ANALYSIS - Cryogenic Operation for LHC Cryogenic System: Criticality analysis, determination of critical ... Given the obtained numerical results, a criticality ranking will be operated,

ANALYSIS

LHCCryogenicSystem:Criticalityanalysis,determinationofcriticalhardwarecomponents

Abstract

The Large Hadron Collider (LHC) superconducting systems require being cooled down below 2K. The cooling power is provided by helium refrigerators. Gas and liquid helium are both supplied through various valves boxes and cryogenic lines to the LHC, with a minimum of substitution solutions in case of failure of a subcomponent.

The reached current overall availability is above 90%, but should be increased to maximise the operation time of the LHC.

A criticality analysis is set up at CERN: the aim of this analysis is to highlight critical components in terms of failure occurrence and severity, and then, to adapt the current maintenance plan and the process layout, in order to avoid incidental and accidental events as well.

In order to quantify occurrence and severity of failure events, a statistical study of operational data from 1st January 2011 to 1st August 2012 is led. This document gathers the statistical results, in terms of failure rates, occurrences and durations of losses of ‘Cryo Maintain’ conditions and mean times between failures (MTBF). This allows drawing conclusions about the cryogenic process’ overall criticality.

Applicability LHC Cryogenic System

Date 29/08/2012

Issued Jimmy MARTIN

Checked Krzysztof BRODZINSKI

Approved Krzysztof BRODZINSKI

CERN CH-1211 Geneva 23 Switzerland

CRYOGENIC OPERATION for LHC

CERN Dep./Group

TE/CRG‐OA

EDMS Document No.

xxxxxxxx

LHC Cryo Document Number

LHC‐CRYO‐OP‐A‐110

LHC Document No: LHC-CRYO-OP-A-110 EDMS Document No: xxxxxxxx

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TableofContents

1. PURPOSE OF THE DOCUMENT ................................................................................................... 4

2. METHODOLOGY OF STUDY .......................................................................................................... 4

2.1 FMECA analysis ......................................................................................................................... 4

2.2 HAZOP technique ...................................................................................................................... 5

2.3 Events analysis ........................................................................................................................... 5

2.4 Reliability computations ............................................................................................................. 5

2.5 The chosen method ................................................................................................................... 6

3. CRITICALITY ANALYSIS AND CRITERIONS .............................................................................. 6

3.1 Definition of criticality ................................................................................................................. 6

3.2 Index of occurrence ................................................................................................................... 7

3.3 Index of severity ......................................................................................................................... 8

3.4 Index of non-detection ............................................................................................................... 9

3.5 Other possible indexes .............................................................................................................. 9

4. STATISTICAL TOOLS ..................................................................................................................... 10

4.1 Weibull’s laws for reliability probabilities ............................................................................... 10

4.2 Failure rate function ................................................................................................................. 11

4.3 Mean Time Between Failures ................................................................................................. 13

4.4 Computing Weibull’s parameters from raw data ................................................................. 14

4.4.1 Sampling ............................................................................................................................ 14

4.4.2 Linear Regression method .............................................................................................. 14

4.5 Kolmogorov-Smirnov’s test ..................................................................................................... 19

5. STATISTICAL ANALYSIS ............................................................................................................... 20

5.1 Sampling raw data ................................................................................................................... 20

5.2 Weibull analysis ........................................................................................................................ 21

5.2.1 Cold compressors ............................................................................................................ 21

5.2.2 Warm compressors .......................................................................................................... 27

5.2.3 Turbines ............................................................................................................................. 34

5.2.4 CV valves ........................................................................................................................... 40

5.2.7 Electric heaters ................................................................................................................. 50

5.2.8 PV valves for nitrogen ...................................................................................................... 53


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5.3 Global synthesis and index ‘O’ attribution ............................................................................ 55

5.4 Severity analysis and index ‘S’ attribution ............................................................................ 57

6. CRITICALITY RANKING ................................................................................................................. 58

7. SOURCES OF BIAS – LIMITS OF THE ANALYSIS .................................................................. 60

8. FOR FURTHER ................................................................................................................................ 60

9. CONCLUSION .................................................................................................................................. 60

10. REFERENCES ............................................................................................................................. 61


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1. PURPOSE OF THE DOCUMENT

The aim of this document is to present results of a statistical study based on operational data, from 1st January 2011 to 1st August 2012, and their consequences in terms of criticality of LHC cryogenic process’ hardware components.

Firstly, the choice of the presented method will be justified, and then the treatment of failure events related to hardware components (i.e. cold compressors, CV valves, etc.) and the calculation of failure rate functions, MTBFs and causes distribution will be developed.

Secondly, the criticality of each hardware component, its various causes of failure and the possible solutions in order to reduce the occurrence probability and severity of such events will be discussed.

2. METHODOLOGY OF STUDY

In terms of criticality analysis, various methods have been applied for many years in industrial sector, such as (non-exhaustive list):

FMECA; HAZOP; Events analysis; Reliability computations.

2.1 FMECA analysis

‘FMECA’ is the acronym for ‘Failure Mode and Effects, Criticality Analysis’. This method focuses on each elementary part of the studied process. Firstly, the ranking of most critical elements is done, based on a ‘criticality’ index (called ‘RPN’ for ‘Risk Priority Number’), which is the multiplication of three sub-indexes:

Index of occurrence of a failure mode, applied on an elementary process’ part; Index of severity of the events; Index of non-detection of the events.

This stage of the analysis is completed by using numerical data from the process (i.e. probabilities of failure by components).

Then, the four following questions are formulated, in order to understand the specific processes which lead to failures, and then avoid them as well:

What kind of failure mode could happen? What could be its effects on the process? What are the causes which lead to this failure mode?


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How to detect the failure occurrence?

However, managing with a sharp level of detail - needed in this analysis - can be hard for a complex process’ layout, such as the LHC cryogenic process.

2.2 HAZOP technique

Given the nominal process’ parameters, the HAZOP technique tries to understand the consequences of certain variations of these parameters, i.e. what could happen if the liquid helium temperature - initially at 1.8 K – suddenly increases and reaches a 5 K value. By ranking the consequences of such process’ parameters variations in order of severity, it is possible to focus on most critical events and then, to find what kind of component can be responsible for variations of process’ conditions.

After highlighting these critical components, their failure modes can be explored by formulating the next questions:

What are the causes which lead to this failure mode? How to detect the failure occurrence?

For a complex process, such as the studied one, the analysis of parameters’ variations can be very difficult. Indeed, many combinations of variables are possible, and each of them can lead to a specific event.

2.3 Events analysis

The events analysis is based on cumulated events description for a certain interval of time (i.e. data from ‘Logbook OA’). By finding the origins of a given event with a ‘tree of causes’, it is possible to determine what components lead to the detected failure, and then apply actions in order to avoid the new occurrence of this event.

This method is appropriate to post-incidental/accidental situations, when an event must be entirely analysed in order to avoid its repetition, but is unsuitable when finding critical elements of a process. Indeed, latent failure modes (for example due to the aging of components) are occulted by this method, whose input is past events.

2.4 Reliability computations

This technique is based on a statistical treatment of numerical data, in order to rank components given their own criticality. Determination of criticality is possible by considering the computed values of failure rate functions (most relevant information is their trends). The classical used statistical tool is the Weibull analysis, described on chapter 4. In addition to the previous stage, the origins of failure modes must be explored, in order to avoid severe events occurrence.


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2.5 The chosen method

The chosen method is based on reliability computations (using the Weibull analysis) for hardware components which act directly on the process, such as warm and cold compressors, valves, turbines, etc. Given the obtained numerical results, a criticality ranking will be operated, using the next FMECA criterions:

Index of occurrence of a failure mode; Index of severity of the events;

Note the non-detection index will not be used here, because of a lack of data, which constitutes the input for its formulation.

The Pareto analysis of causes will be operated in order to associate to critical process’ elements some origins of failure, and to help the future analysis that would be set up to avoid the occurrence of highlighted critical failure modes.

So, we can define the chosen method as a mix between reliability computations and FMECA aspects (at a much large level of details).

3. CRITICALITY ANALYSIS AND CRITERIONS

3.1 Definition of criticality

A component is an elementary part of a complex ensemble such as the LHC’s cryogenic process. It ensures one or several technical functions, and so its failure can lead to severe situations, according to the importance of these functions in the overall process. The probability of occurrence of failures, also called “failure function” F(t) is a function of time. It is proven the age of a component has a direct effect on its probability to be on failure, and as a consequence, on its reliability, which is defined as below:

R(t) = 1 – F(t)

Thus, it is relevant to quantify at any time, increase and then monitor the reliability of an elementary type of components, in order to maintain the overall process’ availability above an acceptability limit (low limit of global efficiency for LHC’s cryogenic process = 90%). Reliability can be improved by calculating and decreasing failure rates of a statistical population of components. However, severity of failures must be taken into account when studying the availability of a process; the calculation of failure rate functions is not enough.

Non-frequent events associated to severe consequences can highly reduce the overall process’ availability, even if the failure rate is low: for example, a cold compressor’s stop generally leads to the process’ stop, i.e. loss of ‘Cryo Maintain’ conditions. Conditions for starting up the process are reached along various durations; this has a direct incidence on the overall availability. So, it is much more efficient to associate to each type of component a criticality index.

The criticality of a population of components is evaluated by calculating its intrinsic failure rate and by taking into account the consequences of a failure event on the overall process in which it is integrated, and the probability monitoring systems (and operators) have to detect such an


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event. For example, a type of components which is defined by a high and growing failure rate and a high non-detection probability of severe failure events will be declared highly critical.

The index C of criticality can be calculated as below:

C O ∗ S ∗ D

With:

O: Index of occurrence of failure events (from 0 to 1); S: Index of severity of these events (idem); D: Index of non-detection (idem).

3.2 Index of occurrence

The computation of index O of occurrence is based on the statistical study of cumulated operation data, from 1st January 2011 to 1st August 2012 (extracted from the ‘Logbook OA’). The input data is the date of failure during the studied period, associated to a process’ hardware component. The output result is the ‘failure rate function’ λ(t), computed by correlating operation data related to a given hardware component to a statistical law (see chapter 5).

Then, given the overall number of failures N cumulated during the studied period and the λ (t) function’s trend of each type of components included in the present analysis, an index O of occurrence of failure is associated to them, as follows:

The following code must be applied in order to associate to each type of component an occurrence index O:

Green 0.25; Yellow 0.5; Orange 0.75; Red 1.

For example, a population of a given hardware component which has been affected by 25 failures from 1st January 2011 to 1st August 2012 and whose failure rate function’s trend is constant will be associated to an index O equal to 0.75, according to the previous table.

These criterions of association can be discussed and re-evaluated if necessary. Their adjustment depends on the limit of acceptability of occurrence previously defined by the ‘Cryogenics - Operation for Accelerators’ section.


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Note that four levels of failure rate function’s trend are specified. The trend is defined by the value of the β parameter of the statistical Weibull distribution (see chapter 5). Recall the following rules in order to associate a β-value to a trend:

β < 1 λ’s trend = ↘ β = 1 λ’s trend = →

1 < β ≤ 2 λ’s trend = ↗ +

2 < β λ’s trend = ↗ ++

3.3 Index of severity

In this criticality analysis, the severity of a failure is evaluated by two criterions:

The loss of ‘Cryo Maintain’ conditions induced by the event (also noted ‘CM=0’); The recovery time of these conditions.

During a failure, if requested physical conditions such as fluid level, temperature, pressure or flow rate are lost, the ‘Cryo Maintain’ (CM) indicator turns into zero. As a consequence, the LHC experiments are interrupted (the beams are dumped), while the ‘CM’ conditions are not reached.

Dates of loss of ‘CM’ conditions and their associated duration before recovery are implemented in an Excel© table by the ‘Cryogenics – Operation for Accelerators’ section, every week (see [8] and [9]). This serves as a statistical basis to the criticality analysis.

Three levels of duration before recovering the ‘CM’ conditions are considered:

T < 8h; 8h ≤ T < 18h;

18h ≤ T. The index S of severity is defined when associating these three previous levels of duration with the overall number of ‘CM’ losses during the statistical study period. Considering the range of occurrence of ‘CM’ losses highlighted through the present statistical analysis, three levels of number N are also set up:

N‘CM’=0 < 2; 2 ≤ N’CM’=0 < 4; 4 ≤ N’CM’=0.

Note that the case of no loss of ‘CM’ conditions during the statistical study period is considered as a (T < 8h; N‘CM’=0 < 2) case. The following table gathers the different cases of association between durations and occurrence of loss of ‘CM’ conditions:


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Recall the next code of colour in order to associate to each failure case an index S of severity:

Green 0.25; Yellow 0.5; Orange 0.75; Red 1.

When events for a studied type of component have different duration before recovering ‘CM’ conditions, intermediate S indexes must be calculated. Then, a mean value of S is considered as the overall severity index. For example, a given type of component led to 2 losses of ‘CM’ conditions with a recovery time of 7 and 19 hours and 5 losses with recovery durations from 8 to 18 hours. Its mean severity index is calculated as below:

S inter1 = 0.25 (T=7h < 8h; N‘CM’=0 = 1 < 2); S inter2 = 0.75 (18h ≤ T=19h ; N‘CM’=0 = 1 < 2); S inter3 = 1 (8h ≤ T < 18h; 4 ≤ N‘CM’=0 = 5).

∗ S inter1 + S inter2 + S inter3) = 0.67

These criterions of association can be discussed and re-evaluated if necessary. Their adjustment depends on the definition of severity previously set up by the ‘Cryogenics - Operation for Accelerators’ section.

3.4 Index of non-detection

The notion of criticality generally includes the non-detection aspect of a failure. Indeed, the overall availability of a process may be highly reduced in some cases when rare and non-severe events are not quickly detected.

As the requested input in order to compute this index is incomplete, D index will be occulted here.

3.5 Other possible indexes

Other indexes are possible to define the criticality of process’ components, such as:

Index of accessibility of the component to maintain in case of failure;


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Index of existence of alternative solutions/spares compared to available time to react:

Available delay before stop (CM=0)

T > 24h 5 < T < 24h 1 < T < 5h 10min < T < 1h 0min < T < 10min Immediate Stop

0 spares ‐ 0 alt. solution

8h < Avail. Spare + set up

3 < Avail. Spare + set up < 8h

Built‐in with delay or spare <

3h

Alt. solution ready to switch

Sometimes, accessibility of a given component is limited, e.g. components inside vacuum vessels. Even if spares are available in order to maintain components, non-accessibility gives severity when they fail. Also, in certain circumstances, alternative solution or spares are available and can be set up with a high accessibility, but the available delay before losing ‘CM’ conditions is too reduced: such events are declared severe. That’s the two previous indexes could take into account.

4. STATISTICAL TOOLS

4.1 Weibull’s laws for reliability probabilities

A Weibull’s law is a continuous law of probability, which is a good approximation of stochastic physical processes. For this reason, it is currently used in fields like maintenance, when components’ lifetime is studied.

In lifetime computations, the ‘reliability probability’ R(t) can generally be approximated by a Weibull’s law, which is expressed as below:

Where:

t = time variable; β = shape parameter; γ = position parameter; η = scale parameter.


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Recall that R(t) represents the reliability in time of a type of components, in a given environment. R(t) varies from 0 to 1. Its opposite is the ‘failure probability’ F(t), which is expressed by:

1 1

The γ position parameter is generally equal to zero. But in some specific cases, it may be useful to offset in time the Weibull’s law with a γ value.

In other cases, successive Weibull’s laws must be considered in order to define the reliability probability of a type of components in time. For example, a bi-Weibull’s law is used when components have two different reliability probability trends in time. The first Weibull’s law defines the first interval of lifetime of a population, from t = 0 to t = T.

Then, a combination of both distinct Weibull’s laws characterises the second interval of lifetime, from t = T to t = ∞.

This is expressed by the following system:

, T

, T

Where:

α = scale parameter for 1st law; θ = shape parameter for 1st law; β = shape parameter for 2nd law; η = scale parameter for 2nd law; γ = position parameter; t = time variable.

From a given Weibull’s law, it is possible to determine two current statistical indicators:

The failure rate function ; The Mean Time Between Failures (MTBF).

4.2 Failure rate function

The failure rate function of a population is given by:

∗


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The three parameters β, η, and γ are related to the Weibull’s law of the studied population. The failure rate function represents a number of failures per unit of time (here, [t] = day), and has three different trends, depending on the β values (e.g. η = 300 and β = [0.8, 1, 1.3]):

When a bi-Weibull distribution describes the cumulated frequency of failures of a given population, two distinct expressions for λ(t) must be considered, as follows:

∗ , T

∗ ∗ , T

It is interesting to note the second expression of λ(t) is simply the sum of two failure rate functions. R(t) for a bi-Weibull distribution has a much more complex expression from t = T to t = ∞, which is not just a sum of two R(t) sub functions, as previously seen on chapter 4.1.

In case of a bi-Weibull distribution, the shape of the overall failure rate function of a population (from t = 0 to t = ∞) is more sophisticated than the previous one. For example, the following typical ‘bathtub’ shape can be obtained (but others can be found, given the bi-Weibull distribution’s parameters):

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121

λ(t)

β < 1

1 < β

β = 1


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Where the red scatter is related to the first failure rate expression, from t = 0 to t = T, and the green to the second expression, from t = T to t = ∞.

4.3 Mean Time Between Failures

The Mean Time Between Failures (MTBF) of a population represents an estimator of its reliability, given its environment and its various solicitations. When the overall failure number N is up to 10, the MTBF can be approximated by:

TBF is the calculated time between two consecutive failures. This approximation assumes the failure rate is constant along the statistical study interval. For a variable failure rate, this expression loses its validity.

If the reliability of a population is given by a Weibull’s law (whatever the trend of λ (t)), it is possible to compute the MTBF as below:

. 11

Where:

η and β are the Weibull distribution’s parameters;

is the Euler’s Gamma function:

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 100 200 300 400 500 600 700

λ(t)

time t (day)


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The values are given by current probability tables and software. Recall that it is possible to compute MTBF values by using the EXP (GAMMALN) function from Excel©.

In case of a bi-Weibull distribution, a conservative way to compute the MTBF is to use the second law’s parameters (if the second law is corresponding to an increasing step of the failure rate function).

4.4 Computing Weibull’s parameters from raw data

4.4.1 Sampling

To be reliable, the sampling of past events must satisfy the two following points:

Include at least 10 events; Be related to a coherent population of components, i.e. solicited by a common failure

mode. If a combination of two dominant failure modes is found, the coherence of the population can also be reached.

4.4.2 Linear Regression method

If we assume reliability probability is given by a Weibull’s law, so:

We can determine β, η and γ parameters as follows:

Computation of cumulated frequency of failures F*(t):

∗ 00 1

Where:

0 is the overall number of failures of the sample, and so, the number of survivors at time t = 0;

is the number of survivors at time t.

Note the ‘*’ superscript index which indicates the expression of F(t) is unbiased. The statistical bias related to the sampling is avoided by using the ‘mean estimator’ of F(t), noted F*(t) (the use of a median estimator for F(t) is also possible, which is not-significantly different from the mean estimator in the present analysis).

Computation of Ln(Ln(1/R*(t)):


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First, assume a Weibull distribution of failures and γ = 0:

∗

1∗

1∗

1∗

∗

11 ∗ ∗

11 ∗ ∗ ∗

By demonstrating the linear relation ∗ between ∗ and Ln(t) terms

which are respectively ‘y’ and ‘x’ variables in the previous expression, it is possible to identify a distribution of failures of a given population as a Weibull’s law. A linear regression applied on

the scatter formed by ∗ and Ln(t) values gives the parameters of the supposed

Weibull distribution, as illustrated below:

The R2 correlation coefficient is approximately equal to 1; this demonstrates the linearity of the previous relation. Weibull’s parameters can be deduced from computed ‘a’ and ‘b’ values:

y = 0.8757x ‐ 3.8033R² = 0.9876

‐2.50

‐2.00

‐1.50

‐1.00

‐0.50

0.00

0.50

1.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Ln(Ln(1/(1‐F*(t))))

Ln(t)


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0

If the scatter isn’t linearly distributed (i.e. R2 far from 1) but follows a convex or concave plot, the Weibull distribution hypothesis is not necessarily rejected: this can reveal parameter is not equal to zero. So, the next step of computation must be completed:

Choice of an arbitrary point {y2;t2} near the middle of the plot; Choice of two other points of the plot {y1;t1} and {y3;t3}, linearly equidistant to point 2; Computation of estimated gamma, as below:

23 2 ∗ 2 13 2 2 1

Application to new variable (t – γ) of the previous linear regression method.

If the R2 correlation coefficient is close to 1, the hypothesis of a Weibull distribution is valid. Then, β and η can be computed by a linear regression. In this case, γ is simply the previous computed value.

If R2 correlation coefficient is far from 1 after this additional step of computation, so the Weibull distribution hypothesis must be definitely rejected.

The case of two distinct linear plots reveals a bi-Weibull distribution, which characterises multiple failure modes:

The parameters of the first Weibull’s law can be deduced from the purple scatter, with the previous linear regression method: the second failure mode’s effects are negligible along the

y = 0.8555x ‐ 5.6952R² = 0.9819

y = 3.5604x ‐ 21.553R² = 0.9649

‐4.7

‐3.7

‐2.7

‐1.7

‐0.7

0.3

1.3

2.2 2.7 3.2 3.7 4.2 4.7 5.2 5.7 6.2 6.7

ln(ln(1/(1‐F*(t))))

ln(t)


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first time interval. However, a significant error is introduced when computing the second Weibull’s parameters directly by applying a linear regression to the green scatter. Along the second time interval, both failure modes are combined: the first one is only negligible when events are close to the present date.

So, for the second scatter, the following method must be applied:

Computation of ‘instant failure rate’, given by:

∆

∆ ∗00 ∗

Where:

N(t) is the number of survivors at time t; N(0) is the number of survivors at time t = 0.

Plotting of Ln(λinst (t)) = f[Ln(t)], as illustrated below:

As presented on chapter 4.2, in case of a bi-Weibull distribution, the failure rate function can be simply expressed as a sum of the first failure rate function and a second one, along the second time interval. So, to compute the second distribution’s parameters, it is necessary to subtract λ1(t) to λinst (t) along the second time interval, where λ1(t) is the previously defined expression of λ(t) along the first interval. Then, the next step consists in plotting Ln(λinst (t) – λ1(t)) = f[Ln(t)]:

y = ‐0.1501x ‐ 5.3897

y = 7.1729x ‐ 48.853R² = 0.9506

‐7

‐6

‐5

‐4

‐3

‐2

‐1

0

0 1 2 3 4 5 6 7

Ln(λinst (t))

Ln(t)


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The ‘cross scatter’ represents Ln(λinst (t) – λ1(t)) = f[Ln(t)] along the second time interval, in other words, the logarithm of the second instant failure rate function less the first distribution’s contribution λ1(t), as a function of Ln(t). The determination of second Weibull distribution by a linear regression is now possible.

Recall the expression of a failure rate function, related to a given Weibull distribution (assuming γ = 0):

∗

The previous expression is equivalent to:

∗∗

Then:

1 ∗

1 ∗

∗ ∗

1 ∗ ∗

Given the coefficients ‘a’ and ‘b’ from the linear regression of the cross scatter:

y = 7.1729x ‐ 48.853R² = 0.9506

y = 9.5356x ‐ 63.768R² = 0.9517

‐9

‐8

‐7

‐6

‐5

‐4

‐3

‐2

‐1

0

0 1 2 3 4 5 6 7


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1

0 (When R2 is close to 1)

The same considerations as previously can be applied if R2 is far from 1.

4.5 Kolmogorov-Smirnov’s test

In order to improve the fitting between the observed distribution of failures and a Weibull’s law, a Kolmogorov-Smirnov’s test can be applied to data. Given Weibull’s parameters, a theoretical Ftheo (t) function can be computed:

1 1

Moreover, to determine Weibull’s parameters by a regression method, F*(t) was previously computed, which is the real cumulated frequency of failures.

The Kolmogorov-Smirnov’s test consists in calculating the maximum absolute value of the difference between Ftheo (t) and F*(t), and then in comparing this value to a limit value given by a Kolmogorov-Smirnov’s table. The limit value is a function of the number N(0) of the studied population and a chosen level of risk α. This level of risk represents the probability to accept the hypothesis H0: Ftheo (t) = F*(t) whereas it is false (risk of first kind). Also, the level of confidence in this acceptation of H0 is given by 1–α.

Mathematically, this is expressed by:

Where:

| ∗ |

Recall the following example to determine the D limit value from a Kolmogorov-Smirnov’s table. If the chosen level of risk α is equal to 5%, and the population size is N(0) = 18, so the table gives the D limit value, equal to 0.309:


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If the computed | ∗ | is equal to 0.2801, so we can accept the H0 hypothesis of a fitting between the observed distribution and a Weibull’s law, with a level of confidence equal to 95%.

5. STATISTICAL ANALYSIS

5.1 Sampling raw data

Operational data from 1st January 2011 to 1st August 2012 constitutes the input of the present statistical study. In terms of events, only those classified on type ‘failure/stop’ and ‘problem solved’ were extracted from ‘Logbook OA’. This represents an amount of approximately 500 events along this period. These categories are related to real failures. Eventually, it is possible to extend the analysis to some events on type ‘pre problem’, which are sometimes related to failures.

The classification of events in coherent groups is necessary to manage with such amount of data. The Weibull analysis keeps its validity for populations of components solicited by at maximum two significant failure modes. So, forming too inclusive groups, which may be solicited by various modes of failures, is a risk for the reliability of the Weibull analysis’ results.

These groups are related to the following hardware components we assume solicited by common failure modes:

Cold compressors; Warm compressors; Turbines; CV valves for helium; Electric heaters; PV valves for N2.

Other categories were rejected because not enough data was available for a reliable statistical study from 1st January 2011 to 1st August 2012.


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5.2 Weibull analysis

For complete numerical data about the Weibull analysis, see the table: Excel© ‘LHC CRYO – Statistics’ [7].

5.2.1 Cold compressors

Along the studied period, 25 events are related to cold compressors. When plotting

∗ as a function of Ln(t), the following scatter is reached:

The R2 correlation coefficient is equal to 0.9433, so the present distribution of failures can be assimilated to a Weibull distribution, whose parameters are:

2.683 (Which means the failure rate function is increasing)

367

0

So:

2.683367

∗367

.

The failure rate function’s trend is given by the previous computed expression for λ(t):

y = 2.683x ‐ 15.847R² = 0.9433

‐3.5

‐3

‐2.5

‐2

‐1.5

‐1

‐0.5

0

0.5

1

1.5

4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4

ln(ln(1/(1‐F*(t))))

ln(t)


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When comparing the theoretical probability of failure Ftheo (t) to the observed F*(t), and applying the Kolmogorov-Smirnov’s test, the validity of the Weibull model is confirmed.

Indeed, for N = 25 and α = 5% a Kolmogorov-Smirnov’s table gives the following limit value:

D limit = 0.2640

The maximum value for | ∗ | is equal to 0.1979, so we can accept the Weibull distribution hypothesis, with a 95% level of confidence.

See below the trends for theoretical and observed F(t) probabilities:

By confirming the validity of the model, it is now possible to compute MTBF for cold compressors, as presented on chapter 4.3. Given the β and η values:

0.0000

0.0020

0.0040

0.0060

0.0080

0.0100

0.0120

0.0140

0 100 200 300 400 500 600

λ(t)

time (day)

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

1.0000

0 100 200 300 400 500 600

F(t)

t (day)

theo

real


23


. 11

327

A Pareto analysis of causes, i.e. failure modes gives the following results:

CAUSE Quantity %

COM Command 21 84.0

UTI Utilities 2 8.0

UNK Unknown 2 8.0

MEC Mechanical 0 0.0

ENV Environment 0 0.0

Total 25 100.0

We can observe the main failure mode is related to command about 84% of the overall.

In details, through the ‘command’ category, the cold compressor’s regulator and frequency converter are mainly responsible to the 84% of command failures, as presented below:

CAUSE Quantity %

CCR CC regulation 10 47.6

VFREQ Frequency converter 4 19.0

PLC QURC regulation 3 14.3

INSTRU Instrumentation 3 14.3

SIGN Links (profibus, etc.) 1 4.8

Total 21 100.0

0

5

10

15

20

25

Command Utilities Unknown Mechanical Environment

Global Pareto histogram for CCs


24


For cold compressors, a unique failure mode is probably dominant and may affect both distinct command systems ‘regulator’ and ‘frequency converter’, that could be why the observed failures distribution fits accurately to a Weibull’s law.

Also, we can distribute to each cold compressor a number of failures, as follows:

VFREQ and CCR failure distributionElement Quantity % %Air Liquide %Linde

CC1 6 42.9 83.3 16.7

CC2 4 28.6 0 100

CC3 2 14.3 0 100

CC4 2 14.3 ‐ 100

Total 14 100.0 % value 35.7 64.3

(Probabilities for Air Liquide and Linde failures computed by using the ‘total probabilities formula’)

Note that cold compressors nearby the most soliciting fluid parameters (i.e. to the lowest condition of pressure and temperature) are mainly responsible for failure events: CC1 and then CC2 are mostly on failure. It could be interesting to try to correlate the physical fluid conditions to this failure mode, even if it is related to command systems. By this way, improvements could probably be reached in terms of reliability.

‘Linde’ cold compressors are statistically the most related with failures, and a similar correlation than the previous one could be set up, in order to improve the overall process’ reliability. Note the particular concentration of events around Ln(t) = 5.4, which corresponds to August 2011. Analysing this singular period in terms of failures’ causes could also be a source of progress.

0

2

4

6

8

10

12

CC regulation Frequencyconverter

QURC regulation Instrumentation Links (profibus,etc)

Pareto histogram for COM


25


The following scatter represents the plotting of ∗ as a function of Ln(t), applied

on the 11 events which are due to cold compressors’ regulators failures:

The low R2 value, equal to 0.8371, is due to the singular concentration of events related to CCR failures around August 2011 (i.e. Ln(t) ~ 5.3). But, we can admit the global trend is linear. So, the computation of Weibull parameters is possible:

2.7631

406

0

Given these parameters, we can compute and plot the failure rate function, as follows:

2.7631406

∗406

.

VREQ and CCR failure distribution

CC1

CC2

CC3

CC4

Linde/A.Liquide failure%

%Air Liquide

%Linde

y = 2.7631x ‐ 16.599R² = 0.8371

‐3

‐2.5

‐2

‐1.5

‐1

‐0.5

0

0.5

1

5.4 5.5 5.6 5.7 5.8 5.9 6 6.1 6.2 6.3

ln(ln(1/(1‐F*(t))))

ln(t)


26


Note the current failure rate function’s expression and plot are very similar to the previous ones. Globally, the overall cold compressors’ trend of failure rate follows the cold compressors regulators one.

The computed value of MTBF related to regulators is:

. 11

362

This MTBF is higher than the previous one, given the new computed η value, also higher.

The Kolmogorov-Smirnov’s test allows accepting the Weibull distribution hypothesis:

For N=11 and α = 5%:

D limit = 0.3912


Note that, excepting the singularity of events around August 2011, the theoretical and measured F probabilities fairly fit:

0.0000

0.0020

0.0040

0.0060

0.0080

0.0100

0.0120

0 100 200 300 400 500 600

Lambda (failu

re/day)

time (day)


27


5.2.2 Warm compressors

Warm compressors were affected by 26 failure events along the studied period. The following

plot represents the ∗ values as a function of Ln(t):

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

0 100 200 300 400 500 600

F(t)

t (day)

theo

real

y = 0.8912x ‐ 5.1033R² = 0.9529

‐3.5

‐3

‐2.5

‐2

‐1.5

‐1

‐0.5

0

0.5

1

1.5

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

ln(ln(1/(1‐F*(t)))) ln(t)


28



0.8912 (Which means the failure rate function is decreasing)

307

0

So:

.

0.8912307

∗307

.


When comparing the theoretical probability of failure Ftheo (t) to the observed F*(t), and applying the Kolmogorov-Smirnov’s test, the validity of the Weibull model is confirmed.

Indeed, for N = 26 and α = 5% a Kolmogorov-Smirnov’s table gives the following limit value:

D limit = 0.2591


See below the trends for theoretical and observed F(t) probabilities:

y = 0.0054x‐0.109

0.0025

0.0027

0.0029

0.0031

0.0033

0.0035

0.0037

0.0039

0.0041

0.0043

0.0045

0 100 200 300 400 500 600

Lambda (failu

re/day)

time (day)


29


By confirming the validity of the model, it is now possible to compute MTBF for warm compressors, as presented on chapter 4.3. Given the β and η values:

. 11

325

A Pareto analysis of causes, i.e. failure modes gives the following results:

CAUSE Quantity %COM Command 8 30.8


LEAK Leakage 5 19.2

UTI Utilities 4 15.4

UNK Unknown 2 7.7


Total 26 100.0

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

1.0000

0 100 200 300 400 500 600

F(t)

t (day)

theo

real


30


The Pareto analysis reveals a multiplicity of failure modes. Events related to ‘command’ systems are dominant. Also, ‘mechanical’ and ‘leakage’ types of failures are approximately equivalent.

However, the Drenick’s law says a combination of at least three failure modes leads to a random distribution of failures, which can be represented both by an increasing or decreasing failure rate function, whatever the β values for each mode. So, in that case, it is hard to draw conclusions about the overall failure rate’s trend.

The observation of January-February of 2012 months’ events reveals a concentration of assimilated ‘leakage’ failure. The discovery of such an amount of events is related to the previous Technical Stop period (winter 2012), when maintenance inspections were applied to warm compressors’ stations. So, it is also difficult to evaluate the precise date of such failures. The leakage may be reached during the inspections days (e.g. when the reassembly of a flange is non-correctly operated), or during an anterior date (for more details, see [7]).

So, the Weibull analysis of warm compressors can be operated by censoring events related to leakages. The following plot represents the linear regression method applied to the cumulated frequency of failures:

0

1

2

3

4

5

6

7

8

9

Command Mechanical Leakage Utilities Unknown Environment


31


The R2 correlation coefficient is equal to 0.9504, so the distribution of failures excluding ‘leakage’ can also be assimilated to a Weibull distribution, whose parameters are:

0.8155 (Which means the failure rate function is decreasing)

301

0

So:

.

0.8155301

∗301

.

The refined failure rate function’s trend is given by the previous computed expression for λ(t). The trend is in that case much more decreasing than the previous one: the reached current λ value is around 0.0025 failures per day.

y = 0.8155x ‐ 4.6538R² = 0.9504

‐3.5

‐3

‐2.5

‐2

‐1.5

‐1

‐0.5

0

0.5

1

1.5

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

ln(ln(1/(1‐F*(t))))

ln(t)


32


The application of a Kolmogorov-Smirnov’s test gives the following results:

For N = 21 and α = 5% a Kolmogorov-Smirnov’s table gives the following limit value:

D limit = 0.2872

The maximum value for | ∗ | is equal to 0.1273, so we can accept the Weibull distribution hypothesis, with a 95% level of confidence. Note we gain much more precision about the fitting between the theoretical and observed distributions by censoring ‘leakage’ events: The current D value is lower than the previous one (which is equal to 0.1642).

Given the new β and η parameters, we can compute the warm compressors’ MTBF:

y = 0.0078x‐0.184

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.0055

0 100 200 300 400 500 600

Lambda (failu

re/day)

time (day)

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

1.0000

0 100 200 300 400 500 600

F(t)

t (day)

theo

real


33


. 1 336 (Superior to the previous one)

The new causes’ distribution is given by the next table:



UTI Utilities 4 19.0

UNK Unknown 2 9.5


Total 21 100.0

Two distinct failure modes can be distinguished in that case (i.e. ‘command’ and ‘mechanical’). The influence of ‘utilities’ cause is reduced here, comparing to the previous analysis in which the ‘leakage’ cause was approximately equivalent to both ‘mechanical’ and ‘command’ modes.

So, we can confirm the overall failure rate’s trend is decreasing. The specific study of command and mechanical systems related to warm compressors can’t give in that case reliable results: These systems lead respectively to 8 and 6 failures, which are too low numbers in order to operate a Weibull analysis.

Not to forget to consider leakage events to improve another statistical indicator of performance, i.e. ‘loss of helium’. This criterion is excluded to the present study, even if leakage data may also give a good description about warm compressors global state (state of seals, etc.), as ‘mechanical’ statistics does.

0

1

2

3

4

5

6

7

8

9

Command Mechanical Utilities Unknown Environment


34


5.2.3 Turbines

A total of 25 failures can be imputed to turbines from 1st January 2011 to 1st August 2012. The

representation of ∗ as a function of Ln(t) is given below:


1.0529~1 (Which means the failure rate function is approximately constant)

415

0

So:

.

1.0529415

∗415

.


y = 1.0529x ‐ 6.3469R² = 0.9292

‐4

‐3

‐2

‐1

0

1

2

2.2 2.7 3.2 3.7 4.2 4.7 5.2 5.7 6.2 6.7

ln(ln(1/(1‐F*(t)))) ln(t)


35


For a studied population number N = 25 and a level of risk α = 5%, a Kolmogorov-Smirnov’s table gives the next limit value:

D limit = 0.2640


The following figure represents the plotting of both theoretical and observed F(t) probabilities:

Since the validity of the Weibull distribution hypothesis is confirmed, the turbines’ MTBF can be computed, given the β and η values:

y = 0.0018x0.0529

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 100 200 300 400 500 600

Lambda (failu

re/day)

time (day)

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

1.0000

0 100 200 300 400 500 600

F(t)

t (day)

theo

real


36


. 11

407

The Pareto analysis of causes highlights the predominance of two distinct failure modes: ‘command’ and ‘mechanical’ sources.



UNK Unknown 2 8.0


UTI Utilities 0 0.0

Total 25 100.0

The overall failure rate’s trend is constant, but individual failure rates for ‘command’ and ‘mechanical’ modes are not necessarily both constant: one can be increasing and the other decreasing, giving a constant overall failure rate for turbines. The specific study for ‘command’ systems related to turbines is possible, given the 15 events cumulated from 1st January 2011 to 1st August 2012. Unfortunately, not enough data is available to analyse failure mode associated to ‘mechanical’ origins.

When plotting ∗ as a function of Ln(t), for ‘command’ events, the following

scatter is reached:

0

2

4

6

8

10

12

14

16

Command Mechanical Unknown Environment Utilities


37


The correlation coefficient R2 is here equal to 0.8721, due to a consequent number of events concentrated around X-coordinate Ln(t) = 6.2 (May 2012). However, the overall trend is approximately linear, occulting this particular amount of events, probably due to a specific failure mode we need to understand.

So, we can consider the observed distribution as a Weibull one, whose parameters are:

1.262 (Which means the failure rate function is slowly increasing)

513

0

So:

.

1.262513

∗513

.

When plotting λ(t) as a function of time t:

y = 1.262x ‐ 7.8751R² = 0.8721

‐3

‐2.5

‐2

‐1.5

‐1

‐0.5

0

0.5

1

3.7 4.2 4.7 5.2 5.7 6.2 6.7

ln(ln(1/(1‐F*(t))))

ln(t)


38


The λ’s trend is slowly increasing and is probably responsible for the slow increasing trend of the λ(t) function, related to turbines. So, we can assume the failure rate of mechanical systems is decreasing.

The Kolmogorov-Smirnov’s test applied on the current distribution gives the following results:

For N = 15 and α = 5% a Kolmogorov-Smirnov’s table gives the following limit value:

D limit = 0.3376


Note the difference between Ftheo(t) and F*(t) is higher when only focussing on data related to ‘command’ failure mode.

This is also probably due to the concentration of events around Ln(t) = 6.2 (t ~ 500 days), as seen on next graph:

y = 0.0005x0.262

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 100 200 300 400 500 600

Lambda (failu

re/day)

time (day)


39


The Pareto analysis for causes gives the following histogram:

CAUSE Quantity %FS Flow Switch 6 40.0

PLC TU regulation 5 33.3

INSTRU Instrumentation 3 20.0

SETTING Settings 1 6.7

Total 15 100.0

Two distinct modes of failure are dominant: Failure related to a flow switch or PLC is intrinsically different.. Also, failures classified on type ‘FS’ are not necessarily imputable to the flow switch itself: this could be either a hardware failure or a lack of water inside the water circuit the flow

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

1.0000

0 100 200 300 400 500 600

F(t)

t (day)

theo

real

0

1

2

3

4

5

6

7

Flow Switch TU regulation Instrumentation Settings


40


switch monitors. This is a source of a bias on the statistics, and must be explored in order to censure non-appropriated events and then refine the Weibull model. Note the last events concentrated around t = 500 days are essentially due to ‘FS’ failures (see table [7]). So, this could explain the observed difference between data and the theoretical distribution.

Moreover, taking into account the previous remarks related to the statistical results’ accuracy, note that all PLC events are associated to Air Liquide 18kW refrigerators (which have their own regulation systems), which means the CERN’s PLCs disposed on Linde refrigerators were not affected by failures during the studied period. Also, all ‘FS’ events are related to Linde refrigerators, as seen below:

PLC and FS failure distributionElement Quantity %Air Liquide %Linde

PLC 5 100.0 0.0

FS 6 0 100

Total 11

In terms of MTBF associated to ‘command’ systems of turbines, taking into account the computed β and η values:

. 1 477

5.2.4 CV valves

5.2.4.1 Globalstudy

CV valves were most affected by failures: 43 events were cumulated from 1st January 2011 to

1st August 2012. When plotting ∗ as a function of Ln(t), two scatters with linear

trends are distinguishable:


41


This reveals the existence of two Weibull distributions with specific failure modes, and will lead to the characteristic ‘bathtub’ trend for the failure rate function λ(t). This study is applied on a large population of CVs, localized at different process’ levels (i.e. with different thermo-hydraulic conditions), so we can assume these failure modes are mainly independent of physical properties, such as temperature, pressure and flow rate of the conveyed fluid. Otherwise, it would have been impossible to correlate failures related to such a large population of CVs solicited by very different ranges of physical conditions with any Weibull distribution.

This existence can be firstly confirmed by the very close to 1 R2 correlation coefficients, respectively equal to 0.9819 and 0.9649. However, a Kolmogorov-Smirnov’s test is also mandatory in order to accept the bi-Weibull distribution hypothesis.

Given the important difference of slopes between the two linear trends, we can consider during the first interval of time, only the first Weibull distribution is dominant, and apply the standard linear regression method to determine Weibull’s parameters from the plotting of

∗ as a function of Ln(t) presented above.

So, with R2 = 0.9819:

0.8555

0.0013

0

And then:

∗ 0.0011 ∗ 0.0013 ∗ .

To determine parameters associated to the second Weibull distribution, it is necessary to apply the specific method, presented on chapter 4.4.2, i.e. plotting Ln(λinst (t)) as a function of Ln(t), where λinst (t) is expressed by:

y = 0.8555x ‐ 5.6952R² = 0.9819

y = 3.5604x ‐ 21.553R² = 0.9649

‐4.7

‐3.7

‐2.7

‐1.7

‐0.7

0.3

1.3

2.2 2.7 3.2 3.7 4.2 4.7 5.2 5.7 6.2 6.7

ln(ln(1/(1‐F*(t))))

ln(t)


42


∆

∆ ∗00 ∗

Along the second distinguishable time interval, it is also necessary to substrate to λinst (t) computed values of λ1(t) - previously determined - and then, to apply the logarithm to this difference, as seen below, where the purple scatter represents the corrected term Ln(λinst (t) – λ1(t)):

Given the linear regression results when applied to the purple scatter, we can compute the second Weibull distribution’s parameters (with a R2 correlation coefficient equal to 0.947):

1 9.0255

525

0

And then:

0.0011 ∗ 0.0013 ∗ . , 356days

0.0011 ∗ 0.0013 ∗ . 9.0255525

∗525

.

, 356days

The failure rate function’s trend is given by the next graph:

y = 7.0939x ‐ 48.394R² = 0.9386

y = 8.0255x ‐ 54.334R² = 0.947

‐8

‐7

‐6

‐5

‐4

‐3

‐2

‐1

0

0 1 2 3 4 5 6 7

Ln(λ(t))

Ln(t)


43


Note the current failure rate is equal to 3.5 failures every 100 days, which corresponds to 1 failure every 30 days. This value is approximately the observed current failure rate (see table [7]).

The Kolmogorov-Smirnov’s test gives the results below:

For N = 43 and α = 5%:

D limit = 0.2020


The following graph represents the theoretical and observed R(t) probabilities’ trends:

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 100 200 300 400 500 600 700

Lambda (failu

re/day)

time (day)


44


See below the plotting of the absolute difference | ∗ | as a function of time:

Note the maximum values of ‘D’ are reached from t = 405 to t = 550 days, said differently, from February 2012 to current date. This is due to an important concentration of events which have happened after the winter Technical Stop. A recrudescence of failures is currently observed after a given Technical Stop, during start up stages.

The Pareto analysis of causes gives the next results:

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400 500 600 700

R(t)

t (day)

theo1

theo2

real

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

10 35 73 130 163 191 203 284 321 349 387 405 416 423 433 437 460 483 497 550 573

D=A

bs(F theo(t) ‐F*(t))

t (day)

Kolmogorov‐Smirnov's test


45


CAUSE N %Command 21 48.8Mechanical 11 25.6Unknown 5 11.6Utilities 5 11.6Environment 1 2.3

Total 43 100

Two failure modes are dominant: ‘command’ and ‘mechanical’ causes both represent around 75% of the overall. Given the large available data for each of them, a much more detailed statistical analysis is possible.

5.2.4.2 ‘Command’failuremode

The specific study of ‘command’ failure mode gives the following scatter of ∗ as

a function of Ln(t):

0

5

10

15

20

25

Command Mechanical Unknown Utilities Environment

Global Pareto histogram for CVs

y = 1.8996x ‐ 11.522R² = 0.9254

‐3.7

‐3.2

‐2.7

‐2.2

‐1.7

‐1.2

‐0.7

‐0.2

0.3

0.8

1.3

4.1 4.6 5.1 5.6 6.1 6.6

ln(ln(1/(1‐F*(t))))

ln(t)


46


The overall trend can be considered as linear, with a correlation coefficient R2 equal to 0.9254. The same concentration of events around X-coordinate Ln(t) = 6 as previously seen for other populations of components is observed. This corresponds to spring 2012 (i.e. from February to May 2012), which follows the winter Technical Stop.

Considering the R2 value, the correlation between the observed distribution and a Weibull one is possible:

1.8996

431

0

Given these parameters, we can compute and plot the failure rate function as below:

1.8996431

∗431

.

The validity to the Weibull distribution hypothesis characterized by the previous parameters is confirmed by the next Kolmogorov-Smirnov’s test:

For N = 21 and α = 5%:

D limit = 0.2872


The following graph represents the theoretical and observed F(t) probabilities’ trends:

0

0.001

0.002

0.003

0.004

0.005

0.006

0 100 200 300 400 500 600

Lambda (failu

re/day)

time (day)


47


A much more detailed Pareto analysis for ‘command’ failure mode gives the following distribution of secondary causes:

CAUSE Quantity %PROFIBUS 8 38.1 SIPART 7 33.3 SETTING 3 14.3 SAFETY COMPONENT 1 4.8 UNKNOWN 1 4.8 INSTRU 1 4.8

Total 21 100

Failures which are more frequent than others are mainly due to ‘Profibus’ system (including cables and DP/AP converters) and then to ‘Sipart’ device. Their occurrence goes increasing as

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500 600

F(t)

t (day)

theo

real

0

1

2

3

4

5

6

7

8

9

PROFIBUS SIPART SETTING SAFETYCOMPONENT

UNKNOWN INSTRU


48


revealed by the β value superior to 1. The third most frequent failure mode is the ‘Settings’ one. But in the worst case, it is a systematic problem which is independent to time (when no corrections are applied): there are no apparent reasons its own failure rate increases.

The computation of MTBF for ‘command’ systems related to CVs gives the next value:

. 11

382

5.2.4.3 ‘Mechanical’failuremode

When plotting ∗ as a function of Ln(t) for ‘mechanical’ failure mode, the observed

scatter is equivalent to a linear distribution, as seen below:

Given the R2 value equal to 0.968, we can correlate the failure distribution to a Weibull’s law, whose parameters are:

0.6494

329

0

Given these parameters, we can compute and plot the failure rate function as below:

0.6494329

∗329

.

y = 0.6494x ‐ 3.7642R² = 0.968

‐2.7

‐2.2

‐1.7

‐1.2

‐0.7

‐0.2

0.3

0.8

1.3

2.2 2.7 3.2 3.7 4.2 4.7 5.2 5.7 6.2 6.7

ln(ln(1/(1‐F*(t))))

ln(t)


49


By applying the Kolmogorov-Smirnov’s test to the observed distribution, we can validate the hypothesis of its fitting to a Weibull distribution with a 95% level of confidence, as presented below:

For N = 11 and α = 5%, a Kolmogorov-Smirnov’s table gives the following limit:

D limit = 0.3912


For ‘mechanical’ failure mode, the fitting between theoretical and measured distributions is illustrated below:

The computed MTBF for ‘mechanical’ failure mode is equal to:

y = 0.0151x‐0.351

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 100 200 300 400 500 600

Lambda (failu

re/day)

time (day)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 100 200 300 400 500 600

F(t)

t (day)

theo

real


50


. 11

450

This MTBF value is higher than the related value for ‘command’ systems. Indeed, the failure rate for ‘command’ is increasing whereas it is decreasing for ‘mechanical’. This means ‘command’ systems are mainly responsible for the overall increasing failure rate’s trend associated to CVs.

5.2.7 Electric heaters

When only considering events linked with electric heaters on tunnels, 8 failures are available to a statistical study. Curiously, these events are concentrated on a time period which extends from February 2012 to June 2012 (censuring a singular event on February 2011, which introduces bias in the statistics).

Note not enough data related to electric heaters from phases separators is available to be studied too.

Plotting of ∗ as a function of Ln(t) gives the following results:

Note the low R2 coefficient’s value, equal to 0.8372. This can be explained by the low number of events available during the studied period. Perhaps two failure modes are combined what can leads to such a distribution of events, difficult to correlate with certain accuracy to a Weibull one.

y = 8.9778x ‐ 55.668R² = 0.8372

‐2.5

‐2

‐1.5

‐1

‐0.5

0

0.5

1

6 6.05 6.1 6.15 6.2 6.25 6.3 6.35

ln(ln(1/(1‐F*(t))))

ln(t)


51


However, taking into account that the precision on Weibull parameters will not be as good as requested, we can compute β, η and γ values:

8.9778

493

0

Then we can compute and plot the λ(t) function, as follows:

8.98493

∗493

.

The Kolmogorov-Smirnov’s test allows accepting the Weibull distribution hypothesis:

For N=8 and α = 5 %:

D limit = 0.4543

The maximum value for | ∗ | is equal to 0.114 so we can accept the Weibull distribution hypothesis, with a 95% level of confidence.

When plotting theoretical and measured F functions, we can note the difference between the prediction and the observed failure probability, at current date:

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

400 420 440 460 480 500 520 540 560

Lambda (failu

re/day)

time (day)


52


Indeed, the current λ value is equal to 4 failures every 100 days, which is equivalent to 1 failure every 25 days. This is not the observed value which is around 1 failure every 35 days (see table [7]).

The MTBF related to electric heaters can be computed, as follows:

. 11

467

Even if the current failure rate value is high, the MTBF which comes from a computation including β but also η value is relatively high, in comparison with other components.

In terms of causes, the ‘command’ failure mode is dominant, as seen below:

CAUSE Quantity %Command 6 77.8Hardware 1 11.1Power 1 11.1Unknown 0 0.0Environment 0 0.0

Total 8 100.0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

400 420 440 460 480 500 520 540 560

F(t)

t (day)

theo

real


53


Given the observed precision between the measured and theoretical distribution, we can assume the ‘command’ mode is a combination of at least two failure sub modes.

5.2.8 PV valves for nitrogen

PV valves conveying nitrogen, mainly those referenced ‘PV409’ are frequently associated to leak of nitrogen, which could be problematic in terms of availability of the stored fluid.

Even if not enough data was cumulated from 1st January 2011 to 1st August 2012, it is interesting to note the particular observed accuracy when correlating the current distribution to a Weibull one. This 5-event distribution can be precisely correlated by a linear regression, as presented below:

0

1

2

3

4

5

6

7

Command Hardware Power Unknown Environment

y = 14.344x ‐ 88.375R² = 0.9562

‐2

‐1.8

‐1.6

‐1.4

‐1.2

‐1

‐0.8

‐0.6

‐0.4

‐0.2

0

6.02 6.04 6.06 6.08 6.1 6.12 6.14 6.16 6.18

ln(ln(1/(1‐F*(t))))

ln(t)


54


The R2 coefficient equal to 0.9562 reveals the accuracy of the linear correlation; it is so possible to compute the Weibull’s parameters:

14.344

474

0

The computed expression and plot for λ(t) are presented below:

14.344474

∗474

.

The Kolmogorov-Smirnov’s test gives the results below:

For N = 5 and α = 5%:

D limit = 0.5653

The maximum value for | ∗ | is equal to 0.069, so we can accept the Weibull distribution hypothesis, with a 95% level of confidence. Note the extremely low value for Max(D), in comparison with all the observed values for previous populations of components.

The following graph represents the theoretical and observed F(t) probabilities’ trends:

y = 6E‐38x13.344

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

410 420 430 440 450 460 470 480

Lambda (failu

re/day)

time (day)


55


The leakage rate for PVs conveying nitrogen is considerably increasing, with a high β value equal to 14.344. However, note the low value of the current failure rate, which is equal to 3 failures every 100 days (said differently: 1 event every 30 days). The MTBF related to PVs - nitrogen is equal to:

. 11

457

This value is approximately equal to the MTBF of CVs - ‘mechanical’ failure mode.

5.3 Global synthesis and index ‘O’ attribution

The table below sums up the previous results. We can associate to each next population of components an occurrence index, noted O, recalling the criterions presented in chapter 3.2:

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

410 420 430 440 450 460 470 480

F(t)

t (day)

theo

real


56


Population type

λ’s trend N of events MTBF (day) Index O

Cold compressors –

general

β = 2.68 ↗ ++

25

327

1

Cold compressors –

regulator

β = 2.76 ↗ ++

11

362

1

Warm compressors –

general

β = 0.89

↘

26

325

0.5

Warm compressors –

without leakage

β = 0.82

↘

21

336

0.5

Turbines – general

β = 1.05

→

25

407

0.75

Turbines – ‘command’

β = 1.26 ↗ +

15

477

0.75

CVs – general

↗ ++

43

-

1

CVs – ‘command’

β = 1.9 ↗ +

21

382

1

CVs – ‘mechanical’

β = 0.65

↘

11

450

0.25

Electric heaters

β = 8.98 ↗ ++

8

467

0.75

PVs for N2

β = 14.3 ↗ ++

5

457

0.75


57


5.4 Severity analysis and index ‘S’ attribution

The following ‘severity analysis’ is based on the previously mentioned criterions (see chapter 3.3). Its input is the statistics of losses of ‘CM’ conditions, for 2011 and 2012 (see tables [8] and [9]). Its output is related to the previous populations.

When exploring the statistics of losses of ‘CM’ conditions, the following results can be drawn:

0 0.2 0.4 0.6 0.8 1 1.2

CV valves ‐ MEC

Warm Compressors (without leakage)

PV valves for N2

Turbines ‐ command

Electric Heatears

CV valves ‐ COM

Cold Compressors ‐ regulator

INDEX OF OCCURENCE OF PROCESS HARDWARE COMPONENTS


58


6. CRITICALITY RANKING

The criticality index - previously defined in chapter 3.1 – is obtained by multiplying the occurrence and severity indexes. This leads to the following results:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

PV valves for N2

CV valves ‐ MEC

Electric Heatears

CV valves ‐ COM




SEVERITY OF PROCESS HARDWARE COMPONENTS


59


In terms of criticality, we can conclude the three populations we must focus on, in order to increase the overall process’ availability, are (in order of importance):

Cold compressors’ regulators; CV valves (command); Turbines (command).

Also, a detailed research for causes showed in chapters 5.2.3 and 5.2.4 the most critical elements for command systems are:

Profibus and Sipart devices for CVs; Flow switches and regulators for TUs.

However, if occurrence has less importance in terms of criticality than severity (i.e. losses of ‘CM’ conditions), it is possible to use the previous ‘severity’ histogram to highlight new populations we must focus on. In that case, the next populations have a particular interest (sorted in order of severity):

Cold compressors’ regulators; Warm compressors; Turbines (command).

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0

CV valves ‐ MEC

PV valves for N2

Electric Heatears



CV valves ‐ COM


GLOBAL CRITICALITY OF PROCESS HARDWARE COMPONENTS


60


7. SOURCES OF BIAS – LIMITS OF THE ANALYSIS

For the studied populations, the level of correlation to a Weibull distribution - both tested by the linear correlation coefficient and a Kolmogorov-Smirnov’s test - is acceptable in order to draw conclusions about the overall process’ reliability. However, keeping in mind the previous results in terms of fitting, perhaps statistical bias have been introduced in the present study. Their sources could be:

Singular events included in treated data (e.g. happened during August 2011 for cold compressors);

Bad inclusion of an event in a studied population, because of a bad interpretation of this event into ‘Logbook OA’;

Apparently, the magnitude of this introduction of bias is limited, according to the results shown in chapter 5.

8. FOR FURTHER

The next step of the present analysis consists in highlighting the intrinsic nature of the previously identified causes of failure, and then, setting up an action plan in order to reduce the magnitude of such events, nay to avoid them.

Perhaps, it could be interesting to extend the present methodology of statistical study to other components, such as electric and electronic ones (transmitters, etc.) which are also responsible for several losses of ‘CM’ conditions, which affect the overall process’ availability.

9. CONCLUSION

To conclude this criticality analysis, basis of a statistical method of study have been set up for hardware components. Perhaps this kind of analysis will be extended to other components, and then will lead to a coherent ensemble of statistical data related to process’ reliability. Then, it would be possible to improve our envisaged action plan by focussing on most critical components, not only most critical hardware components.

For the highlighted most critical hardware components, a refined analysis of origins of failure modes must be set up. These components are:

Cold compressors’ regulators; Profibus and Sipart devices for CVs; Flow switches and regulators for TUs.


61


10. REFERENCES

[1] AMDEC – Guide pratique, G. Landy, Afnor editions

[2] Fiabilite, maintenance et risque, D. Smith, Dunod editions

[3] Ingenierie de la maintenance, J-C. Francastel, Dunod editions

[4] Courses in operational safety and maintenance, Arts et Metiers Paristech, D. Dufrene

[5] Courses in statistics, Arts et Metiers Paristech, P. Bastet

[6] CERN-ATS-2012-006, First assessment of reliability data for the LHC accelerator and detector cryogenic system components, G. Perinic

[7] Excel© Table, LHC CRYO – Statistical study, J. Martin

[8] Excel© Table, 2012_Suivi_perte_CM-STAT-V2-inworks, E. Duret

[9] Excel© Table, CM_ALL_SECTORS-2011-V2 + graph, E. Duret

[10] Logbook OA

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