Seediscussions,stats,andauthorprofilesforthispublicationat:https://www.researchgate.net/publication/224703987

StatisticalAlgorithmforPowerTransmissionLinesDistanceProtection

ConferencePaper·July2006

DOI:10.1109/PMAPS.2006.360210·Source:IEEEXplore

CITATIONS

6

READS

22

3authors,including:

GöranAndersson

ETHZurich

347PUBLICATIONS7,578CITATIONS

SEEPROFILE

AntansSauhats

RigaTechnicalUniversity

69PUBLICATIONS97CITATIONS

SEEPROFILE

Allin-textreferencesunderlinedinbluearelinkedtopublicationsonResearchGate,

lettingyouaccessandreadthemimmediately.

Availablefrom:AntansSauhats

Retrievedon:10May2016

1

Statistical Algorithm for Power Transmission LinesDistance Protection

Marija Bockarjova,Student Member, IEEE, Antans Sauhats,Member, IEEE,and Goran Andersson,Fellow, IEEE

Abstract— This paper is focused on the development of the newprinciples and operational algorithms of the transmission linesdistance protection. The proposed distance protection algorithmsutilize statistical information about the undefined and randomparameters and quantities such as equivalent impedances of thesystems at the unmonitored end of the power transmission lineand current and voltage phasor measurements errors. Knowledgeabout the distribution of these parameters and values results inmore secure and dependable procedure for the decision on theline tripping or restraining in case of the appearance of the faultconditions in the network.

The proposed algorithm is based on the modeling of the faultedline and the Monte-Carlo method. The algorithm computesexpected value of the distance to the fault, fault resistance inthe fault point and the standard deviation of the distance to thefault. This leads to the inherent adaptive features of the approach.

Index Terms— Distance protection, Statistics, TransmissionLines, Adaptive protection

I. I NTRODUCTION

D ISTANCE protection is one of the most widely usedmethods to protect transmission lines [1]. The funda-

mental principle of distance relaying is based on the localmeasurements of voltages and currents where the relay re-sponds to the calculated impedance value between the relayterminal and the fault location in the transmission network.

Due to impact of the several unmonitored parameters thatare uncertain in the physical nature, such as errors of thecurrents and voltage measurements, fault resistance, loadingand topology of the system at fault inception instance, theestimate of the impedanceZ contains some error∆Z thatcan lead to the incorrect protection operation. The operationzones of relays help to prevent financial losses and/or technicaljeopardizing that might occur due to misoperation of the relays[5].

These zones are normally selected based on a worst casescenario [2] of the combination of factors that impact perfor-mance of the relay, seeing that the probabilistic problem defini-tion would introduce complexity. Thus, selecting the operationzones, two main requirements of the protection should becompromised: dependability and security, and simultaneouslyboth should be satisfied.

M. Bockarjova and G. Andersson are with Power Systems Laboratoryof Swiss Federal Institute of Technology (ETH) in Zurich, Physikstrasse 3,CH-8092 Zurich, Switzerland (e-mails: bockarjova@eeh.ee.ethz.ch,anders-son@eeh.ee.ethz.ch).

A. Sauhats is with Faculty of Power and Electrical Engineering, RigaTechnical University, 1 Kronvalda str., Riga, LV-1010, Latvia (e-mail:sauhatas@eef.rtu.lv).

The stochastic nature of the power system parameters orthe unforeseen changes in the topology of power system,the distance protection can operate undesirably. As a recentexample, undesired operation of protection contributed tothedevelopment of many blackouts [1].

The increase of the efficiency of the protection operationis possible utilizing the adaptive approach [2]. For example,the adaptive protection depending on the parameters of themonitored process can change the boundaries of the protectionzones. The implementation of the adaptive features of therelays became possible as a result of breakthrough of themicroprocessor technique. Further fast development of thistechnique insures the potential implementation of the moreand more sophisticated procedures for the decision making ontripping or restraining.

Particularly, it becomes realistic to exchange the determin-istic approach by the more advantageous probabilistic one thatrequires real-time implementation of the Monte-Carlo method.

This paper discusses the theoretical background of the latterapproach and shows its efficiency. As an example, the firstzone protection operation is investigated for the most frequentfault type: the single phase to ground, assuming that for thecontrolled line the phasors of the current and voltage areavailable as the result of analog-digital conversion and thesignal processing.

In previous publications [3], [4] the authors have proposedthe application of a statistical approach to fault location.This paper discusses its extension showing principally newpossibilities in the protections area.

II. T HE THEORETICAL BACKGROUND

Let us consider a symmetrical three-phase transmissionline connecting two power systems with known equivalentimpedances (Fig. 1). To determine the fault point let us applythe method based on power computation.

For known (measured) currents and voltages of the mon-itored end of the transmission line the complex values forpositive, negative and zero sequences of powerSi at the faultedpoint can be expressed:

Si = UiI∗i = (Ui1 − Ii1Zi1 − RF Ii)I

∗i (1)

where Si, Ui1, Ii1 is power, measured voltage and currentof sequencei respectively, currentI∗ is conjugate value ofcurrentI andRF is transient resistance of the fault.

9th International Conference on Probabilistic Methods Applied to Power SystemsKTH, Stockholm, Sweden – June 11-15, 2006

© Copyright KTH 2006

2

E1 E2

2siZ1iZ 2iZ1siZ

FRi1U1I

1iI2iI

Fig. 1. One-line diagram of the transmission system equivalent for the SLGfault conditions.

Taking into consideration that in the fault point, in generalcase behind the fault impedance, there is an equality ofsequences powersSi [5]:

2∑

i=0

Si = U1I∗1

+ U2I∗2

+ U0I∗0

= 0 (2)

On (1) and (2) and assuming that the fault impedance ispurely resistive and therefore consumed power is active, itcan be obtained:

2∑

i=0

Qi = Imag

{

2∑

i=0

Ui1I∗i1

K∗i

−

2∑

i=0

|Ii1|2Zi1

K∗i

− . . .

. . . −U11I

∗L

K∗1

+I11I

∗LZ11

K∗1

}

= 0 (3)

where IL is pre-fault current, while the current distributioncoefficientsKi are determined from the equation:

Ki =Zi2 + Zsi2

Zsi1 + Zi1 + Zi2 + Zsi2

(4)

K∗i is the conjugate value ofKi, Zsi1 and Zsi2 are the

Thevenin equivalents of power systems. The impedancesZi1

andZi2 depend on distance to the fault point:

Zi1 = LF Zisp, Zi2 = (L − LF )Zisp (5)

whereL is the length of the protected line,LF is distance tothe fault point andZisp is the i-sequence impedance per linelength unit.

Taking into account that in case of single phase to groundfault there is an equality of currents in the fault point:

I1 = I2 = I0, (6)

the equation (3) can be rewritten in more convenient for theimplementation form:

2∑

i=0

Qi = Imag

{

I∗01

K∗0

2∑

i=0

[Ui1 − Ii1Zi1]

}

= 0 (7)

Hence, (7) is a quadratic equation for unknown distanceLF that can be solved analytically for the assumed impedanceZsi2.

If the distanceLF is known, the resistance may be de-termined expressing phase voltageUF via currentIF of thefaulted phase as follows:

UF =

(

IF + I01

Z0sp − Z1sp

Z1sp

)

Z1spLF +3RF I01

K0

(8)

In reality if the remote end of the transmission line is unmon-itored, the equivalent impedanceZsi2 can be considered to bea random variable.

III. T HE STOCHASTIC APPROACH

Summarizing the stated above equations, one can declarethat the distance to the faultLF and RF are linked to themeasured phasors of the currentsI and voltagesU and un-known equivalent impedancesZsi2 of the remote transmissionline end system by relation of the following form:

LF = Φ(I, U, Zsi2) (9)

whereΦ is for some procedure of the distanceLF and RF

calculation. The procedure employs the measurement resultsof the monitored currents and voltages and information of theimpedanceZsi2 values.

On the other hand, taking into account that the measuredcurrent and voltage data contain random errors - correspond-ingly ∆I and∆U , and, in general, values of the unmonitoredimpedanceZsi2 can also be treated as random, the equation (9)could be considered as basic one to determine the distributionlaw of the estimate of the distances to the faultLFest andRFest or its numerical characteristics.

To determine distribution densityg(LFest) of theLFest onthe base of (9), it is necessary to know the distribution relativedensity functiong(I, U, Zsi2/Iest, Uest)- the density of thecurrent, voltage and impedance distribution under obtainedmeasurement resultsIest, Uest. Theoretically, Bayes’ theoremcould be involved [6]. Determination of the desired distributionfunction g(I, U, Zsi2/Iest, Uest) is possible on the base offaulted line processes simulation and Monte-Carlo methodutilization. For this purpose, significant number of trialsshouldbe performed, and consequently, notable processing time willbe needed.

However, more effective procedure could be obtained, usingthe linearization method, taking into account the physicalnature of the measured values and relatively small values ofmeasurement errors, and supposing that measurements errorsare additive with the zero value of mathematical expectation,it can be stated that [6]:

E[Φ(I, U, Zsi2)] ∼= E[Φ(Iest, Uest, Zsi2)]σ[Φ(I, U, Zsi2)] ∼= σ[Φ(Iest+∆I, Uest+∆U,Zsi2)]

(10)

whereE(...) is the mathematical expectation andσ(...) is thestandard deviation.

Statement (10) is strictly true for linear functions. In con-sidered non-linear case, it is deemed permissible for practicalapplications. On the other hand it becomes possible to employmore efficient procedure for Monte-Carlo method application.

IV. SIMULATIONS

The algorithm for estimation of the mathematical expec-tation E[LF ], E[RF ] and standard deviationσ[LF ], σ[RF ]values based on the Monte-Carlo simulations are shown inFig. 2.

First, the remote end equivalent impedance, for exampleZS02, is randomly selected. Using the measurements, the fault

9th International Conference on Probabilistic Methods Applied to Power SystemsKTH, Stockholm, Sweden – June 11-15, 2006

© Copyright KTH 2006

3

Start

Random choice of

the remote end

impedance

Computation of the

fault location and

fault resistance

Verifying

compliance with

physical nature

Calculation of the

expectation and

standard deviation

value of LF , RF

NOYES

Normal

distribution law

measurements: U, I

Next trial?

Fig. 2. Algorithm for the distance to fault and fault resistance estimation.

Zline

R

jX

Fig. 3. Illustration of the protection quadrilateral operation zone selection;Zline is the positive sequence impedance of the protected line.

distance is computed, and compliance with the physical natureof the task is verified; namely distance to the fault should beless than the line length, fault resistance should be positiveand ZS12, ZS22, values are within the feasible range that isdetermined from the following relation:

(I11 − IL)∗

K∗1

=I∗21

K∗2

=I∗01

K∗0

(11)

The incompliant results are ignored.Let us show that mathematical expectation of the distance

to the fault and fault resistance as well as standard deviationof these parameters are applicable for the conclusion regardsthe required protection action.

For comparison, next section contains results obtained bythe conventional algorithm based on the following equation:

ZF =Uf

If + I0

Z0sp − Z1sp

Z1sp

(12)

This algorithm is affected by the pre-fault current, fault con-ditions and the impact of the remote end infeed, particularlystrong in case of high resistance faults, and therefore, makes

0 20

R, ohm

40 60 80 100 120 140 160

jX,

ohm

0

10

20

30

40

50

60

70

80

90

Fig. 4. The proposed algorithm computed impedance to the fault, as ”seen”by the relay.

necessary certain restrictions in selection of the operationalzone for the conventional protection.

Commonly, the extent of the characteristic along theXaxis is chosen depending on the parameters of the line. Theimpedance loci projection on theR axis depends strongly onthe computation conditions, such as fault impedance and pre-fault power flow.

The coefficient 0.8 in Fig. 3 is a commonly used [10] safetymargin that is derived from practical experience, the valueofthe fault resistance and the coefficient 1.2 are another examplesof empirical assumptions that, generally, tend to limit instanta-neous protection operation to relatively low resistance faults,thus avoiding sympathetic trips. However, as shown below,protection operation could be improved with the more complexbut simultaneously more adjusted to the particular conditionsapproach.

V. RESULTS

A. real power system

The algorithm performance was investigated on power sys-tem of the New Zealand South Island. The data can be foundin [7], pages 84, 196-197 and the diagram on page 403. Fig. 4shows an example of the computation results. The transmissionline impedance is shown by the straight line. The impedanceto the fault, as would be seen by the relay was computed.The faults were simulated at 0, 20, 50, 80, 100% distance ofthe line length with the fault resistance of 0, 10, 50 ohms.Since the results are quite accurate, the corresponding groupsof points in the complex plane can be clearly identified withthe particular fault conditions. Due to uncertainty of the remoteend impedance at more distant and resistant fault, impedanceis varying within certain limits.

For the simulated fault conditions, the algorithm has notoperated incorrectly - none of the external fault was detectedas an internal one. However, more simulations were run forthe reduced power system model, as presented below.

9th International Conference on Probabilistic Methods Applied to Power SystemsKTH, Stockholm, Sweden – June 11-15, 2006

© Copyright KTH 2006

4

Vs

Vr

Vm

Internalfault

Externalfault

Zs Zr

Protectedline

Adjacentline

Zm

remote system

analysedprotection

Fig. 5. The studied 3-bus test system diagram.

B. 3-bus test system

For the full power system model the combination of the pa-rameters is limited and, in addition, simulations require morecomputational effort. In reality, the conditions of the protectionoperation vary significantly depending on the particular linewithin the power system, and also for each particular linethe conditions are changing within certain limits as the powersystem operation changes. Therefore, we show more results ofthe algorithm performance on the reduced system equivalentsand larger variation of operation conditions, simulated byMonte-Carlo method.

Let us define a fault that occurs on the protected line asinternal, and a fault on the adjacent line as external. Fig. 6- 13 represent the modelling results of the described abovealgorithms. The results are discussed in more details in thesubsequent subsections. The simulations were run for thefollowing test system in Fig. 5:

• The nominal voltage of the modelled system is 330 kV;• Operational voltage of the buses varies from 0.95 to

1.05 pu in magnitude;• The pre-fault power flow through the line varies from 0

to 1000 MW in both directions and determined by thebus voltage angle;

• Fault resistance varies from 0 to 50 ohm;• The line protected line length is 150 km;• The adjacent line length is 100 km;• Line apparent parameters are:

Z1,2 = 0.042 + j0.324 [ohm/km],

Z0 = 0.187 + j0.7838 [ohm/km];

• The system equivalent impedances at the monitored lineend, except for the particularly mentioned cases, vary inmagnitude and angle:

|Zs1,2|∈ [16, 80] ohm, 6 Zs1,2∈ [76, 85] deg,

|Zs0| ∈ [20, 90] ohm, 6 Zs0 ∈ [76, 84] deg;

• The impedances of the systemr and systemm varies inthe same range as the systems; Thus, transformed forthe two bus model as in Fig. 1, these impedances resultin equivalent impedance at the remote line end:

Zei =Zri(ZiL2 + Zmi)

Zri + ZiL2 + Zmi

,

|Ze1,2| ∈ [12.04, 46.9] ohm, 6 Ze1,2∈ [76.24, 84.91] deg,

|Ze0| ∈ [16.67, 59.03] ohm, 6 Ze0 ∈ [76.02, 83.63] deg;

The distance protection algorithms operation is frequentlysignificantly impacted by the relation of the sending (local)and receiving (remote) end impedance relations. Generally,distance protection algorithms produce better results forlineswith the strong (low impedance) local end, since the currentinfeed impact from the remote end is lower. In the follow-ing studies, we will distinguish these two cases, taking thefollowing variation limits:

• sending end is strong:

|Zs1,2| ∈ [16, 20] ohm, |Ze1,2| ∈ [30.1, 36.5] ohm,

|Zs0| ∈ [18, 25] ohm, |Ze0| ∈ [50.7, 38.5] ohm;

• sending end is weak:

|Zs1,2| ∈ [80, 130] ohm, |Ze1,2| ∈ [18.4, 27.6] ohm,

|Zs0| ∈ [90, 150] ohm, |Ze0| ∈ [23.6, 36.2] ohm;

The uniform distribution law was assumed for all theparameters variations.

Let us determine the maximal possible operational zone ofthe relay. The multiple computations of the standard deviationand the mathematical expectation of the fault resistance anddistance to the fault at the various short-circuit conditionson the protected line determine the desired area in theZ-plane. On the other hand, by modelling the external faults andload conditions, we can determine the ensemble of points thatcorrespond to the non-operation requirements.

The knowledge of these ensembles shall be the basis for theselection of the reach zone of the relay.

C. Operation under internal faults

Fig. 6 shows result of the computations for the faults thatare simulated on the protected line. For the comparison resultsof the conventional (squares) algorithm (12) are presented. Itcan be observed that the proposed algorithm (balls) is lessinfluenced by the parameter variations and provides accurateresults on the large line extent. This advantage, the ability toprovide less distorted picture, for particularly high impedancefaults on the large power line part can be used to extend thereach zone of the protection.

In addition to mathematical expectation of the impedanceto the fault, for each particular fault, the new algorithm willestimate standard deviation (small balls) of the impedanceandthus, determine if the fault ”belongs” to the operational zone.

D. Operation under external faults

The conditions for the protection misoperation1can be de-rived for the power system equivalent in Fig. 5. At thecertain parameters combination, namely systems equivalentimpedances, fault resistance, distance to the fault, powerinjection in intermediate and remote busses, line impedances,the currents and voltages measured by the protection willcoincide for the external and internal faults. That is a general

9th International Conference on Probabilistic Methods Applied to Power SystemsKTH, Stockholm, Sweden – June 11-15, 2006

© Copyright KTH 2006

5

0 10 20 30 40 50 60 70 800

20

40

60

80

100

120

R, ohm

jX, o

hm

Fig. 6. Comparison of the conventional and the proposed algorithmsoperation for a number of internal faults occurring on the whole line extent(fault resistance is 5, 10 and 30 ohms). Black line representstransmissionline impedance, green squares show the impedances seen by conventionalalgorithm, gray balls are the mathematical expectations± standard deviation(small balls) of the impedance determined by proposed algorithm.

limitation of the distance protection approach and the use ofthe local measurements only.

The number of parameter combinations is large and whichof those would result in misoperation cannot be determinedanalytically. However, the probability of the occurrence of theunfavorable conditions can be determined numerically for theparticular power line.

Fig. 7 - 10 show the estimated impedances by the con-ventional and proposed algorithm in case of the externalfaults. The conventional algorithm will always determine theimpedance, only the points that are close to the possibleoperational zone are shown. The proposed algorithm woulddetermine the impedance in limited number of cases thatwould satisfy its equations. In these relatively low probabilitycases, the determined impedance would correspond to remotehigh impedance faults.

System equivalent impedances ratio does not significantlyinfluence the mathematical expectation of the impedance deter-mined by the proposed algorithm. It may be noticed however,that it could be influenced by the extent of the variations ofthe equivalent impedances.

Fig. 11 - 12 show the histograms of the fault resistance andthe distance to the fault from the beginning of the adjacentline, exclusively for the faults when the proposed algorithm hasmisoperated. That indicates significantly higher probability ofmisoperation for the high resistance faults occurring close toadjacent line beginning. The analysis of the system equivalentimpedances, bus voltages etc relation to the misoperation wascarried out, but no clear dependency identified.

Commonly, more than two lines are connected to thesubstation. That means that for the analyzed protection (Fig. 5)

1In this context by misoperation we understand the case when the proposedalgorithm identifies an external fault as an internal one. The algorithm realperformance will be determined by the selected operational zone, whichshould avoid such impedances.

0 20 40 60 80 1000

10

20

30

40

50

R, Ohm

jX,O

hm

Fig. 7. Fault impedance seen by the conventional (green) and proposed(black) algorithms in cases of the external faults. Proposedalgorithm sees thefault as internal in 3.2% of cases (total number of trials 30 000).

0 20 40 60 80 1000

10

20

30

40

50

R, ohm

jX, o

hm

Fig. 8. Fault impedance seen by the conventional (green) and proposed(black) algorithms in cases of the external faults, when the sending end systemis weak and the maximal power transfer increased up to 1.5 GW; Probability ofmisoperation of the proposed algorithm 4.2% (total number of trials 10 000).

there could be several adjacent lines, thus several possibleequivalents depending on which one of the lines is faulted.However, this does not significantly affect the results repre-sented in Fig. 7 - 10, as the simulated variations of the remotesystems impedances were already relatively large.

E. Performance under no-fault conditions

The widely recognized restriction of the distance protectionis the possible misoperation under heavy transfers and lowervoltage conditions, when the impedance seen by the relaymay move into the operation zone. Thus, the operational zoneshould be reduced to avoid these occurrences. This drawbackcan be improved by introducing the zero sequence currentcheck that would block protection operation under symmetri-cal (no-ground-fault) conditions. However, under the unsym-metry of the measured currents and voltage or measurement

9th International Conference on Probabilistic Methods Applied to Power SystemsKTH, Stockholm, Sweden – June 11-15, 2006

© Copyright KTH 2006

6

0 20 40 60 80 1000

10

20

30

40

50

R, ohm

jX, o

hm

Fig. 9. Fault impedance seen by the conventional (green) and proposed(black) algorithms in cases of the external faults, when the sending end systemis weak and the maximal power transfer is 1 GW. Probability of misoperationof the proposed algorithm 2.8% (total number of trials 10 000).

0 20 40 60 80 1000

10

20

30

40

50

R, ohm

jX, o

hm

Fig. 10. Fault impedance seen by the conventional (green) andproposed(black) algorithms in cases of the external faults, when the sending end systemis strong and the maximal power transfer is 1 GW. Probability ofmisoperationof the proposed algorithm 2.07% (total number of trials 10 000).

errors the protection may be unblocked, thus misoperation maystill occur.

Fig. 13 shows the algorithms performance under unsym-metrical load conditions (or erroneous measurements). Thecurrents and voltages for each phase are simulated as normallydistributed with the standard deviation of 1 degree and 5% ofmagnitude for the random power transfers from 0 to 1000 MW.The figure does not show higher values of the determined bythe conventional algorithm impedances, focusing on the mostinteresting cases (or closest to the implicit reach zone). For theproposed algorithm, however, all the cases of misoperationare shown. Clearly, the values of the impedance seen bythe proposed algorithm correspond to more distant and highresistance faults, which shall allow the increase the reachzoneof relay.

0 10 20 30 40 500

0.067

0.13

0.2

0.27

0.33

0.4

Fault resistance, ohm

Pro

babi

lity,

%

Fig. 11. Probability function of the external fault resistances causingmisoperation of the proposed algorithm.

0 0.2 0.4 0.6 0.8 10

0.33

0.67

1

1.33

1.67

2

Distance to the fault, % of the line length

Pro

babi

lity,

%

Fig. 12. Probability function of the distances to the external faults from thebeginning of the adjacent line causing misoperation of the proposed algorithm.

0 20 40 60 80 1000

10

20

30

40

50

R, Ohm

jX, O

hm

Fig. 13. Impedance determined by the proposed (black) and the conventional(green) algorithm in no fault conditions. The new algorithmhas misoperatedin 0.033% of the cases (for 100 000 trials) and shows high impedance remotefaults.

9th International Conference on Probabilistic Methods Applied to Power SystemsKTH, Stockholm, Sweden – June 11-15, 2006

© Copyright KTH 2006

7

The presented above simulations were run assuming theuniform distribution laws of the system equivalent impedancesand no correlation between the values of the systems equiv-alent zero and negative impedance. More investigation canbe carried in the direction that could positively influence theresults as the search space would be decreased.

VI. CONCLUSIONS

• Distance protection operation is influenced by a numberof random parameters that may lead to incorrect or failedoperation and unnecessary delay in fault clearing.

• The application of a probabilistic approach that takesinto account parameter variations to distance protectionprovides new opportunities for decision of the relayaction. Simulations confirm reliability and high efficiencyof the proposed method that allows an increase of therelay operational zone for high resistance and remotefaults.

REFERENCES

[1] J. De La Ree, L. Yilu, L. Mili, A.G. Phadke, L. DaSilva, ”Catastrophicfailures in power systems: causes, analyses, and countermeasures.” Pro-ceedings of the IEEE, Vol.93, Iss.5, May 2005.

[2] S.H. Horowitz, A.G. Phadke, ”Boosting Immunity to Blackouts”, Powerand Energy Magazine, IEEE, Volume 1, Issue: 5, Sep - Oct. 2003.

[3] A. Sauhats, A. Jonins, M. Danilova, ”Statistical Adaptive Algorithms forFault Location on Power Transmission Lines based on Method ofMonte-Carlo”, in Proc. 7th Conference on Probabilistic Methods Applied toPower Systems, September 22-26, 2002, Naples, Italy.

[4] M. Bockarjova, A. Sauhats, G. Andersson ”Statistical Algorithms forFault Location on Power Transmission Lines”,in Proc. 2005 IEEE PowerTech Conf., June 27-30, 2005, St.Petersburg, Russia.

[5] G.I. Atabekov,Distant approach in long power transmissions protection,Akademija Nauk Armjanskoj SSR, 1953 (in Russian).

[6] G. Korn, T. Korn,Mathematical handbook, McGraw-Hill Book Company,1968.

[7] J. Arrillaga, C.P. Arnold and B.J. Harker,Computer Modeling of Electri-cal Power systems, John Wiley & Sons Ltd, 1983.

[8] T. Takagi, Y. Yamakoshi, M. Yamuaura, R. Kondow, T. Matsushima,”Development of a New Type of Fault Locator Using One TerminalVoltage and Current Data”,IEEE Trans., vol. PAS-101, No 8, pp.2892-2898, Aug. 1982.

[9] S. Tamronglak, S.H. Horowitz, A.G. Phadke, J.S. Thorp, ”Anatomy ofpower system blackouts: preventive relaying strategies”,IEEE Transac-tion on Power Delivery, Vol. 11. No. 2, April 1996.

[10] N. Zhang, M. Kezunovic, ”Implementing an Advanced Simulation Toolfor Comprehensive Fault Analysis”,2005 IEEE/PES Transmission andDistribution Conference & Exhibition, 15-18 Aug. 2005.

VII. B IOGRAPHIES

Marija Bockarjova graduated from the Riga Tech-nical University, Latvia in 2002. She continued stud-ies at the faculty of Electrical and Power Engineeringas Ph.D. student and in 2000-2005 was a planningengineer at the national power company Latvenergo.In 2005 she started the Ph.D studies at ETH Zurich.

Antans Sauhatsreceived Dipl.Eng., Cand.Techn.Sc.and Dr.hab.sc.eng. degree from the Riga TechnicalUniversity (former Riga Polytechnical Institute) in1970, 1976 and 1991 respectively. Since 1991 he isProfessor at Electric Power Systems. Since 1996 heis the Director of the Power Engineering Institute ofthe Riga Technical University.

Goran Andersson (M’86, SM’91. F’97) was bornin Malmo, Sweden. He obtained his M.S. and Ph.D.degree from the University of Lund in 1975 and1980, respectively. In 1980 he joined ASEA:s, nowABB, HVDC division in Ludvika, Sweden, and in1986 he was appointed full professor in electricpower systems at the Royal Institute of Technology(KTH), Stockholm, Sweden. Since 2000 he is fullprofessor in electric power systems at the Swiss Fed-eral Institute of Technology (ETH), Zurich, where heheads the powers systems laboratory. His research

interests are in power system analysis and control, in particular power systemsdynamics and issues involving HVDC and other power electronics basedequipment. He is a member of the Royal Swedish Academy of EngineeringSciences and Royal Swedish Academy of Sciences and a Fellow ofIEEE.

9th International Conference on Probabilistic Methods Applied to Power SystemsKTH, Stockholm, Sweden – June 11-15, 2006

© Copyright KTH 2006