hier. decomp. dc power 1985

8/12/2019 Hier. Decomp. DC Power 1985

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J.A. StarrykElectrical & Comprter Eng. Dept.Ohio UniversityAthens, OH 45701

ABSTRACTThis paper discusses an algorithm to solve thed.c. lperer flor* problern. I; this rnethod largeporier systens are hierarchically decomposed into

ryull manageable subnetvprks (blocks) using nodedecomposition. These blocks are solved sepa-ratelyand are interconnected to obtain the solulion forthe subnetrrorks on the higher level of hierarchv.Finally the solution for the original netrrorf isobtained after all hierarchy has been rcrkedthrough.I. INTrcU'CTIChI

Power flow solutions of a large system havealways been l imited because of its menorv an dcanputational requirernents. To achieve toad fbwsolution with linited conputational resources,following aspect of the problon were considered:(1 ) Sparsity of the adnittance matrix ofthe por*er system.(2) Decomposition (tearing) of porrer systemsnetrrprk in gnaller subnetr,uorks (blocks)to suit computational requirenent.(3 ) Utilization of iocal effects undercontingency condition.Sparsity of the adnittance matrix of powerqgstem nethrcrk was exploited by storing noizeroelements only and thus avoiOing ieros inmathematical operations [l]. tater netuork tearing[deconrposition] Ied to simplicity, butsimultaneous analysis of decrcnposed- ubnet.ork wasnot possible l?l . A heirarchical deconrrcsitionapproach d iscussed in t3 l cou ld ana lyzL la rgepo$rer system netrork effectively by solvingdecomposed subnetwcrks sinultaneouslf. Thiaapproach r,roUld fwther be vialte whencontingencies are sinnrlated dr-ring operationar andplanning str-rdies of the pobrer systern.In this paper heirarchical decornmsitionapproach has been illustrated to solve d.c. loadflow problem and further extended to solvecontingency load flow problem. This extention*Fornerly with the Electrical andlgin"9 1lg Depa.rtnenr, Ohio University,ohio 4570r

HIERARCHICALDECOTVIPOSITIoIiPPRoACH o D.c. PovilERFIOvf SOLLrIIOI{

ComputerAthens,

S.C. Rastogi*-CorlDrate Planningltbstern Farners Electric CooperativeAnadarko, OK 73005

uses the fact (3) rentioned above and thereforeonly 1oca11y affected subnetworks are solved andsolutions of the other subnetrrorks are retainedfrom the base case - result ing in computionalsavings.f n the follor.ring sections of the paper,firstly brief description of the hierarchicaldecorposition of netvork is discussed. Then loadflow eqr:ations and their solution is described.Iater test result of the algorithm is discussed.

rT. HTERARCHICALECOMPOSITIOIVMatrix decomposition is based on its Coatssignal flow graph representation 1,41 in which asguare rnatrix A=[arr] is represented by a graph

with n nodes and k edges, where k is the nurnber ofnonzero coefficients in A. In a Coates grraph edgedirected from node xi to node Fifs- a-weightequal to r31 Oats glpfr of a oeificient natrix 1 ' l - l-t 2 o-JA- 1 3 4 s l[eo lis sholrtn n Fig. 1.

Fig. I Coates graph of the rnatrix A of Example l.A signal flow graph is deconposed through itsnodes into tvo sr.rbgnaphs ( subnetvorks ) . Each of


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these sr:bgraphs can be decomtrnsed further down toa sufficiently snall size. This kind of graphdecomposition is called hierarchical decomposi-tion. The structure of hierarchical deconpositioncan be illustrated by a decomposition tree. Nodesof the tree correslnnd to subgraphs obtained ondifferent levels of decomposition. If a subgraphaj vras obtalnE- during decunposition of sulcgraphG., then there is an edge from the node1corresponding to G, to the node correspondinga "j. Fig. 3 shrows he decomposition treecorrestrnnding to Fig. 2.

Level of decomposi l ion ,fi rs l

Fig. 3 Deconposition tree for Fig. 2.Ittatrix Reorderings After all sr:bgraphs haveueenffig to the structure of thedecomposit ion tree, the nodes of a graph ar erenrsrbered consecutively in descending orderstarting from the partition nodes of the graph G.,then the partition nodes of graph G,, and G. up to

the last partition and then the int'ernal n6des ofthe proper blocks. After the renurrloering, thenrlrbers associated with the graph nodes are calledoriginal indices of the nodes. Such renr-nberingcorreslnnds to reordering the coefficient rnatrix.An exanple for the matrix partitioned according toF ig . 2 i s shown i n F ig . 4 . The re is a s t r i c tcorrespondence betvpen the set of partition nodesof a middle graph and subrnatrices of the reorderedcoefficient matrix. These submatrices are ca1ledinterconnection matrices as they representinterconnection of trro subwstems.J r sJ t r, J t t.J toJ eJ 6J 5J 7.,1 5J 4J 3J 2, J g

Fig. 4 ldonzero pattern of a reordered coefficientmatr ix.III. PO9IER I.OWEqJAf,IONTS

The power flow equations of a pCI,'rer systemunder steady state condition are obtained byequating the power injected into each node bysource or load and pouer transnitted frorn the nodevia transnission netrrcrk. Thus at ith node,

Gl@second

th i rd

Fig. 2 Three level hierarchical deconposition.fn the deccrnposition Lree vre have one initialnode - the one which is the start ing poff iTFges. Termi.nal nodes are those which -are onlythe end points of edges. A11 nodes that are notterminal nodes are middle 1g]es. Subgrraphsassociated with terminET-iEEili6 called oroo"tblocks. [ limit ourselves to bisection a?ti66ffi-graph partition so that every middle node hasexactly tr,vrcdescendants. If m is the index of amiddle subgraph then tvp of its descendants haveindices 2m and 2rftt' l, respectively. This way ofnr-rr{rering the graphs rnakes the analysis ofinterconnections easier.


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G i k ' B i k

Neix

Pi = Pci*Pt i*Pt i{ i (c i* cosern+Brn sin9ik) (1.a)Qt = 0at+Qli{ t i=vi(Ct* sineiX-Bin cosotn) (f .b )wherePGi, PlirPTi real [Dwers injected into the it hnode by the generator, Ioad andties connected to it ,Pr net real [Dhrer injected into the' ith node,QCi'Qf,i, Qti reactive po\Ners injected into ttreith node by the generator, load andties connected to it,Qj net reactive [pwer injected into^ the ith node, .V.,lei ccmplex voltage at the ith node,

v'lhere H, N, J, and L constitute the Jacobian of(1) ,ae and AV are vectors of corrections to theestimates for node rrcltages and magnitudes andAPrA0 are the mignatches between the respectivefDwers injected into and reroved from the nodes.In practice N and J are much snaller than H and L.Therefore (4 ) rmy be approxinated by,faevl =FlFq -(s)a n d : . 6oru1 = l-ti[ l (6Where B' ?nd -B" ar5'Eon'stant matrices. Th esolution of a.c. load flow equations is obtainedby iterating (5) and (5) until misnatch satisfyspecified tolerance by the user. This nethod iscalled decoupled load floqr method.rv. soLuTIoN oF N LINEAR EpUATIONIS

The set of n linear equations,A Y = b ( 7 )can be so lved e f fect ive ly , i f A is a sparsematrix, by using bifactorization nethod t5l. Theinverse of A can be expressed by a m u 1 t i p I eproduct of 2n factor matrices.e l = R ( l ) R ( 2 ) . . * ( n ) r ( n ) . L ( 2 ) L ( 1 ) ( 8 )fn order to find L and R the following sequence ofinternrediate matrices is introduced:e ( o ) = a: , t , = L ( l ) A ( o

R ( 1 )

a ( j ) = L ( j ) A ( j - r ) R ( j )a a

o ( n ) - " ( n ) o ( n - I ) * ( n )- ,

Where reduced matrix a(j ) has elenents defined bvthe following egtntions:" ( j ) = I ; " ( j ) = a ( j ) = oj j i j j k

" (j - r ) a ( - l )i . i j ku (j )=. ( - l ) : ,________ik ik " ( j - r

)j j

index and for i, k =trices r-( ) "t. very sparsety matrix in colurmn j only.

real and imaginary parts of Yik,the elenents in ith rovr & jthcolr-srrr of the netvork a&nittancematrix,nuunber of nodes in the netrrork,= e i - 9 k 'It is interesting to note that, for the Rgvrersystems, Gik are gnall, eik rarely exceeds 20" and

the nodeAus voltages rarely deviate by nore than108 from their noninal values.If we make approximations GifiOr sinOrp9i*rand cose in : l equat ions ( f .b ) reduces to thedecoupled reactive pohr flow eguationsNoi= - Z,"ix vi vrK= lFurther, in norrnal opreration the per unit voltagenagnitudes V. are close to 1. Therefore,approximatinQ V.,= 1, equation (1.a) beconesthe d.c. load flow equation.

NPi = *Z=r.tt*(ei- ek)If the resistance is nu:ch gnaller than thereactance thenNPi = ?iOr* *(ei-ok)*t.t.ntninis in marrix?3T,l ' ; l :r,* r l" i ll . - l = l r l * ix l l . l e )l . I L J l . IL".i L"^l

This eguation can beresulting power flow onbysolved forthe line can e and thebe obtained

n r l

j being the pirotal1 i + l ) 7 . . . . ; D .The left-hand factor maand differ from the uni

I'IL ( j ) = |II

{ t rcre t( j ) = t/ a( :1 )

1- 1 ( )i , j. 11 ( j ) 1

Pik = lA ix * (e t - eo) (3)andandas

a j j j ja n d l ( j ) = _ " ( j - r ) / a ( j - r ) ,l 1 u) )i = ( j + l ) r . . . 7 o


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The rightsparse andonly.

R( ) =

fr,and factonoatrices n( j ) arediffer from the unitv matrix W. TEST EXAMPLEThe data below is a listing of input data tothe six bus sample pot,rer system used to test therethod discussed in the paper.

IIO. OF BUSES= 5 NO. OF LINES 11NO. OF GEIiIS.= 3 SWI}re BUS NO. I

LINE DAf,AF R O I 4 T O R X B C A PL 2 0 .1000 0 .2000 0 .0200r 4 0 .0500 0 .2000 0 .0200I 5 0 .0000 0 .3000 0 .03002 3 0 . 5 0 0 0 0 . 1 5 0 0 0 . 0 3 0 02 4 0 . 0 5 0 0 0 . 1 0 0 0 0 . 0 1 0 02 5 0 . 1 0 0 0 0 . 3 0 0 0 0 . 0 2 0 02 6 0 .0700 0 .2000 0 .02503 5 0 .1200 0 .2600 0 .02503 6 0 .0200 0 .1000 0 .01004 5 0 .2000 0 .4000 0 .04005 5 0 .1000 0 .3000 0 .0300BUS DTTABUS NO. GEN PU-MtiI VOLTAGE P IOAD Q IOAD

I 0 . 0 0 1 . 0 5 0 0 . 0 0 0 . 0 02 1 . 0 0 r . 0 5 0 0 . 0 0 0 . 0 03 1 . 0 0 1 . 0 7 0 0 . 0 0 0 . 0 04 0 . 0 0 1 . 0 0 0 1 . 0 0 1 . 0 05 , 0 . 0 0 1 . 0 0 0 1 . 0 0 1 . 0 06 0 . 0 0 1 . 0 0 0 1 . 0 0 1 . 0 0The system description and test results of theabove exarple are shor*n in Fig. 5. Po*er flows

0u3

also veryin row j

r , ( j ) . r ( j )i ' i+t j 'n1 .

sub-cor-beV. HIERARCHICALDrcO'{POSITION AIfORITTIM

Following are the steps to solve the po$rerflow equation.1. Decomlnse the por.rer qlstem netrporkhierarchically.2. Renr-urber the nodes.3. Form the tlAifl rnatrix for d.c. solution or

B' and B" for a.c. solution.4. Solve equation (2) for d.c. solution, orequations (5) and (6) for a.c. solution usingrethod described in Section IV.5. Obtain d.c. 6Dvrer flow by using equation(3) and for a.c, solution go to step 6.6. Check power misnatch. ff within specifiedtolerance limit then determine pCInr flowelse update egr:ations (5) and (6) and go tosteP 4.toouwLOAO

5 six bus neLlork base case d.c. load flow

I

Y =Let

v t h e r e , ( j ) = _ u ( j - r ) 6 ( i - r ) k = ( j + 1 ) 7 o o o l Dj k j k j jFor symetric natrix Arstructures of left (L) andright (n) factor matrices are the sane and cantherefore be stored effectively in the matrix F asfollows, K I" =l l I L r , " lSolution to equation (7) can be written asR ( 1 ) R ( 2 . . . * ( n ) , ( n ) . . . r ( 2 ) L ( I ) b ( 9

I,{lrere, individual values of 2., z?r. .znt can bestorecl. If the coefficients in- only' one properblock have been altered ( for example in G., ofFig.3 ) then according to above fonrnrla and= thenonzero pattern of Fig. 4, we have to calculatenew values of z.r t, zt r 21 and z.r to obtain thenew vaLue of z. '= Frofn e{,rations'(9) and (10) 'i ehave,

( 1 0 )["rl, = l?l= , tn) . . . t (2)( r )t ; lL r

lvrly = I t l | = n ( I )R(2 ) . . . * ( . )l . ly"lIf qle are interested in the solution for anynatrix, then only those elernents of y whichresponds to suhnatrix specified are tocalculated.

+ 3 6 . 0 + 6 . 6


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thus obtained are satisfactory and encouraging.Allhough CPU t.ime for the present example was notrecorded and therefore, can not be compared,trcwever, it is expected that for larger networksand for contingency cases evaluation, thepresented algorithm will sfpw substantialcornputational savings.\rII. CONCLUSION

A hierarchical deconposition approach forsolving a d.c. load flow for a large poriter systemhas been presented . F i rs t rana lys is o f p roperblocks is perfonred and then subnetworks arecombined in a h ie rarch ica l manner jo in ing twosubnetworks at any tine. Finally, solution for theoriginal netr.ork is obtained after all hierarchyhas been vorked through.The algorithm discussed was tested on severalnnderate size power systems. The results obtainedvrere encouraging, sfowing feasibility of thisapproach in pori,er qgstern analysis. Similarapproach can be developed for E.c. porder flowsolution as well .Such an algorithm is suitable for large po'.rsystenrs, since ccrnputational time can be reducedby solving the subnetworks in parallel. The rostimportant applicaLion of this approach is inevaluation of contingency cases reguired inplanning and operational studies of the powersystem. Since, dr.rring a contingenry there is achange in coeff icients of a proper block an dcomplete analysis of the whole system is no trequired and thus is gained further savings incomputational effort.

VIII. REFERENCESN. Sato and W. F. Tinney, "Techniques forexploiting the Sparsity of Network AdmittanceMatrix", IEEE Trans. Por*er Apparatus andSysterns, Voh-une82, pp. 944-950, 1963.H. H. Happ, "Multi-level Tearing andApplications', IEEE Tfans. PohrerApparatus and Systerns, Voh-rne 92, W. 725-733,1973 .H. Gupta, J. W. Bandler, J.A. Starryk, J.Sharrna, "A Hierarchical oeconpositionApproach For |det\.Drk Analysis", koc. ISCAS.Rdne, Italy, May 1982.w. K. Chen, Applied Graph Theorv - Graphs andElectrical Netcorks, lbrtlrFlclland, london,mK. Zollenkopf, "Bifactorization - BasicCcrnputational Algorithm and HrogrrarmingTechnigues", in large Sparse Sets of LinearEquations. J. K. Reid, Editor, Acadernic Press- New York, pp. 75-96, L97L.

6 . J. A. Star4lk and J. W. Bandler, CSELE - AFortran Pac for the Solutions of SElectr icalUniversity,

I .

z .

3 .

A

5 .

Hamilton, Canada, 1983.

hier. decomp. dc power 1985

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