Research on fault tracing method of traction drive control system

Abstract

Fault propagation is a common occurrence in traction drive control systems. The propagation of faults among components creates difficulties in traceability. Therefore, this paper focuses on the traction transmission control system and proposes a fault tracing method based on fault propagation. First, establish a fault propagation model that includes spatiotemporal characteristics. Then, extract the fault characteristics and the time it takes for faults to propagate at various observation points throughout system operation. Finally, match the spatiotemporal characteristics of the corresponding observation points in the fault propagation model, so as to determine the fault type and location, allowing for fault traceability. The proposed method’s feasibility is verified using a system example. The effectiveness of the proposed method in the traction drive control system was verified through system examples, which can meet the requirements of rapid fault tracing and is effective for weak faults.

Introduction

Fault diagnosis is a crucial method for monitoring the reliable and safe operation of complex systems. Fault tracing serves as a vital component of fault diagnosis to identify the type of fault and locate the fault location [1, 2]. Fault tracing technology has been extensively researched in various fields, including industrial process control and transportation equipment management. Due to the complex internal wiring of the traction control system, the interweaving of multiple physical fields, and the high coupling of functional and electrical connections between components, a fault occurring in one device will propagate to other locations, causing difficulties in identifying the origin of the fault [3, 4]. However, the current fault tracing of the traction control system of high-speed trains mainly focuses on diagnosing a single location of the traction control system when the device or subsystem malfunctions and lacks the study of the mechanism by which faults propagate. The study of the fault propagation features in the traction drive control system can, on the one hand, trace the root cause of faults and, on the other hand, find out the influence of different faults on the observed values of the position parameters of adjacent subsystems for monitoring [5].

Fault propagation is a common phenomenon in real systems, such as traffic jams, mass power faults in electrical systems, Internet outages, chemical, and chemical system accidents. Thesis [6] investigates on fault propagation by mining different effect trajectories in control loops for transient faults that are difficult to detect accurately in networked control systems. In Thesis [7], in the case of fault propagation among components in air handling systems, such as heating and air-conditioning, a dynamic hidden Markov model is used to identify fault modes, which effectively improves fault diagnosis accuracy. In Thesis [8], damage propagation modeling of aerogenerators is implemented. Thesis [9] proposed an improved symbolic transfer entropy and determination of weight threshold method to explore the fault propagation law and then traced the root cause of the fault by analyzing the information transfer changes between nodes and the fault propagation path. Paper [10] proposed a method that can efficiently generate a complete test set of double faults by analyzing the propagation paths of single faults and selecting undetected double faults to generate new test patterns, thus covering most of the double faults for a given current. Paper [11] described how mechanical disturbances on the drive shaft propagate from the disturbance torque to the current at the drive system supply input for mechanical faults in permanent magnet synchronous motor drive systems. Paper [12] proposed a new framework for a fault propagation path modeling method for power systems based on membrane computing and used an event-reinforced neural system to model the fault propagation path, which has the capability of graphical modeling and parallel knowledge reasoning to intuitively reveal the fault propagation path. By reviewing the literature, the current research on fault propagation is mainly focused on network systems, chemical industry, power systems, electronic circuits, etc., while the research oriented to traction drive control systems and traction motor fault propagation is rare.

Due to the intricate wiring of the traction drive control system, the interweaving of multiple physical fields, and the high coupling and high density of functional and electrical connections between components, it makes the fault propagation characteristics between system component units [13]. There are few studies on fault propagation and traceability for traction drive control systems and traction motors, and almost all studies on fault propagation are only from a spatial perspective without considering the temporal characteristics of fault propagation, while the occurrence, spread, propagation, and accumulation of system faults are time delayed [14]. The introduction of time into the study of fault propagation can provide a more realistic and accurate description of the system fault propagation, and the proposed methods such as fault diagnosis or sensor arrangement related to the time factor will be more reasonable on this basis [15, 16]. Therefore, analyzing the temporal features of fault propagation is imperative.

In addition, fault propagation in traction drive control systems is characterized by its dynamic nature and diversity of fault modes, which presents significant challenges in developing fault propagation models and conducting analyses. The dynamic nature refers to the operating condition of the traction drive control system, that is, the variable speed of the train when a fault occurs and propagates; the diversity of fault modes means that many types of faults can occur in the traction drive control system, which can be divided into traction motor faults, variable flow faults such as device faults, TCU (traction drive control unit) faults, and sensor faults, and each major fault can be divided into several fault types. Different fault types may correspond to different fault causes, and the same fault cause may cause multiple fault types. This increases the difficulty of fault analysis in the traction control system. In addition, conducting fault tracing research on weak faults can effectively avoid the occurrence of major accidents, help organize and arrange maintenance on site, and have significant economic and social benefits. This paper mainly studies the propagation of traction motor faults in the traction drive control system.

Traction drive control system fault propagation modeling

Generic model

Set up $Q$ observation points at various locations of the traction drive control system (the observation points should be placed based on the structural characteristics of the system, utilize the available sensor conditions, and establish them at measurable positions) and establish the signal propagation model of $p$ observation points during normal operation of the system.

$${Z}_{p} \left({t}_{p}\right)= {S}_{p, p-1} \cdot {Z}_{p-1} \left({Z}_{p-1}\right)$$

(1)

where ${S}_{p, p-1} =f\left({G}_{p, p-1}, {t}_{p}\right)$ $Q$ denotes the signal at the ${Z}_{p} \left({t}_{p}\right)$ $p \left(p=\mathrm{1,2},\dots Q\right)$ observation point, which can be various physical quantities such as current and voltage; ${Z}_{p-1} \left({t}_{p-1}\right)$ denotes the signal at the $p-1$ observation point; ${t}_{p}$ and ${t}_{p-1}$ denote the time variables at the $p$ and $p-1$ observation points, respectively; ${t}_{p}= {t}_{p-1} + {\Delta t}_{p, p-1}$ and ${\Delta t}_{p, p-1}$ are the time required for the signal at the $p-1$ observation point to propagate to the $p$ observation point; and ${S}_{p, p-1}$ denotes the transfer function from the signal at the $p-1$ observation point ${Z}_{p-1} \left({t}_{p-1}\right)$ to the signal at the $p$ observation point ${Z}_{p} \left({t}_{p}\right)$, where ${G}_{p, p-1}$ is determined by the system structure between the two observation points; when $p=1$, ${Z}_{1}\left({t}_{1}\right) = {S}_{\mathrm{1,0}}\cdot {Z}_{0}\left({t}_{o}\right)$ is set at the $p=0$ point at the $Q$ th observation point, i.e., ${Z}_{0} \left({t}_{0}\right)= {Z}_{Q} \left({t}_{Q}\right)$ means that the signal propagation forms a closed loop, and ${S}_{\mathrm{1,0}}$ indicates the transfer function from $p=Q$ signal ${Z}_{Q} \left({t}_{Q}\right)$ propagation to the point ${Z}_{1} \left({t}_{1}\right)$ with $p=1$; if the point $p=0$ is set at the power supply or traction motor, it means that the signal propagation from the observation point $p=0$ to the observation point $Q$ occurs in an open-loop format. Equation (1) uses the signal at the $p-1$ observation point to characterize the signal at the $p$ observation point, that is, the signal at the $p$ observation point is propagated from the signal at the $p-1$ observation point. Similarly, ${Z}_{p-1}\left({t}_{p-1}\right)= {S}_{p-1, p}\cdot {Z}_{p}\left({t}_{p}\right)$ and ${S}_{p-1, p}$ represent the transfer function from the signal at the observation point $p$ ${Z}_{p}\left({t}_{p}\right)$ to the signal at observation point $p-1$ ${Z}_{p-1}\left({t}_{p-1}\right)$.

Equation (1) can be described as follows:

$${Z}_{p}\left({t}_{p}\right)= {S}_{p-1, p}\cdots {S}_{\mathrm{2,1}}\cdot {S}_{\mathrm{1,0}}\cdot {Z}_{0}\left({t}_{0}\right)= {S}_{i, i-1}\cdot {Z}_{0}\left({t}_{0}\right)$$

(2)

Equation (2) characterizes the signal at the $p$ observation point by the signal at $=0$ ${Z}_{0}\left({t}_{0}\right)$, i.e., the signal at the $p$ observation point is propagated from ${Z}_{0}\left({t}_{0}\right)$ through ${Z}_{1}\left({t}_{1}\right)$, ${Z}_{2}\left({t}_{2}\right)$, and ${Z}_{p-1}\left({t}_{p-1}\right)$.

When a fault occurs in the traction drive control system, the $p$ observation point signal is expressed as follows: ${\sum }_{fp,h}^{k}\left({t}_{fp}^{k}\right)={z}_{p}\left({t}_{fp}^{k}\right)\oplus {z}_{fp,h}^{k}\left[{f}_{h}^{k}\left(\cdot \right),{S}_{p,h}, {t}_{fp}^{k}\right]$(3).

where ${Z}_{p}\left({t}_{fp}^{k}\right)$ is the signal at the observation point of $p$ during normal operation of the traction drive control system; ${Z}_{p}\left({t}_{fp}^{k}\right)= {Z}_{p}\left({t}_{p}\right)$ ${Z}_{fp, h}^{k}\left({t}_{fp}^{k}\right)$ is the signal at the observation point of when a fault of type occurs at $p$ $k$ $h$, where $k=1, 2, \cdots {n}_{f}$ indicates the type of fault in the traction drive control system (different locations of the same component are considered as different types of faults) and $h$ is the location of the fault (different types of faults may have the same location in the system), $h=\mathrm{1,2}\cdots {n}_{\mathrm{g}}$; ${S}_{p,h}$ is the signal transfer function from the fault location $h$ to the observation point;$p$ ${f}_{h}^{k}\left(\cdot \right)$ is the ${z}_{fp,h}^{k}$ which is the source signal of the fault in the traction drive control system at the point $h$ when a fault of type $k$ occurs; ${t}_{fp}^{k}$ is the time variable at the observation point; $p$ ${z}_{fp, h}^{k} \left[{f}_{h}^{k}\left(\cdot \right), {S}_{p,h}, {t}_{fp}^{k}\right]$ is the evolving fault signal propagated by the source signal ${f}_{h}^{k}\left(\cdot \right)$ from the point $h$ to the observation point $p$ and is a function of ${f}_{h}^{k}\left(\cdot \right)$,${S}_{p,h}$ and ${t}_{fp}^{k}$; and is the signal operation, which can be summing or multiplying, i.e., the signal at the observation point $p$ after a fault occurs in the system is obtained by summing or multiplying the signal without the fault signal part ${Z}_{p}\left({t}_{fp}^{k}\right)$ with the evolving fault signal ${z}_{fp,h}^{k}$.

$${z}_{fp,h}^{k}\left({t}_{fp}^{k}\right)= {Z}_{p}\left({t}_{fp}^{k}\right)+ \left({t}_{fp}^{k}\right)+{z}_{fp,h}^{k}\left[{f}_{h}^{k}\left(\cdot \right),{S}_{p,h}, {t}_{fp}^{k}\right]$$

(4)

or

$${z}_{fp,h}^{k}\left({t}_{fp}^{k}\right)= {Z}_{p}\left({t}_{fp}^{k}\right)*{z}_{fp,h}^{k}\left[{f}_{h}^{k}\left(\cdot \right),{S}_{p,h}, {t}_{fp}^{k}\right]$$

(5)

The $p$ observation point time variable ${t}_{fp}^{k}$ can be expressed as ${t}_{fp}^{k}={t}_{h}^{k}+ {\Delta t}_{fp}^{k}$, ${t}_{h}^{k}$ is the time variable at the system fault point $h$, and ${\Delta t}_{fp}^{k}$ is the time required for the propagation of the type $k$ fault signal from the fault point $h$ to the observation point $p$. According to the composition of the execution time of the system for the signal, the propagation time of the fault signal ${\Delta t}_{fp}^{k}$ includes the signal input processing time ${T}_{1}$, the control strategy operation time ${T}_{2}$, and the control operation output processing time ${T}_{3}$.

For the fault source signal ${f}_{h}^{k}\left(\cdot \right)$ in Eq. (3), according to the traction drive control system fault scenario, the following equation can be expressed as follows:

$${f}_{h}^{k}\left(\cdot \right) = \sum^{N}\sum^{nj}\left[\Gamma \left(t-\left({T}_{tj}+ {T}_{cj} * {\tau }_{j}\right)-{T}_{cj} *\left(l-1\right)\right)-\Gamma \left(t-{T}_{tj}-{T}_{cj} *\left(l-1\right)\right)\right]* TH\left\{{cs}_{k}\right\}$$

(6)

where $\Gamma$ is the step function, $N$ is the number of different types of pulse sequences, $j$ is the number of pulse sequences of $j$, ${n}_{j}$ is the total number of pulse signals of $j$, ${n}_{j}=ceil\left(\frac{\left({T}_{t\left(j+1\right)}- {T}_{tj}\right)}{{T}_{cj}}\right)$ and $ceil$ are rounded to positive infinity, ${T}_{cj}$ is the subperiod (the fault signal of $j$), ${\tau }_{j}$ is the operating period (the pulse sequence), $j$ $TH\{\}$ is the threshold function, ${T}_{tj}$ is the trigger moment (the fault signal of $j$), ${cs}_{j}$ is the fault state (the fault signal of $j$), ${cs}_{j}=0$ is the open-circuit fault, and ${cs}_{j}=1$ is the short circuit fault. The equation can represent various fault conditions, such as transient, intermittent, permanent, and their combinations, where ${\tau }_{j}\to 0$ means the fault is transient, $0<{\tau }_{j}<1$ means the fault is intermittent, and ${\tau }_{j}\to 1$ means the fault is permanent.

Common faults in the traction drive control system

(1) Traction motor fault

When the traction motor faults, $f(\cdot )$ can be given by the following Eq. [17]:

$$F\left(\cdot \right)={cs}_{j} \left(M, {f}_{1}, s\right)={A}_{1}(M)*cos\left(2\pi {f}_{s1}t+{\theta }_{1}\right)+{A}_{2}(M)*cos\left(2\pi {f}_{s2}t+{\theta }_{2}\right), 0\le M\le 1$$

(7)

where $M$ indicates the severity of the fault; $M=0$ indicates no fault in the traction motor; $M=1$ indicates a completely broken rotor guide bar in the traction motor; ${A}_{1}$ and ${A}_{2}$ are the magnitudes of the side frequency current components, whose magnitudes are proportional to $M$; ${f}_{1}$ is the fundamental frequency of the current; $s$ is the rate of rotation; ${f}_{s1}$ and ${f}_{s2}$ are the fault characteristic frequencies (contained in the stator current in case of a broken bar fault); and ${f}_{s1}\approx {f}_{s2}$ and ${f}_{s1}= {\mathrm{g}}_{1}\left({f}_{1}, s\right)$ and ${\theta }_{1}$ and ${\theta }_{2}$ are the initial phase angles of the side frequency components.

According to the traction motor fault mechanism analysis as follows:

1) Broken rotor bar fault, ${f}_{s}= \left(1\pm 2\mathrm{ks}\right){f}_{1}$, is its fault characteristic frequency; in that case, the fault signal $f(\cdot )$ is expressed as follows.

$$f\left(\cdot \right)={cs}_{j} \left(M, {f}_{1}, s\right)= {A}_{1}\left(M\right)*cos\left[2\pi \left(1-2ks\right){f}_{1}t+{\theta }_{1}\right]+{A}_{2}\left(M\right)cos\left[2\pi \left(1-2ks\right){f}_{1}t+{\theta }_{2}\right]$$

(8)

2) Stator turn-to-turn short circuit fault, ${f}_{s}= \left[n\pm 2\mathrm{k}\left(1-2s\right)\right]{f}_{1}$, is its fault characteristic frequency; in that case, the fault signal $f\left(\cdot \right)$ is expressed as follows:

$$f\left(\cdot \right)={cs}_{j} \left(M, {f}_{1}, s\right)= {A}_{1}\left(M\right)*cos\left\{2\pi \left[n+2\mathrm{k}\left(1-2s\right)\right]{f}_{1}t+{\theta }_{1}\right\}+{A}_{2}\left(M\right)cos\left\{2\pi \left[n-2\mathrm{k}\left(1-2s\right)\right]{f}_{1}t+{\theta }_{2}\right\}$$

(9)

(3) Air gap eccentricity fault, ${f}_{s}= \left[n\pm \mathrm{k}\left(1-s\right)\right]{f}_{1}$, is its fault characteristic frequency; in that case, the fault signal $f\left(\cdot \right)$ is expressed as follows:

$$f\left(\cdot \right)= {cs}_{j} \left(M, {f}_{1}, s\right)= {A}_{1}\left(M\right)*cos\left\{2\pi \left[n+\mathrm{k}\left(1-s\right)\right]{f}_{1}t+{\theta }_{1}\right\}+{A}_{2}\left(M\right)cos\left\{2\pi \left[n-\mathrm{k}\left(1-s\right)\right]{f}_{1}t+{\theta }_{2}\right\}$$

(10)

(4) End-loop fracture fault, ${f}_{s}= \left(1\pm 2\mathrm{ks}\right){f}_{1}$, is its fault characteristic frequency; in that case, the fault signal $f\left(\cdot \right)$ is expressed as follows:

$$f\left(\cdot \right)= {cs}_{j} \left(M, {f}_{1}, s\right)= {A}_{1}\left(M\right)*cos\left[2\pi \left(1-2ks\right){f}_{1}t+{\theta }_{1}\right]+{A}_{2}\left(M\right)cos\left[2\pi \left(1+2ks\right){f}_{1}t+{\theta }_{2}\right]$$

(11)

(2) Converter fault

The main converter faults are power device failure faults, power device and passive component electrical characteristic degradation faults, passive component failure faults, etc. The power device fault is as described in (6); for the power device and passive component electrical characteristic degradation fault, $f\left(\cdot \right)=f\left(\xi \right)$ and $\xi$ denote the electrical characteristic degradation rate; for the passive component, fault $f\left(\cdot \right)=conts$ and $conts$ are arbitrary constants.

(3) Sensor fault

The CRH2 high-speed train traction drive control system fault types mainly include deviation, drift, shock, accuracy degradation, periodic disturbance, gain, open circuit, short circuit, stuck and nonlinear dead zone faults of voltage, current, and speed sensors. When a sensor failure occurs, ${cs}_{j} \left(\cdot \right)$ can be given by the following equation, ${cs}_{j} \left(c\right)=c$, where $\mathrm{c}$ is a constant real number.

1) In the case of sensor deviation fault, the expression of the fault signal $f\left(\cdot \right)$ is shown in the following equation.

$$f\left(Conts\right)=Conts$$

(12)

2) In the case of sensor drift fault, the expression of the fault signal $f\left(\cdot \right)$ is shown in the following equation.

$$f\left(a\right)=a*t$$

(13)

3) In the case of sensor shock fault, the expression of the fault signal $f\left(\cdot \right)$ is shown in the following equation.

$$f\left(t\right)=\delta \left(t\right)$$

(14)

where $\delta \left(t\right)$ is the shock signal.

(4) Traction controller fault

Common fault types of traction controllers include faulty logic states or hard damage in analog signal I/O modules, digital signal I/O modules, and memory modules.

1) For analog signal I/O module fault, the expression of the fault signal $f\left(\cdot \right)$ is shown in the following equation.

$$f\left(p,q,A\right)=A\left({e}^{pt}- {e}^{qt}\right)$$

(15)

where $p$ and $q$ denote the time coefficients by which the rising and falling edge times and the width of the pulse are determined and $A$ denotes the magnitude of the amplitude.

2) For digital signal I/O module fault, compare the pin level threshold TH with the instantaneous pulse signal; if the latter is large, then $f\left(\cdot \right)=1$; otherwise, $f\left(\cdot \right)=0$.

3) For memory module fault, $f\left(\cdot \right)$ is the random bit flip value of the speed sensor feedback signal (moment of fault).

Actual fault case analysis

CRH2 high-speed train traction drive control system consists of traction transformer, pulse rectifier, intermediate DC link, traction inverter, traction motor, and controller, and the traction motor is a three-phase squirrel cage asynchronous motor. Four observation points are sequentially established on the system. The system’s structure and observation points are outlined in Fig. 1, while the main circuit topology of the traction drive control system is featured in Fig. 2.

Fig. 1

Diagram of observation points of the traction drive control system

Fig. 2

Main circuit topology of traction drive control system

The current signal analysis method is widely used for analyzing faults in the traction drive control system. Through the examination of the circuit structure and modulation theory, it can derive the current signal model for observation point 2 and observation point 4 in the system.

$$\left\{\begin{array}{c}{Z}_{2}\left({t}_{2}\right){S}_{\mathrm{2,1}}\cdot {Z}_{1}\left({t}_{1}\right)\\ {Z}_{4}\left({t}_{4}\right){S}_{4,e}\cdot {Z}_{3}\left({t}_{3}\right)\end{array}\right.$$

(16)

where ${Z}_{1}\left({t}_{1}\right)={\lceil{i}_{a} {i}_{b} {i}_{c}\rceil}^{-1}$ and ${S}_{\mathrm{2,1}}= \lceil{S}_{ua} {S}_{ub} {S}_{uc}\rceil$;${S}_{ua}$, ${S}_{ub}$, and ${S}_{uc}$ are inverter three-phase switching functions; ${S}_{\mathrm{4,3}}= {S}_{ia}$ are rectifier switching functions, and ${i}_{a}$, ${i}_{b}$, and ${i}_{c}$ are inverter output currents (i.e., traction motor stator currents); ${Z}_{2} \left({t}_{2}\right)= {i}_{d1}$ are inverter input currents; ${Z}_{3} \left({t}_{3}\right)= {i}_{d2}$ are rectifier output currents; and ${Z}_{4} \left({t}_{4}\right)= {i}_{N}$ are rectifier input side currents.

The inverter three-phase switching function can be obtained by utilizing the double Fourier transform as follows [18].

$${S}_{ua}= \frac{1}{2}+ \frac{M}{2} cos\left({w}_{1}{t}_{2}\right)+\frac{2}{\pi }\sum_{m=1}^{\infty }\frac{1}{m}{J}_{0}\left(\frac{\pi }{2}mM\right)sin\left(\frac{\pi }{2}m\right)cos\left(m{\omega }_{c}{t}_{2}\right)+\frac{2}{\pi }\sum_{m=1}^{\infty }\sum_{\begin{array}{c}n=-\infty \\ n\ne 0\end{array}}^{\infty }\frac{1}{m}{J}_{n} \left(\frac{\pi }{2}mM\right)sin\left[\frac{\pi }{2}\left(m+n\right)\right]cos\left(m{\omega }_{c}{t}_{2}+ n{\omega }_{1}{t}_{2}\right)$$

(17)

$${S}_{ub}= \frac{1}{2}+ \frac{M}{2} cos\left({w}_{1}{t}_{2}- \frac{2\pi }{3}\right)+\frac{2}{\pi } \sum_{m=1}^{\infty }\frac{1}{m}{J}_{0} \left(\frac{\pi }{2}mM\right)sin \left(\frac{\pi }{2}m\right)\mathrm{cos}\left({\omega }_{c}{t}_{2}\right)+\frac{2}{\pi } \sum_{m=1}^{\infty }\sum_{\begin{array}{c}n=-\infty \\ n\ne 0\end{array}}^{\infty }\frac{1}{m}{J}_{n} \left(\frac{\pi }{2} mM\right)sin\left[\frac{\pi }{2}\left(m+n\right)\right]\mathrm{cos}\left[m{\omega }_{c}{t}_{2}+n\left({\omega }_{1}{t}_{2}- \frac{2\pi }{3}\right)\right]$$

(18)

$${S}_{uc}= \frac{1}{2}+ \frac{M}{2} cos\left({w}_{1}{t}_{2}- \frac{2\pi }{3}\right)+\frac{2}{\pi } \sum_{m=1}^{\infty }\frac{1}{m}{J}_{0} \left(\frac{\pi }{2}mM\right)sin \left(\frac{\pi }{2}m\right)\mathrm{cos}\left({\omega }_{c}{t}_{2}\right)+\frac{2}{\pi } \sum_{m=1}^{\infty }\sum_{\begin{array}{c}n=-\infty \\ n\ne 0\end{array}}^{\infty }\frac{1}{m}{J}_{n}\left(\frac{\pi }{2} mM\right)sin\left[\frac{\pi }{2}\left(m+n\right)\right]cos\left[m{\omega }_{c}{t}_{2}+n\left({\omega }_{1}{t}_{2}- \frac{2\pi }{3}\right)\right]$$

(19)

where ${\omega }_{1}$ is the modulating wave angular frequency, ${\omega }_{c}$ is the carrier angular frequency, $M$ is the modulation system, $m$ is the carrier frequency multiplier, $n$ is the modulating wave harmonic frequency multiplier, ${J}_{0}$ and ${J}_{n}$ are the first type of Bessel functions, and ${t}_{2}$ is the intermediate DC link observation point 2 time variable.

Rectifier $a$ bridge arm switching function [19].

$${S}_{ia}= \sum_{n=1}^{\infty }\frac{4}{\pi }\left[\frac{1}{2n-1}sin\left(2n-1\right)\theta cos\left(2n-1\right){\omega t}_{4}+ \frac{1}{2n-1}cos\left(2n-1\right)\theta sin\left(2n-1\right){\omega t}_{4}\right]=\frac{4}{\pi }sin\left({\omega t}_{4}-\theta \right)+\frac{4}{\pi }\sum_{n=1}^{\infty }\frac{1}{4n\pm 1}sin\left[\left(4n\pm 1\right){\omega t}_{4}- \left(4n\pm 1\right)\theta \right]\left(n=\mathrm{1,2},3,\cdots \right)$$

(20)

where $\theta$ is the rectifier control angle; $\omega$ is the angular frequency of the power supply, and the corresponding frequency is $f\omega$; and ${t}_{4}$ is the rectifier input side time variable.

The stator three-phase current of the traction motor at normal fault-free time is expressed as follows [20]:

$$\left\{\begin{array}{c}{i}_{a}= {I}_{m} cos\left({\omega }_{1}t-\varphi \right)\\ {i}_{b}={I}_{m} cos\left({\omega }_{1}t-\varphi -\frac{2}{3}\pi \right)\\ {i}_{c}= {I}_{m} cos\left({\omega }_{1}t-\varphi +\frac{2}{3}\pi \right)\end{array}\right.$$

(21)

where ${I}_{m}$ and $\varphi$ are the amplitude and phase of the fundamental component of the stator current, respectively, ${\omega }_{1}$ is the angular frequency of the voltage applied to the motor, and ${\omega }_{1}$ corresponds to the frequency of ${f}_{1}$.

Broken rotor bar happens most frequently in rotor faults with an occurrence probability of 10% of the total fault of the traction motor. Assuming that the fault type is $k=1$ and the fault location is $h=1$, the fault source signal of Eq. (6) when the broken bar fault occurs in the traction motor is expressed as follows [17]:

$${f}_{1}^{1}\left(\cdot \right)= {I}_{bp} cos\left(2\pi {f}_{s1}t-{\varphi }_{bp}\right)+{I}_{bn} cos\left(2\pi {f}_{s2}t-{\varphi }_{bn}\right)$$

(22)

where ${f}_{s1}$ and ${f}_{s2}$ are the fault characteristic frequencies and ${f}_{s1}= \left(1+2s\right){f}_{1}$ and ${f}_{s2}= \left(1-2s\right){f}_{1}$ [21, 22]; ${I}_{bp}$, ${I}_{bn}$, ${\varphi }_{bp}$, and ${\varphi }_{bn}$ are the amplitude and phase of the $\left(1+2s\right){f}_{1}$ frequency component and $\left(1-2s\right){f}_{1}$ frequency component, respectively; and $s$ is the slip rate.

At this point, the three-phase stator current of the traction motor (observation point 1) can be expressed as follows:

$$\left\{\begin{array}{c}{Z}_{f1a,1}^{1}\left({t}_{f1}^{1}\right)={i}_{a}={I}_{m} cos\left({\omega }_{1}{t}_{f1}^{1}-\varphi \right)+{I}_{bp} cos\left[\left(1+2s\right){\omega }_{1}{t}_{f1}^{1}-{\varphi }_{bp}\right]+{I}_{bn} cos\left[\left(1-2s\right){\omega }_{1}{t}_{f1}^{1}-{\varphi }_{bn}\right]\\ {Z}_{f1b,1}^{1}\left({t}_{f1}^{1}\right)={i}_{b}={I}_{m} cos\left({\omega }_{1}{t}_{f1}^{1}-\varphi -\frac{2}{3}\pi \right)+{I}_{bp} cos\left[\left(1+2s\right){\omega }_{1}{t}_{f1}^{1}-{\varphi }_{bp}\right]\\ \begin{array}{c}+{I}_{bn} cos\left[\left(1-2s\right){\omega }_{1}{t}_{f1}^{1}-{\varphi }_{bn}-\frac{2}{3}\pi \right]\\ \begin{array}{c}{Z}_{f1c,1}^{1}\left({t}_{f1}^{1}\right)={i}_{c}={I}_{m} cos\left({\omega }_{1}{t}_{f1}^{1}-\varphi +\frac{2}{3}\pi \right)+{I}_{bp} cos\left[\left(1+2s\right){\omega }_{1}{t}_{f1}^{1}-{\varphi }_{bp}+\frac{2}{3}\pi \right]\\ + {I}_{bn} cos\left[\left(1-2s\right){\omega }_{1}{t}_{f1}^{1}-{\varphi }_{bn}+\frac{2}{3}\pi \right]\end{array}\end{array}\end{array}\right.$$

(23)

Combining with Eq. (3), the a-phase current at observation point 1 is as follows.

$${Z}_{1}\left({t}_{f1}^{1}\right) {I}_{m}\mathrm{cos}\left({\omega }_{1}{t}_{f1}^{1}-\varphi \right)$$

$${Z}_{f\mathrm{1,1}}^{1}\left[{f}_{1}^{1}\left(\cdot \right), {S}_{\mathrm{1,1}}, {t}_{f1}^{1}\right]={I}_{bp} cos\left[\left(1+2s\right){\omega }_{1}{t}_{f1}^{1}-{\varphi }_{bp}\right]+{I}_{bn} cos\left[\left(1-2s\right){\omega }_{1}{t}_{f1}^{1}-{\varphi }_{bn}\right]$$

${Z}_{f\mathrm{1,1}}^{1}\left({t}_{f1}^{1}\right)={Z}_{1}\left({t}_{f1}^{1}\right)+{Z}_{f\mathrm{1,1}}^{1}\left[{f}_{1}^{1}\left(\cdot \right), {S}_{\mathrm{1,1}},{t}_{f1}^{1}\right]$, which is consistent with Eq. (4); the phase b and c currents are similar to the phase a current.

Substituting Eqs. (17), (18), (19), and (23) into Eq. (16) yields the expression for the current signal at observation point 2.

$${Z}_{f\mathrm{2,1}}^{1}\left({t}_{f2}^{1}\right)={i}_{d1}=\frac{3}{r} {MI}_{m} cos \varphi +\frac{3}{r}M\left[{I}_{bp}\mathrm{cos}\left(2s {\omega }_{1}{t}_{f2}^{1}+{\varphi }_{bp}\right)+{I}_{bn} cos\left(2s {\omega }_{1}{t}_{f2}^{1}+{\varphi }_{bn}\right)\right]+\sum_{m=1}^{\infty }\sum_{n=-\infty }^{\infty }\frac{2}{m\pi }{J}_{n}\left(m\frac{\pi }{2}M\right)\mathrm{sin}\left[\left(m+n\right)\frac{\pi }{2}\right]\left\{cos\left(m{\omega }_{c}{t}_{f2}^{1}+n{\omega }_{1}{t}_{f2}^{1}\right){i}_{a}+cos\left[m{\omega }_{c}{t}_{f2}^{1}+n\left({\omega }_{1}{t}_{f2}^{1}-\frac{2}{3}\pi \right)\right]{i}_{b}+cos\left[m{\omega }_{c}{t}_{f2}^{1}+n\left({\omega }_{1}{t}_{f2}^{1}+\frac{2}{3}\pi \right)\right]{i}_{c}\right\}=\frac{3}{4}{MI}_{m}\mathrm{cos}\varphi +\frac{3}{4}{MI}_{2s} cos\left(2s {\omega }_{1}{t}_{f2}^{1}-h\right)+{i}_{h}$$

(24)

where $\frac{3}{4}{MI}_{m} cos\varphi$ is the DC component, ${I}_{2s}= \sqrt{{I}_{bp}^{2}+{I}_{bn}^{2}+ {2I}_{bp}{I}_{bn} cos\left({\varphi }_{bp}+{\varphi }_{bn}\right)}$ is the component amplitude of the frequency ${2sf}_{1}$, $h=\mathrm{arctan}\frac{{I}_{bn}\mathrm{sin}{\varphi }_{bn}- {I}_{bp} sin{ \varphi }_{bp}}{{I}_{bn}\mathrm{cos}{\varphi }_{bn}- {I}_{bp} sin{ \varphi }_{bp}}$ is the component phase of the frequency ${2sf}_{1}$, and ${i}_{h}$ is the high frequency component of the summation symbol.

Combining with Eq. (3), the current signal at observation point 2 is as follows.

$${Z}_{2} \left({t}_{f2}^{1}\right)=\frac{3}{4} {MI}_{m}\mathrm{cos}\varphi +{i}_{h}$$

$${Z}_{f\mathrm{2,1}}^{1}\left[{f}_{1}^{1} \left(\cdot \right), {S}_{\mathrm{2,1}}, {t}_{f2}^{1}\right]= \frac{3}{4} {MI}_{2s} cos\left(2s {\omega }_{1}{t}_{2}-h\right)$$

$${Z}_{f\mathrm{2,1}}^{1}\left({t}_{f2}^{1}\right)= {Z}_{2}\left({t}_{f2}^{1}\right)+{Z}_{f\mathrm{2,1}}^{1}\left[{f}_{1}^{1} \left(\cdot \right), {S}_{\mathrm{2,1}}, {t}_{f2}^{1}\right]$$

${Z}_{f\mathrm{2,1}}^{1}\left({t}_{f2}^{1}\right)= {Z}_{2}\left({t}_{f2}^{1}\right)+{Z}_{f\mathrm{2,1}}^{1}\left[{f}_{1}^{1} \left(\cdot \right), {S}_{\mathrm{2,1}}, {t}_{f2}^{1}\right]$, which is consistent with Eq. (4).

In the traction drive control system, observation point 2 and observation point 3 have essentially the same time variable, i.e., ${t}_{f2}^{1} \approx {t}_{f3}^{1}$. Since the intermediate DC link capacitor actually plays the role of low-pass filtering, the high-frequency component of ${i}_{d1}$ in Eq. (4) will be filtered out, containing only the low-frequency component and the DC component; therefore, the observation point 3 current signal can be known by the observation point 2 signal expressed as follows:

$${Z}_{f\mathrm{3,1}}^{1}\left({t}_{f3}^{1}\right)={1}_{d2}= \frac{3}{4}{MI}_{m}\mathrm{cos}\varphi +\frac{3}{4} {MI}_{2s} cos\left(2s {\omega }_{1}{t}_{f3}^{1}-h\right)$$

(25)

Therefore, at observation point 2 and observation point 3, ${2sf}_{1}$ is the characteristic frequency of a weak fault in the pre-breakage period of the traction motor.

Combining with Eq. (3), the current signal at observation point 3 is as follows:

$${Z}_{3} \left({t}_{f3}^{1}\right)= \frac{3}{4} {MI}_{m} cos \varphi$$

$${Z}_{f\mathrm{3,1}}^{1}\left[{f}_{1}^{1}\left(\cdot \right), {S}_{\mathrm{3,1}}, {t}_{f3}^{1}\right]= \frac{3}{r}{MI}_{2s} cos\left(2s {\omega }_{1}{t}_{f3}^{1}-h\right)$$

${Z}_{f\mathrm{3,1}}^{1} \left({t}_{f3}^{1}\right)= {Z}_{3} \left({t}_{f3}^{1}\right)+{Z}_{f\mathrm{3,1}}^{1} \left[{f}_{1}^{1}\left(\cdot \right), {S}_{\mathrm{3,1}}, {t}_{f3}^{1}\right]$, which is consistent with Eq. (4).

Equations (20) and (25) are substituted into Eq. (16) to obtain the expression for the current signal at observation point 4.

$${Z}_{f\mathrm{4,1}}^{1} \left({t}_{f4}^{1}\right)={i}_{N} \frac{3}{2\pi } M\left\{{I}_{m} \mathrm{sin}\left({\omega }_{1}{t}_{f4}^{1}- \theta \pm \varphi \right)+ {I}_{2s} \mathrm{sin}\left[\left(\omega \pm 2s{\omega }_{1}\right){t}_{f4}^{1}- \theta \mp h\right]+\frac{1}{4n\pm 1}\sum_{n=1}^{\infty }{I}_{m} \mathrm{sin}\left[\left(4n\pm 1\right)\omega {t}_{f4}^{1} -\left(4n\pm 1\right)\theta \mp \varphi \right]+\frac{1}{4n\pm 1}\sum_{n=1}^{\infty }{I}_{2s} \mathrm{sin}\left[\left(4n\pm 1\right)\omega {t}_{f4}^{1} \pm 2s{\omega }_{1}{t}_{f4}^{1}- \left(4n\pm 1\right)\theta \mp h\right]\right\}$$

(26)

Therefore, at observation point 4, $\left(4n\pm 1\right) f\pm 2s{f}_{1}$ is the characteristic frequency of the traction motor for the occurrence of a pre-fault with broken rotor bars.

Combining with Eq. (3), the current signal at observation point 4 is as follows:

$${Z}_{4}\left({t}_{f4}^{1}\right)=\frac{3}{2\pi }M\left\{{I}_{m} sin\left(\omega {t}_{f4}^{1}- \theta \mp \varphi \right)+\frac{1}{4n\pm 1}\sum_{n=1}^{\infty }{I}_{m} \mathrm{sin}\left[\left(4n\pm 1\right)\omega {t}_{f4}^{1}- \left(4n\pm 1\right)\theta \mp \varphi \right]\right\}$$

$${Z}_{f\mathrm{4,1}}^{1}\left[{f}_{1}^{1}\left(\cdot \right), {S}_{\mathrm{4,1}}, {t}_{f4}^{1}\right]=\frac{3}{2\pi } M\left\{{I}_{2s} \mathrm{sin}\left[\left(\omega \pm 2s{\omega }_{1}\right){t}_{f4}^{1}-\theta \mp h\right]+\frac{1}{4n\pm 1}\sum_{n=1}^{\infty }{I}_{2s} \mathrm{sin}\left[\left(4n\pm 1\right)\omega {t}_{f4}^{1}\pm 2s{\omega }_{1}{t}_{f4}^{1}-\left(4n\pm 1\right)\theta \mp h\right]\right\}$$

${S}_{f\mathrm{4,1}}^{1}\left({t}_{f4}^{1}\right)={Z}_{4}\left({t}_{f4}^{1}\right)+{Z}_{f\mathrm{4,1}}^{1} \left[{f}_{1}^{1}\left(\cdot \right), {S}_{\mathrm{4,1}}, {t}_{f4}^{1}\right]$, which is consistent with Eq. (4).

The frequency of fault characteristics at different observation points of the traction drive control system reflects the spatial characteristics of fault propagation.

According to the system parameters and motor parameters of the CRH2 traction drive control system operating at 200 km/h speed, the fault characteristic frequency values of different observation points when the traction motor rotor of the traction drive control system is weakly faulty in the pre-breakage period are calculated by combining the above conclusions, as shown in Table 1.

Table 1 Frequency values of fault characteristics at different observation points

According to the execution time of the system on the signal, Eqs. (23) to (26) are as follows:

$$\begin{array}{c}{t}_{f2}^{1}= {t}_{1}^{1}+ {\Delta t}_{f\mathrm{2,1}}^{1} in, {\Delta t}_{f\mathrm{2,1}}^{1}= {T}_{1}+ {T}_{2}+ {T}_{3}, {t}_{f2}^{1}= {t}_{1}^{1}+\left({T}_{1}+ {T}_{2}+ {T}_{3}\right).\\ {t}_{f3}^{1}= {t}_{1}^{1}+ {\Delta t}_{f\mathrm{3,1}}^{1} in,{\Delta t}_{f\mathrm{3,1}}^{1}= {T}_{1}+ {T}_{2}+ {T}_{3}, {t}_{f3}^{1}= {t}_{1}^{1}+\left({T}_{1}+ {T}_{2}+ {T}_{3}\right). \\ {t}_{f4}^{1}= {t}_{1}^{1}+ {\Delta t}_{f\mathrm{4,1}}^{1} In,{\Delta t}_{f\mathrm{4,1}}^{1}= 2\left({T}_{1}+ {T}_{2}+ {T}_{3}\right),{t}_{f4}^{1}= {t}_{1}^{1}+2\left({T}_{1}+ {T}_{2}+ {T}_{3}\right).\end{array}$$

${\Delta t}_{f\mathrm{2,1}}^{1}$, ${\Delta t}_{f\mathrm{3,1}}^{1}$, and ${\Delta t}_{f\mathrm{4,1}}^{1}$ are the time required for the fault signal to propagate to the corresponding observation point after the occurrence of the fault, and ${\Delta t}_{f\mathrm{4,1}}^{1}=2* {\Delta t}_{f\mathrm{2,1}}^{1}=2* {\Delta t}_{f\mathrm{3,1}}^{1}$, which reflects the temporal characteristics of the fault propagation.

According to the fault classification of the traction drive control system of CRH2 EMU, the observation point settings, and the composition of the execution time of the system on the signal, the time required for the propagation of the fault characteristics to different observation points when different faults of the traction drive control system containing traction motor faults occur can be obtained, as shown in Table 2.

Table 2 Time required for propagation of different faults in the traction drive control system to different observation points

Experiment results and analysis

Experimental platform

Experimental studies are conducted on the semi-physical platform of the CRH2 traction drive control system shown in Fig. 3 [23, 24]. The fault injection simulation platform can be downloaded on the website http://gfist.csu.edu.cn/indexE.html; it includes a real-time simulator, a fault-injection unit (FIU), a physical traction drive control unit (TCU), and a real-time data acquisition and monitoring unit. The controller is a physical object. The dSPACE hardware includes a DS1007 CPU board, a DS5203 field-programmable gate array (FPGA) board, a DS4004 digital I/O board, and a DS2103 multichannel high-precision D/A board. Platform normal operation to 2 s was injected with different failure levels of traction motor rotor broken bar early weak faults.

Fig. 3

CRH2 traction drive control system semi-physical platform

The FIU consists of a fault injection controller (FIC), a signal processing unit, a logic solver, and a noise signal generator. The internal structure of the FIU is shown in Fig. 3. The FIC includes a selection signal module, a fault mode module, a fault signal module (internal), and a fault signal module (external), the signal operation process includes signal superposition and signal conditioning, the logic solver includes a logic operation section and a fault injection signal selection section, and the noise signal generator includes a noise signal generation module and a noise signal selection module.

Considering that the fault of the traction motor is destructive, its occurrence is random and influenced by many uncontrollable factors. Furthermore, the fault is an irreversible process, and the cost of man-made destruction of the motor is high. Currently, the simulation of traction motors for high-speed trains mainly focuses on normal operation, with a lack of in-depth research and simulation of faults. The fault injection method, however, can be employed to examine the faults of traction motors, enabling a more adaptable and cost-effective study.

Results and discussion

Figure 4 shows the current time domain waveforms of four different observation points of the traction drive control system. Each observation point contains two waveforms at the top and bottom, where the top graph shows the overall time domain graph of 1 $s$ -3 $s$, and the bottom graph shows the local magnification of the area 1.97 $s$ -2.04 $s$ in the top graph. The local detail graphs show that after a small defect in the previously weakened rotor bar, the defect features spread to different observation points of the system after a certain period of time.

Fig. 4

Schematic diagram of the internal structure of the FIU

The normalized spectrum analysis of the current signals of observation point 1, observation point 2, observation point 3, and observation point 4 under the fault condition is shown in Fig. 6.

Figure 5 displays the time domain current waveform at different observation points in the traction drive control system; during a weak, fault occurs in the pre-broken rotor of the traction motor. The partial magnification illustrates that fault characteristics propagate to different observation points of the system; after a certain time, the weak fault occurs at the second $s$, observation point 1 finds the abnormality at the second $s$, observation point 2 finds the abnormality after time ${\Delta t}_{f\mathrm{2,1}}^{1}$, observation point 3 finds the abnormality after time ${\Delta t}_{f\mathrm{3,1}}^{1}$ found anomaly, and observation point 4 after time ${\Delta t}_{f\mathrm{4,1}}^{1}$ found anomaly, and ${\Delta t}_{f\mathrm{2,1}}^{1}={\Delta t}_{f\mathrm{3,1}}^{1}=0.002s$ and ${\Delta t}_{f\mathrm{4,1}}^{1}=0.004s$, satisfying ${\Delta t}_{f\mathrm{4,1}}^{1}=2*{\Delta t}_{f\mathrm{2,1}}^{1}=2*{\Delta t}_{f\mathrm{3,1}}^{1}$, in accordance with the description of the time required for the propagation of different faults of the traction drive control system to different observation points in Table 2.

Fig. 5

Time domain waveforms of current at different observation points. a Time domain waveform of current at observation point 1. b Time domain waveform of current at observation point 2. c Time domain waveform of current at observation point 3. d Time domain waveform of current at observation point 4

Figure 6 shows the current spectrum of observation point 1 to observation point 4 when the rotor of the traction motor is broken; it can be seen that there are obvious characteristic frequencies 126.6 $Hz$ and 135.6 $Hz$ on both sides of the stator current fundamental frequency 131.1 $Hz$ of observation point 1, which is the characteristic frequency of the fault $\left(1\pm 2s\right){f}_{1}$; there are obvious characteristic frequencies 4.5 $Hz$ of observation point 2 and observation point 3, which is the characteristic frequency of the fault $2s{f}_{1}$; and observation point 4 has obvious characteristic frequencies 45.5 and 54.5 on both sides of the rectifier input current fundamental frequency 50 $Hz$ and 3 times the fundamental frequency 150 of the rectifier input current respectively. There are distinctive eigenfrequencies 45.5 $Hz$ and 54.5 $Hz$ on both sides of rectifier input current 50, 145.5 $Hz$ and 154.5 $Hz$ on both sides of rectifier input current 3 times the fundamental frequency 150 $Hz$, and 245.5 $Hz$ and 254.5 $Hz$ on both sides of rectifier input current 5 times the fundamental frequency 150 $Hz$, which are the fault eigenfrequencies $\left(4n\pm 1\right)f\pm 2s{f}_{1}$. The results of the spectral analysis of the current signals at different observation points are consistent with the theoretical calculation of the fault characteristic frequency values at different observation points in Table 1.

Fig. 6

Current spectrum of different observation points. a Observation point 1 current spectrum. b Observation point 2 current spectrum. c Observation point 3 current spectrum. d Observation point 4 current spectrum

The fault propagation time and space characteristics are verified in Figs. 5 and 6, respectively. The fault occurrence frequency observed at the fault characteristic point suggests that the problem with the traction drive control system is a weak fault caused by a broken rotor bar in the pre-traction motor. Consequently, the location of the fault occurrence can be traced back to the traction motor, and the fault occurrence element is identified as the traction motor rotor guide bar, resulting in successful fault traceability.

Conclusions

To address deficiencies within the traction drive control system, we have implemented observation points and formulated a model with spatiotemporal characteristics via mechanism analysis. Furthermore, fault characteristics and propagation time were analyzed at different observation points for various types of faults. The fault characteristics and propagation time of the observation points are extracted and compared to the spatiotemporal characteristics of the corresponding observation points in the fault propagation model. This process enables the precise identification of the fault type and location, facilitating fault traceability. The experiment confirms that the technique presented in this paper is suitable for the control of traction transmission systems. It can successfully detect minor faults during system operation and furnish the operator with a dependable reference point for swiftly identifying the fault source and extent.