1997 series compensation of distribution and

University of WollongongResearch Online

University of Wollongong Thesis Collection University of Wollongong Thesis Collections

1997

Series compensation of distribution andsubtransmission linesRobert Arthur BarrUniversity of Wollongong

Research Online is the open access institutional repository for theUniversity of Wollongong. For further information contact ManagerRepository Services: [email protected].

Recommended CitationBarr, Robert Arthur, Series compensation of distribution and subtransmission lines, Doctor of Philosophy thesis, Department ofElectrical and Computer Engineering, University of Wollongong, 1997. http://ro.uow.edu.au/theses/1349

http://ro.uow.edu.au/


http://ro.uow.edu.au

http://ro.uow.edu.au/theses

http://ro.uow.edu.au/thesesuow



SERIES COMPENSATION OF

DISTRIBUTION AND SUBTRANSMISSION LINES

A thesis submitted in fulfilment of the requirements for the award of the degree

of

DOCTOR OF PHILOSOPHY

from

THE UNIVERSITY OF WOLLONGONG

by

Robert Barr, B.E.(Hons),M.E.,C.P.Eng.,F.I.E.(Aust)

Department of Electrical and Computer Engineering 1997

1

Dedication

To m y wife, Linda and children, Jane, Adrian and Karen.

Acknowledgement

I wish to express m y appreciation and gratitude to m y supervisor, Dr D o n Piatt for his

most valuable assistance with this project.

Declaration

I hereby certify that this thesis is entirely my own work and has not been submitted for

the award of a degree to any other university or institution.

RoberBarr

Series Compensation of Distribution and Subtransmission Lines

2

Abstract

Series capacitors can increase the power carrying capacity of subtransmission and

distribution lines by reducing voltage regulation. When considering series capacitor

compensation of distribution lines and subtransmission lines careful consideration needs

to be given to capacitor location, ferroresonance, ohmic reactive value, transient

behaviour, short circuit withstand and capacitor protection. Conventional design

approaches include shunt connected resistors, spark gaps, metal oxide varisters, thyristor

controlled reactors and bypass switches.

This thesis describes the use of a saturating choke and damping resistor to control

ferroresonance, transients, and through fault currents. A small scale laboratory non

linear single phase ferroresonant circuit was constructed with realistic per unit

component values. Both 3rd and 2nd subharmonic ferroresonance modes were predicted

by modelling and generated in practice.

Choke and damping resistor parameters were selected by modelling to eliminate all the

unwanted ferroresonant states. Experimental work confirmed that all the unwanted

ferroresonant states were eliminated from the laboratory circuit. The transient and short

circuit performance of the system is considered. The proposed arrangement offers an

effective countermeasure to ferroresonance for series compensated distribution lines.

The system also allows some control of system fault levels and transient circuit

behaviour. The technique is simple, effective and requires no sophisticated control,

protection or bypass switch systems.


Table of Contents

/. Introduction

2. Compensation Effects on Line Voltage Ptofiles

3. An Overview of the Ferroresonance Phenomenon

4. Time Domain Transient Ferroresonance Model

5. Frequency Domain Ferroresonance Model

6. Laboratory Ferroresonant Circuit "A"

7. Laboratory Ferroresonant Circuit "B"

8. Stability of Frequency Domain Ferroresonant Solutions_

9. Management of Ferroresonance and Short Circuit Withstand

10. Existing Techniques for Managing Ferroresonance and Short Circuit Currents 79

11. A New Technique for Managing Ferroresonance 83

12. Selection Of Component Values 95

13. Scaling Up To Real Distribution Networks 98

14. Future Research Directions 108

15. Conclusions 110

16. References 113


4

List of Appendices

Appendix A

Detailed Experimental and Frequency Domain Model Results for Circuit "A" 128

Appendix B

Modelled Transient R L C Behaviour for Circuit "A" 131

Appendix C

Detailed Experimental and Frequency Domain Model Results for Circuit "B" 134

Appendix D

Modelled Transient R L C Behaviour for Circuit "B" 137

Appendix E

Experimental Circuit "D" 3 Phase 140

Appendix F

Modelled stored Energy in Circuit "A" 142


5

List of Main Symbols

co power frequency - radians per second

X transformer flux linkage - weber turns

X c choke flux linkage - weber turns

e mismatch voltage error

a sine switch on angle - radians

<|) V-I phase angle - radians

fr frequency ratio

I distribution line current - amps

Lci Choke leakage inductance - henrys

R distribution line resistance - ohms

T power frequency period - seconds

V c capacitor voltage - volts

V d line voltage drop - volts

VI load voltage - volts

Vs supply voltage - volts

X distribution line reactance - ohms

X c capacitor reactance - ohms


6

1. Introduction

Series Compensation in Power Distribution Systems

1.1.1 Capacitors have been used for the series compensation of transmission and

distribution lines for many years. Series capacitor installations have been

described in the literature as far back as 1954 102. Pioneer power system

engineers were seriously examining the merits and demerits of series

capacitors and analysing the subharmonic ferroresonance phenomena [401] in

the 1930's.

1.1.2 Series capacitors are now in common use at the transmission level with

hundreds of units in service throughout the world. Although series capacitors

have been in use for a long time they have not found widespread acceptance as

a viable economic power system component at the distribution level. It is the

use of series capacitors at the distribution level that is the subject of this thesis.

1.1.3 Distribution power systems have special characteristics that make them

different from transmission systems. This thesis is aimed at examining ways in

which the viability and effectiveness of series capacitor compensation can be

improved at the distribution level.

The Series Capacitor Compensation Circuit.

1.2.1 Figure 1 shows the typical voltage profile of a very weak distribution line with

a lumped load at the end. The line resistance and reactance is distributed along

the line. As the load varies from light load to full load there is a substantial

voltage difference seen by customers. It is the difference between light load

and full load voltage that in many cases limits the capacity of the line.


AAAA^ -rrrnr^

Distributed line R & L

73 d

o

75 d O

Figure 1 Voltage Profile of a W e a k Distribution Line

1.2.2 Figure 2 shows the effect of series capacitor compensation on the same line.

The power line supplying the transformer is represented by distributed linear

inductance and resistance. The circuit contains a series capacitor to tune out

the effects of the line inductance. W h e n used for series compensation, the

capacitance will normally be chosen so as to tune out all or most of the line

inductance at the power frequency.

r Y M A 7 ™ ^ -yVVVXA-^nnpo^

CAP,

Distributed line R & L

-a a. o. CK

73

a o

Light Load

±0.90

Figure 2 Voltage Profile of a W e a k Distribution Line with Series Capacitor

Compensation


8

1.2.3 The fundamental issues concerning series capacitors are the same today as

they were in the pioneering days of 1930's. Series capacitors offer the

potential to tune out all or part of the series inductance of lines at the power

frequency. This can result in reduced voltage regulation, enhanced power

transfer capability and improved system stability. Series capacitors are

particularly attractive in controlling voltage fluctuations associated with

rapidly varying loads. With such significant potential for enhanced power

transfer capability the question arises as to w h y series capacitors have not

found widespread use and acceptance at the distribution level.

1.2.4 Series capacitors are not in widespread use at the distribution level because of

the generation of ferroresonant overvoltages, fault level problems, problems

associated with capacitor withstand of heavy through fault currents and high

cost.

1.2.5 Series capacitors can produce subharmonic ferroresonant overvoltages and

currents. This phenomenon is generally not well understood by power system

engineers with the result that a series capacitor installation is considered a

high risk option or simply not considered at all. The possibility of serious

damage to capacitors, transformers and customer installations by

ferroresonance is of real concern and requires careful management. The

experimental and modelling work performed in the course of this project has

highlighted h o w destructive ferroresonance can be.

1.2.6 Effective solutions to the problems of ferroresonance and capacitor protection

are of great potential benefit. Analysis of most large power system distribution

networks will identify locations of high voltage regulation where voltage

conditions could be improved by series compensation.


9

1.2.7 This thesis is concerned with gaining a fundamental understanding of

ferroresonance and other series capacitor related problems and developing

new solutions.

Improved Voltage Control with Series Capacitors

1.3.1 Voltage control in electric power systems is of fundamental importance in

achieving desired power flows and maintaining voltage levels within specified

limits.

1.3.2 Great engineering effort and capital expenditure is invested in power systems

to provide sufficient "system strength" to maintain voltage levels within the

required margins. Distribution lines are normally limited in their load carrying

capacity by either thermal current rating considerations or excessive voltage

drop. In general terms distribution systems supplying high load density areas

such as Commercial Business Districts and areas of high density housing

development tend to be current rating limited. Regions of low load density

such as rural areas and regional towns tend to be supplied by distribution

systems that are voltage drop limited.

1.3.3 High voltage regulation in distribution feeders is not the main limitation in

itself. It is the voltage variation between light load and full load that limits the

maximum feeder loads that can be accommodated. With off circuit

transformer taps on distribution transformers, the voltage variation in the high

voltage distribution feeder is directly reflected on to low voltage customers.

Australian Standard AS2926 sets maximum voltage variation at + 6 % to - 6 %

or 226V to 254V in a 240V system.

1.3.4 There are many engineering approaches to overcoming problems of excessive

voltage regulation including augmentation of lines, construction of additional

lines, shunt capacitors, voltage regulators (on load tap changing auto

transformers), on load tap changing transformers and construction of new


10

substations. These approaches often involve large capital costs in areas where

there are low load densities. Development of a low cost series compensation

arrangement for distribution systems could provide significant advantages in

selected situations.

.4 Improved System Stability with Series Capacitors

1.4.1 Series capacitors can increase the stability of power systems by reducing the

effective impedance of lines. Reduced line impedance has the effect of

increasing system fault levels and increasing the strength of interconnection of

a distributed network of generators [512] [513],

.5 Ferroresonance

1.5.1 Ferroresonance is a well documented hazard of series capacitors in

distribution networks 102[103]. Ferroresonance can result in severe

overvoltages in capacitors, distribution transformers and customer

installations.

.6 Subsynchronous Resonance

1.6.1 Subsynchronous resonance is a potential hazard with series capacitors [510],

Subsynchronous resonance involves a low frequency exchange of energy

between a series capacitor and a generator. Subsynchronous resonance can

cause the mechanical failure of generator shafts.

.7 Asynchonous Resonance

1.7.1 Asynchronous resonance is a another potential problem whereby motors can

lock onto subharmonic frequencies on starting and consume abnormally high

currents [102].


11

2. Compensation Effects on Line Voltage Profiles

2.1 Uncompensated Line Voltage Profiles

2.1.1 Figure 3 shows a phasor diagram for the typical uncompensated system with

the corresponding voltage profile.

IR sin(0) + IX cos(0)

VI

Figure 3 Phasor Diagram of an Uncompensated Line

2.1.2 The voltage drop Vd is defined by equation (1) as the difference in magnitude

between the supply voltage Vs and the load voltage VI. As shown on the

phasor diagram, equation (2) is a very good approximation for the voltage

drop in short lines where transmission line effects can be ignored. Equation (2)


12

remains a good approximation for most distribution lines because the angle

between Vs and VI is generally small.

Vd= |Vs|- |V1| (1)

Vd « IRcos(<|>) + IXsin(<|>) (2)

2.1.3 Equation (2) describes the fundamentals of voltage drop performance on

distribution lines and is very useful for analysing the various forms of power

system compensation.

2.1.4 At no load the load current I will be zero and hence both terms IR cos((J>) and

IX sin((p) in equation (2) will be zero. Under these conditions there will be no

line voltage drop. At full load the voltage drop needs to be kept down to a

manageable level (typically 5 % to 1 5 % of the supply voltage Vs). The aim is

to minimise the line voltage drop, Vd. The line resistance, R, is determined by

the line length and conductor size and hence is a fundamental characteristic of

the line. The parameters over which control is possible are the effective

inductive reactance of the line, X, and the effective load power factor (i.e. X

and cos(<|>) ). The effective inductive reactance of the line, X, can be controlled

by series capacitor compensation and the effective load power factor can be

controlled by shunt capacitor compensation.

2.2 Series Compensation Line Voltage Profiles

2.2.1 With series compensation the series capacitors produce a reactive impedance

that typically cancels out all or part of the line reactance, X. Under full

compensation conditions the effective line reactance is zero and hence the

term IX sin(<j>) from equation (2) becomes zero. The voltage drop on the line


13

is n o w a function of the line resistance, load current and load power factor via

the term IR cos(<J>) as shown in equation (3).

V d w IRcos((j>) (3)

2 Series capacitors are most effective in lines with a high X/R ratio. If the X/R

ratio is less than unity then series capacitors will tend to be ineffective. Load

power factor is also a consideration with little or no benefit being gained if the

power factor is close to unity.

3 Figure 4 shows a typical voltage profile of a fully series compensated line

where the capacitor is located at the centre of the line. This figure clearly

shows h o w the installation series capacitors can significantly improve the

voltage variations between full load and light load.

IR sin<0)

Vc

Vs

I

CAP,

Distributed line R 8. L

Light Load

VI

Figure 4 Phasor Diagram of a Fully Compensated Line


14

2.2.4 Series capacitors are not a solution to all line voltage drop problems, however

in the appropriate locations they offer the prospect of solving voltage and

other system problems in a very effective manner. The key requirements for a

successful series compensation scheme is a high X/R ratio (say X/R>=1) and a

power factor below unity (say cos(<|>)<=0.9).

2.3 Shunt Capacitor Compensation Line Voltage Profiles

2.3.1 Shunt capacitor compensation or power factor correction is the most common

form of compensation used in power systems. Large industrial customers are

commonly required by electricity supply authorities to maintain their power

factor above a specified minimum. Power factor control is often achieved in

these cases with shunt connected power factor correction capacitors.

Electricity supply authorities also use shunt connected capacitors in critical

parts of their networks to control reactive power flows.

2.3.2 With shunt compensation the line impedance remains unchanged. The shunt

capacitors increase the effective load power factor towards unity by

generating reactive power for the load. With reference to equation (1) this

means that the effective power factor cos(<|)) moves towards unity and sin((|>)

is driven toward zero. Under conditions of full compensation the term IX

sin((j)) is driven to zero leaving the voltage drop as approximately IR cos(<l)).

2.3.3 The effectiveness of shunt compensation is not limited by the line inductive

reactance. The ultimate limitation in a shunt compensation scheme is the line

resistance. In practical shunt compensation schemes the power factor may be

lifted from say 0.75 lagging (for an industrial plant) up to 0.9. It would be rare

and generally uneconomic to provide sufficient compensation to bring the

power up into the range 0.95 to 1.0.


15

2.3.4 Figure 5 shows a typical voltage profile of a shunt capacitor compensation

scheme. The possible voltage rise effect is shown if the capacitors are left in

service under light or no load conditions. T o overcome this problem

capacitors must be switched in and out of service depending on the reactive

load requirements. This means that voltage changes occur in discrete steps and

switchgear and control equipment is required. Unless specialised thyristor

equipment is used it not possible to react to rapidly fluctuating loads.

& i.o. cs -p

°0.9

0.8

serv'\ce _P 1.1 Light Load capacitors

out of service

ruU Load capacito rs in service

d0.8

Figure 5 Voltage Profile Illustrating Shunt Capacitor Compensation

2.3.5 Harmonic resonances generated by variable speed drives and other non linear

loads are becoming an increasing problem with shunt capacitors [513]. Shunt

capacitors present a low impedance to high frequency harmonics that can

result in abnormally high and destructive capacitor currents.

2.4 Comparison of Series and Shunt Compensation

2.4.1 Both series and shunt capacitor compensation have their advantages and

disadvantages. Both schemes have their place in power systems. Below is a

summary of the advantages and disadvantages.

Series Compensation of Distribution and SubUansmission Lines

16

Series Capacitor Compensation

Advantages

• Compensation naturally regulates with changes in load current.

• L o w risk of problems from load generated harmonics.

• Reduced line currents.

Disadvantages

• Ferroresonance.

• Fault level control.

• Capacitor fault level withstand.

Shunt Capacitor Compensation

Advantages

• No inherent ferroresonance risk.

• Capacitors do not carry line fault currents.

• Reduced line currents.

Disadvantages

• Automatic regulation only possible with expensive control gear.

• Switchgear and control equipment generally required.

• Voltage and V A R changes in discrete steps.

• Inability to respond to rapid load fluctuations.

• Risk of overcurrent damage from load generated harmonics.


17

2.5 Limitations of Series and Shunt Capacitor Compensation Schemes

2.5.1 Both series and shunt capacitor compensation can only provide benefits if the

power load power factor is significantly below unity. Although series and

shunt compensation operate in different ways they both reduce the voltage

drop effect of the line inductive reactance. Lines with a high resistance tend to

be beyond any real scope for improvement with any form of compensation

scheme.

2.5.2 Capacitor compensation techniques are concerned with improving the

electrical performance of the distribution network. They cannot create a

strong system if the existing line is of high resistance. A high line resistance

generally means that a major system augmentation is the only real solution to

severe voltage drop problems.

2.5.3 Compensation schemes in the correct locations can be highly effective in

improving power system performance. If installed without due considerations

to the fundamentals, they are likely to be ineffective.


18

3. An Overview of the Ferroresonance Phenomenon

3.1 The Nature of Ferroresonance

3.1.1 Normal linear circuit resonance is well understood and the conditions required

for resonance are well defined. The most c o m m o n resonances in linear electric

circuits involve series/parallel capacitors and inductances. Resonance(s) occur

at specific and easily predictable values of frequency, inductance and

capacitance.

3.1.2 In circuits consisting of linear resistive, capacitive and inductive elements with

constant voltage power frequency sources only one steady state solution will

exist. In these cases the branch currents and node voltages are single valued

and are predicable using standard circuit theory techniques 511.

3.1.3 Ferroresonance by comparison is more complex and much more difficult to

predict. Ferroresonance results from the interactions of linear circuit elements

in combination with non-linear transformers and chokes. It is the non linear B-

H characteristic of iron cored transformers and chokes that gives

ferroresonance its unique characteristics. Circuits susceptible to

ferroresonance can sustain multiple current waveforms of different frequencies

for a given supply voltage. In these situations the circuits have multiple

solutions to the governing differential equations. It is this aspect of

ferroresonance that makes it particularly interesting to study.

3.1.4 Ferroresonance is one of the fundamental barriers to the widespread use of

series capacitors in distribution and transmission lines. Ferroresonance can

cause dangerous system overvoltages and overcurrents. The conditions

generated by ferroresonance can damage power system equipment and


19

customer installations. Ferroresonance cannot be explained in conventional

linear circuit theory terms and requires sophisticated modelling.

3.1.5 Ferroresonant circuits can exhibit circuit behaviour that is far removed from

conventional linear circuits. Ferroresonant behaviour can have both

symmetrical and non-symmetrical voltage, current and flux waveforms. The

resulting waveforms can have a power frequency fundamental or can have

even or odd subharmonic waveforms. Chaotic circuit behaviour is also

possible with ferroresonant circuits [509].

3.1.6 Series compensation is only one source of power system ferroresonance. The

other major source of power system ferroresonance occurs with single phase

switching involving phase to ground capacitance and iron cored transformers

[301 to 323]. High voltage capacitive voltage transformers feeding iron cored

voltage transformers are also a potential source of ferroresonance [428]. The

ferroresonant models described later in this thesis can be applied to all these

situations. This thesis has concentrated on the ferroresonant effects associated

with series capacitor compensation.

3.1.7 Figure 6 shows the basic ferroresonant circuit configuration. The transformer

has a saturable iron core. The power line supplying the transformer is

represented by a linear inductance and resistance. The circuit contains a series

capacitor to tune out the effects of the line inductance. W h e n used for series

compensation, a capacitance will normally be chosen so as to tune out a

majority or all of the line inductance at the power frequency. This results in

the L C combination having a natural frequency close to the power frequency.

3.2 Linear Circuit Techniques

3.2.1 In the literature [401 to 431] a great deal has been published on mathematical

techniques for analysing the behaviour of ferroresonant circuits. M a n y of the


20

techniques involve making assumptions concerning the non-linear current/flux

linkage relationship.

Vr V c

Figure 6 Series Capacitor Ferroresonant Circuit

2 In many cases the mathematics becomes so complicated and so restricted to

the underlying assumptions that much of the value of the resulting

mathematical expressions is limited. The reality is that ferroresonance is a

complex non-linear phenomena and does not lend itself to clean mathematical

analysis like circuits with linear circuit elements.

3 A great deal can be understood about ferroresonance by studying the step

voltage response of the linear L R C circuit show in Figure 7. The speed at

which the capacitor voltage can change compared to the period of the power

frequency is a critical factor in determining if ferroresonance can be sustained

in the power frequency voltage source circuit shown in Figure 6.


21

VI V r y ^

A L

V(t)

R C

Figure 7 Linear Series L R C Circuit

3.2.4 Determining the rate of change of capacitor voltage in the linear LRC circuit

shown in Figure 7 requires solution of the linear differential equation (4) with

the appropriate initial conditions.

V = L-+R1+-jKlt (4)

3.2.5 There are three (3) fundamental responses possible. Overdamped, critically

damped and underdamped [511].

Case 1 Overdamped (R/2L)2 > 1/(LC)

Figure 8 shows the voltage response of a typically overdamped system.


22

OSO'O

T3

DH

S 7-1

>

o I

(A

o 00 CL, flj <Z)

.&*

CD

U i

•IH ?H

0) CD

swo

SOO'O"

S^OA

23

Case 2 Critically damped (R/2L)2 = 1/(LC)

Figure 9 shows the voltage response of a critically damped system.

Case 3 Underdamped 1/(LC) > (R/2L)2

Figure 10 shows the voltage response of an underdamped system.

Under power frequency supply conditions, overdamped and critically damped LRC

circuits in series with an iron cored transformer are unlikely to produce any

ferroresonant response because of the high level of damping. When studying

ferroresonance the most important and relevant L R C response is the underdamped case.

All the ferroresonant circuits examined in this thesis consist of linear L R C components

that have an underdamped behaviour.

Step Response Analysis of an Underdamped Series LRC Circuit

Below are the key equations that describe the behaviour of an underdamped linear LRC

circuit in response to a step voltage.

let tc=2L/R (5)

let j3=yll/LC-(R/2L)2 (6)

let Zc=^ (7)

Given the definitions given be equations (5),(6) and (7) it can be shown 51 lthat the step

response current is given by equation (8).

V -(-) I = — e *• sin(>ft) (8)


24

osoo

swo

owo

soo-o-

SjpDA

25

u

U

T3 0) OH

S rt

i

a> 03 fi

o </>

0)

0)

o

CD

U i

ci OD

0) •IH

CD

S;JOA

26

u P4 U

\4

OH

S •a

u fi

i 0>

03 fi

o OH 03 0)

_ °* •7" O H H-»

CD

U i

I 03

CU «pH

U 0> CD

0>

SOO'O-

sdure

27

Figure 11 shows the line current associated with the underdamped example

shown in Figure 10. This current response shows the nature of equation 8.

The current has a natural frequency of 0 radians per second and exponentially

decays toward zero with a time constant of tc seconds. O f particular

importance is the rate of rise of capacitor voltage shown in Figure 10 and the

time taken for the capacitor voltage to reach the applied step voltage.

Figure 12 shows the standard series ferroresonant circuit where the linear

L R C circuit elements are in series with a saturable transformer. In order to

gain a basic understanding of ferroresonance behaviour the transformer can be

considered as an open circuit when it is not saturated and a short circuit when

it is saturated. The transformer no load case is being considered. The

transformer flux linkage is governed by equation (9).

Vc

Figure 12 Series Ferroresonant Circuit

*. = Jvpdt (9)


28

Typical Ferroresonant Voltage, Current and Flux Waveforms

3.3.1 Figures 13 and 14 show the voltage, current and flux linkage waveforms of a

circuit in steady state 3rd subharmonic ferroresonance. Analysis of the

waveforms show that when the transformer is not saturated the current flow is

near zero with the result that the capacitor cannot charge or discharge and

hence the capacitor stays at near constant voltage. The voltage applied to the

transformer is the source voltage plus a contribution from the capacitor

voltage. This voltage after a short period of time causes the transformer to

saturate. Transformer saturation results in an effective short circuit across the

primary transformer terminals (Vp=0) and the supply voltage is suddenly

applied across the series L R C elements. The circuit response is similar to

applying the step voltage to the L R C circuit studied earlier. The capacitor will

charge up toward the applied supply voltage.

3.3.2 The rate at which the capacitor voltage can change to form a repetitive pattern

of transformer saturation is a critical factor in determining the possibility of

ferroresonance. During any ferroresonance the transformer can only remain in

saturation for part of a full cycle. If the circuit is capable of changing the

capacitor voltage by a significant amount (in the order of 5 % of the supply

voltage) during part of a cycle then ferroresonance is possible.

3.3.3 In order to predict the possibility of ferroresonance it is useful to examine the

ratio of the natural circuit frequency to the power frequency as defined by

equation 10.

Let Frequency Ratio fr = pV© (10)

3.3.4 The other key indicator is the X/R ratio of the circuit.

X/R = ©L/R (11)

= cotc/2 (12)

= 7ttc/T (13)


>*•

Supp

Itage

<r> O

> >

1

Line

Itage

^H o > >

1

29 2 u

Capa

Itage

U 0 > >

1

P

ransfor

oltage

> H >

s»l°A

30

£ <

8 iT X Oi

1X1 ^ x> x .£ a;

H d ^ J o

m in

o in

in

o

m CO

o co

N

X © ID

CO 0)

u I

6 m

o CN

m

in

o

o o

sjaqa w puB sdmy

31

3.3.5 Equation 13 shows that the X/R ratio is a direct measure of the ratio of the

decay time constant of the natural circuit behaviour (tc) to the power

frequency period (T).

3.3.6 The non linear nature of ferroresonance makes it a difficult and complex task

to predict. Computer modelling is the most c o m m o n way of predicting the

possibility of ferroresonance in a particular circuit.

3.3.7 Modelling and experimental work has shown that the frequency ratio and X/R

ratio can provide a simple method of determining by inspection the possibility

of ferroresonant states. Table 1 below has proved to be a good predictor as to

h o w these two simple ratios influence the risk of ferroresonance. The table

and comments below relate to typical power system conditions where a small

but significant line resistance is in the ferroresonant circuit and the transformer

is operating near its design voltage and hence close to saturation.

X/R«l No

Ferroresonance

X/R«l | No

Ferroresonance

X/R»l | No

Ferroresonance

No

Ferroresonance

Ferroresonance

Possible

Ferroresonance

Possible

No

Ferroresonance

Ferroresonance

Possible

Ferroresonance

Possible

Table 1 - Predictor of Ferroresonace

3.4 Influence of the Frequency Ratio on Ferroresonance Behaviour

3.4.1 For the establishment of sustained ferroresonance the circuit must be capable

of significantly changing the capacitor voltage by charging through the series


32

inductance on resistance over a small part of a cycle. For example if the power

frequency is 50 hertz the capacitor voltage must be able to change its voltage

by a significant amount in a period much less then 0.02 seconds. The

frequency ratio is a key indicator as to the capability of the circuit to exhibit

ferroresonant behaviour.

Case 1 fr « i

Due to the inherent low natural frequency of the circuit the capacitor voltage of the

ferroresonant circuit can change by only a very small amount during any interval the

transformer is in saturation. Under these conditions ferroresonance is not possible.

Case 2 fr «1

Under these conditions the capacitor voltage will change by a significant but limited

amount during any interval the transformer is in saturation. Under these conditions

ferroresonant states are possible. Because of the limited change in capacitor voltage at

each point of transformer saturation, the capacitor voltage tends to change in steps that

can generate repeating wave forms with subharmonic fundamentals. Ferroresonance is

possible.

Case 3 fr»l

Under these conditions the capacitor voltage can change rapidly and track the supply

voltage during any interval the transformer is in saturation. Under these conditions

ferroresonant states are possible.


33

3.5 Influence of the X / R Ratio on Ferroresonance Behaviour

3.5.1 The X/R ratio is a measure of the transient damping characteristic of circuit. A

high line resistance results in a highly damped system with a small X/R ratio.

Case A X/R« 1

Where the X/R ratio is much less than 1 the circuit is highly damped by the line

resistance and the generation of ferroresonance is not possible.

CaseB XZR*1

Under these conditions the circuit is moderately damped and if ferroresonance

establishes it is likely to produce steady state repeating waveforms.

CaseC X/R»l

Under these conditions the circuit is very underdamped. Where ferroresonance

establishes with a large X/R the circuit may create either repeating or non repeating

waveforms. N o n repeating chaotic circuit behaviour is possible [509].

3.5.2 When analysing series compensated circuits it is very useful to calculate the

frequency ratio and the time constant ratio to gain an understanding of the

ferroresonance possibilities.

3.6 Series Compensated Lines

3.6.1 In the case of series compensation, the series capacitor will normally be

chosen so as to tune out all or most of the line inductance at the power

frequency. This means that the frequency ratio fr will typically be unity or

slightly less than unity.


34

3.6.2 For distribution lines, typical X/R ratios are in the range of 0.1 for small

diameter steel conductor lines to 3 for high capacity lines with bundled

conductors [514].

3.6.3 Modelling and experimental work has shown that in series single phase

compensated distribution lines with typical ranges of frequency ratio and X/R

ratio, the most common type of ferroresonance is the subharmonic

fundamental type with stable repeating waveforms.

7 Ferroresonance Cause by Single Phase Switching

3.7.1 Another major source of ferroresonance in power systems is the result of

single phase switching [301 to 323] were there is naturally occurring phase to

earth capacitance. In general the phase to earth capacitance is small creating a

frequency ratio fr much greater than unity.

8 Generation of Chaotic Ferroresonant Waveforms

3.8.1 Generation of chaotic ferroresonance in power systems is rare but has been

identified as possible in the literature [429]. The generation of chaotic

ferroresonance has been reported in an electronic circuit by Deane and Hamill

[509]. The resonance was generated with a square wave voltage generator

operating at high frequency.

9 Conclusions on the Overview of Ferroresonance

3.9.1 A great deal can be understood about ferroresonance by applying linear circuit

techniques to what is a non linear problem. The basic series L R C elements are

linear and their transient behaviour is well understood. The transient behaviour

of the L R C elements can be completely analysed by conventional analytical

means. The saturable transformer is the non linear element and in its simplest


35

form can be thought of as a switch which is open when the core is unsaturated

and closed when the core is saturated.

2 The simple analysis techniques described provide a basic overview of

ferroresonance. The techniques have proved useful in gaining a fundamental

understanding of the ferroresonance phenomenon prior to detailed modelling

and analytical work.


36

4. Time Domain Transient Ferroresonance Model

4.1 The Governing Fundamental Ferroresonant Differential Equation

4.1.1 This model has been developed to operate in the time domain to examine the

behaviour of ferroresonant circuits. The model operates by numerical

integration of the following non-linear differential equation on a step by step

basis:

V(t) = L^ + ^+Ri +I|idt (14) dt dt CJ

where: X (i) is a non-linear function representing the transformer flux linkage.

4.2 The Modelling Process

4.2.1 Given the circuit quantities at t = ti the model calculates a new set of circuit

quantities at t = ti+At where At is a small increment of time. In making the

small step forward in time, an estimate of the new capacitor voltage is made

which by calculation results in a mismatch (error) between the capacitor

current and the transformer current. Applying Newton's method and other

numerical techniques to the error allows the calculation of an improved

estimate of capacitor voltage. Continued improved estimates of the circuit

parameters are achieved by iteration until the error falls below a

predetermined small limit. The model is very robust and able to cope with all

ferroresonant circuits analysed.

4.3 Initial Conditions

4.3.1 The initial conditions are often critical in determining the behaviour of the

circuit. Solution of the equation requires knowledge of the forcing function


37

V(t) and the initial conditions. The initial conditions that need to be defined at

the beginning of each simulation are the capacitor charge and flux linkage. O f

critical importance is the voltage angle at which the circuit is energised.

4.4 Modelling Features Required to find Numerous Ferroresonant Modes

4.4.1 To allow close simulation of experimental procedures, the model allows the

establishment of ferroresonance at a supply voltage of say 200 volts and then

simulating the effect of reducing the voltage down to say 170 volts. Using this

technique, the limits of ferroresonance operation can be found over a range of

supply voltages. The model has been designed to allow simulation of all initial

conditions including variation of turnon angles, residual capacitor charge and

residual flux linkage.

4.4.2 Ferroresonance is such a complex phenomenon that modelling under

conditions of a specified voltage of say 200 volts is not likely to produce the

full ferroresonant picture. There may be three or more ferroresonant states

possible for a specific supply voltage. T o achieve simulation of all the possible

modes requires careful consideration of initial conditions and the path that is

followed to a given supply voltage. Finding all the ferroresonant states

requires more than just a good simulation model, it requires insight by the user

in driving the model to achieve the desired results.

4.5 Modelling the Transient Response

4.5.1 The Time Domain Ferroresonance Model is suited to tracking the transient

response of ferroresonant circuits through until a steady state behaviour is

achieved. The understanding gained by the author of the transient circuit

behaviour has given insight into effective counter measures against

ferroresonance.


38

4.6 Interfacing with the Frequency Domain Model

4.6.1 The Time Domain Ferroresonance Model interfaces with another modelling

system that operates in the frequency domain. The Time Domain

Ferroresonance Model is designed to provide the Frequency Domain

Ferroresonance Model with an initial estimate of harmonic flux linkages. This

technique has proved very useful in mapping out the ferroresonant states in a

quick and effective manner. Both models can accept the transformer non

linear B-H characteristic as a continuously differentiable function.

4.7 Building the Time Domain Ferroresonance Model to suit Experimental

Requirements

4.7.1 Extensive experimental work has given insight into the features required

within the Time Domain Ferroresonance Model. Experimental work showed

that some ferroresonant states were very difficult to achieve in the laboratory.

For example some subharmonic ferroresonant waveforms at a specific supply

voltage required generation of the similar waveforms at a higher voltage,

followed by a slow reduction in the supply voltage. The Time Domain

Ferroresonance Model software was written to allow this type of simulation.

This is particularly important in the area of initial conditions and in being able

to track over a range of supply voltages from high to low or from low to high.

The time domain modelling also allowed the selection of circuit components

that were known to generate ferroresonant responses before the circuits were

constructed.


39

5. Frequency Domain Ferroresonance Model

5.1 The Modelling Process

v(t^ + idf+Ri4/idt <14> 5.1.1 The Frequency Domain Ferroresonance Model is governed by the non-linear

differential equation shown as equation (14). The Frequency Domain

Ferroresonance Model operates by making an estimate of the fundamental and

higher order harmonic flux linkages in the circuit. The method is known as

"harmonic balance" and has been used by researchers studying non linear

systems for many years. The number of harmonics to be analysed can be set by

the user. Typically 9 harmonics are analysed, each with sine and cosine

components giving 18 degrees of freedom. Where even harmonics are

analysed the D.C. level of transformer flux linkage forms an additional 19th

degree of freedom. Using the harmonic flux linkages, loop voltage errors are

calculated for all harmonic components.

5.1.2 The model uses the harmonic loop voltage errors to make an improved

estimate of flux linkages. This process involves making an individual small

change to each of the 19 harmonic flux linkages, one at a time. The 19 sets of

19 errors are used to calculate a 19x19 Jacobian matrix (incremental error

matrix). This matrix is inverted and multiplied by the error vector to produce

an improved estimate of the harmonic flux linkages using Newton's method.

This process is repeated until the error is sufficiently small.

5.1.3 A continuously differentiable B-H function is a fundamental requirement of

the Frequency Domain Ferroresonance Model.


40

5.2 Initial Estimates of Harmonics

5.2.1 An initial estimate of the harmonic flux linkages is required to start the

iteration process. The initial estimate of the harmonic flux linkages can come

from the Time Domain Ferroresonance Model or it can come from a known

nearby Frequency Domain solution.

5.3 Selection of Base Ferroresonant Fundamental Frequency

5.3.1 The base ferroresonant frequency (e.g. power frequency, 2nd subharmonic or

3rd subharmonic) must also be selected by the user prior to starting the

iteration process. Once selected the Frequency Domain Ferroresonance Model

is "blind" to other solutions that are not multiples of the fundamental search

frequency.

5.3.2 For example, if the user set the base ferroresonant frequency to search for

solutions with a 3rd subharmonic fundamental (50Hz/3 = 16.6Hz) it could not

find a 2nd subharmonic solution or a 5th subharmonic solution. The model

could however find a power frequency solution because the power frequency

is the 3rd harmonic of the 3rd subharmonic. Having completed the search for

3rd subharmonic solutions the user could then set the base ferroresonant

frequency to say a 2nd subharmonic frequency and search for additional

solutions.

5.4 Tracking a Locus of Frequency Domain Solutions

5.4.1 The advantage of the Frequency Domain Ferroresonance Model is that it

quickly converges on the steady state solution. Having found a solution in the

frequency domain, new solutions can then be found with small changes to

voltage levels and other variables creating a locus of solutions. This is

achieved by using the nearby solution as the initial condition for the search.


41

.5 Unstable Solutions not Experimentally Achievable

5.5.1 The model operates by identifying locations in state space where the non

linear differential circuit equation is satisfied. Multiple solutions can exist for a

given supply voltage and are found by using different initial conditions and by

searching for different subharmonics.

5.5.2 One of the most interesting findings has been that not all solutions found by

the Frequency Domain Ferroresonance Model are stable solutions that can

exist in practice. Investigation has shown that some solutions exist where the

fundamental ferroresonant differential equation (14) is satisfied but are

unstable and unsustainable in the laboratory. Solutions found in the Frequency

Domain Ferroresonance Model can be tested for stability using the Time

Domain Ferroresonance Model.

5.5.3 In the experimental ferroresonant circuits studied all the unstable solutions

identified have been associated with alternate ferroresonant solutions. In these

cases multiple solutions exist to the governing differential equation but one

solution dominates the other in some sense making it unstable. This is one area

where there is a fundamental difference between linear and non linear circuits.

5.5.4 In linear circuits, a real solution to the governing differential equations is a

necessary and sufficient condition for the solution to exist in practice. In the

non linear ferroresonant circuits studied, solution of the governing differential

equations is a necessary condition but not a sufficient condition for the

solution to exist in practice. Determination of whether a particular solution is

stable or unstable is of considerable interest and is covered later in chapter 8.

6 Interfacing the Frequency Domain Model with the Time Domain Model

5.6.1 The Time Domain Model and the Frequency Domain Model complement each

other and are extremely useful in analysing ferroresonance. Data can be


42

exchanged between models to aid in the search for solutions. This process

allows the mapping of the ferroresonant states with a composite approach.

The process of using both the Frequency Domain Model and the Time

Domain Model in combination is shown in Figure 15.


43

Set initial switch on conditions Vs,Vc, a, X

? — Complete time domain

simulation (typically 100 cycles)

Fourier analyse X into frequency components

Initialise transformer harmonic flux linkages \... \

Use non-linear transformer characteristics to determine in... i_

T

Multiply the inverted Jacobian Matrix by the error vector to determine

incremental change required to XQ ... X„

Determine harmonic error voltages E„ ... e„

I Invert Jacobian Matrix

Determine (n+1) by (n+1) Jacobian Matrix by

analysing individual small increments to X^... \

Use time domain analysis to determine if solution is stable

I M a p and store solution

I Step through required range of Vs (or other key variable) to create a locus of solutions. Use the previous

solution as the initial condition for the next run.

Figure 15 Composite Time Domain and Frequency Domain Model


44

" A " 6. Laboratory Ferroresonant Circuit "A

6.1 Laboratory Circuit "A"- General Arrangement

6.1.1 To test the ferroresonant models, a series compensated circuit was

constructed in the laboratory with the component values as shown in Figure

14. The series capacitor was selected to tune out the line inductance at 50 H z

and demonstrate the generation of both odd and even subharmonics. The

transformer was modelled with no load by equation (15).

L = 0,147H

KZT7P—wv— R=5,6 DHM

V(t)

C=69uF

M A " Figure 16 Ferroresonant Circuit "A

i = 0 . U +0.892 X5 .(15)

6.1.2 The transformer losses were modelled by a 1500 Q resistance across the

primary transformer terminals. The supply frequency was 50 Hz. The

transformer used was a single phase 200V/415V rated at 500 VA. The 200

volt transformer winding was used in the circuit with no connected load. The

supply voltage was sinusoidal. The resulting transformer magnetising curve is


45

shown in Figure 17 and the comparison of measured and modelled

magnetising current over a range of applied transformer voltage is shown in

Figure 18. The detailed results are provided in appendix A.

6.1.3 Circuit "A" represents the series compensation of a distribution line with

approximately 2 5 % line voltage drop at full transformer load. The circuit

characterises the compensation of a very high impedance line. Deviation from

standard per unit distribution line values were used to produce the desired

range of ferroresonant conditions.

6.1.4 This circuit was modelled using both the Time Domain Model and the

Frequency Domain Model and tested under a range of supply voltage

conditions to demonstrate the wide range of ferroresonant conditions possible.

6.2 Comparison of Experimental and Modelled Results

6.2.1 The following figures show the predicted and measured circuit behaviour.

Figure 19 shows the variation in R M S current over a range of applied source

voltages and Figure 20 shows the variation in capacitor voltage over a range

of applied source voltages.

6.2.2 50 Hz Fundamental with Higher Order Harmonics

6.2.2.1 This mode of circuit behaviour is not unusual and simply reflects

increasing transformer magnetising current with increasing applied voltage.

6.2.3 3rd Subharmonic

6.2.3.1 The 3rd subharmonic mode was sustainable only over a range of supply

voltages from 60 volts to 180 volts. Figures 21 and 22 show the Time

Domain Modelled voltage, current and flux waveforms at a supply voltage

of 120 volts R M S .


u is

o X Pu,

47

co

• IH

0)

Ott

Oct

Ol£

ooe

06Z

08Z

ozz

092

osz

OtZ

i- 0€Z

u

D PH

X

•IH

u O (X)

a rt *H

H cu cn cu

s a

h 0 " >

0)

O

o ti

o •IH

U

O

U

T3

(B 33 OJ O

o in p o fN

in

d

oz

01

-o o d

(SPYH sdure) juaurQ SuisfjiuS pv

(sdure) jjuaxinD 'S'WH

6 8

(S»I°A) aSejioA BHpediQ 'S'JMTI

o > o

(0

>

B* w> cn ^ 1/5 O

> >

50

<D 0»

C M •a n d ii — o > >

IH

O u « a. u o > >

o bo

tH 01

S IH O <*. c

^£

01 bo fl "o

> H >

N

O in

EC 01 i-H

u >>

u

s

SIPA

J-H >

o

£

IS) OH

s < 0)

c d

1

51

i

* * tn 3 bo £ ^ - -D ^ c o>

1

SJaclaM PUB sdiny

52

6.2.3.2 The waveforms show perfect symmetry and hence the absence of any

even harmonics. The graphs show that the transformer saturates twice on

the positive side followed by twice on the negative side. The pattern

repeats every 3 cycles hence the generation of the 3rd subharmonic

fundamental waveform of 16.7 Hz.

6.2.3.3 The cycle time used in the following description refer to the time bases in

Figures 21 and 22. W h e n the transformer is being driven into saturation the

transformer voltage falls to near zero causing the source voltage to be

applied to the other circuit elements, namely the series L R C . At the

beginning of the cycle (0.0 cycle time), the capacitor voltage is large and

negative. The capacitor voltage combined with the supply voltage result in

a high positive transformer voltage peaking at 0.25 cycle time. This causes

a high dX/dt which is the cause of the transformer going into saturation.

6.2.3.4 The transformer voltage falls to low values again when saturation occurs

at 0.6 cycle time. At this time the source voltage is effectively applied

across the series L R C elements forcing the capacitor with a large negative

voltage to a smaller negative voltage. The change in capacitor voltage is

determined by the natural frequency of the L and C which in this case is 50

Hz., the damping effect of R, the shape of the transformer A,(i) curve and

the time the transformer remains saturated.

6.2.3.5 When the transformer comes out of saturation at 0.7 cycle time, the

transformer offers a high impedance to the circuit and allows minimal

current flow. During this part of the cycle the capacitor voltage remains

nearly constant. The small negative capacitor voltage combined with the

supply voltage result in a second high positive transformer voltage which

forces the transformer into heavy positive saturation a second time at 1.4

cycle time.


53

6.2.3.6 During the second positive saturation phase, the capacitor charges rapidly

and attains a large positive voltage peaking at 1.6 cycle time. The large

positive capacitor voltage combined with the supply voltage result in a

large negative transformer voltage which forces the transformer into

saturation in the negative direction at 2.1 cycle time. Similarly, a second

heavy saturation occurs in the negative direction at 2.9 cycle time.

6.2.3.7 This process repeats over 3 complete 50 Hz cycles and hence the resulting

waveform has a 3rd subharmonic fundamental (16.7 Hz) with higher order

odd harmonics (e.g. 50 Hz, 83.3 Hz, 116.7 H z etc.).

6.2.4 2nd Subharmonic

6.2.4.1 The ferroresonant circuit also demonstrated the ability to generate 2nd

subharmonic voltages, currents and fluxes. In power systems, the linearity

of most system components and the symmetry of the transformer B-H

loops normally dictates that even harmonics cannot occur.

6.2.4.2 After ferroresonant even subharmonics were found using the Time

Domain Model, the Frequency Domain Model was modified to permit

even subharmonics and the existence of a D.C. component of transformer

flux linkage. Study of the literature revealed that the existence of even

subharmonics had been discovered as far back as 1941 by M c C r u m m

[203].

6.2.4.3 The modelling of the system predicted the existence of a 2nd subharmonic

ferroresonant state and experimental work confirmed the existence of such

a state. A stable 2nd subharmonic state existed for supply voltages in the

range 190 volts to 230 volts. Figures 23 and 24 show the predicted

modelled waveforms for the 2nd subharmonic ferroresonant state at an

applied voltage of 205 volts. These shapes show very good correlation

with experimentally measured wave shapes shown in Figure 25.


54

J3 o >

o CN II cn >

£> §H bo cn fl » o > >

1

c bo .5 «3

d ii « o > >

1

5-

2 ft «

^ rt u o > >

1

u. 0; E O 0)

g fl > H >

CO

o <+H

cu

6 cu bO rt 1—H

o >

ti

u

csi . cu u ti rt ti

o co cu VH

O CU to

cu *H

§> to

ti

o rt

ti

ti

S;I»A

55

X

to VH

o > in

o II >

GO

o, S < <u c

1

OJ bo <a f J X 3 bu

1 VH

J

5

u ts rH

w w

B

CO

VH

o CO

ti rt VH

ti rt

ti

CU ft

ti

U

1 a ° y*^rt cu •>

CJ ti rt ti

O co cu VH

O ft

cu to

ti

o

S VH

rt

43 ti

CD ti

CM

N

X © in

oi 1—4

u >>

U O)

s

bO

sjaqaM P U B sdiny

56

PM3304, FLUKE <S PHILIPS

channel 1 - Transformer Flux Linkage 1.2 webers/division

channel 4 - Current 2 amps/division

time 10 ms/division

supply voltage 205 volts R M S

Figure 25 Measured 2nd Subharmonic Waveforms Circuit "A"

6.2.4.4 Table 2 shows the magnitude and phase of the harmonic transformer flux

linkages. O f special interest is the existence of a D.C. level of transformer

flux linkage.


57

Table 2

Harmonic Components of Transformer Flux Linkage

205 V RMS 50 Hz supply voltage

2nd subharmonic 25 Hz fundamental

Units - Webers peak

Frequency

Hz Tml

Magnitu

0 (D.C.)

25

50

75 '

100

125

150

0.27

4,38;

0.36

•0.02

0.02

•0.02

0.00

-0.26

-0.10

0.01

0.01

0.00

0.00

0.27

1.40

0.37

0.02

002

OM 0.00

6.2.4.5 The key feature of the 2nd subharmonic ferroresonant state is that the

transformer goes into "light" saturation twice in one direction followed by

one "heavy" saturation in the opposite direction. This process repeats over

2 complete 50 Hz cycles and hence the resulting waveform has a 2nd

subharmonic fundamental (25 Hz) with higher order harmonics both odd

and even (e.g. 50 Hz, 75 Hz, 100 Hz etc.).

6.2.4.6 It should be noted that in the 2nd subharmonic ferroresonant state, the

currents generated are in the order of full load current of the transformer

and the capacitor voltage is approximately twice the level expected at full

load. These are extremely high levels of current and voltage in a circuit

with no connected load.


58

6.3 Ferroresonant Modes O f Circuit Behaviour

6.3.1 Using the circuit models, three distinct modes of circuit behaviour were

predicted. W h e n constructed, the circuit displayed all three predicted modes of

behaviour.

6.3.2 The results show clearly how the circuit has up to two (2) stable states for a

single supply voltage. For example, with an applied voltage of 140 volts, the

50 H z fundamental mode produces an R M S current of 0.2 amps, the 3rd

subharmonic mode produces an R M S current of 0.9 Amps. A third unstable

2nd subharmonic state exists at 2.6 Amps. At this unstable position, all the

circuit differential equations are satisfied but the 2nd harmonic state is

unsustainable possibly because of the characteristics of the 3rd subharmonic

state below. The issue of stability is discussed later in Chapter 8.

6.3.3 The ability of the circuit to produce an even second subharmonic fundamental

waveform was experimentally confirmed. In this mode the circuit behaves in a

non-symmetrical manner with a D.C. component of transformer flux. In the

example studied, a stable 2nd subharmonic state cannot be sustained until the

supply voltage exceeds the point where the 3rd subharmonic state has

collapsed. In this circuit the existence of a 3rd subharmonic state appears to

preclude the existence of a stable 2nd subharmonic state at the same supply

voltage. In 2nd subharmonic mode, the circuit displays negative incremental

impedance whereby the current increases as the source voltage is decreased

over the range 230 volts to 190 volts. These characteristics are evident in

Figure 19.

6.3.4 In 3rd subharmonic mode, the circuit displays negative incremental impedance

whereby the current increases as the source voltage is decreased over the

range 180 volts to 140 volts. The currents generated are greater than the


59

transformer magnetising current but significantly less than full load current

levels. These characteristics are evident in Figure 19.

4 Transient Behavioural Characteristics of the Linear RLC Circuit Elements

6.4.1 Circuit "A" has been shown to generate 2nd and 3rd subharmonic waveforms.

It is interesting to examine the linear R L C circuit elements that have allowed

the generation of these characteristics.

6.4.2 The linear RLC elements have the following characteristics:

frequency ratio = 0.998

X/R = 8.25

6.4.3 With reference to Table 1 "Predictor of Ferroresonace" on page 31 this circuit

falls into to the category of fr« 1 and X / R » l in which it is correctly

predicted that ferroresonance is possible. Appendix B shows the modelled

underdamped transient response of the linear L R C circuit elements.

5 Stored Energy in Circuit Components

6.5.1 For circuit "A", appendix F shows details of the stored energy within the

circuit components under number of ferroresonant states.


60

7. Laboratory Ferroresonant Circuit "B"

7.1 Laboratory Circuit 2- General Arrangement

7.1.1 A second laboratory ferroresonant circuit was constructed to allow

comparison of the model with experimental results. The transformer used (and

hence the flux linkage characteristic) was the same as the unit used in the

Laboratory Circuit "A". The circuit parameters are shown in Figure 26.

L = 0,054H C-182uF

VA R=5.6 DHM

V(t)

««>» Figure 26 Ferroresonant Circuit "B

7.1.2 Per unit component values were selected that are more in line with real

distribution lines. The circuit represents the series compensation of a

distribution line with approximately 1 5 % line voltage drop at full transformer

load, 0.9 load power factor and an X/R ratio of 3. The series compensation

reduces the full load voltage drop from 1 5 % to 6%.


61

7.1.3 This circuit was modelled using both the Time Domain Model and the

Frequency Domain Model and tested under a range of supply voltage

conditions.

7.2 Ferroresonant Modes Of Circuit Behaviour

7.2.1 Circuit "B" displayed modes of ferroresonant circuit behaviour similar to

circuit "A" but with some significant differences. Figure 27 shows the

variation in R M S current over a range of applied source voltages and Figure

28 shows the variation in capacitor voltage over a range of applied source

voltages. The detailed results are provided in appendix C.

7.2.2

7.2.3 The results show that circuit "B" is highly susceptible to ferroresonance. The

circuit displays three distinct modes. The first mode is 2nd subharmonic

ferroresonance with capacitor voltages in the order of 200 volts and currents

in the order of 8 amps. The second mode is 3rd subharmonic ferroresonance

with line currents in the order of 3 amps and capacitor voltages in the order of

100 volts. The third mode is the normal 50 H z transformer magnetising

current with small line currents and capacitor voltages.

7.2.4 This analysis illustrates the very high capacitor voltages that ferroresonance

can generate. The possibility of capacitor damage by ferroresonant

overvoltage is very obvious and effective counter measures are essential.

7.3 Existence of Three Circuit Modes of Behaviour over a Range of Supply

Voltages

7.3.1 Of particular interest is the observation that the circuit was able to support all

three modes of behaviour over a narrow range of supply voltages from 230


PQ

3 1 cu VH '• •»H -

ul •v* fi

fi 1 u 1

fa U • £ -. d_

S> 59 £ r«5

+-ti rt * ti O -CO CU i VH t

rro

CU i to -

?y.

-

; _

_ ; , {

; i

' • • ; • - . . . .

fl fl c c 0) Oi

s - .1 01 >H 0) IH rt OJ -rt 0) -r;

S OJ O OH fl 2 X R * £5 fl 41 R <u 53

>nic -

>nic -

nic -

nic -

iel

erim

£ § g g s S 2 5 (B <S 1 1 HH -fl _C _C

•9 -S S S ^ |i 3 P rt 3 *H JH 60 « S £ OJ 0) "S 'S 13 K * ^ C R j_ s_ O O (N (N CO CO LO m

1 I I I

1 ._.

I - -

o o c4

o o

(sdure) juajjro *s*W>I

pa

09Z

.2 u (N

m w

B

ti w VH "H 1

u cu bO rt +•» ^H

o oq > M VH

cu o VH **

P "3 W> rt "H *«. 1

u ti rt ti

O CO cu VH

o ft cu to

: :

' —

1—:

- ™

"3 T3

o S I

u '2 o 2 nfl

,fi 3 U5 T3

s (N

,

fl fl 0)

41 g 1 i

2 '2 2 S *H tj

2 a -5 x •S HO 3 fl cn S "C -Q C u <N CO

fl fl 4)

a EH 41 OH X 4)

'2 o 2 „fl -fl fl cn TJ VH

CO

1

| . .J ... .

fl C 4

a 1 — I -!-H

OJ rH

"g OH 1 1

^ £ SH SH

41 4 -C X O O m in

II

091 £

(SJIOA) aSejioA JOjpudio 'S'W'H

64 o

£H £H ti

rt

rt

cu

co co O

ti -*-

VH '2

o ti

C^ »** CO

CM cu ;** 0> bp O

£ S > *$

VH <+*

O O ."ti CU

ti

cu ti 0)

CM CU

•a ti HH

cu VH

^H

H

in in

o

LT> Ci

o

N X © in *• «j 03 CU ^H

u >» U cu

s

o

O

o

S-JPA JoipBcho X[ddns

65

7.3.2 The ability of circuit " B " to exhibit three independent states over a range of

supply voltages differentiates this circuit behaviour from that of circuit "A".

7.4 Transient Behaviour Characteristics of the Linear RLC Circuit Elements

7.4.1 Circuit "B" has been shown to generate 2nd and 3rd subharmonic waveforms.

Examination of the linear R L C circuit elements provide the following key

characteristics.

frequency ratio fr=1.00

X/R =3.03

7.4.2 With reference to Table 1 "Predictor of Ferroresonace" on page 31 this circuit

falls into to the category of fr« 1 and X / R » l in which it is correctly

predicted that ferroresonance is possible. Appendix D shows the modelled

underdamped transient response of the linear L R C circuit elements.


66

8. Stability of Frequency Domain Ferroresonant Solutions

8.1 Comparison of Circuits "A" and "B"

8.1.1 Circuits "A" and "B" both showed the common characteristic of having both

2nd and 3rd subharmonic ferroresonant states as shown previously in Figures

19, 20, 27 and 28. Circuit "A" showed the unusual characteristic of having an

unstable second subharmonic frequency domain solution over the range of

supply voltage from 120V to 190V.

8.1.2 Over this range of supply voltages, the frequency domain solution represents a

solution to the governing circuit differential equation (14). Despite the

existence of a 2nd subharmonic Frequency Domain solution to circuit "A"

over the range of supply voltages from 120V to 190V, no Time Domain

Model solution was predicted or was found in the actual laboratory circuit

operation. This type of circuit behaviour is unique to non linear circuits. In

linear circuits a solution to the governing differential equations ensures that

the solution state will exist in practice.

8.1.3 Examination of Figures 19 and 20 show that as the supply voltage is

decreased from 230V, a stable 2nd subharmonic ferroresonant state exits until

190V where the 3rd subharmonic state begins. In this circuit the existence of

the 3rd subharmonic state appears to preclude the existence of the 2nd

subharmonic.

8.1.4 The limits of stability of a ferroresonant state are commonly described as

where the determinant of the Jacobian approaches zero. The discussion

contained in the closure on the author's paper [5] refers to this issue.

Modelling and experimental work has shown that if the determinant of the

Jacobian approaching zero is the sole criterion used for predicting the limits of


67

a ferroresonant state then the range of ferroresonant states could be

overestimated. This is definitely the case with circuit "A".

8.2 State Plane Analysis

8.2.1 Three state variables are required to describe behaviour of the series

compensated circuits "A" and "B". These can be selected as:

• line current

• capacitor voltage

• transformer primary current

8.2.2 The transformer current and the line current are almost identical with the small

difference being due to the effect of the 1500 ohm resistor used to represent

the transformer no load losses. Based on the assumption that the line and

transformer currents are approximately equal, circuits "A" and " B " can be

represented by two state variables. The state variables are line current and

capacitor voltage.

8.2.3 Figure 30 shows the phase plane trajectories of circuit "A" at a supply voltage

of 180 volts R M S . Shown are the 50Hz trajectory, 3rd subharmonic trajectory

and the unstable 2nd subharmonic solution from the frequency domain model.

It should be noted that the unstable 2nd subharmonic solution could not be

achieved in the laboratory or in the time domain model. Figure 31 shows the

sweep areas of each of the trajectories.


68

LO

U CN

w w w

>

O oo rH

bfj rt

+•» >

l-H OH

OH

ti (/)

l CO O •!-! CD VH

CU £ VH U

ti CU •IH rt

ti, VH

cu ti

rt l-H

to CU rt

CD

ti

u VH u

r oo

*t

CN

cn

s

3

U cu

c •1-1

HJ

CM

vO

s*l°A Jaipwho

J u

SHJJOA mpvdvr)

70

8.2.4 In circuit "A" it can be observed that the trajectory sweep area of the unstable

2nd subharmonic on the phase plane does not totally enclose the trajectory

sweep area of the 3rd subharmonic. It can also be observed that the 3rd

subharmonic trajectory and the 50Hz trajectory intersect on the phase plane.

Despite the trajectory intersection, the 3rd subharmonic sweep area

completely contains the sweep area of the 50Hz trajectory because the 3rd

subharmonic trajectory loops within itself.

8.2.5 Figure 32 shows the state plane trajectories of circuit "B" at a supply voltage

of 240 volts R M S . Shown are the 50Hz trajectory, 3rd subharmonic trajectory

and the 2nd subharmonic trajectory. Figure 33 shows the sweep areas of each

of the trajectories.

8.2.6 In circuit "B" it can be observed that the trajectory sweep area of the stable

2nd subharmonic on the phase plane totally encloses the trajectory sweep area

of the 3rd subharmonic. It can also be observed that the sweep area of the

50Hz trajectory lies completely within the 3rd subharmonic sweep area.

8.2.7 The state plane trajectories provide an interesting view of the ferroresonance

phenomenon. The state plane does not give any indication of the supply

voltage angle. Hence intersection of different states on the state plane may

occur at completely parts of the supply voltage wave.

8.3 Stability of Frequency Domain Ferroresonant Solutions

8.3.1 It is well documented and understood that a ferroresonant state will cease to

exist when the Jacobian approaches zero [431]. Experimental and modelling

work has shown that a frequency domain ferroresonant solution may be

unstable in the time domain and unachievable in the actual circuit.


71

u CO

w w

B

>

O CM P

bO rt +•» l-H

o > I—H

CH ti

CD i

CM * Qj) VH

a 2 to t?

CU

C rt l-H to

0) rt

Cfl pa

ti

u VH •IH u

o CM

lO

ID

dj

6 <

cu & fl

U cu ti un

o I

LO

o tN

i

SH°A JOjpedlQ

72

u m

a

>

o CM CU bC rt l-H

o > l-H

PH

CH ti

CD CO

£ <

Figu

weep

CD CU ti rt l-H to

OJ 4-i rt +* CD

PH 4) 4 * in in

4 in O u C

j>p

PH

ti

a •IH

u

o CN

LO

LO

03 OH

s O < ti

E fl

U cu ti

LO I

o I

s-(8 -fl fl in

TJ C IN (fl 4 (H

(8 OH

4 4 CD

LO

8 I

sJl°A JO|pBd«3

73

8.3.2 It appears that one frequency domain solution in some way may interfere with

another ferroresonant solution (at a different base frequency) making it

unstable. This is particularly evident in the modelled and measured results for

circuit "A" as shown previously in figure 19 with respect to the 2nd

subharmonic.

8.3.3 The time domain simulation has been used as a reliable test of stability.

Stability is achieved if the time domain simulation produces steady repeating

waveforms over a long period of time.

8.3.4 When the harmonic balance approach is used, the model is blind to all

frequencies that are not multiples of the specified base frequency. It is not

surprising that when different base frequencies are used (e.g. 1/2 and 1/3

power frequency) that conflicting solutions are obtained where sometimes one

solution predominates over the other.

8.3.5 Unstable frequency domain solutions could be related in some way to state

plane trajectory intersections, state plain swept area intersection or other

criteria. Further research could lead to improved methods of stability

determination.


74

9. Management of Ferroresonance and Short Circuit

Withstand

Initiation of Ferroresonance in Series Capacitor Compensated Circuits

9.1.1 In a normal distribution power system operating environment the generation

of ferroresonance in a series compensation scheme under no-load or light load

conditions is likely unless effective counter measures are taken. The previous

results have shown that ferroresonance can generate capacitor overvoltages in

the order of 5 times normal full load operating voltage. During ferroresonant

states currents can be less than full load current or in the order of 5 times load

current and transformer voltage can rise above 2 times nominal.

9.1.2 A substantial transient such as energising a transformer via a switch or circuit

breaker is generally required to initiate a ferroresonant state. In a circuit

without effective countermeasures the ferroresonant state will continue

indefinitely. It is the ongoing nature of ferroresonance that makes it

particularly destructive.

9.1.3 The generation of ferroresonance will only occur if the transformer is driven

into saturation and the transformer is in a no load or light load situation. In the

single phase case where the capacitor holds no initial charge and the

transformer has no residual flux, a voltage switch on time corresponding to

sine 90 degrees (voltage peak) will avoid driving the transformer into

saturation and hence avoid the onset of ferroresonance. A switch on time

corresponding to sine(0) (voltage zero) will drive the transformer into heaviest

saturation and is the switching angle most likely to initiate ferroresonance.

9.1.4 Modelling and experimental work has shown that for single phase circuits with

no stored energy that are susceptible to ferroresonance, there is a critical


75

switching time between sine 0 and sine 90 degrees below which

ferroresonance may be initiated.

5 If the capacitor holds residual charge and/or the transformer holds residual

flux then the critical switching voltage angle can vary considerably. In some

ferroresonant circuits the ferroresonant states cannot be initiated by normal

switch on transient unless there is residual capacitor charge and/or transformer

flux. In experimental circuits "A" and "B" this was the case for the 2nd

subharmonic states.

6 Figure 34 shows a very simple radial series compensation scheme with a

typical arrangement of switchgear. The most likely standard switching

procedures that could provide the transient necessary to drive the system into

a ferroresonant state are:

a) Energising the transformer by closing circuit breaker " X ' with switches "Y"

and "Z " closed.

b) Energising the transformer by closing switch "Y" with circuit breaker " X ' and

switch "Z " closed.

c) Energising the transformer by closing switch " Z " with circuit breaker "X" and

switch " X ' closed.

D cn

CQ

>

X

Circuit Breaker

1 1kV O.H. Line

Y z

Series Capacitor

Transformer

Figure 34 Simplified Series Compensated llkV Line


76

9.1.7 These switching operations are normal day to day power system operations

that are likely to occur during substation commissioning, maintenance and

restoration of supply after faults.

9.1.8 The use of auto reclosing at circuit breaker "X' is a prime candidate for

initiating ferroresonance and is worthy of some attention. Auto reclosing is a

technique used to cater for short duration transient faults. With auto reclosing,

after a protection trip clears a line fault, the circuit breaker is automatically

reclosed after a predetermined time usually in the range 0.1 to 10 seconds. If

the fault remains on the line the circuit breaker trips and locks out. If the fault

is cleared the line remains energised after the reclose.

9.1.9 Auto reclosing is particularly prone to initiating ferroresonance because at the

time of clearing the fault there is a wide range of residual capacitor charge and

transformer flux conditions possible. W h e n the close occurs the voltage angle

can also vary over a wide range creating conditions that in many cases will be

conducive to ferroresonance. Another factor that can be overlooked is that the

feeder load may drop to near zero after a successful reclose. This can occur

because even a very brief interruption of supply can result in motor contactors

dropping out causing significant load shedding in industrial plants and

throughout the supply area.

9.1.10 The transient and load shedding characteristics of auto reclosing make it a

major consideration in the initiation of ferroresonance.

9.2 Short Circuit Fault Considerations

9.2.1 In any series compensation scheme, careful consideration needs to be given to

the effect the series elements have on system fault levels. It is essential that all


77

circuit elements are capable of sustaining the full short circuit conditions for a

reasonable fault clearing time without damage.

9.2.2 The use of series capacitors can greatly increase the system fault levels

because of the reduced overall system impedances. Table 3 shows the effect

on the system fault level of the series capacitors for the experimental circuit

"B".

Table 3

Modelled Short Circuit Fault Currents for the Laboratory

Series Compensated Circuit " B "

Configuration

No series compensation

With capacitor compensation

Cupimtw/LineCurrent

12 amps 6.0 p.u.

35 amps 17.5 p.u.

Notes: - rated load current is 2 amps

- all currents are R M S quantities

9.2.3 The table shows that in the laboratory circuit the fault level at the transformer

primary terminals has increased by almost a factor of three (3). This can have

a significant effect on the fault rating requirement of all power system

components, especially on the fault rating of the series capacitor.


78

9.2.4 The fault rating of the series capacitor is a major consideration. The short

circuit performance of capacitors is restricted in terms of the thermal impact of

short circuit current and the associated voltage stresses [131].


79

10. Existing Techniques for Managing Ferroresonance and

Short Circuit Currents

10.1 ASEA Series Compensation Circuit Arrangements of 1954

10.1.1 Figures 35 and 36 show two series capacitor schemes from the 1954 ASEA

Journal 102. One of the circuits is designed for small series compensation

schemes where the compensation is not more than a few hundred k V A R while

the other is designed for schemes in the order of 1 to 2 M V A R .

10.2 The Simple ASEA Spark Gap Series Compensation Circuit

10.2.1 The spark gap in shown in Figure 35 is set to operate when overvoltages

occur across the capacitor. In this circuit it is difficult to set the spark gap

threshold voltage high enough to cater for normal load currents yet low

enough to provide sufficient capacitor protection for through faults and

damping for subharmonic oscillations. In addition, damage could be caused

by low fault currents through the spark gap that may not be cleared by the

high voltage feeder protection. This circuit is quite simple but it has inherent

limitations.

10.2.2 The series compensation scheme shown in Figure 36 utilises a by-pass circuit

breaker, spark gap, damping resistor and special protection relays.

10.2.3 When subharmonic ferroresonant disturbances occur, the secondary winding

of the voltage transformer contains subharmonic voltages. The subharmonic

voltage is sensed using a low pass filter. After sensing the subharmonic

voltage for a brief period of time the protection scheme closes the circuit

breaker to place the damping resistor in parallel with the capacitor. The

resistor then damps out the subharmonic oscillations. After the protection


80

senses that the subharmonic oscillations are removed the circuit breaker is

opened and the circuit returns to normal operation.

d a m p i n g resistor spark g a p

series capacitor

<^>

bypass switch

ure 35 A S E A Series Capacitor Circuit for not more than few hundred k V A R

bypass switch

-0"O-

damping resistor

spark gap current transformer

Figure 36 A S E A Series Capacitor Circuit for 1-2 M V A R


81

10.3 T h e A S E A By-pass Circuit Breaker Circuit

10.3.1 During line short circuit conditions, the capacitor experiences a large voltage

rise. The voltage rise causes the protective spark gap to arc across. The arc

discharges the capacitor through the damping resistor. The protection senses

the fault current via the current transformer and then closes the circuit

breaker. The closed circuit breaker extinguishes the arc to protect the spark

gap. The short circuit withstand process involves three (3) distinct stages.

Stage 1: All fault current is carried by the capacitor.

Stage 2: Fault current is shared between the capacitor and the spark gap.

Stage 3: Fault current is shared between the capacitor and the circuit breaker.

10.3.2 This circuit is more robust that the previous circuit but it is considerably more

expensive due to the added cost of circuit breakers, voltage transformer,

current transformer and protection relays.

10.4 Modern Day Series Capacitor Arrangements for Distribution Lines

10.4.1 The fundamental techniques developed in 1954 are used by ASEA Brown

Boveri today in their "Minicap" series compensator for distribution [134].

Figure 37 shows the circuit arrangement for a typical "Minicap" installation.

10.4.2 In the modern circuit arrangement the vacuum bypass switch closes on the

loss of supply voltage to ensure the capacitor is bypassed when the circuit is

energised. This minimises the risk of ferroresonance during turn on and feeder

auto reclose. The system incorporates a precision triggered spark gap that is

used to initiate an instantaneous close of the vacuum bypass switch as soon as

an arc develops.


82

-o^b-

series capacitor

-O b- -&^o-

voltage transformer

arc detector

dischage limiting inductor

spark gap

O^O-

vacumm bypass switch - electrically operated

Figure 37 ASEA Brown Boveri Minicap - Typical Arrangement

10.4.3 The "Minicap" system has a discharge limiting inductance to limit the

discharge current of the capacitor when the bypass is initiated. In addition the

system incorporates sophisticated resonance detection equipment to protect

the system from overvoltages generated ferroresonance, self excitation of

motors and power frequency resonance.

10.4.4 The system described tends to be relatively expensive compared to line

augmentations and other solutions to distribution voltage regulation problems.

This thesis is concerned with examining the possibility of achieving series

capacitor compensation at the distribution level using a different approach

involving simpler and less expensive components.


83

11. A New Technique for Managing Ferroresonance

11.1 Areas for Potential Improvement of Existing Series Compensation

Arrangements

11.1.1 Modelling and experimental work with series compensated circuits have

shown what major problems ferroresonance and short circuit currents can be.

The fundamental approach to overcoming these problems has often been to

use protection devices to sense abnormal conditions of ferroresonance and

short circuit and then bypass the capacitor with a switch or circuit breaker to

protect the capacitor.

11.1.2 This approach requires expensive protection sensing equipment,

switches/circuit breakers and other related equipment. During normal

operation the circuit always remains susceptible to ferroresonance and heavy

short circuit current. It is only when abnormal conditions are sensed by the

protection that the circuit is altered to counteract the problems.

11.1.3 New techniques that can address the ferroresonance and short circuit issues

with less protection equipment and reduced hardware requirements offer great

potential benefits for the electricity supply industry.

11.2 Series Compensation With Saturable Choke And Damping Resistor

11.2.1 Figure 38 shows a series compensated circuit configuration with a damping

resistor and saturable choke. The transformer has a saturable iron core. The

power line supplying the transformer is represented by a linear inductance "L"

and resistance "R". The circuit contains a series capacitor to tune out the

effects of the line inductance. The capacitance will normally be chosen so as to

tune out all or most of the line inductance at the power frequency.


84

VI Vr

Saturating Damping Resistor Choke

Rd Xc Vc

^nrvA/w L R

V(-t)

Figure 38 Series Compensated Circuit with Damping Resistor and Saturable

Choke

11.2.2 The use of a saturable choke in conjunction with the damping resistor is the

innovative aspect of the circuit. Extensive modelling of this circuit shows that

it has properties that are well suited to series compensation of distribution

lines.

11.3 Theory of Operation

11.3.1 The fundamental aspect of the series compensation circuit is that all circuit

elements of the series compensator are permanently connected in the circuit.

There are no switches or circuit breakers and no protection relays.

11.3.2 Under emergency full load conditions the voltage across the capacitor will

typically reach 2 0 % of the supply voltage. The saturating choke is designed

such that at full emergency line loading, the knee point of the choke is

sufficiently high so as not to interfere with the normal operation of the

capacitor. Hence under normal operating conditions the choke draws only a


85

small magnetising current and the capacitor effectively carries all the load

current. This provides the line with the desired compensation effect.

11.3.3 Under conditions of subharmonic ferroresonance the capacitor voltage

increases substantially above the knee point driving the choke into saturation.

During saturation the choke looks like a short circuit which effectively places

the damping resistor in parallel with the capacitor. Careful selection of

components can eliminate the undesirable ferroresonant states. The low

frequency components of the subharmonic ferroresonance also assists with

saturating the choke.

11.3.4 Under fault short circuit conditions on the load side of the series compensator

the capacitor voltage rises substantially above normal causing the choke to

saturate. During the parts of the cycle where choke is saturated the damping

resistors are effectively in parallel with the capacitor. This has the effect of

reducing the overall line fault current and substantially reducing the fault

current carried by the capacitor.

11.3.5 One of the significant circuit aspects of the saturable choke is that it provides

a path for D.C. current to be bypassed around the capacitor. The location of

the saturating choke ensures that the series capacitor holds no steady state

D.C. component of voltage/charge. The circuit arrangement forces any D.C.

component of capacitor charge to be discharged via the damping resistor and

saturating choke. Trapped D.C. capacitor charge causes power transformer

saturation which can lead to the onset of ferroresonance.

11.4 Model and Experimental Results with the Series Compensator

11.4.1 Laboratory circuit "B" described earlier in this thesis was used as the base

circuit on which to test the series compensator technique. Figure 39 shows the

series compensated circuit that was modelled and constructed in the

laboratory.


86

Rd=19.1 DHM lllZ?±lnQ

rVvv^T) Xc(ic) L = 0.054H ic >\

R = 5,6 QHM C = 182uF

\A, Vp

V(t)

XCi)

Figure 39 Laboratory Series Compensated Circuit with Saturating Choke

11.4.2 The saturable choke was designed with a 50 hertz knee point voltage

sufficiently high to permit emergency full load operation yet low enough to

provide effective damping. Experimental work found that the saturating choke

could be modelled by equation (16). Details of the modelled and experimental

saturating choke results are shown in Figures 40 and 41 .

ic(^c) = 0.8X,C +28421 Xc9 .(16)

11.4.3 The circuit was modelled to determine the optimum damping resistor value to

eliminate the unwanted ferroresonant states. Modelling showed that if the

ohmic value was too high or too low the ferroresonant states would be

modified but not eliminated. A damping resistor value of 19.1 Q was selected

and inserted into the laboratory circuit.


OJ

U bO fi • IH

CO

&

OJ

O

U

« 3 a> co

fi

OJ

o

2 o a o CO

s o U

co

CO *J 1—(

O KJ

>

bO

o >

o

U

a>

Tx tx

(SIVH sdure) ;uaxin3 SuisfiraSBiv

89

11.4.4 Insertion of the damping resistor and saturable choke was completely

effective in eliminating the ferroresonant states. Figure 42 shows both the

model predicted and laboratory variation in R.M.S. current over a range of

applied source voltages at no load. Both the 3rd subharmonic ferroresonant

state and the 2nd ferroresonant state have been eliminated. The original 3rd

subharmonic ferroresonant and the 2nd ferroresonant states can be clearly

seen in the results of the uncompensated circuit in Figure 27.

11.4.5 The choke and damping resistor prevented the establishment of

ferroresonance following every switching transient attempt to excite the

circuit into a ferroresonant state.

11.4.6 In addition the laboratory circuit was brought into a range of ferroresonant

states without the saturable choke connected. While the circuit was in the

ferroresonant state the saturable choke and damping resistor were connected

to the circuit. The connection of the saturable choke and damping resistor

eliminated the ferroresonance in every case.

11.5 Modelled Transient Response of Laboratory Circuit

11.5.1 The previous experimental and modelling results have shown how the

saturable choke can eliminate the steady state 3rd subharmonic and 2nd

subharmonic ferroresonant modes. The effectiveness of the saturable choke

can be demonstrated by examining switch on transients that would normally

lead to ferroresonance.

11.5.2 When energising transformers, a zero voltage switch on angle is the most

severe starting condition with respect to transformer saturation and inrush

currents. For a series compensated circuit at rest (i.e. no stored energy), a

switch on at voltage zero is the condition most likely to initiate ferroresonance

because of the transformer saturation and inrush current effects.


90

0££

Q3£

OJ

M 0

u bO fi rt rt

CD

OJ *g

5b QQ

•iH

3 u •IH

u 4-» fi

OJ

5 (X)

i

c o

:.;::..::.;

1 :!

'' !

3 c OJ

1 _ s oi C

i o O-; £ a>

| £ £ IH SH

I <V 0J

] X X i o o ' m m 1 1

c 01 X ai t_ C H

09 a

H

o c 0 V) OJ Ci 0 IH IH

Qj

PH "0 HI

ro v. 0

-a c IS 0

2 1

'

- - - - - j - - - j- -; -- -j

I O

o o

1

o o CO

i

a c

\

V

/ 1

1

: : : | /

/

... -..- qj fl SH

- - u

o

~3 TJ

Pi

... d

I -- --

QJ

"o >

|> 1 5-1

S3

s u P H

IH

QJ ~

s S-

o H—

c (0 SH

H TJ aj (8

Pi K -—! \ j —

\

1 \ E

"~~~l>n i o o

O C

o c CN C

1 0l£

r f- 063

083

1 0Z3

093

1 0£3

- 0fr3

1 0£3

033

loi3

r\/x~7

003

\ 061

081

OZl

|09l

lost

Ion |0£t

lo3t

AT T ULL

0UL

06

08

- OZ

- 09

OS

Ofr

0£

03

n u

5

5

(sdine) JjuaxinD •S'pV'H

91

11.5.3 Figure 43 shows the simulated response to a zero voltage switch on without

the saturable choke in place. The figure shows that after a brief transient of

approximately five (5) cycles, a stable 3rd subharmonic ferroresonance is

established and continues indefinitely.

11.5.4 Figure 44 shows the simulation of the same series compensated circuit with

the saturable choke and damping resistor added. The saturable choke and

damping resistor have the effect of damping out the switch on transient in a

way that does not permit the establishment of steady state ferroresonance. At

0.6 of a cycle after switch on the choke current peaks at 4.1 amps and then

returns permanently to near zero amps after two (2) cycles. It is the ability of

the choke to provide a D.C. current path that makes the scheme so effective.

11.5.5 Comparison of the two (2) switch on transients show clearly the effectiveness

of the saturable choke and damping resistor in eliminating ferroresonance.

11.6 Fault Short Circuit Performance

11.6.1 Modelling has shown that the choke and damping resistor characteristics can

be effectively used to control and limit the system fault levels. Table 3

illustrated this point using the laboratory circuit under different configurations.

11.6.2 Table 4 illustrates how the saturable choke and damping resistor can be used

to control system fault levels. The fault currents referred to are for a short

circuit at the transformer terminals with a 200 volt R M S supply. The table

shows that the addition the capacitor to the uncompensated circuited increases

the fault level from 6 p.u. to 17.5 p.u.. This is a very large increase and is the

direct result of the power frequency tuning effect of the capacitor with the line

inductance.


OJ

O

X

u WD fi

•l-H

rt

3 rt

(X) fi

o

b fi u rH

u OJ •IH

CO fi rt

H fi

o <J • IH

CD

sjaqaM PUB sdury

OJ

O X

u bO fi • lH rt rH

fi rt

CD

g)g

•M fi OJ •IH

CO fi rt

u

H fi

o •IH

CD

sduiy

94

Table 4

Modelled Short Circuit Fault Currents for the Laboratory Circuit "B"

Capacitor

Current

P.U.

Choke

Current

P.tL

No series compensation

Capacitor only

Capacitor & choke

6

17.5

5.5

N/A

17.5

6.5

N/A

ill 4.5

Notes:- rated load current is 2 Amps = 1 P.U.

- all currents are R M S quantities

11.6.3 When the choke and damping resistor are added to the series capacitor the

choke heavily saturates under fault conditions causing a substantial circulating

current in the capacitor/choke loop. In this situation the line fault current is

reduced to 5.5 p.u. which is approximates the value of the line fault current in

the uncompensated case.


95

12. Selection Of Component Values

12.1 General Rules for Component Selection

12.1.1 On first inspection the selection of component values appears to be difficult

requiring extensive modelling of each individual situation. Modelling and

analysis of a number of circuit configurations has shown that selection of

component values can be relatively straightforward by the use of the following

rules.

12.2 Capacitor pF Selection

12.2.1 To achieve full compensation the capacitor value "C" should be selected to

give the same reactive impedance as the line inductance at the power

frequency.

i.e. for full compensation C=1/(LQ2) (17)

12.2.2 There are stability advantages in designing for less than full compensation.

Depending on the actual design situation, engineers may elect to less than fully

compensate for the inductive reactance of the line.

12.3 Damping Resistor Ohmic Selection

12.3.1 Modelling and experimental work has shown that the damping resistor ohmic

value is critical to the effective performance of the system. If the ohmic value

is too high then little current flows through the choke/resistor resulting in

ineffective damping under ferroresonant and fault conditions. If the ohmic

value is too low heavy currents flow through the damping resistor under


96

ferroresonant and fault conditions but the I2R loss is too small to provide

effective damping.

12.3.2 Analysis and experimental work on the series compensation scheme has

shown damping resistor ohmic value should be selected to approximately

equal the reactive impedance of the capacitor at the power frequency.

i.e. suggested Rj « 1/(©C) (18)

12.3.3 The reason for this selection is as follows. Circuit damping is provided by

saturating the choke and generating I2R losses in the damping resistor. Under

conditions of choke saturation the choke can be thought of as a short circuit.

Under these conditions the capacitor discharges directly into the resistor with

a time constant of RjC. At the suggested value of R j this time constant is 1/co

or 1/27C (approximately l/6th) of a period. Under conditions of ferroresonance

in the power line circuits the time period of high capacitor voltage is typically

half to one period or 3.5 to 7 R ^ C time constants. This is generally sufficient

time for the damping resistor to sufficiently discharge the capacitor and

prevent the onset of ferroresonance.

12.3.4 If Rd is too large the Capacitor damping resistor time constant is too long to

allow effective damping. If R d is too small the leakage reactance of the

saturating choke becomes significant limiting the circulating current and hence

the I2R effect of the damping resistor.

12.3.5 After selection of a damping resistor value R^ it is important to model the

series compensation scheme to check that all ferroresonant states have been

eliminated.


97

12.4 Saturable Choke

12.4.1 There are a number of aspects of the saturating choke that require careful

design consideration.

12.4.2 Knee Point

12.4.2.1 The knee point of the saturable choke must be sufficiently high to permit

normal and emergency loads without any saturation effects. However for

currents in excess of the emergency load plus a safety margin the choke

must go heavily into saturation.

12.4.3 Leakage Inductance

12.4.3.1 The leakage inductance is the marginal inductance of the choke when it

is in full saturation. The leakage inductance must be sufficiently small so as

not to interfere with the discharge of the capacitor into the damping

resistor. The ideal value would be zero henries.

12.4.3.2 To achieve the desired effect the natural frequency of the capacitor in

combination with the leakage reactance of the choke (Lcj) must be much

greater than the power frequency.

l/Sqrt(LclC) » © ( 19)

or

Lcl« 1/(C©2) (20)


98

13. Scaling Up To Real Distribution Networks

13.1 Limitations of Small Scale Laboratory Experiments

13.1.1 The experimental results have been successful in determining the effectiveness

of the modelling and providing practical insight into the issues associated with

series compensation. The limitation of the experimental work has been one of

scale. The experimental work has been dealing with circuits with a supply

voltage of 240 volts and load currents of 2 amps.

13.1.2 To be effective in practice, the proposed series compensation scheme needs

to operate for a supply voltage of 1 lkV and up and load currents of 10 amps

up to 1000's of amps. Testing and experimenting on this scale has simply not

been possible.

13.1.3 The question arises, are the component values and ratings required for real

series compensation scheme within the realm of engineering practicability? To

answer this question a circuit with realistic distribution values was modelled

and the component values and ratings determined.

13.1.4 The single phase circuit to be modelled is shown in Figure 45. The nominal

supply voltage was llkV and the maximum emergency line load was 262

amps. Worst case conditions were examined including no loss distribution

transformers with sharp saturation curves. The line has 1 1 % full load voltage

regulation at 0.85 power factor.


99

Rd = 6.28 DHM s-aturQ+in9 Choke

Xc(ic)

L = 0,02H IC

R=l,55 DHM C=507uF

V(t) nominal llkV 1 phase

Figure 45 Large Scale Series Compensated Circuit with Saturating Choke

Circuit " C "

13.1.5 The choke and transformer characteristics used are detailed in equations

below. The transformer losses were represented by a 14,400 ohm resistance

across the transformer llkV terminals. The transformer characteristics were

based on commercially available modern power distribution transformers.

i = 0.0165X + 1.429 10"30 X 1 7 (21)

ic=1.0 10-30A,c

15 (22)

13.1.6 This circuit provides full compensation and reduces the line voltage drop

from 1 1 % to 3%. In distribution networks, the length of line, number of

transformers and loading can vary greatly with changes to the switching

configuration and the construction of additions to the system over time.


100

13.1.7 The modelled characteristics of the system are described in the following

figures.

Figure 46 No Load Ferroresonant States with no Saturating Choke Compensation.

Figure 47 Short Circuit Fault level Comparisons with and without Compensation.

Figure 48 Capacitor voltage with and without Compensation under Short Circuit Fault

Conditions.

Figure 49 Choke, Capacitor and Line currents under short circuit fault conditions.

13.1.8 The modelling clearly demonstrates that the damping choke limits the fault

duty of both the line and capacitor. Further modelling also showed that the

ferroresonant states were eliminated.

13.2 Equipment Rating

13.2.1 Based on the modelling, the following ratings were devised for Circuit "C"

Vsrc= 11,000 volts

L=0.02 H

RL=1.55ohm

1 phase

Series Capacitor

C=507 pF

Continuous rating

200 amps rms (252 kvar)

1,256 volts rms

1,775 volts peak


101

co OJ

rt

CD

fi rt fi

O co OJ

u O ft OJ PJH OJ l-H

O

& OJ CO

CD OJ l-H rt <j

CD

SP rt

o o DO

o o vD

O o • < *

o o <N

SPMH sdray

102

IN <*

CO fi O CO •IH

u rt D< & O U

fi rt

o o o

o o o

o o o

SWH sduiy lin«j

103

o o o

8 o

o o o

o o o

o o o

o o o

o o o

o o o

o o o

o o o

o o o CN

o o o

'o > 1—4

CM

Cm

co

SW& aSBj[o A rajped^

0 rt

u • IH

u o X CD h OJ

-a fi

P fi

o •iH * • »

fi

CO

OJ .2 1

oo +j *d

a Si u

I

E OJ +•> CO >>

CD

•a OJ

rt

co fi OJ

&

O

U OJ

o

u

W3 OJ

o >

a CM

s CO

SPVH sdray juauio jpre j

105

3 hour rating

263 amps rms (435 kvar)

1,651 volts rms

2,335 volts peak

3 second fault rating

1,754 amps rms

2,547 amps peak

10,466 volts rms

15,677 volts peak

Damping Resistor

6.28 ohm

3 second rating

1454 amps rms (13.3Mw)

2481 amps peak

Saturable Choke

3 second rating

1,454 amps rms

2,481 amps peak


106

13.3 Comments on the Component Specifications

13.3.1 The required ratings of system components indicate that they are within the

realm of engineering reality.

13.3.2 The continuous rating of the capacitor is readily achievable. The difficulty in

economically constructing such a capacitor is providing for the 3 second fault

rating. This is a fundamental problem of series compensation schemes. It

should be noted that the compensation arrangement not only eliminated the

ferroresonance problem but also reduces the capacitor's 3 second rating from

7,000 amps (uncompensated) to 1,900 amps (compensated). The damping

choke improves the economic feasibility of constructing a suitable capacitor by

a great margin.

13.3.3 There would be no major technical problems in the construction of the

saturating choke or the damping resistor. Pole mounting of all the equipment

is envisaged.

13.4 Ferroresonance Protection

13.4.1 In order to provide additional protection to the distribution network and the

electricity customers, subharmonic ferroresonance protection could be

considered. Figure 50 shows a method of subharmonic ferroresonance

protection which could be provided.

Saturating Choke

3 DO

/

Low Pass Filter Ferroresonant Detection Relay

11 kV O.H. Line A*S*tQ-

Zone Substation Circuit Breaker

metering class current transformer

Tronsformer(s)

Series Capacitor

Figure 50 Proposed Ferroresonant Protection Scheme


107

13.4.2 The subharmonic protection relay would incorporate a low pass filter and

cause the llkV circuit breaker to operate if subharmonic currents were

detected. Such a scheme would use existing feeder circuit breakers and could

be incorporated at relatively low cost.

13.4.3 Subharmonic protection would provide added security to the scheme and

provide some backup protection on the damage to either the saturating choke

or the damping resistor.


108

14. Future Research Directions

14.1 The Need for Field Trials

14.1.1 Further research into the proposed series compensation scheme requires field

trials at 1 lkV or similar voltages. There are limitations and inherent risks in

scaling up the results of small scale laboratory experiments to larger real

power systems. Large scale systems tend to have transformers with lower

losses and sharper saturation characteristics. In many cases real power systems

have high line X/R ratios that are difficult to reproduce at small scale.

14.1.2 For these reasons larger scale field trials are considered the next logical step

in the research and development process.

14.2 The Likelihood of Universal Solutions

14.2.1 Modelling of a number of systems suggests that the use of the component

selection techniques described in section 11 could eliminate the need to model

each individual application. Further modelling and actual field trials are

necessary to draw any definite conclusions on the issue.

14.3 Three Phase Systems

14.3.1 The saturating choke technique in the three phase situation has a great deal of

potential. Analysis of the three phase system has not been attempted in any

depth.

14.3.2 A three phase ferroresonant laboratory circuit was constructed using

component values that model a realistic three phase distribution line. A

number of complex subharmonic waveforms were generated. O f particular

interest was the fact that the saturating choke technique with damping resistor

was able to eliminate all the ferroresonant states. These preliminary results are


109

very encouraging and show that additional modelling of three phase systems is

likely to lead to successful series compensation using the saturating choke

technique on three phase lines. Appendix E shows details of the three phase

laboratory circuit " D " constructed.

14.4 Stability Criteria for Ferroresonance Frequency Domain Solutions

14.4.1 Time domain modelling has been used with success as a test for the stability

of frequency domain solutions. Stability of a frequency domain solutions is

confirmed when the solution can be modelled in the time domain as a steady

repeating waveform over a long period of time.

14.4.2 Opportunities exist for further research into causes of instability and

improved testing for stability.


110

15. Conclusions

15.1 The Benefits of Series Compensation

15.1.1 Increasing the power carrying capacity of transmission and distribution lines

by series compensation offers great potential for electricity supply authorities.

Effective series compensation can reduce voltage regulation in distribution and

transmission systems and provide an effective countermeasure to voltage dips

caused by load fluctuations.

15.2 Modelling the Ferroresonance Phenomenon

15.2.1 The modelling and experimental work presented has highlighted the damaging

ferroresonant overvoltages and overcurrents that can be created by series

capacitors interacting with transformers. A thorough understanding of the

possible ferroresonant modes of behaviour is essential when considering series

compensation of distribution and subtransmission lines.

15.2.2 The Time Domain and Frequency Domain Ferroresonant Models have proved

to be very useful and accurate. The models give the design engineer a valuable

insight into the ferroresonant phenomenon as applicable to series

compensation and other situations. For engineers designing and studying the

feasibility of series line compensation the models offer the ability to:

• determine the possibility of ferroresonant overvoltages and currents.

• determine the possible modes of ferroresonant behaviour.

• determine under what range of operating conditions ferroresonance

can occur.


Ill

• develop and analyse strategies for eliminating ferroresonant

conditions.

15.2.3 The generation of power system even subharmonic ferroresonant states was

demonstrated both experimentally and by modelling in the demonstration

circuits. The generation of even harmonic voltages and fluxes is most unusual

in power systems because of the linear nature of most system components and

the symmetry of transformer B-H loops. Even harmonic generation resulted

from non-symmetrical circuit behaviour, the key element of which was the

existence of a D C component of transformer flux linkage. The models also

predicted the existence of odd ferroresonant harmonics in the experimental

circuits. Experimental work confirmed the existence of all the predicted

ferroresonant states.

15.2.4 The ferroresonant states associated with series compensated transmission and

distribution lines can be modelled and understood. Understanding and

modelling the phenomenon is the key to designing effective countermeasures.

15.3 A New Method of Managing Series Compensation

15.3.1 A method of eliminating ferroresonance in series compensated lines has been

proposed, modelled and found to be effective in a small scale series

compensated laboratory circuit. The modelling and experimental work

presented has highlighted h o w ferroresonant overvoltages and overcurrents

can be eliminated by the use of a saturable choke and damping resistor.

15.3.2 The key to designing a successful scheme is to determine by modelling the

natural ferroresonant states. This modelling is generally required over a wide

range of supply conditions. Having determined the natural ferroresonant

states, additional modelling of the system incorporating the saturable choke


112

and damping resistor are required to ensure that all natural ferroresonant

states are eliminated and that no new states are created.

15.3.3 The performance of the system under transient and short circuit conditions is

of critical importance and needs to be considered at the early design stage.

Fault levels can be controlled within limits to suit the designer's requirements.

15.3.4 The saturable choke technique is simple, effective and requires no

sophisticated control, protection or bypass switch systems. The choke and

damping resistor technique is a non-linear solution to a complex non-linear

problem. With further research the technique opens the way for the more

widespread use of series capacitors in distribution and subtransmission electric

power systems.

15.4 Component Values

15.4.1 Simple techniques for selecting component values have been developed that

with additional modelling and experience may lead to universal general

purpose series compensation solutions. Field trials and additional modelling

are required to make further progress.


113

16. References

16.1 Author's Papers

[1] R.A.Barr, "A Nomogram for the Selection of Matching Design Parameters for the

Underground Residential Distribution of Electricity" Journal of Electrical and

Electronics Engineering, Australia. Vol. 3, No. 2 June 1983, pp. 69-74.

[2] A. Baitch and R.A.Barr, "A Tapping Range and Voltage Level Analysis Chart for

Tap Changing Transformers" IEEE Transactions on Power Apparatus and

Systems, Vol. PAS-104, No. 11, November 1985, pp. 3269-3277.

[3] RABarr and D. Piatt, "Voltage Regulation in Power Transformers by the Control

of Leakage Flux" IEEE Conference paper - Conference on Electromagnetic Field

Computation held at Washington, D.C. U.S.A. December 1988.

[4] R.A. Barr and D. Piatt, "A Temperature Controlled Non-Linear Choke"

Proceedings of the Australasian Universities Power Engineering Conference held

at Wollongong N.S.W. Australia, September 1993, Vol. 2, pp. 687-692.

[5] R.A. Barr and D. Piatt "Modelling and Mapping Ferroresonant States in Series

Compensated Distribution and Subtransmission Lines" IEEE Transactions on

Power Delivery, Vol. 11, No.2, April 1996, pp. 931-939.

[6] R.A. Barr and D. Piatt "Use of a Saturating Choke in the Series Capacitor

Compensation of Distribution Lines" IEE Proceedings on Generation,

Transmission and Distribution, Vol. 143, No.6, November 1996.


114

16.2 Series Compensated Lines

[101] W.G. Shepard, "AC Voltage Regulation with Ordinary Transformers" Electronics

July 1953, pp. 238-244.

[102] S. Smedsfelt and P. Hjertberg, "Series Capacitors for Distribution Networks"

A S E A Journal, Vol. 27, September 1954, pp. 123-136.

[103] E.F. Kratz, L.W. Manning and M. Maxwell, "Ferroresonance in Series Capacitor

Distribution Transformer Applications" Trans. Amer. I.E.E. (P.AS), Vol. 78, Pt

3A, August 1959, pp. 438-449.

[104] V. Madzarevic, "Ease Overvoltages Due to Faults" Electrical World, August 15,

1977, pp. 64-65.

[105] F. Iliceto and E. Cinieri, "Comparative Analysis of Series and Shunt

Compensation Schemes for A C Transmission Systems" IEEE Transactions on

Power Apparatus and Systems, Vol. PAS-96, No. 6, November/December 1977,

pp. 1818-1830.

[106] J.L. Batho, J.E. Hardy and N. Tolmunen, "Series Capacitor Installations in the

B.C. Hydro 500kV System" IEEE Transactions on Power Apparatus and

Systems, Vol. PAS-96, No. 6, November/December 1977, pp. 1767-1776.

[107] V. Madzarevic, F.K. Tseng, D.H. Woo, W.D. Niebuhr and R.G. Rocamora,

"Overvoltages on E H V Transmission Lines Due to Faults and Subsequent

Bypassing of Series Capacitors", IEEE Transactions on Power Apparatus and

Systems, Vol. PAS-96, N o 6, November/December 1977, pp. 1977-1855.


115

[108] A.L. Courts, N.G. Hingorani and G.E. Stemler, "A N e w Series Capacitor

Protection Scheme Using Non-Linear Resistors" IEEE Transactions on Power

Apparatus and Systems, Vol. PAS-97, No. 4, July/August 1978.

[109] E.R. Taylor, "Application of Series Capacitors" Electrical Engineer, October

1978, pp. 7-16.

[110] S.C. Tripathy, K.K. Patel and M.Y. Khan, "Digital Computer Study of Switching

Surges on Series Compensated Lines" Institution of Engineers India Journal

Electrical, Vol. 59, June 1979, pp. 340-346.

[Ill] J.J. Burke, A.P. Engel and S.R. Gilligan, "Increasing the Power System Capacity

of the 50kV Black Mesa and Lake Powell Railroad through Harmonic Filtering

and Series Compensation" IEEE Trans. Power Apparatus and Systems, Vol. PAS-

98, N o 4 July/Aug 1979, pp. 1268-1274.

[112] E.E. Baraket and D.E. Hirst, "Susceptibility of 3-phase Power Systems fo Ferro-

nonlinear Oscillations" IEE Proc. Vol. 126, N o 12, December 1979, pp. 1295-

1300.

[113] C.A. Peterson and J.C. Osterhout, "Metal Oxide Protector for Series Capacitor

Installations" Proceedings of the American Power Conference, Vol. 42, 1980, pp.

692-697.

[114] R.F. Wolff, "Metal Oxide Improves Capacitor Protection" Electrical World,

March 1980, pp. 52-53.


116

[115] G.T. Bellarmine and K. Srikrishna, "Optimum Compensation with Series

Capacitor at Centre of Transmission Line" Institution of Engineers India Journal

Electrical, Vol. 60, June 1980, pp. 275-279.

[116] N.T. Fahlein, "EHV Series Capacitor Equipment Protection and Control" IEE

Proc, Vol. 128, Part C, No. 6, November 1981, pp. 394-401.

[117] K. Murotani, K. Takenaka, M. Asano and M. Inouye, "Development and Testing

of 500kV Series Capacitor", IEEE Transactions on Power Apparatus and

Systems, Vol. PAS-101, No.7, July 1982, pp. 2187-2193.

[118] T. Baitch, "Series Capacitors Boost 1 lkV Line Performance" Australian Electrical

World, Vol. 47, No. 8, pp. 38-48, August 1982.

[119] Y. Mansour, T.G. Martinich and J.E. Drakos, "B.C. Hydro Series Capacitor Bank

Staged Fault Test" IEEE Transactions on Power Apparatus and Systems, Vol.

PAS-102,No.7, July 1983, pp. 1960-1969.

[120] S.P. Seth, "Comparative Analysis of Shunt and Series Compensation Schemes for

400kV Lines" Institution of Engineers India Journal Electrical, Vol. 63, August

1983, pp. 46-50.

[121] C.S. Indulkar, "Series Compensation of EHV Transmission Lines" Institution of

Engineers India Journal Electrical, Vol. 65, October 1984, pp. 85-88.

[122] A. Kalam, "Simulation of Series Compensated EHV Transmission Lines and Their

Protection" Institution of Engineers India Journal Electrical, Vol. 60, December

1985, pp. 178-181.


117

[123] Goldsworthy, "A Linearised Model for MOV-Protected Series Capacitors" IEEE

Transactions on Power Systems, Vol. PWRS-2, No. 4, November 1987, pp. 953-

958.

[124] C.S. Indulkar and B. Viswanatnan, "Maximum Transfer Limited by Voltage

Stability in Series and Shunt Compensated Schemes for A.C. Transmission Lines"

IEEE Transactions on Power Delivery, Vol. 4, No. 2, April 1989, pp. 1246-1252.

[125] Boon-Teck Ooi, Shu-Zu Dai and Xiao Wang, "Solid-State Capacitive Reactance

Compensators" IEEE Trans. Power Delivery, Vol. 7, No. 2 April 1991, pp. 914-

919.

[126] G. G. Karady, "Concept of a Circuit Current Limiter and Series Compensator"

IEEE Trans. Power Delivery, Vol. 6, No. 3 July 1991 pp. 1031-1037.

[127] M. El-Marsafawy, "Application of Series-Capacitor and Shunt-Reactor

Compensation to an existing Practical A C Transmission Line" IEE Proceedings-C

Vol. 138, N o 4, July 1991, pp. 330-336.

[128] R. Vitelli, "Series Capacitors in Distribution Systems" Regional Conference of

the Electricity Supply Engineers Association of N.S.W. held at Narrabri, N.S.W.

Australia 2-3 April 1992.

[129] B.Ooi, S Dai and X. Wang, "Solid State Series Capacitive Reactance

Compensators" IEEE Transactions on Power Delivery, Vol. 7, N o 2 April 1992,

pp. 914-919.

[130] S. G. Helbing and G.G. Karady, "Investigations of an Advanced Form of Series

Compensation" IEEE Transactions on Power Delivery, Vol. 9, No.2, April 1994,

pp. 939-945.


118

[131] "IEEE Standard for Series Capacitors in Power System" IEEE Std 824-1994.

[132] P. Halvarsson and L. Angquist, "Controlled Series Capacitors - Practical and

Economical Solutions" Institution of Engineers Australia, Proceedings of the

Electrical Engineering Congress held in Sydney Australia, November 1994, pp.

147-150.

[133] G. Ledwich and A. Ghosh, "Series Compensation: Steady State Analysis"

Institution of Engineers Australia, Proceedings of the Electrical Engineering

Congress held in Sydney Australia, November 1994, pp. 151-156.

[134] ABB Power Systems, "Minicap Series Compensation of Distribution Lines".

Pamphlet A02-0123 E

16.3 Ferroresonant Subharmonics

[201] I. Travis and CN. Weygandt, "Subharmonics in Circuits Containing Iron-Cored

Reactors" Trans. Amer. I.E.E. (P.A.S.) August 1938, Vol. 57, pp. 423-431.

[202] I. Travis, "Subharmonics in Circuits Containing Iron-Cored Reactors IF'. Trans.

Amer. I.E.E. (P.A.S.), Vol. 58, 1939, pp. 735-742.

[203] J.D. McCrumm, "An Experimental Investigation of Subharmonic Currents" Trans.

Amer. I.E.E. (P.A.S.), 1941, Vol. 60, pp. 533-540.

[204] E. Brenner, "Subharmonic Response of the Ferroresonant Circuit with Coil

Hysteresis" Trans. Amer. I.E.E. (P.A.S.), September 1956, Vol. 75, pp. 450-456.


119

[205] LA. Wright, "Three Phase Subharmonic Oscillations in Symmetrical Power

Systems" Paper 70 TP 625-PWR IEEE Transmission and Distribution Committee

May 1970, pp. 1295-1304.

[206] LA. Wright and K. Morsztyn, "Subharmonic Oscillations in Power Systems

Theory and Practice" Trans. IEEE (PAS) Vol. PAS 89, No. 8, November 1970,

pp. 1805-1815.

[207] T.C. Cheng, "The Effect of Subsynchronous Current on a Static Mho Type

Distance Relay" IEEE Transactions on Power Apparatus and Systems, Vol. PAS-

100, No. 11, November 1981, pp. 4562-4570.

[208] A.S. Akpinar and S.A. Nasar, "Harmonic Balance Analysis of the Subharmonic

Ferroresonance" Electric Machines and Power Systems, Vol. 18, 1990, pp. 409-

428.

16.4 Ferroresonance Associated with High Voltage Switching

[301]E. Clarke, H.A. Peterson and P.H. Light, "Abnormal Voltage Conditions in Three-

Phase Systems Produced by Single-Phase Switching" Trans. Amer. I.E.E.

(P.A.S.), 1941, Vol. 60, pp. 329-339.

[302] G.G. Auer and A.J. Schultz, "An Analysis of 14.4/24.9kV Grounded Wye

Distribution System Overvoltages. Trans. Amer. I.E.E. (P.A.S.), August 1954, pp.

1027-1032.

[303] L.B. Crann and R.B. Flickinger, "Overvoltages on 14.4/24.9kV Rural Distribution

Systems" Trans. Amer. I.E.E. (P.A.S.) , October 1954, pp. 1208-1212.


120

[304] F.C. Van Wormer, "Switching three-phase transformer banks" General Electric

Monograph March 1965.

[305] R.H. Hopkinson, "Ferroresonance During Single-phase Switching of 3-phase

Distribution Transformer Banks" Trans. IEEE, April 1965, pp. 289-293.

[306] A.M. Lockie, "Ferroresonance, a Growing Problem" Monograph of unknown

U S A origin, March 1965.

[307] R.H. Hopkinson, "Ferroresonance During Single-Phase Switching of 3-Phase

Distribution Transformer Banks" (Discussion). Trans. IEEE, June 1965, pp. 514-

517.

[308] J.F. Young, "Ferroresonance - Problems and Applications" Electrical Review, 21

May 1965, pp. 782-785.

[309] R.H. Hopkinson, "Ferroresonant Overvoltage Control Based on TNA Tests on

Three-Phase Delta-Wye Transformer Banks" Trans. IEEE (PAS) Vol. PAS-86,

No. 10, October 1967, pp. 1258-1265.

[310] R.H. Hopkinson, "Ferroresonant Overvoltage Control based on TNA Tests on

Three-Phase Delta-Wye Transformer Banks" IEEE (PAS) Vol. PAS-87, No. 2,

February 1968, pp. 352-361.

[311] F.S. Young, R.L. Schrnid, and P.I. Fergetad, "A Laboratory Investigation of

Ferroresonance in Cable-Connected Transformers" Trans. IEEE (PAS), Vol.

PAS-87, No. 5 May 1968, pp. 1240-1249.


121

[312] R. Archibald, "160kV peaks occur when Switching 13kV Circuits" Electrical

World, August 26, 1968, pp. 52-53, 102.

[313] G.C. Damstra, "Ferroresonance during Single-phase Switching" Electrical

Engineer, 10 April, 1969, pp. 32-34.

[314] G.C. Damastra, "Ferroresonance during Single-phase Switching of Distribution

Transformers" Australian Electrical World, May 1969, pp. 10-11.

[315] A. Clerici and CH. Didriksen, "Dynamic overvoltages and ferroresonance found

in switching. Trans. IEEE, August 1971, pp. 195-203.

[316] E.J. Dolan, D.A. Gillies and E.W. Kimbark, "Ferroresonance in a Transformer

Switched with an E.H.V. line" Trans. IEEE, May 1972, pp. 1273-1280.

[317] G.D. Wale, "Ferroresonance in a Disconnected E.H.V. Power System" GEC

Journal of Science and Technology, Vol. 40, No. 2, 1973, pp. 79-86.

[318] A. Baitch, "Theory and Practice of Ferroresonance due to the Single Phase

Switching of Distribution Transformers" Master of Engineering Thesis University

of N e w South Wales, Australia 1973.

[319] D.R. Smith and S.R. Swanson, "Overvoltages with Remotely-Switched Cable-Fed

Grounded Wye-Wye Transformers" IEEE paper T 75 137-5 November 1974, pp.

1843-1853.

[320] A. Baitch, "Ferroresonance due to the Single Phase Switching of Distribution

Transformers" International Conference on Electricity Distribution Liege, Belgium

1979 Session 3 paper 24.


122

[321] "Dual Prescription for Ferroresonance" Electrical World, May 1, 1979, pp. 68-69.

[322] S. Prusty and M. Panda, "Predetermination of Lateral Length to Prevent

Overvoltage Problems due to Open Conductors in Three Phase Systems" IEE

Proc. Vol. 132, Pt. C, No. 1, January 1985, pp. 49-55.

[323] J.R. Marti and A.C. Soudack, "Ferroresonance in Power Systems, Fundamental

Solutions" IEE Proceedings-C, Vol. 138, N o 4, July 1991, pp. 321-328.

16.5 Ferroresonance Analytical Techniques

[401] C.G Suits, "Studies in Non-Linear Circuits" Trans. Amer. I.E.E. (P.A.S.) June

1931, pp. 724-736.

[402] W.T. Thomson, "Resonant Nonlinear Control Circuits" Trans. Amer. I.E.E.

(P.A.S.) August 1938, Vol. 57, pp. 469-476.

[403] P.P. Odesseyt and E. Weber , "Critical Conditions in Ferroresonance" Trans.

Amer. I.E.E. (P.A.S.)August 1938, Vol. 57, pp. 444-452.

[404] W.T. Thomson, "Similitude of Critical Conditions in Ferroresonant Circuits"

Trans. Amer. I.E.E. (P.A.S.) Vol. 58, March 1939, pp. 127-130.

[405] J.T. Salihi, "Analysis of Instability and Response of Reactors with Rectangular

Hysteresis Loop Core Material in Series with Capacitors" Trans. Amer. I.E.E.

(Comrn. & Electronics), July 1956, pp. 296-307.

[406] G.E. Kelly, "The Ferroresonant Circuit" Trans. IEEE, January 1959, pp. 843-848,

p. 1061.


123

[407] J.T. Salihi, "Theory of Ferroresonance" Trans. IEEE, January 1960, pp. 755-763.

[408] L. Clarke, G.A. Curtis and R.O.M. Powell, "Capacitors in Relation to Transient

Fluctuating and Distorting Loads" Proc. IEE, 1963, pp. 21-32.

[409] L.A. Finzi and A. Lavi, "The Controlled Ferroresonant Transformer" Paper 62-

1022 ATEE Non-Linear Magnetics Committee January 1963, pp. 414-419.

[410] R. Balasubramanian and D.P. Atherton, "Prediction of Jump Resonance in

Systems Containing Certain Multidimensional Nonlinearities" Proceedings IEE,

Vol. 115, No. 9, September 1968, pp. 1369-1372.

[411] G.W. Swift, "An Analytical Approach to Ferroresonance" Trans. IEEE (PAS),

Vol. P A S 88, No. 1, January 1969, pp. 42-46.

[412] S.S. Lamba and R.J. Kavanagh, "Jump-Resonance Criteria for Systems Containing

Double-Valued And Frequency Dependent Non-Linearities" Proceedings IEE,

Vol. 116, No. 7, July 1969, pp. 1225-1228.

[413] L.O. Chua, "Qualitative Analysis of 1st and 2nd-Order Nonlinear Networks"

Proc. IEE, Vol. 118, No. 1, January 1971, pp. 19-28.

[414] S.S. Lamba and R. J. Kavanagh "Phenomenon of Isolated Jump Resonance and its

Applications" Proc. IEE, Vol. 118, No. 8, August 1971, pp. 1047-1050.

[415] A. Semlyen, "Phasor-Trajectory Representation of Near-Resonance Transients In

Quasilinear A.C. Circuits" Proc. IEE. Vol. 118, No. 8, August 1971, pp. 988-992.


124

[416] D.Teodorescu, "Analysis and Synthesis of Nonlinear Control Systems by Means

of a Sampled-Data Nonlinear Matrix" Proc. IEE, Vol. 118, No. 11, November

1971, pp. 1655-1660.

[417] J.J. LaForest, "Program Models Magnetic Saturation" (System engineering)

Electrical World, April 15, 1972, pp. 70-71.

[418] B.S. Ashok Kumar, A.K. Tripathy, K. Parthasarathy and G.C. Kothari "Approach

to the Problem of Ferroresonance in E H V Systems" Proc. IEE, Vol. 119, No. 6,

June 1972, pp. 672-676.

[419] A. Germond, "Computation of the Periodic Overvoltages Due to Ferroresonant

Phenomena in Three-Phase Networks" IEE Conference Publication No. 110,

approx. 1974.

[420] G.C. Kothari, B.S. Ashok Kumar, K. Parthasarathy and H.P. Khincha" Analysis of

Ferro-oscillations in Power Systems" Proc. IEE, Vol. 121, No. 7, July 1974, pp.

616-622.

[421] S. Prusty and S.K. Sanyal, "Some New Solutions to Ferroresonance Problem in

Power System" Proc. IEE, Vol. 124, No. 12, December 1977, pp. 1207-1211.

[422] J.M. Feldman and AL. Cappabianca "On the Accuracy and Utility of Piecewise-

Linear Models of Ferroresonance" IEEE Transactions on Power Apparatus and

Systems, Vol. PAS-97, No. 2, March/April 1978, pp. 469-477.

[423] S. Prusty and S.K. Sanyal "Effect on Core Loss on Multimodal Operation of a

Parallel Ferroresonant Circuit: Some General Conclusions" Proc. IEE, Vol 126,

No. 9, September 1979, pp. 826-836.


125

[424] P.K. Mukherjee and S. Ray, "Computation of Switching Transients for

Ferroresonance Studies" Institution of Engineers India Journal Electrical, Vol. 59,

December 1979, pp. 178-181.

[425] N.L. Diseko and J.P. Bickford, "A Method of Simulating Linear and Non-Linear

Resonant Phenomena Associated with Transformer Feeders" IEE Proc. Vol. 127.

Pt. C, No. 3, May 1980, pp. 169-178.

[426] N. Janssens, A. Even, H. Denoel and P. A. Monfils, "Determination of the Risk of

Ferroresonance in High Voltage Networks. Experimental Verification on a 245kV

Voltage Transformer" Proceeding Vol. 1 Sixth International Symposium on High

Voltage Engineering, N e w Orleans, Louisiana, USA. 28 August to 1 September

1989.

[427] M. Tadokoro, H. Nagata and T. Yamazaki, "Analysis of Abnormal Oscillations of

a Three-Phase Nonlinear Circuit" Electrical Engineering in Japan, Vol. 110, No. 6,

1990, pp. 128-137.

[428] N. Janssens, V. Vanderstock, H. Denoel and P.A. Monfils, "Elimination of

Temporary Overvoltages Due to Ferroresonance of Voltage Transformers : Design

and Testing of a Damping System" Reference 33-204 CIGRE 1990 session 26

August - 1 September 1990.

[429] C. Kieny, "Application of the Bifurcation Theory in Studying and Understanding

the Global Behaviour of a Ferroresonant Electric Power Circuit" IEEE Trans.

Power Delivery, Vol. 6, No. 2 April 1991, pp. 866-872.

[430] C Kieny, G. Le Roy and A. Sabai, "Ferroresonance Study using Galerkin Method

with Pseudo-Arclength Continuation Method", IEEE Trans. Power Delivery, Vol.

6, No. 4, October 1991, pp. 1841-1847.


126

[431] N. Janssens, Th. Van Craenenbroeck, D. Van Dommelen and F. Van De

Meulebroeke, "Direct Calculation of the Stability Domains of Three-Phase

Ferroresonance in Isolated Neutral Networks with Grounded-Neutral Voltage

Transformers" paper 95 S M 420-0 P W R D presented at the Summer Meeting of the

IEEE /PES July 1995, Portland, Oregan USA.

16.6 Miscellaneous

[501] G.N. Patchett, "Automatic Voltage Regulators and Stabilisers" Pitman Publishing

third edition 1970.

[502] Editorial, "Voltage Transformers Have Electronic Ferroresonance Protection"

(Research and Development), Electrical Review, 13 August 1971.

[503] E.T.B. Gross, M.H. Hesse, CM. Summers and AJ.O. Cruickshank, "Approach to

Experimental Electric Power Engineering Education - IF' IEEE Paper T 73 505-5

May 1973, pp. 803-811.

[504] W.K. Macfadyen, R.R.S. Simpson, R.D. Slater and W.S. Wood, "Method of

Predicting Transient Current Patterns in Transformers" Proc. IEE, Vol. 120, No.

11, November 1993, pp. 1393-1396.

[505] W.E. Shula, "Capacitors Help to Start Large Motors" Electrical World, 1

November 1974, pp. 44-47.

[506] A.A. Mahmoud, T.H. Ortmeyer and R.G. Harley, "Effects of Reactive

Compensation on Inductive Motor Dynamic Performance" IEEE Transactions on

Power Apparatus and Systems, Vol. Pas-99, No. 3 May/June 1980, pp. 841-846.


127

[507] Australian Standard A S 2926 "Standard Voltages - Alternating (50 Hz) and

Direct".

[508] IEC Standard IEC 38 "Standard Voltages"

[509] J. H. B. Deane and D.C. Hamill, "Instability, Subharmonics and Chaos in Power

Electronic Systems" IEEE Trans. Power Electronics, Vol. 5, no. 3, July 1990, pp.

260-267.

[510] P.M. Anderson, B.L. Agrawal and J.E. Van Ness, "Subsynchronous Resonance in

Power Systems" Published IEEE Press, IEEE order Number PC0247-7.

[511] J.A. Edminster and J.E. Swann, "Electric Circuits" Published McGraw Hill

International Book Company 1972.

[512] B.M. Weedy "Electric Power Systems" Published John Wiley & Sons 1975.

[513] P.M.Anderson and A.A. Fouad, "Power System Control and Stability" Published

IEEE Press 1994.

[514] The Electricity Authority of New South Wales - "Overhead Line Manual"

Drawing Reference E A S 4 10 2 1977.


128

Appendix A

Detailed Experimental and Frequency Domain Model Results for

Circuit "A"

The following table gives a summary of the model predicted and experimental results for

circuit "A". This data is present in graphical form in figures 19 and 20.


IEEE90

Supply Voltage Vs Volte

RMS

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330

Circuit "A" - Current amps RMS

50 hertz -model

0.01 0.02 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.11 0.12 0.14 0.15 0.16 0.18 0.20 0.23 0.27 0.31 0.36 0.42 0.49 0.58 0.68 0.79 0.93 1.08 1.25 1.44 1.67 1.91 2.19 2.50

50 hertz -experimental

0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.06 0.07 0.09 0.10 0.13 0.15 0.20 0.23 0.29 0.37 0.45 0.53 0.65

2nd subh. -

unstable model solution

2.55 2.58 2.60 2.63 2.62 2.61 2.58

2nd

subharmonic -model

2.58 2.51 2.40 2.22 1.94

2nd

subharmonic -experimental

2.38 2.34 2.28 2.11 1.88

3rd subharmonic -

model

0.74 0.75 0.76 0.78 0.81 0.84 0.88 0.90 0.92 0.93 0.92 0.90 0.86 0.81 0.71 0.51

3rd subharmonic • experimental

0.74 0.81 0.88 0.92 0.94 0.96 0.97 0.98 0.97 0.94 0.88 0.81 0.72

IEEE90

Supply Voltage Vs Volts

RMS

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330

Circuit "A" - Cai

50 hertz •

model

0.0 0.5 1.0 1.4 1.9 2.4 2.9 3.4 3.9 4.4 4.9 5.5 6.1 6.7 7.5 8.3 93

10.5 12.0 13.8 15.9 18.5 21.6 25.4 29.7 34.8 40.7 47.4 55.1 63.8 73.7 84.8 97.2 119.4

50 hertz -

experimental

0.0 03 0.5 0.8 1.1 1.3 1.6 1.9 2.2 2.4 2.7 3.0 3.2 3.5 4.3 4.9 6.3 7.8 9.7 11.2 14.0 18.0 22.4 26.0 31.6

2nd subh. -

unstable model

solution

199.0 197.8 195.8 193.2 189.5 184.5 177.8

pacitor Voltage - Vol

2nd

subharmonic -model

177.8 169.0 158.0 143.5 121.6

2nd

subharmonic -

experimental

166.0 162.0 156.5 143.0 127.0

tsRMS

3rd subharmonic -

model

943 96.1 97.1 97.8 98.0 97.8 97.0 95.6 93.4 90.3 863 81.1 74.8 66.9 56.5 36.4

3rd subharmonic • experimental

101.0 103.2 103.7 103.2 101.6 99.3 95.5 92.4 87.9 83.3 77.6 69.9 62.0

131

Appendix B

Modelled Transient R L C Behaviour for Circuit "A"

Figures 51 and 52 shows the modelled transient behaviour of the series RLC circuit

elements of circuit "A" when excited by a 100 volt step voltage. The transformer

primary winding is short circuited for this simulation. O f particular interest is the near 50

H z response and the rate of decay.


CO O) O) o II o 0) DC fr 0) 3

a>

CM CO

II DC £

132

0) X sr i—

o n _i

E .c o CO m II DC

at co

O

SJTOA

133

in CU U

VM

O CO

0)

3 fi

• IH

u u rt QJ fi

QJ CJ fi

O CO

cu P4 *-fi QJ

tl

u cu aj fi

o & 0) & 0)

U i

I

P4 CO

cu QJ

CD

co CO O)

d u o

1 c Q) zi O"

CM

cci ll DC

V)

2" c X

*T

d ll

i

(A

E O CO in II DC

U. 3

(O II O

0S0"0

swo

fi CJ

u •iH

u

010"0

SOO'O

OOO'O

sooo-

d

sdure

134

Appendix C

Detailed Experimental and Frequency Domain Model Results for

UT>» Circuit "B

The following table gives a summary of the model predicted and experimental results for

circuit "B". This data is present in graphical form in figures 27 and 28.


IEEE92

Supply

Voltage Vs

Volts R M S

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340

Circuit "B" - Current amps RMS

50 hertz -

model

0.00

0.02

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.11

0.12

0.13

0.15

0.16

0.18

0.20

0.23

0.27

0.31

0.37

0.43

0.51

0.60

0.71

0.84

1.00

1.16

1.37

1.59

1.85

50 hertz -

experimental

0.00

0.00

0.02

0.02

0.03

0.03

0.04

0.04

0.04

0.05

0.06

0.06

0.07

0.08

0.09

0,12

0.14

0.17

0.21

0.27

0.33

0.38

0.47

0.58

0.64

0.80

0.95

1.07

1.25

1.49

1.74

2.03

2nd subh. -

unstable

model solution

8.41

8.14

7.91

2nd

subharmonic -

model

7.91

7.66

7.41

7.16

6.95

2nd

subharmonic -

experimental

10.18

10.14

9.88

9.54

9.14

8.66

7.72

3rd

subharmonic -

model

2.49

2.50

2.52

2.56

2.61

2.69

2.77

2.87

2.96

3.04

3.11

3.15

3.17

3.15

3.10

3.01

2.88

2.71

2.47

2.10

3rd

subharmonic -

experimental

2.79

2.82

2.89

2.98

3.11

3.26

3.39

3.54

3.65

3.70

3.71

3.68

3.63

3.57

3.48

3.35

3.17

2.98

2.68

2.46

1.93

IEEE92

I

Supply Voltage Vs

Volts R M S

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340

Circuit "B" - Capacitor Voltage - Volts R M S

50 hertz -

model

0.0 0.2 0.4 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.6 2.8 3.2 3.5 4.0 4.6 53 6.1 7.2 8.4 10.0

11.8

13.9

163 19.2

22.4

26.2

30.4

35.2

40.6

46.6

53.3

50 hertz -

experimental

0.0 0.2 03 0.4 0.5 0.6 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.4 1.6 1.9 2.3 2.9 3.6 4.5 5.5 6.5 8.1 9.7 11.0

13.8

163 18.5

21.4

24.7

29.9

35.8

2nd subh. -

unstable

model solution

231.5

220.0

210.6

2nd

subharmonic -

model

210.6

201.3

192.0

182.8

174.7

2nd

subharmonic -

experimental

253.0

248.0

236.0

222.0

210.0

198.0

176.0

3rd

subharmonic -

model

120.4

1225

123.9

124.8

125.4

125.7

125.7

125.3

124.5

1233

121.4

118.8

115.5

111.4

106.5

100.7

93.8

85.7

75.7

61.9

3rd

subharmonic -

experimental

128.0

128.0

128.0

128.0

127.0

124.0

120.0

118.0

118.0

114.0

111.0

102.0

93.0

90.0

83.0

76.0

70.0

63.0

55.0

48.0

36.0

137

Appendix D

Modelled Transient R L C Behaviour for Circuit "B"

Figures 53 and 54 shows the modelled transient behaviour of the series RLC circuit

elements of circuit "B" when excited by a 100 volt step voltage. The transformer

primary winding is short circuited for this simulation. O f particular interest is the near 50

H z response and the rate of decay. The circuit has a frequency ratio of 1.00 and an X / R

ratio of 3.03. This circuit is clearly underdamped.

Circuit "B" has a similar frequency ratio to circuit "A" with both circuits being tuned to

50 Hz. Circuit "B" has a smaller X / R ratio than circuit "A" indicating it is more damped.

This is particularly evident in the comparison of the step responses. Despite the higher

level of damping, circuit "B" is prone to higher ferroresonant currents and capacitor

voltages than circuit "A".


CO

o o 1 —

II

o flj

DC fr c Oi 3 o-2! u.

CO o CO II

DC

s

138

c Q)

X T If) O

d II

(A £ O CO IT) II DC

CM CO r~ II

O

oeco

swo

owo

StfTO

OttTO

SZO'O

030

cfl 73 O 0) CO

S

0 s

•sxoo

I OWO

S00"0

0000

SOO'0-

SJfOA

139

<4-l

o co +•* fi

cu

g CU l-H

W

fi CJ

u u rt QJ fi

cu CJ fi

O r IT> co =

& u

u cu co fi

O co

cu

cu

u I

CO

cu • l-H

cu Cfl

o o II

o '3 QC c CD -3 O"

3! LL

CO O CO II

or

V)

c <u X m o d n

</)

E

o co ir> ll

DC

LL

OJ

co ll

O

CO

SOO'O

OOO'O

soao-

sdure

140

Appendix £

Experimental Circuit "D" 3 Phase

Figure 55 shows the layout of the three phase circuit "D" constructed in the laboratory.

Rd Xc<^>

L R

^-^ L R

R d >":«c)

^Ttf^AAA L R c

Figure 55 Three Phase Series Compensated Circuit " D "

The circuit parameters were:

L=0.149H

C=69uF

R=1.0ohm

Rd=46

Tx winding resistance=1.2 ohm

The transformer was a three limb iron cored transformer rated at 200 VA 110V phase to

phase on the primary side.

This series compensated circuit was examined in the laboratory both with the saturating

choke damping and without the saturating choke damping.


141

Without damping the circuit was found to be highly ferroresonant and capable of

subharmonic wavefoms of order 1/2 1/3 1/4 1/5. Under some conditions of supply

voltage low frequency oscillations between limbs in the order 5 seconds was observed.

When the damping elements were introduced into the circuit no ferroresonant states

could be generated.


142

Appendix F

Modelled Stored Energy in Circuit "A"

At a supply voltage of 180 volts circuit "A" can operate in two modes. In addition there

is an unstable 2nd subharmonic frequency domain solution. The following figures refer

to the energy stored in the circuit components.

Figure 56 Stored Energy - Circuit "A" 50 Hz Magnetising

Figure 57 Stored Energy - Circuit "A" 3rd Subharmonic

Figure 58 Stored Energy - Circuit "A" Unstable 2nd Subharmonic


>•, bJj SH 01 C w H X o (H OH

JX < D

>, bn S-CU c 0) HJ

•

>, OJJ

u 0) t-w {X (8

u •

143

v© U"> QJ U

S, •»H

be fi

• iH

CO • IH +•»

cu

S> tc*

2 N E O ID i

<3 ^

fi CJ

u u QJ fi

w •a cu u

o CD

Cfl

CJ

0)

sapnof

144

IN in cu u

PH

fi

O

S u rt

X X CD JH

CO

fi CJ *H • iH

u CU fi

PJ

cu

o CD

X M IH CU c w X H X o IH CH H H

< •

>, nn HI

ci; C 01 d •

'H c W O-ns u •

l'£

67

^ ^^^^"^ ^â*t\\ftâ%

^^âtaW\W\WW\\

6T cu

>^

u I

cu

s

CN

=

vO

saynof

145

>, bo u CD e w X

H X o IH H H H H

^ •

>1

Wl u 01 c UJ hJ

•

>, e? QJ

c W CH (8

u •

fi

o &

u rt

X X

CD T3 fi OJ CU l-H

,£> CO

fi

s? 8S* •IH "»H

PH fi CJ

u •IH

u u

QJ fi

PJ T3 CU

u o H-»

CD

sapnof

1997 series compensation of distribution and

Documents