- Email: [email protected]
A practical approach for reliability prediction of pipeline systems
European Journal of Operational Research 198 (2009) 210–214
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
A practical approach for reliability prediction of pipeline systems Yong Sun a,*, Lin Ma a, Jon Morris b a b
CRC for Integrated Engineering Asset Management, School of Engineering Systems, Queensland University of Technology, 2 George Street, Brisbane, Queensland QLD 4001, Australia MPT-Solutions, Industrial Research Limited, P.O. Box 31-310, Lower Hutt, New Zealand
a r t i c l e
i n f o
Article history: Received 8 November 2005 Accepted 28 July 2008 Available online 8 August 2008 Keywords: Pipelines Reliability Maintenance decision support Applied probability Repairable systems
a b s t r a c t Pipelines play an important role in the modern society. Failures of pipelines can have great impacts on economy, environment and community. Preventive maintenance (PM) is often conducted to improve the reliability of pipelines. Modern asset management practice requires accurate predictability of the reliability of pipelines with multiple PM actions, especially when these PM actions involve imperfect repairs. To address this issue, a split system approach (SSA) based model is developed in this paper through an industrial case study. This new model enables maintenance personnel to predict the reliability of pipelines with different PM strategies and hence effectively assists them in making optimal PM decisions. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Pipelines are one of the most applied means to transport gas, oil and water in the world. The failures of pipelines can have huge negative impacts on business, environment and society. Therefore, the reliability of pipelines is a major concern in the field of engineering asset management and has attracted much attention of researchers in the field [1–4]. When modelling the reliability of the pipelines, the existing literature largely focuses on the influences of corrosion [1,2] and creep [3] on the pipeline failures from the material failure mechanism point of view. Very few models have considered the impact of preventive maintenance (PM) on the system reliability of pipelines. In reality, PM is often applied to aged pipelines to reduce unexpected failures and their resultant undesirable impacts. Two commonly used PM policies include the time based preventive maintenance (TBPM) and the reliability based preventive maintenance (RBPM). In the TBPM policy, a pipeline is maintained based on scheduled PM times. The intervals between two PM actions may or may not be the same; whereas in the RBPM policy, a control limit of reliability R0 is defined in advance. Whenever the reliability of a pipeline falls to this predefined control limit, the pipeline is preventively maintained. The purpose of PM is to improve the overall reliability of the entire pipeline. To decide a cost effective PM strategy for a pipeline, accurately predicting the reliability of the pipeline with multiple PM intervals is desired. However, to date, effective models for meeting this need have yet to be devel* Corresponding author. Tel.: +61 7 3138 2442; fax: +61 7 3138 1469. E-mail address: [email protected] (Y. Sun). 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.07.040
oped. The prediction is difficult because pipelines are normally complex systems. Furthermore, imperfect repairs often need to be considered. Imperfect repair in this paper indicates a system after a PM action is between as good as new and as bad as old. A number of papers have been published to consider the influence of imperfect repairs on the reliability of a system [5–8]. In [5,6], Pham and Wang discussed eight categories of methods for treating imperfect maintenance. These methods can be further classified into two groups: (1) state rules based models and (2) improvement factor methods. The former includes (p, q) rule based model, (p(t), q(t)) rule based model, and multiple (p, q) rules based model; whereas the latter includes improvement factor method, virtual age method, shock model method, (a, b) rule and other models reviewed in [5]. In Ref. [7], Rausand and Hoyland reviewed four types of the imperfect repair models: (1) Brown and Proschan’s model; (2) failure rate reduction models; (3) age reduction models; and (4) trend renewal process. The Brown and Proschan’s model is regarded as a (p, q) rule based model in [5,6]. The rest can fall into the group of the improvement factor method. The model given in [8] is also belong to this group although it uses two improvement factors. The state rules based models are largely used for one-component system, and estimating the probabilities for different states is difficult. For using the improvement factor methods, a major difficulty is to estimate the improvement factors, especially for complex systems. In contrast, Ebeling [9] and Lewis [10] presented a heuristic method to predict the reliability of an asset with multiple PM intervals. In this method, PM time is a deterministic variable. This method can produce an intuitive and explicit prediction of reliability and hence very suitable for engineering applications.
Y. Sun et al. / European Journal of Operational Research 198 (2009) 210–214
However, in this model, assets are assumed to have PM actions periodically, i.e., this method was developed based on TBPM only. Besides, this method cannot model the effects of different PM actions on the system reliability. It does not consider system configuration. When modelling imperfect maintenance, it uses a method similar to the improvement factor method. To address the limitations of the existing methods, a split system approach (SSA) has been developed [11,12] (the SSA is termed as SSM (split system model) in [11]). The SSA is different from the other methods in that it models the reliability of a system with multiple PM actions at component level. This approach considers the effects of repaired components on the system reliability and allows the changes of the reliability of the system after each PM action to be calculated rather than estimated by maintenance staff. SSA removes the assumptions on the probability of different states of a system after repairs. These assumptions were used in the state rules based models. This paper aims to predict the reliability of pipelines with multiple imperfect PM actions using SSA through an industrial case study. The outcomes can assist industrial personnel to determine optimal PM strategy for pipelines. Both TBPM and RBPM are considered. Since the effects of PM activities on a system can be modelled using either hazard function (e.g., see [8]) or reliability function (e.g., see [9,10]), the reliability function is used for modelling in this paper. The rest of this paper is organised as follows. In Section 2, the industrial case is described. In Section 3, the concept of SSA is introduced and then formulae are derived for predicting the reliability of pipelines under the condition that the same part is repaired. Section 4 presents prediction results and the corresponding analysis. Conclusions are given in Section 5. 2. Case description A water supply company needed to make an optimal maintenance strategy for a 2224 meter long steel raw water pipeline. Real world data were collected. The observation of failure history indicated that the entire pipeline could be divided into two portions: a total of 200 meter of exposed pipeline composed of pipe bridges and stream crossings and another subsystem comprising of buried pipes. Assume that all pipes in the exposed pipeline had the same failure distribution and the pipes in the 2024 meter of buried pipeline had the same failure distribution. An operational period over 68 years was considered. During this period, the entire pipeline had a 30 years minimum guaranteed life. After 30 years, the exposed pipeline started to wear out with a Weibull failure distribution
" 14:3532 # t 30 ðt P 30Þ: R1 ðtÞ0 ¼ exp 35:6718
ð1Þ
The buried pipeline had an exponential failure distribution
256ðt 30Þ ðt P 30Þ; Rsb ðtÞ0 ¼ exp 1; 000; 000
ð2Þ
where, R1(t)0 and Rsb(t)0 are the original reliability functions of the exposed pipeline and the buried pipeline. Parameter t is the age of the pipeline. The exposed pipeline is connected with the subsystem (buried pipeline) in series, although the subsystem is a complex system. Hence the reliability of the entire pipeline system was
Rs ðtÞ0 ¼ exp½
14:3532 ! t 30 256ðt 30Þ ðt P 30Þ: þ 35:6718 1; 000; 000
ð3Þ
211
When the exposed pipeline was operating in its wear-out stage, conducting PM on the exposed pipeline can improve the overall reliability of the entire pipeline. Since the mean time between failures (MTBF) of the buried pipelines is much longer than the exposed, multiple PM actions over multiple intervals will be implemented on the exposed pipelines. The company needed to decide a cost-effective PM strategy which includes selecting the optimal PM policy and determining the optimal PM parameters. If the TBPM policy is selected, the optimal PM leading times need to be determined; if the RBPM policy is selected, the optimal reliability control limit R0 needs to be determined. To make an optimal decision, the reliability changes of the entire pipeline with different PM strategies have to be predicted accurately. However, as indicated in Section 1, the existing models/methodologies cannot meet industrial need for the explicit prediction of the reliability improvement and the consideration of multiple imperfect repairs. The entire pipeline is normally imperfectly repaired because only partial pipeline will be repaired during a PM action. To address this issue, a SSA based reliability prediction model is developed in the next section. 3. Model development 3.1. Concept of SSA From the above description of the pipeline case, it is noted that when a system is preventively maintained, often only part of its components rather than the whole system is repaired (maintained). Repair information at the component level can often assist in understanding the changes of the reliability of systems after PM actions and hence improve the maintenance outcome of these systems. The information at the component level should be considered when the reliability of a system is being modelled. SSA enables this information to be used in the reliability prediction. The basic concept of SSA is to separate repaired and unrepaired components within a system virtually when modelling the reliability of the system after PM actions. In the following section, the SSA is applied to calculate the reliability of the pipeline with multiple PM intervals. 3.2. Formulation of the pipeline reliability The following analysis is conducted based RBPM policy, i.e., to preventively repair some pipes in the exposed pipeline whenever the reliability of the entire pipeline falls to a predefined control limit of reliability R0. An extension of the outcomes to the TBPM policy is straightforward. In the formulation, the following assumptions were made: (1) Only exposed pipeline is preventively maintained in all PM actions. The reliability functions of repaired pipes are known. (2) The failure of repaired part is independent of unrepaired part. (3) Repair time is negligible. (4) PM time is a deterministic variable. In the TBPM policy, it is scheduled by maintenance personnel; whereas in the RBPM policy, it is calculated based on the predefined reliability control limit of the pipeline. According to the above assumptions, only the reliability of the exposed pipeline changes when the pipeline is preventively maintained. The reliability of the remainder of the pipeline just before and after this PM action does not change.
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Since only the exposed pipeline is repaired in a PM action, the reliability of the entire pipeline after this PM can be improved but lower than its original reliability, leading to an imperfect repair. After imperfect repairs, the reliability of the pipeline declines in a manner shown in Fig. 1. In this paper, the subscript i is used to stand for ‘‘after the ith PM action”. Subscript i = 0 stands for no PM. In Fig. 1, R0 is the predefined control limit of the reliability for the entire pipeline, ti (i = 0, 1, 2, . . . , n, n + 1) is the ith PM time and also the start time for the pipeline to operate again after the ith PM action (note that n is the scheduled number of PM intervals during a given decision horizon which is 68 years in this case study, t0 = 0 and tn+1 = 68 years). Parameter t is absolute time scale (0 6 t < 1) and parameter s is a relative time scale (0 6 s < (ti+1 ti), and i = 0, 1, 2, . . . , n).
t ¼ ti þ s ðt i 6 t < tiþ1 ; and i ¼ 0; 1; 2; . . . ; nÞ:
ð4Þ
When a system receives PM actions, two types of reliability concepts are involved [9,12]: one is the conditional reliability of the system. This reliability indicates the survival probability of a system which has successfully been preventively maintained. It describes the reliability changes between two PM actions. The other is the probability of survivor of the system over its whole life time which takes into account the probability of survival of the repaired components until their individual PM times. It describes the reliability changes of the system over a given period which may cover a number of PM intervals. To distinguish the latter from the conditional reliability, it is termed as overall reliability of the system. According to Ebeling [9] and Lewis [10], for a series system, the relationship between the conditional reliability and the overall reliability is given by
Rs ðtÞi ¼
i Y
R1 ðt k tk1 Þk Rsc ðtÞi
ðt i 6 t < t iþ1
Rsc ðsÞi ¼ R1 ðsÞi Rsb ðsÞi
ði ¼ 0; 1; 2; . . . ; nÞ:
ð6Þ
The initial conditional reliability function of the pipeline is equal to its original overall reliability function, i.e., Rsc(s)0 = Rs(s)0. Therefore, the initial reliability function of the subsystem can be derived from Eq. (6)
Rsb ðsÞ0 ¼
Rs ðsÞ0 : R1 ðsÞ0
ð7Þ
Eq. (7) implies that R1(s)0 – 0. The reliability functions for typical failure distributions such as exponential distribution, normal distribution, lognormal distribution and Weibull distribution all meet this requirement. At time t1, the reliability of the pipeline falls to the control limit R0 and the exposed pipeline is repaired as requested by the PM strategy. After the first PM action, the reliability function of the exposed pipeline becomes R1(s)1, but the reliability function of the subsystem remains the same since it is not repaired. Considering the cumulative effect of time and the definition of the relative time scale, the reliability function of the subsystem after the first PM action, Rsb(s)1, is Rsb(s + t1)0. Hence, the conditional reliability of the pipeline after the first PM action becomes
Rsc ðsÞ1 ¼ R1 ðsÞ1 Rsb ðs þ t 1 Þ0 :
ð8Þ
The start reliability of the exposed pipeline after a PM action, R1(0)i (i = 0, 1,2, . . . , n) is normally equal to one. The conditional reliability function of the pipeline after the ith PM interval can be derived as
Rsc ðsÞi ¼ R1 ðsÞi Rsb ðs þ t i Þ0 ð0 6 s < ðtiþ1 t i Þ; and i ¼ 0; 1; 2; . . . ; nÞ: ð9Þ Substituting Eq. (7) into Eq. (9) with consideration of the cumulative effect of time, gives
and
k¼0
i ¼ 0; 1; 2; . . . ; nÞ;
ð5Þ
where Rs(t)i and Rsc(t)i are the overall reliability function and the conditional reliability function of a system after the ith PM interval, respectively. The value R1(ti+1 ti)i is the reliability of the preventively repaired component just before the (i + 1)th PM interval. Note Q that when i = 0, ik¼0 R1 ðt k t k1 Þk is defined as one. Both reliability concepts are important. The conditional reliability is used to determine PM intervals for the RBPM policy; whereas the overall reliability is used to evaluate PM strategies. Eq. (5) indicates that the overall reliability of the pipeline can be predicted once its conditional reliability has been calculated. Let Rsc(s)i, R1(s)i and Rsb(s)i represent the conditional reliability function of the pipeline, the repaired exposed pipeline and the unrepaired subsystem after the ith PM interval, respectively. According to the reliability theory, the following expression can be obtained:
Rsc ðsÞi ¼
R1 ðsÞi Rs ðs þ t i Þ0 ð0 6 s < ðt iþ1 t i Þ; and i ¼ 0; 1; 2; . . . ; nÞ: R1 ð s þ t i Þ 0 ð10Þ
Eq. (10) can be rewritten using absolute time scale as follows:
Rsc ðtÞi ¼
R1 ðt t i Þi Rs ðtÞ0 R1 ðtÞ0
ðt i 6 t < tiþ1 ; and i ¼ 0; 1; 2; . . . ; nÞ: ð11Þ
Substituting Eq. (11) into Eq. (5) gives
Rs ðtÞi ¼
i R1 ðt ti ÞRs ðtÞ0 Y R1 ðtk t k1 Þk R1 ðtÞ0 k¼0
ðt i 6 t < t iþ1 ; and
i ¼ 0; 1; 2; . . . ; nÞ: In Eq. (12), define that
ð12Þ
Qi
k¼0 R1 ðt k
t k1 Þk ¼ 1, when i = 0.
4. Prediction results and analysis
Rsc(t)
When applying Eqs. (11) and (12) to the case study, R1(t)0 and Rs(t)0 are known as Eqs. (1) and (3). However, the reliability of the exposed pipeline after each PM action needs to be calculated. In principle, this reliability can also be predicted using SSA. However, the derived reliability formulae are complicated. In this paper, for simplification, an approximate formula is used to describe the reliability of the exposed pipeline after a PM action:
Decision horizon
Rsc(t)0
Rsc(t)1 Rsc(t)n-1
Rsc(t)n
τ R0
R1 ðsÞi ¼ R1 ½s þ fc ðti ti1 Þ ði ¼ 1; 2; . . . ; nÞ;
t0
t1
t2
t n t n +1
t
Fig. 1. Changes of the conditional reliability of an imperfectly repaired pipeline with a RBPM strategy.
ð13Þ
where, fc is termed as the recovery coefficient, which is used to represent the degree of the reliability of the exposed pipeline after a PM action recovering to its original reliability. When fc = 0, the state of the exposed pipeline after a PM action is as good as new; when
Y. Sun et al. / European Journal of Operational Research 198 (2009) 210–214
213
fc = 1, the state of the exposed pipeline after a PM action is as bad as old; when 0 < fc < 1, the system has an imperfect repair. The parameter fc can be determined by comparing Eq. (13) with an equation derived based on SSA and some approximation. In this case study, fc = 0.6 0.7. Once the required inputs are determined, Eqs. (11) and (12) can be used to predict the reliability of the entire pipeline with different PM strategies. As an example, Figs. 2 and 3 show the prediction results for two different scenarios (in these figures, MTTF1 and MTTFs are the mean time to failure of the original exposed pipeline and the original entire pipeline, respectively, and T0 is the minimum guaranteed life of the pipeline). In the first scenario, the following two PM strategies were proposed. Strategy one is based on the RBPM policy. In this strategy, whenever the reliability of the entire pipeline falls to 0.988, the exposed pipeline is maintained. The second strategy is based on the TBPM policy. Considering that the minimum guaranteed life of the pipeline, this PM strategy is to start PM at 37 years and then conduct PM on the exposed pipeline each 10.5 years. A recovery coefficient fc = 0.6 was considered. In the second scenario, both RBPM and TBPM polices were also considered. However, for the RBPM strategy, the reliability control limit was changed to 0.975 and the TBPM strategy was changed to start PM at 40 years and then conduct PM every 7 years. A recovery coefficient fc = 0.7 was considered. From these two figures, it can be seen that both TBPM and RBPM strategies can improve the reliability of the entire pipeline. For example, in the first scenario, the overall reliability of the pipeline without PM, at 68 years is lower than 0.1; whereas with the TBPM strategy, the pipeline reliability at the same time is 0.97 and with the RBPM strategy, it is 0.98. It can also be seen that in the first scenario, the RBPM strategy is better than the TBPM strategy from PM effectiveness point of view, because with the same number of PM times (4 times) during the same period, the RBPM strategy enables the pipeline to have a higher overall reliability. For the same reason, the proposed TBPM strategy in the second scenario is better than the proposed RBPM strategy. Comparing Fig. 3 with Fig. 2, one can see that the improved overall reliability in the first scenario is higher than that in the second scenario, whereas during the same period, the required number of PM times in the second scenario (5 times) is more than that in the first scenario (4 times). The reason for this phenomenon is that the PM in the second scenario started later than that in the first scenario and the assumed recovery coefficient in the second scenario was higher than that in the first scenario.
To decide whether to use a RBPM strategy or a TBPM strategy, one also needs to consider the characteristics of these two different strategies. When a RBPM strategy is applied, the reliability of the pipeline between any two PM actions remains above the predefined reliability control limit. However, the PM intervals required by the RBPM strategy are unequal. For example, the minimum required PM interval in the first scenario is 1 years. Since the company needed the PM interval to be greater than 2 years for this pipeline, the proposed RBPM strategy was inapplicable. On the other hand, when a TBPM strategy is applied, PM intervals are scheduled. Nevertheless, the reliability of the pipeline between two PM actions varies. From Fig. 3, it can be seen that during the last interval, the conditional reliability of the pipeline with the TBPM strategy will fall below 0.95. If this reliability is not acceptable, the proposed TBPM cannot be applied. In general, an optimal PM strategy should be determined based on the requirements for reliability and operating (mission) time of the pipeline, cost and PM performance which can be measured by recovery coefficient. However, the details of the optimisation are not discussed due to the limited size of this paper. Figs. 2 and 3 also demonstrate that the pipeline has imperfect repairs. In Section 1, it is indicated that the entire pipeline is
Fig. 2. The reliability of the pipeline with different PM strategies – Scenario 1.
Fig. 4. The changes of the conditional reliability of the pipeline.
Fig. 3. The reliability of the pipeline with different PM strategies – Scenario 2.
214
Y. Sun et al. / European Journal of Operational Research 198 (2009) 210–214
normally imperfectly repaired when only part of it is repaired during PM actions. To further demonstrate this argument, the exposed pipeline was assumed to be replaced by an identical one during each PM action. In this case, fc = 0, i.e., the state of the exposed pipeline after a PM action is as good as new. The reliability prediction results for this case are presented in Fig. 4 (note that all overall reliability curves have been removed from the figure to highlight the changes of conditional reliability). From this figure, it can be clearly seen that the repairs to the entire pipeline was still imperfect even though the exposed pipeline was perfectly maintained. 5. Conclusion A SSA based model has been developed to predict the reliability of pipelines with multiple PM actions. The new model is able to explicitly predict the reliability changes of pipelines over a long term with multiple PM intervals. The new model is hence more suitable for supporting a long term PM decision-making for pipelines. A PM action for a pipeline is often imperfect because normally only part of it are repaired during a PM action. The SSA based pipeline reliability prediction model can deal with multiple imperfect repairs more effectively and more accurately. Furthermore, this prediction model has no restrictions on the forms of failure distribution. The newly developed model has been tested through a case study on a real world system with positive feedback from the company that provided the data. The model promises an effective tool for pipeline PM decision support, although the authors will further study the reliability prediction of pipelines with different repaired
parts and the influence of maintenance costs in their future research. References [1] M.A. Maes, M. Dann, M.M. Salama, Influence of grade on the reliability of corroding pipelines, Reliability Engineering and System Safety – Probabilistic Modelling of Structural Degradation 93 (3) (2008) 447–455. [2] A.P. Teixeira, C. Guedes Soares, T.A. Netto, S.F. Estefen, Reliability of pipelines with corrosion defects, International Journal of Pressure Vessels and Piping 85 (4) (2008) 228–237. [3] R. Khelif, A. Chateauneuf, K. Chaoui, Reliability-based assessment of polyethylene pipe creep lifetime, International Journal of Pressure Vessels and Piping 84 (12) (2007) 697–707. [4] B. Castanier, M. Rausand, Maintenance optimization for subsea oil pipelines, International Journal of Pressure Vessels and Piping – The 16th European Safety and Reliability Conference 83 (4) (2006) 236–243. [5] H. Pham, H.Z. Wang, Imperfect maintenance, European Journal of Operational Research 94 (4) (1996) 425–438. [6] H. Pham (Ed.), Handbook of Reliability Engineering, Springer, London, 2003. [7] M. Rausand, A. Hoyland, System Reliability Theory: Models, Statistical Methods, and Applications, John Wiley and Sons, Hoboken Jersey, 2004. [8] D. Lin, M.J. Zhou, R.C.M. Yam, General sequential imperfect preventive maintenance models, International Journal for Reliability, Quality and Safety Engineering 7 (3) (2000) 253–266. [9] C.E. Ebeling, An Introduction to Reliability and Maintainability Engineering, The McGraw-Hill Company, Inc., New York, 1997, pp. 124–128. [10] E.E. Lewis, Reliability Engineering, second ed., John Wiley and Sons, Inc., New York, 1996, pp. 118–130. [11] Y. Sun, L. Ma, J. Mathew, Reliability prediction of repairable systems for single component repair, Journal of Quality in Maintenance Engineering 13 (2) (2007) 111–124. [12] Y. Sun, L. Ma, J. Mathew, Prediction of system reliability for multiple component repairs, Proceedings of the 2007 IEEE International Conference on Industrial Engineering and Engineering Management, IEEE, Singapore, 2007, pp. 1186–1190.
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
A practical approach for reliability prediction of pipeline systems Yong Sun a,*, Lin Ma a, Jon Morris b a b
CRC for Integrated Engineering Asset Management, School of Engineering Systems, Queensland University of Technology, 2 George Street, Brisbane, Queensland QLD 4001, Australia MPT-Solutions, Industrial Research Limited, P.O. Box 31-310, Lower Hutt, New Zealand
a r t i c l e
i n f o
Article history: Received 8 November 2005 Accepted 28 July 2008 Available online 8 August 2008 Keywords: Pipelines Reliability Maintenance decision support Applied probability Repairable systems
a b s t r a c t Pipelines play an important role in the modern society. Failures of pipelines can have great impacts on economy, environment and community. Preventive maintenance (PM) is often conducted to improve the reliability of pipelines. Modern asset management practice requires accurate predictability of the reliability of pipelines with multiple PM actions, especially when these PM actions involve imperfect repairs. To address this issue, a split system approach (SSA) based model is developed in this paper through an industrial case study. This new model enables maintenance personnel to predict the reliability of pipelines with different PM strategies and hence effectively assists them in making optimal PM decisions. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Pipelines are one of the most applied means to transport gas, oil and water in the world. The failures of pipelines can have huge negative impacts on business, environment and society. Therefore, the reliability of pipelines is a major concern in the field of engineering asset management and has attracted much attention of researchers in the field [1–4]. When modelling the reliability of the pipelines, the existing literature largely focuses on the influences of corrosion [1,2] and creep [3] on the pipeline failures from the material failure mechanism point of view. Very few models have considered the impact of preventive maintenance (PM) on the system reliability of pipelines. In reality, PM is often applied to aged pipelines to reduce unexpected failures and their resultant undesirable impacts. Two commonly used PM policies include the time based preventive maintenance (TBPM) and the reliability based preventive maintenance (RBPM). In the TBPM policy, a pipeline is maintained based on scheduled PM times. The intervals between two PM actions may or may not be the same; whereas in the RBPM policy, a control limit of reliability R0 is defined in advance. Whenever the reliability of a pipeline falls to this predefined control limit, the pipeline is preventively maintained. The purpose of PM is to improve the overall reliability of the entire pipeline. To decide a cost effective PM strategy for a pipeline, accurately predicting the reliability of the pipeline with multiple PM intervals is desired. However, to date, effective models for meeting this need have yet to be devel* Corresponding author. Tel.: +61 7 3138 2442; fax: +61 7 3138 1469. E-mail address: [email protected] (Y. Sun). 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.07.040
oped. The prediction is difficult because pipelines are normally complex systems. Furthermore, imperfect repairs often need to be considered. Imperfect repair in this paper indicates a system after a PM action is between as good as new and as bad as old. A number of papers have been published to consider the influence of imperfect repairs on the reliability of a system [5–8]. In [5,6], Pham and Wang discussed eight categories of methods for treating imperfect maintenance. These methods can be further classified into two groups: (1) state rules based models and (2) improvement factor methods. The former includes (p, q) rule based model, (p(t), q(t)) rule based model, and multiple (p, q) rules based model; whereas the latter includes improvement factor method, virtual age method, shock model method, (a, b) rule and other models reviewed in [5]. In Ref. [7], Rausand and Hoyland reviewed four types of the imperfect repair models: (1) Brown and Proschan’s model; (2) failure rate reduction models; (3) age reduction models; and (4) trend renewal process. The Brown and Proschan’s model is regarded as a (p, q) rule based model in [5,6]. The rest can fall into the group of the improvement factor method. The model given in [8] is also belong to this group although it uses two improvement factors. The state rules based models are largely used for one-component system, and estimating the probabilities for different states is difficult. For using the improvement factor methods, a major difficulty is to estimate the improvement factors, especially for complex systems. In contrast, Ebeling [9] and Lewis [10] presented a heuristic method to predict the reliability of an asset with multiple PM intervals. In this method, PM time is a deterministic variable. This method can produce an intuitive and explicit prediction of reliability and hence very suitable for engineering applications.
Y. Sun et al. / European Journal of Operational Research 198 (2009) 210–214
However, in this model, assets are assumed to have PM actions periodically, i.e., this method was developed based on TBPM only. Besides, this method cannot model the effects of different PM actions on the system reliability. It does not consider system configuration. When modelling imperfect maintenance, it uses a method similar to the improvement factor method. To address the limitations of the existing methods, a split system approach (SSA) has been developed [11,12] (the SSA is termed as SSM (split system model) in [11]). The SSA is different from the other methods in that it models the reliability of a system with multiple PM actions at component level. This approach considers the effects of repaired components on the system reliability and allows the changes of the reliability of the system after each PM action to be calculated rather than estimated by maintenance staff. SSA removes the assumptions on the probability of different states of a system after repairs. These assumptions were used in the state rules based models. This paper aims to predict the reliability of pipelines with multiple imperfect PM actions using SSA through an industrial case study. The outcomes can assist industrial personnel to determine optimal PM strategy for pipelines. Both TBPM and RBPM are considered. Since the effects of PM activities on a system can be modelled using either hazard function (e.g., see [8]) or reliability function (e.g., see [9,10]), the reliability function is used for modelling in this paper. The rest of this paper is organised as follows. In Section 2, the industrial case is described. In Section 3, the concept of SSA is introduced and then formulae are derived for predicting the reliability of pipelines under the condition that the same part is repaired. Section 4 presents prediction results and the corresponding analysis. Conclusions are given in Section 5. 2. Case description A water supply company needed to make an optimal maintenance strategy for a 2224 meter long steel raw water pipeline. Real world data were collected. The observation of failure history indicated that the entire pipeline could be divided into two portions: a total of 200 meter of exposed pipeline composed of pipe bridges and stream crossings and another subsystem comprising of buried pipes. Assume that all pipes in the exposed pipeline had the same failure distribution and the pipes in the 2024 meter of buried pipeline had the same failure distribution. An operational period over 68 years was considered. During this period, the entire pipeline had a 30 years minimum guaranteed life. After 30 years, the exposed pipeline started to wear out with a Weibull failure distribution
" 14:3532 # t 30 ðt P 30Þ: R1 ðtÞ0 ¼ exp 35:6718
ð1Þ
The buried pipeline had an exponential failure distribution
256ðt 30Þ ðt P 30Þ; Rsb ðtÞ0 ¼ exp 1; 000; 000
ð2Þ
where, R1(t)0 and Rsb(t)0 are the original reliability functions of the exposed pipeline and the buried pipeline. Parameter t is the age of the pipeline. The exposed pipeline is connected with the subsystem (buried pipeline) in series, although the subsystem is a complex system. Hence the reliability of the entire pipeline system was
Rs ðtÞ0 ¼ exp½
14:3532 ! t 30 256ðt 30Þ ðt P 30Þ: þ 35:6718 1; 000; 000
ð3Þ
211
When the exposed pipeline was operating in its wear-out stage, conducting PM on the exposed pipeline can improve the overall reliability of the entire pipeline. Since the mean time between failures (MTBF) of the buried pipelines is much longer than the exposed, multiple PM actions over multiple intervals will be implemented on the exposed pipelines. The company needed to decide a cost-effective PM strategy which includes selecting the optimal PM policy and determining the optimal PM parameters. If the TBPM policy is selected, the optimal PM leading times need to be determined; if the RBPM policy is selected, the optimal reliability control limit R0 needs to be determined. To make an optimal decision, the reliability changes of the entire pipeline with different PM strategies have to be predicted accurately. However, as indicated in Section 1, the existing models/methodologies cannot meet industrial need for the explicit prediction of the reliability improvement and the consideration of multiple imperfect repairs. The entire pipeline is normally imperfectly repaired because only partial pipeline will be repaired during a PM action. To address this issue, a SSA based reliability prediction model is developed in the next section. 3. Model development 3.1. Concept of SSA From the above description of the pipeline case, it is noted that when a system is preventively maintained, often only part of its components rather than the whole system is repaired (maintained). Repair information at the component level can often assist in understanding the changes of the reliability of systems after PM actions and hence improve the maintenance outcome of these systems. The information at the component level should be considered when the reliability of a system is being modelled. SSA enables this information to be used in the reliability prediction. The basic concept of SSA is to separate repaired and unrepaired components within a system virtually when modelling the reliability of the system after PM actions. In the following section, the SSA is applied to calculate the reliability of the pipeline with multiple PM intervals. 3.2. Formulation of the pipeline reliability The following analysis is conducted based RBPM policy, i.e., to preventively repair some pipes in the exposed pipeline whenever the reliability of the entire pipeline falls to a predefined control limit of reliability R0. An extension of the outcomes to the TBPM policy is straightforward. In the formulation, the following assumptions were made: (1) Only exposed pipeline is preventively maintained in all PM actions. The reliability functions of repaired pipes are known. (2) The failure of repaired part is independent of unrepaired part. (3) Repair time is negligible. (4) PM time is a deterministic variable. In the TBPM policy, it is scheduled by maintenance personnel; whereas in the RBPM policy, it is calculated based on the predefined reliability control limit of the pipeline. According to the above assumptions, only the reliability of the exposed pipeline changes when the pipeline is preventively maintained. The reliability of the remainder of the pipeline just before and after this PM action does not change.
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Since only the exposed pipeline is repaired in a PM action, the reliability of the entire pipeline after this PM can be improved but lower than its original reliability, leading to an imperfect repair. After imperfect repairs, the reliability of the pipeline declines in a manner shown in Fig. 1. In this paper, the subscript i is used to stand for ‘‘after the ith PM action”. Subscript i = 0 stands for no PM. In Fig. 1, R0 is the predefined control limit of the reliability for the entire pipeline, ti (i = 0, 1, 2, . . . , n, n + 1) is the ith PM time and also the start time for the pipeline to operate again after the ith PM action (note that n is the scheduled number of PM intervals during a given decision horizon which is 68 years in this case study, t0 = 0 and tn+1 = 68 years). Parameter t is absolute time scale (0 6 t < 1) and parameter s is a relative time scale (0 6 s < (ti+1 ti), and i = 0, 1, 2, . . . , n).
t ¼ ti þ s ðt i 6 t < tiþ1 ; and i ¼ 0; 1; 2; . . . ; nÞ:
ð4Þ
When a system receives PM actions, two types of reliability concepts are involved [9,12]: one is the conditional reliability of the system. This reliability indicates the survival probability of a system which has successfully been preventively maintained. It describes the reliability changes between two PM actions. The other is the probability of survivor of the system over its whole life time which takes into account the probability of survival of the repaired components until their individual PM times. It describes the reliability changes of the system over a given period which may cover a number of PM intervals. To distinguish the latter from the conditional reliability, it is termed as overall reliability of the system. According to Ebeling [9] and Lewis [10], for a series system, the relationship between the conditional reliability and the overall reliability is given by
Rs ðtÞi ¼
i Y
R1 ðt k tk1 Þk Rsc ðtÞi
ðt i 6 t < t iþ1
Rsc ðsÞi ¼ R1 ðsÞi Rsb ðsÞi
ði ¼ 0; 1; 2; . . . ; nÞ:
ð6Þ
The initial conditional reliability function of the pipeline is equal to its original overall reliability function, i.e., Rsc(s)0 = Rs(s)0. Therefore, the initial reliability function of the subsystem can be derived from Eq. (6)
Rsb ðsÞ0 ¼
Rs ðsÞ0 : R1 ðsÞ0
ð7Þ
Eq. (7) implies that R1(s)0 – 0. The reliability functions for typical failure distributions such as exponential distribution, normal distribution, lognormal distribution and Weibull distribution all meet this requirement. At time t1, the reliability of the pipeline falls to the control limit R0 and the exposed pipeline is repaired as requested by the PM strategy. After the first PM action, the reliability function of the exposed pipeline becomes R1(s)1, but the reliability function of the subsystem remains the same since it is not repaired. Considering the cumulative effect of time and the definition of the relative time scale, the reliability function of the subsystem after the first PM action, Rsb(s)1, is Rsb(s + t1)0. Hence, the conditional reliability of the pipeline after the first PM action becomes
Rsc ðsÞ1 ¼ R1 ðsÞ1 Rsb ðs þ t 1 Þ0 :
ð8Þ
The start reliability of the exposed pipeline after a PM action, R1(0)i (i = 0, 1,2, . . . , n) is normally equal to one. The conditional reliability function of the pipeline after the ith PM interval can be derived as
Rsc ðsÞi ¼ R1 ðsÞi Rsb ðs þ t i Þ0 ð0 6 s < ðtiþ1 t i Þ; and i ¼ 0; 1; 2; . . . ; nÞ: ð9Þ Substituting Eq. (7) into Eq. (9) with consideration of the cumulative effect of time, gives
and
k¼0
i ¼ 0; 1; 2; . . . ; nÞ;
ð5Þ
where Rs(t)i and Rsc(t)i are the overall reliability function and the conditional reliability function of a system after the ith PM interval, respectively. The value R1(ti+1 ti)i is the reliability of the preventively repaired component just before the (i + 1)th PM interval. Note Q that when i = 0, ik¼0 R1 ðt k t k1 Þk is defined as one. Both reliability concepts are important. The conditional reliability is used to determine PM intervals for the RBPM policy; whereas the overall reliability is used to evaluate PM strategies. Eq. (5) indicates that the overall reliability of the pipeline can be predicted once its conditional reliability has been calculated. Let Rsc(s)i, R1(s)i and Rsb(s)i represent the conditional reliability function of the pipeline, the repaired exposed pipeline and the unrepaired subsystem after the ith PM interval, respectively. According to the reliability theory, the following expression can be obtained:
Rsc ðsÞi ¼
R1 ðsÞi Rs ðs þ t i Þ0 ð0 6 s < ðt iþ1 t i Þ; and i ¼ 0; 1; 2; . . . ; nÞ: R1 ð s þ t i Þ 0 ð10Þ
Eq. (10) can be rewritten using absolute time scale as follows:
Rsc ðtÞi ¼
R1 ðt t i Þi Rs ðtÞ0 R1 ðtÞ0
ðt i 6 t < tiþ1 ; and i ¼ 0; 1; 2; . . . ; nÞ: ð11Þ
Substituting Eq. (11) into Eq. (5) gives
Rs ðtÞi ¼
i R1 ðt ti ÞRs ðtÞ0 Y R1 ðtk t k1 Þk R1 ðtÞ0 k¼0
ðt i 6 t < t iþ1 ; and
i ¼ 0; 1; 2; . . . ; nÞ: In Eq. (12), define that
ð12Þ
Qi
k¼0 R1 ðt k
t k1 Þk ¼ 1, when i = 0.
4. Prediction results and analysis
Rsc(t)
When applying Eqs. (11) and (12) to the case study, R1(t)0 and Rs(t)0 are known as Eqs. (1) and (3). However, the reliability of the exposed pipeline after each PM action needs to be calculated. In principle, this reliability can also be predicted using SSA. However, the derived reliability formulae are complicated. In this paper, for simplification, an approximate formula is used to describe the reliability of the exposed pipeline after a PM action:
Decision horizon
Rsc(t)0
Rsc(t)1 Rsc(t)n-1
Rsc(t)n
τ R0
R1 ðsÞi ¼ R1 ½s þ fc ðti ti1 Þ ði ¼ 1; 2; . . . ; nÞ;
t0
t1
t2
t n t n +1
t
Fig. 1. Changes of the conditional reliability of an imperfectly repaired pipeline with a RBPM strategy.
ð13Þ
where, fc is termed as the recovery coefficient, which is used to represent the degree of the reliability of the exposed pipeline after a PM action recovering to its original reliability. When fc = 0, the state of the exposed pipeline after a PM action is as good as new; when
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fc = 1, the state of the exposed pipeline after a PM action is as bad as old; when 0 < fc < 1, the system has an imperfect repair. The parameter fc can be determined by comparing Eq. (13) with an equation derived based on SSA and some approximation. In this case study, fc = 0.6 0.7. Once the required inputs are determined, Eqs. (11) and (12) can be used to predict the reliability of the entire pipeline with different PM strategies. As an example, Figs. 2 and 3 show the prediction results for two different scenarios (in these figures, MTTF1 and MTTFs are the mean time to failure of the original exposed pipeline and the original entire pipeline, respectively, and T0 is the minimum guaranteed life of the pipeline). In the first scenario, the following two PM strategies were proposed. Strategy one is based on the RBPM policy. In this strategy, whenever the reliability of the entire pipeline falls to 0.988, the exposed pipeline is maintained. The second strategy is based on the TBPM policy. Considering that the minimum guaranteed life of the pipeline, this PM strategy is to start PM at 37 years and then conduct PM on the exposed pipeline each 10.5 years. A recovery coefficient fc = 0.6 was considered. In the second scenario, both RBPM and TBPM polices were also considered. However, for the RBPM strategy, the reliability control limit was changed to 0.975 and the TBPM strategy was changed to start PM at 40 years and then conduct PM every 7 years. A recovery coefficient fc = 0.7 was considered. From these two figures, it can be seen that both TBPM and RBPM strategies can improve the reliability of the entire pipeline. For example, in the first scenario, the overall reliability of the pipeline without PM, at 68 years is lower than 0.1; whereas with the TBPM strategy, the pipeline reliability at the same time is 0.97 and with the RBPM strategy, it is 0.98. It can also be seen that in the first scenario, the RBPM strategy is better than the TBPM strategy from PM effectiveness point of view, because with the same number of PM times (4 times) during the same period, the RBPM strategy enables the pipeline to have a higher overall reliability. For the same reason, the proposed TBPM strategy in the second scenario is better than the proposed RBPM strategy. Comparing Fig. 3 with Fig. 2, one can see that the improved overall reliability in the first scenario is higher than that in the second scenario, whereas during the same period, the required number of PM times in the second scenario (5 times) is more than that in the first scenario (4 times). The reason for this phenomenon is that the PM in the second scenario started later than that in the first scenario and the assumed recovery coefficient in the second scenario was higher than that in the first scenario.
To decide whether to use a RBPM strategy or a TBPM strategy, one also needs to consider the characteristics of these two different strategies. When a RBPM strategy is applied, the reliability of the pipeline between any two PM actions remains above the predefined reliability control limit. However, the PM intervals required by the RBPM strategy are unequal. For example, the minimum required PM interval in the first scenario is 1 years. Since the company needed the PM interval to be greater than 2 years for this pipeline, the proposed RBPM strategy was inapplicable. On the other hand, when a TBPM strategy is applied, PM intervals are scheduled. Nevertheless, the reliability of the pipeline between two PM actions varies. From Fig. 3, it can be seen that during the last interval, the conditional reliability of the pipeline with the TBPM strategy will fall below 0.95. If this reliability is not acceptable, the proposed TBPM cannot be applied. In general, an optimal PM strategy should be determined based on the requirements for reliability and operating (mission) time of the pipeline, cost and PM performance which can be measured by recovery coefficient. However, the details of the optimisation are not discussed due to the limited size of this paper. Figs. 2 and 3 also demonstrate that the pipeline has imperfect repairs. In Section 1, it is indicated that the entire pipeline is
Fig. 2. The reliability of the pipeline with different PM strategies – Scenario 1.
Fig. 4. The changes of the conditional reliability of the pipeline.
Fig. 3. The reliability of the pipeline with different PM strategies – Scenario 2.
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normally imperfectly repaired when only part of it is repaired during PM actions. To further demonstrate this argument, the exposed pipeline was assumed to be replaced by an identical one during each PM action. In this case, fc = 0, i.e., the state of the exposed pipeline after a PM action is as good as new. The reliability prediction results for this case are presented in Fig. 4 (note that all overall reliability curves have been removed from the figure to highlight the changes of conditional reliability). From this figure, it can be clearly seen that the repairs to the entire pipeline was still imperfect even though the exposed pipeline was perfectly maintained. 5. Conclusion A SSA based model has been developed to predict the reliability of pipelines with multiple PM actions. The new model is able to explicitly predict the reliability changes of pipelines over a long term with multiple PM intervals. The new model is hence more suitable for supporting a long term PM decision-making for pipelines. A PM action for a pipeline is often imperfect because normally only part of it are repaired during a PM action. The SSA based pipeline reliability prediction model can deal with multiple imperfect repairs more effectively and more accurately. Furthermore, this prediction model has no restrictions on the forms of failure distribution. The newly developed model has been tested through a case study on a real world system with positive feedback from the company that provided the data. The model promises an effective tool for pipeline PM decision support, although the authors will further study the reliability prediction of pipelines with different repaired
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