Research Article Research on the Structural ...

Research ArticleResearch on the Structural Characteristics of TransmissionGrid Based on Complex Network Theory

Jinli Zhao Hongshan Zhou Bo Chen and Peng Li

Key Laboratory of Smart Grid of Ministry of Education Tianjin University Building 26E 92 Weijin RoadNankai District Tianjin 300072 China

Correspondence should be addressed to Peng Li liptjueducn

Received 19 January 2014 Accepted 7 March 2014 Published 10 April 2014

Academic Editor Hongjie Jia

Copyright copy 2014 Jinli Zhao et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Reasonable and strong structure is an important foundation for the smart transmission grid For vigorously promoting constructionof the smart grid it is of great significance to have a thorough understanding of the complex structural characteristics ofthe power grid The structural characteristics of several actual large-scale power grids of China are studied in this paperbased on the complex network theory Firstly the topology-based network model of power grid is recalled for analyzing thestatistical characteristic parameters The result demonstrated that although some statistical characteristic parameters could reflectthe topological characteristics of power grid from different ways they have certain limitation in representing the electricalcharacteristics of power grid Subsequently the network model based on the electrical distance is established considering thelimitation of topology-based model which reflects that current and voltage distribution in the power grid are subject to Ohmrsquos Lawand Kirchhoff rsquos Law Comparing with the topology-based model the electrical distance-based model performs better in reflectingthe natural electrical characteristic structure of power grid especially intuitive and effective in analyzing clustering characteristicsand agglomeration characteristics of power grid These two models could complement each other

1 Introduction

In recent years the smart grid has become a new trenddevelopment of the power grid in the world Althoughthe definition of the smart grid is different essentiallythe smart grid is based on the original transmission anddistribution grid and electrical equipment using moderninformation communication and control technologies toattain its information intelligence and electricity marketsto solve the issues of renewable energy applied to networkand rational utilization of energy [1 2] The smart gridcan improve the security and reliability of the power gridpromote the development of new and renewable energyand improve the quality of power so as to meet userdemand for the security of the grid and help solve the globalenergy shortages environmental pressures and other majorproblems At the same time operating environment of theelectricity networks becomes more complex due to windsolar renewable resources which are combined to the grid orother new problems obviously the traditional grid structure

and operation mode have not adapted the development[3 4] Reasonable frame structure is closely related to theoperation of the power grid and strong frame structure is animportant foundation for the development of the smart gridStructure of the power grid and system operating mode arefacing another change Many studies have demonstrated theclose relationship between the structure and characteristics ofpower grid [5ndash8] therefore have a thorough understandingof the power grid topology as well as its impact on the secureand stable operation of power grid has important significancefor enhancing the awareness of weak links of power gridstructures ensuring the secure and stable operation of powergrid and promoting the construction of smart grid

The emergence and continuous development of complexnetwork theory enables us to study the power grid as aldquoglobalrdquo complex network [9] General characters of powergrid can be described by comprehensive analysis on thestatistical characteristics of complex network such as degreedegree distribution clustering coefficient and characteristicpath length of the network [10] Based on the comparison of

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 261798 12 pageshttpdxdoiorg1011552014261798

2 Journal of Applied Mathematics

clustering coefficient and characteristic path length betweenthe power grid and its random graph with same nodes andsides [11ndash13] discovered that the power grid in WesternUnited States power grid in Northern Europe and powergrid in Northern China are small-world network while [1314] discovered that Chinarsquos Sichuan-Chongqing power gridand Anhui power grid are more proximate to random net-work Many researches on the characteristics of degree distri-bution concluded exponential distribution [15 16] Howeverpower-law distribution was observed in some researches [517] To explain such different distributions [18] comparedthe statistical characteristic parameters of the topologicalgraphs of large power grids in three North America regionswith the corresponding statistical characteristic parametersof same-size random graph small-world graph and scale-free graph The comparison result demonstrated that thesepower grids are neither small-world network nor scale-freenetwork and their degree distributions aremore proximate toexponential distribution rather than power-law distribution[13] analyzed the impact of small-world characteristics onthe cascading failure propagation qualitatively finding thatthe unique shorter mean distance and higher clusteringcoefficient of small-world power grid facilitated the failurepropagation

Complex network theories and methods are widelyapplied for vulnerability analysis of power grid Howeverpeople discovered in these studies that it is difficult to depictthe operation characteristics of power grid accurately onlythrough the topological structure Based on the centralityindexes of element importance analysis in graph theoryand the electrical characteristic parameters of power gridand [19] put forward the electrical centrality index whichwas used to search important nodes and lines of powergrid Reference [20] established the weighted topologicalmodel of power grid based on the wire reactance whichwas confirmed through practical power grid to be better inreflecting the node importance and operating state of prac-tical electric power system comparing with the unweightedmodel Reference [21] put forward 4 vulnerability indexesof system structure under the consideration of system oper-ation constraint Reference [22] suggested using electricalbetweenness as the criteria of critical line which coveredthe shortage of weighted betweenness model in which itsupposes the currents between buses only flow throughthe shortest path Reference [18] put forward the electricaldistance-based power grid model and compared it with thetopology-basedmodel finding significant difference betweenthe topological structure and electrical structure of powergrid

This paper aims to carry out a deep analysis on thecomplex structure of practical large power grids in Chinathrough the complex network theory and methods In thispaper several typical large-scale transmission grids are sim-plified to different extents and the graph theory model basedon the topology relation as well as the graph theory modelbased on electrical distance is established for discussing therepresentation of their complex network statistical character-istic parameters to the topological structure and operatingcharacteristics of practical power grid

2 Topological StructuralCharacteristics of Power Grid Based onTopology-Based Model

Power grid is a network composed of various electricalelements like power transmission lines and transformersthrough certain way for delivering and distributing electricenergies The topology-based model of power grid is easyto be established according to the topological relationshipbetween its transmission elements In other words all busesare abstracted as the nodes of topology-basedmodel whereasthe transmission lines and windings of transformers areabstracted as edges (in this paper the three-winding trans-former applies 119884-connection with virtual intermediate nodeThe virtual node also is abstracted as the node of topology-based model) All nodes and edges are treated in the sameway Consequently the topology-based model of power gridrepresented by the unweighted and undirected graph (119866 =

(119881 119864)) containing n nodes and e edges was gained where 119881is the node set and119864 is edge setThe adjacencymatrix of nodeis recorded as 119860

119905and the matrix element is

119886119894119895=

1 if (119894 119895) isin 1198640 if (119894 119895) notin 119864

(1)

21 Power Grid Data The analysis objects of this paperare several typical large-scale transmission grids in Chinaincluding NX Grid TJ Grid TJ220 Grid CS Grid NW Gridand NC Grid The topology-based model sizes of these gridsafter data processing are 119866NX = 155 176 119866TJ = 597 648119866TJ220151 167 119866CS = 328 371 119866NW = 1840 2162and 119866NC = 4828 5666 Their nodes vary from hundredsto thousands

22 Topology Analysis of Power Grids The major statis-tical characteristic parameters of network models of theabovementioned power grids were calculated and analyzedincluding various parameters and methods widely appliedin various researches currently such as degree (119896) degreedistribution (119875(119896)) clustering coefficient (119862) and character-istic path length (119871) The results were listed in Table 1 where119899 is the number of nodes and 119890 is the number of edges⟨119896⟩ is the average node degree and 119862random is the clusteringcoefficient of random network with same nodes and sides ascorresponding practical network 119863 is the network diameterwhich is the maximum distance between any two nodes Therepresentation of these parameters to the structure of powergrids will be analyzed one by one in the following text

221 Node Degree (k) Node degree (119896) refers to the edgeamount connected to one node The larger the node degree(119896) is the more important the node is in the networkTake TJ220 Grid for example Its topology-based model is119866TJ220151 167 and its average node degree ⟨119896⟩ values 221Table 2 is the node degree statistics of TJ220 Grid underdifferent voltage levels To ensure the power distributionreliability within the whole city 500 kV nodes are connectedwith each other in the grid to form the backbone ring These

Journal of Applied Mathematics 3

Table 1 Statistical characteristic parameter values of power grids

Grid 119899 119890 119896max ⟨119896⟩ 119862 119862random 119871 119863

TJ 597 648 13 217 00009 000036 10281 20TJ220 151 167 11 221 00135 001463 76037 16NX 155 176 9 227 00212 001464 87566 19CS 328 371 9 226 00181 000689 118725 32NW 1840 2162 11 235 00170 000127 154676 39NC 4828 5666 14 235 00037 000048 163647 43

Table 2 Node degree statistics of TJ220 Grid

Node degree (119896) Voltage level500 kV 220 kV

1ndash3 25 844ndash11 75 16

nodes have large degree and 75 of them have 4ndash6 degreesfully demonstrating the importance of 500 kV substations asload-center substations in the grid In contrast only 16 of220 kV nodes have large node degree (119896 gt 4) while the rest84 have smaller node degree ranging from 1ndash3

222 Degree Distribution (P(K)) Degree distribution (119875(119870))is a general description of node degree in the networkThe basic topological characteristics of power grid can beevaluated by analyzing whether the degree distribution wassubjected to some specific probability distributions Thispaper applies cumulative degree distribution which wasdefined as 119875(119870 ge 119896) = sum

119870ge119896119875(119870) The cumulative degree

distribution curves of power grids are shown in Figure 1which reveals that the degree distributions of all power gridsare approximately subjected to exponential distribution withthe maximum node degree 119896max le 14

If the substations are viewed as nodes during the networkabstract modeling the new networkmodel will involve fewernodes but higher degree of some nodes Take NC Grid forexampleThe new topology-basedmodel after abstracting thesubstations as nodes is1198661015840NC = 871 1199 (the originalmodelis 119866NC = 4828 5666) The cumulative degree distributioncurve is shown in Figure 2 It can be known from Figure 2that the maximum node degree of the new analysis modelincreases (1198961015840max = 16) and the cumulative degree distributionis also subjected to exponential distribution without power-law distribution observed This is because the outlet lines ofsubstation are restricted by the capacity and floor space ofoutlet intervals of the substation thus making it impossibleto involve node with large node degree in the topology-basedmodel

223 Clustering Coefficient (C) Many large-scale complexnetworks in the real world have obvious clustering effect (orknown as aggregation effect) which means that nodes to thesame node in the network are closely related with each otherThe clustering coefficient of network can represent the clusterdegree of nodes Suppose node 119894 has 119896

119894adjacent nodes then

theremay be 119896119894(119896119894minus1)2 edges among these 119896

119894adjacent nodes

to the maximumThe 119864119894(119896119894(119896119894minus 1)2) ratio is defined as the

clustering coefficient (119862119894) of node i where 119864

119894is the practical

edges existed among 119896119894adjacent nodes

119862119894=

119864119894

119896119894(119896119894minus 1) 2

(2)

The clustering coefficient of network is defined as themean clustering coefficient of all nodes

119862 =1

119873

119873

sum

119894=1

119862119894 (3)

According to Table 1 the clustering coefficients oftopology-based model of all power grids are relatively smallIt can be known from the definition of clustering coefficientthat the clustering coefficient of nodes in fact reflects theexistence of delta connection in networkHowever such deltaconnection is rare in practical power grid since the cyclicnetwork is generally connected by more than 3 nodes TakeTJ220 Grid for example It has very low clustering coefficientsince there is only one group of nodes that formed delta con-nectionTherefore the network represents no obvious aggre-gation effect when viewed from the clustering coefficient of itstopology-based model However this conclusion contradictswith the zoning operation of power grid which has obviouscluster effect viewed from the electrical characteristics ofpower grid For example 500 kV wires in TJ220 Grid formeda dual-ring network basically which are connected withexternal wires and accept electric power transmission Onthe other hand the 220 kV Grid forms three power-supplyregions for receiving powers from 500 kV Grid (Figure 3)[23] In such power grid every zone has independent closeelectrical relationships and zones are connected throughinterval connection wires thus achieving the layering andzoning operation of power grid and improving the overallpower distribution reliability and security of urban powergrid maximumly However for topology-based modelingmethod the clustering coefficient (119862) is difficult to reflectsuch structural characteristics of power grid

224 Structural Visualization of Power Grid Based on K-K Layout Algorithm Rational visualization enables us torepresent the power network structure intuitively The geo-graphical wiring layout shown in Figure 3 is a commonvisualization method of power network structure Howeversuch wiring layout is formed by the geographic information


0 5 10 15

NXTJ

NWNC

10minus4

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)

(a) Semilogarithmic coordinates

NXTJ

NWNC

10minus4

10minus3

10minus2

10minus1

100

100

101

P(K

gek)

Node degree (k)

(b) Logarithmic coordinates

Figure 1 Cumulative degree distribution of topology-based model of power grids

0 5 10 15 2010

minus4

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


10minus4

10minus3

10minus2

10minus1

100

100

101

102

P(K

gek)

Node degree (k)


Figure 2 Cumulative degree distribution curves of NC Grid model 119866NC and 1198661015840NC

of substation and wires and fails to reflect the practicalelectrical distance between nodes in the power grid To gaina better observation on the overall power network this paperproposed to using the Kawai-Kamada (K-K) algorithm in thecomplex network theory [24] to achieve the visualization ofpower network structure

The Kamada-Kawai Algorithm is a force directed layoutalgorithm The nodes are represented by steel rings and theedges are springs between themThe basic idea is tominimizethe energy of the system by moving the nodes and changingthe forces between them When the system finally reaches

the optimal layout distances between nodes are close to theirideal distances while the total energy of the system (119864) hitsthe bottom value

119864 =

119899minus1

sum

119894=1

119899

sum

119895=119894+1

1

2119896119894119895(10038161003816100381610038161003816119901119894minus 119901119895

10038161003816100381610038161003816minus 119897119894119895)2

(4)

where 119875119894and 119875

119895are the position coordinates of nodes 119894 and 119895

in the wiring layout 119897119894119895is the ideal distance between nodes

119894 and j 119896119894119895

= 119870119889119894119895represents the elastic coefficient of

the spring between nodes 119894 and j 119870 is a constant and 119889119894119895


Zone2

Zone1

Zone3

220 kV tranmission line500kV tranmission line

220 kV substation500kV substation

Figure 3 Geographical wiring layout of TJ220 Grid

is the shortest distance between nodes 119894 and 119895 Hence theoptimal layout can be gained by seeking the minimum totalenergy of the system

The weighted topology-based model of power grid canbe gained by assigning weights to the connecting edgesbetweennodes in the topology-basedmodel Edgeweights setaccording to the impedance module (|119911

119894119895|) of the wire (119894 minus 119895)

can represent the electrical connection between nodes Thelarger the weight is the weaker the electrical connection willbe In other words

119886119894119895=

10038161003816100381610038161003816119911119894119895

10038161003816100381610038161003816if (119894 119895) isin 119864

0 if (119894 119895) notin 119864(5)

According to the principle of K-K layout algorithm theoptimal layout based on the weighted topology-based modelwill reflect the electrical closeness between different nodesthoroughly Larger geometrical distance between nodes inthe wiring layout represents the weaker electrical connectionwhile shorter geometrical distance represents stronger elec-trical connection

Figures 4(a) and 4(b) represent the K-K optimal layoutof TJ220 Grid based on topology-based model and weightedtopology-based model respectively Comparing with thegeographical wiring layout the visualization of power gridbased on K-K layout algorithm enjoys obvious superioritiesin representing the electrical connection node distributionfully represents the electrical connection between nodeswith evident zoning characteristics especially the 500 kVnodesThese 500 kV nodes have longer geographical distancebetween each other (Figure 3) they have very close electricalconnection practically For this reason 500 kV nodes arepositioned on the core areas in Figure 4 Both geographicalwiring layout and K-K optimal layout based on weightedtopology-based model can represent the overall layout andstructure of power grid from different perspectives Theformer emphasizes on the regional features of electric power

transmission whereas the latter is superior for its intuitiverepresentation of electrical connection between nodes

225 Network Diameter (D) and Characteristic Path Length(L) The distance (119889

119894119895) between nodes 119894 and 119895 is defined as

the number of edges on their shortest connecting path Themaximum distance between any two nodes in the network iscalled as the network diameter (119863)

119863 = max 119889119894119895 (6)

The characteristic path length (L) in the network isdefined as the mean distance between any two nodes

119871 =1

119899 (119899 + 1) 2sum

119894ge119895

119889119894119895 (7)

where L reflects the connecting closeness between nodesThe combination of 119871 and 119863 can reflect the overall layoutof the network Although TJ220 Grid and NX Grid havesimilar nodes and size their structures are significantlydifferent from each other As shown in Figures 4(a) and 5 thebackbone network of TJ220Grid is a compact cyclic structureand supplies powers to zone load whereas the NX Gridrepresents a loose long and narrow dendritic-distributionstructure Their different structures result in their different 119871and D 119871TJ220 = 76037 lt 119871NX = 87566 and 119863TJ220 = 16 lt

119863NX = 19 Radial network has both larger 119871 and 119863 than thecyclic networkTherefore119871 and119863 can reflect the loose degreeof network structure

3 Complexity of Power Grid Based onElectrical Distance Model

According to the analysis in Chapter II the analysis onthe abovementioned statistical characteristic parameters oftopology-based model of power grid enables us to gain alot of information about the structural characteristics of thepower grid However such power grid model fails to repre-sent the electrical characteristics completely Therefore theconclusion about the structural characteristics of power griddrawn from the topological relation has certain limitationsFor example the clustering coefficient (119862) can only declarethe delta connection amount in the power grid but fails torepresent its real zoning characteristic For this reason thepower grid model based on electrical distance is establishedin this paper

31 Graph-Theory Modeling of Power Grid Based on ElectricalDistance The voltage and current distribution in the powergrid follows Ohmrsquos Law and Kirchhoff rsquos Law Thereforestrong or weak electrical connections exist between indirectlyconnected nodes except for that between directly connectednodes The strength of electrical connection between nodescan be measured by electrical distance

Take a power grid with 119899 nodes for example 119868119894and

119894represent the injection current and voltage of node 119894


1

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Zone3

Zone2

Zone1

147

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150151

(a) K-K optimal layout based on topology-based model

Zone1

Zone2

Zone3

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(b) K-K optimal layout based on weighted topology-based model

Figure 4 Optimal layout of TJ220 Grid


Figure 5 Topology layout of NX Grid

respectively According to the Kirchhoff rsquos Law the networkequation expressed by the nodal impedance matrix (Z) is

[[[[[[[

[

11988511

1198851119894

1198851119899

1198851198941

119885119894119894 119885

119894119899

1198851198991

119885119899119894

119885119899119899

]]]]]]]

]

[[[[[[[

[

1198681

119868119894

119868119899

]]]]]]]

]

=

[[[[[[[

[

1

119894

119899

]]]]]]]

]

(8)

When 119868119898

= 0 and 119868119894= 0 (119894 = 1 2 119899 119894 =119898) the mutual

impedance (off-diagonal element of the matrix) between anytwo nodes is

119885119894119898=119894

119868119898

119894 = 1 2 119899 (9)

Elements in the nodal impedance matrix are the open-circuit parameters of power grid including information ofthe whole network In this paper the module of mutualimpedance (|119885

119894119895|) is defined as the electrical distance between

node 119894 and j expressed as119889119890(119894119895)

= |119885119894119895| Higher |119885

119894119895| represents

longer electrical distance and weaker electrical connectionwhereas lower |119885

119894119895| represents shorter electrical distance and

stronger electrical connection The electrical distance matrix(De = |Z|) between nodes in the power grid can be gainedthrough the modules of elements in the impedance matrixFor interconnected networks the impedance matrix is a fullmatrix Therefore the electrical distance matrix (De) is a fullmatrix too During the network modeling based on De theelectrical connection in the network is simplified by settingthe electrical distance threshold (119889

119890(th)) in order to preventthe model from becoming a complete graph In other wordselements (electrical distancegt 119889

119890(th)) in theDe are set as 0 andonly elements (electrical distance le 119889

119890(th)) are kept Underthis circumstance the adjacencymatrix of electrical distance-based power grid model is Ae and its elements are

119886119890(119894119895)

= 119889119890(119894119895)

forall119889119890(119894119895)

le 119889119890(th)

0 forall119889119890(119894119895)

gt 119889119890(th)

(10)

Under different thresholds connection between nodesin the corresponding power grid model based on electricaldistance will be different In the following text different

Table 3 Electrical distance between nodes

Grid |119885average| |119885min| |119885max|

NX 02484 00309 14555TJ220 21073 17394 21331CS 00167 00001 00900

119889119890(th) will be set according to the distribution characteristics

of electrical distance between nodes and then the electricalstructure of power grid will be analyzed based on theestablished different network models

32 Structural Analysis of Power Grid Based on ElectricalDistance Model Table 3 lists the maximum minimum andaverage electrical distances between nodes in TJ220 GridNX Grid and CS Grid Figure 6 represents the probabilitydistribution of electrical distance between nodes in all thesethree grids It can be known fromFigure 6 that although thesethree grids have different electrical distance ranges and distri-butions their electrical distances demonstrate concentratingdistributions Shorter electrical distance may exist betweennodes indirectly connected which may be shorter thanthat between nodes directly connected This indicates thatsimilar electrical characteristics may be observed betweennodes indirectly connected which cannot be reflected by thetopology-based model

According to the above mentioned modeling idea basedon electrical distance the structural characteristics of TJ220Grid are analyzed The cumulative probability distributionof electrical distance between nodes is analyzed to set theelectrical distance threshold reasonably as shown in Figure 7When 119889

119890(th) gt 211 weak connections in the networkdecrease quickly with the decreasing of 119889

119890(th) When 119889119890(th) =

2107 7251 weak connections are deleted and the electricaldistancemodel can justmaintain connectedWith the furtherdecreasing of 119889

119890(th) the network diagram is divided intoseveral independent subblocks accompanied with isolatednodes and slowed decreasing of weak connectionsWhen theelectrical distance model has same edges with the topology-based model (119890 = 167) 9853 relative weak connectionshave to be deleted In this case the corresponding threshold119889119890(th) is 204Table 4 lists some statistical characteristic parameters of

TJ220 Grid under different thresholds Obviously under themodeling based on electrical distance the edges of graphicalmodel will increase which leads to the increase of averageand maximum node degree to different extents Howeverthe cumulative degree distribution of TJ220 Grid still fails torepresent power-law distribution under different thresholds(Figure 8)

Table 4 reveals the biggest difference between the elec-trical distance-based network model and topology-basedmodel The clustering coefficient of the network remains athigher level regardless of the threshold variation indicatingobvious clustering effect of nodes in the power grid Figures9(a) and 9(b) represent the structures of electrical distance-based model of TJ220 Grid at 119889

119890(th) = 2107 and 119889119890(th) = 207where the blue squares represent 500 kV nodes and green


195 197 199 201 203 205 207 209 211 213 2150

005

01

015

02

025

03

035

Electrical distance between nodes

Prob

abili

ty d

istrib

utio

n

(a) TJ220 Grid

018 019 02 021 022 023 024 025 026 0270

002

004

006

008

01

012


Prob

abili

ty d

istrib

utio

n

(b) NX Grid

0Electrical distance between nodes

001 002 003 004 005 006 007 008 0090

Prob

abili

ty d

istrib

utio

n

005

01

015

02

025

(c) CS Grid

Figure 6 Probability distribution of electrical distance between nodes in power grids

195 197 199 201 203 205 207 209 211 213 2150

02

04

06

08

1

(2107 07251)(204 09853)


Cum

ulat

ive p

roba

bilit

y di

strib

utio

n

Figure 7 Cumulative Probability distribution of electrical distance between nodes in TJ220 Grid


0 10 20 30 40 50

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


100

101

102

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


Figure 8 Cumulative degree distribution of electrical distance-based network model of TJ220 Grid

Table 4 Statistical characteristic parameters of electrical distance-based network model of TJ220 Grid

119889119890(th) 119890 ⟨119896⟩ 119896max 119862

2107 3113 4123 49 094162090 2414 3197 44 08785207 1520 2013 38 08469206 816 1081 25 082202055 600 795 21 08203204 167 272 10 03938

triangles represent 220 kV nodes Under different thresholdsthe electrical distance-based models of power grid haveclosely connected clusters with abundant delta connectionsrepresenting close electrical connections Furthermore when119889119890(th) = 2107 the TJ220 Grid forms three obvious clusters

according to the closeness between nodes which echoes withits zones basically In other words the clustering coefficient(119862) of the electrical distance-based power grid model canaccurately and effectively reflect that power grid has obviousclustering effect in electrical connection

To verify the above analysis three typical nodes (V1V2 and V3) are selected in Zone1 Zone2 and Zone3respectively and their electrical distances with nodes indifferent zones are drawn (Figure 10) It can be seen fromFigure 10 that nodes within the same zone have the shortestelectrical distance and two nodes in different zones havelonger electrical distance Therefore with the decreasing of119889119890(th) although the connections between nodes of different

zones are deleted gradually nodes within the same zone stillremain closely connected and finally form the three zoninglayout (Figure 9(a))

When the 119889119890(th) decreases from 2107 to 207 zones

separate from each other as the connections between themare deleted gradually Meanwhile a lot of weak connectionsin same zone are deleted and Zone3 has been separated intotwo sub-zones (Figure 9(b)) Abundant isolated nodes can beobserved in the electrical distance-based model of the powergrids when 119889

119890(th) decreases to 204 Although edges decreasescontinuously with the decreasing of 119889

119890(th) the model stillmaintains several closely connected clusters In other wordsthe network still has evident clustering characteristics thusendowing the network with larger clustering coefficient(Table 4)

4 Conclusions

This paper analyzes the structural characteristics of severallarge-scale practical power grids based on the complex net-work theory and compares the analysis results of their com-plex network parameters under different modeling methodsas well as the power network structural information theseparameters could provide

Firstly the network model based on practical topologi-cal structure is established and the statistical characteristicparameters of topology-based models of all involved powergrids are analyzedThe results demonstrate that these param-eters can reflect the topological characteristics of powergrids to certain extent from different sides (1) The statis-tical analysis of node degree of power grid under differentvoltage levels can disclose different structural characteristicsof grids structure under different voltage levels caused bytheir different role in power supply (2) The comparisonof characteristic path length (119871) and network diameter(119863) of different power grids can reveal the tightness ofpower network structure (3) The node degree distribution


1

2

3

4

5

67

89

10

11

12 13

1415

1617

18

19

20 21

22

23

24

25

26

27

28

29

3031 3233

34

35

3637

38

39

40

41

42

43

44

45 46

47

48

49

50

51

52

53

54

55

56

57

58

5960

61

62

63 64

65

66 67

68

69

70

71

72

73

74

75

7677

7879

80

8182

83

84

8586

87

88

89

9091

92

93

94

95

96

97 9899100101

102

103

104

105

106

107

108109

110111

112

113

114

115

116

117

118

119

120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136137138

139

140

141

142 143

144

145146

147 148

149

150151

Zone3

Zone2

Zone1

(a) 119889119890(th) = 2107

1

2

3

45

6

7

8

9

1011

12 1314

15

1617

18

19

2021

22

23

24

25

26

27

28

29

30

313233

34

35

36

37

3839

40

41

42 43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

6364

65

66

67

68

69

70

71

72

73 74

75

7677

78

79

8081

82

83

84

85

86

87

8889

90

91

92

93

9495

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116117

118

119120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150 151

Zone1

Zone2

Zone3

(b) 119889119890(th) = 207

Figure 9 Structure of electrical distance-based model of TJ220 Grid


0 20 40 60 80 100 120 140 160206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(a) V1

0 20 40 60 80 100 120 140 160204

206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(b) V2

0

Node number

Elec

tric

al d

istan

ce

20 40 60 80 100 120 140 160204

206

208

21

212

214

(c) V3

Figure 10 Electrical distance distribution between V1 V2 and V3 and other nodes Red squares represent the electrical distance betweenthe selected node and nodes in Zone1 Green squares represent the electrical distance between the selected nodes and nodes in Zone2 Bluesquares represent the electrical distance between the selected node and nodes in Zone3

of power grids generally represents exponential character-istic However since the topology-based model neglectsthe electrical characteristics of power grid componentsit fails to reflect the Ohmrsquos Law and Kirchhoff rsquos Law inwhich voltage and current distribution in power grid has tobe followed Consequently the conclusion concerning thestructural characteristics of power grid may have certainlimitations that is clustering coefficient (119862) fails to representthe zoning and clustering characteristics of power gridaccurately

This paper establishes the power grid model based onelectrical distance in order to offset the shortages of topology-based model This model can reflect that current and voltagedistribution in power grid are subject to Ohmrsquos Law andKirchhoff rsquos Law which is in accordance with the practicalrunning of power grid The analysis of several differentpower grids demonstrates a relative concentrate distribution

of electrical distance between nodes in power grids and closeelectrical connections between nodes without direct topo-logical relationship Under different thresholds of electricaldistances electrical distance-based models maintain higherclustering coefficient which indicates the obvious clusteringeffect of power grid in electrical connection Therefore theelectrical distance-based model can analyze the agglomera-tion characteristics and clustering characteristics accuratelyand thoroughly

Both topology-basedmodel and electrical distance-basedmodel can be used to analyze different aspects of powergrid structure effectively thus enabling us to gain a morecomprehensive and accurate understanding of the structuralcharacteristics of power grid Additionally the structuralvisualization of power grid based on K-K layout algorithmcan provide a clear and intuitive representation of the electri-cal connections between nodes in power grid


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The work is supported by the National Key TechnologySupport Program of China (Grant no 2013BAA01B02)

References

[1] L L Li Y B Zhang X L Jin andQ Sun ldquoTracing and learningprobe into the objective and ways of smart grid developmentrdquoEnergy Technology and Economic vol 22 pp 22ndash28 2010

[2] L Zhang and R Huang ldquoEffects of smart grid on electricitymarket development and prospectsrdquo Automation of ElectricPower Systems vol 34 no 8 pp 5ndash71 2010

[3] Y X Yu ldquoIntelli-D-grid for the 21st centuryrdquo Southern PowerSystem Technology Research vol 2 pp 1ndash16 2006

[4] J Yao S Yan S Yang Z Yang and Z Gao ldquoPractice andprospects of intelligent dispatch with Chinese characteristicsrdquoAutomation of Electric Power Systems vol 33 no 17 pp 16ndash482009

[5] A-L Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999

[6] R Albert H Jeong and A-L Barabasi ldquoError and attacktolerance of complex networksrdquo Nature vol 406 pp 378ndash3822000

[7] S Boccaletti V Latora Y Moreno M Chavez and D-UHwang ldquoComplex networks structure and dynamicsrdquo PhysicsReports vol 424 no 4-5 pp 175ndash308 2006

[8] C W Wu ldquoSynchronization in networks of nonlinear dynami-cal systems coupled via a directed graphrdquo Nonlinearity vol 18no 3 pp 1057ndash1064 2005

[9] G A Pagani and M Aiello ldquoThe power grid as a complexnetwork a surveyrdquo Physica A vol 392 pp 2688ndash2700 2013

[10] V Rosato S Bologna and F Tiriticco ldquoTopological propertiesof high-voltage electrical transmission networksrdquoElectric PowerSystems Research vol 77 no 2 pp 99ndash105 2007

[11] D J Watts and S H Strogatz ldquoCollective dynamics of lsquosmall-worldrsquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[12] A J Holmgren ldquoUsing graph models to analyze the vulnerabil-ity of electric power networksrdquo Risk Analysis vol 26 no 4 pp955ndash969 2006

[13] Z-W Meng Z Lu and J Song ldquoComparison analysis of thesmall-world topologicalmodel of Chinese andAmerican powergridsrdquo Automation of Electric Power Systems vol 28 no 15 pp21ndash29 2004

[14] M Ding and P-P Han ldquoSmall-world topological model basedvulnerability assessment algorithm for large-scale power gridrdquoAutomation of Electric Power Systems vol 30 no 8 pp 7ndash402006

[15] R Albert I Albert and G L Nakarado ldquoStructural vulnerabil-ity of the North American power gridrdquo Physical Review E vol69 no 2 Article ID 025103(R) 4 pages 2004

[16] P Crucitti V Latora and M Marchiori ldquoA topological analysisof the Italian electric power gridrdquo Physica A vol 338 no 1-2 pp92ndash97 2004

[17] D P Chassin and C Posse ldquoEvaluating North American elec-tric grid reliability using the Barabasi-Albert network modelrdquoPhysica A vol 355 no 2ndash4 pp 667ndash677 2005

[18] E Cotilla-Sanchez P Hines C Barrows and S BlumsackldquoComparing the topological and electrical structure of theNorth American electric power infrastructurerdquo Systems vol 6pp 616ndash626 2012

[19] Z Wang A Scaglione and R J Thomas ldquoElectrical centralitymeasures for electric power grid vulnerability analysisrdquo inProceedings of the 49th IEEEConference onDecision and Control(CDC rsquo10) pp 5792ndash5797 Atlanta Ga USA December 2010

[20] M Ding and P-P Han ldquoVulnerability assessment to small-world power grid based on weighted topological modelrdquo inProceedings of the Chinese Society of Electrical Engineering(CSEE rsquo08) vol 28 pp 20ndash25 2008

[21] A Mao J Yu and Z Guo ldquoElectric power grid structural vul-nerability assessmentrdquo in Proceedings of the IEEE Power Engi-neering Society General Meeting (PES rsquo06) Montreal CanadaJune 2006

[22] L Xu X-L Wang and X-F Wang ldquoElectric betweenness andits application in vulnerable line identification in power systemrdquoin Proceedings of the Chinese Society of Electrical Engineering(CSEE rsquo10) vol 30 pp 33ndash39 2010

[23] S Y Liu Q Gu L J Zhang and C Liu ldquoResearch onpower supply scheme based on partitioning of 500220 kVTianjin power grid during the 11th five-year planrdquo Power SystemTechnology vol 32 pp 51ndash55 2008

[24] T Kamada and S Kawai ldquoAn algorithm for drawing generalundirected graphsrdquo Information Processing Letters vol 31 no1 pp 7ndash15 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


clustering coefficient and characteristic path length betweenthe power grid and its random graph with same nodes andsides [11ndash13] discovered that the power grid in WesternUnited States power grid in Northern Europe and powergrid in Northern China are small-world network while [1314] discovered that Chinarsquos Sichuan-Chongqing power gridand Anhui power grid are more proximate to random net-work Many researches on the characteristics of degree distri-bution concluded exponential distribution [15 16] Howeverpower-law distribution was observed in some researches [517] To explain such different distributions [18] comparedthe statistical characteristic parameters of the topologicalgraphs of large power grids in three North America regionswith the corresponding statistical characteristic parametersof same-size random graph small-world graph and scale-free graph The comparison result demonstrated that thesepower grids are neither small-world network nor scale-freenetwork and their degree distributions aremore proximate toexponential distribution rather than power-law distribution[13] analyzed the impact of small-world characteristics onthe cascading failure propagation qualitatively finding thatthe unique shorter mean distance and higher clusteringcoefficient of small-world power grid facilitated the failurepropagation

Complex network theories and methods are widelyapplied for vulnerability analysis of power grid Howeverpeople discovered in these studies that it is difficult to depictthe operation characteristics of power grid accurately onlythrough the topological structure Based on the centralityindexes of element importance analysis in graph theoryand the electrical characteristic parameters of power gridand [19] put forward the electrical centrality index whichwas used to search important nodes and lines of powergrid Reference [20] established the weighted topologicalmodel of power grid based on the wire reactance whichwas confirmed through practical power grid to be better inreflecting the node importance and operating state of prac-tical electric power system comparing with the unweightedmodel Reference [21] put forward 4 vulnerability indexesof system structure under the consideration of system oper-ation constraint Reference [22] suggested using electricalbetweenness as the criteria of critical line which coveredthe shortage of weighted betweenness model in which itsupposes the currents between buses only flow throughthe shortest path Reference [18] put forward the electricaldistance-based power grid model and compared it with thetopology-basedmodel finding significant difference betweenthe topological structure and electrical structure of powergrid

This paper aims to carry out a deep analysis on thecomplex structure of practical large power grids in Chinathrough the complex network theory and methods In thispaper several typical large-scale transmission grids are sim-plified to different extents and the graph theory model basedon the topology relation as well as the graph theory modelbased on electrical distance is established for discussing therepresentation of their complex network statistical character-istic parameters to the topological structure and operatingcharacteristics of practical power grid

2 Topological StructuralCharacteristics of Power Grid Based onTopology-Based Model

Power grid is a network composed of various electricalelements like power transmission lines and transformersthrough certain way for delivering and distributing electricenergies The topology-based model of power grid is easyto be established according to the topological relationshipbetween its transmission elements In other words all busesare abstracted as the nodes of topology-basedmodel whereasthe transmission lines and windings of transformers areabstracted as edges (in this paper the three-winding trans-former applies 119884-connection with virtual intermediate nodeThe virtual node also is abstracted as the node of topology-based model) All nodes and edges are treated in the sameway Consequently the topology-based model of power gridrepresented by the unweighted and undirected graph (119866 =

(119881 119864)) containing n nodes and e edges was gained where 119881is the node set and119864 is edge setThe adjacencymatrix of nodeis recorded as 119860

119905and the matrix element is

119886119894119895=

1 if (119894 119895) isin 1198640 if (119894 119895) notin 119864

(1)

21 Power Grid Data The analysis objects of this paperare several typical large-scale transmission grids in Chinaincluding NX Grid TJ Grid TJ220 Grid CS Grid NW Gridand NC Grid The topology-based model sizes of these gridsafter data processing are 119866NX = 155 176 119866TJ = 597 648119866TJ220151 167 119866CS = 328 371 119866NW = 1840 2162and 119866NC = 4828 5666 Their nodes vary from hundredsto thousands

22 Topology Analysis of Power Grids The major statis-tical characteristic parameters of network models of theabovementioned power grids were calculated and analyzedincluding various parameters and methods widely appliedin various researches currently such as degree (119896) degreedistribution (119875(119896)) clustering coefficient (119862) and character-istic path length (119871) The results were listed in Table 1 where119899 is the number of nodes and 119890 is the number of edges⟨119896⟩ is the average node degree and 119862random is the clusteringcoefficient of random network with same nodes and sides ascorresponding practical network 119863 is the network diameterwhich is the maximum distance between any two nodes Therepresentation of these parameters to the structure of powergrids will be analyzed one by one in the following text

221 Node Degree (k) Node degree (119896) refers to the edgeamount connected to one node The larger the node degree(119896) is the more important the node is in the networkTake TJ220 Grid for example Its topology-based model is119866TJ220151 167 and its average node degree ⟨119896⟩ values 221Table 2 is the node degree statistics of TJ220 Grid underdifferent voltage levels To ensure the power distributionreliability within the whole city 500 kV nodes are connectedwith each other in the grid to form the backbone ring These



Grid 119899 119890 119896max ⟨119896⟩ 119862 119862random 119871 119863

TJ 597 648 13 217 00009 000036 10281 20TJ220 151 167 11 221 00135 001463 76037 16NX 155 176 9 227 00212 001464 87566 19CS 328 371 9 226 00181 000689 118725 32NW 1840 2162 11 235 00170 000127 154676 39NC 4828 5666 14 235 00037 000048 163647 43



1ndash3 25 844ndash11 75 16














119862119894=

119864119894

119896119894(119896119894minus 1) 2

(2)


119862 =1

119873

119873

sum

119894=1

119862119894 (3)




0 5 10 15

NXTJ

NWNC

10minus4

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


NXTJ

NWNC

10minus4

10minus3

10minus2

10minus1

100

100

101

P(K

gek)

Node degree (k)



0 5 10 15 2010

minus4

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


10minus4

10minus3

10minus2

10minus1

100

100

101

102

P(K

gek)

Node degree (k)






119864 =

119899minus1

sum

119894=1

119899

sum

119895=119894+1

1

2119896119894119895(10038161003816100381610038161003816119901119894minus 119901119895

10038161003816100381610038161003816minus 119897119894119895)2

(4)

where 119875119894and 119875



119894 and j 119896119894119895




Zone2

Zone1

Zone3






119894119895|) of the wire (119894 minus 119895)


119886119894119895=

10038161003816100381610038161003816119911119894119895

10038161003816100381610038161003816if (119894 119895) isin 119864

0 if (119894 119895) notin 119864(5)







119863 = max 119889119894119895 (6)


119871 =1

119899 (119899 + 1) 2sum

119894ge119895

119889119894119895 (7)









1

2

3

4

5

67

8

9

10

11

1213

14

15

16

17

18

19

20

21 2223

24

25

26

27

28

29

3031

32

33

3435

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

63

64

65

66

67

68

69

707172

73

74

7576

77

78

79

8081

82

83

84

8586

87

88

8990

91

92

93

94

95

96

97

9899

100

101

102

103

104

105

106

107

108

109

110

111112

113

114

115

116

117

118

119

120

121

122

123

124

125

126 127128

129

130

131

132

133134

135136

137

138

139

140

141

142

143

144

145

146

Zone3

Zone2

Zone1

147

148

149

150151


Zone1

Zone2

Zone3

1

2

3

4

5

67

8

9

10

11

12 13

1415

1617

18

19

20

21 2223

24

25

26

27

28

29

30

31

32

33

3435

36

37

38

39

40

4142

43

44

45

46

47

48

49

50

51

52

53

5455

56

57

585960

61

6263

64

65

6667

68

69

70

71

7273

74

75

7677

7879

8081

8283

84

8586

878889 90

91

9293

94

95

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111112

113114

115

116

117

118

119

120

121

122

123

124125

126

127

128129

130

131

132

133134

135136

137

138

139

140

141

142

143

144

145

146

147

148 149

150

151






[[[[[[[

[

11988511

1198851119894

1198851119899

1198851198941

119885119894119894 119885

119894119899

1198851198991

119885119899119894

119885119899119899

]]]]]]]

]

[[[[[[[

[

1198681

119868119894

119868119899

]]]]]]]

]

=

[[[[[[[

[

1

119894

119899

]]]]]]]

]

(8)

When 119868119898

= 0 and 119868119894= 0 (119894 = 1 2 119899 119894 =119898) the mutual


119885119894119898=119894

119868119898

119894 = 1 2 119899 (9)




= |119885119894119895| Higher |119885








119886119890(119894119895)

= 119889119890(119894119895)

forall119889119890(119894119895)

le 119889119890(th)

0 forall119889119890(119894119895)

gt 119889119890(th)

(10)




NX 02484 00309 14555TJ220 21073 17394 21331CS 00167 00001 00900






119890(th) When 119889119890(th) =







195 197 199 201 203 205 207 209 211 213 2150

005

01

015

02

025

03

035


Prob

abili

ty d

istrib

utio

n

(a) TJ220 Grid

018 019 02 021 022 023 024 025 026 0270

002

004

006

008

01

012


Prob

abili

ty d

istrib

utio

n

(b) NX Grid


001 002 003 004 005 006 007 008 0090

Prob

abili

ty d

istrib

utio

n

005

01

015

02

025

(c) CS Grid


195 197 199 201 203 205 207 209 211 213 2150

02

04

06

08

1

(2107 07251)(204 09853)


Cum

ulat

ive p

roba

bilit

y di

strib

utio

n



0 10 20 30 40 50

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


100

101

102

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)




119889119890(th) 119890 ⟨119896⟩ 119896max 119862

2107 3113 4123 49 094162090 2414 3197 44 08785207 1520 2013 38 08469206 816 1081 25 082202055 600 795 21 08203204 167 272 10 03938









4 Conclusions




1

2

3

4

5

67

89

10

11

12 13

1415

1617

18

19

20 21

22

23

24

25

26

27

28

29

3031 3233

34

35

3637

38

39

40

41

42

43

44

45 46

47

48

49

50

51

52

53

54

55

56

57

58

5960

61

62

63 64

65

66 67

68

69

70

71

72

73

74

75

7677

7879

80

8182

83

84

8586

87

88

89

9091

92

93

94

95

96

97 9899100101

102

103

104

105

106

107

108109

110111

112

113

114

115

116

117

118

119

120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136137138

139

140

141

142 143

144

145146

147 148

149

150151

Zone3

Zone2

Zone1

(a) 119889119890(th) = 2107

1

2

3

45

6

7

8

9

1011

12 1314

15

1617

18

19

2021

22

23

24

25

26

27

28

29

30

313233

34

35

36

37

3839

40

41

42 43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

6364

65

66

67

68

69

70

71

72

73 74

75

7677

78

79

8081

82

83

84

85

86

87

8889

90

91

92

93

9495

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116117

118

119120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150 151

Zone1

Zone2

Zone3

(b) 119889119890(th) = 207



0 20 40 60 80 100 120 140 160206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(a) V1

0 20 40 60 80 100 120 140 160204

206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(b) V2

0

Node number

Elec

tric

al d

istan

ce

20 40 60 80 100 120 140 160204

206

208

21

212

214

(c) V3









Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Grid 119899 119890 119896max ⟨119896⟩ 119862 119862random 119871 119863

TJ 597 648 13 217 00009 000036 10281 20TJ220 151 167 11 221 00135 001463 76037 16NX 155 176 9 227 00212 001464 87566 19CS 328 371 9 226 00181 000689 118725 32NW 1840 2162 11 235 00170 000127 154676 39NC 4828 5666 14 235 00037 000048 163647 43



1ndash3 25 844ndash11 75 16














119862119894=

119864119894

119896119894(119896119894minus 1) 2

(2)


119862 =1

119873

119873

sum

119894=1

119862119894 (3)




0 5 10 15

NXTJ

NWNC

10minus4

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


NXTJ

NWNC

10minus4

10minus3

10minus2

10minus1

100

100

101

P(K

gek)

Node degree (k)



0 5 10 15 2010

minus4

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


10minus4

10minus3

10minus2

10minus1

100

100

101

102

P(K

gek)

Node degree (k)






119864 =

119899minus1

sum

119894=1

119899

sum

119895=119894+1

1

2119896119894119895(10038161003816100381610038161003816119901119894minus 119901119895

10038161003816100381610038161003816minus 119897119894119895)2

(4)

where 119875119894and 119875



119894 and j 119896119894119895




Zone2

Zone1

Zone3






119894119895|) of the wire (119894 minus 119895)


119886119894119895=

10038161003816100381610038161003816119911119894119895

10038161003816100381610038161003816if (119894 119895) isin 119864

0 if (119894 119895) notin 119864(5)







119863 = max 119889119894119895 (6)


119871 =1

119899 (119899 + 1) 2sum

119894ge119895

119889119894119895 (7)









1

2

3

4

5

67

8

9

10

11

1213

14

15

16

17

18

19

20

21 2223

24

25

26

27

28

29

3031

32

33

3435

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

63

64

65

66

67

68

69

707172

73

74

7576

77

78

79

8081

82

83

84

8586

87

88

8990

91

92

93

94

95

96

97

9899

100

101

102

103

104

105

106

107

108

109

110

111112

113

114

115

116

117

118

119

120

121

122

123

124

125

126 127128

129

130

131

132

133134

135136

137

138

139

140

141

142

143

144

145

146

Zone3

Zone2

Zone1

147

148

149

150151


Zone1

Zone2

Zone3

1

2

3

4

5

67

8

9

10

11

12 13

1415

1617

18

19

20

21 2223

24

25

26

27

28

29

30

31

32

33

3435

36

37

38

39

40

4142

43

44

45

46

47

48

49

50

51

52

53

5455

56

57

585960

61

6263

64

65

6667

68

69

70

71

7273

74

75

7677

7879

8081

8283

84

8586

878889 90

91

9293

94

95

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111112

113114

115

116

117

118

119

120

121

122

123

124125

126

127

128129

130

131

132

133134

135136

137

138

139

140

141

142

143

144

145

146

147

148 149

150

151






[[[[[[[

[

11988511

1198851119894

1198851119899

1198851198941

119885119894119894 119885

119894119899

1198851198991

119885119899119894

119885119899119899

]]]]]]]

]

[[[[[[[

[

1198681

119868119894

119868119899

]]]]]]]

]

=

[[[[[[[

[

1

119894

119899

]]]]]]]

]

(8)

When 119868119898

= 0 and 119868119894= 0 (119894 = 1 2 119899 119894 =119898) the mutual


119885119894119898=119894

119868119898

119894 = 1 2 119899 (9)




= |119885119894119895| Higher |119885








119886119890(119894119895)

= 119889119890(119894119895)

forall119889119890(119894119895)

le 119889119890(th)

0 forall119889119890(119894119895)

gt 119889119890(th)

(10)




NX 02484 00309 14555TJ220 21073 17394 21331CS 00167 00001 00900






119890(th) When 119889119890(th) =







195 197 199 201 203 205 207 209 211 213 2150

005

01

015

02

025

03

035


Prob

abili

ty d

istrib

utio

n

(a) TJ220 Grid

018 019 02 021 022 023 024 025 026 0270

002

004

006

008

01

012


Prob

abili

ty d

istrib

utio

n

(b) NX Grid


001 002 003 004 005 006 007 008 0090

Prob

abili

ty d

istrib

utio

n

005

01

015

02

025

(c) CS Grid


195 197 199 201 203 205 207 209 211 213 2150

02

04

06

08

1

(2107 07251)(204 09853)


Cum

ulat

ive p

roba

bilit

y di

strib

utio

n



0 10 20 30 40 50

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


100

101

102

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)




119889119890(th) 119890 ⟨119896⟩ 119896max 119862

2107 3113 4123 49 094162090 2414 3197 44 08785207 1520 2013 38 08469206 816 1081 25 082202055 600 795 21 08203204 167 272 10 03938









4 Conclusions




1

2

3

4

5

67

89

10

11

12 13

1415

1617

18

19

20 21

22

23

24

25

26

27

28

29

3031 3233

34

35

3637

38

39

40

41

42

43

44

45 46

47

48

49

50

51

52

53

54

55

56

57

58

5960

61

62

63 64

65

66 67

68

69

70

71

72

73

74

75

7677

7879

80

8182

83

84

8586

87

88

89

9091

92

93

94

95

96

97 9899100101

102

103

104

105

106

107

108109

110111

112

113

114

115

116

117

118

119

120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136137138

139

140

141

142 143

144

145146

147 148

149

150151

Zone3

Zone2

Zone1

(a) 119889119890(th) = 2107

1

2

3

45

6

7

8

9

1011

12 1314

15

1617

18

19

2021

22

23

24

25

26

27

28

29

30

313233

34

35

36

37

3839

40

41

42 43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

6364

65

66

67

68

69

70

71

72

73 74

75

7677

78

79

8081

82

83

84

85

86

87

8889

90

91

92

93

9495

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116117

118

119120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150 151

Zone1

Zone2

Zone3

(b) 119889119890(th) = 207



0 20 40 60 80 100 120 140 160206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(a) V1

0 20 40 60 80 100 120 140 160204

206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(b) V2

0

Node number

Elec

tric

al d

istan

ce

20 40 60 80 100 120 140 160204

206

208

21

212

214

(c) V3









Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15

NXTJ

NWNC

10minus4

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


NXTJ

NWNC

10minus4

10minus3

10minus2

10minus1

100

100

101

P(K

gek)

Node degree (k)



0 5 10 15 2010

minus4

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


10minus4

10minus3

10minus2

10minus1

100

100

101

102

P(K

gek)

Node degree (k)






119864 =

119899minus1

sum

119894=1

119899

sum

119895=119894+1

1

2119896119894119895(10038161003816100381610038161003816119901119894minus 119901119895

10038161003816100381610038161003816minus 119897119894119895)2

(4)

where 119875119894and 119875



119894 and j 119896119894119895




Zone2

Zone1

Zone3






119894119895|) of the wire (119894 minus 119895)


119886119894119895=

10038161003816100381610038161003816119911119894119895

10038161003816100381610038161003816if (119894 119895) isin 119864

0 if (119894 119895) notin 119864(5)







119863 = max 119889119894119895 (6)


119871 =1

119899 (119899 + 1) 2sum

119894ge119895

119889119894119895 (7)









1

2

3

4

5

67

8

9

10

11

1213

14

15

16

17

18

19

20

21 2223

24

25

26

27

28

29

3031

32

33

3435

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

63

64

65

66

67

68

69

707172

73

74

7576

77

78

79

8081

82

83

84

8586

87

88

8990

91

92

93

94

95

96

97

9899

100

101

102

103

104

105

106

107

108

109

110

111112

113

114

115

116

117

118

119

120

121

122

123

124

125

126 127128

129

130

131

132

133134

135136

137

138

139

140

141

142

143

144

145

146

Zone3

Zone2

Zone1

147

148

149

150151


Zone1

Zone2

Zone3

1

2

3

4

5

67

8

9

10

11

12 13

1415

1617

18

19

20

21 2223

24

25

26

27

28

29

30

31

32

33

3435

36

37

38

39

40

4142

43

44

45

46

47

48

49

50

51

52

53

5455

56

57

585960

61

6263

64

65

6667

68

69

70

71

7273

74

75

7677

7879

8081

8283

84

8586

878889 90

91

9293

94

95

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111112

113114

115

116

117

118

119

120

121

122

123

124125

126

127

128129

130

131

132

133134

135136

137

138

139

140

141

142

143

144

145

146

147

148 149

150

151






[[[[[[[

[

11988511

1198851119894

1198851119899

1198851198941

119885119894119894 119885

119894119899

1198851198991

119885119899119894

119885119899119899

]]]]]]]

]

[[[[[[[

[

1198681

119868119894

119868119899

]]]]]]]

]

=

[[[[[[[

[

1

119894

119899

]]]]]]]

]

(8)

When 119868119898

= 0 and 119868119894= 0 (119894 = 1 2 119899 119894 =119898) the mutual


119885119894119898=119894

119868119898

119894 = 1 2 119899 (9)




= |119885119894119895| Higher |119885








119886119890(119894119895)

= 119889119890(119894119895)

forall119889119890(119894119895)

le 119889119890(th)

0 forall119889119890(119894119895)

gt 119889119890(th)

(10)




NX 02484 00309 14555TJ220 21073 17394 21331CS 00167 00001 00900






119890(th) When 119889119890(th) =







195 197 199 201 203 205 207 209 211 213 2150

005

01

015

02

025

03

035


Prob

abili

ty d

istrib

utio

n

(a) TJ220 Grid

018 019 02 021 022 023 024 025 026 0270

002

004

006

008

01

012


Prob

abili

ty d

istrib

utio

n

(b) NX Grid


001 002 003 004 005 006 007 008 0090

Prob

abili

ty d

istrib

utio

n

005

01

015

02

025

(c) CS Grid


195 197 199 201 203 205 207 209 211 213 2150

02

04

06

08

1

(2107 07251)(204 09853)


Cum

ulat

ive p

roba

bilit

y di

strib

utio

n



0 10 20 30 40 50

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


100

101

102

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)




119889119890(th) 119890 ⟨119896⟩ 119896max 119862

2107 3113 4123 49 094162090 2414 3197 44 08785207 1520 2013 38 08469206 816 1081 25 082202055 600 795 21 08203204 167 272 10 03938









4 Conclusions




1

2

3

4

5

67

89

10

11

12 13

1415

1617

18

19

20 21

22

23

24

25

26

27

28

29

3031 3233

34

35

3637

38

39

40

41

42

43

44

45 46

47

48

49

50

51

52

53

54

55

56

57

58

5960

61

62

63 64

65

66 67

68

69

70

71

72

73

74

75

7677

7879

80

8182

83

84

8586

87

88

89

9091

92

93

94

95

96

97 9899100101

102

103

104

105

106

107

108109

110111

112

113

114

115

116

117

118

119

120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136137138

139

140

141

142 143

144

145146

147 148

149

150151

Zone3

Zone2

Zone1

(a) 119889119890(th) = 2107

1

2

3

45

6

7

8

9

1011

12 1314

15

1617

18

19

2021

22

23

24

25

26

27

28

29

30

313233

34

35

36

37

3839

40

41

42 43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

6364

65

66

67

68

69

70

71

72

73 74

75

7677

78

79

8081

82

83

84

85

86

87

8889

90

91

92

93

9495

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116117

118

119120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150 151

Zone1

Zone2

Zone3

(b) 119889119890(th) = 207



0 20 40 60 80 100 120 140 160206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(a) V1

0 20 40 60 80 100 120 140 160204

206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(b) V2

0

Node number

Elec

tric

al d

istan

ce

20 40 60 80 100 120 140 160204

206

208

21

212

214

(c) V3









Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Zone2

Zone1

Zone3






119894119895|) of the wire (119894 minus 119895)


119886119894119895=

10038161003816100381610038161003816119911119894119895

10038161003816100381610038161003816if (119894 119895) isin 119864

0 if (119894 119895) notin 119864(5)







119863 = max 119889119894119895 (6)


119871 =1

119899 (119899 + 1) 2sum

119894ge119895

119889119894119895 (7)









1

2

3

4

5

67

8

9

10

11

1213

14

15

16

17

18

19

20

21 2223

24

25

26

27

28

29

3031

32

33

3435

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

63

64

65

66

67

68

69

707172

73

74

7576

77

78

79

8081

82

83

84

8586

87

88

8990

91

92

93

94

95

96

97

9899

100

101

102

103

104

105

106

107

108

109

110

111112

113

114

115

116

117

118

119

120

121

122

123

124

125

126 127128

129

130

131

132

133134

135136

137

138

139

140

141

142

143

144

145

146

Zone3

Zone2

Zone1

147

148

149

150151


Zone1

Zone2

Zone3

1

2

3

4

5

67

8

9

10

11

12 13

1415

1617

18

19

20

21 2223

24

25

26

27

28

29

30

31

32

33

3435

36

37

38

39

40

4142

43

44

45

46

47

48

49

50

51

52

53

5455

56

57

585960

61

6263

64

65

6667

68

69

70

71

7273

74

75

7677

7879

8081

8283

84

8586

878889 90

91

9293

94

95

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111112

113114

115

116

117

118

119

120

121

122

123

124125

126

127

128129

130

131

132

133134

135136

137

138

139

140

141

142

143

144

145

146

147

148 149

150

151






[[[[[[[

[

11988511

1198851119894

1198851119899

1198851198941

119885119894119894 119885

119894119899

1198851198991

119885119899119894

119885119899119899

]]]]]]]

]

[[[[[[[

[

1198681

119868119894

119868119899

]]]]]]]

]

=

[[[[[[[

[

1

119894

119899

]]]]]]]

]

(8)

When 119868119898

= 0 and 119868119894= 0 (119894 = 1 2 119899 119894 =119898) the mutual


119885119894119898=119894

119868119898

119894 = 1 2 119899 (9)




= |119885119894119895| Higher |119885








119886119890(119894119895)

= 119889119890(119894119895)

forall119889119890(119894119895)

le 119889119890(th)

0 forall119889119890(119894119895)

gt 119889119890(th)

(10)




NX 02484 00309 14555TJ220 21073 17394 21331CS 00167 00001 00900






119890(th) When 119889119890(th) =







195 197 199 201 203 205 207 209 211 213 2150

005

01

015

02

025

03

035


Prob

abili

ty d

istrib

utio

n

(a) TJ220 Grid

018 019 02 021 022 023 024 025 026 0270

002

004

006

008

01

012


Prob

abili

ty d

istrib

utio

n

(b) NX Grid


001 002 003 004 005 006 007 008 0090

Prob

abili

ty d

istrib

utio

n

005

01

015

02

025

(c) CS Grid


195 197 199 201 203 205 207 209 211 213 2150

02

04

06

08

1

(2107 07251)(204 09853)


Cum

ulat

ive p

roba

bilit

y di

strib

utio

n



0 10 20 30 40 50

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


100

101

102

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)




119889119890(th) 119890 ⟨119896⟩ 119896max 119862

2107 3113 4123 49 094162090 2414 3197 44 08785207 1520 2013 38 08469206 816 1081 25 082202055 600 795 21 08203204 167 272 10 03938









4 Conclusions




1

2

3

4

5

67

89

10

11

12 13

1415

1617

18

19

20 21

22

23

24

25

26

27

28

29

3031 3233

34

35

3637

38

39

40

41

42

43

44

45 46

47

48

49

50

51

52

53

54

55

56

57

58

5960

61

62

63 64

65

66 67

68

69

70

71

72

73

74

75

7677

7879

80

8182

83

84

8586

87

88

89

9091

92

93

94

95

96

97 9899100101

102

103

104

105

106

107

108109

110111

112

113

114

115

116

117

118

119

120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136137138

139

140

141

142 143

144

145146

147 148

149

150151

Zone3

Zone2

Zone1

(a) 119889119890(th) = 2107

1

2

3

45

6

7

8

9

1011

12 1314

15

1617

18

19

2021

22

23

24

25

26

27

28

29

30

313233

34

35

36

37

3839

40

41

42 43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

6364

65

66

67

68

69

70

71

72

73 74

75

7677

78

79

8081

82

83

84

85

86

87

8889

90

91

92

93

9495

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116117

118

119120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150 151

Zone1

Zone2

Zone3

(b) 119889119890(th) = 207



0 20 40 60 80 100 120 140 160206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(a) V1

0 20 40 60 80 100 120 140 160204

206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(b) V2

0

Node number

Elec

tric

al d

istan

ce

20 40 60 80 100 120 140 160204

206

208

21

212

214

(c) V3









Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

2

3

4

5

67

8

9

10

11

1213

14

15

16

17

18

19

20

21 2223

24

25

26

27

28

29

3031

32

33

3435

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

63

64

65

66

67

68

69

707172

73

74

7576

77

78

79

8081

82

83

84

8586

87

88

8990

91

92

93

94

95

96

97

9899

100

101

102

103

104

105

106

107

108

109

110

111112

113

114

115

116

117

118

119

120

121

122

123

124

125

126 127128

129

130

131

132

133134

135136

137

138

139

140

141

142

143

144

145

146

Zone3

Zone2

Zone1

147

148

149

150151


Zone1

Zone2

Zone3

1

2

3

4

5

67

8

9

10

11

12 13

1415

1617

18

19

20

21 2223

24

25

26

27

28

29

30

31

32

33

3435

36

37

38

39

40

4142

43

44

45

46

47

48

49

50

51

52

53

5455

56

57

585960

61

6263

64

65

6667

68

69

70

71

7273

74

75

7677

7879

8081

8283

84

8586

878889 90

91

9293

94

95

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111112

113114

115

116

117

118

119

120

121

122

123

124125

126

127

128129

130

131

132

133134

135136

137

138

139

140

141

142

143

144

145

146

147

148 149

150

151






[[[[[[[

[

11988511

1198851119894

1198851119899

1198851198941

119885119894119894 119885

119894119899

1198851198991

119885119899119894

119885119899119899

]]]]]]]

]

[[[[[[[

[

1198681

119868119894

119868119899

]]]]]]]

]

=

[[[[[[[

[

1

119894

119899

]]]]]]]

]

(8)

When 119868119898

= 0 and 119868119894= 0 (119894 = 1 2 119899 119894 =119898) the mutual


119885119894119898=119894

119868119898

119894 = 1 2 119899 (9)




= |119885119894119895| Higher |119885








119886119890(119894119895)

= 119889119890(119894119895)

forall119889119890(119894119895)

le 119889119890(th)

0 forall119889119890(119894119895)

gt 119889119890(th)

(10)




NX 02484 00309 14555TJ220 21073 17394 21331CS 00167 00001 00900






119890(th) When 119889119890(th) =







195 197 199 201 203 205 207 209 211 213 2150

005

01

015

02

025

03

035


Prob

abili

ty d

istrib

utio

n

(a) TJ220 Grid

018 019 02 021 022 023 024 025 026 0270

002

004

006

008

01

012


Prob

abili

ty d

istrib

utio

n

(b) NX Grid


001 002 003 004 005 006 007 008 0090

Prob

abili

ty d

istrib

utio

n

005

01

015

02

025

(c) CS Grid


195 197 199 201 203 205 207 209 211 213 2150

02

04

06

08

1

(2107 07251)(204 09853)


Cum

ulat

ive p

roba

bilit

y di

strib

utio

n



0 10 20 30 40 50

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


100

101

102

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)




119889119890(th) 119890 ⟨119896⟩ 119896max 119862

2107 3113 4123 49 094162090 2414 3197 44 08785207 1520 2013 38 08469206 816 1081 25 082202055 600 795 21 08203204 167 272 10 03938









4 Conclusions




1

2

3

4

5

67

89

10

11

12 13

1415

1617

18

19

20 21

22

23

24

25

26

27

28

29

3031 3233

34

35

3637

38

39

40

41

42

43

44

45 46

47

48

49

50

51

52

53

54

55

56

57

58

5960

61

62

63 64

65

66 67

68

69

70

71

72

73

74

75

7677

7879

80

8182

83

84

8586

87

88

89

9091

92

93

94

95

96

97 9899100101

102

103

104

105

106

107

108109

110111

112

113

114

115

116

117

118

119

120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136137138

139

140

141

142 143

144

145146

147 148

149

150151

Zone3

Zone2

Zone1

(a) 119889119890(th) = 2107

1

2

3

45

6

7

8

9

1011

12 1314

15

1617

18

19

2021

22

23

24

25

26

27

28

29

30

313233

34

35

36

37

3839

40

41

42 43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

6364

65

66

67

68

69

70

71

72

73 74

75

7677

78

79

8081

82

83

84

85

86

87

8889

90

91

92

93

9495

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116117

118

119120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150 151

Zone1

Zone2

Zone3

(b) 119889119890(th) = 207



0 20 40 60 80 100 120 140 160206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(a) V1

0 20 40 60 80 100 120 140 160204

206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(b) V2

0

Node number

Elec

tric

al d

istan

ce

20 40 60 80 100 120 140 160204

206

208

21

212

214

(c) V3









Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












[[[[[[[

[

11988511

1198851119894

1198851119899

1198851198941

119885119894119894 119885

119894119899

1198851198991

119885119899119894

119885119899119899

]]]]]]]

]

[[[[[[[

[

1198681

119868119894

119868119899

]]]]]]]

]

=

[[[[[[[

[

1

119894

119899

]]]]]]]

]

(8)

When 119868119898

= 0 and 119868119894= 0 (119894 = 1 2 119899 119894 =119898) the mutual


119885119894119898=119894

119868119898

119894 = 1 2 119899 (9)




= |119885119894119895| Higher |119885








119886119890(119894119895)

= 119889119890(119894119895)

forall119889119890(119894119895)

le 119889119890(th)

0 forall119889119890(119894119895)

gt 119889119890(th)

(10)




NX 02484 00309 14555TJ220 21073 17394 21331CS 00167 00001 00900






119890(th) When 119889119890(th) =







195 197 199 201 203 205 207 209 211 213 2150

005

01

015

02

025

03

035


Prob

abili

ty d

istrib

utio

n

(a) TJ220 Grid

018 019 02 021 022 023 024 025 026 0270

002

004

006

008

01

012


Prob

abili

ty d

istrib

utio

n

(b) NX Grid


001 002 003 004 005 006 007 008 0090

Prob

abili

ty d

istrib

utio

n

005

01

015

02

025

(c) CS Grid


195 197 199 201 203 205 207 209 211 213 2150

02

04

06

08

1

(2107 07251)(204 09853)


Cum

ulat

ive p

roba

bilit

y di

strib

utio

n



0 10 20 30 40 50

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


100

101

102

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)




119889119890(th) 119890 ⟨119896⟩ 119896max 119862

2107 3113 4123 49 094162090 2414 3197 44 08785207 1520 2013 38 08469206 816 1081 25 082202055 600 795 21 08203204 167 272 10 03938









4 Conclusions




1

2

3

4

5

67

89

10

11

12 13

1415

1617

18

19

20 21

22

23

24

25

26

27

28

29

3031 3233

34

35

3637

38

39

40

41

42

43

44

45 46

47

48

49

50

51

52

53

54

55

56

57

58

5960

61

62

63 64

65

66 67

68

69

70

71

72

73

74

75

7677

7879

80

8182

83

84

8586

87

88

89

9091

92

93

94

95

96

97 9899100101

102

103

104

105

106

107

108109

110111

112

113

114

115

116

117

118

119

120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136137138

139

140

141

142 143

144

145146

147 148

149

150151

Zone3

Zone2

Zone1

(a) 119889119890(th) = 2107

1

2

3

45

6

7

8

9

1011

12 1314

15

1617

18

19

2021

22

23

24

25

26

27

28

29

30

313233

34

35

36

37

3839

40

41

42 43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

6364

65

66

67

68

69

70

71

72

73 74

75

7677

78

79

8081

82

83

84

85

86

87

8889

90

91

92

93

9495

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116117

118

119120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150 151

Zone1

Zone2

Zone3

(b) 119889119890(th) = 207



0 20 40 60 80 100 120 140 160206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(a) V1

0 20 40 60 80 100 120 140 160204

206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(b) V2

0

Node number

Elec

tric

al d

istan

ce

20 40 60 80 100 120 140 160204

206

208

21

212

214

(c) V3









Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










195 197 199 201 203 205 207 209 211 213 2150

005

01

015

02

025

03

035


Prob

abili

ty d

istrib

utio

n

(a) TJ220 Grid

018 019 02 021 022 023 024 025 026 0270

002

004

006

008

01

012


Prob

abili

ty d

istrib

utio

n

(b) NX Grid


001 002 003 004 005 006 007 008 0090

Prob

abili

ty d

istrib

utio

n

005

01

015

02

025

(c) CS Grid


195 197 199 201 203 205 207 209 211 213 2150

02

04

06

08

1

(2107 07251)(204 09853)


Cum

ulat

ive p

roba

bilit

y di

strib

utio

n



0 10 20 30 40 50

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


100

101

102

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)




119889119890(th) 119890 ⟨119896⟩ 119896max 119862

2107 3113 4123 49 094162090 2414 3197 44 08785207 1520 2013 38 08469206 816 1081 25 082202055 600 795 21 08203204 167 272 10 03938









4 Conclusions




1

2

3

4

5

67

89

10

11

12 13

1415

1617

18

19

20 21

22

23

24

25

26

27

28

29

3031 3233

34

35

3637

38

39

40

41

42

43

44

45 46

47

48

49

50

51

52

53

54

55

56

57

58

5960

61

62

63 64

65

66 67

68

69

70

71

72

73

74

75

7677

7879

80

8182

83

84

8586

87

88

89

9091

92

93

94

95

96

97 9899100101

102

103

104

105

106

107

108109

110111

112

113

114

115

116

117

118

119

120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136137138

139

140

141

142 143

144

145146

147 148

149

150151

Zone3

Zone2

Zone1

(a) 119889119890(th) = 2107

1

2

3

45

6

7

8

9

1011

12 1314

15

1617

18

19

2021

22

23

24

25

26

27

28

29

30

313233

34

35

36

37

3839

40

41

42 43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

6364

65

66

67

68

69

70

71

72

73 74

75

7677

78

79

8081

82

83

84

85

86

87

8889

90

91

92

93

9495

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116117

118

119120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150 151

Zone1

Zone2

Zone3

(b) 119889119890(th) = 207



0 20 40 60 80 100 120 140 160206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(a) V1

0 20 40 60 80 100 120 140 160204

206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(b) V2

0

Node number

Elec

tric

al d

istan

ce

20 40 60 80 100 120 140 160204

206

208

21

212

214

(c) V3









Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)


100

101

102

de(th) = 2107

de(th) = 2090

de(th) = 2070

de(th) = 2060

de(th) = 2055

de(th) = 2040

10minus3

10minus2

10minus1

100

P(K

gek)

Node degree (k)




119889119890(th) 119890 ⟨119896⟩ 119896max 119862

2107 3113 4123 49 094162090 2414 3197 44 08785207 1520 2013 38 08469206 816 1081 25 082202055 600 795 21 08203204 167 272 10 03938









4 Conclusions




1

2

3

4

5

67

89

10

11

12 13

1415

1617

18

19

20 21

22

23

24

25

26

27

28

29

3031 3233

34

35

3637

38

39

40

41

42

43

44

45 46

47

48

49

50

51

52

53

54

55

56

57

58

5960

61

62

63 64

65

66 67

68

69

70

71

72

73

74

75

7677

7879

80

8182

83

84

8586

87

88

89

9091

92

93

94

95

96

97 9899100101

102

103

104

105

106

107

108109

110111

112

113

114

115

116

117

118

119

120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136137138

139

140

141

142 143

144

145146

147 148

149

150151

Zone3

Zone2

Zone1

(a) 119889119890(th) = 2107

1

2

3

45

6

7

8

9

1011

12 1314

15

1617

18

19

2021

22

23

24

25

26

27

28

29

30

313233

34

35

36

37

3839

40

41

42 43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

6364

65

66

67

68

69

70

71

72

73 74

75

7677

78

79

8081

82

83

84

85

86

87

8889

90

91

92

93

9495

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116117

118

119120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150 151

Zone1

Zone2

Zone3

(b) 119889119890(th) = 207



0 20 40 60 80 100 120 140 160206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(a) V1

0 20 40 60 80 100 120 140 160204

206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(b) V2

0

Node number

Elec

tric

al d

istan

ce

20 40 60 80 100 120 140 160204

206

208

21

212

214

(c) V3









Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

2

3

4

5

67

89

10

11

12 13

1415

1617

18

19

20 21

22

23

24

25

26

27

28

29

3031 3233

34

35

3637

38

39

40

41

42

43

44

45 46

47

48

49

50

51

52

53

54

55

56

57

58

5960

61

62

63 64

65

66 67

68

69

70

71

72

73

74

75

7677

7879

80

8182

83

84

8586

87

88

89

9091

92

93

94

95

96

97 9899100101

102

103

104

105

106

107

108109

110111

112

113

114

115

116

117

118

119

120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136137138

139

140

141

142 143

144

145146

147 148

149

150151

Zone3

Zone2

Zone1

(a) 119889119890(th) = 2107

1

2

3

45

6

7

8

9

1011

12 1314

15

1617

18

19

2021

22

23

24

25

26

27

28

29

30

313233

34

35

36

37

3839

40

41

42 43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

5859

60

61

62

6364

65

66

67

68

69

70

71

72

73 74

75

7677

78

79

8081

82

83

84

85

86

87

8889

90

91

92

93

9495

96

97

98

99100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116117

118

119120

121

122

123

124125

126

127

128

129

130

131

132

133134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150 151

Zone1

Zone2

Zone3

(b) 119889119890(th) = 207



0 20 40 60 80 100 120 140 160206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(a) V1

0 20 40 60 80 100 120 140 160204

206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(b) V2

0

Node number

Elec

tric

al d

istan

ce

20 40 60 80 100 120 140 160204

206

208

21

212

214

(c) V3









Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100 120 140 160206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(a) V1

0 20 40 60 80 100 120 140 160204

206

208

21

212

214

Node number

Elec

tric

al d

istan

ce

(b) V2

0

Node number

Elec

tric

al d

istan

ce

20 40 60 80 100 120 140 160204

206

208

21

212

214

(c) V3









Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Research on the Structural ...

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