2001TR027 by Math Bollen 1

=

Abstract— This paper addresses the problem of estimating thecharacteristics of three-phase unbalanced voltage dips frommeasured phase voltages. This is important for obtaining statisticson voltage dips and for obtaining information about theunderlying event (e.g. the fault type). Two different algorithmsare compared. The “six-phase algorithm” is computationallysimple and easy to interpret. However large phase-angle jumpslead to wrong estimations. The “symmetrical componentalgorithm” gives a correct value in almost all cases. Exceptionsare events with very severe load influence on the voltages duringthe fault. The latter algorithm is studied in more detail: through ameasurement example and through a study of a range of syntheticevents. The conclusion is that both algorithms have their specificapplication areas.

Index Terms— electromagnetic compatibility (EMC), powerquality, power quality monitoring, power system faults, voltagedips (sags).

I. INTRODUCTION

oltage dips due to non-symmetrical faults will affectthree-phase equipment in a different way than voltage

dips due to symmetrical faults. Therefore additionalcharacterization effort is needed for these so-called three-phase unbalanced dips (i.e. voltage dips due to non-symmetrical faults). A number of proposals for theircharacterization were compared by [1]. Further contributionswere made by [2][3][4][5]. Most of this work was directedtowards obtaining dip characteristics from measurements,without considering the basic circuit theory behind thephenomenon. An alternative approach was followed in [6]:from analyzing basic fault types in idealized systems, aclassification in four dip types was proposed. Although theclassification can be used for stochastic prediction andequipment testing [7], it could not directly be used to classifymeasured voltage dip events. The work presented in [8][9]generalizes the classification and proposes an algorithm forextracting dip type and characteristics from measured voltagewaveforms; a simplified algorithm is presented in [10].

The aim of this paper is to find the limitations of these twocharacterization methods. After a description of thecharacterization method including a discussion on the need forcharacterization, two algorithms are described for extractingcharacteristics from measured voltages. Next these two

M. H. J. Bollen is with the Department of Electric Power Engineering,Chalmers University of Technology, Gothenburg, Sweden (e-mail:m.bollen@ieee.org).

algorithms are applied to a number of synthetic events tocompare their performance. One of the algorithms is alsoapplied to a measured event to show some of theimplementation problems. Finally a range of severe events isused to explain the occurrence of erroneous estimates with thealgorithm.

II. CHARACTERIZATION

A. The aim of CharacterizationVoltage-dip characterization concerns the quantification of

voltage-dip events through a limited number of parameters.The forthcoming power-quality measurement standard IEC61000-4-30 uses two parameters: retained voltage andduration. The retained voltage is the lowest one-cycle rmsvoltage; the duration is the length of time during which the rmsvoltage is below a threshold [11]. This constitutes a loss ofinformation for non-rectangular and multistage events but is anappropriate approximation for most single-channelmeasurements.

Multi-channel measurements are characterized by the lowestretained voltage and the longest duration of all channels. Thisis only an appropriate approximation for balanced dips,whereas the majority of dips are unbalanced. A somewhatstrange consequence of this is that the dip due to a single-phase fault in a resistance-grounded system will be quantifiedthe same (or even more severe) as the dip due to a three-phasefault. However a customer behind a Dy-transformer onlyexperiences a severe event in the second case. Some utilitiessolve this by connecting their monitors phase-to-phase, butthat will limit the amount of information to be obtained fromthe monitors. To overcome this and similar problems animproved dip characterization method has been proposed in[6], which will be summarized below.

The aim of voltage-dip characterization is to presentstatistics in a simple way. These statistics can next be used incompatibility studies, to assess system performance, etc.Another aim is to extract information on the underlyingevent(s). The dip characterization proposed in [6] can be usedto find out the type of fault and the voltage level at which thefault occurred. Such an algorithm can be used as part of anexpert system for the automatic classification of power-qualitydisturbances [12].

Voltage-dip statistics will typically be obtained at the lowervoltage levels, close to the end-users. Automatic analysis of

Algorithms for characterizing measured three-phase unbalanced voltage dips

Math H.J. Bollen, Senior Member, IEEE

V

2001TR027 by Math Bollen 2

power-quality disturbances could find its application at anyvoltage level. At higher voltage levels it could be part ofdisturbance recorders of protection relays.

B. Types of Three-Phase Unbalanced DipsVoltage dips are due to short circuits and earth faults,

transformer energizing and motor starting. By using thevoltage recovery and the voltage unbalance during the dip it ispossible to distinguish between these three types from voltagerecordings [12]. A further subdivision of voltage dips due tofaults is proposed in [6][7]. The basic distinction is betweentypes A, C and D:

• type A is an equal drop in the three phases;• type C is a drop in two phases;• type D is a large drop in one phase with a small drop in

the other two phases.For types C and D a further subdivision is needed to include

the symmetrical phase (the phase with the large voltage dropfor type D, the phase without voltage drop for type C). Theresulting six types of three-phase unbalanced dips are shown inFig.1. Type Db is a drop in phase b; type Cb a drop in phases aand c, etc. Alternatively, type Cb can be interpreted as a dropin the ac voltage difference, etc.

CaCb Cc

Da Db Dc

Fig. 1, Six types of three-phase unbalanced voltage dips: thin arrows indicatenormal voltages; thick arrows voltages during the event.

The two parameters quantifying the dip are thecharacteristic voltage V and the so-called PN factor F, bothcomplex numbers. The phase voltages as a function of thesetwo parameters are, for a type Ca dip:

3

3

21

21

21

21

jVFV

jVFV

FV

c

b

a

+−=

−−=

=

(1)

and for a type Da dip:

3

3

21

21

21

21

jFVV

jFVV

VV

c

b

a

+−=

−−=

=

(2)

The phasor diagrams in Fig. 1 are given for V=0.5 and F=1.The aim of the two algorithms to be discussed below is toobtain the dip type according to Fig.1, the characteristicvoltage V, and the PN factor F, from the complex phase

voltages Va, Vb, and Vc. The latter are obtained from themeasured voltage waveforms. Both characteristic voltage Vand PN factor F are complex numbers. The absolute value andthe argument of the characteristic voltage are referred to as“magnitude” and “phase-angle jump” of the voltage dip,respectively. The magnitude (absolute value of thecharacteristic voltage) is a generalization for three-phaseevents of the retained voltage defined in IEC 61000-4-30.

C. Relation with Fault TypesThe dip type depends on the fault type and the winding

connection of the transformers between the fault and themeasurement location [6]. The characteristic voltage is themain characteristic describing the event. It is determined bythe positive-sequence source and feeder impedance for two-phase and three-phase faults. For single-phase faults the zero-sequence impedance also affects the characteristic voltage ofthe dip [7].

The PN factor is a second characteristic. It is a measure ofthe unbalance of the event. The lower the magnitude of the PNfactor, the more balanced the dip; where the PN factor cannotbe smaller than the characteristic voltage (in absolutevalue)[8]. For single-phase and phase-to-phase faults the PN-factor is close to 1 pu. The deviation from 1 pu is due to theload and will be discussed in more detail later. For two-phase-to-ground faults in solidly-grounded systems, the PN factor isless than 1 pu. The drop in PN factor is at most one third of thedrop in characteristic voltage. For three-phase faults the PNfactor is equal to the characteristic voltage.

The zero-sequence voltage V0 can be used as a thirdcharacteristic to completely describe the event. It givesadditional information for obtaining the location of theunderlying event, but the zero-sequence voltage rarely affectsthe operation of equipment so that it is not needed forstatistics.

III. THE TWO ALGORITHMS

A. The Symmetrical Component AlgorithmThe algorithm proposed in [8] determines the dip type from

the positive-sequence and negative-sequence voltages. From(1), (2) and similar expressions for the other dip types in Fig.1, it can be concluded that the positive-sequence voltage (withreference to a-phase pre-fault voltage) is the same for all diptypes:

( )VFV += 21

1 (3)

The negative-sequence voltage is the same in magnitude butdifferent in argument:

2001TR027 by Math Bollen 3

( )( )

( )( )( )

( )VFaV

VFaV

VFV

VFaV

VFaV

VFV

−−=

−−=

−−=

−=

−=

−=

221

2

21

2

21

2

221

2

21

2

21

2

c

b

a

c

b

a

D type

D type

D type

C type

C type

C type

(4)

where a constitutes a rotation over 120°. If we assume thatF=1 the angle between the drop in positive-sequence positiveand negative-sequence voltage is an integer multiple of 60°.The angle obtained from a measurement can be used to obtainthe dip type:

��

���

−×

°=

1

2

1arg

601

VVT (5)

where T is rounded to the nearest integer:T = 0 � type Ca

T = 1 � type Dc

T = 2 � type Cb

T = 3 � type Da

T = 4 � type Cc

T = 5 � type Db

Knowing the dip type, the other characteristics can beobtained, e.g. from the sum and difference of positive andnegative-sequence voltage according to (3) and (4).

B. The six-phase algorithmA simplified algorithm is described in [10]. After

subtraction of the zero-sequence voltage, the rms voltage isobtained for the three phase voltages and the three phase-to-phase voltages:

( ){ }( ){ }( ){ } rms

rms

rms

313131

cbacC

cbabB

cbaaA

vvvvV

vvvvV

vvvvV

++−=

++−=

++−=

(6a)

��

��� −=

��

��� −=

��

��� −=

3rms

3rms

3rms

acCA

cbBC

baAB

vvV

vvV

vvV

(6b)

The dip parameters are obtained directly: the characteristicvoltage is the lowest of the six rms voltages, the PN factor thehighest of the six. The dip type is determined from the voltageaccording to (6) with the lowest rms value:VA lowest � type Da

VB lowest � type Db

VC lowest � type Dc

VAB lowest � type Cc

VBC lowest � type Ca

VCA lowest � type Cb

The six-phase algorithm can also be used to obtain the

arguments of the complex numbers V and F. The argument ofthe characteristic voltage is the phase angle of the voltage thatgives the lowest rms value: va-v0 for type Da, vb-vc for type Cc,etc. The argument of the PN factor is the phase angle of thevoltage that gives the highest rms value.

IV. NUMERICAL EXAMPLES

To show the performance of the algorithms and theirlimitations, the complex phase voltages are, for a number ofcases, calculated from given dip characteristics according to(1) and (2). The two algorithms are next applied to thecomplex phase voltages and the resulting characteristicscompared with the known input values.

A. Single-phase faultConsider a drop of voltage in phase a down to 50% of its

pre-event value. It is assumed that the voltage in b and cremains as before. According to the classification introducedbefore this is a dip of type Da with F=1, V=0.67, V0=-0.33.The results for the symmetrical component algorithm areshown in the first column of Table 1. The remaining columnsgive the results for a voltage drop in one phase, including aphase-angle jump of -20°, -30°, -40°. As the PN factor equalsexactly one, the angle between drop in positive sequence andnegative-sequence voltage is exactly 180°. The dip type isobtained correctly and so are the other characteristics.

The phenomenon that the characteristic voltage is notexactly equal to the voltage in the faulted phase for single-phase faults is described in [6].

TABLE ISYMMETRICAL COMPONENT ALGORITHM DURING SINGLE-PHASE FAULTS.

50% 0° 50% -20° 50% -30° 50% -40°Va 0.50 0.47-0.17j 0.43-0.25j 0.38-0.32jVb -0.50-0.87j -0.50-0.87j -0.50-0.87j -0.50-0.87jVc -0.50+0.87j -0.50+0.87j -0.50+0.87j -0.50+0.87jV0 -0.17 -0.18-0.06j -0.19-0.08j -0.21-0.11jV1 0.83 0.82-0.06j 0.81-0.08j 0.79-0.11jV2 -0.17 -0.18-0.06j -0.19-0.08j -0.21-0.11j

angle 180° 180° 180° 180°V 0.67 0.65-0.11j 0.62-0.17j 0.59-0.21j

|V| 0.67 0.66 0.64 0.63F 1.00 1.00 1.00 1.00

TABLE IISIX-PHASE ALGORITHM DURING SINGLE-PHASE FAULTS

50% 0° 50% -20° 50% -30° 50% -40°Type 3 (Da) 3 (Da) 3 (Da) 4 (Cc)|V| 0.67 0.66 0.64 0.60|F| 1.00 1.00 1.00 1.02

The same synthetic dips are also applied to the six-phasealgorithm. The results are shown in Table II. For moderatevalues of the phase-angle jump, also the six-phase algorithmgives the correct type and characteristics. For a -40° phase-angle jump the six-phase algorithm results in a type Cc insteadof Da. The rotation in the a-phase voltage is so large that the

2001TR027 by Math Bollen 4

ab-phase difference becomes less in absolute value (afterremoval of the zero-sequence component).

B. Phase-to-phase faultThe same testing has been done for drops in the voltage

between the phases b and c to 50%, thus representing a phase-to-phase fault at the same voltage level or a single-phase faultat another voltage level. For this the algorithm should result ina dip of type Ca with F=1 and V=0.5. The results for bothalgorithms are shown in Table III, where the bottom threerows refer to the results of the six-phase algorithm. Again thesymmetrical component algorithm gives the correct result inall cases, whereas the six-phase algorithm gives a wrong diptype and characteristics for large phase-angle jumps.

TABLE IIITHE TWO ALGORITHMS DURING PHASE-TO-PHASE FAULTS

50% 0° 50% -20° 50% -30° 50% -40°SYMMETRICAL-COMPONENT ALGORITHM

Va 1.00 1.00 1.00 1.00Vb -0.50-0.43j -0.65-0.41j -0.72-0.38j -0.78-0.33jVc -0.50+0.43j -0.35+0.41j -0.28+0.38j -0.22+0.33jV1 0.75 0.73-0.09j 0.72-0.13j 0.69-0.16jV2 0.25 0.27+0.09j 0.28+0.13j 0.31+0.16j

angle 0° 0° 0° 0°V 0.50 0.47-0.17j 0.43-0.25j 0.38-0.32jF 1.00 1.00 1.00 1.00

SIX-PHASE ALGORITHM

Type 0 (Ca) 0 (Ca) 1 (Dc) 1 (Dc)|V| 0.50 0.50 0.47 0.40|F| 1.00 1.00 1.01 1.04

C. Impact of load – drop in voltageMeasurements [7][10][12] as well as simulations [7][9][13]

have shown that the voltages in the three phases drop by anequal factor due to the load. In terms of the classificationpresented in Section II of the paper, the PN factor F drops to avalue less than unity.

This has been modeled for the phase-to-phase faults (typeCa dips) studied before. It is assumed that the voltage drops by15% in all three phases due to the load. The input values of thedip characteristics are Type=Da, V=0.43, F=0.85. The effectof the phase-angle jump is a rotation in V but not in F. Theresults are shown in Table IV. As the PN-Factor is no longerequal to one, the angular difference between the drop inpositive-sequence and the negative-sequence voltage is nolonger exactly an integer multiple of 60°. But after rounding tothe nearest integer multiple of 60° the algorithm still gives inthe correct results.

As all three phases are affected in the same way, the six-phase algorithm shows the same behavior as for unity PN-factor.

TABLE IVSYMMETRICAL COMPONENT ALGORITHM WITH ADDITIONAL DROP IN

VOLTAGE DUE TO LOAD EFFECTS

50% 0° 50% -20° 50% -30° 50% -40°Va 0.85 0.85 0.85 0.85Vb -0.43-0.37j -0.55-0.35j -0.61-0.32j -0.66-0.28jVc -0.43+0.37j -0.30+0.35j -0.24+0.32j -0.19+0.28jV1 0.64 0.62-0.07j 0.61-0.11j 0.59-0.14jV2 0.21 0.23+0.07j 0.24+0.11j 0.26+0.14j

angle 0° 6.9° 8.6° 9.18°V 0.43 0.40-0.15j 0.37-0.21j 0.33-0.27jF 0.85 0.85 0.85 0.85

D. Impact of load – phase shiftThe change in load currents not only leads to a drop in the

three voltages, it typically also causes a phase shift (rotation)of the three voltages. It is again assumed that the load effect isthe same for the three phases: a drop of 15% and a rotation of-20°: thus F=0.85exp(-j20°). The results for this case areshown in Table V. It turns out that the symmetrical componentalgorithm gives an erroneous result for small phase-anglejumps in the voltage. The rotation in PN-factor severely affectsthe angles of (1-V1) and V2, leading to a wrong dip type. Forlarger phase-angle jumps the error in angle becomes less than30° so that the correct dip type results after rounding to thenearest multiple of 60°.

The problem with the symmetrical component algorithm canbe relatively easily solved. The dip type is determined in (5)by rounding off the angle. Modifying the rounding somewhatwill solve the problem: the range -50°, +10° becomes dip type0, etc. The resulting equation, replacing (5) reads as:

°

°+��

���

−=

60

201

arg1

2

VV

T (7)

where T is again rounded to the nearest integer. The results forthis improved symmetrical component algorithm are shown inTable VI.

TABLE VSYMMETRICAL COMPONENT ALGORITHM WITH ADDITIONAL VOLTAGE DROP

AND ROTATION DUE TO LOAD EFFECTS

50% 0° 50% -20° 50% -30° 50% -40°Va 0.80-0.29j 0.80-0.29j 0.80-0.29j 0.80-0.29jVb -0.53-0.20j -0.64-0.14j -0.68-0.09j -0.78-0.04jVc -0.27+0.49j -0.16+0.43j -0.12+0.38j -0.08+0.33jV1 0.60-0.22j 0.56-0.28j 0.54-0.31j 0.51-0.33jV2 0.20-0.07j 0.24-0.01j 0.26+0.02j 0.29+0.04j

angle -48.5° -34.9° -29.8° -26.2°V 0.56-0.01j 0.45-0.07j 0.27-0.33j 0.21-0.37jF 0.64-0.53j 0.67-0.49j 0.80-0.29j 0.80-0.30j

Type 0 (Ca) 0 (Ca) 1 (Dc) 1 (Dc)|V| 0.43 0.43 0.40 0.34|F| 0.85 0.85 0.86 0.89

2001TR027 by Math Bollen 5

TABLE VIIMPROVED SYMMETRICAL COMPONENT ALGORITHM WITH ADDITIONAL

VOLTAGE DROP AND ROTATION DUE TO LOAD EFFECTS

50% 0° 50% -20° 50% -30° 50% -40°angle -28.5° -14.9° -9.8° -6.2°

V 0.40-0.15j 0.33-0.27j 0.27-0.33j 0.21-0.37jF 0.80-0.29j 0.80-0.29j 0.80-0.29j 0.80-0.30j

V. THE SIX-PHASE ALGORITHM

In the previous section, the six-phase algorithm has beenapplied to a small number of synthetic dips with knowncharacteristics. This same process has been repeated for alarger number of synthetic dips covering a wide range ofmagnitude and phase angle of the characteristic voltage. (Asmentioned before, the PN-factor does not affect theperformance of the six-phase algorithm.) The results arepresented graphically in Fig. 2. The black dots indicate thecombinations of magnitude and phase-angle (of thecharacteristic voltage) for which the algorithm gives incorrectresults. Also indicated in the figure, through the solid curves,is the range of magnitude and phase-angle jump that can beexpected. The upper and lower curves are for impedanceangles +10° and -60°, respectively. The black dots aregenerated by varying the impedance angle between -90° and+90°. (See Appendix A for an explanation of the term“impedance angle”.) It follows from Fig. 2 that the six-phasealgorithm gives incorrect results for events with a large(negative) phase-angle jump and moderate drops in voltage.

Fig. 2. Whole-range testing of the six-phase algorithm: the black dotsindicate where the algorithm gives an incorrect result.

VI. CHARACTERISTICS VERSUS TIME

To show some of the potential problems when implementingthe symmetrical component algorithm, it is applied to thethree-phase unbalanced dip shown in Fig. 3. The voltages aremeasured in an 11-kV distribution system during a 132-kVfault. The fault is cleared in about 5 cycles, after which thevoltage recovers. The slow but balanced recovery indicates thepresence of large amounts of induction motor load..

0 2 4 6 8 10 12 14 16-1.5

-1

-0.5

0

0.5

1

1.5

Time [Cycles]

Volta

ge [p

i]

Fig. 3. Example of a three-phase unbalanced dip.

Applying the symmetrical component algorithm to thesemeasurements will result in positive and negative-sequencevoltage phasors V1 and V2, respectively. The symmetricalcomponent voltages are calculated from the complexfundamental phase voltages over a half cycle (10 ms) window.A sliding window is used to obtain the symmetrical componentvoltages as a function of time. The angle between (1-V1) andV2 is used to determine the type of dip. This angle is shown inFig. 4 during the fault. The values of the angle before and afterthe fault are ill-defined because the negative-sequence voltageis very small. The half-cycle window used to extract thesymmetrical component voltages makes that it takes about onehalf-cycle to obtain a value for the angle. Initially (shortly afterfault initiation) the angle is close to 240°: the “ideal value” fora type Cc dip. But gradually the angle decreases. In this case,the maximum deviation is about 25 degrees, just within theerror margin. A slightly larger deviation would have resultedin the estimation of a type Da dip.

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10180

190

200

210

220

230

240

Time [Cycles]

Angl

e [D

egre

es]

Fig. 4. Angle between drop in positive-sequence voltage and negative-sequence voltage for the dip shown in Fig. 3.

The resulting characteristics are shown in Fig. 5 and Fig. 6.Fig. 5 shows the magnitude of characteristic voltage and PN-factor. Initially the PN-factor is close to unity, but decreases

2001TR027 by Math Bollen 6

during the event. Note that the PN-factor is continuous both atfault initiation and at fault clearing. This behavior has beenobserved for all single-phase and phase-to-phase faults. Thedecay and recovery of the PN-factor can be explained from thedecay and recovery of the voltage source behind reactance inthe induction machine model. This also explains that the ratioof characteristic voltage and PN-factor is constant during thefault [9].

Fig. 6 shows the phase angle of the characteristic voltageand the PN-factor. The large transients in the angle at faultinitiation and fault clearing are artifacts due to the Fouriertransform method used to extract the phase-angle information.The figure shows that the angle of the PN-factor is zero at faultinitiation and decreases to about -12° during the 5-cycleduration of the fault. Note that a 12° rotation in PN-factor (Fig.6) gives a 25° rotation in the angle used to detect the dip type(Fig. 4).

0 2 4 6 8 10 12 14 16

0.5

0.6

0.7

0.8

0.9

1

Time [Cycles]

Mag

nitu

de [p

u]

Fig. 5. Absolute value of characteristic magnitude and PN-factor for the dipshown in Fig. 3.

0 2 4 6 8 10 12 14 16-25

-20

-15

-10

-5

0

5

10

Time [Cycles]

Angl

e [D

egre

es]

Fig. 6. Phase angle of characteristic voltage (solid line) and PN-factor (dottedline) for the dip shown in Fig. 3.

As the symmetrical component algorithm uses complexphase voltages as its input, the system frequency must beknown accurately. The discrete-Fourier transform algorithmfor extracting the complex phasors must be synchronized tothe pre-fault voltage. Assuming exactly 50 Hz (60 Hz)

frequency will result in additional errors in the angle used fordetecting the type of dip. An error of 0.1 Hz in frequency willgive in 10 cycles an error of 7.2° in phase angle for the phasevoltages, leading to an error up to 14.4° in the angle used fordetecting the dip type.

VII. THE PERFECT ALGORITHM

To understand why there is no perfect algorithm todetermine the dip characteristics, consider the following set ofsynthetic events:Dip Type = Ca

°−

°−

+= 60

60

1 j

j

xexeV (8a)

( ) ( )VjeVF −°−+= 140 3.07.0 (8b)

with 0<x<∞. The characteristic voltage is given for animpedance angle of -60° (see Appendix A). For the PN-factor,both magnitude and phase angle are assumed to dependlinearly on the magnitude of the characteristic voltage.

This represents dips with a large (characteristic) phase-anglejump, a large drop in PN-factor, and a large phase shift in PN-factor. The symmetrical component algorithm according to (5)has been applied to these events. The resulting values areshown in Fig. 7 and Fig. 8. For x<0.6 the algorithm results inan incorrect dip type (Db in this case). The incorrect dip typein turn results in incorrect values for characteristic voltage andPN-factor.

The estimated magnitudes of characteristic voltage and PN-factor are shown in Fig. 7; the phase angles in Fig. 8. In bothcases the magnitude of the actual characteristic voltage isgiven along the horizontal axis. For low values of thecharacteristic magnitude the algorithm results in significanterrors in both magnitude and phase angle, where themagnitude of the PN-factor and the phase angle of thecharacteristic voltage reach unrealistic values. The presence ofthese unrealistic values can be used as an indication that thealgorithm has given an incorrect result.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Actual Characteristic Magnitude

Estim

ated

Mag

nitu

de

Fig. 7. Estimated magnitude of characteristic voltage (solid line) and PN-factor (dashed line). The actual values are indicated through dotted lines.

2001TR027 by Math Bollen 7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Actual Characteristic Magnitude

Estim

ated

Pha

se A

ngle

Fig. 8. Estimated phase-angle of characteristic voltage (solid line) and PN-factor (dashed line). The actual values are indicated through dotted lines.

The “incorrect values” are mathematically speaking notincorrect. There are simply two combinations of dip type, Fand V, that result in the same phase voltages. For example thevoltage dip, according to (8), with characteristic magnitude50% has the following phase voltages:

°+∠=°−∠=

°−∠=

10753.016268.02085.0

c

b

a

VVV

These phase voltages were generated from a type Ca dipwith the following characteristics:

°−∠=°−∠=

2085.03450.0

FV

But exactly the same phase voltages are obtained for a typeDb dip with characteristics:

°−∠=°−∠=

1072.04268.0

FV

The only way of distinguishing between them is by realisingthat one of the solutions gives non-realistic values of thecharacteristics. Being able to make such a decision requiresknowledge of dip characteristics as they may occur in practice.This calls for a further analysis of available monitoring dataand for detailed simulations of unbalanced faults in realisticsystems. The effect of the load on the complex voltages playsan important role in the dip characteristics so that specialemphasis has to be placed on the load model.

VIII. CONCLUSIONS

It is shown that both algorithms are capable of correctlyobtaining the dip characteristics for most three-phaseunbalanced dips. However the six-phase algorithm results inincorrect dip type and characteristics for shallow and moderateevents with a large phase-angle jump. Voltage dips due tofaults on distribution cables could fall into this category. Morestatistical information on dip characteristics is needed to assessthe severity of this limitation.

The symmetrical component algorithm covers allcombinations of magnitude and phase angle jump, but shows

problems for large rotations in the PN-factor. It is howeverunclear how often such large rotations occur in practice. Buteven when they occur, the initial value of the PN-factor (i.e.shortly after dip initiation) is always close to unity without anyphase shift. Using the information that the dip type cannotsuddenly change without a change in the phase voltages, canbe used to accurately track the changes in the dipcharacteristics. More knowledge of the load effects on voltagedips is needed before this problem can be addressed further.

For applying the symmetrical component algorithm it isessential that accurate values of complex positive-sequenceand negative-sequence voltage are obtained. They can beobtained by applying the Fourier transform on the phasevoltage waveforms, or through a forward and a backward dq-transform. With both methods is it important that the algorithmaccurately tracks the system frequency. A small error infrequency may result in an erroneous dip type.

In this paper, only single-phase and phase-to-phase faultshave been considered. Two-phase-to-ground faults alreadygive a non-unity PN-factor immediately after fault-initiation. Alarge initial phase angle shift in the PN-factor may lead toerrors in the classification in that case.

For extracting additional information on the underlyingevent of a voltage disturbance it is essential that the algorithmgives a correct result, and the symmetrical componentalgorithm should be used. To extract statistics on the supplyperformance erroneous results in a limited number of casesmay be acceptable. The simplicity of the six-phase algorithm isclearly an advantage for statistical studies when large numbersof events need to be processed.

IX. APPENDIX AFor faults on a given feeder the complex characteristic

voltage is found from the standard voltage divider equation:

zZzV

S += (A1)

where the pre-event voltage is assumed to be 1pu, with Zs

the source impedance at the pcc, the distance between thepcc and the fault, and z the feeder impedance per unit length.Expression (A1) can be rearranged into:

α

α

λλ

j

j

eeV

+=

1(A2)

with SZz=λ a measure for the “electrical distance” to the

fault, and ��

���

=SZz�argα the angle between source impedance

and feeder impedance. It easily follows from (A2) thatα=

→)arg(lim

0V

V, so that α is the maximum phase-angle jump

for faults on the given feeder. The angle between source andfeeder impedance (α) is referred to as the “impedance angle”.It determines the relation between magnitude and phase-anglejump of a voltage dip. Expression (A2) is suitable for the

2001TR027 by Math Bollen 8

generation of synthetic voltage dips with realistic magnitudeand phase-angle jump combinations.

The feeder impedance is normally more resistive than thesource impedance, so that α is typically negative. Realisticvalues for the impedance angle are between -10° and +10° fortransmission system faults and between –60° and -10° fordistribution system faults. For faults on distribution cablesvalues between –40° and -60° have been found.

X. REFERENCES

[1] G. Desquilbet, C. Foucher, P. Fauquembergue, Statistical analysis ofvoltage dips, PQA-94 Amsterdam.

[2] J.C. Smith, J. Lamoree, P. Vinett, T. Duffy, M. Klein, The impact ofvoltage sags on industrial plants, Int. Conf. on Power Quality: End-useApplications and perspectives (PQA-91), pp.171-178.

[3] M.F. McGranaghan, D.R. Mueller, M.J. Samotyj, Voltage sags inindustrial power systems, IEEE Transactions on Industry Applications,Vol.29, no.2, pp.397-403, Mar/Apr 1993.

[4] L. Conrad, K. Little, C. Grigg, Predicting and preventing problemsassociated with remote fault-clearing voltage dips, IEEE Transactionson Industry Applications, Vol.27, no.1, Jan. 1991, p.167-172.

[5] D.O. Koval, M.B. Hughes, Canadian national power quality survey:frequency of industrial and commercial voltage sags, IEEE Transactionson Industry Applications, Vol.33, no.3, pp.622-627, May/June 1997.

[6] M.H.J. Bollen, Characterization of voltage sags experienced bythree-phase adjustable-speed drives, IEEE Transactions on PowerDelivery, Vol.12, no.4, pp.1666-1671, October 1997.

[7] M.H.J. Bollen, Understanding power quality problems – voltage sagsand interruptions. New York, IEEE Press, 1999.

[8] L.D. Zhang, M.H.J. Bollen, Characteristic of voltage dips (sags) inpower systems, IEEE Transactions on Power Delivery, Vol. 15, no.2(April 2000), pp.827-832.

[9] L.D. Zhang, Three-phase unbalance of voltage dips, Licentiate thesis,Chalmers University of Technology, Gothenburg, Sweden, November1999.

[10] M.H.J. Bollen, E. Styvaktakis, Characterization of three-phaseunbalanced dips (as easy as one two three?), IEEE Power SummerMeeting 16-20 July, 2000, Seattle, WA, USA, pp.899-904, Vol.2.

[11] IEC 61000-4-30, Power quality measurement methods, CDV 2001.[12] E. Styvaktakis, M.H.J. Bollen, Y.H. Gu, Expert system for classification

and analysis of power system events, IEEE Transactions on PowerDelivery, in print.

[13] G. Yalcinkaya, M.H.J. Bollen, P.A. Crossley, Characterisation ofvoltage sags in industrial distribution systems, IEEE Transactions onIndustry Applications, Vol.34, no.4, July 1998, p.682-688.

XI. BIOGRAPHY

Math Bollen (M’94, SM’96) is professor inelectric power systems in the department ofelectric power engineering at Chalmers Universityof Technology, Gothenburg, Sweden. He receivedhis MSc and BSc from Eindhoven University ofTechnology in 1985 and 1989, respectively.Before joining Chalmers in 1996 he was post-docat Eindhoven University of Technology andlecturer at UMIST, Manchester, UK. Math Bollenleads a team of researchers on power quality,reliability and power-electronic applications to

power systems. His own contribution to research consists of the developmentof methods for voltage dip analysis, which resulted in a text book on powerquality. Math is active in IEEE and CIGRE working groups on voltage dipanalysis and statistics