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Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system
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Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system Baoping Cai a,n, Yonghong Liu a, Yunpeng Ma a, Zengkai Liu a, Yuming Zhou b, Junhe Sun b a b
College of Mechanical and Electronic Engineering, China University of Petroleum, Qingdao, Shandong 266580, China Bureau of Geophysical Prospecting INC., China National Petroleum Corporation, Zhuozhou, Hebei 072750, China
art ic l e i nf o
a b s t r a c t
Article history: Received 29 March 2015 Received in revised form 8 June 2015 Accepted 25 June 2015 This paper was recommended for publication by Prof. Steven Ding
A novel real-time reliability evaluation methodology is proposed by combining root cause diagnosis phase based on Bayesian networks (BNs) and reliability evaluation phase based on dynamic BNs (DBNs). The root cause diagnosis phase exactly locates the root cause of a complex mechatronic system failure in real time to increase diagnostic coverage and is performed through backward analysis of BNs. The reliability evaluation phase calculates the real-time reliability of the entire system by forward inference of DBNs. The application of the proposed methodology is demonstrated using a case of a subsea pipe ram blowout preventer system. The value and the variation trend of real-time system reliability when the faults of components occur are studied; the importance degree sequence of components at different times is also determined using mutual information and belief variance. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Root cause diagnosis Reliability evaluation Real-time Dynamic Bayesian networks Subsea blowout preventer
1. Introduction Complex mechatronic systems for critical applications, such as subsea production systems and spacecraft, require high reliability, i.e., a very low probability of failure. Many methods and tools have been developed, e.g., fault tree [1,2], reliability block diagram [3,4], Petri net [5], and Markov chain [6,7], to evaluate the reliability of a mechatronic system based on its structures and the reliability of each component. However, these traditional techniques have several significant limitations, including binary variable problems [8] and state space explosion problems [9] for complex mechatronic systems. To improve system operation and for preventive maintenance, variation in reliability under different conditions should be determined. Real-time reliability evaluation is ideal because the factors that affect reliability change continuously with time for dynamic systems. Some of the aforementioned traditional methods have been adopted for real-time reliability evaluation. Tanrioven et al. [10] developed a real-time power system reliability analysis method based on a fuzzy Markov model. Ching et al. [11] proposed a new method to update real-time reliability based on data recorded by instruments and sensors installed on a system with
n Corresponding author. Tel.: þ 86 0532 86983505 8616; fax: þ86 0532 86983300. E-mail address: [email protected] (B. Cai).
stochastic simulation. Xu et al. [12] proposed a real-time reliability prediction method for a dynamic system that suffers from a hidden degradation process. In addition, they proposed a realtime reliability prediction approach based on online fault prediction for dynamic systems and adopted exponential smoothing methods for fault prediction and Monte Carlo strategy for reliability prediction [31]. Ma et al. [32] proposed a new layered modeling architecture that consists of dynamic hybrid fault modeling and extended evolutionary game theory for real-time reliability, survivability, and fault tolerance analyses. Wang et al. [33] investigated the issue of real-time reliability evaluation based on a general Wiener process-based degradation model. FaghihRoohi et al. [34] developed a dynamic model for availability assessment of multi-state weighted k-out-of-n systems using the universal generating function and Markov process. Each method has its advantages and disadvantages. For example, Markov models have state space explosion problems [9,10,34]. The major limitation of stochastic simulation is that the monitoring value has to be a scalar; in real situations, however, many monitoring values usually exist [11]. The hidden degradation process identification is unsuitable for a situation in which several degradation processes exist simultaneously, the characteristics of the degradation processes are time varying, or the path function of the fault process is nonlinear [12,31]. On another active research frontier, Bayesian networks (BNs) have attracted considerable attention in the field of system reliability evaluation and fault diagnosis since they were first
http://dx.doi.org/10.1016/j.isatra.2015.06.011 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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proposed by Pearl [13]. Dynamic BNs (DBNs), which are longestablished extensions of ordinary BNs and allow explicit modeling of changes over time, were developed subsequently. In recent years, BNs and DBNs have been applied to study the reliability of multilevel systems [14], two-terminal networks [15], distributed communication systems [16], N-modular redundant systems [17], structural systems [18], human factors [19], and software systems [20]. BNs and DBNs have also been used to study the fault diagnosis of computer numerical control machine tools [21], chillers [22], and ground-source heat pumps [23]. This technique has an advantage over the aforementioned methods in representing uncertain knowledge in dynamic systems. It overcomes binary variable problems and state space explosion problems; however, the prior probability determination and conditional probability table (CPT) elicitation are always challenging and significant issues. Over the past two decades, few studies have reported on the real-time application of BNs and DBNs. Ji et al. [24] proposed a probabilistic framework based on BNs for the modeling and realtime inferring of human fatigue. Armero et al. [25] proposed a probabilistic expert system for real-time prediction of the risk of Legionella from remote information relating to the quality of water in evaporative installations. Hossain et al. [26] proposed a BNbased framework for real-time crash prediction on the basic freeway segments of urban expressways. Murray et al. [27] proposed a BN-based real-time water quality monitoring system. Rigas et al. [28] proposed a BN-based a real-time methodology for the detection of stress events during driving. Straub [29] proposed a generic framework for the stochastic modeling of deterioration on the basis of DBNs; this framework is suitable for near real-time applications in asset integrity management and deterioration control. Codetta-Raiteri et al. [35] proposed DBNs and the software tool RADYBAN for dynamic reliability evaluation when the reliability parameters, such as the failure rates, vary according to the current state of the system. These real-time applications of BNs and DBNs, especially for near real-time and dynamic reliability evaluation, are totally different with the proposed two-phase real-time reliability evaluation methodology in this work. Exactly locating the root cause of a complex mechatronic system failure in real time to increase diagnostic coverage is the precondition and foundation of evaluating real-time reliability values. To resolve this problem, this paper proposes a novel real-time reliability evaluation framework that combines the BN-based root cause diagnosis phase and the DBN-based reliability evaluation phase. The remainder of this is structured as follows: Section 2 presents the DBN-based real-time reliability evaluation methodology. Section 3 analyzes a case to demonstrate the application of the methodology. Section 4 summarizes the paper.
2. Proposed methodology The generic framework of the proposed real-time reliability evaluation is shown in Fig. 1. The framework consists of the BNbased root cause diagnosis phase and the DBN-based reliability evaluation phase. The root cause diagnosis phase is composed of BNs with two layers, namely, fault layer and fault symptom layer. The parent nodes at the fault layer represent the fault of each component in a system. The child nodes at the fault symptom layer represent the symptoms detected using various sensors and even observed information. Root cause diagnosis is performed through backward analysis of BNs, which involves the computation of the posterior probability of any given set of variables from given observations (evidence) represented as the instantiation of some variables to
Fig. 1. DBN-based real-time reliability evaluation framework.
one of their admissible values. As shown in Fig. 1, the inputs of the BNs are the evidence Yi, and the outputs are the posterior probabilities of nodes Xi. Basically, the following two rules can be used to isolate a fault [22]: Rule 1: the one with the largest fault probability and whose fault probability is larger than a certain threshold, ε1, can be isolated. Rule 2: the difference between the largest fault probability and the second one is larger than a certain threshold, ε2, can be isolated. The reliability evaluation phase comprises DBNs with two or more layers. Taking the two-layers DBNs in Fig. 1 as an example, it consists of a component state layer and a system state layer. The parent nodes at the component state layer represent the state of each component in a system. The only child node at the system state layer represents the state of the entire system, and the probability that the system works represents the system reliability value. The DBNs are extended from BNs at set intervals, and the system reliability at different points in time can be calculated through forward analysis of DBNs, which calculates the occurrence probability of nodes on the basis of the prior probability of root nodes and the conditional dependence of each node.
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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Fig. 2. Structure of subsea pipe ram BOP system. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Fig. 3. Structure of BNs for the root cause diagnosis phase.
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According to the sensor data and observed information at a certain time, the root cause of system failure can be located exactly in real time through the root cause diagnosis phase. The component failure is transferred to the corresponding node in the reliability evaluation phase, and then the real-time reliability value can be predicted. Each component normal is not transferred to the corresponding node because normal use can increase continuously the probability of failure of each component.
3. Case study 3.1. Subsea pipe ram blowout preventer system A subsea blowout preventer (BOP) system plays an important role in providing safe working conditions for drilling activities in 3000 m ultra-deepwater regions. Subsea BOP failures could cause catastrophic accidents such as the explosion of the deep-sea petroleum drilling rig Deepwater Horizon and the oil spill off the coast of Louisiana on April 20, 2010. The failure of the BOP to shear the drill pipe and seal the wellbore was conclusively caused directly by the physical location of the drill pipe near the inside wall of the wellbore, which was outside the blind shear ram cutting surface during activation [30]. In the wake of recent disasters in oil and gas exploration and production, the value of the reliability evaluation of subsea BOP systems is becoming recognized. Therefore, in this section, a case of a subsea pipe ram BOP system is studied to demonstrate the application of the proposed real-time reliability evaluation methodology. The structure of the subsea pipe ram BOP system is provided in Fig. 2. Subsea blue pod and subsea yellow pod are designed to form a redundant hydraulic control system. Taking the subsea blue pod as an example, the solenoid valve DDV11 is a selector valve, which provides low pressure pilot fluid from low-pressure accumulators. Solenoid valves DDV12 and DDV13 control the pilot fluid to sub-plate mount valves SPM11 and SPM12. The high-pressure fluid from high-pressure accumulators flows to shuttle valve 1 and shuttle valve 2 via SPM11 and SPM12 to control the closing and opening of the subsea pipe ram BOP, respectively. The subsea yellow pod is capable of performing all the same functions as the subsea blue pod. When a pod fails, the other one will be used to execute all subsea functions and will not be affected by the disabled one. The disabled one will be retrieved to the surface for repair. In the subsea blue pod, five pressure sensors, i.e., PS11, PS12, PS13, PS14, and PS15, are installed in the outlet of solenoid valves DDV11, DDV12, and DDV13, and sub-plate mount valves SPM11 and SPM12 to detect the pressure of pipelines. Similarly, in the subsea yellow pod, five pressure sensors, i.e., PS21, PS22, PS23, PS24, and PS25, are installed in the outlet of solenoid valves DDV21, DDV22, and DDV23, and sub-plate mount valves SPM21 and SPM22 to detect the pressure of pipelines. Two other pressure sensors, i.e., PS31 and PS32, are installed in the outlet of shuttle valve 1 and shuttle valve 2, to detect the pressure of line 5 and line 6, respectively. The displacement sensor DS is installed on one side of the subsea pipe ram BOP system to detect the displacement of the ram. 3.2. Modeling As shown in Fig. 3, the root cause diagnosis of the subsea pipe ram BOP system is established using Netica software. Netica is a powerful, easy-to-use, and complete commercial software program for working with BNs and DBNs and can perform various kinds of inference by using the fastest and most modern algorithms, such as junction trees. Taking the subsea blue pod as an
example, parent nodes DDV11_failure, DDV12_failure, DDV13_failure, SPM11_failure, and SPM12_failure at the fault layer represent the faults of solenoid valves DDV11, DDV12, and DDV13, and subplate mount valves SPM11 and SPM12. Each parent node has two states, i.e. absent and present, which indicate the absence and the presence of the corresponding fault of given observed evidence, respectively. Child nodes PS11, PS12, PS13, PS14, and PS15 at the fault symptom layer represent the information obtained from pressure sensors PS11, PS12, PS13, PS14, and PS15. Each child node has two states, i.e., normal and abnormal, which indicate the normal and the abnormal states of observed evidence of pressure sensors, respectively. The prior probabilities and CPTs of nodes should also be obtained to complete the BNs. The prior probabilities of failure absence for all the components in the root cause diagnosis phase are set to 99%. The CPTs can be obtained from the relationships between faults and fault symptoms. Taking the blue pod as an example, the relationships between faults and fault symptoms are provided in Table 1. The fault of DDV11_failure causes the fault symptoms of abnormal values in pressure sensors PS11, PS12,
Table 1 Relationships between faults and fault symptoms in the blue pod for the root cause diagnosis phase. Fault
Fault symptom
DDV11_failure DDV12_failure DDV13_failure SPM11_failure SPM12_failure
PS11
PS12
PS13
PS14
PS15
0.95 0.04 0.04 0.03 0.03
0.98 0.98 0.02 0.06 0.06
0.98 0.02 0.98 0.06 0.06
0.96 0.98 0.04 0.93 0.05
0.96 0.04 0.98 0.05 0.93
Table 2 CPTs of node PS11 for the root cause diagnosis phase. DDV11 failure
DDV12 failure
DDV13 failure
SPM11 failure
SPM12 failure
P (PS11¼ 1)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0.000000 0.030000 0.030000 0.059100 0.040000 0.068800 0.068800 0.096736 0.040000 0.068800 0.068800 0.096736 0.078400 0.106048 0.106048 0.132867 0.950000 0.951500 0.951500 0.952955 0.952000 0.953440 0.953440 0.954837 0.952000 0.953440 0.953440 0.954837 0.953920 0.955302 0.955302 0.956643
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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PS13, PS14, and PS15 at respective probabilities of 95%, 98%, 98%, 96%, and 96%, but not 100%, because of the uncertainty in system modeling. The existing uncertainty problem is caused by various reasons, with sensor accuracy and measure uncertainty as the primary causes. CPT elicitation is always challenging in BN analysis. Several approaches, such as: direct assessment of the probabilities by one or multiple experts, elicitation of probability ranking on qualitative scale, elicitation of selected model relationships, and filling-up of the remaining relationships with algorithms, have been used [36]. In the current work, multiple expert knowledge and Noisy-OR/AND filling-up algorithm are used. The probabilistic relationships between faults and fault symptoms are determined by two designers in the subsea BOP manufacturer, Rongsheng Machinery Manufacture Ltd. of Huabei Oilfield, Hebei. With the use of relationships and the Noisy-OR model, the CPTs can be computed. For example, the CPTs of node PS11 are calculated using the Noisy-OR model according to the probabilities in Table 1, as provided in Table 2. When solenoid valve DDV11 fails, the probability that pressure sensor PS11 detects pipeline pressure is around 95% but not 100% because of the introduced uncertainty. However, not all the CPTs of nodes can be calculated using the Noisy-OR model. For example, pressure sensor PS31 is used to detect the pressure of line 5, which closes the subsea pipe ram BOP. When either line 1 or line 2 is normal and shuttle value 1 is normal, the line 5 is normal. Line 1 (PS14) and line 2 (PS24) are parallel. Therefore, the Noisy-AND model is used to represent the relationship between line 1, line 2, and PS31.
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Similar to the root cause diagnosis phase, the reliability evaluation phase of the subsea pipe ram BOP system is also established using Netica software, as shown in Fig. 4. Taking the subsea blue pod as an example, parent nodes DDV11, DDV12, DDV13, SPM11, and SPM12 at the component state layer represent the states of solenoid valves DDV11, DDV12, and DDV13, and subplate mounts SPM11 and SPM12, but not the faults of components. Each parent node has two states, i.e., work and fail. The only child node, BOP, at the system state layer represents the state of the entire subsea pipe ram BOP system. Child node BOP has two states, i.e., work and fail, and the probability that the system works represents the system reliability value. As mentioned in Section 1, DBNs are long-established extensions of ordinary BNs and allow the explicit modeling of changes over time. To model the temporal evolution of a system, two time slices for each variable are considered. The current time step is represented by t, and the next time step is represented by t þ Δt. The time interval Δt could be 1 h, 1 week, and even 1 month. A great number of time slices corresponds to a smaller the value of Δt, and, hence, a longer time at which Netica runs. In this work, the DBNs are extended from BNs every four weeks, and the structure of the DBNs extended within the first eight weeks is presented in Fig. 4. No sensors are used for the reliability evaluation phase. Thus, no uncertainty problem is introduced into the reliability evaluation model. Taking the subsea blue pod as an example, solenoid valves DDV11, DDV12, and DDV13, and sub-plate mounts SPM11 and SPM12 are considered to be in a series. Thus, any fault of the components can
Fig. 4. Structure of DBNs for the reliability evaluation phase.
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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Table 3 CPTs of the blue pod for the reliability evaluation phase.
Table 4 Parameters of the Weibull distribution for four components.
DDV11
DDV12
DDV13
SPM11
SPM12
P(blue pod ¼1)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
cause the failure of the subsea blue pod. Therefore, the blue pod node is binary, as shown in Table 3. The lifetime distribution of each component of the subsea BOP system is Weibull distribution, which is one of the most widely used lifetime distributions in reliability engineering. Based on the value of shape parameter β and characteristic life η, it is a versatile distribution that can take on the characteristics of other types of distributions. In this work, the reliability function for the twoparameter Weibull distribution is derived by one minus the cumulative density function. The values of characteristic life η and shape parameter β for each component of the subsea pipe ram BOP system are provided in Table 4. Generally, these values are difficult to collect. In this work, the parameters for Weibull distribution, similar to prior probabilities and CPTs, are determined by the two designers in the subsea BOP manufacturer, Rongsheng Machinery Manufacture Ltd. of Huabei Oilfield, Hebei. In this work, four real-time reliability evaluation cases are investigated using the proposed methodology, as shown in Table 5. Taking Case 1 as an example, the values of pressure sensor PS31 and displacement sensor DS are abnormal, whereas the other sensor values are all normal. These values are input to the root cause diagnosis phase, and the backward analysis of BNs generates the result that shuttle valve 1 fails. The failure state of shuttle valve 1 is input to the subsequent reliability evaluation phase, and the forward analysis of DBNs generates the real-time reliability data, as shown in Fig. 5.
Component
Parameter
DDV SPM Shuttle Ram
η
β
6800 7450 8400 9860
2.3 1.9 2.5 2.9
Table 5 Four real-time reliability evaluation cases. Sensor
PS11 PS12 PS13 PS14 PS15 PS21 PS22 PS23 PS24 PS25 PS31 PS32 DS
Case Case 1
Case 2
Case 3
Case 4
Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Abnormal Normal Abnormal
Normal Abnormal Normal Abnormal Normal Normal Normal Normal Normal Normal Normal Normal Normal
Normal Normal Normal Abnormal Abnormal Normal Normal Normal Normal Normal Normal Normal Normal
Normal Normal Normal Normal Abnormal Normal Normal Normal Normal Abnormal Normal Abnormal Abnormal
reliability evaluation framework is composed of the BN-based root cause diagnosis phase and the DBN-based reliability evaluation phase, the validations of the diagnosis model and the evaluation model are performed separately. The BN model of the root cause diagnosis phase can be validated by comparing diagnosis results with the practical faults. The four cases in Table 5 are practical engineering cases, in which the sensors detect abnormal symptoms and the root cause diagnosis phase performs backward analysis to calculate the posterior probabilities of each parent node, as provided in Table 6. According to the rules presented in Section 2, shuttle valve 1 fails in Case 1, DDV12 fails in Case 2, SPM11 and SPM12 fail in Case 3, and SPM12 and SPM22 fail in Case 4. The fault diagnosis results agree with the practical faults, which indicates that the BN model of the root cause diagnosis phase is correct. A full validation of the DBN model of the reliability evaluation phase is deemed impractical, because the system reliability values are difficult to define. Therefore, a three-axiom-based validation method is applied to the partial validation of the DBN model [37]. At the initial time, the prior probabilities of work for all components are set to 50%. When the probabilities of DDV11, DDV12, DDV13, SPM11, and SPM12 are set to 100% one by one, the reliability of the BOP increases to 1.15%, 1.90%, 3.42%, 6.45%, and 12.5%, respectively. When the probabilities of Shuttle1 and Shuttle2 are set to 100% one by one, the reliability of the BOP system increases to 25% and 50%. Finally, when the last parent node Ram is set to 100%, the BOP system reliability increases to 100%. The exercise of increasing each influencing node satisfies the axioms, thus partially validating the model.
3.3. Results and discussions 3.3.1. Validation of data and model Validation of data and model is an important aspect of the proposed methodology as it will render a reasonable degree of confidence to the results of the model. As the proposed real-time
3.3.2. Real-time reliability evaluation The inferences of fault diagnosis in the root cause diagnosis phase and the reliability prediction in the reliability evaluation phase are performed using Netica software. When no component failure occurs, the sensitivity analysis of time interval, Δt, is
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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Fig. 5. Real-time reliability evaluation model of subsea pipe ram BOP system.
performed by calculating the reliability of the subsea pipe ram BOP system at different time intervals (Δt), including 2 weeks, 4 weeks, and 8 weeks, with the use of the proposed real-time reliability evaluation methodology, as shown in Fig. 6. Time interval, Δt, has little effect on system reliability because the extension of DBNs is based on the Weibull distribution of components and the conditional probability of time slices for each component is obtained from the reliability function of the corresponding two-parameter Weibull distribution. In this work, time interval, Δt, of 4 weeks is selected for the real-time reliability evaluation. As we expected, with an increase in time, the reliability of the subsea pipe ram BOP
system decreases. In the 40th week, the system has a reliability of 99.953%, which indicates that the system failure probability is only 0.047%; therefore, the subsea pipe ram BOP system is sufficiently secure. When several components fail, the root cause diagnosis phase performs backward analysis and diagnosis, the faults in the four cases in Table 5 are input to the reliability evaluation phase at different times, and the real-time reliability values are calculated and plotted, as shown in Fig. 7. As shown in Fig. 7(a), the reliability decreases rapidly before the fault of shuttle valve 1 is detected, and once the fault occurs, the reliability decreases to 0 when the
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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fault occurs immediately after the system starts. Thus, shuttle valve is a fatal weakness of the subsea pipe ram BOP system and the reliability of the shuttle value should be improved as much as possible. As shown in Fig. 7(b), when the fault of solenoid valve DDV12 occurs immediately after the system starts, the reliability decreases, to a level lower than the one when no fault occurs. In the 40th week, the system still has a reliability of 99.371%. When the fault occurs at a certain intermediate time, the reliability
decreases to the value when the fault occurs at t ¼0. These results are due to the fact that the subsea blue pod and subsea yellow pod are redundant, and the failure of DDV12 cannot affect the normal operation of the subsea pipe ram BOP system but can decrease the reliability of the entire system. When the faults of sub-plate mount valves SPM11 and SPM12 occur, the reliability value is similar to the one when the fault of
Table 6 Fault diagnosis results in the four cases. Parent node
DDV11 DDV12 DDV13 SPM11 SPM12 DDV21 DDV22 DDV23 SPM21 SPM22 Shuttle1 Shuttle2 Ram
Posterior probability Case 1
Case 2
Case 3
Case 4
0.0000 0.0000 0.0000 0.0006 0.0006 0.0000 0.0000 0.0000 0.0006 0.0006 1.0000 0.0001 0.0106
0.0000 0.9474 0.0000 0.0607 0.0009 0.0000 0.0000 0.0000 0.0006 0.0006 0.0001 0.0001 0.0001
0.0002 0.0103 0.0103 0.5283 0.5283 0.0000 0.0000 0.0000 0.0006 0.0006 0.0001 0.0001 0.0001
0.0000 0.0000 0.0229 0.0045 0.9734 0.0000 0.0000 0.0229 0.0045 0.9734 0.0001 0.0104 0.0106 Fig. 6. Reliability of the subsea pipe ram BOP system without component failures.
Fig. 7. Reliability of the subsea pipe ram BOP system with component failures at different times: (a) Case 1, (b) Case 2, (b) Case 3, and (b) Case 4.
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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Fig. 8. Sensitivity analysis of subsea pipe ram BOP reliability: (a) mutual information and (b) belief variance.
solenoid valve DDV12 occurs, as shown in Fig. 7(c). When the faults of sub-plate mount valves SPM12 and SPM22 occur, the reliability value is similar to the one when the fault of shuttle valve 1 occurs, as shown in Fig. 7(d). These results may be ascribed to the fact that SPM11 and SPM12 are installed in the same pod, whereas SPM12 and SPM22 are installed in two different pods; therefore, the failure of both pods can cause system failure. 3.3.3. Sensitivity analysis Sensitivity analysis can be used to measure the importance degree to which findings at any node can influence the output at another node. For the purposes of this study, sensitivity analysis can indicate which components can provide the most information for determining system reliability. The sensitivity analysis determines the importance degree sequence using mutual information and belief variance. Mutual information measures the information that two variables share and how much uncertainty about one variable is reduced by knowing one of the other variables. Belief Variance measures the expected squared change in class probabilities in the output node distribution [38]. The relationships measured by differences in the mutual information are not replicated in the belief variance, which examines changes in the probabilities of the target node rather than its estimated value. The mutual information and belief variance of each component on system reliability in the 20th week and the 40th week are calculated using Netica software, as shown in Fig. 8. The absolute values have little individual meaning but are used to rank the indicators from the most to the least important in terms of impacts on the node of interest. For the subsea pipe ram BOP system, the importance degree sequences from high to low in the 20th week and the 40th week are both Shuttle-Ram-SPM-DDV, which indicates that shuttle valve is the most fatal weakness and should be given more attention.
4. Conclusions This paper proposes a novel real-time reliability evaluation methodology that combines the BN-based root cause diagnosis phase and the DBN-based reliability evaluation phase. The root cause diagnosis of faults for components is performed through backward analysis of BNs, and the real-time reliability evaluation is performed through forward analysis of DBNs. The proposed methodology can determine the value and the variation trend of system real-time reliability when faults of components occur and the importance degree sequence of components at different times with the use of mutual information and belief variance. Obtained information can
help engineers to understand the system state at any time and provide suggestions on repair and maintenance. The application of the proposed methodology is demonstrated using a simple case of a subsea pipe ram BOP system. The values and trends of real-time reliability in the four cases with different faults are obtained; the importance degree sequence of components from high to low are as follows: Shuttle-Ram-SPM-DDV, which indicates that shuttle valve is the most fatal weakness and should be given more attention. Future work can be directed toward the development of real-time reliability evaluation software using the proposed methodology for an entire system, such as a subsea BOP system with thousands of components with series, parallel, and even voting relations.
Acknowledgments The authors wish to acknowledge the financial support of National Natural Science Foundation of China (No. 51309240), Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130133120007), China Postdoctoral Science Foundation (No. 2015M570624), Applied Basic Research Programs of Qingdao (No. 14-2-4-68-jch), Science and Technology Project of Huangdao District (No. 2014-1-48) and Fundamental Research Funds for the Central Universities (No.14CX02197A). In addition, the authors would like to thank the anonymous reviewers whose constructive comments were very helpful for strengthening the presentation of this paper. References [1] Dutuit Y, Innal F, Rauzy A, Signoret JP. Probabilistic assessments in relationship with safety integrity levels by using fault trees. Reliab Eng Syst Saf 2008;93:1867–76. [2] Cacheux PJ, Collas S, Dutuit Y, Folleau C, Signoret JP, Thomas P. Assessment of the expected number and frequency of failures of periodically tested systems. Reliab Eng Syst Saf 2013;118:61–70. [3] Guo H, Yang X. A simple reliability block diagram method for safety integrity verification. Reliab Eng Syst Saf 2007;92:1267–73. [4] Catelani M, Ciani L, Luongo V. A simplified procedure for the analysis of safety instrumented systems in the process industry application. Microelectron Reliab 2011;51:1503–7. [5] Kleyner A, Volovoi V. Application of Petri nets to reliability prediction of occupant safety systems with partial detection and repair. Reliab Eng Syst Saf 2010;95:606–13. [6] Liu Y, Rausand M. Reliability assessment of safety instrumented systems subject to different demand modes. J Loss Prev Process Ind 2011;24:49–56. [7] Cai B, Liu Y, Liu Z, Tian X, Li H, Ren C. Reliability analysis of subsea blowout preventer control systems subjected to multiple error shocks. J Loss Prev Process Ind 2012;25:1044–54.
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Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system Baoping Cai a,n, Yonghong Liu a, Yunpeng Ma a, Zengkai Liu a, Yuming Zhou b, Junhe Sun b a b
College of Mechanical and Electronic Engineering, China University of Petroleum, Qingdao, Shandong 266580, China Bureau of Geophysical Prospecting INC., China National Petroleum Corporation, Zhuozhou, Hebei 072750, China
art ic l e i nf o
a b s t r a c t
Article history: Received 29 March 2015 Received in revised form 8 June 2015 Accepted 25 June 2015 This paper was recommended for publication by Prof. Steven Ding
A novel real-time reliability evaluation methodology is proposed by combining root cause diagnosis phase based on Bayesian networks (BNs) and reliability evaluation phase based on dynamic BNs (DBNs). The root cause diagnosis phase exactly locates the root cause of a complex mechatronic system failure in real time to increase diagnostic coverage and is performed through backward analysis of BNs. The reliability evaluation phase calculates the real-time reliability of the entire system by forward inference of DBNs. The application of the proposed methodology is demonstrated using a case of a subsea pipe ram blowout preventer system. The value and the variation trend of real-time system reliability when the faults of components occur are studied; the importance degree sequence of components at different times is also determined using mutual information and belief variance. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Root cause diagnosis Reliability evaluation Real-time Dynamic Bayesian networks Subsea blowout preventer
1. Introduction Complex mechatronic systems for critical applications, such as subsea production systems and spacecraft, require high reliability, i.e., a very low probability of failure. Many methods and tools have been developed, e.g., fault tree [1,2], reliability block diagram [3,4], Petri net [5], and Markov chain [6,7], to evaluate the reliability of a mechatronic system based on its structures and the reliability of each component. However, these traditional techniques have several significant limitations, including binary variable problems [8] and state space explosion problems [9] for complex mechatronic systems. To improve system operation and for preventive maintenance, variation in reliability under different conditions should be determined. Real-time reliability evaluation is ideal because the factors that affect reliability change continuously with time for dynamic systems. Some of the aforementioned traditional methods have been adopted for real-time reliability evaluation. Tanrioven et al. [10] developed a real-time power system reliability analysis method based on a fuzzy Markov model. Ching et al. [11] proposed a new method to update real-time reliability based on data recorded by instruments and sensors installed on a system with
n Corresponding author. Tel.: þ 86 0532 86983505 8616; fax: þ86 0532 86983300. E-mail address: [email protected] (B. Cai).
stochastic simulation. Xu et al. [12] proposed a real-time reliability prediction method for a dynamic system that suffers from a hidden degradation process. In addition, they proposed a realtime reliability prediction approach based on online fault prediction for dynamic systems and adopted exponential smoothing methods for fault prediction and Monte Carlo strategy for reliability prediction [31]. Ma et al. [32] proposed a new layered modeling architecture that consists of dynamic hybrid fault modeling and extended evolutionary game theory for real-time reliability, survivability, and fault tolerance analyses. Wang et al. [33] investigated the issue of real-time reliability evaluation based on a general Wiener process-based degradation model. FaghihRoohi et al. [34] developed a dynamic model for availability assessment of multi-state weighted k-out-of-n systems using the universal generating function and Markov process. Each method has its advantages and disadvantages. For example, Markov models have state space explosion problems [9,10,34]. The major limitation of stochastic simulation is that the monitoring value has to be a scalar; in real situations, however, many monitoring values usually exist [11]. The hidden degradation process identification is unsuitable for a situation in which several degradation processes exist simultaneously, the characteristics of the degradation processes are time varying, or the path function of the fault process is nonlinear [12,31]. On another active research frontier, Bayesian networks (BNs) have attracted considerable attention in the field of system reliability evaluation and fault diagnosis since they were first
http://dx.doi.org/10.1016/j.isatra.2015.06.011 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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proposed by Pearl [13]. Dynamic BNs (DBNs), which are longestablished extensions of ordinary BNs and allow explicit modeling of changes over time, were developed subsequently. In recent years, BNs and DBNs have been applied to study the reliability of multilevel systems [14], two-terminal networks [15], distributed communication systems [16], N-modular redundant systems [17], structural systems [18], human factors [19], and software systems [20]. BNs and DBNs have also been used to study the fault diagnosis of computer numerical control machine tools [21], chillers [22], and ground-source heat pumps [23]. This technique has an advantage over the aforementioned methods in representing uncertain knowledge in dynamic systems. It overcomes binary variable problems and state space explosion problems; however, the prior probability determination and conditional probability table (CPT) elicitation are always challenging and significant issues. Over the past two decades, few studies have reported on the real-time application of BNs and DBNs. Ji et al. [24] proposed a probabilistic framework based on BNs for the modeling and realtime inferring of human fatigue. Armero et al. [25] proposed a probabilistic expert system for real-time prediction of the risk of Legionella from remote information relating to the quality of water in evaporative installations. Hossain et al. [26] proposed a BNbased framework for real-time crash prediction on the basic freeway segments of urban expressways. Murray et al. [27] proposed a BN-based real-time water quality monitoring system. Rigas et al. [28] proposed a BN-based a real-time methodology for the detection of stress events during driving. Straub [29] proposed a generic framework for the stochastic modeling of deterioration on the basis of DBNs; this framework is suitable for near real-time applications in asset integrity management and deterioration control. Codetta-Raiteri et al. [35] proposed DBNs and the software tool RADYBAN for dynamic reliability evaluation when the reliability parameters, such as the failure rates, vary according to the current state of the system. These real-time applications of BNs and DBNs, especially for near real-time and dynamic reliability evaluation, are totally different with the proposed two-phase real-time reliability evaluation methodology in this work. Exactly locating the root cause of a complex mechatronic system failure in real time to increase diagnostic coverage is the precondition and foundation of evaluating real-time reliability values. To resolve this problem, this paper proposes a novel real-time reliability evaluation framework that combines the BN-based root cause diagnosis phase and the DBN-based reliability evaluation phase. The remainder of this is structured as follows: Section 2 presents the DBN-based real-time reliability evaluation methodology. Section 3 analyzes a case to demonstrate the application of the methodology. Section 4 summarizes the paper.
2. Proposed methodology The generic framework of the proposed real-time reliability evaluation is shown in Fig. 1. The framework consists of the BNbased root cause diagnosis phase and the DBN-based reliability evaluation phase. The root cause diagnosis phase is composed of BNs with two layers, namely, fault layer and fault symptom layer. The parent nodes at the fault layer represent the fault of each component in a system. The child nodes at the fault symptom layer represent the symptoms detected using various sensors and even observed information. Root cause diagnosis is performed through backward analysis of BNs, which involves the computation of the posterior probability of any given set of variables from given observations (evidence) represented as the instantiation of some variables to
Fig. 1. DBN-based real-time reliability evaluation framework.
one of their admissible values. As shown in Fig. 1, the inputs of the BNs are the evidence Yi, and the outputs are the posterior probabilities of nodes Xi. Basically, the following two rules can be used to isolate a fault [22]: Rule 1: the one with the largest fault probability and whose fault probability is larger than a certain threshold, ε1, can be isolated. Rule 2: the difference between the largest fault probability and the second one is larger than a certain threshold, ε2, can be isolated. The reliability evaluation phase comprises DBNs with two or more layers. Taking the two-layers DBNs in Fig. 1 as an example, it consists of a component state layer and a system state layer. The parent nodes at the component state layer represent the state of each component in a system. The only child node at the system state layer represents the state of the entire system, and the probability that the system works represents the system reliability value. The DBNs are extended from BNs at set intervals, and the system reliability at different points in time can be calculated through forward analysis of DBNs, which calculates the occurrence probability of nodes on the basis of the prior probability of root nodes and the conditional dependence of each node.
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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Fig. 2. Structure of subsea pipe ram BOP system. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Fig. 3. Structure of BNs for the root cause diagnosis phase.
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According to the sensor data and observed information at a certain time, the root cause of system failure can be located exactly in real time through the root cause diagnosis phase. The component failure is transferred to the corresponding node in the reliability evaluation phase, and then the real-time reliability value can be predicted. Each component normal is not transferred to the corresponding node because normal use can increase continuously the probability of failure of each component.
3. Case study 3.1. Subsea pipe ram blowout preventer system A subsea blowout preventer (BOP) system plays an important role in providing safe working conditions for drilling activities in 3000 m ultra-deepwater regions. Subsea BOP failures could cause catastrophic accidents such as the explosion of the deep-sea petroleum drilling rig Deepwater Horizon and the oil spill off the coast of Louisiana on April 20, 2010. The failure of the BOP to shear the drill pipe and seal the wellbore was conclusively caused directly by the physical location of the drill pipe near the inside wall of the wellbore, which was outside the blind shear ram cutting surface during activation [30]. In the wake of recent disasters in oil and gas exploration and production, the value of the reliability evaluation of subsea BOP systems is becoming recognized. Therefore, in this section, a case of a subsea pipe ram BOP system is studied to demonstrate the application of the proposed real-time reliability evaluation methodology. The structure of the subsea pipe ram BOP system is provided in Fig. 2. Subsea blue pod and subsea yellow pod are designed to form a redundant hydraulic control system. Taking the subsea blue pod as an example, the solenoid valve DDV11 is a selector valve, which provides low pressure pilot fluid from low-pressure accumulators. Solenoid valves DDV12 and DDV13 control the pilot fluid to sub-plate mount valves SPM11 and SPM12. The high-pressure fluid from high-pressure accumulators flows to shuttle valve 1 and shuttle valve 2 via SPM11 and SPM12 to control the closing and opening of the subsea pipe ram BOP, respectively. The subsea yellow pod is capable of performing all the same functions as the subsea blue pod. When a pod fails, the other one will be used to execute all subsea functions and will not be affected by the disabled one. The disabled one will be retrieved to the surface for repair. In the subsea blue pod, five pressure sensors, i.e., PS11, PS12, PS13, PS14, and PS15, are installed in the outlet of solenoid valves DDV11, DDV12, and DDV13, and sub-plate mount valves SPM11 and SPM12 to detect the pressure of pipelines. Similarly, in the subsea yellow pod, five pressure sensors, i.e., PS21, PS22, PS23, PS24, and PS25, are installed in the outlet of solenoid valves DDV21, DDV22, and DDV23, and sub-plate mount valves SPM21 and SPM22 to detect the pressure of pipelines. Two other pressure sensors, i.e., PS31 and PS32, are installed in the outlet of shuttle valve 1 and shuttle valve 2, to detect the pressure of line 5 and line 6, respectively. The displacement sensor DS is installed on one side of the subsea pipe ram BOP system to detect the displacement of the ram. 3.2. Modeling As shown in Fig. 3, the root cause diagnosis of the subsea pipe ram BOP system is established using Netica software. Netica is a powerful, easy-to-use, and complete commercial software program for working with BNs and DBNs and can perform various kinds of inference by using the fastest and most modern algorithms, such as junction trees. Taking the subsea blue pod as an
example, parent nodes DDV11_failure, DDV12_failure, DDV13_failure, SPM11_failure, and SPM12_failure at the fault layer represent the faults of solenoid valves DDV11, DDV12, and DDV13, and subplate mount valves SPM11 and SPM12. Each parent node has two states, i.e. absent and present, which indicate the absence and the presence of the corresponding fault of given observed evidence, respectively. Child nodes PS11, PS12, PS13, PS14, and PS15 at the fault symptom layer represent the information obtained from pressure sensors PS11, PS12, PS13, PS14, and PS15. Each child node has two states, i.e., normal and abnormal, which indicate the normal and the abnormal states of observed evidence of pressure sensors, respectively. The prior probabilities and CPTs of nodes should also be obtained to complete the BNs. The prior probabilities of failure absence for all the components in the root cause diagnosis phase are set to 99%. The CPTs can be obtained from the relationships between faults and fault symptoms. Taking the blue pod as an example, the relationships between faults and fault symptoms are provided in Table 1. The fault of DDV11_failure causes the fault symptoms of abnormal values in pressure sensors PS11, PS12,
Table 1 Relationships between faults and fault symptoms in the blue pod for the root cause diagnosis phase. Fault
Fault symptom
DDV11_failure DDV12_failure DDV13_failure SPM11_failure SPM12_failure
PS11
PS12
PS13
PS14
PS15
0.95 0.04 0.04 0.03 0.03
0.98 0.98 0.02 0.06 0.06
0.98 0.02 0.98 0.06 0.06
0.96 0.98 0.04 0.93 0.05
0.96 0.04 0.98 0.05 0.93
Table 2 CPTs of node PS11 for the root cause diagnosis phase. DDV11 failure
DDV12 failure
DDV13 failure
SPM11 failure
SPM12 failure
P (PS11¼ 1)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0.000000 0.030000 0.030000 0.059100 0.040000 0.068800 0.068800 0.096736 0.040000 0.068800 0.068800 0.096736 0.078400 0.106048 0.106048 0.132867 0.950000 0.951500 0.951500 0.952955 0.952000 0.953440 0.953440 0.954837 0.952000 0.953440 0.953440 0.954837 0.953920 0.955302 0.955302 0.956643
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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PS13, PS14, and PS15 at respective probabilities of 95%, 98%, 98%, 96%, and 96%, but not 100%, because of the uncertainty in system modeling. The existing uncertainty problem is caused by various reasons, with sensor accuracy and measure uncertainty as the primary causes. CPT elicitation is always challenging in BN analysis. Several approaches, such as: direct assessment of the probabilities by one or multiple experts, elicitation of probability ranking on qualitative scale, elicitation of selected model relationships, and filling-up of the remaining relationships with algorithms, have been used [36]. In the current work, multiple expert knowledge and Noisy-OR/AND filling-up algorithm are used. The probabilistic relationships between faults and fault symptoms are determined by two designers in the subsea BOP manufacturer, Rongsheng Machinery Manufacture Ltd. of Huabei Oilfield, Hebei. With the use of relationships and the Noisy-OR model, the CPTs can be computed. For example, the CPTs of node PS11 are calculated using the Noisy-OR model according to the probabilities in Table 1, as provided in Table 2. When solenoid valve DDV11 fails, the probability that pressure sensor PS11 detects pipeline pressure is around 95% but not 100% because of the introduced uncertainty. However, not all the CPTs of nodes can be calculated using the Noisy-OR model. For example, pressure sensor PS31 is used to detect the pressure of line 5, which closes the subsea pipe ram BOP. When either line 1 or line 2 is normal and shuttle value 1 is normal, the line 5 is normal. Line 1 (PS14) and line 2 (PS24) are parallel. Therefore, the Noisy-AND model is used to represent the relationship between line 1, line 2, and PS31.
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Similar to the root cause diagnosis phase, the reliability evaluation phase of the subsea pipe ram BOP system is also established using Netica software, as shown in Fig. 4. Taking the subsea blue pod as an example, parent nodes DDV11, DDV12, DDV13, SPM11, and SPM12 at the component state layer represent the states of solenoid valves DDV11, DDV12, and DDV13, and subplate mounts SPM11 and SPM12, but not the faults of components. Each parent node has two states, i.e., work and fail. The only child node, BOP, at the system state layer represents the state of the entire subsea pipe ram BOP system. Child node BOP has two states, i.e., work and fail, and the probability that the system works represents the system reliability value. As mentioned in Section 1, DBNs are long-established extensions of ordinary BNs and allow the explicit modeling of changes over time. To model the temporal evolution of a system, two time slices for each variable are considered. The current time step is represented by t, and the next time step is represented by t þ Δt. The time interval Δt could be 1 h, 1 week, and even 1 month. A great number of time slices corresponds to a smaller the value of Δt, and, hence, a longer time at which Netica runs. In this work, the DBNs are extended from BNs every four weeks, and the structure of the DBNs extended within the first eight weeks is presented in Fig. 4. No sensors are used for the reliability evaluation phase. Thus, no uncertainty problem is introduced into the reliability evaluation model. Taking the subsea blue pod as an example, solenoid valves DDV11, DDV12, and DDV13, and sub-plate mounts SPM11 and SPM12 are considered to be in a series. Thus, any fault of the components can
Fig. 4. Structure of DBNs for the reliability evaluation phase.
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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Table 3 CPTs of the blue pod for the reliability evaluation phase.
Table 4 Parameters of the Weibull distribution for four components.
DDV11
DDV12
DDV13
SPM11
SPM12
P(blue pod ¼1)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
cause the failure of the subsea blue pod. Therefore, the blue pod node is binary, as shown in Table 3. The lifetime distribution of each component of the subsea BOP system is Weibull distribution, which is one of the most widely used lifetime distributions in reliability engineering. Based on the value of shape parameter β and characteristic life η, it is a versatile distribution that can take on the characteristics of other types of distributions. In this work, the reliability function for the twoparameter Weibull distribution is derived by one minus the cumulative density function. The values of characteristic life η and shape parameter β for each component of the subsea pipe ram BOP system are provided in Table 4. Generally, these values are difficult to collect. In this work, the parameters for Weibull distribution, similar to prior probabilities and CPTs, are determined by the two designers in the subsea BOP manufacturer, Rongsheng Machinery Manufacture Ltd. of Huabei Oilfield, Hebei. In this work, four real-time reliability evaluation cases are investigated using the proposed methodology, as shown in Table 5. Taking Case 1 as an example, the values of pressure sensor PS31 and displacement sensor DS are abnormal, whereas the other sensor values are all normal. These values are input to the root cause diagnosis phase, and the backward analysis of BNs generates the result that shuttle valve 1 fails. The failure state of shuttle valve 1 is input to the subsequent reliability evaluation phase, and the forward analysis of DBNs generates the real-time reliability data, as shown in Fig. 5.
Component
Parameter
DDV SPM Shuttle Ram
η
β
6800 7450 8400 9860
2.3 1.9 2.5 2.9
Table 5 Four real-time reliability evaluation cases. Sensor
PS11 PS12 PS13 PS14 PS15 PS21 PS22 PS23 PS24 PS25 PS31 PS32 DS
Case Case 1
Case 2
Case 3
Case 4
Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Abnormal Normal Abnormal
Normal Abnormal Normal Abnormal Normal Normal Normal Normal Normal Normal Normal Normal Normal
Normal Normal Normal Abnormal Abnormal Normal Normal Normal Normal Normal Normal Normal Normal
Normal Normal Normal Normal Abnormal Normal Normal Normal Normal Abnormal Normal Abnormal Abnormal
reliability evaluation framework is composed of the BN-based root cause diagnosis phase and the DBN-based reliability evaluation phase, the validations of the diagnosis model and the evaluation model are performed separately. The BN model of the root cause diagnosis phase can be validated by comparing diagnosis results with the practical faults. The four cases in Table 5 are practical engineering cases, in which the sensors detect abnormal symptoms and the root cause diagnosis phase performs backward analysis to calculate the posterior probabilities of each parent node, as provided in Table 6. According to the rules presented in Section 2, shuttle valve 1 fails in Case 1, DDV12 fails in Case 2, SPM11 and SPM12 fail in Case 3, and SPM12 and SPM22 fail in Case 4. The fault diagnosis results agree with the practical faults, which indicates that the BN model of the root cause diagnosis phase is correct. A full validation of the DBN model of the reliability evaluation phase is deemed impractical, because the system reliability values are difficult to define. Therefore, a three-axiom-based validation method is applied to the partial validation of the DBN model [37]. At the initial time, the prior probabilities of work for all components are set to 50%. When the probabilities of DDV11, DDV12, DDV13, SPM11, and SPM12 are set to 100% one by one, the reliability of the BOP increases to 1.15%, 1.90%, 3.42%, 6.45%, and 12.5%, respectively. When the probabilities of Shuttle1 and Shuttle2 are set to 100% one by one, the reliability of the BOP system increases to 25% and 50%. Finally, when the last parent node Ram is set to 100%, the BOP system reliability increases to 100%. The exercise of increasing each influencing node satisfies the axioms, thus partially validating the model.
3.3. Results and discussions 3.3.1. Validation of data and model Validation of data and model is an important aspect of the proposed methodology as it will render a reasonable degree of confidence to the results of the model. As the proposed real-time
3.3.2. Real-time reliability evaluation The inferences of fault diagnosis in the root cause diagnosis phase and the reliability prediction in the reliability evaluation phase are performed using Netica software. When no component failure occurs, the sensitivity analysis of time interval, Δt, is
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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Fig. 5. Real-time reliability evaluation model of subsea pipe ram BOP system.
performed by calculating the reliability of the subsea pipe ram BOP system at different time intervals (Δt), including 2 weeks, 4 weeks, and 8 weeks, with the use of the proposed real-time reliability evaluation methodology, as shown in Fig. 6. Time interval, Δt, has little effect on system reliability because the extension of DBNs is based on the Weibull distribution of components and the conditional probability of time slices for each component is obtained from the reliability function of the corresponding two-parameter Weibull distribution. In this work, time interval, Δt, of 4 weeks is selected for the real-time reliability evaluation. As we expected, with an increase in time, the reliability of the subsea pipe ram BOP
system decreases. In the 40th week, the system has a reliability of 99.953%, which indicates that the system failure probability is only 0.047%; therefore, the subsea pipe ram BOP system is sufficiently secure. When several components fail, the root cause diagnosis phase performs backward analysis and diagnosis, the faults in the four cases in Table 5 are input to the reliability evaluation phase at different times, and the real-time reliability values are calculated and plotted, as shown in Fig. 7. As shown in Fig. 7(a), the reliability decreases rapidly before the fault of shuttle valve 1 is detected, and once the fault occurs, the reliability decreases to 0 when the
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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fault occurs immediately after the system starts. Thus, shuttle valve is a fatal weakness of the subsea pipe ram BOP system and the reliability of the shuttle value should be improved as much as possible. As shown in Fig. 7(b), when the fault of solenoid valve DDV12 occurs immediately after the system starts, the reliability decreases, to a level lower than the one when no fault occurs. In the 40th week, the system still has a reliability of 99.371%. When the fault occurs at a certain intermediate time, the reliability
decreases to the value when the fault occurs at t ¼0. These results are due to the fact that the subsea blue pod and subsea yellow pod are redundant, and the failure of DDV12 cannot affect the normal operation of the subsea pipe ram BOP system but can decrease the reliability of the entire system. When the faults of sub-plate mount valves SPM11 and SPM12 occur, the reliability value is similar to the one when the fault of
Table 6 Fault diagnosis results in the four cases. Parent node
DDV11 DDV12 DDV13 SPM11 SPM12 DDV21 DDV22 DDV23 SPM21 SPM22 Shuttle1 Shuttle2 Ram
Posterior probability Case 1
Case 2
Case 3
Case 4
0.0000 0.0000 0.0000 0.0006 0.0006 0.0000 0.0000 0.0000 0.0006 0.0006 1.0000 0.0001 0.0106
0.0000 0.9474 0.0000 0.0607 0.0009 0.0000 0.0000 0.0000 0.0006 0.0006 0.0001 0.0001 0.0001
0.0002 0.0103 0.0103 0.5283 0.5283 0.0000 0.0000 0.0000 0.0006 0.0006 0.0001 0.0001 0.0001
0.0000 0.0000 0.0229 0.0045 0.9734 0.0000 0.0000 0.0229 0.0045 0.9734 0.0001 0.0104 0.0106 Fig. 6. Reliability of the subsea pipe ram BOP system without component failures.
Fig. 7. Reliability of the subsea pipe ram BOP system with component failures at different times: (a) Case 1, (b) Case 2, (b) Case 3, and (b) Case 4.
Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i
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Fig. 8. Sensitivity analysis of subsea pipe ram BOP reliability: (a) mutual information and (b) belief variance.
solenoid valve DDV12 occurs, as shown in Fig. 7(c). When the faults of sub-plate mount valves SPM12 and SPM22 occur, the reliability value is similar to the one when the fault of shuttle valve 1 occurs, as shown in Fig. 7(d). These results may be ascribed to the fact that SPM11 and SPM12 are installed in the same pod, whereas SPM12 and SPM22 are installed in two different pods; therefore, the failure of both pods can cause system failure. 3.3.3. Sensitivity analysis Sensitivity analysis can be used to measure the importance degree to which findings at any node can influence the output at another node. For the purposes of this study, sensitivity analysis can indicate which components can provide the most information for determining system reliability. The sensitivity analysis determines the importance degree sequence using mutual information and belief variance. Mutual information measures the information that two variables share and how much uncertainty about one variable is reduced by knowing one of the other variables. Belief Variance measures the expected squared change in class probabilities in the output node distribution [38]. The relationships measured by differences in the mutual information are not replicated in the belief variance, which examines changes in the probabilities of the target node rather than its estimated value. The mutual information and belief variance of each component on system reliability in the 20th week and the 40th week are calculated using Netica software, as shown in Fig. 8. The absolute values have little individual meaning but are used to rank the indicators from the most to the least important in terms of impacts on the node of interest. For the subsea pipe ram BOP system, the importance degree sequences from high to low in the 20th week and the 40th week are both Shuttle-Ram-SPM-DDV, which indicates that shuttle valve is the most fatal weakness and should be given more attention.
4. Conclusions This paper proposes a novel real-time reliability evaluation methodology that combines the BN-based root cause diagnosis phase and the DBN-based reliability evaluation phase. The root cause diagnosis of faults for components is performed through backward analysis of BNs, and the real-time reliability evaluation is performed through forward analysis of DBNs. The proposed methodology can determine the value and the variation trend of system real-time reliability when faults of components occur and the importance degree sequence of components at different times with the use of mutual information and belief variance. Obtained information can
help engineers to understand the system state at any time and provide suggestions on repair and maintenance. The application of the proposed methodology is demonstrated using a simple case of a subsea pipe ram BOP system. The values and trends of real-time reliability in the four cases with different faults are obtained; the importance degree sequence of components from high to low are as follows: Shuttle-Ram-SPM-DDV, which indicates that shuttle valve is the most fatal weakness and should be given more attention. Future work can be directed toward the development of real-time reliability evaluation software using the proposed methodology for an entire system, such as a subsea BOP system with thousands of components with series, parallel, and even voting relations.
Acknowledgments The authors wish to acknowledge the financial support of National Natural Science Foundation of China (No. 51309240), Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130133120007), China Postdoctoral Science Foundation (No. 2015M570624), Applied Basic Research Programs of Qingdao (No. 14-2-4-68-jch), Science and Technology Project of Huangdao District (No. 2014-1-48) and Fundamental Research Funds for the Central Universities (No.14CX02197A). In addition, the authors would like to thank the anonymous reviewers whose constructive comments were very helpful for strengthening the presentation of this paper. References [1] Dutuit Y, Innal F, Rauzy A, Signoret JP. Probabilistic assessments in relationship with safety integrity levels by using fault trees. Reliab Eng Syst Saf 2008;93:1867–76. [2] Cacheux PJ, Collas S, Dutuit Y, Folleau C, Signoret JP, Thomas P. Assessment of the expected number and frequency of failures of periodically tested systems. Reliab Eng Syst Saf 2013;118:61–70. [3] Guo H, Yang X. A simple reliability block diagram method for safety integrity verification. Reliab Eng Syst Saf 2007;92:1267–73. [4] Catelani M, Ciani L, Luongo V. A simplified procedure for the analysis of safety instrumented systems in the process industry application. Microelectron Reliab 2011;51:1503–7. [5] Kleyner A, Volovoi V. Application of Petri nets to reliability prediction of occupant safety systems with partial detection and repair. Reliab Eng Syst Saf 2010;95:606–13. [6] Liu Y, Rausand M. Reliability assessment of safety instrumented systems subject to different demand modes. J Loss Prev Process Ind 2011;24:49–56. [7] Cai B, Liu Y, Liu Z, Tian X, Li H, Ren C. Reliability analysis of subsea blowout preventer control systems subjected to multiple error shocks. J Loss Prev Process Ind 2012;25:1044–54.
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Please cite this article as: Cai B, et al. Real-time reliability evaluation methodology based on dynamic Bayesian networks: A case study of a subsea pipe ram BOP system. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.011i