Fault detection procedures

Having “pre-processed” the residual data set to reduce the influences of the modelling error, the fault detection can be now implemented via the CUSUM test; 3 steps need to be followed.

1. Calculate the mean values µ0 of the residuals and this should have a value of around zero during the fault free period.

2. Detecting a change in the mean values µ0 based on the selection of δ/2.

3. Determining as soon as possible the time when a fault occurs based on the threshold, λ.

The mean value µ0 of the fault free residuals of ΔTwater, excluding the reset-to-zero data points, shown in Figure 6.13b is calculated to be 0.01 °C and the corresponding standard deviation is 0.8 °C (before the residual resetting), over 3

169 complete cycles. The outputs at various stages of the CUSUM test calculations from Equations 6.1 and 6.2 are given in Table 6.6, which illustrates how the equations are implemented. Due to space limitation, only three blocks of data of a time series measurements of 4 complete cycles were presented. The first and second blocks are both for fault free conditions, and a fault was introduced in the 3rd block of residual data.

Table 6.6 CUSUM test spreadsheet calculations

Time step

ΔTwater(t)

[°C] residual

Increase in mean Decrease in mean

y(t)-(µ0+δ/2) U_n m_n U_n - m_n y(t)-(µ0-δ/2) T_n M_n M_n - T_n 1 0.00 -0.41 -0.41 -0.41 0.00 0.39 0.39 0.39 0.00 2 0.00 -0.41 -0.82 -0.82 0.00 0.39 0.78 0.78 0.00 3 0.00 -0.41 -1.23 -1.23 0.00 0.39 1.17 1.17 0.00 4 0.00 -0.41 -1.64 -1.64 0.00 0.39 1.56 1.56 0.00 5 0.00 -0.41 -2.05 -2.05 0.00 0.39 1.95 1.95 0.00 6 0.38 -0.03 -2.08 -2.08 0.00 0.37 2.72 2.72 0.00 7 0.30 -0.11 -2.20 -2.20 0.00 0.29 3.40 3.40 0.00 8 0.20 -0.21 -2.40 -2.40 0.00 0.19 4.00 4.00 0.00 9 0.13 -0.28 -2.69 -2.69 0.00 0.12 4.51 4.51 0.00 10 0.09 -0.32 -3.01 -3.01 0.00 0.08 4.99 4.99 0.00 11 0.02 -0.39 -3.39 -3.39 0.00 0.01 5.41 5.41 0.00 … 148 -0.01 -0.42 -62.64 -62.64 0.00 0.38 55.76 55.76 0.00 149 -0.83 -1.24 -63.88 -63.88 0.00 -0.44 55.32 55.76 0.44 150 0.00 -0.41 -64.29 -64.29 0.00 0.39 55.71 55.76 0.05 151 0.00 -0.41 -64.70 -64.70 0.00 0.39 56.10 56.10 0.00 152 0.00 -0.41 -65.11 -65.11 0.00 0.39 56.49 56.49 0.00 153 0.00 -0.41 -65.52 -65.52 0.00 0.39 56.88 56.88 0.00 154 0.00 -0.41 -65.93 -65.93 0.00 0.39 57.27 57.27 0.00 155 -0.42 -0.83 -66.77 -66.77 0.00 -0.03 57.23 57.27 0.03 … 190 1.22 0.81 -83.99 -84.81 0.81 1.21 68.01 68.01 0.00 191 0.75 0.34 -83.65 -84.81 1.16 0.74 69.15 69.15 0.00 192 0.75 0.34 -83.32 -84.81 1.49 0.74 70.28 70.28 0.00 193 0.67 0.26 -83.05 -84.81 1.76 0.66 71.35 71.35 0.00 194 0.50 0.09 -82.96 -84.81 1.85 0.49 72.24 72.24 0.00 195 0.18 -0.23 -83.19 -84.81 1.62 0.17 72.81 72.81 0.00 …

170 The first block presents the fault free residuals from the beginning of the data set when the compressor was switched off, noting that the first 5 residual points were set at zero. It is common to set the δ in equation 6.1 as the same value as one standard deviation (Chetouani 2008), i.e. δ/2 was thus set to be ±0.4 °C and as all the data points in the first block are within this range, therefore the CUSUM test results (Un-mn) and (Tn-Mn) are all zero.

For the second block, as before, the residuals for 5 data points were set to zero during the switching moment. However, it is evident that relatively large prediction errors (or residuals) were still encountered before (time step 149) and after (time step 155) the switching moment that were not reset to zero; both exceed a half δ of 0.4 °C. These residuals lead to a positive value of Tn-Mn in the CUSUM test at the corresponding time steps, suggesting that a positive CUSUM output may not always imply a fault (i.e. a false alarm).

The last block in Table 6.6 presents some of the residuals from time step 190 after a fault of flow reduction has been introduced. A series of positive (Un-mn) values suggests that the mean values of the residuals are increasing from the original value of 0.01 °C. Meanwhile, (Tn-Mn) remains at zero as it can only have a value when the residual is negative. Essentially, whenever a series of positive values of (Un-mn) or (Tn-Mn) occur, potentially a fault could be present.

171 The CUSUM test results (Un-mn) and (Tn-Mn) for the entire four cycles are shown in Figure 6.14a and Figure 6.14b respectively. Since only 5 residual data points were set to zero during the on-off/off-on switching moments, it is possible to have some data points in the vicinity of these periods to exceed ±½δ due to modelling errors resulting in some small positive test results (small spikes).

When the fault of the cooling water flowrate reduction occurred at time step 190 in the 4th cycle, the residuals of the ΔTwater increased (as seen the faulty cycle in Figure 6.13b). The CUSUM test output in the faulty zone of Figure 6.14a increased sharply for a short period when compressor shut down and went back to zero (even after the 5-point resetting, followed by a larger peak for a longer period of time when the compressor was re-started at time step 221. The short section between the two peaks suggested that the cooling water restriction had a very little influence on ΔTwater when the compressor was not running.

(a)

(b)

Figure 6.14 CUSUM test results of the ΔTwater residuals after 5-point resetting, (a)Un-mn and (b) Mn-Tn

Fault free Faulty

172 (a)

(b)

Figure 6.15 (a) The residuals of the evaporating temperature Te, (b) The residuals of the Te after 5 data points

being reset

Figure 6.15a shows the residuals of the evaporating temperature for the same four cycles. It was noticed the evaporating temperature also experienced larger prediction errors at the compressor on-off moments though the prediction accuracy of

Te was much better during off-on moments; as a result, as shown in Figure 6.15b, two data points were needed to be set to zero during switching-off moment but none were set to zero during switching-on moment. It could also be seen that the model prediction errors during the compressor off periods were larger than that of the on periods.

From Table 6.3 (Section 6.1), it can be seen that the evaporating temperature was not too sensitive to the cooling water flow reduction, which can also be noted from Figure 6.13b when comparing the fault-free with the faulty cycles, both having the similar patterns of Te residuals, thus suggesting that Te is not a suitable parameter for detecting this fault.

Fault free Faulty

173 Figure 6.16 CUSUM test results for evaporating temperature residual

For the same fault, the CUSUM test procedures were performed for the evaporating temperature of all the 4 cycles. The results are presented in Figure 6.16.

Un - mn remains at zero throughout as there are no large positive residuals and Tn - Mn has large peaks each time corresponding to negative residuals during off-cycle periods. The sizes of the peaks in both fault free and faulty zone are very similar, making it difficult to select a suitable threshold that can detect the fault without causing a false alarm, implying that the off-cycle residuals are not suitable to be used as a fault indicator and the evaporating temperature is not an appropriate choice to detect a cooling water flow restriction fault.

The final step in fault detection is to determine a suitable fault-free threshold λ that can detect fault at a low severity level without causing too many false alarms. As mentioned previously, it is better to choose a small δ/2 at one standard deviation of the fault free residual and adjust the value of λ accordingly. An easier way is to set the threshold according to the CUSUM test results. In Figure 6.14, two thresholds, λ = 0.4 and 0.8, are tested. The λ = 0.4 threshold is able to tolerate the small jumps in Figure 6.14a and at the same time detect the fault at the 191 time step. In Figure 6.14b, using the same λ = 0.4, a false alarm is triggered at 149 time step. However, if the threshold is set to 0.8, the possibility of the false alarm is eliminated. The drawback is that the fault is detected 10s later than with the smaller threshold. In this case, because the

174 delay in detection is rather small, it is more important to avoid the false alarm. Thus λ could be selected as 0.8.

The CuSum tests and the similar analysis had been carried out for all the other specified faults. Table 6.7 shows the results of the threshold selection. The standard deviation after the residual resetting of the fault free residuals is used as δ for both

ΔTwater and Te. λ was selected manually to achieve the quickest detection speed while avoiding any false alarm for individual faults. Therefore, 4.0 was chosen respectively for ΔTwater and Te, i.e. the largest among all the possible λ for individual faults.

In practice, at any instant only one of the two mentioned parameters is needed to raise an alarm for each type of fault. ΔTwater was not sensitive to the binary ice flow reduction fault and the broken scraper fault, therefore the two faults can only be flagged by Te. Similarly, the solution concentration decrease fault could only be flagged by ΔTwater. For the rest three faults, as the sensitivity factors of Te were all larger than those of ΔTwater, the alarms were triggered by the former in most cases.

Table 6.7 The selection of threshold for various faults

Faults ΔTwater Te

δ [°C] λ δ [°C] λ

Binary ice flow reduction 0.01 4.0 0.07 4.0

Cooling water flow reduction 0.01 3.8 0.07 4.0 Initial solution concentration increase 0.01 3.8 0.07 3.7 Initial solution concentration decrease 0.01 4.0 0.07 4.0 Broken scraper 0.01 4.0 0.07 3.8

Scraper motor failure 0.01 4.0 0.07 4.0