A cost driven predictive maintenance policy for structural airframe maintenance 1 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 Chinese Journal of Aeronautics, (2017), xxx(xx) xxx–xxx CJA 776 No of P[.]
Trang 17 aUniversite´ de Toulouse, INSA/UPS/ISAE/Mines Albi, ICA UMR CNRS 5312, Toulouse 31400, France
8 bDepartment of Mechanical & Aerospace Engineering, University of Florida, Gainesville 32611, USA
9 Received 29 June 2016; revised 8 October 2016; accepted 12 December 2016
10
13
14 Extended Kalman filter;
15 First-order perturbation
17 Model-based prognostic;
18 Predictive maintenance;
19 Structural airframe
Abstract Airframe maintenance is traditionally performed at scheduled maintenance stops The decision to repair a fuselage panel is based on a fixed crack size threshold, which allows to ensure the aircraft safety until the next scheduled maintenance stop With progress in sensor technology and data processing techniques, structural health monitoring (SHM) systems are increasingly being considered in the aviation industry SHM systems track the aircraft health state continuously, lead-ing to the possibility of plannlead-ing maintenance based on an actual state of aircraft rather than on a fixed schedule This paper builds upon a model-based prognostics framework that the authors developed in their previous work, which couples the Extended Kalman filter (EKF) with a first-order perturbation (FOP) method By using the information given by this prognostics method, a novel cost driven predictive maintenance (CDPM) policy is proposed, which ensures the aircraft safety while minimizing the maintenance cost The proposed policy is formally derived based on the trade-off between probabilities of occurrence of scheduled and unscheduled maintenance A numerical case study simulating the maintenance process of an entire fleet of aircrafts is imple-mented Under the condition of assuring the same safety level, the CDPM is compared in terms
of cost with two other maintenance policies: scheduled maintenance and threshold based SHM maintenance The comparison results show CDPM could lead to significant cost savings
Ó 2017 Production and hosting by Elsevier Ltd on behalf of Chinese Society of Aeronautics and Astronautics This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/
licenses/by-nc-nd/4.0/ ).
21
22
1 Introduction
23 Fatigue damage is one of the major failure modes of airframe
24 structures Repeated pressurization/depressurization during
25 take-off and landing cause many loading and unloading cycles
26 which could lead to fatigue damage in the fuselage panels The
27 fuselage structure is designed to withstand small cracks, but if
28 left unattended, the cracks will grow progressively and finally
* Corresponding author.
E-mail addresses: yiwang@insa-toulouse.fr (Y Wang), christian.
gogu@univ-tlse3.fr (C Gogu), nicolas.binaud@univ-tlse3.fr
(N Binaud), christian.bes@univ-tlse3.fr (C Bes), haftka@ufl.edu
(R.T Haftka), nkim@ufl.edu (N.H Kim).
Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
Chinese Society of Aeronautics and Astronautics
& Beihang University
Chinese Journal of Aeronautics
cja@buaa.edu.cn www.sciencedirect.com
http://dx.doi.org/10.1016/j.cja.2017.02.005
Ó 2017 Production and hosting by Elsevier Ltd on behalf of Chinese Society of Aeronautics and Astronautics.
Trang 229 cause panel failure It is important to inspect the aircraft
reg-30 ularly so that all cracks that have the risk of leading to panel
31 fatigue failure should be repaired before the failure occurs
32 Traditionally, the maintenance of aircraft is highly
regu-33 lated through prescribing a fixed schedule At the time of
34 scheduled maintenance, the aircraft is sent to the maintenance
35 hangar to undergo a series of maintenance activities including
36 both engine and airframe maintenance Structural airframe
37 maintenance is a subset of airframe maintenance that focuses
38 on detecting the cracks that can possibly threaten the safety
39 of the aircraft In this paper, maintenance refers to structural
40 airframe maintenance while engine and non-structural
air-41 frame maintenance are not considered here Structural
air-42 frame maintenance is often implemented by techniques such
43 as non-destructive inspection (NDI), general visual inspection,
44 detailed visual inspection (DVI), etc Since the frequency of
45 scheduled maintenance for commercial aircraft is designed
46 for a low probability of failure, it is very likely that no safety
47 threatening cracks exist during earlier life of majority of the
48 aircraft Even so, the intrusive inspection by NDI or DVI
49 for all panels of all aircraft needs to be performed to guarantee
50 the absence of critical cracks that could cause fatigue failure
51 Therefore, the inspection process itself is the major driver of
52 maintenance cost
53 Structural health monitoring (SHM) systems are
increas-54 ingly being considered in aviation industry.1–4SHM employs
55 a sensor network sealed inside the aircraft structures like
fuse-56 lage, landing gears, bulkheads, etc., for monitoring the damage
57 state of these structures Once the health state of the structures
58 can be monitored continuously or as frequently as needed, it is
59 possible to plan the maintenance based on the actual or
pre-60 dicted information of damage state rather than on a fixed
61 schedule This spurs the research to predictive maintenance
62 Prognostic is the prerequisite of the predictive maintenance
63 Prognostics methods can be generally grouped into two
cate-64 gories: data-driven and model-based Data-driven approaches
65 use information from previously collected data from the same
66 or similar systems to identify the characteristics of the damage
67 process and predict the future state of the current system
68 Data-driven prognosis is typically used in the cases where
69 the system dynamic model is unknown or too complicated to
70 derive Readers can refer to5,6that give an overview of
data-71 driven approaches Model-based prognostics methods assume
72 that a dynamic model describing the behavior of the
degrada-73 tion process is available For the problem discussed at hand, a
74 model-based prognostics method is adopted since the fatigue
75 damage models for metals have been well researched and are
76 routinely used in the aviation industry for planning the
struc-77 tural maintenance.7–9
78 Predictive maintenance policies that aim to plan the
main-79 tenance activities taking into account the predicted
informa-80 tion, or the ‘‘prognostics index” were proposed recently and
81 attracted researcher’s attention in different domains.10–14The
82 most common prognostics index is remaining useful life
83 (RUL).15–18A large amount of methods on RUL estimation
84 have been proposed such as filter methods (e.g., Bayesian
fil-85 ter,19particle filter,20,21stochastic filter,22,23Kalman filter24,25),
86 and machine learning methods (e.g., classification
meth-87 ods,26,27 support vector regression28) In addition to the
88 numerical solutions for RUL prediction, Si et al.29,30derived
89 the analytical form of RUL probability density function Some
90 of the predictive maintenance policies adopting the RUL as a
91 prognostics index to dynamically update the maintenance time
92 can be found in Refs.12, 14, 31
93
In some situations, especially when a fault or failure is
94 catastrophic, inspection and maintenance are implemented
95 regularly to avoid such failures by replacing or repairing the
96 components that are in danger In these cases, it would be
97 more desirable to predict the probability that a component
98 operates normally before some future time (e.g next
mainte-99 nance interval).32 Take the structural airframe maintenance
100
as an example, the maintenance schedule is recommended by
101 the manufacture in concertation with safety authorities
Arbi-102 trarily triggering maintenance purely based on RUL prediction
103 without considering the maintenance schedule might be
dis-104 ruptive to the traditional scheduled maintenance procedures
105 due to less notification in advance In addition, planning the
106 structural airframe maintenance as much as possible at the
107 scheduled maintenance stop when the engine and
non-108 structural airframe maintenance are performed could lead to
109 cost saving To this end, instead of predicting the remaining
110 useful life of fuselage panels, we consider the evolution of
dam-111 age size distribution for a given time interval, before some
112 future time (e.g next maintenance interval) In other words,
113
we adopt the ‘‘future system reliability” as the prognostics
114 index to support the maintenance decision making This
distin-115 guishes our paper from the majority existing work related to
116 predictive maintenance
117 The motivation developing advance maintenance strategies
118
is to reduce the maintenance costs while maintaining safety
119 Researchers proposed many cost models to facilitate the
com-120 parison of maintenance strategies.10,12,13,33All these cost
anal-121 ysis and comparison share one thing in common The
122 maintenance strategy is independent from unit cost (e.g., the
123 set up cost, the corrective maintenance cost, the predictive
124 maintenance cost, etc.) and the interaction between strategy
125 and unit cost has not been considered, which in fact might
126 affect the maintenance strategy in some situations For
exam-127 ple, in aircraft maintenance, it is beneficial to plan the
struc-128 tural airframe maintenance as much as possible at the same
129 time of scheduled maintenance and only trigger unscheduled
130 maintenance when needed If the cost of unscheduled
mainte-131 nance is much higher than the scheduled maintenance, the
132 decision maker might prefer to repair as many panels as
possi-133 ble at scheduled maintenance to avoid unscheduled
mainte-134 nance That is to say the cost ratio of different maintenance
135 modes could be a factor that affects the maintenance
136 decision-making In this paper, we take a step further from
137 the existing work to take into account the effect of cost of
dif-138 ferent maintenance modes on the maintenance strategy, i.e.,
139 the cost ratio is taken as an input of maintenance the strategy
140 and partially affects the decision-making This is our
motiva-141 tion of developing the cost driven predictive maintenance
142 (CDPM) policy for aircraft fuselage panel By incorporating
143 the information of predicted damage size distribution and
144 the cost ratio between maintenance modes, an optimal panel
145 repair policy is proposed, which selects at each scheduled
146 maintenance stop a group of aircraft panels that should be
147 repaired while fulfilling the mandatory safety requirement
148
As for the process of prognosis, we consider four uncertainty
149 sources The item-to-item uncertainty accounts for the
variabil-150 ity among the population, which is considered by using one
151 degradation model to capture the common degradation
charac-152 teristics in the population, with several model parameters
Trang 3153 following initial distributions across the population to cover the
154 item-to-item uncertainty The epistemic uncertainty refers to
155 the fact that for an individual degradation process the
degrada-156 tion model parameters are unknown due to lack of knowledge
157 This uncertainty can be reduced by measurements, i.e., the
158 uncertainty of parameters can be narrow down with more
mea-159 surements are available The measurement uncertainty means
160 that SHM data could be noisy due to harsh working conditions
161 The process uncertainty refers to the noise during the
degrada-162 tion process This is considered through modeling the loading
163 condition that affect the degradation rate as uncertain To
164 our best knowledge, these four uncertainties cover the most
165 common uncertainties sources that are encountered during
166 the prognostics procedure for fuselage panels
167 To account for the uncertainties mentioned above, a
state-168 space mode is constructed and the Extended Kalman filter
169 (EKF) is used to incorporate the noisy measurements into
170 the degradation model to give the estimates of damage size
171 and model parameters as well as the estimate uncertainty
172 (i.e., the covariance matrix between damage size and model
173 parameters) After obtaining the estimates and its uncertainty
174 from EKF, the straightforward way to predict the future
dam-175 age size distribution is Monte Carlo method, which is
time-176 consuming and gives only numerical approximation Instead,
177 we propose the first-order perturbation method to allow
ana-178 lytical quantification of the future damage size distribution
179 As such, the main contributions of this paper are the
fol-180 lowing four aspects
181 Incorporating the ‘‘future system reliability” as a
prognos-182 tics index to support the maintenance-decision making
183 Considering the cost ratio of different maintenance modes
184 as the input the maintenance strategy
185 Taking into account four uncertainty sources: item-to-item
186 uncertainty, epistemic uncertainty on the degradation
187 model, measurement uncertainty and process uncertainty
188 Utilizing a first-order perturbation method to quantify the
189 future damage distribution analytically
190
191 The paper is organized as follows Section2introduces the
192 crack growth model used for modeling the degradation of the
193 fuselage panels, degradation which induces the requirements
194 for maintenance This degradation process is affected by
195 various sources of uncertainty, which are also described in
Sec-196 tion2 In order to be able to set-up the proposed predictive
197 maintenance strategy we need to be able to predict the crack
198 growth in future time while accounting for the sources of
199 uncertainty present To achieve this we first identify the
200 parameters governing the crack growth based on crack growth
201 measurements on the fuselage panels up to the present time
202 To carry out this identification we use the EKF, which is
sum-203 marized in Section3 Note that due to the various sources of
204 uncertainty we do not identify a deterministic value but a
205 probability distribution Once this probability distribution of
206 the parameters governing the crack growth determined, we
207 need to predict the possible evolution of the crack size in future
208 flights, which is achieved by a first-order perturbation (FOP)
209 method also described in Section3 The FOP method allows
210 to determine the distribution of the crack size at an arbitrary
211 future flight time Based on this information we propose a
212 new maintenance policy, described in Section 5, which
213 minimizes the maintenance cost Section 5 implements a
214 numerical study to evaluate the performance of the proposed
215 maintenance policy Conclusions and suggestions for future
216 work are presented in Section6
217
2 State-space method for modeling the degradation process
218 2.1 State-space model
219 State-space modeling assumes that a stochastic dynamic
sys-220 tem evolves with time The states of the stochastic system are
221 hidden and cannot be observed A set of measurable quantities
222 that are related with the hidden system states are measured at
223 successive time instants Then we have the following
state-224 space model:
225
228
231 where fðÞ and hðÞ are the state transition function and the
232 measurement function respectively xkis the unobserved state
233
at time k.h is the parameter of the state equation f zkis the
234 corresponding measurements that generally contains noise
235
wk and vk are the process noise and measurement noise,
236 respectively Although the parameterh is stationary, subscript
237
k 1 is used because its information is updated with time In
238 the following Sections2.2 and 2.3, we model the equation f and
239
hfor the specific application of fatigue crack growth
240 2.2 Fatigue crack growth model
241 The fatigue damage in this paper refers to cracks in fuselage
242 panels The Paris model7is used to describe the crack growth
243 behavior, as given
244 da
247 where a is the crack size in meters k is the time step, here the
248 number of flight cycles da/dk is the crack growth rate in
249 meter/cycle m and C are the Paris model parameters
250 associated with material properties DK is the range of stress
251 intensity factor, which is given in Eq.(4)as a function of the
252 pressure differential p, fuselage radius r and panel thickness
253
t The coefficient A in the expression of DK is a correction
254 factor compensating for modeling the fuselage as a hollow
255 cylinder without stringers and stiffeners.33
256
DK ¼ Apr
t
ffiffiffiffiffiffi pa
p
ð4Þ 258 259
By using Euler method, Eq (3) can be rewritten in a
260 discrete form and the discretization precision depends on the
261 discrete step Here the step is set to be one, which is the
min-262 imal possible value from the practical point of view, to reduce
263 the discretization error Then the discrete Paris model in a
264 recursive form is given in Eq.(5)
265
ak¼ ak1þ C Apk1r
t
ffiffiffiffiffiffiffiffiffiffiffi
pak1
p
¼ gðak1; pk1Þ ð5Þ 267
268 The pressure differential p can vary at every flight cycle
269 around its nominal valuep and is expressed as
270
Trang 4273 in which Dpkis the disturbance around p and is modeled as a
274 normal distribution random with zero mean and variancer2
p
275 Since uncertainty in pressure is generally small, the first-order
276 Taylor series expansion is used in this paper.34This gives:
277
ak¼ gðak1; pÞ þ@gða@pk1; pÞDpk1 ð7Þ
279
280 where @gðak1; pÞ=@p is the first-order partial derivative of g
281 with respect to p Takingð@gðak1; pÞ=@pÞDpk1as the additive
282 process noise and considering thatp is a given constant, Eq.(7)
283 can be written as
284
286
287 in which fðak1Þ ¼ gðak1; pÞ and
288
290
291 According to Eq.(7)the additive process noise wkfollows a
292 normal distribution with mean zero and variance Qk, given in
293 Eq.(10) Note that Qkcan be calculated analytically
294
Qk¼ ðð@fðak; pÞ=@pÞrpÞ2
¼ ðCmðAr=tÞmðpÞm1ðpakÞm=2rpÞ2 ð10Þ
296
297 2.3 Measurement model
298 Due to harsh working conditions and sensor limitations, the
299 monitoring is imperfect and generally contains noise The
mea-300 surement data is modeled as
301
303
304 Note that Eq.(11)is used to simulate the actual
measure-305 ment data Eqs.(8) and (11)are respectively the state transition
306 function and the measurement function in the state-space
307 model
308 3 Prognostics method for individual panel
309 Prognostic is the prerequisite of the predictive maintenance In
310 this paper, the model-based prognostics method is applied,
311 which is tackled with two sequential phases: (1) estimation of
312 fatigue crack size as well as the unknown model parameters,
313 and (2) prediction of future crack size distribution As
illus-314 trated inFig 1, the true system state is hidden and evolves over
315 time The measurements related to the state are obtained at a
316 successive time step k By using the measurements data up to
317 the current time, the state and parameters of the state equation
318 can be estimated This process is also known as a filtering
319 problem Based on the estimated states and parameters, the
320 state distribution in future time can be predicted In this paper,
321 the filtering problem is addressed by the EKF, and a proposed
322 first-order perturbation method is used to predict the state
323 distribution evolution in future times In this section, the
324 approaches for dealing with the two phases of model-based
325 prognostics are presented respectively in Sections3.1 and 3.2
326 briefly, since the main focus of this paper is the maintenance
327 policy The interested reader could refer to Ref 5for more
328 details on this approach
329 3.1 State-parameter estimation using EKF
330 EKF is used to filter measurement noise based on a given
state-331 space model EKF thus allows to estimate a smooth variation
332
of the state variable (crack size in our case) as well as the
state-333 parameters (m and C in our case) governing these variations
334 When performing state-parameter estimation using the
335 EKF, the parameter vector of interest is appended onto the
336 true state to form a single augmented state vector The state
337 and the parameters are estimated simultaneously In Paris’
338 model, m and C are the unknown parameters that need to be
339 estimated Therefore, a two-dimensional parameter vector is
340 defined as
341
344 Appendingh to the state variable, that is crack size a, the
345 augmented state vector is defined in Eq (13), where the
sub-346 script ‘‘au” denotes the augmented variables
347
350 Then the state transition function and the measurement
351 function in Eqs.(8) and (11)can be extended in a state-space
352 model form as illustrated in Eq.(14) In this way, the
estima-353 tion for Paris’ model parameters and crack size is formalized as
354
a nonlinear filtering problem EKF is applied on the extended
355 system in Eq (14)to estimate the augmented state vector at
356 time k, i.e., xau;k¼ ½ak; mk; CkT
The EKF is used as a black
357 box in the present work and the detail of the algorithm will
358 not be presented here Interested readers are referred to Ref
359
35 for a general introduction to EKF and to Ref 24 for its
360 implementation to state-parameter estimation in Paris’ model
361
By applying EKF, at each flight cycle, the posterior estimation
362
of the augmented state vector, i.e., ^xau;k¼ ½^ak; ^mk; ^CkT, and
363 the corresponding covariance matrix Pk, characterizing the
364 uncertainty in the estimated parameters, are obtained
365
ak
mk
Ck
2 6
3 7
5 ¼
fðak1Þ
mk1
Ck1
2 6
3 7
5 þ
wk1
0 0
2 6
3 7
zk¼ akþ vk
8
>
<
>
:
ð14Þ 367
368 3.2 First-order perturbation (FOP) method for predicting the
369 state distribution evolution
370
We propose the FOP method to address the second phase of
371 model-based prognostics, i.e., the predicting problem, as
372 shown in Fig 1 For the context of crack growth, it allows Fig 1 Illustration of model-based prognostics
Trang 5373 to calculate analytically the crack size distribution at any
374 future cycle Fig 2 illustrates the schematic diagram of the
375 two phases of the discussed model-based prognostics method
376 The noisy measurements are collected up to the current cycle
377 k= S The EKF is used to filter the noise to give estimates
378 for the crack size and the model parameters At time S, the
379 following information is given by the EKF and will be used
380 as initial conditions of the second phase:
381 expected value of the augmented state vector, ^xau;S¼
382 ½^aS; ^mS; ^CST
383 covariance matrix of the augmented state vector PS
384
385 According to the EKF, the state vector xau;Sfollows a
mul-386 tivariate normal distributed with mean ^xau;S and covariance
387 PS, presented as
388
390
391 Based on this information, in the second phase, the FOP is
392 used to calculate analytically the mean and standard deviation,
393 denoted bylkandrk, of the crack size distribution at any future
394 cycle k starting from S + 1 The derivation of the FOP method
395 is detailed inAppendix A The dashed curve in the second phase
396 represents the mean trajectory of the crack size estimated by the
397 first-order perturbation method, i.e.,flkjk ¼ S þ 1; S þ 2; g
398 For illustrative purpose, the crack size distribution at two
arbi-399 trary flight cycles k1 (based onlk andrk ) and k2 (based onlk
400 andrk) are given as examples
401 It should be noted that the cost-driven predictive
mainte-402 nance (CDPM) strategy to be presented in the following
sec-403 tion considers an aircraft being composed of Na panels For
404 each panel, the model-based prognostics process implemented
405 by EKF-FOP method is applied i.e., for each panel, we use
406 EKF to estimate the Paris’ model parameters and crack size
407 from noisy measurements of the crack size at different flight
408 cycles Then we use the FOP method to predict the crack size
409 distribution at a future time based on the information given by
410 EKF (refer to Fig 2) Once the crack size distribution at a
411 future time is available for each panel, this prediction
informa-412 tion is incorporated into the CDPM to help maintenance
413 decision-making The details of CDPM strategy are presented
414 next in Section4
415 4 Cost-driven predictive maintenance (CDPM) policy
416 Currently, aircraft maintenance is performed on a fixed schedule
417 Suppose that the aircraft undergoes the routine maintenance
418 according to a schedule Tn= T1+ (n 1)dT, where n = 1,
419
2, , is the number of scheduled maintenance stop, Tndenotes
420 the cumulative flight cycles at the nth stop, T1is the number of
421 flight cycles from the beginning of the aircraft lifetime to the first
422 scheduled maintenance stop dT is the interval between two
423 consecutive scheduled maintenance stops after T1 Note
424 that T1> dT because fatigue cracks propagate slowly during
425 the earlier stage of the aircraft lifetime With usage and ageing,
426 the aircraft needs maintenance more frequently The schedule
427 {Tn} is determined by aircraft manufacturers in concertation
428 with certification authorities and aims at guaranteeing the safety
429 using a conservative scenario For a given safety requirement this
430 schedule may not be optimal, in terms of minimizing
mainte-431 nance cost Indeed a specific aircraft may differ from the fleet’s
432 conservative properties used in calculating the maintenance
433 schedule and possibly require fewer maintenance stops
434
By employing the SHM system, the damage state can be
435 traced as frequently as needed (e.g every 100 cycles) and the
436 maintenance can be asked at any time according to the
air-437 craft’s health state rather than a fixed schedule This causes
438
an unscheduled maintenance that could happen anytime
439 throughout the aircraft lifetime and generally occurs outside
440
of the scheduled maintenances Triggering a maintenance stop
441 arbitrarily is significantly disturbing to the current scheduled
442 maintenance practice due to no advance notification (e.g., less
443 preparation of the maintenance team), unavailable tools, lack
444
of spare parts, etc These factors lead unscheduled
mainte-445 nances to be more expensive Therefore, we attempt as much
446
as possible to plan the structural airframe maintenance at
447 the time of the scheduled maintenance and avoid the
unsched-448 uled maintenance in order to reduce the cost
449
On the other hand, it makes sense to skip some scheduled
450 maintenance stops Since the frequency of scheduled
mainte-451 nance for commercial aircrafts is designed for a low
probabil-452 ity of failure (107)33, it is very likely that no large crack exists
453 during earlier life of the majority of the aircraft in service
454 Thanks to the on-board SHM system, the damage assessment
455 could be done in real time on site instead of in a hangar,
lead-456 ing to the possibility of skipping unnecessary scheduled
main-457 tenance if there are no life-threatening cracks on the aircraft If
458
a crack missed at schedule maintenance grows large enough to
459 threaten the safety between two consecutive scheduled
mainte-460 nances, an unscheduled maintenance is triggered at once The
461 frequent monitoring of the damage status would ensure the
462 same level of reliability as scheduled maintenance Recall that
463 our objective is to re-plan the structural airframe maintenance
464 while the engine and non-structural airframe maintenance are
465 always performed at the time of scheduled maintenance
466
In summary, it might be beneficial that in civil aviation
467 industry to have the traditional scheduled maintenance work
468
in tandem with the unscheduled maintenance With this
moti-469 vation, the CDPM policy is proposed whose overall idea is
470 described below:
471
The damage states of the fuselage panels are monitored
472 continuously by the on-board SHM system and a damage
473 assessment is performed every 100 flights (which
approxi-474 mately coincides with A-checks of the aircraft)
475
At each assessment, as new arrived sensor data is available,
476 the EKF is used to filter the measurement noise to provide
477 the estimated crack size and parameters of crack growth
478 model for each panel at current flight cycle
Fig 2 Schematic diagram of model-based prognostics
Trang 6479 At the nth scheduled maintenance stop, before the aircraft
480 goes into the maintenance hangar, for each panel, the crack
481 propagation trajectory from maintenance stop n to n + 1 is
482 predicted and the crack size distribution at next scheduled
483 maintenance is obtained by using the first-order
perturba-484 tion method Taking into account this predicted
informa-485 tion of each panel, the cost optimal policy decides to skip
486 or trigger the current nth stop If it is triggered, a group
487 of specific panels is selected to be repaired based on the
pre-488 dicted information to minimize the expected maintenance
489 cost The algorithm of selecting a group of specific fuselage
490 panels is called cost optimal policy and will be described in
491 Section4.5
492 During the interval of two consecutive scheduled
mainte-493 nance stop, if there is a crack exceeding a safety threshold
494 amaint at damage assessment, an unscheduled maintenance
495 is triggered immediately The aircraft is sent to the hangar
496 and this panel is repaired The meaning and calculation of
497 amaintis discussed in Section4.2
498
499
500
501 4.1 Different behavior among individual panels of the population
502 Our objective is an aircraft with Nafuselage panels If all the
503 manufactured panels are exactly the same and these panels
504 work under exactly the same conditions and environment, then
505 the panels will degrade identically However, in practice, due
506 to manufacturing and operation variability there is
panel-to-507 panel variability
508 In this study, the generic degradation model (Paris model)
509 is used to capture the common degradation characteristics
510 for a population of panels while the initial crack size a0and
511 the degradation parameters m and C of each panel follows
pre-512 defined prior distributions across the population to cover the
513 panel-to-panel variability When modeling one individual
514 panel, a0, m and C are treated as ‘‘true unknown draws” from
515 their prior distributions By incorporating the sequentially
516 arrived measurement data, the EKF is used for each panel to
517 estimate the crack size and the material parameters and their
518 distribution at time k Here the superscript is the panel index
519 and the subscript denotes the time instant
520 In this paper, a0is assumed log normally distributed while
521 mand log10Care assumed to follow a multivariate normal
dis-522 tribution with a negative correlation coefficient.36–38
523 4.2 Reliability of system level
524 The critical crack size that causes panel failure can be
calcu-525 lated by the empirical formula in Eq.(16), in which KIC is a
526 conservative estimate of the fracture toughness in loading
527 Mode I and pcris also a conservative estimate of the pressure
528 pgiven its distribution
529
acr¼ KIC
Apcr r
t
ffiffiffi p p
ð16Þ 531
532 Since the damage assessment is done every 100 cycles, if a
533 crack size equals to acris present in a panel in between two
dam-534 age assessments, it will cause the panel failure at once
There-535 fore, another safety threshold amaint, which is smaller than acr
536 is determined to ensure safety between two damage assessments
537 amaintis calculated to maintain a 107probability of failure
538 of the aircraft between two damage assessments (100 cycles),
539 i.e., when a crack size equals to amaintis present on the fuselage
540 panel, its probability of exceeding the critical crack size acrin
541 next 100 cycles is less than 107, hence ensure the safety of
542 the aircraft until next damage assessment At the time of
543 damage assessment, once the maximal crack size among the
544 panel population exceeds amaint, the unscheduled maintenance
545
is triggered immediately and the aircraft is sent to the hangar
546 Since this maintenance stop is unscheduled with very little
547 advance notice only the panel having triggered the stop is
548 replaced in order to minimize operational interruption
549 4.3 Reliability of an individual panel
550
At the nth scheduled maintenance stop (the cumulative cycles
551
is Tn) the crack size distribution of each individual panel before
552 the next scheduled stop is predicted For the ith panel, the
553 probability of triggering an unscheduled maintenance before
554 next scheduled maintenance stops is denoted by P(us|ai) It is
555 approximated by Eq.(17), i.e., the probability that the crack
556 size of the ith panel at next scheduled maintenance ai
Tnþ1 is
557 greater than amaint, given the information provided by EKF
558
at current scheduled maintenance stop, more specifically, the
559 estimated crack size and material property parameters,
560
^ai
T n; ^mi
T n; ^Ci
T n
, and the covariance matrix PiT n
561 PðusjaiÞ ¼ Prðai
Tnþ1> amaintj½^ai
T n; ^mi
T n; ^Ci
T n; Pi
564 The evolution of the crack size distribution from Tnto Tn+1
565
is predicted by the FOP method presented in Section 3.2
566 According to the FOP method, ai
T nþ1 is normally distributed
567 with parametersli
Tnþ1 andri
Tnþ1, which are calculated
analyti-568 cally Thus PðusjaiÞ is computed as
569 PðusjaiÞ ¼
Z 1
a maint
Uðai
T nþ1jli
T nþ1; ri
T nþ1Þdai
571 572 where U is the probability density function of the normal
dis-573 tribution with meanli
T nþ1and standard deviationri
T nþ1
574 Note that the probability of triggering an unscheduled
575 maintenance of a panel is not proportional with its current
576 crack size, i.e., it is not necessarily true that panel with larger
577 crack size is more likely to trigger an unscheduled
mainte-578 nance Due to the variability of crack growth rate among
pan-579 els as well as the uncertainty presented in the crack
580 propagation process, a larger crack size at nth stop may have
581
a lower probability of exceeding amaintbefore next scheduled
582 stop, compared with a smaller crack size
583 4.4 Cost model
584 Some concepts as well as their notations are given firstly before
585 the cost structure is introduced
586
dj
nThe repair decision for the jth panel at the nth scheduled
587 maintenance stop It is a binary value defined as Here the
588 index j is based on the resorted rule that will be introduced
589 Section4.5
590 591
dnj¼ 1 if panel j is repaired
0 if panel j is not repaired
ð19Þ 594
Trang 7595 dnthe decision vector such that dn= [d1
n; d2
n; ; dN a
n ] Nais
597 the total number of fuselage panels in an aircraft
598 c0The set up cost of SHM-based scheduled maintenance,
599 which is a fixed cost that occurs every time the scheduled
600 maintenance is triggered The set up cost is assigned only
601 once even if more than one panel is replaced
602 cun
0 the unscheduled set up cost, which is a fixed cost that
603 occurs when unscheduled maintenance is triggered Due
604 to less advance notification, cun
0 > c0
605 s A variable used to indicate the binary nature of
sched-606 uled maintenance.s = 1 means that the scheduled
mainte-607 nance is triggered and the set up cost is incurred whiles = 0
608 means this scheduled maintenance is skipped thus no set up
610 csthe fixed cost of repairing one panel
611 custhe repair cost at unscheduled maintenance, also called
612 unscheduled repair cost, which is composed of two items,
613 the unscheduled set up cost cun
0 plus the per panel repair cost cs 614
615 The expected maintenance cost at the nth scheduled
main-616 tenance stop, denoted by C(dn), is modeled as the function of
617 the repair decision of each panel, as given in Eq.(20) The first
618 two terms in Eq.(20)represent the scheduled repair cost while
619 the last term represents the unscheduled repair cost Here we
620 assume that the probability for a panel to have more than
621 one unscheduled repair is negligible
622
CðdnÞ ¼ c0s þ cs
XN a j¼1
dnj
!
þ cus
XN a j¼1
ð1 dj
nÞPðusjajÞ
! ð20Þ 624
625 4.5 Cost optimal policy
626 The objective is to find the optimal grouping of several panels
627 to be repaired to minimize the cost when the aircraft is at nth
628 scheduled maintenance stop The algorithm is under the
fol-629 lowing assumptions:
630 The probability for a panel to have more than one
unsched-631 uled repair during the aircraft lifetime is negligible
632 The probability to have more than one unscheduled repair
633 at the same cycle is negligible This means that having more
634 than one panel repaired during unscheduled maintenance
635 do not reduce the average cost of each panel
636
637 At the nth scheduled maintenance, for each panel, the
prob-638 ability of triggering an unscheduled maintenance between stop
639 n and n + 1 is calculated according to tion 4.3 Sort and
640 arrange them in descending order such that
641
Pðusja1Þ > Pðusja2Þ > Pðusjaj1Þ > PðusjajÞ
> Pðusjajþ1Þ > PðusjaN aÞ ð21Þ
643
644 Eq.(21)implies that the panel that is more likely to trigger an
645 unscheduled maintenance is arranged in more front places
646 The motivation is that we are more concerned about the panels
647 with higher probability of having unscheduled repair since
648 unscheduled maintenance is more costly In the following
649 parts, the panel index refers to the order in Eq.(21)
650 Two sets I and J are defined
651
I¼ f1 6 j 6 Njcs6 cusPðusjajÞg ð22Þ
653
654
J¼ f1 6 l 6 Njc0þ lcs6 cus
Xl j¼1
657 For zero set up cost (i.e., c0= 0), the set I contains the
ele-658 ments j such that repairing the j-th panel at current scheduled
659 maintenance cost less than repairing it at an unscheduled
main-660 tenance stop For any value of the set up cost, set J includes the
661 elements j such that repairing all these j panels at scheduled
662 maintenance cost less than at unscheduled maintenance BI
663 and bJare defined as the maximal value and the minimal value
664
of set I and J, respectively Note that BIand bJare scalars
665
BI¼ maxf1 6 j 6 Njcs6 cusPðusjajÞg ð24Þ 667
668
bJ¼ minf1 6 l 6 Njc0þ lcs6 cus
Xl j¼1
671
A simple example is given below to explain the set I and J as
672 well as to illustrate the meaning of BIand bJintuitively
Sup-673 pose there are Nafuselage panels in an aircraft and this aircraft
674
is now at the nth scheduled maintenance stop The objective is
675
to decide whether this aircraft should undergo maintenance or
676 should skip the current maintenance by evaluating the health
677 state for each fuselage panel Firstly, for each panel, its
proba-678 bility of triggering an unscheduled maintenance before next
679 scheduled maintenance is calculated according to the process
680 described in Section4.3 Then these Naprobabilities are sorted
681
in descending order according to Eq (21) Afterward, each
682 probability is multiplied by cusand is compared with cs
Sup-683 pose that we found the following relations:
684
cs6 cusPðusja1Þ
cs6 cusPðusja2Þ
cs6 cusPðusja3Þ
cs6 cusPðusja4Þ
cs> cusPðusja5Þ
cs> cusPðusja6Þ
687 The above case means that for the first 4 panels, the cost of
688 repairing any of them at current scheduled maintenance is less
689 than the cost of repairing it at unscheduled maintenance From
690 the 5th panel to the last panel, it is not economic to repair any
691
of them at current nth scheduled maintenance since their
prob-692 ability of triggering unscheduled maintenance is very low In
693 this case, the set I = {1, 2, 3, 4} and BI= 4
694 The above example considers the situation of repairing one
695 single panel Now we consider the situation of repairing a group
696
of panels Suppose we group the first l panels and then compare
697 the following two costs: (1) the cost of repairing these l panels at
698 current scheduled maintenance, i.e., c0þ lcs, and (2) the expected
699 cost of repairing the l panels at unscheduled maintenance, i.e.,
700
cus
Pl j¼1PðusjajÞ Suppose we found the following relations:
701
c0þ cs> cusðPðusja1ÞÞ
c0þ 2cs> cusðPðusja1Þ þ Pðusja2ÞÞ
c0þ 3cs6 cusðPðusja1Þ þ Pðusja2Þ þ Pðusja3ÞÞ
c0þ Nacs6 cus
XN a j¼1
PðusjajÞ
703
Trang 8704 In the above case, J = {3, 4, , Na} and bJ= 3.
705 From Eqs (22)–(25), the following properties can be
706 deduced straightforward
707
709
710
cs6 cusPðusjajÞ; for j ¼ 1; 2; ; BI ð27Þ
712
713
cs> cusPðusjajÞ; for j¼ BIþ 1; BIþ 2; ; Na ð28Þ
715
716
c0þ lcs> cus
Xl j¼1
PðusjajÞ; for j ¼ 1; 2; ; bJ 1 ð29Þ 718
719
c0þ bJcs6 cus
XB J j¼1
721
722 The proof for Eq (26) is given in Appendix B and
723 Eqs.(27)–(30)can be easily derived from the definitions given
724 in Eqs.(22)–(25) Now we discuss the cost optimal policy at the
725 nth scheduled maintenance stop
726 If set I is empty and the set up cost is zero (i.e., c0= 0), it
727 means that for any panel the expected unscheduled repair cost
728 is smaller than the scheduled one In this case, the optimal
729 repair policy is not to repair any panel at current scheduled
730 maintenance stop, i.e., dn_jðajÞ ¼ 0, for j = 1, 2, , Na Note
731 that djn denotes the optimal repair decision for the jth panel at
732 the nth scheduled maintenance stop
733 If the set I is not empty and the set up cost is zero (i.e.,
734 c0= 0), from Eqs (27) and(28), it can be inferred that for
735 any panel j that j6 BI the expected unscheduled repair cost
736 is larger than the scheduled one, while for any panel j that
737 j> BI, the expected unscheduled repair cost is smaller than
738 the scheduled one In the case of I– £, the set J could be
739 either empty or non-empty Now we discuss these two cases
740 that J¼ £ and J – £, and derive the optimal repair decision
741 in each cases
742 If J is empty, it means that no matter how many panels are
743 paired, the cost of repairing these panels at scheduled
mainte-744 nance stop costs more than at unscheduled maintenance Then
745 the optimal maintenance policy is not to repair any panel at
746 current scheduled maintenance stop, i.e., djnðajÞ ¼ 0, for
747 j= 1, 2, , Na Note that I¼ £ implies J ¼ £ but we can
748 have J¼ £ and I – £
749 If J is not empty (i.e., J– £), from Eqs.(29) and (30), it
750 can be known that for any panel j that j < bJ, repairing the j
751 first panels at scheduled maintenance stop cost more than at
752 unscheduled maintenance, and for j = bJ, repairing the j first
753 panels at scheduled maintenance stop cost less than at
754 unscheduled maintenance As for j > bJ, repairing the j first
755 panels at scheduled maintenance stop can be either better or
756 worse For example, we can have:
757
c0þ cs> cusðPðusja1ÞÞ
c0þ 2cs6 cusðPðusja1Þ þ Pðusja2ÞÞ
c0þ 3cs> cusðPðusja1Þ þ Pðusja2Þ þ Pðusja3ÞÞ or
c0þ 3cs< cusðPðusja1Þ þ Pðusja2Þ þ Pðusja3ÞÞ
759
760 From Eq.(26), it can be known that the range [1, Na] are
761 divided into three intervals by BI and bJ, which are [1, bJ],
762 [bJ+ 1, BI] and [BI+ 1, Na] To determine the optimal policy,
763 it is clear that the bJ-first panels have to be repaired at the
cur-764 rent scheduled maintenance (see Eq.(30)) In addition, since
765 the expected unscheduled maintenance cost of panels in the
766 interval [bJ+ 1, BI] are larger than scheduled maintenance
767 cost (see Eq (27)), they should also be repaired at current
768 scheduled maintenance stop Finally, the optimal repair policy
769
at n-th scheduled maintenance can be summarized as follows:
770
If J¼ £
djn ¼ 0; for j ¼ 1; 2; ; N Else
djn ¼ 1 for j¼ 1; 2; ; BI
0 for j¼ BIþ 1; ; Na
772 773 The above decision implies that when J is empty, the
opti-774 mal decision is not to repair any panel at the nth scheduled
775 maintenance stop The expected cost under this situation is
776
Cðd
nÞ ¼ cus
XN a j¼1
PðusjajÞ
!
ð32Þ 778 779 When J is not empty, the optimal decision is to repair the
780 first BI panels and leave unattended the remaining ones
781 Accordingly, the cost in this case is
782
Cðd
nÞ ¼ c0þ csBIþ cus
XN a j¼B I þ1
PðusjajÞ
!
ð33Þ 784 785 Then the optimized total maintenance cost during the
air-786 craft lifetime, denoted as CðdÞ, is the sum of the cost at each
787 scheduled maintenance Cðd
nÞ
788 CðdÞ ¼X
n
Cðd
790 791 The rigorous mathematical proof regarding Cðd
nÞ < CðdnÞ,
792 i.e., why dnis the optimal decision is given inAppendix B The
793 cost optimal policy is integrated into the predictive policy,
794 whose flowchart is illustrated inFig 3 The above repair
deci-795 sion is made at each scheduled maintenance stop until the end
796 aircraft’s life Then the total maintenance cost during aircraft
797 lifetime CðdÞ can be calculated
798
5 Numerical experiments
799
A fleet of M = 100 aircraft in an airline with each aircraft
con-800 taining Na= 500 fuselage panels is simulated The potential
801 application objective is a short range commercial aircraft with
802
a typical lifetime of 60,000 flight cycles Traditionally, the
803 maintenance schedule for this type of aircraft is designed such
804 that the first maintenance is performed after 20,000 flight
805 cycles and the subsequence maintenance is every 4000 cycles
806 until its end of life, adding up to 10 scheduled maintenances
807 throughout its lifetime, as shown inFig 4
808
To show the benefits of the CDPM, two other maintenance
809 polices are compared with it The first one is traditional
sched-810 uled maintenance and the second is a threshold-based SHM
811 maintenance
812
In traditional scheduled maintenance, at each maintenance
813 stop, the aircraft is sent to the hangar to undergo a series of
814 inspections and all panels with a crack size greater than a
815 threshold arepare repaired The repair threshold arepis
calcu-816 lated to maintain the same reliability as CDPM between two
817 consecutive scheduled maintenance stops over the entire fleet
818 Note that since this strategy seeks to guarantee the same
819 reliability over the entire fleet it is more conservative than
Trang 9820 CDPM, which only has to guarantee the reliability for a single
821 aircraft
822 In threshold-based maintenance, the SHM is assumed to be
823 used and the damage assessment is performed every 100 flights
824 The aim is the same as CDPM to skip some unnecessary early
825 scheduled maintenance while guarantee the safety by triggering
826 unscheduled maintenance Specifically, at each scheduled
827 maintenance stop, if there is no crack size exceeding a
thresh-828 old ath-skip, then the current scheduled maintenance is skipped
829 Between two consecutive scheduled maintenance stops, if a
830 crack grows beyond amaint, the unscheduled maintenance is
831 triggered and all panels whose crack size is greater than arep
832 are repaired The flowchart of threshold-based maintenance
833 is given inFig 5 For additional details on this threshold based
834 maintenance strategy applied to fuselage panels, the reader
835 could refer to Ref.33
836 Three design parameters characterize the threshold-based
837 maintenance First amaintensures the safety It is defined and
838 calculated the same as in CDPM, i.e., to maintain a 107
prob-839 ability of failure between two damage assessments (every 100
840 cycles) for a given aircraft Second ath-skip is calculated such
841 that the probability of one crack exceeding amaintbefore next
842 scheduled maintenance is less than 5% Finally, the repair
843 threshold arep is set the same value as in traditional
844 maintenance
845 Note the difference between threshold-based maintenance
846 and the CDPM In CDPM, the decision of whether or not
847
to repair a panel is treated individually for each panel
848 depending on the relation between the cost ratio (cs/cus) and
849 the probability of triggering unscheduled maintenance While
850
in the threshold-based maintenance, this decision depends on
851 the fixed threshold arep, which is determined for the entire fleet
852 5.1 Input data
853 The values of the geometry parameters defining the fuselage
854 used in the numerical application have been chosen from
855 Ref 33 and are reported in Table 1 These values are
time-856 invariant Recall that we define a correction factor A for stress
857 intensity factor, which intends to account for the fact that the
858 fuselage is modeled as a hollow cylinder without stringers and
859 stiffeners
860
As discussed in Section4.1, we use the Paris model to
cap-861 ture the common degradation characteristics for a population
862
of panels while the initial crack size a0 and the Paris model
863 parameters m and C of each panel are drawn from prior
864 distributions to model the panel-to-panel uncertainty In
addi-865 tion, for each panel, during the crack propagation process, the
866 pressure differential p varies from cycle to cycle and is modeled
867
as a normal random variable See Section2.2for details The
868 uncertainties for a0, m and C and p are given inTable 2 The
869 numerical values of thresholds used are given inTable 3 At
870 the beginning of the simulation, 500 100 samples of a0, m
Fig 3 Flow chart of CDPM
Fig 4 Schedule of the scheduled maintenance process Cycles
represent the number of flights
Trang 10871 and C are drawn and assigned to each panel while p is drawn
872 every cycle during the crack growth process The 50,000
sam-873 ples of m and C are illustrated inFig 6
874 One thing needs to clarify The uncertainties of a0, m and C
875 given inTable 2are the panel-to-panel uncertainty
represent-876 ing the variability among panels population These 500 100
877 samples, denoted as ½ai
0; mi; CiT
, (i = 1, 2, .), are assigned
878 to each panel to form the initial condition of the i-th panel
879 Due to lack of knowledge on single panel, these samples are
880 regarded as ‘‘true unknown draws” that need to be estimated
881 by the EKF During the EKF process, for the ith panel, the
ini-882 tial guess for½ai
0; mi; CiT
are randomly given and is fed to EKF
883 as the start point As the noisy measurements arrive
sequen-884 tially, EKF incorporates the measurements and gives the
opti-885 mal estimates to the crack size and model parameters at time k,
886 denoted as½^ai
k; ^mi
k; ^Ci
kT The estimation uncertainty reduces as
887 time evolves due to more measurements are available Due to
888 limit space, the EKF process will not be detailed here Readers
889 could refer to Ref.24
Fig 5 Flow chart of threshold-based maintenance
Table 1 Numerical values of geometry parameters
Table 2 Numerical values of the uncertainties on a0, m, C and p
Initial crack size/
m
a 0 Lognormal ln N(0.3 10 3 ,
0.0 10 3 ) Paris model
parameters
m, C Multivariate N ( l m , r m , l C , r C , q)
Standard deviation of m
Standard deviation of C
a C.C is correlation coefficient.
b COV means coefficient of variation.