Example of the spent fuel pool

The spent fuel pool system is used in a nuclear power plant to stock the spent fuel which is still heating and radioactive. As this system involves many components it would be quite long to expose all the features of its PDMP model. Therefore we just present the structure of the system and the differential equations ruling its main physical variables. The idea here is simply to give the reader a glimpse of the complexity of an industrial system. We refer the reader to [13] for a more complete presentation of this system.

The fuel is placed in the water, because it contains the remaining radiations produced by the fuel. It is important to control the level of the water in the pool so that the fuel stays immersed, without which radiations would be released in the environment. Also it

Figure 1.20 – The spent fuel pool system

is important to control the temperature of the pool so that the water does not vaporize. To do so a cooling system is attached to the pool. This system is designed to cool down the fuel without rejecting any radioactive matter in the river. As the water of the pool is in contact with radioactive matter it can not be released in the nearby river, and it stays in a circuit of water called the primary circuit. This water is heated to the contact of the fuel, and, further in the circuit, it is cooled down by a secondary circuit of water through a heat-exchanger. The heat gathered by the secondary circuit is then absorbed by a third circuit through an other heat-exchanger. The water of the third circuit is taken directly from the nearby river, and goes back to the river once heated. Each circuit has a pump, a valve, and a clap, if one of these elements breaks the water cannot circulate in the circuit. The combination of these three circuits is called a line.

In case the line would fail, it is backed up by two other lines in passive redundancy. All the lines are connected to the river by a pumping system. In case this pumping system fails the third line has an additional backup pumping system. The lines need current to function, and in case of a power grid loss, each line disposes of a switch system which launches a diesel generator that powers the line. The system is schematized in figure 1.20 without its electric-powering part. When the three lines stay failed, the temperature rises to 100 °C, once this evaporating temperature is reached, the level of the water starts to

decrease2.

The logic diagram in figure 1.21 represent the logic in the systems. In this diagram if there is an uninterrupted path between the bottom line and the top line it means the main function of the system is fulfilled and that the temperature does not rise nor the level decreases. In this diagram each box represents a component, and acts like a switch that breaks the path when the component is failed. The arrows represent conditional ac- tivations, meaning the component pointed by an arrow is activated only if the component at the beginning of the arrow is failed. If a pointed component is not activated it breaks the logic path it is a part of. One can associate a logic function to the figure 1.21 which

Figure 1.21 – reliability diagram of the static approximation of the spent-fuel-pool system is true if the function of the system is fulfilled giving the statuses of the components. We denote Lpool : m ∈ M → {F alse, T rue} the logic function. For each component we define Boolean indicators that are true if they are failed and false otherwise: Grid is the Boolean indicating that the power is no longer available from the grid, Swi indicates if the switch of the i-th line is failed, Gni indicates if the diesel generator of the i-th line is failed, Lni indicates if the i-th line is failed, Exij indicates if the j-th exchanger of the i-th line is failed, Rvi indicates the i-th pump on the river is failed, W cij indicates if the

2. In a real spent-fuel-pool at this stage water would be added to the pool to compensate the loss of level, but we consider in this thesis a spent-fuel-pool system without this function of level management.

j-th water circuit of the i-th line is failed, P pij indicates if the pump of the j-th water circuit of the i-th line is failed, V aij indicates if the valves of the j-th water circuit of the i-th line is failed, Clij indicates if the clap of the j-th water circuit of the i-th line is failed. Considering multiplication of Booleans as an AND operator and the sum as an OR operator, we have: Lpool(M) =Grid [Rv1(Ln3 + Sw3+ Gn3+ Rv2) +(Ln1 + Sw1+ Gn1)(Ln2+ Sw2+ Gn2)(Ln3+ Sw3+ Gn3)] +Rv1(Ln3+ Rv2), +(Ln1)(Ln2)(Ln3), (1.111) where Lni = W ci1+ Exi1+ W ci2+ Exi2+ W ci3 (1.112) and W cij = P pij + V aij + Clij (1.113) We denote by X1

t the temperature of the pool at time t in Celsius degrees, and Xt2 the water level of the pool. P is the residual power of the fuel in the pool, C is the mass heat capacity, ρ is the density of the water, S is the area of the surface of water in contact with the air, Qin is the entering debit water, Tin is the temperature of the entering water, respectively Qout is the debit of water leaving the pull, and Tout is the temperature of the water leaving the pool, l is the mass latent heat of vaporization. When the temperature is below 100 °C or when P + ρC(QinTin− QoutTout) < 0, the differential system ruling the evolution of Xt = (Xt1, Xt2) is: ∀X1 t ∈(0, 100), and X 2 t >0, dX1 t dt = P + ρC(QinTin− QoutTout) ρCSX2 t (1.114) dX2 t dt = 0. (1.115)

When the temperature of the water is at 100 °C and when P + ρC(QinTin− QoutTout) > 0 the differential system is

for X1 t = 100, and Xt2 >0, dX_t1 dt = 0 (1.116) dX2 t dt = − P + ρC(QinTin− QoutTout) ρCSl . (1.117)

Monte-Carlo methods for rare events

As the previous Chapter presented our model for the simulation output, the present chapter deals with the simulation methods. We start this chapter by introducing the naive Monte-Carlo method and its rare event issue, and then we present the importance sampling method and the interacting particle system method. Our goal here is not to review all the methods used in reliability analysis, but rather to introduce the methods that we adapt to PDMP in parts II and IV. We refer the readers to [3, 47, 8, 46, 32] for reviews of methods used in rare event analysis.

2.1

The Monte-Carlo method and its rare event issue

The Monte-Carlo method allows the estimation of an expectation defined by: p= Ef[h(Y )] =

Z

A

h(y)f(y)dζ(y). (2.1)

The estimator of Monte-Carlo takes the following form: ˆpf = 1 Nf Nf X i=1 h(Yi), where Yi iid ∼ f. (2.2)

Theorem 3. If E [|h(X)|] < ∞ the strong law of large numbers implies that the Monte-

Carlo estimator is strongly consistent: ˆpf

a.s. −−−−−−−→

Nf−→+∞

p. (2.3)

Theorem 4. If E [h2(Y )] < ∞, the Monte-Carlo estimator also satisfies a central limit

theorem (CLT): q Nf(ˆpf − p) L −−−−−−−→ Nf−→+∞ N(0, σ2 f), (2.4) where σ2 f = Varf(h(Y )).

Figure 2.1 – Normally distributed estimator and confidence interval quantity p which is defined by:

IC1−α =  ˆp_f − q1−α 2 v u u tˆσ 2 f Nf , ˆpf + q1−α 2 v u u tˆσ 2 f Nf  , (2.5) where ˆσf = _N1 f PNf

i=1(h(Yi) − ˆpf)2 and q1−α

2 is the quantile of level

1−α

2 of a N (0, 1) normal

distribution. According to the Theorem 2.4 and Slutsky’s theorem, IC1−α is an asymptotic

confidence interval of level 1 − α, as lim

Nf→∞P (p ∈ IC

1−α) = 1 − α. (2.6)

Most of the time, one considers a confidence interval of level 95%. In this case qα 2 '1.96, and: IC95% =  ˆp_f −1.96 v u u tˆσ 2 f Nf ,ˆpf + 1.96 v u u tˆσ 2 f Nf  . (2.7)

If we want to get a confidence interval of level 1 − α of size smaller than L > 0, we have to set a number of simulations verifying:

Nf ≥ 4 q2 α 2 σ 2 f L2 (2.8)

The more we want to be accurate and have a narrow confidence interval, the higher the number of simulations needs to be. We can even see in the inequality (2.8), that there is a quadratic relation between the number of simulations required and the inverse of width of the CI 1

2.1.1

Computational burden of the Monte-Carlo estimator