Uncertainty in Transmission Line Parameters: Estimation and Impact ...

1496 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 6, JUNE 2014

Uncertainty in Transmission Line Parameters:Estimation and Impact on Line Current

Differential ProtectionGangavarapu Sivanagaraju, Saikat Chakrabarti, Senior Member, IEEE,

and S. C. Srivastava, Senior Member, IEEE

Abstract— This paper presents a method to estimate the trans-mission line parameters using synchrophasor measurements fromphasor measurement units (PMUs), as well as hybrid measure-ments from PMU and supervisory control and data acquisitionsystem. It also proposes a method to estimate transmission lineparameter uncertainties considering the measurement inaccura-cies. Transmission line protection is important for secure andreliable operation of the power system. With the advent of thesynchrophasor technology and advances in the communicationsystem, current differential relays can be implemented as a reli-able protection system for the transmission lines. In this paper, anew current differential protection scheme has been suggested fortransmission line protection using synchronized measurements.The effect of parameter uncertainty on the protection scheme isalso investigated.

Index Terms— Current differential protection, parameteruncertainties, phasor measurement unit (PMU), transmission linemodeling.

I. INTRODUCTION

ACCURATE determination of transmission line parame-ters is important for all types of system simulation stud-

ies, such as load flow, transient stability, and state estimation.These are also important to improve the accuracy in relaysettings and post-fault location. A method for estimation oftransmission line parameters, using measurements of voltage,current, and power or power factor is discussed in [1]. Estima-tion of positive and zero sequence parameters, using wide areameasurements, is reported in [2]. Reference [3] discussed fourmethods to estimate short transmission line parameters usingsynchronized phasor measurements. All the methods usingsynchronous measurements require large number of phasormeasurement units (PMUs) to get synchrophasor data fromeach and every bus, which may not be economically viable.Due to their relatively high cost, the utilities are providingPMUs at a few selected locations only. It is a well-known

Manuscript received May 5, 2013; revised September 11, 2013; acceptedOctober 15, 2013. Date of publication December 11, 2013; date of currentversion May 8, 2014. This work was supported by the Department ofScience and Technology, India, under Project DST/EE/20100258 and ProjectDST/EE/20100256. The Associate Editor coordinating the review process wasDr. Carlo Muscas.

G. Sivanagaraju is with the Ashok Leyland Technical Centre, Chennai600103, India (e-mail: [email protected]).

S. Chakrabarti and S. C. Srivastava are with the Department of ElectricalEngineering, Indian Institute of Technology, Kanpur 208016, India (e-mail:[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2013.2292276

fact that the PMU measurements have higher resolution andaccuracy compared with the conventional supervisory controland data acquisition (SCADA) measurements. Motivation forPMU placement into the existing conventional measurementsystem is discussed in [4]. The PMU measurements canbe used along with the existing SCADA to improve theaccuracy of various applications. If one can estimate theline parameters using the combination of PMU and SCADAmeasurements, the accuracy will be better as compared withthat with only SCADA measurements. In this paper, a methodis proposed to estimate the transmission line parameters usingonly synchronous (PMU) measurements as well as usinghybrid (combination of PMU and SCADA) measurements.

It is not only sufficient to know the exact value of the lineparameters, but also desirable to know the bounds of theparameter values, within which it will vary because of themeasurement inaccuracies. The range of the line parametervariation is needed to set the bounds of the restrain region forthe current differential relay and to know the uncertainty in thepower system state estimation. A methodology to estimate theuncertainties in the estimated states of the power system dueto measurement inaccuracies is reported in [5]. Reference [6]proposed a methodology to evaluate the uncertainties asso-ciated with the states measured or computed by the PMU.In this paper, a methodology is also proposed to evaluate theuncertainties associated with the estimated transmission lineparameters due to the inaccuracies in the measurements.

Transmission line protection is important to ensure sta-ble operation of the power system. Usually, the distancerelays are used to protect the transmission lines in a powersystem. However, sometimes the distance relays mal-operateunder dynamic conditions such as power swings and voltageinstability. Because of the rapid advancement in the com-munication technology, current differential relays are beingused for protection of the transmission lines [7], [8]. Currentdifferential protection is relatively simple, offers high speedand sensitivity, and is also immune to power swings andexternal faults. Fig. 1 shows the schematic of a line currentdifferential protection. Neglecting the line capacitances, in theabsence of fault, or fault outside the protected zone, the currentflowing into the protected short transmission line is equal tothe current flowing out of the line.

Under normal condition or external fault

I1 = I2. (1)

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SIVANAGARAJU et al.: UNCERTAINTY IN TRANSMISSION LINE PARAMETERS 1497

Fig. 1. Single line diagram of a short transmission line.

Fig. 2. Equivalent pi network of transmission line [5].

If a fault develops inside the protected zone, the currentsI1 and I2 are no longer equal. For the comparison of currents,remote end measurements should be transmitted to the relay.Hence, it requires a high-speed and reliable communicationnetwork. Modern high speed communication networks typi-cally use synchronized optical network or synchronized digitalhierarchies standard for communication, having transmissionrates in the order of 274.2 or 155.5 Mb/s, respectively [9].Reference [9] proposes an adaptive current differential pro-tection scheme with equivalent-π model of the transmissionline. Current differential protection for ultrahigh voltage trans-mission line, using distributed parameter model is discussedin [10]. In this paper, a simple method of current differentialprotection scheme using synchronized phasor measurements isproposed, which is based on the distributed parameter modelof the line.

This paper is organized as follows. Estimation of the trans-mission line parameters using PMU and hybrid measurementsis discussed in Section II. In Section III, a methodology toevaluate the uncertainties associated with the transmission lineparameters due to the inaccuracies in the measurements isdescribed. Section IV explains the new current differentialprotection algorithm using synchronized measurements. Thesimulation results are presented in Section V, and Section VIconcludes this paper.

II. TRANSMISSION LINE PARAMETER ESTIMATION

The equivalent pi network of a long line [11] is shown inFig. 2, which provides

Vs = (1 + ZY )V r + Z I r (2)

Is = (2Y + ZY2)V r + (1 + ZY ) I r (3)

where Vs , Vr are the sending and receiving end voltages andIs , Ir are the sending and receiving end currents, respectively.Z and Y are the series impedance and shunt admittances ofthe equivalent pi network of the long line.

After solving (2) and (3), one can obtain

Z = V 2s − V 2

r

Is Vr + Vs Ir(4)

Y = Is − Ir

Vs + Vr. (5)

Fig. 3. Transmission line with PMUs at the both ends.

Using the distributed parameter model of the long transmis-sion line, the expressions for the sending end voltage Vs andsending end current Is are given by

Vs = cosh(γ L) · Vr + Z0. sinh(γ L). Ir (6)

Is = sinh(γ L)

Z0· Vr + cosh(γ L) · Ir (7)

where Z0 = √z/y is the characteristic impedance, γ = √

z yis the propagation constant, L is the length of the transmissionline, and z and y are the distributed series impedance and shuntadmittance of the transmission line per unit length.

Comparing (2) and (3) with (6) and (7), one can obtain

γ = cosh−1(1) + ZY

LZ0 = Z

sinh(γ L). (8)

Using (8), the distributed impedance and admittance areobtained by the following:

z = Z0·γ (9)

y = γ

Z0. (10)

The line distributed parameters, i.e., resistance r , induc-tance l, and capacitance c of the long transmission line areobtained from (9) and (10).

A. Synchronous Measurements

Consider a transmission line with PMUs installed at both theends, as shown in Fig. 3. Using a PMU, the voltage phasorof the bus and current phasors on the incident lines can bemeasured [12]. The measurements coming from each PMU aretime stamped and synchronized to the universal time coordi-nated. The aim is to estimate the distributed positive sequenceparameters, i.e., resistance, inductance, and capacitance of theline per unit length. The measurements with the same timestamp can be used to estimate the transmission line parameters.The phasors measured are

Vs = Vs � δs = Vs(cosδs + jsinδs)

Vr = Vr � δr = Vr (cosδr + jsinδr )

Is = Is � θs = Is(cosθs + jsinθs)

Ir = Ir � θr = Ir (cosθr + jsinθr ).

Using these measurements, the line parameters can beobtained through the following steps.

1) Substituting Vs , Vr , Is , and Ir measured by PMUs withsame time stamp in (4) and (5) and separating the real


Fig. 4. Transmission line having PMU at one end and SCADA measurementat the other end.

and imaginary parts, the following four equations canbe obtained:

R = ac + bd

c2 + d2 (11)

X = bc − ad

c2 + d2 (12)

G = eg + f h

g2 + h2 (13)

B = f g − eh

g2 + h2 . (14)

The variables a, b, c, d , e, f , g, and h are defined inAppendix A.

2) Calculate γ and Z0 by substituting Z = R + jX andY = G+ jB obtained from previous step in (8).

3) Evaluate distributed parameters z = r+ jx and y =g+ jb using (9) and (10) and calculate the inductanceand capacitance using l = x /2π F and c = b/2π F ,respectively, where F = system frequency.

B. Asynchronous Measurements

Consider the transmission line shown in Fig. 4, where aPMU is installed at bus j . The voltage phasor Vr and currentphasor Ir are, therefore, known. The bus i is provided withSCADA measurement, which is able to measure the powerflow Ps , voltage magnitude Vs , and current magnitude Is atbus i . The phase angle (φS) between the voltage and currentat bus i can be calculated from these SCADA measurements.These measurements are refreshed every 4–5 s. In the caseof PMU measurements, refresh rate can be up to a fewkilohertzs, and each measurement from a PMU is aligned witha time stamp [12]. The conventional measurements used bythe SCADA system also carry local time stamps. Using thetwo time stamps, the synchronous PMU measurements can becombined with the asynchronous conventional measurements.In case there is no common instant corresponding to the mea-surements, one set of measurements can be interpolated [4].

The phasors at the buses i and j are not synchronized.A synchronization angle δ is used to synchronize the voltagephasors at both the terminals. Using the synchronizationangle δ, the phasors at the bus i with respect to globalpositioning system (GPS) reference can be defined as

Vs = Vs � δ = Vs(cosδ + jsinδ) (15)

Is = Is � δ − φs = Is(cos(δ − φs) + jsin(δ − φs)). (16)

The phasors obtained from PMU are

Vr = Vr � δr = Vr (cosδr + jsinδr ) (17)

Ir = Ir � θr = Ir (cosθr + jsinθr ). (18)

Substituting Vs , Vr , Is , and Ir in (5) and separating the realand imaginary parts gives the real part as

G = K M + L N

M2 + N2 (19)

where

K = Is cos(δ − φs) − Ir cos θr , L = Is sin(δ − φs) − Ir sin θr

M = Vs cos δ + Vr cos δr , N = Vs sin δ + Vr sin δr .

Since the power lost in the insulation resistance and coronais extremely small in comparison with other line losses, thevalue of the line conductance parameter is normally very low,and can be neglected, i.e., G = 0. Therefore

K M + L N = 0

(Iscos(δ − φs) − Ir cosθr )(Vscosδ + Vr cosδr )

+(Issin(δ − φs) − Ir sinθr )(Vssinδ + Vr sinδr ) = 0. (20)

Equation (20) is a nonlinear equation with one unknown δ,and it can be easily solved by using any iterative numericalmethod. In this paper, Newton Raphson method has been usedto solve the nonlinear equation. With this synchronizationangle δ, the phasors at bus i are now synchronized withthe phasors at bus j . Thus, the synchronized phasors areobtained at both the terminals of the transmission line. Thesteps described above using synchronous measurements canbe used to estimate the parameters of the transmission lines.

III. TRANSMISSION LINE PARAMETER

BOUNDS ESTIMATION

In this section, estimation of the transmission line para-meter uncertainty bounds by considering uncertainties in themeasurements is discussed. Among the various methods ofevaluation of measurement uncertainty, the one proposed inGuide to the expression of uncertainty in measurement [13] isthe most widely used [14], and adopted in this paper.

A. Synchronous Measurements

The upper and lower bounds, within which the line para-meters lie, are estimated by considering the uncertainty inthe PMU measurements. The uncertainties in the PMU mea-surements are mainly due to the instrument transformers,the A/D converters, the cables connecting them, and thecomputational logic [15]–[17]. The uncertainties due to theinstrument transformers and the cables can be compensatedby calibration of the PMU. The uncertainties due to theA/D converter and the associated computational algorithm aredifficult to compensate, which results in uncertainties in themagnitudes and angles of the voltage phasors and currentphasors computed by the PMU. To evaluate the uncertaintyintervals in the transmission line parameters, the maximummeasurement uncertainties in the voltage magnitude, currentmagnitudes, and phase angle measurements by PMU must be


known. The range of intervals, within which the measuredquantity lies, is usually specified by PMU manufacturers [18].

A nonlinear constrained multivariable optimization problemis formulated, to evaluate the bounds of the transmission lineparameters. The formulation of the optimization problem is asfollows.

To obtain the lower bound of the parameters

minx

f j (x) subject to x − �x ≤ x ≤ x + �x. (21)

To obtain the upper bound of the parameters

maxx

f j (x) subject to x − �x ≤ x ≤ x + �x (22)

where f(x) = [RXG B]T, x = [Vs Vr Is Ir δs δr θs θr ]T,and �x is the maximum uncertainty in the correspondingmeasurement specified by the manufacturer.

The steps to find the bounds on the distributed parametersof the long line are as follows.

1) The minimum and maximum values of equivalent piparameters R, X, G, B of the long line are computedby substituting (11)–(14) in (21) and (22), respectively.

2) Using the maximum values of R, X , G, and B , theupper bounds of the distributed parameters r , l, and care calculated using (8)–(10).

3) Similarly, the lower bounds of the r , l, and c arecalculated using minimum values of R, X , G, and Bin (8)–(10).

A numerical verification based on Monte Carlo simulationis presented in Appendix C to justify the assumption used insteps 2 and 3 above that the maximum and minimum values ofr , l, and c correspond to the maximum and minimum valuesof R, X , G, and B .

B. Asynchronous Measurements

Consider a transmission line provided with PMU at oneend, and with SCADA system on the other end. From SCADAsystem, one gets voltage magnitude Vs , current magnitude Is ,and active power and reactive power flows Ps , Qs , as shownin Fig. 4. The power factor angle φs is given by

φs = sin−1(

Qs

Vs .Is

). (23)

To calculate φs , reactive power Qs is used, because onecan distinguish lagging and leading power factor by the signof Qs . The power factor is lagging if the sign of Qs is −veand it is leading if Qs is +ve.

It is assumed that the multiple measurements by a PMUare independent of each other. The standard uncertainty inthe power factor angle φs and the synchronization angle δis calculated using classical uncertainty propagation theory,and is given by [13]

u(φs) =√√√√ 3∑

k=1

[∂φs

/∂p(k)]2[u(p(k))]2 (24)

u(δ) =√√√√ 7∑

k=1

[∂δ/

∂q(k)]2[u(q(k))]2 (25)

where p = [Vs , Is , Qs ], q =[Vs , Is , φs , Vr , Ir , δr , θr ]and u(p(k)), u(q(k)) are the standard uncertainties in themeasurements p(k), q(k), respectively. The partial derivativeof φs with respect to Vs , Is , and Qs are given in Appendix B.The synchronization angle δ is calculated by solving (20). Thepartial derivative of δ with respect to Vs , Is , φs , Vr , Ir , δr ,and θr are also given in Appendix B.

To evaluate the standard uncertainty in φs , the standarduncertainties in the conventional measurements Vs , Is , and Ps

measured by meters must be known. Usually, the maximummeasurement uncertainty is specified by the meter manufactur-ers. In the absence of any probability distribution of the mea-surement uncertainty specified by the manufacturer, a uniformdistribution may be assumed. The standard uncertainty in themeasurement can then be expressed in terms of the maximummeasurement uncertainty, as given below [13]

u(q(k)) = �q(k)√3

(26)

where �q(k) is the maximum specified uncertainty by themanufacturer in the measurement of q(k).

Similarly, to evaluate the standard uncertainty in δ in (25),the standard uncertainties in Vs , Is , Vr , Ir , δr , θr , and φs mustbe known. The standard uncertainty in φs is already calculatedusing (24) and the standard uncertainties in Vs , Is , Vr , Ir , δr ,and θr can be calculated using the maximum measurementuncertainty specified by the meter manufacturers.

1) Finding the Expressions for R, X, B: Substituting(15)–(18) in (4) and (5) and separating the real part and theimaginary part, the following equations are obtained (shuntconductance is neglected, i.e., G = 0):

R = a1c1 + b1d1

c21 + d2

1

(27)

X = b1c1 − a1d1

c21 + d2

1

(28)

B = f1g1 − e1h1

g21 + h2

1

. (29)

The variables a1, b1, c1, d1, e1, f1, g1, and h1 are definedin Appendix B.

To find the bounds for the transmission line distributedparameters, similar steps can be followed as described usingsynchronous measurements in Section III-A. Only differenceis that, the variable set x = [Vs Vr Is Ir δδrφs θr ]T is used in(21) and (22) and for R, X , and B , (27)–(29) are used (G isneglected).

IV. CURRENT DIFFERENTIAL PROTECTION

A. Principles

In principle, current differential protection scheme using apositive sequence network representation can alone detect allpossible faults [9], as it is excited by both the ground and thephase faults.

Consider an uncompensated three-phase transposed trans-mission line of length L km, as shown in Fig. 5. Assume thattwo PMUs are installed at buses S and R.


Fig. 5. Three-phase transposed transmission line.

Fig. 6. Trip and restrain region in current differential plane.

If there is no internal fault on the line, for an arbitrarypoint k, which is x km away from bus R on the transmissionline

Iskp = Irkp (30)

where p indicates the phase current a, b, and c, Iskp is thecurrent calculated at point k from the sending end data set(Vsp, Isp) using (31) and Irkp is the current calculated atpoint k from the receiving end data set (Vrp , Irp) using (32)

Iskp = −Vsp· sinh(γ )(L − x)

Z0+ Isp · cosh(γ (L − x)) (31)

Irkp = Vrp· sinh(γ x)

Z0+ Irp · cosh(γ x). (32)

From (30), one can define

ratio = R p =∣∣ Iskp

∣∣∣∣ Irkp∣∣ = 1 (33)

angle difference = λp = � Iskp − � Irkp = 0. (34)

If (33) and (34) are satisfied, there is no internal faultor the fault may be outside the transmission line. This canbe visualized in the current differential plane by point � at(00, 1), as shown in Fig. 6. If a fault occurs between busesS and R, any one of the phase currents will not satisfy (33)or (34) and the differential relay will send trip signal to theassociated circuit breaker and isolate the line under fault.In practice, a threshold would be considered for the ratio andangle difference to allow for the tolerance in the measurementand the parameters [9]. In the proposed method, for calculationof the ratio and angle difference in (33) and (34), the currentsat the fault location from the both ends are used. It involvescalculation of the exact fault location.

B. Calculation of Fault Location Index [19]

Consider a three-phase transposed transmission line, asshown in Fig. 5. Assume that a fault occurred at a point k

in the line, which is located x = DL km away from bus R,where D is the per unit fault location index. The voltage atthe fault point k can be calculated using the positive sequencedata sets (Vs , Is) and (Vr , Ir ), as follows:

Vsk = 0.5(Vr + Z0 Ir )eγ x + 0.5(Vr − Z0 Ir )e

−γ x . (35)

Vrk = 0.5(Vs + Z0 Is)e−γ (L−x) + 0.5(Vs − Z0 Is)e

γ (L−x).

(36)

Solving (35) and (36), the per-unit fault location index can beobtained as following [19]:

D = ln(N / M)

2γ L(37)

where M and N are given by

M = 0 · 5e−γ L(Vs + Z0 Is) − 0 · 5(Vr + Z0 Ir )

N = 0 · 5(Vr − Z0 Ir ) − 0.5eγ L(Vs − Z0 Is).

The absolute value of D gives the fault location index D.If there is a fault external to the transmission line (externalfault), then the fault location index D assumes a value outsidethe range (0–1 p.u.) [20], [21]. To find the fault location index,positive sequence symmetrical components of the voltages andcurrents, having the same time stamp, have been used.

C. Proposed Algorithm

The following steps have been used to detect the fault onthe transmission line.

1) Choose the threshold parameters Rpmin, Rp

max, λpmin and

λpmax to detect differential current. This paper has used

±20 phase error and ±20% magnitude error in thecurrent differential plane, as shown in Fig. 6.

2) Compute the fault location index D, with synchronizedpositive sequence measurements coming from PMUs atboth the ends, using (37). If 0 ≤ D ≤ 1, take x = DLotherwise, take x = 0.5L.

3) Compute Iskp and Irkp at the fault location usingsynchronized voltage and current phasors coming fromPMUs (Vsp, Isp) and (Vrp , Irp) and using (31) and (32),respectively.

4) Check if

Rpmin ≤

∣∣ Iskp∣∣∣∣ Irkp∣∣ ≤ Rp

max (38)

and

λpmin ≤ � Iskp − � Irkp ≤ λp

max. (39)

If the above is true, it indicates no fault on the phase-pof the transmission line. Otherwise, there is a fault onthe phase– p of the transmission line and the relay issuestrip signal to the associated circuit breakers, where thephase current violates one of the conditions, given by(38) or (39). The trip decision can be taken if the ratioor the angle difference exceeds the threshold valuesconsecutively for four samples.


Fig. 7. Single line diagram of a simple two-area system [22].

V. SIMULATION RESULTS

The case studies presented in this section are organizedin three parts: 1) the transmission line parameter evaluationusing only PMU and as well as hybrid (combination of PMUand conventional) measurements and comparison of the resultsobtained by both the methods are discussed; 2) the upperand lower bounds of the transmission line parameters arecomputed by considering uncertainty in the measurements;and 3) the proposed current differential protection, with andwithout considering transmission line parameter uncertainties,is investigated.

For reporting the results, a simple two-area system hasbeen taken [22]. The system consists of two similar areasconnected by two weak tie-lines of 220 km each (Fig. 7).Each area consists of two coupled units, each having a ratingof 900 MVA. The transmission system base voltage is 230 kV.The detailed load, line and generator data on a 100-MVA baseare given in [22]. To obtain the results, this two-area systemwas simulated in MATLAB-Simulink. It is assumed that thesamples, coming from MATLAB-Simulink, correspond to thesame instant of time.

A. Estimation of Transmission Line Parameters

The proposed scheme for the estimation of transmission lineparameters using synchronous measurements has been appliedto one of the tie-lines, L1 between buses 7 and 8 in Fig. 7.Table I shows the estimated transmission line parameter valuesusing PMU measurements only and the percentage error in theestimated parameters from the actual values, computed as

% error = estimated value − actual value

actual value× 100. (40)

The method proposed for the estimation of the trans-mission line parameters using asynchronous measurementsor hybrid measurements has been applied to the same lineL1 between buses 7 and 8, and the results are shown inTable II.

It is observed that the parameters estimated using boththe methods are quite close to the actual values. It is alsoobserved that the parameters estimated using only synchronous(PMU) measurements have more accuracy as compared withthe parameters estimated using the hybrid (PMU plus SCADA)measurements.

TABLE I

ESTIMATED VALUES OF TRANSMISSION LINE PARAMETERS

USING PMU MEASUREMENTS ONLY

TABLE II

ESTIMATED VALUES OF TRANSMISSION LINE PARAMETERS USING

HYBRID (PMU AND CONVENTIONAL) MEASUREMENTS

TABLE III

MAXIMUM MEASUREMENT UNCERTAINTIES IN THE MEASUREMENTS

TABLE IV

ESTIMATED TRANSMISSION LINE PARAMETER BOUNDS BECAUSE OF

UNCERTAINTIES IN PMU MEASUREMENTS ONLY

B. Estimation of Transmission Line Parameter Bounds

The proposed methods for estimating the transmission lineparameter bounds have been applied on the same transmissionline L1 between buses 7 and 8. Using the maximum mea-surement uncertainty specified by the manufacturer, the lineparameter bounds are calculated using (21) and (22). Themaximum uncertainties in the measurements assumed inthis paper are shown in Table III [18]. Table IV showsthe estimated lower and upper bounds of the line para-meters with the consideration of uncertainties in the PMUmeasurements.

Using the values of the maximum measurement uncer-tainties, the standard uncertainties in the measurements arecomputed using (26). The standard uncertainties in the powerfactor angle φs and the synchronization angle δ are calculatedusing (24) and (25), respectively. Table V shows the estimatedlower and upper bounds of the line parameters, when conven-tional SCADA measurements are present along with the PMUmeasurements.


TABLE V

ESTIMATED TRANSMISSION LINE PARAMETER BOUNDS

BECAUSE OF UNCERTAINTIES IN

HYBRID (PMU AND CONVENTIONAL) MEASUREMENTS

Fig. 8. Performance of the proposed current differential protection schemefor internal faults on the line.

C. Current Differential Protection

The performance of current differential protection is limitedby Current transformer saturation, sampling asynchronization,and communication channel delay. In this paper, CTs areassumed to be identical and having no saturation. Using ahigh-speed (<1 ms) and reliable communication channel, thedata exchange function can be easily maintained. The problemof sampling synchronization can be overcome if a timingsignal is available from an external source. The GPS providestime synchronization with an accuracy of 1 μs. The proposedcurrent differential protection scheme has been applied tothe line L1 between buses 7 and 8 and simulated for alltype of faults (LG, LLG, LL, and LLLG) with and withoutconsideration of parameter uncertainties. The value of the ratioof the current magnitudes of different phases | Iskp|/| Irkp | andthe angle differences � Iskp − � Irkp are calculated for all typesof faults at different locations on the line and plotted. The+ point indicates the relay operating point in the currentdifferential plane in Figs. 8–11.

1) With Actual Values of Line Parameters: From Fig. 8, forinternal faults on the line, all the operating points fall outsideof the restrain region, so the relay treats it as a fault and sendstrip signal to the associated circuit breakers.

For no fault and external faults, all the operating points fallinside the restrain region, as shown in Fig. 9. As all the pointsare inside the restrain region, the relay will not pick up.

The sensitivity of the current differential protection schemecan be evaluated by its ability to detect a high impedanceinternal fault. The proposed scheme has worked satisfacto-rily up to a fault resistance of 1300 for all types offaults.

Fig. 9. Performance of the proposed scheme for external faults.

Fig. 10. Performance of the proposed current differential protection schemefor internal faults under transmission line parameter.

Fig. 11. Performance of the proposed current differential protection schemefor external faults under transmission line parameter uncertainties.

2) With Consideration of Line Parameter Uncertainties:Fig. 10 shows the performance of the proposed current differ-ential protection scheme for internal faults on the line underconsideration, with transmission line parameter uncertainties.As all points fall outside the restrain region, the relay willpick up and send trip signal to the associated circuit breakers.Similarly, the values of the ratio of the current magnitudesof different phases and the angle differences are calculatedfor no fault or external faults. These points are located inthe current differential plane, as in Fig. 11. The simulationresults show that the proposed scheme is working satisfactorilyunder consideration of line parameter uncertainties up to afault resistance of 1100 for all types of faults.

VI. CONCLUSION

The synchrophasor technology is an emerging technology,which is being now deployed in several utility systems.


This paper has mainly focused on estimating the uncertaintyin the transmission line modeling using synchrophasor tech-nology, and the effect of the uncertainty on the line currentdifferential protection.

An analysis of the uncertainty in the estimated transmissionline parameters with PMU measurements has been presented.The uncertainty is modeled via deterministic upper and lowerbounds on measurement errors. A methodology to determinethe uncertainties in the estimated transmission line parame-ters, when conventional measurements are present along withPMU measurements, is also suggested. Furthermore, a newcurrent differential protection scheme for transmission lineprotection using the PMU measurements at both the endshas been proposed. This method is based on the distrib-uted parameter model of the transmission line. The schemeincludes the effect of the distributed capacitive current. Theproposed scheme has been tested with and without consideringthe line parameter uncertainties. The results show that theproposed scheme is sensitive, robust and can successfullydiscriminate between the internal and external faults. Thescheme does not lose its sensitivity for high resistance faults(up to 1100 ). The scheme is quite fast and accurate asthe phasors can be updated 50–60 times in each secondand the current phasors provided by the PMUs are usedfor comparison. The proposed scheme also works satisfac-torily under the consideration of transmission line parameteruncertainties.

APPENDIX A

EXPRESSIONS FOR THE VARIABLES USED IN SECTION II-A

The expressions for the variables a, b, c, d , e, f , g, and hare as follows:

a = V 2s cos 2δs − V 2

r cos 2δr

b = V 2s sin 2δs − V 2

r sin 2δr

c = Is Vr cos(θs + δr ) + Vs Ir cos(δs + θr )

d = Is Vr sin(θs + δr ) + Vs Ir sin(δs + θr )

e = Is cos θs − Ir cos θr

f = Is sin θs − Ir sin θr

g = Vs cos δs + Vr cos δr

h = Vs sin δs + Vr sin δr .

APPENDIX B

PARTIAL DERIVATIVES AND VARIABLES USED

IN SECTION III-B

The partial derivatives of ‘φs’ with respect to Vs , Is and Qs

are given below

∂φs

∂Vs= 1√

1 − w2· w

Vs

∂φs

∂ Is= 1√

1 − w2· w

Is

∂φs

∂ Qs= −1√

1 − w2· w

Qs.

TABLE A.1

RESULT OF ONE THOUSAND MONTE CARLO SIMULATIONS

The partial derivatives of the synchronization angle δ withrespect to Vs , Is , φs , Vr , Ir , δr , and θr are given below

∂δ

∂Vs= Is · cos φs − Ir · cos(δ − θr )

Is · Vr · sin(δ − φs − δr ) − Ir · Vs · sin(δ − θr )∂δ

∂ Is= Vs · cos φs + Vr · cos(δ − φs − δr )


∂φs= Is · Vr · sin(δ − φs − δr ) − Is · Vs · sin φs


∂Vr= Is · cos(δ − φs − δr ) − Ir · cos(θr − δr )


∂ Ir= −Vs · cos(δ − θr ) − Vr · cos(θr − δr )


∂δr= Is · Vr · sin(δ − φs − δr ) − Ir · Vr · sin(θr − δr )


∂θr= Ir · Vr · sin(θr − δr ) − Ir · Vs · sin(δ − θr )

Is · Vr · sin(δ − φs − δr ) − Ir · Vs · sin(δ − θr ).

The expressions for the variables a1, b1, c1, d1, e1, f1, g1,and h1 are as follows:

a1 = V 2s cos 2δ − V 2

r cos 2δr

b1 = V 2s sin 2δ − V 2

r sin 2δr

c1 = Is Vr cos(δ − φs + δr ) + Vs Ir cos(δ + θr )

d1 = Is Vr sin(δ − φs + δr ) + Vs Ir sin(δ + θr )

e1 = Is cos(δ − φs) − Ir cos θr

f1 = Is sin(δ − φs) − Ir sin θr

g1 = Vs cos δ + Vr cos δr

h1 = Vs sin δ + Vr sin δr .

APPENDIX C

RELATIONSHIP AMONG R, X, G, B AND r, l, c

The steps 2 and 3 in Section III-A assume that the maximumand the minimum values of r , l, and c correspond to themaximum and minimum values, respectively, of R, X , G,and B . As it is difficult to analytically prove such assumption,a numerical verification using Monte Carlo simulation wasperformed. 1000 Monte Carlo trials were run by taking randomsamples of R, X , G, and B within their respective maximumand minimum values obtained by solving (21) and (22), andevaluating r , l, and c using (9) and (10). Table A.1 shows thecomputed maximum and minimum values of r , l, and c, andthe values obtained from Monte Carlo simulations, consideringuncertainties in PMU measurements only. The assumptionused in Section III-A, step 2 and 3 is, thus, validated by theMonte Carlo trials.


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Gangavarapu Sivanagaraju received the B.E.degree from Andhra University, Vishakapatnam,India, in 2009, and the M.Tech. degree from theIndian Institute of Technology Kanpur, Kanpur,India, in 2012.

He is currently a Deputy Manager with AshokLeyland, Chennai, India. His current research inter-ests include synchrophasor technology, power elec-tronics, and power system protection.

Saikat Chakrabarti (S’06–M’07–SM’11) receivedthe Ph.D. degree from the Memorial University ofNewfoundland, St. John’s, NF, Canada.

He is currently an Assistant Professor with theDepartment of Electrical Engineering, Indian Insti-tute of Technology, Kanpur, India. His currentresearch interests include power system dynamicsand stability, state estimation, power system mea-surements and instrumentation, and synchrophasorapplications to power systems.

S. C. Srivastava (SM’91) received the Ph.D. degreein electrical engineering from the Indian Institute ofTechnology Delhi, Delhi, India.

He is currently a Professor with the Departmentof Electrical Engineering, Indian Institute of Tech-nology Kanpur, Kanpur, India. His current researchinterests include synchrophasor technology, stabilityand controls, and technical issues in electricity mar-kets.

Dr. Srivastava is a fellow of INAE, India, IE, India,and IETE, India.

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