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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| References | |
| Download: ML19011A426 (27) | |
Text
Probabilistic Modeling for NPP PRA Lecture 3-1 1
Schedule 2
Course Overview Wednesday 1/16 Thursday 1/17 Friday 1/18 Tuesday 1/22 Wednesday 1/23 Module 1: Introduction 3: Characterizing Uncertainty 5: Basic Events 7: Learning from Operational Events 9: The PRA Frontier 9:00-9:45 L1-1: What is RIDM?
L3-1: Probabilistic modeling for NPP PRA L5-1: Evidence and estimation L7-1: Retrospective PRA L9-1: Challenges for NPP PRA 9:45-10:00 Break Break Break Break Break 10:00-11:00 L1-2: RIDM in the nuclear industry L3-2: Uncertainty and uncertainties L5-2: Human Reliability Analysis (HRA)
L7-2: Notable events and lessons for PRA L9-2: Improved PRA using existing technology 11:00-12:00 W1: Risk-informed thinking W2: Characterizing uncertainties W4: Bayesian estimation W6: Retrospective Analysis L9-3: The frontier: grand challenges and advanced methods 12:00-1:30 Lunch Lunch Lunch Lunch Lunch Module 2: PRA Overview 4: Accident Sequence Modeling 6: Special Technical Topics 8: Applications and Challenges 10: Recap 1:30-2:15 L2-1: NPP PRA and RIDM:
early history L4-1: Initiating events L6-1: Dependent failures L8-1: Risk-informed regulatory applications L10-1: Summary and closing remarks L8-2: PRA and RIDM infrastructure 2:15-2:30 Break Break Break Break 2:30-3:30 L2-2: NPP PRA models and results L4-2: Modeling plant and system response L6-2: Spatial hazards and dependencies L8-3: Risk-informed fire protection Discussion: course feedback 3:30-4:30 L2-3: PRA and RIDM:
point-counterpoint W3: Plant systems modeling L6-3: Other operational modes L8-4: Risk communication Open Discussion L6-4: Level 2/3 PRA:
beyond core damage 4:30-4:45 Break Break Break Break 4:45-5:30 Open Discussion W3: Plant systems modeling (cont.)
W5: External Hazards modeling Open Discussion 5:30-6:00 Open Discussion Open Discussion
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.
4 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.
5 Overview
Sequences 6
LOOP-WR EPS ISO EXT DCL OPR DGR LTC LOOP (Weather-Related)
Emergency Power (EDGs)
Isolation Condenser (IC)
Actions to Extend IC Ops Actions to Shed DC Loads Offsite Power Recovery EDG Recovery Long-Term Cooling 1 hr 1 hr 4 hr 4 hr 8 hr 8 hr 12 hr 12 hr CD CD CD CD CD CD CD CD CD CD CD CD CD 1
2 3
4 5
6 7
8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 Recap and Motivation Risk {si, Ci, pi }
NPP Emergency Power System Example (simplified) 7 EPS Failure Out of Service (Maintenance)
Fail to Run EDG 2 Support Failure Fail to Start EDG 1&2 CCF EDG 2 Fuel Failure EDG 2 Cooling Failure EDG 2 Failure EDG 1 Failure T1 T2 T3 Recap and Motivation
Modeling Random Processes Example: flooding events at Harpers Ferry, WV 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.
8 1880 1900 1920 1940 1960 1980 2000 return period = 12 yr Stochastic Models Ran*dom (adj.):
occurring without definite aim, purpose, or reason
Two Fundamental Stochastic Process Models in NPP PRA
- Poisson
- Used for events occurring over time
- Example PRA uses
- Failures during operation
- Bernoulli (coin flip)
- Used for events occurring on demand
- Example PRA uses
- Failures to start
- Failures to change position 9
Sto*chas*tic (adj.): pertaining to process involving a randomly determined sequence of observations Other stochastic processes of interest to NPP PRA Infant mortality and aging processes (bathtub curve)
Extreme values Gaussian (sums of large numbers of random variables)
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.
The probability of events in an interval depends only on the length of the interval (stationary process)
The probability of an event in an increment Dt is proportional to Dt The probability of more than one event in Dt goes to zero as Dt goes to zero 10 N1 N2 Dt1 Dt2 t
= 1, 2
= = 2 1 Notation
- Random variables are denoted with capital letters N = no. events T = occurrence time
- Specific values are denoted by lower case letters
= 1 D= lD+ D D0 D
D
= 0
> 1 D= D Proportionality constant frequency Stochastic Models
Poisson Process - Distributions
- Poisson probability distribution for number of events in a fixed time interval:
- Probability mass function
- Cumulative distribution function
- Complementary cumulative distribution function 11 Notation pN(nlC) random variable function type value condition
, l = 0, ll = ll
, l 0, ll =
=0
, l =
=0
ll
, l > 0, ll
=
=+1
, l =
=+1
ll Stochastic Models and Distribution Functions
Poisson Process - Distributions
- Poisson probability distribution for number of events in a fixed time interval:
- Mean (aka average, expected value) and variance 12 l, l
=0
, l = l l, l l, l 2 =
=0
l, l 2, l = l Notation E[NlC]
random variable Expectation operator condition Stochastic Models and Distribution Functions
Poisson Process - Distributions
- Exponential probability distribution for event occurrence time:
- Probability density function
- Cumulative distribution function
- Complementary cumulative distribution function 13 l
D0
< + D D
= ll l
l
=
0
l = 1 l l
> l
=
l = l Stochastic Models and Distribution Functions
Poisson Process - Distributions
- Exponential probability distribution for event occurrence time:
- Mean (aka average, expected value) and variance
- Percentiles (value of T for which the cumulative probability equals a specified value). Example (95th percentile):
14 ll
0
l = 1 l
ll
ll 2 =
0
ll 2 l = 1 l2 0.95 =
0 0.95 l = 1 l 0.95 = 1 l 1 0.95 Stochastic Models and Distribution Functions
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.)
15 Stochastic Models
Expected Values - Note on Additivity Consider the joint density function for random variables X and Y:
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.
16
D0,0
< + D< + D D
+
+,,
=
,, +
= +
Distribution Functions
Knowledge Check
- Probability mass/density functions
- Cumulative distribution functions
- Mean (Expected) values
- If l = 10-9/yr, average occurrence time for event = ?
- Mean value of a coin flip?
17
=0
, l = ?
0
l = ?
, l = ?
l = ?
Distribution Functions
Poisson Process - Example (cont.)
18 l = 0.01/yr 0.61 0.30 0.08 0.01 0.37 0.37 0.18 0.06 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 Probability Number of Events t = 50 years (E[N] = 0.5) t = 100 years (E[N] = 1) 0 1
2 3
Stochastic Models
Poisson Process - Example 19 0.E+00 1.E-01 2.E-01 3.E-01 4.E-01 5.E-01 6.E-01 7.E-01 8.E-01 9.E-01 1.E+00 0
100 200 300 400 500 Cumulative Probability Time (year) 0.E+00 2.E-03 4.E-03 6.E-03 8.E-03 1.E-02 1.E-02 0
100 200 300 400 500 Density Function Time (year) 5th percentile 50th percentile 95th percentile Mean 100 200 300 400 500 600 700 800 0
return period 100 years l = 0.01/yr A random sample:
Stochastic Models
Bernoulli Process Independent, identical trials => memoryless coin flip process Binomial probability distribution where Moments 20
, f = lf =
f1 f l, f
=0
, f = f l, f =
=0
l, f 2, f = f 1 f
Stochastic Models and Distribution Functions
Bernoulli Process Example 21 0.605 0.306 0.076 0.012 0.001 0.366 0.370 0.185 0.061 0.015 0.134 0.271 0.272 0.181 0.090 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 Probability Number of Events f = 0.01 0
1 2
3 4
m = 50 trials (E[N] = 0.5) m = 100 trials (E[N] = 1) m = 200 trials (E[N] = 2)
Stochastic Models
Non-Stationary Processes
- Examples
- Extreme weather
- Passive component ageing
- Recovery and repair
- Models
- Parametric
- Multi-parameter (2+ parameters)
- empirical and/or derived
- Simulation 22 h(t) t Stochastic Models
Combining Events - Simple Cases
- OR (U) Gate
- AND () Gate 23 B
A Top B
A Top
= = +
= = l
=
if A and B are independent and if PA and PB are small if A and B are independent Risk concern: situations where l
= +
+
if A and B are mutually exclusive 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
- Probability density function of time to perform overall action Probability calculus can be used to develop distributions for many situations 24 l 1, l 2 =
l 1l 2 l 2 l 1 l 1l 2
= 1 + 2 = 1 l 1
+ 1 l 2 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 25
> =
0
Note: this is an example of a general stress-strength model where failure occurs when the stress exceeds the strength.
Combinations of Random Variables
Time-Reliability Derivation Sketch 26 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.
V U
v v+dv u u+du U > V 2)
The probability of failure is the probability that U and V are in the shaded area.
> =
0
0
3)
If U and V are independent,
,, =
> =
0
Combinations of Random Variables
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 27 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.