A New STATCOM Model for Power Flows Using the Newton-Raphson Method

Transcription

1 > REPLACE THS LNE WTH YOUR PAPER DENTFCATON NUMBER (DOUBLE-CLCK HERE TO EDT) < A New STATCOM Model for Power Flows Using the Newton-Rphson Method Enrique Ach, Senior Member, EEE, nd Behzd Kzemtbrizi, Member, EEE Abstrct The pper presents new model of the STATCOM imed t power flow solutions using the Newton- Rphson method. The STATCOM is mde up of the series connection of oltge Source Converter (SC) nd its connecting trnsformer. The SC is represented in this pper by complex tp-chnging trnsformer whose primry nd secondry windings correspond, notionlly speing, to the SC s AC nd DC buses, respectively. The mgnitude nd phse ngle of the complex tp chnger re sid to be the mplitude modultion index nd the phse shift tht would exist in PWM inverter to enble either rective power genertion or bsorption purely by electronic processing of the voltge nd current wveforms within the SC. The new STATCOM model llows for comprehensive representtion of its AC nd DC circuits this is in contrst to current prctice where the STATCOM is represented by n uivlent vrible voltge source, which is not menble to proper representtion of the STATCOM s DC circuit. One ey chrcteristic of the new SC model is tht no specil provisions within conventionl AC power flow solution lgorithm is ruired to represent the DC circuit, since the complex tp-chnging trnsformer of the SC gives rise to the customry AC circuit nd notionl DC circuit. The ltter includes the DC cpcitor, which in stedy-stte drws no current, nd current-dependent conductnce to represent switching losses. The ensuing STATCOM model possesses unprlleled control cpbilities in the opertionl prmeters of both the AC nd DC sides of the converter. The prowess of the new STATCOM power flow model is demonstrted by numericl exmples where the qudrtic convergence chrcteristics of the Newton-Rphson method re preserved. ndex Terms FACTS, STATCOM, oltge Source Converter (SC), Newton-Rphson method, power flows T. NTRODUCTON HE STATCOM is ey element of the FACTS technology. t is the modern counterprt of the wellestblished Sttic r Compenstor (SC) nd forms the bsic building bloc with which other more dvnced FACTS uipment my be built, such s the UPFC nd the vrious forms of SC-HDC lins. ndeed, the ltter ppliction hs blurred the line between the FACTS nd HDC trnsmission options. n its most bsic form, the STATCOM my be seen to comprise voltge source converter (SC) nd connecting trnsformer which, more Enrique Ach is with the Deprtment of Electricl Energy Engineering t the Tmpere University of Technology (TUT), Tmpere, Finlnd. (emil: enrique.ch@tut.fi). Behzd Kzemtbrizi is with the School of Engineering nd Computer Science of the University of Durhm, Durhm, Englnd, UK (e-mil: bhzdme@me.com ). often thn not, is lod tp-chnging (LTC) trnsformer []-[]. Current models imed t fundmentl fruency studies hve it represented s controllble voltge source behind coupling impednce, very much in the sme vein s the model of synchronous condenser []-[3]. This simple concept represents well the fct tht t the fundmentl fruency, the STATCOM converter s output voltge my be djusted ginst the AC system s voltge to chieve very tight control trgets, cpbility fforded by the switched-mode converter technology []-[8]. By wy of exmple, the rective power flow my be controlled by djusting the converter s output voltge mgnitude ginst the AC system voltge []-[]. The controllble voltge source concept explins the STATCOM s stedy-stte opertion from the vntge of its AC side. However, it fils to explin its opertion from the DC side. A notble exception is the uivlent voltge source model reported in [9], where the STATCOM s AC voltge is expressed s function of the DC voltge nd the mplitude modultion rtio. Nevertheless, incorportion of the switching losses in the DC bus or DC lod would be difficult to represent in this model owing to its uivlent voltge source nture. n most STATCOM models imed t fundmentl fruency power flows there is no esy wy to scertining whether or not the converter s opertion is within the liner region of opertion []. Also, the switching losses tend to be neglected, nd the ohmic losses of the converter, long with the effects of the converter s mgnetics, re normlly lumped together with those of the interfcing trnsformer. To circumvent these shortcomings, new STATCOM model is put forwrd in this pper where the SC is represented by notionl tp-chnging trnsformer nd vrible shunt susceptnce. The primry nd secondry sides of this tp-chnging trnsformer my be interpreted s the SC s AC nd DC sides, respectively. Such SC model tes into ccount, in n ggregted form, the phse shifting nd scling nture of the PWM control. Tht is, its mgnitude nd phse ngle re ssigned to be the mplitude modultion index nd the phse shift tht would exist in PWM inverter to enble either rective power genertion or bsorption purely by electronic processing of the voltge nd current wveforms within the SC. t should be noted tht the SC is designed to operte on constnt DC voltge nd tht reltively smll cpcitor is used to support nd stbilize the voltge t its DC bus. Moreover, this smll rting cpcitor does not contribute per se to the

2 > REPLACE THS LNE WTH YOUR PAPER DENTFCATON NUMBER (DOUBLE-CLCK HERE TO EDT) < rective power exchnge with the power grid []. The new model tes due ccount of the SC switching nd ohmic losses seprtely. t should be noted tht in the new SC model no specil provisions within conventionl AC power flow solution lgorithm is ruired to represent the DC circuit. The reson is tht the complex tp-chnging trnsformer of the SC yields the customry AC circuit nd notionl DC circuit. The SC model is series-connected with the LTC trnsformer model to me up the new STATCOM representtion; model with enhnced control cpbilities in the opertionl prmeters of both the AC nd DC sides of the converter. Such control modelling flexibility ttins specil relevnce when pplied to the relm of SC-HDC or UPFC but these subject mtters re topics of forthcoming publictions. t should be pointed out tht the concept of complex idel trnsformer to model SC hs been pplied elsewhere in connection with the UPFC [, 3]. However, its shunt-connected SC is represented by vrible susceptnce nd it is only its series-connected SC tht is represented by complex idel trnsformer such n pproch represents only n pproximtion to the conventionl two-voltge source model of the UPFC [4, 5]. More importntly, both UPFC models, tht reported in [, 3] nd tht reported in [4, 5], lc DC bus representtion.. NEW SC MODEL A. SC min chrcteristics The STATCOM comprises the series connection of SC nd n LTC trnsformer whose primry winding is shunt-connected with the AC power networ. Physiclly, the SC is built s two-level or multi-level inverter tht uses converter bridge mde up of self-commutting switches driven by PWM control. t uses smll cpcitor bn on its DC side to support nd stbilize the DC voltge to enble converter opertion. The converter eeps the cpcitor chrged to the ruired voltge level by ming its output voltge lg the AC system voltge by smll phse ngle []. The DC cpcitor bn of vlue C DC is shown schemticlly in Fig. (). t should be stted tht C DC is not used per se in the AR genertion/bsorption process. nsted, this process is crried out by ction of the PWM control which shifts the voltge nd current wveforms within the SC to yield either leding or lgging AR opertion to stisfy opertionl ruirements. t is sid tht the SC hs no inerti, its response is prcticlly instntneous, it does not significntly lter the existing system impednce nd it cn internlly generte rective (both cpcitive nd inductive) power []. For the purpose of fundmentl fruency nlysis, the SC s electronic processing of the voltge nd current wveforms is well synthesized by the notionl vrible susceptnce, B, which connects to the AC bus of the idel complex tpchnging trnsformer - see Fig. (b). Note tht B is responsible for the whole of the rective power production in the vlve set of the SC. B. SC nodl dmittnce mtrix representtion The fundmentl fruency opertion of the SC shown schemticlly in Fig. () my be modeled by mens of electric circuit components, s shown in Fig. (b). From the conceptul point of view, the centrl component of this SC model is the idel tp-chnging trnsformer with complex tp which, in the bsence of switching losses, my be seen to ct s nulltor tht constrins the source current to zero, with the source being the cpcitor C DC, nd the ssocited nortor being the vrible susceptnce B [7]. ndeed, in stedy-stte opertion the DC cpcitor my be represented s bttery tht yields voltge E DC nd drws no current [8] this point is ddressed in more detil in Appendix A. Notice tht the winding connected to node is n AC node internl to the SC nd tht the winding connected to node is notionl DC node. Two elements connect to the SC s DC bus, nmely, the source, E DC, nd the current dependent resistor, G sw. Hence, the idel tp-chnging trnsformer is the element tht provides the interfce for the SC s AC nd DC circuits, s illustrted in Fig. (b). t should be emphsized tht no rective power flows through it, only rel power which is in to DC power. + E DC - + E DC - DC circuit = C DC () Figure : () SC Schemtic Representtion; (b) SC uivlent circuit We hve drwn our inspirtion to develop this model, from the following bsic reltionship: j e E () DC where the tp mgnitude m of the idel tp-chnging trnsformer corresponds to the SC s mplitude modultion coefficient where the following reltionship holds for two-level, three-phse SC: m : G sw jb ALE SET, AC circuit jx (b) 3 m, where in the liner rnge of modultion, the mplitude R

3 > REPLACE THS LNE WTH YOUR PAPER DENTFCATON NUMBER (DOUBLE-CLCK HERE TO EDT) < 3 modultion index m tes vlues within bounds: m [9]. The phse ngle is the phse ngle of the complex voltge reltive to the system phse reference, nd E DC is the DC bus voltge which is rel sclr nd on per-unit bsis crries vlue of. Other elements of the electric circuit shown in Fig. (b) re the series impednce which is connected to the idel trnsformer s AC side. The series rectnce X represents the SC s interfce mgnetics. The series resistor R ccounts for the ohmic losses which re proportionl to the AC terminl current squred. Note tht the secondry winding current which is lwys rel quntity, splits into nd. The ltter current is lwys zero during stedy-stte opertion. This is further elborted in Appendix A, where the role of the SC s phse-shifting trnsformer is nlyzed from the vntge of electronic circuits [7]. As one would expect, the complex power conservtion property of the idel trnsformer in Fig. (b) stnds but note tht there is no rective power flowing through it, since ll the rective power ruirements of the SC model (genertion/bsorption) re met by the shunt brnch B connected t node. The power reltionships between nodes nd, which ccount for the full SC model, re: * * *, ( ) jb Re * () The switching loss model corresponds to constnt resistnce (conductnce) G, which under the presence of constnt DC voltge nd constnt lod current, would yield constnt power loss for given switching fruency of the PWM converter. Admittedly, the constnt resistnce chrcteristic my be inccurte becuse lthough the DC voltge is ept lrgely constnt, the lod current will vry ccording to the previling operting condition. Hence, it is proposed tht the resistnce chrcteristic derived t rted voltge nd current be corrected by the qudrtic rtio of the ctul current to the nominl current, ct G nom where G sw would be resistive term exhibiting degree of power behvior. The voltge nd current reltionships in the idel tpchnging trnsformer re: nd G sw The current through the dmittnce connected between nodes nd is: Y m Y (3) (4) Y (5) where Y ( R jx ). At node, the following reltionship holds: m ( ) G m Y G m sw sw Y jb Combining (5) nd (6) nd incorporting constrints from the electric circuit in Fig. (b): (6) Y Y (7) Y Gsw ( Y jb) EDC nd more explicitly: Y cos jsiny cos jsin Y G ( Y jb sw DC ) E (8) Notice tht this expression represents the SC uivlent circuit in Fig. (b) in stedy-stte, with the cpcitor effect represented by the DC voltge E DC. C. SC nodl power utions The complex power model is derived from the nodl dmittnce mtrix where, subsuently, the DC voltge will be referred only s s opposed to E DC : S S Y cos jsin Y cos jsin Y Gsw ( Y jb) (9) Following some rduous lgebr, the nodl ctive nd rective power expressions re rrived t: P Q P Q G G cos B sin B G sin B cos G Gsw G cos B sin ( B B ) G sin B cos () D. SC linerised system of utions These utions re non-liner nd their solution, for pre-defined set of genertion nd lod pttern my be crried out using the Newton-Rphson method. This involves repeted lineriztion of the nodl power utions. Their initil evlution ruires n informed guess of the stte vrible vlues: () () () () () () (,,,,, B ), when the im is to regulte voltge mgnitude t bus using the SC s mplitude modultion rtio (m ) nd eep t constnt vlue. n prctice, the ltter is possible due to the DC cpcitor s ction. The linerized system of utions is: P P Q Q P P Q P P Q Q ( P ( Q P Q P ) ) P Q P P Q P P B Q Q B P P B P P B Q Q B B ()

4 > REPLACE THS LNE WTH YOUR PAPER DENTFCATON NUMBER (DOUBLE-CLCK HERE TO EDT) < 4 Subsuent evlutions of the nodl power utions re crried out using the improved set of vlues being furnished by the itertive process: (,,,,, B ), where (r) is the itertion counter. n this ppliction, the regulted powers P, reg nd Q, reg lso form prt of the control set. The entries ming up. () re given in Appendix B. ) Mismtch power terms nd control vribles: A mismtch power term is the difference between the net power nd the clculted power t given bus, sy, nd. The clculted powers re determined using the nodl power utions (), giving, P P Q P P P Q,net Q,net,net Q Q,net P Q P P,reg,cl Q,reg,cl Q,cl,cl ( P ( P P,gen ( Q,cl Q,gen,cl,gen ( Q,gen P P,lod Q,lod,lod Q,lod ) P ) P,cl ) Q,cl,cl ) Q,cl () The mismtch power flow in brnch - is the difference between the trget power flow t the brnch nd the clculted power. n the SC ppliction, both ctive nd rective power trgets re normlly set to zero. ) Stte vribles nd increments: The stte vrible increments clculted t itertion (r) with the power flow model re: B B ( r ) B ( r ) ( r ) ( r ) (3) 3) Non-regulted solutions: f no voltge regultion t node is pplied, the voltge mgnitude replces m s stte vrible in the linerized power flow ution (). Other control options my be vilble, but some cution needs to be exercised in the SC nd STATCOM pplictions becuse power regultion t node cnnot be chieved since the internl power losses re not nown priori, nd voltge control in the DC node is chieved by virtue of the DC cpcitor. 4) Prcticl implementtions: ) Control Strtegy: As illustrted in Fig. (b), the SC is ssumed to be connected between sending bus,, nd receiving bus,, with the former ten to be the SC s AC bus nd the ltter ten to be the SC s DC bus. The voltge is ept constnt by the ction of smll DC cpcitor bn with rted cpcitnce C DC, which in stedy-stte drws no current. n the Newton-Rphson power flow solution the DC bus will be treted s P-type node with zero nodl power injection nd constnt voltge mgnitude of vlue E DC. Liewise, the voltge mgnitude is regulted within system-dependent mximum nd minimum vlues, fforded by the following bsic reltionship: E R X DC (4) Note tht in the SC s liner rnge of modultion, the index m tes vlues within the bounds: m nd tht 3 m. However, in power systems rective power control pplictions, it is unliely tht vlues of m lower thn.5 will be used. The reson is tht voltge mgnitude t the SC s AC bus must be ept within prcticl limits becuse too high voltge my induce insultion coordintion filure t the point of connection with the power grid nd too low voltge my induce condition of voltge collpse. Note tht with relistic vlues of R =. p.u., X =. p.u. nd E DC = p.u. nd considering lowcurrent opertion, sy. p.u., will te vlue of.64 p.u. with m =.5. n the power flow solution the ctive nd rective powers re regulted on the SC s DC bus the former is set to either zero or to specified DC lod, wheres the ltter is lwys set to zero. b) Simplifying ssumptions: A ey feture of this model is tht the phse ngle vlue t node is independent of circuit prmeters or networ complexity to the left of the phse-shifting trnsformer. The reson is tht the idel phse shifter decouples, ngle-wise, the circuits to the left nd to the right of the idel trnsformer. Moreover, the phse ngle voltge t bus eeps its vlue given t the point of initiliztion. Hence, in the ppliction pursued in this pper, it mes sense to stic to zero phse ngle voltge initiliztion for this bus - when looed t it from the vntge of rectngulr coordintes, its imginry prt does not exist. This my reduce the linerized ution () by one row nd one column since the vlue of is nown priori, i.e., =. c) nitil prmeters nd limits: Three SC prmeters ruire initiliztion. They re the mplitude modultion rtio (m ) nd its phse ngle (). They re normlly set t 3 nd, respectively. The SC is ssumed to operte within the liner region, wheres the phse ngle is ssumed to hve no limits. The third prmeter is the uivlent shunt susceptnce (B ), wich is given n initil vlue tht lies within the rnge B + nd B -. E. SC Test Cses The SC model is pplied in rther contrived test cse where the STATCOM is connected t the receiving end of loded trnsmission line to illustrte its performnce, nd for ese of reproduction. At this point in the pper, it is ssumed tht the STATCOM trnsformer is conventionl trnsformer nd tht its lege rectnce is lumped together with the rectnce of the SC. Hence, we shll refer to it s SC s opposed to STATCOM. Three cses re considered: (i) the SC is used to provide rective power; (ii) the SC is used to drw rective power; nd (iii) the SC is used to supply DC lod.

5 > REPLACE THS LNE WTH YOUR PAPER DENTFCATON NUMBER (DOUBLE-CLCK HERE TO EDT) < 5 ) Test Cse The three-node system shown in Fig. comprises one genertor, one trnsmission line nd one AC/DC converter (SC), which is represented by the elements shown within the broen-line rectngle j Figure : SC providing voltge support t bus The genertor node is ten to be the Slc bus where the voltge mgnitude is ept t p.u. nd its phse ngle provides reference for ll other phse ngles in the networ, excepting bus, where the phse ngle is lwys zero in the STATCOM or SC ppliction. Bus would be interpreted s the DC bus of the SC circuit where the voltge is lwys rel quntity. The following prmeters re used in this system - (i) trnsmission line resistnce nd rectnce:.5 p.u. nd. p.u.; (ii) SC series resistnce nd rectnce:. p.u.,. p.u.; (iii) SC nominl vlues of shunt conductnce nd susceptnce:. nd.5 p.u.; (iv) ctive nd rective power lod t node :.5 p.u. nd. p.u. As lredy stted in Section 4(b), the phse ngle vlue t node is independent of circuit prmeters, networ complexity nd initilizing conditions left of the phse shifter trnsformer - it is not specific to this circuit under test. To prove this point, different initil vlues re given to the Slc bus nd the resulting voltges shown in Tble. t should be noted tht the phse ngle voltge t bus eeps its vlue given t the point of initiliztion nd tht in the ppliction pursued in this pper, we shll stic to zero phse ngle voltge initiliztion for this bus. When looed t it from the vntge of rectngulr coordintes, its imginry prt does not exist. ndeed, n uivlent solution would be obtined by using linerized ution in to () but with no provision for the stte vrible. TABLE POWER FLOW SOLUTON FOR AROUS PHASE ANGLES AT THE SLACK BUS (p.u.) (p.u.) (p.u.) The phse ngle difference between buses nd is, in ech cse: The Newton-Rphson power flow lgorithm converges in 7 itertions in ll three cses, to mismtch tolernce of -. The symbol is used in this tble to signify in to. The SC consumes.7 p.u. of ctive power from the system to ccount for its internl losses whilst supplying.887 p.u. of rective power to the system. The uivlent susceptnce (in cpcitive mode) produces.953 p.u. of rective power nd its cpcitive susceptnce stnds t B =.748 p.u. As one would expect, the SC switching losses re %, corresponding to conductnce G =%. The DC bus voltge is controlled t.44 p.u. nd the voltge mgnitude t bus is ept t.5 with true m =.957. Notice tht m =.87. The phse shifter ngle tes vlue of The line current drwn by the SC is For the se of completeness, the test cse is solved by modeling the SC using its well-nown representtion bsed on the uivlent voltge source []-[4], which, in this cse, hs been extended to incorporte shunt resistor to ccount for the SC s switching losses j. Figure 3: Test circuit using the conventionl voltge source representtion of the SC. Note tht ll the relevnt prmeters for this circuit re the sme s in the circuit in Fig., except tht the resistnce corresponding to the switching losses is connected on the left-hnd side of the complex tp chnger nd, ccordingly, it is ffected by the squre of the off-nominl turns rtio m, i.e., R =./.87. Node is treted s P-type bus with zero ctive power injection nd its voltge mgnitude corresponds to the DC-lie voltge of.44 p.u. in the circuit of Fig., ffected by m, i.e., =.44.87=.338. The results were obtined using conventionl power flow progrm where bus is treted s P bus with zero ctive power contribution nd set to regulte voltge mgnitude t the bus t.338 p.u. As expected, the itertive solutions furnished by both modeling pproches yield similr results but the results t bus merit dditionl nlysis. The complex voltge t the uivlent voltge source corresponds to the cscding of the voltge t bus in Fig. nd its phse shifter complex tp vlue. Furthermore, the rective power contributed by the uivlent susceptnce in the test circuit of Figure uls the rective power generted by the uivlent voltge source in the test circuit of Fig. 3. The following limittions spring to mind in the voltge source model of the SC compred to the new model introduced in this pper: (i) the voltge mgnitude of the voltge source is difficult to determine since only the DC voltge is nown nd the mplitude modultion index (m ) is not nown priori; (ii) by the sme toen, the switching losses will only be nown pproximtely. n this numericl exmple, the switching loss correction given by. (3) ws not pplied in order to be ble to..953.

6 > REPLACE THS LNE WTH YOUR PAPER DENTFCATON NUMBER (DOUBLE-CLCK HERE TO EDT) < 6 compre the response furnished by the two SC models, nmely, the new SC model nd the uivlent voltge source model. n ny cse, little chnge is expected since the current mgnitude (.84 p.u.) is close to the p.u. rted current. Perhps the most noticeble chnge is reduction in the switching loss from % to.4% nd the ensuing djustment in ctive power flows. b) Test Cse The operting conditions of the power circuit in Test Cse re modified to force the SC to drw rective power from the slc genertor connected t bus j corresponds to ohmic loss. The SC contributes.46 p.u. to supply the rective power lod of. p.u. nd the rest being exported to the Slc genertor. The SC uivlent susceptnce with cpcitive vlue of B =. p.u. produces.3634 p.u. of rective power. The SC is set to regulte voltge mgnitude t its AC bus t.5 p.u. nd its ctul complex modultion rtio is: The current drwn by the SC is The solution converges in 7 itertions to tolernce of -.. POWER FLOW STATCOM MODEL For studies t the fundmentl fruency, the STATCOM my be seen to comprise SC nd n interfcing trnsformer, which my be lod tp chnger (LTC). The SC schemtic representtion nd uivlent circuit re given in Fig. nd the uivlent circuit of the LTC trnsformer is given in Fig. 6. t T : t t X l R l Figure 4: The test networ uses the sme circuit prmeters s in Test Cse but the voltge mgnitude t bus is ept t.95 p.u. using m to force the rective power flow into the SC. The SC drws.7 p.u. of ctive power nd.493 p.u. of rective power. The uivlent susceptnce bsorbs.469 p.u. of rective power nd its inductive susceptnce stnds t B =-.68 p.u. The SC switching losses re low,.5%, since the current drwn by the SC is quite smll, i.e p.u. The DC bus voltge is controlled t.44 p.u. nd the voltge mgnitude t bus is ept t.95 with m =.768. The phse shifter ngle tes vlue of c) Test Cse 3 Test Cse is expnded to incorporte lod in the DC side of the SC in the form of bttery system which is ssumed to te constnt power of.5 p.u j. Figure 5: Test networ with bttery lod on its DC bus m =.948 = This test networ uses the sme circuit prmeters s in Test Cse but second lod is dded in the form of bttery which is being supplied through the SC t.5 p.u. of power. The SC is used to eep the voltge mgnitude t.5 p.u. t bus. The totl SC ctive power loss stnds t 4.76% p.u. where 3.8% corresponds to switching loss nd.58%.5 Figure 6: LTC trnsformer uivlent nclusion of the STATCOM model in power flow solution is strightforwrd. t only ruires explicit representtion of the nodl power flow utions of the SC connected between sy, nodes nd R, nd the nodl power utions of the LTC trnsformer connected between sy, nodes R nd K. Alterntively, more compct set of power flow utions my be chieved by relizing tht the interfce point between the SC nd LTC circuits, nmely node, receives zero externl (nodl) current injection. Then mthemticl elimintion of node becomes n option. However, it should be noted tht this reduced model is only ttrctive if we re prepred to lose degree of modeling flexibility, since this bus is not explicitly vilble for regulting ction of either T or m. nsted, the combined regulting ction will te plce in the highvoltge side of the LTC trnsformer. A. Reduced STATCOM nodl dmittnce mtrix The nodl dmittnce mtrix of the LTC trnsformer in Fig. 6, is: Y l TYl TY l T Y l (5) Combining the two individul models yields the compound model representing the SC-LTC or STATCOM model: Y l TY l TY l T Y l Y Y Y G ( Y jb ) sw EDC (6)

7 > REPLACE THS LNE WTH YOUR PAPER DENTFCATON NUMBER (DOUBLE-CLCK HERE TO EDT) < 7 Mthemticl elimintion of node yields the following reduced nodl dmittnce mtrix: YYl T YlY T YlY T YlY ( T Yl Y) Y EDC (7) where T Yl Y nd Y Gsw j B. B. STATCOM nodl power utions Following similr procedure s in section -C for the derivtion of the nodl power utions of the SC, the ctive nd rective power expressions for the STATCOM model re derived: P Q P Q G T G cos B sin l l l B l T G l sin B l cos T G l T G l G T G l cos B l sin T B T B B T G sin l l l B cos l (8) where G T G G B G G B l l l l B T B G B B G B l l l l G T Bl G B BG sw T Gl GG sw B B Gsw G B B T G G B BG T B GG B B B G B l sw l sw G T G G B B BG G B G G G B B l sw l l l sw l sw B l T B Gl Bl Gl BG sw G B Bl GG sw B B T G G T B B l l The numericl solution of ution system (8), for pre-defined set of genertion nd lod pttern, is crried out very efficiently by itertion using the Newton-Rphson method. Similrly to the SC model in Section -C, this involves repeted lineriztion of the nodl power utions nd their initil evlution ruires n informed guess of the stte vribles vlues: ( ) ( ) ( ) ( ) ( ) (,, T, m,, B ). The linerized system of utions my be compcted further by eliminting the row nd column ssocited to the vrible, since this is priori nown vrible tht eeps its vlue t the point of initiliztion, which in this ppliction is zero. The ensuing ution is: P Q P Q P Q P Q P Tm T Q Tm T P Q Tm T Tm T P P B Q Q B Tm Tm P P B Q Q B B where Tm is used to signify the use of either T or m. (9) The ttrction of ution (9) is its rther compct nture in representing the combined opertion of the SC nd the LTC trnsformer with only four vribles. However, this comes t price some modeling flexibility is lost. Notice tht since the connecting bus between the SC nd the LTC is not explicitly vilble in this combined model, it cnnot be controlled by the regulting ction of either T or m. Also, since the DC bus is regulted by the ction of the DC cpcitor nd treted in the power flow solution s P bus then T nd m re vilble solely for the purpose of regulting voltge mgnitude t the high-voltge bus of the LTC trnsformer. Hence, the regulting ction of T nd m is suentil in this model. t should be emphsized tht, from the power flow solution vntge, there is no ctul restriction in ttempting to control the DC bus voltge with either T or m. However, from the uipment opertion point of view, this regulting ction is hrdly ever done. Subsuent evlutions of the nodl power utions re crried out using the improved set of vlues furnished by ( r ) ( r ) ( r ) ( r ) ( r ) ( r ) the itertive process: (,, T, m,, B ), where (r) is the itertion counter. t should be noticed tht in this formultion, the control cpbilities hve been extended compred to tht of the SC in (). t becomes possible to regulte nodl voltge mgnitude t the STATCOM terminl (bus ) using the combined ction of the LTC tp (T) nd the SC mplitude modultion coefficient (m ), one t the time. t should be remred tht in n ctul SC, m tes continuous vlues nd tht in n ctul LTC trnsformer, the tp T tes discrete vlues. Nevertheless, for the purpose of the power flow model using the Newton- Rphson method nd iming t mintining the qudrtic convergence chrcteristic of this itertive lgorithm, the vrible T is ssumed to te continuous vlues. t is t the end of the itertive solution tht the tp T is moved to the nerest physicl tp vlue nd then nodl voltges re redjusted nd power flows nd power losses clculted. The mismtch power terms nd control vribles remin the sme s in (), except tht the subscript replces the subscript. n the stte vribles increments in (3) the subscript is lso replced by the subscript nd the newly introduced stte vrible Tm replces m, T T T () where T nd m re normlly initilized t nd 3, respectively. C. STATCOM Test Cses Two test cses re presented in this section to illustrte the control flexibility fforded by the reduced STATCOM model. The first cse reltes to contrived system which is, essentilly, the sme system s tht used in Test Cse, except tht the STACOM model replces the SC model. The second test cse is modified version of the EEE 3- node system [6] where two STATCOMs regulte voltge mgnitude t two different points in the networ.

8 > REPLACE THS LNE WTH YOUR PAPER DENTFCATON NUMBER (DOUBLE-CLCK HERE TO EDT) < 8 ) Test Cse 4 The power circuit in Test Cse is modified to replce the SC connected t bus by STATCOM, where the LTC s STATCOM figures prominently in Fig j. m =.8945 =-4.59 Figure 7: Upgrded networ used in Test Cse, to include the LTC trnsformer The test networ uses the sme circuit prmeters s in Test Cse except tht the prmeters of the LTC trnsformer re dded to the circuit prmeters: R T =. p.u. nd X T =. p.u. The tp limits re:.8<t<.. The genertor eeps the voltge mgnitude t the slc node t p.u. The STATCOM consumes.34 p.u. of ctive power from the system to ccount for its internl losses whilst supplying.8836 p.u. of rective power to the system. The SC switching losses stnd t G =.4% nd the remining.6% correspond to ohmic losses in the LTC trnsformer nd SC. The DC bus voltge is ept t.44 p.u. by ction of the DC cpcitor nd this bus is treted in the power flow solution s P bus. The voltge mgnitude t bus is ept t.5 p.u. with combintion of selected m of.8945 nd resulting trnsformer tp of T=.335. The current drwn by the STATCOM is b) Test Cse 5 n order to test the performnce of the proposed STATCOM model in lrger power networ, the EEE 3- node system is selected [6]. The fix bns of cpcitors t nodes nd 4 in the originl networ re replced with STATCOMs which re set to regulte voltge mgnitudes t their points of connection with the power grid. Their respective DC voltges re ept t.44 p.u. The relevnt portions of the modified 3-bus system re shown in Fig. 8. The voltge mgnitudes t the compensted buses, nmely, nd 4, re compred in Tble to the cse when conventionl cpcitor bns re connected to these nodes, nd when no compenstion is used. TABLE OLTAGE MAGNTUDES AT THE COMPENSATED BUSES N THE 3-BUS SYSTEM FOR TWO COMPENSATON OPTONS Compenstion Cse T= OLTAGE MAGNTUDE (P.U.) Bus Bus 4 None Fix STATCOMs The two STATCOMs use identicl prmeters nd their LTC trnsformers re set t their nominl tp positions (T=). They re ssumed to contin no resistnce nd their rectnces re X TR =.5 p.u. The SCs series nd shunt prmeters, in per-unit, re: R =., X =.5, G sw =. nd B =.5, respectively. node 6 m =.885 = m =.7979 =-5.3 node j. node ().87+j.67 (b) node node node 9 node node node node node 5 Figure 8: STATCOMs supplying rective power t buses nd 4 of the modified EEE 3-bus system The susceptnce vlues used for the cse with fix compenstion t buses nd 4 re.9 p.u. nd.43 p.u., which re the vlues given in [6]. For the STATCOM cse, the voltges t buses nd 4 re ept t the sme level s those given by the cse with fix compenstion. As expected, one benefit of shunt compenstion is to reduce the system power losses due to n improved voltge profile, nd this trend is shown in the power loss figures presented in Tble. The STATCOM-type compenstion introduces n dditionl ind of power loss which is ssocited with the high-fruency switching of the PWM control used by the SC technology nd ohmic losses. The STATCOM losses re quite low in this cse becuse the currents drwn by the two STATCOMs re low compred to the p.u. rted currents nmely, p.u. nd p.u. TABLE POWER LOSS AT THE COMPENSATED BUSES N THE 3-BUS SYSTEM FOR TWO COMPENSATON OPTONS Compenstion ACTE POWER LOSS (%) Cse Networ STATCOMs None 3. - Fix.89 - STATCOMs.94. The power flow solutions converged in 6 itertions for the first two cses nd in 7 itertions for the STATCOMs, to mismtch tolernce of -.

9 > REPLACE THS LNE WTH YOUR PAPER DENTFCATON NUMBER (DOUBLE-CLCK HERE TO EDT) < 9. CONCLUSONS A new STATCOM model imed t power flow solutions using the Newton-Rphson method hs been introduced. The model represents prdigm shift in the wy the fundmentl fruency, positive suence SC-FACTS controllers re represented. t does not tret the controller s n idelized controllble voltge source but rther s compound trnsformer device to which certin control properties of PWM-bsed inverters my be lined. This rgument is similr to the one dvnced for DC-to-DC converters which hve been lined, conceptully speing, to step-up nd step-down trnsformers [9]. The phse ngle of the complex tp chnger represents the phse shift tht would exist in PWM inverter nd coincides with the phse ngle of the conventionl voltge source model of the SC. More specificlly, this would be the phse ngle ruired by the SC to enble either rective power genertion or bsorption purely by electronic processing of the voltge nd current wveforms within the SC. The switching losses, ohmic losses nd the connecting LTC trnsformer re ll explicitly represented in the new STATCOM model. The complex tp chnger in the SC model nd the rel tp chnger in the LTC model enble n effective voltge regultion t the point of connection with the grid nd t the SC s AC node. The model hs been tested in simple system for ese of reproduction by interested prties. A lrger power system hs lso been used to show tht the new STATCOM power flow model retins its strong convergence chrcteristics. ACKNOWLEDGEMENTS The uthors wish to cnowledge the most vluble nd insightful criticisms mde by the referees of this pper, which hve enhnced the theoreticl bsis on which this STATCOM model hs been developed. REFERENCES [] G.N. Hingorni nd L. Gyugyi, Understnding FACTS: concepts nd technologies of flexible c trnsmission systems. EEE. [] E. Ach, C.R. Fuerte-Esquivel, H. Ambriz-Perez nd C. Angeles- Cmcho, FACTS modeling nd simultion in power networs. John Wiley & Sons, 5. [3] D. J. Gothm nd G. T. Heydt, Power flow control nd power flow studies for systems with FACTS devices, EEE Trns. Power Systems, vol. 3, pp. 6-65, Feb. 998 [4] X. Zhng nd E. J. Hndschin, Optiml power flow control by converter bsed FACTS controllers, presented t Seventh nt. Conf. on AC-DC Power Trnsmission, (Conf. Publ. No. 485), pp. 5-55, 8-3 Nov. [5] D.M. Brod nd D.M. Novotny, Current Control of S-PWM nverters, EEE Trns. on ndustry Applictions, vol., no. 4, pp , 985. [6] H.W. vn der Broec, H.C. Sudelny nd G.. Stne, Anlysis nd Relistion of Pulsewidth Modultor bsed on oltge Spce ectors, EEE Trns. on ndustry Applictions, vol. 4, no., pp. 4-5, 988. [7] R. Wu, S.B. Dewn nd G.R. Slemon, A PWM AC-to-DC Converter with Fixed Switching Fruency, EEE Trns. on ndustry Applictions, vol. 6, no. 5, pp , 99. [8] R. Wu, S.B. Dewn nd G.R. Slemon, Anlysis of n PWM AC-to- DC oltge Source Converter Using PWM with Phse nd Amplitude Control, EEE Trns. on ndustry Applictions, vol. 7, no., pp , 99. [9] C.A. Cñizres, Power Flow nd Trnsient Stbility Models of FACTS Controllers for oltge nd Angle Stbility Studies, EEE PES WM, 3-7 Jn., Singpore, pp ,. [] C. Angeles-Cmcho, O. L. Tortelli, E. Ach nd C. R. Fuerte- Esquivel, nclusion of high voltge dc-voltge source converter model in Newton-Rphson power flow lgorithm, in EE Proc. Gen., Trns. nd Dist.., vol. 5, pp , Nov. 3 [] L. Gyugyi, Dynmic Compenstion of AC Trnsmission Lines by Solid-Stte Synchronous oltge Sources, EEE Trns. on Power Delivery, vol. 9, no., pp. 94-9, April 994. [] S. An nd T.W. Gedr, UPFC del Trnsformer Model, Proc. North Americn Power Symposium (NAPS), pp. 46-5, Oct. 3. [3] S. An, J. Condren nd T.W. Gedr, An del Trnsformer Model, OPF First-Order Sensitivities, nd Appliction to Screening for Optiml UPFC Loctions, EEE Trns. on Power Systems, vol., no., pp , Feb. 7. [4] C.R. Fuerte-Esquivel, E Ach nd H. Ambriz-Perez, A Comprehensive Newton-Rphson UPFC Model for the Qudrtic Power Flow Solution of Prcticl Power Networs, EEE Trns. on Power Systems, vol. 5, no., pp. -9, Feb.. [5] H. Ambriz-Perez, E Ach, C.R. Fuerte-Esquivel, nd A. de l Torre, ncorportion of UPFC Model in n Optiml Power Flow Using Newton s method, EE Proc. Gen., Trns. nd Dist., vol. 45, no. 3, pp , My 998. [6] EEE 3-node Test System. Avilble: [7] C.J.M. erhoeven, A. vn Stveren, G.L.E. Monn, M.H.L. Kouwenhoven nd E. Yildiz, Structured Electronic Design: Negtive Feedbc Amplifiers. Kluwer Acdemic, 3. [8] J.W. Nilsson nd S. Riedel, Electric Circuits (9 th Edition). Prentice Hll,. [9] N. Mohn, T.M. Undelnd nd W.P. Robins, Power Electronics: Converters, Applictions nd Design. John Wiley & Sons, 3. APPENDX A: THE DEAL PHASE SHFTER CRCUT One slient chrcteristic of the new SC model is tht no specil provisions within conventionl AC power flow solution lgorithm is ruired to represent the DC circuit, since the complex tp-chnging trnsformer of the SC my be used with ese to give rise to the customry AC circuit nd notionl DC circuit. However, some further explntion is ruired since the modelling development involves the confltion of AC nd DC circuit concepts t n uivlent node, brought bout by the use of the idel tpchnging trnsformer concept. n order to elborte the explntion from the vntge of electronic circuits, we re going to ssume tht the conductnce ssocited with switching losses, G sw, in Fig. (b), my be referred to the primry side of the idel trnsformer. The relevnt prt of the circuit illustrting such sitution but with cpcitor representtion, s opposed to its uivlent bttery representtion, is shown in Fig. A., + E DC - = C DC Figure A.: Equivlent circuit showing the idel phse-shifting trnsformer Y of Fig. (b) nd neighboring elements, where Y : m e j, G jb. sw

10 > REPLACE THS LNE WTH YOUR PAPER DENTFCATON NUMBER (DOUBLE-CLCK HERE TO EDT) < By invoing. (4), e -j e ( ) (A.) j e m j j e E DC (A.) n stedy-stte, chrged DC cpcitor drws zero current nd it is well-ccepted tht it my be represented s chrged bttery [8] nd, by extension, s DC voltge source feeding no current. These fcts re reflected by utions (A.) nd (A.) nd give the opportunity to interpret the circuit in Fig. A. in terms of electronic circuits concepts. Hence, it my be rgued tht in stedystte this circuit behves s nullor operting on DC source representing the DC cpcitor. The nullor is mde up of nulltor nd nortor [7], represented in this cse by the idel phse-shifting trnsformer nd the uivlent dmittnce, Y, respectively. The circuit in Fig. A. my be re-drwn s follows, + E DC - = nulltor - = + m e Figure A.: nterprettion of the uivlent circuit of Fig. A. in terms of electronic circuit elements The nulltor nd the nortor re sid to be liner, timeinvrint one-port elements. The former is defined s hving zero current through it nd zero voltge cross it. The ltter, on the other hnd, cn hve n rbitrry current through it nd n rbitrry voltge cross its terminls. Nulltors hve properties of both short-circuit (zero voltge) nd open-circuit (zero current) connections. They re current nd voltge sources t the sme time. A nortor is voltge or current source with infinite gin. t tes whtever current nd voltge is ruired by the externl circuit to meet Kirchhoff s circuit lws. A nortor is lwys pired with nultor [7]. Either, by creful exmintion of utions (A.) nd (A.) or by nlysis of the electronic uivlent circuit in Fig. A., it cn be seen tht the idel, complex tp-chnging trnsformer of the SC gives rise to the customry AC circuit nd notionl DC circuit where the DC cpcitor yields voltge E DC but drws no current. n more generl sense nd from the viewpoint of the AC power flow solution, if resistive elements or DC power lods re connected to the notionl DC bus then currents do pss through the idel phse-shifting trnsformer but it would be component of current tht yields nodl voltge with zero phse ngle nd, s one would expect, yields power with no imginry component, hence, no rective power exists in this prt of the notionl DC circuit. j nortor APPENDX B: PARTAL DERATE TERMS FOR THE SC The prtil derivtive terms ming up the Jcobin mtrix in n. () re given below. Note tht these derivtive terms do not include the current dependency in the switching loss term G SW refer to ution (3). P Q B (B.), cl Q P G (B.), cl P m m P G (B.3) Q m m Q B,cl,cl (B.4),cl P Q B (B.5) Q P G (B.6),cl Q B B,cl P m m P G G,cl P Q m B B P (B.7) (B.8) (B.9),cl P Q B (B.), cl Q P G (A.), cl P B (B.) Q B (B.3) P Q,cl B B (B.4) P B (B.5) P P (B.6) P m m P m m (B.7) P P (B.8) P P (B.9) B P (B.) P B Q Q P,cl G m G Q m m Q m m Q,cl m B B Q Q P G m G (B.) (B.) (B.3),cl G m G Q (B.4) Q P,cl Q B Q B m (B.5) Enrique Ach (SM ) ws born in Mexico. He grduted from Universidd Michocn in 979 nd obtined his PhD degree from the University of Cnterbury, Christchurch, New Zelnd, in 988. He ws the Professor of Electricl Power Systems t the University of Glsgow, UK in the period - nd he is now the Professor of Electricl Power Systems t the Tmpere University of Technology (TUT), Finlnd. He is n EEE PES distinguished lecturer. Behzd Kzemtbrizi ws born in Tehrn, rn. He received his BSc in Electricl Power Engineering from Azd University, Tehrn rn in 6. He then received his MSc nd PhD degrees in Electronics nd Electricl Engineering from the University of Glsgow, Scotlnd, UK in 7 nd, respectively. He is now with the School of Engineering nd Computer Science in Durhm University woring s Reserch Associte. He hs been student member of EEE since 7.