1. INT R OD UC T ION 1.1 Maintenance
Maintenance can be the secret of success for an in-dustry and for a ship management.
In the last years, because of the crisis some indus-tries, and even ships maintenance changed, some-times from preventive maintenance to a contingency scenario. Many studies have been carried out aiming the prevention of failures and to enhance the inter-ventions performance intervening equipments when it is imperative and when it is opportunistic.
Opportunistic maintenance can be corrective con-sidering some level of failure allowance (Pham & W ang, 2000). T his can be a good maintenance sys-tem for continuous equipment monitoring which is what we propose. ( Z hou et al, 2006)
T he statistical equipment control (S E C ) can be use-ful to an opportunistic maintenance management, considering that we can try to predict what´s the ten-dency of the equipment to suffer an anomal y and how long it will maintain its operational state.
F or phase 1 of the study, traditional control charts are used and, for phase 2, the modified C umulative S um (C US UM) control charts are proposed for on-line condition monitoring. T he method is illustrated using observations of two variables on a diesel gene-rator.
2. C ONT R OL C HA R T S IN C OND IT ION MONIT OR ING
2.1 Phase 1 Independent D ata
In phase 1 for continuous and independent variables, X and MR control charts are implemented. T o define working parameters charts must be in statistical con-trol. T o parameters calculation for individual obser-vation 200 obserobser-vations must be used on first phase considering the Quesenberry (Pereira & R equeijo, 2012) and L ampreia (2013).
C onsidering
t t
X =me , with e white noise. W here the X is the individual observation and
1
-=
i i i X X
MR
.
T he upper control limit (UC L ) and the center line (C L ) of those charts, are calculated respectively for X chart with X and X
X 3s , and for MR chart with
MR and
4
D MR . T he parameters are estimated by ˆ
μ=X and
2
d MR ˆ =
s .T he constants
3 D ,
4 D and
2 d
depend on the sample dimension and can be con-sulted in a table of Pereira & R equeijo (2012).
Autocorrelated D ata
If the data is autocorrelated the A R IMA (A utore-gressive Integrated Moving A verage) models should
Opportunistic Maintenance B ased on C US UM C ontrol C harts
S . L ampreia, V . V airinhos, V . L obo & R . Parreira
suzana.lampreia@ gmail.com, V alter.vairinhos@ sapo.pt, vlobo@ isegi.unl.pt, ribeiro.parreira@ marinha.pt, C entro de Investigaçã o Naval (C INAV ), Alfeite, 2810 Almada, Portugal
J . G. R equeijo
jfgr@ fct.unl.pt, F aculty of Science and T echnology of the Universidade Nova of L isbon,Mechanical and In-dustrial E ngineer D epartment, L isbon. 2829-516, C aparica, Portugal
be applied. T he techniques to be applied are the same as those used for independent data in tradition-al charts, but applied to the residues from the ad-justment. B ased on the A R IMA model, adjusted by A R (p), MA (q) or A R MA (p,q)), the µ and dispersion σ parameters estimation must be calculated. If there is some outliers those observations must be replaced by the expected values for the corresponding instant; the new residues should be calculated and the same for the reviewed charts. ( Pereira & R equeijo, 2012)
If the model is satisfactory, its residues are esti-mated with
t t
t X
ˆ X
e = - (
t
Xˆ is the expected value for the period t). C ontrol charts are built from these residues, obtaining the mean and standard deviation F or the modelAR
(
p)
, the parameters are estimated from the expressions:( ) p j j1 ˆ ˆ
E X m x 1 j
= = = ∆ - (1) and ( ) p 2
0 j j
j1
ˆ
VÂR X 1
e
g s r j
= = = ∆ - (2)
W here Eˆ(X)is the estimated mean, ξ is mean process determination process from a AR(p) process, and ρj is the lag coefficient correlation, and φj is the order
j parameters from a A R model.
F or more details to independent and autocorre-lated data Pereira & R equeijo (2012) should be con-sulted.
2.2 Phase 2 Independent D ata
“Modified C US UM” charts are built on cumulative sum –C - defined by (Perry & Pignatiello, 2011):
(
)
(
0 C)
; 00
1 - =
=
- Z k C
, max C
t t
t (3)
where
(
(
)
)
X L t
t X T
Z = - s , n
X
s
s = ,
X s d = D , 2 d =
k e
(
)
tan
L L S dard S
T = T - D and d s
1 = D
S , where d1
is constant.
t
X is the sample mean at t, TL is the maximum
admissible value, specified by normative or by the manufacturer, s the process standard deviation, n the sample dimension,
t
Z is the reduced form of
t
X , k the reference value and ∆s is the safety factor.
(B arbosa, 2012)
T he “Modified C US UM” chart has two limits; one is the A lert L evel ( A L ), and the second is the Upper C ontrol L evel (UC L ). T he A L and UC L cal-culations are based on Gan (1991) abacus, in func-tion of A R L (A verage R un L ength) value for both situations and the reference value k. In this study a significance level a =1% (ARL=100) will be
consi-dered to define A L and a =0,2% (ARL=500) to de-fine UC L .
A utocorrelated C harts
F or second phase with autocorrelated data, the Mod-ified C US UM are built based on the C statistic, both calculated by the prediction errors
t
e . T he predic-tion errors are estimated at the timet by
( ) T ( ) ˆT ( )
e T X T X T
t = t - t , where XTt(T) is the value for
that time, and ˆT
(
)
X T
t
the predicted value for timet;
the prediction for the present time is the same as the final value of phase 1.
F or predicted, values the TL expression should
in-clude the mean so it becomes: L ( L)Norma S
T = T - m- D .
( L ampreia, 2013)
T he C US UM charts has a higher and consistent sensibility to parameters values variation. (S ibanda & S ibanda, 2007)
3. ME T HODOL OGY
T he proposed methodology is to be applied to re-pairable systems using statistical control charts, con-sidering both independent and autocorrelated data. S pecificall y, the following steps must be taken:
C ollect 200 samples of each variable with the diesel generator operational and in a good state ( L ampreia, 2013). C heck its indepen-dence, normality and calculate the function-ing parameters:
- F or independent data, build the traditional control charts ( X and MR charts).
- F or autocorrelated data fit an A R IMA Model, build e- MR C harts from the resi-dues.
T o monitor vibration process, build the Mod-ified C US UM chart for independent or auto-correlated data, using the values collected af-ter the anomal y simulation
D efine the standard limits of the variables, and specify the change in the mean to be de-tected.
S et the two control limits - A L and UC L . D efine the intervention rules:
- Proceed to an inspection, when 6 con-secutive points are above the A L .
4. C A S E S T UD Y
F or the case study two variables will be considered, one represents the generator cooling water tempera-ture (V ar1) and the other the L ub oil tempera-ture(V ar2); the maximum allowed values are 89º C and 100º C respectively.
4.1 Phase 1
S tudying the variables, using the S T A T IS T IC A software, we obtain V ar1 autocorrelated and V ar2 independent.
F or example the data normality for V ar1 was ve-rified with the K olmogorov-S mirnov test, F igure 1, where
C rítico
0,886 0,886
D 0,06265
N 200
= = = fora =5%
,and d= 0,04659. B ecause
Crítico
dD the normality condition is accepted.
F igure 1. V ar1 - Normality S tudy
F igures 2 and 3 present the autocorrelation func-tion and partial autocorrelafunc-tion funcfunc-tion for variable 1. W e are in the presence of an A R 4, because we can see a peak in a lag of order 4, the other one´s we´ll be ignored because it has less significance, p= 0,0000.
F igure 2. V ar1 – A C F with review data
F igure 3. V ar1 – PA C F with review data
W e also calculate the residues using S T A T IS T I-C A and built the e –MR C ontrol charts. In figure 4 we can observe the residues of V ar1 without review, and for MR we see two outliers. S o the outliers should be substituted with the values calculated from A R IMA model.
F igure 4. V ar1 - R esidues e –MR C ontrol chart with none
F igure 5. V ar1 - R esidues e –MR C ontrol chart with review
da-ta
T he V ar2 parameters calculated are in table 1:
T able 1. V ar 2 parameters
1 d2
72,84 0,6441 1,128 25,92
F or V ar2 already with the replaced outliers:
F igure 6. V ar2 - e –MR C ontrol chart with reviewed data
F or V ar2 the parameters are: µ =86,282, σ =1,9242.
4.2 Phase 2
T o accomplish this equipment study, because it is operational the data on phase 2 is simulated, consi-dering three anomal y progression steps (1, 2, and 3). F or variable 1 we obtained the following limits:
K = δ /2
0.25 0.5 0.75
A R L 500
UC L
W ith α =0,2%
8.5 5.1 3.5
100
A L
with α =1%
5.51 3.5 2.5
F or progression 1 none observation above zero it is registered, both for V ar1 and V ar2.
F igure 7. V ar1 – Modified C USUM chart with ∆ =0,5σ -
2ªProgressã o
In figure 7, we can see the second progression for variable 1, where there isn´t any need of interven-tion.
F igures 8 and 9 show the third progression where for ∆ =0,5σ the sensibility is higher, and it shows a need for inspection on 23 observation.
F igure 8. V ar1 – Modified C US UM chart with ∆ =0,25σ
-3ªProgressã o
F igure 9. V ar1 – Modified C USUM chart with ∆ =0,5σ
-3ªProgressã o
F igure 10. V ar1 – Modified C USUM chart with ∆ =0,75σ
-3ªProgressã o
F or variable 2 in figure 11 we have some values registered but none above the A L , so there´s no need of intervention.
F igure 11. V ar2 – Modified C U SUM chart with ∆ =0,5σ
-2ªProgression
A nalyzing figures 12 and 13, we can observe that, for third progression for variable 1 with ∆ =1,0σ , the sensibility is higher than for ∆ =0,5σ .
F igure 12. V ar2 – Modified C U SUM chart with ∆ =0,5σ
-3ªProgressã o
F igure 13. V ar2 – Modified C U SUM chart with ∆ =1,0σ
-3ªProgressã o
F igure 12, in observations 21
st
, shows that there is a need for inspection and, on 22
nd
, an intervention action is needed, according to defined rules. A lso according to those rules, F igure 13 shows that, since the 11
th
observation, both an inspection and an inter-vention action are needed, opportunity maintenance should occur accordingl y the equipment availability.
Observing the original data, and the results of the charts, in this case we believe that the ∆ =0,5σ is more accurate than for higher values, not showing excessive sensibility.
5. C ONC L US IONS
F or the same equipment we obtained one variable independent and other dependent.
F or higher ∆ we get higher chart sensibility for this variables monitoring.
F or autocorrelated data the residues should be used in phase 1 and the expected values in phase 2.
T he parameters should be calculated based on a good equipment state data.
T he A R L value should be flexible considering the application and the owner specifications and requi-sites.
T he limits of the C US UM charts and rules for in-tervention should be adequate to the equipment state and fabricant and owner requisites.
T he modified C US UM control chart can be used for online condition monitoring and in a system submitted to an opportunistic maintenance policy.
W e expect that modified C US UM charts allied to an opportunistic maintenance policy can reduce maintenance costs, planning equipment intervention for the right moment.
6. F UT UR E W OR K
T est this equipment data with modified multivariate C US UM charts.
T est other generator variables appl ying modified C US UM charts.
D evelop software that allows the online monitor-ing with various variables.
R E F E R E NC E S
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oppor-tunistic maintenance of a k-out-of-n:g system with
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maintenance policy for continuously monitored systems.
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