PROC. BULK POWER SYSTEM VOLTAGE PHENOMENA{III SEMINAR, DAVOS, SWITZERLAND, AUGUST 1994, PP. 349{358.

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1 PROC. BULK POWER SYSTEM VOLTAGE PHENOMENA{III SEMINAR, DAVOS, SWITZERLAND, AUGUST 99, PP. 9{58. IMPROVING CONTINUATION METHODS FOR TRACING BIFURCATION DIAGRAMS IN POWER SYSTEMS Claudo A. Ca~nzares Antono Z. de Souza Vctor H. Quntana Unversty ofwaterloo Department of Electrcal & Computer Engneerng Waterloo, ON, Canada NL G Abstract: Ths paper proposes several technques for reducng the computatonal burden of tracng bfurcaton dagrams n power systems wth contnuaton methods. The rst technque descrbed n the paper s an mplementaton of the fastdecoupled method n the predctor-corrector contnuaton algorthm. The other proposed technques consst on reducng the system sze. One reducton method s based on the tangent vector calculated durng the predctor step, whereas the second one s based on network partton and detecton of voltage-weak areas. The computatonal ssues related to the actual mplementaton of these methods are also dscussed. The system reducton technques are tested n the 00 bus IEEE system, so that these methods can be analyzed n a realstc envronment. Keywords: contnuaton methods, generc saddle-node bfurcatons, system reducton, fast-decoupled methods.. Introducton Voltage stablty problems n power systems arse from a varety of reasons and operatng condtons, from voltage control problems (taps, automatc voltage regulators, HVDC controls []{[6]) to system bfurcatons (saddle-nodes, transcrtcals, Hopfs [, 6]{[8]). Local bfurcatons are characterzed by one or two system egenvalues lyng on the magnary axs of the complex plane for certan system parameters values,.e., a real egenvalue (saddle-node, transcrtcal, ptchfork bfurcatons) or the real part of a complex conjugate par (Hopf bfurcaton) become zero for gven values of these parameters. Thus, the power system moves from one stable equlbrum pont (s.e.p.) to another as the parameters (load and generaton powers, taps, control settngs, etc.) change slowly, untl certan parameters values are reached and the system \bfurcates,".e., the egenvalues reach the magnary axs. Ths phenomenon can be classed as a steady-state stablty problem []; however the nal result s a system-wde dynamc nstablty [7]{[0]. Bfurcatons of one-parameter dynamc systems have been thoroughly studed n derental equaton models [9, 0]. Several authors have extended ths theory to the analyss of smlar phenomena n power system models [8, 9, 0,, 5, 6, 7, 8], whch are typcally represented by a set of derental equatons and algebrac constrants. Partcularly, n [6, 7] the author thoroughly dscusses the condtons and methods used for detectng saddle-node bfurcatons n these knds of models. Of all local bfurcatons, only saddle-nodes and Hopfs are generc,.e., one expects to encounter these types of bfurcatons unless there are some specal assumptons regardng the system modelng [9, 0]; ths can be demonstrated n power system models for certan system parameters [8]. Therefore, ths paper also concentrates on these two knds of bfurcatons n power networks. Bfurcatons ponts can be located usng drect and/or contnuaton methods [0]. Drect methods are desgned to nd the bfurcaton from a known operatng condton n one soluton attempt, whereas contnuaton methods trace a successon the equlbrum ponts by automatcally changng the parameter value untl the bfurcaton s encountered, and beyond. The manfold resultng from jonng the system equlbra generated by the contnuaton method s known as the bfurcaton dagram (\PV", \QV", or \nose" curves are partcular cases of these dagrams). For power systems models, drect and contnuaton methods have been proven to be numercally well-behaved [6, 7]. Drect methods have been shown to be an ecent way of locatng saddle-nodes when a system s close to these bfurcatons ponts [6]; however, when the system s far from bfurcaton, and especally when all system control and lmts are consdered, ths method tends to fal [6]. Locatng Hopf bfurcatons wth these methods s completely mpractcal as demonstrated n [8]. On the other hand, contnuaton methods do not present these dcultes and have the addtonal advantage of yeldng more nformaton than drect methods, snce complete bfurcaton dagrams for any type of bfurcaton can be generated wth ths technque; however, ths method s tme consumng, partcularly when dealng wth large networks [6]. Ths paper concentrates on dscussng several ssues assocated to the detecton of generc saddle-node bfurcatons wth the contnuaton method. Locatng Hopf bfurcatons does not represent a partcularly dcult numercal problem when contnuaton methods are used, snce these bfurcatons do not produce a sngularty of the system equatons Jacoban or a change n the number of system equlbrum ponts; one can detect these bfurcatons by montorng the egenvalues of system equlbra on the bfurcaton dagram usng typcal egenvalue computatonal technques. Several authors have proposed derent mplementatons of the contnuaton method to detect saddle-node bfurcatons n power systems [6, ]{[6]. These programs can also be used to detect other types of bfurcatons as shown n [8], snce they are slght varatons, based on some typcal characterstcs of power system models, of the technques descrbed n [0]. In [], a tangent vector predctor together wth a perpendcular ntersecton corrector are used to trace part of the bfurcaton dagram, whereas n [, 5] a parameterzaton technque wth a xed parameter corrector are proposed to avod sngulartes of the contnuaton method Jacobans at saddle-node bfurcaton ponts, so that complete bfurcaton dagrams can be traced. The authors n [6, ] use smlar predctor and corrector technques as n [], but ntroduce a step length control technque to avod the need of parameter-

2 zaton when tracng whole bfurcaton dagrams, and to ecently handle system lmts and control mode changes, partcularly when hgh voltage drect current (HVDC) systems are ncluded. In [] an optmal step length control s proposed to deal eectvely wth Q-lmts n generators. Fnally, the authors n [6] propose the use of a secant predctor and an arc-length parameterzaton. Secton dscusses wth detal all of these technques, concentratng on the advantages and dsadvantages of each one of them. The problem of large CPU tmes requred by the contnuaton method when tracng bfurcaton dagrams was dscussed n [6]; lttle has been reported n the lterature, however, on tryng to overcome ths dculty. Ths paper addresses ths ssue by proposng derent technques to reduce CPU tme, so that contnuaton methods can be ecently used n a control center envronment for \real tme" applcatons. Two dstnct technques are proposed and studed, namely fast-decoupled methods and system reductons. The ntroducton of fastdecoupled methods nto the contnuaton method s analyzed n detal n secton, concentratng on the computatonal ssues related to the mplementaton of these technques n exstng programs. In secton, two derent methods to reduce the number of system buses to trace bfurcaton dagrams are proposed and dscussed. The system reducton technques are nally tested n the 00 bus IEEE system, so that several computatonal ssues can be addressed n a realstc envronment.. Contnuaton Method Apower system can be modeled usng a set of nonlnear ordnary derental equatons and algebrac constrants, such as _x = f(x; y; ) () 0 = g(x; y; ) where the vector x < n represents the system state varables, the vector y < m corresponds to the system varables assocated to the nonlnear algebrac constrants, and < + s a slow changng parameter that drves the reduced system _x = s(x;)=f(x; h(x);) () to a local generc saddle-node or Hopf bfurcaton, f along system trajectores of nterest one has a nonsngular Jacoban of the algebrac constrants D yg(). Furthermore, as demonstrated n [6, 7] for ths partcular assumpton, a saddle-node bfurcaton pont of () corresponds to a system equlbrum pont where the Jacoban D zf(z 0; 0) s sngular (F() =[f T () g T ()] T, z =[x T y T ] T, F(z 0; 0)=0). Thus, a bfurcaton manfold of derental equatons () can be traced by montorng the equlbra of (). Voltage proles of power systems have been typcally obtaned by calculatng a seres of equlbrum ponts of equatons () wth successve power ow solutons [, 7]. These proles correspond to parts of the bfurcaton dagram. Hence, by solvng F(z;)=0for ncreasng values of, one can easly trace relevant parts of the bfurcaton dagram; however, snce the system Jacoban becomes sngular at a saddle-node bfurcaton (z 0; 0), and no equlbrum ponts or solutons exst for values of > 0, one cannot accurately compute 0 usng ths smple technque, because dvergence of the Newton- Mn.Mag[e-v] (a) p Mn.s-v (b) p Fg. : (a) Mnmum magntude of the egenvalues and (b) sngular values of the s.e.p.s of a smple generator-lne-load power system, as a functon of the system load P. Raphson based numercal methods does not guarantee nonexstence of solutons of the correspondng equatons. Nevertheless, as shown n [, 0], ths method can be used to obtan solutons rather close to the saddle-node bfurcaton pont due to the hghly nonlnear behavor of the egenvalues of the system Jacoban. Fgure depcts the mnmum magntudes of the egenvalues and sngular values of the steady-state Jacoban for a smple power system consttuted of a generator, a transmsson lne, and a constant power factor PQ load, where the actve load power P s chosen as the slow varyng parameter that drves the system to bfurcaton. Notce the sharp change n the magntude of the egenvalues and sngular values when the system s close to the bfurcaton pont. Smlar behavor has been also observed by several researchers n a varety ofpower systems [7,, 8]. The Voltage Stablty Analyss (Vstab) program dstrbuted by the Electrc Power Research Insttute (EPRI), uses ths technque to successfully trace the stable equlbrum regon of the bfurcaton dagram foravarety ofpower system models [7]. The contnuaton method overcomes the dcultes of the successve power ow solutons method, allowng the user to trace the complete bfurcaton dagram by automatcally changng the value of. The strategy used n ths method s llustrated n Fg., where a known equlbrum pont (z ; ) of the bfurcaton manfold s used to compute the drecton vector z and a change of the system parameter. Ths rst step s known as the predctor, snce t generates an ntal guess (z +z ; + ), whch s then used n the corrector step to compute a new equlbrum pont (z ; )on the bfurcaton dagram. Snce the Jacoban D zfj 0 s sngular at the bfurcaton pont, a parameterzaton s sometmes needed n the predctor and/or corrector steps, dependng on the technques used, to guarantee a well behaved numercal soluton of the related equatons. A detaled descrpton of these technques follows... Predctor and Parameterzaton One way of calculatng the drecton vector z at an equlbrum pont (z ; ) on the bfurcaton manfold, s to compute the tangent vector to the manfold at that pont. Hence, snce F(z ; )=0, then df (z;) =DzF(z;) d d + F ) D zfj =, F d = 0 ()

3 z λ (z, λ ) 0 0 varables, whereas z becomes the new parameter p,.e., for N = n + m p max z z ; z z ;...; z N z N ; (5) λ λ z Fg. : Bfurcaton dagram n state and parameter space depctng the s.e.p.s wth a contnuos bold lne, the u.e.p.s wth a bold dashed lne, and the bfurcaton pont at (z 0 ;0). A contnuaton methodology to move from one equlbrum pont (z ;) to another (z ;) s also shown. Thus, the drecton vector and the parameter step come from the normalzaton of the tangent vector,.e., z k = k =dj k z = d where k < + s a constant that controls the sze of the predctor step. The normalzaton n () results on the reducton of the step sze as the system approaches the saddle-node bfurcaton pont, snce the magntude of the tangent vector ncreases as the system gets closer to ths pont. If the step s too large, then the ntal guess (z +z ; + ) yeld convergence problems n the corrector, whereas f the step s too small, the method takes too many steps to trace the bfurcaton manfold. A technque to determne the optmal value of k was proposed n [], consderng the reactve power lmts of the generators. Good results were reported n [6] for k = n varous system szes, by usng step cuttng when lmts or convergence problems are encountered. Ths predctor technque has the partcular advantage of generatng an approxmaton to the zero rght-egenvector at the bfurcaton pont, snce the tangent vector smoothly converges to ths egenvector [6, 7]. The zero rght-egenvector yelds mportant nformaton regardng voltage senstve areas n the system [, 9] Computng the tangent vector n () does not represent a sgncant computatonal cost, snce one can use the last factored Jacoban matrx D zf(z ; ). However, ths method has dcultes when the equlbrum pont s close to the bfurcaton pont, snce the system Jacoban becomes ll-condtoned. To avod ths problem parameterzaton technques can be used [0]. A relatvely smple technque successfully appled n [6,,, ] s local parameterzaton, whch conssts on nterchangng close to the bfurcaton the parameter wth the system varable z z that has the largest normalzed entry n the tangent vector, so that becomes part of the equatons λ () Nevertheless, when step cuttng and perpendcular ntersecton correctors are used, ths local parameterzaton s not needed n practce [6, ], due to the hghly nonlnear behavor of the Jacoban egenvalues; one must be rather close to the saddle-node bfurcaton pont n order to have an llcondtoned Jacoban matrx. Another type of predctor wth parameterzaton used to take the system around the sngularty of the bfurcaton pont s the arclength method [0, 6]. Ths technque s based on the dea that the system varables and parameter at the equlbrum ponts can be represented as a functon of the arclength s of the bfurcaton manfold,.e., for F(z (s); (s)) = 0, D zfj ds + F d ds where the arclength s must satsfy the condton ds T ds + d ds = 0 (6) = (7) Therefore, by approxmatng z, d, and k =s ds (k < + ), equatons (6) and (7) become D zfj z + F = 0 (8) z T z + = k where k s a scalar constant that denes the length of the arc, and consequently the sze of the predctor step. Equatons (8) can be used to calculate the predctor step nstead of equatons () and (), wth a guaranteed nonsngular Jacoban at the bfurcaton pont. Fnally, a smpler predctor method that does not requre of parameterzaton s the secant method, whch was used n power systems bfurcaton analyss n [6]. Ths technque conssts on approxmatng the tangent vector =d usng two or more prevously determned equlbra on the bfurcaton manfold. Thus, gven two equlbrum ponts (z a ; a ) and (z b ; b ) on the bfurcaton dagram, such that b > a, d z b, z a Equatons () can then be used to calculate the drecton vector and the parameter step. Notce that the closer these two ponts are, wthn reasonable numercal tolerances, the better the approxmaton of the tangent vector; however, more equlbrum ponts have to be computed, takng longer to trace the desred bfurcaton dagram. On the other hand, ponts too far apart generate nadequate approxmatons of =d, yeldng ntal guesses that lead to convergence dcultes durng the corrector step. Usng more ponts on the bfurcaton manfold to better predct ts curvature, requres of greater computatonal resources, but can be used as an alternatve procedure to approxmate the tangent vector when the bfurcaton dagram changes drecton rapdly, partcularly when

4 control lmts are encountered. Hence, dependng on the curvature of the bfurcaton dagram, the secant method presents advantages and dsadvantages wth respect to the two predctors prevously descrbed. A mxed approach s proposed n [0, 6], usng the secant predctor when the tangent vector changes slowly on the \at" part of the bfurcaton dagram, to then swtch to tangent vector or arclength predctors when the manfold presents larger curvatures... Corrector Once an ntal guess (z +z ; + ) s determned n the predctor step, wth or wthout parameterzaton, the actual equlbrum pont (z ; ) on the bfurcaton manfold must be calculated by solvng the followng set of equatons for z and [0]: F(z;) = 0 (9) (z;) = 0 The rst vector equaton n (9) corresponds to the steady-state system equatons, whch have a sngular Jacoban D zfj 0 at the saddle-node bfurcaton pont (z 0; 0). The second scalar equaton represents a phase condton : < N < 7! < that guarantees non sngularty of the corrector equatons Jacoban " D zf D z F # (N +)(N +) at all equlbra on the bfurcaton manfold. Two derent phase condtons () have been successfully used n bfurcaton studes of power systems. The rst condton conssts on denng a perpendcular vector to z, whch starts at (z +z ; + ) and ntersects the bfurcaton manfold at (z;); thus, (z;)=z T (z, z, z )+ (,, ) (0) Ths was ntroduced n [] and successfully appled to varous systems n [6, ]. Ths condton does not requre of any knd of parameterzaton to guarantee non sngularty of equatons (9) for all system equlbra [6, 7]. A smpler phase condton was used n [,, 5, 6], based on the local parameterzaton of the system around the bfurcaton pont. In ths case, a local parameter p ( or z z), s set to a constant value,.e., (z;)=p, p, p The parameter p s chosen based on parameterzaton (5), guaranteeng a nonsngular Jacoban of equatons (9) [0]. Of ths two corrector technques, the perpendcular ntersecton has the advantage of not requrng of parameterzaton. However, t ntroduces an almost full row n the Jacoban matrx, whch must be taken nto consderaton durng the factorzaton process to avod sparsty degradaton.. Fast-Decoupled Technques For certan ac system models, saddle-node bfurcatons can be drectly detected usng load ow equatons P(;V;) = 0 () Q(;V;) = 0 snce these equatons have the same saddle-node transversalty condtons as derental equatons () [6, 7]. Hence, contnuaton methods can be appled to these equatons to obtan the desred bfurcaton dagram. P() and Q() are a subset of F(), and represent the actve and reactve power msmatches at the system buses; the vectors V and stand for the bus phasor voltage magntudes and angles, respectvely. Equatons () can be ecently solved for a gven parameter value usng fast-decoupled load ow technques [0,, ]. Thus, durng a Newton-Raphson teraton one has to solve matrx equaton DPj D V Pj D Qj D V Qj H N M L V V = = P, Q, () for vector [ T V T ] T, where P, =P(,; V,; ), and Q, =P(,; V,; ). Fast-decoupled methods are based rst on the assumpton that V 0 when solvng (), yeldng equaton B 0 =P(,; V,; ) () Assumng then that P 0, leads to B 00 V =Q(, + ; V,; ) () Equatons () and () are solved sequentally (BX verson). B 0 (B matrx) and B 00 (X, matrx) are constant matrces dened as []: B 0 = Hj=0;V = B 00 = Lj=0;V =;R=0 where V 6 stand for all bus phasor voltages, and R represents all system resstances. Smlar assumptons can be used to solve the followng corrector equatons usng the fast-decoupled method: P(;V;) = 0 Q(;V;) = 0 (;V;) = 0 Thus, for the perpendcular ntersecton equaton (0), the matrx equaton of the th Newton-Raphson teraton becomes 6 H N P M L Q T V T 7 5 V 5 = P, Q,, where,v, and are generated by the predctor step. Then, by assumng that 0, V 0, 0, and P 0, respectvely, the followng fast-decoupled corrector equatons can be obtaned: = (,; V,;,) (5) B 0 = P(,; V,;, + ) B 00 V = Q(, + ; V,;, + ) These equatons can be solved sequentally to obtan the desred equlbrum pont on the bfurcaton manfold. For small 5

5 changes and V,.e., small varatons n the system varables z, equatons (5) can be shown to have smlar convergence characterstcs as fast-decoupled load ow equatons () and (). The latter condton can be met by reducng the predctor's parameter step as the system approaches a saddle-node bfurcaton. HVDC systems can be ncluded n these fast-decoupled technques by treatng the ac/dc converters at each teraton as constant PQ loads, whle updatng the system ac bus voltages, and then solvng the DC system equatons wth xed ac converter voltages to calculate new values of the correspondng ac actve and reactve powers; ths process s repeated untl achevng convergence. Although the proposed fast-decoupled corrector has not been mplemented and tested, the prevous analyss just- es ncorporatng ths technque nto an exstng contnuaton power ow program. Work s currently underway at the Unversty ofwaterloo to add these methods to the Pflow program orgnally developed at the Unversty of Wsconsn- Madson [6]. A secant predctor together wth the fast-decoupled corrector descrbed above were proposed for ncluson n future versons of EPRI's program Vstab [7]. The current verson of ths program does not have a true mplementaton of a predctor-corrector contnuaton algorthm; however, the program successfully uses fast-decoupled load ows to determne s.e.p.s on the bfurcaton manfold for xed values of, whch could be vewed as a fast-decoupled mplementaton of the xed parameter corrector descrbed above, wthout local parameterzatons. When the fast-decoupled method fals to converge, the program swtches to a full Newton-Raphson teraton.. System Reducton Another approach to reduce the computatonal burden of tracng bfurcaton dagrams and locatng bfurcaton ponts, s to reduce the system sze whle mantanng acceptable levels of accuracy n the calculatons. A method to reduce the number of system varables for these types of studes was proposed n [8]; n ths case, the system s reduced based on coherency consderatons. Ths paper proposes two derent methods to reduce the system sze, based on certan local characterstcs of saddle-node bfurcatons. A dscusson of these technques follows... Tangent Vector Technque Saddle-node bfurcatons n power systems can be assocated to angle and/or voltage magntude problems at specc system buses through the zero rght-egenvector [6,, 6, 9]. Ths can be easly justed based on the fact that the tangent vector =d converges to ths egenvector at the bfurcaton pont [6, 7]; hence, the largest entres on the zero rghtegenvector pnpont the crtcal areas and buses aected the most by the bfurcaton. On the other hand, small entres on ths vector ndcate that the bfurcaton has lttle steady-state eect on the correspondng system varables or buses. Based on ths observatons, the system varables that suer lttle changes durng the calculaton of the bfurcaton dagram, can be safely elmnated from the computatonal process wthout aectng the nal results,.e., a varable z z can be kept constant at ts last equlbrum value f jz ju where z z n (), and u s a user dened tolerance. Notce that vector z has been already normalzed, whch avods errors of elmnaton of mportant system varables due to small values of all entres n =dj, especally for \at" bfurcaton proles far from the bfurcaton. Ths technque reduces the number of varables and equatons used durng the contnuaton process, decreasng consequently the computatonal burden of tracng bfurcaton dagrams. A problem wth ths method s the possble elmnaton of varables assocated to system controls wth lmts, partcularly bus voltages wth large reactve power support. These voltages change slowly or do not change at all, untl voltage control s lost when mnmum or maxmum lmts are reached. Furthermore, dependng on the system varatons produced by the parameter, an area that ntally s not consdered crtcal due to small changes on the assocated system varables, could become crtcal when close to bfurcaton. Hence, varables that are elmnated n early stages of the contnuaton process, mght become later a source for sgncant computatonal errors. To mnmze ths problem one can allow for the ncluson of these varables as they start changng rapdly; however, ths procedure mples havng to calculate the tangent vector for the full system every few steps, elmnatng some of the computatonal benets of the proposed reducton technque. A smpler but eectve approach s to carry out a system reducton every U steps of the contnuaton method. Ths reducton technque was mplemented nto the Pflow program [6], and tested n the 00 bus IEEE system. The results of these tests and some addtonal computatonal ssues are presented and dscussed n secton 5... Network Partton Several network partton technques have been proposed to reduce the sze of the power system for steady-state voltage stablty analyss [,, 5]. These methods also yeld the crtcal areas and buses n the system, so that control actons can be taken to steer the system away from the bfurcaton pont. A new method based on these technques s proposed here to partton the system and reduce the computatonal burden of calculatng the bfurcaton manfold. The dea s to rst detect the crtcal bus for a gven operatng pont,.e., determne the load bus wth the largest voltage varatons when the system parameter s changed. Ths can be easly done usng the tangent vector generated by the predctor step. The largest entry on that vector determnes the crtcal system varable and ts assocated load bus, as dscussed above. Once the crtcal bus s located, an ntal crtcal area s formed wth the rst neghbors of ths bus (level ). New levels of neghborng buses can be added to ths basc crtcal area to ncrease ts sze. The buses that do not belong to the crtcal area but connect ths area to the rest of the system (border buses), are consdered as part of the reduced system and are treated as constant voltage buses,.e., PV buses, based on the assumpton that ther steadystate voltage magntudes are not sgncantly aected by the parameter changes n the system. Hence, to determne a connecton level that denes an adequate system partton, one of the followng crtera can be used: 5

6 Determne the determnant of the \crtcal matrx" D 0 for the full and parttoned systems, as dened n [],.e., D 0 = D, CA, B (6) ) det (D zfj ) = det (D 0 )det (A) where matrces A, B, C and D are part of the system Jacoban matrx D zfj evaluated at the equlbrum pont (z ; ), A B D zfj = C D D s a matrx partton correspondng to equatons and varables of the crtcal bus l, thus D = 6 P l l Q l l P l V l Q l V l where [P l() Q l()] T F(), and [ l V l] T z. Notce that D 0 for the complete and reduced systems n (6), can be obtaned by factorzng the correspondng Jacoban matrx D zfj. Furthermore, snce D zfj s nonsngular for all equlbrum ponts other than the saddle-node bfurcaton (z 0; 0), det (D 0 ) 6= 0 and det (A) 6= 0 for those equlbra. (At the bfurcaton pont det (D 0 )=0 and det (A) 6= 0, snce det (D zfj 0) = 0 and matrx A s nonsngular, due to the elmnaton from the sngular Jacoban D zfj 0 of the rows and columns wth the largest entres n the zero rght-egenvector.) Hence, by montorng the derence between the determnants of D 0 F for the full system and D 0 R for the reduced system, an adequate partton s assumed to be obtaned when jdet (D 0 R)j,jdet (D 0 F )j (7) where s a postve scalar that denes the desred crteron tolerance. Compare the changes n the tangent vector =dj between the full and the reduced system. Thus, when k R d k, k F d 7 5 k (8) the desred system reducton level s attaned. When the tangent vector predctor s used, the computatonal cost of applyng ths crteron s lower than calculatng matrces D 0, snce one only needs to solve equaton () for each reducton level untl convergence condton (8) s met. Of these two crtera, the tangent vector appears to be more sensble to changes n system sze than the determnant ofd 0 ; however, the norm of =d seems to have a hghly nonlnear behavor wth respect to changes n, smlar to the prevously depcted behavor of the Jacoban egenvalues, whereas the determnant of the reduced matrx assocated to the crtcal bus seems to have an almost lnear behavor as shown n [6]. More tests have to be carred out before a dente concluson can be reached. Ths reducton process can be repeated every few steps of the contnuaton method, and as wth the tangent vector reducton method, one expects to obtan a smaller crtcal area partton as the system approaches bfurcaton. Snce the parameter s usually chosen to smulate varyng load levels throughout the system, the correspondng generator powers must also be allowed to change to supply the desred system load for each value of. For the full system ths does not represent a problem; all system generators are assgned a partcpaton factor of the total avalable power, dependng on ts sze and ntal actve and reactve powers, so that each generator can change ts base power to supply the desred load demand. For the reduced system, where some, or all, border buses have been converted from load (PQ) to generator (PV) buses, one must rst determne the level of power that s beng delvered to or absorbed from the crtcal area through these buses, so that an approprate generaton partcpaton factor can be dened. Hence, these factors can be calculated by comparng the power change on these buses for two derent equlbra on the bfurcaton manfold,.e., for two dstnct values of the power derences on the border buses yeld the desred partcpaton factors. One problem that stll has to be resolved n order to make the proposed method a practcal tool, s the handlng of voltage control lmts, partcularly generator Q-lmts, whch make a sgncant derence n the shape of the bfurcaton manfolds and the locaton of the correspondng bfurcaton ponts, as llustrated n secton 5 and dscussed n [, 6, ]. Studes are underway to nd an approprate answer to ths queston, but so far t does not appear to have a smple soluton. The partton technque presents the same problem as the tangent vector reducton method regardng the possble changes n locaton of the crtcal area wth changes on the parameter. Hence, f the crtcal area at the bfurcaton s not part of the reduced system, the nal results can have sgncant errors as demonstrated for the 00 bus IEEE test system below. Ths s especally true for the partton method, whch, contrary to the tangent vector technque, s based on a local area analyss of the system; therefore, ths method seems to be better suted to handle problems where changes n only aect a partcular area n the system. Due to the unresolved dcultes n the partton method, t was not drectly mplemented n an exstng contnuaton program. The prelmnary results shown n the next secton were obtaned usng the Pflow program [6] and Matlab [7]. 5. Computatonal Results The 00 bus IEEE system was used to test the system reducton technques descrbed above. Ths s an ac system dvded n areas, wth 69 generators and transmsson lnes and transformers, ncludng 5 under-load tap changer (ULTC) transformers. The tests were carred wthout enforcng power ow control between areas and treatng the ULTCs as regular transformers wth xed taps, to smplfy the analyss of the results. The system loads were modeled as constant PQ loads, and the parameter was used to smulate actve and reactve power load ncreases throughout the system,.e., load powers at bus j were smulated as P j = P 0 j ( + ) Q j = Q 0 j ( + ) where P 0 j and Q 0 j represent the base load power. For ths partcular load model, represents the loadng factor n p.u. or total MVA. The generators were assgned a partcpaton fac- 6

7 tor of the load demand based on the ntal generated power, wthout enforcng maxmum actve power lmts. Several tests were run wth and wthout generator Q-lmts, as shown below. The tangent vector reducton technque was ncorporated nto the Pflow program, to obtan the results shown n Tables through for several values of the tolerance u and number of steps U n (8). Fgures {6 depct the bfurcaton dagrams correspondng to each of the test cases n Tables {; the tp of these curves correspond to a saddle-node bfurcaton. No dstncton s made between stable (s.e.p.) and unstable (u.e.p.) equlbrum ponts on these manfolds, snce the egenvalues were only montored to detect a sngularty or saddle-node bfurcaton pont; Hopf bfurcatons were not tracked on these cases. Tables and, and ther correspondng Fgs. and, show the results obtaned for ncreasng load throughout the whole system, wth and wthout enforcng generator Q-lmts, respectvely. On the other hand, Tables and wth ther respectve Fgs. 6 and 5, depct the results obtaned wth and wthout Q-lmts when the load s changed n area only. From these results one can conclude, n general, that the smaller the values of u and U, the smaller the number of equatons and, therefore, the larger the errors. Also, as the crteron tolerance u and the number of steps U decreases, the CPU tme per step of the contnuaton method tends to decrease, as expected, partcularly for the cases where the load s only changng n a small system area. Notce that n some cases the CPU tme tends to ncrease even though the number of equatons decreases, ths s due to the need of havng to refactorze the Jacoban matrx when the equatons change; thus, the smaller the value of U, the more often ths has to be done along the bfurcaton manfold, and consequently the larger the CPU tme. In some cases cyclng and back-trackng problems were detected, especally when the values of u and U are small, ncreasng the overall CPU tme requred to trace the bfurcaton manfolds. Fgure 6 depcts the problem of mmedate Q-lmt nstablty [, 5, 6],.e., the test system becomes mmedately unstable when voltage control s lost at generator bus 9. Table 5 and Fg. 7 depct the results and bfurcaton manfold, respectvely, obtaned wth the network partton method when usng convergence crteron (7) wth =0,. The load powers are assumed to ncrease n all system load buses wthout generator Q-lmt enforcement. The saddle-node bfurcaton max values and crtcal buses shown n Table 5 were obtaned usng a drect method. CPU tmes were not determned n ths case snce the method was not ncorporated nto a contnuaton program; the test shown here was run only to depct some of the problems wth the partton method. Notce that by usng the determnant convergence crteron the system s reduced from 00 buses to 0 (level 9) at the ntal operatng pont,.e., for = 0. Ths reduced system does not contan the crtcal bus at the saddle-node bfurcaton pont (bus 9), consequently yeldng a sgncant error n the bfurcaton manfold depcted n Fg. 5, and n the locaton of the correspondng bfurcaton pont. More tests must be run to determne whether tangent vector crteron (8) yelds better results, and to resolve the problem of how to handle voltage control lmts. Case Reducton Eqns. Tme Tme/step max Error No. Data [sec.] [sec.] [p.u.] [%] u =0 U = u =0,6 u =0, u =0, U =0 5 u =0, U =5 6 y u =0, y Cyclng problems n the u.e.p. regon Table : Results of the tangent vector reducton technque for the 00 bus IEEE system wth no lmts consdered, when the actve and reactve load powers are ncreased throughout the system. The drect method appled to the complete system yelds: max =0:69p.u., and bus 9 as the crtcal bus. Case Reducton Eqns. Tme Tme/step max Error No. Data [sec.] [sec.] [p.u.] [%] 7 u = U = u =0,6 9 u =0, 0 u =0, U =0 y u U =0, =5 y u U =0, = y Back-trackng Table : Results of the tangent vector reducton technque for the 00 bus IEEE system wth lmts consdered, when the actve and reactve load powers are ncreased throughout the system. The drect method appled to the complete system yelds: max =0:0576p.u., and bus 576 as the crtcal bus. Case Reducton Eqns. Tme Tme/step max Error No. Data [sec.] [sec.] [p.u.] [%] u =0 U = u =0,6 5 y u U =0, = 6 u =0, U =0 7 u =0, U =5 8 y u U =0, = y Back-trackng Table : Results of the tangent vector reducton technque for the 00 bus IEEE system wth no lmts consdered, when the actve and reactve load powers are ncreased napartcular area of the system only. The drect method appled to the complete system yelds: max =:0005p.u., and bus 9 as the crtcal bus. Case Reducton Eqns. Tme Tme/step max Error No. Data [sec.] [sec.] [p.u.] [%] 9 u =0 U = u =0, u =0, u =0, U =0 u =0, U =5 u =0, Table : Results of the tangent vector reducton technque for the 00 bus IEEE system wth lmts consdered, when the actve and reactve load powers are ncreased napartcular area of the system only. The system becomes mmedately unstable before the saddle-node bfurcaton when a maxmum Q-lmt s reached at bus 9 when Q lm =0:0p.u. 7

8 V_9 [p.u.] Cases Case Case 5... Case Loadng Factor (lambda) [p.u.] V_9 [p.u.] Cases & --- o Case Case 6 Fg. : Bfurcaton dagrams (PV or nose curves) of a system bus voltage for the orgnal and reduced systems when no lmts are consdered. The actve and reactve powers ncrease n all system load buses. A saddle-node bfurcaton takes place at the maxmum system loadng max =0:69p.u Case 7... Case Loadng Factor (lambda) [p.u.] V_56 [p.u.] Fg. 5: Bfurcaton dagrams (PV or nose curves) of a system bus voltage for the orgnal and reduced systems when no lmts are consdered. The actve and reactve load powers ncrease n a local area only. A saddle-node bfurcaton takes place at the maxmum system loadng max =:0005p.u...5 Cases 9& Cases Case Cases Loadng Factor (lambda) [p.u.] V_9 [p.u.] o Case... Cases & Fg. : Bfurcaton dagrams (PV or nose curves) of a system bus voltage for the orgnal and reduced systems when lmts are consdered. The actve and reactve powers ncrease n all system load buses. A saddle-node bfurcaton takes place at the maxmum system loadng max =0:0576p.u Loadng Factor (lambda) [p.u.] Case Buses Gens. Crtcal det (D 0 ) max Error No. Bus [p.u.] [%] Fg. 6: Bfurcaton dagrams (PV or nose curves) of a system bus voltage for the orgnal and reduced systems when lmts are consdered. The actve and reactve load powers ncrease n a local area only. A Q-lmt mmedate nstablty occurs at Q lm =0:0p.u., before the saddle-node bfurcaton pont Table 5: Results of the network partton reducton technque for the 00 bus IEEE system wth no generator Q-lmts and load ncreasng throughout the system. 8

9 V_90 [p.u.] Case --- Case Loadng Factor (lambda) [p.u.] Fg. 7: Bfurcaton dagrams (PV or nose curves) of a system bus voltage for the orgnal and parttoned systems when no lmts are consdered and the actve and reactve load powers ncrease throughout the system. A saddle-node bfurcaton takes place at the maxmum system loadng max =0:69p.u. 6. Conclusons A comprehensve revew of the current lterature and state of the art of contnuaton methods n power systems bfurcaton analyss s presented. The paper then proposes three dfferent methods to accelerate the computaton of saddle-node bfurcaton ponts and ther assocated manfolds wth the contnuaton method, namely, fast-decoupled correctors, tangent vector system reducton, and network partton. Of these methods, only the tangent vector was ncorporated and tested n a the producton-grade contnuaton program Pflow; however, addtonal tests on larger systems are requred. Work s currently underway to nclude and test the proposed fast-decoupled corrector n Pflow, based on favorable prevous experence wth successve fast-decoupled power ows n Vstab. In the case of system reducton through network partton, several ssues, partcularly the handlng of voltage control lmts, have to be resolved before ths method can be programmed and tested n realstc condtons. Once all these methods are mplemented n a smlar computatonal envronment, a thorough comparson of ther performance wll be carred out to determne ther applcablty n control centers. REFERENCES [] L. H. Fnk, ed., Proceedngs: Bulk Power System Voltage Phenomena Voltage Stablty and Securty, ECC/NSF Workshop, Farfax, VA, Ecc. Inc., August 99. [] Y. Mansour, ed., Suggested Technques for Voltage Stablty Analyss, IEEE/PES Report, 9TH060-5PWR, 99. [] C. C. Lu and K. T. Vu, \Types of Voltage Collapse," n [], pp. {. [] I. Dobson and L. 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10 [5] H. D. Chang, W. Ma, R. J. Thomas, J. S. Thorp, \A Tool for Analyzng Voltage Collapse n Electrc Power Systems," Proc. PSCC, Graz, Austra, August 990, pp. 0{7. [6] H. D. Chang, A. J. Flueck, K. S. Shah, N. Balu, \CPFLOW: A Practcal Tool for Tracng Power System Steady-State Statonary Behavor Due to Load and Generaton Varatons," IEEE/PES 9 WM --PWRD, New York, February 99. [7] Voltage Stablty/Securty Assessment and On-Lne Control, EPRI TR-09, Vol., Aprl 99. [8] M. M. Begovc and A. G. Phadke, \Voltage Stablty Assessment Through Measurement of a Reduced State Vector," IEEE Trans. Power Systems, Vol. 5, No., February 990, pp. 98{0. [9] I. Dobson, \Observatons on the Geometry of Saddle Node Bfurcatons and Voltage Collapse n Electrcal Power Systems," IEEE Trans. Crcuts and Syst.{I, Vol. 9, No., March 99, pp. 0{. [0] B. Stott and O. Alsac, \Fast Decoupled Load Flow," IEEE Trans. Power Apparatus and Systems, Vol. 9, 97, pp. 859{ 869. [] R. A. M. van Amerongen, \A General-Purpose Verson of the Fast Decoupled Load Flow," IEEE Trans. Power Systems, Vol., No., May 989, pp. 760{770. [] A. Montcell, A. Garca, O. R. Saavedra, \Fast Decoupled Load Flow: Hypothess, Dervatons, and Testng," IEEE Trans. Power Systems, Vol. 5, No., November 990, pp. 5{. [] L. Vargas and V. H. Quntana, \Clusterng Technques for Voltage Collapse Detecton," Electrc Power System Research, Vol., No., 99, pp. 5{59. [] A. C. Z. de Souza and V. H. Quntana, \Identcatonof Voltage Collapse Usng Network Parttonng," Proc. NAPS, Washngton, October 99, pp. 9{96. [5] A. C. Z. de Souza and V. H. Quntana, \A New Technque of Network Parttonng for Voltage Collapse Margn Calculatons," paper GTD/9/8 accepted for publcaton on IEE Proc. Generaton, Transmsson and Dstrbuton, June 99. [6] A. C. Z. de Souza, \A Voltage Collapse Approach by Crtcal Bus Determnaton," Ph.D. Research Proposal, E&CE Department, Unversty of Waterloo, Aprl 99. [7] MATLAB, The Math Works Inc., Natck, Massachusetts, 99. of Waterloo, Department of Electrcal and Computer Engneerng, where he s currently a Full Professor and Assocate Charman for Graduate Studes. Hs man research nterests are n the areas of numercal optmzaton technques, state estmaton and control theory as appled to power systems. Dr. Quntana s an Assocate Edtor of the Internatonal Journal of Energy Systems, and a member of the Assocaton of Professonal Engneers of the Provnce of Ontaro. Claudo A. Ca~nzares was born n Mexco, D.F. n 960. In Aprl 98, he receved the Electrcal Engneer dploma from the Escuela Poltecnca Naconal (EPN), Quto-Ecuador, where he was a professor for several years. Hs MS (988) and PhD (99) degrees n Electrcal Engneerng are from the Unversty of Wsconsn{Madson. Dr. Ca~nzares s currently an Assstant Professor at the Unversty of Waterloo, Department of Electrcal & Computer Engneerng, and hs research actvtes are mostly concentrated n the analyss of stablty ssues n ac/dc systems. Antono Z. de Souza was born n Brazl, on December 5, 96. He receved hs B.Sc. n Electrcal Engneerng from the Unversty of the State of Ro de Janero n 987. In September 990, he obtaned hs M.Sc. degree n Electrcal Engneerng at the Catholc Pontcal Unversty of Ro de Janero. He has been workng towards a Ph.D. degree at Unversty of Waterloo, Department of Electrcal and Computer Engneerng, snce September 99, sponsored by the Brazlan agency CNPq. Hs research nterests are n restoraton and voltage control of power systems. Vctor H. Quntana receved the Dpl. Ing. degree from the State Techncal Unversty of Chle n 959, and the M.Sc. and Ph.D. degrees n Electrcal Engneerng from the Unversty of Wsconsn, Madson n 965, and Unversty oftoronto, Ontaro, n 970, respectvely. Snce 97 he has been wth the Unversty 0