- Email: [email protected]

Electric Power Systems Research 33 (1995) 53 61

RU|i OH i i

Effects of dynamic load model parameters on damping of oscillations in power systems Jovica V. Milanovi6, Ian A. Hiskens Department of Electrical and Computer Engineering, The University o/' Newcastle, Newcastle, N S W 2308, Australia Received 10 October 1994

Abstract

The paper discusses the effects that nonlinear dynamic load model parameters have on electromechanical oscillations in power systems. Based on a modified Heffron-Phillips model of a synchronous machine and on a generic model of dynamic loads, analysis shows the dependence of the modal oscillations of the system on load model parameters. A single-machine infinite-bus case has been analyzed. It was found that under some circumstances damping can be improved, but under other conditions dynamic load may cause a decrease in damping.

Keywords: Load dynamics; Electromechanical oscillations; Damping

1. Introduction

Poorly damped electromechanical oscillations in interconnected power systems have been regarded as troublesome for many years [1-3]. Modern power systems are characterized by the concentration of power sources and consumer centers in different, usually distant, areas. These areas are connected via transmission lines of limited transmission capability. Due to the constant growth of demand and the difficulties of construction of power plants near the consumer centers, power systems are subjected to operation close to transmission line capability. Such operation may bring a power system to steady-state instability which can further provoke the building up of poorly damped electromechanical modes and eventually system collapse in the worst case. Poorly damped electromechanical (swing) oscillations are inherent in power systems due to the large number of interconnected synchronous generators. They occur because of synchronous generators swinging against each other. In an n-machine system there will be (n - 1) electromechanical modes. These oscillation modes result from the rotors of machines, behaving as rigid bodies, oscillating with respect to one another, using the transmission system between the machines to exchange the oscillation energy [4]. There are generally two types of oscillatory behavior: localElsevier Science S.A. S S D I 0378-7796(95)00927-A

mode oscillations, which occur between a single machine, or sometimes a small group of machines, and the rest of the system, and inter-area or system oscillations, which occur between large groups of machines. Typical oscillation frequencies of local-mode oscillations range from 0.7 to 2.0 Hz [3,5], while inter-area modes are usually in the range 0.1-0.8 Hz [3,5]. The local modes are fairly well understood and can be analyzed in a satisfactory way, but the inter-area modes, and the factors influencing them, are not fully understood. There are still questions with respect to the underlying dynamic processes. It is very important that the level of damping of significant system modes can be accurately predicted. If this is not the case, it is possible that a system configuration could be proposed that is actually unworkable due to poorly damped oscillations [6]. Different types of flexible AC transmission system devices are being considered to damp oscillations [7 9]. To evaluate the usefulness and tune these devices properly, we must have confidence in damping predictions. Some of the recent studies of interconnected power systems around the world (e.g. Ref. [10]) have found that the measured level of damping is often less than that predicted. It was found that, in response to a disturbance, the system oscillations last for longer than expected. These measurements lead us to question our confidence in the predictions. Generally good agree-

54

J.V. Millanovi( ~, LA. Hiskens /Electric Power Systems Research 33 (1995) 53 61

ment has been obtained between the measured and predicted oscillation frequencies. Where this has not been the case, the system has tended to oscillate more slowly than predicted. So results have been corrected by modeling the motor load around the system to increase the overall system inertia. This adjustment improves the agreement between measured and predicted oscillation frequencies. However, similar techniques have not been helpful in improving damping predictions, The problem is to alter the modeling to reduce damping. In the past, close attention has always been given to modeling generators, automatic voltage regulators, governors, power system stabilizers, and other transmission equipment. The representation of loads in power systems has not traditionally been considered so thoroughly, even though it has been shown that loads can have a significant impact on the results of analysis [11]. The accurate modeling of loads is a difficult task due to many factors [12] such as the large number of diverse load components, ownership and location of load devices in customer facilities that are not directly accessible to the electric utility, changing load composition with time of the day, week, seasons, and weather, lack of precise information on the composition of loads, uncertainties regarding the characteristics of many load components, etc. Power system studies have traditionally used static load models, given in general exponential form by

/ V\"p~

Pd= Po~-~oo) { V\"qs Combinations

of

(1)

constant-impedance

(rips = nqs = 2),

constant-current (nps = rtqs = 1) and/or constant-power (rtps=rtqs=0) load models in polynomial form/[12] have also been used. However, it is known that a large number of loads are not statically dependent on voltage, but actually have some dynamic characteristics. The generic form of response of many loads to a voltage step is shown in Fig. 1. The initial power step, the final power mismatch, and the rate of recovery of the load are parameters which can vary greatly across different load types. A mathematical model of this form of load response is given in Refs. [13,14]. In Ref. [15], the modified Heffron-Phillips model is developed to enable the interaction between the dynamic load and the power system to be studied. The original model was modified by the inclusion of a local nonlinear dynamic load instead of a local impedance load. This modification enabled the mechanism of the interaction between the power system and the dynamic load [16] to be explored in detail. It was shown that variation of the voltage due to load variation at a bus will result in a delayed variation of the load which, due

to feedback behavior, can have a noticeable effect on the damping of electromechanical oscillations. In this paper the effects that parameters of a generic nonlinear dynamic load model have on the damping of modal oscillations are presented. The structure of the paper is as follows. Section 2 describes a nonlinear aggregate dynamic load model and its general properties. Section 3 gives a general description of the system used for analysis. Section 4 provides details of the effects of dynamic load model parameters on electromechanical oscillations in a power system.

2. Nonlinear dynamic load model Measurements of the response of actual power system loads [4] to a step change in voltage show that the response is of the general form shown in Fig. 1. It was found that real and reactive power have qualitatively similar responses, and that they are dynamically related to the voltage. In this study only the active power response will be treated as dynamic, while the reactive power will be considered to be statically related to the voltage. The main characteristics of the measured response are the following: • a step in power immediately follows a step in voltage; • the power then recovers to a new steady-state value; • the power recovery is approximately of exponential form; • the size of the step and the steady-state value are nonlinearly related to the voltage [14]. A model which describes this form of response was proposed in Refs. [13,14] as

T~P,, + ed = es(V) + 'r~o-p(v) ~

(3)

time

P.Q

time

Fig. 1. Typical load response to a step in voltage.

55

J.V. Milanovi& 1.A. Hiskens / Electric Power Systems Research 33 (1995) 53-61

v

Ps(.) - I,t(.)

J-[ Tps*l

Fig. 2. Block diagram representation of the dynamic load model.

where Tp is a time constant which characterizes load response, Pd is the load power demand, P~(V) is the steady-state load characteristic, ap(V) is a nonlinear function of voltage and V is the bus voltage. If the transient load characteristic is defined as t~

P t ( V ) : t a p ( Z ) d r + Co

(4)

td 0

The presented load model has until now been used in the study of voltage collapse behavior [17]. If some disturbance causes a step reduction in system voltage, then the recovery of load from that voltage may lead to further deterioration of the voltage, and ultimately to system collapse. In the investigation presented in this paper, the interest is not in step changes to the voltage, but in sinusoidal variation of the voltage. Referring to Fig. 2, one would expect the load power P~ to vary periodically in response to sinusoidal variation of the voltage. Variation of P~ could then feed back through the system to reinforce the voltage variation. This feedback mechanism is described in detail in Ref. [16]. In these investigations of system damping, only small disturbances are considered. Therefore the load model given by Eqs. (5) and (6) can be linearized around the operating point. Linearization yields Po

where Co is some constant, and if a new state variable Xp is introduced as

Tp AArp = -- AXp -J- ~

Xp :

Po Axp = AP e -- ~o<>nr AV

Pd -- Pt(V)

(5)

then the load model can be written as Tp.~p = --Xp Jr- Ps(V) - Pt(V)

(6)

A block diagram representation of this load model is shown in Fig. 2. The functions P~(V) and P,(V) can be defined as

Po

(Tps -P- 1) AXp : ~

(10)

(rips - r i p , ) AV

(11)

Substituting Eq. (10) into (11) and manipulating yields APd =

where Vo and Po are the nominal voltage of the bus and the corresponding power of the load, respectively, and nps and np, are the steady-state and transient voltage exponents. The voltage exponents tips and npt are generally in the ranges 0 2 and 1-2.5, respectively [12,14]. Time constant Tp, which characterizes the recovery response of the load, can be chosen to represent different types of loads. For loads consisting predominantly of induction motors, such as industrial, agricultural and air conditioning loads, or for industrial plants such as aluminum smelters or power plant auxiliary power systems, Tp is in the range up to one second. For tap changers and other control devices Tp is of the order of minutes, and for heating loads Tp is in hours, In this study the predominant interest is in exploring the effects of the loads that have time constants of the same order of magnitude as electromechanical oscillations, i.e. up to around two seconds. The behavior of reactive power due to a step in voltage could be described by similar equations with different load model parameters, i.e. different time constants and voltage exponents.

(9)

Introducing the Laplace operator, Eq. (9) can be written as

( V\"p~

f V \"~>,

(np~ -- npt ) A V

Po H p s (npt/nr,s)Tps+ 1AV

--

Vo

(12)

Tps+l

This relationship is shown in block diagram form in Fig. 3. It is obvious that the block diagram in Fig. 3 can be transformed into a single lead/lag block. From Eq. (12) it can be seen that the relationship between AP e and A V is influenced by the load time constant Tp and the voltage exponents npt and rips. Bode plots of this relationship for two different values of the ratio /'/pt//Tps and a number of different values of Tp are given in Fig. 4. Also, from Eq. (12) and Fig. 4, it can be seen that at low frequencies the gain approaches (P<,/Vo )nm, i.e. the steady-state load characteristic dominates. At high frequencies, the gain approaches (Po/Vo)ript, so the transient load characteristic dominates. This is consistent with our understanding of the load response. At intermediate frequencies, both the steady-state and transient characteristics have an influence on the behavior. In

AV

,p, Po ~ n

Vo

nps Tp s + 1

ps

A Pd

Tp s + 1

Fig. 3. Block diagram representation of the linearized load model.

J.V. Millanovi(', I.A. Hiskens .; Electric Power Systems Research 33 (1995) 53 61

56

gain [p.u.]

1.4~-

Tp=lOs; ls; 0.5s; 0.1s

.........

~.~". . . . . . .

1.2I

//"

10.2 phase [deg] .~

50

. ~?

....

/'"

"1

10

,

Ea' 8L6..

. ~ : ' ~ . ~= . . . . .

~

"1~:

.~-~,

" ~,(

//

/

\

101

"

j-

~

" " ,¢

....

vt0

vail 3

.1

@

"'"

""

10°

~

.2 .....

102 frequency [rad]

,

.

.

. ......

" ,

x

Fig. 5. Single-machine infinite-bus system. /

1 -2

10-1

/

\

100

\

x

101

102 frequency [rad]

Fig. 4. Bode plots of the magnitude and phase of the dynamic load for different time constants and voltage exponents: broken curves, npt/np~= 15; full curves, npt/np~ = 2 ; from left to right, Tp = 10, 1, 0.5 and 0.1 s.

such cases there is some phase shift through the load. The maximum phase shift is very dependent on the r a t i o /Tpt/nps. As this ratio becomes larger, so does the maximum phase shift. The frequency at which the maximum phase shift through the load occurs depends on the time constant Tp of the load. For smaller values of time constants the maximum phase shift occurs at higher frequencies. For a very large time constant, i.e. a load which responds slowly, the transient load characteristic has the predominant influence on the behavior. For very small values of Tp, i.e. a fast load, the steady-state characteristic is dominant. This can again be explained from the general form of the behavior exhibited by the load model (see Fig. 1). If the time constant is large then the load will take a long time to recover from its transient value to the steady-state value. Therefore, except at very low frequencies, the load will never have time to approach the steady-state characteristic. For small time constants, the load will recover very quickly. So, except for very high-frequency variation of the voltage, the load will always have time to recover to the steady-state characteristic.

adopting the following simplifications. Balanced conditions are assumed; stator winding resistance and stator flux derivatives are neglected; damper windings and saturation effects as well as frequency deviation in speed voltage terms are neglected. Also, the transmission line is assumed to be lossless, i.e. to have zero resistance. Local load is assumed to have only voltage dependent active power. The phasor diagram for the generator in the transient state with a common synchronously rotating reference frame is shown in Fig. 6. A complete derivation of the model used in the analysis is given in Ref. [15]. A block diagram model of the system presented in Fig. 5 is shown in Fig. 7. The block diagram was obtained from linearizing the differential and algebraic equations that describe the dynamic load, synchronous generator and power system. It was built in SIMULINK. The state space model of the complete system was obtained from the block diagram presented in Fig. 7 using MATLAB. The system matrix A has been used for eigenvalue calculation and analysis of the effects that the load model parameters have on the damping and frequency of the electromechanical mode.

j(x d - x'd )Iad

3. Single-machine infinite-bus system description The simple system given in Fig. 5 was used for investigations as it allowed easier exploration of the system-load interaction. A third-order model for the synchronous generator was used so it was identical to that used for the single-machine Heffron-Phillips model [1,18-20]. This model is obtained from the classical Park model of a synchronous machine [21,22] by

Ref

Fig. 6. Phasor diagram for the generator in the transient state.

57

J.V. Milanovi(', I.A. Hiskens / Electric Power Systems Research 33 (1995) 53 61

? r~

....

Beta~"'~l~ i ,~ ~DynLdG '

Tp.s+l DynLdTF

\/3,

Efd

FIx.dec.--'--'--~

I "~

Delta

RoLangl,"IF

Pmech

Fig. 7. System block diagram with only active power dynamics included.

4. Power system-dynamic load interaction It was shown in Ref. [16] that the power system which is more heavily loaded is more sensitive to load variations. This reflects the fact that as systems become more heavily stressed they become more sensitive to parameter changes. The traditional view of the sensitivity of system voltage to load changes relates to step changes in load. Such sensitivity is primarily determined by the fault level of the bus. If load variation is oscillatory, then the dynamics of the power system will also influence the sensitivity. Hence, if resonance occurs between the power system and the dynamic load, the sensitivity can be greatly amplified. Our interest is now in the system behavior when a dynamic load is connected to the power system. Fig. 8 captures the interaction diagramatically. It can be seen that the load provides a feedback path, and hence has the potential to alter the overall system behavior. Depending on the load characteristic, this feedback may improve the damping. But it could also destabilize the system and cause a deterioration in damping [16]. The emphasis here is on the effects of dynamic load model parameter variation on the damping of electromechanical oscillations.

In Figs. 9 and 10 the effects of time constant and voltage exponent ratio variation are shown as root loci of the system electromechanical mode. The arrows on the root locus plots denote the direction of movement of the electromechanical mode of the system with increase of Tp from zero to 100 s. With Tp = 0 s, the static load model is effectively presented. Fig. 9 is related to the case when the generator supplies Pg-----0.9 p.u. and the load absorbs P d = 0.6 p.u. when in the steady state. Active power of 0.3 p.u. flows over the transmission line. In the sequel this case will be denoted as Case A. Fig. 10 is related to the case when the generator supplies Pg = 0.9 p.u. and the load absorbs Pd = 1.2 p.u. in the steady state. The difference in active power of

jw Irad} 6.18

npVnps=2.5/0.2

6.1E

npVnps=2~O.~..~ nptlnps=2/1

6.12 6.1 6.08 6.06

Dynamic Load ~ Power System Fig. 8. Load power system interaction.

&V w

6.04~

6... 2

-0'.,

-0.36'

-0.~

":0.;,,

-0.~

-0:3

-0.26

damping [P.U.I

Fig. 9. Root locus of the system's electromechanical mode for Pg = 0.9 p.u., Pd = 0.6 p.u. and for different %,,'np~ ratios.

J.V. Millanovi(, I.A. Hiskens / Electric Power Systems Research 33 (1995) 53 61

58 Iw [radl

X 10 .3 Pe Lo.u.]

6.6

1

f

6.55

0.5 6.5

C

6.45

6.4

-0.5

6.3~

6.~ 5

I -0,48

I -0.46

I -0.44

I -0.42

i -0 4

I -0,38

I -0.36 -0,34 damping [p.u.]

Fig. 10. Root locus of the system's electromechanical mode for Pg = 0.9 p.u. and Pd = 1.2 p.u.

0.3 p.u. is supplied by the transmission system. In the sequel this case will be denoted as Case B. Consider Fig. 9. It can be seen that as the time constant Tp increases, the damping reduces and reaches its minimum for Tp = 0.1 s. After that, the damping increases with increasing time constant. This behavior can be explained by looking at the Bode plots of the load (Fig. 4). For Tp = 0.1 s the load has a maximum phase shift near the system resonant frequency and the gain is about 0.8 p.u. With further increase of the time constant the phase shift for that particular frequency decreases and the gain is almost constant. In this case the system has a negative phase shift and the load has a positive [15], therefore the system oscillations are being reinforced by the load behavior. Obviously the level of the reinforcement is determined by the gain through the system and the gain through the load. As the gain of the system gets greater, for weaker systems the effect of the feedback through the load increases. It can also be seen from Fig. 4 that for the larger npt///ps ratio the load has a larger phase shift and larger gain, which has as a consequence a larger variation of damping of the electromechanical oscillations of the system. The root locus of the electromechanical mode for the weaker system is given in Fig. 10. The effect of the load dynamics on the damping is more significant in this case, and the change of damping with time constant variation is greater. In this case the system has a positive phase shift and the load too [15]. Therefore, the system oscillations are first being damped by the load behavior for small time constants up to Tp = 0.2 s. For this range of Tp, the load has a large phase shift and smaller gain at the system resonant frequency. As T v increases further, the oscillations begin to be reinforced, so damping deteriorates. In this case the phase shift becomes negligible, and the gain is large. These results

1'0

1'5

20 time [s]

Fig. 11. Power response to a step in field voltage for the system with Pg = 0.9 p.u. and P~ = 0.6 p.u. for static load and for dynamic load with T p = 0 . 1 s and g/pt/llps = 12.5, Pe[p.u.] x 10 "3 1.5

0.! 0 -0.5 -1

-1.5 2.

.

4

.

.6

8

1'0

12 '

1'4

time [s]

Fig. 12. Power response to a step in field voltage for the system with Pg = 0.9 p.u. and Pd - 1.2 p.u. for static load and for dynamic load with Tp =0.1 s and rtpt/rtps = 12.5.

illustrate the discrepancies that can occur if a dynamic load is modeled as being static. The effect of load modeling on the damping of power system oscillations is illustrated in Figs. 11 and 12. A static load model was used for the better damped case in Fig. I l (Case A) and for the worse damped case in Fig. 12 (Case B). The effects of variation of the static voltage exponent nps for constant transient voltage exponent npt and for different values of time constant Tp are presented in Figs. 13 and 14 for Case A. The arrows on the root loci denote the direction of movement of the electromechanical mode of the system with increase in rips. It can be seen that an increase in the static voltage exponent causes an increase in damping. This is in agreement with results given in Ref. [20]. The sensitivity to steady-

59

J.V. Mi/anovi/', I.A. Hiskens /Electric Power Systems Research 33 (/995) 53-61 jw [rad] 6.19

jw [tad] .......

,

nps 6.16

i. . . . .

~

=

i.......

l ........

r----r

0-2

6.E

/Tp=Os /

npl- 1

6.5E

\

/

6.5 6,14

~

'

6.45

/

6.4 6,1;

6.35

6. ~ ~

/

Tp-ls

6.3 6.25 6.2

6,08

6,15

6,

o~

,.

,

-04

-0.39

-0.

~

.

.

-0.37

-0.3e

.

.

.

-0.35

-0.34

-0.33

-0.32 damping [p.u.]

Fig. 13. Root locus of the system's electromechanical mode for Pg = 0.9 p.u. and Po = 0.6 p.u. for different rtps values, with npt = I. jw [rad] 6.1~] 6.1q

,

f

i

,

/

n p s = 0 - 2.5

/

npl - 2.5

,

,

Tp=os

i

J -0.4

-0.45

,__

, -0.3

,

-0.35

-0.25

Ip.u.]

Fig. 15. Root locus of the system's electromechanical mode for Pg = 0.9 p.u. and Pd = 1.2 p.u. for different /'/ps values, with np~ = 1. jw [radl 6.7

---,-nps - o- 2.5

6.6

/

/

6,1

, ,-Tp=ls

, -

, ....

npt - 2.5

6.1, 6,5 6.4,2

.~Tp-0.2s 6.4

6.06

6.3

-Os

6.06 6.2

6.04

/ / Tp-~

6.n~l

'E..42

vp-0.3~ " i ' ~

I

I

I

I

I

I

-0.4

-0.38

-0.36

-0.34

-0.32

-0.3

-0.28 damping [p.u.]

-065

.06

-0.55

-05

i

-0,5

i

-04

i

-035

=

-03

-0.25

damplng [p.u,]

Fig. 14. Root locus of the system's electromechanical mode for Pg = 0.9 p.u. and P~ = 0.6 p.u. for different rtps values, with rtpt ~ 2.5.

Fig. 16. R o o t locus o f the system's e]ectromechanical mode for P~ = 0.9 p,u. and P~ = 1.2 p.u. for different np~ values, with npt = 2.5.

state voltage exponent variation depends on time constant Tp. The electromechanical mode is more sensitive to steady-state voltage exponent variation for time constants in the neighborhood of 0.1 s, i.e. the time constants which give the largest phase shift through the load. Note that the loci intersect at the point where r/ps = rtpt. At such a point the load becomes effectively static, i.e. independent of Tp, with the effective voltage index given by nps. However, the effects of an increase in the steadystate voltage exponent on the damping of oscillations for Case B are different. They are shown in Figs. 15 and 16, where again the arrows on the root locus plots denote the direction of movement of the electromechanical mode of the system with increase in r/ps. Whilst for Tp = 0 s (which corresponds to static load representation), an increase in nps results in an increase in damp-

ing, for other values of Tp an increase in t/ps causes a decrease in damping. This can be explained by considering the different phase shifts through the system for Cases A and B [15]. The sensitivity to r/ps variation is again related to the phase shift through the load. This predicts maximum sensitivity for time constants in the neighborhood of 0.2 s. The effects of variation of the transient voltage exponent npt on the damping of oscillations are presented for Case B in Figs. 17 and 18. It can be seen that an increase in r/pt c a u s e s an increase in damping for fixed T v and different nps (Fig. 17) and for fixed nps and different Tp (Fig. 18). This is due to the increase in the ratio r/pt/nps which is directly related to an increase in phase shift through the load (see Fig. 4). The movement of the root loci toward lower damping with increase in nvs (Fig. 17) is due to a decrease in the ratio npt/np~, and

J.V. Millanovi(', I.A. Hiskens .Electric Power Systems Research 33 (1995) 53 61

60 jw [radJ 6.5E

,

--

Jw [rad] 6.18--

----~

,

,

~r-

, .

,

,

10S

6.16 6.14

,p,.o-26

6.35 6.3 6.08

6.25

6.06

6.04

6.15 f 6"-0

i

I

-0.6

-0,5

i

i

-0.4

i

-0.3

-0.2 -0.1 damping [p.u.]

Fig. 17. Root locus of the system's electromechanJca] mode for Pg = 0.9 p.u. and Pa = 1.2 p.u. for different npt values, with T, = 0.2 s. }w [rad] 6.7 6.6

~ 1 6

6.4

6.3

6.21 nptnps.1 = O- 2.5

64.;6

.0'.5

i

-0,6

£4

-0.35 '

-013

-0.26'

-0'.2

-0.15

damping [p.u.]

Fig. 18. Root locus of the system's electromechanical mode for Pg = 0.9 p.u. and P j = 1.2 p.u. for different npt values, with nps = [.

hence the phase shift through the load. In these figures, the arrows on the root loci denote the direction of movement of the electromechanical mode of the system with increase in npt. The effects of rtpt variation for Case A are presented in Fig. 19. It can be seen that, depending on the value of Tp, an increase in r/pt c a n provoke either an increase or a decrease in damping. This depends on the actual correlation between load and power system gains and the phase shift for particular time constants

Tp. 5. Conclusions

A generic nonlinear dynamic load model has been used to investigate the influence of variation of its parameters on the damping of electromechanical oscillations. It is shown that the load can be represented as a

6.0. 4

i

-0.39

t

-0.38

I

-0.37

i

-0.36

i

-0,35

i

-0.34

-0.33

damping [p.uJ

Fig. 19. Root locus of the system's electromechanical mode for Pg : 0.9 p.u. and Pd = 0.6 p.u. for different npt values, with nps : I.

lead/lag block. The parameters of the load model have a significant influence on the frequency response of the load. The paper further shows that loads which respond dynamically to voltage variations can have an influence on the damping of electromechanical oscillations. It has been found that, depending on the load and system parameters, a dynamic load can reinforce oscillations and so cause a deterioration in damping. It is also possible, though, that the load may oscillate out of phase with the system and so lead to an improvement in damping. A modified Heffron-Phillips model used for this analysis enables, by appropriate choice of load model parameters, simulations to be made which match the classical Heffron Phillips model, as well as simulations which include the effects of load dynamics. The analysis presented was performed on a simple single-machine infinite-bus system with one local nonlinear dynamic load. The results obtained are encouraging as they help to explain the discrepancies that currently exist between measured and predicted levels of damping of electromechanical oscillations in power systems. Further work on multimachine systems with multiple dynamic loads is in progress.

References [1] F.D. deMello and C. Concordia, Concepts of synchronous machine stability as affected by excitation control, IEEE Trans. Power Appar. Syst., PAS-88 (1969) 316-329. [2] J.E. Van Ness, F.M. Brasch, Jr., G.L. Landgren and S.T. Naumann, Analytical investigation of dynamic instability occurring at Powerton station, IEEE Trans. Power Appar. 3),st., PAS-99 (1980) 1386-1393. [3] G.C. Verghese, I.J. Perez-Arriaga and F.C. Schweppe, Selective modal analysis with applications to electric power systems, Part

J.V. Milanovi(, I.A. Hiskens ~Electric Power Systems Research 33 (1995) 53-61

[4[

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[I 2]

lI. The dynamic stability problem, IEEE Trans. Power Appar. Syst., PAS-IOI (1982) 3126-3134. E.V. Larsen and D.A. Swann, Applying power system stabilizers, Parts I, II and llI, IEEE Trans. Power Appar. Syst., PAS-IO0 (1981) 3017-3046. M. Klein, G.J. Rogers and P. Kundur, A fundamental study of inter-area oscillations in power systems, IEEE Trans. Power Syst., 6 (1991) 914-921. T. George, G. Hesse, A. Manglick and C. Parker, Options for an interconnection between the power systems of Queensland and New South Wales, CIGRE Regional Meet. South-East Asia and Western Pae!fic, Gold Coast, Qld., Australia, 1993, Paper No. 7.4. R. Roman-Messina and B.J. Cory, Enhancement of dynamic stability by coordinated control of static VAR compensators, Electr. Power Energy' Syst., 15 (1993) 85-93. L. Wang, A comparative study of damping schemes on damping generator oscillations, IEEE Trans. Power Syst., 8 (1993) 613 619. F.P. de Mello, Exploratory concepts on control of variable series compensation in transmission systems to improve damping of intermachine/system oscillations, IEEE Trans. Power Syst., 9(1) (1994) 102~ 108. B.R. Korte, A. Manglick and J.W. Howarth, lnterconnection damping performance tests on the South-East Australian power grid, Colloq. CIGRE Study Committee 38, Florian6polis, Brazil, 1993, Paper No. 3.7. R.H. Craven and M.R. Michael, Load representations in the dynamic simulation of the Queensland power system, J. Eleetr. Electron. Eng. Aust., 3 (1983) 1-7. IEEE Task Force, Load representation for dynamic performance analysis, IEEE Trans. Power Syst., 8 (1993) 472 482.

61

[13] D.J. Hill, Nonlinear dynamic load models with recovery for voltage stability studies, IEEE Trans. Power Syst., 8 (1993) 166-176. [14] D. Karlsson and D.J. Hill, Modeling and identification of nonlinear dynamic loads in power systems, 1EEE Trans. Power Syst., 9(1) (1994) 157-166. [15] J.V. Milanovi~ and 1.A. Hiskens, A modified Heffron Phillips model with local nonlinear dynamic load, Teeh. Rep. No. EE9378, Department of Electrical and Computer Engineering, University of Newcastle, Australia, Dec. 1993, [16] J.V. Milanovi6 and I.A. Hiskens, Effects of load dynamics on power system damping, 1EEE PES Summer Meet., San Francisco, CA~ USA, 1994, Paper No. 94 SM 578-5 PWRS. [17] D.J. Hill, I.A. Hiskens and D. Popovi6, Stability analysis of load systems with recovery dynamics, Electr. Power Energy Svst., 16 (1994) 277-286. [18] W.G. Heffron and R.A. Phillips, Effect of a modern voltage regulator on underexcited operation of large turbine generators, Trans. A1EE, 71 (1952) 692-697. [19] C.D. Vournas and R.J. Fleming, Generalization of the Heffron Phillips model of a synchronous generator, 1EEE PES Summer Meet., Los Angeles, CA, USA. 1978, IEEE~ New York, pp~ 534-0/1 -7. [20] T.S. Bhatti and D.J. Hill, A multimachine Heffron Phillips model for power system with frequency- and voltage-dependent loads, Eh'etr. Power Energy Syst., 12 (1990) 171 182. [21] P.M. Anderson and A.A. Fouad, Power System Control and Stability, Iowa State University Press, Ames, IA, 1977. [22] A.R. Bergen, Power Systems Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1986.