On the Convexity of the System Loss Function

Transcription

1 O the Covety of the System Loss Fucto Sebasta de la orre, Member, IEEE ad Fracsco D. Galaa, Fellow, IEEE Abstract We show that the system fucto a power etwork s bouded below by ay umber of supportg hyperplaes the space of geeralzed jectos. A supportg hyperplae s defed by the lear aylor seres epaso of the system fucto aroud a gve operatg pot. he supportg hyper-plae property s vald provded that the epaso pot s suffcetly ear the flat-voltage profle (F). We have assessed epermetally the rage of valdty of ths assumpto, called the rage of hyper-plae support (RHS), showg that for typcal etworks, the RHS s broad, partcularly whe the bus voltages are cotrolled ear per ut. he supportg hyper-plae model was also tested as part of a ecoomc dspatch wth trasmsso es to demostrate that ths lear model provdes the same results as whe the es are treated as a o-lear fucto. Ide erms System, load flow, lear appromato, supportg hyper-plaes, covety, formulae. I. OMECLAURE Defto: Flat oltage rofle (F) s the operatg codto uder whch all comple bus voltages are equal to some arbtrary level, e + jf rectagular coordates or δ polar coordates, where e cos( ) f s ( ) = δ ad = δ. Wthout of geeralty, we assume that bus s the referece wth δ = or, equvaletly, wth f =. hs the mples that δ = ad that f =. arameters: umber of etwork buses; umber of load flow equatos ad ukows = -; G Real part of the etwork admttace matr; G ' Sub-matr of G wthout the -th row ad colum; L Block dagoal matr formed wth G ad G' ; B Imagary part of the etwork admttace matr; B ser Compoet of B due to le seres admttaces; B sh Compoet of B due to shut admttaces; ector of s of dmeso ; ector or matr of s; -dmesoal vector of s wth at posto ; hs work was supported by the atural Sceces ad Egeerg Research Coucl (SERC), Caada, by the Fods ature et techologes, Quebec, ad by the Europea Uo project FEDER-CICY FD ad grat FU-A99 of the Mstero de Educacó, Cultura y Deporte of Spa. F. D. Galaa s wth the Departmet of Electrcal ad Computg Egeerg at McGll Uversty, Motreal, Québec, Caada. S. de la orre, was wth the E..S.I. Idustrales, Uversdad de Castlla- La Macha, Cudad Real, Spa, ad s curretly wth the E..S.I. elecomucaoes, Uversdad de Malaga, Spa her e-mals are: galaa@ece.mcgll.ca ad storre@uma.es. Ω Set of odes coected to ode. arables: ector of comple odal voltages; e Real part of ; f Imagary part of ; f ' f ecludg the referece bus ; ector of magtudes squared of ; ector of magtudes of ; δ ector of phase agles of ecludg the referece bus ; -dmesoal vector of rectagular coordate voltage e compoets, ' ; f alue of uder F; e Real part of uder F; Real power jecto at bus ; Q Reactve power jecto at bus ; ector of real power jectos at all buses; ' ecludg the slack bus; Q ector of reactve power jectos at all buses; S Set of buses wth specfed real power jecto;.e., all buses ecept for the slack. Set of buses wth specfed reactve power jecto; S Q S Set of buses where the voltage magtude s specfed; z ector of geeralzed jectos of dmeso comprsed of all S, all Q S Q ad all S ; z() Fucto of characterzg the geeralzed jectos; System ; I ( ) Fucto of characterzg the system. II. IRODUCIO problems such as ut commtmet, resource schedulg, or etwork epaso, the power etwork s ofte modeled a appromate maer or s altogether eglected. hs s doe order to reduce the umber of varables ad the overall computatoal complety, but also to make the problem lear thereby permttg the use of powerful med teger lear programmg tools []. oday, such tools ca relably solve problems wth very large umbers of costrats ad varables, both teger ad cotuous, whle o-lear med-teger programmg tools are stll urelable ad wth lmted applcatos. he smplest etwork models completely gore the trasmsso ad descrbe the etwork as a sgle ode. he et level of complety models the system trasmsso,, the power balace equato as a appromate eplct polyomal fucto of the geerato levels, g,

2 m g = d + ( g) () = Loss appromato formulae were etesvely studed as early as the 96 s, begg wth the B-coeffcets approach [] whch the system fucto was epressed as a costat coeffcet quadratc fucto of the power geerato levels, g. I the early 98 s, takg advatage of specfc aalytc propertes of the load flow equatos rectagular coordates [3], more eact eplct appromato formulae up to thrd order were derved based o aylor seres epasos of terms of z aroud a specfed load flow operatg pot [4]. A smlar result to [4] was publshed recetly wth applcatos to voltage stablty [5]. Other authors have eamed eplct appromatos based o polyomals the geerato levels [6, 7]. It s also possble to model the of every trasmsso le a power etwork by a quadratc appromato of the correspodg le phase agle dfferece. hs leads to a varato of the DC load flow model of the form, = B' δ [ ] = B δ + q( δ) f f f f f whch cludes the le power flows, f, ther lmts () f, ad the appromato of the dvdual le real trasmsso es, q(δ), [8, 9]. otwthstadg the avalablty of the above metoed appromato formulae, whe solvg problems that do ot volve teger decsos, such as a optmal power flow, our eperece t s questoable whether such appromatos offer ay sgfcat advatage over a olear optmzato method that accouts for the full olear load flow model. I problems volvg teger decsos, such as ut commtmet, oe dsadvatage of the more accurate quadratc or hgher order appromato formulae s that order to make use of med-teger lear programmg solvers, t s ecessary to further decompose each o-lear term to pece-wse lear compoets. hs step requres the troducto of ew cotuous ad possbly teger varables, therefore addg to the overall modelg ad computatoal complety. O the other had, the use of lear appromato formulae, although readly compatble wth med-teger lear programmg tools, s subject to greater appromato accuraces. Oe drawback of all appromato formulae, whether lear or hgher order, s that the errors troduced by the appromato comparso wth the eact model caot be easly quatfed or estmated. he ma cotrbuto of ths paper s a theoretcal result that had bee prevously hypotheszed [], but ot rgorously prove or tested, amely to demostrate that, uder certa relatvely weak assumptos, the o-lear system fucto,, s cove the geeralzed power flow jectos, z. As such, a frst order aylor seres epaso ca be show to defe a lower boud appromato of the system lear z, kow as a supportg hyper-plae. Such represetatos are ot appromatos the usual learzato sese, sce they also costtute a lower boud o the o-lear fucto. hs property leads to two mportat results: () As each such equalty s a ecessary codto, ew hyper-plaes ca be added wthout valdatg the estg oes; () he error betwee the o-lear fucto ad the descrbg supportg hyper-plaes s greater tha or equal to zero. hus, the olear system behavour ca be approached wth arbtrary accuracy by creasg the umber of lear ecessary codtos. It s also mportat to ote that the learzed model based o the well-kow cremetal trasmsso (IL) coeffcets s a specal case of a supportg hyperplae. Aother cotrbuto of ths paper s to quatfy the rage of valdty of the covety property of the fucto sde of whch the system es ca be modeled by a set of lear equaltes. umercal results o systems up to 8 buses suggest that the covety property s vald over a wde rage of operatg codtos, cludg typcal operatg pots. he system boudg theorem has also bee appled to solve a y ecoomc dspatch problem to llustrate the accuracy of the hyper-plae appromato. III. SYSEM LOSS FUCIO SUORIG HYER-LAES A. o-lear Load Flow Model I load flow models the decso varables ad goverg equatos are as follows, = ( ); S Q = Q ( ); S Q = ( ); S = = ( ) ( ) = = B. Geeralzed Load Flow Ijectos he vector of geeralzed load flow jectos, z R (3), s defed by the set of, Q ad of equato (3). As show Apped A, Result, the relato betwee z ad s deoted by, z= z( ) J( ) (4) C. System Loss Sestvty ector he system sestvty vector aroud a gve operatg pot, deoted by β, s defed by the sestvtes of the system wth respect to the geeralzed load flow jectos, z, that s, ( z ) z ( ) ( ) ( ) β = = z (5) = J ( ) L

3 3 where we have used the deftos of z() equato (4) ad of () Result 5 of Apped A. ote that the well-kow cremetal trasmsso (IL) coeffcets are the compoets of β correspodg to the real power jectos. he sestvty vector, β, characterzes the frst order aylor seres epaso of the fucto aroud the operatg pot, that s, ( ) = + β z z +ε ( ) ( ) where the quatty ε represets the error ths lear appromato due to hgher order terms. Oe of the propertes of the rectagular load flow formulato derved Apped B s that, ( ) = β z ( ) (7) whch whe substtuted to (6) reduces the frst order epaso of the system to, = β z + ε (8) Apped B also shows that whe s suffcetly ear the F, the appromato error ε s o-egatve for ay feasble z, that s, for ay z for whch there ests a correspodg load flow soluto,. hese results allow us to formally eucate the followg theorem. D. System Loss Boudg heorem Gve ay operatg pot suffcetly ear a flatvoltage-profle such that the load flow Jacoba matr, J ( ), s o-sgular, the, for ay feasble vector of geeralzed load flow jectos, z, the correspodg system satsfes, (6) β z (9) he proof of ths theorem s detaled Apped B. E. Corollary to the System Loss Boudg heorem Uder the codtos of the system boudg theorem, for ay feasble z ot collear wth z ( ), > β z () whle for ay feasble z collear wth z ( ), = β z () he proof of ths corollary follows drectly from Result 9 Apped A. he above lear equato () defes what s called a supportg hyper-plae of the ukow o-lear fucto. Sce power systems ormally operate relatvely ear the F, the promty codto requrg that the epaso pot be suffcetly ear the F s geerally ot a severe lmtato. Epermetal evdece preseted the results secto backs up ths statemet. F. Lear System Loss Model Based o Supportg Hyper- laes Sce (9) s a ecessary codto, ay umber of supportg hyper-plaes based o dfferet epaso pots ca be merged to a more restrctve costrat set. Let SH be a set of β ' s of whch each elemet correspods to a supportg hyper-plae of the system fucto. he, the power balace equato wth es ca be appromated by the followg lear model, = = β z; β SH () I addto, operatoal costrats such as m ma retag ts lear ature. ca easly be added to the model whle stll I. UMERICAL RESULS All quattes ths secto are per ut ecept for the bus voltage phase agles whch are degrees. A. Eamples of Supportg ad o-supportg Hyperplaes Cosder a 5-bus etwork wth the followg le data, ABLE I: ES EWORK LIE DAA; MA, K BASES. From Bus o Bus Seres R Seres X Shut B he F codto polar coordates s gve by δ = [,,,] ad = [,,,,], where the referece agle s δ (5) =. I terms of geeralzed load flow jectos, the F s descrbed by ' = [ ] = [ ],,, ad,,,,. ables II, III ad I below descrbe the supportg hyperplae correspodg to a specfed operatg pot epressed polar coordates. he symbols β ad β these tables deote respectvely the compoets of β correspodg to the geeralzed jectos, ' ad.

4 4 ABLE II: EXAMLE A: SUORIG HYER-LAE. 5-BUS SYSEM ode δ ( ') β ABLE III: EXAMLE A: SUORIG HYER-LAE. 5-BUS SYSEM ode ( ) β ABLE I: EXAMLE A: SUORIG HYER-LAE. 5-BUS SYSEM Egevalue umber λ (L) λ ( L H( β)) I ths case, the epaso pot,, show able II, epressed polar coordates, has phase agles that are ot all close to zero, oe of them beg equal to 43 degrees. hs suggests that the requremet that the epaso pot be suffcetly ear the F s ot very strget. Moreover, as called for by the theory, able I shows that the egevalues of L are all postve but oe. ote, addto, that the egevalues of the error matr, L H( β ), are all close to those of L, ad that the zero egevalue of L remas at zero for the error matr. able, I ad II show a eample of a operatg pot that does ot correspod to a supportg hyper-plae. Here, two of the egevalues of [ L H( β ) ] are egatve. A lear epaso about ths operatg pot would therefore ot be a lower boud for the system over all z. ote that ths o-supportg case, the epaso pot s relatvely far from the F as evdeced from the values of the phase agles. ABLE : EXAMLE B: O-SUORIG HYER-LAE. 5-BUS SYSEM ode δ ( ') β ABLE I: EXAMLE B: O-SUORIG HYER-LAE. 5-BUS SYSEM ode ( ) β ABLE II: EXAMLE B: O-SUORIG HYER-LAE. 5-BUS SYSEM Egevalue umber λ (L) λ ( L H( β)) A etwork wth data based o the IEEE-8-bus test system wth all bus voltages held at per ut s cosdered ow. Due to the large data set, oly some values are provded. able III shows the operatg pot polar coordates, the geeralzed jectos, ad the values of β at two buses. able IX shows a subset cosstg of the smallest ad largest egevalues of L ad of the error matr, all of whch are postve ecept for oe whch s zero. ABLE III: EXAMLE C: SUORIG HYER-LAE. 8-BUS SYSEM z ( ) β ode o o =.975 = = 4.43 = β β 4 4 β β 9 9 =.633 =.44 =.487 =.679 ABLE IX: EXAMLE C: SUORIG HYER-LAE. 8-BUS SYSEM λ (L) λ ( L H( β)) B. Cotuty ad Rage of Hyper-plae Support he rage of hyper-plae support (RHS) descrbes the set of epaso pots vald. Formally, the RHS s a set of for whch the covety property (9) s defed by,

5 5 / β z; β = ( ) ; ( ) RHS J L L H β = (3) = ( ); z= z( ); R Sce the load flow Jacoba,, s o-sgular whe s at or ear, the RHS s a set that cludes ad eteds cotuously outward from boudares are reached whe J( ) L H(β) becomes defte; () J ) all drectos. Its satsfes ether of: () becomes sgular. et, some epermetal tests to estmate the sze of the RHS are coducted. ( C. umercal Estmates of the Rage of Hyper-plae Support I order to quatfy the RHS, a eghborhood aroud the F codto s defed, j ma le δ δ δ ;, j Ω (4) hs codto lmts the agle dffereces across all les. able X shows the results of varous epermets o the 5-bus ma δ le etwork for dfferet values of, each based o, values of the epaso pot,, pcked uformly radomly wth the set defed by (4). As able X shows, the rage of estece of supportg hyper-plaes s qute broad, ecompassg bus agles betwee ±9 degrees ad le agle dffereces up to 76.5 degrees; the latter value beg the most sgfcat. Outsde the rego of support, a small but ozero percetage of the radomly geerated hyper-plaes does ot support the fucto. Moreover, the farther from that rage that a hyper-plae les, the hgher the chaces that t s o-supportg. ABLE X: RAGE OF HYER-LAE SUOR. 5-BUS SYSEM Rato of o-supportg to Supportg hyper-plaes ma δ le /, 76.5 º 3/, 8º /, 9 º he results for the 8-bus etwork wth radomly geerated epaso pots are show able XI. ABLE XI: RAGE OF HYER-LAE SUOR. 8-BUS SYSEM Rato of o-supportg to Supportg hyper-plaes ma δ le /, 68.4 º 7/, 7 º 73/, 8 º hese results dcate that f the absolute le agle dffereces are less tha 68.4º, the, the correspodg hyper-plae s lkely to be supportg. ote that epermets dcate that the rage of hyper-plae support remas essetally uaffected f the bus voltages are kept wth plus or mus 5% of omal.. ALICAIO: ECOOMIC DISACH WIH LOSSES he geeral theory preseted ths paper s sutable for use may dfferet practcal problems where es are a mportat ssue; eamples beg ut commtmet, ecoomcdspatch ad hydro-thermal coordato. ote however that the supportg hyper-plae approach s ot teded to be used to solve problems wth costrats o depedet varables such as le flows, load bus voltage magtudes, or reactve geerato levels. Such problems should be solved wth a optmal power flow For the sake of llustrato, the proposed geeral theory s tested solvg a y ecoomc dspatch problem; the system used s the prevously preseted 8-bus power system; wth 49 geeratg buses, 58 load buses ad 79 les. A. Solvg the ecoomc dspatch. ote that the ecoomc dspatch problem, whe modeled wth supportg hyper-plaes, cossts of the objectve fucto, the bouds for the geerato, oe equalty costrat per hyper-plae ad a addtoal equalty costrat statg that the summato of geerato must equal the summato of demad plus es; thus, o eplct accout for the etwork s eeded. he y ecoomc dspatch may be solved eactly usg hyper-plaes as follows (Algorthm ): Step. Italze a workg set of hyper-plaes wth the trval equalty costrat, that s,. Step. Solve the ecoomc-dspatch problem usg the workg set of hyper-plaes for the geerator outputs. Step. Solve a power flow problem whose put data are the geerato levels of all uts from Step, ecept for the slack. he soluto provdes the values of ad of the slack phase agles of all buses. Step 3. Wth the data obtaed Step compute vector β usg (5). hs vector defes a ew hyper-plae. Step 4. If the dfferece betwee the values obtaed for steps ad s uder a certa threshold, SO; slack otherwse, add the hyper-plae derved Step 3 to the set of workg hyper-plaes ad retur to Step. A alteratve algorthm that s better suted for o-le operato s ow preseted (Algorthm ): Step : Compute a good set of hyper-plaes off-le. Step : Whe o-le, for ay actual value of the vector of demads, solve a ecoomc dspatch usg the equalty costrats obtaed Step. B. Estmatg the error of the method It s clear that Algorthm, the umber of hyper-plaes calculated off-le, H, s the most mportat parameter determg the cost error. hus, a procedure s ow preseted for estmatg the mmum cost error terms of the umber of hyper-plaes used to solve a y ecoomc dspatch, error( H ). (Algorthm 3): he cocept of good depeds o the precso desred; the more hyperplaes cosdered Step, the more accurate the soluto of Step.

6 6 Step : Usg ay sutable algorthm, such as Algorthm above, solve for the eact soluto of a large umber,, of y ecoomc dspatch problems wth dfferet demad levels, ad calculate for each of them the optmal cost: Cost ( ) :.... Step : Compute H supportg hyper-plaes for a set of H operatg codtos dfferet from those foud Step. Step : Solve each of the problems from Step usg oly the H equalty costrats from Step usg Algorthm to obta the appromate costs Cost( ) :.... Comparg these costs wth the optmal costs that were obtaed Step, compute the correspodg 3 errors. Defe ε ) = Cost ( ) Cost ( ) as the error ( obtaed whe tryg to appromate the -th problem wth H hyper-plaes. Defe the mamum of these errors as a estmate of the mamum total error that wll be obtaed wheever H hyper-plaes are used to solve the ecoomc dspatch problem wth Algorthm. I Algorthm 3, ote that H s the umber of hyper-plaes used to create a model. Also, ote that s the umber of problems used to assess the accuracy of that model; the eact soluto for those problems s calculated Step. Fgure shows the mamum total cost error as a percetage of the eact o-lear cost terms of the umber of hyper-plaes used Algorthm. ote that there are four plots Fgure for two dfferet values of ( ad 4) ad two dfferet values of w (% ad 3%). arameter w dcates how ear to the base-case demad were the H radom demads. For stace, w=3% meas that, the radomly geerated demad values le sde a terval of ±3% aroud ther orgal value. Cost error [%],8,7,6,5,4,3,,, =4, w =3% =, w =3% =4, w =% =, w =% Fgure. Cost error as a fucto of the umber of hyper-plaes used Algorthm ote that the effect of addg or removg hyper-plaes as llustrated by Fgure of the paper s ot very sgfcat o the optmum dspatch cost, for eample, as few as hyperplaes gve a error of aroud.3%. I. COCLUSIOS We have show that the system fucto a power etwork s bouded below by ay umber of supportg hyper-plaes the space of geeralzed jectos. A supportg hyper-plae s defed by the lear aylor seres epaso of the system fucto wth respect to the geeralzed jectos aroud a gve operatg pot. hs supportg hyper-plae property s vald over a cotuous rage of epaso pots provded that these are suffcetly ear the flat-voltage profle (F). he requred degree of promty betwee a operatg pot defg a supportg hyper-plae ad the F, called the rage of hyper-plae support (RHS), was assessed epermetally. For typcal etworks, the RHS was foud ot to be very restrctve, partcularly for etworks where the bus voltages are closely cotrolled ear per ut. he supportg hyper-plae model was also tested as part of a ecoomc dspatch wth trasmsso es to demostrate that ths lear model provdes the same results as whe the es are treated as a o-lear fucto. Future research should look at tegratg the supportg hyper-plae model to problems such as ut commtmet wth trasmsso es. he system boudg theorem should also be eamed wth etworks cotag FACS devces or phase-shftg trasformers. I addto, t would be terestg to eted these results to load flow models wth dstrbuted slack geerato. II. AEDIX A For completeess, most of the relevat propertes of the load flow equatos rectagular coordates are preseted ad prove ths apped. For these ad other terestg propertes of the quadratc formulato of the load flow problem, see also [], [] ad [3]. For all the results that follow, we make the realstc assumpto that all trasmsso les have o-zero seres resstace ad zero shut coductace. Result : he system a power trasmsso etwork s postve or zero. roof: hs s self-evdet from eergy coservato ad the fact that the trasmsso etwork s a passve RLC etwork. Result : he system s zero f ad oly f all the seres brach currets are zero. roof: Let I ser j be the seres curret magtude of the le coectg buses ad j wth o-zero resstace R j. he correspodg trasmsso compoet, I ser jrj, s the zero f ad oly f the seres curret magtude s zero. Uder the assumpto that the shut braches are less, the system wll also be zero f ad oly f all the seres brach currets are zero. Result 3: All the seres brach currets are zero f ad oly f the comple bus voltages at all buses are equal. A sutable mmum umber for the IEEE 8-bus system wll be. 3 hs wll always be a postve umber, because the supportg hyperplaes are lower bouds of the actual fucto.

7 7 roof: he seres brach curret through the le coectg buses ad j wth o-zero seres mpedace, R j +jx j, s I ser j j = R + jx j j, whch s zero f ad oly f = j. Result 4: he system s zero f ad oly f the etwork operates at F. roof: hs follows drectly from the defto of F, Results ad 3, ad the zero shut coductace assumpto. Result 5: I rectagular coordates of the comple bus voltages, the system ca be epressed by the followg pure quadratc form, = L ( ) (A) where, = G L (A) G' roof: Epressg the comple ode voltages rectagular coordates, = e+ j f, the total real power cosumed by the etwork, whch defes the system es, s gve by, * * = =R Y jj = = j= * { Y } * {( e jf) ( G jb)( e jf) } =R =R If G s symmetrc, the = ege+ fgf ( ) = ege+ f' Gf ' ' (A3) (A4) However, f G s ot symmetrc, as may occur etworks wth phase-shftg trasformers, the equato (A4), G gets replaced, wthout of geeralty by the symmetrc matr ( G+ G )/. hs s ecessary sce, f G s ot symmetrc, ts egevalues are ot related to those of ( G+ G )/. he latter s always a symmetrc matr whose egevalues are real ad o-egatve. We therefore assume throughout ths apped that G s symmetrc. Recall from the omeclature that, wthout of geeralty, f s here set to zero, thus obtag the secod form equato (A4). Result 6: he symmetrc matr G s postve sem-defte. roof: From Result, for all f ad e. I partcular, by lettg f =, t follows from equato (A) Result 5 that = ege for ay e, whch defes G as postve semdefte. From matr theory, the postve sem-defte codto also mples that all the egevalues of G are real ad o-egatve. Result 7: he coductace matr G has oe ad oly oe zero egevalue wth egevector e e where e s a arbtrary costat. he remag - egevalues must therefore be real ad postve. = roof: o prove that G has oe o-zero egevalue, let the etwork operate at F, that s, = e. From Result 3, all brach currets must be ull, whch mples that the compoet of the et bus curret jectos due to brach currets must also be ull, other words, Y = ( G+ jb ) e = (A5) where ser = + j ser ser ser Y G B s the compoet of the etwork admttace matr due to seres elemets. ote that the shut curret vector compoets, Y = ( jb ) s ot zero at F but uder the assumpto of zero shut coductaces ths compoet does ot cotrbute to the system. Codto (A5) that the seres curret jectos are ull uder F mples that G ( e ) =, whch proves that G has a zero egevalue whose egevector s e. I order to prove that G has oly oe zero egevalue, suppose that there ests aother o-zero vector ê R satsfyg the zero egevalue codto, G ê =, ad ot of the F form e. Sce = e ˆ s a vector of voltages that does ot correspod to the F, accordg to Result 4, the system must be strctly postve, that s, >. A cotradcto the arses sce (A4) states that ˆ ˆ = ege =. Cosequetly, there caot est a vector e ˆ wth the above property ad the oly possble egevector correspodg to the zero egevalue s e. Result 8: he symmetrc matr G ' s postve defte. roof: From Result 6, G s postve sem-defte. hus for all f, f Gf. Moreover, from Result 7, f Gf = for ozero f oly f f s proportoal to. ow, f f =, ( f') G' f' = f Gf. However, wth f =, t s mpossble for f to be proportoal to. hus, wth f =, ( f') G' f' = f Gf > for all f ' ecept for the trval case whe f' =, whch s the codto for postve defteess of G '. Result 9: he matr L defed Result 5 s postve semdefte wth a sgle zero egevalue, whose egevector correspods to the F. roof: From the propertes of block-dagoal matrces, the set of egevalues of L s the uo of the respectve sets of egevalues of G ad G '. As prove Results 7 ad 8, the combed egevalues of these two matrces are all postve ecept for oe whch s zero whose egevector correspods to the F. Result : he vectors of real ad reactve power jectos, ad Q, as well as the vector of bus voltage magtudes,, are quadratc fuctos of the real ad magary comple voltage compoets,. roof: he vector of comple power jectos s, sh sh * { } I dag{ }{ Y} * S = + j Q = dag = (A6) whch rectagular form becomes, S = dag e+ j f G+ j B e+ j f } (A7) { }( )( ) { *

8 8 Separatg S to ts real ad magary parts, ad Q, gves the followg quadratc equatos, = dag{}( e Ge Bf ) + dag{}( f Gf + Be) (A8) Q = dag f Ge Bf dag e Gf + Be {}( ) {}( ) Smlarly, sce = e + f, the vector of voltage magtudes squared ca also be epressed as a quadratc, = dag e e+ dag f f (A9) { } { } Result : Ay scalar lear combato of elemets of Q, ad ca be epressed the form H where H s a real symmetrc matr depedet oly o the etwork parameters. roof: From the defto of the -dmesoal vector the omeclature ad from equato (A8) Result the real power jecto at bus s epressed as, = ( ) ˆ {}( ) dag{}( ) { }( ) dag{ }( ) = ( ) dag e Ge Bf + f Gf + Be = e dag Ge Bf + f Gf + Be Defg ow the symmetrc matr, H = { } G+ G { } { } B+ B { } dag{ } dag{ } dag{ } dag{ } dag dag dag dag B + B G+ G t follows from (A) that ca be epressed as, e ˆ e = H f f (A) (A) (A) ow, defg H as H ˆ wthout the -th row ad colum, = H (A3) Smlarly, for the reactve power jectos we obta, Q Q = H (A4) Q where H s foud by removg the -th row ad colum from, ˆ Q dag{ } B B dag{ } dag{ } G+ G dag{ } H = G dag{ } + dag{ } G dag{ } B B dag{ } (A5) Fally, for bus voltage magtudes squared we obta, where from, H = H (A6) s foud by removg the -th colum ad row dag{ } = dag{ } ˆ H (A7) ow, from equatos (A3), (A4) ad (A6), t follows that ay dvdual geeralzed power flow jecto, z, ca therefore be epressed as a pure quadratc of the form, z = [ H ] (A8) hus, α z = Hα ( ) (A9) = where we defed, H( α) = αh = (A) Result : he load flow problem wth specfed geeralzed jectos, z ; =,,, ca be epressed as a set of equatos of the form, z= J ( ) (A) where J ( ) s a by matr defed by, H J ( ) = M (A) H roof: hs result follows mmedately from equato (A9) Result. Result 3: he matr J() s the Jacoba matr of the load flow equatos rectagular coordates. roof: he load flow Jacoba matr s defed as the sestvty of z wth respect to, whch from equatos (A) ad (A) Result gves, H M H z H = = M = J ( ) (A3) H Result 4: he Jacoba matr J() s lear. roof: hs result follows drectly from (A3). Result 5: For ay, α J( ) = H(α). roof: From Result, Hα ( ) = α = α H = H = H = = α M α J( ) H III. AEDIX B: ROOF OF SYSEM LOSS BOUDIG HEOREM (A4) As show Result 5 of Apped A, the system fucto ca be wrtte as a pure quadratc form, = L (B) Furthermore, from Result 9 Apped A, L has oly oe zero egevalue whose correspodg egevector s the flat voltage profle. We eame ow the behavor of the load flow equatos ad the system fucto terms of devatos from a arbtrary operatg pot,, so that = +. Recallg the epresso for the load flow equatos Result ad the fact that the load flow Jacoba matr s lear from

9 9 Result 4, we have, z = J( + )( + ) (B) = J( )( ) + J( ) + J( ) Smlarly, the system equato (B) ca be rewrtte as, = ( ) L + ( ) L + L (B3) Sce the load flow Jacoba at the operatg pot,, s o-sgular, from (B), = [ J( )] z J( )( ) J( )( ) (B4) Substtutg (B4) to the lear term of (B3) gves, = ( ) L + [ ] L + ( ) L [ J( )] z J( )( ) J( )( ) (B5) [ ( )] z J( )( ) + [ ] L = ( ) L J From the defto of β (5), ad takg ts traspose, the eact system equato (B5) ca ow be epressed as, = β z J( )( ) + [ ] L (B6) = β z β J( )( ) + [ ] L ow, usg Result 5 wth α takg the partcular value of β, t follows from (B6) that, = β z+ [ ] [ L H(β) ] (B7) Comparg (B7) wth (8), we see that the appromato error, ε, s gve by the quadratc form, ε= [ ] [ L H( β) ] (B8) o complete the proof of the System Loss Boudg heorem, we ow show that f F, the real symmetrc matr les suffcetly ear a L H(β) s postve semdefte. hs wll mply that for all R, [ L H( β) ] satsfyg respectvely (B) ad (4), o prove that L H(β) or equvaletly that for all β z ad z (B9) s postve sem-defte frst recall from Result 9 that L =. hus, f, the L L =. Smlarly, gve that the load flow Jacoba s o-sgular at, the J ( ) L =. ow, from equato (A) Result, f β the H( β). β = [ J( ] ) L Sce L ad L H( β ) are arbtrarly close real symmetrc matrces, ther respectve sets of egevalues are real ad arbtrarly close. From Result 9, the egevalues of L are strctly postve ecept for λ ( L ) =. hs meas that all the egevalues of L H( β ) are guarateed to be postve ecept possbly λ ( L H( β )) whch although arbtrarly close to λ ( ) L =, may have shfted to a egatve value. o dscard ths opto, recall from equato (5) ad from Result 5 Apped A that, he, [ J( )] β H( β) L = = [ ] (B) L H( β) = (B) hs mples that L H( β ) stll has a zero egevalue wth egevector ad therefore that λ ( L H( β)) =. hs proves that L H(β) s postve sem-defte ad cocludes the proof of the system boudg theorem. IX. REFERECES [] A. Brooke, D. Kedrck, A. Meeraus ad R. Rama. GAMS/CLEX 7. User otes. GAMS Developmet Corporato. Washgto.. [] E. F. Hll ad W. D. Steveso. A ew Method of Determg Loss Coeffcets. IEEE ras. ower App. ad Syst. ol. AS-87, o. 7, pp , July 968. [3] S. Iwamoto ad Y. amura. A Fast Load Flow Method Retag olearty. IEEE ras. ower App. ad Syst. ol. AS-97, o. 3, pp , March 978. [4] F. D. Galaa ad M. H. Baakar. Appromato Formulae for Depedet Load Flow arables. IEEE ras. ower App. ad Syst. ol. AS-, o. 3, pp. 8-36, March 98. [5] W. Xu et al. Seres Load Flow: A ovel o-teratve Load Flow Method. IEE roc. Ge. ras. Dstrb. ol. 45, o. 3, pp. 5-56, Ja [6] A. Jag ad S. Erter. olyomal Loss Models for Ecoomc Dspatch ad Error Estmato. IEEE ras. ower Syst. ol., o. 3, pp , Aug [7] R. Baldck ad F. F. Wu. Appromato Formulas for the Dstrbuto System: the Loss Fucto ad oltage Depedece. IEEE ras. ower Delvery. ol. 6, o., pp. 5-59, Ja. 99. [8] A. L. Motto, F. D. Galaa, A. J. Coejo ad J. M. Arroyo. etwork- Costraed Mult-erod Aucto for a ool-based Electrcty Market. IEEE ras. ower Syst. ol. 7, o. 3, pp , Aug.. [9] R. A. Jabr. Modellg etwork Losses Usg Quadratc Coes. IEEE ras. ower Syst. ol., o., pp , Feb. 5. [] F. D. Galaa ad M. H. Baakar. Realzablty Iequaltes for Securty Costraed Load Flow arables. IEEE ras. Crcuts ad Syst. ol. CAS-9, o., pp , ov. 98. [] Y.. Makarov, D. J. Hll, ad I. A. Hskes. ropertes of quadratc equatos ad ther applcato to power system aalyss. Iteratoal Joural of Electrcal ower ad Eergy Systems, ol., pp ,. []. Balabaa ad. A. Bckart. Electrcal etwork heory. Joh Wley, 969. [3] J. Jarjs ad F. D. Galaa. Quattatve Aalyss of steady state stablty power etworks. IEEE ras. o ower Apparatus ad Systems, ol. AS-, o., Jauary 98l, pp X. BIOGRAHIES Sebastá de la orre (M 4) was bor Málaga, Spa. He receved the Igeero Idustral degree from the Uversdad de Málaga, Spa, 999 ad the hd degree from the Uversdad de Castlla-La Macha, Spa 3. Hs research terests clude operatos ad ecoomcs of electrcal eergy systems, restructurg of electrc systems ad developmet of ew electrcty markets.

10 Fracsco D. Galaa (F 9) receved hs B.Eg. (Hos.) from McGll Uversty, Motreal, QC, Caada followed by the S.M. ad the h.d. degrees from M.I.. He spet several years at the Brow Bover Research Ceter ad at the Uversty of Mchga. He s presetly the rofessor of Electrcal Egeerg at McGll Uversty. Hs research terest have cluded system securty, optmal power flow, epert systems, ad recetly, ope access ad competto power etworks.