Mohamed Benbouzid

Brice Beltran

Yassine Amirat

Gang Yao

Jingang Han

Hervé Mangel

High-Order Sliding Mode Control for DFIG-Based Wind Turbine Fault Ride-Through

Keywords: Wind turbine (WT), Doubly-Fed Induction Generator (DFIG), Fault-Ride Through (FRT), High-Order Sliding

This paper deals with the Fault Ride-Through (FRT) capability assessment of a Doubly-Fed Induction Generator (DFIG)-based Wind Turbine (WT) using High-Order Sliding Mode (HOSM) control. Indeed, it has been recently suggested that sliding mode control is a solution of choice to the FRT problem. In this context, this paper proposes HOSM as an improved solution that handle the classical sliding mode chattering problem. Indeed, the main and attractive features of HOSMs are robustness against external disturbances (e.g. grid) and chattering-free behavior (no extra mechanical stress on the drive train).

Simulations using the NREL FAST code on a 1.5-MW wind turbine are carried-out to evaluate ride-through performance of the proposed HOSM control strategy in case of grid frequency variations and unbalanced voltage sags.

I. INTRODUCTION

An increasing number of power system operators have implemented technical standards known as grid codes that wind turbines must meet when connecting to the grid [START_REF] Mohseni | Review of international grid codes for wind power integration: Diversity, technology and a case for global standard[END_REF][START_REF] Tsili | A review of grid code technical requirements for wind farms[END_REF]. Generally, these grid codes requirements cover many topics such as, voltage operating range, power factor regulation, frequency operating range, grid support capability, and low fault ride-through capability. Indeed, grid codes dictate Fault Ride-Through (FRT) requirements. Low-Voltage Ride-Through (LVRT) capability is considered to be the biggest challenge in wind turbines design and manufacturing technology [START_REF] Yingcheng | Review of contribution to frequency control through variable speed wind turbine[END_REF]. LVRT requires wind turbines to remain connected to the grid in presence of grid voltage sags.

The Doubly-Fed Induction Generator (DFIG) is one of the most frequently deployed large grid-connected wind turbines. Indeed, when compared with the full-scale power converter WT concept, the DFIG offers some advantages, such as reduced inverter and output filter costs due to low rotor-and grid-side power conversion ratings (25%-30%) [START_REF] Liserre | Overview of multi-MW wind turbines and wind parks[END_REF]. However, DFIG-based WTs are very sensitive to grid disturbances, especially to voltage dips [START_REF] Jadhav | A comprehensive review on the grid integration of doubly fed induction generator[END_REF].

M.E.H. Benbouzid, B. Beltran, and Hervé Mangel are with the University of Brest, EA 4325 LBMS, Rue de Kergoat, CS 93837, 29238 Brest Cedex 03, France (e-mail: Mohamed.Benbouzid@univ-brest.fr, brice.beltran@dga.defense.gouv.fr, Herve.Mangel@univ-brest.fr).

Y. Amirat is with ISEN, EA 4325 LBMS, 20, Rue Cuirassé Bretagne, 29200 Brest, France (e-mail: Yassine.Amirat@isen.fr).

G. Yao and J. Han are with the Shanghai Maritime University, Department of Electrical Automation, 201306 Shanghai, China (email: gangyao@shmtu.edu.cn, jghan@shmtu.edu.cn).

In this context, this paper proposes to address the FRT problems using a so-called active method achieving FRT with no additional devices. The goal is to control rotor voltages and currents, to reduce the rotor overvoltages and/or overcurrents, and therefore avoid the crowbar use/activation in order to keep full DFIG control at all times to meet the FRT requirements. In [START_REF] Cardenas | Overview of control systems for the operation of DFIGs in wind energy applications[END_REF], a critical review of control methods for LVRT compliance with DFIG is given. This state-of-the-art review suggests the need of robust and nonlinear controller. In particular, it has been suggested that sliding mode control is a solution of choice to the FRT problem [START_REF] Martinez | Slidingmode control for DFIG rotor-and grid-side converters under unbalanced and harmonically distorted grid voltage[END_REF]. Therefore and in this particular context, this paper proposes the use of highorder sliding mode control as an improved solution that handle the classical sliding mode chattering problem. Indeed, the main and attractive features of HOSMs are robustness against external disturbances (e.g. grid) and chattering-free behavior (no extra mechanical stress on the drive train) [START_REF] Beltran | Second-order sliding mode control of a doubly fed induction generator driven wind turbine[END_REF][START_REF] Beltran | High-order sliding mode control of variable speed wind turbines[END_REF][START_REF] Benbouzid | A highorder sliding mode observer for sensorless control of DFIG-based wind turbines[END_REF].

The proposed control strategy combines an MPPT using a second-order sliding mode for the DFIG control. This strategy presents attractive features such as chattering-free behavior, finite reaching time, robustness and unmodeled dynamics (generator and turbine). To check the overall control strategy ride-through performance, simulations using the NREL FAST code on a 1.5-MW wind turbine are carried-out in case of grid frequency variations and unbalanced voltage sags.

II. GRID-CODE REQUIREMENTS

Grid-code requirements typically refer to large wind farms connected to the transmission system, rather than smaller stations connected to the distribution network. These new grid codes stipulate that wind farms should contribute to power system control (frequency and also voltage), much as the conventional power stations, and emphasize wind farm behavior in case of abnormal operating conditions of the network (such as in case of voltage dips). The most common requirements include FRT capability, extended system voltage and frequency variation limits, active power regulation, and frequency control, as well as reactive power/power factor and voltage regulation capabilities [START_REF] Mohseni | Review of international grid codes for wind power integration: Diversity, technology and a case for global standard[END_REF][START_REF] Tsili | A review of grid code technical requirements for wind farms[END_REF]. Grid codes main requirements regarding the addressed faults are given below.

A. Frequency Operating Range

Wind power plants are required to run continuously within typical grid frequency variations between 49.5 Hz and 50.5 Hz. Figure 1 gives an example of frequency-grid voltage variations [START_REF] Mohseni | Review of international grid codes for wind power integration: Diversity, technology and a case for global standard[END_REF].

B. Low Voltage Ride-Through

Grid codes invariably require that large wind farms must withstand voltage sags down to a certain percentage of the nominal voltage and for a specified duration. Such constraints are known as LVRT requirements. They are described by a voltage versus time characteristic, denoting the minimum required immunity of the wind power station to the system voltage sags (Fig. 2) [START_REF] Mohseni | Review of international grid codes for wind power integration: Diversity, technology and a case for global standard[END_REF]. Fig. 2. LVRT requirements for different countries [START_REF] Mohseni | Review of international grid codes for wind power integration: Diversity, technology and a case for global standard[END_REF].

III. WIND TURBINE MODELING

The wind turbine modeling is inspired from [START_REF] Beltran | High-order sliding mode control of variable speed wind turbines[END_REF]. In the following, the wind turbine components models are briefly described.

A. The Turbine Model

In this case, the aerodynamic power P a captured by the wind turbine is given by ( )

2 3
1 2

a p P RC v = πρ λ (1)
where the tip speed ratio λ is given by R v

ω λ = ( 2
)
and v is the wind, ρ is the air density, R is the rotor radius, and C p is the power coefficient. The rotor power (aerodynamic power) is also defined by

a a P T = ω
(3) where T a is the aerodynamic torque and ω is the WT rotor speed.

The following simplified model is adopted for the turbine (drive train) for control purposes.

a g J T K T ω = -ω - (4)
where T g is the generator electromagnetic torque, J is the turbine total inertia, and K is the turbine total external damping.

B. The DFIG Model

The control system is usually defined in the synchronous d-q frame fixed to either the stator voltage or the stator flux. For the proposed control strategy, the generator dynamic model written in a synchronously rotating frame d-q is given by ( )

sd
d V R I dt d V R I dt d V R I dt d V R I dt L I MI L I MI L I MI L I MI T pM I I I I φ ⎧ = + -ωφ ⎪ ⎪ φ ⎪ = + +ωφ ⎪ ⎪ φ ⎪ = + -ωφ ⎪ ⎪ ⎪ φ ⎨ = + +ωφ ⎪ ⎪ φ = + ⎪ φ = + ⎪ ⎪ φ = + ⎪ φ = + ⎪ ⎪ = - ⎪ ⎩ (5)
where V is the voltage, I the current, φ is the flux, R is the resistance, L is inductance, M is the mutual inductance, T em is the electromagnetic torque, and p is the pole pair number.

For simplification purposes, the q-axis is aligned with the stator voltage and the stator resistance is neglected. These will lead to

dI d M V R I s L I dt L L dt dI M V R I s L I s dt L L M T p I L ⎧ ⎛ ⎞ φ = - + ωσ - ⎪ ⎜ ⎟ σ ⎝ ⎠ ⎪ ⎪ ⎪ ⎛ ⎞ = - -ωσ -ω φ ⎨ ⎜ ⎟ σ ⎪ ⎝ ⎠ ⎪ ⎪ = - φ ⎪ ⎩ (6)
where σ is the leakage coefficient (σ = 1 -M 2 /L s L r ), ω r is the angular speed, ω s is the synchronous speed, and s is the slip.

IV. DFIG-BASED WIND TURBINE CONTROL

A. MPPT Strategy

The control objective is to optimize the capture wind energy by tracking the optimal torque T opt .

2 opt T k = ω , with max 5 3 1 2 p opt C k R = πρ λ (7)
Details about the adopted strategy are given in [START_REF] Beltran | High-order sliding mode control of variable speed wind turbines[END_REF].

B. HOSM Control Strategy

The DFIG-based WT control objective is to optimize the extracted power by tracking the optimal torque T opt [START_REF] Martinez | Slidingmode control for DFIG rotor-and grid-side converters under unbalanced and harmonically distorted grid voltage[END_REF]. The control is a compromise between conversion efficiency and torque oscillation smoothing.

The reactive power is expressed as follows.

s sq sd sd sq Q V I V I = - (8)
Adapting [START_REF] Beltran | Second-order sliding mode control of a doubly fed induction generator driven wind turbine[END_REF] to our hypotheses, it comes then

s s s s rd s s V V M Q I L L φ = - (9)
As the stator reactive power reference is zero, then

_ s s s r dr e f s s V V I M φ = → = ω ω (10)
Let us consider the following tracking errors.

d M e V RI g L I I L Ld t V R I g L i M e p T M g L L L ⎧ ⎛ ⎞ φ = - + ω σ - - ⎪ ⎜ ⎟ σ ⎝ ⎠ ⎪ ⎪ - -ω σ ⎨ ⎛ ⎞ ⎜ ⎟ ⎪ = - φ - ⎜ ⎟ ⎪ -ω φ σ ⎜ ⎟ ⎪ ⎝ ⎠ ⎩ (12)
If we define the functions G 1 and G 2 as follows.

d M G g LI I L L d t M M G p g L I g T L L L ⎧ ⎛ ⎞ φ = ωσ - - ⎪ ⎜ ⎟ σ ⎪ ⎝ ⎠ ⎨ ⎛ ⎞ ⎪ = - φ -ω σ -ω φ - ⎜ ⎟ ⎪ σ ⎝ ⎠ ⎩
Thus we have

V G RI L L M M e p V G p RI L L L L Γ ⎧ = + - ⎪ σ σ ⎪ ⎨ ⎪ = - φ + + φ ⎪ σ σ ⎩ (13)
Now, lets us consider the following high-order sliding mode controller based on the supertwisting algorithm.

⎧ = + + ⎪ σ ⎪ ⎪ = + ⎪ ⎨ ⎪ = - - φ ⎪ σ ⎪ = - ⎪ ⎩ (14)
Where the constants B 1 , B 2 , B 3 , B 4 , Φ 1 , and Φ 2 satisfy (15). (

)

1 1 1 1 1 2 1 1 2 2 2 1 1 2 2 2 3 2 2 2 3 4 2 2 3 2 4 , 4 , s s r r r r G B M B p B L L L B G B B B L L B ⎧ < Φ ⎪ ⎪ Φ +Φ > φΦ ≥ ⎪ σ σ -Φ ⎪ ⎨ < Φ ⎪ ⎪ Φ +Φ Φ ⎪ > ≥ ⎪ σ σ -Φ ⎩ (15)
Thus, we can assert that there exist finite times t Tem and t Ird leading to (16).

This means that the control objective is achieved.

T T t t = ∀ > ⎧ ⎪ ⎨ = ∀ > ⎪ ⎩ (16)
The above proposed high-order sliding mode (a second) control strategy for a DFIG-based WT is illustrated by Fig. 3.

V. SIMULATIONS USING FAST CODE

Simulation using FAST with Matlab-Simulink ® have been carried out on the NREL WP 1.5-MW wind turbine (Fig. 4) using turbulent FAST wind data shown by Fig. 5 [11]. The wind turbine, the DFIG ratings, and control parameters are given in the Appendix.

An interface has been developed between FAST and Matlab-Simulink ® enabling users to implement advanced turbine controls in Simulink convenient block diagram form (Fig. 6). Hence, electrical model (DFIG, grid, control system, etc.) designed in the Simulink environment is simulated while making use of the complete nonlinear aerodynamic wind turbine motion equations available in FAST (Fig. 7).

A. Frequency Variation

The proposed HOSM-based FRT strategy will be first tested regarding a frequency variation. Indeed, the studied case is a frequency fall from 50 to 48 Hz as illustrated by Fig. 8. This case corresponds to a power generation loss that could lead to a rapid decrease of the grid frequency.

The FRT performances are illustrated by Fig. 9. It clearly shows that the frequency fall has practically no effect on the torque. Indeed, a good tolerance is achieved for this type of fault.

B. Unbalanced Voltage Sags

When unbalanced sags occur (Fig. 10), very high current, torque, and power oscillations appear at double of the electrical frequency, forcing a disconnection. The FRT performances are illustrated by Fig. 11 where an almost constant torque is achieved. Moreover, good tracking performances are also achieved in terms of DFIG rotor current (Fig. 12).

VI. CONCLUSION

This paper dealt with a high-order sliding mode control of a doubly-fed induction-based wind turbine for fault ride through. The HOSM-based FRT strategy has been tested regarding grid frequency variations and unbalanced voltage sags on a 1.5-MW three-blade wind turbine using the NREL wind turbine simulator FAST. The achieved results show promising successful ride-through performance.

Fig. 1 .

 1 Fig. 1. Frequency-voltage variations ranges [© Nordel] [1].

Fig. 3 .Fig. 5 .

 35 Fig. 3. The proposed FRT control structure.

Fig. 6 .

 6 Fig. 6. FAST wind turbine block.

Fig. 7 .

 7 Fig. 7. Simulink model.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Grid voltage.

Fig. 10 .

 10 Fig. 10. Grid voltage.

Fig. 11 .

 11 Fig. 11. Torque tracking performance during unbalanced voltage sags:Reference (blue) and real (green).


Ivanovitch Silva

Luiz Affonso Guedes

Common Cause Failure Analysis for Wireless Sensor Networks

Keywords: Common cause failure, fault tree, reliability, availability, wireless sensor network

Simultaneous failures of multiple devices make the dominant contribution to the unreliability of wireless sensor networks. They can hamper communications over long periods of time and consequently disturb, or even disable, the management algorithms of the network. In this preliminary work, we consider two types of common cause failures: hardware, and the temporary disruption of links. We propose an evaluation methodology based on the fault tree formalism for analyzing the reliability and availability of any wireless sensor network when common cause failures are considered.

I. INTRODUCTION

In wireless sensor networks (WSN), the failure of a single device might not be critical to the applications due to its intrinsic redundancy. However, when simultaneous failures occur with multiple devices, the consequences are likely to be disastrous, particulary for critical applications (patient surveillance, industrial environments, safety monitoring) [START_REF] Dilmaghani | Wireless sensor networks for monitoring physiological signals of multiple patients[END_REF]. This type of failure is known as common cause failure (CCF).

It is very important to measure the impact of CCF as soon as possible, ideally in the early phases of planning and designing the network. When done properly, such an early evaluation can anticipate decisions regarding the topology, criticality of the devices, the levels of redundancy, and network robustness. A tentative effort to evaluate a WSN considering CCF was performed in [START_REF] Shrestha | Infrastructure communication reliability of wireless sensor networks[END_REF], but the technique was focused on a single cluster. By introducing the concept of coverageoriented reliability, the same authors extended the previous work in [START_REF] Shrestha | Infrastructure communication reliability of wireless sensor networks[END_REF] to support a more flexible way to configure failure conditions [START_REF] Shrestha | Modeling and evaluating the reliability of wireless sensor networks[END_REF]. However, it is not possible to create two or more coverage subsets for the same cluster. In [START_REF] Xing | Infrastructure communication reliability of wireless sensor networks considering common-cause failures[END_REF], the effects of CCF on a WSN were also evaluated. The authors proposed a progressive scheme based on binary decision diagrams for evaluating any WSN topology. However, the model supports neither generic network failure conditions nor an importance analysis of the devices.

The main focus here is to investigate the influence of common cause failures on a WSN. It becomes clear from the above discussion that previous research has only provided a partial solution for this problem. We proposed a methodology based on the fault tree formalism in [START_REF] Silva | Reliability and availability evaluation of wireless sensor networks for industrial applications[END_REF] to evaluate the reliability and availability of a WSN, supporting the definition of flexible network failure conditions. Here, we extend this methodology to support CCF for both hardware and links.

II. EVALUATION METHODOLOGY INCORPORATING COMMON CAUSE FAILURES

An overview of the methodology adopted in this preliminary work is given in Fig. 1. The process starts by providing information about the network topology, device types, device failure and repair processes, CCF data, and network failure condition. The latter expresses the conditions that may lead to a network (system) failure, and is defined by a logical expression that combines the failure status of the devices. The next step is to find all the paths between the sink and devices that encompass the failure condition. This is necessary for attaining flexible failure conditions and for supporting selfhealing routing protocols. In the following step, a fault tree is generated using all the previous data. Finally, the metrics of interest (reliability, availability, MTTF, and importance measures) are computed.

A. Assumptions

The main assumptions of this methodology are summarized as follows:

• Faults: only permanent faults are considered. The links, due their wireless nature, are more affected by transient faults (millisecond scale). However, temporary barriers or bad weather conditions can obstruct a link for long hours. This case is also considered as a permanent fault. Failure occurrences are characterized by a failure distribution, by means of a CDF (cumulative distribution function). Any type of CDF can be used to describe their occurrence.

• CCF: the model support two types of CCF: hardware and link. The former is caused by shocks or other actions that damage the device hardware, whereas the latter is caused by temporary interruptions of a link (occurrence milliseconds). Depending on the scenario, the CCF can be fatal or non-fatal to the network.

• Network failure condition: the network failure condition (NFC) defines which combination of devices may lead to a network failure and its equivalent to the TOP event of fault tree. The methodology used in this letter supports any combination that can be expressed using boolean operators (i.e., AND, OR, K-out-of-N).

B. Device Failure Condition

After defining the NFC, it is necessary to define the conditions that may lead to the failure of a device. According to Fig. 2a, a device is considered faulty if all the paths between it and the sink fail (event cp -conectivity problem). On the other hand, as described in Fig. 2b, a path fails if at least one device along that path fails. Finally, a device also can fail if its hardware fails (event hw) or if at least one CCF occurs (Fig. 2c).

Note that it is necessary to exert some effort to find all combinations that may lead to a connectivity failure. In order to attain this, it is necessary to search all paths between the sink and devices that belong to the NFC. The paths are found by performing a depth-first search (DFS) in an adjacency matrix that represents the network. All the paths generated are then stored in a data structure based on a fault tree.

III. PRELIMINARY RESULTS

In this section, we evaluate the common cause failures for wireless sensor networks. The main target of the analysis is to highlight the influence of common cause failures upon network reliability. We assume a wireless sensor network mesh topology typical for condominium monitoring applications to validate the idea (Fig. 3). In this scenario, we consider that the network fails if at least three devices fail. Regarding the failure properties, we assume that device failures occur at a constant rate. In order to simplify the procedure here, we assume that all devices have the same failure rate (λ ≈ 1E -5).

The evaluation measures the influence of CCF on the network reliability. The results are described in Fig. 4. Despite the events CCF 1 and CCF 2 having different configurations, when both events occur the influence on network reliability is similar. This behavior results from the difference in criticality of devices 1, 2, and 7. Note that the network reliability decreases quickly when the events CCF 1 and CC2 are designed together. This scenario can even be pessimistic, but when compared to with the scenario without CCF, we observe that no consideration of CCF for network reliability is a very unrealistic assumption.

IV. DISCUSSION AND FUTURE WORK

In this preliminary work, we proposed an evaluation model for wireless sensor networks considering common cause failures. The proposal is based on the fault tree formalism and it considers hardware and link failures. A mesh topology was used for its validation (network reliability). It is also possible to analyze the network availability, and the criticality of the devices. The result showed the importance of considering CCFs during the design of a network, mainly when multiples CCF can occur. This analysis can be used to design any wireless sensor network. In future research, we plan to extend this methodology to analyze the redundancy, coverage factor, and hierarchical models.

Fig. 1 .Fig. 2 .

 12 Fig.1. Overview of the methodology for common cause failure analysis.

Fig. 3 .

 3 Fig.3. Topology adopted in the evaluation process.

Fig. 4 .

 4 Fig. 4. Network reliability analysis considering common cause failure.

Availability MTTF Component importance


Felipe D Cunha
email: fdcunha@dcc.ufmg.br

Aline Carneiro Vianna

Raquel A F Mini
email: raquelmini@pucminas.br

Antonio A F Loureiro
email: loureiro@dcc.ufmg.br

How effective is to look at a Vehicular Network under a Social Perception?

Vehicular Mobility is strongly influenced by the speed limits and direction of the public roads. At the same time, the driver's behavior produces great influences in vehicular mobility. People tend to go to the same places, at the same day period, through the same trajectories, which le ad them to the appearance of driver's daily routines. These routines lead us to the study of mobility in VANETs under a social perspective and to investigate how effective is to explore social interactions in this kind of network. The work herein proposed presents the characterization and evaluation of a realistic vehicular trace found in literature. Our aim is to study the vehicles' mobility in accordance to social behaviors. With our analysis is possible to verify the existence of regularity and common interests among the drivers in vehicular networks. Finally, we discuss how the social metrics may be used to improve the performance of protocols and services in Vehicular Networks.

I. INTRODUCTION

Vehicular Ad hoc Networks (VANETs) are a special type of Ad hoc networks formed by vehicles with processing and wireless communication ability, traveling on streets or highways. Commonly, the vehicles can communicate directly or by the use of a roadside unit. Through this structure, vehicles can access network services and obtain data from other networks, such as Internet. Due to this nature, VANETs can be established in different environments such as in urban centers and highways [START_REF] Boukerche | Vehicular ad hoc networks: A new challenge for localization-based systems[END_REF], [START_REF] Faezipour | Progress and challenges in intelligent vehicle area networks[END_REF].

VANETs are proposed to assist the drivers in the transit, to avoid collisions, to provide entertainment to drivers and passengers, and to contribute to the creation of an intelligent traffic system. An important application of this type of network is to alert the driver about road conditions and imminent crashes. Recent studies show that 60% of car crashes could be avoided if the drivers were warned some seconds before the collision [START_REF] Faezipour | Progress and challenges in intelligent vehicle area networks[END_REF]. In this context, the interaction among vehicles can strongly contribute to broadcast an alert message and reduce the crash ratio [START_REF] Schoch | Communication patterns in vanets[END_REF].

The communication in VANETs is influenced by mobility patterns of the vehicles: they move and stop according to the speed limits and directions imposed by the public roads. Furthermore, the day period can influence the vehicles mobility. In rush hours, the traffic can be slower and traffic jams can force the driver to stop the car or to change the route. This is a scenario where the vehicles density will be higher. In other hand, weekends and early mornings are considered quiet traffic scenarios, with low vehicles density. This density variation in different hours reinforces the characteristic of dynamic topology of VANETs, making the communication a challenging task in some scenarios.

The mobility is also influenced by the behavior of the driver and its routine. At the weekends, destinations like camping, malls, churches are chosen to leisure and entertainment, and are frequently visited. On the contrary, on weekdays people tend to repeat their paths at the same time and to the same destination, such as: school, work, university, restaurant, etc. In this context, it is likely to find vehicles sharing the same destination and having similar behavior, which suggests studying the mobility in VANET under a social network perspective.

In his paper entitled Small-World Problem [START_REF] Milgram | The small world problem[END_REF], Milgram presents the concept of social networks, showing that people with similar interests tend to relate among themselves. Other important concept is that there are people responsible for connecting groups of people that have a large degree of interaction in the network. In this work, we present a study of the social metrics applied in VANETs. Although similar evaluations can be found in literature [START_REF] Fiore | The networking shape of vehicular mobility[END_REF], [START_REF] Loulloudes | The dynamics of vehicular networks in urban environments[END_REF], [START_REF] Liu | Exploring social properties in vehicular ad hoc networks[END_REF], our work differs from these by the fact that social metrics are used to characterize vehicles' mobility in realistic dataset. We also discuss how these metrics can be applied to improve the performance of some services or existing infrastructures in VANETs. In this context, we focus on the investigation of the following question: How effective is to look to a vehicular network under a social perception, considering the social behavior of the drivers? Through numerical analysis, it is possible to identify peculiar social characteristics in VANETs.

This paper is organized as follows. In Section II, we present a brief survey of the relevant related work. In Section III, we define the dataset used in the analysis, the metrics and the main assumptions of the modeling. In Section IV, we present the quantitative results according to each evaluated metric. In Section V, we discuss the perspective of applying the evaluated metrics on the design and evaluation of the VANETs protocols. Finally, in Section VI, we present the final conclusions of this work and future works.

II. BACKGROUND

In this Section, we survey available traces describing vehicular mobility as well as related works found in literature.

A. Vehicular Traces

Vehicular traces are a special dataset type that contains information about the trajectory of vehicles. Such traces are attractive since they exhibit the real behavior of vehicles in a certain scenario. In literature, there are real and realistic traces. The real traces contain information collected from a group of vehicles, with the use of some localization system device (e.g. GPS). The realistic traces are created by the junction of maps of particular locality, traffic information of this locality, and a mobility' simulator. In this section, we describe traces commonly used to validate protocols.

San Francisco: This real trace describing mobility of taxi cabs in San Francisco [START_REF] Piorkowski | CRAWDAD data set epfl/mobility (v. 2009-02-24[END_REF]. Each taxi has a GPS receiver, and the trace contains GPS coordinates of 500 taxis, collected over 30 days, from May 17 to June 10 of 2008.

Zurich: A multi-agent traffic simulator, developed by ETH Zurich, created a realistic trace for the city of Zurich, Switzerland [START_REF] Naumov | An evaluation of inter-vehicle ad hoc networks based on realistic vehicular traces[END_REF]. This trace contains public and private traffic with high levels of realism, reproducing the behavior of a large city during 24 hours. With 25, 000, 000 records (direction/speed) of 260, 000 vehicles, the trace covers an area of 250 km×260 km.

Shanghai: The SUVnet-trace is a real trace that has the GPS information from more than 4, 000 taxis in the Shanghai city [START_REF] University | Suvnet-trace data[END_REF]. This taxi group has been observed during 28 days, from January 31 to February 27 of 2007.

Cologne: The TAPASCologne Project is a realistic trace that reproduces with the highest realism the car traffic in the Cologne city, in Germany [START_REF] Wagner | Mikroskopische modellierung der personenverkehrsnachfrage auf basis von zeitverwendungstagebchern[END_REF]. The process to make this trace includes important tools of the literature like: OpenStreetMap, SUMO simulator mobility, the information of traffic demand on the city. It brings information of 700, 000 records of individual car trips, during 24 hours in an area of 400 km 2 .

Luxembourg: This is a realistic trace that presents commuting traffic over the area of Luxembourg [START_REF] Pigné | A vehicular mobility model based on real traffic counting data[END_REF]. It has been generated by the VehiLux model and augmented with Gawron's dynamic route assignment algorithm. The trace cover an area of 51km×44km during 11 hours and has 110, 188 records.

B. Related Work

The concepts of Social Networks have been explored in different research areas, mainly to understand the human mobility. In this context, the focus is to analyze physical encounters of the nodes in order to understand the network topology and to improve the performance of the network protocols. Moreover, in VANETs, social analysis can be used to assist the design of protocols, aiming the provision of better services to vehicles. Furthermore, it is important to observe the evolution of the traffic in specific areas and over the time, in order to improve protocols and services performance in VANETs.

In [START_REF] Fiore | The networking shape of vehicular mobility[END_REF], Fiore et al. present a deep analysis of the topological properties of VANETs. The authors use social metrics to investigate the temporal evolution of network topology. Results show how it is possible to take advantage of the vehicular mobility to improve the performance of network protocols. This work is based only in analytical mobility models.

In the same context, in [START_REF] Loulloudes | The dynamics of vehicular networks in urban environments[END_REF], the authors discuss how social metrics can be employed to improve the performance of the routing protocols in VANETs. The analysis, however, is based on a short period of only 2 hours. We argue that short observation period is not enough to extract social behavior in vehicular mobility.

Other social metrics investigation is presented in [START_REF] Liu | Exploring social properties in vehicular ad hoc networks[END_REF]. This work discusses several universal laws of social networks and presents numerical results of the real traces' evaluation. The authors have used two publics dataset San Francisco and Shanghai, describing taxis movements. The social metrics are estimated, and they show the existence of social routine in the VANETs, including the power-law distribution and small world phenomenon. This work however, only considers traces describing mobility of cabs, which has random trajectories with low probability to have routines or similar behaviors.

In [START_REF] Uppoor | Insights on metropolitan-scale vehicular mobility from a networking perspective[END_REF], it is presented a macro and microscopic evaluation of the Cologne trace. The macroscopic analysis involves the evolution of vehicular density and the dynamics of large-scale flows of vehicles through the metropolitan region. In addition, the microscopic analysis considers the distribution of vehicles in the area and the encounters among them. Through the performed evaluation, it is possible to observe the evolution of the traffic over the region and period. Also, the authors show the importance of considering the dynamics of the road traffic in the design of protocols and services to VANETs.

III. METHODOLOGY

This section details the studied trace, the modeling used to create the social graph and the evaluated metrics.

A. Trace Characterization

We decided to use the Zurich Trace [START_REF] Naumov | An evaluation of inter-vehicle ad hoc networks based on realistic vehicular traces[END_REF]. This choice is due to two factors: (i) it represents a trace with records of personal vehicles describing people routines in an urban area; Traces such as San Francisco and Shanghai describe taxis decisions that are strongly dependent on the decisions of passengers that are randomly collected on the streets; (ii) the 24h of records are freely available, contrarily to 2h of the Cologne trace and the 11h of the Luxembourg trace. The Zurich trace represents the evolution of the urban traffic along the day, including the variation like rush hours and daybreaks. We can see this evolution in Figures 1(a) and 1(b). In these figures, it is possible to observe two intervals time that have the major amount of vehicles, called rush hours: 6 -9am and 2 -5pm.

The average speed is 25 km/h and in Figure 1(a) we can see this variation. The minimum speed is around 1 km/h, which represents the presence of traffic jam in some places. Moreover, the maximum speed remains constant, around 30 km/h. This value is attributed to the vehicles far from downtown, which travel through streets with low traffic.

The whole trace has movement information of 260, 000 vehicles, and in Figure 1(b), we can observe the distribution of these vehicles along the day. It is possible to see a concentration of vehicles in rush hours: 173, 470 vehicles at 7am, and 132, 633 vehicles at 3pm. When we compare Figures 1(a) and 1(b) in rush hours, it is possible to perceive a decrease in speed. This happens due to the high number of vehicles in the streets and highways, leading the drivers to reduce the speed or to stop the vehicles. We show in Figures 2(a-d) the heat map of the traffic evolution in the area. We choose to use the traffic from 6 to 7am, and from to 2pm to 3pm, in order to demonstrate the vehicles distribution over the city. In these figures, the black color represents the area with high vehicles density, and the yellow color represents the area with low vehicles density. As can be seen, the streets with black color in the center of figure represent the high traffic density at the center of Zurich. In addition, away from the center, the yellow color represents the low density streets. When we compare Figures 2(a) and 2(b), we observe the increase of traffic in the center. We also note the same behavior with Figures 2(c) and 2(d), which corresponds to the evolution of traffic presented in the Figure 1(b).

B. Modeling

The model considers the encounter among vehicles, i.e., when two vehicles are within communication range of each other. In VANETs, the vehicle-to-vehicle communication might be explored in order to increase the probability of the message to reach the destination and consequently, to increase the delivery ratio. Thus, the better understanding of the evolution of these encounters can help us to improve the network communication.

In our evaluation, we consider communication range of 100 m among vehicles, according to the protocol 802.11p. The Zurich trace describes mobility in urban areas and highways, where the average speed is 25 km/h. Therefore, according to this speed and the communication range, we consider the minimum contact time of one encounter equal 15 seconds(s). Thus, if two vehicles are inside each others communication range, at least for 15 s, we consider that an encounter happened.

Then, we map vehicles mobility and encounters described in the trace into a temporal graph. We first divide the whole trace into discrete time slots of duration t = 1h. Since vehicle traces usually describe fast changes in the topology, short time slots are recommended to better capture traffic changes. In addition, traffic peaks are usually assumed to happen in different hours of the day.

The graph at the time t is an undirected and can be formally defined as a graph G(t) = (V, E), where V represents the set with all vehicles v i and E represents the set of edges e ij . In the temporal graph G(t), an edge e ij (t) exists between the vehicle v i and v j during time t, with i =j. All metrics are evaluated hourly, considering the temporal graph G(t). Considering the fact that the Zurich trace describes 24h of vehicle mobility, 24 graphs are generated, describing the vehicles encounters in time windows of 1 h.

C. Metrics Evaluated

Below the social metrics chosen for this evaluation will be described considering the aspects and characteristics of the mobility of vehicles. As for vehicles mobility, we classify these metrics in macroscopic and microscopic metrics [START_REF] Uppoor | Insights on metropolitan-scale vehicular mobility from a networking perspective[END_REF]. The macroscopic metrics represent measures of the network global state, which can portray the general behavior of all vehicles and the evolution of the temporal graph. The microscopic metrics define individual values for the vehicle representing the behavior of a unique vehicle. We select as macroscopic metrics the Distance, the Density and the Edge Persistence, and as microscopic metrics, we select the Node Degree, the Cluster Coefficient, and the Closeness Centrality.

Macroscopic Metrics:

Distance: this is a metric that comprises the length of a path between a pair of vehicles in terms of the number of hops. In VANETs, the distance can represent the existence of common interest among the drivers. If the distance between the vehicles v i and v j is high, it implies that they are physically separated, and probably they not visit nearby places. Otherwise, if the distance is small, v i and v j may tend to visit close places or have congruent trajectories. Generally, graphs with small world behavior have small distances.

Density: Represents how dense is the network, i.e., the amount of connections existent between the vehicles. In the context of VANETs, urban regions can have higher densities than rural areas. This is particularly correct in downtown region, where the traffic is slow and there are traffic jams. This metric compute the ratio between the number of existing edges e(t) in the graph and the number of edges of a complete graph |V ||V -1|/2.

Edge persistence: is a metric that represents the persistence of an encounter among two vehicles. In this analysis, we consider the edge persistence as the number of times the vehicles v i and v j encounter over the time. We compute this metric in function of the edge weight, which represents the number of times that this encounter e ij (t) happens.

Microscopic Metrics

Node Degree: determines the number of distinct encounters that a vehicle has during a period. This metric can be influenced by the trajectory and the period of the day. If a vehicle passes through a region with higher density traffic or in a rush hour, its degree tends to be high. In other hand, in regions far from the downtown or in hours with low traffic, its degree can be low or null. As a formal definition, a vehicle v i at time t is defined by: Degree i (t) = ||{v j |∃e ij (t)}||.

Cluster Coefficient: this is a metric that evaluates how close are the neighbors of a vehicle v i in the graph, i.e., the probability that two neighbors of v i have already met. This metric can represent regions with traffic jam, describing a high cluster coefficient. Besides, if the network presents the small world phenomenon, its cluster coefficient will also be elevated [START_REF] Watts | Collective dynamics of 'smallworld' networks[END_REF].

Closeness Centrality: This metric measures the centrality of the vehicle according to its distance to the others vehicles in the graph. The more central a vehicle is the lower is its total distance to all others vehicles. Despite being a global metric, its value can also show how central is the trajectory of a vehicle. Likely, a vehicle that travels only in downtown tends to have a low closeness centrality.

IV. NUMERICAL RESULTS

In this section, we focus on the analysis of the metrics discussed in the Section III. All results are computed using the temporal graph G(t), when t = 1h.

A. Distance

It is worth to mentioning this distance metric considers the number of edges that are between two nodes, without ponder their edge weight. In VANETs, this is a metric that can represents if two vehicles have visited common locations in close time, i.e., vehicles with the similar routine tend to have little distance. On the other hand, vehicles that travel in different times, or through distant regions tend to have big distances. The graphic in the Figure 3(a) shows the average of the graphs distance over time. It is possible to observe the behavior of the distance when the number of vehicles increases in network.

In the period where there is no traffic, the value of the distance is represented as 0, and the distance is not considered. From 4am, the distance increases with the growth of the number of vehicles. However, it is possible to note that the measured distances remain short despite of their increase when we compare to the number of vehicles circulating on the streets. For instance, the measured average distance at 7am is of 30 hops, while the number of vehicles is 173, 470. Other important point is that there is a similarity in the distance when we compare the two rush hours. In both hours, the distance increases with the traffic intensify. This happens because, when the number of vehicles in the street increases, there is a tendency to form queues of cars, and consequently increase the distance in the graph. Considering the number of vehicles and the length of distances, it is possible to conclude that this graph presents a small world behavior (i.e. great number of nodes and short distances present in social networks).

B. Density

In this analysis, we evaluated the global density of the graph. Figure 3(b) shows the evolution of the network density over time. It is possible to observe that the evolution of the density does not exceed the value of 2%. Despite of the significantly traffic increase during rush hours, the density remains lower. This happens because the streets have direction restrictions and speed limits, which define the vehicles trajectory and the set of vehicles to possible encounters. Other factor that contributes to the low density is the traffic jam, which reduces the speed of the vehicles and prevents new encounters. In this situation, the vehicle tends to keep the same neighbors while it passes in a specific region. Furthermore, the density evolution shows that the limits imposed by public roads have an influence in the encounter probability of vehicles.

C. Edge Persistence

Persistence encounters happen due to drivers sharing similar routines, e.g. pass through the same path to go to work or back to home at the same hours. Hence, edge persistence may be used to identify mobility similarity among vehicles. We measure the edge persistence of vehicles at the two identified rush periods of the day, due to their higher number of vehicles and encounters. The repeated encounters percentage is then computed based on the comparison of the number of encounters happened among the periods 6 -9am and 2 -5pm.

In the former period, 376, 245 encounters happened while 338, 513 encounters happened in the latter period. The edge persistence for this analysis is 43% of the edges, indicating that a high amount of vehicles tend to have similar mobility in the city, with similar routines. Besides the restrictions imposed by streets, traffic lights, and speed limits, the path decisions of the drivers have a strong influence in the definition of their trip time and trajectory.

D. Node Degree

As previously discussed the traffic density in a metropolitan area varies over the day. In Figure 5(a), we can observe this behavior. The rush hours present an increase in the max, average and median degree of the vehicles. At 7am, the graph presents the highest degree value of 1, 355, which may describe a vehicle that only crosses the streets with high traffic density and makes a big trip. A high degree value can be also seen at 4pm. However, when we look at the average value, at least 50% of vehicles present an increased, due to the traffic daily evolution. This is an acceptable behavior, because with the increase of vehicles it is expected a greater amount of encounters.

Many social networks have been considered as scalefree networks, i.e., networks that have a degree distribution following a power-law behavior. In Figures 4(a-d), we present the degree distribution in rush hours. In these hours, we have approximately 100, 000 vehicles in each period. The curves in the graphs show that a power law describes well the degree distribution, which suggests that the Zurich graph is scale-free. We can see that few vehicles have a high degree, and the major amount of vehicles have a low degree, with values close to 10 2 .

E. Cluster Coefficient

The cluster coefficient is a very important metric in social networks, since it represents the organization of the graph and how close are the neighbors of a node in the graph. In the context of a VANET, the cluster coefficient can aggregate the vehicles that go to the same direction, that have the same interests, or that pass through the same place. Figure 5(b) presents the max, average, and median value of the cluster coefficient for temporal graph G(t) during a day. It is possible to observe that the max cluster coefficient is 1 in rush hours, showing that there are vehicles that all of their neighbors are connected. In metropolitan areas, traffic jams and intersection streets favor the clusters appearance, composed of vehicles sharing the same road and traffic conditions.

When we look at the average and median cluster coefficient values, we observe their increase during the rush hours. In particular, at 7am and at 4pm the average cluster coefficient is 0.5. This represents in average, 50% of the vehicle's neighbors had an encounter over the time. It is interesting to note that a network with a small world behavior has a high value of cluster coefficient, and the Zurich trace tends to perform as a small world network when the density of vehicles is high. This is due to the fact that with more vehicles, more encounters happen and more groups are formed. In addition, traffic jams contribute to form vehicles groups that have common junction points in their trajectories. Finally, it is worth to note the proximity between the average and the median values of the cluster coefficient, which is a good indication that the average is very representative value of the network cluster coefficient.

F. Closeness Centrality

The evaluation of the closeness centrality in a graph is important to understand how close are the nodes. In VANETs, this metric can quantify the proximity between the vehicles, and consequently better understand the mobility pattern. The Figure 5(c) present the max, average and median values of the closeness centrality during a day.

By definition, the higher the value of the closeness of a node, the lower is relative distance to other nodes. Thus, in the graph, we can observe that the curve of closeness increases with the increase of vehicles number. This is due to the fact that with more nodes in the graph, there is a tendency to create more shortcuts between the nodes, and the distance tends to decrease. We can also observe that the maximum closeness value is close to the average and median. As the closeness considers the distance to all nodes, with a high number of nodes, this value tends to be low. This happens mainly in cases where we have many nodes with low degree. On the other hand, in periods of low traffic the value of closeness is very low, which indicates that the graph does not have a central vehicle.

V. SOCIAL PROPERTIES APPLIED

In this work, we have analyzed a vehicular trace under a social perspective, aiming to characterize it according to metrics that lead us to comprehend the vehicular mobility. It is possible to make use of these metrics to improve the network performance of communication protocols and services, aiming to help drivers on their trip. In the following, we will discuss scenarios where the previously analyzed metrics can be used to improve the VANETs performance.

Infrastructure: The communication between vehicles can happen in three modes: vehicle-to-vehicle, vehicle-to-infrastructure, and in hybrid way, i.e., by combining the two others modes. In this context, determine the best locations to deploy roadside units, with the goal of providing continuous connectivity to drivers during their trajectories becomes a challenge. In macroscopic metrics, the information about the vehicles flows in the network can help in this task. For instance, the density metric that measures the number of encounters over time can be used together with the physical position of vehicles to assist in the deployment of new roadsides units, in order to maximize the network coverage. Using this metric, it is possible to detect places demanding high number of connections, which can require more roadside units installed, guarantee a good Internet access.

Network Protocols: Many are the challenges for the communication protocols in this type of network, such as the variation in density and mobility. Thus, providing secure communication and message delivery becomes a challenging task in many scenarios. An important issue for communication is to ensure message delivery in situations where the connection time between vehicles is minimal, and it is necessary to reach the largest number of vehicles as possible. In this case, making use of information provided by microscopic metrics can help deliver the message more efficiently. An example is the use of metric node degree, which describes the number of neighbors of a vehicle. Thus, choosing a vehicle more popular, i.e., with a larger number of neighbors, increases the chance of delivering the message to a wider group. Another important microscopic metric is the closeness centrality, which describes how central is the node in terms of the proximity to other nodes. The choice of central nodes to start the dissemination of information can ensure a wider delivery of the message. The metric density can also be used to promote the routing in the network, avoiding places with low density or disconnected.

Vehicular Applications: Applications in VANETs are proposed in order to assist the driver and guarantee a safe direction. In this scenario, various types of applications can be offered as security alerts, travel assistance in roads, and entertainment. However, every application has the requirement to ensure good performance. Thus, microscopic and macroscopic metrics can be used in different contexts. For instance, the metric edge persistence can be used to identify the drivers who have the same routine and enable resource sharing among them. In other scenarios, the metric cluster coefficient may be used to target a specific group of users, who have a common interest, such as publicity, alert messages and sharing resources. Vehicles belonging to the same group tend to have interests in common, or be physically close. Finally, drivers habitually use different location systems to find places or to create paths to an informed destination. However, such paths are usually built without considering the live information of the vehicles traffic. In this scenario, it is possible to use the metrics density of a region or the node degree of the vehicles, to infer traffic conditions, improving the suggested route.

VI. CONCLUSION

In this work, we presented the social analysis of vehicular trace describing mobility in a large-scale metropolitan area. Using the realistic Zurich trace, we have characterized the mobility of vehicles according to macroscopic and microscopic social metrics. We showed the existence of similar behaviors and routines in the vehicles mobility. Furthermore, it was shown that the vehicles mobility could be characterized as a social network, following the laws of the degree distribution and the small distance between the nodes. It was also possible to verify that the use of social metrics can help in improving the performance of protocols, network infrastructure, and to propose new services to assist the drivers. As future works, we intend to extend the evaluation to traces with longer duration, aiming to analyze drives routine during different days of the week.

  Time of day(h)(b) Daily Traffic.

Fig. 1 .

 1 Fig. 1. Daily evolution of the Zurich trace.

Fig. 2 .

 2 Fig. 2. Heat map of traffic evolution in rush hours.

Fig. 3 .

 3 Fig. 3. Daily evolution of macroscopic metrics.

Fig. 4 .

 4 Fig. 4. Vehicle degree evolution during the day.

Fig. 5 .

 5 Fig. 5. Daily evolution of microscopic metrics.


Piero Baraldi

Michele Compare

E Zio

Component ranking by Birnbaum importance in presence of epistemic uncertainty in failure event probabilities

Keywords: Birnbaum importance measure, possibility theory, importance ranking

teaching and research institutions in France or abroad, or from public or private research centers.   Component ranking by Birnbaum importance in presence of epistemic uncertainty in failure event probabilities

Uncertainty in IMs

Consider a system made up of n components, assumed binary (i.e., they have only two possible states 'working' and 'failed'), and s-independent.

The Birnbaum IM   B j
It of the j-th component of such system at time t is given by [START_REF] Aven | On the use of uncertainty importance measures in reliability and risk analysis[END_REF]:

        B j j Ut It Qt    Q (1)
Simple analytical manipulation yields [START_REF] Zio | Computational Methods for Reliability and Risk Analysis[END_REF]      

B j j j I t U t U t    (2)
       1, jj U t U Q t   Q
is the system failure probability when component j is in failed state. It represents the maximum risk achievement if component j is considered failed [START_REF] Zio | Computational Methods for Reliability and Risk Analysis[END_REF].

       0, jj U t U Q t   Q
is the system failure probability when component j is functioning. It represents the maximum reduction in risk if component j is considered working [START_REF] Zio | Computational Methods for Reliability and Risk Analysis[END_REF]. Consider, as an example, the system in Fig. 1, and assume to know the exact values of the component unreliabilities, Q A , Q B , Q C . The system unreliability is ( , , )

A B C A B C A B C U Q Q Q Q Q Q Q Q Q   
The Birnbaum IMs of the components are then obtained by using (1) as However, the exact values of the component failure probabilities

1 B A B B C BB B A A C B C A B I Q Q Q I Q Q Q I Q Q                I (3)
          12 , ,..., n t t Q t Q t Q  Q are often
not precisely known in practical situations, e.g., because they are assigned by an expert, or estimated through statistical means. Then, our objective is to rank the components according to their Birnbaum IMs, taking into account the epistemic uncertainties in the component failure probabilities. In general, whichever is the framework adopted to represent the epistemic uncertainties, this entails three successive steps:

1. representation the uncertainties on the component"s failure probabilities, In the remaining part of this section, we will consider the example in Fig. 1 assuming that the uncertain unreliabilities j Q , j=A, B, C, are described by the Cumulative Distribution Functions (CDFs), ( ), j Qj Fq whose mean and standard deviations are reported in Table I. Notice that these distributions have been numerically obtained in [START_REF] Baraldi | A method for ranking components importance in presence of epistemic uncertainties[END_REF] by assuming that an expert is able to represent the uncertainties in the component failure rates using lognormal distributions, and by propagating these uncertainties onto the component unreliabilities.

Components importance ranking, under a probabilistic representation of uncertainty

Here we recall the procedure propounded in [START_REF] Baraldi | A method for ranking components importance in presence of epistemic uncertainties[END_REF] to rank the components according to their Birnbaum importance considering the probabilistic uncertainty in their IMs.

Let us consider two generic components k and s; to establish which is the most important, the distribution of the random variable ( , ) Finally, the relation order between components k and s is obtained by comparing ks r to a threshold range

[ ,1 ] l l
T T  , symmetric around 0.5, and considering the following criteria.

 If 1 ks l r T  then k is more important than s.  If ks l rT  then s is more important than k.  If 1 l ks l
T r T    then k is equally important to s. In this case, different kinds of additional constraints or targets should guide the ranking order (costs, repair times, failure impacts on public opinion, etc.).

In practice, the attention is concentrated on component s if the decision maker judges "large" enough (e.g., >0.7) the probability that component s is more important than component k.

Notice that there may be some cases in which the comparison of the Birnbaum importance of three generic components, j, k, s, may lead to a contradictory ranking which does not obey the transitive property,

Components importance ranking, under a possibilistic representation of uncertainty

Experts may not be willing to specify probability distributions of the parameters of the system model such as the component failure probabilities

          12 , ,..., n t t Q t Q t Q  Q
when the available information is incomplete, sparse, conflicting, vague, or non-specific [START_REF] Sentz | Probabilistic bounding analysis in the quantification of margins and uncertainties[END_REF]. Possibility theory has been proposed to deal with epistemic uncertainty in situations characterized by insufficient knowledge on parameter values.

Possibilistic representation of uncertainty in component unreliabilities

Let us consider the generic variable Q; in PT, uncertainty in this variable is represented by means of a possibility function () Q q  , which expresses the degree of possibility of each value q of the variable Q in a set S of being the true (but unknown) value of Q. When ( ) 0 Q q   , it means that the outcome q is considered an impossible situation. When ( ) 1 Q q   , it means that the outcome q is possible, i.e., unsurprising, typical, usual [START_REF] Dubois | Possibility Theory and Statistical Reasoning[END_REF]. These values are mutually exclusive becaue the uncertain variable can assume one true value only. This result also gives the normalization condition : ( ) 1

Q q S q     , which
claims that at least one value is viewed as totally possible, a much weaker statement than when the probability is 1.0 [START_REF] Dubois | Possibility theory, probability theory and multiple valued-logics: A clarification[END_REF].

A possibility distribution may also be viewed as a nested set of confidence intervals, which are the α-cuts

[ , ] { | ( ) } Q q q q q    of  . The degree of certainty that [ , ] qq  contains Q is ([ , ] ) Q N q q  1   if Q  is
continuous [START_REF] Baudrit | Joint Propagation and Exploitation of Probabilistic and Possibilistic Information in Risk Assessment[END_REF].

The possibility and necessity measures ()

Q ΠA , ( )
Q
NA for all subsets AS  are defined by the associated possibility distribution () Q q  through the following maximization and minimization relationships, respectively:

( ) sup{ ( )} Q Q qA Π A q    (4)
( ) 1 ( ) 1 sup{ ( )} Q Q Q qA NA Π notA q       (5)
Let P Q ( ) be a family of probability distributions such that for all events A,  

( ) ( ) Q Q Q N A P A ΠA   .
Then, ( ) inf ( )

QQ N A P A  and ( ) sup ( ) QQ A P A  (6)
where inf and sup are taken with respect to all probability measures in P Q [START_REF] Baudrit | Representing parametric models tainted with imprecision[END_REF]. Hence, the necessity measure is interpreted as a lower limit for the probability, and the possibility measure is interpreted as an upper limit.

Referring to subjective probabilities, the bounds reflect that the analyst is not able or willing to precisely assign his or her probability, and the bounds are the best he or she can do given the information available; in other words, he or she can only describe a subset of P Q which contains his or her probability.

On this basis, we can define the upper () Q Fq and lower ()

Q Fq cumulative distribution functions such that , ( ) ( ) ( ) Q Q Q q S F q F q F q     , with ( ) (] , ]) QQ F q N q    , and
( ) (] , ]) Q Q F q q    
(i.e., the generic set A in (4)-( 6) assumes here the form of ] , ] q  ). For the sake of brevity, in the present work the possibility, and necessity measures (] , ])

Q q    and (] , ]) Q Nq  are indicated with abuse of notation by () Q q  , and
() Q Nq , respectively. For further theoretical details, the interested reader may refer to [START_REF] Baudrit | Representing parametric models tainted with imprecision[END_REF]- [START_REF] Dubois | Possibility theory, probability theory and multiple valued-logics: A clarification[END_REF].

Various approaches for constructing possibility distributions have been proposed depending on the available information [START_REF] Baudrit | Joint Propagation and Exploitation of Probabilistic and Possibilistic Information in Risk Assessment[END_REF], [START_REF] Baraldi | Monte Carlo and fuzzy interval propagation of hybrid uncertainties on a risk model for the design of a flood protection dike[END_REF], [START_REF] Baudrit | Practical representations of incomplete probabilistic knowledge[END_REF]. In this work, we assume that the only available knowledge on the uncertain variable Q j is constituted by its mean * j q , and standard deviation j  . In this case, the Chebyshev inequality can be used to build a possibility distribution [START_REF] Baraldi | A comparison between probabilistic and Dempster-Shafer Theory approaches to Model Uncertainty Analysis in the Performance Assessment of Radioactive Waste Repositories[END_REF]. Such inequality defines a bracketing approximation on the confidence intervals around the known mean * j q of Q j knowing its standard deviation j  :

* * 2 1 ( [ , ]) 1 1 j j j j j P Q q a q a for a a         (7)
The Chebyshev inequality defines a possibility distribution ()

j Qj q  by considering intervals * * [ , ] j j j j
q a q a   as α-cuts of ()

j Qj q
 , and letting

* * 2 ( ) ( ) 1/ j j Q j j Q j j q a q a a        
. This possibility distribution defines a probability family which has been proven to contain all distributions with mean * j q and standard deviation j  , s-independently from the type of probability distribution, i.e, normal, lognormal, gamma, symmetric or not, unimodal or not, etc. [START_REF] Baraldi | A comparison between probabilistic and Dempster-Shafer Theory approaches to Model Uncertainty Analysis in the Performance Assessment of Radioactive Waste Repositories[END_REF].

With regard to the three-components case study of Fig. 1, we now assume that the only available knowledge is constituted by the means and standard deviations of the component failure probabilities (Table I), without any information on the type of distribution. The possibility distributions obtained by using the Chebyshev inequality are shown in Fig. 4 a, b, andc. Π q, and ()

j Q j
N q obtained from the corresponding possibility distributions () j Qj q  of Fig. 4 a, b, and c by using (4), and ( 5) respectively. In details, for a given value j q of j Q , the possibility measure

() j Qj q 
takes the maximum value of ()

j Qj q  
for any jj qq   , whereas the necessity measure takes the minimum value of 1-() j Qj q   for any jj qq   .

j j j Q j Q j Q j F q N q Πq  ,

Possibilistic representation of uncertainty in component Birnbaum IMs

To propagate the epistemic uncertainty from the component unreliabilities onto the component importance measures, the fuzzy extension principle is used [START_REF] Zadeh | Fuzzy Sets[END_REF]. In practice, the following steps are performed.

1. Select a value of α on [0,1], and take as the interval of possible values of the reliabilities of the

components the cut [ , ] { | ( ) } j j j j Q j Q Q q q     , j=A, B, C. For example, if α=1, then the 1-cut of the distributions () A QA q 
, and

) ( B Q q B  are 33 1 [ , ]
9.98e ,19.7e B j I j g  , respectively), when the elements of Q range within the intervals [ , ]

AA QQ       ,
jj QQ  ; that is, calculate , [ , ] inf ( ) B j ll l I j j l q Q Q g g     Q (8) , [ , ] sup ( )
B j l l l I j j l q Q Q g g     Q (9)
These results are the lower, and upper bounds, respectively, of the α-cut of the possibility distributions ()

B j B j I i  , j=A, B, C.
For example, with reference to the component C and (3), we have

B C l l l A B I C C A B l q Q Q q q e g g Q Q e                                Q 33 1
B C ll l A B I C C A B l q Q Q q qe g sup g Q Q e                        Q 3.
Repeat steps 1-2 for another value of α.

Components Importance Ranking

In this section, we present a procedure for ranking the possibility distributions representing the uncertainty in the Birnbaum IM of the system components. If, for example, we consider the possibility and necessity measures of components A, B, C, Figs. 3 a, b, andc show that the upper limit (possibility measure) of the probability that the importance of component C is smaller than 0.9 is 0, whereas the lower limit of the probabilities that the importance of components A and B are larger than 0.06 is 0. Thus, in this case one can conclude that component C is certainly more important than components A and B.

On the other side, ranking the IMs of components A and B is not straightforward, as their possibility and necessity measures overlap. This case calls for the development of a general procedure to rank the possibilistic IMs of the components. To this aim, the ranking procedure discussed in [START_REF] Baraldi | A method for ranking components importance in presence of epistemic uncertainties[END_REF] in the probabilistic framework is modified to take into account that the IMs are not represented by probability distributions but by families of probability distributions whose upper, and lower bounds are the possibility, and necessity measures, respectively. The procedure proposed in this work for the ranking of the IMs is based on the following steps. =1 for AC  ranging within the interval for every k, s, k≠s; in this regard, notice that (0)

1 1 1 1 [ , ] B B B B C A A C I I I I A C A C g g g g           =[
ks    if 0 B B s k I I k s g g     , and (1)
ks    if 0 B B k s I I g g   
 . In the reference example, we can see that the possibility (0

) AC   that AC  =0 is 1, whereas (1) 0 AC    .
2. Considering the lower threshold l T defined in Section 2.1, the relation order between the IMs of components k and s is established on the basis of the following criteria.

i.

If (0) ks l T    , then component k is more important than s.
ii.

If ( 1)

ks l T    , then component s is more important than k.
iii.

Components k and s are equally important in the other cases.

These criteria are justified by the following considerations.

 ( ) 1 ks ks  
 for at least one out of the two values of ks  (i.e., 0 and 1), being the distributions ks   normalized (i.e., there must be at least one point of the Universe of Discourse (UoD) in which the distribution reaches 1). Thus, it is not possible that both the above conditions i and ii are contemporarily verified.

 Let us consider a case in which (0) 0.

P P I I Π        .
To sum up, the probability that component k is more important than component s lies in the interval [0.8,1], whereas the probability of the opposite case is a value between 0 and 0.2. In this situation, in which we are confident of the relevance of k with respect to s, it is reasonable to decide in favor of component k.

According to the proposed criteria, the result of the comparison of the importance of two components, k and s, does not depend on the order in which they are considered because the possibility distributions The results of the application of the proposed procedure to the Birnbaum IMs of the three components A, B, and C of Fig. 1 are reported in Fig. 6. Fig. 6. Component C turns out to be the most important, whereas components A and B are equally important. This final ranking is different from that obtained in the probabilistic framework (Section 2.1), where component B is judged more important than component A. This result is due to the fact that in the probabilistic framework the analyst focuses on one out of the infinite probability distributions encompassed by a possibility distribution. Thus, as expected, the final ranking derived from the uncertain IMs depends on the information available: if the analyst believes that resorting to the probability theory framework is justified, then he or she will be capable of assessing that component B is more important than component A; whereas, in the opposite case, if the analyst is not able to specify a single probability distribution but prefers to use a possibility distribution, then he or she has not enough information to conclude that one component is more important than the other.

Analogously to the probabilistic case, if we consider the problem of ranking the importance of three generic components, j, k, s, one may obtain a contradictory ranking which does not obey the transitive In the case in which the system is made by several components, a sorting algorithm needs to be used to automatically order the components according to their importance. To this aim, a number of algorithms can be found in the literature, which have different computational complexities, memory usage strategies, etc.

[23], [START_REF] Knuth | The Art of Computer Programming[END_REF]. They usually sort the component IMs by performing a limited number of all the possible pairwise comparisons. However, whichever is the sorting algorithm chosen to arrange the components" IMs in ascending order, it needs to be modified to address the case, previously outlined, in which B For example, if the ranking algorithm compares j with k, and k with s, but it does not compare j with s, it will produce a ranking with . To avoid that the obtained final rank depends from which comparisons are made by the ranking algorithm, it is necessary to apply to the components ranking proposed by the sorting algorithm an additional control procedure which checks whether equally important components occupy different positions in the ranking. In this situation, the final ranking is correspondingly modified by assigning the same rank to all the involved components. The Appendix describes one of the most used sorting algorithm, i.e., the Quicksort algorithm [START_REF] Hoare | Quicksort[END_REF], and proposes a control procedure to verify whether equally important components occupy different positions in the ranking.

Industrial case study: Auxiliary Feedwater System

Let us consider an AFWS of a Pressurized Water Reactor (PWR) whose simplified Reliability Block Diagram (RBD) is reported in Fig. 7. The case study is taken from [START_REF] Modarres | Risk Analysis in Engineering: Probabilistic Techniques, Tools and Trends[END_REF], where it is assumed that a) all components are in standby mode, b) all components are periodically tested, and c) the components" unavailabilities Q j are affected by epistemic uncertainties which are described by lognormal probability distributions.

In this work, instead of assumption c, we consider a case in which the only available knowledge on the Q j values is constituted by their mean and standard deviation values (Table ) without any information on the type of probability distributions. To allow a comparison with a probabilistic approach, the values in Table II correspond to the mean and standard deviations of the lognormal distributions used in [START_REF] Modarres | Risk Analysis in Engineering: Probabilistic Techniques, Tools and Trends[END_REF] to represent the uncertainty in the Q j values within a probabilistic approach. The uncertainties in the components" unavailabilities ,..., AN QQ are propagated by the procedure illustrated in Section 3.2, and the possibility and necessity measures of Birnbaum IMs of the components are obtained (Fig. 8). As expected, the largest Birnbaum IM is assigned to component N, which is a single point failure (i.e., its failure results in the loss of AWFS functionality), and thus the overall AWFS reliability is strongly sensitive to the improvement of the reliability of this component.

Because the AFWS has a large number of components, the Quicksort algorithm has been used to automatically order the components on the basis of their importance. Tables III and IV report two different rankings obtained by applying the Quicksort algorithm with two different initial settings. According to Table III, there is a first group of elements (FCDE) whose IMs are sensibly smaller than those of the components of the second group (ABIJM). Another group (GHK) of components with similar importance values has been identified; these are less important than L (the second most important component), which is sensibly less important than N. Differently, in the case of the Quicksort execution reported in Table IV, the direct comparison between the Birnbaum IMs of components K and I is performed, and the two components turn out to be equally important; this result leads to assigning the same importance to the components A B I J M G H K.

Table III

Ranking of the components' IMs obtained in the first execution of the Quicksort algorithm, and confirmed after applying the procedure to verify whether equally important components occupy different positions (possibilistic framework)

Ranking order 14-13-12-11 10-9-8-7-6 5-4-3 2 1

Components FDCE A B I J M G H K L N Table IV
Ranking of the components' IMs obtained in the second execution of the Quicksort algorithm (possibility theory framework)

Ranking order 14-13-12-11 10-9-8-7-6-5-4-3 2 1

Components

FDCE A B I J M G H K L N

To avoid such instability of the importance ranking, according to the procedure described in the Appendix, all the possible comparisons between the components of the system under study are performed (TableV). In the case of Table III, the components" ranking is modified to take into account that

B I B K I I  ,
and the final ranking becomes that of Table IV, whereas in the second run of the Quicksort algorithm (Table IV), the proposed ranking does not need to be modified.

F D C E B A J M I G K H L N F = = = < < < < < < < < < < D = = < < < < < < < < < < C = < < < < < < < < < < E < < < < < < < < < < B = = < = < = < < < A = < = < = < < < J = = < = < < < M = > = > < < I < = < < < G = = < < K < < < H < < L < N
Table reports the results obtained in [START_REF] Baraldi | A method for ranking components importance in presence of epistemic uncertainties[END_REF] applying the probabilistic approach to the same case study. The main difference is that the probabilistic approach results in fewer cases of components with the same importance because the CDFs of the components" important measures appear to be more clearly separated than are the necessity and possibility measures in the possibilistic approach.

Table VI

Final ranking of the components' IMs (probabilistic framework).

Ranking order 14-13 12-11 10-9 8-7 6-5 4 3 2 1

Components C E D F A B I J M K G H L N
With regards to the computational times, when the epistemic uncertainties are described by means of possibility distributions, then the application of the propagation technique, the Quicksort algorithm, and the procedure to verify whether equally important components occupy different positions in the ranking take few seconds. On the contrary, the time required to carry out the same analysis within the probability theory framework is around 20 minutes, with most of the time dedicated to the uncertainty propagation step. This result is due to the fact that uncertainty propagation is performed in the possibilistic case by using the fuzzy extension principle which requires only to apply simple rules of interval algebra, whereas in the probabilistic case it is necessary to resort to MC simulation because analytical approaches are impracticable.

Conclusions

In this work, a procedure has been proposed for ranking system components in order of importance when their failure probabilities are poorly known, and the related uncertainties are described by possibility distributions. The ranking procedure is based on: i) the use of the fuzzy extension principle to propagate epistemic uncertainty from the system parameters to the component importance measures, ii) pairwise comparisons of the obtained component importance measures, and iii) the application of the Quicksort algorithm. Because the method used to compare the Birnbaum importance of three generic components, k, s, j may lead to a ranking which can be partially contradictory ( According to de Morgan"s laws,

B B B B B B B B B B B C B B A C B B A C B A P I I I I P I I P I I P I I I           ( , ) 1 ( ) ( ) ( )
Notice that because both ()

Contradictory ranking (possibilistic framework)

Suppose that (0)

sk l T    , and (0)

Ranking algorithm

Quicksort is a divide-and-conquer algorithm which relies on a partition of the elements based on a quantitative indicator of their "size". To partition the elements, it is required to choose one of them as a pivot, a reference for moving all elements of size smaller before the pivot, and all elements of size larger after it. In the resulting iterative partition procedure, the sublists of smaller and larger elements are recursively sorted.

The pseudo-code of the algorithm can be found in [START_REF] Baraldi | A method for ranking components importance in presence of epistemic uncertainties[END_REF], and [START_REF] Hoare | Quicksort[END_REF]. Notice that the pairwise comparisons also show that H is equally important than G, although the algorithm leaves G in its current position. The sublist of more important components is then sorted. The comparison between N and L shows that the former is more important than the latter.

The less important elements branch contains ABCDEFGMIJK; its middle element, F, is chosen as pivot.

The components ABCDEFIJM are more or equally important than F. In particular, this latter component is equally important as E, C, and D. With reference to the right sublist (more important components), G is the pivot element, and it swaps its position with J; that is, the importance of J is smaller than that of G. The algorithm proceeds as illustrated in Fig. 9.

Once the Quicksort algorithm has been executed, we have to run the control algorithm to verify whether different rank orders have been assigned to equally important components. This need is due to the fact that the algorithm does not perform all the possible direct comparisons.

In details, we consider the l-length vector X of sorted components (i.e., the output of the Quicksort algorithm), and the vector R of the ranking position initially associated to the components (i.e., from 1 to l).

In the case considered in this appendix X=FDCEBAJMIGKHLN.

The rationale of the code is that to have a difference in components importance ranking there must be a column of the comparison matrix whose entries are all "<".

We start from setting the ranking order r=1, and our first objective is to find the set of the least important components (i.e., all those components with rank r=1). Notice that the first component of the list provided by the sorting algorithm which does not belong to this class is the first component of the list characterized by a column of the comparison matrix Comp (TableV) containing only "<", because this is the only condition guaranteeing that all the previous components of the Quicksort list are less important. To identify this component, we set i=2, and check whether the i-th column of the comparison matrix Comp (TableV) contains only "<". In the affirmative case, the rank of the components of X from position 1 to i-1 is set to 1, and the rank of component X(i) is set to i. On the contrary, if column i contains ">" or "=", then X(i) and X [START_REF] Aven | On the ontological status of the concept of risk[END_REF] must be considered of the same importance, and thus of rank 1, because in this case X(i) is equally important to at least one of the previous components. The variable i is updated to i+1, and the procedure is iteratively repeated by considering the new column i of the comparison matrix Comp.

The pseudo-code of the proposed algorithm follows.

r=1; i=2; while i<=l Col=i-th column of Comp V=all(Col=='<'); If V==1 R([j:i-1])=r; r=i; end i=i+1; End For example, consider i=1; then X(1)=F. Flowing over X, we find that the Boolean variable V=1 when i=5.

Then the first group is made up of the components r:i-1=1:4 of X (that is FDCE), which are of rank r=1.

Then r is set to i=5. The algorithm proceeds up to i=l, and the final result is R = [1 1 1 1 5 5 5 5 5 5 5 5 13 14]. His main research efforts are currently devoted to the development of methods and techniques for system health monitoring, fault diagnosis, prognosis, and maintenance optimization. He is a co-author of 41 papers in international journals, and 48 in proceedings of international conferences; and he serves as referee of 5 international journals.

Michele

Fig. 1 .

 1 Fig. 1. System Reliability Block Diagram.

2 .

 2 propagation these uncertainties onto the component"s IM values 1 ,..., BB n II , and 3. ranking the component"s IMs 1 ,..., BB n II taking into account the uncertainties obtained in the previous step.

Fig. 2 (

 2 Fig. 2 (continuous line) shows the Birnbaum IMs distributions () B j B j I Fi obtained by propagating the

I

  is larger than B s I , referred to as "exceedance measure" and given by 1

  However, the demonstration in the Appendix shows that by setting T l lower than 1/3, this contradictory ranking is avoided, and at worst it can happen that B which can be interpreted as a condition in which the three components are equally important.

Fig. 3

 3 Fig.3shows the CDF of the random variables sk  , with s, k=A, B, C obtained by applying the probabilistic

Fig. 2 .Fig. 3 .

 23 Fig. 2. Possibility measure, necessity measure, and CDF of the IMs of the components.

Section 3 .

 3 1 briefly introduces PT, and shows how to obtain a possibility distribution in practical cases; Section 3.2 deals with the propagation of the uncertainty from the component failure probability to the component Birnbaum importance; finally, Section 3.3 illustrates the proposed novel procedure for ranking the component Birnbaum importance.

Fig. 4 d

 4 Fig. 4 d, e, and f show the possibility, and necessity measures () j Q j

Fig. 4 a

 4 Fig. 4 a, b, and c also report the CDFs

  for every j=A, B, C.

Fig. 4 .

 4 Fig. 4. a,b,c: possibility distributions of the unreliabilities of the components A,B, and C, respectively. d,e,f: possibility (dashed lines), and necessity (dotted lines) measures of the corresponding component unreliabilities. The continuous lines refers to the CDF of the component unreliability obtained in Section 2 in the probabilistic framework. The scales of the abscissas of a and b are different from that used in c.

  , respectively (Fig.4 a, and b).

2 .

 2 For every component of the vector I B , compute the smallest, and largest values of the function () j g Q encoding the relationships between the failure probabilities and the Birnbaum IM of the j-th component

4 .

 4 Build the possibility, and necessity measures ()

Fig. 3 a

 3 Fig. 3 a, b, and c show the obtained possibility and necessity measures in the considered example.

  0.00320-0.99995,0.01420-0.99970]=[-0.9950,-0.9860] (in Fig. 3, these values are the smallest values of AC  in which ( ) 1 AC AC    , and the largest in which (

Fig. 5 .

 5 Fig. 5. Possibility distributions of the IMs of two generic components k and s, and corresponding possibility distributions of the variables BB sk s k II    and B B ks k s II    .

  . However, as in the probabilistic framework, setting T l smaller than 1/3 allows avoiding the contradictory ranking with B Appendix for the mathematical details). These situations are addressed by assuming that the three components have the same ranking. i.e.,

  may lead to different rankings depending on which pairwise comparisons are performed.

  if it compares j with k, and j with s, but not k with s, it will rank

Fig. 6 .

 6 Fig. 6. Comparison of the Importance measures: possibility distributions of the variables Ξ kj k, j=A, B, C, and k≠j.

Fig. 7 .

 7 Fig. 7. RBD of the AFWS system [18].

Fig. 8 .

 8 Fig. 8. Possibility, and necessity measures of the components' Birnbaum IMs (reported in different scales.). The continuous line refers to the CDFs obtained in the probabilistic case [14].

  whether equally important components occupy different positions in the ranking has been proposed.The application of the proposed procedure has shown several results. . According to the decision criterion of Section 2., to prove that the contradictory inequality is not possible, we have to show that () on the right-hand side of the equation only introduces an additional constraint on B B I , which is not considered in the left-hand side.

  On this basis, according to the criterion propounded in Section 3.3 we can state that B the probabilistic interpretation of the possibility distributions[START_REF] Kuo | Relations and generalizations of importance measures in reliability[END_REF], from our hypotheses the results showed earlier in this Appendix, we can guarantee that( )

Fig 9

 9 Fig 9 illustrates the application of the Quicksort algorithm in the case study of Section 5, when the

Fig. 9 .

 9 Fig. 9. steps of the Quicksort algorithm.

Table I Component unreliabilities, and IMs (taken from [19]).

 I

		Unreliability:	Unreliability:
		Mean	Standard Deviation
	A	0.015	0.005
	B	0.010	0.005
	C	0.095	0.044

  [START_REF] Zio | Computational Methods for Reliability and Risk Analysis[END_REF]

													 	ks		, and	 	ks	(1) 1  . Equations (4) and (5) allow stating that
	o	N 	ks	(0) 0 	and	(0) 0 2 Π.  ks 	, which can be probabilistically interpreted as
		0		P 	ks	(0)		( P I	BB ks I 	)		(0) 0 2 Π. ks  	; and
	o	N 	ks	(1) 0.8 	, and	Π 	ks	(1) 1  , which can be probabilistically interpreted as
		0.8					ks	(1)	(	BB ks	)	ks	(1) 1

Table II Failure data for the components of the AFWS [18]

 II

	Component Name	Mean unavailability	q	* j	Standard deviation j 
	A	9,963E-05			1,787E-05
	B	9,963E-05			1,7875E-05
	C	70.22E-05			17.750E-05
	D	70.22E-05			17.750E-05
	E	70.22E-05			17.750E-05
	F	70.22E-05			17.750E-05
	G	5,129E-05			1,762E-05
	H	4,080E-05			1,740E-05
	I	0,07088			0,01705
	J	0,07088			0,01705
	K	0,02458			0,001735
	L	23.72E-05			8,875e-05
	M	0,10858			0,053250
	N	3,9778E-05			1,7875E-05

  Compare (BS in mechanical engng., University of Naples Federico II, 2003, PhD in nuclear engng., Politecnico di Milano, 2011) is currently a post-doc at the Politecnico di Milano. He worked as RAMS engineer, and risk manager. His main research efforts are devoted to the development of methods and techniques in support of the maintenance of complex systems. Enrico Zio (BS in nuclear engng., Politecnico di Milano, 1991; MSc in mechanical engng., UCLA, 1995; PhD, in nuclear eng"g., Politecnico di Milano, 1995; PhD, in nuclear engng., MIT, 1998) is Director of the Chair in Complex Systems, and the Energetic Challenge of Ecole Centrale Paris and Supelec, full professor, Rector's delegate for the Alumni Association and past-Director of the Graduate School at Politecnico di Milano, adjunct professor at University of Stavanger. He is the Chairman of the European Safety and Reliability Association ESRA, member of the Korean Nuclear society and China Prognostics and Health Management society, and past-Chairman of the Italian Chapter of the IEEE Reliability Society. He is serving as Associate Editor of IEEE Transactions on Reliability, and as editorial board member in various international scientific journals. He has functioned as Scientific Chairman of three International Conferences, and as Associate General Chairman of two others. others. His research topics include analysis of the reliability, safety and security of complex systems under stationary and dynamic conditions, particularly by Monte Carlo simulation methods; and development of soft computing techniques for safety, reliability and maintenance applications, system monitoring, fault diagnosis and prognosis. He is an author or co-author of five international books, and more than 170 papers in international journals.

Acknowledgment

The authors wish to thank the anonymous reviewers for their comments and suggestions, which have been fundamental for the improvement of the paper.

 We find the relevance of taking into account uncertainties in the computation of the IMs, and the fact that the possibility distributions allow to represent the uncertainties in a way less committed than that offered by the probability theory.

 The final ranking may depend on the investigation framework used to carry out the analysis, which is mainly established on the basis of the quality and quantity of available data. In general, the probabilistic representation and propagation of the uncertainty allows one to get a more refined final ranking, but it calls for a larger amount of available data and more accurate information (which may be lacking in real industrial applications).

The proposed procedure has been applied in this work to rank the components on the basis of their Birnbaum IMs, but it can be easily extended to other IMs (e.g., the Risk Reduction Worth, RRW, Risk Achievement Worth, RAW, Fussel-Vesely, FV). Notice, however, that the propagation of the epistemic uncertainty from the system parameters to the components" importance measure by using the fuzzy extension principle can become computationally more challenging when the importance measure is defined through relationships more complex than those of Birnbaum IM, and which require one to perform divisions between uncertain quantities. Furthermore, the larger the complexity of the system and the associated number of basic events, the larger is expected to be the computational time required to perform the ranking.

Appendix Contradictory ranking (probabilistic framework)

Let us consider a case in which we have to rank the importance of three components: A, B, and C. We have ( ) 1