Train permanent magnet synchronous traction system control method based on fixed time coordination theory

Abstract

The application relates to the technical field of rail train control, in particular to a train permanent magnet synchronous traction system control method based on a fixed time coordination theory, which comprises the following steps: establishing a train permanent magnet synchronous traction system coupling model; defining a dynamic evolution law of the fixed time coordination theory and a permanent magnet synchronous traction system control law; establishing error variables of a speed and a current loop of the permanent magnet synchronous traction system; searching for an optimal slip speed and calculating a wheel speed reference value; establishing an error macro variable, and predicting a maximum convergence time of the system; through the above mode, the train operation control, the wheel-rail adhesion control and the permanent magnet synchronous traction motor control in the train permanent magnet synchronous traction system are simultaneously considered, and the coordination performance in the power unit is ensured.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of rail train control technology, and particularly relates to a train permanent magnet synchronous traction system control method based on fixed time coordination theory. BACKGROUND

[0002] Rail transportation has developed rapidly in recent decades and has become one of the most popular travel modes. The train traction system is a key part of measuring the performance of the train. In recent years, permanent magnet synchronous traction motors have gradually replaced traditional induction asynchronous motors, aiming to improve the power density and efficiency of the train traction system. Therefore, it is of great significance to study the train permanent magnet synchronous traction system control method for the development of rail transportation.

[0003] Most of the current train permanent magnet synchronous traction system control methods only focus on one aspect of permanent magnet synchronous motor control, train operation control or wheel-rail adhesion control, and ignore the coordinated performance of multiple traction motors and driving wheels in the power unit. Some methods focus on the speed synchronization control of the traction motor, while ignoring the interaction of the motor-wheel-rail-train. Some methods emphasize the operation control and adhesion dynamics of the train, but omit the implementation of the speed and torque commands on the motor side.

[0004] In summary, it is necessary to propose a train permanent magnet synchronous traction system control method that simultaneously considers train operation control, wheel-rail adhesion control and permanent magnet synchronous traction motor control, and ensures the coordinated performance in the power unit. SUMMARY

[0005] The present application aims to provide a train permanent magnet synchronous traction system control method based on fixed time coordination theory, which aims to simultaneously consider train operation control, wheel-rail adhesion control and permanent magnet synchronous traction motor control in the train permanent magnet synchronous traction system, and ensure the coordinated performance in the power unit.

[0006] To achieve the above-mentioned purpose, the present application adopts a train permanent magnet synchronous traction system control method based on fixed time coordination theory, which comprises the following steps:

[0007] Establishing a coupled model of the train permanent magnet synchronous traction system;

[0008] Defining the dynamic evolution law of the fixed time coordination theory and the control law of the permanent magnet synchronous traction system;

[0009] Establishing error variables of the speed and current loops of the permanent magnet synchronous traction system;

[0010] Searching for the optimal slip speed and calculating the wheel speed reference value;

[0011] Establishing error macro variables to predict the maximum convergence time of the system.

[0012] wherein, in the step of establishing the coupling model of the train permanent magnet synchronous traction system:

[0013] The matrix form of the coupling model of the train permanent magnet synchronous traction system is:

[0014]

[0015] wherein, subscript ".i" represents the parameters and states of the ith drive shaft, x = [v w ,i d ,i q ] T represents the state variable; u = [T m ,u d ,u q ] T represents the control variable; d = [-T L / J eq ,0,0] T represents the system disturbance variable;

[0016]

[0017] v w represents the train wheel speed; i d and i q represent the d and q axis currents of the motor respectively; T m represents the electromagnetic torque output by the motor; u d and u q represent the d and q axis voltages of the motor respectively; T L = RF μ / (η g i g ) represents the equivalent load on the motor side; J eq = J m i g / R + J w / (η g i g R) represents the equivalent rotational inertia on the motor side; B m and J m represent the viscous damping coefficient and rotational inertia of the motor respectively; ω m represents the mechanical angular speed of the motor; R s represents the stator resistance of the motor; P n represents the number of pole pairs of the motor; L d and L q represent the d and q axis inductances of the motor respectively; φ f represents the permanent magnet flux linkage of the motor; R represents the train wheel radius; F μ represents the train traction force; η g and i gJ and J represent the transmission efficiency and transmission ratio of the gearbox, respectively; J w represents the moment of inertia of the train wheel.

[0018] wherein, in the step of defining the dynamic evolution law of the fixed-time consensus theory and the control law of the permanent magnet synchronous traction system:

[0019] The dynamic evolution law is:

[0020]

[0021] The control law is:

[0022]

[0023] T = diag [T1, T2, T3] is a positive definite diagonal matrix; ψ = [ψ1, ψ2, ψ3] T is a macro variable; α2 and β2 are both normal numbers; m2 > n2 > 0 and q2 > p2 > 0 are odd integers; is the partial derivative of the macro variable with respect to the state variable.

[0024] wherein, in the step of establishing the error variables of the speed and current loops of the permanent magnet synchronous traction system:

[0025] The error variables are e .i = [e v.i , e d.i , e q.i ] T :

[0026]

[0027] wherein, subscript ".i" represents the parameters and states of the ith drive shaft, e v , e d and e q are the wheel speed error, motor d-axis current error and motor q-axis current error, respectively; e t and e s represent the wheel speed tracking and synchronization errors, respectively; k t and k s represent the wheel speed tracking and synchronization coefficients, respectively; and represent the motor d-axis and q-axis reference currents, respectively; N M = 4 represents the number of traction motors in a single motor car; represents the reference speed of the train wheel.

[0028] wherein, in the step of searching for the optimal slip speed and calculating the wheel speed reference value:

[0029] The optimal slip speed search method is to judge dμ / dvs the symbol:

[0030]

[0031] wherein v s and respectively represent the slip speed and the slip speed reference; c s represents the slip speed search step; μ represents the wheel-rail adhesion coefficient;

[0032] The wheel speed reference value is expressed as:

[0033]

[0034] wherein v t represents the train body speed.

[0035] wherein, in the step of establishing the error macro variable and predicting the maximum convergence time of the system:

[0036] The macro variable is:

[0037]

[0038] wherein σ, α1 and β1 are all normal numbers; m1>n1>0 and q1>p1>0 are odd integers;

[0039] The maximum convergence time of the permanent magnet synchronous traction system can be expressed as:

[0040]

[0041] wherein, ρ=(p1+q1) / (2q1), β1′=2 ρ β1 / σ, γ=(m2+n2) / (2n2), λ=(p2+q2) / (2q2), α′ 2j =2 γ α1 / T j , β′ 2j =2 λ β2 / T j , j=1,2,3.

[0042] The application is a train permanent magnet synchronous traction system control method based on fixed time coordination theory, which establishes a train permanent magnet synchronous traction system coupling model; defines dynamic evolution law of fixed time coordination theory and permanent magnet synchronous traction system control law; establishes permanent magnet synchronous traction system speed and current loop error variables; searches for optimal slip speed and calculates wheel speed reference value; establishes error macro variable, and predicts maximum convergence time of the system; realizes simultaneous consideration of train operation control, wheel-rail adhesion control and permanent magnet synchronous traction motor control in the train permanent magnet synchronous traction system, and guarantees the coordination performance in the power unit. BRIEF DESCRIPTION OF DRAWINGS

[0043] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or the prior art description. Obviously, the drawings in the following description only constitute some embodiments of the present application, and for those skilled in the art, other drawings can also be obtained without creative labor based on these drawings.

[0044] Figure 1 is a step flow chart of the train permanent magnet synchronous traction system control method based on fixed time coordination theory of the present application.

[0045] Figure 2 is a principle schematic diagram of the train permanent magnet synchronous traction system control method based on fixed time coordination theory of the present application.

[0046] Figure 3 is a force analysis schematic diagram of the transmission system between the permanent magnet synchronous motor and the wheel of the present application.

[0047] Figure 4 is a wheel-rail adhesion characteristic curve schematic diagram used in the present embodiment of the present application.

[0048] Figure 5 is a train body speed and wheel speed curve schematic diagram of the permanent magnet synchronous traction control system of the present application.

[0049] Figure 6 is a wheel speed tracking error and synchronization error curve schematic diagram of the permanent magnet synchronous traction control system of the present application.

[0050] Figure 7 is a current line optimal slip speed and slip speed curve schematic diagram of the permanent magnet synchronous traction control system of the present application.

[0051] Figure 8 is an equivalent load torque and output electromagnetic torque curve schematic diagram of the permanent magnet synchronous traction motor of the present application. DETAILED DESCRIPTION

[0052] The exemplary embodiments will be described in detail herein with reference to the attached drawings. The description herein relates to the drawings, in which the same numbers represent the same or similar elements, unless otherwise represented. The implementations described in the following exemplary embodiments do not represent all implementations consistent with the present application.

[0053] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the present application. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms "comprises" and / or "comprising," when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof.

[0054] It is to be understood that the singular forms "a", "an", and "the" include plural referents unless the context clearly dictates otherwise. It will be further understood that the terms "comprises" and / or "comprising," when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof.

[0055] Please refer to Figures 1-3 The present application provides a train permanent magnet synchronous traction system control method based on fixed time coordination theory, comprising the following steps:

[0056] S100: Establishing a train permanent magnet synchronous traction system coupling model.

[0057] Further, in the step of establishing a train permanent magnet synchronous traction system coupling model:

[0058] The matrix form of the train permanent magnet synchronous traction system coupling model is:

[0059]

[0060] Wherein, the subscript ".i" represents the parameters and states of the ith drive shaft, x = [v w ,i d ,i q ] T represents the state variable; u = [T m ,u d ,u q ] T represents the control variable; d = [-T L / J eq ,0,0] T represents the system disturbance variable;

[0061]

[0062] v w denotes the train wheel speed; i d and i q denote the d and q axis currents of the motor; T m denotes the electromagnetic torque output by the motor; u d and u q denote the d and q axis voltages of the motor; T L = RF μ / (η g i g ) denotes the motor side equivalent load; J eq = J m i g / R + J w / (η g i g R) denotes the motor side equivalent moment of inertia; B m and J m denote the viscous damping coefficient and moment of inertia of the motor; ω m denotes the mechanical angular speed of the motor; R s denotes the stator resistance of the motor; P n denotes the number of pole pairs of the motor; L d and L q denote the d and q axis inductances of the motor; φ f denotes the permanent magnet flux linkage of the motor; R denotes the train wheel radius; F μ denotes the train traction force; η g and i g denote the transmission efficiency and transmission ratio of the gearbox; J w denotes the moment of inertia of the train wheel.

[0063] In the present embodiment, the wheel-rail adhesion relationship is:

[0064]

[0065] where v w = Rω w denotes the train wheel linear speed; R denotes the train wheel radius; ω w denotes the train wheel angular speed; v t denotes the train body speed; μ denotes the wheel-rail adhesion coefficient; F μ denotes the train traction force; W denotes the axle load; g denotes the gravitational acceleration; a1, a2, a3 and a4 are all Polach model parameters.

[0066] The single axle dynamic model of the train is:

[0067]

[0068] where M represents the car mass; N M = 4 represents the number of motors inside the car; J m and J w represent the moment of inertia of the motor and drive shaft, respectively; T wm represents the torque applied by the drive shaft on the motor shaft through the gearbox; T mw represents the torque applied by the motor shaft on the drive shaft through the gearbox; i g = r g2 / r g1 and η g represent the gear ratio and transmission efficiency, respectively; r g1 and r g2 represent the radii of the driving and driven gears, respectively; B m = 0 and B w = 0 represent the viscous damping coefficients of the motor and drive shaft, respectively; T m represents the electromagnetic torque output by the motor; ω m and ω w represent the angular velocities of the motor and wheels, respectively; F f represents the basic running resistance of the train; b1, b2, and b3 are resistance coefficients.

[0069] The permanent magnet synchronous motor equation in the d-q synchronous rotating coordinate system is:

[0070]

[0071] where i d and i q represent the motor d and q axis stator currents, respectively; u d and u q represent the motor d and q axis stator voltages, respectively; L d and L q represent the motor d and q axis stator inductances, respectively; R s represents the motor stator resistance; P n represents the number of pole pairs; φ f represents the permanent magnet flux linkage.

[0072] Define x .i = [v w.i , i d.i , i q.i ] T , u .i = [T m.i , u d.i , u q.i ] T .

[0073] The train permanent magnet synchronous traction system coupling model is:

[0074]

[0075] Wherein, subscript ".i" represents the parameters and states of the ith drive shaft.

[0076]

[0077] T L = RF μ / (η g i g );

[0078] J eq = J m i g / R+ J w / (η g i g R).

[0079] S200: defining dynamic evolution law of fixed time coordination theory and permanent magnet synchronous traction system control law.

[0080] Further, in the step of defining dynamic evolution law of fixed time coordination theory and permanent magnet synchronous traction system control law:

[0081] The dynamic evolution law is:

[0082]

[0083] The control law is:

[0084]

[0085] T = diag [T1, T2, T3] is a positive definite diagonal matrix; ψ = [ψ1, ψ2, ψ3] T is a macro variable; α2 and β2 are both normal numbers; m2 > n2 > 0 and q2 > p2 > 0 are odd integers; is the partial derivative of the macro variable to the state variable.

[0086] In this embodiment, the macro variable ψ is defined to construct an invariant manifold as follows:

[0087] ψ = ψ (x, t) = 0;

[0088] The dynamic evolution law of the original coordination control theory is improved as follows by introducing the fixed time theory:

[0089]

[0090] ​wherein the positive definite diagonal matrix T = diag [T1, T2, T3] is related to the convergence speed; ψ = [ψ1, ψ2, ψ3] T is the macro variable to be designed; α2 and β2 are normal numbers; m2 > n2 > 0 and q2 > p2 > 0 are odd integers. The selection of the positive definite diagonal matrix T = diag [T1, T2, T3] needs to satisfy the following relationship:

[0091]

[0092] Substituting the above equation into the equation (1) can obtain:

[0093]

[0094] The above equation can be rearranged to obtain the control law of the permanent magnet synchronous traction system based on the fixed time theory as follows:

[0095]

[0096] wherein, is the partial derivative of the macro variable to the state variable.

[0097] S300: Establishing the error variables of the speed and current loops of the permanent magnet synchronous traction system.

[0098] Further, in the step of establishing the error variables of the speed and current loops of the permanent magnet synchronous traction system:

[0099] The error variables are e .i = [e v.i , e d.i , e q.i ] T :

[0100]

[0101] wherein the subscript ".i" represents the parameters and states of the ith drive shaft, e v , e d and e q are the wheel speed error, the motor d-axis current error and the motor q-axis current error respectively; e t and e s represent the wheel speed tracking and synchronization errors respectively; k t and k s represent the wheel speed tracking and synchronization coefficients respectively; and represent the motor d-axis and q-axis reference currents respectively; N M = 4 represents the number of traction motors in a single motor car; and represents the wheel reference speed of the train.

[0102] In the embodiment, the permanent magnet synchronous traction system speed and current loop error is defined, the tracking error and synchronization error of each wheel are controlled to be small, and each motor current reaches the reference value, so that the corresponding electromagnetic torque can be output. The speed and current loop error e .i v.i d.i q.i T

[0103]

[0104] S400: Search for the optimal slip speed and calculate the wheel speed reference value.

[0105] Further, in the step of searching for the optimal slip speed and calculating the wheel speed reference value:

[0106] The optimal slip speed searching method is to judge the sign of dμ / dv s

[0107]

[0108] Wherein, v s and respectively represent the slip speed and the slip speed reference; c s represents the slip speed search step; μ represents the wheel-rail adhesion coefficient;

[0109] The wheel speed reference value is represented as:

[0110]

[0111] Wherein, v t represents the train body speed.

[0112] In the embodiment, the optimal slip speed searching method judges the slip speed change trend according to the sign of dμ / dv s

[0113]

[0114] Wherein, represents the slip speed reference; c s represents the slip speed search step;

[0115] The train wheel speed reference value is the average value of the slip speed reference and the train body speed, as follows:

[0116]

[0117] S500: Establish error macro variable and predict the maximum convergence time of the system. ​​​​​​​

[0118] Further, in the step of establishing the error macro variable and predicting the maximum convergence time of the system:

[0119] The macro variable is:

[0120]

[0121] wherein σ, α1 and β1 are normal numbers; m1>n1>0 and q1>p1>0 are odd integers;

[0122] The maximum convergence time of the permanent magnet synchronous traction system can be expressed as:

[0123]

[0124] wherein, ρ=(p1+q1) / (2q1), β1′=2 ρ β1 / σ, γ=(m2+n2) / (2n2), λ=(p2+q2) / (2q2), α′ 2j =2 γ α1 / T j , β′ 2j =2 λ β2 / T j , j=1, 2, 3.

[0125] In the embodiment, the macro variable based on the fixed-time coordination theory is designed as:

[0126]

[0127] wherein σ, α1 and β1 are normal numbers; m1>n1>0 and q1>p1>0 are odd integers.

[0128] The Lyapunov function used in the Lyapunov theory is:

[0129]

[0130] The first-order derivative of the Lyapunov function is calculated as:

[0131]

[0132] Through the above formula, the following can be finally derived:

[0133]

[0134] wherein, ρ=(p1+q1) / (2q1), β1′=2 ρβ1 / σ, γ = (m2 + n2) / (2n2), λ = (p2 + q2) / (2q2), α' 2j = 2 γ α1 / T j , β' 2j = 2 λ β2 / T j , j = 1, 2, 3.

[0135] According to the fixed time theory, the time T max2 from any state to the invariant manifold ψ = 0 can be obtained max1 , and the time T s from the invariant manifold to the error e = 0:

[0136]

[0137] The maximum convergence time of the permanent magnet synchronous traction system can be expressed as:

[0138]

[0139] Embodiment:

[0140] The parameters of the four motors of the permanent magnet synchronous traction system are selected as follows: the stator resistance R d = 0.641 Ω; the d-axis inductance L q = 1.29 mH; the q-axis inductance L f = 0.659 Wb; the number of pole pairs P n = 4; the moment of inertia J m = 6 kg·m 2 .

[0141] The other parameters of the permanent magnet synchronous traction system are selected as follows: the radii of the front two wheels R .1 = R .2 = 0.41 m; the radii of the rear two wheels R .3 = R .4 = 0.42 m; the moment of inertia of the drive shaft J w = 80 kg·m 2 ; the number of motors in the motor car N M = 4; the mass of the motor car M = 51.06 t; the mass of the axle M = 13.5 t; the gear transmission ratio i g = 2.355; the gear transmission efficiency η g = 0.95; the acceleration of gravity g = 9.8 m / s 2; train running basic resistance parameters b1 = 6.796, b2 = 0.062, b3 = 0.00143; line dry adhesion Polach model parameters a1 = 0.25, a2 = 0.089, a3 = 0.25, a4 = 0.54; line wet adhesion Polach model parameters a1 = 0.3, a2 = 0.1, a3 = 0.3, a4 = 1.2.

[0142] Based on the above permanent magnet synchronous traction system parameters, hardware-in-the-loop simulation experiment is carried out, and the controller parameters are set as: T1 = T2 = T3 = 0.1, k t = 10, k s = 20, σ = 0.5, α1 = 1, β1 = 1, α2 = 2, β2 = 2, m1 = 7, n1 = 5, m2 = 5, n2 = 3, p1 = 3, q1 = 5, p2 = 1, q3 = 3.

[0143] Figure 4 The line adhesion curve used in this embodiment is that the adhesion coefficient increases first and then decreases with the increase of the slip speed. The left stable region of the dotted line is less than the maximum adhesion coefficient, and the right unstable region represents wheel idling or slipping. In addition, the maximum adhesion coefficient of the wet track is lower than that of the dry track. The carriages originally travel on the dry track, the front wheels transition from the dry track to the wet track at 20s, and return to the dry track at 50s, and the rear wheels are delayed by 0.5s compared with the front wheels.

[0144] The control results of the train permanent magnet synchronous traction system based on the fixed time coordination theory are shown in Figure 5 The line speed of each wheel and the vehicle body speed are shown in Figure 5 Obviously, the speed of the wheel is slightly higher than the speed of the train, and the difference is the slip speed. The control method of the application realizes anti-slip traction control, wheel speed tracking and synchronization control, and the wheel speed error convergence performance is excellent. When the adhesion condition changes or high-speed cruising, the method of the application can gradually track the reference wheel speed without appearing the chattering phenomenon.

[0145] Figure 6 The tracking error of each wheel and the synchronization error between the wheels are shown in When the adhesion condition changes at 20s and 50s, the maximum error value is not more than 0.1m / s, and the maximum error value in the stable state is only 0.025m / s.

[0146] Figure 7 The slip speed of each wheel and the reference slip speed under different conditions are shown in Figure 4The actual adhesion working point variation trend is shown by the arrows in the figure. Initially, on the dry track, the adhesion working point gradually moves from the origin to point A. At 20 s, the wheel transitions to the wet track, resulting in a jump from point A to point B, followed by a gradual move from point B to point C. At 50 s, when the car returns to the dry track, a jump from point C to point D occurs, followed by a gradual move from point D back to point A. Figure 7 The slip velocity shown is consistent with the above variations, indicating that the multi-permanent magnet synchronous traction system effectively implements the anti-slip function and ensures the maximum traction efficiency. According to:

[0147]

[0148] The maximum convergence time T of the algorithm used in the present application max The conservative estimate is 8.0343 s. The actual convergence time used during the start-up phase, the transition from the dry track to the wet track, and the transition from the wet track to the dry track is 1.82 s, 1.76 s, and 1.67 s, respectively.

[0149] Figure 8 The load torque and output electromagnetic torque on the motor side, mainly used to reflect the mechanical output characteristics of the permanent magnet synchronous traction system. During the start-up phase, the electromagnetic torque exceeds the load torque, resulting in a rapid increase in train speed. At 20 s, the car enters the low adhesion region, and the control electromagnetic torque is quickly reduced, thereby preventing wheel slip. At 50 s, when the car returns to the high adhesion region, the electromagnetic torque quickly recovers, allowing the traction process to continue until the maximum speed is reached. At about 100 s, the electromagnetic torque and the load torque are almost equal, indicating that the train has entered the high-speed cruise state and remains. The entire process shows that the method used in the present application can achieve traction control of the train while ensuring the anti-slip performance and high-efficiency traction of the train.

[0150] Other embodiments of the application will be apparent to those skilled in the art from consideration of the specification and practice of the application disclosed herein. It is intended that the application be limited only by the scope of the claims, including any amendments thereof, and including equivalents thereof.

[0151] It should be understood that the application is not limited to the precise construction that has been described above and shown in the accompanying drawings, and that various modifications and changes can be made by those skilled in the art without departing from the scope of the application.

Claims

1. A control method of a train permanent magnet synchronous traction system based on fixed-time coordination theory, characterized in that, The method comprises the following steps: A coupled model of the permanent magnet synchronous traction system of the train is established: ； Among them, the subscript Indicates the i Parameters and status of each drive axis, Represents state variables; Indicates the control quantity; represents the system disturbance; ； ； represents the train wheel speed; and respectively represent the motor's d and q shaft current; represents the motor output electromagnetic torque; and respectively represent the motor's d and q shaft voltage; represents the motor side equivalent load; represents the motor side equivalent moment of inertia; and respectively represent the motor's viscous damping coefficient and moment of inertia; represents the motor's mechanical angular speed; represents the motor's stator resistance; represents the motor's number of pole pairs; and respectively represent the motor's d and q shaft inductance; represents the motor's permanent magnet flux linkage; R represents the train wheel radius; represents the train tractive effort; and respectively represent the gearbox's transmission efficiency and transmission ratio; represents the train wheel moment of inertia; A dynamic evolution law of the fixed-time cooperative theory and a control law of the permanent magnet synchronous traction system are defined, including: The dynamic evolution law is: ； The control law is: ； is a positive definite diagonal matrix; is a macro variable; and are both positive constants; and is an odd integer; is the partial derivative of the macro variable with respect to the state variable; An error variable of a speed loop and a current loop of the permanent magnet synchronous traction system is established, and the error variable is: ： ； ； wherein the subscript denotes the i th driving axle parameter and state, , and are the wheel speed error, motor d shaft current error and motor q shaft current error, respectively; and denote the wheel speed tracking and synchronization error, respectively; and denote the wheel speed tracking and synchronization coefficients, respectively; and denote the motor d and q shaft reference current, respectively; denotes the number of traction motors within a single motor coach; denotes the train wheel reference speed; Search for the optimal slip speed and calculate the wheel speed reference value, determine the symbol: ； wherein and respectively denote the slip velocity and the slip velocity reference; denotes the slip velocity search step size; denotes the wheel-rail adhesion coefficient; A wheel speed reference value is expressed as: ； wherein, denotes the train body speed; An error macro variable is established, and a maximum convergence time of the system is predicted: The macro variable is: ； wherein , and are normal numbers; and are odd integers; The maximum convergence time of the permanent magnet synchronous traction system is expressed as: ； wherein , , , , , , , , .

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