IEEE TRANSACTIONS ON, VOL., NO. -, 201-1 Optiml Power Flow OPF) Moel with Unifie AC-DC Lo Flow n Optiml Commitment for n AC-ctenry Rilwy Power Supply System RPSS) fe by High Voltge DC HVDC) trnsmission line Lrs Abrhmsson, Stuent Member, IEEE, Stefn Östlun, Senior Member, IEEE, n Lennrt Söer, Member, IEEE Abstrct In this pper n lterntive rilwy power systems esign bse on n HVDC feeer is stuie. The HVDC feeer is connecte to the ctenry by converters. Such n HVDC line is lso pproprite for DC-fe rilwys n AC-fe rilwys working t public frequency. A unit commitment optiml power flow moel hs been evelope n is pplie on test system. In this pper, the moel is presente in etil. The moel, in the form of n MINLP progrm, uses unifie AC-DC power flow to minimize the entire rilwy power system losses. Simultions of the propose solution show cler vntges regring trnsmission losses n voltges compre to conventionl systems, especilly for cses with long istnces between feeing points to the ctenry, n when there re substntil mounts of regenertion from the trins. I. INTRODUCTION No moel for unifie AC-DC power flow combine with unifie optiml power flow n optiml commitment of converters hve been presente for rilwy pplictions before. This ws completely new. THIS pper presents n suggests moels for new wy of feeing rilwy power supply systems RPSS) tht woul improve voltge qulity n reuce power losses without moifying the ctenry line impences, ing conuctors, or ing extr connections between ctenry n feeing lines. A feeer concept comprising multi-terminl HVDC MTDC) supply line is investigte [1], [2]. Inste of connecting the rectifier n inverter bck-to-bck s in conventionl sttic converter sttion, rectifiers n inverters re interconnecte through istribute DC-bus the HVDC feeer line. Typiclly, rectifiers re plce t firly lrge istnce, compre to the inverters. Determining the optimum rting n istnce of the inverters is outsie the scope of this pper. Moels for etermining the optiml loction of RPSS feeers, pplie on clssicl converter sttions re presente in [3]. L. Abrhmsson n L. Söer re with the Deprtment of Electricl Power Systems, Royl Institute of Technology KTH), Stockholm, Sween, e-mil: lrs.brhmsson@ee.kth.se & lennrt.soer@ee.kth.se). S Östlun is with with the Deprtment of Electricl Energy Conversion, Royl Institute of Technology KTH), Stockholm, Sween, e-mil: stefn.ostlun@ee.kth.se). Mnuscript receive August, 2012; revise - -, 201-. This pper presents etile still-stning-lo mthemticl moels of n optimlly-controlle HVDC-fe RPSS n of clssiclly controlle rotry-mimicking-converter fe systems. The ltter my lso inclue n HVAC trnsmission line, which is consiere n compre to the HVDC line in one of the exmples. The optimlity of the HVDC-feeing is with respect to overll system losses n consiers the unit commitment of the converters besies the power flow irections n mounts. The numericl exmple in Section IV is pte to Sweish conitions, but the ie in itself is with some generliztion pplicble to ny kin of RPSS tht is; DC, lowfrequency AC, n public gri frequency AC ctenries. The mthemticl moels re if further generlize pplicble lso without DC-gri, or with the DC-gri exchnge to n ACgri. AC rilwy gris with public frequency often just hve trnsformers inste of converter sttions, using two of the public gri s three phses. The rwbcks of irect trnsformer feeing re further trete in [2], [4] [8]. 5 10 MVA 15 kv 16 2 3 Hz 25 100 km Public gri 50 Hz +60 kv 60 kv DC DC Fig. 1: Visuliztion of the propose solution comprising n HVDC feeer line.
IEEE TRANSACTIONS ON, VOL., NO. -, 201-2 II. BACKGROUND AND CONTRIBUTION Ril trnsporttion in the worl hs in recent yers risen consierbly ue to energy-efficiency n climte chnge, n increse expecte to continue. This plces greter emns on energy supply both regring instlle power n power/voltge qulity. In generl, the power emn is criticl in res of ense trffic, such s in metropolitn res, wheres mintining the voltge level is the min problem on rurl prts of the min lines s well s on some inustril freight lines [2], [9] [11]. A. AC-DC-lo flow Mny AC-DC-lo flow methos hve been presente, sequentil or unifie, ech metho hve its pros n cons. For public power systems sequentil moels hve been presente in [12], [13]. An erly unifie AC-DC power flow metho for HVDC-fe public AC gris ws presente in [14] where, in orer to voi binry vribles, severl control moes re efine, of which some llows ecouple computtion moel. A unifie AC-DC power flow for DC rilwys is presente in [15]; wheres sequentil one cn be foun in [16]. In [17], literture review of rilwy AC-DC power flow is presente. The moel presente in this pper is unifie, n the methos of solving re hnle by stnr lgorithms. Nonliner solver choices mtters [18]. Accoring to [19] sequentil AC-DC power flows re esier to coe, wheres simultneous unifie) re fster to compute. A simplifie sequentil metho is presente in [19] tht only nees one itertion for ech system, given tht tps re within their limits. This pper uses exct methos, n unifie power flow. B. HVDC-fe rilwys In [20], HVDC-feeing of n LVDC light-ril system is suggeste n presente using moving los. An optiml control of the converters is presente, minimizing the LVDC ctenry losses. The optiml control is compre with HVDC structure without optimiztion which is bit unclerly efine. In this pper, n HVDC-fe AC ctenry is moele, consiering the no-lo converter losses, justifying unit commitment. This pper, oes not consier moving los. C. Benefits with suggeste HVDC solution Using HVDC cbles shoul in most cses be possible without ny itionl ln usge. The effects of lternting mgnetic fiels on the humn boy re uncler. Thus, s precution, new overhe lines re usully not permitte in populte res. Burie cbles re lso less expose wether. D. OPF Optiml Power Flow) The term optiml power flow ws introuce s erly s in 1968 [21], n propose methos for solving the OPF for rilwys hve been foun since the 1990 s [22]. The term contins wie rnge of problems from quite economicl stuies, through losses minimiztions, to conitions on mgnetic fiels, voltge rops, n such [21]. This pper uses clssicl optimiztion moeling n solver softwre. The lterntive to clssicl moels n solvers is using met-heuristic OPF methos [21]. Mny ppers focus on the solution methos, wheres this pper focus on the moeling, n use stnr lgorithms for the optimiztion n equtions solving. The moels presente still hve to be esigne to suit the solvers use. The OPF problem is thoroughly stuie for public gris, for both trnsmission n istribution. In the rilwy fiel, there re less publictions vilble regring OPF. Therefore much work still remins in fining relevnt n efficient moels n interesting problem formultions. An OPF, or generlly, power flows, cn minly be expresse in rectngulr or polr coorintes, n computtionlly they re equivlent, c.f. [23]. This pper uses polr coorintes. In [23] hybri formultion for NLP OPF is presente. Power mismtch equtions re expresse in power for nonzero injection noes n in current for zero injection noes, wheres the power flow equtions re written in voltge rectngulr coorintes. The hybri moel is fst for systems with mny zero injection noes. 1) Public gris: OPF cn be stuie in mny wys, e.g. in [24] the ul of the OPF problem, which cn be expresse s semi-efinite problem SDP) is erive. SDPs cn be solve to globl optimum in polynomil time, c.f. [24]. It cn be verifie when the ulity gp is zero. The theory of [24] cn be pplie for mny relevnt OPF problems. The theory oes however not pply for problems contining integer vribles, s the problem of this pper oes, llowing switching on n off of converters. The moel presente in [24] is not pplicble for unit commitment problems or other mixeinteger optimiztions. Moreover, it is ssume tht ctive power loss is nonzero, but smll, in prctice, so it is uncler if the metho is pplicble on generl RPSSs even if leving the binries behin. Another SDP moel for OPF is presente in [25]. The generl OPF is hr to solve [26], so sometimes when solving multi-objective problems, met-heuristic methos hve to be use. In this pper, there is cler objective, n it is lso the uthors belief tht clssicl optimiztion methos re to prefer when possible. The security constrine OPF is in [27] presente s n NLP using fst ecouple lo flow n solve s n SQP sequentil qurtic progrmming). The solutions re suppose to mnge single outge. In [28], OPF is one in GAMS for istribution systems. RPSSs re often ril ue to the nture of the lo they re supplying. There is however no esire to keep the gri un-meshe, ctully the esire is the opposite. In istribution systems, the gris re esire to be ril, so the plnning is bit ifferent. In [28], the problem is solve two-stge, by Beners ecomposition. The mster problem is MIQCP [29], wheres the slve problem is NLP [29]. The system network configurtion involves ecision vribles opening n closing connections. In this pper, the loss function escribes the rel losses,
IEEE TRANSACTIONS ON, VOL., NO. -, 201-3 wheres in [28], the losses function is pprent-power epenent. In [30], prllelizble methos for lrge-scle istribute OPF re presente. These clssify the system into regions, n convexify the problem to mke it mngeble. Another coorinte ecentrlize multi-re OPF problem with economic interest n slightly simplifie power system escription [31], uses DC-lo-flow with nonliner terms n cosine pproximtion of the losses. Lgrnge relxtion ecomposition proceures re pplie. Normlly, RPSS re not region-bse in tht sense. It hppens tht minline RPSSs re connecte to metropolitn or suburbn power systems, but it is rre. OPF moels cn lso inclue n focus on electricity prices n mrket moels. In the minly euction-oriente GUIbse open-source softwre [32], such OPF is trete. In such stuies, the equtions escribing the power system re not rrely hevily simplifie. In [33] Newton s metho is use for public trnsmission gri voltge/rective power control, moele s n NLP where tp chnger levels re roune to possible vlues fter simultion. A suboptiml power flow ws etermine reucing the set of control vribles, suppose to be use by opertors. Both losses n the ifferences of rective prouction reserves were minimize. 2) Rilwy gris: A rilwy OPF moel where the rective power consumption of trins is controlle is presente in [34]. It is shown tht rective control of moern trins is very beneficil if there re bout hlf moern n hlf ol-fshione trins in the fleet. An erly rilwy optiml commitment stuy [35] trets the converter sttions s slck buses, seprting the system into power sections, llowing fst power-flow clcultion. The iscrepncies from exct power flows cn however rech up to 40 %, so more etile moel like the one presente in this pper cn efinitely be motivte. In relity, the number of rotry converters committe to sttion impcts the power output from neighboring sttions. Rilwy OPF with linerize function controlling the ctive power injections by controlling the voltge phse ngles is presente in [22], where one of the converters in the RPSS constitutes slck bus. Rilwy OPF results, minimizing energy costs for DC-fe systems, re presente in [36], the moels re however not presente in etil. In [37] successive LP OPF lgorithm, esigne for rilwy usge, is presente. DC rilwys n the impct of letting vrious numbers of substtions inclue rectifiers, n/or inverters re stuie in [38]. Rilwy OPF cn lso consier the ifference in electricity prices between ifferent in-feeing public gri opertors [39]. III. MODELLING A. Suggeste DC Feeer Lines The feeer line my be implemente s groun cble. The cble coul be 60 kv ipole with 240 mm 2 luminium conuctors which yiels power hnling cpbility of pproximtely 50 MW. Such cble hs the impence R line = 0.1175 Ω/km. 1) B. Converter sttions 1) Propose Converters: The converter losses for inverter moe re moele s secon orer polynomil ccoring to eqution 43), where I g is the per-unit current on the ACsie of the converter, n P c,i is the per-unit power. It is norml tht converter losses re moele s secon orer polynomils of the AC-sie current, c.f. [12], [13]. The qurtic term represents the resistive losses of the converter, wheres the liner term represents constnt-voltge-type losses, n the constnt term correspons to iling losses. The polynomil is erive ssuming inverter opertion t unity power fctor, which is when the losses re the highest. This moel is use for inverter moe of the converter, regrless of cos φ. Operting the converter s rectifier will yiel slightly lower losses, in this pper moele s 0.9 times the inverter losses, c.f. eqution 44), lso here regrless of cos φ. The per-unit losses re for simplicity ssume to be inepenent of the converter rtings. It is however ssume tht ll the converters use in the cse stuies re of the sme rting, n in Cse B the loss functions for ech HVDC feeing point re thus ouble. All hrmonics, tht cn be substntil epening on the type of vehicles, re neglecte. The propose converter solution uses meium frequency trnsformer technology [2], [40]. 2) Clssic Converters: If sttic converters re use, the terminl voltge n phse shift cn be controlle freely [4]. However, for esy replcement n inter-operbility, sttic converters re often me to mimic the behvior of rotting converters, i.e. in Sween, Norwy, n prts of Estern Germny. The chrcteristics of rotry n rotry-mimicking converters re given in e.g. [41]. The voltge is controlle to be slightly ecresing with incresing rective lo ccoring to eqution 68). In this pper, lossless moel of the converters is use. If the converters woul be moele with losses, P G;g in the first nonliner term corresponing to the motor sie) of eqution 70) shoul be replce by P M;g. If P M;g n P G;g woul not be the sme, some moeling corresponing to equtions 41), 43), n 44) woul be neee. The motoring power woul lso hve to be introuce in eqution 71). C. Ctenry The ctenry is typicl rrngement in Sween enote 120 mm 2, 2Å, which mens tht the cross section of the contct wire is 120 mm 2 with two luminum BT-system return conuctors of ech 212 mm 2 cross section re, prllel connecte to the trck. The impence Z line = 0.20 + j0.20 Ω/km. 2) This rrngement is reltively wek. D. Lo In this work, the vehicles re ssume to be equippe with moern voltge source converters VSC) cpble to operte t unity power fctor n negligible hrmonic currents.
IEEE TRANSACTIONS ON, VOL., NO. -, 201-4 E. Gri Topology The supplying DC gri is comb-shpe, with converter sttion in ech comb-pin en. The MTDC gri is connecte to the public gri in the upper left prt of the own-pointing comb, c.f Figure 1. Tht is the explntion for tht the power inflows re slightly bigger in CE2 thn in CE3 in Tble VIII. As consequence, in orer to even out the losses between the converters, this is compenste by slightly greter rective power prouction in CE3 thn in CE2. CE is in this pper n bbrevition for Connecting Equipment, generlize RPSS feeer enottion. The AC-gris both the one connecte to n MTDC feeing gri, n the clssicl one connecte irectly to the public gri hve the comb pointing upwrs. In the propose solution, the impences between ctenries n converters results in extr noes, wheres in the clssicl cses, these smll impences re inclue in the converter moeling. F. Moeling escription The MTDC problem is formulte s n optimiztion problem where the system ctive power losses re to be minimize, wheres the stuies of the present RPSS is system of equtions to be solve. The mthemticl formultions re presente Section III-G. Prmeters, continuous) vribles n binry vribles re efine in Tbles IV n VI respectively, wheres numericl vlues of most prmeters re presente in Tble V. 1) The HVDC-fe OPF system: The objective function, the totl losses, minp L, to be minimize is stte in eqution 67). Minimiztion is one subject to the bouns of equtions 5) 24) n to the constrints of equtions 33) 67). The initil vrible levels re not presente in this pper ues to less importnce n spce limittions. The converters re moele s twin-noes one ACnoe n one DC-noe strongly relte to ech other), c.f. eqution 41), where the ctive) power consume on one sie equls the ctive) power injecte on the other sie, plus the converter losses. Tht is specil kin of power mismtch eqution. The converter losses for inverting moe re escribe by eqution 43), wheres the losses when rectifying re escribe by eqution 44). Converter losses re boune below by P L,mx if turne on, n by 0 if turne off, c.f. eqution 51). Converter losses equl the inverting losses if turne on n inverting, c.f. equtions 52) n 53). Converter losses equl the rectifying losses if turne on n not inverting, c.f. equtions 54) n 55). Converter pprent power is efine by eqution 46), wheres the converter currents re efine by eqution 42). Apprent power is limite on the AC sie, 45) where voltge level rop t the converter terminl, forces the mximum converte power to rop ccoringly, n eqution 62) forces the converter pprent power own to zero if switche off; rective power is limite on the AC sie, by voltge level in 49) n 50) n by unit commitment in 60), n 61); n ctive power is limite on the DC sie by currents in 47), 48), n in sign by 63) n 64), n by commitment in 65), n 66); n on the AC sie irectionwise by 56) n 57), n by commitment in equtions 58) n 59). Some of these limits re inspire by n concretize from the ones presente in [7], [12]. Detils like converter filters, n/or insie-converter trnsformers re not consiere. The DC system is fe by n infinitely strong lossless three-phse-ac-to-dc converter. Finlly, 33), 34), n 35) re power flow equtions; 36), 37), n 38) re the power mismtch equtions; n 39) n 40) re conuctor losses equtions. 2) The clssiclly controlle system: When solving the clssicl problem, the system of equtions 34), 35), 37), 38), 68) voltge control of RPSS sie of converter, 69), 70) voltge phse ngle shifts cuse by motor n genertor, n 71) moeling the public gri phse shift ue to rilwy loing is solve. The losses re clculte mnully fter solving tht system of equtions. In the clssicl feeing solution, converter losses re not stuie, but moels tht coul be implemente re present in e.g. [41]. G. Mthemticl Moelling The set efinitions for the clssicl converter control, re presente in Tble III. For the MTDC stuies, they re presente in two ifferent tbles. For Cse A, Tble I, n for for Cse B Tble II. Unitless electricl prmeters n vribles re, if presente s imensionless, expresse in perunit p.u.). TABLE I: Sets, Cse A, MTDC problem Set, 2 {1,2,...,8,17}, 2 {9,10,...,16} g p ) {13} g ) {5,6,7,8} g ) {9,10,11,12} θ ) {5} n l, ) {13,14,15,16} n g, ) {9,10,11,12,14,15,16} n g, ) {1,2,3,4,17} AC noes inex DC noes inex Public gri to MTDC gri converter The HVDC-AC converter noes, AC sie The HVDC-AC converter noes, DC sie The reference ngle noe, AC sie The DC noes with no los The DC noes with no genertion The AC noes with no genertion TABLE II: Sets, Cse B, MTDC problem Set, 2 {1,2,3,4,9,10}, 2 {5,6,7,8} g p ) {7} g ) {3,4} g ) {5,6} θ ) {3} n l, ) {7,8} n g, ) {5,6,8} n g, ) {1,2,9,10} AC noes inex DC noes inex Public gri to MTDC gri converter The HVDC-AC converter noes, AC sie The HVDC-AC converter noes, DC sie The reference ngle noe, AC sie The DC noes with no los The DC noes with no genertion The AC noes with no genertion Prmeters re efine numericlly in Tble V where I mx is the mximl converter current of ech converter in sttion, clculte such tht the converter shoul be ble to eliver rte power t voltge levels of t lest 0.8 p.u., n where
IEEE TRANSACTIONS ON, VOL., NO. -, 201-5 TABLE III: Sets, All cses, Clssicl converter control. Set, 2 {1,2,...,9}, 2 {1,2,3,4} g,g 2 ) {1,9} g,g 2 ) {1,4} g n ) {2,3,...,8} g n ) {2,3} t ) {5} t ) {2,3} Prmeter AC noes inex, Cse A AC noes inex, Cse B The rotting converter noes, Cse A The rotting converter noes, Cse B The non-rotting-converter noes, Cse A The non-rotting-converter noes, Cse B Noes with trins, Cse A Noes with trins, Cse B TABLE IV: Prmeters G, 2 Rel prt of AC-sie mittnce mtrix conuctnce) B, 2 Imginry prt of AC-sie mittnce mtrix susceptnce) G, 2 DC-sie conuctnce mtrix PD; Active power los on AC-sie, i.e. trins Q D; Rective power los on AC-sie, i.e. trins U G;gp Fixe voltge level on DC-sie of public gri to MTDC gri converter C g,g Binry mtrix connecting DC-sies n AC-sies of converter sttions S b Bse power # conv Number of converters per sttion I mx Mximl converter current per converter unit) P L,mx Upper boun on converter sttion losses S mx Upper boun on converter sttion pprent power P mx Upper boun on converter sttion ctive power Q mx Upper boun on converter sttion rective power Ug 0 The terminl no-lo voltge level on ctenry sie of clssicl converter k q Constnt for clssicl converter voltge control θ 50;g The no-lo ngle of the public gri x qm Motor-sie inner rectnce of rotry converter Q48/Q49 x qg Genertor-sie inner rectnce of rotry converter Q48/Q49 Xg 50 The short-circuit rectnce of the public gri Q 50;g Rective power consumption on the motoring sie U M;g Motor-sie voltge level P L,mx is clculte ccoring to eqution 43). Since the loss function is efine for one converter, in sttion with mny converters, the losses hs to be multiplie with the number of converters. Trin los re given from the cse escriptions, conuctnces n susceptnces re given from equtions 1) n 2), n from the cse escriptions. The mtrix C g,g is constructe such tht it is only one for the right combintion of converter noes, e.g. for Cse A tht woul be n C 5,9 = C 6,10 = C 7,11 = C 8,12 = 1 3) C g,g = 0 4) TABLE V: Given numericl vlues of prmeters Prmeter n numericl vlue S b 5 MVA S mx # conv P mx # conv 1 I mx 0.8 U G;gp 1 Ug 0 1.1 Q mx # conv k q 20 MVAr/kV x qm 49 % x qg 53 % Xg 50 15 % Q 50;g 0 U M;g 1 P L,mx # conv 0.0135Imx 2 + 0.0097I mx + 0.015 ) for ll other combintions of g,g. This is use in eqution 41) to ensure tht power flowing out of the DC gri comes in t the right point in the AC gri, n vice vers. The constnt k q of Tble V is efine uner the ssumption tht U n U 0 re given in kv n Q G is given in MVAr in 68). Vrible U U θ P P Q PD; P G; Q G; P G; P L PL; PL; PL;g c P c,i P c,r I g S g Ψ 0;g Ψ G;g Ψ M;g α g γ g TABLE VI: Vribles Voltge level in AC noe Voltge level in DC noe Voltge ngle in AC noe Net injecte ctive power t AC bus Net injecte power t DC bus Net injecte rective power t AC bus Power los on DC-sie, i.e. converter sttions connecting DC gri to ctenry. Active power genertion on AC-sie, i.e. converter sttions connecting DC gri to ctenry. Rective power genertion on AC-sie, i.e. converter sttions connecting DC gri to ctenry. Power genertion on DC-sie, i.e. converter sttions connecting DC gri to public gri. Totl losses in the entire rilwy power supply system Mrginl losses for ech noe in the AC gri conuctors Mrginl losses for ech noe in the DC gri conuctors Losses in the converters connecting DC gri to ctenries Losses in the converters connecting DC gri to ctenries, ue to inverting Losses in the converters connecting DC gri to ctenries, ue to rectifying The AC-sie currents of the converters connecting the DC gri the ctenries. The AC-sie pprent power of the converters connecting the DC gri the ctenries. The phse ngle of the public gri in the clssicl moel The phse ngle shift ue to genertor of converter in the clssicl moel The phse ngle shift ue to motor of converter in the clssicl moel Tell whether the converter sttion connecting MTDC gri to ctenry is inverting 1) or rectifying 0), binry Tells whether the converter sttion is committe 1) or not 0), binry These vribles upper, n lower bouns re liste in equtions 5) 32). Typiclly, goo col-strt level vlues shoul be in the mile of the rnge of possible or probble vlues of vrible [23]. In this pper, some level vlues re in the mile, n some close to the expecte result. 0 P L 5 5) P mx P G;g P mx 6) 0 P G;n g, 0 7) 5 P G;g p 5 8) 0 P G;n g, 0 9) P mx P D;g P mx 10) 0 P D;n l, 0 11) Q mx Q G;g Q mx 12) 0 Q G;n g, 0 13) 0 S g S mx 14) π θ π 15)
IEEE TRANSACTIONS ON, VOL., NO. -, 201-6 0 θ θ 0 16) 0.4 U n g, 1.3 17) 0.4 U g 1.1 18) 0.95 U n g, 1.05 19) 1 U g p 1 20) 0.0 I g I mx 21) 0.0 P c P L,mx 22) 0.0 P c,i P L,mx 23) 0.0 P c,r 0.9 P L,mx 24) 0.4 U 1.2 25) π 2 θ π 18 26) 2 PG;g 2 27) 0 P G;g n 0 28) 2 Q G;g 2 29) 0 Q G;g n 0 30) π 2 Ψ 0;g π 9 11π 18 Ψ G;g + Ψ M;g π 9 31) 32) where, the constrints to be use in optimiztion, or the equtions to be use in eqution solving, re in 33) 71). P = U U 2 G, 2 33) 2 P = U U 2 G, 2 cosθ θ 2 )+ 2 +B, 2 sinθ θ 2 ) ) 34) Q = U 2 U 2 G, 2 sinθ θ 2 ) B, 2 cosθ θ 2 ) ) 35) P = P G; P D; 36) P = P G; P D; 37) Q = Q G; Q D; 38) PL; = U G, 2 U 2 cosθ θ 2 ) U ) 39) 2 ) PL; = U G, 2 U 2 U 40) 2 0 = PG;g PD;g + g +PL;g c ) 41) Cg,g SG;g = I g # conv Ug 42) P c,i = # conv 0.0135 ) 2 I g + 0.0097Ig + 43) + 0.015) P c,r = # conv 0.0135 ) 2 I g + 0.0097Ig + +0.015) 0.9 44) S g U g I mx # conv 45) 0 = P G;g ) 2 + Q G;g ) 2 Sg ) 2 46) P D;g U g I mx # conv 47) P D;g U g I mx # conv 48) Q G;g U g I mx # conv 49) Q G;g U g I mx # conv 50) PL;g c P L,mx γ g 51) PL;g c P c,i ) P L,mx 2 αg γ g 52) PL;g c P c,i ) + P L,mx 2 αg γ g 53) PL;g c P c,r ) P L,mx 1 + αg γ g 54) PL;g c P c,r ) + P L,mx 1 + αg γ g 55) PG;g P mx α g 56) PG;g ) P mx 1 αg 57) P G;g P mx γ g 58) P G;g P mx γ g 59) Q G;g Q mx γ g 60) Q G;g Q mx γ g 61) SG;g S mx γ g 62) PD;g P mx α g C g,g 63) g PD;g ) P mx 1 αg Cg,g 64) g PD;g P mx γ g C g,g g 65) PD;g P mx γ g C g,g g 66) P L = PL; + PL; + PL;g c 67) g U g = Ug 0 Q G;g k q # conv 68) θ g = Ψ 0;g + Ψ M;g + Ψ G;g 69) 0 = Ψ M;g + Ψ G;g + + 1 3 rctn x qm PG;g # conv UM;g 2 + x qm Q50;g # conv x qg PG;g # conv + rctn U 2 G;g + x qg QG;g # conv, ) Ψ 0;g = θ 50 g 1 3 rctn Xg 50 P G;g UM;g 2 + X 50 Q 50;g ) + IV. COMPARATIVE CASE STUDIES 70) 71) A comprison hs been me between existing solutions n the suggeste one. Two ifferent test cses hve been simulte n compre. The test cses were chosen to illustrte importnt properties of the system, like commitment of converters n the bility for regenertion. The numericl exmples re inspire by the 15 kv 16 2 3 Hz RPSS of Sween. A comprison is me how the system cts now n how it cts when replcing present converters n trnsformer substtions with HVDC feeing n controlling it optimlly rther thn following the clssicl control schemes. For etils of how the Sweish RPSS is constitute, plese refer to [42] [44].
IEEE TRANSACTIONS ON, VOL., NO. -, 201-7 The moeling softwre GAMS [29] hve been use. For the cses where systems of equtions were bout to be solve, CNS problem clss ws efine, solve using the CONOPT [45] lgorithm. For the OPF cses the problem ws moele s n MINLP problem using the locl solver lgorithm BONMIN [45]. Globl solvers like COUENNE [45] hve been trie for smll test cses when the respective solutions correspone very well. Due to extensive computtionl times, globl solving cn not be one for ll the stuie cses. In the OPF problems, voltge level n ngle on the converters were controllble with regr to instlle pprent power, n other physicl limittions. The converters coul be either online or offline, n rectifying or inverting; giving rise to ifferent loss functions. This ll ws subject to minimiztion of the totl power losses in the system. TABLE VII: Summry of results Cse A Cse A Cse B Cse B CCC OPF CCC OPF Bse voltge 15 kv Bse power 5 MVA Min. voltge 1.051 1.057 0.982 1.012 Mx. voltge 1.099 1.100 1.147 1.162 Conversion losses 0.060 0.087 Trnsmission losses AC 0.071 0.064 0.454 0.223 Trnsmission losses DC 0.004 0.006 Totl losses 0.128 0.316 TABLE VIII: Excerpt of results, Cse A. CE1 CE2 CE3 CE4 PD;g ) 0.000 0.866 0.858 0.000 PG;g ) 0.000 0.836 0.828 0.000 Q G;g ) 0.000 0.029 0.036 0.000 P G;1,P3,4,P 7,6,P G;9 0.835 0.709 0.709 0.835 Q G;1,Q 3,4,Q 7,6,Q G;9 0.048 0.107 0.107 0.048 P1,2,,,P 9,8 0.711 - - 0.711 P1,3,,,P 9,7 0.125 - - 0.125 OPF voltges U 1,U 2,U 3,U 4 1.100 1.100 1.100 1.100 U 5,U 6 U 7 U 8 1.100 1.100 1.100 1.100 CCC voltges U1,U 3,U 7,U 9 1.099 1.094 1.094 1.099 Trin Lo OPF: P D;17 = 1.600 CCC: P D;5 = 1.600 Trin Voltge OPF: U 17 = 1.057 CCC: U 5 = 1.051 ) Cse A: Single motoring trin n 25 km between CEs: This cse ws chosen in orer to be ble to compre two kins of centrlize RPSS. The clssicl with trnsformers connecting the trnsmission line n the ctenry between the converters, n the propose one where both converters n trnsformers re replce by HVDC converter sttions. The trin is locte in the mile of the section. The power consume by the trin is 8 MW t unity power fctor. The two centrlly locte trnsformers re rte to 16 MVA, wheres the leftmost n rightmost trnsformers re rte to 25 MVA. The clssicl converters re connecte to the ctenry. The solutions for both OPF n clssicl control re isplye in Tble VIII. The voltge t the pntogrph is bove 15.7 kv which mens tht the performnce of the locomotive is unffecte [46]. The losses re ominte by trnsmission losses but both trnsmission losses n conversion losses re of the sme mgnitue. The trnsmission losses re slightly bigger in the clssicl feeing solution, wheres the pntogrph voltge levels typiclly re the sme, c.f. Tbles VII n VIII. TABLE IX: Excerpt of results, Cse B. CE1 CE2 CE1 CE2 PD;g ), P G;g ) 0.900 1.210 0.939 1.162 Q G;g ) 0.077 0.145 - - P G;1,P G;4,Q G;1,Q G;9 0.346 0.800 0.272 0.726 OPF U 1,U 2,U 3,U 4 1.100 1.099 1.100 1.100 Trin Lo P D;9 = 1.600 P D;10 = 1.600 Trin Voltge U 9 = 1.162 U 10 = 1.012 CCC U1,U 4,, 1.105 1.088 - - Trin Lo PD;2 = 1.600 P D;3 = 1.600 Trin Voltge U2 = 1.147 U 3 = 0.982 b) Cse B: One motoring trin n one brking trin n 100 km between CEs: This cse illustrtes power flow uring regenertive brking on typicl line with CEs locte 100 km from ech other. Normlly, feebck to the ntionl gri is not pprecite by public gri owners. Therefore, in the MTDC cse the impct is stuie when it is not llowe to fee bck to the public gri from the MTDC gri. The istnce between the trins is 66.7 km n the istnce to the closest converter sttion is 16.7 km, for both of the trins. The ctive n rective power consumptions of the two locomotives re the sme n equl to Cse A, with the exception tht one of them regenertes, n the other one is motoring. About 60 % of the power is regenerte to the HVDC line n fe bck to the consuming trin. The remining regenerte power is trnsmitte through the ctenry line. The AC losses re lmost hlve in the OPF cse compre to the clssicl one, c.f. Tble VII. In the OPF cse, the converter losses re less thn hlf of the trnsmission losses. The minimum voltge levels re bove 15 kv for OPF stuy, n not fr from it in the clssicl feeing stuy. V. CONCLUSIONS & DISCUSSION There is significnt ifference in the power flow results between the simulte system with converter control ccoring to Sweish regultions n the HVDC OPF solution. The ifference is most pronounce in the cse with regenertive brking n comprtively long inter-converter istnce, Cse B, where the clssicl AC trnsmission losses re more thn twice the OPF losses. The ifference is cuse by significnt power flow through the ctenry when no controllble HVDCfeeer is present. With HVDC-supplie rilwys, the interconverter istnces cn be fr without cusing unberble losses. As cn be seen compring cses A n B, the centrlize solution is better thn without ny trnsmission lines t ll [22]. One key issue for the optiml solution is the exct
IEEE TRANSACTIONS ON, VOL., NO. -, 201-8 moeling of the converter losses. The propose system hnles regenertive brking well even over lrge istnces. In rel-life, it is not possible to exctly know ll momentry lo levels, lo positions n levels of power conversion. Therefore, truly optiml solution is not chievble in future ppliction. The benefit of etermining the optiml solution is however tht it sets theoreticl upper boun on how smll the losses possibly cn be using smrt control strtegy. Tht is, the minimum losses shows the potentil of the technology but not how to implement the solution. A quest for future stuies is to exmine control rules bse upon trffic in the power system sections connecte to the converter being controlle, n power system mesurements from jcent converters. Robust pproches coul be stochstic optimiztion moels, e.g. chnce constrine OPF [47] but pplie to the rilwy, but lso simpler vrints such s creting control lw for converters by stuying vrious OPF solutions. 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IEEE TRANSACTIONS ON, VOL., NO. -, 201-9 [44] M. Olofsson, Power Flow Anlysis of the Sweish Rilwy Electricl System, tech. rep., Royl Institute of Technology KTH), Stockholm, Sween, 1993. Licentite Thesis. [45] GAMS, Solver escriptions, July 2011. [46] Norwegin ntionl rilwy ministrtion Jernbneverket), Simultions Report, Rilwy Electric Power Supply, Ofotsbnen originl title in Norwegin), tech. rep., Norwegin ntionl rilwy ministrtion Jernbneverket), 2007. [47] H. Zhng n P. Li, Chnce constrine progrmming for optiml power flow uner uncertinty, Power Systems, IEEE Trnsctions on, vol. 26, pp. 2417 2424, Nov. 2011. Stefn Östlun Stefn Östlun ws born in Gävle, Sween, in 1961. He receive the M.Sc. n the Ph.D. egrees from the Royl Institute of Technology KTH), Stockholm, Sween, in 1985 n 1992, respectively. From 1993 to 1999, he ws Senior Lecturer t the KTH. In 2000, he ws ppointe Professor in electric power engineering/electric rilwy trction. His reserch interest inclues electricl mchines, power electronics, electric rilwy trction, hybri electric vehicles n engineering euction. He hs been in chrge of the EE curriculum t KTH since 1999 n is presently cting s Vice Den t the School of Electricl Engineering, KTH. Lrs Abrhmsson Lrs Abrhmsson ws born in Luleå, Sween in 1979. He receive his M.Sc. egree in Engineering Physics from the Luleå University of Technology, Luleå, Sween in 2005. He is currently octorl cnite t the eprtment of Electric Power Systems t the Royl Institute of Technology, Stockholm, Sween. Lennrt Söer Lennrt Söer ws born in Soln, Sween in 1956. He receive his M.Sc. n Ph.D. egrees in Electricl Engineering from the Royl Institute of Technology, Stockholm, Sween in 1982 n 1988 respectively. He is since 1999 professor in Electric Power Systems t the Royl Institute of Technology. He lso works with projects concerning eregulte electricity mrkets, istribution systems, power system restortion, risk nlysis n integrtion of solr n win power.