Voltage Collapse Dynamics and Control in a Sample Power System

Copyright © IFAC 12th TrielUlial World Congress, Sydney, Australia, 1993

VOLTAGE COLLAPSE DYNAMICS AND CONTROL IN A SAMPLE POWER SYSTEM H.O. Wang·, E.H. Abed· and A.M.A. Hamdan u ·Depart~nt

of Elutrical Enginuring and the Institute for Systems Research, University of Maryland, College Park, MD 20742, USA ··Depart~1Il of Elutrical Engmuring, University of Science and Technology, Irbid. Jordan

Abstract. Nonlinear phenomena, including bifurcations, chaos and crises, have been determined to be crucial factors in the inception of voltage collapse in power system models. The issue of controlling voltage collapse in the presence of these nonlinear phenomena is addressed in this paper. The work employs a example power system model akin to one studied in several recent papers. The bifurcation control approach is employed to modify the bifurcations and to suppress chaos and crises. The control law is shown to result in improved performance of the system for a greater range of parameter values. Keywords. Voltage collapse; electrical power systems; bifurcations; power system control; control nonlinearities.

Chiang (1989), but differs from it in several ways detailed in Section 2.

1. INTRODUCTION Voltage collapse in electric power systems has recently received significant attention in the literature. This has been attributed to increases in power demand which result in operation of an electric power system near its stability limits. A commonly held view is that voltage collapse arises at a saddle node (static) bifurcation of equilibrium points (see, e.g., Kwatny et al., 1986; Dobson and Chiang, 1989) . Dobson and Chiang (1989) introduced a simple example power system to illustrate the saddle node bifurcation mechanism. The presence of a saddle node bifurcation in a dynamical system does not preclude the presence of other, possibly more complex, bifurcations. Thus, the recent papers (Abed et al., 1990, 1992a; Chiang et al., 1992; Ajjarapu and Lee, 1992) have shown that indeed other bifurcations occur in the example power system model studied in Dobson and Chiang (1989) . Other papers have also studied bifurcations in voltage dynamics in other power system models (Abed et al. , 1984; Rajagopalan et al., 1989; Venkatasubramanian et al., 1992).

Upon revealing the various bifurcations and the associated rich dynamics, one would naturally ask the question : what. can we do about voltage collapse? what is the possible role of feedback control in such situation? In the present paper, we report some positive results in this direction. In previous work (Abed and Fu, 1986, 1987; Abed et al., 1992b) problems of bifurcation control have been addressed . This involves design of feedback cont.rols to modify the stability and amplitude of bifurcated solutions in general nonlinear control systems. The latter work has been applied to control problems in high incidence flight, stall of jet engines, oscillatory behavior of tethered satellites and chaotic dynamical systems. The present paper shows the utility of the bifurcation control approach to the control of voltage collapse phenomenon . It is demonstrated that voltage collapse can be postponed through modifying the bifurcations and suppressing the chaos and crises.

The fact has therefore now been established that a variety of bifurcations, static and dynamic, occur in power system models exhibiting voltage collapse. The purpose of this paper, which continues the work reported in (Abed et al., 1992a; Wang et al., 1992) is to determine the implications of these bifurcations for the vol tage collapse phenomenon and to address the issue of voltage collapse control. The present paper is based on our paper (Wang et al., 1992) where the role of a boundary crisis of a strange attractor in voltage collapse was first reported. The power system example used in the present paper and in Wang et al. (1992) is similar to that of Dobson and

2. A POWER SYSTEM MODEL Dobson and Chiang (1989) employed a simple power system which includes a capacitor in parallel with a nonlinear load. (See Fig . 3 of Dobson and Chiang, 1989.) It is found that the value of a reactive power loading parameter (Qd at the saddle node bifurcation is approximately 11.41 per unit. This is a rather high value , and is a consequence of inclusion of the capacitor in the example system. It seems that it is rather difficult to reach this level of reactive load at normally encountered power factors. For this reason, we modify the power syst.em example of Dobson and 825

Chiang (1989), mainly through deletion of the capacitor from the system . It follows from Dobson and Chiang (1989) that the system dynamics (with no capacitor) is governed by (P(8 m , 8, V), Q(8 m , 8, V) are specified below) :

5m = W Mw = -dmw + Pm - Em VYmsin( 8m . 2,

-

0.•

(1) (2)

8)

K qw 8 = -Kqv2V - RqvV +Q(8 m ,8, V) - Qo - Q1 (3) TKqwKpvV = KpwKqv2 V 2+(KpwKqv-KqwKpv)V +Kqw (P(8m, 8, V) - Po - Pd -Kpw(Q(8 m , 8, V) - Qo - Qd (4)

o. •

.---_"""T'"_ _

O. l"+-_ _ 1 . 10

P(8 m , 8, V) = -Eo VYo sin(8) +EmVYmsin(8m-8) Q( 8m , 8, V) = Eo VYo cos( 8) +Em VYm cos(8 m - 8) - (Yo + Y m )V 2

= 0.01464,

= 0 and

9m

Qo

= O.

= 0.3,

Em

=

1.05, Yo

2:.)0

~-__,_--_.__-~

1 . "0

l.SO

1 . 60

Q,

1 . 70

v

0 . . . . . . ._ _ _ _ _ _ _ _ _ _ _ _ _ _- - ,

(5) Fig. 2. Magnificd bifurcation diagram for boxed region in Fig. 1

(6)

Most of the parameter values used in the present study agree with those of Dobson and Chiang (1989) . The parameters given there correspond to a large generator . Our choice of parameter values corresponds to a medium sized generator (500MW). Of the parameter values used here, those which differ from that of Dobson and Chiang (1989) are as follows: M

3.30

Fig. 1. V vs. Q1 at systcm cquilibria

The notation is basically identical to that of Dobson and Chiang (1989), with the caveat that in the present paper there is no need for primed quantities since the capacitor is no longer included in the system. (See Dobson and Chiang (1989) for details.) The load includes a constant PQ load in parallel with an induction motor. The real and reactive powers supplied to the load by the network are

90

v 0.,-----------------,

• PDB: Period doubling bifurcation • BSKY: Blue sky bifurcation For ease of reference, we denote the values of the parameter Ql at which the bifurcations CD-CD occur by QC{J-Q~, respectively. Assuming the parameter Ql is quasist.at.ically increased, for the 'usual' values of tile parameter Ql < QC{J, the system operates at the stable equilibrium. As the parameter is increased, the equilibrium loses stability at the Hopf bifurcation point HBCD, giving rise to an unst.able periodic orbit . Since this orbit gains st.abilit.y at the cyclic fold bifurcation CFB@, a st.able periodic orbit surrounds the equilibrium at and slightly beyond the Hopf bifurcation point.. This result in a hysteresis loop near IIECD. The system then can operate at this stable periodic orbit through a jump from the stable equilibrium as the parameter Ql crosses QC{J . For greater values of the parameter Ql, this periodic solution also loses stability at PDB@), but in doing so gives birth to a new stable (period doubled) periodic orbit (which loses stabilit.y at PDB@) . This scenario repeats itself in a cascading fashion, each time making available a stable periodic orbit, until a strange attractor emerges. The system operates on the strange attractor until the strange attractor disappears ("crisis") . After this crisis, there is no stable invariant set in the vicinity of the nominal equilibrium at which to operate. Thus , the system must now undergo a large transient excursion . In the next subsection, this excursion ("voltage collapse") is tied to the disappearance of the strange attractor. Not.e Fig. 2 shows the

= 3.33,

3. DYNAMICS OF VOLTAGE COLLAPSE 3.1. Bifurcation Analysis In this section, the results of a bifurcation analysis of the model (1)-(6) are summarized (Wang et al., 1992). Fig. 1 shows the dependence of the voltage magnitude V at system equilibrium points as a function of the bifurcation parameter Ql . A solid line corresponds to stability of an equilibrium, while a dashed line corresponds to instability. Fig . 2 depicts a blown-up bifurcation diagram, detailing some of the bifurcations which occur in the boxed region of Fig. 1. In Fig. 2 circles indicate the minimum of the variable V for periodic solutions. Open circles indicate instability and solid circles indicate stable periodic solutions. Note that Fig. 1 depicts two bifurcations, and Fig. 2 depicts a total of five additional bifurcations . These seven bifurcations are labeled HBCD, SNB~, CFB@, PDB@), PDB@, BSKY@ and BSKYCD . The acronyms are: • HB: Hopf bifurcation • SNB: Saddle node bifurcation • CFB: Cyclic fold bifurcation 826

continuation of the periodic orbits appearing at the cyclic fold bifurcation CFB@ and the period doubling bifurcation PDB@). Each of these periodic orbits disappears in a collision with the unstable (saddle) low voltage equilibrium point . The disappearance of these orbits is indicated by BSKY@ and BSKY(7) in Fig. 2.

bounded by the stable llIanifold of the unstable limit cycle, and so this region shrinks as criticality is approached . These factors motivate the design offeedback control laws directed at the Hopf bifurcation which reduce the negative effects and increase the stability margin of the system in parameter space. As shown in Abed et al. (1992b) , such control action can also suppress the chaos and crises by 'squeezing' the period doubling cascades. This is an approach to influence the various aspects of the global bifurcations (e.g. blue sky catastrophes) through local control.

3.2 . Boundary Crises and Voltage Collapse The bifurcations discussed in the foregoing analysis, and especially the sudden disappearance of the strange attractor, are crucial to the understanding of voltage collapse for the model power system under consideration . We claim that, for the model under study, voltage collapse is triggered by the boundary crisis of the strange attractor (Grebogi et al. , 1982), i.e., its sudden destruction through collision with the low voltage saddle point . An attracting invariant set exists for parameter values Ql up to the critical value, Qi, at which the boundary crisis takes place.

4.1. Voltage Collapse Control We carry out the deSign of controllers for voltage collapse control for the model (1 )-( 4) subj ect to control u which is iriserted additively at the right hand side ofEq. (4) . Note that the control occurs in the excitation system and involves a purely electrical control system . Feedback signals which are some dynamic function of the speed ware widely used in power system stabilizers (PSS) . The speed signal needs no washout since it does not affect the syst.em equilibrium structure at steady state. Note also that such a controller does not affect the position of the saddle node bifurcation SNB~.

Voltage collapse occurs precisely at the parameter value Ql = Qi. Note that one may view

the Hopf bifurcation as an signal of impending voltage collapse. 4. BIFURCATION CONTROL OF VOLTAGE COLLAPSE

The objectives of cont.rol is 1) to prevent the occurrence of the jump behavior , 2) to increase the region of attract.ion of the stable equilibrium point, and 3) to delay t.he collapse (in parameter space) . One control law design transforms the subcritical Hopf bifurcation into a supercritical bifurcation and ensllfes a sufficient degree of stability of the bifllfcatcd periodic solutions over a range of parameter values of interest. These control laws allow stable operation close to the point saddle node bifllfcation. Another control design involves changing the critical parameter value at which the Hopf bifurcations occur by a linear feedback cont.rol. The voltage collapse can be delayed by such a linear control. Note that controllability of the crit.ical mode facilitates the design.

In this section, we consider local control of voltage collapse at its inception. That is, we design controllers which can delay the occurrence of voltage collapse, as opposed to controllers for recovery from voltage collapse. The controllers we seek do not involve forced system operation in parameter ranges where voltage collapse does not occur, but are designed to work in the parameter ranges of difficulty. In order to control voltage collapse in power system models such as the one studied in the precedi ng section, one has to design control laws to deal with bifurcations, chaos and crises. In our work (Wang and Abed, 1992; Abed et al., 1992b), we have shown that control laws which significantly reduce the amplitude of a bifurcated solution, or significantly enhance its stability over a nontrivial parameter range, are viable tools in the taming of chaos. Here similar techniques will be employed to control the bifurcations, chaos, and crises. In doing so we expect to increase the stability margin of the system in parameter space. In other words, voltage collapse will be 'postponed' so that stable operation of the system will be allowed beyond the point of impending collapse in the open loop system.

Stabilizing the Hopf Bifurcation To render the Hopf bifurcation IIBCD supercritical, we employ a cubic feedback with measurement of w . Control function u is of t.he form

(7) where k n < 0 is the (scalar) cubic feedback gain. Fig. 3 shows superimposed bifurcation diagrams for the closed loop system with various control gains k n . This along with simulation evidences indicate that larger values of the gain Ik n I result in a reduced amplit.ude of the stable limit cycles. Note that the boundary crisis is delayed and the possible operat.ing range of the system in parameter space is increased .

In this particular model, the stable equilibrium point loses its stability through the sub critical Hopf bifurcation (local), and the boundary crisis (global) of the strange attractor triggers the voltage collapse. The subcriticality of the Hopf bifurcation has several negative effects on the system. The system may exhibit a jump to the coexisting attractor- limit cycle from the stable equilibrium under perturbation. Moreover, the region of attraction of the stable equilibrium is

Delaying the Hopf Bifurcation A linear control eXists for delaymg the 110pf [)Ifurcation point: u

827

= -k/w.

(8)

v

6. REFERENCES

o ...c t , - - - - - - - - - - - - - - - - - - ,

Abed, E.H., and P.P. Varaiya (1984). Nonlinear oscillations in power systems . Int. J. Elec. Power and Energy Syst., 6, 37-43. Abed, E.H., and J .11. Fu (1986). Local feedback stabilization and Lifllfcation control, I. Hopf bifurcation . Systcms and Control Letters, 7, 11-17. Abed, E.H ., and J .ll . Fu (1987) . Local feedback stabilization and bifllfcation control, 11. Stationary bifurcation . Systems and Control Letters, 8, 467-473 . Abed, E.H., A.M .A. Hamdan, H.-C . Lee, and A.G. Parlos (1990) . On bifurcations in power system models and voltage collapse. Proc. 29th IEEE Conf Dec. Contr., 3014-3015 . Abed, E.H., J .C. Alexander, H. Wang, A.M.A. Hamdan, and H.-C. Lee (1992a) . Dynamic bifurcations in a power system model exhibiting voltage collapse . Proc. 1992 IEEE Int. Symp . on Circuits and Systems, 2509-2512 . Abed, E .H., H.O. Wang, and R.C. Chen (1992b). Stabilization of period doubling bifurcations and implications for control of chaos. Proc. 31st IEEE Conf Dec. Contr., 2119-2124. Full version to appear in Physica D . Ajjarapu, V., and n. Lee (1992). Bifurcation theory and its applicat.ion to nonlinear dynamical phenomena in all electrical power system. IEEE Trans. on Power Systems, 7, 424-431. Chiang, H.-D., C.-W. Liu, P.P. Varaiya, F .F . Wu, and M.G. Lauby (1992) . Chaos in a simple power system. Paper No. 92 WM 151-1 PWRS, IEEE Winter Power Meeting. Dobson, I., and H.-D . Chiang (1989) . Towards a theory of voltage collapse in electric power systems. Systems amI Control Letters, 13, 253262. Grebogi, C., E . Ott, and J .A . Yorke (1982). Chaotic attractors ill crisis . Physical Review Letters, 48,1507-1510 . Kwatny, H.G., A.K. Pasrija, and L.Y. Bahar (1990). Static bifllfcation in electric power networks: Loss of steady-state stability and voltage collapse. IEEE Trans . Circuits Syst., CAS·33,981-991. Rajagopalan, G ., P.W . Sauer, and M.A . Pai (1989) . Analysis of volt.age control systems exhibiting Hopf bifllfcat.ion. Proc. 28th IEEE Conf Dec. Contr., 332-335. Venkatasubramaniall, V., 11. Schiittler, and J . Zaborszky (1992). Volt.age dynamics: Study of a generator wit.h voltage control, transmission and matched I\HV load . IEEE Trans. Automatic Control, 37,1717-1733. Wang, H., E.H. Abed, and A.M.A . Hamdan (1992) . Is voltilge collapse triggered by the boundary crisis of a st.rallge attractor? Proc. 1992 American Control Cord., 2084-2088. Wang, H., and E.II . Abed (1992) . Bifurcation control of chaotic dynamical syst.ems. Froc. of NOLCOS'92: Nonlincar Control System Design Symposium, (M. Fliess, Ed.), 57-62.

0.500. O • .,~-___.-__r-___,r_-,..._-._-_r_--t

2.550

2.560

2.570

2 . 580

2 . 5'0

a.600

2.610

c,

2.610

Fig. 3. Superimposed bifurcation diagrams for cubic control (7) with different gains: a. -0.1; b. -0.5; c. -1.0; d. -5.0

c,

•. 7 ' 0 , - : - - - - - - - - - - - - - - - - , •. 6 0 . 1 - - - - - - - - - - - - _

'.50.

'.,0.

'.10.

•. '''4----.----,--.---,----r---r----'-t -.060

-.050

-.oco

-.0)0

-.020

- . 010

0.000

kl

0.010

Fig. 4. Relationship between critical 01 at which the Hopf bifurcation occurs and gain k, of linear control (8)

where k/ < 0 is the (scalar) linear feedback gain. Such a control is found to be able to delay the sub critical Hopf bifurcation HB
Qrp

The two types of control law given above, namely the cubic control (7) and the linear control (8), can be combined to result in a composite control law which both delays and stabilizes the Hopf bifurcation. 5. CONCLUSIONS Voltage collapse is triggered by a boundary crisis of a strange attractor for sample power system model. For the issue of voltage collapse control, it is demonstrated that the voltage collapse can be postponed by feedback control of the various nonlinear phenomena. Although the relative importance of the effects of these nonlinear phenomena in general power systems under stressed conditions is still a topic for further research, the bifurcation control approach appears to be a viable technique for control of these systems.

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