 s, meeting the punctuality requirements.

6. Conclusions

Based on the following consistency theory, without changing the original train diagram and adding additional energy storage equipment, firstly, the urban rail train is regarded as a multi-agent system, the train acceleration is regarded as a negotiation variable, and the coordination coefficient is introduced. The following consistency protocol is designed based on the event-triggering mechanism. By using linear matrix inequality and Lyapunov stability theory, the consistency analysis is carried out, and it is proved that there is no Zeno phenomenon in the system. Then, a parameter adaptive adjustment strategy is designed. By taking the punctuality of the train as the constraint condition, the number and time of consistent coordination of follower trains are increased as much as possible to improve the utilization rate of regenerative braking energy of the leader train. Finally, the train diagram of a certain line of Jinan Metro Operation Company in China is taken as an example for simulation. The results show that, based on the method proposed in this paper, the follower train coordination variable realizes coordination and ensures the punctuality of the train. At the same time, the total utilization ratio of the follower trains to the leader train was increased to 23.7%, 23.45%, and 18.81%, respectively, and the actual traction energy consumption was reduced by 5.73%. The research results of this paper verify the effectiveness of the following consistency method in the energy-saving operation of urban rail trains, but the method studied in this paper only improves the regenerative braking energy utilization rate of the leader train and only considers the energy-saving effect of the follower train. Therefore, based on this paper, the following consistency of energy-saving operation of urban rail trains under switching topology can be further studied to further save actual traction energy consumption.