Interaction Graphs for Cascading Failure Analysis in Power Grids: A Survey

1 Interaction Graphs for Cascading F ailure Analysis in Po wer Grids: A Surv e y Upama Nakarmi, Student Member , IEEE , Mahshid Rahnamay-Naeini, Member , IEEE , Md Jakir Hossain, Student Member , IEEE , Md Abul Hasnat, Student Member , IEEE Abstract —Understanding and analyzing cascading failur es in power grids have been the focus of many researchers f or years. Howev er , the complex interactions among the large number of components in these systems and their contributions to cascading failur es ar e not yet completely understood. Therefor e, various techniques have been developed and used to model and analyze the underlying interactions among the components of the power grid with respect to cascading failures. Such methods are important to rev eal the essential information that may not be readily a vailable fr om power system physical models and topologies. In general, the inﬂuences and interactions among the components of the system may occur both locally and at distance due to the physics of electricity gover ning the power ﬂow dynamics as well as other functional and cyber dependencies among the components of the system. T o infer and capture such interactions, data-driven approaches or techniques based on the physics of electricity ha ve been used to develop graph-based models of interactions among the components of the power grid. In this survey , various methods of developing interaction graphs as well as studies on the r eliability and cascading failure analysis of power grids using these graphs ha ve been reviewed. Index T erms —interaction graphs; cascading failures; data- driven; electrical distance; po wer grids; system modeling I . I N T RO D U C T I O N A. An Overview of the Revie w and Overall Signiﬁcance Cascading failures in power grids are successi ve interde- pendent failures of components in the system, which are usually initiated by few outages due to internal or exogenous disturbances and are propagated in a relativ ely short period of time leading to large blackouts [1], [2]. Examples of blackouts that resulted from cascading failures are the case of US Northeast blackout in 2003 [3], Italian blackout in 2003 [4], Brazilian blackout in 2009 [5], and Indian blackout in 2012 [6]. While large blackouts are infrequent, their occurrence still has substantial risks associated with the signiﬁcant economic losses and social impacts that they cause. Understanding and mitigating cascading failures in power systems remain a challenge due to the large size of these systems as well as complex and sometimes hidden interactions among the components. V arious studies and models have been de veloped to understand and control these comple x phenomena including methods based on power system simulation [7], [8], deter - ministic analytical models [9], probabilistic models [10]–[12], and graph-based models [13]–[63]. For a survey of various The authors are with the Department of Electrical Engineering, Uni versity of South Florida, T ampa, FL, 33620. E-mail addresses: unakarmi@mail.usf.edu, mahshidr@usf.edu, mdjakir@mail.usf.edu, hasnat@mail.usf.edu methods for studying cascading failures, see [64]–[69]. Each of these approaches shed light on dif ferent aspects of these phenomena. Among these categories of approaches, graph-based meth- ods have attracted a lot of attention due to the simplicity of the models and their ability to describe the propagation behavior of the failures on the graph of the system [70], [71]. Many initial graph-based models were dev eloped based on the physical topology of the power system, where the connections among the nodes represent the actual physical connections among the components of the system [72], [73]. Howe v er , the studies in [62], [74] showed the lack of strong connection between the physical topology of the system and failure propagation in cascading failures in power grids. In general, inﬂuences and interactions among the components of the system during cascade process may occur both locally and at distance due to the physics of electricity governing the power ﬂow dynamics as well as other functional and cyber dependencies among the components of the system. For instance, historical as well as simulation data verify that failure of a critical transmission line in the po wer grid may cause overload/f ailure of another transmission line that may or may not be topologically close. Therefore, graph models based on the physical topology of the system are not adequate in describing the propagation behavior of failures in power grids. Hence, new methods are emerging to re veal the complex and hidden interactions that may not be readily av ailable from physical topology of the po wer system. These new approaches are focused on extracting and modeling the underlying graph of interactions among the components of the system. While the focus of this survey is on graph-based methods, other modeling approaches such as probabilistic, risk analysis-based, and agent-based approaches [75] can also be used for modeling interactions among the components of the system. B. Revie w Methodology In this surve y , various techniques for building the inter - action graphs are re viewed. While the main focus of this surve y is on the methods for constructing interaction graphs, reliability studies, and analyses performed using such graphs are also brieﬂy discussed.The beneﬁt of interaction graphs is that the interactions among the components are topologically local, which simpliﬁes the study and analysis of propagation behavior of failures and properties/roles of various compo- nents in the system during the cascade process. Note that as cascading failures are attributed to the transmission network 2 of power grids, the focus of these studies are mainly on the transmission netw ork. Moreov er , the majority of studies based on the interaction graphs are focused on cascading failure analyses in power grids; ho wev er , interaction graphs can be used for other applications in power grids (e.g., analyzing reliability to targeted attacks) or even other networked systems such as transportation networks [76], [77]. The earliest research study in ﬁnding interactions between components can be traced back to the 1989 study in [78], in which interactions between the buses in the system represented a measure of electrical distance between components based on changes in voltage magnitude sensitivities. While the power grid analyzed in this study was not modeled as a graph, the concept in this study has been widely adapted by numerous graph-based models to represent interactions between components. In this survey , methods for constructing interaction graphs are broadly categorized into two main classes: data-driven approaches and electric distance-based approaches. As the name implies, the data-driven approaches for building interaction graphs rely on data collected from the system (historical and real data or simulation data) for inferring and characterizing interactions among the compo- nents of the system. Further , three categories are deﬁned for data-driv en interaction graphs based on the method used for analyzing the data.These include: (1) methods based on outage sequence analysis [13]–[37], (2) risk-graph methods [38]– [41], and (3) correlation-based methods [29], [30], [42], [79]. The category of outage sequence analysis is further divided into four sub-categories including (i) consecuti ve failure-based methods [13]–[20], (ii) generation-based methods [21]–[26], (iii) inﬂuence-based methods [27]–[30], and (i v) multiple and simultaneous failure-based methods [31]–[37]. This nov el taxonomy is used to classify thirty detailed research studies, including conference and journal publications, in the data- driv en category into various subcategories. On the other hand, electric distance-based approaches ex- ploit properties based on the physics of power and electricity gov erned by Kirchoff ’ s laws to deﬁne interactions among the components. Thus, the interactions are represented by electri- cal distances , which illustrates the properties of the electrical interactions based on power ﬂo ws among the components. T wo sub-categories are deﬁned for electric distance-based interaction graphs based on the power grid conditions that are considered for creating the graphs. These include: (1) methods that deﬁne the interactions based on changes in the po wer ﬂow due to changes in physical attributes of components caused by outage conditions [43]–[47] and (2) methods that deﬁne the interactions among components during normal or non- outage operating conditions [48]–[63], [78]. The category of deﬁning interactions during non-outage operating conditions is classiﬁed into two sub-categories: (i) impedance-based methods, which deﬁne interactions by considering a single impedance measure among the components connected over multiple paths [48]–[59] and (ii) sensiti vities in components’ states due to changes in voltage magnitudes and voltage phase angles [60]–[63], [78]. This novel taxonomy is used to classify twenty-one detailed research studies, including conference and journal publications, in the electric distance-based cate gory into v arious subcategories. Figure 1 sho ws the taxonomy of the revie wed methods for constructing v arious types of interaction graphs. The ﬁgure also speciﬁes section numbers for the categories and subcategories, in which the methods hav e been discussed. Research studies included in this revie w have been found using databases such as IEEE Explore, Elsevier , AIP (American Institute of Physics) Publishing, Springer, APS (American Physical Society) Physics, and MDPI. Most studies in this revie w are fairly recent and have been published in the past decade. In addition to the revie w of various methods for constructing interaction graphs, v arious reliability analysis and studies performed using these graphs are also brieﬂy revie wed. Some studies of interaction graphs are focused on identifying critical components in the cascade process of power grids [14]–[24], [27]–[31], [38]–[40], [48], [56]–[59]. These studies can have different purposes such as (1) identifying the vulnerable or most inﬂuential components of the system in the cascade process by utilizing standard centrality metrics [14]–[20], [50], [54] or deﬁning new centrality metrics [21]–[24] and (2) identifying the set of components whose upgrade (for instance, by increasing the power ﬂow capacity of transmission lines) or protection can help in mitigating the risk of cascading failures and large blackouts [27], [28], [31], [58] or quantifying the performance of the grids after addition of ne w transmission lines [58], [59]. T o characterize the latter, some works [16], [38]–[40], [59] focus on the response of power grids to attacks and failure scenarios using metrics that quantify the efﬁciency of the grid before and after the attacks. Furthermore, to characterize the role of components in the cascade process, some efforts are focused on characterizing the patterns and structures in interaction graphs using community detection ap- proaches [29], [30] or tree structures [43], [44]. Moreo ver , the work in [44] uses interaction graphs to identify transmission lines that when switched of f create partitions that limit the propagation of failures in the power grid. Some studies [31], [42] also use the structures and patterns in interaction graphs to characterize and predict the distrib ution of cascade sizes. Similarly , structures in interaction graphs hav e been used for reliability analysis of zonal patterns [51] and partitioning into voltage control regions [78]. C. K e y Contributions and Review Structur e The key contributions of this revie w work are as follows: 1) A no vel classiﬁcation of methods for constructing in- teraction graphs into two categories: data-driv en and electric distance-based approaches as well as multiple sub-categories as shown in Figure 1. 2) A comprehensive study and detailed discussion on the techniques used in the construction of interaction graphs. Moreov er , key properties and limitations of each type of interaction graph is discussed. Suggestions on addressing the limitations and possible future directions are pre- sented. 3) A brief overvie w of cascading failure analysis in power grids using the constructed interaction graphs are pre- sented. 3 Interac ti on Gra p h s Data - dr iv en (Sec t ion 3. 1) Ou t a ge Seq u en c e (Sec t ion 3. 1. 1) Con s ec uti ve Fa i l u res (Sec t ion 3. 1. 1. 1) Ge n e ra t i o n - bas ed Fa i l u re s (Sec t ion 3. 1. 1. 2) Infl uenc e - bas ed (Sec t ion 3. 1. 1. 3) Multi ple and Simult a ne o us Fa i l u res (Sec t ion 3. 1. 1. 4) Ri s k Gra p h (Sec t ion 3. 1. 2) Corr el a ti on - bas ed (Sec t ion 3. 1. 3) Elec t r ic Di s tan c e - bas ed (Sec t ion 3. 2) Ou t a ge Con di ti on - base d (Sec t ion 3. 2. 1) Non - outage Con di ti on - base d (Sec t ion 3. 2. 2) Impeda nc e - bas ed (Sec t ion 3. 2. 2. 1) Ja co b i a n (Sec t ion 3. 2. 2. 2) F I G . 1 : T axonomy of methods for constructing interaction graphs with section numbers shown for the categories and sub- categories. The organization of the rest of the revie w is as follows. In Section II, a brief discussion on the preliminary terms pertinent to this re view is presented including: (1) deﬁnition and causes of cascading f ailures, (2) well established cascading failure models in literature, and (3) physical topology-based graphs of po wer grids. In Section III, a comprehensive discussion on the various methods for constructing interaction graphs is provided, and in Section IV, a brief discussion is presented on the reliability analysis and cascading failure studies performed using the interaction graphs constructed in Section III. Finally , in Section V, the revie w is summarized and concluded. I I . D E FI N I T I O N S A. Cascading F ailures Cascading failures are the leading cause of wide area blackouts [3]. While large blackouts are infrequent, the power law behavior exhibited by the blackout size distribution (e.g., measured in terms of unserved energy , numbers of customers with no service, number of transmission lines tripped) w arrants the need to study such ev ents [7]. A cascading failure can be deﬁned as a sequence of interdependent outage e vents, initi- ated by few outages or disturbances [1], [64]. The initiating ev ents can be attributed to various factors such as natural disasters, vegetation disturbances (e.g., tree contact), human errors, software/hardware errors, and so on. In recent years, cyber/physical attacks on po wer grids, such as the case of the Ukrainian cyber attack of 2015 [80], are also precursors to cascading failures. After the occurrence of the initiating ev ents, the dependent sequence of outages results from various internal events such as voltage and angular instabilities, line ov erloads, hidden failures caused due to the misbehavior of protection de vices as well as errors related to maintenance, operation, and human factors [64]. Further , various operating conditions of the power grid, such as the initial loading conditions of the components, also affect the behavior of the ov erall power grid during cascade processes. B. Models of Cascading F ailur es Modeling and studying cascading failures include a div erse ﬁeld of techniques and approaches. They include topological models, high lev el statistical models, deterministic and prob- abilistic models, simulation-based models for analyzing quasi steady and dynamic behavior of the system, or hybrid models, interdependent models with other systems (e.g., communica- tion systems), and so on (see [66]). Many of these meth- ods hav e been bench-marked and validated as well as cross validated [81], [82]. Since this revie w is focused on graph- based methods, the upcoming sections begin by discussing the physical topology-based graphs of power grids and their limitations in representing interactions among components during cascading failures, which motiv ates the subsequent discussion of interaction graphs for power grids. C. Physical T opology-Based Graphs of P ower Grids As mentioned in Section I, initial graph-based studies of power grids, such as [70]–[73], were based on the physical topology of the power grid. In general, a power grid can simply be represented by a graph, G = ( V , E ) , where V represents the set of generator , transmission, substation, or load b uses, and E represents the set of po wer lines [83]. Such a graph shows the physical connectivity among the components of the system. V arious studies have been performed on the physical topology of the power grids by analyzing their global structural properties [72], [73], such as av erage path length, clustering coefﬁcient, and degree distribution, for analyzing power grids with respect to standard comple x networks such as small world, random, and scale-free graphs. Particularly , in the study in [73], the a verage path length and clustering coef ﬁcient of real-world power grids were compared to their equi valent random and scale-free network models. Ho wev er , the study concluded that real-world power grids dif fered signiﬁcantly from standard network models as the clustering coefﬁcient and the average path length of real-world grids were signiﬁcantly greater than that of their comple x network model counterparts. Some studies performed on the physical topology also focused 4 on properties of the electrical connections [84] identiﬁed using centrality measures such as degree, eigenv ector , closeness, and betweenness (for a revie w and deﬁnition of centrality measures refer to [85]). Ho wev er , it has also been discussed that physical graphs may be inadequate in representing and capturing the interactions among the components of the po wer grid [62], [74], speciﬁcally for analyzing cascading failures. This limitation is due to the inability of the physical graphs to capture the dynamics of interactions at-distance in cascading failures, for instance, due to Kirchoff ’ s and Ohm’ s laws. In recent literature, some studies consider the physical as well as electrical properties of power grids. The focus of such studies is on the generation of synthetic power grid networks that consider the heterogeneity of the components in terms of their operating voltages [83], [86]. In such types of graphs, each verte x is associated with a v oltage rating such that transmission lines are represented as edges between vertices of the same voltage le vel and v oltage transformers are represented as edges between vertices with different voltage le vels. Ho wev er , exten- siv e analysis of such graphs for cascading failure scenarios is still an open research problem. I I I . G R A P H O F I N T E R AC T I O N S In this section, the methods of modeling power systems by graphs of interactions are revie wed in two distinct categories: data-driv en methods and electric distance-based methods. These methods b uild a graph of interactions for the system, denoted by G i = ( V i , E i ) in which the set of v ertices V i are the components of the system whose interactions are of interest, such as the set of b uses or transmission lines. Further, the set E i represents the set of interactions/inﬂuences among the components, which may be directed, undirected, weighted (representing the strength of interactions or inﬂuences), or unweighted depending on the analysis of interest. A. Data-Driven Methods for Interaction Graphs V arious data-driven approaches have been proposed for inferring and modeling interactions among the components of the power grid. These approaches rely on data from simulation or historical outage datasets. As the historical datasets are limited, the majority of studies use simulation data. Howe ver , the focus of this re vie w is not on re viewing the mechanism for generating cascade data; for example, from power system simulations. Rather , the focus is on modeling the cascade data into interaction graphs and the subsequent reliability analysis performed on such interaction graphs. The cascade datasets used in the various methods will be mentioned as necessary when discussing the methods. In this paper , ﬁ ve classes of data-driv en methods hav e been identiﬁed and revie wed for modeling interaction graphs for studying cascading failures in power grids as shown in T able I. Next, each class of method is discussed in detail. 1) Interaction Graphs Based on Outage Sequences in Cas- cading F ailur es: This class of methods rely on cascading failure data in the form of a sequence of failures in each cascade. For instance, the sequence l 5 → l 7 → l 3 → l 6 represents an e xample of sequence of transmission line failures in a cascade scenario, where l i represents outaged transmission lines and the arrow represents the order in which the lines failed during cascades. These methods are based on analysis of sequence of failures for e xtracting interactions and focus on the cause and effect interactions among failure of components. Methods in this category use various techniques and statistics to analyze such data as discussed next. a) Interaction Graph Based on Consecutive F ailur es: In this cate gory of outage sequence analysis, only direct consecutiv e failures in a sequence are used for deri ving the interaction links among the components of the system. In other words, two components in the system have an inter- action link, e i,j ∈ E i , only if they appear as successiv e outages in the order l i → l j in a cascade scenario in the dataset. The order of the outages represents the direction of the links in the interaction graph, e.g., outage sequence l 5 → l 7 suggests an outgoing link from node l 5 to node l 7 . Further , if the example sequence l 5 → l 7 → l 3 → l 6 represents a longer sequence of transmission line failures in a cascade scenario, the following directed interaction edges will belong to graph G i , i.e., { e 5 , 7 , e 7 , 3 , and e 3 , 6 } ∈ E i . The strength of interactions among the components in this case can be characterized using the statistics of occurrences of pairs of successive outages in cascade scenarios in the dataset. For instance, the work in [13] assigns weights to the interaction edges by statistical analysis of the number of times that a pair of successiv e line outages occurs in the cascade dataset. For instance, the weight of the interaction link from node l a to node l b can be characterized as | l a → l b | / ( total number of successiv e pairs in the overall cascade dataset ) , where | l a → l b | is the number of times failures l a and l b occurred successi vely in the cascade dataset. These weights can be interpreted as the probability of occurrence of each pair of successive line outages. Examples of studies using this method to dev elop the power grid’ s graph of interactions include [13]–[20], where transmission lines in the system are the vertices V i of the interaction graph G i . In the study presented in [87], the sequences of consecutive failures are called fault chains . For creating the dataset of fault chains, in the ﬁrst step, a single transmission line is tripped as the initiator of cascading failure in the simulation, and in the subsequent steps, the most overloaded component due to power ﬂow re-distrib utions is considered as the next failure in the overall sequence of consecutiv e failures. In the studies in [14]–[16], [18]–[20], for a power grid with n transmission lines, n fault chains are created, and the edges among consecutiv e failures in each chain are weighted based on power ﬂow changes in a line after the failure. In the work in [17], fault chains are created by considering multiple initial failures such that a system with n transmission lines may hav e more than n fault chains. In addition to power ﬂow re- distributions, the work in [19] also considers the temperature ev olution process of transmission lines during cascades while constructing fault chains. Thus, the fault chains are capable of reﬂecting thermo-physical ef fects of transmission lines during cascades. Finally , a fault chain graph is dev eloped by combining all fault chains together into a single graph where the vertices are all the components that ha ve f ailed in the 5 T A B L E I : Classiﬁcation of existing studies using the data-driv en taxonomy . Category Subcategory Further Subcategory W orks Data-driv en Interaction Graphs Outage Sequence Consecutiv e Failures [13], [14], [15], [16], [17], [18], [19], [20] Generation-based Failures [21], [22], [23], [24], [25], [26] Inﬂuence-based [27], [28], [29], [30] Multiple and Simultaneous Failures [31], [32], [33], [34], [35], [36], [37] Risk-graph [38], [39], [40], [41] Correlation-based [29], [30], [42] fault chains, and the edges between the vertices exist if the outages have successiv ely occurred in the fault chains. F or pairs of outages ( l i → l j ) that hav e reoccurred in multiple fault chains, their combined edge weight in the fault chain graph is averaged. b) Interaction Graph Using Generation-based Analysis of F ailur es: The method based on the consecuti ve failures discussed in Section III-A1a focuses on one to one impact that the outage of a line has on the outage of another line. Ho wev er , in cascading failures, instead of pair-wise interactions among successiv e failures, a group of failures may contribute to failures of other components. Therefore, it is important to consider the effects of groups of failures and characterize interactions among the components based on the ef fects among groups of components. The works presented in [21]–[25] deﬁne such groups as generation of failures within a cascade process, which are failures that occur within short temporal distance of each other .In these works, the sequence of failures in the cascade are divided into sequence of generations, and the failure induced cause and effect relationships are considered between consecutiv e generations. Speciﬁcally , outages occur- ring in generation m + 1 are assumed to be caused by outages in generation m . The interactions based on successi ve generations are deﬁned in dif ferent ways in the literature. For instance, the authors in [27] assume that all components in generation m hav e interactions with all components in generation m + 1 , i.e., if generation m has n 1 number of components and generation m + 1 has n 2 number of components, then the number of interactions between generation m and m + 1 will be n 1 × n 2 . Howe v er , some studies argue that considering all possible pairs of interactions among components of two consecutive generations ov erestimates the interactions among components [21]–[26]. Speciﬁcally , all line outages in one generation may not be the cause of a line outage in the next generation. Therefore, in the works presented in [21], [23], the cause of failure of a line k in generation m + 1 is considered to be due to the failure of a line in generation m with the maximum inﬂuence value on the line k . The inﬂuence value for component j in generation m is deﬁned as the number of times that the component j has failed in generation m before the failure of line k in the successi ve generation m + 1 in the cascade dataset. For cases where two or more lines in generation m have the same maximum inﬂuence values on line k in generation m + 1 , all such components are assumed to interact with line k . In the works discussed so far in this section, the interaction among component j in generation m and component k in generation m + 1 will be represented by a directed link e j,k . While the works in [21], [23] limit the interactions by only considering the maximum inﬂuence values in current generation as the probable cause of component failures in the ne xt generation, the work in [22] giv es an estimate of the interactions between successive generations using the expectation maximization (EM) algorithm. Initially , all failed components in generation m are assumed to be the causes of failure of all components in generation m + 1 . Howe ver , the actual components in generation m (hidden v ariables) that cause failure of components in generation m + 1 are found using the iterativ e process of updating the probabilities of failures. Thus, after the iterativ e update process is completed, some probabilities of failures between components may be zero, which remov es the overestimated interactions of the initial assumption. The weight of the interaction links can also be deﬁned in various ways. For instance, the weight of the link can be deﬁned as the ratio of number of times that the pair of components appeared in two successiv e generations over the total number of times that component k has appeared in the dataset. This weight can be interpreted as the likelihood of failure of component k in the next generation given the failure of component j in the current generation. The work in [24] also generates the graph of interactions based on the maximum inﬂuences among generations in cascades, similar to the method used in [21], [23]. Howe v er , the graph of interactions has two additional layers: one layer to capture the weight of interactions based on the amount of load shed that has occurred after the failures in generation m (therefore, the dataset requires additional information about the amount of load shed during the cascade process) and the other layer to capture the weights of interactions based on the electric distance between transmission lines during the cascade process. W e will present the detailed discussion of the electric distance-based interactions in Section III-B2. The study in [26] considers both statistical properties as well as the amount of load shed that has occurred between successi ve generations to assign interaction link weights. Howe ver , the study in [26] identiﬁed islands formed in the power grids during outages and then selectiv ely assigned links between components of successiv e generations only if the generations were located in 6 islands that were direct consequences of one another . c) Inﬂuence-Based Interaction Gr aph: In this method, the interactions among the components are deri ved based on successiv e generations in cascades; howe ver , the weights of the interactions are characterized based on the inﬂuence model and the branching process probabilistic framew ork. The inﬂuence model is a networked Markov chain framew ork, originally introduced in [88] and was ﬁrst applied to cascade dataset in the work presented in [89]. This survey revie ws the studies that use the inﬂuence model in the context of po wer grids to develop the graph of interactions. In these studies, the transmission lines in the system are considered as the vertices V i of the interaction graph G i and the inﬂuences/interactions between the lines as the edges E i . In [27], authors consider interaction links among all pairs of lines in two successiv e generations in a cascade using the inﬂuence model. The weights of the directed links are deriv ed in two steps. In the ﬁrst step, a branching process approach is used in which each component can produce a random number of outages in the next generation. The number of induced outages by each component is assumed to have a Poisson distribution based on the branching process model. Parameter λ i speciﬁes the propagation rate (mean number of outages) in generation m + 1 for the outage of component i in generation m . In other words, this step deﬁnes the impact of components on the process of cascade by describing how many failures their failure can generate [12]. In the second step, it is assumed that giv en that component i causes k outages in the next generation, some components are more likely to outage than others. Therefore, they calculate the conditional probability g ( j | i ) , which is the probability of component j failing in generation m + 1 , given the failure of component i in generation m . If only g ( j | i ) values based on the statistical analysis of data are considered, then the probability of failure of component j giv en component i failure will be kno wn; howe v er , the expected number of failures from failure of component i is not known. Hence, both steps are important in characterizing the inﬂuences among components. The ﬁnal step consists of combining the information from the ﬁrst and second steps into a single inﬂuence matrix H (representing the links of graph of interactions and their weights). The elements of the matrix are deﬁned based on the conditional probability that a particular component j fails in the next generation m + 1 , giv en that component i has failed in generation m and that generation m + 1 includes exactly k failures. This probability can be deﬁned as P ( j | i, k ) = 1 − (1 − g [ j | i ]) k . Then, the conditional probability h i,j,m that component j fails in generation m + 1 , given that component i failed in generation m , ov er all possible values of k represents the actual elements of H and is found by multiplying P ( j | i, k ) with the probability of k failures occurring as follows: h i,j,m = ∞ X k =0 (1 − (1 − g [ j | i ] k )) λ k i,m k ! e − λ i,m . (1) Based on the inﬂuence graph, cascading failures can start with a line outage at a node of the graph and propagate prob- abilistically along the directed links in the graph. Examples of other works, that have used the inﬂuence-based approach to deriv e the graph of interactions for power grids include [28]– [30]. d) Interaction Gr aph of Multiple and Simultaneous F ail- ur es: This class of methods also use the sequence of failures to model interactions among components of the system; howe ver , they consider the interactions among multiple simultaneous failures. Speciﬁcally , in the study presented in [31], a Markovian graph was dev eloped with the goal of addressing the problem of capturing the effect of multiple simultaneous outages within generations on the characterization of the interactions among the components of the successi ve generations in a cascade. In this case, the nodes of the graph represent the states of the Markov chain deﬁned as the set of line outages in a generation of the cascade, and the links represent the transition among the states (i.e., interactions between successiv e generations of outages). Hence, each node in the graph may represent the outage of a single line or multiple lines. Mark ovian interaction graphs differ from generation-based and inﬂuence-based inter- action graphs as edges are the interactions between successive generations of sets of line outages instead of the individual interactions between line outages in successive generations. Markovian interaction graphs also consider a node with a null state, which represents the state where the cascade stops. This state occurs at the end of all cascade scenarios. The transition probabilities among the states (i.e., the weight of the links) from state i to state j can be estimated by counting the number of consecutiv e states in which state i and state j occur in all the cascades and di viding by the number of occurrences of state i . Other studies that consider the interaction among multi- ple failures at the same time are presented in [32], [33]. In these works, the state of each component in the po wer grid, including b uses and transmission lines/transformers, is represented by 0 for a failed condition and 1 for a working condition. Then, states of all components in the power grid are aggregated and represented by a vector with size n + l , where n is the number of buses and l is the number of transmission lines/transformers. A single state vector can be regarded as a node in the interaction graph, and there are exactly 2 n + l number of nodes in the graph, representing all possible states. Initially , all components in the power grid are in working condition such that the initial state consists of a state vector i represented by all ones. After a component failure occurs, the state of the component changes to 0 , and the corresponding entry of the component in the initial state vector i is also updated, and thus, a ne w state vector j is formed. This transition is represented in the interaction graph by a directed link from the initial state vector i to the updated state vector j . This process is continued for all sequences of failures in the cascade dataset. Finally , some nodes in the state transition graph may be highly connected whereas some nodes may be isolated. Note the obtained interaction graph is directed but unweighted. Other examples of methods that consider interactions of multiple failures at the same time include [34]–[37]. In these works, fault chains are used to construct a state failur e 7 network. Each fault chain is also associated with its ﬁnal load loss v alue. Note that the deﬁnition of a fault chain has already been discussed in Section III-A1a. Similar to the studies in [32], [33], each component is represented by binary values 0 or 1 for working and failed conditions, respectively . The working/failed conditions for all components in the system are aggregated to form a state. Each state is an n -dimension v ector , where n is the number of transmission lines in the power grid. Howe v er , in contrast to the studies in [32], [33], where all pos- sible states, i.e., 2 n , are considered as nodes in the interaction graph, states are enumerated from the fault chain data. For example, for a fault chain { l 5 → l 7 → l 3 → l 6 } → l oad loss , four states are used for representing the four line faults in the chain, and a single state is used to represent the total load loss associated with the fault chain. All such fault chains in the cascade dataset are combined together into the state failure network. Nodes in the network represent the states enumerated from the complete dataset, and links in the network represent the failed component occurring immediately after a particular state. Thus, the links can be regarded as records of component failures that occur after a state. Next, nodes and links in the network are assigned weights by back-propagation of the ﬁnal load losses based on the probability of occurrence of the links. Thus, each node (state) and each link (failed component) is associated with an expected load loss after their occurrence. 2) Risk Gr aphs for Interaction Graph: The work presented in [38] introduces the risk-based interaction graphs, which describe the interactions or relationships among the nodes (i.e., buses/substations) of the po wer grid based on the effects of their simultaneous failures in causing damage in the system. This graph is not solely focused on analysis of interactions among components during cascading failures. Instead, it is focused on the vulnerability analysis of the po wer grid, and the effect of failures is assessed using metrics such as net-ability , which measures the effecti veness of a power grid subjected to failures, based on power system attributes including power injection limitation and impedance among the components. Construction of risk graphs is done in two steps. The ﬁrst step includes generating and tracking the sets of strongest node combinations whose simultaneous failures have signiﬁcant effects on the power grid. Identiﬁcation of such sets of strong node combinations can be done by reducing the search space or exhausti ve search methods [38], [39]. Reduced search space strategy is the preferred method for computational purposes. For instance, in [39], the search works as follows: given m - node combination of components, which causes damage in the network, m + k -node combination (where k represents additional components) should cause an ev en greater damage. In the second step, these sets of strong node combinations are used to form the risk graphs . If a node appears at least once in the sets of strong node combinations, then the node becomes a verte x of the risk graph. Links among nodes in the risk graph exist if they appear in the same set of strong node combinations. Both nodes and links in the risk graph are weighted based on the frequency of their appearance in the sets of strong node combinations. This approach results in a weighted but undirected node risk graph, where higher weight v alues on the links suggest stronger node combinations. Node risk graphs are dependent on the system parameters such as ratio of capacity to the initial load of the nodes in the system. T o remove dependencies on system parameters, node risk graphs can be constructed for multiple parameter values and combined together to form the node integrated risk graph using the risk graph additivity property [38], [39]. The aforementioned risk graph can also be extended to a directed risk graph, where the remo v al of components in a speciﬁc order in strong node combinations are considered. The study in [40] constructs the directed node risk graphs and the directed node integrated risk graph with the same concept as its undirected counterpart in the studies in [38] and [39]. Another similar concept to risk graph is the double contin- gency graph introduced in [41]. While m contingency com- binations of attack scenarios for the po wer grid was studied in the risk graphs, many methods focus on N-2 contingency analysis as the power grid is considered to be N-1 protected [90]. In the double contingenc y graph, the vertices of the graph are the transmission lines, and the links between v ertices show pairs of transmission lines whose simultaneous f ailure as initial triggers can af fect the reliability of the system by , for instance, violating the thermal constraint rules in the po wer grid. Similar to the risk graph, double contingency graph only considers combinations of initial triggers and lacks information about the components, which will be affected due to the outage of the initial triggers. Therefore, the work in [41] uses a combination of the double contingency graph with inﬂuence graph for reliability analysis of the power grid. 3) Corr elation-Based Interaction Gr aph: The work in [42] presents a graph of interactions for power grids based on the correlation among the failures of the components. In the correlation-based interaction graph in [42], vertices represent the transmission lines, and the edges represent the pairwise correlation between line failures in the cascade dataset. The correlation dependence between failures are captured in the correlation matrix, whose ij th elements are positiv e Pear- son correlation coefﬁcient between the failure statuses of components i and j in the cascade dataset. The resulting correlation matrix is symmetric and can be interpreted as an undirected and weighted interaction graph, where the nodes are the failed lines, the edges are the interactions between the lines, and the weights are the correlation v alues among the components. Similarly , the studies in [29] and [30] also construct correlation-based interactions graphs from simulated cascade dataset consisting of sequences of transmission line failures. 4) Comparison of Data-Driven Methods for Constructing Interaction Graphs: In the past decade, cascading failures hav e been activ ely modeled and analyzed using data-driv en interaction graphs. These graphs can be considered as abstract models of power grids for studying cascading failures; as the ph ysical details of the system such as generation, load consumption, line power ﬂows, capacity constraints, etc., are not explicitly considered; instead, they are implicitly captured through the cascade data. This section provides a brief com- parison among the data-driv en interaction graphs and discusses their key properties and limitations (summarized in T able II). One key difference among the data-driven methods for 8 T A B L E I I : Ke y properties and limitations of the methods for building data-dri ven interaction graphs. Category Ke y Properties Limitations Outage Sequence: Consecutiv e Failures • Simple deriv ation of interactions based on consecutiv e order of failures, resulting in directed graphs. • W eights of interactions are assigned using various methods, e.g., statistical analysis or physical/electrical properties of the system. • Only considers direct one-to-one consecutiv e interactions in cascades. • Does not consider interactions between groups of components. Outage Sequence: Generation- based Failures • Interactions are considered between groups of components (based on the concept of generations) and are used to deﬁne one-to-one interaction links between components of consecutive generations. • Considers order of failures/interactions, resulting in directed graphs. • Interactions between groups of components of two consecutiv e generations characterized using various methods (may overestimate/underestimate the interactions). Outage Sequence: Inﬂuence- based • Generation-based interactions are considered to deﬁne one-to-one interaction links between components based on the inﬂuence framework. • Considers order of failures/interactions, resulting in directed graphs. • Considers propagation rate of line outages as well as their probability of causing further outages through inﬂuence framework. • Inﬂuence-based framework provides mathematical tractability for certain analysis of the dynamics of failures. • Interaction links among all pairs of components of two consecutiv e generations may overestimate the interactions. Outage Sequence: Multiple and Simultaneous Failures • Characterizes simultaneous interactions between groups of components instead of only one-to-one interactions between pairs of individual components. • Considers order of failures/interactions, resulting in directed graphs. • Considers group interactions, eliminating the challenge of overestimation or underestimation of pair-wise interactions between individual components. • The number of nodes in the resulted graphs (to capture possible states of group interactions) will be large. Risk-graph • Characterizes the interactions based on targeted failures of the components, resulting in directed graphs. • Requires small dataset for small number of targeted failures. • Interactions between groups of components are not considered. • Reduced performance when number of targeted failures increase. Correlation-based • Uses simple statistical correlation measure to construct the graph. • Does not consider the order of failures in interactions. • Does not consider group interactions. • Resulted graphs are usually dense. constructing interaction graphs is the type of data that they need to b uild the graph. For instance, inﬂuence-based methods can be applied to both simulation data as well as historical data. Ho wev er , for the consecuti ve f ailure-based methods, such as fault-chains, as well as risk graphs, the sequence of failures needs to be generated by targeted failure of components to create a comprehensiv e list of failures to build the interaction graph. Moreover , as the combinations of the initial targeted failures increase, the computational complexity of generating the list of failures and building the risk graphs also increases. Another difference among the data-dri ven methods is their ability in considering group interactions; e.g., the consecutiv e failures-based methods can only consider one-to-one interac- tions based on direct consecutive orders of failures. Howe ver , generation-based and inﬂuence-based methods can consider the inﬂuence of a group of components on another group of components (using the concept of generations) and use it to deﬁne one-to-one interactions among pairs of components. Multiple and simultaneous failures interaction methods not only consider the inﬂuence of a group of components on an- other group but also allow for interaction links among groups of components instead of indi vidual pairs of components. The data-driv en methods for constructing interaction graphs result in directed graphs except for the correlation-based method. The correlation-based method cannot capture the order and direction of interactions due to symmetric properties of the correlation measure. Another challenge with the correlation- based method is that the method generally results in dense graphs, with many links sho wing small correlations among the components, which may need to be thresholded or applied 9 with other techniques to be able to focus on more signiﬁcant correlations. In general, one of the key limitations of the data-driv en methods is the heavy dependency of the deri ved graphs and inferred interactions on the dataset and the applied method, which may lead to overestimation/underestimation of the interactions. As shown in [29], [30], interaction graphs deriv ed from different outage data are different and can lead to different conclusions; e.g., the operating setting of the power grid (such as its loading level) will affect the cascade process and the cascade dataset [91] and, consequently , lead to different interaction graphs. Using a single interaction graph to model power grids and capture all possible operating setting conditions that can af fect the system is still a challenge to address. There is also a need to test the interaction graphs to historical cascade data [64]. Historical data shows the heavy- tailed distribution of blackout sizes and can be used to bench- mark the data-driven interaction graphs [64]. Similarly , cross validating the various techniques with each other may also provide insights into the most useful techniques in extracting interactions from cascade data. B. Electric Distance-Based Interaction Graphs While there is abundant literature focused on data-driven interaction graphs, various methods hav e been proposed for modeling interactions among the components using the dy- namics of power ﬂow as well as the physical/electrical prop- erties of the system and components. In this revie w , such meth- ods are called electric distance-based methods. In a po wer grid, electricity does not ﬂow through the shortest path between two nodes i and j . Instead, it can ﬂow through parallel paths between nodes i and j based on the physical properties of the system and its components as well as the physics of electricity (i.e., Ohm’ s law). Thus, the electrical interactions between the components may extend beyond the physical topology and the direct connections in the power grid. The concept of electric distance was ﬁrst introduced by Lagonotte et al. [78] in 1989 as a measure of coupling between buses in the power system and was based on sensitivities in the power system due to changes in voltage magnitudes. There are various methods in the literature for characterizing the electrical distances between components. These methods can use distribution factors in power grids including power transfer distribution factor (PTDF) [92], which indicates the change in real power on transmission lines due to changes in real po wer injection at dif ferent nodes of the system, and line outage distribution factor (LODF) [93], which measures the changes in the po wer ﬂow of transmission lines due to the outage of another line. Next, the electric distance-based methods for constructing interaction graphs are discussed in two categories: outage condition-based and non- outage condition-based as shown in T able III. 1) Outage Condition-Based Interaction Graph: This class of methods for characterizing electric distance-based interac- tion graphs are focused on interactions among the components of the po wer grid during outage conditions. F or instance, in the study presented in [44], interactions among the components as well as their weights are derived using the changes in the power ﬂows in transmission lines during outage conditions. Thus, the outage induced interaction graph G i consists of the set of vertices V i that represent the transmission lines and the set of edges E i that represent the impact of outage of one line on another . This impact is characterized using LODF [93], where LODF for line e i,j ∈ E i is calculated based on the ratio of the impact of outage of line i on line j based on the reactance of all possible spanning tree paths between the lines, ov er the impact of outage of line i on line j using the reactance of all alternati ve spanning tree paths where the po wer can ﬂow (i.e., excluding the spanning tree path of line i ) [43]. Howe v er , during cascading failures, the impact of a failed line on the remaining lines is not limited to changes in power ﬂows. In a power grid, if tw o or more lines share a b us, outage of one line may expose the remaining lines (connected through the same bus) to incorrect tripping due to malfunctioning of the protection relays. The exposed lines are prone to failure and an increase in po wer ﬂow in the exposed lines e xacerbates their tripping probability causing further outages. Such failures are known as hidden failures. In the studies in [45]–[47], vertices V i represent transmission lines as well as a hidden failure state, and edges E i represent inter-line interactions as well as interactions between lines and the hidden failure state. Thus, the interaction graph G i will hav e n + 1 nodes where n is the number of transmission lines, and the extra one node represents the hidden failure state. The hidden f ailure node has bidirectional links from itself to e very other node in the power grid. Ho wev er , the hidden failure node does not have any inﬂuence on itself. The inter-line interaction e i,j ∈ E i shows the increase of po wer ﬂow in line j due to outage of line i . The interaction from the hidden failure node to a line i reﬂects the likelihood of failure of line i due to the failure of other nodes when the power ﬂo w in line i exceeds a ﬂow limit mar gin. Interaction from line i to the hidden failure node reﬂects the av erage tripping probability of all the other lines due to the outage of line i . This interaction can be regarded as the aggregated inﬂuence that the outage of line i has on the tripping probability of all the other remaining lines. 2) Non-Outage Condition-Based Interaction Graphs: This class of methods for constructing the electric distance-based interaction graphs is focused on interactions among the com- ponents of the power grid during normal operating conditions. For instance, measures to capture the characteristics of the power ﬂo w paths between components, such as impedance of the transmission lines, can be used to form electric distance- based interaction graphs. Moreover , po wer system sensiti vities based on PTDF and the ones showing changes in voltage magnitudes and voltage phase angles, deriv ed from Jacobian matrices during normal conditions, can also be used to form electric distance-based interaction graphs. In this paper , non- outage condition-based interaction graphs are broadly cate- gorized into two categories: impedance-based and Jacobian. Next, both categories are discussed in detail. a) Impedance-Based Interaction Graph: Impedance- based electric distance interaction graphs G i consist of v ertices V i that represent buses and edges E i that represent elec- trical interactions between pairs of buses weighted by their corresponding impedance-based electrical distances. Inv erse 10 T A B L E I I I : Classiﬁcation of existing studies using the electric distance-based taxonomy . Category Subcategory Further Subcategory W orks Electric Distance-based Interaction Graphs Outage Condition-based [43], [44], [45], [46], [47] Non-outage Condition-based Impedance-based [24], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59] Jacobian [52], [60], [61], [62], [63] admittance matrix, more commonly known as the impedance matrix Z , is one of the simplest forms of representing electrical interactions between pairs of b uses in the system and is found by in verting the system admittance matrix Y , i.e., Z = Y − 1 . Matrix Z shows the relationship between the nodal bus voltage vector and the nodal current injection vector . Howe ver , unlike matrix Y , which is sparse, impedance matrix Z is non-sparse as it represents the changes in nodal voltage throughout the system due to a single nodal current injection between a pair of nodes in the system. Therefore, edges in the impedance-based interaction graph are the connections between the elements in the Z matrix with weights between buses i and j correspond- ing to their absolute value of the impedance, i.e., | Z ij | [48]– [52]. Smaller magnitudes of impedance represent shorter elec- tric distance between buses. Note that the individual elements Z ij in matrix Z are comple x v alued. The studies in [53]– [55] also adopt the concept of representing electrical distances between buses using the Thev enin equiv alent impedance be- tween buses but apply the condition that power only ﬂows from generator buses to load buses such that impedance v alues between generator buses and load buses suggest edges in the interaction graph. Similarly , in the study in [56], in addition to the impedance between pairs of generator and load buses in the system, the power ﬂows through the lines along the path between the pairs of buses is considered. Therefore, electrical connections between generator and load buses i and j along path k , are weighted by the impedance between buses i and j as well as the PTDF of the lines along the path k of po wer ﬂows between the buses. In the studies in [53]–[56], matrix Z has dimension Z G × Z L , where Z G and Z L are the number of generator and load buses, respecti vely . Thus, matrix Z may be asymmetric, depending on the number of generators and load as well as the entry of the elements suggesting that the interaction graph is directed. Note that studies in [48]–[56] are not focused on cascading failures, but their concept of formulating impedance-based electrical distances can be e xtended for studying cascade processes. For example, in the study in [24], transmission lines are considered as the nodes of the interaction graph, and thus, impedance-based electric distances between pairs of transmission lines are assigned as weights of the interaction links. T o ﬁnd the electric distance between transmission lines i and j , where line i connects bus i s to i d and line j connects bus j s to j d , the minimum of the four possible Thev enin equiv alent impedances between the pairs of buses, i.e., Z i s j s , Z i s j d , Z i d j s , and Z i d j d is taken. This interaction graph is also the third layer of the multi-layered interaction graph discussed in Section III-A1b used for modeling cascading failures. Cascading failures can also be studied by representing interactions between pairs of nodes by effecti ve resistances between the nodes. Effecti ve resistance, R ij , between nodes i and j , also known as Klein resistance distance [94], is the equiv alent resistance of all parallel paths between the nodes. It was initially introduced in the study in [94] as a measure of distance in graph theory , and in the context of po wer systems, it shows the potential difference between nodes i and j due to unit current injection at node i and withdrawal at node j . While impedance-based electrical distances between nodes account for non-linear approximations of power ﬂow , effecti ve resistances only consider linear approximations of power ﬂo ws in the grid. For cascading failure analysis, as the impedance of a transmission line in a high voltage transmission network is dominated by the imaginary part of impedance, i.e., reactance, the effecti ve resistance between nodes can be formulated in terms of their reactance. Thus, in the studies in [57]– [59], ef fectiv e resistance between nodes i and j is found as R ij = Q ii − 2 Q ij + Q j j where, Q ij is the row i and column j element of Q + , which is the Penrose pseudo-inv erse of the Laplacian matrix Q . Matrix Q is deﬁned as the difference between the weighted diagonal degree matrix and the weighted adjacency matrix deri ved from the physical topology and shows the relationship between the buses and transmission lines in the grid. In the studies in [57]–[59], the weights of the edges in the physical topology required for ﬁnding the weighted diagonal degree matrix and adjacency matrix are the susceptance (i.e., the imaginary part of impedance) v alues between the nodes. Note that matrix Q constructed using susceptance v alues is equiv alent to matrix Y . Thus, edges E i in the effecti ve resistance interaction graph reﬂect the electrical connections between the buses with weights between nodes i and j being the corresponding R ij values. b) J acobian Interaction Graph: Electric distance-based interaction graphs can also be constructed using the sensiti vity matrix of the power grid during normal operating conditions. In such interaction graph G i , vertices V i represent the buses, and the edges E i represent the electrical interactions in terms of sensitivities between the buses. These sensitivities can be found using the Jacobian matrix, which is obtained during Newton–Raphson-based load ﬂo w computation. Jacobian sen- sitivity matrix J shows the effect of complex power injection at a bus on the v oltage magnitude and voltage phase angles of other b uses. It consists of four sub-matrices: matrix J P θ , which shows the relationship between nodal active power injections and voltage phase angle changes; matrix J PV , which shows the relationship between nodal active power 11 injections and voltage magnitude changes; matrix J Q θ , which shows the relationship between nodal reactiv e po wer injections and voltage phase angle changes; and matrix J QV , which shows the relationship between nodal reactiv e po wer injections and voltage magnitude changes. The inv erse of any of these Jacobian sub-matrices, denoted as J − 1 , can be used to ﬁnd the sensitivity matrix by using the Klein resistance distance [94], whose individual element is calculated as x ii + x j j − x ij − x j i , where x ij represents the element in row i and column j of the in verted Jacobian sub-matrix J − 1 in consideration. In the study in [52], all of the four Jacobian sub-matrices are applied to Klein resistance distance to form sensitivity matrices for the purpose of visualizing power grids, which in turn can be used to form interaction graphs whose edges E i represent the electrical interactions between the components of the sensitivity matrices weighted by their corresponding elements. Howe v er , the literature has re vealed that most studies are focused on two of the four Jacobian sub-matrices, i.e., matrix J P θ and matrix J QV . The remaining sub-matrices J Q θ and matrix J PV are not used in the literature due to un-intuiti ve interpretations. The seminal work of electrical distances by Lagonotte et al. in [78] focuses on using the Jacobian sub- matrix ( J QV ), also known as the voltage sensitivity matrix ∂ V / ∂ Q , to ﬁnd the electric distance between b uses. Similarly , the study in [60] also uses the v oltage sensiti vity matrix. In both studies, the matrix of maximum attenuations is found, which consists of columns of voltage sensitivity matrix divided by the diagonal v alues. Finally , electrical interactions e i,j ∈ E i between buses i and j weighted by their electric distance is deriv ed as the logarithm of the individual elements of the attenuation matrix. The study in [61] also uses the v oltage sensitivity matrix to ﬁnd electric distance between buses, but instead of ﬁnding matrix of attenuations, the study applies the sensiti vity matrix to Klein resistance distance formulation. Note that studies in [60], [61] analyze risk of cascading failures by studying the voltage collapse phenomenon, which is a sequential process during which large parts of the power grid may suffer due to low voltages [95]. Similarly , the studies in [62] and [63] apply the Jacobian sub-matrix J P θ or the sensiti vity matrix ∂ P / ∂ θ to Klein resistance distance and ﬁnd electrical distances between buses. Howe ver , the studies [62], [63] are not focused on cascading failure analysis but are included in this revie w as their electric distance- based interaction graphs can ha ve potential implications for analyzing cascading failures. 3) K e y Properties and Limitations of Electric Distance- Based Interaction Graphs: In this section, key properties and limitations of the electric distance-based interaction graphs are brieﬂy discussed. Howe v er , v arious methods are not directly compared as different methods have different applications as well as concepts in their models. Electric distance-based methods consider the fundamental characteristics of a po wer grid and the power ﬂo w while deﬁn- ing interactions among components. These graphs differ from data-driv en interaction graphs as they directly use the inherent electrical properties of po wer grids based on Kirchoff ’ s and Ohm’ s laws. They are also not speciﬁc to cascading failure studies and can be used for a v ariety of applications such as contingency analysis [49], reliability studies for deﬁning zones for load deli verability analysis [51], response to tar geted attacks [55], and so on. Speciﬁcally , grid attrib utes such as ad- mittance/impedance matrices, Jacobian matrices, and PTDF’ s reﬂect the operational features of power grids and can be used for the mentioned applications. Since these graphs do not rely on extensi ve simulation data, computational complexity of ﬁnding such graphs is relatively small. For cascading failure analysis, LODF-based interaction graphs [43], [44] as well as interaction graphs sho wing the impact of hidden failures [45]–[47] on lines during cascades hav e been used. These methods usually work well for cascade scenarios inv olving a small number of contingencies and smaller sized system. Howe v er , constructing such interaction graphs for cascade studies in v olving a large number of con- tingencies is computationally expensi ve. T o this end, effecti ve resistance-based interaction graphs can be used for cascade studies, as the y can be easily produced by ﬁnding the ef fectiv e resistance [57]–[59] between components without extensi ve power ﬂow simulations. In addition to the abov e mentioned applications, electric distance-based interaction graphs can also be used for visualizing distinct electrical structure of power grids in two dimensions [52]. I V . R E L I A B I L I T Y A N A L Y S I S U S I N G I N T E R AC T I O N G R A P H S O F P OW E R G R I D S Interaction graphs constructed in Section III can be used for v arious analysis; speciﬁcally related to the reliability of the power grid, including analyzing the role of components and ﬁnding critical ones that contrib ute heavily in a cascade process, predicting distrib ution of cascades sizes, and studying patterns and structures that reveal connections and properties of the components in the power grid that extend beyond physical topology-based graphs. Thus, reliability studies per- formed using these interaction graphs are divided into v arious categories as discussed below . A. Critical Component Analysis The studies that identify and analyze the role of critical components in power grid’ s reliability are classiﬁed into three broad categories that include (1) using pre-e xisting as well as novel measures to ﬁnd critical buses/transmission lines, (2) ev aluating attack strategies that cause signiﬁcant damage in the power grid, and (3) employing mitigation measures such as upgrading transmission lines or adding new components to protect the identiﬁed critical components. 1) Critical Component Identiﬁcation: This class of reliabil- ity analyses focus on ﬁnding critical buses/transmission lines by analyzing structural properties of interaction graphs using standard centrality measures such as degree, betweenness, etc. (for a revie w of standard centrality measures, refer to [85]) or by deﬁning novel interaction graph based metrics. a) Critical Component Identiﬁcation Using Standar d Centrality Measur es: In the studies presented in [14]–[16], [18]–[20], fault chain-based interaction graphs are found to be scale free graphs, indicating that most nodes possess lo w 12 degrees but a limited number of nodes possess high in and out degrees. Thus, in the fault chain-based interaction graphs, vertices with higher degrees are assumed to be the critical components of the system. Similar conclusions are obtained by authors in the studies in [50], [54], where the impedance-based interaction graph is observed to be scale-free and consisting of a limited number of nodes with high degrees, which are considered as the critical components of the system. These are examples of works that consider the degree centrality measure to identify critical components of the system. Other centrality measures such as betweenness, eigenv ector , and PageRank ha ve also been considered on interaction graph- based representations of power grids including [45], [47], [61] to ﬁnd critical components of the system. b) Critical Component Identiﬁcation Using New Central- ity Measur es: In addition to the studies that rely on stan- dard centrality measures; some works dev elop new centrality measures in the conte xt of power grids and the dev eloped interaction graphs to analyze criticality of the components. For instance, in the generation-based interaction graphs [21]– [24], [26], out-strength measure, which is the sum of the weights of the interaction links originating from a node, is used to ﬁnd critical transmission lines. Such lines are the ones whose failure at any stage of the cascade including the initial stage or the propagation stage induces failure in a signiﬁcant number of other transmission lines. Outages in the initial stages are caused by e xternal factors such as bad weather conditions, improper vegetation management, and exogenous e vents, whereas outages in the propagation stage is due to power ﬂow re-distributions, hidden failures, and other interactions between components as discussed in Section II-A. Inﬂuence-based [27], [28] and multiple and simultaneous failure [31] interaction graphs are also used to ﬁnd critical transmission lines, but they explicitly focus on lines whose failure during the propagation stage of cascading failures causes large cascades. Particularly , the studies in [27], [28] use a cascade probability vector derived using the inﬂuence- based interaction graph to quantify the probability of failure of lines during the propagation stage of cascades and deﬁne critical lines as the ones whose corresponding entries in the probability vector hav e higher values. Similarly , the study in [31] ﬁnds the probability distribution of states of the multiple and simultaneous failure-based interaction graph and deﬁnes critical lines as the ones that belong to states with higher probability of occurrence. Inﬂuence-based and correlation- based interaction graphs constructed in the studies in [29], [30] are also used to ﬁnd the critical transmission lines during cascade processes by using a community-centrality measure. As the name suggests, the measure quantiﬁes the criticality of transmission lines based on their community membership, where critical lines are the ones that belong to multiple communities or act as bridges between communities. Note that communities are deﬁned as groups of vertices with strong connections among themselves and few connections outside (for deﬁnition of communities and a revie w of community detection methods on graphs, refer to [96]). Identiﬁcation of critical lines is not limited to data-dri ven interaction graphs. Multiple studies use electric distance-based interaction graphs for such analysis as well. Effecti ve resis- tance between components in the effecti ve resistance-based interaction graph can be summed for all node pairs in the graph to ﬁnd the effective graph resistance metric of the power grid. Effecti ve graph resistance metric was initially deﬁned in the study in [94] as Kirc hhoff index and used in the study in [97] as a robustness metric. Lo wer values of this metric suggest that the power grid is robust to cascading failures. Effecti ve graph resistance can also be found using the eigen v alues and eigenv ectors of the Laplacian matrix of the grid [98]. In the study in [57], critical transmission lines are found by measuring the changes in effecti v e graph resistance before and after the removal of the line. In a similar manner , impedance-based interaction graphs constructed in the study in [56] and [53] are also used to ﬁnd critical transmission lines by measuring the changes in net-ability metric before and after the removal of a line. Net-ability reﬂects the performance of a network by quantifying the ability of a generator to transfer power to a load within the power ﬂow limits. 2) Studying the Effect of Line Upgrades and Line Addi- tions on Reliability of P ower Grids: While identiﬁcation of critical components in the power grid is necessary , assessing the impact of modiﬁcations and protection of such critical components in the overall power grid is the next step in the study . In the studies presented in [27] and [28], an inﬂuence interaction graph-based metric is used to quantify the impact of upgrading the critical lines (for example, by improving vegeta- tion management around the lines or by improving protection systems) on cascade propagation. The work in [31] uses the multiple and simultaneous failure-based interaction graph to do a similar study . In both interaction graphs, the authors conclude that upgrading lines that take part in the propagation of failures during cascades reduces the risk of large cascades compared to the upgrade of lines that initiate cascades. While the studies in [27], [28] inv estigate the performance of the power grid networks after line upgrades, the studies in [58] and [59] use ef fectiv e graph resistance metric to study the impact of adding transmission lines in optimal locations of the power grid. Ho wev er , in [59] the authors warn that placing an additional line between a pair of nodes does not necessarily imply increased robustness of the grid. In fact, grid robustness may decrease after adding additional lines (due to Braess’ s paradox [99]), if the additions are done haphazardly . 3) Analyzing Response to Attack/F ailur e Scenarios: In ad- dition to identifying critical components and characterizing the impact of their modiﬁcations in the reliability of power grids, the study of the response of power grids to attacks and failures is also necessary . Such studies can be used to ﬁnd critical components and attack strategies that threaten the reliability of the overall po wer grid. In the studies in [38]–[40], node integrated risk graphs are used to ﬁnd groups of transmission lines whose remov al from the graph causes the largest drop in net-ability of the power grid, as discussed in Section III-A2. These groups can be found in real-time independent of system parameters. The study in [59] also analyzes the robustness of power grids to deliberate attacks using the effecti ve graph resistance metric, as discussed in Section IV -A1b. 13 B. Pr ediction of Cascade Sizes Forecasting cascade sizes is another challenge in the relia- bility analysis of power grids. In the study in [42], a correlation graph-based statistical model, known as the co-susceptibility model, is used to predict cascade size distrib utions in trans- mission network of power grids using individual failure prob- abilities of transmission lines as well as failure correlations between transmission lines found from the correlation matrix. The study exploits the idea that groups of components that hav e higher correlations are likely to fail together and uses the correlation matrix to ﬁnd such co-susceptible groups, which is an approximate estimate of the cascade size giv en an initial trigger failure. A similar idea is used in the studies in [29], [30], where components within the same community are assumed to be likely to fail together, and the size of communities gi ves an approximation of cascade sizes. The study in [31] also characterizes size of cascades by using the states of the Markov chain to ﬁnd the probability distribution of the number of generations in a cascade. C. Studying P atterns and Structures in Interactions Structures and patterns in networks are important in de- scribing the spread of various processes such as infectious diseases, behaviors, rumors, etc. [100]–[102]. For instance, in the studies in [29], [30], [43], [44], structures present in the interaction graphs of power grids are used to study the impact of cascading failures in the transmission network and utilize the graph structure to mitigate large cascades. The graph structure considered in the studies in [29], [30] are communities whereas the graph structure considered in the studies in [43], [44] are tree partitions. In the study in [43], tree structures present in the outage condition-based interaction graph showed that transmission line failures could not propagate across common areas of tree partitions. Further , the extended work of [43] in [44] found the critical components of the tree partitions, known as bridges. The failure of bridge lines plays a crucial role in the propagation of cascading failures. Howe v er , failure of non- bridge components does not propagate failures and the impact is more lik ely to be contained inside smaller regions/cells. This important property of bridge lines is used in mitigation of cascading failures by switching off transmission lines that cause negligible network congestion as well as improv e the robustness of the system. Similarly , in the studies in [29], [30], inﬂuence-based and correlation-based interaction graphs were used to limit propagation of cascading failures inside community structures. Particularly , the authors used the idea that failures can be trapped within communities by protecting the bridge/overlap nodes, which connect multiple communities together . Purposes of analyzing structures present in the interaction graphs are not limited to mitigation of cascading failures. For instance, in the study in [49], network structures were used for contingency analysis. The impedance-based interaction graph in [49] was pruned by removing edges above an operator deﬁned threshold. Then, the common structure between the pruned impedance-based interaction graph and the topological graph of the power grid was analyzed and veriﬁed to be the contingencies that violate transmission line limits and cause ov erloads. Similarly , the study in [51] identiﬁed zonal pat- terns in the impedance-based interaction graph for reliability assessment of zones for load deliverability analysis. V . S U M M A RY Analyzing interactions among the components of power grids during cascading failures can re veal important informa- tion about the failure propagation process in these systems and the role of the components in the process. It was discussed that the physical topology of the power grids have limitations in modeling cascading failure process in power grids. As such, new methods are emerging to capture non-local inﬂu- ences/interactions among the components of the po wer grid. In this survey , v arious techniques for constructing the graph of interactions (be yond the physical topology of the power grid) were re vie wed. A nov el taxonomy was presented for classifying the existing research studies into various categories and subcategories based on the type of data and techniques used in inferring the interactions and creating the graph model of the system. Finally , a brief overvie w of various reliability analysis studies based on the interaction graphs was presented. Speciﬁcally , the presented taxonomy introduced two main categories for the methods of constructing interaction graphs: data-driv en and electric distance-based approaches. The key properties and limitations of various techniques in each cate- gory as well as an overvie w of the methods in each category hav e been discussed. While for data-dri ven methods, a vail- ability of cascade data is a necessity , the electric distance- based methods rely on accurate physical models of the system (which are not required for data-driv en methods). Therefore, depending on the availability of the data or the physical model of the system, one can choose between these categories of methods. Moreo ver , as the existing methods use various infor- mation for inferring and modeling interactions, they can shed light on various aspects of the cascade process and reliability and vulnerability of the components. As such, depending on the analysis of interest, different techniques can be selected. Studies in both areas of data-driven and electric distance-based approaches are ongoing, and it can be e xpected that these methods will greatly contribute in providing new understand- ings about the cascade process in power grids. Furthermore, in the area of cascading failure reliability analysis based on interaction graphs, recent literature is more inclined to wards identifying structures and patterns in the interaction graphs to study the vulnerabilities in po wer grids. Howe ver , other analyses, such as predicting the path and size of the cascades, can also utilize the interaction graphs of the system. Next, future directions of research related to the interaction graphs for cascading failure analysis are brieﬂy revie wed. One of the ke y areas of research in this domain would be to relax the dependence of the constructed interaction graph on the applied method, data, and operating characteristics of the system. As discussed in this survey , the method used for inferring the interactions; the av ailable data; and the operating setting of the system, such as the loading le vel of its 14 components, all af fect the interaction graphs. It is important to be able to deriv e uniﬁed graphs of interactions that implicitly capture the ef fect of such factors in the model. Moreov er , the computational complexity of constructing interaction graphs (which increases with the size and scale of power grids) has limited their application to off-line analysis of reliability . This limitation suggests the need for further research in de veloping more effecti ve data-dri ven and electric distance-based methods for dev eloping real-time analysis based on interaction graphs for time sensitiv e analysis during cascading failures. In addition, most of the existing studies using the interaction graphs are observed to be focused on the identiﬁcation of the critical components of the power grid. Such analyses can be a step towards dev eloping mitigation strategies that can be applied to create more reliable power grids. As such, it is important that practical and economically feasible mitigation strategies be studied and dev eloped, based on the insights provided by the interaction graphs. 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Interaction Graphs for Cascading Failure Analysis in Power Grids: A Survey

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