**2. Resilience enhancement of traction motor with coupled faults**

In order to achieve resilience enhancement with high accuracy and fast speed, the empirical mode decomposition (EMD), fault coding, SVM multiclassification model and IGWO are employed in this chapter, as shown in **Figure 1**.

### **2.1 Fault feature extraction**

For the nonlinear electromagnetic torque signals, the EMD is introduced to decompose the signal into different linear components. Then, the intrinsic mode functions (IMF) obtained from the EMD are utilized to calculate the IMF entropy, which is used to determine the fault feature matrix with the help of fault attribute knowledge coding.

### **2.2 Fault mechanism analysis for traction motor control system**

As shown in **Figure 2**, the traction control unit achieves control of the traction motor speed and current by adjusting the gating signal of the converter at a given traction/braking command. However, the existing control units only detect faults by comparing the measured values of the sensors with the threshold values, which makes it difficult to effectively diagnose early fault (bearing fault, broken rotor bars fault, stator interturn short circuit fault, air gap eccentricity fault, rotor broken bar and interturn short circuit coupled fault). During the small fault injection phase, the traction motor current, magnetic flux and speed still satisfy the fifth-order model in the stator-side (*a*, *b*) coordinate system. Related studies have shown that the variation of parameters such as rotational inertia, rotor resistance, stator resistance and mutual inductance in the motor model can respectively describe the severity of bearing fault, broken rotor bars fault, stator interturn short circuit fault, air gap eccentricity fault.

**Figure 1.** *Flowchart of the proposed method.*

**Figure 2.** *High-speed train traction system.*

$$\begin{aligned} \frac{dw}{dt} &= \frac{1}{(l+\Delta l)} (\rho\_{sa} i\_{sb} - \rho\_{sb} i\_{sa} - T\_L) \\ \frac{di\_{sa}}{dt} &= -\frac{(R\_r + \Delta R\_r)}{\sigma} i\_{sa} - \frac{(R\_r + \Delta R\_r)}{L\_r} (1 + \theta(M + \Delta M)) i\_{sa} \\ -\alpha i\_{sb} &+ \frac{(R\_r + \Delta R\_r)}{L\_r \sigma} \rho\_{sa} + \frac{\alpha \sigma}{\sigma} \rho\_{sb} + \frac{1}{\sigma} u\_{sa} \\ \frac{di\_{sb}}{dt} &= -\frac{(R\_r + \Delta R\_r)}{\sigma} i\_{sb} - \frac{(R\_r + \Delta R\_r)}{L\_r} (1 + \theta(M + \Delta M)) i\_{sb} \\ -\alpha i\_{sa} + \frac{(R\_r + \Delta R\_r)}{L\_r \sigma} \rho\_{sb} + \frac{\alpha}{\sigma} \rho\_{sa} + \frac{1}{\sigma} u\_{sb} \\ \frac{d\rho\_{sa}}{dt} &= -(R\_i + \Delta R\_i) i\_{sa} + u\_{sa} \\ \frac{d\rho\_{sb}}{dt} &= -(R\_i + \Delta R\_i) i\_{sb} + u\_{ab} \\ \frac{d\rho\_{sb}}{dt} &= -(R\_i + \Delta R\_i) i\_{sb} + u\_{ab} \\ \tau\_t &= \rho\_{sa} i\_{sb} - \rho\_{gi} i\_{sa} \end{aligned} \tag{1}$$

where *usa* and *usb* are the stator side *a* phase and *b* phase voltages; *isa* and *isb* are the stator side *a* phase and *b* phase currents; *φsa* and *φsb* are the stator side *a* phase and *b* phase fluxes; *w* is the motor speed, *J* is the motor inertia, *ΔJ* represents the change in inertia in case of bearing fault; *Rr* is the rotor resistance, *ΔRr* is the change in rotor resistance in case of broken rotor strip fault; *Rs* is the stator resistance, *ΔRs* means the relative change of stator resistance; *Ls*, *Lr* and *M* are stator self-inductance, rotor selfinductance and mutual inductance, respectively; is the mutual inductance change during air gap eccentricity fault, *TL* is the load torque; *<sup>ϑ</sup>* <sup>¼</sup> *<sup>M</sup>*þΔ*<sup>M</sup> <sup>σ</sup>Lr* ,

$$\sigma = L\_{\mathfrak{g}} \left( \mathbf{1} - \frac{\left( \mathbf{M} + \Delta \mathbf{M} \right)^{2}}{L\_{\mathfrak{s}} L\_{\mathfrak{r}}} \right).$$

From the above equation, it can be seen that the electromagnetic torque *Te* contains the interaction of magnetic chain *φsa*, *φsb* and current *isa*, *isb*, and also describes the variation law of motor speed *w*. It is sensitive to the early motor fault variation quantities *ΔJ*, *ΔRr*, *ΔRs* and *ΔM*, which can best characterize the traction motor fault features. According to [10, 11], the motor early bearing fault severity *η*1, broken rotor bars fault severity *η*2, stator interturn short circuit fault severity *η*3, air gap eccentricity fault severity *η*<sup>4</sup> can be quantified and described as follows:

$$\begin{aligned} \eta\_1 &= \frac{\Delta f}{J} \times 100\text{\%}; \quad \eta\_2 = \frac{\Delta R\_r}{R\_r} \times 100\text{\%} \\ \eta\_3 &= \frac{\Delta R\_s}{R\_s} \times 100\text{\%}; \quad \eta\_4 = \frac{\Delta M}{M} \times 100\text{\%} \end{aligned} \tag{2}$$

### **2.3 Fault severity levels coding**

Based on the available fault features obtained by EMD and IMF entropy, the current analysis methods often ignore the difference in fault levels. In order to make use of the distributed feature more effectively, the fault features are first coded in group mode, and then the fault feature bases that reflects various fault severity levels can be obtained. According to [12], the grouping fault features can be divided into five subblocks in terms of normal state, bearing fault, rotor broken strip, interturn short circuit and air gap eccentricity. The corresponding fault codes are described as follows:

*High-Speed Train Traction System Reliability Analysis DOI: http://dx.doi.org/10.5772/intechopen.111911*

$$\begin{cases} f\_1(H\_1, H\_j, B\_1) = \gamma\_1 \\ \vdots \\ f\_5(H\_1, H\_j, B\_5) = \gamma\_5 \end{cases} \tag{3}$$

where *f* <sup>1</sup>, ⋯, *f* <sup>5</sup> are the code mapping functions characterizing the motor operation conditions, *γ*1, ⋯, *γ*<sup>5</sup> refer to the coded fault feature matrix; *B*1, ⋯, *B*<sup>5</sup> are the five binary matrices describing the fault severity levels, which can be determined by the maintenance experience. As for *Bi*, the corresponding binary bits are defined in **Table 1**.

For the single fault, **Table 1** summarizes the number of potential classes is 32. Since the real fault that occurred is limited, the constructed fault coding matrix will cover 160 classes that can satisfy fault attribute transfer with various levels.

As mentioned before, the key to coding is to determine the attributes of newly extracted features only by these bases. Then, the fault feature-based agile diagnosis problem is naturally done in a classification way that transfers from training faults to target faults [13].

### **2.4 Gray wolf optimization algorithm**

The gray wolf optimization algorithm simulates the behaviors of searching and tracking, encircling and attacking the prey based on the cooperative behavior of the gray wolf pack to achieve the purpose of optimal solution. The details are as follows:

Step 1: Social hierarchy. The gray wolf population has an extremely strict social dominance hierarchy, according to which the gray wolves can be divided into four classes, from high to low, namely *α*, *β*, *δ* and *μ*. *α*, *β*, *δ* are the three wolves with the best adaptation in the pack, and the remaining gray wolves are *μ*. The pursuit is launched by *α*, *β*, *δ* to perform a prey tracking roundup, and the location of the prey corresponds to the optimal global solution of the SVM parameter optimization problem.

Step 2: Surrounding the prey. When searching for prey, the gray wolf will gradually approach and then encircle, and this behavior can be expressed as:

$$\begin{cases} D = \left| \mathbf{Z} \cdot \mathbf{X}\_p(\tau) - \mathbf{X}(\tau) \right| \\ \mathbf{X}(\tau + \mathbf{1}) = \mathbf{X}\_p(\tau) - \mathbf{A} \cdot \mathbf{D} \\ \mathbf{A} = 2\mathbf{a} \cdot r\_1 - \mathbf{a} \\ \mathbf{Z} = \mathbf{2} \cdot r\_2 \end{cases} \tag{4}$$

where *τ* is the number of iterations, *A*, *Z* is the coefficient vectors, *Xp* is the position vector of the prey, *X* is the position vector of a gray wolf, and *r*1,*r*<sup>2</sup> are random vectors in [0,1]. The value of the convergence factor *a* is linearly decreased from 2 to 0 as follows:


**Table 1.** *Binary code for fault severity levels.*

$$a = 2 - 2 \cdot \frac{\pi}{T} \tag{5}$$

where *T* is the maximum number of iterations.

Step 3: Hunting. After encircling the prey, wolves *β* and *δ* will hunt under the leadership of wolf *α*. In order to simulate hunting behavior, it is assumed that *α*, *β*, *δ* have a better understanding of the potential location of the prey. The formula is expressed as follows:

$$\begin{cases} D\_a(\tau) = |\mathbf{Z}\_1 \cdot \mathbf{X}\_a(\tau) - \mathbf{X}(\tau)| \\ D\_\beta(\tau) = \left| \mathbf{Z}\_2 \cdot \mathbf{X}\_\beta(\tau) - \mathbf{X}(\tau) \right| \\ D\_\delta(\tau) = |\mathbf{Z}\_3 \cdot \mathbf{X}\_\delta(\tau) - \mathbf{X}(\tau)| \end{cases} \tag{6}$$

$$\begin{cases} X\_1(\tau) = X\_a(\tau) - A\_1 \cdot (D\_a(\tau)) \\ X\_2(\tau) = X\_\beta(\tau) - A\_2 \cdot \left(D\_\beta(\tau)\right) \\ X\_3(\tau) = X\_\delta(\tau) - A\_3 \cdot (D\_\delta(\tau)) \end{cases} \tag{7}$$

$$X(\tau+\mathbf{1}) = \frac{X\_1(\tau) + X\_2(\tau) + X\_3(\tau)}{3} \tag{8}$$

where *Dα*, *Dβ*, *D<sup>δ</sup>* denotes the distance between *α*, *β*, *δ* and other individuals, respectively, Z1, Z2, Z3 are random vectors, *Xα*,*Xβ*, *X<sup>δ</sup>* denotes the current position of *α*, *β*, *δ*, *X*ð Þ *τ* þ 1 is the optimal solution for the current iteration.

### **2.5 Improved gray wolf optimization algorithm**

### *2.5.1 Initialization strategy based on chaotic tent mapping*

The traditional gray wolf algorithm solves optimization problems often based on randomness to generate initial populations, which makes the initial populations unevenly distributed and leads to a reduced speed of finding the best. In contrast, chaotic motion has the properties of randomness, regularity and ergodicity, and using these advantages can generate a better diversity of initial populations and improve the global search ability of the algorithm. The Tent mapping is as follows:

$$\mathbf{x}\_{t} + \mathbf{1} = \begin{cases} 2\mathbf{x}\_{t}, \mathbf{0} \le \mathbf{x}\_{t} \le \mathbf{0}.5\\ 2(\mathbf{1} - \mathbf{x}\_{t}), \mathbf{0}.5 < \mathbf{x}\_{t} \le \mathbf{1} \end{cases} \tag{9}$$

where *xt*, *xt*þ<sup>1</sup> denotes the position of the *t th* and ð Þ *<sup>t</sup>* <sup>þ</sup> <sup>1</sup> *th* gray wolves in the onedimensional space individual.

### *2.5.2 Adaptive adjustment strategy for control parameters*

The implementation of the gray wolf algorithm mainly lies in prey localization and wolf pack movement, and the position update of individual gray wolves is influenced by the parameters. When j j *A* < 1, the wolf pack narrows the search range and conducts local search, and when j j *A* > 1, the wolf pack expands the search range and conducts global search to find a better preferred solution. The value of *A* in turn varies with the convergence factor *a*, which can balance the global and local search ability for the gray wolf algorithm. In the traditional gray wolf

optimization algorithm. Meanwhile, the parameter *a* varies using a linear adjustment strategy, which ignores the diversity of optimization problems to be solved and makes it difficult to reach the global optimum. In this chapter, the new convergence factor *a* that varies nonlinearly with the number of iterations can be described as follows:

$$a(t) = (a\_{\text{initial}} - a\_{\text{final}}) \cdot \left(\frac{T - \tau}{T}\right)^{\ell} \tag{10}$$

where *a*initial is 2, *a*final is 0, *τ* is the current number of iterations, *T* is the maximum number of iterations, ℓ is the nonlinear adjustment coefficient. In this chapter, ℓ is chosen as 0.2. At the beginning of the iteration, the *a* value can be reduced more slowly to increase the search range, and at the end of the iteration, the *a* value can be reduced faster to increase the convergence speed of the algorithm.

### *2.5.3 Inertia weight position update*

As can be seen from Eq. (7), the traditional gray wolf algorithm position update mechanism lacks weights related to the number of iterations and is prone to fall into local optimum. In this paper, a new gray wolf position updating formula based on inertia weights, which empowers the gray wolf to jump out of local extremes, as follows:

$$
\kappa(\tau) = \kappa\_{\text{max}} - (\kappa\_{\text{max}} - \kappa\_{\text{min}}) \cdot \frac{\tau}{T} \tag{11}
$$

where *κ* is the inertia weight; *κ*max denotes the maximum value of inertia weight, generally taken as 0.9; *κ*min denotes the minimum value of inertia weight, generally chosen as 0.4.
