2013 08 30 TB C4 307 Resonance and Ferroresonance in Power Networks Final for SC

RESONANCE ANDFERRORESONANCE INPOWER NETWORKS

WG C4.307

Members

Zia Emin, Convenor (GB)

Manuel Martinez-Duro, Task Force Leader (FR), Marta Val Escudero, Task Force Leader (IE)

Robert Adams (AU), Herivelto S Bronzeado (BR), Bruno Caillault (FR), Nicola Chiesa (NO), David Jacobson (CA),

Lubomir Kocis (CZ), Terrence Martinich (CA), Stephan Pack (AT), Juergen Plesch (AT), Michel Rioual (FR), Juan A

Martinez-Velasco (ES), Yannick Vernay (FR), Francois Xavier Zgainski (FR)

Contributions have also been received from Tim Browne (NZ).

Copyright 2013

Ownership of a CIGRE publication, whether in paper form or on electronic support only infersright of use for personal purposes. Are prohibited, except if explicitly agreed by CIGRE, totalor partial reproduction of the publication for use other than personal and transfer to a thirdparty; hence circulation on any intranet or other company network is forbidden.

Disclaimer notice

CIGRE gives no warranty or assurance about the contents of this publication, nor does itaccept any responsibility, as to the accuracy or exhaustiveness of the information. All impliedwarranties and conditions are excluded to the maximum extent permitted by law.

ISBN : (To be completed by CIGRE)

Resonance and Ferroresonance in Power Networks

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ISBN : (To be completed by CIGRE)


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AcknowledgementsThe convenor wishes to express his thanks and gratitude to Nicola Chiesa, Manuel Martinez Duro, Marta ValEscudero, and Terrence Martinich for their enduring hard work during the preparation of this technical brochure.


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Resonance and Ferroresonance inPower Networks

EXECUTIVE SUMMARY ................................................................................................................................. 6

CHAPTER 1 INTRODUCTION TO RESONANCE AND FERRORESONANCE ........................................... 8

CHAPTER 2 UNDERSTANDING RESONANCE AND FERRORESONANCE ........................................... 112.1 Introducing Resonance ..................................................................................................................... 11

2.1.1 Resonance in Electric Circuits ........................................................................................................... 112.1.2 Series and Parallel Resonance ......................................................................................................... 112.1.3 Duality of Series and Parallel Resonant Circuits ................................................................................ 122.1.4 Free Oscillations in Electric Circuits ................................................................................................... 172.1.5 Ideal Series Resonant Circuit ............................................................................................................ 192.1.6 Damped Series Resonant Circuit ...................................................................................................... 21

2.2 Introducing Ferroresonance .............................................................................................................. 232.2.1 Effect of circuit capacitance ............................................................................................................... 262.2.2 Effect of source voltage ..................................................................................................................... 272.2.3 Effect of circuit losses ....................................................................................................................... 28

2.3 Physical Description of a Ferroresonant Oscillation ........................................................................... 292.4 Types of Ferroresonance Oscillations................................................................................................ 33

CHAPTER 3 TYPICAL NETWORK TOPOLOGIES LEADING TO RESONANCE IN TRANSMISSIONCIRCUITS ............................................................................................................................ 35

3.1 Introduction ....................................................................................................................................... 353.1.1 Typical circuit capacitances ............................................................................................................... 363.1.2 Typical circuit reactances .................................................................................................................. 37

3.2 Potentially Risky Configurations ........................................................................................................ 373.2.1 Shunt-Compensation and Uneven Phase Operation .......................................................................... 373.2.2 Shunt-Compensation and Three-Phase Switching in Multi-Circuit Rights of Way ............................... 393.2.3 Distribution Embedded Generation Islanded with Transmission Circuit .............................................. 43

CHAPTER 4 RESONANCE IN SHUNT COMPENSATED TRANSMISSION CIRCUITS ............................. 444.1 Introduction ....................................................................................................................................... 444.2 Line Resonance in Uneven Open-Phase Conditions.......................................................................... 45

4.2.1 Physical description .......................................................................................................................... 454.2.2 Approximate Steady State Analytical Solution ................................................................................... 474.2.3 Effect of Various Design Parameters ................................................................................................. 56

4.3 Detailed Analysis of Line Resonance in Uneven Open-Phase conditions using Time-DomainSimulation ......................................................................................................................................... 61

4.3.1 Steady State Analysis ....................................................................................................................... 614.3.2 Temporary Overvoltage Analysis ....................................................................................................... 644.3.3 Summary of Parameters Affecting Line Resonance in Open-Phase Conditions ................................. 70

4.4 Line Resonance in Multiple-Circuit Rights of Way .............................................................................. 714.4.1 Physical description .......................................................................................................................... 71


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4.4.2 Discussion on Circuit Parameters ...................................................................................................... 754.4.3 Case Study ....................................................................................................................................... 764.4.4 Summary and commentary of resonance issues associated with shunt-compensated multiple-

circuit rights of way............................................................................................................................ 834.5 Practical Consequences of Line Resonance ...................................................................................... 844.6 Mitigation Options ............................................................................................................................. 85

CHAPTER 5 NETWORK CONFIGURATIONS LEADING TO FERRORESONANCE ................................. 885.1 Ferroresonance in voltage transformers (VT)..................................................................................... 88

5.1.1 VT and Circuit Breaker Grading Capacitors ....................................................................................... 895.1.2 Line VTs ........................................................................................................................................... 895.1.3 VT and Double Circuit Configuration ................................................................................................. 905.1.4 VT in Ungrounded Neutral Systems with Low Zero-Sequence Capacitance ....................................... 90

5.2 Ferroresonance in power transformers .............................................................................................. 935.2.1 Transformer Terminated Transmission Line in Multi-Circuit Right of Way ........................................... 935.2.2 Lightly Loaded Transformer Energized via Cable or Long Line from a Low Short-Circuit Capacity

Network ............................................................................................................................................ 945.2.3 Transformer energized in one or two phases ..................................................................................... 955.2.4 Transformer connected to a series compensated line ........................................................................ 97

CHAPTER 6 MODELLING AND STUDYING ........................................................................................... 996.1 Analytical Solution Methods .............................................................................................................. 996.2 Digital Simulation Methods .............................................................................................................. 1006.3 Modelling of Network Components .................................................................................................. 102

6.3.1 Extent of the Network Model ........................................................................................................... 1026.3.2 Overhead Line Model ...................................................................................................................... 1026.3.3 Transformers .................................................................................................................................. 1036.3.4 Shunt Reactors ............................................................................................................................... 1046.3.5 Other Substation Equipment ........................................................................................................... 104

6.4 Sensitivity to Parameters ................................................................................................................. 1056.4.1 Effect of Magnetising Curve ............................................................................................................ 1056.4.2 Influence of Circuit Breaker Closing Times ...................................................................................... 1086.4.3 Influence of the Damping in the Circuit ............................................................................................ 108

CHAPTER 7 MITIGATION OF FERRORESONANCE ............................................................................ 1097.1 Mitigation of VT Ferroresonance ..................................................................................................... 109

7.1.1 Secondary Open Delta Resistor ...................................................................................................... 1097.1.2 Secondary Wye Resistor............................................................................................................... 1107.1.3 Secondary Wye Resistor in Series with a Saturable Reactor ......................................................... 1117.1.4 Other Mitigation Options .................................................................................................................. 1127.1.5 Mitigation of VT Ferroresonance in Ungrounded Neutral Systems ................................................... 112

7.2 Mitigation of Power Transformer Ferroresonance ............................................................................ 114

CHAPTER 8 CONCLUSIONS................................................................................................................. 116

ANNEX A RESONANCE EXAMPLES ................................................................................................. 124A1. Resonance Associated with Single-phase Autoreclose Switching of 275 kV Shunt Reactor ............. 124A.2 Line Resonance experienced in 275 kV Double Circuit as a result of System Expansion ................. 127A.3 Line Resonance Experienced in 400 kV and 225kV Subnetwork De-energized for Black-Start

Test ................................................................................................................................................ 133A.4 High Temporary Overvoltages When A Distribution-Connected Generator Energizes An Isolated

Ungrounded & Faulted High Voltage System ................................................................................... 140


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ANNEX B FERRORESONANCE EXAMPLES ...................................................................................... 146B.1 Power Transformer Terminated Line Ferroresonance ...................................................................... 146B.2 Power Transformer Ferroresonance Teed from a Multi-Circuit Right of Way .................................... 150B.3 Ferroresonance of a VT in Ungrounded Neutral Configuration ......................................................... 152B.4 Ferroresonance with Power Transformer Connected to Series Compensated Line .......................... 156B.5 Ferroresonance of a Line VT with Circuit Breaker Grading Capacitors ............................................. 162B.6 Ferroresonance on Transformer Energization from a Weak Network ............................................... 167


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EXECUTIVE SUMMARY

Resonance and ferroresonance are a subset of a broad phenomena group that can cause temporary overvoltages(TOV) in power systems. Common causes of TOV include system faults, load rejection, line energization, linedropping/fault clearance, reclosing, and transformer energization. Other special cases of TOV include parallel lineresonance, uneven breaker poles in shunt-compensated circuits, ferroresonance and back-feeding [1]. TheseTOVs have detrimental effects on power quality and can lead to dielectric or thermal failure of equipment. CigreWG 33.11 originally covered this subject in a series of publications between 1990 and 2000 [1] - [4].

Significant research has been carried out since the previous work from Cigre WG 33.11 was published, especiallyon numerical analysis techniques. Hence, a new Cigre WG C4.307 was established with the objective of expandingthe research and documenting in detail special cases of TOV. This particular Technical Brochure (TB) concentrateson resonance conditions at power frequency and on the special case of ferroresonance. Harmonic resonancesexcited by transformer energization have also been studied by WG C4.307 and are the subject of a companion TB.

This TB presents a comprehensive review of the main aspects related to two special sources of TOV: (i) resonanceassociated with the use of shunt compensation and (ii) ferroresonance. Neither resonance nor ferroresonance arenew phenomena, and a comprehensive list of technical references is provided in this document. However thisinformation is very scattered and not always readily available to practising power system engineers. The objectiveof this TB is to compile that knowledge in a simple and concise document that can serve as a guideline for planningengineers and technical consultants to identify potentially dangerous network topologies, to carry-out detailedstudies and to assess mitigation options. Comprehensive theory background and methods of analysis are providedwithin the document as well as a list of typical topologies prone to each phenomenon and practical examples ofrecent incidents experienced in power systems. A companion TB produced by the same WG is devoted to therelated topic of transformer energization phenomena.

The scope of this TB is:

1. Compilation of technical documentation related to the phenomena of resonance and ferroresonance inpower networks.

2. Provide a detailed physical explanation of the phenomena of resonance and ferroresonance in powernetworks.

3. Highlight typical network topologies that present high risk of resonance or ferroresonance.4. Illustrate methods of analysis for resonance and ferroresonance.5. Provide mitigation options.6. Present practical examples of resonance and ferroresonance incidents in transmission networks.

The document is structured in eight Chapters, followed by two Annexes the first one containing the resonancecases and the second one for ferroresonance cases. Each Chapter is self-contained and provides different degreesof technical detail and complexity, in such a way that it is not necessary to read all chapters sequentially to acquirean overview of all the issues covered in the TB. For instance, if the reader is only interested in an overview oftypical network topologies with a high risk of resonance (Chapter 3), then Chapter 4 may be skipped as it covers avery comprehensive theoretical treatment of the phenomenon. On the other hand, the detailed theoretical analysesmay be of great interest to electrical engineering students or to planning engineers tasked with the analysis andsolution of resonant or ferroresonant problems.

The topics covered by each Chapter are as follows:

Chapter 1 provides a high-level introduction to the phenomena of line resonance and ferroresonance andhighlights some changes that are shaping the development of modern power networks, which may increase therisk of resonance and ferroresonance conditions if these phenomena are overlooked at the design stages. Acomprehensive review of technical literature is provided in this Chapter and some past incidents are highlighted.Finally, consequences on operational reliability, costs, safety, and stress on equipment are also discussed.

Chapter 2 introduces the theory behind the phenomena of linear resonance and ferroresonance. The Chapterstarts with a review of the well known series and parallel R-L-C circuits and gradually builds the theoretical analysis


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to cover more complex combinations of series and parallel elements, as representative of many practical cases inactual power systems. The complex topic of ferroresonance is subsequently introduced by considering the seriesresonant R-L-C circuit driven by a voltage source, where the inductor is nonlinear. A distinctive characteristic of aferroresonant circuit is the existence of several stable solutions. Graphical solutions are provided as usefulvisualisation tools of the steady state behaviour of the linear resonant case as well as the ferroresonant circuit. TheChapter concludes with a description of the different types of ferroresonant oscillations, including the representativevoltage waveforms and frequency spectrum.

Chapter 3 gives an overview of typical network topologies that can give rise to resonance associated with the useof shunt-reactors in transmission circuits or substations. It is emphasised that the risk of resonance is not restrictedto normal operating conditions and special precautions should be taken during unusual network configurationssuch as black-start restoration operations or maintenance/testing of sectionalised parts of the network. Byincreasing awareness of the potential risky topologies it is expected that hazardous incidents of resonance can beprevented or mitigated.

Chapter 4 provides a comprehensive theoretical and practical treatment of the resonant phenomena encounteredin shunt-compensated transmission circuits. Approximate steady state equations are presented, which enable initialscreening of the risk of resonance for a particular topology and set of parameters. A comparison of the results ofthe approximate steady state analysis to detailed EMT-type simulations shows very good agreement. The effects ofdesign parameters such as tower design, neutral reactor, magnetic core construction, line transposition, magneticcore saturation, etc are explored and a summary table is included ranking their importance. A practical example isprovided of surge arrester failure due to over-voltage on a long 500kV transmission line having 72% shuntcompensation which, due to breaker failure, resulted in a prolonged two open-pole condition. This topology resultedin a resonant condition almost perfectly tuned to power frequency. Finally, practical consequences of lineresonance are discussed and a range of active and passive mitigation options is given as a tool-box to assistpower systems engineers in the selection of the most cost-effective scheme for each particular application.

Chapter 5 gives an overview of typical network topologies that can give rise to ferroresonance. By increasingawareness of the potential risky topologies, it is expected that hazardous incidents of ferroresonance can beprevented or mitigated. The Chapter is divided in two main sections, one devoted to inductive voltage transformersand another one dedicated to power transformers.

Chapter 6 introduces various analytical approaches as well as digital simulation techniques for the study offerroresonant circuits. A brief discussion is provided on various nonlinear dynamic analysis tools such as phase-space, Poincar section, and bifurcation diagram techniques. Equally, comprehensive suggestions and guidelinesare presented for the modelling of various electrical items of plant in EMT-type simulation tools that can be usedboth in resonance and ferroresonance studies. The sensitivity of the study results to various model and simulationparameters is discussed.

Chapter 7 is devoted to presenting and discussing possible mitigation techniques for ferroresonance. Thesemeasures range from selection of design parameters to avoid risky scenarios (for instance magnetic cores with lowflux density), special relaying schemes, special switching procedures, introduction of damping, etc.

Chapter 8 summarises the conclusions of this work, followed by a list of technical references.

Finally, the Annexes include a compilation of resonance and ferroresonance examples in transmission networks.These comprise four cases of resonance and six cases of ferroresonance. Field measurements and simulationsare given in combination with descriptions of the investigations carried out and the adopted solutions. Theseexamples will provide practising power system engineers with a broad picture of the hazards associated withresonance and ferroresonance and options to deal with them in a cost-effective manner.


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CHAPTER 1 INTRODUCTION TO RESONANCE AND FERRORESONANCE

Ambitious targets for CO2 emissions reductions and integration of renewable generation in power systems aredriving the need for significant reinforcement of existing transmission grids worldwide, in particular new highcapacity corridors are required to transfer large amounts of power from remote areas with high natural resources(i.e. wind, wave, tidal, etc) to the demand centres. At the same time, increasing public opposition to theconstruction of new overhead transmission infrastructure is driving the need for new pylon designs that minimisevisual impact resulting, in many cases, in smaller structures with reduced clearances. Where possible, existingcorridors are being upgraded and operated at higher voltage levels with minimum modifications to the towers, thusincreasing its transfer capability. Furthermore, the use of underground cable circuits at HV and EHV transmissionlevels is steadily increasing, not only in congested urban areas, but also in remote rural locations in order to reducethe environmental impact of new circuits in specific designated zones and to accelerate the connections of windfarms to the transmission grids. These fundamental changes in the design and technology used for newtransmission circuits are resulting in an increased system capacitance that is shifting the network natural resonantfrequencies closer to the power frequency (50/60 Hz).

Generally, resonance occurs in electric circuits that are able to periodically transform energy from an electric fieldinto a magnetic field and vice versa. It is the characteristic of such a circuit that if some single energy is deliveredinto it (either of electric or magnetic type), the circuit then starts to oscillate with the so called free oscillations.Generally, electric circuits are more complex, consisting of many capacitances and inductances that can exchangeenergy between them via various paths and their free oscillations are composed from several frequencies.

It is important to note that resonance referred to in this document applies to fundamental frequency resonance onlyand that if harmonics are present, either due to saturation of transformers or reactors, the resonance conditionsmay change significantly.

A large section of this document is dedicated to resonance conditions in shunt compensated transmission circuits.This is not a new phenomenon, described in technical publications as early as 1962 [68]. However, availableliterature dealing with this type of resonance, reporting field experiences and assessing or recommendingmitigation actions is very scattered and not always readily available to utility planning engineers and technicalconsultants. This Technical Brochure aims to compile that knowledge in a simple and concise document that canserve as a guideline for planning engineers and consultants to identify dangerous topologies associated with theuse of shunt compensation in transmission circuits, to carry-out detailed studies and to assess mitigation options.

A second phenomenon covered in this document is ferroresonance. In its simplest terms ferroresonance can bedescribed as a non-linear oscillation due to the interaction of an iron core inductance with a capacitance.Ferroresonance is a harmful low frequency oscillation where a non-linear reactance can be driven into saturationand oscillate with the circuit capacitance giving rise to severe overvoltages, with almost no damping when theamplitude is moderate, and in some circumstances, excessive overcurrents. If enough energy provided by thesource is coupled to compensate for the circuit losses, this oscillation can be sustained indefinitely.

The phenomenon of ferroresonance came to light in 1920 when it was first reported by P. Boucherot [5] to describean oscillation between a power transformer and a capacitance. Ferroresonance became a problem in the early partof the century when small isolated systems were interconnected by long transmission lines [6] [7], but at that timethe cause of the problem was not understood. In the 1940's and 1950's the phenomenon recurred as the electricitysupply industry expanded and longer overhead distribution systems were introduced into service. The termsneutral instability [8] and voltage displacement [9] were also used in the 1940s referring to the same or verysimilar phenomenon, although the term ferroresonance has prevailed. In 1966 it was discovered that, for cableconnected transformers, ferroresonance can occur even on circuits as short as 200 metres [10], [11]. Since thattime many studies and investigations have been carried out and a number of papers have been published on thesubject.

Ferroresonance has focussed the attention of numerous researchers over the years with the outcome of extensiveliterature addressing the subject, proposing analysis methods and reporting cases experienced by various utilities.However, despite the vast amount of research and technical documentation available, it still remains widelyunknown today and is somehow misunderstood by many power network utilities. It is especially feared by power


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systems operators, as it seems to occur randomly, normally resulting in the catastrophic destruction of electricalequipment and the consequent adverse effect on the reliability of power network. This general lack of awarenessmeans that ferroresonance is, by and large, overlooked at the planning and design stages or, at the other extreme,held responsible for inexplicable equipment failures [12]. However, use of non linear tools enabled a betterunderstanding of the behaviour and these networks [87] and the determination of the different solutions (harmonic,pseudo-periodic and even chaotic) along with the importance of the magnetic flux as a crucial state variable, even ifsome areas have to be investigated further, especially when transformers are highly non linear.

Sustained overvoltages seen under resonance or ferroresonance conditions could stress equipment such astransformers and breakers, and would cause surge arresters to conduct over extended period of time exceedingtheir energy dissipation capabilities. A catastrophic failure of a surge arrester for example could damage other keyequipment in a substation and could also cause injury to personnel if they happen to be around at the time.Therefore resonance and ferroresonance primarily pose a health and safety hazard to the substation personneldue to the risk of explosion in the work place. An example of such threat is reported in [13], where a 230 kV voltagetransformer failed catastrophically due to ferroresonance causing damage to equipment up to 33 meters away.Nobody was injured in this instance but the experience illustrates the danger that site operators are exposed to.

Although not very common, some cases of line resonant incidents can be found in literature. A recent example isdiscussed in section 4.2.2.4 of this document, where two 500kV surge arresters failed as a result of resonance in ashunt-compensated circuit following an uneven circuit breaker operation. A similar case was reported in thediscussions of [82], where surge arrester failures were observed in a shunt-compensated 765 kV line also followinguneven circuit breaker operations. Many more examples of plant equipment destruction caused by ferroresonancehave been documented in the literature. A very interesting case is reported in [14] where 72 voltage transformerswere destroyed in a 50 kV network in Norway. An investigation revealed that all the damaged voltage transformerswere from the same manufacturer whereas voltage transformers from other two manufacturers which were also inservice survived the incident. The catastrophic destruction of a 230 kV voltage transformer in a cogenerationsubstation is reported in [15]. The failure of a 275 kV voltage transformer in UK is reported in [16]. Other typicalexamples include the explosive failure of a 115 kV voltage transformer in Canada [17], the explosive failure ofvoltage transformers in France [12] and the total destruction or partial damage of six 345 kV voltage transformersas reported by a USA utility [18].

From an operational point of view, resonant and ferroresonant oscillations represent a potential threat to powernetwork plant integrity. The large current pulses caused by transformer saturation may overheat the transformerprimary winding and might, eventually, cause insulation damage. The large voltage oscillations, temporary orsustained, can also cause severe stresses on the insulation of all the equipment connected to the same circuit.Surge arresters are normally the most vulnerable apparatus in substations due to their low TOV withstandcapabilities [19].

Resonance and ferroresonance can also have an adverse effect on the reliability of the power network. The forcedoutage of part of a substation due to an equipment failure can cause severe overloading in other parts of thenetwork that could evolve into a cascade tripping [20] or result in extended outage of major power network assets.

From an economic perspective, resonance and ferroresonance could represent unaccounted costs to electricpower utilities. The cost could be twofold: on the one hand, there is an explicit cost associated with the replacementof damaged or destroyed electrical plant, and on the other hand, there are high or perhaps even severe costsassociated with a reduced network security and possible disconnection of some customers. Quantification of thelatter is not a straightforward task and could only be fully quantified if performed on an individual case basis.

Ferroresonant waveforms are highly distorted, with a large content of harmonics and sub-harmonics. This in turnresults in a degraded power quality and possible misoperation of some protection relays [21]. Transformeroverheating may also occur under Ferroresonant conditions due to excessive flux densities. Since the core issaturated repeatedly, the magnetic flux finds its way into the tank and other metallic parts. This can cause charringor bubbling of paint in the tank [22].

In general, it is possible to distinguish temporary overvoltages from ferroresonance; in the former, the amplitudemay be very high initially but decreases rapidly in most cases. As harmonics are involved, the fluxes circulating in


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the iron core may lead to overheatings in the core, and especially affecting the insulation between laminations.These points are not covered by the IEC 60071-1, describing the standard tests to be performed, when addressingstresses linked to insulation coordination issues. IEC 60071-1 enables the specification and subsequent purchaseof transformers for new installations, but does not address particular aspects related to the behaviour of theequipment under operating conditions such as transformer energization.

As resonance and ferroresonance may induce a long duration phenomena, the overvoltages may affect the agingof the insulation through overheating of the iron core, but may not lead to the insulation breakdown of the bushing,as an example, in the case when the amplitude of the overvoltage is moderate.

It is interesting to note that ferroresonance is normally accompanied by a very loud and characteristic noise causedby magnetostriction of the steel and vibrations of the core laminations. This noise has been described in [22] asthe shaking of a bucket of bolts or a chorus of thousand hammers pounding on the transformer from within.Although difficult to describe, the noise is definitely different from and louder than that heard under normaloperating conditions at rated voltage and frequency.


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CHAPTER 2 UNDERSTANDING RESONANCE AND FERRORESONANCE

2.1 Introducing Resonance

2.1.1 Resonance in Electric CircuitsThe phenomenon of resonance exists in a large diversity of physical systems and arises when the system isaffected by periodical excitation with a frequency similar to its natural frequency of oscillation. When a system isexcited, it tends to oscillate at its natural frequency. If the excitation source has the same frequency as thesystems natural frequency, the systems response to that excitation can be very large. In order to have resonancein a system, it is necessary to have two forms of energy storage, with energy being periodically transformed fromone form to the other, and vice versa: in mechanical systems these are kinetic and potential energy, in electricalsystems these are electrical and magnetic energy. Thus, electrical circuits with magnetic and electric fields havethe capability of resonating. Electrical resonance occurs when the magnetic and electric energy requirements areequal, just as a mechanical system resonates when kinetic and potential energy requirements are balanced.

The phenomenon of resonance has very useful applications in some fields. For instance, in telecommunications,resonant circuits are used to select a group of frequencies from a broader group. Such application, as an example,can be part of a radio filter that selects one station for reception, rejecting all others, by means of a variablecapacitor.

A useful application of resonance in electrical power systems is the design of filters for the suppression of harmfulharmonics. However, the phenomenon of resonance can also be very destructive in power systems. Specialcaution is required in the design and operation of the power network to avoid the occurrence of resonance at thepower frequency (50/60 Hz). Such resonance occurrence would lead to uncontrolled system overvoltages thatcould stress and damage equipment.

Electrical resonance occurs in a circuit when the capacitive reactance (1/ZC) equals the inductive reactance (ZL) atthe driving frequency. This frequency, also called natural frequency, is given by Eq. 2-1.

=

Eq. 2-1

2.1.2 Series and Parallel ResonanceThere are two types of resonance: series and parallel. A basic scheme of series resonance is in Figure 2-1 (a) andparallel in Figure 2-1 (b).

For every combination of L and C, there is only one frequency (in both series and parallel circuits) that causes XL toexactly match XC; this frequency is known as the natural or resonant frequency (Eq. 2-1). When the resonantfrequency is fed to a series or parallel circuit, XL becomes equal to XC, and the circuit is said to be resonant at thatfrequency.

In the case of series resonance all circuit elements are in one branch with common current (Figure 2-1 (a)). Thecircuit impedance is given by Eq. 2-2. At low frequencies, the reactance of the capacitor dominates and the phaseangle approaches 90, with current leading voltage. As the frequency increases, the inductive reactance becomessignificant and, at the resonant frequency (Eq. 2-1), it grows to the point of cancelling the reactance of thecapacitor. At the resonant frequency, the inductor and capacitor series combination becomes invisible and R is thetotal impedance of the circuit. Voltages ULS and UCS reach high amplitudes but have opposing phase angles andcancel each other out. Note that series resonance must be excited by an alternating voltage source. At seriesresonance, the circuit current is limited only by the resistor R up to a value IS = US/R. At frequencies aboveresonance, the inductor dominates the circuit characteristics and the phase angle approaches 90 lagging.Z = R + j L 1C Eq. 2-2


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In the case of parallel resonance, all circuit elements are in parallel and they have the same voltage (Figure 2-1(b)). The circuit admittance is given by Eq. 2-3. At low frequencies, the susceptance of the inductor is large anddominates the circuit admittance. As frequency is increased, the inductive susceptance diminishes and thecapacitive susceptance grows until they become equal at the resonant frequency (Eq. 2-1). This resonancefrequency is the same for parallel and series circuits. Thus, series and parallel resonance occur at the samefrequency for the same combination of inductor and capacitor. At the resonant frequency, the inductor andcapacitor parallel combination becomes invisible and G is the total admittance of the circuit. Currents ICP and ILPreach high amplitudes but have opposing phase angles and cancel each other out. Note that parallel resonancemust be excited by an alternating current source. At parallel resonance, the circuit voltage is limited only by theconductance G. As the frequency increases above resonance, the capacitive susceptance dominates the circuitcharacteristics. Thus, the circuit admittance reaches its minimum at resonance and becomes very large at low andhigh frequencies. In other words, at low and high frequencies, the parallel circuit impedance is very small but itreaches a maximum at the frequency of resonance. This behaviour is the opposite from the series circuit, wherethe impedance reaches its minimum at resonance.Y = G + j C 1L Eq. 2-3

Figure 2-1 Series (a) and parallel (b) resonant circuits

2.1.3 Duali ty of Series and Parallel Resonant Circuit sIn practice, it is very rare to find perfect series or parallel circuits like those illustrated in Figure 2-1. It is morecommon to find mixed parallel and series combinations of inductors and capacitors forming series-parallel circuits.An example of this scheme with a series-parallel combination of two capacitors and one inductor is shown in Figure2-2 below. The figure to the left - Figure 2-2 (a) is excited with a voltage source whereas the figure to the right -Figure 2-2 (a) is excited with a current source. In both cases the circuit parameters and topology are the same. Itwill be illustrated next that this circuit has the ability to resonate in two modes: series when excited by a voltagesource and parallel when excited by a current source at the relevant frequencies.

a) Series b) Parallel


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Figure 2-2 Series-parallel resonant circuit with voltage or current source excitation

To facilitate analysis, the series-parallel circuit shown in Figure 2-3 (a) can be converted into the series circuit ofFigure 2-3 (b) using Thevenin theorem. The series circuit is equivalent to the original series-parallel circuit if itscapacitance is equal to the sum of the series and parallel capacitances and the amplitude of the voltage source isdecreased in ratio of the capacitance divider. This circuit has one series resonant frequency given by Eq. 2-4,which can only be excited by a voltage source. The same original circuit has one parallel resonant frequency givenby Eq. 2-5, which can only be excited by a current source.

(series) = ( + ) Eq. 2-4

(parallel) =

Eq. 2-5

Figure 2-3 Series-parallel resonant circuit with voltage or current source excitation

The examples presented below illustrate the duality behaviour of the series-parallel circuit, depending on the typeof excitation. The following parameters have been assumed for the series-parallel circuit (Figure 2-3 a): L = 1 H,CS = 1 nF, CP = 1 nF. For these parameters, the resonant frequencies calculated with Eq. 2-4 and Eq. 2-5 arefn(series) = 3558.81 Hz and fn(parallel) = 5032.92 Hz.

a) Excitation by Voltage Source b) Excitation by Current Source


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The results of a frequency scan are included in Figure 2-4, which shows the driving impedance of the circuit as afunction of frequency. This circuit presents two resonant modes: series resonance at 3.5 kHz and parallelresonance at 5.03 kHz. At low frequencies, the circuit impedance is very large and capacitive. As the frequencyapproaches the first resonant mode (3.5 kHz), the impedance drops and the phase angle starts increasing until theimpedance reaches its minimum value and the phase angle reaches zero (i.e. the circuit behaves in resistivemode). After that, the impedance starts increasing and becomes inductive. At the second resonant mode (5.03kHz), the impedance reaches its maximum value and the phase angle drops to zero again. With increasingfrequency, the circuit impedance decreases and the phase angle approaches -90 again, returning to capacitivemode.

Figure 2-5 illustrates the dual behaviour of this series-parallel circuit when supplied by a voltage or a currentsource. In this case, the same circuit has been supplied with a voltage or a current source of variable frequencyand the voltage across the inductor has been monitored. It can be seen that, when the circuit is supplied with avoltage source (green curve), the voltage across the inductor reaches a maximum at a frequency of 3.5 kHz, whichcorresponds to the resonance of (CS + CP ) = 2 nF with L = 1 H. Alternatively, when the circuit is supplied with acurrent source (red curve), the voltage across the inductor reaches a maximum at a frequency of 5.03 kHz, whichcorresponds to the resonance of CP = 1 nF with L = 1 H. Even though the series-parallel circuit has the capability ofresonating in two different modes, only the series mode is excited by a voltage source and only the parallel mode isexcited by a current source.

It can be argued that ideal voltage and current sources do not exist in practice. Ignoring harmonic current sources,power systems are entirely supplied by voltage sources. A voltage source placed behind a small source impedancebehaves like an ideal voltage source. However, a voltage source placed behind a high source impedancebehaves more like an ideal current source. Therefore, the required excitation for series or parallel resonances canbe present in a circuit, depending of the network topology and parameters.

The effect of the source impedance on the response of the series-parallel circuit is examined in Figure 2-6. Thesesimulation results have been obtained by placing a resistive source impedance between the ideal voltage sourceand CS in the series-parallel circuit of Figure 2-3 (a) with L = 1 H, CS = 1 nF and CP = 1 nF. The resistive sourceimpedance has been varied from 0.1 to 5 M and the voltage across the inductor has been plotted as a functionof the source frequency. The simulation results show a clear series resonant peak for low values of sourceimpedance (i.e. between 0.1 and 10 k). The amplitude of the series resonant peak decreases as the sourceimpedance is increased due to its current limiting effect. For a range of source impedance between 10 k and 100N the circuit experiences a transition with a very flat curve and a shifting resonant frequency. Finally, in the 500N to 5 M range, the circuit reaches a new resonant condition at its parallel resonant frequency of 5.03 kHz.Further increases in the resistive source impedance only improve the quality factor of the parallel resonance, butthere are no more frequency shifts in the resonant behaviour. Thus, this circuit sees the supply as a voltage sourcein the 0.1 and 10 k source impedance range and as a current source in the 500 k to 5 M source range,exciting the two different resonant modes accordingly.

The effect of the series capacitance CS on the response of the series-parallel circuit is examined in Figure 2-7. Inpractice, the series capacitance in a circuit, CS, can come from series capacitors, grading capacitors in circuitbreakers, coupling capacitance with parallel circuits or inter-phase capacitances in transmission circuits. Thesimulation results shown in Figure 2-7 have been obtained by changing the value of CS in the series-parallel circuitof Figure 2-3 (a) with L = 1 H and CP = 1 nF. The circuit is supplied by an ideal voltage source, without sourceimpedance. The value of CS has been varied from 1 nF to 5 pF and the voltage across the inductor has beenplotted as a function of the source frequency. This circuit exhibits a series resonant point at 3.5 kHz for CS = 1 nF.As the value of series capacitance, CS, is reduced, the resonant frequency increases and converges towards theparallel resonant frequency of 5.03 kHz. This behaviour can be interpreted in two ways: first, as CS is smaller, (CS +CP ) # CP , therefore the series resonant frequency calculated with Eq. 2-4 converges to the parallel resonantfrequency calculated with Eq. 2-5 and the circuit has only one resonant mode. Second interpretation is that thesmall series capacitance CS creates a high source impedance for the ideal voltage source, which then behaves likea current source exciting the parallel combination of L and CP. Thus, series or parallel resonance is just anaming tag for the behaviour of this circuit with a single mode of oscillation.


Page 15

And finally, for completeness, the effect of the parallel capacitance CP on the response of the series-parallel circuitis examined in Figure 2-8. In practice, the parallel capacitance in a circuit, CP, can come from shunt capacitors,stray capacitance of equipment or phase-to-ground capacitances in transmission circuits. The simulation results ofin Figure 2-8 have been obtained by changing the value of CP in the series-parallel circuit of Figure 2-3 (a) with L =1 H and CS = 1 nF. The circuit is supplied by an ideal voltage source, without source impedance. The value of CPhas been varied from 1 nF to 5 pF and the voltage across the inductor has been plotted as a function of the sourcefrequency. This circuit exhibits a series resonant point at 3.5 kHz for CP = 1 nF. As the value of parallelcapacitance, CP, is reduced, the resonant frequency increases and converges towards the parallel resonantfrequency of 5.03 kHz. The amplitude of the resonant voltage naturally increases as CP drops and converges to afixed value, in contrast to the behaviour shown in Figure 2-7 for the reduction of CS. This behaviour can again beinterpreted in two ways: first, as CP is smaller, (CS + CP ) # CS , therefore the series resonant frequency calculatedwith Eq. 2-4 converges to the parallel resonant frequency calculated with Eq. 2-5 and the circuit has only oneresonant mode. Second interpretation is that the small shunt capacitance CP creates a high impedance (1/ZCP),approaching an open circuit, in parallel with L, therefore the circuit effectively becomes a series combination of CSand L, with only one resonant mode at f = 1/(2S(L.CS). Again, series or parallel resonance is just a naming tagfor the behaviour of this circuit with a single mode of oscillation.

Figure 2-4 Driving impedance of series-parallel circuit as a function of frequency (CS = CP = 1 nF, L = 1 H)

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Figure 2-5 Response of the Series-Parallel circuit to a voltage source excitation (green trace) and current sourceexcitation (red trace) Inductor voltage - (CS = CP = 1 nF, L = 1 H)

Figure 2-6 Transition between series and parallel resonance in series-parallel circuit with increased sourceimpedance Voltage source excitation - (CS = CP = 1 nF, L = 1 H)

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Figure 2-7 Shifting in resonance frequency in a series-parallel circuit as a function of Cs (CP = 1 nF, L = 1 H)

Figure 2-8 Shifting in resonance frequency in a series-parallel circuit as a function of CP (CS = 1 nF, L = 1 H)

2.1.4 Free Oscil lations in Electric CircuitsElectromagnetic resonance can occur in electric circuits that are able to periodically transform energy from anelectric field into a magnetic field and vice versa. Such circuits experience free oscillations when energy isdelivered to them, either to the electric field or to the magnetic field.

Free oscillations are also called natural oscillations because their frequency is given by passive parameters of acircuit. For example the circuit shown in Figure 2-9(a) with C = 100 nF and L = 100 H starts to oscillate in anundamped fashion following the switching (at t = 0.1 sec). In this example the frequency of free oscillation (Eq. 2-1)

voltage_injection_1nF_cs.pl4: v:L_____-voltage_injection_500pF_cs.pl4: v:L_____-voltage_injection_200pF_cs.pl4: v:L_____-voltage_injection_100pF_cs.pl4: v:L_____-voltage_injection_50pF_cs.pl4: v:L_____-

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Page 18

is fn = 50,33 Hz and the voltage and current waveforms measured in the ideal L-C circuit are shown in Figure2-9(b).

Figure 2-9 Behaviour of undamped circuit (a) Ideal L-C Circuit (b) Free oscillations in ideal L-C circuit

In practice, free oscillations are typically damped since part of the electromagnetic energy exchanged between theinductor and the capacitor is transferred into thermal energy and dissipated in the resistive elements. Resistivelosses in circuits come from the resistance of conductors, corona effect, lossy polarisation in dielectrics or fromalternating magnetisation in ferromagnetic cores (hysteresis and eddy currents). An example of a lightly dampedfree oscillation is shown in Figure 2-10 (b), obtained with a resistor R value of 1 k. Figure 2-11 (a) shows anotherexample with higher losses, illustrating a free oscillation lasting for a few cycles. However, if the circuit losses arevery high, a free oscillation will not occur because all the energy in the circuit is dissipated in the first cycle and thetransient becomes aperiodical, as shown in Figure 2-11 (b).

Figure 2-10 Behaviour of damped circuit (a) R-L-C Circuit (b) Damped free oscillations

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Figure 2-11 Examples of damped oscillations (a) strong damping (b) aperiodical transient

The following sections introduce various concepts of series resonance in its transient form from zero initialconditions to the final resonant state, rather than straight into the steady state form, as the former is of moreconcern in power networks. In understanding series resonance it is appropriate to choose either the voltage acrossthe inductor (UL) or the capacitor (UC) as the circuit parameter to monitor. Both are of equal magnitude but withphase angle shift of 180 between them. UL has been selected in this document.

2.1.5 Ideal Series Resonant CircuitFigure 2-12 shows the transition of an ideal lossless series oscillatory circuit with natural frequency fn = 50 Hz toresonance following the connection of a 50 Hz voltage source. As a function of time, the rise of the resonantvoltage amplitude UL (as an envelope) is given by

() = Eq. 2-6If we consider the network frequency as a constant, the rise time of resonant voltage on this basic circuit isindependent of the circuit parameters, except for the magnitude of the excitation voltage US. In this particularexample, the resonant voltage rate of rise is 1570.8 kV/s based on US value of 10 kV and a source frequency of 50Hz. The capacitor and inductor values used in this resonant circuit were 101.32 nF and 100 H respectively, but thesame result could be obtained for different combinations of capacitor and inductor values with the same product,such as 1013.2 nF and 10 H respectively.

Figure 2-12 Ideal series resonance oscillation with n S

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Page 20

The rate of rise of the resonant current amplitude can be obtained as US/2L and the current is given by

() = Eq. 2-7which is independent of frequency but inversely proportional to inductance L. This implies that various 50 Hz seriesresonant circuits with various combinations of LC parts have the same rise time of resonant voltage under similarexcitation conditions, but the currents fed from the voltage source and their rise is inversely proportional toresonant inductance. In the above example the two LC combinations that give the same resonant voltages UL,would therefore result in different rise time for the resonant current (50 A/s and 500 A/s). It is of interest to note thatthe active impedance of the resonant circuit is also changing in time according to

() = () = Eq. 2-8Introducing a small difference between the voltage source frequency fs and the free oscillation natural frequency fn,will result to a phase shift between voltage phasors that will change slowly as shown in Figure 2-13, and theresonant overvoltage will fluctuate within a sine wave envelope in accordance with:

() = [ () + ()] Eq. 2-9and for sufficiently small differences between S and n it can be simplified to

() = ( +) ( ) Eq. 2-10In the above equation the cosine term represents the main resonant oscillation whereas the sinus term determinesthe envelope (or low frequency beat) of the oscillation resulting from the interaction between the source and theresonant circuit. If fn fs, the angle between phasors moves from 90 to 0 and then to 270 (Figure 2-13 a). For fn !fs the angle between phasors moves from 90 to 180 and then to 270 (Figure 2-13 b). In both cases the energyexchange has the same periodic time evolution. Initially an energy pump from the source to the resonant circuit isapparent and as the resonant current starts lagging this exchange decreases and at 90 phase shift it stops. At thispoint the exchange of energy is reversed and it flows back to the source as can it can be seen in Figure 2-14 wherea pulsed power is flowing into the resonant circuit (red with +ve polarity) and then back to the source (red with vepolarity). Integral value of the pulsed power gives the accumulated energy in the oscillatory circuit (green) whichperiodically reaches a maximum and then returns back to zero.


Page 21

Figure 2-13 Ideal series resonance oscillation with nzS

Figure 2-14 Energy exchange between source and resonant circuit

2.1.6 Damped Series Resonant CircuitIn real power networks resonant circuits are never lossless and hence it is important to visualise the effect of losseson the resonant cases explained in the previous section.

In a damped resonant circuit with fn = fs, the resonant voltage will not increase indefinitely as in ideal losslesscircuits, because the resonant current is limited by the resistance R and the maximum resonant voltage is given by:

() = Eq. 2-11

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An example of this behaviour is given in Figure 2-15, obtained for a series L-C resonant circuit with a 101.32 nFcapacitor and a 100 H inductor with a source voltage of 10 kV, for two different resistance values (300 : and1000 :). In this example the natural frequency of the circuit is the same as the voltage source frequency (fn = fs). Itcan be observed that the higher resistance value (green waveform) provides higher limitation on the resonantvoltage developed across the circuit elements.

Introducing a small difference between the excitation frequency fs and the frequency of free oscillation fn results in amodulated wave. Figure 2-16 illustrates this behaviour for two values of resistance and the following circuitparameters: L = 114.63 H, C = 101.32 nF, Us = 10kV, fn = 46.70 Hz. The small damping case presents a modulatedwave similar to an ideal lossless circuit but slowly decaying to reach the steady-state resonant voltage (Figure 2-16(a)). In the higher loss case (Figure 2-16 (b)), the modulated oscillation is quickly damped.

If fn fs, the angle between the source voltage and the resonant voltage phasors moves from 90 to 0 and then,due to losses, it cant reach 270 but settles to a value between 0and 90 following the non-zero minimum point ofthe modulation as seen in Figure 2-17 (a). Similarly, Figure 2-17 (b) shows that for fn ! fs the angle betweenphasors moves from 90 to 180 and then to a value between 0and 90 following the non-zero minimum point ofthe modulation.

Figure 2-15 Damped resonant voltage with n S

Figure 2-16 Damped series resonance oscillation with nzS (two different dampings)

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Figure 2-17 First over-swing and steady state of damped resonance (R = 1000 :) with nzS

2.2 Introducing FerroresonanceIn simplest terms, ferroresonance can be described as a non-linear oscillation arising from the interaction betweenan iron core inductance and a capacitor. In this section, the description of ferroresonance follows from the previoussections with a basic analysis of a series resonant circuit and gradually increases the level of complexity to providea comprehensive explanation of the physical mechanism driving the nonlinear oscillation of ferroresonance. In thisinitial description, a very simplified model of the magnetic core is used for a better understanding of the basicmechanisms driving the oscillation.

As with linear resonance, ferroresonant circuits can be either series or parallel, albeit only series configurations aretypically encountered in transmission networks. It should be noted that parallel ferroresonant configurations arecommon in distribution systems with ungrounded or resonant neutral connections. For simplicity and betterunderstanding, the analysis and explanation that follows is based on series resonant and ferroresonant circuitsonly.

A basic series R-L-C circuit is shown in Figure 2-18 which includes the series connection of a voltage source US, toa capacitor C, an inductor L, and a resistor R. All circuit elements are linear.

Making use of phasor analysis, the equation describing the steady-state behaviour of the above circuit expressedas:

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= + ( ) 1 = + ( ) Eq. 2-12where Zs is the angular frequency of the voltage source.

Figure 2-18 Linear Series R-L-C circuit

Resonance occurs when the capacitive reactance equals the inductive reactance at the driving frequency. Underthis condition the circuit impedance becomes purely resistive.

= Eq. 2-13The most characteristic feature of a linear R-L-C circuit is that there is only one natural frequency, fn, at which theinductive and capacitive reactances are equal. This frequency is given in Eq. 2-1.

A graphical solution of Eq. 2-12 is presented in Figure 2-19 [22]. The circuit resistance has been ignored forsimplicity. The voltage-current representation results in two straight lines with slopes equal to the inductive andcapacitive reactances respectively. The intersection of both lines yields the current in the circuit. Figure 2-19 (a)shows the operating point for a source frequency fS below the circuit natural frequency fn. It can be seen that thecapacitive reactance, XC, exceeds the inductive reactance, XL, resulting in a leading current and a high voltageacross the capacitor. Similarly, Figure 2-19 (c) shows the operating point for a source frequency above the circuitnatural frequency, fn. It can be seen that in this case the inductive reactance, XL, exceeds the capacitive reactance,XC, resulting in a lagging current and a high voltage across the inductor. Finally, Figure 2-19 (b) shows that, for asource frequency equal to circuit natural frequency fn, the inductive and capacitive reactances are equal and thetwo lines become parallel, yielding a solution of infinite current and voltages.

In practice all circuits have some sort of losses, even if in small amounts. These resistive losses have the effect oflimiting the amplitude of current and voltages in resonance as follows:

=

Eq. 2-14

= = Eq. 2-15 =

= 1

Eq. 2-16

Q is normally referred as the circuit quality factor, which gives an indication of the resistive losses and the circuitgain. It becomes apparent that low circuit losses lead to high capacitor and inductor voltages under resonantconditions.


Page 25

Figure 2-19 Graphical Solution of Linear Series L-C circuit

Replacing the inductor L of the linear series R-L-C circuit of Figure 2-18 with a saturable magnetic core, a seriesferroresonant circuit can be obtained as shown in Figure 2-20. What differentiates ferroresonance from linearresonance is that the inductance is not constant; therefore the ferroresonant frequency calculated with Eq. 2-1becomes a moving target. This means that a range of circuit capacitances can potentially lead to ferroresonance ata particular source frequency. Another characteristic of ferroresonance is the existence of several solutions. Thisdistinctive behaviour will be illustrated next.

Figure 2-20 Series Ferroresonant Circuit

An in-depth analysis of the circuit shown in Figure 2-20 is complex and requires the solution of nonlinear differentialequations. The circuit analysis, however, can be simplified considerably and yet provide a thorough conceptualdescription of ferroresonance by limiting the calculations to power frequency and steady state [23]. It should benoted that the presence of the non-linearity introduces harmonics in the current and voltage waveforms. However,for simplicity, the description that follows assumes perfect sinusoidal voltage and current waveforms oscillating atpower frequency. Furthermore, the resistive losses are also ignored. Under these particular conditions, theequation describing the steady-state circuit behaviour at power frequency can be expressed as:

= () 1 = () Eq. 2-17where XC is the circuit capacitive reactance at power frequency, ZS is the source angular frequency and XL(I) is thevariable reactance of the saturable magnetic core. This voltage across the non-linear inductance [UL(I) = I XL(I)] is afunction of the current, which is characteristic of the ferromagnetic inductance and is solely dependent on thenumber of turns and the dimensions of the iron core.


Page 26

Eq. 2-17 has been solved graphically in Figure 2-21 [22]. The intersection of the US + I.XC line with the non-linearI XL(I) curve gives the solution for the current in the circuit. The first distinctive characteristic of this graphicalvisualisation is that there are three possible solutions:

x Point 1 represents a normal operating point in which the circuit is working in an inductive mode, withlagging current and low voltages. Voltage and current related by a linear expression. The inductive voltageis greater than the capacitive voltage by the source voltage. This is a stable solution.

x Point 3 represents a ferroresonant state in which the circuit is working in a capacitive mode, with leadingcurrent and high voltages. Voltage and current are related by a non-linear expression. The capacitivevoltage is greater than the inductive voltage by the source voltage. This is also a stable solution.

x Point 2 is another circuit solution but it represents an unstable state.

The stability of solutions 1 and 3 can be demonstrated with the following considerations: at point 1, a smallincrease or decrease of the current will result in a small linear change in the capacitor voltage. However, thecounteracting inductive voltage changes more intensely with current due to its steeper slope, therefore the currentwill return to its original value to find equilibrium. Similarly, at point 3, a small variation in current will result in asmall variation in inductive voltage. The counteracting capacitive voltage changes more intensely due to its steeperslope, and therefore the current will return to its original value again.

The instability of point 2 can be demonstrated by slightly increasing the current, which results in a small increase ininductive voltage and a large increase in the capacitive voltage. In this case, the steepness of the capacitivevoltage is higher than the opposing inductive voltage and therefore the current will continue increasing away frompoint 2. A similar consideration can be made for a small decrease in current.

Figure 2-21 Graphical Solution of the Series Ferroresonant Circuit

2.2.1 Effect of circuit capacitanceFigure 2-22 illustrates the effect of the circuit capacitance on the onset of ferroresonance. It can be seen that as thecapacitance value is reduced, the slope of the US + I.XC line increases and the three possible solutions movetowards the vertical axes. Figure 2-22 (a) shows that there is a critical capacitance value, Ccritical, for which theoperating points 1 and 2 disappear and the only possible solution is a ferroresonant state, point 3. Similarly, Figure2-22 (b) shows that higher capacitances result in a reduced slope in the US + I.XC line. It is inferred that, for alarge enough capacitance value, the operating points 2 and 3 disappear and the only possible solution is a normal

-2000

-0.06

IXL(I)U

I

US + IXC

2

1

3

ULUS

UCU0

UC

US

UL XC

US

UL

UC

I

Solution atpoint 1

UL

UC

I

Solution atpoint 2

(unstable)

UL

UC

I

Solution atpoint 3

US

US


Page 27

state, point 1. This result has practical implications in transmission substations since it suggests thatferroresonance can be avoided by the connection of a large capacitance.

Figure 2-22 Graphical Solution Illustrating the Effect of Circuit Capacitance

2.2.2 Effect of source voltageThe effect of the source voltage is illustrated in Figure 2-23. As this voltage is increased, the US + I.XC line movesupwards to a point in which there is no intersection in the first quadrant. Operating points 1 and 2 disappear andthe only possible solution is point 3, which is a ferroresonant state. Note also that the disconnection of the sourcevoltage, U, may not result in the elimination of ferroresonance, as illustrated with state 3. As U is removed, theoperating point simply slides to the right, but remains in the saturated region. This statement assumes that thecircuit has no losses, which is not true in reality, but it serves to illustrate the fact that, in theory, the ferroresonantoscillations can be self-sustained.

Figure 2-23 Graphical Solution Illustrating the Effect of the Source Voltage


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2.2.3 Effect of circuit lossesA more thorough analysis of series ferroresonance can be performed by introducing the damping effect of thecircuits resistive elements. If these losses are considered, the equation describing the steady-state behaviour ofthe circuit shown in Figure 2-20 can be written as:

= + (() ) Eq. 2-18 = ( ) + ( () ) Eq. 2-19

| () | = ( ) Eq. 2-20The first term of Eq. 2-20 is plotted in Figure 2-24 (a). It is shown that multiple solutions are possible when the I.XCline intersects the UL(I)= IXL(I) curve in the saturation region. To the left of IC the circuit operates in an inductivemode whereas the region to the right of IC corresponds to a capacitive mode. The second term of Eq. 2-20 is anellipse that crosses the horizontal axis at I = US/R and the vertical axis at US. This is plotted in Figure 2-24 (b). Theintersection of this ellipse with the |UL(I)- I.XC| curve gives the current in the circuit. Figure 2-24 (b) shows threepossible solutions for a circuit resistance R1, which represents a low loss scenario. As previously demonstrated,solution 2 is an unstable state, solution 3 is a ferroresonant state and solution 1 corresponds to a normal state. Ifthe circuit losses are increased, Figure 2-24 (c) shows that the multiplicity of solutions can disappear. In particular,if IC > US/R there is only one possible solution which corresponds to a normal operating state. This illustrates thatthe onset of ferroresonance can be avoided by increasing the circuit losses.

Figure 2-24 Graphical Solution Illustrating the Effect of Circuit Resistance

It should be noted that the above qualitative description is an over simplification of the complex ferroresonantbehaviour that has been limited to steady state and power frequency. It is emphasized that this analysis is not validfor operation in the saturated region of the inductance, i.e. under ferroresonance, due to the high harmonic content.The analysis, however, is perfectly valid in the linear region operation and can be used to find the boundary limiting

00.00 0.01

220 IRE IXV CL

IC

I

V

2

0RE

0E1

00.00 0.01

220 IRE

IXV CL

IC

I

V

1

0RE

0E1 2 3

00.00 0.01

IXV CL

VL(I)VC

I

V

Capacitive

Zone

InductiveZone

Xc

IXU CL

IXU CL IXU CL 22 IRUs

22 IRUs


Page 29

parameters, where the circuit ceases to operate in a linear mode. Despite its limitations, the graphical analysis hasprovided a very good intuitive insight into the key features governing a ferroresonant oscillation, as follows:

x Multiple steady-states are possible in a ferroresonant circuit.x Jump phenomena, called bifurcations, can occur where the operating point changes drastically for a small

change in circuit parameters, supply voltage or frequency.

x The circuit capacitance is critical for the occurrence of a ferroresonant state.- Low values of capacitance favour the onset of ferroresonance.- As the capacitance is increased, either ferroresonance or normal operating conditions may arise.- Very high values of capacitance can prevent the onset of ferroresonance.

x As the source voltage is increased, the risk of ferroresonance is also increased.x As the circuit losses are increased, the risk of ferroresonance is reduced.x Unlike linear resonance, there is not a single natural frequency of oscillation in a ferroresonant circuit.

Rather, such frequency is variable depending on the intensity of the current and the magnitude ofmagnetisation.

x The frequency of the ferroresonant waveforms may differ from the source voltage.

2.3 Physical Descript ion of a Ferroresonant Oscillat ionThe description of ferroresonance presented in the previous section, although good enough as a firstapproximation, does not provide a real understanding of the mechanisms driving a ferroresonant oscillation.Various explanations of the physical behaviour of ferroresonant circuits can be found in [12], [17], [23] and [24]. Areview of those descriptions, expanded for an enhanced understanding of this complex phenomenon, is presentednext.

Figure 2-25 shows a series R-L-C circuit with a nonlinear inductor and a switch. A two-segment piecewise linearrepresentation is used for the magnetizing impedance. The circuit losses are initially ignored for simplicity. Theprospective current and voltage waveforms under this simplification are presented in Figure 2-26. Initially, thecapacitor charge is equal to U0. At t = 0 sec the switch is closed and the capacitor C starts discharging through theinductor working in its linear region, Lunsat. The frequency of this oscillation is:

1= 1LunsatC Eq. 2-21

This is a very slow discharge process due to the large value of Lunsat. Nevertheless, the flux linkage slowly builds upin the magnetic core until saturation is reached. This is shown in Figure 2-26 at t = t1, when the magnetizingreactance drops to its saturated value, Lsat.

As Lsat is a few orders of magnitude smaller than Lunsat the capacitor discharges very rapidly. The frequency of thisnew oscillation is Z2:

2= 1LsatC Eq. 2-22


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Figure 2-25 Basic Ferroresonant Circuit

Between t1 < t < t2 all the energy stored in the electric field of the capacitor is transferred into the magnetic field ofthe coil. At t = t2 the voltage has dropped to zero and the current reaches its peak. The magnetic field thencollapses and starts charging the capacitor in the opposite polarity. At t = t3 the current through the inductor fallsinto the linear region and the capacitor starts charging through Lunsat. As Lunsat is a few orders of magnitude higherthan Lsat, the frequency of this oscillation Z1 is much lower than the previous one. The current decreases veryslowly and, consequently, very little variation can be appreciated in the capacitor voltage. At t = t4 the voltage in thecapacitor reaches U0 and the discharge process starts again. It can be observed that a full ferroresonant periodcomprises two full charge-discharge cycles.

Using Faradays law, the flux linkage at any time can be calculated as the area under the voltage-time curve. Assuch, the flux linkage from t3 to t5 is equal to the shaded area in Figure 2-26 (a). This can be expressed as:

= 2 =

Eq. 2-23

2 ( ) Eq. 2-24Eq. 2-24 can be used to calculate the period of the ferroresonant oscillation as follows:

= 2 ( ) = 4

Eq. 2-25

= 1

= 2 Eq. 2-26 = + = 4

+ 2 Eq. 2-27

I

)

Lunsat

Lsat)sat

IsatC L

R

U0


Page 31

Figure 2-26 Physical Behaviour of a Ferroresonant Circuit without Losses

Eq. 2-27 indicates that the frequency of a ferroresonant oscillation is directly linked to the circuit capacitance, C, theinitial charge of the capacitor, U0, and the non-linear characteristics of the magnetic core: Lsat and )sat.

It has been shown that the basic ferroresonant circuit of Figure 2-25 behaves like a two-state oscillator switchingbetween two frequencies: low frequency during the unsaturated state and high frequency during the saturatedstate. In the absence of losses, this process will repeat indefinitely with a period Tferro. In reality, the circuit losseswill cause the amplitude of the oscillation to decay. It is a direct consequence of Faradays law that, the lower thevoltage amplitude applied to the magnetic core, the longer it will take to reach saturation. As a result, the frequencyof the ferroresonant oscillation will decrease gradually until the process dies out. Figure 2-27 illustrates aferroresonant oscillation affected by circuit losses. It is shown that the voltage magnitude decreases with eachtransition of polarity. This is due to the high (I2R) losses occurring during the saturated state. These losses are verylow during the unsaturated period due to the low current flow and, hence the voltage remains almost constant.

It has been illustrated that the introduction of losses makes the system dissipative, which causes the amplitude ofthe oscillations to decay. In order for the ferroresonant oscillations to be maintained, energy needs to be suppliedexternally to counteract the losses. This is shown in Figure 2-28, where a voltage source has been introduced torepresent an external source of energy. It is shown that the combined effect of the source voltage and theoscillatory trapped charge is to raise the voltage at the reactor terminals just before each transition. If this voltagerise is enough to compensate for the voltage drop caused by the resistive losses during the transition in polarity,the oscillations is maintained indefinitely.

(a) Voltage, Flux and Current Waveforms (b) Flux-Current relationship

t

t

t

U0

-U0

) sat

-) sat

Isat

-Isatt1

t2t5t3 t4

Lsat Lunsat

I

)

t = 0

t1

t2

t3

t4

t5

Lsat

Lunsat

Charge LDischarge C

Charge CDischarge L

Charge LDischarge C

Charge CDischarge L


Page 32

Figure 2-27 Physical Behaviour of a Ferroresonant Circuit with Losses

With regards to the voltage source two situations could arise in a ferroresonant circuit [17]:

1) If the initial ferroresonant frequency calculated with Eq. 2-27 is higher than the source frequency, there is achance that the decaying frequency of the oscillations will lock at the source frequency. This will result infundamental frequency ferroresonance, as illustrated in Figure 2-29 (a) where TL-C=TS, or fL-C=fS.

2) If on the other hand the initial oscillation frequency calculated with Eq. 2-27 is lower than the sourcefrequency, there is a chance that it will lock at an odd sub-multiple of the power frequency. This will resultin sub-harmonic ferroresonance, as illustrated in Figure 2-29 (b) where TL-C=3TS, or fL-C=fS/3.

Figure 2-28 Effect of Coupled Voltage on Ferroresonant Waveform

Figure 2-29 Derivation of Ferroresonant Modes

U0

U

t

U1

U2

U3

U0 > U1 > U2 > U3 > .

T0 < T1 < T2 < .

T0 T1 T2

-5

0

5

0

-5

0

5

0 0.035

-5

0

5

0 0.035

(a) Fundamental Frequency Ferroresonance (b) Sub-Harmonic Ferroresonance

ULUC

USUS

UC UL

TS

TL-C

TS

TL-CU

tt

U


Page 33

2.4 Types of Ferroresonance Oscil lationsFerroresonant waveforms are categorised according to their periodicity. Based on field experience, experimentalobservations and extensive numerical simulations, ferroresonance has been categorised into the following modes.

x Periodic Ferroresonance ModesPeriodic ferroresonance is characterised by waveforms that repeat themselves. These waveforms are highlydistorted, presenting a dominant frequency that can be either fundamental or sub-harmonic.

In the case of fundamental frequency ferroresonance, the oscillations are mainly at the same frequency as thedriving source. Although the supply frequency is dominant, a large number of harmonics is normally present. Incase of sub-harmonic ferroresonance, the oscillations normally arise at frequencies that are integral odd sub-multiples of the fundamental frequency. Two examples of typical periodic ferroresonant waveforms and frequencyspectrum are shown in Figure 2-30 (fundamental frequency) and Figure 2-31 (sub-harmonic).

Figure 2-30 Typical Fundamental Ferroresonant Voltage Waveform and Frequency Spectrum

Figure 2-31 Typical Sub-Harmonic Ferroresonant Voltage Waveform and Frequency Spectrum

x Quasi-Periodic Ferroresonance ModesThe quasi-periodic regimes are characterised by non-periodic oscillations having, at least, two main frequencies.The fundamental frequency is normally present along with lower sub-harmonic frequencies. A distinctivecharacteristic of these waveforms is the presence of a discontinuous frequency spectrum.

This ferroresonant mode has not been reported very frequently as a stable state. It was first observed in France[30] during a black-start restoration test in a 400 kV system. It has also been referred to as transitional chaos in[31] to describe a state that has no indication of periodicity but still shows features of fundamental and sub-harmonic ferroresonance. This behaviour suggests that the operation is continuously shifting between

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