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Advanced Course on Key Elements of Risk Quantification and PSA

**Next Event: NINE Headquarters, LUCCA (Italy),****22-26 September 2025****Early Registration 25 July 2025**

**Day 1**

• Boolean Algebra, Probability Theory, Random Variables

o Basic rules and properties of Boolean algebra. Laws of absorption and idempotency

and why are they important for PSA. De Morgan’s theorem.

Definition of probability. Probability of logical sum of mutually exclusive events.

Probability of logical sum of any events. Probabilities of logical product and logical

sum of independent events. Conditional probability and event sequences.

(Why are these things important in PSA or quantitative risk assessment in general?)

Random variables, distinctive and continuous. Probability distributions. Cumulative

functions and density functions. How we determine or “judge out” a probability distribution.

• Definition of “Risk” for an Engineer and Its Role in Considerations of Safety

o “Risk curve” in the probability-consequence space. Frequency (probability)

of exceedance versus frequency (probability) of occurrence. Theoretical definition of risk.

(Why is risk in the PSA guidebooks presented by “complementary cumulative distribution

function”?) Presentation of “risk curve” in practice: risk matrix. Simplification by means of

representatives or substitutes for consequences (e.g. “reactor core damage”, “large (early)

releases"…). Margin assessment and risk assessment. What is the real meaning of

“combined" use of deterministic and probabilistic safety analyses”?

**Day 2**

• Definition of “Risk” for an Engineer and Its Role in Considerations of Safety (continued)

o Risk modeling: logic risk equation and quantitative (numeric) risk equation. Methods and

approaches. FMEA, HAZOP and similar techniques for risk evaluation. Event trees (ETs)

and fault trees as tools to develop the risk equation. Elements of ETs and FTs. “Basic events”.

Structure function.

• Basics of Reliability Engineering As Used in PSA

o Basic differential reliability equations. “Repair-to-failure” cycle. “Failure-to-repair” cycle.

Combined cycle. Basic terms: failure (repair) density function, failure (repair) rate,

and others. (What does it really mean “reliability” and what really is “availability”?)

Types of equipment (“components”) modeled in PSA. Repairable versus non-repairable.

Standby versus normally operating. Failure-on demand versus time-related failure.

Reliability models for “basic events” representing failures of systems, structures and

components (SSC) in PSA model. (Failure during mission time. Standby, periodically tested

component. Frequency-type basic events associated with initiators.) Parameters required

to quantify the reliability models (basic events) in PSA.

**Day 3**

• Common Cause Failure (CCF) Modeling in PSA

o Independent versus dependent failure events. Unconditional versus conditional failure

probability. Importance of CCF modeling in PSA model fault trees. CCF group. Basic CCF

failure probability model. Beta Factor Model. Multiple Greek Letters (MGL) Model. Alpha

Factors Model. Estimators for MGL and alpha factors. Availability of CCF data. Mapping

(“specialization”) of CCF events from source to target systems or plants. MGL vs. Alpha

Factors: which one is “better” to use in a PSA?

• Data Analysis in PSA

o Point estimate. Likelihood function. Maximum likelihood estimator (MLE). Estimate with

quantitative characterization of uncertainty. Two basic concepts: confidence intervals and

uncertainty distributions. Bayesian inference based on prior (“generic”) knowledge and

observed evidence (“plant-specific”. Basic terms: Prior distribution. Likelihood function.

Posterior distribution. Bayesian inference as applied to the two most basic reliability

parameters in a PSA: probability of failure on demand and failure rate (Prior distributions

usually found in the “generic” databases; Establishing of likelihood functions; Posterior

distributions: numerical integration or analytical calculation.) Estimating or assessing

other parameters for PSA model: Initiator frequencies. Unavailability due to test or

maintenance (TM). Others. How to estimate failure rate or failure probability in the case

of zero failures? How to establish uncertainty distribution for other types of parameters?

**Day 4**

• Human Reliability Analysis (HRA)

o Main types of human errors: pre-initiator errors, initiator-inducing errors and post-initiator

errors. Identification of human interfaces, characterization of human failure events (HFE),

definition of success criteria and other relevant shaping factors. Overview of techniques

for the assessment of human error probabilities (HEP) which are nowadays used in PSAs,

including: THERP, ASEP, SLIM, ATHEANA, SPAR-H and others. Dependency among human

failure events. Unconditional and conditional HEPs. Incorporation of HFEs into the PSA model.

Screening. Quantification.

• Risk Quantification

o “Risk” as a frequency (probability, likelihood) of the top event considered by the PSA.

What is a “minimal cutset”? PSA-model integration principles: “master FT” versus

combining system-level MCSs. Quantification of top-event based on the generated “list” of

MCSs. (“Inclusion-exclusion principle” and its approximations. What is “Mincut Upper Bound”

(MCUB) approximation?) Issues with “success events”. Propagation of parameter uncertainty.

Moment propagation. “Monte Carlo” sampling. Uncertainty distribution for top event

probability (frequency). Mean of the distribution versus point estimate.

**Day 5**

• Special Topics 1: Probability of Phenomena and Other Probabilities Needed for PSA and

Quantitative Risk Assessments

o Concept of “Accident Progression Event Tree” (APET) or event tree for phenomena

and how it compares to event trees for safety functions.

Structural failure probability. Consideration of stress versus strength (load versus capacity).

Convolution of stress and strength distributions. Other similar concepts. Time required

versus time needed. Probability of recovering power or other supporting functions or

systems (“recovery probability”). Containment fragility. Establishing probability distribution

based on the results of deterministic analyses or some other concepts.

• Special Topics 2: Quantification of Risk from External Hazards with Focus on Seismic Events

o Characterization of seismic hazard for seismic PSA. Seismic hazard frequency curve.

Seismic fragility curves. Convolution of seismic hazard with seismic fragility. Quantitative

screening of SSCs based on low seismic risk. Seismically-induced failures, random failures

and their combinations. Seismic risk quantification.