Risk based decision analysis Uncertainty modeling
John Dalsgaard Sørensen
Aalborg University, Denmark
Content:
• Risk analysis
• Decision theory
• Uncertainty modeling
• Example
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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Risk-based decision process
Def. of connections
and criteria
Risk analysis support
decisions under uncertainty
Define system
Damage scenarios
are identified
What can go wrong
and why?
The theoretical basis for risk
analysis is a decisionanalysis
Analysis of
consequences
The main task is to optimize
costs and risks
Analysis of
probabilities
Identify
risk scenarios
Analysis of
sensitivities
Risk
assessment
Monitoring
Control
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
Risk
treatment
2
Risk based decision process
Analysis of probabilities
Analysis of failure probabilities can
be based on different information on
the considered systems and
components:
-
Mechanical and electrical
components
observed failure rates
Structural components
probabilistic models for
failure probabilities
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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Power supply
lost
Risk based decision process
+
Engine
fails
Analysis of probabilities
Power
cables
Fail: 0.01
Safe: 0.99
•
Methods to estimate probabilities:
- Failure trees
-
Bayesian nets
Backup
fuel
Fail: 0.01
Safe: 0.99
Reliability methods: FORM/SORM,
simulation
BackUpFuel
-
Main Fuel
Fuel
Engine
Pf =
Fail: 0.01
Safe: 0.99
∫{ f
( x ) dx
X
Ω f = g ( x ) ≤ 0}
Sicher
g( x ,t) = 0
Cables
Logisches System
PowerSuppl
1
FailDayHour
Versagen
1
1
1
N
1
M1
M2
MN-1
MN
ProdLoss
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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Risk based decision process
Mechanical Failure
(Data/Fault trees etc.)
Operational Failure
(Eng. Judgement/
data/experience)
Structural Failure
(SRA)
Environmental Risk
Fatality Risk
Economic Risk
Decision
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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Uncertainty modeling
Types of uncertainty:
• Inherent natural variability
• Model uncertainty
• Statistical uncertainty
- Aleatory uncertainty
- Epistemic uncertainty
Treated within framework of Bayesian probability theory and
Bayesian decision theory
JCSS (Joint Committee on Structural Safety):
• WP1: Probabilistic Model Code
• WP2: Risk analysis
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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Uncertainty modeling
Aleatory uncertainty (inherent natural variability)
• uncertainty that cannot be reduced by means of collection of additional information
• depends on models applied in formulation of problem: Scale & Time
Observed annual
extreme wave height
Model for annual
extremes
Aleatory
uncertainty
Epistemic
uncertainty
Regression model to
predict future extremes
Predicted future
extreme wave height
Aleatoric uncertainty:
Epistemic uncertainty:
annual extreme wave height
stochastic model for annual extreme wave height
statistical uncertainty related to statistical parameters
extrapolation to extreme wave height for larger return periods
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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Decision theory
Optimal decision _ highest expected utility (expected benefits, costs, fatalities, …)
Prior decision analysis:
Optimal decision:
E [U ] = max E X [U (Z, X )]
Z
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
E [Total Cost ] = min E X [C (Z, X )]
Z
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Decision theory
Posterior decision analysis:
Information available: measurements (monitoring, inspection), observations, …
Optimal decision:
E [U ] = max E X'' [U (Z, X )]
E [Total Cost ] = min E X'' [C (Z, X )]
Z
Z
Bayesian updating:
f Q'' (q xˆ ) =
f Q' (q )L(xˆ q )
∫
f (q )L(xˆ q )dq
'
Q
f X'' (x xˆ ) = f XU (x xˆ ) = ∫ f X (x q )f Q'' (q xˆ )dq
P(F ∩ I ) P(I F )P(F )
P(F I ) =
=
P(I )
P(I )
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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Decision theory
Preposterior decision analysis:
Planning of: measurements, observations, …
Optimal decision:
[
]
E [U ] = max ES' E X'' [U (Z, E, S, d(S), X )]
Z , E ,d
[
]
E [Total Cost ] = min ES' E X'' [C (Z, E, S, d(S), X )]
Z , E ,d
d(S):
decision rule
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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Decision problems
Prior decision analysis:
• Risk assessment for verification and design
• Calibration of risk acceptance criteria; partial safety factors in codes
Posterior decision analysis:
• Re-qualification
• Service life extension
Preposterior decision analysis:
• Planning of
– Tests
– Monitoring
– Maintenance
– Inspections, …
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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Decision problems – updating of uncertainty
Decision problem at time t:
Strength:
epistemic uncertainty (realization at _=0)
Extreme load
if extreme load events are conditional independent:
uncertainty associated with future load events cannot be
updated based on observations of past load events
if extreme load events are dependent: part of uncertainty
becomes epistemic after realization of past load events and
this epistemic part of the uncertainty associated with future
load events can be updated.
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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Example – Maximum Likelihood Method
- Extreme significant wave heights
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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Example – Maximum Likelihood Method
- Extreme significant wave heights
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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Example – Maximum Likelihood Method
- Extreme significant wave heights
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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Summary
•
•
Risk analysis
Decision theory
•
•
Uncertainty modeling
– Aleatory uncertainty
– Epistemic uncertainty
Uncertainties change type with time and scale
Only epistemic uncertainty can be updated
•
Statistical uncertainty: Maximum Likelihood Method
IEA Annex XXIII Workshop, Risø, December 12-13, 2005
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