Online Optimal Transmission Line Parameter

Estimation For Relaying Applications

Yuan Liao1, Senior Member, IEEE Mladen Kezunovic2, Fellow, IEEE

actual values of these factors may differ from those employed

in the estimation process, considerable discrepancies between

Abstract Transmission line protective relaying algorithms

actual line parameters and estimated values could occur.

usually require transmission line parameters as inputs and thus

accuracy of line parameters plays a pivotal role in ensuring the An alternative approach for line parameter estimation is

reliable performance of relaying algorithms. Online estimation of discussed in [10], where the line parameters are derived based

line parameters is highly desirable and various methods have on impedances calculated at one terminal using acquired

been proposed in the past. These methods perform well when the voltage and current phasors assuming the other end of the line

measurements utilized are accurate; they may yield erroneous

is open or short-circuited. Such required measurements may

results when the measurements contain considerable errors.

be difficult to obtain. Moreover, inaccuracies in the

Based on non-linear optimal estimation theory, this paper puts

measurements may lead to appreciable errors in parameter

forward an optimal estimator for deriving the positive sequence

estimation.

line parameters, capable of detecting and identifying the bad

To obtain the most prevailing line parameters, online

measurement data, minimizing the impacts of the measurement

errors and thus significantly improving the estimation accuracy. estimation approaches would be highly desirable and

The solution is based on the distributed parameter line model especially beneficial to protection applications. This type of

and thus fully considers the effects of shunt capacitances of the techniques take advantage of online voltage and current

line. Case studies based on simulated data are presented for measurements [11-15]. In [11], two sets of synchronized

demonstrating the effectiveness of the new approach.

voltage and current phasors from the two terminals of the line

are utilized to obtain the ABCD parameters of the line. How

Index Terms Transmission line parameter estimation, Non-

distributed parameter per unit length can be obtained is not

linear estimation theory, Bad measurement detection and

covered. The estimation accuracy highly relies on

identification, Distributed parameter line model.

synchronization precision [12]. The authors of [13] suggest a

method for deriving the line characteristic impedance and

I. INTRODUCTION

propagation constant by making use of online voltage and

current phasors captured at sending-end and receiving-end of

T the line. Reference [14] solves for the line parameters based

ransmission line parameters including series resistance,

on Laplace transform technique by utilizing three sets of

series reactance and shunt susceptance are critical inputs

synchronized voltage and current phasors. Another approach

to diverse power system analysis programs. Precision of line

based on synchronized voltage and current phasors obtained

parameters is thus essential in ensuring the accuracy of the

from both ends of the line is introduced in a recent paper,

obtained analysis results. Especially in areas of power system

which emphasizes the benefits of online parameter estimation

protection, many distance relaying algorithms entail line

to distance relaying [15]. All these algorithms assume that

parameters for determining appropriate relay settings,

highly precise synchronization is available and may yield

calculating fault distance, and making a sound tripping

significant errors if this does not hold.

decision [1-3]. It has been established that the value of line

Generally speaking, the existing online estimation

resistance, inductance and capacitance is frequency dependent

algorithms perform well when the measurements utilized are

[4-6]. This paper is concerned only with estimating line accurate. In practice sometimes the measurements may

parameters at the fundamental frequency, which are usually contain errors due to various reasons like the current

required by relaying applications.

transformer saturation, data conversion errors or

Various algorithms for computing transmission line

communication device abnormalities [16-17]. Such

parameters have been presented in the past literature. Classical

measurement errors as well as possible synchronization errors

approaches as described in [7-9] utilize factors such as tower

could result in substantial inaccuracy in estimates.

and conductor geometric parameters, conductor type, assumed

Therefore, as pointed out in [18], it will be very desirable if

ambient conditions, etc. for estimating line parameters. Since

an online parameter estimation approach can be designed

capable of detecting and identifying measurement errors and

synchronization inaccuracy. In this way, the bad

1

Yuan Liao (abphjl@r.postjobfree.com) is with the Department of electrical and

measurements would be removed and only the sound

computer engineering, University of Kentucky, Lexington, KY 40506, USA.

2

Mladen Kezunovic (abphjl@r.postjobfree.com) is with the Department of measurements exploited to achieve a more precise estimate of

electrical and computer engineering, Texas A&M University, College Station,

the line parameters. Detection of synchronization inaccuracy

TX 77843-3128, USA.

can alert us to avoid synchronization assumption in parameter

2

estimation and meanwhile take timely actions to remedy the improve the estimation accuracy and increase the robustness

problems leading to synchronization inaccuracy. of the algorithm by minimizing the impacts of possible

This paper proposes an online algorithm for estimating the measurement errors. More accurate estimates of the line

positive sequence line parameters able to make the most of all parameters can lead to more accurate calculation of relay

the measurements available and minimize the impacts of settings and more reliable operation of relaying algorithms.

measurement and synchronization errors. The proposed Note that zero sequence components in the circuit during

solution is based on the fundamental frequency voltage and normal operations are usually negligible, thus the proposed

current phasors, which can be calculated from recorded algorithm mainly aims at estimating positive sequence line

waveforms or directly obtained from measuring devices such parameters instead of zero sequence parameters. Nonetheless,

as phasor measurement units. We employ a two terminal if significant zero sequence components arise due to various

transmission line model for illustrating the solution. reasons such as unbalanced load conditions, existence of faults

The equivalent circuit based on distributed parameter external to the studied line, etc., the proposed algorithm will

be equally applicable for estimating zero sequence parameters

line model is harnessed to automatically and fully consider

of the line.

shunt capacitances and distributed parameter effects of long

The proposed approach for estimating positive sequence

lines.

parameters is delineated in the following sections.

Section II illustrates the proposed method. The evaluation

studies are reported in Section III, followed by the conclusion.

B. Proposed optimal estimator

II. PROPOSED OPTIMAL ESTIMATOR FOR LINE PARAMETER

The following discussion assumes that synchronized voltage

ESTIMATION

and current measurements at P and Q during normal operation

are available. Figure 2 depicts the positive sequence network

Section A presents the overall description of the proposed of the system during normal operation.

method, with elaborations given in Sections B. Section C

explains the procedure for detecting and identifying possible

I qi

I pi

bad measurements. P Q

Z c sinh( l )

A. Overall description of the method

V qi

V pi

Q tanh( l / 2) tanh( l / 2)

P

Zc Zc

EG EH

Figure 1. Transmission line considered for analysis

Figure 2. Positive sequence network of the system during normal operation

Consider the line, assumed to be a transposed line, between

terminals P and Q as shown in Figure 1, where EG and E H In the figure, the following notations are adopted.

represent the Thevenin equivalent sources. i th phasor measurement of positive sequence

V pi, I pi

The proposed algorithm draws on the steady state voltage

voltage and current at P;

and current phasors at terminal P and Q that are measured at

Vqi, I qi i th phasor measurement of positive sequence voltage

different moments such as every hour during normal

operations. The aim is to estimate the positive sequence series and current at Q; i = 1,2 N, N being the total number of

resistance and reactance and shunt susceptance of the line per

measurement sets, with each set consisting of V pi, I pi, Vqi

unit length.

Based on one set of voltage and current phasors at P and Q, and I qi ;

two complex equations can be obtained based on the positive

Zc characteristic impedance of the line;

sequence equivalent circuit, which link the phasors with the

unknown line parameters. Separating the two complex propagation constant of the line;

equations into four real equations and solving these equations l length of the line in mile or km.

gives rise to the solution of unknown variables.

However, since the phasor estimates may contain errors,

Based on Figure 2, we can derive the following equations

estimation of line parameters based on a single set of

[3, 7]:

equations could be unreliable. Fortunately, a set of redundant

equations may be derived by utilizing voltage and current

V pi Z c sinh( l ) I pi + sinh( l ) tanh( l / 2)V pi Vqi e j = 0 (1)

measurements taken at different moments such as each hour

[14]. Then, the non-linear estimation theory may be utilized to

obtain an optimal estimate of the line parameters and each of

the phasors. Statistical approaches can also be adopted to

detect, identify and remove any possible bad data, and thus

3

I pi tanh( l / 2)V pi / Z c + I qi e j = ( x8 N + 2 + jx8 N +3 )( jx8 N + 4 ) (12)

(2)

j

tanh( l / 2)Vqi e / Zc = 0

Introduce S and F ( X ) as the measurement vector and

function vector, respectively, with their elements shown in the

Z c = z1 / y1 (3) Appendix. The measurement vector and function vector are

related by:

= z1 y1 (4)

Where,

S = F(X ) + (13)

the synchronization angle between measurements at

P and Q, representing any possible synchronization

Where, is determined according to meter characteristics.

error..

The optimal estimate of X is obtained by minimizing the

z1, y1 positive-sequence series impedance and shunt

cost function defined as:

admittance of the line per mile or km, respectively.

The phasors in (1-2) and synchronization angle are

J = [S F ( X )]T R 1[S F ( X )] (14)

considered as known measurements. Define

M = [V p1, I p1, V q1, I q1 V pN, I pN, VqN, I qN, ] (5)

The solution to (14) can be derived following the Newton-

will be assigned a value of zero since synchronized Raphson method [3]. After X is obtained, (6) can be applied

measurements are utilized. Modeling in the system to compute the estimated values of the measurement phasors.

equations could detect potential synchronization error due to For estimating positive-sequence line parameters, the

synchronizing device failures, as shown in Section III. M i, proposed method is also applicable to complex topology such

as parallel lines as long as the voltage and current

i = 1 (4 N + 1), designates the i th element of M .

measurements at two ends of the line are available, since there

We define the measurement functions for each measurement is no mutual coupling between positive-sequence circuits.

as

C. Detection and identification of bad measurements

jx 2 i

Yi ( X ) = x 2i 1e, i = 1 4 N (6)

To detect the presence of bad measurement data, the method

Y4 N +1 ( X ) = x8 N +1 (7)

based on chi-square test as illustrated in [7, 19] can be utilized.

In this method, the expected value of the cost function is

Where X denotes the unknown variable vector, defined as calculated first, which is equal to the number of degrees of

freedom designated as k . Then the estimated value of the cost

X = [ x1, x 2 x8 N, x8 N +1, x8 N +2, x8 N +3, x8 N + 4 ]T (8) 2

function C J is obtained. If C J k,, then the presence of

2

bad data is suspected with probability (1 - ). Value of k,

Where,

vector or matrix transpose operator;

T can be calculated for a specific k and based on chi-square

x1, x2 x8 N variables required to represent the distribution. If bad data exists, the measurement corresponding

to the largest standardized error will be identified as the bad

4 N complex measurements;

data. In our study, we choose to be 0.01, indicating a 99%

synchronization angle;

x8 N +1

confidence level on the detection. More details are referred to

x8 N +2, x8 N +3 and x8 N +4 positive-sequence transmission line pages 655-664 of [7].

series resistance, series reactance and shunt

susceptance per unit length, respectively. III. CASE STUDIES

By employing the defined variables, (1-2) can be written

as f 2i 1 ( X ) = 0 and f 2i ( X ) = 0, respectively as shown below.

This section presents the case studies demonstrating the

procedure and effectiveness of the proposed solution for

f 2i 1 ( X ) = x8i 7 e jx 8i 6 Z c sinh( l ) x8i 5e jx 8i 4 + detecting and identifying possible bad measurements and thus

(9) deriving a more accurate estimate of the line parameters.

sinh( l ) tanh( l / 2) x8i 7e jx 8i 6 x8i 3e jx 8i 2 e jx 8 N +1 = 0 A steady state analysis program has been developed to

generate steady state phasors during normal operations. A 500

f 2i ( X ) = x8i 5 e jx8i 4 tanh( l / 2) x8i 7 e jx8i 6 / Z c + kV, 200 mile transmission line system with configuration as

(10) shown in Figure 1 is modeled, with line parameters and source

x8i 1e jx8i e jx8 N +1 tanh( l / 2) x8i 3e jx8i 2 e jx8 N +1 / Z c = 0 impedances being referred to [3]. The distributed parameter

line model is used to calculate the equivalent series impedance

and shunt admittance of the line, which is used in the steady

Where,

state analysis program. Varying the angle difference between

i = 1,2 N, representing the index of the measurement set;

the voltage sources EG and E H will change the power flow

Z c = ( x8N + 2 + jx8N +3 ) /( jx8N + 4 ) (11)

transferred from terminal P to Q, and accordingly different

4

sets of voltage and current phasors at P and Q can be 2

less than 9,0.01, it is concluded based on Section II-C that no

produced. These phasors are then distorted with Gaussian

bad data exists.

noises with specified variances and then utilized as inputs to

It can be seen that quite accurate estimates have been

the developed algorithm for estimating the line parameters.

achieved by the proposed method.

The algorithm has been implemented in Matlab.

Representative results are reported in this section. The per unit

system is utilized in the following discussions, with a base B. Cases with bad synchronization

voltage of 500 KV and base voltampere of 1000 MVA. All the Although synchronization based on global positioning

cases utilize 3 sets of measurements ( N = 3 ) and the system is normally highly precise, synchronization errors still

following starting values: one for phasor magnitude, zero for may arise due to various reasons such as improper hardware

phasor angle, zero for synchronization angle, 1E-3 for line wiring, unavailability of the GPS time reference and

resistance per unit length, 1E-3 for line reactance per unit communication problems. This subsection illustrates how such

length, and 1E-3 for line shunt susceptance per unit length. errors may be pinpointed by the proposed method.

The estimator achieves the optimal estimate of line parameters We first generate the measurement phasors. Next a

quickly, around five iterations for all the cases. For the chi- synchronization error is applied to the measurements at

square test, we choose to be 0.01. terminal Q to emulate the synchronization error. Then the

estimator is applied to obtain the results and compute the

value of the cost function C J . Figure 3 depicts the calculated

A. Cases without bad measurements

value of the cost function versus the synchronization error.

One curve is obtained by using a value of 1E-6 for

This subsection studies the behavior of the algorithm when

synchronization variance, and the other curve using 1E-4.

there are no bad measurements.

Variances for other measurements are set to 1E-4. It can be

seen that the cost function becomes considerably larger when

Table I. Optimal estimates of line parameters and phasors without bad

the synchronization error reaches 6 degrees, which can be

measurements

Quantity Measured values Optimal estimates utilized to detect the presence of bad measurement data. As

0.82189 + j0.53537 0.82057 + j0.53457

V (p.u.) expected, Figure 3 also reveals that a smaller variance value of

p1

the synchronization angle makes the estimator more sensitive

0.51781 + j0.7094 0.52166 + j0.71375

I p1 (p.u.)

to synchronization errors. A more detailed analysis is shown

0.97005 + j0.1433 0.96981 + j0.14534

Vq1 (p.u.) below.

-0.65313 - j0.38769 -0.64951 - j0.38421

I q1 (p.u.) 40

Syn. var.=1e-4

0.85496 + j0.48538 0.85912 + j0.48732

V p 2 (p.u.) Syn. var.=1e-6

35

0.49938 + j0.65946 0.49677 + j0.65723

I p 2 (p.u.)

30

Value of the cost function

0.99233 + j0.13169 0.98917 + j0.12903

Vq 2 (p.u.)

25

-0.60678 - j0.30728 -0.60846 - j0.30943

I q 2 (p.u.)

20

0.92162 + j0.41448 0.91792 + j0.41406

V p3 (p.u.)

15

0.43283 + j0.55017 0.4338 + j0.54963

I p3 (p.u.)

10

0.9951 + j0.10816 0.99895 + j0.1091

Vq3 (p.u.)

-0.53077 - j0.19337 -0.52993 - j0.19302 5

I q3 (p.u.)

(degrees) 0 0.027705 0

0 1 2 3 4 5 6 7 8 9 10

line resistance 0.00099667 0.00094316 Synchronization error (degrees)

(p.u./mile)

Figure 3. Cost function versus the synchronization errors

line reactance 0.0023566 0.0023762

(p.u./mile)

Table II lists the optimal estimates when the

line susceptance 0.0018349 0.0018315

(p.u./mile) synchronization has an error of 10 degrees using a value of

1E-6 for synchronization variance.

Assuming that all the measurements have the same error The estimated value of the cost function C J is computed as

variance value of 1E-4, the optimal estimates of the line 2

38.6139. Since C J exceeds 9,0.01, presence of bad data is

parameters and phasors are shown in Table I. The measured

values and the optimal estimates are shown in the 2nd and 3rd suspected, based on Section II-C.

column respectively. The line parameters in the 2nd column

indicate the actual line parameters.

2

We have k = 9, 9,0.01 = 21.666, and the estimated value

of the cost function C J is calculated as 2.3955. Since C J is

5

After removing from the measurement set, a new optimal

Table II. Optimal estimates of line parameters and phasors with bad

synchronization

estimate can be obtained as shown in Table III, which

indicates that a more accurate estimate of the parameters has

Quantity Measured values Optimal estimates

been reached. N/A means the corresponding measurement is

0.82189 + j0.53537 0.81899 + j0.54556

V p1 (p.u.) 2

not available. In this case, k is 8.0, 8,0.01 = 20.09, and the

0.51781 + j0.7094 0.53716 + j0.71177

I p1 (p.u.)

estimated value of the cost function C J is 2.372. Since C J is

0.9802 - j0.027324 0.98241 - j0.026031

Vq1 (p.u.) 2

less than 8,0.01, all the data are considered fairly accurate,

-0.71053 - j0.26839 -0.68662 - j0.26274

I q1 (p.u.) and the estimates are regarded as acceptable.

Therefore, the optimal estimator is able to successfully

0.85496 + j0.48538 0.85777 + j0.49139

V p 2 (p.u.)

detect and identify the possible synchronization errors, and

0.49938 + j0.65946 0.51108 + j0.6618

I p 2 (p.u.) obtain a more accurate estimate by employing only the reliable

measurements.

1.0001 - j0.042627 0.99751 - j0.040914

Vq 2 (p.u.)

-0.65092 - j0.19724 -0.63507 - j0.19519

I q 2 (p.u.)

C. Cases with bad voltage or current measurements

0.92162 + j0.41448 0.91694 + j0.40859

V p3 (p.u.)

Large errors in voltage or current measurements can lead to

0.43283 + j0.55017 0.4489 + j0.56397 considerable inaccuracy in parameter estimates. This

I p3 (p.u.)

subsection illustrates how such bad measurements can be

0.99877 - j0.066278 0.99817 - j0.052813

Vq3 (p.u.)

detected by the optimal estimator. A value of 1E-6 for

-0.55628 - j0.098266 -0.54378 - j0.093658 synchronization variance and 1E-4 for other measurements are

I q3 (p.u.)

utilized.

(degrees) 0 0.011675

line resistance 0.00099667 0.0014293

(p.u./mile)

Table IV. Optimal estimates of line parameters and phasors with bad

line reactance 0.0023566 0.0034997

voltage measurement

(p.u./mile)

line susceptance 0.0018349 0.0024303

Quantity Measured values Optimal estimates

(p.u./mile)

0.98627 + j0.64245 0.95422 + j0.62266

V p1 (p.u.)

Table III. Optimal estimates of line parameters and phasors after 0.51781 + j0.7094 0.52723 + j0.71686

I p1 (p.u.)

synchronization angle data is removed

0.97005 + j0.1433 0.99817 + j0.16352

Vq1 (p.u.)

Quantity Measured values Optimal estimates

0.82189 + j0.53537 0.82063 + j0.53433 -0.65313 - j0.38769 -0.65767 - j0.379

V p1 (p.u.) I q1 (p.u.)

0.51781 + j0.7094 0.52125 + j0.71374 0.85496 + j0.48538 0.87891 + j0.49956

I p1 (p.u.) V p 2 (p.u.)

0.9802 - j0.027324 0.98027 - j0.02521 0.49938 + j0.65946 0.4919 + j0.65531

Vq1 (p.u.) I p 2 (p.u.)

-0.71053 - j0.26839 -0.70684 - j0.26561 0.99233 + j0.13169 0.97071 + j0.11704

I q1 (p.u.) Vq 2 (p.u.)

0.85496 + j0.48538 0.85915 + j0.48725 -0.60678 - j0.30728 -0.60473 - j0.31174

V p 2 (p.u.) I q 2 (p.u.)

0.49938 + j0.65946 0.49642 + j0.65708 0.92162 + j0.41448 0.93511 + j0.42323

I p 2 (p.u.) V p3 (p.u.)

1.0001 - j0.042627 0.99652 - j0.044755 0.43283 + j0.55017 0.43057 + j0.54731

Vq 2 (p.u.) I p3 (p.u.)

-0.65092 - j0.19724 -0.65339 - j0.19914 0.9951 + j0.10816 0.98324 + j0.10014

I q 2 (p.u.) Vq3 (p.u.)

0.92162 + j0.41448 0.9179 + j0.41423 -0.53077 - j0.19337 -0.52722 - j0.1947

V p3 (p.u.) I q3 (p.u.)

0.43283 + j0.55017 0.43348 + j0.54927 (degrees) 0 -0.0024721

I p3 (p.u.)

line resistance 0.00099667 0.0014317

0.99877 - j0.066278 1.0028 - j0.066347

Vq3 (p.u.) (p.u./mile)

line reactance 0.0023566 0.0023126

-0.55628 - j0.098266 -0.55571 - j0.098164

I q3 (p.u.) (p.u./mile)

line susceptance 0.0018349 0.0017765

(degrees) N/A 10.2775

(p.u./mile)

line resistance 0.00099667 0.00093137

(p.u./mile)

Suppose that there is an error of 20% in the magnitude of

line reactance 0.0023566 0.0023485

(p.u./mile) V p1, then the optimal estimates will be obtained as shown in

line susceptance 0.0018349 0.0018164

Table IV.

(p.u./mile)

The estimated value of the cost function C J is calculated as

To identify the possible bad data, the standardized errors are 2

79.6378, which is larger than 9,0.01 = 21.666 . Therefore,

calculated and the largest standardized error is 6.0173, which

corresponds to . Hence, is identified as the bad data.

6

Table VI. Optimal estimates of line parameters and phasors with bad

presence of bad measurements is suspected and V p1 is

current measurement

successfully identified as the bad data.

After the bad measurement is removed, a new set of optimal Quantity Measured values Optimal estimates

0.82189 + j0.53537 0.82374 + j0.53603

estimates are calculated as shown in Table V. In this case, V p1 (p.u.)

2

k = 7, 7,0.01 = 18.475, and the estimated value of the cost 0.62137 + j0.85128 0.58756 + j0.81357

I p1 (p.u.)

2

7,0.01,

function C J is 2.2706. Since C J is less than all the 0.97005 + j0.1433 0.98272 + j0.1302

Vq1 (p.u.)

data are considered fairly accurate and the estimates are -0.65313 - j0.38769 -0.68713 - j0.41756

I q1 (p.u.)

regarded as satisfactory. Comparison between Tables IV and

0.85496 + j0.48538 0.86043 + j0.48057

V manifests that the line parameter estimation accuracy is V p 2 (p.u.)

considerably enhanced. 0.49938 + j0.65946 0.49637 + j0.66786

I p 2 (p.u.)

Table V. Optimal estimates of line parameter and phasors with bad voltage 0.99233 + j0.13169 0.98345 + j0.13429

Vq 2 (p.u.)

measurement being removed

-0.60678 - j0.30728 -0.61089 - j0.30494

I q 2 (p.u.)

Quantity Measured values Optimal estimates

0.92162 + j0.41448 0.91818 + j0.4081

V p3 (p.u.)

N/A 0.81473 + j0.53124

V p1 (p.u.)

0.43283 + j0.55017 0.43481 + j0.56013

I p3 (p.u.)

0.51781 + j0.7094 0.52155 + j0.71368

I p1 (p.u.)

0.9951 + j0.10816 0.99361 + j0.11324

Vq3 (p.u.)

0.97005 + j0.1433 0.96858 + j0.14463

Vq1 (p.u.)

-0.53077 - j0.19337 -0.53226 - j0.18952

I q3 (p.u.)

-0.65313 - j0.38769 -0.64918 - j0.38447

I q1 (p.u.)

(degrees) 0 -0.0049759

0.85496 + j0.48538 0.85825 + j0.48687

V p 2 (p.u.)

line resistance 0.00099667 0.00090433

0.49938 + j0.65946 0.49695 + j0.65726 (p.u./mile)

I p 2 (p.u.)

line reactance 0.0023566 0.002237

0.99233 + j0.13169 0.98999 + j0.12948

Vq 2 (p.u.) (p.u./mile)

line susceptance 0.0018349 0.0019697

-0.60678 - j0.30728 -0.60854 - j0.30929

I q 2 (p.u.) (p.u./mile)

0.92162 + j0.41448 0.91717 + j0.4137

V p3 (p.u.)

Table VII. Optimal estimates of line parameter and phasors with bad

current measurement being removed

0.43283 + j0.55017 0.43393 + j0.54972

I p3 (p.u.)

0.9951 + j0.10816 0.99962 + j0.10946

Vq3 (p.u.) Quantity Measured values Optimal estimates

0.82189 + j0.53537 0.82081 + j0.53486

V p1 (p.u.)

-0.53077 - j0.19337 -0.52998 - j0.19291

I q3 (p.u.)

N/A 0.52873 + j0.72348

I p1 (p.u.)

0 0.00038015

(degrees)

line resistance 0.00099667 0.00092429 0.97005 + j0.1433 0.97106 + j0.14371

Vq1 (p.u.)

(p.u./mile)

-0.65313 - j0.38769 -0.65343 - j0.38749

line reactance 0.0023566 0.0023836 I q1 (p.u.)

(p.u./mile)

0.85496 + j0.48538 0.85926 + j0.48665

V p 2 (p.u.)

line susceptance 0.0018349 0.0018355

(p.u./mile)

0.49938 + j0.65946 0.49679 + j0.65825

I p 2 (p.u.)

Similarly, bad current measurements may also be 0.99233 + j0.13169 0.98863 + j0.12958

Vq 2 (p.u.)

successfully detected and identified. Suppose that there is an

-0.60678 - j0.30728 -0.60866 - j0.309

I q 2 (p.u.)

error of 20% in the magnitude of I p1, then the estimates will

0.92162 + j0.41448 0.91796 + j0.41344

be derived as shown in Table VI. The estimated value of the V p3 (p.u.)

cost function C J is calculated as 81.312, which is greater than 0.43283 + j0.55017 0.43395 + j0.55067

I p3 (p.u.)

2

9,0.01 = 21.666 . Therefore, presence of bad measurements is 0.9951 + j0.10816 0.99844 + j0.10957

Vq3 (p.u.)

suspected and I p1 is identified as bad data. -0.53077 - j0.19337 -0.53012 - j0.19268

I q3 (p.u.)

After the bad current measurement is removed, a new set of

0 -0.00028832

(degrees)

optimal estimates are calculated as shown in Table VII. In this

line resistance 0.00099667 0.00094006

2

case, k = 7, 7,0.01 = 18.475, and the estimated value of the (p.u./mile)

line reactance 0.0023566 0.0023649

2

cost function C J is 1.3604. Since C J is less than 7,0.01, all (p.u./mile)

line susceptance 0.0018349 0.001846

the data are considered fairly accurate and the estimates are

(p.u./mile)

regarded as satisfactory. Comparing Tables VI and VII

evinces that the line parameter estimation accuracy is

significantly improved.

7

[17] J. R. Linders, C. W. Barnett, J. W. Chadwick, et al, Relay performance

IV. CONCLUSION

considerations with low-ratio CTs and high-fault currents, IEEE

This paper presents an algorithm for estimating the positive Transactions on Industry Applications, Vol. 31, No. 2, March-April

sequence parameters of a transmission line by utilizing online 1995, pp. 392-404.

[18] Yuan Liao, Algorithms for power system fault location and line

voltage and current phasors measured at different moments

parameter estimation, the 39th Southeastern Symposium on System

from two terminals of the line during normal operations. This Theory, Mercer University, Macon, Georgia, March 4-6, 2007.

paper demonstrates that it may be feasible to design an [19] A. Abur and A. G. Exposito, Power System State Estimation Theory

approach for detecting, identifying and removing possible bad and Implementation, Marcel Dekker, Inc., New York, USA, 2004.

measurements and thus improving the estimation accuracy.

When synchronized measurements are employed, possible

VI. APPENDIX

synchronization errors can also be detected, thus enhancing

the line parameter estimation accuracy. The developed Elements of vector S are:

algorithm is based on distributed parameter line model and

thus fully considers the effects of shunt capacitance and S i = 0, i = 1 4 N (A.1)

distributed parameter effects of long lines. Quite encouraging

S 2i + 4 N 1 = abs( M i ), i = 1,2 4 N (A.2)

results have been obtained by simulation studies.

S 2i + 4 N = angle( M i ), i = 1,2 4 N (A.3)

V. REFERENCES S12 N +1 = M 4 N +1 (A.4)

Where abs and angle yield the magnitude and angle of

[1] J. L. Blackburn, Protective Relaying Principles and Applications,

the input argument, respectively.

Marcel Dekker, Inc., New York, USA, 1998.

[2] S.H. Horowitz and A. G. Phadke, Power System Relaying, Research

Studies Press Ltd., Taunton, Somerset, England, 1995.

Elements of function vector F ( X ) are:

[3] Y. Liao, Fault location utilizing unsynchronized voltage measurements

during fault, Electric Power Components & Systems, vol. 34, no. 12,

December 2006, pp. 1283 1293. F2i 1 ( X ) = Re( f i ( X )), i = 1 2 N (A.5)

[4] H.W. Dommel, EMTP Theory Book, Vancouver, BC, Microtran Power

F2i ( X ) = Im( f i ( X )), i = 1 2 N (A.6)

System Analysis Corporation, May 1992.

[5] J. R. Marti, Accurate modeling of frequency-dependent transmission

F2i + 4 N 1 ( X ) = abs(Yi ( X )) = x2i 1, i = 1 4 N (A.7)

line in electromagnetic transient simulations, IEEE Transactions on

Power Apparatus and Systems, Vol. PAS-101, No. 1, January 1982, pp. F2i + 4 N ( X ) = angle(Yi ( X ) = x 2i, i = 1 4 N (A.8)

147 155.

F12 N +1 ( X ) = Y4 N +1 ( X ) = x8 N +1 (A.9)

[6] M. C. Tavares, J. Pissolato, and C. M. Portela, Mode domain

multiphase transmission line model Use in transient studies, IEEE

Transactions on Power Delivery, Vol. 14, No. 4, October 1999, pp.

1533 1544.

VII. BIOGRAPHY

[7] John Grainger and William Stevenson, Power System Analysis,

McGraw-Hill, Inc., New York, USA, 1994.

[8] S. M. Chan, Computing overhead line parameters, Computer Yuan Liao (S 98-M 00-SM 05) is an Assistant

Applications in Power, Vol. 6, No. 1, 1993, pp. 43 45. Professor with the Department of electrical and

[9] H. Dommel, Overhead line parameters from handbook formulas and computer engineering at the University of Kentucky,

computer programs, IEEE Transactions on PAS, Vol. PAS-104, No. 4, Lexington, KY, USA. He was a R&D Consulting

February 1985, pp. 366 370.

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