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ARTICLE IN PRESSG ModelEPSR-3456; No. of Pages 5Electric Power Systems Research xxx (2012) xxx xxx

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Electric Power Systems Research

jou rn al h om epage: www.elsev ier .co

Short communication

On Kro

Fabian MThe Center for 8757,

a r t i c l

Article history:Received 27 SeReceived in reAccepted 25 JaAvailable onlin

Keywords:MeshNodePartitioningParallelPowerSimulationDiakopticsKronMulticoreSystem

f teaverloorks;ulatiludedints tg no

1. Introduction

Kron [1,works formsubsystems

Aorigx = b

A1 A2

DT1 DT2

Aorig = originx = networkb = excitatioi; xi = netwvector of suis internalpartitions; vector; u = b

Tel.: +1 51E-mail add

Expressing (2) in compound matrix form results in (3), where

0378-7796/$ doi:10.1016/j. this article in press as: F.M. Uriarte, On Krons diakoptics, Electr. Power Syst. Res. (2012), doi:10.1016/j.epsr.2012.01.016

2] derived from physical principles that electric net-ulated in the form of (1) could be partitioned into p

and reformulated as (2):

(1)

D1 D2. . .

... Ap Dp

DTp Q

x1x2...xpu

=

b1b2...

bp0

(2)

al (unpartitioned) network coefcient matrix; variables vector (node voltages or mesh currents);n vector; Ai = network coefcient matrix for subsystemork variables vector for subsystem i; bi = excitationbsystem i; Di = connection matrix linking subsystem

variables to its boundary variables; p = number ofQ = boundary immittance matrix; 0 = zero matrix oroundary variable vector (branch voltages or currents).

2 232 8079; fax: +1 512 471 0781.ress: f.uriarte@cem.utexas.edu

each compound matrix is expanded in (4) and (5). In (5), thesubscript r represents the number of boundary variables that areformed when partitioning the original network into p partitions.[Ablock DDT Q

] [xu

]=[b0

](3)

Ablock=

A1A2

. . .Ap

; x=

x1x2...xp

; b=

b1b2...

bp

; D=

D1D2...

Dp

(4)

u =

u1u2...

ur

; Q =

Q1Q2

. . .Qr

(5)

D(i, j) { = 1, if xi is positively coupled to uj

= 1, if xi is negatively coupled to uj= 0, if xi is not coupled to uj.

Expanding the rows of (3) results in

x = A1blockb A1blockDu (6)

Qu = DTx. (7)

see front matter 2012 Elsevier B.V. All rights reserved.epsr.2012.01.016ns diakoptics

. Uriarte

Electromechanics of The University of Texas at Austin, 10100 Burnet Road, Austin, TX 7

e i n f o

ptember 2011vised form 25 January 2012nuary 2012e xxx

a b s t r a c t

Diakoptics is a well-known method otems. This paper exposes two often-obranches are not required to tear netwdependent on the power system forming software development. It is concnodes) offers more disconnection poin less boundary variables than tearinboundary network.m/locate /epsr

United States

ring electric networks into computationally smaller subsys-oked, important properties related to diakoptics. One is that

the other is that the order of the boundary network is stronglyon variablea choice commonly made too prematurely dur-

that, rst, tearing zero-immittance branches (meshes andhan branch tearing; second, that tearing meshes can resultdes, and, hence, reduce the computation effort of solving the

2012 Elsevier B.V. All rights reserved.

Please cite r Syst. Res. (2012), doi:10.1016/j.epsr.2012.01.016

ARTICLE IN PRESSG ModelEPSR-3456; No. of Pages 52 F.M. Uriarte / Electric Power Systems Research xxx (2012) xxx xxx

Substituting (rst) x from (6) into (7) and (then) u from (7) backinto (6) results in (8). (Eq. (8) is also known in mathematics asWoodburys method for inverting modied matrices [35].) Substi-tuting the expanded matrices of (4) and (5) in (8) produces (9) and(10), whichby Kron.

x = A1blockb

x =

x1x2...xp

u = (DTA

In the cocomplex po(10) at eacdue to the solution tecsuch proble

This secof solution.important p

2. First pro

In relatiothe boundawas apparethe immittaing a netwo

Two sitrst is whevariables. Iculate arouwhenanalages as varijunctions oit constitutQ = 0. Hencmesh or no

Substituthe derivatisolution fordifference bin the solut[Ablock D

DT 0

x =

x1x2...xp

u =

The adva

iab

F

iakop is nero-iero-ire: e-phices mallyems

isadvd out

-imile th-zero

fairles inesh

nter- intu

tota occodalte nundrix srse o

not d oute sens wiles a

esh f

sides of ub1, ian pois can

vbc ma

block =

im

bc ]T

trices mesh resistance matrices, mesh current vectors, and EMFs in subsystems 1 and 2, respectively. After forming D, whichds on a networks topology, the prior matrices and vectors

substituted in (12) and (13) to obtain a partitioned-networkn. As noted, boundary branches do not exist in Fig. 1: thises the condition Q = 0. this article in press as: F.M. Uriarte, On Krons diakoptics, Electr. Powe

is the diakoptical solution to (1) as originally proposed

A1blockD

u (DTA1blockD Q)

1(DTA1blockb) (8)

=

A11 b1A12 b2

...A1p bp

A11 D1A12 D2

...A1p Dp

u (9)

1blockD Q)

1(DTA1blockb) (10)

ntext of electromagnetic transient simulations of large,wer systems [6,7], the sequential solution of (9) andh time step is more efcient than the solution of (1)parallelization opportunities offered by (9) [3,8]. Thishnique has seen application to power systems sincems were solved longhand.tion summarized Krons diakoptics and its equations

The following sections expose two often-overlooked,roperties related to diakoptics.

perty: zero-immittance tearing

n to (3), this section shows that it is not necessary forry immittance matrix Q to exist, i.e., Q = 0. (This propertyntly not recognized by Kron.) The matrix Q representsnce (if any) of all boundaries that form when partition-rk, where branch tearing is only a special case.uations where Q = 0 occurs are presented next. Then power systems are formulated in mesh currents asn this case, it can be said that its mesh currents cir-nd open spaces of zero conductance. The second isogouslypower systems are formulated in node volt-ables, where it can be said that nodes appear as galvanicf zero impedance. If said meshes or nodes are bisected,es a tearing of zero-immittance branches which makeseforth, zero-immittance tearing may be referred to asde tearing, or as topological tearing.ting Q = 0 in (3) results in (11), where after followingons from (6) and (7) through (9) and (10), results in them shown as (12) and (13) instead. Apparently the onlyetween solution sets (9) and (10) and (12) and (13) ision of ubut there are more.] [

x

u

]=[b

0

](11)

=

A11 b1A12 b2

...A1p bp

A11 D1A12 D2

...A1p Dp

u (12)

(DTA1blockD)

1(DTA1blockb) (13)

ntages of zero-immittance tearing are:

1) In dThisin z

2) In zwhethrechonorsyst

The dpointe

1) ZeroWhnonthistancin mcouand

2) TheThisin ncreaa bomatspa

It istearingis morsystemexamp

2.1. M

Consourceinto ianectiosourcevab and

Theare A

[ imesh1[ vab v

Maare thevectordepencan besolutioproducibc

ig. 1. Original oating unpartitioned electrical network.

tics, Q exists only if boundary branches can be dened.ot always possible and is a limitation that does not existmmittance tearing.mmittance tearing, a network can be partitioned any-inside or outside power apparatus, or at single- orase buses. This exibility increments the number ofavailable to graph-theoretic algorithms [9], which are

used to search for the disconnection points of power.

antages of zero-immittance tearing should also be:

mittance tearing retains the non-zero structure of Aorig.is is not detrimental, it is preferable to decrease the

count when using sparse solvers [10]. Diakoptics doesy wellbut only if (intentionally) tearing mutual induc-

nodal formulations [11], or shunt-impedances at buses formulations [12]. The former is a rather obscure andintuitive approach to tearing; the latter, is self-evidentitive [7].l system order increases for every zero-immittance tear.urs in mesh formulations due to mesh bisections, and

formulations due to node bisections. The bisectionsew variables internal to each subsystem interfaced atary. This consideration is only important if using full-olution techniques, which are rarely preferable overnes.

ifcult to show that the advantages of zero-immittanceweigh its disadvantages as computational performancesitive to good network segregations (i.e., balanced sub-th minimal tears) than it is to a few matrix ll-ins. Twore provided next to illustrate zero-immittance tearing.

ormulation example

r the ungrounded network in Fig. 1. Placing two voltagenknown value across the open spaces bisects iab and ibcb2 and ibc1, ibc2, respectively. Fig. 2 shows the discon-nts after the bisection. From circuit theory, the voltage

be torn as shown in Fig. 3, as the boundary variablesare common to both subsystems.trices and vectors for the torn network of Fig. 3

diag(A1, A2) = diag(Rmesh1, Rmesh2), x = [ x1 x2 ]T =esh2 ]

T, imesh1 = [ iab1 ibc1 ]T, imesh2 = [ iab2 ibc2 ]T, u =, and b = [b1 b2 ]T = [ emesh1 emesh2 ]T.

Rmesh1, Rmesh2, vectors imesh1, imesh2 and emesh1, emesh2

Please cite

ARTICLE IN PRESSG ModelEPSR-3456; No. of Pages 5F.M. Uriarte / Electric Power Systems Research xxx (2012) xxx xxx 3

Add unknown

voltage sources

iab1

ibc1

iab2

ibc2

vab

vbc

Fig. 2. Bisection of open circuits from adding voltages sources forms a boundary or disconnection point.

Subsystem 1 Subsystem 2iab1 iab2

i

vab

Tear voltage sources

havin

2.2. Nodal f

Considerof unknownand vc, intoFig. 5 showFrom circuias boundary[1318].

The mFig. 6 ar[ x1 x2 ]

T =vnodal2 = [ v[ inodal1 inovnodal2 and ibc1 vbc

Fig. 3. Tearing the voltage sources creates two subsystems this article in press as: F.M. Uriarte, On Krons diakoptics, Electr. Power Syst

Fig. 4. Original grounded unpartitioned electri

ormulation example

the grounded network in Fig. 4. Placing current sources values in line with the galvanic junctions bisects va, vb,

va1 and va2, vb1 and vb2, and vc1 and vc2, respectively.s the current sources added to the disconnection point.t theory, current sources can be torn as shown in Fig. 6,

variables ia, ib, and ic are common to both subsystems

atrices and vectors for the torn network ofe Ablock = diag(A1, A2) = diag(Gnodal1, Gnodal2), x =

[ vnodal1 vnodal2 ]T, vnodal1 = [ va1 vb1 vc1 ],

a2 vb2 vc2 ], u = [ ia ib ic ]T,and b = [b1 b2 ]T =dal1 ]

T. Matrices Gnodal1, Gnodal2, and vectors vnodal1,inodal1, inodal2 are the nodal conductance matrices, node

voltage vec2, respectiv

Similar vectors afopartitionedexist in Fig.

3. Second

A boundsubsystemsthese bounsolveyet atem.

Fig. 5. Addition of unknown current sources between short circuits formbc2

g common boundary variables.. Res. (2012), doi:10.1016/j.epsr.2012.01.016

cal network.

tors, and current injection vectors for subsystems 1 andely.to the mesh case, after forming D, the matrices andre can be substituted in (12) and (13) to obtain a-network solution. As noted, boundary branches do not

4. This also produces the condition Q = 0.

property: formulation choice

ary or disconnection point is where two or more interface. Determining the number and location ofdaries to produce p partitions is a difcult problem to

solution is required prior to partitioning a power sys-

s a boundary or disconnection point.

Please cite

ARTICLE IN PRESSG ModelEPSR-3456; No. of Pages 54 F.M. Uriarte / Electric Power Systems Research xxx (2012) xxx xxx

havin

[A1]

5]

No

y variables r depends on the formulation choice.

The numboundary vthe computa formulatioas p increasis unknownsoftware mwithout shnarios.

Consideconnectiondisconnectisame boundon the left ilated in mesis the samewhich resudifferent.

The valuthe (1) freqthe forwardof O(r2) eacif p is to scal

As a supformulationdifcult to cby Table 1. Iary variablepower systrgndmesh

is give

formulationin (16).

Comparilations can

isconnection points for an arbitrary partitioning case of p = 10 partitions.

nectioni (di)

Number ofconductors at di (Ni)

Number of subsystemsinterfaced at di (si)

3 53 103 2Fig. 6. Tearing the current sources creates two subsystems

i1

[A3]

[A4]

[A2]

[A1]

[A5]

i 2

i3

i4i5 u1

[A

p = 5r = 1

Mesh Formulation

Fig. 7. A partitioning situation that shows how the number of boundar

ber of boundary variables r (i.e., the total number ofoltage or current sources introduced) adversely affectsation time of u in (13); hence, it is important to choosen variable (nodes or meshes) such that r remains smalles past two. It appears that the importance of this choiceor deliberately ignoredin much of the literature asanufacturers and books advocate nodal formulations

Table 1List of d

Disconpoint

1 2 3 this article in press as: F.M. Uriarte, On Krons diakoptics, Electr. Power Sys

owing how it performs in partitioned simulation sce-

r a grounded network where one (of many possible) dis- points has been identied in Fig. 7. At this particularon point, there are p = 5 partitionsall interfaced at theary. This is not unusual in situations where p > 2. Showns the disconnection point when the network is formu-h currents. In this formulation, r = 1. Shown on the right

disconnection point, but formulated in node voltages,lts in r = 4. As noticed, the values of r in each case are

e of r has a negative impact to in (13). Since is dense,uent LU re-factorization (e.g., due to switching) and (2)backward solutions at each time step (two algorithmsh) suggests that it is prudent to keep r smallespeciallye with the number of available parallel processing units.porting case to demonstrate that r depends on the

choice, consider a hypothetical power system (ratheronvey visually) with the disconnection points describedf such power system is grounded, the number of bound-s rgnd

meshin a mesh formulation is given by (14); if the

em is ungrounded, the number of boundary variablesn by (15). For the same grounded power system, a nodal

results in a number of boundary variables rgndnodal

given

ng the values of r in (14), (15), and (16), mesh formu-demand less time to solve the boundary equations in

45

partitionedtant, and imatrices ar

rgndmesh

=d

i=1

rungndmesh

=d

i=1

rgndnodal

=d

i=1

4. Conclus

Tearing as original(zero-immible disconthis is aalgorithms,as ne-graig common boundary variables.

[A2]

u1

[A4]

[A3]

u2

u3

u4v

p = 5r = 4

dal Formulationt. Res. (2012), doi:10.1016/j.epsr.2012.01.016

3 63 3

simulations. This nding is rarely recognized as impor-s commonly obfuscated by the ease in which nodale formed instead [12].

Ni = (3 + 3 + 3 + 3 + 3) = 15 (14)

(Ni 1) = (2 + 2 + 2 + 2 + 2) = 10 (15)

Ni(si 1) = 3(4 + 9 + 1 + 5 + 2) = 63 (16)

ions

networks does not require boundary branchesly proposed by Kron. Tearing meshes and nodesttance tearing) increases the number of possi-nection points (or boundaries) to choose from;n important consideration for graph-theoreticwhichfurthermorepermits graph segregationsned as one power apparatus per subsystem.

Please cite r Syst

ARTICLE IN PRESSG ModelEPSR-3456; No. of Pages 5F.M. Uriarte / Electric Power Systems Research xxx (2012) xxx xxx 5

There are several situations where mesh formulations resultsin less computational effort (i.e., a smaller r) to solve the boundarynetwork: a well-known bottleneck in diakoptics-based partition-ing. This is an important consideration that is often overlooked, andmerits attention early in the software design stage when decidingbetween nodal and mesh formulations.

The results presented by this work are applicable to multicoreelectromagnetic transient simulations of large, complex power sys-tems [6]. Tearing meshes and nodes (instead of branches) resultsin a better selection of disconnection points when using graph the-ory to determine where to partition a power system. Choice of thecorrect (or best-suited) formulation method for a power systemsimulation problem results in faster multicore (parallel) simula-tion results due to the reduced number of boundary variables (r).While no one formulation method is best-suited for all problems,the formulation choice should be assessed by computer programsbefore running parallel simulations.

References

[1] G. Kron, Diakoptics, The Piecewise Solution of Large-Scale Systems, MacDonald& Co., London, 1963.

[2] A. Brameller, M.N. John, M.R. Scott, Practical Diakoptics for Electrical Networks,Chapman & Hall, London, 1969.

[3] I.S. Duff, A.M. Erisman, J.K. Reid, Direct Methods for Sparse Matrices, OxfordUniversity Press, Oxford, 1986.

[4] H.V. Henderson, S.R. Searle, On deriving the inverse of a sum of matrices, SIAMRev. 23 (1981) 5360.

[5] A. Klos, What is diakoptics? Int. J. Elec. Power 4 (1982) 192195.[6] F.M. Uriarte, R.E. Hebner, A.L. Gattozzi, Accelerating the simulation of shipboard

power systems, in: Grand Challenges in Modeling & Simulation, The Hague,Netherlands, 2011.

[7] F.M. Uriarte, Multicore simulation of an ungrounded power system, IET Elec.Syst. Trans. 1 (2011) 3140.

[8] Z. Quming, S. Kai, K. Mohanram, D.C. Sorensen, Large power grid analysis usingdomain decomposition, in: Design, Automation and Test in Europe, 2006. DATE06. Proceedings, 2006, pp. 16.

[9] G. Karypis, V. Kumar, hMETIS: a Hypergraph Partitioning Package, Departmentof Computer Science & Engineering, University of Minnesota, Minneapolis, MN,1998.

[10] T.A. Davis, Direct Methods for Sparse Linear Systems, SIAM, Philadelphia, 2006.[11] F.M. Uriarte, K.L. Butler-Purry, Diakoptics in shipboard power system sim-

ulation, in: North American Power Symposium (NAPS), Southern IllinoisUniversity, Carbondale, 2006, pp. 201210.

[12] F.M. Uriarte, A tensor approach to the mesh resistance matrix, IEEE Trans. PowerSyst. 26 (2011) 19891997.

[13] P.W. Aitchison, A. Klos, Network tearing an alternative method, Int. J. Elec.Power 13 (1991) 308320.

[14] A. Sangiovanni-Vincentelli, C. Li-Kuan, L.O. Chua, A new tearing approach node tearing nodal analysis, in: IEEE International Symposium on Circuits andSystems, 1975, pp. 143147.

[15] C. Yue, X. Zhou, R. Li, Node-splitting approach used for network partition andparallel processing in electromagnetic transient smiulation, in: InternationalConference on Power System Technology, Singapore, 2004.

[16] B.M. Zhang, N.A. Xiang, S.Y. Wang, Unied piecewise solution of power-systemnetworks combining both branch cutting and node tearing, Int. J. Elec. Power11 (1989) 283288.

[17] A. Kalantari, S.M. Kouhsari, An exact piecewise method for fault studies ininterconnected networks, Int. J. Elec. Power 30 (2008) 216225.

[18] S. Esmaeili, S.M. Kouhsari, A distributed simulation based approach for detailedand decentralized power system transient stability analysis, Elec. Power Syst.Res. 77 (2007) 673684. this article in press as: F.M. Uriarte, On Krons diakoptics, Electr. Powe . Res. (2012), doi:10.1016/j.epsr.2012.01.016

On Kron's diakoptics1 Introduction2 First property: zero-immittance tearing2.1 Mesh formulation example2.2 Nodal formulation example

3 Second property: formulation choice4 ConclusionsReferences