Transmission Line’s Parameter Estimation UsingState Estimation Algorithms

Andres Olarte and Hernando Diaz

Abstract—Some experiences on the use of state estimationmethods for identifying the parameters of transmission lines aredescribed. As a result, an algorithm to estimate the parametersof transmission lines and transformers is proposed. It uses aset of measurements for different operating conditions; thesemeasurements can be obtained from historical records. Usingmeasurements of voltage and active and reactive power flows onboth ends of a given line, an estimator is built that can computethe parameters of a line, independently of the condition of the restof the system. In a similar way, it can compute the parametersand, more importantly, the position of the tap of a transformer.The method has been used to identify the parameters of all linesand transformers for the IEEE 30-node system. The performanceof the estimation process, as proposed, is independent of thenumber of lines to be identified as long as there is enoughvariability in the line’s loading, so the size of the system is notrelevant, as long as there are enough measurements.

I. INTRODUCTION

State estimators are currently used on a routine basisfor secure and economic operation of power systems. Theycompute bus voltage magnitudes and phase angles based ona noisy and redundant set of measurements including voltagemagnitudes at some nodes and active and reactive power forsome transmission lines and transformers. (Weighted) leastsquares estimators were first proposed and used in the 1970’sand currently are the most widely applied [2], [4], [18]. Sincethat time, a great research effort has been directed towardsresolving the problems inherent to the estimation process suchas reducing the solution times, detection and correction ofsystematic errors and using information of the power systemto increase the reliability and speed of the estimation [1], [2],[6], [10], [11], [20], [21], [23].

During the first experiences in the implementation of stateestimators, it was recognized the importance of having ac-curate values of the parameters of the transmission system.Since the estimates are obtained from equations relating mea-surements to state variables, any parameter errors entering intothat model are bound to affect the state estimates obtained. Theestimation process assumes that all parameters are known anduses them, iteratively, to estimate a state vector.

In practice, theoretical models are used to calculate trans-mission line equivalent circuit parameters. Estimates ofimpedance values of power transformers are obtained duringfactory tests. However, parameter values for transmission linesare very rarely, if ever, measured. Thus, those parameters are

Andres Olarte (Email: faolarted@unal.edu.co) and Hernando Diaz (Email:hdiazmo@unal.edu.co) are with the Department of Electrical Engineering,National University of Colombia, Bogota

often in error [3], [7], [12], [18], [22]. Kusic [12] argues that,due to deviation from the ideal conditions assumed duringcalculations, the values encountered in utilities’ data baseshave errors that may be as high as 25% to 30%. Since thetransmission system parameters have a critical influence onmost economic and security-related decisions, it is evidentthe need for a means to accurately measure or estimate theparameters of transmission lines. In [13], it is shown that amore accurate line model may have dramatic effects on theeconomics and security of the system operation.

Technical literature shows several techniques proposed todeal with the problem of parameter estimation. Some authorsuse sensitivity analysis on the residues during the state esti-mation process to identify a group of lines whose parametersmight be in error [14], [15], [22]. However, those techniquescan be applied only if the set of lines whose parameters needto be determined is small; results deteriorate rapidly as thenumber of uncertain parameters increases. Some proposalsaugment the state vector with the parameters to be estimatedand do a simultaneous state and parameter estimation. An earlyattempt to refine the parameter estimates was that of Debs[8] who used several measurement samples, sequentially intime, to increase redundancy; this, combined with a sequential(Kalman-based) estimator, allowed the estimation of states andparameters simultaneously. Some techniques use tools fromrobust control theory to determine a region of confidencewhere the parameter values are expected to be, based onthe knowledge of the uncertainty of the model [16]. Thesemethods provide more information than the traditional estima-tion process. However, they require far more information andcomputations than traditional techniques which makes themless attractive for on-line implementation.

Although the literature on parameter estimation is not veryextensive, it can be said that a common characteristic is thatthey only attempt, either directly or sequentially, to estimatethe parameters of a few transmission lines [22]. Thus, a prac-tical method to estimate in a systematic way the parametersof all elements of the transmission system is not availablenowadays.

This paper is organized as follows: In section II, the stateestimation problem is formulated in a very general form asa minimum norm problem. Section III presents the problemof estimating state variables and parameters, simultaneously,within the framework of minimum norm estimation. On sec-tion IV a method is described that allows the estimation of allparameters of a line, based on the estimation of parametersand states on a fictitious power system. The application of the

©2008 IEEE.

method is then described on section VI; it is used to estimatethe parameters of all lines for the IEEE 30-node system.The parameter estimation was extended to the estimationof the parameters (impedance and tap position) of a powertransformer as exposed on section V. The paper ends with aconclusions section.

II. MINIMUM NORM STATE ESTIMATION

Almost all methods for calculating a state estimate canbe formulated as a problem of minimizing a norm in an n-dimensional vector space.

Given a measurement set z ∈ Rm, related to the state

variables x ∈ Rn by:

z = F (x) + ε (1)

where F is some nonlinear function and ε is a measurementerror term.

The state estimation (SE) problem can be defined as

x = arg{min

x‖z − F (x)‖p

}where ‖·‖p is any (fixed) norm in R

m.This formulation includes the most common estimation

problems; i. e., if the norm is the usual Euclidean norm,

‖x‖2 :=√∑

|xi|2 = xT x

then the minimum norm problem becomes the standard leastsquare (LS) problem. The weighted least square (WLS) prob-lem can also be stated this way:

‖x‖2w :=∑

wi |xi|2 = xT Wx

where W is a diagonal matrix whose Wii term gives themeasurement weight wi of the ith measurement.

The weighted absolute value (WLAV) estimator, anothermethod used in power systems, minimizes the objective func-tion:

‖x‖1w :=m∑

i=1

wi |zi − Fi(x)|

and it can also be expressed as a minimum norm problem,using the 1-norm.

The minimum norm formulation leaves aside a number ofquestions about the probability distribution of measurementerrors. However, noise and error characteristics are seldom, ifever, known. Therefore, this problem formulation can be ofgreat practical value.

A. Solution of the Minimum Norm State Estimation

The solution of the state estimation problem is obtained [4]through an iterative process as follows:

j ← 0xj ← x0 {Select an initial point}repeat

H ← [∂F∂x

]xj {Linearize around xj}

∆z ← z − F (xj) {Compute residuals}

∆xj+1 ← arg{

min∆x ‖∆z −H∆x‖p}

xj+1 ← xj + ∆xj+1 {Update estimate}j ← j + 1

until maxi

∣∣∣xji − xj−1

i

∣∣∣ < ε

x← xj

The algorithm above seeks a solution to the nonlinearSE problem through a sequence of linearization-linear SEsubproblems. This a well established and tried procedure [19].

B. State Estimation in Power Systems

The state estimation problem in a power system can bedefined as follows: Given a set of measurements that includes:

• Bus measurements: voltage magnitude V Mk and net active

and reactive power injections Pk , Qk for bus k.• Line and transformer measurements: active and reactive

power flows Pij , Pji, Qij and Qji for an elementconnected between nodes i and j.

And a state vector that includes bus voltage magnitude Vk andphase angle φk for node k

Find the components of the state vector that minimize thenorm of the error.

In this case, the mathematical model relating the measuredand calculated values is given by the line flow and powerinjection equations [5], [17].

1) Redundancy: An important issue when doing state esti-mation in power systems (and other applications as well) is thequestion of having enough measurements to guarantee that asolution can be obtained. A Redundancy Index, ρ is definedas

ρ =No. of measurements

No. of state variables to be estimated

A high value of ρ usually means a low chance of matrix ill-conditioning.

C. Inclusion of pseudo measurements

In addition to meter readings, other information is availableabout a power system. Several types of non-metered informa-tion are easily obtained about the system. Most noticeable isthe existence of zero injection nodes: i. e., switching nodeswhere the net power is always zero. Other pseudo measure-ments include load forecasts at some nodes. The informationcontained in the pseudo measurements can be incorporatedinto the model (1) by giving them a weight that depends onhow reliable they are. Zero injection conditions could be givena very high weight since they constitute an equality constraintand, hence, they are arbitrarily accurate.

Pseudo measurements increase the redundancy of the prob-lem. However, experience has shown that the inclusion of mea-surements with widely different weights may lead to numericaldifficulties. Therefore, it is more convenient to include zeroinjection pseudo measurements as equality constraints. The SEproblem then becomes a Constrained State Estimation (CSE)problem. The CSE problem may be formulated as

x = arg{

minx‖z − F (x)‖p : G(x) = 0

}

where G is a nonlinear function relating the state variables tothe nodal power at zero injection buses.

III. SIMULTANEOUS STATE AND PARAMETER ESTIMATION

Often the parameters of the transmission system are notknown exactly. This becomes an additional source of uncer-tainty besides the usual ones related to the measurement errors.When the measurements are redundant enough, it is possibleto include some of the parameters as pseudo state variablesand obtain an estimate of both states and parameters.

Let µ be a vector of uncertain parameters of a power system,and let the measurement model (1) be modified to include theeffects of states and parameters as:

z = F (x, µ) + ε (2)

then an estimate of states and parameters can be obtained as:

(x, µ) = arg{

min(x,µ)‖z − F (x, µ)‖

}The estimation process can be summarized in the followingalgorithm:

j ← 0xj ← x0 {Select an initial state}µj ← µ0 {Initial parameter estimates}repeat

H ←[

∂F∂x

∂F∂µ

](xj ,µj)

{Linearize around xj , µj}∆z ← z − F (xj , µj) {Compute residuals}[∆xj+1

∆µj+1

]← arg

{min∆x

∥∥∥∥∆z −H

[∆x∆µ

]∥∥∥∥p

}

xj+1 ← xj + ∆xj+1 {Update state}µj+1 ← µj + ∆µj+1 {Update parameters}j ← j + 1

until maxi

∣∣∣xji − xj−1

i

∣∣∣ < ε & maxi

∣∣∣µji − µj−1

i

∣∣∣ < ε

µ← µj

A. Application to a Power System

In power system applications, some kind of residual analysisis used to determine the presence of bad measurements orparameter errors [15], [22]. A joint parameter-state estimationcan then compute the parameters of a few elements [14]. How-ever, as the number of parameters to be estimated increases,the convergence of the process deteriorates.

Inclusion of equality constraints in the presence of parame-ter uncertainty may have a damaging effect on the performanceof the method. Errors in the impedance of one or several linesconnected to a zero injection node can lead to convergenceslowing or failure if an equality constraint estimation modelis used. This will be described in the results section.

IV. TRANSMISSION LINE PARAMETER ESTIMATION

The parameters of a single line can be estimated frommultiple power flow and voltage magnitude measurementsobtained from operational data. Consider a power line forwhich k measurement sets are available. Let us consider the(fictitious) power system consisting of k identical copies of

V S1 ∠0◦ V R

1 ∠δ1P S

1 , QS1 P R

1 , QR1

V S2 ∠0◦ V R

2 ∠δ2P S

2 , QS2 P R

2 , QR2

V Sk ∠0◦ V R

k ∠δk

P Sk , QS

k P Rk , QR

k

Fig. 1. Fictitious power system used for estimating line parameters

the same transmission line, connected between nodes Sk andRk, as shown in Fig. 1.

It is therefore assumed that all the lines have the sameparameters: series resistance R, and reactance X and shuntsusceptance Y and that all the S nodes have the same phaseangle (0◦). Each measurement set includes P S , QS , P R,QR, V S and V R data. Simultaneous state and parameterestimation for this fictitious system allows the identificationof the line’s parameters. In this process, the estimates ofstate variables are disregarded for all purposes, except tocheck the convergence of the process and for ensuring thephysical relevance of the solution (e.g., rejecting solutions withnonphysically meaningful angles or voltage magnitudes).

A. Resampling to improve estimates

The application of the line parameter estimation algorithmas described above provides accurate estimates as long asthere is enough variability in the measurement data, as willbe discussed in the results section. SCADA records used forthe estimation typically include the readings corresponding tonumerous load conditions.

In order to ensure that there is enough variation in the inputdata to the estimation process, a resampling scheme may beadopted in such a way that a number nS of random samples,each with a fixed size k is selected. The parameter estimationis performed for each of the samples and the average of the n S

parameter estimates is utilized as an estimate of the parametersbeing identified.

During the calculations, non-physically relevant values ofstate variables are rejected and the corresponding samples areeliminated from the process. Further refinements of the methodare possible: e. g., the number nS of samples may be chosenadaptively to stop computations when the estimates reach asteady state.

The resampling process can be described by the followingalgorithm:

Start with Measurement Set Z = {Zi, i = 1 . . . , N}for l = 1 to nS do

Select a random sample of size k from ZEstimate line parameters µl from sample

end forCompute µ = average{µl}

V. TRANSFORMER PARAMETER ESTIMATION

The estimation procedure devised for transmission lines canbe extended to power transformers. Transformer impedancesare usually measured to a high accuracy using a short-circuittest. However, for units with tap changing under load capabil-ity, the position of the tap may not be exactly known duringnormal operating conditions. Knowing the position of the tapaccurately is very important from the operating point of view.

A three-phase transformer with an off-nominal tap positioncan be represented approximately [5], [9], in per unit, as anideal transformer with complex transformation ratio α : 1, inseries with an impedance zt = 1/yt, as shown on Fig. 2. It isassumed that the tap is located on the R side.[

IS

IR

]=

[αyt −yt

− |α|2 yt α∗yt

] [V S

V R

](3)

From this equation, complex power flow expressions can beobtained as:

SS = V SIS∗ = α∗y∗t

∣∣V S∣∣2 − y∗

t V SV R∗ (4)

SR = V RIR∗ = − |α|2 y∗t V S∗V R + αy∗

t

∣∣V R∣∣2 (5)

Using a simultaneous state and parameter estimation for thismodel, using several measurement sets, the admitance y t aswell as the tap position α may be obtained. The procedure isanalogous to the one applied to transmission lines.

VI. RESULTS

The study started by using a conventional state-parameterestimation. Our state estimation program relies on the use ofzero injection nodes to guarantee a redundancy high enoughto ensure convergence. However, as the combined estimationwas attempted for estimating the parameters of any of the linesconnected to a zero injection constrained node, convergencewas never achieved. Even after increasing local and globalredundancy, the equality constrained state estimation neverconverged. When the equality constraint was eliminated, theestimation was possible as long as the redundancy was highenough. We conjecture that the uncertainty in the parametersassociated with an equality constraint makes it very difficultto satisfy the constraint while trying to adjust the parameter.Actually, this was the starting point for the development of anestimation algorithm for individual lines.

In order to test the performance of the algorithm designedto estimate the parameters of individual lines, the IEEE 30node test system was used. The parameters of all lines andtransformers of the system were estimated. Bus loads andgenerations were changed to simulate a load curve operation

V S

ISyt

V x

α : 1

IRV R

Fig. 2. Transformer with off-nominal tap α

of the system for one day. For each load condition simulated,active and reactive power flows for all lines were recordedas well as voltage magnitudes for all nodes. Transformer tapswere maintained in a fixed position throughout the study. Anadditive random noise was simulated by adding a randomvariable to the recorded values of power and voltage to accountfor measurement errors. Two different probability distributionswere employed to represent the measurement errors: The firstone was a gaussian (normally) distributed noise with 95% ofthe values within ±0.5% of the true variable being measured,designed to approximate the behavior of an accuracy class 0.5instrument. As an alternative, in order to analyze the smooth-ing effect of the resampling process, a uniform distribution oferrors was considered.

Application of the estimation process showed that theestimation errors were smaller when the measurement datacorresponded to heavily loaded lines than lines under lightload. The best results were obtained when there were datawith widely varying loading conditions. Depending on themeasurement data, the estimates could be off by as much as40%, although this occurred only rarely. Observation of thepatterns of estimation errors, depending on loading conditions,provided the motivation for the resampling scheme that wasfinally implemented.

A resampling scheme was executed for the WLS estimationof each element. For each line, nS = 1000 samples, each ofsize k = 24, were obtained at random from the simulatedconditions database. For each of the 1000 cases, parameterestimates were obtained and the results were averaged toprovide estimated values of all parameters of each line. Noticethat, since the estimation process for a given line only involvesa small estimation problem, the whole repeated sampling andestimation process can be performed in a few seconds on a3GHz computer. All the programming was done in Matlab.

As illustration, the results for the line connecting nodes 6and 7 will be presented. In the first place, gaussian distributederrors were considered. Figs. 3 and 4, show histograms ofthe distribution of the results of estimating X and R for line6—7. Also shown on those graphs are the true value of theparameter and the average value of the estimates obtained fromeach sample. Vertical scale shows the number of cases. Theestimated values of the parameters are very close to the actualparameters in all cases.

In order to characterize the statistical performance of theestimates, the results of 100 runs of the resampling processwere analyzed. The results show that the estimation processis very robust: All 100 cases gave estimates that were within±1.1% of the real, exact value. 79% of the cases yielded anestimate with error less than 0.5%. A histogram summarizingthe results of the process are presented in Fig. 5.

Another test of the method was performed, assuming thatthe measurement errors have a uniform distribution with zeromean and distributed between ±0.5% of the magnitude beingmeasured. Of course, this error variable has a larger variancethan the normally distributed one considered before. However,based on the central limit theorems of statistics, it is expected

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.260

50

100

150

X [pu]

Fre

quen

cy

Fig. 3. Histogram of line reactance estimates. Normally distributed errors.Vertical lines show the estimated (average) and true values

−0.01 0 0.01 0.02 0.03 0.04 0.05 0.060

50

100

150

R [pu]

Fre

quen

cy

Fig. 4. Histogram of line resistance estimates. Normally distributed errors.Vertical lines show the estimated (average) and true values

−1.5 −1 −0.5 0 0.5 1 1.50

2

4

6

8

10

12

14

X Error [%]

Fre

quen

cy

Fig. 5. Histogram of errors in X: 100 runs

that the average values of many samples of the same size,be normally distributed, regardless of the distribution of themeasurement errors. All the histograms show that this isindeed very approximately the case. See Fig. 6.

Table I summarizes the results for the estimation of param-eters of line 6—7. As shown there, the estimation errors arenegligible.

It would be most desirable to test the algorithm with reallife data and compare the estimated values with the real lineimpedance data. However, all available line parameters havebeen obtained from theoretical models and they have not beenexperimentally tested. In this case, that has not been possible.Nevertheless, direct testing can be done if the algorithm isextended to the estimation of the parameters of a transformer,as explained in section V. Simulated tests have been performedon the transformers of the IEEE 30-Node Test System.

A resampling process identical to the one described abovefor transmission lines was done for transformers, with com-parable results. As illustration we consider the estimationof parameters for the transformer connected between nodes4—12. The parameters were modified slightly to include aresistance term to test if it can be estimated. The results ofthe estimation of the tap position are shown in Fig. 7. Finally,the results of the estimation are summarized in table II. Onceagain, the results are very good.

Parameter estimation using the WLAV estimator were alsocalculated. However, the results were less robust than theWLS, and they cannot be recommended at this time.

VII. CONCLUSION

An algorithm was created that estimates the parameters ofindividual lines and transformers for which there are measure-ments at both sides of the element. The method uses standardoperating data collected during a period of time. The algorithmis based on simultaneous Weighted Least Square state andparameter estimation for a fictitious system, made up ofidentical copies of the line being studied. It uses measurementsof active and reactive powers and voltage magnitudes at bothends of the element at different instants of time. The sameprocedure allows the estimation of the (complex) tap positionof transformers.

Estimation results are better when the line is heavily loadedor when measurements at peak and light load conditions are

TABLE IPARAMETER ESTIMATION RESULTS FOR LINE 6—7

Distribution R X Y/2

Normal 0.0228 0.1806 0.0075Uniform 0.0230 0.1803 0.0075Exact value 0.0230 0.1804 0.0075

TABLE IITRANSFORMER PARAMETER ESTIMATION

Parameter R X α

Real 0.0200 0.2560 0.9320Estimated 0.0201 0.2562 0.9319

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.260

50

100

150

X [pu]

Fre

quen

cy

Fig. 6. Histogram of line reactance estimates. Uniformly distributed errors.Vertical lines show the estimated (average) and true values

0.929 0.93 0.931 0.932 0.933 0.934 0.935 0.9360

50

100

150

α

Fre

quen

cy

Fig. 7. Estimation of transformer tap position histogram. Normally distributederrors. Vertical lines show the estimated (average) and true values

combined. Using the algorithm on a single set of measure-ments, it was found that estimation errors for series reactancewere usually lower (typically by as much as 50%) than errorsin resistance. A resampling scheme was designed to ensurethat there is enough variation in the input data. The averageof all estimated values was used as the best parameter estimatewith excellent results.

Performance of the estimation method was tested on sim-ulated measurements including normally and uniformly dis-tributed errors, to account for typical instrument operation.The estimated values were virtually identical. The results ofthe process on the uniformly distributed error case were morewidely spread, as was to be expected.

It was found that the inclusion of equality constraints asso-ciated with elements with uncertain parameters in the standardstate estimation process was very risky. When combined stateand parameter estimation was attempted for any of the linesconnected to a zero injection constrained node, convergencewas never achieved for our problem.

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Andres Olarte received a B.S. in Electronics Engineering from the Univer-sidad Distrital, Bogota, Colombia in 2005. He was awarded an M.S. Degreein Industrial Automation from National University of Colombia in 2007. Heis now a doctoral student in Electrical Engineering at the same institution.

Mr. Olarte’s research interests are state estimation for power systems andmodelling of complex nonlinear systems.

Hernando Diaz received a B.S. in Electrical Engineering from the NationalUniversity of Colombia in 1978. He was awarded an M.S. Degree in ElectricPower Engineering and a Ph. D. in Electrical Engineering from RensselaerPolytechnic Institute, Troy, NY in 1984 and 1986, respectively.

He has been with the Department of Electrical Engineering at the NationalUniversity of Colombia, Bogota, where he is a Professor and Director of thegraduate program.

Prof. Diaz’s main research interest is in modeling nonlinear dynamical

systems with application to complex phenomena, including power systems.

He has recently been involved in the application of engineering modeling and

control to biological problems.