(1) L. are all the parameter variables in the polynomials above. Wu Elimination intends to reduce the number of

Transcription

1 Wu Elmnaton Method Appled n Statc Analyss of the Power System QING ZHANG, and CHEN CHEN Department of Electrcal Engneerng Shangha Jao Tong Unversty 800 Dong Chuan oad, Shangha 0040 CHINA chchen@sjtu.edu.cn Abstract: - Wu Elmnaton Method s appled to obtan analytcal soluton of power flow equatons wthout ether extraneous roots or mssng roots. The symbolc process of Wu Elmnaton s mplemented by software MAPLE. A mnor group of solutons s obtaned under low load other than the major group of solutons. Based on computaton, analytcal V P, V Q curves as well as the steady state operaton area. The results are compared wth the results of Homotopy Method. Dscusson on unstable equlbrum ponts s presented whch s potentally valuable for research on voltage stablty and angle stablty of complex power systems. Key-Words: - Wu Elmnaton Method, Power system, V P curve, Voltage stablty, Angle stablty approach s appled to determne the Hopf bfurcaton pont of dynamc voltage stablty [6]. On bass of results obtaned by WEM, the V P s dscussed, and the voltage stablty s analyzed n ths paper. Introducton Wu Elmnaton Method (WEM s proposed to solve smultaneous polynomal equatons, and provde effcent algorthm to calculate not only wth numbers but also wth symbols. So far, WEM, also called the mechanzed mathematcs, has already been appled n the proof of theorems by machne n many theoretc aspects (mathematcs, physcs, etc.. WEM s able to enumerate all the solutons of polynomal equatons wthout ether extraneous or mssng roots. The solutons are analytcal and superor to numercal methods. Papers [],[] were on WEM appled n power system, and others n constraned dynamcs [3] and n rotor dynamcs [4]. Ths paper presents WEM appled n solvng power flow equatons. Newton-alfson method [5] and fast decoupled method [6], etc. are well known n solvng power flow equatons. Those numercal methods depend on the ntal condtons and are not possble to obtan the unstable parts snce the Jacoban matrx does not converge under those condtons. Power flow equatons whch are nonlnear exhbt multple solutons [7]-[9]. Homotopy method was employed n [9]-[] and genetc algorthms was appled n [] to obtan all the possble solutons. An teraton method was newly proposed n [3] for low-voltage power flow soluton. WEM can obtan all the solutons to smultaneous nonlnear equatons, so as solutons to the power flow equatons. Voltage stablty s the most wdespread power qualty ssue. esearch on propagaton of voltage fluctuaton along practcal power dstrbuton systems s presented n [4].Contnuaton power flow s used n statc voltage stablty [5]. A novel Basc Concept of WEM The basc concept of WEM s presented n ths secton. Comparson between WEM and Gaussan Elmnaton Method s dscussed for clear comprehenson of WEM. Furthermore, the process to apply WEM s gven n ths secton.. Wu Elmnaton Method and Gaussan Elmnaton Method Gaussan Elmnaton method s well known n solvng lnear equatons. A set of trangular equatons s obtaned by elmnaton and equvalent to the orgnal equaton set. Then the trangular equaton set s easy to be solved by smple substtuton. The Wu Elmnaton s mplemented by addton or subtracton of polynomals, after the process a set of trangular equatons s obtaned. The format of the trangular polynomal equaton set s as follows: Pxuu (,, Kun r = 0 P( x, x, u, ukun r = 0 ( L Pr( x, xkxr, u, ukun r = 0. where u, u, L, u n r are all the parameter varables n the polynomals above. Wu Elmnaton ntends to reduce the number of ISSN: Issue, Volume 7, December 008

2 parameters smlar to Gaussan Elmnaton. However, the set of trangular equatons s not exactly equvalent to the orgnal set of equatons durng Wu Elmnaton whch s dfferent from the Gaussan Elmnaton. The solutons of the set of orgnal equatons are the solutons of the set of trangular equatons, but not necessarly vce versa. The Wu Elmnaton s mplemented by calculatng the remander of polynomals.. Calculaton of Characterstc Set n Wu Elmnaton The kernel of WEM s calculatng the characterstc set ( CS. The process of calculatng the CS s actually the process to elmnate the varables. PS = + ( PS, S No PS = PS Classfy PS Form BS S = e md( PS / AS S = Yes, =k CS = BS k Fg. Flow chart of calculaton of characterstc set. The procedure s composed by fnte repeats of the steps as follows [7]: PS = { P, PL P k } s denoted as PS. th In round of elmnaton, the operaton on PS s as follows: Step : Classfy the polynomals n PS by class whch s determned by the varables n the polynomals. Polynomals wth the same class are n the same group. Step : Form the basc set ( BS. Select the polynomal wth the lowest order exponent of the man varable n each group to form a trangular set and then transform the set nto an ascendng set. Step 3: Calculate the remander set ( S of PS about BS, whch s denoted by S = e md( PS / BS. Then put S nto PS to get PS +. That means PS + = ( PS, S. If S =, whch means the remanders of PS about BS are zeroes, then BS s the characterstc set of PS, denoted by CS ; Otherwse, contnue to repeat the steps above. The process of elmnaton s descrbed by the flow chart n Fg.. 3 WEM Appled n Solvng Power Flow Equatons A power system contans n + nodes, of whch n s are PV nodes, n ns are PQ nodes and one s the slack node. The power flow equatons are denoted n Cartesan coordnaton. The voltage of th node s denoted as V = e + jf ( j =. All the power flow equatons are n nd order as follows: PQ node: n+ n+ ( ( 0 (a P e G e B f f G f + B e = s k k k k k k k k k= k= n+ n+ ( ( 0, (b Q f G e B f + e G f + B e = s k k k k k k k k k= k= where =,, n ns. PV node: n+ n+ ( ( 0 (3a P e G e B f f G f + B e = s k k k k k k k k k= k= Vs ( e + f =0, (3b where = n n +, n n +, n, G and B are s s th the real part and magnary part of node s admttance respectvely. The power njecton at PQ node s denoted as Ss = Ps + jq s, n whch P s s the actve power, Q s s the reactve power. For PV node, the voltage magntude s a constant, denoted by Vs θ and the actve power njecton s denoted by P s. The procedure of applyng WEM n solvng power flow equatons s a process to solve the CS of the power flow equatons. Accordng to the calculaton flow chart, power flow equatons should be classfed accordng to the class. For a power system wth n + nodes, there are n power flow equatons contanng varables whch are denoted by (e, f,, e f, L e n, f n. For convenence, suppose the sequence of elmnatng varables s (e, f, e, f, L e n, f n, then all the power flow equatons are classfed nto r groups accordng to the class of each equaton. The polynomal seres s denoted by PS = { P, P, LP n ; P n+, LP ; ;, } n+ n L P n+ n+ L+ n P r + L, n+ n+ L+ nr where k k ISSN: Issue, Volume 7, December 008

3 CLS( P = L = CLS( Pn = n r CLS( Pn + = L = CLS( Pn + n = n r + L (4 CLS( Pn + n + L+ n CLS( P r n n n n + = L = + + L+ = r CLS( Pn + n + L+ n CLS( P r n n n n + = L = + + L+ =, r n+ n + L + nr =. n The man varable n the frst group whch contans n polynomals s f r. The man varable th n the r group whch contans nr polynomals s e. Accordng to the calculaton flow n Fgure, select a polynomal wth the lowest power exponent of the man varable n the r groups respectvely to form the BS of PS after classfyng all the equatons, then solve the remander set whch s denoted by S of PS about BS, and then S together wth PS s denoted by PS, whch means PS S = ( PS,. epeat the elmnaton operaton untl S =, that means the remander of any polynomal n PS about BS s zero. At last, there are n polynomals n BS whch s denoted as follows: B ( fn = 0 B ( fn, en = 0 L (5 Bn ( fn, en, K fr, er, K f = 0 Bn( fn, en, K fr, er, K f, e = 0 BS s a set of trangular equatons whch s easy to solve. However, BS s not equvalent to the orgnal set of equatons PS. In order to obtan the true solutons of PS, t s necessary to deal wth the solutons of BS accordng to the theorems n WEM. A three-node system below s an example used to explan the applcaton of WEM specfcally. 4 Case Study Fg. shows a three-node system [0], [8]. The power njecton of each node and reactant of each branch are also showed n the fgure. All the parameters are per unt value. Node s a PQ node. The voltage of node s unknown, denoted by V = e + jf. Node s a PV node. The voltage of node s denoted by V = e + jf. WEM s appled to solve the power flow equatons of the system shown n Fg Fg. Three-node system. 4. The calculaton procedure Accordng to (-(3, four equatons about e, f,, f are as follows: e p = 0..74e ee +.587e f e p f fe.587 f f.0407 f.74 f f f.587 fe.855 f = e e f +.587ee +.855e p = ee fe e f f.587e f.35 f p =.00 e f 4 (6 By calculaton, the sequence of elmnatng varable s f, f, e, e.the process of solvng Eq. (6 by WEM s as follows: CLS( p = CLS( p = CLS(p 3 = 4, CLS( p 4 = 3. Select p 3 and p4 to form basc set BS, and calculate the remander set (, ; Then PS = ( p,,, p,,. Select and p 4 p p3 4 to form basc set BS, and calculate the remander set of PS about BS. epeat the procedure to obtan the CS of ( p, p, p3, p4. It s necessary to menton that the kernel of the procedure s formng the basc set. As mentoned before, polynomal wth low power exponent of man varable s the frst choce to form basc set. So t s unnecessary to do some calculaton whch would create remander wth very hgh power exponent of man varable. The Table below shows the process to calculate the CS of (6, n whch PS has been classfed nto groups separated by parentheses. The CS of ( p, p, p, p 3 4 s (5, 4, p 3. Accordng to WEM theorems, the zero set of the equatons can be expressed by the formula Zero( PS = Zero( CS / I + Zero( PS, I [7]. Under accurate calculaton condton, the zero set s obtaned by elmnate extra roots from zero set of CS. However, due to the complexty of power flow ISSN: Issue, Volume 7, December 008

4 equatons and lmt calculatng ablty of the software Maple, t s dffcult to elmnate the extraneous roots under fnte precson and the remanders of orgnal equatons on CS are not vod. In order to obtan the solutons of orgnal equatons, the followng method s used n the paper: frst obtan the solutons of the CS, and then back substtute the solutons nto the orgnal equatons to calculate the errors. Extraneous roots are taken off by settng a certan error. Analyss of the results shows that the results obtaned by WEM n the way are correct and hghly precse. Table Solvng CS of power flow equatons of three-node system PS BS S ( p, p, p3,( p 4 ( p4, p 3 = e md( p / BS = e md( p / BS ( p, p, p3,( p4,, (, p 3 = e md( p4 / BS = e md( / BS ( p, p, p3,( p4,, (, (,, p 3 3 = e md( / BS3 ( p, p, p3,( p4,, (,, 3 ( 3,, p 3 4 = e md( / BS4 ( p, p, p3,( p4,, (,, 3, 4 ( 4,, p 3 5 = e md( 4 / BS 5 Table Solutons of three-node system by WEM e f e f j j j j j j j j Table 3 Solutons of three-node system by Homotopy method E E* E E * Power system solutons (E and E* are conjugate pars j j j j j j j j j j j j j j j j None power system solutons (E and E* are not conjugate pars j j j j j j j j Table 4 Comparson of complex results by WEM and Homotopy method 5 6 V V V V Wu j j j j Homotopy j j j j Dfference 0+j j j j CS 4. Analyss of esults Set the dgts as 50 when solvng ( p, p, p3, p4 by WEM. 5 n the s a polynomal wth the only varable e, the hghest order exponent beng 40. So solve CS frst to obtan the 40 groups of solutons. Back substtute all the 40 groups of solutons to ISSN: Issue, Volume 7, December 008

5 obtan the solutons of the orgnal set. Table shows all the 6 groups of solutons obtaned by WEM, n whch all the solutons are dsplayed n 6 dgts for convenence. Homotopy method s used to calculate power flow of the three-node system [0]. As shown n Table 3, there are 4 groups of power system solutons and groups of non power system solutons. By comparng results n Table and Table 3, the 4 groups of power system solutons are almost the same. As for the two groups of complex solutons n [0], the real part e and the magnary part f of the voltage can not be separated from the results by Homotopy method. So add e and f obtaned by WEM by the formula V = e+ jf to calculate V and V. Table 4 shows the comparson of complex results by the two methods. By comparson, t can be concluded that the results obtaned by WEM and Homotopy method are almost the same. Fg. 3 V P curve wth Q = Plottng analytcal V P and V Q curves WEM can obtan all the solutons of power flow equatons. Change the system operaton condton to calculate the power flow by WEM whle removng the complex results whch do not satsfy the power system. Plot the analytcal V P and V Q curves wth all the results obtaned by WEM. System operaton stablty can be analyzed through V P and V Q curves. The procedure of plottng the V P and V Q curves at PQ node n the three-node system s as follows. The actve and reactve power njectons are negatve at the PQ node n the three-node system. In the paper, the absolute values of the actve and reactve powers are used for convenence to plot and explan. In the three-node system, change P wth Q = 0. to obtan all the power flow solutons to plot V P curve at node, as shown n Fg. 3. When P s small, there are four groups of real solutons; as P ncreases, two groups of the solutons tend to converge. When P = 0.78, the upper branch and the lower branch of curve converge at one pont. As P contnues to ncrease, the remanng two groups of solutons also tend to converge. When P =.46, the upper branch and the lower branch of curve converge at one pont. Curve 3 and curve 4 n Fg. 3 are the mean value of voltage (MVV of curve and curve, respectvely, as changng. P Fg. 4 V Q curves wth P = 0.. By analyss of V P curve n Fg. 3, the upper branches of curve and curve has smlar changng trend. Further dscusson should be made to make sure whether operatonal ponts n curve are stable or unstable. The lower branches are clearly the unstable operaton condtons. In the three-node system, change Q wth P = 0. to obtan all the power flow solutons to plot V Q curve at node, as shown n Fg. 4. The V Q curve s smlar to V P curve. When Q = 0.447, the upper branch and the lower branch of curve converge at one pont. When Q =.78, the upper branch and the lower branch of curve converge at one pont. Curve 3 and curve 4 n Fg. 4 are the MVV of curve and curve respectvely as changng Q. The fttng curves n Fg. 5 show the relatonshp between Qmax and Pmax of operaton lmt condtons whle P > 0, Q > 0.The shadow area A n Fg. 5 s the set of all the steady state operaton condtons whle the shadow area B s the set of ISSN: Issue, Volume 7, December 008

6 possble stable operaton condtons wth relatvely hgh voltage under low voltage wth lght load. Fg. 5 Curves of Q P. 6 Propertes of Jacoban matrx The relatonshp between power and voltage as well as angle could be expressed n the lnearzed equatons as follows: ΔP JPθ JPV Δθ Δθ = J Q JQ θ J = Δ QV ΔV ΔV (7 where ΔP s the ncremental change n bus actve power; ΔQ s the ncremental change n bus reactve power; Δθ s the ncremental change n bus voltage angle; ΔV s the ncremental change n bus voltage magntude. J = [ JQV JQ θjpθ JPV ] s the reduced Jacoban matrx. By Schur s formula[9], we could obtan the followng equaton: det( J = det( JP θ det( J (8 The precondton of Schur formula s that J P θ exsts. J P θ s proved to be strctly nvertable under general power system condton and an nreducble dagonally domnant matrx n [0]. As a result, Jacoban matrx J s sngular when the reduced Jacoban matrx J s sngular. The concluson s to judge the voltage stablty by Jacoban matrx or by the reduced Jacoban matrx usng V Q senstvty method are equvalent. The rank of the Jacoban matrx for the three-node system s three. Keep reactve power constant whle changng actve power to calculate the operatonal parameters V, θ, θ. When P s small, there are four groups of solutons; when P s large, there are only two groups of solutons. max max As shown n Table 5, when P = (labeled as Pmax wth an arrow lne n Fg. 3, det( J correspondng to the two groups of solutons are almost zero; when P =.46 (labeled as P max wth an arrow lne n Fg. 3, correspondng to the two remanng groups of solutons are almost zero. The frst, second, thrd and fourth columns n Table 5, Table 6 and Table 7 are correspondng wth the I, II, III, IV branches n Fg. 3, respectvely. V Q senstvty analyss s used to measure the degree of system stablty. Let Δ P = 0, then Δ Q = JΔV, where J = [ JQV JQ θjpθjpv]. So V J Δ = ΔQ. J s th the reduced V Q Jacoban matrx n whch dagonal element s the V Q senstvty at bus. When V Q senstvty s postve, the system s stable; the smaller the senstvty, the more stable the system. Oppostely, when V Q senstvty s negatve, the system s unstable. The J of three-node system s frst order matrx. So the V Q senstvty equals the determnant of the matrx. The results are shown n Table 6 by whch the degree of stablty under dfferent operaton condtons can be measured. Table 5 Determnant of Jacoban matrx wth Q = 0. P As shown n Table 6, when P > 0.78, the number of the groups of solutons changes from 4 to ; when P >.46, there s no soluton of the power flow equatons. The V Q senstvtes n the st and nd column are postve, so the system s stable. The V Q senstvtes n 3rd and 4th column are negatve, so the system s unstable. The senstvty s very large when reachng the stablty lmt. Even very small Δ Q would lead to large change of the voltage. Table 7 shows the sgn of det( J, det( J P θ and det( J under dfferent operatonal condtons n ISSN: Issue, Volume 7, December 008

7 Fgure 3. Analyze the changng process of det( J frst whch s correspondng to operaton condtons, as shown n curves n Fgure 3: det( J > 0, correspondng to the upper branch of curve, as P ncreases; det( J = 0, when P reaches the lmt. In other words, J s sngular; det( J < 0, correspondng wth the lower branch of curve. The changng process of det( J s smlar wth that of det( J, whle det( J P θ remans postve durng the process. The changng process of det( J whch s correspondng to operatonal condtons, as shown n curves n Fgure 3 s that: det( J < 0, correspondng wth the upper branch of curve, as P ncreases; det( J = 0, when P reaches the lmt. In other words, J s sngular; det( J > 0, correspondng wth the lower branch of curve. The changng process of det( J s smlar wth that of det( J, whle det( J P θ remans negatve durng the process. Table 6 Determnant of red uced V Q Jacoban matrx wth Q = 0. Table 5 and Table 7 are not the same, ndcatng dfferent operatonal mode of power system. 7. Analyss of Equlbrums Analyss n sectons above focuses on statc voltage stablty. On bass of calculaton, V P curve wth four branches s obtaned n secton 5, showng that upper branches and lower branches wth smlar changng trend respectvely. Through analyss of the Jacoban n secton 6, further dscusson should be made on these equlbrums. Voltage stablty and angle stablty are two aspects of power system stablty. The relatonshp between the two aspects s dscussed []. A general energy functon frame s presented for voltage and angle stablty []. Snce all the solutons of power flow equatons could be obtaned by WEM, t s convenent to analyze the operatonal mode of system. On the bass of the three-node system n Fg., PV node connects wth a generator denoted by a smple model compromsng an nternal voltage behnd an effectve reactance, as shown n Fg. 6. The results are denoted n polar coordnate form shown n Table 8, where V g and θ g represent the ampltude and angle of the nternal voltage. P Table 7 The sgn of det( J, det( J P θ,and det( J wth Q = det (J + + det( J P θ + + det( J + + By WEM, propertes of Jacoban matrx are thoroughly analyzed above. The V Q senstvtes correspondng wth the I and II branches n V P curve are postve. However, the propertes shown n.0 0 Fg. 6 Three-node system wth a generator Tab le 8 System equlbrums obtaned by WEM V θ /( θ /( V g θ g /( Analyss of the relatonshp between voltage and angle stablty s presented n [3], showng that the relatonshp could be lnked by the unstable equlbrum ponts (UEP whch are classfed nto voltage stablty mode wth low voltage- small angle and angle stablty mode wth hgh voltage-large ISSN: Issue, Volume 7, December 008

8 angle. The frst Equlbrum n Table 8 s voltage stablty mode UEP, correspondng wth pont n IV branch n V P curve; the second equlbrum s wth low bus voltage-large angle, correspondng wth pont n III branch n V P curve; the thrd equlbrum s angle stablty mode UEP, correspondng wth pont n II branch n V P curve; the fourth equlbrum s the stable operatonal mode correspondng wth pont n I branch n V P curve P max 330 Fg. 7 θ P curve Fg. 7 dsplays the locus curves of varaton n the PQ bus angle wth load. The arrows ndcate the changng drecton. The II branch s wth large angle, correspondng wth II branch n V P curve, ndcatng angle nstablty (b Fg. 8 θ V curve n polar coordnates The θ V curves clearly show that the bus voltage decreases quckly after the bfurcaton pont denoted by an arrow lne, whle the bus voltage wth large angle correspondng wth the II branch n V P curve begns to decrease quckly before the bfurcaton pont. Fg. 9 shows the varaton n nternal angle wth the load. The equlbrums on the upper branch of curve n V P curve are wth large angle despte relatvely hgh voltage, smlarly wth θ P curve Fg. 9 θg P curve P max (a Fg. 0 dsplay the varaton n angular separaton between buses of the three lnes and nject actve power wth the load respectvely, where θ = θ θ θ3 = θ3 θ θ 3 = θ 3 θ θ 3 = 0.The power drectons are from bus to bus, from bus 3 to bus, and from bus 3 to bus respectvely. In an dealzed two-node model, the relatonshp between power and angle s snusodal. The varaton n power wth angular separaton between buses of the three-node system s dfferent from the snusodal relatonshp. However, valuable nformaton could be obtaned from the analyss of the relatonshp between power and angle. ISSN: Issue, Volume 7, December 008

9 θ /( θ 3 /( P (a θ 3 /( P (b P Pf - > Pf 3-> Pf 3-> (a P (b P (c (c Fg.0 a θ P curve, a Pf curve; P b θ 3 P curve, b Pf curve; 3 P c θ 3 P curve, c Pf curve. 3 P Fgures above show the locus of actve power n each lne of the three-node system wth the change of the load. Table 9 shows the change trend of the actve power n the three lnes. The actve power n the three lnes ncreases wth the load under operatonal mode correspondng wth the I branch n V P curve, whle there s fluctuaton of actve power n the lnes under other operatonal mode correspondng to the other branches n V P curve. Table 9 Change of actve power n each lne Curve Curve I branch IV branch II branch III branch Lne - Lne Lne The practcal power system s far more complex than the three-node system. However, the locus curves of operatonal parameters (e.g. the nternal voltage of generaton could provde potentally valuable nformaton for research on complex systems. P 8. Concluson WEM can obtan all the analytcal solutons of smultaneous polynomal equatons wthout ether extraneous roots or mssng roots. As a result, WEM can solve power flow equatons to plot analytcal V P, V Qcurves as well as the steady operaton area. The propertes of power-flow Jacoban matrx are also analyzed, as well as testfyng the propertes of the actve power/angle sub-matrx n the power flow Jacoban. Analyss of equlbrums s presented, showng that stable equlbrum pont wth hgh voltage-small angle, one UEP wth low voltage-small angle, one UEP wth hgh voltage-large angle, one UEP wth low voltage-large angle. WEM provdes a new way to study the characterstc of power system by obtanng all the solutons of power flow equatons. However, t s dffcult to solve the power flow equatons by WEM when the number of nodes ncreases, for the exponent of varable n the equatons are much hgh durng the calculaton procedure. Acknowledgment Ths work was supported by Natonal Natural Scence Foundaton of Chna ( The authors would lke to thank Prof. Sh H. of Academy of Mathematcs and System Scence, Chnese Academy Scence and Prof. Wang J. for ther help. eferences: [] Chen, C., Zhao, X.Q., and Cao, G.Y., Mechanzed Mathematcs Wu Elmnaton Method Appled n Power System, Proc. of the CSEE, Vol. 8, No., 998, pp [] Chen, C., and Cao, W., The Applcaton of Wu Method n Electrc-magnetc Transent Analyss durng Power System Short-crcut, Proc. of the CSEE, Vol., No., 00, pp [3] Ja, Y. F., Chen, Y. F., and Xu, Z. Q., Applcaton of Wu Elmnaton Method to constraned Dynamcs, Appled Mathematcs and Mechancs(Englsh Edton, Vol. 7, No. 0, 006, pp [4] Wang, L.G., Hu, C., Huang, W. H., Applcaton of Wu Elmnaton Method n otor Dynamcs, Chna Mechancal Engneerng, Vol. 3, No., 00, pp [5] Tnney, W.F., and Hart, C.E., Power Flow Soluton by Newton s Method, IEEE Trans. on ISSN: Issue, Volume 7, December 008

10 Power Apparatus and System, Vol. PAS-86, NO., 967, pp [6] Scott, B., and Alsac, O., Fast Decoupled Load Flow, IEEE Trans. Power Apparatus and System, Vol. PAS-93, No.3, 974, pp [7] KIos, A., and Kemer, A., The non-unqueness of load flow soluton, Proc. of PSCC V, 975, pp. 3./8. [8] Korsak, A. J., On the queston of unqueness of stable load-flow solutons, IEEE Trans. on Power Apparatus and Systems, Vol. PAS-9, No. 3, 97, pp [9] Salam, F. M. A. et al., Parallel processng for the load flow of power systems: the approach and applcatons, Proc. of 8th IEEE Conf. on Decson and Control, Vol. 3(3-5, 989, pp [0] Guo, X.G., and Salam, F. M. A., The number of (equlbrum steady-state solutons of models of power systems, IEEE Trans. on Crcut and Systems, Vol. 4, NO. 9, 994, pp [] Ma, W., and Thorp, J. S., Solvng for all the solutons of power systems equatons, Proc. 99 IEEE ISCAS, Vol. 5(3-6, 99, pp [] Yn, X., Applcaton of Genetc Algorthms to Multple Load Flow Soluton Problem n Electrcal Power Systems, Proceedng of the 3nd IEEE Conference on Decson and Control, Vol. 4(5-7, 99, pp [3] Klump,.P., Overbye, T.J., A new method for fndng low-voltage power flow solutons, IEEE Power Engneerng Socety Meetng, Vol. (6-0, 000, pp [4] A, Q, Gu, D., Jang, C., et al, esearch on propagaton of voltage fluctuatons along practcal power dstrbuton systems, WSEAS Trans. on Crcuts and Systems, Vol. 4, No. 9, 005, pp [5] Gu, C., A, Q., Wu, J., Contnuaton power flow (CPF and ts usage n statc voltage stablty analyss consderng dfferent statc load models and system operatng constrants, WSEAS Trans. on Crcuts and Systems, Vol. 4, No. 0,005, pp [6] L, H., Cheng, H., Wang, C., A novel approach for determnng the Hopf bfurcaton pont of dynamc voltage stablty n power system, WSEAS Trans. On Crcuts and System, Vol. 4, No. 7, 005, pp [7] Wu, T. Mathematcs Mechanzaton, Scence Press/ Kluwer Pub., 000. [8] Wang, W. L., and Chen, C. Applcaton of Wu Elmnaton Method n Solvng for All the Steady-State Power Flow Solutons, Journal of Shangha Jao Tong Unversty, Vol. 38, No. 8, 004, pp [9] Horn,. A., and Zhang, F. The Schur Complement and Its Applcaton, New York: Sprnger, 005. [0] Cao, G.Y., and Lu, L.X. Analyss on the Actve Power/Angle Submatrx n Power Flow Jacoban Matrx, Power System Technology, Vol. 3, No. 5, 007, pp [] Vournas C.D., Sauer P. W.. elatonshps between Voltage and Angle Stablty of Power System. Electrcal Power and Energy System, Vol. 8, NO. 8, 996, pp [] Overbye T. J., PaM A., Sauer PW. Some Aspects of the Energy Functon Approach to Angle and Voltage Stablty Analyss n Power Systems. Proceedngs of the 3st Conference on Decson and Control Tucson (Artzons, 99, pp [3] WU H., HAN Z.. Analyss of the relatonshp between voltage stablty and angle stablty through equlbrums. Automaton of Electrcal Power Systems, Vol. 7, No., 003, pp ISSN: Issue, Volume 7, December 008