The $\backslash (\backslash$mathrm$\{$I$\}^\{2\}\backslash$mathrm$\{$~S$\}\backslash )-$LWR design considered in P$.$S$.\backslash$#$5,$ Prob. $4,$ consists of four trains of the DHRS shared by one accumulator $($ACC$).$ Thermal hydraulic analysis indicates that operation of three DHRS trains will be required to remove decay heat after reactor shutdown without over$-$pressurizing the reactor pressure vessel $($RPV$).$ Thus, to allow adequate decay heat removal and avoid RPV over$-$pressurization, even with only one or two DHRS trains operating, we propose to install an automatic depressurization system $($ADS$)$ with two valves in parallel although only one valve is illustrated in P$.$S$.\backslash$#$5.$ If two DHRS trains function properly, opening only one ADS valve suffices to relieve primary coolant pressure and avoid core damage $($CD$),$ while opening both ADS valves is required to avoid $\backslash ($ C D $\backslash )$ if only one DHRS train functions.

$($a$)(20$ points$)$ Construct an event tree representing the actuation of two specific DHRS trains $($A and B $)$ and ADS following a postulated LOCA, with the assumption that reactor scram occurs properly following the LOCA. Use the same failure probabilities and hazard rate for

the three valves in the DHRS secondary loop and ACC as in P$.$S$.\backslash$#$5,$ while the failure probability for each of the two ADS valves to open when required is estimated as $0\mathrm{.}001.$ Given an estimated LOCA frequency of $0\mathrm{.}02/$reactor$-$year and with the requirement for passive decay heat removal for at least $72$ hours as in AP$1000$ plants, obtain the CD frequency for the LOCA scenario.

$($b$)(20$ points$)$ Determine minimal cut sets resulting in CD as the end states and obtain the frequency for each of the CD end states.