101027939 Chapter 3 Slow Transients[1]

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MODELING GUIDELINES FOR LOW FREQUENCY TRANSIENTSReport Prepared by the Low-Frequency Transients Task Force

of the IEEE Modeling and Analysis of System Transients Working Group

Contributing Authors: R. Iravani (Chair)A.K.S. ChandhuryI.D. Hassan, J.A. Martinez, A.S. Morched,

B.A. Mork, M. Parniani, D. Shirmohammadi, R.A. Walling

Abstract: The objective of this report is to provide guidelinesfor modeling and analyses of low-frequency (approximately 5 to1000 Hz) transients of electric power systems, based on the useof digital time-domain simulation methods. For the ease of ref-erence, the low-frequency transients are divided in seven dis-tinct phenomena. This report (1) briefly describes the physicalnature of each phenomenon, (2) identifies those power systemcomponents/apparatus which either contribute to or areaffected by the phenomenon, (3) provides guidelines for digitaltime-domain simulation and analyses of the phenomenon and(4) provides sample study-system and typical digital time-domain simulation results corresponding to each phenomenon.A comprehensive list of reference is also included in this reportto provide further in-depth information to the readers.

Keywords: Low-Frequency Transients, ElectromechanicalTransients, Modeling, Time-Domain Analysis, TorsionalDynamics, Turbine Vibrations, Bus-Transfer, ControllerInteractions, Harmonic Interactions, Ferroresonance

1. INTRODUCTION

An interconnected power system can experience undesirableoscillations and transients as a result of small-signal perturba-tions, large-signal disturbances, and nonlinear characteristicsof the system components. The oscillations cover a wide fre-quency range approximately from 0.01 Hz to 50 MHz. Oscil-lations in the frequency range of 0.01 to 1000 Hz are definedin this report as low-frequency (slow) transients. We inter-changeably use the terms ìslow transientsî, ìlow frequen-cy(LF) dynamicsî, and ìLF oscillationsî throughout thisreport. All the issues relevant to low-frequency inter-areaelectromechanical oscillations (approximately 0.1 to 1 Hz)and classical turbine-generator swing modes (approximately1 to 2.5 Hz) are discussed by other IEEE working groups, andare not discussed here. A general guideline for representationof network elements for electromagnetic transient studieshave been previously published [1.1]. The mandate of theIEEE Low-Frequency Transients Task Force is to providemodelling guidelines for time-domain analysis of LF oscilla-tions within the frequency range of 5 to 1000 Hz. Low fre-quency dynamics are of concern with respect to power systemstability issues and/or temporary overvoltages.

phenomena of 60 Hz power systems in the LF range are di-vided into the following categories:

1.Torsional oscillations (5 to 120 Hz)

2.Transient torsional torques (5 to 120 Hz)

3.Turbine blade vibrations (90 to 250 Hz)

4.Fast bus transfer (1 to 1000 Hz)

5.Controller interactions (10 to 30 Hz)

6.Harmonic interactions and resonances (60 to 600 Hz)

7.Ferroresonance (1 to 1000 Hz)

For each of the above phenomenon this report provides (1) abrief explanation of the physical phenomenon, (2) modelingguidelines for time-domain simulation and analyses, and (3)typical sample systems and simulation results.

This report is intended for practicing power system engineerswho are involved in system analysis, system control, and sys-tem planning. To use the report efficiently, adequate under-standing of the physical phenomenon of interest andfamiliarity with the concepts and techniques of digital com-puter simulation approaches are necessary.

Section 2 of the report deals with low-frequency transientswhich involve both electrical and mechanical dynamics, i.e.,torsional oscillations, transient torsional torques, turbine-blade vibrations and fast bus-transfer. Section 3 discusseslow-frequency electrical dynamics, as a result of control sys-tems interactions. Section 4 provides analysis guidelines forharmonic interactions and resonance phenomena. The phe-nomenon of ferroresonance is discussed in Section 5.

2. LOW-FREQUENCY ELECTROMECHANICAL DYNAMICS

This section provides modeling and analysis guide-lines for low-frequency dynamics which involve electrome-chanical oscillations. The phenomena which are covered inthis section are torsional oscillations, transient torques, tur-

DRAFT July 98: Note: Figure numbering to be fixed. Some figures are too light

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bine-blade vibrations, and bus-transfer.

2.1 DEFINITIONS

2.1.1 Torsional Oscillations [2.1, 2.2, 2.3, 2.4, 2.5]

Shaft system of a steam turbine-generator experiences tor-sional oscillations when one or more of its natural oscillatorymodes, usually at subsynchronous frequencies, are excited.Sustained or negatively damped torsional oscillations occurwhen a turbine-generator shaft system exchanges energy withan electrical system at the shaft oscillatory modes. This ex-change of energy can exist if the electrical system is equippedwith either series capacitors or HVDC converter stations. Thephenomenon of torsional oscillations can also exist as a resultof interaction between the shaft system of a steam turbine-generator and

the generator excitation systems through either AVR or PSScontrol loops,

electronically controlled governor system,

voltage control loop of an electrically close static VAR. compen-sator (SVC)

large electric arc furnaces.

Although AVR, PSS and governor system can excite torsionaloscillations, the excitation is primarily due to inadequate con-trol design considerations and can be avoided by introducingfilters in the control circuitry. Thus, this report does not con-sider the generator controls as the main contributors to thephenomenon of torsional oscillations (Table 2.1).

The phenomenon of torsional oscillation is referred to as sub-synchronous resonance (SSR) when it is a result of interactionbetween a shaft system and a series capacitor compensatedtransmission line. The problems associated with the phenom-enon of small-signal torsional oscillations are:

i ) Sustained or even negatively damped oscillations whichare considered as small-signal instability problems, and

ii ) ( loss of life of turbine-generator shaft segment(s) due to thefatigue induced in the shaft segment(s) as a result of eachoscillatory cycle.

2.1.2 Transient Torsional Torques [2.1, 2.2, 2.3, 2.4, 2.5]

The shaft segments of turbine-generator units are exposed tolarge-amplitude, oscillatory, mechanical stresses as a result ofelectric network faults, and planned and unplanned switchingincidents. Frequencies of the shaft mechanical stresses are

the natural frequencies of the shaft torsional oscillatorymodes. Usually, the oscillatory mode at the first torsional fre-quency dominates the shaft transient oscillations. The majorincidents which result in severe shaft stresses are: line-to-linefaults, three-phase faults, fault clearing, automatic reclosures,and out-of-phase synchronization. The amplitudes of theshaft transient stresses can be particularly large when the net-work is equipped with series capacitors.

High amplitude shaft mechanical stress can induce significantfatigue in the shaft segments and result in noticeable shaftlife-time reduction during each oscillatory cycle. Such oscil-lations may even result in catastrophic shaft failure. The pri-mary purpose of time-domain investigation of turbine-generator shaft mechanical stresses is to identify the peaktorques imposed on the shaft segments, after system distur-bances. Transient shaft mechanical stresses calculated basedon time-domain simulation methods also can be used to esti-mate shaft loss of life as a result of system disturbances.

2.1.3 Turbine-Blade Vibrations [2.6]

Frequencies of turbine-blade vibrational modes areusually within 90 to 250 Hz, and constitute supersynchro-nous frequency modes. Identification of supersynchronousfrequency modes and their corresponding frequencies is bestcarried out by solving elasticity equation of the shaft systemas a continuum, based on the use of finite element methods.This approach is beyond the scope of this report and usuallycarried out by turbine manufacturers.

In this report, the objective is to investigate the impact oflarge-signal disturbances on those supersynchronous frequen-cy natural modes which are the reason for turbine-blade vibra-tions. Thus the required model is tailored to representparticular supersynchronous modes and not all of them.

The concern with turbine-blade vibrations is fractureand loss-of-life of the blades due to the fatigue induced in theblades by repetitive or sustained oscillations. Vibrations of tur-bine-blades can be excited by large-signal electrical distur-bances, e.g. faults, fault clearing, line switching, reclosure, andout-of-phase synchronization.

2.1.4 Fast Bus Transfer [2.7, 2.8, 2.9]

Motors and other loads in utility and heavy industrial applica-tions are supplied during normal operation from a preferredpower source. An alternate power source is normally provid-ed to supply such motors and other loads during planned shut-downs and upon loss of normal power from the preferredpower source. The process of disconnecting the motors andother loads from one source and reconnecting to an alternatesource is commonly defined as ìbus transferî. Manual transfer

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means are normally provided to allow transferring the motorsand other loads from one power source to the other. However,upon loss of the preferred power source, the motors and otherloads are automatically transferred to the alternate powersource. This automatic transfer is necessary to allow uninter-rupted operation of the motors and other loads important topersonnel safety and process operation. This report does notaddress the concept of bus transfer by means of semi-conduc-tor switches [2.23].

The normal and alternate power source connections are al-ways selected such that they are in phase. Therefore, manualtransfers can be accomplished in a make-before-break, i.e.,the motors and loads are connected to the second powersource before the first power source is disconnected. In thisoverlapping transfer, the power supply is not interrupted andthe motors are not subjected to transients. However, duringautomatic transfers, the motors may be disconnected fromboth power sources for a short duration depending on the typeof transfer and the associated circuit breakers operating times.The time during which the motors are disconnected from bothpower sources is termed the ìdead timeî. Dead time is usuallybetween two cycles to 12 cycles. If the relative angle betweenthe motor residual voltage and the power source voltage be-comes large enough at the time of reconnection with signifi-cant residual voltage remaining, the resultant voltage

between the power source and the motor will produce an in-rush current. The inrush current may be significantly largelythan the normal full voltage staging current. Such high inrushcurrents cause high winding stresses and transient shafttorques which can damage the motor and/or the driven equip-ment.

The most common bus transfer scheme is the fast bus transferscheme. In this scheme, opening of the normal power sourcebreaker initiates closing of the alternate power source breakerwithout intentional time delay. Fast bus transfer operationsresult in the motors being disconnected from both powersources for a duration of as short as two cycles to as long as12 or more cycles.

Presently, there are no generic criteria to ensureacceptable fast bus transfer operations. Therefore, it is nec-essary to analyze the transient behavior of motors during fastbus transfer operations. The analysis should be on a case bycase basis to ensure that the motors will not be subjected toexcessive inrush currents and/or shaft transient torques.

2.2 MODELING GUIDELINES

2.2.1 Study Zone

In contrast to an inter-area, electromechanical, oscillatory

mode which propagates almost through the entire of an inter-connected electric network, the phenomena described in Sec-tion 2.1 are experienced only within a limited part of thenetwork. The section of the network which experiences thephenomenon of interest, and must be represented in adequatedetail for the study of the phenomenon, is referred to as the“Study Zone” The rest of the network is referred to as the “ex-ternal system” The external system is represented by anequivalent model. Identification of border nodes of the studyzone for a meshed network requires significant familiaritywith the network, as well as engineering judgment. As ofnow, there is no straightforward and systematic approach toidentify the border nodes. One approach involves multipleharmonic analyses of the system under investigation asboundaries are extended to identify if new resonant frequen-cies (at the frequency range of interest) with low dampings ex-ist.

Proper determination of the study zone can exert a major im-pact on the investigations of torsional dynamics and transienttorques. Comparatively, the impact of the study zone on thevibrations of turbine blades is less significant. Identificationof the study zone for bus transfer studies is relatively straight-forward.

2.2.2 Component Model

Table 1 identifies the study zone components and their equiv-alent models for investigations of slow transient phenomena.Further explanation of the system components are given in thefollowing sections.

2.2.2.1 Synchronous Generator Electrical System [2.10]

Figure 2.1 shows a second-order and a third-ordermodels of a synchronous machine. Inclusion of the differen-tial leakage inductance Lf1d in the second-order model isrecommended. The differential leakage inductance hasnoticeable influence on the damping, and the range of insta-bility of each torsional mode, (with respect to series compen-sation level), part icularly for a salient pole machine.However, Lf1d does not influence the phenomenon of bladevibrations.

Representation of machine electrical system based onthe third-order model, Fig. 2.1, is more accurate. Inclusion ofthe differential leakage inductance Lf12d in the third-ordermodel has the same impact as that of Lf1d for the second-ordermodel. Magnetic saturation of a synchronous machine, both ond-axis and q-axis, does not have any significant impact on thephenomenon of small-signal torsional oscillations, but has pro-

Table 1: Component Model

Component TorsionalOscillations

TransientTorques

Turbine-B ladeVibrations

Fast BusTrans fer

SynchronousGenerator’s

Electrical System

Second-OrderModel and

Preferably Third-Order Model (d-q-o

Model)

Third-OrderModel (d-q-o

Model)IncludingSaturation

Third-Order Model(d-q-o Model)

IncludingSaturation

Notapplicable

Turbine-GeneratorShaft System

Mass-Spring-Dashpot Model

Mass-Spring-Dashpot Model

DetailMass-Spring-

Dashpot Model

NotApplicable

PowerTransformer

ConventionalLow-FrequencyModel including

SaturationCharacteristic

ConventionalLow-FrequencyModel including

SaturationCharacteristic

ConventionalLow-FrequencyModel including

SaturationCharacteristic

ConventionalLow-

FrequencyModel

includingSaturation

Characterist icTransmiss ion Line Equivalent-%

ModelEquivalent-%

ModelEquivalent-%

ModelNot

ApplicableSeries/Shunt

CapacitorIdeal Capacitor Ideal Capacitor Ideal Capacitor Ideal

CapacitorSeries/Shunt

ReactorSeries R-L Series R-L Series R-L Series R-L

Static Load Fixed ImpedanceLoad

Fixed ImpedanceLoad

Fixed ImpedanceLoad

FixedImpedance

LoadLarge Motor Load d-q-o Model of

Electrical System,Mass-Spring-

Dashpot Model ofShaft System

Voltage SourceBehind Fixed

Impedance

Voltage SourceBehind FixedImpedance

d-q-o Modelof

ElectricalSystem,

Mass-Spring-DashpotModel of

Shaft SystemHVDC Converter

Stat ionDetailed Model of

Converter andLinearized

(Simplified) Model of Controls

Detailed Models ofConverter and

Controls

Detailed Models ofConverter and

Controls

NotApplicable

SVC Detailed Model ofPower Circuitry and

Linearized(Simplified)

Model of Controls

Detailed Model ofPower Circuitry

and Controls

Detailed Model ofPower Circuitry

and Controls

NotApplicable

Circuit Breaker Ideal Switch Ideal Switch Ideal Switch Ideal SwitchGeneratorControls

Unimportant Unimportant Unimportant NotApplicable

Protection System` Unimportant Series CapacitorOvervoltages

Protection System

Series CapacitorOvervoltages

Protection System

NotApplicable

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nounced impact on transient torques and blade vibrations.

Fig. 2.1. Synchronous machine 2nd-order and 3rd-order models

2.2.2.2 Turbine-Generator Mechanical System [2.11, 2.12, 2.13]

Figure 2.2 shows a six-mass shaft system and its equivalentmass-spring-dashpot model. The mass-spring-dashpot modelof Fig. 2.2 assumes that (1) the high-pressure turbine (HP), theintermediate-pressure turbine (IP), the low-pressure turbines(LPA and LPB), the generator rotor (G), and the excitor(EXC) are rigid masses, and (2) each shaft section is com-posed of a spring constant (Kij ) and a cyclic damping (Dij ).The main shortcoming of the model is that neither the shaftcyclic dampings (Dijîs) nor the viscous dampings (Diís) canbe directly measured or calculated. Neglecting the dampingsprovides the most pessimistic dynamic response, which is of-ten the objective of an investigation. The discussion of [2.11]provides further description of the mass-spring-dashpot mod-el. Figure 2.3 shows a mass-spring-dashpot model of the tur-bine-generator set of Fig. 2.2 for investigation of turbine-blade vibrations. This model represents blades of turbine sec-tions as lumped masses [2.6].

In most studies, the power plant under consideration iscomposed of more than one turbine-generator unit. If all the tur-bine-generator units are nominally identical, and under almostequal loading conditions, they can be represented by a single,equivalent turbine-generator unit. Otherwise, each turbine-gen-

erator unit must be separately represented.

In most studies, the power plant under consideration is com-posed of more than one turbine-generator unit. If all the tur-bine-generator units are nominally identical, and under almostequal loading conditions, they can be represented by a single,equivalent turbine-generator unit. Otherwise, each turbine-generator unit must be separately represented.

Fig. 2.1. Turbine-generator shaft system and its mass-spring-dashpot mode

Fig. 2.2. Mass-spring dashpot model of the turbine-generator for turbine-blade vibrational studies (mechanical damping is neglected)

2.2.2.3 Power Transformer

Classical low frequency transformer model with proper con-nections at both HV and LV sides is adequate for representa-tion of each power transformer within the Study Zone. Figure2.4 shows the classical model of a single-phase transformerfor simulation of low frequency dynamics. No-load V-I mag-netic saturation characteristic can be used as a fair approxima-tion of core saturation for the phenomena of interest. A three-phase transformer model is developed based on proper con-nections of primary and secondary windings of the single-phase model of Fig. 2.4.

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Fig. 2.1. Low frequency model of a single-phase transformer.

2.2.2.4 Transmission Line

Equivalent- is an accurate model for representation of a longor medium length transmission line for the phenomena underinvestigation. In many reported studies, the shunt capacitivebranches of the line model are also neglected. Shunt capaci-tive branches of the line model do not have any major impacton the

system subsynchronous frequency resonant modes, but theireffect on supersynchronous oscillatory modes can be notice-able. Shunt capacitive branches, particularly in the case oflong lines, have a significant effect on the system steady-stateconditions, e.g. the magnitude of generator power angle.Therefore, depending on the operating conditions, they mayhave a noticeable impact on the dampings of low frequencyoscillatory modes.

2.2.2.5 Series and Shunt Capacitor Banks

Series capacitors are the main cause of severe shaft torsionaloscillations and their presence in each transmission section isaccurately represented by three lumped, ideal, capacitorbanks. Similar to the shunt capacitive branches of a transmis-sion line, shunt capacitor banks do not have any direct impacton the shaft dynamics. However, since shunt capacitors alterthe voltage profile of the system, they may noticeable impacton the dampings of the oscillatory modes depending on theoperating condition. Thus, representation of shunt capacitorsin the system model, particularly under heavy loading condi-tions, is recommended.

2.2.2.6 Shunt Reactor

Shunt reactors can have a noticeable impact on the steady-state operating conditions, e.g. voltage profile, which can im-pact the dampings of the low frequency dynamics. Thus, rep-resentation of shunt reactors, particularly under light loadingconditions, is recommended.

2.2.2.7 Loads

ìFixed Impedanceî model is an adequate load represen-tation when turbine-generator shaft dynamics are of concern.However, if an induction motor load or a synchronous motorload is comparable to the MVA rating of the turbine-generatorunder consideration, fixed impedance representation of the loadmay result in erroneous conclusions. Under such conditions, theload is best represented by either an equivalent induction motoror an equivalent synchronous motor.

Motor loads must be represented in details for fast bus transferphenomenon. For these studies, parallel identical motor loadscan be lumped in an equivalent motor load.

2.2.2.8 HVDC Converter Station

Shaft dynamics of a turbine-generator can be excited as a re-sult of interaction between the turbine-generator and eitherrectifier current-control or the inverter extinction angle (volt-age) control of an HVDC link. Thus, if both the rectifier andthe inverter stations are within the study zone, both converterstations, dc line, and the associated controls, with adequatelevel of sophistication, must be represented in the systemmodel.

Each arm of a six-pulse converter is modelled by an idealswitch including series and parallel snubber circuits. Theswitch represents a group of series/parallel connected diodesor thyristor valves. The three-phase transformer model ofSection 2.2.2.3 can adequately represent converter transform-er of a six-pulse HVDC converter for low frequency studies.Connection of two six-pulse converter models with propertransformer models constitutes a 12-pulse HVDC convertermodel. The model of each pole of an HVDC converter stationis realized by assembling an adequate number of 12-pulseconverter models. If small-signal dynamics are of concern,e.g. torsional oscillations, a bipole HVDC link can be approx-imated by an equivalent monopolar link. Otherwise, e.g. forinvestigation of transient torques, bipolar representation isnecessary. Models of smoothing reactors and ac/dc filters aredeveloped by proper connections of lumped RLC elements.Multiple -sections is the recommended model of an HVDCline.

Block diagram of the controls of a bipole Hvdc system fortime-domain simulation is given [2.14]. Further details of thecontrol blocks are available in Chapter 8 of [2.15].

When the inverter station is not within the Study Zone, the in-verter station and the dc line can be represented by an equiv-alent controlled voltage source, and only the rectifier stationand its controls must be modelled in details. Similarly, therectifier station and the dc line can be modelled as an equiva-lent controlled current source and only the inverter station andits control system be represented in detail, if the rectifier sta-tion is not within the Study Zone.

An HVDC installation may have multiple auxiliary controlsfor various purposes, e.g. damping inter-area oscillations, fre-quency control, and reactive power/voltage modulations. It isrecommended to represent such auxiliary controls in the sys-tem model to identify their possible adverse impacts on thetorsional oscillatory modes.

2.2.2.9 Static VAr Compensator (SVC)

Field experience and theoretical studies indicate that possible

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adverse effect of an SVC on the shaft torsional dynamics arenot as severe when compared with that of an HVDC converterstation [2.16]. However SVCs have been recognized as ef-fective countermeasures for shaft torsional dynamics. A con-ventional SVC is composed of thyristor-switched capacitors(TSCs) and thyristor-controlled reactors (TCRs) [2.17]. Dur-ing small-signal dynamics, e.g. torsional oscillations, an SVCcan be approximated as fixed capacitors (FCs) and TCRs.thy-ristor valves in each arm of either the TCR or the TSC aremodelled as two equivalent ideal switches including the par-allel snubber branch. The three-phase transformer model ofSection 2.2.2.3 can adequately represent an SVC transformerfor low frequency studies. Controlled reactor, switched ca-pacitor and the SVC filter components are represented in thetime-domain simulation model by proper combinations oflumped RLC elements. Chapter 9 of [2.15] and reference[2.18] provide details of the controls of an SVC for time-do-main simulation. Similar to an HVDC converter station, anSVC may be equipped with auxiliary controls, e.g. supple-mental SSR damping control. Thus, all the closed-loop con-trols must be represented in the simulation model to attain arealistic time-response of an SVC.

2.2.2.10 Generator Controls

Conventional generator controls, i.e. automatic voltage regu-lator (AVR), power system stabilizer (PSS), and governorsystem generally do not have major (positive or negative) ef-fects on turbine-generator shaft dynamics. Although there arereports of torsional excitation as a result of PSSs and electron-ically controlled governors, the adverse effect can be prevent-ed by introducing filters in the control circuitry. Thus, thedynamics of excitation and governor systems are neglected,and the input mechanical power and the generator field volt-age are considered as constant values for time-domain inves-tigation of shaft dynamics. For those particular cases whereeither AVR, PSS or governor may aggravate torsional oscil-lations [2.1, 2.2, 2.3, 2.4, 2.5], they can be represented by theirlinearized models in the system model.

2.2.2.11 Protection System

Overvoltage protection system of series capacitor can have asignificant impact on large-signal torsional torques and tur-bine-blade vibrations following network transients. Thus, forthe simulation of these two phenomena, the series capacitorovervoltage protection scheme including Zn0 varistor andthe associated bypass logic and power circuitry must be rep-resented in the system model.

2.3 TEST SYSTEMS

2.3.1 Torsional Oscillations

The IEEE Working Group on Subsynchronous Resonance hasintroduced two benchmark models for time-domain simula-tion of turbogenerator torsional oscillations [2.12, 2.13]. The

benchmark models have been extensively used for time-do-main as well as frequency-domain investigation of the phe-nomenon of torsional oscillations. Numerous study results,using the benchmark model, have been published in the IEEEPES Transactions [2.1, 2.2, 2.3, 2.4, 2.5].

Time-domain simulation and frequency-domain eigen analy-sis are widely used as complementary approaches for recipro-cal verification of torsional studies.

2.3.2 Transient Torques

The first and the second IEEE benchmark models for Small-Signal torsional studies introduced in Section 2.3.1 also havebeen extensively used for transient torque studies. Due to thenonlinear nature of large-signal torsional oscillations, digitaltime-domain simulation is the only approach to investigatethe phenomenon. There are no measurement results regardingtransient torques in the widely circulated technical literature.Thus, simulation results cannot be readily compared with ac-tual field tests. At this stage, a general verification rule is toensure that the simulation results satisfy the well understoodbehavioral patterns and immediately after switching inci-dents.

2.3.3 Turbine-Blade Vibrations

The radial power system of [2.6] is recommended as the testsystem. The system is composed of a multi-mass tubine-gen-erator which is connected to an infinite bus through two par-allel lines. The system can be used to study blade vibrationsof low-pressure turbine sections. It should be noted that incontrast to the shaft torsional oscillations (either small-signalor large-signal), the blade vibrations are not readily quantifi-able from time-domain responses. Thus, a frequency spec-trum analysis, e.g. FFT should be conducted on the timeresponse to obtain the relative amplitudes and frequencies ofthe blade dominant oscillatory modes.

A qualitative verification of the simulation results can be ob-tained based on the comparison of the frequencies of the bladevibrations, deduced from FFT of the simulation results, withthose provided by the turbine manufacturer.

2.3.4 Bus-Transfer

The simplified system introduced in [2.7], is recommended asthe test system for bus transfer studies. Typical motor loaddata for simulation studies are available in [2.19].

Ideally, validating a model of a fast bus transfer operationshould include validating the individual motor models and thecircuit breakers operating times. Individual motor models can

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be validated by simulating motor starting and running condi-tions and comparison of other simulation results to data re-corded during an actual motor instantaneous current, power,apparent power (VA), and speed. However, since a typicalbus transfer model may include 15 or more motors, it may notbe practical to validate individual motor models.

To establish the dead time and a range of the expected accu-racy, it is recommended to perform a fast bus transfer test witha few motors connected and simulating the test conditions us-ing motor models based on the manufacturer supplied data.Since measuring the transient variations in the motor shafttorque is a complex task, it is suggested to monitor, simulateand compare the following parameters:

bus instantaneous voltage

individual motors instantaneous currents

total instantaneous currents through the alternate source circuitbreaker

individual motors instantaneous power and apparent power

motor speed

errors can then be determined by comparison of the test datawith simulation results. A statistical measure of the expectedmodel accuracy may be based on the method of the root of thesum of the squares of the individual errors (RSS). The expect-ed error in the actual bus transfer analysis would be less thanthe RSS of the errors derived due to the larger number of mo-tors included. References [2.7, 2.8, 2.9] provide some test re-sults which can be used as general guidelines to verify thepattern of behaviour of the system variables due to the bustransfer phenomenon.

Appendix A provides further information regarding fast bus-transfer and typical time-domain simulation results.

3. CONTROL SYSTEM INTERACTIONS

3.1 DEFINITION

Closed-loop controls associated with various power systemapparatus, e.g. SVC controls, HVDC converter controls, con-trols of adjustable series capacitors, generator automatic volt-age regulators (AVRs), and generator power systemstabilizers (PSSs) have natural oscillatory modes at frequen-cies in the subsynchronous frequency range of 1 to 35 Hz.Depending upon the ìelectrical distanceî between the appara-tus, the associated closed-loop controls can interact and resultin either unsatisfactory operation of the device(s), sustainedoscillations, or even small-signal instability. Another type ofcontroller interactions is the interaction between a closed-

loop control system and a natural oscillatory mode of an ap-paratus. One practical case of controller interaction phenom-enon is that of multiple SVCs [3.1]. The problem of controllerinteractions attracts more attention as the number of powerelectronic based devices increases.

3.2 STUDY ZONE

When two or more interacting controls are identified, thestudy zone encompasses those system components whichmust be represented with adequate details to investigate theinteraction phenomenon. Since the frequencies of interest arein the subsynchronous frequency range, the study zone is usu-ally identified based on the criteria used for the study zone oftorsional oscillations, Section 2.2.1.

3.3 DEVICE MODELS

3.3.1 Generator Electrical System

If a turbine-generator controls system, i.e. governor system,AVR, PSS, and its torsional mechanical modes do not partic-ipate in the interaction phenomenon, then the generator elec-trical system can be modelled as an ideal, fixed-frequency,three-phase, voltage source behind a three-phase inductance.Otherwise, the second-order model or the third-order modelfor Section 2.2.2.1 should be used.

3.3.2 Turbine-Generator Mechanical System

When the generator electrical system is represented either bythe second-order model or the third-order model, the shaftsystem should be represented by the mass-spring-dashpotmodel of Section 2.2.2.2. Otherwise, the shaft dynamics andconsequently its oscillatory modes can be ignored.

3.3.3 Power Transformer

When a generator is represented by a voltage source behind aninductance, the generator step-up transformer is representedby a series RL branch in each phase. Otherwise, the low-fre-quency transformer model of Section 2.2.2.3 should be usedto represent the transformer in the overall system model. Ingeneral, the low-frequency transformer model is an adequaterepresentation of a power transformer for investigation ofcontroller interaction phenomenon. The harmonics generatedas a result of transformer saturation have much higher fre-quencies than those of controller interactions. Thus the satu-ration does not have any major role in the controllerinteraction phenomenon.Transmission Line

Per-phase equivalent- model is an adequate representation ofa line for investigation of the phenomenon of control interac-tions.

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3.3.4 Series and Shunt Capacitor Banks

Presence of series capacitors in a transmission line can alterthe level of controller interactions or even excite the interac-tion mode(s) [3.2]. Impacts of parallel (shunt) capacitors onthe controller interactions is significantly less than that of a se-ries capacitor. Both series and shunt capacitors can be ade-quately represented by three-phase lumped capacitor banksfor investigation of controller interactions.

3.3.5 Shunt Reactor

Similar to shunt capacitors, fixed, shunt inductors do not havea major impact on controller interactions. Nevertheless, shuntreactors are adequately represented by three-phase lumped in-ductances for investigation of controller interactions.

3.3.6 Loads

ìFixed Impedanceî model of loads within the study zone pro-vides accurate representation of the loads for investigation ofcontroller interaction phenomenon. Very large load areas canalso be represented by an ìinfinite busî with proper phase an-gle to draw the required power at the fundamental frequency.The impacts of various load models on the phenomenon ofcontroller interactions have been neither adequately investi-gated nor reported in the literature.

3.3.7 HVDC Converter Station

Rectifier or inverter firing angle controls can interact withother system controllers, e.g. SVC controls, and excite controlinteraction phenomenon. Contribution of an HVDC convert-er station to the controller interaction phenomenon is primari-ly as a result of the natural oscillatory modes of its controlloop(s) and not due to the harmonics generated by the valveswitchings. If both inverter and rectifier are within the studyzone, both converter stations, the connecting dc link, and allthe associated controls must be represented in the study mod-el. Further details on representation of each 12-pulse convert-er are given in Section 4.3.8.

All the steady-state continuous controls of rectifier and invert-er stations, e.g. DC current control, DC voltage control, ACvoltage control or reactive power control, real power control,and frequency control must be represented in the model. Thecontrol model must adequately represent firing and synchro-nization schemes used for the converter values.

When the inverter station is not within the Study Zone, the in-verter station and the dc line can be represented by an equiv-alent controlled voltage source, and only the rectifier stationand its controls be modelled in detail. Similarly, the rectifierstation and the dc line can be modelled as an equivalent con-

trolled current source and only the inverter station and its con-trol system be represented in details, if the rectifier station isnot within the Study Zone.

3.3.8 Static VAR Compensator (SVC)

A conventional SVC, which is composed of thyristor-con-trolled reactor (TCR) and fixed capacitor (FC), can interactwith an HVDC converter station or other SVCs through theirclosed-loop controls and excite the phenomenon of controllerinteraction. An SVC model for control interaction studiesshould accurately represent the SVC and its control system inthe frequency range of 5 to about 45 Hz. The steady-statecontinuous controls including all the auxiliary loops, e.g. SVCvoltage control and SSR damping control, must be represent-ed in the simulation model. Further details of an SVC small-signal model are available in Section 4.3.9.

3.3.9 Generator Controls

Conventional synchronous generator controls, i.e. governorsystem, AVR, and PSS are designed to perform correspond-ing tasks at very low frequencies (0.1 to 2.5 Hz), and are notthe prime cause of controller interactions. Thus the dynamicsof the generator controls often can be neglected for the inves-tigation of controller interaction phenomenon. However, iftheir presence in the overall system model is required, theirconventional low-frequency, linearized models would suf-fice.

3.3.10 Harmonic Filters

Harmonic filters of SVCs are adequately represented bylumped RLC circuits. Similarly, ac side and dc side harmonicfilters of HVDC converter stations are represented by lumpedRLC circuits.

3.4 TEST SYSTEM

Figure 3.1 shows the recommended test system for the inves-tigation of controller interactions [3.1] of multiple SVCs. De-pending upon the operating conditions and parameters, thevoltage control loops of the SVCs can interact and exhibitsmall-signal instability. Inclusion of control limits in themodel is not necessary since the control interaction consti-tutes a linear phenomenon and nonlinearities are not involved.icomp in Fig. 3.1 is the total current of each TCR and the as-sociated capacitor bank. The systems data and initial condi-tions are given in [3.3, 3.4].

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Fig. 3.1. Test systems for investigation of controller interaction phenomena

3.5 VERIFICATION OF SIMULATION RESULTS

Small-signal controller interactions also can be investigatedbased on the linearized model of the system under investiga-tion, using eigen analysis approaches [3.6, 3.7. 3.8]. Bothtime-domain simulation and the eigen analysis of controllerinteractions are conducted for qualitative comparison of theresults and their mutual verifications.

4. HARMONIC INTERACTION AND RESONANCE

4.1 DEFINITION

Operation of power electronic converters, e.g. an HVDC con-verter station, is characterized by generation of current and/orvoltage harmonics. These harmonics are classified as charac-teristic and noncharacteristic harmonics. In contrast to char-acteristic harmonics, amplitudes and orders ofnoncharacteristic harmonics cannot be accurately predictedby conventional analytical techniques, e.g. Fourier analysis.Time-domain simulation methods provide an alternative ap-proach for the analysis of noncharacteristic harmonics. Ref-erences [4.1] and [4.2] provide a comprehensive descriptionof the physical phenomena resulting in harmonic interactions.

The main concerns with the presence of noncharacteristic har-monics are (1) harmonic interactions and/or resonance [4.1],and (2) the interference phenomenon [4.2].

Radio and telephone interference as a result of dc side har-monics of HVDC converters is a well known phenomenon.Also, second and third harmonic instability of ac systems dueto harmonic modulation characteristic of HVDC converterhas been encountered in the existing installations [4.1].

4.2 STUDY ZONE

Those system apparatus which either generate or interact withthe frequencies of interest must be represented in adequate de-tails, and they identify the study zone. Also transmission lineswhich connect the apparatus within the study zone must berepresented with adequate accuracy in the frequency range ofinterest in the system model. The remainder of the systemwhich neither generates nor interacts with the harmonics canbe simplified and represented by its frequency dependentequivalent model [4.3].

4.3 DEVICE MODEL

4.3.1 Generator Electrical Model

Rotating machines within the study zone do not contribute tothe harmonic interaction phenomenon and can be representedby equivalent voltage sources behind fixed RL elements.

4.3.2 Turbine-Generator Mechanical System

dynamics do not play any noticeable role in the harmonic in-teraction phenomenon. Thus, the shaft model can be readilydiscarded from the overall system model.

4.3.3 Power Transformer

Both stray capacitances and magnetic saturation characteris-tics of power transformers within the study zone can have sig-nificant impact on power system harmonics. The magneticsaturation characteristic has a deterministic impact on the sec-ond harmonic instability and can be fairly represented by theno-load V-I characteristic in the magnetization branch of thetransformer. The winding stray capacitances to the tank havea noticeable effect on the interference phenomenon [4.2]. Thestray capacitance can be adequately modelled by a single ca-pacitance from the winding terminal to the ground [4.2].

4.3.4 Transmission Lines

Transmission lines within the study zone are best representedas distributed parameter lines including parameter frequencydependency. However, if the frequency range of interest doesnot cover high frequencies (more than 300 Hz), each trans-mission line can be represented by multiple sections.

4.3.5 Series and Shunt Capacitor Banks

Series and shunt capacitors have deterministic impacts on se-ries and parallel resonant frequencies of the system and mustbe represented in the overall system model for harmonic stud-ies. Both series and shunt capacitors are adequately repre-sented by lumped three-phase capacitor banks.

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4.3.6 Shunt Reactor

Similar to series and shunt capacitors, shunt reactors also in-fluence the system natural resonant frequencies and must berepresented in the system model. A shunt reactor is adequate-ly represented by a three-phase lumped reactor bank.

4.3.7 Loads

ìFixed Impedanceî model is a valid representation for loadswithin the study zone, unless the load is known to have partic-ular resonant frequency or generates particular harmonic(s)which can affect the harmonic phenomenon of interest.

4.3.8 HVDC Converter Station

The HVDC converter station is one of the majorsources for generation of harmonics which cause interferenceand/or instability of electrical power networks [4.1]. Therequired model of an HVDC converter station for studying inter-ference and harmonic interaction phenomena is the same as themodel described in Section 3.3.8.

Fig. 4.1. Lumped equivalent of the stray capacitances of a 12-pulse HVDC converter and the converter transformers

• Exact parameters of the snubber circuits of each valve chain should be included in the model [4.2]. It should be noted that in some transients programs, the exact param-eters of snubber circuits cannot be used. Unrealistic snubber circuits are required by these programs to avoid numerical problems.

• The model used for the valve firing circuitry should gener-ate actual firing instants. Otherwise, the amplitudes and orders of noncharacteristic harmonics will be noticeably distorted as a result of improper firing instants [4.4].

• Stray capacitances of the converter transformers, valve structure, and smoothing reactor must be adequately rep-resented in the system model [4.2]. The impact of stray capacitances can be represented by a set of lumped

capacitors, Fig. 4.2.• Magnetic saturation characteristics of converter trans-

formers must be included in the model [4.4].

4.3.9 Static VAR Compensator (SVC)

Static VAR compensators have not been reported as a sourceof interference phenomenon and harmonic interactions.However, in the vicinity of HVDC converter stations andFACTS devices, a static VAR compensator can aggravateharmonic related issues [4.4]. The required SVC model fortime-domain investigation of harmonic problems is the sameas the model described in Section 3.3.9, except for the follow-ing differences:• Snubber circuits of each valve chain must be included in

the simulation model.• The model of valve firing circuitry must be capable of gen-

erating exact firing instants.

Operating point and parameter values of a SVC can readilyinfluence series/parallel resonant frequencies of a networkand consequently tune the system for resonant conditions, e.g.second harmonic resonance [4.4]. The above model can alsobe used for this class of resonant conditions which normallyoccur at noncharacteristic harmonics generated by powerelectronic circuits.

4.3.10 Generator Control

Automatic voltage regulator, power system stabilizer, andgovernor system do not influence harmonic related problem.Thus, their model can be excluded from the system model fortime-domain harmonic studies.

4.3.11 Harmonic Filters

SVC and HVDC harmonic filters must be modelled as de-scribed in Section 3.3.11.

4.4 TEST SYSTEM

The HVDC-AC system of Fig. 4.3 is proposed as the test sys-tem for the investigation of harmonic interactions phenomenaand the second harmonic instability issues.

Fig. 4.1. HVDC-AC test systems for time-domain simulation of harmonic interaction phenomena, and the second harmonic instability

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The HVDC link is a ±450-kV, 936-km, 2000-MW, 12-pulse,bipole configuration. Each pole is equipped with ac side anddc side filters. The inverter neutral is equipped with a neutralfilter. The rectifier neutral is solidly grounded close to the sta-tion. Parameters and control system of the Manitoba HydroísBipole-2 HVDC system [4.5, 4.6] are adopted for the test sys-tem of Fig. 4.3.

The rectifier ac system, Fig. 4.3, is composed of an equivalent26-kV source which is connected to the rectifier ac busthrough a 26/235-kV transformer and a short 230-kV line.The effective short circuit ratio (ESCR) of the rectifier ac sys-tem is 3.6.

The inverter ac system consists of a 230-kV ac source whichis connected to the inverter station through a 500-kV, 832-kmtransmission system. The transmission line is equipped with240/525-kV Y - Y connected transformer at the source side.The ac line is divided in three sections, Fig. 4.3. Each inter-mediate station is equipped with a 400 MVA capacitor bankfor voltage profile improvement. Loads #1, #2, and #3 are rat-ed at 920-MVA, 400-MVA and 360-MVA respectively. Theinverter station is also equipped with an SVC which can ad-just its reactive power from 180-MVAR inductive to 510-MVAR capacitive. Electrical parameters of the inverter acsystem are given in [4.7]. The ESCR of the inverter ac side is2.2.

References [4.8] and [4.9] provide various HVDC/ac bench-mark models that also can be used for the analyses of harmon-ic interactions and resonance phenomena. The first HVDCbenchmark model [4.8] proposed by CIGRE WG 14-02 alsoexhibits second harmonic resonance and can be adopted forinvestigation of harmonic instability phenomenon. This sys-tem is less complicated as compared with that of Fig. 4.3.Reference [4.10] provides a very simple circuit configurationwhich exhibits instability due to switching characteristic ofthyristor-controlled reactor (TCR). A set of time-domainsimulations results of the test systems of Fig. 4.3 is given in[4.4].

4.5 VERIFICATION OF SIMULATION RESULTS

There are several technical papers which deal with analysisand measurement of noncharacteristic harmonics of HVDCconverter stations [4.2, 4.11, 4.12, 4.13]. The primary con-cern in these papers is the dc side triplen harmonics whichcause interference and not the second harmonic instabilityproblem. Reference [4.15] provides a modelling approach forrepresentation of a six-pulse converter with respect to the sec-ond harmonic for eigen analysis. Such eigen analysis ap-proach can be used as an alternative technique for validationof simulation results. Eigen analysis studies based on the

modelling approach of [4.15] are reported in [4.14]. Refer-ences [4.16, 4.17, 4.18] provide a comprehensive and funda-mental description of the harmonic interaction phenomenon.However, there are not that many measurements and investi-gation of the harmonic interaction phenomenon to establish amethod for verification of time-domain simulation studies.Reference [4.19] introduces an alternative approach based onfrequency scanning method for identification of harmonic in-stabilities in HVDC systems. This approach may be used forqualitative verification of digital time-domain simulation ap-proach.

5. FERRORESONANCE

In this section, ferroresonance is introduced and a generalmodeling approach is given. An overview of available litera-ture and contributors to this area is provided. A simple caseof ferroresonance in a single phase transformer is used to il-lustrate this ìphenomenonî. Three phase transformer corestructures are discussed. Ferroresonance in three phasegrounded-wye distribution systems is described and illustrat-ed with waveform data obtained from laboratory simulations.Representation of the study zone is discussed, modeling tech-niques are presented, and implementation suggestions aremade. Three case studied are presented. Transformer repre-sentation is critical to performing a valid simulation. The di-rection of ongoing research is discussed, and the reader isadvised to monitor the literature for ongoing rapid improve-ments in transformer modeling techniques.

5.1 INTRODUCTION TO FERRORESONANCE

Research involving ferroresonance in transformers has beenconducted over the last 80 years. The word ferroresonancefirst appears in the literature in 1920 [5.7], although papers onresonance in transformers appeared as early as 1907 [5.4].Practical interest was generated in the 1930s when it wasshown that use of series capacitors for voltage regulationcaused ferroresonance in distribution systems [5.9], resultingin damaging overvoltages.

The first analytical work was done by Rudenberg in the 1940s[5.36]. More exacting and detailed work was done later byHayashi in the 1950s [5.17]. Subsequent research has beendivided into two main areas: improving the models used topredict the behavior of the transformers, and studying fer-roresonance involving transformers installed in power sys-tems.

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An understanding of the nonlinear parameters describing atransformer core is prerequisite to dealing with ferroreso-nance. Swift [5.47] and Jiles [5.20] have provided insight intotransformer core behavior and the separation of hysteresis andeddy current losses. Frame [5.15] and others have developedpiecewise-linear methods of modeling the nonlinearities insaturable inductances.

Hopkinson [5.19] performed system tests and simulations onthe effect of different switching strategies on the initiation offerroresonance in three phase systems. Smith [5.38] catego-rized the modes of ferroresonance in one type of three phasedistribution transformer based on the magnitude and appear-ance of the voltage waveforms. Arturi [5.2] and Mork [5.29]have demonstrated the use of duality transformations to ob-tain transformer equivalent circuits. Mork [5.27] and Kieny[5.21] have shown that the theories and experimental tech-niques of nonlinear dynamics and chaotic systems can be ap-plied to better understand ferroresonance and limitationsinherent in modeling a nonlinear system. Developments inthe near future are expected to be in the areas of developingimproved transformer models and applying nonlinear dynam-ics to the simulation of ferroresonance.

5.2 FERRORESONANCE IN A SINGLE PHASE TRANS-FORMER

In simple terms, ferroresonance is a series "resonance" in-volving nonlinear inductance and capacitances. It typicallyinvolves the saturable magnetizing inductance of a transform-er and a capacitive distribution cable or transmission line con-nected to the transformer. Its occurrence is more likely in theabsence of adequate damping. A simple case of ferroreso-nance is presented here as an illustration.

When rated voltage is applied to an unloaded single phasetransformer, only a very small excitation current flows (Fig.5.1). In this case, the 120-volt winding of a 120-240 volt 1.5kVA dry-type transformer is energized, resulting in an excit-ing current, whose peak amplitude is 0.05 per unit. Referringto the equivalent circuit shown, it is seen that this current con-sists of two components: the magnetizing current and the coreloss current. The magnetizing current, which flows throughthe nonlinear magnetizing inductance LM, is required to in-duce a voltage in the secondary winding of the transformer.The core loss current, flowing through RC, makes up the eddycurrent losses and hysteresis losses in the transformer's steelcore.

Fig. 5.1. Unloaded single phase transformer with rated voltage applied.Solid waveform is applied voltage; dashed waveform is exciting current

Although usually assumed linear, RC is dependent on voltageand frequency. The excitation current contains high order oddharmonics, due to transformer core saturation. RW and LLare the winding resistance and winding leakage inductance,respectively. They are assumed to be linear parameters. Theirmagnitudes are relatively small compared to LM and RC andso are usually ignored in no-load situations [5.3,5.24].

If a capacitor is placed between the voltage source and the un-loaded transformer, ferroresonance may occur (Fig. 5.2). Anextremely large exciting current (1.92 per unit peak) is drawnand the voltage induced on the secondary may be much largerthan rated (1.44 per unit peak). The high current here is dueto resonance between CS and LM; ferroresonance in mostpractical situations results in smaller exciting currents. Anyoperating "modes" which result in a significantly distortedtransformer (inductor) voltage waveform are typically re-ferred to as ferroresonance, although the implication of reso-nance in a classical sense is arguably a misnomer. Eventhough the "resonance" occurring does involve a capacitanceand an inductance, there is no definite resonant frequency,more than one response is possible for the same set of param-eters, and gradual drifts or transients may cause the responseto jump from one steady-state response to another.

High-order odd harmonics are characteristic of the wave-forms, whose shapes might be conceptually explained interms of the effective natural frequency 1 LMCS as LM goesin and out of saturation. Steep slopes (fast changes) occurwhen LM is saturated, and flat slopes occur when LM is op-erating in its linear unsaturated region.

Due to nonlinearity, two other ferroresonant operating modes

-2 00. 0

-1 50. 0

-1 00. 0

-5 0 .0

0 .0

50 .0

100 .0

150 .0

200 .0

0 .0s 10 .0m s 20 .0m s 30 .0m s 40 .0m s 50 .0m s-1 .0

-0 .75

-0 .5

-0 .25

0 .0

0 .25

0 .5

0 .75

1 .0

T IME

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are possible, depending on the magnitudes of source voltageand series capacitance. In this case, all modes are seen to pro-duce periodic voltage waveforms on the transformer second-ary [5.26,5.29]. In general, gradual changes in source voltageor capacitance will cause state transitions. A reversal to con-ditions that caused a transition will not reverse the transition,due to nonlinearity of LM [5.36]. Transients can also triggertransition from mode to mode.

In modern terms, these jumps are referred to as bifurcations[16,27,29,45], and may be better understood by applying thetheory of nonlinear dynamics and chaos. A long-used intui-tive explanation of these jumps, based on a graphical method,is given by Rudenberg [5.36]. However, this method is not agood analytical tool since it is based only on the fundamentalfrequency and neglects harmonics.

Damping added to the circuit will attenuate the fer-roresonant voltage and current. Some damping is alwayspresent in the form of resistive source impedance, transformerlosses, and also corona losses in high voltage systems, but mostdamping is due to the load applied to the secondary of the trans-former.

Fig. 5.2. Same transformer as in Fig. 5.1, fed through a 75µF capaci-tance,operating in ferroresonance. Solid waveform is terminal voltageof

transformer; dashed waveform is the current.

Damping added to the circuit will attenuate the ferroresonantvoltage and current. Some damping is always present in theform of resistive source impedance, transformer losses, andalso corona losses in high voltage systems, but most dampingis due to the load applied to the secondary of the transformer.Therefore, a lightly-loaded or unloaded transformer fedthrough a capacitive source impedance is a prime candidatefor ferroresonance.

This elementary type of ferroresonance is similar to thatwhich occurred in the series capacitor compensated distribu-tion systems of the 1930s. It can also occur, from differentsources of capacitance, in today's single phase distributiontransformers and voltage instrument transformers [5.1,5.18].It can also occur in series-compensated transmission lines.

Ferroresonance can lead to heating of transformer, due to highpeak currents and high core fluxes. High temperatures insidethe transformer may weaken the insulation and cause a failureunder electrical stresses. In EHV systems, ferroresonancemay result in high overvoltages during the first few cycles, re-sulting in an insulation coordination problem involving fre-quencies higher than the operating frequency of the system.

Because of nonlinearities, analytical solution of the ferrores-onant circuit must be done using time domain methods. Typ-ically, a computer-based numerical integration method isapplied using time domain simulation programs such as theEMTP.

5.3 MAGNETIC BEHAVIOR OF THREE PHASE TRANS-FORMERS

It is incorrect to assume that a three phase transformercore is magnetically equivalent to three single phase transform-ers, i.e. that the three phases have no direct magnetic coupling.Such an assumption can lead to serious errors, especially if oneis investigating a transformer's behavior under transient orunbalanced conditions.

Fig. 5.1. Core configurations commonly used in three phase transform-ers.Only one set of windings is shown.

The only type of core that displays magnetic characteristicssimilar to three single phase transformers is the triplex core.Although the cores share the same tank, they are magneticallyisolated (except for leakage fluxes). Core laminations can bestacked or wound. Zero sequence fluxes will circulate indi-vidually in each core, and tank heating is not a problem. Un-der normal balanced operation, exciting currents in eachphase are identical, except for their 120 shift in phase angle.

All of the other core configurations provide direct flux linkag-es between phases via the magnetic core. Simply stated, ap-plying a voltage to any one phase will result in voltages being

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induced in the other phases (only in the adjacent phase(s) inthe case of the five-legged wound core). Further, the degreeof saturation in each limb of the core affects the way fluxflows divide. The apparent reluctance seen by each of thewindings changes depending on the degree of saturation ineach of the limbs of the transformer core. Therefore, excitingcurrents vary from phase to phase, even under balanced oper-ation. A brief discussion of each of these core types follows:

Core-form transformers require the least amount of core ma-terial to manufacture. Laminations are stacked. Their worstproblem is that unbalanced operation results in zero sequencefluxes which cannot circulate in the core. These zero se-quence fluxes are forced through the insulation surroundingthe core and through the transformer tank. Tank steel is notlaminated like the core is, so eddy currents can heat the tankand cause damage. Therefore, this type of core should only beused where load currents are balanced.

The shell-form core provides a magnetic path for zero se-quence flux, and is much better-suited for unbalanced opera-tion. Laminations are stacked. There is a large base oftransformers with this type of core (about half of the installedthree phase power transformers in the US).

The four-legged core also provides a magnetic path for zerosequence flux. This type of core design is not very common.It is the only type of core whose outer phases do not exhibitlike behavior.

The five-legged stacked core also provides a magnetic pathfor zero sequence flux, but has a more symmetric core. Thistype of core is often specified where a low-profile is desirablefor shipping or for visual appearance in urban substations.

The five-legged wound core is made up of four concentrical-ly-laminated cores. The unique feature of this core is that onlyadjacent phases are directly linked via a magnetic path. As-suming no flux leakage between cores, the two outer windingassemblies are not magnetically coupled. Tank heating isminimized, since there are zero sequence flux paths in thecore. Because of its low cost, this type of transformer core iswidely used in distribution systems.

The winding configuration used does not have any effect onthe transformer core model. Delta, wye, or zig-zag windingconnections are made outside of the model of the core equiv-alent. However, behavior of the transformer is strongly de-pendent on the winding configuration.

5.4 FERRORESONANCE IN THREE PHASE SYSTEMS

Ferroresonance in three phase systems can involve large pow-er transformers, distribution transformers, or instrumenttransformers (VTs or CVTs). The general requirements forferroresonance are an applied (or induced) source voltage, asaturable magnetizing inductance of a transformer, a capaci-tance, and little damping. The capacitance can be in the formof capacitance of underground cables or long transmission

lines, capacitor banks, coupling capacitances between doublecircuit lines or in a temporarily-ungrounded system, and volt-age grading capacitors in HV circuit breakers. Other possibil-ities are generator surge capacitors and SVCs in longtransmission lines. Due to the multitude of transformer wind-ing and core configurations, system connections, varioussources of capacitance, and the nonlinearities involved, thescenarios under which ferroresonance can occur are seeming-ly endless [5.5].

System events that may initiate ferroresonance include singlephase switching or fusing, or loss of system grounding. Theferroresonant circuit in all cases is an applied (or induced)voltage connected to a capacitance in series with a transform-er's magnetizing reactance.

Fig. 5.4 gives three examples of ferroresonance occurring in anetwork where single phase switching is used. A wye-con-nected capacitance is paralleled with an unloaded wye-con-nected transformer. The capacitance could be a capacitorbank or the shunt capacitance of the lines or cables connectingthe transformer to the source. Each phase of the transformeris represented by jXm, since ferroresonance involves only themagnetizing reactance.

Fig. 5.1. Three examples of ferroresonance in three phase systems.

If one or two poles of the switch are open and if either the ca-pacitor bank or the transformer have grounded neutrals, thena series path through capacitance(s) and magnetizing reac-

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tance(s) exists and ferroresonance is possible. If both neutralsare grounded or both are ungrounded, then no series path ex-ists and there is no clear possibility of ferroresonance. In allof these cases, the voltage source is the applied system volt-age. Ferroresonance is possible for any of the core configura-tions of Fig. 5.3 (even for triplexed or a bank of single phasetransformers).

Depending on the type of transformer core, ferroresonancemay be possible even when there is no obvious series pathfrom the applied voltage through a capacitance and a magne-tizing reactance. This is possible with three phase core typeswhich provide direct magnetic coupling between phases,where voltages can be induced in the open phase(s) of thetransformer. To illustrate, a grounded-wye to grounded-wyetransformer typical of modern distribution systems is consid-ered. A recent survey in the US showed that 79% of under-ground rural distribution systems use this configuration, soferroresonance problems in this type of installation are of spe-cial interest [5.23,5.25,5.40,5.41]. A simplified schematic ofsuch a system is shown in Fig. 5.5. The distribution line isrepresented by its RLC pi equivalent, with no interphase cou-pling. Three phase circuit breakers and gang-operatedswitches are used at the substation where distribution linesoriginate, but single phase switching and interrupting devicesare used outside of the substation.

Fig. 5.2. Typical distribution system supplying a three phaseload through a grounded-wye to grounded-wye transformer.

Either overhead lines or underground cables connect trans-formers to the system. Cables have a relatively large shunt ca-pacitance compared to overhead lines, so this type offerroresonance most often involves underground cables, but isalso possible due solely to transformer winding capacitance.

Three phase or single phase transformers can appear at theend of a distribution line or at any point along the line. Threephase transformers may have any one of the several core typesdiscussed in the previous section.

Whether ferroresonance occurs depends on the type ofswitching and interrupting devices, type of transformer, theload on the secondary of the transformer, and the length andtype of distribution line. A long underground line is muchmore capacitive than a short overhead line. However, due tononlinearities, increased capacitance does not necessarilymean an increased likelihood of ferroresonance. Operatingguidelines based on linear extrapolations of capacitance maynot be valid. Also, as mentioned previously, the smaller theload on the transformer's secondary, the less the system damp-ing is and the more likely ferroresonance will be. Therefore,a highly capacitive line and little or no load on the transformerare prerequisites for ferroresonance. Binary loads (either fullload or no load) such as irrigation, are essentially zero most ofthe time and cannot be relied upon to damp ferroresonance.

Ferroresonance is rarely seen provided all three source phasesare energized, but may occur when one or two of the sourcephases are lost while the transformer is unloaded or lightlyloaded. The loss of one or two phases can easily happen dueto clearing of single phase fusing, operation of single phasereclosers or sectionalizers, or when energizing or deenergiz-ing using single phase switching procedures.

If one of the three switches of Fig. 5.4 were open, only twophases of the transformer would be energized. If the trans-former is of the triplex design or is a bank of single phasetransformers, the open phase is simply deenergized and theenergized phases draw normal exciting current. (Existence ofcapacitor banks or significant phase to phase capacitive cou-pling could still result in ferroresonance, but that possibility isnot addressed here).

However, if the transformer is of the three-, four- or five-legged core type, a voltage is induced in the "open" phase.This induced voltage will "backfeed" the distribution lineback to the open switch. If the shunt capacitance is signifi-cant, ferroresonance may occur. The ferroresonance that oc-curs involves the nonlinear magnetizing reactance of thetransformer's open phase and the shunt capacitance of the dis-tribution line and/or transformer winding capacitance. It hasbeen shown that the ferroresonant circuit is a series combina-tion of the shunt cable capacitance and the magnetizing induc-tance of one of the transformer's wound cores [5.23]. Theequivalent circuit for this transformer is derived later in thispaper.

An example of ferroresonant voltage and current waveformsoccurring under this scenario is shown in Fig. 5.6. In thiscase, rated voltage was applied to X2 and X3, while X1 wasunenergized and had 9µF attached to simulate a length of un-derground distribution cable.

Whether in ferroresonance or not, this backfeed situation canbe dangerous, as operating personnel may assume that theload side of the open switch is deenergized and safe to workon, when in fact a high voltage is present. Also, it can be seenthat single phase loads connected along this backfed phase

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will continue to be supplied, although with dangerously highor low voltage levels and with poor power quality.

Therefore, use of single phase interruption and switchingpractices in systems containing the five-legged core trans-formers is the main operating tactic responsible for initiatingferroresonance. Replacement of all single phase switchingand interrupting devices with three phase devices would elim-inate this problem, although economics discourages suchlarge scale upgrades. An alternate solution would be to re-place all five-legged core transformers with single phasebanks or triplex designs wherever there is a small load factor.System wide operation and design implications of this prob-lem have been more fully addressed in prior work [5.25].

5.5 NONLINEAR DYNAMICS AND CHAOS APPLIED

TO FERRORESONANCE

Ferroresonant circuits can be analyzed as damped nonlinearsystems driven by sinusoidal forcing function(s) [5.27]. Thenonlinear behavior of ferroresonance falls into two main cat-egories. In the first, the response is a distorted periodic wave-form, containing the fundamental and higher-order oddharmonics of the fundamental frequency. The second type ischaracterized by a nonperiodic, or chaotic, response. In bothcases the response's power spectrum contains fundamentaland odd harmonic frequency components. In the chaotic re-sponse, however, there are also distributed frequency harmon-ics and subharmonics. A good conceptual introduction tochaos and nonlinear dynamics is given by [5.16], and a goodtheoretical introduction can be found in [5.45].

At least 2 different periodic responses are possible fora single phase transformer [5.26], similar to that of Fig.5.1. Fer-roresonance in the above three phase five-legged core distribu-tion transformer can be periodic or nonperiodic. "Lower energymodes" [5.1] (involving relatively low energy oscillationsbetween the inductance and capacitance, similar to the wave-forms shown in Fig. 5.5) produce periodic voltages on the sec-ondary. Some of the periodic modes of ferroresonance maycontain subharmonics, but still have strong power frequencycomponents, but take longer than one fundamental cycle torepeat. This occurs more typically for very large values of C.

Fig. 5.1. Measurement of ferroresonance in a three phase grounded-wye to grounded-wye five-legged core transformer. Voltage waveform is solid; cur-

rent waveform is dashed.

The "higher energy modes" [5.1] of ferroresonance involvingrelatively large capacitances and little damping can produce anonperiodic voltage on the open phase(s). These voltagewaveforms can be quite similar to those of Duffing's equation[5.45], which describes a nonlinear forced oscillator common-ly used to illustrate the behaviors of a nonlinear dynamicalsystem. Transitions between periodic and nonperiodic modesoccur due to gradual changes in circuit parameters or to tran-sients. And as with Duffing's equation, initial conditions de-termine the mode that operation stabilizes in after thetransients die down.

The recognition that ferroresonance is a nonlinear and some-times chaotic process opens up many possibilities. The new-ly-developed techniques for analysis of nonlinear dynamicalsystems and chaos are being evaluated for use with ferroreso-nance [5.27,5.21]. Use of geometric graphical methods likephase plane projections and PoincarÈ sections can be appliedto obtain a better understanding of ferroresonance.

5.6 MODELLING AND ANALYSIS OF FERRORESO-NANCE

5.6.1 Overview

Ferroresonance has never been well-understood. Therefore,there is a great deal of misinformation on ferroresonance inthe literature. A good example of this concerns the applica-tion of grounded-wye to grounded-wye five-legged core dis-tribution transformers. As recently as 1989, specification ofthis type of transformer was recommended to eliminate orminimize the possibility of ferroresonance [5.14,5.35]. Thismisinformation is gradually being corrected [5.25,5.32], butengineers must be cautious and continue to update them-selves.

Efforts in past years seem focused on refining equivalent cir-cuit models for transformers and performing simulations us-ing a transient circuit analysis program such as EMTP.Although these programs use fairly robust methods of numer-ical integration, such as the trapezoidal rule, results are onlyas good as the models used (and the initial conditions if theonset of ferroresonance is a concern). Simulation results havea great sensitivity to the model used and errors in nonlinearmodel parameters. Unfortunately, determining the model'snonlinear parameters is probably the biggest modeling diffi-culty. Three phase transformer modeling has not progressedas far as single phase modeling. A different model is requiredfor each type of core, and a different means of determining themodel parameters.

Ideally, use of a correct transformer model would allow an en-gineer to simulate situations where ferroresonance is likely.

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Simulation results could then be used to avoid this problemwhen designing a distribution system. Difficulties in deter-mining an adequate model and in simulating every possiblecombination of initial condition and transient make predictionless than certain.

5.6.2 The Study Zone

Parts of the system that must be simulated are the source im-pedance, the transmission or distribution line(s), the trans-former, and any capacitance not already included. Sourcerepresentation is not generally critical. Unless the source con-tains nonlinearities, it is sufficient to use the steady-state thev-enin impedance and open-circuit voltage. The distributionline or transmission line can be assumed to be an RLC cou-pled pi-equivalent, cascaded for longer lines. Shunt or seriescapacitors may be represented as a standard capacitance, par-alleled with the appropriate dissipation resistance. Stray ca-pacitance may also be incorporated either at the corners of anopen-circuited delta transformer winding or midway alongeach winding. Other sources of capacitance are transformerbushings and interwinding capacitances, and possibly busbarcapacitances.

One of the most critical parts of any ferroresonance study isthe transformer model. The transformer contains the nonlin-earities, and modeling results are most sensitive to correctrepresentation of magnetic saturation and core loss. The restof this discussion focuses mainly on how the transformershould be modeled. Many are dissatisfied with the transform-er modeling capabilities in today's modeling packages. Therehas been much discussion recently as to what improvementscan be made in modeling techniques [5.6,5.13,5.46].

5.6.3 More on Single Phase Transformer Models And Parameters

Single phase transformers are typically modeled as shown inFig. 5.2. This model is topologically correct only for the casewhere the primary and secondary windings are not concentri-cally wound. LL2 is essentially zero for concentric coils. Er-rors in leakage representation are not significant, however,unless the core saturates. Obtaining the linear parameters forthis 2-winding transformer may be difficult. Short circuittests give total impedance (R1 + R2) + j(X1 + X2). A judge-ment must be made as to how it is divided between the prima-ry and secondary windings.

If the transformer has three or more windings, the Rs and Xsfor the individual windings of each phase may be separated.Sometimes one of Xs is negative, but this will not usuallycause a problem in the time domain transient simulation. Thisapproach satisfactorily separates the winding resistances, butmay not correctly account for mutual inductive coupling. Tosolve this problem, a coupled L representation for the shortcircuit inductances is recommended [5.11]. Binary short cir-cuit (shorting two windings at a time while leaving all others

open) tests for all possible combinations of windings must beperformed to obtain the inductance matrix. Additional devel-opments are still needed, however, since the core equivalentcannot be correctly incorporated with this representation (theonly place it can be connected is on one of the external trans-former terminals).

Model performance depends mainly on the representation ofthe nonlinear elements RC and LM. RC has traditionally beenmodeled as a linear resistance. Such a core loss representa-tion, if it represents the average losses at the level of excita-tion being simulated, may in fact yield reasonable results.Due to eddy current losses and hysteresis losses being nonlin-ear, calculation of a linear core loss resistance RC gives dif-ferent values for each level of excitation. Using the value ofRC closest to rated voltage may be a good enough estimate.Past research has shown low sensitivities to fairly largechanges in RC [5.29] for single phase transformers, but a highsensitivity for three-phase cores.

LM is typically represented as a piecewise linear -i character-istic [5.22], or perhaps as a hysteretic inductance[5.15,5.20,5.33]. The linear value of LM (below the knee ofthe curve) does not much affect the simulation results [5.8],although great sensitivities are seen for the shape of the kneeand the final slope in saturation.

Factory test data provided by the transformer manufacturer isoften insufficient to obtain the core parameters. Open circuittests should be made for 0.2 to 1.3 pu (or higher) instead of thetypical 0.8 to 1.14 pu range. It is important that open circuittests be performed for voltages as high as the conditions beingsimulated, or the final -i slope of LM must be guessed. Somethought should be given to the requirements of test reportswhen specifying new transformers.

A method proposed by Dommel [5.11,5.22] is often used toconvert the RMS V-I open circuit characteristic to the -i char-acteristic of LM. To successfully use this method, the first(lowest) level of excitation must result in sinusoidal current,or errors will result in the form of an S-shaped -i curve. Also,the V-I characteristic must extend as high as the highest volt-age that will be encountered in the simulation. An extensionon this method has been proposed to obtain a nonlinear v-irepresentation of RC [5.31], but the resulting flux-linked vs.IEX loop does not seem to correctly represent the core losses.

Modern low-loss transformers have comparatively large in-ter-winding capacitances which can affect the shape of the ex-citation curve [5.47]. This can cause significant errors whenthe above method is being used to obtain core parameters. Inthese cases, factory tests must be performed to get the -i curvebefore the coils are placed on the core. A means of removingthe capacitive component of the exciting current has also beendeveloped [5.29].

5.6.4 Three Phase Transformer Models And Model Parame-ters

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For three phase transformers, it is possible to make a simpli-fied model by connecting together three of the above singlephase models. If this is done, a triplex core configuration isassumed (see Fig. 5.3). A delta-wye transformer of this typeis shown in Fig. 5.7. It is postulated that zero sequence (ho-mopolar) effects are included almost entirely by the leakageinductance of the delta windings [5.11,5.22].

Fig. 5.1. Model of a delta to wye transformer bank made up of three single phase transformer models [22].

If the transformer does not have any delta windings, zero se-quence effects may be included by adding a set of delta wind-ings to the model whose total leakage impedance is equal tothe transformer's zero sequence inductance. This may workfor a three-legged core transformer that has an air path forzero sequence flux, but is highly questionable in the case oftransformers having a saturable zero sequence flux path.

Factory three-phase excitation test reports will not provide theinformation needed to get the magnetizing inductances forthis model. Note that standards require the exciting current tobe stated as the "average" value of the RMS exciting currentsof the three phases. Unless it is a triplexed core, this is mean-ingless, since the currents are not sinusoidal and they are notthe same in every phase. Therefore, the waveforms of the ap-plied voltage and exciting currents in all three phase should begiven by the manufacturer for all levels of applied voltage.

The model might be improved by using a coupled inductancematrix to model the short circuit characteristics of three phasetransformers. Binary short circuit tests involving all windingsof all phases must be performed. Problems can arise for RMSshort circuit data involving windings on different phases,since the current may be nonsinusoidal. A problem also existswith connecting the core equivalent. Three single phase coreequivalents are often attached to the windings closest to thecore, and may provide acceptable results in some cases, espe-cially in the case of the three-legged stacked core. Questionsexist as to this method's validity, especially depending on thetype of core being analyzed. The most important question is,however, what is the topology of the core equivalent? Amethod of obtaining topologically correct models is presentedin the next section.

5.6.5 Use of Duality Transformations to Obtain Equivalent Circuits

This method is based on the duality between magnetic andelectrical circuits. It was originally developed by Cherry[5.10] in 1949 and Slemon [5.37] in 1953. Using dualitytransformations, equivalent circuit derivations reduce to exer-cises in topology. These methods did not receive much atten-tion at first, presumably since computers were not available.Researchers have recently begun to use duality to provideequivalent circuit models which are more topologically cor-rect [5.2,5.29,5.30,5.34,5.39,5.42,5.44]. This approach re-sults in models that include the effects of saturation in eachindividual leg of the core, inter-phase magnetic coupling, andleakage effects. Results are promising, and ongoing workseems most focused on developing and improving duality-based models.

To illustrate the method, a duality derivation used to obtainthe model for the five-legged wound core transformer [5.28]is done here and a case study is presented later in this paper.A section view of this type of transformer is shown in Fig. 5.8.The magnetic flux paths and assumed leakage flux paths arelabeled. In the equivalent magnetic circuit, windings appearas MMF sources, leakage paths appear as linear reluctances,and magnetic cores appear as saturable reluctances.

The next step is the duality transformation itself. Using thesymbol to denote the transformation between electrical andmagnetic circuit elements, MMF I (MMF = NI), d /dt V, andL (L = N2/ ). In terms of topology, meshes and nodes in themagnetic circuit transform into nodes and meshes respective-ly in the electrical circuit. The resulting equivalent circuit isgiven in Fig. 5.9.

To make the model practically useful, each current source re-sulting from the transformation has been replaced with an ide-al transformer to provide primary-to-secondary isolation andcoupling to the core, while preserving the overall primary tosecondary turns ratio. Turns ratios are chosen so that core pa-

3-20

rameters are referenced to the low voltage windings. The por-tion of the model inside the coupling transformers representsthe core and leakages. Winding resistance and interconnec-tion of the windings appears external to the coupling trans-formers. The advantage to this is that the derived coreequivalent can be used independently of winding configura-tion (delta, wye, zig-zag, etc.). Winding resistance, core loss-es, and capacitive coupling effects are not obtained directly,but can be added to this topologically-correct equivalent elec-trical circuit.

Fig. 5.1. Development of magnetic circuit for grounded-wye togrounded-wye five-legged wound core transformer. At top, transformer core sectional view used as a basis for duality derivation. Leakageflux paths are labeled. Bold dividing lines mark division in corereluctances. Equivalent magnetic

circuit is shown at bottom.

Tests have been developed to determine the parameters forthis model [5.28].

5.7 CASE STUDIES

5.7.1 Case Study #1: VT Ferroresonance on Floating Sys-tems

It is possible that parts of a power system can be oper-ated for short times without system grounding. One commonexample is the no-load energization of the wye side of a wye todelta power transformer.

The delta side will "float" with respect to earth, until someload or other source of grounding is connected. If there is a

voltage transformer (VT) connected to the delta side of thepower transformer, ferroresonance can occur (see figure).The capacitance in this case comes from whatever "stray"coupling capacitance exists between the delta windings andearth. Adding a resistive burden to the VT can eliminate theproblem.

Fig. 5.1. Duality derived equivalent circuit with current sources replaced by ideal coupling transformers. Winding resistances have also been added

A recent problem occurring in a 50-kV network in the Hafs-lund area near Moss, Norway, serves as an excellent example[5.18]. The clearing of a short circuit removed the only re-maining source of grounding on the system. After the faultwas cleared, the only remaining zero sequence impedancewas due to capacitive coupling to earth. After operating inthis way for only 3 minutes, ferroresonance had destroyed 72of the VTs used for measurement and protective relaying. All72 of the damaged VTs were from the same manufacturer.The VTs of two other manufacturers that were also in serviceduring this time were not damaged.

Fig. 5.10 shows the typical VT arrangement used in this sys-tem. The VTs have two low voltage windings. The secondaryis used for measurement and protective relaying purposes.The burden on that winding has a very high impedance and itseffects can be ignored when considering ferroresonance. It isthe tertiary windings which are shown in Fig. 5.10. Thesewindings are connected in open delta and loaded with a damp-

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ing resistance RO. The purpose of this damping resistance isto damp out ferroresonance, and this design has been com-monly used for many years.

Fig. 5.2. Typical VT connection in 50-kV Norwegian subtransmission sys-tem.

Since some of the VTs were damaged and the others weren't,the VTs of different manufacturers obviously must have dif-ferent characteristics. The problem at Hafslund thereforeforced a re-evaluation of the specification and application ofvoltage transformers. EMTP was used to simulate the systemconditions that caused the VT failures. VT model parameterswere obtained from the manufacturers. Parameters are shownin Table 1. Saturation characteristics were calculated basedon core material B-H data, core dimensions, and number ofprimary turns. Data for the damaged VTs are listed as VT #1.

The designed flux densities BM at rated voltage vary. As amore uniform basis of comparison, the flux densities wereconverted to flux-linked values (Fig. 5.11). Note that VT #1will saturate out at lower levels than the other VTs, and onemight guess this to be one of the reasons these failed and theothers didn't. But this can only be confirmed from simulationresults.

Fig. 5.12 shows the reduced equivalent used in the

EMTP model. System positive and negative sequence imped-ances were found to be very small compared to the primaryimpedances of the VTs, and could be neglected. The zerosequence impedance ZO consists almost entirely of the straycapacitance of the floating system, and is therefore very impor-tant. Values of ZO varied from 0.6 - j219 to 0.2 - j221 , depend-ing where in the system. ZO therefore becomes the only systemimpedance needed in the model, and the positive sequence volt-age sources can be modeled as stiff sources. The core losses ofthe VTs were also neglected, their values being much higherthan the damping resistance RO.

Fig. 5.3. Comparison of the saturation characteristics of the three VTs. Note the much lower saturation level of VT #1, the ones that were damaged.

Fig. 5.4. Reduced system equivalent, neglecting line impedances and lump-ing all VTs in each phase into an aggregate jXM.

Many simulations were run, with various combinations ofVTs and values of R0. It was found that ferroresonance oc-curred in most cases where RO was set to the 60 value typi-cally used in system design. It was also seen that the highmagnetizing currents drawn by VT #1 while in ferroresonancecaused high IðR losses in the windings, which thermally de-stroyed those VTs. If all of the VTs from manufacturer #1were replaced with different VTs and if RO was reduced to 10, ferroresonance would not occur. It was therefore recom-mended that the failed VTs be replaced with those of eitherVT #2 or VT #3. A decrease in the value of RO standardlyTable 1: Linear parameters used to model the VTs at Hafslund

RP X P X T N1:N 3 BMAX

VT #1 32506 25006 0.0 16 20k:23 1.05T

VT #2 32186 30946 0.0 16 ~3 6k:42 0.77T

VT #3 75886 48336 0.0 16 25k:29 0.83T

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being used was also recommended.

5.7.2 Case Study #2: Ferroresonance in Distribution Systems

This case involves the verification of the 75-kVA five-leggedwound-core distribution transformer model developed earlier.Ferroresonance was staged on the secondary windings in thelaboratory. Balanced 3-phase voltage was applied to the sec-ondary windings, and then one or two phases of the supplywere removed and replaced with various values of shunt ca-pacitance. Scenarios investigated were loss of one sourcephase to the center or an outer winding, and loss of two sourcephases to either the two outer windings or to the center wind-ing and one outer winding [5.27].

Measured waveforms were then compared to EMTP simula-tions. The transformer equivalent circuit used was essentiallythat of Fig. 5.9. Details of model development and parametervalues are given in [5.29].

Since many ferroresonant modes are possible, bifurcationsimulations were first run. A bifurcation is essentially a jumpfrom one mode of ferroresonance to another. A simulationtechnique was developed to very slowly ramp the capacitance[5.12,5.28] and record jumps from one mode to another. Fig.5.13 gives one bifurcation diagram for the case where aramped capacitance is connected to unenergized winding X1and rated positive sequence voltage is applied to X2 and X3.Due to nonlinearities, it is important to ramp the capacitanceboth upward and downward, to ensure that as many ferrores-onant modes are discovered as possible.

Using the bifurcation diagram as a road map, ferroresonancefor capacitances of 5µF, 10µF, 22.5µF, 14.6µF, and 18 µFwas simulated. This corresponds to waveforms of periods 1,2, 3, 5, and chaotic (nonperiodic). "Period 3" simply meansthat the waveform takes three periods of the forcing functionto repeat -- it contains 1/3 harmonics.

Fig. 5.1. Sample bifurcation diagram. Shunt capacitance on X1 is ramped from 0 to 30 µF. Blurred areas correspond to chaos.

Fig. 5.14 shows the result of one of the EMTP simula-tions and compares it to the actual measurements. The model

correctly predicts the existence of all modes of ferroresonance atthe correct values of capacitance. The actual waveforms simu-lated are very close for the periods one, two, and three. Periodfive is generally correct, with slightly lower than actual peakamplitudes predicted. The chaotic response predicted is slightlyhigher than actual. The model used a simplistic linear resistanceto represent the core losses of each core. The model's accuracycould be improved by implementing a more correct (complex)core loss representation.

5.7.3 Case Study #3: Ferroresonance of Autotransformer

This case is taken from the Ontario Hydro systemwhere the Cataraqui 230/115-kV autotransformer T2, fed by lineX3H, was experiencing ferroresonance upon deenergization ofline X3H and the 115-kV bus (Fig. 5.15). The deenergizing cir-cuit breaker was also experiencing a high recovery voltage. Itwas deduced that capacitive coupling between line X3H and thestill-energized lines X4H and X522A was driving the autotrans-former into ferroresonance. Damping resistors were added tothe tertiary of T2, but it was not certain whether the resultingdamping was sufficient to limit the duration of ferroresonanceand the related recovery voltage.

Fig. 5.1. Period 3 ferroresonance, 22.5µF connected X1

Fig. 5.2. Ontario Hydro 230-kV system. Ferroresonance involving line X3H and connected transformer at Cataraqui Transformer Station.

Several EMTP simulations were run, with Y-connected resis-tive loads of zero, 133 kW/phase, and 266 kW/phase attachedto the tertiary of T2. In each case, the 115-kV breaker of T2was assumed to open last. Two double-circuit 230-kV lines,an existing 500-kV line, and a future 500-kV line were includ-

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ed in the corridor, resulting in an 18-phase coupled-circuittransmission equivalent (Fig. 5.16).

Fig. 5.17 shows the circuit breaker recovery voltagefor one of the cases.

It is interesting to note that a 133-kW/phase load did aneffective job of damping ferroresonance in T2, but resulted in ahigher recovery voltage than no damping at all. The circuitbreaker was marginally able to handle the recovery voltagewhen the load was doubled to 267 kW/phase. Simulations werealso performed for deenergization of T1, with similar but lesssevere behaviors noted. Recommendations were made to add267 kW/phase loads to both transformers, and add surge arrest-ers to the high and low voltage terminals of both transformers.

Fig. 5.3. Sequence of development of the transmission right-of-way

It is interesting to note that a 133-kW/phase load did aneffective job of damping ferroresonance in T2, but resulted in ahigher recovery voltage than no damping at all. The circuitbreaker was marginally able to handle the recovery voltagewhen the load was doubled to 267 kW/phase. Simulations werealso performed for deenergization of T1, with similar but lesssevere behaviors noted. Recommendations were made to add267 kW/phase loads to both transformers, and add surge arrest-ers to the high and low voltage terminals of both transformers.

Fig. 5.4. Cataraqui (T2) Autotransformer Ferroresonance. HV terminal volt-age on Phase C is 2.0 per unit, with 133 kW/phase of damping.

5.8 RECOMMENDATIONS

Is seen that many different types of ferroresonance can anddo occur. Because of the nonlinear nature of ferroresonance,it is difficult to predict if and where it might next occur. Thepower system engineer should be aware, however, that it ispossible for lightly-loaded transformers operating in the pres-ence of source or shunt capacitance to experience ferroreso-nance. Capacitance can be present in the form of cables,series or shunt capacitor banks, or even stray capacitances ininadequately-grounded portions of the system.

Transient simulations are helpful in confirming or predict-ing the likelihood of ferroresonance, but only if a correct mod-el is used. Per phase simulations of three phase systems willnot give correct results, due to various possible transformercore configurations and winding connections. A completethree phase model must be used. Therefore, the key to tran-sient modeling is use of the proper transformer model. Devel-opment and use of acceptable transformer models should be apriority task.

The development of improved topologically correct models isa significant advancement, but model performance still de-pends on improving the way in which the cores are represent-ed. Transformer core configuration must be considered andsaturation characteristics must be accurately known to operat-ing levels well above rated voltage.

At this time, it is seen that modeling of ferroresonance is asmuch an art as a science. As such, it is important if possibleto verify the results by checking the simulations against sys-tem measurements. It is highly recommended that anyone ac-tive in this area must continually monitor the literature forimprovements in modeling techniques.

6. SUMMARY

This document provides a set of general guidelines for digital-computer time-domain simulation of low-frequency (approx-imately 5 to 1000Hz) transients of electric power systems.

-4.0E+5

-3.0E+5

-2.0E+5

-1.0E+5

0.0

1.0E+5

2.0E+5

3.0E+5

0.0s 100.0ms 200.0ms 300.0ms 400.0ms

CATA RAQUI T2 FERRORESONANCEEXTERNAL DAMPING = 133 kW / phase

402 kV peak

TIME

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The report is intended for practicing engineers who are in-volved in analysis, control and system planning issues relatedto electronic power systems. It is assumed that the reader has(1) a fair understanding of the physical phenomena and (2) anadequate knowledge of digital simulation techniques. Theguidelines are provided for seven transient torsional torques,(3) turbine-blade vibrations, (4) fast bus transfer, (5) control-ler interactions, (6) harmonic interactions and resonance, and(7) ferroresonance. For those phenomena which have exten-sively discussed in the literature, i.e. (1) to (4), general guide-lines are provided and the reader is frequently referred to thetechnical literature for further in-depth modeling and simula-tion issues. The emphasis of this document is on phenomena(5), (6) and particularly (7).

7. REFERENCES

[1.1]CIGRE, “Guidelines For Representation of Network El-ements When Calculating Transients”, CIGRE WorkingGroup 33.02, 1990.

[2.1] IEEE Torsional Issues Working Group, “Fourth Supple-ment To A Bibliography For The Study of SubsynchronousResonance Between Rotating Machines and Power Systems”,IEEE Trans., Vol. PWRS-12, No. 3, pp. 1276-1282, August1997.

[2.2] IEEE SSR Working Group, “A Bibliography for theStudy of Subsynchronous Resonance Between Rotating Ma-chines and Power Systems”, IEEE Trans., Vol. PAS-95, No.1, pp. 216-218, Jan. - Feb. 1976.

[2.3]IEEE SSR Working Group, “First Supplement to a Bib-liography for the Study of Subsynchronous Resonance Be-tween Rotating Machines and Power Systems”, IEEE Trans.,Vol. PAS-98, No. 6, pp. 1872-1875, Nov. - Dec. 1979.

[2.4]IEEE SSR Working Group, “The Second Supplement toa Bibliography for the Study of Subsynchronous ResonanceBetween Rotating Machines and Power Systems”, IEEETrans., Vol. PAS-104, No. 2, pp. 321-327, Feb. 1985.

[2.5]IEEE SSR Working Group, “Third Supplement to a Bib-liography for the Study of Subsynchronous Resonance Be-tween Rotating Machines and Power Systems”, IEEE Trans.,Vol. PWRS-6, No. 2, pp. 830-834, May 1991.

[2.6]T.P. Tsao, C. Chgn, “Restriction of Turbine Blade Vibra-tions in Turbogenerators”, IEE Proceedings, Vol. 137, Part C,No. 5, pp. 339-342, 1990.

[2.7]J.D. Gill, “Transfer of Motor Loads, Between Out-of-Phase Sources”, IEEE Trans. on Industrial Applications, Vol.IA-15, No. 4, pp. 376-381, 1979.

[2.8]T.A. Higgins, P.L. Young, W.L. Snider, H.J. Holley,“Report on Bus Transfer: Part II, Computer Modelling for

Transfer Studies”, IEEE Trans. on Energy Conversion, Vol.EC-5, No. 3, pp. 470-476, 1990.

[2.9]P.L. Young, W.L. Snider, T.A. Higgins, H.J. Holley,“Report on Bus Transfer: Part III, Full Scale Testing andEvaluation”, IEEE Trans. on Energy Conversion, Vol. EC-5,No. 3, pp. 477-484, 1990.

[2.10]P.L. Dandeno, M.R. Iravani, “Development and Appli-cation of Third-Order Turbogenerator Electrical StabilityModels With Particular Reference To Subsynchronous Reso-nance Studies”, IEEE Trans., Vol. EC-10, No. 1, pp. 78-86,March 1995.

[2.11]G. Gross, M.C. Hall, “Synchronouos Machine and Tor-sional Dynamic Simulation in Computation of Electromag-netic Transients”, IEEE Trans. on Power Apparatus andSystems, Vol. PAS-97, No. 4, pp. 1074-1086, 1978.

[2.12]IEEE SSR Working Group, “First Benchmark Modelfor Computer Simulation of Subsynchronous Resonance”,IEEE Trans. on Power Apparatus and Systems, Vol. PAS-96,No. 5, pp. 1565-1572, 1977.

[2.13]IEEE SSR Working Group, “Second Benchmark Modelfor Computer Simulation of Subsynchronous Resonance”,IEEE Trans. on Power Apparatus and Systems, Vol. PAS-104, No. 5, pp. 1057-1066, 1985.

[2.14]D.A. Woodford, “Validation of Digital Simulation ofDC Links”, IEEE Trans. on Power Apparatus and Systems,Vol. PAS-104, No. 9, pp. 2588-2595, September 1985.

[2.15]Manitoba HVDC Research Centre, “EMTDC User’sManual”, 1988.

[2.16]N. Rostamkolai, et al, “Subsynchronous Torsional In-teractions with Static VAR Compensators - Influence ofHVDC”, IEEE Trans. on Power Systems, Vol. PWRS-6, No.1, pp. 255-261, Feb. 1991.

[2.17]L. Gyugyi, “Fundamentals of Thyristor-ControlledStatic VAr Compensators in Electric Power System Applica-tions”, IEEE Publication 87TH0187-5 PWRS, pp. 8-27, 1987.

[2.18]A.N. Vasconcelos, et al, “Detailed Modeling of an Ac-tual Static VAR Compensator for Electromagnetic TransientStudies”, IEEE Trans. on Power Systems, Vol. PWRS-7, No.1, pp. 11-19, 1992.

[2.19]H.J. Holley, T.A. Higgins, P.L. Young, W.L. Snider,“A Comparison of Induction Motor Models for Bus TransferStudies”, IEEE Trans. on Energy Conversion, Vol. EC-5, No.2, pp. 310-319, 1990.

[2.20]EPRI Report, “Bus Transfer Studies for Utility Mo-tors”, EERI EL-4286, Vol. 2, Project 1763-2, October 1986.

[2.21]G.J. Rogers, D. Shirmohammadi, “Induction MachineModelling For Electromagnetic Transients Program”, IEEETrans., Vol. EC-2, No. 4, pp. 622-628, December 1987.

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APPENDIX A

FAST BUS TRANSFER TRANSIENTS

Introduction

Motors and other loads in utility and heavy indus-trial applications are supplied during normal operation from apreferred power source. An alternate power source is nor-mally provided to supply such motors and other loads duringplanned shutdowns and upon loss of normal power from thepreferred power source. The process of disconnecting themotors and other loads from one source and reconnecting toan alternate source is commonly defined as “bus transfer”.

Manual transfer means are normally provided toallow transferring the motors and other loads from one powersource to the other. However, upon loss of the preferredpower source, the motors and other loads are automaticallytransferred to the alternate power source. This automatictransfer is necessary to allow uninterrupted operation of themotors and other loads important to personnel safety and pro-cess operation.

The normal and alternate power source connections

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are always selected such that they are in phase. Therefore,manual transfers can be accomplished in a make-before-break, i.e., the motors and loads are connected to the secondpower source before the first power source is disconnected.In this overlapping transfer, the power supply is not inter-rupted and the motors are not subjected to transients. How-ever, during automatic transfers, the motors may bedisconnected from both power sources for a short durationdepending on the type of transfer and the associated circuitbreakers operating times. The time during which the motorsare disconnected from both power sources is termed the“dead time”. It is commonly longer than two cycles and canbe as long as 12 cycles. While motors are disconnected fromboth power sources they decelerate. The deceleration ratedepends on the motor-load inertia and the synchronizingpower flowing between motors due to their differing charac-teristics. As the motor decelerates, the relative anglebetween the motor internal voltage and the power sourcevoltage changes. Also, the motor residual voltage decays at arate which depends on the motor magnetic characteristics,speed and initial loading. If the relative angle between themotor residual voltage and the power source voltagebecomes large enough at the time of reconnection with sig-nificant residual voltage remaining, the resultant voltagebetween the power source and the motor will produce aninrush current. The inrush current may be significantlylarger than the normal full voltage starting current. Suchhigh inrush currents cause high winding stresses and tran-sient shaft torques which can damage the motor and/or thedriven equipment.

The most common bus transfer scheme is the fastbus transfer scheme. In this scheme, opening of the normalpower source breaker initiates closing of the alternate powersource breaker without intentional time delay. The fast bustransfer operations result in the motors being disconnectedfrom both power sources for a duration of as short as 2 cyclesto as long as 12 or more cycles.

Presently, there are no generic criteria to ensureacceptable fast bus transfer operations. Therefore, it is nec-essary to analyze the transient behavior of motors during fastbus transfer operations. The analysis should be on a case bycase basis to ensure that the motors will not be subjected toexcessive inrush currents and/or shaft transient torques.

A.2Modeling and Analysis

A three-phase model of the motors and the powerdistribution system is required. This is to permit simulatingthe breaker individual pole interruption at separate currentzeros and analyzing the effect of unbalanced faults on themotor behavior. The model must simulate the motor statorand rotor dynamics, the load dynamics, and the power sourcedynamics when available. The larger motors should be indi-vidually modeled; smaller motors unless for the motor beingstudied, if any, may be lumped together and modeled by one

equivalent motor with typical characteristics. The distinctionbetween large and small motors should be made on a case bycase basis.

A.2.1 Motor Electrical System

The motor electrical system may be modeled by thedifferential equations describing the stator and rotor quanti-ties and flux linkages [2.20] or by the two-axis model [2.21].A single rotor motor model may be adequate since the motorspeed usually does not drop significantly during the time afast transfer is accomplished. The model should account forsaturation in the magnetizing, stator and rotor leakage reac-tances.

A.2.2Loads

The mechanical load should be modeled by itstorque-speed characteristics and moment of inertia. Com-mon centrifugal and axial pumps and fans may be modeledby a quadratic torque-speed characteristic.

Non-motor loads may be lumped together and mod-eled by an equivalent resistance-inductance circuit. Non-motor loads would be included in the model to account fortheir damping effects on the motors during the dead time.

A.2.3Motor-Load Shaft Torsional Model

The shaft system should be modeled by the motorrotor mass connected to the load rotor mass by a flexiblespring representing the shaft [2.22]. The motor air gaptorque excites the mass representing the motor rotor whilethe load torque excites the mass representing the load rotor.The shaft torsional model should include the effect of damp-ing and shaft flexibility. The effect of shaft flexibility is par-ticularly important in applications where loads have a largeinertia relative to that of the motor. An example of this appli-cation is torsional study of a large boiler fan. Under suchconditions, the shaft flexibility may cause the shaft torque tobe higher than the motor air gap torque [2.22].

A.2.4Circuit Breakers

The circuit breakers should be modeled as a threepole switch which can be opened or closed at a preset time.The three poles of the circuit breaker connecting the alternatesource must be modeled to close simultaneously. the individ-ual poles of the circuit breaker disconnecting the normalsource must be modeled to open only at the respective cur-rent zero. In analysis involving transfers caused by highlevel electrical faults, the individual poles may be modeled toopen at the respective first current zero following the end ofthe breaker arcing time. This is a conservative approachwhich, in effect, models a zero resistance arc.

A.2.5Power Sources

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Generally, events which initiate bus transfers suchas the loss of the generator in a generating station also initiatedisturbances to the connected power system. The dynamicvariations of the power system voltage magnitude and phaseangle are normally determined as part of system stabilitystudies.

Ideally, the normal and alternate power source mod-els should reflect the dynamic variations in the voltage mag-nitude and phase angle following the initiating event. Thiscan be accomplished by a point to point representation of thesystem voltage magnitude and phase angle profiles. Alter-nately, the system voltage magnitude and phase angle may bemodeled by polynomials fitting their profiles for the shortduration of interest. However, in the event that data on thedynamic behavior of the power system is not available, thenormal and alternate power sources may be modeled as idealsources in phase.

A.2.6Transformers

Transformer dynamics have a little or no effect onbus transfer operations. Therefore, a transformer may bemodeled as an ideal transformer in series with a lumpedresistance in series with a lumped inductance representingthe transformer equivalent impedance.

A.2.7 Cables/Lines

Cable and Lines may be modelled by their -equiva-lents.

A.2.8Simulation and Analysis

In selecting conditions to be analyzed, the followingshould be taken into consideration:

The motor initial loading (prior to the transfer) has asignificant effect on the rate of change of the motor internalvoltage phase angle. Higher loads cause faster drop in rotorspeed and faster rates of change in the phase angle. There-fore, the worst case transfer results when the motors are oper-ating at their highest loading.

The motor-load inertia also has a significant effecton the internal voltage phase angle rate of change. Motorswith a high inertia have a slower rate of change than motorswith low inertia. Therefore, the effect of fast bus transferoperation with and without such high inertia motors shouldbe evaluated.

Bus transfers are initiated by low and high levelelectrical faults. The motor residual voltage decays at a highrate until the fault is cleared by opening the source breaker.This causes the transient shaft torque produced upon closingthe alternate source breaker to be relatively low. However,faults such as line to ground faults cause the motor to experi-ence a high oscillatory torque before the fault is cleared. Theeffect of the torsional stress caused by such high oscillatory

torques should be investigated.The motor air gap torque at the instant of closing the

alternate source breaker is determined by the motor residualvoltage magnitude and phase angle. The magnitude of themotor residual voltage decreases with time while the phaseangle increased with time. This causes the magnitude of theair gap torque to be cyclic. It has a minimum value at someshort bus dead time, peaks as the dead time increases andthen decreases as the dead time increases further. The dura-tion of the dead time at which the air gap torque attains mini-mum and maximum values are system specific and dependson the connected motors characteristics and load levels.

The account for the above considerations, the fol-lowing fast bus transfer operations should be simulated andanalyzed.

Transfers caused without high level electrical faultswith motors operating at their highest loadings when thealternate source voltage is at its maximum level.

Transfers caused without high level electrical faultsas in 1) above except without the largest high inertia motorrunning.

Transfers caused by high level electrical faults (lineto ground and three line to ground faults).

The following parameters should be monitored dur-ing the simulations:

• bus instantaneous voltages• bus voltage phase angle• individual motors instantaneous currents• individual motor air gap torques• individual motors shaft torques (when modeled)• individual motors speed

A.3 Model Validation

Ideally, validating a model of a fast bus transferoperation should include validating the individual motormodels and the circuit breakers operating times. Individualmotor models can be validated by simulating motor startingand running conditions and comparison of the simulationresults to data recorded during an actual motor starting test.Parameters be compared include motor instantaneous cur-rent, power, apparent power (VA), and speed. However,since a typical bus transfer model may include 15 or moremotors, it may not be practical to validate individual motormodels. As an alternative, the bus transfer model can bebased on modeling motors using unadjusted manufacturerssupplied data and establishing a range of the expected accu-racy.