# Robust Fault Detection in Monitoring Chemical Processes Using Multi-Scale PCA with KD Approach

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Modeling Based on PCA

**F**apturing the information discarded during dimensionality reduction.

#### PCA for Fault Detection

#### 2.2. Kantorovich Distance Indicator

**G**and

**H**. The process involves the following steps:

- The distribution
**G**is partitioned into r segments ${G}_{1}$, ${G}_{2}$ …${G}_{r}$, each containing j data points. For instance, ${G}_{1}$ comprises $[{g}_{11},{g}_{12}\dots {g}_{1j}]$, ${G}_{2}$ comprises $[{g}_{21},{g}_{22}\dots {g}_{2j}]$, and so on until ${G}_{r}$ which includes $[{g}_{r1},{g}_{r2}\dots {g}_{rj}]$. - Similarly, the distribution
**H**is divided into r segments ${H}_{1}$, ${H}_{2}$ …${H}_{r}$, with each segment also containing j data points. For instance, ${H}_{1}$ comprises $[{h}_{11},{h}_{12}\dots {h}_{1j}]$, ${H}_{2}$ comprises $[{h}_{21},{h}_{22}\dots {h}_{2j}]$, and so on until ${H}_{r}$ which includes $[{h}_{r1},{h}_{r2}\dots {h}_{rj}]$. - The evaluation of the KD metric compares each segment of
**G**with that of**H**. This comparison entails matching each segment of the source distribution ${G}_{1}$ with all segments ${H}_{1}$ through ${H}_{r}$. Specifically, the first sample of ${G}_{1}\left({g}_{11}\right)$ is compared with the first sample of ${H}_{1}\left({h}_{11}\right)$, ${H}_{2}\left({h}_{21}\right)$, up to ${H}_{r}\left({h}_{r1}\right)$. Subsequently, next comparison takes place until all the segments of**G**and**H**are covered. - Once the comparison of all samples of ${G}_{1}$ with corresponding samples of ${H}_{1}$ through ${H}_{r}$ is completed, the next segment ${G}_{2}$ undergoes comparison with all segments ${H}_{1}$ through ${H}_{r}$ of distribution
**H**. This iterative process continues until all segments of distribution**G**are compared with all segments of distribution**H**. - The distances resulting from the segment comparisons between the two distributions are recorded and used to evaluate the KD index, as described in Equation (9). A minimum value of the KD index indicates relative similarity between distributions
**G**and**H**, while a larger value suggests dissimilarity between them.

#### 2.3. Multi-Scale Filtering Using Wavelets

**x(t)**on mathematical functions that is represented as follows [46]:

#### 2.4. Multiscale Representation of Data

#### 2.5. Multiscale Data Filtering Algorithm

- Decompose the noisy signal on a set of orthonormal wavelet basis functions to transform it into the time-frequency domain.
- To apply thresholding to wavelet coefficients, suppress any coefficients that are smaller than a designated threshold value.
- Reconstruct the signal by applying the inverse wavelet transform, which results in a noise-free signal that retains the important features of the original signal.

#### 2.6. Multi-Scale PCA Modeling

#### 2.7. The Proposed MSPCA-KD Fault Detection Strategy

**Step 1:**For normally operating data, perform the data pre-processing.**Step 2:**Decompose the data in wavelets for optimal decomposition depth.**Step 4:**Develop a reference threshold for KD-statistical detector using kernel density estimation (KDE) approach.**Step 5:**When new data (Testing data that is possibly having an fault scenario) is available, perform the data processing.**Step 6:**From the reference multi-scale PCA model, generate residuals R2.**Step 7:**Generate the KD metric for R1 and R2 using the segmentation process described in Section 2.2.**Step 8:**Declare a Fault if the KD metric crosses the reference threshold.

## 3. Results and Discussion

#### 3.1. Monitoring Faults in Distillation Column Process

**Bias fault:**A bias fault involves a constant offset in the readings of a particular sensor or variable. In this scenario, a 7% bias fault is introduced into temperature variable 5 from sampling time instant 250 until the end of the testing data. Introducing this fault in temperature variable 5 implies a persistent distortion in the measurements of this specific temperature parameter. This distortion persists throughout the latter part of the testing data, affecting the accuracy of the readings.**Drift fault:**A drift fault signifies a gradual change or drift in the sensor readings over time. In this scenario, a drift sensor fault with a slope of 0.01 is introduced into temperature variable 1 during the same time frame. This introduced drift on testing data implies a continuous and gradual shift in the measurements of this temperature parameter. Such a fault can mimic the effect of changing conditions in the distillation column, potentially impacting process control.**Intermittent fault:**An intermittent fault involves sporadic variations or disruptions in sensor readings. In this scenario, intermittent faults having a small magnitude with 8% of the total variation are inserted in the concentration variable of the bottom stream between the sampling time instants [100, 200] and [350, 450], respectively. Monitoring intermittent faults is crucial for capturing irregular disturbances in the system.

#### 3.2. Monitoring Faults in CSTR Process

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**A schematic overview of the distillation column process, highlighting structural components, RTD sensors, and the entry conditions for a binary mixture of propane and isobutene.

**Figure 4.**Correlation matrix heatmap depicting the Pearson correlation among variables in the fault-free distillation column dataset.

**Figure 5.**RadViz visualization illustrating the influence of different factors on (

**a**) ‘Propane’ and (

**b**) ‘Isobutene’ concentrations in the distillation column. Each point on the circular plot represents a data point, and the positioning of points along the circumference reflects the values of various factor.

**Figure 7.**Intermittent fault monitoring in the DC process by PCA based methods under SNR level of 15: (

**a**) PCA-${T}^{2}$, (

**b**) PCA-Q, (

**c**) PCA-KD (Red line indicates significance threshold).

**Figure 8.**Intermittent fault monitoring in the DC process by MSPCA based methods under SNR level of 15: (

**a**) MSPCA-${T}^{2}$, (

**b**) MSPCA-Q, (

**c**) MSPCA-KD (Red line indicates significance threshold).

**Figure 9.**Intermittent fault monitoring in the DC process by PCA based methods under SNR level of 5: (

**a**) PCA-${T}^{2}$, (

**b**) PCA-Q, (

**c**) PCA-KD (Red line indicates significance threshold).

**Figure 10.**Intermittent fault monitoring in the DC process by MSPCA based methods under SNR level of 5: (

**a**) MSPCA-${T}^{2}$, (

**b**) MSPCA-Q, (

**c**) MSPCA-KD (Red line indicates significance threshold).

**Figure 13.**Bias fault monitoring by PCA based methods in the CSTR process under SNR level of 15: (

**a**) PCA-${T}^{2}$, (

**b**) PCA-Q, (

**c**) PCA-KD (Red line indicates significance threshold).

**Figure 14.**Bias fault monitoring by MPCA based methods in the CSTR process under SNR level of 15: (

**a**) MSPCA-${T}^{2}$, (

**b**) MSPCA-Q, and (

**c**) MSPCA-KD (Red line indicates significance threshold).

**Figure 15.**Bias fault monitoring by PCA based methods in the CSTR process under SNR level of 5: (

**a**) PCA-${T}^{2}$, (

**b**) PCA-Q, (

**c**) PCA-KD (Red line indicates significance threshold).

**Figure 16.**Bias fault monitoring by MPCA based methods in the CSTR process under SNR level of 5: (

**a**) MSPCA-${T}^{2}$, (

**b**) MSPCA-Q, and (

**c**) MSPCA-KD. (Red line indicates significance threshold).

**Table 1.**Fault detection performance of PCA and MSPCA-based monitoring methods in the distillation column process under an SNR of 15.

Fault | Index | PCA-${\mathit{T}}^{2}$ | PCA-Q | PCA-KD | MPCA-${\mathit{T}}^{2}$ | MPCA-Q | MPCA-KD |
---|---|---|---|---|---|---|---|

Bias | FDR | 46.18 | 58.49 | 82.06 | 94.56 | 83.59 | 100.00 |

FAR | 0.80 | 7.60 | 0.00 | 0.04 | 1.60 | 0.00 | |

Precision | 98.37 | 88.95 | 100.00 | 99.59 | 98.20 | 100.00 | |

F1-score | 63.09 | 66.27 | 90.14 | 97.09 | 90.70 | 100.00 | |

Interm | FDR | 22.50 | 60.00 | 87.50 | 95.00 | 97.00 | 100.00 |

FAR | 0.00 | 6.41 | 0.00 | 3.52 | 11.00 | 0.00 | |

Precision | 100.00 | 95.23 | 100.00 | 85.46 | 85.46 | 100.00 | |

F1-score | 36.98 | 75.03 | 93.30 | 94.50 | 91.09 | 100.00 | |

Drift | FDR | 84.73 | 75.57 | 85.22 | 88.02 | 90.85 | 96.12 |

FAR | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

Precision | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | |

F1-score | 91.73 | 86.08 | 92.02 | 93.62 | 95.20 | 98.02 |

**Table 2.**Fault detection performance of PCA and MSPCA-based monitoring methods in the distillation column process under an SNR of 5.

Fault | Index | PCA-${\mathit{T}}^{2}$ | PCA-Q | PCA-KD | MPCA-${\mathit{T}}^{2}$ | MPCA-Q | MPCA-KD |
---|---|---|---|---|---|---|---|

Bias | FDR | 10.78 | 51.38 | 71.09 | 20.94 | 71.76 | 93.51 |

FAR | 1.20 | 8.40 | 0.00 | 0.40 | 4.40 | 0.00 | |

Precision | 86.73 | 88.95 | 100.00 | 99.23 | 94.47 | 100.00 | |

F1-score | 19.59 | 65.05 | 87.60 | 34.36 | 81.68 | 96.64 | |

Interm | FDR | 8.41 | 15.50 | 57.50 | 38.00 | 75.50 | 98.75 |

FAR | 0.00 | 8.00 | 0.00 | 5.28 | 6.42 | 0.00 | |

Precision | 100.00 | 55.30 | 100.00 | 82.10 | 88.40 | 100.00 | |

F1-score | 16.66 | 24.74 | 73.01 | 52.00 | 81.42 | 99.31 | |

Drift | FDR | 63.36 | 46.56 | 65.75 | 80.15 | 83.21 | 86.50 |

FAR | 0.00 | 0.40 | 0.00 | 0.00 | 0.00 | 0.00 | |

Precision | 100.00 | 99.17 | 100.00 | 100.00 | 100.00 | 100.00 | |

F1-score | 77.57 | 62.82 | 79.33 | 88.98 | 88.80 | 92.62 |

SNR | Fault | PCA-${\mathit{T}}^{2}$ | PCA-Q | PCA-KD | MPCA-${\mathit{T}}^{2}$ | MPCA-Q | MPCA-KD |
---|---|---|---|---|---|---|---|

15 | Bias | 0.003 | 0.003 | 0.094 | 0.007 | 0.001 | 0.038 |

Intermittent | 0.450 | 0.010 | 0.110 | 0.010 | 0.010 | 0.060 | |

Drift | 0.229 | 0.235 | 0.2175 | 0.133 | 0.114 | 0.076 | |

5 | Bias | 0.420 | 0.038 | 0.129 | 0.312 | 0.038 | 0.091 |

Intermittent | 0.650 | 0.675 | 0.175 | 0.250 | 0.010 | 0.050 | |

Drift | 0.362 | 0.427 | 0.312 | 0.229 | 0.190 | 0.160 |

Variables | Symbol | Units |
---|---|---|

Reactor Temperature | T | K |

Reactor Concentration | ${C}_{A}$ | kmol/m^{3} |

Flow rate of feed stream | F | m^{3}/min |

Flow rate of coolant flow | ${F}_{c}$ | m^{3}/min |

Inlet concentration | ${C}_{Ao}$ | kmol/m^{3} |

Inlet temperature of reactant A | ${T}_{o}$ | K |

Inlet temperature of the coolant | ${T}_{cin}$ | K |

Fault | Index | PCA-${\mathit{T}}^{2}$ | PCA-Q | PCA-KD | MPCA-${\mathit{T}}^{2}$ | MPCA-Q | MPCA-KD |
---|---|---|---|---|---|---|---|

Bias | FDR | 12.45 | 40.36 | 94.32 | 25.14 | 75.23 | 97.27 |

FAR | 7.84 | 1.00 | 0.00 | 10.75 | 1.64 | 0.00 | |

Prec | 70.51 | 98.38 | 100.00 | 77.08 | 98.16 | 100.00 | |

F1-score | 21.30 | 57.42 | 97.07 | 37.62 | 84.98 | 98.61 | |

Interm | FDR | 14.50 | 55.50 | 75.00 | 16.50 | 79.50 | 95.00 |

FAR | 5.42 | 2.65 | 0.00 | 3.85 | 2.25 | 0.00 | |

Prec | 50.00 | 90.24 | 100.00 | 61.11 | 92.44 | 100.00 | |

F1-score | 22.48 | 68.84 | 85.71 | 26.19 | 85.45 | 97.43 | |

Drift | FDR | 50.78 | 70.44 | 73.11 | 60.72 | 75.22 | 86.09 |

FAR | 8.15 | 1.00 | 0.00 | 6.78 | 0.00 | 0.00 | |

Prec | 90.40 | 99.06 | 100.00 | 93.20 | 100.00 | 100.00 | |

F1-score | 65.18 | 82.09 | 84.46 | 73.99 | 85.85 | 92.52 |

Fault | Index | PCA-${\mathit{T}}^{2}$ | PCA-Q | PCA-KD | MPCA-${\mathit{T}}^{2}$ | MPCA-Q | MPCA-KD |
---|---|---|---|---|---|---|---|

Bias | FDR | 11.12 | 32.95 | 71.73 | 18.86 | 62.05 | 95.15 |

FAR | 9.18 | 2.30 | 0.00 | 10.43 | 1.00 | 0.00 | |

Prec | 63.63 | 95.39 | 100.00 | 72.17 | 98.74 | 100.00 | |

F1-score | 19.12 | 49.11 | 83.53 | 30.00 | 76.11 | 97.51 | |

Interm | FDR | 13.25 | 48.00 | 62.00 | 13.00 | 75.00 | 88.50 |

FAR | 7.71 | 3.00 | 0.00 | 4.59 | 2.02 | 0.00 | |

Prec | 38.97 | 85.71 | 100.00 | 50.90 | 93.16 | 100.00 | |

F1-score | 19.86 | 61.84 | 76.54 | 21.00 | 83.18 | 94.00 | |

Drift | FDR | 44.11 | 68.67 | 71.39 | 55.45 | 73.11 | 84.75 |

FAR | 5.41 | 0.00 | 0.00 | 6.45 | 0.00 | 0.00 | |

Prec | 92.32 | 100.00 | 100.00 | 92.93 | 100.00 | 100.00 | |

F1-score | 59.45 | 81.42 | 83.30 | 69.63 | 84.46 | 91.74 |

SNR | Fault | PCA-${\mathit{T}}^{2}$ | PCA-Q | PCA-KD | MPCA-${\mathit{T}}^{2}$ | MPCA-Q | MPCA-KD |
---|---|---|---|---|---|---|---|

15 | Bias | 0.025 | 0.006 | 0.079 | 0.020 | 0.005 | 0.054 |

Intermittent | 0.140 | 0.007 | 0.180 | 0.250 | 0.007 | 0.100 | |

Drift | 0.311 | 0.288 | 0.266 | 0.300 | 0.233 | 0.155 | |

5 | Bias | 0.068 | 0.056 | 0.193 | 0.031 | 0.005 | 0.061 |

Intermittent | 0.160 | 0.007 | 0.200 | 0.280 | 0.007 | 0.105 | |

Drift | 0.362 | 0.427 | 0.312 | 0.351 | 0.255 | 0.166 |

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**MDPI and ACS Style**

Kini, K.R.; Madakyaru, M.; Harrou, F.; Vatti, A.K.; Sun, Y.
Robust Fault Detection in Monitoring Chemical Processes Using Multi-Scale PCA with KD Approach. *ChemEngineering* **2024**, *8*, 45.
https://doi.org/10.3390/chemengineering8030045

**AMA Style**

Kini KR, Madakyaru M, Harrou F, Vatti AK, Sun Y.
Robust Fault Detection in Monitoring Chemical Processes Using Multi-Scale PCA with KD Approach. *ChemEngineering*. 2024; 8(3):45.
https://doi.org/10.3390/chemengineering8030045

**Chicago/Turabian Style**

Kini, K. Ramakrishna, Muddu Madakyaru, Fouzi Harrou, Anoop Kishore Vatti, and Ying Sun.
2024. "Robust Fault Detection in Monitoring Chemical Processes Using Multi-Scale PCA with KD Approach" *ChemEngineering* 8, no. 3: 45.
https://doi.org/10.3390/chemengineering8030045