66- DeMello and Cocordia Paper

IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, VOL. PAS-88, NO. 4, APRIL 1969

Concepts of Synchronous Machine Stability as

Affected by Excitation ControlFRANCISCO P. DEMELLO, SENIOR MEMBER, IEEE, AND CHARLES CONCORDIA, FELLOW, IEEE

Abstract-The phenomena of stability of synchronous machinesunder small perturbations is explored by examining the case of asingle machine connected to an infinite bus through externalreactance.The analysis develops insights into effects of thyristor-type

excitation systems and establishes understanding of the stabilizingrequirements for such systems. These stabilizing requirementsinclude the voltage regulator gain parameters as well as the transferfunction characteristics for a machine speed derived signal super-posed on the voltage regulator reference for providing dampingof machine oscillations.

INTRODUCTION

THE PHENOMENON of stability of synchronous machineoperation has received a great deal of attention in the past

and will receive increasing attention in the future. As economiesin system design are achieved with larger unit sizes and higherper unit reactance generating and transmission equipment de-signs, more emphasis and reliance is being placed on controls toprovide the required compensating effects with which to offsetthe reductions in stability margins inherent from these trendsin equipment design [1].

Concurrent with these trends are improvements in calculatingmethods and computing capability which permit predictingcomplex dynamic effects [2]-[5], providing the means for de-signing control equipment with the proper characteristics.Among several aspects of stability of synchronous machine

operation, an important one is the mode of small perturbationstability referred to as steady-state, dynamic or conditionalstability. Increasing attention has been focused recently on theeffects of excitation control on the damping of oscillations whichcharacterize the phenomena of stability. In particular, it hasbeen found useful and practical to incorporate transient sta-bilizing signals derived from speed, terminal frequenicy, or power[6], [7] superposed on the normal voltage error signal of voltageregulators to provide for additional damping to these oscillations.

This paper deals with an analysis of the phenomena of stabilityof synchronous machines under small perturbations by examiningthe case of a single machine connected to a large system throughexternal impedance. The object of this analysis is to developinsights into effects of excitation systems and to establish anunderstanding of the stabilizing requirements for such systems.By examining a wide range of system and machine parametersand probing into causes and effects, a logical set of guide rulesand concepts is developed to explain the nature of the problemand to arrive at a set of recommendations for stabilizing throughexcitation control. In order to limit the scope of this paper, theexcitation system investigated is one which could be character-

Paper 68 TP 129-PWR, recommended and approved by thePower Generation Committee of the IEEE Power Group for presen-tation at the IEEE Winter Power Meeting, New York, N. Y.,January 28-February 2, 1968. Manuscript submitted September18, 1967; made available for prirnting November 29, 1967.The authors are with the General Electric Company, Schenec-

tady, N. Y.

Fig. 1. Linearized small perturbation relations of a single generatorsupplying an infinite bus through external impedance.

ized by a small time constant of between 0.03 and 0.05 secondwhich is typical of thyristor-type systems.The method of analysis of course can easily be extended to

excitation systems with different dynamic characteristics. It isbelieved that from this analysis covering a wide range of con-ditions for the single machine case, one can project recommen-dations for stabilizing machines in multimachine systems.

DYNAMICS OF A SINGLE MACHINE CONNECTED TO ALARGE SYSTEM-FUNDAMENTAL CONCEPTS

Block Diagranm Relations

The phenomena of stability and damping of synchronousmachines for the mode of small perturbations can be examinedwith the aid of block diagrams relating the pertinent variables ofelectrical torque, speed, angle, terminal voltage, field voltage,alnd flux linkages. The relations in the block diagrams discussedin this paper apply to a 2-axis machine representation with afield circuit in the direct axis but without amortisseur effects.Although a more rigorous representation should include amortis-seur or solid iron eddy current effects in both axes, this simplerrepresentation is sufficient to establish the basic effects anddevelop concepts.The basic phenomenon in question is the stability of the torque-

angle loop, i.e., the behavior of the rotor angle and speed follow-ing a small disturbance such as a mechanical torque disturbance.Although the whole subject can be and has been explored byvarious stability analysis techniques such as Routh's criterion[8], eigenvalue analyses [4], etc., considerable value is found inanalyzing the phenomena in the light of elementary servo-mechanism and frequency response theory, thereby developingan insight into the basic elements that cause various effects.We will attempt to relate the familiar concepts of small

perturbation stability of a single machine supplying an infinitebus through external impedance to the elements and relationsshown in Fig. 1. These relations and block diagram have beentreated previously in [9]. The parameters in these relations arelisted in the Appendix. They are derived by small perturbation

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DEMELLO AND CONCORDIA: SYNCHRONOUS MACHINE STABILITY AND EXCITATION CONTROL

analysis on the fundamental synchronous machine equations,and hence are functions of machine and system impedances aswell as operating point.The parameters of Fig. 1 are defined as follows:

change in electrical torque for a change inrotor angle with constant flux linkages in thed axis

change in electrical torque for a change ind-axis flux linkages with constant rotor angle

impedance factor

for the case where the external impedance isa pure reactance Xe

1 AEqKK4 = q demagnetizing effect of a change in rotor angle

K5 Aet|Ae

AEQ'

change in terminal voltage with change inrotor angle for constant E '

change in terminal voltage with change inEq' for constant rotor anglefield open circuit time constant

Td8' = K3T&' effective field time constant under load.

It is important to recognize that, with the exception of K3which is only a function of the ratio of impedances, all otherparameters change with loading, making the dynamic behaviorof the machine quite different at different operating points.Since these parameters change in rather complex manner, it isdifficult to reach general conclusions based on parameter valuesfor one operating point only.

Performance with Constant d-Axis Flux LinkagesReturning to Fig. 1, let us first look at the portion shown in

Fig. 2, which is pertinent for the condition of constant fluxlinkages in the d axis.

This portion shows a torque-speed-angle loop which gives riseto a natural period of oscillation co,, = a/377 K1/M rad/s. Ne-glecting damping due to electrical load characteristics, we notethat this torque-speed-angle loop is a pure oscillator with zerodamping much like a spring-mass system. The lighter the ma-chine (smaller M) and the higher the transient synchronizingpower coefficient K1, the higher is the oscillation frequency.From the relations in the Appendix, we note that K1 is decreasedby increasing system and machine reactances. It is also affectedby machine loading in a somewhat peculiar manner due totransient saliency effects.

In most practical cases K1 is positive giving rise to oscillations.It is possible in very unusual cases of very long ties and relativehigh loading on these ties to have K1 go negative. In these cases,even if we were to maintain constant flux linkages in the d axis,the machine would be unstable in a real root stability sense,i.e., the machine angle would run away monotonically and expo-nentially with time. Although with special stabilizing in thecontrol of excitation, it is possible to render some of thesesystems stable, they are primarily of academic rather thanpractical interest.

Fig. 2. Linearized torque-angle relationships for the condition ofthe constant flux linkages in the d axis.

ms ~ A

Ln D -rATD L-.J

Fig. 3. Torque-angle loop defining synchronizing torques AT. anddamping torques ATD.

Damping Torques, Synchronizing Torques-Damping RatioBefore introducing the effects of field losses and excitation

control, it is desirable to develop a few simple concepts relatingthe behavior of the second-order system of Fig. 3 which intro-duces a damping term D.

In this system, damping is provided by developing a negativetorque proportional to and in time phase with speed pa. Theresulting characteristic equation is

S2 + (D/M) S + (Ki 377/M) = 0 (1)

giving rise to damped oscillations with frequency ',, Vi -tand damping ratio r where

Wn = VK1 377, v = 2D/VK1M 377.

One form of instability is approached as the damping ratiogoes to zero. For normal values of damping ratios, the frequencyof oscillation is not materially different from the value for zerodamping ratio. For conceivable ranges of inertia, impedances,and loading values, this frequency of oscillation will be in theneighborhood of 0.5 to 2 Hz with the possibility of extremevalues of up to 4 Hz at the high end and 0.1 Hz at the low end.At this point it is appropriate to introduce the concept of

synchronizing and damping torques. At any given oscillationfrequency, braking torques are developed in phase with themachine rotor angle and in phase with the machine rotor speed.The former are termed synchronizing torques and the latterdamping torques. The torque oscillations developed by anyparticular means can be broken down into these components foran insight as to their effect on stability.

Stability can be endangered by a lack of either or both syn-chronizing and damping torques. The traditional stability cri-terion with which industry is most acquainted concerns the testsfor positive synchronizing torques which determine whether ornot forces will be set up to restore the rotor angle of the machinefollowing an arbitrarily small displacement of this angle. Nocorrespondingly simple guide rules have been developed to testfor the possibility of another form of instability which wouldarise for cases where damping torques became negative. Thislatter form of instability is becoming increasingly important asmachine stable operation is more than ever dependent on auto-matic control of excitation.

K =TeAb| Eqf

KIAE, I

K3

Xd'+XeK3 = Xd+Xe

Xd+Xe

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IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, A PRIL 1969

Performance with Constant Field VoltageIntroducing now the effects of field losses, we refer to Fig. 4

which adds the armature reaction terms expressed as a de-magnetizing influence with increasing rotor angle through theblock K4.Examine now the contribution of torque due to this effect

shown by the dashed lines in Fig. 4.The effect of this componenit of torque can be described by the

expression

AT K2K4K3AS due to AEqf 1 + STd,jK3

(2)

These coefficients are always positive, hence we note that atsteady state or zero oscillation frequencies, this demagnetizingcomponent of torque is - (K2K4K3)Ab which is opposed in signto the transient synchronizing component discussed in theprevious section, +K1IA.The familiar steady-state stability criterion with constant

field voltage defines the stability limit as the condition for whichthe steady-state synchronizing power coefficient K1- K2K4K3is zero.

We note that at very high oscillation frequencies, w >> 1/K3 T&'(note that l/K3Ts' = 1/ Td,'), the phase angle of the torquecomponent in question given by (2) is plus 90 degrees whichmeans that at these high frequencies, these torques are almostentirely damping torques. The magnitude of the torques, ofcourse, gets attenuated with increasing frequency. For typicalmachine and system parameters resulting in oscillation fre-quencies of about 1 Hz, the field would only contribute a dampingratio of between 0.03 and 0.05.

In terms of time domain behavior, the series of traces in Fig. 5help describe the significance of the various effects discussed so

far. The plots are the time response of machine angle following a

small step change in mechanical torque.Fig. 5(a) shows the case of constant Eq' and K1> 0 es-

tablishing the characteristic oscillations with zero damping. Asmentioned before, damping due to load characteristics is ne-

glected in these discussions.Fig. 5 (b) shows the effect of the field demagnetizing resulting

in the same type of oscillations with slight damping superposedon a monotonic drift as the machine establishes new steady-stateflux level conditions. This case is for K1- K2K4K3 > 0, i.e.,the machine has steady-state stability or stability under closemanual control in the traditional sense (positive zero frequencysynchronizing power coefficient).

Fig. 5(c) illustrates the angle response for the case where themachine is operating with constant field voltage beyond thesteady-state limit. Here we continue to see the oscillations;however, they are superimposed on a monotonically unstablecomponent. For such an operating point, stable operation can

only be achieved by superimposing the effects of excitationcontrol which cancel out the unstable monotonic component.Operation in this region under regulator control has been re-

ferred to as operation with dynamic stability or conditionalstability. Providing stability by excitation control requires can-

celling out the monotonic instability without deterioratingdamping or giving rise to negative damping. These effects willbe discussed in the next section.

Finally, Fig. 5(d) shows the behavior for the rare case whereK1 is very small or negative and the system manifests instabilitywithout oscillations even under assumption of constant directaxis flux linkages Eq,'.

I

IIIII

- --i

Fig. 4. Torque-angle loop including direct axis field effects.

Eq CONSTANT Efd = CONSTANT Efd= CONSTANT Eq'-CONSTANTrK1 > 0 Kr-KI K3 K4>O Ki -K2K3K4<0 Ki <O

t _-

(a) (b) (c) (d)Fig. 5. Typical responses of rotor angle following small step increases

in mechanical torque.

EFFECTS OF VOLTAGE REGULATORS

Block Diagram Relations

The effect of the voltage regulator on damping and syn-

chronizing torques will now be analyzed by referring to theblock diagram of Fig. 6 where the additional necessary relationsbetween Eq' and et, 6 and et, and Efd and et have been included.

In this treatment, we are considering a simple voltage regu-

lator-excitation system, i.e.,

AEfd/Aet = -K/ (1 + STe) (3)

with a gain K6 and small time constant Tf as might be typicalof thyristor-type exciters.By means of small perturbation analysis, the terminal voltage

deviations Aet can be related to changes in angle AS through theproportionality factor K5 and to changes in flux linkages AEq'by the proportionality factor K6.From the relations in the Appendix, it can be seen that K6

is always positive although its magnitude gets reduced withloading (large machine internal angles) and with small externalimpedances. The constant K5, on the other hand, can haveeither sign and considerable range in magnitude depending on

the impedances and the operating condition. This variation inthe parameter K5 has considerable bearing on the effects on

damping of voltage regulators. Before developing insights intothese effects, let us analyze regulator gain constraints.

Regulator Gain Constraints

One important criterion of voltage regulator performance isits operation with the machine on open circuit for which case

LT m

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DElMELLO AND CONCORDIA: SYNCHRONOUS MACHINE STABILITY AND EXCITATION CONTROL

One more point of interest is in order. The regulating loop, asdescribed in Fig. 7, also has significance with the machine underload. This loop by itself describes the closed-loop effects ofvoltage as affected by flux changes only, i.e., with no changes inangle. Under loaded conditions, this loop is given by the blockdiagram of Fig. 8. The machine function Aet/AEfd is now

lAet K6K3AEfd 1+ STd,'K3

which can also be expressed as

Aet K6(Tdz'/Tdo')AEfd 1 + STd,

(4)

(5)

Fig. 6. Single machine supplying an infinite bus through externalimpedance including effects of voltage regulator-excitation system.

Fig. 7. Voltage regulating loop with machine on opent circuit.

AG t ~~~AE fdAtref K6K3IS T [IFSK3T'do

Fig. 8. Voltage regulating loop with machine connected to systemassuming no changes in rotor angle.

machine angle has no significance. The regulator-machine loopis therefore described by Fig. 7.For well-damped performance of this regulating loop [10],

it is desirable to maintain the crossover frequency less than1/2Te which would mean that the gain K6 should be approxi-mately less than Td,'/2 T,. For an exciter time constant of0.05 second and open-circuit field time constant To,' of 5 seconds,we note that it would be desirable to hold the regulator gain toless than 50 due to these open-circuit performance considerations.

It should be noted that the gain we speak of can be interpretedalso as the transient regulator gain. Since it may be desirable,from a steady-state regulation or droop consideration, to haveconsiderably higher regulator steady-state or dc gain, the desiredtransient gain reduction can be obtained with a series lag-leadnetwork (1 + Tis)/ (1 + T2s) where

Til T2 = (transient gain)/ (steady-state gain)

or alternately with rate feedback from exciter voltage.By selecting 1/ T1 as considerably lower than the crossover

frequency, the effects of concern are not significantly differentif the assumption is made that the regulator is represented by(K61)! (1 + ST,) rather than by

[wK2 (1+ ST1)]/[(1+StTi (1+Sga

where K,i = K,2 TilT2 is the transient gain.

Disregarding the effect of K6 which can vary from about 1 to aconsiderably lower value (occasionally for very high externalreactances combined with heavy shunt resistance loads K6 canbe slightly greater than the open-circuit value of 1), we notethat the effect of the shorter field time constant under loadTdz' is offset by the lower gain Td//Td0' so that the closed-loopresponse of this voltage component regulating loop under loadconditions is not materially different from the closed-loop re-sponse on open circuit.As a first approximation using familiar frequency response

analysis techniques [10], this closed-loop response between theterminal voltage component due to flux changes and change involtage reference can be described by two time constants inseries, one equal to T'I/K,K6 and the other equal to Te.

Aet KeK6

Aet ref I constant (1/K3+ K6K6) [1 + S (Td'/KeK6K) ] (1 + S T6.)

[1+ S(To'/KK6) ](1+ ST)for large K,. (6)

Similarly

,AE6' K,Aet ref constant (1/K3+ K,K6) [1+ S (Tdo'/KeK6)] (1 + S Tfe)

K6[1 + S(Tdo'/K,K6)I](1 + STe)for large Kf. (7)

Voltage Regulator Effects on Machine StabilityComparing Fig. 6 with Fig. 4, we note first that, with the

voltage regulator in operation, the demagnetizing componentK4A6 causes a greatly attenuated effect relative to that whichit had with no voltage regulator. That is, if we compare theexpressions for change in electrical torque for a change in angledue to this demagnetizing action, we have

AT -K4K2K3Ab 1+STd,'K3 (8)

in the case of no voltage regulator, and we previously describedits effects in terms of field damping and negative synchronizingtorques.For the case with voltage regulator, this particular component

of torque becomes approximatelyA/T -K4K2AS "KK6[1 + S(Tdt'/KI6K)] (9)

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IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, APRIL 1969

AsATCOM PONENT

Aet

Fig. 9. Component of torque produced by voltage regulator actionresponse to voltage deviationts produced by rotor-angle deviations.

When comparing (8) and (9), we note that at low frequencies(9) is considerably smaller than (8) by the factor 1/K,K6K3,while at very high frequencies, the two expressions approacheach other. The significance of these facts is that the steady-statenegative component of synchronizing power coefficient is practi-cally eliminated with a voltage regulator with fairly high gain.Meanwhile, since the effective time constant has been reducedfrom Td5' to Td,'/K6K6, the damping component of torque dueto this effect is correspondingly reduced, i.e., the phase lagapproaches 90 degrees for considerably higher oscillation fre-quencies.Now that we have disposed of the component of torque

produced by the K4 branch in Fig. 6 as rather negligible andcontributing only slightly to damping, let us analyze the contri-bution coming through the branch K5 accounting for the effectof angle on terminal voltage. This terminal voltage in turnaffects the flux and torque through the closed voltage regulatorloop.The component in question is described by the block diagram

of Fig. 9.The exact expression for this component of torque due to a

change in angle and its effect on voltage isAT K2KeK5A5 (1/K3 + K6K3) + S (T,,/K3 + Td5') + SA Td,'T

Using the approximate expressions for the closed loop whichapply for the usual range of constants in question, this becomes

AT-KK

AS K [1 + S (Tdo'I/KK6) ] (1 + STe) (11)

Finally, including the previous component produced by the K4branch, the demagnetizing effect of angle, we have the completeexpression for changes in torque due to changes in angle viatheir effect on E.'AT -K2 KeK5+ K4]+ STeK4}

A due to AEq 1/K3 + K6K6 + S (T7/K3 + Tdo') + S2TTds' Tf

(12)

Examine now the synchronizing and damping torque contri-butions from this expression for different oscillation frequencies.

Synchronizing Torques

This torque is the component in time phase with angle.Neglecting the demagnetizing effect through K4, i.e., using thesimpler expression (10), we note that this synchronizing torquecomponent TS at oscillation frequency co is

-K2K6K5 A (13)1/K3 + K6Ke WIo2T&'TE

For very low frequencies

T - -K2K6K5A81/K3 + K6Ke

-K2K5sAK8

for high values of KR.We note that with K5 positive this synchronizing component

is negative. This is of no particular concern as for those caseswhere K5 is positive (low to medium external impedance andlow to medium loadings) K1 is usually high so that the netsynchronizing component K,- K2K5/K6 is still significantlygreater than zero.For K5 negative, which occurs for moderate to high system

transfer impedances, and heavy loadings, we note that thesynchronizing torque component is positive at low frequencies.This is beneficial in those cases where the transient synchronizingcoefficient K1 is low or negative, or where the steady-statesynchronizing coefficient without regulator K1 - K2K4K3 isnegative.As oscillation frequencies increase and w2 approaches

(1/K3 + K6K,)/ (Td0'T,), we note from (13) that the synchro-nizing torque becomes infinite and for even larger frequenciesit reverses sign. Actually before this can happen, the oscillationfrequency becomes very high and instability develops becauseof negative damping.

Damping TorquesFrom (11), we see that the damping torque component TD

due to voltage regulator action is

K2K,K5(T,/K3+ Td&')w AO. (14)(1/K3 + K6KE - c2 Tdo'T )2 + (T6/K3 + T&')2w2

This component gives positive damping whenever K5 is positivebut for a large number of cases where K5 is negative, it con-tributes negative damping. Further, we note that, with K5negative, increasing Kf increases the magnitude of negativedamping causing instability. However, at the other extremewith zero regulator gain, we only get the small amount of fielddamping described before.We thus have a conflicting problem. In those cases where K5

is negative and which are generally the cases involving stabilityproblems, a voltage regulator is of major help in providingsynchronizing torques and curing that part of the stabilityproblem. However, in so doing it destroys the natural dampingof the machine which is small to start with. The recourse hasbeen to have just enough regulator gain to provide synchronizingpower coefficient without cancelling all of the inherent machinedamping.

This can be a satisfactory solution in most cases; however,there can be instances where stability is provided by the regu-lator wvith very poor damping, making operation extremely oscil-latory. In some special cases of very long lines requiring operationnear the line limit, the solution is to have a fairly high regulatorgain to provide the necessary synchronizing power coefficient.In these cases, one effective way to solve the damping problemis to provide a special stabilizing signal derived from machinespeed, terminal frequency, or power.

Recognizing that the problem of damping can be solved byauxiliary means removes some of the constraints placed onregulator gain. However, as was noted before, there remain theregulator gain constraints explained in the Section RegulatorGain Constraints. Since there are fixed relationships between the

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variables, speed, terminal frequency, and power which can beused to relate the requirements of one signal in terms of therequirements of another, we will confine our analysis to use ofsignals from machine rotor speed. It is recognized that the useof any particular signal in preference to another will depend onhardware and implementation considerations outside the scopeof this paper.

STABILIZING SIGNALS FROM SPEED

Phase and Magnitude RelationsThe stabilizing signal requirements to provide damping through

transient manipulation of the voltage reference in response tospeed can be derived as follows. Fig. 10 shows the elements ofthe system in question relating the effect of speed through thestabilizing function G(S) through the voltage regulator loopsaffecting AEq', which produces a component of torque ATsig.The question is: what should be the nature of G (S) to provide

adequate damping over a wide range of machine and systemparameters and operating conditions?The over-all function between the component of torque pro-

duced by the stabilizing signal and speed is

ATsig G(S)K2K, (15)ps (1/K3 + KeK6) + S (TI/K3 + Td,1) + S2Td'T, (

For the usual range of constants, this function can beapproximated by

ATsig G(S)K2KEp6 (1/K3 + K,K6)[1+ S(Td4,'/K,K6)1(1 + STE)

For large values of K, it can be further approximated by

A Tsig K2 G (S) (17)p6 K6 [1 + S (Td,'/K6K,) ] (1 + ST.)

If we wish AT8ig/pb to provide pure damping throughout thefrequency range, then G (S) ideally should be a pure lead functionwith two zeros, i.e., G(S) = K(1 + ST&'/K6KE)(1 + STe).Such a function is not physically realizable and further we donot wish to have a permanent steady-state effect of speed onthe voltage reference. Hence, the stabilizing function should besome compromise function which provides damping over thespectrum of expected frequencies of oscillation, i.e., a functionwith enough phase lead to compensate a significant part of thephase lag contributed by the machine and regulator.

Reverting back to the concepts of synchronizing and dampingcomponents of torque, we note the following. If at a givenfrequency of oscillation co, G (jw) has more lead angle than thelag angle of the machine function

AT K2KeAet ref (1/K3 + KsK6) + j1w (TI/K3 + T') -2

which for large K, is approximately equal to

K2KG (1 + jo Tdo'/K6KE) (1 + jco T,)

(this difference being less than 900), then in addition to thedamping component, we have a negative synchronizing compo-nent, the magnitude of these two components being related tothe sine and cosine of this difference in angle. That is, let thelead angle of G (S) be 6 at a given frequency of oscillation c, andthe lag angle of the machine regulator function AT/IAeref be y atthis same frequency. Then the relative magnitude of the syn-

Ps

AT Sig

AEfd

I I Aet Component

Fig. 10. Component of torque produced by voltage regulator actionin response to a speed-derived signal.

chronizing torque Ts to damping torque TD at that frequencyproduced by the stabilizing signal is

Ts sin(-y-0) (18)TD cos (,y - 0)

Since it is very difficult to synthesize a function which willprovide exact angle cancellation for all conditions, i.e., having0= y at all points, and further since the stability problem isoften helped by providing some synchronizing component, wecome to the following conclusions relating to the nature of thestabilizing signal function.

1) At low frequencies of oscillation which are indicative ofweak synchronizing torques, one should strive to hold 0 lessthan -y, or if this is not possible, at least keep 6 close to y. Atthese low frequencies, the machine regulator function ATTe/Aethas small lagging phase angles, hence the stabilizing functionshould correspondingly have small leading phase angles. As wasmentioned before, a medium-high-regulator gain provides syn-chronizing torques through the regulator by virtue of the effectof angle on terminal voltage K5. Hence, what should be attemptedat the low-frequency range is to provide damping withoutappreciably hurting the synchronizing torque.

2) At the high extreme of the spectrum of frequencies ofoscillation, there is no particular danger to subtracting from thesynchronizing component. As a matter of fact, since the machineregulator function's phase angle becomes increasingly laggingwith increasing frequency whereas, because of hardware limi-tations, the stabilizing function ceases to provide increasingphase leads at these higher frequencies, it is natural to expecto to be less than y. However, in this range of frequencies (y-0)should not be so large as to add significantly to the synchronizingcomponent with its attendant effect on increasing the frequencyof oscillation. As long as the difference between the machineregulator lag and the phase lead for the stabilizing function iswithin ±:30' for the frequency spectrum of concern, acceptableresults are obtained.

Stabilizing Signal ConstraintsThere are several constraints that have to be observed when

prescribing a stabilizing signal design that will be physicallyrealizable and desirable from other considerations. These con-siderations and constraints are as follows.

1) The signal should not produce a steady-state offset ofvoltage reference with frequency hence it must approach a purerate signal at very low frequencies.

2) Lead functions can only be realized with lead-lag pairs,and there is a limit as to how small the lag time constant can be.It is conservative to assume that this lag time constant cannotbe smaller than 0.05 second.

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SEARCH FOR UNIVERSALLY APPLICABLESTABILIZING FUNCTION

General ApproachThe insights that have been developed into the problem of

providing damping from stabilizing signals may convince onethat there is no way of developing a universally applicablestabilizing function that would be optimal for a wide range ofmachine and system parameters, and operating conditions. How-ever, even if not optimal, it would be highly desirable to havea signal transfer function which would provide damping in thosecases where it was badly needed without deteriorating dampingin those instances where there was no particular need for pro-viding supplementary damping.The single machine against infinite bus analysis outlined in

this paper has been used to explore a variety of conditions of asingle machine with different parameters and loading conditions.For each of these conditions there are any number of functionswhich provide adequate damping simply by having the rightrange of magnitude and phase at the particular frequency ofoscillation of concern.By examining the phase and magnitude of these functions as

related to the frequency of oscillation that was pertinent for theparticular case, we arrive at bands of phase angles and magni-tudes as functions of frequency of oscillation. It turns out thatthis band falls within ranges for which a general function witha rather well defined range of parameters serves reasonably well.

Heuristic reasoning permits casting the results of this singlemachine analysis to the general multimachine case. It is knownthat in a multimachine situation there is a wide band of modesof oscillation ranging from those between tightly coupled lightmachines to those between groups of machines separated bylong ties. It is reasonable then to expect that to be effective inthe general power system environment, the stabilizing signal onany given machine must acconmmodate this wide range of possi-bilities. Further, from a practical standpoint, it is essential thatthe range of adjustment not be critical since systems change frommonth to month as do operating conditions.

Range of Single Machine Cases StudiedThe parameters K1, K2, K3, K4, K5, and K6, defined in the

Appendix, were calculated for a number of cases.Two system configurations were studied. One was described

by a single machine supplying an infinite bus through an externalimpedance, Fig. 11 (a). Table I describes all the different combi-nations of parameters and loading explored with this systemconfiguration.The machine constants that were used are

Hydro Steam

Xd 1.14 1.6Xd 0.24 0.32Xq 0.66 1.55Tdo' 12 6

The second configuration attempting to simulate the conditionof a system supplying local load connected to a large systemthrough a weak tie is shown on Fig. 11 (b).

Table II lists the different combinations of parameters ex-plored for this configuration.

Xe

(a)

(b)Fig. 11. External impedance system configurationis for single

machine against infinite bus.

TABLE ISINGLE MACHINE AGAINST INFINITE Bus THROU(CH EXTERNAL

REACTANCE-TERMINAL VOLTAGE = 1.0

ExternalMlachine Inertia Reactance Machinie LoadinigType H Xe P +jQ

Hydro, 1.5, 5 0.1, 0.4, 0.1 + jO, 0.5 + jO,steam 0.7, 1.0 1 + jO.5, 1 + jO,

1 -jO.5

TABLE II

ExternalMachine Inertia Impedance Machine LoadingType H XE RE P +jQ

Steam, 1.5, 5.0 1.0 1.0 1.0 + jO,hydro 1.5 + jO.3,

1.5 - jO.3,0.5 + jO.3,0.5 - jO.3

5.0 1.0 1.0 + jO,1.2 + jO.1,1.2 - jO.1,0.8 + jO.3,0.8 - jO.3

Guide Rules forRequirements

Stabilizing Function Phase and Milagnitude

As developed in the Section Phase and Magnitude Relations,the lead phase angle of the signal G (S) at a particular oscillationfrequency w required to provide pure damping at that frequencywould be equal to the phase lag of the expression

AT K2K,Aet ref 1/K3 + KeK6 + S (TI/K3+ Tdo') + S2 Tdo' T (

The phase angle of (19) as well as the reciprocal of its magnitudeA for the frequency of oscillation w = V/K1377/1M radians wascalculated for the various cases listed above using the expressionsfor the coefficients K1 to K6 in the Appendix.As far as a criterion for the required magnitude of the signal

is concerned, we could arbitrarily select a magnitude of thissignal that would result in a damping ratio of 0.5.

Referring to the torque-angle characteristic equation

S2+ (D/M)S+ (K1377/M) = 0

322

(1)



Fig. 12. Plot of magniitude of stabilizing functions versus frequencyof oscillation. Scatter points are values of AMw for a variety ofsteam machine system parameters and loading conditions aslisted in Tables I and II.

Illll IIIAS REQUIRED FOR PURE DAMPING

12C X- VARIOUS CONDITIONS FOR CONFIGURATION WITHEXTERNAL REACTANCE ONLY

10 *- VARIOUS CONDITIONS FOR CONFIGURATION WITHSHUNT RESISTANCE AND SERIES REACTANCE _k

AS PRODUCED BY TRANSFER FUNCTION x,90 KS (I+ S/8 + 2/64 -_

98 (I+3S) + S/20+S2/400) 2AJ57p_KS ([ S/8)2 | S= j-Atw7

70 (I43S)0+S/20)2

.50

40-

20~ - _

O 0.4Z 0o.6 o.8 2 4 6 8 10t = RADS / SEC

120 4

Fig. 12 is a scatter plot of the desired magnitude AlMo. of thesignal versus frequency of oscillation co as dictated by the abovecriterion. Points for cases with the system configuration ofFig. 11 (a) are marked with X, whereas those with the systemconfiguration of Fig. 11 (b) are marked with dots.A composite plot of phase angles dictated by the above idealized

criterion for all cases incorporating the two configurations isshown on Fig. 13. Again X are points corresponding to caseswith the configuration of Fig. 11(a), whereas dots correspondto those for the configuration of Fig. 11 (b).

It is interesting to note the ranges of the scatter plots for thetwo different configurations. Most stability problems are char-acterized by oscillations at the low end of the spectrum and atthis end, the stabilizing function needs little phase lead. Thisconfirms the results of several studies which were concerned withlong distance transmission problems.On Figs. 12 and 13 we have superposed plots of magnitude

and phase, respectively, for two stabilizing functions discussedin the next Section.

Stabilizing Functions Studied-Analog Computer ResultsFrom the general shape of the scatter plots of the ideal

stabilizing function by the above criterion, and guided by theunderstanding of the many effects developed in previous sections,several stabilizing functions were tried on the system of Fig. 1simulated on an analog computer.

These analog computer studies confirmed with time domainresults the general findings of the frequency response analysis.In particular, these results were valuable in obtaining a cali-bration on how far from the ideal values of AMco and q5 themagnitude and phase angle of a stabilizing function could bewithout adverse effects.The nature of the scatter plots shows that ideally the function

should have little phase angle in the frequency range of 1 to 2rad/s. and then have a rapidly rising phase angle as the fre-quency of oscillation increases to about 15 rad/s.The magnitude and phase angle of the function

Aet ref

pa

60S(1 + S/8 + S2/64)(1 + 3S) (1 + S/20 + S2/400)

are shown in Figs. 12 and 13, respectively. The function inquestion is made up of a rate term with long washout of 3seconds in series with a complex pair of zeros (cn = 8, ¢ = 0.5)and a complex pair of poles (cod = 20, r = 0.5).In Figs. 12 and 13 we also have plotted the magnitude and

phase of the function

Fig. 13. Phase angle versus frequency of oscillation. Scatter plotpoints are phase angles required to cancel the lagging phase anglesof the machine function (AE8')/(Aee ref) for a variety of steammachine system parameters and loading conditions as listed inTables I and II.

where D describes the gain of the damping term ATD/pI, wenote that co. = vKK1377/M and D= \D/V377KM--.For a constant damping ratio, say = 0.5, we have D=

/377 K1M = MVIK1377/M = Alco.The magnitude of the signal that would be required would

then be A\V377K1M or AlMeS where A is the reciprocal of themagnitude of the function AT/\Aet ref for co = VK1377/M.The voltage regulator transient K gain K2, was taken at a

good compromise value of 4Td0'.

Aet ref

Pa60S (1+ S/8)2

(1 + 3S) (1 + S/20)2

which is similar to the previous function except for a dampingratio v = 1.0 which turns the zeros and poles to real values.

Figs. 14 and 15 are sample results of the analog study illu-strating some significant aspects of the problem. The traces aredeviations in speed for a small mechanical torque disturbance.Where pertinent, values of AMw L4, as ideally desired by theabove criterion, have been listed, and noted alongside are theactual phase and magnitude values of the stabilizing functionsS LO at frequency w. Parameters K1 through K6 have also beenlisted.Among these parameters describing machine dynamics, the

most significant ones are c = K1377/M, K5, and K6. e indi-

323

6

LL

C.,2.4

uv"a:1a


IEEE TRANSACTIONS ON POWVER APPARATUS AND SYSTEMS, APRIL 1969

KfT= o K/T =5 K/T=io K/T 20 K/T=O K/T= 1I K/T. 20 K/T =30

LoS HJ~=.=1-.5 .- .AVPwAjvv=- NAA- AAH55 ---b~- -1H =5

(a) (b)

KiT=1O0 K/T=K5 K/T= KT 30 K/T= 10 KT=15 K/T'20 K/T=30

.T -

HH= 1.5 -- *- H =5 --O

(a) (b)K/T=O. K4=IO K/TgO K/T=30 K/r=O K/T= O K/T=20

H= 1.5 ---

H= 5

4 SEC

(C)

For (a) and (b):

(d)Stabilizing Function

KS(1 + S/8 + S2/64)(1 + TS)(1 + S/20 + S2/400)

For (c) and (d) T = 3 seconds:

KS(1 + S/8)2(1 + TS)(1 + S/10 + S2/400)2

Xe = 0.1, P + jQ = 1.0 + j0.5,

Ki = 1.01, K2 = 1.149, K3 = 2.78,

K4 = 1.47, KG = -0.097, K6 = 0.419.

Fig. 14. Speed deviations of a steam unit following a small stepdecrease in mechanical torque. Effect of stabilizing with speed-derived signal. Conditions: regulator-exciter gain Ke = 25, T4 =0.05 second.

cates the frequency of oscillation of concern. K5 by its sign andmagnitude indicates the extent to which the voltage regulatorper se helps or hinders damping. Finally, K6, as explained in theSection Voltage Regulator Effects on Machine Stability, influencesthe phase lag contributed by the voltage regulator loop.

Fig. 14 (a) and (b) shows the effects of stabilizing with variousstrengths of the complex function

KS(1+ S/8+S2/64)(1 + 3S) (1 + S/20 + S2/400)

for the case of a fully loaded overexcited steam unit supplyinga stiff system (X0 = 0.1). Cases are shown for inertia constantsof H = 1.5 and H = 5.0. All pertinent constants are listed inthe captions.

Fig. 14(c) and (d) contains similar results for the case wherethe stabilizing function is composed of real poles and zeros

KS (1 + S/8)2(1 + 3S) (1 + S/20)2

Table III lists the values of cw, AM1wL 4, and S L 0 for thevarious cases of Fig. 14.These values show that for the light inertia cases [Fig. 14(a)

and (c)], the first of the stabilizing functions is more effectivethan the second, and this is confirmed by the analog computerresults.

Fig. 15 contains results of similar nature for the configurationof local load with high tie reactance (RE = 1, XE = 5) with arelatively high tie loading of 0.2 + jO.1.The two stabilizing functions are almost equally effective.

This rather extreme loading case shows that it is possible tostabilize with a voltage regulator and supplementary speedsignal even though the constant flux linkage synchronizingcoefficient K1 is negative.

K/T = 10 K,T=15 KT=o20 K/Tr30 K/T=IO KT = 15 K/T 20 K/T.30

*4 H = 1. 5 *0 -4 H =5 --4

4 SEC

(c) (d)Stabilizing Function

For (a) and (b):

KS(1 + S/8 + S2/64)(1 + TS)(1 + S/20 + S2/400)

For (c) and (d):

KS(1 + S/8)2(1 + TS)(1 + S/20)2

T = 3 seconds

RE = 1.0, XE = 5.0, P + jQ = 1.2 + j0.1,

K= -0.191, K2 = 2.35, Ka = 2.227,

K4 = 0.046, K5 = -0.136, K6 = 1.062.

Fig. 15. Speed deviation of a steam unit following a small stepdecrease in mechanical torque. Effect of stabilizing with speed-derived signal. Configuration of Fig. 11(b). Conditions: regulator-exciter gain Ke = 25, Tf = 0.05 second.

TABLE III

co AMw /O S Z for K =60

Fig. 14(a) 11.27 81.4 Z980 50 Z860Fig. 14(b) 6.17 81.8 /80.30 18 /44.50Fig. 14(c) 11.27 81.4 Z980 50 /520Fig. 14(d) 6.17 81.8 L80.30 29 L440

In order to illustrate the effect of voltage regulator gain ondamping, we have included in Fig. 16(a) and (b) results ofangle deviations for a step change in mechanical torque for thecase where no stabilizing is used. Fig. 16(c) is for a case wherethe parameter Ks is positive, leading to some benefit in dampingwith voltage regulator gain, whereas Fig. 16(b) shows theadverse effects of voltage regulator gain on damping for the casewhere K5 is negative.

CONCLUSIONThe small perturbation stability characteristics of a single

machine supplying an infinite bus through external impedancehave been explored by means of frequency response analysesgiving insights into effects of machine and system parameters,voltage regulator gain, and stabilizing functions derived fromspeed and working through the voltage reference of the voltageregulator.The study has explored a variety of machine loadings, machine

inertias, and system external impedances with a determinationof the oscillation and damping characteristics of voltage or speedfollowing a small disturbance in mechanical torque. An attempthas been made to develop some unifying concepts that explainthe stability phenomena of concern, and to predict desirablephase and magnitude characteristics of stabilizing functions.

324



KSc - KE = 25 KE= 50 KE =100

AB \

4 SEC

t

(a)

3) A stabilizing function operating on speed and developinga transient offset of voltage regulator reference can be selectedto be close to universally applicable for a wide range of machineand system parameters. This function could be of the form

Aet ref (pu)

pa (pu)KS(1 + ST1)2

(1+ ST)(l+ ST2)2

whereKE =25

KC37J2g

Xe = 0.4, P +jQ = 0.5 + jO, H = 5,

K1 = 1.05, K2 = 1.07, K3 = 2.78,

K4 = 1.37, K1 = 0.0544, K6 = 0.439.

X6 = 0.4, P + jQ = 1.0 -jO.5, H = 5,

K1 = 1.18, K2 = 1.73, K3 = 2.78,

K4 = 2.21, K5 = -0.255, K6 = 0.080.

Fig. 16. Angle deviations of a steam unit following a small stepdecrease in mechanical torque. Effect of voltage regulator-excitergain K, at two different loading conditions.

It is evident that for every combination of machine andsystem parameters, and loading conditions, there are a varietyof stabilizing signal transfer functions operating on speed whichgive essentially the right phase and magnitude relation for thefrequency of concern. It is a challenge, however, to find a

universal function which would be adequate for the wholespectrum of possibilities. By examining this spectrum of possi-bilities over a credible band of machine and system parametervalues, as well as loading conditions, it appears that a faircompromise function can be recommended that will be almostuniversally applicable.

In the general case of a machine in a power system, there are

many modes of oscillation between machines and groups ofmachines; these modes can change from day to day with loadingconditions and from year to year with changes in power systemconfiguration. Accordingly, the search for a universally applicablestabilizing function has a great deal of merit.

For a thyristor-type excitation systems characterized by a

time constant in the neighborhood of or smaller than 0.05second, certain general conclusions and recommendations can

be made to cover the majority of applications. These are as

follows.1) A transient voltage regulator gain of about 25 pu AEfd/pu.

Aet is a good compromise for a machine with an open-circuittime constant Td,' = 6. An examination of the block diagramrelations and the constraints set by voltage regulator loopdamping on open circuit convinces us that the most direct wayof compensating for differences in machine open-circuit timeconstant is to vary the transient voltage regulator gain in directproportion to this time constant.

2) The transition from steady-state gain to transient gaincan be obtained with rate feedback from exciter voltage or witha lag-lead network (1 + T2S)/ (1 + TIlS) in series with theregulator where T2/IT1= (transient gain)/ (steady-state gain)and T2 should be larger than 1 second.

T 2to4

T1 0.1 to 0.2

T2 0.05

K/T lOto40.

This function usually provides a significant amount of dampingto the rotor angle oscillations. It should not be concluded,however, that other functions are not equally or better suitedfor particular cases, or that stabilizing is mandatory in all cases.Normal damping forces provided by load characteristics andamortisseurs are entirely adequate in many instances.

4) There appears to be some correlation between the strengthof the signal required and the inertia of the machine. Fromexamination of several sets of runs where the only differencewas inertia, we conclude that the strength of the signal shouldvary approximately as the square root of inertia.

5) The need for stabilizing increases with increasing de-pendence on the excitation to provide synchronizing powerthrough a high response excitation system.These conclusions and recommendations have been checked

out in detailed digital multimachine simulations. Among severalproblems being presently studied are the dynamics under largedisturbances and design requirements of signal limiting. Alsobeing studied are the relative merits of stabilizing with terminalfrequency or with a signal derived from electrical power insteadof speed.

APPENDIX

For a machine connected to an infinite bus with voltage Ethrough an external impedance r6 + jX6, Fig. 11 (a), assumingas negligible amortisseur effects, armature resistance, armaturep, terms, and saturation, the following relationships apply.

et=2=2 + e 2

-ed = 4jbq =-Xqiqe,q = {ld = Eq' - Xd'id

E1 = E8' + (Xe- Xd') id

Te= Eqiq

id = [Eq-E cos 1E{[Xe+ X]/[re2 + (Xe+ Xq)2]}

-E sin b{r6/[r 2 + (Xe+ Xq)2]}

iq [-Eq -E cos8]{re/[r.2 + (Xe + X2)2]}

+ E sin 6{ [Xe + Xg]/[re2 + (Xe + X.)2]}

EJq' = XadIfd- (Xd - Xd') id

Ti' (dEq'/dt) =Efd - Xalfd

Tm- Te = MEd (pb)/dt].

KE =O

AB

Kce - 6.25

4 SEC

t(b(b)

For (a):

For (b):

325



Expressing all equations in small oscillation form and pre- For the special case of zero external resistance, the expressionsserving the basic variables Ae1, AEq', and Ab, one can derive becomethe following relations. -

Aet = K5A6 + K6AEEq'

SE' K3/AEfd K3K4A3Q 1 + STdo'K3 1 + STdO'K3

ATe = K1/A + K2AEq'

where

E= E sK,= ~~[re sin 6,o+ (Xe.+ Xd') COS 80]

A

ig.Eo± aE(Xq- Xd') (Xe+ Xq) sin68

A

- re (Xq - Xd') COS bo]

K2 = [r-Eo +E qo(+ (Xe + Xq) (Xq - Xd'))

1K3= [1+ (Xe + Xq) (Xd Xd-)}

4 = (Xd Xd)[ (Xe + Xq) sin O. - re cos bo]

K5=edo X reEo sin 6. + (Xe+ Xd ) 0EoCos0oet, A

eqoXt [reEo cos 5- (Xe + Xq)Eo sin 8o1

eeo dl_d (Xe + Xq) edo re

eto A et, A

A = [r.2+ (Xe+ Xd') (X + Xe)].

The steady-state operating values of 86, Ego EO, e&,, and e.,are derived from the standard machine vector diagram. Ex-pressed as a function of steady-state terminal voltage eto, andsteady-state real and reactive load currents Ip and Iqo, thesevalues are

Eqo = (etO + IqoXq)2 + (IpoXq)2

E= (et, - Inore- IqoXe)2 + (IpoXe- Iqore)2

sin =etolpo(Xq+ Xe) - reXq(IlP2+ Iqo2)EqoEo

eto (eto-Iqo (Xq - Xe) - Ipore) XeXq (Ipo2 + Iqo2)COS 80 =--

EqoEo EqoEo

iqo = [Ipo (eto + IqXq) - IgolpoXg]/Eqo

ido = [Ipo2Xq + Iqo (eto + IqoXq)]/Eqo

ego= [(eto+IqoXq)/Eqo]eto

edo= iqoXq.

K Aq- Ad .id qol2Po COS 00

K=Xe + Xd O Xe +Xq

E. sin 60K2=

Xe + Xd'

K3 = Xd + xeXd + Xe

Xd- Xd'K4 = -, EO, sin 60

Xe + Xd

KO = x9 edEoR-cos6- Xd -e E sin °0Xe, + Xq et. Xe, + Xd' et,0

K6 = Xe eqo=Xe + Xd' et'

The configuration of Fig. 11(b) is conlverted to the form ofFig. 11 (a) by making a Thevenin equivalent where

re = REX21/ (RE2 + XE2)and

Xe = RBXEB (RE2 + XE2).

NOMENCLATURE

All quantities in per unit on machine base.

d, iq armature current, direct and quadrature axis com-ponents

ed, eq armature voltage, direct and quadrature axis com-ponents

et terminal voltageEq' voltage proportional to direct axis flux linkagesEfd generator field voltage (one per unit is the value for

1 per unit terminal voltage on the air gap line, opencircuit)

Xe, XE equivalent system reactancesRE system shunt resistanceS Laplace operator, or magnitude of stabilizing signial8 angle between quadrature axis and infinite busp8 per unit speed deviation from synchronousTe electrical torqueH inertia constant, seconds111 inertia coefficient = 2H, secondsD damping coefficientE infinite bus voltage.

Subscript o means steady-state value.Prefix A indicates small change.

REFERENCES[1] F. P. deMello, D. N. Ewart, and M. Temoshok, "Stability of

synchronous machines as affected by excitation systems, ma-chine and system parameters," Proc. American Power Conf.,vol. XXVII, pp. 1150-1159, 1965.

[2] D. N. Ewart and F. P. deMello, "A digital computer programfor the automatic determination of dynamic stability limits,"IEEE Trans. Power Apparatus and Systems, vol. PAS-86, pp.867-875, July 1967.

[31 D. N. Ewart and F. P. deMello, "FACE, a digital dynamicanalysis program," presented at the Power Industry ComputerApplication Conference, Pittsburgh, Pa., May 15-17, 1967.

326


DEMEI,1.O AND CONCORDIA: SYNCHRONOUS MACHINE STABILITY AND EXCITATION CONTROL

AEr + Kt LVf I I+S 022 ASt1+ 5015 I+ S5 I+S046

Fig. 17. Typical transfer function for hydro generator on opencircuit including amortisseur effects.

TM

Fig. 18. Synchronous machine and excitation system.

[4] J. M. Undrill, "Dynamic stability calculations for an arbitrarynumber of interconnected synchronous machines," IEEE Trans.Power Apparatus and Systems, vol. PAS-87, pp. 835-844, March1968.

[5] M. S. Dyrkacz, C. C. Young, and F. J. Maginniss, "A digitaltransient stability program including the effects of regulator,exciter, and governor response," AIEE Trans. Power Apparatusand Systems), vol. 79, pp. 1245-1257, 1960 (February 1961 sec.).

[6] P. L. Dandeno, A. N. Karas, K. R. McClymont, and W.Watson, "Effect of high-speed rectifier excitation systems ongenerator stability limits," IEEE Trans. Power Apparatus andSystems, vol. PAS-87, pp. 190-201, January 1968.

[71 F. R. Schleif, H. D. Hunkins, G. E. Martin, and E. E. Hattan,"Excitation control to improve power line stability," IEEETrans. Power Apparatus and Systems, vol. PAS-87, pp. 1426-1434, June 1968.

[8] C. Concordia, "Steady-state stability of synchronous machinesas affected by voltage regulator characteristics," AIEE Trans.,vol. 63, pp. 215-220, May 1944.

[9] W. G. Heffron and R. A. Phillips, "Effect of modern amplidynevoltage regulators on underexcited operation of large turbinegenerators," AIEE Trans. (Power Apparatus and Systems).vol. 71, pp. 692-697, August 1952.

[10] H. Chestnut and R. W. Mayer, Servomechanisms and RegulatingSystem Design, vol. 1. New York: Wiley, 1951.

Discussion

G. Manchur, K. McClymont, and W. Watson (The Hydro ElectricPower Commission of Ontario, Toronto, Canada): The authors areto be congratulated for their paper which, with careful study, can

Manuscript received February 16, 1968.

give an excellent insight into transfer function requirements forstatic excitation systems.

In our experience, the technique of developing linear transferfunctions for a synchronous machine connected through reactanceto an infinite bus, and the study of these functions to determine therequirements of the excitation system, should be the first step insynthesizing excitation system transfer functions. This techniquewas used in developing the speed stabilizing signal described in [6]using block diagrams similar to those described by the authors.We question whether the gain KE should be limited to less than

Tdo'/2TE as suggested by the authors. It appears that this recom-mendation is based on achieving a heavily damped system having adamping factor of 0.707 for the two time constant system considered.In practice, we think the amortisseur windings should also be con-sidered in establishing limits of gain for the controlled-rectifierexcitation system.The effect of amortisseurs is to reduce the permissible value of

voltage regulator gain to maintain the same relative stability of theclosed-loop system. For example for the system shown in Fig. 17,in order to maintain a phase margin of 65°, which the relationshipKB = Tdo'/2TE would give neglecting amortisseurs, the gain mustbe reduced to approximately KE = 0.5 (Tdj/2TE) when includingamortisseur effect. However, if the design criterion for the controlloop were chosen to require 300 phase margin, a commonly acceptedvalue, instead of 650, the permissible value of voltage regulator gain,including amortisseur effects would be KE = 4 (Tdo,/2TE), an increasein gain of eight times. We have used gains in the range Tdo'/2TR to4 (Tdo'/2TE) on a number of hydro machines and the higher valueshave not created any difficulties.

Equation (17) relates the torque component AT,jw to generatorspeed variations. It shows that the requirement for phase lead inthe function G (S) will be minimized if the excitation system gainKE is high. For example, with KE equal to 200, K6 equal to 0.5, andTdo,' 5 seconds, the time constant T4'I/K.KR is only 0.05 secondand phase lead would not be required on the usual hydro generator.This is in accordance with our experience since phase lead of the

327



stabilizing signal has not been required on aniy of our hydro gener-ators. On the other hand for thermal units, closely coupled to thesystem, K6 will be significantly smaller (about 0.2) so that phaselead networks will usually be required for optimum system per-formance. The use of high gain and a small time constant TE wherepossible would appear to be superior to the use of phase lead, sincethe optimum amount of phase lead required varies with systemconditions and as noted by the authors, it is essential that theexcitation system perform satisfactorily over a wide range of systemconditions.The block diagrams in the paper demonstrate why all successful

stabilizing signals developed to date use a speed or frequency devia-tion signal or derive a speed signal from generator power variations.

In our work we have used block diagrams based on the same smallperturbation analysis presented by the authors but having a differentform as shown in Fig. 18. This diagram shows the relationship forbalanced conditions of all the currents and voltages which can bemeasured on a synchronous machine. By including saturation andamortisseur windings, this model has provided a basis for comparingcalculated transfer functions with those measured in field tests usingfrequency response techniques.

M. H. Kent (Southern California Edison Company, Los Angeles,Calif. 90053): In their introduction, the authors state that ". . . more

emphasis and reliance is being placed on controls to provide therequired compensating effects with which to offset the reductionsin stabilitymargins inherent from these trends in equipment designs."It should be pointed out that the concurrent improvements in calcu-lating methods would be tending to emphasize response of controlequipment regardless of the trends in equipment designs, and that ablind reliance on assumed margins built into equipment should never

substitute for a more complete understanding of the problem. Thispaper clearly and significantly contributes to the needed under-standing of system damping in the area of small perturbationstability.

In striving to derive a universal stabilizinig signal, the analysisshould include a large perturbation mode of stability since saturationeffects and limits on regulator voltage, field voltage, and stabilizingsignals can produce surprising results.

In view of the important fundamental concept involved in theapplication of supplemental excitation control as given by theauthors, it may be helpful to examine this concept a little morequalitatively. The response of a generator and its excitation systemin a manner such that it contributes negative damping can beillustrated as in Fig. 19 (a). This figure shows the vector diagram ofa generator that has experienced a decrease in electrical power out-put, P8. The generator rotor position will change directly in time-phase with P., and the generator terminal voltage Vt will increase.Due to the rotor position change there will result a velocity, (thederivative of a with respect to time), accompanied by an inducedfield voltage, Efd induced, and an induced field current will lag thisvoltage by something less than 90 degrees. Accelerating power P8 ofthe machine will be directly proportional to the second derivativeof with respect to time, as given by the basic machine equations inthe paper. Thus, the time-phase relationships of the major machineparameters are established, and the effect of automatic excitationcontrol will now be discussed.The positive change in terminal voltage Vt will result in a negative

error signal VTerror being applied to the automatic voltage regulator.This error signal will act through the regulator and exciter to producea change in field voltage to buck the increased terminal voltage. Thechange in field current Ifdv due to the change in field voltage will lagby nearly 90 degrees because of the large main field time constant.The two field current vectors, Ifd,, and Ifd induced} can be added usingsuperposition to obtain the net change in field current. The net fieldcurrent vector can now be resolved into components along the rotor-position and rotor-speed co axes. Since torque is directly propor-tional to the net field current, it can be seen that there is a torquecomponent (the synchronizing component) in phase with rotor posi-tion, but that the component reflected onto the axis of rotor speed(the damping component) is negative.

Zfd net

e , S -.-

11d induced

Net Ifd from Fig. Io

6

New fdnet I --'

'fd s

Pa

Efd induced

tw~

(a)

SupplementalSignal

Pa

\EfdasPositive Damping Torque Component

(b)

Fig. 19. (a) Negative damping. (b) Positive damping usingsupplemental control.

The effect of a selectively chosen supplemental regulator controlsignal is shown in Fig. 19 (b). In this figure, the result of adding a

supplemental signal is superimposed on Fig. 19 (a). For this illustra-tion, the supplemental signal is chosen so that it is in time-phasewith accelerating power, and since it acts through the voltageregulator and its associated time delay, a change in field voltagewill follow which in turn will produce a change in field current, Ifdsv.Adding the net field current vector obtained in Fig. 19(a) to thatresulting from the supplemental signal produces a new net fieldcurrent vector. Resolving this vector into components along therotor-position and rotor-speed axes, shows that the damping torquecomponent has become positive. Thus, supplementary regulatorcontrols can eliminate negative damping emanating from the excita-tion system and introduce positive damping.As noted in the paper, the time-phase relationships shown in

Fig. 19 are determined by the frequency of oscillation and the timeconstants involved, and the supplemental signal itself could originatefrom speed or frequency or from accelerating power. Because dif-ferent systems oscillate at different frequencies and have differenttime constants, it would ordinarily not be desirable to have thesupplemental signal in exact time-phase with accelerating power, butthe illustration shows the principle. It can be seen in Fig. 19 (b) thatan improperly chosen supplemental signal can result in significantreduction in synchronizing torque, which would not be desirable,while maximizing damping torque.The authors show that under many system circumstances, excita-

tion systems can be a major source of negative damping on systemoscillations. Since the circumstances required for this to occur are

not especially unique, perhaps consideration should be given toincluding stabilizing signals as an integral part of all conventionaland high-speed excitation systems.

F. R. Schleif and W. B. Gish, (U. S. Bureau of Reclamation, Denver,Col.): This is a most welcome exploration and documentation of theprinciples pertinent to the practical application of supplementaryexcitation control. Preliminary recommendations concerning applica-tion of supplementary control to rotating exciters were proposed on

the basis of explorations described in [71 with the remark that


Negative Damping Torque Component

Efdv f.4dv

vI'rror.: - Vt

328




application of such control to higher speed static systems should besoon forthcoming from the manufacturers. This paper contributesgenerously to the fulfillment of that expectation and the authors areto be congratulated on an excellent contribution which is not onlythorough but timely.As a result of extensive studies, the Pacific Intertie Engineering

Guidance Committee has recommended supplementary control notonly on important new machines but also to a large number ofstrategically located existing machines having the more conventionalrotating exciter systems. Although these systems must contend withadditional limitations such as nonlinearity, lower speed of response,and relationship to loading conditions, some of the guides offered inthis paper may be extended to the rotating exciter systems. However,applications of such recommendations are complex and there is muchroom for improvement. We hope further extensions into this fieldwill be forthcoming soon.The paper promises further investigation of signal limiting require-

ments. Further considerations of these necessary functions will bewelcome. Concerning abnormal under- and overvoltage conditions,our own studies have indicated that rapid removal of supplementarysignals upon crossing a voltage limit of ±5 percent and rapid restora-tion of signal upon returning within limits serve this purpose withoutadverse influence. Our pilot application at Glen Canyon, which hasbeen operating for five months to date, has performed satisfactorilywith this system.The paper does not give support for its conclusion which states

"The signal should not produce a steady-state offset of voltagereference with frequency ...." In our own study of the problem asstated in [7], it is not clear that this conclusion is warranted in allcases. It has been proposed [11] that a substantial benefit canbe accomplished by reducing voltage as a function of frequency underdisaster conditions. The frequency supplementary signal wouldautomatically accomplish this type of action where desirable, suchas near load centers. On the other hand, generating stations servinglong transmission lines may not benefit by this action. The point ofthis remark is that reset in the supplementary signal need not bemandatory. It may be advantageous to make it optionally available.

REFERENCES[11] K. L. Hicks, "Disaster control coordination for large intercon-

nected svstems," IEEE Spectrum, vol. 4, pp. 52-55, November1967.

F. P. deMello and C. Concordia: We wish to thank all of the dis-cussers for their comments, which add considerably to the valueof this paper.

In regard to the various points raised by Mr. Manchur, Mr.McClymont, and Mr. Watson, we would comment as follows.The optimum value of the gain KE is a matter of judgment and

the value of Tdo'/2Ts suggested in the paper was considered a goodcompromise considering its effect on performance under small andlarge perturbations. It is quite true that from the point of view ofstability under the linear small perturbation mode, a considerably

Manuscript received March 15, 1968.

larger value of KE is perfectly satisfactory. Also, it is true that withthis larger value of KE, the effective response of flux to voltagereference has less phase lag and therefore the lead requirements forthe stabilizing signal are correspondingly less. However, the higherKE, the more likely is the mode of operation where the exciter hitslimits and, under these conditions, the effective phase lag of flux tosignal can be drastically different and considerably greater than thatunder the linear mode of operation. Use of describing functionmethods of analysis of the mode of operation where the exciter ishitting limits will show that in such a mode, the stabilizing signalwill need considerably greater phase lead to produce damping.An effective means of preserving the beneficial damping charac-

teristics of the stabilizing signal under large disturbance conditions,such as occur following fault conditions, as well as under the smallperturbation mode, is to have moderate values of KE which do notcause the exciter to continually hit limits under typical large dis-turbance conditions, thereby avoiding the drastic change in phaseangle characteristics under these conditions.We do appreciate the need for more complete machine representa-

tions including effects of amortisseurs, saturation, etc. As a matterof fact, the concepts and recommendations developed in the paperhave been checked out on very complete digital representations ofmachines in multimachine situations. For purposes of clarity, how-ever, it was felt that the basic effects and concepts should be estab-lished with the simplest possible 2-axis representation, excludingamortisseur effects.We appreciate Mr. Schleif and Mr. Gish's comments on their

studies for the Pacific Intertie system and agree that much is yet tobe learned in terms of application requirements considering thenonlinear aspects of the problem. Our own investigations of largedisturbance conditions have confirmed the general recommendationscontained in the paper, particularly as regards effects of maintainingmoderate regulator transient gains as explained above.

In regard to the value of a permanent offset of voltage with respectto speed, although this may be desirable from the point of view ofbalancing generation with load, it should be noted that the gains ofvoltage offset per unit of frequency deviation are in the order of 20times what could be tolerated on the basis of a constant volts percycle criterion. As long as the stabilizing signal effect is limited to asafe excursion in terminal voltage, we agree that there may be caseswhere a permanent offset voltage with frequency may be desirable.

It should be noted that this offset would be more effective at theload utilization points than at the generator terminals, since reduc-tion of generator voltage would also cause a reduction of transmissionvoltage and intertie capacities.Mr. Schleif and Mr. Gish suggest removing the stabilizing signal

when the voltage deviation exceeds a certain limit, rather thanlimiting the signal itself. We do not think this is a good idea, andindeed there may well be cases where removing the signal mightdrive the voltage even farther away from normal. On the other hand,when the signal itself is simply limited, it is at least always con-tributing toward damping regardless of the magnitude of the voltagedeviation.We appreciate Mr. Kent's endorsement of the need for calculating

methods for a more complete understanding of the problem. Hisqualitative description of the phenomena of damping and synchron-izing powers as affected by voltage regulation is an interesting addi-tion to this subject. However, we must point out that it neglects theinteraction of the two signals, i.e., the voltage error vector in Fig.19 (a) should be modified by voltage regulator action.

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66- DeMello and Cocordia Paper

Documents

constant flux linkages

single machine connected

synchronous machine operation

rotor angle

small perturbations

stabilizing requirements

synchronous machines

power apparatus