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Research Papers

Analysis of the Robustness and Recovery of Critical Infrastructures by Goal Tree–Success Tree: Dynamic Master Logic Diagram, Within a Multistate System-of-Systems Framework, in the Presence of Epistemic Uncertainty

[+] Author and Article Information
E. Ferrario

Chair on Systems Science and the Energetic Challenge, European Foundation for New Energy, Electricité de France, École Centrale Paris–Supelec, Grande Voie des Vignes, 92295 Chatenay Malabry, France e-mail: elisa.ferrario@ecp.fr

N. Pedroni

Chair on Systems Science and the Energetic Challenge, European Foundation for New Energy, Electricité de France, École Centrale Paris–Supelec, Grande Voie des Vignes, 92295 Chatenay Malabry, France e-mails: nicola.pedroni@ecp.fr, nicola.pedroni@supelec.fr

E. Zio

Chair on Systems Science and the Energetic Challenge, European Foundation for New Energy, Electricité de France, École Centrale Paris–Supelec, Grande Voie des Vignes, 92295 Chatenay Malabry, France; Politecnico di Milano, 20133 Milano, Italye-mails: enrico.zio@ecp.fr, enrico.zio@supelec.fr, enrico.zio@polimi.it

Manuscript received July 29, 2014; final manuscript received January 14, 2015; published online July 1, 2015. Assoc. Editor: Alba Sofi.

ASME J. Risk Uncertainty Part B 1(3), 031001 (Jul 01, 2015) (14 pages) Paper No: RISK-14-1032; doi: 10.1115/1.4030439 History: Received July 29, 2014; Accepted April 27, 2015; Online July 01, 2015

In this paper, we evaluate the robustness and recovery of connected critical infrastructures (CIs) under a system-of-systems (SoS) framework taking into account: (1) the dependencies among the components of an individual CI and the interdependencies among different CIs; (2) the variability in component performance, by a multistate model; and (3) the epistemic uncertainty in the probabilities of transitions between different components states and in the mean values of the holding-times distributions, by means of intervals. We adopt the goal tree success tree–dynamic master logic diagram (GTST–DMLD) for system modeling and perform the quantitative assessment by Monte Carlo simulation. We illustrate the approach by way of a simplified case study consisting of two interdependent infrastructures (electric power system and gas network) and a supervisory control and data acquisition (SCADA) system connected to the gas network.

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Figures

Grahic Jump Location
Fig. 1

Top: Interdependent gas (solid lines) and electric (dashed lines) infrastructures and SCADA system (dotted lines) [1]; the quantities demanded by the end-nodes D1, D2, L1, and L2 are reported in bold. Bottom: deterministic and stochastic arc capacities (1 cu. ft≈0.028 m3).

Grahic Jump Location
Fig. 2

Holding-time distributions (matrices Tc) for the arcs described by semi-Markov processes: each element of the matrix represents a normal distribution with uncertain (interval) mean and fixed standard deviation. State transition probability matrices (Pc) for the arcs described by Markov and semi-Markov processes: each element of the matrix represents an interval for the corresponding transition probability.

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Fig. 3

Conceptual sketch of GTST–DMLD: the filled dots indicate the possible dependencies between the objects (filled dot on the left) and between the objects and functions (filled dot on the right), the logic gates indicate how a given function depends on the input values

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Fig. 4

Example of an element C that depends on two elements A and B by an “AND” gate

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Fig. 5

Examples of direct and indirect dependencies with respect to possible graph representations

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Fig. 6

Examples of constraint-based dependencies with respect to possible graph representations

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Fig. 7

GTST–DMLD of the case study in Sec. 2 corresponding to the graph of Fig. 1

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Fig. 8

Right (dotted line) and left (solid line) cumulative distribution functions of the product delivered to the nodes D1, D2, L1, and L2 at the steady-state (1 cu. ft.≈0.028 m3)

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Fig. 9

Right (dotted line) and left (solid line) cumulative distribution functions of the recovery time of the supply of the demand nodes, starting from the worst scenario

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Fig. 10

Exemplification of step B1 for the row i=2 of the probability matrix Pc, c=S2_DS2, to identify Zc,i combinations of transition probability values

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Fig. 11

Exemplification of step B1 for the row i=2 of the holding-time distribution matrix Tc, c=S2_DS2, to identify Mc,i combinations of mean values

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Fig. 12

Exemplification of step B2 to identify a set transition probability matrix P̲̲c,k, k=1,…,Kc, for component c=S2_DS2, given the ∑i=1ScZc,i vectors obtained at step B1

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Fig. 13

Exemplification of step B4 to identify the steady-state probability vectors for Markov and semi-Markov processes

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