For respective Base Case Plant, the LOCA frequencies are determined for three time periods: T= = 25 years after plant startup (corresponding to today’s state-of-knowledge), T = 40 years after plant startup (corresponding to original design life), and T = 60 years after plant startup (corresponding to end-of-life extension). The time-dependent analysis is performed in two different ways. First a ‘prospective analysis’ is performed based on a Markov model of piping reliability (Figure D.15). Second, a ‘retrospective analysis’ is performed by using Bayesian statistics.

D.6.4.1 Use of Markov Model to Determine Time-Dependency — According to the Markov model diagram in Figure D.15, a piping component can be in four mutually exclusive states: S (= Success), C (= Cracked), F (= Leaking, non-active leakage, or active leakage with leak rate within Technical Specification Limit) or L (= Leaking, with leak rate in excess of Technical Specification Limit). The time-dependent probability that a piping component is in each state S, C, F, or L is described by a differential equation. Under the assumption that all the state transition rates are constant the Markov model equations will consist of a set of coupled linear differential equations with constant coefficients. The reliability term needed to represent LOCA frequency is the system failure rate or hazard rate h{t}, which is time-dependent. The hazard rate is defined as:

h{t} = (1/(1- L{t})) x dL{t}/dt (D.9)

Where:

1 — L{t} = S{t} + C{t} + F{t} (D. 10)

The hazard rate is a function of time and the parameters of the Markov model; h{t} is the time-dependent frequency of pipe rupture. Reference [D.14] provides solutions to the Markov model and derives an expression for h{t} as a function of the six parameters associated with the 4-state Markov model: An occurrence rate for detectable flaws (ф), a failure rate for leaks given the existence of a flaw (Д) two rupture frequencies including one from the initial state of a flaw (pF) and one from the initial state of a leak (pL), a repair rate for detectable flaws (ft), and a repair rate for leaks (p). The latter two parameters dealing with repair are further developed by the following simple models.

P p

ft = —FI FD (D.11)

(TFI + TR)

Where:

PFI = probability that a piping element with a flaw will be inspected per inspection interval. This parameter has a value of 0 if it is not in the inspection program and 1 if it is in the inspection program. For the inspected elements, a value of 1 is used for any ISI inspection case and 0 for the case of no ISI. The element may be selected for inspection directly by being included in the sections sampled for ISI inspection, or indirectly by having a rule such that if degradation is detected anywhere in the system, the search will be expanded to include examination of that element.

PFD = probability that a flaw will be detected given this element is inspected. This is the reliability of the inspection program and is equivalent to the term used by NDE experts, “Probability of detection (POD).” This probability is conditioned on the occurrence of one or more detectable flaws in the segment according to the assumptions of the model. Also note that

TFI = mean time between inspections for flaws, (inspection interval).

TR = mean time to repair once detected. Depending on the location of the weld to be repaired, the actual weld repair could take on the order of several days to much more than a week. Accounting for time to prepare for repair, NDE, root cause evaluation, etc., the total outage time attributed to the repair of a Class 1 weld is on the order of 1 month or more. However, since this term is always combined with TFI, and TFI could be 10 years, in practice the results are insensitive to assumptions regarding TR

Similarly, estimates of the repair rate for leaks can be estimated according to:

Where:

PLD = probability that the leak in the element will be detected per leak inspection or detection period

TLI = mean time between inspections for leaks. For RCPB piping the time interval between leaks can be essentially instantaneous if the leak is picked up by radiation alarms, to as long as the time period between leak tests performed on the system.

TR = as defined above but for full power applications, this time should be the minimum of the actual repair time and the time associated with cooldown to enable repair and any waiting time for replacement piping.

A summary of the root input parameters of the Markov model and the general strategy for estimation of each parameter is presented in Table D.18.

Table D.18 Four-State Markov Model Root Input Parameters

Parameter	Assumed or Estimated Value	Basis
CO	2.1 x 10-2/year {=(.25) x (.90)/(10+(200/8760))}	Element assumed to have a 25% chance of being inspected for flaws every 10 years with a 90% detection probability. In the given example detected flaws will be repaired in 200 hours
A	7.92 x 10-1/ year {=(.90) x (.90)/(1+(200/8760))}	Element is assumed to have a 90% chance of being inspected for leaks once a year with a 90% leak detection probability
Pc	Table D.13, D.14 and D.15	The basis is developed in Sections D.4 and D.5.
Лс	Table D.13 and D.14	The basis is developed in Sections D.4 and D.5.
pF	2.0 x 10-2/year	If the element is already leaking, the conditional frequency of ruptures is assumed to be determined by the frequency of severe overloading events; the given value is equal to the frequency of severe water hammer (from PIPExp database).
Ф	Variable (for IGSCC Ф = 7.58 x. (Лс + Pc))	The occurrence rate of a flaw is estimated from service data. As an example, IGSCC in the BWR operating environment will create ca. 7.58 flaws for every through-wall leak that is observed.
PFI	1 or 0	Probability per inspection interval that the pipe element will be included in the inspection program.
PFD	Variable (see text above for details)	Probability per inspection interval that an existing flaw will be detected. A chosen estimate is based on NDE reliability performance demonstration results and difficulty and accessibility of inspection for particular weld.
Pld	Variable (0 — no leak detection to 0.9 for leak detection using current methods/technology)	Probability per detection interval that an existing leak will be detected. Estimate based on system, presence and type of leak detection system, and locations and accessibility.
TFI	10 years (per ASME XI)	Flaw inspection interval, mean time between in­service inspections.
Tld	Variable (1.5 — once per refueling outage / 1.92E-2 — weekly / 9.13E-4 — each shift)	Leak detection interval, mean time between leak detections. Estimate based on method of leak detection; ranges from immediate/ continuous to frequency of routine inspections for leaks (incl. hydrostatic pressure testing).
Tr	Variable (see text above for details)	Mean time to repair the affected piping element given detection of a critical flaw or leak. Estimate of time to tag out, isolate, prepare, repair, leak test and tag into service.

In addition to generating a time-dependent LOCA frequency, the Markov model provides a basis for investigating the sensitivity of LOCA frequency to different in-service inspection and leak detection strategies. The Markov model determines the inspection effectiveness factor, I, which is the ratio of the LOCA frequency with credit for inspections to that given no credit for inspections:

h25{inspprog’ j’}
h25{noinsp}

Where:

h25{inspprog ‘j’} = hazard rate at T = 25 given inspection strategy ‘j.’ h25{noinspj = hazard rate given no inspections.

The solutions to the Markov model for time dependent hazard rates are developed in terms of closed form analytical solutions using an Excel spreadsheet. In this study the time-dependent LOCA frequencies are determined for twelve cases that are defined by varying the following parameters (Table D.19):

• Whether or not the piping segment is subjected to any ISI program;

• The extent of the ISI program (‘Caused-Based’ vs. ‘Extensive’, all encompassing);

• The inspection interval of the ISI program;

• Type and frequency of leak detection. The different leak detection methods include primary system mass balance calculations, visual observation (through video monitor), (PWR) containment sump level and flow rate monitors, airborne particulate radioactivity and gaseous radioactivity monitors, and different main control room monitors for primary system temperature, pressure, etc.

Table D.19 Inspection Cases Evaluated for Selected Pipe Segments

Leak Inspection Strategy	In-Service Inspection Strategy
None	Cause-Based [Pfd = 0.50]	Comprehensive [Pfd = 0.90]
None	Case 1	Case 5	Case 9
Refueling Cycle (Hydro Test)	Case 2	Case 6	Case 10
Weekly	Case 3	Case 7	Case 11
8 Hour Shift	Case 4	Case 8	Case 12

The time-dependent LOCA frequencies associated with the five Base Cases are summarized in Figures D.26 through D.40. Figure D.26 is assumed to be representative of Base Case 1; the LOCA frequency at T = 25 years is equal to the calculated point estimate of 8.24E-06 per reactor-year under an assumption of “caused — based” ISI with POD = 0.5 and leak detection (e. g., hydrostatic pressure testing prior to exiting a refueling outage). This assumption is applied to the other base cases as well (Figures D.29, D.32, D.35, and D.38).

It is noted that the service data input to the calculation is associated with piping that has been subjected to different inspection strategies. In some cases flaws have been detected fortuitously and in other cases the flaw detection has resulted from augmented IGSCC inspection programs. The results in Figure D.26 are based on an assumed ‘cause-based’ inspection strategy whereby the inspection sample is determined by an initial discovery of a flaw. If a flaw is found, the inspection is immediately expanded to cover other similar locations. The combination of inspection sample and rules for expanded search for flaws are sufficient to result in an average probability of detection (POD) of 0.50. The analysis also considers what in this study is

termed “comprehensive” ISI, which implies 100% ISI coverage using state-of-the-art NDE technology. Such a program is assumed to result in an average probability of detection of 0.90.

Figure D.26 Time-Dependent BWR-1 Cat 1 LOCA Frequency Given ‘Cause-Based’ ISI

Figure D.27 Time-Dependent BWR-1 Cat 1 LOCA Frequency Assuming no ISI

Figure D.28 Time-Dependent BWR-1 Cat 1 LOCA Frequency Given ‘Comprehensive ISI’

Figure D.29 Time-Dependent BWR-2 Cat 1 LOCA Frequency Given ‘Cause-Based’ ISI

Figure D.30 Time-Dependent BWR-2 Cat 1 LOCA Frequency Assuming no ISI

Figure D.31 Time-Dependent BWR-2 Cat 1 LOCA Frequency Given ‘Comprehensive ISI’

Figure D.33 Time-Dependent PWR-1 Cat 1 LOCA Frequency Assuming no ISI

Figure D.34 Time-Dependent PWR-1 Cat 1 LOCA Frequency Given ‘Comprehensive ISI’

Figure D.35 Time-Dependent PWR-2 Cat 1 LOCA Frequency Given ‘Cause-Based’ ISI

Figure D.36 Time-Dependent PWR-2 Cat 1 LOCA Frequency Assuming no ISI

Figure D.37 Time-Dependent PWR-2 Cat 1 LOCA Frequency Given ‘Comprehensive ISI’

Figure D.39 Time-Dependent PWR-3 Cat 1 LOCA Frequency Assuming no ISI

Plant Age [Years]