A Generlal Method for Small Signal Stability Analysis 1998 Ie

8/2/2019 A Generlal Method for Small Signal Stability Analysis 1998 Ie

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IEEE Transactions on Power Systems, Vol. 13, No . 3, August 1998A Generlal Method for Small Signal Stability Analysis

Yuri V. Makarov Zhao Yang Dong David J . Hill

979

Department of Electrical EngineeringThe University of Sydney

NSW 2006, AustraliaA b s t r a c t - This paper presents a new general method forcomputing the different specific power system small sig-

nal stability conditions. The conditions include the pointsof minimum and max imum damping of oscillations, saddlenode and Hopf bifurcations, and load flow feasibility bound-aries. All these chara cteristic points are located by optimiz-ing an eigenvalue objective function along the rays specifiedin the space of system parameters. The set of constraintsconsists of the load flow (equations, and r equirem ents appliedto the dynamic state matrix eigenvalues and eigenvectors.Solutions of the o ptimization problem correspond t o specificpoints of interest mentioned above. So, the proposed gen-eral method gives a Comprehensive characterization of thepower system small signal stability properties. T he specificpoint obtained depends upon the initial guess of variablesand numerical methods used to solve the constrained op-timization problem. The technique is tested by analyzingthe small signal stability propert ies for well-known examplesystems.

I . I N T R O D U C T I O NModern power grid:; are becoming more and more

stressed with the load demands increasing rapidly. Thevoltage collapses which occurred recently have again drawnmuch at tention to the issue of stability security margins inpower systems [I] . The small signal stability margins arehighly dependent upon such system factors as load flowfeasibility boundaries, minimum and maximum dampingconditions, saddle node and Hopf bifurcations, etc. Un-fortunately, it is very difficult to say in advance which ofthese factors will make a decisive contribution to instabil-ity. Despite the progress achieved recently, the existingapproaches deal with these factors independently - see [a],[3] for example, and additional attempts are needed to geta more comprehensive view on small-signal stability prob-lem.To study the power system small signal stability prob-lem, an appropriate model for the machine and load dy-PE-153-PWRS-16-09-1997 A paper recommended and approved bythe IEEE Power System Engineering Committee of the IEEE PowerEngineering Society for publication in the IEEE Transactions on PowerSystems. Manuscript submlitted May 27, 1997; made available forprinting September 30, 1997.

namics is required. For example, the models given in [4] -[ll]can be used. They include generator and excitationsystem differential equations, sta tor and network algebraicequations. These equations build up the set of differential-algebraic equations (1)

X l = F ( Z l 1 2 2 , Y , T )0 = G ( X I , Z Z , Y , T ) (1)

In the equation ( l ) , 1 is the vector of state (differential)variables, x~ is the vector of algebraic variables, y is thevector of specified system parameters, and T is a param-eter chosen for bifurcation analysis. In many cases y is afunction of T.

In the small signal stability analysis, the set (1) is thenlinearized at an equilibrium point to get the system Jaco-bian and state matrix . The structure of the system Ja-cobian J is shown in Fig. 1(which follows the structuregiven in [12]) , where J l f stands for the load flow Jaco-bian, J11 = d F / d x l , Jl a = d F / d x z , J z ~ d G / d x l andJzz = d G / d x z are different parts of J corresponding to dif-ferential and algebraic variables. In Fig. 1, Qgen standsfor the reactive power at generator buses, Psb is the activepower at the swing bus, 6 s the vector of machine rotorangles, w is the vector of machine speeds, K is the vector ofthe s tate variables except 6 nd w (such as E ; , E&, f d , VR,and RF;oad bus voltages K o a d and angles @ l o a d should beconsidered as dynamic state variables in cases where loaddynamics is considered [la]), d and i, are vectors of d-axisand q-axis currents; v,,,and N o a d stand for generator andload bus voltages; d s b is the swing bus voltage angle, andB denotes voltage angles at all buses except the swing bus.The prefix A means a small increment in correspondingvariables.

The problem addressed here is that these different smallsignal stabili ty conditions correspond to different physi-cal phenomena and mathematica l descriptions [13]. Saddlenode bifurcations happen where the state matrix

J" = J11 - J lzJ ,- , lJzlbecomes singular and, for example, a static (aperiodic)type of voltage collapse or angle instabili ty may be observedas a result. Hopf bifurcations occur when the system stateiiiatrix J has a pair of conjugate eigenvalues passing theimaginary axis while the other eigenvalues have negativereal parts, and the unstable oscillatory behavior may beseen. Singularity induced bifurcat ions are caused by sin-gularity of the algebraic submatrix Jz z - see Fig. 1, and

0885-8950/98/$10.00 0 1997 IEEE


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980Differentialequations

for A6Differentialequations

AK A&Algebraic

Stiltillequations

. .....__.

Fig. 1. Structu re of th e system Jacobian.they result in fast collapse type of instability [14]. The loadflow feasibility boundary corresponds to a surface where theload flow Jacobian matrix J i f is singular, and it restrictsa region in the space of power system parameters whereload flow solutions exist. Under certain modeling simplifi-cations, this boundary coincides with the saddle node bi-furcation conditions [15], [16], but in general case it shouldbe taken into account separately.

To locate the saddle node and Hopf bifurcations along agiven ray in the space of y , the following equation can beemployed PI, 31, ~ 7 1 , 8 1 :

f (z ,y o + r a y ) = 0 (2)P ( Z , yo + rAy)l + W l = 0 (3)F(z,o + 7Ay)l - wl = 0 (4)

1;-1 = 0 (5)1; = 0 (6)

where w is the imaginary part of a system eigenvalue; 1and 1 are real and imaginary parts of the correspondingleft eigenvector 1; 11 + l: is the i - h element of the leftejgenvector 1; yo + r A y specifies a ray in the space of y;J stands for the sta te matrix obtained from the linearizedmodel given in Fig. 1provided that the algebraic subma-trix J 2 2 is nonsingular.

In the above set, (2) is the load flow equation and condi-tions (3)-(6 ) provide an eigenvalue with zero real part andthe corresponding left eigenvector.

Solutions of the system (2 ) - (6 ) correspond to either sad-dle node (w = 0) or Hopf (w f 0) bifurcations. Neverthe-less the ext reme load flow feasibility conditions (if they donot coincide with the saddle node bifurcations) can not belocated by means of this system. Actually, if the load flowfeasibility boundary is met on the ray yo + rA y but thereis no an eigenvalue with zero real part, the system (2)-(6)becomes inconsistent and has no a solution.

Therefore, if the system (2)-(6 ) is used, it is necessaryto analyze the load flow feasibility conditions additional ly.The corresponding procedures a re well known - see [a], [19],[20] for instance. The general idea behind these procedures

I

Fig. 2 . Different solutions of the problem (10)-(15):1 ,2 - minimum and maximum damping3 - saddle (w = 0) or Hopf (w # 0) bifurcations4 - load flow feasibility bounda ry

is illustrated by the following system:f ( z , y o + TAY) = 0 (7 )

Ji f (z,o + rAy)l = 0 (8 )l i - 1 = 0 (9 )

where 1 is the left eigenvector corresponding t o a zero eigen-value of the load flow Jacobian matrix J i f . The system(7)- (9) gives the load flow feasibility boundary points alongthe ray yo + rAy by a similar way as the system (2)-(6)generates saddle node and Hopf bifurcation poin ts.

By subsequent solution of both the problems (2)- (6) and(7)- (9) the general issue may be resolved, but a challengingtask is to find a procedure which can generate all small-signal characteri stic points by itself.

11. GENERALETHODTo locate the saddle node and Hopf bifurcations as well

as the load flow feasibility boundary points within one pro-cedure, the following constraint optimization problem isproposed

a2j maxlmin (10)

f ( ~ ,O + r a y ) = 0 (11)?(x, yo + rAy)l - al + wl 0 (12)?(z , yo + rAy)l - al - wl = 0 (13)

( - 1 = 0 (14)1; = 0 (15)

subject to

where a is the real part of an eigenvalue of interest.The problem may have a number of solutions, and all of

them presents different aspects of the small-signal stabil ityproblem as shown in Fig. 2.

The minimum and maximum damping points 1 and 2correspond to zero derivative d a l d r . The constraint set(11)-(15) gives all unknown variables a t these points. Theminimum and maximum damping, determined for all oscil-latory modes of interest, provides an essential informationabout damping variations caused by a directed change ofpower system parameters.


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98 1The saddle node or IHopf bifurcations 3 correspond to

a = 0. They indicate the small-signal stability limits alongthe specified loading tra.jectory yo + r a y . Besides revealingthe type of instability (aperiodic for w = 0 or oscillatoryfor w # 0 ) , the constraint set (11)-(15) gives the frequencyof critical oscillatory mode. The left eigenvector l = l+jl(together with the right eigenvector r = r+jr which canbe easily computed in its turn) determine such essentialfactors as sensitivity of a with respect to y, the mode,shape, participation factors, observability and excitabilityof the critical oscillatory mode [21] -[23].

The load flow feasibility boundary points 4 reflect themaximal power transfer capabilities of the power system.Those conditions play a decisive role when the system isstable everywhere on th- ray yo + r A y up to the load flowfeasibility boundary. The optimization procedure stops atthese points as the constraint (11)can not be satisfied any-more.A strict proof of the optimality conditions in the loadflow feasibility points requires a complicated mathematicalanalysis, and we will not give this proof in the paper. Someinitial ideas of this proof are briefly reported in the sequel.Consider the trajectory of ~ ( r ) ,+ 03 satisfying (11).At the load flow feasibility point, parameter r can not beincreased anymore beyond its limit value r*. Nevertheless,the trajectory x ( r ) can be smoothly continued by furtherdecreasing r see [24], for example. Suppose that the func-tion a[z( r ) ] s monotonous and continuous in vicinity ofr,, say, along the trajectory Z(T), he increment da is pos-itive. As the increment rlr changes its sign at the point r,,this means that the derivative daldr changes its sign atr* . Thus the enough optimality conditions (the constantsign of the second derivative of the objective function withrespect to r ) are met at the point r, [33].

The problem (10) -(15 ) akes into account only one eigen-value each time. The procedure must be repeated for alleigenvalues of interest. The choice of eigenvalues dependsupon the concrete task l,o be solved. The eigenvalue sensi-tivity, observability , excitability and controllability factors[ a l l , [22] can help to determine the eigenvalues of interest,and trace them during optimization. For example, the in-terarea oscillatory modes can be identified and then ana-lyzed using (10)-(15).

The result of optimization depends on the initial guessesfor all variables in (10)-(15). To get all characteristic pointsfor a selected eigenvalue, different initial points may becomputed for different values of T. At each point, the loadflow conditions, state matrix eigenvalues and eigenvectorscan be obtained, and then a particular eigenvalue selectedto start the optimization procedure. More effective ap-proaches for finding all characteristic points require addi-tional development. At the moment we are analyzing thepossibility to use th e Genetic Algorithms for this purpose.

111. TESTINGN D VALIDATION O F T H E M E T H O DOur purpose here is to demonstrate whether the pro-

posed method is able to locate all these characteristicpoints depending on the initial guesses of r , x, a , w , E

cT Pd+jQdFig. 3 . The single-machin e infinite-bus power system model

and 1 . The results will be then validated by comparingsome of them with the results obtained in other papers,and by transient simulations conducted at the characteris-tic points.

The single machine infinite bus power system model [25]presented in Fig. 3, and 3 machine 9 bus power system [26]shown in Fig. 1 2 will be studied here. Similar models werestudied in [4] -[6], [17], [26] -[30].

The standard Gauss -Newton procedure from M a t l a bwas used here for optimization [31].A. Sangle Machane I n jna te B u s Power System Modelcover both generator and load dynamics [25].

The model consists of four differential equations, whichThe mathemat ical model of the system is the following:

ICqws

+dKiw K ~ , [ - - E ~ y ~ V c o s ( S BO - h ) --F,y,Vcos(S - + 8, - h ) ++ ( Y ~ C O S ( Q O - h)+ ycos(Q - h ) ) V 2 ]- 1 < p w ( p 0 + PI) y p w ( Q ~+ Q I ) (19)

where k4 = TI


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1 5 -

1 -

O 5 -a2 0 -Dc?05 -

-1 5 --

- Reactive Load Power Q1 p u 10 12

Fig. 4. a = ReX computed for PI = 0 and & I =variable

Fig. 5 . Subcritical (I) and supercritical (11)bifurcations

The system (16)-(19) depends on four stat e variables S,S,, w , V . Their values at the initial load flow point arethe following: S = 2.75, 6, = 11.37,w = 0 , and V = 1.79.Note that the initial point is not a physical solution as thevoltage V is too high as &1 is zero.

The results of numerical simulations are presented in inFig. 4-11.

The dependence of the real part a = ReX of criticaleigenvalue X upon QI is shown in Fig. 4. It is seen thatthere are both subcritical (point I) and supercritical (point11) Hopf bifurcations along the chosen loading direction.Fig. 5presents the root locus for the cri tical eigenvalue con-jugate. The subcritical (point I) and supercritical (point11) Hopf bifurcation points are displayed. Both these char-acteristic points were successfully located by the proposedmethod, (10 -15) .Fig. 6. shows the load flow feasibility and bifurcationboundaries on the plane of the Ioad parameters PI and &I.The boundaries were obtained by the proposed optimiza-tion method when the loading direction was changed bysubsequent rotation of Ay in the plane PI and &I . Exactlythe same curves were computed in [17] by separate solutionof the problems (2)- (6) and (7)- (9). To verify the results,transient simulations were performed at several points inthe plane PI - Q1 . Point A with PI = 0 and & I = 10.88

- 1 5 12020 -15 -10 -5 0 5 10 15 20Active Load Power P1 p u

Fig. 6. The feasibility and Hopf bifurcation boun daries

-203074 03076 0.3078 0308 03082 03084 0Machine angle

Fig. 7. Phase portra it at point A near subcritical bifurcation

was placed within the load flow feasibility region close tosubcritical bifurcation boundary. Then a small disturbancewas applied. The corresponding phase por trai t is shown inFig. 7 . In a complete correspondence with the theoreticalexpectations, the system has experienced sustained oscilla-tions.

Those oscillations can be viewed in Fig. 8where the loadbus voltage against time is presented.

Point B with PI = 0 and Q1 = 11.4 was placed withinthe load flow feasibility region close to the supercritical bi-furcation boundary. (Note th at , in the vicinity of points Aand B , the Hopf bifurcation boundary as shown in Fig. 6actually consists of internal subcritical and ex ternal super-critical boundaries located very closely). Al l eigenvaluesat point B have small negative real parts. The phase por-trait for a small disturbance applied at point B is given inFig. 9. The system undergoes decreasing oscillations. Thecorresponding voltage behavior is shown in Fig. 10 .

The next point was taken close to the poin t B but outsidethe load flow feasibility boundary. The system experiencesvoltage collapse as illustrated by Fig. 11.

By solving the optimization problem, the system smallsignal stability boundaries were obtained.

The minimum and maximum damping conditions on theplane Pl Q1 were studied as well. Table 1presents the


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1.112

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$1 1116rBP2 1.11141

1.1112

1

-

1.111

Fig. 8. Transient process near subcritical bifurcation (point A )

- ' I034 47 0.3448 0.3449 0. 345 03451 0.345 2 0.3453 0.3454Machine angleFig. 9. Phase portr ait at point B near supercritical bifurcation

Voltage transient process0 9 3 6 5 1 . 1

0 1 2 3 4 5 6 7 8 9 10Time, s

Fig. 10. Load bus voltage transient s near supercritical bifurcation

:::I , , , , ,0 0.5 1 1.5 2 2.5Time s

Fig. 11. The system load bus voltage transients near feasibilityboundaryTABLE

MINIMUMN D MAXIMUM AMPING URVESFOR THE SINGLE-MACHINENFINITE-BUSYSTEMImin-d;mping max-dymping I214.5537 -0.9679

curves of minimum and maximum damping for differenteigenvalues.B . Three-Machine Nine-Bus Power System Model

The power system model is composed of 3 machines and9 buses [26] as shown in Fig. 12 . Stability studies for asimilar system can be found in [4], [30]. The machines ofthe system are modeled by using the classical model formachine 1 and two-axis model for machines 2 and 3 - seeequations (22) -(27).

Unlike [4] and [30], in our tests we neglected the excitationsystem dynamics, so our state variables were the following:6, , E; , and E&.Algebraic variables were I d , I,, 6 , and V ;bifurcation parameters were Pld and Q l d . . The notations,paramete r values and description of the system (22) -(27)can be found in [as].


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1s t minimum 2nd minimumBu s damping point damping point Bifurcation point

Fig. 1 2 . The 3-machine %bus systembdc23ps

0.02, I , , I 1 , , I I I

-002si -14 -12 -1 -08 -06 -04 -02 0 02 04 06Reactive load power a1bus 5, p uFig. 13 . Real pa rts of system eigenvalues vs reactive power

Unlike the used single machine infinite bus model, whichincludes an induction motor load, this 3 machine 9 busmodel considers constant load models only. In the generalcase of small signal stabili ty analysis, the load dynamicsshould be definitely taken into considerat ion, but neverthe-less some stabil ity aspects can be stud ied with the constantload model [32].

In the preliminary examination, the loads Pld and Q l dat buses 5, 6 and 8 were increased in proportion with theload size. The increase of load was followed by the cor-responding increase of generation in proportion with thegenerator size. The total increment in load was equal tothe tot al increment in generation. The loading was re-peated till the point where load flow did not converge. Ateach step, eigenvalues of the state matrix were computed toreveal the bifurcation points and minimum and maximumdamping conditions.

The system eigenvalue behavior along the chosen loadingdirection is shown in Fig. 13.

There are several points of interest which can be clearlyseen in Fig. 13. They include the maximum damping.minimum damping, bifurcation points and points close t o

0.0000-0.8125-0 850

7 0.0000S -0.6500

0.0000

-0.0000 0.0000 -0.0000-0.3250 -0.5000 -0.2000-0.1950 -0,3600 -0.1200-0.0000 0.0000 -0.0000-0.2275 -0 000 -0.14000.0000 0.0000 0.0000

PI d0000.0000

-0.5781-0.41630.0000-0.46250.0000

-0.0000-0.2313-0 388-0.0000-0 619

the load flow feasibility boundary. Some of the obtainedcharacteristic points are summarized in Table 2.

In the examination of the proposed method based on theoptimization problem (10) -(15), the method has been ap-plied t o locate these poin ts of interest. The results showedthat the constrained optimization procedure converged to-ward all the points of interests depending the init ial guessof variables. The initia l guesses were chosen using the routhestimates of the characteristic points obtained in prelimi-nary examination.

I V . CONCLUSIONA new method which computes the minimum and maxi-

mal damping, saddle node and Hopf bifurcations and loadflow feasibility boundary points as part of a common pro-cedure has been developed in the paper. The method hasbeen tested and validated by numerical simulat ions, com-parison with the previous results obtained for the used testsystems, and by transient simulations conducted at thecharacteristic points. Further work is required to developtechniques for obtaining the initial guesses of variables, fastand reliable solving the constrained optimization problem,and handling of large power systems. Various practicalapplications of the new method await for further develop-ments as well.

V. ACKNOWLEDGEMENTThis work was sponsored in part by an Australian Elec-

tricity Supply Indus try Research Board grant Voltage Col-lapse Analysis and Control.

Z. Y. Dongs research work was supported by SydneyUniversity Electrical Engineering Postgraduate Scholar-ship.

REFERENCESV . Ajjarapu and B. Lee, Bibliog-raphy on Voltage Stability, Iowa Sta te University, see WWWsite http://www.ee.iastate.edu/u en ka t a r / V o l t a g eS t a b i l i t g B i b .I. Dobson, Computing a Closest Bifurcation in Stability Mul-tidimensional Parameter Space, Journal of Nonlinear Science,Y . V. Makarov, D. J. Hill an d J. V . Milanovic, Effect of Load Un-certainty on Small Disturbance Stability Margins in Open-AccessPower Systems, Proc. Hawaii International Conference on Sys-t e m Sciences HICSS-30, ihei, Maui, Hawaii, Janua ry 7-10, 1997,P. W . Sauer, B . C . Lesieutre an d M . A . Pai, Maxim um Loadabil-ity an d Voltage Stability in Power Systems, International J o u r -

Vol. 3 . , NO . 3 . , pp.307-327, 1993.

Vol. 5, pp. 648-657.
http://www.ee.iastate.edu/http://www.ee.iastate.edu/


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985nal o f Electrical Power and Energy Systems, Vol. 15, pp. 145-154,June 1993.M. A. Pai Structural Smbility in Power Systems, J. H.Chow,P. V. Kokotovic and R. -1. Thomas edit, in Systems and ControlTheory for Power Sy stems, Springer-Verlag, 1995, pp. 259-281.P. W . Sauer and B. C . Lesieutre, Power System Load Modeling,J. H.Chow, P. V . Kokotovic and R. J. Thoma s edit, in Systemsand Control Theory for Power Systems, Springer-Verlag, 1995,P. W. Sauer and M. A. Pai , Power System Steady-state Stabilityand the Load-Flow Jacobian, IEEE Trans. on Power Systems,Vol. 5, No. 4, Novem ber 1990, pp . 1374-1383.M. A. Pai, A. Kulkarni and P. W. S auer, Parallel Dynamic Sim-ulation of Power System s, IEEE CH2868-8/90/0000, pp. 1264-1267.C. Gajagopalan, B. Leisieutre, P. W. Sauer and M. A. Pai, Dy-namic Aspects of Voltage/Power Characteristics, IEEE/PES 91

pp. 283-313.

SM L I Y - 2 PW RS.[lo] C. Rajagopalan, P. W. !Sauer an d M. A. Pai, Analysis of Volt-age Control Systems Exhi biting Hopf Bifurcation, Proceedings o fthe 28th Conference on Decision and Control, Tampa, Florida,December 1989, pp. 332-335.[ I l l C. Rajagopalan, P. W. Sauer an d M. A. Pai, An Integrated Ap-proach to Dynamic a nd Static Voltage Stability,A CC89 Pitts-burgh, June 2 1 - 2 3 , 1989.[12] B. Lee and V. Ajja rapu, LBifurcation low: A Tool to StudyBoth St atic an d Dynamic Aspects of Voltage Stability , Proceed-ings of the Bulk Power System Voltage Phenomena III. VoltageStability, Security and Control, Davos, Switzerland, Aug. 22--26,1994, pp. 305-324.[I31 H. G. Kwatny, R. F. Fischl and C. Nwankpa, Local Bifurca-tion in Power Systems: Theory, Computation and Application,D. J. Hill (ed), Special Issue on Nonlinear Phe nomena in PouerSystems: Theory and Practical Implications, IEEE Proceedings,Vol. 83, No. 11, November, 1995, pp. 1456-1483.

[14] V. Venkatasubramanian, H. Schattler an d J. Zaborszky, Dy-namics of Large Constrained Nonlinear Systems -A TaxonomyTheory, D. J. Hill (ed), Special Issue on Nonlinear Phenom-ena in Power Systems: Theory a n d Practical Implications, IEEEProceedings, Vol. 83, No. 11, November, 1995, pp. 1530-1561.[15] V.A. Venikov, V.A. Stroe v, V.I. Idelchik and V.I. Taraso v, Es-timation of Electric Power System Steady-St ate Stability in LoadFlow Calculation, IEEL: Trans. on Power Apparatus and Sys-tems, Vol. PAS-94, May-.June 1975, pp. 1034-1041.

[16] P. W. Sauer and M. A. Pai, Power System Steady-State Sta-bility and the Load-Flow Jacobian , IEEE Trans. on Power Sys-tems, Vol. 5, N o. 4, Nove mber 19 90, pp. 1374-1383.

[17] 2. Y. Dong,Y. V. Makai-ov an d D. J. Hill, Compu ting the Ape-riodic and Oscillatory Small Signal Stability Boundaries in theModern Power Grids, PI-oc. Hawaii Internationa l Conference onSystem Sciences HICSS-30, Kihei, Maui, Hawaii, January 7-10,1997.

[18] Y. V. Makarov , V. A. blaslennikov and D. J. Hill, Calculationof Oscillatory Stability Margins in the Space of Power SystemControlled Para meters , Proc. o f the International Symposiumon Electric Power Engineering Stockholm Power Tech: PowerSystems, Stockholm, Sweden, 18-22 Jun e, 1995, pp. 416-422[19] Y. V. Makarov and I. A. Hiskens, A Continuation Method Ap-proach to Finding the Closest Saddle Node Bifurcation Point,Proc. NSF/ECG Workshop on Bulk Power System Voltage Phe-nomena 111, Davos, Switzerland, August 1994.[20] I. A. Hiskens and R. J. Davy, A Technique for Explo ringthe Power Flow Solution Space Boundary, Proc. of the Inter-national Symposium o n Electric Power Engineering StockholmPower Te c h , Vol. Power Sy stems, Stockholm, Sweden, June 1995,pp. 478-483.

[all P. Kundur, Power System Stability and Control, New York:McGraw-Hill, 1994.[22] D. J. Hill, V. A. Maslennikov, S . M. Ustinov, Y. V. Makarov,A. Manglick, B. Elliott, N. Rotenko and D. Conroy, AdvancedSmall Disturbance Stability Analysis Techniques and MATLABAlgorithms A final Report of the work Collaborative ResearchProject Advanced Sys tem Analy sis Techniques wzth the NEW

South Wales Electricity Transmission Authority and The Depart-ment of Electrical Engineering, The Univers ity of Sydney, April1996.

[23] I. A. Gruzdev, V. A. Maslennikov and S. M. Ustinov, Devel-opment of Methods and Software for Analysis of Steady-StateStability and Damping of Bulk Power Systems, In: Methodsand Software fo r Power Sys tem Oscillatory Stability Comp uta-tions, Publishing House of the Federation of Power a nd Electro-Technical Societies, St. Petersburg, Russia, 1992, pp. 66-88 (inRussian).[24] Y.V. Makarov, A.M. Kontorovich, D . J. Hill and I. A. Hiskens,Solution Characteristics of Q uadratic Power Flow Problems,Proc. 12-th Power Syste m Computat ion Conference, Vol. 1,Dres-den, Ge rmany, 19-23 August, 1996, pp. 460-467.[25] H . -D. Chiang, I. Dobson, et al. On Voltage Collapse in ElectricPower System s, IEEE Trans. Power Systems, Vol. 5, No. 2, May1990.[26] P. M. Anderson and A. A. Fouad, Power System Control andStability, Iowa S tate University Press, Ames, IA. 1977.

[27] J . H. Chow, editor, Time-Scale Modeling of Dynamic Networkswith Applications to Power Systems , Springer-Verlag, Berlin,1982.[28] J . H. Chow and K. W. Cheung, A Toolbox for Power SystemDynamics and Control Engineering Education and Research,

IEEE/PES Trans. on Power Systems, Vol. 7, N o. 4, November1992.[29] H. G. Kwatny, X . -M. Yu and C. Nwankpa, Local BifurcationAnalysis of Power Syste ms Using MATLA B, Y5CH35764, Pro-ceedings of the 4th IEEE Conference on Control Applications,Albany, N. Y. Septemb er 28-29, 1995, pp. 57-62.[30] M. K. Pal, Voltage Stability Analysis Needs, Modeling Re-quirement, and Modeling Adequacy, I E E Proceedings of PartC., Vol. 140, pp. 279-286, July 1993.[31] A. Grace, Op timization TOOLBOX Users Guide, The Math-Works, Inc. October 1994.[32] J . H. Chow and A. Gebreselassie, Dynamic Voltage StabilityAnalysis of a Single Machine Constant Power Load System,Proc. 29th Conference o n Decision and Control, Honolulu Hawaii,December 1990, pp. 3057-3062.

[33] . P.E. Gill, W. Murray and M. H. Wright, Practical Optimiza-tion, Academic Press, 1981.Y u r i V . Makarov received the Ph. D. in Electrical Networks an dSystems (1984) from the St. Petersbu rg Sta te Technical University(former Leningrad Polytechnic Ins titu te), Russia. He is an Associate

Professor a t Departm ent of Power Systems an d Networks in the sameUniversity. Now he is conducting his research work occupyinga seniorresearch position a t the De partmen t of Electrical Engineering in theUniversity of Sydney, Australia. His research inte rests are mainly inth e field of power system analysis, stability and co ntrol.

Zhao Yang Dong was born in China, 1971. He received hisBSEE degree as first class hono r in July, 1993. Since 1994, he ha dbeen stu dying in Tianjin University (Peiyang University), Tianjin,China for his MSEE degree. Now, he is continuing his study as apostgr aduate research student in the D epartme nt of Electrical Engi-neering, the University of Sydney, Australia. His research interestsinclude power system ana lysis an d control, electric machine a nd drivesystems areas.

David J . Hill (M76, SM91, F93) received his B.E. and B.Sc.degrees from the University of Queensland, Australia in 1972 and1974 respectively. In 1976 he received his Ph.D. degree in ElectricalEngineering from the University of Newcastle, Australia. He cur-rently occupies the Chair in Electrical Engineering at the Universityof Sydney. He is also a Head of Electrical Engineering Departmentat the sam e University. Prev ious appointm ents include research po-sitions at t he University of California, Berkeley and the D epartme ntof Automa tic Control, Lund In stitute of Technology, Sweden. From1982 to 1993 he held various academic positions at the Universityof Newcastle. His research interests are mainly in nonlinear systemsand control, stability theory, power system dynamics and security.His resent applied work consists of various projects in power systemstabilization and power plant con trol carried out in collabora tion withutilities in Australia an d Sweden.

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A Generlal Method for Small Signal Stability Analysis 1998 Ie