Dependability analysis of systems modeled by non-homogeneous Markov chains

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Reliability Engineering and System Safety 61 (1998) 235–249 q 1998. Published by Elsevier Science Limited All rights reserved. Printed in Northern Ireland 0951–8320/98/$19.00

Dependability analysis of systems modeled by non-homogeneous Markov chains Agapios Platis a,b, Nikolaos Limnios b & Marc Le Du a a Dept PEL, Electricite´ de France, 1, av. du Ge´ne´ral de Gaulle, B.P. 408-92141 Clamart, Cedex France Div. Mathe´matiques Applique´es, Universite´ de Technologie de Compie`gne, B.P. 529-60205 Compie`gne, Cedex France

b

(Received 2 January 1997; accepted 24 May 1997)

The case of time non-homogeneous Markov systems in discrete time is studied in this article. In order to have measures adapted to this kind of systems, some reliability and performability measures are formulated, such as reliability, availability, maintainability and different time variables including new indicators more dedicated to electrical systems like instantaneous expected load curtailed and the expected energy not supplied on a time interval. The previous indicators are also formulated in the case of cyclic chains where asymptotic results can be obtained. The interest of taking into account hazard rate time variation, is to get more accurate and more instructive indicators but also be able to access new performability indicators that cannot be obtained by classical methods. To illustrate this, an example from an Electricite´ De France electrical substation is solved. q 1998 Elsevier Science Limited.

NOMENCLATURE NHMC/HMC ENS IN/IR X ¼ {X n, n [ IN}}

Non-homogeneous/homogeneous Markov chain Energy not supplied Set of integer/real numbers a Markov chain with E ¼ {0,..., s} its system’s state space. State probability vector

1{Xn } ¼ 1

P(n) ¼ {P i(n) ¼ P(X n ¼ i),i ¼ 0,..., s} S, j, T, m Sojourn time, Exit distribution, Hitting time, expected hitting time pn Transition probability matrix at time n Transition probability from state i to state pnB (i,j) j, at time n Sub-matrix of p n obtained by eliminating pn g the rows & columns of p n not belonging to B U/D Set of Up/Down states T 1 s,v ¼ (1,...,1, 0,...,0) s-dimensional column vector whose v first elements are ones and the others are zeros v-dimensional column vector whose ele1 v ¼ (1,...,1) T ments are ones a Initial distribution of the chain X Sub-vector of a corresponding to the Up/ a 1/a 2 Down states subset U/D L(t) Demanded load at time t z Interrupted customer ratio er first state in the class E r visited by the chain g E 3 IR → IR states reward, i.e. g(i,t) ; g i(t) is the ‘efficiency’ of the state i, at

time t. In fact, it is the load supplied by the substation at time t and while being in state i if the chain at time n is in state i, 0 else

1 INTRODUCTION Recent developments in the modeling and analysis of highly dependable systems require the use of some more sophisticated models-rather than the blind homogeneous Markovian models. For instance Electricite´ De France (EDF) used this for the homogeneous Markov approach to model an electrical substation in order to evaluate some measures of performance of great interest. However, the assumptions made using this model, like the fact that hazard rates are time constant, are quite obsolete and they do not really represent the underlying physical process. Take for example the system restoration delay, following an incident, this delay will be much longer if the event takes place during the night (alarm at the operator’s house) rather than if it happens during working hours (operator in front of his control panel). But this is also the case for reward rates, the load, for instance, varies during the day (peak hours in the morning and in the evening), but it also fluctuates with the season (load is maximum in winter). And it is precisely because the 235

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load is weaker in the summer that scheduled maintenance is more frequent during this period. Unfortunately, it is also the time during which, the failures affecting electrical lines, caused in the majority by lightning, are the most frequent. From this example, the necessity of a non-homogeneous modeling becomes apparent. Therefore, the model we propose takes into consideration the time dependence of the hazard rates but also the time dependence of the reward rates. From this modeling, we can evaluate the classical dependability measures such as the reliability and availability but also the corresponding performability indicators, including the expected load curtailed and the expected energy not supplied on a time interval, which are currently used by EDF (in the homogeneous case). In our previous articles, we have studied non-homogeneous Markov chains for modeling reliability indicators10 and performability indicators10,11 for electrical power systems, and more specifically electrical substations. We also studied the asymptotic availability concerning cyclic non-homogeneous Markov chains12. In this article, we give a unified framework of non-homogeneous Markov chain analysis where we formulate reliability and performability indicators for NHMC but also for cyclic NHMC where asymptotic results are also available. The major difficulty in the non-homogeneous Markov chains (NHMC) is to compute the transition probabilities and therefore the state probabilities. A way to avoid computing a non-homogeneous Markov chain is to transform this chain into an homogeneous one. If X ¼ {X n; n [ IN} is a NHMC defined on a finite state space E and N a INvalued random variable independent with X. Then the chain {(XNn , N n);n [ IN} defined on a non-finite state space E9 ¼ E 3 IN is homogeneous. By eliminating the non-homogeneous problem, another one was introduced, notably, the chain is no longer defined on a finite state space making the computation hardly feasible. A semi-Markov chain is a natural generalization of a Markovian chain. The future evolution of the system depends only on the time spent since the last transition. This time, called local time (or backward recurrence time), is measured by a clock which is reset at every state transition. It is clear that if the clock becomes an integrated part of the system (which comes to consider E 3 IR þ as state space with E the state space of the semi-Markovian system), then the system becomes Markovian. This model seems suitable for systems whose components are replaced or totally renovated when a failure occurs. Here, the hazard rates depend more on the components of the system, that is the material and less on the external environment. In contrast, a non-homogeneous Markov chain is a Markov chain whose transition function depends on the total time. This model seems to fit systems whose component hazard rates depend essentially on the external environment, like load variation. Considering the electrical substation as the system to be studied, one can see that the fluctuation of the hazard rates are only caused by external environment reasons (that is

lightning frequency and operators’ working hours). This explains our motivation to study non-homogeneous chains. In addition to this, some systems may require to use models ‘mixing’ semi-Markov chains and non-homogeneous Markov chains called Semi-Markov non-homogeneous chains. Recently, more and more complex models appear, but also more elaborated measures that interpret reality much closer. Even with a classical Markov model, we can access now to a very detailed level of information: the joint distribution of the sojourn-times or the number of visits until absorption2, the distribution of the total time spent in a subset of states14. Another complicated model, the SemiMarkov model6, whose main advantage is to permit nonexponential distributions for transitions between states, is also used in the modeling of fault-tolerant systems13. All these models and measures are based on more complex algorithms—compared with the classical matrix inversion for the state probability of a Markov system—however, it is possible nowadays and affordable to compute them. This article is organized as follows. In Section 2, basic developments of non-homogeneous Markov chains are defined, then reliability and performability indicators are defined by means of these chains. In Section 3, some additional properties are studied: cyclicity and ergodicity and again reliability and performability indicators that derive from this modeling are formulated. Finally, in Section 4, an example from an EDF electrical substation is solved in order to illustrate our modeling.

2 NON-HOMOGENEOUS MARKOV CHAINS In this section, the transient analysis of NHMC is presented in order to establish the basic concepts and to facilitate the introduction of the reliability and performability indicators. Some other classical measures are also expressed by these chains such as the sojourn time and the hitting time. Let E ¼ {1,2,..., s} , IN be a set of points that represent the system state space and X ¼ {X n; n [ IN} an E-valued stochastic process on a probability space, whose probability measure is P. X is a Markov chain if, for all j [ E and all n . 0, we have; P(Xn þ 1 ¼ jlX0 , X1 , …, Xn ) ¼ P(Xn þ 1 ¼ jlXn )

(1)

This is the well known ‘memoryless property’ of the Markov systems, that is the future evolution of the system does not depend on the history of that system, but only on the present. We define for each i,j [ E, p n(i,j) ¼ P(X nþ1 ¼ jlX n ¼ i) the transition probability from state i to state j in the time-interval [n,n þ 1]. The probability (p n(i,j); n [ IN, i,j [ E) is called transition function of the chain X. It is a one step transition probability. Multiple step transition functions can also be defined by p n,m(i,j) ¼ P(X m ¼ jlX n ¼ i). We have p n(i,j) ¼ p n,nþ1(i,j) and p n,n ¼ I the identity matrix. The transition function has the following essential property, called the Chapman-Kolmogorov equation, for

Analysis of Markov systems all i,j [ E; pm, n (i, j) ¼

X

pm, r (i, k)pr, n (k, j) for all rm , r , n

(2)

k[K

237

n the  states  subset A is defined by mTi (n) ¼ Ei ½T ÿ ¼ E TlXn ¼ i and m(n) ¼ (m 1(n),..., m dim(n)) , with (dim ¼ lBl). For n ¼ 0, m is the entrance time, for the first time in A. ! ` X B p0, k 1dim (7) m(0) ¼ I þ k¼1

2.1 Remark If p n(i,j) does not depend on n, the Markov chain is time homogeneous. Let a be the initial probability distribution on E, in other words a(i) ¼ P(X 0 ¼ i), and p n the transition matrix of the Markov chain. X is completely defined by its initial distribution a and its transition function (p n;n [ IN). Indeed, we have; ! nY ¹1 pk (j) (3) Pj (n) ¼ P(Xn ¼ j) ¼ (ap0, n )(j) ¼ a

The cumulative sojourn time is the total expected time that the system spends in state j, in the interval of time [1,n]. Let j be a fixed state and X 0(q) ¼ i, the initial state. The random variable, Njn , is defined as the total time spent in state j, in the interval of time [1,n], that is; Njn ¼

Pni (S ¼ K þ 1) ¼ pn (i, i)…pn þ k ¹ 1 (i, i){1 ¹ pn þ k (i, i)} (4) On the details of the obtention of this equation, see Appendix A. The expected sojourn-time at time m, in state i (expectation corresponding to Pni (:)), is then; X Ein (S) ¼ 1 ¹ pn (i, i) þ 1#k#` (k þ 1)pn (i, i)…pn þ k ¹ 1 (i, i) {1 ¹ pn þ k (i, i)} Ein (S) ¼ lim

 l→`

ð5Þ

1 þ pn (i, i) þ pn (i, i)pn þ 1 (i, i) þ …

þ pn (i, i)pn þ 1 (i, i)…pn þ 1 (i, i) ¹ lpn (i, i)pn þ 1 (i, i)…pn þ l þ 1 (i, i)ÿ The exit distribution, at time n, is defined as the probability to enter in state j, when the system leaves the state i. That is, for i Þ j, the following probability; jn (i, j) ¼ P(Xn þ 1 ¼ jlXn ¼ i, Xn þ 1 Þ i) ¼

pn (i, j) 1 ¹ pn (i, i) (6)

2.2 Remark Relations (4) and (6) give us the necessary elements for simulating (Monte-Carlo simulation) NHMC, given the initial law a and the transition probability p n. Let T be the hitting time, for the first time after n, in a states subset A , E, A Þ E, A Þ B , when the chain is in the state i [ B (B ¼ E\A), at time n, i.e.: T(q) ¼ `m . n; Xm (q ) [ A ¹ n. The mean hitting time in

  1{Xk ¼ j} :Hence Ei Njn

k¼1

¼

n X

Pj (k) ¼

k¼1

k¼0

P j(n) is called state probability at time n. The vector P(n) ¼ (P 1(n),...P s(n)) is called state probability vector at time n. An analog formulation is P(n) ¼ ap0, n . The sojourn-time S, at time n is defined as the time spent in a state i [ E, when the system enters in that state at time n (between n ¹ 1 and n), that is Pni (S ¼ 1) ¼ P(Xn þ 1 Þ ilXn ¼ i, Xn ¹ 1 Þ i) ¼ 1 ¹ pn (i, i) and for all k $ 1

n X

n X

! p0, k (i, j)

k¼1

The probability of entering state i at time n (n $ 1) is defined as the probability of being in a state 1 (1 Þ i) at time n ¹ 1 and transiting to state i between n ¹ 1 and n. This probability is given by; X i}Pl (n ¹ 1)pn ¹ 1 (1, i) (8) 1[Ef{

The interest in that probability is that it will lead us to compute the interruption frequency, i in that case will be a down state. 2.3 Reliability analysis If the state space E is decomposed into two subsets U and D, U containing the Up states of the system and D containing the set of Down states with v ¼ lUl, and the following properties: U ∪ D ¼ E, U Þ B and U Þ E. The transition matrix is given now by 3 2 U UD 7 6 6 p pn 7 7 U 6 n 7 , pn pn ¼ 6 7 6 6 DU D 7 5 4 pn

pn

is the transition probabilities matrix inside the subset of operational states U and pD n is the transition probabilities DU matrix inside the non-operational subset D, pUD n , pn represent the transition matrices from the subsets U to D and from D to U. The reliability is defined as the survivor function of the hitting time in the subset D, whereas the maintainability is defined as the cumulative distribution function of the hitting time in the subset U. There are enough elements, now, to formulate the basic reliability indicators. The reliability at time n, is given by; R(n) ¼ P(;i # n, Xi [ U) ¼ a1 pU 0, n 1n Under the condition that the series of

(9) pU 0, k

converges, the

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A. Platis et al. To evaluate this random variable, the first two moments are calculated. Relation (14) can also be expressed;

MTTF can easily be given by; ! ` X U p0, k 1n MTTF ¼ a1 I þ k¼1

WN ¼

The maintainability at time n is given by the following relation; M(n) ¼ 1 ¹ P(;i # n, Xi [ D) ¼ 1 ¹ a2 pD 0, n 1s ¹ n (n $ 1) (10) Under an analog condition (series of pD 0, k converges), the MTTR can also be given by; ! ` X D p0, k 1s ¹ n MTTF ¼ a2 I þ k¼1

¹1 X NX

gj (n)1{Xn ¼ j}

This quantity when divided by the cumulated demand is also called throughput availability3. The result, for the expectation, is; E[WN ] ¼

NX ¹1

g(n)T PT (n)

(15)

n¼0

And for the variance; X XN ¹ 1 Vari [WN ] ¼ j[E n ¼ 0 gj (n)2 p0, n (i, j)(1 ¹ p0, n (i, j)) (16)

The instantaneous availability at time n, is given by; A(n) ¼ P(Xn [ U) ¼ ap0, n 1s, n

(14)

j[E n ¼ 0

(11)

þ2

X

X

gj (n)gk (m)p0, n (i, j)

j, k[E 0#n,m#N ¹ 1

3 (pn, m (j, k) ¹ p0, m (i, k))

2.4 Performability analysis The performability is defined as a measure combining reliability and performance measures. This notion was introduced by Meyer7 and was developed and mainly applied to problems of fault-tolerant systems. For Meyer7 and Pattipati9, the performability of a system over a mission time interval [0,T] is defined as the probability density function of the random variable W T where; ZT g(Xs )ds (12) WT ¼ 0

and g(i), i [ E is the performance rate (reward rate) of the system in state i and g(X t) is the performance rate (reward rate) of the system in configuration state X t. This performability notion is also found in Smith, Trivedi, Ramesh15 and Limnios5 under another definition where performability or cumulated reward are synonyms of cumulative distribution function of the random variable W T. One may also be interested by the complementary distribution of W T which in other words means what is the probability of meeting one desired performability level within a specified time interval. The previous definitions concern the continuous time. In discrete time, the performability or cumulated reward until fixed time N [ IN* by analogy will be defined, as the probability density function (or cumulative distribution function/complementary Cdf) of the following random variable; WN ¼

NX ¹1

g(Xn , n)

(13)

n¼0

where g is the reward time-dependent bounded function defined in the notations: g(i,n) ; g i(n). For each state i from E at time n, a corresponding reward rate g i(n) is associated: it is the reward rate of the system.

¼2

X

NX ¹1

gj (n)gk (n)p0, n (i, j)p0, n (i, k)

0#j,k#s n ¼ 0

2.5 Remark The bounds can be estimated by the Bienayme´ – Tchebycheff inequality. The expectation and the variance being calculated, and with the previous inequality, an estimation of the bounds can be obtained. The performability indicators described in the following sections are modeled with NHMC, they are useful for electrical power systems, particularly for Electricite´ De France. Most of these indicators already exist in the homogeneous case, they were derived from traditional power system indicators1. The following indicators were initially defined in Ref. 11. The introduction of classes will allow to better understand the system but also to simplify the problem. These classes contain states that are common to one kind of interruption. Fig. 1 represents a partition of the states space E into a set E r(r ¼ 0,...,k). Class E 0 represents the nominal operating states while the other classes E 1,...,E k represent interruption (non-operational) states. We denote by e r the first state in the class E r visited by the chain. The performability indicators are defined by means of a reward function. This reward function is a function of the instantaneous load and the interrupted customer ratio z which is defined by; g i(t) ¼ z iL(t) ¼ (1 ¹ d i)L(t). The load curtailed at time t is measured by L(t) ¹ g i(t) which is the difference between the demanded load and the supplied load (when the system is in state i). The energy not supplied at time t is given by (L(t) ¹ g i(t))Dt where Dt is the interruption duration.

Analysis of Markov systems

239

These two indicators are important to evaluate non-quality events. The IELC indicators is more representative of the number of interruptions, while the EENSTI is more representative of the interruption duration. 2.8 Three dimensional indicators We can also access to three-dimensional indicators that can bring a concise information; this information is given by means of the interruption frequency. In fact, we can give the interruption frequency depending on the interruption duration (k) and the day (n) of the year where occurs this interruption. Therefore, the summation of all the interruption frequencies that are between k and k þ 1, will give;

Fig. 1. Partition of states space E.

2.6 Instantaneous expected load curtailed (IELC) The instantaneous expected load curtailed is defined as the mean value of load curtailed at time t. In other words, this indicator represents the number of times a customer will be interrupted weighted by the quantity of load interrupted each time11. In discrete time, it is the probability of the following event: At time n ¹ 1, the system is in an operational state E 0 and at time n (next unit of time), the system is in a nonoperational state E\E 0. At time n the load delivered is g(X n,n) and the load curtailed is [L(n) ¹ g(Xn , n)]. IELC(n) ¼

k X X X X

a(x)p0, n ¹ 1 (x, i)pn ¹ 1

x[E0 i[E0 l ¼ 1 j[El

3 (i, j)dj L(n)

t

X Zt þ v i[E

t

di L(u)Pi (Xu ¼ i)du

ð18Þ

In discrete time, if v is an integer and h the digitization step, then; " # v X X di Pi (k þ n) L(k þ n)h (19) EENSTI(n, v) ¼ k ¼ 0 i[E

if D ¼ (di )i[E then EENSTI(n, v) ¼

v X   P(k þ n)DT L(k þ n)h k¼0

ð21Þ Another interesting indicator is the interruption frequency depending on the ENS (k) and the day (n) where occurs this interruption. Therefore, at time n, the summation of all the interruption frequencies whose ENS is between k and k þ 1, will give; Frdi_ENS (k, n) " # r X X z l 1{Xn ¹ 1 [E0 , Xn ¼ el )} 1{k#ENS(n, j),k þ 1} ¼E

ð22Þ

ð17Þ

The expected energy not supplied on a time interval (EENSTI) is defined as the mean value of the energy not supplied in a fixed time interval [t,t þ v] caused by one or more interruptions in that interval11. X Zt þ v [L(u) ¹ gi (u)]1{Xu ¼ i} du EENSTI(t, v) ¼ E ¼

l ¼ 1 j[El

l ¼ 1 j[El

2.7 Expected energy not supplied on a time interval (EENSTI)

i[E

Frdi_interrupt (k, n) " # r X X z l 1{Xn ¹ 1 [E0 , Xn ¼ el )} 1{k# duration(n, j) , k þ 1} ¼E

ð20Þ

3 CYCLIC NHMC In Section 2, systems with varying hazard rates were studied and modeled by NHMC. However such systems may have another interesting property which is that they have cyclic hazard rates. In other words, rates may vary from hour to hour but remain identical from year to year. This special property will lead us to study Cyclic NHMC and the benefits of these chains is to simplify computational complexity but also to ‘forget’ the initial state of the system. The transition function of (p n(i,j); n [ IN, i,j [ E) is called cyclic (or periodical) of period d (d . 1), if d is the smallest integer verifying P mdþr(i,j) ¼ p r(i,j); m, r [ IN, i,j [ E. 3.1 Remark It is preferred to use the terminology cyclic NHMC instead of periodic NHMC in order to not confuse with the periodicity defined by classical HMC. Let Y n be an embedded homogeneous chain from X n such as for all n $ 0, we have Y n ¼ X nd. If p n is the transition matrix of the NHMC X n, then p 0,d is the constant transition matrix of the HMC Y n.

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A classical result is that if a HMC is irreducible and aperiodic then the chain is ergodic. In other words, the transition matrix from time 0 to time n, tends to an ergodic matrix (a matrix whose rows are identical) as n tends to infinity. This result will be used for the embedded HMC Y n in order to obtain the asymptotic results concerning the NHMC X n. The basic probabilistic indicators as well as the reliability and performability indicators are therefore easily formulated. From eqn (3), the expression of the state probability vector can easily be obtained. Since the transition matrix is cyclic with a period of length d(d . 1) and n ¼ md þ r, then; !m ! dY ¹1 rY ¹1 pk pk (j) Pj (md þ r) ¼ P(Xmd þ r ¼ j) ¼ a   ¼ a(p0, d )m p0, r (j)

k¼0

The hitting time can easily be obtained: if (I ¹ pB0, d ) is invertible and for n ¼ 0, m is the entrance time, for the first time in A and with B ¼ E\A. ! d X ÿ
¹ 1 pB0, k I ¹ pB0, d (30) m(0) ¼ I þ 1dim k¼1

We might be interested in computing the total expected time that the system spends in a state j, in the interval of time [1,n] where n ¼ md þ r, " # md þr X  md þ r  ¼ p0, k (i, j) Ei Nj k¼1

" ¼(

mX ¹1

#" (p0, d )

k¼0

"

k¼0

ð23Þ

þ (p0, d )

m

k

d X

# p0, k

k¼1 r X

#

p0, k (i, j)

ð31Þ

k¼1

If the HMC Y n is irreducible and aperiodic then its transition matrix from time 0 to time n tends to an ergodic matrix. The transition matrix from time 0 to time n, of this chain is: p0, n 9 ¼ p0, nd ¼ (p0, d )n Therefore; lim

[p0, d ] ¼ Ep ¼ 1s p n

n→`

(24)

Therefore asymptotically, the steady-state probabilities inside a cycle are; pr (j) ¼ lim m→` Pj (md þ r)  ÿ
 ¼ a lim m→` [p0, d ]m p0, r (j) ¼ [a1s pp0, r ](j) ¼ [pp0, r ](j)

ð25Þ

The sojourn-time S, at time n ¼ md þ r is defined as the time spent in a state i [ E, when the system enters in that state at time n, that is Pni (S ¼ 1) ¼ 1-p r(i,l) and for all k $ 1. Pni (S ¼ k) ¼ pr (i, i)…pr þ k ¹ 1 (i, i){1 ¹ pr þ k (i, i)}

(26)

The expected sojourn-time at time m, in state i (expectation corresponding to Pni (:)), is then; X (k þ 1)pr (i, i)…pr þ k ¹ 1 (i, i) Eni (S) ¼ 1 ¹ pr (i, i) þ 1#k#`

3 {1 ¹ pr þ k (i, i)}

ð27Þ

1 [1 þ pr (i, i) þ pr (i, i)pr þ 1 (i, i) þ … Eni (S) ¼ 1 ¹ prodi þ pr (i, i)pr þ 1 (i, i)…pr ¹ 2 (i, i) þ prodi ]

ð28Þ

where prod i ¼ p r(i,i)p rþ1(i,i)...p d¹1(i,i)p 1(i,i)...p r¹1(i,i). The exit distribution, at time n-md þ r, is defined as the probability to enter in the state j, when the system leaves the state i. That is, for i Þ j, the following probability; jn (i, j) ¼

pr (i, j) 1 ¹ pr (i, i)

(29)

but it might be also interesting to compute the cumulative sojourn time asymptotically that is in a interval of time [qd þ r, qd þ u] where q tends to infinity, u ¼ md þ s, s , d and r , d. In this case the cumulative sojourn time does not depend on the initial state i anymore, and we have; dX ¹1 sX ¹1   pk þ pk E Njmd þ s ¼ m k¼0

(31)

k¼0

The probability of entering state i asymptotically at time r (1 # r # d) is defined as the probability of being in a state 1 (1 Þ i) at time r ¹ 1 and transiting to state i between r ¹ 1 and r. This probability is given by S l [ E\{i} p l(r ¹ l)p r¹1(l,i). The Reliability at time md þ r, is given by;  m U R(md þ r) ¼ P(;i # md þ r, Xi [ U) ¼ a1 pU 0, d p0, r 1n (32) Under the above conditions (see 30), the MTTF can easily be given by; ! d X ÿ
¹1 U U p0, k I ¹ p0, d MTTF ¼ a1 I þ 1n k¼1

The Maintainability at time md þ r is given by the following relation; m(md þ r) ¼ 1 ¹ P(;i # md þ r, Xi [ D)  m D ¼ 1 ¹ a 2 pD 0, d p0, r 1s ¹ n (r $ 1)

ð33Þ

Under the same conditions, the MTTR can also be easily given by; ! d X ÿ
¹1 D D p0, k I ¹ p0, d (34) 1s ¹ n MTTF ¼ a2 I þ k¼1

The instantaneous Availability at time md þ r, is given by; A(md þ r) ¼ P(Xmd þ r [ U) ¼ a[p0, d ]m p0, r 1s, n

(35)

Analysis of Markov systems

241

Fig. 2. Electrical substation diagram.

before, that is, as the mean value of load curtailed at time n. For n ¼ md þ r, we have the following asymptotic result;

Hence, lim A(md þ r) ¼ Ar (`) ¼ lim a[p0, d ] p0, r 1s, n m

m→`

m→`

¼ aEp p0, r 1s, n

ð36Þ

Since the matrix E p is ergodic, E pP 0,r is also ergodic, that means that all the rows of this matrix are identical. Let L r be one row of this matrix, we have aE pP 0,r ¼ [S ia(i)]L r ¼ L r. Hence, for the asymptotic availability we have d values corresponding to each r inside the cycle and they are given by; n X Lr (i) (37) Ar (`) ¼

IELC(r) ¼

l¼0

pir pr (i, j)dj L(r)

The Expected energy not supplied on a time interval is defined as the mean value of the energy not supplied in a fixed time interval [r1 , r1 þ md þ r2 ] because of one or more interruptions in that interval. We obtain the following result. EENSTI(r1 , md þ r2 ) ¼

dX ¹1

1pk DT L(k)h þ (m ¹ 1)

k ¼ r1

þ

r2X ¹1

dX ¹1

1pk DT L(k)h

k¼0

1pk DT L(k)h

ð41Þ

k¼0

The formulation, by means of Cyclic NHMC, of the former indicators gives; Frdi_interrupt (k, n) ¼

r X X X

z l pn ¹ 1 (i)pn ¹ 1 (i, el )1{k#duration(n, j),k þ 1}

i[E0 l ¼ 1 j[Elk

the expectation of the Performability over an interval [0, md þ r], is; E[Wmd þ r ] ¼

dX ¹1

g(l)T Mm ¹ 1 PT (l) þ

l¼0

rX ¹1 l¼0

ð42Þ

and, Frdi_ENS (k, n)

g(l)T Mm PT (l) ¼ (38)

r X X X

z l pn ¹ 1 (i)pn ¹ 1 (i, el )1{k#L(n):duration(n, j),k þ 1}

i[E0 l ¼ 1 j[El

ð43Þ

And asymptotically over a period [md þ r, (m þ 1)d þ r], the result becomes; E[W0, d ] ¼

(40)

i[E0 l ¼ 1 j[E1

i¼1

for all r such that 0 # r , d, and A r(`) denoting the asymptotic availability at time r. One can remark that the asymptotic availability does not depend on the initial distribution a. For the performability analysis, in addition to the cyclic property of the chain, it is considered that the reward rate is also cyclic here, some additional results are obtained then. If m h X ÿ
l i T p0, d ; Mm ¼

k X X X

dX ¹1

g(l þ r)T pTlþ r

4 NUMERICAL APPLICATION (39)

l¼0

The instantaneous expected load curtailed is defined like

In this section, we consider the case of an electrical substation of EDF, whose role is to transform high voltage into

242

A. Platis et al. Table 1. Operational configurations of the primary substation

Configuration

CB DJL1

CB DJL2

CB DJT1

CB DJT2

Conf. 1

Close

Close

Close

Close

Conf. 2

Close

Open

Close

Close

Sc. Maint. L

Close

Open

Close

Close

Sc. Maint. T

Close

Close

Close

Open

Fig. 3. State transition diagram (37 states).

Substation diagram

Analysis of Markov systems

243

Fig. 6. Load (three-day evolution). Fig. 4. Overhead line failure rate.

medium voltage, but also protects the electrical network and dispatches the energy on the network. This is the case of a real electrical substation composed of a certain number of components that we will not describe here. Fig. 2 represents a primary substation that is supplied in high voltage by two lines LINE 1 and LINE 2, and delivers medium voltage to the customers. This substation contains 6 measure transformers, 2 earthing switches, 5 circuit breakers, 4 isolating switches, 2 voltage transformers, 4 surge arresters, 2 transformers, 1 busbar, 2 busbar switches but also distance protections, auxiliary distance protections and directional protections. The general functioning of an electrical substation is the following: in case a fault takes place on LINE 1 that causes the opening of the circuit-breaker DJL1, a restoration system closes the circuit breaker DJL2. This restoration goes along, however, with a brief supply interruption of the customers. Four operational configurations described by Table 1, can be obtained, by arranging the positions of the circuitbreakers (position open/close). In addition to these operational states, there are 33 failure states corresponding to 120 failure scenarios (numbers inside the boxes in Fig. 3). These scenarios were grouped into 33 generic scenarios (scenarios that lead to the same failure states). Each operational state can lead to its own failure states which is illustrated by the state transition diagram of Fig. 3.

There are 4 types of repairs (transition from the failure state to the operational state). The faster is the automatic restoration which lasts less than a minute and is performed automatically by an automatism. The remote restoration lasts 5 or 20 min. and is accomplished by an operator whose delay depends on the hour of the day, that is whether the operator is in front of his control panel or not. The third restoration type is when the operator has not sufficient elements to decide, in that case a local intervention is needed, that generally lasts more in the winter (difficult access conditions) and during the night. Finally, the last type is the complete repair of the line which lasts approximately 3 h. On Fig. 3, we can distinguish the four operational states corresponding to different system configurations (B, D, LC, TC). Each operational state is connected to a certain amount of non-operational states (corresponding to interruption states). Each arc connecting the states is considered double way (operational → non-operational: failure rate and non-operational → operational: repair rate). The hazard rates were omitted deliberately from the graph, for simplification reasons (see Appendix A). States (B, D, LC, TC) can be grouped in the class of operational states: class E 0 while the rest of the states are considered belonging to one type of class except D6 which is grouped with D96. The time evolution of some rates is given later, along with the load variation with time. Fig. 4 represents the overhead line failure rate. It has two peaks: one corresponding to the higher amount of lightning

Fig. 5. Electrical isolation for maintenance rate.

Fig. 7. Instantaneous availability.

244

A. Platis et al.

Fig. 8. Reliability.

Fig. 10. Instantaneous expected load curtailed.

in the summer while the second corresponds to the hard climatic conditions of the winter (snow, ice, wind,...). Fig. 5 represents the electrical isolation for maintenance rate. This rate is higher in the summer because of the higher amount of isolation in that season which corresponds to a period of low load. The restoration delay when the restoration system has failed, evolves during the day. The restoration is accomplished in 5 min during working hours, 20 min otherwise. For simplification, we will assume that the restoration rates are given by inverting the restoration delays. Fig. 6 represents the load demanded by the customers supplied by the substation, which varies with time. This substation is supposed to supply domestic customers and not industrial ones. The evolution of load is seasonal, essentially caused by temperature and luminosity variations. The load evolves also according to the hour of the day. The consumption peaks of the morning and the evening, also the dead hours of the night are observable. Since all the hazard rates of this system are cyclic (most rates have a 24 h cycle but all have a 1 year cycle), it can be modeled by cyclic NHMC with a cycle d ¼ 1 year. The transition rates matrix obtained by the graph (Fig. 3) and the failure and repair rates corresponding to that graph (see Appendix A), is transformed into a transition probability matrix by p n ¼ I þ A(n)h where A(n) is the transition rates matrix at time n, h is the digitization step, I is the

identity matrix and p n the transition probability matrix which is the basic component of the dependability indicators. The graphical representations of some indices are given here. Fig. 7 shows the instantaneous availability computed with NHMC (only averaged daily values are shown) and the instantaneous availability computed with HMC and average values for the hazard rates. The interest with the NHMC is that a clearer vision of the availability of the system is given. With an average value of 17 3 10 ¹7 for the unavailability (1-Availability), the unavailability varies from 25 3 10 ¹7 in the summer to 5 3 10 ¹7 in the spring and autumn (a factor 5 variation). In Fig. 8, reliability computed with NHMC is slightly different from the reliability computed with HMC and average values for the hazard rates. In the spring, NHMC reliability is higher than HMC reliability while HMC reliability is higher than NHMC reliability in the autumn. Fig. 9 represents the interruption frequency (sum of the frequencies of entering interruption states) obtained by NHMC and the interruption frequency obtained by HMC. In the homogeneous cases the stationary frequency of state i is given by freq i ¼ (¹a ii)p i with p i the stationary probability of state i and a ii the diagonal element of the transition rates matrix. The advantage of using NHMC is shown here again, by obtaining more information. The critical period, in

Fig. 9. Interruption frequency.

Fig. 11. Energy not supplied on a 10 day period.

Analysis of Markov systems

Fig. 12. Distribution of the interruption duration.

Fig. 13. Distribution of energy not supplied.

245

246

A. Platis et al.

that case, is the summer, improvements can therefore be orientated for that season. In Fig. 10, the instantaneous expected load curtailed is represented (only averaged daily values are shown). The peak appearing in the summer period is mainly caused by the high amount of lightning and therefore a high interruption frequency. The large peak in the winter is because a high amount of load is curtailed each time a customer is interrupted. Fig. 11 represents the energy not supplied on a 10 day time interval. Two peaks are distinguished in that figure: one in the winter and the other in the summer. The peak in the winter is explained by the fact that the interruption frequency and the load are high in that period. The load is however lower in the summer but the interruption duration are longer because of the maintenance, more frequent during that period, this explains the peak in the summer. We can distinguish some facts such as that the most important amount of interruptions are the smallest ones. Another information is that the interruptions have a two peaks: one in the summer caused by lightning and one in the winter caused by hard climatic conditions (Fig. 12). Lastly, we can see that the variation summer/winter of the longest interruptions are more important than the other interruptions. This can be explained by the accumulation of two phenomena: more scheduled maintenance associated to an increasing line failure rate in the summer (Fig. 13). We can observe here that the most important incidents have the smallest impact (weak ENS). The peaks in the summer and in the winter are always observable.

author is also grateful to Mrs. Lajoie-Mazenc for her helpful suggestions, and also to Mr. Desquilbet, Substation group manager and Mr. Mignard, head of Substations and Lines department of EDF, that constantly supported his research work. This research is realized with the financial support of EDF and the French government.

APPENDIX A OBTAINING EQN (4) From the event {S(q) ¼ k þ 1} ¼ {Xm þ 1 (q) ¼ i, …, Xm þ k (q) ¼ i, Xm þ k þ 1 (q) Þ i} on {Xm ¹ 1 (q) Þ i, Xm (q) ¼ i} the following relation, Pm i (S ¼ k þ 1) ¼ P(Xm þ 1 ¼ i, …, Xm þ k ¼ i, Xm þ k þ 1 Þ ilXm ¹ 1 Þ i, Xm ¼ i) is obtained and consequently (4). Obtaining eqn (7) X From Ein ½T ÿ ¼ 1 þ j[B Ejn þ 1 ½T ÿpn (i, j)where m(n) ¼ 1 þ p nm(n þ 1) and for n ¼ 0, the result is given. Obtaining the expectation of the Performability E[WN ] ¼

X

a(i)Ei [WN ] ¼

i[E

5 DISCUSSION

¼

X i[E

Taking into account time variation and therefore using NHMC leads firstly to obtain more information. Indeed, some indicators such as the frequency are known in stationary conditions and with constant hazard rates5 as a result, only a constant value of the frequency can be obtained whereas using NHMC provides an instantaneous indicator, this is also the case for performability indicators such as the energy not supplied or the load curtailed. It is important also to notice that this information can be aggregated into 3D diagrams which provides much more concise information. Secondly, more accurate information can be obtained since all the variation of the entry data is taken into account. Important differences may be observed when computing the ENS and the load curtailed with average values of the hazard rates and load, and when using NHMC. These differences can be also observed in other numerical examples11

ACKNOWLEDGEMENTS The authors wish to thank an anonymous referee for valuable comments that helped to improve this article. The first

X i[E

a(i)

¹1 X NX

a(i)

¹1 X NX

gj (n)Ei [1{Xn ¼ j} ]

j[E n ¼ 0

gj (n)Pi [Xn ¼ j]

j[E n ¼ 0

in the case of the cyclic chain the corresponding result can be obtained easily. Obtaining the variance of the Performability; Vari [1{Xn ¼ j} ] ¼ Ei [1{Xn ¼ j} 1{Xn ¼ j} ] ¹ Ei [1{Xn ¼ j} ]2 ¼ Pi [Xn ¼ j](1 ¹ Pi [Xn ¼ j]) and, covi [1{Xn ¼ j} 1{Xm ¼ k} ] ¼ Ei [1{Xn ¼ j} 1{Xm ¼ k} ] ¹ Ei [1{Xn ¼ j} ]Ei [1{Xm ¼ k} ] ¼ Pi [Xn ¼ j, Xm ¼ k] ¹ Pi [Xn ¼ j]Pi [Xn ¼ k] For n Þ m and 1 # j, k # s,   covi 1{Xn ¼ j} 1{Xn ¼ k}  ÿ    
¼ Pi Xm ¼ j Pi Xm ¼ klXn ¼ j ¼ Pi Xm ¼ k For n ¼ m and j Þ k,       covi 1{Xn ¼ j} 1{Xm ¼ k} ¼ ¹ P ¹ i Xn ¼ j ¼ Pi Xn ¼ k The variance of W N can be computed now.

Analysis of Markov systems Table 2. Transition rates. The transition rates are given by the following tables and formulae Transition

Rate

B . B1

2*{0·3*l L1*g PX þ 0·3*l L2*g PX*g PS þ 0·3*[l L5 þ l L6]g PW} m REENC 2*{0·7*l L1*g PX þ 0·7*l L2*g PX*g PS þ 0·7*[l L5 þ l L6]g PW þ [l L1 þ l L2 þ l L5 þ l L6]*g DJ} m REENC 2*{0·3*l L3*g PX þ 0·3*l L4*g PX*g PS þ 0·3*[l L7 þ l L8]g PW} m TELE 2*{0·7*l L3*g PX þ 0·7*l L4*g PX*g PS þ 0·7*[l L7 þ l L8]g PW þ [l L3 þ l L4 þ l L7 þ l L8]*g DJ þ [l Pa þ l CONN]*[g PX þ g DJ] þ l TCM} m TELE 2*[l TCM þ 3*l CONN þ l SMALT þ l DJ]*g PX m DEP 2*{9*l CONN þ l SMALT þ 2*l DJ þ 3*l SEC þ l TT þ l Pa þ l TR*g DJ} m TELE 2*[6*l CONN þ 3*l SEC þ l TT þ l DJ þ l Pa]* g PX þ l DJ m DEP l DEB m DEB l L1 þ l L2 þ l L5 þ l L6 m REENC [(l L3 þ l L4 þ l L7 þ l L8) þ l Pa þ l CONN þ l TCM]*(1 ¹ g ARS) 2(1 ¹ g DJ) 2 m BASC [(l L3 þ l L4 þ l L7 þ l L8) þ l Pa þ l CONN þ l TCM]*2*g ARS*(1 ¹ g DJ) 2 m TELE [(l L3 þ l L4 þ l L7 þ l L8) þ l Pa þ l CONN þ l TCM]*(1 ¹ g ARS) 2g DJ m DEP [(l L3 þ l L4 þ l L7 þ l L8) þ l Pa þ l CONN þ l TCM]*(1 ¹ g ARS) 2g DJ m DEP [9*l CONN þ l SMALT þ 2*l DJ þ 3*l SEC þ l TT þ l Pa](1 ¹ g ARS)(1 ¹ g DJ)(1 ¹ g PX) þ l TR*g DJ m BASC m TELE [9*l CONN þ l SMALT þ 2*l DJ þ 3*l SEC þ l TT þ l Pa]*g ARS*(1 ¹ g DJ)(1 ¹ g PX) m TELE [7*l CONN þ l SMALT þ l DJ þ 3*l SEC þ l TT]* g DJ*(1 ¹ g ARS) m DEP [2*l CONN þ l SMALT]*(1 ¹ g ARS) 2 (1 ¹ g DJ) 2*g PX m BASC [l CONN þ l DJ]*(1 ¹ g ARS)(1 ¹ g DJ)*g PX m DEP [4*l CONN þ l SMALT þ 2*l DJ þ 3*l SEC þ l TT þ l Pa]*(1 ¹ g ARS)(1 ¹ g DJ)*g PX m DEP l DJ þ [2*l CONN þ l DJ þ l Pa]*(1 ¹ g ARS) (g DJ þ g PX) m DEP [7*l CONN þ 3*l SEC þ l TT þ 2*l DJ þ l Pa]* (1 ¹ g PX) þ l TR*g DJ

B1 . B B . B2 B2 . B B . B3 B3 . B B . B4

B4 . B B . B5 B5 . B B . B6 B6 . B B . B7 B7 . B B.D D.B D . D1 D1 . D D . D2 D2 . D D . D3 D3 . D D . D4 D4 . D D . D5 D5 . D D . D6 D6 . D69 D69 . D D . D7 D7 . D D . D8 D8 . D D . D9 D9 . D D . D10 D10 . D D . D11 D11 . D D . D12 D12 . D D . D13

247

Table 2. (continued ) Transition

Rate

D13 . D D . D14 D14 . D B . LC LC . B LC . LC1 LC1 . LC LC . LC2

m TELE [7*l CONN þ 3*l SEC þ l TT þ 2*l DJ þ l Pa]*g PX m DEP l CONSL m DECL l L1 þ l L2 þ l L5 þ l L6 m REENC l L3 þ l L4 þ l L7 þ l L8 þ l Pa þ 7*l CONN þ l TCM þ l SMALT þ l DJ þ 3* l SEC þ l TT m REPAR l CONN þ l DJ m DEP 2*l CONN þ l DJ þ l Pa þ l TR*g DJ m DEP [7*l CONN þ 3*l SEC þ l TT þ 2*l DJ þ l Pa]* (1 ¹ g PX) þ l TR*g DJ m TELE [7*l CONN þ 3*l SEC þ l TT þ 2*l DJ þ l Pa]*g PX m DEP l CONST m DECT (2*0·3 þ 0·7)*[l L1*g PX þ l L2*(g PX þ g PS) þ (l L5 þ l L6)*g PW] þ [l L1 þ l L2 þ l L5 þ l L6]*g DJ m REENC 2*0·3*[l L3*g PX þ l L4*(g PX þ g PS) þ (l L7 þ l L8)*g PW] m TELE 0·7*[l L3*g PX þ l L4*(g PX þ g PS) þ (l L7 þ l L8)*g PW] þ [l L3 þ l L4 þ l L7 þ l L8 þ l Pa þ l CONN]*g DJ þ [l Pa þ l CONN]*gPX þ 3*l CONN þ l TCM þ l SMALT þ l DJ m DEP [3*l SEC þ 6*l CONN þ l TT þ 2*l DJ þ l Pa þ l TR] m REPAR [2*l SEC þ 4*l CONN þ l TT]*g PX m DEP

LC2 . LC LC . LC3 LC3 . LC LC . LC4 LC4 . LC LC . LC5 LC5 . LC LC . LC6 LC6 . LC B . TC TC . B TC . TC1 TC1 . TC TC . TC2 TC2 . TC TC . TC3

TC3 . TC TC . TC4 TC4 . TC TC . TC5 TC5 . TC " DL (t) ¼ B1 e

¹ (t ¹ m1 )2 j1

" þ B2 e

þe

¹ (t ¹ m2 )2 j2

¹ (t ¹ m1 ¹ 8760)2 j1

þe

þe

¹ (t ¹ m2 ¹ 8760)2 j2

¹ (t ¹ m1 þ 8760)2 j1

þe

#

¹ (t ¹ m2 þ 8760)2 j2

# þD

B 1 ¼ 1·44694023; m1 ¼ 4704; j 1 ¼ 4 3 10 6 with B 2 ¼ 0·38077374; m2 ¼ 200; j 2 ¼ 2 3 10 5 where t is the hour (from 0 to 8760). D ¼ 0·38077374.

Appendix A.1 Obtaining the IELC The IELC is defined as the probability of the following event: At time n ¹ 1, the system is in an operational state E 0 and at time n (next unit of time), the system is in a nonoperational state E\E 0. At time n the load delivered is g(Xn, n ) and the load curtailed is [L(n) ¹ g(Xn , n)]. Equivalently, IELC is the expectation of C n, where C n is defined

248

A. Platis et al.

Table 3. Failure rates

Coverage

Repair rates

l L1(t) ¼ 1·04*d L(t) year l L2(t) ¼ 2·23 3 10 ¹1*d L(t) year ¹1 l L3(t) ¼ 3·05 3 10 ¹2*d L(t) year ¹1

g DJ ¼ 1·4 3 10 g PX ¼ 8·5 3 10 ¹3 g PS ¼ 1·4 3 10 ¹2

l L4(t) ¼ 9·99 3 10 ¹3*d L(t) year ¹1

g PW ¼ 3·3 3 10 ¹2

¹1

¹3

l L5(t) ¼ 1·67 3 10 ¹2*d L(t) year ¹1

g ARS ¼ 4·1 3 10 ¹2

l L6(t) ¼ 3·58 3 10 ¹3*d L(t) year ¹1 l L7(t) ¼ 3·89 3 10 ¹4*d L(t) year ¹1 l L8(t) ¼ 3·88 3 10 ¹4*d L(t) year ¹1 l PA(t) ¼ 2 3 10 ¹4*d L(t) year ¹1 l TCM(t) ¼ 1·4 3 10 ¹3*d L(t) year ¹1 l TT(t) ¼ 1·4 3 10 ¹3*d L(t) year ¹1 l TC(t) ¼ 1·4 3 10 ¹3*d L(t) year ¹1 l SMALT(t) ¼ 3 3 10 ¹4*d L(t) year ¹1 l SEC(t) ¼ 2 3 10 ¹4*d L(t) year ¹1 l DJ(t) ¼ 1·7 3 10 ¹3*d L(t) year ¹1 l CONN(t) ¼ 2·89 3 10 ¹5*d L(t) year ¹1 l TR(t) ¼ 8·3 3 10 ¹2 year ¹1 (constant rate)

m REENC ¼ 1·5768 3 10 year ¹1 m BASC ¼ 525 3 600 year ¹1 m repar ¼ 2920 year ¹1 ( 105120 year ¹ 1 for8h00 # t # 17 h00 mTELE (t) ¼ 26280 year ¹ 1 else 8 525600 ¹1 > > > < Tdep (t) þ 28·5 year for8h00 # t # 17h00 mDEP (t) > 525600 > > year ¹ 1 else : Tdep (t) þ 43·5

m DECL ¼ 176 year ¹1 m DECT ¼ 40·6 year ¹1 m DEB ¼ 4 year ¹1

lCONST (t) "

as follows; Cn ¼ 1{Xn ¹ 1 [E0 , Xn [EromanE 0}[L(n) ¹ g(Xn , n)] r X X X 1{Xn ¹ 1 ¼ i, Xn ¼ j} [L(n) ¹ g(Xn , n)] ¼

¼ B1 e

¹ (t ¹ m1 )2 j1

þe

¹ (t ¹ m1 ¹ 8760)2 j1

þe

¹ (t ¹ m1 þ 8760)2 j1

#

i[E0 k ¼ 1 j[Ek

with B1 ¼ 2:4; m1 ¼ 4704; j1 ¼ 5 3 106

Therefore, Ex [Cn ] ¼ IELC(x, n)

Tdep (t) "

¼ 1{Xn ¹ 1 [E0 , Xn [EromanE 0}[L(n) ¹ g(Xn , n)] ¼

r X X X

¼ B1 e P(Xn ¹ 1 ¼ i, Xn ¼ j)[L(n) ¹ g(Xn , n)]

i[E0 k ¼ 1 j[Ek

and the result is given. The derivations for the asymptotic cyclic case can be easily obtained by introducing eqn (23), eqn (24) and eqn (25). Table 2 and Table 3. This data has to be used exclusively as example for checking the results or building other reliability models l DEB ¼ 1 year ¹1 lCONSL (t) " ¼ B1 e

¹ (t ¹ m1 )2 j1

þe

¹ (t ¹ m1 ¹ 8760)2 j1

þe

¹ (t ¹ m1 þ 8760)2 j1

with B1 ¼ 2 3 66; m1 ¼ 4704; j1 ¼ 5 3 106

#

¹ (t ¹ m1 )2 j1

þe

¹ (t ¹ m1 ¹ 8760)2 j1

þe

¹ (t ¹ m1 þ 8760)2 j1

#

and B1 ¼ 20; m1 ¼ 360; j1 ¼ 107

REFERENCES 1. Allan, R. N., and Billinton, R., Reliability evaluation of power systems, Plenum Press, 1990. 2. Csenki, A., Dependability for Systems with a Partitioned State Space, Lecture Notes in Statistics, Springer–Verlag, New York, 1994. 3. Gerontidis, I. Cyclic strong ergodicity in non-homogeneous Markov systems. SIAM J. Matrix Anal. Appl., 1992, 13(2), 550–566. 4. Limnios, N., Processus Stochastiques et Fiabilite´ (Stochastic processes and reliability), CS-61 course-DEA-System Control-UTC, 1994.

Analysis of Markov systems 5. Limnios, N. Throughput availability in Markov systems. IEEE Transactions on Reliability, 1992, C-41(2), 219– 224. 6. Limnios, N. Dependability analysis of semi-Markov systems. Reliability Engineering and System Safety, 1995, 55, 203– 207. 7. Meyer, J. F., On evaluating the Performability of Degradable Computing Systems, IEEE Transactions on Computers, C-29 No. 8, 1980. 8. Page`s, A. and Gondran, M., Fiabilite´ des syste`mes (System Reliability), Eyrolles, Paris, 1980. 9. Pattipati Krishna, R., Li, Y. & Blom Henk, A. P. A unified framework for the performability evaluation of fault-tolerant computer systems. IEEE Transactions on Computers, 1993, 42(3), 312–326. 10. Platis, A., Limnios, N. and Le Du, M., Electrical Substation Performability & Reliability Indicators modeled by NonHomogeneous Markov Chains. In ESREL’96-PSAM-III International Conference, Crete, June 1996. 11. Platis, A., Limnios, N. & Le Du, M. Performability of

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electrical power systems modeled by non-homogeneous Markov chains. IEEE Transactions on Reliability, 1996, 45(4), 605–610. Platis, A., Limnios, N. and Le Du, M., Asymptotic Availability of Systems Modeled by Non-Homogeneous Markov Chains. In Annual Maintainability & Reliability Symposium, Philadelphia, January 1997. Rubio, G. & Sericola, B. Sojourn times in Semi-Markov reward processes: application to fault-tolerant systems modeling. Reliability Engineering and System Safety, 1993, 41, 1–4. Sericola, B. Closed-form solution for the distribution of the total time spent in a subset of states of a homogeneous Markov process during a finite observation period. Journal of Applied Probability, 1990, 27, 713–719. Smith, R.M., Kishor, S., Trivedi and Ramesh, A.V., Performability Analysis: Measures, an Algorithm, and a Case Study, IEEE Transactions on Computers, 37(4) 1988. Feller, W., An Introduction to Probability Theory and its Applications, vol. 1, Wiley, 1968.