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DETECTION AND IDENTIFICATION OF NETWORK PARAMETER ERRORS USING

CONVENTIONAL AND SYNCHRONIZED PHASOR MEASUREMENTS

A Dissertation Presented

by

JUN ZHU

to

The Department of Computer and Electrical Engineering

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Electrical Engineering

Northeastern University Boston, Massachusetts

December 2008

ii

ABSTRACT

Detection and Identification of Network Parameter Errors Using

Conventional and Synchronized Phasor Measurements. (December 2008)

Jun Zhu, B.S., Shanghai Jiao Tong University;

M.S., Texas A&M University

Chair of Advisory Committee: Ali Abur

The successful control and operation of the power system are performed based on the

correct results of state estimation. The sources of errors of state estimation include the

measurement noise and gross errors, circuit breaker status and switch information, and

network parameter errors. Among them, the network parameter errors are the most

challenging ones to be identified since generally network parameters are assumed to be

known in state estimation. The inconsistency detected during state estimation may be

blamed on measurement errors and the network parameter errors may affect the results of

state estimation for a long time.

The existing parameter error identification methods have the common limitations that

they cannot perform correct identification with the presence of measurement error and a

suspect parameter set is required a priori state estimation.

This dissertation is dedicated to developing a network parameter error identification

method to overcome the limitations of the existing methods. This method can be

expanded by including PMUs in the power system to identify the errors appearing in the

critical k-tuples of parameters.

The existing parameter error identification methods are review first. Then an algorithm of

network parameter error identification and correction is developed. This method can

differentiate the measurement errors from the parameter errors even when they appear

simultaneously. This method does not need to select a suspect parameters set a priori. It is

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illustrated that the parameters in some particular topology and measurement

configuration form a critical k-tuple. The error of parameters in such critical k-tuple can

be detected but cannot be identified. It is demonstrated that such critical k-tuples are

caused by the possible multiple solutions of parameters and cannot be eliminated by only

conventional measurements. After reviewing the state estimation algorithm with PMUs,

the method proposed in the second part is expanded. By installing PMUs are some

strategic locations, the method is able to identify the parameter errors in the critical

k-tuples.

iv

To My Uncle Geyao Chu and My Parents

v

ACKNOWLEDGEMENTS

First of all, I would like to express my sincere gratitude to my advisor Dr. Ali Abur, for

his guidance, patience and encouragement during my Ph.D. studies at Texas A&M

University and Northeastern University. Many thanks to Dr. Abur for enlightening the

way of my research and critical technical comments on the completion of this

dissertation.

I am also very grateful to Dr. Alex Stankovic and Dr. Bahram Shafai for serving as

members of my committee.

Thanks to the department of Electrical and Computer Engineering Department of Texas

A&M University and Northeastern University for their financial support.

I would express my special thanks to my uncle Geyao Chu for his encouragements and

support on my decision on studies abroad.

Finally, I would like to thank my parents for their love and support all these years.

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TABLE OF CONTENTS

CHAPTER Page

I INTRODCUTION............................................................................................................................................1

A. BACKGROUND ............................................................................................................................................1 B. CONTRIBUTION OF THIS DISSERTATION ......................................................................................................3

1. Network Parameter Error Identification ..............................................................................................3 2. State Estimation with PMUs ................................................................................................................3 3. Parameter Error Identification with PMUs..........................................................................................3

C. ORGANIZATION OF CHAPTERS....................................................................................................................4

II REVIEW OF THE EXISTING PARAMETER ERROR IDENTIFICATON METHODS........................5

A. INTRODUCTION ...........................................................................................................................................5 B. METHODS BASED ON RESIDUAL SENSITIVITY ANALYSIS ...........................................................................7 C. METHODS AUGMENTING THE STATE VECTOR...........................................................................................10

1. Solution using Conventional Normal Equations...............................................................................11 2. Solution based on Kalman Filter Theory...........................................................................................13

D. CONCLUSIONS...........................................................................................................................................15

III PARAMETER ERROR IDENTIFICATION.............................................................................................16

A. INTRODUCTION .........................................................................................................................................16 B. LAGRANGE MULTIPLIERS METHOD .........................................................................................................18

1. Lagrangian Model ...............................................................................................................................18 2. Computation of Normalized Lagrange Multipliers...........................................................................22 3. Correction of the Parameter in Error .................................................................................................23

C. PARAMETER IDENTIFICATION ALGORITHM..............................................................................................24 1. Overall Process....................................................................................................................................24 2. Flow Chart ...........................................................................................................................................26

D. SIMULATION RESULTS ..............................................................................................................................27 1. Line Impedance or Measurement Error.............................................................................................27 2. Transformer Tap or Measurement Error ............................................................................................29 3. Errors in Shunt Capacitor/Reactor .....................................................................................................31 4. Simultaneous Errors............................................................................................................................33 5. Critical K-tuples of Parameters ..........................................................................................................34

E. CONCLUSION .............................................................................................................................................37

IV STATE ESTIMATION USING PHASOR MEASUREMENTS ..............................................................38

A. INTRODUCTION .........................................................................................................................................38 B. STATE ESTIMATION METHOD WITH PHASOR MEASUREMENTS...............................................................40

1. Problem Formulation without Using a Reference Bus .....................................................................40

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CHAPTER Page

2. Observability of Network ...................................................................................................................44 3. Rules of Observability and Error Identification ................................................................................46

C. SIMULATION RESULTS ..............................................................................................................................47 1. State Estimation with and without a Reference.................................................................................47 2. Merging Observable Islands Using Phasor Measurements ..............................................................49 3. Phasor Measurement Error Identification..........................................................................................50 a. One Phasor Measurement in an Observable Island.........................................................................................50 b. Two Phasor Measurements in an Observable Island.......................................................................................52 c. Three Phasor Measurements in an Observable Island ....................................................................................54

D. CONCLUSION ............................................................................................................................................56

V NETWORK PARAMETER ERROR IDENTIFICATION USING PHASOR MEASUREMENTS ......57

A. INTRODUCTION .........................................................................................................................................57 B. MULTIPLE SOLUTIONS OF PARAMETER ERRORS .....................................................................................59 C. ALGORITHM USING PHASOR MEASUREMENTS .......................................................................................64 D. SIMULATION RESULTS ..............................................................................................................................68

1. Critical K-tuples in Radial Lines........................................................................................................68 2. Critical K-tuples in Separated Radial Lines ......................................................................................71 3. Critical K-tuples in Radial Lines Formed by Missing Measurements.............................................72 4. Simultaneous Errors in Network Parameters and Phasor Measurements........................................74

E. CONCLUSION .............................................................................................................................................75

VI CONCLUSIONS AND FUTURE WORK.................................................................................................77

REFERENCES.....................................................................................................................................................79

VITA .....................................................................................................................................................................82

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LIST OF TABLES

TABLE Page Table 1. Simulated Parameter and Measurement Errors .......................................................................27 Table 2. Results of Error Identification - 14-bus System ......................................................................28 Table 3. Results of Error Identification - 30-bus System ......................................................................28 Table 4. Results of Error Identification - 57-bus System ......................................................................29 Table 5. Estimated and True Parameters of Line Impedances............................................................... 29 Table 6. Simulated Parameter and Measurement Errors .......................................................................30 Table 7. Tap and Measurement Error Identification - 14-bus System................................................... 30 Table 8. Tap and Measurement Error Identification - 57-bus System................................................... 31 Table 9. Estimated and True Parameters of Taps .................................................................................. 31 Table 10. Shunt Susceptance Errors......................................................................................................32 Table 11. Estimated and True Parameters of Shunt Susceptances......................................................... 32 Table 12. Multiple Error Identification Results..................................................................................... 33 Table 13. Estimated and True Parameters of Multiple Errors ............................................................... 34 Table 14. Simultaneous Estimation of All Identified Parameters.......................................................... 34 Table 15. Objective Function Values for Tests A and B ........................................................................35 Table 16. Error Identification of Series Lines ....................................................................................... 36

Table 17. the Structure of nH .............................................................................................................42

Table 18. Simulated Measurement Errors and PMU measurement....................................................... 47 Table 19. Results of Error Identification-voltage phasor measurements...............................................48 Table 20. Results of Error Identification- current phasor measurements ..............................................48 Table 21. Results of parameter error detection (fail).............................................................................52 Table 22. Results of parameter error detection-voltage phasor measurements .....................................53 Table 23. Results of parameter error detection-current phasor measurements......................................54 Table 24. Results of parameter error identification-voltage phasor measurements............................... 55 Table 25. Results of parameter error identification-current phasor measurements ............................... 56 Table 26. Objective Function Values for Tests A and B ........................................................................62 Table 27. Estimated States for Tests A and B........................................................................................ 63

Table 28. the Structure of pxH ............................................................................................................66

Table 29. Parameter Error Identification - 14-bus system (I)................................................................ 69 Table 30. Parameter Error Identification - 30-bus system (I)................................................................ 70 Table 31. Parameter Error Identification - 57-bus system.....................................................................71 Table 32. Parameter Error Identification - 30-bus system (II) .............................................................. 72 Table 33. Parameter Error Identification - 14-bus system (II) .............................................................. 74 Table 34. Phasor Measurement Error Identification - 14-bus system ................................................... 75

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LIST OF FIGURES

FIGURE Page Figure 1. Flow chart of parameter error identification method............................................................. 26 Figure 2. IEEE 14-bus system...............................................................................................................35 Figure 3. 5 bus system with 2 observable islands ................................................................................. 45 Figure 4. Phase Angle determination with phasor measurements ......................................................... 45 Figure 5. PMUs with common reference .............................................................................................. 46 Figure 6. 14 bus system with 3 observable islands ...............................................................................49 Figure 7. 118-bus system with 9 Areas .................................................................................................51 Figure 8. Radial lines in the system ......................................................................................................59 Figure 9. Multiple solutions of voltage and current phasors.................................................................61 Figure 10. IEEE 14-bus system.............................................................................................................62 Figure 11. Critical k-tuples in the system.............................................................................................. 64 Figure 12. Part of the IEEE 30-bus system ........................................................................................... 69 Figure 13. Part of the IEEE 57-bus system ........................................................................................... 70 Figure 14. New Observable Island in IEEE 14-bus system ..................................................................73

CHAPTER I

INTRODCUTION

A. Background

Power system state estimation (SE) is one of the essential functions in modern Energy

Management System (EMS). During the power system operation the measurements are

collected through the Supervisory Control and Data Acquisition System (SCADA) from

remote terminal units (RTU) and sent to the control center. The measurements are

composed of line power flows, bus voltages and power injections, generator outputs,

loads, circuit breaker status information, and transformer tap positions. Those raw data

and measurements together with the network model saved in the database are processed

by the state estimator in order to provide an optimal estimation of the system state. Based

on this solution provided by SE the other EMS application functions such as the

contingency analysis, economic dispatch, and load forecasting can be correctly

performed.

The performance of the SE depends on the accuracy of the received measurements data

as well as the network parameter data of the system model. The measured data are subject

to noise or errors in the metering system or communication process. With proper

redundancy in the measurement configuration the state estimator can perform the bad

data processing to detect, identify and eliminate the bad measurements in the

measurement set.

The circuit breaker and switch status included in the measurements are processed by the

topology processor. A bus/branch model will be generated by it. If there is any error in

the information, a wrong model will be created and this will lead to large errors in the

estimated state variables.

2

Another source of errors is the network parameter database. Once the power grid is built

the corresponding network parameters are determined and saved in the database for

future use. Those data are supposed to be known and error-free in the state estimation and

will be used for a long time. However, network parameter data may also be incorrect as a

result of inaccurate calculations during initialization, failure to update the database at

sufficient frequency, error in meter calibrations or various other reasons. If that is the

case, the state estimator may ascribe the problem to the related measurements and such

error in the database may not be detected for a long time.

Recently a new monitoring device called the Phasor Measurement Unit (PMU), has been

introduced into the power systems. Popularity of these devices triggered two major

projects conducted in the North America for the utilization and application of PMUs:

WAMS in the western area and EIPP in the eastern area. More than a hundred utility

companies, ISOs, national labs, and vendors are participating in these projects.

Phasor measurement units are devices that can provide direct measurements of

synchronized phase angles with respect to the time reference provided by the global

positioning system (GPS) satellites. The sampling rate of PMUs is much faster than the

SCADA scan rate and the phasor measurements provided are more accurate than the

conventional measurements. With the availability of the phase angle measurement, which

is the key state variable in state estimation, PMUs are very helpful for EMS applications.

Based on the high sampling rate, a real-time state calculation can be performed with

enough number and proper location of PMUs in the system. With numbers of PMUs

installed in their strategic position, the redundancy level of the system is improved to

cover the measurement loss or line outage, to decrease the condition number of state

estimation, or to improve the error detection and identification. PMUs are considered as

the next generation monitoring devices that will improve the performance of power

system for their high sampling rate and accuracy. The applications of PMUs in diverse

aspects by utilizing their advantages are being studied.

3

B. Contribution of this dissertation

The objectives of this research work are to develop a network parameter error

identification method that overcomes the limitations of the existing methods and further

enhance that method so that it can account for PMUs as well as the conventional

measurements in the system. The essential contributions of this dissertation are briefly

outlined below.

1. Network Parameter Error Identification

The Lagrange multipliers corresponding to the equality constraints of network parameters

are used to detect and identify the parameter error. Those Lagrange multipliers are

calculated based on the results of the conventional weighted least-square (WLS) state

estimation solution; so existing code of state estimation can be used directly for this part.

This method can identify the network parameter error as well as the analog measurement

error even when they appear simultaneously. Another important advantage of this

method is that there is no need to specify any suspect set of parameters a priori.

2. State Estimation with PMUs

Phasor measurements are integrated in the conventional state estimation formulation.

Requirements to ensure robustness of state estimation in the presence of single PMU

errors or losses are identified. Network observability analysis is revisited to modify the

rules so that they also account for phasor measurements and detection and identification

of their errors.

3. Parameter Error Identification with PMUs

Benefits of having PMUs in detection and identification of network parameter errors are

investigated. The existence of critical k-tuples in network parameters is illustrated and the

causes of such critical k-tuples are analyzed. In the case of an error in one of the

parameters in the critical k-tuple, it is shown that such an error can be detected but cannot

4

be identified only with traditional measurements. Installing PMUs in proper locations is

shown to facilitate identification of such errors. This is a case which demonstrates the

advantages of having PMUs over other conventional measurements.

C. Organization of Chapters

This dissertation is organized as below.

Chapter II reviews the existing methods of network parameter error identification and

estimation.

Chapter III describes the new algorithm of parameter error identification.

Chapter IV investigates the application of PMUs in the state estimation and concludes the

rules to detect and identify the single phasor measurement error.

Chapter V analyzes the cause of critical k-tuples of parameters and presents a solution to

eliminate them, which involves the install of PMUs at proper locations.

Chapter VI is the summary of the research work in this dissertation.

5

CHAPTER II

REVIEW OF THE EXISTING PARAMETER ERROR

IDENTIFICATON METHODS

A. Introduction

State Estimation (SE) is an important function of Energy Management System (EMS). It

processes redundant measurements and provides steady-state operating state for advanced

EMS application programs. Once receiving field measurement data, network parameter,

network topology, and other information, state estimation filters incorrect data to ensure

that the estimated state is correct. Weighted Least Squares (WLS) approach is usually

used to solve state estimation problem due to its statistical prosperities.

Successful execution of state estimation is dependent on the validity of the measurements,

the topology data and the assumed network parameters. Given enough measurement

redundancy, the state estimator can be used to identify such errors in the measurements,

network topology or parameters. Bad data processing for the measurements is a common

carried out by state estimators since analog bad data appear quite frequently in the

measurements. Topology errors typically lead to large errors in the estimated states and

their identification may be slightly easier compared to the parameter errors, which may

go undetected for long periods of time leading to permanent errors in state estimation

results.

Almost all application functions in energy control centers use the network model that is

built based on the topology processor. Topology processor uses the real-time as well as

static information about the status of all the circuit breakers in the system in order to form

the real-time network model for the system. It implicitly assumes that all parameters

associated with the power system elements such as transmission lines, transformers,

shunt capacitors or reactors, etc. are perfectly known and the inconsistencies detected in

6

state estimation will be blamed on the related measurements. However, the network

parameter values stored in the static database may also have errors caused by inaccurate

manufacturing data, miscalibration, tap changer being locally modified without

knowledge of the control center, etc. This error of the network parameters may not be

identified for a long time, which will lead to permanent errors in the results of state

estimation and affect other application functions. Hence, detection, identification and

correction of network parameter errors are very important.

Several researchers investigated these problems and published their proposed solutions in

different venues. In general, the proposed methods can be classified under two major

categories [1]:

Methods based on residual sensitivity analysis. These methods are performed

following a converged state estimation solution and require the user to identify a set

of suspected parameters.

Methods augmenting the state vector. The suspected parameters are included in the

state vector and both the state and parameters are simultaneously estimated. Two

different bus related techniques have been proposed to deal with the augmented

model:

* Solution using normal equations. Except for some observability and numerical

issues this method is a straightforward extension of conventional SE model.

* Solution based on Kalman filter theory. Under this approach several measurement

samples are radially processed in order to recursively improve existing parameter

values.

These methods will be briefly reviewed in the following sections.

This chapter is organized as below.

Section A is introduction part.

7

Section B reviews the methods based on residual sensitivity analysis.

Section C reviews the methods augmenting the state vector. This section

including two parts with solutions using normal equations and Kalman filter

theory.

The last section is the conclusion part.

B. Methods based on residual sensitivity analysis

In this section, a brief review of the methods of parameter estimation and error

identification based on residual sensitivity analysis is presented.

The method presented in [2-5] is based on the sensitivity relationship between residuals

and measurement errors:

r S e (2.1)

where S is the residual sensitivity matrix

1 TS I HG H W (2.2)

and

TG H WH (2.3)

is the gain matrix. Just like a set of errors on all related measurements of the target branch,

a parameter error on it could have the same effect on the estimated state. A basic

measurement model can be expressed this way:

0 0, , , ,s s s s s s sz h x p e h x p h x p h x p e (2.4)

where

z is the measurement vector

x is the true state vector

h is the nonlinear function relating measurements and state vectors

8

se is the measurement error vector

p and 0p are respectively the true and erroneous values of network parameter

subscript s refers to the involved adjacent measurements.

The term 0, ,s sh x p h x p in (2.4) is the parameter error, which may be observed

as equivalent measurement errors. If that parameter error is large enough, a bad data will

be detected and the related measurements will most likely be identified as in error. The

equivalent measurement error can be written as:

0, , ss s p

hh x p h x p ep

(2.5)

where 0pe p p is the parameter error. A linear relationship between the residuals of

related measurements sr and the parameter error pe can be derived from equations

(2.4), (2.5) and (2.1):

ss ss p s

hr S e rp

(2.6)

where ssS is the s by s matrix of S corresponding to the s involved measurements and

sr is the residual vector that would be obtained when the parameter is correct.

Equation (2.6) gives the linear relationship between the measurement residuals sr and

an unknown parameter error pe in the presence of noise sr . Therefore it makes the

determination of pe a local estimation problem if every residual are weighted according

to its normal distribution.

Papers [2-4] use this linear model particularly on transformer tap but it can be extended

to other types of parameters. Some publications made some revisions to make it

9

numerically more efficient but the basic idea of the method remains the same. The

procedure proposed in [5] is composed of two steps: In the first step a bias vector is

estimated and the parameter errors are obtained in the second step from the computed

bias vectors. The main difference between [5] and [2-4] is that the bias vector is

expressed in terms of the line flows.

The general procedures of the methods based on residual sensitivity analysis are:

(1) Identify all the involved measurements (power flows on the branch, power injections

at the ending nodes).

(2) Perform traditional state estimation and the residuals of relevant measurements sr .

(3) Calculate ˆpe based on sr using the equations above.

(4) Correct the suspect parameters p̂ by the equation 0ˆ ˆpp p e .

(5) Perform a new state estimation and new detection tests; if the latter are still positive,

go to 3; otherwise STOP.

The main advantage of this method is that the identification and parameter estimation

procedures constitute additional and separate routines and there is no need to modify the

main SE code. However, by investigating the whole process, there are three main

limitations of this approach:

1) The suspect parameter sets must be pointed out before the parameter estimation is

performed.

2) The measurements in the system are assumed to be correct or the linear relationship

of measurement residuals sr and the parameter error pe will hold true.

3) Each time the suspect parameter changes, the related measurements vector s have to

be adjusted.

10

C. Methods augmenting the state vector

In this method the suspected parameters p are treated as additional state variables.

Therefore, the objective function can be written as:

2

1, ,

m

i i ii

J x p w z h x p

(2.7)

where iw is the weight for measurement i.

p is the vector of suspected parameters.

If the suspect parameter is line parameter or transformer tap, it can be added to the model

as a pseudo-measurement using the value 0p provided in the database. In such a case,

the objective function can be expressed with a new term:

2

20

1, ,

m

i i i pi

J x p w z h x p w p p

(2.8)

where pw is the arbitrary weight assigned to the pseudo-measurement.

Due to the study of [6] the introduction of the pseudo-measurement may not help on the

parameter estimation. If p is not observable with existing measurements the new term in

(2.8) is critical and useless. On the other hand, if this term is redundant, the value

assigned to pw may significantly influence the estimated value p̂ . Due to the analysis

of the influence of pw on p̂ in [6], it is suggested that the pseudo-measurement 0p

should not be added to the model. Even if it is added for observability purposes, a rather

small value should be assigned to pw .

There are two types of solutions related to this augmented model, the solution using

conventional normal equations and the solution based on Kalman filter theory. They will

be proposed in the following sections.

11

1. Solution using Conventional Normal Equations

The augmented state vector approach was applied in [7-9] using the conventional state

estimation formulations. The state vectors can be solved by WLS state estimation:

TG x H W z (2.9)

Generally this solution involves the expansion of the Jacobian matrix H , which will be

enlarged to accommodate the extra columns and extra rows of suspect parameters as new

state variables and new pseudo-measurements, respectively. The expanded Jacobian

matrix will have the following structure:

0 ...... 0

... ... ...

... ... ...0 ... 0

0 ... ... 0 1 ... 0... ... ... ... ... ... ...0 ... ... 0 0 ... 1

eH

(2.10)

where the upper leftmost part corresponds to the conventional Jacobian. The right part is

the extra columns corresponding to the suspect parameters. The lower part is the extra

rows of pseudo-measurements of the suspect parameters. The nonzero elements in

the new columns correspond to the relevant measurements.

After this new Jacobian matrix is built, the state estimation is performed similarly to the

traditional one and the suspect parameter values are obtained when SE converges. This is

the general idea of the solution but it is applied in different forms. In some early papers [7,

8] the augmented state variables are the suspect parameters themselves but in the latter

approaches it is not the case. For instance, in [9] the extended state variables are the

incremental power flows originated by parameter errors rather than the parameters. After

the traditional state variables and the state variables of the power flows are obtained, the

12

suspect parameters can be calculated subsequently. This indirect approach is used to

prevent the numerical problems caused by lack of pseudo-measurements.

At the flat start, the extra columns of the Jacobian matrix in (2.10) will be virtually null if

the corresponding pseudo-measurements are not included. This will lead to a nearly

singular gain matrix during the first iteration and failure of state estimation if such gain

matrix is used. One way to avoid this problem is to use the other related variables rather

than the parameters as the extra state variables as the approach proposed in [9]. The other

way is to include the parameters in the state vector from the second iteration.

The general procedures of this approach are listed below:

(1) Read the input data that include network parameter data and real-time measurements,

suspect parameter set.

(2) Form measurement Jacobian matrix H with additional state variables.

(3) Perform the state estimation.

(4) Update the state variables.

(5) Repeat steps (3) and (4) until it converges. Calculate the final parameters.

This approach provides a straightforward but efficient way to estimate a parameter value.

Comparing to the methods based on residual sensitivity analysis, it does not need to apply

the state estimation repeatedly to obtain the same accuracy. Note that a preliminary

regular state estimation may be needed to identify the suspect parameter set. Since such

suspicious parameters are selected by observing the normalized residuals of the related

measurements, it is not so reliable. It may be misled by the measurement errors and

generate an incorrect suspect parameter set. The limitations of this approach are listed

below:

1) The suspect parameter set must be determined before the state estimation.

13

2) When the suspect measurement set changes, the Jacobian matrix H has to be rebuilt

accordingly.

3) The measurements in the system are supposed to be correct or the estimated

parameter will be biased.

2. Solution based on Kalman Filter Theory

A typical parameter error identification method applying the Kalman filter theory is

proposed in [10]. This method can estimate multiple types of variables including network

parameters, transformer taps, measurement bias and standard deviations of measurement

errors.

At every time sample k, the measurements can be expressed as:

, ,z k h x k k p e k (2.11)

where time sample k is included in measurement equation h to show the possibility of

network changes form one time sample to the next. The parameters are assumed constant

for the entire time sample period.

To estimate the state vectors, the following equation should be optimized:

1

, , , ,Tm

k i i i ii

J z k h x k k p W z k h x k k p

(2.12)

Let 0p be the a-priori value assigned to p , the estimates of parameters p can be

updated every new time sample and be used as a pseudo-measurement:

1k k pp p e k (2.13)

where the error vector pe k is assumed to have zero mean and covariance matrix

pR k . Then the objective function can be written in the following equation with

additional pseudo-measurements of suspected parameters:

14

11 1

Tk k p k k kJ p p R k p p J (2.14)

At the k-th time sample, only the vector z k and the updated estimates of parameters

and their covariances are considered. In this sense this algorithm is a recursive one.

The work in [11] presents two important differences with respect to [10]:

(1) The problem is localized into several small observable sub-networks containing the

unknown parameters.

(2) Parameters are modeled as Markov processes, thereby allowing estimation of

time-varying parameters.

The procedure of the Kalman filter method starts by estimating only the parameters of a

few branches with maximum redundancies. As the parameters of those branches are

estimated, they are used to extend the process to less metered branches, and so on. The

solution will eventually include all network branches with adequate local redundancy,

excluding only those for which the process can not be reliably performed.

Comparing to the other methods, the solutions based on Kalman filer theory is more

appropriate for the estimation of time-varying parameters in a localized area since it

makes use of several measurement samples in continuous time stamps. But it still has

some kind of limitations:

1) The need to update the covariance matrix of parameter errors, as well as other related

overheads, make this approach more cumbersome and costly, especially when the

number of parameters is high.

2) The measurement error must be identified and rejected before the parameter estimation,

or it may corrupt.

3) Although a suspect parameter set may not be necessary, this approach still can only

perform localized parameter estimation.

15

D. Conclusions

By reviewing the existing methods of parameter error identification and estimation, it can

be observed that they have some common limitations.

The first limitation is a suspect parameter set is required to be selected before the

parameter error identification. Considering the dimension of the power system, it is not

mathematically practical to include all the network parameters in the system in the

estimation matrix. Sometime the suspicious parameters are arbitrarily selected by

experienced network operators. This suspect parameter set could also be generated by the

algorithm based on measurement residuals. However, such algorithm may be biased

because of the measurement error.

The second limitation is that the bad data in measurements have to be removed from the

network before performing the parameter error identification or estimation. For the

methods based on residual sensitivity analysis, the measurement error will bias the

parameter error identification results. For the methods with augmenting state variables,

the algorithm used to generate the suspect parameter set are severely influenced by the

bad measurements. Thus the parameter error estimation based on such suspect parameter

set may be incorrect.

The existing methods are not able to identify the network parameter error if a suspicious

set of parameters are not provided. In the mean time, those methods are vulnerable to the

analog measurement errors. Those are the two major defects of the existing methods. The

algorithm presented in the next chapter will aim on them and try to overcome those two

limitations.

16

CHAPTER III

PARAMETER ERROR IDENTIFICATION

A. Introduction

All the energy management system (EMS) applications make use of the network model

in the mathematical formulation of their problem. Transmission line resistances,

reactances and charging capacitances, transformer reactances and tap values, and shunt

capacitor/reactor values are examples of network parameters that are required to build the

network model. Among the EMS applications, state estimation plays an important role

since it provides the network model for all other applications.

Traditionally, state estimation is carried out assuming that the correct network model is

known. Therefore, any inconsistencies detected during the estimation process will be

blamed on the analog measurement errors. However, this is not always true in the

practical system. Errors in the network model may be due to topology and/or parameter

errors.

As reviewed in the previous chapter, the existing methods of parameter error

identification are of two types. The first type is based on residual sensitivity analysis,

where the sensitivities of the measurement residuals to the assumed parameter errors are

used for identification. The second type used a state vector augmented by additional

variables, which are the suspected parameters. This approach can be implemented in two

different ways: one using the static normal equations and the second the other using

Kalman filter theory.

The existing methods have two common limitations: A suspect parameter set is required

before parameter error estimation; the bad data in measurement set has to be removed

before parameter error identification. Without the first condition, the methods of

augmenting state vector can not be processed and the methods based on residual

17

sensitivity analysis may have trouble on computation for a large system. With the

measurement error in the system, the existing methods of parameter error identification

and estimation may be biased or corrupted.

Topology errors, on the other hand, involve incorrect status information for circuit

breakers, and several methods are proposed so far for their detection and identification

[12-16]. Among these methods, a recent one that is based on reduced system model and

the use of Lagrange multipliers [15], [16] addresses the main shortcoming of the previous

proposed methods by eliminating the need to identify a suspect substation before

topology error identification.

In this chapter, a new parameter error identification method that complements the

topology error identification method of [16] is proposed. This method is based on the

Lagrange multipliers of the parameter constraints. A set of additional variables that

correspond to the errors in the network parameters is introduced into the state estimation

problem. However, direct estimation of these variables is avoided by the proposed

formulation. Following the traditional state estimation solution, measurement residuals

are used to calculate the Lagrange multipliers associated with the parameter errors. If

these are found to be significant, then the associated parameter will be suspected of being

in error.

The main advantage of this method is that the normalized measurement residuals and

parameter error Lagrange multipliers can be computed, allowing their identification even

when they appear simultaneously. The first part of the proposed procedure is based only

on the conventional weighted least-squares (WLS) state estimation solution; however, the

subsequent error identification and correction procedures will have to be implemented

and integrated into the existing code. There is no need to specify a suspect set of

parameters a priori, since the method will readily identify the erroneous parameters along

with any existing bad measurements.

18



Section B describes the method used for parameter error identification. First the

Jacobian matrix with the presence of parameter errors and equality constraints

are formulated. The Lagrange multipliers are obtained based on the results of

state estimation using the new Jacobian matrix. Then the way to normalize the

Lagrange multipliers and the residuals are presented. The method to correct the

parameter in error is also given in the section.

Section C gives the overall procedures of the parameter error identification

method. A flow chart of the method is also presented.

Simulation results of typical systems are shown in section D.


B. Lagrange Multipliers Method

In this section, a brief presentation of the Lagrange multipliers method will be given. It is

composed of three parts: (1) the formulation of the state estimation, (2) how to compute

the normalized Lagrange multipliers, and (3) how to correct the bad parameter.

1. Lagrangian Model

Consider the following measurement model:

( , )ez h x p e (3.1)

where

z measurement vector;

( , )eh x p nonlinear function relating the measurements to the system states and

network parameter errors;

19

x system state vector, including voltage magnitudes and phase angles;

ep vector containing network parameter errors;

e vector of measurement errors.

Buses with no generation or load will provide free and exact measurements as zero power

injections. These can be treated as equality constraints given by:

, 0ec x p (3.2)

Network parameter vector will be modeled as:

t ep p p (3.3)

where p and tp are the assumed and true network parameter vectors. Network

parameter errors are normally assumed to be zero by the state estimator. Therefore, for

error free operation, the following equality constraint on network parameter errors will be

used:

0ep (3.4)

The weighted least squares (WLS) state estimation problem in the presence of network

parameter errors and equality constraints can then be formulated as the following

optimization problem:

12Minimize

Subject to ( , ) 00

t

e

e

J(x) r Wrc x pp

(3.5)

where:

( , )er z h x p is the measurement residual vector,

20

W is the diagonal matrix whose inverse is the measurement error covariance matrix,

cov(e).

Applying the method of Lagrange multipliers, the following Lagrangian can be defined

for the optimization problem of (3.5):

1,

2tt tL r Wr c x p pe e (3.6)

Applying the first order optimality conditions:

0L t tH Wr Cx xx

(3.7)

0L t tH Wr Cp pp

(3.8)

, 0L

c x pe

(3.9)

0L

pe

(3.10)

where:

, e

x

h x pH

x

(3.11)

, ex

c x pC

x

(3.12)

, ep

e

h x pH

p

(3.13)

, e

pe

c x pC

p

(3.14)

21

and are the Lagrange multipliers for the equality constraints (3.2) and (3.4).

Equation (3.8) can be used to express in terms of and :

r

S

(3.15)

where:

t

WH pS

Cp

(3.16)

is the parameter sensitivity matrix.

Equality constraint (3.4) allows substitution of pe in (3.7) - (3.9). Denoting

, 0h x and ,0c x by 0h x , 0c x respectively, the measurement equations will

take the following form:

0 ( )z h x e (3.17)

0 0c x (3.18)

Note that (3.17) and (3.18) are the conventional measurement and zero injection

equations used by the state estimators. They do not include parameter errors as explicit

variables. Substituting the first order Taylor approximations for 0h x and 0c x , the

following linear equations will be obtained:

xH x r z (3.19)

0 0( )xC x c x (3.20)

where:

00 , xxxx vectorstate system for the guess initial thebeing )( 00 xhzz

22

Using (3.7), (3.19), and (3.20), the following equation will be obtained:

0 0

00 0 ( )0 0

t tH W C xx xH I r zxC c xx

(3.21)

This equation is the same equation used for iterative solution of the conventional WLS

state estimation problem. Hence, the solution for the measurement residuals r and the

Lagrange multipliers for the zero injections can be obtained first by iteratively

solving (3.21). Once the state estimation algorithm successfully converges, (3.15) can be

used to recover the Lagrange multiplier vector associated with the parameter errors.

2. Computation of Normalized Lagrange Multipliers

Since the main aim of this work is to identify parameter errors, the validity of the

constraint (3.10) will have to be tested. This can be done based on the Lagrange

multiplier vector associated with the parameter error vector ep . In order to test the

significance of a given i value, it will be normalized using its covariance matrix

cov , which can be obtained in [17] and also shown below.

Letting [ ]Tu r and using (3.15):

cov cov( ) tS u S (3.22)

The covariance of u, cov u can be calculated by first expressing r and in terms of

the measurement mismatch. To do that, let the inverse of the coefficient matrix in (3.21)

be given in partitioned form as follows:

10 1 2 3

0 54 60 0 7 8 9

t t E E EH W Cx xH I E E ExC E E Ex

(3.23)

23

Noting that 0 0c x at the solution, (3.21) will yield the following expressions for r and

:

5r E z (3.24)

8E z (3.25)

Let [ ]5 8TE E , then:

u z (3.26)

1cov( ) tu W (3.27)

The Lagrange multipliers for the parameter errors can then be normalized using the

diagonal elements of the covariance matrix defined in (3.22):

( , )

N ii i i

(3.28)

for all i = 1…k, where k is the total number of network parameters whose errors are to be

identified.

Note that the denominator in (3.28) will be zero for cases where local measurement

redundancy does not allow detection of errors in parameter. One such case is when all

measurements that are functions of a parameter are critical. The other obvious one is

when there are no measurements that are functions of a parameter.

3. Correction of the Parameter in Error

After the parameter in error is identified, this specific parameter can be corrected by

estimating its true value simultaneously with the other state variables [1]. In order to

accomplish this, the state vector is augmented by the suspicious parameter p , yielding

the following new state vector, v :

24

1 2, ,..., |nv x x x p (3.29)

where v is the set of state variables.

1x ,…, nx conventional state variables

p parameter preciously identified as erroneous.

The solution of the state estimation problem will yield not only the state estimates but

also the estimated value of the suspect parameter.

C. Parameter Identification Algorithm

1. Overall Process

The above formulation can be used to develop an algorithm to detect and identify

network parameter errors. Such an algorithm is proposed below:

Step 1. Normal State Estimation

This is the WLS state estimation problem as currently solved by existing

software. In addition to the measurement residual vector r , the solution will

provide the Lagrange multiplier vector of zero injections if they are treated

as equality constraints in the state estimation formulation. The solution involves

repeated solution of (3.21) until convergence. Note that all parameter errors are

assumed to be zero and therefore ignored at this step.

Step 2. Bad Data and Parameter Error Identification

Compute the normalized residuals for the measurements, and the normalized

Lagrange multipliers for the parameter errors, as in (3.28). Section B.2 illustrates

the steps leading to (3.28). Choose the larger one between the largest normalized

residual and the largest normalized Lagrange multiplier.

25

If the chosen value is below the identification threshold, then no bad data or

parameter error will be suspected. A statistically reasonable threshold to use

is 3.0, which is the one used in all simulations presented in the next section.

Else, the measurement or the parameter corresponding to the chosen largest

value will be identified as the source of the error.

Step 3. Correction of the Parameter Error

If a measurement is identified as bad, it is removed from the measurement

set. Equivalently, its value can be corrected using a linear approximation for

the estimated measurement error [18].

If a parameter is identified as erroneous, it is corrected by estimating its

value by the method described in Section B.3 using the augmented state

vector defined as (3.29). Substitute the estimated parameter value for the old

one and go to Step 1.

Note that bad data and parameter errors are processed simultaneously. This is possible

provided that there is sufficient measurement redundancy and the parameter errors are not

strongly correlated with the bad data. Since parameter errors are persistent whereas bad

data usually appear in a single scan, the likelihood of simultaneously having strongly

interacting bad data and parameter errors is small. Furthermore, using this approach,

there is no need to specify which parameter is to be tested for errors, a priori state

estimation. Those three steps are separated from each other. Step 2 uses the results of the

normal state estimation done in Step 1, and the set of suspicious parameters can be easily

changed in Step 2 and without requiring re-estimation of the system states.

26

2. Flow Chart

The flow chart of the algorithm is shown below.

Figure 1. Flow chart of parameter error identification method

Normal S.E.

Read Data

Start

Compute Nr and N

max( N )>

No Error

max( N )max( Nr )>

3.0?

Correct the

Parameter in error

Delete the

Bad measurement

27

D. Simulation Results

The parameter error identification procedure described in the previous section is

implemented and tested on IEEE 14-, 30- and 57-bus test systems. Different cases are

simulated where errors are introduced in transmission line parameters, transformer taps,

shunt capacitors, and analog measurements. Both single errors and simultaneously

occurring errors in analog measurements and parameters are simulated. The performance

of the method as well as its limitations is illustrated through these examples.

1. Line Impedance or Measurement Error

This section presents single errors in transmission line impedances or analog

measurements. The method is shown to differentiate between these different types of

errors and to correctly identify the error. The simulated errors for the three test systems

are listed in Table 1, where tests A and B are carried out as follows.

Test A) An error is introduced in the line parameter listed in Table 1; all analog

measurements are error free.

Test B) No parameter errors are introduced; all measurements are error free, except

for the listed flow in Table 1.

Table 1. Simulated Parameter and Measurement Errors

Bad Parameter/Meas. Test System

Test A Test B

14-bus 54r 54q

30-bus 75x 75p

57-bus 64r 46q

28

Tables 2 to 4 show the sorted normalized residuals and normalized Lagrange multipliers,

obtained during the tests of Table 1. The correction of the parameter error is shown in

Table 5.

Table 2. Results of Error Identification - 14-bus System

Test A Test B Measurement/

Parameter Normalized residual /

Measurement/ Parameter

Normalized residual /

54r 7.88 54q 12.02

42r 5.98 5q 8.61

52r 4.84 4q 6.57

54q 4.81 54x 5.35

65t 4.59 42x 4.18






75 x 25.47 75p 19.50

67x 22.01 75r 12.34

52x 21.92 5p 10.56

67r 15.78 6q 9.97

52r 15.42 67x 9.86

29






64r 14.82 64q 8.78

64q 9.65 64r 5.96

43r 7.37 65x 4.22

54r 7.09 4s 4.01

64p 6.79 4q 4.01

Table 5. Estimated and True Parameters of Line Impedances

Test system Bad

Parameter Corrected Parameter

Parameter without error

14-bus 54r 0.01355 0.01355

30-bus 75x 0.11593 0.11600

57-bus 64r 0.04295 0.04300

As evident from the above, single line impedance errors as well as single analog

measurement errors can be identified and corrected by this approach.

2. Transformer Tap or Measurement Error

This section presents single errors in transformer taps or analog measurements. The

simulated errors for the two test systems are listed in Table 6, where tests A and B are

carried out as follows.

Test A) A 1% error is introduced in the transformer tap value; all analog

measurements are error free.

30

Test B) No parameter errors are introduced; all measurements are error free,

except for the flows listed in Table 6.

Tables 7 to 8 show the sorted normalized residuals and normalized Lagrange multipliers,

obtained during the tests of Table 6. The correction of the parameter error is shown in

Table 9.

Table 6. Simulated Parameter and Measurement Errors

Bad Parameter/Meas. Test System

Test A Test B

14-bus 4 9t 9 4q

57-bus 4913t 4913p

Table 7. Tap and Measurement Error Identification - 14-bus System


Parameter Normalized residual / N


Normalized residual / N

4 9t 24.10 9 4q 19.67

9 4q 18.04 4 9t 14.71

4 7t 11.74 4 7t 7.05

7 9r 11.17 7 9r 6.83

4q 8.78 5 6t 2.75

31

Table 8. Tap and Measurement Error Identification - 57-bus System





4913t 63.19 4913p

18.52

4913q 53.48 4913x 6.71

4913x 48.69 4948r 6.37

4948x 25.60 49p 6.17

4746r 20.03 4614x 5.58

Table 9. Estimated and True Parameters of Taps

Test system Bad Parameter Corrected Parameter


14-bus 4 9t 0.96000 0.96000

57-bus 4913t 0.89502 0.89500

As evident from the above, the method successfully identifies and corrects transformer

tap errors while maintaining its ability to identify any errors appearing in analog

measurements.

3. Errors in Shunt Capacitor/Reactor

Errors in the parameters of shunt devices such as capacitors or reactors can be detected

but not identified. The reason is the lack of redundancy, i.e., there is only one

measurement, namely, the reactive power injection at the corresponding bus, whose

expression contains this parameter. Hence, when there is an error in this injection

measurement or an error in the shunt device parameter, this error will be detected, but its

32

source cannot be identified. The injection measurement and the parameter constraint

constitute a critical pair. This case illustrates two examples of this limitation for 14- and

30-bus test systems.

Errors are introduced in the shunt susceptances at bus 9( 9s ) and at bus 24( 24s ) of 14-

and 30-bus systems, respectively. The normalized residuals and Lagrange multipliers are

given in sorted form in Table 10. Note that the reactive injection measurements and shunt

susceptances have identical normalized values, indicating that they constitute a critical

pair whose errors cannot be identified.

The estimated and true parameter values are shown in Table 11.

Table 10. Shunt Susceptance Errors

14-bus system 30-bus system Measurement/




9s 5.80 24s

12.72

9q 5.80 24q 12.72

109q 3.05 2422q 5.78

94t 2.51 22q 5.23

14q 2.05 2423q

4.65

Table 11. Estimated and True Parameters of Shunt Susceptances

Test system Bad Parameter Corrected Parameter


14-bus 9s 0.1900 0.1900

30-bus 24s 0.0432 0.0430

33

4. Simultaneous Errors

This section shows the identification of multiple errors occurring simultaneously in the

14-bus system. Errors are simulated in the reactance of the transmission line 2–4, tap of

the transformer 4–9, and the power flow measurement in line 4-2. The largest normalized

value test is used to identify these errors one at a time. Results of normalized value tests

for each error identification cycle are presented in Table 12.

Table 12. Multiple Error Identification Results

Error identification cycle 1st 2nd 3rd

z/ p Nr / N z/ p Nr / N z/ p Nr / N

42x 60.56 94t 23.87 24p 5.07

24p 46.48 49p 17.99 3p 3.75

54x 40.49 74t 10.00 4p 3.02

52x 30.24 97r 9.78 42r 2.86

94t 25.00 4p 9.68 54p 2.25

Identified and Eliminated error

42x 94t 24p

When corrected, the parameter values are found, as shown in Table 11. Notice that when

there are multiple errors in the network parameters as well as analog measurements;

repeated application of the largest normalized value test can identify errors one by one, as

shown in Table 10. However, due to the interaction between multiple parameter errors,

radial correction of parameter errors may yield approximate values, as in Table 11. This

approximation error can be minimized by executing an extra estimation solution, where

all identified parameters are included simultaneously in the augmented state vector. The

34

results for this case are shown in Table 14. Note that the results in Table 14 are more

accurate than those given in Table 13.

Table 13. Estimated and True Parameters of Multiple Errors

Step Bad

Parameter Corrected Parameter

Parameter without

error

1st 42x 0.17400 0.17632

2nd 94t 0.96015 0.96000

Table 14. Simultaneous Estimation of All Identified Parameters

Bad Parameter Estimated Parameter

True Parameter

42x 0.17633 0.17632

94t 0.96000 0.96000

Similar to the case of the multiple interacting and conforming bad data, there may be

situations where strongly interacting parameter and analog measurement errors cannot be

identified due to error masking. Such cases are, however, rare and cannot be handled by

this method.

5. Critical K-tuples of Parameters

It is found that in some cases the normalized Lagrange multipliers of the parameters are

so close that the error among them cannot be correctly identified. The parameters with

the similar normalized Lagrange multipliers are defined forming a critical k-tuple.

One such situation is illustrated by the following test that is carried out on the IEEE

14-bus system whose diagram and measurements are shown in Figure 2.

Test A) The reactance 6 12x for line 6–12 is incorrect; all measurements are exact.

35

Test B) The reactance 12 13x for line 12–13 is incorrect; all measurements are

exact.

Figure 2. IEEE 14-bus system

The incorrect parameters for the two neighboring lines are chosen as shown in Table 15.

These two parameter errors will be detectable but not identifiable. Either one of the

parameters can be identified as incorrect, depending upon the initial conditions used in

the iterative solution of the state estimation problem. The results of Tests A and B are

shown in Table 16.

Table 15. Objective Function Values for Tests A and B

Erroneous Parameter

Assumed Value

True Value

xJ

Test A 126x 0.23656 0.25581 14.7064

Test B 1312x 0.29988 0.19988 14.7068

36

Table 16. Error Identification of Series Lines





6 12x 3.8291 126x 3.8280

12 13x 3.8250 1312x

3.8148

6 13x 2.8902 136x 2.7479

6 12p 2.4126 1312p 2.5182

12 13p 2.3390 126p 2.4759

As shown in Table 16, in Test A, the proposed method correctly identified 6 12x as the

erroneous parameter, while in Test B, the same algorithm still identified the same

parameter instead of the incorrect parameter 12 13x as bad data. The failure of parameter

error identification may be caused by the almost identical objective function values

corresponding to the two tests in Table 15. It suggests that there may be multiple

solutions of the parameter sets of 6 12x and 12 13x since the two solutions have the

‘same’ objective function value. So the state estimator fails to find which parameter is in

error.

This is the case cannot be solved by the current method. The cause of such critical

k-tuples of parameters will be analyzed in chapter V and a solution to this problem will

be presented. Before that the basic rules of including PMUs in the state estimation will be

introduced in the next chapter.

37

E. Conclusion

This chapter presents a method for identifying network parameter errors, even in the

presence of bad analog measurements. The parameter error identification is accomplished

by formulating the parameter errors as zero equality constraints and then testing the

significance of the associated Lagrange multipliers. These are computed from the

normalized measurement residuals obtained by the WLS state estimation. The method

can deal with mixed-type multiple errors in measurements and network parameters. There

is also no need to specify a set of suspect parameters before state estimation. Once the

parameter error is identified, its correct value is estimated using the augmented state

estimation method. Several examples are simulated to illustrate the effectiveness of the

method. This paper also shows the inherent limitations of error identification for certain

special cases. The method can be readily implemented as a user-defined option by

modifying an existing WLS state estimation code.

38

CHAPTER IV

STATE ESTIMATION USING PHASOR MEASUREMENTS

A. Introduction

State estimation plays a key role in the power system secure operation. It provides an

optimal estimation of current power system states based on the received measurements

and network topology. The system states are defined as the voltage phasors at each and

every bus in the system. While the magnitudes of these phasor voltages could be

measured, their phase angles have become available as measurements only very recently,

through the introduction of synchronized phasor measurements. Phasor measurement

units (PMU) are devices, which use synchronization signals from the global positioning

system (GPS) satellites and provide the positive sequence phasor voltages and currents

measured at a given substation.

The accuracy and reliability of state estimation is improved when phasor measurements

are introduced into the power system while there are sufficient number of other types of

measurements to estimate the system state [19, 20]. Since in most cases it is not possible

to install enough PMUs in the system to perform a strictly phasor-based state estimation,

they are used as extra measurements. Hence, the problem remains nonlinear as before and

solution will have to be iterative. Previous work on phasor measurements looks at issues

related to utilization and placement of PMUs [21, 22]. A two-step procedure, which

allows linear recursive estimation for the PMU measurements, is also developed in [23].

Irrespective of the existence of phasor measurements, state estimation problem has so far

been formulated by using an arbitrarily selected bus as the reference and its voltage phase

angle is set equal to zero. In the absence of any phase angle measurements, this practice

presents no problems and provides a suitable framework to define the system state where

the actual value of the reference bus voltage phase angle is irrelevant. After the

39

introduction of PMU measurements, the choice of a reference bus will no longer be an

arbitrary decision. There are two possibilities:

1. Choose a bus where no PMU exists:

This will create inconsistencies between the arbitrarily assigned reference angle at

the chosen bus and actual phase angle measurements provided by PMUs at other

buses.

2. Choose one of the buses with PMUs as the reference bus:

This will work as long as the PMU at the chosen bus functions perfectly. If the

measurements provided by this PMU contain errors, then these errors will not be

detectable and will bias the estimated state.

This issue has been recognized early on and alternative approaches were considered.

Among them is a document [24] which is produced by the Eastern Interconnection

Phasor Project (EIPP) group. In this document a virtual bus angle reference, which is

computed as the average of several phase angle measurements by PMUs located in the

vicinity of a chosen bus is introduced. This approach still remains vulnerable to errors

in individual PMU measurements despite the use of averaging.

In this chapter, state estimation problem is formulated without using a reference bus. This

formulation assumes that there will be at least one phase angle measurement in the

system in order to make it observable. If no phase measurements exist, then an arbitrary

bus phase angle will be assigned a zero value, reducing the problem formulation to the

traditional one with arbitrarily selected reference bus. In this approach, phasor

measurements are treated just like the traditional measurements.

Furthermore, an observability analysis is developed by using the whole gain matrix when

the voltage and current phasor measurements are in presence and no reference bus is

selected. Without the phasor measurements, the power system is assumed to be composed

of multiple observable islands. The theory to make such system observable by

40

introducing the phasor measurements is presented. On its basis, the rules of the

observability and to detect and identify the phasor measurement in error will be

concluded. However, there is a special case cannot be included in those rules. When there

is a current phasor measurement connecting two observable islands, the state estimation

will have multiple solutions and the observability analysis will lose uniqueness due to

such measurement configurations. In this case, additional phasor measurements must be

introduced in the system to obtain the unique observability.



Section B presents the proposed formulation of state estimation with phasor

measurements along with study of observability analysis. The theory of

observability according to number of phasor measurements is derived. The

rules to detect and identify the error in phasor measurements are concluded.

Simulation results of typical systems are shown in section C.

The last section provides the main conclusions for this chapter.

B. State Estimation Method with Phasor Measurements

1. Problem Formulation without Using a Reference Bus

Consider the measurement model:

z h x e (4.1)

where:

z is the measurement vector,

h x is the nonlinear function relating the measurements to the system states,

x is the system state vector including bus voltage magnitudes and phase angles,

41

e is the vector of measurement errors.

Applying linear approximation, equation (4.1) can be written as

z H x e (4.2)

where:

mxxx

mxhzz

mx is the system state vector at the m-th iteration.

h xH

x

(4.3)

State estimation problem is normally formulated by selecting a reference bus and setting

its phase angle as zero. However, after introduction of PMU measurements, keeping a

reference bus will only lead to undetectable error in that reference PMU. One of the

reasonable solutions is to use the actual phase angle measurements provided by PMUs

directly and not to select an arbitrary reference bus. Hence the measurement Jacobian has

to include an extra column corresponding to the previously designated reference bus

angle. Also the phasor measurements of voltages and line currents will be included and

those rows will be added to the Jacobian. The resulting new Jacobian matrix has the form

shown in Table 17 and is labeled as nH to differentiate it from the conventional H

matrix.

42

Table 17. the Structure of nH

V

iP VPi

iP

iQ iQV

iQ

fP VPf

fP

fQ fQV

fQ

V 1 ---

--- 1

I VI

I

V

where:

iP , iQ , fP , fQ are real and reactive power injections and flows, respectively.

V and are bus voltage magnitudes and phase angles.

I and are current magnitudes and phase angles.

The partial derivative of iP , iQ , fP , fQ are the same with normal state estimation.

However, the equations of I and need to be derived.

The definition of current magnitude from bus i to bus k is

i

SIV

(4.4)

where S is the magnitude of complex power from bus i to bus k, the following can be

written:

43

2 2 2S P Q (4.5)

S P P Q Qx S x S x

(4.6)

where x represents the variable of interest. Then:

21

i i i i i

I P P Q Q SV V S V S V V

(4.7)

1

k i k k

I P P Q QV V S V S V

(4.8)

1

i i i i

I P P Q QV S S

(4.9)

1

k i k k

I P P Q QV S S

(4.10)

The phase angle of current is defined as:

i

iQarctgP

(4.11)

where is the phase angle of complex power flow S. Differentiating with respect

to the state variables:

21

i i i

Q PP QV S V V

(4.12)

21

k k k

Q PP QV S V V

(4.13)

211

i i i

Q PP QS

(4.14)

2

1

k k k

Q PP QS

(4.15)

44

If the state estimation begins from flat start, which assumes that all the voltage

magnitudes to be ones and all the phase angles to be zeroes, then the function of current

phasor measurements in the Jacobian matrix will be undefined. To overcome this

problem, small artificial line charging susceptances will be introduced into the system

during the first iteration of state estimation and removed afterwards [25].

2. Observability of Network

One normally used method to check the network observability is by identifying the zero

pivots during the factorization of gain matrix, the details of which can be found in [18].

In the normal state estimation with reference bus, this G matrix can be written as

1TAA AA A AAG H R H (4.16)

where

AAA

hH

is the decoupled Jacobian for the real power measurement.

By doing Cholesky factorization of AAG , if only one zero pivot is encountered, the

system is observable.

When using PMU measurements, since the current measurements cannot be readily

decoupled like in the case of real and reactive power measurements, the gain matrix has

to be formed in full as shown below:

1Tn n nG H R H (4.17)

In this case due to the presence of phasor measurements, the system will be declared

observable if no zero pivots are encountered while factorizing nG . When there is more

than one observable island in the system excluding the phasor measurements, then there

has to be at least one phase angle measurement in every observable island to make the

overall system observable.

Assume there is a 5-bus system that is shown in Figure 3.

45

Figure 3. 5 bus system with 2 observable islands

This system is composed of two observable islands excluding phasor measurements. Two

phasor measurements are introduced to make the system observable. They are voltage

phasor measurements 1V , 1 and current phasor measurement 3 5I , 3 5 .

With the traditional measurements, the bus voltage magnitudes and the angle differences

within one observable island are determined. With the voltage phasor measurement at bus

1, the exact phase angle of bus 1 is determined and so as the other buses in this island.

When the bus voltage magnitudes and the angle differences in one observable island are

determined, the angle differences between those buses and the current phasors are also

determined by the traditional measurements. When the current phasor measurement from

bus 3 to bus 5 is introduced, the exact phase angles of bus 3 and 5 are determined. The

diagram is shown in Figure 4.

Figure 4. Phase Angle determination with phasor measurements

46

The voltage phase angles are determined due to the phasor measurement in that

observable island as shown in the figure above. The phase angle measurements provided

by PMUs are synchronized with respect to the time reference provided by GPS. This way

the GPS reference can be treated as an artificial reference bus and both observable islands

are connected to this reference through the phasor measurements. This way the whole

system is made observable by introducing those phasor measurements, which is shown in

Figure 5.

Figure 5. PMUs with common reference

3. Rules of Observability and Error Identification

Assuming that the power system is composed of multiple observable islands excluding

phasor measurements, the rules of observability and error identification of power system

can be concluded as follows due to the theory presented above.

Rule 1: If there is one phase angle measurement per observable island, irrespective of

whether it is a voltage or a current phase angle measurement, the system will be

declared as observable. In such a case all the observable islands are connected

via the phase angle measurements. However, that particular phase angle

measurement is a critical measurement of that observable island and the error in

that measurement cannot be detected.

Rule 2: In order to detect and identify the errors in phasor measurements, their

47

redundancies are to be increased in their respective observable island. If there

are two phase angle measurements in one observable island, the error in either

of the phasor measurement can be detected but cannot be identified. The two

phase angle measurements in that observable island form a critical pair.

Rule 3: Only if there are at least three phase angle measurements in one observable island,

the error in any of the phasor measurement can be identified. This is the

minimum requirement to ensure identification of errors appearing in phasor

measurements.

C. Simulation Results

The above described reference-free rectangular state estimation formulation is

implemented and tested on IEEE test systems having 14 and 118 buses. Both issues of

observability and solution without a reference angle are considered in setting up the

simulations.

1. State Estimation with and without a Reference

This section presents a single error in phase angle measurement in a 14-bus system that is

fully observable. The simulated errors and the detailed PMU measurement for the two

test systems are listed in Table 18, where the tests A and B are carried out as follows:

Test A: Using the normal state estimation method with the reference bus at bus 1.

Test B: Using the reference-free method.

Table 18. Simulated Measurement Errors and PMU measurement

Error PMU measurement

1 1V , 8V ,

12V , 1 ,

8 , 12

1 1V , 1 5I

, 1 2I

,1 ,

1 5 ,

1 2

48

Tables 19-20 show the sorted normalized residuals Nr obtained during the tests of Table

18. Note that a statistically reasonable threshold of Nr is 3.0, which is the one used in

all simulations presented in the following. Note that test A results fail to identify the error

and falsely suspect the phase angle of bus 8 and reactive power injection at bus 1 as a bad

phase measurement, while test B correctly identifies the erroneous phase measurement at

bus 1.

Table 19. Results of Error Identification-voltage phasor measurements

Test A Test B

Measurement Normalized

residual Measurement

Normalized residual

8 7.50 1 10.53

74p 5.73 8 7.20

65p 5.60 74p 5.48

12 5.20 65p 5.35

1V 4.53 12 4.96

Table 20. Results of Error Identification- current phasor measurements

Test A Test B

Measurement Normalized

residual Measurement

Normalized residual

1q 4.24 1 6.06

21 4.00 1q 4.18

21q 3.55 21 3.95

51 1.94 21q 3.50

51q 1.75 51 1.91

49

As evident from the above, continuing to use the existing state estimators by simply

adding phasor measurements will lead to unidentifiable errors which will in turn corrupt

the estimated state. This undesirable situation can be avoided by a simple revision in the

problem formulation where no explicit phase angle is used as the reference. The

following cases will elaborate on this further by studying observability and bad data

processing issues for different phasor measurement configurations.

2. Merging Observable Islands Using Phasor Measurements

This section verifies the condition to acquire observability of the system with several

observable islands. These observable islands can be merged into one by strategically

placing PMUs. However, care should also be exercised in order to ensure reliability

against possible errors in any one of these newly introduced phasor measurements.

The system for testing is shown in Figure 5. Note that phasor measurements are not

included yet in this system. The system is composed of three observable islands that are

indicated by the lines in the figure.

The tests A and B are created as follows:

Test A: Four phasor measurements 1V , 1 , 8V , 8 are in the system.

Test B: Six phasor measurements 1V , 1 , 8V , 8 , 12V and 12 are in the system.

Figure 6. 14 bus system with 3 observable islands

50

The results of test A show the system is not observable and test B is observable. The

system can be made observable by introduction of a phase angle measurement per

conventional observable islands.

3. Phasor Measurement Error Identification

To detect and identify the phasor measurement error, three PMUs, including that one in

error, are needed in that particular observable island, which is defined without the

existence of phasor measurements.

a. One Phasor Measurement in an Observable Island

The results of simulations will be presented when there is only one phasor measurement

in one observable island. This section illustrates two examples of this limitation for 14-

and 118-bus test systems.

The 14 bus system used for testing is shown in Figure 6. There are 3 voltage phasor

measurements in the system, which are at bus 1, 8 and 12.

The 118 bus system is shown in Figure 7. This 118-bus system is divided into 9 areas.

Among them, area 8, area 9 and the rest part of the system form three observable islands

without existence of phasor measurements. There are 9 voltage phasor measurements in

the system, which are at bus 3, 18, 35, 27, 76, 47, 103, 93, 55.

14-bus system: Voltage phase angle measurement of bus 12 is in error.

118-bus system: Voltage phase angle measurement of bus 93 is in error.

Note that in those tests, every observable island is assigned one phasor measurement, so

all the phase angle measurements are critical measurements. Current phasor

measurements are not introduced into the system. The simulation results are shown in

Table 21.

51

Figure 7. 118-bus system with 9 Areas

52

Table 21. Results of parameter error detection (fail)





7 8p 0.0113 6 0.0180

8p 0.0112 104q 0.0153

2q 0.0082 18 0.0151

1V 0.0076 104 105q 0.0149

3q 0.0064 76 0.0118

As is evident from Table 21 having a single phasor measurement in every observable

island will not allow detection of bad data appearing in any one of the phasor

measurements.

b. Two Phasor Measurements in an Observable Island

The results of simulations will be presented when there are two phasor measurements in

one observable island. This case illustrates two cases of this limitation for 14- and


Case 1.

14-bus system: Comparing to Case 1, one more phasor measurements 13V , 13 are

introduced into the system. Voltage phase angle measurement of bus 12 is in error.

118-bus system: Comparing to Case 1, one more phasor measurements 97V , 97 are


53

Note that in those tests, every observable island is assigned two phasor measurements,

those phase angle measurements form a critical pair. Current phasor measurements are

not introduced into the system. The simulation results are shown in Table 22.

Table 22. Results of parameter error detection-voltage phasor measurements





12 4.81 93 4.85

13 4.81 97 4.85

6 12p 0.874 96 97p 4.82

6 13p 0.867 93 94p 4.80

12 13q 0.795 94 96p 4.79

Case 2.

14-bus system: Comparing to Case 1, one more phasor measurements 1 2I , 1 2 are


118-bus system: Comparing to Case 1, One more phasor measurements 103 100I , 103 100

are introduced into the system. Voltage phase angle measurement of bus 103 is in error.

Note that in those tests, every observable island is assigned two phasor measurements,

those phase angle measurements form a critical pair. Current phasor measurements are

introduced into the system. The simulation results are shown in Table 23.

As is evident from Table 22 and 23 having two phase angle measurements in a single

observable island will allow detection but not identification of bad data appearing in any

one of them since those two phase angle measurements form a critical pair.

54

Table 23. Results of parameter error detection-current phasor measurements





1 2 4.55 100103 3.86

1 4.55 103 3.86

1q 1.53 103100q 2.40

1 2q 1.33 103q 1.56

2q 1.08 3635p 0.90

c. Three Phasor Measurements in an Observable Island

The results of simulations will be presented when there are three phasor measurements in

one observable island. This section illustrates two cases of this limitation for 14- and


Case 1.

14-bus system: Comparing to Case 1, two more phasor measurements 13V , 13 , 11V , 11

are introduced into the system. Voltage phase angle measurement of bus 12 is in error.

118-bus system: Comparing to Case 1, two more phasor measurements 97V , 97 , 90V ,

90 are introduced into the system. Voltage phase angle measurement of bus 93 is in error.

Current phasor measurements are not introduced into the system. The simulation results

are shown in Table 24.

55

Table 24. Results of parameter error identification-voltage phasor measurements





12 5.50 93 5.65

13 2.80 90 2.90

11 2.66 96 97p 2.88

6 12p 1.26 93 94p 2.87

12 13q 0.93 94 96p 2.87

Case 2.

14-bus system: Comparing to Case 1, two more phasor measurements 1 2I , 1 2 , 1 5I ,

1 5 are introduced into the system. Voltage phase angle measurement of bus 1 is in error.

118-bus system: Comparing to Case 1, two more phasor measurements 103 100I , 103 100 ,

103 104I , 103 104 are introduced into the system. Voltage phase angle measurement of bus

103 is in error.

Current phasor measurements are not introduced into the system. The simulation results

are shown in Table 25.

As is evident from Table 24 and 25 having three phase angle measurements in the

observable island, error in one of these phase angle measurements can be detected and

identified.

56

Table 25. Results of parameter error identification-current phasor measurements





1 5.12 103

4.23

1 2 2.77 100103 3.00

1 5 2.34 103100q 2.35

1q 1.96 103 104 1.72

1 2q 1.41 104q 1.42

D. Conclusion

This chapter presents a reference phase angle free formulation of the state estimation

problem which requires at least one phase angle measurement. It is shown that bad data

associated with PMU measurements can be detected and identified by using this

formulation. Furthermore, state estimation can act as an initial filter before the raw

phasor data are sent to the phasor data concentrators (PDC) enabling gross error detection

and elimination for these new devices. It is expected that the general rules derived for

testing observability and bad data identifiability for PMUs will assist system planners in

deciding the number and location of PMUs to be placed for full observability and error

identifiability.

57

CHAPTER V

NETWORK PARAMETER ERROR IDENTIFICATION USING

PHASOR MEASUREMENTS

A. Introduction

State estimation is one of the essential energy management system (EMS) application

functions which allow secure operation of power systems. SE provides a complete and

reliable real-time database for analysis and economy-security functions carried out by

EMS. Generally, SE functions ideally with the assumptions including small measurement

noise, adequate measurement redundancy and correct network configuration and

parameters. Unfortunately, these assumptions do not always hold true.

Measurement redundancy is the key in successful processing of measurement and

parameter errors. While methods such as the largest normalized residual tests exist and

can effectively detect and identify measurement errors, they become useless in the

presence of critical measurements and critical k-tuples. These are the types of

measurements whose errors can not be detected and identified due to lack of local

redundancy. The bad data in critical measurements cannot be detected and those in

critical k-tuples cannot be identified.

Similar limitations apply to the case of network parameter error detection and

identification. There are a lot of existing parameter error identification methods, among

which the one presented in chapter III illustrates the existence of the critical k-tuples of

network parameters. Similar to the critical measurement k-tuples, those critical parameter

k-tuples are the result of poor measurement redundancies, but also of certain topologies.

Moreover, these critical parameter k-tuples cannot be eliminated by introducing more

conventional measurements. They can only be removed with the availability of more

phasor information.

58

Recently phasor measurement units (PMU) are introduced into power system due to

various benefits they may bring, which include the improvement on state estimation

performance. The PMUs are suggested to be installed in the system to improve the

measurement redundancy so as to maintain observability during contingencies or to

eliminate existing critical measurements [26, 27].

In this chapter, the parameter error identification method proposed in chapter III for

conventional measurements is extended to the case where phasor measurements are also

available. Using only with the conventional measurements, the parameters of certain

branches that are connected in a radial manner form a critical parameter k-tuple. Such

parameters can assume more than one set of values that will satisfy all incident

conventional measurements. For such configurations, while the parameter error can be

detected, it will not be possible to identify the branch with which it is associated. By

introducing phasor measurements at strategic buses, possibility of such multiple solutions

can be eliminated and hence the errors in those parameters can be correctly identified.



Section B illustrates the existence and cause of the critical k-tuples of network

parameters. And the conditions of forming the critical k-tuples of network

parameters appear are also presented.

Section C presents the proposed phasor measurements included formulation

without a reference bus.

The results of simulations showing the improvements of parameter error

identification capability after introducing PMUs are given in section D.


59

B. Multiple Solutions of Parameter Errors

As shown in chapter III, under particular circumstances, some of the line parameters form

critical k-tuples in parameter error identification.

When there is a critical parameter k-tuple and there is an error on any one of the

parameters in this k-tuple, this error can be detected but not identified. Multiple possible

solutions exist for the parameters of the k-tuple that will satisfy the existing conventional

measurements. Consider two network parameters 1p , 2p and their erroneous values

1 'p , 2 'p . If two different solutions x , 'x yielding the same objective function value

can be found such that

1 2 1 2, ', ', , 'J x p p J x p p (5.1)

then the WLS state estimator will equally likely converge to either one of these solutions.

Hence, it will not be possible to identify which of these two parameters is actually in

error. Such error can be detected but cannot be identified. In fact, multiple possible

solutions of the parameters are the cause of the critical k-tuple. As an illustration,

consider the system with radial lines shown in Figure 8.

Figure 8. Radial lines in the system

60

Bus k, l and m are connected in a radial manner. The rest of the system is an observable

island. There are three power measurements on the radial lines, klP , lmP and lP . The

linear equations of the power flows and injection can be expressed as:

k lkl

kl

Px

(5.2)

l mlm

lm

Px

(5.3)

l lm klP P P (5.4)

If the line impedances have been changed to 'klx and 'lmx , the following equations can

be satisfied with the conditions that k and m remain the same.

'''

k lkl

kl

Px

(5.5)

'''

l mlm

lm

Px

(5.6)

and 'kl klP P , 'lm lmP P .

The relationship between the impedances klx and 'klx , lmx and 'lmx can be derived:

' 'kl k l

kl k l

xx

(5.7)

' 'lm l m

lm l m

xx

(5.8)

By equations (5.7) and (5.8) if 'klx is known, 'lmx can be obtained. This means

whatever 'klx is, theoretically there will be a corresponding 'lmx , which lead to the

61

same power flow solutions with the parameter set klx and lmx . In this case, there will be

two sets of parameter solutions that will satisfy all the measurements. Since the voltage

phasors in the observable island remain the same, the only differences between the two

solutions are the voltage phasor of bus l and the current phasors in the radial lines, which

is illustrated graphically in Figure 9. However, the phase information is not available in

the traditional measurement set, those changes will not be detected.

Figure 9. Multiple solutions of voltage and current phasors

A more comprehensive example will be given to demonstrate the case of multiple

solutions using IEEE 14-bus system whose diagram and measurements are shown in

Figure 10:

Test A: a set of impedances 6 12x and 12 13x are 0.25581 and 0.19988, respectively.

Test B: another set 6 12'x and 12 13'x are 0.23370 and 0.29988, respectively.

62

Figure 10. IEEE 14-bus system

The measurement set is assumed to include at least one power flow on each branch and

one power injection at each bus. There are 3 bus voltage magnitude measurements at bus

1, 8 and 10. The results of the tests are shown in Table 26.

Table 26. Objective Function Values for Tests A and B

6 12x 12 13x J x

Test A 0.25581 0.19988 0.0380

Test B 0.23370 0.29988 0.0578

The estimated states for the two test cases are shown in Table 27. Note that the two

estimates differ only for the phase angle of bus 12.

63

Table 27. Estimated States for Tests A and B

Test A Test B Bus No: V θ V θ

1 1.0600 0 1.0600 0

2 1.0450 -5.2382 1.0450 -5.2383

3 1.0100 -13.1669 1.0100 -13.1671

4 1.0159 -10.8858 1.0159 -10.886

5 1.0180 -9.2401 1.0179 -9.2403

6 1.0700 -14.8838 1.0700 -14.8841

7 1.0678 -14.636 1.0678 -14.6363

8 1.0900 -16.4762 1.0900 -16.4765

9 1.0605 -16.003 1.0605 -16.0033

10 1.0547 -16.0926 1.0547 -16.0929

11 1.0587 -15.6255 1.0587 -15.6257

12 1.0556 -15.7629 1.0561 -15.675

13 1.0510 -15.8768 1.0510 -15.877

14 1.0384 -16.9445 1.0384 -16.9447

As shown in the above case, the two set of parameters will lead to nearly similar

objective function value and state vector. The only apparent difference is the phase angle

of bus 12. Since the phase angle information is not available unless phasor measurements

are available, both of those parameter sets are equally acceptable to the state estimator.

In the actual IEEE 14-bus system, the values of 6 12x and 12 13x are 0.25581 and

0.19988, respectively. The case above demonstrates that if there is an error in one of the

parameters 6 12x and 12 13x , such as the combination 0.25581 and 0.29988, or 0.23370

and 0.19988, respectively, the system cannot identify which parameter is in error. Since

there are two possible solutions of parameters could satisfy the system measurement set

in normal state, those two parameters are equally likely to be in error and they form a

critical pair.

64

A critical parameter k-tuple is defined as the set of parameters whose errors are

detectable but not identifiable. Based on the examples illustrated above and in chapter III,

the network topology and measurement configurations to form a critical k-tuple of

parameters can be concluded as: If the system is composed of multiple sub-networks

connected in a radial manner, the impedances of radial lines connecting those

sub-networks may form a critical k-tuple, which is shown in Figure 11.

Figure 11. Critical k-tuples in the system

In Figure 11, the system is composed of three sub-networks, part I, part II and part III.

Those three sub-networks are connected by the branches line 1, 2 and 3 radially to form a

loop. This case the corresponding parameters such as resistances or reactances on lines 1,

2 and 3 form a critical k-tuple.

C. Algorithm Using Phasor Measurements

Since the multiple solutions differs on the phase angle of the particular bus, if the phase

information of that bus is included in the measurement set then there will only be one

solution left. By eliminating the multiple solutions of the parameter values the critical

k-tuple of the parameters will also be removed. The basic algorithm of the method is the

same with the one presented in chapter III. However, the phasor measurements will be

incorporated in the Jacobian matrix. The main equation of state estimation can be

derived:

65

0 00

0 0 ( )0 0

t tH W C xn xH I r znC c xx

(5.9)

where nH is the measurement Jacobian matrix whose structure is shown in Table 17 in

chapter IV. Note that no reference bus is selected so nH includes voltage magnitudes

and phase angles of all the buses in the state vector.

The matrix xC corresponding to the zero injections can be built in the same structure

with nH .

And the Lagrange multipliers , which corresponds to parameters in error, can be

expressed as

xr

S

(5.10)

where:

px

pxx

tS

WHC

(5.11)

is the new parameter sensitivity matrix. pxH are the gradients of H with respect to

network parameters instead of voltage magnitudes and phase angles, whose structure is

shown in Table 28.

66

Table 28. the Structure of pxH

r x t s

iP iPr

iPx

iPt

0

iQ iQr

iQx

iQt

iQs

fP fPr

fP

x

fP

t

0

fQ fQr

fQ

x

fQ

t

0

V 0 0 0 0

0 0 0 0

I Ir

Ix

It

0

r

x

t

0

where:

iP , iQ , fP , fQ are real and reactive power injections and flows, respectively.

V and are bus voltage magnitudes and phase angles.

I and are current magnitudes and phase angles.

r , x , t , s are line resistances, reactances, transformer taps and shunt capacitances,

respectively.

The matrix pxC are the gradients of C with respect to network parameters. It has the

same structure with pxH .

67

Based on the equations (5.9) and (5.10), the normalized Lagrange multipliers N can be

derived using the equations illustrated in chapter III. The algorithm of the network

parameter error identification with PMUs can be briefly presented below:

Step 1. State Estimation.

This is the state estimation using the new Jacobian matrix including the phasor

measurements. There is no reference bus in the system.

Step 2. Computation of rN and λN.

Compute the normalized residuals rN for the measurements and the normalized

Lagrange multipliers λN for the parameter errors. Computational details for

normalizing the residuals can be found in chapter III.

Step 3. Bad Data and Parameter Error Identification.

Choose the larger one between the largest normalized residual and the largest

normalized Lagrange multiplier.

If the chosen value is below the identification threshold, then no bad data or

parameter error will be suspected. A statistically reasonable threshold to use

is 3.0, which is the one used in all simulations presented in the next section.

Else, the measurement or the parameter corresponding to the chosen largest

value will be identified as the source of the error.

Considering the rules concluded in chapter IV, there should be at least three phasor

measurements introduced in each observable island to assure enough redundancies to

identify the error in those phasor measurements.

In the following section, case studies will be presented in order to verify the detectability

and identifiability of the error in the parameters by the method with and without the

phasor measurements.

68

D. Simulation Results

The above described parameter error identification procedure is implemented and tested

on IEEE 14, 30 and 57 bus test systems. Different cases are simulated with different

combinations of erroneous parameters in particular critical k-tuple. The results of the

cases with and without the phasor measurements are compared.

1. Critical K-tuples in Radial Lines

This case illustrates the critical k-tuple of the parameters in radial lines. A single error in

one of the parameter in the critical k-tuple is introduced. The results of parameter error

identification with and without phasor measurements are compared. Two test cases are

created as follows:

Case 1. A single error is introduced to 12 13x in IEEE 14-bus system as shown in Figure

10. The parameters in lines 6-12 and 12-13 form a critical k-tuple.

Test A: No phasor measurements.

Test B: Using phasor measurements. In addition to the traditional measurements used in

test A, three voltage phasor measurements are introduced at bus 1, 8 and 12. The test

results are shown in Table 29.

69

Table 29. Parameter Error Identification - 14-bus system (I)





6 12x 6.7902 12 13x 6.9650

12 13x 6.7667 6 12x 5.0262

6 13x 4.6628 6 13x 4.8206

12 13p 4.6505 12 13p 4.7497

6 12p 4.3490 6 12p 4.0811


12. The parameters in lines 15-18, 18-19, 19-20 and 20-10 form a critical k-tuple.

Figure 12. Part of the IEEE 30-bus system

70



test A, four voltage phasor measurements are introduced at bus 1, 18, 19 and 20. The test


Table 30. Parameter Error Identification - 30-bus system (I)





15 18x 15.4741 18 19x 17.6994

18 19x 15.4470 19 20x 11.2100

20 10x 15.3713 15 18x 10.1697

19 20x 15.2465 18 19p 10.3255

18 19p 9.0775 20 10x 9.3565


13. The parameters in lines 29-52, 52-53, 53-54 and 54-55 form a critical k-tuple.

Figure 13. Part of the IEEE 57-bus system

71





Table 31. Parameter Error Identification - 57-bus system





54 55x 30.6234 53 54x 33.4402

53 54x 30.4777 53 54p 23.9776

29 52x 27.4314 54 55x 18.4719

52 53x 25.3114 33 32x 18.2441

53 54p 21.7044 54 55p 15.9108

As evident from the above, in some cases the error in one of the parameters in a critical

k-tuple cannot be correctly identified, where those parameters in critical k-tuple are in

radial lines. By introducing the phasor measurements on the buses inside that radial route,

the critical k-tuple does not exist any more and those errors can be correctly identified.

2. Critical K-tuples in Separated Radial Lines

This case illustrates the critical k-tuple of the parameters in radial lines separated by some

sub-networks. A single error is introduced to 27 25x in IEEE 30-bus system. The

parameters in lines 28-27, 27-25 and 25-24 form a critical k-tuple. Those are not purely

connected radial lines. There are some other sub-networks connected between them, such

as line 27-29, 29-30, 30-27 and 25-26. The results of parameter error identification with

and without phasor measurements are compared in tests A and B as follows:

72



test A, three voltage phasor measurements are introduced at bus 1, 25 and 27. The test


Table 32. Parameter Error Identification - 30-bus system (II)





24 25r 24.8360 27 25x 30.7918

27 25x 24.7589 25 16.9824

28 27x 21.4546 25 27p 16.1264

25 27p 13.9257 24 25r 11.1870

24 25p 12.6138 28 27x 10.2282

As shown in the case, in some cases the error in one of the parameters in a critical k-tuple

cannot be correctly identified, where those parameters in critical k-tuple are in radial lines

separated by other observable islands. By introducing the phasor measurements on the

buses inside that radial route, the critical k-tuple does not exist any more and those errors

can be correctly identified.

3. Critical K-tuples in Radial Lines Formed by Missing Measurements

This case illustrates the critical k-tuple of the parameters in the observable island, which

is formed because of missing of measurements. A single error is introduced to 13 14x in

IEEE 14-bus system as shown in Figure 14.

73

Figure 14. New Observable Island in IEEE 14-bus system

The power injection measurements on buses 5 and 6 and power flow measurement 5-6

are missing. The area indicated by the red lines as shown in the figure contains radial

sections. The parameters in lines 13-14, 14-9, 6-11, 11-10 and 10-9 are candidates to

form critical parameter k-tuples. Two tests are run as follows:





In this case critical k-tuple is formed not only by topology but by missing of

measurements. In test A the normalized Lagrange multipliers of the top three parameters

are very close, making it difficult to identify the incorrect one. Test B results show that

such critical k-tuples can be handled by introducing phasor measurements and the

incorrect parameter can be easily identified.

74

Table 33. Parameter Error Identification - 14-bus system (II)





10 11x 27.7765 13 14x 28.6870

13 14x 27.7706 9 14x 27.8853

6 11x 27.5534 13 14p 18.8854

9 14x 26.8986 9 14p 15.0538

9 10r 20.5636 6 11x 13.2814

4. Simultaneous Errors in Network Parameters and Phasor Measurements

This case illustrates the performance of the identification method when phasor

measurements and parameters have simultaneous errors. The test system is 14-bus system

shown in Figure 11. The following tests are performed:

Test A: The phase angle measurement 13 is in error; there is no parameter error in the

system.

Test B: The phase angle measurement 13 is in error; and the parameter 13 14x is in

error.

The test results are shown in Table 34.

75

Table 34. Phasor Measurement Error Identification - 14-bus system





13 37.4070 6 11x 38.0247

6 11x 22.4019 13 29.6729

11 15.2374 6 13x 25.6998

10 13.5740 10 11x 25.5696

6 13x 13.1220 13 14x 18.2823

As shown in the table, when there is no other error in the system the phasor measurement

error can be identified. However, when the parameter in a critical k-tuple and a phasor

measurement nearby are both in errors, they cannot be identified correctly. This is

because the phasor measurement is the crucial information for the parameter error

identification in critical k-tuple. When it is in error, there is no way to perform the

identification.

E. Conclusion

With only traditional measurements in the system, sometimes the parameters in error may

not be identified correctly. The appearance of the critical k-tuples and the failures of

parameter error identification caused by it are analyzed in this chapter. One convenient

and efficient way to solve this problem is suggested. Introducing PMUs to the nearby

buses, the critical k-tuples of the parameters are eliminated. By including redundant

phasor information, any error in the parameters can be correctly identified. The

simulation results of IEEE 14-bus, 30-bus, and 57-bus systems are illustrated to compare

76

the different results of parameter error identification with and without phasor

measurements.

77

CHAPTER VI

CONCLUSIONS AND FUTURE WORK

This dissertation is concerned with the detection, identification and elimination of errors

which exist in the databases used by the energy management system applications.

Specifically, it investigates the errors associated with network parameters, such as the

resistance, reactance and line charging capacitance of transmission lines, taps of

off-nominal tap transformers, susceptances of static capacitors and reactors, etc. The

challenge in dealing with network parameter errors is twofold:

There are a very large number of parameters to monitor, so choosing a small subset

of parameters at a time as suspect and trying to estimate their values may be

prohibitively time consuming.

Measurement errors may occur and interact with parameter errors, making it

difficult to distinguish between them.

Existing parameter error identification and correction methods have the common

limitation that they can be correctly performed only if there is no measurement error in

the system. Furthermore, they require the user to identify a suspect parameter set a priori

state and parameter estimation. This dissertation presents a practical solution to address

the above challenges by formulating the state and parameter estimation problem

Essential contributions of the proposed approach are the ability to detect and identify

errors appearing simultaneously in analog measurements as well as network parameters

and also being able to do this without the need to select a suspect set of parameters a

priori state estimation. Furthermore, this dissertation also shows that the errors in a

certain set of parameters can only be detected but not identified with only conventional

measurements in the system. Such parameters are defined as critical k-tuples. The

topologies that create such critical k-tuples of parameters are analyzed. These are also

78

shown to lead to multiple solutions of parameter combinations which satisfy all the

available measurements and these multiple solutions can only be avoided via the use of

phasor measurements. Parameter error identification is therefore found to be one of the

problems that can significantly benefit from the existence of phasor measurements which

are shown to enable identification of errors which can not be identified when using only

conventional measurements.

The dissertation also investigates the state estimation formulation and associated

observability issues when phasor measurements are incorporated into the measurement

set. It is shown that with minor reformulation of the state estimation problem, phasor

measurement errors can be processed, bad data can be processed and removed and a

reliable set of phasor data can be provided to the phasor data concentrators. One of the

benefits of using phasor measurements is shown to be in merging observable islands,

where PMUs can be placed at arbitrary locations within individual islands without the

being restricted to the boundary buses as in the case of conventional injection

measurements.

This dissertation uncovers a set of topologies that lead to multiple solutions for certain

parameter errors and illustrates how they can be avoided via PMU placement using

examples. A possible next step could be to derive a general algorithm to detect the

potential critical k-tuples of parameters based on the network topology and related

measurements. This may be followed by the development of an optimal PMU

placement method to eliminate the critical k-tuples of parameters. These studies are

suggested as follow-up work to this dissertation.

79

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82

VITA

Jun Zhu received B.S. degree in electrical engineering from Shanghai Jiaotong University,

Shanghai, China in 2000. During the following two years, he worked as a system

engineer dealing with the control systems of power plants for Shanghai Automation

Instrumentation Co. Ltd.. He received his M.S. degree in Electrical Engineering from

Texas A&M University in May 2004. After that he began purchasing his Ph.D. degree in

the Department of Computer and Electrical Engineering at Texas A&M University and

Northeastern University.

The typist for this thesis was Jun Zhu.

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