Vulnerability Assessment of Power Grids Based on Both Topological and Electrical Properties

1 V ulnerabilit y Assessment of Po wer Grids Based on Both T opological and Electrical P rop erties Cunlai Pu and Pang W u Abstract —In modern po wer grids, a local failure or attack can trigger catastr oph ic cascading failures, which ma ke it chal- lenging to assess the attack v ulnerability of power grids. In this Brief, we define the K -link attack problem and study th e attack vulnerability of po wer grids under casca ding failures. Particularly , we propose a link centrality measure based on both topological and electrical properties of p ower grids. According to thi s centrality , we propose a greedy attack algo rithm and an optimal attack algorithm. S imulation results on standard IEEE bus test data sho w that the optimal attack is b etter than the gr eedy attack and the traditional PSO-based attack in fracturing power grid s. Moreo ver , the greedy attack has smaller computational complexity than the optimal attack and the PSO- based attack with an adeq uate attack efficiency . Our work helps to understand the vulnerability of power grids and pro vides some clues f or securing power grids. Index T erms —V ulnerability assessment, power grid, l ink cen- trality , cascading failur e, n etwork attacks. I . I N T RO D U C T I O N Now a d ays, power grids in the r eal world face various kinds o f risk s such as natural disasters an d attacks. Even worse, a lo c al failure in power grids ca n result in lar ge- scale blackouts [ 1], [2]. A recent example is the nationw id e recurrin g elec tr ical blackouts in V enezuela began in Mar c h 2019, w h ich was supposed to be cau sed by a local vegetation fire a nd cyber a ttacks. The cata stro phic ca scad es o f failures pose a great threat to human life and national security . Th us, it is of g reat impo rtance to und erstand and con tr ol the cascad in g failures of power gr ids. In the past d ecade, the complex network theor y has be en widely applied to the study of cascading failures [ 3]. On the m odelling side, power grids can b e ab stracted as in ter- depend ent networks, and then th e perc o lation theo ry has been used to explor e the dy n amics of cascading failures [4]. Rich behaviors ha ve been observed when taking the power grid as interdepen dent network. For instance, Buldy rev et al. [5] foun d a hybr id phase transition (HPT) , whe r e the or der par a meter has both a jump and a critical scaling. On th e co ntrolling side, research e r s propo sed many opti- mization strategies ag a inst ca scading failures in p ower grids. T u e t al. [ 6] used the simulated a nnealing metho d to op timize the network topology , and found that it is better to make the network sparsely connected, and p lace the generators as decentralized h ubs. They fur ther inv estigated the w e ak interdepen dency between network s of c y ber-physical systems C. Pu and P . W u are with the School of Computer Scienc e and Engineerin g, Nanjing Uni versity of Science and T echnology , Nanj ing 210094, China. Email: pucunla i@njust.ed u.cn. (CPS) and d iscussed how the failure p ropag a tio n probab ilities affect the robustness of CPS [7]. Chen et al. [ 8] performed the critical node analysis to iden tify th e v ital n odes in ter ms of network robustness. T h ey foun d that assortativ e couplin g of no d e destructiveness is more robust in den sely coupled n et- works, wh e reas disassortati ve coup ling of no de robustness and node destructiveness is better in spar sely coupled networks. Zhong et al. [9] stud ied the rep air process against ca scad ing failures by considering the o ptimization of repair resou rce, timing and load tolerance, for d ifferent coupling strength and network topolo gies of interd ependen t n e twork s. Zhu et al. [10] established two multiob jectiv e op timization mod els that consider both th e operational co st of links and the robustness of networks. Zhang e t al. [11] emp loyed the p article swarm optimization (PSO) algorithm to op timize the defense resou rce allocation to improve the n etwork rob ustness. T o better understand and contro l cascadin g failure s, we need to explore the role o f individual n odes and links in po we r grids. When q uantifying the impo r tance of nodes or links, the complex ne twork the o ry only focuses on network topo logy informa tio n [3], [12], [ 13]. Howe ver, the electrical featur e s of no des an d links ar e prof ound [1 4]. Particularly , a lin k of small topological importance might ha ve large current lo ad. The bro ken of this kind of link h as significan t im pact on the function of p ower gr ids. It is th us more reason able to co nsider both topo logical and electrical featu res of power g rids when characterizin g the im portance of nodes and links. In this brief , we study vulner a bility assessment of power grids under cascading failures. W e define the importance of links b ased o n both top ological and electrical features, a nd remove a few impor tant link s as the in itial attack that trigg ers cascading failures. Our main contributions ar e as follows: • W e prop ose a link centrality measure, which co mbines link degree and link current. The weights of the two features are tuna ble and represented b y t wo v ariables. This centrality is better than link degree and lin k current in quan tif ying th e importance of links in power grids. • According to the lin k centrality , we prop ose a greed y attack alg orithm an d an optimal a ttac k algo rithm. Th ese attack algo rithms are design ed to cause large-scale cas- cading failures, based on which we can assess the vul- nerability of power grids by simulation. In the next section, we p r esent the cascading failure m odel and related metrics. I n section I II, we introduce ou r link centrality measure and the parameter tu ning me thod. I n section IV , we define the link attack pro blem and provide our attack algorithm s. I n section V , we show the simulation results and 2 related analysis. Section VI is our conclu sion. I I . M O D E L A N D M E A S U R E M E N T In a power grid, power stations an d transmission lin es can be abstracted as nodes and links, re spectiv e ly . Then, we obtain the network top ology of power grid, den o ted by G = ( V , E ) , where V is the node set with N nod es, and E is the link set consisting of M link s. A. Current model Usually , there are g enerator no de, con sumer nod e , distri- bution node, an d tran sf o rmer node in a power grid . Her e, following Ref. [6] and fo r the purp ose of simplification, we only co n sider two kinds of n odes: generator no de i and consumer node j . Then, the Kirchhoff ’ s current law equ ation for a power g rid is written as Y ∗ [ · · · v i v j · · · ] = [ · · · v i I j · · · ] , (1) in which Y =     . . . . . . . . . . . . . . . y i 0 . . . . . . − Y ji Y j j . . . . . . . . . . . . . . .     , (2) and v i and v j represent the voltages of ge nerator no de and consumer n ode, respectively . I j is the curren t of consumer node j . I n matrix Y , y i = 1. Y i j is the admittan ce of link ( i , j ) , and Y i j = 0 if nodes i and j are not con nected. Also, we have Y j j = − ∑ i 6 = j Y ji . When the admittances of transm ission lines, the v oltag es o f generator nod es and the current con sumptions of consum er nod es a r e given, th e voltage o f each nod e can be computed by Eq. (1). Then, the curren t flowing thr ough lin k ( i , j ) can be calculated as I i j = ( v i − v j ) ∗ Y i j . B. Cascading failur e model Assume the load of node i is L ( i ) = u i ∗ I oi , where I oi is the total curren t flo win g out of node i , and th e load of link ( i , j ) is the cur rent flowing th rough it, i.e. , I i j . The max imum load a node can b ear is set to be 1 + α times of its o riginal load , and th e maximu m load o f link ( i , j ) is assumed to be 1 + β times of its original cur rent, wher e parameters α and β are the safety margins of nodes and links, respectively . Note that the original state o f nodes o r links corresp onds to the case whe n the power grid oper a tes no r mally , that is there is n o attacks or failures. In th e cascading failure model [5], there is usually an initial attack , e.g. ran domly removing a node or link. Th is initial ev ent will cause the load change of th e oth er n odes and links, especially for tho se close to the area of initial attack. When the load of a n o de or link exceeds its max imum allowed value, it will break, which further causes th e lo ad change of the other nod es and links. The detailed steps in th e ca scad ing failure simulation process are as follows [6]: i) Calcula tin g the initial loa d and ma x imum load of each compon ent (n ode and link) in the power grid. ii) Randomly removing a compone n t. iii) T he gr id topo logy chang e s due to the remov al, r ecalcu- lating the loa d of each c o mpone nt. A component is set to be broken if its load exceeds its maximum. iv) After removing the failed co m ponen ts, the n etwork splits into several small subgr aphs. If a subgraph d oes not con tain a generato r n ode, all nodes in this subgrap h are set to be in valid nodes. v) Rep eating steps iii) -iv) until th e load of all rema ining compon ents is n o greater than the maximum . C. V ulnerability measur ement In power systems, the outage scale is usually measured by the num ber of failed nodes. Following Ref . [6], the d a mage that is caused by compo nent set i is quantified as Φ ( i ) = N unserved ( i ) N , (3) where N unserved ( i ) is the nu mber of total failed nod es after th e cascading failure caused by the initial r e moval of comp onent set i . Note that the failed n odes contain the overloade d ones and those in the subnetwork o f no generator nodes. Obviously , the larger the damag e, the more critical the compo nent set is to the power gr id. I I I . L I N K C E N T R A L I T Y M E A S U R E A N D I T S S O L U T I O N Nodes and links are the main co mpone nts in power gr ids. Node centrality ha s been wid ely discussed in th e literatu re. Here we stud y th e link centrality which is relati vely less discussed, but critical to the vulner a bility assessment of power grids. Previously , many topolo gy based lin k cen tralities were developed [3]. Howe ver, the joint effect of top o logical an d electrical p roperties of links on the v ulnerab ility o f p ower gir d s is still not well under stood. A. Our link centrality measur e W e quan tify the c e ntrality of link s b ased on both top ological and electrical features. Specifically , we consider the link degree and link cur r ent. Th e d egre e of a link is th e n u mber of link s (except itself) incid ent to its two end nod es. T he current of a link is th e rate of flow of electric charge pasting it. Note that dur ing the cascading failure, the degree and current of a link might change du e to the broken of overloaded compon ents. Sin c e we focus on the initial attacks, we use th e original state of power g rid to quantify the ce n trality of links. Then, the centrality of link ( i , j ) is defined as Θ i j = h 1 D i j + h 2 I i j , (4) where D i j and I i j are the initial degree and curren t of link ( i , j ) , respectively . For link c urrent, w e ign ore its directio n and use its absolute v alue. h 1 and h 2 , ranging in ( − ∞ , ∞ ) , are th e weights of link d egree and link cu rrent. In real ap plications, we usually need to find the optim al values of h 1 and h 2 , which is a NP-hard problem. B. P a rameter tuning b ased on PSO W e use the particle swarm optimizatio n algorithm (PSO) [15], to search the optimal parame te r s of ou r lin k centrality measure. Compar ed with other heuristic algorithms, PSO has powerful globa l sear ch ability an d is easy to implement. In this 3 algorithm , there are m ( > 0 ) particles moving on the ne twork . Particle i is a candidate solution of p arameters, X i = [ h 1 , h 2 ] , and corr esponds to a local op timal value p i best . For all the particles, ther e is a glo bal o ptimal value g best = ma x { p i best } . The particle p o sition is continuously u pdated ac cording to the following e q uations: v i k + 1 = wv i k + c 1 r 1 ( p i best − x i k ) + c 2 r 2 ( g best − x i k ) (5) x i k + 1 = x i k + v i k (6) w = w 0 − it er / it er max (7) In Eq . ( 5), r 1 and r 2 are ran dom number s in [0,1]. c 1 is the cognitive coefficient and c 2 is the social learn ing co efficient. In Eq. ( 6), v i k and x i k are the velocity and position of the i th p article in the k th iteration. In Eq. (7 ) , w is the inertia coefficient. Large inertia co efficient he lp s particles jump out of the local optim u m, w h ile small inertia coefficient is b eneficial to the local accur ate search and convergence of th e algorithm. w 0 is the initial inertia coefficient. it er an d it er max respectively represent the curren t n umber of iter a tions and the maximu m allowed n umber of iterations. I V . A P P L I C AT I O N O F L I N K C E N T R A L I T Y I N V U L N E R A B I L I T Y A S S E S S M E N T W e apply our link ce n trality to the vulner ability assessment of power gr ids. In the assessment, we usually simulate net- work attacks and the consequ ent cascading failures, and then calculate the d a mage caused by the attack s. Different kind s of attacks result in different d a mage. Here, we consider the lin k attacks. The simplest lin k attack strategy is ran dom attack, in which we rand omly remove a certain fraction of link s. More efficient link attack strategies are desire d in the vu lnerability assessment. A. Pr oblem Definition Giv en an integer K , the pro blem is to fin d a set of K link s (1 ≤ K ≤ M ) , the removal of which will cause the maximum damage to th e power g rid (abbreviated as K LS prob lem). Note that the damag e is measur ed as the p ercentage of failed nod e s after the cascading failure (see Eq. (3)), which is triggered by the initial removal of the K -link set. Theorem 1. The K LS pr oblem is N P- C om pl et e. Pr oof. Given a set of K link s, we can calcu late the per c entage of failed nodes after th e cascading failure trigg ered by the removal o f the set of links in polyn o mial time, which means that the K LS pro blem is N P . Moreover , th e K LS problem can be r educed to the 0/1 knap sack prob lem [16]. I n this pr o blem, giv en a set of items, each with a weigh t an d a value, we determine the nu mber o f ea c h item, 0 or 1, to inc lu de in a collection so that the total weight is less than or equal to a giv en limit and the to tal value is a s large as possible. In the K LS problem , each link can cause som e damage, and can be selected or not. The number of selected link s is fixed to be K . Th e task is to deter mine the K -link set that achieves the largest damage. Since 0 /1 kn apsack prob le m is N P - C om pl e t e , so is the K LS problem.  B. Optimal attack b ased on PSO Since the K LS p roblem can be reduced to the 0 /1 knapsack problem and is N P - Co m pl et e , we e m ploy the PSO to search the optima l K -link set. In the m particles, par ticle i is set to be X i = [ x 1 , x 2 , . . . , x M ] , where x j correspo n ds to th e j th link of the p ower grid . If this link is selected to be removal, x j = 1; otherwise, x j = 0. The constra in t is ∑ M j = 1 x j = K . Th e pa rticles update their velocity an d position iteratively based on Eqs. (5 )- (7) until find ing the appr oximately optimal solution of the K LS problem . Since the duratio n of single ca scad e failure is u nable to estimate, we only consider the number of cascading failures when discussing the time co mplexity . The PSO b ased o ptimal attack (PSO-OA) has the time complexity of O ( m ∗ i t er max ) . The pseudoco de of this alg o rithm is in Algorithm 1. Algorithm 1 PSO based optimal attack (PSO-O A) Require: Adjacency m a trix G , power gene r ator node set Q , and integer K Ensure: The total num ber of selected links is fixed to be K . Initialisation : Rand omly generate m particles; each particle contains M elements; r andomly set K elements to be 1 and the rest to 0. for i = 1 , it er max do for j = 1 , m do Calculate Φ ( X j ) for each particle end for Update the optimal solution for e a ch particle p i best and the global optimal solution for the particle swarms g best for s = 1 , m do if Φ ( X s ) > Φ ( p s best ) then Φ ( p s best ) , p s best ← Φ ( X s ) , X s end if if Φ ( X s ) > Φ ( g best ) then Φ ( g best ) , g best ← Φ ( X s ) , X s end if if Φ ( X s ) = g best then Re-initialize the particle end if end for Update speed v and position x Select th e to p K po sitions of each particle based on the ranking of their c urrent value and reset them to 1; the other positions are reset to 0. end for return the K links o f the largest dam a ge C. Greedy attack ba sed on link centrality In a simple gr e edy attack, we can calculate th e damag e of each link based o n Eq. (3) and then select the K link s of the largest damage. Howev er , it is time consuming to calculate the damage of all links. Therefore, we use our link centrality measuremen t to filter the links so th at the relatively impo rtant links ar e left fo r consider ation. Specifically , we ran k all o f the links based on their link centr a lity values. T hen, w e calcula te the dam a ge of the top L ( M L % ≥ K ) percent of links in th e ranking , and select the K lin ks of the largest damage. 4 Note that the p arameters of our link centrality m easure affect the ranking of lin k s and ther efore th e selection of the K - link set. Th e better param eters co r respond to a better r a n king, and then L can be much smaller . The time complexity of the link centrality based gree d y attac k ( LC-GA) is O ( M L % ) with - out considerin g the d uration of single cascading f ailure. Note that we can also use PSO to search the optimal param eters in this alg o rithm, but the compu tational cost is very large. Th e pseudoco de of LC-GA is g iven in Algorithm 2. Algorithm 2 Link centrality based greedy attack (LC-GA) Require: Adjacen cy m atrix G , power gen erator nod e set Q , and integer K Ensure: The total num ber of selected links is fixed to be K . Initialisation : Giv e a par ameter set Select the top L % links based on the ranking o f link centrality values for j = 1 , M L % do Calculate the damage Φ for each selected link end for return the K links o f the largest damage D. Lin k centrality based optimal attac k The PSO-O A algorithm does not use th e topolo gical and electrical prop erties of power g rids. Thus, it is supposed to be not efficient. The LC-GA algo r ithm does use the top o logical and electr ical prop erties, but it need s to calcu late the lin k centrality a nd the damag e o f many links, which h as large computatio nal cost. Here, we employ PSO to search the optimal parameters so that removing the top K links in the ranking of link centrality leads to the maximum d amage. Th e set of the top K lin ks is thu s an ap proxim ate solution o f th e K LS pr oblem. The time comp lexity o f this link centra lity based optimal attack (LC-OA) is O ( m ∗ it er max ) without con sid ering the duration of single cascading failure. The pseudoco de o f this algorithm is provided in Algorithm 3. Algorithm 3 Link centrality based optimal attack (LC-O A) Require: Adjacen cy m atrix G , power gen erator nod e set Q , and integer K Ensure: The total num ber of selected links is fixed to be K . Initialisation : Randomly gene r ate m particles; eac h particle contains 2 elements w h ich are r andomly set to values in the range [-1, 1 ]. for i=1 , i t er max do Calculate the centrality of each link based on Eq. (4) Calculate the damage of removing the K links of the largest centrality Update p best and g best based on the PSO algorith m end for return the K links o f the largest centrality V . S I M U L A T I O N R E S U LT S Based on MA TLAB, we do simulation exper iments to v a li- date o ur link centrality measure and co mpare th e per forman c e of proposed attack algorithm s. W e use th e stan d ard I EEE bus test data [17 ] including IEEE 1 18 b u s, 1 45 b u s, and 162 bus. W e rando mly set 10 percent o f nodes to be gen erator no des. The related par a meters of the experiments are giv en in T able 1. T ABLE I Parame ter settin gs in the experi m ents m 10 c 2 0.7 i t er max 30 α 0.2 w 0 0.96 β 0.2 c 1 0.7 L 50% A. Single link attack I E E E 118 bus I E E E 145 bus I E E E 162 bus 0.0 0.2 0.4 0.6 R a ndo m C ur r e nt D e gr e e O ur c e nt r a l i t y Fig. 1: Th e resu lts of d amage Φ caused by the rem oval of the most cr itical link associate d with different centrality measur es on three IEEE bus test data. First, we study the single link attack, K = 1, which is re m ov- ing the most critical link in te r ms of damage. Th e link degree, link curr ency , and our lin k centrality are used respectively to determine the critical lin k. For ou r lin k ce ntrality measure, we u se PSO to find the o ptimal solution (see LC-OA). The results are shown in Fig . 1, which are the average of 100 indepen d ent ru ns. W e can see that fo r all the thr ee IEEE bus test data, the random removal has the smallest dama g e, and the d amage of our lin k cen trality is larger than the link degree and lin k cur rent. Moreover , link current is more efficient th an link degree, which in d icates that we should focu s on electrica l features m ore th a n topo logical featur es in the vulne r ability assessment of power gr ids. B. Multiple link attack Furthermo re, we study the mu ltiple link attack, k > 1 , to compare th e p e rforman ce of PSO-O A, LC-OA, and LC-GA. For LC-GA, pa r ameters h 1 and h 2 are bo th set to 1. The results of d amage are p r ovided in Figs. 2 and 3. In Fig. 2 , for all the IE E E bus d ata, the damage gen erally increases with the n umber of r e moved links for a ll the algorith m s. This is because th e more links rem oved at the beginn ing r esult in a wider range of cascading failure. An excep tion is that for LC- GA, the dam a g e decreases wh en the numb e r of rem oved links increases f rom 1 to 2. The reason is that LC-GA is essentially a greedy algorithm so that its so lu tion of K = 2 is not the optimal one, while for K = 1, the solution could be the optimal one. 5 0 5 10 15 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 0.5 0.6 0.7 0.8 0.9 1.0 K P S O - O A L C - G A L C - O A I E E E 118 bus K P S O - O A L C - G A L C - O A I E E E 162 bus Fig. 2: The results o f dam a g e Φ v s. numb er o f rem oved links K for the three attack algor ithms on IEE E 118 bus and 16 2 bus data. 0 200 400 600 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0 200 400 600 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 N um be r o f i t e r a t i o ns P S O - O A , K = 3 L C - O A , K = 3 P S O - O A , K = 4 L C - O A , K = 4 I E E E 162 bus N um be r o f i t e r a t i o ns P S O - O A , K = 5 L C - O A , K = 5 P S O - O A , K = 6 L C - O A , K = 6 I E E E 118 bus Fig. 3: The results of damag e Φ v s. num ber of iterations for LC-O A an d PSO-O A on IEE E 118 b u s and 162 bus d ata. Moreover , we see that the damage o f LC-OA is much larger than PSO-OA and LC-GA. In real situations, the n umbers of particles and iter ations a r e limited , as the par ameter settings in our experimen ts. In this case, LC-OA can find a be tter solution th an PSO-OA for the multiple link attack problem. The damag e of LC-GA is smaller than LC-OA, since th e former is a greed y algor ith m, while the latter is based on g lobal optimization essentially . Note that LC-GA is sometim es better than PSO-OA as shown in th e results of K > 6 on IEEE 162 bus data. This further validates our link cen trality measu r e. In Fig. 3, we see that f o r a given K , the d amage of LC-O A increases and conv e rges faster than PSO-OA with the growth of numbe r of iterations. This fur th er d emonstrates that LC-O A is more efficient than PSO-GA. Moreover, we see that PSO- O A is p rone to fall in to th e loc a l optimum an d needs many iterations to jump out of it. V I . C O N C L U S I O N In summ a r y , we study the vulnera bility assessment of power grids under cascading failures. W e define th e K -link attack problem and prove that it is NP-comp le te . Particularly , we propo se a link cen trality m easure by com bining the link degree with link current. W ith this cen tr ality , we develop two attac k algorithm s, which are the link ce ntrality b ased greedy attack (LC-GA) and the link cen trality b ased optimal attack (LC- O A). W e evaluate ou r link centrality measur e and its relate d attack alg orithms on stand a rd IEEE bus test data. Sim ulation results show that ou r lin k centrality measur e p erforms better than the link degree and link curre nt in identifying the optimal link in the single link attack scena r io. Furthe r more, in the mul- tiple link attack problem, LC-OA is much more ef ficient than LC-GA and the traditional PSO based optimal a ttack (PSO- O A) algorithm. 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