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Electric Power Systems Research 92 (2012) 20– 28

Contents lists available at SciVerse ScienceDirect

Electric Power Systems Research

jou rn al h om epa ge: www.elsev ier .com/ locate /epsr

distance relay algorithm based on the phase comparison principle

inisa J. Zubic a,∗, Milenko B. Djuric b

Faculty of Electrical Engineering, University of Banjaluka, Patre 5, 78000 Banjaluka, Bosnia and HerzegovinaFaculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11120 Belgrade, Serbia

r t i c l e i n f o

rticle history:eceived 20 May 2011eceived in revised form 7 May 2012ccepted 14 May 2012

a b s t r a c t

In this paper we propose a distance relay algorithm based on a time-domain phase comparator with asmaller computational burden than the traditional phasor-domain based algorithms. The phase com-parator used in the time-domain algorithm is based on the average power on a half and full-cycle data

vailable online 29 June 2012

eywords:istance relaysime-domain phase comparatorhasor-domain phase comparator

window. The algorithm includes a filter which mitigates decaying DC-offset in the current as well asthe capacitive coupled voltage transformer transients. The algorithm is compared to an algorithm basedon the phasor-domain approach with full-cycle DFT. Both algorithms are tested on various fault loca-tions, fault resistances, load directions, inception angles and SIR. This paper also includes analysis of thecomplex frequency response of the complete time-domain algorithm, and how the protected zone for achosen operating characteristic depends on the line load and SIR.

. Introduction

Power transmission lines are the vital links between generat-ng plants and customers. These lines are exposed to various typesf faults, and distance relays are widely used for transmission linerotection. The distance protection is based on the principle of the

inear dependence between the value of the impedance measuredy the distance relay and the distance between locations of theelay and the fault [1]. Distance relay algorithms are usually basedn using phasors at the fundamental frequency. This is the phasor-omain approach where voltage and current phasors are eitherompared [2,3], or used for measuring the fault impedance [4–12].hasors are computed by using some of the well-known methodsf signal processing, such as: Fourier method [13–15], least errorquares [16–19], wavelet method [20–23], and Newton’s method24–26].

In modern numerical distance relays phasors do not need toe calculated. Instead of the phasor-domain approach, the relayperation is evaluated in the time-domain using instantaneousoltages and currents. Time-domain based relaying algorithms arentroduced in [27], where a time-domain comparator is based onharacteristic of an electromechanical cylinder unit, and a cross-olarized MHO characteristic is used. Results of this approach areresented in [2,3,28] where it is shown that the comparator based

n the cylinder unit has an inverse characteristic that allows forast clearing of close-in faults, but clearing faults at the end of therotected zone is very slow (up to four fundamental cycles).

∗ Corresponding author. Tel.: +387 65 861 441.E-mail addresses: zubic@etfbl.net (S.J. Zubic), mdjuric@etf.rs (M.B. Djuric).

378-7796/$ – see front matter © 2012 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.epsr.2012.05.017

© 2012 Elsevier B.V. All rights reserved.

This paper compares a distance relay algorithm based on thetime-domain phase comparator and a distance relay algorithmbased on the phasor-domain phase comparator. In order to achievefast operating time, in the time-domain approach we used a simplephase comparator based on the average power (AP [29]) instead ofthe comparator based on the cylinder unit [2,3,27,28]. The AP phasecomparator uses a half-fundamental cycle data window that allowsfor fast operating time, but this phase comparator is affected byeven-order harmonics in the current when CT saturation occurs. Inorder to eliminate impact of even-order harmonics we used the sec-ond version of the time-domain algorithm with a full-cycle phasecomparator. We have shown that the proposed algorithm is as fastas the phasor-domain algorithm based on DFT, yet it requires asmaller computational burden.

Many tests with different fault locations, fault resistances,inception angles, source impedance ratios (SIR) and line loads areconducted to show the algorithm’s security. We have also analyzedvery complex frequency response of the complete algorithm basedon the time-domain phase comparator. Each result is comparedto the phasor-domain algorithm based on the full-cycle discreteFourier transformation, and both algorithms use the same digitalfilter to eliminate decaying DC-offset [30].

2. MHO distance relay operating characteristics

It is convenient to display the operating characteristic for dis-tance protection in the complex R–X plane. We used an MHO

operating characteristic (Fig. 1) that can be achieved with one phasecomparator with border angles ±90◦, because the average-power(AP) phase comparator used in the time-domain algorithm has bor-der angles of ±90◦. The trip zone for the relay is the region inside

S.J. Zubic, M.B. Djuric / Electric Power Systems Research 92 (2012) 20– 28 21

tilgracticfieintst

d

wfidcdf

Table 1Inputs to a cross-polarized MHO relay.

Type of fault Ur Ir Upol

a-g Ua Ia + k0 I0 kbc(Ub − Uc)b-g Ub Ib + k0 I0 kca(Uc − Ua)c-g Uc Ic + k0 I0 kab(Ua − Ub)a-b Ua − Ub Ia − Ib kc · Uc

b-c Ub − Uc Ib − Ic ka · Ua

c-a Uc − Ua Ic − Ia kb · Ub

a,b,c – phases; g – ground; k0 is a relay setting called residual compensation factor;ka,b,c are chosen that Upol be in phase with Ur.

Table 2The relationship between the time delay Tϕ in voltage signals and the angle of theMHO characteristic ϕ (m = 32).

Fig. 1. Z – plane representation of the MHO characteristics.

he circle. The line impedance is shown as a solid line stretch-ng from the origin to the point labeled ZL, and the angle of theine impedance is labeled ϕL. The MHO characteristic is set toive the relay a reach of 85% of the total line length (ZP). Fig. 1aepresents a self-polarized characteristic while Fig. 1b represents

cross-polarized MHO characteristic. We use the self-polarizedharacteristic only to explain relay’s reach, but we implementedhe cross-polarized characteristic in proposed algorithms becauset has better fault resistance coverage and better behavior whenlose-in faults occurs. The self-polarized characteristic has twoxed dots, A and B in Fig. 1a, the line between A and B is the diam-ter of the MHO circle (d). The angle of the operating characteristics labeled ϕ. In order to achieve better fault resistance coverage it isecessary to use an MHO characteristic with a smaller angle thanhe angle of the protected line ϕL (angle ϕ < ϕL). It should be con-idered that for the constant relay reach Zp, diameter d depends onhe angles ϕL and ϕ:

= Zp

cos(ϕL − ϕ)= Zp

cos(�ϕ). (1)

The cross-polarized characteristic has only one fixed dot, A,hile position B depends on pre-fault load, fault location, SIR and

ault resistance. Fig. 2 shows that in a case when line load increasesn a forward direction we can expect overreach, while for reverseirection we can expect underreach. That means that for a chosenross-polarized characteristic we have to test the relay’s reach for

ifferent pre-fault conditions. Results for different angles ϕ, SIR,ault resistances and load directions are presented in Section 5.

Fig. 2. Z – plane representation of the cross-polarized MHO characteristics.

Delay 3 4 5 6 7ϕ (◦) 33.75 45 56.25 67 78.75Tϕ (ms) 1.875 2.5 3.125 3.75 4.375

The cross-polarized MHO operating characteristic can be imple-mented in the phasor-domain using phasors V1 and V2:

V- 1 = −e−jϕU- pol, (2)

V- 2 = −e−jϕU- r + I-rd. (3)

The phasor V1 is the polarizing quantity, while the phasor V2is the operating quantity that depends on the current, as well asthe voltage signal. Phasors Ur, Upol and Ir are functions of phasevoltages and currents on the protected transmission line and thefunctions are dependent on the type of fault according to Table 1.The self-polarized characteristic uses Ur instead of Upol as a polar-izing quantity.

In the phasor-domain approach, a distance relay calculates pha-sors V1 and V2 from phasors Ur, Upol and Ir, then calculates thephase difference = arg(V1) − arg(V2). If a fault occurs (i.e. the faultimpedance Z = Ur/Ir falls into the region inside the circle from Fig. 1)then the phase difference between V1 and V2 is:

|˛| > 90◦. (4)

In the other case, if the impedance falls out of the circle fromFig. 1, the phase difference between V1 and V2 is:

|˛| < 90◦. (5)

Eqs. (2) and (3) can be written in the time-domain as:

v1(k) = −upol(k-delay), (6)

v2(k) = −ur(k-delay) + d · ir(k), (7)

where k represents kth sample at time t, delay is an integer thatchanges the angle of the MHO characteristic ϕ. That indicates thatin the time-domain approach based on the phase comparator withborder angles ±90◦ we can choose only a few MHO characteristicswith angle ϕ from the equation:

ϕ = delay · 360◦

m, (8)

where m represents number of samples per fundamental cycle T(Table 2 is given for m = 32). Parameter ‘delay’ also represents atime shift T� that is caused by the angle ϕ in Eqs. (2) and (3):

Tϕ = delay · T

m= T

ϕ

360◦ . (9)

The time shift Tϕ can cause a time delay in the fault detection

because the algorithm does not use the latest voltage samples. Wecould achieve the same operating characteristic making the timeshift appears in the current signal ir, but that would have a big-ger impact on the operation speed since the rate of the change in

22 S.J. Zubic, M.B. Djuric / Electric Power Systems Research 92 (2012) 20– 28

cv(

tp

2

aviftsss|

˛io

C

Flvuw

T

Eos

T

ImTra

28 dB (F1 in Fig. 5).In order to reduce decaying DC offset in ir and impact of the tran-

sient response of capacitive coupled voltage transformers (CCVTs)[31], we propose using Characteristic Harmonic Digital Filter

Fig. 3. Time-domain phase comparator based on the average power.

urrent at the moment of a fault is larger than the change in theoltages. The tests in Section 5 show that the proposed Eqs. (6) and7) provide a good operation speed for the time-domain algorithm.

The proposed distance relay algorithm is based on the use ofhese Eqs. (6) and (7), and on a simple time-domain phase com-arator that detects if |˛| > 90◦.

.1. Time-domain phase comparator

In this paper we applied a phase comparator based on the aver-ge power, presented in [29]. The phase comparator uses an averagealue of the product of input signals p(k) = v1(k) · v2(k) for determin-ng their phase difference. It is well-known that the product of twoundamental harmonics has two components: a component withhe double frequency (the reactive power component), and a con-tant component that depends on the phase difference betweenignals (the active power component). Negative values of the con-tant (active power) component indicate that the phase difference˛| > 90◦, and distance relay must operate.

In Fig. 3 are shown signals v1(t) and v2(t) with a phase difference = 30◦, and their product p(t) that has a positive average value C on

ts cycle Tp = T/2. The average value C is determined by an integralf p on the cycle Tp, or on the cycle T:

(t) = 1Tp

∫ t

t−Tp

v1(�) · v2(�) d�. (10)

or the practical implementation of Eq. (10), two data windows ofength m are needed. These data windows contain discrete values1(k) and v2(k) calculated from the raw samples ir(k), upol(k) andr(k) through Eqs. (6) and (7). The average power of v1(k) · v2(k),ithin one cycle Tp, can be determined by the sum:

DHC = 2m

m/2∑k=1

v1(k) · v2(k). (11)

q. (11) represents the time-domain half-cycle algorithm. The sec-nd version of the time-domain algorithm is based on the full-cycleum within one cycle T:

DFC = 1m

m∑k=1

v1(k) · v2(k). (12)

n the time-domain approach the MHO characteristic is realized

aking use of Eqs. (6), (7) and (11), or alternatively (6), (7) and (12).

he same operating characteristic in the phasor-domain approachequires computing the phasors Ur, Upol and Ir, then calculating V1nd V2 through Eqs. (2) and (3), and finally their phase difference

Fig. 4. Block diagram of the time-domain and the phasor-domain algorithms.

= arg(V1) − arg(V2). Computing each of the phasors requires thesame computational burden as the time-domain phase comparisonin Eq. (12). This indicates that the time-domain approach allows usto realize a distance relay algorithm with a smaller computationalburden than in the phasor-domain approach.

3. Block diagram

The proposed algorithms are based on the time-domainapproach utilizing Eqs. (6), (7), (11) and (12). Characteristics of theproposed algorithms are compared to characteristics of a phasor-domain algorithm, based on the well-known full cycle discreteFourier transformation and Eqs. (2) and (3) in the phasor-domain.The block diagram of the complete system for testing the algo-rithms is shown in Fig. 4.

Input signals from the protected line are obtained from a sim-ulator that calculates electromagnetic transient behavior of thepower system. In our testing we used a sampling rate fs = 1600 Hz,or m = 32 samples per fundamental cycle. Fault type detection isimplemented according to Table 1.

The anti-aliasing filter (F1) is an analog low-pass filter that isused to minimize aliasing effects as well as to attenuate the highfrequency components. We used simulation of an analog secondorder Butterworth low-pass filter with cutoff frequency of 93.6 Hz,stop-band cutoff frequency of 380 Hz and stop-band attenuation

Fig. 5. Frequency responses of the anti-aliasing filter (F1) and CharmDF (F2).

S.J. Zubic, M.B. Djuric / Electric Power Systems Research 92 (2012) 20– 28 23

Table 3The relationship between the sampling rates m and the phase differences ˇ.

(a

s

wrss

Dsstb

K

we

K

poias

ra

4

ptdttt

m 16 20 32 64ˇ 78.75◦ 81.00◦ 84.37◦ 87.19◦

CharmDF), presented in [30], in current ir and voltage signals ur

nd upol.CharmDF is based on the equation:

f(k) = s(k) − s(k − 1) (13)

here s(k) represents kth signal sample at time t, s (k − 1) rep-esents (k − 1)th signal sample at time (t − TS), TS represents aampling cycle (TS = T/m) and sf(k) represents the kth filtered signalample.

The CharmDF significantly decreases the level of the decayingC component in the current and voltage signals with only one

ample time delay, and also makes a positive phase shift in theignal sf which can be seen in Fig. 6. The phase shift depends onhe sampling rate m, which is shown in Table 3. The relationshipetween sf and s can be shown by their phasors Sf and S:

1 · S-f = eiˇ · S-, (14)

here magnitudes of the phasors are related through the followingxpression [30]:

1 = 12 · sin(�/m)

= |S-||S-f|

. (15)

The CharmDF is a high-pass filter that attenuates the DC com-onent, while amplifying harmonic components with frequenciesver the fundamental frequency. Its frequency response is shownn Fig. 5 (F2). The frequency responses of the sums (11) and (12),nd complete time-domain algorithms are presented in the nextection.

For easy comparison of transient behavior of the analyzed algo-ithms, the output of the phasor-domain based algorithm PDFC islso set to detect a fault when its value becomes negative.

. Frequency responses of the proposed algorithms

For obtaining the most realistic frequency response of the pro-osed algorithms, we have to consider frequency responses ofhe anti-aliasing filter, decaying DC-offset filter, and the time-

omain phase comparators. The anti-aliasing filter is an LP filterhat attenuates harmonic components (F1 in Fig. 5), but passeshe DC component. The decaying DC-offset filter is a high-pass fil-er that attenuates the DC component, while amplifying harmonic

Fig. 6. Original (s) and wave (sf) filtered by CharmDF.

Fig. 7. Frequency responses of the half-cycle sum (F3) and the full-cycle sum (F4).

components with frequencies over the fundamental frequency (F2in Fig. 5). The half-cycle sum in Eq. (11) is a filter whose frequencyresponse is shown in Fig. 7 (F3), while the full-cycle sum in Eq. (12)is a filter whose frequency response is shown in Fig. 7 (F4). Thefull-cycle sum can filter out all harmonic components while thehalf-cycle sum can filter out only even-order harmonics. In order todetermine the frequency response of the whole time-domain algo-rithms we have to analyze how the harmonic components in inputsignals ir(k), upol(k) and ur(k) appear in the algorithms’ outputsTDHC and TDFC.

The phase comparator has two inputs (v1 and v2) and the mainproblem in the time-domain approach is that a harmonic compo-nent in one input appears as a different harmonic component intheir product p. For example, in Fig. 8 we show that the fifth har-monic in v2, after multiplying with v1, appears as two components,the fourth and the sixth harmonic. The integral of p will filter out thesecond, the fourth and the sixth harmonic components, and keeponly the DC component containing the information of the phasedifference between v1 and v2. In a different case, if an even-orderharmonic appears in v2, for example the fourth harmonic, it willappear in p as the third and the fifth harmonic component. Thehalf-cycle sum in Eq. (11) cannot filter out odd-order harmoniccomponents and they will appear in output TDHC as a fluctuat-ing component, but the full-cycle sum in Eq. (12) is able to filterout these harmonics.

This short analysis shows that, in contrast to the phasor-domainapproach, for time-domain algorithms it is not possible to plot aclassical frequency response with gain on the vertical axis becausean input harmonic component appears as a few different harmoniccomponents in the outputs TDHC and TDFC. This is the reason whywe decided to use error as the variable in our frequency response,instead of gain. We compared the accurate value C, without har-monic components in input signals, to the outputs TDHC and TDFCwhen a harmonic component appears in one input signal. The valueerror represents the maximal deviation from the accurate value C(Fig. 9). In order to compare the proposed time-domain algorithms

to the phasor-domain algorithm PDFC, we used the same criteriato obtain the frequency response of the phasor-domain algorithmwhen harmonic components are presented in signals ir(k) and ur(k).

Fig. 8. Signals v1, v2 and p in the frequency domain.

24 S.J. Zubic, M.B. Djuric / Electric Power Systems Research 92 (2012) 20– 28

F

rapsmcpot(ogFw

at

5

at

ictac

cr

Fp

Table 4The relationship between the protected zone and the angle ϕ for different faultresistances Rf .

Rf 0 � 10 � 20 � 30 � 40 �

ϕ = 33.75◦ 85% 99% 89% 81% 74%ϕ = 45.00◦ 85% 88% 81% 72% 64%ϕ = 56.25◦ 85% 84% 75% 65% 56%ϕ = 67.00◦ 85% 80% 70% 59% 48%

Table 5Impact of the load direction to the distance relay protected zone (SIR = 0.3, ϕ = 45.00◦ ,Rf = 0 �).

LOAD � = 5◦ � = 10◦ � = 15◦ � = 20◦

Forward (� = �) 85% 85% 85% 85%Reverse (� = −�) 84% 83% 83% 82%

Table 6Impact of the load direction to the distance relay protected zone (SIR = 5, ϕ = 45.00◦ ,Rf = 0 �).

LOAD � = 5◦ � = 10◦ � = 15◦ � = 20◦

ig. 9. Evaluation of the harmonic components’ impact on the algorithms’ accuracy.

In Fig. 10 we show the frequency response of the proposed algo-ithms, as well as the frequency response of the phasor-domainlgorithm. In our analyses we considered all explained filters andut harmonic components in ir(k) and ur(k) from 0 to 8 withtep 0.1. For this testing each harmonic component has the sameagnitude as the fundamental component of ir or ur, and each

omponent is added to the input signal 200 times with randomhase angle. We can see from Fig. 10 that the frequency responsef the full-cycle time-domain algorithm (TDFC) is very similar tohe frequency response of the full-cycle phasor-domain algorithmPDFC) based on the FCDFT. The half-cycle TDHC is able to filterut only odd-order harmonics in input signals. All algorithms haveood attenuation for DC components. The frequency responses inig. 10 do not have a sense for the fundamental harmonic becausee do not calculate gain.

In the next section we will compare the behavior of the proposedlgorithms with the phasor-domain algorithm using numerousests.

. Testing of the algorithms

Performances of the relaying algorithms are evaluated through series of tests using a simulator that calculates electromagneticransient behavior of the power system.

Many tests with different fault locations, fault resistances Rf,nception angles, source impedance ratios (SIR) and line loads areonducted to show the algorithms’ performances. The power sys-em parameters are given in Fig. 11. Line load is controlled by thengle �. In our testing we used m = 32 samples per fundamental

ycle.

The first task is choosing an optimal angle of the MHO operatingharacteristic for all further tests. In Table 4 we showed how theelay’s reach depends on the fault resistance for different angles

ig. 10. Frequency responses of the proposed time-domain algorithms and thehasor-domain algorithm.

Forward (� = �) 84% 85% 86% 87%Reverse (� = −�) 81% 79% 78% 76%

of the MHO characteristic ϕ. For each angle ϕ we chose a cross-polarized MHO characteristic that offers 85% relay reach for forwardline load (in Fig. 11 load flows from ES to ER, � = 10◦), fault resistanceRf = 0 � and SIR = 0.3. We can notice that for some fault resistancesit is possible overreach, but for increasing ϕ and Rf reach is smaller.According to Table 4 we chose the cross-polarized MHO character-istic with ϕ = 45◦ (delay = 4) for all further tests.

Tables 5 and 6 show how the relay’s reach for a chosen operatingcharacteristic depends on different load directions and differentSIR. We can notice that for small SIR and forward load direction, therelay’s reach does not depend on the line load, yet for the reversedirection, we can notice a small underreach. Bigger underreach andsmall overreach is noticeable for SIR = 5.

The algorithms are tested on phase-to-ground fault, phase-to-phase fault, and three-phase fault. Voltage and current waveformsare shown in Fig. 12 for a-g fault, Fig. 14 for a-b fault and Fig. 16for a-b-c fault at 40 km and with SIR = 0.3. The waveforms contain alot of noise after the fault inception. The outputs of the algorithmsare showed in Fig. 13(a-g fault), Fig. 15(a-b fault) and Fig. 17(a-b-cfault). It can be observed that the time-domain full-cycle algorithmTDFC has very similar output to the phasor-domain algorithm PDFC.The half-cycle algorithm TDHC has a smaller time delay but in caseof the current transformer saturation (Fig. 18) we can observe thatthe output of the TDHC algorithm fluctuates due to the presence ofeven-order harmonics in input current.

For faults that cause very low line voltages, the transient out-put voltage from capacitive coupled voltage transformers (CCVT)may be different from its input voltage waveform. Fig. 19a showsthe voltage waveform in the primary system (ratio voltage) and

the transient voltage waveform in the secondary of CCVT. If we didnot use CharmDF, then the algorithms’ outputs would be affectedas shown in Fig. 19b. After the fault occurs the output in Fig. 19b

Fig. 11. Single line diagram of the power system model.

S.J. Zubic, M.B. Djuric / Electric Power Systems Research 92 (2012) 20– 28 25

Fig. 12. Voltage and current waveforms for phase-to-ground fault at 40 km distance.

drtC

l

F

Fig. 15. Outputs from compared algorithms for phase-to-phase fault.

Fig. 13. Outputs from compared algorithms for phase-to-ground fault.

rops below its steady value and that can cause the incorrect over-each. In order to reduce CCVT transients we applied CharmDF to

he voltage signals. Fig. 20 shows that applying this filter mitigatesCVT transients and TDFC does not drop below its steady value.

We also compared operating times for different SIR and faultocations. From Fig. 21 it can be observed that the TDHC algorithm

ig. 14. Voltage and current waveforms for phase-to-phase fault at 40 km distance.

Fig. 16. Voltage and current waveforms for three-phase fault at 40 km distance.

has an operating time under 0.6 fundamental cycle for all fault loca-tions and SIR. The TDFC algorithm has an operating time up to 1.1fundamental cycles, and its speed is practically the same as thespeed of the PDFC algorithm.

The operating time is also tested for a phase-to-ground fault

at different fault locations, at various inception angles, and faultresistances, while SIR = 0.3. The results of simulations are summa-rized in Tables 7–15. Inception angles of the voltage waveform are90◦, 45◦ and 0◦. The operating time of the algorithms is provided in

Fig. 17. Outputs from compared algorithms for three-phase fault.

26 S.J. Zubic, M.B. Djuric / Electric Power Systems Research 92 (2012) 20– 28

Fig. 18. Impact of saturation of the current transformer to the time-domain algo-rithms’ outputs.

Fig. 19. Impact of the CCVT transient on the algorithm’s output when CharmDF isnot applied.

Table 7Operating time in cycles for an inception angle of 90◦ and Rf = 0 �.

Fault location Time domain algorithm Phasor domainPDFC

TDHC TDFC

10% 0.08 0.38 0.3530% 0.41 0.53 0.5350% 0.42 0.58 0.5870% 0.52 0.98 0.9880% 0.6 1.07 1.06

Fig. 20. Impact of the CCVT transient on the algorithm’s output when CharmDF isapplied.

Fig. 21. Operating time of the compared algorithms for different source impedanceratios (SIR).

S.J. Zubic, M.B. Djuric / Electric Power Systems Research 92 (2012) 20– 28 27

Table 8Operating time in cycles for an inception angle of 45◦ and Rf = 0 �.

Fault location Time domain algorithm Phasor domainPDFC

TDHC TDFC

10% 0.1 0.16 0.1630% 0.15 0.56 0.5550% 0.25 0.6 0.670% 0.55 0.74 0.7480% 0.6 1.05 1

Table 9Operating time in cycles for an inception angle of 0◦ and Rf = 0 �.

Fault location Time domain algorithm Phasor domainPDFC

TDHC TDFC

10% 0.21 0.28 0.2630% 0.28 0.62 0.650% 0.35 0.74 0.7270% 0.63 0.82 0.8280% 0.72 1.15 1.1

Table 10Operating time in cycles for an inception angle of 90◦ and Rf = 20 �.

Fault location Time domain algorithm Phasor domainPDFC

TDHC TDFC

10% 0.1 0.39 0.3830% 0.4 0.52 0.5250% 0.42 0.78 0.770% 0.51 0.96 0.9680% 0.92 1.42 1.41

Table 11Operating time in cycles for an inception angle of 45◦ and Rf = 20 �.

Fault location Time domain algorithm Phasor domainPDFC

TDHC TDFC

10% 0.12 0.27 0.2630% 0.22 0.55 0.5550% 0.48 0.65 0.6470% 0.6 1 180% 0.7 1.15 1.2

Table 12Operating time in cycles for an inception angle of 0◦ and Rf = 20 �.

Fault location Time domain algorithm Phasor domainPDFC

TDHC TDFC

10% 0.23 0.37 0.3730% 0.3 0.65 0.6550% 0.57 0.76 0.7670% 0.68 1.1 1.0880% 0.82 1.3 1.3

Table 13Operating time in cycles for an inception angle of 90◦ and Rf = 40 �.

Fault location Time domain algorithm Phasor domainPDFC

TDHC TDFC

10% 0.28 0.42 0.4130% 0.41 0.75 0.6850% 0.45 0.9 0.970% nop nop nop80% nop nop nop

Table 14Operating time in cycles for an inception angle of 45◦ and Rf = 40 �.

Fault location Time domain algorithm Phasor domainPDFC

TDHC TDFC

10% 0.15 0.48 0.4730% 0.45 0.62 0.6250% 0.52 0.95 0.9170% nop nop nop80% nop nop nop

Table 15Operating time in cycles for an inception angle of 0◦ and Rf = 40 �.

Fault location Time domain algorithm Phasor domainPDFC

TDHC TDFC

10% 0.28 0.61 0.630% 0.52 0.72 0.7250% 0.66 1.06 1.04

70% nop nop nop80% nop nop nop

fundamental cycles, while nop is used for cases when an algorithmdid not operate. Tables 7–15 show that the nop cases occurred forthe biggest Rf according to Table 4 for ϕ = 45◦. The operating timealso increases for larger Rf. Those results also show that the pro-posed time-domain algorithm has nearly the same operating speedas the phasor-domain algorithm.

6. Conclusions

Two versions of a distance relay algorithm based on the time-domain approach are presented and compared to an algorithmbased on the phasor-domain approach with full-cycle DFT. Thereare two main differences between the time-domain and the phasor-domain algorithms. The first is that it is not possible to get classicalfrequency response for the time-domain algorithms because itsphase comparator multiplies input voltages and currents. We useda different criterion and showed that the full-cycle version of thetime-domain algorithm has nearly the same frequency response asthe phasor-domain algorithm based on FCDFT. The half-cycle ver-sion of the time-domain algorithm could not filter out even-orderharmonics, and that is the reason why its algorithm cannot be usedin a case where current transformer saturation is expected. Thesecond difference is that the time-domain algorithm uses delayedinput signals in order to achieve phase shifts for different cross-polarized MHO operating characteristics. Tests showed that thesedelayed signals practically did not affect the operation speed.

After all tests we can make a few conclusions. The proposedtime-domain algorithm has nearly the same frequency responseand operating speed as the traditional phasor-domain based algo-rithm. The main advantage is that the time-domain approachallows for realization of an MHO operating characteristic with asmaller computational burden than the phasor-domain approach,which requires calculating the phasors for all inputs. The main dis-advantage of the time-domain approach is that the angle of theoperating characteristic can be set in only discrete steps that aredependent on the sampling frequency.

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