Reliability Assessment of Repairable System based on Performance Data

MATERIALS AND METHODS

The proposed steps in performing the analysis are as shown in Fig. 1 which is based on the method proposed by Louit et al. (2009) and is summarized as follows:

(1)

where I(.) is an indicator variables taking the value of 1 whenever the condition is met.

Fig. 1:	Proposed process flow for the TTFF (time to functional failure) data to apply the appropriate model

As M can be approximated by normal distribution, the calculated M value is then compared to the standard value to decide whether the null hypothesis can be accepted. The equation in relating M to z factor is as shown in Eq. 2:

(2)

where, r is the number of repair (or failure).

(3)

where, T_j is the time to jth failure and [a,b] is the interval of the observation. on the other hand is given by:

Case study
System description: The system chosen to illustrate the applicability of the model is one of the gas powered turbines operating in a gas district cooling. The capacity of the turbine is 4.2 MW. The daily system performance is based on peak hour demand between 8 am to 5 pm during weekdays. A total of 1290 days of data were gathered and used in the analysis. Figure 2 shows the plot system performance data versus days of production. The minimum production limit is set based on research work done by Majid and Nasir (2011) on a similarly configured system. The system is assumed to experience functional failure whenever the system performance falls below the minimum production limit. Based on this assumption, the TTFF is shown in Table 1.

Fig. 2:	Production output (KW) versus days of production used to calculate the TTFF (time to functional failure)

Table 1:	TTFF (time to functional failure) Data calculated based on the time the system falls below the minimum production limit

RESULTS AND DISCUSSION

The results provided in the following sections prove the possibility of using performance data in reliability analysis in cases where failure data is not available. This is evident from statistical analysis comparing actual function failures to the predicted failures. However, each TTFF needs be tested according to the proposed process flow so that appropriate analysis can be applied.

Test for independence: Test for independence was performed by using serial correlation test (Kumar and Klefjso, 1992) which will detect the presence of dependent data. In this test, the TTFF data was plotted against a one lag time data as shown in Table 2 and Fig. 3. Since the data was randomly scattered based on Fig. 3, it can be concluded that one failure to the next was independent. In other words, the current failure did not have any influence over immediate subsequent failure.

Trend test: The second step in the analysis is to perform a graphical trend test based on the plot of cumulative time versus number of failure of the data in Table 1.

Fig. 3:	Result of dependency test for TTFF (time to functional failure) which shows no correlation, thus failures are independent

Fig. 4:	Graphical trend test for TTFF (time to functional failure) which shows good fit to the straight line indicating no trend in the failure data

The result is shown in Fig. 4 which indicates a good fit for linear regression (R² = 0.95). This means that the data did not indicate any trend and thus can be assumed to be identically distributed.

Analytical trend test: To further validate the graphical test of IID, two analytical tests were conducted namely Mann test and Laplace test. Both tests were performed based on 95% confidence limits and the results are shown in Table 3. Since the calculated values for both tests fall between the z_cr values at 95% confidence interval, the data can be assumed to be identically distributed.

Model assumptions: Based on both graphical and analytical tests, the TTFF data can be safely assumed to be IID and exponentially distributed. As such, the system failure rate can be estimated by fitting exponential distribution to the TTFF data. The Probability Distribution Function (PDF) is shown in Fig. 5. Rank regression method was subsequently used to estimate the failure rate, λ = 0.0097 failure day^-1.

Fig. 5:	PDF plot for TTFF (time to functional failure) which show a good fit to exponential distribution

Table 2:	TTFF (time to functional failure) and lag-time TTFF used to test for independence between two subsequent failures

Table 3:	Mann and Laplace test results showing no trend in failure data at 95% confidence interval

With the failure rate, the reliability function for the system with respect to time can be represented as:

(4)

The plot of reliability function is shown in Fig. 6.

The reliability plots shows that system reliability is approaching zero at 640 days (or the probability of failure equals 1). This means that, there will be at least 1 system failure when the system operates continuously for 640 days. As mentioned above, the failure is when the system fails to meet the minimum requirement and not the total system failure. This result can be used to schedule a shutdown for planned maintenance.

Fig. 6:	Reliability versus time plot showing the expected functional failure to occur at 640 days

Fig. 7:	Effect to reliability as demand changes. As the demand increases, the probability of not meeting the demand also increases

The timing for the shutdown can also be based on system reliability target as required by the organization. For example, a preventive maintenance should be performed when the reliability figure falls below 20% which in this particular case, between 5 to 6 months of operation time. Bear in mind that the reliability is for the whole system with all modes of failure. Further analysis using the maintenance data should be performed to understand failure modes so that proper preventive action can be done during the planned shutdown.

The analysis can be further extended to look at the impact to reliability as demand or the minimum requirement changes. Obviously, the higher the demand, the more likely the system will not be able to meet the requirement. Figure 7 shows the changes in reliability as requirement changes.

Model validation: The proposed approach was validated by comparing the predicted failure data to the actual system failure.

Fig. 8:	Comparison between predicted failure and the actual occurrence of functional failures

Fig. 9:	Statistical comparison between actual failures and predicted failure

The predicted failure is based on mean time between failures which is the reciprocal of failure rate. The result shown in Fig. 8 shows the predicted failure time based on mean time between failures compared with the actual occurrence of failure. A statistical comparison using Dunnett’s test (Walpole et al., 2007) shows statistical equivalent proving the validity of the approach as shown in Fig. 9.

CONCLUSION

This study presents an alternative approach in assessing system reliability based on abundantly available system performance data instead of failure data. Based on the approach, the reliability result is actually measuring the probability that the system is able to meet the minimum requirement for a given operating condition for a stated period of time. The result is valuable in planning a system shutdown depending on the organization’s reliability target.

However, this approach is only valid for a system level reliability encompassing all modes of failures. As such, other information such as component failures or types of failure are not apparent. Thus, maintenance data and condition monitoring data are still essential in assessing the component level of reliability performance.

ACKNOWLEDGMENT

The authors wish to thank Universiti Teknologi PETRONAS for providing the necessary support for this research.