# Nonlinear Hybrid Multipoint Model of High-Speed Train with Traction/Braking Dynamic and Speed Estimation Law

Authors: Chao Jia; Hongze Xu; Longsheng Wang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper establishes a NHMPM (Nonlinear Hybrid Multipoint Model) for HST (High-Speed Train) with the traction/braking dynamic and speed estimation law. Firstly, a full-order flux observer is designed using regional pole assignment theory to calculate the electromagnetic torque. The traction and braking forces are obtained according to this electromagnetic torque. Then the basic running resistance force is reformulated by considering the aerodynamic drag distribution characteristics, and the nonlinear in-train coupling force is analyzed as well. Next, the NHMPM including integer variables of running status and car types is built, where an adaptive parameter estimation algorithm and a speed estimation law are proposed to estimate unknown resistance coefficients and train speed, respectively. The effectiveness of the proposed algorithm, law, and NHMPM is verified through numerical simulations last.


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## Body

## 1. Introduction

HST can operate safely and efficiently mainly depending on the performance of ATO (Automatic Train Operation) system, and the accurate dynamic model is the first step-stone for designing ATO control law [1]; therefore, many researchers have paid more attention to it during the past few decades [2–13]. There are mainly two types of models: single-point model and multipoint model. The former model ignores in-train dynamics and considers the train as a single mass point [2], which is widely used in energy saving optimization [3], automatic driving [4], and precise stopping [5]. However, the train is connected by couplers to transmit large traction and braking force, and this nonlinear force has a strong effect on the longitudinal motion of HST [6]. Therefore, it is necessary to consider the in-train dynamics that leads to the latter one.The multipoint model was first introduced in 1990 [7] and then several scholars began to research on it [8–11]. In [8], a longitudinal multipoint model was established, in which the couplers were modelled as nonlinear springs, and the aerodynamic drag was assumed to act on the first car only. Reference [9] proposed a longitudinal model for heavy-haul trains, which took the coupler system as a linear spring with damping to simplify the calculation, and the aerodynamic drag was processed in the same manner as [8]. Reference [10] proposed a single coordinate model to solve the problem that the in-train coupling force was difficult to measure and directly control. In [11], a hybrid integer train model that includes running status integer variable was proposed after piecewise linearization of the train running resistance force.It is noted that in most existing work, the traction and braking force are treated as control variables; the dynamics of these two forces are ignored, and therefore, the actual dynamic characteristics of the train cannot be accurately reflected [12]. In a recent work, an attempt has been made in accounting for these two forces, where the traction and braking force are linked with the motor current by nonlinear relations and two single-point models about traction and braking dynamics are proposed, respectively, in [13]. It is worth noting that the traction motor is a high-order, strong coupling, multivariable nonlinear system, and a simply nonlinear function cannot describe the traction and braking dynamics comprehensively. In addition, the basic resistance force is obtained by DAVIS formula and the resistance coefficients are available in most existing works. Also, in order to facilitate the design of the controller, the basic resistance force is linearized. However, the resistance coefficients are variable and unknown due to the change of the line condition and the external environment [14], and the aerodynamic drag is related not only to the speed and resistance coefficients of the train, but also to the location of the car in practice [15].Furthermore, train speed is the premise of completing ATO control. However, the installation of speed sensors increases the system cost, reduces the reliability of the system, and is not suitable for operating in a harsh environment [16]. By contrast, speed sensorless technology can identify the motor speed from easily measured physical quantities (stator voltage or stator current, for example) [17] and the train speed is obtained through the linear relation between these two speeds. In recent years, full-order flux observer method has received the widespread attention [18–22]. Generally, in order to make the observer converge faster than the traction motor, the observer gain matrix is designed such that the poles of the observer are k times more than the motor model. Reference [18] presented a method to estimate the motor speed based on adaptive flux observer, and let k=1, but it caused the system to be unstable in low-speed range. Thus, [19] linearized the observer at the equilibrium point and increased the speed term; the range of k that can guarantee the stability of the system was obtained using the Routh stability criterion. An adaptive speed identification scheme for induction motor based on the hyperstability theory was proposed and k=1.2 was chosen in [20]. In [21], the adaptive speed estimation for IM was analyzed, the necessary and sufficient conditions for stability of the speed estimation system were derived as well. And, furthermore, a novel observer gain was designed to guarantee the stability over the whole speed range by Lyapunov theory in [22]. However, if the observer gain is designed by traditional exact pole assignment method, it will limit the damping speed, deteriorate robustness, and be hard to satisfy multiple performances in some practical applications [23], such as robotic arm movement model [24], uniform damping control of low-frequency oscillations in power system [25], aircraft carrier landing control system [26], and flexible buildings [27]. Therefore, circularly regional pole assignment can be adopted to design the observer gain, which can make the observer have more degrees of freedom.In this paper, traction and braking dynamics are described in detail, the basic resistance force is reformulated by considering aerodynamic drag distribution characteristics, and the nonlinear in-train coupling force is analyzed as well; the NHMPM including integer variables of running status and car types (locomotives or carriages) is built. An adaptive parameter estimation algorithm and a speed estimated law are proposed to provide unknown resistance coefficients and train speed required in the model, respectively.The rest of this paper is organized as follows. In Section2, traction and braking dynamics are discussed. Section 3 reformulates the basic resistance force and analyzes the nonlinear in-train coupling force. In Section 4, integer variables are introduced to represent running status and car types, and the NHMPM is derived, where an adaptive parameter estimation algorithm and speed estimation law are proposed to provide unknown resistance coefficients and train speed required in the model, respectively. Simulation is conducted in Section 5, and Section 6 draws the conclusion.


## 2. Traction and Braking Dynamics

### 2.1. Full-Order Flux Observer

The traction motor nominal model is described as the following equation in the stationary reference frame.(1)x˙s=Am0xs+Bmus,ys=is=Cmxs.Am0=-λRs0Lr+Rr0LsI+ωr0JλRr0I-λLrωr0J-Rs0I0=A11A12A210=a11I+a11′Ja12I+a12′Ja21I0,Bm=b1II=λLrII,Cm=I0,I=1001,J=0-110,0=0000.Here, xs=isψsT; is=isαisβT is stator current; ψs=ψsαψsβT is stator flux; us=usαusβT is stator voltage; Rs is stator resistance; Rr is rotor resistance; Ls is stator inductance; Lr is rotor inductance; Lm is mutual inductance; λ=1/(LsLr-Lm2); ωr is motor angular speed.


### 2.2. Observer Gain Matrix

Definition 1.
For the sake of simplicity, let us introduce the following notations.(2)Ama=Am0-cIr,Cmr=Cmr.(3)Amae=Am0+GCm-cIr.Here, c,0 is the center of Dc,r and r is radius of Dc,r as shown in Figure 1. G is observer gain matrix.Figure 1
Circular regionDc,r.Takex^s=i^sψ^sT as the state variable of the observer; the full-order flux observer of traction motor can be written as(4)x^˙s=Am0x^s+Bmus+Gy^s-yswhere “^” denotes observed quantities; y^s=i^s=Cmx^s. Using (1) and (4), the actual state estimation error system is (5)e˙=Am0+GCme.Here, e=x^s-xs.G is designed such that all the poles of the observer are assigned in circular region Dc,r; the circular region can be expressed as(6)kmaxσi+1T2+kmaxΩi2≤1T2.Here, σi, Ωi are the real part and the imaginary part of the ith pole of the motor model, and they satisfy(7)si=σi+jΩi∣Ωi=max⁡Ω1,Ω3,i=1,3.T is the sampling period, and it satisfies the following inequality.(8)T≤Tc≤-2σiσi2+Ωi2.And kmax=-2σi/Tc(σi2+Ωi2).Theorem 2.
LetR be a positive definite symmetric matrix of appropriate dimensions. All the eigenvalues of Am0+GCm belong to Dc,r and the state estimation error system (5) is stable under an observer gain matrix G if and only if there exists a positive P satisfying the following inequality(9)-PPAmaAmaTP-P-CmrTR-1Cmr<0.Then the observer gain matrix G is given by (10)G=-AmaP+CmrTR-1Cmr-1CmrTR-1.Proof.
By Schur Complement Lemma in [28], (9) implies(11)PAmaP+CmrTR-1Cmr-1AmaTP-P<0.Multiplying P-1 by both sides of inequality (11), we can get (12)AmaP+CmrTR-1Cmr-1AmaT-P-1<0.And there exists a positive definite symmetric matrix Q such that (13)AmaP+CmrTR-1Cmr-1AmaT-P-1+Q=0.Let(14)V=AmaeP-1AmaeT-P-1.According to (13), we have (15)P-1=AmaP+CmrTR-1Cmr-1AmaT+Q.Substituting (15) into (14) leads to(16)V=AmaeP-1AmaeT-AmaP+CmrTR-1Cmr-1AmaT-Q.Now denote U=AmaP+CmrTR-1Cmr-1 and notice that (17)AmaP+CmrTR-1Cmr-1AmaT=UCmrTR-1CmrUT+UPUT.Equation (16) is now given by (18)V=AmaeP-1AmaeT-UCmrTR-1CmrUT-UPUT-Q.When (19) is satisfied, (19)G=-AmaP+CmrTR-1Cmr-1CmrTR-1=-UCmrTR-1,we have(20)AmaeP-1AmaeT=UPUT.And then (21)V=UPUT-UCmrTR-1CmrUT-UPUT-Q=-UCmrTR-1CmrUT-Q=-GRGT+Q<0.That is, (22)V=AmaeP-1AmaeT-P-1<0.Or equivalently(23)AmaeTPAmae-P<0.That is, by Lemma 1 of [23], all the eigenvalues of Am0+GCm belong to D(c,r), the error system is stable, and the proof is completed.Then the electromagnetic torque is obtained by the following equation.(24)Te=npψ^sTJis=npψ^sβisα-ψ^sαisβ.Here, np denotes the number of pole-pairs of traction motor; the electromagnetic torque is different when the train run in traction mode or braking mode, and we denote them as Tet and Teb, respectively.


### 2.3. Traction Force and Braking Force

In this paper, we assume that the traction force and braking force of HST are all from the electromagnetic torque of traction motor, and they are linked with the motor stator voltage by a nonlinear relation. This nonlinear relation can be written as(25)Fti=2NmaηDTeti=Ktinpψ^sβisα-ψ^sαisβ=Kti·ht′usi,Fbi=2NmaηDTebi=Kbinpψ^sβisα-ψ^sαisβ=Kbi·hb′usi,i=1,2,3…p.Here, Fti, Fbi are traction force and braking force of ith car, respectively; Nm is the number of traction motors; a is gear ratio; η is transmission efficiency; p is the number of the locomotives; D is the diameter of half-worn wheel rolling circle.


## 2.1. Full-Order Flux Observer

The traction motor nominal model is described as the following equation in the stationary reference frame.(1)x˙s=Am0xs+Bmus,ys=is=Cmxs.Am0=-λRs0Lr+Rr0LsI+ωr0JλRr0I-λLrωr0J-Rs0I0=A11A12A210=a11I+a11′Ja12I+a12′Ja21I0,Bm=b1II=λLrII,Cm=I0,I=1001,J=0-110,0=0000.Here, xs=isψsT; is=isαisβT is stator current; ψs=ψsαψsβT is stator flux; us=usαusβT is stator voltage; Rs is stator resistance; Rr is rotor resistance; Ls is stator inductance; Lr is rotor inductance; Lm is mutual inductance; λ=1/(LsLr-Lm2); ωr is motor angular speed.


## 2.2. Observer Gain Matrix

Definition 1.
For the sake of simplicity, let us introduce the following notations.(2)Ama=Am0-cIr,Cmr=Cmr.(3)Amae=Am0+GCm-cIr.Here, c,0 is the center of Dc,r and r is radius of Dc,r as shown in Figure 1. G is observer gain matrix.Figure 1
Circular regionDc,r.Takex^s=i^sψ^sT as the state variable of the observer; the full-order flux observer of traction motor can be written as(4)x^˙s=Am0x^s+Bmus+Gy^s-yswhere “^” denotes observed quantities; y^s=i^s=Cmx^s. Using (1) and (4), the actual state estimation error system is (5)e˙=Am0+GCme.Here, e=x^s-xs.G is designed such that all the poles of the observer are assigned in circular region Dc,r; the circular region can be expressed as(6)kmaxσi+1T2+kmaxΩi2≤1T2.Here, σi, Ωi are the real part and the imaginary part of the ith pole of the motor model, and they satisfy(7)si=σi+jΩi∣Ωi=max⁡Ω1,Ω3,i=1,3.T is the sampling period, and it satisfies the following inequality.(8)T≤Tc≤-2σiσi2+Ωi2.And kmax=-2σi/Tc(σi2+Ωi2).Theorem 2.
LetR be a positive definite symmetric matrix of appropriate dimensions. All the eigenvalues of Am0+GCm belong to Dc,r and the state estimation error system (5) is stable under an observer gain matrix G if and only if there exists a positive P satisfying the following inequality(9)-PPAmaAmaTP-P-CmrTR-1Cmr<0.Then the observer gain matrix G is given by (10)G=-AmaP+CmrTR-1Cmr-1CmrTR-1.Proof.
By Schur Complement Lemma in [28], (9) implies(11)PAmaP+CmrTR-1Cmr-1AmaTP-P<0.Multiplying P-1 by both sides of inequality (11), we can get (12)AmaP+CmrTR-1Cmr-1AmaT-P-1<0.And there exists a positive definite symmetric matrix Q such that (13)AmaP+CmrTR-1Cmr-1AmaT-P-1+Q=0.Let(14)V=AmaeP-1AmaeT-P-1.According to (13), we have (15)P-1=AmaP+CmrTR-1Cmr-1AmaT+Q.Substituting (15) into (14) leads to(16)V=AmaeP-1AmaeT-AmaP+CmrTR-1Cmr-1AmaT-Q.Now denote U=AmaP+CmrTR-1Cmr-1 and notice that (17)AmaP+CmrTR-1Cmr-1AmaT=UCmrTR-1CmrUT+UPUT.Equation (16) is now given by (18)V=AmaeP-1AmaeT-UCmrTR-1CmrUT-UPUT-Q.When (19) is satisfied, (19)G=-AmaP+CmrTR-1Cmr-1CmrTR-1=-UCmrTR-1,we have(20)AmaeP-1AmaeT=UPUT.And then (21)V=UPUT-UCmrTR-1CmrUT-UPUT-Q=-UCmrTR-1CmrUT-Q=-GRGT+Q<0.That is, (22)V=AmaeP-1AmaeT-P-1<0.Or equivalently(23)AmaeTPAmae-P<0.That is, by Lemma 1 of [23], all the eigenvalues of Am0+GCm belong to D(c,r), the error system is stable, and the proof is completed.Then the electromagnetic torque is obtained by the following equation.(24)Te=npψ^sTJis=npψ^sβisα-ψ^sαisβ.Here, np denotes the number of pole-pairs of traction motor; the electromagnetic torque is different when the train run in traction mode or braking mode, and we denote them as Tet and Teb, respectively.


## 2.3. Traction Force and Braking Force

In this paper, we assume that the traction force and braking force of HST are all from the electromagnetic torque of traction motor, and they are linked with the motor stator voltage by a nonlinear relation. This nonlinear relation can be written as(25)Fti=2NmaηDTeti=Ktinpψ^sβisα-ψ^sαisβ=Kti·ht′usi,Fbi=2NmaηDTebi=Kbinpψ^sβisα-ψ^sαisβ=Kbi·hb′usi,i=1,2,3…p.Here, Fti, Fbi are traction force and braking force of ith car, respectively; Nm is the number of traction motors; a is gear ratio; η is transmission efficiency; p is the number of the locomotives; D is the diameter of half-worn wheel rolling circle.


## 3. Running Resistance Force and In-Train Coupling Force

In this section, the basic running resistance force is reformulated by analyzing the aerodynamic drag distribution characteristics, and the nonlinear in-train coupling force of each car is described simultaneously.


### 3.1. Running Resistance Force with Aerodynamic Resistance Distribution

The running resistance forcefri of each car consists of the basic running resistance force fbi and additional running resistance force fai; the empirical formula (DAVIS formula) is(26)fri=fbi+fai=mic0+cvvi︸rollingresistanceforce+micavi2︸aerodynamicdrag+fai,i=1,2,3…n.Here, mi,vi are the mass and speed of the ith car, respectively; c0,cv,ca denote the resistance coefficients, which are related to the HST type; fai includes gradient, tunnel, and curve resistance force. The rolling resistance force is dominant in low-speed range and as the speed increases, the aerodynamic drag becomes dominant.In fact, the aerodynamic drag consists of friction drag and pressure drag, which account for 24.7% and 75.3% of the total aerodynamic drag, respectively. Moreover, the aerodynamic drag of each car is not only related to its own mass and speed, but also related to the location of the car (the first car, the middle car, or the last car, for example) and whether to install air conditioning fairing, pantograph, and compartment connections, bogies, or other parts [15]. Therefore, the running resistance force can be reformulated as(27)fri=fbi+fai=mic0+cvvi+kmiMcavi2+fai.Here, M is the total mass of the train; kmi is the percentage of aerodynamic drag of ith car in the total aerodynamic drag.Take CRH3 (China Railway High-Speed 3 series Electric Multiple Unit) as an example; the parameters of CRH3 are shown in Table1. The basic running resistance force of each car is shown in Figure 2 when the speed varies from 0 m/s to 100 m/s (it is equivalent to the actual train speed from 0 km/h to 350 km/h). The dotted and solid lines represent the forces calculated by the first two items of (26) and (27), respectively.Table 1
The parameters of CRH3.
Symbol Value Unit m 1 - m 8 67.2,69.6,68.8,63.2, 60.8,68.8,69.6,68 t k1-k8 16.1, 19.7, 7.5,9.6, 10.6,7.0,14.1,15.4 % c 0 , c v , c a 0.42,0.0016,0.000132 -Figure 2
Basic running resistance force for CRH3.In addition, the force deviation between these two formulas is shown in Figure3. From Figure 3, we know that the basic running resistance force of each car has a large deviation before and after the formula modification, and the greater the speed, the larger the deviation. Moreover, the second car has the maximum deviation. Since the second and the seventh car are equipped with pantographs (see Figure 4), the pressure drag of these two cars is relatively greater than other cars without pantographs. Moreover, these two pantographs are installed opposite to each other, which results in two different wake flow fields; the intensity of the tail vortex induced by the second car’s pantograph is stronger, which leads to the greater pressure drag than the seventh [15]. Thus, the second car has the largest aerodynamic drag and maximum deviation.Figure 3
Basic running resistance force deviation between (26) and (27) of each car for CRH3.Figure 4
The framework of CRH3 series EMU and force analysis for one car.


### 3.2. In-Train Coupling Force

A train is made up of many cars through springs; the coupling force is divided into two parts: spring part and damping part, and its dynamic equation can be written as(28)finii+1=ksixi-xi+1︸springpart+kdvi-vi+1︸dampingpart.Here, finii+1 is the coupling force between the ith car and the (i+1)th car; xi,xi+1 are displacement of the ith car and (i+1)th car, respectively; ksi,kd are elastic coupling coefficient and damping coupling coefficient, respectively. ksi is the nonlinear function of displacement deviation, that is,(29)ksi=k01+μxi-xi+12i=1,2,…,n-1.Here, k0 is constant; μ=0 denotes linear spring, that is, ksi=k0, which does not exist in practice; μ<0 and μ>0 denote softening spring and hardening spring, respectively [8].


## 3.1. Running Resistance Force with Aerodynamic Resistance Distribution

The running resistance forcefri of each car consists of the basic running resistance force fbi and additional running resistance force fai; the empirical formula (DAVIS formula) is(26)fri=fbi+fai=mic0+cvvi︸rollingresistanceforce+micavi2︸aerodynamicdrag+fai,i=1,2,3…n.Here, mi,vi are the mass and speed of the ith car, respectively; c0,cv,ca denote the resistance coefficients, which are related to the HST type; fai includes gradient, tunnel, and curve resistance force. The rolling resistance force is dominant in low-speed range and as the speed increases, the aerodynamic drag becomes dominant.In fact, the aerodynamic drag consists of friction drag and pressure drag, which account for 24.7% and 75.3% of the total aerodynamic drag, respectively. Moreover, the aerodynamic drag of each car is not only related to its own mass and speed, but also related to the location of the car (the first car, the middle car, or the last car, for example) and whether to install air conditioning fairing, pantograph, and compartment connections, bogies, or other parts [15]. Therefore, the running resistance force can be reformulated as(27)fri=fbi+fai=mic0+cvvi+kmiMcavi2+fai.Here, M is the total mass of the train; kmi is the percentage of aerodynamic drag of ith car in the total aerodynamic drag.Take CRH3 (China Railway High-Speed 3 series Electric Multiple Unit) as an example; the parameters of CRH3 are shown in Table1. The basic running resistance force of each car is shown in Figure 2 when the speed varies from 0 m/s to 100 m/s (it is equivalent to the actual train speed from 0 km/h to 350 km/h). The dotted and solid lines represent the forces calculated by the first two items of (26) and (27), respectively.Table 1
The parameters of CRH3.
Symbol Value Unit m 1 - m 8 67.2,69.6,68.8,63.2, 60.8,68.8,69.6,68 t k1-k8 16.1, 19.7, 7.5,9.6, 10.6,7.0,14.1,15.4 % c 0 , c v , c a 0.42,0.0016,0.000132 -Figure 2
Basic running resistance force for CRH3.In addition, the force deviation between these two formulas is shown in Figure3. From Figure 3, we know that the basic running resistance force of each car has a large deviation before and after the formula modification, and the greater the speed, the larger the deviation. Moreover, the second car has the maximum deviation. Since the second and the seventh car are equipped with pantographs (see Figure 4), the pressure drag of these two cars is relatively greater than other cars without pantographs. Moreover, these two pantographs are installed opposite to each other, which results in two different wake flow fields; the intensity of the tail vortex induced by the second car’s pantograph is stronger, which leads to the greater pressure drag than the seventh [15]. Thus, the second car has the largest aerodynamic drag and maximum deviation.Figure 3
Basic running resistance force deviation between (26) and (27) of each car for CRH3.Figure 4
The framework of CRH3 series EMU and force analysis for one car.


## 3.2. In-Train Coupling Force

A train is made up of many cars through springs; the coupling force is divided into two parts: spring part and damping part, and its dynamic equation can be written as(28)finii+1=ksixi-xi+1︸springpart+kdvi-vi+1︸dampingpart.Here, finii+1 is the coupling force between the ith car and the (i+1)th car; xi,xi+1 are displacement of the ith car and (i+1)th car, respectively; ksi,kd are elastic coupling coefficient and damping coupling coefficient, respectively. ksi is the nonlinear function of displacement deviation, that is,(29)ksi=k01+μxi-xi+12i=1,2,…,n-1.Here, k0 is constant; μ=0 denotes linear spring, that is, ksi=k0, which does not exist in practice; μ<0 and μ>0 denote softening spring and hardening spring, respectively [8].


## 4. Nonlinear Multipoint Model of HST

In this section, NHMPM including integer variables of running status and car types is established, where an adaptive parameter estimation algorithm for estimating the unknown resistance coefficients is proposed and a train speed estimated law is derived to get train speed.


### 4.1. Nonlinear Hybrid Multipoint Model of HST

Take CRH3 as an example; it consists of four locomotives equipped with traction units (see Figure4, the black wheels) and four carriages, and we define the output of traction units as ue. Let γ=1 and γ=0 denote locomotives and carriages, respectively. In addition, we define “traction and cruise” states as “traction mode” because both of these two states require traction force; let δ=1 denote this mode; define “coasting” and “braking” states as “coast mode” and “braking mode”; let δ=0 and δ=-1 denote these two modes, respectively. Therefore, the output of the execution unit of the ith car is described as(30)uei=γδFti+Fbi+δFti-Fbi2=γδKtiht′usi+Kbihb′usi+δKtiht′usi-Kbihb′usi2i=1,2,3,…,n;δ=-1,0,1.Now we assume that the displacement of each car is the same when the coupler is not stressed. The force analysis of the other cars is the same except the first and the last car which lack coupling force; the force analysis is shown for the fifth car in Figure 4. According to the Newton’s second law, NHMPM can be expressed as (31).(31)m1v˙1=ue1-fin12-fr1=γδKt1ht′us1+Kb1hb′us1+δKt1ht′us1-Kb1hb′us12-k0x1-x2-k0μx1-x23-kdv1-v2-m1c0+cvv1-km1Mcav12-fa1,miv˙i=uei+fini-1i-finii+1-fri=γδKtiht′usi+Kbihb′usi+δKtiht′usi-Kbihb′usi2+k0xi-1-xi+k0μxi-1-xi3+kdvi-1-vi-k0xi-xi+1-k0μxi-xi+13-kdvi-vi+1-mic0+cvvi-kmiMcavi2-fai,i=2,3,…,n-1;mnv˙n=uen+finn-1n-frn=γδKtnht′usn+Kbnhb′usn+δKtnht′usn-Kbnhb′usn2+k0xn-1-xn+k0μxn-1-xn3+kdvn-1-vn-mnc0+cvvn-kmnMcavn2-fanγ=0,1;δ=-1,0,1x˙i=vii=1,2,3,…,nSequentially, input variable u≔us1,us2…usn2n×1T and state variable x≔v1,v2…vn,x1,x2…xn2n×1T are defined for transforming (31) to a nonlinear function form, where usi=usαiusβiT is traction motor stator voltage of the ith car. In addition, the additional resistance force of the train is not easy to describe in a mathematical form such that it will be regarded as an unknown and bounded disturbance term d; furthermore the speed and displacement of the first car are taken as the output variable y≔v1x1T. Therefore, (31) can be rewritten as the nonlinear function form.(32)x˙=f′x,u,d,y=Cx.Here, C=10…000…000…010…02×2n.


### 4.2. Adaptive Parameter Estimation

Resistance coefficients are unknown and time varying, which will affect the accuracy of the model in practice. In this section, an adaptive parameter estimation algorithm is proposed to estimate these coefficients online for NHMPM, and the algorithm is based on the idea of [29].The nonlinear model (32) is modified as the following form.(33)x˙=f0′x,u,d+C0+Cvx+Cax2.Here, Km=km1km2…kmn, MΛ=m1m2…mn, C0=C0p0n×n, Cv=Cvp0n×n0n×n0n×n, Ca=KmM/MΛCap0n×n0n×n0n×n, C0p=-c0c0…c0T, Cvp=-diag(cv,cv,…cv), Cap=-diag(ca,ca,…ca).Now, we define the following vector.(34)O≔C0,Cv,CaT.(35)Φ≔1,x,x2T.Then (33) can be rewritten as(36)x˙=f0′x,u,d+ΦTO.In order to simplify the adaptive parameter estimated algorithm and improve the engineering practicability, the resistance coefficients of each car are assumed to be equal as the train is in the same environment most of the time. Thus, we use the equivalent single-point model to estimate the resistance coefficients. It can be described as(37a)v˙e=f0′′us,d0+φTθ(37b)x˙e=ve.Here, d0 is bounded disturbance term; φ=1,ve,ve2T; θ=c0,cv,caT, θ∈Rnθ is an unknown time-varying parameter. However, we can obtain a rough range from wind tunnel testing. In addition, we give the following assumptions.Assumption 3.
There exists an initially known nominal compact setΘ0≜B(θ0,zθ0), which is described by an initial nominal estimate θ0 and associated error bound zθ0=sups∈Θ0s-θ0 such that θ∈Θ0.Let the state predictor for (37a) be denoted as v-˙e and define (38)v-˙e=f0′′us,d0+φTθ-+kaev+yωθ-˙.Here, θ-≔c-0,c-v,c-aT is parameter estimate vector; ka>0 is constant matrix; ev=ve-v-e is prediction error; yω is the output of the filter and it can be expressed as(39)y˙ω=φ-kayω,yωt0=0.Let θ~=θ-θ-; from (37a) and (38), (40)e˙v=φTθ~-kaev-yωθ-˙.Define(41)ς=ev-yωθ~.From (40) and (41), we can get (42)ς˙=-kaς,ςt0=evt0.The adaptive estimation law of parameter θ- is given by (43).(43)θ-˙=κyωTev-ς,θ-t0=θ0with κ=κT>0.Definition 4.
(1)
Y ∈ R n θ × n θ is generated from(44)Y=yωTyω,Yt0=0. (2)
κ _ = ϖ _ ( κ ); excitation index ε(t)=ϖ_(Y(t)); and contraction factor is(45)0<αt=11+κ_εt⩽1.
The uncertainty setΘ≜B(θ-,zθ) is updated as the updating of the parameter estimate θ- and its associated error bound zθ=sups∈Θs-θ-. And zθ is given as (46)zθ=zθε=Vε.Here, Vε(t)=α(t)Vε(t0), Vε(t0)=(1/2)(zθ0)2.Algorithm 5 (adaptive estimation algorithm of the parameterθ-).
Step 1. Initialize k a, φ, κ, θ0, ve(t0), yω(t0), ς(t0),andzθ(t0)=zθ0, θ-(t0)=θ-0, Θ0≜B(θ0,zθ0) at time t0.
Step 2. Measure (or estimate, see Section 4.3) the current value of ve(ti), and obtain yω,ev,ς by (39)-(42).
Step 3. Update θ-(ti) and zθ(ti) according to (43) and (46).
If the following conditions are met,(47)zθti⩽zθti-1-θ-ti-θ-ti-1.Otherwise, keep the value of the last time, which is(48)θ-ti,zθti=θ-ti-1,zθti-1.
Step 4. Iterate back to Step 2, incrementing i=i+1.


### 4.3. Train Speed Estimation

In this section, an adaptive speed estimated law for HST is proposed to get the train speed information instead of speed sensors.Assumption 6.
ω r 0 is constant in the control period; that is, dωr0/dt=0.In this paper, we assume that the motor parameters are constant except motor speed. Thus, the full-order flux observer of traction motor can be rewritten as(49)x^˙s′=A^mx^s′+Bmus+Gy^s′-ys=Am0+ΔAmx^s′+Bmus+Gy^s′-ysA^m=A^11′A^12′A^21′0=a11I+a^11′Ja12I+a^12′Ja21I0=-λRs0Lr+Rr0LsI+ω^rJλRr0I-λLrω^rJ-Rs0I0ΔAm=ΔωrJ-b1ΔωrJ00where x^s′=i^s′ψ^s′T; y^s′=i^s′=Cmx^s′; Δωr=ω^r-ωr0.According to (49) and (1), the state estimation error system is(50)e˙1=Am0+GCme1+ΔAmx^s′.Here, e1=x^s′-xs.Now we define a Lyapunov function candidate.(51)V1=e1Te1+ω^r-ωr02λ1.Here, λ1 is a positive constant. The time derivative of V1 by Assumption 6 becomes(52)dV1dt=e1TAm0+GCme1+ΔAmx^s′+Am0+GCme1+ΔAmx^s′Te1+2Δωrλ1dω^rdt=e1TAm0+GCmT+Am0+GCme1+2e1TΔAmx^s′+2Δωrλ1dω^rdt=e1TAm0+GCmT+Am0+GCme1+2ΔωrΔis′TJi^s′-b1ψ^s′+2Δωrλ1dω^rdt.By Theorem 2, the first term on the right side of (52) is a seminegative matrix. When (53) is satisfied, (53)dω^rdt=λ1Δis′TJb1ψ^s′-i^s′.The last two terms on the right side of (52) are equal, which leads to dV1/dt≤0. Therefore, it satisfies the Lyapunov stability law and guarantees the stability of the speed estimation system. And (53) is the angular speed identification adaptive law; then, PI adaptive law is adopted to satisfy the rapidity of identification, and it can be expressed as (54)ω^r=kwp+kwI∫dtΔis′TJb1ψ^s′-i^s′.Here, kwp, kwI are PI parameters, respectively. Then HST speed is obtained by (55).(55)v^=60πD1000a·30πω^r=1.8Dakwp+kwI∫dtΔis′TJb1ψ^s′-i^s′.Here, v^ is the speed of HST.


## 4.1. Nonlinear Hybrid Multipoint Model of HST

Take CRH3 as an example; it consists of four locomotives equipped with traction units (see Figure4, the black wheels) and four carriages, and we define the output of traction units as ue. Let γ=1 and γ=0 denote locomotives and carriages, respectively. In addition, we define “traction and cruise” states as “traction mode” because both of these two states require traction force; let δ=1 denote this mode; define “coasting” and “braking” states as “coast mode” and “braking mode”; let δ=0 and δ=-1 denote these two modes, respectively. Therefore, the output of the execution unit of the ith car is described as(30)uei=γδFti+Fbi+δFti-Fbi2=γδKtiht′usi+Kbihb′usi+δKtiht′usi-Kbihb′usi2i=1,2,3,…,n;δ=-1,0,1.Now we assume that the displacement of each car is the same when the coupler is not stressed. The force analysis of the other cars is the same except the first and the last car which lack coupling force; the force analysis is shown for the fifth car in Figure 4. According to the Newton’s second law, NHMPM can be expressed as (31).(31)m1v˙1=ue1-fin12-fr1=γδKt1ht′us1+Kb1hb′us1+δKt1ht′us1-Kb1hb′us12-k0x1-x2-k0μx1-x23-kdv1-v2-m1c0+cvv1-km1Mcav12-fa1,miv˙i=uei+fini-1i-finii+1-fri=γδKtiht′usi+Kbihb′usi+δKtiht′usi-Kbihb′usi2+k0xi-1-xi+k0μxi-1-xi3+kdvi-1-vi-k0xi-xi+1-k0μxi-xi+13-kdvi-vi+1-mic0+cvvi-kmiMcavi2-fai,i=2,3,…,n-1;mnv˙n=uen+finn-1n-frn=γδKtnht′usn+Kbnhb′usn+δKtnht′usn-Kbnhb′usn2+k0xn-1-xn+k0μxn-1-xn3+kdvn-1-vn-mnc0+cvvn-kmnMcavn2-fanγ=0,1;δ=-1,0,1x˙i=vii=1,2,3,…,nSequentially, input variable u≔us1,us2…usn2n×1T and state variable x≔v1,v2…vn,x1,x2…xn2n×1T are defined for transforming (31) to a nonlinear function form, where usi=usαiusβiT is traction motor stator voltage of the ith car. In addition, the additional resistance force of the train is not easy to describe in a mathematical form such that it will be regarded as an unknown and bounded disturbance term d; furthermore the speed and displacement of the first car are taken as the output variable y≔v1x1T. Therefore, (31) can be rewritten as the nonlinear function form.(32)x˙=f′x,u,d,y=Cx.Here, C=10…000…000…010…02×2n.


## 4.2. Adaptive Parameter Estimation

Resistance coefficients are unknown and time varying, which will affect the accuracy of the model in practice. In this section, an adaptive parameter estimation algorithm is proposed to estimate these coefficients online for NHMPM, and the algorithm is based on the idea of [29].The nonlinear model (32) is modified as the following form.(33)x˙=f0′x,u,d+C0+Cvx+Cax2.Here, Km=km1km2…kmn, MΛ=m1m2…mn, C0=C0p0n×n, Cv=Cvp0n×n0n×n0n×n, Ca=KmM/MΛCap0n×n0n×n0n×n, C0p=-c0c0…c0T, Cvp=-diag(cv,cv,…cv), Cap=-diag(ca,ca,…ca).Now, we define the following vector.(34)O≔C0,Cv,CaT.(35)Φ≔1,x,x2T.Then (33) can be rewritten as(36)x˙=f0′x,u,d+ΦTO.In order to simplify the adaptive parameter estimated algorithm and improve the engineering practicability, the resistance coefficients of each car are assumed to be equal as the train is in the same environment most of the time. Thus, we use the equivalent single-point model to estimate the resistance coefficients. It can be described as(37a)v˙e=f0′′us,d0+φTθ(37b)x˙e=ve.Here, d0 is bounded disturbance term; φ=1,ve,ve2T; θ=c0,cv,caT, θ∈Rnθ is an unknown time-varying parameter. However, we can obtain a rough range from wind tunnel testing. In addition, we give the following assumptions.Assumption 3.
There exists an initially known nominal compact setΘ0≜B(θ0,zθ0), which is described by an initial nominal estimate θ0 and associated error bound zθ0=sups∈Θ0s-θ0 such that θ∈Θ0.Let the state predictor for (37a) be denoted as v-˙e and define (38)v-˙e=f0′′us,d0+φTθ-+kaev+yωθ-˙.Here, θ-≔c-0,c-v,c-aT is parameter estimate vector; ka>0 is constant matrix; ev=ve-v-e is prediction error; yω is the output of the filter and it can be expressed as(39)y˙ω=φ-kayω,yωt0=0.Let θ~=θ-θ-; from (37a) and (38), (40)e˙v=φTθ~-kaev-yωθ-˙.Define(41)ς=ev-yωθ~.From (40) and (41), we can get (42)ς˙=-kaς,ςt0=evt0.The adaptive estimation law of parameter θ- is given by (43).(43)θ-˙=κyωTev-ς,θ-t0=θ0with κ=κT>0.Definition 4.
(1)
Y ∈ R n θ × n θ is generated from(44)Y=yωTyω,Yt0=0. (2)
κ _ = ϖ _ ( κ ); excitation index ε(t)=ϖ_(Y(t)); and contraction factor is(45)0<αt=11+κ_εt⩽1.
The uncertainty setΘ≜B(θ-,zθ) is updated as the updating of the parameter estimate θ- and its associated error bound zθ=sups∈Θs-θ-. And zθ is given as (46)zθ=zθε=Vε.Here, Vε(t)=α(t)Vε(t0), Vε(t0)=(1/2)(zθ0)2.Algorithm 5 (adaptive estimation algorithm of the parameterθ-).
Step 1. Initialize k a, φ, κ, θ0, ve(t0), yω(t0), ς(t0),andzθ(t0)=zθ0, θ-(t0)=θ-0, Θ0≜B(θ0,zθ0) at time t0.
Step 2. Measure (or estimate, see Section 4.3) the current value of ve(ti), and obtain yω,ev,ς by (39)-(42).
Step 3. Update θ-(ti) and zθ(ti) according to (43) and (46).
If the following conditions are met,(47)zθti⩽zθti-1-θ-ti-θ-ti-1.Otherwise, keep the value of the last time, which is(48)θ-ti,zθti=θ-ti-1,zθti-1.
Step 4. Iterate back to Step 2, incrementing i=i+1.


## 4.3. Train Speed Estimation

In this section, an adaptive speed estimated law for HST is proposed to get the train speed information instead of speed sensors.Assumption 6.
ω r 0 is constant in the control period; that is, dωr0/dt=0.In this paper, we assume that the motor parameters are constant except motor speed. Thus, the full-order flux observer of traction motor can be rewritten as(49)x^˙s′=A^mx^s′+Bmus+Gy^s′-ys=Am0+ΔAmx^s′+Bmus+Gy^s′-ysA^m=A^11′A^12′A^21′0=a11I+a^11′Ja12I+a^12′Ja21I0=-λRs0Lr+Rr0LsI+ω^rJλRr0I-λLrω^rJ-Rs0I0ΔAm=ΔωrJ-b1ΔωrJ00where x^s′=i^s′ψ^s′T; y^s′=i^s′=Cmx^s′; Δωr=ω^r-ωr0.According to (49) and (1), the state estimation error system is(50)e˙1=Am0+GCme1+ΔAmx^s′.Here, e1=x^s′-xs.Now we define a Lyapunov function candidate.(51)V1=e1Te1+ω^r-ωr02λ1.Here, λ1 is a positive constant. The time derivative of V1 by Assumption 6 becomes(52)dV1dt=e1TAm0+GCme1+ΔAmx^s′+Am0+GCme1+ΔAmx^s′Te1+2Δωrλ1dω^rdt=e1TAm0+GCmT+Am0+GCme1+2e1TΔAmx^s′+2Δωrλ1dω^rdt=e1TAm0+GCmT+Am0+GCme1+2ΔωrΔis′TJi^s′-b1ψ^s′+2Δωrλ1dω^rdt.By Theorem 2, the first term on the right side of (52) is a seminegative matrix. When (53) is satisfied, (53)dω^rdt=λ1Δis′TJb1ψ^s′-i^s′.The last two terms on the right side of (52) are equal, which leads to dV1/dt≤0. Therefore, it satisfies the Lyapunov stability law and guarantees the stability of the speed estimation system. And (53) is the angular speed identification adaptive law; then, PI adaptive law is adopted to satisfy the rapidity of identification, and it can be expressed as (54)ω^r=kwp+kwI∫dtΔis′TJb1ψ^s′-i^s′.Here, kwp, kwI are PI parameters, respectively. Then HST speed is obtained by (55).(55)v^=60πD1000a·30πω^r=1.8Dakwp+kwI∫dtΔis′TJb1ψ^s′-i^s′.Here, v^ is the speed of HST.


## 5. Numerical Simulation

In this section, CRH3 is chosen for the simulation to verify the validity of speed estimation law, adaptive parameters estimation algorithm, and NHMPM. In order to simplify the simulation and not to lose the generality as well, three cars are chosen (the first and last are locomotives and the middle is carriage). The simulation parameters are shown in Table2 [15, 16, 30, 31].Table 2
The simulation parameters.
Symbol Value Unit U d, UL, Pm, f, np 3200, 2750, 560, 138, 2 V , V , k W , H z , - Nm,D,Rs0,Rr0 16,0.875,0.1065,0.0663 - , m , Ω , Ω Ls,Lr,Lm,c0,cv,ca, 1.31,1.93,53.6,0.42,0.0016,0.000132 m H , m H , m H , - , - , - kd,k0,μ,a,η 5 × 1 0 6 , 2 × 1 0 7 , - 0.1 ,  2.788,0.975 N·s / m , N / m , - , - , - k1,k2,k3,m1,m2,m3 31.4,38.5,30.1,67.2,69.6,68 % , % , % , t , t , tIn Table2, UL is rated line voltage, Pm is rated power of traction motor, and f is rated frequency of traction motor.The verification framework is shown in Figure5, and the detailed process is as follows.Figure 5
The verification framework.Firstly, the target speed (see black line in Figure8) is given which is 15m/s (0-1s), 70 m/s (1-2s), 30 m/s (2-3s), 3 m/s (3-4s). The real speed (see blue line in Figure 8) and the stator current are measured by sensors after traction motor vector control. In addition, the estimated speed is obtained by (55) (see red line in Figure 8). Compare the estimated speed and real speed to verify the validity of the speed estimation law.Secondly, according to measured stator current and observed stator flux by full-order flux observer, the electromagnetic torque is obtained by (24), so we can get traction force and braking force by (25) and these forces are also taken as input for TMPM (Traditional Multipoint Model).Lastly, the calculated speed of NHMPM is obtained by (31). Compare the real speed and calculated speed to prove the validity of NHMPM. And determine whether the calculated speed of NHMPM is closer to the real speed than TMPM to verify the accuracy of NHMPM.In addition, determine whether the resistance coefficients estimated by Algorithm5 converge to the real values to verify the effectiveness of the adaptive parameter estimation algorithm.


### 5.1. The Validity of Speed Estimation Law

According to Table2, we can get (56)Am0=-54.23-650.313802.0496×105-650.31-54.23-2.0496×105380-0.10650000-0.106500,Bm=315.16700315.1671001.When ωr=1.5p.u., the poles of the motor model are -20.64+649.3i, -20.64-649.3i, -33.59-1.01i, -33.59+1.01i. We can get T≤9.78×10-5 by (8); let T=9×10-5 and then kmax=1.08. Therefore, the circular region Dc,r is obtained by (6) with center (c,0)=-1/9×105,0 and r=1/9×105. Figure 6 shows the trajectories of observer poles when ωr changes from 0 to1.5p.u. and the circular region Dc,r. From the figure, we know that all of the poles of observer belong to the circular region Dc,r. The poles of the motor model and observer are shown in Figure 7 when ωr=1.5p.u.Figure 6
The trajectories of observer poles and circular regionDc,r.Figure 7
The poles of the motor model and observer.Figure 8
The estimated speed of the train.According to the simulation parameter, solving the linear matrix inequality (9), the feasible solution of P and observer gain matrix G are(57)P=10-6×0.00650-2.0561-0.239200.00650.2392-2.0561-2.05610.23921832.8-2.87487×10-9-0.2392-2.0561-2.87487×10-91832.8,G=-0.02150.0008-0.00080.0215-2.3978×10-52.8138×10-62.8138×10-6-2.3978×10-5.And then the train speed can be estimated by (55). Figure 8 displays the estimated result which indicates that the estimated speed can converge to real speed after 0.3s at each period, and the speed estimation system is stable. The error curve between real speed and estimated speed is shown in Figure 9; the max, min, and mean error value are given in Table 3. From Figure 9 and Table 3, we know that the proposed speed estimation law in this paper can estimate the train speed accurately; the mean error is less than 0.1m/s at each period. In addition, the max error is 2.32m/s and the error rate is 3.32% (the target speed is 70m/s at this period).Table 3
The error value between real speed and estimated speed.
Value (m/s) Period (s) 0s-1s 1s-2s 2s-3s 3s-4s Max 0.1394 1.3407 2.3227 0.6965 Min -0.4277 -0.1374 -0.8565 -1.0835 Mean -0.0015 0.0910 -0.0128 -0.0378Figure 9
The error between real speed and estimated speed.


### 5.2. The Validity of NHMPM and Adaptive Parameter Estimation Algorithm

Figure10 shows the estimated resistance coefficients using the proposed adaptive parameter estimation algorithm. The unknown resistance parameters can converge to the real value accurately even when the real value changes from c0=0.42, cv=0.0016, ca=0.000132 to c0=0.47, cv=0.0026, ca=0.001132. It verifies the validity of the adaptive estimation algorithm.Figure 10
The estimated values of resistance coefficients.Figure11 and Table 4 show the verification results of NHMPM; it can be seen that the speed of NHMPM (green line) is basically consistent with the real speed that verifies the effectiveness of NHMPM. Moreover, compared with the TMPM, the NHMPM speed is closer to the real speed at the same traction and braking force. And the mean error between real speed and NHMPM speed is smaller than the mean error between real speed and TMPM speed at all periods especially at the periods of 2s-3s and 3s-4s, which indicates that NHMPM is more accurate than TMPM.Table 4
The max, min, and mean error between real speed and TMPM speed, real speed and NHMPM speed.
Period 0s-1s 1s-2s 2s-3s 3s-4s Value (m/s) Types TMPM NHMPM TMPM NHMPM TMPM NHMPM TMPM NHMPM  Max 0.1561 0.1550 0. 6063 0. 5372 0.9741 0.6317 -0.1487 0.2647 Min -0.5214 -0.5214 -0.7588 -0.7434 -0.8481 -0.4356 -1.2354 -0.8210 Mean 0.0763 0.0325 0.2871 0.2658 0.714 -0.2734 1.1202 -0.7064Figure 11
Train speed trajectory.The traction motor stator voltageusα,usβ and the electromagnetic torque of the first car are shown in Figures 12 and 13, respectively. Figure 14 displays the traction and braking force of three cars. As the second car is a carriage, its traction and braking force are zero. In contrast, the first and the third car are equipped with traction motor; therefore, they can supply traction and braking force in all modes.Figure 12
Traction motor stator voltage of the first car. (a)usα, (b) usβ.
(a) (b)Figure 13
The traction motor electromagnetic torque of the first car.Figure 14
Traction and braking force of each car.


## 5.1. The Validity of Speed Estimation Law

According to Table2, we can get (56)Am0=-54.23-650.313802.0496×105-650.31-54.23-2.0496×105380-0.10650000-0.106500,Bm=315.16700315.1671001.When ωr=1.5p.u., the poles of the motor model are -20.64+649.3i, -20.64-649.3i, -33.59-1.01i, -33.59+1.01i. We can get T≤9.78×10-5 by (8); let T=9×10-5 and then kmax=1.08. Therefore, the circular region Dc,r is obtained by (6) with center (c,0)=-1/9×105,0 and r=1/9×105. Figure 6 shows the trajectories of observer poles when ωr changes from 0 to1.5p.u. and the circular region Dc,r. From the figure, we know that all of the poles of observer belong to the circular region Dc,r. The poles of the motor model and observer are shown in Figure 7 when ωr=1.5p.u.Figure 6
The trajectories of observer poles and circular regionDc,r.Figure 7
The poles of the motor model and observer.Figure 8
The estimated speed of the train.According to the simulation parameter, solving the linear matrix inequality (9), the feasible solution of P and observer gain matrix G are(57)P=10-6×0.00650-2.0561-0.239200.00650.2392-2.0561-2.05610.23921832.8-2.87487×10-9-0.2392-2.0561-2.87487×10-91832.8,G=-0.02150.0008-0.00080.0215-2.3978×10-52.8138×10-62.8138×10-6-2.3978×10-5.And then the train speed can be estimated by (55). Figure 8 displays the estimated result which indicates that the estimated speed can converge to real speed after 0.3s at each period, and the speed estimation system is stable. The error curve between real speed and estimated speed is shown in Figure 9; the max, min, and mean error value are given in Table 3. From Figure 9 and Table 3, we know that the proposed speed estimation law in this paper can estimate the train speed accurately; the mean error is less than 0.1m/s at each period. In addition, the max error is 2.32m/s and the error rate is 3.32% (the target speed is 70m/s at this period).Table 3
The error value between real speed and estimated speed.
Value (m/s) Period (s) 0s-1s 1s-2s 2s-3s 3s-4s Max 0.1394 1.3407 2.3227 0.6965 Min -0.4277 -0.1374 -0.8565 -1.0835 Mean -0.0015 0.0910 -0.0128 -0.0378Figure 9
The error between real speed and estimated speed.


## 5.2. The Validity of NHMPM and Adaptive Parameter Estimation Algorithm

Figure10 shows the estimated resistance coefficients using the proposed adaptive parameter estimation algorithm. The unknown resistance parameters can converge to the real value accurately even when the real value changes from c0=0.42, cv=0.0016, ca=0.000132 to c0=0.47, cv=0.0026, ca=0.001132. It verifies the validity of the adaptive estimation algorithm.Figure 10
The estimated values of resistance coefficients.Figure11 and Table 4 show the verification results of NHMPM; it can be seen that the speed of NHMPM (green line) is basically consistent with the real speed that verifies the effectiveness of NHMPM. Moreover, compared with the TMPM, the NHMPM speed is closer to the real speed at the same traction and braking force. And the mean error between real speed and NHMPM speed is smaller than the mean error between real speed and TMPM speed at all periods especially at the periods of 2s-3s and 3s-4s, which indicates that NHMPM is more accurate than TMPM.Table 4
The max, min, and mean error between real speed and TMPM speed, real speed and NHMPM speed.
Period 0s-1s 1s-2s 2s-3s 3s-4s Value (m/s) Types TMPM NHMPM TMPM NHMPM TMPM NHMPM TMPM NHMPM  Max 0.1561 0.1550 0. 6063 0. 5372 0.9741 0.6317 -0.1487 0.2647 Min -0.5214 -0.5214 -0.7588 -0.7434 -0.8481 -0.4356 -1.2354 -0.8210 Mean 0.0763 0.0325 0.2871 0.2658 0.714 -0.2734 1.1202 -0.7064Figure 11
Train speed trajectory.The traction motor stator voltageusα,usβ and the electromagnetic torque of the first car are shown in Figures 12 and 13, respectively. Figure 14 displays the traction and braking force of three cars. As the second car is a carriage, its traction and braking force are zero. In contrast, the first and the third car are equipped with traction motor; therefore, they can supply traction and braking force in all modes.Figure 12
Traction motor stator voltage of the first car. (a)usα, (b) usβ.
(a) (b)Figure 13
The traction motor electromagnetic torque of the first car.Figure 14
Traction and braking force of each car.


## 6. Conclusions

This paper established a NHMPM including the integer variables of running status and car types for HST. As the basis of research, the traction/braking dynamics was discussed and the running resistance force was reformulated. The author also analyzed the nonlinear in-train coupling force. Besides, an adaptive parameters estimation algorithm and a train speed estimation law were proposed. The proposed algorithm and law provide the unknown resistance coefficients and train speed required in the model, respectively. At last, numerical simulations are conducted to verify the effectiveness of the proposed algorithm, law, and NHMPM. The results of the verification are shown as follows.(1) The estimated speed using the proposed estimation law can converge to the real speed accurately. The mean error between real speed and estimated speed is less than 0.1m/s at each period, and the maximum error is 2.32m/s (the target speed is 70m/s at this period). (2) The estimated unknown resistance coefficients can converge to the real value accurately even when the real value changes. (3) The speed of NHMPM is basically consistent with the real speed. Compared with TMPM, it is closer to the real speed at all periods especially at the periods of 2s-3s and 3s-4s.



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*Source: 1364657-2019-01-17.xml*