4.1. Data Source
In August 2017, 578 electricity meters were installed, and the field operation data between 2017 and February 2022 were collected. The failure time and cumulative failure rate of 35 failed electricity meters by the end of 2019 are shown in
Table 1.
Combined with the data in the table above, scale parameters
and shape parameters
can be calculated according to Equations (3) and (4). According to Equations (9) and (10), the failure rate
and the failure number
of each future prediction time interval
can be obtained. The confidence interval of
can be calculated by Equation (16), and the relative dispersion degree of failure number
can be calculated according to Equation (35). The future prediction time interval
in the table below represented 365 days from the end of 2019 to the end of 2020, 730 days from the end of 2019 to the end of 2021, and 790 days from the end of 2019 to the end of February 2022, respectively. The responding calculation results are shown in
Table 2 and
Table 3 below.
The actual failure of electricity meters was 9, 23, and 27 in the following 365 days, 730 days, and 790 days, respectively.
Table 3.
Prediction interval and relative dispersion ratio.
Table 3.
Prediction interval and relative dispersion ratio.
| Bilateral Interval | |
---|
365 | [7.295, 19.48] | 0.9151 |
730 | [16.66, 35.34] | 0.7193 |
790 | [18.18, 37.86] | 0.7027 |
As seen from the above table, among the three future time intervals, when the future time interval was 365 days, the relative dispersion ratio was the largest and close to 1. The relative dispersion ratio was the smallest when was 790 days.
4.3. Analysis of Relative Dispersion Degree of Prediction Results
The observation time was 487 days, 852 days, and 1217 days, respectively. The relationship between the prediction failure number
and the future prediction time interval
was analyzed, as shown in
Figure 3.
As shown in the figure above, increased with the growth of , and went into a stable trend in the later stage gradually. With the continuous increase of , all the electricity meters would finally fail. On the whole, as the time increases, more meters need to be replaced, and we should increase the storage of meters accordingly.
According to the parameters of the calculation of the data in three observations, the relationship between the prediction failure number in the next 365 days and the observation time was analyzed, as shown in the following figure.
As shown in
Figure 4,
in the next 365 days decreased with the increase in
, and
predicted by the observed time of the previous 487 days was a temporary increase by the time of
, and then decreased with the rise of
, and the decline was faster than the other two. On the whole, we can see that the number of electricity meters to be replaced in the next year decreased with the increase in the observation time.
The observation time was 487 days, 852 days, and 1217 days, respectively. The relationship between the number of the predicted failure in the next 365 days and the total batch quantity
was analyzed, as shown in
Figure 5.
The figure above shows that in the next 365 days increased with the increase in when and were unchanged, and in the prediction of the data in the longer observed period increased more slowly with the rise in . Therefore, as the number of meters installed in batches increases, the number of meters to be replaced in the next year will also increase, and we will need to reserve more meters accordingly.
The relationship between the relative dispersion ratio
and the future prediction time interval
was analyzed, as shown in
Figure 6.
As shown in the figure above, decreased with and finally exhibited a stable trend. It can be seen that the value of was first close to 1 and then away from 1 with the increase in , which indicated that the dispersion degree of the number of meter failure decreased first and then increased.
The observation time was 487 days, 852 days, and 1217 days, respectively. The relationship between the relative dispersion ratio
in the next 365 days and the total quantity
was analyzed, as shown in
Figure 7.
As shown in the figure above, decreased with the increase of when the time of and were all unchanged, and finally exhibited a stable trend. It can be seen that the value of dispersion ratio first approached 1 and then moved away from 1 with the increase in the total number of electricity meters in batches, which indicated that the dispersion degree of the number of electricity meter failure decreased first and then increased.
4.4. Bayesian Fault Point Estimation and Interval Estimation Case Analysis
After multisource data fusion, the reliability life was 2920 days to 5840 days when the reliability
was 0.9. Since the data used for Bayesian prediction in this chapter were the field data from August 2017 to the end of 2019 in
Table 1, the parameter
was 0.91697 calculated from August 2017 to the end of 2019 without previous information.
The previous distribution parameters were obtained according to Equations (25) and (29):
= 95.17269,
= 1778004.98, and
= 0.0000728898. The corresponding Bayes point estimates and interval estimates of failure number were obtained according to Equations (31), (32) and (34), as shown in
Table 5 and
Table 6 below.
After verification, the Bayesian prediction result of the number of failures in this example was slightly less than the actual number of on-site failures.
Table 6.
Bayesian interval prediction and relative dispersion ratio.
Table 6.
Bayesian interval prediction and relative dispersion ratio.
| Bayes Bilateral Interval | |
---|
365 | [7.496, 19.828] | 1.589 |
730 | [17.180, 36.204] | 1.249 |
790 | [18.765, 38.820] | 1.222 |
In this example, the Bayesian interval estimation results were consistent with the prediction results without previous information fusion.
The prediction results of the Bayes method and those without fusion of previous information change over time, as shown in the following figures (
Figure 8,
Figure 9,
Figure 10 and
Figure 11).
Above, the prediction results of were increased with the increase in , and the failure number prediction results of the Bayes failure were smaller than those that do not merge the prior information. As was increasing, the final two kinds of failure number were expected to be close to the number of batches. Based on the Bayesian prediction results, the number of meters to be replaced in the predicted time interval was relatively small. Corresponding to the Bayesian prediction case, we only need to reserve fewer meters.
Based on the data of the previous 852 days of observation time, the relationship between the prediction failure number
of the next 365 days and the observation time
was analyzed by using no prior information fusion and the Bayes method, as shown in
Figure 9.
Figure 9.
The relationship between Bayesian and classical prediction of failure times and the observation time variation.
Figure 9.
The relationship between Bayesian and classical prediction of failure times and the observation time variation.
Above, according to the data in the previous 852 days of observation time, predicted by the two methods decreased with the increase in when the prediction period was the coming 365 days. The expected number of failures by the Bayesian method decreased more slowly with the increase in . In the end, indicated by the two methods was stable with increasing observation time. With the rise in observation time, compared with the number of electricity meters to be replaced in the next year without prior information prediction, the number of electricity meters to be replaced in the next year under Bayesian prediction is first small and then large.
Based on the data of the previous 852 days of observation time, the relationship between the prediction failure number
of failures of the next 365 days and the total batch quantity
were analyzed using no prior information fusion and the Bayes method, as shown in
Figure 10.
Figure 10.
The relationship between Bayesian and classical prediction of failure times and the total quantity variation.
Figure 10.
The relationship between Bayesian and classical prediction of failure times and the total quantity variation.
As seen from the above figure, in the next 365 days increased with the increase in the number of batches when the time and the future prediction time interval were all unchanged, and predicted by the Bayesian method increased more slowly with the increase in . As the total number of meters installed in the same batch increases, the number of meters that will need to be replaced in the next year under the Bayesian prediction is less than that under the prediction without prior information.
Based on the data of the previous 852 days’ observation time, the relationship between the relative dispersion ratio
of the next 365 days and the prediction time interval
was analyzed using no prior information fusion and the Bayes method, as shown in
Figure 11.
Figure 11.
Bayesian and classical methods predict the relative dispersion of results with future time intervals.
Figure 11.
Bayesian and classical methods predict the relative dispersion of results with future time intervals.
As seen from the above figure, decreased with the increase in and finally exhibited a stable trend, and the Bayesian dispersion ratio prediction result was more significant than the dispersion ratio prediction result without prior information. On the whole, it can be seen that the dispersion ratio of Bayesian prediction results was closer to 1, which indicated that the dispersion degree of Bayesian prediction results was lower.
Based on the data of the previous 852 days of observation time, the relationship between the relative dispersion ratio of the next 365 days and the total batch quantity was analyzed using no prior information fusion and the Bayes method, as shown in the following figure.
As seen from
Figure 12, the Bayes prediction dispersion ratio curve was initially higher than the dispersion ratio curve without prior information, and then slowly approached consistency. The value of
calculated by the two methods decreased with the increase in
when the time of observation and the prediction time interval were all unchanged, and finally exhibited a stable trend. On the whole, it can be seen that the dispersion degree of Bayesian prediction results was consistent with that of prediction results without prior information.
It can be seen from the above figures that the Bayes prediction of the number of electric meter faults after information fusion was consistent with that without previous information fusion. In the prediction of the relative dispersion ratio, the relative dispersion degree of Bayesian prediction was lower and the effect was better.