aguidetosteady-statevoltagestabilityanalysis

7/28/2019 AGuidetoSteady-StateVoltageStabilityAnalysis

1/26

A Guide to Steady-State Voltage Stability Analysis

Anthony B. Morton

November 2007

Contents1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 DC Power Flow Across Series Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . 13 AC Power Flow Across Series Impedance: Two-Bus Load Flow . . . . . . . . . . . . . 34 Approximate Solution for Voltage in terms ofP, Q . . . . . . . . . . . . . . . . . . . . 65 Sensitivity of Voltage to Reactive Power . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Graphical Representation of Stability Limit . . . . . . . . . . . . . . . . . . . . . . . . 77 Relation to Power-Angle Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Operation at Constant Power Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Remote Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1310 Effect of Long Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1611 Effect of Shunt Admittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

12 Shunt Susceptance, Limits and Voltage Sensitivity . . . . . . . . . . . . . . . . . . . . 1913 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Appendix: Derivation of the Two-Bus Load Flow Solution . . . . . . . . . . . . . . . . 23

1 Introduction

Voltage stability, broadly speaking, refers to the ability of a power system to sustain appropriatevoltage levels through large and small disturbances. Kundur [2] discusses many aspects of voltagestability and techniques for stability analysis.

The purpose of this paper is to explain in some detail one aspect of voltage stability mostcommonly encountered in discussions with utilities. This is termed steady-state voltage stabilitybecause it is assessed using steady-state load flow solutions of the network, rather than dynamical

considerations. (Despite this, it does have important implications for power system dynamics.)Steady-state voltage stability is concerned with limits on the existence of steady-state operating

points for the network. The nonlinearity of the load flow equations for an AC network placesconstraints on the maximum power flow that can occur on network branches before the equationscome to possess no solution and the network enters voltage collapse. These power flow constraintsare additional to the more familiar static constraints based on thermal limits and voltage drop, andin transmission networks are often more stringent.

This paper has been written to provide some background to the voltage stability problem, andto document exact solutions in the simplest cases. While these exact solutions are not technicallydifficult to obtain, they appear to be omitted from most discussions of steady-state voltage stability.

2 DC Power Flow Across Series Resistance

A simple picture of the basic voltage stability problem is provided by considering power transfer ina DC circuit, where there are no complex numbers to complicate matters. In Figure 1, power is

1


2/26

+VS

R

?

Load,power P

+

V

Figure 1: Transfer of power in a DC circuit

delivered from a DC voltage source to a DC load via series resistance R. If VS denotes the sourcevoltage and V the load voltage, one can immediately write the current in the circuit as

I =VS V

R(1)

and the power P transferred to the load as

P = V I =V(VS V)

R. (2)

It is convenient to express this in terms of the ratio V /VS of load voltage to source voltage magnitude,abbreviated to throughout this paper. The power equation that results is

P =V2SR

(1 ), = VVS

. (3)

Figure 2 plots the power flow P against the voltage ratio . The left hand plot illustrates thefundamental relationship (3) in the way it is usually taught to first-year engineering studentsasthe maximum power transfer law. This way of viewing it emphasises the principle of optimising thepower transfer and only the power transferas for example when transmitting a signal down a wire

in an audio or telecommunications circuit. It is then true that one maximises the signal power bymatching the input impedance of the receiver to the impedance of the line, thereby making = 1/2and attaining the peak of the power curve.

(Usually when this circuit is presented to students, the load is taken to be resistive, so that theload voltage is given by a simple voltage-divider rule. Thus the impression is sometimes gained that(3) is simply the maximum power that can be developed in a resistor. This misses the importantpoint that the relation (3) holds regardless of the nature of the load: the only assumption we used inderiving it via (1) and (2) was that the load draws real power P. So (3) is a fundamental maximisingprinciple with broad application.)

In the power system context, however, it is more instructive to view the relationship as onthe right hand side of Figure 2, with power on the horizontal axis. This is because the technical

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

Voltage ratio

PowerP,asproportionofVS2/

R

0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Voltageratio

Power P, as proportion of VS

2/ R

Figure 2: Power-voltage relationships in a DC circuit

2


3/26

VS

R jX

?

P, Q

+

V

Figure 3: Transfer of power in an AC circuit

objective in a power system is to satisfy a given demand for power while keeping both losses andvoltage fluctuations to a minimum. Viewed this way, what equation (3) does is establish a hard limitof V2S/4R on the amount of power that can be delivered across the resistance R. But it is also clearthat operating at or near this transmission limit is undesirable, because:

1. Its too inefficient. When = 1/2 the voltage drop across R is VS/2, and the I2R loss in the

line is equal to the power actually delivered to the load.

2. There is too much voltage fluctuation. This is particularly clear from the right hand side ofFigure 2: we see that the sensitivity /P of load voltage to power becomes arbitrarily largeas P approaches the limit. In practice load power always fluctuates, but large fluctuations involtage give rise to flicker in lamps and other undesirable effects.

3. The system is liable to collapse into an undesirable operating condition. Again, this is evidentfrom the right hand side of Figure 2. IfP is less than the limiting value, there are two possibleoperating points for , and these approach each other rapidly as P tends toward the limit. Ifthe load continues to draw the same power at any voltage level, operation close to the limitincreases the likelihood of dynamic fluctuations causing a transition from a desirable high-voltage, low-current operating point to an undesirable low-voltage, high-current operatingpoint.

These three considerationsparticularly the thirdsum up the reasons why power system engineersconcern themselves with steady-state voltage stability. Notice again that all three stem from thenature of the steady-state operating conditions (Figure 2 is a steady-state characteristic), and notfrom the dynamical characteristics of connected plant. Plant dynamics may exacerbate the condi-tions that drive a system into instability, but the instability is still considered to originate in thesensitivity of steady-state voltage to steady-state power flow.

3 AC Power Flow Across Series Impedance: Two-Bus Load

Flow

Voltage stability in conventional AC power systems depends on the vagaries of AC power flow, where

we must switch to phasor analysis and consider reactive power in addition to real power.Figure 3 presents the AC analogue of Figure 1: an AC voltage source feeding an AC load (with

real power P and reactive power Q) across an AC series impedance R +jX.In power system terms, this is a 2-bus load flow problem. Load flow equations are highly nonlinear

and except in the very simplest cases it is not practical to seek an analytical solution. Even in the2-bus case the exact solution is rarely discussed, except in the special case where R = 0, when it isstated as

P =VSV

Xsin (4)

with VS and V the magnitudes of the terminal voltages, and the phase difference between them(the power angle). Equation (4) immediately gives the limit VSV /X for the maximum power transferin terms of the terminal voltage magnitudes. However, it suffers from the drawback that it does

not provide any direct information on the way the load voltage V varies with P, in the way thatequation (3) does in the DC case. And, of course, it fails to apply when the resistance R is asignificant fraction of the reactance X, as is the case with many real AC lines.

3


4/26

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Real power P (MW)

Voltageratio

30 MVAr

20 MVAr

10 MVAr

No Q

-10 MVAr

-20 MVAr

-30 MVAr

Figure 4: Power-voltage relationships in an AC circuit (r = 0.001, x = 0.005)

It appears to be little known that the two-bus load flow, for resistive as well as lossless lines, hasa fairly straightforward analytical solution. It takes the form of a quadratic equation for V2, whereV is the magnitude of the load voltage:

V4 +

2(RP + XQ) V2S

V2 + (R2 + X2)(P2 + Q2) = 0. (5)

(See the Appendix for a derivation.)As in the DC case, this can be written more conveniently in terms of , the ratio V /VS of load

to source voltage magnitude. For convenience we adopt the notational shorthand

r =R

V2S, x =

X

V2S. (6)

It is best to regard this simply as a notational device rather than as a per-unit representation, sinceVS will not always be equal to the nominal system voltage. A convenient scale of units in most powersystem studies is to express VS and V in kV, R and X in ohms, P in MW and Q in MVAr. Theunits ofr and x are then inverse MVA, so that when multiplied by P or Q they give dimensionlessquantities.

In terms of , r and x, and the load power P and Q, equation (5) reads

4 + (2(rP + xQ) 1) 2 + (r2 + x2)(P2 + Q2) = 0. (7)

This can be solved as an ordinary quadratic equation for 2. The solution is most convenientlystated by defining the two dimensionless variables

= rP + xQ (8)

= xP rQ (9)

and the solution then reads

2 =1

2

1 2

1 4( + 2)

. (10)

Figure 4 illustrates the solution for in terms of P, for the values r = 0.001, x = 0.005 and severalvalues of Q.

Figure 4 for AC power transfer is broadly similar to Figure 2 for DC, with an additional effectdue to the reactive power Q. As one may expect, a positive Q (VAr import) at the load tends todepress the voltage V relative to VS, while a negative Q (VAr export) raises the voltage. But at anyfixed value of Q there are again two branches to the solution for the voltage in terms of P (the

4


5/26

PV curve), and these branches meet at a limiting value of P (the voltage collapse point), justas in the DC case.

Note that the maximum power transfer for stable operation is highly sensitive to the reactivepower Q: in the example of Figure 4, a 30MVAr export at the load more than doubles the transfercapacity compared with a 30MVAr import. But while injecting reactive power boosts the transfercapacity, it also increases the critical voltage at the turning point of the PV curve. For a 30MVArimport the critical voltage is around 55% of VS, while for a 30MVAr export the critical voltage is

around 75% of VS.(Important: For simplicity we assume for the time being that the reactive power is associated

with a constant-Q source, such as would be provided by a Statcom. Section 8 deals with the case ofa load having a fixed power factor, while Sections 11 and 12 deal with reactive compensation usingpassive shunts such as capacitor banks.)

Clearly it is important to be able to identify the voltage collapse point on the PV curve, whereP reaches a maximum (Pmax, say) and the steady-state solution breaks down. Equation (10) allowsthis to be done fairly easily, since this is just the point where the quantity under the square rootfalls to zero. In other words, where

4( + 2) = 1. (11)

Substituting the definitions of and and expanding, this results in a quadratic equation for Pmax:

4x2

P2max + 4r(1 2xQ)Pmax + 4r

2

Q2

+ 4xQ 1 = 0 (12)with the single positive solution

Pmax =

(r2 + x2)(1 4xQ) r(1 2xQ)

2x2. (13)

Equation (13) is the exact formula for the real power flow limit in an AC circuit that absorbsconstant reactive power Q at the receiving end. The formula has a number of useful approximationsand special cases. For example, if xQ 1 one has the approximate formula

Pmax z r2x2

(1 2xQ) (14)

where z = r2 + x2 is the line impedance divided by V2S. (See the following section for the square-root approximation used.) Thus, the power transfer limit is a roughly linear function of the reactivepower, with VAr absorption reducing the limit and VAr injection increasing it.

If the circuit is lossless (r = 0) the formula (13) takes the simpler form

Pmax =

1 4xQ

2x. (15)

This can also be well approximated when xQ 1 as

Pmax 12x

Q, xQ 1. (16)

In other words, in a lossless circuit there is an effective limit on the sum of the real and reactive power

given by 1/2x, or equivalently V2S/2X. So each 1MVAr of reactive power injected at the receivingend increases the power transfer capacity by around 1MW. This however is an approximation andbecomes invalid when Q is large.

The other quantity important for steady-state stability is the critical voltage VC, or the voltageat the point P = Pmax. Again, this is the point where the quantity under the square root in equation(10) becomes zero, so the voltage ratio is

=VCVS

=

1

2 =

1

2 rPmax xQ

=

z2(1 2xQ) rz1 4xQ

2x2. (17)

If xQ 1, an approximation similar to that in (14) gives the approximate formulaVCVS

z(z r)2x2

(1 2xQ). (18)

5


6/26

-50 -40 -30 -20 -10 0 10 20 30 40 500

20

40

60

80

100

120

Reactive power Q absorbed at load (MVAr)

Powertransfer(MW),Volta

ge(%

ofVS)

Power transfer limit Pmax

Critical voltage

Figure 5: Transfer limit and critical voltage as function of Q (r = 0.001, x = 0.005)

This confirms analytically that the critical voltage decreases with increasing VAr absorption at thereceiving end, and increases with increasing VAr injection. This imposes a limit on the strategy ofincreasing circuit capacity by injecting large amounts of reactive power, because this injection alsoraises the critical voltage until it becomes similar in magnitude to VS. It is undesirable to operatethe receiving end close to the critical voltage because this carries a greater risk of voltage collapse;on the other hand, it is not practical to run the receiving end at a voltage V significantly greaterthan VS.

Figure 5 depicts the power transfer limit and the critical voltage as a function of Q, for a circuitwith the same parameters as in Figure 4.

4 Approximate Solution for Voltage in terms ofP, Q

While it is not particularly relevant to the topic of voltage stability, it is worth noting that many com-mon rule of thumb approximations to the voltage in a two-bus load flow can be easily derived fromthe exact solution (10). The technique involved is the so-called Taylor or binomial approximationfor the square root:

1 x 1 x2

, x 1. (19)This can be applied to the square root in (10) in the case where and are small: that is, wherethe line impedances (r, x) are small, or the load powers (P, Q) are small, or both.

Taking just the conventional positive solution in (10) and using the approximation (19) gives

2 12

1 2 + (1 2 22)

= 1 2 2. (20)Then, taking one more square root and using the approximation (19) a second time gives

1 2

2

= 1 rP xQ (xP rQ)2

2. (21)

For a rule-of-thumb approximation one usually ignores the 2/2 term entirely (as it is an order of

magnitude smaller than ). Written in terms of the original voltages, the resulting approximationreads

V VS RPVS

XQVS

. (22)

6


7/26

Equation (22) provides a simple first approximation for small voltage drops on an AC circuit.When the circuit is lossless or nearly lossless (R 0), and per-unit quantities are used with thesource voltage close to nominal (VS 1) the voltage drop is given approximately by the productXQ. For resistive circuits the product RP also contributes to the voltage drop.

The validity of this approximation relies on the quantities RP/V2S, XQ/V2S, XP/V

2S and RQ/V

2S

all being much smaller than 1. As is evident from Figure 4, it is a very poor guide to voltage dropwhen this condition fails.

5 Sensitivity of Voltage to Reactive Power

An alternative criterion for steady-state voltage stability is the sensitivity V/Q of voltage toreactive power, for a fixed level of real power flow. According to this criterion, the circuit is consideredstable if an incremental reactive power draw reduces the voltage, and unstable if it increases thevoltage.

In Figure 4 it can be seen that at a fixed value of P (that is, on a vertical line), the voltage willalways reduce with increasing VAr demand provided the circuit is operating on the desirable upperhalf of the PV curve. However, if the circuit operates on the undesirable lower half of Figure4, this relationship is inverted and an increasing VAr demand will increase the voltage. (Strictlyspeaking, this is not true in some cases near the bottom left of the diagram, but this operating region

is of much less practical importance than the region near the turning points of the PV curves.)This sensitivity criterion has a fairly clear practical rationale. Voltage control in transmission

and distribution networks is usually accomplished using reactive shunt devices that adjust theirreactive power, on the assumption that positive Q draw (more inductive) will suppress the voltageand that negative Q draw (more capacitive) will boost it. If the system enters an unstable operatingcondition where the reverse is the case, the voltage controller enters a positive-feedback situation,where voltage is continually adjusted in the wrong direction until a protective trip of networkelements results, leading to the possibility of cascading failure.

The sensitivity of voltage to reactive power can be analysed with the help of equation (10).Provided VS is held fixed, incremental changes in V are equivalent to incremental changes in =V /VS, and so the criterion can be evaluated using the partial derivative /Q. Taking the positivesolution in (10) gives

Q=

1

2

Q

2

= 12

x +

x + 2r2Q 2rxP1 4 ( + 2)

. (23)

Now, we are interested primarily in the sign of this quantity. So the multiplicative factor 1/2 neednot greatly concern us, since is always positive. Likewise, the quantity 1 4( + 2) under thesquare root is always positive provided P is less than Pmax.

It is seen that given our underlying assumptions (the circuit is a plain series impedance, sourcevoltage is held fixed, receiving end power is fixed at P and Q, and the circuit operates in thepositive branch with P < Pmax), the stability criterion V/Q < 0 can be violated only in circuitswith non-negligible resistance. Furthermore, instability requires either that Q be very large andnegative (of order 1/r) or that P be very large and positive (again of order 1/r). In the latter caseP is likely to exceed Pmax in any event.

When the effect of shunt admittance must be considered, the picture is more complicated, al-though the overall conclusion is the same. See Section 12 below for further discussion.

6 Graphical Representation of Stability Limit

In Section 3 a quadratic equation (12) was derived for the power transfer limit Pmax in terms of thereactive power Q and the impedance parameters r and x. Since quadratic equations can be graphedrelatively easily, this raises the possibility of illustrating the stability limit graphically, giving apictorial summary of the variation of Pmax with Q, r and x.

A convenient visualisation starts from equation (11), rewritten as

=

1

4 2

. (24)

This is the equation of a vertical parabola in the (, ) plane, shown on the left hand side of Figure6. The limit curve crosses the axis at = 1/4 and is symmetric about this axis. It crosses the

7


8/26

-0.5 0 0.5

0

0.25

Stable region

Unstable region

-150 -100 -50 0 50 100 150

-100

-50

0

50

100

Stable region

Unstable region

- / z

/ z

P

Q

VS

2/ 4Z

VS

2/ 2Z

Figure 6: Graphical representation of stability limit equation (12)

axis at = 1/2 and = 1/2. Since the distance from the origin to these intercepts is twice thedistance to the apex, the origin is the focus of the parabola, speaking geometrically. The region of

stability is the interior of the parabola, where + 2 < 1/4; the contrary holds for the exteriorregion.

To make this meaningful in terms of the power P and Q, notice that the definition of andequations (8) and (9)can be written in the matrix form

=

r xx r

PQ

. (25)

The matrix appearing here is the matrix of a rotation in the plane, albeit slightly disguised. Itbecomes more explicit upon dividing through by z =

r2 + x2 and reversing the sign of :

/z

/z =

cos sin

sin cos

PQ , cos =

r

z, sin =

x

z, z = r

2 + x2. (26)

Equation (26) says that if one takes the axes in a PQ chart and rotates them anticlockwise throughthe angle , one obtains the and axes as in the left hand side of Figure 6only with a scalingby z and a reflection in the axis.

The result is shown on the right hand side of Figure 6. The stability limit curve is a parabola,having its focus at the origin of the PQ plane and its axis of symmetry inclined at an angle = arctan(X/R) to the P axis. The apex of the parabola is located a distance V2S/4Z along thisaxis, where Z =

R2 + X2 is the circuit impedance in the original units. The parabola passes

through two other points located symmetrically at a distance of V2S/2Z along an axis perpendicularto the axis of symmetry (the latus rectum, in technical terms).

The importance of this graphical visualisation is not as a method to construct the limit curves(since we have computer software to do this), but rather as a tool to help build engineering intuitionabout the limits of operation for AC circuits. By picturing how the parabola of Figure 6 movesthrough PQ space as parameters are varied and how Pmax varies along the curve as Q changes,initial conclusions about the capacity of circuits can be drawn which can then be rigorously testedin computer simulations.

7 Relation to Power-Angle Analysis

Lets now return to equation (4), which is the way the voltage constraint on power flow is mostcommonly expressed:

P =VSV

Xsin .

The usefulness of this equation (at least in the lossless case) is that it establishes a relationship

between the power flow on the one hand, and the power angle on the otherand this depends onlyweakly on other system variables when the system operates as intended. Since voltages in a powersystem are usually controlled within a narrow range90% to 110% in most transmission systemsthe quantity VSV /X varies only slightly from its nominal value V

2b /X, where Vb is the nominal system

8


9/26

ZI

V

V

I I

S

Figure 7: Angle relationships for two-bus AC circuit

voltage. Accordingly, changes in power flow P are intimately connected with changes in the powerangle . Furthermore, since sin can never exceed 1, this equation establishes an absolute upperlimit on power flow in lossless circuits, which holds regardless of how much reactive compensationis available.

How does our analysis in terms of and variables relate to this more familiar power-angle

analysis, and how might it be extended beyond the lossless case? Figure 7 helps answer this question:it shows the phasor relationships between the voltages VS and V, the load current I, and the voltagedrop ZI = (R +jX)I. Aside from these magnitudes, there are four angles appearing in Figure 7:

The power angle between VS and V. The power factor angle , being the angle by which the load current I lags the load voltage

V. (If the current leads the voltage, is defined to be negative.) The load powers P and Qhave their familiar expressions in terms of :

P = V Icos , Q = V Isin . (27)

The circuit impedance angle , which is just the phase of the complex number Z = R + jX(or the normalised z = r +jx).

The associated power angle . This can be thought of as the amount by which the voltagedrop differs in phase from the ideal value /2, which occurs when the load has unity powerfactor and the circuit is lossless. The value of can be derived by inspection of Figure 7 as

=

2+ . (28)

Equations for the power angle are found by resolving the phasor VS into components in phase andin quadrature with V. From Figure 7 and equation (28), the component in quadrature with V hasmagnitude

VSsin = ZI cos = ZI sin( ). (29)

Multiply both sides by V and use (27), and there results

V VSsin = ZV I(sin cos cos sin )= XP RQ. (30)

Lastly, normalise both sides by V2S to obtain

sin = . (31)

The component of VS in phase with V is, from Figure 7

VScos = V + ZI sin = V + ZI cos( ). (32)Multiply both sides by V to obtain

V VScos = V2 + ZV I(cos cos + sin sin )

= V2 + RP + XQ (33)

9


10/26

and then normalise by V2S to obtain cos = 2 + . (34)

So, one sees that knowing , and the ratio = V /VS immediately gives the power angle using

cos = +

, sin =

. (35)

This establishes the relationship between power-angle analysis using and the above analysis using and .

The importance generally ascribed to power-angle analysis stems from the use of as a proxyfor the voltage stability margin. By (4), the point of maximum power flow occurs when reaches/2: accordingly, the proximity of to /2 is taken as an approximate margin of stability. Thusmany transmission networks have operational criteria stipulating that for any transmission linknot exceed some threshold value (say, 30 degrees).

We can, however, use the preceding analysisspecifically, equation (10)to derive a more precisestability margin and relate this to the angle . Consider again the quantity under the square rootsign in (10):

=

1 4 ( + 2). (36)This makes an excellent candidate for a margin of stability: its square root is equal to the difference

between the two solutions for 2; it is equal to 1 when the network is quiescent ( P = Q = 0); and itis by definition equal to 0 at the point where voltage collapse occurs. Now, use (35) to write and in terms of and substitute in (36):

=

1 4(cos ) 42 sin2

=

1 4 cos + 42(1 sin2 )

=

1 4 cos + 4( cos )2= 2 cos 1. (37)

In particular, consider the case when = 0. Then is equal to the critical voltage ratio C = VC/VS,

and is equal to thecritical angle

C. From (37) the two are related by the identity

Ccos C =1

2. (38)

This relationship between C and C holds for all AC circuits that can be represented as in Figure3: that is, with a series impedance and power (P, Q) independent of voltage. Just as importantly, itdemonstrates that the voltage ratio and the angle are of equal importance as stability indicators.If the angle criterion for stability gains more attention, it is only because one tends to operate withina narrow range of values, so that it is more common to hold steady and allow to increase withincreased loadings than it is to hold steady and allow to drift.

Of course, equation (38) can also be used to convert a critical voltage formula into a criticalangle formula. For example, taking equation (17) for critical voltage leads to the following formulafor the critical angle in terms of Q and the circuit impedances:

C = arccos

x

2z2(1 2xQ) 2rz1 4xQ

. (39)

The critical angle is plotted in Figure 8, for the same example as used in Figures 4 and 5. It isseen that the critical angle declines steeply as reactive power is absorbed at the load, falling below30 degrees when Q equals roughly 20MVAr. Of course, equation (38) guarantees that the criticalvoltage and the critical angle rise and fall together, and that there is a trade-off in maintaining ahigh margin between operating voltage and critical voltage on the one hand, and maintaining a highangle margin on the other.

For reference, Figure 9 depicts the identity relation (38) between critical voltage and angle. Itcan be seen that C declines precipitously as the critical voltage ratio falls below 75 per cent or so.

10


11/26

-50 -40 -30 -20 -10 0 10 20 30 40 500

10

20

30

40

50

60


Criticalangle(

degrees)

Figure 8: Critical power angle as function of Q (r = 0.001, x = 0.005)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

10

20

30

40

50

60

70

80

Critical voltage ratio C

= VC

/ VS

CriticalangleC(

degrees)

Figure 9: Relationship between critical voltage and critical angle

11


12/26

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Real power P (MW)

Voltage

ratio

0.90 lead, = -0.48

0.95 lead, = -0.33

0.98 lead, = -0.2

unity pf, = 0

0.98 lag, = +0.2

0.95 lag, = +0.33

0.90 lag, = +0.48

Figure 10: Power-voltage relationships for fixed power factor (r = 0.001, x = 0.005)

8 Operation at Constant Power Factor

Textbook discussions of voltage stability, such as those of Kundur [2] or Gonen [1], present the basicPV curves somewhat differently to Figure 4. This is because they derive these curves not byvarying P with the reactive power held constant, but instead by varying P with the power factorheld constant.

This latter assumption is appropriate for transmission lines supplying a load centre, in the casewhere the reactive power comes primarily from natural load variation rather than from auxiliarycompensation equipment. This was a valid assumption in the classical power systems of the early

to mid 20th century, but is increasingly inaccurate given the prevalence of capacitor banks, SVCsand other compensating equipment in modern transmission systems. The presence of compensationequipment means that reactive power is controlled more or less independently of real power, so thatcurves such as those in Figure 4 may be more relevant.

Nonetheless, the equations for two-bus load flow can be solved just as well on the assumptionof fixed power factor as on the assumption of fixed reactive power. The equation used (7) and itssolution (10) are the same, only now we make the substitution

Q = P, =

1

pf2 1. (40)

(The parameter is defined for convenience: it is just another way of expressing the power factor.

Note however that unlike the power factor, is given the same sign as Q: positive for VAr absorptionand negative for VAr injection. Thus for a load, 0.95 lagging power factor is equivalent to 0.33,while 0.95 leading power factor gives 0.33.)

With the substitution (40), the solution (10) is still used, only with the parameters and calculated as

= (r + x)P k1P (41) = (x r)P k2P. (42)

Then the solution from (10) reads

2 =1

2 1 2k1P

1 4k1P 4k22P2(43)

and is plotted in Figure 10 for various different power factors. A notable effect of operation at aconstant power factor is seen at some leading power factors, where the voltage can at first riseslightly with increasing power flow, before falling again as power approaches the stability limit. By

12


13/26

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.520

30

40

50

60

70

80

90

100

110

120

Power factor coefficient

Powertransfer(MW),Voltage(%ofVS),Angle(degrees)


Critical voltage

Critical power angle

Figure 11: Transfer limit, critical voltage and angle at fixed power factor (r = 0.001, x = 0.005)

comparison, as Figure 4 shows, when a load operates with constant reactive power irrespective ofreal power, the voltage falls with increasing power flow in virtually all practical cases.

As before, the power limit occurs when the quantity under the square root becomes zero, that iswhen

4k22P2 + 4k1P 1 = 0. (44)

The solution is

Pmax = k21 + k

22 k1

2k22

=

(1 + 2)(r2 + x2) r x

2(x r)2 (45)

and the corresponding critical voltage VC and critical angle C are given by

VCVS

=

1

2 (r + x)Pmax, cos C = 1

2(VC/VS). (46)

The transfer limit, critical voltage and angle for our example circuit ( r = 0.001, x = 0.005) are givenin Figure 11. This is the analogue of Figure 5 for operation with constant power factor at the load.

9 Remote Generation

So far the discussion has focussed on the case shown in Figure 3, where real power flows fromthe source VS to an AC load. Just as important is the case where real power flows in the reversedirection, from a remote generator toward the source (or infinite bus).

It is of course possible to apply all the previous formulae, such as equation (10) for voltage andequation (12) for limiting power flow, and simply insert negative values of P to represent generation.But generation normally follows a sign convention where generation quantities are positive, and itis convenient to recast the formulae to reflect this.

For a generator, define the generated real power Pgen to be positive for net export (thus equivalentto P for a load). Likewise, define Qinj to be positive for reactive power export (equivalent to Qfor loads), as is usual practice for a generator. Then for the voltage solution, we replace the variables and with their sign-reversed equivalents (subscripted with g to avoid confusion with the originalvariables):

g = rPgen + xQinj (47)

g = xPgen rQinj (48)

13


14/26

0 20 40 60 80 100 120 140 1600

0.2

0.4

0.6

0.8

1

1.2

Generated power Pgen

(MW)

Voltage

ratio

30 MVAr

20 MVAr

10 MVAr

No Q

-10 MVAr

-20 MVAr

-30 MVAr

Figure 12: Power-voltage relationships for AC generator (r = 0.001, x = 0.005)

and equation (10) now reads

2 =1

2

1 + 2g

1 + 4(g 2g)

. (49)

Figure 12 illustrates the solution for remote generation, again using the values r = 0.001 andx = 0.005. This is simply the negative half of Figure 4: it can be reflected in the vertical axis andadjoined to the left of Figure 4, whereupon the voltage traces become closed curves. Note againthat we are using the opposite sign convention for Q as well as P, so the 30MVAr curve for a load

is equivalent to the -30MVAr curve for a generator.The power limit Pmax for generation occurs when the quantity under the square root in (49)

vanishes:1 + 4g 42g = 0, (50)

that is, when4x2P2gen 4r(1 + 2xQinj)Pgen + 4r2Q2inj 4xQinj 1 = 0. (51)

The solution, Pmax, is given by

Pmax =

(r2 + x2)(1 + 4xQinj) + r(1 + 2xQinj)

2x2. (52)

Comparing (52) with (13) and noting that Qinj =

Q, it is seen that the only difference is the sign of

the term r( 1 + 2xQinj). This reflects the fact that the Pgen figure for a remote generator is inclusiveof the circuit losses, but the P figure for a load excludes losses. For a lossless circuit (r = 0), thepower transfer limit Pmax is the same for generated power as for consumed power.

(This asymmetry in the transfer limits for load and for generation can also be seen in thegraphical visualisation of Figure 6. When the circuit is lossless, the angle in the construction ofthe diagram is 90 degrees, and the limit curve is symmetrical as between positive and negative P.The apparently greater capability in the negative P region in Figure 6, which depicts a relativelylossy circuit, reflects the fact that at points of large P and/or large Q, much of the generated poweris consumed in circuit losses.)

The critical voltage VC is obtained from (49) when the quantity under the square root is zero:

VC

VS= 1

2

+ rPmax + xQinj

=

z2(1 + 2xQinj) + rz

1 + 4xQinj

2x2. (53)

14


15/26

-50 -40 -30 -20 -10 0 10 20 30 40 500

20

40

60

80

100

120

140

160

180

Reactive power injecton Qinj

(MVAr)

Powertransfer(MW),Voltage(%ofVS),Angle(degrees)


Critical voltage

Critical power angle

Figure 13: Transfer limit, critical voltage and angle for remote generator (r = 0.001, x = 0.005)

Comparing (53) with (17), it is seen that the two formulae differ only in the sign of the secondterm in the numerator. As with the formulae for Pmax, this difference arises from the accounting forcircuit losses, and is absent in lossless circuits.

Figure 13 depicts the power transfer limit, critical voltage and critical angle as a function ofQinj for the example circuit used above. (The relation (38) between C and C is unchanged forgeneration: the key difference is that V now leads VS in phase.) Figure 13 may be compared withFigure 5 for an equivalent load. The power transfer limit for the generator is significantly higher,and again, this is due to the fact that the circuit losses are supplied from the generator, whereas in

the load case the losses are supplied from the source and not counted in the transferred power. Thecritical voltage is also higher, and for large reactive power injections approaches close to VS itself.This reflects the fact that large reactive power injections will significantly boost the local voltage ata generator, and only when the stability limit is approached will this boosting effect not be seen. Italso shows, however, that there is a limit to the ability to boost the transfer capacity by injectingreactive power at a generator: eventually it becomes impossible to sustain nominal voltage at thegenerator without risking instability.

Lastly, it will be seen that the only difference in the sensitivity of voltage to reactive power(see Section 5) when moving from remote load to remote generation arises from the change in signconvention:

Qinj=

1

2

x +

x + 2rxPgen 2r2Qinj

1 + 4 g 2g

. (54)

Just as with a remote load, an injection of reactive power causes a rise in voltage under virtuallyall circumstances, provided the network operates on the normal upper branch of the voltage curvein Figure 12. This condition is violated only for a very large VAr injection of order 1/r, which isgenerally too large to be practical.

The differences between the network solutions for load and generation can be summed up asfollows:

1. Equation (10) gives the voltage ratio for a remote load fed through a series impedance (e.g.Figure 4), while equation (49) gives the voltage ratio for a remote generator (e.g. Figure 12).The only difference is in the sign convention used for real and reactive power.

2. Equations (13) and (17) give the power transfer limit and critical voltage, respectively, for aremote load. For a generator it is necessary only to change the sign of the term proportionalto r, and to pay attention to the sign convention used for reactive power. This results inequations (52) and (53) respectively. For lossless circuits there is no difference between thetransfer limit and critical voltage for loads and generators.

15


16/26

10 Effect of Long Lines

There are two more important generalisations to consider, which are dealt with in this and thefollowing section. All our analysis to this point has been for the simple circuit model of Figure 3,where the transmission circuit is a simple lumped impedance, and the remote load or generator hasfixed real power P, and either fixed reactive power or fixed power factor.

Real transmission lines and cables do not behave as lumped impedances. Cables in particular

have a relatively high capacitance per unit length and must be treated as distributed circuits if theyare more than a few kilometres long. Most overhead lines up to around 80km long can be adequatelymodelled as series impedances, but longer lines must include the effect of capacitance. For mediumlength overhead lines up to about 300km it usually suffices to aggregate the shunt capacitance ateach end, but above this length the full distributed circuit representation must be used.

In any case, it does not greatly change any of the calculations above to use the full long-lineformulae in place of a series impedance. As the Appendix shows, for long lines there is an exactanalogue to the two-bus load flow equation (7), where r and x are replaced with coefficients thatdepend on the line parameters. The parameters for an exact representation of transmission linesare:

The line length l;

The resistance R and reactance X per unit length; The shunt susceptance B and conductance G per unit length; The surge impedance ZC =

(R +jX)/(G +jB ), split into real and imaginary components

as ZC = RC +jXC; and

The propagation constant =

(R +jX)(G +jB), split into real and imaginary componentsas = +j.

From these parameters, define three coefficients as follows:

r =RC sinh2l XC sin2l

2V2S, (55)

x = RC sin2l + XC sinh2l2V2S

, (56)

a =cosh2l + cos 2l

2. (57)

Aside from the normalisation of r and x by V2S (as with r and x in (6)), these coefficients dependonly on the line constants and not on the source or load conditions.

Having defined these coefficients, let the variables l and l be defined (for a remote load)analogously to (8) and (9):

l = rP + xQ (58)

l = xP rQ. (59)

Then the voltage formula is only slightly modified from (10):

2 =1

2a

1 2l

1 4(l + 2l )

. (60)

See the Appendix for a derivation of this formula. The main change in the form of (60) is the factor1/2a in place of 1/2: this is the Ferranti effect whereby lightly-loaded lines and cables can have ahigher voltage at the remote end than at the source, due to the voltage boosting effect of the linecapacitance.

The condition for a short line is that l 1 and l 1. Under this condition it can be shownthat r and x become approximately equal to the equivalent lumped impedances r = (Rl)/V2Sand x = (Xl)/V2S, and that a

is very close to 1. Equation (60) then reduces to equation (10), as

expected.Figure 14 illustrates the long-line effect. It shows the power-voltage relationships for a load atunity power factor, where the line length is varied but the conductor impedance varied inverselywith length, such that the total line impedance in the absence of long-line effects is always equal to

16


17/26

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Real power P (MW)

Voltage

ratio

10 km

1000 km

2000 km

3000 km

4000 km

5000 km

Figure 14: Power-voltage relationships with a long AC line

that used in Figure 4. In this way, the effect of the long-line behaviour is isolated from that dueto multiplication of impedance as circuit length increases. The capacitance is as appropriate foran overhead line, giving a wavelength in the order of several thousand kilometres: accordingly, thelong-line effect only becomes significant at 1000km or greater in this example. Below this lengththe short-line approximation holds, and the curve is then essentially identical to the Q = 0 curve inFigure 4.

One important conclusion from Figure 14 is that the short-line approximation is conservative:even where it is not strictly valid, the transfer limit given by the short-line approximation is lessthan the actual transfer limit. However, the use of the approximation may still underestimate thecritical voltage.

For a remote generator, one uses the formulae of Section 9, with r and x in place of r and x:

lg = rPgen + x

Qinj, (61)

lg = xPgen rQinj, (62)

2 =1

2a

1 + 2lg

1 + 4(lg 2lg)

. (63)

With either remote load or remote generation, or for operation at constant power factor, thetransfer limit formulae are identical to those in Section 3, Section 9 or Section 8 respectively, withr and x in place ofr and x. The critical voltage formulae, and the sensitivity of voltage to reactivepower, are also the same with one proviso: the factor a now multiplies the formulae. This means

the critical voltage VC is scaled by the factor 1/a

, and the sensitivity d/dQ by the factor 1/a

.Since a is typically less than 1, this means that VC can be slightly higher, and the sensitivity to Qslightly greater, for long lines than for short lines of equivalent series impedance.

With long lines, however, one loses the neat phasor diagram of Figure 7 and with it the convenientlink between the power angle and the and variables given by (35). Similarly, the relation(38) between the critical voltage and critical angle no longer holds, except approximately. It is,nonetheless possible to define a different angle by analogy with (35):

cos = +l

, sin =l

, (64)

and this angle does indeed attain a critical value C satisfying the relation

Ccos

C =

1

2 . (65)

The angle does bear an approximate relation to the true angle between V and VS, but there isno longer an exact correspondence with the other stability variables.

17


18/26

11 Effect of Shunt Admittance

A similar generalisation to that in the previous section covers the case where there is shunt impedanceconnected at the remote load or generation site: for example, where reactive compensation takesthe form of a passive capacitor bank rather than a constant- Q device. A capacitor bank injectsless reactive power under low voltage conditions than under nominal voltage conditions, and so willresult in a steady state voltage different from that indicated for a constant Q source in equation (10)

and Figure 4. This in turn will affect the power transfer limit.Assume than that in addition to the load with fixed real power P and reactive power Q, there

are shunts connected with shunt conductance G and susceptance B (either of which may be zero).Define the nominal shunt power as the real and reactive power that would be drawn by the shuntimpedances if the voltage V were equal to the source voltage VS, that is,

Psh = GV2S, Qsh = BV2S. (66)

(Notice the minus sign for Qsh: capacitors for example have positive susceptance B but negativeVAr absorption by the load sign convention.)

Then to take account of the shunts, the load flow equation (7) with normalised circuit impedancesmust be expanded to the following:

ash4 + (2(rshP + xshQ)

1) 2 + (r2 + x2)(P2 + Q2) = 0 (67)

where

ash = 1 + 2(rPsh + xQsh) + (r2 + x2)(P2sh + Q

2sh), (68)

rsh = r + (r2 + x2)Psh, (69)

xsh = x + (r2 + x2)Qsh. (70)

(See the Appendix for details.) The solution for again has almost the same form as previously:the solution variables and are now defined as

sh = rshP + xshQ (71)

sh = xshP rshQ (72)and = V /VS is found from the formula

2 =1

2ash

1 2sh

1 4(sh + 2sh)

. (73)

The important thing to note here is that the P and Q occurring in these formulae do not include theshunt components. The shunt power instead enters through the definition of ash, rsh and xsh. Noticethat the effect of a shunt is very similar (aside from a constant scale factor) to that of increasing theline resistance parameter by an amount (r2 + x2)Psh, and the line reactance by (r

2 + x2)Qsh. (Theeffective reactance is reduced when the shunt is capacitive.)

Figure 15 illustrates the effect of compensating with a capacitor bank, and how this differs froma constant VAr source. The solid blue line shows the uncompensated case, which is identical to thatin Figure 4. The blue lines show the effect of adding 30MVAr of constant Q (dashed) or capacitors

(solid). (The dashed blue line is also identical to the 30MVAr curve in Figure 4.) Notice that thesolid and dashed lines intersect at the point where V = VS ( = 1), since this is the point where thetwo forms of compensation provide the same actual reactive p ower. Also shown as a green dashedline is the PV curve when no compensation is used but the line reactance is set equal to xsh; thisgives the same behaviour as the solid red line apart from a scaling by the factor 1 /ash. Accordingly,it will be noticed that the capacitive compensation scenario (red solid line) and the reduced lineimpedance scenario (dashed green line) give the same power transfer limit. This limit is lower thanfor the constant Q compensation scenario (red dashed line).

Power-angle analysis is less helpful in the presence of shunt admittances than with pure constant-power load. By substituting the voltage-dependent formulae for load power in (30) and (33), onecan show that

cos = (1 + rPsh + xQsh) +

1

(rP + xQ), sin = (xPsh rQsh) +1

(xP rQ), (74)but these expressions are difficult to relate to sh and sh. The critical-angle relation (38) is notvalid in general for voltage-dependent load.

18


19/26

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Real power P (MW)

Voltageratio

No VArs-30MVAr: constant Q-30MVAr: shunt capNo VArs, equiv X

Figure 15: Power-voltage relationships: capacitor bank versus constant-Q source

In certain situations, one must take into account both long-line effects and constant-impedanceload. The solution is then found by combining elements of the above formulae with those in Section10. As the Appendix shows, the parameters analogous to (68) through (70) are

ash = a + 2(rPsh + x

Qsh) +1

a

(r)2 + (x)2

(P2sh + Q2sh), (75)

rsh = r +

1

a

(r)2 + (x)2

Psh, (76)

x

sh = x

+

1

a

(r

)2

+ (x

)2

Qsh, (77)

where r, x and a are as defined above for long lines. The resemblance to equations (68) through(70) is plain: one just uses the long-line equivalent parameters r and x in place of r and x, andinserts a factor a or 1/a where indicated to account for the Ferranti effect.

With these definitions the solution formulae for = V /VS are again analogous to those for theplain series impedance. The intermediate solution variables are

sh = r

shP + x

shQ (78)

sh = x

shP rshQ (79)and the solution for is

2 =1

2ash

1 2sh 1 4 (sh + (sh)2) . (80)

With remote generation, the reverse sign convention gives

shg = r

shPgen + x

shQinj (81)

shg = x

shPgen rshQinj (82)with the solution formula

2 =1

2ash

1 + 2shg

1 + 4

shg (shg)2

. (83)

12 Shunt Susceptance, Limits and Voltage SensitivityThe equations of the previous section show that with shunts, as with long lines, there is no need toradically revise the formulae for power transfer limit, critical voltage or sensitivity to reactive power,

19


20/26

-50 -40 -30 -20 -10 0 10 20 30 40 500

20

40

60

80

100

120


Powertransfer(MW),Volta

ge(%

ofVS)

Transfer limit, constant QTransfer limit, shunt ind/capCrit. voltage, constant QCrit. voltage, shunt ind/cap

Figure 16: Transfer limit and critical voltage for passive versus constant- Q compensation

provided the shunt powers Psh and Qsh are considered to be excluded from the transferred powerand from the reactive power variable in the sensitivity analysis. In this case, one can simply bundlethe shunts into the equivalent impedance parameters rsh and xsh (or r

sh and x

sh) in the formulaeand scale by ash (or ash) where appropriate, just as for long lines.

There are, on the other hand, cases where the shunt powers Psh and (in particular) Qsh mustbe treated as variables in their own right and not bundled in with other quantities. The mostimportant such case is where switched capacitor banks or shunt reactors are used to provide reactivecompensation. This is particularly common where large amounts of reactive p ower are required,either to b oost (capacitively) the transfer capability of high-impedance lines, or to compensate(inductively) charging currents on long lines or cables. It can then be important to analyse thesensitivity of transfer limits or voltage to the shunt reactive power Qsh, rather than to the intrinsicreactive power Q of the load or generator. (Once again, Qsh is really just an alias for the amount ofcapacitance connected: the amount of actual reactive power from a given capacitance will vary.)

In this situation, the transfer limit with a load is the usual equation (13) but with xsh in placeof x:

Pmax =

(r2 + x2sh)(1 4xshQ) r(1 2xshQ)

2x2sh. (84)

Here P and Q are the fixed real and reactive power associated with the load, and we assumePsh = 0. If for simplicity it is assumed the load itself operates at unity power factor, then Q = 0and substituting formula (70) for xsh gives (after some algebra)

Pmax =

(r2 + x2)(1 + 2xQsh + (r2 + x2)Q2sh) r

2 (x2 + 2x(r2 + x2)Qsh + (r2 + x2)2Q2sh). (85)

Figure 16 depicts the limit given by (85) in comparison with the limit for constant-Q compensationdepicted in Figure 5. There is a clear difference: in general a given amount of passive inductive orcapacitive VArs changes the transfer limit by less than the same amount of VArs from a constant-Qsource. This is not clear from equation (85), but if one assumes xQsh to be small one can apply theapproximations

1 + + O(2) 1 +

2,

1

1 + + O(2) 1 , 1 (86)

and after some algebra, there results the following approximate form of (85) (withz

=

r

2 +x

2):

Pmax z r2x2

1

1 r

z r +2r2

x2

xQsh

. (87)

20


21/26

Comparing this with the corresponding approximate formula (14), the difference in sensitivity toQsh is apparent. In particular, when r/x is small, a small amount of reactive power Qsh causes Pmaxto change by about half as much as an equivalent small amount of fixed reactive power Q.

Figure 16 also shows the variation of critical voltage with reactive power. Surprisingly, it isfound that even though Pmax is significantly affected by the change from constant-Q to passiveshunt compensation, the critical voltage in this example is virtually unchanged except for largecapacitive values. To see why, note that the formula for critical voltage with a shunt is

VCVS

=

1 2rshPmax 2xshQ

2ash. (88)

In this example we assume that rsh = r and Q = 0. Now notice that equation (85) for Pmax can bewritten in terms of ash and z as

Pmax =z

ash r2(z2ash r2) . (89)

Substituting this into (88) now gives

VCVS

=

1

2ash

1 r z

ash r

z2ash r2

=

1

2ash z

2ash rzashz2ash r2

=

1

2ash z

2 z(r/ash)z2 (r/ash)2 .

So far, all the working is exact. We now approximate, setting r/

ash r, giving

VCVS

1

2ash z

2 zrz2 r2 =

z(z r)2ashx2

z(z r)

2x2(1

2xQsh). (90)

Comparing (90) with the approximation (18) for critical voltage, the similarity is evident. Theapproximation works best for low-resistance circuits, since if r is much smaller than z, the quantities(z r) and (z r/ash) will be approximately the same over many values of ash.

The above formulae for Pmax and VC are, again, not greatly changed if one places a generatorinstead of a load at the end of the circuit. As already discussed in Section 9, certain terms willchange sign, either through the use of an opposing sign convention or as a result of the generatornow having to supply the circuit losses. Again, one may use the formulae in Section 9 and substitutersh and xsh for all occurrences of r and x.

The analysis of sensitivity to reactive power requires some attention, as the behaviour withrespect to Qsh looks very different to that with respect to Q. Looking again at Figure 15 for a load,however, reveals nothing unexpected: as long as the circuit operates on the upper half of the PV

curve, the injection of capacitive VArs always raises the voltage.One can confirm this analytically by differentiating equation (73) with respect to Qsh. The

algebra is somewhat involved but eventually results in:

Qsh=

1

2

Qsh

2

= z2

2ash

Q +

Q(1 2rshP) + 2xshP21 4(sh + 2sh)

ash

x + z2Qsh

. (91)

Consider first the case of a unity-power-factor load (Q = 0). Provided Qsh is not too large (thatis, smaller than V2S/X), it is not hard to see that the sign of /Qsh is always negative and theexpected behaviour results: absorption of VArs reduces the voltage and injection of VArs raises it.

The same will be true if the load is inductive (Q > 0): the quantity in the large parenthesesremains positive and the overall sensitivity is negative. On the other hand, if the load is capacitive

(Q < 0) it is theoretically possible for the quantity in parentheses to change sign, even with Qsubstantially less than V2S/X. Nonetheless, in most practical cases the second term, on the far rightof (91), dominates the first term and keeps the overall sensitivity negative. This is because the firstterm is of order z2 while the second term is of order x and so is generally larger by a factor of 1/z.

21


22/26

The equivalent sensitivity equation for a generator is found by reversing the sign convention forP and Q, as follows:

Qsh=

z2

2ash

Qinj + Qinj(1 + 2rshPgen) 2xshP2gen

1 + 4(shg 2shg)

ash

x + z2Qsh

. (92)

Note that although we adhere to the generator sign convention for Pgen and Qinj, we are using thesame load sign convention for Qsh as before. This is a common convention for shunts, whetherthey appear in conjunction with loads or generators, and it is important not to be confused by this.Accordingly, we expect that for a stable system /Qsh will be negative, as for a load.

The change in sign convention does not greatly affect the overall sensitivity result. If Qinj is zero,then (92) is easily seen to be negative as long as Qsh is not impractically large (of order V

2S/X).

The same is true if Qinj is negative (the generator absorbs VArs). Only if Qinj is positive (generatorinjects VArs) is it possible for the first term in (92) to change sign and become positive. Yet in mostpractical situations, the second term dominates and keeps the overall sensitivity negative.

13 Summary and Conclusion

Steady-state voltage stability refers to the constraint imposed on power flow in electrical circuitsby the nonlinear load-flow equations. Given any circuit conditions, there is a limit on the amount ofreal power that can flow from one point to another in a circuit. In DC circuits this limit is given bythe maximum power transfer rule, while in AC circuits the analysis is more complicated and thelimit also depends on the reactive power flow. As this limit is approached, the likelihood of voltagecollapse increases due to the convergence of the dual solutions for voltage and the high sensitivityof voltage to power flow.

Although the equations for general AC circuits have no useful analytical solution, usable formulaeare available for the case of power flow between two buses, given the voltage magnitude at one bus(the source) and the power P and Q at the other (the load). Studies of larger power systemsoften focus on one critical transmission link which can be studied in this manner.

The exact formula for voltage at the load has the same general form whether the circuit is mod-elled as a simple series impedance or as a long transmission line. One first defines two intermediatevariables and as

= rP + xQ

= xP rQ

where P and Q are the fixed power components of the load, and r and x are impedance parametersfor the circuit. In terms of and the voltage formula is

V

VS

2=

1

2a

1 2

1 4 ( + 2)

where VS is the source voltage and a is a parameter dependent on the circuit. Because the formula

is derived from a quadratic equation, there are two branches to the voltage solution correspondingto the sign. The positive branch is held to be the normal or stable operating regime, whilethe negative branch is held to be unstable as it results in abnormally low voltages.

If the load is actually a generator, an alternative sign convention is generally used for the power:instead of the power consumption P and Q one has the generated power Pgen and injected reactivepower Qinj which are sensed positive in the opposite direction, back toward the source. The aboveequations are then modified to

g = rPgen + xQinj

g = xPgen rQinjV

VS

2=

1

2a

1 + 2g

1 + 4

g 2g

.

The definition of the circuit parameters r, x and a depends on the nature of the circuit, and ofany shunts connected at the remote end. If the circuit can be modelled as a plain series impedance,then a = 1 and r and x are just the impedance normalised by the square of the source voltage:

22


23/26

r = R/V2S, x = X/V2S. If the circuit must be modelled as a long line, then equations (55), (56) and

(57) are used for r, x and a. These impedance parameters are also normalised by V2S, but otherwisedepend only on the characteristics of the line.

If there are shunt admittances connected at the remote end, so that changes in voltage affectthe power at the remote end, this is reflected through the parameters r, x and a as indicated inequations (68), (69) and (70). The effect of a shunt at the remote end is much the same as that ofincreasing the line resistance and reactance by (r2 + x2)Psh and (r

2 + x2)Qsh respectively, where Psh

and Qsh are the powers consumed by the shunt at V = VS. (Note that if the shunt is capacitive, theeffect is to reduce the effective line reactance, in accordance with intuition.)

The steady-state stability margin is related to the size of the quantity under the square root inthe voltage equation. As the real power transfer P increases, this quantity falls to zero and the loadflow equation ceases to have a solution: one says that voltage collapse has occurred. Depending onthe sign convention used (load or generation), the equation for the stability limit is = 1/4 2 org =

2g 1/4, and has a convenient graphical visualisation.

The form of the limit equation is the same for long lines as for short lines, and also in the presenceof shunts, provided the appropriate impedance parameters r and x are used. However, the effect onthe limit of a fixed reactive power Q is quite different to that of a nominal reactive power Qsh from apassive shunt, because of the way the shunt power varies with voltage. This difference is illustratedin Figure 16.

The critical voltage is the voltage at the point where the solution collapses. It is therefore foundsimply by omitting the square-root term in the voltage equation, since it corresponds to the pointwhere this term vanishes. Generally, the injection of VArs will increase both the transfer limit andthe critical voltage.

One may also relate the analysis based on and to the more traditional type of analysis basedon power angle . It is thereby possible to extend power-angle analysis to circuits with significantresistance; however, the analysis based on is only approximate in the case of long lines or whenthere is constant-admittance load. When load or generator power is constant, it is found that thecritical angle C at which voltage collapse occurs is intimately related to the critical voltage ratioC by the relation Ccos C = 1/2, independent of other circuit parameters or conditions.

Sometimes an alternative stability criterion is spoken of, based on the sensitivity of voltage toreactive power. By this criterion, the system is voltage stable whenever the injection of VArs raisesthe voltage and absorption of VArs lowers the voltage, and voltage unstable when the reverse is

the case. In almost all practical cases, this voltage stability criterion is equivalent to operation inthe positive branch of voltage: if the system is stable by one criterion it will be stable by theother. For this rule to be violated requires that either P or Q be very large, similar in magnitudeto V2S/Z where Z is the circuit impedance. Other considerations generally preclude operating thecircuit at such high power for a given circuit impedance.

Appendix: Derivation of the Two-Bus Load Flow Solution

Plain Series Impedance

In this section an equation is derived for the voltage magnitude V in Figure 3, given the sourcevoltage magnitude VS, the power P and Q absorbed at the load, and the series impedance R +jX.

In the following, a tilde (e.g. V) denotes a complex phasor quantity, while the same symbolwithout a tilde (e.g. V) denotes the magnitude of the phasor quantity. Complex conjugation isindicated with an asterisk, thus Z = R +jX, Z = R jX.

Our starting point is Ohms Law, whereby the complex voltage difference across the seriesimpedance equals the impedance times the current I:

VS V = ZI. (93)

Now multiply this through by the conjugate voltage V to obtain

VSV V2 = ZIV. (94)

The complex power S is defined as VI, and its conjugate is S = VI. So the above can be

rewritten asVSV

= V2 + ZS. (95)

23


24/26

We can now multiply this whole equation by its own conjugate, to obtain

V2SV2 =

VSV

VSV

=

V2 + ZS

V2 + ZS

= V4 +

ZS + ZS

V2 + Z2S2

= V4 + 2ReZSV

2 + Z2S2. (96)

Finally, taking Z = R +jX and S = P +jQ gives

ZS = (R jX)(P +jQ) = (RP + XQ) +j(RQ XP) (97)Z2S2 = (R2 + X2)(P2 + Q2) (98)

and substitution in (96) gives

V4 +

2(RP + XQ) V2S

V2 + (R2 + X2)(P2 + Q2) = 0. (99)

This immediately leads to the load flow equation (7) given in the text, upon dividing through byV4S and setting r = R/V

2S, x = X/V

2S.

Series Impedance with Shunt LoadsIf the circuit includes a shunt admittance Y = G + jB in addition to the constant-power elementP +jQ, then the complex power in the above derivation must be modified to

S = P +jQ + YV2 = (P + GV2) +j(Q BV2). (100)With this change, the derivation goes through as above. We now have

ZS = (R jX) P + GV2 +j(Q BV2)= RP + XQ + (RG XB )V2 +j RQ XP (RB + XG)V2 (101)

Z2S2 = (R2 + X2)(P2 + Q2 + 2(GP BQ)V2 + (G2 + B2)V4) (102)and substituting into (96) gives, after rearranging

1 + 2(RG XB ) + (R2 + X2)(G2 + B2)V4

+

2(RP + XQ) + 2(R2 + X2)(GP BQ) V2S

V2

+ (R2 + X2)(P2 + Q2) = 0. (103)

Now we normalise by V2S, setting

r =R

V2S, x =

X

V2S, Psh = GV

2S, Qsh = BV2S, (104)

and divide through by V4S, resulting in

1 + 2(rPsh + xQsh) + (r

2

+ x2

)(P2

sh + Q2

sh)

4

+

2(rP + xQ) + 2(r2 + x2)(PshP + QshQ) 1

2

+ (r2 + x2)(P2 + Q2) = 0. (105)

Finally, collecting the coefficients of P and Q in the 2 term and defining rsh, xsh and ash as inSection 11 results in equation (67) as given in the text.

Long Lines

The derivation for a long line follows a similar process: only the algebra is slightly more involved.Start with the voltage equation for a long transmission line [1, 3]:

VS = V cosh l + ZCIsinh l. (106)

For brevity, we will write this as

VS = WV + ZI, W = cosh l, Z = ZC sinh l . (107)

24


25/26

As before, multiply through by V:

VSV = W V2 + ZIV = W V2 + ZS (108)

and multiply this equation by its conjugate:

V2SV2 = W V

2 + ZSWV2 + ZS

= W2V4 + 2Re

WZS

V2 + Z2S2. (109)

Before we substitute the actual expressions for W and Z, we will work through to the actual solutionto show how the general form is preserved. First we normalise Z, setting

z =Z

V2S. (110)

Then dividing (109) through by V4S and rearranging, we get

W24 +

2Re

WzS 1

2 + z2S2 = 0. (111)

For the solution of this quadratic equation, resolve the quantity WzS into its real and imaginarycomponents as

WzS = j. (112)Then the solution is

2 =1

2W2

1 2 +

(1 2)2 4W2z2S2

=

1

2W2

1 2 +

1 4 + 42 4(2 + 2)

=

1

2W2

1 2 +

1 4( + 2)

. (113)

We see that the solution has the form (60) if we define a

= W

2

.It remains only to substitute the actual line parameters. For a, we have

a = W2 = |cosh l|= sinh2 l + cos2 l =

cosh 2l + cos 2l

2. (114)

For the remaining parameters, first calculate

Wz =ZCV2S

sinh l cosh l

=1

2V2S(RCjXC)(sinh 2l j sin2l)

= RC sinh2l XC sin2l

2V2Sj RC sin2l + XC sinh2l

2V2S(115)

= r jx

and finally multiply by S:

j = WzS = (r jx)(P +jQ) = rP + xQ j(xP rQ). (116)

We see that this matches the definitions of a, r, x, l and l in the text.

Long Lines with Shunt Loads

To account for shunts, we repeat the long-line derivation, but replacingS with

S +

Y

V2

. Thus(108) becomesVSV

= W V2 + ZS + ZY V2 (117)

25


26/26

and (109) in its turn becomes

V2SV2 =

W + ZY

V2 + ZS

W + ZY

V2 + ZS

=

W2 + 2Re

WZY

+ Z2Y2

V4 + 2Re

W + ZY

ZS

V2 + Z2S2. (118)

As before, we normalise Z using (110), and we also normalise Y along the lines of (104), setting

y = Y V2S = GV2S +jBV2S = Psh jQsh. (119)After rearranging and dividing by V4S, equation (118) now reads

W2 + 2Re

Wzy

+ z2y2

4 +

2Re

W + zy

zS 1

2 + z2S2 = 0. (120)

Now, if one defines W + zy

zS = j (121)

ash = W2 + 2Re

Wzy

+ z2y2 (122)

then it follows by multiplying (121) by its conjugate that

2 + 2 =

W2 + 2Re

Wz

y

+ z2y2

z2S2 = a

shz2S2 (123)

and hence the solution of this (real) quadratic equation again agrees with the formula in the text:

2 =1

2ash

1 2 +

1 4 + 42 4ashz2S2

=1

2ash

1 2 +

1 4 + 42 4(2 + 2)

=1

2ash

1 2 +

1 4( + 2)

. (124)

Finally one substitutes the line and load parameters, noting that W and z are the same as inthe long-line derivation above. Hence

W2 = a

Wz = r jx

z2 =W2z2

W2=

(r)2 + (x)2

a

Wzy = (r jx)(Psh +jQsh) = rPsh + xQsh j(xPsh rQsh)WzS = rP + xQ j(xP rQ)

zyzS = z2yS =(r)2 + (x)2

a(PshP + QshQ j(QshP PshQ))

and so

a

sh = a

+ 2 (r

Psh + x

Qsh) +1

a

(r

)2

+ (x

)2

P2sh + Q

2sh

(125)

= rP + xQ +1

a

(r)2 + (x)2

(PshP + QshQ) (126)

= xP rQ + 1a

(r)2 + (x)2

(QshP PshQ) (127)which again are seen to match the formulae in the text.

References

[1] Turan Gonen. Electric Power Transmission System Engineering: Analysis and Design. Wiley,1988.

[2] P. Kundur. Power System Stability and Control. McGraw-Hill, 1994.

[3] W.D. Stevenson. Elements of Power System Analysis. McGrawHill, fourth edition, 1982.

aguidetosteady-statevoltagestabilityanalysis

Documents