semi-markov models with an application to power-plant reliability analysis.pdf

8/14/2019 Semi-Markov models with an application to power-plant reliability analysis.pdf

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526 IEEE TRANSACTIONS ON RELIABILITY, VOL. 46, NO. 4 997 DECEMBER

Semi-Markov Models with an Application toPower-Plant Reliability Analysis

Mihael PermanUniversity of Ljubljana, LjubljanaAndrej SenegaEnikUniversity of Ljubljana, Ljubljana

Matija TumaUniversity of Ljubljana, Ljubljana

Key Words Markov process, Semi-Markov process,Transition probability, Weibull holding time, Power-plant op-eration, Availability.

Summary Conclusions ystems with, 1) a finitenumber of states, and 2) random holding times in each state,are often modeled using semi-Markov processes. For generalholding-time distributions,closed formulas for transition prob-abilities and average availability are usually not available. Re-cursion procedures are derived to approximate these quantitiesfor arbitrarily distributed holding-times; hese recursion proce-dures are then used to fit the semi-Markov model with Weibulldistributed holding-times to actual power-plant operating data.The results are compared to the more familiar Markov mod-els; the semi-Markov model using Weibull holding-times fits thedata remarkably well. In particular comparing the transition

es shows that the probability of the system being inthe s tate of refitting converges more quickly to its limiting valueas compared to convergence in the Markov model. This couldbe because the distribution of the holding-times in this stateis rather unlike the exponential distribution. The more flexiblesemi-Markov model with Weibull holding-times describesmoreaccurately the operating characteristics of power-plants, andproduces a better fit to the actual operating data.

1. 1NTR.ODUCTIONAcronymPP power-plant.

Continuous-time Markov chains are widely used in thestate-space modeling of reliability and system-availabilityfor electric power systems [l 21. The state-space ap-proach applies also to modeling PP operation and to an-alyzing its reliability availability. In Markov modelsthe holding-times are exponentially distributed which isoften too restrictive and might not fit the a ctual operatingda ta well. Considering more general holding-times leads tosemi-Markov processes which are less amenable t o analyt ictrea tment but provide more flexible models. Semi-Markov

lThe singular plural of an acronym are always spelled the same.

processes have been extensively studied; see [3 ] for a re-view.

Section 2 considers the problem of computing the tran-sition probabilities for semi-Markov processes. Analyticformulas in [4]are in te rms of infinite series of convolutionsof th e semi-Markov kernel; Laplace transforms are used togive closed form expressions. Unfortunately, these Laplacetransforms can rarely be inverted analytically. Numericalcomputation of inverse Laplace transforms for a relatedproblem concerning the distribution of the number of u pstates in a given time interval is in [5]. To avoid Laplacetransforms, we derive a recursion formula for the Markovrenewal kernel when the holding-times have a discrete lat-tice distribution, and then use this kernel to compute thetransition probabilities. We obta in a recursion formula forthe average system availability.

Section 3 applies the recurrence formulas to approxi-mate the transition probabilities and related quantitieswhen the holding-times have a Weibull distribution withparameters depending only on the present state. The con-tinuous Weibull distribution is first replaced by a latticedistribution closely resembling it. This latti ce distributionis a slight modification of the discrete Weibull distribution[6]. The parameters in the semi-Markov model are esti-mated with Weibull holding-times from recorded holding-times in stat es for the Slovenian coal-firedPP Sostanj (294MW unit) and examine the fit of the model. (The Markovmodel for this data-set hm been considered in [7 ] . Theparametersof the Weibull holding-times and the transitionprobabilities of the embedded Markov chain are estima tedand used in the recursion. The fit is then compared to thefit of the Markov model.Notation-A average unavailability

E,{.} E{.IYo i}F,( t ) Cdf{holding-time in state i}G set of system statesG'i , j system states

set of system u p states

0018-9529/97/$10.0001997 IEEE


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PERMAN ET A L SEMI-MPLRKOVMODELS W ITH AN APPLICATION TO POWER-PLANT RELIABILITY ANALYSIS 527

lattice stepSf{holding-time in state j }indicator function: Z(True) = 1, Z(Fa1se) = 0number of transitionstransition matrix for the embedded Markovchain with entries p z , jconditional probability Pr{.IYo = i}Pr,{Y, = j } : transition probability at time t[defined in (2)]limtAw (t)]: transition probabilityfrom i to j for {X,}semi-Markov kernel with entriestime-dependlent Q with entries, Q a , j ( t )Markov renewal matrix with entries Ra,3time-instant of transition ntime, a r.v.timeWeibull distributed r.v.embedded discrete Markov chainr.v. with values in Gtrajectory of the embedded Markov chainlimiting availabilitylimt-tw[At~,or j E G[scale, shape] parameter of Weibull distributionZ ( i = j )equilibrium measurescardinality o set A .

EO, t]) = Ra, j t ) :Markov renewal measure

Other, standard notation is given in Information forR,eaders Authors at the rear of each issue.

Figures2 3 were produced with MATLAB [13].2. TRANSITION PR.OBABILITIES

Following Cinlar [4], z semi-Markov process is defined asa sequence of 2-dimens ional r.v. , { (X,, T,) : n E 1 , 2 , .. }with the properties:X , is a discrete-time Markov chain taking values ina countable set of possible states G of the system andrepresents its state after transition n.

The holding-timles, T, T,-1, between two tran-sitions are r.v. whose distribution depends only on thepresent state X , and the state after the next transition:Pr{X,+1 = j,T,+l T, 1X0, . . . ,X,; TO, .. ,T,}

= Pr{X,+1 = j ,Tn+l T, tlX,}. (1)A s s u m p t i o n s

1. The process is homogeneous in the sense that thereis a family of generalized Q i , j for i j E G (a semi-Markovkernel) such that:Q i , j ( t ) = Pr{X,+1 = j ,%+I T, lx, = i} (2)for any two states i , j G.2. The holding-times,T,+1 -Tn,re positive with prob-

ability 1. While in (2) the holding-times can depend onthe present following states, this paper mostly assumes

that the semi-Markov kernel is of the form:Q z , j ( t ) = Pa,j .Fz t); (3)thus the distribution of the holding-time depends only

on the present sta te. For Weibull holding-times, used insection 3, the semi-Markov kernel is:Q a , j ( t ) = Pa,j * weif(t/~z;z ) . (4)

3. The continuous-time process {y : t 2 0) whichdescribes the system state that is evolving according tothe semi-Markov model is:Y = X , on the interval T, 5 t < T,+l. 5 )4. The holding-times are as defined in (2).We compute Pr{Y, = jlY0 = i } , the average availability

of such a system in a finite time interval [O,t], and theasymptotic behavior of th e system as time O.To compute the availability or average availability of asystem that evolves as a semi-Markov process, the transi-tion probabilities need to be computed. The formulas forthese transit ion probabilities have been studied extensively[4]. A brief derivation is given here.D ef i n i t i on 1. The R(t )over G is:Ri,j(t)=

COE,{ xZ(X, =j).Z T, [O,t])};i,jE G;t 2 0. ( 6 )

n=OThe transition probabilities are given by theorem 1.T h e o r e m 1 [4]:Pr{yt = jlY0 = i } =

The proof is in appendix A . 1 .R.oughly, (7) comes from adding the probabilities of

events of the form: {process Y visits j for occasion nin s,s + ds) , then stays in j until time t} . Given thata transition to j occurs at time s, what happens after sis conditionally s-independent of what happens before s.Then factor out the first term on the left in (7 ) , viz, prob-ability that no transition occurs in (t , ]; sum over n toobtain Ri,j(t); nd integrate over s.

The average system-availability is:

To simplify the notat ion, use:H j ( s ) = Prj{Tl > s } = 1 Q j , k ( s ) . (9)

k E G


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528 IEEE TRANSACTIONS ON RELIABILITY, VOL. 46, NO. 4, 1997 DECEMBER

Corollary 1comes from theorem 1.Corollary 1. For YO= i

jEG '

The proof is in appendix A.2.Compute the limits of the average unavailability as t

increases. Appendix A.3 trea ts the special case wherethe holding-time distribution depends only on the presentsta te, The formula can be derived from [4];a short, simplederivation is given here.DerivationEq (7) for transition probabilities is used for Weibullholding-times. Since the Weibull distribut ion does nothave a simple Laplace transform, the kernel R(t) cannotbe computed analytically. For computation, th e Weibulldistribution is replaced by an appropria te discrete distri-bution (see section 3).For discrete holding-times that take only values in thelattice { k .h : > 0}, (7) becomes:

prz{yk ,h = j } =k

[R2,J(Z. Rz,J((Z 1 ) . )] H ( ( k + 1 ) .h ) . 11)z=o

Because R,,J is a jump function for any i , j , the integralbecomes a sum.

Let Rz,J(-h)= 0. R(t) can be computed recursively:00

R,,j(t)= E , { Z qxn = Z( Tn E [O,tl)n = O

00

= E , { E i { C Z ( X , = j .x Tn [O,t])ITi,Xi}}n=O

00

= , J +Ez{CZ(Xn =j).Z Tn [O,t])lTl,X1}n =l

The last line of th e equations follows from the preced-ing one by the semi-Markov proper ty of the sequence{ ( X n , T n ) n 2 0). For an alternative approach see [8]where integral equations for the transition probabilitiescan be obtained using the case where the set of up statescontains only one stat e, and point availabilities rathe r thaninterval availabilities are considered. However,a, differentset of integral equations would have to be solved for eachPZJ ( t ) .

R,ewrite (12) for discrete holding-times on the lattice:R ( m . h ) =

mI + C [ Q ( l h ) I 1) .h ] R ( ( m ) h ) (13)

1= 1R(0) = I;

I identity matrix = c , ~ ) .This recursion is used in section 3 to approximate

transition probabilities when the holding-times have theWeibull distribut ion. The continuous Weibull distribut ionis replaced by a discrete lattice distribution which closelyresembles it.

3. WEIBULL HOLDING-TIMES2This section presents a model for PP operation with 6

states and Weibull holding-times. The transition proba-bilities are computed with (ll), and the parameters forthe Weibull distribution and the transition probabilitiesfor the embedded Markov chain have been estimated fromreal operating data.3.1 Model Assumptions for This Example

Figure 1. PP Model with Possible TransitionsA. The system state s and the possible transitions be-B. The 6 system sta tes are [9, IO ] :

tween the m are in figure l .I. Operating sta te (up).a. Stoppage due to low power-demand (up).3. Boiler failure (down).4, Turbine failure (down),5.(down).6. Refitting (down).C. The only transitions from down states, with the ex-

ception of the transition from sta te 3 to 2, are intostate 1. (This simplification is chosen on the basis of theoperating data, ie, other transitions rarely occur.) How-ever, the formulas from section 2 do apply to arbitrarytransit ions.

Stoppage due to states other than 2 - 4

2Thenumber of significant figures is not intended to imply any ac-curacy in the estimates, but t o illustrate the arithmetic. T he Pascalcode used for numerical computations is available from the authors.


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PERMAN ET A L SEMI-MARKOV MODELS WITH AN APPLICATION TO POWER-PLANT RELIABILITY ANALYSIS 529

D. The holding-times are Weibull distributed withparameters depending only on the current state. The Cdfof the holding-time between states i j is therefore:weif(u/qi; Pi , i E G. (14)The mean of a r.v. with this Cdf is p i =-vi .r(l+ 1/&).E. The Slovenian coal-powered PP SoStanj 294 MWis the demonstrative example. Operating data for 1979 -1996 are available, but the analysis is only for 1979 - 19913.2 Model Evaluationchain are estimated by (15) and are shown in table 1.

1101.The transition probabilities of the embedded Markov

@ i , j = ni , j /n i* (15)Notation

FT Fisher information matrixn,,3

72%

number of transitions from state i tostate j in the period in assumption Enumber of all transitions from state i intoany other state .

Table 1 State-Transition Probabilities of theEmbedded Markov Chain [PP Sostanj 294 MW]

l j1 ,2 = 0.43 lj 1, ~ 0.36 lj1,4= 0.06 51,s = 0.11lji,6 0.01 $3,1 0.91 l j 3 ~ 0.08The parameters of th,e Weibull holding-times were esti-

mated by maximum likelihood assuming the observationsof the holding-times n at given stat e are i.i.d. The standarddeviations were estimated from the asymptotic variancesgiven by the inverse of FI, hich for the Weibull distri-bution is (the prime implies the derivative):

Table 2 gives the point estimates and their estimatedstandard deviations [ll:i.

Table2. Estimated Pairameters and Standard Deviationsof the Weibull Distribution [PP Sostanj 294 MW]

[the estimated standard deviation is in ( )]

1 0.78 (0.031) 235.6 (16.2)2 0.94 (0.057) 59.5 ( 5.1)3 0.88 (0.057) 31.2 3.1)4 0.41 (0.065) 8.7 ( 4.3)5 0.59 (0.068) 3.2 ( 0.82)6 4.68 (1.381) 1174.0 (99.6)

state .,8 4

For the Weibull holding-times, closed formulas for R( t )are not available. As an approximation, the Weibull dis-tribution has been replalced by a discrete distribution that

closely resembles it. The Weibull r.v., W, is replaced by adiscrete r.v., W, taking values in h, 2h, . . . such that:Pr{W = k . h} = Pr{(k 1 ) . < W 5 k . h},

fo rk= 1 ,2 , . . . . (17)This approach is often used for numerical computation.The distribution (17) is suggested in [12] where further

references are given. The distribution of W is also re-lated to the discrete Weibull distribution as defined in [6];((Wlh) h ) has the discrete Weibull distribution withsuitable parameters. It is plausible to anticipate that , forsmall h, the transition probabilities obtained with this ap-proximation closely resemble the transition probabilitiesfor the original Weibull distributions. Indeed, one can eas-ily prove that:lim [Pri{y,= } ] = Pri{x = j } , for all i , j .h-0

With this discrete approximation, R t) is computed viathe recursion formula (13), and the transition probabilitiesare computed from (11). Figures 2 3 show the computedtransition probabilities along with t he analogous quantitiesfor the Markov model with exponential holding-times. See[7] for the details of the exponential case.

ProbabilityP(l ,l)I

Ilo00 20007510 l hl[forP 1,1)1Weibull (solid), Exponential (dashed)Figure 2. Transit ion Probabil i ty

The convergence of Pr,{Yt = j} to the equilibrium val-ues is faster than in the exponential model. At t = 2000hours there is a remarkable closeness between the theoret-ical limits and the transition probabilities in [4]. Table 3gives numerical results. The Weibull model fits the actualda ta slightly better than the exponential model. For exam-ple, the availability of the PP in question over the 12 years


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530 IEEE TRANSACXIONS ON RELIABILITY, VOL. 46, NO. 4, 1997 DECEMBER

was 86.6%. The semi-Markov model predicts the aver-age availability computed from (10) as 89.7% whereas theMarkov model predicts 90.5%. Formulas for average un-availability in the semi-Markov case are in appendix A.3.

Table 3. Transition Probabilities and Theoretical Limits[PP SoEtanj 294 MW]

= E,{z(T, 10 t ] )z(xn jn_>O

(A-1)Prz{Tn+l Tn > t TnIX Tn}}.However, given X, the holding-times are, by assumption,state Prl{Yzooo = j } Prl{Y, = } conditionally s-independent; hence the conditional prob-1 0.81 0.81 ability in the last line in (A-1) becomes:

2 0.08 0.083 0.03 0.044 0.00 0.005

1 Q x , ~ , I ~ ( ~Tn). (A-2)LEG

Use (A-2) in (A-1):5 0.002 0.0026 0.06 0.06 Prz{yt = j } = E,{Z(Tn E [0,t ] ) Z(Xn = j

n > OProbabilities P 1,i) for 1=2 3 6 Q~, , , k ( tTn))}

k G= E,{Z(Tn E [O, tl) Z ( X n = 3 )

n20

(A-3)pL Js a measure on [ O , o o such that for every measur-

able A [0,CO :P ~ , ~ ( A ) E,{z(x,= > l ~ .S } .Pr,{T, E d s )for P 1, ) , P 1,3) ,P(1,6)1Weibull (solid), Exponential (dashed)Figure 3. Transition Prob ability nT0= E;{Z(X, = j Z(Tn A)}CKNOWLEDGMENT n2O

We are pleased to thank the referees whose remarks con- = E I { E Z ( X n = j Z(Tn E A ) } .siderably improved an earlier version of the paper. In par- n > O

Q.E.D.icular we thank Dr. A. Csenki for his remarks and forpointing out to us a wealth of references in the field. The R'2 ,J ( t ) ~ i , ~ ( [ O , t ] )s the Cdf of pz,j.A.2 Proof of (10)-A t =


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PERMAN ET A L SEMI-MARKOV MODELS WITH AN APPLICATION TO POWER-PLANT RELIABILITY AN ALYSIS 531

All the integration changes are justified because the func-tions integrated are positive. Q.E.D. On the other hand, the occupation measures in (A-8)(A-9) are proportional to the equilibrium measure T for

all starting states k :A.3 Limiting Average Unavailability

The asymptotic availability of a system modeled by asemi-Markov process is important. It is sufficient to findthe limit:

lim [: Z X 8 i ) d s , for all i E G. (A-5)t+03The average availabilities computed in appendix A.3.2

equal the limiting transition probabilities, if these exist.The asymptotic availabilities in table 3are computed from(A-9).A.3.1 Theorem 2

Let:{ X,,,) :n 2 0) be a semi-Markov process;the embedded Markov chain {X,} e irreducible,the semi-Markov kernel Q i , 3 ( ~ )epend only on the

recurrent, and aperiod,ic with equilibrium measure T ;initial state i;Then, with probability 1,

v, = So s . F , ( d s ) .

s-independently of X O or every a.A.3.2 Proof of theorem 2N o t a t i o n

7 0r,D ,Sn Cr1L-1 0 for all n,which is true for models with continuous holding-timeswith finite s-expectations.E{D1} < 03 E{Sn} < c a ,

Individual holding-times depend on the state, but giventhat the states are conditionally s-independent, then~ ( ~ 1 1E{ ~ mTm-1))

m


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532 IEEE TRANSACTIONS ON RELIABILITY,VOL. 46. NO. 4, 1997 DECEMBER

[12] A. Csenki, Cumulative operational time analysis of finitesemi-Markov reliability models, Relaabilaty EnganeenngtY S y s te m S a fe t y , vol 44, num 1, 1994, pp 17-25.

[13] M A T L A B 4 . 2 ~ sers Guide, 1994 Dec; Mathworks[14] S. Asmussen, Applied Probability and Queues, 1987; John

Wiley Sons.AUTHORS

Dr. Mihael Perman; Faculty of Mechanical Engg; Univ. ofLjubljana; ABkerEeva 6, 1000 Ljubljana, SLOVENIA.Internet (e-mail): mihael [email protected]

Mihael Perman (born 1961) received his PhD (1990) fromthe Statistics Department a t the University of California a tBerkeley. Since then he has held positions at the University ofLjubljana, University of Southern California, and CambridgeUniversity. He now teaches at the Faculty of Mechanical En-gineering, University of Ljubljana. His research interests in-clude stochastic processes, Brownian motion, and mathemati-cal statistics.Dr. Andrej Senegainik; Faculty of Mechanical Engg; Univ. ofLjubljana; ASkerEeva 6, 1000 Ljubljana, SLOVENIA.Internet (e-mail): [email protected]

Andrej SenegaEnik (born 1961, in Ljubljana) received hisBSc (1984), MSc (1991), and his doctoral degree (1995) at theFaculty of Mechanical Engineering Science at the UniversityofLjubljana. He teaches at the Faculty of Mechanical Engineeringin Ljubljana. His research interests are in thermodynamics,power systems, and reliability of power plants.Dr. Matija Tuma; Facultyof Mechanical Engg; Univ. of Ljubl-jana; ASkerEeva 6, 1000 Ljubljana, SLOVENIA.Matija Tuma (born 1938, in Ljubljana) received his BSc(1962) from the Faculty of Mechanical Engineering, Universityof Ljubljana, and his Dr.Tech.Sci (1978) from the Federal Insti-tute of Technology (ETH) Zurich, Switzerland. He worked forseveral companies in Slovenia Switzerland for 14 years beforejoining the Faculty of Mechanical Engineering of the Univer-sity of Ljubljana as a Full Professor of Thermal Power PlantEngineering Thermodynamics in 1982. His research inter-ests include gas-steam cycles, cogeneration of heat power,(in)organic Rankine cycles, and reliability of power plants. Hehas (co)authored over 50 technical papers.Manuscript TR96-077 received 1996 May 28;revised 1997 February 12, 1997 July 22Responsible editor: O.G. QkogbaaPublisher Item Identifier S 0018-9529(97)08545-X

semi-markov models with an application to power-plant reliability analysis.pdf

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