Optimal Inspection Policy based on Measurement Quality Degradation: Case of Pitot Sensors System

Transcription

1 Preprints, IFAC Conference on Manufacturing Modelling, Management and Control Optimal Inspection Policy based on Measurement Quality Degradation: Case of Pitot Sensors System Wajih Ezzeddine Jérémie Schutz Nidhal Rezg Université de Lorraine, LGIPM, EA 3096, Metz, F-57045, France ( - - Abstract: In this paper, an inspection policy is proposed for a multi-unit system which is during his mission, is assumed to be irreparable. The system is subject to random deterioration which impacts his measurement accuracy. The proposed approach aims to assess the degradation in such a way to increase the efficiency and the accuracy of the whole system. To control this deterioration, inspections are performed and after which we decide how to compute system measurements. According less importance to measurements given by a component occurs whenever the level of its deterioration reaches a specific level threshold. The objective is to determine an optimal periodic inspection policy which minimizes the average total measurement quality degradation per unit of time in a finite time horizon. Our study is applied to a specific case which is the Pitot sensors system. However, the approach could be generalized for every irreparable multiple-unit system subject to random deterioration. A numerical example is provided to illustrate the proposed inspection model. Keywords: Inspection policy, random deterioration, Pitot sensors, optimal inspection dates 1. INTRODUCTION Inspection activities are usually investigated to assess the degradation of the system. Indeed, inspection allows to assess the degradation process of the production system and to collect crucial reliability data.this data provides the different information about the degradation level of the system. In our case, we assume that after an inspection, there are two decisions that have to be made. One decision is to determine what kind of computation to be made, whether the studied component measurement should be given less importance than other component measurements or completely ignored or whether should be left as it is before the inspection activity. The other decision is to determine when the next inspection should be performed.during flight, the plane passes generally through two areas: area outside radar and area inside radar. In the last area, radar gives all information needed by the pilot like air density or plane speed with high accuracy. The following figure shows the example of the two different areas in the MH370 fight.(fig.1). Our study corresponds to the period at which the plane is situated in the area outside radar when all information is not given by the radar and we have to use other instruments to measure the speed of the plan. The system used for this is the Pitot system composed of fix sensors giving the information about speed and a retractable sensor used to verify the accuracy of fix sensors measurements. Fig. 1. Example for areas outside and inside radar 2. LITTERATURE REVIEW Many proposals for single component deteriorating system have been established under simplifying assumptions. Markov and semi-markov models have been the principal approach in modelling condition based maintenance. Works of Bérenguer et al. (1997) and Grall et al. (2002) can be cited in this contest. The majority of the models appeared in the literature assume that the systems degradation level can only be known through periodic inspection as typical in safety systems of nuclear plants, Hontelez et al. (1996). For the continuously inspected systems recently investigated by Kopnov (1999), the two level policies from the inventory theory have been adapted to the CBM problem of degrading systems. A common Copyright 2016 IFAC 1281

2 feature of the models discussed is that the state of the system is described as a state of a Markov process and then the analysis proceeds to finding analytically the probabilities of the various states. However, if the system is made of several multi-state components, the analysis becomes excessively complicated. Barata et al. (2002) applies a stochastic degradation model for a single, non-repairable components to study a multi-component system. Many proposals deal with system availability optimization. Chelbi et al. (2008) extended the work of Badıa et al. (2002). In Chelbi et al. (2008), numerical solutions have been presented for Normal and Weibull failure distributions. Sarkar and Sarkar (2000) expressed the system availability function as well as the limit average availability in order to determine the inspection period. In Aït-Kadi and Chelbi (2010), the suggested inspection strategy aims to reduce the frequency of the random failures and to increase the system availability. Recently, many proposals deal with systems subjected to continuous monitoring, i.e. the preventive maintenance is performed according to the exceeded threshold. For this reason, the preventive maintenance threshold is considered as the only decision variable in the works by Liao et al. (2006) and Tian et al. (2012). Liao et al. (2006) investigates a maintenance policy in order to achieve the maximum availability value of a system subjected imperfect maintenance actions and short-run availability constraint. The author determines an optimum preventive maintenance threshold for a degrading system modelled by a Gamma process. Chouikhi et al. (2014) proposes a maintenance model taking into account environmental deterioration. The system considered is a single unit system subject to random deterioration. The authors proposed model aims to reduce the environmental degradation by assessing the system deterioration. For more details about inspection and maintenance models, the reader may refer to the review of literature in Chelbi and Ait-Kadi (2009) and Sharma et al. (2011). In the present paper, an inspection optimization model is proposed for the Pitot sensors system. The considered system is a multiple-units of speed measurement during a flight. The system is composed of components which the measurement quality degradation of each component is assumed to be subject to random deterioration which impacts the measurement accuracy. The proposed inspection policy aims to assess the measurement quality degradation of the system. After inspection, each component measurement is either given less importance than the rest of other components, completely ignored or left as it was before inspection. A numerical example is investigated to illustrate the proposed inspection policy. Conclusions and perspectives are drawn in final section. it is important to mention that the degradation investigated in this paper designs the degradation of the measurements accuracy. Therefore, in the numerical example, the degradation level of a sensor expresses of the percentage of the measurement incertitude compared to the referential measurements (those of the Fix sensor). 3. DEGRADATION MODEL 3.1 Single non-repairable component Case of Fix Pitot Sensor Consider a single Fix Sensor which supposed to be non-repairable component during a flight. The sensor is considered degrading randomly in time.the degradation process described in Barata et al. (2002) is adopted. The following assumptions are assumed: Sensors are subject to a defined mission time T miss, the time variable is divided into successive time steps. The sensor i has a specified initial degradation level, d i (0). At each time step t, the sensor i can fail with probability q i (d i (t)) which depends on the degradation level d i (t) reached by the sensor at that time.(fig. 2). Each sensor i is non-repairable during the mission time; this means that its degradation level cannot decrease. There is a maximum allowed degradation level, d max, that the sensor i may reach within its mission time and in correspondence of which the probability of failure, q i (d max ), is q max,i. If at any given time t < T miss, d i (t) > d max, the sensor i is considered failed. Fig. 2. Probability of a Fix sensor failure q as a function of the degradation level normalised to its maximum value d max During the mission time, the sensor i may degrade by a random amount proportional to the degradation level exhibited at that time. It means that at each time step, the sensor i can remain in this actual state or can be incremented by a random quantity, which represents a percentage of the degradation level currently achieved. The following equation is here adopted to represent this process: d i (t) = d i (t 1) (1 + Ψ) Where d i (t), is the degradation level at time step t and Ψ 0,is the random fractional increase in degradation. Case of retractable Sensor Retractable sensor is used to inspect the exactitude the Fix sensors measurements accuracy. This sensor has a different work cycle to fix sensors. During the system good working time (K out of N sensors give correct measurements), the retractable sensor is permanently inside and comes outside just in the inspection dates. During the system fail, the sensor is permanently outside to replace failing sensors. The 1282

3 following assumptions are assumed for Retractable sensor degradation Model: The retractable sensor is designed to function for a defined mission T miss, the time variable is divided into successive time steps. The sensor has a specified initial degradation level, d(0). For the system good working time, the sensor sensor degrades by a fixed quantity at each inspection. The number of inspections will be a subject for optimization in our Inspection policy.. From the date of system failure (i.e. K-out-of-N sensors are failed) to the end of the mission, retractable sensor degrades continuously as the same degradation model of Fix sensors. 3.2 Multiple non-repairable system Consider now the whole system components (a multicomponent non repairable system) which components are inspected periodically in order to detect as soon as possible failure dates. Then, system have to ignore failing components indications. Each Fix sensor has the same degradation behavior described by the stochastic model illustrated in the paragraph 3.1 and the retractable sensor degrades as illustrated in section 3.2. The specific assumptions are the following: The system is made of N Fix Pitot sensors and a retractable sensor. Each sensor i has a specified initial degradation level d 0,i. At each time step t, a fix sensor can fail with probability q i (d i (t)). There is a maximum allowed degradation level, denoted by d max,i, that the fix sensor i may reach within its mission time and in correspondence of which the probability of failure, q i (d max,i ) is q max,i. If the ith sensor reaches a given degradation threshold d m,i, its degradation is incremented by an additional fix quantity µ at each time step between the reaching date and the next inspection date. This parameter will be subject of optimization. Failures are assumed to break the sensor. Thus, in case of failure at time t, the sensor cannot be repaired and its measurements have to be ignored in the rest of the mission. If sensor i reaches d max,i : it is considered failed and its measurements will be ignored during the rest of the mission (δ = 0). If sensor i reaches d m,i,respectively d max,i,and not inspected: the system measurement quality degrades as quickly as it is not yet inspected. This will be expressed by an additional quantity of quality degradation incremented for each unit of time between the time at which d m,i,respectively d max,i, is exceeded and it s inspection date. If a K out of N sensors are failing: The retractable sensor is permanently out to take measurements for the rest of the mission. None of the above: sensor i continues its normal operation. The optimization problem concerns the choice of the optimal periodic inspection policy which aims to detect N sensors failures dates as soon as possible and consequently maximize the system measurements accuracy. The objective is then to minimize the expected total system measurement quality degradation during system operation over the defined mission time. The degradation function uses the following factors: µ t : designs the total expected degradation rate. µ s : designs the degradation quantity of the retractable sensor at each inspection. µ 1 : designs the degradation quantity of a sensor per a unit of time between the date of reaching d m,i and the date of its detection. µ 2 : designs the degradation quantity of a sensor per a unit of time between the date of reaching d max,i and the date of its detection. τ 1,i corresponds to the time at which the degradation rate d m,i is exceeded. τ 2,i corresponds to the time at which the degradation rate d max,i is exceeded.(fig. 3, Fig. 4). 4. INSPECTION POLICY 4.1 Optimal inspection policy At each time step, one of the following events takes place exclusively for each component of the system: If sensor reaches d m,i : its measurements will be taken into account but not with the same importance of other sensors. For example, if we consider that the speed is calculated as the weighted average (noted as AVG) of the three sensors measurements when their degradations are inferior to d m,i ; AVG (V1+V2+V3). If, for example, the third sensor reached d m,3, speed will be calculated as AVG (V1+V2+δ*V3) with δ < 1. Fig. 3. Stochastic degradation rate for a Fix Pitot sensor Fig. 4. Example of measurement quality degradation due to the first Pitot sensor degradation 1283

4 { 1 if i τ1,i > (k 1)T and τ m k = 1,i < kt 0 otherwise { 1 if i τ2,i > (k 1)T and τ max k = 2,i < kt 0 otherwise NC: number of inspections during the mission. T miss : Duration of the mission. T : Length of the period between two successive inspections. The total measurement quality degradation of the system is given by: µ t =NC µ s + N NC µ 1 (kt τ 1,i ) i m k+ N NC µ 2 (kt τ 2,i ) i max k NC N µ 1 (kt τ 2,i ) i m k i max k 4.2 Proposed algorithm to compute the optimal inspection sequence The numerical algorithm used to compute the optimal inspection sequence is presented in (Fig.5). 5. NUMERICAL EXAMPLE A numerical example is proposed to illustrate the investigated inspection model. It is assumed that different rates of degradations are well estimated and known. Furthermore, probability distributions of system degradation composed of three sensors and time to failure are assumed to be well estimated. In the present numerical example, these input parameters are set according to what follows. T miss = 6 hours Time step: 0.5 hour µ s = 0.2 µ 1 = 0.1 µ 2 = 0.15 Failure and reaching d m dates are deduced from the Ψ values at each time step for each sensor and the (d m,i,d max,i ) values. The following values are used in this example: τ 1,1 = 2; τ 1,2 = 3.5; τ 1,3 = 4.5 τ 2,1 = 4.5; τ 2,2 > T miss ; τ 2,3 > T miss. When applying this values, the following results are obtained (Table. 1): Table 1. Average total degradation for different periodic inspection policies NC µ t The following figure (Fig. 6) illustrates the evolution of the total system degradation in function of the number of periodic inspections realized. Fig. 6. Total system degradation based on the number of periodic inspections For this example, the optimal periodic inspection policy is NC = 3; T = 2hours. 6. CONCLUSION Fig. 5. Algorithm for computation of the optimal inspection sequence In this paper, we have proposed a periodic inspection policy for a multiple unit irreparable system. Our approach 1284

5 discusses the case of Pitot sensors system during a flight. It is assumed that sensors are subject to random deterioration. Our aim was to determine inspection dates which minimize the average total degradation of measurement quality and assess to the system degradation. As a future work, we are currently working on application of other inspection policies. Other future works can deal with the development of other optimization methods. Obtained results should be analysed and compared.we aim to ameliorate the expression of the proposed function for the periodic policy and study the case where the inspection policy can be sequential. A computation methods like Nelder-Mead Algorithm or Genetic algorithm...) will be studied to define these inspection policies. Sharma, A., Yadava, G., and Deshmukh, S. (2011). A literature review and future perspectives on maintenance optimization. Journal of Quality in Maintenance Engineering, 17(1), Tian, Z., Lin, D., and Wu, B. (2012). Condition based maintenance optimization considering multiple objectives. Journal of Intelligent Manufacturing, 23(2), REFERENCES Aït-Kadi, D. and Chelbi, A. (2010). Inspection strategy for availability improvement. Journal of Intelligent Manufacturing, 21(2), Badıa, F., Berrade, M.D., and Campos, C.A. (2002). Optimal inspection and preventive maintenance of units with revealed and unrevealed failures. Reliability Engineering & System Safety, 78(2), Barata, J., Soares, C.G., Marseguerra, M., and Zio, E. (2002). Simulation modelling of repairable multicomponent deteriorating systems for on condition maintenance optimisation. Reliability Engineering & System Safety, 76(3), Bérenguer, C., Chengbin, C., and Grall, A. (1997). Inspection and maintenance planning : An application of semi-markov decision processes. Journal of intelligent manufacturing, Chelbi, A. and Ait-Kadi, D. (2009). Inspection strategies for randomly failing systems. In Handbook of maintenance management and engineering, Springer. Chelbi, A., Ait-Kadi, D., and Aloui, H. (2008). Optimal inspection and preventive maintenance policy for systems with self-announcing and non-self-announcing failures. Journal of Quality in Maintenance Engineering, 14(1), Chouikhi, H., Khatab, A., and Rezg, N. (2014). A condition-based maintenance policy for a production system under excessive environmental degradation. Journal of Intelligent Manufacturing, 25(4), Grall, A., Bérenguer, C., and Dieulle, L. (2002). A condition-based maintenance policy for stochastically deteriorating systems. Reliability engineering systems safety, Hontelez, J.A., Burger, H.H., and Wijnmalen, D.J. (1996). Optimum condition-based maintenance policies for deteriorating systems with partial information. Reliability Engineering & System Safety, 51(3), Kopnov, V. (1999). Optimal degradation processes control by two-level policies. Reliability Engineering & System Safety, 66(1), Liao, H., Elsayed, E.A., and Chan, L.Y. (2006). Maintenance of continuously monitored degrading systems. European Journal of Operational Research, 175(2), Sarkar, J. and Sarkar, S. (2000). Availability of a periodically inspected system under perfect repair. Journal of Statistical Planning and Inference, 91(1),