Complex systems reliability: A complex challenge for reliability analysts

Transcription

1 Complex systems reliability: A complex challenge for reliability analysts Enrico Zio Chair on Systems Science and the Energy Challenge Ecole Centrale Paris and Supelec, EuropeanFoundationforNew Energy-Electricitéde France

2 Complex Systems 2

3 Complex Systems 3

4 Complex Systems 4

5 Complex Systems Network of many interacting components Components of heterogeneous type Hierarchy of subsystems Interactions across multiple scales of space and/or time Dependences (uni-directional) and interdependences (bi-directional) 5

6 Engineered Complex Systems 6

7 Engineered Complex Systems 7

8 Engineered Complex Systems 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. 8

9 Dependences and interdependences W. Kroger and E. Zio, Vulnerable Systems, Springer,

10 Dependences and interdependences: Connectivity link: - Connectivity link = structural connection between two nodes. - For a network who only has connectivity links, a node fails only if it loses all links, i.e. if it becomes isolated from the connected part. - Failure in a purely connectivity network is a percolation process. 10

11 Dependences and interdependences: Dependency link (2 types): 1. Local dependency link: failure of a node causes all its linked neighbors (who depend on it) to fail. 2. Load redistribution dependency: failure due to node overload and redistribution to its neighbors, which could be led to failure. - Failure in a purely dependency network is a cascading failure process. 11

12 Engineered Complex Systems Dynamic complexity : emergence of system behavior in response to changes in the environmental and operational conditions of parts of the system. 12

13 Modelling Engineered Complex Systems system logic representation system mathematical model system model quantification uncertainty analysis and quantification 13

14 Engineered Complex Systems 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, } 14

15 Systems of Systems 15

16 Reliability Analysis of Systems of Systems Power transmission Communication Railway Cyber Dependency, p cr Physical Dependency Physical Dependency Cyber Dependency, p cp 16

17 Reliability Analysis of Systems of Systems 17

18 Reliability Analysis of Systems of Systems Infrastructures affected by the mini telecommunication blackout in Rome,

19 Reliability analysis 19

20 Reliability analysis 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 W. Kroger and E. Zio, Vulnerable Systems, Springer, 2011 Application for system improvements (optimization): - design - operation - interdiction/protection 20

21 Reliability analysis Reliability analysis of Engineered Complex Systems APPROACHES Topological Flow Phenomenological Logical OUTPUTS System indicators Critical elements 21

22 Reliability Analysis of Engineered Complex Systems: The Dual Analysis Engineered complex systems: structure + dynamics+ failure/recovery process Aggregation Challenge Direct Problem Evaluating Global Indicators Inverse Problem Identifying Vulnerabilities at the Components Level Disaggregation Challenge Engineered complex systems modeling: topological, flow, phenomenological, logic Detail Computational cost 22

23 Reliability analysis Reliability analysis of Engineered Complex Systems APPROACHES Topological Flow Phenomenological Logical OUTPUTS System indicators Critical elements 23

24 Reliability Analysis of Networked Engineering Systems Hierarchical network representation framework and vulnerability analysis 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

25 Reliability analysis Reliability analysis of Engineered Complex Systems APPROACHES Topological Flow Phenomenological Logical OUTPUTS System indicators Critical elements 25

26 Modelling the cascading failure (topological method) Node load: Reliability Analysis of Networked Engineering Systems L 1 = k N N j V, j V, k V i S C S C, Node capacity: C k = ( 1 + α ) L k n ij n ij (k) N S, N C V S, V C α number of shortest paths between generators and distributors number of shortest paths between generators and distributors passing through node k number of generator, distributor set of generator, distributor Network tolerance (robustness) j k n ij ( k) n ij Initialize load, capacity Initial failure load redistribution more failures occur? NO cascading end loss evaluation YES betweenness based cascading failure model 26

27 Reliability Analysis of Networked Engineering Systems Optimal network design against cascading failure Objectives: maximize the resilience of the network in resisting to cascading failures with limited construction cost min min { Vul( G) } i V S, j V ϕx C ij Network cost Cascading failure loss X ij > 0 j VC i V s.t. X ij > 0 i VS j V C Variables: generator distributor links X ij Improve network resilience by adding redundant links in a suitable way Tradeoff between cost and gained network resilience Fang Y.-P., Zio E., Optimal Production Facility Allocation for Failure Resilient Critical Infrastructures, ESREL cascading vulnerability cascading vulnerability cost α original network Pareto solution 3 Pareto solution

28 Reliability Analysis of Systems of Systems: Topological Analysis (Cascading failures) Spreading rules: fixed load (5%) transferred after a failure to neighboring nodes fixed load, I, (10%) transferred after a failure to interdependent nodes 61% 87% 105% 87% 65% 103% 106% 70% 101% 85% 49% 106% 58% 32% 93% 100% 48% 105% 96% 91% 22% 67% Propagation follows until no more working component can fail 38% 21% 100% = component relative limit capacity Initiating event: uniform disturbance (10%) 28

29 Reliability Analysis of Systems of Systems: Topological Analysis (Cascading failures) Average Cascade Size, S S cr = 15% Average initial load, L L cr = L cr = E. Zio and G. Sansavini, "Modeling Interdependent Network Systems for Identifying Cascade-Safe Operating Margins", IEEE Transactions on Reliability, 60(1), pp , March

30 End-to-end delay: main performance indicator for the information flow of an application T = Ts + Tb + Tp + Tr T Reliability Analysis of Systems of Systems: Topological Analysis (Cascading failures) = T N = P D s s r TP = l v λ ( i) ( ) r i i i i= 1 ( ) ( ) ( ) λ µ λ The Smart Grid must operate in a region of the system parameters that is far from the catastrophic phase transition point Zio, E. and Sansavini, G., Vulnerability of Smart Grids with Variable Generation and Consumption: a System of Systems Perspective, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, ISSN

31 Reliability analysis Reliability analysis of Engineered Complex Systems APPROACHES Topological Flow Phenomenological Logical OUTPUTS System indicators Critical elements 31

32 Earthquake Analyzing Vulnerability and Failures in Systems of Systems: Safety and Resilience Analysis + Aftershocks Time (t) Magnitude (M) t0 M0 t1 t2 M1M2 t3 M3 t4 M4 t5 M5 t6 M6 t NPP state 3: Healthy 2: Marginal 1: At Risk Recovery t Ferrario E., Zio E., Reliability Engineering & Systems Safety, 114, Pages OBJECTIVE: Estimate the safety and physical resilience of a Nuclear Power Plant (NPP) exposed to risk from seismic events in a system-ofsystems framework. 32

33 Main inputs: System-of-Systems approach to external events risk assessment Case study 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) 33

34 Modelling Engineered Complex Systems system logic representation system mathematical model system model quantification uncertainty analysis and quantification 34

35 System logic representation: GTST-DMLD 35

36 Modelling Engineered Complex Systems system logic representation system mathematical model system model quantification uncertainty analysis and quantification 36

37 System mathematical model: multistate At component level Structure 3: No damages Function 3: Fully working 2: Slight damages 2: Partialy working 1: Strong damages 1: Not working Combinations of structural and functional multistates considered Structure Function Structure Function 3 1 Structure 3 1 Function 3 1 At system level e.g., water pipe e.g., power pole e.g., automatic Structural Functional depressurization system State Structural Functional damage[%] output [gpm] State damage[%] output [%] Structural Functional 3 0 State 0 10 (small damage[%] output [%] /intermediate leaks) > 10 < > > 0 0 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. 37

38 Modelling Engineered Complex Systems system logic representation system mathematical model system model quantification uncertainty analysis and quantification 38

39 Quantitative evaluation: procedural steps Probabilistic Seismic Hazard Analysis: Ground motion at a site of interest for any magnitude Fragility evaluation: Conditional probability of exceeding a level of damage, given a ground motion level Safety 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 Repeat steps 1 2 n times Estimated probability of the NPP to be in the functional state 1, 2 or 3 Resilience 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 reached 3. Sort the RT in increasing order and carry out the analysis from the smallest RT 4. Evaluate the occurrence of aftershocks before the restoration of the component with smallest RT 5. 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 if the NPP is in state 3, stop the algorithm; else, proceed with the analysis of the component with the next smallest RT Repeat steps 1 6 k times Probability density function of the RT of the safety of the NPP (states 2 and 3) 39

40 Analyzing Vulnerability and Failures in Systems of Systems: Safety and Resilience Analysis Safety Estimate of the probability that the NPP enters in a risk (1), marginal (2) or healthy (3) state after the occurrence of an earthquake (binary vs. multistate models). Multistate Binary Multistate model allows identifying marginal safety conditions that could degrade to risky 40

41 Analyzing Vulnerability and Failures in Systems of Systems: Safety and Resilience Analysis Resilience Probability 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 Multistate Binary state PDF μ = 2.6 d PDF μ = 4.3 d μ = 72.9 d Multistate Binary state Recovery time [d] μ = 22.5 d Recovery time [d] Multistate model shows that a faster recovery to a marginal state is possible, but a longer time is needed to reach a healthy state 41

42 Reliability analysis Reliability analysis of Engineered Complex Systems APPROACHES Topological Flow Phenomenological Logical OUTPUTS System indicators Critical elements 42

43 Simulation and Optimization under Uncertainty: Energy Management Microgrid agents: district (D), middle-size train station (TS) and urban wind power plant (WPP) Elizaveta Kuznetsova, Yan Fu Li, Carlos Ruiz, Enrico Zio, Graham Ault, Keith Bell (accepted in Energy Journal/under submission) 43

44 Simulation and Optimization under Uncertainty: Energy Management Agent representation as open system with inlet and outlet power flows from continuous interactions with other agents of the microgrid Power balance equations: +, + h,,, + =0 + =0, + h,,, =0 44

45 Simulation and Optimization under Uncertainty: Energy Management Hierarchical decision adopted here: power producers (TS & WPP) decide the portion to sell to the D In case of difference between predicted and real power output, producers obliged to supply committed power to the D Previsions of the operational conditions (i) agent personal previsions (ii) previsions received from other agents through ISO Decision variables based on the previsions D WWP TS

46 Simulation and Optimization under Uncertainty: Energy Management Minimize s.t. + +, h,,, + + Power balance, 1 + h,,, Power flow through the battery ( $ =0 ) & + Cost function for the time period T PV power production ' ( * +, * + h,-,, 0, h +, 1 0, h 1 0, 1 0,./0 0 0 Export to the District 46

47 Simulation and Optimization under Uncertainty: Energy Management Minimize s.t. + +, h,,, + 0 : :+2 + < -, Γ -, +$ +$ 0 < -, +$ >? < -, +$ >? ( $ =0 )+ $ +$ +$ +$ ) $ +$ $ $? $? $? =0 & +$???, 1 + h,,, + 0 :+1 +< B Γ B +$ 0, < B +$ >? ' ( * +, * + h,-,, 0 0,./0, 0, 0, +$??? Uncertainty sources in the microgrid: power output from renewable generators 1 23 and 1 422, power demand of consumers 1 56 and 1 7 and electricity prices 1 8, 1 9 and Similar for other agents 47

48 Reliability analysis Reliability analysis of Engineered Complex Systems APPROACHES Topological Flow Phenomenological Logical OUTPUTS System indicators Critical elements 48

49 System of Systems Resilience Modeling and Analysis Research Context Interconnected infrastructures Linear time-invariant (LTI) dynamic systems Vulnerability and resilience Robust design and protection 49

50 System of Systems Resilience Modeling and Analysis Objectives Model the dynamics behavior of the interconnected systems Describe and analyze the system performances: Resilience region: The part of the state-space for which the transient behavior of the system evolves into the operation region (operation mode). Non-resilience region: The part of the state-space for which the transient behavior of the system evolves into the out-ofoperation region (out-of-operation mode). 50

51 System of Systems Resilience Modeling and Analysis System modeling Consider a linear time-invariant (LTI) dynamic system 0C()=((0,-#%%, where 0 E is the system state andw#t% E is the disturbance. Set-theoretic methods [Blanchini 2008]: A set E is said to be robustly positively invariant (RPI) for the above system if for all 0#0% and all w#% the solution is such that0#% fori0. S F. Blanchini and S. Miani, Set-Theoretic Methods in Control, Springer,

52 System of Systems Resilience Modeling and Analysis System dynamic modeling ConsiderK 9 interconnected systems descried by: L 0C M()= N M 0 M O& PM 0 P - M #%; ( 0 M T M,,U 1,,K 9 ; WU, M Q R 0CM#%X M 10 M #% ; ( 0 M IT M,,U 1,,K 9 ; WU, Where: -0 M (t) Z0,1[isthecontinuousinternalstateofthe 1h system; -T M Z0,1[istheexternalthresholdofthe 1h system; -- M #% Z0,1[representsaboundeddisturbanceofthe 1h system; -X M = opq #rst R% isthefailurerate,andx 55u M Z0,1]; R -N M opq t R istherecoveryrate,andn 55v M Z0,1]; R -& MP Z0,1]isthecouplingfactor 52

53 System of Systems Resilience Modeling and Analysis Case Study (1) Consider a system of 2 interconnected systems where the system response is described by the switching dynamics: Mode 1: 0 r T r,0 y T y z 0C r#% N r 0 r - r 0Cy#% N y 0 y - y Mode 2: 0 r IT r,0 y T y z 0C r#% X r 0 r X r 0Cy#% N y 0 y Mode 3: 0 r T r,0 y IT y 0Cr#% N z r 0 r 0Cy#% X y 0 y X y Mode 4: 0 r IT r,0 y IT y z 0C r#% X r 0 r X y 0Cy#% X y 0 y X r 53

54 Case Study (2) Steps for describing the resilience region: Find the geometric locus of the equilibrium point 0 {. System of Systems Resilience Modeling and Analysis 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). 54

55 Case Study (3) Steps for describing the resilience region: Find the geometric locus of the equilibrium point 0 {. System of Systems Resilience Modeling and Analysis 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). 55

56 Case Study (4) Steps for describing the resilience region: Find the geometric locus of the equilibrium point 0 {. System of Systems Resilience Modeling and Analysis 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). 56

57 Case Study (5) Steps for describing the resilience region: Find the geometric locus of the equilibrium point 0 {. System of Systems Resilience Modeling and Analysis 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). 57

58 System of Systems Resilience Modeling and Analysis Case Study (6) Case scenarios Resilience region Non-resilience region Deadlock behavior Algebraic conditions of the system paramters: 0 - r T r N r 0 - y T y N y Algebraic conditions of the system paramters: - r IT r N r - y IT y N y Find conditions on the system parameters. 58

59 Conclusions 59

60 The complexity of analyzing the vulnerability and failures in engineered complex systems Structural complexity: heterogeneity, dimensionality, connectivity Dynamic complexity : emergent behavior Uncertainty: aleatory, epistemic, perfect storms, black swans 60

61 The complexity of analyzing the vulnerability and failures in engineered complex systems System analysis: - hazards and threats identification - physical and logical structure identification - dependencies and interdependences identification and modeling - dynamic analysis (cascading failures) Vulnerability and failure analysis of Engineered Complex Systems APPROACHES Quantification of system safety indicators Identification of critical elements Topological Flow Phenomenological Logical Application for system improvements: - design - operation - interdiction/protection System indicators Critical elements OUTPUTS Systems of systems W. Kroger and E. Zio, Vulnerable Systems, Springer,

62 The complexity of analyzing the vulnerability and failures in engineered complex systems Structural Complexity + Dynamic Complexity Modeling, Simulation, Optimization and Computational Challenges Detail Topological Computational cost Phenomenological Detail Computational cost Uncertainty Detail Logic Computational cost Flow Detail Computational cost Risk + Control Theory Integrated Approach 62

63 Acknowledgments Chair SSDE (ECP+Supelec, EDF): Yiping Fang, Elisa Ferrario, Elizaveta Kuznetzova, Yanfu Li, Rodrigo Mena, Nicola Pedroni Politecnico di Milano : Giovanni Sansavini 63

64 More info Research ssde.fr (Ecole Centrale Paris and Supelec) lasar.cesnef.polimi.it (Politecnico di Milano) Application Aramis3d.com 64