[ieee ieee eurocon 2013 - zagreb, croatia (2013.07.1-2013.07.4)] eurocon 2013 - transmission line...

Transmission Line Impedance Estimation Based on PMU Measurements

Renata Matica #1, Vedran Kirincic *2, Srdjan Skok *3, Ante Marusic “4 # HEP Transmission System Operator

Kupska 4, 10000 Zagreb, Croatia 1 [email protected]

* Department of Power Systems, Faculty of engineering, University of RijekaVukovarska 58, 51000 Rijeka. Croatia

2 [email protected] 3 [email protected]

“Department of Energy and Power Systems, Faculty of electrical computing and engineering, University of ZagrebUnska 3, 10000 Zagreb, Croatia

4 [email protected]

Abstract— State estimation is one of the most important network management functions, responsible for determining the actual state of the power network based on the network model, available telemetry (analog measurements and digital device statuses) and pseudo measurements. Phasor measurement units (PMUs) provide valuable instantaneous information regarding power system stability and security, and thus have the potential to improve the reliability of the state estimator. Also, measured synchrophasors can be used for transmission line impedance estimation, which results in more accurate power system state estimation, in comparison with the traditional approach that uses catalogue values of the transmission line impedance.

Keywords: power system, state estimation, impedance estimation, phasor measurement unit, synchrophasors.

I. INTRODUCTION

All network management functions such as the state estimator, contingency analysis and optimization programs use network models in the mathematical formulation of the problem. The state estimator is the core application since it provides an input for all other applications. Among other information needed, transmission line parameters, such as resistance and reactance, are required to build the network model. The error in the parameter value, which is ordinarily assumed as known and correct, may introduce a large error in the state estimator solution.

Phasor measurement units (PMU) were introduced in the early 80s and became the measurement of choice for modern power systems. They provide positive sequence voltage and current measurements synchronized to within a microsecond, which is possible with the availability of the global positioning system (GPS) and the sampling data processing techniques developed for relay protection devices. The Wide Area Monitoring (WAM) system based on PMU devices has been implemented in the Croatian transmission system

operator HEP since 2003. In this operational time the system was enhanced and extended in different stages and timing. The first stage was the implementation of the WAM system for the online monitoring of a transmission corridor between two important nodes in the 400 kV network. After the first convincing experience of this system, the next step was the extension of the system with new PMUs to observe the complete 400 kV transmission network with frequency and positive sequence phasor recordings of 400 kV node voltages and currents in outgoing transmission lines [1]-[2]. The synchrophasors are mainly used for post mortem analysis and disturbance recordings [3]-[4], as well as in finding an optimal solution to incorporate the synchrophasor measurements to enhance the state estimation procedure [5]. Since the best features of the synchrophasors are their high accuracy, synchronicity and reliability, these key features might be of great use in a different approach to the state estimation enhancement by calculating parameter values of transmission lines used in the state estimator power system model. This approach was presented in [6]. Different methods of calculating parameters of transmission lines are used ([7]-[8]), in this paper a new approach of estimating transmission lines impedance based on [9] and [10] with multiple synchrophasor measurement scans is presented. The method is tested on actual PMU data from the devices installed on 400 kV lines. As high data reliability is a main precondition for using synchrophasors in online energy management system (EMS) applications such as the state estimator, the reliability of the PMU device data delivery was analyzed for actual PMU devices installed in the Croatian transmission power system.

The paper is organized as follows: section II gives an overview of the PMU data reliability, which gives reasonable adequacy of using synchrophasors in many EMS applications. The section III gives a theoretical background of the state estimation problem and the most commonly used method for solving it, the Weighted Least Squares (WLS). In section IV the influence of the transmission line parameter error on the

EuroCon 2013 • 1-4 July 2013 • Zagreb, Croatia

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state estimation is assessed by giving an real data from PMU devices. In section Vthe algorithm for transmission line parametsynchrophasors is presented and in sectiontesting results are given. The section VII giv

II. RELIABILITY OF SYNCHROPHASOR D

Reliability of the measurement fromdepends of the possibility of the PMU continuously in the central phasor data cand of the telecommunication system to dtime, without errors or data loss [11].

To determine the performance of the dshould be defined the task of each layer intransmission. The PMU device performs analog to digital conversion of the meatelecommunication system delivers the mePDC, which correlates, stores and diapplications and systems.

Availability of the PMU measured dafollowing factors, which define the measure

1) Availability of the telecommunication sydevice is switched off (Factor 1): The PMoff, or the PMU device is turned on and Gbut the measured data is not sent to the centelecommunication error.

2) GPS system availability (Factor 2):sends the data to the central server throtelecommunication system, but the PMsynchronized to the GPS system, the data time tag and therefore is unusable.

3) PMU measurement system error (Faccommunicates with the central server and with the GPS system, but the measured datflag.

The measured data from the PMU devifor the PMU devices installed in the CroaSystem Operator (TSO), as given in Tableanalyzed for a period of one year. The analyfor 5 PMUs installed on 400 kV nodes in tPMUs are of the same manufacturer, using and software version.

TABLE I PMU DATA RELIABILITY

Station # Reliability (%

Factor 1 Factor 2 PMU #1 99.94 99.99 PMU #2 99.97 100 PMU #3 99.96 100 PMU #4 99.97 99.99 PMU #5 99.97 99.99 Total 99.96 99.99

Fig. 1 gives total error participation in P

example using the V the framework of

er estimation using n VI the associated ves conclusions.

DATA DELIVERY

m a PMU device to send the data

oncentrator (PDC) deliver the data on

deployed system, it nvolved in the data

measurement and sured values. The

easured data to the istributes data to

ata depends of the ement chain:

ystem or the PMU U device is turned GPS synchronized,

ntral server due to a

The PMU device ough the operating

MU device is not does not have the

ctor 3): The PMU it is synchronized

ta has a bad quality

ice were inspected atian Transmission e I. The data were ysis was performed the system. All the the same firmware

%) Factor 3

99.95 99.98 99.97 99.98 99.97 99.97

MU unavailability.

Fig. 1. Total error participation in PM

From the results it can be sedisruption was mainly causesystem error or the PMU meaverage unavailability time fosystem for one year period is total PMU unavailability time fowere not delivered to the centrathe period of one year.

Fig. 2. Average PMU unavaila

Fig. 3 presents total time oyear period.

Fig. 3. Total time of PMU una

Factor 348%

0:00:000:02:530:05:460:08:380:11:310:14:240:17:170:20:100:23:02

PMU #1 PMU #2

Tim

e hh

:mm

:ss

0:00:00

2:24:00

4:48:00

7:12:00

9:36:00

12:00:00

14:24:00

16:48:00

Factor 1

Tim

e hh

:mm

:ss

MU unavailability for one year period

een that the PMU data delivery d by the telecommunication

easurement system error. The or each PMU installed in the given in Fig. 2. Summing the

for all the PMUs, the PMU data al PDC only 0.29 % of time in

ability time for one year period

of PMU unavailability for one

availability for one year period

Factor 148%

Factor 24%

2 PMU #3 PMU #4 PMU #5

Factor 2 Factor 3


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III. THEORETICAL BACKGROUND AND CLASSICAL APPROACH TO WEIGHTED LEAST SQUARES PROBLEM

The most frequently used method for solving an over determined set of nonlinear equations in the state estimation is the WLS method, where the errors are set to the same physical dimension, by dividing every error with its standard deviation �. The standard deviation is a measure of data scattering, i.e. average deviation from a mean value. The weight factors are determined according to the measurement units’ precision, so more precise measurements, with larger weight factors, influence more on the state estimation result.

For the power system with n unknowns and m measurements, relationship between the measurements and the state vector is given by:

exhz += )( (1)

where: z – the measurement vector (m,1), x – the state vector (n,1), h(x) – a non-linear functions vector (m,1), e – the error vector (m,1).

An initial assumption is that all the errors are mutually independent and normally distributed e ~ N(0, �2). The state estimation problem is then represented as a minimization of the weighted sum of squares of residuals, i.e. differences between the measured and estimated values:

��==

��

��

� −=

−=

m

j j

m

j j

xhzxhzxJ1

2

12

2 )())(()(σσ

(2)

[ ] [ ])()()( 1 xhzRxhzxJ T −⋅⋅−= − (3)

where:

J(x) – an objective function,

R – Measurement Covariance Matrix.

If all the errors are mutually independent, R is a diagonal matrix filled with �2, while W = R-1 is a diagonal Weight Matrix with the weight factors 1/�2. The condition for the objective function minimum is:

( ) [ ] 0)()()( 1 =−⋅⋅−=∂

∂= − xhzRxHxxJxg T (4)

where H(x) is the Jacobian matrix of partial derivatives of h(x) with respect to the state variables. The system is observable when the number of unknowns is equal to rank of H(x).

The linearization of g(x) by expanding it into the Taylor series gives:

( ) ( ) 0...)()( =+−⋅+= kkk xxxGxgxg (5)

where G(xk) is the Gain Matrix:

( ) ( ) ( )kkTk

k xHRxHxxgxG ⋅⋅=

∂∂= −1)( (6)

Because of neglecting higher order terms, the solution can be found with an iterative process. The change of the state vector in the k-th iteration is:

( ) ( ) [ ])(1 kkTkk xhzRxHxxG −⋅⋅=Δ⋅ − (7)

kkk xxx −=Δ +1 (8)

The calculation ends when the maximal change in the state vector becomes smaller than a previously chosen tolerance:

ε<Δ kxmax (9)

Fig. 4 gives a state estimation flow diagram [12].

kxkk ,,,0 maxε=

?maxkk <

)(xh

( ) ( ) ( )kkTk xHRxHxG ⋅⋅= −1

( ) ( ) [ ])(1 kkTkkk xhzRxHxxGfromx −⋅⋅=Δ⋅Δ −

?max ε<Δ kx

1+= kkkkk xxx Δ+=+1

Fig. 4. A state estimation flow diagram [12]


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IV. INFLUENCE OF PARAMETER ERROR ON STATE ESTIMATION RESULTS

Branch parameter values stored in the fixed database, which is available at the network management system central data engineering server, may be incorrect due to several reasons, such as: • Inaccurate manufacturing data or bad line length

estimation; • Network changes not updated in the central database;• Temperature changes.

It is of quite interest to assess how far a single parameter error influences the results of the state estimation problem. The influence was tested on actual synchrophasors measured by the PMU devices installed on 400 kV lines in the Croatian transmission power system. The simplified network model is shown in Fig. 5.

Fig. 5. The simplified network model used for testing

The measurements at each node were monitored and estimated. The line impedance ZL,34 between nodes 3 and 4 was changed in steps of 5 % and the influence on the estimated values of all the nodes was monitored. The results are shown in Fig. 6.

Fig. 6. Influence of parameter error on estimated values

Since the actual (true) value of the voltage magnitude and angle is not known, the estimated values ��are compared to the measured values��, hence the residuals (�� are used to assess the effect of parameter change on estimation of the voltage values. Fig. 6 shows on the y axes the ratio between the residual value when the series impedance of the line is changed and the ratio when the parameter is correct. The ratio is calculated for each node. The major impact of the change can be seen on the voltage estimated values of the adjacent nodes to the line with changed parameters.

V. PARAMETER ESTIMATION BASED ON SYNCHROPHASORS

Calculation of sequence impedances based on synchronized readings from the PMUs, which are deployed one at each end of the line, allows for a simple method for obtaining the results. A positive sequence impedance of the transmission line can be derived from positive sequence currents and voltages. The sequence impedances are subject to changes depending on the weather conditions and the power flow. Considering the high synchrophasors data sampling frequency and large amount of the data arriving in the PDC, it can be assumed that on each measurement stamp the measurement value of the voltage and current does not change much.

With this assumption, the measurement model with multiple scans of measurements can be used:

�� (10)

where: z – the measurement vector containing the measured currents and voltages for each measurement scan ti, h – the nonlinear vector function relating the unknown variables to the measurements, x(ti) – the vector containing the state variables, p – vector contacting the unknown transmission line parameters, e – the measurement error for each measurement scan ti, with mean of zero and variance Rz.

It is assumed that both the initial estimate of the parameter vector p and its covariance matrix Rp are known.

Using the given measurement model, in each estimation step the state variables and the unknown parameters can be calculated.

The used line transmission model is a � model, shown in Fig. 7. For the numerical solution it is assumed that the synchrophasors from both sides of the transmission line are available.

Fig. 7. � model of the transmission line

For the mathematical model of the line the series admittance g+jb and the shunt susceptance jy are used. Using rectangular coordinates the following equations can be written applying Kirchhoff’s laws:

�� (11)

�� (12)

0 5 10 15 20 25 30 35 40 450

0.2

0.4

0.6

0.8

1

1.2

1.4

Change of impedance of line ZL,34 (%)

Estim

ated

val

ue c

hang

e in

%

Node 1Node 2Node 3Node 4Node 5Node 6


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�� (13)

�� (14)

Separating the real and imaginary parts, the following equations can be derived:

��!��" � �� (15)

�#�!��" � �� (16)

��!��" � �� (17)

�#�!��" � �� (18)

�!$��" � �� (19)

�#!$��" � �� (20)

�!$��" � �� (21)

�!$��" � �� (22)

The measurement vector, which contains voltage and current measurements on each side of the line, in rectangular format is defined as:

�%�� & �'�!��"��#'�!��"�� '�!��"�#'�!��"� () (23)

The set of eight equations from number (15) to (22) represents the measurement model for the transmission line for each measurement scan at time ti. For number n of measurement sets, the measurement vector will have 8 x n elements.

In the presented approach the measurement vector zx(ti) will be augmented with the vector of unknown parameters g, b and y:

�* � &��() (24)

The final measurement vector has the following form:

�� &�%��*��() (25)

The residual vector is defined as:

+�� (26)

where h(ti) is the vector function relating the measurement vector and the unknown variables.

The estimated values are the series admittance and shunt susceptance; and simultaneously the voltage magnitude and angle at each end of the line. The augmented state vector has the following form:

�� &�� () (27)

The Jacobian matrix is then given:

,�� -,%�� ,*��. � / (28)

where Hx is the submatrix of partial derivatives of the measurements with respect to the state variables, and Hp is the submatrix of partial derivatives with respect to the parameters.

The augmented gain matrix is:

0�� 1,%2 �� 34��,%�� ,%2 �� 34��,*��

,*2 �� 34��,%�� ,*2 �� 34��,*�� *��5 (29)

The covariance matrix of the augmented state vector can be written as:

�� 1 34�� .. *4��5 (30)

The normal equation can be written as:

6 7�� 0��4�,��) ��4� 8 +�� (31)

The normal equation is solved for the measurement states and parameters and the procedure is iterated until the required accuracy � is reached.

VI. RESULTS

The parameter estimation method presented in the previous section was tested using the actual synchrophasors from the PMU devices installed on 400 kV lines in the Croatian transmission power system. The PMU devices measure the current and voltage phasors on each side of the transmission line, and the data is sampled every 20 ms.

The algorithm was tested for different number of measurement samples in each parameter estimation cycle. The input parameters for the test sets are given in Table III.

In the first set of tests the transmission line parameters were estimated using one data sample per parameter estimation cycle. This approach is similar to on-line parameter calculation. Fig. 8 shows the estimated values for each estimation cycle, where Xcal_1 are the calculated values and Xcat are the adopted catalogue values of the transmission line reactance. The values on the abscissa show the number of consecutive estimations. The results show considerable difference of the estimated from the catalogue values.


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TABLE III PARAMETER ESTIMATION INPUTS

Test no.

Number of samples

Synchrophasors sampling period [ms]

Number of successive estimations

Measurement set time included in one estimation cycle [s]

1. 1 20 20 0.02 2. 100 20 20 2 3. 200 20 20 4 4. 500 20 20 10 5. 700 20 20 14

Fig. 8. Successive estimation of line reactance using one measurement set per estimation cycle

The histogram of the estimated parameter value and catalogue value difference is given in Fig. 9. The difference is high as 2.6 %.

Fig. 9. Histogram of estimated impedance value and catalogue value difference using one measurement sample

In the second test, the same number of successive estimations was used, but the number of measurement samples used in the algorithm was increased to 100. Hence, the time span for each measurement window, i.e. measurement vector zx is 2 seconds. The results of this test are shown in Fig. 10.

Fig. 10. Successive estimation of line reactance using 100 measurement sets per estimation cycle

The histogram of the difference is given in Fig. 11. The difference is high as 1.7 %.

Fig. 11. Histogram of estimated parameter value and catalogue value difference using 100 measurement samples

The influence of the number of measurement samples used in the estimation cycle was tested by calculating the variance of the estimated values in each estimation step. The results can be seen in the Fig. 12.

Fig. 12. Variance of parameter values as a function of number of measurement samples used in estimation cycles

From Fig.12 it can be seen that the variance changes rapidly until the measurement sample number reaches 200, hence 4 seconds of measurement data at 20 ms sampling rate, after which the variance does not change much. From this it can be concluded that increase of the measurement span

2 4 6 8 10 12 14 16 18 200.0464

0.0466

0.0468

0.047

0.0472

0.0474

0.0476

0.0478

Number of succesive estimations

X [p

u]

X calculated

X catalogue

2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.5 2.55 2.60

5

10

15

20

25

Difference in estimated and catalogue impedance value (%)

Freq

uenc

y

2 4 6 8 10 12 14 16 18 200.0464

0.0465

0.0466

0.0467

0.0468

0.0469

0.047

0.0471

0.0472

0.0473

Number of succesive estimations

X [p

u]

X calculated

X catalogue

1.5 1.55 1.6 1.65 1.70

2

4

6

8

10

12

14

16

18

20

Difference in estimated and catalogue impedance value (%)

Freq

uenc

y

0 100 200 300 400 500 600 700 800 900 10000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Number of used measurement samples

Var

ianc

e


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window above 200 measurement scans does not affect the accuracy of the estimation cycle.

VII. CONCLUSION

In this paper an algorithm for estimating parameters of a transmission line is presented. The algorithm uses synchrophasors of currents and voltages at each end of the line and it is based on the WLS method. The estimated values are state variables (voltage magnitudes and angles at each end of the line) and the line impedance. The algorithm was tested using the actual synchrophasors derived from the PMU devices installed on 400 kV lines in the Croatian transmission power system.

The testing results show that the parameters can be estimated quite accurately using a sampling window of 2 seconds (100 measurement sets). Larger sampling windows do not affect much the improvement of the algorithm results.

The impedance calculated in this manner reflects realistic power system conditions, such as the loading of the line or ambient temperature, since it is derived from real time measurements. The algorithm might be of use in a non-invasive improvement of the state estimation process by online modifying of the network admittance matrix. The online parameter calculation can also be used in protection systems by modifying the relay setting parameters accordingly to the impedance change.

REFERENCES [1] S. Skok, R. Matica, I. Sturlic, “Enhanced open architecture of phasor

data concentrator,“ European Transactions on Electrical Power, 2010. [2] S. Skok, I. Sturlic, R. Matica, „Multipurpose Open System

Architecture Model of Wide Area Monitoring,“ PowerTech 2009, Bucharest, Romania, June 2009.

[3] S. Skok, D. Brnobic, V. Kirincic, "Croatian Academic Research Wide Area Monitoring System - CARWAMS", The International Journal on Communications Antenna and Propagation - IRECAP, vol. 1, no. 4, pp. 72-78, Aug. 2011

[4] S. Skok, V. Kirincic, K. Frlan, “Dynamic Analysis of Wind Farm Operation Integrated in Power System Based on Synchronized Measurements,” Engineering review, 30 (2010), no. 1, pp. 73-83, 2010.

[5] V. Kirincic, S. Skok, I. Pavic, „Power system state estimation based on PMU measurements vs SCADA measurements,” International Review on Modelling and Simulations - IREMOS, accepted for publication , ISSN 1974-9821, 2012.

[6] S. Skok, I. Pavi�, A. Barta, I. Ivankovi�, N. Baranovi�, Z. �erina, R. Matica, “Hybrid State estimation model based on PMU and SCADA measurements”, 2nd International Conference “Monitoring of Power System Dynamic Performance”, Saint Petersburg, Russia, 28-30 April 2008

[7] G.Valverde, D.Cai, J.Fitch, V.Terzija, “Enhanced State Estimation with Real-time Update Network Parameters using SMT”, Power & Energy Society General Meeting, 2009. PES ’09 IEEE, 26 – 30 July, 2009 , Calgary, Canada

[8] D.Shi, D.J. Tylavsky, N.Logic, M.Koellner, “Identification of Short Transmission-Line parmateres from Synchrophasor Measurements”, Power Symposium, 2008. NAPS ’08. 40th North America, 28-30 Sept. 2008.

[9] C. Borda, A. Olarte, H. Diaz, “PMU based line an transformer parameter estimation”, Power Systems Conference and Exposition, 2009. PSCE '09. IEEE/PES,pp 1-8, 15-18 March 2009.

[10] A. Monticelli, State estimation in electric power systems, A generalized Approach, Springer, 1st Edition, 1999.

[11] Easter Interconnection Phasor Project, Performance Requirements Task Team, Part II, Targeted Applications: State Estimation

[12] Y.J. Yoon, Study of the utilization and benefits of phasor measurement units for large scale power system state estimation, M.S. Thesis, Texas A&M University, 2005. Online: http://repository.tamu.edu/handle/1969.1/3345 (16.2.2011).


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