On the Use of Graph Theory for Railway Power Supply Systems Characterization

Transcription

1 Intll Ind yst (0) : 9 DOI 0.00/s ORIGIAL PAPER On th Us of Graph Thory for Railway Powr upply ystms Charactrization Manul Coto Pablo Arbolya Cristina Gonzálz-Morán Rcivd: January 0 / Rvisd: March 0 / Accptd: March 0 / Publishd onlin: 8 May 0 pringr cinc+businss Mdia ingapor 0 Abstract Ovr th yars, graph thory has provn to b a ky tool in powr systms modling and analysis. In this papr, th authors propos a systmatic mthod for railway powr supply systms (RP) dscription that can b applid to any systm. This mthod rprsnts th diffrnt lmnts of th RP with a st of subgraphs. Mrging ths subgraphs, th rprsntativ graph of th whol RP and its associatd adjacncy and incidnc matrics will b obtaind. Onc ths matrics ar obtaind, Kirchhoff s laws can b asily implmntd. In this work, th mthod is applid to a DC light traction systm. Th systm that fds th traction ntwork through powr transformrs combind with rctifirs is also includd. With th proposd approach, th variability problms in th systm topology and dimnsions ar ovrcom, obtaining an invariant systm, vn whn th trains chang thir rlativ position, or whn a nw train ntrs into or xits th systm. Introduction In 900, Poincar stablishd th principls of algbraic topology introducing th dscription of graphs by using th incidnc matrix. Thn, in 9, Vbln showd how th Kirchhoff s laws could b formulatd by applying Poincar thory [8]. B Pablo Arbolya arbolyapablo@uniovi.s Manul Coto cotomanul@uniovi.s Cristina Gonzálz-Morán gonzalzmorcristina@uniovi.s Dpartmnt of Elctrical Enginring, Univrsity of Ovido, Ovido, pain This was just th bginning of multipl improvmnts and innovations in graph thory and its application to powr systms modling and analysis. Th bulk of ths improvmnts took plac in th dcads of 90 s and 90 s, whn th classical topological formulas wr modifid to fit passiv ntworks containing mutual couplings and activ ntworks. For xampl, mthods for solving a systm of indfinit linar algbraic quations by applying a dirctd wightd graph, known as a signal-flow graph, wr proposd in [,,,,8,]. uccssiv fforts wr don to dvlop th graph dscription by mans of matrix tchniqus [,]. owadays, th graph thory is still in vogu, but nw advancs do not only li on th graph thory itslf, but also on its applications to a wid rang of diffrnt problms. Rgarding powr systms analysis and modling, thr main graph thory applications hav bn idntifid: Modling and analysis of non linar ntworks [0]. Graph thory is applid to formulat and dscrib non-linar ntworks for dynamic analysis, rvaling that graph thory is a vry usful tool for complx ntwork dscription. Th mthod, known as two-graph modifid nodal analysis was usd by th Korrs t al. in [] for th snsitivity analysis of priodically switchd linar ntworks instad of th stat spac modling. In this way, th dfinition of stat-variabls and th quation formulation ar avoidd. In [], th us of switching signal flow graph mthod was applid to larg signal analysis of non linar circuits in supr-lift convrtrs. Obsrvability analysis in powr systms.in[], th graph thortic obsrvability analysis is xtndd to mak it capabl of procssing switching branchs in lctric powr systms. Th authors in [0] idntify th maximum numbr of obsrvabl islands in a masurd powr systm. Th optimal placmnt of phasor masurmnt units to

2 8 Intll Ind yst (0) : 9 b combind with powr flow tchniqus in obsrvability analysis of powr systms is dtrmind in []. Powr flow tchniqus []. Graph thory is applid to dcompos a larg-dimnsion optimum powr flow problm, with a larg numbr of thrmal limit constraints into a st of mdium-dimnsion problms. In [9], th graph thory is usd to analyz th ffct of phas-shifting transformrs ovr th powr flow, dtrmining th optimum numbr of this kind of dvics and thir location to rach a crtain control in th whol powr systm. A nw mthod to xtract all possibl radial configurations in a wakly mshd powr systm using graph basd sarch mthods is proposd in [].In [] a thr-phas distribution powr flow solution for unbalancd radial distribution systms has bn dvlopd basd on graph thory. A solution for coordinating larg-scal multi-agnt systms is prsntd in [], and in [] graph thory is usd to dploy multipl layrs of multi-agnt systms to protct sub-systms around idntifid loads in a powr systm. Th RP rprsnts a vry important part of th traction systm infrastructur, so in ordr to mak proprly dimnsioning, tak corrct dcisions about futur invstmnts, or just mak an accurat stimation of th opration costs, w nd a good stimation of nrgy consumption and pak powr at ach lin and substation conncting th to th DC systm. For this rason, larg fforts hav bn don to obtain an accurat, robust, asy to implmnt and computationally light powr flow mthod [,, 8,0]. In [], an unifid AD/DC powr flow mthod to simultanously solv th whol quation systm was proposd. This mthod includs svral faturs such as: matrix basd formulation in dq coordinats, compact form, a tchniqu basd in graph thory to simulat th train motion and a nw procdur to obtain powr flows, losss and injctd/absorbd powrs in all nods. Th main advantag is that th systm topology and dimnsion rmain constant, vn if a nw train coms into stag or xits th systm. Th prsnt work is focusd on th powr flow problm. Th ntwork dscription procdur is xpandd and th us of th graph thory basd mthod applid to RP dscription is xplaind at lngth. Th structur of this papr is as follows. Thortical Background sction prsnts a short thortical background; th obtaining of th matrics dscribing a graph that ar usd in succssiv sctions is xplaind. Railway Powr ystm Associatd Graph sction dfins th RP associatd graph, nods and lins numration critria, and matrics construction considring th and DC subsystms and th trains as static loads (without movmnt). In Train Movmnt and Influnc in ystm Dimnsion and Topology sction th authors xplain how to ovrcom th problm of th topology variation du to th train movmnt. such that: V = {,,, } E = {,,, } = (, ) = (, ) = (, ) = (, ) Fig. a Digraph G = (V, E); b subgraph of G In this sction th trains ar considrd lik dynamic loads that can vary thir position, but thy can also vary in numbr sinc on train can b activatd or dactivatd in th systm. With th us of th graph thory this problm can b solvd kping constant th ntwork dimnsion and topology. In Rsults sction, a cas of study is dscribd and analyzd. Finally, in Applications sction a st of conclusions ar statd. Thortical Background A graph G consists of two finit sts (V, E), whr V is th st of lmnts calld vrtics or nods and E is th st of lmnts calld dgs. ods will b rprsntd as,,...,n and dgs as,,... k. Each dg l is idntifid with a pair of nods (i, j).g is adirctd, orintd graph or digraph if its dgs ar idntifid with ordrd pairs of nods. Othrwis Gisan undirctd or a non orintd graph. In addition, a graph is a simpl graph if it has no slf-loops or multidgs [,9]. Th digraph G = (V, E) is dpictd in Fig. a. Hr in aftr, simpl digraphs will b usd. Whr: V ={,,, } E ={,,, }

3 Intll Ind yst (0) : 9 9 ordr) th matrix with dimnsion (n, n) that rlats th connctions btwn nods is formd as follows [,9]: { numbr of dgs btwn i and j i = j i, j = () numbr of slf-loops in i i = j Fig. Complt bipartit graph V V Fig. Complmntary subgraphs of graph dpictd in Fig. a.a Complmntary subgraph. b Complmntary subgraph such that: = (, ) = (, ) = (, ) = (, ) A subgraph H = (V h, E h ) of a graph G = (V g, E g ),isa graph whr V h and E h ar substs of V g and E g rspctivly [9]. In Fig. b a subgraph of th digraph dpictd in Fig. a is rprsntd. A bipartit graph G = (V, E) is a graph whos vrtics (st V ) can b dividd into two substs V and V, such that vry dg of E has a vrtx in V and othr in V.Ifitxists an dg (i, j) for vry vrtx i of V and vry vrtx j of V, th graph is calld complt bipartit graph, s Fig.. This kind of graphs can b also simpl digraphs and will b vry usful for dscribing th connctions btwn trains and DC nods, as it will b xplaind in furthr sctions. In this papr th authors will us th union of subgraphs to form th complt graph of th systm. Thrfor, bing G = (V, E ) and G = (V, E ) subgraphs of G = (V, E), th union G G calld sum graph, has V V nods and E E dgs. If G G = G, thn G and G ar complmntary subgraphs of G [9], s Fig.. Evry graph can b dfind by its adjacncy ( ) and its incidnc matrix (Ɣ). For a graph G with n nods (graph In this papr, as it was mntiond, all graphs will b simpl digraphs so () can b rformulatd as: { adjacncy btwn i and j i > j i, j = () 0 othr cass ormally, adjacncy matrics ar dfind as symmtric. Howvr, according to (), th matrics ar dfind as uppr triangular, thus th rdundant information storag is avoidd allowing gratr fficincy in th us of spars matrics. In th sam way, th graph G with n nods and k dgs is fully dfind by th (k, n) dimnsion incidnc matrix (Ɣ) [9,], dfind as: i dg incidnt and dirctd from j Ɣ i, j = i dg incidnt and dirctd to j () 0 i dg not incidnt on j Railway Powr ystm Associatd Graph Th RP will b rprsntd by a graph, rplacing ach lmnt by an dg or nod, indpndntly of th lmnt natur. A typical ntwork as shown in Fig., will b dscribd as a digraph, charactrizd by th aformntiond matrics. Th whol RP systm can b dividd into thr diffrnt subsystms: DC subsystm. subsystm. substations. Link subsystm. Each subsystm can b rprsntd by a subgraph of th complt systm graph. Each subsystm has th following nods: DC subsystm: Trains, DC trminals in rctifir substations and cross points in catnaris. Th numbr of nods in this subsystm will b n DC = n t + n DC. Whr n t is th numbr of trains and n DC th numbr of nods for th basic DC topology. subsystm: nods and scondary of powr transformrs ( substation or link nods). Th numbr of nods for this subsystm will b n = nl +n. Whr n L is th numbr of link nods and n is th numbr of ntwork nods. Link subsystm: It dos not hav own nods. It only has vrtics rprsnting th powr convrtrs.

4 0 Intll Ind yst (0) : 9 Fig. Rail powr systm G G Link DC Thrfor, th graph dscribing th whol systm will b a n ordr graph, whr: n = n DC + n () Th (n, n ) adjacncy matrix rprsnting this systm, can b calculatd as: TOT = DC + L + () whr DC, L, ar th DC, th link and th subgraphs adjacncy matrics, rspctivly. Ths matrics hav to b total dimnsion (n, n ), in ordr to comput TOT according to (). DC, L, rprsnt ach subgraph adjacncis, but considring that all nods will appar in ach subgraph. For xampl, th graph dscribd by DC will hav all nods rprsntd in it. Howvr, nithr th or th link nod will prsnt adjacncis. From now on, DC, L and will dnot subgraphs with thir particular dimnsion. Thrfor, DC and will rprsnt th subgraphs with thir subsystm nods. Howvr, th link subsystm dos not hav own nods, so its dimnsion will vary with th numbr of no-train typ DC nods and with all link nods in th subsystm.. od Enumration Critria and ubgraphs Matrics Th critria usd to list nods bgins with th trains and gt on with th rst of nods in DC, thn th link nods, and finally th nods. Thus th vctor with all nods in th systm will b as follows: vn =[ vn DC vn ] () vn DC =[ t s DC ] () vn =[ l s ] (8) whr vn DC and vn ar th nod vctors of DC and subsytm rspctivly; t, s DC, l and s ar th vctors containing th train typ nods, DC topology nods, link nods and powr systm nods. Th systm of Fig. can b dpictd following this critria in Fig.. According to this numration critria and th adjacncy matrix dfinition in (),th dscribing th RP will b a block matrix (s Fig. ) whr lind zons corrspond to non zro valus. Th (n DC, n DC) dimnsion matrix DC will b dfind as: DC = whr: tt (n t,n t ) ts (n t,n DC ) 0 ss (n DC,n DC ) (9) tt is th adjacncy matrix rprsnting connctions btwn trains. ts is th adjacncy matrix rprsnting connctions btwn trains and DC topology nods. ss is th adjacncy matrix rprsnting ral connctions btwn DC nods. Ths matrics will dfin thr DC subsystm subgraphs. If only th train positions in Fig. wr considrd, tt would b a null matrix bcaus thr is no adjacncy btwn

5 Intll Ind yst (0) : 9 Fig. Edg and nod numration critria 0 G 8 G 0 link 8 link 9 link 8 9 DC DC DC DC DC DC n DC n n t n DC n L n n t n DC DC n DC Fig. Whol systm block adjacncy matrix n L link n n trains. All cass must b considrd as it will b xplaind in Train movmnt and influnc in systm dimnsion and topology sction bcaus it may b th cas in which two trains wr in th sam lin, rsulting in an adjacncy btwn thm. To formulat TOT following (), DC will b compltd to total dimnsion: DC = ( DC 0 (n DC 0 (n,n DC ) 0 (n,n ),n ) ) (0) Th formulation of L for any systm is shown in (). As it can b obsrvd this matrix has only non-zro trms in th positions corrsponding to DC topology nods (n DC ) and link nods (n L ). Thn L will b a (n DC, n L ) dimnsion matrix. ) 0 (n t,n L ) 0 (n t,n ) L = 0 (n DC,n DC 0 (n,n DC L 0 (n DC,n ) ) 0 (n,n ) (), with dimnsion (n, n ) dscribing subsystm graph, could b formulatd as follows: = 0 (n L,nL ) traf o 0 (n,n L ) s () whr traf o, with dimnsions (n L, n ), is th adjacncy matrix dscribing transformrs, and s is th adjacncy matrix dscribing lins btwn nods, and its dimnsions (n, n ). Th total dimnsion matrix dscribing th subgraph will b: 0 (n DC =,n DC) 0 (n DC,n ) () 0 (n,n DC) ow Eq. () can b compltd as: TOT = DC + L DC = L ()

6 Intll Ind yst (0) : basd on th nd nod. Thn all th outgoing nod dgs and so on. Thraftr, with sam critria, link dgs and dgs will b numratd. This critria can b obsrvd in Fig., applid to th xampl of Fig.. As it was discussd in prvious sctions, th authors usd digraphs for dscribing th systm, howvr th dg dirction ar not displayd in Fig.. Ths dirctions allow to st a rfrnc dirction for th diffrnt physical variabls involvd on ach dg (i.. currnts or powrs). With th nod and dg numration critria xposd, th vctor containing all systm dgs is as follows: v = [ DC L ] () 0 whr DC, L and ar th vctors containing DC, link and dgs rspctivly. At th sam tim can b dividd in transformr dgs ( traf o ) and lins dgs ( lin ): Fig. systm graph = [ traf o lin] () Th total numbr of dgs (n ) in th graph dscribing th RP will b:. Edg Enumration Critria and Incidnc Matrix Formulation With th sam critria usd for nods, th dgs will b numratd. First th DC subsystm dgs, thn th link dgs and finally th subsystm dgs. Thus starting with th DC subgraph, th dg numration critria starts numbring all outgoing nod dgs following an ascnding ordr n = n DC + n L + n () whr n DC,n and n L ar th numbr of dgs in DC, and link subsystm rspctivly. For th DC subsystm, th numbr of dgs can b computd as th sum of th dgs of th thr subgraphs that form it: n DC = n tt + nts + nss (8) (c) Fig. 8 ystm subgraphs. a DC subsystm subgraph. b subsystm subgraph. c Links subsystm subgraph

7 Intll Ind yst (0) : 9 Fig. 9 Constant dimnsion rail powr systm G G 8 link 9 link 0 link DC DC DC DC DC DC DC Fig. 0 Edg variation xampl whr n tt ar dgs btwn trains, nts dgs btwn trains and DC topology nods and n ss ar dgs corrsponding to DC lins without trains. Th link subsystm dgs (n L ) rprsnt th substation rctifirs. Finally, in th subsystm, th dgs rprsnt lins ) and transformrs (n traf o ). (n lin n = n traf o + n lin (9) From ach subgraph, th incidnc matrix is obtaind from its adjacncy matrix. Thn th Ɣ TOT with dimnsion (n, n ), is computd: Ɣ DC 0 Ɣ TOT = 0 Ɣ L 0 0 Ɣ (0) Bing Ɣ DC of dimnsion (n DC (n L, n DC, n DC + n L ) and Ɣ of dimnsion (n ), ƔL of dimnsion, n ). Ɣ TOT could b obtaind from TOT dirctly, howvr by doing this, a link dg could b numratd bfor a DC dg. Basd on Fig., if th incidnc matrix is dirctly computd from TOT, all dgs outgoing will b first numratd, thn all outgoing and so on. But nod shows an adjacncy with a link nod, so an dg from th link subsystm would b numratd bfor th rst of th DC dgs outgoing nods and. Consquntly in Fig., corrsponding to adjacncy (, ) would b (, ), changing th Ɣ TOT structur substantially. In Fig. th graph rprsnting th whol systm with th trains positiond as in th xampl of Fig. is dpictd. In Fig. 8 th subgraphs rprsnting th thr subsystms ar shown.

8 Intll Ind yst (0) : priod of tim. Th main drawbacks of th traditional solving mthod can b summarizd as: A procdur to dtrmin which trains ar in th systm at ach stp of simulation and its position should b dvlopd. A nw topology must b dfind at ach instant, varying th numbr of nods, thir rlativ positions and th lins conncting thm. An numration critria should b dsignd to idntify ach ntwork lmnt (nod or lin) at ach instant. inc th sam critria is applid to diffrnt instants, diffrnt indxs can b assignd to th sam lmnt at two diffrnt instants. Du to this last point, thr may b changs in th nod and dg vctors, both in dimnsion and componnts, making vry difficult to compar diffrnt instants. Evn if th vctors corrsponding to diffrnt instants had th sam dimnsion, a givn position of th vctor may blong to diffrnt lmnts. Hnc, a tracking subroutin should b ncssary to find an lmnt in th vctor at diffrnt instants. (c) 8 On account of this, th authors propos a partial modification of this procdur to ovrcom th aformntiond difficultis. Fig. t of subgraphs that rprsnts th DC ntwork topology. a ub-graph dscribing all possibl connctions btwn trains (complt typ graph). b ub-graph dscribing connctions btwn trains and DC topology nods (complt bipartit graph). c ub-graph dscribing th ral DC ntwork topology Train Movmnt and Influnc in ystm Dimnsion and Topology Until now, it was xplaind how RP can b rprsntd with a graph, and how this graph is compltly dfind through a st of matrics. Howvr, th RP is not a static ntwork. Th dynamic variation is drivd from th movmnt of loads (trains) in th DC subsystm. In ordr to analyz th DC subsystm proprly, two main problms must b solvd. Th first on has to do with th numbr of trains in th systm at a givn instant. This numbr is not constant and provoks a chang in th problm dimnsion among diffrnt instants. Th scond issu lis on trains movmnt, which producs changs in th rlativ position of nods during th simulation. Traditionally, to ovrcom ths difficultis a nw problm is st out for ach instant. Ths problms do not lad to major issus whn th systm is small and th tim intrval is not too long. Howvr, th disadvantags of th traditional way of solving th problm aris whn a ral systm must b studid during a significant. olution to ystm Dimnsion Variability Th solution to kp constant th dimnsion of th systm is to considr all th trains apparing in th tmporal intrval of study to b rprsntd in th graph, rgardlss of whthr thy ar physically or not in th systm. Throughout this sction, th authors will work with a thr trains cas, as shown in Fig. 9. Th numration critria is th sam xplaind in od numration critria and subgraphs matrics sction. For systm dpictd in Fig. 9, n t is, thn DC topology nods will bgin with indx.. olution to ystm Topology Variability With th nod numration critria abov dscribd a particular nod will always rprsnt th sam train or th sam nod rgardlss th instant. Howvr, th train movmnt continus causing changs in th systm topology. Assuming for instanc on instant in th systm dpictd in Fig. 9, in which only train is locatd btwn nods and, with th dg numration critria xplaind in Edg numration critria and incidnc matrix formulation sction, th dgs will b = (, ), = (, ), = (, ), = (, ) and = (, ) (graph shown in Fig. 0a). upposing that th train movs toward nod, thr will b an instant whn th train will b situatd btwn nods and,

9 Intll Ind yst (0) : (c) (d) Fig. DC ubsystm graphs. a DC subsystm graph. b ubgraph tt. c ubgraph ts. d ubgraph ss thrfor = (, ) and = (, ). Also, for this scond instant if a nw train (train ) appars btwn nods and, two nw dgs = (, ) and = (, ) will mrg. Th dg btwn nod and would bcom = (, ) and will connct nods and ( = (, )), (s Fig. 0b). Thrfor, th influnc of train movmnt in th dimnsion of th dgs vctor and th chang in th adjacncy rprsntd by th sam dg i in two diffrnt instants is provd. Th dynamic variation of th systm only affcts th DC subsystm. In ordr to construct an invariant dimnsion systm, w considr that all trains ar connctd among thm and with all DC topology nods, thus covring all th possibilitis. Thraftr, only dgs rprsnting trains that ar physically in th systm will b activatd. o from now on, DC and Ɣ DC construction will tak into account all possibl connctions. Th graph rprsnting th whol DC systm will b constructd considring all possibl connctions. It is comprisd of thr subgraphs, as it is rprsntd in Fig. : ubgraph dscribing all possibl connctions btwn trains, Fig. a. It must b noticd that th cas whr all trains ar running btwn two DC nods can xist. To covr all possibilitis, this graph is a complt typ graph; that is, a simpl on in which vry pair of distinct nods ar connctd by a uniqu dg [9]. ubgraph dscribing connctions btwn trains and DC topology nods, Fig. b. Evry train can b connctd to vry DC topology nod at diffrnt simulation stps. In this cas th subgraph symbolizing ths connctions is a complt bipartit graph btwn trains and DC nods.

10 Intll Ind yst (0) : 9 Tabl DC subsytm dgs ubsystm Adjacncy Edg Adjacncy Edg DC subsystm (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) 8 (, ) 8 (, ) 9 (, ) 9 (, ) 0 ubgraph dscribing th ral DC ntwork topology, Fig. c. It rprsnts th ral topology of th DC systm without trains. Th adjacncy matrix corrsponding to th DC subsystm can b calculatd as th sum of th abovmntiond subgraphs adjacncy matrics. DC = tt + ts + ss ( tt (n = t,n t ) 0 ) ( ) ss (n DC,n DC ) tt (n t,n t ) ts (n = t,n DC ) 0 ss (n DC ( 0 ts (n t,n DC ) 0 0,n DC ) ) () Th numbr of dgs of DC subsystm can b calculatd as: n DC = n tt + nts + nss () n tt = n t(n t ) n ts () = n t n DC () Th numbr of dgs in th third subgraph (n ss ) only dpnds on th numbr of ral lins conncting DC topology nods. Th Eq. () rprsnts th numbr of dgs of a complt graph of n t nods. Equation () rprsnts th numbr of dgs in a complt bipartit graph of n t nods in V and n DC nods in V,[9]. Onc th DC subsystm is computd considring all possibl connctions, th invariant part of RP (links and subsystms) is addd. In Fig. th graph rprsnting th whol DC subsystm and th thr subgraphs ar rprsntd. In Tabls and th dgs ar listd for ach subsystm. Tabl Links and subsytm dgs ubsystm Adjacncy Edg Links subsystm (, 8) 0 (, 9) (, 0) subsystm (8, ) Tabl Train position Train Location Instant t (9, ) (0, ) (, ) (, ) (, ) 8 (, ) 9 (, ) 0 (, ) Instant t + t od Distanc (%) od Distanc (%) (, ) 90 (, ) 0 (, ) 0 (, ) 0 (, ) 0 (, ) 80 Fig. DC subsytm schm for instants dfind in Tabl. a Instant t. b Instant t + t Rsults For th xampl dpictd in Fig. 9 two diffrnt instants will b studid. Th trains ar locatd as dscribd in Tabl, whr th column Distanc rprsnts th train position as a prcntag of th total lin lngth. Th DC subsystm schms for th considrd positions ar dpictd in Fig..

11 Intll Ind yst (0) : 9 Fig. DC subsystm graphs for instants dfind in tabl. a Instant t. b Instant t + t Comparing both schms, it can b obsrvd that dg (, ) is invariant. Edgs (, ), (, ) and (, ) also appar in both instants. Figur shows th graphs corrsponding to th two instants. Th dgs xisting for ach cas ar dpictd. It can b obsrvd that th dg rprsnting th adjacncy (, ), is 9 in both cass. Furthrmor, th adjacncis apparing in both cass ((, ), (, ) and (, )) ar rprsntd by th sam dgs (, and ). Tabl shows th activ and non activ dgs at ach instant and Fig. shows th TOT and Ɣ TOT for th xampl. Applications On of th possibl applications of ths thory is th fast formulation of all Kirchhoff s currnt and voltag laws in al lins and nods of a railway systm. For instanc, by mans of th Ɣ matrix, all currnt and voltag Kirchhoff laws can b xprssd in a compact form as follows [,]: g(z) = Mz T = 0 () whr z is th vctor rprsnting voltag and currnt magnituds. And M can b xpandd as follows. Ɣ DC and Ɣ rprsnt th DC and subsystms topology rspctivly. R DC B is th branch rsistanc matrix of th DC subsystm. RB and XB ar th rsistanc and ractanc matrics rspctivly, rprsnting th impdanc btwn nods. I idntity matrix. And is a block diagonal matrix. Th first Tabl Activ DC subsystm dgs in instants dpictd in Fig. ubsystm Instant t Instant t + t Adjacncy Edg Adjacncy Edg DC ubsystm (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) 8 (, ) 8 (, ) 9 (, ) 9 (, ) 0 (, ) 0 (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) 8 (, ) 8 (, ) 9 (, ) 9 block is an idntity matrix, th scond block is a diagonal matrix dnotd as L (n s,n s ). Elmnt s ii blonging to L is if th DC substation i is connctd to th ntwork and s ii is 0 whn th DC substation i is not connctd to th grid.

12 8 Intll Ind yst (0) : 9 Fig. Matrics for schm Fig. 9. Whr+ symbol implis and opn circl symbol.a TOT. b Ɣ TOT Γ DC Λ DC Λ L 0 Γ L 0 Λ 0 Γ M = R DC B RB XB XB RB (Ɣ DC ) T (Ɣ ) T I Ɣ DC Ɣ Ɣ () (Ɣ ) T Conclusions Rfrncs As it was dmonstratd, with th proposd mthod, a complx RP can b asily charactrisd and dscribd vn whn it has multipl loads that chang thir rlativ position with th tim. By mans of th dscribd tchniqu th Kirchhoff Currnt laws and th Kirchhoff voltag laws can b formulatd in a xtrmly asy compact form. Th proposd mthod has bn dvlopd for a railway traction systm, but it could b also applid to any powr systm, bing spcially usful for thos systms in which loads and gnrators vary thir rlativ positions. Anothr application might b a systm with combination of and DC systms as for xampl High Voltag Dirct Currnt (HVDC) systms.. Andri, H., Chicco, G.: Idntification of th radial configurations xtractd from th wakly mshd structurs of lctrical distribution systms. IEEE Trans. Circuits yst. I: Rgul. Pap. (), 9 8 (008). I9-88. Arbolya, P., Diaz, G., Coto, M.: Unifid ac/dc powr flow for traction systms: a nw concpt. IEEE Trans. Vh. Tchnol. (), 0 (0). Abrahamsson, L., odr, L.: Fast stimation of rlations btwn aggrgatd train powr systm data and traffic prformanc. IEEE Trans. Vh. Tchnol. 0(), 9 (0). I Brg, C.: Thori ds graphs t ss applications. Dunod, Paris (9). Braunagl, D.A., Kraft, L.A., Whysong, J.L.: Inclusion of dc convrtr and transmission quations dirctly in a nwton powr flow. IEEE Trans. Powr Appar. yst. 9(), 88 (9). I008-90

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