Bayesian analysis for event tree for nuclear power plant

Homework Assignment 5

Problem 1 (50 points)

Consider a task described by the HRA event tree in Figure 1. The task consists of two subtasks. Success of either subtask will lead to the success of the overall task. Failure of both subtasks will lead to the failure of the overall task. The human error probabilities for the two subtasks are denoted by p1 and p2, respectively.

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Figure 1: The HRA event tree for problem 1.

Our prior knowledge of p1 can be described by a prior probability distribution π1(p1): ln p1 ∼ N (lnµ1 = ln 0.01, σ2

1 = 0.6682). Our prior knowledge of p2 can be described by a prior probability distribution π2(p2): ln p2 ∼ N (lnµ2 = ln 0.02, σ2

2 = 0.9782). Both prior distributions are truncated between 0 and 1.

Suppose we have collected the following operator performance data D: x1 = 1, xi = 0, i = 2, . . . , 10. i = 1, . . . , 10 denotes operator i. xi = 0 means that operator i is successful in completing the task (consisting of two sub-tasks), and xi = 1 means that operator i is failed in completing the task. No observation of individual sub-tasks is available.

Using the above information, perform a Bayesian analysis for p1 and p2. After the analysis, plot the histogram for the joint posterior distribution for p1 and p2, as well as the histograms for the marginal posterior distributions for p1 and p2, respectively. Also provide the means and 5th and 95th percentiles of the marginal posterior distributions for p1 and p2.

Based on the posterior distribution of p1 and p2, update the estimate of the human error probability for the overall task described in the HRA event tree in Figure 1. Provide the mean and 5th and 95th percentiles of the updated human error probability for the overall task.

Note that in the posterior distribution p1 and p2 are not independent of each other.

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Problem 2 (50 points)

Consider the diagnosis part of the SPAR-H method in this problem. Suppose we have performed experiments with licensed reactor operators for a specific

task and collected operator performance data from the experiments. We had n = 10 trials of experiments and collected 10 data points. Each data point specifies whether the operator was successful or failed in the corresponding trial. Among the 10 data points, we observed nF = 3 failures.

For this specific task, the performance shaping factors (PSFs) are assigned the following levels. The multipliers corresponding to the assigned levels are shown in parentheses.

• Available time: nominal (1)

• Stress: high (2)

• Complexity: nominal (1)

• Experience/training: nominal (1)

• Procedures: available, but poor (5)

• Ergonomics/HMI: nominal (1)

• Fitness for duty: nominal (1)

• Work processes: nominal (1)

Based on the SPAR-H method, the human error probability for diagnosis, denoted by pd, can be obtained as

pd = 0.01× 1× 2× 1× 1× 5× 1× 1× 1 = 0.1 (1)

In the SPAR-H method, we can consider the multipliers corresponding to different PSF levels as the parameters in the method. Typically, we are confident in multiplier values of 1 for “nominal” PSF levels since that is how a “nominal” level is defined (neither positive nor negative effect on operator reliability). However, we are much less confident in multiplier values for other PSF levels. For example, in this problem we are less confident in the multiplier value of 2 for “high” “stress” and the multiplier value of 5 for “available, but poor” “procedures”.

In this problem, let us consider the multiplier value for “high” “stress” and the multiplier value for “available, but poor” “procedures” as two uncertain parameters in the SPAR-H diagnosis model. Let us denote the two parameters by msh and mpp , respectively. s and h in msh denote “stress” and “high”, respectively. The two p’s in mpp denote “available, but poor” and “procedures”, respectively.

Suppose based on our prior knowledge, we developed the following prior distributions πsh(msh) for msh and πpp(mpp) for mpp : msh ∼ N (µsh = 2, σmh

= 0.61); mpp ∼ N (µpp = 5, σpp = 2.44). msh is truncated between 1 and 5, and mpp is truncated between 1 and 20.

Perform a Bayesian analysis using the information given above. After the analysis, plot the histogram for the joint posterior distribution for msh and mpp , as well as the histograms

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for the marginal posterior distributions for msh and mpp , respectively. Also provide the means and 5th and 95th percentiles of the marginal posterior distributions for msh and mpp .

Based on the joint posterior distribution of msh and mpp , update the estimate of the human error probability for diagnosis in the task considered in this problem. Provide the mean and 5th and 95th percentiles of the updated human error probability for diagnosis in this task.

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