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

RESONANCE AND FERRORESONANCE IN POWER 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).

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ISBN : (To be completed by CIGRE)

Resonance and Ferroresonance in Power Networks

ISBN : (To be completed by CIGRE)

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Resonance and Ferroresonance in Power Networks

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

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Resonance and Ferroresonance in Power Networks

Resonance and Ferroresonance in Power Networks EXECUTIVE SUMMARY ................................................................................................................................. 6 CHAPTER 1

INTRODUCTION TO RESONANCE AND FERRORESONANCE ........................................... 8

CHAPTER 2

UNDERSTANDING RESONANCE AND FERRORESONANCE ........................................... 11

2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.2 2.2.1 2.2.2 2.2.3 2.3 2.4

Introducing Resonance ..................................................................................................................... 11 Resonance in Electric Circuits ........................................................................................................... 11 Series and Parallel Resonance ......................................................................................................... 11 Duality of Series and Parallel Resonant Circuits ................................................................................ 12 Free Oscillations in Electric Circuits................................................................................................... 17 Ideal Series Resonant Circuit ............................................................................................................ 19 Damped Series Resonant Circuit ...................................................................................................... 21 Introducing Ferroresonance .............................................................................................................. 23 Effect of circuit capacitance ............................................................................................................... 26 Effect of source voltage ..................................................................................................................... 27 Effect of circuit losses ....................................................................................................................... 28 Physical Description of a Ferroresonant Oscillation ........................................................................... 29 Types of Ferroresonance Oscillations................................................................................................ 33

CHAPTER 3 3.1 3.1.1 3.1.2 3.2 3.2.1 3.2.2 3.2.3

Introduction ....................................................................................................................................... 35 Typical circuit capacitances ............................................................................................................... 36 Typical circuit reactances .................................................................................................................. 37 Potentially Risky Configurations ........................................................................................................ 37 Shunt-Compensation and Uneven Phase Operation.......................................................................... 37 Shunt-Compensation and Three-Phase Switching in Multi-Circuit Rights of Way ............................... 39 Distribution Embedded Generation Islanded with Transmission Circuit .............................................. 43

CHAPTER 4 4.1 4.2 4.2.1 4.2.2 4.2.3 4.3 4.3.1 4.3.2 4.3.3 4.4 4.4.1

TYPICAL NETWORK TOPOLOGIES LEADING TO RESONANCE IN TRANSMISSION CIRCUITS ............................................................................................................................ 35

RESONANCE IN SHUNT COMPENSATED TRANSMISSION CIRCUITS ............................. 44

Introduction ....................................................................................................................................... 44 Line Resonance in Uneven Open-Phase Conditions.......................................................................... 45 Physical description .......................................................................................................................... 45 Approximate Steady State Analytical Solution ................................................................................... 47 Effect of Various Design Parameters ................................................................................................. 56 Detailed Analysis of Line Resonance in Uneven Open-Phase conditions using Time-Domain Simulation ......................................................................................................................................... 61 Steady State Analysis ....................................................................................................................... 61 Temporary Overvoltage Analysis ....................................................................................................... 64 Summary of Parameters Affecting Line Resonance in Open-Phase Conditions ................................. 70 Line Resonance in Multiple-Circuit Rights of Way .............................................................................. 71 Physical description .......................................................................................................................... 71

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Resonance and Ferroresonance in Power Networks 4.4.2 Discussion on Circuit Parameters...................................................................................................... 75 4.4.3 Case Study ....................................................................................................................................... 76 4.4.4 Summary and commentary of resonance issues associated with shunt-compensated multiplecircuit rights of way............................................................................................................................ 83 4.5 Practical Consequences of Line Resonance ...................................................................................... 84 4.6 Mitigation Options ............................................................................................................................. 85

CHAPTER 5

NETWORK CONFIGURATIONS LEADING TO FERRORESONANCE................................. 88

5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.2 5.2.1 5.2.2

Ferroresonance in voltage transformers (VT)..................................................................................... 88 VT and Circuit Breaker Grading Capacitors ....................................................................................... 89 Line VTs ........................................................................................................................................... 89 VT and Double Circuit Configuration ................................................................................................. 90 VT in Ungrounded Neutral Systems with Low Zero-Sequence Capacitance ....................................... 90 Ferroresonance in power transformers .............................................................................................. 93 Transformer Terminated Transmission Line in Multi-Circuit Right of Way........................................... 93 Lightly Loaded Transformer Energized via Cable or Long Line from a Low Short-Circuit Capacity Network ............................................................................................................................................ 94 5.2.3 Transformer energized in one or two phases ..................................................................................... 95 5.2.4 Transformer connected to a series compensated line ........................................................................ 97

CHAPTER 6 6.1 6.2 6.3 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 6.4 6.4.1 6.4.2 6.4.3

Analytical Solution Methods .............................................................................................................. 99 Digital Simulation Methods .............................................................................................................. 100 Modelling of Network Components .................................................................................................. 102 Extent of the Network Model ........................................................................................................... 102 Overhead Line Model ...................................................................................................................... 102 Transformers .................................................................................................................................. 103 Shunt Reactors ............................................................................................................................... 104 Other Substation Equipment ........................................................................................................... 104 Sensitivity to Parameters................................................................................................................. 105 Effect of Magnetising Curve ............................................................................................................ 105 Influence of Circuit Breaker Closing Times ...................................................................................... 108 Influence of the Damping in the Circuit ............................................................................................ 108

CHAPTER 7 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.1.5 7.2

MODELLING AND STUDYING ........................................................................................... 99

MITIGATION OF FERRORESONANCE ............................................................................ 109

Mitigation of VT Ferroresonance ..................................................................................................... 109 Secondary Open Delta Resistor ...................................................................................................... 109 Secondary “Wye” Resistor............................................................................................................... 110 Secondary “Wye” Resistor in Series with a Saturable Reactor ......................................................... 111 Other Mitigation Options.................................................................................................................. 112 Mitigation of VT Ferroresonance in Ungrounded Neutral Systems ................................................... 112 Mitigation of Power Transformer Ferroresonance ............................................................................ 114

CHAPTER 8

CONCLUSIONS................................................................................................................. 116

ANNEX A

RESONANCE EXAMPLES ................................................................................................. 124

A1. A.2 A.3 A.4

Resonance Associated with Single-phase Autoreclose Switching of 275 kV Shunt Reactor ............. 124 Line Resonance experienced in 275 kV Double Circuit as a result of System Expansion ................. 127 Line Resonance Experienced in 400 kV and 225kV Subnetwork De-energized for Black-Start Test ................................................................................................................................................ 133 High Temporary Overvoltages When A Distribution-Connected Generator Energizes An Isolated Ungrounded & Faulted High Voltage System ................................................................................... 140

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Resonance and Ferroresonance in Power Networks

ANNEX B B.1 B.2 B.3 B.4 B.5 B.6

FERRORESONANCE EXAMPLES...................................................................................... 146

Power Transformer Terminated Line Ferroresonance ...................................................................... 146 Power Transformer Ferroresonance Teed from a Multi-Circuit Right of Way .................................... 150 Ferroresonance of a VT in Ungrounded Neutral Configuration ......................................................... 152 Ferroresonance with Power Transformer Connected to Series Compensated Line .......................... 156 Ferroresonance of a Line VT with Circuit Breaker Grading Capacitors ............................................. 162 Ferroresonance on Transformer Energization from a Weak Network ............................................... 167

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Resonance and Ferroresonance in Power Networks

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, line dropping/fault clearance, reclosing, and transformer energization. Other special cases of TOV include parallel line resonance, uneven breaker poles in shunt-compensated circuits, ferroresonance and back-feeding [1]. These TOVs have detrimental effects on power quality and can lead to dielectric or thermal failure of equipment. Cigre WG 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, especially on numerical analysis techniques. Hence, a new Cigre WG C4.307 was established with the objective of expanding the research and documenting in detail special cases of TOV. This particular Technical Brochure (TB) concentrates on resonance conditions at power frequency and on the special case of ferroresonance. Harmonic resonances excited 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) resonance associated with the use of shunt compensation and (ii) ferroresonance. Neither resonance nor ferroresonance are new phenomena, and a comprehensive list of technical references is provided in this document. However this information is very scattered and not always readily available to practising power system engineers. The objective of this TB is to compile that knowledge in a simple and concise document that can serve as a guideline for planning engineers and technical consultants to identify potentially dangerous network topologies, to carry-out detailed studies and to assess mitigation options. Comprehensive theory background and methods of analysis are provided within the document as well as a list of typical topologies prone to each phenomenon and practical examples of recent incidents experienced in power systems. A companion TB produced by the same WG is devoted to the related topic of transformer energization phenomena. The scope of this TB is: 1. 2. 3. 4. 5. 6.

Compilation of technical documentation related to the phenomena of resonance and ferroresonance in power networks. Provide a detailed physical explanation of the phenomena of resonance and ferroresonance in power networks. Highlight typical network topologies that present high risk of resonance or ferroresonance. Illustrate methods of analysis for resonance and ferroresonance. Provide mitigation options. 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 resonance cases and the second one for ferroresonance cases. Each Chapter is self-contained and provides different degrees of technical detail and complexity, in such a way that it is not necessary to read all chapters sequentially to acquire an overview of all the issues covered in the TB. For instance, if the reader is only interested in an overview of typical network topologies with a high risk of resonance (Chapter 3), then Chapter 4 may be skipped as it covers a very comprehensive theoretical treatment of the phenomenon. On the other hand, the detailed theoretical analyses may be of great interest to electrical engineering students or to planning engineers tasked with the analysis and solution 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 and highlights some changes that are shaping the development of modern power networks, which may increase the risk of resonance and ferroresonance conditions if these phenomena are overlooked at the design stages. A comprehensive 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 Chapter starts with a review of the well known series and parallel R-L-C circuits and gradually builds the theoretical analysis Page 6

Resonance and Ferroresonance in Power Networks to cover more complex combinations of series and parallel elements, as representative of many practical cases in actual power systems. The complex topic of ferroresonance is subsequently introduced by considering the series resonant R-L-C circuit driven by a voltage source, where the inductor is nonlinear. A distinctive characteristic of a ferroresonant circuit is the existence of several stable solutions. Graphical solutions are provided as useful visualisation tools of the steady state behaviour of the linear resonant case as well as the ferroresonant circuit. The Chapter concludes with a description of the different types of ferroresonant oscillations, including the representative voltage waveforms and frequency spectrum. Chapter 3 gives an overview of typical network topologies that can give rise to resonance associated with the use of shunt-reactors in transmission circuits or substations. It is emphasised that the risk of resonance is not restricted to normal operating conditions and special precautions should be taken during unusual network configurations such as black-start restoration operations or maintenance/testing of sectionalised parts of the network. By increasing awareness of the potential risky topologies it is expected that hazardous incidents of resonance can be prevented or mitigated. Chapter 4 provides a comprehensive theoretical and practical treatment of the resonant phenomena encountered in shunt-compensated transmission circuits. Approximate steady state equations are presented, which enable initial screening of the risk of resonance for a particular topology and set of parameters. A comparison of the results of the approximate steady state analysis to detailed EMT-type simulations shows very good agreement. The effects of design parameters such as tower design, neutral reactor, magnetic core construction, line transposition, magnetic core saturation, etc are explored and a summary table is included ranking their importance. A practical example is provided of surge arrester failure due to over-voltage on a long 500kV transmission line having 72% shunt compensation which, due to breaker failure, resulted in a prolonged two open-pole condition. This topology resulted in a resonant condition almost perfectly tuned to power frequency. Finally, practical consequences of line resonance are discussed and a range of active and passive mitigation options is given as a tool-box to assist power 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 increasing awareness of the potential risky topologies, it is expected that hazardous incidents of ferroresonance can be prevented or mitigated. The Chapter is divided in two main sections, one devoted to inductive voltage transformers and another one dedicated to power transformers. Chapter 6 introduces various analytical approaches as well as digital simulation techniques for the study of ferroresonant circuits. A brief discussion is provided on various nonlinear dynamic analysis tools such as phasespace, Poincaré section, and bifurcation diagram techniques. Equally, comprehensive suggestions and guidelines are presented for the modelling of various electrical items of plant in EMT-type simulation tools that can be used both in resonance and ferroresonance studies. The sensitivity of the study results to various model and simulation parameters is discussed. Chapter 7 is devoted to presenting and discussing possible mitigation techniques for ferroresonance. These measures range from selection of design parameters to avoid risky scenarios (for instance magnetic cores with low flux 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 simulations are given in combination with descriptions of the investigations carried out and the adopted solutions. These examples will provide practising power system engineers with a broad picture of the hazards associated with resonance and ferroresonance and options to deal with them in a cost-effective manner.

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Resonance and Ferroresonance in Power Networks

CHAPTER 1

INTRODUCTION TO RESONANCE AND FERRORESONANCE

Ambitious targets for CO2 emissions reductions and integration of renewable generation in power systems are driving the need for significant reinforcement of existing transmission grids worldwide, in particular new high capacity 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 the construction of new overhead transmission infrastructure is driving the need for new pylon designs that minimise visual impact resulting, in many cases, in smaller structures with reduced clearances. Where possible, existing corridors are being upgraded and operated at higher voltage levels with minimum modifications to the towers, thus increasing its transfer capability. Furthermore, the use of underground cable circuits at HV and EHV transmission levels is steadily increasing, not only in congested urban areas, but also in remote rural locations in order to reduce the environmental impact of new circuits in specific designated zones and to accelerate the connections of wind farms to the transmission grids. These fundamental changes in the design and technology used for new transmission circuits are resulting in an increased system capacitance that is shifting the network natural resonant frequencies closer to the power frequency (50/60 Hz). Generally, resonance occurs in electric circuits that are able to periodically transform energy from an electric field into a magnetic field and vice versa. It is the characteristic of such a circuit that if some single energy is delivered into 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 exchange energy 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 only and that if harmonics are present, either due to saturation of transformers or reactors, the resonance conditions may 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, available literature dealing with this type of resonance, reporting field experiences and assessing or recommending mitigation actions is very scattered and not always readily available to utility planning engineers and technical consultants. This Technical Brochure aims to compile that knowledge in a simple and concise document that can serve as a guideline for planning engineers and consultants to identify dangerous topologies associated with the use 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 be described 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 saturation and oscillate with the circuit capacitance giving rise to severe overvoltages, with almost no damping when the amplitude is moderate, and in some circumstances, excessive overcurrents. If enough energy provided by the source 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 describe an oscillation between a power transformer and a capacitance. Ferroresonance became a problem in the early part of the century when small isolated systems were interconnected by long transmission lines [6] [7], but at that time the cause of the problem was not understood. In the 1940's and 1950's the phenomenon recurred as the electricity supply industry expanded and longer overhead distribution systems were introduced into service. The terms “neutral instability” [8] and “voltage displacement” [9] were also used in the 1940’s referring to the same or very similar phenomenon, although the term “ferroresonance” has prevailed. In 1966 it was discovered that, for cable connected transformers, ferroresonance can occur even on circuits as short as 200 metres [10], [11]. Since that time many studies and investigations have been carried out and a number of papers have been published on the subject. Ferroresonance has focussed the attention of numerous researchers over the years with the outcome of extensive literature 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 widely unknown today and is somehow misunderstood by many power network utilities. It is especially feared by power

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Resonance and Ferroresonance in Power Networks systems operators, as it seems to occur randomly, normally resulting in the catastrophic destruction of electrical equipment and the consequent adverse effect on the reliability of power network. This general lack of awareness means 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 better understanding 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 if some 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 as transformers and breakers, and would cause surge arresters to conduct over extended period of time exceeding their energy dissipation capabilities. A catastrophic failure of a surge arrester for example could damage other key equipment 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 personnel due to the risk of explosion in the work place. An example of such threat is reported in [13], where a 230 kV voltage transformer 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 is discussed in section 4.2.2.4 of this document, where two 500kV surge arresters failed as a result of resonance in a shunt-compensated circuit following an uneven circuit breaker operation. A similar case was reported in the discussions of [82], where surge arrester failures were observed in a shunt-compensated 765 kV line also following uneven circuit breaker operations. Many more examples of plant equipment destruction caused by ferroresonance have been documented in the literature. A very interesting case is reported in [14] where 72 voltage transformers were destroyed in a 50 kV network in Norway. An investigation revealed that all the damaged voltage transformers were from the same manufacturer whereas voltage transformers from other two manufacturers which were also in service survived the incident. The catastrophic destruction of a 230 kV voltage transformer in a cogeneration substation is reported in [15]. The failure of a 275 kV voltage transformer in UK is reported in [16]. Other typical examples include the explosive failure of a 115 kV voltage transformer in Canada [17], the explosive failure of voltage transformers in France [12] and the total destruction or partial damage of six 345 kV voltage transformers as reported by a USA utility [18]. From an operational point of view, resonant and ferroresonant oscillations represent a potential threat to power network plant integrity. The large current pulses caused by transformer saturation may overheat the transformer primary winding and might, eventually, cause insulation damage. The large voltage oscillations, temporary or sustained, 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 withstand capabilities [19]. Resonance and ferroresonance can also have an adverse effect on the reliability of the power network. The forced outage of part of a substation due to an equipment failure can cause severe overloading in other parts of the network 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 electric power utilities. The cost could be twofold: on the one hand, there is an explicit cost associated with the replacement of damaged or destroyed electrical plant, and on the other hand, there are high or perhaps even severe costs associated with a reduced network security and possible disconnection of some customers. Quantification of the latter 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 turn results in a degraded power quality and possible misoperation of some protection relays [21]. Transformer overheating may also occur under Ferroresonant conditions due to excessive flux densities. Since the core is saturated repeatedly, the magnetic flux finds its way into the tank and other metallic parts. This can cause charring or bubbling of paint in the tank [22]. In general, it is possible to distinguish temporary overvoltages from ferroresonance; in the former, the amplitude may be very high initially but decreases rapidly in most cases. As harmonics are involved, the fluxes circulating in

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Resonance and Ferroresonance in Power Networks 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 addressing stresses linked to insulation coordination issues. IEC 60071-1 enables the specification and subsequent purchase of transformers for new installations, but does not address particular aspects related to the behaviour of the equipment under operating conditions such as transformer energization. As resonance and ferroresonance may induce a long duration phenomena, the overvoltages may affect the aging of 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 caused by magnetostriction of the steel and vibrations of the core laminations. This noise has been described in [22] as “the 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 normal operating conditions at rated voltage and frequency.

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Resonance and Ferroresonance in Power Networks

CHAPTER 2

UNDERSTANDING RESONANCE AND FERRORESONANCE

2.1 Introducing Resonance 2.1.1 Resonance in Electric Circuits The phenomenon of resonance exists in a large diversity of physical systems and arises when the system is affected by periodical excitation with a frequency similar to its natural frequency of oscillation. When a system is excited, it tends to oscillate at its natural frequency. If the excitation source has the same frequency as the system’s natural frequency, the system’s response to that excitation can be very large. In order to have resonance in a system, it is necessary to have two forms of energy storage, with energy being periodically transformed from one form to the other, and vice versa: in mechanical systems these are kinetic and potential energy, in electrical systems these are electrical and magnetic energy. Thus, electrical circuits with magnetic and electric fields have the capability of resonating. Electrical resonance occurs when the magnetic and electric energy requirements are equal, 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 variable capacitor. A useful application of resonance in electrical power systems is the design of filters for the suppression of harmful harmonics. However, the phenomenon of resonance can also be very destructive in power systems. Special caution is required in the design and operation of the power network to avoid the occurrence of resonance at the power frequency (50/60 Hz). Such resonance occurrence would lead to uncontrolled system overvoltages that could stress and damage equipment. Electrical resonance occurs in a circuit when the capacitive reactance (1/ C) equals the inductive reactance ( L) at the driving frequency. This frequency, also called natural frequency, is given by Eq. 2-1.

=

Eq. 2-1

2.1.2 Series and Parallel Resonance There are two types of resonance: series and parallel. A basic scheme of series resonance is in Figure 2-1 (a) and parallel 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 to exactly match XC; this frequency is known as the natural or resonant frequency (Eq. 2-1). When the resonant frequency is fed to a series or parallel circuit, XL becomes equal to XC, and the circuit is said to be resonant at that frequency. In the case of series resonance all circuit elements are in one branch with common current (Figure 2-1 (a)). The circuit impedance is given by Eq. 2-2. At low frequencies, the reactance of the capacitor dominates and the phase angle approaches 90°, with current leading voltage. As the frequency increases, the inductive reactance becomes significant and, at the resonant frequency (Eq. 2-1), it grows to the point of cancelling the reactance of the capacitor. At the resonant frequency, the inductor and capacitor series combination becomes invisible and R is the total impedance of the circuit. Voltages ULS and UCS reach high amplitudes but have opposing phase angles and cancel each other out. Note that series resonance must be excited by an alternating voltage source. At series resonance, the circuit current is limited only by the resistor R up to a value I S = US/R. At frequencies above resonance, the inductor dominates the circuit characteristics and the phase angle approaches 90° lagging.

Z

=R+j

L

1 C

Eq. 2-2

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Resonance and Ferroresonance in Power Networks 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 and dominates the circuit admittance. As frequency is increased, the inductive susceptance diminishes and the capacitive susceptance grows until they become equal at the resonant frequency (Eq. 2-1). This resonance frequency is the same for parallel and series circuits. Thus, series and parallel resonance occur at the same frequency for the same combination of inductor and capacitor. At the resonant frequency, the inductor and capacitor parallel combination becomes invisible and G is the total admittance of the circuit. Currents I CP and ILP reach high amplitudes but have opposing phase angles and cancel each other out. Note that parallel resonance must be excited by an alternating current source. At parallel resonance, the circuit voltage is limited only by the conductance G. As the frequency increases above resonance, the capacitive susceptance dominates the circuit characteristics. Thus, the circuit admittance reaches its minimum at resonance and becomes very large at low and high frequencies. In other words, at low and high frequencies, the parallel circuit impedance is very small but it reaches a maximum at the frequency of resonance. This behaviour is the opposite from the series circuit, where the impedance reaches its minimum at resonance.

Y

=G+j

C

1 L

Eq. 2-3

a) Series

b) Parallel

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

2.1.3 Duality of Series and Parallel Resonant Circuits In practice, it is very rare to find perfect series or parallel circuits like those illustrated in Figure 2-1. It is more common 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 Figure 2-2 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. It will be illustrated next that this circuit has the ability to resonate in two modes: series when excited by a voltage source and parallel when excited by a current source at the relevant frequencies.

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Resonance and Ferroresonance in Power Networks

a) Excitation by Voltage Source

b) Excitation by Current Source

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 of Figure 2-3 (b) using Thevenin theorem. The series circuit is equivalent to the original series-parallel circuit if its capacitance is equal to the sum of the series and parallel capacitances and the amplitude of the voltage source is decreased 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 given by Eq. 2-5, which can only be excited by a current source.

(series) =

(parallel) =

(

+

Eq. 2-4

)

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 type of 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 are f n(series) = 3558.81 Hz and fn(parallel) = 5032.92 Hz.

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Resonance and Ferroresonance in Power Networks The results of a frequency scan are included in Figure 2-4, which shows the driving impedance of the circuit as a function of frequency. This circuit presents two resonant modes: series resonance at 3.5 kHz and parallel resonance at 5.03 kHz. At low frequencies, the circuit impedance is very large and capacitive. As the frequency approaches the first resonant mode (3.5 kHz), the impedance drops and the phase angle starts increasing until the impedance reaches its minimum value and the phase angle reaches zero (i.e. the circuit behaves in resistive mode). After that, the impedance starts increasing and becomes inductive. At the second resonant mode (5.03 kHz), the impedance reaches its maximum value and the phase angle drops to zero again. With increasing frequency, the circuit impedance decreases and the phase angle approaches -90° again, returning to capacitive mode. Figure 2-5 illustrates the dual behaviour of this series-parallel circuit when supplied by a voltage or a current source. In this case, the same circuit has been supplied with a voltage or a current source of variable frequency and the voltage across the inductor has been monitored. It can be seen that, when the circuit is supplied with a voltage source (green curve), the voltage across the inductor reaches a maximum at a frequency of 3.5 kHz, which corresponds to the resonance of (CS + CP ) = 2 nF with L = 1 H. Alternatively, when the circuit is supplied with a current source (red curve), the voltage across the inductor reaches a maximum at a frequency of 5.03 kHz, which corresponds to the resonance of CP = 1 nF with L = 1 H. Even though the series-parallel circuit has the capability of resonating in two different modes, only the series mode is excited by a voltage source and only the parallel mode is excited 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 impedance behaves like an “ideal” voltage source. However, a voltage source placed behind a high source impedance behaves more like an “ideal” current source. Therefore, the required excitation for series or parallel resonances can be 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. These simulation results have been obtained by placing a resistive source impedance between the ideal voltage source and CS in the series-parallel circuit of Figure 2-3 (a) with L = 1 H, CS = 1 nF and CP = 1 nF. The resistive source impedance has been varied from 0.1 to 5 M and the voltage across the inductor has been plotted as a function of the source frequency. The simulation results show a clear series resonant peak for low values of source impedance (i.e. between 0.1 and 10 k ). The amplitude of the series resonant peak decreases as the source impedance is increased due to its current limiting effect. For a range of source impedance between 10 k and 100 the circuit experiences a transition with a very flat curve and a shifting resonant frequency. Finally, in the 500 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, but there are no more frequency shifts in the resonant behaviour. Thus, this circuit sees the supply as a voltage source in 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. In practice, the series capacitance in a circuit, CS, can come from series capacitors, grading capacitors in circuit breakers, coupling capacitance with parallel circuits or inter-phase capacitances in transmission circuits. The simulation results shown in Figure 2-7 have been obtained by changing the value of CS in the series-parallel circuit of Figure 2-3 (a) with L = 1 H and CP = 1 nF. The circuit is supplied by an ideal voltage source, without source impedance. The value of CS has been varied from 1 nF to 5 pF and the voltage across the inductor has been plotted as a function of the source frequency. This circuit exhibits a series resonant point at 3.5 kHz for C S = 1 nF. As the value of series capacitance, CS, is reduced, the resonant frequency increases and converges towards the parallel 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 resonant frequency calculated with Eq. 2-5 and the circuit has only one resonant mode. Second interpretation is that the small series capacitance CS creates a high source impedance for the ideal voltage source, which then behaves like a current source exciting the parallel combination of L and C P. Thus, “series” or “parallel” resonance is just a naming tag for the behaviour of this circuit with a single mode of oscillation.

Page 14

Resonance and Ferroresonance in Power Networks And finally, for completeness, the effect of the parallel capacitance CP on the response of the series-parallel circuit is 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 of in 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 CP has been varied from 1 nF to 5 pF and the voltage across the inductor has been plotted as a function of the source frequency. This circuit exhibits a series resonant point at 3.5 kHz for CP = 1 nF. As the value of parallel capacitance, CP, is reduced, the resonant frequency increases and converges towards the parallel resonant frequency of 5.03 kHz. The amplitude of the resonant voltage naturally increases as CP drops and converges to a fixed value, in contrast to the behaviour shown in Figure 2-7 for the reduction of CS. This behaviour can again be interpreted in two ways: first, as CP is smaller, (CS + CP ) CS , therefore the series resonant frequency calculated with Eq. 2-4 converges to the parallel resonant frequency calculated with Eq. 2-5 and the circuit has only one resonant mode. Second interpretation is that the small shunt capacitance C P creates a high impedance (1/ CP), approaching an open circuit, in parallel with L, therefore the circuit effectively becomes a series combination of CS and L, with only one resonant mode at f = 1/(2 (L.CS). Again, “series” or “parallel” resonance is just a naming tag for the behaviour of this circuit with a single mode of oscillation. 800 [ohm] 700

Magnitude

600 500 400 300 200 100 0 0.5

2.4

4.3

6.2

8.1

[kHz]

10.0

4.3

6.2

8.1

[kHz]

10.0

(file current_injection.pl4; x-var f ) v:C_S

80

[°] 46

Angle

12

-22

-56

-90 0.5

2.4

(file current_injection.pl4; x-var f) a:C_S

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

Page 15

Resonance and Ferroresonance in Power Networks 800

16

[kV]

[V]

700

14

600

12

500

10

400

8

300

6

200

4

100

2

0 0.010

1.675

3.340

5.005

6.670

8.335

[kHz]

0 10.000

current_injection.pl4: v:L_____-

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

16 4.0

14

3.5 0.6

3.0

12

[V] 0.5 0.4

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0.1 0.0 4400

1.5

8

4600

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1.0 0.5

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0.0 3000

3500

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4500

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[Hz]

0.1ohm_source_impedance.pl4: v:L_____-

4 2 0 2000

3000

0.1ohm_source_impedance.pl4: v:L_____2000 3000 10kohm_source_impedance.pl4: v:L_____0.1ohm_source_impedance.pl4: v:L_____50kohm_source_impedance.pl4: 10kohm_source_impedance.pl4:v:L_____v:L_____100kohm_source_impedance.pl4: 50kohm_source_impedance.pl4: v:L_____v:L_____500kohm_source_impedance.pl4: v:L_____-

4000

5000

50kohm_source_impedance.pl4: v:L_____100kohm_source_impedance.pl4: v:L_____500kohm_source_impedance.pl4: v:L_____1Mohm_source_impedance.pl4: v:L_____-

6000

[Hz]

7000

500kohm_source_impedance.pl4: v:L_____1Mohm_source_impedance.pl4: v:L_____3Mohm_source_impedance.pl4: v:L_____5Mohm_source_impedance.pl4: v:L_____-

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

Page 16

6000

Resonance and Ferroresonance in Power Networks 16 1.2

14 1.0

12 0.8

10

0.6

8

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6

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4

0.0 4400

4600

4800

5000

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[Hz]

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2 0 2000

3000

4000

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6000

[Hz]

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2000 3000 voltage_injection_1nF_cs.pl4: v:L_____voltage_injection_1nF_cs.pl4: v:L_____-voltage_injection_200pF_cs.pl4: v:L_____voltage_injection_20pF_cs.pl4: v:L_____voltage_injection_500pF_cs.pl4: v:L_____voltage_injection_100pF_cs.pl4: v:L_____voltage_injection_500pF_cs.pl4: v:L_____voltage_injection_10pF_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_____-voltage_injection_5pF_cs.pl4: v:L_____voltage_injection_20pF_cs.pl4: v:L_____voltage_injection_50pF_cs.pl4: v:L_____-

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

25 UL

f (resonance) = 5.02 kHz for CP = 5 pF

20

f (resonance) = 3.55 kHz for C P = 1 nF

15 Reducing C P

10

5

0 3000

3500

4000

4500

5000

5500

[Hz]

6000

voltage_injection_1nF_cp.pl4: voltage_injection_1nF_cp.pl4:v:L_____v:L_____-voltage_injection_200pF_cp.pl4: v:L_____-voltage_injection_20pF_cp.pl4: v:L_____voltage_injection_500pF_cp.pl4: voltage_injection_500pF_cp.pl4: v:L_____v:L_____voltage_injection_100pF_cp.pl4: v:L_____-voltage_injection_10pF_cp.pl4: v:L_____voltage_injection_200pF_cp.pl4: voltage_injection_200pF_cp.pl4: v:L_____v:L_____voltage_injection_50pF_cp.pl4: v:L_____- voltage_injection_5pF_cp.pl4: v:L_____voltage_injection_100pF_cp.pl4: v:L_____voltage_injection_20pF_cp.pl4: v:L_____voltage_injection_50pF_cp.pl4: v:L_____-

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

2.1.4 Free Oscillations in Electric Circuits Electromagnetic resonance can occur in electric circuits that are able to periodically transform energy from an electric field into a magnetic field and vice versa. Such circuits experience free oscillations when energy is delivered 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 a circuit. For example the circuit shown in Figure 2-9(a) with C = 100 nF and L = 100 H starts to oscillate in an undamped fashion following the switching (at t = 0.1 sec). In this example the frequency of free oscillation (Eq. 2-1) Page 17

Resonance and Ferroresonance in Power Networks is f n = 50,33 Hz and the voltage and current waveforms measured in the ideal L-C circuit are shown in Figure 2-9(b). 10.0 [kV] 7.5

1.0 [A] 0.6

5.0 2.5

0.2

0.0 -0.2

-2.5 -5.0

-0.6 -7.5 -10.0 0.08

0.10

0.12

(file Fig_2-1.pl4; x-var t) v:U_L -

0.14

0.16

0.18

[s]

-1.0 0.20

c:U_C -U_L

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 the inductor and the capacitor is transferred into thermal energy and dissipated in the resistive elements. Resistive losses in circuits come from the resistance of conductors, corona effect, lossy polarisation in dielectrics or from alternating magnetisation in ferromagnetic cores (hysteresis and eddy currents). An example of a lightly damped free oscillation is shown in Figure 2-10 (b), obtained with a resistor R value of 1 k . Figure 2-11 (a) shows another example with higher losses, illustrating a free oscillation lasting for a few cycles. However, if the circuit losses are very high, a free oscillation will not occur because all the energy in the circuit is dissipated in the first cycle and the transient becomes aperiodical, as shown in Figure 2-11 (b). 10.0

1.0

[kV]

[A]

7.5

0.6 5.0 2.5

0.2

0.0 -0.2

-2.5 -5.0

-0.6 -7.5 -10.0 0.08

0.10

(f ile Fig_2-2.pl4; x-v ar t) v :U_L

0.12 -

c:U_R

Figure 2-10 Behaviour of damped circuit – (a) R-L-C Circuit

Page 18

0.14

0.16

0.18

-U_L

(b) Damped free oscillations

[s]

-1.0 0.20

Resonance and Ferroresonance in Power Networks

10.0

1.00

[kV]

[A]

6.6

0.66

3.2

0.32

-0.2

1.0 [A]

8

0.8

6

0.6

4

0.4

2

0.2

0

0.0

-0.02

-3.6

-7.0 0.08

10 [kV]

-0.36

0.10

(f ile f ig_2-3a.pl4; x-v ar t) v :U_L -

0.12 c :U_R

0.14

0.16

0.18

[s]

-0.70 0.20

-2 0.08

0.10

0.12

(f ile f ig_2-3b.pl4; x-v ar t) v :U_L

-U_L

-

c:U_R

Figure 2-11 Examples of damped oscillations – (a) strong damping

0.14

0.16

0.18

[s]

-0.2 0.20

-U_L

(b) aperiodical transient

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

2.1.5 Ideal Series Resonant Circuit Figure 2-12 shows the transition of an ideal lossless series oscillatory circuit with natural frequency f n = 50 Hz to resonance following the connection of a 50 Hz voltage source. As a function of time, the rise of the resonant voltage amplitude UL (as an envelope) is given by

( )=

Eq. 2-6

If we consider the network frequency as a constant, the rise time of resonant voltage on this basic circuit is independent of the circuit parameters, except for the magnitude of the excitation voltage US. In this particular example, the resonant voltage rate of rise is 1570.8 kV/s based on US value of 10 kV and a source frequency of 50 Hz. The capacitor and inductor values used in this resonant circuit were 101.32 nF and 100 H respectively, but the same 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. 20 [kV] 15

1.6 [MV] 1.2

10

0.8

5

0.4

0

0.0

-5

-0.4

-10

-0.8

-15

-1.2

-20

-1.6 0.0

0.2

(file Fig_2-5.pl4; x-var t) v:U_L -

0.4

Figure 2-12 Ideal series resonance oscillation with

Page 19

0.6

0.8

v:U_S

n

S

[s]

1.0

Resonance and Ferroresonance in Power Networks

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

( )=

Eq. 2-7

which is independent of frequency but inversely proportional to inductance L. This implies that various 50 Hz series resonant circuits with various combinations of LC parts have the same rise time of resonant voltage under similar excitation conditions, but the currents fed from the voltage source and their rise is inversely proportional to resonant 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 that the active impedance of the resonant circuit is also changing in time according to

( )=

( )

=

Eq. 2-8

Introducing a small difference between the voltage source frequency f s and the free oscillation natural frequency f n, will result to a phase shift between voltage phasors that will change slowly as shown in Figure 2-13, and the resonant overvoltage will fluctuate within a sine wave envelope in accordance with:

( )=

[

(

)+

(

)]

and for sufficiently small differences between

( )=

(

+

)

S

(

and

Eq. 2-9

n

it can be simplified to

)

Eq. 2-10

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

Page 20

Resonance and Ferroresonance in Power Networks

150

150

[kV]

[kV]

100

100

50

50

0

0

-50

-50

-100

-100

-150 0.0

-150

0.1

(f ile Fig_2-6-a.pl4; x-v ar t) v :U_L

0.2 -

0.3

0.4

0.5

[s]

v :U _S

0.6

0.0

0.1

(f ile f ig_2-6-b.pl4; x-var t) v:U_L -

0.2

0.3

0.4

0.5

[s]

0.6

v:U_S

Figure 2-13 Ideal series resonance oscillation with

n

S

40 [kW] 30

1500 [J] 1200

20 10

900

0 600

-10 -20

300 -30 -40 0.05

0.10

0.15

(file fig_2-6-b.pl4; x-var t) p:U_S -XX0001

0.20

0.25

0.30

0.35

0.40

[s]

0 0.45

e:U_S -XX0001

Figure 2-14 Energy exchange between source and resonant circuit

2.1.6 Damped Series Resonant Circuit In real power networks resonant circuits are never lossless and hence it is important to visualise the effect of losses on the resonant cases explained in the previous section. In a damped resonant circuit with f n = f s, the resonant voltage will not increase indefinitely as in ideal lossless circuits, because the resonant current is limited by the resistance R and the maximum resonant voltage is given by:

( )=

Eq. 2-11

Page 21

Resonance and Ferroresonance in Power Networks An example of this behaviour is given in Figure 2-15, obtained for a series L-C resonant circuit with a 101.32 nF capacitor and a 100 H inductor with a source voltage of 10 kV, for two different resistance values (300 and 1000 ). In this example the natural frequency of the circuit is the same as the voltage source frequency (f n = fs). It can be observed that the higher resistance value (green waveform) provides higher limitation on the resonant voltage developed across the circuit elements. Introducing a small difference between the excitation frequency fs and the frequency of free oscillation fn results in a modulated wave. Figure 2-16 illustrates this behaviour for two values of resistance and the following circuit parameters: L = 114.63 H, C = 101.32 nF, Us = 10kV, fn = 46.70 Hz. The small damping case presents a modulated wave 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 can‘t reach 270° but settles to a value between 0°and 90° following the non-zero minimum point of the modulation as seen in Figure 2-17 (a). Similarly, Figure 2-17 (b) shows that for fn fs the angle between phasors moves from 90° to 180° and then to a value between 0°and 90° following the non-zero minimum point of the modulation.

1.2 [MV] 0.8 0.4 0.0 -0.4 -0.8 -1.2

0

1

2

3

4

5

6

Figure 2-15 Damped resonant voltage with UL

[s]

8

S

UL

150

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[kV]

[kV]

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50

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n

7

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0.5

1.0

a) R = 300

1.5

2.0

b) R = 1000

Figure 2-16 Damped series resonance oscillation with

Page 22

n

S

(two different dampings)

2.5

[s]

3.0

Resonance and Ferroresonance in Power Networks 120

80

[kV]

[kV] 60

80

40 40 20 0

0 -20

-40

-40 -80 -60 -120 0.1

0.2

(f ile f ig_2-10-a.pl4; x-v ar t) v :U_L -

0.3

0.4

0.5

[s]

0.6

v :U_S

-80 2.95

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(f ile f ig_2-10-a.pl4; x-v ar t) v :U_L -

2.97

2.98

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[s]

3.00

v :U_S

A) fn < fs = 50 Hz

120

80

[kV]

[kV]

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60 40

40 20 0

0 -20

-40

-40 -80 -60 -120 0.1

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(f ile Fig_2-10-b.pl4; x-v ar t) v :U_L -

0.3

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v :U_S

0.5

[s]

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(f ile Fig_2-10-b.pl4; x-v ar t) v :U_L -

2.97

2.98

2.99

[s]

3.00

v :U_S

B) fn > fs = 50 Hz Red waveform: Voltage across inductor L Green waveform: Source Voltage

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

) with

n

S

2.2 Introducing Ferroresonance In simplest terms, ferroresonance can be described as a non-linear oscillation arising from the interaction between an iron core inductance and a capacitor. In this section, the description of ferroresonance follows from the previous sections with a basic analysis of a series resonant circuit and gradually increases the level of complexity to provide a comprehensive explanation of the physical mechanism driving the nonlinear oscillation of ferroresonance. In this initial description, a very simplified model of the magnetic core is used for a better understanding of the basic mechanisms driving the oscillation. As with linear resonance, ferroresonant circuits can be either series or parallel, albeit only series configurations are typically encountered in transmission networks. It should be noted that parallel ferroresonant configurations are common in distribution systems with ungrounded or resonant neutral connections. For simplicity and better understanding, the analysis and explanation that follows is based on series resonant and ferroresonant circuits only. A basic series R-L-C circuit is shown in Figure 2-18 which includes the series connection of a voltage source US, to a 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 expressed as:

Page 23

Resonance and Ferroresonance in Power Networks

=

where

(

+

s

)

1

=

+

is the angular frequency of the voltage source.

(

)

Eq. 2-12

Figure 2-18 Linear Series R-L-C circuit Resonance occurs when the capacitive reactance equals the inductive reactance at the driving frequency. Under this condition the circuit impedance becomes purely resistive.

=

Eq. 2-13

The most characteristic feature of a linear R-L-C circuit is that there is only one natural frequency, f n, at which the inductive 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 for simplicity. The voltage-current representation results in two straight lines with slopes equal to the inductive and capacitive 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 f S below the circuit natural frequency f n. It can be seen that the capacitive reactance, XC, exceeds the inductive reactance, XL, resulting in a leading current and a high voltage across the capacitor. Similarly, Figure 2-19 (c) shows the operating point for a source frequency above the circuit natural frequency, f n. 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 a source frequency equal to circuit natural frequency f n, the inductive and capacitive reactances are equal and the two 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 of limiting 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 circuit gain. It becomes apparent that low circuit losses lead to high capacitor and inductor voltages under resonant conditions.

Page 24

Resonance and Ferroresonance in Power Networks

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 series ferroresonant circuit can be obtained as shown in Figure 2-20. What differentiates ferroresonance from linear resonance is that the inductance is not constant; therefore the ferroresonant frequency calculated with Eq. 2-1 becomes a moving target. This means that a range of circuit capacitances can potentially lead to ferroresonance at a particular source frequency. Another characteristic of ferroresonance is the existence of several solutions. This distinctive 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 differential equations. The circuit analysis, however, can be simplified considerably and yet provide a thorough conceptual description of ferroresonance by limiting the calculations to power frequency and steady state [23]. It should be noted 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 at power frequency. Furthermore, the resistive losses are also ignored. Under these particular conditions, the equation describing the steady-state circuit behaviour at power frequency can be expressed as:

=

( )

1

=

( )

Eq. 2-17

where XC is the circuit capacitive reactance at power frequency, S is the source angular frequency and XL(I) is the variable reactance of the saturable magnetic core. This voltage across the non-linear inductance [UL(I) = I XL(I)] is a function of the current, which is characteristic of the ferromagnetic inductance and is solely dependent on the number of turns and the dimensions of the iron core.

Page 25

Resonance and Ferroresonance in Power Networks Eq. 2-17 has been solved graphically in Figure 2-21 [22]. The intersection of the “US + I.XC” line with the non-linear “I XL(I)” curve gives the solution for the current in the circuit. The first distinctive characteristic of this graphical visualisation is that there are three possible solutions: Point 1 represents a normal operating point in which the circuit is working in an inductive mode, with lagging current and low voltages. Voltage and current related by a linear expression. The inductive voltage is greater than the capacitive voltage by the source voltage. This is a stable solution. Point 3 represents a ferroresonant state in which the circuit is working in a capacitive mode, with leading current and high voltages. Voltage and current are related by a non-linear expression. The capacitive voltage is greater than the inductive voltage by the source voltage. This is also a stable solution. 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 small increase or decrease of the current will result in a small linear change in the capacitor voltage. However, the counteracting inductive voltage changes more intensely with current due to its steeper slope, therefore the current will return to its original value to find equilibrium. Similarly, at point 3, a small variation in current will result in a small variation in inductive voltage. The counteracting capacitive voltage changes more intensely due to its steeper slope, 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 in inductive voltage and a large increase in the capacitive voltage. In this case, the steepness of the capacitive voltage is higher than the opposing inductive voltage and therefore the current will continue increasing away from point 2. A similar consideration can be made for a small decrease in current.

I·XL(I)

U US + I·XC

UL

2

UL UL

1 U0

UC US

US US

US

UL

I I

UC

UL

US

UC Solution at point “1”

-0.06

I

I

UC

XC Solution at point “2” (unstable)

UC 3

-2000

Solution at point “3”

Figure 2-21 Graphical Solution of the Series Ferroresonant Circuit

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

Page 26

Resonance and Ferroresonance in Power Networks state, point 1. This result has practical implications in transmission substations since it suggests that ferroresonance 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 sou rce voltage The effect of the source voltage is illustrated in Figure 2-23. As this voltage is increased, the “US + I.XC” line moves upwards to a point in which there is no intersection in the first quadrant. Operating points 1 and 2 disappear and the only possible solution is point 3, which is a ferroresonant state. Note also that the disconnection of the source voltage, U, may not result in the elimination of ferroresonance, as illustrated with state 3’. As U is removed, the operating point simply slides to the right, but remains in the saturated region. This statement assumes that the circuit has no losses, which is not true in reality, but it serves to illustrate the fact that, in theory, the ferroresonant oscillations can be self-sustained.

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

Page 27

Resonance and Ferroresonance in Power Networks

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

+ (

=

=(

|

) +(

( )

)

()

( )

|=

Eq. 2-18

(

)

Eq. 2-19

)

Eq. 2-20

The first term of Eq. 2-20 is plotted in Figure 2-24 (a). It is shown that multiple solutions are possible when the I.XC line intersects the UL(I)= I·XL(I) curve in the saturation region. To the left of IC the circuit operates in an inductive mode whereas the region to the right of IC corresponds to a capacitive mode. The second term of Eq. 2-20 is an ellipse that crosses the horizontal axis at I = US/R and the vertical axis at US. This is plotted in Figure 2-24 (b). The intersection of this ellipse with the “|UL(I)- I.XC|” curve gives the current in the circuit. Figure 2-24 (b) shows three possible 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. If the 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 that the onset of ferroresonance can be avoided by increasing the circuit losses. V

Xc

VC

Inductive Zone

VL(I)

Capacitive Zone

VULL X XCC I I

I

0 0.00

0.01

V

V

VULL XXCCI I

E0 0 0.00

1

2

3

2 2 E 02U s RR II 2

E0

I IC

E0 R1

2 2 E 20U s RR II 2

0.01

0 0.00

VULL XXCCI I

1 I

E0 R2

IC

0.01

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 ferroresonant behaviour that has been limited to steady state and power frequency. It is emphasized that this analysis is not valid for 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

Page 28

Resonance and Ferroresonance in Power Networks parameters, where the circuit ceases to operate in a linear mode. Despite its limitations, the graphical analysis has provided a very good intuitive insight into the key features governing a ferroresonant oscillation, as follows: Multiple steady-states are possible in a ferroresonant circuit. Jump phenomena, called bifurcations, can occur where the operating point changes drastically for a small change in circuit parameters, supply voltage or frequency. 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.

As the source voltage is increased, the risk of ferroresonance is also increased. As the circuit losses are increased, the risk of ferroresonance is reduced. 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 of magnetisation. The frequency of the ferroresonant waveforms may differ from the source voltage.

2.3 Physical Description of a Ferrores onant Oscillation The description of ferroresonance presented in the previous section, although good enough as a first approximation, 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]. A review of those descriptions, expanded for an enhanced understanding of this complex phenomenon, is presented next. Figure 2-25 shows a series R-L-C circuit with a nonlinear inductor and a switch. A two-segment piecewise linear representation is used for the magnetizing impedance. The circuit losses are initially ignored for simplicity. The prospective current and voltage waveforms under this simplification are presented in Figure 2-26. Initially, the capacitor charge is equal to U0. At t = 0 sec the switch is closed and the capacitor C starts discharging through the inductor working in its linear region, Lunsat. The frequency of this oscillation is: 1=

1

Eq. 2-21

Lunsat C

This is a very slow discharge process due to the large value of Lunsat. Nevertheless, the flux linkage slowly builds up in the magnetic core until saturation is reached. This is shown in Figure 2-26 at t = t1, when the magnetizing reactance 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 this new oscillation is 2: 2=

1

Eq. 2-22

Lsat C

Page 29

Resonance and Ferroresonance in Power Networks

R

Lsat

sat

C

Lunsat

U0

L

Isat

I

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 of the coil. At t = t2 the voltage has dropped to zero and the current reaches its peak. The magnetic field then collapses and starts charging the capacitor in the opposite polarity. At t = t3 the current through the inductor falls into the linear region and the capacitor starts charging through Lunsat. As Lunsat is a few orders of magnitude higher than Lsat, the frequency of this oscillation 1 is much lower than the previous one. The current decreases very slowly and, consequently, very little variation can be appreciated in the capacitor voltage. At t = t 4 the voltage in the capacitor reaches –U0 and the discharge process starts again. It can be observed that a full ferroresonant period comprises two full charge-discharge cycles. Using Faraday’s law, the flux linkage at any time can be calculated as the area under the voltage-time curve. As such, the flux linkage from t3 to t5 is equal to the shaded area in Figure 2-26 (a). This can be expressed as:

=2

= (

2

Eq. 2-23

)

Eq. 2-24

)=4

Eq. 2-25

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

=2 ( =

1

=

=2

Eq. 2-26

+

=4

+2

Eq. 2-27

Page 30

Resonance and Ferroresonance in Power Networks

Lsat U0

Lunsat

t

Lsat

t2

t1 -U 0

Charge L Charge C Charge L Discharge C Discharge L Discharge C

t3

Charge C Discharge L

Lunsat

sat

t -

t4

t= 0

I

sat

t5

Isat -Isat

t t1 t3 t2

t4

t5

(a) Voltage, Flux and Current Waveforms

(b) Flux-Current relationship

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, the initial 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 switching between two frequencies: low frequency during the unsaturated state and high frequency during the saturated state. In the absence of losses, this process will repeat indefinitely with a period T ferro. In reality, the circuit losses will cause the amplitude of the oscillation to decay. It is a direct consequence of Faraday’s law that, the lower the voltage amplitude applied to the magnetic core, the longer it will take to reach saturation. As a result, the frequency of the ferroresonant oscillation will decrease gradually until the process dies out. Figure 2-27 illustrates a ferroresonant oscillation affected by circuit losses. It is shown that the voltage magnitude decreases with each 2 transition of polarity. This is due to the high (I R) losses occurring during the saturated state. These losses are very low 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 of the oscillations to decay. In order for the ferroresonant oscillations to be maintained, energy needs to be supplied externally to counteract the losses. This is shown in Figure 2-28, where a voltage source has been introduced to represent an external source of energy. It is shown that the combined effect of the source voltage and the oscillatory trapped charge is to raise the voltage at the reactor terminals just before each transition. If this voltage rise is enough to compensate for the voltage drop caused by the resistive losses during the transition in polarity, the oscillations is maintained indefinitely.

Page 31

Resonance and Ferroresonance in Power Networks

U U0 > U1 > U 2 > U3 > …. T0 < T1 < T2 < ….

U0

U2 T1

T0

T2

t

U3

U1

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 a chance that the decaying frequency of the oscillations will “lock” at the source frequency. This will result in fundamental frequency ferroresonance, as illustrated in Figure 2-29 (a) where TL-C=TS, or f L-C=fS. 2) If on the other hand the initial oscillation frequency calculated with Eq. 2-27 is lower than the source frequency, there is a chance that it will “lock” at an odd sub-multiple of the power frequency. This will result in sub-harmonic ferroresonance, as illustrated in Figure 2-29 (b) where TL-C=3TS, or fL-C=f S/3. 5

0 0

-5

Figure 2-28 Effect of Coupled Voltage on Ferroresonant Waveform

TL-C

U5 UC US 0 0

-5

TL-C

5 U

UC

UL

UL

US

t

0

0.035

0

TS

TS

-5

(a) Fundamental Frequency Ferroresonance

(b) Sub-Harmonic Ferroresonance

Figure 2-29 Derivation of Ferroresonant Modes

Page 32

t 0.035

Resonance and Ferroresonance in Power Networks

2.4 Types of Ferrores onance Oscillations Ferroresonant waveforms are categorised according to their periodicity. Based on field experience, experimental observations and extensive numerical simulations, ferroresonance has been categorised into the following modes. Periodic Ferroresonance Modes Periodic ferroresonance is characterised by waveforms that repeat themselves. These waveforms are highly distorted, 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 the driving source. Although the supply frequency is dominant, a large number of harmonics is normally present. In case of sub-harmonic ferroresonance, the oscillations normally arise at frequencies that are integral odd submultiples of the fundamental frequency. Two examples of typical periodic ferroresonant waveforms and frequency spectrum 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

Quasi-Periodic Ferroresonance Modes The 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 distinctive characteristic 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 subharmonic ferroresonance. This behaviour suggests that the operation is continuously shifting between various periodic modes without stabilising into any particular one. An example of a quasi-periodic waveform and frequency spectrum is given in Figure 2-32.

Page 33

Resonance and Ferroresonance in Power Networks

Figure 2-32 Typical Quasi-Periodic Ferroresonant Voltage Waveform and Frequency Spectrum

Chaotic Ferroresonant Modes Chaotic ferroresonance waveforms show an irregular and apparently unpredictable behaviour and a broadband power spectrum with a sharp component at system frequency. This ferroresonant mode is characterised by a nonperiodic waveform with a continuous frequency spectrum. Although the possibility of chaotic ferroresonant modes has been widely described in literature, [30] to [37], this mode has only been predicted in EHV substations for unrealistic values of source voltage, circuit capacitance or losses [33] to [36]. For instance, reference [33] reported that chaotic ferroresonance could only be obtained for a source voltage in excess of 25.26 pu when realistic values of transformer losses were employed. It is noteworthy that no practical experience of a sustained chaotic ferroresonance in an EHV substation has been reported to date. An example of a chaotic waveform and frequency spectrum is given in Figure 2-33.

Figure 2-33 Typical Simulated Chaotic Ferroresonant Voltage Waveform and Frequency Spectrum

Page 34

Resonance and Ferroresonance in Power Networks

CHAPTER 3

TYPICAL NETWORK TOPOLOGIES LEADING TO RESONANCE IN TRANSMISSION CIRCUITS

3.1 Introduction The most common case of power frequency resonance in transmission circuits is related to the application of shunt compensation (Figure 3-1). Line resonance associated with single phase (or unbalanced) operation of circuit breakers has been described in the literature [68] - [73] and is explained in detail in section 4.2 of this document. As a rule of thumb, shunt compensation degrees in excess of 70% can lead to high temporary overvoltages following single-phase switching operations or a result of circuit breaker malfunctioning. The resonant condition arises from the interaction between the shunt-reactor and the phase capacitance in the “disconnected” phase(s), with energy coupled from the remaining “energized” phases via the inter-phase capacitances. The key elements required to form a series resonant circuit are: 1. 2. 3. 4.

Shunt reactors directly connected to a transmission circuit Inter-phase capacitive coupling At least one phase is disconnected At least one phase is energized

Series line resonance arising from unbalanced switching operations can affect single circuits and multi-circuit rights of way. A similar phenomena can also occur in multi-circuit rights of way when one of the circuits is fully de-energized (i.e. three phase disconnection). This phenomenon has been described in the literature [77]-[82], and is also explained in detail in Section 4.4 of this document. The resonant condition occurs when a de-energized shunt-compensated circuit is in close proximity to another energized circuit. As a rule of thumb, shunt compensation degrees in excess of 60% can lead to high overvoltages for typical inter-circuit capacitive coupling. The parallel resonant condition arises from the interaction between the shunt-reactors and the line capacitance in the “disconnected” circuit, with energy coupled from the nearby parallel circuit(s). The key elements required to form a parallel resonant circuit are: 1. Shunt reactors directly connected to a de-energized transmission circuit 2. Inter-circuit capacitive coupling with another energized transmission circuit

Phenomena

Circuit Topology

Switching Operation

Single Circuit

Unbalanced (section 4.2)

Line Resonance in Shunt-Compensated Circuits

Unbalanced (section 4.2) Muti-Circuit Right of Way Balanced (section 4.4)

Figure 3-1 Summary of Line Resonance in Shunt Compensated Circuits

Page 35

Resonance and Ferroresonance in Power Networks Typical network topologies with risk of resonance at power frequency are presented in the next sections. This list is not exhaustive and additional topologies can also result in resonance during unusual network configurations such as blackstart restoration operations (see ANNEX A for an example). Furthermore, it is important to note that shuntcompensation is not an essential requirement for power frequency resonance, as the required reactance can be present in other circuit components, such as transformers (see ANNEX A for an example) or a weak feeding network impedances.

3.1.1 Typical circuit capacitances When assessing possible resonant conditions, series and parallel circuit capacitances are equally important. Parallel capacitances are due to the phase-to-ground capacitance of the lines or cables, shunt capacitor banks, and – to a lesser extent – stray capacitances in all apparatus. Series circuit capacitances appear in the grading capacitors of circuit breakers, phase-to-phase capacitances in single-circuit lines and inter-circuit capacitance in multi-circuit rights of way. Typical transmission circuit capacitances are listed below for illustration purposes: a) France : The typical phase-to-ground capacitance (C0) of overhead-lines is in the range of 10-13 nF/km for 400 kV lines and 8-9 nF/km for 225 kV and 90 kV lines. The inter-circuit capacitance of 400 kV double circuit-lines is in the range 0.2-1.2 nF/km. The cable capacitance to ground is in the range of 100200 nF/km for 400 kV and 225 kV XLPE cables and 150-350 nF/km for 90 kV XLPE cables. b) Ireland : 400 kV overhead-line (single circuit): C+ = 11.59 nF/km, C0 = 7.77 nF/km c) Canada (BC Hydro) 500 kV overhead-line (single circuit): C+ = 12.9 nF/km, C0 = 7.44 nF/km d) 750kV overhead line between Hungary and USSR [71]: (Hungarian section): C+ = 13.25 nF/km, C0 = 9.72 nF/km e) Saudi Arabia : 380kV double circuit line [83]: C+ = 13.76 nF/km, C0 = 7.78 nF/km f)

500 kV circuits in Thailand [74]: o o

Single circuit construction: Cph-gr = 8.55 nF/km, Cph-ph = 1.64 nF/km (i.e. C+ = 13.47 nF/km, C0 = 8.55 nF/km Double circuit construction:

Configuration

Cph-gr

Cph-ph

Ccct-cct (perfect transposition)

Both circuits in service One circuit in service with the other circuit grounded

5.39 nF/km

1.76 nF/km

1.05 nF/km

0.74 nF/km

Ccct-cct (unlike phases in incomplete transposition) 1.21 nF/km

8.55 nF/km

1.76 nF/km

---

---

---

Ccct-cct (like phases in incomplete transposition)

g) 400 kV circuit constructions in Hungary [75]. Line configuration Conventional 400 kV flat arrangement Conventional 400 kV delta arrangement Compact 400 kV (2 x 500mm2 phase conductors) Compact 400 kV (3 x 300mm2 phase conductors)

C0 [nF/km] 8.23 5.95

C+ [nF/km] 10.96 8.77

Cph-ph [nF/km] 0.91 0.94

7.03

12.55

1.83

7.46

13.95

2.16

Series capacitances in HV and EHV circuit breakers range between 100 pF and 800 pF for 225kV to 400kV applications.

Page 36

Resonance and Ferroresonance in Power Networks

3.1.2 Typical circuit reactances Circuit reactances are present in the form of shunt reactors, series reactors, transformers and, to a smaller extent, line series reactances. It is not possible or practical to give typical values as these are highly dependent on the intended application. For example, a series reactor installed for fault level mitigation purposes will be tailordesigned for its specific location. Rather that providing typical reactance values, the following sections will highlight network topologies with risk of resonance, which will justify a more detailed analysis.

3.2 Potentially Risky Configurations 3.2.1 Shunt-Compensation and Uneven Phase Operation This phenomenon can occur in single and multi-circuit rights of way if one circuit is operated under un-even phase switching conditions. Uneven phase operation in transmission circuits can be: a) Intentional: single-phase tripping schemes applied to transmission circuits for various reasons, such as (i) improve system transient stability, (ii) improve system reliability and availability, (iii) reduce switching overvoltages and/or (iv) reduce shaft torsional oscillations in large thermal units [73] or b) Unintentional: misoperation of circuit breakers or POW control o during an opening operation: one (or two poles) may get stuck, resulting in two (or one) phases being de-energized while one (or two) phase remains energized (see Figure 3-2 A and B). o during a closing operation: one (or two) poles my fail to close, resulting in two (or one) phases being energized while one (or two) phases remain de-energized (see Figure 3-2 B and C).

(A) One stuck circuit breaker pole during opening operation

˜ ˜ ˜

(B) Two stuck circuit breaker poles during opening operation

Us

A

Disconnected phase

Us

B

Disconnected phase

Us

C

Energised phase

Stuck Pole

(C) One stuck circuit breaker pole during closing operation

˜ ˜ ˜

Us

A

Us

B C

Us

A

Disconnected phase

B

Energised phase

Stuck Pole

Us

C

Energised phase

Stuck Pole

(D) Two stuck circuit breaker poles during closing operation

Disconnected phase

Pole fails to close

Us

˜ ˜ ˜

Us

Energised phase Energised phase

˜ ˜ ˜

Us

A

Disconnected phase

Pole fails to close

Us

B

Disconnected phase

Pole fails to close

Us

C

Energised phase

Figure 3-2 Uneven phase operation due to circuit breaker misoperation Regardless of how the uneven phase operation arises, a series resonant circuit can be formed between the deenergized phase(s) shunt reactor(s) and the circuit capacitance. This resonant condition will last as long as the uneven phase operation remains. Typically, protection or automated control systems will limit the duration of the uneven phase operation to less than 1 – 2 seconds, however large resonant overvoltages – even for short duration

Page 37

Resonance and Ferroresonance in Power Networks - can cause damage in HV equipment. This phenomenon can occur in both, single circuit and multi-circuit rights of way and it is discussed in detail in section 4.2. This type of resonance can only occur if a shunt reactor remains electrically connected to the disconnected phase(s), as would normally happen with fixed shunt compensation directly applied to the circuit. However, some unusual situations may also arise during commissioning or blackstart restoration paths where the inductance comes from busbar shunt reactors rather than line shunt reactors. This is illustrated in Figure 3-3 and Figure 3-4. Special attention must be paid to identify all possible sources of reactances likely to interact with the circuit capacitance. (A) One stuck circuit breaker pole during Busbar + Line De-Energisation

A

B

C Disconnected phase

A

Disconnected phase

B

Energised phase

C Stuck Pole

˜ ˜ ˜

Substation-A

Substation-B Busbar Shunt-Reactors

(B) Two stuck circuit breaker poles during Busbar + Line De-Energisation

A

B

C Disconnected phase

A

Energised phase

B Stuck Pole

Energised phase

C

Stuck Pole

˜ ˜ ˜

Substation-A

Substation-B Busbar Shunt-Reactors

Figure 3-3 De-energization of Line and Busbar with shunt-reactors connected to the Busbar

Page 38

Resonance and Ferroresonance in Power Networks (A) One stuck circuit breaker pole during Busbar + Line Energisation

B

A

C A

Disconnected phase

Pole fails to close Energised phase

B

Energised phase

C Substation-A

˜ ˜ ˜

Substation-B Busbar Shunt-Reactors

(B) Two stuck circuit breaker poles during Busbar + Line Energisation

B

A

C Disconnected phase

A Pole fails to close

Disconnected phase

B

Pole fails to close Energised phase

C

˜ ˜ ˜

Substation-A

Substation-B Busbar Shunt-Reactors

Figure 3-4 Energization of Line and Busbar with shunt-reactors connected to the Busbar

3.2.2 Shunt-Compensation and Three-Phase Switching in Multi-Circuit Rights of Way This phenomenon can only occur in multi-circuit rights of way, when one circuit is de-energized while a parallel circuit remains energized. Figure 3-5 shows a typical double circuit tower with one circuit in service (I) and another circuit out-of-service (II). Due to inter-circuit capacitive coupling, voltage is induced in an open (not earthed) line if the parallel circuit is energized. The normal induced voltage in the de-energized circuit (Ucircuit_II) can be estimated as: _

=

Eq. 3-1 _

+

where Cs is the inter-circuit capacitance between circuits I and II and Cp is the capacitance to ground of circuit II (see Figure 3-5).

Page 39

Resonance and Ferroresonance in Power Networks

CS CP

Figure 3-5 Capacitances in Double-Circuit Transmission Line This normal induced voltage in the de-energized circuit is typically just a small fraction of the inducing voltage. As an illustrative example, the normal induced voltage in the 500kV double circuit referred to in Section 3.1.1 is approximately 16.3% of the inducing 500kV source, assuming Cs = 1.05 nF/km and Cp = 5.39 nF/km. However, the installation of shunt reactors can introduce resonant conditions at (or near to) power frequency for certain operating topologies and degrees of shunt compensation. Under resonance (or near resonance) conditions, the induced voltages on the de-energized circuit are several orders of magnitude higher than those calculated with Eq. 3-1 and can over-stress the line connected equipment. Early identification of these topologies will allow implementation of cost-effective mitigation solutions at the design stage. When assessing these scenarios, it is essential to consider both aspects related to the resonant overvoltages: (i) amplitude and (ii) duration. Figure 3-6 illustrates three operating scenarios in which resonance can be observed in a shunt-compensated deenergized circuit, for certain size of shunt reactors. c) Case 1 reproduces a possible situation where one circuit is energized while the parallel circuit is out of service. A resonant circuit can be formed in the de-energized circuit depending on the size of the installed shunt reactors. This is a steady-state condition – i.e. the resonant condition will be present in the deenergized circuit as long as the parallel circuit is energized. d) Case 2 reproduces a possible situation where a fault occurs in the energized circuit while the parallel circuit is out of service. A resonant circuit can be formed in the de-energized circuit depending on the size of the installed shunt reactors. This is a temporary condition excited by the fault in the parallel circuit– i.e. the resonant condition will be present until the fault is cleared. e) Case 3 reproduces a scenario where there is a fault in the de-energized circuit while the parallel circuit is in service. This scenario could arise as follows: o

During the maintenance outage of one circuit with the other parallel circuit still in service (or energized), earths are applied to the disconnected circuit. A resonant circuit can be formed if one or two phases of the earthing switch fail to close (i.e. effectively creating a SLG or LLG fault on the de-energized circuit) resulting in high overvoltages on un-earthed phase(s) of the disconnected

Page 40

Resonance and Ferroresonance in Power Networks circuit. The resonant condition will last as long as the unbalanced earthing remains or as long as the parallel circuit is energized. or o During normal operation of both circuits, a SLG or LLG fault occurs in one of them and it is cleared by three-phase tripping – i.e. the faulted circuit is now de-energized. A resonant circuit can be formed resulting in high overvoltages on the healthy phase(s) of the disconnected circuit. This is a temporary condition – i.e. if resonance occurs, high overvoltages will be present in the deenergized circuit only for the duration of the fault (i.e. until extinction of secondary arc) or until the auto-recloser brings the circuit back into service. An example of typical amplitude and location of resonant overvoltages on a 765kV double circuit construction is presented in Figure 4-34 for the three cases described above as a function of the shunt compensation degree.

1

2

3

Closed

Circuit #1

Closed

Open

Circuit #2

Open

Closed

Circuit #1

Closed

Open

Circuit #2

Open

Closed

Circuit #1

Closed

Open

Circuit #2

Open

Risk of resonance for shunt compensation degrees of 60-70% and 100%

Resonant condition in steady-state

Risk of resonance for shunt compensation degrees of 60-70% and 100%

Resonant condition for duration of fault

Risk of resonance for shunt compensation degrees of 60-100%

Resonant condition for duration of fault

Figure 3-6 Risk of resonance in shunt compensated double-circuit lines The scenarios shown in Figure 3-6 can electrically arise under various network topologies, other than the standard double-circuit construction with shunt reactors directly connected to the line. A few examples are illustrated in the following subsections. These examples may seem unrealistic during normal operating conditions, but they can arise as a result of extraordinary switching operations during commissioning, maintenance of equipment or during emergency situations as part of a blackstart restoration path. Identification of these critical topologies is essential to guarantee that the equipment is not overstressed.

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Resonance and Ferroresonance in Power Networks

3.2.2.1 Busbar Shunt Reactors + Double Circuit Tr ansmission Line This example presents two credible topologies leading to resonance in a double-circuit transmission line due to the interaction with busbar shunt reactors. The dangerous topology arises when the busbar (with the shunt reactor) and one of the circuits are de-energized while the parallel circuit remains energized from a remote end, thus coupling energy to the reactor + de-energized circuit combination.

cct #1

cct #2

cct #1

cct #2

Figure 3-7 Double-Circuit Line and Busbar Shunt Reactors Topology 1: Figure 3-7 (a) shows a busbar section in substation B with two line feeders and one shunt reactor connected to it. Cct ii is energized from substation A and open at substation B. A resonant circuit can be formed upon opening the parallel cct#1 circuit breaker in Substation A. This topology effectively leaves the busbar shunt reactor directly connected to the de-energized circuit (cct#1). Resonance occurs between the busbar shunt reactor and the capacitance of the de-energized circuit (cct#1), with energy coupled from cct ii, via inter-circuit capacitive coupling. Topology 2: Figure 3-7 (b) shows another situation where resonance can occur in a similar network topology. In this case, cct#2 is energized from substation A and open at substation B while cct#1 is connected to Substation B (without voltage) but open at Substation A. A resonant circuit can be formed upon closing the shunt-reactor circuit breaker. The resonant circuit is identical to the previous topology.

3.2.2.2 Power Transformer, Tertiary Shunt Reactors and Double Circuit Transmission Line This example presents two possible topologies leading to resonance in a double-circuit transmission line due to the interaction with shunt reactors connected to the tertiary winding of a power transformer. The dangerous topology arises when the transformer (with the tertiary shunt reactor) and one of the circuits are de-energized while the parallel circuit remains energized from a remote end, thus coupling energy to the transformer/reactor + deenergized circuit combination. Similar to the example described in section 3.2.2.1 for busbar shunt reactors, Figure 3-8 shows the network topology where a resonant circuit can be formed. The description of the switching scenarios and topologies is the same as in section 3.2.2.1, with the circuit reactance arising from the series combination of tertiary reactors and power transformer reactance.

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Resonance and Ferroresonance in Power Networks

cct #1

cct #2

cct #1

cct #2

Figure 3-8 Double-Circuit Line and Transformer Tertiary Shunt Reactors

3.2.3 Distribution Embedded Gen eration Islan ded with Transmission Circuit A series resonant circuit can be formed between a distribution connected generator, its step-up transformer, the feeding distribution network, the transmission transformer and the capacitance of a transmission circuit, should this subsystem become islanded. An example of this topology is illustrated in Figure 3-9, where the opening of the circuit breaker at the remote end of the transmission circuit is the trigger for the creation of the resonant circuit. There is a series resonant condition when the inductive source, distribution network and transmission transformer match the capacitance of the transmission circuit at power frequency. Upon resonance, high voltages will be imposed on the transmission circuit and the distribution network. An example of this topology is illustrated in detail in ANNEX A .

LV

MV

Distribution Network

MV

HV

HV Long Transmission Circuit

Embedded Generator

or

Figure 3-9 Distribution Generator Islanded with Long Transmission Circuit

Page 43

Open CB

Resonance and Ferroresonance in Power Networks

CHAPTER 4

RESONANCE IN SHUNT COMPENSATED TRANSMISSION CIRCUITS

4.1 Introduction The application of shunt reactors to long transmission circuits has been common practice for many years as a passive and economical means to compensate for the effect of distributed line capacitance. The shunt reactors compensate for the reactive power surplus in case of reduced power transfer, load rejection or an open transmission line end, limiting steady-state over-voltages. Shunt reactors are usually required in EHV overhead lines longer than 200 km [77]. The degree of shunt compensation, k, provided by a reactor bank is quantified as a percentage of the positive sequence susceptance of the circuit to which it is applied: k [%]

BL 100 BC

1( s L ) 100 ( s C )

1 2 s

L

100

Eq. 4-1

C

where L+ is the shunt reactor inductance per phase (positive sequence), C+ is the positive sequence line capacitance and s is the system angular frequency. Notwithstanding the main objective of limiting steady-state over-voltages in lightly loaded or open transmission circuits, the installation of shunt reactors can result in phase-to-ground voltages above nominal values under certain abnormal or temporary operating conditions: 1. Uneven open-phase conditions in a shunt compensated transmission circuit – i.e. at least one phase is disconnected while the other phase(s) remain energized. This condition can arise from the use of single-phase tripping and autoreclosing schemes (SPAR) or from the misoperation of circuit breakers with independent operating mechanisms on each phase. During line energization, one phase could be left open while the other two phases are still energized due to a stuck pole in the circuit breaker. Similarly, two phases could be left open while the other phase is still energized as a result of a stuck pole during line de-energization. Energy is coupled into the resonant circuit via the phase-to-phase capacitances. Reference [72] provides a very good insight into this resonant condition, which is expanded in section 4.2 of this document. This phenomenon can arise in single and double circuit line constructions alike and it is a temporary abnormal condition – i.e. protection relays or control systems will act to restore the circuit to a balanced operation by either connecting the de-energized phase(s) (auto-reclosing) or disconnecting the energized phase(s) (tripping the circuit). In either case, the resonant condition will disappear when the uneven phase operation condition is removed. 2. Three-phase disconnection of one circuit in a shunt compensated double-circuit construction, while the parallel circuit remains energized. Energy is coupled into the resonant circuit via the circuit-to-circuit capacitances. This phenomenon has sometimes been referred to as “parallel line resonance” in the technical literature. References [77] to [82] deal with this resonant condition in great level of detail. Also, section 4.4 of this document provides an overview of the issues affecting this resonant problem and possible solutions. This phenomenon arises only in multi-circuit rights of way as it necessitates the capacitive coupling from an energized parallel circuit. The resonant condition can be permanent (circuit intentionally out of service) or temporary (circuit tripped by protection and reclosed after a dead-time period). Typical degrees of shunt compensation used for overhead transmission circuits are in the range of 60%-80% when single-phase autoreclosing (SPAR) is used, although higher values of compensation can be found in conjunction with 3-phase autoreclosing or mixed overhead/underground circuits. Shunt compensation degrees close to 100% are normally required for EHV cable circuits due to their higher capacitance.

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Resonance and Ferroresonance in Power Networks

4.2 Line Resonance in Uneven Open-Phase Conditions 4.2.1 Physical description The following assumptions and simplifications are made in order to describe the basic mechanisms of line resonance in a shunt compensated circuit operated with one or two phases open (de-energized): 1. 2. 3. 4. 5.

The transmission circuit is fully transposed and without losses. All circuit elements are linear. The circuit series impedance is neglected. Shunt reactors are applied to compensate for k of the circuit capacitance (Eq. 4-1) There is no inter-phase magnetic coupling in the shunt reactors. This is the same as saying that the positive and zero sequence reactances are equal. 6. The neutral point of the shunt reactors is directly connected to ground. Given the above assumptions and simplifications, a shunt-compensated transmission circuit, at no load, can be represented by the parallel combination of a lumped capacitance and inductance, as shown in Figure 4-1.

Figure 4-1 Connection of shunt reactors in Transmission Circuit The equivalent phase-to-ground impedance per phase (Zeq) is given by the following expression: 1

Z eq

j

s

C0

|| j

j

L 1

s 2 s

L L

j C0 1

s

L C0 C

Eq. 4-2

k 100

where k is the degree of shunt compensation defined in Eq. 4-1, L+ is the shunt reactor inductance per phase (positive sequence), C+ is the positive sequence capacitance of the circuit, C0 is the zero sequence capacitance of 1 the circuit and S is the angular frequency of the voltage source. Three situations can occur depending on the degree of shunt compensation (k): 1)

C0 k 100 C

The equivalent phase to ground impedance, Zeq, is capacitive.

2)

C0 k 100 C

The equivalent phase to ground impedance, Zeq, is inductive.

3)

C0 k 100 C

The equivalent phase to ground impedance, Zeq, is infinite.

1

Note that the zero sequence capacitance of a symmetrical transmission circuit (C0) is the same as the capacitance of the phase conductors to ground (Cph-gr)

Page 45

Resonance and Ferroresonance in Power Networks

150

150

*10 3

*10 3

Magnitude Z

Magnitude Z

Figure 4-2 shows the frequency scan of the equivalent phase-to-ground impedance per phase, Zeq, of a 400 kV transmission line assuming two degrees of shunt compensation: 60% and 70%. The C0/C+ ratio of this circuit is 0.67. Figure 4-2 (a) shows that with shunt compensation degree of 60% (i.e. k < C0/C+), the phase-to-ground impedance is capacitive at 50 Hz. Increasing the degree of shunt compensation to 70% (i.e. k > C0/C+), Figure 4-2 (b) shows that the phase-to-ground impedance becomes inductive at power frequency. Although not shown in the figure, it is clear that a shunt compensation degree of 67% would result in infinite impedance to ground at 50 Hz.

120

120

90

60

30

60

30

0

0 35

40

45

47.350

55

60

65

Frequency [Hz]

35

70

50

51.1

55

60

65

Frequency [Hz]

70

50

Phase Z

Inductive Capacitive

25 0 -25 -50

Capacitive

Inductive

25 0 -25 -50

-75 -100 35

45

75

75 50

40

100

100

Phase Z

90

-75

40

(a)

45

50

47.3

55

60

65

Frequency [Hz]

60% Shunt Compensation Degree

70

-100 35

40

(b)

45

50

51.1

55

60

65

70

Frequency [Hz]

70% Shunt Compensation Degree

Figure 4-2 Equivalent line-to-ground impedance (Z eq ) in a transmission line with C 0 /C + =0.67 If we assume that one phase conductor is disconnected while the other two phases remain energized (for example following a single phase trip), the equivalent phase to ground impedance of that phase - Z eq (Eq. 4-2) - becomes series connected with the inter-phase capacitances to the energized phases. This is illustrated in Figure 4-3 below. As previously discussed, Zeq can be capacitive or inductive depending on the degree of shunt compensation applied to the circuit. For low degrees of shunt compensation (i.e. k < C0/C+ ) Zeq is capacitive. The series connection of two capacitances will not give rise to resonance issues. However, high degrees of shunt compensation (i.e. k > C0/C+) will result in Zeq becoming inductive. The series connection of inductive and capacitive elements will result in series resonance if both reactance values become equal. This series resonant circuit is excited by the voltage source on the energized phases and gives rise to high currents and voltage across the reactor. It is concluded that series resonance can occur during uneven open-phase conditions when k > C0/C+. Series resonance arises from the parallel combination of the shunt reactor and line-to-ground capacitance connected in series with the inter-phase capacitances. In practice, typical C0/C+ ratios in standard transmission line constructions are in the 0.6 – 0.7 range. This means that, under the assumptions made above, there is a risk of series resonance following open-phase conditions when the degree of shunt compensation exceeds 60-70%. The source of the series resonance is the uneven compensation of positive and zero sequence line capacitance provided by the shunt reactors.

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Resonance and Ferroresonance in Power Networks

Figure 4-3 Simplified Equivalent Circuit during Single-Phase Opening

4.2.2 Approximate St eady State Analytical Solution Given the potential damage to line connected equipment, such as surge-arresters, instrument transformers, shunt reactors and circuit breakers, the circuit configurations leading to excessive over-voltages need to be identified. The key questions to be resolved for any transmission line construction requiring shunt compensation are: 1. What are the particular reactor sizes that give rise to resonant conditions? 2. What is the induced open-phase voltage for any particular degree of shunt compensation? A high level answer to those questions can be given using the simple formulae presented in sections 4.2.2.1 and 4.2.2.2 next. It should be noted that this is a steady-state analysis and higher temporary over-voltages can be expected during transient conditions. For clarity, the equations are presented in terms of both, positive and zero, sequence capacitances as well as phase-to-ground and inter-phase capacitances. The relationship between these magnitudes (assuming symmetrical line construction) is as follows: C

C ph

C0

C ph

gr

3 C ph

Eq. 4-3

ph

Eq. 4-4

gr

The equations presented next (sections 4.2.2.1 and 4.2.2.2) are based on the assumptions made in section 4.2.1. In particular, the assumptions of symmetrical line parameters, equal positive and zero sequence reactance for the shunt reactors and solidly earthed reactor neutral connection apply (see section 4.2.3.2 for the effect of a neutral reactor). Furthermore, it must be emphasised that losses and saturation effects have been ignored at this stage for simplicity. In practice, the theoretical steady-state over-voltages calculated with this approach may be limited by corona losses and/or reactor core saturation.

4.2.2.1 One Open-Phase It is assumed that phases B and C are energized while phase A is disconnected (Figure 4-4 (a)). This circuit, as seen from disconnected phase A, can be simplified as Figure 4-4 (b). By applying the Thevenin theorem, this circuit can be reduced further as Figure 4-4 (c), which is a common series L-C circuit with a natural frequency of oscillation equal to f n_(1 open-phase): fn (1 open

1

phase ) 2

L (C ph

gr

2 C ph

Eq. 4-5

ph )

Using circuit analysis to the equivalent shown in Figure 4-4 (c), the following expressions are derived: Shunt compensation degree that causes series resonance at power frequency:

Page 47

Resonance and Ferroresonance in Power Networks

1 2 k1 1 3

C ph

ph

C ph

gr

C ph

ph

C ph

gr

1

2 C 3 C0

1

Eq. 4-6

C C0

Induced open-phase voltage for a compensation degree k: U1

1 C ph

gr

C ph

ph

1

3 (1 k )

1

3 1 k C0 1 C

˜ ˜ ˜

(a)

Eq. 4-7

1

Us(t)

A

a2.Us(t)

B

a.Us(t)

C

Cph-ph Cph-ph

L+

C0

L+

L+

Cph-ph

C0 = Cph-gr

C0

a

˜ ˜

(b)

a2.Us(t)

B

a.Us(t)

C

a

Cph-ph

Cos 120 2

Cos 240

j Sin 120 j Sin 240

A Cph-ph

C0 = Cph-gr

L+

Thevenin

Uthev(t)

(c)

UThev (t )

2 Cph-ph + Cph-gr

˜

A L+

C ph 2 C ph

ph

ph

C ph

U s (t ) gr

Figure 4-4 Simplified circuit for the analysis of Line Resonance

4.2.2.2 Two Open-Phases A similar approach can be used with the two open-phases scenario, resulting in another equivalent L-C circuit with a natural frequency of oscillation equal to fn_(2 open-phases): f n (2 open

1

phases ) 2

L (C ph

gr

C ph

Eq. 4-8

ph )

Similarly to the one open-phase condition, the following expressions are derived: Shunt compensation degree that causes series resonance at power frequency:

Page 48

Resonance and Ferroresonance in Power Networks

1 k2

C ph

ph

C ph

gr

1 3

1

C ph

ph

C ph

gr

1 C 3 C0

1

Eq. 4-9

C C0

Open-phase voltage for a compensation degree k: U2

1 C ph

gr

C ph

ph

3

1 (1 k )

2

3 1 k C0 1 C

Eq. 4-10

2

4.2.2.3 Practical Example As an illustrative example, the analytical “approximate” method presented above has been used to estimate the resonant conditions in a standard 400 kV transmission line design used in Ireland, as a function of the degree of shunt compensation. For this construction, the circuit capacitances are C+=11.59 nF/km and C0=7.77 nF/km. The line is assumed to be fully transposed and the neutral point of the shunt reactors is directly connected to ground. Power frequency is 50 Hz. Figure 4-5 shows the natural frequencies of oscillation for one and two open-phase(s) conditions, as a function of the degree of shunt compensation. It can be seen that the natural frequency increases with the degree of compensation. These frequencies reach values within ±0.5 Hz of power frequency for compensation degrees between 77% and 79% during operation with two open phases and between 88% and 91% during operation with one open-phase. Figure 4-6 presents the steady-state open-phase voltages as a function of the shunt compensation degree, calculated using Eq. 4-7 and Eq. 4-10. These curves clearly show resonant conditions for shunt compensation degrees of 78% and 89% for the two open-phases and the one open-phase conditions, respectively. Shunt compensation degrees from 68% to 99% bring near-resonant conditions with steady-state open-phase voltages in excess of 1 pu. It should be noted that this illustrative example is based on a number of stated simplifications and that the calculated voltages refer to steady-state conditions only. In practice, temporary conditions may lead to voltages in excess to those calculated using this analytical method. On the other hand, saturation or circuit losses may limit these over-voltages. Notwithstanding its limitations, this “approximate” method enables the engineer to carry-out a speedy estimation of the risk of power frequency resonance for a particular circuit configuration and degree of shunt compensation. Further detailed studies are required when it is envisaged to operate close to a resonant peak. This is typically done using time domain simulation, as shown in section 4.3. The following can be concluded from this example: 1. A symmetrical shunt-compensated transmission circuit exhibits two resonant peaks: one for one open-phase and a second one for two open-phases conditions. 2. The two open-phases condition presents a resonant peak at a lower degree of shunt compensation than the one-open-phase condition. 3. Steady-state voltages in excess of 1 pu can be expected for a wide range of shunt compensation degrees.

Page 49

Resonance and Ferroresonance in Power Networks 70 [Hz] 60

52.0 [Hz] 51.5

50

50.5

51.0

50.0

40

49.5

30

49.0 48.5

20

48.0 70%

10

75%

80%

85%

fn_1open-phase

0 10%

20% 30% 40%

50% 60% 70%

fn_1open-phase

90%

95%

100% [k]

fn_2open_phases

80% 90% 100% 110% 120% [k]

fn_2open_phases

Figure 4-5 Natural oscillation frequencies of a 400 kV shunt-compensated line under one and two open phase conditions 5.0

V [pu] 4m

4.5 6.0 m

4.0 3.5

Two open-phases 4.1 m 10.25

One open-phase

10.25

3.0

26.0 m

2.5 2.0 1.5 1.0 0.5 0.0 10%

68% 20%

30%

40%

50%

60%

70%

78%

89%

80%

90%

99%

k

100% 110%

Figure 4-6 Steady-State open-phase voltage (approximate analytical solution) in a 400 kV line as a function of the Shunt Compensation Degree, k.

4.2.2.4 Field Measurements Showing 500kV Surg e Arrester Failures During a Two-OpenPhase Condition In April of 2012 a Canadian utility experienced failures of a 500 kV line terminal breaker and two 500 kV surge arresters on two different phases of a long EHV circuit during a prolonged two open-phase condition. Very high TOVs occurred from induced voltages and a resonant condition in the shunt compensated circuit, as discussed in section 4.2.2. The incident occurred during routine maintenance of protection at one terminal resulting in an inadvertent three-phase trip of the unfaulted line, initiated by line protection, followed by an automatic reclose, and then immediately followed by a protective re-trip of the line. This case provides a good example of hazardous TOVs that can occur due to capacitive coupling in the presence of series resonance on open phases of an EHV line equipped with shunt reactors that provide a high degree of shunt compensation.

Description of the case

Page 50

Resonance and Ferroresonance in Power Networks Figure 4-7 shows a simplified single-line diagram of the 277 km 500 kV shunt (and series) compensated and transposed circuit 5L2 from GMS Station to WSN Station. The series capacitor bank does not play a role in this case because the bypass breaker was closed before the open phase condition occurred. The 500 kV shunt reactors (2040 Ohms/phase) 5RX2 at GMS end of the line and 5RX4 at WSN end provide 72.2% compensation of the positive sequence capacitance of the line. These reactors have solidly grounded neutrals hence the line is operated in three pole trip and reclose mode. The line originally went into service before single pole operation could be reliably achieved with the 500 kV breaker technology existing at the time. The GMS bus (part of a major hydroelectric installation) is the master end for high speed auto-reclose and the associated breakers 5CB5 and 5CB11 are equipped with POW closing. At the WSN (follow) end 5CB3 and 5CB4 are not equipped for POW closing. There is a set of surge arresters 5SA26 protecting 5RX2 and another set of arresters 5SA34 at 5RX4, as indicated in the diagram. Figure 4-8 shows a typical guyed-V tower for this flat-configuration circuit. The average height of the conductor above ground at the tower and the average conductor sag are 26.7 m and 10 m, respectively. Each phase comprises a bundle of four 316.1 mm2 ACSR conductors in a 45.7 cm by 45.7 cm square arrangement.

Transmission Line Parameters:

Z1 = 7.26 + j92.54 Y1 = 1356 µMho Z0 = 56.86 + j342.4 Y0 = 780.5 µMho

Figure 4-7 500 kV Circuit Details For the Two Open-Phase Event

Page 51

(uncomp.)

Resonance and Ferroresonance in Power Networks

Figure 4-8 Details of Circuit 5L2 Typical 500 kV Guyed-V Structure

Sequence of events On 16 April, 2012, during routine protection maintenance at GMS, one phase of a CT connected to 5L2 line protection was inadvertently shorted and isolated under load but without blocking the line protection. During a sequence of events (see Table 4-1) which lasted 7.6 seconds from initiation of the inadvertent trip to complete line isolation, one breaker (WSN 5CB4 Phase A) and two surge arresters (WSN 5SA34 Phase B and GMS 5SA26 Phase A) failed. The former arrester failed because of excessive and prolonged TOV while the latter arrester failed due to repeated high switching surges due to restriking within the failed breaker. The unintended isolation of the CT when 5L2 was under load initiated a three-phase trip of the line, first at GMS followed by the WSN end. Table 4-1 Sequence of Events – 16 April 2012 Order

Time-stamp - PST

Event

1

15:24:27:93 hrs

Unintentional Trip of 5L2 at GMS and WSN terminals

2

15:24:28:61 hrs

Phases B and C automatically reclose at GMS

3

15:24:28:70 hrs

Suboptimal Phase A POW reclose at GMS

4

-

Reclose at WSN

5

15:24:28:71 hrs

Trip initiation at GMS and DTT to WSN

6

WSN 5CB4 Pole A stuck closed on trip

7

WSN 5SA34 Phase B failure

8

15:24:33.75 hrs

GMS 5SA26 Phase A failure causing ground fault

9

15:24:34:59 hrs

Trip initiation at WSN by timed ground fault protection

10

15:24:35:71 hrs

WSN 5CB4 breaker trip

Page 52

Resonance and Ferroresonance in Power Networks Following the line trip, 5L2 line protection initiated automatic high speed reclose. The POW controller at GMS successfully reclosed phases B and C but delayed reclosing Phase A by about 6 cycles. Shortly after Phase A closed at the lead end, line protection transiently picked up and initiated a line re-trip and also sent a direct transfer trip (DTT) to the follow terminal, which, by this time had successfully reclosed three-phase. However, during the second line tripping operation, Pole A of WSN 5CB4 became stuck and failed to open. As a result, the circuit became single-phase energized from WSN. Phases B and C of 5L2, with the line-end reactors and associated surge arresters, were open but capacitively coupled to the energized Phase A. The induced overvoltages on the open phases caused failure of WSN 5SA34 Phase B. Pole A of 5CB4 in the stuck condition sustained uncontrolled multiple restrikes which, in about 6 seconds, led to the second surge arrester failure – Phase A of 5SA26 at GMS. Figure 4-9 shows the instantaneous phase-to-ground voltages and 5L2 line currents at GMS (the upper three traces are the voltages, followed by the corresponding phase currents) and at WSN for a period of 680 ms, starting about 5 cycles before the auto-reclose of phases B and C at GMS. These traces were recorded by digital fault recorders at these two stations on 16 April. In examining these traces, the following two points must be kept in mind: 1. The voltage waveforms are NOT plotted on the same scale. The scaling of the voltage-axis for each plot is independently determined based on the maximum and minimum instantaneous values so that the entire plot fits within the bounds of the plotting area. The same applies for the line currents. 2. Inspection of the Phase B and Phase C voltage waveforms at WSN, during the two open-phase condition, indicates “flat topping”. This is NOT due to surge arrester conduction (or other non-linear phenomenon) during the excessive TOV but is because the overvoltages are so high that they have exceeded the pre-set range of the digital fault recorder. The recordings at GMS do not have this problem. Prior to time T = 0, circuit 5L2 was isolated and all three phase-to-ground voltages at both line terminals exhibit ringdown oscillations as stored energy oscillates between the line capacitance and the shunt reactors as a result of the prior trip-out of the line. At time T = 10 ms the POW controller at GMS auto-reclosed Phases B and C (but not yet Phase A) while the remote end of the line was open. Thus, there was a one open-phase condition for about 5 cycles before Phase A reclosed (late). During these 5 cycles, the amplitude of the open Phase A voltage escalated dramatically and reached 669.5 kVp (1.64pu). When the POW controller reclosed Phase A at T = 110 ms, it did so near a voltage zero, a non-optimal point on the voltage wave, thereby initiating a large DC offset in the current due to the re-energization of the line-end reactors. However, about 2 cycles later there was a re-trip of the line but, since there was no zero crossing of the Phase A current at that time, this current continued uninterrupted. However, at T = 190 ms the WSN end of the line reclosed three-phase. The disturbance created by closing the follow end of 5L2 Phase A created a current zero at GMS and the slowly decaying DC component in the current was transferred to the WSN Phase A breakers. A re-trip of the line at WSN occurred about 2 cycles later (T = 240 ms) but the Phase A current could not be interrupted until the next zero crossing, which occurred about 13 cycles later. Subsequently, there were multiple restrikes, which can be seen in the WSN Phase A line current indicating that one of the two breakers 5CB3 or 5CB4 had failed. Therefore, for more than 13 cycles, there was a two open-phase condition on 5L2 resulting in high TOVs on B and C phases of the line. The instantaneous Phase B voltage was about 668 kVp (1.64 pu) and Phase C voltage was about 697 kVp (1.71 pu), as measured at GMS. The failure of the surge arrester on Phase B at WSN and on Phase A at GMS would have occurred some time beyond the time frame of Figure 3. Unfortunately, there were no time stamps for the switchings recorded at WSN because event logs were overwritten due to the large number of events.

Page 53

Resonance and Ferroresonance in Power Networks One open-phase condition

Auto-reclosure occurs near voltage zero

V-A GMS

V-B GMS

V-C GMS

DC Offset DC Offset Transferred to WSN Terminal

Phase A recloses late

Two open-phase condition V-B 1.64 pu V-C 1.71 pu

Phase B recloses Line Re-trip at GMS Phase C recloses

V-A WSN

V-B WSN

V-C WSN

WSN Phase A Recloses

No Current Zero for 13 Cycles

WSN Phase B Recloses Line Re-trip at WSN

WSN Phase C Recloses

WSN 5CB4 restriking

Figure 4-9 Field Recordings of 16 April 2012 Two Open-Phase Event. Upper Three Traces are the Phase A, B, and C Voltages on 5L2 at GMS End Followed by the Corresponding Phase A, B and C Line Currents. The Corresponding Voltages and Currents at WSN End Appear Below

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Resonance and Ferroresonance in Power Networks Steady State “Approximate” Analysis of the TOV It is useful to calculate, analytically, the expected steady state TOVs during the open-phase conditions for 5L2 having two shunt reactors and compare these results to the field recordings. The positive and zero phase sequence line impedances and shunt susceptances are shown on Figure 4-7. At each 5L2 terminal there is a set of shunt reactors, each having a reactance of 2040 per phase (L+ = 2.706 H for two reactors in parallel per phase). Power frequency is 60 Hz. From Eq. 4-3 and Eq. 4-4, Cph-ph = 1/3(C+ - C0) = 0.5087 µF and Cph-gr = C0 = 2.072 µF From Eq. 4-1, the degree of shunt compensation, k, provided by two reactors on circuit 5L2 is 72.27% One Open-Phase Condition: From Eq. 4-5, the natural frequency of oscillation, f n, for 5L2 having two shunt reactors and one open phase is 55.0 Hz. From Eq. 4-6, the shunt compensation degree that causes series resonance at fundamental frequency, k1, is 0.8586, which is well above the actual compensation level of 0.7227. From Eq. 4-7, the induced open-phase voltage for 72.27% shunt compensation, k, for steady-state conditions is theoretically 1.04 pu. Two Open-Phase Condition: From Eq. 4-8, the natural frequency of oscillation, f n, for 5L2 having two shunt reactors and two open phases is 60.2 Hz, which is almost precisely the fundamental frequency. From Eq. 4-9, the shunt compensation degree that causes series resonance at fundamental frequency, k2, is 0.7172, which is very close to the actual compensation level of 0.7227. From Eq. 4-10, the induced open-phase voltage for 72.27% shunt compensation, k, for steady-state conditions is theoretically 26.0 pu, which is extremely high. Discussion For circuit 5L2, the degree of shunt compensation required to obtain series resonance at fundamental frequency for one open phase is 85.9% which is significantly more than the actual shunt compensation of 72.2%. This can be compared to the 89% shown on Figure 4-5 for the example 400 kV overhead circuit. For the two open-phase condition of 5L2 the natural frequency of this configuration is 60.2 Hz, indicating that this case is almost precisely resonant at fundamental frequency and high voltages on the two open phases can be expected. Alternatively, 71.72% shunt compensation is required to produce resonance at fundamental frequency, which is almost identical to the actual compensation of 72.2%. This can be compared to 78% compensation required for the 400 kV example circuit. From a theoretical perspective, a one open-phase condition of 5L2 having two shunt reactors will result in only a negligible induced steady state overvoltage whereas a two open-phase condition can be expected to produce a potentially hazardous temporary overvoltage. The theoretical calculations of open-phase induced voltages assume steady-state conditions and ignore the nonlinear effects of surge arrester conduction, magnetic saturation of the shunt reactors, and corona losses on the conductors. They also assume that the line is balanced (i.e. the phase-to-phase capacitances are all identical). It should therefore not be surprising to find differences between the induced voltages calculated analytically and the voltages actually observed on 5L2 during the unusual open-phase conditions in April of 2012. For the two openphase condition, both theory and field measurement indicated very high induced TOVs on the two open phases, but with only the (very understandable) disagreement in the severity of the overvoltage. It is also quite understandable that there was a surge arrester (WSN 5LA34) failure during the prolonged TOVs, although it might have been expected to occur on Phase C rather than Phase B. It is possible that the arrester on Phase B had a different V – I characteristic than the Phase C arrester. During the 4 to 5 cycles when Phase A was open but Phases B and C were energized from GMS the actual induced overvoltage was higher than what would be expected from simple calculation for a one open-phase condition. This was likely because conditions were not in steady state.

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Resonance and Ferroresonance in Power Networks

4.2.3 Effect of Various Design Pa rameters The effect of the following design parameters on the occurrence of resonance is discussed in the next subsections: Mixed Overhead Line and Underground Cable Transmission Circuits – section 4.2.3.1 Neutral Reactors – section 4.2.3.2 Shunt reactor core construction – section 4.2.3.3

4.2.3.1 Mixed O verhead Line and Cable Circuits There are two main characteristics of underground cables that have a direct impact on line resonance: 1. 2.

The capacitance of an underground cable is typically in the order of 20 – 30 times the capacitance of an equivalent overhead line circuit. = (20 30). HV and EHV cables have screens on each phase, therefore there is no inter-phase capacitive coupling. =

The addition of a section of underground cable to an overhead transmission line increases the overall C0/C+ ratio of the circuit. This ratio changes rapidly from approximately 0.6-0.7 (no cable section) to 1 (no overhead line section). The main implication of a higher C0/C+ ratio is that the resonant peaks shift towards higher levels of shunt compensation. This is illustrated with an example in Figure 4-10. In this example it has been assumed that the overhead line construction is as per Figure 4-6 and that the capacitance (per km) of the cable section is 25 times the capacitance (per km) of the overhead line section. The results are plotted as a function of the proportion of cable length in the entire length of circuit n°= length_UGC/(length_UGC+length_OHL). It is shown that the introduction of an underground cable section, even if small (for example 10% of circuit length), has a dramatic effect on moving the resonant peaks towards regions of high degrees of shunt compensation (i.e. higher than 90%). In this example, both resonant peaks are above 99% of shunt compensation when the section of cable exceeds 50% of the total circuit length. k[%] 100% 98% 96% 94% 92% 90% 88% 86% 84% 82% 80% 78% 76% 0

0.1

0.2

0.3

0.4

0.5

k(1_open Phase)

0.6

0.7

0.8

0.9

1

n[pu]

k(2_open phases)

Figure 4-10 Location of Resonant Peaks in a Mixed Overhead/Cable Circuit vs proportion of cable length section The effect of increasing the proportion of cable length into a mixed transmission circuit is illustrated with an example in Figure 4-11. In this example, the section of cable circuit has been increased from 0% to 70% of the total circuit length. It can be seen that the resonant peak (only one open-phase condition is shown for clarity) moves towards 100% as the proportion of cable length in the circuit is increased. Also, the amplitude of the resonant voltages drops with the increased proportion of cable due to the reduced inter-phase capacitive coupling.

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Resonance and Ferroresonance in Power Networks

5.0

U [pu]

4.5

n=0% UGC k=89%

4.0 3.5

n=10% UGC k=97%

3.0 2.5 2.0

n=30% UGC k=99.06%

1.5

n=70% UGC k=99.81%

1.0 0.5

k

0.0 40%

50%

60%

U1 (n=0)

70%

80%

U1 (n=0.1)

90%

U1 (n=0.3)

100%

110%

U1 (n=0.7)

Figure 4-11 Steady-state open-phase voltage in a Mixed Overhead/Cable Circuit as a function of the shunt compensation degree, k, and increasing length of cable (one open-phase)

4.2.3.2 Effect of Neutral Reactors It has been demonstrated in section 4.2.1 that the origin of the series resonant circuit during uneven open-phase operation is linked to the unequal compensation of positive and zero-sequence line capacitances (i.e. C0/C+ ratio). This resonant circuit can be detuned by the introduction of a properly dimensioned neutral reactor affecting the zero-sequence compensation. Neutral reactors are normally used in conjunction with phase reactors in long transmission lines to reduce the amplitude of the secondary arc current and to increase the reliability of SinglePhase Auto-Reclosing (SPAR) schemes [68], [73]. A typical neutral reactor connection is shown in Figure 4-12.

Figure 4-12 Four Reactor Compensation Scheme – Equivalent line-to-ground impedance

Eq. 4-11 below gives the equivalent line-to-ground impedance per phase (Z eq-0) for the four-reactor bank configuration shown in Figure 4-12. Z eq

1 0

j

s

C0

|| j

s

L0

j 1

s 2 s

L L

3LN 3LN C0

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Eq. 4-11

Resonance and Ferroresonance in Power Networks The minimum value of neutral inductance - LN - that can prevent the formation of a series resonant circuit at power frequency is such that the equivalent line-to-ground impedance, Zeq_0, becomes capacitive. That minimum neutral reactor - LN_min - is calculated using Eq. 4-12 below:

LN _ min

1 3

1 2 s

L

Eq. 4-12

C0

Various approaches can be adopted to optimise the size of the neutral reactor for a particular circuit configuration. Two examples are: 1.

Even compensation of positive and zero sequence line capacitance [84]

LN _ even _ k

2.

L 3

C C0

1

Compensating inter-phase capacitive coupling to minimise secondary arc current [68]-[70] C

LN _ min

Eq. 4-13

sec arc

L 3

k

C0 C C

Eq. 4-14

C0 C

It should be noted that the application of Eq. 4-12 or Eq. 4-14 results in negative value of neutral reactors (i.e. need for a neutral capacitor) for k < C0/C+ and k < 1-(C0/C+), respectively. In practice, these are very low degrees of shunt compensation, which are very distant from resonant peaks. The installation of a neutral reactor for mitigating resonance is not justified in these cases. Figure 4-13 compares the size of the neutral reactors calculated using Eq. 4-12, Eq. 4-13 and Eq. 4-14 for a practical range of shunt compensation degrees (i.e. between 70% and 100%). In this comparison, the ratio of the neutral to phase reactors is plotted as a function of the shunt compensation degree. It can be seen that there is a significant difference in the size requirements for the neutral reactor in the lower range of shunt compensation degrees, with Eq. 4-14 requiring the largest neutral reactors. This implies that, even though smaller neutral reactors may be effective in detuning a potential resonant circuit, they may not be adequate for the purposes of secondary arc extinction and successful SPAR. Both phenomena should be analysed simultaneously in order to achieve the most cost-effective solution. Furthermore, Figure 4-13 also shows that, as the degree of shunt compensation is increased, the three approaches converge to the same neutral reactor size.

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Resonance and Ferroresonance in Power Networks

LN/L+ 0.30 0.25 0.20 0.15 0.10 0.05

k

0.00 70%

75%

80%

LN(min-sec-arc)/L+

85%

90%

LN(even_k)/L+

95%

100%

LN(min)/L+

Figure 4-13 Ratio of Neutral Reactor to Phase Reactor as a function of the Shunt Compensation degree (C 0 /C + = 0.67)

4.2.3.2 .1 Practical Ex ample with Neutral Reactors (S te ady-S tate ) This example illustrates the effect of connecting a neutral reactor in the 400kV line construction described in section 4.2.2.3. Shunt compensation degree of 78% has been selected because it results in series resonance for the two open-phases scenario when the reactors’ neutral is directly grounded (see Figure 4-6). Figure 4-14 compares the steady-state open-phase voltages for the “no neutral-reactor” case with the corresponding voltages arising from the connection of neutral reactors of different sizes – Eq. 4-12, Eq. 4-13 and Eq. 4-14. It can be seen LN_Min (calculated with Eq. 4-12,) effectively detunes the circuit from the resonance condition, however open-phase voltages close to 1 pu are still observed. Further reductions of induced voltage are achieved with LN_even_k (Eq. 4-13) and LN_min-sec-arc (Eq. 4-14).

6 V [pu]

5 4 3 2 1 2op en ph 1op en ph

a ses

a se

SoS l iodli dN

N e ue utrt aral LN l _LN( m(M inin) LN ) _(L eNv(e e vne LNL _nk k _N ((m )) Cain r-lsse sconar)c )

0

Figure 4-14 Open-phase voltages in a Flat Line construction with 78% Shunt Compensation degree. Effect of Neutral Reactors In practical terms, the installation of a neutral reactor shifts the resonant peaks to higher levels of shunt compensation degrees. This is illustrated in Figure 4-15 below which shows resonant peaks at 91% and 97%

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Resonance and Ferroresonance in Power Networks compensation degrees when a neutral reactor is connected to the circuit. This compares to 78% and 89% compensation degrees for the same resonant peaks in the absence of a selected neutral reactor. 5

5

4.5

4.5

4

4

3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100%105%110%115%120% 1 open-phase

1 open-phase (Xn)

50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100%105%110%115%120% 2 open-phase

2 open-phase (Xn)

Figure 4-15 Steady-state open-phase voltages with a neutral reactor as a function of shunt compensation degree, k

4.2.3.2 .2 Insulation Level Con side rations associa ted wi th N eutral Reac tors When assessing the connection of a neutral reactor, it should be kept in mind that its size has direct implications on the insulation requirements – i.e. the larger the neutral reactor, the higher the voltage at the neutral point of the phase reactors. For economic reasons, it is desirable to keep the insulation class of the reactor neutral as low as possible. In the absence of detailed insulation co-ordination studies, the required neutral point Basic Insulation Level (BIL) can be roughly determined using Eq. 4-15 [70]:

Neutral Point BIL Phase Reactor BIL

1

1 L phase

Eq. 4-15

LN

The following points summarise the considerations related to the use of neutral reactors to mitigate resonance conditions: 1. A wide range of neutral reactor sizes can be selected to detune the resonant circuit during open-phase conditions. 2. The installation of a neutral reactor shifts the resonant peaks towards higher degrees of shunt compensation. 3. Neutral reactors are usually required to minimise secondary arc current. Both, resonance and SPAR performance must be assessed simultaneously. 4. The size of the selected neutral reactor has direct implications on the required insulation levels for phase and neutral reactors.

4.2.3.3 Effect of Reactor Core Construction The design of the magnetic core has a large effect on the reactors’ behaviour during unbalanced open-phase conditions. The only type of reactor core that does not have direct magnetic coupling between phases is the singlephase unit2. Three-phase reactors present different levels of zero sequence coupling depending on the core design, as follows [82].

2

Note that air core reactors do not have magnetic coupling between phases either. However, their use is generally limited to low and medium voltages due to their high intensity external magnetic fields when energized.

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Resonance and Ferroresonance in Power Networks Type I.

Shell-type and four/five-legged core type reactors provide a magnetic path for the zero sequence flux, hence, the coupling between phases is very small and can normally be neglected [76]. In this type of reactor X0/X+=1.

Type II.

Three-legged core type reactors present a strong magnetic coupling between phases. In this type of reactors X0/X+= 0.5 – 0.7.

The resonance analysis presented in the previous sections has assumed 3-phase shunt reactors with X0/X+=1. However, if type II reactors are used for line shunt compensation, the effect of the magnetic coupling is equivalent to connecting a “negative” neutral reactor (i.e. X0/X+< 1), therefore, the location of the resonant peaks is shifted towards lower degrees of shunt compensation.

4.3 Detailed Analysis of Line Resonance in Uneven Open-Phase conditions using Time-D omain Simulation The analytical “approximate” solution presented in section 4.2 is a very useful tool to obtain an initial high-level estimation of the risk of resonance for a particular shunt-compensated line construction. However, this approach is based on a number of simplifications and does not capture some practical effects such as circuit asymmetries, nonlinearities or complex circuit topologies. When the analytical approximate method suggests proximity to a resonant peak, a more rigorous analysis is required. Time domain simulation using EMT software provides the right tools to carry-out this type of detailed analysis. An example of time-domain simulation illustrating the effect of some key design parameters is presented next. This parametric analysis starts with steady-state voltage calculations and continues with an assessment of temporary over-voltages arising from switching operations. Modelling guidelines to represent each circuit element are discussed in CHAPTER 6.

4.3.1 Steady State Analysi s This section illustrates the effect of selected design variables on the induced voltages under uneven open-phase conditions. Only steady-state voltages are assessed in this section. For this purpose, the relevant circuit breaker poles are represented in open position for the entire duration of the simulation.

4.3.1.1 Effect of Tower Design The line capacitances (phase-to-phase and phase-to-ground) play an important role in the occurrence of resonance at power frequency. These capacitances are mainly defined by the conductor geometry. In order to illustrate the effect of the line geometry, four typical transmission line constructions are compared. These are shown in Figure 4-16: (a) flat construction, (b) vertical construction, (c) delta construction and (c) inverted delta construction. For the purposes of this analysis a linear model has been used to represent the shunt reactors, keeping the neutral directly grounded. As shown in Figure 4-16, the inter-phase distances have been set to 10m and a minimum ground clearance of 15m has been used for each line construction. One conductor per phase and full line transposition has been assumed. The simulated open-phase voltages are plotted in Figure 4-17 (a) and Figure 4-17 (b) respectively. It can be seen that the flat line construction leads to resonance at the highest shunt compensation degrees, hence providing the largest safety margin for the typical values used in practice. The main reason for this behaviour is the higher zero sequence capacitance of the flat configuration due to the closer proximity of conductors to ground. The delta and vertical configurations present similar resonance performance while the inverted delta configuration results in resonance at the lowest shunt compensation degrees.

Page 61

Resonance and Ferroresonance in Power Networks C 10m B 10m

10m A

B

C

10m

A

C

A

A

10m

C 10m

10m 10m

B

B

10m

10m 15m

15m (a)

15m

15m

(b)

(c)

(d)

Figure 4-16 Transmission Line constructions: (a) Flat Configuration, (b) Vertical Configuration, (c) Delta Configuration, (d) Inverted Delta Configuration.

10 U [pu] 9 8 7 6 5 4 3 2 1 0 60%

k 65%

70%

75%

80%

85%

90%

95%

100%

10 U [pu] 9 8 7 6 5 4 3 2 1 0 60%

k 65%

70%

75%

80%

85%

90%

95%

100%

Figure 4-17 Steady-State Open-Phase Voltages in Shunt Compensated Transmission Lines. Effect of Line Construction Type.

4.3.1.2 Effect of Line Transposition The effect of line asymmetry is illustrated using the flat line configuration shown in Figure 4-16 without any phase transposition. This geometry has been selected because it leads to the highest degree of asymmetry. For the purposes of this analysis, a linear model has been employed to represent the shunt reactors and the neutral point has been directly grounded. The simulation results are shown in Figure 4-18(a) and Figure 4-18(b) for the one open-phase and two openphases scenarios, respectively. The following effects can be observed: 1.

For the one open-phase scenario there are two resonant peaks corresponding to the external phases and the central phase respectively. The central phase presents resonance at a slightly higher compensation degree due to the higher inter-phase capacitances with respect to the external phases.

2.

Three resonant peaks are observed for the two open-phases scenario. Two peaks appear when one of the open phases is the central one whereas only one peak appears when the two external phases are open.

The results shown in Figure 4-18 can be compared with Figure 4-17 to analyse the effect of the line asymmetry. The main difference can be seen in the behaviour of each phase for the un-transposed scenario. The three phases have identical performance when the line is fully transposed, whereas they present different numbers and location of resonant peaks in the un-transposed case. This effect is not very significant for the one open-phase scenario since the two resonant peaks are in very close proximity. However, the three resonant peaks arising from the two open-phases scenario are quite dispersed, increasing the range of shunt compensation degrees that could lead to harmful over-voltages.

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Resonance and Ferroresonance in Power Networks 10 U [pu]

10 U [pu] 9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2 1

1 0 60%

65%

70%

75%

80%

85%

90%

95%

100%

105%

110%

115%

k 120%

0 60%

k 65%

70%

75%

80%

85%

90%

95%

100% 105%

110%

115% 120%

Figure 4-18 Steady-State Open-Phase Voltages for Untransposed Shunt Compensated Transmission Line – Flat Construction.

4.3.1.3 Effect of Reactor Saturation The effect of the reactor’s core saturation is illustrated by comparing the steady-state performance of a flat line configuration using linear and non-linear reactors. A full line transposition has been assumed and the shunt reactors’ neutral has been solidly earthed. The phase reactors’ saturation knee-point has been assumed at 1.25 pu The simulation results are included in Figure 4-19 (a) and Figure 4-19 (b) for the one open-phase and two openphases scenarios, respectively. To facilitate the comparison, the figures include the results for both linear and nonlinear reactors. It can be seen that the only effect of saturation is to limit the open-phase voltage to a value close to the saturation knee-point. For compensation degrees leading to voltages below 1.25 pu, both curves are identical since the shunt reactors are working in the linear region. 2.00

U [pu]

2.00

1.75

1.75

1.50

1.50

1.25

1.25

1.00

1.00

0.75

0.75

0.50

0.50

0.25

U [pu]

0.25 k

0.00

50%

60%

70%

80%

90%

100%

110%

120%

k

0.00

50%

60%

70%

80%

90%

100%

110%

120%

Figure 4-19 Steady-State Open-Phase Voltages for Shunt Compensated Transmission Line. Effect of Shunt Reactors Saturation A sensitivity analysis has been carried out to investigate the effect of different saturation knee points on the steadystate performance of the circuit described above. The results obtained with 1.25 pu, 1.5 pu and 1.75 pu are presented in Figure 4-20. Only the “one open-phase” scenario is presented for simplicity. This graph illustrates again that reactor core saturation will limit the steady-state resonant overvoltages to a value close to the kneepoint. Below that value, the reactor is operated in the linear region and there are no differences between any of the models. In practical terms, a higher saturation knee point means that the equipment will be exposed to higher overvoltages when the shunt reactors operate in the saturated region. Typical saturation knee points used in industry are in the range 1.25 to 1.75 pu.

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Resonance and Ferroresonance in Power Networks

2.25

U [pu]

2 1.75 1.5 1.25 1 0.75 0.5 0.25 0 50%

k 60%

70%

80%

90%

100%

110%

120%

130%

140%

Figure 4-20 Steady-State Open-Phase Voltages for Shunt Compensated Transmission Line. Sensitivity of saturation knee-point

4.3.2 Temporary Overvoltage Analysis The steady-state voltages computed in sections 4.2.2 (approximate method) and 4.3.1 (detailed EMT analysis) represent the final continuous values that can be reached in the absence of remedial actions – i.e. if the uneven open-phase condition is left in the circuit indefinitely. These steady-state voltages will be rarely reached in practice due to the long time-constant of the resonant circuit and to the operation of protection or control systems. However, during the transient process following a switching operation, temporary voltage oscillations will exceed the steady state values and may stress the equipment insulation for a short period. A detailed time domain simulation of those events allows accurate assessment of the TOV. This analysis is illustrated in the next subsections by simulating the effect of selected design variables. Assessment of the equipment withstand capabilities to the computed TOVs is beyond the scope of this Technical Brochure.

4.3.2.1 Effect of Tower Design This section illustrates the temporary over-voltages arising from uneven open-phase conditions in each of the four line configurations shown in Figure 4-16 (section 4.3.1.1). For the purposes of this illustrative analysis, the shunt compensation degree has been fixed to 65%, the line has been made fully transposed and the shunt reactors’ neutral has been directly grounded. The voltage waveforms obtained from the time-domain simulations are shown in Figure 4-21. The steady-state voltages reached after the temporary oscillations have dampened out are also included for illustration purposes. The simulations started from a steady-state solution with the circuit breaker closed and the line energized at 1 pu voltage. After 5 power frequency cycles, one or two circuit breaker poles were open to simulate an unbalanced open-phase operation. The phase-to-ground voltage at the line sending end was recorded and it is plotted in Figure 4-21. It can be clearly seen that a transient voltage oscillation follows the circuit breaker operation. This waveform is characterised by a beat process with low modulation frequencies, which are determined by the capacitance and shunt reactance of the disconnected phase(s). A couple of seconds after the switching, the temporary oscillations are fully dampened and the open-phase voltages converge to the values computed in the steady-state analysis (Figure 4-17 in section 4.3.1.1). For each scenario shown in Figure 4-21, the two open-phases condition leads to higher TOVs than the one openphase condition. This is due to the closer proximity of a resonant pole for the selected degree of shunt compensation (65%), as demonstrated in the steady-state analysis of Figure 4-17. The inverted delta configuration resulted in the most severe TOV. The highest voltage obtained in the simulations was 3.2 pu, corresponding to the two open-phases scenario. This, again, is due to the closest proximity to a resonant pole as shown in Figure 4-17 and reinforces the need for an ample safety margin in order to minimize stress on equipment insulation.

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Resonance and Ferroresonance in Power Networks

3

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(file 2PH_VERTICAL.pl4; x-var t) v:SEND_A

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v:SEND_B

0.6 v:SEND_B

0.6

(file 2PH_INVERTED_DELTA.pl4; x-var t) v:SEND_A

v:SEND_B

Figure 4-21 Temporary Open-Phase Voltages in Transmission Lines with 65% Shunt Compensation. Effect of Tower Construction.

4.3.2.2 Effect of Line Transposition As illustrated in the steady-state analysis of section 4.3.1.2, the main consequence of an incomplete line transposition is that each phase behaves differently due to asymmetries in the inter-phase and phase-to-ground capacitances. The temporary voltage oscillations arising from uneven switching operations are illustrated in this section. For the purposes of this analysis, a flat line configuration has been selected because it leads to the highest degree of asymmetry. A 75% shunt compensation degree has been used, which provides a very low margin to

Page 65

Resonance and Ferroresonance in Power Networks resonance in the two open-phases condition. The shunt reactors have been assumed to be linear and the neutral is directly grounded. The simulation results for the two open-phases scenario are shown in Figure 4-22. The voltage waveforms obtained with a fully transposed line configuration are also shown for comparison. The simulations started from a steady-state solution with the circuit breaker closed and the line energized at 1 pu voltage. After 5 power frequency cycles, two circuit breaker poles were opened to simulate the unbalanced operation. Each possible combination of disconnected phases was simulated. It can be seen that, for the untransposed line case, the amplitude and frequency of temporary voltage oscillations depends on the relative position of the disconnected phases. When disconnecting the central phase and one of the external phases, Figure 4-22 (a) shows that the untransposed configuration leads to higher TOVs than the fully transposed case. This is due to the closer proximity to a resonant pole: 79.8% in the untransposed configuration vs. 82.2% in the fully transposed case. On the other hand, the disconnection of the two external phases results in higher TOVs for the fully transposed case, as shown in Figure 4-22 (b). This again, is due to the closer proximity to a resonant pole: 87.23% in the untransposed configuration vs. 82.2% in the fully transposed case. 4

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1.0

AC_UNTRANSPOSED.pl4: v :SEND_A

Figure 4-22 Temporary Open-Phase Voltages in Transmission Lines with 75% Shunt Compensation. Effect of Line Transposition

4.3.2.3 Effect of Neutral Reactors It is discussed in section 4.2.3.2 that the resonant circuit formed during uneven switching operations can be detuned by the introduction of an adequate neutral reactor. The influence of this reactor on resonant TOVs is evaluated in this section. In this example, the size of the neutral reactor has been calculated according to Eq. 4-13. For the purposes of this illustrative analysis, a fully transposed flat line construction as described in section 4.2.2.3 and 80% shunt compensation degree have been selected. This is just below a resonant peak for the two openphases condition with a directly grounded neutral connection, as shown in Figure 4-6 (section 4.2.2.3). The simulation results are shown in Figure 4-23. The results obtained with a directly grounded neutral have also been included in the graphs for comparison. The simulations started from a steady-state condition with the circuit breaker closed and the line energized at a voltage of 1 pu. After 5 power frequency cycles, one or two circuit breaker poles were opened to simulate an unbalanced open-phase operation. The phase-to-ground voltage at the line sending end was recorded and it is plotted in Figure 4-23. It can be seen that the presence of a neutral reactor greatly reduces the amplitude of voltages induced on the open phase(s). When the neutral reactor, as calculated from Eq. 4-13, is inserted in the circuit the open-phase voltages quickly decay in an oscillatory manner with a dominant frequency of 50 Hz and a low frequency modulation of 6 Hz. A final steady-state of 0.08 pu is reached within 1 second. If the neutral is directly grounded, the amplitude of the low frequency oscillations is considerably higher, reaching a theoretical value of nearly 5 pu for the two open-phases scenario. Warning: this example is presented for illustrative purposes only in order to assess the impact of installing a neutral reactor. In practice, corona losses, saturation of magnetic cores and/or surge arrester operation will limit the

Page 66

Resonance and Ferroresonance in Power Networks induced open-phase voltage well below the waveform presented in Figure 4-23 (b) for the solid neutral grounding case. All these voltage limiting elements should be considered when studying the performance of a specific shuntcompensated transmission line. 2.500

5.00

1.875

3.75

1.250

2.50

0.625

1.25

0.000

0.00

-0.625

-1.25

-1.250

-2.50

-1.875

-3.75

-2.500 0.0

0.2

0.4

0.6

0.8

1.0

-5.00 0.0

0.2

0.4

0.6

0.8

1.0

Figure 4-23 Temporary Open-Phase Voltages in Transmission Lines with 80% Shunt Compensation. Effect of Neutral Reactor

4.3.2.4 Effect of Reactor Saturation The steady-state analysis discussed in section 4.3.1.3 has illustrated that the effect of reactors’ saturation is to limit the open-phase voltages to a value close to the “knee-point”. A switching condition has been introduced in this section to investigate the effects of saturation on the amplitude and duration of TOVs. The simulation results corresponding to one open-phase condition in a fully transposed flat line configuration are shown in Figure 4-24. Other line configurations follow the same trend and are not included for clarity. This graph shows the highest voltage recorded during the simulation (“max TOV”) as well as the voltage decay reached after 5 seconds (“TOV decayed after 5 seconds”). The results of a steady-state analysis, without a switching transient, are also included for comparison. It can be seen that the maximum TOVs arising from the switching operation are significantly higher than their corresponding steady-state voltages for the entire range of shunt compensation degrees. The TOV decayed after 5 seconds matches the steady-state solution in most cases, except for a range between 57.4% and 84.3% shunt compensation degrees. The high TOVs observed in this range are caused by ferroresonance, which can be explained with the use of Figure 4-25. In this figure, k1 represents a compensation degree where the reactor normally operates in a linear mode in point-1 (i.e. between 57.4% and 84.3%). A second stable point with high voltage and current (point 3) is also possible, and it can be reached as a result of a switching transient. For higher degrees of shunt compensation, k2 and k3 above 84.3%, there is no possible operating point in the linear region and the only stable solution resides in the saturated region. This is a stable ferroresonant state with high current and voltage. It can also be observed that the ferroresonant voltage drops as the degree of shunt compensation is increased. For a more detailed explanation of ferroresonance, please refer to section 2.2. An example of a voltage waveform for a shunt reactor driven into ferroresonance is shown in Figure 4-26. This figure compares the simulation of a single-phase trip for two degrees of shunt compensation: 56% and 60%. A saturation knee point of 1.25pu has been assumed in this example. Similar results can be obtained with higher knee points. The graphs on the left (i and iii) show steady-state simulations without a switching operation (i.e. phase-A open in steady-state). For the graphs on the right (ii and iv), the simulations started from a steady-state condition with the circuit breaker closed and the line energized at a voltage of 1 pu, followed by a trip of phase-A at t = 0.1 s. Figure 4-26 (ii) shows that, for the 56% shunt compensation degree, the open-phase voltage reaches 1.75 pu just after the trip. This open-phase voltage decays in an oscillatory manner to reach the “no-trip” steadystate solution of 0.27 pu (i.e. stable point 1) in approximately 2 seconds. On the other hand, Figure 4-26 (vi) shows the equivalent results for a shunt compensation degree of 60%. In this case, the open-phase voltage after the trip reaches 2.7 pu and decays to reach a new steady-state of 2.05 pu (stable point 3), significantly higher than the “no-

Page 67

Resonance and Ferroresonance in Power Networks trip” steady state of 0.3 pu (stable point 1). The switching transient causes the operating point to jump from the linear region (point 1) and lock into a saturated state (point 3), which represents a ferroresonant condition. 3.5

V [pu]

3 2.5 2 1.5 1 0.5

k 0 40%

45%

50%

55%

60%

65%

70%

75%

80%

85%

90%

95%

100%

Figure 4-24 One Open-Phase Voltage in Fully Transposed Line as function of shunt compensation degree. Saturated reactors model (1.25pu saturation knee). Effect of switching transient

Figure 4-25 Illustration of ferroresonance associated with increased shunt compensation degree

Page 68

Resonance and Ferroresonance in Power Networks 2.0

2.0

[pu]

[pu]

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

-1.0

-1.0

-1.5

-1.5

-2.0

-2.0 0

1

2

3

4

5

[s]

6

(f ile steady _state_56%. pl4; x-v ar t) v :SEND_A

0

1

(i) Steady-state simulation – 56% Shunt Compensation 3

3 [pu]

2

2

1

1

0

0

-1

-1

-2

-2

-3 1

2

3

4

5

[s]

6

(ii) 1ph trip simulation – 56% Shunt Compensation

[pu]

0

2

(f ile special_cases_56%.pl4; x-v ar t) v :SEND_A

3

4

5

(f ile steady _state_60%. pl4; x-v ar t) v :SEND_A

[s]

6

-3 0

1

2

3

4

5

[s]

6

(f ile special_cases_60%.pl4; x-v ar t) v :SEND_A

(iii) Steady-state simulation – 60% Shunt Compensation

(iv) 1ph trip simulation – 60% Shunt Compensation

Figure 4-26 Simulation Waveforms of one open-phase overvoltages for 56% and 60% shunt compensation degrees. Reactor saturation included (1.25pu knee-point)

Page 69

Resonance and Ferroresonance in Power Networks

4.3.3 Summary of Para meters Affecting Line Resonance in Open- Phase Conditions Design Parameter Network strength (MVA Short Circuit power) Tower Design

Impact on Resonance No

Marginal

Comment The resonant circuit is disconnected from the main network, only linked by capacitive coupling. The tower design affects the circuit capacitances. But the effect is not significant for practical line designs. It may have higher impact in “compact” line designs.

High

The level of unbalanced circuit capacitance can be significant in long circuits, presenting different resonant points for each phase.

High

Magnetic saturation of shunt reactors will affect the amplitude of resonant overvoltages. However, it will not affect the size of reactors causing resonance. Ferroresonance is also possible.

Medium/High

The saturation knee point will affect the amplitude of resonant overvoltages. However, it will not affect the size of reactors causing resonance. The effect is not very significant for the practical range typically used in industry.

Reactor core construction (X0/X+ )

High

Magnetic coupling between phases has a large impact on the location of resonant peaks.

Installation of Neutral Reactor

High

The installation of a neutral reactor can detune the resonant circuit.

Installation of Neutral Resistor

High

The installation of a neutral resistor can add damping to the resonant circuit.

Installation of cable sections

High

The installation of an underground cable section in a mixed circuit can effectively detune the resonant circuit. Resonance moves towards higher levels of shunt compensation degrees.

Size of shunt reactors (i. e. % shunt compensation degree)

High

This is the most critical parameter determining the occurrence of resonance.

Circuit asymmetries/transposition

Magnetic saturation of reactor core

Reactor core saturation knee point

Circuit Breaker Grading Capacitors

No

No impact because circuit breaker grading capacitors are too small compared with the circuit capacitance.

Flows on energized phases

No

The impact of inductive coupling on resonance is negligible compared to the capacitive coupling.

Page 70

Resonance and Ferroresonance in Power Networks

4.4 Line Resonance in Multiple-Circuit Rights of Way 4.4.1 Physical description De-energized shunt compensated transmission lines can form resonant circuits for a definite number of shunt compensation degrees. Under resonance conditions, severe overvoltages can develop across the shunt reactors, caused by capacitive coupling from nearby energized circuits. This network topology is illustrated in Figure 4-27. Unlike the phenomena described in section 4.2 (Line Resonance in Uneven Open-Phase Conditions), where only one or two phases are de-energized, this section deals with a full three-phase de-energization of the circuit under study. It can be demonstrated that, for a fully transposed circuit, there are four degrees of shunt compensation that can result in resonance affecting the de-energized circuit ([81], [82]). Two of those resonant peaks can be excited under normal operating conditions in the energized circuit or during fault conditions on either circuit. The other two resonant peaks can only be excited during faults on the de-energized circuit. Table 4-2 summarises the location of the resonant peaks as a function of the degree of shunt compensation. A discussion of each resonant peak follows in the next sub-sections. It should be noted that up to nineteen resonant peaks can be observed in untransposed double circuits. Approximate equations to derive them can be found in [81]. In practical terms, some of the resonant peaks are very close to each other or even overlap. For simplicity, this document will assume fully transposed circuits. A detailed EMT analysis is recommended in cases where significant circuit asymmetries are expected and degrees of shunt compensation are close to those described in Table 4-2. Similarly, it emphasized that the expressions included in Table 4-2 are approximations derived from electrostatic equations ([78] - [82]) and should only be used to obtain an initial estimation of the dangerous ranges of shunt compensation degrees for a particular circuit design. A detailed EMT analysis is recommended to assess the prospective overvoltages on the de-energized circuit and to evaluate mitigation options. Closed CB

Closed CB

Energized Circuit Mutual Capacitive Coupling De-Energized Circuit

Open CB

L+ L0

C+ C0

Open CB

Figure 4-27 Shunt-compensated double circuit line for analysis of resonance It can be observed from Table 4-2 that the location of the resonant points in a transposed circuit are only determined by the parameters of the de-energized circuit; namely the C0/C+ ratio of the de-energized circuit and the X0/X+ ratio of the shunt reactor connected to the de-energized circuit. The amplitude of the resonant voltages is determined by the exciting circuit – i.e. capacitive coupling (mainly zero sequence) with parallel energized circuit and the zero sequence equivalent voltage source due to the flow of zero sequence current in the energized circuit.

Page 71

Resonance and Ferroresonance in Power Networks Table 4-2 Approximate location of resonant peaks in a de-energized transposed transmission circuit

Resonant Peak

#1

k [%]

Fault on de-energized circuit SLG

LLG

LL

LLL

N h 100

#2

1 2 N 3

#3

2 N 3

#4

No fault

Any fault on energized circuit

3 h 100 2 h 3 h 1 2 h

100

100

where: k is the degree of shunt compensation of the de-energized circuit, as defined in Eq. 4-1. N

C0 C

h

X0 X

; with C0 and C+ are the zero and positive sequence capacitances of the de-energized circuit. L0 with X0 (L0) and X+ (L+) are the equivalent zero and positive sequence reactance (inductance) L

of the shunt reactor(s) connected to the de-energized circuit.

4.4.1.1 First Resonant Point The degree of shunt compensation resulting in the first resonant peak, k1, in a fully transposed circuit is given in Eq. 4-16. This resonant point can be excited during any fault on the energized circuit, no fault condition, or during any of the following faults on the de-energized circuit: ungrounded line-to-line or three-phase.

k1

N h 100

Eq. 4-16

This is a zero-sequence parallel resonant condition with the zero-sequence capacitance of the de-energized circuit matching the zero-sequence inductance of the shunt reactor (see Figure 4-28). The excitation is provided by zero sequence current coupled into the de-energized circuit via the mutual capacitive coupling with the parallel energized circuit. In practical terms, the zero sequence voltage in the energized circuit during normal operating conditions (i.e. no fault) is close to zero and, therefore, the amplitude of the coupled voltages on the de-energized circuit are not dangerous (see blue dotted line in Figure 4-34-(i) for an example). However, single-line-to-ground faults on the energized circuit can produce very high zero sequence voltage (close to the positive sequence voltage) which will excite the parallel resonant circuit. This was demonstrated in Section 2.1.3 for the equivalent series-parallel resonant circuit where the series capacitance CS is relatively small to provide a high source impedance. A single-line-to-ground fault on the energized circuit is the most onerous condition, with very high voltages coupled onto the de-enegized circuit for the duration of the fault (see red continuous line in Figure 4-34-(i) for an example). Other faults produce lower levels of unbalance, resulting in lower resonant voltages, although they can still be high enough to damage equipment (see red continuous lines in Figure 4-34-(ii and iii)).

Page 72

Resonance and Ferroresonance in Power Networks Us Cph-ph

Us Cph-ph

Cph-ph

Us Us L

L

Cm

L C0

C0

C0

Cm

C0

L0

Cph-ph

Equivalent parallel resonant circuit #1 Cph-ph

L0

L0

Cph-ph

L0

C0

C0

C0

Figure 4-28 Equivalent circuit describing resonant point #1

4.4.1.2 Second Resona nt Point The degree of shunt compensation resulting in the second resonant peak, k2, in a fully transposed circuit is given in Eq. 4-17. This resonant point can be excited only during single-line-to-ground faults on the de-energized circuit.

k2

1

2 N 3

3 h 2 h

100

Eq. 4-17

This is a zero-sequence parallel resonant condition. The shunt-reactor size determining this resonant condition is the same as for the “two-open-phase” condition described in section 4.2 (see Figure 4-29). The excitation is provided by zero sequence current flowing through the fault on the de-energized circuit, which is sustained via zero sequence capacitive coupling with the parallel energized circuit. The amplitude of the resonant voltages across the healthy phases on the de-energized circuit can be very high (see red line in Figure 4-34-(iv) for an example). Us C ph-ph

Us Cph-ph

Cph-ph

Us Us L

L

L C0

C0

Cm

C0

Cm

L0

C0

C ph-ph

C ph-ph

Equivalent parallel resonant circuit #2 C ph-ph

L0

L0

L0

C ph-ph

C0

C0

C0

Figure 4-29 Equivalent circuit describing resonant point #2

Page 73

Resonance and Ferroresonance in Power Networks

4.4.1.3 Third Resonant Point The degree of shunt compensation resulting in the third resonant peak, k3, in a fully transposed circuit is given in Eq. 4-18. This resonant point can be excited only during line-line-ground faults on the de-energized circuit.

k3

N

2 3

3 h 1 2 h

100

Eq. 4-18

This is a zero-sequence parallel resonant condition. The shunt-reactor size determining this resonant condition is the same as for the “one-open-phase” condition described in section 4.2 (see Figure 4-30). The excitation is provided by zero sequence current flowing through the fault on the de-energized circuit, which is sustained via capacitive coupling with the parallel energized circuit. The amplitude of the resonant voltages across the healthy phases on the de-energized circuit can be very high (see red line in Figure 4-34-(v) for an example).

Figure 4-30 Equivalent circuit describing resonant point #3

4.4.1.4 Fourth Resonant Point The degree of shunt compensation resulting in the fourth resonant peak, k4, in a fully transposed circuit is given in Eq. 4-19. This resonant point can be excited during any fault on the energized circuit, no fault condition, or during any of the following faults on the de-energized circuit: single-line-to-ground or ungrounded line-to-line.

k4

100

Eq. 4-19

This is a positive-sequence parallel resonant condition with the positive-sequence capacitance of the de-energized circuit matching the positive-sequence inductance of the shunt reactor (see Figure 4-31). The excitation is provided by positive sequence current coupled into the de-energized circuit via the mutual capacitive coupling with the parallel energized circuit. In practical terms, the positive sequence voltage in the energized circuit is not affected to a great extent by the operating condition, therefore the resonant overvoltages imposed on the de-energized circuit are very similar in all cases (see Figure 4-34 (i) to (vi) for an example). This can be a very onerous configuration, with typical steady state voltages exceeding equipment ratings. Shunt compensation degrees close to 100% should be avoided by design when positive sequence excitation is available from nearby circuits.

Page 74

Resonance and Ferroresonance in Power Networks

Figure 4-31 Equivalent circuit describing resonant point #4

4.4.2 Discussion on Circuit Para meters It has been illustrated in Section 4.4.1 that the only parameters affecting the location of the resonant points are associated with the geometry of the de-energized circuit, while the parallel energized circuit provides the excitation to sustain the resonant condition. In particular, the main factors determining the formation of a resonant circuit are the C0/C+ ratio of the de-energized circuit and the X0/X+ ratio of the shunt reactor connected to the de-energized circuit. Typically (C0/C+) varies within a very narrow range for typical constructions. For overhead transmission lines, (C0/C+) is normally in the range of 0.5 – 0.7, whereas underground cable circuits have a (C0/C+) ratio of 1. Mixed overhead/underground circuits have (C0/C+) ratios lying between these two ranges. Furthermore, the (X0/X+) ratio of the shunt reactors is dependent on the core construction. Typically, the following ranges can be assumed for solid grounded neutral arrangements [82]: Three-phase shell type, four or five legged core, single-phase units or air core reactors: (X0/X+)=1 Three-phase three legged core: (X0/X+) = 0.5 to 0.7 The installation of neutral reactors increases the above ratios as X’0 = X0 + 3. XN To illustrate the effect of each parameter, the equations presented in Table 4-2 have been plotted as a function of the (C0/C+) or the (X0/X+) ratios for typical ranges – see Figure 4-32. The following can be observed: 1. Low values of (C0/C+) or (X0/X+) result in resonant peaks that are quite distant apart. This is the case for typical overhead line constructions and three-legged core shunt reactors. These configurations require special attention to guarantee operation in a safe mode. 2. High values of (C0/C+) result in resonant peaks at higher levels of shunt compensation degrees and reduces the distance between the peaks. This is the case for typical mixed overhead line / underground cable constructions. Special attention needs to be paid in these cases as the cable section normally drives the need for high degrees of shunt compensation, close to 100%.

Page 75

Resonance and Ferroresonance in Power Networks 3. High values of shunt reactor (X0/X+) ratio shift resonance to degrees of compensation above 100% (i.e. overcompensation). This can be achieved with the introduction of neutral reactors.

Shunt Compensation Degree [%]

Shunt Compensation Degree [%]

100.0 90.0 80.0 70.0 60.0 50.0 40.0 30.0 20.0 10.0 0.0

120.0 100.0 80.0 60.0 40.0 20.0 0.0

0.5

0.6

0.7

0.8

0.9

1.0

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Czero / Cpos Resonant peak 1

Resonant peak 2

Resonant peak 3

2.2

Xzero / Xpos

resonant peak 4

Resonant peak 1

Resonant peak 2

Resonant peak 3

resonant peak 4

Figure 4-32 Location of resonant points in a transposed transmission circuit as a function of X and C

4.4.3 Case Study 4.4.3.1 System Description A practical example of parallel resonance in a 765 kV system is presented in this section. The system data is shown in Figure 4-33 [77]. It consists of two 765 kV transmission lines, flat construction, coupled along 193.1 km. One circuit (cct #2) is opened at both ends and shunt compensated at the receiving end only. Shunt-compensation on the energized circuit is not relevant for this analysis. Both circuits are fully transposed. The shunt reactors have (X0/X+) = 1 and the neutral point is solidly grounded. The power system frequency is 60 Hz. 20

765kV

~ SENDING

Closed

193.1km Circuit #1

Closed

Open

193.1km Circuit #2

Open

765kV

15.24

~

A1

15.24 B1

C1

Phase conductors: 4x1.385in

RECEIVING

20.4

15.24

15.24

15.24 A2

B2

at 18in spacing

C2 21.0

dc Resistance = 0.085 /mile

Shunt reactor

All dimensions are in metres

Figure 4-33 One-Line Diagram and Double Circuit Geometry The analysis will seek to answer the following practical questions: 1. What sizes of shunt reactors will lead to resonance on the de-energized circuit? 2. What is the amplitude of the induced voltages at any operating condition? 3. How sensitive are the location of the resonant points to circuit or system design parameters?

Page 76

Resonance and Ferroresonance in Power Networks

4.4.3.2 Comparative Analysis of Solution Methods The shunt-reactor values leading to resonance in the de-energized circuit have been calculated with two independent methods: (i) electrostatic approximate equations included in Table 4-2 and (ii) a detailed EMT model. The inputs for the electrostatic analysis are: C0 = 1.75 F ; C+ = 2.54 F, calculated with a standard power frequency line constant tool for the circuit geometry shown in Figure 4-33. N = C0 / C+ = 0.69 The shunt reactor is a three-phase shell-type unit. Typically X0 = X+ can be assumed for this reactor core type, therefore h = X0/X+= 1. For the EMT analysis, the following was assumed: Both transmission circuits were modelled using Bergeron model (i.e. distributed line parameter at power frequency). One full line transposition was assumed. For that purpose, the circuits were modelled as three untransposed line sections (64.36 km each) and the phase transposition was carried out explicitly at the connection nodes between each section. A linear representation was adopted for the shunt reactors for simplicity. Circuit breakers in circuit #1 were represented in a closed position. Circuit breakers in circuit #2 were represented in an open position. Steady-state simulations, without any switching, were carried out by changing the size of the shuntreactors. Line-to-ground voltage at the terminal of the shunt reactors was monitored as a function of the reactor size. Table 4-3 shows the location of the resonant peaks calculated with each method. It can be seen that both sets of results compare very well, confirming that the electrostatic “approximate” method is robust enough for an initial risk assessment of resonance (i.e. identifying the size of shunt reactors leading to resonance). If a problem is suspected, detailed EMT simulations are recommended to determine the severity of the problem and to investigate mitigation options. In this example, the amplitude of induced voltages on the de-energized circuit were calculated using the EMT model described above. These voltages were plotted against shunt reactor Mvar (positive sequence) to identify trends, as shown in Figure 4-34. The following can be observed from this steady-state analysis: 1. The de-energized circuit (cct #2) exhibits two resonant points in a steady-state normal condition, without any circuit faults. The first peak, close to a compensation degree of 70%, leads to voltages well below 1 pu. The 3 second resonant peak, at 100% shunt compensation degree, leads to voltages of up to 8 pu . In particular, shunt compensation degrees in the range from 98% to 104% lead to induced voltages in excess of 1 pu on the de-energized circuit (cct #2). 2. Faults on the energized parallel circuit (cct #1) do not affect the location of the resonant points; however the amplitude of the induced voltages on the de-energized circuit (cct #2) increases significantly with respect to the “no fault” condition. See Figure 4-34 (i), (ii) and (iii). 3. Faults on the de-energized circuit (#2) affect the amplitude and location of the resonant peaks with respect to the “no fault” condition. See Figure 4-34 (iv), (v) and (vi). 4. The most onerous case arises from a single-line-to ground fault on the energized circuit (cct #1) – see Figure 3 4-34 (i). This condition results in induced voltages of up to 21 pu at the resonant point. For this fault scenario, steady-state voltages on the de-energized circuit (cct #2) exceed 1 pu for a range of shunt compensation degrees from 61% to 77% and from 98% to 104%.

3

Note that shunt reactor saturation has been ignored in this example for simplicity. In reality, saturation will typically limit the voltage to a value close to the knee point (i.e. in the order of 1.5pu)

Page 77

Resonance and Ferroresonance in Power Networks Table 4-3 Calculated location of Resonant Peaks EMT Simulations

Resonant Peak

Approximate Electrostatic Method

No fault

#1

68.2% (386 Mvar)

69.5% (390 Mvar)

Fault on Energized Circuit 69.5% (390 Mvar)

#2

79.2% (444.3 Mvar)

---

#3

89.6% (502.6 Mvar)

#4

100.0% (561 Mvar)

LL

LLL

---

---

69.5% (390 Mvar)

69.5% (390 Mvar)

---

80.21% (450 Mvar)

---

---

---

---

---

---

90.02% (505 Mvar)

---

---

100.7% (565 Mvar)

100.7% (565 Mvar)

100.7% (565 Mvar)

---

100.7% (565 Mvar)

---

Resonant Peak #4

Resonant Peak #1

1pu

100

1000

1pu

100

400 71%

500 89%

600 107%

700 125%

Reactor MVAr (at 765kV)

% Shunt Compensation

200 36%

300 53%

400 71%

500 89%

600 107%

Resonant Peak #4

Resonant Peak #1

Resonant Peak #1

10000

Resonant Peak #2

RMS Line to Ground Voltage [kV]

SLG Fault on Circuit #2 1000

1pu

100

200 36%

300 53%

Resonant Peak #1

Resonant Peak #4

600 107%

700 125%

LLG Fault on Circuit #2 1000

1pu

100

Reactor MVAr (at 765kV)

% Shunt Compensation

100 18%

600 107%

700 125%

Resonant Peak #4

10000

1000

1pu

100

3PH Fault on Circuit #2 No Fault

10 500 89%

500 89%

Reactor MVAr (at 765kV)

No Fault

10 400 71%

400 71%

(iii) Resonant Peak #3

No Fault 300 53%

100 18%

% Shunt Compensation

(ii)

10000

200 36%

100

700 125%

Reactor MVAr (at 765kV)

% Shunt Compensation

(i)

100 18%

1pu

10 100 18%

RMS Line to Ground Voltage [kV]

300 53%

1000

No Fault

10 200 36%

Resonant Peak #4

Resonant Peak #1

No Fault

10 100 18%

3PH Fault on Circuit #1

10000

No Fault

RMS Line to Ground Voltage [kV]

Resonant Peak #4

Resonant Peak #1

LL Fault on Circuit #1

10000

1000

SLG

RMS Line to Ground Voltage [kV]

RMS Line to Ground Voltage [kV]

LLG

RMS Line to Ground Voltage [kV]

SLG Fault on Circuit #1

10000

Fault on De-Energized Circuit

10 200 36%

300 53%

400 71%

500 89%

600 107%

700 125%

Reactor MVAr (at 765kV)

% Shunt Compensation

(iv)

(v)

100 18%

200 36%

300 53%

400 71%

500 89%

600 107%

700 125%

Reactor MVAr (at 765kV)

% Shunt Compensation

(vi)

Figure 4-34 Steady-State Line-Ground Voltages on circuit #2 (assumed both circuit transposed and shunt reactors with (X 0 /X + ) = 1)

4.4.3.3 Considerations about fault conditions 1. Faults on Energized Circuit (circuit #1)

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Resonance and Ferroresonance in Power Networks It should be noted that a fault on the energized circuit is not a steady-state situation as, in reality, the protection relays will trip the faulted circuit in a definite time (either in primary or back-up protection times). Therefore, the line equipment will not be subjected to the level of stress computed in the steady-state analysis shown in Figure 4-34. A detailed simulation of a single-line-to-ground fault at the first resonant peak (69.5%) is illustrated in Figure 4-35 below. These EMT simulation results relate to the double circuit topology shown in Figure 4-33. Figure 4-35 (i) shows the theoretical steady-state solution assuming a permanent fault on the energized parallel circuit (circuit #1). A steady-state voltage of 21 pu3 is coupled on the de-energized circuit (circuit #2). Figure 4-35 (ii) shows the simulation of a fault on the energized circuit (cct #1) without a subsequent trip. For the first 100 ms of simulation, the coupled voltage on circuit #2 is just 0.4 pu, due to the normal operation of circuit #1. A t = 100 ms, a slg fault is applied to circuit #1. It can be seen that the voltage coupled on the deenergized circuit (cct #2) rises slowly to reach the “theoretical” steady state value in approximately 3 seconds. Figure 4-35 (iii) and (iv) show a realistic scenario in which a fault in the energized circuit (cct #1) is followed by the trip of that circuit. Two tripping times are illustrated: 100ms and 500ms, as typical primary and back-up times. It can be seen that, due to the tripping of the parallel circuit (cct #1), the induced voltages on the deenergized circuit (cct #2) do not reach the theoretical steady-state value. A primary fault clearance time of 100ms results on a maximum induced voltage of 2.1 pu3. In contrast, fault clearance by back-up protection in 500ms results on a maximum induced voltage of 8.3 pu3. Both maximum values are reached immediately after the trip of the parallel energized circuit (cct #1) – i.e. removal of the exciting source stops the resonant condition and results in a slow decay of the voltages on the de-energized circuit (cct #2). It should be noted that the coupled voltage on each of the three phases of circuit #2 is in phase, indicating that this is a zero-sequence phenomena.

2. Faults on De-Energized Circuit (circuit #2) Faults on the de-energized circuit (cct #2) can reach steady state in some circumstances. This can happen as a result of an actual fault (un-intentional) or by closing earth-switches (intentional) while the circuit is out-of-service. In this case the protection relays will not clear the fault as the circuit breakers are already open, therefore the fault and resonant condition can remain indefinitely. Detailed simulations of single-line-to-ground faults on circuit #2 at the second resonant peak (80.21%) are illustrated in Figure 4-36 below. Figure 4-36 (i) shows the theoretical steady-state solution assuming a permanent SLG fault on the deenergized circuit (circuit #2). A steady-state voltage of 4.4 pu3 is coupled on the other two phases of the deenergized circuit (circuit #2). Figure 4-36 (ii) shows the simulation of a SLG fault on circuit #2 while being out-of-service. For the first 100 ms of simulation, the coupled voltage on circuit #2 is just 0.16 pu. A t = 100 ms, a SLG fault is applied to phase-A of circuit #2. It can be seen that the voltage coupled on the other two phases rises slowly to reach the “theoretical” steady state value in approximately 4 seconds. This is a possible scenario and the line equipment could be subjected to this stress until some remedial action is implemented. This fault could possibly be caused by the closing of earthing switches in only one phase. Figure 4-36 (iii) shows the simulation results for the following scenario: both circuit #1 and circuit #2 are initially in service, a SLG fault occurs on circuit #2 at t = 100 ms and the protection relays trip that circuit in 100 ms. It can be seen that the voltage coupled on the other two phases of circuit #2 (now de-energized) rises slowly to reach the “theoretical” steady state value in approximately 4 seconds. A low frequency oscillation of 6 Hz can also be observed. This is a possible scenario and the line equipment could be subjected to this stress until some remedial action is implemented. Furthermore, the high coupled voltages will make extinction of the secondary arc difficult. Figure 4-36 (iv) shows a similar scenario to (iii) but in this case autoreclosing is applied to circuit #2. It can be seen that closing the circuit breakers in circuit #2 (i.e. re-energizing the circuit) returns the voltages to normal. This scenario assumes that the secondary arc extinguishes successfully during the autoreclosing dead time. It should be noted that the coupled voltages on circuit #2 are perfectly balanced before the fault occurrence. However, they move “in-phase” during the resonant condition caused by the fault, indicating that this is a zerosequence phenomenon.

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Resonance and Ferroresonance in Power Networks 25.00

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Figure 4-35 Temporary Overvoltages on de-energized circuit (cct #2) during SLG fault and trip in parallel energized circuit (cct #1) – 390 Mvar shunt reactor / 69.5% shunt compensation 5.00

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Figure 4-36 Temporary Overvoltages on de-energized circuit (cct #2) during SLG fault and trip in circuit #2 – 450 Mvar shunt reactor / 80.21% shunt compensation

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Resonance and Ferroresonance in Power Networks

4.4.3.4 Effect of reactor’s zero sequence parameters The sensitivity of the results to the zero sequence parameters of the shunt reactors is analysed by assuming the introduction of a neutral reactor. In this example, the ratio between neutral reactor and phase reactors (Xn/X+) is assumed to be 0.35. This results in h = (X0/X+) = (X+ + 3Xn)/ X+= 2.05. Keeping the other system parameters unchanged, the resonant points obtained with the electrostatic “approximate” method and the EMT model are shown in Table 4-4. It can be seen that both methods of analysis produce very similar results. Detailed line-toground voltages coupled on de-energized circuit #2 were obtained with the EMT model and are illustrated in Figure 4-37. The following can be observed from this sensitivity analysis: 1.

The introduction of a neutral reactor shifts the zero sequence resonant points (#1, #2 and #3) to higher degrees of shunt compensation.

2.

The positive sequence resonant point (#4) is not affected by the neutral reactor.

3.

The introduction of a neutral reactor does not affect the amplitude of the resonant over-voltages on the deenergized circuit.

Table 4-4 Calculated Resonant Peaks (Reactor X 0 /X + = 2.05) EMT Simulations

Approximate Electrostatic Method

No fault

#1

141.1% (791 Mvar)

142.6% (800 Mvar)

Fault on Energized Circuit 142.6% (800 Mvar)

#2

120.3% (674.7 Mvar)

---

#3

108.0% (606.2 Mvar)

#4

100.0% (561 Mvar)

Resonant Peak

Fault on De-Energized Circuit SLG

LLG

LL

LLL

---

---

142.6% (800 Mvar)

142.6% (800 Mvar)

---

122.1% (685 Mvar)

---

---

---

---

---

---

109.6% (615 Mvar)

---

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100.7% (565 Mvar)

100.7% (565 Mvar)

100.7% (565 Mvar)

---

100.7% (565 Mvar)

---

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Resonance and Ferroresonance in Power Networks

4.4.4 Summary and commentary of resonance issues ass ociated with shuntcompensated multiple-circuit rights of way 1. Parallel resonance can occur in a de-energized shunt-compensated transmission circuit for certain degrees of shunt compensation. Energy is coupled from a nearby energized circuit through capacitive coupling. 2. The location of the resonant peaks is only dependant on the de-energized circuit parameters (i.e. deenergized circuit capacitances and shunt-reactor parameters). 3. The main factor determining the location of the resonant peaks is the ratio between the inter-phase and phase-to-phase capacitances in the de-energized circuit and the X0/X+ ratio of the shunt reactors. 4. The parameters of the parallel energized circuit affect the amplitude of the resonant voltages on the open circuit, but not the location of the resonant peaks. 5. Under normal operating conditions, there are three values of shunt-compensation degree leading to resonance in an un-transposed circuit. Only two values are observed if the circuit is transposed. 6. Under fault conditions, the number of shunt-compensation degrees leading to resonant conditions increases to nineteen in un-transposed circuits and to four in transposed circuits. 7. Out of four possible resonant conditions in a fully transposed shunt-compensated line, three of them are zero sequence and one is positive sequence. 8. The positive sequence resonant condition is independent of the energized or de-energized circuit characteristics. – i.e. it always arises at 100% shunt compensation. 9. Single-Line-to-Ground Faults (SLG) on the energized circuit provide the highest excitation for the zerosequence resonant conditions, leading to the highest overvoltages on the de-energized circuit. 10. Short circuit impedance of the feeding energized circuit does not affect the resonant location or amplitude of resonant overvoltages. 11. The magnetic coupling between phases in the shunt reactors has a large impact on the prospective resonant points. It shifts the zero-sequence resonant conditions (#1, #2 and #3) towards lower degrees of shunt compensation. The positive-sequence resonant condition (#4) is not affected. 12. The installation of neutral reactors shifts the zero-sequence resonant conditions (#1, #2 and #3) towards higher degrees of shunt compensation. The positive-sequence resonant condition (#4) is not affected. 13. The X0/R0 (Q0) and X+/R+ (Q+) quality factors in the shunt reactor affect the amplitude of the resonant overvoltages, but not the location of the resonant peaks. 14. It is important to assess the magnitude and duration of the resonant overvoltages considering the type of event (i.e. fault or normal operation) and automatic or manual actions (i.e. operation of protection relays or network controllers). 15. Although not explicitly covered in this document, the analysis of resonance can be extended to include transformer terminated lines, with or without shunt compensation. The presence of a delta tertiary winding reduces the equivalent X0/X+ ratio of the circuit and it can prevent the occurrence of zero-sequence resonances in some configurations [82]. 16. Sensitivity analysis must be carried out taking equipment tolerances and frequency deviations into account. Uncertain parameters such as soil resistivity, conductor sag, shunt conductance and zero-sequence damping must be carefully analysed. Furthermore, electromagnetic effects due to line loading also need to be included as they can cause small shifts in the resonant peaks. Reactor saturation will limit the amplitude of resonant overvoltages to values close to the saturation knee point. A very good example where some of these factors have been included in the studies of a real system expansion is reported in [83].

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4.5 Practical Consequences of Line Resonance The analysis described in this Chapter has illustrated the severity of prospective overvoltages arising from resonant conditions in shunt-compensated circuits. Losses have been ignored for simplicity, leading to a conservative, worstcase scenario, of the expected overvoltages appearing on the line. In practice, the measured overvoltages tend to be lower than calculated due to the presence of higher than predicted damping in real systems. This approach adds certain safety margins to the designs and it is generally accepted as good engineering practice. The main area of concern when dealing with shunt compensation and possible resonant conditions is related to excessive over-voltages and over-currents in the affected circuit, stressing insulation and causing damage to circuit equipment such as shunt reactors, surge arresters, instrument transformers and circuit breakers. This condition represents a threat to the integrity of the line equipment, but also a safety hazard to staff working in the vicinity of this equipment. The availability and reliability of the circuit is also compromised. Examples of surge arresters and shunt reactor failures attributed to parallel resonant conditions have been reported in [82]. The capability of the line equipment to withstand the prospective resonant overvoltages is dependent on the amplitude and duration of those overvoltages. Equipment withstand capabilities are outside the scope of this Technical Brochure and the reader is recommended to consult with the manufacturer or to check the relevant standards defining the required capabilities of each apparatus. Typical events resulting in short-duration stress to equipment include SPAR and circuit breaker failure (assuming that breaker failure protection is in place) in single circuits or faults in double circuits while one of them is out-ofservice. In contrast, the normal operation of a circuit running parallel to a second circuit which has been removed from service (for instance, to carry-out maintenance) can result in long-term stress to the out-of-service circuit equipment. Even in cases where the expected induced voltages during resonant or near-resonant conditions do not represent a threat to the equipment insulation, they need to be taken into account while planning maintenance outages in order to guarantee the safety of personnel. Attempting to use a ground stick or portable earth could result in arcs as the line is approached by the operator. Moreover, earthing switches must be rated adequately to interrupt the large shorting currents arising from the induced voltages. The saturation of shunt reactors can be an issue. Despite the “detuning” effect caused by the reduced saturated reactance, prolonged operation in this region is undesirable. As saturation becomes more pronounced, heating becomes a problem. In addition, core vibrations at higher than normal flux densities is particularly undesirable for gapped reactors. The harmonic distortion introduced during saturation is another issue to consider. Damage to several EHV shunt reactors due to resonance has reported in several publications ([80]-[82]). Zero sequence current flows can also damage the reactors even when voltages are below the rated values. It has been reported in [81] that reactor zero-sequence currents for three-legged, core form reactors can be in excess of their rating even for low lengths of coupling with parallel circuits – i.e. less than 10% of its length. Special attention must be paid for adequate zero sequence current rating specification. Line resonance can also affect the performance of SPAR by increasing the recovery voltage and compromising secondary arc extinction. Secondary arc current and recovery voltage need to be investigated in conjunction with line resonance in applications where SPAR or high-speed autoreclosing needs to be deployed. Longitudinal stress in circuit breakers (i.e. across open poles) must be assessed to ensure that there will be no internal flashovers due to excessive recovery voltage or steady-state stress in the circuit breakers. This stress builds-up slowly during resonant conditions.

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4.6 Mitigation Options A list of possible actions to mitigate power frequency resonance in shunt compensated transmission circuits is presented in Table 4-5. This list is not exhaustive and it deals only with the most generic topological problems. It is possible that other customised solutions may be more effective in specific situations. The reader is advised to carry-out detailed studies to understand the nature of the resonant circuit, so that the most cost effective solution can be applied taking into account not only resonance but also other design and operational constraints like system stability, voltage rise, line loadings, protection performance, operation and maintenance procedures, etc. An iterative approach is normally required to capture all technical constraints. In this assessment, the events leading to resonant conditions need to be evaluated in the context of probability of occurrence and impact on system safety and reliability. A HILP (High Impact – Low Probability) assessment is recommended in order to avoid overdesigning mitigating solutions. The mitigation options proposed in Table 4-5 are classified into two main groups: a)

Passive Methods: to prevent the occurrence of network conditions likely to result in resonance.

b)

Active Methods: to detect the conditions that could lead to resonant overvoltages and introduce actions (manual or automatic) to minimise the stress on equipment.

Furthermore, the type of resonant condition addressed by each mitigation solution is described as: Type 1: Unbalanced phase switching in shunt-compensated line in a single circuit construction or in a multiple circuit corridor. Type 2: Planned long-term disconnection of a shunt-compensated line in a multiple circuit corridor (for example to carry-out maintenance) Type 3: Unplanned short-term disconnection of a line in a multiple circuit corridor (for example to clear a fault followed by automatic reclosing) Type 4: Unplanned long-term disconnection of a shunt-compensated line in a multiple circuit corridor (for example due to a permanent fault) Table 4-5 Mitigation Options for Resonance in Shunt Compensated Circuits Passive

Mitigation Option

Comment

Install Neutral Resistors to damp zerosequence resonances

The required amount of shunt compensation can be split between the line and the station (busbar or transformer tertiary winding) to limit the amount of compensation attached to a de-energized circuit. The installation of Neutral Reactors must be co-ordinated with the performance of SPAR schemes. Adequate insulation levels must be specified for phase and neutral reactors. There is a requirement to adequately specify the energy duty of surge arresters across the phase and neutral reactors. Adequate insulation levels must be specified for phase reactors and neutral resistors.

Unground neutral point of shunt reactors

Adequate insulation levels must be specified for phase reactors. There is a requirement to

Avoid shunt compensation degrees leading to resonance or near-resonance

Install Neutral Reactors to detune zerosequence resonances

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Type of Resonance

1, 2, 3, 4

1, 2, 3, 4

1, 2, 3, 4 1, 2, 3, 4

Resonance and Ferroresonance in Power Networks

Select the type of Shunt Reactors depending on magnetic coupling between phases

Phase/circuit transposition

Design the circuit geometry to modify its C0/C+ ratio

Avoid the use of single-phase switching

adequately specify the energy duty of surge arresters across the phase reactors. Three legged core type reactors have strong magnetic coupling which shifts zero sequence resonances to lower degrees of shunt compensation. Single phase, 3-phase shell-type or 3-phase 4/5 legged core reactors will shift zero sequence resonances to lower degrees of shunt compensation. Certain phase transposition schemes can help reducing resonant overvoltages. It is reported in [77] that transposing the energized circuit only (not the de-energized one) can reduce the resonant voltages by a factor of approximately 80. C0/C+ of the shunt compensated circuit is the most critical parameter affecting resonance for a defined size of shunt reactor. This measure will mitigate resonance associated with SPAR, however it does not prevent resonance arising from circuit breaker misoperation or stuck poles.

1, 2, 3, 4

1, 2, 3, 4

1, 2, 3, 4

1

Active Mitigation Option

Install grounding switches to by-pass shunt reactors when the line is de-energized

Install circuit breakers to disconnect the shunt reactors when the line is deenergized

Comment Operational procedures can specify that grounding switches (3-ph) must be closed immediately after line de-energization to deal with Type-2 resonance. Special Protection Schemes can be implemented to trip the line and automatically close the grounding switches if a permanent fault or failure to reclose is detected. A timed overvoltage relay can trigger this remedial action. This mitigation option deals with Type-4 resonance. Have to consider the transients imposed on the reactor when the switch across it is opened Operational procedures can specify that the line shunt reactors must be disconnected before or immediately after line deenergization to deal with Type-2 resonance. Special Protection Schemes can be implemented to trip the line and automatically disconnect the shunt-reactors if a permanent fault or failure to reclose is detected. A timed overvoltage relay can trigger this remedial action. This mitigation option deals with Type-4 resonance.

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Type of Resonance

2, 4

2, 4

Resonance and Ferroresonance in Power Networks

Install Shunt Reactors with tapped windings and on-load tap changers

Install circuit breaker failure and pole discrepancy protection schemes.

Operational procedures can specify that the line shunt reactors must be tapped before or immediately after line de-energization to deal with Type-2 resonance. Special Protection Schemes can be implemented to trip the line and automatically tap the shunt-reactors if a permanent fault or failure to reclose is detected. This action deals with Type-4 resonance. The objective is to quickly disconnect the remaining energized phases in the same circuit in order to remove the excitation in the resonant circuit. Timing must be co-ordinated between SPAR dead-time and TOV withstand of line apparatus. The CB failure protection will guarantee that the energized phase(s)/circuit is disconnected in a timely manner – i.e. removing the excitation to the resonant circuit.

2, 4

1, 3

Install Special Protection Scheme to disconnect the selected phase reactor(s) upon detection of an uneven phase operation.

It requires installation of a circuit breaker with independent phase operation on the shunt reactors.

1

Install Special Protection Scheme to close the selected phase grounding switch(es) in parallel with phase reactor(s) upon detection of an uneven phase operation.

The objective is to by-pass the phase reactor in the open phase(s), therefore removing the resonant circuit.

1

Fast fault clearance

This measure deals with temporary overvoltages coupled to a de-energized circuit during a fault on a parallel energized circuit. Due to the large time constant associated with the resonant circuit, a fast fault clearance (i.e. removal of the excitation) will prevent excessive coupled voltage to develop. An additional benefit is a limitation of the duration of the resonant condition.

3

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Resonance and Ferroresonance in Power Networks

CHAPTER 5

NETWORK CONFIGURATIONS LEADING TO FERRORESONANCE

Field experience and extensive research have shown that for an electrical circuit to exhibit ferroresonance the following elements are needed [12]: Nonlinear inductance Capacitance. Low Losses. An external source of energy The nonlinear inductance can be due to the magnetic core of a voltage transformer or it may have the complex structure of a three-phase power transformer. These are made of saturable ferromagnetic materials, hence the term “ferroresonance”. The quality of these ferromagnetic materials is continuously improving, with a significant reduction in losses, which is a desirable feature from an economical aspect of power system operation. Consequently, power transformers, instrument transformers and reactors are normally specified by the networks owners with the minimum achievable losses, which ironically contributes to an increased risk of ferroresonance. Circuit capacitance can be due to a number of elements, such as line-to-line capacitance or conductor to earth capacitance of long transmission lines, underground cables, circuit breaker grading capacitance, busbar capacitance, bushing capacitance, series capacitors and shunt capacitor banks. Furthermore, significant capacitances are present in GIS. As a result, the suitable conditions for ferroresonance in electrical power systems can arise under many diverse configurations. Identifying ferroresonant conditions in a power system is not a straightforward task due to the complexity of interphase and inter-circuit capacitances in countless possible configurations. Fortunately, studies and field experience have shown that certain power system configurations are more susceptible to ferroresonance than others. The following sections list the most common situations in which voltage and power transformers become involved in ferroresonance. In addition a further section is included to show possible configurations leading to ferroresonance in distribution systems. Although these are normally applicable to utilities, reports of occurrences of ferroresonance have also been reported in IPPs and industrial plants [25] to [27].

5.1 Ferroresonance in voltage transformers (VT) There are two basic types of voltage transformers: capacitive voltage transformers and electromagnetic (wound) voltage transformers. A capacitive voltage transformer (CVT) is essentially a capacitive voltage divider with an electromagnetic transformer connected to the lower capacitor in the capacitance string. It is quite common practice to include a series reactor with the transformer and the capacitance in order to provide improved regulation and to compensate phase shifting introduced by the capacitances. This forms the classical circuit required for ferroresonance initiation. However, in order to minimise occurrences of ferroresonance, manufacturers include ferroresonance suppression circuits on the secondary side of CVTs. There are many ways of introducing this suppression. One option is the installation of a load in series with a circuit tuned to fundamental frequency where the fundamental frequency is blocked from the load. Another option is the installation of saturable reactors where more load is added depending on the voltage level and hence the saturation of the reactor. A third option is to introduce surge suppressors that clamp abnormally high voltage due to ferroresonance. It should be noted that all these ferroresonance suppression techniques can adversely affect the performance of the CVT especially with rapid voltage magnitude and phase angle changes. As CVTs are inherently not prone to ferroresonance due to the suppression circuits employed, they are not considered any further when looking into ferroresonance.

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5.1.1 VT and Circuit Breaker Grading Capacitors This section describes the ferroresonant circuit formed by a set of isolated electromagnetic (inductive) VTs that remain energized through the grading capacitors of open EHV circuit breakers. Grading capacitors are normally installed in EHV circuit breakers to meet high breaking duty requirements by the use of multiple breaking chambers. The use of grading capacitors started in the 1950’s with air blast breakers and has continued through different technologies and interrupting medium. Nowadays, SF6 circuit breakers are almost exclusively installed in EHV transmission systems and the trend in design is to reduce the number of interrupting chambers and to increase the size of the grading capacitors. These capacitors are placed in parallel with each braking chamber to achieve an even recovery voltage distribution. In this case, an oscillation occurs between the non-linear magnetizing inductance of the VT and the circuit stray capacitance. The energy required to sustain the ferroresonant condition is provided through the grading capacitors of open circuit breakers.

5.1.1.1 Busbar VTs VT ferroresonance can occur when a busbar is de-energized by opening the circuit breakers in all the feeders. This can be the result of clearing a busbar fault or a maintenance outage. The necessary conditions for ferroresonance to occur are: Presence of inductive VTs in the isolated busbar section. The feeder circuit breakers (all or some of them) are equipped with grading capacitors. This configuration is shown in Figure 5-1 (a) with the reduced equivalent circuit in Figure 5-1 (b). The capacitance Cg represents the grading capacitance of each open circuit breaker and n is the number of line bays. Cs represents the capacitance to ground of the disconnected section of busbar. R and L represent the VT losses and magnetizing inductance respectively.

Figure 5-1 (a) Network configuration leading to busbar VT ferroresonance and (b) the Equivalent Circuit

5.1.2 Line VTs VT ferroresonance can occur when line VTs are de-energized. The necessary conditions for ferroresonance to occur are: The line circuit breaker is equipped with grading capacitors The bus disconnector is closed and the line disconnector is open Presence of inductive VTs between the circuit breaker and the line disconnector. This configuration is shown in Figure 5-2 (a) with the reduced equivalent circuit in Figure 5-2 (b). The capacitance Cg represents the grading capacitance of the circuit breaker, Cs represents the capacitance to ground of the disconnected line bay, R and L represent the VT losses and magnetizing inductance, respectively. Ferroresonance

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Resonance and Ferroresonance in Power Networks may occur as a result of a line de-energization by opening the circuit breaker with the bus disconnector DA closed and DL open. (a)

Busbar

DA

CB

P1

CT

P2 VT

DL

Line

Cg DE

DEM Cg (b)

CS

E

R L

Figure 5-2 (a) Network Configuration leading to Line VT ferroresonance and (b) the Equivalent Circuit

5.1.3 VT and Double Circuit Configuration This type of ferroresonance can arise when two high voltage lines are strung on the same pylons as a doublecircuit line construction. Ferroresonance occurs when one of the lines is disconnected while the other one remains energized as shown in Figure 5-3 (a). The reduced equivalent circuit is shown in Figure 5-3 (b) where capacitance CC represents the coupling with the energized circuit, CS represents the capacitance to ground of the disconnected line bay, R and L represent the VT losses and magnetizing inductance, respectively. The ferroresonant condition is sustained by the capacitive coupling with the energized circuit. A ferroresonant incident in this topology has been experienced in the transmission network in Belgium [28]. It has been reported in [21] that the risk of this type of ferroresonance is almost negligible when the length of coupling of the two lines exceeds 5 to 10km.

5.1.4 VT in Ungrounded Neutral S ystems with Low Zero-Sequenc e Capacitance This type of ferroresonance condition has been mainly observed in distribution systems. A typical network topology is shown in Figure 5-4. This topology comprises a set of inductive VTs connected between phases and ground in a system with no direct neutral grounding. The VTs are in parallel with the zero sequence capacitance of the distribution network. In general this phenomenon occurs in unloaded system (connection of a load normally increases the zero-sequence capacitance of the network), however, transient disturbances arising from faults or switching operations can drive the VTs into saturation and, depending on the relative values between network capacitance and VT magnetising reactance, ferroresonance can be initiated. This is a parallel ferroresonant event. Due to the ungrounded neutral connection, only phase-to-phase voltages are fixed. During ferroresonance, the neutral point is displaced and the voltage on one or two phases rises with respect to ground, giving the false impression of single phase to ground fault. This neutral displacement is illustrated in Figure 5-5. The resulting phase to ground overvoltages can cause excessive stress on the equipment insulation. A very interesting incident experienced in a 50kV network in Norway is reported in [14]. In this case, the clearing of a fault disconnected the only source of grounding in the 50kV system. As a result of this event, 72 VTs were destroyed. All VTs damaged by ferroresonance were from the same manufacturer, while VTs from two other manufacturers, also in service, and not damaged. An investigation into the ferroresonant incident revealed that the saturation characteristics of the three VT types were very different and that the damaged VTs had a significantly lower saturation point. The early saturation of these VTs and the high magnetising currents drawn while in 2 ferroresonance caused excessive I R losses in the windings, which thermally destroyed them. This incident drove a re-assessment of the specification and application of Voltage Transformers.

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Resonance and Ferroresonance in Power Networks

Figure 5-3 (a) Network Configuration leading to Double Circuit Line VT ferroresonance and (b) the Equivalent Circuit

Figure 5-4 Network Configuration leading to Ungrounded VT Ferroresonance The risk of ferroresonance is highly dependent on the characteristics of the distribution network therefore it is difficult to quote, in general terms, the conditions under which ferroresonance can exist. However, as a rule of 4 thumb, the following formula can be used [38]. It can be estimated that there is a risk of ferroresonance when the connected zero sequence capacitance expressed as s km of overhead line circuit is

s<

42000 [km ] U

Eq. 5-1

where Un is the system nominal voltage expressed in kV.

4

This formula is applicable to normal voltage transformers with the ratio

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¦

.

¦

.

kV

Resonance and Ferroresonance in Power Networks A similar ferroresonance condition can occur on a slightly different configuration. This is shown in Figure 5-6 (a) with ferroresonance occurring between the voltage transformer and the capacitance of a supply transformer between high voltage and medium voltage sides. The HV side neutral is insulated from earth, although the HV side is earthed at other points. The MV side, also insulated from earth, is connected to a set of three voltage transformers but does not supply any load. Following an earth fault on the HV side of the transformer, its HV neutral potential can be raised temporarily to a high voltage. The circuit formed by this voltage En (Figure 5-6 (b)), the coupling capacitance Cn and, on each phase, the zero-sequence capacitance Co in parallel with Xs can thus be brought to a state of ferroresonance giving rise to an overvoltage on the MV side. After clearance of the fault on the HV side, ferroresonant conditions may be maintained by the normal voltage existing on the HV neutral point.

Figure 5-5 Neutral Displacement and Phase-Ground Overvoltages during Ferroresonance

(a)

(b)

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Resonance and Ferroresonance in Power Networks Figure 5-6 (a) Configuration for VT ferroresonance due to HV/MV capacitance and (b) Simplified circuit

5.2 Ferroresonance in power transformers 5.2.1 Transformer Terminated Transmission Li ne in Multi-Circuit Right of Wa y A ferroresonant circuit can be formed when a transformer terminated transmission line is de-energized while there is an energized circuit running in parallel in the same right of way. A typical topology is illustrated in Figure 5-7 (a). It shows a power transformer directly connected to a transmission line in “Substation 1”. Circuit breakers are only installed on the LV side of the transformer in “Substation 1” (CB1), whereas the HV side circuit breakers are installed in a remote location, at “Substation 2” (CB2). The transmission line between “Substation 1” and “Substation 2” (cct A) shares a right of way with other transmission line(s) (cct B). Ferroresonance can occur after opening circuit breakers CB1 and CB2 if the parallel circuit remains energized. In this case, an oscillation occurs between the non-linear magnetising inductance of the transformer and the capacitance of transmission line it is connected to. The energy required to sustain the ferroresonant condition is provided from the capacitive coupling with the parallel energized circuit. This topology can occur when circuit breakers are not installed on both sides of a transformer, for instance to reduce costs in the early stages of system development. It should be noted that even when circuit breakers are installed on both sides of the transformer, this topology can still arise if a circuit breaker fails to trip after a transformer fault and the back-up protection trips the circuit breaker at the remote end of the transmission circuit. Ferroresonance has also been reported to occur following uneven breaker pole operation (i.e. stuck circuit breaker pole). A similar network topology that has been reported to exhibit ferroresonance is a “tee” or “tapped” step-down transformer as illustrated in Figure 5-7 (b) where the transformer is tapped from one of the two parallel circuits. A simplified equivalent circuit is shown in Figure 5-7 (c), where CA represents the capacitance to ground of Cct A, CA-B represents the inter-circuit capacitance between Cct A and Cct B, R represents the circuit losses and L represents the transformer magnetising inductance. It should be noted that this simplified equivalent circuit is only valid for banks of single phase transformers, where there is no coupling between phases. Otherwise, a three-phase equivalent circuit must be used. Notwithstanding the simplification, the equivalent circuit assists in the visualisation of the key parameters determining the likelihood of ferroresonance in this topology. Various modes of ferroresonance, typically fundamental frequency and sub-harmonic ferroresonance can arise depending on the circuit parameters, initial conditions or switching sequence. Field measurements of a 150km, 400 kV double circuit line terminating in a 500MVA transformer are described in [29]. This reference work relates the length of the parallel coupling with the type of ferroresonant modes. Very short lines (i.e. less than 20km) do not experience ferroresonance. Fundamental frequency ferroresonance is observed for medium line lengths, while sub-harmonic ferroresonant oscillations are dominant in long circuit lengths (i.e. >150km). The result of the oscillations is normally an overvoltage stressing the line and transformer insulation and transformer overflux causing excessive heating and possible damage to the core and windings. It should be noted that no destructive failure of a power transformer due to ferroresonance has been reported to date, however sustained operation of a transformer in this condition is a main contributor to its accelerated aging.

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Resonance and Ferroresonance in Power Networks

Capacitive coupling with parallel energized circuit

Capacitive coupling with parallel energized circuit

Figure 5-7 Transformer terminated transmission line

5.2.2 Lightly Loaded Transformer Energized v ia Cable or Long Line from a Low Short-Circuit Capacity Net work A ferroresonant circuit can be formed when a large power transformer is energized from a weak source via a long overhead line or cable. This is a parallel ferroresonant topology where an oscillation arises between the non-linear magnetising inductance of the transformer and the capacitance of transmission line or cable it is connected to. The voltage source is located behind a large inductive source impedance. A typical network topology and its equivalent circuit are shown in Figure 5-8. This type of network topology can arise as a result of unusual switching operations Page 94

Resonance and Ferroresonance in Power Networks such as during emergency switching operations or system restoration following a blackout. It is best avoided by proper system design and planning. An example of ferroresonance in this topology was experienced in France during a black-start restoration test where two 1080 MVA transformers were energized from a 90 MW hydro unit via a 360 km long overhead line [30]. Sustained non-periodic ferroresonant oscillations have been recorded in this incident.

Figure 5-8 Transformer energized from weak source via long transmission circuit

5.2.3 Transformer energiz ed in one o r two ph ases Power transformer ferroresonance in this topology is more common in distribution systems due to the use of singephase switching or the deployment of fuses. The system neutral can be either grounded or ungrounded, although ungrounded neutrals are more susceptible to the phenomena. It is a series ferroresonant phenomenon with the voltage source coming from the back feed voltage induced on the open phase of the transformer (this is highly dependent on the core construction) as illustrated in Figure 5-9 for typical topologies. The non-linear inductance is a combination of the individual limbs in the transformer core and the capacitance comes from the connected network. It can be seen that series connections of a capacitance with the transformer’s reactance are possible when one or two phases are disconnected. This can be the result of fuses blowing, sequential operation of singlephase switches or accidental break in a phase conductor. It should be noted that the capacitance involved in this series ferroresonant circuit is not necessarily just the feeding network between the transformer and the open switch/fuse but it can also include other circuits still connected to the transformer terminals, such as capacitor banks and the stray capacitance of the transformer winding. This concept is illustrated in Figure 5-10 where for instance opening of one phase in S1 can leave a significant amount of circuit capacitance connected in series with the reactance of TX1 and TX2. It should be noted that, in the case of a capacitor bank, if both neutrals (cap bank and transformer) are grounded or both are ungrounded, then no series path exists and there is no clear possibility of ferroresonance. The result of this series ferroresonant phenomenon is typically an overvoltage of 2 to 3 p.u. on the transformer windings and on the feeding network. This overvoltage imposes stress on the transformer insulation and on any feeding cable. The following three conditions must be met for ferroresonance to occur: At least one phase must be energized. Sufficient capacitance must be present between the transformer and the open point. The loading and losses in the transformer must be low.

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Resonance and Ferroresonance in Power Networks The maximum length of MV cable that can be connected to a distribution transformer energized in one or two phases can be calculated using the following formula [39]:

L

=

0.6 I C 1.58 + C

%

KVAr 1000

62.8 (kV )

C

[m ]

Eq. 5-2

where Lcritical is the critical cable length in meters, Imag% is the transformer magnetising current (typically in the order of 0.8%), KVAr is the transformer rating in kva, kVR is the rated voltage in kV, C’CC is the cable’s core to core capacitance in F/km and C’CS is the cable’s core to sheath capacitance in F/km.

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Resonance and Ferroresonance in Power Networks

Figure 5-9 Transformer accidentally energized in one or two phases

Figure 5-10 Distribution Network Topologies Prone to Ferroresonance in single-phase switching

5.2.4 Transformer connected to a series compensated line The installation of series capacitors in long transmission networks for voltage regulation purposes can increase the risk of ferroresonance in certain configurations. A load rejection on a long transmission/distribution line with series compensation could initiate ferroresonance. This is a series ferroresonant circuit. If the load rejection was caused by opening the low voltage side circuit breakers, the temporary overvoltages would cause transformers to operate into the saturation region. With series compensation in-service, this line open condition could initiate a ferroresonance condition that would stress dielectric and thermal withstand capability of equipment. A typical topology is illustrated in Figure 5-11. An example is described in Appendix B4. Ferroresonance results in large voltages and currents in the capacitor and the transformer.

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Resonance and Ferroresonance in Power Networks

Figure 5-11 – Series Compensated Distribution Circuit

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Resonance and Ferroresonance in Power Networks

CHAPTER 6

MODELLING AND STUDYING

Due to the nonlinear nature of the ferroresonant phenomenon, analytical solutions are not simple to implement. Several approaches as highlighted in the next section have been adopted in the past with significant limitations. Time-domain digital simulation has emerged as the most powerful tool for the description of the phenomena in multiphased networks. However, the use of non linear tools, complementary to the EMT, has been very helpful in order to understand the phenomena involved and especially the main variables of interest. In general EMT type simulation tools provide explicit representation of nonlinear elements and facilitate the modelling of complex circuit or substation configurations to assess whether or not ferroresonance can occur. It should be noted that due to the sensitivity of the phenomenon to the circuit parameters and initial conditions, a large number of simulations is required to assess the likelihood of ferroresonance in a particular network topology or to gain confidence in a particular mitigation scheme. Analytical solution methods associated with linear resonance were covered in Chapter 4 and hence it is not repeated in this chapter.

6.1 Analytical Solution Methods Different analytical solution methods have been applied to circuits exhibiting ferroresonant behaviour. The most widely applied method is based on the principle of harmonic balance which is a special case of the Galerkin method in which the state equations are represented by a Fourier series. Although analytical methods are extremely powerful in identifying fundamental frequency ferroresonant regimes for a given circuit configuration, the identification of sub-harmonic ferroresonance is quite difficult due to the complexity of the mathematical derivations. The Galerkin method, based on the search of the harmonic solutions of a given network, is not capable of identifying non-periodic or chaotic solutions which can occur in ferroresonant circuits [31], [33], [35] and [36].[87] Harmonic balance method has been utilised successfully by Janssens et al [40] to calculate directly the existence and stability zones of ferroresonant oscillations on a 245kV electromagnetic voltage transformer. Likewise the same method of analysis has been used in [41] to study the failure of an electromagnetic voltage transformer when the 220kV busbar was de-energized by a circuit breaker fitted with grading capacitors across its contacts. Others [42] have tried to use the method when looking to provide certain criteria for the stability of sustained sub-harmonic ferroresonance but omitting higher harmonics in their calculations which resulted in discrepancies between their experimental and calculated results. The pseudo-arc-length continuation method as an addition to the Galerkin method was suggested by Kieny [43], [44]. Using the two methods in conjunction they were able to predict all periodic solutions including sub-harmonic and higher harmonic cases but they failed to identify any pseudo-periodic or chaotic solution.[88], [89] Ritz's method of harmonic balance was used in [45] to formulate an analytical solution to the problem of fundamental frequency ferroresonance. Using this method it was possible to map the boundaries between safe and fundamental ferroresonance regions as a function of the system parameters. The second type of analytical solution utilises the ‘incremental describing function method’ usually applied to nonlinear control systems to examine transformer ferroresonance. The method is based on the use of a dual input describing function to analyse nonlinear systems [46]. It involves evaluating the gain of one of the frequency components in passing through the nonlinear element when the input to the element consists of two sinusoidal waves of differing amplitudes, frequencies and phases. When applied to ferroresonance, the nonlinear element represents the transformer saturation characteristic and the two sinusoidal inputs are the system voltage applied to the transformer, and the disturbance, such as a temporary overvoltage. The primary advantage of this method is that it allows the critical jump values to be directly calculated. Since its introduction for application to transformer ferroresonance [47] the method has been subjected to considerable investigation [48] to [51]. As an analytical tool the method is very powerful but it is not very easy to implement and requires very accurate system data to produce useful results, especially in the case of complicated power system networks.

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Resonance and Ferroresonance in Power Networks It is worth mentioning that Andrei and Halley [52] developed a different analytical approach to determine whether ferroresonance can occur, based on the energy transferred from the system to the voltage transformer during the switching transient.

6.2 Digital Simulation Methods Since analytical solution methods are not capable of predicting non-periodic ferroresonance occurrences and even sometimes sub-harmonic ferroresonance, researchers have used different digital simulation techniques including EMT analysis and chaos theory to analyse ferroresonant circuits. It was Kieny [30], [44] who first suggested applying chaos to the study of ferroresonance in electric power systems by studying the possibility of ferroresonance in power transformers, particularly in the presence of long capacitive lines as highlighted by occurrences in France in 1982, and produced a bifurcation diagram indicating stable and unstable areas of operation. Kieny [53] was also able to present a non-periodic (pseudo-periodic), but not chaotic waveform, using Lyapunov exponents and was able to prove the stability of this waveform and postulated that the bifurcation theory was the right mathematical framework to study ferroresonance phenomenon. A fourth-order Runge-Kutta formula was introduced [33] to solve numerically the differential equation of a basic ferroresonant circuit consisting of a capacitor in series with a nonlinear inductor. The same circuit was later used [54] in a different software toolkit package [55] to solve the nonlinear equations of the circuit and produced bifurcation diagrams with source voltage E being the bifurcation parameter for different exponent of the nonlinearity. Nonlinear dynamical techniques were used [56] in an attempt to establish a methodical approach for identifying all possible initial conditions and, consequently, the different types of ferroresonant oscillations that can occur in a capacitor voltage transformer. EMT type program was used [57] to model the behaviour of an 8.33 MVA power transformer with construction of Poincaré maps, phase plane trajectories, frequency spectrum and bifurcation diagrams. An add-on computer simulation was developed [58] which was subsequently incorporated into an EMT type software to study the problem of ferroresonance in a three-phase five-legged grounded wye-wye power transformer to reveal different types of ferroresonant wave-forms. EMT type software was also used in [15] to simulate the conditions present during a voltage transformer failure concluding that the method provided an accurate and inexpensive means to simulate the potential for ferroresonance. Modelling nonlinear dynamical systems generally leads to a set of differential equations where some control parameters influence explicitly the solution type. That can be the voltage supply of an electric circuit, the flux circulating in the iron core of the transformer, the temperature of a chemical reaction or the gain of a feedback system. When critical values of the parameters induce an abrupt change in the type of solution, there is a bifurcation for the system. Two main aspects have to be addressed when using such approaches, linked to the study of dynamic systems: a) Simplification of the electrical system, from a real multiphase network, to a set of ODEs, describing the main characteristics of the network, especially the source, the resonant circuit, including its non linearity, and the dissipative elements. b) Searching for solutions of the system (harmonics, pseudoperiodic, chaotic) using adaptive numerical methods when varying the main state variables.

The following sections briefly describe the different mathematical tools used in the study of dynamical systems.

Phase Space The phase space of a dynamical system is a mathematical space in which the instantaneous state of the system is represented by the movement of a point representing the state variables of the system. As time evolves, the initial state point follows a trajectory which closes onto itself if the response is periodic and is called a cycle.

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Resonance and Ferroresonance in Power Networks For a chaotic solution, the phase space will have a very complex trajectory never closing onto itself and is called a strange attractor.

Poincaré Section A Poincaré section is a tool invented by Henri Poincaré as a means of simplifying phase space diagrams of complicated systems. It is constructed by recording the phase space trajectory as a sequence of discrete points at constant time intervals. If this sampling is done at intervals corresponding to the system’s forcing frequency (in the case of ferroresonance the power frequency) then, for a periodic waveform, with the same frequency as the forcing function, the Poincaré section will show only one point. Likewise a sub-third harmonic waveform will produce three points. However a chaotic waveform will produce a Poincaré section with a random set of points confined to a particular region of the plane as can be seen in Figure 6-1. For a dynamical system such any ferroresonance configuration, the Poincaré section provides a simplification of the phase space diagram while retaining the essential features of the dynamics.

Bifurcation Diagram Phase space diagrams and Poincaré section provide information about the dynamics of the system for specific parameter values. The dynamic behaviour may also be viewed more globally over a range of parameter values, thereby allowing simultaneous comparison of regions of periodic and chaotic behaviour. A change in the type of solution to a set of ordinary differential equations when a parameter is varied, is called a bifurcation. A bifurcation diagram provides a summary of the essential dynamics and is therefore a useful method of acquiring this overview. It is an important tool for discovering interesting parameter regimes for a dynamic system. A bifurcation diagram is actually a collection of many Poincaré sections each calculated for a different value of a particular parameter in the system. A typical bifurcation diagram for a ferroresonance circuit is shown in Figure 6-2.

1.2

derivative of state variable

1

0.8

0.6

0.4

0.2

-2.7

-2.65

-2.6

-2.55

-2.5

-2.45

-2.4

-2.35

-2.3

state variable

Figure 6-1 Poincaré section of a chaotic waveform

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-2.25

-2.2

0 -2.15

Resonance and Ferroresonance in Power Networks 1.5

1

state variable

0.5

0 0

20

40

60

80

100

120

140

160

180

-0.5

-1

-1.5

bifurcation parameter

Figure 6-2 Bifurcation diagram for ferroresonance circuit

6.3 Modelling of Network Co mponents 6.3.1 Extent of the Network Model Ferroresonance is a localised phenomenon and, as a general rule, large network models are not necessary. Only the main elements directly involved in to ferroresonant circuit (i.e. non-linear reactance, capacitances and voltage source) need to be represented in detail. Therefore both in ferroresonance and resonance studies the feeding network can be represented as a Thevenin source equivalent calculated at power frequency.

6.3.2 Overhead Line Model An accurate representation of the line parameters at resonant and near-resonant frequencies is essential. The most critical elements that need to be reproduced accurately are the circuit capacitances, therefore the model has to be based on accurate geometrical configuration of conductors at the towers and along the spans. Circuit phase transpositions, if present, need to be represented explicitly. The frequency dependency of the model parameters is not critical since the phenomena of interest is resonance at (or near) power frequency. A multi-phase distributed parameter model, such as Bergeron, calculated at power frequency normally yields reasonable results. A frequency dependant model like JMARTI can also yield accurate results when the transformation matrix is calculated at (or near) power frequency. Corona losses can reduce the amplitude of resonant overvoltages when the critical corona onset voltage is exceeded. These losses are dependent on a large number of random variables, atmospheric conditions among them. Although there is significant literature dealing with corona losses under normal voltage operating conditions, there is very limited published experimental data on corona at power frequency above critical voltage [72]. In practical terms, corona will only contribute to the attenuation of Temporary OverVoltages and most studies tend to ignore these losses to add a safety margin to the computed results.

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Resonance and Ferroresonance in Power Networks Depending on the ferroresonant topology, it may be necessary to represent overhead lines in detail, for example in cases where energy is coupled from an energized parallel circuit or, alternatively, to assess the detuning effect of switching a long circuit. Given that ferroresonance is a low frequency phenomenon, frequency dependency is not a critical feature of the line model. Therefore, a multi-phase distributed parameter line model like Bergeron is sufficient for most ferroresonant studies. The exception is when ferroresonance is initiated by a line switching transient where a frequency dependant model of the circuit under study is recommended. Other circuits not involved in the switching can be represented with lumped or distributed parameter models. If inter-circuit capacitances are part of the ferroresonant circuit, both parallel circuits must be represented in detail using accurate tower geometry. Phasetransposition, if employed, must be explicitly modelled. If the overhead line employs any series capacitors, these can be modelled as lumped capacitive elements connected in series between two line sections. Special attention must be paid if frequency dependant line models are used in order to avoid non-passivity at low frequency (from 0 Hz to 100 Hz). This problem has been reported in [66]. When excited at low frequencies, these models “create” power due to apparent negative resistivity. The situation can be dangerous when dealing with resonance and ferroresonance because non-passive line models can give incorrect results that can be confused with regular ferroresonant shapes. A general procedure to check passivity in a frequency dependent line model is to calculate the admittance matrix [Y] and check that the eigenvalues of the real part of [Y] are positive.

6.3.3 Transformers Power transformers must be represented with a three-phase model in order to reproduce correctly the coupling between phases. Voltage transformers on the other hand can be represented with single-phase models, with the secondary and tertiary winding connections represented externally to the model. Transformer stray, bushing and inter-winding capacitances can be represented as lumped elements in parallel with the appropriate windings. The nonlinear behaviour of the magnetic core of the transformer is the most critical aspect of the model and therefore correct representation of the saturation effects along with losses is a key factor for the accuracy of the simulation results. Some transformer models available in commercially available EMT type software packages do not support inclusion of this data. In such cases the magnetic core data has to be represented externally. Some models include the hysteretic behaviour of the magnetic core including losses where this is normally the area inside the hysteresis loop. Normally, it is difficult to implement this type of model due to unavailability of data to the user. To overcome this, some models employ a single-valued representation of a nonlinear inductor with a damping resistor added externally to account for the losses. The nonlinear inductor provides a smooth computation since flux is the integral of voltage and performs satisfactorily provided that the curve is not defined by too many segments. In some stand alone cases, a Preisach type mode [59] or a Preisach-Biorci-Pescetti hysteresis model [60] has been utilised to include the hysteresis effect in the study of ferroresonance. It should be mentioned that although this type of saturation modelling provides very good results it has the drawback of being extremely difficult to implement due to the unavailability of data without specialised equipment testing. The location of the saturation curve is also important in three phase power transformers. To obtain reasonable results, the saturation curve must be represented in parallel with the closest winding to the magnetic core (this is normally the LV winding). This approximation gives accurate results for frequencies below 1 kHz. The magnetic core losses are critical in any simulation involving saturation. Various representations can be found in the literature: hysteresis loop, non-linear resistor and linear resistor. The advantages and limitations of each representation are discussed in [14], [61], [62]. Those technical publications conclude that the most accurate representation is the hysteresis reactor; however the parameters required for its developments are normally not available to the user and involve special testing. The non-linear resistor representation, on the other hand, can have serious limitations since hysteresis losses depend on flux and not voltage. A linear resistance is the most common representation for the magnetic core losses. It is reported in [14] that this core loss representation, if it represents the average losses at the level of excitation being simulated, yields reasonable results.

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Resonance and Ferroresonance in Power Networks For more detailed information on the modelling of transformers for this type of studies the reader is advised to refer to CIGRE TB on Transformer Energization – A Study Guide.

6.3.4 Shunt Reactors The core construction has a large effect on the reactors’ behaviour during unbalanced open-phase conditions. The only type of reactor that does not have direct magnetic coupling between phases is the single-phase unit. Threephase reactors present different levels of zero sequence coupling depending on the core design. The shell-type and four/five-legged core-type reactors provide a magnetic path for the zero sequence flux, hence, the coupling between phases is very small and can normally be neglected. The three-legged core construction, however, presents a strong magnetic coupling between phases, which must be accurately represented when this reactor type is used for line shunt compensation. Shunt reactors are normally specified to remain linear up to a knee point of around 125% to 175% of nominal voltage. For a preliminary analysis of line resonance, the shunt reactors can be represented as linear lumped elements. However, operation near a resonant peak may drive the reactor into saturation and initiate a ferroresonant oscillation. Therefore, a detailed analysis of the circuit must include reactors’ saturation. Reactor losses affect the amplitude of near-resonant overvoltages. Typical quality factors for modern reactors built with low loss materials are in the order of 1000. This can be represented by a lumped resistor connected in series with the reactor.

6.3.5 Other Substation Equipment The main aspect of the substation equipment that needs to be modelled accurately is the capacitive component. The capacitances of all pieces of plant contributing to the ferroresonant circuit need to be represented accurately. These elements include the busbars, bay conductors, disconnectors, current transformers, voltage transformers, surge arresters, circuit breakers, power transformers, shunt capacitors, etc. Series capacitances are of major importance when the capacitance is connected to the target transformer; i.e. the case of series ferroresonance when the series capacitance is formed by the capacitances between the lines of a double circuit line, with the first line being energized, and the second one is de-energized with the transformer connected to it. A distinction is made between capacitances to ground of conductors and stray capacitances of plant apparatus:

Capacitances to Ground of Busbars and Bay Conductors An accurate model of the busbars and bay conductors is required. These conductors must be modelled based on the geometrical layout of the station. Lumped parameter line models are normally adequate for busbars or typical conductor lengths within the station unless very long busbars are involved, in which case distributed models may be employed. However, this approach restricts the size of the integration time-step and increases the computation time. While this restriction may not be important for one single simulation, it may be impractical when carrying out a large number of parametric analysis simulations. In practice, for normal lengths of conductors within a substation, a lumped parameter model (i.e. multi-phase “pi” representation) calculated at power frequency yields adequate results.

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Resonance and Ferroresonance in Power Networks

Stray Capacitances of Apparatus The stray capacitance of each apparatus involved in the ferroresonant circuit can be represented as lumped elements connected between the phase conductors and ground. Accurate values of these capacitances must be obtained from the equipment manufacturers and test reports. In addition to the above, certain equipment may need to be modelled explicitly, as follows: Surge arresters may need to be represented if there is a concern about their energy absorption capabilities being exceeded during the ferroresonant oscillations. Surge arresters can also influence the ferroresonant oscillation mode [85]. Current transformers and PLC line traps do not need to be represented. Circuit breakers can be represented as ideal switches. If the circuit breakers are equipped with grading capacitors, these need to be represented in parallel with the switch. Capacitor banks can be represented as lumped capacitive elements with the appropriate connections (i.e. wye or delta).

Circuit Breakers Circuit Breakers can be represented as ideal time-controlled switches. Circuit breaker grading capacitors should be represented explicitly as a parallel capacitance across the ideal time-controlled switch. Stray capacitances of the circuit breaker are of major importance when the phenomena appearing at the transformer de-energization are of importance. In particular, they have an impact on the final value of the residual flux remaining in the iron core of the transformer following the opening of the circuit breaker poles.

6.4 Sensitivity to Para meters 6.4.1 Effect of Magnetising Curve The saturation curve is one of the key parameters for the occurrence of ferroresonance, whether the non linearity is pronounced or not. One of the key components of the saturation curve is the slope in the fully saturated region normally referred to as Lsat. This is equal to Lair, the air core inductance of the transformer, minus the inductance corresponding to stray losses. Value of Lair can be estimated with a very good accuracy either by analytical formulas [64] or by 3D electromagnetic calculations [65], assuming that the relative permeability of the iron core is equal to 1. Figure 6-3 shows three measured magnetizing curves for three different VTs all used in a 400 kV network. They are presented in the same scale, to demonstrate differences between them. Figure 6-4 shows voltage - frequency dependencies for the above VTs for capacitances of 1 and 10 nF for a voltage range of 0 to 1600 kV based on EMT type software simulation. VT3, with the highest magnetisation knee point, connected to a capacitor of more than 5 nF is not able to oscillate on the fundamental power frequency for realistic levels of voltages, so fundamental frequency ferroresonance is impossible for this particular VT. On the contrary VT1, with its easy saturability, can exhibit ferroresonance for a wide range of frequencies.

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VT3

4500

4000

3500

VT2 3000

3000

2500

2500

2500

2000

2000

2000

1500

1500

1500

1000

1000

1000

500

500

500

VT1

0

0 0

0,2

0,4

0,6

0,8

1

0

0

0,2

0,4

0,6

0

0,8 1 Current [A]

Current [A]

0,2

0,4

0,6

0,8

1

Current [A]

Figure 6-3 Magnetizing curves (Wb - I) for three different 400 kV VTs

C=1 nF

C=10 nF

180

140

VT3

160

VT3

VT2

VT2 VT1

VT1 100

120

Frequency (Hz)

Frequency (Hz)

140

120

100 80

80 60

60 40

40 20

20

0

0 0

200

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600

800

1000

1200

1400

1600

0

200

400

600

800

1000

1200

1400

1600

Voltage (kV)

Voltage (kV)

Figure 6-4 Frequency of oscillations as a function of initial voltage for three types of VTs Ferroresonance is a highly nonlinear phenomenon which is very sensitive to the circuit parameters and initial conditions for the transformer and the power system. The choice of representation for the magnetising curve was investigated using EMT type software and is illustrated below. For this analysis, two approaches were compared: (i) piecewise linear representation and (ii) two – term polynomial curve representation. The data used in this analysis is shown in Figure 6-5. Both saturation curves were tested on the same network model for the two typical ferroresonant configurations described below. In both cases, the simulation results were dependent on the selected representation for the magnetising curve. These two examples illustrate the high sensitivity of ferroresonance to small variations in the circuit parameters or initial conditions.

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Figure 6-5 Current-Flux magnetising curve of a transformer approximated with (a) piecewise linear model and (b) polynomial curve model Case 1: Transformer energized through grading capacitance of circuit breaker The circuit shown in Figure 6-6 is assumed to operate in a steady state no-load condition and a temporary threephase short circuit fault is applied to the secondary side of the transformer. The simulation results are shown in Figure 6-7 (a) and (b). Based on the piecewise linear magnetization characteristic, the simulated voltage show a normal operating condition for the transformer voltage whereas in case of the polynomial saturation curve, the transient state is followed by a fundamental mode of a ferroresonance oscillation.

Figure 6-6 Fault clearance leaving transformer energized through the grading capacitance of a circuit breaker

a) Piecewise linear saturation curve

b) Polynomial saturation curve

Figure 6-7 Simulation of fault clearance leaving transformer energized through the grading capacitance of a circuit breaker (a) normal response using piecewise linear representation of saturation curve (b) ferroresonant response using a polynomial representation for the saturation curve Case 2: Transformer connected to a double circuit transmission line The configuration shown in Figure 6-8 is very well documented to be favourable for ferroresonance. The simulation results are shown in Figure 6-9 (a) and (b). It can be seen that, in this simulation test, the piecewise linear (a)

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Resonance and Ferroresonance in Power Networks characteristic results in a dangerous ferroresonance overvoltage, whereas the polynomial saturation curve (b) does not show any overvoltage.

Figure 6-8 Line disconnection leaving transformer energized through the coupling with parallel circuit

Figure 6-9 Simulation of line disconnection leaving transformer energized through the coupling with parallel circuit (a) ferroresonant response using piecewise linear representation of saturation curve (b) normal response using a polynomial representation for the saturation curve It should be noted that the polynomial function representation has the problem that it has only three parameters and has therefore limited flexibility to accurately represent the core nonlinearity in the linear part around the knee point and in the saturation region. An inaccurate magnetization characteristic can result in erroneous ferroresonance simulations; therefore it is not very useful for ferroresonance analysis.

6.4.2 Influence of Circuit B reaker Closing Tim es The closing times of a circuit breaker are a key parameter for the suppression of ferroresonance; the reason is mainly due to the fact that the major state-variable in the electrical circuit is the flux circulating in it, which is constituted mainly by the magnetic flux circulating in the iron core of the transformer. It may also be shown, in the case of discrepancies in the closing times for circuit breaker poles, that the phenomena may be generated due to it, and stopped when re-synchronizing the poles of the circuit breaker which operates.

6.4.3 Influence of the Damping in the Circuit Damping factors are of major importance. In other parts of physics, dealing with non linear systems, especially in chemistry, dissipative aspects are of major importance, linked to thermodynamic aspects, entropy [17], which characterises the balance between the energy brought into the system (source as an input), and the dissipative parts (losses of the system). In electrical network, when the energy is injected in the electrical system from one side, through the up-stream network, the resonance phenomena may be damped through the dissipative components.

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Resonance and Ferroresonance in Power Networks

CHAPTER 7

MITIGATION OF FERRORESONANCE

Ferroresonance can happen at any voltage level with the appropriate combination of capacitance, non-linear inductance and low losses. The first line of defence against its harmful effects is an understanding of the phenomena so that the risky network conditions can be avoided.

7.1 Mitigation of VT F erroresonance A wide range of remedial actions have been used worldwide for the mitigation of VT ferroresonance in EHV substations. These are classified into two main groups: a)

Passive Methods: To prevent the occurrence of network conditions likely to result in ferroresonance.

b) Active Methods: To allow the network conditions that could lead to ferroresonance to exist but to introduce losses to quickly dampen out any harmful oscillation. The “passive mitigation methods” comprise the design of the physical layout of substations to minimise inter-circuit couplings, the installation of VTs in the line side of the disconnectors, the selection of inductive VTs with lower flux density, the use of capacitive VTs and the implementation of switchgear interlocking or revised switching procedures. Passive methods provide the best protection against ferroresonance. However, even when an installation is carefully planned with the risk of ferroresonance in mind, the number of unusual circuit configurations that may arise during commissioning or maintenance procedures is countless. Furthermore, substation refurbishments or expansions may modify the circuit capacitance, leading to an increased risk of ferroresonance. The example of a substation in Canada is reported in [13] where, due to circuit breaker upgrades, the circuit capacitance drastically increased after several years of equipment upgrades and resulted in the destruction of a 230kV inductive VT. The “active mitigation methods” comprise the use of resistors connected to “Wye” secondary windings, the use of resistors connected across open-delta windings, the use of air-core reactors connected in parallel with the HV winding, the use of a series combination of a resistor and a saturable reactor in the secondary winding, and the closing of grounding switches upon detection of ferroresonance. A combination of passive and active mitigation methods will provide an optimum protection against ferroresonant overvoltages. Given the low cost of most of these methods and the severe consequences of ferroresonance, the combined application of active and passive mitigation options is recommended as the most cost-effective solution.

7.1.1 Secondary Open Delta Resi stor This is the most widely used mitigation method due to its simplicity. The connection of a damping resistor to the open-delta winding configuration is shown in Figure 7-1.

V

Rdamping-

Figure 7-1 Connection of damping resistor to the VT open-delta secondary winding

This is an “active method” since ferroresonance is mitigated by the introduction of resistive losses. The advantage of this connection is that the required losses are only introduced during unbalanced operating conditions.

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Resonance and Ferroresonance in Power Networks Consequently, the accuracy of readings obtained with other secondary windings, for metering or protection, is not affected. The sizing criteria for the resistor are as follows: a) The resistance value should be as low as possible in order to dampen the ferroresonant oscillations very rapidly. b) The resistor needs to cater for the possibility of earth-faults in the system, which will impose a zero sequence voltage across its terminals. The thermal burden of the VT secondary must not be exceeded under any circumstances, including for circuit breaker failure contingencies. It can be observed that the two requirements for the ohmic value of the resistor are exclusive; hence a balance needs to be found. Ferracci recommends the use of the following expressions [12]:

R=

3 3 U P

P =

Eq. 7-1

(3 U ) R

Eq. 7-2

where US is the rated secondary voltage, Pe is the rated thermal burden of the secondary and PR is the rated VA of the resistor. The reader must be warned that the above expressions are empirical and should only be used as a starting point of detailed analysis into the optimum size of resistor. Furthermore, the reader must also be aware that this solution is not effective in all cases of VT ferroresonance. Karlicek reported in [39] that sub-harmonic ferroresonant modes require smaller resistors than fundamental frequency modes to be suppressed, suggesting that a prior knowledge of the expected mode of ferroresonance is required in order to mitigate it. Experience in Ireland has shown that a resistance value as low as 0.5 was not efficient in suppressing fundamental ferroresonance. Analogue computer simulations reported in [23] indicated that the open-delta resistor was only successful in eliminating ferroresonance when the zero sequence voltage exceeded 15% of the system voltage. These uncertainties in the effectiveness of open-delta resistors justify the need to carry-out detailed analysis when considering them as a valid mitigation option.

7.1.2 Secondary “ Wye” Resistor The connection of damping resistors in parallel with the secondary windings of a VT is shown in Figure 7-2. Some methods of calculating the damping resistance can be found in literature: Price [23] recommends the following expression:

R =X

E E

Eq. 7-3

where RC is the critical resistance seen from the primary VT winding, Xb is the open circuit breaker capacitive reactance, ES is the voltage at the knee-point of the VT saturation curve and E is the rated system voltage. Ferracci recommends the use of the following expressions [12]:

R=

U k P P

Eq. 7-4

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Resonance and Ferroresonance in Power Networks

P =

U R

Eq. 7-5

where: US is the rated secondary voltage, k is a factor between 0.25 and 1 to guarantee that errors and service conditions remain within the limits specified by the applicable standards (i.e. IEC 186 or IEEE C57). Pt is the rated VA output of the secondary and Pm is the VA required for measurement. Li et al [63] recommend a theoretical resistance value based on the critical damping of an RLC circuit.

R

1 L 2 C r

Eq. 7-6

where Lunsat is the unsaturated VT inductance, C is the circuit capacitance, and r is the VT transformation ratio.

Rdamping-y

VT

Figure 7-2 Connection of damping resistors to the VT “wye” secondary windings The main drawback of this connection is that continuous losses are introduced in the circuit under normal operating conditions. Furthermore, unless a dedicated secondary winding is used for this purpose only, the damping resistor will be connected in parallel with the measuring instruments or protection relays, affecting the accuracy of measurements. The specification of an additional secondary winding for the sole connection of a damping resistor increases considerably the cost of the VT and may not be justified in most circumstances. Furthermore, spare secondary windings may not be available in existing installations. Some solutions to overcome these difficulties have been suggested in the literature. For example, [23] recommended switching the resistors into the secondary circuit only when the conditions for ferroresonance are met. This is done by connecting an interposing relay that only operates when all the breakers feeding a busbar VT are open or when a line VT is de-energized.

7.1.3 Secondary “Wye” Resistor in Series with a Saturable Reacto r This connection is shown in Figure 7-3. Under normal operating conditions, the non-saturated inductance presents such a large impedance that it guarantees no interference with the measuring or protecting devices sharing the same circuit. Under ferroresonant conditions, the inductance is driven into saturation inserting the resistor in the circuit. The additional load dampens the ferroresonant phenomenon. It should be noted that the proper design of this damping circuit requires a prior knowledge of the ferroresonant modes to be expected, which means that the design has to be evaluated on a case-by-case basis.

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Figure 7-3 – Damping resistors and saturable reactors on the VT “wye” secondary windings

7.1.4 Other Mitigation Options Other mitigation options worth exploring are listed below: Specification of EHV circuit breakers without grading capacitors. Optical VT technology. Capacitive VTs. Installation of inductive VTs in the line side of a line disconnector. Interlocking schemes to avoid risky circuit configurations. Specification of VTs with a higher saturation point. Optical transformer technology is a promising emerging alternative that eliminates the concern for ferroresonance. Further operating experience is required, however, before they can be generally installed in EHV substations. It is standard for manufacturers to include “anti-ferroresonant” measures in the design of Capacitive Voltage Transformers (CVT). These voltage transformers are only an option when fast line discharge is not required. If automatic reclosing is employed, as it is generally the case in EHV transmission lines, the CVTs cannot discharge the trapped line voltage and, consequently, the level of switching transient overvoltages at reclosing will be very high. This issue has large implications in the selection of line surge arresters and the insulation level, which increases significantly the cost of the installation. Consequently, CVTs are very rarely used to avoid line VT ferroresonance in EHV substations. On the other hand, if the ferroresonant problem to be mitigated is due to busbars VTs, the use of capacitive VTs provides the simplest and cheapest solution since automatic reclosing is not deployed for busbar faults. The use of inductive VTs with higher saturation knee, i.e. lower flux density, can reduce the risk of ferroresonance. An example of VT magnetising curves from three different manufacturers is shown in Figure 6-3. It should be noted, however, that the risk is not entirely eliminated. It just makes the onset of ferroresonance more difficult. Furthermore, if ferroresonance occurs, the level of overvoltages will be higher. The increased cost must be balanced against the achieved performance. The mitigation options discussed above are applicable to both cases of VT ferroresonance; VTs energized through grading capacitors of circuit breakers and VTs in double line configuration topology.

7.1.5 Mitigation of VT Ferroresonance in Ungrounded Neutral Systems For the mitigation of VT ferroresonance in ungrounded neutral systems or in arc-suppressed neutral systems the following options can be considered:

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Operate the VT at a low induction value The first design countermeasure against ferroresonance is to dimension the core for low induction. This is an effort to ensure that, on occurrence of transient overvoltages, the core does not saturate and cause ferroresonance. A first precaution is to ensure that nominal inrush current will not cause saturation. The induction of a VT will reach nearly twice its steady-state value if the switch-in occurs at the zero crossing of the applied voltage. Therefore, to avoid going into saturation in these circumstances, the rated induction must be less than half of the saturated value. On an arc-suppressed or ungrounded neutral system, it is possible for the applied voltage to be 3 times the rated value, so the induction of the VT on such as system should be less than 0 . 5 induction.

3 times the saturated

Introduce resistive losses to damp-out ferroresonance. As described in previous sections a damping circuit connected to the secondary winding is the usual way to mitigate ferroresonance in an inductive VT. This damping circuit may be connected in for as long as required to damp out ferroresonance or it may be connected permanently. The advantage of the former is that the damping resistor does not need to be rated for long time voltage application as it is only in the circuit when ferroresonance is likely to occur. However, the introduction of extra circuitry is a disadvantage and raises the question of physical mounting of extra components and connection of same as well as the possibility of misoperation due to failure of a component. Whichever arrangement is used, the damping resistor is normally connected to the open delta secondary winding of the VT.

Add more capacitance in parallel with the VTs This phenomenon is normally limited to small values of zero sequence capacitance. The connection of a long circuit or a capacitor bank can be enough to prevent the onset of ferroresonance. One large North American utility has, for many years, successfully avoided incidents of ferroresonance with VTs used in many ground fault detection applications by using the arrangement shown in Figure 7-4. Two bus VTs are connected from phase A to phase B and phase C to phase B. These VTs are rated for the full line-to-line voltage on the bus. The third VT is connected from phase B to ground and is rated 2 times the line-to-line voltage. The two different kinds of VTs are used to reduce the risk of ferroresonance. The auxiliary VTs, to step down the voltage to the relay, are connected grounded wye-open corner delta. When a fault occurs, the neutral shifts and a voltage of 3 times normal secondary voltage may occur across the open-corner delta. A resistive burden of around 125 Ohms is connected across the open-corner of the delta. The utility has adopted this scheme as standard for use with ground fault detection with autotransformer tertiaries. Some utilities employ bus VTs that are fully fluxed for line-to-line voltage but do not use resistive loading across the secondary windings or a resistive burden across the open-corner delta secondary. It should be noted that modern digital relays that are used to detect a ground fault on ungrounded busses, such as delta-connected tertiary windings of autotransformers, do not use auxiliary VTs. They connect directly to the bus VT phase-to-ground secondary windings. Thus, there is no opportunity to apply damping resistance across an open-corner delta VT secondary winding unless the VTs have dual secondary windings where one set can be configured accordingly. Ground fault detection schemes applied to floating neutral systems, based on modern digital relays having reduced burdens may be more susceptible to ferroresonance problems than schemes employing the older electro-mechanical relays.

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Resonance and Ferroresonance in Power Networks A

B

C

C

A

BUS PTs C

A

C

B

A

AUX PTs C B

A1

A2

YV XV

250

YV1

250 RELAY

IAV

XV1

Figure 7-4 Interconnection of VTs and Auxiliary VTs

7.2 Mitigation of Power Transformer Ferroresonance Mitigation options applicable to power transformer ferroresonance and especially in the case of transformer terminated EHV lines in multi-circuit configuration can be grouped into three basic approaches:

Avoid circuit parameters or operating conditions favouring ferroresonance One option to consider is to limit the length of the de-energized transmission line that can be left connected to the transformer. This could be achieved by the installation of circuit breakers on both sides of the transformer when the critical circuit length is exceeded. It is also possible to specify transformers especially for new developments with low flux density. This measure tends to be very uneconomical and impractical as it increases significantly the cost and size of the transformer. A third alternative is to disconnect any parallel circuit before the transformer is de-energized.

Minimise the energy transfer that is required to sustain the ferroresonant oscillations Losses could be increased artificially during switching by means of series resistors inserted in the switching devices to avoid sustained ferroresonance. Another possibility to increase the losses is by loading the delta connected tertiary windings with a suitable resistor. In some cases where a de-energized circuit and accompanying transformer are in proximity to a parallel live circuit, the energy transferred through the inter-circuit coupling to the ferroresonance can be significantly reduced by the introduction of, or modification to, phase transpositions in one or both circuits.

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Control the duration of ferroresonance by operational switching In this category ferroresonance is allowed to occur but its duration is controlled by careful management of switching operations such that ferroresonance oscillations are quenched. One possible way is to open the disconnectors used for the isolation of the transformer to remove it from the transmission line. However, special attention must be given to this measure as the ferroresonant current could be sizable, resulting in large arcing and possible damage to disconnector contacts. Another option is the employment of appropriately rated earth switches whereby earth switches are closed automatically following detection of ferroresonance. It is also possible to initiate disconnection of the parallel circuit to remove the source of energy. However this action should be assessed in the context of the entire system integrity and stability as it results in a double circuit outage. In the case of a distribution system power transformer ferroresonance due to the use of singe-phase switching or the deployment of fuses, the most obvious mitigation measure is to avoid the use of single-phase switches or fuses. Alternatively the length of cable connected to the transformer could be limited by installing a circuit breaker cubicle as close as possible to the transformer terminals. It is also possible to introduce a resistive load to the circuit. Following a general network disturbance or black-out, a hydroelectric or a gas turbine power unit with black start capability may be used with the primary objective of restoring power to the auxiliaries of a nuclear power plant [86]. This usually involves energization of the auxiliary transformer of a nuclear power unit so that it can participate in the restoration of the network. In order to avoid resonance or ferroresonance due to the magnetization of the target transformer, it is possible to use the hydroelectric or gas turbine power units to perform a "progressive voltage" reenergizing, with the objective of applying a voltage to the target transformer that increases from 0 to its final value in several seconds. In an automatic sequence, the source unit is brought up to its rated rotation speed, unexcited, some protections having been inhibited so that on request at the voltage is increased following closure of the units’s circuit breaker and excitation contactor. The increase of voltage on the lines and transformers is performed under the control of the automatic voltage regulator (AVR), for example by means of a ramp, to its final value in several seconds or even several tens of seconds. The AVR voltage set-point is normally set to its minimum value, corresponding in general to 90% of the rated generator voltage. In some cases with long lines where the level of compensation is low, a lower value may be required to comply with the maximum authorised levels on the lines. This reduced voltage set-point helps to reduce the voltage level at the target transformer terminals, thus avoiding its saturation. By this progressive way of re-energizing transformers on weak networks, transient saturation is avoided and there are no inrush currents. Thus there is a low risk of appearance of ferroresonance or resonance phenomena characterized by temporary harmonic overvoltages that could present a danger to the equipment.

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CHAPTER 8

CONCLUSIONS

This comprehensive study guide provides information on, and suggests tools and/or methodologies for, the analysis and resolution of problems created by power frequency resonance and ferroresonance in power networks. The main problems of interest are the overvoltage stresses imposed on power system components resulting from temporary overvoltages which, as a result of these phenomena, can be high enough to be hazardous. An introduction of the topics of resonance and ferroresonance begins by considering resonance, firstly by considering the ideal series L-Cs circuit driven by a voltage source and secondly the ideal parallel L-Cp circuit driven by a current source. The discussion is then extended to the series-parallel Cs-L-Cp circuit connected to a voltage source, which is representative of many practical cases of power frequency resonance that can occur in power systems. When the series capacitance is relatively small, it is demonstrated that the combination of a voltage source in series with Cs behaves as an equivalent current source, thereby resulting in the familiar topology for parallel resonance. The topic of ferroresonance is then introduced by considering the series resonant R-L-Cs circuit driven by a voltage source, where the inductor is now nonlinear. A graphical solution for the steady state response of the linear case is extended to the simple nonlinear circuit where the effects of varying the source voltage, circuit capacitance and circuit losses are considered. Although an oversimplification of the actual phenomenon, equations are provided describing the steady state performance. A more rigorous treatment is given for the case where energy is exchanged between a capacitor and a lossy nonlinear inductor when the inductor is switched onto the capacitor having a trapped charge. The more general case of a voltage source driving a series nonlinear L-Cs ferroresonant circuit is next explored. The types of ferroresonant oscillations are presented, including the representative voltage waveforms. An overview is given of the typical network topologies that can give rise to power frequency resonance in shuntcompensated transmission circuits. For problems to occur, the shunt reactors do not necessarily have to be directly connected to the line but could be on the bus. If only a single circuit is considered, then a one (or two) open-phase condition is required for a resonant topology, leaving two (or one) energized phases and phase reactors in parallel with the line’s phase-to-ground capacitance. This unbalanced condition could be the result of a switching misoperation. The resonance is excited by capacitive coupling between the energized phase(s) and the deenergized phase(s). Considering double circuit lines or multiple circuits mutually coupled on the same right-of-way, the resonant topology requires that one shunt-compensated circuit has been disconnected. The excitation for the resonance is provided by the inter-circuit capacitive coupling. In addition to the resonances possible for normal (nofault) conditions, faults on either the de-energized or the energized circuit can lead to more possible resonant conditions. If imperfectly transposed or untransposed double circuits or multi-circuits are considered, many more resonant topologies are possible. An entirely different power frequency resonance can occur when an embedded generator isolates with a relatively long line or cable, effectively producing a series L-C circuit driven by a voltage source. If the circuit is tuned to a frequency near the fundamental, large circulating currents and high phase-toground voltages can result. A very comprehensive and detailed treatment is presented of the resonant topologies encountered in shuntcompensated transmission circuits due to a one or two open-phase condition and, in the case of double-circuits, when a shunt compensated “de-energized” circuit is mutually coupled to an adjacent energized circuit. For the single-circuit topologies, approximate steady state equations are developed which enable straight forward calculation of (1) the critical degrees of shunt compensation where resonances can be expected and (2) the temporary overvoltages on the open phases for a particular degree of shunt compensation. The effects on the resonance of tower design, neutral reactor rating, and core construction of the phase reactors are explored. A practical example is provided of surge arrester failure due to overvoltage on a long 500kV transmission line having 72% shunt compensation which, due to breaker failure, resulted in a prolonged two open-pole condition. This topology resulted in a resonant condition, as indicated by the simplified steady state analysis, almost perfectly tuned to power frequency. For double circuit lines where there is capacitive coupling between a shunt compensated de-energized circuit and an adjacent energized circuit, simple formulas are given to estimate the degree of shunt compensation resulting in resonance for cases with and without a fault on either circuit. A comparison of the results of the approximate steady state analysis to detailed EMT-type simulations is provided. A careful choice of phase reactor rating to avoid critical degrees of shunt compensation will prevent resonant

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Resonance and Ferroresonance in Power Networks overvoltage problems due to unintentional uneven phase operation or intercircuit coupling. A range of possible mitigation options is given as a tool-box for the analysis of resonance in shunt-compensated transmission circuits. The network configurations involving electromagnetic VTs and power transformers which can lead to ferroresonance are considered. VTs connected to an isolated bus or line which remains energized through the grading capacitors of an open circuit breaker can go into ferroresonance. VTs applied to ungrounded neutral systems and to double-circuit lines or having low zero sequence capacitance can also be prone to ferroresonance. Ferroresonance can occur with power transformers, such as when a transformer-terminated transmission line is deenergized but remains capacitively coupled to a parallel energized circuit. The ferroresonance can be power frequency or subharmonic, depending on line and exposure lengths as well as initial conditions. A lightly loaded transformer energized from a long line in one or two phases, as could happen due to breaker misoperation, fuse operation, or sequential single-phase switch operation, can also be a candidate for ferroresonance. Considering that one or two phases of the feeding source could be open, a comprehensive diagram is provided for various three-phase transformer winding configurations, which illustrates the possible ferroresonance topologies in power transformers. Analytical solution methods, such as the harmonic balance method or the incremental describing function method, have been applied to study ferroresonant circuits. These somewhat esoteric approaches, however, are not capable of predicting non-periodic ferroresonance and sometimes not even subharmonic ferroresonance. EMT-type time domain programs are a suitable tool for ferroresonance studies since they enable explicit representation of non-linear elements and facilitate modelling of complex systems sometimes required for such studies. Usually a large number of simulations are required to explore sensitivity of the solution to changes in parameters and initial conditions. A brief discussion is provided on various nonlinear dynamic analysis tools such as phase-space, Poincaré section, and bifurcation diagram techniques. Since ferroresonance is generally a localized phenomenon, large network models are not required for its study. Modelling of network components in an EMT-type program is discussed. For overhead lines, frequency dependence is not critical but line capacitance must be accounted for accurately and phase transpositions should be explicitly represented. For transformers, the correct representation of the magnetic core characteristics and core losses is important for ferroresonance simulations. A linear resistor in parallel with a hysteretic inductor model connected across the winding closest to the core provides a reasonable representation of the magnetic core. For shunt reactors, the construction of the core must be accounted for in the reactor model. Suggestions are given for the modelling of other substation components, with emphasis on the need to accurately account for the stray capacitances associated with these components. Techniques to mitigate ferroresonance are presented and described. For ferroresonance involving magnetic VTs, mitigation is either accomplished passively or actively. Passive methods are designed to prevent occurrence of network conditions conducive to ferroresonance and provide the best protection. Examples of passive mitigation include the design of the physical layout to minimize inter-circuit coupling, selection of VTs having a lower flux density (higher knee point), and the implementation of special switching procedures. Active mitigation of ferroresonance with VTs includes application of damping resistors to wye-connected VT secondary windings and connection of a suitably rated resistor across the VT open-corner delta winding. Formulas to estimate the values of these resistors are provided. For VTs that are applied to ungrounded neutral situations, some special mitigative methods are discussed. Often, switching procedures can be used to avoid occurrence of ferroresonance. Finally, several case studies and examples of actual resonance and ferroresonance are provided in this document.

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Bibliography/References [1]

Temporary Overvoltages: Causes, Effects and Evaluation. Paper presented in the name of Cigre Working Group 33.10 and IEEE Task Force on TOV. Cigre 1990 session.

[2]

Temporary Overvoltage Withstand Characteristics of Extra High Voltage Equipment. Electra No 179, August 1998.

[3]

Temporay Overvoltages – System Aspects. Electra No 185, August 1999.

[4]

Temporary Overvoltages – Test Case Results. Electra No 188, February 2000.

[5]

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BEN DRISS Khaled « Application de la théorie des bifurcations à la prédiction des résonances nonlinéaires à l'enclenchement de transformateurs à vide » Thèse préparée et soutenue à Supélec, EEI (Université Paris VI) le 13/06/1994 [« Application of the bifurcation theory to the prediction of non linear resonances in case of transformer energization » PhD thesis, SUPELEC-PARIS VI, 13-06-1994]

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Resonance and Ferroresonance in Power Networks

ANNEX A

RESONANCE EXAMPLES

A1. Resonance Ass ociated with Single-phas e Autoreclos e Switching of 275 kV Shunt Reactor On one circuit of a 275 kV double-circuit line, an existing 30 Mvar fixed shunt reactor with a neutral earthing reactor was reaching the end of its life. System studies had identified that this fixed shunt reactor should be replaced with a 50 Mvar switched shunt reactor.. Separate studies identified that the line was sufficiently short that secondary arc currents would self-extinguish if the shunt reactor were to be solidly earthed, with no requirement for a neutral reactor. This case study examines whether neutral earthing is required to avoid resonance in the recovery voltage under single-phase autoreclose conditions. The system under study is shown in Figure A-1.

Figure A-1 Overview of double-circuit line and shunt reactor studied under single-phase autoreclose For the 50 Mvar shunt reactor when solidly earthed, the ultimate steady-state recovery voltage at the Circuit A line exit from Bus X (coupled from healthy conductors), is around 40kVrms phase-earth with one phase of Circuit A out5, as indicated in Figure A-2. This value is well below the maximum continuous overvoltage (MCOV) of 191kV for the surge arrester intended for this application and is not of concern. Whilst the MCOV is exceeded when the breakers open, the situation persists for well under a second and is also not of concern. However, Figure A-3 indicates that substitution with a 30 Mvar solidly earthed reactor (equivalent to removal of the neutral reactor from the existing shunt reactor on Circuit A) yields a recovery voltage of around 330 kV, after slow oscillations arising from the switching have settled down. This recovery voltage exceeds the 191 kV arrester MCOV and could lead to equipment damage.

5

If, instead, three phases of Circuit A are out, representing three-phase autoreclose, the steady state recovery voltage drops to around 20kVrms phase-earth. Page 124

Resonance and Ferroresonance in Power Networks

Recovery Voltage (kVrms L-G) 800

VCOV

600 400 200 0 -200 -400 -600 -800 x

0.0

1.0

2.0

3.0

4.0

5.0

... ... ...

Figure A-2 Recovery voltage for 50 Mvar solidly earthed shunt reactor, blue phase breakers opening at 0.5s [local end] and 0.6s [remote end] (x axis is time in seconds)

Recovery Voltage (kVrms L-G) 800

VCOV

600 400 200 0 -200 -400 -600 -800 x

0.0

1.0

2.0

3.0

4.0

5.0

... ... ...

Figure A-3 Recovery voltage for 30 Mvar solidly earthed shunt reactor, blue phase breakers opening at 0.5s [local end] and 0.6s [remote end] (x axis is time in seconds) Figure A-4 shows the effect upon the recovery voltage (blue phase open only, which is the worst of the three in this instance) with varying shunt reactor sizes. For a solidly earthed shunt reactor, a peak occurs around 27 Mvar, suggesting that a 50 Hz resonance would be likely to be excited, with recovery voltages potentially exceeding several hundred kV, depending on resistive damping. Adding a 1000 neutral earthing reactor moves the resonant condition to occur for a shunt reactor of around 33.5 Mvar. However, the intended 50 Mvar shunt reactor yields fully acceptable recovery voltages. In practice, recovery voltages as high as those suggested by the peak of Figure A-4 would not be seen; rather, this is a prospective recovery voltage. Surge arresters on the line would tend to reduce

Page 125

Resonance and Ferroresonance in Power Networks

actual voltages, as would flashovers when surge arrester capabilities are exceeded. For illustration purposes, nonlinear effects such as surge arrester conduction and saturation of the shunt reactors were not modelled.

Recovery voltage (kVrms L-G)

3500 3000 Solid earthing 2500 2000 1500

1000

NEX

1000 500 0 0

20

40

60

80

100

120

140

160

Shunt Q (MVAr)

Figure A-4 Effect of shunt neutral reactor size on recovery voltage on line end (blue phase only) For the system studied in this case, the 50 Mvar reactor was found to avoid recovery voltage resonance problems, with solid earthing or with a 1000 neutral earthing reactor. However, smaller reactors risk resonance between the reactor and capacitive coupling from adjacent conductors during single-phase autoreclose operations. Simulation of recovery voltages with varying reactor sizes under single- and three-phase autoreclose allows a potentially relatively narrow resonant peak to be identified and then mitigated, either by a different size of reactor or by altering the neutral earthing arrangement.

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Resonance and Ferroresonance in Power Networks

A.2 Line Resonance experienced in 275 kV Doub le Circuit as a result of System Expansion Network Topology The 30 Mvar line reactors installed at Daven on the Daven to Robert and the Daven to Cano 275 kV transmission lines where designed to prevent over-voltages at the Daven end of these lines when they were originally energized from the Lakeside Substation further away. Since this time the Roberts 275/132 kV substation has been established, significantly “shortening” the lines. The original line design also ensured that the transmission line was fully transposed between Daven and the present location of Roberts substation, between Roberts to the proposed site of Billo, and again between Billo and Lakeside.

Daven 30MVAr Reactors

133.7km

Cano

212.5km

Roberts

376.5km

Lakeside

Figure A-5 Daven – Roberts – Lakeside 275 kV Right of Way

Resonance Arising from Planned Network Expansion Now that Cano substation has been established (cut into the Daven to Roberts #1 line), the effective line length has been again reduced (and the transmission line at that location is in mid-transposition). In terms of impedance, the transmission line has a capacitive reactance of approximately -j2600 whereas the 30 Mvar reactor has an inductive reactance of approximately +j 2500 . A line length of about 5 km longer, or a system frequency of 51 Hz, or if the line reactor had an actual reactance of about 29.1 Mvar, would all result in a true resonant condition occurring with an apparent impedance in excess of 3.2 M resulting in possibly rated phase-to-ground voltages appearing on one phase, lower voltages on the other two phases, and unbalanced resonant currents flowing through the reactor and reactor neutral. The presence of these voltages and currents make it impossible to open the reactor isolator.

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Resonance and Ferroresonance in Power Networks

In the short term, to switch the line reactor in and out of service, the Daven to Roberts 275 kV transmission line must also be de-energized to remove the source of excitation voltage.

Calculations, On-site Tests and Results This section details the analysis carried out to identify whether a parallel resonant condition could conceivably occur between the 275 kV 30 Mvar line reactor at Daven and the line capacitance on the Daven to Cano 275 kV transmission line. This section also discusses the results of tests carried out to verify this parallel resonant condition.

Transmission Line Parameters Table A-1 below contains the line parameters for the Daven – Roberts and Daven – Cano – Roberts 275 kV line sections.

Table A-1 Daven – Roverts 275 kV Line Section Parameters Positive Sequence Parameters

Zero Sequence Parameters

From Bus

To Bus

Line Distance

R1 (pu)

X1 (pu)

B1 (pu)

R0 (pu)

X0 (pu)

B0 (pu)

Daven

Roberts

212.5

0.0160

0.1139

0.4606

0.0662

0.3053

0.3272

Daven

Cano

133.7

0.01006

0.07164

0.28972

0.04164

0.19203

0.20581

Cano

Roberts

78.8

0.00594

0.04226

0.17088

0.02456

0.11327

0.12139

Daven

Lakeside

376.5

0.0264

0.19370

0.85800

0.1127

0.5309

0.5435

Note: Impedance values are in pu on 100 MVA base.

Daven 275 kV Line Reactor Parameters Table A-2 below contains the Daven 275 kV line reactor parameters.

Table A-2 Daven 275 kV Line Reactor Parameters at Fundamental Frequency (50 Hz) Line Reactors

Neutral Reactors Reactance ( /phase)

From Bus

To Bus

Rating (Mvar)

XLu

XLv

XLw

Daven

Roberts

30.30

2502

2491

2508

Daven

Cano

30.19

2512

2505

2509

Inductance (H)

R

3.36

26.06

)

XL ) 1056.2

Investigation of Parallel Resonant Condition Table A-3 below compares the capacitive reactance of the transmission line with the inductive reactance associated with the corresponding line reactor.

Table A-3 Reactance comparison between line and corresponding line reactor assuming nominal system parameters From Bus

To Bus

Line Capacitive Reactance ( )

Daven

Roberts

1641.88

Daven

Cano

2610.28

Daven

Lakeside

881.41

Page 128

Line Reactor Inductive Reactance ( )

2504.97

Resonance and Ferroresonance in Power Networks

From Table A-3, it can be seen that for the Daven – Cano 275 kV line, the capacitive line reactance of 2610 close proximity to the line reactor inductive reactance of 2505 .

is in

Making use of the Pi Equivalent Model of a transmission line (shown below in Figure A-6), the total apparent impedance of a transmission line can be calculated.

Figure A-6 Pi Equivalent model of a transmission line

Comparing this total line impedance with the inductive reactive component associated with the line reactor will provide an indication of whether a true resonance condition exits. Using the line parameters shown in Table 1, the total apparent line impedance values (Ztotal) were calculated (note that all calculations have assumed nominal system parameters): 1. Daven – Roberts 275 kV line,

Z total

4586

2. Daven – Cano 275 kV line,

Z total

70920

3. Daven – Lakeside 275 kV line,

Z total

1271

; ; .

To enable visualisation of the resonance condition, total apparent line impedance versus line length plots were generated for the Daven – Cano – Roberts – Lakeside 275 kV line section; these are shown below in Figure A-7 and Figure A-8. From Figure A-7, it can be seen that the total apparent line impendence is of a considerable magnitude for a line length in the range of 130 – 150 km. For all other line lengths the total apparent line impedance is negligible. From Figure A-8, it can be seen that by expanding the scale for line length over the 130 – 150 km range shows that a true resonant condition occurs with an apparent impedance of 3.2 M at a line length of approximately 138.5 km. Figure A-7 illustrates that the total apparent line impedance is negligible at and near the locations of the Roberts and Lakeside substations along the line section. When considering the implications of the total line impedance versus line length plots shown below in Figure A-7 and Figure A-8, the following points need to be considered: 1.

These plots are based on calculations performed using nominal system parameters such as unity system voltage and fundamental frequency;

2.

While great care is taken in ensuring all parameters used to model lines and devices within its transmission network are as accurate as possible, never-the-less errors are always inherent and need to be considered and suitable allowances made;

3.

A Daven – Cano 275 kV line length increase of approximately about 5 km, or a system frequency of 51 Hz, or if the line reactor had an actual reactance of approximately 29.1 Mvar, would all result in a true resonant condition occurring with an apparent impedance in excess of 3.2 M resulting in possibly rated phase to ground voltage appearing on one phase, lower voltages on the other two phases, and unbalanced resonant

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Resonance and Ferroresonance in Power Networks

currents flowing through the reactor and reactor neutral. The presence of these voltages and currents make it impossible to open the reactor isolator.

Hence, variations in any of the above discussed system or modelling parameters has the potential to move the location of the resonance condition towards the Cano substation. Based on this analysis, the potential for a resonance condition is highly likely and appropriate measures need to be undertaken to ensure this condition can be abated.

3.5 3

True Resonance Condition

Total Line Impedance, Ztotal (M ) 2.5 2 1.5 1 0.5 0 0

25

50

75

100

125

150

175

200

225

250

275

300

325

350

375

Line Length [km] Daven

Cano

Roberts

Lakeside

Figure A-7 Total Line Impedance versus line length for the Daven – Cano – Roberts – Lakeside 275 kV section

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Resonance and Ferroresonance in Power Networks

Total Line Impedance, Ztotal (M )

3.5

True Resonance Condition

3 2.5 2 1.5 1

Cano

0.5 0 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150

Line Length [km] Figure A-8 Total Line Impedance versus line length at Cano 275 kV substation

Test Results Tests were carried out designed to measure the near parallel resonant condition, which was suspected to be occurring between the 275 kV 30 Mvar line reactor at Daven and the line capacitance on the Daven - Cano 275 kV transmission line. This near resonant effect is excited by the closely coupled parallel Daven to Roberts 275 kV line, when the Daven to Cano 275 kV line is isolated and unearthed at both ends. The resultant effect is (differing) steady-state high voltages on the unearthed lines (e.g. 140 kV, 100 kV and 60 kV phase to ground, whereas, nominal phase to ground voltage is 159 kV). This network configuration commonly occurs when switching the line reactor in and out of service using its motorised isolator. Prior to the commissioning of Cano substation the steady state induced voltages were close to 0 kV. The following Table A-4 contains a summary of the test results.

Table A-4 Daven – Cano 275 kV Line Resonance Test Results Condition

Phase Voltage (kV)

Phase Current (A)

Daven Line Reactor Neutral Current (A)

Vu

Vv

Vw

Iu

Iv

Iw

In

System Normal

166.8