ML19011A426
| ML19011A426 | |
| Person / Time | |
|---|---|
| Issue date: | 01/16/2019 |
| From: | Office of Nuclear Regulatory Research |
| To: | |
| Nathan Siu 415-0744 | |
| Shared Package | |
| ML19011A416 | List:
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| Download: ML19011A426 (27) | |
Text
Probabilistic Modeling for NPP PRA Lecture 3-1 1
Course Overview Schedule Wednesday 1/16 Thursday 1/17 Friday 1/18 Tuesday 1/22 Wednesday 1/23 3: Characterizing 7: Learning from Module 1: Introduction Uncertainty 5: Basic Events Operational Events 9: The PRA Frontier L3-1: Probabilistic L5-1: Evidence and L9-1: Challenges for NPP 9:00-9:45 L1-1: What is RIDM?
modeling for NPP PRA estimation L7-1: Retrospective PRA PRA 9:45-10:00 Break Break Break Break Break L1-2: RIDM in the nuclear L3-2: Uncertainty and L5-2: Human Reliability L7-2: Notable events and L9-2: Improved PRA using 10:00-11:00 industry uncertainties Analysis (HRA) lessons for PRA existing technology L9-3: The frontier: grand W1: Risk-informed W2: Characterizing W6: Retrospective 11:00-12:00 thinking uncertainties W4: Bayesian estimation Analysis challenges and advanced methods 12:00-1:30 Lunch Lunch Lunch Lunch Lunch 4: Accident 6: Special Technical 8: Applications and Module 2: PRA Overview 10: Recap Sequence Modeling Topics Challenges L8-1: Risk-informed L2-1: NPP PRA and RIDM: regulatory applications 1:30-2:15 early history L4-1: Initiating events L6-1: Dependent failures L8-2: PRA and RIDM L10-1: Summary and closing remarks infrastructure 2:15-2:30 Break Break Break Break L2-2: NPP PRA models L4-2: Modeling plant and L6-2: Spatial hazards and L8-3: Risk-informed fire Discussion: course 2:30-3:30 and results system response dependencies protection feedback L6-3: Other operational L2-3: PRA and RIDM: W3: Plant systems modes 3:30-4:30 point-counterpoint modeling L6-4: Level 2/3 PRA:
L8-4: Risk communication Open Discussion beyond core damage 4:30-4:45 Break Break Break Break W3: Plant systems W5: External Hazards 4:45-5:30 modeling (cont.) modeling Open Discussion Open Discussion 5:30-6:00 Open Discussion Open Discussion 2
Overview Key Topics
- Characteristics of basic stochastic models used in NPP PRA: Poisson and Bernoulli processes
- Distribution functions and expected values:
concepts and notation
- Combinations of random variables
- Useful results
- Underlying theory 3
Overview Resources
- A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, 1965.
- R.L. Winkler and W.L Hays, Statistics: Probability, Inference and Decision, Second Edition, Holt, Rinehart and Winston, New York, 1975.
- G. Apostolakis, The concept of probability in safety assessments of technological systems, Science, 250, 1359-1364, 1990.
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Overview Other References
- N.D. Singpurwalla, Reliability and Risk: A Bayesian Perspective, Wiley, Chichester, 2006.
- A.E. Green and A.J. Bourne, Reliability Technology, Wiley-Interscience, London, 1972.
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Recap and Motivation Risk {si , Ci , pi }
Sequences LOOP Emergency Isolation Actions to Actions to Offsite (Weather- Power Condenser Extend Shed Power EDG Long-Term Related) (EDGs) (IC) IC Ops DC Loads Recovery Recovery Cooling LOOP- EPS ISO EXT DCL OPR DGR LTC WR 1 2 CD 3
4 CD 5
12 hr 6 CD 12 hr 7 CD 8
9 CD 10 8 hr 11 CD 8 hr 12 CD 13 14 CD 15 4 hr 16 CD 4 hr 17 CD 18 19 CD 20 1 hr 21 CD 1 hr 22 CD 6
Recap and Motivation NPP Emergency EPS Failure Power System Example (simplified)
EDG 1 EDG 2 Failure Failure T1 EDG 2 Support Failure EDG 1&2 Fail to Fail to Out of CCF Start Run Service (Maintenance)
EDG 2 EDG 2 Fuel Failure Cooling Failure T2 T3 7
Stochastic Models Ran*dom (adj.):
occurring without Modeling Random Processes definite aim, purpose, or reason
- Example: flooding events at Harpers Ferry, WV return period = 12 yr 1880 1900 1920 1940 1960 1980 2000
- For long-term planning purposes, its useful to treat the flood generating process as a random process. (Of course, as a major storm approaches, behavior is much less random.)
- Even for processes involving definite aim/purpose/reason (e.g.,
operator actions), important contextual features can be viewed as random, resulting in an overall random process.
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Stochastic Models Two Fundamental Stochastic Process Models in NPP PRA
- Poisson
- Sto*chas*tic (adj.): pertaining
- Used for events occurring over to process involving a time randomly determined sequence of observations
- Example PRA uses
- Other stochastic processes of
- Initiating events interest to NPP PRA
- Failures during operation Infant mortality and aging processes (bathtub curve)
- Bernoulli (coin flip) Extreme values
- Used for events occurring on Gaussian (sums of large numbers of random variables) demand
- Example PRA uses
- Failures to start
- Failures to change position 9
Stochastic Models Poisson Process - Assumptions
- For non-overlapping time intervals, the (random) number of events in each interval are independent (independent increments).
Example: N1 is independent of N2. Notation N1 N2
- Random variables t
Dt1 Dt2 are denoted with capital letters
- The probability of events in an interval depends only on the length N = no. events of the interval (stationary process)
T = occurrence time
= 1 , 2 = = 2 1
- Specific values are denoted by lower
- The probability of an event in an increment Dt is proportional to Dt case letters D
= 1 D = lD + D =0 D0 D
- The probability of more than one event in Dt goes to zero as Dt goes to zero Proportionality constant frequency
> 1 D = D 10
Stochastic Models and Distribution Functions Notation Poisson Process - Distributions function value type
- Poisson probability distribution for pN(nlC) number of events in a fixed time interval: random condition variable
- Probability mass function l l
, l = 0, ll =
- Cumulative distribution function l l
, l 0, ll = , l =
=0 =0
- Complementary cumulative distribution function l l
, l > 0, ll = , l =
=+1 =+1 11
Stochastic Models and Distribution Functions Notation Poisson Process - Distributions Expectation operator
- Poisson probability distribution for E[NlC]
number of events in a fixed time random condition interval: variable
- Mean (aka average, expected value) and variance l, l , l = l
=0 l, l l, l 2 = l, l 2 , l = l
=0 12
Stochastic Models and Distribution Functions Poisson Process - Distributions
- Exponential probability distribution for event occurrence time:
- Probability density function
< + D l = l l D0 D
- Cumulative distribution function l l = l = 1 l 0
- Complementary cumulative distribution function l > l = l = l 13
Stochastic Models and Distribution Functions Poisson Process - Distributions
- Exponential probability distribution for event occurrence time:
- Mean (aka average, expected value) and variance 1
ll l =
0 l 1
ll ll 2 = ll 2 l =
0 l2
- Percentiles (value of T for which the cumulative probability equals a specified value). Example (95th percentile):
0.95 1 0.95 = l = 1 l 0.95 = 1 0.95 0
l 14
Stochastic Models Poisson Process - Notes
- The model parameter l has units of inverse time and is called frequency. This does not imply regular occurrence.
- The mean value of T (i.e., 1/ l) is often called the return period. Again, this does not imply regularity.
- If lt < 0.1, FT(tll) lt (rare event approximation)
- Poisson process is memoryless - the conditional probability of an event in the interval (t + Dt), given the system state at time t, is independent of past history (i.e., how the system arrived at its current state).
- Characteristic time trace: clusters of events with intervening large gaps. (See earlier flooding example.)
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Distribution Functions Expected Values - Note on Additivity
- Consider the joint density function for random variables X and Y:
< + D < + D D0,0 D
- The expected value of X + Y is the sum of the expected values for X and Y, regardless of the uncertainty in X and Y, and regardless of the dependence between X and Y.
+ + , ,
= , , + , ,
= +
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Distribution Functions Knowledge Check
- Probability mass/density functions
, l = ? l = ?
0
=0
- Cumulative distribution functions
, l = ? l = ?
- Mean (Expected) values
- If l = 10-9/yr, average occurrence time for event = ?
- Mean value of a coin flip?
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Stochastic Models Poisson Process - Example (cont.)
l = 0.01/yr 0.70 0.61 0.60 t = 50 years (E[N] = 0.5) 0.50 t = 100 years (E[N] = 1)
Probability 0.40 0.37 0.37 0.30 0.30 0.18 0.20 0.10 0.08 0.06 0.01 0.00 0 1 2 3 Number of Events 18
Stochastic Models Poisson Process - Example 1.E-02 1.E+00 9.E-01 l = 0.01/yr 1.E-02 8.E-01 Cumulative Probability 5th percentile Density Function 7.E-01 8.E-03 6.E-01 6.E-03 5.E-01 50th percentile 4.E-01 4.E-03 Mean 3.E-01 2.E-01 2.E-03 95th percentile 1.E-01 0.E+00 0.E+00 0 100 200 300 400 500 0 100 200 300 400 500 Time (year) Time (year) return period A random sample:
100 years 0 100 200 300 400 500 600 700 800 19
Stochastic Models and Distribution Functions Bernoulli Process
- Independent, identical trials => memoryless coin flip process
- Binomial probability distribution
, f = lf = f 1f where
- Moments l, f , f = f
=0 l, f = l, f 2
, f = f 1 f
=0 20
Stochastic Models Bernoulli Process Example 0.70 0.605 0.60 f = 0.01 m = 50 trials (E[N] = 0.5) 0.50 m = 100 trials (E[N] = 1)
Probability 0.40 0.366 0.370 m = 200 trials (E[N] = 2) 0.306 0.30 0.271 0.272 0.185 0.181 0.20 0.134 0.076 0.090 0.10 0.061 0.012 0.0010.015 0.00 0 1 2 3 4 Number of Events 21
Stochastic Models Non-Stationary Processes
- Examples
- Extreme weather
- Passive component ageing h(t)
- Recovery and repair
- Models t
- Parametric
- Multi-parameter (2+ parameters)
- empirical and/or derived
- Simulation 22
Combinations of Random Variables Combining Events - Simple Cases
- OR (U) Gate Top
= = +
= + if A and B are mutually exclusive
+ if A and B are independent and if A B PA and PB are small
- AND () Gate Top
= = l
= if A and B are independent A B Risk concern: situations where l 23
Combinations of Random Variables More Complex Situations: Functions of Random Variables
- PRA models can involve combinations of random variables.
Expected values might behave intuitively, but full distributions might not.
- Example: an operator action requires the performance of two tasks in sequence. The time to perform each task is exponentially distributed with rate li (i = 1, 2).
- Mean time to perform overall action 1 1
= 1 + 2 = +
l1 l2
- Probability density function of time to perform overall action l 1l 2 l 1 , l 2 = l 1 l 2 l2l1
- Probability calculus can be used to develop distributions for many situations 24
Combinations of Random Variables Results for Three Situations of Interest
- Event tree sequences. The occurrence of a sequence, which involves a Poisson-distributed initiating event (frequency l) followed by a string of subsequent Bernoulli events (probability fi), is a Poisson process with frequency lf1f2
- Event tree end states. The occurrence of an end state (e.g., core damage) that can be reached by one or more event tree sequences is a Poisson process with frequency equal to the sum of the sequence frequencies.
- Time-reliability. If Ta is the (random) time available to perform required actions, Tn is the (random) time needed to perform these actions, and Ta and Tn are independent, the probability of failure is
> =
0 Note: this is an example of a general stress-strength model where failure occurs when the stress exceeds the strength. 25
Combinations of Random Variables Time-Reliability Derivation Sketch
- 1) For notational simplicity, use U to represent Tn and V to represent Ta. Recall fX (*) is the probability density function for random variable X and FX(*) is the cumulative distribution function; fX,Y(*, *) is the joint density function for X and Y.
- 2) The probability of failure is the probability that U and V are in the shaded area.
v+dv
> = , , v 0 0 V
- 3) If U and V are independent, U>V
, , =
> = u u+du 0
U 26
Closing Comments
- Mathematical details given in this lecture provide
- Basis for parameter estimation procedure (Lecture 5-1)
- Conditions where standard results break down (large l, non-stationary processes)
- Partial basis for confidence in PRA foundation
- Basis for approaches to concerns (e.g., adding mean values when uncertainties are large)
- Additional details are provided in the background slides and are the subject of numerous texts on probability, statistics, stochastic processes, and reliability engineering Probability math is essential the consistent application of logic throughout the analysis (If X then Y, given C), but is far from the entirety of NPP PRA.
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