new principle for transmission line protection using phase portrait plane

8/8/2019 New Principle for Transmission Line Protection Using Phase Portrait Plane

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Published in IET Generation, Transmission & Distribution

Received on 11th October 2007

Revised on 10th September 2008

doi: 10.1049/iet-gtd:20080456

ISSN 1751-8687

New principle for transmission line protection

using phase portrait planeM.M. Eissa*Department of Electrical Engineering, Helwan University, Helwan, Cairo, Egypt

*Projects Department, King Abdulaziz University, Jeddah, P.O. Box 80200, The Kingdom of Saudi ArabiaE-mail: [email protected]

Abstract: Phase portraits are a powerful mathematical model for describing oriented textures. An isotangent-

based approach in a phase portrait is introduced to discriminate between internal and external faults. The

scheme described is new and is a different approach to the problem of relaying ground faults on transmission

lines. The geometrical theory of differential equations is used to drive a symbol set on the basis of the visual

appearance of phase portraits. The phase portrait of the instantaneous rate of change of current against

voltage gives valuable information on the transient and stability characteristics of system configuration with

high fault resistance and shunt susceptance of the transmission lines. Hundred percent of the protected zone

is successfully discriminated rather than the external zone. An additional parameter that is considered, which

is sometimes neglected in protection studies, is the shunt susceptance of the transmission lines.

1 Introduction

Transmission line protection is generally regarded as themost complex of all relaying applications. This is because ofthe distances involved and the variety of possibilities thatexist in system configuration and opportunities for bothcorrect and incorrect operations. A distance relay isdesigned to only operate for faults occurring between therelay location and the selected reach point, and remainsstable for all faults outside this region or zone [1].

In developing distance relay equations, the fault underconsideration is assumed to be ideal (i.e. zero resistance)[210]. In reality, the fault resistance will be between twohigh-voltage conductors, whereas for ground faults, the faultpath may consist of an electrical arc between the high-voltageconductor and a grounded object. The fault resistanceintroduces an error in the fault distance estimate and, hence,may create an unreliable operation of a distance relay [11].

The impedance seen by the relay is not proportional to thedistance between the relay and the fault, in general, becauseof the presence of resistance at the fault location.

Some techniques for arcing faults detection and faultdistance estimation are introduced in [12, 13]. The

techniques are based on the voltage and current at oneterminal in the time domain. The overhead line parametersand arc voltage amplitude during fault are given. Thetechniques have optimal application in the medium-voltagenetworks and symmetrical faults. Some techniques [1419]are suggested for enhancing the high fault resistanceproblem. These techniques accommodate this problem byshaping the trip zone of the distance relay, to ensure thatthe apparent impedance is included inside the trip zone.New techniques using high-frequency components of thefault-generated transient signals are also studied in [2022].

Distance-based techniques are applied using the currentand voltage at the relaying point by differential equations.

The fundamental assumption used in this approach is thatthe transmission line can be represented by either a lumped-series impedance or a single PI-section. An additionalparameter that is sometimes neglected in protection studiesis the shunt susceptance of the transmission lines. Forthe EHV lines, the shunt susceptance is rather large. If thesusceptance is omitted from the study, the results areconsidered to be errors.

In this paper, a new isotangent-based approach in a phaseportrait to discriminate between an internal and external faultis introduced. Phase portrait analysis is done to obtain full

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doi: 10.1049/iet-gtd:20080456 & The Institution of Engineering and Technology 2008

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insight into the fault trajectory during the steady state andtransients concerning the effect of high fault resistance andshunt susceptance. The geometrical theory of differentialequations to describe the transmission line during faultis obtained. The instantaneous rate of change of currentagainst voltage in a phase portrait is given. Hundred percent

of the protected zone is successfully discriminated.

2 Phase portraits

Phase portraits provide an analytical tool to study systems offirst-order differential equations [23]. The method hasproved to be useful in characterising oriented texture [24,25]. Let _x and _y denote two differentiable functions oftime t, related by a system of first-order differentialequations as

_x F(x, y) cx dyf (1)

_y G(x, y) ax by e (2)

to describe a set of oriented textures comprising saddles,star-nodes, nodes, improper nodes, centres and spirals(Fig. 1). Equations (1) and (2) can be represented in matrixnotation as

_~X ~A~X ~B (3)

where

~X xy ,

~Aca

db and

~Bfe

~A is the characteristic matrix of the system.

A point at which both _xand _yare zero is said to be a criticalpoint (xo, yo). It means that there is no flow orientation at thispoint. If there is only one critical point, it carries the position

information about the flow pattern. The critical points of asystem with differential equations (1) and (2) are theintersection points of the following equations

F(x, y) cx dyf 0 (4)

G(x, y) ax by e 0 (5)

The purpose of the phase portrait analysis is to obtain fullinsight for the fault trajectory during the steady state andtransients concerning the effect of high fault resistance andshunt susceptance.

3 Phase portrait classification

The elements of the characteristic matrix are used todetermine six flow patterns if the characteristic matrix isnon-singular. The two classifications may be added forsingular characteristic matrices. The eight classifications

derived from the characteristic matrix are no-flow, constantflow, star-node, node, improper node, saddle, centre andspiral. Some of these shapes are shown in Fig. 1 and canbe obtained as follows:

Node: the components x(t) and y(t) are exponentials thateither simultaneously converge to, or diverge from, thefixed point coordinates x0 and y0.

Saddle: the components x(t) and y(t) are exponentials, whereas one of the components [either x(t) or y(t)]converges to the fixed point, and the other diverges from it.

Spiral: the components x(t) and y(t) are exponentiallymodulated sinusoidal functions the resulting streamlineforms a spiral curve.

Throughout the paper, a set of differential equations, whichdescribe the transmission line during the fault, is given. Thedifferent events of fault condition can be described bythe phase trajectory, which passes through the phaseportraits. The transition between different phase portraitsis determined since phase trajectories are continuous lines.

The isotangent-based approach in the phase portrait willsuccessfully discriminate between the internal and external

faults, as will be explained in Section 7.

4 Conventional problems

Fig. 2 shows the tripping quadrilateral characteristic in thecase of high fault resistance. It will be noted that theresistance of the fault arc takes the fault impedance outsidethe relays tripping characteristic and, hence, it does notdetect this condition. Alternatively, it is only picked upeither by zone 2 or zone 3 in which case tripping will beunacceptably delayed.

The infeed effect can be seen from Fig. 3, where there areother lines and sources feeding current to a fault atFfrom busR. The relays at bus Sare set beyond this fault point to F0. As

Figure 1 Phase portraits for a system of linear first-order

differential equations

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theinfeed increases in proportionaltoIS, theapparentimpedanceincreases. Since this impedance, as measured by the distancerelay, is larger than the actual, the reach of the relay decreases;that is, the relay protects less of the line as infeed increases.

The following equation proves that the voltage at bus Sis related to the current at bus Sby the equation ES ZLISZR(ISIR) and the apparent impedance seen by the relay at Sis Zapp ES=IS ZL ZR(ISIR)=IS the current IR, thecontribution to the fault from the tap, is known as the infeedcurrent and causes error in measurements.

The paper introduces a new approach on the basis of the

instantaneous rate of change of current against voltage usingthe phase portrait. The relay uses one end transmission linedata taking in consideration all fault events such as highfault resistance and remote-end infeed. The new approachdiscriminates precisely between internal and external faults onthe basis of the isotangent lines.

5 Transmission line model

The technique takes a more fundamental approach withregard to the inclusion of high-frequency oscillations thatcan occur during faults. This can be achieved by including

the effect of capacitance of the transmission line using asingle PI-section transmission model.

Fig. 4 shows the protection equivalent for an EHVline. The length of the transmission and the effect of theline susceptance will be variables in this case. Thetransmission line is divided into two sections, separated at

the point of fault application. Both segments of thetransmission line are represented as a long line equivalentPI-section.

6 Development of the algorithm

As explained above, a set of differential equations to describethe transmission line with the susceptance effect during thefault will be given. Fig. 4 shows the transmission lineconfiguration with the shunt capacitance and the faultresistance. It is assumed that the magnitude of any arc

resistance Rf is small and that the effect of the branchedcapacitance can be neglected. Therefore the circuit equationcan be described as follows

v nR(is ic) nLd(is ic)

dt Rf(is ic ir) (6)

Rearranging (6)

d(is ic)

dt

v

nL

R

L

RfnL

(is ic)

RfnL

(ir) (7)

The current ic is the capacitive current flowing through thelumped equivalent capacitance at relay location (S), and itis related simply to the voltage at the relaying point bynoting that

ic C

n

dv

dt(8)

Rearranging (8)

dv

dtn

C(is ic)

n

C(is) (9)

Figure 3 Effect of the infeed on impedance measured by

distance relays

Figure 2 Under reach of the distance relays (RF , fault

impedance; RS , Z1, quadrilateral characteristics and ZL line

impedance)

Figure 4 Protection equivalent for an EHV line (n fault

location on protected line as portion of the total)



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From (7) and (9) in the matrix form

d(is ic)dt

dv

dt

0

B@

1

CA

R

L

RfnL

1

nLn

C 0

0

B@

1

CA

(is ic)

v

0

B@

1

CA

RfnL

ir

n

Cis

0BB@

1CCA

(10)

Equation (10) describes the characteristic of the transmissionline. To study the dynamics of the transmission line, Rf, L, Rand n are changed. For every given Rf, the correspondingphase portrait is ic v is shown in Fig. 5. Also, for everygiven n, the corresponding phase portrait is ic v is

shown in Fig. 6. The phase portrait of the external andinternal faults is seen in Fig. 7.

As shown in Figs. 5 7, the instantaneous current against voltage of (10) gives valuable information on the transientand stability characteristics at different fault conditions. Afocus on the transient part is circled to show that valuableinformation and features can be obtained from the phaseportrait.

From the recognisable shapes in the portrait, some fault

events can be easily identified. However, other fault eventshave valuable information, but the phase portrait is not easilyidentified. For this reason, another phase portrait producedfrom is ic v is used to assist the discrimination betweendifferent fault events, specifically the internal and externalfaults. The isotangent-based approach in a phase portraitto discriminate between the internal and external faults isused.

7 Isotangent-based approach

An isotangent curve is the locus of points in a flow field,which share the same flow orientation [26, 27]. All curvesare always straight lines for non-singular, two-dimensionalfirst-order phase portraits. Also, the relationship betweenthe flow orientation and the slope of the isotangent lines isunique for every fault condition along the transmission

lines. The following section can prove these facts.

Since the systems of (9) and (7) corresponding to (2) and(1), respectively, are linear in the parameter space, parameterasset is unique for any given flow pattern. For a

Figure 5 Phase portrait of is ic v for different R (R 0

and R 100) and SLG fault on phase-A

Figure 6 Phase portrait of is ic v for different fault

location and SLG fault on phase-A

Figure 7 Phase portrait illustration between internal and

external faults and SLG fault on phase-A



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two-dimensional first-order system, the tangent is given by

tan(w(x, y)) tan(w(is ic, v)) _y

_x

_v_is _ic

G(x, y)

F(x, y)

ax by e

cx

dy

f

(11)

if tan w b (11) becomes

(a cb)x (b db)y (efb) 0 (12)

which is a straight line for any given w. Since the lineequation is a function ofb, every straight line is unique fora given point set in a given flow pattern. For the matrix ~Agiven in (10), there is a unique straight line with a uniqueisotangent, which depends on the system parameters.

The paper presents a phase portrait on the basis of the

instantaneous rate of change of current against voltage, thatis, (d(is ic))=dt dv=dt rather than the instantaneouscurrent and voltage (is ic v). The phase portrait givesrecognisable different isotangents for different cases ofinternal faults.

The X/R ratio of the transmission line is fixed. The(d(is ic))=dt dv=dt portrait can be considered as a locusof the instantaneous _Z. The portrait gives more visualisationof the instantaneous part of the _Z. The portrait in such acase is concerned with the rate of change of the impedancein the transient part, which is highly affected by the

PX

and PR of the transmission line configuration at a point(i.e. equivalent of X/equivalent of R at the point of fault).

This ratio can be calculated from the input impedance at thefault point as will be explained later.

In case of external faults, the phase portrait gives an almostfixed isotangent. Thus, the process of discrimination betweeninternal and external faults can be easily identified.

The isotangent given in (11) depends on theP

X=P

Rratio (i.e. equivalent of X/equivalent of R at the point offault) of system configuration. This can be easily calculatedas the input impedance at the fault point. The isotangent

or theP

X=P

R ratio is changed during internal faultsbased on the fault location. During external faults, theisotangent or the

PX=

PR ratio seen at the fault point is

almost fixed. Consequently, the isotangent in the phaseportrait during internal faults is different from theisotangent in case of external faults.

To prove this fact, a simplified configuration given inFig. 8 is considered. For a three-phase transmission lineoperating at a certain voltage and having an impedanceconnected to the generating station busbar, for a certainfault at Fi, the input impedance is calculated using

Thevenins as

ZinFi (n(RjX)jXg1)==((1 n)(RjX)jXg2) (13)

Equation (13) has theP

X=P

R ratio determined as

Xt(n(1 n)R2X0X00)Rt(2n(1 n)RXX

000)

Rt(n(1 n)R2 X0X00)Xt(2n(1 n)RXX

000)(14)

where Rt nR (1 n)R, Xt nXXg1 (1 n)XXg2, X

0 (nXXg1), X

00

((1 n)XXg2) and X000

2n(1 n)RX (1 n)RXg1 (1 n)RXg2. TheP

X=P

R

ratio given above becomes totally different if the faultlocation is changed along the transmission line, and thatcan be easily recognised using the phase portrait based onthe rate of change of the current against voltage.

Similarly, the input impedance Zin for an external fault atFe can be calculated as follows

ZinFe (RjX) jXg1)==( jXg2) (15)

Equation (15) has theP

X=P

R ratio determined as

R2

(XXg1)(XXg1 Xg2)RXg2

(16)

TheP

X=P

R ratio given in (16) is different from thePX=

PR ratio given in (14). For extra and ultra high-

voltage transmission lines, the installed source capacitieslocated at the sending and receiving ends of the TL arealmost constant. For an internal fault along the transmissionline, the value of the (14) changes according to the faultlocation and, consequently, the isotangent is also changed.

However, for an external fault along the transmission line,

the source capacity at the far end (Fig. 8) will have very highsource capacity compared with the transmission lineparameters. The rate of change of the impedance in thephase portrait during the transient part depends on theP

X=P

R ratio given in (16), which is totally differentfrom that in (14) and almost with a fixed value.Consequently, the isotangent in the phase portrait is alsodifferent and fixed for external faults.

Figs. 911 show the instantaneous rate of change ofcurrent against voltage using a phase portrait for the samefault conditions as shown in Figs. 5 7. As shown in these

figures, the isotangent for different cases of fault events isdifferent, and the process of discrimination is easy to beidentified.

Figure 8 Input impedance seen at fault point for internaland external faults



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The criterion for the protection relay to initiate a trip signalis set. For the studied configuration system given in Fig.8, (16)is used to set up the relay. Thus, the isotangents in the case ofinternal faults respond up or down away from the thresholdboundary THR. In this respect, the isotangent for externalfaults has a unique value equal to THR. This criterionsuccessfully recognises the internal and external faults, and100% of the transmission line is protected. THR can be

easily calculated as given in (11) from the slop of the lineafter fault inception.

Fig. 12 shows the difference between the conventionaldistance relay with three zones and the new techniquebased on the phase portrait principle.

8 Power system configuration

The power system used for testing the proposed new methodis a part of 500 kV power systems shown in Fig. 13. Thesystem includes two generating stations. A relay is locatedat Bus S as shown in Fig. 13. The voltage and current

Figure 9 Phase portrait based on rate of change fordifferent fault resistances (R 0 and R 100)

Figure 11 Phase portrait illustration between internal and

external faults in case of rate of change-based portrait

Figure 10 Phase portrait based on rate of change for

different fault locations

Figure 12 Zone of operation between the distance relay

and rate of change using phase portrait

a Distance relayb The rate of change using phase portrait

Figure 13 Studied configuration system and different

events of faults



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signals are the inputs to the relays, and 250 km is the linelength. The results are described on a phase portrait using

the (d(is ic))=dt dv=dt diagram. The relay is set toprotect 100% of the line (250 km); it forms the first zoneof the relay. The power system is modelled and differentsymmetrical and unsymmetrical faults with solid and faultresistance are simulated using the ATP-EMTP.

The performance of the proposed technique was evaluatedfor different types of internal and external faults, faultlocation and fault resistance. Results showed faults aretaken with a fault resistance ranged from 0 to 300 V.

Table 1 shows the different cases of the fault conditions. The THR is adjusted using (16). The parameters of the

studied configuration system is given in the Appendix.

Fig. 14 shows the phase portrait of the rate of change of thecurrent against voltage for different cases of fault conditionsmentioned in ( Table 1). The isotangent in the case of anexternal fault is almost fixed. The suggested technique givesthe solutions for the symmetrical and unsymmetrical faults

with solid and fault resistances. The threshold boundary

THR for 100% of the transmission line is set. As shown inthe figure, all internal faults rather than the external faultshave different isotangents (i.e. the isotangent in the case ofinternal faults lies up and down away from the THRboundary). This indicates that the relay is able to distinguishsuccessfully between internal and external faults.

9 Conclusions

The paper presented a technique for discrimination betweeninternal and external faults using a phase portrait. The paperused the geometrical theory of differential equations to drive a

symbol set on the basis of the visual appearance of the phaseportraits. The phase portrait of the instantaneous rate ofchange of current against voltage gives valuable informationon the transient and stability characteristics of the systemconfiguration with high fault resistance. The paper showedthat for non-singular, two-dimensional first-order phaseportraits, every isotangent curve is a unique straight line. Theresults showed the flow patterns for different internal faults

with different isotangent lines. In case of external faults, aunique isotangent line is given. The discrimination betweeninternal and external faults can be identified. Furthermore, theimpact of high fault resistance and remote-end feed isassessed. The results indicated different events with types offault condition, fault resistance and fault location using thephase trajectory which passes through the phase portrait.Hundred percent of the protected zone is successfullydiscriminated rather than the external zone.

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Figure 14 Phase portrait discrimination between internal

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Table 1 Different cases of fault conditions

Case Fault resistance,V Fault location, km

F1 (SLG) 200 50

F2 (SLG) 0 150

F3 (SLG) 100 150

F4 (SLG) 0 240

F5 (SLG) 0 10 (external)


F7 (2LG) 0 50 (external)




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11 Appendix

The parameters of the three-phase 500 kV transmission lineR0 0:1896V=km

R1 0:018V=km

L0 3:456mH=km

L1 0:936mH=km

C0 0:00828mF=km

C1 0:01134mF=km

The source capacity rating is

GVA 10 (sending end)

GVA 6 (receiving end)



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