SUBSYNCHRONOUS RESONANCE IN SERIES COMPENSATED TRANSMISSION LINES

John W. Ballance

Southern California Edison Company

Rosemead, California

ABSTRACT

The concept of subsynchronous resonance in series compensatedtransmission lines is presented. Contributions of synchronous generatorrotor motion and induction generation to sustained subsynchronousoscillation are discussed. Computer simulation studies of a 500 kvtransmission system are shown to closely correspond to actual systemtest data. Adverse effects of subsynchronous resonance on system com-ponents are described.

Saul Goldberg

California State Polytechnic University

San Luis Obispo, California

SERIES RESONANCE

The series impedance of a series compensated transmission lineand generator may be approximately calculated as shown below iflumped-parameter equivalents are used for lines and shunt elementsare neglected [101.

Z (fn) = Re.t + j27Tf. Lext + Ri,ee + j27rf. LII,.

(1)+ 12.C, +Rgien+j27rLgen

INTRODUCTION

Literature available, mostly through AIEE, prior to the latterpart of 1970 [1-91 describes the concept of series resonance possiblein power transmission lines utilizing series capacitors. Some of thereferences have described the phenomenon as one form of self-excitation. Series capacitors have been extensively applied in EHVtransmission lines in the Western United States for the purpose of com-pensating the series inductance of lines as an aid to stability.

Symbol

Rext, Lext

RI Ine, LI,ine Cline

Interpretation

Equivalent load resistance and inductance

Transmission line resistance, series inductance and

series capacitance.

To the authors' knowledge, the phenomenon as previouslydescribed has not been regarded as serious enough on these systems to

warrant special design considerations. Recent studies are now suggest-ing that the existence of a series resonant frequency between 0 and 60hertz combined with the characteristics of a synchronous machinecreate the possibility of sustained or poorly damped subsynchronousoscillations. With the flow of non-60 hertz frequencies in the machinearmature, oscillating torques and torsional vibration may result.

This paper presents the results of an extensive study undertakento identify the cause of electrical subsynchronous oscillations and to

develop a method of predicting the frequency and magnitude of theseoscillations.

Rgen* Lgen Generator resistance and inductance as viewed

from transmission system at f..

At resonance,

1Im(Z(fn)) 0= j27rfn (Lent + LIine+ Lgen) + j27JfnC,ine (2)

It follows then that the line has a natural resonant frequency fn where

fn is approximately

I

27r \/ Cline (Lext + Litne+ Lgen)

Paper T 73 167-4, recommended and approved by the Transmission andDistribution Committee of the IEEE Power Engineering Society for presentation atthe IEEE PES Winter Meeting, New York, N.Y., January 28-February 2, 1973.Manuscript submitted October 2, 1972; made available for printing December 13,1972.

A more accurate solution for fns including the effects of line shunts

and charging is obtained by a network technique. Figure 1 is a plot of

fn as a function of series compensation, for a typical 500 kV systemwith radial generator. The Appendix describes a computer program

called RES which has been written to calculate the network impedanceat subsynchronous frequencies.

1649

(3)

50145

40

35

30Ra 25

20

15

10

5

PERCENT COMPENSATION = (100) Y XC SERIES CAPACITORSI XL SYSTEM COMPONENTS

10 20

PERCENT SERIES COMPENSATION OF TOTAL SYSTEM

30

FIGURE 1. Resonant Frequency of Radial Transmission Line

If a value of f equal to fn is substituted into (1),

Z (fn) = Rest + Riine

It is clear from (6) that when fn < f0, the value for slip is negative andthe synchronous generator operates as an induction generator present-ing an apparent negative resistance to the system. A plot of generatorresistance, Rg,, as a function of the subsynchronous resonant fre-quency, f., is included as Figure 2.

c r 11

~c ZZI- 0

wc:i) °

<Cr 0L

uiI

(4)

For normal power systems Z (fn) is always positive.

INDUCTION GENERATOR EFFECTS

For resonant oscillations to be sustained, the net system resistancemust be nearly zero which could not be attained in a simple seriesresonant circuit. The concept of sustained subsynchronous resonance

for a salient pole machine has been presented in [10]. The followingmethod for calculating the impedance of a large synchronous round-rotor generator at subsynchronous frequencies is outlined in [11].

At subsynchronous frequencies, the apparent generator impedanceis approximately:

R, j

Zgen ±+i60 Xd" per unit (5)

where S, the slip, is

RESONANT ELECTRICALFREQUENCY -fn

(HERTZ)10 20 30 40 50 60

RGEN = RrotorS

S = Rn -60f%

Rrotor = 2%

FIGURE 2. Negative Resistance Characteristics of a Synchronous Generatorat Subsynchronous Frequencies

When the value of apparent negative generator resistance is equalin magnitude to the system resistance external to the machine, thesubsynchronous oscillations are undamped.

As an example, consider the radial transmission system shown inFigure 3.

EOUIVALENT T-TRAN SMISSION LINE -.-( STEP UP GENERATORSYSTEM TRANSFORMERLOAD I

l~~-j.013 .0014+j.038 j.013 .000+ j.K~~~~ I

i.0002 -jl)oy I jl70y

jI

AC

I 1170y ji1oy,

lPEDANCES IN PER UNIT ON 100MVA 525KV BASEDMITTANCES 'yIN MVAR AT 525 KV

S-= f. fandR = K (r2-rl) (6)Figure 3. A Typical Series Compensated

Radial Transmission System

subsynchronous electrical frequency.

f. synchronous electrical frequency or rotor electrical speed=60 hertz.

R, per unit rotor resistance.

KR correction factor = 1.7

r, per unit positive sequence rotor resistance.

per unit negative sequence rotor resistance.

per unit subtransient generator reactance.

The apparent generator impedance can be modeled as

Zgen (f) =- S 60 Xd"

Values for Rr and X '" suggested in references are as follows:

Rr= 1.7 (.02)=.034

Xd" =- 0.10

Ref. [11]

Ref. [13]

1650

z

0

'J0"I

cLLI0

'C

0uz

Symbol

fn~

Interpretation

r2

X "d

Both parameters are expressed in per unit on the machine base. Assum-ing the machine base is 900 MVA, then the generator impedance onthe line base of 100 MVA, 525 KV, yields,

R,= .034 (100/900) = .0038 per unit

Xd"=0.1 (100/900) = 0.01 1 per unit

The level of series compensation in Figure 3 is approximated by

Percent Total Compensation PTC

100 xe (7)

Xc is the reactance of the series capacitors and XL is the inductivereactance of the generator, line and load. Thus

100 (.0134 + .0134)PTC =

(.006+ .0384 +.0196 +.011)

36%

The resonant frequency of the system, f,, was calculated by theRES program to be 36 hertz. From (5), the generator impedance is

Zg, 30 .0038 36 .1)n(30) - (36-60) +i 60 (.011)

36

--.0057 + j .0066 per unit

The resistance of the system external to the generator was calcu-lated by RES to be

Rexternal .002 per unit

Note that this solution would predict increasing oscillations be-cause the sum of the external resistance and the generator resistance isnegative at resonance.

GENERATOR ROTOR MOTION EFFECTS

Positive sequence subsynchronous currents flowing in the gener-ator stator cause oscillating torques on the generator rotor at frequen-cies equal to 60+f,f and 60-fn, where f. is the subsynchronousresonant frequency. From a review of the shaft torsional response,

1651

apparently the oscillations at f =60 + fn are reasonably well damped.This is not the case, however, at fm 60 - f., where f,, is the rotoroscillation frequency. If fm corresponds to a natural torsional vibrationfrequency of the turbine-generator shaft system, the magnitude of therotor oscillations are greatly amplified. It is reasoned that this rotormotion contributes an additional apparent negative resistance to theelectrical system at frequencies f,= 60 + fm and f2= 60 - = fn.As the electrical system is inductive at f1, there is no resonance and therotor resistance may be neglected. At f2 the system is in resonance, andthe negative rotor resistance must be included in the calculations for Z.Thus, if fm is sufficiently close to a shaft natural frequency, the generatorapparent negative resistance is more negative than predicted by calcu-lations including only the induction generator effects. A thoroughtreatment of the torsional motion possibilities is included in [14].

Equation 5 predicts that subsynchronous oscillations will be sus-tained if the level of series compensation is greater than that valuewhich would yield

Z,y8 (fn) + Zgen (60-f.) =0 (8)

With the additional effects of rotor motion, it becomes possible foroscillations to be sustained within a range of compensation whichis lower than the level predicted by (8).

SIMULATION RESULTS

System measurements were taken at the Mohave Generating Sta-tion in January, 1972, to determine the susceptibility of the system to

oscillations when the line is series compensated. System configurationfor the measurements is shown in Figure 4. Small disturbances were

introduced into the system by insertion of series capacitor modules intothe Mohave-Lugo line. Results of Case 6.2, the simultaneous insertionof four series capacitor modules into the Mohave-Lugo line, are sum-

marized in Figure 5. The Mohave-Eldorado line was open during thiscase. Subsynchronous oscillations are observable in both the high andlow pressure stator current oscillograms. The measured resonant fre-quency is 22.8 hertz with a damping time constant of approximately.43 seconds. As illustrated, the subsynchronous oscillations are posi-tively damped.

Using the RES program, the calculated resonant frequency is23.4 hertz, with a damping time constant of approximately .63 seconds.The error between calculated and observed values is attributed to theuse of an equivalent impedance at Lugo which is not equal to the actualimpedance on the day of the test.

4) ELDORADO(3 of)4) pMOENKOPIH( )f

FOUR CORNERS

EIII345 kV

* SERIES CAPACITOR BANK AT LUGO ON MOHAVELINE WITH BYPASS SWITCH INITIALLY CLOSED.TOTAL SERIES CAPACITOR BANK REACTANCE =

35% LINE REACTANCE.

FIGURE 4. System Configuration for January 7,1972 Measurements

HP GENERATORB40 StatorCurrent1100 Amp / Div

LP GENERATORB0 StatorCurrent1100 Amp/Div

--.-+ ;-~V:t1 it-1 fI,.1tIHt.

TIME (40 MSEC Per Division)

FIGURE 5. January 7,1972 Case 6.2 ResultsInsertion Of 35% Series CompensationInto Mohave-Lugo Line

TRANSIENT SIMULATION RESULTS

The effect of rotor motion on system response is being investi-gated by the Advanced Systems Technology group of the WestinghouseElectric Company. This study was initiated by an Edison contract withWestinghouse [12] for this purpose. Results of the study are basedon a Westinghouse computer simulation program which represents thecomplete rotor circuit, the turbine-generator mechanical system andthe three-phase electrical transmission system.

A simulation study based on assumed system conditions has beenmade using this program. The turbine-generator shaft system describedin [14] has been included in the simulation.

The study results shown in Figure 6 illustrate that with 31.8%total series compensation, the simulation would predict increasing oscil-lations. Limiting of the oscillations at a constant magnitude could beexpected if saturation effects of the machine are included in theprogram.

Simulation results for 36.2% series compensation are shown inFigure 7. Because the calculated slip frequency no longer matches anatural torsional mode, better damping is anticipated and demonstratedby the simulation, thus confirming the hypothesis that rotor motionand torsional resonance have significant effects on apparent generatorimpedance at subsynchronous frequencies. It should be noted thatEquation (5) predicts more negative rotor resistance than is indicatedby either the tests or simulations.

HP GENERATORELECTRICAL TORQUE

LP GENERATORELECTRICAL TORQUE

LINE CURRENT

- ____ .il.I .i ii']

TIME

HP GENERATORELECTRICAL TORQUE

LP GENERATORELECTRICAL TORGUE

LINE CURRENT

(50 MSEC PER DIVISION) TIME (50 MSEC PER DIVISION)

FIGURE 6. Simulation Of Subsynchronous Resonance31.8 % Total Series Compensation

FIGURE 7. Simulation Of Subsynchronous ResonanceSeries Compensation Increased To 36.2%

1652

As a further check, the system conditions corresponding to theJanuary 1972 Case 6.2 have been simulated. The results of this simu-lation as shown in Figure 8, correspond very closely with actualoscillograms.

zw

cc

cc

cc&

cc

w

w

0

V\\il '\lA fl'0.'I__- i-- --

TIME 150 MSEC PER DIVISION)

SIMULATION RESULTS

MEASURED VALUES

FIGURE 8. January 7, 1972 Case 6.2 Simulation Result

TRANSIENTS

If subsynchronous oscillations cause pulsating torques on therotor, then these torques must also exist elsewhere on the turbine-

generator shaft. In fact, if the applied mechanical oscillation frequency

fmr corresponds to a natural mode frequency of the shaft, the torquesin the shaft are amplified due to resonance phenomena. Results ob-

tained from simulations of the turbine-generator shaft response to

subsynchronous frequencies raise concern for shaft integrity followingtransients which stimulate subsynchronous oscillations. A review of

these effects on the turbine-generator shaft system is included in a

companion paper [14].

A digital transients program [151 and the G.E. Transient Net-

work Analyzer [16] have been used to perform studies on the Mohave500 kV transmission system to predict the magnitudes, frequenciesand decay rates of torques caused by faults. Results of these studies

are being analyzed to determine the effects possible on the generatorsor other transmission components.

EFFECTS OF SUBSYNCHRONOUS RESONANCE

It has been shown that subsynchronous resonance can cause

torques in the shafts of synchronous machines. Under certain circum-

stances subsynchronous currents may also appear in lower voltagecircuits, and may cause similar stresses on generator auxiliary motors

as well as noticeable light flicker near the generating station.

The effects of subsynchronous resonance can theoretically lead

to system instability if the currents flow in major series compensatedtransmission lines. At subsynchronous frequencies, the increased re-

actance of the series compensation causes a proportionally higher volt-

age to appear across the capacitor bank, leading to possible incorrect

capacitor gap flashover. No instance of a voltage high enough to bypasscapacitors has been calculated.

1653

CONCLUSIONS

Resonance in series compensated transmission lines coupled withthe induction generator effects of a synchronous generator, and theeffects of rotor motion near shaft resonant frequencies, has been shownto make possible sustained electrical system oscillations. Theoreticalexplanations have been shown to correlate very closely with actualsystem observations and computer simulation studies.

These relatively new discoveries of capacitor side-effects must beconsidered where capacitors are used and may, under the most unfavor-able circumstances require special precautions in the use of seriescapacitors near generating stations.

REFERENCES

(1) M. I. Alimansky, "The Applications and Performance of SeriesCapacitors" General Electric Review (Schenectady, N.Y.), Vol.33, No. 11, November 1930, pp. 616-625.

(2) J. W Butler, C. Concordia, "Analysis of Series Capacitor Appli-cation Problems;' AIEE Transactions, Vol. 56, 1937, pp. 975-978.

(3) C. Concordia, G. K. Carter, "Negative Damping of ElectricalMachinery" AIEE Transactions, Vol. 60, March 1941, pp. 116-120.

(4) C. F Wagner, "Self-Excitation of Induction Motors with SeriesCapacitors;' AIEE Transactions, Vol. 60, 1941, pp. 1241-1247.

(5) Edith Clarke, S. B. Crary, "Stability Limitations of Long-Distance AC Power Transmission Systems" AIEE Transactions,1941, pp. 1051-1059. Discussion by C. Concordia, AIEE Trans-actions, 1941, Vol. 60, p. 1299.

(6) E. C. Starr, R. D. Evans, "Series Capacitors for TransmissionCircuits' AIEE Transactions, Vol. 61, 1942, pp. 963-973.

(7) R. D. Bodine, C. Concordia, G. Kron, "Self-Excited Oscillationsof Capacitor Compensated Long-Distance Transmission Sys-tems;' AIEE Transactions, Vol. 62, 1943, pp. 41-44 discussionpp. 371-372.

(8) B. M. Jones, J. M. Arthur, G. M. Stearns, A. A. Johnson, "A10,000 kVA Series Capacitor Improves Voltage in 66 kV LineSupplying Large Electric Furnace Load, AIEE Transactions,Vol. 67, 1948.

(9) R. L. Witzke, E. L. Michelson, "Technical Problems Associatedwith the Application of a Capacitor in Series with a SynchronousCondenser;' AIEE Transactions, Vol. 70, 1951, pp. 519-525.

(10) H. M. Rustebakke, C. Concordia, "Self-Excited Oscillations ina Transmission System Using Series Capacitors;' IEEE Trans-

actions, PAS-89, No. 7, September/October 1970, pp. 1504-1512.

(11) L. A. Kilgore, L. C. Elliott, E. R. Taylor, "The Prediction andControl of Self-Excited Oscillations Due to Series Capacitors inPower Systems' IEEE Transactions, PAS-90, Vol. 3, May/June1971.

(12) Southern California Edison Company, Subsynchronous Reso-

nance Phenomenon Study, Advanced Systems Technology,

Westinghouse Electrical Corporation, East Pittsburgh, Penn-sylvania.

(13) Electrical Transmission & Distribution Book, WestinghouseElectric Corporation, 1964.

(14) S. Goldberg, W R. Schmus, "Torsional Stresses in Turbine-Generator Shaft Due To Subsynchronous Resonance;' sub-mitted to IEEE- 1973 Power Engineering Society Meeting.

(15) Herman W Dommel, "Digital Computer Solution of Electro-

magnetic Transients in Single and Multiphase Networks;' IEEETransactions, PAS-88, No. 4, April 1969.

(16) "Transient Network Analyzer Study of the Sub-SynchronousCurrents for Southern California Edison Company-MohavePlant;' Electric Utility Engineering Operation, General ElectricCompany, Schenectady, New York, June 9, 1972.

(17) G. W. Stagg, A. H. El-Abiad, Computer Methods In Power Sys-tem Analysis, McGraw-Hill, 1968.

APPENDIX

From the equivalent pi-section transmission line model illustratedbelow, the driving point impedance at A may be calculated by Equa-tion 1.

A B C

where Z. is impedance of shunt reactors

Zc is impedance of series capacitors

ZB is impedance of line charging (B/2)ZL is impedance of transmission.line

ZEXT is impedance of system beyond bus C

ZGEN is impedance of generator at subsynchronous frequencies,including the step-up transformer impedance

ZS ZBA + ZS ZC (A+ BZB )ZBA+ (Z S+ Za) (A + BZB) r GEN

A ZEXT ZS (ZB + ZL) + (ZEXT+ ZS) (ZB ZC + ZI,[ZC + ZB1)B --ZEXT ZS + (ZEXT + ZS) (ZB + ZC)

The above equations are limited in application to radial transmis-sion elements. To generalize the approach, a method to calculate theZ..S matrix [17] has been written into a computer program namedRES. ZBUS is calculated at each frequency and an iterative approach isused to converge on the resonant frequency, fn, such that

Im Z(fn) =0

1654

Discussion

C. E. J. Bowler, C. Concordia, and J. B. Tice (General Electric Company,Schenectady, New York. 12305.): This paper has presented the facts ofan occurrence of subsynchronous instability which seemed, at least tosome people, to be at variance with traditional theories on the subject.Since this occurrence, much work has been done to describe the nature'of the problem analytically, and it has now been shown to fit an ex-tension of the previous work that incorporates additional details in theproblem description. As the paper discusses the problem and results ina general way, it seems desirable to clarify certain statements.

1.) Regarding the statement that positive (phase) sequence sub-synchronous currents at frequency fn in the stator produce oscillatingtorques at frequencies of both 60 - fn and 60 + fn, we believe this is notquite correct. The physical circumstances are approximately as follows:

If we consider the case of a balanced armature and network circuitand a rotor with some moderate degree of saliency, all natural rotor cur-rents of frequency flr, f2r, etc., will have corresponding stator currentsof 60 - flr and 60 + flr, 60 - f2r and 60 + f2r, etc. That is, each natur,lfrequency will have two frequencies in the stator but only one in therotor. The higher frequency is much the smaller, because the saliency isnot great.

On the other hand, if we consider the case of a single-phase statorcircuit (for example, a single-line-to-ground transmission fault, whichcorresponds to a line-to-line load on the generator), and for simplicityneglect the slight rotor saliency, a single-phase (not positive-(phase)-sequence) stator current of natural frequency fs will have correspondingrotor current of frequency 60 - fs and 60 + fs.

For example, a rotor natural-frequency current of 29 Hz would bemanifested by stator currents of 31 Hz with positive phase sequences,and also of 89 Hz. For example, again, a single-phase stator current of31 Hz would produce rotor currents of 29 Hz and 91 Hz.

Thus, although the apparent good damping of this 60 + fn fre-quency mentioned in the paper is better explained by the absence ofthe current in the first place, there is the very significant possibility of60 + fn current and torques on single-phase faults.

2.) The paper refers to "reasoning" that rotor motion can con-tribute to negative resistance, and to the "hypothesis" that rotor motionhas significant effects. It should be pointed out that references 3 and 10of the' paper had already demonstrated this point, by showing the com-pletely overriding effect of rotor motion near resonance of the rotoras a whole with the power system. Moreover, paper T 73 218-5, beingpresented at a different session at this same meeting, shows a similaroverriding effect of rotor motion at any torsional natural frequencyunless there is an appreciable amount of mechanical damping.

Returning to the facts presented, namely the presence of an in-stability with 31.8% compensation but the existence of a stable statewith a higher compensation of 36.2%, this can be explained as the ef-fect of the nearness of the slip frequency currents to a torsional reso-nance. This phenomenon is a critical function of the net system re-sistance and the torsional mechanical damping. It is conceivable that insome cases a time simulation might not predict this conditionalstability at 36.2% compensation if the wrong choice of damping weremade, since with sufficiently low damping it is possible to be unstableeven with 36.2% compensation. On the other hand, assuming a largevalue of damping would mean that instability would not occur for anyvalue of compensation up to 36.2% oer more.

The failure of the MOHAVE units in this mode has provided acertain amount of quantitative detail regarding the performance ofmachines when subjected to these phenomena that will be valuable inassessing the importance of mechanical torsional resonance and thevery important contribution of mechanical damping, which apparentlyvaries considerably with machine loading. It is evident that additionaltests and correlation with calculated data on torsional natural frequen-cies and damping are urgently needed in view of the increasing use ofseries compensation, especially in the systems of the western utilities.In many instances it is likely that utility system modifications, includingthe addition of resistance and/or tuned filters will be required in orderto achieve a combination of machine-system characteristics that iscompatible with continued reliable service in the future.

Manuscript received February 12, 1973.

G. A. Fischer, R. Quay, and R. L. Winchester (General ElectricCompany, Schenectady, N.Y. 12305.): The problem' described in thispaper is real. Two shaft failures occurred on Southern CaliforniaEdison's Mohave No. 1 high pressure turbine-generator during similaroperational circumstances,_the first in late 1970 and the second ap-proximately a year later. The failures have since been related to elec-

Manuscript received February 20, 1973.

trical subsynchronous resonance involving the turbine-generator andthe series compensated transmission system, where the system elec-trical resonant frequency coincided very nearly with the secondtorsional critical of the rotating shaft system.

In this discussion we will describe the nature of thesMohave No. 1failures, and comment on turbine-generator characteristics which havean important bearing on the subsynchronous resonance problem.

Mohave No. 1 is a cross-compound machine. The high pressureelement consists of high pressure and intermediate pressure turbineshafts, the generator, and a shaft-driven alternator exciter.

The failure occurred in the shaft section between the generatorand the shaft driven alternator, at the main generator collector, and isshown in Figure 1. A conducting path was. established from each col-lector ring through the insulating sleeves to the shaft. Heavy currentflow through the double ground then eroded large pockets of metalfrom the shaft and collector rings.

Analysis of line current oscillograms taken during the disturbanceindicated the presence of appreciable magnitudes of currents of sub-synchronous frequency. Subsynchronous current produces subsyn-chronous torque on the generator of approximately the same per unitmagnitude, but at slip frequency. It was possible to very accurately de-termine the subsynchronous current frequency from the oscillograms,and it turned out that the corresponding slip frequency was very nearlyequal to the second torsional critical of the shaft system. The secondtorsional critical essentially involves the exciter alternator shaft oscil-lating mechanically against the remaining shaft inertial elements, asshown in Figure 3 of the companion paper1. Maximum twist occurs inthe shaft section between the generator and alternator.

Although the failure first appeared to be electrical in nature, de-tailed study indicated that mechanical phenomena precipitated thefailure of the electrical insulation. Figure 2 shows a polished andmagnified cross-section from the shaft in the region of the failure. Thecollector insulating sleeve was originally along the outer surface, andthe bore contained the insulated collector connections. The lightregion at the center of the cross-section, extending in about a ½/2-inchband between inner and outer surfaces, was determined, metallurgically,to be a heat affected zone in which temperatures in excess of 650°Cwere required to producQ the observed change in metal structure.

We have established that the temperatures required to produce theheat affected zone were the result of mechanical strain cycling, wherethe stress magnitude and frequency were large enough to produce sub-stantial heat generation in the metal, resulting in the very hightemperatures.

The phenomena have been duplicated in the laboratory. Figure 3shows a I / 10-scale model of the shaft, which was subjected to torsionalstrain cycling at a frequency corresponding to the slip frequency atMohave. The discolored sections on the shaft indicate abnormal heating,and were produced in a matter of seconds after the start of the test.

We have concluded that the failure was initiated when abnormalheating iin the shaft caused heating of the collector insulation, whicheventually failed and resulted in a double ground.

The mechanical data used to analyze the simptified model ofpower system and turbine-generator interaction considered in this, andin a companion paperl, is not routinely supplied and must be obtainedfrom the equipment manufacturer. The shaft inertias and spring con-stants are regularly used by our company in calculating short circuit,synchronizing out-of-phase or other fault torques on the machine. Inthese cases, the primary frequencies of the driving torques are the wellknown values of zero, and 60 and 120 Hertz which are far enough re-moved from the shaft torsional natural frequencies that the simplifiedlumped mass model may be used with sufficient accuracy. This may notbe true for the subsynchronous currents which occur very close to themechanical resonance.

In the lumped mass model, the inertias of the turbine andgenerator rotors, including all assembled components such as wheels,buckets, couplings, retaining ring and windings, are considered as con-nected together by the effective spring constants of the smaller diam-eter shafts. In actual fact, when the large diameter wheels and bucketsof the last stages of the low pressure turbine rotors and the couplingsare included as additional mass points, not only is the number of de-grees of freedom and hence the number of critical speeds increased,but there is also a slight change in the calculated criticals for the lowermodes. This slight change becomes important when the forcing fre-quency is close to the critical and the mechanical damping is low. Thispoint is well covered in the companion paperl.

It is well known that the response of a mechanical spring-mass sys-tem at or close to a critical frequency is limited by the damping. Wewould like to emphasize that the damping values are low, and are notvery well known. Torsional damping in large turbine-generators hassimply not been a matter of primary concern in the past, and only asmall amount of test data is available as a result of some very limitedtests. Additional tests are necessary to determine the machine dampingvalues before they can be used with confidence, and these are beingundertaken.

1655

Fig. 1. Photograph of damaged collector, MohaveNo. 1.

Fig. 2. Cross section from damaged shaft.

g-. °7/A¢ (SIN 27TfttI 1.5 \ L5 +2°

.0

60 50 40 30 20 10 0LINEFREQ

Fig. 4. TypicalA vs. °Ccurves.

2I

0

CDz

Z I

UJ

j:5. n10 .1 10 100% LIFE EXPENDED PER INCIDENT

Fig. 5. Cyclic life expenditure curve.

Fig. 3. Photograph of model shaft test specimen.

1 656

Although, the companion paper1 does mention the high torquegain that occurs at resonance, it should be repeated that for actual sub-synchronous current stimulus that can occur, the response torque at thecritical (or even several cycles off the critical) can be higher than theshaft can withstand. The curves shown in the companion paperl werederived from data furnished by our company which defined shaftcapabilities in terms of acceptable stimulating torque levels, even at thecriticals. It should not be assumed that these conditions can be met onany series compensated transmission system without extensive applica-tion studies and possible system design modifications. Of primary in-terest then to the system designers will be the values of subsynchronouscurrents that are permissible, since these will determine the degree ofsystem modification which is required.

The present large turbine-generator set is a very complex machinedesigned for maximum efficiency, consistent with long life and high re-liability, with the steam conditions and electrical parameters specifiedby the user. The torsional characteristics tend to be fixed by other de-sign criteria such as short steam paths, minimum foundation length, oilfilm stability, long bearing life, short circuit torques, etc. There appearslittle likelihood that any practical designs will significantly change thefirst three or four torsional resonances below 60 Hertz. Therefore, theproblem becomes one of determining the torsional fatigue life capa-bility of the machine, and of specifying it in a manner that is useful tothe electrical system designer.

One of our first attempts to do this was the use of A vs. a curves,as described in the companion paperl, in which allowable subsyn-chronous current at each frequency from 10 to 60 Hertz was shown forthree different system electrical time constants which seemed possibleto attain with various system modifications. A typical set of curves isshown in Figure 4.

This approach has the advantage of separating the problem at thegenerator terminals into the electrical system problem and the turbine-generator mechanical response. This separation has, in the past, servedthe industry well and has permitted both the utility system and themachine designers to progress in their well defined areas of responsibilityas rapidly as was economically feasible. In the present situation, how-ever, it appears at this time as though the interaction between the elec-trical system resonance and the turbine-generator mechanical resonanceis sufficient to make it impossible to uniquely define the terminal cur-rent components and system stability criteria without considering themachine-system response at the same time. This is discussed in refer-ence 2.

More recently, machine capability has been defined by curves ofcyclic life expenditure per subsynchronous transient incident, as shownin Figure 5. This curve is necessary if torsional duty above the highcycle fatigue life of the machine is to be permitted. At the same timewe are gathering data to improve the accuracy of fatigue life calcula-tions, it behooves the utility owner and the power generation industryto determine and define the frequency of occurrence that is to be ex-pected. This includes both the frequency and severity of transients onany one system and a projection of the number of systems which mightbe using series compensation in the near future.

We would like to re-emphasize that there is practically no directlyapplicable test information to support current estimates of turbine-generator cyclic torque capability. Our estimates are based on smallspecimen correlations between torsional and bending fatigue phenomenawith extrapolations for large sizes, actual geometry details, operatingenvironment, etc. At least a few years will be required to build the sub-stantial test machines required, and to run the tests necessary to provideaccurate data on machine capability.

With the increased knowledge obtained from such critical speed,damping and fatigue life tests, the turbine-generator manufacturer maybe able to make moderate machine design changes and provide cycliclife calculations with the same confidence that has characterized otherdevelopments in the history of power generation. But when all is saidand done, we believe it will become clear that this problem cannot besolved in the turbine-generator alone. Other ways will have to be foundto alleviate the effects by system design and modification.

REFERENCESl. Subsynchronous Resonance and Torsional Stresses in Turbine-

Generator Shafts, Saul Goldberg, W. R. Schmus, C73 135-1, 1973PES Winter Meeting.

2. Self Excited Torsional Frequency Oscillations with Series Capacitors,C. E. J. Bowler, D. N. Ewart, C. Concordia, T 73 218-5, 1973 PESWinter Meeting.

E. R. Taylor, L. A. Kilgore, and D. G. Ramey (Westinghouse ElectricCorporation, East Pittsburg, Pa. 15112.): This paper and a companion

Manuscript received February 13, 1973.

paper describe Southern California Edison's experience with thephenomena of self excitation. The authors are to be complemented forbringing this potential problem to the attention of the industry and fortheir observations on the causes of such difficulties. The data collectedduring their tests is of great value to the industry in verifying the ex-istence of these oscillations in operating power systems and in verifyingthe accuracy of computing methods used to analyze these problems.

It should be noted that the simulation shown in Figure 8 was madeusing a digital transient analysis program 1 that represented the sourcesby a voltage source behind subtransient reactance. The close match withfield test results shows that the torsional interaction is not a significantfactor in the electrical currents following switching transients until therotor motion has had time to build up. This allows problems associatedwith switching transients and faults to be studied separately from theself excitation problems.

Although these studies have shown that a serious problem exists atthe Mohave station, certain other points need to be brought out toavoid the conclusion that one must abandon the very real economicadvantages of series compensation. It is interesting to note that this isthe first and only case reported where the phenomenon has appeared inturbine generators connected to series compensated lines, although anumber of such transmission systems with series compensation havebeen in operation for some time. It is instructive to examine these othersystems and to note the differences from that of the author's system.These other large systems have most likely not exhibited thesephenomena basically because of the divergence of loads and because ofthe numerous machines with different torsional frequencies in thesesystems. The subsynchronous currents in such systems are damped anddissipated reducing the subsynchronous current component in any onemachine to zero or to a very low magnitude. Also, remote water wheelgenerators, even though directly feeding into series compensated lines,are generally not susceptible to the electromechanical aspects of thisproblem since their shaft mechanical torsional frequencies are usually10 Hz or lower, corresponding, to an electrical system frequency of50 Hz or higher. Such a high electrical frequency requires higheramounts of compensation than have to this time been used2. The in-duction generator action of future plants, with EHV or UHV lines,however, should be examined, and where there is the possibility of thisphenomenon being a problem, consideration should be given to rotorconnected damper windings to reduce the rotor resistance. This is arelatively inexpensive addition.

With remote turbine generation feeding into one or two similarlines it may be practical to avoid the exact dlectrical frequencies thatcorrespond to the mechanical torsional frequencies of the units, and soavoid this difficulty. With multiple dissimilar compensated lines it maynot be possible to avoid' all the frequencies of concern, and in suchcases, it may be necessary to provide relatively inexpensive electricaldamping circuits3.

REFERENCES

1. Feero W. E., Juves, J. A., and Long R. W. - "Digital Models of LargeSystems for Transient Analysis." Proceedings of the American PowerConference Vol. 32, 1970 pp. 1076-1081.

2. L. W. Lloyed, E. R. Taylor, F. G. Berg, S. M. Merry, "A Study ofReinsertion Transient Voltages and Currents for Series Capacitor'son USBR Glen Canyon - Flagstaff 345-kV Lines" Presented at the1971 Summer Meeting of the IEEE Power Engineering Society.

3. L. A. Kilgore, L. C. Elliott, E. R. Taylor, "The Prediction and Con-trol of Self Excited Oscillations Due to Series Capacitors in PowerSystems," IEEE Transactions, PAS - 90, Vol. 3, May/June 1971.pp. 1305-131 1.

Walter H. Croft (Arizona Public Service Company, Phoenix, Ariz.85036.): The authors are to be complimented on their presentation ofa straightforward approach in the analysis of the effects of subsyn-chronous resonance and its application to a practical system wherecause of a failure of a generator has been traced to subsynchronousresonance.

The system equations contain the quantity Rline. Since Rline is aquantity that varies with temperature to a great degree as well as fre-quency to a small degree, it would appear meteorological conditionsexisting at a given time could influence the answers given by the calcu-lation. It appears in an after-the-fact analysis the meteorological con-ditions existing at the time of the incidence would have to be con-sidered. Likewise, an analysis establishing a threshhold to limit themagnitude of the subsynchronous oscillations would require thelowest possible value of _Rline or basically the lowest tempe-ratures towhich the line will be exposed. For long lines, meteorological con-

Manuscript received February 20, 1973.

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ditions may vary considerable over the length of the line at any giventime so a composite value of Rline is necessary.

I would appreciate the authors' comments on the effect of thisvarying value of Rline caused by the lines' meteorological environment.

Also since both R and L external quantities appear representingadditional system and/or load, it can be assumed that the magnitude ofthe load and its characteristics' could have an influence on subsyn-chronous resonance values obtained. Would the authors please commenton this?

J. W. Ballance and S. Goldberg: The authors thank the discussors fortheir comments.

Messrs. Bowler, Concordia and Tice are correct in noting thatpositive sequence stator currents transform to 60-fn torques in therotor. Their second comment states that previous papers have described

Manuscript received April 25, 1973.

the effect of rotor motion. These papers considered only frequencies inthe very low range which has been called hunting. Our paper and otherrecent papers have extended the rotor motion effects to the range of20 to 50 Hz.

As Messrs. Taylor, Kilgore, and Ramey have noted, sustained sub-synchronous resonance has not been encountered at any' othergenerating station. However, recent studies have indicated that manyplants in the West, both existing and planned, may be susceptible tohigh shaft torques as a result of either sustained or transient subsyn-chronous currents. 'The discussors suggestions of methods to avoidexact resonance are welcome.

Messrs. Fischer, Quay, and Winchester are to be complimented fortheir thorough presentation of the mechanical aspects of the Mohavefailure. Their past research has contributed greatly to the explanation ofthe failure. It is hoped that future research and testing may suggest ameans to prevent future occurrences of this type.

The resistance of a transmission line varies both as a function ofthe temperature, as Mr. Croft has noted, and as a function of electricalfrequency (skin effect). Load changes also affect the'net system re-sistance and reactance. These effects can be significant in a study ofsusceptibility to subsynchronous resonance, as Mr. Croft has suggested.

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