1. Introduction

2. Traction Motor Stator ITSC Fault Diagnostic Condition Control

Figure 1. Structural diagram of the EMU traction system.

Figure 1. Structural diagram of the EMU traction system.

2.1. Working Status of Two-level Traction Inverter

The main circuit topology of the two-level traction inverter is shown in Figure 1, which mainly consists of six IGBTs, T₁, T₂, T₃, T₄, T₅ and T₆, forming a three-phase full bridge inverter circuit. For the convenience of analysis, three ideal switching functions are usually defined [6]: S_A={1-T₁ switch on; 0-T₄ switch on }, S_B={1-T₃ switch on; 0-T₆ switch on }, S_C={1-T₅ switch on; 0-T₂ switch on }. There are eight combinations of S_A, S_B and S_C, and their switch states in various modes are shown in Table 1.

Table 1. Working status of two-level traction inverter.

Table 1. Working status of two-level traction inverter.

Mode	0	1	2	3	4	5	6	7
SA	0	0	0	0	1	1	1	1
SB	0	0	1	1	0	0	1	1
SC	0	1	0	1	0	1	0	1
Voltagevector	$\overrightarrow{U_0}$	$\overrightarrow{U_1}$	$\overrightarrow{U_2}$	$\overrightarrow{U_3}$	$\overrightarrow{U_4}$	$\overrightarrow{U_5}$	$\overrightarrow{U_6}$	$\overrightarrow{U_7}$

2.2. ITSC Fault Diagnostic Condition Control

Traction motor ITSC fault diagnostic conditionIis shown in Figure 2. The driving signals of T₁, T₂ and T₆ are the same, and switched on and off simultaneously; the driving signals of T₃, T₄ and T₅ are the same, and switched on and off simultaneously. When T₁, T₂ and T₆ are switched on, and T₃, T₄ and T₅ are switched off simultaneously, this is equivalent to applying a voltage vector $\overrightarrow{U_4}$to the traction motor; when T₃, T₄ and T₅ are switched on, and T₁, T₂ and T₆ switched off simultaneously, this is equivalent to applying a voltage vector $\overrightarrow{U_3}$to the traction motor. The directions of the three currents i_a, i_b and i_c in Figure 2 represent the reference directions of the currents, not the actual flow directions. Suppose the IGBTs driving signals Ugs1, Ugs2, Ugs6, and Ugs3, Ugs4, Ugs5 drive the IGBTs in a bipolar sine pulse width modulation mode. The intermediate DC voltage applies single-phase SPWM voltages to the stator windings of the traction motor. Under diagnostic condition I, the b-phase winding and the c-phase winding of the traction motor are connected parallel and in series with the a-phase winding.

Figure 2. Diagnostic condition I inverter control.

Figure 2. Diagnostic condition I inverter control.

Traction motor ITSC fault diagnostic condition II is shown in Figure 3. The driving signals of T₂, T₃ and T₄ are the same, and switched on and off simultaneously; the driving signals of T₁, T₅ and T₆ are the same and switched on and off simultaneously. When T₂, T₃ and T₄ are switched on, and T₁, T₅ and T₆ are switched off simultaneously, this is equivalent to applying a voltage vector $\overrightarrow{U_5}$to the traction motor; when T₁, T₅ and T₆ are switched on, and T₂, T₃ and T₄ are switched off simultaneously, this is equivalent to applying a voltage vector $\overrightarrow{U_2}$to the traction motor. The directions of three currents i_a, i_b, and i_c in Figure 3 represent the reference directions of the currents, not the actual flow directions. Suppose the IGBTs driving signals Ugs1, Ugs5, Ugs6, Ugs2, Ugs3, and Ugs4 drive the IGBTs in a bipolar sine pulse width modulation mode. The intermediate DC voltage applies single-phase SPWM voltages to the stator windings of the traction motor. Under diagnostic condition II, the a-phase winding and the c-phase winding of the traction motor are connected parallel and in series with the b-phase winding.

Figure 3. Diagnostic condition II inverter control.

Figure 3. Diagnostic condition II inverter control.

Traction motor ITSC fault diagnostic condition III is shown in Figure 4. The driving signals of T₁, T₂ and T₃ are the same, and switched on and off simultaneously; the driving signals of T₄, T₅ and T₆ are the same, and switched on and off simultaneously. When T₁, T₂ and T₃ are switched on, and T₄, T₅ and T₆ are switched off simultaneously, this is equivalent to applying a voltage vector $\overrightarrow{U_6}$to the traction motor; when T₄, T₅ and T₆ are switched on, and T₁, T₂ and T₃ are switched off simultaneously, this is equivalent to applying a voltage vector $\overrightarrow{U_1}$to the traction motor. The directions of three currents i_a, i_b and i_c in Figure 4 represent the reference directions of the currents, not the actual flow directions. Suppose the IGBTs driving signals Ugs1, Ugs2, Ugs3, Ugs4, Ugs5, and Ugs6 drive the IGBTs in a bipolar sine pulse width modulation mode. The intermediate DC voltage applies single-phase SPWM voltages to the stator windings of the traction motor. Under diagnostic condition III, the a-phase winding and the b-phase winding of the traction motor are connected parallel and in series with the c-phase winding.

Figure 4. Diagnostic condition III inverter control.

Figure 4. Diagnostic condition III inverter control.

3. SPWM Excitation Voltage and Goertzel Algorithm

3.2. Calculation of Current Fundamental Component Using Goertzel Algorithm

TCU sets the frequency of excitation voltage input for ITSC fault diagnosis, and the current fundamental frequency can be obtained accurately. Under the condition of knowing the exact current fundamental frequency, the current signals can be truncated throughout the entire cycle to avoid spectral leakage. The Goertzel algorithm can calculate only the amplitude and phase angle of the fundamental current component to reduce computational complexity and improve computational speed [33,34,35,36]. x(n) is a sampling sequence of length N, n∈[0, N-1], where n is an integer, then the DFT of x(n) is:

$$X(k)={\displaystyle \sum _{n=0}^{N-1}x}(n)W_N^{nk}$$

(8)

In Formula (8), taking$W_N={\mathrm{e}}^{-j\frac{2\pi }{N}}$ into the following Formula to obtain:

$$W_N^{-kN}={\mathrm{e}}^{\mathrm{j}2\pi k \mathrm{N}/\mathrm{N}}={\mathrm{e}}^{\mathrm{j}2\pi k}=1$$

(9)

Multiply Formula(9) to the right of Formula(8):

$$X(k)=W_N^{-kN}{\displaystyle \sum _{n=0}^{N-1}x}(n)W_N^{kn}={\displaystyle \sum _{n=0}^{N-1}x}(n)W_N^{-k(N-n)}$$

(10)

Define two sequences as:

$$x_e(n)=\left\{\begin{array}{ll}x(n),\hfill & 0⩽n⩽N-1\hfill \\ 0,\hfill & else\hfill \end{array}\right.$$

(11)

$$h_k(n)=\left\{\begin{array}{ll}{W}_N^{-kn},\hfill & n⩾0\hfill \\ 0,\hfill & n<0\hfill \end{array}\right.$$

(12)

Formula (10) can be expressed as the convolution of the two sequences:

$$y_k(n)={\displaystyle \sum _{l=0}^{N-1}x}(l)W_N^{-k(n-l)}=x_e(n)*h_k(n)$$

(13)

Perform a Z-transform on Formula (13), where the Z-transform of y_k(n) is Y_k(z), the Z-transform of x_e(n) is X_k (z), and the Z-transform of h_k(n) is H_k(z). The time-domain convolution of the two sequences is equal to the frequency domain multiplication. From Formula(12), it can be obtained:

$$H_k(z)=\frac{1}{1-W_N^{-k}{z}^{-1}}$$

(14)

Multiply$\left(1-W_N^k{z}^{-1}\right)$the numerator and denominator of Formula(14) simultaneously:

$$H_k(z)=\frac{1-W_N^k{z}^{-1}}{\left(1-W_N^{-k}{z}^{-1}\right)\left(1-W_N^k{z}^{-1}\right)}=\frac{1-W_N^k{z}^{-1}}{1-2\cos (2\pi k/N)z^{-1}+z^{-2}}$$

(15)

The corresponding filter structure is shown in Figure 5, and the input-output relation of the filter is as follows:

$$\left.\begin{array}{l}{v}_k(n)=2\cos (2\pi k/N)v_k(n-1)-v_k(n-2)+x(n)\hfill \\ y_k(n)=v_k(n)-W_N^k{v}_k(n-1)\hfill \end{array}\right\}$$

(16)

The filter output y_k(N) in Figure 5 is the transformation coefficient X(k) of the N-point DFT at point k. If it is necessary to calculate the amplitude and phase angle of a specific frequency, the corresponding k value is:

$$\frac{k}{N}=\frac{f_{\mathrm{int}}}{f_s}$$

(17)

Calculate the amplitude and phase of a specific frequency find using Goertzel as shown in Formulas (18) and (19):

$$|y(N){|}^2=v^2(N-1)+v^2(N-2)-2\cos \left(\frac{2\pi k}{N}\right)v(N-1)v(N-2)$$

(18)

$$\theta =\arg \{y((N)\}=\arctan \frac{\sin (2\pi k/N)\cdot v(N-2)}{v(N-1)-\cos (2\pi k/N)\cdot v(N-2)}$$

(19)

Figure 5 shows that the Goertzel algorithm can calculate a y_k(n) for every x(n) collected. Data collection and computation can be carried out simultaneously, overcoming the disadvantage of DFT that it needs to wait for all N data to be collected before processing and to improves the calculation speed. Meanwhile, when calculating DFT, it is necessary to compute the values of all N spectral lines. If only the amplitude and phase angle of a single frequency spectral line need to be calculated, the calculation of the remaining N-1 spectral lines will be wasted. From Formulas(18) and (19), it can be seen that Goertzel can only calculate the amplitude and phase angle of a specific spectral line, which also saves computational costs.

Figure 5. Goertzel algorithm flowchart.

Figure 5. Goertzel algorithm flowchart.

4. Fault Features and Diagnostic Model of Traction Motor ITSC Fault

4.1. Fault Features of Traction Motor ITSC Diagnostic Conditions

When an ITSC fault occurs in a-phase winding of the traction motor, the symmetry relation of the three-phase winding changes. As shown in Figure 2, when diagnostic condition I is implemented while the traction motor is stationary, the b-phase and c-phase are in parallel. The inductance and resistance asymmetries exist between b-phase or c-phase winding due to an ITSC fault. The amplitude difference between the b-phaseand c-phase current Δi_bc will occur, and the phase angle difference Δθ_bc will also occur. Calculating the current amplitude difference Δi_bc and phase angle difference Δθ_bc between the b-phase and c-phase can detect the winding asymmetries caused by the ITSC fault on b-phase or c-phase winding. Similarly, diagnostic condition II can be applied to detect the asymmetries between the c-phase and a-phase; diagnostic condition III can be applied to detect the asymmetries between the a-phase and b-phase. Table 1 defines six ITSC fault diagnosis fault features under three diagnostic conditions when the traction motor is stationary.

Table 2. Definition of fault features.

Table 2. Definition of fault features.

Diagnostic condition I	Diagnostic condition II	Diagnostic condition III
Δibc=ibmax-icmax	Δica=icmax-iamax	Δiab=iamax-ibmax
Δθbc=θb-θc	Δθca=θc-θa	Δθab=θa-θb

4.3. Fault Diagnosis of Traction Motor ITSC Fault

Figure 6 shows the ITSC fault flowchart for the traction motor stators based on the Goertzel algorithm and random forest. It mainly includes two parts: random forest model training and online diagnosis. In two parts, the fault diagnostic excitation SPWM voltage parameter setting and diagnostic condition control can be embedded in the EMU TCU control program. After training the ITSC fault diagnosis model, the TCU controls the inverter to perform three fault diagnostic conditions before the traction motor is driven. The current amplitude features Δi and phase angle features Δθ need to be calculated through Goertzel. The six fault features are input into the fault diagnosis model to diagnose the ITSC fault of the traction motor and output the diagnostic results. This diagnosis method can ensures that the traction motor is free of ITSC faults before the TCU controls the inverter to drive the traction motor.

Figure 6. Flowchart of fault traction motor ITSC diagnosis.

Figure 6. Flowchart of fault traction motor ITSC diagnosis.

5. Diagnosis For Traction Motor ITSC Fault Simulation Experimental Platform

5.1. Traction Motor ITSC Fault Diagnosis Simulation Experimental Platform

Figure 7 shows a schematic of the experimental simulation platform for the EMU’s traction motor ITSC fault diagnosis. The fault diagnostic conditions control program on the PC was loaded into the DSP28335. The silicon rectifier equipment output a DC voltage to simulate the intermediate DC link voltage of the EMU. The power electronics development platform is composed of a DSP28335, IGBTs driver circuits, and a three-phase bridge inverter circuit. The power electronics development platform output the SPWM voltages required for the three diagnostic conditions. The ITSC fault simulation motor adopted a three-phase squirrel cage induction motor, and the three-phase winding tap was led to the junction box during fabrication, as shown in Figure 7. The metal short circuit fault was simulated by direct short-circuiting of connectors, and the non-metallic short circuit was simulated by connecting resistors between the connectors.

Figure 7. Traction motor ITSC fault diagnosis simulation experimental platform.

Figure 7. Traction motor ITSC fault diagnosis simulation experimental platform.

The real equipments of the ITSC fault diagnosis simulation experimental platform of the traction motor is shown in Figure 8. The voltage and current signals were measured using voltage and current probes, and a DL850E oscilloscope was used to store the voltage and current signals. During the experiment, the DC voltage, the motor phase voltage, the short-circuit inter-turn current, and the three-phase current were measured. The DL850E oscilloscope was used to collect and record the above signals. The DL850’s LPF(low pass filter)was used for hardware filtering voltage and current signals. The LPF cutoff frequency was 400Hz. The sampling frequency f_s was set to 2000Hz for DL850E.

Figure 8. Photographs of the traction motor ITSC fault simulation experimental platform.

Figure 8. Photographs of the traction motor ITSC fault simulation experimental platform.

5.2. SPWM Excitation Voltage Parameters of Experimental Platform

The parameters of the ITSC fault experimental motor and the ITSC fault diagnosis control parameters are shown in Table 3. The silicon rectifier provided a DC300V voltage for the power electronics development platform. The SPWM reference modulation wave frequency f_r was 100Hz, the carrier frequency f_c was 5000Hz, and the modulation ratio N was 50. Considering both the ITSC fault simulation motor nominal current and the power electronics development platform’s output capability, the output current RMS value of the power electronics development platform was adjusted by changing the modulation index M. When the modulation index M was 0.4, RMS value of the equivalent parallel phase winding current of each diagnostic condition was about 3.5A, and the RMS value of the series winding current was about 7A.

Table 3. ITSC fault motor and diagnostic condition control parameters.

Table 3. ITSC fault motor and diagnostic condition control parameters.

Parameters	Values	Parameters	Values
Nominal power	5.5kW	Nominal frequency	50Hz
Nominal voltage	380V	Connection mode	Y
Nominal current	11.7A	Nominal speed	1445rpm
Poles	4	Turns per phase	164
Modulation frequency	100Hz	Modulation index	0.4
Carrier frequency	5000Hz	DC-link voltage	300V

5.3. Analysis of ITSC Fault Diagnosis Signals

The measurement signals of a-phase ITSC fault with 1Ωtransition resistance and 39 short-circuit turns under diagnostic condition II were analyzed. Figure 9(a) shows the DC voltage waveform output by the silicon rectifier device. Figure 9(b) shows the motor phase voltage waveform after LPF filtering the SPWM voltage output from the power electronics development platform. The experimental induction motor phase voltages contain specific harmonic components. Under ITSC fault diagnostic condition II, as shown in Figure 3, the induction motor is equivalent to the a-phase winding in parallel with the c-phase winding and then connected in series with the b-phase winding. The phase voltage relationship is shown in Formula (20), and the waveforms of the a-phase voltage and the c-phase voltages in Figure 9(b) overlaps.

$$u_{an}=u_{cn}=-\frac{1}{2}{u}_{bn}$$

(20)

Figure 9. DC voltage and experimental motor phase voltages: (a) Output voltage of silicon rectifier equipment; (b) Diagnostic condition II experimental motor phase voltages.

Figure 9. DC voltage and experimental motor phase voltages: (a) Output voltage of silicon rectifier equipment; (b) Diagnostic condition II experimental motor phase voltages.

Figure 10 (a) shows the short-circuit inter-turn current of the a-phase winding during an ITSC fault. After setting an ITSC fault in the a-phase, a sinusoidal current with a specific harmonic component is generated between the short-circuit turns. The DL850E trigger function was used to record the occurrence time of the short circuit faults, there was no ITSC fault in the a-phase before 24.99 seconds in recording time. Figure 10 (b) shows the three-phase current of the experimental motor with a fundamental frequency of 100Hz. When there is no ITSC fault, the phase current relation of the experimental motor under ideal conditions is shown in Formula (21), i_a and i_c are almost equal, and their waveforms overlap. After the ITSC fault occurs in the a-phase, there is some difference between i_a and i_c, and their waveforms no longer overlap.

$$i_{\mathrm{a}}=i_{\mathrm{c}}=-\frac{1}{2}{i}_{\mathrm{b}}$$

(21)

Figure 10. ITSC fault short-circuits inter-turn current and motor three-phase current: (a) ITSC fault short-circuit inter-turn current; (b) ITSC fault three-phase currents.

Figure 10. ITSC fault short-circuits inter-turn current and motor three-phase current: (a) ITSC fault short-circuit inter-turn current; (b) ITSC fault three-phase currents.

5.4. The Impact of ITSC Fault Extent on Features

The ITSC fault damage to induction motors is related to shout-circuit turns and transition resistance. Full period data truncation was used to eliminate the current fundamental frequency leakage. The Goertzel algorithm was used to calculate the six fault features under the three diagnostic conditions defined in Table 1. Table 4 shows the fault setting parameters during the experimental process. In the case of metal short- circuiting, the transition resistance value is 0. In the case of non-metallic short circuits, four resistance values were selected as the transition resistance. The second and seventh turns of each stator winding were used as the short-circuit ends. A total of 9 different short-circuit turns were obtained.

Table 4. ITSC fault setting parameters.

Table 4. ITSC fault setting parameters.

Parameters	values
Transition resistance (Ω)	0, 1, 2, 4, 8
Short-circuit turns	5, 7, 12, 20, 25, 34, 39, 47, 52

Figure 11 shows the variation of fault features with different numbers of short-circuit turns, the transition resistance of the experimental motor was 1Ω. Figure 11 (a) shows that after the ITSC fault occurs in the a-phase, Δi_bc feature remains almost unchanged with the change of short-circuit turns under diagnostic condition I. But, Δi_ca and Δi_ab significantly increases and decreases respectively with the increase of short-circuit turns under diagnostic condition II and III. Figure 11(b) shows that the Δθ_bc feature remains almost unchanged with the number of short-circuit turns under diagnostic condition I. But, Δθ_ca and Δθ_ab significantly increases and decreases respectively with the increase of short-circuit turns under diagnostic condition II and III.

Figure 11. Current fault features change with different numbers of short-circuit turns: (a) Amplitude features Δi change with the numbers of short-circuit turns; (b) Phase angle features Δθ change with the numbers of short-circuit turns.

Figure 11. Current fault features change with different numbers of short-circuit turns: (a) Amplitude features Δi change with the numbers of short-circuit turns; (b) Phase angle features Δθ change with the numbers of short-circuit turns.

Figure 12 shows the fault features change with the transition resistance between short-circuit turns, the number of short-circuit turns was 39. Figure 12 (a) shows that after the ITSC fault occurs in the a-phase, Δi_bc feature remains almost unchanged with the change of the transition resistance under diagnostic condition I. But, Δi_ca and Δi_ab significantly increases and decreases respectively with the transition resistance increase under diagnostic condition II and III. Figure 12(b) shows that Δθ_bc feature remains almost unchanged with the transition resistance under diagnostic condition I. But, Δθ_ca and Δθ_ab significantly increases and decreases respectively with the increase of the transition resistance under diagnostic condition II and III.

Figure 12. Current fault features change with the transition resistance: (a) Amplitude features Δi change with the transition resistance; (b) Phase angle features Δθ change with the transition resistance.

Figure 12. Current fault features change with the transition resistance: (a) Amplitude features Δi change with the transition resistance; (b) Phase angle features Δθ change with the transition resistance.

5.5. Fault Detection and Location Based on Random Forest Model

According to the fault experimental motor settings for ITSC simulation in Table 4, ITSC faults were set on each phase of the experimental motor. The six fault features as in Table 2 were measured and calculated. The train and test sets include 45 samples of ITSC faults at different extents in each phase and 45 healthy samples. A total of 180 samples were obtained in the experiment. The samples were labeled as healthy, a-phase ITSC fault, b-phase ITSC fault, and c-phase ITSC fault in the four types. Common machine learning algorithms were used to classify the data set and randomly selected 35 samples as train samples and 10 as test samples from various types. Table 5 shows that the various common classification algorithms have relatively high classification accuracy, indicating that the proposed fault feature extraction based on Goertzel algorithm for traction motors under the stationary state is effective. This method can provide reliable fault features for ITSC fault detection and location. In several common classification algorithms, random forest have a classification accuracy of 100% on both train and test samples. The ITSC fault can be accurately detected and located based on the Goertzel algorithm and the random forest for the ITSC fault simulation motor on the experimental platform . At the same time, the misclassified samples of the other classification algorithms were analyzed, all of which were fault samples with high transition resistance and few short-circuited turns were misclassified as healthy samples.

Table 5. Diagnosis results of common classification models.

Table 5. Diagnosis results of common classification models.

Models	Accuracy of the train sets	Accuracy of the test sets
BP neural network	97.86%	97.5%
KNN	98.57%	100%
SVM	95%	97.5%
Naive Bayes	99.29%	100%
Rondom Forest	100%	100%

6. Conclusions

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