Piecewise Deterministic Markov Processes and Dynamic Reliability

Transcription

1 Piecewise Deerminisic Markov Processes and Dynamic Reliabiliy H Zhang 1, K Gonzalez 1, F Dufour 1, and Y Duui 2 1 IMB, UMR 5251, Universié Bordeaux 1,Talence, France 2 IML-LAPS, UMR 5218, universié Bordeaux 1, Talence, France Absrac: If he reliabiliy communiy sill remains ineresed in dynamic reliabiliy heory, i is no really convinced by he abiliy of already available approaches o rea curren problems coming from operaional domain, even if heir mehodological qualiy is undeniable. This paper is in kipping wih he opic of wo papers presened in previous conferences. Is aim is o show he poenialiies of a mehod ha combines he high modeling capaciy of he piecewisedeerminisic processes and he grea compuing power inheren o he Mone-Carlo simulaion. This mehod has been applied o a well-known es-case example o es is abiliy o solve common dynamic reliabiliy problems. Two ses of resuls have been obained. The firs one has been compared o hose coming from a Peri ne model o ge a preliminary validaion of he proposed mehod. The second one, relaed o a more complex case, has been compared o already published resuls found in he lieraure. Conrary o already exising mehods, our approach is an exac Mone Carlo sampling mehod, i does no need ime-space discreizaion. Keys words: dynamic reliabiliy, piecewise deerminisic Markov processes, Peri nes 1 INTRODUCTION A curren challenge in reliabiliy analysis oday is o ake ino accoun he dynamic behavior of sysems. The modeling is a key sep in order o sudy he properies of he involved physical process. I appears now necessary o ake ino accoun explicily and in a realisic way he dependencies (in oher words) he dynamic ineracions exising beween he physical parameers (for example : pressure, emperaure, flow rae, level) of he process suppored by he sysem and he funcional and dysfuncional behavior of is componens. For a large class of indusrial processes, he layou of operaional or accidenal sequences generally comes from he occurrence of wo ypes of evens: The firs ype is direcly linked o a deerminisic evoluion of he physical parameers of he process, The second ype of evens is purely sochasic. I usually corresponds o random demands or failures of sysem componens I is well-known ha he classical mehods used in sysems reliabiliy field, such as combinaory approaches (faul rees, even rees, reliabiliy diagrams) or Markov and semi-markov models are no able o correcly model physical processes involving deerminisic behavior. To overcome his difficuly, he auhors propose o carry ou an approach hey have already applied o a simple hybrid sysem [3]. This approach is inroduced in he nex secion. In 198, M.H.A. Davis [2] inroduced in probabiliy heory he Piecewise Deerminisic Markov Processes (PDMP) as a general class of models suiable for formulaing opimizaion problems in queuing and invenory sysems, mainenance-replacemen models, invesmen scheduling and many oher areas of operaion research. The noion of piecewise deerminisic process is very inuiive and simple o describe. The sae space of his sysem is given, for 1

2 example, by a subse E of he se of real numbers R. Saring from x in E, he process follows a deerminisic rajecory (given, for example, by he soluion of an ordinary differenial equaion) unil he firs jump ime T 1, which occurs eiher sponaneously in a random manner, or when he rajecory his he boundary of E. In boh cases, a new poin is seleced by a random operaor and he process resars from his new poin. Consequenly, if he physical parameers of he physical process under consideraion are described by he sae x of a piecewise deerminisic process, beween wo jumps he sysem follows a deerminisic rajecory. As menioned before in he case of evens, here exis wo ypes of jump: The firs one is deerminisic. From he mahemaical poin of view, i is given by he fac ha he rajecory his he boundary of E. From he physical poin of view, i can be seen as a modificaion of he mode of operaion when a physical parameer reaches he criical value. The second one is sochasic. I models he random naure of some failures or inpus modifying he mode of operaion of he sysem. The aim of his paper is o show he abiliy of he PDMP approach o solve common dynamic reliabiliy problems by applying i o a specific bu no rivial example of hybrid sysems known as he heaed ank sysem (HTS o be brief) [5, 6, 7]. The remainder of he aricle is organized as follows. Secion 2 presens he mahemaical model relaed o our PDMP approach. The heaed ank sysem is described in secion 3.2. The general implemenaion of his model is developed in secion 3.3. In secion 4 he PDMP approach is applied o he basic case of he HTS problem, in which he effecs of emperaure are no considered. The same case in hen reaed by means of a PN-model and he resuls obained by boh mehods are compared in view of a reciprocal validaion. Finally, only he PDMP approach is able o handle he more elaboraed case where he consrains induced by emperaure are aken ino accoun. This is he aim of secion 5, which precedes a shor conclusion. 2 MATHEMATICS MODEL Le d be an applicaion of K in N, where K is a counable se associaed wih he lis of possible regimes of operaion of he physical process. Le (E! )! "K a family of open subses of R d(! ). For! "K,!E " denoes he boundary of E!. A piecewise deerminisic Markov processes is deermined by is local characerisics (! ",# ",Q " ) "$K where E!! " is a Lipschiz coninuous vecor field in deermining a flow! " (x,). Now define he following family of ses indexed by K {! # E : x = "( y, ). y! E, > } $ + : = x { x! $ E : x = #( y, " ). y! E, > } " : = The se! " + # $E " represens he boundary poins a which he flow! " (x,) exis from E!, while! " # $ E " is characerized by he fac he flow saring from a poin in E! will no leave E! immediaely. Therefore, i is naural o define he sae space by {(, x) : K e x $ "!! "! # " + } E : E = and he boundary of he sae space is given by + #" : E! R is he jump rae of he process. Q! : E " # + $ [,1] is a ransiion measure saisfying he following propery (!(", x) #K $ E + ), Q " (x,e ' {(", x) }) = 1 I is shown by Davis (1993) ha here exiss a probabiliy space (!,F,{ F },P X ) on which he X = ( m, x ) can : moion of he PDMP (X ) wih { } be defined ieraively as follows. Saring from! "K and x, he firs jump ime T 1 is given by 2

3 { } P X (T 1 > ) := I { " <! (x )} exp # $! ((x,s))ds Then he rajecory of (X ) for [,T 1 ) is given by # x =! " (x,) $ m = " A ime T 1, he process jumps o a new locaion and o a new regime defined by he random variable X 1 = (! 1, x 1 ) wih probabiliy disribuion Q! ("(x,t 1 ),#). Saring from X 1 = (! 1, x 1 ), he nex iner-jump ime T 2! T 1 and pos-jump locaion X 2 = (! 2, x 2 ) are seleced in a similar way. Under some echnical hypoheses, he process so defined is Markovian. I has piecewise deerminisic coninuous rajecories wih jump ime T, T, 1 2 L and pos-jump locaion X, X, 1 2 L. The sae space of his process is defined by he produc of an Euclidean space and a discree se. Therefore, his processes belongs o he class of hybrid models. The variable m 2 models he regime of he physical sysem and influences he flow of sae variable x. 3 THE HEATED TANK PROBLEM AND IMPLEMENTATION This problem has been reaed and solved by Marseguerra and Zio. [4, 5]. They have esed various Mone Carlo approaches o reliabiliy and safey analysis. Tombuyses [6] have used he same sysem o presen coninuous cell-o-cell mapping Markovian approach (CCCMT). The sysem is no rivial because of exisence of wo processes variables (liquid level and emperaure). 3.1 The heaed ank sysem Fig. 1 The heaed holdup ank The sysem consiss of a ank conaining a fluid whose level is conrolled by hree componens: wo inle pumps (Uni 1 e 2) and one oule valve (uni 3) (Fig. 1). Each componen has four saes: OFF, ON, Suck OFF, and Suck ON. Fig. 3 schemaizes he ransiion among differen saes. I is a inhomogeneous Poisson jumps process. A hermal power source heas up he fluid, he failure raes of he componens depends on he emperaure.! c = a(") ˆ! c, c = 1,2,3 (1) a(!) = (b 1 e b c (! "2) + b 2 e "b d (! "2) ) / (b 1 + b 2 ) Where a(!) is a funcion of emperaure (Fig. 2) As menioned before, he sysem is described in Secion 3.1. I will be modeled by a PDMP process in Secion 3.2, and in Secion 3.3 we presen he numerical implemenaions of his model. Fig. 2 The funcion a(!) 3

4 wih b 1 = 3.295, b 2 =.7578, b c =.5756, b d =.231 ˆ! 1 = h "1 ˆ! 2 = h "1 ˆ! 3 = h "1 Conrol laws are used o modify he sae of he componens o keep he liquid beween wo limis : 6 meers and 8 meers Law 1: If he liquid level drops under 6 meers, he componens 1,2,3 are pu respecively in he sae 1, 1 and (if hey are no suck ON or OFF) Law 2: if he liquid level rises above 8 meers, he componens 1,2,3 are pu respecively in he sae, and 1 (if hey are no suck ON or OFF). Fig. 3 Saes and ransiions of he componens The wo coninuous variables are he liquid level h and he emperaure! ha are boh funcions of he sae of he componens. A =, he sysem is assumed o be in he equilibrium sae, i.e. he componens are in he sae (1,,1) and he emperaure is C and he liquid level is 7 meers. The variables (h(),!()) saisfy he following differenial equaions dh / d =! 1 (") (2) ' d# / d = (! 2 (") $! 3 (")#) / h Where! = (! 1,! 2,! 3 ) and c!{1,2,3} wih " if c is OFF or suck OFF! c = # $ 1 if c is ON or suck ON! 1 (") = (" 1 + " 2 # " 3 )G! 2 (") = (" 1 + " 2 )G$ in ! 3 (") = (" 1 + " 2 )G, G = 1.5, $ in = 15 Physically, he discree variables! denoe he differen regimes of he sysem, and! i, i!{1,2,3} are consans. The sysem (2) is deduced from he mass and energy conservaion laws. We are ineresed in hree possible Top Evens: drayou (h! 4 meers), overflow (h! 1 meers) and ho emperaure (! " 1 C), he p 1 (), p 2 (), p 3 () are he cumulaive probabiliies of hese Top Evens a ime. 3.2 Soluion of differenial equaion sysem Le x = (h,! ) be he iniial condiion of he process variables a ime =,! = (! 1,! 2,! 3 ) he componen saes, we will give here he analyical soluion of he sysem defined by (2). According o he configuraion of he sysem, he coefficiens of he differenial equaion sysem can be zero, and here exis four differen cases. #! 1 (") =, $ dh / d = If $ hen and! 3 (") = d! / d = " 2 (#) / h $ h() = h!() =!() + " (#) 2 ' h $! 1 (") =, dh / d = If hen! 3 (") # ' d! / d = (" 2 (#) $ " 3 (#)!) / h h() = h '!() =! e "# 3 ($ )/h + # 2($) # 3 ($) " # # 3 ($ ) 2($) # 3 ($) e" h ' (! If $ 1 (") #, $ dh / d =! hen 1 (")! 3 (") = d# / d =! 2 (") / h * h() =! 1 (") + h, + #() = # +! (") 2! 1 (") ln $! (") ' 1, + 1 h ( ) -! If $ 1 (") #, dh / d =! hen 1 (")! 3 (") # ' d# / d = (! 2 (") $! 3 (")#) / h h() =! 1 (") + h ( ' #() = h $ # h $ () +! (") 2! 3 (")! (") 2! 3 (") h$ $ ( ()h ) Where! = " 3 (#) / " 1 (#) $. We can see ha in he hree firs cases, he sysem can be considerae as ``degeneraes'' because he wo variables evolve independenly, while in he las case, he emperaure depends no only on he ime bu also on he liquid level h(). Given an iniial condiion x and he componens saes! = (! 1,! 2,! 3 ), one can easily 4

5 deduce he values of! i ("), i = 1,2,3and calculae!, he nex insance ha he couple (h(),!()) will reach heir physical boundary. Thanks o exisence of analyical soluion, we do no need o discreize he physical space and do numerical inegraion. 3.3 Implemenaion Wih he noaion given in Secion 2, he sae of he sysem can be defined by X = (!, x ). Le X T = (!, x ) = ((1,,1),(7, )) be he iniial condiion of he sysem, before he firs sopping ime, he sysem saisfies following equaion # X = (!,"! (, x )) if < T 1 $ (3) (! 1, x 1 ) if = T 1 Where he jumping ime T 1 is a random number wih following surviving funcion P X (T 1 > ) := I { " <! (x )} exp # { $! ((x,s))ds} The jump rae! " (#(x,s)) is ime-dependen and { } is he ime ha v! (x) := inf >,"(x,s) = #E $ he flow ouches he boundary. A Mone Carlo mehod can be applied o simulae X T1, we can describe he algorihm by 4 seps. A firs, we simulae he jumping ime of he 3 componens, each componen has is own failure rae defined by (1). In he second, we calculae! he ime for he flow o exi E!, by aking he minimum of hese imes,! = min{ ",! c } we obain he nex sopping ime c T 1 = T +!. In he hird sep, we updae he flow value o he ime T 1, o ge x 1 =! " (x,t 1 ). In he las sep, we calculae he new sae! ha he sysem will ake afer he jump, following a ransiion measure Q!. Taking X T1 = (! 1, x 1 ) as new iniial condiion, his procedure will hen be repeaed o obain X X,L unil a fixed final ime is reached and { } T, 2 T3 his complees a whole MC hisory. This is a general procedure for simulaion of PDMP processes. I was applied wih success in [3]. However here are wo specificiies in his ank sysem. The firs one is ha he sysem is nonrepairable, so in he firs sep of he procedure, when a componen is in he sae 2 or 3, is failure rae is zero, we jus se he corresponding jumping ime! c o be!. The second one is he fac ha he failure rae! c () is inhomogeneous, i depends on he emperaure, which is soluion of wo dimensions ODE, we use he algorihm presened in pages in [1]. The ransiion measure Q is easy o be implemened. I is deerminaed by conrol laws when he level of liquid is 6 or 8 meer, or by a Bernoulli draw (p =.5) when one of hree componens is in failure. As menioned in [6] he Mone Carlo soluion of he problem and is programming was made easier by he following facors: he conrol laws are deerminisic, he componens are nonrepairable, he dependence on he emperaure is idenical for all he failure raes. However, he presened model can handle more complex configuraions such as reconfigurable componens wih differen failure raes. 4 CASE-STUDY 1: HTS WITHOUT TEMPE- RATURE 4.1 The PDMP approach This approach is firs applied. The general model presened in secion 3 has been simplified for his case by canceling relaions beween emperaure! and boh fluid level h and failure raes λ c. Some numerical resuls hus obained are gahered in able 1 of secion The PN approach Peri nes are powerful ool for modeling in a concise way he behavior of any sysem. Because of he lack of space, he main feaures of his approach can no be described here, bu hey can be found in many papers already published (see, for insance, ref [7, 8]). However some informaion is given hereafer o explain how he PN-model can describe he dynamic evoluion of he sudied sysem. Some PN-models dedicaed o our problem are presened in figures 4 and 5. In fac, we need seven PNs o model he HTS problem. Only four PNs are depiced. The one presened in figure 4a models he behaviour of pump1 (P1) and he PN locaed in figure 4b is used o updae he iniial condiions and o compue he differen ime delays o reach he specific levels defined in secion 3.1. The PN presened in figure 5a is relaed o he conrol laws, and he oher one in figure 5b indicaes he possible occurrence of each considered op evens, i.e., he overflow and he dryou. 5

6 4.3 Comparison of he obained resuls A small sample of resuls is presened in able 1. They correspond o he occurrence probabiliies of he above op-evens esimaed from boh PDMP and PN models. (a) (b) Fig. 4 (a) PN-model for he pump1; (b) PN-model used o compue ime delays Table 1 Overflow and dry-ou probabiliies Time Overflow Dry-ou (hour) PDMP PN PDMP PN Afer some previous successful experimens [9], we consider ha he good agreemen beween he ses of resuls of able 1 enables us o claim PDMP and PN models are suiable o model he behaviour of any hybrid sysem exhibiing only one coninuous process variable, such as fluid level in a ank, and o esimae heir indicaors of ineres, such even probabiliies. This being said, wha happens if wo coninuous variables mus be simulaneously aken ino accoun? Unil now and as far as we know, he PN approach has no proved is abiliy o solve his kind of problems, conrary o he PDMP mehod, as shown in he nex secion. 5 CASE-STUDY 2: HTS WITH TEMPERATURE CONSIDERATION (a) (b) Fig. 5 (a) PN-model for he conrol laws ; (b) PN-model relaed o he occurrence of he wo op evens In his case he overall model previously presened is used. The cumulaive probabiliies p 1 (), p 2 (), p 3 () of he Top Evens can be esimaed by using large number of ime hisories N. We use he resuls from 1 7 hisories as our reference soluion. Figure 6 gives he resuls from a 1 3 and 1 4 hisories sampling, we can observe he convergence of he mehod wih respec o N. The lef picure of figure 7 gives he PDMP resuls from 1 5 hisories while he righ one gives he soluion obained by CCCMT and Mone Carlo. Every curve is superposed by he PDMP reference soluion (dashed line). Table 2 gives he CPU ime (on an AMD Operon Processor 275) for PDMP Mone Carlo mehod. The efficiency of our approach is due o he fac ha neiher ime discreizaion nor space discreizaion is used. This reduces considerably he compuaional ime and memory requiremens. 6

7 Fig. 6 PDMP resuls for N=1e3 and N=1e4 compared wih reference soluion Fig. 7 PDMP resuls for N=1e5 and comparison of reference soluion wih CCMT soluion problems. REFERENCES Number of Hisories N Table 2 CPU Time CPU ime 2.2 GHz s s m37s m37s.13 6 CONCLUSIONS Esimaion of relaive error p 1 (!) Through a simple bu no rivial es-case, i seems o us ha he PDMP approach is an alernaive way o correcly model he behaviour of hybrid sysems and o evaluae heir performance in erms of reliabiliy and availabiliy. Our nex challenge is wofold. I consiss in performing RAMS analyses of realisic size sysems by means of he presened PDMP mehod and by using an exended PN model able o handle such 1 Cocozza-Thiven, C. Processus sochasiques e Fiabilié des sysèmes, Springer, Davis, M. Markov models and opimizaion. London: Chapman and Hall, Dufour, F. and Duui, Y. Dynamic reliabiliy : A new model. In Proceedings of ESREL 22 Lambda-Mu 13 Conference, 22 4 Marseguerra, M. and Zio, E. The cell-oboundary mehod in Mone Carlo based dynamic PSA. Reliabiliy Engng Sysem Safey, 1995, 45, Marseguerra, M. and Zio, E. Mone Carlo approach o PSA for dynamic process sysems. Reliabiliy Engng Sysem Safey, 1996, 52, Tombuyses, B. and Aldemir, T. Coninuous cello-cell mapping and dynamic PSA. In Proceedings of ICONE 4 conference, Duui, Y., Châele, E., Signore, J. P. and Thomas, P., Dependabiliy modeling and evaluaion by using sochasic Peri nes: 7

8 applicaion o wo es cases. Reliabiliy Engng and Sysem Safey, 1997, 55, Malhora, A. and Trivedi, K. S., Dependapiliy modeling using Peri nes. IEEE Trans. Reliab., 1995, 44, Zhang, H., Dufour, F., Innal, F. and Duui, Y. Dynamic reliabiliy and piecewise-deerminisic processes. In Proceedings of Lambda-Mu 15 Conference, 26 8

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