Maintenance It

Probabilistic Evaluation of the Effect of Maintenance

Parameters on Reliability and Cost

Mohsen Ghavami Mladen Kezunovic

Electrical and Computer Engineering Department Electrical and Computer Engineering Department

Texas A&M University Texas A&M University

College Station, TX 77843-3128, USA College Station, TX 77843-3128, USA

*********@***.****.*** *******@***.****.***

In the literature, it is also analyzed whether these models are

Abstract Preventive maintenance is performed to extend the

realistic or not, especially when there is a non-periodic

equipment lifetime or at least the mean time between failures.

Cost-effective maintenance scheduling is important due to budget inspection [3]. In this paper, the inspection/maintenance

constraints in the current situation where reduction of the strategy is evaluated by a proposed model. In most of

operating and capital cost is the focus of the power industry. In maintenance strategies, the inspection is non-periodic and

order to establish a cost-effective maintenance, quantitative increased at the end of life cycle of the component. Also, the

evaluation of maintenance parameters is critical. In this paper, a inspection intervals are deterministic, and the duration of the

probabilistic model to achieve cost-effective maintenance inspection is a constant number. The model proposed in this

strategies is presented. Reliability indices such as mean duration,

paper follows this kind of maintenance strategy.

state probability and visit frequency of each state, are computed

using Monte Carlo simulation and demonstrated using a This paper is focused on the way the life cycle of the

numerical example. Further, cost analysis is performed by component is implemented. Although representing the life time

computing all associated costs including inspection, maintenance of the component in several discrete stages according to the

and failure costs based on the reliability indices. deterioration levels is well known concept, this paper is

different from the rest in the way the life cycle of the

Keywords; State diagrams, Deterioration, Maintenance, component is simulated in a selected algorithm. Specifically

Inspection, Monte Carlo simulation

the way the transition between the life time stages and

inspection stages has been handled is explored. This model

I. INTRODUCTION would be suitable particularly in the case of non-periodic

inspection strategies. The transition time distribution between

The utilities perform regular inspection, planned

deterioration stages are assumed to be exponential, where as

maintenance at a selected working state of components and on-

the transition from deterioration stage to inspection stage is

demand repair or replacement at the failure state of component.

assumed to be a constant number.

They have always utilized maintenance programs to keep their

equipment in desirable working condition for as long as it is The paper is organized as follows. Section II discusses

feasible [1]. Probabilistic maintenance models and reliability maintenance models using state diagram and transition rates

centered maintenance have been presented to optimize between the states. In section III, a model is proposed to

maintenance and reliability costs [2]-[10]. A risk based simulate the life cycle of the component. Cost analysis is

approach is proposed for maintenance scheduling of circuit discussed in section IV. In section V, a numerical example is

breaker in [5]. This approach is different from the other risk presented and solved using Monte Carlo simulation to extract

based approaches in the way the risk is calculated. It utilizes the reliability indices of the probabilistic maintenance model,

the maintenance quantification models developed earlier to followed by conclusions in section VI.

quantify the circuit breaker maintenance [12]-[13]. These

approaches are working pretty well when there is a continuous

II. MAINTENANCE MODELING USING STATE DIAGRAMS

monitoring or the inspection rate is so high, which results in a

lot of available data about the condition of the component. It is a matter of common knowledge that component

failures are divided into two categories: either random failures

In the literature, a state diagram is used to represent the

or those arising as a consequence of deterioration. Note that

deterioration process of the component [1]-[2]. It is assumed

these are state models, not Markov model as there are no

that the remaining time in each state is a random variable

assumptions made about the time distributions of the individual

exponentially distributed [1]-[2], [4], [13]-[14]. With this

transitions [1]. The process of deterioration can be thought of a

assumption, the state diagram can be represented by a Markov

sequence of deterioration stages shown in Fig. 1. In most

process and there are some analytical solutions for this model.

applications, considering three deterioration stages such as

initial stage (D1), a minor (D2) and a major (D3) deterioration

Mohsen Ghavami and Mladen Kezunovic are with the Department of

Electrical and Computer Engineering, Texas A&M University, College stage, is sufficient [2]. If no maintenance is performed, a new

Station, TX 77843-3128, USA(emails: *********@***.****.***,

component will run through all the stages, respectively. It is

*******@***.****.***).

978-1-4244-5721-2/10/$26.00 2010 IEEE PMAPS 2010

Figure 1. State diagram for modeling the life cycle of the component

(without maintenance)

Figure 3. State diagram for modeling the life cycle of the component with

inspection/maintenance strategy, the transition duration between states is

supposed to be exponentially distributed, so the transition rate is a

constant number.

The most important assumption in the model, shown in Fig.

3, is that the maintenance actions are not carried out in a

Figure 2. State diagram with adding the maintenance state

predefined schedule. Based on regular inspections, it can be

decided if and what kind of maintenance should be done. The

reasonable that these stages can be defined by specific signs decision after inspection can be either doing nothing (where the

that appeared in the component because of aging and realized condition of the equipment is in deterioration stage D1) or

by inspections. It is a good assumption (near reality) that the carrying out specific kind of maintenance denoted by M2 or

failure probability of the component is arisen by these M3. As seen in Fig. 3, the inspection rates 1- 3 can be equal

consequences of deterioration of stages, and the remaining time which means this approach holds for either periodic or non-

of the component in each stage is independent of the time for periodic inspection. It is obvious that the probability of

which the component has been in that stage. detecting a critical situation at the end of the component s life

cycle is increased and returning the component to the previous

The negative exponential probability distribution is the only

situation needs more effort. So, it is reasonable that the

one that has the memory less property [15], and it is used to

inspection rate is higher if the equipment is deteriorated more.

represent the probability of such event. In the real world, most

There is a good discussion and model about non-periodic

of utilities conduct maintenance actions based on periodic

inspection in [3].

inspection or maybe non-periodic inspection. It means that the

state of the system is completely unknown unless inspection is

III. EXRACTING RELIABILITY INDICES USING MONTE

performed [16].

CARLO SIMULATION

Although more discussions are needed about inspection

The goal of this section is to devise a model which follows

models, the previous model shown in Fig. 2 can be improved to

the maintenance strategy in the real world and afterward

the model shown in Fig. 3 which includes inspection states. It

solving the model using Monte Carlo simulation. To have

is shown that there is an inspection state instead of dotted-line

compatible results with the maintenance strategy in the real

for maintenance in state D1. In this model, based on inspection

world, two assumptions are considered in this model. First,

results, two kinds of maintenance, M2 (minor maintenance) or

inspection rates should be increased along with aging of the

M3 (major maintenance), can be performed, or the component

component. The most of maintenance strategies utilized by

will be left without any kind of maintenance if it is in state D1.

utilities have non-periodic inspection rates increasing at the end

The expected result of all maintenance actions is only

of life cycle of the component. Second, the remaining time in

improvement to the previous stage [6], [17]-[18]. In some

the inspection state and also inspection duration are

literature, waiting periods after inspections are considered.

deterministic and they cannot be modeled by exponential

Also some contingencies where no improvement is achieved,

distributions. Thus, the model is not still Markov process and

or even some damage is done by maintenance activities are

the answer is hard to derive through analytical solutions.

reported [2]. For the sake of simplicity, these cases are not

Therefore, the best way to solve the proposed model in this

considered in this paper. If one assumes that remaining life

section is Monte Carlo simulation. In general, inspections are

time in each state has an exponential probability distribution

performed which leads to three kinds of decisions followed:

and the transition rate between states are constant numbers, the

state diagram will turn into Markov process. There are some do nothing, if the component is still in initial stage D1;

analytical methods to solve this probabilistic model and extract

reliability indices such as mean durations, visit frequencies and Carry out minor maintenance M2, if the component is

mean time between failures [15], [19]. Monte Carlo simulation in stage D2. This will return the device to stage D1;

can be used to solve this probabilistic model when the answer

Carry out major maintenance M3, if the component is

is difficult to drive through analytical solutions. Maintenance

in state D3. This will improve the component

models with this structure are discussed in the literature [2],

condition to stage D2;

[4], [13]-[14], [20].

In order to establish a maintenance strategy including these

assumptions, the model shown in Fig. 3 should be improved to

the model shown in Fig. 4. The main idea of this model is that

the deterioration process and inspection strategy are two

parallel processes. It means the deterioration process does not

change the next inspection time determined by last inspection.

It is more realistic because the state of component is

completely unknown if the inspection is not performed [16]. In

the model seen in Fig. 3, the inspection rate will change from

1 to 2 if the component is deteriorated single step. In the real

world, there is a non-periodic inspection which means 1 is not

equal to 2, and the next inspection time is determined at the

last inspection.

The main goal of this section is to develop an algorithm to

simulate the life cycle of a component in a probabilistic model

and find out the reliability indices by using the Monte Carlo

simulation. To understand the model and the proposed

algorithm to solve that, some iteration is followed. The

parameters in this model are:

inspection intervals [years];

inspection duration [years];

repair rate [1/years];

deterioration rate [1/years];

Suppose the component is at state Dx D1 at time t0=0 and it

will transit either to state Dx D2, or inspection state I (there is

the same story for the other situations; it means in each step, it

transits either to the next deterioration stages or to the

inspection state). The simulation of transition from state D1 to

state D2 can be modeled with a random number generated by

an exponential distributed random number generator with rate

1, denoted by d12. For transition from State Dx D1 to

inspection state I, it is a constant number equal to the

inspection interval 1 for the first inspection. Similarly in Fig.

3, 1 is the inspection rate and 1/ 1 is equal to inspection

period, but here the inspection intervals are not exponentially

distributed random variables. Thus, the leaving time of the state

Dx D1, is either (t0 + d12) or (t0 + 1). As seen in the Fig. 4, the

inspection interval denoted by x, which is similar to the

inverse of the transition rate to inspection states in Fig. 3, is

varying (it can be equal to 1, 2 or 3) and it is determined at

the last inspection period.

Assume that (t0+d12) > (t0+ 1), so first there is a transition

from state Dx D1 to inspection state I. This inspection will

show us that the component is still in state Dx D1 and Figure 4. A model to simulate the life cycle of the component with

according to the maintenance strategy, nothing will be inspection/maintenance strategy. This model is solved by Monte Carlo

performed and the system will return to the state Dx D1. simulation.

Approximately, the inspection duration is neglected in

comparison to inspection intervals but it can be considered,

from state Dx D1 to state Dx D2 will be (t0+d12). This is the

which is equal to 1 (inspection duration). Thus, the component

main difference between this model and conventional Markov

is returned to state Dx D1 at time (t0+ 1+ 1). The next transition

process based model shown in Fig. 3. In that model, after

can be either to state Dx D2 or inspection state as before. The

returning from inspection state I1, a new number will be

most important point in this algorithm is that another random

regenerated for the deterioration time from State D1 to state D2

number for transition time from state Dx D1 to state Dx D2 has

which is not compatible with the situation in the reality.

not been generated because the time of deterioration does not

change when inspection is performed (inspection does not Therefore, the next leaving time of state Dx D1 is either

make any improvement). Therefore, the next transition time (t0+d12) or (t0+ 1+ 1+ 1).

Suppose that (t0+d12)

deterioration time from state Dx D1 to state Dx D2 is less than and failure. These notations are followed:

the inspection time. At time (t0+d12), the component will

CT=total expected annual cost

deteriorate to state Dx D2; afterward there are two possible

ways to leave the state Dx D2. Obviously, it can transit either CF=the cost of repair or replacement paid after failure

to state Dx D3, or to inspection state. The time of transition

CMx=the cost of maintenance action type x (M2 or M3)

from state Dx D2 to state Dx D3 can be obtained by a random

number generated by an exponentially distributed random CI=the cost of inspection

number generator with rate 2, denoted by d23. As the

CT=CF (frequency of failure state) + CM2 (frequency of state

deterioration transition from state Dx D1 to state Dx D2 is not

M2) + CM3 (frequency of state M3) + CI(frequency of state I)

predetermined (it is only a probabilistic model), the next

inspection time will not change, which means the next The cost paid after the failure of the component may not

inspection time is still (t0+ 1) as before. As it is mentioned include only the repair or replacement cost, but also the cost of

already, this is the main difference between this algorithm for event consequences and damages to the entire system should be

Monte Carlo simulation and the other algorithms based on involved if the supply is interrupted by that failure. There is the

Markov process model. In the Markov process models, a new same scenario for the maintenance and inspection. Maybe, the

random number will be generated for the next inspection time cost is needed to take the component out of service for

in this case. Therefore, there are two possible transition time, maintenance or inspection. At these situations, the mean

(t0+d12+d23) and (t0+ 1). duration time of component being at each state is critical and

the time can contribute to the cost of that state. In some

If (t0+d12+d23) > (t0+ 1), it will transit to inspection state

maintenance strategies, there is some waiting period before it is

where it will be revealed that the component condition is in

suitable time for doing maintenance or inspection to reduce the

state Dx D2, and the specific kind of maintenance M2 is

costs. If the duration of these periods are not comparable to

required. Maintenance is done immediately after inspection and

inspection intervals, it can be neglected, and that is why it is

the component s condition will return to the state Dx D1. The

not mentioned in the model shown in the previous section.

maintenance action duration can be neglected in comparison

with the inspection intervals, but it is an exponential random Another element in the cost estimation is the visit

number with rate 2, denoted by m2. At time (t0+ 1+ 2+m2), the frequency, which is calculated based on the maintenance

condition of the component is repaired as new and returns to strategy. In Markov process, the visit frequency of state j is the

state Dx D1. To continue the modeling of the life cycle of the frequency of encountering state j from the other states. In the

component, the new numbers for transition times have to be model in Fig. 3, all the times, when there is a transition from

derived using random number generators as before. Thus, there state D1 to state I, are counted in the visit frequency of state D1.

will be two numbers; (t0+ 1+ 2+m2+d12new) for transition to Although there is a transition to inspection state, the component

state Dx D2, and (t0+ 1+ 2+m2+ 1) for transition to inspection is still in deteriorating stage D1. So, it should not be considered

state. in the visit frequency and mean duration of state D1 [3]. In the

model shown in Fig. 4, there is the same scenario to figure out

Returning to the first assumption in the last paragraph, if

the reliability indices using Monte Carlo simulation. Therefore,

(t0+d12+d23) (t0+ 1), it leads to

detection of the component condition which is in state Dx D3.

After inspection, it will be decided to perform the major V. NUMERICAL EXAMPLE

maintenance denoted by M3. This kind of maintenance will This section presents, a numerical example based on the

return the component condition single step back. Finally at time ideas in the past two sections. The input data is referred to in

(t0+ 1+ 3+m3), it returns to state Dx D2. Be careful that for the [2]. The data is obtained from the analysis of a number of 230

next iteration, the inspection interval would be 2. It could be KV air-blast circuit breakers with a total operating history of

(t0+d12+d23+d3F)

Contact this candidate