TECHNICAL REPORT

A-3270 6-21-91

DEGRADATION MODELING: EXTENSIONS AND APPLICATIONS

F. Hsu, W.E. Vesely*, E. Grove, M. Subudhi, and P.K. Samanta

Risk & Reliability Analysis Group

Engineering Technology Division

Department of Nuclear Energy

Brookhaven National Laboratory

Upton, New York 11973

*Science Applications International Corporation

June 1991

Prepared for

Nuclear Plant Aging Research (NPAR) Program

Under Contract No. DE-AC02-76CH00016

NRC FIN A-3270
EXECUTIVE SUMMARY

Component degradation modeling includes modeling of occurrences of component degradations

and analyses of these occurrences to understand the degradation process and its implications. The

degradation modeling that we discuss focuses on the analysis of times of degradation and failure

occurrences to understand the aging degradation in components. Our previous paper' discusses the

basic concepts and the mathematical development of a simple degradation model. Using the

degradation modeling methodology, failure data on residual heat removal (RHR) pumps and service

water (SW) pumps were analyzed to detect indications of aging and to infer the effectiveness of

maintenance in preventing age-related degradations from transforming to failures. In this paper, further

applications and extensions of degradation modeling are discussed.

Additional applications of degradation modeling are carried out for air compressors, a

continuously operating component. We demonstrate that the analysis of degradation occurrences is

useful in understanding the aging process and the role of maintenance in that process. For the air

compressors, the failure rate and degradation rate show an early decreasing trend followed by a

significant increasing trend that indicates effects of aging. The failure rate, which is significantly lower

than the degradation rate in the first three years, increases faster in the later years, reaching

approximately the same value as the degradation rate at the end of the ten years of operation. This

behavior indicates the ineffectiveness of maintenance in preventing degradation from transforming into

failures as the air compressors age.

Another important application of degradation modeling approaches is to predict aging-failure

rate from degradation rate. Since aging-related failures, in general, pass through a degradation state

first, the degradation rate serves as a precursor of the failure rate. Increasing aging trend in the

degradation rate can signal future increasing aging trends in the failure rate. We study a simple linear

relationship between these two parameters considering any delayed effect that degradations may have

iii
on failure occurrences. An example of an application using the data on RHR pumps shows a time-lag

of 2 years for degradation to affect failure occurrences.

For additional applications, extensions of degradation modeling are presented. The extended

models, which we are developing, will explicitly show the reliability effects of different maintenance and

test intervals, different maintenance and test efficiencies, and different repair times. Thus, the extended

model will allow us to evaluate the reliability effects of different maintenance programs.

iv
CONTENTS

EXECUTIVE SUMMARY .............................................

LIST OF FIGURES ................. 

LIST OF TABLES .................................................

ACKNOWLEDGEMENT .................. ;.;

l. INTRODUCTION ..............................................

2. DEGRADATION ANALYSIS USING AIR COMPRESSOR AGING DATA..

2.1 Overview of Degradation Modeling Approaches ...................

2.1.1 State Representation of Degradation Modeling ..............

2.1.2 Transition Probabilities ...............................

2.1.3 Frequency of Degradation, Frequency of Failure,

and Transition Probabilities ............................

2.1.4 Aging Effects on Degradation Rate and Failure Rate .

2.1.5 Assumptions and Limitations in the Methodology ............

' '' * ,'1' '. ' . .

2.2 Definitions of Degradationst ..... ...

23 Analysis Approach .. .................

2.4 Aging Effects on Degradation. .................................

2.5 Aging Effects on Failure .....................................

2.6 Evaluation of Maintenance Effectiveness ........................

3. SENSITIVITY ANALYSES OF DEGRADATION MODELING RESULTS ...

3.1 Sensitivity Analysis on the Partitioning of

Aging-Failure Data ............. ;.-;.:.i

3.1.1 Analysis Approach ....................-

3.1.2 Analysis Results on Sensitivity Partitioning -

of Aging-Failure Data ................................

3.1.3 The General Conclusion on Sensitivity of

Aging-Data Partitioning .................-

3.2 Sensitivity Analysis on Uncertainty of

Degradation Occurrence Times.

3.2.1 Analysis Approach .

33 Sensitivity of Test Frequency to the Estimation of

Degradation Frequency.

Pan

iii

vii

viii

Lix

1-1

2-1

2-1

2-2

2-3

2-4

2-5

2-6

2-8

2-9

2-10

2-12

2-13

3-1

3-1

- 3-2

3-2

3-6

3-7

3-7

3-10

v
CONTENTS (Cont'd)

4. ANALYSIS OF DEGRADATION-FAILURE RELATIONSHIP - EVENT COUNT

BASED APPROACH .......... ................................. 4-1

4.1 A Distribution-Free Statistical Test Approach for

Data Combining ............. ............................. 4-2

4.2 Estimation of Degradation and Failure Frequency

Using Event Counts ........... ............................ 4-5

43 Analysis of Correlation Between Degradation and

Failure Frequencies - Time-Lag Considerations ..... .............. 4-6

4.4 Estimation of Failure Rate from Degradation Data -

Time-Lag Regression .......... ............................ 4-9

45 Applications of the Degradation Rate-Failure

Rate Relationship ........... ............................. 4-11

5. EXTENSIONS OF DEGRADATION MODELING: THE ANALYSIS OF

PROGRESSIONS OF DEGRADED STATES ......................... 5-1

5.1 Background and Basic Concepts ................................ 5-1

5.2 Basic Approach ................. 5-2

53 Incorporation of a Component Degraded State .5-5

5.4 Incorporation of a. Maintenance State .5-10

5.5 Resolving Additional Degradation States .5-12

5.6 Incorporating Effects of Surveillance Tests .5-12

5.7 Calculation of Component Reliability Characteristics

Including the Component Unavailability .5-14

5.8 Incorporation of Aging Effects .5-15

6. SUMMARY .6-1

REFERENCES ................................................. R-1

APPENDIX A AGING DATA EVALUATION OF AIR COMPRESSORS ........ A-1

APPENDIX B: STATISTICAL RESULTS FOR SENSITIVITY ANALYSIS IN

PARTITIONING OF AIR COMPRESSOR DATA .... ........... B-1

APPENDIX C: STATISTICAL RESULTS FOR SENSITIVITY ANALYSIS ON

UNCERTAINTY IN DEGRADATION OCCURRENCE TIMES .... C-1

vi
FIGURES

Pare

2.1 Alternatives for degraded state definitions ............................. 2-2

2.2 Markov state diagram component degradation modeling ................... 2-3

2.3 Age-dependent degradation rate .................................... 2-11

2.4 Age-dependent failure rate ........................................ 2-14

2.5 Estimated maintenance effectiveness ................................. 2-14

3.1 Sensitivity of age dependent failure rate due to

subjectivity in data partitioning ..................................... 3-5

3.2 Sensitivity of estimated maintenance effectiveness

in data partitioning ............ ................................. 3-5

3.3 Age-dependent degradation rate with uncertainty

of degradation times .3-9

3.4 Sensitivity of test frequency to component degradation rate .3-12

4.1 Degradation & failure rate estimation ................................ 4-6

4.2 Degradation & failure rate estimation ................................ 4-10

vii
TABLES

Page

2.1 Typical Examples of Compressor Degradation Levels and

Failure Mode and Effect ......................................... 2-9

3.1 Sensitivity of Failure Time Trend on Data Partitioning .................... 3-3

3.2 Sensitivity Comparison of Data Partitioning on

Maintenance Effectiveness ........................................ 3-6

3.3 Sensitivity of Degradation Time Trend on the Uncertainty of

Degradation Occurrences ......................................... 3-9

4.1 Degradation & Failure Data Based on Counts .......................... 4-2

4.2 Cross-tabulation of Age by Plants for Data on RHR Pumps .... ............ 4-3

4.3 Kendall's Rank Correlation Analysis Results for RHR Pumps

at 3 plants ................................................. 4-8

4.4 Kendall's Rank Correlation Analysis Using Time-Lag Considerations .... ..... 4-9

4.5 Estimated Parameters on Time-Lag Regression Analysis .................. 4-11

viii
ACKNOWLEDGEMENT

The authors wish to acknowledge Mr. Satish Aggarwal, NRC Technical Monitor for his technical

support and encouragement in this project. The report also benefitted from reviews by J. Higgins, J.

Taylor and R. Hall of BNL.

We are very thankful to Ms. Donna Storan for an excellent job in preparing this manuscript.

ix
1. INTRODUCMION

This report presents the status of degradation-modeling'development in understanding the

component aging process and the role of maintenance in mitigating that process. Component

degradation modeling as defined here includes modeling of occurrences of component degradations and

analyses of these occurrences to understand the degradation process and its implications.

Our earlier report' presents the basic concepts and the mathematical development of a simple

degradation model. In this modeling approach, dividing the operational performance of a component

into three states, normal operating state, degraded state, and failure state, we established relations

among these states using rates of degradation and failure occurrences. The relations were used to define

estimates of the effectiveness of maintenance in preventing degradations from becoming failures.

Specific applications of the theoretical model were performed which resulted in quantitative models of

component degradation rates and component failure rates, all of them derived from plant-specific data.

Degradation analyses were carried out for residual heat removal (RHR) pumps and emergency service

water (SW) pumps - standby active" components. Analyses of degradation data for both these pumps

showed a 'bathtub" curve for the degradation rate where a distinct, increasing aging trend was observed

as time progressed. Interesting, pump failure rates did not show any increasing trend for the same time

period, thus demonstrating the need to identify aging trends through analyses of component

degradations. The applications also demonstrated how this modeling approach can be used to analyze

and assess the effectiveness of maintenance.

To explore further the applicability of degradation modeling approaches, we analyzed a different

component, air compressor, using the methodoiogy'defined in our earlier report . Air compressors,

continuously operating component, are different from standby active components studied in our previous

applications. Thus, the degradation modeling analysis of air compressors shows the applicability'of the

approach for an active component under different operating conditions. Also,- because they are

1-1
continuously operating component, air compressors are expected to suffer degradations which are

detected and corrected; thus, making them ideal candidates for degradation modeling analysis.

The degradation modeling approaches is an extension of standard reliability approaches where

only two states, normal operating state and failure state, are considered. By including an additional

state, degradation state, we obtain useful information on component aging and on the role of

maintenance in preventing age-related failures. However, this modeling requires additional data which

can have large associated uncertainties. In this report, sensitivity analyses are performed to address the

uncertainties in the degradation modeling results due to inclusion of information on component

degradations.

The promising results obtained from the application of simple degradation models to standby

and continuously operating components (RHR pumps, SW pumps, and air compressors) encouraged us

to expand this concept to study additional related aspects. The additional aspects studied here include:

a) relationship between degradation and failures,

b) extension of modeling to show effects of different maintenance and test intervals, and

different repair times, and

c) extension of degradation modeling to include multiple degraded states.

Understanding the relationship between degradations and failures is important in aging studies

and can result in significant benefits in defining maintenance strategies for controlling aging and in

conducting aging reliability and risk studies particularly when aging failure data are sparse. In terms of

maintenance strategies, if degradation-failure relationship is known, then effective maintenance/

repair/refurbishment can be performed through monitoring of degradations, thus avoiding component

failures. For aging reliability studies, relationship between degradations and failure is important since

it can be used to estimate failure rates from degradation.rates when failure data are sparse. In this

report, this important correlation between degradations and failures is statistically studied and the

concept of a delayed effect of degradation on failures is explored.

1-2
Finally, the simple degradation model studied and applied so far uses only degradation and

failure occurrence times. The usefulness of the model can be enhanced by including additional relevant

test and maintenance related information whereby the effect of test and maintenance strategies, in terms

of test and maintenance frequencies, duration, and efficiencies, can be determined. Theoretical

development which extends the basic degradation model to include different test and maintenance

interval, different test and maintenance efficiencies, and different repair times is discussed. Basic

framework for extending the model to multiple degraded states is also presented.

The report is organized as follows, Chapter 1 is the introduction. Analysis of air compressors

is present in Chapter 2. Chapter 3 discusses the uncertainty issues brought in by including degradation

information and the sensitivity analyses performed to study the impact of the issues on degradation

modeling results. Chapter 4 explores the relationship between degradations and failures and the

extension of the degradation modeling is presented in Chapter 5. The results and insights of this study

are summarized in Chapter 6. Appendices present air compressor data used in the evaluations and the

statistical evaluations conducted for the results presented in the main body of the report.

1-3
2. DEGRADATION ANALYSIS USING AIR COMPRESSOR AGING DATA

; ~ ~ ~ -> .. . ' .

In this section, we present an application of age-related degradation and failure data analysis

based on the component degradation modeling approach described in our earlier report'. Here, our

objective is to explore the applicability of degradation modeling approaches for a continuously operating

component, different from standby components studied previously. We selected an air compressor for

this analysis. The analysis approach is similar to that followed for the components studied in our

previous report'.

2.1 Overview of Deeradation Modellnr Appronches

In this section, we a briefly summarize the degradation modeling approaches. Basically, we

present the relationships to be used in applying degradation modeling to component degradation and

failure data, the assumptions of degradation modeling, and basic formulations of the modeling

approaches. Detailed mathematics of specific degradation modeling can be obtained from our earlier

report'.

To understand degradation modeling, we study a repairable component, i.e., a component that

is being repaired and maintained. The active" components, pumps, valves, circuit breakers, and

compressors, are repairable components and are the focus of this study.

For the simple degradation model studied, we make the following assumptions:

1. Degradation always precedes failure.

2. When a component is repaired after a failure, the operational state of the component

reflects more restoration than when on-line maintenance is performed.

3. When maintenance is performed following detection of a degraded condition, the

component is restored to a maintained state, which reflects less restoration than when

repair is performed after a failure.

We call the state after repair of a failure the "o" state; the state after failure the "fC state; and

the one after maintenance the "n" state.

2-1
We use the Markov process approaches for degradation modeling, because with these

approaches simple models can be constructed first and then expanded later to yield more complex

models. Statistical analysis is coupled to the models to estimate unknown parameters from degradation

data. The simplest model we present considers only one degraded state. Expanded modeling can

include multiple-degraded states (Figure 2.1).

A

Measure of

Performance

Operating Stats Operating State

Degraded State 1

Degraded State 2

Degraded Stat.

Degraded State n

Failure State Failure State

Shne Degraded Muttipe Degraded

State Defirition State Defnition

Figure 2.1. Alternatives for degraded state definitions

2.1.1 State Representation of Degradation Modeling

The Markov approaches of degradation modeling can be described by the state diagram for a

component (Figure 2.2). Based on our assumptions, the component can be in a degraded state (d-state)

through three processes:

a. the component reaches its first degraded state from a restored state (o-state),

2-2
b. the component undergoes recurring degradation with no intermediate failure (it is

assumed that the component is in a maintained state (m-state) following a degradation),

and

c. the component undergoes degradation following restoration resulting from a failure (fstate).

! ' ' ; . 'l , .' '- .., 

The component can fail only from a degraded state (d-state). However, it is assumed that maintenance

is performed every time a degraded state is detected. Thus, a maintained state (m-state) is reached

following a degraded state (d-state). For Markov modeling considerations, these two states are

equivalent in this analysis.

c~~iitD ( 3 .~ - PDM{ P~DM 1 IMD P.D . (i

BFO

o-state: restored state d-state: degraded state

m-state: maintained state f-atate: failed state

P : transition probability from i-state to J-state

Figure 2.2. Markov state diagram component degradation modeling

(single degraded state)

2.1.2 Transition Probabilities

The transition probabilities among the various states are as follows:

POD = probability that degradation occurs after the component is restored, with

no failure before a degradation

2-3
PDM

PMD

= PMF

POF

- 1 because we assume degradation always precedes failure

= probability that maintenance is carried out once a degraded state is

identified

= probability that degradation occurs after maintenance before a failure

occurs

= probability that failure occurs after maintenance (performed following

detection of a degraded state), with no intermediate degradation

= probability that component is restored following failure

= 1

PDF

Our interest lies in obtaining PMD and PDF* Principally, PDF describes the effectiveness of

maintenance and the probability of transferring to a failed state once a degraded state is reached.

Similarly, PMD, expresses the probability of recurring degradation before failure.

2.1.3 Frequency of Degradation, Frequency of Failure, and Transition Probabilities

Frequency of degradation defines the frequency of degraded state, i.e., the number of degraded

states observed for a component per unit time. Similarly, the frequency of failure represents the failure

states observed per unit time.

Let

WD(t) = the degradation frequency at t

WF(t) = the failure frequency at t

Developing balance equations from the renewal theory2 , one can obtain the steady-state solution

that relates the frequency of degradation, the frequency of failure, and the transition probabilities.

(Mathematical derivation is described in our earlier report'.) WD and WF represent the steady state

degradation and failure frequencies.

WD = WF + WDP.MD (2.1)

WF = WDPDF (2.2)

24
Expressed in terms of transition probabilities,

PDF = wF/ WD. (23)

PMD = 1 - WF / WD = 1- PDF (2.4)

These expressions define how the steady-state transition probabilities (DF and PMD) can be

obtained from the frequency of degradation and the frequency of failure. Using component reliability

data bases such as the Nuclear Plant Reliability Data System (NPRDS) or a plant-specific data base, one

can determine WF and WD, and hence, WF / WD for various components. These ratios can also be

determined for various failure modes of a component to evaluate the type of maintenance carried out

for a component.

The interpretations of the steady-state solutions are as follows:

1. The larger the ratio of frequency of failure and frequency of degradation (WF / WD) the

larger is the probability that a failure will occur after degradation, PDF.

2. For a given degradation frequency, WD, the larger the probability, PDFI the larger is the * ~~~~ .~~~~~~~~- 

failure frequency, WF.

3. The ratio WF / WD is a measure of ineffectiveness of maintenance in that it is equal to

PDF. However, smaller values of PDF can result in larger values of WF, if WD is larger.

4. Another measure of effectiveness of maintenance is the failure frequency, WF itself,

which is equal to WDPDF.

The approaches presented above define how information on degradation can be used to obtain

the characteristics of degradation (frequency, the transition probabilities from degraded to failure state

and from maintained to degraded state), and how the frequency of component failure relates to such

characteristics.

2.1.4 Aging Effects on Degradation Rate and Failure Rate * ;$ s . e! . * . s v;1 ' .

The effect of aging on component reliability may be manifested through either increased .ga . . . , .. -:r. , arlier . v ;,

degradation or incased failures, or both. Generally, earlier studies have focussed on increased failures

2-5
due to aging. Here, the focus has been on degradations, along with an analysis of failures to seek a

relationship between the two.

The degradation rate, AMD, is defined as the rate of degradation occurring after maintenance

given that no previous degradation has occurred. Similarly, the failure rate, ADF is the rate of a failure

occurring after a degradation given that no previous failure has occurred.

The age-dependent AMD can be obtained by observing the times of degradation. The time of

degradations, t, t2,...,t is used to estimate the parametric form of 1MD(t). Similarly, time of failures is

used to estimate the parametric form of ADF(t).

When times of failure of the aged component are also present, along with the information on

degradation, the former can be used to develop the age-dependent ADF, which can then be compared

to MD. The different behavior of ADF(t) and AMD(t) signify different types of effectiveness of

maintenance in the component's aging process. If AMD(t) shows a significant aging effect as opposed to

ADF(t), then the maintenance averts component failure. Conversely, maintenance is ineffective if the

transition probability, PMD' in the aging process is higher than the steady-state value.

2.1.5 Assumptions and Limitations In the Methodology

The degradation modeling presented in this section is the first step in developing the component

aging reliability model using data on degradation. The specific analyses of examples presented in the

next section also demonstrate the applicability of the methodology and show how useful insights can be

derived from this approach. Nevertheless, at this time, several assumptions for this simple model are

made, many of which will be dealt with as we make future extensions to the model and gain more

experience with the analyses. In this section, we discuss the assumptions and limitations in the

methodology and their implications in our results.

1. In the modeling presented, the component degradation is represented by a singledegradation state. Degradations are generally continuous and not discrete as treated

in the model. For this simplest model, the assumption is that a degradation state occurs

2-6
I

when the degradation, which can be continuous, exceeds some threshold. Our objective

is to demonstrate how important insights relating to aging and maintenance can be

obtained by using degradation information in its simplest form. As we stated, more

extended models can be developed that allow multiple states of degradation.

2. The model assumed that maintenance is performed every time a degraded state is

detected. A degraded state as used in the model is a state in which degradation has

exceeded a threshold requiring maintenance. Thus, a degraded state is associated with

a requirement for maintenance. The data used in the analyses are delineated so that

the identified degraded states are associated with maintenance. However, we recognize

that component degradations can be identified where no maintenances are performed.

Extended models with multiple degraded states will be able to distinctly treat degraded

states which are not necessarily associated with maintenance requirements.

3. Maintenance as used in the model is corrective maintenance, not preventative

maintenance. More frequent corrective maintenances are associated with more frequent

degradation occurrences exceeding some threshold. Nondetected degradations and

scheduled maintenances are not explicitly treated by the model.

4. Data requirements for applications of degradation modeling are more comprehensive

because degradation data are required. However, degradation data are often

unavailable, and if available, they are often incomplete. The interpretation of available

data for degradation modeling application also needs to be systematized. Realizing the

difficulty in obtaining comprehensive data, one of the objectives of this paper is to

develop models which show how degradation data can be specifically used for

maintenance. If these specific benefits and uses of degradation data are presented, then

there will be more of an incentive to collect more accurate degradation data.

2-7
2.2 Definitions of Deifradations

To analyze degradations, the degraded state of the component must first be defined so it can

be identified and analyzed. Definitions of the degraded state can be at a gross level or at a detailed

level. At a gross level, a component is described as degraded whenever any deterioration occurs which

does not cause loss of function. The operational performance of the component is divided into three

states: the normal operating state, the degraded state, and the failure state. An example of a gross

definition of degradation is that a component degradation occurs whenever corrective maintenance is

required, but the component has not failed.

More detailed modeling of degradations involves dividing the degradation space into multiple

degraded states. A given degraded state is then associated with a given range of characteristics of the

component or performances of the component. For example, detailed degraded states for circuit

breakers can be defined based on defined ranges for the pick-up/drop-out voltage, inrush/holding

current, and other measurable degradation characteristics.

The advantage of defining more detailed degradation states is that we can accurately predict

impacts on the failure rate of the component. When aging occurs, the component generally progresses

through a series of degradation states before failure occurs. By analyzing and modeling this progression,

we can more accurately predict when failure will occur. For initial work, the gross definition of

degradation can be used, which basically equates the degradation state occurring whenever corrective

maintenance is required. Figure 2.1 illustrates the basic alternative for defining the degradation state.

Table 2.1 presents an example of component data analyses identifying degraded states, along

with failure states of the component. In this example, derived from the analyses of data for air

compressors, failure states and degraded states of air compressors are distinguished based on engineers'

judgement using the information on failure effect and the identified effected subcomponent. In some

situations, judgements were used to determine whether the degradation was of the magnitude to be

defined as a failure. For example, in general, an oil leak at the piston rod seal can be a degraded state

2-8
. .i. . 1 I .. . n., ; :, i; , I . : 

for an air compressor, but in the example in Table 2.1, the leak was of sufficient magnitude to be called

a. failure of the air cm eo......................................................... .- . .

a failure of the air compressor.

1 Table 2.1. Typical Examples of Compressor Degradation Levels and Failure Mode and Effect

Compressor Failure Degradation

Subcomponent Classification FailureEffect , .Failure Mode Level

Bearings D Monthly preventive Low D

maintenance -

bearings greased .

Filter D Monthly P.M. - filter Low D

cleaned

Gasket D . Oil leak by gasket Gasket Intermediate D

deterioration

Jacket Heat D Corrosion deposits Mechanical Intermediate D

Exchanger - built up by aftercooler debris; poor water

. .cheumiistry'

Bolts and D Fractured stud on Mechanical High D

Fasteners spacer vibration

Pistons D Brass filings in high Mechanical wear High D

and low pressure

regions found during

P.M. .,_,,_ .

Piston F Oil leak at piston rod Mechanical wear F

seal

Lube Oil F Pump seized and 'High temperature, F

System became inoperable mechanical wear

23 Analysis Approach

The main objective of the analysis was to obtain the aging failure rate and degradation rate

based on component age-related failure and degradatio'n data, respectively.- These two parameters are

used to obtain the"effectiveness of maintenance-in preventing age-related failure.

For the analysis of air compressors, aging data from only one of two BWRs were used. Based

on the statistical test, the aging data in the two available units were not compatible with each other

2-9
(Appendix A). Therefore, the aging data from unit one was used to provide a data base from four

similar compressors. Similarly, statistical tests were also conducted to justify the data pooling across the

four components.

The process of data collection (Appendix A) provides specific degradation and failure times of

four similar compressors from one BWR. The data for each of the compressors individually were

insufficient to determine the parameters (degradation rate and aging failure rate). Therefore, we

analyzed data from the group of components (i.e., four compressors). Similar to the analysis on RHR

pumps1 , statistical tests were conducted to justify the use of data across components.

(1) Mann-Whitney U Test

The Mann-Whitney U test was used in the analysis to identify components belonging to the same

population.

The four components in unit studied showed statistically identical distributions of times between

degradations (and failures).' Thus, the aging data from across the four components in unit one is

coiibined for the analysis.

(2) Trend Testing and Identification of Age-Group with Degradation and Failure Times

The data obtained by the "data combining" method' were tested for time trends before

developing age-related degradation and failure rates. Statistical tests were used to define component

age groups showing similar aging behavior. We observed that the first three years of the compressors

life showed a decreasing trend, and the last five years showed a increasing trend on both degradation

rate and failure rate.

2.4 . Aine Effects on Degradation

We analyzed the degradation data for the compressors with the following objectives:

(a) To identify age-groups where statistically significant time trends exist, and

(b) To determine time trends and degradation rates, using regression analysis.

2-10
The details of the statistical analysis are presented in the appendices. The results and the

characteristics of estimated degradation rate are summarized in Figure 23, which shows both the

degradation rate (d) and the logarithm of the degradation rate (dAd) that characterized the air

compressors over ten years (presented as 40 quarters). Statistical tests (Appendix A) showed that the

degradation behavior across these components are similar, and accordingly, a generic degradation

characteristic was studied. The age-dependent degradation rate is based on approximately 240

degradation occurrences observed for four compressors over the ten years of operation.

Degradation Rate -D (# per quarter) nA 

7. 39 2

Estimated Degradation Rate - .48 - 7~wlth-95% -ionfidence brounds 1.5

Data Method - Combining

2.72 _ _ Ouasi. Cubic spline it.- 1

1.65 _ 0.5

1o . . 1-+- , 1 , o0

0.61 7 _ ___ _-0.5

0.37 _ -1

0.22 _ ___ -1.5

0.14 , , , , , , . ,-2

0 5 10 15 20, 25 30 35 40

* Age In quarters (3 month period)

Figure 23. Age-dependent degradation rate (data on 4 air compressors)

The following observations can be made from the age-dependent degradation rate for the

underlying air compressors.

(a) The degradation rate shows significant age-dependence; the early life of the

compressors (the first five years) shows a statistically significant decreasing

trend, and the later life (last five years) shows a statistically significant increasing

trend with the age of the compressors.

2-11
(b) The increase in degradation rate, which is of interest in aging studies, is

significant.

(c) The 95% confidence bounds for the degradation rate show that the uncertainty

in the estimation is reasonably small. The large number of degradations

observed in the component contributed to this lower uncertainty.

2.5 Agin! Effects on Failures

The aging-failure data for the compressors were also analyzed with the following objectives:

(a) To identify age-groups where statistically significant time trends exist, and

(b) To determine aging-failure rates in the age-groups where time trends exist.

Figure 2.4 shows both the failure rate (Af) and the logarithm of the failure rate (InA1 ) for the air

compressors over ten years. The age-dependent failure rate presented is based on 25 failures observed

for four compressors over ten years of operation.

The following observations can be made from the aging-failure rate obtained for the air

compressors:

(a) The aging-failure rate shows significant decreasing trend in the first two and a

half years (in 10 quarters), and an increasing trend for the last five years of the

component's ten-years life.

(b) The behavior of aging-failure rate is similar to the degradation rate in the early

two and one-half years, but differs after that. The aging-failure rate was

generally lower (factor of 2 to 8) than the degradation rate and the difference

decreased with increasing age. The aging failure rate reached about the same

level as the degradation rate at the end of the component's ten years of

operation.

2-12
(c) The 95% confidence bounds associated with aging-failure rate show higher

uncertainty compared iothe degradation rate due to the lower number of

failures observed.

2.6 Evaluation of Maintenance Effectiveness

As discussed in our earlier reportI, the degradation modeling approach estimates the

effectiveness of maintenance in preventing age-related failures. The transition probability from a

maintenance state to a failure state signifies the ineffectiveness of maintenance in the simplified model

studied. The complement of maintenance ineffectiveness is maintenance effectiveness.

The maintenance effectiveness for the air compressors is obtained for each ten quarters of age.

The maintenance effectiveness (1 = excellent maintenance, 0 = poor or no maintenance), as plotted

in Figure 2.5, varies between 0.3 and 0.9 for the first 30 quarters, but is significantly lower (about 0.1)

in the last 10 quarters, which signifies the small difference maintenances made in preventing

degradations of components from transferring to failures.

2-13
IlAt

-2

1.5

_ 1

_- 0.5

-0

, -0.5

_-1

_- -1.5

- -2

0 5 10 15 20 25 30 35 40

Age in quarters (3 month period)

Figure 2.4. Age-dependent failure rate (component: 4 air compressors)

1

0.8

0.8

0.4

0.2

n

Maintenance Effectiveness

-Estimated Maintenance Effectiveness

Data Method - Combining

_ 10 20 . 30 40__, 

v

0 10 20 30 40

Age in Quarters ( month periods)

Figure 2.5. Estimated maintenance effectiveness (component: air compressors)

2-14
3. SENSITIVITY ANALYSES OF DEGRADATION MODELING RESULTS

The degradation modeling approach depends on the available data on component performance

and the interpretation of the data for use in this approach. The limitations in obtaining the data and

the subjectivity involved in interpreting it have the potential to affect the results used in judging aging

effects. In this section, we present sensitivity analyses to study the impact of several issues on

degradation modeling results. Sensitivity analyses were performed for three aspects:

i) partitioning of component reliability records (test and maintenance data) into

degradation and failure states,

ii) uncertainty in degradation occurrence times, and

iii) impact of component test frequency to detect degradation and failure occurrences.

3.1 Sensitivity Analysis on the Partitioning orAcina-Fnilure Data

The partitioning of aging-failure data, from the engineering standpoint, is the fundamental issue

of whether a component's state is classified as degraded or failed. Analysis of engineering data for

defining component degradation state and failure state are discussed in Appendix A.

The main objective of this sensitivity analysis is to analyze the sensitivities of degradation

modeling results to the uncertainties in aging-data partitioning into degradation or failure. As stated,

because clear, detailed information is not always available to define the component state, subjectivity

will be involved in defining the state. If this process for defining the component state can cause

significant variation in the results obtained through degradation modeling analyses, then approaches

should be defined to account for uncertainties in using the results of the analyses. Processes can also

be defined to limit the uncertainties in defining the component state from the available data base. In

the next section, aging data on air compressors are used as a case study for this issue.

3-1
3.1.1 Analysis Approach

Sensitivity analysis was performed by determining aging failure rate, degradation rate, and the

maintenance effectiveness parameter for different data classifications obtained from the data evaluation

process. The analysis was performed based on the initially obtained data set.

The partitions of existing aging-data obtained by redefining the degraded and failure states for

sensitivity evaluations were somewhat judgmental. For instance, the gasket leaking problem of the air

compressors was classified as a degraded state of the component in the original data base. Ambiguity

exists, however, in the sense that the gasket leakage problem up to a certain degree of severity will not

affect the required functional performance of the air compressor, but beyond that level of severity the

component will not be able to perform the required function, i.e., the component is in a failed state.

Due to lack of detailed information, clearly identifying the component states is difficult. Therefore, in

this sensitivity study, component degraded states were re-evaluated and defined as failure state where

clear judgement could not be made. The aging-data set obtained after this partitioning and the

description of related partitioning criteria are presented in the Appendix B.

3.1.2 Analysis Results on Sensitivity Partitioning orAging-Failure Data

The same statistical analysis method was used on the data set obtained by repartitioning the

aging-data for sensitivity evaluation. The details of data analysis and statistical test results are presented

in Appendix B. Since it was assumed that failure occurs through the process of degradations, the times

between degradation occurrences remain unchanged regardless of the failure time partitions. Therefore,

throughout this sensitivity study the degradation rate is not affected by this repartitioning of component

states.

3-2
Results of the analysis and general findings on the sensitivity of data partitioning are summarized

as follows:

(a) Statistical Test for Data Combinin2

The statistical test for data combining across components was conducted on the newly

partitioned data set. The results showed no significant difference for the Mann-Whitney U-Test on the

new data set compared to the initial data set. The statistical tests still justified using data from unit one

across the four air compressors; Table B.3 in Appendix B gives the details of these results.

(b) Trend Test Results and Identification of Aee-Groups with Failure Times

The data obtained for this sensitivity evaluation were tested 'for' time trends. 'No significant

differences in time trends between corresponding age-groups compared to the original evaluations were

observed. Table 3.1 presents the comparison of statistical parameters representing the failure rate trends

showing the effect of data partitioning.

Table 3.1.' Sensitivity of Failure Time Trend on Data Partitioning

Aging Rate Constant Model

Partition Age - - b - - Ina -

Status Group n ____ (%er) Pero Befoue Grop juncertainty uncertainty standard ----l b p..(5% error) ' na (595 erro r error

0-15 .0.233 -. 024 CL0.409 0.43 '0.386 CL.0.934 0.025 *; 0.584

CU:-0.055 CU: 1.804

15-40 0.1012 0.035 CL-O.0078 -3.696 0.007 CL-6.22 0.035 1.398

CU:0.1964 - CU:-1.163

0-12 0.349 0.005 CL3-0.551 1.079 0.07 CL-4.118 0.005 0.878

CU:-0.147 . CU:2.276 '

After

1240 0.085 0.02 CLO.015 -2.842 0.005 CU-4.743 0.02 1.322

- CU:0.1S4 _ . CJ.0.939

P: Significance level

3-3
The results presented in Table 3.1 can be interpreted as follows. In the early age group (about

0-15 quarters), the component shows negative aging effects in both cases (before and after data

partition). However, the component shows a slightly lower negative aging factor before data partitioning

rather than after partitioning. In the later age group (around 15-40 quarters), the aging factor before

data partitioning (b = 0.1012) is slightly higher than the factor after data partitioning (b = 0.085). This

higher factor shows that the aging effect may be slightly over estimated due to subjectivity in datapartitioning.

(c) Aging Effect on Degradation

There were no changes found in aging effects over degradation rate due to data partitioning

because the degradation state remained unchanged following repartitioning for sensitivity evaluations.

(d) Aging Effect on Failures

The following sensitivity observations can be made from the aging failure rate obtained for the

air compressors, based on the newly partitioned data base:

i. The aging-failure rate shows slightly different downward time trend in the first

10 quarters in that the failure rate obtained using the newly partitioned data

shows a relatively smaller downward time trend (a factor of 4.5 decrease)

compared to the previously obtained time-dependant failure rate (a factor of 7

decrease).

ii. In the last 15 to 20 quarters, no differences were found on the increasing time

trends between the two data sets obtained by sensitivity failure time partitioning.

Figure 3.1 shows the sensitivity of aging failure rate to the subjectivity in data partitioning.

(e) Sensitivity Effects of Data Partitioning on Maintenance Effectiveness

The estimated maintenance effectiveness based on the newly partitioned aging data was obtained

(Figure 3.2) by the same approach as before; the maintenance effectiveness varies between 0.7 to 0.85

for the first 20 quarters and then reaches approximately 0.1 in the last 10 quarters.

3-4
Failure Rate )A (# per quarter) inAt

,.4.

2.

1.

1.

0.

0.

O.

Estimated Failure Rate with

48 95%_confidence bounds ,a't M- thQd - Combining Ruai Cubic sp ine lit

72 ftr Dats -Ro-partitioning

,0 \\ {Bass Case Data Partitioning /

61

22 . _ ___ _ _ _ _ _ _ _ _ _ _ _ _ __.. 

Iz

1.5

0

0.5

0

-0.5

-1

-t.6

-2

40

I .0

0 5 l1 1 '20 25 30 35

Age in quarters (3 month period)

Data Source: 4 air compressors combined

Figure 3.1. Sensitivity of age dependent failure rate

due to subjectivity in data partitioning

4

Maintenance Effectiveness ': . I 

0.8

0.51-

E rtisuated Maintenance Effectivenese

Base Case Data Partitioning

Eltinated Maintenance Effectiveness

Alter Date Re-partitioning

Data Method - Conbining

.~ ~ '- . ,. -

0.4

0.2 

II

U-

. O - 10 : 20 30 40

Age, in Quarters (3 month periods)

Data Source: 4 Air Compressors Combined

Figure 3.2. Sensitivity of estimated maintenance effectiveness

in data partitioning

3-5
The maintenance effectiveness has almost the same time trend as the one obtained from the

initial data set, except that the magnitude of the maintenance effectiveness (newly obtained) is decreased

about 25%. Such a decrease is apparently caused by the increase in component failure rate obtained

from repartitioning of the data. Table 3.2 gives the details of comparison of the maintenance

effectiveness parameter.

Table 3.2. Sensitivity Comparison of Data Partitioning

on Maintenance Effectiveness

Maintenance Effectiveness

Age Group/

Data 0-10 10-20 20-30 3040

Partitioning (quarters) (quarters) (quarters) (quarters)

Before 0.781 0.912 0.587 0.345

After 0.684 0.843 0.371 o.146

3.1.3 The General Conclusion on Sensitivity of Aging-Data Partitioning

The following conclusions can be made from the results of this sensitivity evaluation:

i. relatively small uncertainties on aging-data partitions will not significantly affect

the results obtained by using the degradation modeling approach,

ii. sensitivity of failure-data partitions is reflected on the magnitude of the

estimated failure rate, whereas the effect of the time trends on both degradation

and failure rate was minimal. However, the engineering standard used in data

partitioning can have impact on the degradation modeling results, and

iii. the maintenance effectiveness was sensitive to data partitioning. Thus, defining

engineering criteria for data partitioning is important for applying degradation

modeling in maintenance decisions.

3-6
3.2 Sensitivity Analysis on Uncertainty of Degradation Occurrence Times

In this section, we present a sensitivity study on the uncertainty of degradation occurrence times.

The primary focus of this analysis is to incorporate the uncertainties present in the component

degradation times used in the analysis. In practice, particularly for standby components where the

component is tested at specified intervals to detect failures or degradation, there can be differences

between the time at which degradation is detected and the time at which it actually occurred. This

uncertainty was investigated by incorporating the undetected degradation occurrence times into data

analysis of the degradation modeling method.

The sensitivity study was carried out based on the data on RHR pumps (as used in our earlier

,:. ~ .. .. .. . . , . . ; .. - -

report). The following assumptions were made to calculate undetected degradation occurrence times:

(a) On the average, the uncertainty in degradation occurrence time is one-half of the test

interval.

(b) Degradation occurrence is a Poisson process in nature, and consequently, the undetected

times between each degradation follow an exponential distribution with a constant mean,

which is assumed to be the same as the distribution of test intervals.

(c) A Monte-Carlo simulation technique was used, and a computer program was written to

generate the random sample for the undetected degradation occurrence time intervals.

321 Analysis Approach

In this sensitivity analysis, the time-dependent degradation rate was determined based on data,

recalculated by incorporating the undetected degradation occurrence times. The data were modified

according to the assumptions stated above, that is, the time intervals between degradation occurrences

in the original data base were subtracted by the undetected times, which were generated randomly by

a computer program using Monte-Carlo simulation techniques.

*5 3 -- * - * , .~ , , ;.

The statistical analysis method used in our earlier report' was applied on the new data set for

RHR pumps [Appendix C, Table C.lI, which was obtained by imposing the uncertainty in times of

3-7
degradation occurrences. The uncertainty times (undetected degradation occurrence times) were

randomly generated by a computer program, which assumed an exponentially distributed random

variable with a mean of (30/2=) 15 days, i.e., approximately 0.166 quarter, or half month. In other

words, an average of 15 days (half of the test interval of one month) undetected time was imposed on

the initial data set as the uncertainty in the degradation occurrence time. Details of data modification

and statistical test results are presented in Appendix C.

Since this sensitivity analysis was primarily focused on degradation occurrence times, only the

time-dependent degradation rate and its aging effect were analyzed and compared with the initial results,

which do not include uncertainties in degradation occurrence times. Results of sensitivity analysis on

the uncertainty of degradation occurrence for RHR pumps are summarized as follows:

(a) Trend Test Results and Identification Of Age-Group with Degradation Times

The data obtained by including of uncertainty in degradation occurrence times were tested for aging

trends. We found no significant differences of time trends due to including uncertainty in degradation

occurrence times. The comparison of statistical parameters showing the impact of uncertainty in

degradation occurrence times is presented in Table 3.3.

The statistical parameters in Table 3.3 show almost the same time trending results in both cases

which shows that the sensitivity of uncertainty in degradation occurrences is not statistically significant.

(b) Effect on Degradation Rate, Failure Rate, and Maintenance Effectiveness

No changes were found in aging effect over the time-dependent degradation rate, because of the

imposition of uncertainty in degradation occurrence times. Figure 3.3 gives the time-dependent

degradation rate over 10 years for RHR pumps in 3 plants. The imposed uncertainty on degradation

occurrence times resulted in shifting the occurrence intervals which moved in the same direction (i.e.,

the original time intervals were all subtracted by a certain amount of uncertainty time, signifying that

a degradation may have occurred before the detected time). However, this behavior, does not affect

the time-dependent degradation rate.

3-8
Similarly, the aging effect as the failure rate remained unchanged, and thus, the maintenance

effectiveness parameter was also insensitive to uncertainty in degradation occurrence times.

Degradation Rate Ax( # per quarter) InEk

- .. ,. 54.

33.1.

20.0,

12. 1

7.31

4.41

, 0.3i

1.6x

1.0d

0.6

0.3i

*0.2.

0. I.

* 0.0d

0.0

0.0

0.0.

2 _ _

Estimated Degradation Rate 

8 _ _ / 95% confidence bounds _

Data Method I - Combining

9 ----- O Quasi Cublc-spIlne fit---- 2

7I -t unceftanty In dngradatlon times

r~~~~~~~~~~~~~~~~~~~~

WIthOut UnCei'aintr0ylndegratenjle__ _-_-_ _

Uncertainty times - exponentiai with mean of 30 days

%.,

I

0

1

-2

.3

.4

0 10 - .->. 20 30 40

Age in Quarters (3 month periods)

Data source: RHR pumps In 3 plants

Figure 3.3.' Age-dependent degradation rate

with uncertainty of degradation times

Table 3.3. Sensitivity of Degradation Time Trend on the

Uncertainty of Degradation Occurrences

Aging Rate Constant Model

Age 

Data Group b Ina

Status (quarters) ^uncertainty uncertainty | standard

b i P. (5% error) Ina P (5% eor) P | error

0-20 -0.095 0.0006 CU:-0.0508 0.541 0.025 CU: 1.015 0.0001 1.234

Ignore - . C.:-0.1395 ;. . .. . CL 0.066

Occurrence

Uncertainty 20-40 0.105 0.046 CU: 0.207 -4.161 0.012 CU:-7.325 0.045 1.287

CL: 0.0022 CL.-0.997

0-20 .0.098 0.0001 CU:-0.0508 0.586 0.018 CU: 0.1027 0.0001 1.259

Incorporate CL-0.144 . CL: 1.068

Occurrence

Uncertainty 2040 0.1037 0.05 CU: 0.204 4.091 0.015 CU:-7.132 0.053 1.327

CL: 0.0013 CL:-0.913

P: Significance level

3-9
3.3 Sensitivity of Test Freguencv to the Estimation of Degradation Frequency

The degradations observed in a time period is dependent on the number of surveillance tests

performed on the component. However, certain degradations can also be observed via operating

parameters. By conducting surveillance testing at a frequency lower than the occurrence of degradation

frequency, occurrencesof degradations'can remain undetected. Therefore, we studied the correlations

between the observed degradation frequency and the test frequency. The analysis presented focusses

on investigating the sensitivity and functional relationship between these two parameters. The following

assumptions were made in deriving the correlation between degradation rate and test rate.

i. The number of degradations observed in a given time period follows a poisson process.

ii. Probability of observing a number of degradations in a total number of N tests follows

a binomial distribution.

iii. Degradation frequency is assumed to be time invariable in a fixed time period.

Let,

Pd: probability of observing a degradation in each test.

T: assumed test interval.

A: time invariant degradation rate (piecewise constant).

Nd: number of times tested for degradation in time period (tj, t2 ).

a>,: test frequency (or test rate).

L. total time period of observations.

D^: number of degradations observed in time period of T.

Based on the above assumptions, the Cumulative Distribution Function (CDF) of Nd can be

derived as follows:

3-10
Since,

= 1 (3.1) t T

so,

N=Int( t =t {(t2-ti)cot} .-

R . 2 . -

Int(L w ) ; (3.2)

E(D.) = Pd Nd = (1-e )L

T

= [T (3.3)n 

or,

E(D) = (1-e T).L cot -;

.~~~~~~~~~~~~~~~~~~~~~~~34

= (-e )Int (t2_lt 34

Thus, the probability of observing number of D. degradations given Pd, Nd, and L will be:

P,(DD Pd, Nd, L)

N D, -I(-.

= (D )(l -TjD.e e T( D)

3-11
Hence,

the cumulative distribution function (CDF) of D. can be obtained as:

P,(k < D.IPd, N, L)

(N) (1-eT)k e-AT(N-k) (3.6)

The above expression (3.6) shows the probability of observing the number of degradations less

than and equal to D. in the time interval T with given degradation rate l; probability of observing a

degradation in each test (Pd); and the test frequency (t).

The results of the sensitivity correlations between A and wt are plotted in Figure 3.4, where

complementary cumulative distribution function (CCDF) for observing degradations are plotted for six

different values of test frequencies. The figure shows that the estimated degradation frequency can be

considerably affected by the test frequency, which indicates the need for incorporating the test frequency

directly into our degradation modeling approach.

CCDF of observing degradations (1-Pr)

1 Test freq.

0.8 _ _ -- -_----.

Tealr freq.

0.8 -W 3/r-----_ _

0.2 _ __ --- 

0

0 5 10 15 20 25 30

# of Degradations

- Wt-3/year - Wt-e/year- Wt-9/year

- Wt-12/year - Wt-15/year Wt-20/year

Figure 3.4. Sensitivity of test frequency to component degradation rate

3-12
4. ANALYSIS OF DEGRADATION-FAILURE RELATIONSHIP - EVENT-COUNT BASED

APPROACH

In this chapter, we present an event-count based approach for data analysis to study the

relationship between degradations and failures. This approach uses non-parametric statistical methods

to estimate and seek relations between degradation and failure rates based solely on the number of

observed degradations and failures in each unit time or age interval.

This approach provides a simple framework for exploring the relationship between degradation

and failure rate. Since aging-related failures, in general, pass through a degraded state first, the

degradation rate serves as a precursor to the failure rate. Increasing aging trend in the degradation rate

can signal future increasing aging trends in the failure rate. We study simple linear relationship between

these two parameters considering any delayed effect that degradations may have on failure occurrences.

In general, disciplines that can be used to 'develop' relationships between the degradation rate

and the failure rate include engineering, reliability, and statistical disciplines. Engineering and reliability

disciplines are required to develop the theoretical models between the degradation rate and failure rate.

Statistical disciplines are required to estimate unknown parameters and to validate the theoretical

models. The relationships between the degradation rate and failure rate, which are studied here,-are

among the simplest models to develop; they are consistent with reliability and engineering

considerations. In the relationships which are developed, the degradation rate is related to the failure

rate by appropriate transition probabilities. These transition probabilities are obtained by studying the

correiations between occurrences of degradation and failures. They also include the effectiveness of

maintenance in controlling the degradations before becoming failures.

The data analysis is the first step'in obtaining the necessary parameters for-these relationships.

The relationships can be applied in several different ways.

4-1
4.1 A Distribution-Free Statistical Test Approach for Data Combinin!

In this section, a distribution-free alternative of two-way Anova F test is demonstrated based on

the event-count data analysis method. The analysis was carried out using the data on RHR pumps

presented in our earlier report.l

Since the non-parametric statistical test is used to check the data discrepancy for data combining

across the three plants, the original aging data set for the RHR pumps was reconstructed as given in

Table 4.1 and 4.2, where the number of degradations for each plant within each year were counted and

grouped into a plant-by-age contingency table.

Table 4.1. Degradation & Failure Data Based on Counts

(12 RHR pumps - 3 units)

Age Plant 1 Plant 2 Plant 3 Total

(Years) - i --. (Years) ndl nfl nd2 nf2 nd3 nf3 Nd Nf

1st 5 0 13 1 1 3 19 4

2nd 2. 0 4 7 2 0 8 7

3rd 9 1 2 1 4 0 15 2

4th 4 0 3 0 0 0 7 0

5th 6 2 1 0 2 0 9 2

6th 4 0 3 1 0 0 7 1

7th 2 0 0 1 0 0 2 -1

8th 1 1 :0 .0 2 0 3 1

9th 1 0 0 0 14 0 15 0

10th 0 0 0 0 2 0 2 0

11th 0 0 0 0 5 0 5 0

4-2
Table 4.2. Cross-tabulation of Age by Plants for Data on RHR Pumps

(A Distribution-Free Alternative of 2-Way Anova F Test)

--,,Plant

2PI.

2

RowTotal

19

8

15

.7

:7

2

3

15

2

5

96

Column

Total 32 38 26

In Table 4.2, the variable Pl G=1,2,3) for each column represents data from different plants,

and the variable t (i = 1, 11) for each row represents data from different age groups. The effect

of the time unit selected for the-age group on the sample size depends on the sparsity of the data set.

In this analysis, the aging data over the component life span of eleven years were divided into eleven

groups, where data within the time unit of one year were combined.

4-3
To test the homogeneity of aging data in different plants, in order to combine them to increase

the sample density, we used a chi-sqare test to check the following hypotheses:

(a) Do the plants (in terms of the number of events) appear to differ in the composition of

their age groups?

(b) If differences exist between numbers in any one row, would they be only due to the

random chances?

(c) Are the distributions of population from plants 1 through 3 identical?

The statistical hypothesis listed above, and the related test can be expressed mathematically as:

Ho: P(x = tilPli.) = P,(x = t) (4.1)

where,

H. is the null hypothesis representing the homogeneity of aging behavior in different plants.

The test statistic:

Q c (0- EE 2 (4.2)

j j Ei

where,

Ei is the expected frequency for the ijth cell in Table 4.2

Oi is the observed number of degradations in the jth sample (i.e., jth plant) belonging to

category i (i.e., age group i).

ni+ denotes the total number of items of age group i in the combined sample,

n+j denotes the total number of items for plant j in the combined sample, and

n denotes the total number of observations.

4-4
c Y

C r ~~~~~~~~~~~~~~~~~~~~~~(

n = n,j= nix 44

where r and c represent the number of rows and columns in Table 4.2.

For a given type 1 error a, H will be rejected at significance level a if:

(Oj -Ej)2

S X2(r-t)(c-It > a

ii '3E

Based on data in Table 4.2, the above test was conducted at a significance level of a=0.05, which gives:

x2 ~ ~ 2 (4.6)

(r-t). (c-I). a - io.ZO.05 = 81.41 > Q = 61.03

that is,

Ho can not be rejected at a=0.05 level, which justifies combining data from RHR pumps from

3 plants for estimating degradation rate.

4.2 Estimation of Degradation and Failure Frequency Using Event Counts

Using the event-count based data set obtained by combining across the data from the 3 plants,

a point estimate method was applied to obtain the estimation of degradation and failure rate over each

year of the 10-year age period. We used a non-linear regression technique on the yearly point estimates

to obtain the approximate time-dependent degradation and. failure requency. The, estimated

degradation and failure frequency are plotted on Figure 4.1.-The degradation and failure rate estimates

shown are consistent compared with the results obtained by the parametrical estimates given in our

earlier report'.

4-5
out.l DogdaaIonFrallre Rato Total # of Events

1.04 25

. De.,adatioe isty Eatimted

By *~ealso rege.

20

Obseretd Degrader$*e

*:- - " "#"e ... -;p° 1 5

1.04

0. 

'.

0.11 Estimated Failure Rate

0 2 4 8 8 10 12

Age (year)

Figure 4.1. Degradation & failure rate estimation

(event-counts based approach - Data on RHR pumps from 3 plants)

4.3 Analvsis of Correlation Between Degradation and Failure Frequencies - Time-Lag

* Considerations

As we stated, the objective of degradation modeling is to develop relationships between the

component degradation rate and the component failure rate. These relationships involve predicting how

the component failure rate will change based on observations of the component degradation rate. Of

most interest is predicting aging trends in the failure rates based on observed aging trends and patterns

in the degradation rate.

If A denotes the failure rate, and Ad denotes the degradation rate, then the objective of

degradation modeling can be interpreted as developing relationships between A and A. If the symbol

'R' denotes these relationships then we may write:

4-6
If = R(ld) (4.7)

Thus, the objective of degradation modeling is to find the relationship R.

Since aging-related failures, in general, pass through a degradation stage first, the degradation

rate serves as a precursor to the failure rate. Increasing aging trends in the degradation rate can signal

future increasing aging trends in the failure rate. Also, by recording the characteristics of the

degradations, the severity of the degradation rate can be determined. Increasing severities of the

degradation rate can also signal future increases in the failure rate. We, however, focused on relating

occurrence rates and did not study the impact of increased severity of the degradation to failure rate

at this time. Effect of increased degradation severity can be studied by expanding the Markov modeling

approach to multiple degraded states supported by engineering criteria and data to obtain the necessary

information from tests on component, maintenance, and operability records.

For our study, the relationship (5.7) is expressed as:

Af (t+l) = Cdf 1d(t) (4.8)

where,

1d(t) is the degradation rate at time (t)

At+l) is the failure rate at time (t+l)

I is the time-lag at which degradations impact failure occurrences

Cdf is the correlation coefficient between degradation occurrences and failure occurrences

The above expression assumes a linear relationship where Cdf, to be estimated from data

analyses, is similar to the parameter of maintenance ineffectiveness. The parameter I represents the

delayed effect because the component generally passes through a degraded state before experiencing

failures, and is also estimated from data.

4-7
We used the event-count based data analysis to determine the correlation coefficient as well as

the lagged time between degradation and failure frequencies. Using the data in Table 4.1, the Kendall's

Rank Correlation analysis method was employed to estimate the correlation coefficient for each

individual plant data, as well as the combined data of the 3 plants. A statistical software package

(STATGRAPH) was used to calculate the correlation coefficient for a large number of possible time-lag

values. Among all the calculated time-lag correlation coefficients, the correlation coefficient using a

time-lag of 2 years reached the maximum value at a significance level of a=0.029. The statistical results

of Kendall's Rank Correlation coefficients are summarized in Table 4.3 and Table 4.4.

Table 4.3. Kendall's Rank Correlation Analysis Results

for RHR Pumps at 3 Plants

Correlation Analyses Between Nf and Nd.

Plant 1: Correlation Coefficient: 0.3570

Significance Level: 0.0139

Plant 2 Correlation Coefficient: 0.5429

Significance Level: 0.0005

Plant 3: Correlation Coefficient: -0.2067

Significance Level: 0.5134

3 Plant Combined:

Nfvs. Nd: Correlation Coefficient: 0.3721

Significance Level: 0.0692

N= number of failures

Nd = number of degradations

4-8
Table 4.4. Kendall's Rank Correlation Analysis Using Time-Lag Considerations

(Data on RHR Pumps from 3 Units)

No Time-Lag

Correlation Coefficient: 0.3721

Significance Level: *'0.0692

One-Year Time-Lag

Correlation Coefficient: 0.1826

Significance Level: 0.3966 -

Two-year Time-Lag

Correlation Coefficient: 0.505

Significance Level: 0.0294

4.4 Estimation of Failure Rate from Detradation Data - Time-Lae Regression

One of the applications of degradation modeling is to estimate the failure rate from the

degradation rate of a component. Here, using the time-lag correlation coefficients obtained in the

previous section, the failure counts are estimated from degradations counts. The lagged regression

technique was used to estimate the failure' frequency based on the correlation coefficient and estimated

time-lag. A linear regression model was usedalthough time-lag regression methods can use exponential

or other non-linear models depending on the data distribution properties. :

- Analysis of data on RHR pumps' is presented as an example of this application. Figure 4.2

presents the estimated failure frequency from the degradation frequency, and Table 4.5 shows estimated

parameters from the data used in obtaining the failure frequency.

4-9
Quarterly Oegatdionlrallure State Total # of Events

2.00 25

6.8e10rlea Riteg1.te Use of Lag-time Correlation

Rae,...Iea t~~or Failure Rate Estimate

20

l.4G ~~~~~~~Observed DOOCSAdatOe

1.04

0 ,,__ - - .--. _ 10

OTeurved ales at.

0.21 Estimated Failure Rate

0.00 I 0

0 2 4 6 8 10 12

Age (year)

Figure 4.2. Degradation & failure rate estimation

(event-counts based approach - data on RHR pumps from 3 units)

In Figure 4.2, the estimated failure rate in the last two years (age 9 to 11) is obtained from the

degradation rate. This estimated failure rate is obtained by using equation 4.8, where both the

correlation between degradation and failures and the delayed effects are incorporated. The correlation

coefficient and the 2-year time-lag were estimated using the first 9 years of data. The estimated failure

rate shows the increasing trend similar to the degradation rate, but lagged by 2 years. This estimated

failure rate is comparable to the failure rate obtained by assuming no failures during this period (age

9 to 11). However, because of the increasing trend, the estimated rate starts becoming larger than that

obtained otherwise. As we discussed previously, estimating failure rate from degradation data can

significantly help risk-evaluation of aging, but the results need to be validated further.

4-10
Table 4.5. Estimated Parameters on Time-Lag Regression Analysis

(Combined Data of RHR Pumps from 3 Units)

Obtained Regression Model: Nf = -0.112 + 0.09 4 2Nd

Coefficient: 0.0942 at significance level: 0.05

Constant: -0.112 at significance level: 0.453

-- . R Squared: 42.2% (indicates 42.2% of the total sample

variation is explained by the model)

Correlation Coefficient (Nf vs. Nd): 0.649

Model is Significance at Level:. 0.0582

Standard Error of Estimation: 0.677

4.5 Applications of the Degradation Rate-Failure Rate Relationship

Once the relationship between the degradation rate and the failure rate is determined, it can

be applied in several important ways. We studied one application (estimation of component failure rate

from degradation rate), but other important applications can be studied with potential advantages. Some

are summarized below. . -

1. The component failure rate can be estimated from degradation data. This estimation

greatly increases the accuracy of the failure rate estimation for reliability and risk

evaluations, and allows the failure rate to be estimated even if there is no failure data.

If failure data exists, the estimate of the failure rate from the failure data can be

optimally combined with the estimate from the degradation data.

2. Aging trends in the component failure rate can be estimated from aging trends in the

degradation rate. This estimation is one of the most powerful applications of the

degradation rate-failure rate relationship. Aging trends, which are identified in

4-1l
I

degradation data, can be input into the relationship to predict the aging trend in the

failure rate. The determined aging-dependent failure rate in turn, can, be input to

reliability and risk models to predict the resulting, impact on the reliability and risk.

3. Degradations can be monitored for their reliability and risk impacts. Alert levels and

warning levels can be designed to monitor degradation to indicate when the failure rate

is too high or is significantly increasing.

4. Maintenance can be monitored for its reliability and risk effectiveness. This again

immediately follows from the degradation rate-failure rate relationship. The degradation

rate-failure rate relationship which is determined through degradation modeling, is a

function of the maintenance program. If the degradation rate as determined from the

data on corrective maintenance and preventative maintenance implies that the failure

rate is too high or is significantly increasing, then the maintenance is ineffective. If the

failure rate is maintained at an acceptable level, then the maintenance is effective from

a reliability and risk standpoint.

The accuracy and extent to which degradation rate-failure rate relationship can be determined

are critical in demonstrating these applications. These applications can provide important inputs in

maintenance decisions and aging evaluations, because in the past, degradations and maintenances have

not been explicitly related to the failure rate, except in special cases.

4-12
S. EXTENSIONS OF DEGRADATION MODELING: THEANALYSIS OF PROGRESSIONS OF

DEGRADED STATES

5.1 Backsyround and Basic Concepts

The previous degradation models' that were developed involved integral equations. The

integral equations expressed the degradation rate and failure rate (failure frequency) of a component

at a given time in terms of the degradation rates and failure rates at earlier times multiplied by

appropriate transition probabilities. These integral equations, which are basically balance equations,

were used to obtain solutions for the degradation rate and failure rate for given cases. The

degradation rate and failure rate were also used to define measures of maintenance effectiveness.

Our present work extends the previous degradation modeling by presenting models which

explicitly show the reliability effects of different maintenance and test intervals, different

maintenance and test efficiencies, and different repair times. This work also allows the reliability

effects of different maintenance programs including minimal maintenance programs and

comprehensive condition monitoring programs to be evaluated. The extended degradation models

also allow multiple degraded states instead of a single degraded state. For example, the multiple

degraded states can be associated with different degrees of degradation or a given degraded state

can be associated with a particular piecepart of the component failing or degrading. A given

degraded state can also be associated with a particular degradation mechanism or failure mechanism.

The extended degradation models, which are presented, include the previous models as

special cases. However, the extended models are however presented in terms of differential

equations. The previous models can also be presented in terms of differential equation models by

simply differentiating the integral equations. The differential equation approach allows the models

to be extended to cover multiple degraded states and to include detailed test, maintenance, and

repair characteristics. The differential equation approach also allows standard and powerful Markov

model approaches to be applied to obtain specific results when particular data are input.

5-1
5.2 Basic Approach

A straightforward, yet powerful approach for analyzing aging and degradation is to analyze

degraded state progressions. This analysis consists of two steps: 1) classifying components' degraded

states and 2) identifying transition rates between the individual states. The transition rates give the

rate of transfer from one state to another. From the transition rates, component reliability

characteristics as a function of age can then be determined by standard Markov state modeling

techniques 2 -

4

, but applied to degradation analyses. This report describes the approaches as applied

to degradation analyses.

In the simplest case, which is generally modeled in probabilistic risk analyses (PRAs), the

component is assumed to have only two states, a failed state and a successful operating state;

intermediate degraded states are not explicitly modeled. Let 0 denote the operating state and 1

denote the failed state. The transition matrix, which contains the transition rates by which the

component can transfer from 0 to 1 and vice versa, is then constructed.

For the two-state model (1=failed, O=operating), the transition matrix is the following.

. Final

- State 0 1

Initial

State

0 0 X

. 1 0

R~ 

5-2
The quantity, A, is the component failure rate, which is the rate for transferring from an

operating state (0) to a failed state (1) in a given unit of time. The quantity, , is the repair rate,

which is the rate for transferring from a failed state (1) back to an operating state (0) in a unit of

time. The entries in the (0, 0) and (1, 1) positions of the transition matrix are zero since there is no

change in state.

The failure rate, A, and repair rate, p, or their equivalents, are estimated to determine the

associated component reliability characteristics. The failure rate, A, can be estimated as one over the

mean time to failure TFp

(5.1)

TF

where TF is estimated as the average of the observed times to failure. The repair rate, p, can be

estimated as one over the mean time to repair TR,

A T - . (52)

TR

where TR is estimated as the average of the observed repair times.

Once the failure rate and repair rate are estimated, the component unavailability and other

reliability quantities can be determined using state balance equations (which are termed

Kolmogorov's equations). For example, the probability, p(t), that the component is in a failed state

at time t+dt, which is the component unavailability, is given by the balance equation,

p(t+dt) = p(t)(1-jpdt) (5.3)

+ (1-p(t)) dt

5-3
This balance equation can be expressed in words as:

Probability that

the component is

down at t+dt

- Probability that

the component is

down at t

Probability that

repair is not

completed in t to t+dt

+ Probability that

the component is

up at t

Probability that

the component fails

in t to t+dt (5.4)

As observed in the above equation, the repair rate, IL, is such that when multiplied by a small

increment of time, dt gives the probability of repair being completed in that increment of time when

the component is down. One minus this quantity, i.e. 1-ILdt, gives the probability of the repair not

being completed. The failure rate, A, is such that when multiplied by a small increment of time, i.e.,

Adt, gives the probability of the component failing in the increment of time given the component is

initially up.

Expanding p(t+dt) one then obtains:

p(t) dP dt = p(t) - p(t )Idt +Adt - p(t)Adt dt

or

(5.5)

(5.6) dp +(P+A)p=A

dt

54
which is a standard first-order differential equation and can be solved by standard techniques to give:

pWt = A (1-e~(+ )).. . ....................... (5.7)

In PRAs, the steady-state or asymptotic value is used, where t goes to infinity ():

Xco) A -:--(5.8) i+T

53 Incorporation of a Component Dezraded State

In addition to the success state and failed state, assume that one degraded stated also is

considered for the component. Let the three states be denoted as 0,1,2:

0= operating state (good as new state) (5.9)

1= degraded state (a single degraded state) (5.10)

2= failed state (5.11)

The transition matrix can then be constructed as:

Initial Final

State State 0 1

0 0 1 X02

1 ~1o 11 12

*2 20 01 

5-5
The transition rates are defined by:

Ao: the transition rate of going from an operating state (0)

to a degraded state (1) (5.12)

A02: the transition rate of going from an operating state (0)

to a failed state (2) without intermediate degradation

being detected (5.13)

A10: the transition rate of maintenance correcting a degraded

state (1) to an operating state (0) (5.14)

Al1 : the transition rate of going from a degraded state (1) to

another degraded state () without intermediate failure (5.15)

A1 2: the transition rate of going from a degraded state (1)

to a failed state (2) (5.16)

A20: the transition rate of repair from a failed state (2) to

an operating state (0) (5.17)

A21: the transition rate of repair from a failed state (2) to

a degraded state (1) (5.18)

5-6
.:Note that we included a transition rate, ,I which allows transitions from a degraded state to

another degraded state; for example after maintenance has been performed.

For application, the transition rates need to be translated into quantities which can be

interpreted by engineers and can be estimated from data or engineering experience. One such

translation is to define the transition rates in times of average times of occurrences:

A1 I o, average' time to a degraded state from an operational state (5.19)

.l2= . T ;T = average time to catastrophic failure from a operational state without

T02

intermediate degradation occurring (or being detected) (5.20)

110 -;Tio = average maintenance detection time plus duration time for restoring the

TIO

degraded state (1) to as good as new state (0) (5.21)

ll j-iTtl = average time from a degradation occurrence to another degradation Til

occurrence (without an intervening catastrophic failure or replacement

occurring) (5.22)

~~1i7 ~ ~ .- 

A12 T ' 12 = average time from a degradation occurrence to a catastrophic failure

T12

occurrence without the degradation being corrected (5.23)

120 T20 = average repair time in which as failed state is restored to as good as new

T20

state (5.24)

5-7
A21 =-1 ;T21 = average repair time in which a failed state is restored to a degraded

T21

state. (5.25)

These average times, (To,, T02, etc.,) can then be estimated from data or from engineering

experience. The above transition rate definitions can also be specialized for given situations, such as:

A02 = 0:

A10 = 0:

A1 = 0:

A21 = 0:

all catastrophic failures are preceded by a degradation

maintenance restores the component only to a degraded state

(as good as old)

maintenance restores the component to as good as new

repair restores the component to as good as new.

(5.26)

(5.27)

(5.28)

(5.29)

.. . . .

Also, the average times (Tol, TM, etc.,) can be expanded to identify more detailed factors and

contributions:

A PMO (5.30)

TIM +TM

TIM = average maintenance detection time

TM = average maintenance duration

PMO = the fraction of time that the maintenance restores the degraded state to as good as new

11 TIM+TM+T.MI (5.31)

5-8
TIM

TM

TMI

= average time from a degradation occurrence to a maintenance action

= average maintenance duration

= average time from a maintenance completion to a degradation occurrence without

intervening catastrophic failure

.12= 1

TIM + TM + TM2

(532)

''. - ' .. S .. ::'' . 2 .';' . :I ',- . , o - , X , 

TM2 = average time from a maintenance to a catastrophic failure without another degradation

occurring or being detected. (TIM and TM defined as above)

P2D

T20=-

TR

TR = average repair time

P20= fraction of time that repair restores the component to as good as new

P21

TR

P21 =fraction of time that the repair restores the component to a degraded state

(5.33)

(534)

The above quantities can be estimated from maintenance and failure data and from

maintenance procedures. The reliability characteristics of the component, including the age

5-9
dependent failure rate and age-dependent unavailability, can then be determined using balance

equations and standard differential equation techniques. These reliability characteristics, in turn, can

be used in reliability and risk evaluations. Because the transition rates are expressed in terms of

basic degradation and maintenance characteristics, there is a direct relationship between the

degradation characteristics, the maintenance program characteristics, and the resulting reliability and

risk implications.

S.4 Incorporation of a Maintenance State

To explicitly identify when the component is in maintenance, a maintenance state can be

defined and can be added to the transition matrix. Let the maintenance state be denoted by M.

The transition matrix is given below showing the added transition rates involving the maintenance

state M.

Initial Final

State State 0 1 M 2

0 XOM

1 XEM

M XMO AMl XM2

2

5-10
The additional maintenance transition rates are given by:

AOM = transition rate for an operational component being placed in maintenance (this could be

considered as scheduled preventive maintenance, or can be set to zero if such occurrences are rare)

(535)

AIM = transition rate for going from a degraded state to a maintenance state (5.36)

= T ;TIM = average time from a degradation occurrence to a maintenance action

IM

AMO = transition rate for going from a maintenance state to an operational state (5.37)

= PMO (5.38)

TM

PMO = fraction of time the maintenance restores the component to as good as new

TM = average maintenance duration

A E M. (539) TM

PMI = fraction of time that the maintenance restores the component to a degraded state

A M2 (5.40)

T M

PM2 = fraction of time the maintenance causes the component to be failed,

5-11
5.5 Resolvin2 Additional Dearadation States

For additional degraded states 1, 2... k, one must now define the transition rates from the

other states to 1, 2,...,k, the transition rates among 1, 2,...,k, and the transition rates to the other

states. In terms of the transition matrix, this amounts to incorporating appropriate additional

columns and rows for the degraded states:

Initial Final

State State 0 1 2 .... k k+1

0 0 X01 x02 .... XOk XOk+1

1 10 11 12 . k1 Xlk+1

2 >20 21 k22 .... 2k %2k+1

k XkO %c1 ?9.2.*. Xkc kkk+1

- k+1 +kB10 4k+11 ,Xk+12 .... Xk+lk 0

In the above matrix, 0 is the operational state with no degradation, and k+1 is the failed

state. The additional transition rates need to be estimated by expressing them in terms of average

times of occurrences of events and fractions of times specific outcomes occur.

5.6 Incorporatinz Effects or Surveillance Tests

When failures are not detected until surveillance tests are performed, then the transition

rate, 120, from a failed state (2) to an-operational state (0) can explicitly account for surveillance tests

5-12
which are performed. If the transition rate, L2., is defined in terms of the average time from a failed

state to an operation state then:

-

1 A20 = T2;T0

T20

= the average time from a failed state to an operational state (5.41)

Now T20 can be expressed as

T20 = TU + TR (5.42)

where,

TU = average time to detect the failure (the average undetected time)

and

TR = average repair time

(5.43)

(5.44)

For a constant failure rate (i.e. constant transition rate), when the comj

equally likely to fail in the interval, T, between tests. Then:

ponent does fail, it is

A, I . .

TU = T/2

,; ,; ,. .: , .

(5.45) " . ' I.I I ' '';; - .,, ! I , I

5-13
- - -A*~~~~~~~~

Hence,

T20 = T + TR (5.46)

and

T2m TR (5.47)

Thus, the transition rate, A.., explicitly contains the surveillance test interval. Consequently,

the component failure probabilities and component unavailabilities, which are determined from the

transition rates, contain the effects of the surveillance test. It should be noted that the above

formulas, including the test intervals, also apply to a general multistate model as described in the

previous section, where A20 is replaced by the more general transition rate, Ak+lO, and where k+1 is

the failed state.

5.7 Calculation of Component Reliability Characteristics ncludine the ComDonent

Unavailability

As we indicated, using the transition matrix, the probability that the component is in any

given state at a given time can be determined. These state probabilities allow all the component

reliability characteristics to be determined including the component failure rate and the component

unavailability. This section reviews the basic approaches that are used to determine the state. Let

p1(t) be the probability that the component is in state i at a given time, where i is any state such as

the operational state, a degraded state, or a failed state. Then, the general balance equation for

p1(t+dt) is:

5-14
p,(t+dt) pj(t)Ajdt+p(t)(1-dt) . . (5.48)

where Aji is the transition rate from state j to state i and the sum is over all states j that can transfer . . ...--.C-;........;*...;

to i. The rate Ai is the sum of all transition rates out of state i,

l = L Aik , .~ ., ,;. (5.49)

k

.* . , .. ,- i ; i . , , ,

Expanding p,(t+dt) to first order, Equation (2.49) becomes the first order differential equation,

...... .... (5.50).

dt '~pfl ' p~)j 

, . , - ., - -, - ...- . . . -;. .: , 

This set of differential equations for all the states i can be solved by standard differential

equation techniques. The results are the state probabilities p1(t) as a function of time t for all states.

The unavailability of the component, in particular, is the state probability p2(t) (or pk+I(t) in

the more general case), which is the probability that the component is in a failed state at time t.

The component failure frequency is the probability that the component is in an operational state and

then fails per unit time. Hence, the failure frequency is given by p0 (t)102, where p0 (t) is the

probability of being in an operation state, and A02 is the transition rate of going to a failed state.

The probability that the component is in a degraded state of time, t, is given by qd(t), where d is the

degraded state identifier. Other reliability characteristics can be determined similarly and can then

be used as inputs to reliability and risk evaluations.

5.8 Incorporation of Aping Effects

Whenever degraded states are identified and included in the transition matrix, the resulting

component failure rate, which includes the transitions through the degraded states, is age-dependent.

5-15
The failure rate is age-dependent even when the individual transition rates are constant and time

independent. Thus, by classifying degraded states and defining the associated transition rates, one

has a straightforward way of analyzing aging effects on the component failure rate and other

reliability characteristics. The transition rates also include degradation characteristics and test and

maintenance program characteristics; thus, reliability is directly related to these characteristics.

The transition rates can also be modified to analyze the effects of different aging

management programs. For example, if the test and maintenance program does not replace

degraded parts, but only assures that the component is operational, then the transition rate for

restoring the component to as good as new can be taken to be zero (i.e., Ato = 0, where 1 is the

degraded state, and 0 is the good as new, operational state). This situation represents a minimal

maintenance policy and resulting component reliability characteristics can be determined for given

degradation characteristics. Additional maintenance activities can be analyzed by modifying the

maintenance characteristics and associated transition rates. The effects of maintenance can be

modeled in more detail by classifying different degraded states and defining the transitions

associated with the maintenance procedures. Furthermore, pieceparts of the component can

furthermore be individually modeled by defining transition rates for each piece or part and obtaining

the resulting state probabilities.

5-16
6. SUMMARY '

Our previous report' describes the basic concepts and applications of degradationmodeling for

aging analysis of standby active components. This report presents more applications and extensions

of degradation modeling approaches'to study the effects of aging in components and the role of

maintenance in controlling the effects of aging. Here, degradation modeling approaches are applied in

studying the aging effects and maintenance effectiveness of continuously operating components (air

compressors). ''Sensitivity evaluations are performed to study the effects of uncertainties on the

degradation modeling results, and also; the mathematical 'modeling is extended for studying reliability

effects of 'maintenance' strategies and for interfacing with probabilistic risk assessment '(PRA)

evaluations.' '

a) Application of Degradation Modeiing to a' Continuously Operating Component

The application of degradation modeling approaches to a continuously operating component (air

compressors) shows the usefulness of this modeling approach in studying aging effects and the role of

maintenance in'this type of component. Analyses of degradation and failure data of air compressors

using degradation modeling approaches show'that aging effects are evident in both degradation'and

failure occurrences. In this case, both rates show aging effects; however, the faster increase in the

failure rte compared to the degradation rate indicates the'ineffectiveness of maintenance, which is

reflected in the evaluation of maintenance effectiveness. 'The decline in maintenance effectiveness with

age signifies that maintenance' is effective in preventing age-reiaied degradations from failures.

b) Sensitivity Analyses of Degradation Modeliig Results

Sensitivity evaluations were performed to evaluate the'effect of three factors: a) engineering

evaluation of failure data, i.e., subjectivity in classifying degraded vs. failure state of a component, b)

uncertainty in degradation occurrence time available from plant records, and c) the effect of test

frequency. Results of sensitivity evaluation show that the effects of data partitioning is not significant.

The subjectivity involved in the data evaluation does not change the overall trend in the results.

6-11
Similarly, the effect of uncertainty in degradation occurrence time is also not significant. However, we

observed that the estimated degradation rates can be influenced by the frequency at which tests are

performed to detect degradations or failures. The effect of test frequency is probably pronounced

because of the single degraded state in the modeling. The extensions of degradation modeling presented

in this report includes test frequency and also, multiple degraded states.

c) Relation between Degradation and Failure Frequency

Understanding the relationship between degradations and failures is an important aspect in the

degradation modeling approaches. Knowledge of relationships between degradations and failures will

help define the maintenance activities necessary for preventing degradation-caused failures and can be

used in risk-evaluations of aging. In this report, an event-count based approach to data analysis is

presented to study correlations between degradation and failure frequencies. We used this approach

to discover if there were delayed effects of degradations on failures. For the specific component studies

(RHR pumps), a lag-time of 2 years was observed between degradation and failure occurrences.

Existence of such lag-times, which are expected to be component specific, can be beneficial for deciding

the maintenance activities that are necessary to mitigate the effects of aging. Additional applications

will be needed to demonstrate the validity of the existence of time-lag between degradations and failures.

d) Extensions of Degradation Modeling

Our present work extends the previous degradation modeling by presenting models which

explicitly show the reliability effects of different maintenance and test intervals, different maintenance

and test efficiencies, and different repair times. This extension will allow us to evaluate the reliability

effects of different maintenance programs.

6-2
REFERENCES

1. P.K. Samanta et al., Degradation Modeling with Application to Aging and Maintenance

Effectiveness Evaluation," NUREG/CR-5612, BNL-NUREG-52252, March 1991.

2. A.T. Bharucha-Reid, Elements of the Theory of Markov Processes and Their Applications.

McGraw Hill, New York, 1960.

3. H.L Tijms, Stochastic Modeling and Analysis John Wiley and Sons, New York, 1986.

4. L. Kleinrock, Queueing Systems. Vols 1 and 2, John Wiley and Sons, New York, 1975.

5. M. Villaran et al., Aging Assessment of Instrument Air Systems in Nuclear Power Plants,

NUREG/CR-5419, BNL-NUREG-52212, January 1990.

R-1
APPENDIX A: AGING DATA EVALUATION OF AIR COMPRESSORS

A.1 Brief Description of Air Compressors

Air compressors used in instrument and service air systems in nuclear power plants are either

positive displacement or nonpositive displacement types, called continuous flow and dynamic

compressors. The reciprocating-piston compressor is the most common positive displacement type

because of its high-pressure capability, ability to dissipate the heat of compression, and versatility. Air

is compressed by the alternate filling and compression of a cylinder by the reciprocating motion of a

' ~ ~ , ~ . ~ .,. ~ ., ( - , .4 . . --

piston. The rotary motion of the crankshaft, driven by and electric motor, diesel, or some other prime

mover, is translated via the connecting rod into the reciprocating motion of the piston within the

cylinder. On the intake stroke, the piston moving downward in the cylinder creates a negative pressure

across the spring-loaded intake valve causing it to open. Intake air is drawn through the filter/silencer

into the cylinder. When the piston reaches the bottom of its stroke, the differential pressure across the

intake valve is less than the spring force pushing to close the valve. Therefore, the intake valve closes

and the compression portion of the cycle begins. As the piston moves upward into the cylinder bore,

reducing the volume as it travels, air pressure and temperature increase. When the pressure differential

across the discharge valve exceeds the spring pressure holding it closed, the valve opens. The volume

of hot compressed air is then driven into the system via the discharge manifold as the piston continues

to the top of its stroke. Once the piston reaches the top of its stroke, the differential pressure across

the discharge valve drops below the closing force of the spring and the valve closes, completing the cycle.

A.2 Engineering Evaluation of Air Compressor Aging Data

Testing and maintenance records on air compressors were analyzed to obtain the aging data for

degradation modeling. In this evaluation, the first step was to identify whether the condition of the

component indicated an age-related problem. Every test and maintenance record was inspected and

'The description of air compressors is reproduced from Villaran et al.5.

A-1
categorized as defined in Table A.1. The age-related problems are classified as N, D, or F; non-aging

problems include human errors in performing the maintenance or testings (H), affecting component

performance (E). Table A.2 illustrates typical examples of each category for an instrument air

compressor, and Table A.3 presents a typical failure mode and effect analysis defining various levels of

degradation (low, intermediate, and high).

Table A.1. Categorization of Component Failure or Maintenance Data

Aging-Related Problems

N: No component degradation. No maintenance was performed.

D: Definite degradation in the component. Maintenance was performed to repair the

degraded condition.

F: Severe degradation in the component. Immediate maintenance was required.

Non-Aging-Related Problems

H: Component degradation due to a human error.

E: Degradation of components other than the prime component being evaluated.

Maintenance was performed to correct the degradation, which if not corrected, may

have had a deleterious effect on the prime component.

A-2
Table A.2. Analysis of Maintenance Log for an Instrument Air Compressor

Primary Failure

Equipment No. Date - - Description Subcomponent -Classification

1K107A 4/07/80 Quarterly PM: outer bearings Bearings D

greased with EP #2

1K107A 12/10/87 Inst. air cylinder leaking. ' Mechanical F

Replaced worn oil wiper

. ring. The shaft was also _

worn.

1K107A - -4/20/87 Semi-annual I.A. P.M. - ----- Motor - N

inspected and cleaned motor.'

No loose bblts,-cracks,'worn

parts, or excessive grease. All . satisfactory.

2K108B 5/6/85 .Compressor has excessive oil Mechanical H

leak. Removed and

reassembled packing, it was

installed backwards.

Operates satisfactorily. - _ 

2K107B ' 5/4/83 Monthly P.M: Replaced Filter E

'intake filter.'

.---  1. . I - . - I I ---  - - I -_ - - - 1. . .1 ..

I . .. . .. . . 1 . .

- .-.- . - - . ._. - - I  . ... . . .

. I I . , ,

A-3
Table A3. Typical Examples of Degradation Levels and Failure Mode and Effect on Compressors

Compressor Failure Degradation

Subcomponent Classification Failure Effect Failure Mode Level

Bearings D Monthly preventive Low D

maintenance -

bearings greased'

Filter D Monthly P.M. - filter Low D

cleaned

Gasket D Oil leak by gasket Gasket Intermediate D

deterioration

Jacket Heat D Corrosion deposits Mechanical Intermediate D

Exchanger built up by aftercooler debris; poor water

chemistry

Bolts and D - Fractured stud on Mechanical High D

Fasteners spacer vibration

Pistons D Brass filings in high Mechanical wear High D

and low pressure

regions found during

P.M.

Piston F Oil leak at piston rod Mechanical wear F

seal

Lube Oil F Pump seized and High temperature, F

System became inoperable mechanical wear

A.3 Agina Data on Air Compressors

Aging data on air compressors are obtained by analyzing plant maintenance records for two

units. Since the units are different ages, data for each unit covered different periods: 36 quarters for

unit one and 20 quarters for unit two (Table A.4). This table contains the observed dates for

degradations and failures. Table A.5 presents the failure data where only the observed failure times are

recorded. As these tables show, significantly more information on component aging is obtained by

focussing on degradation data.

A-4
i

Table A.4. Air Compressor Aing Data: Degradation and Failure Times

(2 Nuclear Units)

Mo Dv 'Yr Pit Comp SEVTY DSCP

1 24 80 1 1 D BRGS

4 7 80 1 1 D' BRGS

7 25 80 1 1 D BRGS

9 2 80 1 1 D BRGS

12 3 80 1' 1' D MECH

12 30 80 1 1 D BRGS

3 17 81 I1 D JKTHX

4 24 81 1 1 D BRGS

6 22 81 1 1" D: BRGS

8 24 '81 1 : 1 D: COMPR

9 29 81 1 .:,1- D. BRGS

11- S 81 1 1 D FLTR

1 4 82 1 1- D' BRGS

1 25 82 1 -1 D COMPR

4 14 82 1 1 D MECH

1 20 83 1' -1- F COMPR

3 9 83 1 '11 D COMPR.

1 19 84 11 F COMPR

4 30 84 1 I D COMPR

11 16 84 1:1 D MECH

I 17 85 1:. 1 F COMPR

7 12 85 1 -1 D UNLOA

7 15 85 1 1- F UNLOA

12 27 85 1. 1 F COMPR

4 17 -. '86 1 1 D MOTOR

iV- 24 86 1 1 D MECH

1 5 87 1-:1 D LUBOIL

3 9 87 1 1 F LUBOIL

7 2 :87 1.1 D COMPR

i2 10 ; 87 1 1; F COMPR

12 17 87 1.IL1 D COMPR

2 15 ^88 1-1 D GASKE

4 1 88 1 I D GASKE

6 21 88 1- 1 D UNLOA

I 14 88 1 1 D GASKE

11 26 79 1 ;2 D COMPR

1 13 80 1 2 D MECH

1 24 -80 1 2 D BRGS

1 31 >80 1 ':2 D COMPR

4 7 80 1 2..2 'D BRGS

7 25 80 1i" 2 D *BRGS

12 30 So 1.;2 D -BRGS

1 27 81 D 2 F LUBOIL

4 24 81 1 2 D BRGS

T Ti Ap

0.64 0.64 1

0.81 1.46 1

1.20 2.66 1

0.41 --3.07 1

1.01 4.08 1

,0.30 4.38 1

0.86 5.23 1

0.41 5.64 1

:0.64 6.29 1

0.69 6.98 1

:0.39 7.37 1

0.40 7.77 1

0.66 8.42 1

0.23 8.66 1

0.88 9.53 1

3.07 -12.60 1

*054 13.14 1

3.44 16.59 1

1.12 17.71 1

2.18 19.89 1

0.68 20.57 2

.1.94 2251 2

-0.03 22.54 2

1.80 24.34 2

1.22 2557 2

2.41 .27.98 2

0.46 -28.43 2

-0.71 29.14 2

1.26 30.40 2

1.76 32.16 2

0.08 32.23 2

0.64 328 2

0.51 33.39 2

0.89 34.28 2

1.59 35.87 2

1.29 1.29 1

052 1.81 1

0.12 1.93 1

0.08 2.01 1

0.73 2.74 1

1.20 3.94 1

1.72 5.67 1

0.30 5.97 1

0.97 6.93 1

A-S
I

Table A.4. (Cont'd)

Mo Dv Yr PIt Comp SEVTY DSCP Ti Ti Acp

5 S 81 1 

2

6 22 81 1 

2

8 26 81 1 

2

9 29 81 1 

2

1 5 81 1 

2

1 4 82 1 

2

2 12 82 1 

2

2 24 82 1 

2

4 23 82 1 

2

6 21 82 1 

2

4 25 84 1 

2

5 28 84 1 

2

7 25 84 1 

2

10 17 85 1 

2

4 17 86 1 

2

8 18 86 1 

2

11 24 86 1 

2

6 11 87 1 

2

12 17 87 1 

2

9 8 88 1 

2

9 17 88 1 

2

10 2 88 1 

2

10 6 88 1 

2

10 7 88 1 

2

8 10 79 1 

3

8 14 79 1 

3

10 22 79 1 

3

2 27 80 1 

3

3 27 80 1 

3

4 7 80 1 

3

6 27 80 1 

3

7 25 80 1 

3

8 1 80 1 

3

10 18 80 1 

3

10 20 80 1 

3

12 30 80 1 

3

3 17 81 1 

3

3 27 81 1 

3

4 24 81 1 

3

6 22 81 1 

3

9 29 81 1 

3

11 5 81 1 

3

1 4 82 1 

3

2 24 82 1 

3

7 21 82 1 

3

10 21 82 1 

3

D COMPR

D BRGS

D MECH

D BRGS

D FLTR

D BRGS

F MECH

D COMPR

D MOTOR

D COMPR

D COMPR

D COMPR

F JKTHX

D COMPR

F BRGS

D MECH

D MECH

D LUBOIL

F COMPR

D COMPR

F COMPR

F UNLOA

F COMPR

D MECH

D BRGS

D COMPR

D GASKE

D BRGS

D JKTHX

D BRGS

D BRGS

D BRGS

D COMPR

D JK-IX

D MECH

D MOTOR

D LUBOIL

D FLTR

D BRGS

D MECH

D BRGS

D FLTR

D BRGS

D COMPR

D BRGS

D COMPR

0.12

0.52

0.71

0.37

0.40

0.66

0.42

0.13

0.66

0.64

738

0.37

0.63

4.91

2.00

1.34

1.07

2.19

2.07

2.90

0.10

0.17

0.04

0.01

0.10

0.04

0.76

1.39

0.33

0.11

0.89

0.31

0.07

0.86

0.02

0.78

0.86

0.11

0.30

0.64

1.08

0.40

0.66

0.56

1.63

1.00

7.06 

1

7.58 1

8.29 

1

8.66 1

9.06 

1

9.71 

1

10.13 

1

10.27 1

10.92 

1

11.57 1

18.94 1

19.31 1

19.94 1

24.86 2

26.86 2

28.20 

2

29.27 

2

31.46 

2

33.52 2

36.42 2

36.52 

2

36.69 

2

36.73 

2

36.74 

2

0.10 1

0.14 

1

0.90 

1

2.29 1

262 1

2.73 1

3.62 

1

3.93 1

4.00 1

4.86 1

4.88 1

5.66 1

6.51 

1

6.62 1

6.92 

1

7.57 1

8.64 1

9.04 1

9.70 1

10.26 1

11.89 1

12.89 1

A-6
Table AA. (Cont'd)

Mo Dv Yr Pt Cmp SEVTY DSCP -- Tij - - - Agp

.4 4

4 22

10 21

12 12

1 20

1 26

11 16

1 27

2 .27

4 30

4 17

4 27

7 23

10 2

3 31

-10, 6

7 31

8 10

10 2

10 11

2 2

2 27

3 27

4 -7

6. 24

6 27

7 .25

11 6

12 30

3 3

3 17

3 27

4 24

6 22

7 28

7 29

8 14

9 29

10 6

11 5

1 4

1,25

4.. ¢23

5 24

7 .1

7 21

.83

.83

:83

:83

84

84 :84

.85

85

,85

86

87

87

87

88

88

79

79

79

79

80

.80

80

80

80

80

80

80

*80

81

81

81

81

81

81

81

81

81

81

81

82

82

82

.82

82

82

1 3 D MECH

1 3. D COMPR

1 3 F COMPR

1 3 D MECH

1 3 D ;GASKE

1 3.- D MECH

1 3, D- FLTR

1 3 D GASKE

1 3 D MECH

1 3, , D COMPR

1 ;-.3 F BRGS

1 ,,,3 D UNLOA

1 3 D_ COMPR

1 3 . D COMPR

1 -3,g,- D MECH

1 3 D MECH

1 4 F, COMPR

1. 4 D. BRGS

1 .4 . F < LUBOIL

1 .4 D COMPR

1 -4 D COMPR

1 -4 D,- BRGS

1 4. D. JKTHX

1 4,. D, BRGS

I .4.. D LUBOIL

1 .4 - D BRGS

1 4. D. BRGS

1 4i D ,, MECH

I,,4-; D, - BRGS

I -4- D, FLTR

1 .- 4 D, LUBOIL,

1 .- 4 D, FLTR

1 -;. 4 , D,-. BRGS

1 4., D, BRGS

1 4 . D,: GASKE

1 4 D COMPR

1 .,4, D. COMPR

1 .4-, D.: BRGS

14i. D MECH -

I D; FLTR

I 4,- D. BRGS

I _4i D:. COMPR,

I .4.4 D, MOTOR

1 .4, D, MECII

I -, 4. D- MECII

1 4 D BRGS

1.81 14.70 . 1

0.20 14.90 1

1.99 16.89 1

0.57 17.46 1

0.42 17.88 1

0.07 17.94 1

3.22 21.17 2

0.79 21.96 2

033 22.29 2

0.70 22.99 2

3.86 26.84 2

4.11 30.96 2

0.96 31.91 2

0.77 32.68 2

1.99 34.67 2

2.06 36.72 2

0.68 0.68 1

0.10 0.78 1

0.58 136 I

0.10 1.46 1

1.23 2.69 1

0.28 2.97 1

0.33 3.30 1

0.11 3.41 1

0.86 4.27 . 1

0.03 4.30 1

0.31 4.61 1

1.12 5.73 1

0.60 6.33 1

0.70 7.03 1

0.16 , 7.19 1

0.11 7.30 I

0.30 7.60 1

0.64 8.24, 1

0.40 8.64 1

0.01 8.66 1

0.17 .8.82 1

0.50 932 1

0.08 9.40 1

032 9.72 I

0.66 1038 1

0.23 10.61 1

0.98 11-59 I

0.34 11.93 1

0.41 .12.34 1

0.22 12.57 I

A-7
Table A.4. (Cont'd)

Mo Dy Yr Pit Comp SEVTY DSCP ri T Agp

1 20

3 15

4 4

7 25

7 20

11 16

3 28

6 10

12 1

12 6

1 i0

3 29

4 21

1 5

7 2

2 3

7 1

7 15

3 2

5 4

7 13

8 16

10 2

4 25

6 4

6 27

12 19

7 8

1 7

10 27

2 5

2 12

6 22

11 29

2 28

3 31

3 2

6 8

12 21

3 5

5 30

10 22

2 6

7 8

11 1

5 27

83 1 4

83 1 4

83 1 4

83 1 4

84 1 4

84 1 4

85 1 4

85 1 4

85 1 4

85 1 4

86 1 4

86 1 4

86 1 4

87 1 4

87 1 4

88 1 4

88 1 4

88 1 4

83 2 1

83 2 1

83 2 1

83 2 1

83 2 1

84 2 1

84 2 1

84 2 1

84 2 1

85 2 1

86 2 1

86 2 1

87 2 1

87 2 1

87 2 1

87 2 1

88 2 1

88 2 1

83 2 *2

83 2 2

83 2 2

84 2 2

84 2 2

84 2 2

85 2 2

85 2 2

85 2 2

86 2 2

D COMPR

F BRGS

D MECH

F COMPR

D COMPR

D MECH

F MECH

D GASKE

D GASKE

D MECH

D COMPR

F MOTOR

F UNLOA

D COMPR

D UNLOA

F GASKE

D UNLOA

D COMPR

F COMPR

D COMPR

D JKT-IX

D JKTHX

D MECH

D MECH

F MECH

F MECH

F FLTR

D JKTHX

D COMPR

D UNLOA

F JKTHX

D MECH

D UNLOA

D JKTHX

D MECH

D UNLOA

F COMPR

D COMPR

D COMPR

D MECII

F COMPR

D COMPR

D MECII

D MECH

D GASKE

F GASKE

1.99

0.61

0.21

1.23

3.94

1.29

1.47

0.80

1.90

0.06

0.38

0.88

0.24

Z82

1.97

2.34

1.64

0.16

0.56

0.69

0.77

0.37

0.51

Z26

0.43

0.26

1.91

Z21

1.99

3.22

1.09

0.08

1.44

1.74

0.99

0.37

0.56

1.07

Z14

0.82

0.94

1.58

1.16

1.69

1.26

Z29

14.56 1

15.17 1

1538 1

16.61 1

20.56 2

21.84 2

23.31 2

24.11 2

26.01 2

26.07 2

26.44 2

27.32 2

27.57 2

30.39 2

32.36 2

34.70 2

36.34 2

36.50 2

0.56

1.24

ZO

2.38

2.89

5.14

5.58

5.83

7.74

9.96

11.94

15.17

16.26

16.33

17.78

19.52

20.51

20.88

0.56

1.62

3.77

4.59

5.53

7.11

8.27

9.96

11.21

13.50

A-8
Table A.4. (Cont'd)

Mo Dv Yr Pit Comp SEVTY DSCP T1 Ti Agp

1 13 87 2 2 F MECH 2.51 16.01

2 12 87 2 2 D COMPR 032 16.33

3 31 87 2 2 D MOTOR 0.54 1688

4 30 87 2 2 D UNLOA 032 17.20

7 6 87 2 2 D COMPR 0.73 17.93

4: -14 88 2- 2 -D -UNLOA -- 3.09 ---21.02

10 26 88 2 2 D. GASKE 2.13 23.16

11 8 88 2 2 D UNLOA 0.13 23.29

11 14 88 2 2 D COMPR 0.07 23.36

2 29 84 2 3 D COMPR 0.36 0.36

4 16 84 2 3 D -COMPR 0.52 0.88

7 20 84 2 3 D. MECH 1.04 1.92

7 8 85 2 3 D. MECH. 3.87 5.79

1 9 88 2 3 D GASKE 10.01 15.80

6 4 88 2 3 D -GASKE 1.61 ' 17.41

11 26 88 2 3 D _,MECH. 191 1932

10 23 84 2 4 F- - COMPR 2.72 2.72

-3 24 85 2 4 F MOTOR 1.68 4.40.

4 19 85 2 4 D -UNLOA 0.28 4.68

5 6 85 2 4 D .CtMPR* 0.19 4.87

-7 8 85 2 4 D UNLOA 0.69 5.56

1 9 86 2 4 D GASKE 2.01 7.57

1 14 86 2 4 D UNLOA 0.06 7.62

8 8 86 2 4 D COMPR 227 9.89

1 29 *87 2 4 D MOTOR 1.90 11.79

10 3 87 2 '4 D UNLOA 2.71 14.50

- 1 27 88 2 4 D -COMPR 1.27 15.77

7 27 88 2 4 D UNLOA 2.00 17.77

NOTE:

BRGS - bearings

MECH - mechanical

JKTHX - jacket heat exchanger

COMPR - compressor - -.

UNLOA - unloader

GASKE - gasket .-. - .

LUBOIL - lubrication oil

FLTR - filter r- - -.

A-9
T- Time intervals of observed events

r - Age at which an event is observed

Yi - Reciprocal of T4

Table A.5. Compressor Aging Failure Data (Unit 1; 4 air compressors)

Mo Dv Yr Pit Com SEVTY DSCP Fij FtI Yi LaYi

7 31 79 1 4

10 2 79 1 4

1 27 81 1 2

2 12 82 1 2

1 20 83 1 1

3 15 83 1 4

1 19 84 1 1

7 25 83 1 4

10 21 83 1 3

7 25 84 t 2

1 17 85 1 1

7 15 85 1 1

3 28 85 1 4

12 27 85 1 1

4 17 86 1 3

4 17 86 1 2

3 29 86 1 4

4 21 86 1 4

3 9 87 1 1

12 10 87 1 1

12 17 87 1 2

2 3 88 t 4

9 17 88 1 2

10 2 8 1 2

10 6 88 1 2

F COMPR

F LU3OIL

F LUBOIL

F MECH

F COMPR

F BRGS

F COMPR

F COMPR

F COMPR

F JKTHX

F COMPR

F UNLOA

F MECH

F COMPR

F BRGS

F BRGS

F MOTOR

F UNLOA

F LUBOIL

F COMPR

F COMPR

F GASKE

F COMPR

F UNLOA

F COMPR

0.68

0.68

5.97

4.16

12.6

13.81

3.99

1.44

16.89

9.81

3.98

1.97

6.7

18

9.95

6.92

4.01

0.25

4.8

3.02

6.66

7.13

3

0.17

0.04

0.68

1.36

5.97

10.13

12.6

15.17

16.59

16.61

16.89

19.94

20.57

22.54

23.31

24.34

26.84

26.86

27.32

27.57

29.14

3216

33.52

34.7

36.52

36.69

36.73

1.47

1.47

0.17

0.24

0.08

0.07

0.25

0.69

0.06

0.10

0.25

0.51

0.15

0.56

0.10

0.14

0.25

4.00

0.21

0.33

0.15

0.14

0.33

5.88

25.00

0386

0.386

.1.787

-1.426

-2534

-2.625

-1384

-0.365

-2.827

*2.283

*1.381

-0.678

-1.902

-0-588

-2.298

-1.934

-1.339

1386

-1.569

-1.105

-1.896

*1.964

-1.099

1.772

3.219

FT*, - Time intervals between observed failures

FT; - Age at which an event (failure) is observed

FY; - Reciprocal of FT6

A-10
AA Stntisticul Test Results for Combining of Air Compressors Awns D ta

The Mann-Whitney test for the aging data of air compressors of two available nuclear units

rejected the null hypothesis of identical samples (based on comparison of average ranks and significance

level of 0.05).' Table A.6 presents the results for'the-air compressors in the two nuclear units. Similarly, , - ~I - -

the test was carried out for. the four air compressors within unit one, where the null hypothesis of

identical samples was not rejected.

Therefore, the degradation behavior of the four air compressors in unit one belonged to the

;same populatio'n, and'they' iWere combined'to build'the data base for air compressors. -Table A.7

presents the test. results for combining the four components in unit one.'

Table A.6. M-W Test Results for 2 Units

Air Compressor Aging Data

Average # of Average #, of

Rank of Valves of Rank of' Valves of

Comparison 1st 1st . .2nd. . 2nd Total Test

of Samples Samples Sample ' Samples Sample obs Statistic Z P

,____________ __ ________ ._____ _ x__ . 1.m rx X,,Value

plant 1 97.617 154 127.18 56 210 3.116 0.00183

plant 2 . ' 

A-11
-------------

Table A.7. Statistical Test Results for Air Compressors in Unit 1

Average # of Average # of

Rank of Valves Rank of Valves of Test

Comparison - 1st of 1st 2nd 2nd Total Statistics a.

of Samples Sample Sample Sample Samples obs z Value

component 1- 36.89 35 31.97 33 68 -1.019 0.3083

component 2

component 1- 39.21 35 34.96 38 73 -0.85 0.395

component 3

component 1- 48.96 35 36.93 48 - 83 -1.241 0.25

component 4

component 2- 35.73 33 36.24 38 71 0.098 0.92

component 3

component 2- 43.89 33 39.01 48 81 -0913 0.361

component 4

component 3- 46.71 38 40.96 48 86 -1.057 0.291

component 4

AS Regression Annlvsis to Ohtnin Aing Rates

For the age-groups showing significant trend with time, regression analysis are performed to

obtain the aging rates. For degradation data, decreasing trend is defined for the 0-20 quarters, and

increasing trend is defined for the remaining life: 20-40 quarters. The degradation and failure rate

parameters, a and b, are presented in Table A.8 and A.9, respectively.

A-12
Table A.8. Estimated Results for Degradation Rate Analysis

(4 compressors; data combined)

Data Use Age Intervals Aging Rate b | Constant na _ Model

Method

Estimated Significant Uncertainty Estimated Significant Uncertainty Significant Standard

Parameter Level (5% error) Parameter Level (5% error) Level Error of

Estimate

Data 0-20 -0.071 0.0003 CL: -0.107 1.33 0.0001 CL: 0.986 0.0003 0.945

Combining (quarters) _ CU: -0.0337 CU: 1.679

20-40 0.06 0.073 CL: 0.005 -1.626 0.074 CL: -3.423 0.0735 1.046

(quarters) . CU: 0. 15 CU: 0.17

Table A.9. Estimated Results for Failure Rate Analysis

Data Use Age Aging Rate b Constant na | Model l

Method Intervals

Estimated Significant Uncertainty Estimated Significant Uncertainty Significant Standard

Parameter Level (5% error) Parameter Level (5% error) Level Error of

Estimate

Data 0-15 -0.233 0.024 CL: -0.409 0.435 0.386 CL: -0.934 0.025 0.584

Combining (quarters) CU: -0.055 CU: 1.804 |

15-20 0.1012 0.035 C- 0.0078 -3.696 0.007 CL -6.22 0.035 1.398

(quarters) CU: 0.1946 CU: 1.163

4-

w~
APPENDIX B: STATISTICAL RESULTS FOR SENSITVITY ANALYSIS IN

PARTITIONING OF AIR COMPRESSOR DATA

B.1 Database Obtained from Sensitivity Partitioning of Comuressor Azine Data

Table B.1 presents the aging data on air compressors, based on the sensitivity data partitioning

of failure severity. The failure data partitioned from degradations were expressed by a character variable

"CRITC" (the "" shows a partition from degradation to failure, and the "&" shows a partition from

failure to degradation). Table B.2 gives the aging failure data set, obtained from the data in Table B.1.

B.2 Statistical Test Results for Data Combining Usins partitioned Aine Failure Data

The Mann-Whitney test for the aging data of the four compressors in unit one was conducted

using the data set obtained after the sensitivity data partitioning. The null hypothesis of identical

samples (based on comparison of average ranks and significance level of 0.05) was not rejected. Table

B3 presents the results for the statistical tests.

.4.. . .

B-i~~~~~~~~~~~
Table B.1. Compressor Aging Data for Sensitivity Partitioning

Mo Dv Yr Pit Como SEVTY CRTC DSCP i3 Ti Frij Fi

1 24

4 7

7 25

9 2

12 3

12 30

3 17

4 24

6 22

8 24

9 29

11 5

1 4

1 25

4 14

1 20

3 9

1 19

4 30

11 16

1 17

7 12

7 15

12 27

4 17

11 24

1 5

3 9

7 2

12 10

12 17

2 15

4 1

6 21

11 14

11 26

1 13

1 24

1 31

4 7

7 25

12 30

1 27

4 24

5 5

80 1 1

80 1 1

80 1 1

80 1 1

80 1 1

80 1 1

81 1 1

81 1 1

81 1 1

81 1 1

81 1 1

81 1 1

82. 1 1

82 1 1

82 1 1

83 1 1

83 1 1

84 1 1

84 1 1

84 1 1

8S 1 1

85 1 1

85 1 1

85 1 1

86 1 1

86 1 1

87 1 1

87 1 1

87 1 1

87 1 1

87 1 1

88 1 1

88 1 1

88 1 1

88 1 1

79 1 2

80 1 2

80 1 2

80 1 2

80 1 2

80 1 2

80 1 2

81 1 2

81 1 2

81 1 2

D BRGS

D BRGS

D BRGS

D BRGS

D MECH

D BRGS

D JKTHX

D BRGS

D BRGS

D COMPR

D BRGS

D FLTR

D BRGS

F * COMPR

D MECH

F COMPR

D COMPR

F COMPR

D COMPR

D MECH

F COMPR

D UNLOA

F UNLOA

F COMPR

D MOTOR

D MECH

D LUBOIL

F LUBOIL

D COMPR

F COMPR

D COMPR

F * GASKE

D GASKE

D UNLOA

F * GASKE

D COMPR

D MECH

D BRGS

D COMPR

D BRGS

D BRGS

D BRGS

F LUBOIL

D BRGS

D COMPR

0.64 0.64

0.81 1.46

1.20 2.66

0.41 3.07

1.01 4.08

0.30 4.38

0.86 5.23

0.41 5.64

0.64 6.29

0.69 6.98

0.39 7.37

0.40 7.77

0.66 8.42

0.23 8.66 8.66 8.66

0.88 9.53

3.07 12.60 3.94449 12.6

0.54 13.14

3.44 16.59 3.99 16.59

1.12 17.71

2.18 19.89

0.68 20.57 3.98 20.57

1.94 22.51

0.03 22.54 1.97 22.54

1.80 24.34 1.8 24.34

1.22 25.57

2.41 27.98

0.46 28.43

0.71 29.14 4.8 29.14

1.26 30.40

1.76 32.16 3.02 32.16

0.08 32.23

0.64 32.88 0.72 32.88

0.51 3339

0.89 34.28

1.59 35.87 2.99 35.87

1.29 1.29

0.52 1.81

0.12 1.93

0.08 2.01

0.73 2.74

1.20 3.94

1.72 5.67

0.30 5.97 5.97 5.97

0.97 6.93

0.12 7.06

B-2
Table B.1. (Cont'd)

''Mo Dv Yr P-'-PIt Comp' SEVTY CRTC--DSCP T - Flri i-- rti

. 6 22 81 .1 2

8 26 81 1 2

9 29 81 1 2

11 5 81 -1 2

1 4 82 ,1 2

2 12 82 1 2

2 24 82 1 .2

4 23 82 1 2;

6 21 82 1 2

4 25 84 1 2.

5 28 84 1 2

7 25 84 1 2

10 17 85 1 2

4 17 86 1 2-

8 18 - 86 1 2

11 24 86 1 ,2

6 11 87 1 2

12 17 87 1 2

9 8 88 1 2

9 17 88, 1 2

10 2 88. 1 2

10 6 88 1 2

10 7 88 1 ,2

8 10 79 1 3

8 14 79 , ., 1 .3,

10 22 79, 1 3

2 27 80 - 1 .3,

3 27 80 1 3

4 7 80 1 3

6 27 80,-- I 3

7 25 80, 1 3

8 1 80 1 .3

*.10 18 80 1 ,3

10 20 80 1 3

12 30 80 1 3

3 17 81 1 -3

3 27 81 1 3

4 24 81--. -I 3

6 22 81- ,1 ,3

9 29 81, 1 3

11 5 81 1 3

1 4 82 1 -3

2 24 82 1 3

7 21 82 1 .3

10 21 82 1 3

4 4 83 1 3

-i.

D BRGS

F -. MECH

D BRGS

D .; FUR ,

D . BRGS

F ;:- MECH

F - COMPR

D . MOTOR

D . COMPR

D - COMPR,

D , COMPR

F - JKHX

D COMPR

D & BRGS

D p ,- MECH

D MECH

D -. LUDOIL

F , COMPR

F . COMPR ,

F . COMPR

F UNLOA

F , COMPR

D MECH,

D BRGS

D . COMPR,

F GASKE

D -;. BRGS

D - JK1IX

D BRGS

D BRGS

D BRGS

D COMPRD , .JKTIIX

D , .J; ', MECIH

D ,,; -,MOTOR

D LUBOIL

D; , FLTR

D . BRGS

F. , * MECII.

D - BRGS

D FLTR

D ., BRGS

D - COMPR

D. BRGS

D,' COMPR

F ' MECII

0.52 7.58

0.71 829 2.32 8.29 *

037 8.66

0.40 9.06

0.66 9.71

0.42 10.13 4.16 10.13

0.13 10.27 0.14 10.27

0.66 10.92

0.64 11.57

738 18.94

037 19.31 .

0.63 19.94 - 9.81 19.94

4.91 24.86

2.00 26.86:

1.34 28.20

1.07 29.27

2.19 31.46

2.07 3352, 6.66 33.52

2.90 36.42 2.9 36.42

0.10 36.52 0.1 36.52

0.17 36.69 0.17 36.69

0.04 '36.73 0.04 36.73

0.01 36.74

0.10 . 0.10 '

0.04 0.14,

0.76 0.90 0.9 0.90

1.39 2.29

0.33 2.62,

0.11 2.73

0.89 3.62

0.31 3.93

0.07 4.00

0.86 4.86

0.02 4.88

0.78 5.66

0.86 6.51

0.11 6.62'

0.30 6.92'.

064 7.57. 6.67 7.57

1.08 .8.64''

0.40 9.04': '

0.66 9.70'

0.56 10.26:'

1.63 11.89

1.00 :12.89'

1.81 14.70' 7.13 14.70

B-3
Table B.1. (Cont'd)

Mo Dv Yr Pit Comp SEVTY CRTC DSCP Tij Ti Fri Fti

4 22

10 21

12 12

1 20

1 26

11 16

1 27

2 27

4 30

4 17

4 27

7 23

10 2

3 31

10 6

7 31

8 10

10 2

10 11

2 2

2 27

3 27

4 7

6 24

6 27

7 25

11 6

12 30

3 3

3 17

3 27

4 24

6 22

7 28

7 29

8 14

9 29

10 6

11 5

1 4

1 25

4 23

5 24

7 1

7 21

1 20

83 1 3

83 1 3

83 1 3

84 1 3

84 1 3

84 1 3

85 1 3

85 1 3

85 1 3

86 1 3

87 1 3

87 1 3

87 1 3

88 1 3

88 1 3

79 1 4

79 1 4

79 1 4

79 1 4

80 1 4

80 1 4

80 1 4

80 1 4

80 1 4

80 1 4

80 1 4

80 1 4

80 1 4

81 1 4

81 1 4

81 1 4

81 1 4

81 1 4

81 1 4

81 1 4

81 1 4

81 1 4

81 1 4

81 1 4

82 1 4

82 1 4

82 1 4

82 1 4

82 1 4

82 1 4

83. 1 4

COMPR

COMPR

MECH

GASKE

MECH

FLR

GASKE

MECH

COMPR

& BRGS

UNLOA

COMPR

COMPR

MECH

MECH

COMPR

BRGS

LUBOIL

* COMPR

COMPR

BRGS

JIKTlIX

BRGS

LUBOIL

BRGS

BRGS

MECH

BRGS

FLTR

LUBOIL

FLR

BRGS

BRGS

* GASKE

COMPR

COMPR

BRGS

MECH

FLTR

BRGS

COMPR

MOTOR

MECHI

MECII

BRGS

COMPR

0.20

1.99

0.57

0.42

0.07

3.22

0.79

0.33

0.70

3.86

4.11

0.96

0.77

1.99

2.06

0.68

0.10

0.58

0.10

1.23

0.28

0.33

0.11

0.86

0.03

0.31

1.12

0.60

0.70

0.16

0.11

0.30

0.64

0.40

0.01

0.17

0.50

0.08

0.32

0.66

0.23

0.98

0.34

0.41

0.22

1.99

14.90

16.89 2.19 16.89

17.46

17.88

17.94

21.17

21.96

22.29

22.99

26.84

30.96

31.91

32.68

34.67

36.72

0.68 0.68 0.68

0.78

1.36 0.68 1.36

1.46 0.1 1.46

2.69

2.97

3.30

3.41

4.27

4.30

4.61

5.73

6.33

7.03

7.19

7.30

7.60

8.24

8.64 7.18 8.64

8.66

8.82

9.32

9.40

9.72

1038

10.61

11.59

11.93

12.34

12.57

14.56
Table B.1. (Cont'd)

Mo Dy Yr Pit Comp SEVTY CRTC DSCP Tii nl Fri Fli

3 15 83 1 4 4 .F BRGS 0.61'15.17 -6.53 15.17

4 4 83 1 4 D:, 'MECH 021 15.38 ,

7 25 83 ,1.4 F~ \ ,COMPR 1.23 16.61 1.44 16.61

7 20 84 1 -4 D COMPR 3.94 20.56

11 16 84 1 4 D . MECH - 1.29 21.84 

3 28 85 1 4 F .- MECH - 1.A7 23.31 6.7; 23.31

6 10 85 1 4 F * -GASKE 0.80 24.11 0.8 24.11'

12 1 85 1 4 D GASKE 1.90 26.01

12 6 85 1 4 D! ;MECH 0.06 26.07

1 10 86 1 4 D. ' ;-COMPR 0.38 26.44

3 29 86 1 4 F MOTOR 0.88 2732. 3.21 27.32

4 21 86 1 4 F- *,UNLOA 0.24 27.57 0.25 27.57

I' S 87 1 ,-4 D- : COMPR 2.82 30.39

7 2 87 1 4 D UNLOA 1.97 32.36

2 3 88 1 4 F ,FGASKE 2.34 34.70 7.13 34.7

7 1 88 1 4 D -UNLOA 1.64 36.34

7 15- 88 1 4 D -COMPR . 0.16 36.50

f

.~~~ . . .. 

B-5
- --~~~~~

Table B.2. Failure Data after Sensitivity Modification

(4 Components Combined)

Mo Dv Yr PIt Comp SEVTY DSCP Fti Ag Md

1 25

1 20

1 19

1 17

7 15

12 27

3 9

12 10

2 15

11 14

1 27

8 26

2 12

2 24

7 25

12 17

9 8

9 17

10 2

10 6

10 22

6 22

4 4

10 21

7 31

10 2

10 11

7 28

3 15

7 25

3 28

6 10

3 29

4 21

2 3

82 1 1

83 1 1

84 1 1

8 5 1 1

85 1 1

8S 1 1

87 1 1

87 1 1

88 1 1

88 1 1

81 1 2

81 1 2

82 1 2

82 1 2

84 1 2

87 1 2

88 1 2

88 1 2

88 1 2

88 1 2

79 1 3

81 1 3

83 1 3

83 1 3

79 1 4

79 1 4

79 1 4

81 1 4

83 1 4

83 1 4

85 1 4

85 1 4

86 1 4

86 1 4

88 1 4

F COMPR

F COMPR

F COMPR

F COMPR

F UNLOAD

F COMPR

F LUBOIL

F COMPR

F GASKET

F GASKET

F LUBOIL

F MECH

F MECH

F COMPR

F JKTHX

F COMPR

F COMPR

F COMPR

F UNLOAD

F COMPR

F GASKET

F MECH

F MECH

F COMPR

F COMPR

F LUBOIL

F COMPR

F GASKET

F BRGS

F COMPR

F MECH

F GASKET

F MOTOR

F UNLOAD

F GASKET

8.66

3.94

3.99

3.98

1.97

1.80

4.80

3.02

0.72

2.99

5.97

2.32

4.16

0.14

9.81

6.66

2.90

0.10

0.17

0.04

0.90

6.67

7.13

2.19

0.68

0.68

0.10

7.18

6.53

1.44

6.70

0.80

3.21

0.25

7.13

8.66 1

12.60 1

16.59 1

20.57 2

22.54 2

24.34 2

29.14 2

32.16 2

3288 2

35.87 2

5.97 1

8.29 1

10.13 1

10.27 1

19.94 1

33.52 2

36.42 2

36.52 2

36.69 2

36.73 2

0.90 1 *

7.57 1

14.70 1 *

16.89 1

0.68 1

1.36 1

1.46 1

8.64 1

15.17 1

16.61 1

23.31 2

24.11 2

27.32 2

27.57 2

34.70 2

B-6
Table B3. M-W Test Results for Data Counting of the Four Air Compressors

(aging failure data was portioned)

Average # of Average # of

Rank of Valves Rank of Valves

Comparison of 1st of 1st 2nd of 2nd Total Test cc

Samples Sample Sample Sample Sample obs Statistical Z Values

Component 1- 113 10 9.7 10 20 -0.566 0.571

Component 2 l

Component 1- 7.3 10 8 4 14 0.212 0.8

Component 3 .Il

Component 1- 12.2 10 9.91 11 -0.81 21 0.417

Component 4

Component 2- 6.9 10 9 4 0.777 16 0.436

Component 3 l

Component 2- 10.25 10 11.682 11 0.493 21 0.622

Component 4 l

Component 3- 9.625 4 7.41 11 15 -0.784 0.433

Component 4 l

B-7
APPENDIX Q STATISTICAL RESULTS FOR SENSITIVITY ANALYSIS ON

UNCERTAINTY IN DEGRADATION OCCURRENCE TIMES

C.1 Database for the Sensitivity Analysis of Uncertainty in Degradation Occurrence Times

Table C.1 and C.2 present the'aging data for the RHR pumps and air compressors. The data

was obtained after imposing the uncertainty of degradation occurrence times. The uncertainty time

intervals were assumed to be an exponentially distnbuted random variable with a mean of 15 days.

6. -

1

. 1,i

..

.

.

. .

. .

.1

C-1
Table C.1. Units RHR Pump Data Combined with

Uncertainty of Degradation Times

Mo Dv Yr Plant Comp Svtv Ti T Uncrt r, T ir

5 1 80

I IS 81

3 16 82

10 28 82

9 8 83

2 17 84

7 1 84

7 26 85

5 12 80

1 1S 81

3 16 82

10 28 82

3 17 83

4 18 84

7 26 85

3 10 86

1 9 87

5 10 88

6 7 80

1 15 82

3 16 82

10 28 82

9 8 83

6 8 84

8 7 84

7 26 85

2 2 87

4 25 80

5 12 80

3 16 82

10 28 82

12 15 82

3 17 83

4 18 84

5 5 84

6 29 84

7 26 85

7 28 86

1 4 83

8 25 83

11 8 83

2 2 84

8 7 84

5 8 85

1 16 86

4 19 88

1 4 83

7 28 83

11 8 83

6 19 84

8 2 84

saIl

SaSI

sasl

sail

sasl

sasI

sasI

sasl

sasl

sasl

sasl

sasI

53S I

sasl

sasI

sasl

sasl

sasl

sasl

sasl

sasl

sasi

sasI

sasl

sasl

sast

sasl

sasI

sail

s3si

sasl

saSl

sasl

sail

sasl

sasl

sasl

saSl

sas2

sas2

sas2

s=s2

s. 2

sas2

sas2

sas2

s=s2

sas2

sas2

sas2

.sas2

a

a

a

a

a

a

a

a

b

b

b

b

b

b

b

b

b

Cb

C

C

C

C

C

C

C

C

d

d

d

d

d

d

d

d

d

d

d

a

a

a

a

a

b

a

a

a

b

b

b

b

b

D 1.33 1.33 0.257

D 2.88 4.21 0.017

D 4.73 8.94 0.025

D 2.47 11.41 0.522

D 3.50 14.91 0.148

D 1.82 16.73 0.165

D 1.49 18.22 0.233

D 4.33 22.56 0.181

D 1.46 1.46 0.033

D 2.76 4.21 0.132

D 4.73 8.94 0.008

D 2.47 11.41 0.266

D 1.60 13.01 0.328

F 4.40 17.41 0.037

D 5.14 22.56 0.101

D 2.54 25.10 0.142

F 3.38 28.48 0.262

D 5.40 33.88 0.469

D 1.73 1.73 0.334

F 6.53 8.27 0.364

D 0.68 8.94 0.033

D 2.47 11.41 0.200

D 3.50 14.91 0.450

D 3.06 17.97 0.126

D 0.66 18.62 0.039

D 3.93 22.56 0.682

D 6.18 28.73 0.141

D 1.27 1.27 0.030

D 0.19 1.46 0.047

D 7.49 8.94 0.019

D 2.47 11.41 0.389

D 0.52 11.93 0.002

D 1.08 13.01 0.120

F 4.40 17.41 0.042

D 0.19 17.60 0.055

D 0.60 18.20 0.445

D 4.36 2256 0.079

D 4.08 26.63 0.065

D 0.03 0.03 0.000

1 2.57 2.60 0.162

D 0.81 3.41 0.406

D 0.99 4.40 0.083

F 2.06 6.46 0.161

F 3.07 9.52 0.026

D 2.81. 12.33 0.113

F 9.14 21.48 0.105

D 0.03 0.03 0.000

D 2.27 2.30 0.005

D 1.11 3.41 0.358

F 2SI 5.92 0.345

F 0.48 6.40 0.354

1.08

4.19

8.92

10.89

14.76

16.57

17.99

2238

1.42

4.08

8.94

11.15

12.68

17.37

22.45

24.96

28.22

33.41

1.40

7.90

8.91

11.21

14.46

17.84

18.58

21.87

28.59

1.24

1.41

8.93

11.02

11.93

12.89

17.37

17.54

17.76

22.48

26.57

0.03

2.44

3.01

4.32

6.29

9.50

12.22

21.37

0.03

2.9

3.05

5.58

G.05

1.08

3.12

4.73

1.97

3.87

1.81

1.42

4.39

1.42

2.66

4.86

2.21

1.54

4.69

5.08

2.50

3.26

5.19

1.40

6.50

1.01

2.30

3.25

3.38

0.74

3.29

6.72

1.24

0.17

7.52

2.10

0.91

0.96

4.48

0.18

0.21

4.72

4.09

0.03

2.41

0.57

131

1.98

3.20

2.72

9.15

0.03

2.26

0.76

2.52

0.47

C-2
TableC.1. (Cont'd)

Mo Dv Yr Plint -Comp Svt Tmj~~ fl- Uncrt . * . Tiir

1 30 86 sas2

2 11r t86 sas2

3 24 S87 sas2

12 17 87 -sas2

2 4 88 sas2

1 4 83 sas2

2 1 83 sas2

3 4 83 sas2

5 25 83 sas2

9 27 83 sas2

2 16 84 sas2

5 16 84 sas2

8 15 84 sas2

3 7 85 sas2

2 3 89 sas2

1 4 83 sas2

1 11 83 sas2

4 12 83 ss2

3 5 84 sas2

8 2 84 sas2

8 15 84 sas2

9 20 84 sas2

3 7 85 sas2

12 17 87 sas2

8 1 74 duan

12 5 74 duan

12 15 75 duan

9 20 76 duan

11 21 76 duan

12 26 76 duan

1 16 79 duan

3 16 82 duan

6 3 82 duan

10 23 82 duan

2 25 83 duan

3 3 85 duan

7 1 86 duan

4 23 75 duan

12 18 78 duan

3 10 82 duan

4 4 82 duan

5 1 82 duan

6 8 82 duan

8 1 82 duan

10 23 82 dunn

2 9 83 duan

3 1 85 dunn

4 23 82 duan

10 23 83 duan

3 1 85 duan

9 14 74 duan

3 18 76 dunn

b D

b; D

:b D

b D

*b D

c. DC D

C D

C D

o D

C D

C D

o F

C D

o F

d D

d D

d D

d F

d D

d F

d F

d D

d D

a D

a F

a D

a D

a D

a D

a D

a D

a D

a D

a D

a D

a D

b F

b D

b D

b D

b D

b D

b D

b D

b D

b D

C D

C D

C D

d F

d D

6.09

0.12

4.53,

2.92

0.58

0.03

0.30

0.37

0.90

1.36

1.60

1.00

0.99

230

15.84

0.03

0.08

1.01

3.64

1.63

0.14

0.39

1.91

11.22

1.00

1.38

4.17

3.11

0.68

0.39

8.39

12.83

0.86

1.56

1.41

8.20

5.37

3.97

14.78

13.13

0.27

0.30

0.41

0.59

0.91

1.23

8.36

32.36

6.06

5.53

1.48

6.16

12.49 10.213

1261 0.166

17.14 0.491

20.07 0.151

20.64 iO.015

0.03 0.014

0.33 0.129

0.70 0.038

1.60 0.124

2.96 0.057

4.56 0.028

5.56 0.004

6.54 0.367

8.84 0.067

24.69 0.147

0.03 0.000

0.11 0.013

1.12 0.131

4.77 0.180

6.40 0.274

6.54 0.045

6.93 0.219

8.84 0.234

20.07 0.008

1.00 0.268

2.38 0.084

6.54 0.013

9.66 0.316

10.33 0.096

10.72 0.073

19.11 0.005

31.94 0.020

32.80 0.623

34.36 0.217

35.77 0.299

43.97 0.041

49.33 0.150

3.97 0.047

18.74 0.019

31.88 0.389

32.14 0.002

32.44 0.120

32.86 0.042

33.44 0.055

34.36 0.445

35.59 0.079

43.94 0.065

32.36 0.152

38.41 0.162

43.94 0.406

1.48 0.161

7.63 0.026

12.28

12.45

16.65

19.92

20.63

0.02

0.20

0.66

1.48

2.90

453

5.55

6.18

8.78

24.54

0.03

0.10

0.99

4.59

6.13

6.50

6.71

8.61

20.06

0.73

2.29

6.53

9.34

10.24

10.65

19.11

31.92

32.18

34.14

35.47

43.93

49.18

3.92

18.73

31.49

32.14

32.32

32.81

33.39

33.91

35.51

43.83

32.20

38.25

13.54

1.32

7.61

6Z3

0.17

4.21

3.26

0.71

0.02

0.19

OA6

0.81

IA2

1.63

1.02

0.63

2.60

15.76

0.03

0.07

0.89

3.60

1.54

0.37

0.21

1.90

11.45

0.73

1.56

4.24

2.81

0.90

0.41

8.46

12.82

0.25

1.96

1.33

8.46

5.26

3.92

14.81

12.76

0.65

0.18

0.49

0.58

0.52

1.60

8.37

32.20

6.05

5.29

1.32

6.29

C-3
Table C.1. (Cont'd)

Mo Dv Yr Plant Comp Svtv Tij nI Uncrt nol Tii' , ... . . _ .. . .

11 4 76 duan

S 1 82 duan

10 23 82 duan

12 1 82 duan

1 1 83 duan

1 1 84 duan

3 1 85 duan

d D 2.51 10.14 0.113

d D 22.30 32.44 0.105

d D 1.91 34.36 0.170

d D 0.42 34.78 0.184

d D 0.39 35.17 0.005

d D 4.06 39.22 0.358

d D 4.72 43.94 0.345

10.03 2.42

32.34 22.31

34.19 1.85

34.59 0.41

35.16 0.57

38.86 3.70

43.60 4.73

Ca4
I - 'I , 1

Tablc C.2. Aging Data with Uncertainty Times of Degradation Occurrences

(4 Compressors Combined)

Mo Dv

1 24

4. 7

7 25

9 2

12 3

12 30

3 17

4 24

6 22

8 24

9 29

11 5

I 4

1 25

4 14

1 20

3 9

1 19

4 30

.11 -16

1 17

7 12

7 15

12 27

4 17

11 24

1 5

3 9

7 2

12 '10

12 17

2 ,15

4 1

6 '21

11 14

, 11 26

1 13

,, 1'.' 24

1 31

4 7

7 25

12 30

1 27

4 24

5 5

6 22

8 26

9 29

11 5

1 4

Yr PIt Comp SEVTY DSCP' I Tfl

80 1

80 1

80 1

80 1

80 1

80 1

81 1

81 A

81 1

81 1

81 I

81 -1

82 1

82 1

82 1

83 1

83 1

84 1

84 1

84 1

85 1

85 1

85 1

85 1

86 1

86 1

87 1

87, 1

87 1

87 1

87 1

88 1

88 1

88- 1

88 1

79 1

80 1

80, 1

80 1

80 1

80 1

80 1

81 1

81' 1

81 1

81 1

81 1

81 1

81 1

82 1

1 D BRGS* 0.64 0.64

I D ' BRGS '-0.81 1.46

1 D 'BRGS 1.20 2.66

1 D BRGS 0.41 3.07

1' D MECH 1.01 4.08

1 D BRGS '030 438

1 D J CHX 0.86 5.23

I D BRGS 0.41 5.64

1, D 'BRGS'. 0.64 6.29

1 D COMPR ', 0.69 6.98

1 , D BRGS 0 39 737

1 D FLTR' 0.40 7.77

1' D BROS 0.66 8.42

1 F - kCOMPR 0.23 8.66

1 D MECH 0.88 9.53

1 ' F COMPR 3.07 12.60

1 D COMPR -0.54 13.14

1 F COMPR 3.44 16.59

1 D COMPR 1.12 17.71

I , D -' MECH '2.18 19.89

1 F COMPR 0.68 20.57

1 D UNLOA, 1.94 22.51

I F UNLOA'' 0.03 22.54

1 F COMPR 1.80 24.34

1 D 'MOTOR 122 25.57

1 D MECH - 2.41 27.98

I D 'LUBOIL '- 0.46 28.43

1 F LUBOIL '0.71 29.14

I D COMPR 1.26 30.40

1' F COMPR ' 1.76 32.16

1 D COMPR 0.08 32.23

I F GASKE 0.64' 32.88

I D GASKE 0.51 3339

1 D"- UNLOA 0.89 t 34.28

1 F GASKE -1.59' 35.87

2 D COMPR- 1.29' 1.29

2 D MECH' 0.52' 1.81

2 D BRGS 0.12 1.93

2 D COMPR 0.08 '- 2.01

2 D BRGS ' 0.73 2.74

2 D BRGS - 1.20' 3.94

2 D BRGS ; 1.72 5.67

2 F LUBOIL' 0.30 5.97

2 D.. BRGS 0.97:' 6.93

2 D COMPR 0.12- 7.06

2 D BRGS 052 .' 7.58

2 F' MECII ' 0.71 8.29

2 D ''BRGS' 037 8.f6

2 D I LTR 0.40' 9.06

2 D BRGS 0.66 9.71

C-5

Uncrt

0.043

0.003

0.004

0.087

0.025

0.027

0.039

0.030

0.024

0.005

0.022

0.001

0.044

0.055

-0.006

0.017

0.024

0.044

0.078

0.013

'0.056

0.061

0.005

0.033

0.075

.0.021

0.007

0.114

0.023

0.019

0.005

0.008

0.003

' 0.065

0.000

0.020

0.007

0.009

1 0.014

0.013

0.011

0.007

0.025

0.027

0.028

0.014

0.027

0.004

0.019

0.018

Ti Tip

0.601 0.387

1.453 0.851126

2.651 1.198648

2.900 0.248753

4.053 1.152969

4.350 0.297311

5.194 0.844084

5.614 0.419928

6.265 0.650769

6.972 - 0.707233

7.345' 0.372376

7.765 0.420662

8.378 0.612544

8.601 0.222923

9.527 0.926321

12.583 3.056038

13.121 0.537528

16.545 3.424386

17.633 1.08788

19.876 2.24299

20.511 0.63507

22.450 1.93941

22.539 0.088485

24.311 1.77219

25.492 1.180518

27.957 2.465128

28.427 0.469974

29.031 0.604018

30.377 1.34576

32.137 1.76038

32228' 0.091432

32.870' 0.64149

33.386' 0.515809

34.213' 0.827231

35.866 1.653388

- 1.269 1.2689

1.804 0.535133

' 1.924 0.120063

1.928 0.003653

2.731 0.803461

3.934 1.202385

5.660 1.726497

5.941 0.281125

6.906 0.965109

6.889 -0.01743

7.564 0.675057

8.262 0.69801

8.651 039204

-9.037 0.385473

9.694 0.656876
Table C.2. (Cont'd)

Mo Dv Yr PIt Comp SEVTY DSCP iT nf Uncrt ri 

T

2 12

2 24

4 23

6 21

4 25

5 28

7 25

10 17

4 17

8 18

11 24

6 11

12 17

9 

8

9 17

10 

2

10 

6

10 

7

8 10

8 14

10 22

2 27

3 27

4 

7

6 27

7 25

8 1

10 18

10 20

12 30

3 17

3 27

4 24

6 22

9 29

11 

5

1 

4

2 24

7 21

10 21

82 1 2 F MECH 0.42

82 1 2 F COMPR 0.13

82 1 2 D MOTOR 0.66

82 1 2 D COMPR 0.64

84 1 2 D COMPR 738

84 1 2 D COMPR 0.37

84 1 2 F JKITHX 0.63

85 1 2 D COMPR 4.91

86 1 2 D BRGS 2.00

86 1 2 D MECH 1.34

86 1 2 D MECH 1.07

87 1 2 D LUBOIL 2.19

87 1 2 F COMPR 2.07

88 1 2 F COMPR. 2.90

88 1 2 F COMPR 0.10

88 1 2 F UNLOA 0.17

88 1 2 F COMPR 0.04

88 1 2 D MECH 0.01

79 1 3 D BRGS 0.10

79 1 3 D COMPR 0.04

79 1 3 F GASKE 0.76

80 1 3 D BRGS 1.39

80 1 3 D JKTHX 0.33

80 1 3 D BRGS 0.11

80 1 3 D BRGS 0.89

80 1 3 D BRGS 0.31

80 1 3 D COMPR 0.07

80 1 3 D JKTHIX 0.86

80 1 3 D MECH 0.02

80 1 3 D MOTOR 0.78

81 1 3 D LUBOIL 0.86

81 1 3 D FLTR 0.11

81 1 3 D BRGS 0.30

81 1 3 F MECH 0.64

81 1 3 D BRGS 1.08

81 1 3 D FLTR 0.40

82 1 3 D BRGS 0.66

82 1 3 D COMPR 0.56

82 1 3 D BRGS 1.63

82 1 3 D COMPR 1.00

10.13

10.27

10.92

11.57

18.94

19.31

19.94

24.86

26.86

28.20

29.27

31.46

33.52

36.42

36.52

36.69

36.73

36.74

0.10

0.14

0.90

2.29

2.62

2.73

3.62

3.93

4.00

4.86

4.88

5.66

6.51

6.62

6.92

7.57

8.64

9.04

9.70

10.26

11.89

12.89

14.70

14.90

16.89

17.46

17.88

17.94

21.17

21.96

22.29

22.99

26.84

30.96

0.028

0.031

0.001

0.060

0.058

0.059

0.036

0.136

0.082

0.025

0.002

0.002

0.002

0.021

0.006

0.021

0.009

0.005

0.001

0.061

0.011

0.025

0.034

0.028

0.002

0.022

0.030

0.046

0.00

0.037

0.039

0.001

0.023

0.045

0.014

0.002

0.053

0.016

0.012

0.001

0.003

0.028

0.036

0.050

0.007

0.014

0.005

0.003

0.003

0.065

0.000

0.020

10.105 0.411369

10.236 0.131136

10.921 0.685238

11.507 0.585759

18.887 7.37981

19.252 0.365176

19.909 0.656851

24.719 4.810583

26.774 2.054185

28.175 1.401126

29.264 1.089402

31.453 2.189168

33.520 2.066468

36.401 2.880877

36.516 0.115179

36.668 0.152238

36.724 0.0557

36.740 0.015898

0.099 0.0993

0.144 0.045084

0.889 0.744387

2.264 1.375531

2.588 0.323765

2.567 -0.02139

3.620 1.053332

3.911 0.291321

3.970 0.058566

4.810 0.839903

4.870 0.060325

5.619 0.748785

6.472 0.853134

6.621 0.148775

6.899 0.27817

7.522 0.622927

8.630 1.108383

9.042 0.411818

9.647 0.605071

10.240 0.592317

11.877 1.637091

12.888 1.011333

14.697 1.808582

14.733 0.036787

16.853 2.11924

17.406 0.553112

17.871 0.465188

17.861 -0.0098

21.162 3.300546

21.948 0.785952

22 2S6 0.338031

22.924 0.63S342

26.844 3.920055

30.936 4.091558

4 4 83 1 

3

4 22 83 1 

3

10 21 83 1 

3

12 12 83 1 

3

1 20 84 1 

3

1 26 84. 1 

3

11 16 84 1 

3

1 27 85 1 

3

2 27 85 1 

3

4 30 85 1 

3

4 17 86 1 

3

4 27 

. 87 1 

3

F MECII 1.81

D COMPR 0.20

F COMPR 1.99

D MECII 0.57

D GASKE 0.42

D MECII 0.07

D FLTR 3.22

D GASKE 0.79

D MECII 0.33

D COMPIt 0.70

D BRGS 3.86

D UNLOA 4.11

C-6
Table C.2.' (Cont'd)

Mo Dv Yr PIt Camp SEVTY DSCP. Tij Ti Uncr Ti' TR

7 23 87 1 3 D COMPR 0.96 31.91 0.007 31.904 0.968467

10 2 87, 1 3 D COMPR 0.77 32.68 0.009 32.669 0.764507

3 31 88 1 3 D -- MECH - 1.99 34.67 0.074 34.593 1.923948

10 6 88 1 3 D MECH 2.06 36.72 0.013 36.709 2.116499

7 31 79 1 4 F COMPR 0.68 0.68 0.011 - ' 0.667 0.6669

8 10 79 1 4 D - BRGS 0.10 0.78 0.007 0.771 0.104275

10 2 79 1 4 F LUBOIL 0.58 !1.36 0.025 1.330 0558903

10 11 79 1 4 F COMPR ,'0.10 1.A6 0.027 1A29 '0.098442

2 2 80 .1. 4 D .. COMPR 1.23. 2.69 0.068 2.621 1.192614

2 27 80 1 4 D -BRGS 0.28 2.97 0.014 2.953 0331677

3 27 80 1 4 D JK=HX 0.33 3.30 0.027 3.273 0320232

- - 4 7 80 1 4 D - BRGS - 0.11 3.41 0.004' 3.407 0.133649

6 24 80 1 4 D LUBOIL -0.86 427 0.019 4.248 0.841029

: - 6 27 80 1 4 D BRGS 0.03 4.30 0.018 4.282 0.034654

7 25 '-80 1 4 D --'BRGS '031 4.61 0.028 4.583 0.300258

11 t' 6 80 1 4 D MECH '1.12 5.73 0.031 5.703 1.120025

12 30 80 1 4 D BRGS 0.60 6.33 0.001 6.332 0.629682

- ' 3 ' 81 1 4 D FLTR' '0.70 7.03 0.060 6.974-0.641314

3 :17 81 1 4 D ' LU801L -0.16 7.19 0.058 7.131 0.157588

* 3 27 81 1 4 D FLTR 0.11 7.30 0.059 ' 7.241 0.109621

4 -24.81 1 4 D 'BRGS 0.30' -7.60 '0.036 ' 7.564 0323517

6 22 81 1 4 D BRGS 0.64 8.24 0.136 8.108 0.543917

7 28 81. 1 4 F GASKE 0.40 8.64 0.082 8.563 0.454185

7 29 81 1 4 D COMPR 0.01 8.66 0.025 8.630 0.067793

8 14 81 1 4 D COMPR 0.17 8.82 0.002 8.820 0.19402

9 29 -81 1- 4 D BRGS 0.50 9.32 0.002 9.320 0.500279

10 6 81 1 4 D MECII 0.08 9.40 0.002 9398 0.07758

11 5 81 1 4 D FLTR 0.32 9.72 0.021 9.701 0.303099

1 4 82 1 4 D BRGS 0.66 10.38 0.006 10.371 0.670734

1 25 82 1 4 D COMPR 0.23 10.61 0.021 10.590 0.218904

4 23 82 1 4 D MOTOR 0.98 11.59 0.009 11.579 0.989033

5 24 82 1 4 D MECH 0.34 11.93 0.005 11.929 0.349231

7 1 82 1 4 D MECH 0.41 12.34 0.001 12.344 0.415178

7 21 82 1 4 D BRGS 0.22 12.57 0.028 12.400 0.056262

1 20 83 1 4 D COMPR 1.99 14.56 0.011 14544 2.144321

3 15 83 1 4 F BRGS 0.61 15.17 0.025 15.142 0.597753

4 4 83 1 4 D MECII 0.21 15.38 0.034 15344 0.201542

7 25 83 1 4 F COMPR 1.23 16.61 0.072 16.539 1.195037

7 20 84 1 4 D COMPR 3.94 20.56 0.002 20.553 4.014729

11 16 84 1 4 D MECH 1.29 21.84 0.022 21.823 1.269099

3 28 85 1 4 F MECH 1.47 23.31 0.030 23.2S1 1.458566

6 10 85 1 4 F GASKE 0.80 24.11 0.046 24.065 0.784347

12 1 85 1 4 D GASKE 1.90 26.01 0.008 26.004 * 1.938102

12 6 85 1 4 D MECII 0.06 26.07 0.014 25.983 -0.0202

1 10 86 1 4 D COMPR 0.38 26.44 0.039 26.405 0.422115

3 29 86 1 4 F MOTOR 0.88 27.32 0.001 27.321 0.915442

4 21 86 1 4 F UNLOA 0.24 27.57 0.023 27.544 0.222614

1 5 87 1 4 D COMPR 2.82 30.39 0.045 30.344 2.800704

7 2 87 1 4 D UNLOA 1.97 32.36 0.014 32.342 1.997271

2 3 88 1 4 F GASKE 2.34 34.70 0.002 34.693 2.356262

7 1 88 1 4 D UNLOA 1.64 36.34 0.053 36.292 1.59396

7 15 . 88 1 4 D COMPR 0.16 36.50 0.016 36.484 0.192317

C-7
C2 Comparison of Statistical Test Results for Data Combinine on RHR PumD Aping Data

The Mann-Whitney test for the aging data with uncertainty times incorporated were conducted

for the RHR pumps and air compressors, respectively. Table C3 and C.4 present the test results.

Table C.3. Comparison of Test Results on Data Combining for RHR Pumps

Average # of Average # of

Uncertainty Rank of Valves Rank of Valves

Times Comparisons 1st of st 2nd of 2nd Total Test R

Considerations of Samples Sample Sample Sample Sample obs Statistical Z Value

plant - 36.30 38 37.75 35 73 0.287 0.77

plant 3 _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _

Before plant 2 - 31.14 37 42.17 35 72 2.23 0.03

plant 3 36.13 38 37.94 ___3 ____ .7

plant I - 36.13 38 37.94 35 73 0.359 0.71

plant 2 _ _ _ _ _ _ _ _ _ _

plant I - 36.15 38 37.91 35 73 0.347 0.727

plant 3 . l

After plant 2 - 31.5 34 42.25 35 69 2.07 0.082

plant 3 3_ 7 . _ 7 3 - 0.0

plant I - 44.5 36 31.2 7 73 -1.63 0.07

plant 2 __ _ _ ___ _ __ _ __ _ _ _

C-8
Table C.4. Comparison of Test Results on Data Combining for Air Compressors

Kruskal-Wallis analysis of CMPUNCRT.Ti, by Comp

-= i.. Level Sample Size Average Rank

1 35 88.2857

2 33 78.2727

' ,, l 3 38 78.0789

4 48 68.6458

0 Test statistic = 3.95496 Significance level = 0.266369

| Kruskal-Wallis analysis of CMPUNCRT.Tj by Comp 0

Level Sample Size Average Rank

B .s |1 35 88.9571

2 33 77.2727

3 38 78.5921

4 48 67.5938

0 Test statistic = 1.54586 Significance level = 0.671726

C-9