Modeling the Complexity of Critical Infrastructures
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Transcript of Modeling the Complexity of Critical Infrastructures
MODELING THE COMPLEXITY OF CRITICAL
INFRASTRUCTURESEnrico Zio
Chair on Systems Science and the Energy Challenge – Ecole Centrale Paris and Supelec,
European Foundation for New Energy-Electricité de France
Energy Department, Politecnico di Milano, Italy
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Statement 1:Critical Infrastructures are
(Engineered) Complex Systems
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•Network of many interacting components
•Components of heterogeneous type
•Hierarchy of subsystems
•Interactions across multiple scales of space and/or time
Complex Systems
Dependences (uni-directional) and interdependences (bi-directional)
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Critical Infrastructures are Engineered Complex Systems
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Structural complexity :• heterogeneity of components across different technological domains due to increased integration among systems
• dimensionality: large number of nodes highly interconnected also with other systems (dependences and interdependences)
• scale of connectivity demands for increased amount and quality of information to describe the state of the system.
Critical Infrastructures are Engineered Complex Systems:
Structural complexity
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Dynamic complexity :• emergence of system behavior in response to changes in the environmental and operational conditions of parts of the system.
Critical Infrastructures are Engineered Complex Systems:
Dynamic complexity
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Statement 2:To protect Critical Infrastructures, we must
model them to know their behavior
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system logic representation
system mathematical model
system model quantification
uncertainty analysis and quantification
Modeling Engineered Complex Systems
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physical attributes{structure, dynamics, dependencies and interdependencies, …}
operation and management attributes{communication, control, human and organizational factors, logistics…}
performance and safety attributes{reliability, availability, maintainability, risk, vulnerability, …}
economic attributes{life-cycle costs, costs-benefits, market drivers…}
social attributes{supply-demand, active players, …}
environmental attributes{pollution, sustainability, …}
Modeling Engineered Complex Systems
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Systems of Systems
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Power transmission
Railway
Communication
Cyber Dependency, pcr
Physical Dependency
Physical Dependency
Cyber Dependency, pcp
Systems of Systems
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Corollary to statement 2:To protect Critical Infrastructures, we must
model their response to hazards, failures and
threats to analyze their
Reliability/Risk/Vulnerability/Resilience/…
characteristics
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Reliability/Risk/Vulnerability/Resilience/…
analysis
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System analysis:
- hazards and threats identification
- physical and logical structure identification
- dependencies and interdependences identification and modeling
- dynamic analysis (cascading failures)
Quantification of system indicators
Identification of critical elements
Application for system improvements (optimization):
- design
- operation
- protection
W. Kroger and E. Zio, “Vulnerable
Systems”, Springer, 2011
Reliability/Risk/Vulnerability/Resilience/… analysis
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Statement 3:To model the (engineered) complex systems (of
systems) which make our Critical
Infrastructures, there is not one single modeling
approach that “captures it all”
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Modeling
Critical
Infrastructures
Phenomenological LogicalTopological
APPROACHES
System indicators
Critical elements
OUTPUTS
Modeling the complexity of Critical Infrastructures
Flow
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Modeling the complexity of Critical Infrastructures:
The Dual Analysis
Direct Problem
Evaluating Global
Indicators
Detail
• Critical Infrastructures are engineered complex systems: structure + dynamics+
failure/recovery process
Computational cost
Aggregation
Challenge
Inverse Problem
Identifying
Vulnerabilities at
the Components
Level
Disaggregation
Challenge
• Critical Infrastructures modeling: topological, flow, phenomenological, logic
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Phenomenological LogicalTopological
APPROACHES
System indicators
Critical elements
OUTPUTS
Modeling the complexity of Critical Infrastructures
Flow
Modeling
Critical
Infrastructures
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Hierarchical Hierarchical Hierarchical Hierarchical network representation framework network representation framework network representation framework network representation framework and vulnerability analysis and vulnerability analysis and vulnerability analysis and vulnerability analysis
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31
34
59
60 61 62 64
7176 78
7983
107 109
110
111
112114
86
119
2340
� Criticality of the inter-cluster components
� Multi-level reliability analysis based on the hierarchical network representation
Fang Y.-P., Zio E. “Unsupervised spectral clustering for hierarchical modelling and criticality analysis of complex networks,” Reliability Engineering & System Safety, Volume 116, 2013, Pages 64-74.
Modeling the complexity of Critical Infrastructures
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Phenomenological LogicalTopological
APPROACHES
System indicators
Critical elements
OUTPUTS
Flow
Modeling the complexity of Critical Infrastructures
Modeling
Critical
Infrastructures
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Modelling the Modelling the Modelling the Modelling the ccccascading failure (topological method)ascading failure (topological method)ascading failure (topological method)ascading failure (topological method)
betweenness–based cascading failure model
∑ ≠≠∈∈∈= kjiVkC
VjS
Vjij
n
kij
n
CN
SN
Lk ,,,
)(1
kLkC )1( α+=
Node load:
Node capacity:
)(kij
n
number of shortest paths between generators and distributorsij
n
number of shortest paths between generators and distributors passing through node k
NS, NC number of generator, distributor
VS, VC set of generator, distributor
α Network tolerance (robustness)
Initialize load, capacity
Initial failure
load redistribution
more failures occur?
YESYESYESYES
NONONONO
cascading end
loss evaluation
Modeling the complexity of Critical Infrastructures
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Optimal network design against cascading failureOptimal network design against cascading failureOptimal network design against cascading failureOptimal network design against cascading failure
Objectives: maximize the resilience of the network in resisting to cascading failures with limited construction cost
{ }
∈∀>
∈∀>
∑
∑
∑
∈
∈
∈∈
0
0
s.t.
)(min
min,
SVj
ij
CVi
ij
VjViij
ViX
VjX
GVul
X
C
CS
ϕ Network cost
Cascading failure loss0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.00 2000.00 4000.00 6000.00 8000.00
casc
adin
g vu
lner
abili
ty
cost
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
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0.4
0.5
0.6
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0.9
1
α
casc
adin
g vu
lner
abili
ty
original networkPareto solution 3 Pareto solution 5
2
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171
Improve network resilience by adding redundant links in a suitable way
Tradeoff between cost and gained network resilience
Variables: generator distributor links ijX
Fang Y.-P., Zio E., “Optimal Production Facility Allocation for Failure Resilient Critical Infrastructures,” ESREL 2013.
Modeling the complexity of Critical Infrastructures
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87%
38%
103%
105%93%
70%
48%
106%
101%
100%
61%
65%
Spreading rules:
• fixed load (5%) transferred after a failure to neighboring nodes
• fixed load, I, (10%) transferred after a failure to interdependent nodes
87%
21%
49%
67%96%
58%
22%
106%
32%
91%
105%
85%
Propagation
follows until no
more working
component can
fail 100% = component relative limit capacity
Initiating event: uniform disturbance (10%)
Modeling the complexity of Critical Infrastructures
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0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10
5
10
15
20
25
Average initial load, L
Ave
rage
Ca
sca
de S
ize
, S
Scr = 15%
Lcr = 0.8662Lcr = 0.7266E. Zio and G. Sansavini, "Modeling Interdependent Network Systems for Identifying Cascade-Safe Operating
Margins", IEEE Transactions on Reliability, 60(1), pp. 94-101, March 2011
Modeling the complexity of Critical Infrastructures
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Phenomenological LogicalTopological
APPROACHES
System indicators
Critical elements
OUTPUTS
Flow
Modeling the complexity of Critical Infrastructures
Modeling
Critical
Infrastructures
26
Main inputs:• Main Feedwater system
Internal barriers:• Water systems:
- High Pressure Coolant Injection (HPCI) System - Low Pressure Coolant Injection (LPCI) System
• Depressurization system:- Automatic Depressurization system (ADS)
• Power system: - Diesel Generator (DG)
External supports:• Water system:
- Water from the river• Power system:
- Offsite power
Recovery supporting elements:
• Road transportation system: - Road access (R)
Modeling the complexity of Critical Infrastructures
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system logic representation
system mathematical model
system model quantification
uncertainty analysis and quantification
Modeling the complexity of Critical Infrastructures
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System logic representation: GTST-DMLD
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system logic representation
system mathematical model
system model quantification
uncertainty analysis and quantification
Modeling the complexity of Critical Infrastructures
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At component level
At system level
Combinations of structural and functionalmultistatesconsidered
Structure
1: Strong damages
2: Slight damages
3: No damages
Function
1: Not working
2: Partialy working
3: Fully working
Structure
1
2
3
1
2
3
Function Structure
1
3
1
2
3
Function Structure
1
3
1
3
Function
StateStructural
damage[%]Functional
output [gpm]3 0 5000
20 ÷ 10 (small
/intermediate leaks)4625
1 > 10 < 4625
StateStructural
damage[%]Functionaloutput [%]
3 0100
2 0 ÷ 12
1 > 10 0
StateStructural
damage[%]Functionaloutput [%]
3 0 1001 > 0 0
e.g., water pipe e.g., power pole e.g., automaticdepressurization system
State 3 (Healthy): Safety of the Nuclear Power Plant (NPP) given by two water systems: one of them is in state 3 and the other one is at least in state 2.
State 2 (Marginal): Safety of the NPP given by one water system that is at least in state 2.
State 1 (At Risk): No safety of the NPP: all the water systems are in state 1.
System mathematical model: multistate
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system logic representation
system mathematical model
system model quantification
uncertainty analysis and quantification
Modeling the complexity of Critical Infrastructures
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Quantitative evaluation: procedural steps
1. Evaluate the structural (and corresponding functional) state of each component by MC simulation
2. Compute the functional state of the NPP by GTST – DMLD
Safety
Probability density function of the RT of the safety of the NPP (states 2 and 3)
Probabilistic Seismic Hazard Analysis: Ground motion at a site of interest for any magnitudeFragility evaluation: Conditional probability of exceeding a level of damage, given a ground motion level
1. Sample the recovery time (RT) of the state 2 and/or 3 of each component from the corresponding pdfs
2. Determine the next structural state that will be reached3. Sort the RT in increasing order and carry out the analysis
from the smallest RT4. Evaluate the occurrence of aftershocks before the
restoration of the component with smallest RT5. If the component with the smallest RT is not affected by
aftershocks (i.e., it reaches the next state determined at step 2.), evaluate the functional state of the NPP; otherwise sample a new RT for the components affected by the aftershocks and go to step 3.
6. if the NPP is in state 3, stop the algorithm; else, proceed with the analysis of the component with the next smallest RT
Resilience
Estimated probability of the NPP to be in the
functional state 1, 2 or 3
Repeat steps 1 – 2 n times
Repeat steps 1 – 6 k times
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Analyzing Vulnerability and Failures in Systems of
Systems: Safety and Resilience Analysis
ResilienceProbability density functions (PDFs) of the time necessary to restore the marginal (2) and healthy (3) states of the NPP from a risk state (1), after the occurrence of an earthquake and its aftershocks, in the case of multistate and binary state model.
• From state 1 to state 2 • From state 1 to state 3
0 20 40 60 80 1000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
PD
F
Recovery time [d]
μ = 2.6 d
Multistate
Binary state
0 20 40 60 80 1000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Recovery time [d]
Multistate
Binary state
μ = 4.3 d
μ = 22.5 d
μ = 72.9 d
PD
F
Multistate model shows that a faster recovery to a marginal state is possible, but a longer time is needed to reach a healthy state
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Phenomenological LogicalTopological
APPROACHES
System indicators
Critical elements
OUTPUTS
Reliability analysis
Flow
Modeling
Critical
Infrastructures
35
Mode 1: �� � ��, �� � ������� � ����� ������� � ����� ���
Mode 2: �� � ��, �� � ������� � ����� � ������ � �����
Mode 3: �� � ��, �� � ��� ���� � ����� ���� � ����� ���
Mode 4: �� � ��, �� � ������� � ����� � ������ � ����� � ��
Consider a system of 2 interconnectedsystems where the system response isdescribed by the switching dynamics:
Modeling the complexity of Critical Infrastructures
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� Find the geometric locus of the equilibrium point ‘��’.� Describe the invariant set which contains the equilibrium point.� Find the reachable regions for the invariant set (i.e. the invariant
set is a basin of attraction for the resilience region).
Steps for describing the resilience region:
Modeling the complexity of Critical Infrastructures
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Conclusions
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Structural complexity: heterogeneity, dimensionality, connectivity
Dynamic complexity : emergent behavior
Uncertainty: aleatory, epistemic, perfect storms, black swans
The complexity of analyzing the Reliability/Risk/ Vulnerability/
Resilience/… in Critical Infrastructures
39
System analysis:
- hazards and threats identification
- physical and logical structure identification
- dependencies and interdependences identification and modeling
- dynamic analysis (cascading failures)
Quantification of system safety
indicators
Identification of critical elements
Application for system improvements:
- design
- operation
- interdiction/protection
W. Kroger and E. Zio, “Vulnerable
Systems”, Springer, 2011
Systems of systems
Modeling
Critical
Infrastructures
PhenomenologicalLogical
Topological
APPROACHES
System indicators
Critical elements
OUTPUTS
Flow
The complexity of analyzing the Reliability/Risk/ Vulnerability/
Resilience/… in Critical Infrastructures
40
Modeling, Simulation, Optimization and Computational Challenges
Detail Computational cost
Integrated Approach
Topological
Logic
Detail Computational cost
FlowDetail Computational cost
Structural Complexity + Dynamic Complexity
Uncertainty
Risk + Control Theory
Detail Computational cost
Phenomenological
The complexity of analyzing the Reliability/Risk/ Vulnerability/
Resilience/… in Critical Infrastructures
41
Acknowledgments
Chair SSDE (ECP+Supelec, EDF): Yiping Fang, Elisa Ferrario, Elizaveta Kuznetzova, Yanfu Li, Rodrigo Mena, Nicola Pedroni
Politecnico di Milano (ex): Giovanni Sansavini
42
Research
www.ssde.fr (Ecole Centrale Paris and Supelec)
lasar.cesnef.polimi.it (Politecnico di Milano)
Application
www.aramis3d.com
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