Natural connectivity and robustness in the topological structure of Eletrosul transmission system

Transcription

1 Trabalho apresentado no CMAC-Sul, Curtba-PR, 4. atural connectvty and robustness n the topologcal structure of Eletrosul transmsson system M.S. Onetta, T. Carlotto, J.M.V. Grzybows Unversdade Federal da Frontera Sul - UFFS Av. Dom João Hoffmann, 33, CEP , Erechm, RS. E-mal: jose.grzybows@uffs.edu.br Abstract: The robustness of networ structures under falure or malfuncton of a subset of ts nodes s recognzed to be related to the number of redundant paths between nodes, whch facltate the redstrbuton of flows and contrbutes to the mantenance of ts functons n the perod post-falure. Ths paper nvestgates robustness on the bass of the varaton of natural connectvty nduced by the addton of lns n a model of Eletrosul transmsson system. The results provde nformaton regardng optmal strateges for the expanson of the electrcal grd from the vewpont of maxmzng the avalablty of redundant closed paths wthn the networ structure, whch s beleved to be related to the structural robustness of networ systems. Introducton Electrcal networs are large-scale systems whose elements feature hghly nterdependent behavor. Ths s so manly to nduce nerta whch, n ths context, s the nclnaton to reman n unformly-paced global movement thus dsspatng local nstabltes. A central matter regardng electrcal networs and networ structures n general s the extent to whch ther topology can support the contnuty of the processes they perform and reman operatonal under the falure or malfunctonng of a subset of ts nodes or lns. Such robustness conssts of an nnate structural ablty to handle shortcomngs, and t s recognzed to be related to the avalablty of redundant paths connectng nodes, whch facltate the redstrbuton of flows after a falure. Consder prevous studes whch found that ncreased number of alternatves reduces damage n the dameter a networ under attac []. Also, the addton of redundant lns was shown to mprove the tolerance aganst coordnated attacs [2], such that robustness s argued to ncrease wth connectvty[3][4]. Further, dverse networ systems such as water dstrbuton networs were found to have ts robustness assocated to the exstence of redundant paths [5]. In ths regard, a measure of redundant paths n a networ structure s found n the concept of natural connectvty, whch s regarded as a computatonally affordable and physcally ntutve framewor to access the number and mportance of closed paths between nodes [6]. The ntuton behnd such relaton s that the greater number of alternatve paths s lely to facltate the communcaton between vertces as some lns or nodes are damaged [6]. As nvestments n power grds are subject to economcal restrctons, ncreasng structural robustness n such networs shall obey optmalty crtera. In the context of structural robustness, a queston arses: how to add lns such that connectvty ncreases the most? That s to say, how to maxmze the number of redundant closed paths wth a mnmum addton of lns? Ths paper studes the mpact of the addton of lns upon the natural connectvty of the graph of Eletrosul transmsson system [7]. In the context of ths nvestgaton, the queston beng ased translates nto an objectve crtera,.e., the varaton of natural connectvty (VC). The results are gven n terms of VC maps and they reveal optmal strateges for maxmal connectvty ncrease. They also provde potentally nsghtful nformaton regardng the expanson of the electrcal grd towards a more and more robust and redundant topologcal structure. 2 Materals and methods As modeled accordng to ts topology, a power grd can be represented by a graph n whch nodes (generators, substatons, transformers) communcate wth other nodes by means of lns (transmsson lnes). Due to the nature of ts functonng, a power grd can be modeled as a smple DOI: / SBMAC

2 undrected graph G( V, E ) composed by nodes from a set V whch are lned by edges from a set E. The resultng graph nduces a symmetrcal adjacency matrx wth entres A = A = f there s a ln between nodes and j and A = A = 0, otherwse for, j =,...,, wth A = 0. From the j j adjacency matrx, the networ Laplacan matrx can be defned as G = D A, where D s a dagonal matrx for whch D = A. In such a graph, a wal of length s defned as an j= j alternatng sequence of nodes and lns v0e... ve2 ev wth v V e, = v v ; further, t s closed f the wal ends n the same ln as t started, that s, v = v0. In the calculaton of the natural connectvty, the weghted sum of numbers of closed wals gves shorter closed wals ncreased mportance by dvdng the number of closed wals of a gven length by the factoral of the length, as gven by + n S = () = 0! and ( ) where n the number of closed wals of length can be obtaned from the adjacency matrx, that s, n ( ) j j th power of the λ trace A (2) = = = Equatons () and (2) yeld n λ λ λ S = = = = e (3) = 0! = 0 =! = = 0! = that s named Estrada ndex [6]. By scalng and averagng S, the natural connectvty of the graph s defned as λ λ = ln e (4) = whch represents an average of the egenvalues of graph the adjacency matrx. We consder the measure n equaton (4) and the adjacency matrx nduced by the graph of Eletrosul transmsson system [7] to evaluate the mpact of the addton or removal of nodes. Towards that end, we defne the VC λ as where C λ λ λ C λ s the natural connectvty of the graph ( ) connectvty of the unchanged graph. 3 Results and dscusson = (5) G V, E C and we tae λ to be the natural Consder the orgnal topology of the transmsson system, as shown n Fgure. Let us consder the stuaton n whch 0 lns (arbtrary) are added to the orgnal topology n two dfferent manners: () by choosng the ones that nduce largest ncrease n VC and then () by choosng them at random. Fgure 2 shows the values of the VC correspondng to the addton of one edge between nodes and j n the orgnal graph, shown n Fgure. Hghlghted colored spots ndcate lns that, as created, would boost connectvty the most. Frst, the lns ten lns that nduce hghest ncrease n VC were added to the orgnal networ and the natural connectvty of the new networ reached 2.22, thus 0,70 hgher than the orgnal value, whch demonstrates a consderable ncrease n the number of alternatve closed paths wthn the networ structure, especally short ones, whch are regarded as more mportant snce they denote 'proxmty' between nodes. The result s shown n Fgure 3. ote that the new lns encompass a small regon of the networ and they connect geographcally close nodes, thus mang them sutable for an electrcal networ. It can be observed that the evoluton of electrcal networs (as much as other nfra-structure networs) conform to mnmum-dstance rules,.e., new lns are most lely created between nodes that are geographcally close, thus mang the soluton feasble. DOI: / SBMAC

3 Fgure - Orgnal topology of Eletrosul transmsson system: nodes wth major potental for connectvty ncrease are ndcated n color gradent from red (lowest) to yellow (hghest), λ Fgure 2 - Varaton of natural connectvty as a functon of the addton of one node: colored spots dentfy the poston of new edges causng hgher mpacts upon networ s natural connectvty. To the,2,...,,2,..., ; to the rght, hghlght over left, the contour plot for values of VC over the set { } { } the regon wth the hghest values of VC DOI: / SBMAC

4 Fgure 3 - Altered topology of Eletrosul transmsson system after the addton of 0 new nodes that ncrease natural connectvty the most (n green), 2.22 λ. Second, ten random lns are added to the orgnal networ and the natural connectvty of the new graph s calculated. To guarantee the unformty n the results of the second approach, the process s repeated.000 tmes, and the results are presented as averages. The natural connectvty of the new networ accordng to ths process reached the value.58, wth mnmum.55 and maxmum.65 among all realzatons, thus well below those obtaned by drectng addton by VC ncrease. Other than maxmzng the number of alternatve routes wthn the networ structure, the proposed VC can also be used to evaluate the mpact of the removal of nodes, thus allowng dentfcaton of fragle spots wthn the networ structure. Such nformaton can be taen nto account n prorty nvestments regardng the mprovement of the grd's structural robustness, to the extent that the results are confrmed by nvestgatons that tae nto account dfferent dmensons of the same problems, e.g., electrcal, economcal, techncal, socal, among others. Whle the topology of a transmsson system s an oversmplfcaton of ts general functonng, t can provde nsghts nto the role of structure n the overly complex mechansms that rule the functonng and falng of power grds, towards a refnement of current nowledge and understandng. From the topologcal vewpont, the message conveyed by the VC map s that ncreased connectvty n wll mprove the overall robustness of the entre networ aganst events that may dsturb the structure of the networ, such as the loss of a transmsson lne. The authors than FAPERGS for the support, grant number 2/ Keywords: natural connectvty, power-grds, robustness, Eletrosul transmsson system. References [] B. Shargel, H. Sayama, I. Epsten, Y. B. Optmzaton of Robustness and Connectvty n Complex etwors. Physcal Revew Letters, volume 90, number 6, 0.03, 03. [2] Y. Zhuo, Y. Peng, K. Long, Y. Lu. On allocatng redundancy lns to mprove robustness of complex communcaton networ, Procedgs of the SPIE 7633, etwor Archtectures Management and Applcatons VII, 7633A, do:0.7/2.8538, 09. DOI: / SBMAC

5 [3] Y. Zhuo, Y. Peng, K. Long, and Y. Lu, Improvng Robustness of Complex Communcaton etwors by Allocatng Redundancy Lns, n Asa Communcatons and Photoncs Conference and Exhbton, Techncal Dgest (CD) (Optcal Socety of Amerca), paper FD3, 09. [4] G. Ranjan, Z. Zh-l, A Geometrc Approach to Robustness n Complex etwors. Dstrbuted Computng Systems Worshops, number ,. [5] Yazdan, A. and Jeffrey, P. Applyng etwor Theory to Quantfy the Redundancy and Structural Robustness of Water Dstrbuton Systems. J. Water Resource Plannng and Management, 38(2), 53 6, 2. [6] J. Wu, M. Barahona, Y.J. Tan, H.Z. Deng. atural connectvty of complex networs. Chnese Physcs Letters, volume 27, number 7, , 0. [7] Map of the Electrcal Transmsson System, Eletrosul Centras Elétrcas. Avalable onlne at DOI: / SBMAC