**4. PCA method application on WRIM**

### **4.1. Simulation conditions**

Nine state variables (m=9) have been chosen to be monitored and 10000 measures (N=10000) during 4s are considered. The WRIM faults are introduced from the initial time (t=0s) to the final time (t=4s) of the different simulations. The machine is coupled to a mechanical load torque (10Nm) at t=2s. The considered faults are respectively, increases from 10% to 40% of the resistance value of both stator and rotor coils.

### **4.2. Choice of the number of principal components**

The Figure 6 and the Figure 7 represent the residues variation of the WRIM stator current versus time and show impact of the « *l* » number in the diagnosis approach.

**Figure 6.** Stator current residue for *l* = 5

Figure 6 show that the chosen number of components is too high then the residual space dimension is reduced. Some faults are projected in the principal space and the stator current residues can not be detectable.

However, with the Figure 6, the number of components is well chosen. Faults can be detected and localized and the PCA model is well reconstructed.

Generally, the detection approach in the case of diagnosis based on analytical model is linked with the residues generation step. From these residues analysis, the decision making step must indicate if faults exist or not. The residues generation approach can be the state estimation approach or the parameter estimation approach.

The residue indicates the information losses given by the matrix dimension reduction of the state variables matrix data to be monitored. Indeed, a small residue means that the estimated value tends to approach the exact value in healthy operation case.

In our case, the eigenvalues corresponding to the number of the retained principal components represent 93% of the total sum of eigenvalues. 0nly 7% of the total represent the residues subspace. One can conclude that the PCA model has been well constructed.

**Figure 7.** Stator current residue for *l* = 6

#### **5. Simulation results**

78 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

eigenvalues <sup>1</sup> ,..., *l m* 

information that it conveys.

**4.1. Simulation conditions** 

**Figure 6.** Stator current residue for *l* = 5




0

Phase "A" stator current

0.5

1

1.5

**4. PCA method application on WRIM** 

the resistance value of both stator and rotor coils.

**4.2. Choice of the number of principal components** 

*Xp* is the principal estimated matrix and *E* the residues matrix which represents information losses due to data matrix *X* reduction. It represents the difference between the exact and the approached representations of *X*. This matrix is associated with the lowest

Nine state variables (m=9) have been chosen to be monitored and 10000 measures (N=10000) during 4s are considered. The WRIM faults are introduced from the initial time (t=0s) to the final time (t=4s) of the different simulations. The machine is coupled to a mechanical load torque (10Nm) at t=2s. The considered faults are respectively, increases from 10% to 40% of

The Figure 6 and the Figure 7 represent the residues variation of the WRIM stator current

0 0.5 1 1.5 2 2.5 3 3.5 4

Time [s]

versus time and show impact of the « *l* » number in the diagnosis approach.

. Therefore, in this case, the data compression preserves all the best

*XX E <sup>p</sup>* (30)

The different simulation results have been performed with respect to the simulation conditions mentioned earlier.

Figure 8 to Figure 13 and Figure 16 represent the real variations without PCA method, and Figure 14, Figure 15 and Figure 17 represent the residue variations with PCA application of the faulted WRIM state variables in considering the stator defaults.

With the WRIM state variables, other quantities obtained by their transformations have been calculated:


**Figure 8.** Real variations versus time of the stator current of the healthy and faulted WRIM

**Figure 9.** Real variations versus time of the rotor current of the healthy and faulted WRIM

the faulted WRIM state variables in considering the stator defaults.

quadrature axis and direct axis currents with Park transformation,

axis currents with Concordia transformation.

**Figure 8.** Real variations versus time of the stator current of the healthy and faulted WRIM

3.2 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.3

Time [s]

**Figure 9.** Real variations versus time of the rotor current of the healthy and faulted WRIM

2.5 3 3.5 4

Time [s]

calculated:

axis and -8 -6 -4 -2 0


Rotor phase "a" current [A]

Stator phase "A" current [A]

Figure 8 to Figure 13 and Figure 16 represent the real variations without PCA method, and Figure 14, Figure 15 and Figure 17 represent the residue variations with PCA application of

With the WRIM state variables, other quantities obtained by their transformations have been

**Figure 10.** Real variations versus time of the shaft rotational speed of the healthy and faulted WRIM

**Figure 11.** Real variations versus time of the electromagnetic torque of the healthy and faulted WRIM

**Figure 12.** Real variations of axis current versus the phase axis current of the stator phase

**Figure 13.** Real variations of the quadrature axis current versus the phase direct axis current of the stator phase

**Figure 12.** Real variations of

stator phase





0

Quadrature axis current [A]

5

10

15




0

Beta axis current [A]

5

10

15

20

axis current versus the phase


Alpha axis current [A]

**Figure 13.** Real variations of the quadrature axis current versus the phase direct axis current of the


Direct axis current [A]

axis current of the stator phase

**Figure 14.** Variations of the stator phase "A" current residues versus time of the healthy and faulted WRIM

**Figure 15.** Variations of the rotor phase "a" current residues versus time of the healthy and faulted WRIM

**Figure 16.** Real variations of electromagnetic torque versus the shaft rotational speed of the WRIM

**Figure 17.** Variations of electromagnetic torque residues versus the shaft rotational speed residues of the WRIM
