Quantitative common cause failure modeling for auxiliary feedwater system involving the seismic-induced degradation of flood barriers

Figures & data

Figure 1. Fault tree for possible flood.

Figure 2. Additional flood barriers in turbine building as parallel Layout 1.

Figure 3. Additional flood barriers in turbine building as sequential Layout 2.

Figure 4. Seismic-induced one-barrier failure (Layout 2).

Figure 5. Seismic-induced two-barrier failure (Layout 2).

Figure 6. Seismic-induced three-barrier failure (Layout 2).

Table 1. Possible failure types for two layouts.

	Possible failure	Possible failure
Scenario	events for Layout 1	events for Layout 2
(1)	Independent failure (1/3)	No failure (0/3), or independent failure (1/3)
(2)	Partial CCF (2/3)	No failure (0/3), or independent failure (1/3), or partial CCF (2/3)
(3)	Complete CCF (3/3)	Complete CCF (3/3)

Table 2. Description of four Markov states.

State	Description
S	Success: no water leakage.
L	Leak: the leakage is less than the drainage ability.
D	Deformation: the possible water flow rate is larger than the drainage ability but not reaching maximum.
F	Failure: the possible water flow rate is the same as doors without flood barriers.

Table 3. Description of four transition parameters.

Transition
parameters	Description
φSL	Leak occurrence probability given the state of Success.
λLD	Deformation occurrence probability given the state of Leak.
ρLF	Failure occurrence probability given the state of Leak.
ρDF	Failure occurrence probability given the state of Deformation.

Figure 7. Four-state Markov model for the degradation of flood barriers.

Table 4. Uncertainty distribution assumption for Markov parameters.

	Uncertainty treatment
	Likelihood	Prior
Parameters	function	distribution
Initiating states	Multinomial	Dirichlet
Leak	Poisson	Lognormal
Deformation	Binomial	Lognormal
Failure given Leak	Binomial	Lognormal
Failure given Deformation	Binomial	Lognormal

Table 5. Assumed water flow rate through barriers for Markov states.

States	Flow rate (m^3/min)
Success	0
Leak	0.25
Deformation	1
Failure	2

Figure 8. Time-dependent probabilities of Markov states.

Figure 9. The posterior distributions for transition parameters in Markov model.

Table 6. Calculation assumption for water flow rate through a barrier.

Parameters	Measurements
Single AFW pump room	80 m^2
Critical water h eight (CWH)	0.15 m
Drainage ability per room	0.5 m^3/min
Most severe flood	2 m^3/min
Minimal time to reach CWH (without flood barrier)	8 min

Table 7. Hypothetical experiments with states account for the estimation of transition parameters.

	Number of state records				Number of state records after certain
	after seismic shock				duration of flood propagation
States	S	L	D	F	S	L	D	F
Experiment (1)	80	16	3	2	72	23	3	2
Experiment (2)	85	12	2	1	75	21	3	1
Experiment (3)	83	15	1	1	71	23	4	2

Figure 10. Water flow rate and height in AFW pump rooms (Layout 1).

Figure 11. Water flow rate though flood barriers (Layout 2).

Figure 12. Water height in AFW pump rooms (Layout 2).

Figure 13. Time-dependent failure probability percentage (Layout 1).

Figure 14. Time-dependent failure probability percentage (Layout 2).

Table 8. Hypothetical flow rate of flood water sources.

	Flow rate through
Water sources	Barrier 1 (m^3/min)	Probability
Circulating water (CW)	1.2	1.30E−4
Service water (SW)	0.75	1.80E−5
Fire protection water (FPW)	0.5	5.10E−4
Three water sources (CCF)	2	1.40E−5

Table 9. Failure probability distributions for different scenarios.

		Probability
Failure types		CW	SW	FPW	CCF
Layout 1	0/3	0	0	1 (5.10E−4)	0
	1/3	0.2 (2.60E−5)	0.2 (3.60E−6)	0	0.2 (2.80E−6)
	2/3	0.2 (2.60E−5)	0.2 (3.60E−6)	0	0.2 (2.80E−6)
	3/3	0.6 (7.80E−5)	0.6 (1.08E−5)	0	0.6 (8.40E−6)
Layout 2	0/3	0.2 (2.60E−5)	0.2 (3.60E−6)	1 (5.10E−4)	0.2 (2.80E−6)
	1/3	0.133 (1.73E−5)	0.8 (1.44E−5)	0	0.133 (1.86E−6)
	2/3	0.667 (8.67E−5)	0	0	0.067 (9.38E−7)
	3/3	0	0	0	0.6 (8.40E−6)

Table 10. The estimated decomposed alpha factors for internal flood.

Failure types		Probability	Percentage	Decomposed alpha factors
Layout 1	0/3	5.10E−4	Screened out	–
	1/3	3.24E−5	20.00%	α^C11 = 0.2
	2/3	3.24E−5	20.00%	α^C12 = 0.2
	3/3	9.72E−5	60.00%	α^C13 = 0.6
Layout 2	0/3	5.42E−4	Screened out	–
	1/3	3.36E−5	25.89%	α^C11 = 0.2589
	2/3	8.76E−5	67.63%	α^C12 = 0.6763
	3/3	8.40E−6	6.48%	α^C13 = 0.0648

Figure 15. The update of global alpha factors.