Branch-and-Bound algorithm applied to uncertainty quantification of a Boiling Water Reactor Station Blackout

doi:10.1016/j.nucengdes.2015.07.029

Nuclear Engineering and Design

Volume 295, 15 December 2015, Pages 283-304

https://doi.org/10.1016/j.nucengdes.2015.07.029 Get rights and content

Highlights

•
Dynamic Event Tree solutions have been optimized using the Branch-and-Bound algorithm.
•
A 60% efficiency in optimization has been achieved.
•
Modeling uncertainty within a risk-informed framework is evaluated.

Abstract

Evaluation of the impacts of uncertainty and sensitivity in modeling presents a significant set of challenges in particular to high fidelity modeling. Computational costs and validation of models creates a need for cost effective decision making with regards to experiment design. Experiments designed to validate computation models can be used to reduce uncertainty in the physical model. In some cases, large uncertainty in a particular aspect of the model may or may not have a large impact on the final results. For example, modeling of a relief valve may result in large uncertainty, however, the actual effects on final peak clad temperature in a reactor transient may be small and the large uncertainty with respect to valve modeling may be considered acceptable. Additionally, the ability to determine the adequacy of a model and the validation supporting it should be considered within a risk informed framework. Low fidelity modeling with large uncertainty may be considered adequate if the uncertainty is considered acceptable with respect to risk. In other words, models that are used to evaluate the probability of failure should be evaluated more rigorously with the intent of increasing safety margin. Probabilistic risk assessment (PRA) techniques have traditionally been used to identify accident conditions and transients. Traditional classical event tree methods utilize analysts’ knowledge and experience to identify the important timing of events in coordination with thermal-hydraulic modeling. These methods lack the capability to evaluate complex dynamic systems. In these systems, time and energy scales associated with transient events may vary as a function of transition times and energies to arrive at a different physical state. Dynamic PRA (DPRA) methods provide a more rigorous analysis of complex dynamic systems. Unfortunately DPRA methods introduce issues associated with combinatorial explosion of states. This paper presents a methodology to address combinatorial explosion using a Branch-and-Bound algorithm applied to Dynamic Event Trees (DET), which utilize LENDIT (L – Length, E – Energy, N – Number, D – Distribution, I – Information, and T – Time) as well as a set theory to describe system, state, resource, and response (S2R2) sets to create bounding functions for the DET. The optimization of the DET in identifying high probability failure branches is extended to create a Phenomenological Identification and Ranking Table (PIRT) methodology to evaluate modeling parameters important to safety of those failure branches that have a high probability of failure. The PIRT can then be used as a tool to identify and evaluate the need for experimental validation of models that have the potential to reduce risk. In order to demonstrate this methodology, a Boiling Water Reactor (BWR) Station Blackout (SBO) case study is presented.

Introduction

Uncertainty quantification in modeling of dynamic systems with respect to risk analysis presents a series of unique challenges. Modeling of such systems can be extremely complex and modeling simplifications to improve computational efficiency can often increase uncertainty into the model. In addition, the dynamic nature of a nuclear power plant and the changes in the state space of the plant and the timing of those changes adds additional complexity. Dynamic Probabilistic Risk Assessment (DPRA) allows for modeling of complex dynamic systems as well as allowing for uncertainty quantification with respect to risk. As systems and transients become more complex, DPRA suffers from state or combinatorial explosion. The number of required simulations grows exponentially if one aims to evaluate timing of events in the transient as well as sensitivity and uncertainty analyses of the models themselves. Monte-Carlo methods and Dynamic Event Trees (DET) have historically been used for DPRA. Nielsen et al. (2014) documented a method for optimization of DETs. Using the Branch-and-Bound algorithm (Land and Doig, 1960), the solutions for DET are achieved much more efficiently to evaluate the timing of events within the transient. The optimal solution within this framework is to find the branches with the highest probability of failure. In the case of nuclear power plants (NPP), this is typically exceeding peak clad temperature of the fuel cladding.

Identification of the highest probability of failure conditions, can lead to quantification of uncertainties that can allow for analysts to determine important aspects of modeling that have the greatest impact on risk. By evaluating these aspects of the model, modeling fidelity and validation experiments can be designed to provide for an increase in safety margin, which can potentially allow plants to operate at a higher power, resulting in an increase in revenue for utility companies.

In this paper, we extend the optimization algorithm for solutions of DETs to include sensitivity and uncertainty analysis. This methodology is applied to commercial BWR and an evaluation of parameters that have previously been evaluated with regards to a Phenomenological Identification and Ranking Table (PIRT) are evaluated (Gertman and Messina, 2012). The resulting analysis yields a PIRT methodology that is based on risk-informed analysis as well as expert-based judgment.

The analysis was performed by using the DET option within the RAVEN framework (Rabiti and Alfonsi, 2012, Alfonsi et al., 2013), with the RELAP5-3D thermal-hydraulics code (RELAP5-3D). The DAKOTA code (Adams et al., 2013), which was coupled with RELAP5-3D, was used to perform the sensitivity analysis and uncertainty quantification (SA/UQ) at each branching point in the DET simulation. Using this information and the Branch-and-Bound algorithm, the DETs were pruned to obtain the highest probability of failure for each credible plant state. The SA/UQ results can then be used to provide analysts with risk informed decision analysis with regards to modeling and validation.

For completeness, the Branch-and-Bound (BB) algorithm is described in the next section. The BB algorithm relies on bounding functions to constrain the unrolling of unexplored nodes in the DET. The selections of bounding functions are important as they define the physical model to be evaluated. As a consequence, LENDIT (L – Length, E – Energy, N – Number, D – Distribution, I – Information, and T – Time) metrics and S2R2 (state, system, resource, and response) sets (Tokuhiro, 2001) were develop to address this need. In Section 4.2 the application of the BB method, LENDIT and the S2R2 sets to the DET framework is presented for the SA/UQ parameter evaluations as described above. This approach was implemented in RAVEN (Rabiti and Alfonsi, 2012, Alfonsi et al., 2013), which is part of the Multi-physics Object-Oriented System Environment (MOOSE) framework (Gaston et al., 2009).

Section snippets

Branch-and-Bound algorithm

The Branch-and-Bound algorithm is a technique that is used to reduce the search space in combinatorial optimization problems. The algorithm utilizes various system constraints to evaluate which branching paths to follow or which branching paths to disregard (prune) (Land and Doig, 1960).

In general, the problem is to minimize (maximize) a function f(x) of ‘state’ variables (x₁,…, x_n) over a region of feasible solutions, S: $\min_{x \in S} f (x)$ where the function f is called the objective function. The set

LENDIT scales

The LENDIT (Nielsen et al., 2014, Tokuhiro, 2001) scales are a collection of relevant physical quantities such as time, energy, and length here applied to nuclear power plant operator-action depends. These scales reflect, for example, the length (L) of the liquid level in the core; the amount of (thermal) energy (E) in the reactor; the number (N) of NPP operators; and approximate anticipated time (T), from core damage, under insufficient cooling of the decay heat. Although all LENDIT scales

Implementation of Dynamic Event Trees in RAVEN

Dynamic Event Tree, as opposed to classical event trees, allows for the incorporation of dynamic plant state transitions at various points in time as dictated by the cumulative distribution functions (Alfonsi et al., 2013, Catalyurek et al., 2010). At pre-determined points in time, typically defined as the inverse of the Cumulative Distribution Function (CDF), the SBO simulation is stopped and a branching condition in the DET is evaluated. The branching condition determines a possible change in

Uncertainty quantification methods

Evaluating the uncertainty within this DET framework was performed by randomly sampling parameters that may have a large impact on success or failure. The intent is to determine which parameters greatly influence the final results with respect to risk. For accident sequences determined in the DET to result in clad failure, a formal PIRT process is used to determine the affects of the parameters that may have large impacts on the simulation.

The PIRT consisted of a method described in Gertman and

BWR description

In order to demonstrate the capabilities of the Branch-and-Bound algorithm for addressing highly complex DET simulations, a BWR SBO model is developed and analyzed. The BWR SBO was modeled using the RELAP5-3D code and is representative of the General Electric Mark I BWR, similar to the Fukushima Nuclear Power Plant. The BWR model consists of 3440 MWt 1152 MWe BWR reactor. The reactor containment vessel consists of a Mark I pressure suppression pool design with drywells light bulb shaped. The

BWR Station Blackout event tree

The BWR SBO involves the following sequence of events:

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LOSP occurs as the result of loss of both switch yard supplies – t = 0 s
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LOSP results in the following actions
- ∘
  Operators successfully scram the reactor
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  Main steam isolation valves automatically close and provide primary containment
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  Emergency diesel generators start to provide emergency AC power
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  Decay heat from the reactor is provided by the Residual Heat Removal System
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  Batteries are assumed to be initially functional
- ∘
  Pressure is maintained by a

Dynamic Event Tree analysis of BWR SBO

In order to optimize the DETs, the LENDIT metrics and S2R2 sets are combined to initialize the constraining functional parameters for the Branch-and-Bound algorithm. The initial LENDIT metrics as described above begins with establishing a sequence of possible scenarios and transitions. The DET Response sets or branching conditions are presented in Table 2. Fig. 14 provides a diagram of the LENDIT metrics for the BWR SBO. The metrics are established from operating procedures as well as the

Results of Dynamic Event Tree with optimization

Evaluating the DET for a BWR SBO, the highest probability of failure occurred with a failure of the DGs at 3.5 min, followed by battery failure at 63 min. Without recovery, cladding damage is reached at 235 min, or 182 min after battery failure. The second case resulting in failure, has DG failure at 11.4 min with battery failure at 195 min. Cladding failure occurs in this simulation at 359 min or 164 min after battery failure. The time-scale for recovery in both conditions is approximately 2–3 h after

Sensitivity and uncertainty analysis

With an optimized DET established and performing a sensitivity and uncertainty analysis on parameters that can be modeled or adjusted in a RELAP5-3D input file, a PIRT analysis can be conducted. In reviewing reference Sakai et al. (2014), a small subset of the parameters is chosen. Using the DAKOTA simulation tool described above, a correlation matrix can be obtained to determine various phenomenological effects associated with the actual model relative to the results of interest (i.e., peak

Conclusions

This paper provides a methodology for optimizing DET and applies that methodology to create a risk informed PIRT based on the dynamics of the system. The PIRT developed using a Dynamic Event Tree method can be used to design experiments and modeling parameters that are important to safety. The capabilities to evaluate the uncertainty in models with regards to a risk informed decision analysis allows for modeling decisions and validation experiments to be designed to address needs to increase

Acknowledgements

The authors would like to thank the Laboratory Directed Research and Development (LDRD) program at the Idaho National Laboratory (INL) [release number 00119 under blanket master contract 00042246] for supporting this work. The INL is operated by the Battelle Energy Alliance (BEA) for the Department of Energy under DOE contract DE-AC07-05ID14517.

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Branch-and-Bound algorithm applied to uncertainty quantification of a Boiling Water Reactor Station Blackout

Highlights

Abstract

Introduction

Section snippets

Branch-and-Bound algorithm

LENDIT scales

Implementation of Dynamic Event Trees in RAVEN

Uncertainty quantification methods

BWR description

BWR Station Blackout event tree

Dynamic Event Tree analysis of BWR SBO

Results of Dynamic Event Tree with optimization

Sensitivity and uncertainty analysis

Conclusions

Acknowledgements

Reliab. Eng. Syst. Saf.

Nucl. Eng. Des.

DAKOTA: A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis, Version 5.3.1

Dynamic event tree analysis through RAVEN

Uncertainty Analysis of RELAP5-3D

An automatic method of solving discrete programming problems

Econometrica