STC Small Signal Stability Power System

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Vol. 44 No.2 SCIENCE IN CHINA (Series E) April 2001

Chaotic phenomena and small signal stability region ofelectrical power systemsYU Yixin ( ~ 9 J ; ) , JIA Hongjie ( % , ~ & WANG Chengshan CI..A J-J )School of Electrical Automation and Energy Engineering, Tianjin University, Tianjin 300072, ChinaReceived September 6, 2000

Abstract In this paper, chaos phenomena and its influence on the power system small signal stability region (SSSR) are studied. We first review the studies on the SSSR, and point out that it is veryimportant to make clear whether there exist some chaotic components on the boundary of the SSSR.Next, with some analytic skills of nonlinear dynamic system, we give a complete bifurcation diagram ofa chaos existing in a sample power system, from a limit cycle (period-I ) to chaotic state through cascading period-doubling bifurcation. The characteristics of the system energy varying in the continuousbifurcation are also shown. Thus a conclusion that chaos is always out of the Hopf bifurcation components (HB) on the SSSR' s boundary. Based on this conclusion and some further studies, we confirmthat, from the viewpoint of power system engineering, we do not need to consider the existence ofchaos in the SSSR and its boundary. Therefore greatly simplifying the study of SSSR. Moreover,some aspects of the attractive regions of chaos and limit cycle are also studied, which is helpful to understanding some mistakes in some previous articles.Keywords: power systems, chaos, small signal stability region, nonlinear system.

With the fast development of the power system, its deregulation and the environment protection constraint aggravation, power system instability problems have become increasingly concernedtopics and have attracted more and more attention. The voltage instability and collapse have so farbeen a hotspot. According to the IEEE suggestions [I], power system stability (including voltagestability and angle stability) analysis can be classified into 3 categories: transient stability analysis (TSA) , small signal stability analysis (SSSA) and static stability analysis (SSA). In this paper, we only study some problems of power system SSSA and power system small signal stabilityregion (SSSR). Without confusion, the 'stability' in the following of this paper refers to thesmall signal stability.

It is commonly accepted that the system small signal instability is associated closely to somebifurcation of power systems[2]. There are three kinds of bifurcation in literature: ( i) saddlenode bifurcation (SNB), first presented by Kwatny[3], which is relative to the power systemmonotonous instability phenomena, (ii) Hopf Bifurcation (HB), which, as first pointed out byAbed[ 4] , is relative to the power system oscillatory instability phenomena, (iii) singularity inducedbifurcation (SIB), which is first defined by Zaborszky'


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viewpoint of power system engineering. And, some researchers[7-IO] have found that in the powerinjection space, the power load required for the appearance of chaos may be less than that for theHopf bifurcation. If this is true, the boundary of SSSR should consist of a chaotic component besides the well-known three bifurcation sub-boundaries of the SNB, HB and SIB, thus making thestudy of SSSR very complicated due to the complexity of chaotic behavior.

In this paper, chaos phenomena and its influence on the power system SSSR are studied.We first review the studies on the SSSR, and point out that it is very important to make clearwhether there exist some chaotic components on the boundary of the SSSR or not. Next, withsome analytic skills of nonlinear dynamic system, we give a complete bifurcation diagram of achaos existing in a sample power system, from a limit cycle (period-L) to chaotic state throughcascading period-doubling bifurcation. The characteristics of the system energy varying in thecontinuous bifurcation process are also shown, which gives a conclusion that chaos always appearoutside the HB boundary of SSSR. Based on the conclusion and some further s tudies , we confirmthat, from the viewpoint of power system engineering, there is no need to consider the existenceof chaos in the SSSR and its boundary. This conclusion excludes the unnecessary complication inthe study of SSSR. Moreover, some aspects of the attractive regions of chaos and limit cycle arealso studied, which helps to understand some mistakes in some previous articles.1 Power system small signal stability and power system small signal stability region(SSSR)

Power system small signal stability is the ability of a power system to remain in a state of theoriginal operating equilibrium under small and instantaneous disturbances, or to transfer to a newstate of operating equilibrium similar (i . e. differential homeomorphism) to the original state' ifthe small disturbances keep constant. That is to say, when a power system is small signal stable,it will never diverge monotonously or lose the operating equilibrium through oscillation of increasing amplitude under any small disturbances . Obviously, the studies on the power system smallsignal stability only concern the conditions close to a system operating equilibrium.

Commonly, a power system can be described by the following differential-algebraic equations(DAE)[13] : = f ( x , y , p ) , (1)o = g (x , y , p ) ,where x E R" is the system state variable vector, y E R" is the system algebraic variable vectorand p E RP is the system control variable vector. For a given vector p , the system operating equilibrium set EPs(p) can be defined as

EPs(p) j (x ,y) I f ( x , y ,p ) = 0 & g(x ,y ,p ) = of. (2)Power system small signal stability is just defined on the system EPs. Given an equilibrium

point (xo, Yo) E EPs, linearize the system of eq. (1) near the point of (xo, Yo):= fx I (x ,y ) 6x + fy I ( ) . 6y ,o 0 xo'Yo (3)o = s. I (xo'yo) 6x + gy I (x ) . 6y,0 ' Yo

where, L f y' s., gyare abbreviations of af lox , aflay, aglax and agloy, respectively,Further define A ( p ) ~ f x l ( x o ' y o ) ' B ( p ) ~ f y l ( x o ' y o ) ' C ( p ) ~ g x l ( x o ' y o ) and D (p )gy I (xo'yo)' Then system of eq. (3) can be described as


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No.2 CHAOTIC PHENOMENA & SSSROF ELECTRICAL POWER SYSTEMS 189

{ 6 = A (p) . 6x + B (p) . 6y , (4 )0= C(p ) 6 x+D (p ) 6 y .If the matrix D (p ) of eq. (4) is nonsingular, 6 y can be eliminated and eq . (4) can be simplified into AX = A(p ) . ~ x (5)where A (p ) =A (p ) - B (p )D (p ) - 1C (p ) .

Theorem 1. Suppose that matrix D (p ) of eq. (4) is nonsingular, the system of eq. (1)is small signal stability if and only if all the eigenvalues of matrix A(p ) have negative real parts.

Note that all the matrices A(p ) , B(p) , C(p) , D(p) and A(p ) are functions of thecontrol variable p. And the eigenvalues of A (p) are also functions of parameter p . According tothe variation in eigenvalues of A(p ) , we can define the following two types of instability modes:

Definition 1. Suppose that matrix D (p ) of eq . (4 ) is nonsingular, if there is an eigenvalue A whose real part turns from negative to positive with the continuous variation of p, andthen define the point p' where A=0 as the saddle-node bifurcation (SNB) point of system (1) .After the point of SNB, system (1) will lose its small signal stability through a monotonous divergence mode.

Definition 2. Suppose that matrix D (p ) of eq . (4) is nonsingular. If there is a couple ofconjugate eigenvalues A, A* = a j b (b 0) whose same real part turns from negative to positive with the continuous variation of p , then define the point p' where A,A * = j b as the Hopfbifurcation (HB) point of system (1). After the point of HB, system (1) will lose its small signal stability through an oscillatory mode of increasing amplitude.

The assumption of the nonsingularity of matrix D (p ) is not always true. Under some condition, matrix D (p ) can turn singular. So the algebraic vector 6y cannot be eliminated, and eq.(5) cannot be obtained. Ref. [5] has defined such a phenomenon as one type of bifurcation,singularity induced bifurcation (SIB) as follows.

Definition 3. Define the point p ' where matrix D (p) turns singular as the SIB point ofsystem (1). At this point , algebraic eq. (4) turns unsolvable. The following theorem describesthe characteristics of system SIB.

Theorem 2[6,14J With the variation in parameter p , a real eigenvalue f1. of matrix D (p )comes across the origin and turns its signal. At the same time, a real eigenvalue A of matrixA(p ) also changes its s ignal , and will come from one infinite end to the other ( i. e. + 00 -- 00 or - 00 -- + 00 ). For the case of an eigenvalue A of matrix, A (p ) turns from - 00 to+ 00 , system (1) loses its small signal stability and the instability mode is monotonous divergence.

Furthermore, in the control parameter space RP, the small signal stability region n sss ofsystem (1) can be defined as follows.

Definition 4.Q sss 1p EW I all the eigenvalues of A(p ) have negative real par ts f (6)

Note that when matrix D (p ) is singular, A(p ) does not exist . So the boundary dn sss of systemSSSR consists of three types of point sets (bifurcation curves) [13 J :

da sss = ISNBs I U 1HBs f U 1SIBs I . (7)In practical system power calculation, SIB can hardly appear.

In recent years, some researchers[7-12J have captured some chaotic phenomena in power


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system simulation. Chaos exhibits a continuous and random oscillation . So it is forbidden in thepower system operation. According to refs. [7-10 J, in parameter RPspace, chaos can exist inthe Q ", just as shown in fig. 1, where chaotic region is within the HB sub-boundary and in thesystem SSSR. As discussed above, if this conclusion is true, it will cause much trouble in thestudies of SSSR .

2 Evolution of the bifurcation under smalldisturbances and chaostable region ( SSR)

Fig. 1 . The case th at chaos in th e sys tem SSSR ( QI d ispower system reac tive load and d is sys tem damp ing fac tor ) .

0.080.1 0

Chaos is a completely undeterminist ic phenomenon without any randomness generated by acompletely deterministic system itself IS] . From theabove description , we may catch a glimpse of itscomplexity. The chaotic phenomenon is so complexthat we cannot give it a precise definition, even anappropriate description. For a long time, peoplecan only give some typical properties about a chaotic set denoted by '1i'[IS,16 ] as follows: ( i) '1i' isstable and bounded (in some references, it is often

called a strange attractor}: (ii) any trajectory in '1i' is unstable; (iii ) there are uncountable trajectories in '1i', which are all dense in '1i'; (iv) all of the periodic trajectories in '1i' make up adense set; (v) the trajectory in '1i' is very sensitive to the initial conditions, i . e. no matter howclose to each other they are at the beginning, any two trajectories will be diverged and becomequite different things; (vi) '1i' is a Cantor set, which implies that '1i' is a fractal, etc . In the following part of this section, we will use a simple power system to illustrate the evolution of the bifurcation when the system comes into chaos .

0.06"t l0.04

A simple power system.ig . 2 .

2 . 1 A sample chaotic attractor in power systemConsider the power system model with 3 buses

shown in fig. 2. which has been used in ref. [ 8 J. Itcan be regarded as one generator supplying power to a local load. The external system is treated as an infinitebus. And the line connecting the generator to the infinitebus can be regarded as a weak tie . The detailed equations are given in Appendix 1. All of the system data are the same as those in ref. [ 12 J exceptwhat we emphasize especially .

We can get the following ordinary differential equation ( ODE) :x = I( x ,p ) , (8 )where p is the bifurcation variable. When a two-axis generator model with a fast excitation systemis used, the state variables of eq. ( 8 ) will be x =[() , Sm , E' d, E' s : Efd, ()L , VLF. When aclassical generator model is used (see sec . 5), the state variables then change into x =[(), Sm '()L , VLF, which is the same model as that in refs. [7- 10J except Y3 equals zero and a correction is made to the generator inertia M in this paper . We still choose Q1d as the bifurcation variable p in the following.

In fig . 2, when we choose the two-axis generator model with a fast excitation system , and


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set the system parameters as follows: T4 = 0.05, KA = 140. Suppose that the initial state vectorxo= [0.7611155,0,1.332678, -0.3283270,4. 198358,0.2396075,0.7795296]T. Select thefollowing two values of Q1d (there is only 5 X 10 - 7 difference between these two variables) as bifurcation variables:Case 1 . Q1d = 1 .2030000 (plotted with a solid line in the diagram) ;

Case 2. QI d = 1 .2030001 (plotted with a dashed line in the diagram) .Other unmentioned variables are chosen the same as Case a of ref. [12] . Integrating the

system equations, we can get the results of fig. 3 ( a) - ( c). Fig. 3 ( a) illustrates the curve ofthe generator phase angle 0 vs time t. Fig. 3 (b) gives the zoom-in result in the time period t =1460 - 1500s. And fig. 3 ( c) is the o-sm phase graph of Case 1 . We can observe from these figures that the results have all the properties of chaotic attractor, such as boundedness (note thatwe can always get the same results even with different integration algorithms and integrating up to30000 s) , sensitivity.to the initial condition, density.and instability.of all the trajectories , etc.

1490480l Is(b)

1470

I . : I f.::.:: ,', c : 'iii l.'11''H'H " . : : ~ ,r ,:, I, "'i , , ,, I I ,, , I ,, , , , , , , ,', : I , I ' , , , , I , ", , " I , ' , I, I , , , , ,, , , , I :i . , I i. , , ,i I14601.0

0.0

'.0;) 0.

1500000lI s(a)

1.0

0.5

Fig. 3. (a) Two ii-I curves with respect to the two ini ti al points; (b ) the zoom-in ii-I curves with 1= 1460-1500s; (c) the'm-ii phase curve of Case I .2.2 Evolution of the system bifurcation to chaos

Using the technique of Piocare section and Piocare mapping' IS], we give the full bifurcationdiagram of the system in fig. 4, where the Piocare section is selected as . ISm =0, and 0< 8c s 8c denotes the oscillating center of the angle 0 f , where 1J is the 0 coordinate of the crosspoint of the Piocare section and system trajectories. The top frame in fig. 4 is a zoom-inbranchof the period-3 screen . And, the evolution of the system periods is given in figs. 5 ( a)and (b) .


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Fig. 4 . System bifurcation graph in the (b, Q d) plane.1.200.1 95

o

1.190

0.20.4

- 0.2- 0.2

When Qld is less than 1 .1 9 1 5 p. u . ,there is only a stable period-one solution in thesystem (fig. 5 ( b ) -0. Then a period-2 trajectory appears, and keeps up until the Q I d equalsabout 1.1970 p . u . (fig. 5 ( b ) - 2 ) . From thenon , a period-4 trajectory floats up (fig. 5 ( b )3) , then period-8 occurs at QI d = 1 . 19808 p.u , and so on. Until Q I d is equal to about1 . 19845 p. u. , the chaotic phenomenon comesinto light. Now, the system is controlled by aso-called strange attractor (fig. 5 ( b ) -4 ) . Inthe study of famous logistic mapping, it isknown that there exist some exquisite structuresin the chaotic region[15J. The system behaviorof this paper is very similar. There are also

exquisite structures of these type in the chaotic region as shown in fig. 4 . When QI d is equal to1.20136 p . u . approximately, a distinct period-3 is popping out (fig. 5 ( b ) - 5 ) . This period-3trajectory is kept up till Q I d = 1.201525 p . u. Then system comes to period-6 (fig. 5 ( b ) -6 ) ,period-12 (where Qld = 1.201559 p . u . ) , and so on. Until Qld is about 1.20160 p . u . , thesystem comes again to a chaotic condition through a series of period-doubling bifurcation. At thispoint, another stable strange attractor appears (fig. 5 ( b) -7). Throughout this evolution, the system oscillatory amplitude becomes larger and larger. At last, when QI d is about 1 .2035 p . u. ,the stable condition of the strange attractor is broken. At this point, the system has already approached the critical state of the chaotic condition.

Another property of the chaotic system is that its bifurcation graph is a fractal['5J. Whensome small local screens of the whole graph zoom in , one can get a similar graph with the samestructure just like the original one. It is true in our system. They all come from a regular stateand go to an irregular condition through the continuous period-doubling bifurcation as shown infig. 4 . Due to the difficulty in illustration, we only give the zoom-in result of a typical period-3branch. Of course, there are also more exquisite and smaller screens in the zoom-in part and theother diagram part. For example, there are obviously two screens of period-5 and period-6 justbefore the screen of period-3 in fig. 4 . But the capture of their trajectories is more difficult.

Chaos is another type of oscillating phenomenon which is near instability. But it is very difficult to precisely capture the chaos surface in practice. In real power system operation, chaos isnot allowed to exist. Moreover, from the above discussion, we can find that when the load of thesystem is monotonically increasing, it will meet the PDB prior to the chaos in the parameterspace. At the same time, they are very close. So it is reasonable to use the boundary of PDB asan approximate boundary of chaos from the viewpoint of power system engineering.3 Rules of system energy variation in the evolution of bifurcation to chaos

Two famous theorems for chaotic phenomena, i. e. Sarkovskii Theorem[ 15J and Li-Yorke[16JTheorem, make use of the Sarkovskii series (see Appendix 2 ) . This series, in the real physicalsystem, is always related to the energy needed for the appearance of each periodic cycle, such asthe critical temperature for Lorenz system' 17J, the critical temperature difference for the varied


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Period-l T P c r i o - 2 t P e r i o d - 4 t P e r i o d - ~ orChaosQJ.f 1.1915 1.1970 1.19808 1.198301.19845

eriod-3T Period-6r Period-12-.--. TChaos1.20136 1.201525 1.201559 1.201566 1.20160(a)

193

s.;

s'"

0.00

- 0.02- 0.5 o 0.5 1.0s

7

(b)

Fig. 5. (a ) Periods evolution of the system. (b) System phase graph of different periods. 1-6 . Periods-I - 6 ; 7. chaos.

Benard cells[12] structures, the critical stress for the fracture phenomenon[13], etc. In an electrical power system, the evolution of system periods also obeys the following

Proposition 1. In the periodic evolution of a system (fig. 4) , the required energy of thesequent period is incremental, i ,e 0 E 1 (energy of period-L) < E 2 (energy of period-Z) < E4 0 0 0 24 l> 23 t>22 t>21 l> 2 t>1; finally, rank all the rest number of N in a descending order as

the last l ine: 23 , 22 , 21 , 2, 1, which is in fact all the powers of 2 ordered descendingly. Nowthe Sarkovskii series consists of all the natural numbers N.

Acknowledgements This work was supported by the National Key Basic Research Special Fund of China (Grant No.G1998020303).

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STC Small Signal Stability Power System