Node ranking for network topology-based cascade models – An Ordered Weighted Averaging operators' approach

Reliability Engineering and System Safety 155 (2016) 115–123

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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress

Node ranking for network topology-based cascade models – An Ordered Weighted Averaging operators' approach Elvis Hernández-Perdomo a, Claudio M. Rocco b, José E. Ramirez-Marquez c,d,n a

Business School, University of Hull, United Kingdom Universidad Central de Venezuela, Caracas, Venezuela c School of Systems & Enterprises, Stevens Institute of Technology, Hoboken, NJ, United States d Tecnológico de Monterrey, School of Science and Engineering, Campus Guadalajara, México b

art ic l e i nf o

a b s t r a c t

Article history: Received 10 September 2015 Received in revised form 6 May 2016 Accepted 24 June 2016 Available online 25 June 2016

The importance of network components under fault conditions has been assessed by different techniques. However, the indicators analyzed in the literature do not consider that some isolated events, such as component outages may trigger other events. For example, in a power system, the outage of transmission equipment (e.g., a power line or a transformer) may cause the redistribution of the power flow and could cause overloading of neighboring elements. These potential cascade effects have been analyzed using several models. Based on different assumptions, these models are able of determining more precisely, the important elements of the network. In this paper, the authors extend a previous non-parametric multicriteria aggregation approach to include the decision-maker preferences. The new approach, based on the use of aggregation rules that relies on parametric Ordered Weighted Averaging (OWA) operators to support the decision-making process, is able to produce a unique ranking of components. The aggregation rule is based on the classic OWA operator that considers decision-maker preferences associated with risk perception, compensation, entropy of information, among other aspects, and the weighted OWA operator (WOWA) for assessing the relative importance of the criteria. To illustrate the approach, the effects of the additional information provided by the decision-maker as well as their variations are evaluated using a real electric power grid under three cascade models. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Cascade events Hasse diagram OWA operators Partial order sets Power system Sensitivity analysis

1. Introduction During the last decades, interest in understanding how the performance of networks diminishes as a function of component failures (nodes and links) has significantly grown. Several authors have analyzed this problem from different perspective (see [1–4] among others). For example, Ref. [3] analyzes the problem of determining the minimum set of nodes to be removed that produces the maximum reduction of cohesion (i.e., the network fragmentation problem). Other models that have analyzed the effects of deleting nodes and arcs are: the Most Vital Arcs Problem [1]; the k-edge Survivability Problem [2]; the Key Player Problem/Negative [3], and the Critical Node Problem [4], to name a few. Other authors (see for example [5] and references therein) have used concepts associated with the centrality of a graph. For example, Freeman, as mentioned in [5] defines the betweenness of a n Corresponding author at: School of Systems & Enterprises, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030, United States. E-mail address: [email protected] (J.E. Ramirez-Marquez).

http://dx.doi.org/10.1016/j.ress.2016.06.014 0951-8320/& 2016 Elsevier Ltd. All rights reserved.

node or a link as the degree of participation of each component in all possible paths between pairs of nodes. Independent of the main concept used, these approaches attempt to determine the set of elements which, when taken out of service, by a fault or an intentional attack, causes network disruption. The magnitude of these disruptions is used subsequently to determine the classification of components or groups of components from the most to least important. However, such models do not consider that some isolated events, such as the failure of a component, may trigger other events. For example, the outage of a transmission component in an electric power system could produce a redistribution of the power flow and could cause overloading of neighboring elements. Protection devices would disconnect those overloaded components causing the possible start of “the first stage of a cascade” [6]. Indeed, it is possible that the power flow would need to be redistributed again, leading to potential further overload of additional components and their consequent disconnection. Each cascade stage could worsen the system performance. The process can continue until finally stabilizes (quiescent state): there

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are no longer cascading effects or the system could not perform its intended function. Several cascade models have been presented in the literature (e.g., [7–9], among others). In general, the failure of a component or set of components could start a cascading sequence. At the quiescent state, the performance of the system, such as the load curtailed or an index related to the network cohesion, is associated to the failed component. Since these cascade models are based on different assumptions, the consequence in the system due to a component failure could be different. Thus, the component importance assessment must be considered as a multi-indicator system: each component is characterized simultaneously by several criteria (or attributes) that represents the consequence in the system. In order to determine the overall importance of each components based on the results of several cascade models, the decisionmaker (DM) could select the decision-making problem defined as “Problematique γ” in [10]: ranking a set of alternatives from the best to the worst ones. For systems modeled as networks, different aggregation approaches have been proposed in the literature. In general, such studies only consider centrality-based importance measures (e.g., [11–13]). Recently Rocco et al. [14] presented an approach for the ranking of “components derived from three simple cascade models using a non-parametric technique based on partial order theory” but without considering DM preferences. This paper proposes an extension of the approach in [14] by considering the use of Ordered Weighted Averaging (OWA) operators [15,16] for identifying critical nodes. The proposed approach structures an aggregation rule, which allows assessing risk appetite, compensatory aggregations, entropy of information (dispersion), and the embedding of experts' preferences over the criteria selected (relative importance) [17–20]. The effects of the relative importance are also analyzed. The paper makes use of the Hasse diagram (HD) technique as a preliminary analysis tool for assessing the importance of the components of the system under study. HD is a fully non-compensatory graphic technique, based on the mathematical concepts of partial order [21], that produces an aggregated picture of the components to be ranked, and is capable to highlight possible conflicts among them, suggesting a set of components that could be partially ordered. HD is a non-parametric technique that does not require information regarding DM preferences. The paper is structured in five sections. In Section 2, three cascade models are briefly reviewed. Section 3 describes the fundamental ideas of partial order set, Hasse diagrams and the OWA models for ranking. Section 4 illustrates the proposed approach using a real electric power grid. The final Section 5 presents conclusions and future work.

2. Cascade models This section briefly reviews the basic concepts of a cascade model and describes the three cascade models to be used in Section 4. It is assumed that the topology of the network is known and could be represented by a graph G (N, A), where N is the set of N nodes and A is the set of A links connecting nodes. A cascade model evaluates if the outage of a selected component (i.e., the event) is able to trigger additional events and determines the possible cascade sequence. Once the cascade failure terminates (the quiescent state), the performance of the final network topology is assessed. As in Rocco et al. [14] three simple cascade models are described and used to illustrate the proposed approach. However, it is important to realize that any other cascade model could be considered, since the approach presented in this paper assumes

that the consequences of the outage of a component are properly evaluated no matter the cascade models selected. 2.1. Model 1: Pepyne et al. [7] This model simulates cascading effects by defining a probability of propagation for each link as follows. A node is randomly selected for failure forcing the flow to be redistributed among the nearby links, causing potential overload of specific components and triggering protecting devices to disconnect the lines with a probability of failure propagation pij (this probability is a function of protection enhancements or maintenance actions). 2.2. Model 2: Crucitti et al. [8] This model considers the N
N adjacency matrix e, where eij ϵ [0,1] is a measure of the efficiency in the “communication” along the arc between node i and node j. Initially, at time t¼ 0, eij=1 for all the existing arcs. The average network efficiency E is defined as:

E=

1 1 1 = ∑ ∑ eij N ( N − 1) i, j ∈ N, i ≠ j dij N ( N − 1) i , j ∈ N , i ≠ j

where dij and eij are the geodesic distance and the efficiency between nodes i and j, respectively. Each node is characterized by a capacity defined as the maximum load that node can handle. An efficient path is defined as the shortest path between any two nodes. Then, the number of all of the shortest paths of a network that passes through node i is used as a proxy of the load L i (t ) on node i at time t. The capacity Ci of node i is assumed proportional to its initial load L i (0), i.e., Ci=αL i (0), where α 41 is the tolerance parameter of the network. The removal of a specific node “starts the dynamics of redistribution of flows on the network” and affects the most efficient paths between nodes. At each time t, the model defines a rule for the time evolution of eij “that mimics the dynamics of flow redistribution following the breakdown of a node”:

⎧ Ci ⎪ eij ( 0) L ( t ) if L i ( t ) > Ci i eij ( t + 1) = ⎨ ⎪ eij ( 0) if L i ( t ) ≤ Ci ⎩ where j extends to all the first neighbors of i. Therefore, if at time t if a node i is congested, the efficiency of all of the arcs passing through is reduced and eventually the new most efficient paths appear. 2.3. Model 3: Wu et al. [9] In this model, each node i has a weight βi=k iΘ , where k i is the node degree and Θ is a selected parameter that controls the strength of the node weight. The cascade effects is simulated by redistributing the flow through a broken node i among its nearest neighboring nodes. The additional flow ΔFj received by the neighboring node j is:

ΔFj = Fi

βj ∑l ∈ Ω βl i

where Ωi is the set of neighboring nodes of i. Each node i in the network can handle a maximum flow Φi , which is assumed proportional to its weight Φi=ck iΘ . If ( Fj+ΔFj )>Φj then the node j will be broken, and further redistribution is induced. 3. Ranking techniques Let P define a set of m objects (for example, nodes) to be

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analyzed and let the descriptors, g1,… ,gk,… ,gn define the different criteria selected (for example, the load curtailed, the network cohesion or the average network efficiency assessed by different cascade models). In general, each descriptor has associated a preferred direction of improvement. For example, a criterion with a low value corresponds to a low rank [22].

3.1. Hasse diagram Prior to implementing any technique for ranking components, a preliminary analysis can be performed with a Hasse diagram. The Hasse diagram is a visualization technique based on the mathematical concept of partial order sets, where a set of objects are related to each other by a binary relation, including the incomparability among objects, defined by the axioms of reflexivity, antisymmetry, and transivity [22]. A total order in P can be easily derived if only one descriptor is used. Two elements a, b ∈ P are comparable if gk ( a)≤gk ( b) ∀ k . In the case of two descriptors g1 and g2, it is possible that g1 ( a)≥g1 ( b) and g2 ( a) < g2 ( b). In such case, a and b are said to be incomparable, denoted by a b. If several objects are mutually incomparable, set P is called a partially ordered set, or poset. The poset is denoted as ( P, ≤ ), where P is a set of objects and ≤ is a subset of P × P which is an order relation on P . That is, a relation that obeys the following axioms of order [22]: (i) reflexivity: a ∈ P ⇒ a ≤ a , (ii) antisymmetry: a, b ∈ P , a ≤ b and b≤ a ⇒ a = b, and (iii) transivity: a, b, c ∈ P , a ≤ b and b ≤ c ⇒ a ≤ c . The objects in a poset can be represented by a Hasse Diagram, a graph whose vertices are the objects in P, and there is an edge between two objects only if they are comparable and one covers the other, i.e., when no other element is in between the two. In this diagram, higher-ranked objects are given higher vertical positions (or level). Operational details on how drawing effectively Hasse diagrams could be found in [22]. An example of such diagram and its interpretation is presented in the next section. Note that HD does not require any information regarding the preferences of a decision-maker. Additionally, since the construction of the HD is based on the comparison of the objects using a descriptor at a time, no normalization procedures are required. A Hasse Diagram is a useful tool for showing the relations among objects, but it could not provide a total order of objects. However, in the case that an object dominates the rest of objects in P, the HD will clearly show this situation. In this case, the best object is directly determined and there is no need to perform additional evaluations. An interesting result that could be easily derived from a HD are the chains, a subset of objects where a complete ranking is defined, without the use of any aggregation procedure (i.e., without additional information from the DM). The concept of the “average rank” has been used to approximate the rank of the objects, such as the Local Partial Order Model (LPOM) [23], the extended LPOM (LPOMext) [24] or the approximation suggested in [25]. It is important to realize that these approaches do not consider DM preferences, i.e., they are nonparametric approaches.

3.2. The basics of the Ordered Weighted Averaging method The OWA operators were proposed by Yager [15]. An OWA operator F is a mapping F: Rn → R given by:

117

n

F ( a1, a2, …, an ) =

∑

wj a ( j)

j=1

(1)

where: n W¼(w1,w2,...wn)T, wjϵ[0,1]; ∑ j = 1 wj is an associated weighting vector;. ai is the score or the rank of component i when evaluated by g1,… ,gk,… ,gn and a(j ) is the jth largest aj (that is, the elements of ( a1, a2,… ,an ) are ordered decreasingly and the sum is taken over this ordered set). Note that if ai are scores or qualitative measurements, a normalization procedure could be required before the aggregation. The weight wj reflects the importance of the ith ordered position rather than a particular descriptor [15]. The DM preferences are modeled in the OWA approach through the weighing vector W. Three special cases are [16]: T

Max : Wmax = ( 1, 0, …, 0) and Fmax ( a1, a2, …, an ) = max { a1, a2, …, an }

Min : Wm in = (0, 0, …, 1)T and Fmin ( a1, a2, …, an ) = min { a1, a2, …, an }

⎛1 1 1 ⎞T Average: WAve = ⎜ , , …, ⎟ and FAve ( a1, a2, …, an ) ⎝n n n⎠ ( a1 + a2 + … + an ) = n The first case selects the best rank and corresponds to the maximax criterion or the purely optimistic decision. The second case considers that there are no criteria ranking the object in a lower position (the maximin criterion or the purely pessimistic decision (no compensation)). The Average operator is equivalent to the conventional average linear combination method [26] and corresponds to the Laplace decision criterion. This three cases show that W can model different types of combination rules, from non-compensatory to compensatory.

3.3. The Ordered Weighted Averaging and decision-maker preferences In this section, two basic aspects are considered in order to define the ranking assessment: 1) how to quantify the weighting vector that mimics the DMs' preferences, (i.e., optimistic, pessimistic, compensatory, and so on) [15,16] and 2) how to consider the relative importance of the attributes. This latter aspect has been generalized by defining the Weighted OWA (WOWA), a weighting schema that allows the preponderance of different information sources in an aggregation process [18,19,27,28]. The use of such additional information defines these techniques as parametric techniques. 3.3.1. Obtaining the weighting vector The most common approach to obtaining the weighting vector is the one based on the use of linguistic quantifiers allowing DMs to translate their preferences in terms of the proportions, or numbers, of attributes that must be met to obtain a final aggregation [16,20]. For example, the proportions are set as relative quantifiers: most of the criteria should be satisfied, all the criteria must be satisfied, and at least half of the criteria should be satisfied, among others. However, several additional methods have been proposed to obtain the OWA weights [29] and to mimic the preferences of the DM (e.g., from risk prone to risk averse) but

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considering additional aspects described later. In this paper, the relative linguistic quantifiers are taken into consideration because they can be represented by an interval of [0,1] and are easier to implement in practical terms [30]. To assess the aggregate quality of a component, the OWA method uses a regularly increasing monotonic (RIM) quantifier Q , an approach for computing the weights wj according to the following formula [16]:

⎛ j⎞ ⎛ j − 1⎞ ⎟, 1 ≤ j ≤ n wj = Q ⎜ ⎟ − Q ⎜ ⎝ n⎠ ⎝ n ⎠

(2)

(3)

For example, if the case of three attributes, then the weights are determined as:

⎡ 1 ⎤α ⎡ 3 ⎤α ⎡ 2 ⎤α ⎡ 2 ⎤α ⎡ 1 ⎤α w1 = ⎢ ⎥ , w2 = ⎢ ⎥ − ⎢ ⎥ , w3 = ⎢ ⎥ − ⎢ ⎥ ⎣ 3⎦ ⎣ 3⎦ ⎣ 3⎦ ⎣ 3⎦ ⎣ 3⎦ If α = 0, the quantifier at least one (which corresponds to the logical ‘‘oring’’ of all the arguments and represents a risk-seeking T or risk-prone DM) is derived, which produces Wmax=( 1, 0, … , 0) . If α ¼1, the for all quantifier (which corresponds to the logical ‘‘anding’’ of all the arguments and represents a risk-averse DM) is T selected, which produces Wmin=( 0, 0, … , 1) . If α ¼1, the quantifier some (which corresponds to a compensatory operator), represents a risk-neutral DM, and produces the

(

weight vector WAve=

1 , n

wj ( a) a ( j)

(5)

2 2 2 2 2 w1 = ⎡⎣ 0.20⎤⎦ , w2 = ⎡⎣ 0.53⎤⎦ − ⎡⎣ 0.20⎤⎦ , w3 = ⎡⎣ 0.83⎤⎦ − ⎡⎣ 0.53⎤⎦ ,

w4 = [1]2 − [0.83]2 or W ( a) = ( 0.04, 0.24, 0.41, 0.31) The aggregate quality for a, associate with the rule most (α = 2) can be calculated as follows: 4

F ( a) =

∑

wj ( a) bj = ( 0.04)( 1) + ( 0.24)( 0.7)

j=1

)

+ ( 0.41)( 0.6) + ( 0.31)( 0.5) = 0.610 For b, the decreasing order is (C4, C3, C1, C2 ) , meaning b() = ( 1.0, 0.9, 0.6, 0.3) . The order of the relative importance values that matches with Bb is (0.30, 0.17, 0.33, 0.20), and the accumulated importance values are (0.30, 0.47, 0.80, 1.00). The positional weights associated with this object are 2 2 2 2 2 w1 = ⎡⎣ 0.30⎤⎦ , w2 = ⎡⎣ 0.47⎤⎦ − ⎡⎣ 0.30⎤⎦ , w3 = ⎡⎣ 0.80⎤⎦ − ⎡⎣ 0.47⎤⎦

2 2 w4 = ⎡⎣ 1⎤⎦ − ⎡⎣ 0.80⎤⎦ or W ( b) ( 0.09, 0.13, 0.42, 0.36)

3.3.2. Including the relative importance of attributes The quantification of the relative importance of the selected attributes [31], or weighting the information sources [18,19,27] with the WOWA operator, deals when the DMs tend to give more importance to some attributes over others. For example, DMs could be more interested in those components with a higher rank on the cascade model proposed by Wu et al. [9] in relation to the other models. In order to incorporate this information into the OWA operators, some modifications are made to Eq. (3) because each criterion can now have a particular relative importance [18,19,27,34]. Let RI = {RI1, RI2…RIn } be a set of importance values (DMs' preferences quantified) over the selected criteria, where n ∑ j = 1 RI j = 1, RI j ≥ 0. Then the WOWA operator [19] depends on both RI and the performances of each project a considered, as follows:

⎛ j ⎞ ⎛ j−1 ⎞ wj ( a) = Q ⎜⎜ ∑ RIka ⎟⎟ − Q ⎜⎜ ∑ RIka ⎟⎟, 1 ≤ j ≤ n ⎝ k=1 ⎠ ⎝ k=1 ⎠

∑

As an example [28], let us consider two objects a and b, and four criteria C1, C2, C3, C4 . The importance values associated with these criteria are RI1= 0.33, RI2=0. 20, RI3=0. 17, and RI4=0. 30. In addition, the impacts of each object on the multiples criteria are a¼ (0.7, 1, 0.5, 0.6) and b ¼(0.6, 0.3, 0.9, 1.0). Let us assume that the quantifier selected is defined by Q ( r )=r 2 (i.e., α = 2, the quantifier most). For object a, the decreasing order of its components produces a() = (1, 0.7, 0.6, 0.5), that is, the order considered is (C2, C1, C4, C3 ). This rearrangement must be considered to order the relative importance components (i.e. ,RI2, RI1, RI4, RI3 ) to produce (0.20, 0.33, 0.30, 0.17). From here, the accumulated values are (0.20, 0.53, 0.83, 1.0). The positional weights associated with object a are

1 1 T ,… , n . n

There are other α parameters that can be used and interpreted. For instance, α ¼ 2 can be used as the quantifier most [16], which semantically means “most” of the criteria are being satisfied (between 30% and 80%); or α ϵ[0,0.5] can be used as the quantifier at most half [32], which means that “at most half” of criteria are being satisfied; α ¼0.1 is identified as the “at least a few” quantifier while α ¼0.5 is the “a few” quantifier. Additional properties for selected values of α can be found in [15,17,33].

j

n

F ( a1, a2, …, an ) =

j=1

where j is a position, n is the total positions (or attributes) considered, and Q(0)¼0. As suggested by Yager [31], in terms of decision-making analysis, this approach is more suitable to model attributes that are generally maximized. This expression is adjusted by a parameter α ∈ [0, ∞] in order to obtain a family of RIM quantifiers as follows:

Q α ( r ) = r α with α ≥ 0

those criteria that satisfy the rearrangement defined by the vector and a(j) in Eq. (1) for project a . Since Eq. (4) depends on each possible project, each project will be evaluated using a specific set of weight. Finally, the RIM quantifier remains as in Eq. (3), and the aggregation rule (1) is adjusted to obtain the following equation:

(4)

where ∑k = 1 RIka = r aj is the accumulated relative importance of

The aggregate qualitative for b, associated with the rule most ( α = 2) is 4

F ( b) =

∑ wj ( b) a ( j) = ( 0.09)( 1.0) + ( 0.13)( 0.9) j=1

+ ( 0.42)( 0.6) + ( 0.36)( 0.3) = 0.566 Hence, object a has the highest aggregate quality and is the preferred. It is important to observe that because the ordering of impacts by alternative is different, leading to a different ordering in relative importance, the weighting schemes produced are different for each alternative [16,28]. 3.3.3. Characterizing the Ordered Weighted Averaging method Yager [15,16] introduces a set of complementary indices to provide additional information to DMs to justify their actions and decision. They rely on the linguistic quantifiers and weighting schemes to characterize not only the OWA operators but also the DMs' preferences in term of risk attitude, compensation, and use of information. Subsequently, Malczewski [33] included another

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complementary index to measure the trade-off between compensation and risk. These indices are the “orness”, the “andness”, the “dispersion”, and the “trade-off”, and they can be interpreted and represented as follows: The “orness” measures the propensity of the OWA operator to accentuate lower or higher individual performances (or rankings) in order to compute the final aggregation. If the “orness” is closer or equal to 1, DMs can be seen to be optimistic (risk-prone) in terms of the final ranking or aggregation. Therefore, if the “orness” is closer or equal to 0, DMs can be seen to be pessimistic (riskaverse) in terms of the final ranking or aggregation. The maximum value is obtained when α = 0. n

orness ( W ) =

⎛

n−1

j=1

+

n−1 n−2 w1 + w2 + … n−1 n−1

n−n wn n−1

n− 1

=

⎞

∑ ⎜⎝ n − j ⎟⎠ wj = ⎛

∑ ⎜⎝ j=1

1 ⎞ ⎛ j⎞ ⎟Q ⎜ ⎟ n − 1⎠ ⎝ n ⎠

(6)

The “andness” is the complement of “orness”. If the “andness” is closer or equal to 1, DMs can be viewed as pessimistic in terms of the final ranking or aggregation. Therefore, if the “andness” is closer or equal to 0, DMs can be seen to be optimistic in terms of the final ranking or aggregation. The maximum value is obtained when α = ∞.

andness ( W ) = 1–orness ( W )

1 ln ( n)

WOWA operator does not depend much on the RI. As mentioned in Hernandez [38], the most important difference when considering relative importance occurs for the neutral-risk behavior.

4. Example: The Italian 380 kV power grid The Italian power grid (HVIET) presented in Fig. 1, is modeled as an undirected graph with N ¼m ¼310 nodes (objects) and A¼361 lines (Fig. 1). The performance of the system for each node outage is assessed by using the general approach suggested by Rocco et al. [14]: Let NCM be the number of cascade models selected. Let Pij represents the performance of the network when node i is selected for failure, under cascade model j. For j¼1, NCM { For i¼ 1,m { Select node i as the failed node Perform the cascade model j Once the cascade failure terminates (the quiescent state), determine Pij } }

(7)

The “dispersion” measurement, also known as the Shannon’s entropy, evaluates the degree to which the aggregation takes into account all information in the attributes. For example, dispersion ( Wmax )¼dispersion( Wmin )¼0. That means that the aggregation rules corresponding to these vectors only consider a single piece of information, which is, respectively, the first (highest) and last (lowest) information. The maximum value is obtained when α = 1. The “dispersion” measurement, also known as the Shannon’s entropy, evaluates the degree of information considered. For example, dispersion( Wmax ) ¼dispersion( Wmin )¼0, that is, these two operators consider only a single value: the highest and the lowest respectively. The maximum value is obtained when α = 1.

dispersion ( W ) = −

In this example, NCM¼3 and the performance of the network is evaluated as: For models 1 and 3, by determining the fragmentation at the quiescent state while for model 2, by calculating the average efficiency. Table 1 shows the ranking of each node for each cascade model. A value of 1 means that the node represents the most important nodes, i.e., the node with the most severe effects after the cascading reaches the quiescent state. For example, considering the first cascade model, node 183 is ranked as the most important.

n

∑ wj

ln ( wj ) (8)

j=1

Finally, the “trade-off” index measures the degree to which “a poor performance on one criterion may be offset by a high performance in another” [33]. This measurement takes values between 0 (no compensation) and 1 (perfect compensation). The maximum value is obtained when α = 1. n

trade − off ( W ) = 1 −

119

(

n ∑ j = 1 wj − n−1

1 2 n

)

(9)

Some of these indexes have been used to propose methods to derive the set of weights. For example, O’Hagan [35], determines a special class of OWA operators having maximal entropy of the OWA weights for a given level of orness. Fullér and Majlender [36] suggested a method to derive OWA operator weights with minimal variability. The indexes previously derived depend on the weights selected. In the case of the WOWA operator, Torra [37] concludes that “the greatest the dimension (i.e., more elements are considered), the more similar is the WOWA orness to the orness of the corresponding quantifier”. In other words, the orness of the

Fig. 1. The Italian high voltage power grid.

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Table 1 Ranking of each node, for the three cascade model analyzed.

Table 2 Ranking of nodes, using the LPOM method.

Node

Model 1

Model 2

Model 3

Node

Model 1

Model 2

Model 3

Node

Position

Node

Position

Node

Position

184 117 144 219 128 191 158 161 194 226 109 154

8 20 9 17 23 16 13 22 7 21 19 6

2 9 17 24 8 18 1 7 16 23 6 13

12 15 21 9 20 8 6 19 24 7 11 18

168 183 159 214 201 204 207 103 153 178 148 2

3 1 5 14 12 10 4 15 2 18 11 24

3 5 21 15 11 4 12 22 14 20 19 9

17 10 23 5 4 22 16 3 13 2 1 14

183 158 168 184 201 148 207 214

1 2 3 4 5 5 7 8

153 109 103 178 154 191 161 204

8 10 11 12 13 14 15 15

117 159 2 194 144 219 128 226

17 17 17 20 21 21 21 21

Nodes ranked simultaneously after the 10th positions in all cascade models are not considered, so the total number of nodes to be evaluated is 24. 4.1.1. Preliminary Hasse diagram Fig. 2 shows the nine-level Hasse diagram for the case under study (derived using the PyHasse software [39]). The higher level represents the most important nodes. From Fig. 2, the following conclusions are derived: a) The relation of dominance among elements (i.e., the more important nodes) goes from a higher level to a lower level. For example, node 183 dominates node 109 and node 109 dominates node 2 (according to the transitivity rule, node 183 is also more important than node 2). b) Nodes 144, 219, 128 and 194 are located in the bottom of the diagram, meaning that those nodes are the least important node. c) Incomparable nodes are located at the same level, for example, nodes 183 and 168 at the first level. This situation is clearly explained by looking at Table 2. The ranks of node 183 for the cascade models evaluated are 1st, 5th and 20th while for node 168 are 3rd, 3rd and 17th. That means that node 183 dominates node 168 only for cascade model 1. This example shows a possible disadvantage of the Hasse Diagram, i.e., two objects could be considered as incomparable “even if most of the indicators in one object are higher than the other” [40]. d) Incomparability also occurs when there are no paths between nodes. For example, nodes 183 and 178 are incomparable. e) The chains detected in the Hasse Diagram allow defining particular rankings. For example, nodes 183, 153 and 194 are the elements of a chain. This fact is very important because, in the case of using a weighting schema to derive an aggregate quality, the relation of dominance is preserved. f) In special cases, the Hasse Diagram could detect isolated

Fig. 2. Hasse diagram.

objects: “objects that have neither an upper nor a lower neighbor” [41]. In this example, no isolated objects are detected. The fact that at the first level there are incomparable elements means that there is no dominant node, i.e., the simultaneous assessment by three cascade models concludes that none of the nodes could be considered as the most important node. This fact also means that a high variation on the final ranking of the nodes could be expected due to the OWA weight selections. In this situation, the DM could be interested in deriving an aggregate quality and hence producing a global ranking of the nodes under analysis. As previously described, the LPOM methods could be used to obtain such ranking, even if no preferences are considered. Table 2 shows the ranking of the nodes using LPOM. Note that the ranking positions are in concordance with the previous dominance relations. 4.2. OWA operator Fig. 3 shows the rankings of the 24 nodes for selected values of

α in [0, 10]. Note that depending on the value of α, some nodes

change its position. In some case the rank improves (e.g., node 2), in some case is worse (e.g., node 194) and in some cases it fluctuates (e.g., node 168). For values of α 44, the position of the nodes becomes stable, at least for the four most important nodes: 183, 184, 201 and 158. Table 3 shows the result for selected values of α. For α ¼ 0, nodes 148, 158 and 183 are ranked in the first position (tied) because ‘at least one’ time they are ranked in the first position by the cascade models. Similarly, since nodes 153, 178 and 184 are ranked ‘at least one’ time in the second position, OWA rank them in the fourth position (tied). For α ¼1 node 183 is considered the most important followed by nodes 158 and 184. Note that node 183 is always considered the most important node no matter the value of α. For α ¼ 1 (the for all quantifier), nodes 184 and 201 are tied in the second most important position. This is because the worst

Fig. 3. OWA ranking for different values of α.

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Table 3 OWA ranking for different values of α. Node

184 117 144 219 128 191 158 161 194 226 109 154 168 183 159 214 201 204 207 103 153 178 148 2

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Table 5 Effects on the ranking for RI vectors.

α

Node

0

0.5

1

1.5

1

4 21 21 21 19 19 1 16 16 16 14 14 7 1 12 12 9 9 9 7 4 4 1 21

3 18 21 24 23 15 2 19 17 22 13 14 4 1 16 10 7 9 8 12 6 11 5 20

3 16 17 22 24 15 2 20 17 23 10 12 4 1 21 9 5 10 8 14 6 13 7 17

3 16 17 21 23 13 2 20 19 24 10 11 4 1 22 9 5 12 7 15 6 14 8 18

2 13 15 22 19 9 4 16 22 19 11 9 8 1 19 6 2 16 7 16 5 13 11 22

position occupied by the nodes is the same (i.e., the 12th position). Note that the dominance relations detected using the Hasse diagram, are maintained under the OWA operator (for example, node 184 dominates node 144, node 144 is more important than node 2, and so on). Table 4 shows the Spearman correlation coefficient among the LPOM and OWA rankings, for different values of α. The best correlation value corresponds to α ¼0.7. This means that for this example the LPOM could be considered as a quasi-full-compensatory operator. 4.3 WOWA operator Table 5 shows an example of the effects on the ranking of the nodes when considering that RI values are equals (i.e., the basic OWA) or for the selected RI ¼ {0.50, 0.30, 0.20}. The highlighted cells represent the ranking of the nodes that are equal in both cases, considering the same value of α. For example, there are no differences when α ¼0. To illustrate how RI could affects the ranking of the nodes, 1000 set of RI weights simulated from a uniform distribution between (0,1) (i.e., without any preference information provided by the Table 4 Spearman correlation coefficient among LPOM and OWA for different α values. α

Corr (LPOM,OWA(α))

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.839 0.885 0.920 0.934 0.945 0.952 0.952 0.957 0.949 0.949 0.942

All RIi equals

RI = { 0.50, 0.30, 0.20}

α

α

Node

0

0.5

1

1.5

1

0

0.5

1

1.5

1

184 117 144 219 128 191 158 161 194 226 109 154 168 183 159 214 201 204 207 103 153 178 148 2

4 21 21 21 19 19 1 16 16 16 14 14 7 1 12 12 9 9 9 7 4 4 1 21

3 18 21 24 23 15 2 19 17 22 13 14 4 1 16 10 7 9 8 12 6 11 5 20

3 16 17 22 24 15 2 20 17 23 10 12 4 1 21 9 5 10 8 14 6 13 7 17

3 16 17 21 23 13 2 20 19 24 10 11 4 1 22 9 5 12 7 15 6 14 8 18

2 13 15 22 19 9 4 16 22 19 11 9 8 1 19 6 2 16 7 16 5 13 11 22

4 21 21 21 19 19 1 16 16 16 14 14 7 1 12 12 9 9 9 7 4 4 1 21

4 19 17 23 24 18 3 22 16 21 14 10 2 1 13 11 9 8 6 15 5 12 7 20

3 19 15 23 24 17 4 21 13 22 12 8 2 1 14 10 7 9 6 18 5 16 11 20

3 17 14 24 23 16 4 21 13 22 11 8 2 1 15 10 7 9 6 19 5 18 12 20

2 13 15 22 19 9 4 16 22 19 11 9 8 1 19 6 2 16 7 16 5 13 11 22

decision maker) were generated by using the procedure in Tervonen and Lahdelma [42]. As a result, the probability of the occurrence of each node in each ranking position is derived. This uncertainty propagation analysis (i.e., the effects of the uncertainty of the weights) is similar to the Stochastic Multiobjective Acceptability Analysis approach (SMAA) (i.e., methods “developed for situations where criteria values and/or weights or other model parameters are not precisely known” [43]). Figs. 4–6 show the heat map for selected values of α. In the graphs, the more highly probable ranked nodes appear in the upper left, and the darker the intersection the higher the probability that the ranking should be applied to the nodes. As detected by the Hasse diagram, the weight selection produces different rankings. For example, Fig. 4 shows that, for α ¼0.5, the most probable rank for node 183 is the first position but it could be also ranked in the second position. Note that the most probable rank for nodes 183 and 158 is the first but the probability for node 183 is higher than for node 158. The same behavior is also observed from Figs. 5 and 6. The three selected values of α show that nodes 128, 148, and 226 are likely to remain in the worst positions. Note that in general, the most or the less important nodes are directly derived from the Hasse diagram. However, the nodes 184, with one of the lowest probability in the first positions, and 226, with a relevant probability in the worst position, are not easily detected. Hence, the uncertainty analysis based on the weights used in WOWA gives information to the decision maker about the robustness of the decision. In terms of the decision-making analysis, it can be observed, from Figs. 4–6, that for any linguistic quantifiers and relative weights, nodes 158, 168, and 183 are located around the first four best positions. On the other side, nodes 128, 226 and 159 are highly likely to be located in the last three worst positions. Node 184 remains always between the second and the sixth position when changes in linguistic quantifiers and the uncertainty analysis are considered. Nodes 117, 144, 148, 154, 191,

122

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Fig. 4. Heat map showing the probability of ranking for each node, for α¼ 0.5.

Fig. 6. Heat map showing the probability of ranking for each node, for α ¼2.

5. Conclusions

Fig. 5. Heat map showing the probability of ranking for each node, for α ¼ 1.

among other, are highly dispersed in relation with the initial rankings. Node 117 varies between the rakings 16th and 19th considering uncertainty and changes in the linguistic quantifiers. Finally, the rest of the nodes have high dispersion on their likely rankings. It can be observe that without the uncertainty analysis, the decision maker can make wrong decision based on the simple outputs of the Hasse diagram or any other decision model. In other words, the global ranking of the selected nodes are high sensitive to the uncertainty analysis. Note that in this example, there is no a single probability larger than 60% in the scale presented in the heat maps.

This paper proposes a valid approach for identifying critical nodes, when different cascade model are used. Cascade models allow assessing whether the outage of a component is able to trigger additional events. The approach presents a more elaborate version of [14], which is based on the use of non-parametric aggregation techniques that do not consider the preference of the analyst. The approach proposed in this paper includes several interesting features. First, it structures an aggregation rule, based on the use of Ordered Weighted Averaging operators, which allows assessing risk appetite and compensatory aggregations. For example, based on the cascade models selected, the analyst could choose the most important node considering: a) only the best ranking of a node in any model; or b) its average ranking; or c) the best ranking on at most half of the models, among other possible considerations. Second, it allows considering the importance of the cascade model’ weights, thus providing additional flexibility to the decision-maker. Third, it includes the use of the Hasse diagram as a preliminary tool, based on the mathematical concepts of partial order [21], for detecting possible dominance relations. Finally, the proposed approach shows how the assessment of the robustness of the decision could be evaluated, by considering variations on the decision makers’ preferences (e.g., due to uncertainty) as well as on the weights importance. The hypothetical quantitative comparison between the two approaches would be based on different assumptions. The fact that the ranking derived by one approach could be different from the ranking obtained by another approach does not mean that one approach is better [44].

References [1] Ball MO, Golden BL. Finding the most vital arcs in a network. Oper Res Lett 1989;8:73–6. [2] Myung Y, Kim H. A cutting plane algorithm for computing k-edge survivability of a network. Eur J Oper Res 2004;156:579–89.

E. Hernández-Perdomo et al. / Reliability Engineering and System Safety 155 (2016) 115–123

[3] Borgatti SP. Identifying sets of key players in a social network. Comput Math Org Theor 2006;12:21–34. [4] Arulselvan A, Commander CW, Elefteriadou L, Pardalos PM. Detecting critical nodes in sparse graphs. Comput Oper Res 2009;36:2193–200. [5] Hanneman RA, Riddle M. Introduction to social network methods Riverside, CA: University of California, Riverside; 2005 [published in digital form at 〈http//faculty.ucr.edu/  hanneman/〉]. [6] Pahwa S, Hodges A, Scoglio C, Wood S. Topological Analysis of the Power Grid and Mitigation Strategies Against Cascading Failures. 2010. arXiv1006.4627v1. [7] Pepyne DL, Panayiotou CG, Cassandras CG, Ho Y-C. Vulnerability assessment and allocation of protection resources in power systems. In: Proceedings of the american control conference arlington, VA; June 25–27, 2001. [8] Crucitti P, Latora V, Marchiori M. Model for cascading failures in complex networks. Phys Rev E 2004;69. [9] Wu Z, Peng G, Wang W, Chan S, Wo EW. Cascading failure spreading on weighted heterogeneous networks. J Stat Mech Theory Exp 2008. http://dx. doi.org/10.1088/1742-5468/2008/05/P05013. [10] Roy B. Méthodologie multicritère d'aide à la decision.París: Economica; 1985. [11] Rocco CM, Ramirez-Marquez JE. Innovative approaches for addressing old challenges in component importance measures. Reliab Eng Syst Saf 2012;108:123–30. [12] Du Y, Gao C, Hu Y, Mahadevan S, Deng Y. A new method of identifying influential nodes in complex networks based on TOPSIS. Phys A: Stat Mech Appl 2014;399:57–69. [13] Almoghathawi Y, Barker K, Nicholson CD, Rocco CM. A multi-criteria decision analysis approach for importance identification and ranking of network components. Submitt Reliab Eng Syst Saf 2015. [14] Rocco CM, Ramirez-Marquez JE, Yajure C. A non-parametric aggregation technique for identifying critical nodes in a network, using three topologybased cascade models. In: second international conference on vulnerability and risk analysis and management (ICVRAM2014). Liverpool. p. 668–76. [15] Yager RR. On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans Syst Man Cybern 1988;18:183–90. [16] Yager RR. Quantifier guided aggregating using OWA operators. Int J Intell Syst 1996;11(1):49–73. [17] Fullér R. OWA operators in decision making. In: Carlsson C, editor. Exploring the limits of support systems. Turku Centre for Computer Science, Finland General Publication; 1996. p. 3. [18] Llamazares B. On generalizations of weighted means and OWA operators. In: 7th conference of the european society for fuzzy logic and technology 2011. Atlantis Press. p. 9–14. [19] Torra V. The WOWA operator and the interpolation function W* Chen and Otto’s interpolation method revisited. Fuzzy Sets Syst 2000;113(3):389–96. [20] Yager RR, Filev DP. Essential of fuzzy modeling and control.New York, USA: John Wiley & Sons; 1994. [21] Bruggemann R, Münzer B. A graph-theoretical tool for priority setting of chemicals. Chemosphere 1993;27:1729–36. [22] Bruggemann R, Patil G. Ranking and prioritization for multi-indicator systems. Dordrecht: Springer; 2011. [23] Bruggemann R, Sorensen P, Lerche D, Carlsen L. Estimation of averaged ranks by a local partial order model. J Chem Inf Comp Sci 2004;44:618–25. [24] Bruggemann R, Carlsen L. An improved estimation of averaged ranks of partial

123

orders. MATCH: Commun Math Comput Chem 2011;65:383–414. [25] De Loof K, De Baets B, De Meyer H. Approximation of average ranks in posets. Match: Commun Math Comput Chem 2011;66:219–29. [26] Badea AC, Rocco C, Tarantola S, Bolado R. Composite indicators for security of energy supply in Europe using ordered weighted averaging. Reliab Eng Syst Saf 2011;96:651–62. [27] Torra V, Godo L. Averaging continuous distributions with the WOWA operator. In: Proceedings of the second european workshop on fuzzy decision analysis and neural networks for management, planning and optimization (EFDAN 1997). Dortmund, Germany. p. 10–9. [28] Yager RR, Kacprzyk J. The ordered weighted averaging operator. Theory and applications.New York, USA: Springer Science þ Business; 1997. [29] Zarghami M, Szidarovszky F. Fuzzy quantifiers in sensitivity analysis of OWA operator. Comput Ind Eng 2008;54:1006–18. [30] Zadeh LA. A computational approach to fuzzy quantifiers in natural languages. Comput Math Appl 1983;9(1):149–84. [31] Yager RR. Induced aggregation operators. Fuzzy Sets Syst 2003;137(1):59–69. [32] Diaz E. Selective Merging of Retrieval Results for Metasearch Environments [Ph.D. Dissertation]. University of Louisiana at Lafayette. The Center of Advanced Computer Studies; 2004. [33] Malczewski J. Ordered weighted averaging with fuzzy quantifiers GIS-based multicriteria evaluation for land-use suitability analysis. Int J Appl Earth Obs Geoinf 2006;8(4):270–7. [34] Yager, R. R. Hierarchical Framework for Constructing Intelligent Systems Metrics. NIST Special Publication SP; 2001, 423–430. [35] O’Hagan M, Aggregating template or rule antecedents in real-time expert systems with fuzzy set logic. In: Proceedings of the 22nd annual IEEE asilomar conference signals, systems, computers. Pacific Grove, CA; 1988. p. 81–689. [36] Fuller R, Majlender P. An analytic approach for obtaining maximal entropy OWA operator weights. Fuzzy Sets Syst 2001;124(1):53–7. [37] Torra V. Empirical analysis to determine Weighted OWA orness. In: FUSION 2001: Proceedings of the fourth international confrence on information fusion, international society of information fusion, vol. II. 2001. p. 11–6. [38] Hernandez E. A Multicriteria Methodology to Rank Investment Projects. MSc. Thesis in Operational Research. Universidad Central de Venezuela (in Spanish). 2006. [39] Bruggemann R, Restrepo G, Voigt K. Towards a new and advanced partial order program: pyhasse. In: Owsiński JW, Bruggemann R, editors. Multicriteria ordering and ranking: partial orders, ambiguities and applied issues. Warsaw, Poland: The Hasse Diagram Workshops; 2008. [40] Al-Sharrah G. Ranking using the copeland score a comparison with the hasse diagram. J Chem Inf Model 2010;50:785–91. [41] Restrepo G, Weckert M, Bruggemann R, Gerstmann S, Frank H. Ranking of refrigerants. Environ Sci Technol 2008;42:2925–30. [42] Tervonen T, Lahdelma R. Implementing stochastic multicriteria acceptability analysis. Eur J Oper Res 2007;178:500–13. [43] Corrente S, Figueira JR, Greco S. The SMAA-PROMETHEE method. Eur J Oper Res 2014;239:514–22. [44] Zio E, Golea L, Rocco CM. Identifying groups of critical edges in a realistic electrical network by multi-objective genetic algorithms. Reliab Eng Syst Saf 2012;99:172–7.