Integration of Reliability, Robustness and Resilience for Engineered SystemShuoqi Wang, Industrial & Systems EngineeringDr. Kailash Kapur, Professor, Industrial & Systems EngineeringDr. Dorothy Reed, Professor, Civil & Environmental Engineering

University of Washington, Seattle, WA, USA

Outline of Presentation• Motivation• Reliability Models• Resilience Models• Summary and Future Work

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Motivation

The greatest glory in living lies not in never falling, but in rising every time we fall.

– Nelson Mandela

Motivation• We rely heavily on our civil infrastructures to maintain normal

operations of social systems• Reliability and Robustness are two concepts linked to

improving performance of a system• Vugrin, Warren and Ehlen (2011) pointed out that neither

natural nor manmade disruptive events could be completely prevented

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Motivation

United Airlines Flight 175 flies low toward the South Tower of the WorldTrade Center, shortly before slamming into the structure. The north towerburns after an earlier attack by a hijacked airliner in New York City, onSeptember 11, 2001. (Reuters/Sean Adair)

A truck drives through water pushed over a road by Hurricane Sandy inSouthampton, New York, on October 29, 2012. Hurricane Sandy, themonster storm bearing down on the East Coast, strengthened on Mondayafter hundreds of thousands moved to higher ground, public transport shutdown and the stock market suffered its first weather-related closure in 27years. (Reuters/Lucas Jackson)

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Motivation• In addition to Reliability and Robustness, Resilience is also

becoming very important for improving the quality of infrastructures.

• Department of Homeland Security has made critical infrastructure resilience the top-level strategic objective.

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Motivation

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Reliability

RobustnessResilience

• Integrating Reliability, Robustness and Resilience will build a system that can withstand, absorb, adapt, and recover from disturbance

• We define System Effectiveness as a measurement of the combined effect of Reliability, Robustness and Resilience (R3)

Motivation• The System Effectiveness

(𝑅𝑅3) is illustrated in the figure as the colored area under system performance function, defined by 𝑋𝑋(𝑡𝑡).

• 𝑋𝑋 𝑡𝑡 is a normalized indicator of the system’s performance over time 𝑡𝑡

– 𝑋𝑋 𝑡𝑡 = 1 indicates a perfectly functioning system

– 𝑋𝑋 𝑡𝑡 = 0 means a totally failed system

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𝑡𝑡

𝑋𝑋(𝑡𝑡)

0

1

𝑡𝑡0 𝑡𝑡𝑡 𝑇𝑇

𝑓𝑓𝑋𝑋(𝑡𝑡)

𝑓𝑓𝑋𝑋(𝑡𝑡)

Reliability/Robustness

(Degradation) (Recovery)

Resilience

Motivation

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System of

Systems

Electric Power

DeliveryTransportation

Utilities

Building Support Systems

Business Operations

Emergency Services

Financial Services

Food Supply

Government

Health Care

Telecommunications

• The infrastructure in this research are consist of eleven interconnected lifelines (Chang, McDaniels, and Reed 2004, Reed, Powell and Westerman 2010)

• Electric power delivery is especially a crucial lifeline which supports the operation of other lifelines

Motivation

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• The figure shows electric power delivery restoration after some major hurricanes in U.S.

• 𝑄𝑄(𝑡𝑡) is the percentage of customers without power at time 𝑡𝑡/𝑇𝑇 (normalized time, 𝑇𝑇 is the total time length of the restoration process after each hurricane)

• We are very interested in reducing the impact of each storm on the infrastructure and increase the speed of restoration process

Reliability Models

Physics of Failure

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• To determine the probability of failure of a system, one important approach is to study the physics of failure

• The basics of the concept is that a given component has a certain stress resisting capacity; if the stress induced by the operating conditions exceeds this capacity, failure results

• Therefore, instead of studying the whole distribution of both stress and strength, we should focus on the tail portion of each distribution where they interfere

Stress-Strength Reliability Model• The state of a system 𝑈𝑈 can be determined by its general

random stress 𝑋𝑋 and general random strength 𝑌𝑌– 𝑈𝑈 = 𝑌𝑌 − 𝑋𝑋

• The reliability of the system can be expressed as

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𝑈𝑈

𝐴𝐴

1

0

Success

Failure

Binary-State Reliability

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• Traditionally, the system under consideration is assumed to only operate in two states, either ‘success’ or ‘failure’, in other words, a binary-state view

• Let 𝐴𝐴 be a set of system state variable 𝑈𝑈, 𝐴𝐴 is a binary set where

Fuzzy Set Theory• The introduction of fuzzy set theory made it possible to extend the stress-

strength reliability model from binary-state to fuzzy-state• Let Ω be a space of objects, and 𝜔𝜔 be an element of Ω. According to Zadeh,

a fuzzy set �̃�𝐴 in Ω was characterized by a membership function 𝜇𝜇 �𝐴𝐴 𝜔𝜔which associated with each point in Ω a real number in the interval 0, 1 , with the value of 𝜇𝜇 �𝐴𝐴 𝜔𝜔 at 𝜔𝜔 representing the grade of membership of 𝜔𝜔 in �̃�𝐴

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Fuzzy Set Theory

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• An important concept related to fuzzy set, also the foundation of many fuzzy operations is ‘α-cut’, defined as

𝜇𝜇 �𝐴𝐴(𝑢𝑢)

𝑢𝑢

Fuzzy Reliability• One approach to make the

distribution fuzzy is to introduce fuzzy numbers as parameters of the p.d.f.

• Buckley (2003) gave a example where 𝑌𝑌~𝑁𝑁 �𝜇𝜇𝑌𝑌,𝜎𝜎𝑌𝑌 and 𝑋𝑋~𝑁𝑁 �𝜇𝜇𝑋𝑋,𝜎𝜎𝑋𝑋

• �𝜇𝜇𝑌𝑌 and �𝜇𝜇𝑋𝑋 are two triangular fuzzy number (12/15/18) and (6/10/14) respectively, while 𝜎𝜎𝑌𝑌 = 3 and 𝜎𝜎𝑋𝑋 = 2are deterministic

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Fuzzy Reliability• Let 𝑈𝑈 = 𝑌𝑌 − 𝑋𝑋 and compute 𝑃𝑃 𝑈𝑈 > 0 , the resulting α-cuts of the fuzzy probability are shown in the table

• The figure at the bottom right illustrates these α-cuts

• The system fuzzy reliability is

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α α-cuts0 [0.2895, 0.9996]

0.2 [0.4339, 0.9984]0.4 [0.5878, 0.9946]0.6 [0.7291, 0.9847]0.8 [0.8410, 0.9621]1.0 [0.9172, 0.9172]

Resilience Models

Definition of Resilience

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Field Definition of Resilience

Ecology a measure of the persistence of systems and of their ability to absorb change and disturbance and still maintain the same relationship between populations or state variables (Holling 1973)

Safety the intrinsic ability of a system to adjust its functioning prior to, during or following changes and disturbances, so that it can sustain required operations under both expected and unexpected conditions (Hollnagel 2010)

Community Seismic the ability of social units (e.g., organizations, communities) to mitigate hazards, contain the effects of disasters when they occur, and carry out recovery activities in ways that minimize social disruption and mitigate the effect of future disasters (Bruneau, Chang et al. 2003)

Civil Engineering the ability to “bounce back” after a major disturbance (Reed, Powell et al. 2010)

Enterprise the ability and capacity to withstand systemic discontinuities and adapt to new risk environment (Starr, Newfrock et al. 2003)

Network the ability of an entity to tolerate, endure and automatically recover from challenges under the conditions of the network, coordinated attacks and traffic anomalies (Aggelou 2009)

Healthcare the process of negotiating, managing and adapting to significant sources of stress or trauma (Windle, Bennett et al. 2011)

Food Security the capability of households to absorb the negative effects of unpredictable shocks or disasters, rather than predicting the occurrence of a crisis (Sibrian 2008)

Resilience Models• Mechanical Analogy

– To model the recovery process of electric power delivery, a over-damped harmonic oscillator is used as a mechanical analogy, proposed by Cimellaro et al (Cimellaro, Reinhorn, & Bruneau, 2010);

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Resilience Models• Here, 𝑄𝑄(𝑡𝑡) is the displacement of the mass from equilibrium point, which

represents the loss of performance. 𝑋𝑋 𝑡𝑡 = 1 − 𝑄𝑄(𝑡𝑡) on the other hand, represents level of performance.

• Given 𝑚𝑚 = mass; 𝑐𝑐 = damping; and 𝑘𝑘 = stiffness• When c2-4mk > 0, we say the motion is overdamped;

• Let 𝛼𝛼 = 𝑐𝑐2𝑚𝑚

and 𝛽𝛽 = 𝑐𝑐2−4𝑚𝑚𝑚𝑚2𝑚𝑚

• We have

• Given Q(0) = q0, and Q’(0) = v0,

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Resilience Models for Power Restoration

• Mechanical Analogy– To check the fitness of the proposed model regarding actual

recovery process, we need to evaluate the parameters used in the model using actual data.

– Both linear and nonlinear regression method can be used to estimate the parameters.

• Electric power outage data were obtained from situation reports published by Office of Electricity Delivery & Energy Reliability of U.S. Department of Energy( http://www.oe.netl.doe.gov/emergency_sit_rpt.aspx)

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Power outage data collection

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Event Year Data points Event Year Data points

Hurricane Isabel 2003 9 Hurricane Katrina 2005 49

Hurricane Charley 2004 10 Hurricane Rita 2005 24

Hurricane Frances 2004 12 Pacific Northwest Storm

2006 5 - 10

Hurricane Ivan 2004 9 Hurricane Gustav 2008 15

Hurricane Jeanne 2004 9 Hurricane Ike 2008 21

Hurricane Dennis 2005 6 Hurricane Isaac 2012 42

Hurricane Wilma 2005 21 Hurricane Sandy 2012 19

Linear Regression

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0.800

0.850

0.900

0.950

1.000

0 5 10 15 20 25

X(t)

t/days

Rita X(t)

Actual Q(t)

Estimate Q(t)

0.000

0.200

0.400

0.600

0.800

1.000

1.200

0 5 10 15 20 25 30 35 40 45 50

X(t)

t/days

Katrina X(t)

Actual Q(t)

Estimate Q(t)

R2=0.9842

R2=0.7956

X(t)X(t)

X(t)X(t)

Nonlinear Regression

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Model comparison for various hurricanes

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Linear Regression Nonlinear Regression

Hurricane ω ζ 𝑹𝑹𝟐𝟐 ω ζ 𝑹𝑹𝟐𝟐

Isabel 1.05 2.03 0.8743 0.42 1.01 0.9852

Charley 0.55 1.06 0.8921 0.49 1.00 0.9851

Frances 0.88 1.20 0.9551 0.75 1.25 0.9876

Ivan 0.78 1.30 0.9957 0.75 1.33 0.9988

Katrina 0.74 5.36 0.7956 0.05 1.48 0.9938

Rita 0.61 1.58 0.9842 0.33 1.13 0.9987

Gustav 0.61 1.06 0.9231 0.54 1.06 0.9952

Ike 0.70 2.46 0.8783 0.28 1.01 0.9969

Isaac 8.32 9.12 0.7511 0.76 1.00 0.9912

Summary and Future Work

Summary• We reviewed the traditional definition of ‘reliability’, as well

as new perspectives towards it, e.g. multi-state, and fuzzy reliability

• The term ‘resilience’ was introduced and its various definitions were discussed

• Quantitative models of reliability was discussed with focus on fuzzy reliability models

• Quantitative models of resilience was discussed and a model based on mechanical analogy was presented with an application related to power delivery restoration

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Future Work• Expand the reliability model using stress and strength

interference theory where the p.d.f. of stress and strength random variables contains fuzzy parameters

• Expand the mechanical analogy for modeling recovery in general systems

• Incorporate robust design theory into the modeling of recovery• Combine the fuzzy reliability model and the resilience model,

as well as robust design theory, for an integrated measurement of system effectiveness.

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Thank you!