To verify the low false alarm rate and fault diagnosis performance of this method,
and
are chosen as instances. As described in
Section 3.1, different groups for
and
have been obtained and listed in
Table 3 when different grouping thresholds are used. As can be seen, the number of the sensors has decreased along with the increase of the grouping thresholds. A high correlation between sensors will contribute to the constructing of a virtual sensor in the same group. All chiller sensors are divided into the same group, if the grouping threshold takes 0.
4.2.1. Verification of Low False Alarm Rate
We assume that only one sensor is considered as a fault state. Input sensors
and
with minimum biases are used to verify the low false alarm rate of this method. Sensor grouping threshold takes 0.8. In the figure below, the first half time series are normal samples and the last half are fault samples.
Figure 6 shows the absolute deviation of
and
when the corresponding input sensors
and
are faulty. As can be seen in
Figure 6a, if only one virtual sensor is used, the absolute deviation will increases after fault occurs. It means that the fault diagnosis performance will be affected by the input sensor. However, another sensor fault occurs if only one absolute deviation exceeds the fault diagnosis threshold, as depicted in
Figure 5 and
Figure 6b,c. The false alarm rate of
is reduced from 28.0% to 0.0%. In the same way, the false alarm rate of
is reduced from 41.0% to 0.0%.
To further verify the performance of this method, the false alarm rate of all chiller sensors is listed in
Table 5. As we can see, the false alarm rate is close to 0 when two virtual sensors with different input sensor are used. For instance, the false alarm rate of
will reduce from 18.0% to 3.0%, if two virtual sensors for
are used. The high false alarm rate will occur if only one virtual sensor is used. Therefore, two virtual sensors with different input sensors can achieve a low false alarm rate.
4.2.2. Fault Diagnosis Performance
As a deep learning model, the LSTM model, which is developed to address sequential data with its ability to encode temporal information, has a better performance than the linear regression and nonlinear regression models such as the artificial neural network model. Linear regression is a linear approach to modeling the relationship among different sensors. The artificial neural network model, which has great nonlinear curve fitting capability, can achieve a better performance than the linear regression model. Due to the high correlation between sensors in the same group and the strong feature extract ability of the LSTM model, either of the two sensors can achieve a better performance on fault diagnosis. Virtual sensors are used to diagnose faults in this paper, which are constructed by the linear regression (LR) model, the artificial neural network (ANN) model and the LSTM model. and with minimum biases are chosen as instances.
The performance of virtual sensors can be verified by the different grouping thresholds. In
Figure 7, the absolute deviation and fault diagnosis ratios obviously increase along with increase of grouping threshold. The absolute deviation of the LSTM-based virtual sensor has significant performance than FC-based and LR-based virtual sensor, when the fault samples occur. For instance, the average absolute deviation between pressure sensor
and virtual sensors will decrease from 0.027 to 0.021, if LSTM-based virtual sensor is replaced by FC-based and the grouping threshold takes 0.8. The average absolute deviation is decreased from 0.027 to 0.019 and the fault diagnosis ratios is decreased from 100.0% to 95.0% if the LSTM-based virtual sensor is similarly replaced by LR-based. When the fault samples occur, increasing absolution deviation can be obtained along with the increase of the grouping thresholds. For instance, the average absolute deviation is decreased from 0.021 to 0.027, if the grouping threshold takes 0 instead of 0.8. It reveals that LSTM-based virtual sensors have an excellent performance along with the increase of the grouping threshold.
As shown in
Figure 8, average fault diagnosis ratio are compared in different biases. For these two sensors, the fault diagnosis ratios of three methods increases with the increase of the introduced biases fault in
Table 4. All methods easily reach 100% when the magnitude of biases is the maximum of each sensor. However, as the magnitude of biases decreases, the fault diagnosis rates show a difference. For instance, when the biases of the temperature sensor
is tiny, such as 0.5, FC-based and LR-based virtual sensors have a poor performance while the fault diagnosis ratio of the LSTM-based virtual sensor reaches 98%. It means that LSTM-based virtual sensors have a better performance in the case of the positive or negative biases. When the biases are negative, all methods are influenced to various extents. For instance, average fault diagnosis ratios of the pressure sensor
are 98.5%, 94% and 89.75% respectively, provided that magnitude of biases is changed from 0.5 to −0.5. However, fault diagnosis ratios of LSTM-based virtual sensors always achieve a better performance than other methods.
To further verify the performance of another virtual sensor during training and testing, the mean absolute error (MAE) is used in this paper. MAE is frequently used to measure the differences between values predicted by a model and the values observed. The MAE between the physical sensor and virtual sensor is defined as
where
N represents sampling steps,
represents value of virtual sensor and
represents value of physical sensor. In actual implementation, the MAE of the corresponding sensor is obtained by calculating the average of the two virtual sensors.
As shown in
Table 6,
p-values between the traditional methods and the proposed method were calculated. The
t test can easily be adapted to testing the following hypotheses at a specified level of significance
:
,
.
will be rejected at any level
, which means there is not a significant difference between the two methods. As we can see,
p-values between the traditional methods and the proposed method are always smaller than 0.05. This means that LSTM-based virtual sensors are significantly different to the two other methods.
In addition to
and
,
Figure 9 shows MAE of another nine sensors during training when the grouping threshold takes 0.8. Smaller value of MAE indicates better training performance of the virtual sensors. As we can see, due to the different reading range of different sensors, virtual sensors show different performances for different physical sensors. However, LSTM-based virtual sensors always have a lower MAE value compared with LR-based or FC-based virtual sensors. It reveals that LSTM-based virtual sensors always have a better training performance.
Table 7 summarizes MAE of another nine sensors during testing, to which minimum positive biases are added. The grouping threshold also takes 0.8. On the contrary, the maximal value of MAE indicates better training performance of the virtual sensors. When the minimum negative bias occurred, virtual sensors based on different models show a different performance. Therefore, LSTM-based virtual sensors have a better performance for both positive and negative biases.
In order to verify the performance of different physical sensors, fault diagnosis ratios of all physical sensors with minimum biases are listed in
Table 8 and
Table 9.
Table 8 lists fault diagnosis ratios when the grouping threshold takes 0.8 and the minimum positive biases are added. It can be clearly seen that LSTM-based virtual sensors have a better performance than another two methods. This conclusion is consistent with the results of
Figure 9 and
Table 7. For instance, fault diagnosis rate with positive bias is 91.00% and 96.25% rather than 98.75%, when the LSTM-based virtual sensor is similarly replaced by LR-based and FC-based virtual sensor for physical sensor
. The fault diagnosis ratios are reduced by 7.75% and 2.5% compared to LR-based and FC-based virtual sensor, respectively. When the minimum negative biases occur, LSTM-based virtual sensors still have a better performance in
Table 9.