**5.1 Validation on the steady-state process**

A typical steady-state numerical example [54], mimicking a typical industrial system, is considered to evaluate the performance of the proposed method in steady-state operation. The model of the process is:

$$
\begin{bmatrix} \mathbf{x}\_1 \\ \mathbf{x}\_2 \\ \mathbf{x}\_3 \\ \mathbf{x}\_4 \\ \mathbf{x}\_5 \\ \mathbf{x}\_6 \end{bmatrix} = \begin{bmatrix} -0.2310 & -0.0816 & -0.2662 \\ -0.3241 & 0.7055 & -0.2158 \\ -0.2170 & -0.3056 & -0.5207 \\ -0.4089 & -0.3442 & -0.4501 \\ -0.6408 & 0.3102 & 0.2372 \\ -0.6408 & 0.3102 & 0.2372 \\ \end{bmatrix} \begin{bmatrix} t\_1 \\ t\_2 \\ t\_3 \end{bmatrix} + \text{noise} \tag{39}
$$

where *t*1, *t*2, and *t*<sup>3</sup> are zero-mean random variables with standard deviations of 1, 0.8, and 0.6, respectively. The noise included in the process is zero-mean with a standard deviation of 0.2, and is normally distributed. To build the model, 1000 samples are generated using such process. The number of simulated faults is 2000, with the data samples generated according to the model above and the fault magnitude being a random number uniformly distributed between 0 and 5. The signal variable in fault is also random uniformly sampled among the six possible variables, as simulated in [59].

Although this application represents a typical steady-state operation, we assume a set of sequential time-series data, assigning a constant time interval of *η* ¼ 1*s* with *r* = 3. To measure the performance of the propose methods, we employ different measures of the performance metrics (e.g., missed alarm rate (MAR), missed and false alarms rate (M&FAR), true and false alarms rate (T&FAR), true alarm rate (TAR), and fault detection rate (FDR)) as proposed in [54] and briefly defined in Appendix A.3. The purpose of this application is to verify the performance of the proposed method in monitoring during steady-state operation and to compare the results with those of AAKR (see Appendix A.1) and AABKR.

**Table 1** and **Figure 11** show the alarm rates of AAKR, AABKR, and the modified AABKR computed from the prediction of the simulated faults. It is interesting to note that although the MAR of AAKR is a bit higher than that of AABKR, the performance of the two models does not differ significantly and both models have suffered from the spillover effects (i.e., the effect that a faulty signal has on the predictions of the fault-free signals) as evident from the values of T&FAR (i.e., the detection of faults in both faulty signal and at least one fault-free signal). Conversely, the performance of the modified AABKR is better than those of the other two methods in terms of TAR (i.e., the detection of fault only in a signal that actually has the fault, without false alarm in other fault-free signals) and T&FAR; hence, the modified AABKR is more resistant to spillover and more robust than both AAKR and AABKR. It can be observed that even though the TAR value of the modified AABKR is larger than those of the other two models, FDR values of the three methods did not differ significantly because of the larger values of T&FAR for AAKR and AABKR (53.4 and 61.8%, respectively). Therefore, it is important to further examine the rate of correct fault diagnosis of the three methods using absolute residual values of the faults successfully detected, which produced the FDR values. **Figure 12** shows the rate of correct fault diagnosis of the three methods. We


**Table 1.** *Alarm rates (%) of validation on the numerical steady-state process.* *Fault Detection by Signal Reconstruction in Nuclear Power Plants DOI: http://dx.doi.org/10.5772/intechopen.101276*

**Figure 11.** *Alarm rates in a steady-state numerical process.*

**Figure 12.** *Rate of correct fault diagnosis in a steady-state numerical process.*

observe that the performance of the modified AABKR is comparable to that of AAKR, and the method can, thus, also be used effectively for signal validation during steady-state process operation.
