Hierarchical markov reliability / availability models for energy & industrial infrastructure systems conceptual design

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Hierarchical Markov Reliability / Availability Models for Energy & Industrial Infrastructure Systems Conceptual Design A.N. Ajah,a,b P. M. Herder,a J. Grievink,b and M. P.C. Weijnen,a a

Energy and Industry Group, Faculty of Technology, Policy and Management, b Process Systems Engineering Group, Faculty of Applied Science, Delft University of Technology, 2600 GA, Delft, Netherlands.

Abstract Modelling infrastructure system’s deterioration and repair behaviours through a markov model is not only essential for accurately predicting the system’s future reliability condition but also act as key inputs for effective infrastructure systems maintenance. During the conceptual (re)design of these systems, myriads of components are usually involved. A multi-state markov modelling of this component is emphasised in this work. However, the exponential explosion in the size of the markov model as the number of such components increase may pose great limitation in its application at this stage of design. We also present a hierarchical modelling approach that could aid the designer in not only overcoming this limitation but also in the detailed screening and analysis of the reliability and availability of such infrastructure systems’ components. The application effectiveness and utility of the proposed approach is tested by means of a case study, the reliability modelling of a proton exchange membrane (PEM) fuel cell power plant. Keywords: Reliability & Availability, Markov model, Conceptual Design, Infrastructure systems.

1. Introduction Infrastructure system’s (energy, gas, water) reliability, just like the reliability of any engineered system, is increasingly becoming an important performance indicator. This increasing importance and the associated pressure on the system designers calls for a drastic change in the ways the infrastructure systems are currently conceptually designed. Most of the common practices of estimating infrastructure system’s reliability and overall availability at the conceptual design stage are either done on an ad hoc basis or basically rely on some predefined and assumed component availability (usually 80-95%). Two major reliability analysis can be distinguished; measurement and model-based (Sathaye et al., 2000). During systems design, measurement based analysis may be infeasible and expensive, hence model based approaches are often relied upon. However, most of such reliability models utilize the non-state-space model which assumes that the components are independent of one another in their failure and repair behaviours. The markov model could well predict the dependencies and future conditions of these infrastructure components, systems and networks through the characterization of such deterioration and repairs in a probabilistic continuous or discrete-state manner. Capturing such infrastructure system’s deterioration and repair behaviours and their effects on the overall system, through markov model is not only essential for accurately predicting the system’s future reliability condition but act as key inputs for effective system maintenance (corrective and preventive). However, the explosion in the size of the markov model

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as the number of such systems, components or networks increases is being perceived as a major limitation in its application during the early phase of conceptual design where time is often a constraining factor. Due to this exponential growth, it may not be easy to markov model complex infrastructure systems reliability as a once-through entity. What is needed is an approach that can decompose the reliability problem into manageable levels of abstraction, address the reliability issues separately at the decomposed levels and then aggregate the results at each level into final system reliability. This work builds on this concept as it dwells on a hierarchical markov modeling approach to circumvent this limitation and also aid the designer in the proper screening and analysis of the reliability and availability of infrastructure systems at the early phase of the conceptual design process.

2. Hierarchical markov modeling approach In the hierarchical markov modeling being proposed, the reliability problem is decomposed into three manageable markov levels (components, units and system), based on the structural and behavioral complexity of the system. This decomposition reduces the number of state space problems to a size that could be easily handled. At the component level, each component is markov-modeled separately. This gives the designer, extra degree of insight into the dynamic performance of the components and hence in the selection of components to feature in the design. At the unit (sub-system) level, for a given flow diagram of an infrastructure system, an aggregation of the equipment based on functional and structural similarities is carried out. This aggregation reduces the number of equipment to be markov-modeled. Lastly, at the higher (system) level, the evaluated subsystems markov reliability and availability are combined into the total system reliability and availability. In this way, the size of the problem as well as the computational efforts is significantly reduced. The attributes of decomposition levels is as shown in table 1. Table 1: attributes of decomposed levels Attributes Analysis type

Level of decomposition Component

Unit

System

Markov

Markov

RBD + Markov

States

Multi-states

Reduced

Reduced

Structural model

No connectivity

State space diagram connectivity

Series-parallel

Behavioral model

Independent

Dependence of units

Dependence of units

Reliability and availability modeling at the component level: At the component level, the essence of reliability and availability analysis being proposed is to assess the reliability characteristics and states of each of the components with a view to selecting the most reliable and promising ones. However, the conventional attitude is to assume that a component has binary basic states-operable and failed states normally designated as 1 and 0 respectively. In most infrastructure systems, there are some groups of system failures that may not be immediately observed upon occurrence. Such system disorder do manifest as small errors or defects such as pump fouling, pipeline partial blockage etc that do not cause immediate total or catastrophic failure. Nonetheless, if left undetected and unrepaired, such failures still grow to cause a larger failure that result in unscheduled downtime. Before the catastrophic or total system downtime, occurrence of such menial disorder may force the system into a state of reduced functionality vis-à-vis its incipient operational state. These sorts of failures that do not result into immediate catastrophic failure of the system but can lead to diminishing functionality have been modeled as transient failures and the states at which they occur, as transient states, states r and d in

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figure1. Considering and modeling these transient states using the markov approach will give a more realistic steady state and dynamic reliability characteristics of the components and will help in the proper screening and selection of components to be featured in the system being designed. The transition diagram as depicted below covers possible states and transitions that a component can realistically experience. Transition from state u (fully operational) to transient states r or d may be caused by environmental and human induced problems respectively while the transition from state r or d back to state u is caused by the removal of such a failures (repairs). If at states r or d, no action is taken, the component might finally transit to the state l (permanent failure) which may be restored to either states u (full restoration) or to states r and d (partial restoration), depending on the nature of the maintenance action taken. The direct transition from states u to state l depicts mechanical and other unknown problems which lead to instantaneous permanent failures.

Figure 1: Transition diagram for a multi-state markov modeled component

It has been reported that human errors and environmental factors contribute about 40% of total equipment downtimes (www.plantweb.emersonprocess.com); hence, we have differentiated between states r and d to highlight the effects of operators and environmental factors on the availability of components of especially energy infrastructures. Assuming there is enough data, we think that considering them early in the design process; will aid the designers in the critical assessment of these factors and thus the screening, differentiation and better choice of the more reliable components. The probability of components (Pi) to be in any of the M states could be obtained from the solutions of sets of equation 1. λ and μ are components failure and repair rates.

⎞ ⎞ ⎛ ⎛ dPi (t ) = − ⎜⎜ ∑ λ i → j ⎟⎟ Pi (t ) + ⎜⎜ ∑ μ j → i ⎟⎟ P j (t) dt ⎠ ⎠ ⎝ j ≠i ⎝ j

i = 0,..., M ; M ≥ 2

(1)

Reliability and availability modeling at the unit level: Having identified the most reliable components to feature in the design based on the detailed component level markov modeling, at the unit (sub-system) level, the combinatorial dependencies and redundancies associated with these components is also markov-modeled. It is envisaged that for complex infrastructure systems, involving units with components in the order of tens and hundreds, state space explosion problem will be imminent. To circumvent this, we propose unit’s component aggregation. The basic idea behind this is to reduce the number of equipment in the units of such complex large-scale systems by substituting some sets of equipment whose individual unavailability does not clearly affect the system performance, or whose failure and repair rates are similar, with a single representative component. For a unit with N number of components, the state space size is estimated by 2N, but with the approximate state space aggregation technique (Lefebvre, 2002) the size reduces to: k

∏ (N i =1

i

+ 1)

(2)

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where Ni is the number of components of the ith family, k is the total number of different aggregate families. Assuming there are a total of 6 components (i.e. N=6 with 2 components aggregated into each family (Ni=2) then k becomes 3); the total number of unreduced state space is 64 while the approximate reduced state space is 27. However, such aggregation is valid for set of equipment which have single (or multiple) input(s) and single output to the remaining components of the system, and for certain types of multiple input-output subsystems (Van Rossen, 1994). Since aggregation policies and rules are often based on dedicated structural and behavioral heuristics, caution, experience and sound engineering judgment are needed of the designer in the application of these aggregation techniques.

Reliability and availability modeling at the system level: At the system level, using a reliability block diagram depicting the connections(series, parallel, series-parallel etc) and other reliability characteristics the aggregated individual units of the system is drawn and the overall system structure and reliability, analyzed using the network reduction (Sahner and Trivedi, 1986; Knegtering and Brombacher,2000) approach. From the network reduction technique, given the combined markov models and RBD constructs, if a system has a series-parallel structure, its overall reliability can be obtained using : Rsys(t)=

{1− (1− R

unit a

}

(t))parallel* (1− Runitb (t))parallel*...(1− Runit n (t))parallel * (Runit m (t))series

(3)

Where Rsys (t) is the system overall reliability at time t, Ruint i (t) is the reliability of unit i at time t as obtained from the unit markov reliability modeling. If all the units in the system have parallel connection structure, the last factor of equation 3 can be neglected.

3. Illustrative Case study As an illustration, the proposed decomposed markov reliability model is applied to the analysis of the reliability characteristics of a proton exchange membrane (PEM) fuel cell power plant as shown in a condensed block diagram of figure 2.

(a)

(b)

Figure 2: a) Condensed block diagram of a PEMFC power plant. b :) Aggregated units of the PEMFC

Fuel cell power plant is a heterogeneous system featuring the interactions between chemical, mechanical, thermal and electrical components and subsystems for converting fuels such as gasoline and natural gas to alternating current. Natural gas flows into the reformer where it is converted into hydrogen. The hydrogen produced together with oxygen from air are then routed into the (PEM) cell stack assembly where the O2 and H2 streams are electrochemically converted into electrical power, steam and water. The steam is recycled to the reformer for the reforming process. The electrical power generated, in the form of DC power flows to the power inverter for inversion into an AC power before being routed to grid.

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Component level: The case study described above has more than ten components such as compressors, pumps, humidifiers, reactors, transformers, membrane, control equipments etc. For the compressor, table 2 shows the state probabilities. Steady state is assumed at 250 hrs. Table 2: compressor steady-state probabilities Time(hrs) 0

Compressor 1&2

State u

State i

State j

State l

1

0

0

0

250

Compressor 1 0.9679 0.0093 0.0152 0.0076 Compressor 2 0.9736 0.0085 0.0113 0.0066 From the foregoing, it could be deduced that the compressor 2, with higher chances of being operational and lower chances of environmental and human failures (states i & j) outperforms compressor 1 and thus would win the selection process.

Unit (subsystem) level: Using the concept of components aggregations as discussed in section 2, and assuming redundancy, the components of the PEMFC power plant have been lumped to form the basic unit as shown in figure 2b. In all, five units with 32 state and 80 transition paths as depicted in figure 4a is identified. If a second degree aggregation is carried out, the number of states reduces to about 18, with 32 transition paths (figure 4b). From figure 4a, sets of markov differential equations (equation 5) results, assuming different failure and repair rates for the various transitions to and fro states. The solution of these differential equations, gives the probability of the system to be in each depicted state. Such solution could be obtained using any of the numerical methods of Euler, Runge Kutta and LU decomposition or with software such as MATLAB. Table 3 shows the results of the state probabilities for the second degree aggregated units of 18 state spaces.

(a)

(b)

Figure 4: a) Ist degree reduced state space [Si denotes State i, μi,j and λi,j denote repair and failure transition rates from state i to state j]; b) 2nd degree reduced state space of the illustrative case study.

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⎡ n ⎢ − ∑ λ 1, i ⎡ P1 ( t ) ⎤ ⎢ i = 2 ⎢ ⎥ d ⎢ P2 ( t ) ⎥ ⎢ λ 1, 2 = ⎢ ⎥ ⎢ dt ⎢ ⎢ ⎥ ⎣ Pn ( t ) ⎦ ⎢⎢ λ 1, n ⎣⎢

μ 2 ,1 n

∑ (λ

−

i =1 , i ≠ 2

2 ,i

⎤ ⎥ ⎥ ⎡ P1 ( t ) ⎤ ⎢ ⎥ μ n , 2 ⎥ ⎢ P2 ( t ) ⎥ ⎥ ⎥ ⎥⎢ ⎥ ⎢⎣ Pn ( t ) ⎥⎦ n −1 − ∑ μ n ,i ⎥ i =1 ⎦⎥

μ n ,1

; μ 2 ,i )

…

λ 2 ,n

(5)

Table 3: State probabilities of the 2nd order reduced units Time 0hrs 250hrs S10 0 4.2E-5

S1 1 0.9725 S11 0 1.12E-6

S2 0 0.0491

S3 0 0.01452

S12 0 1.812E-6

S4 0 1.91E-6

S13 0 4.13E-7

S14 0 7.02E-5

S5 0 3.4E-6

S6 0 0.0081

S15 0 9.73E-9

S7 0 2.7E-6

S16 0 5.63E-7

S8 0 3.8E-6

S17 0 1.705E-8

S9 0 2.7E-6 S18 0 1.52E-10

System level: At this level, the components and subsystems, initially abstracted are integrated using the network reduction techniques, to obtain the system reliability. Using equation 3, and the second order reduced units of figure 2b (with Rreformer 1 =Rreformer 2 = 0.9725; RFCP Stack1 = RFCP Stack 2 = 0.95 and Rtransformer = 0.8) the overall system reliability at steady state have been estimated at 0.79998. However, if another redundant transformer (same reliability of 0.8) is introduced into the network, the system reliability goes up to 0.99999, thus reflecting the effect of redundancy on reliability.

4. Conclusions: Most reliability models assume a binary component or system reliability (up and down states). However, many real-world components of energy and water infrastructure systems are often of multi-state nature, with different performance levels and several failure modes that induce degradation effects on the entire system performance. We present a multi-state component reliability model and the hierarchical decomposition of infrastructure system markov-reliability modeling into three levels. This decomposition approach, apart from its utilization in the management of the state space explosion also helps the designer in the proper use of reliability as performance criteria for components selection. The utility of the proposed approach is tested by means of a case study, proton exchange membrane (PEM) fuel cell power infrastructure system.

References. Knegtering B and A.C. Brombacher (2000), A method to prevent excessive numbers of markov states in markov models for quantitative safety and reliability assessment, ISA Trans, 39, pp 363-369 Sahner S.A. and K.S.Trivedi, (1986), A hierarchical, combinatorial-markov method of solving complex reliability models, in proceeding of joint computation conference pp 817-825, Sathaye,A., Ramani S., K.S. Trivedi (2000) Availability models in practice, Conf. proceeding intl, workshop on fault tolerant control and computing. Van Rossen J.C.P. (1994 ) Criticality Rating and Safety Analysis in FRAMS. Center for process Systems Engineering, London. Yannick lefebvre (2002), Approximate aggregation and application to reliability, 3rd International Conference on Mathematical Methods in Reliability

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