Effects of load modelling on analysis of power system voltage stability

Effects of load modelling on analysis of power system voltage

stability T J Overbye Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana. IL 61801. USA

Voltage stability is a major concern for viable system operation. Much of the work in this area has focused on the use of the P - V curve as a tool for system voltage security assessment. This paper examines the effects of different load models upon this analysis method and discusses some of its limitations. Both static and dynamic load models are considered. The relationship between the nosepoint of the curve, the point of system bifurcation, and the stability of different regions of the P - V curve are also examined. It is shown that the load model has a significant effect in the determination of system voltage security. In particular, the paper shows that whether a system loses stability by a Hopf bifurcation is very dependent upon the load dynamics. Additionally, an analysis technique is discussed for quantifying the voltage security of a power system. Keywords: dynamic voltage stability, load modelling, power system bifurcation

I. I n t r o d u c t i o n The problem of voltage instability in electric power systems has received much attention over the last several years, with much of the current work summarized in References 1 and 2 and references therein. While much progress has been made in the understanding of the voltage instability problem and its simulation, the development of analytic tools to aid in the quantification of voltage stability has proven challenging. This is partly due to the many different time frames of the underlying system dynamics. These dynamics range from the relatively fast dynamics of, for example, the generator excitation system and some loads, to the mid-range dynamics of LTC transformers and generator reactive power limiters, to the longer term dynamics associated with thermostatically controlled loads and average system load variation. Received 26 April 1993; accepted 12 January 1994

Volume 1 6 Number 5 1 994

One of the traditional methods for analysing the voltage security of a system has been the use of the P - V curve 3'2. A P - V curve is typically generated by performing a series of power flow solutions in which the real power load at a bus or buses is specified as an independent variable and the bus voltage magnitude as a dependent variable. The voltage is then plotted as a function of the real power load. Because of widespread familiarity with power flow analysis techniques, the P - V curve has proven to be a convenient and useful analysis tool, particularly when dealing with static, constant power loads. The need to include the voltage dependency and dynamics of the loads when studying the voltage stability problem has been increasingly recognized 1'4. However the inclusion of such models has often resulted in confusion concerning topics such as system bifurcation, stability and the use of energy methods, particularly when relating such topics back to the familiar P - V curve. Therefore, the goal of this paper is to show how different load models affect voltage stability and subsequently how they affect P - V type analysis. Additionally, it is shown that the mechanism by which an increasingly stressed system loses stability (i.e., by either a saddle node or Hopf bifurcation) is very dependent upon the load dynamics, even when detailed generator dynamics are modelled. Finally, there is a brief discussion of an analytic technique for quantifying system voltage security and to provide the user with an easily interpreted security margin.

II. P o w e r system I o a d a b i l i t y - s t a t i c load models Consider the simple two-bus system shown in Figure l. The real and reactive power balance equations at bus 2 are then given by 1

0 = --PL(2, I/"2)-- -- V2V1 sin(02 -- 01)

(la)

X

1

0 -- --QL(2, V2) + - V2V1 cos(02 - 01) - 1 (V2)2 (lb) X

0142-061 5/94/05/0329-10 Q 1994 Butterworth-Heinemann Ltd

X

329

Effects of load modelling on analysis of voltage stability. T. J. Overbye performing a series of power flow solutions as 2 is gradually increased. The resultant curve would then be identical to Figure 2. However there are a number of significant differences between the two cases. First, with a purely resistive load model the power flow equations no longer have two solutions for a given 2. This can be seen by noting that with purely resistive load the system is linear and therefore has a single, unique solution. Thus, for example, while points x H and x L are still associated with the same MW load, they are not both power flow solutions for the same system. They require a different number of the constant resistive devices to be connected to the system and are therefore associated with different system topologies. The second difference is that with resistive loads there is no maximum number of devices which can be connected to the system; the nonphysical loss of solution situation is avoided. For a more general static load model of the form

x=01

/-01 =

10/--0"

,Vz)+ I

Qk{k,Vz)

Figure 1. Two-bus system diagram

0.8 !~ 0.6 "-6 > 0.4 .~ 0.2

100

200

300

400

500

P(V) + j Q ( V ) = PL V kp +

600

Real power load (MW)

Figure 2. P - V curve

with the real and reactive load a function of both the voltage magnitude at the load bus and an independent parameter 2. Initially assume that QL(2, 1/2) = 0 and that PL(2, I/2) has no voltage dependence and is a linear function of 2 PL(2, 1/2) = 2P0

(2)

where Po could be thought of as the amount of power consumed by a single constant power device, and 2 as the number of such devices connected to the system. The P - V curve for the system, shown in Figure 2, is then generated by performing a series of power flow solutions for a gradually increasing value of 2. The P - V curve shows the long recognized fact that for a given 3~, the power flow equations (1) may have two solutions. For example, for a load of 250 M W the two solutions are shown as points x H and x L. As the load is increased these two points move closer together, eventually coalescing for some 2 = 2*. Thus when the load is modelled as consisting solely of devices with constant power demand Po, Figure 2 implies that there is a finite limit, 2", on the number of such devices which can be connected to the system. The presence of such a limit raises the question of what would happen if one were to try to connect more than 2* of these constant power devices to the system. The lack of a real solution of the algebraic power flow equations (1) for a case that is certainly physically plausible implies a shortcoming in the load model. Additionally, at 2* the power flow Jacobian is singular, and hence the convergence characteristics of a standard Newton-Raphson power flow would be poor in the vicinity of this singularity. Now consider the case in which the load is purely resistive PL(2, V2) = ;.(V2)2Go

(4)

it is shown in Reference 5 that, provided PL > 0 and kp, kq > l, the power flow will exhibit at least one solution. Of course, once the nose point is passed, the switching in of additional loads results in a net decrease in total demand. While this may not be a desirable operating point, its occurrence is not precluded by the model. The third difference between the cases is that with the resistive load, the power flow Jacobian (assumed here to incorporate the voltage dependent load model) is not singular at the nose point, or any other point (except in the limiting case as 2 -* oo). A Newton-Raphson type power flow should have no convergence difficulty. Before covering the last difference between the two load models, it is important to note that much of the ambiguity associated with using a P - V curve for static loads models other than constant power can be mitigated if the voltage is plotted as a function of 2, rather than power. This is shown in Figure 3 for the cases of constant power load (kp = 0), constant current load (kp = 1), resistive load (kp = 2), and a mixed case of 50% constant power load and 50% resistive load: PL()., I/2) =

2[O.5(V2)2Go + 0.5Po]

(5)

Note that even though the curves shown in Figure 3 for the various static load models are quite different, they have exactly the same P - V curve (i.e., Figure 2). A diagram showing the variation of a dependent variable (e.g. V) as a function of the independent variable

Constant power /~- Constant current ] ~ Resistive

0.8

~.dc- 50/50 power/resistive,

a~

/i

0.6 ~

0.4

/': a

/

/ I

0.2

(3)

where GO could be thought of as the conductance of a single device and 2 is again the number of such devices connected to the system. The P V curve for this new load model could then also be generated by again

330

jQLVkq

5O0

1000

1500

2000

L o a d c o e f f i c i e n t /~

Figure 3. Load coefficient 2

voltage

as

a

function

of

load

Electrical Power & Energy Systems

Effects of load modelling on analysis of voltage stability: T. J. Overbye

~

0.8

m I~ 0.6 o

> 0.4 ..... ~....... :,.c~..................+................

,-.1 (1.2

:J.i./

±'~",~ 0

, , , t , , , ,

100

200

300

...... 400

~, , , 500

600

L o a d bus M W load

Figure 4. P-V curve and bifurcation point for mixed load case

2 is often referred to as a bifurcation diagram 6. The presence of multiple solutions for a given 2 (that is, for the same number of devices connected to the system) can then be readily determined from the bifurcation diagram. The point where two solutions join together is known as a saddle node bifurcation (or turning point) 6:'8. At this point the power flow Jacobian is singular. Thus for the constant power load there are two solutions for 2 < 500; at 2 = 500 the system experiences a saddle node bifurcation (labelled as point 'A' on the figure) at which both solutions vanish. For the constant current load there is just a single solution for 2 ~< 1000 and no solution for larger 2, while for the resistive load there is a single solution for all values of 2. The mixed load case resembles the constant power case in that there are two solutions for 2 less than some critical value, with these two solutions coalescing in a saddle node bifurcation (labelled as 'B' in Figure 3). However this saddle node bifurcation does not occur at the point of maximum real power load. Rather, as shown in Figure 4, the saddle node bifurcation point is now on the lower portion of the P - V curve. Thus it is incorrect to equate the nose point of the P - V with the point of saddle node bifurcation, unless one is only considering a constant power load. For load models other than constant power, it is quite likely that the bifurcation point will be somewhere on the lower half of the curve. It is, however, possible that for some realistic load models the bifurcation point can even be on the upper half of the curve. For example in Reference 9 it is shown that recently-developed air conditioning equipment, with load characteristics that decreasing voltage results in increased power demand, contributed to the 1987 voltage collapse in Tokyo. Thus this type of load would be modelled using equation (4) with a negative value of kp. With, for example, kp = - 0.5, Figure 5 shows that the bifurcation point (labelled as point 'C' in the figure) is actually on the upper half of the curve at a value of about 490 MW. The last difference between the various static load models concerns the stability of the power flow solution. There has often been a lot of ambiguity concerning the stability of different regions of the P - V curve, particularly the bottom half of the curve. This is in part due to two different definitions for stability commonly applied to this problem. The first definition, referred to here as voltage stability, is mostly concerned with the static aspects of the problem. In Reference 2 voltage stability

Volume 16 Number 5 1994

is defined as 'the ability of a system to maintain voltage so that when load admittance is increased, load power will increase, and so that both power and voltage are controllable'. This type of voltage stability is often determined using power flow based sensitivity measures 1°'11. Based upon this definition, the upper portion of the P - V curve is considered stable, while the entire lower portion of the curve is usually declared to be unstable. The second type of stability usually associated with the voltage security problem is small signal (or steady state) stability 12. The small signal (SS) stability of a power system equilibrium point is determined by linearizing the system dynamic model about an equilibrium point and then calculating the system eigenvalues. If there are no eigenvalues with positive real parts the system is stable, otherwise it is unstable. In contrast to the definition of voltage stability, in order to discuss SS stability it is imperative to model at least some system dynamics. Typically the power system is modelled by a set of differential-algebraic equations (DAEs) of the form f =

f(x,

0 =

g(x,

y)

(6)

y)

(7)

where x is the vector of the state variables and y is the vector of the algebraic variables. The steady state stability of an equilibrium point is determined by first linearizing (6) and (7) about the equilibrium point

(8)

D ILAyJ and then eliminating the algebraic variables A~ = ( A - B D - XC)Ax --

(9)

AAx

(10)

The eigenvalues of ~, then determine the small signal stability of the equilibrium point. For the case considered in this section of static load models (dynamic load models are introduced in the next section), all of the dynamics and state variables are associated with the generators. If the generator at bus 1 is modelled using the two-axis model with an IEEE type I exciter 13, it is shown in Reference 13 that with a constant impedance load the system is SS stable over the entire P - V curve. It is important to note, however, that with static load models the load voltage magnitude and angle

•~ 0.8 i " ......... ~

.

C

I~ 0.6 i. l--Highvoltagesolution ..... /;i. I ""L°w v°Itages°luti°~ // ¢¢ 0.4

)........... .. ! <.

0.2 0

0

100

200

300

400

500

600

L o a d bus M W load

Figure 5. P-V

curve

for

real

power

load

with

kp = - 0 . 5

331

Effects of load modelling on analysis of voltage stability. T. J. Overbye

,

B

-*-

0.8

0.6

i

0.4

0.2 , , .... ........ 100 200 300

, 400

, ,; .... 500 600

Load power (MW)

Figure 6. P-V curve stability regions for constant power loads

When the load is 477 MW (point 'C') the system regains stability when the other eigenvalue returns to the left hall plane. The entire bottom portion of the P V curve is stable. Thus SS stability results obtained using eigenvaluc analysis differ substantially from the results obtained based on the definition of voltage stability. While the presence of a Hopf bifurcation on the upper portion of the P - V curve is not unexpected 15, its appearance at such a relatively low loading is somewhat disconcerting and does not appear to match the results obtained in actual operation. This, coupled with the apparent SS stability of the lower portion of the P - V curve and the nonphysical nature of the constant power load models, suggests that the use of dynamic load models needs to be examined.

III. Dynamic load models

i,

2 :oj

"~

0

ta

c

-1

0

1

2

3

4

E i g e n v a l u e real component

Figure 7. Variation of critical eigenvalue in complex plane

In general dynamic load representation has received relatively little attention, with most of the industry still using static models 16. This is partly because in many transient stability studies the inclusion of dynamic load models with moderately loaded systems has little impact upon stability iv, and partly because of the difficulty in developing and verifying dynamic models for an aggregate, time-varying load. In this section it will be demonstrated that for the voltage stability problem the use of dynamic load models has a significant impact on the assessment of system voltage stability. Two different types of dynamic models are considered. First, let the real power load be represented with both voltage magnitude and frequency dependence ls'19 and the reactive load with dependence upon voltage magnitude and its range of change ~20,z~:

are algebraic rather than state variables. Thus one cannot talk about the system response to a small perturbation in, for example, load voltage magnitude; the load bus voltages are constrained by the algebraic power balance equations. This restriction is removed when dynamic load models are considered. When a constant power load model is used, the P - V curve contains (as shown in Figure 6) a number of different stability regions which can be determined by eigenvalue analysis. For the upper half of the curve, Figure 7 shows the variation in the critical eigenvalues as the load is varied (with the arrows indicating the direction of eigenvalue movement as the load is increased), while Figure 8 shows the variation in just the real component of the critical eigenvalues (for high and low voltage solutions) as a function of the load; these figures are similar to those shown in Reference 13. The system is stable from an initial load of 0 MW (point 'O') until a load of about 236 MW (point 'A'). At 'A' the system becomes unstable when a pair of complex eigenvalues cross the jo~ axis (i.e., a Hopf bifurcation). Participation factor analysis indicates that this instability is primarily associated with the E'q/Rf pair of the generator/exciter. These two eigenvalues then rejoin on the real axis when the load is 327 MW (point 'B'). The two eigenvalues then travel in opposite directions on the real axis, with the right-most eigenvalue re-entering the left half plane by passing through infinity when the load is 377 MW (i.e., a singularity induced bifurcation1*). 332

~.(Vi, 2)

dp,(2)O,

(lla)

Qe, = Q,(V,, 2) + do,(),)(/,

(llb)

eLi =

-}-

This model takes into account the fact that the real power load is often dependent upon bus frequency, while the reactive load model gives a simple, though somewhat approximate, representation of the behaviour of a third-order induction motor model with flux dynamics. In Reference 22 a similar model is derived for modelling large (greater than 150 h.p.) induction motors, while in Reference 23 a model is presented which also includes a term dependent upon the time derivative of P,(V,, 2). The

g

--Low volta,go: solution}

1

-

0

0

100

200

300

400

500

600

Load power (MW)

Figure 8. Variation eigenvalue

in real component

of critical

Electrical Power & Energy Systems

Effects of load modelling on analysis of voltage stability." T. J. Overbye values of de/(2) and doi(2 ) are assumed to be strictly positive and would normally be dependent upon 2 (the amount of load connected to the system). This model is identical to the load dynamics which have been used for energy function analysis of voltage instability 24. With this dynamic model, the voltage magnitude and angle at the load bus are now state variables. For the two bus system with no generator dynamics model (i.e., bus 1 is an infinite or slack bus) the system differential equations (6) can be written by substituting (11) into (1) 1 VzV1 sin 021 (12a) d p 2 ( ) , ) O 2 = _ P ( V 2 , ~.) X

dQ2(2)l;"2 = -Q(V2, 2) + 1 V2V1 cos 021 - 1 ([72) 2 X

X

(lZb) and written in matrix form

as

2)

L

-Q(V2,

X

1 V2V1

)t) +

V2VI sin 021 cos 021

X

1 (V2)2 X

(13)

where D is the positive diagonal matrix

D=

de2(;.)

0

do2( )

The small signal stability of the system can then be found as before, equations (6)-(10), by linearizing the system dynamic model about an equilibrium point and calculating the eigenvalues of,~. However since the model now has no algebraic equations, ~ = A, with A defined for a fixed value of 2 as A = DJ(x)

•

~....

0.8

(14)

where J(x) is the power flow Jacobian. Since D is a positive diagonal matrix, premultiplying J(x) by D will not change the signs of the eigenvalues of J(x). Therefore the SS stability of an equilibrium can be determined from just the eigenvalues of the power flow Jacobian. This concurs with the result that for certain modelling assumptions (see, for example, Reference 28) the stability of a power flow solution can be determined from the eigenvalues of the Jacobian of the algebraic power flow equations 25'26'27'28. With this model it is straightforward to show that with no voltage dependence in the Pi(2)/Qi(2) terms the upper half of the P V curve is stable and the lower half unstable, and that with voltage dependence the high voltage solutions in Figures 4 and 5 are stable while the low voltage solutions unstable. However because of the uncertainty in the actual values of the elements of D, the magnitudes of the eigenvalues of the power flow Jacobian cannot be directly related to the magnitudes of the eigenvalues of A. This dynamic load model can also be applied with the two-axis generator model used in the last section. For example, Figure 9 shows the P - V curve for the case of a constant power load with dp(2) = 0.1 and dQ(2) = 0.1. As was the case with the static load model, the upper portion of the curve is initially stable. At 'A' (with load of 226 MW versus 236 MW for the static case) the system

Volume 16 Number 5 1994

again becomes unstable through a Hopf bifurcation. These two eigenvalues rejoin on the real axis at point 'B' (at a load of 343 MW versus 327 MW for static case) and then travel in opposite directions along the real axis. The left-most of the pair still re-enters the left half plane by passing through the origin at point 'C' (at a load 479 MW versus 477 for static case). However, in contrast to the static case, the right-most eigenvalue never reaches infinity, but remains in the right half plane not only until the nost point is reached but also for the entire bottom half of the curve. Thus the system loses stability at point 'A' and it is never regained. Table 1 shows the variation in point 'A' with respect to changes in de and d o. This point is relatively insensitive to changes in de and somewhat sensitive to changes in the value of dQ. For low values of dQ, the Hopf instability mode is primarily due to the generator/exciter E'q/Ry pair. However as the value of d o is increased, the contribution from the load bus voltage magnitude also becomes significant. A method for analytically calculating these sensitivities has recently been presented in Reference 29. Regardless of the value of dQ, the bottom half of the P-V curve remains unstable due to a single positive eigenvalue. The mode associated with this eigenvalue is dominated by the load bus voltage magnitude. The second type of dynamic load model is motivated by the fact that many loads are equipped with regulators which, when modelled in aggregate, vary the load impedance in order to maintain constant power consumption 3°. Examples of such loads include thermostatically controlled loads, motor drives, computer power supplies, and the aggregate effects of otherwise

E,El

0.6

~

0.4

.... ! ............i . . . . .

! ......:.........

....:: s d ~ b l ~ ...... ., :1711. . . . . . . . . . i

.,"

.1 0.2

0

100

Figure 9. Dynamic generator model

200 300 400 L o a d power (MW)

load

stability

500

600

with

two-axis

Table 1. Variation in point'A' (in MW) with respect to dp and dQ dp

dQ

0.05

0.10

0.50

1.00

2.00

0.05 0.10 0.50 1.00 2.00

231 227 208 196 189

229 226 207 195 189

228 224 206 194 188

228 224 206 194 188

228 224 206 194 188

333

Effects of load modelling on analysis of voltage stability. T. J. Overbye

. ~ .~_

.,

( ) ....

o

~> -1 ._~ -2

. . . .

-1

i

.

.

.

.

-0.5

.

.

.

.

.

.

0

.

0.5

E i g e n v a l u e real component

Figure 10. Variation TG2 = 0.4 s

in

critical

eigenvalues

for

unmodelled distribution LTC transformers. These types of loads can be modelled as a variable conductance and susceptance with a first order delay3°'3t'4:

T~,C, = P,

(15a)

-- (Vi)2Gi

T~i[3i = Qi - (Vi)2Bi

(15b)

where TG, and TB, are the time constants associated respectively with the conductance and the susceptance. For static analysis, where one is only concerned with the equilibrium points for which the left-hand side of (15) is zero, this is identical to a constant power model. However for dynamic analysis, the system response will be highly dependent upon the values of the time constants. This dependence is to be expected, since as the time constants approach zero the loads tend toward constant power, while as they approach infinity the loads tend toward constant impedance. Their actual values are dependent on the load being modelled and can vary substantially from significantly less than a second for converter controlled loads to many minutes for thermostatic loads and for the response of LTC transformers. For an actual load test reported in Reference 31 the time constant is on the order of 100 seconds. For the case without generator dynamics, the equations for the two-bus system consist of two differential and two algebraic equations: T G 2 d 2 = P2 - - ( V 2 ) 2 G 2

(16a)

TB2B2 = Q 2 -

(16b)

(V2)2B2

1 0 =

--(V2)2G2

--

V2V 1

sin 021

power consumed at the bus, with the subsequent result that (16a) causes G2 to increase back towards the original value; likewise an upward perturbation results in an increase in power with a subsequent decrease in G2 to the original equilibrium point. However, on the bottom half of the curve a perturbation causing G2 to decrease (move to the right) results in an increase in real power which further decreases G2, with the dynamic trajectory following the P - V curve around to the upper half equilibrium. An increase in G2 causes a decrease in real power which results in a further increase in G2, moving the operating point toward zero. Thus the SS stability results using this model are consistent with the definition of voltage stability from Reference 2. Next, the effects of this load model are discussed when it is used with the detailed two-axis generator/exciter model with an IEEE type I exciter (i.e., the dynamic generator model from Reference 13). Of particular concern are the model's effects upon the initial loss of stability through the Hopf bifurcation and the stability of the lower half of the curve. As the value of T(~2 is increased, the onset of the Hopf bifurcation is pushed farther out on the P - V curve, from 236 M W for the case with T~2 = 0 (i.e., the constant power model shown in Figure 7) to 372 MW for To2 = 0.59 s. The maximum positive real value attained by this eigenvalue pair as the load is varied also decreases as Tc2 is increased. For example, Figure 10 plots the variation in the critical eigenvalues of the system as the load is increased for To2 = 0.4 (with the arrows again indicating the direction of eigenvalue movement as the load is increased). Note the decrease in the maximum positive real value obtained by comparing Figure 10, in which the maximum real value of the eigenvalues is about 0.25, with the constant power model used with Figure 7 where the maximum value is greater than 3. For sufficiently larger TG2 (greater than about 0.6) the system no longer experiences a Hopf bifurcation; the critical eigenvalue pair never reaches the right half plane. Rather, the variation in the critical eigenvalues is as shown in Figure 11, with the eigenvalues initially approaching the right-hand plane as the load is increased, but then looping back to the left for further increases in load. Stability is now eventually lost at a much higher load (479 MW for T~2 = 0.6) when the eigenvalue which is most associated with the load dynamics passes through the origin. This load level is very close to the saddle node bifurcation of the power flow equations (i.e., singularity of the power flow

(16c)

X

= -(V2)2B2 +

1 X

V2V 1 cos 021 --

1

(V2)2 (16d)

The equilibrium points of (16) are then just the power flow solutions for a constant power load of Pz/Qz. The steady state stability of this set of algebraic-differential equations is then determined using (16) and (8) to (10). Eigenvalue computation indicates that the upper portion of the P - V curve is stable, while the lower half is unstable. Intuitively this stability can be verified (with reference to the P - V curve in Figure 2) by first noting that the algebraic equations constrain the operating point to always being on the P V curve. At an equilibrium point on the upper half of the curve, a perturbation causing G 2 to decrease (move to the left) results in a decrease in

334

2

X

g --~ rz~

-2

-3

. . . .

-1

i

,

~

-0.5

. . . . .

0

t

,

,

,

0.5

E i g e n v a l u e real component

Figure 11. Variation

in

critical

eigenvalues

for

[G2 ~--" 0 . 6 S

Electrical Power & Energy Systems

Effects of load modelling on analysis of voltage stabifity." T. J. Overbye 18kV

230kV

230 kV

jo.tM,25

I

I

o.mm.io.o I0. I045

(~)le/2ao

13.8 kV

,0rl(~)G

rJo/~ 3.a

® o

g o o

o .

.2

-

Load A

i

23o'.,,

o

16.5k / ~ )

Figure 12. Nine-bus three-machine system

Jacobian) with a constant power load (500 MW). The bottom portion of the P-V curve is unstable with a single positive eigenvalue for To2 > 0 ; the mode of this instability is also associated with the load dynamics. The diversity of an actual aggregate load is approximated by modelling the load as two components: one with a very fast time constant power loads (TG2 = 0.05 s), and the second with a much slower time constant (T~2 = 5 s) to represent the slower responding portions of the load, such as thermostatically controlled devices. Whether the system experiences a Hopf bifurcation and at what load level stability is lost is then shown to be dependent upon the load mix. For a ratio of 90% fast to 10% slow a Hopf bifurcation occurs at 250 MW. This increases to 270 MW for an 80/20 mix, to 303 MW for 70/30, and to 358 for 60/40% fast to slow. For a 50/50 load mix the system no longer experiences a Hopf bifurcation; rather, stability is lost at 479 MW by a single eigenvalue passing into the right-hand plane through the origin. This is again close to the power flow saddle node bifurcation load level. This section has examined the effects of two different dynamic load models on system voltage stability. It has been shown that load dynamics can have a tremendous impact on system stability. For both dynamic models the lower half of the P - V curve has been shown to be unstable. The upper half of the curve is stable for sufficiently low loading. Whether the system experiences a Hopf bifurcation at higher loading depends upon the load characteristics. For the first model, the Hopf bifurcation occurred at a relatively low loading, regardless of the values of dp and dQ. For the second model, the presence of a Hopf bifurcation is quite dependent upon the value of TG and the load mix. For cases without a Hopf bifurcation the system loses stability very close to the point of singularity of the power flow Jacobian. The absence of Hopf bifurcations at low loadings in practice along with actual load tests 3x suggests the first order variable admittance as a viable load model for at least a class of loads. This provides justification for the assumption that voltage instability is primarily due to network and load considerations, rather

Volume 16 Number 5 1994

than due to the effects of generator dynamics. In the next section the voltage stability of a larger system is analysed using this model.

IV. Voltage security assessment Similar results can be obtained using the first order variable admittance model on larger systems. In this section the load model is used with the nine-bus, three-machine system from References 32 and 13; all machines are represented using the detailed two-axis with IEEE type I exciter model. It is shown that for sufficiently large values of T~, and Tm the system loses stability through an eigenvalue passing through the origin with a loading very close to the point of power flow Jacobian singularity. For such a scenario it is demonstrated that energy methods provide an excellent method of assessing proximity to voltage collapse. For the nine-bus system shown in Figure 12 assume the three loads (at buses 5, 6 and 8) are modelled using the variable admittance model (15) and both real and reactive loads are parameterized as a linear function of 2 (i.e., uniform load participation). Figure 13 shows the variation in bus 5 voltage magnitude for the high and low voltage solutions as a function of total system load. To determine the effect the values T~, and Ta, have upon the mechanism and load at which the system loses stability, eigenvalue analysis was performed for various combinations of T~, and Tm as the total system load was increased. The results are summarized in Table 2, which shows that the dependence of system stability on the values of TG, and Tat is similar to that of the two-bus case. For low values of T~t and Tat the system loses stability through a Hopf bifurcation, with this point getting pushed further out on the P - V curve as their values are increased. For example Figure 14 shows the variation in the critical eigenvalues for the case of TGt/TB, = 0.3. The system experiences a Hopf bifurcation at a loading of 704 MVA as the eigenvalue pair crosses the j~ axis into the right half plane (these eigenvalues cross back into the left half plane slightly before the system experiences a saddle node bifurcation at 795 MVA). Eventually, for sufficiently high values of TGt in particular, the critical eigenvalues never reach the right half plane. An example of this is shown in Figure 15 for the case of TGt/TB,--0.5. Now the critical complex eigenvalue pair reaches a maximum real part of -0.075 (at a load of approximately 760 MVA) before decreasing back to the left. Rather the system loses stability with a

1.0 0.8

B

0.6 .........I--High

voltage solueonl I--Low voltage solut/on J

~

-

•

0.4 tf"

0.2

...........

x.~,~..~ f y

0.0 3 5 0 4 0 0 4 5 0 5 0 0 550 6 0 0 6 5 0 7 0 0 7 5 0 8 0 0 Total system MVA load

Figure 13. Variation in bus 5 voltage magnitude

335

Effects of load modelling on analysis of voltage stability." T. J. Overbye Table 2. Variation in total MVA loading at which stability is lost

TGi T~

0.10

0.30

0.50

1.00

5.00

0.10 0.30 0.50 1.00 5.00

661H 674 H 698 H 735 H 758 n

684 H 704 ~ 735 ~ 788 ~ 788"

751H 795 795 795 795

795 795 795 795 795

795 795 795 795 795

H Hopf bifurcation.

1.5

1.0 t

8 0.5

0.0 ..~

-0,5 . . . .~----L..~ . . . . . . u~

-1.0

'

"''~

-"Q

-1.5 -0.5

-0.4

Figure 14. Variation TG/TB = 0.3 S

-0.2

-0.1

Eigenvalue

-0.3

real

in

0.0

0.1

0.2

component

critical

eigenvalues

power flow solution needed to calculate the low voltage solution. An advantage of using energy methods is that the determination of the measure itself only requires information based upon the current operating point of the system (e.g., the determination of the low voltage solution); there is no need to make assumptions about future load variation. This frees the method from being dependent upon a particular assumed path from the current operating point to the onset of voltage instability. Since the future variation in the loads is never known precisely (particularly in the unusual operating states usually associated with systems close to voltage collapse), the practical benefit of this independence is that it frees the method from possible errors introduced in path dependent methods when the assumed future load variation differs from the actual variation. However it is often useful to be able to translate an energy measure into an actual MW, Mvar or MVA margin to voltage instability. In Reference 35 it is shown that using a computationally trivial transformation (maximum of order n) the energy measure can be used to approximate loadability margins for a whole family of load increase patterns starting from the current operating point at which the energy is computed. For example the left-hand axis of Figure 14 shows the variation in the energy measure for the nine-bus system as the load is increased. It should be stressed that the calculation of the energy measure at a particular loading requires no assumption about future load participation

for 1.5

o

single eigenvalue passing through the origin; this point is practically identical to the value where the two power flow solutions coalesce in a saddle node bifurcation (795 MVA). Similar analysis was performed on the system using varying combinations of fast (TG,/TB, = 0.05) and slow responding load (TG,/TB, = 5.0). As was the case with the two-bus system, the requirement to avoid the Hopf bifurcation is again that a sufficiently larger percentage of the load responds slowly. For this case a mixture of 40/60% slow to fast load is sufficient. Thus for a large class of load models the onset of voltage instability can be related directly to the point of saddle node bifurcation of the power flow equations. The voltage security of such systems can be quantified by using a suitable measure to determine proximity to this point. A number of different techniques have been proposed, with most of the current work contained in Reference 1. One technique which is particularly efficient numerically in providing such a measure is the use of energy methods 24'33. In this method voltage security is quantified by using a scalar energy function to determine the distance between the high voltage solution and a low voltage solution. As the system approaches the point of saddle node bifurcation where these two solutions coalesce, this energy measure goes to zero. In Reference 34 it is shown that the voltage security of a large system can be assessed with computation equivalent to a few power flow solutions; if one is just concerned with the voltage security at a particular bus or small area, this measure can be determined with just the one additional

336

8

0.5

..~ =~

-0.5

.

-1.5 -0.5

.

.

.

.

.

. . ~ . . . . ,. -0.4 - 0 . 3

.

.

.

.

,,J ......... - 0 . 2 -0.1

.

I .... J .... , .... 0 0.1 0.2

Eigenvalue real component

Figure 15. Variation TG/TB = 0.5 s

in

critical

eigenvalues

5.0

500

)~ ?'~,~ ,o

~

for

:

[~ Energy measure

I--Estimated MVA margin

}...... ! .....

400

3.0

300 "t

2.0

zoo ~

1.0

100

0.0

0 350

400

450

500

550

600

650

700

750

800

Total s y s t e m l o a d ( M V A )

Figure 16. Variation in energy measure and M V A

margin

Electrical Power & Energy

Systems

Effects of load modelling on analysis of voltage stability: T. J. Overbye and can thus be considered an absolute measure of voltage security. The energy measure can then be used to determine an estimated MVA margin by making an assumption about load participation. For the assumption of uniform load participation, the right-hand axis compares this estimated MVA margin with the actual MVA margin. The close correlation between the curves shows that the energy method can be used to provide an accurate indication of how much additional load the system can tolerate before experiencing voltage instability by a saddle node bifurcation.

V. Conclusion The type of model used to represent the loads has been shown to have a tremendous effect upon the assessment of system voltage stability. Results such as the location of the saddle node bifurcation point, the number of power flow solutions, the presence of Hopf bifurcations on the upper half of the P - V curve, and the stability of the lower half of this curve have all been shown to be dependent upon the load model. This certainly suggests that more research is needed in the area of load modelling, particularly of aggregate loads under large voltage a n d / o r frequency deviations. More practically, the paper suggests that with the first order variable admittance load model (with a sufficient quantity of slowly responding load), the stability results obtained from the definition of voltage stability nearly match the results obtained using eigenvalue analysis. Thus, for this model the upper portion of the P - V curve is stable to (nearly) the nose point, while the lower half is unstable, with voltage collapse occurring by a saddle node bifurcation. A computationally feasible technique to assess the voltage stability of such a system has also been discussed. V l . Acknowledgements The author would like to acknowledge the support of NSF through its grant NSF ECS-9209570 and the Power Affiliates program of the University of Illinois at Urbana-Champaign. V I I . References 1 Proc. Bulk Power System Phenomena I I - Voltage Stability and Security McHenry, MD, August 1991

2 Mansour, Y (ed) 'Voltage stability of power systems: concepts, analytical tools, and industry experience' IEEE 90 THO359-2-PWR (1990) 3 Fink, L (ed) Survey of the voltage collapse phenomenon, North American Electric Reliability Council, Princeton, NJ (August 1991) 4 Pal, M K 'Voltage stability conditions considering load characteristics' IEEE Trans. Power Syst. Vol 7 (February 1992) pp 243-249 5 Lesieutre, B C, Sauer, P W and Pai, M A 'Sufficient conditions on static load models for network solvability' Proc. 1992 North American Power Symposium, Reno, NV, October 1992

6 Seydel, R From equilibrium to chaos: Practical bifurcation and stability analysis Elsevier, New York (1988) 7 Araposthatis, A, Sastry, S and Varaiya, P 'Analysis of the power-flow equation' Int. J. Electr. Power Energy Syst. Vol 3 (July 1981) pp 115-126

Volume 16 Number 5 1994

8 Kwatny, H G, Pasrija, A K and Bahar, L Y 'Loss of steady state stability and voltage collapse in electric power systems' Proc. 24th IEEE Conf. on Decision and Control (December 1985) pp 804-811

9 Kurita, A and Sakurai, T 'The power system failure on July 23, 1987 in Tokyo' Proc. 27th IEEE Conf. on Decision and Control, Austin, TX, December 1988 10 Flatabo, N, Ognedai, R and Carlsen, T 'Voltage stability condition in a power transmission system calculated by sensitivity methods' IEEE Trans. Power Syst. Vol 5 (November 1990)pp 1286-1293 11 Aiiarapu, V and Christy, C 'The continuation power flow: a tool for steady state voltage stability analysis' IEEE Trans. Power Syst. Vol 7 (February 1992) pp 416-423 12 IEEE Committee Report 'Proposed terms and definitions for power system stability' IEEE Trans. Power Appar. Syst. Vol 101 (July 1982)pp 1894-1898 13 Rajagopalan, C, Lesieutre, B, Sauer, P W and Pai, M A 'Dynamic aspects of voltage/power characteristics' IEEE Trans. Power Syst. Vol 7 (August 1992) pp 990-1000 14 Venkatasubramanian, V, Schattler, H and Zaborszky, J 'Voltage dynamics: Study of a generator with voltage control, transmission, and matched MW load' IEEE Trans. Autom, Control Vol 37 (November 1992) pp 1717 1733 15 Abed, E H and Varaiya, P P 'Nonlinear oscillations in power systems' Int. J. Electr. Power Energy Syst. Vol 6 (January 1984) pp 37-43 16 IEEE task force on load representation for dynamic performance 'Load representation for dynamic performance' IEEE PES 1992 Winter Power Meeting Paper 92 WM 126-3 PWRD (January 1992)

17 Vaahedi, E, El-Din, H M Z and Price, W W 'Dynamic load modeling in large scale stability studies' [EEE Trans. Power Syst. Vol 3 (August 1988) pp 1039-1045 18 Bergen, A R and Hill, D J 'A structure preserving model for power system stability analysis' IEEE Trans. Power Appar. and Syst. Vol 100 (January 1981) pp 25-35 19 Van Cutsem, Th. and Pavella, M R 'Structure preserving direct methods for transient stability analysis of power systems' Proc. 24th IEE Conf. Decision and Control Fort Lauderdale, FL, December 1985 20 Vu, K T 'An analysis of mechanisms of voltage instability' Proc. IEEE 1992 ISCAS San Diego, CA, May 1992, Vol 5, pp 2259-2532 21 Jimma, K, Tomac, A, Vu, K and Liu, C C 'A study of dynamic load models for voltage collapse analysis' Proc. Bulk Power System Voltage Phenomena H Voltage Stability and Security, McHenry, MD, August 1991, pp

423-429 22

Lesieutre, B C 'Network and load modeling for power system dynamic analysis' PhD Dissertation, University of Illinois, Urbana-Champaign, IL (1992)

23 Hill, D J 'Nonlinear dynamic load models with recovery for voltage stability studies' IEEE PES Winter Meeting WM 102-4 PWRS, New York, NY (January 1992) 24 DeMarco, C L and Overbye, T J 'An energy based security measure for assessing vulnerability to voltage collapse' IEEE Trans. Power Syst. Vol 5 (May 1990) pp 419-427 25

Venikov, V A, Stroev, V A, Idelchick, V I and Tarasov, V I 'Estimation of electrical power system steady-state stability' [EEE Trans. Power Appar. Syst. Vol 94 (May/June 1975) pp 1034-1041

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Effects of load modelling on analysis of voltage stability. 1-. J. Overbye 26 Tamura, T, Mori, H and lwamoto, S 'Relationship between voltage instability and multiple load flow solutions in electric power systems' IEEE Trans. Power Appar. Syst. Vol 102 (May 1983) pp 1115-1123

31 Taylor, C W 'Voltage stability part 1: Introduction, definitions, time frames/scenarios, and incidents' Surt,ev q~ the voltage collapse phenomenon NERC (August 1991) pp 51-65

27 Tiranuchit, A and Thomas, R J 'A posturing strategy against voltage instabilities in electrical power systems' IEEE Trans. Power Syst. Vol 3 (February 1988) pp 87-93

32 Anderson, P M and Fouad, A A Power system control and stability, The Iowa State University Press, Ames, IA (1976)

28 Sauer, P W and Pal, M A 'Power system steady-state stability and the load-flow Jacobian' IEEE Trans. Power Syst. Vol 5 (November 1990) pp 1374-1383

33 Overbye, T J and DeMarco, C L 'Improved techniques for power system voltage stability assessment using energy methods' IEEE Trans. Power Syst. Vol 6 (November 1991 ) pp 1446-1452

29 Dobson, I, Alvarado, F and DeMarco, C L 'Sensitivity of Hopf bifurcations to power system parameters' Proc. 31st IEEE Conf. Decision and Control Tucson, AZ, December 1992, pp 2928-2933

34 Overbye, T J 'Use of energy methods for on-line assessment of power system voltage security' IEEE PES Winter Meeting WM 121-4 PWRS, New York, NY (January 1992)

30 Graf, K-M 'Dynamic simulation of voltage collapse processes in EHV power systems' Proc. Bulk Power System Voltage Phenomena - Voltage Stability and Security EPRI EL-6183 (January 1989) pp 6.45-54

35 Overbye, T J, Dobson, 1 and DeMareo, C L 'Q-V curve interpretations of energy measures for voltage security' IEEE PES Winter Meeting WM 184-2 PWRS, Columbus, OH (February 1993)

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Electrical Power & Energy Systems