Sensitivity and Uncertainty Analysis of LargeScale Systems

1
Sensitivity and Uncertainty Analysis of
Large-Scale Systems

• Dan G. Cacuci
• Commissariat a lÉnergie Atomique, France
  University of Karlsruhe, Germany
• ACE Workshop, NCSU, Raleigh, NC, May 31 June
  1, 2006

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OUTLINE

• Sensitivity and Uncertainty Analysis of Models
  and Data Basic Concepts
• Paradigm Applications of ASAP to Large-Scale
  Systems
• QUENCH (Reactor Safety RELAP5/MOD3.3)
• IFMIF (Reliability Markov Chains)
• Global Adjoint Sensitivity Analysis Procedure
  (GASAP) for Nonlinear Systems
• On-Going EURATOM Projects
• NURESIM (NUclear REactor SIMulation)
• SNF-TP (Sustainable Nuclear Fission Technology
  Platform)
• Open Problems (Grand Challenges ?) For Audience
  Discussion

3
Sources of Uncertainties

• 1. stochastic uncertainty
• Arises because the system under investigation can
  behave in many different ways
• 2. subjective (epistemic) uncertainty
• Arises from the inability to specify an exact
  value for a parameter that is assumed to have a
  constant value in the respective investigation.
• Epistemic (subjective) uncertainties characterize
  a degree of belief regarding the location of the
  appropriate value of each parameter.
• In turn, these subjective uncertainties lead to
  subjective uncertainties for the response, thus
  reflecting a corresponding degree of belief
  regarding the location of the appropriate
  response values as the outcome of analyzing the
  model under consideration.

4
Example PSA of Nuclear Power Plants Involve
Both Stochastic and Epistemic Uncertainties

• Stochastic uncertainties arise due to the
  hypothetical accident scenarios which are
  considered in the respective risk analysis
• Epistemic uncertainties arise because of
  uncertain parameters that underlie the estimation
  of the probabilities and consequences of the
  respective hypothetical accident scenarios

5
SCOPE of Sensitivity Uncertainty Analysis

• Scope of local analysis to analyze the behavior
  of the system response locally around a chosen
  point (for static systems) or chosen trajectory
  (for dynamical systems) in the combined phase
  space of parameters and state variables.
• Scope of global analysis to determine all of the
  system's critical points (bifurcations, turning
  points, response maxima, minima, and/or saddle
  points) in the combined phase space formed by the
  parameters and dependent (state) variables, and
  subsequently analyze these critical points by
  local sensitivity and uncertainty analysis.

6
METHODS Statistical and Deterministic

• Statistical Methods sampling-based methods
  (random sampling, stratified importance sampling,
  and Latin Hypercube sampling), first- and
  second-order reliability algorithms (FORM and
  SORM, respectively), variance-based methods
  (correlation ratio-based methods, the Fourier
  amplitude sensitivity test, and Sobols method),
  and screening design methods (classical
  one-at-a-time experiments, global one-at-a-time
  design methods, systematic fractional replicate
  designs, and sequential bifurcation designs).
• Deterministic Methods brute-force method
  based on recalculations, the direct method
  (including the decoupled direct method), the
  Greens function method, the forward sensitivity
  analysis procedure (FSAP), and the adjoint
  sensitivity analysis procedure (ASAP).

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Important Distinction Statistical vs.
Deterministic Methods

• Statistical uncertainty and sensitivity analysis
  methods first commence with the uncertainty
  analysis stage, and only subsequently proceed to
  the sensitivity analysis stage.
• The above conceptual procedural path is the
  reverse of the path underlying the deterministic
  methods of sensitivity and uncertainty analysis,
  where the sensitivities are determined prior to
  using them for uncertainty analysis

8
Uses of local sensitivities

• Understand the system by highlighting important
  data
• Eliminate unimportant data
• Determine effects of parameter variations on the
  systems behavior
• Design and optimize the system (e.g., maximize
  availability/minimize maintenance)
• Reduce over-design
• Prioritize the improvements to be effected in the
  respective system
• Prioritize introduction of data uncertainties
• Perform local uncertainty analysis by using the
  method of propagation of errors (also known as
  the propagation of moments, or the
  Taylor-Series). Note the propagation of
  errors method is used both for processing
  experimental data obtained from indirect
  measurements and also for performing uncertainty
  analysis of computational models.

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Measurements Basic Concepts and Terminology (1)

• A measurement is the process of finding the value
  of a physical quantity experimentally with the
  help of special devices called measuring
  instruments.
• The result of a measurement is a numerical value,
  together with a corresponding unit, for a
  physical quantity. Note that a measurement has
  three features
• The result of a measurement must always be a
  number expressed in sanctioned units of
  measurements. The purpose of a measurement is to
  represent a property of an object by a number.
• A measurement is always performed with the help
  of some measuring instrument measurement is
  impossible without measuring instruments.
• A measurement is always an experimental procedure.

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Measurements Basic Concepts and Terminology (2)

• The true value of a measurable quantity is the
  value of the measured physical quantity, which,
  if it were known, would ideally reflect, both
  qualitatively and quantitatively, the
  corresponding property of the object.
• The theory of measurement relies on the following
  three postulates
• The true value of the measurable quantity exists
• The true value of the measurable quantity is
  constant (relative to the conditions of the
  measurement)
• The true value cannot be found.
• Since measuring instruments are imperfect, and
  since every measurement is an experimental
  procedure, the results of measurements cannot be
  absolutely accurate.

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Classification of Measurement Errors

• Methodological errors are caused by
  unavoidable discrepancies between the actual
  quantity to be measured and its model used in the
  measurement.
• Instrumental measurement errors are
  caused by imperfections of measuring instruments.
  
• Personal errors are caused by the
  individual characteristics of the person
  performing the measurement.
• The general form for the absolute measurement
  error is
• Since the true value of a measurable quantity is
  always unknown, the errors of measurements must
  be estimated theoretically, by computations,
  using a variety of methods, each with its own
  degree of accuracy.

12
Direct and Indirect Measurements

• Direct measurement the quantity to be measured
  interacts directly with the measuring instrument,
  and the value of the measured quantity is read
  directly from the instruments indications.
• Indirect measurement the value of the unknown
  quantity is calculated by using matched
  measurements of other quantities, called measured
  arguments or, briefly, arguments, which are
  related through a known relation to the measured
  quantity.
• For example, the density of a homogeneous solid
  body is inferred from an indirect measurement, in
  three steps (i) measuring directly the bodys
  mass (ii) measuring directly the bodys volume
  (iii) taking the ratio of the measurements
  obtained in the steps (i) and (ii).

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The Measurement Equation

• In an indirect measurement, the true but unknown
  value of the measured quantity or response,
  denoted by R, is related to the true but unknown
  values of the arguments
  
  
• by a known relationship (i.e., function) f.
• This relationship is called the measurement
  equation, and can be generally represented in the
  form
• NOTE the measurement equation can be
  interpreted to represent not only results of
  indirect measurements but also results of
  computations.

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Propagation of Errors (1)

• In practice, the true values
  are not known they are considered to be random
  variables distributed according to a joint
  probability density function
• with expectation values
  and covariances
• The measurement equation becomes

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Propagation of Errors (2)

• Expanding R in a Taylor series gives
• For uncorrelated parameters

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Propagation of Errors (3)

• Variance (uncorrelated parameters)
• When only linear terms are retained, with
  correlated parameters

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Consistent Combination of Computational and
Experimental Information Data Adjustment/Assimila
tion

• Computed responses
• Parameters , with
  covariances
• To first order
• Hence, covariance for computed response is
• Measured responses ,
  with
• Response deviations
• Response-parameters covariances

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Data Adjustment/Assimilation (2)

• Define adjusted parameters
• adjusted responses
• To first order
• Bayes Theorem yields
• adjusted parameters
• adjusted responses
• adjusted (reduced) parameter covariances
• adjusted (reduced) response covariances
• Consistency check test

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Adjoint Sensitivity/Uncertainty Analysis
Procedure (ASAP)

• Fundamental Goal of ASAP use Adjoint Operators
• to compute deterministically the response
  sensitivities
• exactly and efficiently.
• ASAP circumvents the need to perform repeatedly
  the expensive Forward Sensitivity calculations.

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Sensitivity Analysis Implementation Conceptual
Flow Chart
Mathematical Model Select Base-Case
Problem Select Responses
Discretized FSM
Differential FSM
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Differential FSM
Discretized FSM
Differential Adjoint Sens. Model (ASM)
Discretized Adjoint Sens. Model (DASM)
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Test Bundle in the QUENCH Facility
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QUENCH Fuel Rod Bundle RELAP Model
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THE RELAP5/MOD3.2 Two-Fluid Model (REL/CDE)

• RELAP5/MOD3.2 simulates a wide variety of
  hydraulic and thermal transients in nuclear and
  non-nuclear systems, concentrating on simulation
  of design basis accidents in LWRs.

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Sensitivity Analysis Implementation

• For nominal values Go, solve the RELAP5
  original system to obtain
• the base-case values co and Ro(co , Go)

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RELAP5/MOD3.2 Discretized Model

• Staggered spatial mesh RELAP volumes and
  junctions
• Time-discretization semi-implicit or nearly
  implicit

• 13 coupled nonlinear difference equations

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Adjoint Sensitivity Model for RELAP5/MOD3.2

• The (Discrete) Adjoint Sensitivity Model for the
  Two-Fluid Equations (ASM-REL/TF)

• Note The Adjoint System does not depend on G, so
  it must be solved only once for each R.
• The sensitivity DR becomes

Once F(n) is obtained from the Adjoint System,
DR can be calculated most efficiently for any G.
Thus, ASAP should be used in practice for
large-scale systems with many parameters
variations.
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Heat Structure Model in RELAP5/MOD3.2

• 1 heat conduction equation for each heat
  structure

• Forward Sensitivity System (FSS) for the heat
  conduction equations

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• Implement Adjoint Sensitivity Analysis Procedure
  (ASAP)
• Obtain the Adjoint Sensitivity Model
  (ASM-REL/TFH)
• where

• Note The Adjoint System does not depend on
  parameter variations G, so it must be solved only
  once for each response R.
• The sensitivity DR is

Once FA(n) is obtained from the Adjoint System,
DR can be calculated most efficiently for any G.
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RELSEN Input Processing
RELAP ADJSEN

• Schematic representation of the ASAP
  implementation in RELAP

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• Quench-04 Sequence of Events

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Time-evolution of the relative sensitivities of
the inner ring heated rod at 1.3m to the most
important parameters
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Nomenclature

• ?1 Nominal power factor 0.7
• ?2 Nominal power from 0 to 121s 4279W
• ?3 Nominal power up to 2088.6s 16350W
• ?4 Nominal power up to 2103s 3874W
• ?6 Nominal internal source multiplier (axial
  peaking factor) for power of heat structure of
  the heated rod at 0.7m 0.05255. This value is
  multiplied by the power to obtain the total power
  generated in this heat structure
• ?8 Zircaloy, nominal volumetric heat capacity at
  640K 2168KJ/m³K
• ?11 ZrO2 Pellets, nominal volumetric heat
  capacity at 700K 3510KJ/m³K
• ?23 Nominal surface area of volume 0.003007m²,
  volume centre at 1.2m of the pipe height

34
IFMIF Plant
Accelerator D, 32-40MeV, 125mA x 2
Target Liquid Li Jet, Beam Foot Print 5 x 20
cm2 Test Cell 0.5 L (20-50dpa/fpy), 6 L (1-20
dpa/fpy)
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Accelerator System
RF Tube Cavity Circulator
Crowbar RF Driver Rectifier
Switchgear Transformer Rectifier
50m
Li-Target
5m
100 keV Injector
5-8 MeV RFQ
40 MeV DTL
46 m
High Energy Beam Transport

• Duty Factor CW
• Availability gt 88
• Maintainability hands-on
• Design Lifetime gt20 years

Ion Source Filament or ECR Type RF Tube and
Windows Low Beam Divergence
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D-Li Stripping Neutron Source
Typical Reactions 7Li(d,2n)7Be
6Li(d,n)7Be 6Li(n,T)4HeDeuterons 32,
36, 40 MeV 2x 125 mA Beam footprint 5x20
cm2
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Overall IFMIF System Availability Requirements

• Availability goal 70 of calendar time
• 365 days x 24 hrs x 0.70 8760 hrs x 0.70 6132
  hrs
• Scheduled maintenance 1160 hrs
• 1 mo shutdown 31x 24 744 hrs
• 8 hr maint./wk 52 x 8 416 hrs
• Scheduled operating time 7600 hrs
• Required inherent availability 6132/7600
  0.8068
• (This number is a budget, which assumes that on
  the average, over its design life, the machine
  may be down 1468 hours per year in unscheduled
  repairs due to randomly occurring failures that
  cannot be predicted deterministically neither
  from monitoring the wear and tear nor in any
  other way - for these failure modes only a
  statistical estimate is possible. Also, every
  repair leaves the system in exactly the same
  state as it was before the failure, i.e. no
  better or worse)

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Reliability/Availability/Maintainability/Inspectab
ility RAMI
RAMI
LIFE-CYCLE MANAGEMENT
PRA
Consequences Hazards Releases
Radiological Chemical Exposure Security L
osses Production Budget
Schedule Quality
Event Trees
Fault Trees
Frequencies Initiators Preventors
Mitigators
Specifications
RELIABILITY I/MTBF
AVAILABILITY I-MTTR/MTBF
MAINTAINABILITY I-MTTM/MTBM
INSPECTABILITY I-MTTI/MTBI
TECHNOLOGICAL DEVELOPMENT
EQUIPMENT
OPERATIONS/ COMPLIANCE
41
Illustrative Example Risk analysis process for a
plant, showing relationship of pinch points,
frequency vectors, event trees and transition
matrices.
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Schematic of an Event Tree (Horizontal), shown
with Fault Trees (Vertical) used to evaluate
probabilities of different events
43

• The parameters xi, i1,,m, that enter a RAMI
  model are considered random variables distributed
  according to given probability distribution
  function (PDFs)
• Hence, the Reliability Availability of a system
  will itself be a random variable (since it is a
  function of random variables).

44
RAMI assists designers towards optimum system
design by

• Establishing reliability and maintainability
  requirements at the subsystem and component
  levels,
• Identifying system sensitivities to RAMI
  uncertainties,
• Influencing the level of design redundancy,
• Estimating the contribution of maintenance to the
  life cycle cost (spares and replacements),
• Identifying the areas for potential technology
  development.

45
RAMI Modeling Steps

• a. Top Down Analysis
• Identify the major subsystems
• Break up each major subsystem in its constitutive
  assemblies
• Break up each assembly in sub-assemblies
• Break up each sub-assembly in its components

• b. Bottom Up Synthesis
• Determine RAMI values for individual components
• Calculate RAMI values for the sub-assembly from
  its components
• Continue RAMI calculations at successively higher
  order structures, by considering the structures
  at each level as the components of a higher level
  structure
• Obtain the TOP-level RAMI values for the highest
  level system

NOTE At each level, Markov-type models are
usually used to calculate Dynamic RAMI values for
the respective structures.
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The First Level of the Fault-Tree for the
Accelerator System
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The Markov Chain for the First Level of
Accelerator System
50
The Transition Rate Matrix Structure for the
First Level of Accelerator System
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Calculation of the derivatives (sensitivities)

• The system under consideration (including event
  and fault trees,
• Markov-Models, constraints, correlations) can be
  represented as
• a single system of K equations of the form
• subject to boundary and/or initial conditions
  represented as
• where
• System response (reliability, failure
  probability, etc.)

52

• First-order derivatives (sensitivities) of Rsys
  with respect to xi are
• The functions are obtained by
  solving m-times the following differentiated
  Forward Sensitivity System (of K-equations)
• subject to boundary and/or initial conditions
• NOTE the above Forward Sensitivity System must
  be solved anew for each of the m system
  parameters. This is computationally very
  expensive!

53
Conceptual Mathematical Procedure underlying ASAP

• (i) Rewrite the sensitivities
  in inner-product form as
• where
• (ii) Construct the adjoint of the Forward
  Sensitivity System by forming the inner product
  of the adjoint function
  with Eq. (4) to obtain

54

• (iii) Use the mathematical definition of the
  adjoint operator, namely
• (iv) Obtain the Adjoint Sensitivity Equations
  for ? by setting
• with ? subject to adjoint boundary and/or
  initial conditions
• (v) Obtain, finally, the sensitivities
  in terms of the adjoint function ? as

55
Transient Availability of the Accelerator System
and its Main Subsystems
56
Conceptual Framework for Global Optimization and
Sensitivity Analysis

• Mathematical Model
• m linear and/or nonlinear equations
• parameters
• dependent variables (pressures, temperatures,
  etc.)
• inequality and/or equality constraints for
  parameters
• responses to be optimized

57
Objectives of Global Optimization and Sensitivity
Analysis

• Find all critical (bifurcation, limit, turning)
  points underlying the nonlinear model
• Find all critical (maxima, minima, saddle points)
  points of responses
• Perform local sensitivity and uncertainty
  analysis around selected design and critical
  points

58
Global Aims Cannot be Attained by Concepts Based
on Taylor-Series !

• A functional Taylor-Series
• is valid only within its convergence radius!
• Even the second-order terms are impractical to
  compute (the corresponding adjoints would
  depend on the perturbations !)

59
Global Optimization and Sensitivity Analysis
(Cacuci, 1990) Homotopy Path Computed by
Pseudo-Arc-Length Continuation

• Homotopywhere
• 
  
• 
  
• 
  
• 
  are the adjoint
  functions.

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Pseudo-arc-length continuation

• Impose
• which implies
• Thus s becomes the arclength parameter on the
  path in the inflated space
• Note determines the critical
  points of P
• determines the
  bifurcation and turning points
• Computations along the homotopy path are
  performed efficiently using a combined
  Newton-Secant (with regula falsi) method, which
  determines all critical points, globally, with
  probability one.

61
EURATOM Nuclear Reactor Simulations (NURESIM)
Software Platform

• EU Integrated Project 36 months (2005 - 2007
  7 630 500)
• Objective to provide a European Standard
  Software Platform for modeling, recording, and
  recovering computer data for the next-generation
  nuclear reactors simulations
• 18 Partners CEA EDF (F), F. Z. Rossendorf
  GRS Uni-Karlsruhe (D), PSI (CH), ASCOMP (CH),
  TU-Delft (N), KTH (S), Uni-Pisa (I), U.P. Madrid
  (S), Uni-Louvain (B), JSI (Slovenia), VTT LUT
  (Finland), INRNE (Bulgaria), NRI (Czech Rep.),
  KFKI (Hungary)
• Coordinator CEA
• 5 Sub-Projects Core Physics, Thermal-Hydraulics,
  Multi-Physics, Sensitivity Uncertainty
  Analysis, Integration
• Common Software Platform SALOME (CEA)

62
NURESIM Organization
63
The NURESIM Software Platform Architecture
Coupling Solvers through SALOME
64
DESCARTES Architecture
65
KALIF Platform for Sensitivity Uncertainty
Analysis
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EURATOM Sustainable Nuclear Fission Technology
Platform (SNF-TP)

• Coordination Action 24 months
• 22 Partners CEA, CNRS, JRC, PSI, SCK/CEN, FZR,
  FZK, KFKI, NRI, JSI, NRG, CIEMAT, ENEA, U.P.
  Madrid, Uni-Karlsruhe, U-Rome, EDF, AREVA,
  ANSALDO, VTT, Vattenfall, NEXIA
• Coordinator CEA

68
Sustainable Nuclear Fission Technology Platform
(SNF-TP)

• Objectives
• Establish a sustainable, closed fuel cycle for
  electricity production using innovative
  (Generation IV) fast neutron reactor systems in
  conjunction with partitioning and transmutation
  (PT) technologies
• Establish a commercially viable Very High
  Temperature Reactor (VHTR) for process heat and
  hydrogen production
• Improve the performance of currently operating
  (Generation II) and future near-term (Generation
  III) LWRs while maintaining a high degree of
  safety, assessment of novel designs (e.g., SCWR),
  and establishing a unified approach of LWR life
  time extension
• Assure adequate training to preserve and enhance
  the human competence in the nuclear field
  maintain renew the infrastructure necessary for
  achieving sustainability of nuclear energy
  cooperate with other EU-Projects, especially the
  hydrogen platform, geological waste disposal, and
  fusion materials activities.

69
Sustainable Nuclear Fission Technology Platform
(SNF-TP)
LWR (current Gen-3) Competitiveness and Safety
Optimization
VHTR Process Heat, Electricity H2
Materials Fuel Development
Reactor Design Safety
Training and RD Infrastructures
SRA Platform Deployment
System Integration (Economy, non proliferation
)
Fast Neutron Systems Closed Fuel Cycle, PT
Critical Reactors ADS
Geological Disposal Technologies, design, safety
assessment
70
Sustainable Nuclear Fission Technology Platform
(SNF-TP)
71
Road Map for Establishing a Technology Platform
GoP
Vision 2020
Report of the Group of Personalities
CASNF-TP
SRA
The Strategic Research Agenda - Revision every 2
years -
Stakeholders
Research Programs
Public (EU, National, Euro-control,
etc.) and Private (Industry)
Research Projects