Availability of an intermittently required system : application to a fossil fuel power plant Marc Bouissou(1),(2)

(1)

EDF Recherche et D´eveloppement 1, av. du G´en´eral De Gaulle 92141 Clamart cedex, France

Yannick Lefebvre(1),(2)

(2)

Universit´e de Marne-la-Vall´ee 5, bd Descartes, Champs-sur-Marne ´e cedex 2, France 77454 Marne-la-Valle

Abstract: Electricit´e de France produces about 75% of its electricity with nuclear power plants. Some hydraulic plants and most fossil fuel power plants are used to pass electricity consumption peaks. Therefore these facilities are required only intermittently. For such systems, a breakdown does not result in a production loss during a standby period. This particular feature is not taken into account by conventional availability evaluation methods. The objective of this paper is to introduce a definition of availability which holds in such a context, and to describe a mathematical method suited to the calculation of this new definition of availability.

Keywords: Phased mission system, availability, productivity, Markov process

1

Introduction

In a context of growing competition between electricity producers, it becomes more and more important to be able to predict the availability of power plants, in order to optimize the design of new facilities, and the exploitation of existing plants. This is why several methods and tools [3] have been set up in order to assess the availability of nuclear power plants. These methods are applicable only to systems which are expected to work continuously. But some power plants, like hydraulic and fossil fuel plants, are required only to pass consumption peaks. This explains the need for new methods able to take such a feature into account. 1

The objective of this paper is to introduce a definition of availability which holds in such a context, to give the principles of a mathematical method suited to the calculation of this particular definition of availability, and to give some results obtained with this method. The paper is organized as follows: • we first give an example of an intermittently required system: it is one of the main systems in a fossil fuel plant, which must adapt to the power the plant has to produce. We give a simplified description, limited to its main components: three pumps, with a complex exploitation policy. This example is sufficient to illustrate the difficulties that can be encountered in such a study. • then we give a definition of availability which is suited to this kind of system, taking into account the fact that there is a production loss only when the system is failed during a period when it is required. • thirdly, we present the method we propose, in a general framework. • finally, we apply this method to our example, and give the results of a few sensitivity studies, which show the influence of the particular features of this system on its availability.

2

System example

The system we are interested in is the feedwater system in a fossil fuel power plant. This system is composed of three identical motor-driven pumps (denoted MDP1, MDP2 and MDP3). Each one can provide 50% of the total production. The operation of the feedwater system is only required during the week, with a difference between day and night: • the maximum production is expected at daytime, which means that two MDP must be in working state, • only 50% of the maximum production is expected during the night: one operating MDP is therefore sufficient. Let us define now the maintenance policy of this system. If a failure occurs during the week, but if at least one operating MDP remains, the repair of the failed component doesn’t start immediately. In fact, the maintenance operations only begin when the system is stopped, that is to say during the next week-end. On the other hand, if the three MDP are failed, the system is of course stopped immediatly and a “maintenance period” begins. This period comes to an end when the repair of each component is completed. As soon as the feedwater system is stopped (for the week-end or for a maintenance period), it starts to cool. Different on demand failures may occur when

the system is restarted. The probability of such failures depend on the cooling duration d. Three cases have to be distinguished: • d < 10h : the system is said to be in a “warm stopping state”, • 10h ≤ d < 72h : the system is said to be in a “lukewarm stopping state”, • d ≥ 72h : the system is said to be in a “cold stopping state”. In the sequel of the paper, we will consider a more general framework. The system is supposed to have a finite state space E. The subset of working states (respectively failure states) is denoted M (respectively P). The evolution of the system is described by a stochastic process (Xt )t≥0 . The succession of working periods and standby periods is modelled by a stochastic process (Zt )t≥0 , called the operating process, with values in a finite state space F which can be divided in two parts: • the subset S, gathering the states in which the system’s operation is required. • the subset R, gathering all the others states of F. The example we described above is an instance of this general framework.

3

Availability definition

From the electricity producer point of view, what is important is the amount of energy which cannot be produced over a long period of time, because this is the major part of financial losses. For an intermittently required system, the production loss due to a breakdown depends on two factors: of course the consequences of the breakdown on the system, but also the production required. For instance, there is no outage if the feedwater system described in section 2 is failed during the week-end. Therefore, we use a special definition for the average availability of the system over the period [0, t]: Z At = 1 −

X η∈E, ξ∈S

c(η, ξ)

0

t

1{Xs =η, Zs =ξ} ds Z t 1{Zs ∈S} ds

(1)

0

where c(η, ξ) ∈ [0, 1] denotes the proportion of production loss when (Xt )t≥0 is in the state η, and (Zt )t≥0 is in the state ξ.

We are interested only in the limit of this quantity when time tends to infinity, because problems that can occur at the beginning of the system life are not significant. To study the asymptotic average availability of the system, we need to calculate the asymptotic distribution of the process (Xt , Zt )t≥0 . Note that if the system is never required, the definition is no longer valid, but who could be interested in the availability of a system that is never required? At the other extreme, if the system is continuously required, our definition becomes equivalent to the so-called EFOR (Equivalent Forced Outage Rate) which is conventionally used by electricity companies. Remark : It is also possible to give an appropriate definition for the availability et , and define it as at time t of such a system. We denote the point availability A follows: X et = 1 − A c(η, ξ) pt (η, ξ) (2) η∈E, ξ∈S

where:

 pt (η, ξ) =



P Xt = η Zt = ξ if P Zt = ξ
> 0 0 if P Zt = ξ = 0

Using the definitions 1 and 2 given in this paragraph, we showed that the average availability and the point availability converge (3) to the same value when t tends to +∞, under reasonable mathematical assumptions (see [6]). es , E At ) −→t→+∞ lim A es At −→t→+∞ lim A s→+∞

4

s→+∞

(3)

Method description

The two main difficulties in such a case are on the one hand the cooling phenomenon, and on the other hand the fact that the system is not expected to work continuously. It is quite difficult to calculate the asymptotic average availability when both problems are taken into account. That is why we will present two models: • in the first one, the exploitation calendar of the system is simplified, in order to be able to take into account the cooling phenomenon. The operating process (Zt )t≥0 can only be in two states: “the maximum production is required”, and “no production is required”. • in the second model, the operating process can be in intermediate states (for example the difference between day and night for the feedwater system). But in this case, we will not consider the cooling phenomenon. Both models are instances of so called phased mission systems (see [7]).

4.1

The first model

We suppose here that the operating process (Zt )t≥0 is an alternating renewal process. Let 1 be the state in which the maximum production is expected, and 0 be the state in which no production is required. The system (Xt )t≥0 evolves according to two different Markov processes, depending on the fact that (Zt )t≥0 is in the state 1 or in the state 0. The system can be stopped for two reasons. First, if (Zt )t≥0 enters the state 0, no production is required: the system doesn’t have to operate, and will be restarted when (Zt )t≥0 enters the state 1. If the system breaks down, in other words if (Xt )t≥0 enters the subset of failure states P, the system is also stopped, until (Xt )t≥0 enters the subset of working states M. When restarting the system in the state η ∈ M, on-demand failures may occur: the process (Xt )t≥0 can enter the state ρ with a probability depending on the cooling duration.   γ1 (η, ρ) ∈ ]0, 1[ for a warm stopping state γ2 (η, ρ) ∈ ]0, 1[ for a lukewarm stopping state  γ3 (η, ρ) ∈ ]0, 1[ for a cold stopping state If the system is successfully restarted, the coooling period comes to an end. Note that if the operating process (Zt )t≥0 stays in 1 and 0 during an exponentially distributed time, and if we forget about the cooling phenomenon, then (Xt , Zt )t≥0 is a Markov process with a finite state space. This well-known case has been studied for a long time in the literature (see for example [4]). In order to bring our model closer to this simple case, we use the phase method to replace the “true” operating process by a Markov process with finite state space. This method, also called the fictitious states method in the literature, makes use of the fact that any probability distribution on R+ can be approximated by a structured combination of exponentially distributed transitions (see [1] for a detailed presentation of this method, [8] and [2] for a few examples of applications to the reliability field). For instance, a deterministic transition can be approximated by an Erlang distribution, as shown on figure 1.   δd=⇒ E F  



   Exp(n/d) E2 En E F     Exp(n/d)

Figure 1: Approximation of a Dirac δd by an Erlang(n, n/d) In this example, the quality of the approximation increases with the value of the parameter n, which is equal to the number of fictitious states in the transition.

We suppose hereafter that (Zt )t≥0 is the Markov process constructed using the phase method. The subsets S and R gather respectively the fictitious states introduced to replace the states 1 and 0. Two mathematical approaches can be envisaged to calculate the asymptotic distribution of (Xt , Zt )t≥0 and obtain the asymptotic average availability of the system. We only give here the main idea of each approach (see [6] for details). 4.1.1

First approach: introduction of an extra continuous variable

Because of the cooling phenomenon, (Xt , Zt )t≥0 is not a Markov process. But we can obtain a Markov process by introducing an extra variable. This one is denoted Lt and is equal to the time elapsed since the beginning of the cooling period at time t (it equals −1 if the system is not cooling). Thus, (Xt , Zt , Lt )t≥0 is a Markov process, but its state space is non discrete because of this new variable. To study its stationary distribution, we will use the theory of Markov processes with general state space (see [1]). The approach used to calculate the stationnary distribution π is due to Cocozza (see [5]). 4.1.2

Second approach: a renewal property of the system

The aim of this second approach is to calculate the asymptotic distribution of (Xt , Zt )t≥0 , without introducing any continuous variable. When the system is succesfully restarted, the cooling period comes to an end. At this moment, the process (Xt , Zt )t≥0 forgets its past: the future of the system only depends on the state of its components, and on the production required. Consequently, the process (Xt , Zt )t≥0 is semi-regenerative (see [4]). It is then possible to show (see [4]) that the asymptotic availability can be obtained from the behaviour of the system before the first successful restart.

4.2

The second model

From now on, we suppose that the system does not cool when it is stopped. On the other hand, we will be able to calculate the asymptotic availability with a more complicated operating process (Zt )t≥0 . The only assumptions made on this process is that its state space F is finite, and that one state of F is a regeneration point for (Zt )t≥0 . The elements of F are denoted ξ1 , ξ2 , . . . , ξm . We consider m Markov processes with the same finite state space E. These processes are respectively denoted M1 , . . . , Mm . When the operating process (Zt )t≥0 is in the state ξi , the system (Xt )t≥0 evolves according to the process Mi .

On-demand failures may still occur when restarting the system, but with a probability which does not depend any more on the break duration. The assumptions made on (Zt )t≥0 and on the system behaviour ensure once again that (Xt , Zt )t≥0 is a semi-regenerative process. Indeed, the process (Xt , Zt )t≥0 forgets its past each time the operating process enters its regeneration state. This allows us to use the second approach, given in §4.1.2: it is possible to show that the asymptotic average availability of the system only depends on the system life before the first return of (Zt )t≥0 in its regeneration state. The first approach (§4.1.1) is no longer valid in this model, in particular because there could be some deterministic transitions in the process (Zt )t≥0 .

5

Application to our example

In this section, we apply the methods described above to the feedwater system. We consider two different models. In the first one, we suppose that the maximum output is expected during the week: no difference is made between day and night. Because of the numerical limits we encountered, we considered a simplified system structure: each MDP is seen as a single component. In the second model, the different output levels required during the week are taken into account, but the three MDP are not supposed to cool when the feedwater system is stopped. Note that a far more realistic model of the system has been used (each MDP is composed of about ten components).

5.1

The first model

Let us first consider the operating process of this system. The maximum production is expected during the week (108h), and the system is stopped for the week-end (60h). The process (Zt )t≥0 is constructed using the phase method: each deterministic duration is approximated by an Erlang distributed duration (see figure 2). The subsets S and R are the following: n o n o S = W1 , . . . , W s , R = W E1 , . . . , W E r Each motor-driven pump is seen as a single component, with constant failure and repair rates (denoted respectively λ and µ). Refer to [6] for the construction of the Markov processes which characterize the evolution of the system. The different states of the process (Xt )t≥0 are denoted (η1 , η2 , η3 ), where ηi equals 0 if the MDPi is failed and 1 in the other case. Let M1 be the subset gathering all the states in which the system is able to produce its maximum output: n o M1 = (1, 1, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1)

 δ108 - =⇒ WE W   δ60 

  s/108 - W2 W1   6 r/60  W Er  

 - Ws  s/108

?   r/60 W E2  W E1  

Figure 2: Construction of the operating process (Zt )t≥0 If two MDP are failed, the system still works but can produce only 50% of the maximum output. Such states are gathered in the subset M2 . n o M2 = (1, 0, 0), (0, 1, 0), (0, 0, 1) Finally, the subset of failure states P includes of course the state (0, 0, 0), but also all the intermediate states between (0, 0, 0) and the total repair. In these intermediate states, the system could produce a part of the required output, but as we said in §2, the system is restarted after completion of all repairs. n o P = (0, 0, 0), (1, 0, 0)R , (0, 1, 0)R , (0, 0, 1)R , (1, 1, 0)R , (1, 0, 1)R , (0, 1, 1)R Let us consider now the assumptions made on the cooling phenomenon. When the system is restarted, an on demand failure may affect each component with a probability depending on the cooling duration.   γ1 ∈ ]0, 1[ γ2 ∈ ]0, 1[  γ3 ∈ ]0, 1[

for a warm stopping state for a lukewarm stopping state for a cold stopping state

Our aim is now to give a few sensitivity studies which show the influence of the particular features of this system on its availability. Note that the average availability over the period [0, t] is defined in this case as follows: Z At = 1 −

0

t

Z

1{Xs ∈P, Zs ∈S} ds 50 − Z t 100 1{Zs ∈S} ds 0

0

t

1{Xs ∈M2 , Zs ∈S} ds Z t 1{Zs ∈S} ds 0

As we have already said, the quality of the approximation used to construct (Zt )t≥0 increases with the number of fictitious states introduced (s + r in this

case). But it is not hard to understand that from a computational point of view, the problems also increase with this parameter. Therefore, it is interesting to study the influence of the number of fictitious states on the asymptotic average availability of the system. Figure 3 shows the evolution of the asymptotic average unavailability with respect to s + r (with s = r). The parameters used are the following: λ = 4 × 10−4 , µ = 2 × 10−2 , γ1 = 1.3 × 10−4 , γ2 = 2.6 × 10−4 , γ3 = 5.2 × 10−4

13e-5 12e-5 11e-5 10e-5 9e-5 8e-5 7e-5 6e-5 5e-5 4e-5 3e-5 2

6

10

14

18

22

26

30

34

38

Figure 3: Influence of the fictitious states (1 − A(∞) with respect to s + r) This curve shows that the unavailability decreases when the number of fictitious states increases. It seems sufficient to use about 20 fictitious states to give a good (and slightly pessimistic) approximation for the asymptotic average availability of the feedwater system. Another problem for such a model is the estimation of three different on demand failure rates. Most of the time, only one value is available for a component. Since the system is reliable enough, most of the cooling periods are due to normal stops during week-ends (60h). So, γ2 is the most influential parameter. This is why we choose to use the feedback of experience value for this parameter, and to assume the existence of a constant c such that: γ1 =

1 × γ2 , γ3 = c × γ2 c

But the problem is to choose the value of this constant. Figure 4 shows

the importance of this choice, through the evolution, with respect to c, of the quantity: ∆=

c A (∞) − A1 (∞) 1 − A1 (∞)

× 100

where Ac (∞) denotes the asymptotic average availability of the system if we suppose γ1 = γ2 /c , γ3 = c × γ2 . The values used for the parameters λ, µ and γ2 are the same as above.

9

8

7

6

5

4

3

2

1

0 1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

Figure 4: Influence of the cooling phenomenon (∆ with respect to c) So, the constant c is not a sensitive parameter. Even for c = 3, which makes a big difference between the three on demand failure rates, the relative difference between the asymptotic average unavailabilities indexed by c and 1 is less than 10%. In other words, the influence of the cooling phenomenon is not really significant. Therefore, it seems more judicious to use the second model, which does not take into account the cooling phenomenon, but in which the operating process and the system structure are more realistic.

5.2

The second model

In this model, we take into account the difference between night and day for the operating process. During the week, the maximum production is expected at daytime, but only 50% is expected during the night.

The process (Zt )t≥0 is here totally deterministic. In fact, Zt indicates the current day at the time t, and also the moment of the day (that is to say day or night). Let us split the set S in two subsets: S =D∪N where D and N gather respectively the days and the nights of the week. The subset R represents the week-end. Each pump is a series assembly made of ten components with constant failure and repair rates. Because of this particular structure and the fact that the three MDP are identical, we could build a lumped graph of 76 states (which is still too big for the first model, but reasonable for this one). Each component has a single on demand failure probability We still consider the subsets M1 and M2 to make a difference between the “good” and the degraded working states. The system evolves according to three different Markov processes, when the operating process is respectively in D, N or R. Report to [6] for the construction of these three processes. The definition of availability that we will use takes into account the different output levels required during the week: Z At = 1−

0

t

Z 1{Xs ∈P, Zs ∈D} ds 50 − Z t 100 1{Zs ∈S} ds 0

0

t

Z t 1{Xs ∈M2 , Zs ∈D} ds + 1{Xs ∈P, Zs ∈N } ds 0 Z t 1{Zs ∈S} ds 0

Indeed, when the system is failed (Xt ∈ P), all the production is lost at daytime, but only 50% is lost during the night. Moreover, if the feedwater system is in a degraded working state (Xt ∈ M2 ), 50% of the production is lost during the day. From a computational point of view, this model is far more attractive than the first one. This can be easily understood, noting that the second model does not include fictitious states, and that the CPU time necessary to calculate the asymptotic average availability increases significantly with the number of fictitious states.

6

Conclusion

We have presented in this paper the example of the feedwater system in a fossil fuel power plant. Then, we have introduced a definition of the availability which takes into account the particular features of this system, and developed two models in order to calculate its asymptotic value.

The results we have given show that the cooling phenomenon does not influence significantly the asymptotic average availability of the feedwater system. As a consequence, the second model we have presented seems more interesting: the system is not supposed to cool when it is stopped, but the exploitation calendar modelling is more realistic. Moreover, a more complex modelling of the feedwater system seems possible in this model, which is far more efficient than the first one from a computational point of view. Consequently, this model could be useful to assess the availability of intermittently required systems.

References [1] Asmussen, S., Applied Probability and Queues, Wiley, 1992 [2] Aven, T., Jensen, U., Stochastic models in reliability, Springer, 1998 [3] Bouissou, M., Bourgade, E., Unavailability Evaluation and Allocation at the Design Stage for Electric Power Plants : Methods and Tools, Proceedings of the RAMS’97 conference, Philadelphia,jan. 1997 [4] Cocozza-Thivent, C., Processus stochastiques et fiabilit´e des syst`emes, Springer, 1997 [5] Cocozza-Thivent, C., A model for a dynamic preventive maintenance policy, Journal of Applied Mathematics and Stochastic Analysis, 13:4 (2000) 321-346 [6] Lefebvre, Y., Availability of an intermittently required system, Pr´epublications de l’´equipe d’Analyse et de Math´ematiques Appliqu´ees, Universit´e de Marne-la-Vall´ee, to be published [7] Y.Ma, K.S.Trivedi, “An algorithm for reliability analysis of phasedmission systems”, Reliability Engineering and System Safety n◦ 66, 1999, pp 157-170 [8] Ngom, L., Cabarbaye, A., Geffroy, J.-C., Interests and implementation of the fictitious state method in markovian modelling, Proceedings of the PSAM4 conference, New York City,sept. 1998

About the authors Marc Bouissou is a senior engineer of EDF R & D, with over 17 years of experience in the reliability engineering field. He also holds an appointment at CNRS (University of Marne la Vall´ee) as an associate research director. He has led the development of highly innovative tools, based on AI techniques, to support the activities of reliability engineering, and PSAs for nuclear power plants. He has published more than 30 technical papers, and gives RAMS lectures in several universities. His recent work is about RAMS allocation, computer controlled systems, architecture optimization. He is the vice-president of the ”Methodological Research” working group of the French ISDF (RAMS Institute) association. He received a degree of the ”Ecole Nationale Sup´erieure des Mines de Paris” engineering school in 1980. Y. Lefebvre is a Ph.D candidate of Marne la Vall´ee University, who works in collaboration with EDF. His research interest includes reliability and availability of complex ageing systems.